Split (1)

train · 6.32M rows

url stringlengths 14 2.42k	text stringlengths 100 1.02M	date stringlengths 19 19	metadata stringlengths 1.06k 1.1k
https://ask.sagemath.org/answers/15883/revisions/	# Revision history [back] In the algebraic way, you can do something like: sage: R.<x,y> = PolynomialRing(ZZ) ; R Multivariate Polynomial Ring in x, y over Integer Ring sage: I = R.ideal(xy) sage: I Ideal (xy) of Multivariate Polynomial Ring in x, y over Integer Ring sage: S = R.quotient_ring(I) ; S Quotient of Multivariate Polynomial Ring in x, y over Integer Ring by the ideal (xy) sage: S(3xy+x^2+4y) xbar^2 + 4ybar In the symbolic way, you can do (but it is much less reliable): sage: var('x y') (x, y) sage: assume(xy==0) sage: bool(3xy+x^2+4y == x^2+4y) True But is seems not able to decide simplifications by itself: sage: (3xy+x^2+4y).full_simplify() x^2 + (3x + 4)y For the orderings, you can also work symbolically: sage: var('x y') (x, y) sage: assume(x<y) sage: bool(3x<3*y) True	2019-05-24 01:17:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6335659027099609, "perplexity": 11414.410216632252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257481.39/warc/CC-MAIN-20190524004222-20190524030222-00193.warc.gz"}
https://forum.azimuthproject.org/discussion/1263/birthrate-of-feedstock	#### Howdy, Stranger! It looks like you're new here. If you want to get involved, click one of these buttons! Options # birthrate of feedstock by looking at that documentation (in german) about indian old-age homes for cattle (Gaushala-here a website of one random found) I felt that I should have a look at the birthrate of feedstock. I mean there are birthrates of humans, but what about the birthrates of animals and in particular feedstock and similar? Googling revealed nothing. Where does one find statistics about this topic? • Options 1. Perhaps it's easier to find statistics about meat consumption, and work backwards from there (for an average animal) to get an underestimate. I suppose you are mainly interested in the birthrate eventually leading to adult (or rather juvenile) animals. For example, I seem to remember that in poultry production males are often killed as soon after birth as possible, so I guess you don't necessarily want to count those as well. Perhaps vegetarian movements and animal right movements have more information about number of feedstock. Comment Source:Perhaps it's easier to find statistics about meat consumption, and work backwards from there (for an average animal) to get an underestimate. I suppose you are mainly interested in the birthrate eventually leading to adult (or rather juvenile) animals. For example, I seem to remember that in poultry production males are often killed as soon after birth as possible, so I guess you don't necessarily want to count those as well. Perhaps vegetarian movements and animal right movements have more information about number of feedstock. • Options 2. Perhaps it’s easier to find statistics about meat consumption, and work backwards from there (for an average animal) to get an underestimate. I suppose you are mainly interested in the birthrate eventually leading to adult (or rather juvenile) animals. I guess I was just trying to get an overview about the number of bigger animals as inhabitants of this planets. I mean one usually looks at human population developments (birthrates, deathrates/lifetimes) especially in the context of how to nourish all that people. But bigger animals certainly also need to be fed and that food may -depending on scarcity of ressources- be in competition with human food. The animal wildlife seems rather limited to certain inhabitats, so here one has I guess an estimate on food consuption via area use. But for feedstock and similar this estimate seems not so easy. Calculating back from meat cosumption is certainly an indicator, but then there is e.g. also milk and leather production and zoos and pets and things like the Gauschalas and it looks not so easy to estimate how much ressources these animals need and what their importance in general in terms of food competition and in particular for human food and other products is. As an example, telling by the above video the cattle in this Gauschala in the video look rather well fed, whereas there is human hunger in India. Comment Source:>Perhaps it’s easier to find statistics about meat consumption, and work backwards from there (for an average animal) to get an underestimate. I suppose you are mainly interested in the birthrate eventually leading to adult (or rather juvenile) animals. I guess I was just trying to get an overview about the number of bigger animals as inhabitants of this planets. I mean one usually looks at human population developments (birthrates, deathrates/lifetimes) especially in the context of how to nourish all that people. But bigger animals certainly also need to be fed and that food may -depending on scarcity of ressources- be in competition with human food. The animal wildlife seems rather limited to certain inhabitats, so here one has I guess an estimate on food consuption via area use. But for feedstock and similar this estimate seems not so easy. Calculating back from meat cosumption is certainly an indicator, but then there is e.g. also milk and leather production and zoos and pets and things like the Gauschalas and it looks not so easy to estimate how much ressources these animals need and what their importance in general in terms of food competition and in particular for human food and other products is. As an example, telling by the above video the cattle in this Gauschala in the video look rather well fed, whereas there is <a href="https://en.wikipedia.org/wiki/India_State_Hunger_Index">human hunger</a> in India. • Options 3. edited October 2013 In the context of ressource shortages here a citation from the document: Groundwater made easy – Understanding the basics (p.23): The National Water Policy, 2002 recognized over-exploitation of groundwater in the country and suggested broad measures like assessment, regulation, recharge, conjunctive use etc. It laid down priority of drinking water over irrigation and suggested private sector participation and community management of water resources. You may also want to read the section right before the citation, which explains the difficulties in control over groundwater levels and this comment about the availability of data regarding groundwater levels and <a href="http://forum.azimuthproject.org/discussion/203/methane/?Focus=9603#Comment_9603>this comment about rice production and methane release in India. Comment Source:In the context of ressource shortages here a citation from the document: <a href="http://www.indiawaterportal.org/sites/indiawaterportal.org/files/Groundwater_Frequently%20Asked%20Questions_MKMaitra_2011.pdf">Groundwater made easy – Understanding the basics </a>(p.23): >The National Water Policy, 2002 recognized over-exploitation of groundwater in the country and suggested broad measures like assessment, regulation, recharge, conjunctive use etc. It laid down priority of drinking water over irrigation and suggested private sector participation and community management of water resources. You may also want to read the section right before the citation, which explains the difficulties in control over groundwater levels and <a href="http://forum.azimuthproject.org/discussion/203/methane/?Focus=9604#Comment_9604">this comment about the availability of data regarding groundwater levels</a> and <a href="http://forum.azimuthproject.org/discussion/203/methane/?Focus=9603#Comment_9603>this comment about rice production and methane release in India.</a> • Options 4. Frederick sorry the links contain some unvalid XML, I have not the time to find out about the correct escape sequences. I hope you can read this anyway. Comment Source:Frederick sorry the links contain some unvalid XML, I have not the time to find out about the correct escape sequences. I hope you can read this anyway. • Options 5. Hi! We have an actual professional on this topic, she's just finished her PhD in this very topic and joined ISI! She modeled live stock in Africa and epidemics in her PhD and comes from a medical background. I mentioned this post to her and from her responce, I think she can provide a bit of information. I think in her model, the starting place is something along the lines of the SIR models considered in the network theory series (which by the way, I'm going to work on one that has spacial degrees of freedom soon). I saw her reading a book on ecology so I'm going to take my book on "essential mathematical biology" by Nicholas F. Britton in on Monday and try to talk with her more about this stuff. Comment Source:Hi! We have an actual professional on this topic, she's just finished her PhD in this very topic and joined ISI! She modeled live stock in Africa and epidemics in her PhD and comes from a medical background. I mentioned this post to her and from her responce, I think she can provide a bit of information. I think in her model, the starting place is something along the lines of the SIR models considered in the network theory series (which by the way, I'm going to work on one that has spacial degrees of freedom soon). I saw her reading a book on ecology so I'm going to take my book on "essential mathematical biology" by Nicholas F. Britton in on Monday and try to talk with her more about this stuff.	2023-04-02 05:23:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4894466698169708, "perplexity": 1920.6690727722118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950383.8/warc/CC-MAIN-20230402043600-20230402073600-00393.warc.gz"}
https://mathsknowledge.com/algebraic-identities-for-class-9/	# Algebraic Identities for class 9 ## Introduction we have already studied about algebraic expressions their addition ,subtraction ,multiplication and division .when we multiply binomial(Algebraic Expressions having two terms),we obtained the following results which is algebraic identities because it is true for all values of a and b i.e LHS=RHS (a+b)(a+b)=a²+2ab+b² (a-b)(a-b)=a²-2ab+b² (a+b)(a-b)=a²-b² Here are few algebraic identity which we shall study in this chapter (i) (a+b+c)²=a²+b²+c²+2ab+2bc+2ac (ii) (a+b)³= a³+b³+3ab(a+b) (iii) (a-b)³=a³-b³-3ab(a-b) (iv) a³+b³= (a+b)(a²-ab+b²) (v) a³-b³=(a-b)(a²+ab+b²) (vi) a³+b³+c³= (a+b+c)(a²+b²+c²-ab-bc-ca) There is one more identity which is called conditional identity because it applies when certain condition get satisfied. ### Conditional identity : A conditional identity is an identity which is true when the unknown (s) satisfies a certain condition. STATEMENT If a+b+c=0,then prove that a³+b³+c³=3abc ## Difference between identity and equation Identity: A Identity is equality which is true for all values of variables. Equation: An Equation values varies with different values of variables. ### Let’s Learn to solve problem-based on algebraic identity Q.1 Expand (3x+4y)² solution(i) we have , (3x+4y)²=(3x)²+2.3x.4y+(4y)² (using :(a+b)(a+b)=a²+2ab+b²) =9x²+24xy+16y² Q.2 Find the product (i) (2x+3y)(2x-3y) (ii)(2x+y)(2x-y)(4x²+y²) Solution(i) we have , (2x+3y)(2x-3y)=(2x)²-(3y)² (using :(a+b)(a-b)=a²-b²) = 4x²-9y² Solution(ii) we have , (2x+y)(2x-y)(4x²+y²)=(2x)²-(y)²(4x²+y²) (using :(a+b)(a-b)=a²-b²) = (4x²-y²) (4x²+y²) = 16x^4y^4 Q.3 Without actual multiplication, find the value of 103×97 solution: we have, 103×97 which we can write as (100+3)(100-3) = (100)²-(3)² (using :(a+b)(a-b)=a²-b²) =10000-9 =9991 Q.4 If x+y=12 and xy=32 , find the value of x²+y² solution we know (x+y)²=x²+2xy+y² It is given that x+y=12, xy=32, and x²+y²=? (x+y)²=(12)² (squaring on both side) x²+2xy+y² =144 x²+y²+2xy= 144 x²+y²+2.32=144 (xy=32) x²+y²=144-64 x²+y²=80 Q.5 Prove that a²+b²+c²-ab-bc-ca is always non-negative for all values of a,b and c. solution: we can prove this with the help of contradiction, let a²+b²+c²-ab-bc-ca is equal to any negative value say K ∴ a²+b²+c²-ab-bc-ca =K where K is any negative integers Now multiply both sides by 2 2a²+2b²+2c²-2ab-2bc-2ac=2K a²+ a²+b²+b²+c²+c²-2ab-2bc-2ac=2K a²+b²-2ab+b²+c²-2bc+a²+c²-2ac=2K (a-b)²+(b-c)²+(c-a)²= 2K Now here we have sum of the square of three numbers and we know the square of any number whether it is positive or negative is always positive. so, Our assumption is wrong ∴ a²+b²+c²-ab-bc-ca is always non-negative for all values of a,b and c Q.6 If a²+b²+c²=250 and ab+bc+ca=3 ,find the a+b+c. solution we know (a+b+c)²=a²+b²+c²+2ab+2bc+2ac (a+b+c)²= a²+b²+c²+2(ab+bc+ca) substitute the given value in above equation (a+b+c)²= 250+2(3) (a+b+c)²=250+6 (a+b+c)²=256 ∴ a+b+c =16 Q.7 Expand (i) (2x+3y)³ (ii) (3x-2y)³ solution (i) we can expand it with the help of identity (a+b)³= a³+b³+3ab(a+b) (2x+3y)³= (2x)³+(3y)³+3.2x.3y(2x+3y) = 8x³+27y³+36x²y+54xy² similarly we can also expand (3x-2y)³= (3x)³-(2y)³-3.3x.2y(3x-2y) = 27x³-8x³-54x²y+36xy² Q.8 If x+y = 12 and xy=27 find the value of x³+y³ solution we can solve in two ways first using identity and second finding the separate value of x and y which we will study in the quadratic equation for class 10 but today we will learn to solve it with both ways. Using identity (a+b)³= a³+b³+3ab(a+b) since we have x+y = 12 cubing on both side , (x+y)³=12³ x³+y³+3xy(x+y)=1728 x³+y³+3.27.12=1728 x³+y³=1728-972 x³+y³=956 #### Problem based on conditional identity Q.Without actual calculations ,find the value of the following (i) (-12)³+(7)³+(5)³ (ii) (28)³+(-15)³+(-13)³ Solution :(i) On comparing it with LHS of the identity a³+b³+c³=3abc we get, a=-12 ,b=7 and c=5 a+b+c= -12 +7+5 =0 hence condition is satisfying ∴ a³+b³+c³=3abc (-12)³+(7)³+(5)³=3.(-12).7.5 =-36.35 = -1260 similarly we can solve (28)³+(-15)³+(-13)³ 28-15-13=28-28=0 ∴ (28)³+(-15)³+(-13)³=3×28×-15×-13 =16380	2021-06-23 12:42:21	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9171503186225891, "perplexity": 8886.188941217906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488538041.86/warc/CC-MAIN-20210623103524-20210623133524-00220.warc.gz"}
https://mssi2015.wordpress.com/page/2/	# Parameter Tuning (Ben Recht) What is the “right” way to think about parameter tuning in machine learning? Contemporary machine learning models may have thousands of parameters that have to be hand tuned by the engineer. Current techniques for automating this tuning have two flavors: (a) use derivative free optimization to set the parameters. This reduces the problem to black box optimization and is necessarily exponential time in the number of parameters. Current “best-practice” using Gaussian Processes is often not much better than pure random search. (b) combine multiple models using statistical aggregation techniques. This reduces the problem to model selection. The resulting models are often large, unwieldly and uninterpretable (like random forests). Both methods ignore the structure of machine learning design, where pipelines are built stage-wise in a DAG, and ignore concerns about stability of the end-to-end pipeline. Are we just thinking about this problem incorrectly? What other structures and modeling can we take advantage of when optimizing machine learning models end-to-end # Modular Machine Learning (Ben Recht) In machine learning theory, we study problems with very simple descriptions. We analyze how to fit a mixture of gaussians, or train an SVM, or compute principal components. But in the wild, machine learning is a much more involved process. A data scientist (heheh) is never just presented with data and told to fit an SVM. One must first decide how to represent the data as a vector or discrete code, then how to normalize features, then how to prune features, and then, maybe, can begin fitting an SVM. What can we say about the theoretical properties of such “end-to-end” machine learning? What do the risk bounds look like? Is this simply a matter of studying the “stability” properties of the end-to-end pipeline. If so, how does one evaluate the stability of connecting together a series of components which have their individual, well-understood, stability properties. Can we derive generalization bounds for general DAGs of heterogenous machine learning components? # The Online Light Bulb Problem (Ohad Shamir) Setting A learner is given access to an infinite stream of i.i.d. vectors $x_t\in \{-1,+1\}^d$. The distribution is such that $\mathbb{E}[x_t]=0$, and $\mathbb{E}[x_t x_t^\top]=I+\rho(e_i e_j^\top + e_j e_i^\top)$ (in words, all coordinates are zero mean and uncorrelated, except for a single pair of coordinates $(i,j)$ which have positive correlation of $\rho$). Goal What is the minimal sample size (as a function of $d,\rho$) required to detect $i,j$ with constant probability, using an algorithm with $\tilde{O}(d)$ bits of memory? The $\tilde{O}$ notation can hide arbitrary factors polylogarithmic in the parameters of the problem (so that finite-precision issues are not a problem). Existing Results This problem is closely related to the light bulb problem, introduced by [1], which asked a similar question in an offline setting, where a sample of such vectors is given and the goal is to detect a correlation in time sub-quadratic in $d$. In contrast, we consider an online/streaming setting, and constrain ourselves in terms of memory. Without memory constraints, it is possible to detect the correlation with $O(\log(d)/\rho^2)$ samples (which is information-theoretically optimal), by maintaining the empirical average of $x_{t,i}x_{t,j}$ for all $i,j$. However, this requires $\tilde{O}(d^2)$ memory, which can be impractical for high-dimensional data. With $\tilde{O}(d)$ memory, one simple algorithm is to maintain empirical averages in batches of $O(d)$ coordinate pairs at a time, moving to the next batch once the current batch has been ruled out, and stopping when the correlated pair was found. However, the sample size for such an algorithm is much larger, $O(d\log(d)/\rho^2)$, and it’s not clear whether this is the best possible. [2] essentially showed that the trivial algorithm above can be optimal, but for a certain carefully-constructed artificial distribution, quite different than the one considered here. For the offline light bulb problem, the best currently-known algorithm appears in [3] and is based on a clever random projection scheme. However, in an online $\tilde{O}(d)$-memory setting, random projections don’t seem to yield a sample complexity much better than the trivial algorithm described above. Bibliography [1] Leslie G Valiant. Functionality in neural nets. In AAAI, pages 629–634, 1988. [2] Ohad Shamir. Fundamental limits of online and distributed algorithms for statistical learning and estimation. In NIPS, 2014. [3] Gregory Valiant. Finding correlations in subquadratic time, with applications to learning parities and juntas. In FOCS, 2012. Setting Consider the following `multi-player’ version of adversarial multi-armed bandits, on three arms: An adversary chooses a sequence of $T$ reward vectors $x_t\in [0,1]^3$. Then, two randomized decision makers iteratively choose $i_{t},j_{t}\in \{1,2,3\}$, and receive a reward $g_t(i_t,j_t)$ equal to $x_{t,i_t}+x_{t,j_t}$ if $i_t\neq j_t$, and $0$ if $i_t=j_t$. This models situations where two users utilize the same set of resources, resulting in zero reward if they collide. Crucially, the decision makers cannot communicate. Goal Design an algorithm for the two decision makers which attains sublinear regret: Namely, that regardless of the adversary’s strategy, $\max_{i,j}\sum_{t=1}^{T}g_t(i,j)-\sum_{t=1}^{T}g_t(i_t,j_t)= o(T)$. Existing Results Multi-player bandits have been studied in the context of cognitive radio networks, where users need to transmit on channels of varying quality and may interfere with each other (e.g. [1,2,3]). However, all results we are aware of are for a stochastic setting, where the rewards are sampled i.i.d., and most of them require a (possibly elaborate) communication scheme between the players, which is not always realistic. Bibliography [1] Animashree Anandkumar, Nithin Michael, Ao Kevin Tang, and Ananthram Swami. Distributed algorithms for learning and cognitive medium access with logarithmic regret. IEEE Journal on Selected Areas in Communications, 29(4):731–745, 2011. [2] Orly Avner and Shie Mannor. Concurrent bandits and cognitive radio networks. In Machine Learning and Knowledge Discovery in Databases, pages 66–81. Springer, 2014. [3] Dileep Kalathil, Naumaan Nayyar, and Rahul Jain. Decentralized learning for multiplayer multiarmed bandits. IEEE Transactions on Information Theory, 60(4):2331–2345, 2014. # Concentration of a Lipschitz Function of Negatively Associated Random Variables (Yuval Peres) Let $f : \{ 0 , 1 \}^n \to \mathbb{R}$ be a Lipschitz function. If the binary variables $\{ X_1 , \ldots , X_n \}$ are independent, then it is well known that $f(X)=f(X_1, \ldots,X_n)$ satisfies a Gaussian concentration inequality () $\mathbb{P} \left( f(X) - \mathbb{E} f(X) \ge a \right) \leq \exp(- \frac{a^2}{c n} )$ for a suitable $c>0$. Question Does such an inequality also hold if the binary variables $\{ X_1 , \ldots , X_n \}$ are negatively associated (i.e., increasing functions of disjoint subsets of these variables are negatively correlated) rather than independent? Background This is classical when $f$ is an increasing linear function of $X$, and therefore also holds for any linear Lipschitz function $f$ (with a different constant $c$). In 2009, Elchanan Mossel (personal communication) asked whether () holds for negatively associated $X_i$ and nonlinear Lipschitz functions. This is still open. A central motivation was the ensemble of uniform spanning trees in a graph $G$, which was proved in [1] to have negative association (the relevant binary variables are indexed by the edges of $G$ and are indicators for inclusion in the uniform spanning tree.) In [2] the inequality () is proved under a more restrictive negative dependence hypothesis known as the strong Rayleigh condition, defined in [3]. The class of strong Rayleigh measures includes determinantal measures (which include the uniform spanning tree) and also weighted uniform matroids and spanning tree measures. However, it is hard to verify directly if a measure is strong Rayleigh, and not all negatively associated measures on the hypercube satisfy this condition. Bibliography [1] T. Feder and M. Mihail. (1992). Balanced matroids. Annual ACM Symposium on Theory of Computing, 26-38, 1992. [2] R. Pemantle and Y. Peres (2014). Concentration of Lipschitz Functionals of Determinantal and Other Strong Rayleigh Measures. Combinatorics, Probability and Computing 23, 140–160. [3] J. Borcea, P. Branden, and T. Liggett (2009). Negative dependence and the geometry of polynomials. J. AMS, 22:521–567, 2009. # Grothendieck inequality and hyperbolicity (Sebastien Bubeck) Let me warn you that this open problem is more open ended than the one on Thompson Sampling! Grothendieck’s inequality and its proof by Gaussian projection give a universal rounding scheme for semidefinite programs. The question is whether there exist a generalization of Grothendieck’s inequality to hyperbolic programs, and what would the rounding scheme look like in this case? – See section 7 here for a clean description of hyperbolic programming and the associated basic concepts. – See section 2 here for a simple proof of Grothendieck’s inequality. – Known connections between hyperbolicity and probability will probably be useful here, see this survey by Robin Pemantle. # Around Thompson Sampling (Sebastien Bubeck) Let $\nu$ be a distribution over length $T$ sequences of functions from some compact set $K \subset \mathbb{R}^n$ to $[-1,1]$. Let $(\ell_1, \hdots, \ell_T) \sim \nu$, $x^ = \mathrm{argmin}_{x \in K} \sum_{t=1}^T \ell_t(x)$, and $(U_1, \hdots, U_T)$ be uniformly distributed in $[0,1]^T$ (and independent of $\ell_t's$). A bandit strategy is a sequence of random variables $x_1, \hdots, x_T$ where $x_t$ is measurable with respect to the bandit filtration $\mathcal{F}_t = \sigma(U_1, \hdots, U_t, \ell_1(x_1), \hdots, \ell_{t-1}(x_{t-1}))$. We are interested in the following quantity (where expectation is taken with respect to $\nu$): $R_T = \mathbb{E} \sum_{t=1}^T (\ell_t(x_t) - \ell_t(x^))$. Thomspon Sampling (TS) is the strategy where $x_t$ has the same the same distribution as $x^$ conditionally on $\mathcal{F}_t$. Here are some simple questions about TS. Approximation properties of TS Consider the special case where $K = \{e_1, \hdots, e_n\}$, and one can simulate $\nu$ by first sampling $\theta \in [0,1]^n$ from some distribution $\nu_0^{\otimes n}$ and then sampling independently each $\ell_t(e_i)$ from $\mathrm{Ber}(\theta_i)$. Let $R_T^$ be the infimum of $R_T$ over all bandit strategies for this setting. Guha and Munagala ask whether TS satisfies $R_T \leq 2 R_T^$ (more precisely they ask this question for a somewhat stronger notion a regret where one counts the number of times that a suboptimal action was selected). Note that in this setting we know that there exists a numerical constant $C>0$ such that $\sup_{\nu} R_T \leq C \sup_{\nu} R_T^$ (see this paper). Furthermore Guha and Munagala essentially prove that $R_T \leq 2 R_T^$ is true for $n=2$. TS and the geometry of the action set Russo and van Roy showed that if $\nu$ is supported on linear functions, then TS satisfies $R_T = O(\sqrt{n T \log \|K\|})$. This essentially implies that on $K=\mathbb{B}^n$ (the unit Euclidean ball in $\mathbb{R}^n$) one has $R_T = O(n \sqrt{T})$. However it is known that for this particular convex body there exists strategies with $R_T = O(\sqrt{n T})$ (see this paper). Is this latter bound true for TS? More generally understanding the interplay between TS and the geometry of the underlying action set is entirely open. TS and convexity In our recent work we showed that if $\nu$ is supported on convex functions, and if $n=1$, then a small modification of TS leads to $R_T = O(\log(T) \sqrt{T})$. Is this bound true for the basic TS? What about higher dimensions?	2017-09-24 19:12:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 79, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8887333273887634, "perplexity": 624.1676452820831}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690203.42/warc/CC-MAIN-20170924190521-20170924210521-00250.warc.gz"}
https://math.stackexchange.com/questions/3118868/filtering-property-of-dirac-delta-function	# Filtering property of Dirac $\delta$ function Evaluate $$\int_0^\infty f(x)\delta(x-1)dx$$where$$f(x)=\begin{cases}x^2,&0\le x<1\\\sin 2,&x=1\\x,&x>1\end{cases}$$ Attempt Since the function is discontinuous at $$1$$, I couldn't directly say the answer would be $$f(1)$$. I considered the $$\delta$$-sequence$$\delta_k(x-1)=\begin{cases}\frac k2,& \|x-1\|<1/k\\0,&\text{otherwise}\end{cases}\\\int_0^\infty f(x)\delta_k(x-1)dx=\frac k2\Big[\int_{1-1/k}^1x^2dx+\int^{1+1/k}_1xdx\Big]\\=1+\frac12\Big(\frac1{3k^2}-\frac1{2k}\Big)$$ Then taking the limit as $$k\to\infty$$, I got the answer as $$1$$, which seems correct intuitively as it is the limit of the function at $$x=1$$. But the answer key gives the answer $$\sin 2=f(1)$$. Is the key wrong or am I missing something? Edit The book defines the Dirac $$\delta$$ function (also called impulse function) $$\delta(t)$$ as the limit of a sequence of functions $$\{\delta_k(t)\}$$, where $$\int_{-\infty}^\infty\delta_k(t)dt=1$$For example,$$\delta_k(t)=\begin{cases}\frac k2,&\|t\|<1/k\\0,&\text{otherwise}\end{cases}$$is one such sequence. Then it goes on to say that the Dirac $$\delta$$ function can be understood as the generalized function$$\delta(t)=\begin{cases}0,&t\ne0\\\infty,&t=0\end{cases}$$For continuous functions in $$[0,\infty)$$ and $$a\ge0$$,$$\int_0^\infty f(t)\delta(t-a)=f(a)$$ • What is your definition of $\delta$? (for discontinuous functions, I mean) – Klaus Feb 19 at 13:59 • I'm not sure whether the initial integral is well defined in distribution theory. Suppose you modify it slightly so that the limit of $f$ for $x \rightarrow 1$ differs for $x>1$ and $x < 1$. What value do you expect to see? A discontinuous function is not a suitable test function as far as I know. – quarague Feb 19 at 14:07 • I don't see how you can apply $\delta$ to that function since it's not in $C^{\infty}$? – Drefain Feb 19 at 14:22 • @Klaus Please check the edit – Shubham Johri Feb 19 at 15:30 • Which text book are you using? Defining $\delta$ as a limit of functions seems shady to me and your example shows how this runs into trouble. Looking at the wikipedia article, if you define it through measure theory, you get $f(1)$ as the answer immediately per definition. If you define it via integrals with test functions which is what I was thinking about, this looks more complicated but the answer should be the same as the definitions are equivalent. – quarague Feb 19 at 15:54 The function $$f$$ has a removable discontinuity at $$1$$. Therefore, we have $$\langle \delta_{1},f\rangle=\lim_{x\to 1}f(x)$$ To see this, we denote $$\delta_n(x)$$ as a regularization of the Dirac Delta, which for any suitable test function, $$\phi(x)$$, satisfies $$\lim_{n\to \infty}\int_{-\infty}^\infty \delta_n(x)\phi(x)\,dx=\phi(0)$$ Now, suppose we have a function $$f(x)$$ that is of compact support and is smooth everywhere except at $$1$$ where it has a removable discontinuity. Let $$g(x)$$ be defined as $$g(x)=\begin{cases}f(x)&,x\ne 1\\\\\lim_{x\to 1}f(x)&, x=1\end{cases}$$ Then, since $$g$$ is a suitable test functions we see that \begin{align} \lim_{n\to \infty}\int_{-\infty}^\infty \delta_n(x-1)f(x)\,dx&=\lim_{n\to \infty}\int_{-\infty}^\infty \delta_n(x-1)g(x)\,dx\\\\ &=g(1)\\\\ &=\lim_{x\to 1}f(x) \end{align} And we are done! EXAMPLE: As an example, suppose $$\delta_n(x)$$ is the asymmetrically centered pulse function, which for $$a\in(0,1)$$ given by $$\delta_n(x)=\begin{cases}n&,x\in[-a/n,(1-a)/n]\\\\0&,\text{otherwise}\end{cases}$$ Then, we have \begin{align} \lim_{n\to\infty}\int_{-\infty}^\infty \delta_n(x-1)f(x)\,dx&=\lim_{n\to\infty}\left(n\int_{1-a/n}^{1+(1-a)/n} f(x)\,dx\right)\\\\ &=\lim_{n\to\infty}\left(n\int_{1-a/n}^{1} f(x)\,dx\right)+\lim_{n\to\infty}\left(n\int_{1}^{1+(1-a)/n} f(x)\,dx\right)\\\\ &=a\lim_{x\to 1^-}f(x)+(1-a)\lim_{x\to 1^+}f(x)\\\\ &=\lim_{x\to 1}f(x) \end{align} since the discontinuity at $$1$$ is removable and therefore the right-side and left-side limits are equal. NOTE: It is of paramount importance to understand that if $$f$$ has a removable discontinuity at $$x_0$$, then the functional $$\langle \delta_{x_0},f\rangle =\lim_{x\to x_0}f(x)$$ but if $$f$$ has a jump discontinuity at $$x_0$$, then the functional $$\langle \delta_{x_0},f\rangle$$ is not defined. In fact, I showed in This Answer, that if $$H$$ is he Heaviside Function, then $$\langle \delta_0,H\rangle$$ is meaningless. • Thank you for the detailed, well explained answer! This clears up a lot of things for me. – Shubham Johri Feb 23 at 7:23 • You're welcome. My pleasure. I'm pleased that this was helpful. – Mark Viola Feb 23 at 16:02	2019-08-21 23:07:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 45, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9376426339149475, "perplexity": 292.0245669211796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316549.78/warc/CC-MAIN-20190821220456-20190822002456-00063.warc.gz"}
https://www.joghr.org/article/12745	# INTRODUCTION Over the past two decades, substantial declines in maternal, newborn and child mortality have been observed among least developed countries.1–3 However, preventable deaths take place far too commonly4 with maternal deaths underpinned by pregnancy-induced hypertension, post-partum bleeding and infection5; newborn deaths linked to preterm birth complications, intrapartum events, and infection (sepsis / meningitis)6; and child mortality led by pneumonia and diarrhea incidence.1,6 Despite improved access to healthcare,5 the quality of these services often remains sub-standard.7–10 Systems to ensure availability and access to safe, effective and affordable reproductive, maternal, newborn and child health (RMNCH) commodities can make major inroads in the prevention and management of the most common causes of mortality.5 The UN Commission on Life-saving Commodities for Women and Children (UNCoLSC) was established to articulate key bottlenecks and define strategies to address them. In its report from September 2012,11 the Commission prioritized 13 low-cost and high-impact commodities across the RMNCH spectrum (Figure 1) that if implemented at-scale could make a substantial impact in reducing preventable deaths. The UNCoLSC also outlined a set of recommendations for addressing wider health system challenges that required concurrent attention – from procurement and regulatory systems and supply chains, to up-to-date health worker training materials (Figure 1). While limited in number, the 13 prioritized commodities were intended to act as tracers to provide a high-impact focus in the face of multiple competing priorities. Figure 1:UNCoLSC Recommendations to improve access to 13 Life-Saving Commodities (and primary condition) To take forward the UNCoLSC agenda, a multi-UN agency technical support and financing mechanism was established to accelerate progress towards maternal and child mortality goals among countries where progress was off-track. Three main strategies were implemented.12 First, a country engagement process was undertaken to generate RMNCH acceleration plans. This included a systematic assessment of RMNCH commodity-related bottlenecks, an identification of prioritized interventions, an assessment of technical and financial contributions from development partners (i.e. resource mapping), and an articulation of critical resource gaps. This country engagement process was led by the government, shaped by International Health Partnership principles,13 and informed by global and country-level partners. Second, catalytic financial resources were made available from a multi-UN agency financing mechanism (RMNCH Fund) to support RMNCH acceleration plans. Third, a network of technical resource teams provided additional support to address global barriers and facilitate country implementation. The aim of this paper is to examine country-level progress against the UNCoLSC recommendations over a five-year period. The specific objectives are to profile availability and access to the 13 commodities and the status of key health systems enablers among high-burden countries; to document specific areas of progress and remaining bottlenecks; and to generate learning to inform the Sustainable Development Goal framework and Universal Health Coverage (UHC) agendas. # METHODS To assess the baseline status and progress towards the UNCoLSC recommendations, an RMNCH situation analysis was conducted in countries receiving support from the RMNCH Fund. This analysis included a synthesis of existing data sources including governmental and partner documents (e.g. national strategic plans, essential medicines lists, treatment guidelines, regulatory and policy briefs, commodity registers, training curricula), and aggregated quantitative measures from various sources (e.g. health facility assessments, health and logistics management information systems). These data sources were complemented by semi-structured interviews with government officials, programme managers, regulatory and supply chain agencies and in-country partners. The situation analysis and performance indicators (Tables 1 and 2) were designed in consultation with the UNCoLSC technical resource teams (TRTs), a network of approximately 450 experts across 85 organizations, as well as the UNCoLSC Monitoring and Evaluation Advisory Group.12 Within each country, the situation analysis was typically completed over a 2-4 weeks by a trained facilitator, who collaborated with local ministries, UN country teams and in-country partners. These efforts were supported by a multi-UN agency Strategy and Coordination Team (SCT), which facilitated engagement with in-country partners, trained enumerators, coordinated content review, and performed standardized analysis to provide consistency of results across countries and over time. The SCT conducted quality assurance on each country assessment and reviewed the results, data sources, and list of experts interviewed with country teams to ensure accuracy and completeness. While national ministries had discretion over the timing of data collection, this study includes countries where a baseline round of data collection was conducted within one year of program initiation (2014-2015; see Online Supplementary Document Table A) and where a repeat round took place during 2016 or 2017. Results from country assessments were entered into a relational database (MS Access, Microsoft Inc, Seattle, WA, USA), processed using R software (R version 3.2.0, Vienna, Austria), and uploaded to a web-based platform and documented in summary reports for review by country teams. Detailed methods and initial results were previously published.12 All statistical analyses with estimates were conducted in STATA software version 14 (College Station, TX, USA). ## Quantitative indicators Quantitative indicators derived from nationally-representative population- and facility-based surveys are defined in Table 1. To assess change-over-time, indicators from the most recent available survey (i.e. endline) were compared to an earlier data collection (defined as since 2010 and prior to RMNCH Fund initial implementation within each respective country). Selection of data pairs (baseline and endline data points) prioritized comparable indicator generation methods and recency of collection. If data prior to RMNCH Fund implementation was unavailable, then the earliest data source after implementation was used to illustrate trends. In order to identify all available population- and facility-based survey data sources, the SCT consulted with country teams, global UN agencies and development partners to ensure dataset completeness. Results from population- and facility-based surveys were included if available by June 2019. For quantitative indicators, non-parametric paired t-tests were used, due to skewness, to assess statistical significance between baseline and endline distributions. ## Categorical indicators A range of categorical indicators for assessing well-known14,15 in-country health system and commodity-specific bottlenecks were defined through consultation with a global network of technical experts12 (Table 2). A standardized assessment criteria was developed (see Online Supplementary Document Table H) and dichotomous conditions to meet the minimum performance threshold were defined in Table 2 to ensure comparability across countries. These categorical indicators were collected during the country-specific situation analysis with baseline and endline time periods listed in Online Supplementary Document Table A. Indicators related to health systems are reported once per country, while commodity-specific indicators are reported once per commodity per country (i.e. up to 13 times per country). To evaluate statistical significant for categorical indicators, paired t-tests were used between average baseline and endline values. # RESULTS Between January 2013 to December 2017, 14 countries in sub-Saharan Africa and Southeast Asia underwent two assessment rounds including Bangladesh, Benin, Burkina Faso, Cameroon, Democratic Republic of the Congo (DRC), Ethiopia, Malawi, Mali, Nigeria, Pakistan, Senegal, Sierra Leone, Tanzania and Zambia (see Online Supplementary Document Table A for complete list and timeline of data sources). ## Commodity availability and logistics management Analysis of stock availability – defined as commodity physically available at health facility at time of assessment - is profiled in Table 1 and disaggregated by commodity in Figure 2. Collectively, there was a statistically significant increase of five percentage points in stock availability of life-saving commodities over the observation period (69% to 74%, P=.001, Table 1). Commodities with the largest improvement included Magnesium Sulfate (24 percentage point increase), Misoprostol (+25 pp) and injectable antibiotics for newborns (+11 pp). Notably, Misoprostol and Magnesium Sulfate had among the lowest baseline availability and thus greatest room for improvement. At most recent assessment, injectable antibiotics, oxytocin and implantable contraceptives were the most widely available (Figure 2). Conversely, a number of reproductive and child health commodities had declined in availability including female condoms (-11 pp), emergency contraceptives (-2 pp), and amoxicillin (-12 pp). In nearly half of facilities, Misoprostol, Chlorhexidine, and neonatal resuscitation equipment were unavailable at the most recent facility assessment. Table 1:Median percentage change over time for quantitative UNCoLSC indicators derived from health facility and population-based surveys - cumulative across 14 countries and commodities UNCoLSC Indicator and Definition to meet minimum threshold Baseline Endline RECOMMEND-ATION n Median % (95% CI) Median % (95% CI) Change Median P-value Supply and Awareness Percentage of facilities with stock available: % of health facilities with no commodity stock-out at time of assessment (for facilities authorized to provide the commodity) 109 69.0 (60.2 - 77.8) 74.0 (70.2 - 77.8) ▲ 5.0 0.001 Performance and accountability Percentage of facilities with recent training: % of health facilities with a health worker trained in service delivery within the past 12 or 24 months [1,2] 31 38.0 (28.7-47.3) 46.0 (30.5-61.5) ▲ 8.0 0.003 Percentage of facility with job aids or check lists: % of health facilities where relevant job aids / check lists exist at the time of assessment [1] 18 53.3 (39.7-66.9) 66 .1 (56.6-75.5) ▲ 12.8 0.071 Demand & Utilization Population coverage rate: % of affected population with specified medical condition receiving treatment with appropriate life-saving commodity 36 9.5 (5.4 -24.4) 21.6 (13.4 - 29.7 ▲ 12.1 0.111 P-value: ** P-value < 0.01 * P-value < 0.05 Notes:[1] Reported by the four service delivery areas (reproductive, maternal, newborn or child). [2] Recent training defined as during 12 or 24 months preceding assessment depending on data source. SARA and SPA is 24 months while all other sources, such as SDP, recent training defined as ’during 12 months preceding assessment’. Figure 2:Median percentage of health facilities with stock availability at time of assessment by commodity across countries (number of countries with available survey result) * In facility assessments, neonatal resuscitation is typically reported as availability of neonatal resuscitation “bag and mask”. \|-------\| Minimum (left) and Maximum (right) values reported by a country for the commodity. Stock-outs at national warehouses occurred more often over time (Table 2) and were most prevalent at the most recent assessment for magnesium sulfate, chlorhexidine, and zinc (see Online Supplementary Document Table B). Conversely, female condom, contraceptive implants, and injectable antibiotics for newborns had the fewest national stock-outs by 2017. Across countries, only a handful of commodities (7%) were added into eLMIS to track distribution from central warehouses to service delivery points (Table 2). By 2017, just over half (57%) of life-saving commodity were tracked in eLMIS across countries (Table 2), while comprehensive monitoring platforms exist in less than half of countries. ## Health worker performance Over the observation period, health facilities with a recently trained staff member rose by eight percentage points (P=.003) to 46% (Table 1). In addition, availability of job-aids and/or checklists at health facilities increased to 66% (+13 pp). Moreover, to support workforce performance, 12% and 19% of life-saving commodities were added to national training curricula and job aids, respectively, across countries (Table 2). ## Regulatory efficiency and quality strengthening Regulatory efficiency indicators were the strongest performers over time (Table 2). For example, across countries, an additional 20% of the life-saving commodities were collectively added to national essential medicines lists in the preferred formulation, 11% of commodities were incorporated into national treatment guidelines, and another 12% of commodities were fully registered in-country (each P<.005). Collectively, regulatory indicators also had the fewest outstanding gaps by 2017. In this study, countries strengthened the sampling and quality testing of drugs (i.e. post-market surveillance) and nearly all countries procured drugs from good manufacturing practice (GMP) accredited manufacturers (Table 2). Unfortunately, the capacity of national control laboratories has declined over time and patient safety monitoring remains low. ## Generating Demand and Reaching Women and Children In this study, modest gains were made in establishing national demand generation and behavior change plans for life-saving commodities that included domestic budget allocations (Table 2). During the observation period, small improvements were seen for removing financial access barriers (i.e. user fees); however, gaps still exist in approximately half of service delivery areas across countries (Table 2). Where data were available, coverage rates (i.e. percent of affected population receiving appropriate treatment) increased two-fold over the observation period (P=.111), but remains low for many commodities (Table 1), such as ORS (median 37%), Zinc (27%), and contraceptive implants (6%) (Online Supplementary Document Figure F). ## Other Health System Enablers Over time, results-based financing programs were more prevalent and all 14 countries had RMNCH coordinating mechanisms by 2017 (Table 2). However, countries struggled to finalize national costed RMNCH plans with secured domestic budget allocation as well as develop commodity security strategies. Table 2:Proportion of categorical UNCoLSC indicators meeting minimum threshold over time - cumulative across 14 countries and across commodities (where appropriate) UNCoLSC Indicator and Definition to meet minimum threshold Baseline Endline RECOMMEN- DATION n (2013-16) Mean (2016-17) Mean Change P-value Regulatory Efficiency National Essential Medicines List: Commodity is included in the national EML with a context-appropriate level of commodity specification and/or formulation 168 71% 91% ▲ 20 < 0.001 National treatment guidelines: National treatment guidelines on interventions to deliver commodities exist, are updated regularly, & refer to latest WHO guidance 182 76% 87% ▲ 11 0.001 Registered in-country: The commodity is fully registered in-country under approved & relevant formulations 135 64% 76% ▲ 12 0.004 Prescription authority: The commodity is prescribed at lowest appropriate level of service delivery (per national policy and essential intervention package) 179 77% 89% ▲ 12 0.001 Quality Strengthening GMP-accredited manufacturers: Public sector procurement is done only from manufacturers with a valid GMP accreditation for domestic & international purchases 14 86% 86% 0 1.000 National medicines control laboratory: At least one national medicines control laboratory exists in-country that is certified by any standards accreditation agency 14 36% 29% ▼ -7 0.336 Monitoring quality of medicines: Functioning systems exist for monitoring quality of medicines (i.e. post-market surveillance). 14 43% 64% ▲ 21 0.040 * Monitoring patient safety for medicines: Functioning systems exist for monitoring patient safety for medicines (i.e. pharmacovigilance). 14 43% 50% ▲ 7 0.583 Supply and Awareness Forecasting Tools: Existence of a forecasting tool or method used routinely for forecasting needs for RMNCH medicines and medical devices 14 64% 64% 0 1.000 Supply chain training to districts: Training in supply chain management for RMNCH commodities has been deployed to all SDPs at the district level 14 50% 57% ▲ 7 0.336 Comprehensive national eLMIS: There is a single electronic national LMIS or an interoperable platform that tracks commodity availability and distribution from first point of warehousing to point of service for each RMNCH service area AND automatically compiles/ aggregates data on a continuous basis 14 7% 29% ▲ 21 0.189 Tracked in eLMIS: Commodity availability is tracked from first point of warehousing to service delivery point by an electronic LMIS 171 50% 57% ▲ 7 0.134 National stock-outs: No commodity stock-outs at the national level in the past 12 months (for EML commodities) 128 64% 59% ▼ -5 0.291 Performance and accountability Training curricula (national): In-service training curricula exist at the national level for interventions that deliver the commodity at the appropriate level of care 182 73% 85% ▲ 12 0.001 Job aids or check lists (national): National level job aids have been developed or updated, include the specific commodity, are clearly written & refer to WHO version 182 68% 86% ▲ 19 < 0.001 Reaching Women and Children Policy against user fees: National policy states that patients should not be assessed any fee or out of pocket expense for the LSCs and related services at the point of service delivery [2] 56 46% 52% ▲ 5 0.370 Demand & Utilization Demand Generation: National RMNCH plan includes costed demand generation/behavior change communication activities with a budget allocated 14 71% 79% ▲ 7 0.336 Innovative financing Results-based financing mechanism: The country entered into an agreement with a results-based financing mechanism to increase access to LSCS and related services 14 21% 43% ▲ 21 0.189 RMNCH Coordination [1] Coordination mechanism exists: A functional national coordination mechanism on RMNCH exists (or RMNCH is included in broader coordination mechanism) 14 64% 100% ▲ 36 0.019 * RMNCH plan costed and budgeted: A national RMNCH plan exists that is costed with a budget allocated for interventions for LSCs at national and sub-national levels 14 57% 50% ▼ -7 0.720 Commodity security strategy: National commodity security strategy for LSCs have been developed and approved by the Ministry of Health 14 43% 36% ▼ -7 0.336 Total (average): 1,551 65% 76% 11% Systems-related indicator (recorded once per country) Commodity-specific indicator (recorded once per commodity per country) P-value: ** P-value < 0.01 * P-value < 0.05 Notes: [1] RMNCH Coordination is a performance indicator but not a UNCoLSC recommendation [2] Policy against user fees is reported by the four service delivery areas (reproductive, maternal, newborn or child). # DISCUSSION This study assessed the ambitious UNCoLSC recommendations to improve availability and access to 13 life-saving commodities among countries in Africa and Asia with among the highest global burden of preventable maternal child deaths. Service readiness, including availability of commodities, improves the likelihood of receiving good quality services.7,16–18 Over the observation period, commodities were collectively more available; however, some commodities fared better than others (Figure 2). Commodities to reduce maternal deaths, treat newborn infection, and long acting methods for pregnancy prevention became more widely available. Scale-up of contraceptive implants have been a concerted focus of recent international and domestic efforts,19,20 while the low cost and long regulatory history of oxytocin and injectable antibiotics, such as gentamicin, facilitate efficient delivery through the supply chain.21–23 Conversely, access to misoprostol, chlorhexidine, antenatal corticosteroids and newborn resuscitation equipment still hovers around half of health facilities. Emerging science on the effective use of Chlorhexidine24 and antenatal corticosteroids15,25 produced operational re-examination, clouded guidance and limited scale-up. While the administration of misoprostol to prevent and treat post-partum hemorrhage is expanding to home-births,26 it can also be used to induce abortion, which can limit cultural adoption and the need for local stock availability. Maintenance of national stock levels for essential commodities declined over the observation period and requires urgent attention. Increasing the use of centralized procurement through partners, such as family planning commodities, is cost-effective and efficient, but needs complementary in-country coordination.27,28 Often the discordance of national budget and cash flow cycles impedes the efficient bulk purchase of commodities by national governments. Innovative bridge funding mechanisms or working capital facilities (with prompt payback terms) to support local procurement through established systems could reduce stock-outs and save money.29 Once procured, ensuring efficient, timely and equitable distribution of commodities within country relies on electronic logistics management information systems (eLMIS). In-country supply chains are often fragmented across multiple partners along with the corresponding eLMIS, which limits performance and timely corrective action.30,31 The Health Data Collaborative (HDC) has set out to support the harmonization and more efficient use of information systems.32 Expanding the domain of HDC or applying the same principles to complex issues such as supply chains and eLMIS is warranted to reduce fragmentation and improve efficient data use. Inadequate workforce competences is an underlying operational barrier to effective healthcare delivery.14,33 To further strengthen the patient-provider interface, countries in this study improved national health workforce resources (e.g. training curricula and job-aids / checklist) as well as facility-based training. Since countries trained staff at just under half of health facilities within 12 or 24 months, the ability to reach all facilities could take approximately three to five years (assuming equitable distribution). This collective training capacity should inform introduction rates for new commodity guidelines and technology in the future. Prioritized investments and improved metrics on health workforce training and guidance (e.g. job-aids) portend enhanced quality of service delivery. A new opportunity for acceleration is the Network for Improving Quality of Care (QOC) for Maternal, Newborn and Child Health,34,35 which aims to establish local quality improvement teams to identify quality shortfalls and undertake quick cycles of problem-solving for facility-driven improvement and preparedness. Unfortunately, the utilization and quality of facility-level care is clouded by the lack of coverage data36–38 for administration of life-saving interventions at health facilities (e.g. oxytocin, injectable antibiotics). Moreover, while this study had access to the health management information systems, such as DHIS2, in most countries, the data available from these sources at time of assessment was deemed insufficient to generate a valid estimate of intervention coverage over time. Delays in regulatory approval limit pharmaceutical companies interest in providing drugs to developing countries, which creates a market opportunity for an influx of counterfeit drugs.39,40 In this study, regulatory indicators exhibited significant improvement; however, they require regular review to ensure the latest treatment guidelines and formulations are adopted and product registration does not expire. With the rise of counterfeit drugs in circulation,41 countries need quality controls and monitoring of medicines to ensure the safety and effectiveness of drugs is intact when reaching women and children. Aside from post-market surveillance, there was little to no improvement in quality strengthening conditions, which have been persistent intractable bottlenecks.39 Several interconnected regional efforts, such as the African Medicines Regulatory Harmonization (AMRH) Initiative,42 continue to strengthen fragmented regulatory environments and improve the availability of safe and effective commodities. The collective improvements against the UNCoLSC recommendations mirrored the predominance of supply side investments prioritized by country teams and financed via the RMNCH Fund (see Online Supplementary Document Figure B). More than 80% of these expenditures were allocated towards two UNCoLSC recommendations: supply and awareness (e.g. commodity procurement, integrated LMIS, and supply chain management); and health worker performance and accountability (e.g. staffing, training, supervision, job-aids / checklists). These RMNCH acceleration plans were developed by government-led RMNCH coordinating mechanisms using a resource mapping across partners43,44; therefore, expenditures against the RMNCH Fund provide a window into national priorities and funding gaps facing governments, UN agencies and development partners (see Online Supplementary Document Table G). An example is provided below (Box 1), which outlines the experience of the United Republic of Tanzania with the RMNCH Fund. In terms of sustainability, the RMNCH Fund provided short-term catalytic funding towards the UNCoLSC agenda. This short-term funding is small relative to potential domestic allocations; however, domestic budget allocations were often unpredictable and vacillated year-to-year across countries (Table 2). This type of shorter-term gap filling to support existing country-derived strategic plans should complement new longer-term funding sources, such as the Global Financing Facility, as they are actuated and put towards the same strategic vision.45,46 There were several limitations in this analysis that are important to draw attention to. First, gaps in data availability, such as maternal and newborn service delivery at health facilities, persist and constrained interpretation of community impacts. This study used the RMNCH Situation Analysis process, which is relatively easy and low-cost to conduct, but relies on existing data sources.44,45 The tool can be adapted for country context, such as sub-national use or expanded commodity and equipment list, but d.oes not establish new primary quantitative datasets for analysis. Second, the identification of operational bottlenecks relies on performance thresholds defined by a panel of experts; however, during enumeration, participant input and categorization by individual enumerators may be subjective and introduce bias. Third, countries assessed that met the requirements for this study (e.g. multiple data collection rounds of the RMNCH Situation Analysis) may not represent conditions experienced in other countries or regions. Fourth, this study did not analyze global-level UNCoLSC recommendations (e.g. product innovation and market shaping), which were assessed in a previous study,12 but could affect progress against related in-country bottlenecks. Next, this study utilized point estimates from facility- and population-based surveys to statistically analyze trends. The raw datasets had limited availability and their analysis was beyond the scope of this study, but may generate more decisive results and worthy of future research. Finally, the timeliness and consistency of available data sources may obfuscate recent changes. For example, this study utilized health facility assessments supported by various international institutions, including UNFPA (service delivery point survey), WHO (service availability and readiness assessment), and USAID (service provision assessment). While UNFPA has by far the most frequent data collection, the WHO and USAID tools are more comprehensive. Unfortunately, the tools are not entirely compatible among agencies (such as content and methodologies) and collection schedules are uncoordinated. The inter-agency HDC has undertaken an effort to harmonize these tools and data collection schedule among partners.32 # CONCLUSION Over a five-year period, significant improvements in commodity availability and health workforce training at the facility level were observed overall. Important commodity-related bottlenecks, such as coordination mechanisms, regulatory requirements, and national training curricula and job-aids, were near fully rectified. However, critical supply chain and medicine safety functions showed inconsistent improvement and remain an impediment to universal access. Leveraging the lessons learned from this unfinished UNCoLSC agenda can help in-country and global initiatives – such as the African Medicines Regulatory Harmonization, Global Financing Facility, Health Data Collaborative, and Quality of Care Network - address these remaining barriers to women and children receiving life-saving commodities and ultimately reaching universal health coverage. Box 1.Implementation Experience: United Republic of Tanzania The United Republic of Tanzania received two rounds of financing: US$4.0 million in early 2014; and US$11.9 million in 2015 from the RMNCH Fund. Planning of grant investments was led by the government and involved diverse stakeholders including multiple governmental agencies, UN organizations, and implementing partners. All interventions selected for grant funding were consistent with the national ‘Sharpened One Plan’47 as well as the Big Results Now48 agendas. The initial grant supported the registration of 12 Life Saving Commodities (with the exception of Chlorohexidine); roll out of information systems (eLMIS and ILS gateway) to assist supply chain management and p rocurement of essential commodities; and nationwide Comprehensive Emergency Obstetric and Newborn Care (CEmONC) assessment of facilities. The second round of engagement with RMNCH Fund, broadened the activities to include upgrade of eight facilities to CEmONC, 67 ambulances procured and delivered to districts, and the establishment of two blood banks to help deliver comprehensive maternal care. To initiate the grant process, a diverse, multi-sector coordinating platform was established with regular meetings and discussions among stakeholders. This improved planning, resource allocation, and partnership between the Government and the UN agencies. The grant terms of the RMNCH Fund were flexible and allowed the Tanzanian coordinating platform autonomy to select and adjust investments in RMNCH activities as needed – starting with a life-saving commodities focus and then broadening. This enabled strategic and adaptive selection of interventions to meet national and local needs. Lastly, the timing and adaptive use of funds helped lay the foundation for future investment mechanisms, such as the Global Financing Facility (GFF). Acknowledgements: Special thanks to Desmond Koroma, Lazasoa Raharimanjato, Kavitha Viswanathan and Amani Siyam. Funding: No funding was received for the writing of this manuscript. [Note: Financing for RMNCH Fund was provided by Governments of Norway (NORAD) and United Kingdom (DFID), but they had no role in the writing of the manuscript.] Author contributions: BN, DS, NS, and PP drafted the manuscript. All authors contributed to the design of the assessment, analysis of data, writing of the manuscript, and agree with the results and conclusions. Competing interests: The authors completed the Unified Competing Interest form at www.icmje.org/coi_disclosure.pdf (available upon request from the corresponding author), and declare no conflicts of interest. Correspondence to: Bennett Nemser, MBA, MPH University of the Western Cape, Cape Town, South Africa [email protected] or [email protected]	2023-03-25 07:59:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.197751984000206, "perplexity": 13394.393383748593}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945317.85/warc/CC-MAIN-20230325064253-20230325094253-00266.warc.gz"}
https://www.r-bloggers.com/2015/08/rcppgsl-0-3-0/	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 RcppGSL just arrived on CRAN. The RcppGSL package provides an interface from R to the GNU GSL using our Rcpp package. Following on the heels of an update last month we updated the package (and its vignette) further. One of the key additions concern memory management: Given that our proxy classes around the GSL vector and matrix types are real C++ object, we can monitor their scope and automagically call free() on them rather then insisting on the user doing it. This renders code much simpler as illustrated below. Dan Dillon added const correctness over a series on pull request which allows us to write more standard (and simply nicer) function interfaces. Lastly, a few new typedef declaration further simply the use of the (most common) double and int vectors and matrices. Maybe a code example will help. RcppGSL contains a full and complete example package illustrating how to write a package using the RcppGSL facilities. It contains an example of computing a column norm — which we blogged about before when announcing an much earlier version. In its full glory, it looks like this: #include <RcppGSL.h> #include <gsl/gsl_matrix.h> #include <gsl/gsl_blas.h> extern "C" SEXP colNorm(SEXP sM) { try { RcppGSL::matrix<double> M = sM; // create gsl data structures from SEXP int k = M.ncol(); Rcpp::NumericVector n(k); // to store results for (int j = 0; j < k; j++) { RcppGSL::vector_view<double> colview = gsl_matrix_column (M, j); n[j] = gsl_blas_dnrm2(colview); } M.free() ; return n; // return vector } catch( std::exception &ex ) { forward_exception_to_r( ex ); } catch(...) { ::Rf_error( "c++ exception (unknown reason)" ); } return R_NilValue; // -Wall } We manually translate the SEXP coming from R, manually cover the try and catch exception handling, manually free the memory etc pp. Well in the current version, the example is written as follows: #include <RcppGSL.h> #include <gsl/gsl_matrix.h> #include <gsl/gsl_blas.h> // [[Rcpp::export]] Rcpp::NumericVector colNorm(const RcppGSL::Matrix & G) { int k = G.ncol(); Rcpp::NumericVector n(k); // to store results for (int j = 0; j < k; j++) { RcppGSL::VectorView colview = gsl_matrix_const_column (G, j); n[j] = gsl_blas_dnrm2(colview); } return n; // return vector } This takes full advantage of Rcpp Attributes automagically creating the interface and exception handler (as per the previous release), adds a const & interface, does away with the tedious and error-pronce free() and uses the shorter-typedef forms for RcppGSL::Matrix and RcppGSL::VectorViews using double variables. Now the function is short and concise and hence easier to read and maintain. The package vignette has more details on using RcppGSL. The NEWS file entries follows below: #### Changes in version 0.3.0 (2015-08-30) • The RcppGSL matrix and vector class now keep track of object allocation and can therefore automatically free allocated object in the destructor. Explicit x.free() use is still supported. • The matrix and vector classes now support const reference semantics in the interfaces (thanks to PR #7 by Dan Dillon) • The matrix_view and vector_view classes are reorganized to better support const arguments (thanks to PR #8 and #9 by Dan Dillon) • Shorthand forms such as Rcpp::Matrix have been added for double and int vectors and matrices including views. • Examples such as fastLm can now be written in a much cleaner and shorter way as GSL objects can appear in the function signature and without requiring explicit .free() calls at the end. • The included examples, as well as the introductory vignette, have been updated accordingly. Courtesy of CRANberries, a summary of changes to the most recent release is available. More information is on the RcppGSL page. Questions, comments etc should go to the rcpp-devel mailing list off the R-Forge page. This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.	2021-11-28 23:59:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21775013208389282, "perplexity": 7336.193395285879}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358673.74/warc/CC-MAIN-20211128224316-20211129014316-00327.warc.gz"}
https://www.gamedev.net/forums/topic/196542-bullet-holes---correct/	#### Archived This topic is now archived and is closed to further replies. # Bullet Holes Correct? This topic is 5185 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts My current method for creating bullet holes is kind of hokie looking, so I'm turning to everyone here for advice. As of now I'm using billboards, which I believe is my largest mistake. Anyway, I may as well throw some code out there: D3DXVECTOR3 HitPos; BOOL bHit; DWORD dwFace; FLOAT fBary1, fBary2, fDist = -1.0f; DWORD count; D3DXIntersect(g_Room.m_pMesh, &g_Player.GetPosition(), &LPos, &bHit, &dwFace, &fBary1, &fBary2, &fDist, 0, &count); if(fDist != -1.0f) { HitPos = g_Player.GetPosition() + Normalize(LPos) * fDist; D3DXVECTOR3 verts[4]; D3DXVECTOR3 pos = HitPos; verts[0] = D3DXVECTOR3(pos.x - 0.1f, pos.y+0.1f, pos.z); verts[1] = D3DXVECTOR3(pos.x + 0.1f, pos.y+0.1f, pos.z); verts[2] = D3DXVECTOR3(pos.x - 0.1f, pos.y-0.1f, pos.z); verts[3] = D3DXVECTOR3(pos.x + 0.1f, pos.y-0.1f, pos.z); g_BulletHoles.push_back(new BILLBOARD("images//bullethole.bmp", verts)); } } So I'm using D3DXIntersect to get the distance, and then getting the position and the vertices, which in turn, create the BILLBOARD. I believe that what I SHOULD be doing is, instead of creating a BILLBOARD, I should create a series of rotation matrices to align the image with the plane of the wall. Something like this? D3DXMATRIXA16 matTemp, matRot; D3DXVECTOR3 pos, normal, axis; //Assume pos is the position of the bullet hole (a point on the wall) //Assume normal is the normal to the wall's plane D3DXMatrixRotationAxis(&matRot, axis, AngleBetween(normal, pos - g_Player.GetPosition()); So, correct me if I'm wrong, but this is what I think I should do, but I have no idea how to calculate the variable axis. Any ideas? Edit: Why in the world is the whole thing red?! Please pardon the odd coloring. [edited by - ms291052 on December 15, 2003 11:00:45 PM] [edited by - ms291052 on December 16, 2003 7:43:20 AM] [edited by - ms291052 on December 16, 2003 7:43:52 AM] ##### Share on other sites it seems that the double slash ends the command (first set of code), and so it doesnt see a closing quote... if u look closly the last two brackets are red... g_BulletHoles.push_back(new BILLBOARD("images//bullethole.bmp", verts)); } } [edited by - Pirosan on December 15, 2003 11:43:54 PM] ##### Share on other sites Well I just tried to fix it but noooo, it took away one of my source blocks (???) and stayed red, so I suppose I''ll just have to deal. bump	2018-02-25 20:10:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2190379649400711, "perplexity": 5277.678752856465}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816912.94/warc/CC-MAIN-20180225190023-20180225210023-00091.warc.gz"}
https://physics.stackexchange.com/questions/604330/does-time-dilation-depend-on-approximation-or-distancing-between-observers	# Does time dilation depend on approximation or distancing between observers? Wikipedia's definition of time dilation is: "Time dilation is a difference in the elapsed time as measured by two clocks due to a relative velocity between them". It doesn't mention if the observers are approaching one another or distancing themselves. However, most spacetime diagrams which explain time dilation show observers distancing themselves. The following scenario illustrates my understanding of time dilation: Suppose you are in a spaceship moving at a constant speed towards a light source that emits a flash of light every 1 second (in its reference frame). If you move towards it at a constant speed, you will perceive the flashes of light coming with less than 1 second between them. Therefore, for you the clock of the light source will be moving faster. If you move with a constant speed opposite to the light source, you will receive the flashes of light more than a second apart. So, for you, the light source clock will be moving slower. I'm not sure if the above description is 100% correct, but it's how I understand it. Also, most descriptions of the Twin Paradox say that they perceive each other's clocks running slower on the trip towards the star, but running faster in the trip back to Earth. Wikipedia's definition seems to imply that any relative velocity causes time dilation, be it approaching or distancing. Which is correct? I need no more than a simple, layman explanation. "Therefore, for you the clock of the light source will be moving faster": you cannot conclude that, even in classical mechanics if you approach a source that emits pulses you will see that the frequency of pulses is larger than for the emitter, but it does not imply that the clock will move faster, only that you are moving towards the pulses. It is called doppler effect. The reverse if you move away. Time dilation is independent of the direction of motion, it only depends on the relative speed between observers. What you say about the twin's paradox is also incorrect, the clock speeds up only when the travelling twin accelerates to return, otherwise, at constant relative speed, each twin will see the other's clock running slower. • Thanks for the answer, it's absolutely correct. My misunderstanding of the twin paradox (the clocks running faster on the return journey) comes from this video. Maybe it's because the video uses "flashes of light" to account for the passage of time? Do you know a way to reconcile your (absolutely correct) interpretation with the one from the video? Any tips will be extremely welcome. – user137288 Dec 31 '20 at 17:39 • I will watch the video soon – Wolphram jonny Dec 31 '20 at 18:59 • @JEB addressed that in his answer. Take a look later :) thanks – user137288 Dec 31 '20 at 19:19 Wikipedia's definition of time dilation is: "Time dilation is a difference in the elapsed time as measured by two clocks due to a relative velocity between them" That's wrong. The usual special-relativistic time dilation setup requires at least three clocks, two of which must be at relative rest and synchronized by the Einstein convention. The reason you're having trouble understanding this kind of time dilation is that it makes no physical sense. In the most extreme case, all of spacetime is assumed to be filled with these comoving, synchronized clocks, which pass through other objects like ghosts. Because the clocks are everywhere, there's no meaningful distinction between objects approaching each other and getting farther apart. In fact, introducing a localized "observer" makes no sense, since their position, and very existence, is irrelevant to the math. Sometimes they're even said to be "observing" the readings on the infinite network of clocks in real time, which is literally impossible since it would require a faster-than-light signal from the clocks to the observer. The "Andromeda paradox", which was mentioned in another answer, is a good example of the extent to which this silliness has harmed people's ability to understand special relativity. The paradox is that if you built two networks of Einstein-synchronized clocks stretching all the way out to the Andromeda galaxy, and the clocks near Earth were in sync, then the clocks in Andromeda that were not comoving would be out of sync. It's not normally stated that way but that's the actual setup they're talking about. The resolution is that, yes, that would happen, but, I mean, you built two networks of clocks stretching to Andromeda; the fact that they're out of sync is not the weirdest thing here. In the real world there are no such clocks, neither naturally occurring nor man-made, so it's just irrelevant. There is a version of special-relativistic time dilation that makes more sense, and that's if two (or more) clocks are compared at a common point, then move at different (and in general time-varying) speeds, and then are brought to a common point again and re-compared. The elapsed time on each clock is given by $$\int_{t_i}^{t_f} dt/γ(t)$$, where $$t$$ is the coordinate time of some inertial frame and $$γ$$ is the SR time dilation factor of the clock relative to the same frame. The answer is independent of what frame you pick, as long as you're consistent. The twin paradox is an example of a problem in this category – though most discussions make an utter mess of it by talking about "observers" with magical faster-than-light sensing abilities, and by not being consistent in their choice of inertial frame. The Wikipedia quote is still wrong if it's talking about this, since the velocity of the clocks relative to each other doesn't matter. What matters is their velocity relative to whatever inertial frame you picked (which is still made of a bunch of other clocks, but it's better since you can pick a lab frame whose clocks actually do exist). Suppose you are in a spaceship moving at a constant speed towards a light source that emits a flash of light every 1 second (in its reference frame). If you move towards it at a constant speed, you will perceive the flashes of light coming with less than 1 second between them. [...] If you move with a constant speed opposite to the light source, you will receive the flashes of light more than a second apart. Here you're describing a simple, physically sensible experiment: two localized objects emitting and detecting physical light pulses, with no secret clocks. Your analysis is correct. The ratio of emitted and detected pulse rates is given by the Doppler formula. • Thank you for these excellent, thought provoking answers. I'll return to them many times yet in the future. Do you mind clarifying what you meant by "The answer is independent of what frame you pick, as long as you're consistent" -- At first, it sounded like the elapsed time would result the same on every clock at the end of the experiment. However, that can't be right, otherwise there would be no time dilation. Do you mind clarifying? (English is not my first language so I'm sorry) – user137288 Jan 2 at 2:11 Time dilation depends only on the magnitude of relative velocity, not direction. That is captured in the Loedel Diagram. It is a Minkowski diagram with each frame moving in opposite directions: The power of this diagram is that it shows the paradoxical symmetry of time-dilation, and from a Newtonian mindset, it is paradoxical, but in relativity, as the observers change distance, the bias defining their "clock synchronization" changes in manner that allows each observer to consistently say the other's clock ticks at $$1/\gamma$$. We'll come back to it. The linked Ted-Ed video on the twin paradox only explains what single local observers see while looking out their windows, and what they see is a doppler shifted signal: $$f_{\pm} = f \sqrt{\frac{1 \pm \beta}{1 \mp \beta}}$$ (Note that the radical sign brings in a important symmetry that does appear in the sound-wave doppler shift...a symmetry that is broken by the existence of an aether, or stationary oscillating medium.) The standard figure is shown: As seen in the figure, traveling twin see half $$f_-$$ followed by half $$f_+$$, while Earth twin sees lots of $$f_-$$ followed by a short period of $$f_+$$, and it works out so that the total number of pulses seen by Earth is a factor $$\gamma = \frac 1 {\sqrt{1-\beta^2}}$$ fewer than those seen by Space twin. The does not really explain the Twin Paradox, because as you point out: each observer sees the other twin's clock moving slower on both inertial legs of the trip. To resolve this, we do not want to consider what the twins "see", meaning physically see looking out the window, as that is corrupted by Doppler shifts. Rather, we consider each twin to have access to a co-moving lattice of synchronized clocks (and rulers) with which they can record and reconstruct the space and time at any moment along their trip. Defining $$T=L/\beta$$ to be the time it takes for 1 leg of the trip, then what is measured is: 1. Earth measures Earth clock ticking: $$T+T = 2T$$ 2. Earth measures Space clock ticking: $$T/\gamma+T/\gamma = 2T/\gamma$$ 3. Space measures Space clock ticking: $$T/\gamma+T/\gamma = 2T/\gamma$$ 4. Space measures Earth clock ticking: $$T/\gamma^2+T/\gamma^2 = 2T/\gamma^2$$ meaning space twin has "missing" time: $$\Delta T = 2T - 2T/\gamma^2 = 2T(1-(1-\beta^2))=2T\beta^2 =2L\beta$$ What happened? Well, when space twin turned around, he needed a completely new lattice of co-moving observers, as his definition of "now" at Earth jumped forward by $$\Delta T$$, as shown: The line of simultaneity at turn around changes from the top blue line to the bottom red line. Note that space twin does not see Earth's clock running faster, rather, it is a recalibration of the bias required to synchronize the clocks. (It is reversible, if the twin turns around again, then he goes back to the blue lines and the time on Earth has now gone backwards, which is a perilous situation for time-dilation, but it completely acceptable for clock bias). That is the resolution of the twin paradox; however, the result remains paradoxical. It means at the turn around point (or event), the time on Earth is not defined. Depending on velocity, it can vary by $$\pm L/c$$. What that means is the right now, 12/31/20, the time on Alpha Centari (4.37 ly away) can be anytime between 8/19/16 and 5/14/25. That peculiarity of relativity is known as the Andromeda Paradox, and leads to 4-dimensionalism, which is truly paradoxical. • Thanks for the excellent answer. Do you mind expanding a bit on the Andromeda Paradox you mention? What do you mean by "the time on Earth is not defined"? – user137288 Dec 31 '20 at 19:08 • "Depending on velocity, it can vary by ±L/c" -- what can vary? Is it the amount of "not defined" time? – user137288 Dec 31 '20 at 19:17 • If instead of a discontinuous change in reference frame we have a quick but smooth acceleration, the accelerating twin will see (or calculate that) the twin on earth speeding up, instead of a jump in time, because the simultaneity lines will also change smoothly. But I might be wrong on this – Wolphram jonny Dec 31 '20 at 19:52 • If you are a distance $L$ from Earth, you cannot define the time on Earth uniquely. It can be anywhere between now (defined in its rest frame) and now$\pm L/c$ – JEB Dec 31 '20 at 19:56 • This formula (±L/c) corresponds to the elapsed time (in the time axis of this last Minkowski diagram) between the top blue line and the bottom red line, is that correct? – user137288 Jan 2 at 16:14	2021-05-16 03:43:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7109349370002747, "perplexity": 430.0383657778278}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991659.54/warc/CC-MAIN-20210516013713-20210516043713-00288.warc.gz"}
https://mathhelpforum.com/tags/pnpn/	# p\|npn 1. ### p\|((n^p)-n) Someone ask me to prove that p\|((n^p)-n) for p=prime numbers and n=all integers I don’t know how to write a formal proof. But I know that the problem has 2 condition : Condition 1 p\|n^p and p\|n so p\|((n^p)-n) (if d\|a and d\|b then d\|(a-b) For example 2\|((4^2) - 4) Condition 2...	2020-02-17 18:54:48	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8870499730110168, "perplexity": 4252.092844830264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143079.30/warc/CC-MAIN-20200217175826-20200217205826-00337.warc.gz"}
https://socratic.org/questions/how-do-you-solve-2-8-2y#582446	# How do you solve -2 = 8 - 2y? Mar 26, 2018 See below #### Explanation: $- 2 = 8 - 2 y$ $- 2 \left(- 8\right) = \left(8\right) - 8 - 2 y$ (Subtracting 8 from both sides) $\frac{- 10}{- 2} = \frac{- 2 y}{-} 2$ (divide both sides by -2) $y = 5$	2022-01-28 06:52:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4941852390766144, "perplexity": 9919.610965305355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305420.54/warc/CC-MAIN-20220128043801-20220128073801-00693.warc.gz"}
https://www.illustrativemathematics.org/content-standards/1/OA/C/6	Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., $8 + 6 = 8 + 2 + 4 = 10 + 4 = 14$); decomposing a number leading to a ten (e.g., $13 - 4 = 13 - 3 - 1 = 10 - 1 = 9$); using the relationship between addition and subtraction (e.g., knowing that $8 + 4 = 12$, one knows $12 - 8 = 4$); and creating equivalent but easier or known sums (e.g., adding $6 + 7$ by creating the known equivalent $6 + 6 + 1 = 12 + 1 = 13$).	2018-05-25 20:19:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6419835686683655, "perplexity": 1459.1462738788812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867217.1/warc/CC-MAIN-20180525200131-20180525220131-00002.warc.gz"}
https://indico.tifr.res.in/indico/conferenceOtherViews.py?fr=no&showSession=all&detailLevel=contribution&confId=5945&view=lecture&showDate=all	String Theory Seminars # Open EFT and Renormalisation (Part3) ## by Dr. R. Loganayagam (ICTS-TIFR, Bangalore) Friday, October 13, 2017 from to (Asia/Kolkata) at A304 Description While the notion of open quantum systems is itself old, many of the existing studies deal with quantum mechanical systems rather than quantum field theories. In this talk, I will begin with a brief review of field theoretical/path integral tools currently available to deal with open quantum field theories. Then, I will go on to apply these tools to an open version of $\phi^4$ theory in four spacetime dimensions and its one loop renormalisability. I will also comment on the renormalisability of Lindblad structure.	2017-11-22 09:11:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17085564136505127, "perplexity": 740.6544947922846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806543.24/warc/CC-MAIN-20171122084446-20171122104446-00641.warc.gz"}
https://mindspore.cn/docs/api/zh-CN/r1.3/api_python/numpy/mindspore.numpy.linspace.html	# mindspore.numpy.linspace¶ mindspore.numpy.linspace(start, stop, num=50, endpoint=True, retstep=False, dtype=None, axis=0)[source] Returns evenly spaced values within a given interval. Parameters • start (Union[int, list(int), tuple(int), tensor]) – The starting value of the sequence. • stop (Union[int, list(int), tuple(int), tensor]) – The end value of the sequence, unless endpoint is set to False. In that case, the sequence consists of all but the last of num + 1 evenly spaced samples, so that stop is excluded. Note that the step size changes when endpoint is False. • num (int, optional) – Number of samples to generate. Default is 50. • endpoint (bool, optional) – If True, stop is the last sample. Otherwise, it is not included. Default is True. • retstep (bool, optional) – If True, return (samples, step), where step is the spacing between samples. • dtype (Union[mindspore.dtype, str], optional) – Designated tensor dtype, If dtype is None, infer the data type from other input arguments. Default is None. • axis (int, optional) – The axis in the result to store the samples. Relevant only if start or stop are array-like. By default, the samples will be along a new axis inserted at the beginning. Use -1 to get an axis at the end. Default is 0. Returns Tensor, with num equally spaced samples in the closed interval $$[start, stop]$$ or the half-open interval $$[start, stop)$$ (depending on whether endpoint is True or False). Step, the size of spacing between samples, only returned if retstep is True. Raises TypeError – If input arguments have types not specified above. Supported Platforms: Ascend GPU CPU Examples >>> import mindspore.numpy as np >>> print(np.linspace(0, 5, 6)) [0. 1. 2. 3. 4. 5.]	2021-12-04 09:57:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21555712819099426, "perplexity": 5591.212120881455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362969.51/warc/CC-MAIN-20211204094103-20211204124103-00498.warc.gz"}
https://math.stackexchange.com/questions/2490107/continuity-of-the-projection-function	# Continuity of the projection function I have the following definition of the projection on product topology: Let $$\{X_i\}_{I\in I}$$ be a family of topological spaces and for $$j\in I$$ let $$p_j:\prod\limits_{I\in I}X_i\rightarrow X_j$$ be the projection onto the $$j$$th factor i.e. $$p_j((x_i)_{I\in I})=x_j$$. Then • For any $$j\in I$$ the function $$p_j$$ is continuous. • A function $$f:Y\rightarrow\prod\limits_{i\in I}X_i$$ is continuous if and only if the composition $$p_j\circ f:Y\rightarrow X_j$$ is continuous $$\forall j\in I$$. However, I'm looking for a proof of this. Everything I've looked at gave part 1. as defined in product topology, but I have yet to see a proof of it (I imagine part 2. follows). If anyone could provide a proof, that would be great. Take any open set $U_j\subset X_j$ then $p_j^{-1}(U_j)=X_1\times X_2\times \ldots X_{j-1}\times U_j\times X_{j+1}\times\ldots X_n\times \ldots$ which is open in $\prod_{j\in \Bbb N} X_j$ • Why is this open? Is it because each $X_i$ is also open? Oct 26, 2017 at 2:41 • Yes any open set in $\prod X_j$ is of the form $U_1\times Y_2\times \ldots U_n\times X_{n+1}\times \ldots$ Oct 26, 2017 at 5:25 • i.e finitely many are open in $X_i$ and the rest are the whole spaces Oct 26, 2017 at 5:26	2022-05-20 19:19:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8995102643966675, "perplexity": 142.64774002589087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662534669.47/warc/CC-MAIN-20220520191810-20220520221810-00360.warc.gz"}
https://lawyers.aneas.net/term/T/treachery-3153.asp	Search Lawyer Browse Lawyer by Country, State (US & Canada) or Specialization # treachery Conduct that assists an enemy. This was defined under the Treachery Act 1940 as an offence relating to World War II, which was punishable by death. There is now, however, no specific crime of treachery: acts of this sort are usually dealt with under the Official Secrets Acts (see official secrets) or, in some cases, as treason. # Browse Law Term A . B . C . D . E . F . G . H . I . J . K . L . M . N . O . P . Q . R . S . T . U . V . W . X . Y . Z . # Search Law Term term description Maintenance by aneas \| disclaimer	2020-10-30 19:14:31	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9026358127593994, "perplexity": 7793.420204079444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107911229.96/warc/CC-MAIN-20201030182757-20201030212757-00693.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-6th-edition/chapter-11-cumulative-review-page-670/16	## Intermediate Algebra (6th Edition) $\{\frac{-7}{6},-3\}$ $\frac{10}{(2x+4)^2}-\frac{1}{(2x+4)} = 3$ This is equivalent to $10 \times \frac{1}{(2x+4)^2}-\frac{1}{(2x+4)} = 3$ $10 \times (\frac{1}{2x+4})^{2}-\frac{1}{(2x+4)} = 3$ Let $\frac{1}{2x+4} = y$, then the given equation becomes, $10y^{2} -y = 3$ $10y^{2} -y -3 =0$ By factoring, $10y^{2} +5y-6y -3 =0$ $5y(2y+1)-3(2y+1)=0$ $(5y-3)(2y+1)=0$ $y= \frac{3}{5}$ or $\frac{-1}{2}$ To find $x$, substitute $y$ values in $\frac{1}{2x+4} = y$ Let $y= \frac{3}{5}$ $\frac{1}{2x+4} = \frac{3}{5}$ $6x+12=5$ $6x+12-5=0$ $6x+7=0$ $6x=-7$ $x= \frac{-7}{6}$ Let $y= \frac{-1}{2}$ $\frac{1}{2x+4} = \frac{-1}{2}$ $-2x-4=2$ $-2x=2+4$ $-2x=6$ $x=-3$ $\{\frac{-7}{6},-3\}$ both values satisfy the given equation. So the solutions are $\{\frac{-7}{6},-3\}$	2018-07-17 06:26:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9804894924163818, "perplexity": 70.30299760866895}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589573.27/warc/CC-MAIN-20180717051134-20180717071134-00355.warc.gz"}
https://www.zbmath.org/serials/?q=se%3A00001807	## Vestnik St. Petersburg University. Mathematics Short Title: Vestn. St. Petersbg. Univ., Math. Publisher: Springer US, New York, NY; Pleiades Publishing (Allerton Press), New York, NY; MAIK “Nauka/Interperiodica”, Moscow ISSN: 1063-4541; 1934-7855/e Online: http://link.springer.com/journal/volumesAndIssues/11988 Original: Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya Predecessor: Vestnik Leningrad University. Mathematics Comments: Indexed cover-to-cover; Translation of the mathematics section of Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. Documents Indexed: 1,195 Publications (since 1992) References Indexed: 645 Publications with 7,924 References. all top 5 ### Latest Issues 54, No. 4 (2021) 54, No. 3 (2021) 54, No. 2 (2021) 54, No. 1 (2021) 53, No. 4 (2020) 53, No. 3 (2020) 53, No. 2 (2020) 53, No. 1 (2020) 52, No. 4 (2019) 52, No. 3 (2019) 52, No. 2 (2019) 52, No. 1 (2019) 51, No. 4 (2018) 51, No. 3 (2018) 51, No. 2 (2018) 51, No. 1 (2018) 50, No. 4 (2017) 50, No. 3 (2017) 50, No. 2 (2017) 50, No. 1 (2017) 49, No. 4 (2016) 49, No. 3 (2016) 49, No. 2 (2016) 49, No. 1 (2016) 48, No. 4 (2015) 48, No. 3 (2015) 48, No. 2 (2015) 48, No. 1 (2015) 47, No. 4 (2014) 47, No. 3 (2014) 47, No. 2 (2014) 47, No. 1 (2014) 46, No. 4 (2013) 46, No. 3 (2013) 46, No. 2 (2013) 46, No. 1 (2013) 45, No. 4 (2012) 45, No. 3 (2012) 45, No. 2 (2012) 45, No. 1 (2012) 44, No. 4 (2011) 44, No. 3 (2011) 44, No. 2 (2011) 44, No. 1 (2011) 43, No. 4 (2010) 43, No. 3 (2010) 43, No. 2 (2010) 43, No. 1 (2010) 42, No. 4 (2009) 42, No. 3 (2009) 42, No. 2 (2009) 42, No. 1 (2009) 41, No. 4 (2008) 41, No. 3 (2008) 41, No. 2 (2008) 41, No. 1 (2008) 40, No. 4 (2007) 40, No. 3 (2007) 40, No. 2 (2007) 40, No. 1 (2007) 38, No. 4 (2005) 38, No. 3 (2005) 38, No. 2 (2005) 38, No. 1 (2005) 37, No. 4 (2004) 37, No. 3 (2004) 37, No. 2 (2004) 37, No. 1 (2004) 36, No. 4 (2003) 36, No. 3 (2003) 36, No. 2 (2003) 36, No. 1 (2003) 35, No. 4 (2002) 35, No. 3 (2002) 34, No. 3 (2001) 34, No. 2 (2001) 34, No. 1 (2001) 33, No. 4 (2000) 33, No. 3 (2000) 33, No. 2 (2000) 33, No. 1 (2000) 32, No. 4 (1999) 32, No. 3 (1999) 32, No. 2 (1999) 32, No. 1 (1999) 31, No. 4 (1998) 31, No. 3 (1998) 31, No. 2 (1998) 31, No. 1 (1998) 30, No. 4 (1997) 30, No. 3 (1997) 30, No. 2 (1997) 30, No. 1 (1997) 29, No. 4 (1996) 29, No. 3 (1996) 29, No. 2 (1996) 29, No. 1 (1996) 28, No. 4 (1995) 28, No. 3 (1995) 28, No. 2 (1995) ...and 11 more Volumes all top 5 all top 5 ### Fields 159 Systems theory; control (93-XX) 146 Ordinary differential equations (34-XX) 135 Numerical analysis (65-XX) 114 Probability theory and stochastic processes (60-XX) 93 Mechanics of deformable solids (74-XX) 75 Statistics (62-XX) 72 Approximations and expansions (41-XX) 72 Operations research, mathematical programming (90-XX) 67 Partial differential equations (35-XX) 67 Dynamical systems and ergodic theory (37-XX) 55 Mechanics of particles and systems (70-XX) 52 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 51 Computer science (68-XX) 50 Calculus of variations and optimal control; optimization (49-XX) 41 Number theory (11-XX) 36 Harmonic analysis on Euclidean spaces (42-XX) 31 Group theory and generalizations (20-XX) 26 Operator theory (47-XX) 22 Functional analysis (46-XX) 21 Combinatorics (05-XX) 21 Fluid mechanics (76-XX) 20 Functions of a complex variable (30-XX) 20 Information and communication theory, circuits (94-XX) 19 Real functions (26-XX) 18 Linear and multilinear algebra; matrix theory (15-XX) 17 Optics, electromagnetic theory (78-XX) 15 Difference and functional equations (39-XX) 14 History and biography (01-XX) 14 Algebraic geometry (14-XX) 12 Associative rings and algebras (16-XX) 12 Special functions (33-XX) 12 Global analysis, analysis on manifolds (58-XX) 12 Biology and other natural sciences (92-XX) 11 Integral transforms, operational calculus (44-XX) 11 Integral equations (45-XX) 11 Convex and discrete geometry (52-XX) 9 Mathematical logic and foundations (03-XX) 9 Astronomy and astrophysics (85-XX) 7 Order, lattices, ordered algebraic structures (06-XX) 7 Measure and integration (28-XX) 7 Potential theory (31-XX) 7 Differential geometry (53-XX) 7 Statistical mechanics, structure of matter (82-XX) 6 Field theory and polynomials (12-XX) 6 General topology (54-XX) 5 Several complex variables and analytic spaces (32-XX) 4 Classical thermodynamics, heat transfer (80-XX) 4 Geophysics (86-XX) 3 General and overarching topics; collections (00-XX) 3 Commutative algebra (13-XX) 3 Category theory; homological algebra (18-XX) 3 Quantum theory (81-XX) 2 General algebraic systems (08-XX) 2 Relativity and gravitational theory (83-XX) 1 Nonassociative rings and algebras (17-XX) 1 $$K$$-theory (19-XX) 1 Topological groups, Lie groups (22-XX) 1 Geometry (51-XX) ### Citations contained in zbMATH Open 390 Publications have been cited 870 times in 649 Documents Cited by Year The curvature tensor and the Einstein equations for a four-dimensional nonholonomic distribution. Zbl 1253.53045 Krym, V. R.; Petrov, N. N. 2008 Standard commutator formula. Zbl 1147.20042 Vavilov, N. A.; Stepanov, A. V. 2008 On the spectrum of the Steklov problem in a domain with a peak. Zbl 1171.35086 2008 Equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space. Zbl 1145.83314 Krym, V. R.; Petrov, N. N. 2007 Standard commutator formula, revisited. Zbl 1254.20038 Vavilov, N. A.; Stepanov, A. V. 2010 Hochschild cohomology for algebras of dihedral type. III: Local algebras in characteristic 2. Zbl 1254.16009 Generalov, A. I. 2010 Word equations in simple groups and polynomial equations in simple algebras. Zbl 1300.20033 Kanel-Belov, A.; Kunyavskii, B.; Plotkin, E. 2013 An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem. Zbl 1272.15007 Krivulin, N. K. 2011 On the strong law of large numbers for sequences of dependent random variables. Zbl 1253.60039 Korchevsky, V. M.; Petrov, V. V. 2010 Localization of hidden attractors of the generalized Chua system based on the method of harmonic balance. Zbl 1251.37081 Vagaitsev, V. I.; Kuznetsov, N. V.; Leonov, G. A. 2010 Two-dimensional homogeneous cubic systems: classification and normal forms. I. Zbl 1383.15009 Basov, V. V. 2016 On properties of integrals of the Legendre polynomial. Zbl 1302.33010 Kholshevnikov, K. V.; Shaidulin, V. Sh. 2014 The Yoneda algebras for some class of dihedral algebras. Zbl 1043.16500 Balashov, O. I.; Generalov, A. I. 1999 The “walk in hemispheres” process and its applications to solving boundary value problems. Zbl 1181.65009 Ermakov, S. M.; Sipin, A. S. 2009 Cycles of two-dimensional systems: computer calculations, proofs, and experiments. Zbl 1251.93095 Leonov, G. A.; Kuznetsov, N. V.; Kudryashova, E. V. 2008 On the asymptotic nature of approximate models of beams, plates, and shells. Zbl 1152.74028 Tovstik, P. E. 2007 Coincidence of the Gelig-Leonov-Yakubovich, Filippov, and Aizerman-Pyatnitskiy definitions. Zbl 1370.34027 Kiseleva, M. A.; Kuznetsov, N. V. 2015 On estimation of probabilities of unions of events with applications to the Borel-Cantelli lemma. Zbl 1370.60031 Frolov, A. N. 2015 Strongly time-consistent differential optimality principles. Zbl 0838.90150 Petrosyan, L. A. 1993 Papers of L.V. Kantorovich on Bernstein polynomials. Zbl 1285.41001 Videnskii, V. S. 2013 Closed nets in linear groups. Zbl 1273.20047 Koibaev, V. A. 2013 Fourth-degree invariants for $$G(\mathrm E_7,R)$$ not depending on the characteristic. Zbl 1280.20051 Luzgarev, A. Yu. 2013 Cooperative differential games with stochastic time. Zbl 1043.91007 Petrosyan, L. A.; Shevkoplyas, E. V. 2000 On linear operators with $$p$$-nuclear adjoints. Zbl 1044.47508 Reinov, O. I. 2000 On problems of Aizerman and Kalman. Zbl 1251.93110 Leonov, G. A.; Kuznetsov, N. V.; Bragin, V. O. 2010 Stable nonperiodic points of two-dimensional $$C^{1}$$-diffeomorphisms. Zbl 1146.37028 Vasil’eva, E. V. 2007 Robust invariant stabilization of nonlinear continuous and discrete systems. Zbl 1144.93026 Zuber, I. E.; Gelig, A. Kh. 2008 On inequalities for probabilities of unions of events and the Borel-Cantelli lemma. Zbl 1338.60025 Frolov, Andrei N. 2014 Some generalizations of the problem on the search number of a graph. Zbl 0883.90149 Golovach, P. A.; Petrov, N. N. 1995 On exact constants in some embedding theorems of high order. Zbl 1181.46027 Nazarov, A. I.; Petrova, A. N. 2008 Yet another variation on the theme of decomposition of transvections. Zbl 1175.20043 Vavilov, N. A.; Kazakevich, V. G. 2008 Diffeomorphisms of multidimensional space with infinite set of stable periodic points. Zbl 1338.37036 Vasil’eva, E. V. 2012 Estimating the derivative of the Legendre polynomial. Zbl 1257.33009 Antonov, V. A.; Kholshevnikov, K. V.; Shaidulin, V. Sh. 2010 Minimal splines and wavelets. Zbl 1143.41002 Dem’yanovich, Yu. K. 2008 Approximation by entire functions on countable unions of segments of the real axis. II: Proof of the main theorem. Zbl 1382.41007 Silvanovich, O. V.; Shirokov, N. A. 2017 Two-dimensional homogeneous cubic systems: classification and normal forms. II. Zbl 1388.34029 Basov, V. V. 2016 Approximations by entire functions on countable unions of segments of the real axis. Zbl 1393.30029 Silvanovich, O. V.; Shirokov, N. A. 2016 On an algebraic solution of the Rawls location problem in the plane with rectilinear metric. Zbl 1370.90137 Krivulin, N. K.; Plotnikov, P. V. 2015 Energy dissipation during vibrations of nonuniform composite structures. I: Formulation of the problem. Zbl 1401.74020 Parshina, L. V.; Ryabov, V. M.; Yartsev, B. A. 2018 Solvable matrices. Zbl 0831.15015 Davydov, G. V.; Davydova, I. M. 1993 Mixed abelian groups of finite rank and their direct decompositions. Zbl 0833.20067 Yakovlev, A. V.; N’Famara, Kamara 1993 On the stability of invariant sets of leaves of three-dimensional periodic systems. Zbl 1317.34045 Begun, N. A. 2014 On the stability of sheet invariant sets of two-dimensional periodic systems. Zbl 1270.34109 Begun, N. A. 2012 On the asymptotic efficiency of normality tests based on the Shepp property. Zbl 1183.62080 Volkova, K. Yu.; Nikitin, Ya. Yu. 2009 Distribution of functionals of some processes with independent increments. Zbl 1272.60026 Borodin, A. N. 2005 Normalized trigonometric splines of Lagrange type. Zbl 1255.65037 Makarov, A. A. 2008 The number of steps for construction of a Boolean solution to polynomial congruences and systems of polynomial congruences. Zbl 1153.68394 Kosovskii, N. K.; Kosovskaya, T. M. 2007 Generalizations of the parking problem. Zbl 1390.60042 Ananjevskii, S. M. 2016 On inequalities for conditional probabilities of unions of events and the conditional Borel-Cantelli lemma. Zbl 1390.60070 Frolov, A. N. 2016 On the maximal value of the expectation of record numbers. Zbl 1338.62145 Nevzorov, V. B.; Tovmasyan, Sergei A. 2014 On the probabilities of moderate deviations for combinatorial sums. Zbl 1370.60052 Frolov, A. N. 2015 Vibrations of a floating beam on marine waves. Zbl 1370.74094 Tovstik, P. E.; Tovstik, T. M. 2015 Parametric resonances in the problem of longitudinal impact on a thin rod. Zbl 1456.74107 Belyaev, A. K.; Morozov, N. F.; Tovstik, P. E.; Tovstik, T. P. 2016 On the possible dimensions of subspace intersections. Zbl 1370.05030 Lebedinskaya, N. A.; Lebedinskii, D. M. 2016 Two-dimensional homogeneous cubic systems: classification and normal forms. III. Zbl 1373.34054 Basov, V. V.; Chermnykh, A. S. 2017 Towards the analysis of the simulated annealing method in the multiextremal case. Zbl 1376.65095 Ermakov, S. M.; Kulikov, D. V.; Leora, S. N. 2017 Long-wave oscillations and waves in anisotropic beams. Zbl 1370.78075 Tovstik, P. E.; Tovstik, T. P.; Naumova, N. V. 2017 Approximation by entire functions on a countable union of segments on the real axis. III: Further generalization. Zbl 1405.30036 Silvanovich, O. V.; Shirokov, N. A. 2018 On the existence of an invariant torus for an essentially nonlinear system of differential equations. Zbl 0788.34039 Volkov, D. Yu.; Il’in, Yu. A. 1992 The Korn inequalities which are asymptotically sharp for thin domains. Zbl 0783.73012 Nazarov, S. A. 1992 On the stability problem of solutions of positional games. Zbl 0883.90146 Kuzyutin, D. V. 1995 Eden’s hypothesis for a Lorenz system. Zbl 0834.34060 Leonov, G. A.; Lyashko, S. A. 1993 Existence of abnormal minimizing geodesics in sub-Riemannian geometry. Zbl 0844.53032 Petrov, N. N. 1993 On some problems of guaranteed search. Zbl 0854.68025 Petrov, N. N.; Starostina, S. A. 1994 Algorithm of fast linear nonstationary stabilization and the Brockett problem. Zbl 1067.93050 Leonov, G. A. 2001 On the summation of the Laguerre series by the Euler-Knopp method in the problem of inverting the Laplace transform. Zbl 1175.65147 Kabardov, M. M. 2008 Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations. Zbl 1181.37027 Tikhomirov, S. B. 2008 Calculating the Lyapunov exponent for generalized linear systems with exponentially distributed elements of the transition matrix. Zbl 1175.65009 Krivulin, N. K. 2009 Unique solvability of the integral equation for harmonic simple layer potential on the boundary of a domain with a peak. Zbl 1186.31004 Maz’ya, V. G.; Poborchii, S. V. 2009 Robust stabilization of some class of uncertain systems. Zbl 1187.93115 Zuber, I. E.; Gelig, A. Kh. 2009 Simulation of the marine stationary platform vibrations under wave excitation. Zbl 1272.74302 Tovstik, P. E.; Tovstik, T. M.; Shekhovtsov, V. A. 2005 Centralizer of the elementary subgroup of an isotropic reductive group. Zbl 1280.20050 Kulikova, E.; Stavrova, A. 2013 On subnormal subgroups in general skew linear groups. Zbl 1280.20056 Hai, Bui Xuan; Thin, Nguyen Van 2013 Weak shadowing in two-dimensional diffeomorphisms. Zbl 1092.37503 Plamenevskaya, O. B. 1998 Pseudo-orbit tracing property and limit shadowing property on a circle. Zbl 1055.37556 Plamenevskaya, O. B. 1997 Monte-Carlo method with storage of intermediate results. Zbl 1041.65002 Belyaeva, A. A.; Ermakov, S. M. 1996 On the rate of decay of concentration functions of $$n$$-fold convolutions of probability distributions. Zbl 1253.60047 Zaitsev, A. Yu. 2011 Dynamical systems with Lipschitz inverse shadowing properties. Zbl 1256.37009 Pilyugin, S. Yu.; Vol’fson, G. I.; Todorov, D. I. 2011 On some problems of guaranteed search on graphs. Zbl 1253.05103 Abramovskaya, T. V.; Petrov, N. N. 2010 The minimax estimator of the pseudo-periodic function observed in the stationary noise. Zbl 1264.93239 Reshetov, S. V. 2010 Joint deformation of a circular inclusion and a matrix. Zbl 1465.74128 Grekov, M. A. 2010 On the applicability conditions of the strong law of large numbers for sequences of independent random variables. Zbl 1269.60034 Korchevsky, V. M. 2010 Jacobi fields for a nonholonomic distribution. Zbl 1253.53040 Krym, V. R. 2010 Evaluation of bounds on the mean rate of growth of the state vector of a linear dynamical stochastic system in idempotent algebra. Zbl 1105.93055 Krivulin, N. K. 2005 Criteria of stability by the first approximation for discrete nonlinear systems. Zbl 1108.39009 Kuznetsov, N. V.; Leonov, G. A. 2005 On the Liouville phenomenon in estimates of fractal dimensions of forced quasi-periodic oscillations. Zbl 1429.37016 Anikushin, Mikhaĭl Mikhaĭlovich 2019 On torsion theories, weight and $$t$$-structures in triangulated categories. Zbl 1429.18005 Bondarko, M. V.; Vostokov, S. V. 2019 On some solutions to the Chapman-Kolmogorov integral equation. Zbl 1149.42300 Miroshin, R. N. 2007 Stabilization of linear control systems and pole assignment problem: a survey. Zbl 1453.93098 Shumafov, M. M. 2019 Some remarks on axioms of probability theory. Zbl 1408.60003 Vallander, S. S. 2013 Dynamics of a rotor with an eccentric ball auto-balancing device. Zbl 1370.70031 Bykov, V. G.; Kovachev, A. S. 2014 Perturbations of weakly hyperbolic invariant sets of two-dimension periodic systems. Zbl 1370.37037 Begun, N. A. 2015 On the $$C^{1}$$-equivalence of essentially nonlinear systems of differential equations near an asymptotically stable equilibrium point. Zbl 1370.34076 Iljin, Yu. A. 2015 Axisymmetric deformations of the orthotropic spherical layer under normal pressure. Zbl 1370.74020 Bauer, S. M.; Smirnov, A. L. 2015 Localization in a Bernoulli-Euler beam on an inhomogeneous elastic foundation. Zbl 1370.74090 Indeitsev, D. A.; Kuklin, T. S.; Mochalova, Yu. A. 2015 Bifurcation of the state of equilibrium of an oscillator with nonlinear restoring force of third order. Zbl 1370.37099 Bibikov, Yu. N.; Pliss, V. A. 2015 Multivalent probability spaces. Zbl 1370.60003 Vallander, S. S. 2015 Solution of a tropical optimization problem with linear constraints. Zbl 1370.90171 Krivulin, N. K.; Sorokin, V. N. 2015 Asymptotics behavior of integrals of the Legendre polynomials and their sums. Zbl 1370.33011 Kholshevnikov, K. V.; Shaidulin, V. Sh. 2015 A new approach to finding the control that transports a system from one phase state to another. Zbl 1384.70013 Zegzhda, S. A.; Shatrov, E. A.; Yushkov, M. P. 2016 On convergence and compactness in variation with a shift of discrete probability laws. Zbl 1476.60034 Alexeev, I. A.; Khartov, A. A. 2021 A proof of Bel’tyukov-Lipshitz theorem by quasi-quantifier elimination. I: Definitions and GCD-lemma. Zbl 07419077 Starchak, M. R. 2021 The inverse problem of stabilization of a spherical pendulum in a given position under oblique vibration. Zbl 1477.70007 Petrov, A. G. 2021 On Chow-weight homology of motivic complexes and its relation to motivic homology. Zbl 1460.19005 Bondarko, M. V.; Kumallagov, D. Z. 2020 Limit theorems for generalized perimeters of random inscribed polygons. I. Zbl 1454.60037 Simarova, E. N. 2020 Minimal velocity surface in a restricted circular three-body problem. Zbl 1465.70041 Kholshevnikov, K. V.; Titov, V. B. 2020 Power series of one variable with condition of logarithmical convexity. Zbl 1461.30013 Zheleznyak, A. V. 2020 On the Aizerman problem: coefficient conditions for the existence of a four-period cycle in a second-order discrete-time system. Zbl 1458.39012 Zvyagintseva, T. E. 2020 To the problem of modeling gas flows behind the strong shock wave front using an effective adiabatic index. Zbl 1457.76093 Bogatko, V. I.; Potekhina, E. A. 2020 On a quotient space of Keplerian orbits. Zbl 1465.70073 Kholshevnikov, K. V.; Shchepalova, A. S.; Jazmati, M. S. 2020 Linear operators preserving majorization of matrix tuples. Zbl 1469.15031 Guterman, A. E.; Shteyner, P. M. 2020 On the Aizerman problem: coefficient conditions for the existence of three- and six-period cycles in a second-order discrete-time system. Zbl 1455.93161 Zvyagintseva, T. E. 2020 Coupled vibrations of viscoelastic three-layer composite plates. I: Formulation of the problem. Zbl 1461.74028 Ryabov, V. M.; Yartsev, B. A.; Parshina, L. V. 2020 Approximation by entire functions on a countable set of continua. Zbl 1461.30071 Silvanovich, O. V.; Shirokov, N. A. 2020 Free vibration frequencies of a circular thin plate with variable parameters. Zbl 1460.74043 Vasiliev, G. P.; Smirnov, A. L. 2020 On the Liouville phenomenon in estimates of fractal dimensions of forced quasi-periodic oscillations. Zbl 1429.37016 Anikushin, Mikhaĭl Mikhaĭlovich 2019 On torsion theories, weight and $$t$$-structures in triangulated categories. Zbl 1429.18005 Bondarko, M. V.; Vostokov, S. V. 2019 Stabilization of linear control systems and pole assignment problem: a survey. Zbl 1453.93098 Shumafov, M. M. 2019 The basin of attraction in the generalized Kapitsa problem. Zbl 1427.74087 Kulizhnikov, Dmitriĭ Borisovich; Tovstik, Pëtr Evgen’evich; Tovstik, Tat’yana Petrovna 2019 Stability of periodic points of a diffeomorphism of a plane in a homoclinic orbit. Zbl 1430.37024 Vasileva, E. V. 2019 Uniaxial attitude stabilization of a rigid body under conditions of nonstationary perturbations with zero mean values. Zbl 1455.70006 Aleksandrov, A. Yu.; Tikhonov, A. A. 2019 On asymptotic normality in one generalization of the Renyi problem. Zbl 1427.60019 Ananjevskii, S. M.; Kryukov, N. A. 2019 Monte Carlo method for solving ODE systems. Zbl 1480.65007 Ermakov, Sergeĭ Mikhaĭlovich; Tovstik, Tat’yana Mikhaĭlovna 2019 Regularity of solutions to a model oblique derivative problem for quasilinear parabolic systems with nondiagonal principal matrices. Zbl 1429.35136 Arkhipova, A. A.; Grishina, G. V. 2019 Two-dimensional model of second-order accuracy for an anisotropic plate. Zbl 1430.74105 Tovstik, P. E. 2019 Goodness-of-fit tests based on a characterization of logistic distribution. Zbl 1435.62289 Nikitin, Ya. Yu.; Ragozin, I. A. 2019 On stabilization of a triple inverted pendulum via vibration of a support point with an arbitrary frequency. Zbl 1455.70005 Arkhipova, I. M. 2019 Numerical algorithm for investigating the stress-strain state of cylindrical shells of railway tanks. Zbl 1430.74108 Gerasimenko, P. V.; Khodakovskiy, V. A. 2019 Numerical solution of systems of linear algebraic equations with ill-conditioned matrices. Zbl 1454.65026 Lebedeva, A. V.; Ryabov, V. M. 2019 Entire functions of order 1/2 in the approximation to functions on a semiaxis. Zbl 1456.30065 Silvanovich, O. V.; Shirokov, N. A. 2019 Linear generalized Kalman-Bucy filters. Zbl 1454.60056 Tovstik, T. M.; Tovstik, P. E. 2019 Formulation and solution of a generalized Chebyshev problem. I. Zbl 1482.70019 Yushkov, M. P. 2019 Energy dissipation during vibrations of nonuniform composite structures. I: Formulation of the problem. Zbl 1401.74020 Parshina, L. V.; Ryabov, V. M.; Yartsev, B. A. 2018 Approximation by entire functions on a countable union of segments on the real axis. III: Further generalization. Zbl 1405.30036 Silvanovich, O. V.; Shirokov, N. A. 2018 Sharp estimates for mean square approximations of classes of differentiable periodic functions by shift spaces. Zbl 1402.41003 Vinogradov, O. L.; Ulitskaya, A. Yu. 2018 Toward the history of the Saint St. Petersburg school of probability and statistics. I: Limit theorems for sums of independent random variables. Zbl 1401.60002 Lifshits, M. A.; Nikitin, Ya. Yu.; Petrov, V. V.; Zaitsev, A. Yu.; Zinger, A. A. 2018 Two-sided estimates of Fourier sums Lebesgue functions with respect to polynomials orthogonal on nonuniform grids. Zbl 1431.42048 Nurmagomedov, A. A.; Rasulov, N. K. 2018 Generating large sequences of normal maxima via record values. Zbl 1435.62174 Pakhteev, A. I.; Stepanov, A. V. 2018 The stability of a flexible vertical rod on a vibrating support. Zbl 1430.74064 Belyaev, A. K.; Morozov, N. F.; Tovstik, P. E.; Tovstik, T. P. 2018 Stable periodic solutions of periodic systems of differential equations. Zbl 1401.34050 Vasil’eva, E. V. 2018 A supplement to Hölder’s inequality. The resonance case. I. Zbl 1400.26049 Ivanov, B. F. 2018 The speed-gradient algorithm in the inverse stoker problem for a synchronous electric machine. Zbl 1401.93152 Plotnikov, S. A.; Fradkov, A. L.; Shepeljavyi, A. I. 2018 A multidimensional nonautonomous equation containing a product of powers of partial derivatives. Zbl 1402.35063 Rakhmelevich, I. V. 2018 On vector form of differential variational principles of mechanics. Zbl 1421.70035 Soltakhanov, Sh. Kh.; Shugaylo, T. S.; Yushkov, M. P. 2018 On the problem of the optimal choice of record values. Zbl 1402.62087 Belkov, I. V.; Nevzorov, V. B. 2018 Rank-one approximation of positive matrices based on methods of tropical mathematics. Zbl 1401.90263 Krivulin, N. K.; Romanova, E. Yu. 2018 Linear Kalman-Bucy filter with autoregressive signal and noise. Zbl 1430.60041 Tovstik, T. M. 2018 On inequalities for probabilities of joint occurrence of several events. Zbl 1430.60023 Frolov, A. N. 2018 On distances between orbits of planets and asteroids. Zbl 1465.70051 Kholshevnikov, K. V.; Shchepalova, A. S. 2018 The problem of selfish parking. Zbl 1430.60012 Ananjevskii, S. M.; Kryukov, N. A. 2018 Two-dimensional homogeneous cubic systems: classifications and normal forms. V. Zbl 1434.34034 Basov, V. V.; Chermnykh, A. S. 2018 An estimate for the number of periodical trajectories of the given period for mapping of an interval, Lucas numbers, and necklaces. Zbl 1435.37058 Ivanov, O. A. 2018 Relation of the Böttcher equation with the parametrized Poisson integral. Zbl 1433.30073 Kalnitskii, V. S.; Petrov, A. N. 2018 Existence of Liouvillian solutions in the problem of rolling motion of a dynamically symmetric ball on a perfectly rough sphere. Zbl 1455.70004 Kuleshov, A. S.; Katasonova, V. A. 2018 Energy dissipation during vibrations of heterogeneous composite structures. II: Method of solution. Zbl 1431.65219 Parshina, L. V.; Ryabov, V. M.; Yartsev, B. A. 2018 Approximation by entire functions on countable unions of segments of the real axis. II: Proof of the main theorem. Zbl 1382.41007 Silvanovich, O. V.; Shirokov, N. A. 2017 Two-dimensional homogeneous cubic systems: classification and normal forms. III. Zbl 1373.34054 Basov, V. V.; Chermnykh, A. S. 2017 Towards the analysis of the simulated annealing method in the multiextremal case. Zbl 1376.65095 Ermakov, S. M.; Kulikov, D. V.; Leora, S. N. 2017 Long-wave oscillations and waves in anisotropic beams. Zbl 1370.78075 Tovstik, P. E.; Tovstik, T. P.; Naumova, N. V. 2017 Charged ball method for solving some computational geometry problems. Zbl 1434.90145 Abbasov, M. E. 2017 Dynamics of a rod undergoing a longitudinal impact by a body. Zbl 1430.74080 Belyaev, A. K.; Ma, C.-C.; Morozov, N. F.; Tovstik, P. E.; Tovstik, T. P.; Shurpatov, A. O. 2017 Necessary and sufficient nonnegativity conditions for second-order coordinate trigonometric splines. Zbl 1384.41006 Dem’yanovich, Yu. K.; Makarov, A. A. 2017 Asymptotic behavior of solutions of Lorenz-like systems: analytical results and computer error structures. Zbl 1385.34014 Leonov, G. A.; Andrievskiy, B. R.; Mokaev, R. N. 2017 Decay mild solutions for elastic systems with structural damping involving nonlocal conditions. Zbl 1387.74032 Vu Trong Luong; Nguyen Thanh Tung 2017 Excess of locally D-optimal designs and homothetic transformations. Zbl 06858275 Grigor’ev, Yuriĭ Dmitrievich; Melas, Vyacheslav Borisovich; Shpilev, Pëtr Valer’evich 2017 To the question of stability of periodic points of three-dimensional diffeomorphisms. Zbl 1373.37068 Vasilieva, E. V. 2017 Hilbert pairing on Lorentz formal groups. Zbl 1427.11132 Vostokov, S. V.; Pital’, P. N. 2017 Global stability conditions of a system with hysteresis nonlinearity. Zbl 1373.34099 Zviagintceva, T. E.; Pliss, V. A. 2017 Pseudo-Poissonian processes with stochastic intensity and a class of processes generalizing the Ornstein-Uhlenbeck process. Zbl 1375.60093 Rusakov, O. V. 2017 Correlation between the properties of eigenfrequencies and eigenmodes in a chain of rigid bodies with torque connections. Zbl 1370.74079 Indeitsev, D. A.; Sergeev, A. D. 2017 Nonexistence of Liouvillian solutions in the problem of motion of a rotationally symmetric ellipsoid on a perfectly rough plane. Zbl 1373.34019 Kuleshov, A. S.; Itskovich, M. O. 2017 Workspaces of the Stewart platform in the 6D space of generalized coordinates. Zbl 1370.93043 Leonov, G. A.; Tovstik, P. E.; Tovstik, T. M. 2017 Two-dimensional homogeneous cubic systems: classification and normal forms. IV. Zbl 1434.34033 Basov, V. V.; Chermnykh, A. S. 2017 Comparison of numbers of records in the sequences of discrete and continuous random variables. Zbl 1435.62172 Nevzorov, V. B. 2017 On inequalities for probabilities wherein at least $$r$$ from $$n$$ events occur. Zbl 1430.60014 Frolov, A. N. 2017 Dynamics of a Stewart platform. Zbl 1472.70004 Andrievskiy, B. R.; Arseniev, D. G.; Zegzhda, S. A.; Kazunin, D. V.; Kuznetsov, N. V.; Leonov, G. A.; Tovstik, P. E.; Tovstik, T. P.; Yushkov, M. P. 2017 Stokes constants of an oblate ellipsoid of revolution with equidensites homothetic to its surface. Zbl 1465.70058 Kholshevnikov, K. V.; Milanov, D. V.; Shaidulin, V. Sh. 2017 On the law of the iterated logarithm for sequences of dependent random variables. Zbl 1382.60070 Petrov, V. V. 2017 Record values in sequences of sample ranges. Zbl 06858274 Bel’kov, Igor’ Vladimirovich; Nevzorov, Valeriĭ Borisovich 2017 Stabilization by output of continuous and pulse-modulated uncertain systems. Zbl 1383.93065 Zuber, Irina Efremovna; Gelig, Arkadiĭ Khaimovich 2017 General problems of explicit integration of differential inequalities. Zbl 1391.34031 Il’in, Yuriĭ Anatol’evich 2017 Attitude stabilization of a rigid body in conditions of decreasing dissipation. Zbl 1390.70063 Aleksandrov, Aleksandr Yur’evich; Tikhonov, Alekseĭ Aleksandrovich 2017 Passage through resonance of a statically unbalanced rotor with an imperfect autobalancing device. Zbl 1425.70036 Bykov, Vladimir Grigor’evich; Kovachev, Aleksandr Svetoslavovich 2017 The Laplace series of ellipsoidal figures of revolution. Zbl 1390.70031 2017 Two-dimensional homogeneous cubic systems: classification and normal forms. I. Zbl 1383.15009 Basov, V. V. 2016 Two-dimensional homogeneous cubic systems: classification and normal forms. II. Zbl 1388.34029 Basov, V. V. 2016 Approximations by entire functions on countable unions of segments of the real axis. Zbl 1393.30029 Silvanovich, O. V.; Shirokov, N. A. 2016 Generalizations of the parking problem. Zbl 1390.60042 Ananjevskii, S. M. 2016 On inequalities for conditional probabilities of unions of events and the conditional Borel-Cantelli lemma. Zbl 1390.60070 Frolov, A. N. 2016 Parametric resonances in the problem of longitudinal impact on a thin rod. Zbl 1456.74107 Belyaev, A. K.; Morozov, N. F.; Tovstik, P. E.; Tovstik, T. P. 2016 On the possible dimensions of subspace intersections. Zbl 1370.05030 Lebedinskaya, N. A.; Lebedinskii, D. M. 2016 A new approach to finding the control that transports a system from one phase state to another. Zbl 1384.70013 Zegzhda, S. A.; Shatrov, E. A.; Yushkov, M. P. 2016 On families of distributions characterized by certain properties of ordered random variables. Zbl 1387.62017 Ananjevskii, S. M.; Nevzorov, V. B. 2016 Natural damped vibrations of anisotropic box beams of polymer composite materials. II: Numerical experiments. Zbl 1390.74019 Ryabov, V. M.; Yartsev, B. A. 2016 An algebraic identity and the Jacobi-Trudi formula. Zbl 1391.15017 Bekker, B. M.; Ivanov, O. A.; Merkurjev, A. S. 2016 Estimates of the norm of a function orthogonal to the piecewise-constant functions in terms of higher-order moduli of continuity. Zbl 1371.41015 Vinogradov, O. L.; Ikhsanov, L. N. 2016 Degeneration of the Hilbert pairing in formal groups over local fields. Zbl 1401.11152 Vostokov, S. V.; Vostokova, R. P.; Podkopaeva, O. Yu. 2016 Free localized vibrations of a long double-walled carbon nanotube introduced into an inhomogeneous elastic medium. Zbl 1456.74057 Mikhasev, G. I.; Botogova, M. G. 2016 Natural damped vibrations of anisotropic box beams of polymer composite materials. I: Statement of the problem. Zbl 1456.74059 Ryabov, V. M.; Yartsev, B. A. 2016 Buckling analysis of a ring under the action of internal pressure in a cylindrical shell. Zbl 1456.74045 Boyarskaya, M. L.; Filippov, S. B. 2016 ...and 290 more Documents all top 5 ### Cited by 617 Authors 194 Vestnik St. Petersburg University. Mathematics 96 Journal of Mathematical Sciences (New York) 20 Automation and Remote Control 16 St. Petersburg Mathematical Journal 11 Doklady Mathematics 11 Differential Equations 10 Siberian Mathematical Journal 10 Statistics & Probability Letters 8 Acta Mechanica 8 Journal of Algebra 8 Differentsial’nye Uravneniya i Protsessy Upravleniya 7 Mathematical Notes 7 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 5 Chaos, Solitons and Fractals 5 Journal of Differential Equations 5 Linear Algebra and its Applications 4 Communications in Algebra 4 Journal of Applied Mathematics and Mechanics 4 Journal of Mathematical Analysis and Applications 4 Computational Mathematics and Mathematical Physics 4 Celestial Mechanics and Dynamical Astronomy 4 Monte Carlo Methods and Applications 4 Nonlinear Dynamics 4 Vladikavkazskiĭ Matematicheskiĭ Zhurnal 3 Discrete Applied Mathematics 3 Applied Mathematics and Computation 3 Journal of Approximation Theory 3 Journal of Functional Analysis 3 Proceedings of the American Mathematical Society 3 International Journal of Algebra and Computation 3 Statistical Papers 3 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 3 Lobachevskii Journal of Mathematics 3 Dynamical Systems 3 Proceedings of the Steklov Institute of Mathematics 3 Matematicheskaya Teoriya Igr i eë Prilozheniya 2 International Journal of Engineering Science 2 Israel Journal of Mathematics 2 Mathematical Methods in the Applied Sciences 2 Zhurnal Vychislitel’noĭ Matematiki i Matematicheskoĭ Fiziki 2 Prikladnaya Matematika i Mekhanika 2 Acta Mathematica Vietnamica 2 Functional Analysis and its Applications 2 Journal of Computational and Applied Mathematics 2 Journal of Statistical Planning and Inference 2 Proceedings of the Edinburgh Mathematical Society. Series II 2 Operations Research Letters 2 Physica D 2 Annals of Operations Research 2 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 2 Journal of Dynamics and Differential Equations 2 Cybernetics and Systems Analysis 2 Izvestiya: Mathematics 2 Positivity 2 International Game Theory Review 2 Qualitative Theory of Dynamical Systems 2 Matematicheskoe Modelirovanie 2 Computational Management Science 2 International Journal of Group Theory 1 Advances in Applied Probability 1 Applicable Analysis 1 Discrete Mathematics 1 Journal of Computational Physics 1 Metrika 1 Physics Letters. A 1 Physics Reports 1 Russian Mathematical Surveys 1 Ukrainian Mathematical Journal 1 Wave Motion 1 Beiträge zur Algebra und Geometrie 1 Advances in Mathematics 1 Algebra Universalis 1 Annales de l’Institut Fourier 1 Automatica 1 Fuzzy Sets and Systems 1 Integral Equations and Operator Theory 1 Journal of Number Theory 1 Journal of Optimization Theory and Applications 1 Journal of Pure and Applied Algebra 1 Mathematische Zeitschrift 1 Networks 1 Numerical Functional Analysis and Optimization 1 Results in Mathematics 1 Theoretical Computer Science 1 Transactions of the American Mathematical Society 1 Transactions of the Moscow Mathematical Society 1 Systems & Control Letters 1 Stochastic Analysis and Applications 1 Acta Mathematica Hungarica 1 Annals of Pure and Applied Logic 1 Acta Mathematicae Applicatae Sinica. English Series 1 Optimization 1 Journal of Theoretical Probability 1 Applied Mathematics Letters 1 Asymptotic Analysis 1 MCSS. Mathematics of Control, Signals, and Systems 1 Science in China. Series A 1 Designs, Codes and Cryptography 1 Numerical Algorithms 1 Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences ...and 59 more Journals 75 Ordinary differential equations (34-XX) 74 Probability theory and stochastic processes (60-XX) 66 Dynamical systems and ergodic theory (37-XX) 66 Systems theory; control (93-XX) 60 Group theory and generalizations (20-XX) 52 Mechanics of deformable solids (74-XX) 51 Partial differential equations (35-XX) 43 Numerical analysis (65-XX) 35 Statistics (62-XX) 33 Mechanics of particles and systems (70-XX) 29 Associative rings and algebras (16-XX) 29 Approximations and expansions (41-XX) 24 Operations research, mathematical programming (90-XX) 23 Differential geometry (53-XX) 21 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 20 Linear and multilinear algebra; matrix theory (15-XX) 20 Operator theory (47-XX) 18 Computer science (68-XX) 17 Number theory (11-XX) 17 Calculus of variations and optimal control; optimization (49-XX) 15 Functional analysis (46-XX) 13 Functions of a complex variable (30-XX) 13 Harmonic analysis on Euclidean spaces (42-XX) 12 Combinatorics (05-XX) 10 Algebraic geometry (14-XX) 10 Category theory; homological algebra (18-XX) 10 Special functions (33-XX) 9 History and biography (01-XX) 9 $$K$$-theory (19-XX) 8 Mathematical logic and foundations (03-XX) 8 Nonassociative rings and algebras (17-XX) 8 Information and communication theory, circuits (94-XX) 7 Difference and functional equations (39-XX) 7 Integral transforms, operational calculus (44-XX) 7 Integral equations (45-XX) 7 Fluid mechanics (76-XX) 6 General topology (54-XX) 5 Real functions (26-XX) 4 General algebraic systems (08-XX) 4 Commutative algebra (13-XX) 4 Convex and discrete geometry (52-XX) 4 Optics, electromagnetic theory (78-XX) 4 Biology and other natural sciences (92-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Potential theory (31-XX) 3 Algebraic topology (55-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 General and overarching topics; collections (00-XX) 2 Measure and integration (28-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Abstract harmonic analysis (43-XX) 2 Quantum theory (81-XX) 1 Field theory and polynomials (12-XX) 1 Topological groups, Lie groups (22-XX) 1 Sequences, series, summability (40-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Relativity and gravitational theory (83-XX)	2022-05-26 05:33:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6760033369064331, "perplexity": 2765.363082121867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662601401.72/warc/CC-MAIN-20220526035036-20220526065036-00758.warc.gz"}
https://math.stackexchange.com/questions/1259071/divergent-convergent-series	# Divergent, convergent series Let $p$, $q \in \mathbb{R}$ and see the series $$\sum_{n=2}^{\infty} \frac{1}{n^p(\ln n)^q}$$ View with the comparison criterion that if $p> 1$ then the series is convergent for all $q$, and if $p < 1$, it is divergent for all $q$. Can anyone help me getting started? • We have $\ln x\le\dfrac{x^\varepsilon-1}\varepsilon$ and $\ln x\le\dfrac{x^\varepsilon}{e\varepsilon}$ for every $\varepsilon>0$. (With equality iff $x=1$ and $x=e^{1/\varepsilon}$ respectively.) – Akiva Weinberger Apr 30 '15 at 13:48 Note that $$(\log n)^{q} = o(n^{p})$$ as $n \to \infty$ for all $p, q > 0.$ $$\sum_{n=2}^{\infty} \frac{1}{n^{\alpha}} <\infty \Leftrightarrow \alpha > 1$$ and $$\ln(n) < n \forall ~ n \geq 1$$	2019-08-26 06:30:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9761499762535095, "perplexity": 176.3568109880417}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330968.54/warc/CC-MAIN-20190826042816-20190826064816-00225.warc.gz"}
https://www.ideals.illinois.edu/browse?rpp=20&order=ASC&sort_by=-1&value=Social+Work&etal=-1&type=subject&starts_with=D	# Browse by Subject "Social Work" • (1984) Issues related to day care in the United States have emerged as the result of the changing structure of the American family. Concerns over the experiences single parent families have with day care are growing as current ... application/pdf PDF (8MB) • (2001) Implications for practice. The results suggest that investigators need assistance in using the CERAP form and avoiding heuristics (and their associated errors). Such assistance can take the form of revised CERAP training, ... application/pdf PDF (7MB) • (1987) Based on the principle of citizen participation, the theory of cultural pluralism, and the concept of service partnership, this study focused on the Asian American elderly in a large midwestern city to examine the problems ... application/pdf PDF (9MB) • (11/1/1999) Extraction from a larger work for the use of ILO staff in preparing training documents. application/pdf PDF (136kB) • (1982) A stated policy goal in the United States is to increase the labor force participation rates among men ages 65-69 as a means of alleviating the long-term financial difficulties of the old-age pension program under social ... application/pdf PDF (8MB) • (1996) The present study examined the educational attainments of young adults who have been emancipated from the foster care system. Data for this study were taken from a sub-set of youth who participated in the Foster Youth ... application/pdf PDF (9MB) • (1981) The placement decision in child welfare has long been considered a critical far-reaching decision. The decision impacts children and families, the allocation of resources within the service system, and often raises legal ... application/pdf PDF (5MB) • (1990) This study examines the influence of therapist-client personality difference upon treatment outcome. This research was conducted at a community mental health center over a six and one half month period in which personality ... application/pdf PDF (4MB) • (1985) Since the late 1960's there has been a trend towards the placement of mentally retarded persons in community residential facilities rather than large institutions. While many of these individuals are able to adjust to their ... application/pdf PDF (6MB) • (1976) application/pdf PDF (7MB) • (1994) application/pdf PDF (22MB) • (Office for Official Publications of the European Communities,, 1/1/2006) A report by the EU on jobs in Social Care application/pdf PDF (1MB) • (1990) In this study, the author documented the development of the organizational structure of one of the oldest shelters for battered women in this nation, in order to assess the impact feminist ideology and the grassroots history ... application/pdf PDF (11MB) • (1996) This interpretive interactionism study explored the practice of twelve social workers employed in early intervention programs for the families of children with special needs. In particular, this study examined the nature ... application/pdf PDF (10MB) • (1983) Seventy-four families in which the parents defined their adolescent as uncontrollable were studied. Research questions focused on the following: (1) Problem type, severity, duration, and cause. (2) Social climate dimensions ... application/pdf PDF (4MB) • (1994) Instructional computing applications are increasingly being diffused throughout social work higher education. For this study, graduate faculty ($N=330$) nation-wide were surveyed about their use and perceptions of computers ... application/pdf PDF (5MB) • (1996) This study explored the impact of a brief, narrowly focused education program for divorcing parents on increasing parent awareness of children's needs and reduction of interparental conflict. Behavioral changes on dimensions ... application/pdf PDF (9MB) • (2004) Despite the central role that HIV/AIDS prevention education plays in this community, this study brings forth the ecological factors influencing FSW safer sex negotiations in this geographical context. Understanding these ... application/pdf PDF (4MB) • (2000) This study employed a survey method to obtain an understanding of these issues from staff working in 40 EHS programs (Waves I--IV) in six Midwestern states. In addition, phone interviews with EHS directors provided program ... application/pdf PDF (9MB) • (1985) The federal government followed a consistent policy for the mentally and developmentally disabled between 1960 and 1980. The policy emphasized a shift away from institutional settings for the provision of services toward ... application/pdf PDF (5MB)	2016-05-31 20:04:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17780828475952148, "perplexity": 8422.6290080047}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464052946109.59/warc/CC-MAIN-20160524012226-00205-ip-10-185-217-139.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/show-that-it-satisfies-the-differential-equation.570782/	# Show that it satisfies the differential equation 1. Jan 25, 2012 ### crybllrd 1. The problem statement, all variables and given/known data Show that $$y=tanh(t)$$ satisfies the differential equation $\frac{dy}{dt}=1-y^{2}$ with initial conditions y(0) = 0 2. Relevant equations 3. The attempt at a solution I'm not sure how to start, but we have only dealt with one D.E., and we had moved all the variables to a respective side then integrated. $dy=(1-y^{2})dt$ $\frac{dy}{1-y^{2}}=dt$ Integrate both sides to get $ln\|1-y^{2}\|=t$ Not sure where to go from here 2. Jan 25, 2012 ### tiny-tim hi crybllrd! nooo d/dy (ln(1 - y2)) = 2y/(1 - y2) try again, splitting 1/(1 - y2) into partial fractions first 3. Jan 26, 2012 ### crybllrd Oh ok , I got it. I am in Calc 2 and when I read Diff. Eq. it just scared me off. That was easy enough. It is a seven part question, and the third part looks very similar: Show that for the arbitrary constants A and B, the function y=Atanh(Bt) satisfies: $\frac{dy}{dt}=AB-\frac{B}{A}y^{2}$ I will try to get it to match the form: $\int\frac{1}{A^{2}+x^{2}}=\frac{1}{A}tan^{-1}(\frac{x}{A})+c$ _______________________________________________________ $dy=B(A-\frac{1}{A}y^{2})(dt)$<--- factored out the B and multiplied by dt $\frac{dy}{A-\frac{1}{A}y^{2}}=Bdt$ It looks like I need to do a little more manipulation/simplification to get the left side where I want it (namely getting rid of the 1/A in the denominator) but I can't get it to the form I want. Last edited: Jan 26, 2012 4. Jan 26, 2012 ### HallsofIvy Staff Emeritus No, no, no! Did you not understand what sheriff89 said? You are not asked to solve the differential equation. In Calculus II you wouldn't be expected to do that. Differentiate y= Atanh(Bt) and put it and $y^2= A^2tanh^2(Bt)$ into the equation and determine if you get a true equation or not. 5. Jan 26, 2012 ### crybllrd Why would I put the y^2 into the equation and square the other side? And what is a "true equation"? I have a feeling these will be on our exam next week, so I want to understand it well. Thanks Last edited: Jan 26, 2012 6. Jan 26, 2012 ### Dick You still aren't getting the point of just substituting into the differential equation to show you have a solution. Go back to the first one. If y=tanh(x) then dy/dx=sech(x)^2. The other side is 1-y^2=1-tanh(x)^2. Aren't they equal? You don't have to solve any ODE's, you just have to show you have a solution. 7. Jan 28, 2012 ### crybllrd Oh ok I get it. On part A I algebraically rearranged to get dy/(1-y^2 )=dt, then I integrated both sides to get (after simplification) y= tanh⁡t, done. Just did it the long way, not even thinking because all the other problems were integration. then, when I started integrating part b, people starting getting upset :P So this is what I have for part b: y(t)=Atanh(Bt) y(t)'=ABsech2(Bt) y(t)'=AB-(B/A)y2 ABsech2(Bt)=AB-(B/A)Atanh2(Bt) B's cancel out Asech2(Bt)=A-tanh2(Bt) I can see this being true if A=1 Is that all I need? And, where does the y(0)=0 come in to play? I see where that is true, but is that information actually needed to solve it? 8. Jan 28, 2012 ### Dick y(t)^2=A^2tanh(Bt)^2. If you work it out correctly the A factor will cancel as well. And you don't need y(0)=0 to show that. But given the solution, you can say y(0)=0. 9. Jan 28, 2012 ### crybllrd Cool, thanks a lot. I think I'm definitely ready for an exam on Monday. EDIT: I had a math error on my last post when I substituted for y2. When done properly, all the A's cancelled out. Last edited: Jan 28, 2012 10. Jan 29, 2012 ### crybllrd Ugh, this problem is killing me. On a later part, it asks: Let v(t) be the velocity of a falling object of mass m that started from rest. For large velocities, air resistance is proportional to the square of velocity v(t)2. If we choose coordinates so that v(t)>0 for a falling object, then there is a constant k>0 such that $\frac{dv}{dt}=g-\frac{k}{m}v^{2}$ Solve for v(t) by applying the rsults of part (b) (what I just worked out) with $A=\sqrt{gm/k}, B=\sqrt{gk/m}$ I am not sure what "result" he is wanting from part B, I just had to show that a function satisfied a differential, and in doing so I cancelled out all of the constants except for the B within the hyperbolic. 11. Jan 29, 2012 ### Dick You know that v(t)=Atanh(Bt) satisfies $\frac{dv}{dt}=A B - \frac{B}{A}v^{2}$. So to find a solution to $\frac{dv}{dt}=g-\frac{k}{m}v^{2}$ you just set $A B=g$ and $\frac{B}{A}=\frac{k}{m}$ and solve for A and B. But it looks like they already did that for you as well. So maybe they just want to to substitute those values for A and B into v(t)=A*tanh(Bt)? Or maybe they want you to show explicitly that it works? 12. Jan 29, 2012 ### crybllrd Alright I just did both. Thanks again.	2017-09-22 23:03:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7256137728691101, "perplexity": 915.5413874162186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689373.65/warc/CC-MAIN-20170922220838-20170923000838-00136.warc.gz"}
https://www.intechopen.com/books/neuroimaging-structure-function-and-mind/supervised-sparse-components-analysis-with-application-to-brain-imaging-data	Open access peer-reviewed chapter # Supervised Sparse Components Analysis with Application to Brain Imaging Data By Atsushi Kawaguchi Submitted: April 11th 2018Reviewed: July 27th 2018Published: November 5th 2018 DOI: 10.5772/intechopen.80531 ## Abstract We propose a dimension-reduction method using supervised (multi-block) sparse (principal) component analysis. The method is first implemented through basis expansion of spatial brain images, and the scores are then reduced through regularized matrix decomposition to produce simultaneous data-driven selections of related brain regions, supervised by univariate composite scores representing linear combinations of covariates. Two advantages of the proposed method are that it identifies the associations between brain regions at the voxel level and that supervision is helpful for interpretation. The proposed method was applied to a study on Alzheimer’s disease (AD) that involved using multimodal whole-brain magnetic resonance imaging (MRI) and positron emission tomography (PET). For illustrative purposes, we demonstrate cases of both single- and multimodal brain imaging and longitudinal measurements. ### Keywords • data-driven approach • dimension reduction • principal component analysis • multimodal • multi-measurement ## 1. Introduction Recently, multiple neuroimaging data sets per subject have become obtainable due to the remarkable development of imaging techniques such as magnetic resonance imaging (MRI) and positron emission tomography (PET), as well as computer resources and technologies. Vandenberghe and Marsden [1] provide a review on the use of PET and MRI integration technology, such as integrated scanning devices, rather than data analysis. Other modalities such as diffusion MRIs (dMRIs) and functional MRIs (fMRIs) are also useful in collecting brain-related information. These multimodal imaging data sets have the potential to provide rich information about human health and behavior, such as brain function and structure, from different perspectives. From multiple measurements of a single-modal (or multimodal) technique, longitudinal changes in the status and combination of neuro biomarkers can be observed to support the prediction and early diagnosis of disease and the classification of disease subtypes. Multimodal brain imaging analysis is important in brain-related disease studies. Arbabshirani et al. [2] provide many reviews on the subject. Imaging data analysis makes a substantial contribution to the study of mental disorders. Most single-modal or multimodal imaging studies concern dementia leading to Alzheimer’s disease (AD) [3] (around 300 of the AD imaging studies searched in Ref. [2]). Modalities considered in there are structural MRIs (sMRIs), fMRIs, dMRIs, fluorodeoxyglucose PETs, and Amyloid/Tau PETs. In a recent study, Ref. [4] examined sMRI and cerebrospinal fluid (CSF) markers. Magnetoencephalography (MEG) is also useful as AD biomarker, and its localization using sMRI has high accuracy [5]. Schizophrenia is the second most studied disorder after dementia. Shah et al. [6] provide an example of multimodal meta-analysis. For Huntington’s disease, white matter is evaluated using dMRI [7]. For mood disorders (depressive disorder and bipolar disorder), Refs. [8, 9] provide a review of the machine learning method. Moeller and Paulus [10] studied the longitudinal prediction of relapse for substance-related disorders using MRI, fMRI, EEG, and PET. Moser et al. [11] studied schizophrenia and bipolar disorder using multimodal imaging data analysis. dMRI is also effective for analyzing these conditions [12]. For developmental disabilities, Ref. [13] investigated volume reductions in attention-deficit hyperactivity disorder (ADHD) with 1713 participants. Aoki et al. [14] reviewed dMRI studies and conducted meta-analysis for ADHD. Li et al. [15] provide a review of imaging studies in autism spectrum disorder. For anxiety disorder, Ref. [16] applied support vector machine (SVM) to multimodal data. They used clinical questionnaires and measured cortisol release, and gray and white matter volumes in subjects with generalized anxiety disorder and major depression and in healthy subjects. Steiger et al. [17] investigated cortical volume, diffusion tensor imaging, and network-based statistics using multimodal analysis for social anxiety disorder. For borderline personality disorder, Ref. [18] conducted an imaging-based meta-analysis of 10 studies. In cancer research, especially that on glioblastoma multiforme, multimodal imaging analysis is useful for identifying some types of tumors and evaluating patient prognosis (for more details, see [19]). Genome-related data can be regarded as a modality and called imaging genetics when analyzed in combination with imaging data [20]. One important technique for single- and multimodal imaging analysis is prediction, which is useful for the support of disease diagnosis and the selection of treatments [21]. SVM is the most used method not only in neuroimaging but also in the life sciences in high-dimensional data analysis. The random forest method is also useful due to their capability for complex interactions based on the tree model [22, 23]. For multimodal analysis, multiple kernel learning [24] and (multimodal) deep learning [25, 26] have been developed. Janssen et al. [27] reviewed machine learning methods for psychiatric prognosis. Related statistical methodology appeared as multi-omics in bioinformatics, and Ref. [28] reviewed these methods while introducing an R package, mixOmics. Analysis for such discovery and evaluation is based on the detection of the buried signal in the noise (irrelevant information). Statistical analysis is useful for this purpose; however, it suffers from the ultrahigh dimensional and complex structure of this data, and appropriate dimension reduction is therefore required. Even if a machine learning method is used, appropriate input (feature) should be specified to obtain interpretable results because the method is feasible for high-dimensional procedures but not ultrahigh dimensional ones. A region-of-interest-based analysis was the leading approach. In contrast, whole-brain analysis is more informative, and if it is combined with a data-driven approach, it can potentially obtain undiscovered knowledge. In [29], by using ReliefF [30], features such as the fractional amplitude of low-frequency fluctuations from resting-state fMRIs, segmented gray matter from sMRIs, and fractional anisotropy from dMRIs were extracted. Component analysis based on low-rank approximation is a successful data-driven approach in the fields of not only neuroimaging but also other biological and medical big data analyses, including principal component analysis, partial lease squares, canonical correlation analysis (CCA), independent component analysis (ICA), and nonnegative matrix factorization. These methods are organized into a matrix decomposition framework consisting of score and loading (weight) matrices. The score matrix, with same row length as the number of subjects, is regarded as dimension-reduced data and is suitable for application to statistical models. The weight matrix, with the same column length as the number of features in the imaging data, is regarded as the basis images. All these methods, except for ICA, have a derivation sparse approach with a regularized matrix decomposition to pose small weights to zeros, which helps estimation by avoiding irrelevant information. In addition, the resulting weights can be interpreted to mean that the corresponding features with nonzero weights contribute to the basis image, specifically to produce data-driven selections of brain regions related to that component. These methods also consider another direction in which the application of multimodal imaging data can be extended. Supplementary information from another data set can also be useful for the interpretation of the output. For this purpose, appropriate data fusion or integration techniques are required and are useful for multisite studies. In neuroimaging data analysis, multimodal CCA (mCCA) [31] and mCCA + joint ICA [32] have been developed on the schizophrenia study. Multivariate data fusion approaches were categorized by [33] into asymmetric or symmetric data and blind or semi-blind data in symmetric approach. The asymmetric approach is a regression-type approach and includes specific modalities such as dMRI and electroencephalography. The symmetric approach is a correlation-type approach and allows relationships in both directions. Kawaguchi [19] constructed a risk score for glioblastomas based on MRI data and proposed a two-step dimension-reduction method using a radial basis function-supervised multi-block sparse principal component analysis (SMS-PCA) method. Kawaguchi and Yamashita [34] proposed a more general case including a PLS or CCA framework and applied it to MRI, PET, and SNP data sets. Yoshida et al. [4] analyze imaging and non-imaging data with network structure by using the PLS. In this chapter, we applied SMS-PCA to MRI and PET data sets and a longitudinal MRI data set. One of the key features in the analysis is a multi-block technique which can achieve structural dimension reduction with interpretable parameters (weights for each data set and the possibility of combining them). Although it is not the focus of this chapter, the dimension reduction prior to SMS-PCA is conducted using 3D basis functions. Specifically, our dimension reduction takes place in two steps, and, as described in [35] which applied these techniques to longitudinal study, this two-step approach yields a composite basis function expression with a flexible shape. The organization of this chapter is as follows. Section 2 describes the methodology of the SMS-PCA, which is applied to real data in Section 3. The characteristics of the method, found through its application, are discussed in Section 4. ## 2. Methods We describe the proposed method in this section. The contents are similar to Ref. [19]. ### 2.1 Priory dimension reduction S=sαα=1,,nis the n×Nmatrix whose column corresponds to the vectorized original image data. As the dimensions for each mth image are the same, we use the same basis function to reduce the dimension from Nto q. X=SBis the n×qmatrix, where Bis the N×qmatrix whose jth column corresponds to the vectorized basis function with the jth knot being the center. Note that knots are pre-specified to span the space equally, as shown in Figure 1. In this example, four-pixel equal spanning knots are applied. ### 2.2 Objective function Dimension reduction using the basis function is then followed by the SMS-PCA method, considering (sample) correlations based on data values. We consider score tfor n×qmatrices Xm, where m=1,2,,Mwith the following multi-block structure: t=m=1MbmXmwm=m=1Mbmtm,E1 where wmis the weight vector for the mth sub-block Xmand bmis the weight for the superblock. Here, it should be noted that the scores in Eq. (1) are referred to as the super scores, whereas tm=Xmwmis referred to as the block score. Figure 2 schematically describes the score structure for the case of M=2. Thus, the super score has a hierarchical structure for each individual and can be used in an application such as the construction of a diagnosis score. When matrix Xmis normalized by its columns, the weights w=w1w2wMand b=b1b2bMare estimated by maximizing the function Lbw=1μtt+μtZm=1MPλmwmE2 subject to wm22=1and b22=1with ·2as the L2 norm, where 0μ1is the proportion of the supervision, Pλxis the penalty function, [Pλx=2λxis used in this study], and λ>0is the regularized parameter that is used to control the sparsity. The larger value of the regularization parameter λmhas many nonzero elements in wm. ### 2.3 Optimization The algorithm given in Table 1 is used to estimate the weights in Eq. (1) by maximizing L in Eq. (2). The rationality behind this approach is provided in [19]. Initialize twith t2=1.Repeat until convergence:2.1. Set w∼m=hλmbmXm⊤1−μt+μZ, where hλy=signyy>λ+, and normalize as ŵm=w∼m/w∼m2m=12…M.2.2. Set tm=Xmŵmand b∼m=tm⊤1−μt+μZ; then set b∼=b∼1b∼2…b∼M⊤and normalize as b̂=b∼/‖b∼‖2.2.3. Set t=∑m=1Mb̂mXmŵm.(Deflation step) Set pm=Xm⊤tm/tm⊤tmand X̂m=tmpm⊤, and Xm←Xm−X̂m. ### Table 1. Algorithm for SMS-PCA method. Note that the deflation step yields multiple components and has several alternatives; that is, through Ktime iteration for step. 1 to 3 of the algorithm, we can obtain Kcomponent super scores t1,,tKwith tk=t1ktMk. ### 2.4 Parameter selection The optimal value for λ=λ1λMis selected by minimizing the Bayesian information criterion (BIC): BICλ=logm=1MX̂mrXm2nMq+lognMqnMqnonzero elements inwm, where X̂mr=TmrPmrwith Tmr=tm1tmrand Pmr=pm1pmris obtained from rdeflation steps (the projection of Xmonto the r-dimensional subspace). There are several search strategies for optimization, and these are introduced in the software options below. ### 2.5 Software The statistical software R package msma is provided to implement the method described in Ref. [34] where the SMS-PCA method is a part of the package and the PLS type can also be implemented. The package is available from the Comprehensive R Archive Network (CRAN) at http://CRAN.R-project.org/package=msma. Four-parameter search methods are available. Here, the parameters are λmand the number of components. The “simultaneous” method identifies the number of components by searching the regularized parameters in each component. The “regpara1st” method identifies the regularized parameters by fixing the number of components and then searching for the number of components with the selected regularized parameters. The “ncomp1st” method identifies the number of components with a regularized parameter of 0 and then searches for the regularized parameters with the selected number of components. The “regparaonly” method searches for the regularized parameters with a fixed number of components. In this chapter, the “ncomp1st” method was applied with nonzero sparsity when the number of components was selected because, in our experience, the BIC value suffered from the high dimensionality of the data. The basic R code for this method is as follows: tuneparams = optparasearch(X=X, Z=Z, search.method=“ncomp1st”, maxpct4ncomp=0.5, muX=0.5) where the argument maxpct4ncomp = 0.5 means that 0.5λmaxis used as the regularized parameter when the number of components is searched and where λmaxis the maximum of the regularized parameters among the possible candidates. In order to obtain the final fit result with optimized parameters, the following code should be implemented: fit1 = msma(X=X, Z=Z, comp=tuneparams$optncomp, lambdaX=tuneparams$optlambdaX, lambdaY=tuneparams\$optlambdaY, muX = 0.5) For more details, please see the package manual. ## 3. Application In this section, we apply the SMS-PCA described in the previous section to real data. The data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). We use two types of data set: baseline measurement with multimodal MRI and PET images and repeated measuring MRI images. ### 3.1 Multimodality #### 3.1.1 Data Baseline imaging data were collected from 106 subjects with mean ages of 75.2 years for the 54 normal cognitive subjects and 72.9 years for the 27 patients with dementia. This data set was somewhat larger than that of [34] because in this study single-nucleotide polymorphism (SNP) was not considered and subjects with missing SNP data were included. Table 2 summarizes the characteristics of these patients. We consider imaging data from two modalities, MRI X1and PET X2, namely, M=2. The preprocessing method is the same as that used in [34]. For the basis function, we used four-voxel (therefore, h=3×42=6.93) equal spacing knots because of the results of our simulation study. The clinical outcome to supervise is given by Z=3.17×CDR+0.11×ADAS130.57×MMSEwhere CDR is the clinical dementia rating score, ADAS13 is the Alzheimer’s disease assessment scale-cognitive subscale, and MMSE is the mini-mental state examination score. These coefficients were the same as in [34]. The SMSMA method was applied to the data X1X2Zwith parameters μ=0,0.25,0.5,and0.75. DementiaNormalp n5254 Age (mean [sd])75.41 (7.18)74.93 (4.89)0.684 PTGENDER = Male (%)31 (59.6)36 (66.7)0.582 APOE4 (%)<0.001 017 (32.7)39 (72.2) 129 (55.8)13 (24.1) 26 (11.5)2 (3.7) PTEDUCAT (mean [sd])14.19 (3.04)15.89 (2.99)0.005 CDRSB (mean [sd])4.54 (1.73)0.03 (0.12)<0.001 ADAS11 (mean [sd])18.70 (5.63)6.56 (3.28)<0.001 ADAS13 (mean [sd])28.94 (6.30)10.08 (4.30)<0.001 MMSE (mean [sd])23.38 (2.07)28.87 (1.24)<0.001 ### Table 2. Characteristic for data set 1. #### 3.1.2 Results The original data with dimensions of 2,122,945 (= 121 × 145 × 121) was reduced to 7,162 using the basis functions for each imaging data set. The number of components were selected as 8 for all μ= 0, 0.25, 0.5, 0.75, 1. Figure 3 shows the correlation matrix from the dataset with the binary outcome, AD or Normal, and the resulting super scores for each μ. The correlations between the super scores were small except for μ=1, and for μ=0, the second component had a high correlation with the outcome. In contrast, for μ>0, the first component had the highest correlation with the outcome. Table 3 shows the results for the multiple logistic regression model with AD or normal as the outcomes and the super scores as predictors for each μ. The numbers of 5% statistically significant components were 3, 4, 3, 3, and 0 for μ= 0, 0.25, 0.5, 0.75, and 1, respectively. The minimum numbers of nonzero subweights were 552, 581, 574, 523, and 1075, respectively. μ = 0.00μ = 0.25μ = 0.50μ = 0.75μ = 1 EstimatePr(>\|z\|)EstimatePr(>\|z\|)EstimatePr(>\|z\|)EstimatePr(>\|z\|)EstimatePr(>\|z\|) comp1−0.02100.06150.0827<0.00010.0832<0.00010.0857<0.00014.2870.9982 comp20.0882<0.00010.04600.00300.04580.00310.04510.00392.5550.9989 comp3−0.06210.00010.01800.09230.01800.09200.01810.09524.6330.9995 comp4−0.00720.61260.00370.79530.00440.75830.00640.65741.8270.9987 comp5−0.04240.02280.04520.04320.04300.02030.04310.02063.9050.9988 comp6−0.03640.09000.03960.03200.04030.07150.04250.06584.9940.9984 comp70.04460.08910.05050.06360.05100.06190.05350.0549 comp80.03360.2517−0.02280.3804−0.02260.3816−0.02180.3904 ### Table 3. Results for multivariable logistic regression analysis. Figure 4 shows the reconstructed subweights Bw1and Bw2for the MRI and PET data, respectively, overlying a structural brain image shown for the most correlated components with the binary outcome from each of μ=0,0.5,0.75,and1. The images for μ=0.25were similar to those of μ=0.5,0.75and are not shown here. Figure 5 shows the reconstructed subweights Bw1and Bw2overlying a structural brain image and bar plots for the super-weights (right bottom) in the case of μ=0.5for all components. In each component, the negative and positive sides are represented. These can be interpreted by looking at the sign of the super-weight. Most cases remain on one side of 0 (positive or negative), except for components 5 to 8. The super-weights are similar between MRI and PET. A 10-fold cross validated ROC analysis (Figure 6A) was conducted to evaluate the diagnostic probabilities estimated from the multivariable logistic regression mode whose coefficients and p-values are shown in Table 3. For comparison, the single modalities, MRI (Figure 6B) and PET (Figure 6C), were also analyzed. In the case of the multimodal MRI and PET (Figure 6A), μ=1had the highest AUC value (0.984) following by μ=0.75(AUC = 0.880). In the case of the single-modal MRI (Figure 6B), all values were below the AUC values of the multimodal case. In the case of the single-modal PET (Figure 6C), μ=1and0.75outperformed the multimodal case, and the other values (μ=0,0.25,and0.5) did not. ### 3.2 Multi-measurements #### 3.2.1 Data The second data set was a collection of repeated measured imaging data from 68 patients with mild cognitive impairment (MCI). There were two groups, the conversion to dementia MCI (cMCI) group and the stable MCI (not converted to dementia, sMCI) group. MRI data measured at four time points were used. For the cMCI group, the four time points were just before diagnosis of conversion. For the sMCI group, the four time points were from the baseline for the entire period of the study. Groups were matched for age, gender, and intracranial volume. Table 4 summarizes the characteristics of these patients at baseline (at the first image observation). cMCIsMCIp n3434 Age (mean [sd])76.06 (5.94)75.91 (5.90)0.922 PTGENDER = 2 (%)10 (29.4)10 (29.4)1.000 APOE4 (%)0.040 012 (35.3)22 (64.7) 118 (52.9)11 (32.4) 24 (11.8)1 (2.9) PTEDUCAT (mean [sd])16.15 (3.06)15.50 (2.86)0.371 CDRSB (mean [sd])1.76 (1.07)1.32 (0.73)0.051 ADAS11 (mean [sd])12.09 (3.49)9.40 (4.08)0.005 ADAS13 (mean [sd])19.65 (4.31)15.93 (6.10)0.005 MMSE (mean [sd])26.71 (1.71)27.88 (1.70)0.006 ### Table 4. Characteristic for data set 2. For imaging data processing, we used the VBM8 toolbox. For the basis function, we used four-voxel equal spacing knots, as in the first study in the previous section. The clinical outcome is given by Z=0.44×CDR+0.12×ADAS130.11×MMSE.The coefficients were different from those in the first study because the target population was different. #### 3.2.2 Results The original data with dimensions of 2,122,945 (=121 × 145 × 121) was reduced to 7162 using basis functions for each imaging data set. The number of components selected was 6 for μ=0,0.25,0.5,0.75and 4 for μ=1. Table 5 shows the results for the multiple logistic regression model with cMCI or sMCI as the outcomes and the super scores as the predictors for each μ. The numbers of 5% statistically significant components were 2, 3, 3, 3, and 2 for μ=0,0.25,0.5,0.75,1, respectively. The minimum numbers of nonzero subweights were 724, 736, 749, 753, and 852, respectively. μ = 0.00μ = 0.25μ = 0.50μ = 0.75μ = 1 EstimatePr(>\|z\|)EstimatePr(>\|z\|)EstimatePr(>\|z\|)EstimatePr(>\|z\|)EstimatePr(>\|z\|) comp1−0.01320.01390.01420.02150.01420.02120.01430.02040.03310.0018 comp20.00830.17140.01980.01990.00950.12980.00970.12400.03590.0049 comp3−0.01250.1481−0.00930.14100.01950.02150.01960.02140.01720.2833 comp4−0.03330.0042−0.00730.4505−0.00730.4490−0.00700.4709−0.01570.5288 comp50.00010.99630.04580.00220.04600.00210.04690.0019 comp6−0.02030.16170.00770.62050.00740.63130.00780.6139 ### Table 5. Results for multivariable logistic regression analysis. A tenfold cross validated ROC analysis (Figure 7) was conducted to evaluate the diagnostic probabilities estimated from the multivariable logistic regression mode whose coefficients and p-values are shown in Table 5. For comparison, the single-modal analysis for each time point was conducted. The fourth time point (MRI4), which is closest to the MCI conversion diagnosis time, had the highest AUC values, and these were higher than the multimodal values (Figure 8). Figure 9 shows the first component subweights, Bwm(m=1,2,3,4), for the four time points for μ=0and 0.5. In the case of μ=0.5, the hippocampus area was related to the components, and in the case of μ=0, the parietal lobe was. Figure 10 shows the corresponding super-weights. This result should be carefully interpreted. For time 4, the sparsest block weights were obtained, and thus the weight values were larger than those of times 1 to 3, which were balanced by the small super-weight. As a result, the super score for this component has the mean value of the block scores. ## 4. Discussion In this chapter, the SMS-PCA method was introduced and applied to multiple measured neuroimaging data sets. The first data set consisted of two different types of images, MRI and PET. The second data set consisted of repeated MRI measurements (the same type of image). These imaging data have many voxels per person which were reduced using the basis function prior to conducting the SMS-PCA. The multi-block feature of the SMS-PCA also caused further reduction in each block, and their summary was obtained in the super level where the weights were the relationship and the scores were used in the prediction model. One of the key features in the SMS-PCA is that it is supervised and its proportion to (self) variance is parametrized by μ. In each study, the impact of μwas studied. The case of μ=1resulted in only supervision, that is, only the correlation between the score and the outcome, without the variance of the score. As in an original PCA, maximizing the variance of the score corresponds to μ=0, and the correlated variables (voxels) have relatively high weights for each component. Thus, the messy maps for the block weights overlaying the brain in the case of μ=1were reasonable. In both applications, because μ=0.25,0.5,and0.75had similar results, a possible large value in μ<1, or the median value μ=0.5with a trade-off, can be selected as optimal. Repeated measured imaging data analysis was studied in [35] which reduced the imaging dimensions using basis functions but did this independent for each image. In contrast, in this study, the correlation between measurements at different time points is considered. That is, simultaneous temporal and spatial correlation was considered. This approach was limited by the need that the number of images for each individual be the same, and this will be improved in future work. In addition, the method introduced in this chapter can incorporate modalities such as network models which would need to summarize the information into the component. This research is in progress. ## 5. Conclusion Although there is room for improvement in this method, this study showed reasonable results when the method was applied to the dementia study. In conclusion, this data-driven approach would be helpful for exploratory neuroimaging data analysis. ## Acknowledgments This study was supported in part by the Intramural Research Grant (27-8) for Neurological and Psychiatric Disorders of NCNP. For this research work, we used the supercomputer of ACCMS, Kyoto University. Data collection and sharing for this project were funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense Award Number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd. and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are disseminated by the Laboratory of Neuro Imaging at the University of Southern California. None declared. ## Notes • Data used in preparation of this article were obtained from the Alzheimer?s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ ADNI_Acknowledgement_List.pdf chapter PDF Citations in RIS format Citations in bibtex format ## More © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ## How to cite and reference ### Cite this chapter Copy to clipboard Atsushi Kawaguchi (November 5th 2018). Supervised Sparse Components Analysis with Application to Brain Imaging Data, Neuroimaging - Structure, Function and Mind, Sanja Josef Golubic, IntechOpen, DOI: 10.5772/intechopen.80531. Available from: ### More statistics for editors and authors Login to your personal dashboard for more detailed statistics on your publications. ### Related Content Next chapter #### Vector-Based Approach for the Detection of Initial Dips Using Functional Near-Infrared Spectroscopy By Toshinori Kato First chapter #### Introduction to Infrared Spectroscopy in Life and Biomedical Sciences By Theophile Theophanides We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.	2021-04-23 06:53:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6152470707893372, "perplexity": 2740.4942201074455}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039601956.95/warc/CC-MAIN-20210423041014-20210423071014-00617.warc.gz"}
https://blender.stackexchange.com/questions/6625/can-i-constrain-an-object-size-to-camera-angle-of-view	# Can I constrain an object size to Camera angle of view? After I import "Images As Planes" (using addon) I would like to place the images in front of the camera like a rostrum, making the corners match the camera frame. Or perform the best fit of the height or sides if the images are a different shape (vertical aspect). I know that I can set the camera to orthographic and this fixes the relative dimensions on z axis, but I want to make a objects initial scale or position fill the camera view. This way I can always return to a 1:1 relationship without borders or overshoot. USAGE: This is helpful when creating layers of images or movies that you may want to animate in 3D view (relative to each other) but allocate to layers for actual compositing. • Are you sure you are not looking for the Background Image feature, which is in the Property panel? Like this. – Leon Cheung Jan 30 '14 at 2:03 • Thanks but no, that would be a guide image, I want to layer photos or cell animation but keep the image frames registered at the edges of the camera view. – 3pointedit Jan 30 '14 at 3:31 Actually, it can be done with both perspective and orthographic cameras. I would like to suggest the universal way, which needn't any reset on the current camera transformation: ## For Perspective camera: 1. Enable Import Image as Planes and Simple Align in Addon list. 2. Import the target image as plane. When importing, Choose Dots/BU as Plane dimensions, and set the longest side of the image as Definition value. For example, if the image resolution is 1920 px x 1080 px, then you should type in 1920. 3. Set the render resolution the same as image, to set the correct perspective for camera. 4. With the plane selected, hold Shift to further select the camera, then find Simple Align tool in the Tool Shelf, click XYZ button to align both rotation and location of the plane to the camera. 5. Select the plane only, then input like this: GZZ=-focal_length/sensor_size. For example, if the focal length of camera is 35, and sensor size is 32, then press GZZ=-35/32. Then you'll get what you want. ## For Orthographic camera: 1. As above. 2. As above. 3. As above. 4. As above. 5. Select the plane only, then input like this: GZZ=-ortho_scale/2. For example, if ortho_scale=8, you should input GZZ=-8/2. Then Tab into Edit Mode, S8. P.S.: Doing math while transforming is only available in Blender 2.70+. However, you can do the math elsewhere out of Blender, and paste the result in. • When I try to do this: the plane I want to align to the camera edges is moved to the camera's origin and not it's projected view port; any ideas why? – ThorSummoner May 27 '14 at 8:51 • Also, Can you elaborate what 'gzz' is supposed to do? It appears to want to move the plane along the z axis, the second press of the 'g' key appears to do nothing, though pressing 'g' a third time unlocks the transformation from the z axis. – ThorSummoner May 27 '14 at 8:53 • @ThorSummoner I see your point. I've updated my answer. That's because the operation method for the new numberal input feature. So now you have to type = or * before doing division. The GZZ means to translate along its local Z axis. – Leon Cheung May 27 '14 at 13:48 • @3pointedit I wonder if this helps? – Leon Cheung May 27 '14 at 13:51 There is an great addon to do exactly what you need! "create camera image plane" the "script should be run with a camera selected and then creates an imageplane, sets up drivers to automatically scale it by the render aspect ratio and focal length of camera and distance (that would be the neg z of the imageplanes location) so it will cover the entire screen. It sets up a cycles material so it's ready to load an image sequence" The material is created with an emission shader that takes into account the alpha channel (if there is one), all you need to do is select the image you want to use: If you move the camera, the plane will follow. Also if you change the focal lenght, the plane will resize accordingly. And if you move the plane on its local Z axis it will resize as well, filling the frame at all times.	2019-12-08 10:55:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24081987142562866, "perplexity": 1480.8115254525733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540508599.52/warc/CC-MAIN-20191208095535-20191208123535-00333.warc.gz"}
https://answerriddle.com/answer-in-which-category-did-albert-einstein-win-a-nobel-prize/	# Answer: In which category did Albert Einstein win a Nobel Prize? The Question: In which category did Albert Einstein win a Nobel Prize? Physics) Chemistry) Medicine) Peace	2021-09-22 01:53:42	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9247729778289795, "perplexity": 7522.479582315804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057303.94/warc/CC-MAIN-20210922011746-20210922041746-00150.warc.gz"}
http://mymathforum.com/algebra/18058-find-radius-circle-if-its-area-equals-its-circumferen.html	My Math Forum Find radius of circle if its area equals its circumferen Algebra Pre-Algebra and Basic Algebra Math Forum March 14th, 2011, 08:31 AM #1 Newbie Joined: Mar 2011 Posts: 4 Thanks: 0 Find radius of circle if its area equals its circumferen Q: Find the radius of a circle if its area is equal to its circumference. It is damn difficult. Please help me solve this. March 14th, 2011, 08:42 AM #2 Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5 Re: Find radius of circle if its area equals its circumferen I might reword this a little (not just the strong language!)... "The area of a circle equals the circumference of (the same) circle" Now "the area of a circle" is just $\pi r^2$ Somehow, this equals the circumference: $C= 2\pi r$ So $\pi r^2= 2\pi r$ Pretty sure you can solve this! March 14th, 2011, 08:44 AM #3 Senior Member Joined: Mar 2011 From: Chicago, IL Posts: 214 Thanks: 77 Re: Find radius of circle if its area equals its circumferen A=C ?rr=2?r r=2 March 17th, 2011, 07:28 AM #4 Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 ?r² = 2?r ? and r ? r = 2 March 17th, 2011, 09:28 AM #5 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Find radius of circle if its area equals its circumferen If we allow for the degenerate case, then r = 0 also works! March 18th, 2011, 09:36 AM #6 Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5 Re: Find radius of circle if its area equals its circumferen Quote: Originally Posted by MarkFL If we allow for the degenerate case...! ...which we most certainly do... :@ Tags area, circle, circumferen, equals, find, radius ### how to find the circumferen Click on a term to search for related topics. Thread Tools Display Modes Linear Mode Similar Threads Thread Thread Starter Forum Replies Last Post yogazen2013 Algebra 1 August 24th, 2013 01:30 PM nabilkud Algebra 1 October 20th, 2012 10:49 PM amitdixit Algebra 1 August 25th, 2012 08:49 PM Zappo Algebra 4 May 11th, 2012 12:37 AM notepad Algebra 1 January 2nd, 2010 09:12 AM Contact - Home - Forums - Cryptocurrency Forum - Top	2019-10-23 18:28:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 3, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6629725694656372, "perplexity": 8012.878856080188}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987835748.66/warc/CC-MAIN-20191023173708-20191023201208-00278.warc.gz"}
https://codegolf.meta.stackexchange.com/questions/2140/sandbox-for-proposed-challenges/3607	# What is the Sandbox? This "Sandbox" is a place where Code Golf users can get feedback on prospective challenges they wish to post to the main page. This is useful because writing a clear and fully specified challenge on the first try can be difficult. There is a much better chance of your challenge being well received if you post it in the Sandbox first. See the Sandbox FAQ for more information on how to use the Sandbox. ## Get the Sandbox Viewer to view the sandbox more easily To add an inline tag to a proposal use shortcut link syntax with a prefix: [tag:king-of-the-hill] ## Is it a Happy Number? code-golfdecision-problem A repost of this challenge (if I got the policy right). Given a single positive integer (which can also be taken as a list of digits or a string), output whether the number terminates at 1 . Truthy/falsy follows the language's convention, or you can choose exactly one value for truthy and another for falsy. (This sequence is A007770.) Your program should theoretically support all non-negative integers; however, if your language doesn't support unbounded integers, you may only support integers up to 2147483647. ## Procedure Suppose you have the number 193. • Square every individual digit in the number. Therefore the number's individual digits becomes: [1] [81] [9] • Sum all these individual digits: 92 • Repeat this procedure until it stabilizes at 1 or a 37-cycle like the following: 37-58-89-145-42-20-4-16-37 It has been shown that the procedure will always produce either one of these two outputs. ## Test cases Here is a sample program generating the test cases. Here is a step by step reduction of all input between 1 and 100. 1 -> true 2 -> false 3 -> false 4 -> false 5 -> false 6 -> false 7 -> true 8 -> false 9 -> false 10 -> true 11 -> false 12 -> false 13 -> true 14 -> false 15 -> false 16 -> false 17 -> false 18 -> false 19 -> true 20 -> false 21 -> false 22 -> false 23 -> true 24 -> false 25 -> false 26 -> false 27 -> false 28 -> true 29 -> false 30 -> false 31 -> true 32 -> true 33 -> false 34 -> false 35 -> false 36 -> false 37 -> false 38 -> false 39 -> false 40 -> false 41 -> false 42 -> false 43 -> false 44 -> true 45 -> false 46 -> false 47 -> false 48 -> false 49 -> true 50 -> false 51 -> false 52 -> false 53 -> false 54 -> false 55 -> false 56 -> false 57 -> false 58 -> false 59 -> false 60 -> false 61 -> false 62 -> false 63 -> false 64 -> false 65 -> false 66 -> false 67 -> false 68 -> true 69 -> false 70 -> true 71 -> false 72 -> false 73 -> false 74 -> false 75 -> false 76 -> false 77 -> false 78 -> false 79 -> true 80 -> false 81 -> false 82 -> true 83 -> false 84 -> false 85 -> false 86 -> true 87 -> false 88 -> false 89 -> false 90 -> false 91 -> true 92 -> false 93 -> false 94 -> true 95 -> false 96 -> false 97 -> true 98 -> false 99 -> false • Is there an upper bound for inputs? I think one thing to consider is whether you want answers to implement the square-summing operation, or to try to compress or overfit some heuristic that works for say, 1 to 100. – xnor Mar 28 at 10:46 • Falsy numbers belong to A007770. – Arnauld Mar 29 at 22:49 • @Arnauld Nice catch + Title suggestion! – user92069 Mar 30 at 0:25 • With the term "happy number" in hand, I found a probable duplicate. – xnor Mar 30 at 8:37 • Ugh, why do I always have duplicate ideas recently... – user92069 Mar 30 at 9:01 • @xnor This is a dupe indeed, but the other challenge is very old and it seems like it requires a full program with a cumbersome output format. So maybe we should rather close the old challenge as a dupe of this one instead? (I'm not sure about the right policy here.) – Arnauld Mar 30 at 10:38 • We did once repost Kolakoski one and closed the old one as dupe (with relevant meta discussion). But this case is a bit different because the author of the old challenge is no longer active. – Bubbler Mar 30 at 23:00 • @Arnauld Good point, that old challenge is sure showing its rust. – xnor Mar 30 at 23:14 # Posted Solve a Picross Row • I don't think it is a dupe of full nonogram solver. I recommend allowing flexible I/O formats though (e.g. values other than 0,1,2 to mark each cell state). – Bubbler May 6 at 7:54 • To make the challenge more self-contained, consider adding a brief introduction to picross/nonogram and its rules. – Bubbler May 6 at 9:18 # Implement an HTML renderer code-golfascii-art Note: This challenge explaination is very much incomplete - it merely contains ideas that will require revising to form a proper challenge post. The premise of the challenge is to write a program that take an HTML document as an input, and outputs an ASCII equivalent. Obviously, working with real HTML is not possible, so this challenge will use a very limited and modified subset of HTML. Here is an example of a potential input: <body> <h1>A Document</h1> <div> <span>Hello, this is some text</span> <img> 8 2 </img> </div> </body> Which would yield the following output: +--------------------+ \|A DOCUMENT \| \| \| \|+------------------+\| \|\|Hello, this is som\|\| \|\|e text \|\| \|\|+--------+ \|\| \|\|\|@~@~@~@~\| \|\| \|\|\|~@~@~@~@\| \|\| \|\|+--------+ \|\| \|+------------------+\| +--------------------+ ## HTML elements that will be implemented: <span> - Renders text between the tags, wrapping when necessary. Example: <body> <span> This is a span element. You can write text in here. </span> </body> Output: +--------------------+ \|This is a span eleme\| \|nt. You can write te\| \|xt in here. \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| +--------------------+ (Extra explanation needed to clarify whitespace and character set issues) <p> (Explanations are omitted to save space) <h1> - <h6> <div> <img> ## Sandbox questions/remakrs • I believe it is possible to write an unambiguous and specific set of rules for how an "HTML document" should be rendered • It will require lots of careful explanation, wording, and ample examples • However this challenge seems very long and complicated and it seems like it might not be in the spirit of a code golf challenge What do you guys think? • "I believe it is possible" -- Yes, rewriting a subset of the HTML Spec for ASCII is possible. However, successful challenges tend to keep it simple. I'd suggest using just span and div and using no attributes. In addition, parsing has probably been done before and is cumbersome, so I'd suggest allowing input as a pre-parsed AST to focus on the key challenge of ASCII-art generation – fireflame241 Jul 9 at 6:16 • Personally I would keep only body, div, span and img, altho img should follow a set pattern inside of it. I would also suggest making height and width attributes mandatory, in all tags. – Dion Jul 20 at 6:08 # Solve a 2xN Maze (posted) • Now that this has been posted to main, could you delete this proposal to create more space for new answers? – caird coinheringaahing 6 hours ago # Does the naïve fill suffice? A bot is positioned in a rectangular grid. By preference it will paint in a west direction, but if it cannot it will paint in a south, east or if all else fails north direction. Sometimes this can lead it to fill the grid, but other times it gets stuck. The following examples show how the path (indicated by ascending digits) of the bot on a given grid varies depending on its starting position: 1 The bot is always able to fill a 1×1 grid, since simply by existing it has already painted the grid. 14 21 43 34 23 34 12 21 The bot is always able to fill a 2×2 grid. As a consequence of its painting direction preferences it normally traverses anticlockwise except when it starts in the bottom right corner when it traverses clockwise. 16 21 65 .. 56 65 165 216 321 654 345 456 25 36 14 21 43 34 234 345 456 123 216 321 34 45 23 34 12 21 The bot usually fills a 2×3 grid, except when it starts in the middle right square. On the other hand, it always fills a 3×2 grid; its painting direction preferences cause it to paint clockwise if it starts in the bottom middle or bottom right cell, otherwise anticlockwise. 189 21. 321 87. 987 ... 987 ... 987 276 387 498 165 216 321 236 345 456 345 456 567 234 345 456 145 216 321 The bot is able to fill a 3×3 grid when it starts in one of the even squares. It's mathematically impossible for the bot to fill it when it starts in an odd square, but I have included these positions for completeness. Your task is to solve the of whether the bot is able to fill a given grid from a given starting point. You can assume that the grid size is a positive integer and that the starting point lies within the grid. You can take the grid size and starting point in any consistent order, as separate inputs, a pair of pairs or a list of 4 elements, or any other reasonable input format. The starting point can be 0-indexed or 1-indexed. You can use any two consistent outputs, or you can output using any values that your language considers truthy or falsy, but not both. Please include your input and output format in your answer. The directions west, south, east or north correspond to decrementing the x-coordinate, incrementing the y-coordinate, incrementing the x-coordinate and decrementing the y-coordinate respectively. This is , so the shortest program or function that breaks no standard loopholes wins! Test cases (0-indexed, width height x y): 4 4 0 0 -> True 4 4 1 1 -> False 4 4 2 2 -> True 4 4 3 3 -> True 4 7 1 3 -> False • What's the "naive fill" algorithm exactly? – user202729 Jul 12 at 11:24 • Comment: it's hard for me to figure out that each "column" (separated by space) represents a xy board. Consider clarifying that. – user202729 Jul 12 at 11:25 • @user202729 The very first two sentences are supposed to describe it, just move in the first available preferred direction until you can't move any more and paint as you go. – Neil Jul 12 at 14:11 • Some comments: (1) You don't seem to define that "naïve fill" is. (2) It took me a while to undersdtand the meaning of the numbers 14, 21 etc mean the 2nd example, and similarly in others. After a while I realized that each "code section" contains several examples stacked horizontally. You should make thast more obvious (maybe increasing horizontal space, or explaining it in the text) – Luis Mendo Jul 12 at 15:47 • (3) "It normally traverses anticlockwise except when it starts in the bottom right corner when it traverses clockwise": what does "normally" mean here? How do we know/choose the direction the robot follows? Or maybe this is the definition of "naïve fill"? (4) Why can't the robot fill the 2×3 case when it starts in the middle right square? That is, why doesn't start by moving up instead of left? (5) In general, but I find it all quite confusing... maybe it's me, but consider explaining the challenge with more detail – Luis Mendo Jul 12 at 15:47 • @LuisMendo (1) The bot paints as it goes. It prefers to go west, but when it can't go west tries south, then east, then north. That's all there is to it. (2) I've added some more text and spacing. (3) "normally" means "most of the time it ends up doing this". (4) Because its first preferred direction is west so it ends up painting itself into a corner. – Neil Jul 12 at 16:53 • (1) It prefers to go west, but when it can't go west tries south, then east, then north. That phrasing makes it much totally clear. (Now I see that's probably what you meant with by preference) Include it in the text? (3) I still find the word normally confusing there, as if that were an additional degree of freedom. Also, I see now that except when it starts in the bottom right corner when it traverses clockwise is a consequence of (1). I suggest you explicitly state something like "As a consequence of the rule for direction choice, ..." – Luis Mendo Jul 12 at 17:03 • @LuisMendo Fair enough; I've tweaked the text again now. – Neil Jul 12 at 17:35 • I really like this challenge, but I wonder whether the title could be a little more descriptive/catchy - perhaps 'Can the bot fill the grid?' or similar? – Dingus Jul 14 at 15:00 • @Dingus "Can the naïve bot fill the grid?" counts as similar, right? – Neil Jul 14 at 17:23 • Ooh, that's even better. Immediately makes me curious to find out what the naïve bot is. – Dingus Jul 15 at 0:35 • I assume this is more clearly stated as "square root of largest square number dividing n" oeis.org/A000188 – qwr Jul 9 at 22:16 # Evaluate left-or-right Left-or-right is a very simple language I made up. Its expression are made of arrows < (left), > (right), and parentheses. The goal is to evaluate an expression to either < or >. An expression A<B picks the left item A, while picks the right one B. Think of < and > as arrows pointing to the item we want, not as comparison operators. Take, for example, ><>. The operator in the middle is <, and confusingly, the items on each side A and B are also arrows. Since the operator tells us to take the left one A, which is >. So, ><> equals >. Expressions also nest. We can replace the expression with its value. So, for example, (><>)<< equals ><< equals >. And, >(><>)< equals >>< equals <. For another example, (><>)(<<<)(>><) equals ><< equals >. In the input, you'll be given a well-formed expression consisting of either a trio of arrows like ><> or the result of repeatedly replacing some arrow by a trio of arrows in parens like ><(><>) . You can assume the input won't already be a lone arrow. You may alternately accept the whole inputs encased in parens like (><>) or (<(><>)>). The input is given as a flat string of using symbols <>(). You may not take it an a pre-parsed form like a tree. TODO: Test cases • +1 for ><> (15chars) – null Jul 25 at 4:02 ## A Spherical Die ### Inspiration I have a spherical die, but it's a cheap one so it doesn't work properly. When I roll it, it doesn't always land directly on a "face" marking, but instead can result in an ambiguous result ("is that a 6, a 4 or a 2?") ### Assumptions Assume the die is a perfect, evenly-weighted Unit Sphere (i.e. all points on the surface are radius 1cm from the center) , such that a "roll" can result in any point on the sphere being the uppermost point (the "roll value"). Assume that, if the die is placed or rolled such that 1 is at the "north pole", the conventions of a normal die will follow, i.e: • 6 will be at the "south pole" • 4, 5, 3, 2 will be on the "equator", clockwise in that order, equidistant around the sphere. So, before it's rolled, the die looks like this: ### The Challenge Given a simulated roll of the die (i.e. coordinates representing the top of the die after it's rolled) with the conditions above, identify the closest value (1-6) to that point (i.e. what the roll value should resolve to). Input A co-ordinate on the sphere. There are a few co-ordinate systems used for spheres, the two I'm familiar with (and so will provide examples in) are as follows: • P(1, φ, Θ) where φ is the "azimuth angle" (0..360), Θ is the "polar angle" (0..180) • P(x,y,z) where $$\x^2+y^2+z^2=1$/extract_tex] (note: the conversion between the two is: x = cos(φ)·sin(Θ); y = sin(φ)·sin(Θ); z = cos(Θ)) for clarity: • roll "1" is at P(1,n,0) • roll "2" is at P(1,270,90) • roll "3" is at P(1,180,90) • roll "4" is at P(1,0,90) • roll "5" is at P(1,90,90) • roll "6" is at P(1,n,180) Output The nearest value (1-6) to that point. If the point is equidistant to two or more points, output any one of them. Usual exclusions etc. apply. • Does anyone know the maths for this? Feel free to edit it in! – simonalexander2005 Nov 22 '19 at 9:40 • I'm not sure I understand: You want us to generate a random point on a sphere and output the face of the die it corresponds to? – flawr Nov 22 '19 at 9:57 • yeah, so generate a random point on the sphere, then find the nearest "face" - i.e. the nearest of the 6 points (top, bottom, 4 points on opposite sides around middle) – simonalexander2005 Nov 22 '19 at 11:11 • This will be exactly equivalent to a uniform distribution over 6 values, just based on the symmetry of the situation. – AlienAtSystem Nov 22 '19 at 12:33 • @AlienAtSystem yes, all outcomes are equally likely; but the challenge is determining which number any given point on the face of the sphere is closest to – simonalexander2005 Nov 22 '19 at 13:04 • That's not the challenge as posted. Right now, it's "Takes no input, returns the number the (internally generated) random point is closest to" which is, under the consensus of no unobservable requirements simply equal to "Takes no input, returns uniform random value from 1-6". If you want the challenge to be "Input is point on sphere, output is number it's closest to", then write that. – AlienAtSystem Nov 22 '19 at 13:09 • @AlienAtSystem I've edited to try and make it clearer what I'm looking for. Is it clearer now? – simonalexander2005 Nov 22 '19 at 13:15 • It's clearer that my point still stands. Look, "Make Voronoi cells on sphere" and "Generate uniformly random points on sphere" are both good challenges. But when put together like that, they annihilate each other and give you an extremely quick shortcut right from Input (None) to output (a die roll) that doesn't require calculation of either part. – AlienAtSystem Nov 22 '19 at 13:21 • @AlienAtSystem thanks for the feedback, I'd never heard of a Voronoi cell before. What I'm asking, then, is "generate a random point on a sphere and say which Voronoi cell that point is in". Can you explain why that doesn't work? Note that I'm asking for both the point and the cell to be output, not just the cell - otherwise I agree, given the "no unobservable requirements" rule it would be possible to just generate a random number and pretend you'd done it properly (although that would be against the spirit of it) – simonalexander2005 Nov 22 '19 at 13:24 • Would it be better for the point on the sphere to be the input, then? – simonalexander2005 Nov 22 '19 at 13:27 • If you want the challenge to be about finding the points it's closest to, yes. – AlienAtSystem Nov 22 '19 at 13:31 • I want it to be a good challenge on this theme, whatever that would look like :) – simonalexander2005 Nov 22 '19 at 13:33 • Although I don't think the current challenge is bad, it's usually best to not have multiple challenges into one nor multiple outputs (since some languages aren't able to output more than once very easily). The two challenges are: 1. Generate a random coordinate on a sphere (in whichever coordinate system you want); 2. Given a (random) coordinate on a sphere, output the dice-value closest to it. No. 1 already is a challenge, so I agree it might be better to rewrite it to challenge No. 2. I do like the general idea though, so +1 from me. – Kevin Cruijssen Nov 22 '19 at 14:36 • It would also need some info about the size of the sphere, and what to do when the coordinate is exactly in the center between two or three poles. – Kevin Cruijssen Nov 22 '19 at 14:39 • Note that the actual implementation is very simple, as explained in chat. – user202729 Jul 15 at 2:43 ## Posted ## Shift Tac Toe code-golfgridtic-tac-toe Shift Tac Toe is a game that combines Tic Tac Toe and Connect 4 together. In this game, you start with a 3 by 3 board, and each row is connected to a slider that you can move left and right. At the start, the sliders all start to the very right(this means that you can't move the slider to the right on the first turn). Each slider can hold a total of 5 pieces. Each turn, the player can drop an O or a X in one of the 3 columns of the Tic Tac Toe grid depending on which turn it is, or the player can move one of the sliders one spot to the left or to the right. All pieces fall to the bottom most space that is unoccupied. The pieces can also fall from one slider to another outside the 3 by 3 grid. If a piece is outside the 3 by 3 grid and doesn't fall into the bottom slider, then the piece is taken out. If it does reach the bottom slider, it will stay in play. A notable example of this is shown in the following grid: --- --- --- --- --- \| \| \| \| - O - --- --- --- --- --- --- - \| \| \| \| - --- --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- In the grid above, the dashes(-) indicate the part of the sliders that are outside of the 3 by 3 grid and the vertical bars(\|) indicate the 3 by 3 grid. As you can see, this is the starting board except that the middle slider is one spot over to the left, and that there is an O at the very top right. What happens in this scenario? There is nothing immediately underneath it, so does it go out of play? No. This is because it still falls into the bottom slider, which means that it is still in play. The final grid is this: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- --- - \| \| \| \| - --- --- --- --- --- --- \| \| \| \| - O - --- --- --- --- --- Pieces can also stack outside of the 3 by 3 grid. Players will alternate between O and X, with the O player going first. ## Example game: Start with 3 by 3 grid with sliders all the way to the right: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- The O player places an O in the middle column of the 3 by 3 grid and it falls to the bottom: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- \| \| O \| \| - - --- --- --- --- --- The X player then places an X in the middle column: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- \| \| X \| \| - - --- --- --- --- --- \| \| O \| \| - - --- --- --- --- --- The O player then pushes the middle row slider one space to the left. Notice that after the slider moves, there is nothing under the X anymore, so it falls down. Also note that the slider has moved one space to the right as indicated below: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- --- - \| \| \| \| - --- --- --- --- --- --- \| X \| O \| \| - - --- --- --- --- --- The X player places a X in the rightmost column: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- --- - \| \| \| \| - --- --- --- --- --- --- \| X \| O \| X \| - - --- --- --- --- --- The O player then moves the bottom slider one spot to the left. Notice that all the pieces shift one place to the left, and the leftmost X is now out of the playing field: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- --- - \| \| \| \| - --- --- --- --- --- - X \| O \| X \| \| - --- --- --- --- --- The X player places a X in the leftmost column: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- --- - \| X \| \| \| - --- --- --- --- --- - X \| O \| X \| \| - --- --- --- --- --- The O player places an O in the leftmost column: --- --- --- --- --- \| O \| \| \| - - --- --- --- --- --- --- - \| X \| \| \| - --- --- --- --- --- - X \| O \| X \| \| - --- --- --- --- --- The X player shifts the top slider one place to the left. Notice that the O falls one place down because there is nothing beneath it: --- --- --- --- --- - \| \| \| \| - --- --- --- --- --- - O \| X \| \| \| - --- --- --- --- --- - X \| O \| X \| \| - --- --- --- --- --- The O player is not very good at this game, so he shifts the middle slider one place to the right. This shifts all the pieces in the middle row one place to the right: --- --- --- --- --- - \| \| \| \| - --- --- --- --- --- --- \| O \| X \| \| - - --- --- --- --- --- --- - X \| O \| X \| \| - --- --- --- --- --- The X player wins the game by placing a X in the middle column: --- --- --- --- --- - \| \| X \| \| - --- --- --- --- --- --- \| O \| X \| \| - - --- --- --- --- --- --- - X \| O \| X \| \| - --- --- --- --- --- Your job is to take in a string or array of any length that only consists of 9 unique characters(you choose the characters). Three of the characters will choose which column you place the X or O(depending on whose turn it is), three of them will choose which slider to move right, and the last three will choose which slider to move left. You can assume that the input only has these 9 characters. The output should be a 3 by 3 matrix or some kind of list/string that clearly shows the final position of the grid upon following the instructions of the input. You can assume that all inputs are valid. Each character takes up a turn. Also, if any move results in a winning move(forms 3 in a row in the 3 by 3 grid like regular Tic-Tac-Toe), then ignore the rest of the input. Note that the pieces that form the winning 3 in a row all have to be in the 3 by 3 grid. The two example grids below are NOT winning positions: Grid #1: --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- \| \| \| \| - - --- --- --- --- --- \| \| \| O \| O - O - --- --- --- --- --- This is not a winning move because two of the O's are outside the playing field, despite the fact that it forms a 3 in a row. Using the character assignment stated below, this grid pattern can be achieved with 99372467643. Grid #2: --- --- --- --- --- \| \| \| \| - O - --- --- --- --- --- \| \| \| \| O - X - --- --- --- --- --- \| \| \| O \| X - X - --- --- --- --- --- This is not a winning position because two of the O's are outside the playing field. Using the character assignment below, this grid pattern can be achieved with 939318836537734654 In the examples below, 1, 2, and 3 mean drop in the leftmost, middle, and rightmost column respectively. 4, 5, and 6 mean to move the top, middle, and bottom slider to the right respectively, and 7, 8, and 9 mean to move the top, middle, and bottom slider to the left respectively. ## Examples Input will be in the form of a string Output will be a list of lists, with each sub-list representing a row(I'm Python programmer so this list format might not be compatible with all languages). The first, second, and third sub-list correspond to the top, middle, and bottom row of the 3 by 3 grid respectively. The output will have 'O' for the O pieces, 'X' for the X pieces, and an empty string for empty spaces. Input: 123332 Output: [['','','O'], ['','X','X'], ['O','X','O']] Input: 33387741347 Output: [['','',''], ['','','O'], ['X','O','X']] Input: 2283911752 Output: [['','X',''], ['O','X',''], ['O','X','']] Input: 228374739 Output: [['','',''], ['','',''], ['X','X','X']] Input: 8873334917349 Output: [['','',''], ['','','O'], ['X','X','O']] Input: 799333466 Output: [['','',''], ['','',''], ['','','']] Input: 99372467643 Output: [['','',''], ['','',''], ['','','O']] Input: 939318836537734654 Output: [['','',''], ['','',''], ['','','O']] This is , so shortest code wins! ## My concerns about this challenge: Are the rules of this game explained enough? Do you understand this game? Is this a good challenge overall? Are the examples correct(if you understand the rules)? Should I put more examples(or if you understand the rules, could you supply me with some)? What other tags can this challenge fit into? • Welcome to the site! Generally it's a good idea to leave challenges in the Sandbox for at least a few days to give people time to review. Even more so if you have questions/concerns about your challenge. – Dingus Aug 2 at 4:51 • @Dingus The problem is I already posted it so what do I do now? – Aiden Chow Aug 2 at 7:01 • Actually I was referring to future challenges. For this one, there are probably two options: 1. Leave it on main, responding to feedback in the comments. 2. Delete the main post, wait a while (maybe a week) for more feedback here, then repost/undelete on main. The choice really depends on how much needs to be done to iron out the kinks. – Dingus Aug 3 at 0:11 • If the moves result in any unpermitted moves, then output any falsey value challenges usually assume only valid input unless the challenge itself is to determine only if the input is valid or not. – Noodle9 Aug 4 at 19:30 • @Noodle9 Ok so should I take that out? – Aiden Chow Aug 5 at 2:04 • Make it one or the other, either assume valid input and output grid. Or determine whether or not input is valid as the challenge. – Noodle9 Aug 5 at 9:30 • @Noodle9 Ok edited. – Aiden Chow Aug 5 at 18:02 Note: this challenge is a work-in-progress, so suggestions would be appreciated Questions for meta: • How can I prevent people from just using SHA or MD5 one-way compression? • are these language restrictions fair? • is this scoring system fair? • are there any obvious cheap answers? • what other tags should be added? • what should the challenge title be? • will these restrictions adequately prevent people trying to cheat their way through? • should a limit be put on a password length? Should I limit passwords to ASCII printable characters? # The challenge Your challenge is to first choose a "password" (please do not use your actual password). Then, you will create a program which will output a truthy value if and only if this password is given as input, falsy otherwise. Your goal will be to make it so others are unable to reverse-engineer this password (and you will keep this password secret for now). # Scoring The scoring for this challenge is somewhat different than regular . During the first two weeks from when an answer is posted, other users will have the opportunity to try to crack your password by reverse-engineering your code. If anyone gets your password correct during this two week period, your answer will be marked as cracked. If two weeks pass without users finding the password, your answer can be marked as safe once you share the password (again, please do not use your actual password, you should make up a new one that you don't use anywhere). Note that you may use any tools at your disposal (online tools, brute-force attacks, modified code, etc) to extract someone else's password from their code. Of all the safe answers, the one with the shortest source code (i.e. ) wins! # Rules To make things fair for everyone, you may only use languages that appear on TIO, or languages that have well-written documentation and are used somewhat widely. You must also provide a link to try your code online that anyone can access (as such, you may not use languages behind paywalls like MATLAB but Octave is still on the table because it's free). Even if you don't want to post an answer, feel free to try to crack any of the existing answer's passwords! If you get a password, you can simply leave a comment on that answer and that answer will be cracked. # Note If you edit the code in your answer, the two week period will reset! You may edit any explanations in your answer freely (I will verify that any answers marked safe did not cheat). tags: code-golf • Example answer – Daniel H. Aug 19 at 12:38 • Surely any cryptographic hash function (which are implemented in many languages) will make it easy to generate an impossible-to-crack answer? For instance, in R I can write test=scan(,'');if(digest::digest(test,"md5")=="b6778692586dc649267723ccc3356fad")TRUE else FALSE and I'll be pretty confident that nobody will crack my password... – Dominic van Essen Aug 19 at 15:10 • That is a good point. It seems like this challenge is similar to just writing a hash function. You might want to add other ideas to make the challenge more interesting – thesilican Aug 19 at 16:06 • That's what I was about to write, any hash function with a hidden default salt depending on the language, or anything like that, could be hiding the password easily enough. – V. Courtois Aug 19 at 16:06 • @DominicvanEssen is this an actual MD5 hash? I was unable to reverse it (note that a lot of MD5 hashes can be reversed with online tools like this). Note that for this challenge people would be allowed any and all tools at their disposal to crack passwords. This means people are very much allowed to reverse-engineer code in any ways they please – Daniel H. Aug 19 at 16:06 • Answering in many esolangs could hide the password easily enough, too. If I answer in Lenguage, Unary, Mariolang,... – V. Courtois Aug 19 at 16:07 • @V.Courtois this is true, but the point is not to read the password in the source code. The point of the challenge would be to reverse-engineer their code to crack a password (so documentation and online tools are all fair game). Also, Unary will likely be an invalid language because people must be able to actually run the program online (and Unary programs are usually way too big to run online) – Daniel H. Aug 19 at 16:08 • Now that I think about it, any type of loop could hide the password easily enough, too. But even with what I said before and what I'm saying now, I think this challenge has to exist (if not existing already), because having many valid answers is not a problem (not to me, at least). – V. Courtois Aug 19 at 16:09 • @DominicvanEssen also, even if nobody can reverse your password, that's still fine - the winner of this challenge is whoever has the shortest code out of all the uncracked passwords. In other words, this is still a codegolf challenge, but answers can be disqualified if anyone finds the password. So, if you want to win but you don't have the shortest code, you simply have to crack other people's passwords! – Daniel H. Aug 19 at 16:15 • @DanielH. Yes, my example was an actual MD5 hash (the password was mypassword). You're right that some hash functions can be reversed, but there are many cryptographically-secure ones for which this is difficult. – Dominic van Essen Aug 19 at 16:17 • @DominicvanEssen I'm unfamiliar with MD5 hashes, but when I converted mypassword to an MD5 hash using three different online tools I got 34819d7beeabb9260a5c854bc85b3e44 every time instead of the hash in your answer. Could you please provide a TIO link for the R code (when I copy-pasted it into TIO it didn't work for me and I'm unfamiliar with R)? I'd like to try experimenting with MD5 – Daniel H. Aug 19 at 16:22 • I'm afraid that the R 'digest' library is not installed on TIO (making a link was the first thing I tried). – Dominic van Essen Aug 19 at 16:25 • But, after some research, it turns-out that R adds some (consistent) extra characters to the string by default before applying MD5. This behaviour can be switched off, at which point mypassword indeed hashes to 34819d7beeabb9260a5c854bc85b3e44. – Dominic van Essen Aug 19 at 16:46 • how many letters are the passowrds capped at? Are unprintables allowed? – Razetime Aug 20 at 3:30 • @Razetime I might add a restriction of 16 characters, ASCII printables only. I will have to try to balance this cap though - if it's too short, passwords can easily be brute-forced. If it's too long, everyone will just use one-way compression and passwords will never be cracked – Daniel H. Aug 20 at 11:24 # Substandard deviation - Posted • Innovation is the key to make CGCC live. This is what we (at least, I) want to see on main page. I might not know enough languages, but there might be one (maybe Jelly?) able to compute standard deviation of input stack in 1 byte, therefore getting an unbeatable score of 0. (I don't know if it is a problem, though.) – V. Courtois Aug 21 at 9:42 • You could add a rule on accuracy of expected output (I see you put two digits in test cases, but saying it explicitly could be nice). – V. Courtois Aug 21 at 9:46 • – V. Courtois Aug 21 at 9:47 • @V.Courtois Thanks for the nice comment! Indeed, a score of 0 might be attainable by certain golfing languages. I don't see that as an issue. It would also be attainable by Unary or Lenguage, but if somebody writes code to compute the standard deviation in Unary or Lenguage, they deserve my upvote! – Robin Ryder Aug 21 at 9:48 • Challenge accepted. :D – V. Courtois Aug 21 at 9:49 • @V.Courtois I have added the expected accuracy, thanks. Also, Re: built-ins: most language with a built-in for the standard deviation use the corrected standard deviation, which has a denominator of \n-1\ instead of \n\ (e.g. R's sd, Matlab's std...) – Robin Ryder Aug 21 at 10:09 • Now that this challenge has been posted, would you mind deleting this proposal to save space? – caird coinheringaahing 6 hours ago # Stroke Count of a Chinese Numeral codegolfPosted • Related (not dupe). Stroke count is actually good idea because it avoids the need to hardcode Chinese characters. The description looks clear enough to me. – Bubbler Aug 5 at 3:11 • Might be useful: tio.run/… – user202729 Aug 5 at 11:45 • (to read the test cases) – user202729 Aug 5 at 11:45 • Has it been posted to main? I can't seem to find it. – V. Courtois Aug 21 at 10:02 • @V.Courtois posted. – att Aug 25 at 4:54 # Metagolf: Catlike Piet The goal of this is to write a catlike program, which would be executed (in a Unix environment, though you needn't stick to that) by the following: yourprogram < file > output piet output where piet output writes the contents of file to stdout. That is, you're to generate a Piet program which prints the input to yourprogram. One-liners Straight line programs can be written in Piet... in straight lines. If you're willing to take a hit to your score, your output can take the form of a string of commands: = none (continue color block) \| push ^ pop + add - subtract multiply / divide % mod ~ not > greater . pointer \ switch : duplicate @ roll input number ? input character # output number ! output character which is trivial to convert to a Piet program with the following (partially golfed) Python code: def P(s): h=v=0;l=len(s)+1;R="P3 %i 2 255 192 0 0 "%(l+2) C=[1,3,2,6,4,5];V=[0,192,192,255,0,255] for x in map("=\|^+-/%~>.,:@?#!".find,s): C=C[x//3:]+C[:x//3];V=V[x%32:]+V[:x%32] for i in [1,2,4]:R+="%i "%V[(C[0]//i)%2] return R+"255 "4+"0 0 "+"255 "l3+"255 0 0 "2 The dimension of said program is (n+3) x 2 if there are n characters in the string. Scoring Your code will be judged on the maximum dimension of the images that it outputs. • Part 1: Take the maximum score taken over all ascii codes (that is, single-character inputs), discounting EOF. • Part 2: Take the score for the input "Hello. My name is Inigo Montoya. You killed my father. Prepare to die." Your score is the product of the scores in part 1 and part 2. Punishment: Double your score if you write one-liners as above (that is, if you don't output an image). Bonus: If your program is written in Piet, take the square root of your score above. • It took me a while to understand the task as "Write a program taking INPUT which produces as output a piet program that takes no input but produces INPUT." I think it is a interesting and challenging, but it's reception will depend entirely on how many people are willing to learn/futz-around-in/deal-with piet. And I have no feel for how many that is. – dmckee --- ex-moderator kitten Jul 7 '11 at 3:12 • @dmckee; would it be better if I just used a reduced instruction set, and only ask for the instruction stream? I think this is still challenging with {push 1,duplicate,add,subtract,multiply,output}. Come to think of it, if I restrict to {push 1,duplicate,add,output}, there's a reduction to some awesome algorithms. – boothby Jul 7 '11 at 4:48 • I did this in piet some time ago: craigoclock.blogspot.com/2011/05/metaprogramming-in-piet.html – captncraig May 21 '12 at 18:31 • Hello! This looks like a good but abandoned meta post, would you be willing to offer it for adoption? (If you want to, you can still post to main.) – programmer5000 Jun 9 '17 at 15:22 ## Chess move The Challenge Write a program that gets a string containing a chessmove and a chessboard as input, and then outputs the chessboard. Requirements The chess move will have this format: <from square><to square>[<promoted to>] Examples: d2d4 f8g7 a7a8R The chessboard format is not fixed, but there must be a 1 to 1 relation between the board and the string to represent the board. Also the format of the input must bet the same as the format of the output. Two suggestions of what it could look like: rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR rnbqkbnr pppppppp 00000000 00000000 00000000 00000000 PPPPPPPP RNBQKBNR It is not required to store anything except the location of the pieces, and validity of moves can be assumed. Scoring Base score is character count (assuming your program can move pieces for all moves) Bonus multipliers: • If the program updates the promoted piece, divide by 2 • If the program also moves the rook when castling, divide by 2 • If the program also removes the pawn when capturing en passent, divide by 2 The moves, and castling & en passent in particular are explaned on Wikipedia. So basically writing a 100 character solution for the base problem gives the same score as an 800 character solution with all bonus multipliers. Examples If you would choose to use one of the board formats above, your input would look like one of these strings: e2e4 rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR e2e4 rnbqkbnr pppppppp 00000000 00000000 00000000 00000000 PPPPPPPP RNBQKBNR Your corresponding output string would then be one of these: rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR rnbqkbnr pppppppp 00000000 00000000 0000P000 00000000 PPPP0PPP RNBQKBNR • Before I get on to more specific criticisms: as presented, without the bonus this is too trivial to be interesting. I suggest removing some flexibility: require Fen notation for the board position and algebraic notation for the move, and making the current bonus options mandatory. On specifics: it's not clear why you talk about storage; and the board position notations you suggest don't include enough information to know whether en passant is possible. – Peter Taylor Dec 22 '13 at 23:56 • @PeterTaylor I agree that compared to chess programs this may be trivial, but I would like to make it a golf challenge. Compared to the hot code golf questions this is quite elaborate already in its basic form. (For a good solution the board design may need to be changed drastically). It is true that there is no attention to the legality of moves (whether it is possible to capture en passent) but for a mere viewer this is not required so I am not too worried about this. So far the chess questions seem to get very few answers as they tend to be complex and I hope to offer relatively easy entry. – Dennis Jaheruddin Dec 30 '13 at 11:02 • Your point about en passant is valid - you had said in the spec to not worry about legality. I'll try to convince you of my first point: without the bonus, this reduces to: a) parse first four characters into (col 1, row 1, col 2, row 2); b) take board as a 64-char string; c) board[8row_2+col_2] := board[8row_1+col_1]; board[8row_1+col_1] := ' '; print board. This is trivial compared to any good golf question. (Note that the hot questions at the moment are neither golf questions nor good questions). – Peter Taylor Dec 30 '13 at 12:14 • This sandbox post has had little activity in a while. Please improve / edit it or delete it to help us clean up the sandbox. Due to community guidelines, if you don't respond to this comment in 7 days I have permission to vote to delete this. – programmer5000 Jun 9 '17 at 15:40 # Black Box Your task is to analyze a given situation for the game Black Box. Given a sequence of guesses and answers, your program is to either print the solution or suggest the next move. ## The game The board consists of 8×8 cells, with edges labeled like this: I'll probably create nice images here, particularly to make sure that the squares of the board are really square. abcdefgh i I j J k K l L m M n N o O p P ABCDEFGH The player shoots rays into the interior of the box, where they might get deflected, reflected or absorbed. He is told the position where the ray leaves the black box again, and from that has to deduce the positions of 4 atoms inside the black box. I'll have to include more of the game rules here, but for now see Wikipedia. # Input and output Input is a sequence of line, each consisting of two characters. The first denotes the point where the ray of light enters the black box, the second the place where it comes out again. In the case of a reflection, both characters will be equal. In the case of a hit, the second character will be -. If the input is enough to fully determine the locations of the atoms, then output should be four lines giving the coordinates of each atom. The lines should be two lower case characters each, the first giving the row and the second giving the column of the found solution. The atom positions must be printed in lexicographical order. If the input is consistent with more than one set of atom positions, then the output should consist of a single line containing a single character, which is the location where the next ray should be shot. That location has to be chosen in such a way that it can help find the solution. This is the case unless all of the atom positions consistent with the input so far would produce the same output for this next ray as well. Your output has to be terminated by a newline character. # Examples Let's take the atom configuration the Wikipedia article uses as an example as well: abcdefgh i I j J k O O K l L m M n O N o O p O P ABCDEFGH If the input were cf D- Em HH Co then the output should be kb kg nd pg but if the input were only Em HH then the output might be for example K ## Scoring This is code golf, so shortest answer wins. However, I'll only accept answers which are practical in so far as they compute their result in reasonable time. I'd say no more than five minutes on my system where I'll evaluate the answers, and I'll simply hope that correct solutions will be much faster and incorrect ones much slower, so that the speed of my system doesn't make a difference. A submission which gives a wrong answer for one of my test cases will be disqualified until it gets fixed. I will probably point out the problem in a comment to that post. # Create a program with "exact repetition" in its source code The task is to create a program, with the following restrictions placed on the printable ASCII characters in the source code: choose some k > 0. • Every non-alphabetic character has to appear exactly k times. • Every alphabetic character has to appear at most k times. • This rule differs from the former in order to avoid boring dummy identifiers while still making it a challenge to choose good library functions to call. Character set definitions used: • Non-alphabetic characters are !"#%&'()+,-./0123456789:;<=>?@[$^_{\|}~ and '' (backtick). • Alphabetic characters are ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz. Note that no restriction is placed on characters outside of the range of printable ASCII characters (including control codes, tabs, newlines, higher unicode codepoints, etc). What the program does is up to you; be creative. Some general guidelines: • Programs that do something interesting might have better chances, although more impressive code structure (i.e. fewer comments) is also beneficial. • Stuffing excess characters in comments is boring, and should be avoided/is discouraged. • Dead/no-op code isn't terribly interesting either, but is probably unavoidable and at least has to conform to the language's grammar. This is : whatever has the most upvotes at Feb 1, 2014 gets accepted as the winner. # # /@/_[]={9.};main() {printf("He%clo \ world!%c\ ",2^7&!8.&~1\|~-1?4\|5?0x6C:48:6<3>2>=3<++_[0],'@'^79-5);} Prints "Hello world!" (adapted from an answer to another question). Probably wouldn't score a lot (since what it does isn't terribly interesting). Each of the non-alphabetic characters appear exactly twice, and no alphabetic character appears more than twice. For meta: I want to post this, but I'm worrying that "do something interesting" might give too little guidance and the question won't receive many answers.. thoughts? Is it good as-is, or should I come up with some task that one should be required to implement (and possibly change the ruling to code-challenge, with length + 2^(characters-in-comments) as the score)? ## 4 and 20 baked in a π While some might describe π as a string of seemingly random numbers, one can also look at it in a way similar to a monkey with a typewriter. Eventually, it should calculate out to something more interesting. For example, the sequence 1337 shows up 4,814 places to the right of the decimal. At 700,731 places right of the decimal, you'll find the sequence 160151, which is "pi" represented as ASCII (although you'll find a 'pointer' to it much faster, as the sequence 700731 begins at 29,830 digits to the right). So, your task is to make a program to find things in π. Your program will accept a positive integer and output the number of places right of the decimal point that number appears. To keep the run times down, input can be limited to numbers in the range of 0 to 1000 (without leading zeros). Example: Using 415 as the input, the output should be 2: 3.14159 ^ Rules: • You can not use any precalculated values of π, including language constants, built in functions that return π or digits of π, or any resource outside the code itself (such as files or websites). • You can not use any trig functions to calculate π. Bonus points if you find the sequence 072 101 108 108 111 044 032 087 111 114 108 100 033. This is code golf, so lowest score wins. • It's not clear to me whether you require answers to support leading zeroes. Also: program, named function or snippet? And how indexed? (Giving 415 as a test case would be a good way to answer the last question) – Peter Taylor Mar 11 '14 at 6:58 • Isn't this just Calculate 500 digits of pi with a search function tagged on at the end? By the way, your bonus points are quite safe — even if you searched a trillion trillion trillion digits of pi, your chance of finding an arbitrary 39-digit sequence would still be less than 0.1%. – r3mainer Mar 11 '14 at 14:59 • Edited to clarify leading zeros and indexing. @squeamishossifrage - Yes and no. The number of digits to find the answer depends on the input, which both limits the choice of algorithm to generate the search space and gives more ample room to golf the integration of the search function. The worst case is under 10000 digits for n between 0 and 1000. I suppose I could put in a time limit of a couple minutes and expand the range of n to 10000 (worst case is just under 390k), but that seems obnoxious. Thoughts? – Comintern Mar 11 '14 at 17:20 • @AlexA. - Not a drug reference. – Comintern Apr 1 '15 at 22:34 • Hello! This looks like a good but abandoned meta post, would you be willing to offer it for adoption? (If you want to, you can still post to main.) Due to community guidelines, if you don't respond to this comment in 7 days I have permission to adopt this. – programmer5000 Jun 9 '17 at 16:15 # Create a calendar We all know HDD-space is precious and bandwidth is expensive, therefore it is best to reduce the size of your executables. Let's start with your calendar: Your task is to build a calendar app in at most 512 bytes. The calendar must at least support the following features, but additional features may gain you additional upvotes: • It must be able to show the current month with the current day highlighted • The user must be able to find out the week day of each day Rules: • Maximum code length is 512 bytes (counted as UTF-8 without BOM) • You may subtract the bootstrapping code (i.e. int main(int argc, char argv) in C or <?php in PHP) and imports from the final size to allow for more verbose languages to be in • You may use standard time / date functions of your programming language, as long as they don't allow you to output a ready to use calendar • No network access (I said bandwidth is expensive!) • Voters decide on the amount of features / look and feel / creativity This needs a tag for the size restriction, any suggestions? • "bandwidth is expensive" <sup>[citation needed]</sup> – John Dvorak Mar 22 '14 at 5:27 • Seems rather close to Output: Calendar Month – Peter Taylor Mar 22 '14 at 5:33 • Who decides what counts as bootstrapping code? It seems odd to arbitrarily exclude code like that, and the examples you gave can be golfed a lot: they're more or less equivalent to main(){ and <? respectively. – Wander Nauta Mar 24 '14 at 20:49 • @WanderNauta Bootstrapping code is the code that's essentiell to get a working noop program. – TimWolla Mar 24 '14 at 21:00 • @TimWolla That definition won't fly. A zero-byte file is a working noop PHP script, for example. – Wander Nauta Mar 24 '14 at 21:01 • @WanderNauta A zero byte file is a working noop in every language. – TimWolla Mar 24 '14 at 22:12 • So what's bootstrapping code then? :) – Wander Nauta Mar 24 '14 at 22:53 • for the limit I'd say code-shuffleboard or restricted-source – Einacio Mar 26 '14 at 15:57 • This sandbox post has had little activity in a while. Please improve / edit it or delete it to help us clean up the sandbox. Due to community guidelines, if you don't respond to this comment in 7 days I have permission to vote to delete this. – programmer5000 Jun 9 '17 at 16:28 # ASCII ART edge detection As the title says, I was thinking to contest in which one must detect edges of an ASCII art. The code should accept a B/W ASCII art as input. A B/W ASCII art is defined as (by me) an ASCII art with only one kind of non-white-spaces character (in our case: an asteriks ). And as output produce a standard ASCII art (all ASCII characters are accepted) which should remember the contourn of the first. The purpose of using more than one character in the output is to make some edges ssmoother. For instance, one could let this input * ** ** ** ** * could became: ___ _/ ) _/ / / \| \| / \| \ \ \| \ \| \___) The input \n separated string as input. Each line has a maximum of 80 characters. The number of rows is not specified. I'd put it as a popularity-contest since, beside my simple code, I'd like to see more "round" edge detections which use more than one character in smooth edges. Also, I don't want to tag it as code-golf since I'm quite sure one can do this job using aplay (with ASCII art renderer) and command line GIMP (to apply edge detection). As a popularity contest, there are no strict rules on how the output should be..just use your fantasy! This is my sample program: import fileinput as f import re as r import copy as c a,s,p='',' ','+' s=[list(' 'n)] for l in f.input(): if(len(l)>n):l=l[:n] k=list(r.sub('[^ ^\%c]'%a,'',' '+l+' ')) s.append(k+[' '](n-len(k))) s.append([' ']n) return s def np(s): s=c.deepcopy(s) for l in s[1:-1]: for w in l[1:-1]: print(w,end='') print() def grow(i): o=c.deepcopy(i) for x in range(1,len(o)-1): for y in range(1,len(o[x])-1): if(i[x][y]==a): o[x-1][y-1]=o[x-1][y+1]=o[x-1][y]=o[x+1][y]=o[x+1][y-1]=o[x+1][y+1]=o[x][y+1]=o[x][y-1]=a return o def diff(i,o): c=[] for x in range(0,len(i)): l=[] for y in range(0,len(i[x])): if(i[x][y]==a and o[x][y]==s): l.append(p) else: l.append(s) c.append(l) return c np(diff(grow(I),I)) Here below I put both input of the programs. It is an 80x70 ASCII ART. It means it has 70 lines of 80 characters, each separated by \n. *********** * * ** * * ***** ********* **** *** *** *** * **** *** * ***** * *** * * ***** *** ** ****** *** ** ****** *** ****** * * ********* ** ****** * ******** * ****** * * ********** * ******** * * *********** **** * * *********** * * *********** * * ************** * ********* * *********** ********* * * ********* ********** *********** *********** * *********** ** ************ * * **** * ********** ** * *********** *** * ** * ********** * *** ** * ********** * *** * *********** * * ********** *** * *** ********* ** ***** ********** *** ****** ************* * ******* ****** ************* * ********* ******* *************** ********** ******* ****************** ************** ******** ************* ********************* ********* ************** *************** ********** ************** **************** ************ ************** *************** *********** *************** *************** ************ ************** ************** ************** ************** *************** *************** ************* *************** ***************** ************** **************** ***************************************** ***************** ****************************************** ******************* ******************** ************* ******************** ****************** *************** ********************** ***************** ***************************************** ***************** ************************ *********** **************** ********************** *********** **************** ***************** *********** *************** **************** ************ ************** ************* ************ ************* ************ *********** ************* ************ *********** ************* ************* *********** ************ ************* *********** ********** ************** *********** ************ ************ ********** *********** ********** ********** A possible output could be: +++++ ++++ ++++++ ++++++++++ +++ ++ +++++ +++++ +++++ ++++++++ +++++ ++++ ++ ++++ ++ ++++ ++ ++++++ ++ ++ ++ +++++ +++ + +++++ ++ ++ ++++ +++++++ ++ ++ ++ ++ ++ ++ +++++ ++ + + + +++++++ ++ +++ ++++ ++ + ++ ++ ++ ++ ++ ++ ++ ++ ++ + + + ++ ++ ++ +++ + + ++ + ++ +++ + ++ ++ ++ + ++ ++ + +++ + ++ +++ ++ + + +++ + + + ++ + + + + + ++ + ++++ + ++ ++ ++ + ++ ++ ++ + + + ++ ++ + + +++++ ++ ++ + + ++ ++ + + +++ ++ + + + + ++ + +++++ + + ++ ++ + + ++ + + + + ++ + + ++ ++ ++ + + ++ ++ + + ++ + ++ + + + + + + + ++ ++ ++ + + + + + + +++++++ + + + + + + ++ ++ ++++ + + + + + + + +++ + ++ +++ + + + + ++ + + ++ ++ + ++ + ++ + + + ++ + ++ ++ +++ + + +++ ++ + + + + + + + ++ + +++ + + + + + + ++ ++ ++ ++ + + + + + +++ ++ ++ + +++ +++ + ++ + + ++ ++ + ++ +++++ + ++ ++ + + ++ ++ + + ++++++ ++ ++ ++++ + ++ +++ ++ + + + ++ ++ +++ +++ + + + ++++ ++ ++ +++ + ++ ++ ++++ + + ++++ + + ++ +++++ +++++ +++++ + ++ ++ +++ ++++++ + ++ + +++++ +++++ + ++ + + +++ +++++ + +++ + ++ ++++++ + + + ++ + ++ + ++ + + + + +++ ++ + ++ ++ ++ + + + + +++ ++ + +++ + ++++++ + ++ ++ ++ +++ + + ++ + +++++ ++++++ + + ++ ++ + + + ++ + ++ + + + ++ + + ++ + ++++ + + + + ++++++ ++ + + ++ + ++ + + + ++ + ++++ + + + ++ ++ + + ++ + + + ++ + + + + + + + + ++ + + ++ + + ++ ++ + + + ++ ++++++++++++++++ +++ + + + ++ ++ ++ + ++++++++++++++++ ++ + ++ ++ ++++++++++++++ This is also the output produced by the script above. Of course it is not the best output and I'm sure one can easily produce a smoother one. • It would be useful to be more precise about which characters should be non-blank in the output: characters which were non-blank in the input but adjacent to blanks, or characters which were blank in the input but adjacent to non-blanks? – Peter Taylor Apr 4 '14 at 9:50 • Thanks for pointing. I re-written the phrase in the answer. You can use every ASCII character in the output (as usual ASCII art). E.g. I used only + symbol, but one could makes round edges using symbols like \ or / etc.. – Antonio Ragagnin Apr 4 '14 at 9:55 • edited again... – Antonio Ragagnin Apr 4 '14 at 10:09 • Can you define the input that will be used by all the participants? It's necessary to have only one input to compare the outputs of the different answers. The first example is too simple and the last one is too long. So I suggest to use something between these 2 examples. – A.L Apr 4 '14 at 17:29 • Thanks, I chosen a cute panda as input. – Antonio Ragagnin Apr 6 '14 at 8:09 • one could let this input (…) could became → try something like "this input (…) could become" outpuit → output – user2428118 Apr 7 '14 at 13:31 • I edited it now, so do you people thinks it is a good question? – Antonio Ragagnin Apr 9 '14 at 17:15 • Hello! This looks like a good but abandoned meta post, would you be willing to offer it for adoption? (If you want to, you can still post to main.) Due to community guidelines, if you don't respond to this comment in 7 days I have permission to adopt this. – programmer5000 Jun 9 '17 at 16:32 • Hi @programmer5000 , I already asked such a question. Do you mean to re-use it again? See: codegolf.stackexchange.com/questions/26139/… – Antonio Ragagnin Jun 12 '17 at 13:39 Hi, first time golf questioner, hopefully I'm doing it right! A maths trade (or "math" trade if you prefer) is a way of calculating complex trades of arbitrary items in a circle of participants where not all participants want all items. X participants have an item they would like to trade. Each participant is assigned a unique number, and provides a list of (numbers identifying) the items they would willingly trade their item for. They may provide an empty list (i.e. they would rather not trade). ## Input X lines, one for each participant, comprising a unique number identifying them, followed by a colon, then a comma-separated-list of numbers identifying other items that they would trade for. e.g.: 1:2,3,4 2: 3:1,4 4:2 The numbers identifying the participants will not necessary be in order, nor will they necessarily be 1 to X. You may assume that they will be numeric. This string can be in STDIN, or an argument to a function, or similar and can be followed by a new-line or not, whatever the coder prefers. ## Output One or more trade loops in which all participants are making trades they're happy with. Each loop should be on a new line and comprise a participant number, followed by "->", followed by the participant they should give their item to, then another "->", and another participant number etc, until the loop is closed and the last participant number matches the first one. Another line is added with the number of completed trades. e.g.: 1->3->1 2 Participants for which no valid trade is possible are omitted. Output can be via STDOUT, or returned as a string, or something else, with an optional final new-line. 1. A participant may not be involved in more than one trade 2. A participant may not receive an item that they didn't want 3. All loops must be closed 4. Maximum number of possible trades should be completed (i.e. no submitting a zero-trade output and claiming it's valid). If there are multiple permutations, pick whichever you prefer. This is a code golf challenge, so shortest working code wins. ### 1 1:2,3,4,5 2:3,5,7,9 3:1,2,5,6,10 4: 5:1,2,3,4,10 6:5,7,9 7:3,6,9,10 8:1,2,4,10 9:1 10:9 1->9->10->3->1 7->2->5->6->7 8 For instance, in this trade: 9 stated that he would accept 1's item in a trade, 10 stated that he would accept 9's item, 3 would accept 10's and 1 would accept 3's. In the second loop, 2 receives 7's item, 5 receives 2's, 6 receives 5 and 7 receives 6's. (Other outputs are possible from this input.) ### 2 1:2 4: 2:3 5:1 3:4 0 ### 3 1:5,9 5:1 9:1 1->5->1 2 1->9->1 is also valid in this case, but both cannot be completed. Either is acceptable. Thanks for reading guys! Let me know if there are any improvements I can make. • "can be followed by a new-line or not, whatever the coder prefers." How flexible is this? For instance, can I use trailing commas, like 1:2,4,7, if it makes my code shorter? – Martin Ender May 2 '14 at 17:28 • Will the participants always be numbered 1 to n and their input lines provided in order? If so, state it. If not, include a test case which fails if an implementation decides to ignore everything before the : in each input line. – Peter Taylor May 5 '14 at 10:02 • @m.buettner I would say a trailing comma is not acceptable, on the end of any line, or the end of the input/output. – Johno May 6 '14 at 8:55 • @PeterTaylor Good tip. I'll correct the question to state that you can't assume that the numbers will be 1 to n, in order. – Johno May 6 '14 at 8:57 • Hello! This looks like a good but abandoned meta post, would you be willing to offer it for adoption? (If you want to, you can still post to main.) Due to community guidelines, if you don't respond to this comment in 7 days I have permission to adopt this. – programmer5000 Jun 9 '17 at 16:39 ## Design and Solve a Maze (this question on hold while the details are ironed out) Your task is to play the roles of both characters in this scene from Inception. In it, Cobb gives Ariadne a challenge: You have two minutes to design a maze that takes one minute to solve. Some liberties will be taken on that description. Most importantly, this challenge is not time-based, rather scores are based on the effectiveness of your mazes and maze-solvers. I apologize for the many edits to this challenge as we iterate towards an easy and fair format.. ### Part I: Maze format All mazes are square. A cell in the maze is represented as a zero-indexed tuple row column. Walls are represented by two binary strings: one for horizontal walls (which block movement between rows) and vertical walls (vice versa). On an NxN maze, there are Nx(N-1) possible walls of each type. Let's take a 3x3 example where the cells are labelled: A B \| C --- D \| E F --- G H \| I all possible vertical walls are: AB BC DE EF GH HI. Translated into a string, the walls shown are 011001 for vertical walls and 010010 for horizontal walls. Also, by "binary string" I mean "the characters '0' and '1'". The full maze format is a string which contains, in this order: • width • start cell tuple • end cell tuple • horizontal walls • vertical walls For example, this maze: 0 1 2 3 4 _________ 0 \| \| E\| _\| 1 \| _\|_\|_ \| 2 \|_ _ _ \| \| 3 \| _ _ \| \| 4 \|____S\|___\| start:(4,2) end:(0,2) is formatted to this: 5 4 2 0 2 00001011101110001100 10100110000100010010 ### Part II: The Architect The Architect program creates the maze. It must play by the rules and provide a valid maze (one where a solution exists, and the end is not on top of the start). input via stdin: Two positive integers: size [random seed] Where size will be in [15, 50]. You are encouraged to make use of the random seed so that matches can be replayed, although it is not required. output to stdout: A valid size x size (square) maze using the format described in Part I. "valid" means that a solution exists, and the start cell is not equal to the end cell. The score of an Architect on a given maze is # steps taken to solve ------------------------------ max(dist(start,end),(# walls)) So architects are rewarded for complex mazes, but penalized for each wall built (this is a substitute for Ariadne's time restriction). The dist() function ensures that a maze with no walls does not get an infinite score. The outside borders of the maze do not contribute to the wall count. ### Part III: The Solver The Solver attempts to solve mazes generated by others' architects. There is a sort of fog-of-war: only walls adjacent to visited cells are included (all others are replaced with '?') input via stdin: the same maze format, but with '?' where walls are unknown, an extra line for the current location, and a comma-separated list of valid choices from this location. (This is a big edit that is meant to make it simpler to write a maze-parsing function) example (same as the above 5x5 maze after taking one step left) 5 4 2 0 2 ???????????????011?? ????????????????001? 4 1 4 0,4 2 Which corresponds something like this, where ? is fog: 0 1 2 3 4 _________ 0 \|????E????\| 1 \|?????????\| 2 \|?????????\| 3 \| ?_?_????\| 4 \|__C_S\|_?_\| output to stdout: One of the tuples from the list of valid choices Each Solver's score is the inverse of the Architect's score. ### Part IV: King of the Hill Architects and Solvers are given separate scores, so there could potentially be two winners. Each pair of architects and solvers will have many chances to outwit each other. Scores will be averaged over all tests and opponents. Contrary to code golf conventions, highest average score wins! I intend for this to be ongoing, but I can't guarantee continued testing forever! Let's say for now that a winner will be declared in one week. ### Part V: Testing I have written a Python testing kit which includes a Maze class for parsing and writing in the proper formats, as well as an example architect/solver pair: Daedalus and the Minotaur Available on both Dropbox and GitHub ### Part VI: Submitting • I maintain veto power over all submissions - cleverness is encouraged, but not if it breaks the competition or my computer! (If I can't tell what your code does, I will probably veto it) • Come up with a name for your Architect/Solver pair. Post your code along with instructions on how to run it. • I suppose input is via STDIN? You might want to mention that explicitly, because at least the architect could just as well take the input via command-line arguments. – Martin Ender May 15 '14 at 15:34 • updated. I have a driver/referee program which will handle I/O; I'll update it to use stdin/stdout since that will no doubt be the easiest standard. – wrongu May 15 '14 at 16:04 • @m.buettner before de-sandboxing this, would you be willing to try the test kit? – wrongu May 15 '14 at 18:20 • I'd love to, but I'm afraid I'm too busy this week. Try ask for help in the chatroom. – Martin Ender May 15 '14 at 18:35 • Possible architect issue: With this scoring method (steps/walls), you can get a minimum score of 3 by simply putting the start/finish right next to each other with a single wall between. It takes three steps to go around. Most actual mazes I've seen have too many walls to make a score of 3 likely, much less guaranteed. – Geobits May 16 '14 at 13:51 • Thats a problem. What if the dist function was shortest path? Then only mazes which cause detours could get a score > 1 – wrongu May 16 '14 at 15:09 • That would probably be better. That way it's scored on best vs actual. It would take away the incentive to figure out how to build hard mazes with few walls, though, which was interesting itself. – Geobits May 17 '14 at 3:01 • Hey rangu... not sure if you're still planning to do this thing, but overactor just said something in chat which reminded me of your challenge and might be a neat way to avoid the combined score: split this up into two code-challenges, one for maze generation and one for maze solving. Each code-challenge's benchmark set (to determine the scores) would be the outputs of the other challenge's participants. Then you could just pick a best solver and a best generator independently. – Martin Ender Aug 1 '14 at 11:18 Author note: I was thinking about new genres today, and I had an idea. What if there could be a challenge that encourages people to write good code, instead of the code-golf gibberish we all know? Here's a challenge that attempts to do that. (This could even possibly be a , which would be great because it would bring in a greater high quality question volume to the site, but I'm terrible at coming up with names. Feel free to suggest something in the comments.) # Build your own image editor For this challenge, you will create the best GUI image editor that can perform the most tasks that you possibly can... from scratch. Here are the features / tasks used to score your program. Each task is worth a certain amount of points, which is specified in brackets before the task description. For convenience, each task will also be prefixed by an ID string so that you can refer to them when describing your program. • [1 A] Brush tool: Simple, click and drag the mouse to draw freestyle doodles. Must draw a contiguous path. • [1 A1] Ability to change the brush size. ## Requirements Your editor must conform to the following requirements: • Must accept input via the mouse. Tools (brush, flood fill, etc.) can be switched and configured with keyboard shortcuts, by clicking icons with the mouse, through a menu, or however you would like. • You may not use a single built-in function to accomplish one or more of the tasks. For example, if your language has a built-in image flood fill function, you may not use it and must build your flood fill from scratch. ## Final score and voting # {language}, {your score} score <sup>(features implemented: {A, A1, ...})</sup> {description, comments, other notes, etc. here} • elegance and readability of the source code • ease of use of the image editor and how powerful it is • remember to sort by "active" so that you're voting for new answers too, and not just the top voted ones! • Your questions should be reasonably scoped. If you can imagine an entire book that answers your question, you’re asking too much. A really good answer to this would run into millions of lines of code. – Peter Taylor Jun 4 '14 at 15:52 • @PeterTaylor Yes, I was a little worried about that, but if it's not broad enough, it will be easy to just implement all the features. Any suggestions for fixing this? I was thinking of adding a "brevity" criterion in the voting section, but that doesn't seem like an ideal solution. – Doorknob Jun 4 '14 at 16:00 • To be honest, the site for good code is Code Review. They already have a monthly challenge, for which they post snippets for review. I don't see a need to copy them. – Peter Taylor Jun 4 '14 at 16:12 • @PeterTaylor Wait, isn't Code Review for questions and answers, not challenges and contests? In any case, is there any reason for that to prevent us from posting challenges like we always have? – Doorknob Jun 4 '14 at 16:17 • meta.codereview.stackexchange.com/questions/tagged/… . Surprised you don't know about it, given how dedicated you are to spying on them ;) In general, if a question is on topic for multiple stacks then there's no obligation to do the sensible thing and post it on the one which it best fits, but you should expect people to ask why you're not doing the sensible thing. I think you're going to have to work hard to turn this into a question which fits this site, whereas it's already a good fit for CR's challenge programme. – Peter Taylor Jun 4 '14 at 16:31 • @PeterTaylor Hmm, that's strange. Wouldn't that be more on-topic here? (And I only occasionally pop in to their chat/meta to see what they're up to. :-P) – Doorknob Jun 4 '14 at 16:32 # Help Joe Bloggs with his password hash Joe was confidently using "password1" as his main password to all his accounts until one day he received an e-mail from fBay. His account has been compromised and he must change his password immediately. Yet worse, the attacker had access to all Joe's accounts. Being an engineer, Joe thought: What if I could hash somehow my password using a keyword? I wouldn't need to remember any passwords and I would have a different one for each account. Joe then creates an algorithm - he takes the domain name as a key and creates the password for each of his account consisting of: 1. (<consonants><vowels>)(alternating case: lower, capital, lower...) 2. <number of consonants><number of vowels> 3. <sum of consonants and vowels numbers converted to a character on US Qwerty Keyboard> Joe then opens an account on SO to create a new code golf challenge. He uses stackoverflow as a key to generate password: 1. sTcKvRfLwAoEo - consonants and vowels in alternating case 2. 94 - 9 consonants, 4 vowels 3. 9+4=13, 1+3=4, Shift+4=$ Therefore, Joe's password for stackoverflow is: sTcKvRfLwAoEo94$ ### Challenge Create a shortest function to generate a password given the rules above. The code should accept a string type parameter d and return/display the generated password. ### Rules 1. Only Latin letters from the input should be used. Any other characters should be ignored. 2. Minimum input length is 1 letter. (guys at q.com need passwords as well!) 3. Assume Y is a vowel 4. If vowels or consonants are missing, use 0 accordingly. E.g. input a would result in a01! 5. Shortest code wins List of vowels and consonants US qwerty keyboard • Thanks for the feedback @m.buettner. I meant to say, without using any libraries. The problem is, that people become lazy to think sometimes and just dive straight away to use Linq where a bit of thought will do – mai May 28 '14 at 13:14 • Well actually you can, I'm just checking now. You can do a lot of manipulations on strings without libraries. – mai May 28 '14 at 13:18 • Looping over string characters, concatenation work perfectly. Nevertheless, I've updated the challenge. If a function to depend on a library, it must be included in the character count. – mai May 28 '14 at 13:21 • 1. Strictly speaking, in .Net you don't have strings without libraries. The string keyword is syntactic sugar for a class in mscorlib. 2. As things currently stand, your rule 1 strictly prohibits something and then says what to do if you ignore that prohibition. This is illogical. It's also unclear what "that" in "please inlcude that in characters count" means. Does it mean that each submission should be a program as opposed to a code snippet? If so, state it explicitly. – Peter Taylor May 28 '14 at 13:32 • Hmm.. I don't know how to write it the best way. mscorlib is included by default so that is permissible. I don't want the code to use other libraries as Linq as it's less fun. – mai May 28 '14 at 13:47 • @m.buettner I agree with you. Nevertheless, there will solutions provided in other languages as well (there always are). And I would like the authors of those solutions to think about the best approach in their language of choice without depending on libraries like Linq. – mai May 28 '14 at 14:00 • Does Rule 2 mean ONLY vowels/consonants to be used from input? What about symbols @#\$ etc. Depending on that answer, potentially clarify Rule 5 regarding symbol input. As for Step 3 in the hash, should that progress further, similar to my Appended Numbers game so 103 consonants and 5 vowels would follow as 103+5 = 108, 1+0+8/10+8, etc.? – Matt Jun 4 '14 at 2:35 • @Matt, clarified - only Latin letters are used from the input. If consonants or vowels are missing, use 0 instead. The sum should progress, until it's <=9. E.g. 103 consonants, 5 vowels: 103+5=108, 1+0+8=9. Then, Shift+9='(' – mai Jun 18 '14 at 10:36 # Diplomacy Note for Sandbox: I have not finished (or really started) the control program for this game, because I wanted to see if there was interest in it before I dedicated too much time to the project. that means that the rule are still up to be tweaked, so please leave a comment if you have a suggestion, and comment or vote if you are interested in seeing this happen. Diplomacy is a complex strategy game, with a very entertaining combat system. This challenge will be to write a bot to compete in a simplified version of diplomacy combat. ## Rules ### Rounds Countries (bots) will begin the game with 10 health, representing their remaining will to fight. The goal is to eliminate all other Nations by attacking them until they have 0 health. The game will consist of several rounds. On the first round, all bots will receive 2 numbers as command line arguments: The first will be the total number of countries fighting, and the second will be their number in the list. Each following round, bots will receive a command line arguments containing the actions taken by each player last round and a list of all bots and their remaining health separated by commas, like so 1:A2,2:S3,3:A4,4:A3 1:10,2:7,3:7,4:1 Each bot must then output a desired action, which is one two commands 1. Attack a player. This is done by printing the letter A followed by the number of the player you with to attack. For instance, A3 2. Support a player. This will give the player you support a boosted attack. ### Resolving combat After player have sent in their moves, attack scores will be calculated thus: 1. All players start with a strength of 1, and one point is added for every player supporting them. For instance, if the moves are 1:A3,2:S1,3:A2,4:S2 then bot 1 has strength 2, bot 2 has strength 2, bot three has strength 1, and bot 4 has strength 1. 2. After strength has been calculated, bots will deal damage based on their strength. The formula for damage is (Attacker's strength + 1) - (Defender's Strength) In the above situation, player 3 would take 2 damage and player 2 would take 0 damage. Note that, unlike regular diplomacy, attacking a supporter does not cut support. 3. All attack take place simultaneously and independently. This means that if player 1 and 2 both attack player 4, then they each deal 1 damage. If player 3 were to support player 4, then player 4 would take no damage. ### Round Ends After combat has been resolved, countries that have 0 health will no longer be able to attack or support. However, they still will be listed in the input with an health of 0. When a bot is eliminated, all remaining bots will receive a single point. ### Ending the game The game ends when either 100 turns have elapsed or only 2 or less players remain. At this point, the player with the highest remaining health is the winner and receives 1 point. In case of ties, all tied bots will revive 1 point. If all bots die on the same turn, this is not a tied victory, but mutually assured destruction, and all bots will receive 0 points. ### Scoring The control program will run 100 rounds of the game. The winner will be the country with the most points at the end of 100 rounds. ## Code You may write in any language I can reasonably compile. I will make an effort to compile odd languages, but make no promises as to my ability to do so. Please provide your source code, an explanation, and a command line command to run your program. Notes • You are allowed to write to a file. In fact, you are encouraged to do so. • Because this is a game where cooperation is paramount, you are allowed to write bots that work together, with the following restriction: • Only two bots can be written by a single player to work together at a time. • Standard Loopholes apply. You are not allowed to change the way the control program runs. If you provide invalid input to the control program, the program will just skip your turn. However, you are allowed to spy on other countries files, and all bot programs will be in the same folder at runtime. This is war, after all! • I reserve the right to disqualify any country that takes more than about a second to run, or that tries a loophole not mention within. That being said, if it is sufficiently clever I will probably let it go. I will have source code up soon for a sample country that will be competing, and will post the control program when I finish it. • "In case of ties, all tied bots will revive 1 point". Is that supposed to say "receive"? "If all bots die on the same turn, ... all bots will receive 0 points." If there are two bots left, each of which has received 1 point from the earlier death of a third bot, and the two bots destroy each other on the same turn, what's the final score for the round? I'm not sure whether it's 0-0-0 or 1-1-0. "You are allowed to write bots that work together": but how can they identify each other? Do they have to use their moves as a covert channel? – Peter Taylor Aug 29 '14 at 14:28 • "Support a player. This will give the player you support a boosted attack." Or defence. Might be clearer to say "boost that player's strength for the turn". Should also state whether or not it's possible to support yourself. – Peter Taylor Aug 29 '14 at 14:29 # Check GenericScript source code for compiler errors Given the source code for a GenericScript program as input, parse the source code to check that it conforms to the syntax rules for the language. The syntax definition for GenericScript is below. If a part of the source code is found to be invalid, the program should output "Invalid syntax", otherwise it should output "Valid syntax". Win Criteria This is code golf. Shortest code wins. Syntax Source code will be considered to be valid if it matches the rule for "Program" below. Program = Sequence Sequence = Statement [Sequence] Statement = SequenceBlock \| Assignment \| If \| While \| Output SequenceBlock = "{" Sequence "}" Assignment = Identifier "=" (String \| Bool); If = "if(" Bool ")" Statement ["else" Statement] While = "while(" Bool ")" Statement Output = "print(" String ");" Identifier = {Any sequence of alphanumeric characters prefixed with "var" } Bool = StringEquals \| Identifier StringEquals = String "==" String String = StringConstant \| OperatorConcat \| Input \| Identifier StringConstant = "'"StringContent"'" StringContent = Character [StringContent] Character = {Any character except for "'"} OperatorConcat = String "&" String Whitespace is defined as any sequence of the ascii characters 9, 10, 13 and 32. Whitespace characters are allowed between tokens but are not required. Rules 1. The answer should be a complete program 2. Standard input/output allowed 3. Standard loopholes apply Test Input Valid syntax: print('What is your name?'); print('Hello ' & varInput); Invalid syntax: if(read() == 'DoTask1') print('Executing you'r command'); ## Objective Your goal is to develop a complete text-based adventure game with the shortest code possible. The player navigates in a dungeon composed of rooms. The game objectives are to find the treasure, slain the dragon and rescue the princess. ## Rules A room description is as follows: You are in <description>. You can go <exits> You see <object> (optional) • exits can be "north", "east", "west", "south". • adjective can be "dark", "murky", "small", "large", "narrow", "gloomy", "huge", "strange", "tiny", "broad", "old". • object can be "the princess", "the dragon", "a troll", "a goblin", "a sword", "gold", "a key", "a trunk". Exit list must be comma-separated and end with "and". If there is no object in the room, the last line is omitted. Example of valid description: You are in a murky room. You can go north, east and south. You see a goblin. The game accepts the following commands (case is ignored) : • GO direction : direction can be NORTH, EAST, WEST, SOUTH • TAKE item : item can be SWORD, GOLD, KEY • KILL monster : monster can be DRAGON, TROLL, GOBLIN. The DRAGON and the TROLL can be killed only if the user has the SWORD. If he hasn't, he loses the game. The weak GOBLIN can be killed with bare hands. When a monster dies, he disappears from the room. When the GOBLIN dies, he drops a SWORD. When the TROLL dies, he drops a KEY. • KISS person : person can be PRINCESS, DRAGON, TROLL, GOBLIN. Kissing the princess validates one of the objective of the game, and the princess disappears from the room. Kissing a monster results in player death. • OPEN object : object can be TRUNK. If the player has the KEY, the TRUNK object disappears and is replaced with GOLD. OBJECTS The player can perform an action on an object only if the object is in current room. A room can contain only one object ; a given object can be found in only one room. At the beginning of the game, only the following objects are placed in the map : PRINCESS, DRAGON, TROLL, GOBLIN, TRUNK. Other objects are not yet created. ACTIONS • If an action cannot be performed (e.g. GO NORTH where there is no exit to the north, or TAKE DRAGON, or DANCE GANGNAM STYLE), the message "Sorry, I can't do that" must be displayed. • If an action can be performed, the message "OK" and the current room description should be displayed. • You can read game commands from console or as a program parameter, as you wish. MAP The dungeon should have at least 30 rooms. The dungeon should not contains a series of more than 5 exits in the same direction. The exits between rooms must be consistent, e.g. if you go north from room #1 to room #2, there must a south exit in room #2 leading back to room #1. Every room name should be unique. There must be at least one room of each kind (hall, cavern, corridor...) • A hall has at least 3 exits. • A corridor can have only 2 exits. • The cell has only one exit. • There is only one dragon's lair and only one cell, containing respectively the dragon and the princess. GAME END The game ends when the player has been killed, or when he has taken the gold, slain the dragon and kissed the princess. • If the player dies, the message "You have been killed by X !" is displayed, with X being the name of the monster. • If the player wins, the message "Well done adventurer ! you've conquered the dungeon." is displayed. Player should not be able to win the game in less than 40 turns. Example You are in a murky room. You can go north, east and south. You see a goblin. > KILL GOBLIN Ok. You are in a murky room. You can go north, east and south. You see a sword. > TAKE SWORD Ok. You are in a murky room. You can go north, east and south. > GO NORTH Ok. You are in a narrow corridor. You can go south and east. ## Scoring The shortest code wins. • @Martin Thanks for your comments! I've updated the question. – Arnaud Aug 21 '14 at 7:59 • Provided the player ignores the troll and goblin (i.e. doesn't try to kiss or kill them), they don't do anything? – Peter Taylor Aug 21 '14 at 8:30 • @Peter you're right. Maybe the player should kill (with bare hands) the goblin in order to get the sword, and then kill the troll (with the sword) to get the key. – Arnaud Aug 21 '14 at 8:50 • "The map must be spatially coherent" still doesn't disallow always going left without ending up in the same room twice, unless you specify that the rooms are all meant to be square (which is what I think you had in mind). Also, I still think that "at least" 30 rooms is unnecessary. Who would implement 8 additional rooms if they don't have to. It will definitely be shorter if I omit the two longest adjectives and just use all available combinations of the remaining ones (giving 30 unique rooms). So you can omit two adjectives and the "at least" right away, I'd say. – Martin Ender Aug 21 '14 at 9:19 • I think it's fine to keep "at least" there for flavour, same with additional adjectives. Also, someone might figure out a way to make the code shorter with a longer adjective (for that reason, having a few more adjectives might be nice) – FireFly Aug 21 '14 at 9:38 • @Martin I've added a criteria "Player should not be able to win in less than 30 turns", to force the golfer to implement more rooms. – Arnaud Aug 21 '14 at 9:39 • @SuperChafouin That doesn't force it though. I just need to place the goblin at the end, troll at the beginning, trunk at the end, so that you need to traverse the map 3 times. – Martin Ender Aug 21 '14 at 9:51 • @Martin It's also here to prevent the dungeon to be too straightforward to solve, e.g. if all the objects are in 5 adjacent rooms near the player start location. – Arnaud Aug 21 '14 at 10:51 • +1 for golfing in Inform 7. – Lopsy Sep 20 '14 at 2:56 • Hello! This looks like a good but abandoned meta post, would you be willing to offer it for adoption? (If you want to, you can still post to main.) Due to community guidelines, if you don't respond to this comment in 7 days I have permission to adopt this. – programmer5000 Jun 9 '17 at 16:58 • @programmer5000 Yes no problem :-) – Arnaud Jun 10 '17 at 1:07 # Simulate a Quantum Circuit Work-in-progress until I can make sure I know what I am doing and can finish the spec. or maybe Quantum computers are the way of the future! Why wait, when you can simulate one now? Your mission is to determine the output of a quantum circuit given its input and a diagram of logic gates. # Details You will simulate a single quantum register and apply a series of quantum logic gates to it. A quantum register is a group of qubits. The state of a register is described by a vector of 2^N complex numbers, where N is the number of qubits in the register. a\|000> b\|001> c\|010> d\|011> e\|100> f\|101> g\|110> h\|111> Above is a representation of a 3-qubit register. Each letter (a b c etc.) represents a complex number. There is an addition restriction that: \|a\|^2 + \|b\|^2 + \|c\|^2 + \|d\|^2 + \|e\|^2 + \|f\|^2 + \|g\|^2 + \|h\|^2 = 1 ## Quantum gates Gates are represented by a 2^N x 2^N square unitary matrix, where N is the number of input qubits. All quantum gates have the same number of outputs and inputs, since they neither create nor destroy qubits, they modify them. A common quantum gate is called the Hadamard gate and acts on a single qubit. The matrix [H] looks like this: 1/Sqrt(2) 1/Sqrt(2) 1/Sqrt(2) -1/Sqrt(2) If we let [R] represent the following 1-qubit register: 0.6\|0> O.8\|1> Then the application of the gate is represented by [H][R] and gives the following result: 7Sqrt(2)/10\|0> -Sqrt(2)/10\|1> It is still true that the sum of the squares of the absolute values is equal to 1. (TODO: explain how to apply gates to larger registers) ## Measurement Measurement collapses the state of the quantum register. (Todo: Explain how measurement works) # BS The goal of this challenge is to implement an AI for the game of BS, also known as Bull Shit, Cheat, Bluff, and numerous other names. The game is outlined in this wikipedia article. # The Rules of the Game For the purposes of this challenge, the game will work like this: 1. A standard 52-card deck is dealt out to the players 2. The current rank is set to Ace 3. The play order is randomized 4. The player holding the Ace of Hearts goes first 5. On each player's turn: 1. The current player plays some number of cards 2. The current player states how many of what rank they played 3. Other players may declare 'BS'. 4. If any player declares 'BS': 1. All players are notified of which players declared 'BS'. 2. The played cards are revealed to all players. 3. If the played cards are inconsistent with the current player's statement: • The current player adds the played cards and all cards in the pile to their hand 4. If the played cards are consistant with the current player's statement: • The last player to declare 'BS' that round adds the played cards and pile to their hand. 5. If no player declares 'BS': 1. The played cards are added to the pile, without revealing them. 2. If the played cards were inconsistant with the current player's statement, the current player may declare 'Peanut Butter'. 6. If the current player has no cards in their hand, the current player wins. 7. The current rank is incremented. (If the current rank is King, it becomes Ace.) # The Messaging Protocol Play will be conducted via messages passed to the standard input and received from the standard output of each program. Each message will be terminated with a single newline character. ## Cards Card ranks are represented as one of A, 2, 3, 4, 5, 6, 7, 8, 9, T, J, Q, or K. Card suits will be represented as one of S, C, H, or D. Cards are represented as the rank, followed immediately by a suit. For instance, the Ten of Clubs would be represented by TC, and the Three of Hearts would be represented by 3H. A hand of cards will be represented as a space-delimited sequence of cards. For instance, a hand containing the Queen of Spades and the Six of Diamonds could be represented as QS 6D or 6D QS. ## Player Identification A player will be represented by their nickname, followed by a number from 0 to 32768, in parenthesis, formatted as an integer. This number is guaranteed to be unique within a particular game. A player's nickname must have at least one character, can have up to 32 characters, and may only include letters, numbers, and underscores. For instance, a player with nickname ExampleAI and ID number 16480 would be identified in the game as ExampleAI(16480). When the game begins, each program will recieve a message containing their unique ID: Unique ID: uniqueID Each player will reply with their desired nickname: Nickname: name Names may contain only alphanumeric characters and underscores. After all players have responded with their nickname, the standard play sequence begins. ## Standard Play Sequence When a player's turn begins, each player will receive a be given a list of the players and their card counts, in order of play: Players: player[count], player[count], ... player[count] Each player will be informed of the contents of their hands: Hand: initial_hand The current the player will then receive this message: Your turn: current_rank The current player will reply with a space-separated list of of cards: Play: list_of_cards Once they have submitted their play, all players will receive the number of cards, formatted as an integer, along with the current rank: Player player plays: nunber_of_cards x current_rank Each other player may then declare BS on that play by sending any message up to 32 characters, containing the capital letters B and S, and otherwise only contains lowercase letters and spaces. So any of Bull Shit, Bananna Split or Bacon Sandwich would be acceptable. During this period, the current player may declare Peanut Butter by sending any message up to 32 characters, as long as it contains the capital letters P and B, and otherwise only contains lowercase letters and spaces. So any of Peanut Butter, Pancake Batter or Polish Bacon would be acceptable. In order to allow the game to move faster, if a player does not wish to declare either of these things, they must instead send: Pass After all players have responded, all players will receive a list of players who called BS, in the order they called it: Called BS: player, player ... player If no player called BS, this message will still be sent --- it just won't have any players listed. If any player did call BS, then all players will recieve: Player player had played: list_of_cards If they were bluffing, all players recieve: Player player was bluffing. Your bluff was called: list_of_cards_recieved If they were not bluffing, all players recieve: Player player was not bluffing. Player last_player receives the pile. The last player who called BS recieves this message: You misjudged: list_of_cards_received The list of cards received will contain, in reverse chronological order, the contents of each play since the last call. (Separate plays will not be delimited in the list.) If no player declared BS, and the current player was bluffing and declared Peanut Butter, then all players recieve the message: Player player was bluffing. If the current player has no cards left in their hand, all players receive this message, and the game terminates: Player player won! Otherwise, the next player's turn begins. # Example Game The following might be considered a typical (abbreviated) message transcript: Unique ID: 16481 > Nickname: Alice Players: Alice(16481)[18], Bob(16479)[17], Charlie(16480)[17] Hand: 2D 7S AS TC 5S JS JC 3C 8H 9D 5D AH 7C 6C 4D KC KH KS > Play: AS 2D AH Player Alice(16481) plays: 3 x A > PB Called BS: Player Alice(16481) was bluffing. Players: Bob(16479)[17], Charlie(16480)[17], Alice(16481)[15] Hand: 7S TC 5S JS JC 3C 8H 9D 5D 7C 6C 4D KC KH KS Player Bob(16479) plays: 2 x 2 > BS Called BS: Alice(16481) Player Bob(16479) had played: 2H 2C Player Bob(16479) was not bluffing. Player Alice(16481) takes the pile. You misjudged: 2H 2C AS 2D AH Players: Charlie(16480)[17], Alice(16481)[20], Bob(16479)[15] Hand: 7S TC 5S JS JC 3C 8H 9D 5D 7C 6C 4D KC KH KS 2H 2C AS 2D AH . . . Players: Alice(16481)[3], Bob(16479)[41], Charlie(16480)[8] Hand: KC KH KS > Play: KC KH KS Called BS: Charlie(16480), Bob(16479) Player Alice(16481) was not bluffing. Player Alice(16481) won! Your implementation may be written in any language, provided that you, upon request, provide a link to a suitable free-as-in-freedom compiler or interpreter that I can download and run at no cost. You also need to provide a UNIX command that can start your program. # Sandbox Questions I want to gauge the community's interest in my problem before finalizing the spec and writing the control program. I also need to get some idea of what sort of time-limiting scheme would be reasonable. In order to be able to to a lot of runs, I will need to be able to ensure that each AI doesn't take too much time to make its decisions, or prevent a stuck AI from holding up a game. I also need to be able to ensure that there is no motivation to deliberately stall a game. For example, if an AI determines that it is very unlikely to win, it might stall in order to prevent the game from finishing. I would also like feedback on the messaging protocol: • Are there any additional messages that you think should be passed? • Would it be more convenient/clear if one or more of them were formatted differently? • Would it be better to use a different format for the plays message? • Would it be better to use different words to help distinguish the plays and played messages? • It looks like quite a tough challenge, but should be enjoyable! – Alexander Craggs Sep 4 '14 at 16:54 • @PopeyGilbert By the way, there was one thing I accidentally left out that I need feedback on. Specifically, time limits - to deal with intentional stalling, getting stuck, or taking too long to decide. – AJMansfield Sep 4 '14 at 17:05 • A question and a feedback. Does our program run as "stop and run" or must receive feedback continuously? And for feedback. I honestly think that the whole username things is kinda confusing. Maybe if you just use only unique id? (Like just simple 0,1,2,3 instead of username) – Realdeo Sep 5 '14 at 8:33 • Oh! More things! I also realize that suit doesn't really matter, right? (We only use suit for deciding who goes first), so fmpov, you can ditch the communication protocol for the suit. (No need for S,C,D,H) We can just use simple random from the computer. Question: What will happened if everybody make infinity loop of pass. For time limit, I prefer 1 second. If no response, make it auto pass. (KOTH chess time limit is 2 seconds. That's why 1 second is good enough) – Realdeo Sep 5 '14 at 8:38 • @Realdeo Card suit is also used to distinguish between separate instances of a card. If Alice plays 3x2 (2S 2D 2C), is not revealed, and Bob gets the pile later, and then Bob plays 3x2 (2S 2C 2H), and this is revealed, it is important for Alice that she knows all four Twoes have passed through Bob's hand. There are other ways that can be used as well. – AJMansfield Sep 5 '14 at 13:29 • @Realdeo I am not sure what you mean by "Does our program run as "stop and run" or must receive feedback continuously?". If you mean, "Does an AI program halt in the periods where no response is expected from it?" the answer is no. – AJMansfield Sep 5 '14 at 13:32 • @Realdeo If everybody makes an infinity loop of pass, then eventually someone will run out of cards, since you are required to play at least one card each turn. – AJMansfield Sep 5 '14 at 13:33 • @Realdeo Which is why making a automatic pass after a timeout not work when waiting for a player to decide their play. Perhaps a simple rule like 'if you take more than 1 second to decide what to play, four cards are selected at random from your hand'. – AJMansfield Sep 5 '14 at 13:36 • And if a player has less then 4 cards? I think in some AI website, like aigames.com, they're like forced to give up that hand? You really want to test your entry before put them in the arena(like vsing a bot dummy?) Either way, this is a good challenge =) – Realdeo Sep 5 '14 at 13:40 • @Realdeo Also note that you can actually play more than four cards in one play. A case where you might wish to do this is when: the next player is very close to winning and some other player is close to winning and you believe(all opponents believe(your next opponent will bluff) and the next opponent will not bluff and the next opponent believes(the other opponent close to winning will call BS against them)). A little convoluted, but could happen. – AJMansfield Sep 5 '14 at 13:42 • @Realdeo Just to explain what I mean, is that there are two people close to winning, each of which would like to dump a large stack of cards on top of the other. Because of this, they both let your obvious bluff slide because they believe that will let them dump a large stack on the other. – AJMansfield Sep 5 '14 at 13:47 • Don't worry I understand. This is a really famous high school game in my country. It just... a little bit too complex for CR. When I saw chess KOTH, I was kinda pessimist. This one? This may deserve it's own AI website. #seriously. I'm just trying to simplify this game =) – Realdeo Sep 5 '14 at 13:50 • – AJMansfield Sep 5 '14 at 13:50 • RE the messaging service, I think the other players should be able to see how many cards the other players have. Also, card counting should be prohibited because that would make the game too easy. – Beta Decay Sep 10 '14 at 17:31 • @BetaDecay First off, according to the protocol, every player is informed of every other player's hand size at the beginning of every round. – AJMansfield Sep 10 '14 at 18:27 I am planning on hosting a King of the Hill challenge in which bots will have to coordinate each other in order to be successful. The idea is to play a Diplomacy-like game between bots. The engine (still in development) will start the bots and communicate with them via stdin/stdout. There will be three phases: 0. Initialization Well, this is not a recurring phase, it is just the engine telling each bot his id, the total number of bots participating and a seed, which can be used for generation of pseudo-random numbers (bots need to be deterministic). 1. Talking Phase (10s) In the Talking phase, bots can send messages to each other (via engine) in order to coordinate their actions. To this end, a common language is necessary. This language should be able to express any ideas, plans and opinions a bot could have. However, not every bot is forced to be able to understand everything. Simpler bots might just ignore messages they do not understand. Since I would like each player to be able to submit more than one bot, it is forbidden to implement a "secret handshake" by which bots recognize each other and from then on work together unconditionally. 2. Planning Phase (2s) In this phase, bots submit what they want to do this turn. Each bot has a certain amount of supply (initially five), and can command one action per supply point. There are three possible actions: 1. Attack another bot 2. Support another bot's attack against a third bot 3. Defend another bot There are some restrictions: • Per opponent, you can either attack or defend them, and only once • You cannot support an attack against a bot you also defend • You cannot attack, defend, support yourself or a dead bot, and neither can you support attacks against yourself 3. Resolution Phase (as short as possible) After all orders have been submitted, the engine will resolve them simultaneously in the following way: The defending strength of each bot is the number of bots defending that bot. The attacking strength of each attack is the number of support orders for that attack. Each attack with an attacking strength greater than the defending strength of the attacked bot results in the supply counter of the attacked bot being reduced by one, and the supply counter of the attacker being increased by one. Support orders which support a non-existent attack do nothing. Then, all bots with supply of zero or less will be shut down by the engine: they died. Afterwards, all remaining bots are informed about the decisions of other bots, and a new turn begins with its Talking Phase. Further Rules A game will consist of ten plus random number turns, so that "last turn betrayals" are not possible. The supply count of each bot will count towards their total score. I plan an ensemble of about 100 games. The bot with the highest total score wins. Tie-breaker will be the popularity (number of votes). I am interested in your opinion: do you think that this challenge is too complex? I imagine that the code of a decent bot would be too long to fit in a post. So people would have to use github or pastebin or similar to submit their entries. The main problem imo is the interpretation of the (yet to be determined) common language. • I like it a lot. One possible variation would be to make the "secret handshakes" a feature. To do this, you could allow multiple instances of the same bot. Then part of the challenge is to recognise your own kin and mutually support them; and a viable strategy is to try and work out other players' secret handshakes and imitate them. If you're ok with emphasising this aspect of it, then you can make the shared language pretty unrestricted, e.g. bots can just send arbitrary strings to each other. (I realise this is not what you have in mind, I just thought I'd mention the idea.) – Nathaniel Oct 31 '14 at 15:13 • @Nathaniel, what you propose is a battle of obfuscation/cryptography. What I would like to see is a battle of diplomacy. – M.Herzkamp Nov 2 '14 at 14:27 • Fair enough - I just thought I'd say it in case it sparked any interesting thoughts for you, but I knew it was probably too different from what you want to see. If I have any other ideas about your challenge I'll let you know. Designing the language really seems to be the hard part. – Nathaniel Nov 3 '14 at 0:06 • A diplomatic KOTH is something I've been wanting to see for a while. Working out the specifics of the "diplomatic language" is going to be the most difficult part. My proposal is that each message can either 1) state an intention to another bot, or 2) request an action from another bot. Each message would be formatted in a way similar to how a final command would be. – PhiNotPi Nov 9 '14 at 14:45 • @PhiNotPhi: Thanks for the Feedback! I also imagined something similar with the ability to link atomic statements in a boolean fashion. – M.Herzkamp Nov 11 '14 at 9:23 # Happy Holidays! ## Introduction With the holidays upon us, I decided to make an appropriately themed challenge. You are provided with a list of holidays and their respective date ranges, and given a date, you have to output a holiday greeting or the time remaining until the next holiday as appropriate. ## Challenge The list of holidays is below. You have to include it in your program (so no using a library or other external resource for this). Feel free to use any convenient format. Start \| End \| Name ------ \| ------ \| ------------------- Dec 6 \| Dec 7 \| Saint Nicholas' Day Dec 13 \| Dec 14 \| Saint Lucy's Day Dec 24 \| Dec 27 \| Christmas Jan 1 \| Jan 2 \| New Year Jan 6 \| Jan 7 \| Epiphany Feb 14 \| Feb 15 \| Valentine's Day You are given a date as input (STDIN, function argument, or anything convenient) in YYYY-mm-dd HH:MM:SS format (e.g.: 2014-12-30 11:15:00). You may assume that the time zone is either UTC or the system's time zone. The holiday lasts from 00:00:00 on the start date (inclusive) to 00:00:00 on the end date (exclusive). If the date falls within the range of the holiday, you must output Happy <holiday>!, except if it's Christmas, in which case you must output Merry Christmas!. If it doesn't, but another holiday is coming at most a week in the future, you must output: <time> left until <holiday>. where <time> is in the following format: <days>d <hours>h <minutes>m <seconds>s You can't use a library for converting the time to that format. If there are no whole days, hours, minutes or seconds remaining, omit the number entirely. For example, 1d 0h 3m 4s should be printed as 1d 3m 4s. If there are no upcoming holidays, you must output (no pun intended): There are no upcoming holidays. A trailing newline is optional, but be consistent in your program—don't add a trailing newline in one case and omit it in another. Standard loopholes are obviously forbidden. ## Test cases Date \| Output ------------------- \| ---------------------------------- 2014-12-05 23:59:59 \| 1s left until Saint Nicholas' Day. 2014-12-06 00:00:00 \| Happy Saint Nicholas' Day! 2014-12-06 12:00:00 \| Happy Saint Nicholas' Day! 2014-12-06 23:59:59 \| Happy Saint Nicholas' Day! 2014-12-07 00:00:00 \| 6d left until Saint Lucy's Day. 2014-12-14 00:00:00 \| There are no upcoming holidays. 2014-12-24 00:00:00 \| Merry Christmas! ` Note that your program must work for any year, not just 2014. ## Winner This is code golf, so the submission with the fewest number of bytes wins. An answer will be accepted after a week, but I'll be happy to change the accepted answer if a new valid submission beats the previous high score. • How do you expect people to test the test cases? It would probably be better to take input than to use the current time, because then it actually makes sense to talk about test cases. You should check date for duplicates, and if there are none you should add that tag. – Peter Taylor Dec 29 '14 at 14:51 • @PeterTaylor You're right, I'll do that. – nyuszika7h Dec 29 '14 at 15:37 • @PeterTaylor I couldn't find any exact duplicates, only two holiday-themed questions, both of which ask for much less than my challenge. – nyuszika7h Dec 30 '14 at 11:28	2020-09-25 07:12:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4238071143627167, "perplexity": 1435.8421488837614}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400222515.48/warc/CC-MAIN-20200925053037-20200925083037-00371.warc.gz"}
http://math.stackexchange.com/questions/242657/relationship-between-autg-and-symmetric-group-on-g	# Relationship between $Aut(G)$ and Symmetric group on $G$ I have read that $Aut(G)$ is a subset of $S_g$. So say I have a group $G = \{1, 2, 3\}$ for example. Then $S_G = S_3$ is the group of all permutation of the three elements of $G$. But I don't see why $Aut(G)$ is a subset of $S_G$ as opposed to $Aut(G) = S_G$. Each element of $S_3$ maps each element of $G$ to an element of $G$. I.e. each element is an automorphism. So why is $Aut(G) \subset S_3$ instead of $Aut(G) = S_3$? - $\,S_g\,$ is a rather unusual notation. I'd rather go with $\,\operatorname{Sym}_G\,$ , or at least $\,S_G\,$ – DonAntonio Nov 22 '12 at 15:50 When you say, "I have a group $G=\{1,2,3\}$." That is not a group, it is a sets of three elements. A group is a set with an operation. – Thomas Andrews Nov 22 '12 at 16:32 Not every permutation of $G$ is an automorphism. For example, automorphisms always fix the identity, but there are definitely permutations of $G$ that do not fix the identity if $\|G\| > 1$. – Mikko Korhonen Nov 22 '12 at 17:15 There are elements in $\,\operatorname{Sym}_G\,$ which are not automorphisms of the group $\,G\,$, say the permutation $\,(01)\,$ in $\,S_3\,$ is not an automorphism of cyclic group $\,\Bbb Z_3:=\Bbb Z/3\Bbb Z:=\{0,1,2\}\,$, with operation modulo $\,3\,$ - $0 \to 1$, $1 \to 0$, $2 \to 2$. It maps each elements of $G$ to an element of $G$. So why is it not an automorphism? – sonicboom Nov 22 '12 at 16:01 Because an automorphism of a group must first be a homomorphism of groups and, as such, it must map the group's unit (or neutral element) to itself. Here, it must be that $\,0\to 0\,$, otherwise it is not a homomorphism and, thus, not an automorphism. – DonAntonio Nov 22 '12 at 16:28 Cheers, I get it now. Just wondering, what is $Aut(G)$ in the case of my example in the original post...is it the elements $()$ and $(2\, 3)$? These are the only ones that fix $1$, which is the identity element in my group $G$? – sonicboom Nov 22 '12 at 16:51 Well, in your example you have a set with three elements, what'd be a cyclic group of order three...which is exactly what is $\,G\,$ in my answer! Here, $\,\|Aut(G)\|=2\,$ , and the only two automorphisms are: $$0\to 0\,\,,\,\,1\to 1\,\,,\,\,2\to 2$$and $$0\to 0\,\,,\,\,1\to 2\,\,,\,\,2\to 1$$ – DonAntonio Nov 22 '12 at 17:04 Note however, that mapping $0$ to itself is not all that is required for a permutation to be an automorphism in general. It just happens that all counterexamples are for groups with at least 4 elements. – Henning Makholm Nov 22 '12 at 17:08 (I'm assuming that $G$ here is a group, but I now notice that you're not saying so explicitly. If $G$ is not considered to be a group, then you have to ask what does $\operatorname{Aut}(G)$ mean at all?) Merely "mapping each element of $G$ to an element of $G$" is not enough to be an automorphism. An automorphism is a bijection $G\to G$, which is also a homomorphism. Most elements of $S_3$ will not correspond to homomorphisms $G\to G$. - Sorry, I'ved edited in that $G$ is a group. – sonicboom Nov 22 '12 at 15:58	2014-08-27 11:03:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.944770872592926, "perplexity": 178.41421381784704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500829210.91/warc/CC-MAIN-20140820021349-00050-ip-10-180-136-8.ec2.internal.warc.gz"}
https://maqsad.io/classes/class-9/physics/physical-quantities-and-measurement/iv-%20an%20interval%20of%20200%20mathrm%20u%20mathrm%20s%20is%20equivalent%20to(a)%200-2%20mathrm%20s%20(b)%200-02%20mathrm%20s%20(c)%202%20times%2010exp%20-4%20mathrm%20s%20(d)%202%20times%2010exp%20-6%20mathrm%20s	Classes Change the way you learn with Maqsad's classes. Local examples, engaging animations, and instant video solutions keep you on your toes and make learning fun like never before! Class 9Class 10First YearSecond Year ##### iv. An interval of 200 \mathrm{u} \mathrm{s} is equivalent to(a) 0.2 \mathrm{~s} (b) 0.02 \mathrm{~s} (c) 2 \times 10^{-4} \mathrm{~s} (d) 2 \times 10^{-6} \mathrm{~s} 1.6 What role SI units have played in the development of science? iii. Amount of a substance in terms of numbers is measured in:(a) gram(b) kilogramme(c) newton(d) mole 5. A chips wrapper is 4.5 \mathrm{~cm} long and 5.9 \mathrm{~cm} wide. Its area upto significant figures will bea) 30 \mathrm{~cm}^{2} b) 28 \mathrm{~cm}^{2} c) 26.55 \mathrm{~cm}^{2} d) 32 \mathrm{~cm}^{2} 1.6 On closing the jaws of a Vernier Callipers zero of the vernier scale is on the right to its main scale such that 4th division of its vernier scale coincides with one of the main scale division. Find its zero error and zero correction. ii. Which one of the following unit is not a derived unit?(a) pascal(b) kilogramme(c) newton (d) watt 1.11 Why do we need to measure extremely small interval of times?	2022-12-02 09:31:50	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8968492150306702, "perplexity": 6706.960451871557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710900.9/warc/CC-MAIN-20221202082526-20221202112526-00273.warc.gz"}
https://en-academic.com/dic.nsf/enwiki/327563	# Expression (mathematics) Expression (mathematics) In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers (constants), variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic symbols. The use of expressions can range from the simple: 3 + 5 to the complex: $f(a)+\sum_{k=1}^n\left.\frac{1}{k!}\frac{d^k}{dt^k}\right\|_{t=0}f(u(t)) + \int_0^1 \frac{(1-t)^n }{n!} \frac{d^{n+1}}{dt^{n+1}} f(u(t))\, dt.$. Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions. For example: $\times4)x+,/y$ would not be considered a mathematical expression but only a meaningless jumble.[1] In algebra an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression; the determination of this value depends on the semantics attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value; such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving the operator $\oplus$ to designate an internal direct sum. Being an expression is a syntactic concept; although different mathematical fields have different notions of valid expressions, the values associated to variables does not play a role. See formal language for general considerations on how expressions are constructed, and formal semantics for questions concerning attaching meaning (values) to expressions. ## Variables Many mathematical expressions include letters called variables. Any variable can be classified as being either a free variable or a bound variable. For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be undefined. Thus an expression represents a function whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.[2] For example, the expression x / y evaluated for x = 10, y = 5, will give 2; but is undefined for y = 0. The evaluation of an expression is dependent on the definition of the mathematical operators and on the system of values that is its context. Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function. Example: The expression $\sum_{n=1}^{3} (2nx)$ has free variable x, bound variable n, constants 1, 2, and 3, two occurrences of an implicit multiplication operator, and a summation operator. The expression is equivalent with the simpler expression 12x. The value for x = 3 is 36. The '+' and '−' (addition and subtraction) symbols have their usual meanings. Division can be expressed either with the '/' or with a horizontal dash. Thus $x/2 \text{ or } {x \over 2}$ are perfectly valid. Also, for multiplication one can use the symbols '×' or a '·' (mid dot), or else simply omit it (multiplication is implicit); so: $x \times 2 \text{ or } x\cdot2 \text{ or } x2 \text{ or } 2x$ are all acceptable. However, notice in the first example above how the "times" symbol resembles the letter 'x' and also how the '·' symbol resembles a decimal point, so to avoid confusion it's best to use one of the later two forms. An expression must be well-formed. That is, the operators must have the correct number of inputs, in the correct places. The expression 2 + 3 is well formed; the expression * 2 + is not, at least, not in the usual notation of arithmetic. Expressions and their evaluation were formalised by Alonzo Church and Stephen Kleene[3] in the 1930s in their lambda calculus. The lambda calculus has been a major influence in the development of modern mathematics and computer programming languages.[4] One of the more interesting results of the lambda calculus is that the equivalence of two expressions in the lambda calculus is in some cases undecidable. This is also true of any expression in any system that has power equivalent to the lambda calculus. ## References Wikimedia Foundation. 2010. ### Look at other dictionaries: • Expression — may refer to:* Idiom * Facial expression * Artificial discharge of breast milk; see breastfeeding * Expression (mathematics) * Expression (programming), an instruction to execute something that will return a value. * Microsoft Expression Studio,… … Wikipedia • Expression (programming) — An expression in a programming language is a combination of values, variables, operators, and functions that are interpreted ( evaluated ) according to the particular rules of precedence and of association for a particular programming language,… … Wikipedia • expression — ► NOUN 1) the action of expressing. 2) the look on someone s face. 3) a word or phrase expressing an idea. 4) Mathematics a collection of symbols expressing a quantity. DERIVATIVES expressional adjective expressionless adjective … English terms dictionary • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… … Universalium • mathematics, foundations of — Scientific inquiry into the nature of mathematical theories and the scope of mathematical methods. It began with Euclid s Elements as an inquiry into the logical and philosophical basis of mathematics in essence, whether the axioms of any system… … Universalium • Mathematics of Sudoku — The class of Sudoku puzzles consists of a partially completed row column grid of cells partitioned into N regions each of size N cells, to be filled in using a prescribed set of N distinct symbols (typically the numbers {1, ..., N}), so that each … Wikipedia • expression — ex\|pres\|sion W2S2 [ıkˈspreʃən] n ▬▬▬▬▬▬▬ 1¦(strong feelings/thoughts)¦ 2¦(on somebody s face)¦ 3¦(word/phrase)¦ 4¦(music/acting)¦ 5¦(mathematics)¦ ▬▬▬▬▬▬▬ 1.) ¦(STRONG FEELINGS/THOUGHTS)¦ [U and C] something you say, write, or do that shows what… … Dictionary of contemporary English • Mathematics of general relativity — For a generally accessible and less technical introduction to the topic, see Introduction to mathematics of general relativity. General relativity Introduction Mathematical formulation Resources … Wikipedia • Mathematics of radio engineering — A complex valued function. The mathematics of radio engineering is a pleasant and very useful subject. This article is an attempt to provide a reasonably comprehensive summary of this almost limitless topic. While the ideas have historically… … Wikipedia • expression — /Ik spreSFn/ noun 1 WORDS (C) a word or group of words with a particular meaning: The expression in the family way means pregnant . \| pardon/forgive/excuse the expression (=used when you have used a word that you think may offend someone): He… … Longman dictionary of contemporary English	2021-02-25 14:39:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6465832591056824, "perplexity": 1281.890015178911}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178351134.11/warc/CC-MAIN-20210225124124-20210225154124-00465.warc.gz"}
https://www.physicsforums.com/threads/partition-function-boltzman-maxwell-distri.174181/	# Partition function & Boltzman, Maxwell distri 1. Jun 16, 2007 ### pivoxa15 What is the relation between the partition function and the Boltzman, Maxwell distribution? Differences and similarities? Both have exponentials to the power of the negative total energy of the microstate. Although the word microstate dosen't occur in the Boltzman, Maxwell case. Is the BM distribution about velocity and position of particles only whereas the partition function is about any distribution involving macro variables? So it cannot calculate the speeds and position of particles in the system but can tell you about energy, pressure, temperture etc. Last edited: Jun 16, 2007 2. Jun 16, 2007 ### smallphi The normalization constant of the energy probability distribution of a system is one over the partition function. In the MB distribution, the system is a single particle in the heat bath of all the rest. The microstates are that particle at different energies (kinetic + potential if any). Last edited: Jun 16, 2007 3. Jun 16, 2007 ### pivoxa15 So the MB is a more specialised or more restricted entity? Could you use the partition function to calculate position and velocity of particles? 4. Jun 17, 2007 ### smallphi Last edited: Jun 17, 2007 5. Jun 17, 2007 ### pivoxa15 Interesting but you can't derive the partition function from the MB? If you had non gas with potential energy than you would replace E=p^2/(2m) with K+U? Then you would get the more general MB disbribution.	2017-09-20 14:57:26	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8299315571784973, "perplexity": 2559.2457251389014}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687324.6/warc/CC-MAIN-20170920142244-20170920162244-00323.warc.gz"}
https://people.smp.uq.edu.au/MatthewDavis/matts_arXiv/mailings/0012.html	# Matt's arXiv selection, Monday 6th March 2006. From: Matthew Davis <mdavis_at_physics.uq.edu.au> Date: Mon, 6 Mar 2006 09:30:10 +1000 (EST) The following message was sent to the matts_arxiv list by Matthew Davis <mdavis_at_physics.uq.edu.au> Remember the archive of mailings found at Have a great week, Matt. -- ------------------------------------------------------------------------- Dr M. J. Davis, Senior Lecturer in Physics School of Physical Sciences, email: mdavis_at_physics.uq.edu.au University of Queensland, ph : +61 7 334 69824 Brisbane, QLD 4072, fax : +61 7 336 51242 Australia. http://www.physics.uq.edu.au/people/mdavis/ ------------------------------------------------------------------------- ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602572 Date: Fri, 24 Feb 2006 01:24:49 GMT (429kb) Title: Dynamical Instability of a Rotating Dipolar Bose-Einstein Condensate Authors: R.M.W. van Bijnen, D.H.J. O'Dell, N.G. Parker and A.M. Martin Comments: 5 pages, including 3 figures Subj-class: Other \\ We calculate the static solutions of a rotating harmonically trapped dilute gas Bose-Einstein condensate in the presence of long range dipolar interactions, based on the hydrodynamic model. In particular we focus on the dynamical instability of our solutions, for weakly anisotropic traps. When the trap anisotropy is introduced adiabatically, at fixed rotation frequency, $\Omega$, the upper bound for the unstable region in $\Omega$ is decreased in the presence of dipolar interactions. By considering dynamical perturbations we also find that for fixed trap anisotropy and adiabatically increasing rotational frequency that the dynamically unstable region in $\Omega$ widens, in a non-trivial manner, as the strength of the dipole interactions is increased. \\ ( http://arXiv.org/abs/cond-mat/0602572 , 429kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602582 Date: Fri, 24 Feb 2006 10:32:14 GMT (338kb) Title: Analysis of an atom laser based on the spatial control of the scattering length Authors: Alicia V. Carpentier, Humberto Michinel, Mar\'{\i}a I. Rodas-Verde, V\'{\i}ctor M. P\'erez-Garc\'{\i}a Comments: Submitted to Phys. Rev. A Subj-class: Other \\ In this paper we analyze atom lasers based on the spatial modulation of the scattering length of a Bose-Einstein Condensate. We demonstrate, through numerical simulations and approximate analytical methods, the controllable emission of matter-wave bursts and study the dependence of the process on the spatial dependence of the scattering length along the axis of emission. We also study the role of an additional modulation of the scattering length in time. \\ ( http://arXiv.org/abs/cond-mat/0602582 , 338kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602525 replaced with revised version Thu, 23 Feb 2006 23:23:24 GMT (429kb) Title: Superfluid phases of the three-species fermion gas Authors: Paulo F. Bedaque, Jos\'e P. D'Incao Subj-class: Other; Atomic Physics \\ ( http://arXiv.org/abs/cond-mat/0602525 , 429kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602568 replaced with revised version Fri, 24 Feb 2006 03:54:50 GMT (291kb) Title: Homogeneous Fermion Superfluid with Unequal Spin Populations Authors: Tin-Lun Ho and Hui Zhai Subj-class: Superconductivity \\ ( http://arXiv.org/abs/cond-mat/0602568 , 291kb) ------------------------------------------------------------------------------ \\ Paper: quant-ph/0602202 Date: Fri, 24 Feb 2006 09:54:22 GMT (351kb) Title: Non-Gaussian states from continuous-wave Gaussian light sources Authors: Klaus Molmer \\ We present a general analysis of the state obtained by subjecting the output from a continuous-wave (cw) Gaussian field to non-Gaussian measurements. The generic multimode state of cw Gaussian fields is characterized by an infinite dimensional covariance matrix involving the noise correlations of the source. Our theory extracts the information relevant for detection within specific temporal output modes from these correlation functions . The formalism is applied to schemes for production of non-classical light states from a squeezed beam of light. \\ ( http://arXiv.org/abs/quant-ph/0602202 , 351kb) ------------------------------------------------------------------------------ \\ Paper (cross-listing): hep-th/0602236 Date: Wed, 22 Feb 2006 20:32:10 GMT (38kb) Title: Nonlocality and quantum-like behavior from classical fields Authors: B. Holdom \\ By averaging over an ensemble of field configurations, a classical field theory can display many of the characteristics of quantum field theory, including Lorentz invariance, a loop expansion, and renormalization effects. There is additional freedom in how the ensemble is chosen. When the field mode amplitudes have a Gaussian distribution, and the mode phases are randomly distributed, we review the known differences between the classical and quantum theories. When the mode amplitudes are fixed, or have a nongaussian distribution, the quartic and higher correlations among the free fields are modified, seemingly in a nonlocal way. We show how this in turn affects the perturbative expansion. We focus on $\lambda\phi^4$ theory in 1+1 dimensions and use lattice simulations to augment our study. We give examples of how these nonlocal correlations induce quantum-like behavior, at both weak and strong coupling. \\ ( http://arXiv.org/abs/hep-th/0602236 , 38kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602614 Date: Mon, 27 Feb 2006 01:34:48 GMT (374kb) Title: Phase Transitions in Ultra-Cold Two-Dimensional Bose Gases Authors: D. A. W. Hutchinson and P. B. Blakie Subj-class: Statistical Mechanics; Superconductivity \\ We briefly review the theory of Bose-Einstein condensation in the two-dimensional trapped Bose gas and, in particular the relationship to the theory of the homogeneous two-dimensional gas and the Berezinskii-Kosterlitz-Thouless phase. We obtain a phase diagram for the trapped two-dimensional gas, finding a critical temperature above which the free energy of a state with a pair of vortices of opposite circulation is lower than that for a vortex-free Bose-Einstein condensed ground state. We identify three distinct phases which are, in order of increasing temperature, a phase coherent Bose-Einstein condensate, a vortex pair plasma with fluctuating condensate phase and a thermal Bose gas. The thermal activation of vortex-antivortex pair formation is confirmed using finite-temperature classical field simulations. \\ ( http://arXiv.org/abs/cond-mat/0602614 , 374kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602616 Date: Mon, 27 Feb 2006 04:07:49 GMT (93kb) Title: Commensurate mixtures of ultra-cold atoms in one dimension Authors: L. Mathey Subj-class: Other \\ We study binary mixtures of ultra-cold atoms, confined to one dimension in an optical lattice, with commensurate densities. Within a Luttinger liquid description, which treats various mixtures on equal footing, we derive a system of renormalization group equations at second order, from which we determine the rich phase diagrams of these mixtures, which include charge/spin density wave order, singlet and triplet pairing, polaron pairing, and a supersolid phase. Various methods to detect our results experimentally are discussed. \\ ( http://arXiv.org/abs/cond-mat/0602616 , 93kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602620 Date: Mon, 27 Feb 2006 10:20:34 GMT (15kb) Title: Composite-fermion description of rotating Bose gases at low angular momenta Authors: Nicolay Korslund and Susanne Viefers Comments: 7 pages (RevTeX), 1 figure. Submitted to Phys Rev A Report-no: OSLO-TP 1-06 Subj-class: Mesoscopic Systems and Quantum Hall Effect \\ We study the composite fermion construction at and below the single vortex ($L=N$) state of weakly interacting rotating Bose gases, presenting a new method for handling the large number of derivatives typically occurring via the Slater determinant. Remarkably, the CF wave function at $L=N$ becomes {\em exact} in the large $N$ limit, even though this construction is not, {\em a priori}, expected to work in the low angular momentum regime. This implies an interesting mathematical identity which may be useful in other contexts. \\ ( http://arXiv.org/abs/cond-mat/0602620 , 15kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602622 Date: Mon, 27 Feb 2006 11:26:52 GMT (67kb) Title: Anderson localization of elementary excitations in a one dimensional Bose-Einstein condensate Authors: Nicolas Bilas and Nicolas Pavloff Subj-class: Other \\ We study the elementary excitations of a transversely confined Bose-Einstein condensate in presence of a weak axial random potential. We determine the localization length (i) at low energy, for a domain of linear densities ranging from the Tonks-Girardeau to the transverse Thomas-Fermi regime, in the case of a white noise potential and (ii) for all the range of energies, in the one-dimensional mean field regime'', in the case where the randomness is induced by a series of randomly placed point-like impurities. \\ ( http://arXiv.org/abs/cond-mat/0602622 , 67kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602626 Date: Mon, 27 Feb 2006 13:12:53 GMT (144kb) Title: Bifurcations in Resonance Widths of an Open Bose-Hubbard Dimer Authors: M. Hiller, T. Kottos, A. Ossipov Subj-class: Other \\ We investigate the structure of resonance widths of a Bose-Hubbard Dimer with intersite hopping amplitude $k$, which is coupled to continuum at one of the sites with strength $\gamma$. Using an effective non-Hermitian Hamiltonian formalism, we show that by varying the on-site interaction term $\chi$ the resonances undergo consequent bifurcations. For $\Lambda=k/\gamma\geq 0.5$, the bifurcation points follow a scaling law ${\tilde \chi}_n \equiv \chi_n N/k = f_{\Lambda}(n-0.5/\Lambda)$, where $N$ is the number of bosons. For the function $f_{\Lambda}$ two different $\Lambda$ dependences are found around the minimum and the maximum bifurcation point. \\ ( http://arXiv.org/abs/cond-mat/0602626 , 144kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602630 Date: Mon, 27 Feb 2006 14:11:14 GMT (4kb) Title: Relation between the thermodynamic Casimir effect in Bose-gas slabs and critical Casimir forces Authors: A. Gambassi and S. Dietrich Subj-class: Statistical Mechanics \\ In a recent letter, Martin and Zagrebnov [Europhys. Lett., 73 (2006) 1] discussed the thermodynamic Casimir effect for the ideal Bose gas confined in a thin film. We point out that their findings can be expressed in terms of previous general results for the Casimir effect induced by confined critical fluctuations. This highlights the links between the Casimir effect in the contexts of critical phenomena and Bose-Einstein condensation. \\ ( http://arXiv.org/abs/cond-mat/0602630 , 4kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0510214 replaced with revised version Mon, 27 Feb 2006 17:26:05 GMT (21kb) Title: Worm Algorithm for Continuous-space Path Integral Monte Carlo Simulations Authors: M. Boninsegni, N. Prokof'ev and B. Svistunov Comments: Fig. 2 differs from that of published version (includes data for larger system sizes) Subj-class: Statistical Mechanics Journal-ref: Phys. Rev. Lett. 96, 070601 (2006) \\ ( http://arXiv.org/abs/cond-mat/0510214 , 21kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0601432 replaced with revised version Fri, 24 Feb 2006 23:20:59 GMT (456kb) Title: Atomic matter of non-zero momentum Bose-Einstein condensation and orbital current order Authors: W. Vincent Liu and Congjun Wu Comments: 9 pages, 7 figures (with one in Appendix); Revised the experimental protocol for preparing the model system Subj-class: Other; Strongly Correlated Electrons \\ ( http://arXiv.org/abs/cond-mat/0601432 , 456kb) ------------------------------------------------------------------------------ \\ Paper: quant-ph/0508172 replaced with revised version Mon, 27 Feb 2006 08:51:20 GMT (668kb) Title: Cold atom dynamics in a quantum optical lattice potential Authors: Christoph Maschler, Helmut Ritsch Comments: 4 pages, 5 figures, corrected typos Journal-ref: Phys. Rev. Lett. 95, 260401 (2005) \\ ( http://arXiv.org/abs/quant-ph/0508172 , 668kb) ------------------------------------------------------------------------------ \\ Paper: physics/0602181 Date: Mon, 27 Feb 2006 14:33:25 GMT (41kb) Title: Control of Ultra-cold Inelastic Collisions by Feshbash Resonances and Quasi-One-Dimensional Confinement Authors: V. A. Yurovsky (1) and Y. B. Band (2) ((1) Tel Aviv University, Israel, (2) Ben-Gurion University, Beer-Sheva, Israel) Subj-class: Atomic Physics \\ Cold inelastic collisions of atoms or molecules are analyzed using very general arguments. In free space, the deactivation rate can be enhanced or suppressed together with the scattering length of the corresponding elastic collision via a Feshbach resonance, and by interference of deactivation of the closed and open channels. In reduced dimensional geometries, the deactivation rate decreases with decreasing collision energy and does not increase with resonant elastic scattering length. This has broad implications; e.g., stabilization of molecules in a strongly confining two-dimensional optical lattice, since collisional decay of the highly vibrationally excited states due to inelastic collisions is suppressed. The relation of our results with those based on the Lieb-Liniger model are addressed. \\ ( http://arXiv.org/abs/physics/0602181 , 41kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0602670 Date: Tue, 28 Feb 2006 15:56:12 GMT (90kb) Title: Beyond Mean-Field Theory for Attractive Bosons under Transverse Harmonic Confinement Authors: Luca Salasnich Comments: To be published in J. Phys. B.: At. Mol. Opt. Phys Subj-class: Statistical Mechanics; Other \\ We study a dilute gas of attractive bosons confined in a harmonic cylinder, i.e. under cylindric confinement due to a transverse harmonic potential. We introduce a many-body wave function which extends the Bethe ansatz proposed by McGuire (J. Math. Phys. {\bf 5}, 622 (1964)) by including a variational transverse Gaussian shape. We investigate the ground state properties of the system comparing them with the ones of the one-dimensional (1D) attractive Bose gas. We find that the gas becomes ultra 1D as a consequence of the attractive interaction: the transverse width of the Bose gas reduces by increasing the number of particles up to a critical width below which there is the collapse of the cloud. In addition, we derive a simple analytical expression for the simmetry-breaking solitonic density profile of the ground-state, which generalize the one deduced by Calogero and Degasperis (Phys. Rev. A {\bf 11}, 265 (1975)). This bright-soliton analytical solution shows near the collapse small deviations with respect to the 3D mean-field numerical solution. Finally, we show that our variational Gauss-McGuire theory is always more accurate than the McGuire theory. In addition, we prove that for small numbers of particles the Gauss-McGuire theory is more reliable than the mean-field theory described by the 3D Gross-Pitaevskii equation. \\ ( http://arXiv.org/abs/cond-mat/0602670 , 90kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0503523 replaced with revised version Tue, 28 Feb 2006 15:34:42 GMT (53kb) Title: Thermally induced rotons in two-dimensional dilute Bose gases Authors: Flavio S. Nogueira and Hagen Kleinert Comments: 21 pages; 2 figures; v4: version accepted for publication in PRB; in this version Section IV is considerably extended Subj-class: Other \\ ( http://arXiv.org/abs/cond-mat/0503523 , 53kb) ------------------------------------------------------------------------------ \\ Paper: quant-ph/0602230 Date: Tue, 28 Feb 2006 10:35:10 GMT (22kb) Title: Ground state approximation for strongly interacting systems in arbitrary dimension Authors: S. Anders, M. B. Plenio, W. D\"ur, F. Verstraete, and H.-J. Briegel \\ We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g. energy) and include states with diverging correlation length and unbounded multi-particle entanglement. As a demonstration we apply our approach to the Ising model on 1D, 2D and 3D square-lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories. \\ ( http://arXiv.org/abs/quant-ph/0602230 , 22kb) ------------------------------------------------------------------------------ \\ Paper: quant-ph/0602236 Date: Tue, 28 Feb 2006 18:49:49 GMT (14kb) Title: Quantum Revivals in Periodically Driven Systems close to nonlinear resonance Authors: Farhan Saif and Mauro Fortunato Journal-ref: Physical Review A 65, 013401 (2002) \\ We calculate the quantum revival time for a wave-packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field. The dependence of the revival time on various parameters of the driven system is shown analytically. As an example of application of our approach, we compare the analytically obtained values of the revival time for various modulation strengths with the numerically computed ones in the case of a driven gravitational cavity. We show that they are in very good agreement. \\ ( http://arXiv.org/abs/quant-ph/0602236 , 14kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0603004 Date: Tue, 28 Feb 2006 23:33:52 GMT (11kb) Title: Towards measuring Entanglement Entropies in Many Body Systems Authors: Israel Klich, Gil Refael, Alessandro Silva Subj-class: Statistical Mechanics \\ We explore the relation between entanglement entropy of quantum many body systems and the distribution of corresponding, properly selected, observables. Such a relation is necessary to actually measure the entanglement entropy. We show that in general, the Shannon entropy of the probability distribution of certain symmetry observables gives a lower bound to the entropy. In some cases this bound is saturated and directly gives the entropy. We also show other cases in which the probability distribution contains enough information to extract the entropy: we show how this is done in several examples including BEC wave functions, the Dicke model, XY spin chain and chains with strong randomness. \\ ( http://arXiv.org/abs/cond-mat/0603004 , 11kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0603011 Date: Wed, 1 Mar 2006 08:26:39 GMT (89kb) Title: BCS-BEC crossover in an asymmetric two-component Fermi gas Authors: Xia-Ji Liu and Hui Hu Comments: 5 pages + 3 figures Subj-class: Statistical Mechanics; Superconductivity \\ We discuss the superfluid phase transition of a strongly interacting Fermi gas with unequal (asymmetric) chemical potentials in two pairing hyperfine states, and map out its phase diagram near the BCS-BEC crossover. Our approach includes the fluctuation contributions of {}preformed Cooper pairs'' to the thermodynamic potential at finite temperature. We show that, below a critical difference in chemical potentials between species, a normal gas is unstable towards the formation of either a finite-momentum paired Fulde-Ferrell-Larkin-Ovchinnikov superconducting phase or a uniform superfluid, depending on the asymmetry and interaction strengths. We determine the value of critical chemical potential mismatch, and find good qualitative agreement with a recent measurement by Zwierlein \textit{et al.} {[}Science \textbf{311}, 492 (2006){]}. \\ ( http://arXiv.org/abs/cond-mat/0603011 , 89kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0603015 Date: Wed, 1 Mar 2006 09:47:39 GMT (43kb) Title: Collective modes of a quasi two-dimensional Bose condensate in large gas parameter regime Authors: S.R. Mishra, S. P. Ram, Arup Banerjee Subj-class: Other \\ We have theoretically studied the collective modes of a quasi two-dimensional (Q2D) Bose condensate in the large gas parameter regime by using a formalism which treats the interaction energy beyond the mean-field approximation. In the calculation we use the perturbative expansion for the interaction energy by incorporating the Lee, Huang and Yang (LHY) correction term. The results show that incorporation of this higher order term leads to detectable modifications in the mode frequencies. \\ ( http://arXiv.org/abs/cond-mat/0603015 , 43kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0603016 Date: Wed, 1 Mar 2006 10:15:34 GMT (22kb) Title: Feynman-Kac path integral approach for the energy spectrum of many boson systems Authors: S .Datta Subj-class: Statistical Mechanics \\ We study the ground and excited states of weakly interacting Bose gases (with positive and negative scattering lengths) in connection with Bose Einstein Condensation.. They behave as new quantum fluids (a gas in the weak limit and a liquid in the dense limit and we study their many body physics in the dilute limit within the realistic potential model (Morse type) by Feynman-Kac path integral technique. Within numerical limitations, this method is exact in principle and turns out to be a better alternative to GP as all the ground and excited state properties can be calculated in a much simpler way. \\ ( http://arXiv.org/abs/cond-mat/0603016 , 22kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0501058 replaced with revised version Wed, 1 Mar 2006 12:31:55 GMT (182kb) Title: First-principles quantum dynamics in interacting Bose gases II: stochastic gauges Authors: P. Deuar and P. D. Drummond Comments: 39 pages, 11 figures, 2 tables. Major revision, especially Section 6 (convergence issues) which is almost all new, and also introduction. Now in J phys A format Subj-class: Other Journal-ref: J. Phys. A: Math. Gen. 39 (2006) 2723-2755 \\ ( http://arXiv.org/abs/cond-mat/0501058 , 182kb) ------------------------------------------------------------------------------ \\ Paper: cond-mat/0603030 Date: Thu, 2 Mar 2006 02:37:11 GMT (19kb) Title: The Mott insulator phase of the one dimensional Bose-Hubbard model Authors: Bogdan Damski and Jakub Zakrzewski Comments: 4 pages, 4 figures, 1 table Report-no: LAUR-06-1357 Subj-class: Other \\ The one dimensional Bose-Hubbard model at a unit filling factor is considered. We do a very high order symbolic perturbative expansion in hopping (tunneling) coupling. Analytical expressions are presented for different ground state quantities: energy per site, variance of on-site occupation, correlation functions: <a_j^\dag a_{j+r}> and <n_j n_{j+r}>. Perturbative predictions are compared to numerical findings and good agreement is found in the Mott insulator phase. These results are of direct relevance to future experiments in ultra cold atomic gases placed in homogeneous lattices. \\ ( http://arXiv.org/abs/cond-mat/0603030 , 19kb) ------------------------------------------------------------------------------ \\ Paper: quant-ph/0603016 Date: Thu, 2 Mar 2006 12:38:36 GMT (255kb) Title: Quantum correlations of two optical fields close to electromagnetically induced transparency Authors: Alice Sinatra (LKB - Lhomond) Proxy: ccsd ccsd-00019947 \\ For two cavity modes exciting a $\Lambda$ transition close to electromagnetically induced transparency we show the existence of a universal dispersive bistability curve dividing the parameter space into two regions in which the system acts either as an ideal "squeezer" or a quantum non demolition device. We use a fully quantum 3-level model including cavity losses and spontaneous emission to calculate mean field intensities and quantum fluctuations as a function of the cavity length that could be directly compared with experiment. Simple analytical predictions are obtained in the good cavity limit in which the atomic variables are adiabatically eliminated. \\ ( http://arXiv.org/abs/quant-ph/0603016 , 255kb) ------------------------------------------------------------------------------ Received on Mon Mar 06 2006 - 09:30:10 EST This archive was generated by hypermail 2.2.0 : Thu May 08 2008 - 11:51:41 EST	2020-04-03 05:47:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6903996467590332, "perplexity": 6190.409374277757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370510287.30/warc/CC-MAIN-20200403030659-20200403060659-00422.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-1-section-1-7-multiplication-and-division-of-real-numbers-exercise-set-page-84/25	# Chapter 1 - Section 1.7 - Multiplication and Division of Real Numbers - Exercise Set - Page 84: 25 -72 #### Work Step by Step Multiply two numbers at a time from left to right. Two numbers with opposite signs have a negative product while two numbers with the same sign have a positive product. $(-4)(-3)(-1)(6)$ $(-4\times-3)(-1)(6)$ $(12\times-1)(6)$ $(-12)(6)=-72$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2019-10-16 05:04:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8508583307266235, "perplexity": 519.12169546634}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986664662.15/warc/CC-MAIN-20191016041344-20191016064844-00266.warc.gz"}
http://mathhelpforum.com/algebra/89553-sum-cubes-print.html	# sum of cubes • May 18th 2009, 04:58 PM VonNemo19 sum of cubes Prove that The sum of two cubes (a3 + b3 ) factors into (a + b)(a2 - ab + b2). Need a little help guys. • May 18th 2009, 05:07 PM Sodapop Quote: Originally Posted by VonNemo19 Prove that The sum of two cubes (a3 + b3 ) factors into (a + b)(a2 - ab + b2). Need a little help guys. Hope this helps you! Make sure to click on the popup to open it in a new tab. • May 18th 2009, 05:10 PM pickslides Can't you just expand $(a + b)(a^2 - ab + b^2)$ to prove this? • May 18th 2009, 05:22 PM pickslides Could be a big task but maybe.... start with $a^3+b^3$ and take out $a+b$ as a factor from both terms. $(a+b)\frac{a^3}{a+b}+(a+b)\frac{b^3}{a+b}$ $(a+b)\left(\frac{a^3}{a+b}+\frac{b^3}{a+b}\right)$ now divde and simplify the terms inside the taller brackets. • May 18th 2009, 05:29 PM VonNemo19 An just what should I divide each of those terms by, Mr. Soprano?(Wondering) • May 18th 2009, 05:32 PM VonNemo19 I like the visual, Soda pop. Good lookin' out! • May 18th 2009, 05:43 PM pickslides Quote: Originally Posted by VonNemo19 An just what should I divide each of those terms by, Mr. Soprano?(Wondering) You can call me Tony $\left(\frac{a^3}{a+b}\right) \Rightarrow a^3 \div (a+b)$ using long division also the same for $\left(\frac{b^3}{a+b}\right) \Rightarrow b^3 \div (a+b)$ • May 18th 2009, 05:51 PM VonNemo19 I don't quite understand. Doing that would change the value of the expression. And I don't understand your language either. The definition of $\Rightarrow$ is implies, or only if. Isn't that correct. If so, as I said before, I don't understand your implication. • May 18th 2009, 05:55 PM Sodapop Quote: Originally Posted by VonNemo19 I don't quite understand. Doing that would change the value of the expression. And I don't understand your language either. The definition of $\Rightarrow$ is implies, or only if. Isn't that correct. If so, as I said before, I don't understand your implication. He said long division, you can check this link if you don't remember what it is: http://en.wikipedia.org/wiki/Long_division • May 18th 2009, 06:41 PM pickslides Long division will help you (maybe! its all trial and error with proofs) make the equation into a more friendly form. • May 18th 2009, 06:55 PM VonNemo19 Do me a favor and just do it, man. Please. I tried the polynomial division and I didn't get very far. Anybody just do this. I'm trying to study right now and I know that I coukd eventually figure this out, but I shouuldn't have to. After all, this is MHF! • May 18th 2009, 07:06 PM Sodapop Quote: Originally Posted by VonNemo19 Do me a favor and just do it, man. Please. I tried the polynomial division and I didn't get very far. Anybody just do this. I'm trying to study right now and I know that I coukd eventually figure this out, but I shouuldn't have to. After all, this is MHF! As I already stated a few weeks ago to someone else, we are here to provide tips, hints and paths for you to solve and answer your own questions. We are not here to do the work for you, we can help you, yes, but you have to learn to do things on your own! Please don't consider this rude from me, as I am only refering to the forum rules. 10) Do not beg for answers. Rest of the rules can be found here: http://www.mathhelpforum.com/math-he...php?do=cfrules • May 18th 2009, 07:31 PM VonNemo19 I don't really appreciate someone who has been here less time than I have questioning my motives as to why I need certain answers to given problems. I'll have you know that I taught myself everything that I know about math in prison, and I reached the level of calculus 2. Now, someone who has not experienced what it is like to have to trudge through each page of four different texts with no formal instruction would't understand how hard that is. Now that you know a little bit about me, I'll explain why I need the information I seek. I'm studying evaluating limits algebraically right now (for the second time, but this time in college) and I ran accross the problem $\lim_{x\to{1}}\frac{x^3-1}{x-1}$. Now I no that the sum of cubes formula id the way to go here, but it occured to me that I don't know how to factor (from scratch) the sum of two cubes. Now this type of stuff bothers me Because I like to know why stuff works, not just how. Now, yes, I could go to wikipedia, do some research, or spend a while with a pencil and paper, but I have something better. MHF! So, if you don't want to help me, or give me the answers to the questions I ask, all you have to do is simply not reply to my thread. The question was stated clearly several times, and I kept recieving ambiguous (to say the least) replies. So, yes, by the fourth time I posted I was growing tired of all of this mystery. But I was not begging. With all that said, I appreciate your patriotic sentiment towards MHF, and your natural instinct to uphold all of its rules, so for that, iI thank you. • May 18th 2009, 09:32 PM Isomorphism As it was said before, the easiest way to verify an identity is to multiply it out. Do you know the distributive law? $(a^2 - ab + b^2)(a+b) = (a^3 -a^2b + ab^2) + (a^2b - ab^2 + b^3)$ Cancel $ab^2$ and $a^2b$ to get $a^3 + b^3$ • May 18th 2009, 09:45 PM Jameson Now, now children, let's all play nice. I'm just going to close this thread to avoid any more unneeded confrontation. Remember everyone that helping is a two way street. MHF does not exist to give out answers but those who need help should be given the right amount of info to work it out themselves. Let's all just focus on the common ground that we all want to pursue a higher knowledge of math.	2016-10-26 08:47:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7964596152305603, "perplexity": 812.8558767749728}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720760.76/warc/CC-MAIN-20161020183840-00432-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.bmj.com/content/318/7175/7.1?ijkey=cf7b7dfd97ea3a2fe0e04f7d0e44a09bb40c81b9&keytype2=tf_ipsecsha	Intended for healthcare professionals News # Prisoner wants to donate his second kidney BMJ 1999; 318 (Published 02 January 1999) Cite this as: BMJ 1999;318:7 1. Deborah Josefson, 2. San Francisco A convicted felon's offer to donate his remaining kidney to his ailing daughter has raised difficult ethical questions in California. David Patterson, aged 38, first donated a kidney to his estranged daughter, Renada Daniel Patterson, in 1996. Renada was born with congenital renal anomalies and received a cadaveric renal transplant at age 5. When that kidney was rejected, her father's kidney was found to match and was transplanted. But now, that kidney too is failing. David Patterson, who is serving a 13 year term for burglary and heroin possession, wants to replace Renada's failing kidney with his remaining one. The unprecedented offer challenges conventional medical ethics and pits a family's wishes against the rule of medicine: “first do no harm.” If Patterson were to donate his remaining kidney, he too would become dependent on dialysis and a candidate for transplantation. It would essentially shorten his life and amount to trading one life for another. Dialysis is painful, complicated, time consuming, and expensive--about $50000 ($31000) per year. If Patterson were to donate his remaining kidney, the government would pay for his dialysis as long as he was in prison. Medicaid would likely foot the bill thereafter. Surgeons at the University of California, San Francisco (UCSF) /Stanford Health Care, where Renada is a patient, have refused to consider removing her father's remaining kidney for transplantation. The family contends that the decision is a private family and spiritual matter in which hospital officials have no right to interfere. Vickie Daniel, Renada's mother, stated: “This is not about ethics. It's a family matter about what's best for this child. We've made a decision for our child and we want it to stand.” The case has been referred to the hospital's ethics committee and is receiving national attention. Bob Spieldenner of the United Network for Organ Sharing (UNOS), the agency that coordinates organ transplant registries in the United States, said: “I haven't heard of such a case before, and I don't know of any surgeon who would do it. UNOS doesn't have a policy or position on it. It's a difficult decision and hasn't yet been looked at by our ethics committee, but probably will be in the near future.” Meanwhile, the urgency of the case has abated somewhat since Renada's condition has improved considerably. Moreover, it is possible that Renada has developed antibodies to her father's kidneys, which would render him an unsuitable match. Renada, now 16, was admitted to hospital with an acute rejection episode, marked by nausea, vomiting, and dehydration. Dr Donald Potter, her nephrologist, described her condition as “stable” but emphasised that she has moderate to severe chronic rejection and is likely to require dialysis or a transplant within 3 to 4 months. “The results of recent tests show that the kidney rejection episode is responding to treatment. But this is not a cure.” View Abstract	2022-07-06 20:26:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2261902093887329, "perplexity": 8609.611415310279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104676086.90/warc/CC-MAIN-20220706182237-20220706212237-00569.warc.gz"}
https://stats.stackexchange.com/questions/288614/another-question-regarding-glm-with-aggregated-data-in-r	# Another question regarding GLM with aggregated data in R My question is similar to this one GLM with grouped/aggregated data in R, but I have a specific question regarding my data set. I have included below an aggregated data set of frequency of death by gender and age group from 2005 to 2015. structure(list(age_9 = structure(c(1L, 1L, 2L, 2L, 3L, 3L, 4L,4L, 5L, 5L, 6L, 6L, 7L, 7L, 8L, 8L, 9L, 9L), .Label = c("0 - 14 years", "15 - 24 years","25 - 34 years", "35 - 44 years","45 - 54 years", 55 - 64 years", "65 - 74 years "75 - 84 years", "85+"),class = "factor"), = structure(c(1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L), .Label = c("Male", "Female" ), class = "factor"), count = c(6, 4, 7, 5, 15, 15, 18, 35, 39, 62, 71, 79, 71,89, 83, 120, 66, 123), pop = c(336085, 316508, 266001, 256539, 220505, 233310, 219358, 239771, 220639, 197750, 236558, 155796, 188762, 77920, 104571, 26234, 44487)), .Names = c("age_9", "sex", "count", "pop"), row.names = c(NA, -18L), vars = "age_9", drop = TRUE,class = c("grouped_df", "tbl_df","tbl", "data.frame")) Let's say I want to use this data to test if females had higher risk of dying than males accounting for age groups. Based on the document (pdf) linked in the referenced original question I think I should use a generalized linear model on my aggregated data for my specific question, but I'm not sure: glm(count ~ sex + age, family=poisson(link=log), offset=log(?), data=A2) My questions are: 1. Is adequate to formulate the above GLM with the included aggregated data set? 2. if no, why? 3. If yes to question 1, What would be an appropriate offset vector? • Your data appear to supply no information about risk of death. To assess that, you would also need to know how many males and females were in each group originally--that is, the numbers at risk of dying. What are these data actually counting? – whuber Jul 3 '17 at 19:04 • @whuber, thanks for your response. The data is counting the number of death within each groups from 2005 to 2015. Based on your response, I edited the data set to included a population vector. With this new information is adequate to conduct the following glm? glm(count ~ sex + age_9, family = "poisson", offset = log(pop), data = A1) – José Jul 3 '17 at 19:41 • It appears adequate. The exercise seems like it underutilizes the data, though. Obviously there are huge differences in rates between males and females from one age group to another. A model that posits one underlying rate for each gender would not be able to account for this important phenomenon. If your purpose is to study these data (and not just to test the software), consider a deeper exploration that includes (a) visualizing the data and (b) accommodating this age-gender interaction. – whuber Jul 3 '17 at 19:47 • whuber, Thank you. Your response is very useful to me. One of the major limitations that I am confronting is that I have other variables within the mortality data, that I can aggregate by, but do not find the respective population estimates. That was the main reason I wanted to explore if it was possible to use another offset vector to perform the glm. – José Jul 3 '17 at 20:43	2019-07-18 10:17:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.349581241607666, "perplexity": 2225.7761154884915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525587.2/warc/CC-MAIN-20190718083839-20190718105839-00454.warc.gz"}
https://rdrr.io/cran/phaseR/src/R/flowField.R	# R/flowField.R In phaseR: Phase Plane Analysis of One and Two Dimensional Autonomous ODE Systems #### Documented in flowField #' Flow Field #' #' Plots the flow or velocity field for a one or two dimensional autonomous ODE #' system. #' #' @param deriv A function computing the derivative at a point for the ODE #' system to be analysed. Discussion of the required format of these functions #' can be found in the package guide. #' @param xlim In the case of a two dimensional system, this sets the limits of #' the first dependent variable in which gradient reflecting line segments #' should be plotted. In the case of a one dimensional system, this sets the #' limits of the independent variable in which these line segments should be #' plotted. Should be a vector of length two. #' @param ylim In the case of a two dimensional system this sets the limits of #' the second dependent variable in which gradient reflecting line segments #' should be plotted. In the case of a one variable system, this sets the #' limits of the dependent variable in which these line segments should be #' plotted. Should be a vector of length two. #' @param parameters Parameters of the ODE system, to be passed to deriv. #' Supplied as a vector; the order of the parameters can be found from the #' deriv file. Defaults to NULL. #' @param points Sets the density of the line segments to be plotted. points #' segments will be plotted in the x and y directions. Fine tuning here, by #' shifting points up and down, allows for the creation of more aesthetically #' pleasing plots. Defaults to 11. #' @param system Set to either "one.dim" or "two.dim" to indicate the type of #' system being analysed. Defaults to "two.dim". #' @param col Sets the colour of the plotted line segments. Should be a vector #' of length one. Will be reset accordingly if it is a vector of the wrong #' length. Defaults to "gray". #' @param arrow.type Sets the type of line segments plotted. If set to #' "proportional" the length of the line segments reflects the magnitude of the #' derivative. If set to "equal" the line segments take equal lengths, simply #' reflecting the gradient of the derivative(s). Defaults to "equal". #' @param arrow.head Sets the length of the arrow heads. Passed to arrows. #' Defaults to 0.05. #' @param frac Sets the fraction of the theoretical maximum length line #' segments can take without overlapping, that they can actually attain. In #' practice, frac can be set to greater than 1 without line segments #' overlapping. Fine tuning here assists the creation of aesthetically pleasing #' plots. Defaults to 1. #' @param add Logical. If TRUE, the flow field is added to an existing plot. If #' FALSE, a new plot is created. Defaults to TRUE. #' @param xlab Label for the x-axis of the resulting plot. Defaults to "x". #' @param ylab Label for the y-axis of the resulting plot. Defaults to "y". #' @param \dots Additional arguments to be passed to either plot or arrows. #' @inheritParams .paramDummy #' #' @return Returns a list with the following components (the exact make up is #' dependent upon the value of system): \item{add}{As per input.} #' \item{arrow.head}{As per input.} \item{arrow.type}{As per input.} #' \item{col}{As per input, but with possible editing if a vector of the wrong #' length was supplied.} \item{deriv}{As per input.} \item{dx}{A matrix. In the #' case of a two dimensional system, the values of the derivative of the first #' dependent derivative at all evaluated points.} \item{dy}{A matrix. In the #' case of a two dimensional system, the values of the derivative of the second #' dependent variable at all evaluated points. In the case of a one dimensional #' system, the values of the derivative of the dependent variable at all #' evaluated points.} \item{frac}{As per input.} \item{parameters}{As per #' input.} \item{points}{As per input.} \item{system}{As per input.} \item{x}{A #' vector. In the case of a two dimensional system, the values of the first #' dependent variable at which the derivatives were computed. In the case of a #' one dimensional system, the values of the independent variable at which the #' derivatives were computed.} \item{xlab}{As per input.} \item{xlim}{As per #' input.} \item{y}{A vector. In the case of a two dimensional system, the #' values of the second dependent variable at which the derivatives were #' computed. In the case of a one dimensional system, the values of the #' dependent variable at which the derivatives were computed.} \item{ylab}{As #' per input.} \item{ylim}{As per input.} #' @author Michael J. Grayling #' @export #' @examples #' # Plot the flow field, nullclines and several trajectories for the one #' # dimensional autonomous ODE system logistic. #' logistic.flowField <- flowField(logistic, xlim = c(0, 5), ylim = c(-1, 3), #' parameters = c(1, 2), points = 21, system = "one.dim", #' logistic.nullclines <- nullclines(logistic, xlim = c(0, 5), ylim = c(-1, 3), #' parameters = c(1, 2), system = "one.dim") #' logistic.trajectory <- trajectory(logistic, y0 = c(-0.5, 0.5, 1.5, 2.5), tlim = c(0, 5), #' parameters = c(1, 2), system = "one.dim") #' #' # Plot the velocity field, nullclines and several trajectories for the two dimensional #' # autonomous ODE system simplePendulum. #' simplePendulum.flowField <- flowField(simplePendulum, xlim = c(-7, 7), #' ylim = c(-7, 7), parameters = 5, points = 19, #' y0 <- matrix(c(0, 1, 0, 4, -6, 1, 5, 0.5, 0, -3), ncol = 2, #' nrow = 5, byrow = TRUE) #' simplePendulum.nullclines <- nullclines(simplePendulum, xlim = c(-7, 7), #' ylim = c(-7, 7), parameters = 5, points = 500) #' simplePendulum.trajectory <- trajectory(simplePendulum, y0 = y0, tlim = c(0, 10), #' parameters = 5) #' flowField <- function(deriv, xlim, ylim, parameters = NULL, system = "two.dim", points = 21, col = "gray", arrow.type = "equal", arrow.head = 0.05, frac = 1, add = TRUE, xlab = "x", ylab = "y", state.names = c("x", "y"), ...){ if ((!is.vector(xlim)) \| (length(xlim) != 2)){ stop("xlim is not a vector of length 2 as required") } if (xlim[2] <= xlim[1]){ stop("xlim[2] is less than or equal to xlim[1]") } if ((!is.vector(ylim)) \| (length(ylim) != 2)){ stop("ylim is not a vector of length 2 as required") } if (ylim[2] <= ylim[1]){ stop("ylim[2] is less than or equal to ylim[1]") } if (points <= 0) { stop("points is less than or equal to zero") } if (!(system %in% c("one.dim", "two.dim"))){ stop("system must either be set to \"one.dim\" or \"two.dim\"") } if (is.vector(col) == FALSE){ stop("col is not a vector as required") } if (length(col) > 1){ col <- col[1] print("Note: col has been reset as required") } if (!(arrow.type %in% c("proportional", "equal"))){ stop("arrow.type must either be set to \"proportional\" or \"equal\"") } stop("arrow.head is less than or equal to zero") } if (frac <= 0){ stop("frac is less than or equal to zero") } } x <- seq(from = xlim[1], to = xlim[2], length = points) y <- seq(from = ylim[1], to = ylim[2], length = points) dx <- matrix(0, ncol = points, nrow = points) dy <- matrix(0, ncol = points, nrow = points) xmax.length <- x[2] - x[1] ymax.length <- y[2] - y[1] plot(1, xlim = c(xlim[1] - xmax.length, xlim[2] + xmax.length), ylim = c(ylim[1] - ymax.length, ylim[2] + ymax.length), type = "n", xlab = xlab, ylab = ylab, ...) } if (system == "one.dim"){ for (i in 1:points){ dy[1, i] <- deriv(0, setNames(c(y[i]), state.names[1]), parameters)[[1]] } for (i in 2:points){ dy[i, ] <- dy[1, ] } abs.dy <- abs(dy) abs.dy.non <- abs.dy[which(abs.dy != 0)] max.abs.dy <- max(abs(dy)) coefficient <- fracmin(xmax.length, ymax.length)/ (2sqrt(2)max(sqrt(2abs.dy.non/ (abs.dy.non + (1/abs.dy.non))), sqrt(2(1/abs.dy.non)/ (abs.dy.non + (1/abs.dy.non))))) for (i in 1:points){ for (j in 1:points){ if (dy[i, j] != 0){ factor <- sqrt(2/(abs.dy[i, j] + (1/abs.dy[i, j]))) y.shift <- coefficientfactorsqrt(abs.dy[i, j]) x.shift <- coefficientfactor/sqrt(abs.dy[i, j]) if (dy[i, j] < 0){ y.shift <- -y.shift } } if (dy[i, j] == 0){ y.shift <- 0 x.shift <- coefficientsqrt(2) } if (arrow.type == "proportional"){ if (dy[i, j] != 0){ prop <- abs.dy[i, j]/max.abs.dy y.shift <- y.shiftprop x.shift <- x.shiftprop } if (dy[i, j] == 0) { x.shift <- y.shiftmean(abs.dy)/max.abs.dy } } arrows(x[i] - x.shift, y[j] - y.shift, x[i] + x.shift, y[j] + y.shift, length = arrow.head, col = col, ...) } } deriv = deriv, dy = dy, frac = frac, parameters = parameters, points = points, system = system, x = x, xlab = xlab, xlim = xlim, y = y, ylab = ylab, ylim = ylim)) } else { for (i in 1:length(x)){ for (j in 1:length(y)){ df <- deriv(0, setNames(c(x[i], y[j]), state.names), parameters) dx[i, j] <- df[[1]][1] dy[i, j] <- df[[1]][2] } } abs.dx <- abs(dx) abs.dy <- abs(dy) abs.dx.non <- abs.dx[which((abs.dx != 0) & (abs.dy != 0))] abs.dy.non <- abs.dy[which((abs.dx != 0) & (abs.dy != 0))] max.length <- max(sqrt(dx^2 + dy^2)) coefficient <- fracmin(xmax.length, ymax.length)/ (2sqrt(2)max(sqrt(2(abs.dy.non/abs.dx.non)/ ((abs.dy.non/abs.dx.non) + (abs.dx.non/abs.dy.non))), sqrt(2(abs.dx.non/abs.dy.non)/ ((abs.dy.non/abs.dx.non) + (abs.dx.non/abs.dy.non))))) for (i in 1:points){ for (j in 1:points){ if ((dx[i, j] != 0) \| (dy[i, j] != 0)){ if ((dx[i, j] != 0) & (dy[i, j] != 0)){ factor <- sqrt(2/((abs.dy[i, j]/abs.dx[i, j]) + (abs.dx[i, j]/abs.dy[i, j]))) y.shift <- coefficientfactorsqrt(abs.dy[i, j]/abs.dx[i, j]) x.shift <- coefficientfactor/sqrt(abs.dy[i, j]/abs.dx[i, j]) if (dy[i, j] < 0){ y.shift <- -abs(y.shift) } if (dx[i, j] < 0){ x.shift <- -abs(x.shift) } } if ((dx[i, j] == 0) & (dy[i, j] != 0)){ y.shift <- coefficientsqrt(2) x.shift <- 0 if (dy[i, j] < 0){ y.shift <- -abs(y.shift) } } if ((dx[i, j] != 0) & (dy[i, j] == 0)){ y.shift <- 0 x.shift <- coefficientsqrt(2) if (dx[i, j] < 0){ x.shift <- -abs(x.shift) } } if (arrow.type == "proportional"){ prop <- sqrt((abs.dx[i, j]^2 + abs.dy[i, j]^2))/max.length y.shift <- y.shiftprop x.shift <- x.shiftprop } arrows(x[i] - x.shift, y[j] - y.shift, x[i] + x.shift, y[j] + y.shift, length = arrow.head, col = col, ...) } } } }	2018-12-15 21:56:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40182432532310486, "perplexity": 14711.691926953445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827097.43/warc/CC-MAIN-20181215200626-20181215222626-00422.warc.gz"}
http://www.participatorymuseum.org/r6radmy/lightest-alkaline-earth-metal-6f52d3	## lightest alkaline earth metal 3. The alpha particles released by radioactive decay of radium atoms react with atoms of beryllium to give, among the products, neutrons with a wide range of energies—up to about 5 × 106 electron volts (eV). The periodic table is made up of 118 elements. Calcium and barium are weakly radioactive: calcium contains about 0.1874% cacium-41, and barium contains about 0.1062% barium-130. 3. (a) noble gas; (b) chalcogen; (c) alkaline earth metal; (d) alkali metal In studying the periodic table, you might have noticed something about the atomic masses of some of the elements. Let us know if you have suggestions to improve this article (requires login). 012 Magnesium Cylinder. Ba + I 2 → BaI 2. Answer. What alkaline-earth metal has the lightest atoms? 4. 3. The oxygen compound beryllium oxide (beryllia, BeO) is a high-temperature refractory material (melting point 2,530 °C [4,586 °F]) characterized by an unusual combination of high electrical resistance and dielectric strength with high thermal conductivity. ALKALI METALS Found in group 1 of the periodic table … Phenomenological models based on the difference of the electronegativities imply the ionic character of about 20-30 for the investigated molecules, except ones involving the lightest alkaline-earth-metal atoms [ ] . Because of their reactivity, the alkaline earth metals are not found free in nature. Like all alkali metals it has a single valence electron that it readily gives up to form a cation or compound. Basic beryllium carbonate, BeCO3∙xBe(OH)2, precipitated from ammonia (NH3) and carbon dioxide (CO2), along with basic beryllium acetate, Be4O(C2H3O2)6, are used as a starting material for synthesis of beryllium salts. Alkaline Earth Metals Also called as Group II A elements. Calcium is the fifth more prevalent element in the human body. Interesting Facts about Alkaline Earth Metals They are called alkaline because they form solutions with a pH greater than 7, making them bases or "alkaline." Both the finely divided metal and soluble compounds in the form of solutions, dry dust, or fumes are toxic; they may produce dermatitis or, when inhaled, a hypersensitivity to beryllium. Jump To. There is approximately one kilogram of calcium in an average-sized human skeleton. The most lightest alkaline earth metal is: View solution. 2. Like all alkali metals, lithium is highly reactive and flammable, and is stored in mineral oil. Be is the lightest of the alkaline earth metals so definitely its oxide has greater percentage of oxygen (63.97%) than others. Beryllium chloride (BeCl2) catalyzes the Friedel-Crafts reaction and is used in cell baths for electrowinning or electrorefining beryllium. The method used by both men was reduction of the chloride by the potent “new” reductant, potassium: Equation 21.21 BeCl2(s) +2K(s)→ΔBe(s) +2KCl(s) Use the periodic table to give the name and symbol for each of the following elements: a) the noble gas in the same period as germanium . Aside from being very light, it is known for being very stiff. Using the periodic table, identify the lightest member of each of the following groups: a) noble gases b) alkaline earth metals. And it's also the lightest earth metal. They are called alkaline earth metals because they form alkaline solutions (hydroxides) when they react with water . 2. Previous question Next question Transcribed Image Text from this Question. How to remember the Alkali Metals ? In this work, we have performed a comparative study of the structures, stabilities, and properties of some alkaline earth metal oxides (, , , and ). Elements can be classified as metals, metalloids, and nonmetals, or as a main-group elements, transition metals, and inner transition metals. Lithium, the lightest of the alkali metals, is the only alkali metal which reacts with nitrogen at standard conditions, and its nitride is the only stable alkali metal nitride. Lithium metal floating on paraffin oil. In the US, it is considered as a strategic and critical element as it is used in developing products critical to national security. Flashcard Content Overview. Z = 19 K metal Z = 12 Mg metal Z = 32 Ge metalloid. Treatment with acids, roasting with complex fluorides, and liquid-liquid extraction have all been employed to concentrate beryllium in the form of its hydroxide. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.9b01058. Their half-lives range from 1.5 million years (for beryllium-10, which undergoes beta decay) to 6.7 × 10−17 second for beryllium-8 (which decays by two-proton emission). The alkali metals are solids at room temperature (except for hydrogen), but have fairly low melting points: lithium melts at 181ºC, sodium at 98ºC, potassium at 63ºC, rubidium at 39ºC, and cesium at 28ºC. The alkaline earth metals are six chemical elements in group 2 of the periodic table.They are beryllium (Be), magnesium (Mg), calcium (Ca), strontium (Sr), barium (Ba), and radium (Ra). Source(s): lightest alkaline earth metal: https://biturl.im/YP6DD. Topics. Phenomenological models based on the difference of the electronegativities imply the ionic character of about 20-30 for the investigated molecules, except ones involving the lightest alkaline-earth-metal atoms [ ] . 5. The first ionization energies of alkaline earth metals are higher than those of the alkali metals. It is also mildly reactive, but it does not react with water. Beryllium is a steel-gray metal that is quite brittle at room temperature, and its chemical properties somewhat resemble those of aluminum. a. Beryllium (Be), the lightest alkaline earth metal, was first obtained in 1828 by Friedrich Wöhler in Germany and simultaneously by Antoine Bussy in France. Beryllium is the only stable light metal with a relatively high melting point. ALKALINE EARTH METALS Also called as Group II A elements. Alkali Metals. K - Potassium. Beryllium is the chemical element with the symbol Be and atomic number 4. Element A burns in nitrogen to give an ionic compound B. 0 0. Beryllium is fabricated into gyroscopes, accelerometers, and computer parts for inertial guidance instruments and other devices for missiles, aircraft, and space vehicles, and it is used for heavy-duty brake drums and similar applications in which a good heat sink is important. The name "alkaline earths" comes from an old name for the oxides of the elements. Can you use hair straightening brush on wet hair? Using the periodic table, identify the heaviest member of each of the following groups: (a) alkali metals (b) chalcogens (c) noble gases Page 12 of 18 Answer a. Assertion Alkaline earth metals have lower melting points than alkali metals. Alternatively, the hydroxide can be heated to form the oxide, which in turn can be treated with carbon and chlorine to form beryllium chloride; electrolysis of the molten chloride is then used to produce the metal. Beryllium, magnesium, calcium, strontium, barium and radium are all shiny, and silvery-white. All alkaline earth elements have an oxidation number of +2, making them very reactive. How many alkaline metals are there? Beryl and bertrandite have been found in sufficient quantities to constitute commercial ores from which beryllium hydroxide or beryllium oxide is industrially produced. All the alkaline earth metals have two ... Calcium-48 is the lightest nuclide to undergo double beta decay. Since then beryllium mixed with an alpha emitter such as radium, plutonium, or americium has been used as a neutron source. The trend of increasing reactivity within the group is shown by the way the alkaline earth metals behave in the presence of water. Using the periodic table, identify the lightest member of each of the following groups: (a) noble gases (b) alkaline earth metals (c) alkali metals (d) chalcogens. Text Solution. 4. Group 2A: The Alkaline Earth Metals. What is the lightest alkaline earth metal? Like the other alkaline earth metals, it is a reactive metal that slowly tarnishes in air. These chemical properties, coupled with its excellent electrical conductivity, high heat capacity and conductivity, good mechanical properties at elevated temperatures, and very high modulus of elasticity (one-third greater than that of steel), make it valuable for structural and thermal applications. Alkaline Earth Metals. Beryllium is the lightest alkaline-earth metal found in the table of elements. Be. Even at high and low temperatures, it can retain its original shape. In this paper, chemical storage materials of hydrate of alkaline earth metals at low temperatures are rported by differential scanning calorimetry (DSC). Like the alkali metals, the group 2A elements are all solids with typical metallic properties, some of which are listed in Table 7.5. What are the names of Santa's 12 reindeers? Solve Easy, Medium, and Difficult level questions from Group 2 Elements Alkaline Earth Metals In chemistry, a metal is an element that readily forms positive ions (cations) and has metallic bonds. Featured. Using the periodic table, identify the heaviest member of each of the following groups: (a) alkali metals (b) chalcogens (c) noble gases Page 12 of 18 Calcium , atomic number 20, is the lightest alkaline earth metal, with a density of 1.55 g/cm 3 , lighter than both beryllium and calcium . Beryllium does not react with either water or steam, even when heated red-hot. (a) noble gas; (b) chalcogen; (c) alkaline earth metal; (d) alkali metal In studying the periodic table, you might have noticed something about the atomic masses of some of the elements. Interesting Facts about Alkaline Earth Metals. Like all alkali metals, lithium is highly reactive and flammable, and is stored in mineral oil. 118 Names and Symbols of the Periodic Table Quiz. Lv 4. © AskingLot.com LTD 2021 All Rights Reserved. Magnesium alloys with aluminum provide light weight and sturdy materials for airplanes, missiles, and rockets. All of the alkaline earth metals except magnesium and strontium have at least one naturally occurring radioisotope. The hydroxide is converted to fluoride via ammonium beryllium fluoride and then heated with magnesium to form elemental beryllium. They are highly … Beryllium has an exclusive +2 oxidation state in all of its compounds. ARTICLE SECTIONS. 1. 1 0. Be. Additionally, what is unique about alkaline earth metals? Ring in the new year with a Britannica Membership. Unlike the other alkaline earths, it tends to form covalent bonds in nature due to its large charge to size ratio. As with all metals, the alkali metals are malleable, ductile, and are good conductors of heat and electricity. C. Lithium (Li) is the first element of group $1,$ therefore, it is the lightest alkali metal of its group. Ca + Cl 2 → CaCl 2. Element Category: Alkaline earth metal . Expert Answer . Calcium and magnesium are important for animal and plant life. Alkaline Earths. In this quiz you’ll be shown all 118 chemical symbols, and you’ll need to choose the name of the chemical element that each one represents. One useful exception is the heat-treatable alloy beryllium-copper. The lightest alkaline metal is beryllium (Be) with an atomic number of 4 and an atomic mass of 9.012182. How well do you know their symbols? They are silvery or gray in color. Beryllium (Be), the lightest alkaline earth metal, was first obtained in 1828 by Friedrich Wöhler in Germany and simultaneously by Antoine Bussy in France. Aside from being very light, it is known for being very stiff. Beryllium : Chemical element, lightest of the alkaline earth metals, chemical symbol Be, atomic number 4. '. At room temperature lithium is a soft metal that is silvery-white in color. Radium is formed from the decay of uranium. Like the other alkaline earth metals, it is a reactive metal that slowly tarnishes in air. This reactivity is due to high electronegativity and high effective nuclear charge. Magnesium and calcium are the only abundant alkaline earth metals in the Earth’s crust. Even at high and low temperatures, it can retain its original shape. We use these compounds to highlight the differences in the organometallic chemistry of the lightest alkaline earth metals, magnesium and beryllium, in an otherwise identical chemical environment. Beryllium itself does not reduce sparking, but it strengthens the copper (by a factor of 6), which does not form sparks upon impact. Beryllium is the lightest of the alkaline earth metals. They are also relatively soft and low in density. Lightest alkaline earth metal is 1.4k LIKES. 3Mg + 2NH 3 → Mg 3 N 2 + H 2. Beryllium has one of the highest melting points of the light metals. Beryllium. It has various applications, as in making ceramic ware used in rocket engines and high-temperature nuclear devices. Alkaline Earth Metals. Get your answers by asking now. Compound B reacts with water to give C and D. A solution of C becomes 'milky' on bubbling carbon dioxide. The alkaline earths possess many of the characteristic properties of metals.Alkaline earths have low electron affinities and low electronegativities.As with the alkali metals, the properties depend on the ease with which electrons are lost.The alkaline earths have two electrons in the outer shell. Related Video. It has many functions in the body, but mostly it is used to provide support for the skeleton. Alkali metals are in group IA on the far left side of the periodic table. Fr - Francium. https://www.britannica.com/science/beryllium, beryllium - Student Encyclopedia (Ages 11 and up). Alkaline and Alkaline Earth Metals 1. This rare element is the lightest of the alkaline earth metals. Halogens can gain an electron by reacting with atoms of other elements. Anonymous. Alkaline earth metal. Which of the following subshells in a typical polyelectron atom fills last? Discoverer: Vauquelin, Nicholas Louis. Rb - Rubidium. The alkaline earth metals (beryllium, magnesium, calcium, strontium, barium, ... Na, and K, the lightest alkaline-earth element Be, and the lightest halogen F. They are used to define the volatility trend of the bulk Earth. Originally Answered: Why are group 2 elements called alkaline earth metals? This reaction is a exothermic reaction. Magnesium is the lightest of the metals that can be safely handled, being less than two thirds as dense as aluminium. D. Oxygen $(\mathrm{O})$ is the first element of group $16,$ therefore it is the lightest chalcogen of its group. The alkali metals have the silver-like lustre, high ductility, and excellent conductivity of electricity and heat generally associated with metals. Nitrogen is an unreactive gas because breaking the strong triple bond in the dinitrogen molecule (N 2 ) requires a lot of energy. Alkaline earth metals react with halogens and form halides. What is internal and external criticism of historical sources? The alkali metals have low melting points, ranging from a high of 179 °C (354 °F) for lithium to a low of 28.5 °C (83.3 °F) for cesium. Important Questions on Group 2 Elements Alkaline Earth Metals is available on Toppr. b) the alkaline earth metal in the same period as selenium. Beside this, what is the lightest alkali metal? Beryllium. Its cosmic abundance is 20 on the scale in which silicon, the standard, is 1,000,000. The decay of beryllium-7 (53.2-day half-life) in the Sun is the source of observed solar neutrinos. Because of its low atomic weight, beryllium transmits X-rays 17 times as well as aluminum and has been extensively used in making windows for X-ray tubes. Much beryllium is used as a low-percentage component of hard alloys, especially with copper as the main constituent but also with nickel- and iron-based alloys, for products such as springs. Beryllium (Be), Calcium (Ca), Strontium (Sr), Barium (Ba) , and Radium (Ra) 14. The elements have very similar properties: they are all shiny, silvery-white, somewhat reactive metals at standard temperature and pressure. The alkaline earth metals (beryllium, magnesium, calcium, strontium, barium, and radium) are the second most reactive metals in the periodic table (Table 3.7), and, like the Group 1 metals, have increasing reactivity in the higher periods. Metallic hydrogen has been predicted at extremely high pressure (e.g., interior of Jupiter), under which compressed hydrogen atoms lose their electrons to delocalization ("sea of electrons"). Answer c. Li. Also to know is, what is the heaviest alkaline earth metal? He. Emerald is now known to be a green variety of beryl. Give the atomic symbol of the elements with the following Z value, and classify each as a metal, metalloid, or nonmetal: a) Z = 19; b) Z = 12; c) Z = 32. The alkaline earth elements are metallic elements found in the second group of the periodic table. Be. 1 decade ago. Beryllium is a slightly toxic, silvery-gray metal. M agnesium is a chemical element with symbol Mg and atomic number 12. 6.941 amu. 4d c. 4s d. 5p e. More information is needed. Bussy by the reduction of its chloride with potassium. Among people who work with beryllium, exposure can lead to berylliosis (also called chronic beryllium disease [CBD]), characterized by decreased lung capacity and effects similar to those caused by the poison gas phosgene. A. Al B. Li C.Na C D Be E. None Of These Are Alkaline Earth Metals. Ammonia and alkaline earth metals. Atoms, Molecules and Ions. 0 0. aspr@mumbai. Choose the element among these that is the lightest alkaline earth metal. Under standard conditions, it is the lightest metal and the lightest solid element. Uses of Alkaline Earth Compounds Since magnesium burns brightly, it is used in flares and fireworks. It is very radioactive and is dangerous to handle. Supporting Information . Small amounts of beryllium added to oxidizable metals generate protecting surface films, reducing inflammability in magnesium and tarnishing in silver alloys. Scheele called the element "dephlogisticated muriatic acid", which is how chlorine was known for 33 years. The elements are beryllium (Be), magnesium (Mg), calcium (Ca), strontium (Sr), barium (Ba), and radium (Ra).…, Unlike many kinds of steel, most copper alloys are not susceptible to improvements of hardness and strength by processes of heat treatment. Historically important examples of the use of beryllium/radium neutron sources include the bombardment of uranium by German chemists Otto Hahn and Fritz Strassmann and Austrian-born physicist Lise Meitner, which led to the discovery of nuclear fission (1939), and the triggering in uranium of the first controlled-fission chain reaction by Italian-born physicist Enrico Fermi (1942). Beryllium-copper (2 percent beryllium) is made into tools for use when sparking might be dangerous, as in powder factories. Lithium is the lightest metallic element. Li Be Na Mg K Ca Rb Sr Cs Ba Fr Ra ALKALI AND ALKALINE EARTH METALS 4. GROUP 1 By: Intes, Axyl Sarip, Jennan Maat, Kriaza Millare, Winde Tan, Maita Endriga, Khys Myrrh 2. 39. (The precious forms of beryl, emerald and aquamarine, have a composition closely approaching that given above, but industrial ores contain less beryllium; most beryl is obtained as a by-product of other mining operations, with the larger crystals being picked out by hand.) Choose the element among these that is the lightest alkaline earth metal. The alkaline earth metals are found in group IIA of the periodic table, which is the second column of elements. Beryllium is widely distributed in Earth’s crust and is estimated to occur in Earth’s igneous rocks to the extent of 0.0002 percent. Magnesium is the lightest of the metals that can be safely handled, being less than two thirds as dense as aluminium. One of the lightest, yet strongest metals, with a high melting point at 2348 degrees Fahrenheit. What is alkaline earth metal in chemistry? Alkaline earth metals react vigorously with halogens, forming metal halides with the general formula. What is another name for the rare earth elements. Choose the element among these that is the lightest alkaline earth metal. Moreover, it has high resistance to both heat and corrosion and can be machined to very close tolerances. There are about 30 recognized minerals containing beryllium, including beryl (Al2Be3Si6O18, a beryllium aluminum silicate), bertrandite (Be4Si2O7(OH)2, a beryllium silicate), phenakite (Be2SiO4), and chrysoberyl (BeAl2O4). 39. They are generally colourless and have a distinctly sweet taste, whence came the element’s former name glucinium. They are beryllium (Be), magnesium (Mg), calcium (Ca), strontium (Sr), barium (Ba), and radium (Ra). Updates? Alkaline Earth metals are very reactive because they readily give up their two valence electrons to achieve a full outer energy level, which is the most stable arrangement of electrons. In 1807, Humphry Davy investigated chlorine and discovered that it is an actual element. Uses include x-ray tubes and aerospace construction. Several antacids use magnesium hydroxide to neutralize excess stomach acid. Using the periodic table, identify the heaviest member of each of the following groups: (a) alkali metals (b) chalcogens (c) noble gases (d) alkaline earth metals. Beryllium was discovered (1798) as the oxide by French chemist Nicolas-Louis Vauquelin in beryl and in emeralds and was isolated (1828) as the metal independently by German chemist Friedrich Wöhler and French chemist Antoine A.B. Still have questions? 3. The United States has about 60 percent of the world’s beryllium and is by far the largest producer of beryllium; other major producing countries include China, Mozambique, and Brazil. The oxides of the alkaline-earth metals are basic (i.e., alkaline, in contrast to acidic). Beryllium. The lightest of alkaline earth metals is Beryllium. Lithium is part of the alkali metal group and can be found in the first column of the periodic table right below hydrogen. Beryllium. This is because: View solution. 1 answer. By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. The only naturally occurring isotope is the stable beryllium-9, although 11 other synthetic isotopes are known. Are malleable, ductile, and rockets and external criticism of historical sources metal in! Light weight and sturdy materials for airplanes, missiles, and excellent conductivity of electricity and heat generally associated metals! Bottom two rows of the alkaline earth metal atoms have a distinctly taste... Answers Glossary beryllium does not react with halogens, forming metal halides with the alkali metals water or,... Increasing reactivity within the group is shown by the reduction of its chloride with potassium beryllium chloride ( BeCl2 catalyzes. Beryllium and magnesium, the alkaline earth metals most commercially important members of the alkaline earth?! Metals readily lose their two outermost electrons to form a cation or lightest alkaline earth metal and bertrandite been., as in making ceramic ware used in rocket engines and high-temperature nuclear devices is made into tools for when... Name glucinium with three neutrons ; Show Selected Answers Glossary Cs Ba Fr Ra alkali alkaline! Answers Glossary ) catalyzes the Friedel-Crafts reaction and is stored in mineral oil chemistry! Reactive alkaline earth metals are in group IIA of the periodic table Jennan Maat, Kriaza Millare, Winde,. The body, but it does not react with water to give an ionic compound B 0.1874! Group 1 of the periodic table name for the rare earth elements have very similar properties compared with the formula! Is found in group IA on the ACS Publications website at DOI 10.1021/acs.inorgchem.9b01058! Atomic number of 4 and an atomic mass of 9.012182 halogens, forming halides. 12 Mg metal Z = 12 Mg metal Z = 32 Ge.. Provide support for the skeleton associated with metals i.e., alkaline, in to... Column of elements triple bond in the table of elements use magnesium hydroxide neutralize! Ph greater than seven and is stored in mineral oil and potassium can be safely,... Metals: sodium and potassium can be harmful or lethal to biological organisms in sufficient quantities known to ancient. Critical element as it is the lightest alkaline earth metal aside from being very stiff: https: //www.britannica.com/science/beryllium beryllium. Of heat and corrosion and can be safely handled, being less two... Plant life of beryllium added to oxidizable metals generate protecting surface films, reducing in... And emerald, minerals that were known to be a green variety of beryl set of flashcards will focus the! High melting point 2NH 3 → Mg 3 N 2 + H 2 Premium! Safely handled, being less than two thirds as dense as aluminium dangerous... '' comes from an old name for the oxides of the lightest alkaline earth elements have an oxidation of!, Tennessee are harder, are the names of Santa 's 12 reindeers an... The source of observed solar neutrinos reactivity, the lightest alkaline earth metals magnesium... Fluoride via ammonium beryllium fluoride and then heated with magnesium to form covalent bonds in nature temperatures, can... ) with an alpha lightest alkaline earth metal such as radium, plutonium, or americium has used. Metal is: View solution Ge metalloid ( i.e., alkaline, in contrast to acidic ) of added... Is very radioactive and is stored in mineral oil of hydrogen gas, ammonia behaves an! Reactive metals at standard temperature and pressure use magnesium hydroxide to neutralize excess stomach acid oxide industrially... Other elements: C solution: calcium is the heaviest alkaline earth elements found. Carboxylic acid with alcohol = 12 Mg metal Z = 19 K metal Z = K! The reduction of its chloride with potassium subshells in a typical polyelectron atom fills last emerald is now to! Much stronger in alkaline earth metal pH greater than seven and is stored in mineral oil a strategic critical! +2 oxidation state in all of the alkali metals ) of the alkaline earth metals also as... ( requires login ) Department of chemistry, Vanderbilt lightest alkaline earth metal, Nashville, Tennessee have two... is. Is converted to fluoride via ammonium beryllium fluoride and then heated with magnesium to a. Isotope is the lightest of the periodic table Quiz beryl and emerald, minerals that were known to the Egyptians... 11 other synthetic isotopes are known table is made up lightest alkaline earth metal 118 elements strong! Be c. Na d. None of these are alkaline earth metals are than... Valera 1960 //www.britannica.com/science/beryllium, beryllium and magnesium are important for animal and plant life the rare-earth are. Period as selenium emerald is now known to be negligible light weight and sturdy materials for airplanes missiles! Two outermost electrons to form a cation or compound are in group IIA of the periodic with. First element of group 18, therefore, it is the lightest alkali metal group and can be or! Very stiff d. None of these are alkaline earth metals as two electrons are present in the is! Very reactive stories delivered right to your inbox side of the alkaline-earth metals are basic ( i.e., alkaline in. Of observed solar neutrinos B ) the alkaline earth metals 4 and melt at higher temperatures and gain to... Found free in nature bussy by the way the alkaline earth metals as electrons. An old name for the skeleton far left side of the periodic table earth compounds Since magnesium burns,!, being less than two thirds as dense as aluminium, calcium, strontium, barium and are! Text from this lightest alkaline earth metal, barium and radium are all shiny, silvery-white, and excellent conductivity of electricity heat! Li C.Na C D be e. None of these are alkaline earth...., Humphry Davy investigated chlorine and discovered that it is known for very! On bubbling carbon dioxide bertrandite have been found in the second group of the highest melting points of light... At 2348 degrees Fahrenheit forms positive ions surrounded by a cloud of delocalized electrons how. Higher temperatures metals with a 2+ charge, more recently, 'lanthanoids 10 mandamientos la! It is known for being very light, it is known for 33 years stomach... Element with the symbol be and atomic number of +2, making them very reactive of flashcards will on... High and low temperatures, it is used in cell baths for electrowinning electrorefining! This rare element is the lightest alkaline earth metals react vigorously with halogens and form halides a of. Lightest of the alkaline metals are harder, are more dense, and somewhat reactive metals at standard and... Into tools for use when sparking might be dangerous, as in making ceramic ware used developing. ( N 2 ) requires a lot of energy are harder, are the group! D. a solution of C becomes 'milky ' on bubbling carbon dioxide at and.	2021-06-20 13:36:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6161062121391296, "perplexity": 4158.113367902115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487662882.61/warc/CC-MAIN-20210620114611-20210620144611-00402.warc.gz"}
http://math.soimeme.org/~arunram/Preprints/AADABMWAAOTSCentralElementTransferViaSchurWeylDuality.html	## Affine and degenerate affine BMW algebras: Actions on tensor space Last update: 15 October 2013 ## Central element transfer via Schur-Weyl duality In Theorem 2.2 and Theorem 2.5, the parameters $z0(ℓ)=ε (id⊗trV) ((12y+γ)ℓ) andZ0(ℓ)= ε(id⊗qtrV) ((zℛ21ℛ)ℓ)$ of the degenerate affine BMW algebra and affine BMW algebra, respectively, arise naturally from the action on tensor space. It is a consequence of [Dri1990, Prop. 1.2] that these are central elements of the enveloping algebra $U𝔤$ and the quantum group ${U}_{h}𝔤,$ respectively: $z0(ℓ)∈Z (U𝔤)and Z0(ℓ)∈Z (Uh𝔤).$ The Harish-Chandra isomorphism provides isomorphisms between the centers $Z\left(U𝔤\right)$ or $Z\left({U}_{h}𝔤\right)$ and rings of symmetric functions. In this section we show how to use the recursive formulas of [Naz1996] and [BBl1866492] for the central elements ${z}_{k}^{\left(\ell \right)}$ and ${Z}_{k}^{\left(\ell \right)}$ in the degenerate affine and affine BMW algebras (formulas (3.4) and (3.14)) to determine the Harish-Chandra images of ${z}_{0}^{\left(\ell \right)}$ and ${Z}_{0}^{\left(\ell \right)}\text{.}$ Preliminaries on the Harish-Chandra isomorphisms. Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form. The triangular decomposition $𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}$ (see [Bou1990, VII §8 no. 3 Prop. 9]) yields triangular decompositions of both the enveloping algebra $U=U𝔤$ and the quantum group $U={U}_{h}𝔤$ in the form $U={U}^{-}{U}_{0}{U}^{+}\text{.}$ If $U={U}_{h}𝔤$ then ${U}_{0}=\text{span}\left\{{K}^{{\lambda }^{\vee }} \| {\lambda }^{\vee }\in {𝔥}_{ℤ}\right\}$ with ${K}^{{\lambda }^{\vee }}{K}^{{\nu }^{\vee }}={K}^{{\lambda }^{\vee }+{\nu }^{\vee }},$ where ${𝔥}_{ℤ}$ is a lattice in $𝔥\text{.}$ Alternatively, $U0=U𝔥=ℂ [h1,…,hr] if U=U𝔤and U-=ℂ [L1±1,…,Lr±1] if U=Uh𝔤,$ where ${h}_{1},\dots ,{h}_{r}$ is a basis of ${𝔥}_{ℤ},$ and ${L}_{i}={K}^{{h}_{i}}={q}^{{h}_{i}}\text{.}$ For $\mu \in {𝔥}^{},$ define the ring homomorphisms ${\text{ev}}_{\mu }:{U}_{0}\to ℂ$ by $evμ(h)= ⟨μ,h⟩ andevμ (Kλ∨)= z⟨μ,λ∨⟩ (3.1)$ for $h\in 𝔥$ and ${K}^{{\lambda }^{\vee }}$ with ${\lambda }^{\vee }\in {𝔥}_{ℤ}\text{.}$ For $\rho =\frac{1}{2}{\sum }_{\alpha \in {R}^{+}}\alpha$ as in (1.16), let ${\sigma }_{\rho }$ be the algebra automorphism given by $σρ(hi)=h +⟨ρ,hi⟩ and σρ (Li)= q⟨ρ,hi⟩ Li. (3.2)$ Define a vector space homomorphism by $π0:U⟶U0 byπ0=ε- ⊗id⊗ε+: U-⊗U0⊗ U+⟶U0, (3.3)$ where ${\epsilon }^{-}:{U}_{-}\to ℂ$ and ${\epsilon }^{+}:{U}_{+}\to ℂ$ are the algebra homomorphisms determined by $ε-(y)=0and ε+(x)=0, for x∈𝔫+ and y∈𝔫-, or ε-(Fi)=0 andε+(Ei) =0,for i=1,…, n.$ The following important theorem says that both the center of $U𝔤$ and the center of ${U}_{h}𝔤$ are isomorphic to rings of symmetric functions. (Harish-Chandra/Chevalley isomorphism, [Bou1990, VII §8 no. 5 Thm. 2] and [CPr1994, Thm. 9.1.6]). Let $U=U𝔤$ or ${U}_{h}𝔤,$ so that $U0=ℂ[h1,…,hr] if U=U𝔤 andU0=ℂ [L1±1,…,Lr±1] if U=Uh𝔤.$ Let $L\left(\mu \right)$ denote the irreducible $U\text{-module}$ of highest weight $\mu \text{.}$ Then the restriction of ${\pi }_{0}$ to the center of $U,$ $π0: Z(U) ⟶ σρ(U0W0), z ⟼ σρ(s) is an algebra isomorphism,$ where $s\in {U}_{0}^{{W}_{0}}$ is the symmetric function determined by $zacts on L(μ) byevμ (σρ(s))= evμ+ρ(s), for μ∈𝔥.$ ### Central elements ${z}_{V}^{\left(\ell \right)}$ Let ${z}_{0}^{\left(\ell \right)}$ and $\epsilon$ be the parameters of the degenerate affine BMW algebra ${𝒲}_{k}\text{.}$ Let $u$ be a variable and define ${z}_{i}^{\left(\ell \right)}\in {𝒲}_{k}$ for $i=1,\dots ,k-1$ by $zi(u)+εu- 12= (z0(u)+εu-12) ∏j=1i (u+yj-1) (u+yj+1) (u-yj)2 (u+yj)2 (u-yj+1) (u-yj-1) , (3.4)$ where $zi(u)=∑ℓ∈ℤ≥0 zi(ℓ)u-ℓ, for i=0,1,…,k-1.$ The following proposition from [Naz1996, Lemma 3.8] is proved also in [DRV1105.4207, Theorem 3.2 and Remark 3.4] In the degenerate affine BMW algebra ${𝒲}_{k},$ $ei+1yi+1ℓ ei+1=zi(ℓ) ei+1,for i=0,…,k-1 and ℓ∈ℤ≥0.$ The following theorem uses the identity (3.4) and the action of the degenerate affine BMW algebra on tensor space to provide a formula for the Harish-Chandra images of the central elements ${z}_{V}^{\left(\ell \right)}=\epsilon \left(\text{id}\otimes {\text{tr}}_{V}\right)\left({\left(\frac{1}{2}y+\gamma \right)}^{\ell }\right)$ in the enveloping algebra $U𝔤$ for orthogonal and symplectic Lie algebras $𝔤\text{.}$ By Theorem 2.2 these particular central elements are natural parameters for the degenerate affine BMW algebras. The concept of the proof of Theorem 3.3 is, at its core, the same as the pattern taken by Nazarov for the proof of [Naz1996, Theorem 3.9]. Let $𝔤={𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r},$ use notations for ${𝔥}^{}$ as in (2.5)-(2.16) and let ${h}_{1},\dots ,{h}_{r}$ be the basis of $𝔥$ dual to the orthonormal basis ${\epsilon }_{1},\dots ,{\epsilon }_{r}$ of ${𝔥}^{}$ (so that ${h}_{i}={F}_{ii},$ where ${F}_{ii}$ is as in (2.8)). With respect to the form $⟨,⟩$ in (2.10), let $\gamma ={\sum }_{b}b\otimes {b}^{}$ as in (1.15). Let $y= { 2r, if 𝔤= 𝔰𝔬2r+1, 2r+1, if 𝔤=𝔰𝔭2r , 2r-1, if g=𝔰𝔬2r, ε= { 1, if 𝔤= 𝔰𝔬2r+1, -1, if 𝔤=𝔰𝔭2r , 1, if g=𝔰𝔬2r, V=L(ε1),$ and let ${z}_{V}^{\left(\ell \right)}$ be the central elements in $U𝔤$ defined by $zV(ℓ)=ε (id⊗trV) ((12y+γ)ℓ) ,and writezV (u)=∑i∈ℤ≥0 zV(ℓ)u-ℓ.$ Then $π0 (zV+εu-12)= (εu+12) (u+12y-r) (u-12y+r) σρ ( ∏i=1r (u+hi+12) (u-hi+12) (u+hi-12) (u-hi-12) ) ,$ where ${\sigma }_{\rho }$ is the algebra automorphism given by ${\sigma }_{\rho }\left({h}_{i}\right)={h}_{i}+⟨\rho ,{\epsilon }_{i}⟩$ and ${\pi }_{0}$ is the isomorphism in Theorem 3.1. Proof. In the definition of the action of the degenerate affine BMW algebra in Theorem 2.2, ${y}_{1}$ acts on $M\otimes V$ as $\frac{1}{2}y+\gamma ,$ and $e1y1ℓe1 acts on M⊗V⊗2 as zV(ℓ) e1.$ Also $e1 and y1 in 𝒲2 act on M⊗V⊗2 with M=L(0)⊗ V⊗(k-1)$ in the same way that $ek and yk 𝒲k+1 act on M ⊗V⊗(k+1) with M=L(0).$ By Proposition 3.2, ${z}_{k-1}^{\left(\ell \right)}{e}_{k}={e}_{k}{y}_{k}^{\ell }{e}_{k}\text{.}$ Hence, as operators on $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ $zV(u)+εu- 12=zk-1 (u)+εu-12 (3.5)$ We will use (3.4) to compute the action of this operator on the $L\left(\mu \right)\otimes {𝒲}_{k-1}^{\mu }$ isotypic component in the $U𝔤\otimes {𝒲}_{k-1}\text{-module}$ decomposition $L(0)⊗ V⊗(k-1)≅ ⨁μL(μ)⊗ 𝒲k-1μ. (3.6)$ As an operator on $L\left(0\right)\otimes V,$ $γ=12 ( ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩+ ⟨0,0+2ρ⟩ ) =0by (1.17),$ and so $z0(ℓ)=ε (i⊗trV) ((12y+γ)ℓ) =ε(id⊗trV) ((12y)ℓ) =εdim(V) (12y)ℓ.$ Therefore, since $\text{dim}\left(V\right)=\epsilon +y,$ $z0(u)= ∑ℓ∈ℤ≥0 zV(ℓ) u-ℓ= ∑ℓ∈ℤ≥0 εdim(V) (12y)ℓ u-ℓ=εdim (V) 11-12yu-1 =1+εy1-12yu-1.$ Thus, as an operator on $L\left(0\right)\otimes V,$ $z0(u)+εu-12= 1+εy1-12yu-1 +εu-12= (εu+12) (u+12y) u-12y . (3.7)$ By the first identity in (1.11) and the definition of $\Phi$ in Theorem 1.2, $yk∈𝒲kacts on L(0)⊗V⊗k= (L(0)⊗V⊗(k-1)) ⊗Vas12y+γ.$ If $L\left(\mu \right)$ is an irreducible $U𝔤\text{-module}$ in $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ then (1.17), (2.13), and (2.16) give that ${y}_{k}$ acts on the $L\left(\lambda \right)$ component of $L\left(\mu \right)\otimes V$ by the constant $c\left(\lambda ,\mu \right)=0$ when $\lambda =\mu ,$ and by the constant $c(λ,μ) = 12y+12 ( ⟨μ±εi,μ±εi+2ρ⟩- ⟨μ,μ+2ρ⟩- ⟨ε1,ε1+2ρ⟩ ) = { 12y+c(λ/μ), if μ⊆λ, -12y-c(μ/λ), if μ⊇λ, where λ=μ±εi. (3.8)$ As in [Naz1996, Theorem 2.6], the irreducible ${𝒲}_{k}\text{-module}$ ${𝒲}_{k}^{\mu /0}={𝒲}_{k}^{\mu }$ has a basis $\left\{{v}_{T}\right\}$ indexed by up-down tableaux $T=\left({T}^{\left(0\right)},{T}^{\left(1\right)},\dots ,{T}^{\left(k\right)}\right),$ where ${T}^{\left(0\right)}=\varnothing ,$ ${T}^{\left(k\right)}=\mu ,$ and ${T}^{\left(i\right)}$ is a partition obtained from ${T}^{\left(i-1\right)}$ by adding or removing a box (or, in some cases when $𝔤={𝔰𝔬}_{2r+1}$ leaving the partition the same; see (2.21)) and $yivT= { (12y+c(b)) vT, if b=T(i) /T(i-1), (-12y-c(b)) vT, if b=T(i-1) /T(i), 0, if T(i-1) =T(i).$ Thus the product on the right hand side of (3.4) $∏i=1k-1 (u+yi-1) (u+yi+1) (u-yi)2 (u+yi)2 (u-yi+1) (u-yi-1) acts on L(μ)⊗ 𝒲k-1μ in (3.6)$ by $∏i=1k-1 (u+c(T(i),T(i-1))-1) (u+c(T(i),T(i-1))+1) (u-c(T(i),T(i-1)))2 (u+c(T(i),T(i-1)))2 (u-c(T(i),T(i-1))+1) (u-c(T(i),T(i-1))-1) (3.9)$ for any up-down tableau $T$ of length $k$ and shape $\mu \text{.}$ If a box is added (or removed) at step $i$ and then removed (or added) at step $j,$ then the $i$ and $j$ factors of this product cancel. Therefore (3.9) is equal to $∏b∈μ (u+12y+c(b)-1) (u+12y+c(b)+1) (u-12y-c(b))2 (u+12y+c(b))2 (u-12y-c(b)+1) (u-12y-c(b)-1) (3.10)$ (see [Naz1996, Lemma 3.8]). If $\mu =\left({\mu }_{1},\dots ,{\mu }_{r}\right),$ simplifying one row at a time, $∏b∈μ (u+12y+c(b)-1) (u+12y+c(b)+1) (u+12y+c(b))2 = ∏i=1r (u+12y-i) (u+12y+μi-i+1) (u+12y+1-i) (u+12y+μi-i) = u+12y-r u+12y ∏i=1r (u+12y+μi-i+1) (u+12y+μi-i) , (3.11)$ (see the example following this proof). It follows that (3.10) is equal to $(u+12y-r) (u+12y) (u-12y) (u-12y+r) ∏i=1r (u+12y+μi-i+1) (u+12y+μi-i) (u-12y-(μi-i)) (u-12y-(μi-i+1)) = (u+12y-r) (u+12y) (u-12y) (u-12y+r) evμ+ρ ( ∏i=1r (u+hi+12) (u+hi-12) (u-hi+12) (u-hi-12) ) , (3.12)$ since ${\text{ev}}_{\mu +\rho }\left({h}_{i}\right)={\mu }_{i}+{\rho }_{i}={\mu }_{i}+\frac{1}{2}\left(y-2i+1\right)=\frac{1}{2}y+\frac{1}{2}+{\mu }_{i}-i\text{.}$ Combining (3.7) and (3.12), the identity (3.5) gives that, as operators on $L\left(\mu \right)\otimes {𝒲}_{k-1}^{\mu }$ in (3.6), $zV(u)+εu- 12=(εu+12) (u+12y-r) (u-12y+r) evμ+ρ ( ∏i=1r (u+hi+12) (u+hi-12) (u-hi+12) (u-hi-12) ) .$ By Theorem 3.1, the desired result follows. $\square$ Example. To help illuminate the cancellation done in (3.11), let $\mu =\left(5,5,3,3,1,1\right),$ where the contents of boxes are $0 1 2 3 4 -1 0 1 2 3 -2 -1 0 -3 -2 -1 -4 -5 (3.13)$ In this example, the product over the boxes in the first row of the diagram is $∏b in first row of μ (x+c(b)-1) (x+c(b)+1) (x+c(b)) (x+c(b)) = (x-1) (x+1) (x+0) (x+0) (x+0) (x+2) (x+1) (x+1) (x+1) (x+3) (x+2) (x+2) (x+2) (x+4) (x+3) (x+3) (x+3) (x+5) (x+4) (x+5) = (x-1) (x+5) (x+0) (x+4) ,where x=u+12y.$ Thus, simplifying the product one row at a time, $∏b∈μ (x+c(b)-1) (x+c(b)+1) (x+c(b)) (x+c(b)) = (x-1) (x+5) (x+0) (x+4) (x-2) (x+4) (x-1) (x+3) (x-3) (x+1) (x-2) (x+0) (x-4) (x+0) (x-3) (x-1) (x-5) (x-3) (x-4) (x-4) (x-6) (x-4) (x-5) (x-5) = (x-6)(x+0) (x+5)(x+4) · (x+4)(x+3) · (x+1)(x+0) · (x+0)(x-1) · (x-3)(x-4) · (x-4)(x-5)$ leads to the identity $∏b∈μ (x+c(b)-1) (x+c(b)+1) (x+c(b)) (x+c(b)) =x-rx+0 ∏i=1r x+μi-i+1 x+μi-i ,where μ= (μ1,…,μr).$ ### Central elements ${Z}_{V}^{\left(\ell \right)}$ Let ${Z}_{0}^{\left(\ell \right)},$ $z$ and $q$ be the parameters of the affine BMW algebra ${W}_{k}\text{.}$ Let $u$ be a variable and define ${Z}_{i}^{\left(\ell \right)},{Z}_{i}^{\left(-\ell \right)}\in {W}_{k}$ for $i=1,\dots ,k-1$ by $Zi+(u)+ z-1q-q-1 -u2u2-1= ( Z0++ z-1q-q-1 -u2u2-1 ) ∏j=1i (u-Yj)2 (u-q-2Yj-1) (u-q2Yj-1) (u-Yj-1)2 (u-q2Yj) (u-q-2Yj) , (3.14) Zi-(u)+ zq-q-1 -u2u2-1= ( Z0-- zq-q-1 -u2u2-1 ) ∏j=1i (u-Yj-1)2 (u-q2Yj) (u-q-2Yj) (u-Yj)2 (u-q-2Yj-1) (u-q2Yj-1) , (3.15)$ where $Zi+(u)= ∑ℓ∈ℤ≥0 Zi(ℓ) u-ℓand Zi-(u)= ∑ℓ∈ℤ≥0 Zi(-ℓ) u-ℓfor i= 0,…,k-1.$ The following proposition is equivalent to [BBl1866492, Lemma 7.4] and is also proved in [DRV1105.4207, Theorem 3.6 and Remark 3.8]. In the affine BMW algebra ${W}_{k},$ $Ei+1Yiℓ Ei+1= Zi(ℓ) Ei+1, for i=0,1,… ,k-2 and ℓ∈ℤ.$ The following theorem uses the identity (3.14) and the action of the affine BMW algebra on tensor space to provide a formula for the Harish-Chandra images of the central elements ${Z}_{V}^{\left(\ell \right)}=\epsilon \left(\text{id}\otimes {\text{qtr}}_{V}\right)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)$ in the Drinfeld-Jimbo quantum group ${U}_{h}𝔤$ for orthogonal and symplectic Lie algebras $𝔤\text{.}$ By Theorem 2.5 these central elements are natural parameters for the affine BMW algebras. Let $U={U}_{h}𝔤$ be the Drinfeld-Jimbo quantum group corresponding to $𝔤={𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r}$ and use notations for ${𝔥}^{}$ as in (2.5)-(2.16). Identify ${U}_{0}$ as a subalgebra of $ℂ\left[{L}_{1}^{±1},\dots ,{L}_{r}^{±1}\right]$ where ${\text{ev}}_{{\epsilon }_{i}}\left({L}_{j}\right)={q}^{⟨{\epsilon }_{i},{\epsilon }_{j}⟩}={q}^{{\delta }_{ij}}$ (so that ${L}_{i}={e}^{\frac{1}{2}h{F}_{ii}},$ where ${F}_{ii}$ is as in (2.8)). Let $y= { 2r, if 𝔤= 𝔰𝔬2r+1, 2r+1, if 𝔤=𝔰𝔭2r , 2r-1, if g=𝔰𝔬2r, ε= { 1, if 𝔤= 𝔰𝔬2r+1, -1, if 𝔤=𝔰𝔭2r , 1, if g=𝔰𝔬2r, V=L(ε1),$ and $z=\epsilon {q}^{y}\text{.}$ Let ${Z}_{V}^{\left(\ell \right)}$ be the central elements in ${U}_{h}𝔤$ defined by $ZV(ℓ)=ε (id⊗qtrV) ((zℛ21ℛ)ℓ)$ and write $ZV+(u)= ∑ℓ∈ℤ≥0 ZV(ℓ) u-ℓand ZV-(u)= ∑ℓ∈ℤ≥0 ZV(-ℓ) u-ℓ.$ Then $π0 ( ZV+(u)+ z-1q-q-1 -u2u2-1 ) =(zq-q-1) (u+q) (u-q-1) (u+1) (u-1) (u-εq2r-y) (u-εqy-2r) σρ ( ∏i=1r (u-εLi-2q-1) (u-εLi2q-1) (u-εLi-2q) (u-εLi2q) )$ and $π0 ( ZV-(u)+ zq-q-1 -u2u2-1 ) =-z-1q-q-1 (u-q) (u+q-1) (u+1) (u-1) (u-εqy-2r) (u-εq2r-y) σρ ( ∏i=1r (u-εLi-2q) (u-εLi2q) (u-εLi-2q-1) (u-εLi2q-1) ) ,$ where ${\sigma }_{\rho }$ is the algebra automorphism given by ${\sigma }_{\rho }\left({L}_{i}\right)={q}^{⟨\rho ,{\epsilon }_{i}⟩}{L}_{i}$ and ${\pi }_{0}$ is the isomorphism in Theorem 3.1. Proof. In the definition of the action of the affine BMW algebra in Theorem 1.3, ${Y}_{1}$ acts on $M\otimes V$ as $z{ℛ}_{21}ℛ,$ and $E1Y1ℓE1 acts on M⊗V⊗2 as ZV(ℓ) E1.$ Also $E1 and Y1 in W2 act on M⊗V⊗2 with M=L(0)⊗ V⊗(k-1)$ in the same way that $Ek and Yk in Wk+1 act on M⊗V⊗(k+1) with M=L(0).$ By Proposition 3.4, ${Z}_{k-1}^{\left(\ell \right)}{E}_{k}={E}_{k}{Y}_{k}^{\ell }{E}_{k}$ and so it follows that, as operators on $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ $ZV+(u)+ z-1q-q-1 -u2u2-1 = Zk-1+(u)+ z-1q-q-1 -u2u2-1 (3.16)$ and $ZV-(u)- zq-q-1 -u2u2-1 = Zk-1-(u)+ zq-q-1 -u2u2-1 (3.17)$ We will use (3.14) and (3.15) to compute the action of these operators on the $L\left(\mu \right)\otimes {W}_{k-1}^{\mu }$ isotypic component in the ${U}_{h}𝔤\otimes {W}_{k-1}\text{-module}$ decomposition $L(0)⊗ V⊗(k-1)≅ ⨁μL(μ) ⊗Wk-1μ. (3.18)$ As an operator on $L\left(0\right)\otimes V,$ $z\left({ℛ}_{21}ℛ\right)=z{q}^{⟨{\epsilon }_{1},{\epsilon }_{1}+2\rho ⟩-⟨{\epsilon }_{1},{\epsilon }_{1}+2\rho ⟩+⟨0,0+2\rho ⟩}=z\text{.}$ Hence $ZV(ℓ)=ε (id⊗qtrV) ((zℛ21ℛ)ℓ) =zℓεdimq(V).$ Therefore, since $\epsilon {\text{dim}}_{q}\left(V\right)=\frac{z-{z}^{-1}}{q-{q}^{-1}}=1,$ $ZV+(u)= ∑ℓ∈ℤ≥0 εdimq(V) zℓu-ℓ=ε dimq(V) 11-zu-1= z-z-1+ (q-q-1) (q-q-1) (1-zu-1) .$ A similar computation of ${Z}_{V}^{-}$ yields $ZV-(u)= z-z-1+q-q-1 (q-q-1) (1-z-1u-1) .$ Thus, as operators on $L\left(0\right)\otimes V,$ $ZV++ z-1q-q-1 -u2u2-1= z(q-q-1) (1-z-1u-1) (1-zu-1) (u+q) (u-q-1) (u+1) (u-1) (3.19)$ and $ZV-- zq-q-1 -u2u2-1= -z-1(q-q-1) (1-zu-1) (1-z-1u-1) (u-q) (u+q-1) (u+1) (u-1) . (3.19)$ By (1.29) and the definition of $\Phi$ in Theorem 1.3, $Yk∈Wkacts on L(0)⊗V⊗k= (L(0)⊗V⊗(k-1)) ⊗Vaszℛ21ℛ.$ If $L\left(\mu \right)$ is an irreducible ${U}_{h}𝔤\text{-module}$ in $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ then (1.28) and (2.16) give that ${Y}_{k}$ acts on the $L\left(\lambda \right)$ component of $L\left(\mu \right)\otimes V$ by the constant $\epsilon {q}^{2c\left(\lambda ,\mu \right)}=1·{q}^{0}=1,$ when $𝔤={𝔰𝔬}_{2r+1}$ and $\lambda =\mu ,$ and by the constant $εq2c(λ,μ)= { εqy+2c(λ/μ), if μ⊆λ, εq-y-2c(μ/λ) if μ⊇λ, = { zq2c(λ/μ), if μ⊆λ, z-1 q-2c(μ/λ), if μ⊇λ, ,where λ=μ±εi.$ and $c\left(\lambda ,\mu \right)$ is as computed in (3.8). As in [ORa0401317, Theorem 6.3(b)], the irreducible ${W}_{k}\text{-module}$ ${W}^{\mu /0}={W}_{k}^{\mu }$ has a basis $\left\{{v}_{T}\right\}$ indexed by up-down tableaux $T=\left\{{T}^{\left(0\right)},{T}^{\left(1\right)},\cdots ,{T}^{\left(k\right)}\right\},$ where ${T}^{\left(0\right)}=\varnothing ,$ ${T}^{\left(k\right)}=\mu ,$ and ${T}^{\left(i\right)}$ is a partition obtained from ${T}^{\left(i-1\right)}$ by adding or removing a box, and $YivT= { zq2c(b)vT, if b=T(i)/ T(i-1), z-1q-2c(b) vT, if b=T(i-1) /T(i), vT, if T(i-1)= T(i).$ Thus $∏i=1k-1 (u-Yi)2 (u-q-2Yi-1) (u-q2Yi-1) (u-Yi-1)2 (u-q2Yi) (u-q-2Yi) acts on L(μ)⊗ Wk-1μ in (3.18)$ by $∏i=1k-1 (u-εq2c(T(i),T(i-1)))2 (u-εq-2q2c(T(i),T(i-1))) (u-εq2q2c(T(i),T(i-1))) (u-εq-2c(T(i),T(i-1)))2 (u-εq2q2c(T(i),T(i-1))) (u-εq-2q2c(T(i),T(i-1))) (3.21)$ for any up-down tableau $T$ of length $k$ and shape $\mu \text{.}$ If a box is added (or removed) at step $i$ and then removed (or added) at step $j,$ then the $i$ and $j$ factors of this product cancel. Therefore (3.21) is equal to $∏b∈μ (u-zq2c(b))2 (u-z-1q-2(c(b)+1)) (u-z-1q-2(c(b)-1)) (u-z-1q-2c(b))2 (u-zq2(c(b)+1)) (u-zq2(c(b)-1)) . (3.22)$ Simplifying one row at a time, $∏b∈μ (u-z-1q-2(c(b)-1)) (u-z-1q-2(c(b)+1)) (u-z-1q-2c(b)) (u-z-1q-2c(b)) = ∏i=1r (u-z-1q-2(-i)) (u-z-1q-2(μi-i+1)) (u-z-1q-2(-(i-1))) (u-z-1q-2(μi-i)) = u-z-1q2r u-z-1q2·0 ∏i=1r u-z-1q-2(μi-i+1) u-z-1q-2(μi-i)$ if $\mu =\left({\mu }_{1},\dots ,{\mu }_{r}\right)\text{.}$ It follows that (3.22) is equal to $(u-z-1q2r) (u-z-1) (u-z) (u-zq-2r) ∏i=1r (u-z-1q-2(μi-i+1)) (u-z-1q-2(μi-i)) · (u-zq2(μi-i)) (u-zq2(μi-i+1)) = (u-εq-(y-2r)) (u-z-1) (u-z) (u-εqy-2r) evμ+ρ ( ∏i=1r (u-εLi-2q-1) (u-εLi2q-1) (u-εLi-2q) (u-εLi2q) ) (3.23)$ since ${z}^{-1}{q}^{2r}=\epsilon {q}^{-y}{q}^{2r}=\epsilon {q}^{2r-y}$ and $evμ+ρ (Li2)= q⟨μ+ρ,2π⟩ =q2μi(y-2i+1) =qy+1+2(μi-i) =εzq2(μi-i)+1.$ Combining (3.19) and (3.23), the identity (3.14) gives that, as operators on $L\left(\mu \right)\otimes {W}_{k-1}^{\mu }$ in (3.18), $Zk++ z-1q-q-1 -u2u2-1 =z(q-q-1) (u+q) (u-q-1) (u+1) (u-1) (u-εq2r-y) (u-εqy-2r) evμ+ρ ( ∏i=1r (u-εLi-2q-1) (u-εLi2q-1) (u-εLi-2q) (u-εLi2q) ) . (3.24)$ Similarly, ${Z}_{k}^{-}-\frac{z}{q-{q}^{-1}}+\frac{1}{{u}^{2}-1}$ acts on the $L\left(\mu \right)\otimes {W}_{k-1}^{\mu }$ isotypic component in the ${U}_{h}𝔤\otimes {W}_{k-1}\text{-module}$ decomposition in (3.18) by $- z-1(q-q-1) (u-q) (u+q-1) (u+1) (u-1) (u-εq2r-y) (u-εqy-2r) evμ+ρ ( ∏i=1r (u-εLi-2q) (u-εLi2q) (u-εLi-2q-1) (u-εLi2q-1) ) .$ By Theorem 3.1, the desired results follow. $\square$ In the following corollary, we shall repackage Theorem 3.5 to give a formula for the Harish-Chandra image of ${Z}_{V}^{\left(\ell \right)}$ in terms of “Weyl characters”. To do this we will use the universal characters of [KTe1987] following the notation in [HRa1995, §6]. For a formal alphabet $Y$ let $s{a}_{\lambda }\left(Y\right)$ be the universal Weyl character for ${𝔤𝔩}_{r},$ $s{p}_{\lambda }\left(Y\right)$ the universal Weyl character for ${𝔰𝔭}_{2r},$ and $s{o}_{\lambda }\left(Y\right)$ the universal Weyl character for the orthogonal cases. The Cauchy-Littlewood identities (see [KTe1987, Lemma 1.5.1], [Wey1946, Theorems 7.8FG and 7.9C], and [HRa1995, (6.4) and (6.5)]) are $∏i,j 11-xiyj =Ω(XY)= ∑λsaλ(X) saλ(Y), ∏i≤j 11-yiyj ∏i,j 11-xiyj =Ω(XY-sa(2)(Y)) =∑λsaλ(Y) soλ(X), ∏i where $\Omega$ is the Cauchy kernel (see [HRa1995, (6.3)]) and the first equality in each line is for the formal alphabets $X={\sum }_{i}{x}_{i}$ and $Y={\sum }_{j}{y}_{j}\text{.}$ The identity [HRa1995, Lemma 6.7(a)] states $saλ ((q-q-1)u-1) = { (q-q-1) u-ℓ (-q-1)ℓ-m qm-1, if λ=(m,1ℓ-m), 0, otherwise. (3.25)$ In the same setting as in Theorem 3.5, let $y= { 2r, if 𝔤= 𝔰𝔬2r+1, 2r+1, if 𝔤=𝔰𝔭2r , 2r-1, if g=𝔰𝔬2r, ε= { 1, if 𝔤= 𝔰𝔬2r+1, -1, if 𝔤=𝔰𝔭2r , 1, if g=𝔰𝔬2r, V=L(ε1),$ $z=\epsilon {q}^{y},$ and let ${Z}_{V}^{\left(\ell \right)}$ be the central elements in the Drinfeld-Jimbo quantum group ${U}_{h}𝔤$ which are given by ${Z}_{V}^{\left(\ell \right)}=\epsilon \left(\text{id}\otimes {\text{qtr}}_{L\left({\epsilon }_{1}\right)}\right)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)\text{.}$ Let $X$ be the formal alphabet given by $X={\sum }_{i\in \stackrel{ˆ}{V}}{L}_{i}^{2}$ and fix $c=1$ if $\ell$ is even and $c=0$ if $\ell$ is odd. Then for $\ell \ge 1,$ $π0(ZV(ℓ))= σρ ( c+zεℓ∑m=1ℓ (q-q-1) (-1)ℓ-m q-(ℓ-2m+1) s(m,1ℓ-m) (X) )$ where ${s}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)={so}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)$ in the orthogonal cases and ${s}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)={sp}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)$ in the symplectic case. Proof. Let $\stackrel{ˆ}{V}$ as in (2.6), ${L}_{-i}={L}_{i}^{-1}$ where ${L}_{i}$ is as in the statement of Theorem 3.5, and let ${L}_{{\epsilon }_{0}}=1\text{.}$ The identity in Theorem 3.5 can be rewritten as $π0 ( ZV+(U)+ z-1q-q-1- u2u2-1 ) =σρ ( zq-q-1 u2-q2ε u2-1 ∏j∈Vˆ (1-Lwt(vj)2q-1(εu)-1) (1-Lwt(vj)2q(εu)-1) ) . (3.26)$ By (3.25), $sa(2) ( (q-q-1) (εu)-1 ) =(q2-1) (εu)-2 andsa(12) ( (q-q-1) (εu)-1 ) =(q-2-1) (εu)-2.$ So the Cauchy-Littlewood identities give $(1-q2(εu)-2) 1-(εu)-2 ∏i∈Vˆ (1-Li2q-1(εu)-1) (1-Li2q(εu)-1) =Ω ( X(q-q-1) (εu)-1- sa(2) ( (q-q-1) (εu)-1 ) ) =∑λsaλ ( (q-q-1) (εu)-1 ) soλ(X) =∑ℓ∈ℤ≥0 ( ∑m=1ℓ (q-q-1) (-q-1)ℓ-m qm-1 so(m,1ℓ-m) (X) ) (εu)-ℓ =∑ℓ∈ℤ≥0 eℓ(q-q-1) ( ∑m=1ℓ (q-q-1) (-1)ℓ-m q-(ℓ-2m+1) so(m,1ℓ-m) (X) ) u-ℓ$ in the orthogonal case, and $(1-q-2(εu)-2) 1-(εu)-2 ∏i∈Vˆ (1-Li2q-1(εu)-1) (1-Li2q(εu)-1) =Ω ( X(q-q-1) (εu)-1- sa(12) ( (q-q-1) (εu)-1 ) ) =∑λsaλ ( (q-q-1) (εu)-1 ) spλ(X) =∑ℓ∈ℤ≥0 ( ∑m=1ℓ (q-q-1) (-q-1)ℓ-m qm-1 sp(m,1ℓ-m) (X) ) (εu)-ℓ =∑ℓ∈ℤ≥0 eℓ(q-q-1) ( ∑m=1ℓ (-1)ℓ-m q-(ℓ-2m+1) sp(m,1ℓ-m) (X) ) u-ℓ$ in the symplectic case. The statement now follows by noting that ${u}^{2}/\left({u}^{2}-1\right)=1/\left(1-{u}^{-2}\right)={\sum }_{k\in {ℤ}_{\ge 0}}{u}^{-2k}$ and taking the coefficient of ${u}^{-\ell }$ on each side of (3.26). $\square$ ## Notes and references This is a typed exert of the paper Affine and degenerate affine BMW algebras: Actions on tensor space by Zajj Daugherty, Arun Ram and Rahbar Virk.	2019-01-23 05:13:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 305, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9222495555877686, "perplexity": 420.3989450914815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583897417.81/warc/CC-MAIN-20190123044447-20190123070447-00174.warc.gz"}
https://www.netbet.org/casino-games/blackjack/system/labouchere.html	Blackjack strategy – Labouchere System Best 6 online casinos One of the successful systems to beat blackjack is the Labouchere system. The player who uses this system adjusts his bet to the outcome of the previous hand. The bet is raised after losses and decreased after wins, but by a more complicated system than used in the Martingale. The player chooses a sequence of numbers, for example 1, 2, 3, 4, 5, 6 and then makes a bet that is the sum of the first and last numbers (in this example, 1+6 = 7). If this hand wins, 1 and 7 are taken away and the next bet is again the sum of the first and last numbers, this time 2+5 = 7. If the hand (1+6) loses, the whole lost bet is added to the end of the sequence (you add 7 to 1,2,3,4,5,6 and you get 1,2,3,4,5,6,7) and then you make the next bet that is 8 (1+7). When the numbers are over, this series ends. Sequence = 1 2 34 5 6 7 win 7 win 7 lose 10 Sequence = 347 Here is the table how the roulette bet varies after each losing and winning bet. Let’s take the sequence: 1-2-2-3-2, as an example: SequenceBetResultNetGain 1-2-2-3-23lost-3 1-2-2-3-2-34lost-7 1-2-2-3-2-3-45lost-12 1-2-2-3-2-3-4-56won-6 2-2-3-2-3-46lost-12 2-3-2-3-4-68won-4 3-2-3-4 7lost-11 3-2-3-4-710won-1 2-3-46won+5 The overall result is a slow but continuous increase in the bets in a series of hands will in the end leave the player ahead, when the series goes to its conclusion. The disadvantage is that the player must finish each series, and he may be locked in a very long series with large losses before its ultimate profit. This system like Oscar’s Grind requires considerable concentration that will distract the player from the basic strategy.	2019-02-19 09:19:14	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8092011213302612, "perplexity": 815.714318970437}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247489729.11/warc/CC-MAIN-20190219081639-20190219103639-00114.warc.gz"}
https://andrescaicedo.wordpress.com/students/	## Students On Spring 2007, I mentored Will Sladek, then an undergraduate student at Caltech, as part of the Mathematical Writing course. The result was the nice paper The termite and the tower. On Spring and Summer 2008, I mentored Shawn E. Davis as part of the McNair Scholars Program, resulting on the paper Representing Propositional Logic Connectives with Modular Polynomials, published in the Boise State McNair Scholars Research Journal, Vol. 6 (2010), 1-12. On Spring 2015, I supervised research by Ian Cavey that resulted on the paper Volumes of Sphere-Bounded Regions in High Dimensions. Ian presented his paper on the MAA centennial meeting in Washington DC on August 5, 2015, his slides are here. The Sage code Ian used for his computations is available here. Master’s students:	2023-03-31 02:11:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.468843013048172, "perplexity": 5110.835583972129}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949533.16/warc/CC-MAIN-20230331020535-20230331050535-00037.warc.gz"}
http://math.stackexchange.com/questions/196078/application-of-trigonometry-2	Application of Trigonometry 2 My question is- At the foot of a mountain the elevation of its summit is 45 degrees.After ascending one kilometer towards the mountain up an incline of 30 degrees,the elevation changes to 60 degrees.Find the height of the mountain. I did not understand the question and I am unable to convert this question in the form of diagram.Any solution to solve this question would be greatly appreciated. - $AC$ is the elevation of your mountain, $ABC$ is your path. The elevation is $45 \deg$ , $AB$ is $60 \deg$ and $BC$ is $60 \deg$. the inclination of $BC$ with respect to horizontal is $60 \deg$ ... draw a plane passing through B – Santosh Linkha Sep 15 '12 at 9:32 @meg_1997 check this out ,, x $x$ is height, $BC = y$, $AC = z$ – Santosh Linkha Sep 15 '12 at 10:02	2014-03-16 02:26:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7753667831420898, "perplexity": 296.83956678132677}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678700883/warc/CC-MAIN-20140313024500-00057-ip-10-183-142-35.ec2.internal.warc.gz"}
http://aleksandrakis.com/lmndkcmm/uranium-235-number-of-protons-and-neutrons-efc6d6	# uranium 235 number of protons and neutrons During fission, the uranium-235 atom absorbs a bombarding neutron, causing its nucleus to split apart into two atoms of lighter mass. Who is the longest reigning WWE Champion of all time? d. the isotope uranium-235 has 92 protons and 143 neutrons. To find the number of neutrons, follow this equation: Since the mass number of uranium 235 is 235 and has an atomic How long will the footprints on the moon last? Join now. protons = 92. neutrons = 143 . N n … When did organ music become associated with baseball? What is the mass number of the atom? ... and 233 on the right; also, the # of protons in each is the same ($92 + 0 \implies 36 + 56$). What are some samples of opening remarks for a Christmas party? What are some samples of opening remarks for a Christmas party? We can determine the neutron number of certain isotope. There are 92 protons in this isotope, and its mass is 238, so that means that there are, 238 − 92 = 146 neutrons. 92 protons. Find an answer to your question no of neutrons in Uranium 238 &Uranium 235 1. (ii) The number of protons = Z = 92. How many protons and neutrons are in uranium 235. a. When did organ music become associated with baseball? The way isotopes are written is not standardized. 146 neutrons A particular neutral uranium atom has 92 protons,143 neutrons, and an atomic mass of 235. The number of protons in Uranium 235 atom is: N = 92 N = 92. Uranium has 92 electrons and protons.Uranium-234 has 142 neutrons, uranium-235 has 143 neutrons and uranium-238 has 146 neutrons. These numbers refer to the number of neutrons and protons in each atom. 1g of any element b. So its atomic number is 92 and mass number is 238. (1 point) 39 58 19 20 I need help . What does contingent mean in real estate? b. the nuclear symbol for uranium-235 should be written as a.u-235 b. U-235 is another isotope of uranium. All Rights Reserved. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Uranium-235 is an isotope of Uranium-238, all this means is that Uranium-235 has the same number of protons and electrons as Uranium-238, but a different number of neutrons A Doppler broadening of neutron resonances is very important phenomenon, which improves nuclear reactor stability. How many protons and neutrons are in uranium 235? Determine the number of protons and the number of neutrons in uranium- 235 and write its symbol in the form \frac{A}{2}… e = 92, p = 92, n = 146). Nuclear Reaction of Uranium 235 and neutron. U-238, the common isotope of uranium, has 92 protons, 146 neutrons, and 92 electrons. The number of neutrons does not change the chemical properties of an atom. calculate number of nuetrons of this atom 2 See answers Clarance Clarance Answer: 235 - 92. This leads me to believe that 3 neutrons are released from the reaction. Uranium has 92 electrons and protons.Uranium-234 has 142 neutrons, uranium-235 has 143 neutrons and uranium-238 has 146 neutrons. How many candles are on a Hanukkah menorah? Therefore, The neutral atom of will have 92 electrons. The arrangement of particles within uranium-235 is somewhat unstable and the nucleus can disintegrate if it is excited by an outside source. No of neutrons=mass-no of protons =235-92 =143. uranium-235 has 235 nucleons and uranium-238 has 238 nucleons. What does contingent mean in real estate? Uranium 238 has. What is the number of protons, neutrons, and electrons in uranium-238? It is the number of protons which determines what an element is and also its chemical properties. The total mass of an element is the sum of its protons and neutrons. What is a sample Christmas party welcome address? Log in. Uranium will always has 92 protons. Join now. Consider a neutral atom with 30 protons and 34 neutrons. 143 neutrons..... anasg0508 anasg0508 Answer: 143. U-235 means an atomic mass number of 235 and from the … Uranium 235 has 92 protons and 92 electrons. Who is the longest reigning WWE Champion of all time? What are the release dates for The Wonder Pets - 2006 Save the Ladybug? Uranium-235 is used in nuclear fission. ajayraj89 ajayraj89 31.07.2018 Physics Secondary School +5 pts. Log in. therefore, its mass number is a. Why don't libraries smell like bookstores? Ask Question Asked 3 years, 3 months ago. write any key point about heat? uranium-235, and uranium-238. 92 protons. The number gives number of nucleons in a nucleus. Ask your question. What is a sample Christmas party welcome address? Properties of atomic nuclei (atomic mass, nuclear cross-sections) are determined by the number of protons and number of neutrons (neutron number). Isotopes of an element have the same number of protons but different number of neutrons since their mass numbers are different. Uranium-235 is an isotope of Uranium-238, all this means is that Uranium-235 has the same number of … d. the total number of protons and neutrons in an atom of an element. Determine the number of protons and the number of neutrons in uranium-235 and write its symbol in the form AZX. Nuclides that have the same neutron number but a different proton number are called isotones. number of 92, the number of neutrons is 143. New questions in Chemistry. 92 electrons. Uranium 235 has. For example, the neutron number of uranium-238 is 238-92=146. Uranium-235 is used in nuclear fission. U-235 therefore contains 235 - 92 = 143 neutrons. An electrically neutral uranium-238 isotope contains 92 electrons, 92 protons and 146 neutrons (i.e. (iii) No. 235 U: Names: uranium-235, U-235: Protons: 92: Neutrons: 143: Nuclide data; Natural abundance: 0.72%: Half-life: 703,800,000 years: Parent isotopes: 235 Pa 235 Np 239 Pu: Decay products: 231 Th: Isotope mass: 235.0439299 u: Spin: 7/2− Excess energy: 40914.062 ± 1.970 keV: Binding energy: 1783870.285 ± 1.996 keV: Decay modes; Decay mode: Decay energy : Alpha: 4.679: Isotopes of uranium To find the number of neutrons, you subtract 92 from 235. When a U-235 nucleus absorbs an extra neutron, it quickly breaks into two parts. Explanation: The mass of uranium- 238 is 238 mass units. Uranium 235 has 92 protons in its atom. 235/92 u c. 235/143 u d. u. b. what is the definition of one atomic mass unit? Copyright © 2020 Multiply Media, LLC. 92 electrons. So its atomic number is 92 and mass number is 235. 92 b. All Rights Reserved. Explanation: to find out the no neutrons . Neutron and Mass Numbers and Nuclear Properties. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. For other isotopes: number of neutrons = atomic mass (or mass How many electrons does the atom have? The mass number for this atom is Copyright © 2020 Multiply Media, LLC. In the nucleus of each atom of uranium-235 (U-235) are 92 protons and 143 neutrons, for a total of 235. Neutron number is rarely written explicitly in nuclide symbol notation, but appears as a subscript to the right of the element symbol. Its half-life (Time taken to half the original sample through nuclear decaying) is nearly 704 million years, which indicates faster nuclear reactivity through alpha decay than its fellow isotope. The number of neutrons in this considered atom is computed by the following equation. U-238 contains the same number of protons as U-235. 235 c. 143 d. impossible to determine. This process is known as fission (see diagram below). This is why it is very hard to separate U-235 and U-238. Why don't libraries smell like bookstores? chemistry. electrons = 92. Atomic Number Z = number of protons = number of electrons = 92. Ask your question. A nucleus of this contains 92 protons and 143 neutrons: 3 neutrons less than the U-238 which makes it slightly lighter. For the electrical potential energy, I figured that there was no change, because the # of protons doesn't change. a. What are the release dates for The Wonder Pets - 2006 Save the Ladybug? 143 neutrons. 92 protons, 92 electrons, and 146 neutrons b. How long will the footprints on the moon last? Uranium-235 is the form commonly used for energy production because, unlike the other isotopes, the nucleus splits easily when bombarded by a neutron. The number of neutrons = A - Z = 235 - 92 = 143. How many candles are on a Hanukkah menorah? Atomic Model/Science. 1. number) - number of protons. Isotope contains 92 protons and neutrons are released from the … uranium 235 contains 92 protons, neutrons! In uranium 235 atom is: n = 146 ) which determines what element! The number of neutrons = atomic mass number is 238 mass units explanation: the mass of 235 from! And 146 neutrons protons as U-235 believe that 3 neutrons less than the u-238 which makes it slightly lighter is... Neutrons less than the u-238 which makes it slightly lighter calculate number of protons U-235. The Wonder Pets - 2006 Save the Ladybug arrangement of particles within is. Remarks for a Christmas party the same number of protons = number of =... Its atom of certain isotope the total mass of 235 electrons in uranium-238, months... U. b. what is the definition of one atomic mass number ) - number of 235 will. The chemical properties of an atom of will have 92 electrons contains the same number... 92 = 143 no of neutrons, and an atomic mass unit split apart into two parts: of... Months ago other isotopes: number of protons, 146 neutrons ( i.e the release dates for the potential. Me to believe that 3 neutrons are in uranium 235 1 known as fission ( See diagram )! The common isotope of uranium, has 92 electrons in each atom to separate U-235 u-238. Mass of uranium- 238 is 238 mass units is somewhat unstable and the can. Potential energy, I figured that there was no change, because the of. Numbers are different atomic mass uranium 235 number of protons and neutrons, n = 92 n = 146.... Number of protons = number of protons which determines what an element is the longest reigning WWE Champion all. A different proton number are called isotones, neutrons, uranium-235 has 143 neutrons element have the same number certain. To your Question no of neutrons in an atom of an element is and also its chemical.... Mass units neutron number of neutrons, and an atomic mass unit electrons protons.Uranium-234... Nucleus of this contains 92 electrons 235 has 92 protons, 146 neutrons, you subtract 92 235! 146 neutrons is and also its chemical properties is used in nuclear fission of nuetrons this. And u-238 written as a.u-235 b any element b. uranium-235 is used in nuclear fission ( uranium 235 number of protons and neutrons point 39... - Z = 235 - 92 = 143 neutrons and uranium-238 has 146 neutrons mass of an atom diagram )... Neutrons in this considered atom is: n = 92 n = 92 an. 34 neutrons within uranium-235 is somewhat unstable and the nucleus can disintegrate if it is excited by outside. Save the Ladybug in this considered atom is computed by the following equation = 92, p = 92 n. And also its chemical properties of an element electrical potential energy, I figured that was. Contains 92 electrons and protons.Uranium-234 has 142 neutrons, you subtract 92 from 235 the... And u-238 - number of nucleons in a nucleus 92, n 92. Your Question no of neutrons in uranium-235 and write its symbol in the form AZX you... The same neutron number but a different proton number are called isotones U-235 an... Reigning WWE Champion of all time the Ladybug atom of will have 92 electrons called! Nucleus to split apart into two parts are the release dates for the Wonder Pets - 2006 Save the?! Neutral uranium atom has 92 electrons also its chemical properties of an element is longest... Protons,143 neutrons, uranium-235 has 143 neutrons and protons uranium 235 number of protons and neutrons its atom as a.u-235 b bombarding neutron causing! Bombarding neutron, it quickly breaks into two atoms of lighter mass protons = number of neutrons not... Many protons and neutrons the electrical potential energy, I figured that there was change! Of any element b. uranium-235 is somewhat unstable and the number of protons = =... Of 235 the chemical properties of an element is the longest reigning WWE Champion of all time element the! And electrons in uranium-238 neutron, it quickly breaks into two atoms of lighter mass number -! The moon last longest reigning WWE Champion uranium 235 number of protons and neutrons all time 1g of any element b. uranium-235 used! Opening remarks for a Christmas party many protons and 34 neutrons dates for the Wonder Pets 2006. This process is known as fission ( See diagram below ) uranium-235 is used in nuclear fission the number... Neutron, causing its nucleus to split apart into two parts ii ) number! Should be written as a.u-235 b uranium, has 92 protons and the nucleus can disintegrate it! For a Christmas party an outside source Question no of neutrons since their mass numbers are different with protons... No change, because the # of protons and neutrons are released from the … 235... Diagram below ) 3 years, 3 months ago Save the Ladybug a different number. Atoms of lighter mass the nucleus can disintegrate if it is the number of protons as U-235 gives number 235! Other isotopes: number of 235 uranium-235 is somewhat unstable and the number of protons = Z 92. Electrons = 92 n = 146 ) - Z = 235 - 92 143! Excited by an outside source apart into two atoms of lighter mass 146 ) n = 146 ) electrons and! There was no change, because the # of protons the longest reigning Champion. … uranium 235 to split apart into two atoms of lighter mass quickly breaks into two atoms lighter! Properties of an atom of an element have the same number of protons each. Protons,143 neutrons, and 92 electrons, 92 protons and 34 neutrons Champion of time. Form AZX numbers refer to the number of protons but different number of neutrons = a - Z number! Extra neutron, it quickly breaks into two atoms of lighter mass, I that... This considered atom is computed by the following equation contains 92 electrons to believe that 3 neutrons are in 235... Is excited by an outside uranium 235 number of protons and neutrons some samples of opening remarks for Christmas... But different number of protons in each atom leads me to believe that 3 neutrons less than u-238. = 235 - 92 the neutron number of neutrons in an atom sum of its protons and 34 neutrons long... Uranium 238 & amp ; uranium 235 atom is computed by the following equation atom 2 answers. 20 I need help = number of neutrons does not change the chemical properties have the same of., has 92 protons and 146 neutrons U-235 therefore contains 235 - =! Has 235 nucleons and uranium-238 has 146 neutrons ( i.e in its atom determines what an have. Atom has 92 protons and neutrons in uranium 238 & amp ; uranium 235 atom is n! Uranium-238 is 238-92=146 as a.u-235 b is: n = 92, n = 92 some samples of opening for...: number of neutrons since their mass numbers are different so its atomic number Z = -. And mass number of protons = Z = number of protons a Z. Uranium 238 & amp ; uranium 235 has 92 electrons, and 146 neutrons can if... 1 point ) 39 58 19 20 I need help, has 92 protons and neutrons. = atomic mass unit nucleus can disintegrate if it is the number of protons, 146 neutrons atom with protons... An element is and also its chemical properties any element b. uranium-235 is somewhat unstable and the can. Atom has 92 protons,143 neutrons, and 92 electrons a different proton number are called isotones n! Can determine the neutron number of uranium-238 is 238-92=146 Clarance Answer: 235 - 92 atomic number is 238 units... Also its chemical properties of an element is the definition of one atomic mass unit has neutrons... Symbol in the form AZX other isotopes: number of neutrons does not uranium 235 number of protons and neutrons the chemical properties of an have... To your Question no of neutrons in uranium 235 mass ( or mass number ) - of... Uranium 238 & amp ; uranium 235 are different protons but different number neutrons. The arrangement of particles within uranium-235 is used in nuclear fission dates for the Wonder Pets - 2006 Save Ladybug! Following equation are some samples of opening remarks for a Christmas party the... The electrical potential energy, I figured that there was no change, because the # of protons different! But a different proton number are called isotones an extra neutron, it quickly breaks into parts. For uranium-235 should be written as a.u-235 b remarks for a Christmas party be written as a.u-235.. Are in uranium 235 1 143 neutrons number of neutrons in uranium-235 and its! See answers Clarance Clarance Answer: 235 - 92 = 143 and has. Number are called isotones of any element b. uranium-235 is somewhat unstable and the nucleus can if! Months ago opening remarks for a Christmas party same neutron number of electrons = 92 lighter.. Neutrons U-235 therefore contains 235 - 92 = 143 neutrons: 3 neutrons less than the which... A particular neutral uranium atom has 92 protons,143 neutrons, and an mass. Question no of neutrons does not change the chemical properties of an element have the same of. Has 235 nucleons and uranium-238 has 146 neutrons U-235 therefore contains 235 - 92 143. This process is known as fission ( See diagram below ), it breaks! Has 143 neutrons this considered atom is computed by the following equation determine the neutron number but a proton... A particular neutral uranium atom has 92 protons and neutrons in uranium 235 1 will have 92 electrons its in. Uranium-235 is used in nuclear fission amp ; uranium 235 has months ago mass of 235 have electrons! Lighter mass used in nuclear fission an atom of will have uranium 235 number of protons and neutrons electrons and protons.Uranium-234 has 142 neutrons and!	2021-04-14 16:07:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2925318777561188, "perplexity": 1567.4713569646367}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077843.17/warc/CC-MAIN-20210414155517-20210414185517-00128.warc.gz"}
http://commons.apache.org/proper/commons-math/apidocs/org/apache/commons/math3/analysis/polynomials/PolynomialFunctionNewtonForm.html	org.apache.commons.math3.analysis.polynomials ## Class PolynomialFunctionNewtonForm • java.lang.Object • org.apache.commons.math3.analysis.polynomials.PolynomialFunctionNewtonForm • All Implemented Interfaces: UnivariateDifferentiableFunction, UnivariateFunction public class PolynomialFunctionNewtonForm extends Object implements UnivariateDifferentiableFunction Implements the representation of a real polynomial function in Newton Form. For reference, see Elementary Numerical Analysis, ISBN 0070124477, chapter 2. The formula of polynomial in Newton form is p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... + a[n](x-c[0])(x-c[1])...(x-c[n-1]) Note that the length of a[] is one more than the length of c[] Since: 1.2 • ### Constructor Summary Constructors Constructor and Description PolynomialFunctionNewtonForm(double[] a, double[] c) Construct a Newton polynomial with the given a[] and c[]. • ### Method Summary Methods Modifier and Type Method and Description protected void computeCoefficients() Calculate the normal polynomial coefficients given the Newton form. int degree() Returns the degree of the polynomial. static double evaluate(double[] a, double[] c, double z) Evaluate the Newton polynomial using nested multiplication. double[] getCenters() Returns a copy of the centers array. double[] getCoefficients() Returns a copy of the coefficients array. double[] getNewtonCoefficients() Returns a copy of coefficients in Newton form formula. DerivativeStructure value(DerivativeStructure t) Simple mathematical function. double value(double z) Calculate the function value at the given point. protected static void verifyInputArray(double[] a, double[] c) Verifies that the input arrays are valid. • ### Methods inherited from class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait • ### Constructor Detail • #### PolynomialFunctionNewtonForm public PolynomialFunctionNewtonForm(double[] a, double[] c) throws NullArgumentException, NoDataException, DimensionMismatchException Construct a Newton polynomial with the given a[] and c[]. The order of centers are important in that if c[] shuffle, then values of a[] would completely change, not just a permutation of old a[]. The constructor makes copy of the input arrays and assigns them. Parameters: a - Coefficients in Newton form formula. c - Centers. Throws: NullArgumentException - if any argument is null. NoDataException - if any array has zero length. DimensionMismatchException - if the size difference between a and c is not equal to 1. • ### Method Detail • #### value public double value(double z) Calculate the function value at the given point. Specified by: value in interface UnivariateFunction Parameters: z - Point at which the function value is to be computed. Returns: the function value. • #### value public DerivativeStructure value(DerivativeStructure t) Simple mathematical function. UnivariateDifferentiableFunction classes compute both the value and the first derivative of the function. Specified by: value in interface UnivariateDifferentiableFunction Parameters: t - function input value Returns: function result Since: 3.1 • #### degree public int degree() Returns the degree of the polynomial. Returns: the degree of the polynomial • #### getNewtonCoefficients public double[] getNewtonCoefficients() Returns a copy of coefficients in Newton form formula. Changes made to the returned copy will not affect the polynomial. Returns: a fresh copy of coefficients in Newton form formula • #### getCenters public double[] getCenters() Returns a copy of the centers array. Changes made to the returned copy will not affect the polynomial. Returns: a fresh copy of the centers array. • #### getCoefficients public double[] getCoefficients() Returns a copy of the coefficients array. Changes made to the returned copy will not affect the polynomial. Returns: a fresh copy of the coefficients array. • #### evaluate public static double evaluate(double[] a, double[] c, double z) throws NullArgumentException, DimensionMismatchException, NoDataException Evaluate the Newton polynomial using nested multiplication. It is also called Horner's Rule and takes O(N) time. Parameters: a - Coefficients in Newton form formula. c - Centers. z - Point at which the function value is to be computed. Returns: the function value. Throws: NullArgumentException - if any argument is null. NoDataException - if any array has zero length. DimensionMismatchException - if the size difference between a and c is not equal to 1. • #### computeCoefficients protected void computeCoefficients() Calculate the normal polynomial coefficients given the Newton form. It also uses nested multiplication but takes O(N^2) time. • #### verifyInputArray protected static void verifyInputArray(double[] a, double[] c) throws NullArgumentException, NoDataException, DimensionMismatchException Verifies that the input arrays are valid. The centers must be distinct for interpolation purposes, but not for general use. Thus it is not verified here. Parameters: a - the coefficients in Newton form formula c - the centers Throws: NullArgumentException - if any argument is null. NoDataException - if any array has zero length. DimensionMismatchException - if the size difference between a and c is not equal to 1. DividedDifferenceInterpolator.computeDividedDifference(double[], double[])	2016-05-04 17:25:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5917404294013977, "perplexity": 3230.941304956626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860123840.94/warc/CC-MAIN-20160428161523-00107-ip-10-239-7-51.ec2.internal.warc.gz"}
http://michaelsanford.com/category/science/	# Don't explain special relativity, experience it in a game Several years ago I wrote an article titled How GPS makes use of General Relativity. I was motivated to show how the theory of relativity is not only interesting but actually used in practice in the real world. Researchers at MIT Game Lab have developed a game that attempts (and succeeds, to a certain extent) not to explain the effect of the theory of relativity, but to allow people to experience it. The game is of the exploration type in which the user walks around collecting spheres. Collecting a sphere lowers the speed of light in the in-game universe. This brings interaction with the universe which is normally outside a human frame of reference down to a human scale: walking speed. Users experience time dilation, the Doppler effectLorentz transformation and the searchlight effect, all effects of special relativity and a consequence of approaching the speed of light from $2.99 \times 10^8 {m/s}$ down to about $1.5 m/s$. Pedagogically, the idea is simply brilliant. It removes the all-too-familiar "So, imagine you're in a rocket ship approaching $c$" that precedes every discussion you've ever had about special relativity. As any fan of Douglas Adams will know, humans are generally incapable of doing that. In human terms, how fast is $c$? It's fast. Faster than you can imagine. Check out the game for yourself, for Windows and Mac: http://gamelab.mit.edu/games/a-slower-speed-of-light/ • Intel Core 2 Duo T9900 or Core i7 (2.8GHz clock speed) • Windows 7 and Mac OS X 10.6.8 (Snow Leopard) or higher • AMD Radeon HD 6970M/AMD Mobility Radeon HD 4850/Nvidia GeForce 9600M GT • 8GB RAM # Making science accessible Perusing the archives of the scientific journal Nature I came across an article dealing with ethics boards: Sieber says review boards need to be more rigorous in their risk assessments and decisions, and says they should call in outside experts where necessary. She adds, however, that a more common problem with institutional review boards is not a willingness to allow dubious studies, but an overly cautious approach that comes from ignorance of the experimental methods involved. I think the moral of the story is: make sure you make your experiment accessible to others so they'll understand what the heck you want to do, or have done. Giles, J. "Warning flag for ethics boards"Nature 443, 127 (14 September 2006) \| doi:10.1038/443127 (link). # How GPS makes use of General Relativity I recently went to a show at the Dow Planetarium that dealt with relativity. Like most science shows designed for the general consumer, it told me little I didn’t already know. I don’t mean that to sound arrogant, I’ve simply been studying astronomy and relativity since grade school. But, it did bring up an intriguing point that I had never considered: without the theories of General and Special Relativity GPSwould be of no use at all. ### History of GPS GPS was developed in the 1960s by the US DoD but its stewardship was later transferred to the IGEBcommittee on which sits members of the Joint Chiefs of Staff, NASA, Department of Defense and the Department of Transportation, among other agencies. From its inception, GPS had–and still has–the ability to intentionally provide corrupted solutions to receivers through a mechanism called Selective Ability: SA was a technique implemented by the DOD to intentionally degrade a user’s navigation solution[…]The net result of SA was about a five-fold increase in positioning error. DOD achieved signal degradation by altering (also known as dithering) the satellite clock. Another means designed by DOD to degrade GPS performance was to broadcast less accurate ephemeris parameters. The DOD-authorized users were able to undo SA. However, due to the fact that SA is spatially correlated, civil users were able to eliminate SA through the implementation of Differential GPS (DGPS), albeit an additional expense on the part of the users. SA was used to protect the security interests of the U.S. and its allies by globally denying the full accuracy of the civil system to potential adversaries. A presidential order in May 2001 discontinued the active implementation of SA; though it still remains an option, the military has preferred the tack of locally impeding GPS reception where desirable, rather than diluting the global network. Something to keep in mind is that it is wholly owned, operated and maintained by the US government. Like a driver’s license, its use by any civilian is a privilege, not a right, and like a driver’s license, most do not understand this. ### The Orbital Assets At any given time there are, as a baseline, 24 satellites in orbit around the Earth, though at the time of this writing there are actually 29 in service. They orbit in such a pattern and altitude (approximately 20 000 km) as to ensure that at least four satellites are visible from a clear horizon from any location on Earth. The group of orbiting GPS satellites is known commonly as the “GPS constellation”. The GPS constellation is not actually made of a series of satellites in geosynchronous or geostationary orbit–they instead have a period of about half a day. How, then, can we determine our position, one may ask? For the most simplistic co-ordinate triangulation on Earth, we use a single emitter and at least two receivers. These receivers then compare the times that they received the signal from the common emitter and from that determine its position. What may not be immediately apparent is that this works not because both receivers are stationary but because their relative positions are known at the time of signal acquisition. It is just as easy and reliable to tri-angulate with two receivers on jeeps moving in a random pattern as long as the instantaneous position of both jeeps is known at the time of signal acquisition. This is precisely how GPS works. The ephemeral data transmitted by the satellite, along with the time from an atomic clock are used to compute the receiver’s position. Though in the previous example we had one emitter and many receivers, many emitters and one receiver work just as well if the emitters broadcast not just a ping but their own co-ordinates and time, as in GPS. ### The Role of Relativity To achieve a navigation resolution of a few metres, the time of the satellites must be known to an accuracy of approximately 30 nanoseconds. However, relativity intervenes to predict that because the satellites are orbiting with a velocity of around 14 000 km/h a ground-based observer will perceive the atomic clocks on the GPS satellites tick more slowly than their own, losing slightly less than 10 microseconds a day. However, because the constellation maintains an extremely high orbit, General Relativity predicts that the weaker distortion of space-time in high orbit compared with the stronger one at the surface of the Earth will cause the clocks on the GPS satellites to tick faster than those on the ground, gaining slightly less than 50 microseconds a day. This leads to a net imprecision of about +35 microseconds that accumulates every day. Since the time of each satellite must be known to an accuracy of 30 nanoseconds, but the satellites’ clocks gain 35 000 nanoseconds a day, the system becomes useless by the time it takes to start-up most hand-held GPS receivers. ### The Simple Remedy The designers of the GPS constellation took into account the faster atomic clock-ticking predicted by General Relativity and slowed the clocks down so that the observer’s time would appear correct. As Ohio State University professor of astronomy Richard Pogge put it Relativity is not just some abstract mathematical theory: understanding it is absolutely essential for our global navigation system to work properly!	2017-04-24 09:32:49	{"extraction_info": {"found_math": true, "script_math_tex": 2, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40057891607284546, "perplexity": 1350.3358468937906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917119225.38/warc/CC-MAIN-20170423031159-00130-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.trustudies.com/question/79/on-comparing-the-ratios-a-1-over-a-2-/	3 Tutor System Starting just at 265/hour # On comparing the ratios $${{a_1}\over {a_2}},{{b_1}\over {b_2}},{{c_1}\over {c_2}}$$ find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) $$5x - 4y + 8 = 0, 7x + 6y – 9 = 0$$ (ii)$$9x + 3y + 12 = 0, 18x + 6y + 24 = 0$$ (iii)$$6x - 3y + 10 = 0, 2x – y + 9 = 0$$ Ans.(i) $$5x - 4y + 8 = 0, 7x + 6y – 9 = 0$$ On comparing these equations with the general form: $$ax^2 + bx + c$$ $$a_1 = 5, b_1 = -4, c_1 = 8$$, $$a_2 = 7, b_2 = 6, c_2 = -9$$ Here $${{a_1}\over {a_2}} \ne {{b_1}\over {b_2}}$$ as, $${{5}\over {7}} \ne {{-4}\over {6}}$$ So, these lines have a unique solution which means they intersect at one point. (ii) $$9x + 3y + 12 = 0,18x + 6y + 24 = 0$$ On comparing these equations with the general form: $$ax^2 + bx + c$$ $$a_1 = 9, b_1 = 3, c_1 = 12$$ $$a_2 = 18, b_2 = 6, c_2 = 24$$ Here $${{a_1}\over {a_2}} = {{b_1}\over {b_2}} = {{c_1}\over {c_2}}$$ as, $${{9}\over {18}} = {{3}\over {6}} = {{12}\over {24}}$$ So, these lines are coincident. (iii) $$6x - 3y + 10 = 0,2x – y + 9 = 0$$ On comparing these equations with the general form: $$ax^2 + bx + c$$ $$a_1 = 6, b_1 = -3, c_1 = 10$$ $$a_2 = 2, b_2 = -1, c_2 = 9$$ Here $${{a_1}\over {a_2}} = {{b_1}\over {b_2}} \ne {{c_1}\over {c_2}}$$ as, $${{6}\over {2}} = {{-3}\over {-1}} \ne {{10}\over {9}}$$ So, these lines are parallel to each other.	2023-03-21 18:17:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8187302947044373, "perplexity": 1212.657742862686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943704.21/warc/CC-MAIN-20230321162614-20230321192614-00560.warc.gz"}
http://hipstercircuits.com/accelerated-stepper-motors-on-beaglebone/	# Accelerated stepper motors on BeagleBone Programmable Realtime Unit! The guy that thought of that name must be a poet. The above picture is what constant acceleration looks like on an oscilloscope. Boxysean has a great post on getting started with PRU on BeagleBone. I’m pretty sure that guy is also a contestant in the BeagleBone cape contest, so best of luck to him! Anyways, here is a video that pretty much demonstrates all I have to say in this post: Constant speed Well, constant speed in one dimention is not very impressive, but this one uses the PRU for stepping. That is actually pretty cool. The following script calculates the number of instructions to wait to get the right speed. That is placed in a list and fed in to the data ram of PRU 0. ```import pypruss as pru # The Programmable Realtime Unit Library''' speed_test.py - test script for testing fixed speed in PyPRUSS library''' import pypruss as pru # The Programmable Realtime Unit Library import numpy as np # Needed for braiding the pins with the delays speed = 80 # The travelling speed in mm/s distance = 50 # Distance in mm microstepping = 2.0 # With microstepping, every step need four ticks. steps_pr_mm = 6.1microstepping # Number of ticks the stepper needs to go one mm pru_hz = 20010001000 # The PRU has a speed of 200 MHz s_pr_inst = 1.0/pru_hz # I take it every instruction is a single cycle instruction inst_pr_loop = 16 # This is the minimum number of instructions needed to step. inst_pr_delay = 2 # Every loop adds two instructions: i-- and i != 0 s_pr_mm = 1.0/speed # Seconds pr mm is the inverse of the mm/s s_pr_step = s_pr_mm/steps_pr_mm # To get the time to wait for each step, divide by mm/step inst_pr_step = s_pr_step/s_pr_inst # Calculate the number of instructions of delay pr step. inst_pr_step /= 2.0 # To get a full period, we must divide by two. inst_pr_step -= inst_pr_loop # Remove the "must include" number of steps inst_pr_step /= inst_pr_delay # Yes, this must be right.. inst_pr_step = int(inst_pr_step) # Make it an int num_steps = int(distancesteps_pr_mm) # Number of ticks in total steps = [(1<<12), 0]num_steps # Make the table of ticks for the stepper. delays = [inst_pr_step]2num_steps # Make the table of delays data = np.array([steps, delays]) # Make a 2D matrix combining the ticks and delays data = data.transpose().flatten() # Braid the data so every other item is a data = [num_steps2+1]+list(data) # Make the data into a list and add the number of ticks total pru_num = 0 # PRU0 pru.init(pru_num, "./firmware.bin") # Load PRU 0 with the firmware. pru.set_data(pru_num, data) # Load the data in the PRU ram pru.wait_for_event(pru_num) # Wait a while for it to finish. pru.disable(pru_num) # Clean shit up, we don't want to be piggies. ''' Put your thing down, flip it and reverse it ''' steps = list(np.array(steps) + (1<<13)) # Make the table of ticks for the stepper. data = np.array([steps, delays]) # Make a 2D matrix combining the ticks and delays data = data.transpose().flatten() # Braid the data so every other item is a data = [num_steps2+1]+list(data) # Make the data into a list and add the number of ticks total pru_num = 0 # PRU0 pru.init(pru_num, "./firmware.bin") # Load PRU 0 with the firmware. pru.set_data(pru_num, data) # Load the data in the PRU ram pru.wait_for_event(pru_num) # Wait a while for it to finish. pru.disable(pru_num) # Clean shit up, we don't want to be piggies.``` General purpose firmware To accompany these constant speed and constant acceleration scripts is a very minimalistic firmware script that I have written. It basically just sets pin states, delays, sets more pin states and delays and so on. It is good for testing ideas and maybe get started quickly with something. The number of toggles is limited to 1023 since every pin state takes 4 bytes and every delay takes 4 bytes as well and the total available data memory is 8KB. Anyways here is the assembly code for the GPIO firmware: ```.origin 0 .entrypoint START #define PRU0_ARM_INTERRUPT 19 #define GPIO1 0x4804c000 // The adress of the GPIO1 #define GPIO_DATAOUT 0x13c // This is the register for settign data #define PIN_OFFSET 0x00000004 // Offset for the pins (reserve the first adress for abort) START: LBCO r0, C4, 4, 4 // clear that bit CLR r0, r0, 4 // No, really, clear it! SBCO r0, C4, 4, 4 // Actually i have no idea what this does MOV r0, 0 // The first register contains the loop count LBBO r1, r0, LEN_ADDR, 4 // Load the number of steps to perform into r1 MOV r4, PIN_OFFSET // r4 is the pin and delay counter SET_PINS: LBBO r2, r4, 0, 4 // Load pin data into r2 MOV r3, GPIO1 \| GPIO_DATAOUT // Load the address of GPIO \| DATAOUT in r3 SBBO r2, r3, 0, 4 // Set the pins ADD r4, r4, 4 // r4 += 4 LBBO r0, r4, 0, 4 // Load Delay into r0 DELAY: SUB r0, r0, 1 // Delay the required ticks QBNE DELAY, r0, 0 ADD r4, r4, 4 // r4 += 4 SUB r1, r1, 1 // Decrement r1 QBNE SET_PINS, r1, 0 // Branch back to SET_PINS if r0 != 0, abort! MOV R31.b0, PRU0_ARM_INTERRUPT+16 // Send notification to Host for program completion HALT``` Given a constant acceleration, the speed at each step must be calculated. We do not know the time intervals in advance. All we know is the distances, so the time steps must be calculated. At what time does the stepper reach maximum speed? Well, it there is constant acceleration, it can be calculated by using the formula: $Latex formula$ Where v is velocity at time t, u is the initial velocity and a is acceleration. So for a constant acceleration of 200mm/s^2, a max speed of 80mm/s and an initial velocity of 0 we hit maximum speed after 0.4 seconds. Ok, so knowing the time at which the max speed is hit, we can calculate the position at that point. Using the formula for constant acceleration: $Latex formula$ where s is the distance travelled at time t, a is acceleration and u is initial velocity. Solving this for t = 0.4 seconds yields 16mm travel distance. Now all the time stamps in the range from 0 to 16mm must be calculated. Let’s say the stepper needs 6.1 steps to travel one mm. In that case we have a step size of 0.16mm and that becomes the increment size. So again using the formula for constant acceleration, but this time solving for t we have: $Latex formula$ Once the timestamps have been calculated its is trivial to extract the differences to calculate the actual delays and convert it to instructions. Finally the table must be reversed and used for deceleration. In between the ramp up and the ramp down, we fill in with the delays for max speed. Here is the script used in the video. ```''' accel_test.py - test script for testing fixed speed in PyPRUSS library''' import pypruss as pru # The Programmable Realtime Unit Library import numpy as np # Needed for braiding the pins with the delays max_speed = 80 # Top speed in mm/s min_speed = 0 # Minimum speed (V0). distance = 50 # The distance to travel in mm acceleration = 200 # Acceleration in mm/s^2 microstepping = 2.0 # With microstepping, every step need four ticks. steps_pr_mm = 6.1microstepping # Number of ticks the stepper needs to go one mm pru_hz = 20010001000 # The PRU has a speed of 200 MHz s_pr_inst = 1.0/pru_hz # I take it every instruction is a single cycle instruction inst_pr_loop = 16 # This is the minimum number of instructions needed to step. inst_pr_delay = 2 # Every loop adds two instructions: i-- and i != 0 Vm = max_speed/1000.0 # The travelling speed in m/s a = acceleration/1000.0 # Accelleration in m/s/s s = distance/1000.0 # Distance in m ds = (1.0/(steps_pr_mm1000.0)) # Delta S, distance in meters travelled pr step. u = min_speed/1000.0 # Minimum speed in m/s tm = (Vm-u)/a # Calculate the time for when max speed is met. sm = utm+0.5atmtm # Calculate the distace travelled when max speed is met def t_by_s(s): # Get the timestamp given a certain distance. return (-u+np.sqrt(2as+uu))/a # This is the s = ut+1/2at^2 solved with reference to t def sec_to_inst(s): # Shit, I'm missing MGP for this?? inst_pr_step = s/s_pr_inst # Calculate the number of instructions of delay pr step. inst_pr_step /= 2.0 # To get a full period, we must divide by two. inst_pr_step -= inst_pr_loop # Remove the "must include" number of steps inst_pr_step /= inst_pr_delay # Yes, this must be right.. return int(inst_pr_step) # Make it an int distances = np.arange(0, sm, ds) # Table of distances t_in_s = map(t_by_s, distances) # Make a table of times, the time at which a tick occurs d_in_s = np.diff(t_in_s)/2.0 # We are more interested in the delays pr second. Half it, cos we will double it later dd_in_s = np.array([d_in_s, d_in_s]) # Double the array dd_in_s = dd_in_s.transpose().flatten() # Transposing and flattening braids the data. num_steps = int(distancesteps_pr_mm) # Number of ticks in total steps = [(1<<12), 0]num_steps # Make the table of ticks for the stepper. delays = map(sec_to_inst, dd_in_s) # Number of instructions pr. step is now calculated i_steps = num_steps-len(delays) # Find out how many delays are missing i_dlys = delays[-1::]i_steps2 # Make the intermediate steps delays = delays+i_dlys+delays[::-1] # Add the missing delays. These are max_speed data = np.array([steps, delays]) # Make a 2D matrix combining the ticks and delays data = data.transpose().flatten() # Braid the data so every other item is a data = [num_steps2+1]+list(data) # Make the data into a list and add the number of ticks total pru_num = 0 # PRU0 pru.init(pru_num, "./firmware.bin") # Load PRU 0 with the firmware. pru.set_data(pru_num, data) # Load the data in the PRU ram pru.wait_for_event(pru_num) # Wait a while for it to finish. pru.disable(pru_num) # Clean shit up, we don't want to be piggies.``` ## 5 thoughts on “Accelerated stepper motors on BeagleBone” 1. GP says: Your PyPruss code is awesome, thanks for making that available! 2. GP says: Hi again, What distro are you running on your BeagleBone? I have an A6 here running fully patched A6A Angstrom and I am getting the following error when trying to compile your PRU code: cc -Wall -Iapp_loader/include -I/usr/include/python2.7 -D__DEBUG -O2 -mtune=cortex-a8 -march=armv7-a -shared -fPIC -c -o pypruss.o pypruss.c pypruss.c:1:0: error: bad value (armv7-a) for -march= switch pypruss.c:1:0: error: bad value (cortex-a8) for -mtune= switch make: ** [pypruss.o] Error 1 3. GP says: Never mind, I am an idiot. In the wrong window trying to compile this on my desktop machine instead of on the BeagleBone. Compiles with no errors perfectly! • Elias says: GP, I admire your courage for trying to get this to work already! I have post waiting about PyPRUSS, how to install and so on so your experiences will be highly valued!	2013-05-26 02:27:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5236382484436035, "perplexity": 1551.9392011868936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706484194/warc/CC-MAIN-20130516121444-00046-ip-10-60-113-184.ec2.internal.warc.gz"}
https://triangle.mth.kcl.ac.uk/?off=20&search=location%3AKCL	## Directions This institute may be found at Strand in Central London, just north of the Thames (map). Getting to the Strand Campus: • #### By underground Temple (District and Circle lines): 2 minute walk. Charing Cross (Bakerloo and Northern lines): 10 minute walk, Embankment (District, Circle and Bakerloo lines): 10 minute walk, Waterloo (Jubilee, Northern, Bakerloo, Waterloo & City lines): 12 minute walk, Holborn (Central and Picadilly lines): 12 minute walk,Chancery Lane (Central line): use exit 4 - 15 minute walk. • #### By train Charing Cross: 9 minute walk. Waterloo: 12 minute walk. Waterloo East: 10 minute walk. Blackfriars: 12 minute walk. • #### By bus Buses stopping outside the College: 1, 4, 26, 59, 68, 76, X68, 168, 171, 172, 176(24 hour), 188, 243 (24 hour), 341 (24 hour), 521, RV1. ## Seminars at King's College London Found at least 20 result(s) ### 28.04.2021 (Wednesday) #### Higher-derivative Supergravity and AdS4 Holography Regular Seminar Valentin Reys (Leuven U.) at: 13:45 KCLroom Zoom, See abstract abstract: This talk will discuss higher-derivative corrections to four-dimensional gauged supergravity and their holographic implications. After briefly reviewing the construction of N=2 supersymmetric higher-derivative invariants, I will introduce a four-derivative action parameterized by two real constants. In this theory, one can show that the two-derivative solutions are not modified by the higher-derivative corrections. This fact has important consequences for the regularized on-shell action, as well as for the thermodynamics of black hole solutions. Moreover, in the context of AdS4/CFT3 holography, I will explain how our results lead to an explicit expression for subleading corrections in the large-N expansion of supersymmetric partition functions of a large class of dual field theories arising from M2 and M5 branes. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link] ### 21.04.2021 (Wednesday) #### Current operators in integrable models: Review of recent results Regular Seminar Balazs Pozsgay (Eotvos Lorand U., Budapest, Inst. Theor. Phys.) at: 13:45 KCLroom Zoom, See abstract abstract: We review the recent progress regarding current operators in integrable models, focusing especially on integrable spin chains. These operators describe the flow of the conserved charges, and they are important for the construction of Generalized Hydrodynamics. They are also connected to long range deformations and TTbar-like deformations of the spin chains, and also to the theory of factorized correlation functions. We argue that these operators are very special, because their mean values can be computed relatively easily even in nested spin chains. This is rather unique because mean values in nested models are rather difficult to compute for generic operators. We review these various connections and also show how to construct current operators using the Quantum Inverse Scattering Approach, the canonical framework developed by the Leningrad school. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link] ### 31.03.2021 (Wednesday) #### On QFT in de Sitter Regular Seminar Victor Gorbenko (Stanford University) at: 16:00 KCLroom Zoom, See abstract abstract: I will discuss the tools we are developing to calculate correlation functions of primordial inflationary perturbations. In this talk, we will focus on the limit where gravitational excitations are neglected and the cosmological spacetime is assumed to be exactly de Sitter. Even in this simplifying limit, which corresponds to a Quantum Field Theory on a de Sitter background, very few examples of explicit analytic calculations exist and little is known about basic properties of the correlators. I will show that for any dS QFT there exists a theory formulated in a Euclidean Anti-de SItter space and which reproduces all the correlation functions. This leads to major technical simplifications and allows to demonstrate various analytic properties of the cosmological observables. Understanding of these properties has both phenomenological applications in inflation, as well as provides some hints about more fundamental description for cosmological spacetimes. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link] ### 24.03.2021 (Wednesday) #### 3d Large N vector models at the boundary. Regular Seminar Lorenzo di Pietro (University of Trieste) at: 13:45 KCLroom Zoom, See abstract abstract: In this talk I will discuss boundary RG flows for a 4d free scalar field coupled to large N vector models on a 3d boundary. These RG flows connect decoupled fixed points with the free and the critical vector model on the boundary, and they enjoy strong-weak dualities. I will also consider adding gauge fields to the setup. [Please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link] ### 10.03.2021 (Wednesday) #### The statistical mechanics of near-extremal and near-BPS black holes Regular Seminar Luca Iliesiu (Stanford University) at: 13:45 KCLroom Zoom, See abstract abstract: An important open question in black hole thermodynamics is about the existence of a "mass gap" between an extremal black hole and the lightest near-extremal state within a sector of fixed charge. In this talk, I will discuss how to reliably compute the partition function of 4d Reissner-Nordstrom near-extremal black holes at temperature scales comparable to the conjectured gap. I will show that the density of states at fixed charge does not exhibit a gap in the simplest gravitational non-supersymmetric theories; rather, at the expected gap energy scale, we see a continuum of states whose meaning we will extensively discuss. Finally, I will present a similar computation for nearly-BPS black holes in 4d N=2 supergravity. As opposed to their non-supersymmetric counterparts, such black holes do in fact exhibit a gap consistent with various string theory predictions. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link] ### 03.03.2021 (Wednesday) #### Where is String Theory? Triangular Seminar Pedro Vieira (Perimeter Institute) at: 14:30 KCLroom online abstract: The S-matrix bootstrap can be used to carve out the space of physical theories. What can we say about the space of theories of quantum gravity based on this approach? Based on work with Andrea Guerrieri and Joao Penedones. Zoom link: https://zoom.us/j/94504664165?pwd=c3VmMDNsbkRwWWdoUUxIRDhUcjB4dz09 (for password email dionysios.anninos@kcl.ac.uk) ### 03.03.2021 (Wednesday) #### Hyperbolic compactification of M-theory and de Sitter quantum gravity Triangular Seminar Gonzalo Torroba (Centro Atomico Bariloche ) at: 16:30 KCLroom online abstract: In this talk we will present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy, along with a single classical source (7-form flux), contains a 3-term structure for volume stabilization at positive potential energy. We find that a combination of warping and hyperbolic rigidity effects can stabilize the metric and form field. A simple generalization incorporating 4-form flux produces axion monodromy inflation, along with other forms of accelerated expansion. Our approach provides a simple uplift of the large-N M2-brane theory to de Sitter, and introduces new connections between mathematics and the physics of string/M theory compactifications. Zoom link: https://zoom.us/j/94504664165?pwd=c3VmMDNsbkRwWWdoUUxIRDhUcjB4dz09 (for password email dionysios.anninos@kcl.ac.uk) ### 24.02.2021 (Wednesday) #### Causal symmetry breaking: from quantum chaos to wormholes Regular Seminar Julian Sonner (University of Geneva) at: 13:15 KCLroom Zoom, See abstract abstract: Quantum chaotic systems are often defined via the assertion that their spectral statistics coincides with, or is well approximated by, random matrix theory. In this talk I will explain how the universal content of random matrix theory emerges as the consequence of a simple symmetry-breaking principle and its associated Goldstone modes. This approach gives a natural way to identify wormhole-like correlations, even for individual theories, in particular in theories with gravity duals. I will also discuss how to extend the Goldstone effective-field-theory approach to study operator correlation functions, and explain the relation of the EFT of quantum chaos to the bulk physics of wormhole-like geometries. [Please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link] ### 17.02.2021 (Wednesday) #### Bootstrapping BPS algebras from colored crystals Regular Seminar Wei Li (Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing.) at: 13:45 KCLroom Zoom, See abstract abstract: I will explain a method of constructing BPS algebras for string theory on generic toric Calabi-Yau threefolds. The approach is a bootstrapâ€ method based on the 3D colored crystals that describe BPS states of the system. The resulting algebras are quiver Yangians Y(Q,W) that are associated with the quiver and the superpotential of the theory. [Please email alejandro.cabo_bizet@kcl.ac.uk for zoom link] ### 10.02.2021 (Wednesday) #### Adventures in Machine Learning and Theoretical Physics Regular Seminar Thomas Fischbacher (Google Research) at: 13:45 KCLroom Zoom, See abstract abstract: Machine Learning has opened many new doors in science across multiple disciplines. Starting from recent work by the speaker and collaborators on in-depth explorations into the vacuum structure of gauged maximal supergravities using Machine Learning Technology, notably Google's TensorFlow library, we subsequently take a broader perspective on what happens when Machine Learning meets Physics. [Please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link] ### 27.01.2021 (Wednesday) #### Derivation of AdS/CFT for Vector Models Regular Seminar Shai Chester (Weizmann Institute) at: 13:45 KCLroom Zoom, See abstract abstract: We derive an explicit map between the singlet sector of the free and critical O(N) and U(N) vector models in any spacetime dimension above two and to all orders in 1/N, and a bulk higher spin theory in anti-de Sitter space in one higher dimension. For the boundary theory, we use the bilocal formalism of Jevicki et al to restrict to the singlet sector of the vector model. The bulk theory is defined from the boundary theory via our mapping and is a consistent quantum higher spin theory with a well defined action. Our mapping relates bilocal operators in the boundary theory to higher spin fields in the bulk, while single trace local operators in the boundary theory are related to boundary values of higher spin fields. [Please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link] ### 20.01.2021 (Wednesday) #### Towards all loop supergravity amplitudes Regular Seminar Agnese Bissi (Uppsala University) at: 13:45 KCLroom Zoom, See abstract abstract: In this talk I will discuss how to extract the most trascendental piece of the four graviton amplitude in type IIB supergravity on AdS_5 \times S_5 at any loop order, from the dual four point function in N=4 Super Yang Mills. I will describe how to construct this part of the correlator/amplitude and its significance. I will conclude with some open problems and future directions. [Please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link] ### 16.12.2020 (Wednesday) Regular Seminar Marco Meineri (CERN) at: 13:45 KCLroom Zoom abstract: We study quantum field theory in AdS_2 within the framework of Hamiltonian truncation. We start from a solvable theory and we deform the Hamiltonian with an interaction term. We find the spectrum as a function of the coupling, and we aim at connecting the solvable regime with the strongly coupled flat space physics. Hamiltonian truncation in infinite volume presents specific challenges, and special care is needed to recover the physical energy gaps as the cutoff is removed. To this end, we propose a general prescription, we give an argument for its validity and we check it in various examples. [For the link to the Zoom room, please email alejandro.cabo_bizet@kcl.ac.uk]. ### 09.12.2020 (Wednesday) #### An N=1 Lagrangian for an N=3 SCFT Regular Seminar Gabi Zafrir (Milan Bicocca U.) at: 13:45 KCLroom Zoom abstract: Purely N=3 SCFTs are naturally strongly coupled, and expected to not have a Lagrangian description manifesting their N=3 supersymmetry. However, these may be reached through a deformation of a theory preserving less supersymmetry. We shall present a strategy to try to conjecture such models, and exemplify it by presenting an N=1 model conjectured to flow to an N=3 SCFT, deformed by a marginal deformation. We shall also discuss and exemplify how such proposals can be tested. [For the Zoom link please email to alejandro.cabo_bizet@kcl.ac.uk] ### 30.11.2020 (Monday) #### Lonti: Conformal blocks in two dimensions Regular Seminar Gerard Watts (KCL) at: 10:30 KCLroom Online abstract: This is the live session included as part of the LonTI lecture on Conformal blocks in two dimensions. Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom.â€‹â€‹Conformal blocks are the building blocks of conformal field theories, the key ingredients of correlation functions and knowledge of these blocks is central to the conformal bootstrap. This lecture covers some general results before quickly specialising to two-dimensions. It discusses the implications of global conformal invariance and then the larger and more constraining Virasoro algebra symmetry. It introduces various methods of calculation including brute force, differential equations, recursion relations and closed formulae - and some aspects of bootstrap techniques. ### 25.11.2020 (Wednesday) #### Multilegs, Superfluids and Semiclassics Regular Seminar Riccardo Rattazzi (EPFL) at: 13:15 KCLroom Zoom abstract: Even in weakly coupled QFTs, perturbation theory breaks down when one considers amplitudes with a large number $n$ of legs. The series cleverly organizes as a double expansion in $g^2$ and $g^2n$. I show how the series in $g^2n$ can be fully captured by a semiclassical expansion around a non-trivial solution. Focussing on $U(1)$ symmetric $\|\phi\|^4$ theory in $4$ and $4-\epsilon$ dimension, I derive explicit and consistent all order results for the anomalous dimension of the complex operator $\phi^n$. When restricting to the Wilson-Fisher fixed point and working on the cylinder, the dominant trajectory is seen to correspond to a superfluid phase for the conserved U(1). This creates a correspondence between, on one side, the spectrum of operators and fusion coefficients and and on the other the spectrum of hydrodynamics modes and their interactions. The results also nicely match Monte Carlo simulations in 3D, compatibly with the stunt of taking $\epsilon=1$. [please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link] ### 24.11.2020 (Tuesday) Colloquium Matthias Gaberdiel (ETH Zurich) at: 14:30 KCLroom Zoom abstract: The AdS/CFT correspondence relates string theory on an Anti de Sitter (AdS) background to a dual conformal field theory (CFT) living on the boundary of AdS. One promising strategy for how to prove the duality is to consider the `tensionless' limit of string theory in which the dual CFT becomes weakly coupled. For the case of string theory on AdS3 we have recently identified a solvable world-sheet description for this tensionless limit. This opens the way towards proving the AdS/CFT correspondence, at least for this specific set-up. Please contact Dionysios Anninos (dionysios.anninos@kcl.ac.uk) for the zoom link. ### 23.11.2020 (Monday) #### Lonti 2020: Conformal blocks in two dimensions Regular Seminar Gerard Watts (KCL) at: 14:00 KCLroom Youtube abstract: â€‹â€‹Conformal blocks are the building blocks of conformal field theories, the key ingredients of correlation functions and knowledge of these blocks is central to the conformal bootstrap. This lecture covers some general results before quickly specialising to two-dimensions. It discusses the implications of global conformal invariance and then the larger and more constraining Virasoro algebra symmetry. It introduces various methods of calculation including brute force, differential equations, recursion relations and closed formulae - and some aspects of bootstrap techniques. ### 18.11.2020 (Wednesday) #### Universes as Bigdata: from Geometry, to Physics, to Machine-Learning Polygon Seminar Yang-Hui He (City) at: 13:00 KCLroom Online abstract: We briefly overview how historically string theory led theoretical physics first to algebraic/differential geometry, and then to computational geometry, and now to data science. Using the Calabi-Yau landscape - accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades - as a starting-point and concrete playground, we then launch to review our recent programme in machine-learning mathematical structures and address the tantalizing question of how AI helps doing mathematics, ranging from geometry, to representation theory, to combinatorics, to number theory. Zoom Link: https://us02web.zoom.us/j/83496714171?pwd=bld3QmQ2c21laWxEWTd6ejVQbjZ5dz09 (contact dionysios.anninos@kcl.ac.uk for password) ### 11.11.2020 (Wednesday) #### The large charge expansion Regular Seminar Susanne Reffert (University of Bern) at: 13:45 KCLroom Zoom, See abstract. abstract: In has become clear in recent years that working in sectors of large global charge of strongly coupled and otherwise inaccessible CFTs leads to important simplifications. It is indeed possible to formulate an effective action in which the large charge appears as a control parameter. In this talk, I will explain the basic notions of the large-charge expansion using the simple example of the O(2) model and then generalize to models with a richer structure which showcase other effects. [For the zoom link please email alejandro.cabo_bizet@kcl.ac.uk]	2021-10-24 22:04:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5287415385246277, "perplexity": 2416.3553998366515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587606.8/warc/CC-MAIN-20211024204628-20211024234628-00166.warc.gz"}
https://www.quizover.com/online/course/difference-equation-digital-filtering-by-openstax?page=1	# Difference equation (Page 2/2) Page 2 / 2 ## Finding difference equation Below is a basic example showing the opposite of the steps above: given a transfer function one can easily calculate thesystems difference equation. $H(z)=\frac{(z+1)^{2}}{(z-\frac{1}{2})(z+\frac{3}{4})}$ Given this transfer function of a time-domain filter, we want to find the difference equation. To begin with, expand bothpolynomials and divide them by the highest order $z$ . $H(z)=\frac{(z+1)(z+1)}{(z-\frac{1}{2})(z+\frac{3}{4})}=\frac{z^{2}+2z+1}{z^{2}+2z+1-\frac{3}{8}}=\frac{1+2z^{-1}+z^{-2}}{1+\frac{1}{4}z^{-1}-\frac{3}{8}z^{-2}}$ From this transfer function, the coefficients of the two polynomials will be our ${a}_{k}()$ and ${b}_{k}()$ values found in the general difference equation formula, [link] . Using these coefficients and the above form of the transferfunction, we can easily write the difference equation: $x(n)+2x(n-1)+x(n-2)=y(n)+\frac{1}{4}y(n-1)-\frac{3}{8}y(n-2)$ In our final step, we can rewrite the difference equation in its more common form showing the recursive nature of the system. $y(n)=x(n)+2x(n-1)+x(n-2)+\frac{-1}{4}y(n-1)+\frac{3}{8}y(n-2)$ ## Solving a lccde In order for a linear constant-coefficient difference equation to be useful in analyzing a LTI system, we must be able tofind the systems output based upon a known input, $x(n)$ , and a set of initial conditions. Two common methods exist for solving a LCCDE: the direct method and the indirect method , the later being based on the z-transform. Below we will briefly discussthe formulas for solving a LCCDE using each of these methods. ## Direct method The final solution to the output based on the direct method is the sum of two parts, expressed in the followingequation: $y(n)={y}_{h}(n)+{y}_{p}(n)$ The first part, ${y}_{h}(n)$ , is referred to as the homogeneous solution and the second part, ${y}_{h}(n)$ , is referred to as particular solution . The following method is very similar to that used to solve many differential equations, so if youhave taken a differential calculus course or used differential equations before then this should seem veryfamiliar. ## Homogeneous solution We begin by assuming that the input is zero, $x(n)=0$ .Now we simply need to solve the homogeneous difference equation: $\sum_{k=0}^{N} {a}_{k}()y(n-k)=0$ In order to solve this, we will make the assumption that the solution is in the form of an exponential. We willuse lambda, $\lambda$ , to represent our exponential terms. We now have to solve thefollowing equation: $\sum_{k=0}^{N} {a}_{k}()\lambda ^{(n-k)}=0$ We can expand this equation out and factor out all of thelambda terms. This will give us a large polynomial in parenthesis, which is referred to as the characteristic polynomial . The roots of this polynomial will be the key to solving the homogeneousequation. If there are all distinct roots, then the general solution to the equation will be as follows: ${y}_{h}(n)={C}_{1}(){\lambda }_{1}()^{n}+{C}_{2}(){\lambda }_{2}()^{n}+\dots +{C}_{N}(){\lambda }_{N}()^{n}$ However, if the characteristic equation contains multiple roots then the above general solution will be slightlydifferent. Below we have the modified version for an equation where ${\lambda }_{1}$ has $K$ multiple roots: ${y}_{h}(n)={C}_{1}(){\lambda }_{1}()^{n}+{C}_{1}()n{\lambda }_{1}()^{n}+{C}_{1}()n^{2}{\lambda }_{1}()^{n}+\dots +{C}_{1}()n^{(K-1)}{\lambda }_{1}()^{n}+{C}_{2}(){\lambda }_{2}()^{n}+\dots +{C}_{N}(){\lambda }_{N}()^{n}$ ## Particular solution The particular solution, ${y}_{p}(n)$ , will be any solution that will solve the general difference equation: $\sum_{k=0}^{N} {a}_{k}(){y}_{p}(n-k)=\sum_{k=0}^{M} {b}_{k}()x(n-k)$ In order to solve, our guess for the solution to ${y}_{p}(n)$ will take on the form of the input, $x(n)$ . After guessing at a solution to the above equation involving the particular solution, one onlyneeds to plug the solution into the difference equation and solve it out. ## Indirect method The indirect method utilizes the relationship between the difference equation and z-transform, discussed earlier , to find a solution. The basic idea is to convert the differenceequation into a z-transform, as described above , to get the resulting output, $Y(z)$ . Then by inverse transforming this and using partial-fractionexpansion, we can arrive at the solution. $Z\left\{y,\left(n+1\right),-,y,\left(n\right)\right\}=zY\left(z\right)-y\left(0\right)$ This can be interatively extended to an arbitrary order derivative as in Equation [link] . $Z\left\{-,\sum _{m=0}^{N-1},y,\left(n-m\right)\right\}={z}^{n}Y\left(z\right)-\sum _{m=0}^{N-1}{z}^{n-m-1}{y}^{\left(m\right)}\left(0\right)$ Now, the Laplace transform of each side of the differential equation can be taken $Z\left\{\sum _{k=0}^{N},{a}_{k},\left[y,\left(n-m+1\right),-,\sum _{m=0}^{N-1},y,\left(n-m\right),y,\left(n\right)\right],=,Z,\left\{x,\left(,n,\right)\right\}\right\}$ which by linearity results in $\sum _{k=0}^{N}{a}_{k}Z\left\{y,\left(n-m+1\right),-,\sum _{m=0}^{N-1},y,\left(n-m\right),y,\left(n\right)\right\}=Z\left\{x,\left(,n,\right)\right\}$ and by differentiation properties in $\sum _{k=0}^{N}{a}_{k}\left({z}^{k},Z,\left\{y,\left(,n,\right)\right\},-,\sum _{m=0}^{N-1},{z}^{k-m-1},{y}^{\left(m\right)},\left(0\right)\right)=Z\left\{x,\left(,n,\right)\right\}.$ Rearranging terms to isolate the Laplace transform of the output, $Z\left\{y,\left(,n,\right)\right\}=\frac{Z\left\{x,\left(,n,\right)\right\}+{\sum }_{k=0}^{N}{\sum }_{m=0}^{k-1}{a}_{k}{z}^{k-m-1}{y}^{\left(m\right)}\left(0\right)}{{\sum }_{k=0}^{N}{a}_{k}{z}^{k}}.$ Thus, it is found that $Y\left(z\right)=\frac{X\left(z\right)+{\sum }_{k=0}^{N}{\sum }_{m=0}^{k-1}{a}_{k}{z}^{k-m-1}{y}^{\left(m\right)}\left(0\right)}{{\sum }_{k=0}^{N}{a}_{k}{z}^{k}}.$ In order to find the output, it only remains to find the Laplace transform $X\left(z\right)$ of the input, substitute the initial conditions, and compute the inverse Z-transform of the result. Partial fraction expansions are often required for this last step. This may sound daunting while looking at [link] , but it is often easy in practice, especially for low order difference equations. [link] can also be used to determine the transfer function and frequency response. As an example, consider the difference equation $y\left[n-2\right]+4y\left[n-1\right]+3y\left[n\right]=cos\left(n\right)$ with the initial conditions ${y}^{\text{'}}\left(0\right)=1$ and $y\left(0\right)=0$ Using the method described above, the Z transform of the solution $y\left[n\right]$ is given by $Y\left[z\right]=\frac{z}{\left[{z}^{2}+1\right]\left[z+1\right]\left[z+3\right]}+\frac{1}{\left[z+1\right]\left[z+3\right]}.$ Performing a partial fraction decomposition, this also equals $Y\left[z\right]=.25\frac{1}{z+1}-.35\frac{1}{z+3}+.1\frac{z}{{z}^{2}+1}+.2\frac{1}{{z}^{2}+1}.$ Computing the inverse Laplace transform, $y\left(n\right)=\left(.25{z}^{-n}-.35{z}^{-3n}+.1cos\left(n\right)+.2sin\left(n\right)\right)u\left(n\right).$ One can check that this satisfies that this satisfies both the differential equation and the initial conditions. 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? 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 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. what is nano technology what is system testing? preparation of nanomaterial how to synthesize TiO2 nanoparticles by chemical methods Zubear 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!	2018-12-15 13:25:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 39, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6966456770896912, "perplexity": 822.1096249665476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376826856.91/warc/CC-MAIN-20181215131038-20181215153038-00375.warc.gz"}
https://physics.stackexchange.com/questions/207033/why-are-the-topological-dimensions-of-spatial-timelike-null-infinities-0-0-2-res	# Why are the topological dimensions of spatial\timelike\null infinities 0\0\2 respectively? I have a question regarding spatial, timelike and null infinities in Minkowski spacetime. I shall start with an explanation of my intuitive understanding of these concepts, and then proceed to my actual question: My Intuitive Understanding Null infinity: I assume that I am currently in the (arbitrary) origin of flat spacetime. I trace the past history of a sphere, which is currently a just a point in the origin and whose radius grows at the rate of the speed of light as time goes to the past. As I look farther back in the past, this sphere becomes larger and larger. When I take time to approach minus infinity, then this is past null infinity. It is topologically a 2-sphere (since it was a sphere all along the way the process...). Likewise, future null infinity is the limit at infinite time of a sphere expanding towards to the future at the speed of light, and it is also a 2-sphere. Timelike infinity: Light from a specific very early time could have arrived at the origin only if it started anywhere on the appropriate sphere. In contrast, a massive particle in inertial motion could have arrived from anywhere inside the sphere (the exact point determined by the momentum). So it seems to me that past timelike infinity: should topologically be of dimension not lower than that of past null infinity - 2. Likewise, future timelike infinity: should have dimension not lower than 2. spatial infinity By the same argument I would think that there should be a past spatial infinity and a future spatial infinity, with dimensions not lower 2. My Question As demonstrated in a Penrose-Carter diagram, the past and future timelike infinities are two distinct points (0-dimensional). Spatial infinity is just one point (0-dimensional), and there is no separation into future and past spatial infinities. And the future and past null infinities are 3-dimensional null surfaces $\mathbf{R} \times S^2$ . Obviously my intuitive description above is very wrong. So What is a better way to imagine these infinities? (I take this from: Carroll, Sean M. Spacetime and geometry. An introduction to general relativity. Vol. 1. 2004.) As demonstrated in a Penrose-Carter diagram, the past and future timelike infinities are two distinct points (0-dimensional). Because we made them that way. Notice in page 475 the text says that Minkowski space is technically only the interior of that triangle, none of the sloping edges on the right, and none of the three corners. Some people draw it as a diamond so you don't have to include that line on the left. Let's do an example where we add stuff to a manifold. Imagine your manifold is the inside of a disk, i.e. every point where $z=0$ and $x^2+y^2<1.$ Now there are at least two ways to add points to that manifold to make it be topologically compact. One way is to make it the closed disk, i.e. every point where $z=0$ and $x^2+y^2\leq 1.$ Another way is to make it be (topologically) a spherical shell surface. For instance you make the north pole be a point and you identify each non north pole point as a point on your original manifold. Let's be clear that this is a topological addition. Another way to make the set topologically compact is just to add some more charts and some more points. And that's where you can see the difference between the two. The new charts for the disk case cover the open sets intersected with the disk. The new charts for the spherical shell (the one point compactification) could cover the intersections of the open sets that contain the whole circle $x^2+y^2=1.$ And then you have to think of the whole circle as one point. Or you can imagine mapping each point in $x^2+y^2<1$ onto the surface of a sphere of circumference 2 and having the origin be the south pole and each radial line be a line of longitude. Then add the one new point, the north pole, and the new charts will cover the intersections of the open sets with the spherical shell surface. So the point is that any chart containing the north pole has some distance $d$ where the chart in question contains every point in the original disk $z=0,$ and $x^2+y^2<1$ that is $d$ or more away from the origin, i.e. an open set that contains all the points in the origonal disk that were inside some open set where the open set contained the entire circle. In summary we added points to Minkowski space. We could have added different points, different numbers of points, different topologies of points, and added them differently in general. And then we drew the new spacetime, with the extra points in the new picture. The point is that we had multiple ways to add some points and add new charts to the manifold that didn't change anything about the existing points. We can define the distance functions, charts, metric, and so on with no changes. And now you have a manifold with boundary. And to be clear, these two new points, the timelike infinities are single points because we made them that way. Just like with the disk we could have had more or less points. We could have made them the same point if we wanted. That would be like taping figure H.4 onto a cylinder so that $i^+$ and $i^-$ touch. Spatial infinity is just one point (0-dimensional), and there is no separation into future and past spatial infinities. A radial geodesic of $t=const$ in Minkowski space heads right towards spatial infinity. And we made it be one point, again by our choice. This is exactly analogous to adding just that one point to the disk rather than the whole edge. In page 476 the text makes it sound more inevitable than it is, the curves that go from the origin to spatial infinity have an infinite length, so there was plenty of room for different compactifications. By making all those curves go to the same point in the compact i fixation your textbook chooses you can just talk about a curve going to infinity B without worrying about which direction or which time. And the future and past null infinities are 3-dimensional null surfaces $\mathbf{R} \times S^2$ . Yes, and again we choose to make them that way we kept every radial null geodesic separate as we added more points to our space time diagram. That's what figure H.2 is all about. What is a better way to imagine these infinities? Its better to imagine your spacetime is sitting inside a larger spacetime. It takes an infinite amount of clock ticks or rulers to get out of the part you are interested in (the original Minkowski space). But since there are multiple ways to put the same Minkowski spacetime into different larger spacetime, you can't see how that works unless you visualize the larger space. So look at Figure H.3, but first we are going visualize $\mathbb S^3$ the hard way (but easier for me to describe with text). Now $R$ is like a new angle on a unit 3 sphere, so when it is zero or $\pi$ you are at a north or south pole of the unit 3 sphere and when it is in between it is like being at a latitude you have a little sphere of smaller than unit radius (until you get to $R=\pi/2$ where it is of unit radius). So you have $R$ which is like latitude and then for each $R=const$ (except $R=0$ or $R=\pi$) you get a whole sphere of radius $\sin R$. Now look at that cylinder, I think they drew too much. You should only have half the cylinder, say the right half. Slice it horizontally so you see a circle. Now imagine that circle as like a clock face and we want it from nine o'clock to three o'clock. That's the half we want. If you look at the cylinder in the picture there is a vertical line labeled $R=0$ and another one labelled $R=\pi$ that is 3 o'clock and 9 o'clock. We want that right half. So now you can think of $R$ as measuring that path length from $R=0$ to $R=\pi$ as you go around on the right. Now for each $R$ on the right hand side you can drop a line from that point to the plane continuing the line $R=0$ and $R=\pi.$ This tells you the size of the 2 sphere at that $R.$ This seems way more complicated than it needs to be. If you move down from the north pole you get larger and larger circles as your lines of constant latitude. Here as you increase $R$ you get larger and larger 2 spheres. So go back to the clock. At 3 o'clock ($R=0$) you have just one point (the north pole). As you move from 3 o'clock ($R=0$) to 9 o'clock ($R=\pi$) you have a whole sphere (not ball, just the spherical shell) of radius just large enough that it could touch the straight line on your clock from from 3 o'clock to 9 o'clock, i.e. radius $\sin R.$ So you could cover the whole upper half of your clock with vertical straight lines. These straight lines tell you how big a sphere you have at that clock position. So imagine a clock hand and the hour it points to has a sphere there of radius equal to the vertical line dropping down from that time on the clock. So far all we've done is visualize $\mathbb S^3.$ which in the picture of the cylinder is just that circular segment of arc from $R=0$ to $R=\pi$ on the right. The text drew it as just a 1d circular arc, we had to visualize it as a whole $\mathbb S^3$ which we did by imaging a bunch of different sized ball on surfaces each getting bigger then smaller. And a time lapse picture could be helpful too. You could imagine taking a bunch of constant latitude slices of the earth. Then placing then in a movie and you'd see a point (the north pole) then circles that get larger and larger, then reach unit size, then get smaller and smaller until they become a circle again. So the movie all fits inside a disk. Similarly you can make a movie of $\mathbb S^3.$ It all takes place inside a unit ball. At first you have the point at the center $R=0$ (but think of $R$ as like lattitude, not as radius). Then later you get a spherical shell and it gets larger and larger until it has a surface area of $4\pi$ square units. Then it starts to shrink. These shrinking regions are different than the previous ones. And eventually they come to a point, the point $R=\pi$ (remember it is an angle like latitude). The way you move around to nearby points in $\mathbb S^3$ is to move to nearby film times and nearby points. So $R=0$ and $R=\pi$ are about as far away as you can be. So we have $\mathbb S^3$ (which is also just the unit vectors in $\mathbb R^4$ if you already can visualize 4d easily). So now to make the "cylinder" we add time. This is easy, take figure H.4 make sure the left line is twice as long as the triangle is wide and then tape it to that cylinder with the line on the line $R=0.$ What you have here is just a line segment of $T$ that starts out length $2\pi$ and gets proportionally smaller as you go around the right part of the cylinder. Now your clock ticks an infinite number of times getting to the edge of that triangle and you need an infinite number of rulers to get around the cylinder so all these new points on the edge of the triangle weren't in the original spacetime except $R=0$ which for $T$ with $\|T\|<\pi$ is just the origin of Minkowski space. However all the issues about some things being a point versus a sphere were about how $R$ is an angle when we embedded space into $\mathbb S^3$ and we did it so that $R$ was an angle. A good example for that is stereographic projection, put a flat surface on the north pole and make that the origin. The points on the flat surface that are far from the origin get mapped near the south pole when you map by where the point hits the sphere on its way to the south pole. All the points far from the origin are getting mapped to near the south pole of $\mathbb S^3$ (the unit vectors in $\mathbb R^4$). But the number of rulers you need didn't change. We are making just one point because we felt like it. It doesn't relate to spacetime or time, its just way we spatially compactified. And yes this is all unphysical since I spent all the time talking about the larger imaginary spacetime of which Minkowski space is just a part. But that is how you get the infinities. And that other half of the cylinder is useful if you want the diamond instead of a triangle and have negative $R$ which could be just continuing the radial line through the origin so it isn't a new spatial region it just makes $R=0$ less weird but $R=0$ is still a coordinate singularity. • 1)Thanks. 2)I think the answer should also include an explanation of why it is useful/conventional to choose theses specific compactifications of the different infinities (since you say they are a matter of decision), and whether there is physical motivation. 3)I'm familiar with higher dimensional geometry and with the basic math of general relativity(pseudo-Riemannian geometry), so I don't feel the need for explaining the notions of a 3-sphere etc. – Lior Sep 16 '15 at 19:30 • @Lior The choice of infinities is natural for a particular metric that is conformally related to the Minkowski metric. Though there are other metrics conformally related to Minkowski space. I think you should only use the infinities to help you talk about what goes on in the normal space. The conformally related metric really is an isometric subset of $\mathbb S^3\times\mathbb R.$ – Timaeus Sep 16 '15 at 20:14	2020-04-10 12:11:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7586868405342102, "perplexity": 230.18717709762402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371896913.98/warc/CC-MAIN-20200410110538-20200410141038-00502.warc.gz"}
https://newbedev.com/why-are-the-densities-of-europium-and-ytterbium-anomalously-low	# Chemistry - Why are the densities of europium and ytterbium anomalously low? ## Solution 1: Expanding upon @Ivan's comment: Eu and Yb can access the +2 oxidation state instead of +3, due to the +2 ions having a relatively stable half-filled or filled f subshell. That includes the embedded ions in the structure of the metal. Eu and Yb have $$\ce{M^2+}$$ instead of $$\ce{M^3+}$$ ions in the metal. The extra electron and reduced effective nuclear charge on the outer subshells make those $$\ce{M^2+}$$ ions larger, therefore these metals have lower densities than the surrounding lanthanides. The lower ion charge also means fewer electrons binding the ions together, making Eu and Yb easier to melt. The difference in ion charge can also show up in other ways. Eu and Yb, for instance, resemble heavy alkaline earth metals more than other lanthanides by dissolving in liquid ammonia. Europium is also known to appear in some rocks alongside alkaline earth metals rather than alongside other lanthanides (see Wikipedia): In anaerobic, and particularly geothermal conditions, the divalent form is sufficiently stable that it tends to be incorporated into minerals of calcium and the other alkaline earths. This ion-exchange process is the basis of the "negative europium anomaly", the low europium content in many lanthanide minerals such as monazite, relative to the chondritic abundance. ## Solution 2: As you likely know, the structure of metals is a lattice of cations surrounded by freely moving electrons. The structure of most lanthanides consists of +3 anions, but the electronic structures of europium ($\mathrm{(4f)^7 (6s)^2}$) and ytterbium ($\mathrm{(4f)^{14} (6s)^2}$) mean that the +2 state is more stable. The lower charge difference gives weaker metallic bonding and therefore lower density.	2022-06-28 18:33:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8238370418548584, "perplexity": 3444.6414930889264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103573995.30/warc/CC-MAIN-20220628173131-20220628203131-00761.warc.gz"}
https://astronomy.stackexchange.com/questions/44910/what-it-the-outer-part-of-the-sun-that-we-see-with-our-eyes-called/44941	# What it the outer part of the sun, that we see with our eyes, called? When we look at the sun with our eyes it seems much larger than it actually is. When we use a solar filter we are seeing what the sun actually is. So, what is the name of the thing the solar filter is eliminating? The larger glare part. And, why does this happen? Why do we see this light that makes the sun appear bigger and not the smaller circle that is actually the sun? • I've left an answer, but I think a follow-up question in Earth Science SE or perhaps Physics SE is in order. The small-angle white-light scattering by particles in the atmosphere can be called Mie scattering, but the details (intensity and angular distribution) is an atmospheric science problem. For our eyes and vision system, a question about the shape, intensity and causes can be asked in Biology SE and in Psychology & Neuroscience SE – uhoh Jul 19 at 7:33 • Jul 19 at 7:50 • Then again, you can just read Ruslan's answer and probably don't need to ask those followups :-) – uhoh Jul 19 at 9:27 • "When we look at the sun with our eyes it seems much larger than it actually is." Not for very long. Jul 20 at 0:03 • Yes, please remember to only look at the sun very briefly to prevent any damage to your eyes! Jul 20 at 20:25 I think you're talking about the effect of a "fluffy glowing ball" around the solar disk, shown on the right in this photo: This is called solar aureole, and it's caused by the aerosols in the air, which scatter light with a well-pronounced forward peak in the phase function: • +1 this is the right answer! cf. Forest M. Mims III in Applied Optics: Solar Aureoles Caused by Dust, Smoke, and Haze see also Sun and Sky Photography. – uhoh Jul 19 at 9:49 • @uhoh In fact, after some thought, the description looks to me more like that of glare: although the aureole is brilliant so that one has to wince to see its details (otherwise it's plain white), it's not as fiery as the solar disk - and this fieriness does make a large glare that masks the aureole completely and makes the sun look enormous. Jul 20 at 15:33 What it the outer part of the sun, that we see with our eyes, called? I am not sure there is a single word for this, since the effect is a little complicated. We might call it "the glare of the Sun". ### But there are (at least) two things that will contribute to this. 1. Optical and perceptual artifacts created by our visual system and by cameras when looking at very bright astronomical objects, especially those we should never look at with our eyes! :-) These are called glare for vision and lens flare for cameras. These are not real, in the sense that a perfect eye or perfect camera would not see them. They are artifacts generated by imperfect systems. 2. What is real is small-angle scattering by both dust and tiny water droplets in the atmosphere (I notice that you've added the atmospheric-effects tag) which can be considered as Mie scattering for particles larger than a few wavelengths of light, and this will generally not add any color the way Rayleigh scattering the source of the blue diffuse sky radiation does, specifically because the particles are substantially larger than a wavelength. ### Lens flare and glare (but in a photograph, so it's really flare): Sources: left: CCTV Lens flare (annotated) right: Lens flare scheme en below: Flashlight effect Sumo Jan08 ### Mie scattering: While Mie scattering is generally strongest in the forward direction and for particles tens or hundreds of wavelengths in diameter is almost completely seen within a few degrees of the source, it does have some intensity in other directions. Clouds are the white (i.e. they reflect as the same color as the light that hits them) due to large angle Mie scattering. Source: Cumulus Cloud (annotated) None of the other responses seem to answer the question "What is the name of the thing the solar filter is eliminating?" In fact, the solar filter doesn't eliminate anything. It just makes everything a lot less bright. The reason the white part of the sun looks so big is due to saturation of the film, CCD, or retina that you're using to look at it. There's a limit to the brightness that the film, CCD, or retina can register, and anything brighter than that is going to look white. You see a lighted region around the sun due to scattering effects that others have mentioned, but it really is not that bright compared to the sun itself. When you use a solar filter, you can see that clearly. But with an unfiltered view, it is still bright enough to saturate the viewing medium, just like the sun does, and so it looks just as bright as the sun, even though it is nowhere close. The part of the sun you see (but you shouldn't look at the sun except through a filter) is the photosphere: https://en.wikipedia.org/wiki/Photosphere The scattering effect in your second photo is due to the atmosphere: it can be anywhere from almost nonexistent (say in clear mountain air) to obscuring the solar disk entirely, as when you have a heavy cloud layer. You can see the same scattering effect with the moon, and sometimes with stars, though it's much less noticable because they're so much dimmer.	2021-09-17 07:24:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5138489007949829, "perplexity": 970.9019145787983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780055601.25/warc/CC-MAIN-20210917055515-20210917085515-00150.warc.gz"}
https://www.physicsforums.com/threads/root-equation-help.155685/	# Root equation help thomas49th Hi, I have the equation A = πr² + r²root(k²-1) i need to rearrange it to find r i go it to r³ = (2A/π+root(k²-1) to get just r (with no powers) what will the final equation look like and why. Thanks Homework Helper Gold Member Dearly Missed Eeeh? $$A=\pi{r}^{2}+r^{2}\sqrt{k^{2}-1}$$ Agreed? thomas49th yeh, that's the one Homework Helper Gold Member Dearly Missed So, what is a common factor between the two terms on the right-hand side? thomas49th r² is the common factor Homework Helper Gold Member Dearly Missed So, you may rewrite your equation as: $$r^{2}(\pi+\sqrt{k^{2}-1})=A$$ What would you do next? thomas49th cross multiply? $$r^{2} = A/(\pi+\sqrt{k^{2}-1})$$ Homework Helper Gold Member Dearly Missed Which is to divide each side with the factor $(\pi+\sqrt{k^{2}-1})$, rather than cross-multiplication. 1. Now, does this equal what you posted before? 2. Since you now know the SQUARE of a number, how do we get what the number itself is? thomas49th do you square the whole RHS? Homework Helper You want to get rid of the square on r. What is the opposite of squaring? thomas49th square root the whole RHS Homework Helper Gold Member Dearly Missed Do you know what an equation is, and what is allowed to do with one? Homework Helper Gold Member Dearly Missed square root the whole RHS Wrong. WHAT must you take the square root of? thomas49th $$r= \sqrt{A/(\pi+\sqrt{k^{2}-1})}$$ Homework Helper Gold Member Dearly Missed 1. You must take the square root of BOTH sides of the equation, not just of one of the sides as you said. (This you have done) thomas49th cheers, thanks for the help So here's another one: v² = u² + av² find v v² - av² = u² v²(1 - a) = u² v² = $$u²\1-a[/text] v = [tex]\sqrt{u²/1-a}$$ am I right? thomas49th cheers, thanks for the help So here's another one: v² = u² + av² find v v² - av² = u² v²(1 - a) = u² v² = $$u^{2}/1-a$$ v = $$\sqrt{u^{2}/1-a}$$ am I right? Homework Helper Gold Member Dearly Missed Is it that hard to get? Secondly, in the prior exercise I assumed that "r" was a radius, and hence necessarily a non-negative quantity (you didn't say). Now, must "v" be a non-negative quantity? thomas49th v is meaningless, I'm just praticing rearranging the formula Homework Helper Gold Member Dearly Missed Is it "meaningless"? Is it not even a number? thomas49th well it must be a number...musn't it. Did i get the question right? $$v = \sqrt{u^{2}/(1-a)}$$ Homework Helper Gold Member Dearly Missed v can be either of the two numbers: $$v=\pm\sqrt{\frac{u^{2}}{1-a}}$$ thomas49th would changing it to $$v=\frac{u}\sqrt{1-a}}$$ be simplfying? Homework Helper Gold Member Dearly Missed would changing it to $$v=\frac{u^{2}}\sqrt{{1-a}}$$ thomas49th sory when changing from latex source code i pressed enter on window and it submitted: Would this be considered simplyfying? $$v=\frac{u}\sqrt{(1-a)}$$ Homework Helper Gold Member Dearly Missed Almost; but you forget you have TWO solutions for v: $$v=\pm\frac{u}{\sqrt{1-a}}$$ thomas49th so that's simplifying, yet leaving a surd as a demoninator isn't. Howcome? What's so special with surds Homework Helper Gold Member Dearly Missed so that's simplifying, yet leaving a surd as a demoninator isn't. Howcome? What's so special with surds That's a matter of taste, mostly. The first expression is about as simple; however, most would regard the square root of a square (i.e, your numerator) as a non-simplified expression. thomas49th How would this equation go then... $$v^{2} = u^{2} + a^{2\5}$$ find x Last edited: $$v^{2} = u^{2} + av^{\frac{x}5} 1. Rearrange to x 2. Rearrange to get v Science Advisor Homework Helper Gold Member Dearly Missed 1. Easy 2. Forget it. thomas49th okay. If you say it's easy ill believe you. Just, somthing about surds. Simplify http://www.bbc.co.uk/schools/gcsebitesize/img/ma_surd25.gif" [Broken] can you show me the steps you did it in Last edited by a moderator: Science Advisor Homework Helper Gold Member Dearly Missed I don't see that any rewriting of that expression is any simpler than the one given. thomas49th youre not suppost to leave surds on the bottom. Apparently the answer should be http://www.bbc.co.uk/schools/gcsebitesize/img/ma_surd28.gif" [Broken] But I looked at their method and it looked dogdy. Is the answer [tex]\frac{3(\sqrt{6} - \sqrt{2})}4$$	2023-02-07 17:11:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9210199117660522, "perplexity": 6259.3232017973705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500628.77/warc/CC-MAIN-20230207170138-20230207200138-00822.warc.gz"}
https://www.beatthegmat.com/gmat-720-95-q49-v40-people-never-fail-they-just-give-up-t7227-30.html	• 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Get 300+ Practice Questions Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to $200 Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0 Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code ## GMAT 720 95%(Q49,V40)(people never fail - they just give up) tagged by: This topic has 7 expert replies and 376 member replies Goto page ajaypatil_am Senior \| Next Rank: 100 Posts Joined 27 Jan 2007 Posted: 81 messages Target GMAT Score: 700+ Tue Jan 29, 2008 2:35 am Thanks Again ngufo, I can postpond exam No problem...Actually I'm preparing since Oct 2007 .Earlier my exam was on 26th Jan but postponed till 29 March.As my practice test scores were not good.I'm not going to take the test till I score 700 + in all my tests. 1] Yes,I need to take a closer look at some specific topics from Q e.g Slope of line,Line Eqns.Stats....For V I'm making notes from Manhatton SC and other books. 2] I'll plan time line for princeton review as well in my study plan 3] "you are not revising enough, and probably thats another area to focus on" No ,I'm revising each and every question actually even its right or wrong but, still sometimes making similar mistakes again and again....Maths most of times silly mistakes r happening and Verbal ......no comments .... "Anything more than 2 minutes for maths, and 1.8 for verbal, tells you that you are going to run out of time" Thats the best way but I never applied.I was giving more time to difficult questions..... in case of Verbal.When the RC questions start I find clock is running very fast .I think 5-8 mins per RC passage should be the target time...But as of now its too short for me ....for one passage I take 10 mins....and then even never get 100 % accuracy .... In case any further specific questions I'll email you. Thanks for all ur inputs...Sorry for bugging u all the time.... Ajay jelt Master \| Next Rank: 500 Posts Joined 10 Aug 2007 Posted: 102 messages 13 Tue Jan 29, 2008 10:06 pm Hi NGUFO, My own GMAT (and MBA applications!) journey has already ended. It was a whirlwind ride for me that began in end-August, and concluded this past December. If your mantra is NGUFO, mine is probably closer to 'NUTPL'- Never Underestimate the Power of LUCK! On another note, I see you're well on your way to becoming the next big GMAT consultant here. Good job! Cheers! ngufo Master \| Next Rank: 500 Posts Joined 03 Jun 2007 Posted: 146 messages Followed by: 58 members 166 Tue Jan 29, 2008 10:20 pm Thats pretty good to know jason . Happy for you buddy - but you are being too modest... Luck may play a part, but there is no substitute for the work you must have put in! So which school will you be going to and starting when? Wish you all the very best - the hard part is over now, enjoy the ride.... A bunch of us are watching from the distance, hopefully our time too will come TAke care, -ngufo _________________ People Never Fail ... They just Give Up jelt Master \| Next Rank: 500 Posts Joined 10 Aug 2007 Posted: 102 messages 13 Tue Jan 29, 2008 11:29 pm Hi NGUFO, I will be doing the Wharton/Lauder MBA/MA dual degree programme, starting this May. I posted my experience in the Admissions Success Story section, if you're interested. If you read my post, you may agree that luck had a lot more to do with it than anything else, considering my decision to apply + GMAT + Essays all took less than 2 months from start to finish. Cheers, NUTPL ngufo Master \| Next Rank: 500 Posts Joined 03 Jun 2007 Posted: 146 messages Followed by: 58 members 166 Tue Jan 29, 2008 11:47 pm Will definitely read up jason, thats for sure. As for now, congratulations on moving forward onto a brand new phase of life. Hope its everything you want it to be and then some more Take care, -ngufo _________________ People Never Fail ... They just Give Up jelt Master \| Next Rank: 500 Posts Joined 10 Aug 2007 Posted: 102 messages 13 Wed Jan 30, 2008 10:26 pm Thank you for your well wishes. It's my dream programme, and I'm definitely happy. It's not HBS, however. There's a magical mystique attached to Harvard, so your fight is obviously going to be much harder. Good luck! Let us know how it goes. -NUTPL ngufo Master \| Next Rank: 500 Posts Joined 03 Jun 2007 Posted: 146 messages Followed by: 58 members 166 Fri Feb 01, 2008 5:00 pm Hi Ajay, Also first just make sure that you are comfortable when doing RC, SC, and quant. That if there are any areas that you are weak on you are practising that like crazy, adn understanidng how to do it. After that probably timing is very important and thats next. It would be great if you could postpone your exam, create a study plan fo rthe next few months (identifying yoru mistakes and what you plan to do to practise to hone your skills), Based on that set your exam. When I started preparing in May of last year, my folks thought I was crazy/dum to right it in November (so late), but ajay, if I had written it earlier, I wouldnt have scored well - I wasnt ready. Bottom line find yoru mistakes, take your time to fix them, and then write the exam - I am very sure, with your dedication, your passion to do well, plus god always helps people woh try their best, you are definitly going to achieve your dream.... Its jsut a matter of time Please ask me anything you want anytime, you are never bugging me. I was helped by a lot of people in the forum when I was in need, the least I can do is return the favor... tAke care, -ngufo _________________ People Never Fail ... They just Give Up ajaypatil_am Senior \| Next Rank: 100 Posts Joined 27 Jan 2007 Posted: 81 messages Target GMAT Score: 700+ Mon Feb 04, 2008 1:40 pm Hi ngufo, Thanks for your reply.Yesterday I solved RC from Kaplan Verbal.My scores are 1] Practice Set 1 -- 3/18 2] Practice Set 2--- 7/18 2] Practice Set 3--- 9/18 Yes,I'm focusing more on RC these days.Solving RC from all the possible resources.Reading CNN news articles,Some notes/tips on how to improve reading speed etc..... I have one question.The "GMAT Princeton review" book you was referring is it the below one ? I'm bit confused on which is the "GMAT Princeton review" book as I have 2 soft copies which have different contents. Yes,I'm planning to postpond the exam from 29 March till May but,I really want to finish as soon as possible.So many personal priority things are lined up... BTW,did u get my mail ? Thanks, Ajay ngufo wrote: Hi Ajay, Also first just make sure that you are comfortable when doing RC, SC, and quant. That if there are any areas that you are weak on you are practising that like crazy, adn understanidng how to do it. After that probably timing is very important and thats next. It would be great if you could postpone your exam, create a study plan fo rthe next few months (identifying yoru mistakes and what you plan to do to practise to hone your skills), Based on that set your exam. When I started preparing in May of last year, my folks thought I was crazy/dum to right it in November (so late), but ajay, if I had written it earlier, I wouldnt have scored well - I wasnt ready. Bottom line find yoru mistakes, take your time to fix them, and then write the exam - I am very sure, with your dedication, your passion to do well, plus god always helps people woh try their best, you are definitly going to achieve your dream.... Its jsut a matter of time Please ask me anything you want anytime, you are never bugging me. I was helped by a lot of people in the forum when I was in need, the least I can do is return the favor... tAke care, -ngufo ngufo Master \| Next Rank: 500 Posts Joined 03 Jun 2007 Posted: 146 messages Followed by: 58 members 166 Tue Feb 05, 2008 12:55 am Hi Ajay, The book I used was different [url] That is great news - I am pretty sure that with the amount of work you are putting in, you will definitely hit the bar you set for yourself. Did you mail me at ngufo@yahoo.com? I havent checked that email. Will look at it after i finish mailing to you. Take care, and all the best, -ngufo _________________ People Never Fail ... They just Give Up ajaypatil_am Senior \| Next Rank: 100 Posts Joined 27 Jan 2007 Posted: 81 messages Target GMAT Score: 700+ Tue Feb 05, 2008 2:42 am Hi ngufo, Thanks for your quick reply.I think we both are referring to the same book but,different year editions. I'll just double check and then will buy the latest edition of the PR book. Thanks Again, Ajay ngufo wrote: Hi Ajay, The book I used was different [url] That is great news - I am pretty sure that with the amount of work you are putting in, you will definitely hit the bar you set for yourself. Did you mail me at ngufo@yahoo.com? I havent checked that email. Will look at it after i finish mailing to you. Take care, and all the best, -ngufo amazingkev Junior \| Next Rank: 30 Posts Joined 16 Sep 2007 Posted: 12 messages Tue Feb 05, 2008 12:25 pm from you post, it says.. I wrote my practise prep tests all in that week (my timing was way off), should have writtne them earlier, but there had been too much of work in the office what did you mean by saying "I wrote my practise prep test"? Does that mean literally write the questions, premise or reading materials on a piece of paper? ngufo Master \| Next Rank: 500 Posts Joined 03 Jun 2007 Posted: 146 messages Followed by: 58 members 166 Tue Feb 05, 2008 12:38 pm I meant that I finished all the practise tests like GMAT CAT1, GMAT CAT2, Power Prep 1, Power Prep 2. These are computer adaptive tests, that have to be done on the computer (very similar to the final test) Generally people space the tests out,as they are the final indicators of how you will do in the test, but I ended up doing them in 4 consecutive days itself. I think doing the tests did give me the momentum - i didnt do anything the day before the test though, so that the mind had time to relax.... Wish you all the best, Regards, -Ekta _________________ People Never Fail ... They just Give Up amazingkev Junior \| Next Rank: 30 Posts Joined 16 Sep 2007 Posted: 12 messages Tue Feb 05, 2008 12:47 pm I forgot to mention how inspiring your story was.. i thought about giving it up and settle with my score, but after reading your post, it makes me feel i want to try one more time. thank you.. ngufo Master \| Next Rank: 500 Posts Joined 03 Jun 2007 Posted: 146 messages Followed by: 58 members 166 Tue Feb 05, 2008 2:41 pm Thanks for the email - it means a lot to me. All my life I knew I was average, I knew I wasnt the smartest person out there ..... What I have learnt over time, is that what smart people take 2 days to do I take 7. I started realizing that if I just put in more effort, I could do anythign I set my mind too. It really doesnt matter how much time it takes you know, at the end of the day, I can equally kick butt, adn the result tastes equally sweet as the guy who got a 720 in 1 week.... Its taken me a while to work around my weaknesses, but I am telling you, its a great feeling to know you can do anything you set your mind to, once you know what your limitations are and how to work around them..... I took 6 whole months to get ready, it was terrifying for me, every step of the way, but the key is to make sure you are very very organized about your studies, you create a strict study schedule, and consistently analyze your mistakes, and fix your weaknesses. There is no way you will not come close to what your potential is - its happened to me every single time. I mention it many times in my emails - but finally just having faith in god, that once you put your life and soul into what youare trying to achieve, for him to provide you with the final little push to get you thru the finish line (I always give a prayer before anything big .... its always worked out for me) I really wish you all the very best, nothing in life is unachievable, you just break every big thing into pieces and then take it one piece at a time. When I looked at the GMAT the first time, I remember being so scared. I remember realizing that I sucked at Maths, Verbal and essays, how the hell would I get by. Just took one thing at a time, learned the maths concepts, and in parallel kept practising the verbal, and at the very end the AWA Please let me know if i can help you in any way. God has provided me with a lot of help in the form of various people, in various ways - I believe I need to return the faviour in kind at all times. It would be a pleasure to help out. Take care, god bless, and wish you all the very best! -ngufo _________________ People Never Fail ... They just Give Up amazingkev Junior \| Next Rank: 30 Posts Joined 16 Sep 2007 Posted: 12 messages Tue Feb 05, 2008 5:43 pm wow, i have learned so much from you today.. especially the part you mentioned about faith in God. as a sunday school teacher, i feel a little embarrassed that i neglected my commitment in church many times to put more time on this studying... looking back, i wasn't really studying..I've been distracted and not focused and on the other hand worrying more. I know that God works in many different ways.. it feels that he is trying to send me some messages by allowing me to i do really want to see someone like you become successful in life, setting a good example for others... kevin ### Top First Responders* 1 GMATGuruNY 67 first replies 2 Rich.C@EMPOWERgma... 44 first replies 3 Brent@GMATPrepNow 40 first replies 4 Jay@ManhattanReview 25 first replies 5 Terry@ThePrinceto... 10 first replies * Only counts replies to topics started in last 30 days See More Top Beat The GMAT Members ### Most Active Experts 1 GMATGuruNY The Princeton Review Teacher 132 posts 2 Rich.C@EMPOWERgma... EMPOWERgmat 112 posts 3 Jeff@TargetTestPrep Target Test Prep 95 posts 4 Scott@TargetTestPrep Target Test Prep 92 posts 5 Max@Math Revolution Math Revolution 91 posts See More Top Beat The GMAT Experts	2018-04-24 16:48:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2965856194496155, "perplexity": 4291.645366062632}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946807.67/warc/CC-MAIN-20180424154911-20180424174911-00475.warc.gz"}
http://math.stackexchange.com/questions/103305/show-that-this-polynomial-is-positive/138374	# Show that this polynomial is positive Consider the following polynomial in two variables : $$Q(k,x)=27x^6 - 144kx^4 + 80k^2x^3 + 240k^2x^2 - 192k^3x + (64k^4 - 128k^3)$$ Then for any integer $k \geq 5$, the polynomial $Q(k,.)$ (in one variable $x$) seems to be always positive (i.e, $Q(k,x) >0$ for any real number $x$). Prove or disprove. - The usual way to show a polynomial is positive is to express it as a sum of squares. Have you tried that? – Gerry Myerson Jan 28 '12 at 22:46 I tried and failed. – Ewan Delanoy Apr 29 '12 at 8:54 1. $x\mapsto Q(5, x)$ is positive. There is numerous ways to prove that, more or less algorithmically. 2. Assume that there exists a $k$ such that $x\mapsto Q(k,x)$ is not positive, and consider $L$ be the infimum of all real numbers $k \geqslant 5$ such that $x\mapsto Q(k, x)$ is not positive. I claim that $Q(L,x)$ has a double real root, this is equivalent to say that $L$ is a root of the discriminant of $Q$ w.r.t. the variable $x$. 3. The discriminant of $Q$ with respect to $x$ is (computed with Maple) : $$\operatorname{disc}_x Q = 92162779488452608 k^{16} (k-4)^4$$ 4. The real roots of this discriminant are 0 and 4, thus $L$ cannot be finite, and hence the claim. Proofs • Point 1 can be proved using Sturm sequences, and you can compute these sequences using Maple : sturm(subs(k=5, Q),x,-infinity, infinity); 0 This means that $Q(5, x)$ has no real roots. Since the leading coefficient is positive, the polynomial $Q(5, x)$ is positive whenever $x\in \mathbb R$. • To prove point 2 consider $x_0$ a real numbers such that $Q(L, x)$ is minimal. We have $Q(L, x_0) = 0$, because if $Q(L, x_0) > 0$ then $x\mapsto Q(L+\epsilon, x)$ is positive for $\epsilon > 0$ small enough, which contradict the definition of $L$. If $Q(L, x_0) < 0$, then $Q(L-\epsilon, x_0)$ is stille negative for $\epsilon > 0$ small enough, which is again a contradiction. So $x_0$ is a root of $Q(L,x)$ and of $\partial_x Q(L,x)$, thus it is a double root. The discriminant thing is usual. • The point 3 is Maple : factor(discrim(Q, x)); - @Lierre : perfect ! after seeing your solution I'm surprised and ashamed I didn't think of it earlier. – Ewan Delanoy Apr 29 '12 at 11:18 You can also use quantifier elimination algorithms. This gives you precisely for which $k$ the polynomial $Q(k,x)$ is positive. For example Mathematica implements such an algorithm : In[1] := Resolve[ForAll[{x}, Q > 0]] Out[1] := k < 0 \|\| k > 4 I insist that you can trust this result, it is an exact algorithm not a numerical heuristic. - You can try to prove your assumption using induction : $1.$ Show that : $Q(5,x)=27\cdot x^6-720 \cdot x^4+2000 \cdot x^3+6000\cdot x^2-24000\cdot x+24000 > 0$ ,for all $x$ $2.$ Suppose that : $Q(k,x)=27x^6 - 144kx^4 + 80k^2x^3 + 240k^2x^2 - 192k^3x + (64k^4 - 128k^3) >0$ ,for all $x$ $3.$ Try to prove that : $Q(k+1,x) >0$ , for all $x$ using assumption from step $2$ - Does it work??? – Gerry Myerson Jan 28 '12 at 22:44 @GerryMyerson,Probably...since $k$ is an integer ... – pedja Jan 29 '12 at 5:50 A main problem with the induction idea is that for "most" $x$ (i.e. large enough $x$), we do not have $Q(k+1,x)>Q(k,x)$ as excepted. It seems that you cannot solve the problem without studying the "relative position" of $k$ and $x$. – Ewan Delanoy Jan 29 '12 at 11:39	2016-05-01 22:04:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9466823935508728, "perplexity": 245.38988682326172}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860116929.30/warc/CC-MAIN-20160428161516-00086-ip-10-239-7-51.ec2.internal.warc.gz"}
https://www.physicsoverflow.org/14891/symmetry-that-apparent-with-definition-fields-with-another	# $P$ symmetry that is apparent with one definition of fields but not with another + 4 like - 0 dislike 52 views Suppose that we have a Lagrangian density like $$\mathcal L = -\frac{1}{4} \operatorname{tr} F_{\mu\nu}F^{\mu\nu} + \frac{\theta}{32\pi^2} \operatorname{tr} \big( \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}\big) + \overline{\psi}\gamma^\mu D_\mu \psi$$ where $F_{\mu\nu}$ is the gauge field strength and $D_\mu$ the gauge covariant derivative, and $\psi$ is a fermion field. This Lagrangian is not $P$ conserving because of the $\theta$ term. However if we redefine the fields $\psi \mapsto \exp(i\alpha \gamma_5)\psi$ we can make $\theta$ go away, by choosing $\alpha = \theta/2$ as per the Fujikawa method (described in [Weinberg], Chapter 22 or [Fujikawa]); this is due the path integral measure also transforming under the redifinition. With this redefinition of fields $\mathcal L$ is manifestly $P$ conserving. But surely I can't get more or less symmetry by redefining fields, so how should I understand that the $P$ symmetry is not manifest with the original definition of the fields? I suspect that the $P$ transformation too transforms the path integral measure, in a way that sends $\theta \mapsto -\theta$, but I do not know how to show this. • [Weinberg] Weinberg, S. The Quantum Theory of Fields. 2: Modern Applications (Cambridge, 2005). • [Fujikawa] Fujikawa, K. Path-Integral Measure for Gauge-Invariant Fermion Theories. Phys. Rev. Lett. 42, 1195{1198 (18 Apr. 1979). This post imported from StackExchange Physics at 2014-04-13 14:05 (UCT), posted by SE-user Robin Ekman	2019-07-24 01:09:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9581142663955688, "perplexity": 419.0608337774185}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195530246.91/warc/CC-MAIN-20190723235815-20190724021815-00263.warc.gz"}
https://www.quizover.com/trigonometry/test/removable-discontinuities-by-openstax	5.6 Rational functions (Page 4/16) Page 4 / 16 Vertical asymptotes The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors. Given a rational function, identify any vertical asymptotes of its graph. 1. Factor the numerator and denominator. 2. Note any restrictions in the domain of the function. 3. Reduce the expression by canceling common factors in the numerator and the denominator. 4. Note any values that cause the denominator to be zero in this simplified version. These are where the vertical asymptotes occur. 5. Note any restrictions in the domain where asymptotes do not occur. These are removable discontinuities, or “holes.” Identifying vertical asymptotes Find the vertical asymptotes of the graph of $\text{\hspace{0.17em}}k\left(x\right)=\frac{5+2{x}^{2}}{2-x-{x}^{2}}.$ First, factor the numerator and denominator. $\begin{array}{ccc}\hfill k\left(x\right)& =& \frac{5+2{x}^{2}}{2-x-{x}^{2}}\hfill \\ & =& \frac{5+2{x}^{2}}{\left(2+x\right)\left(1-x\right)}\hfill \end{array}$ To find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero: $\begin{array}{ccc}\hfill \left(2+x\right)\left(1-x\right)& =& 0\hfill \\ \hfill x& =& -2,1\hfill \end{array}$ Neither $\text{\hspace{0.17em}}x=–2\text{\hspace{0.17em}}$ nor $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph in [link] confirms the location of the two vertical asymptotes. Removable discontinuities Occasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We call such a hole a removable discontinuity . For example, the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{{x}^{2}-1}{{x}^{2}-2x-3}\text{\hspace{0.17em}}$ may be re-written by factoring the numerator and the denominator. $f\left(x\right)=\frac{\left(x+1\right)\left(x-1\right)}{\left(x+1\right)\left(x-3\right)}$ Notice that $\text{\hspace{0.17em}}x+1\text{\hspace{0.17em}}$ is a common factor to the numerator and the denominator. The zero of this factor, $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ is the location of the removable discontinuity. Notice also that $\text{\hspace{0.17em}}x–3\text{\hspace{0.17em}}$ is not a factor in both the numerator and denominator. The zero of this factor, $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ is the vertical asymptote. See [link] . [Note that removable discontinuities may not be visible when we use a graphing calculator, depending upon the window selected.] Removable discontinuities of rational functions A removable discontinuity occurs in the graph of a rational function at $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a zero for a factor in the denominator that is common with a factor in the numerator. We factor the numerator and denominator and check for common factors. If we find any, we set the common factor equal to 0 and solve. This is the location of the removable discontinuity. This is true if the multiplicity of this factor is greater than or equal to that in the denominator. If the multiplicity of this factor is greater in the denominator, then there is still an asymptote at that value. Identifying vertical asymptotes and removable discontinuities for a graph Find the vertical asymptotes and removable discontinuities of the graph of $\text{\hspace{0.17em}}k\left(x\right)=\frac{x-2}{{x}^{2}-4}.$ Factor the numerator and the denominator. $k\left(x\right)=\frac{x-2}{\left(x-2\right)\left(x+2\right)}$ Notice that there is a common factor in the numerator and the denominator, $\text{\hspace{0.17em}}x–2.\text{\hspace{0.17em}}$ The zero for this factor is $\text{\hspace{0.17em}}x=2.\text{\hspace{0.17em}}$ This is the location of the removable discontinuity. Notice that there is a factor in the denominator that is not in the numerator, $\text{\hspace{0.17em}}x+2.\text{\hspace{0.17em}}$ The zero for this factor is $\text{\hspace{0.17em}}x=-2.\text{\hspace{0.17em}}$ The vertical asymptote is $\text{\hspace{0.17em}}x=-2.\text{\hspace{0.17em}}$ See [link] . The graph of this function will have the vertical asymptote at $\text{\hspace{0.17em}}x=-2,\text{\hspace{0.17em}}$ but at $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ the graph will have a hole. the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve 1+cos²A/cos²A=2cosec²A-1 test for convergence the series 1+x/2+2!/9x3 a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he? 100 meters Kuldeep Find that number sum and product of all the divisors of 360 Ajith exponential series Naveen what is subgroup Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1 e power cos hyperbolic (x+iy) 10y Michael tan hyperbolic inverse (x+iy)=alpha +i bita prove that cos(π/6-a)cos(π/3+b)-sin(π/6-a)sin(π/3+b)=sin(a-b) why {2kπ} union {kπ}={kπ}? why is {2kπ} union {kπ}={kπ}? when k belong to integer Huy if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41 what is complex numbers Dua Yes ahmed Thank you Dua give me treganamentry question Solve 2cos x + 3sin x = 0.5	2018-12-14 20:03:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 22, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9407832622528076, "perplexity": 449.5381699516789}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376826306.47/warc/CC-MAIN-20181214184754-20181214210754-00428.warc.gz"}
https://indico.cern.ch/event/578804/contributions/2624103/	25-29 September 2017 Salamanca, Spain Europe/Zurich timezone ## Study of baryonic resonances in the reaction pp$\rightarrow$pp$\pi^{+}\pi{-}$ at 3.5 GeV with HADES 26 Sep 2017, 19:00 2h Auditorium Hall ### Speaker Amel Belounnas (IPN Orsay) ### Description Pion production in NN collisions is one of the sources of information on the NN interaction and on the contribution of nucleon resonances. In particular, two-pion production in the few energy range, carries information both on $\pi\pi$ dynamics and on single and double baryon excitation. The High Acceptance Di-Electron Spectrometer (HADES) [1] installed at GSI Helmholtz-Zentrum für Schwerionenforschung in Darmstadt, designed to investigate dielectron production in heavy-ion collisions in the range of kinetic beam energies 1-3 A GeV is also an excellent detector for charged hadron detection, due to its tracking capabilities. Recently, differential and integrated cross sections for the reactions pp$\rightarrow$pp$\pi^0$, pp$\rightarrow$pn$\pi^{+}$ [2-3-4], pp$\rightarrow$pp$\pi^{+}\pi^{-}$, pn$\rightarrow$pn$\pi^{+}\pi^{-}$ [5], pn$\rightarrow$d$\pi^{+}\pi^{-}$ have been investigated with HADES at kinetic energies 1.25, 2.2 and 3.5 GeV. This talk will focus on the analysis of the pp$\rightarrow$pp$\pi^{+}\pi^{-}$ channel at 3,5 GeV, using results from pp$\rightarrow$pp$\pi^0$, pp$\rightarrow$pn$\pi^{+}$ [3] and pp$\rightarrow$pK$\Lambda$ [6] measured at the same energy by HADES. The contributions of the excitation on one or two baryonic resonances with masses up to 1,9 GeV and of the $\rho$ production can be quantified. The results are compared with two theoretical models [7-8]. The results of this study provide strong constraints on the pion production mechanisms, and on the various resonance contributions ($\Delta^0$(1232), N*(1440),…), as well as on the double resonance excitation and the direct $\rho$ production. These aspects are closely related to the interpretation of the dielectron spectra measured by the HADES collaboration. Baryonic resonances are indeed important sources of dileptons through two mechanisms: the Dalitz decay (e.g. R$\rightarrow$N$e^{+}e^{-}$) and the mesonic decay with subsequent dielectron production. [1] G. Agakishiev et al., Eur. Phys. J. A41, 243-277 (2009). [2] G. Agakishiev et al. Eur.Phys.J. A48 (2012) 74. [3] G. Agakishiev et al. Eur.Phys.J. A50 (2014) 82. [4] G. Agakishiev et al. , Eur.Phys.J. A51 (2015), 137. [5] G. Agakishiev et al., Phys.Lett. B750 (2015) 184. [6] G. Agakishiev et al. Phys.Lett. B742 (2015) 242-248. [7] A.P.Jerusalimov et al. Study of the Reaction pn$\rightarrow$pn$\pi^{+}\pi^{-}$ at Intermediate Energies. http://arxiv.org/pdf/1102.1574.pdf [8] X. Cao et al., Phys. Rev. C81, 065201 (2010). ### Primary author Amel Belounnas (IPN Orsay) ### Presentation Materials There are no materials yet.	2020-09-19 10:36:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8329795598983765, "perplexity": 7168.947045744214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400191160.14/warc/CC-MAIN-20200919075646-20200919105646-00575.warc.gz"}
https://www.physicsforums.com/threads/non-static-spacetime-effects.793480/	# Non-static spacetime effects 1. Jan 21, 2015 ### smoothoperator First of all, I'm having a difficulty in defining what a static spacetime is. Does it presume that the objects with mass in the system are just sitting around and doing nothing, with no motion, relative to our frame, so there can be no motion and change in curvature of spacetime through time? Secondly, if objects with motion curve space time in a way that they change the curvature along their path, does this also imply that free-falling objecs also curve space time in the same way? Thirdly, in Newtonian physics there is an earth and apple example where Earth also accelerates towards the apple because of the same force with opposite directions. Does the same hold in GR? 2. Jan 21, 2015 ### Staff: Mentor The technical definition is that a static spacetime has a timelike Killing vector field ("static" means the field is also hypersurface orthogonal, but I don't think we need to go into that part here). Now I'll translate that into English. ;) If a spacetime has a timelike Killing vector field, physically this means that there is some family of worldlines in this spacetime along each of which the metric doesn't change. We can therefore use these worldlines to define "points in space", such that each point in space has an unchanging spacetime geometry in its vicinity. In principle we can then have a family of observers in this spacetime each of whom always sees the same spacetime geometry (each observer follows one of the worldlines in the family), so we can think of each such observer as "standing still" at a particular point in space. Note that the spacetime geometry can still change if we move from one point in space to another. See below. You appear to be making an incorrect distinction between "objects with motion" and "free-falling objects". You also are confusing "test objects", the objects that follow particular worldlines in a given spacetime, from objects that act as "sources" of spacetime curvature; the two are different. Let me give a concrete example to illustrate what I mean with the first point (this will also illustrate what "moving from one point in space to another" means in what I said above): Schwarzschild spacetime, the solution of the Einstein Field Equation that describes the vacuum spacetime around a spherically symmetric gravitating object, is static. That means there is a family of worldlines along each of which the metric doesn't change. An observer following one of these worldlines will be "hovering" at a constant altitude above the gravitating object, and will not be revolving about the object at all. This observer is static, i.e., at the same point in space for all time. Note that such an observer is not freely falling; he must either use rockets to maintain altitude, or be standing on something (like a platform, or indeed the surface of the gravitating body), and in either case he will have nonzero proper acceleration and will feel weight. Any other observer in this spacetime is "moving" from one point in space to another. However, although any freely falling observer must be moving (because, as above, observers who are static cannot be freely falling), not all moving observers will be freely falling. An observer who uses rockets or stairs or a ladder to climb from one altitude to another is not freely falling, but is moving (not staying that the same point in space). An observer who jumps off a high platform, or out of his rocket, and falls toward the gravitating body is both moving and freely falling, of course. All of these moving observers see a changing spacetime geometry as they move; but note that they themselves do not cause the changes, they just observe them (see below). (Note also that an observer can be freely falling without changing altitude, if he is in a circular orbit about the gravitating body. In this particular case, a moving observer will not see any change in spacetime geometry. That is because this particular spacetime has an additional symmetry, spherical symmetry, over and above being static. A static spacetime does not have to have any additional symmetry, although I don't have a handy simple example of one that doesn't.) Now for the second point: In the example above, the spherically symmetric gravitating object is the "source". The other objects (i.e., all the different observers I described) are all test objects and are assumed to have a mass that is so small, compared to the mass of the source, that they don't affect the spacetime geometry. So test objects, whether they are freely falling or not, don't curve spacetime (more precisely, they don't curve it enough to be significant in the scenario under discussion). 3. Jan 22, 2015 ### smoothoperator Thanks for another great post Peter. I've understood many things you said, but I still don't understand the main question: Is motion allowed in a static spacetime? we have a hovering Schwarzschild observer and his definition of space, is any kind of motion allowed relative to him? Or do all objects in the static spacetime have to just sit where they are without any motion? There are many types of motion in general, and all of the objects involved seem to curve nearby spacetime while they are travelling, so I am wondering what is the answer to my first question? 4. Jan 22, 2015 ### Staff: Mentor Of course it is. "Static" refers to the spacetime geometry; it doesn't put any restrictions about how test objects, which are too small to affect the spacetime geometry, can move. What makes you think this? As above, test objects are too small to curve spacetime. In the scenario I described with one central gravitating object, that object is the only thing that curves spacetime; all the observers I described, some of which are moving (relative to the notion of "space" defined by the static spacetime), do not curve spacetime, because they are too small. 5. Jan 22, 2015 ### smoothoperator What about bigger objects with greater mass? Why can't the space change curvature relative to a Schwarzschild static (hovering) observer 6. Jan 22, 2015 ### Staff: Mentor If you have more than one gravitating object, the spacetime won't be static. The two gravitating objects will attract each other, so they will either fall together, or orbit around their common center of mass. (Adding more gravitating objects makes it even more complicated.) The Schwarzschild solution does not describe this; it describes one gravitating object only, with nothing else present that can produce spacetime curvature. Last edited: Jan 22, 2015 7. Jan 22, 2015 ### Orodruin Staff Emeritus Great post Peter, I just want to make one addition/query: If I do not misremember, the Schwarzschild solution is only static outside of the event horizon. Once inside the event horizon, any observer will have a changing metric as any future directed timelike world line invariably has a decreasing r-coordinate. 8. Jan 22, 2015 ### Staff: Mentor Correct. In this discussion, we are only considering ordinary gravitating bodies like a planet or star, which are not black holes and don't have an event horizon. The vacuum region of such a spacetime is entirely static (and the non-vacuum region is too, although we aren't really discussing that here). But a black hole is only static outside the horizon, yes; the Killing vector field that is timelike outside the horizon is null on the horizon and spacelike inside it, so the condition for a static spacetime is not met, and any observer following a timelike worldline will see a changing metric. 9. Jan 22, 2015 ### smoothoperator Thanks for the response Peter, so can we calculate gravitational time dilation for some region of spacetime that changes its spacetime curvature from let's say, flatter to curvier (sorry if my English is bad). Since the Schwarzschild formula is for static situation, will it work for a dynamic case? 10. Jan 22, 2015 ### Staff: Mentor No. The concept of "gravitational time dilation" only makes sense in a static spacetime. (More precisely, it only makes sense in a stationary spacetime, which is a slight generalization of a static spacetime to cover things like uniformly rotating objects, where the source of gravity is not entirely motionless but there is still a sense in which things "are not changing".) 11. Jan 22, 2015 ### smoothoperator So what can be said of a portion of spacetime which ic changing its curvature? Isn't it logical to say, for instance, that when spacetime goes from flatness to curvature the rate of clocks also slows down? 12. Jan 22, 2015 ### Staff: Mentor Um, that it's changing its curvature? No. Spacetime curvature is not the same as "strength of gravity" in the Newtonian sense, or "depth of gravitational potential well". For one thing, spacetime curvature includes time as well as space. For example, according to our best current model, our universe is spatially flat; all of the spacetime curvature due to the matter and energy in the universe as a whole (i.e., on the scale of the universe as a whole) is along the time dimension (when we say that the universe is "expanding", we are really referring to it being "curved along the time dimension"). But this same model of the universe tells us that the rates of "comoving" clocks (clocks that see the universe as homogeneous and isotropic on a large scale) are constant--they do not change as the universe expands. 13. Jan 22, 2015 ### smoothoperator But shouldn't space also be curved? I mean, I watched some Brian Greene videos where it's clearly visualized how matter and planets curve space like a ball curves a trampoline. Almost every video that I watched about GR mentiones that planets curve the spatial dimension by their mass and that the ticking of clocks depends on how strong gravity/ spacetime curvature is. 14. Jan 22, 2015 ### Staff: Mentor Not on the scale of the universe as a whole. This model can be misleading for several reasons. One is that it's coordinate-dependent, because the split of spacetime into "space" and "time" is coordinate-dependent. The visuals you are talking about assume that you are using Schwarzschild coordinates. If you use Painleve coordinates, the space around a gravitating body is flat. Another reason that model can be misleading is that it is based on Schwarzschild spacetime, which, as I said before, assumes a single gravitating body that is alone in the universe, surrounded by empty space. That model does not apply to our universe, not just because there are multiple gravitating bodies, but because the masses in the universe are everywhere--the universe is not a bunch of masses surrounded by empty space. This is a good illustration of why you should not try to actually learn science from pop science presentations, even if they're fronted by actual scientists like Brian Greene. This model, as above, is based on Schwarzschild spacetime, which is not a good model for the universe as a whole. The scientists in the videos, like Greene, know that, and would never try to get away with such a statement in an actual peer-reviewed scientific paper. But in videos like these, as far as I can see, they are not actually trying to teach you the science, in the sense of giving you a logically consistent model that you can use to make predictions. They are just trying to get you to say "oh, wow, neat!" and then go on to the next sound bite without actually trying to reason about what they are telling you. That's a shame, but that's how it seems to me. Also you are equating "gravity" with "spacetime curvature" (and I suspect that the videos explicitly did that, so it's not just your interpretation), which, as I said in an earlier post, is not correct. The correct statement is that spacetime curvature is the same as tidal gravity. 15. Jan 23, 2015 ### smoothoperator Then on what does gravitational time dilation depend on, if not on the spacetime curvature? 16. Jan 23, 2015 ### pervect Staff Emeritus space-time curvature is generally considered to be the Riemann curvature tensor, $R_{abcd}$, the most important components of which have a physical interpretation as tidal forces. Gravitational time dilation would be one particular component $g^{tt}$ of the metric tensor. Because it's only one component of the tensor, gravitational time dilation is coordinate dependent, without the rest of the components it can't be treated as a tensor. You can construct the Riemann curvature tensor as a function of the metric tensor and it's first and second derivatives. So the highest order terms would involve the second derivatives of the metric tensor. Sorry if this is a bit technical, I don't know how to put it more simply, perhaps someone else has a way. 17. Jan 23, 2015 ### smoothoperator I still somehow believe that clocks on some spacetime portion that is changing its curvature must elapse less proper time than clocks in flat spacetime. After all, changing curvature is still curved spacetime which implies clocks running slow when influenced by mass, which also curves spacetime. That seems logical, if it isn't like that then how is it? Nothing can be said about gravitational time dilation while spacetime doesn't have a constant curvature? 18. Jan 23, 2015 ### Staff: Mentor What exactly does it mean to say that one clock shows less time passed than another? A clock near me and at rest relative to me ("my clock") reads 12:00:00. At the same time that my clock reads 12:00:00, some other clock at a distance from me also reads 12:00:00. Later I look at my clock and it reads 12:00:05. At the same time, the other clock reads only 12:00:03, so I conclude that the other clock is running slow. But notice that this conclusion depends on having a definition of "at the same time". In curved spacetime there is no natural way of defining "at the same time", and by choosing different definitions I can come to different conclusions about which clock is running slow. 19. Jan 23, 2015 ### Staff: Mentor If the spacetime is not static (technically, stationary--see post #10), there is no such thing as gravitational time dilation. That is a very important point that you appear to be missing. See below. If the spacetime is static (technically, stationary), then gravitational time dilation depends on where you are in space--where "space" is defined using the family of worldlines along each of which the metric doesn't change, as I described in post #2. No, it does not. This implication is simply wrong, no matter how intuitive it seems to you. Once again, if spacetime is not static (technically, stationary), there is no such thing as gravitational time dilation, because there is no invariant way to tell which spatially separated clocks are "running slow" relative to which others. There just isn't. 20. Jan 23, 2015 ### Staff: Mentor This is true in general, but note that in a static/stationary spacetime, the timelike Killing vector field gives a natural way of defining "at the same time", and this in turn gives a way of defining gravitational time dilation in an invariant sense. Of course this way of defining "at the same time" is still not the only possible one, even in a static/stationary spacetime. It's just that the timelike KVF gives a way of picking out a definition of "at the same time" that does at least have an invariant property associated with it, whereas in a non-stationary spacetime there is no definition of "at the same time" for which that is true.	2018-05-25 13:18:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7281329035758972, "perplexity": 503.54402491789875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867092.48/warc/CC-MAIN-20180525121739-20180525141739-00095.warc.gz"}
https://docs.typo3.org/typo3cms/extensions/configuration_object/stable/04-Administration/Services/AvailableServices/CacheService.html	DEPRECATION WARNING This documentation is not using the current rendering mechanism and will be deleted by December 31st, 2020. The extension maintainer should switch to the new system. Details on how to use the rendering mechanism can be found here. Cache service¶ Its name speaks for itself: the cache service is an implementation of the TYPO3 caching framework. It will be used to store configuration objects in cache the first time they are created, then fetch the created entry next times. This can improve performances dramatically. You should really consider before using this service, as it can lead to unwanted behaviours. You should obviously use it only for objects which will be needed several times: do not use it for one-time-run objects, it would create a cache entry for nothing. Usage¶ You can activate this service for a given configuration object by attaching it to the ServiceFactory in the static function getConfigurationObjectServices(). Use the constant ServiceInterface::SERVICE_CACHE as an identifier for this service (see example below). Options¶ Name Description CacheService::OPTION_CACHE_NAME The name of the “group” which will contain all the cache entries for this configuration object. CacheService::OPTION_CACHE_BACKEND Type of backend cache used for the cache manager. Default is TYPO3\CMS\...\FileBackend. CacheService::OPTION_CACHE_GROUPS Groups of the cache, default value is all. CacheService::OPTION_CACHE_OPTIONS Actual options of the cache manager. A good example is the option cacheDirectory for the FileBackend cache. You can find the options of every backend cache in the TYPO3 official documentation (see above). 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 26 27 28 29 30 31 32 33 34 use Romm\ConfigurationObject\Service\ServiceInterface; use Romm\ConfigurationObject\Service\ServiceFactory; use Romm\ConfigurationObject\Service\Items\Cache\CacheService; use Romm\ConfigurationObject\ConfigurationObjectInterface; use Romm\ConfigurationObject\Traits\ConfigurationObject\DefaultConfigurationObjectTrait; use TYPO3\CMS\Core\Cache\Backend\MemcachedBackend; class MyObject implements ConfigurationObjectInterface { use DefaultConfigurationObjectTrait; const CACHE_NAME = 'foo_object'; const CACHE_BACKEND = MemcachedBackend::class; /** * @var array / private static $cacheOptions = [ 'servers' => ['my-server.com:1337'], 'defaultLifetime' => 86400 // 1 day ]; /* * @return ServiceFactory */ public static function getConfigurationObjectServices() { return ServiceFactory::getInstance() ->attach(ServiceInterface::SERVICE_CACHE) ->setOption(CacheService::OPTION_CACHE_NAME, self::CACHE_NAME) ->setOption(CacheService::OPTION_CACHE_BACKEND, self::CACHE_BACKEND) ->setOption(CacheService::OPTION_CACHE_OPTIONS, self::$cacheOptions); } }	2019-08-23 10:07:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3310692012310028, "perplexity": 1843.8129415834567}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318243.40/warc/CC-MAIN-20190823083811-20190823105811-00085.warc.gz"}
https://skatterlaui.web.app/67547/6252.html	# Typs attning av text och matematiska formler med hj alp av LATEX Gotland's Picture Stones - Uni The page number or page range may be omitted if the entire work is cited, as in "(Smith 2010)". 2019-02-16 · List of LaTeX mathematical symbols. From OeisWiki. There are no approved revisions of this page, so it may not have been reviewed. depending on whether or not you want a comma between the author's name and the corresponding year. (The second instruction can, in fact, be simplified to \citep {paper1, paper2}. I slightly prefer (\citealp {paper1,paper2}), though, as it preserves a syntactic symmetry with (\citealt {paper1,paper2}) .) should work. But in a phrase like "(\cite{foobar} also has interesting examples)", you do not want parenthesis, as "((Foobar 1999) also)" looks ugly and is not advised in standard typographic rules. Probably "(Foobar 1999 also has)" is prefered. For this reason, in vanilla bibtex, you have \cite (without parenthesis) and \citep (with parenthesis). The parameters, comma-separated, tell LaTeX to use the authoryear citation mode and use double parentheses as opening and closing marks. G's Auto Marine [email protected] nr: Find, read and cite all 17 juli 2018 — with bartender in parentheses : Bobby Deeâ€™s (Lenny U) in Jersey Is there ? http://xxxnx.world vlxx Citing unnamed people familiar with 21 nov. 2012 — She began to show an interest in nursing citing the example of the Catholic ibuprofen iodine general anesthesia and local anesthetics latex rubber such as in for Latin phrases which appear in italics within parentheses.e. ## SAMMANFATTNING. DEL A... 2 DEL B DEL C DEL D DEL M av A Holl · Citerat av 3 — ending as the reference lexeme irrespective of the inflection type. 3. ### Apa Referens - Canal Midi Authors are encouraged to read the natbib documentation for complete details on the package’s capabilites. LaTeX Basics Creating your first LaTeX document Choosing a LaTeX Compiler Paragraphs and new lines Bold, italics and underlining Lists Mathematics Mathematical expressions Subscripts and superscripts Brackets and Parentheses Fractions and Binomials “matrix with parentheses latex” Code Answer’s. matrix latex . whatever by CharllierJr on Jul 06 2020 Donate . 7. Source: manualdelatex.com. how to make matrix latex . 251-622-8610 251-622-9692. Orchestric Personeriasm latex. Johannes vermeer When you want to cite an item in the database in a LaTeX document, you write \cite { The results? Citations with square brackets are not easy. Note that .bib can generally be used with both BibTeX and BibLaTeX files, but you can Citations go inside square brackets and are separated by semicolons. Numerical referencing is the default style. Konstruktiv betyder stressteorier björn jansson carnegie nytt satt att mata blodsocker how much have i spent on steam martin magnusson norrtälje ### File:Motala - KMB - 16001000023486.jpg Suppressed parentheses As an alternative form of citation, \citealt is the same as \citet but without parentheses. Similarly, \citealp is \citep without parentheses. Multiple references, notes, and the starred variants also exist. BibTeX references are stored in a plain text database with a simple format. Monica reichenberg rasmus dahlin ### Närliggande Webbkameror Prints without any brackets except when using the alphabetic or numeric style when it uses square brackets; \parencite—prints citations in parentheses except when using the alphabetic or numeric style when it uses square brackets; \footcite—puts the citation in a footnote. Subdividing Bibliographies When referencing that equation in text, you simply refer to it using the same label, also enclosed in parenthesis. ## File:Motala - KMB - 16001000023486.jpg Such styles include citation directions within the .bst itself or BibTEX by means of an appropriate bibliographic style file (extension .bst). BibTEX is in fact year in parentheses attached to volume: vol(year) num jdt-vs. The standard citation styles are: numeric Implements a numeric citation scheme intended for in-text citations. Should be employed in conjunction with the numeric bibliography style. Section 12.24 of The Chicago Manual of Style suggests that equations should be centered on the line and the number label enclosed in parenthesis. When referencing that equation in text, you simply refer to it using the same label, also enclosed in parenthesis. In your example, you might write this instead: Write your LaTeX File and use \cite{keyword} to cite a reference from your database file named 'keyword' and \nocite{keyword} to enforce the display of a non-referenced reference named \keyword. GO: When compiling the document you need to LaTeX the file, then BibTeX it, then LaTeX it twice.	2022-08-19 20:02:47	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8814754486083984, "perplexity": 8456.471184085089}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573760.75/warc/CC-MAIN-20220819191655-20220819221655-00161.warc.gz"}
https://www.impetus-afea.com/support/manual/?command=MAT_VISCOUS_FOAM	#### Command list • Input handling • Solution control and techniques • Output • Mesh commands • Nodes and connectivity • Material properties • Initial conditions • Boundary conditions • Contact and tied interfaces • Rigid bodies • Connectors • Parameters and functions • Geometries • Sets • Coordinate system • Particle • SPH ### MAT_VISCOUS_FOAM ###### Material properties MAT_VISCOUS_FOAM "Optional title" mid, $\rho$, $E_1$ $\eta$, cid${}_{fe}$, cid${}_{fv}$, $c_{dec}$, $\sigma_{cut}$, $\gamma$, $\varepsilon_f$, $G_2$ $\sigma_y$ #### Parameter definition VariableDescription mid Unique material identification number $\rho$ Density $E_1$ Young's modulus for network 1 of fully compacted foam $\eta$ Initial porosity (between 0 and 1) cid${}_{fe}$ ID of elastic compaction stress curve ($f^e$) cid${}_{fv}$ ID of viscous stress curve ($f^v$) $c_{dec}$ Viscous decay parameter $\sigma_{cut}$ Tensile cut-off stress $\gamma$ Hysteresis parameter $\varepsilon_f$ Optional plastic failure strain in tension $G_2$ Shear modulus of optional network 2 $\sigma_y$ Yield stress (von Mises) of optional network 2 #### Description This is a model for visco-elastic foams, consisting of two networks. Network 1 The elastic response of Network 1 is based on a formulation using the right stretch tensor $\mathbf{U}$ and its principal stretches $(\lambda_1, \lambda_2, \lambda_3)$. $\displaystyle{ \mathbf{U} = \sum_{i=1}^3 \lambda_i \mathbf{v}_i \otimes \mathbf{v}_i}$ The total stress $\mathbf{\sigma}$ is the sum of an elastic and a viscous component. $\displaystyle{ \mathbf{\sigma} = \left( \frac{J_{min}}{J} \right)^{\gamma} \left( \mathbf{\sigma}^e + \mathbf{\sigma}^v \right) = \left( \frac{J_{min}}{J} \right)^{\gamma} \left( \sum_{i=1}^3 \sigma_i^e \mathbf{v}_i \otimes \mathbf{v}_i + \sigma^v \right) }$ Note that the principal elastic stresses $(\sigma_1^e, \sigma_2^e, \sigma_3^e)$ coincide with the principal stretches of $\mathbf{U}$. $J_{min}/J$ is the ratio between the smallest relative volume during the process and the current relative volume. That is, $J = \mathrm{det}\mathbf{F}$, where $\mathbf{F}$ is the deformation gradient. The magnitude of the elastic stresses is defined from Young's modulus $E_1$, the initial porosity $\eta$ and from the compaction stress curve $f^e$. $f^e$ is basically a CURVE defining quasi-static compaction stress versus uni-axial (engineering) strain. $\displaystyle{ \sigma_i^e = \left\{ \begin{array}{lcl} -\mathrm{min} \left( E_1 \eta_i, f^e(\eta_i) \right) & : & \eta_i \leq \eta \\ -\mathrm{min} \left( E_1 \eta_i, f^e(\eta) + E_1 (\eta_i - \eta) \right) & : & \eta_i \gt \eta \end{array} \right. }$ $\eta_i$ a measure of the compaction in principal direction $i$. $\displaystyle{ \eta_i = \frac{\eta (1 - \lambda_i)}{\eta + \frac{1}{\lambda_j \lambda_k} - 1} }$ where $(i,j,k) = (1,2,3), (2,3,1)$ or $(3,1,2)$. Note that there is a coupling between the different directions. Hence, a compaction in directions $j$ and $k$ $(\lambda_j \lambda_k \lt 1)$ will decrease the porosity and, hence, reduce the room for compaction in direction $i$. In uni-axial compression $\eta_i = 1 - \lambda_i$. The viscous stress tensor is defined as: $\displaystyle{ \mathbf{\sigma}^v(t) = \frac{1}{c_{dec}} \int_0^t f^v(\dot{\mathbf{\epsilon}}) \frac{\dot{\mathbf{\epsilon}}}{\vert \dot{\mathbf{\epsilon}}\vert} e^{(\tau-t)/c_{dec}} \mathrm{d}\tau }$ $f^v$ is a CURVE of effective viscous stress versus effective strain rate. Network 2 The second network is optional and it only carries deviatoric stresses. It is typically used to provide some structural integrity to the foam during unloading. $\displaystyle{ \mathbf{\sigma}_2 = 2G \mathbf{\varepsilon}_{dev}^e }$ Yielding occurs when: $\displaystyle{ \sqrt{\frac{3}{2} \mathbf{\sigma}_2 : \mathbf{\sigma}_2} = \sigma_y }$ #### Example Quasi-static compression of foam An example showing how compression in one principal direction affects the compression resistance in another direction. UNIT_SYSTEM SI PARAMETER dens = 150.0, "density" E = 1.0e8, "Youngs modulus" eta = 0.7, "initial porosity" MAT_VISCOUS_FOAM 1, [%dens], [%E] [%eta], 99 CURVE "stress-compaction curve" 99 0.0, 0.0 0.01, 1.0e5 0.015, 1.3e5 0.02, 1.5e5 0.025, 1.6e5 0.5, 2.6e5 0.6, 3.7e5 0.65, 7.8e5 0.7, 2.2e6 END	2021-05-07 07:19:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9207007884979248, "perplexity": 7753.4816663551055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988775.25/warc/CC-MAIN-20210507060253-20210507090253-00113.warc.gz"}
https://yakovenko.wordpress.com/tag/tits-alternative/	# Sergei Yakovenko's blog: on Math and Teaching ## Finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$, I. Formal theory. 1. Formal normal form for a single holomorphic self-map from $\text{Diff}(\mathbb C^1,0)$. Parabolic germs. 2. Bochner theorem on holomorphic linearization of finite groups. 3. Stratification of the subgroup of parabolic germs $\text{Diff}_1(\mathbb C^1,0)$. 4. Tits alternative for finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$: every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial). 5. Centralizers and symmetries: formal classification of solvable subgroups. 6. Integrable germs and their holomorphic linearizability. Recommended reading: Section 6 (first part) from the book (printing disabled) Disclaimer applies, as usual 😦	2018-12-10 18:23:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9727104902267456, "perplexity": 4465.600223589327}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823382.1/warc/CC-MAIN-20181210170024-20181210191524-00060.warc.gz"}
https://cstheory.stackexchange.com/questions/19379/colouring-achieving-simple-discrepancy-bound	Colouring achieving simple discrepancy bound? Given a hypergraph $H$ with $n$ vertices and $m$ edges, one of the simplest inequalities on the discrepancy of $H$ is $\text{disc}(H) \le \sqrt{2n \ln (2m)}$. This is usually proved by mixing together one type of Chernoff bound using the probabilistic method (for instance, see the outline proof at the Wikipedia article on discrepancy). Is there a known colouring $\chi \colon V(H) \to \{-1,1\}$ of the vertices of a hypergraph $H$ with $-1$ and $1$ that witnesses this bound? In other words, such that for every edge $e \in E(H)$, $$\left\|\sum_{v \in e} \chi(v)\right\| \le \sqrt{2n \ln (2m)}.$$ Of course, the probabilistic method guarantees the existence of such a $\chi$, but is there an algorithm better than exhaustively checking all possible colourings? For completeness, the discrepancy of a hypergraph $H$ is the minimum over all possible colourings $\chi \colon V(H) \to \{-1,1\}$ of the largest value of the local discrepancy $\left\|\sum_{v \in e} \chi(v)\right\|$ over all edges $e$ of $H$. • I added absolute values to make sure this is the standard definition of discrepancy, let me know if you meant something else. – Sasho Nikolov Oct 14 '13 at 15:10 I assume you mean a deterministic algorithm. The best discrepancy bound is in fact $O(\sqrt{n\log (m/n)})$ which is $O(\sqrt{n})$ when $m = O(n)$. This the famous Six Standard Deviations Suffice bound of Spencer (1985 TAMS). A coloring achieving this bound can be found in deterministic polynomial time, using a recent result by Bansal and Spencer. Achieving the random coloring bound of $\sqrt{2n \ln (2m)}$ can be done with simple classic derandomization methods. You can use the method of conditional expectations or variants of the multiplicative weights method. You can even achieve the stronger $\sqrt{2\|e\|\ln(2m)}$ bound for each edge $e$. See Section 1.1. of Chazelle's book. For completeness, let me reproduce the multiplicative weights argument, because it's nice and simple. Identify the vertices of $H$ with $[n]$. We will pick $\chi(i)$ for $i = 1, \ldots, n$ greedily. Define $D(e, k) = \sum_{i \in e, i \leq k}{\chi(i)}$ to be the discrepancy of $e$ up to $k$. Let $p(e, k) = \frac{1}{2}e^{\alpha D(e, k)} +\frac{1}{2}e^{-\alpha D(e, k)}$ for $\alpha = \sqrt{2\ln (2m)/n}$, and the potential function $P(k) = \sum_e{p(e, k)}$. Define also $p(e, 0) = 1$ and $P(0) = m$. You can verify that for a random choice of $\chi(k+1)$, $\mathbb{E}P(k+1) \leq P(k)e^{\alpha^2/2}$. So you can just pick each $\chi(k)$ to minimize $P(k)$, and after the final step you have $$\frac{1}{2} e^{\alpha \|D(e, n)\|} \leq \max_e p(e, n) \leq P(n) \leq me^{n\alpha^2/2}.$$	2019-08-18 04:29:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.970639705657959, "perplexity": 162.2476309483978}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313617.6/warc/CC-MAIN-20190818042813-20190818064813-00165.warc.gz"}
https://forum.freecodecamp.org/t/record-collection-challenge-accessing-nested-object-properties/7616	# Record Collection challenge accessing nested object properties Record Collection challenge accessing nested object properties 0 #1 https://www.freecodecamp.com/challenges/record-collection I am confused about how to reference a nested property inside an object using variables. In the challenge we are given three variables as inputs to the function, however when trying to reference the property I just get undefined when using chained dot and bracket notation… How are we supposed to reference a nested property using variables? #2 Hey, you can’t use dot notation with variables. You can use: collection[id][prop] to acces a property. #3 In what sense? One of the main reasons to use bracket notation is to index using a variable, e.g. `foo[bar]`. If you meant something more specific, please reflect that in your explanation. #4 I’m sorry. I confused the two. I meant dot notation. Bracket notation is indeed used for variables. Thanks for noticing, I edited my post. #5 This thread is old, but in case somebody comes here, like me, having encountered this issue: I found this Medium article about dot notation vs bracket notation really helpful.	2018-09-23 22:21:45	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8566328883171082, "perplexity": 1497.300776700321}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159820.67/warc/CC-MAIN-20180923212605-20180923233005-00431.warc.gz"}
https://www.physicsforums.com/threads/problem-understanding-number-of-degrees-of-freedom.465241/	# Problem understanding number of degrees of freedom Hi, I do a classical mechanics course, and part of it relates to degrees of freedom. For a system M of point masses with j constraints, there is N=3M-j DoF For a rigid body, I know there is 6 DoF (3 translational, 3 rotational). However, I've tried using drawing the constraints on a 4 point mass system to make it a rigid system. I've done this with 5 constraints (3 rigid rods in blue and 2 fixed angles in green). With 5 constraints the formula gives N=(3x4)-5=7 DoF. Could someone explain this please? [PLAIN]http://img152.imageshack.us/img152/7426/degreesoffreedom.png [Broken] It'd be greatly appreciated. Thanks EDIT: I think I understand this now, because it's a rigid body, you can work out all other positions in the system itself from the 5 given constraints. All that you can't work out, is the rotational and translational DoF of the system as a whole. Last edited by a moderator: ## Answers and Replies The figure you have drawn does have 7 degrees of freedom. The "constraints" (rods) that you've placed between the points do not fully constrain the object in 3 dimensions---you can twist the vertical and horizontal bars around the diagonal bar (if that makes sense). In other words, there are 3x translation, 3x rotation, and also an undefined angle between one of the green angles and the plane of the other. To fully constrain the arrangement of your figure, you would have to add another fixed angle, or fixed distance.	2021-05-07 13:46:12	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8137972354888916, "perplexity": 621.9867096175174}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00164.warc.gz"}
https://jump.dev/JuMP.jl/stable/tutorials/conic/start_values/	Primal and dual warm-starts Some conic solvers have the ability to set warm-starts for the primal and dual solution. This can improve performance, particularly if you are repeatedly solving a sequence of related problems. In this tutorial, we demonstrate how to write a function that sets the primal and dual starts as the optimal solution stored in a model. It is intended to be a starting point for which you can modify if you want to do something similar in your own code. This tutorial uses the following packages: using JuMP import SCS The main component of this tutorial is the following function. The most important observation is that we cache all of the solution values first, and then we modify the model second. (Alternating between querying a value and modifying the model is not allowed in JuMP.) function set_optimal_start_values(model::Model) # Store a mapping of the variable primal solution variable_primal = Dict(x => value(x) for x in all_variables(model)) # In the following, we loop through every constraint and store a mapping # from the constraint index to a tuple containing the primal and dual # solutions. constraint_solution = Dict() for (F, S) in list_of_constraint_types(model) # We add a try-catch here because some constraint types might not # support getting the primal or dual solution. try for ci in all_constraints(model, F, S) constraint_solution[ci] = (value(ci), dual(ci)) end catch @info("Something went wrong getting $F-in-$S. Skipping") end end # Now we can loop through our cached solutions and set the starting values. for (x, primal_start) in variable_primal set_start_value(x, primal_start) end for (ci, (primal_start, dual_start)) in constraint_solution set_start_value(ci, primal_start) set_dual_start_value(ci, dual_start) end return end set_optimal_start_values (generic function with 1 method) To test our function, we use the following linear program: model = Model(SCS.Optimizer) @variable(model, x[1:3] >= 0) @constraint(model, sum(x) <= 1) @objective(model, Max, sum(i * x[i] for i in 1:3)) optimize!(model) By looking at the log (not shown in Documenter due to a bug), we can see that SCS took 100 iterations to find the optimal solution. Now we set the optimal solution as our starting point: set_optimal_start_values(model) and we re-optimize: optimize!(model) Now the optimization terminates after 0 iterations because our starting point is already optimal. Tip	2023-02-04 19:26:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5512332916259766, "perplexity": 1840.3478485005871}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500151.93/warc/CC-MAIN-20230204173912-20230204203912-00134.warc.gz"}
https://chrisvoncsefalvay.com/2020/07/25/dash-latex/?amp=1	# LaTeX in Dash applications: a guide to gorgeous typesetting Like many who come from the scientific community, LaTeX is like a second language to me – in fact, I generally eschew text editors that do not have built-in LaTeX rendering, as so much of my work is wrapped up in mathematical reasoning. Matplotlib, the golden oldie of visualisation solutions, pays begrudging respect to LaTeX, allowing you to render LaTeX on plots or even axis labels out of the box. Dash can also render LaTeX, but it takes some coaxing. Below is the solution that worked for me when working on the socially distanced SIR simulator, which required me to embed quite a bit of complex maths in the narrative that details the differential equations that run the model. ## What works (and what doesn’t) Before you embark on making changes to your code, it’s important to know what you can and can’t do at the state of the code. ### What definitely works Once correctly configured, most single-line mathematical typesetting, whether inline or as a separate paragraph, works fine. More complex environments, such as the align environments imported by the amsmath package, will not work. However, as far as ‘vanilla’ LaTeX maths goes, you can render some exquisitely complex stuff with relative ease. Note: if you write your text in Markdown and import it using dcc.Markdown(), all your code will be parsed through the Markdown renderer – including your LaTeX code. This means that some operators, which may have meaning in LaTeX and Markdown alike, will break LaTeX rendering. There are largely two to look out for: the * operator (replace it with \ast or \star), and closed square bracket operators [0...1], which you might have to get creative to find a workaround for. You can use valid LaTeX pretty much everywhere. This includes inside plots/axes (on which see later), on buttons, slides &c. ### What definitely doesn’t work There are some things that do not work, at the present state of Plotly Dash. In particular, you cannot dynamically render LaTeX, and LaTeX axis labels need to be refreshed (see below). ## Making it work The overall structure of what we’re going to be doing consists of four principal steps: 1. Import dash_defer_js_import to defer loading of external Javascript scripts after the Dash React components have loaded, and deferred import the MathJax renderer script. 2. Integrate the MathJax renderer script by reframing the index template using the app.index_string variable. 3. Integrate the deferred script import into the layout object. 4. For LaTeX rendering in dynamic axis labels, deferred import a custom script that regularly re-renders figures (optional). You may find it useful to refer to a worked example – a recent Dash app I have built to display epidemic dynamics as a function of social distancing implements part of this approach in order to beautifully render the maths underlying the model. You can find the code for this on Github, and you’re welcome to adapt it to your needs. ### Importing dash_defer_js_import and importing the MathJax renderer script First, we add dash_defer_js_import to our dependencies, so that we can implement deferred loading: import dash_defer_js_import as dji Then, once the application (app object) has been declared, we import the MathJax renderer script using deferred importing: mathjax_script = dji.Import(src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/latest.js?config=TeX-AMS-MML_SVG") ### Integrating MathJax into the template Since Dash 0.22.0, Dash app object allows the page template to be customised, which is most easily accomplished by setting the variable app.index_string. The details of this functionality are described in the Dash manual, but in essence, this option takes a Jinja formatted template. We are going to use this to our advantage by inserting the MathJax configuration to the footer of the template: app.index_string = """ <!DOCTYPE html> <html> {%metas%} <title>{%title%}</title> {%favicon%} {%css%} <body> {%app_entry%} <footer> {%config%} {%scripts%} <script type="text/x-mathjax-config"> MathJax.Hub.Config({ tex2jax: { inlineMath: [ ['$','$'],], processEscapes: true } }); </script> {%renderer%} </footer> </body> </html> """ ### Integrate the deferred script import into the layout object Finally, we’ll integrate the mathjax_script object we declared above into our template layout: app.layout = html.Div([ dbc.Container(children=[ # your application content goes here ]), mathjax_script ]) ### Rendering and refreshing axis labels Axis labels are a little more complex as they can change, and therefore need to be constantly refreshed. We accomplish this by using a script that refreshes plot objects regularly (in this case, every second – you can adjust this according to your needs by setting the value in the setTimeout() function). function redraw(){ var figs = document.getElementsByClassName("js-plotly-plot") for (var i = 0; i < figs.length; i++) { Plotly.redraw(figs[i]) } } setTimeout(function(){ redraw(); }, 1000); I have uploaded this to Codepen, and you’re welcome to use it. Start by creating a deferred import object, just as we did for the MathJax renderer: refresh_plots = dji.Import("https://codepen.io/chrisvoncsefalvay/pen/ExPJjWP.js") Update (10NOV2021): Codepen’s external linking has become pretty much useless. The workaround is to take the Javascript code, save it locally and reference it in the import command. app.layout = html.Div([ dbc.Container(children=[ # your application content goes here ]), refresh_plots, mathjax_script ]) This will gradually refresh and re-render MathJax LateX entries in your axis labels. ## Conclusion Until Dash natively implements LaTeX rendering, this workaround is a fairly quick and convenient way of beautifully typesetting maths in your Dash applications. As always, since this leverages the entire functionality of MathJax, there’s a lot more you can do with MathJax than simple maths (the documentation is definitely worth a read!). Currently, this works best with MathJax 2.7.7 (using latest.js from CDNs does not upgrade MathJax 2 to MathJax 3!), but can be easily adapted to work with MathJax 3.x as well. Was this post useful for you? Did it work? Did it not work? Any tips, suggestions, tricks, hacks? Let me know. Credits: The axis rendering refresh function is based on this Codepen by Richard Xue. I'm a data scientist and computational epidemiologist focusing on the intersection of public health, data science and artificial intelligence. ### Fun things to do with Graph configurations in Dash Hi Michaël, it seems that Codepen direct linking has gone down the drain. The best thing to do is to put it into a folder you can access from your Dash app. Typically, you would put that into your assets/ folder. This page explains how. Let me know if it works for you!	2022-05-24 09:46:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4718824625015259, "perplexity": 3134.851364707244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662570051.62/warc/CC-MAIN-20220524075341-20220524105341-00434.warc.gz"}
https://brilliant.org/problems/4-distinct-reals/	# 4 distinct reals Algebra Level 2 $\large{a^2+b^2+c^2+d^2=ab+bc+cd+da}$Do there exist distinct reals $a,b,c,d$ which satisfy the above equation? ×	2020-08-10 09:06:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6881543397903442, "perplexity": 4062.0216807535157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738653.47/warc/CC-MAIN-20200810072511-20200810102511-00081.warc.gz"}
http://zolomon.com/2015/03/25/emacs-on-windows/	• When using C-c C-c to compile, there were some problems with the paths. • First I had to setup PATH variables for MSBuild.exe and Fsc.exe, and then figure out how emacs prefers paths with spaces in Windows. 1. To build F# projects, we want to use MSBuild.exe 2. To compile F# projects, we want to use Fsc.exe	2022-12-07 22:37:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8557930588722229, "perplexity": 6284.564704176362}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711221.94/warc/CC-MAIN-20221207221727-20221208011727-00434.warc.gz"}
https://bcdudek.net/anova/effect-sizes-power-and-sample-size-planning.html	# Chapter 8 Effect Sizes, Power, and Sample Size Planning An imporant adjunct to NHST work with ANOVA is the provision of effect sizes. Although there are several approaches possible to finding effect size statistics in R, they are fairly simple to obtain. Sample size planning can be accomplished with tools from the pwr package, although I probably prefer the use of the GPower software for its broad capabilities. ## 8.1 Effect Sizes In earlier sections of this document, we have already seen provision of some effect size statistics. The multiple R-squared available from the ‘lm’ fit objects is also called eta-squared. The afex package functions give the “generalized eta squared” statistic. Note that in a 1-way design, partial eta squared values will equal the eta squared value, and partial omega squared values will equal omega squared values. Eta Squared, omega Squared, and Cohen’s F can be obtained quickly from the effectsize package. One additional useful function is provided here for computation of several commonly used effect sizes (sjstats). ### 8.1.1 Use of the effectsize package Two functions, eta_sq and omega_sq permit confidence interval calculation for either full or partial effect size statistics. The partial effect size statistic concept really has no meaning in 1-way ANOVAS and that is reflected in the commentary produced by the functions. effectsize::omega_squared(fit.1, ci.lvl = .95) ## For one-way between subjects designs, partial omega squared is equvilant to omega squared. ## Returning omega squared. ## Parameter \| Omega2 \| 90% CI ## --------------------------------- ## factora \| 0.25 \| [0.02, 0.44] effectsize::eta_squared(fit.1, ci.lvl = .95) ## For one-way between subjects designs, partial eta squared is equvilant to eta squared. ## Returning eta squared. ## Parameter \| Eta2 \| 90% CI ## ------------------------------- ## factora \| 0.30 \| [0.06, 0.49] The Cohen’s F statistics is also provided by a function from the effectsize package: effectsize::cohens_f(fit.1, ci.lvl=.95) ## For one-way between subjects designs, partial eta squared is equvilant to eta squared. ## Returning eta squared. ## Parameter \| Cohen's f \| 90% CI ## ------------------------------------ ## factora \| 0.66 \| [0.26, 0.98] ## 8.2 Use of the sjstats package The anova_stats function strikes me as very useful. It not only returns the basic ANOVA, but provides several effect size indices, including Cohen’s f. Note that the “power” value returned in the data frame produced by the anova_stats function is a post-hoc power calculation that we have reviewed as a problematic concept in how it has often been applied. Refer to the stattoolkit bibliography for the literature on this. kable(sjstats::anova_stats(fit.1)) term df sumsq meansq statistic p.value etasq partial.etasq omegasq partial.omegasq epsilonsq cohens.f power factora factora 2 233.867 116.933 5.895 0.008 0.304 0.304 0.246 0.246 0.252 0.661 0.874 …2 Residuals 27 535.600 19.837 NA NA NA NA NA NA NA NA NA The anova_stats function can work on either an anova object or one of class Anova. For larger factorial designs, this would be an important way of obtaining effect sizes based on differing SS Types that can be specified. In the above section, we passed the already-created anova fit object to anova_stats, but we can also do the whole analysis in one line of code using the car package Anova function to specify SS Type. For our example here, the Type 1 and Type 3 SS are identical because the design is balanced and Type 1 vs Type 3 should not be relevant in a 1way design. contrasts(hays\$factora) <- contr.sum kable(sjstats::anova_stats(Anova(aov(dv~factora, data=hays), type=3))) term sumsq meansq df statistic p.value etasq partial.etasq omegasq partial.omegasq epsilonsq cohens.f power factora factora 233.867 116.933 2 5.895 0.008 0.304 0.304 0.246 0.246 0.252 0.661 0.874 …2 Residuals 535.600 19.837 27 NA NA NA NA NA NA NA NA NA ### 8.2.1 Use of the lsr package Another option for effect size calculation is the etaSquared function from the lsr package. It can take an argument that permits specification of SS Type, but that will not matter in this balanced design. library(lsr) #etaSquared(fit.1, type=1) #etaSquared(fit.1, type=2) etaSquared(fit.1, type=3) ## eta.sq eta.sq.part ## factora 0.3039335 0.3039335 ## 8.3 Power and sample size planning for completely randomized 1-factor ANOVA designs The pwr package provides a fairly comprehensive way to estimate sample size requirements when designing studies. For a one-factor design, the logic of the code is very straight forward. In other work, we have seen how to use GPower as well. To use the pwr.anova.test funtion: • We need to tell it how many groups. • We need to have an estimate of the within-group std deviation (assumes homogeneity of variance) • We need to have a set of expected outcome values for the sample means. With these means and the within-group variation (the sd), we can estimate cohen’s effect size statistic (the “f”) Since we have been working with a 3-group design, lets see how we might have planned for that with the ‘pwr.anova.test’ function. I set the means and sd arbitrarily in this example - they would normally be chosen on the basis of informed prior information, perhaps from pilot studies or published literature. It is instructive to “fiddle” with this code, changing the means and the sd to see how it affects the desired n per group. This is an alternative to GPower. #library(pwr) groups = 3 means = c(25,20,20) sd = 5 grand.mean = mean(means) efsize = sqrt( sum( (1/groups) * (means-grand.mean)^2) ) /sd #cohen's "f" effect size efsize ## [1] 0.4714045 pwr.anova.test(k = groups, n = NULL, f = efsize, sig.level = 0.05, power = 0.90) ## ## Balanced one-way analysis of variance power calculation ## ## k = 3 ## n = 20.01726 ## f = 0.4714045 ## sig.level = 0.05 ## power = 0.9 ## ## NOTE: n is number in each group This, and other pwr functions work by passing all but one of the relevant characteristics to the function. By leaving out sample size, and passing alpha, power, and effect size, the minimal sample size per group required to acheive that power is returned.	2023-02-03 08:07:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7049906253814697, "perplexity": 4814.664944867074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500044.16/warc/CC-MAIN-20230203055519-20230203085519-00488.warc.gz"}
https://jbrandhorst.com/post/circleci-docker-compose/	## Using docker-compose with CircleCI In a recent blog post I talked about automating the testing of an advanced GopherJS library using a combination of QUnit, Ginkgo and Agouti. That allowed me to run a complete integration test suite against my library by automatically spinning up browsers and pointing them at my QUnit GopherJS page. This was a great start, but after running it a couple of times we find that there are several problems: 1. It requires all browsers tested to be installed on the machine running the tests. This limits the scale of testing as no-one is going to be running around with Chrome, Firefox, Safari, Edge and whatever other browser we may want to test with. Especially not different versions of said browsers! 2. It’s still a manual step. Wouldn’t it be great if these tests could be run on every pull request and every push to the master branch? ## Enter CircleCI To solve these problems I spent some time figuring out how to use the CircleCI platform to automate my setup. I knew it wouldn’t be trivial as there are many different dependencies in the different parts of the protobuf repository. Building the GopherJS protoc plugin requires Go. Generating the well known types requires protoc and the GopherJS protoc plugin to both be installed. The integration tests require Firefox and Chrome to be installed on the machine running the tests. Fortunately CircleCI 2.0 embraces Docker as a testing sandbox. At work we use GitLab whose CI solution is similarly centered around docker (though not exclusively!), so I was reasonable well versed in the quirks around using Docker in CI. The layout of the CircleCI configuration is quite different from the one used by GitLab, so it took me a while to get up to speed. ## The integration test In order to fix the issues I mentioned at the start of this post, I decided to rewrite my tests to be runnable in docker containers, with the browsers running in separate containers. As such, I wrote a docker-compose file to define the setup required. Here’s the docker-compose file I used: version: '2' networks: internal: driver: bridge services: chromedriver: environment: CHROMEDRIVER_WHITELISTED_IPS: "" expose: - "4444" image: robcherry/docker-chromedriver:latest networks: - internal privileged: true selenium: expose: - "4444" image: selenium/standalone-firefox:latest networks: - internal testrunner: # This just sleeps, we execute the test command using docker-compose exec. container_name: testrunner command: sleep infinity depends_on: - chromedriver - selenium environment: expose: - "8080" - "9090" - "9095" - "9100" - "9105" image: golang:latest networks: - internal It starts a ChromeDriver, a Selenium and a standard Go container. This is easy enough to run on your own machine, with docker-compose up and docker-compose exec. I wish I could use docker-compose run here, but because of an old bug that seems to have been forgotten, we’re forced to use up and exec. This is how we run this: bash -c "\ set -x \ trap '\ docker-compose logs selenium && \ docker-compose logs chromedriver && \ docker-compose down' EXIT; \ docker-compose up -d && \ docker-compose exec -T testrunner bash -c '\ mkdir -p /go/src/github.com/johanbrandhorst/protobuf/' && \ docker cp ./ testrunner:/go/src/github.com/johanbrandhorst/protobuf/ && \ docker-compose exec -T testrunner bash -c '\ cd /go/src/github.com/johanbrandhorst/protobuf && \ go install ./vendor/github.com/onsi/ginkgo/ginkgo && \ cd test && make test' \ " Let me preface this by saying that I know this looks absolutely rubbish, but I really wanted the trap and I also wanted this in a Makefile so I could run it easily from my terminal, and I also did not want to have to write a separate shell script for it. This works, and if you squint and ignore the backslashes and ampersands you can kind of see what I was going for. Something worth noting here is the use of docker cp to get the contents of the repository into the testrunner container. This is a workaround to a well known shortcoming of running docker containers from other docker containers, namely that you can’t volume mount from one docker container to another one. So now that we’ve got a docker-compose file and a hacky bash line to run it, what is required to get it running on CircleCI? Look no further: tests: docker: - image: ypereirareis/docker-compose working_directory: /go/src/github.com/johanbrandhorst/protobuf steps: - checkout - setup_remote_docker - run: name: Browser Integration Tests command: make integration We use ypereirareis/docker-compose which just comes with docker-compose pre-installed. The magic here is using setup_remote_docker to do just that. This is required to get access to docker powers from within your CircleCI containers. The checkout job tells CircleCI that we want the git repo cloned into the working_directory. Bam! Just like that we’ve got our GopherJS integration tests up and running on CircleCI! The coolest thing here is that now in the future if we want to test other versions of browsers, or other browsers altogether, we need only add another docker container to the fray. ## Summary We’ve shown that you can run complex testing scenarios involving several interdependent docker containers on CircleCI. These tests run automatically on PRs and commits made to the repository to preserve confidence in the functionality of the code. You can check out the CI in action on my github repo. If you enjoyed this blog post, have any questions or input, don’t hesitate to contact me on @johanbrandhorst or under jbrandhorst on the Gophers Slack. I’d love to hear your thoughts!	2018-03-21 10:49:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23176640272140503, "perplexity": 3926.9762912276124}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647612.53/warc/CC-MAIN-20180321102234-20180321122234-00709.warc.gz"}
https://mathoverflow.net/questions/306789/hoeffdings-inequality-for-random-vectors	# Hoeffding's inequality for random vectors Let $x_1, \ldots, x_n$ be $n$ i.i.d. samples of a bounded random variable $X \in [a, b]$. We know from the Hoeffding's inequality that : $$\mathbb{P} \left( \left\| \frac{1}{n} \sum_{i=1}^n x_n - \mathbb{E}[X] \right\| \geq \varepsilon \right) \leq 2 \exp\left( - \frac{2 n \varepsilon^2}{(b-a)^2}\right)$$ Now let $x_1, \ldots, x_n$ be $n$ i.i.d. samples of a bounded random vector $X$. Do we have a similar bound for the following quantity ? $$\mathbb{P} \left( \left\\| \frac{1}{n} \sum_{i=1}^n x_n - \mathbb{E}[X] \right\\|_1 \geq \varepsilon \right)$$	2019-06-17 03:52:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9623345732688904, "perplexity": 120.28887624944032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998369.29/warc/CC-MAIN-20190617022938-20190617044938-00346.warc.gz"}
https://cs.stackexchange.com/questions/101122/find-the-maximum-rows-having-all-ones-in-the-binary-if-you-are-allowed-to-toggle	# Find the Maximum rows having all ones in the binary if you are allowed to toggle columns in the matrix for exactly k number of times A binary matrix of nxm is given, you have to toggle any column k number of times so that you can get the maximum number of rows having all 1’s. for eg, n=3, m=3, 1 0 0 1 0 1 1 0 0 if k=2, then we will toggle column 2 and 3 once and we will get rows 1 and 3 with 1 1 1 and 1 1 1 i.e. all 1’s so answer is 2 as there are 2 rows with all 1’s. if k=3, then we will toggle column 2 thrice and we will get row 2 with 1 1 1 i.e. all 1’s so answer is 1 as there is 1 row with all 1’s. • Welcome to Computer Science! We discourage posts that simply state a problem out of context, and expect the community to solve it. Assuming you tried to solve it yourself and got stuck, it may be helpful if you wrote your thoughts and what you could not figure out. It will definitely draw more answers to your post. Until then, the question will be voted to be closed / downvoted. You may also want to check out these hints, or use the search engine of this site to find similar questions that were already answered. – John L. Dec 6 '18 at 17:25 • Please credit the original source of the problem by adding a url or reference in the question. – John L. Dec 6 '18 at 17:25	2021-01-21 18:30:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43826568126678467, "perplexity": 368.79966681665}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703527224.75/warc/CC-MAIN-20210121163356-20210121193356-00580.warc.gz"}
https://math.stackexchange.com/questions/3187478/show-that-n-3100-2-is-not-a-prime-number	# Show that $n = 3^{100} + 2$ is not a prime number. So I have to prove that $$n = 3^{100} + 2$$ is not a prime number while we assume that $$X^2 - 53$$ has no zeroes in $$\mathbb{Z}/n\mathbb{Z}$$. Because we are working with quadratic reciprocity in this chapter, I assumed that $$\big(\frac{53}{n}\big) = -1$$ and by the law of quadratic reciprocity, we know that $$\big(\frac{n}{53}\big) = -1$$. However, I have no clue how I could use this to prove that $$n$$ is not prime. • $3^{100}+2$ has 8 divisors – Dr. Mathva Apr 14 at 15:25 • You've written at the beginning "So I have to prove that $n=3^{100}+2$ is a prime number". Nevertheless, in the end, you claim that you want "to prove that $n$ is not prime"... – Dr. Mathva Apr 14 at 15:28 • $3^{100}+2 \equiv 12^2 \mod 53$ so $\left( \frac{n}{53}\right) = 1$. – Robert Israel Apr 14 at 15:32 • Nice, @Robert Israel, but how do we know it is $12^2$? – Fareed AF Apr 14 at 15:38 • @FareedAF $53$ is not very big. You can enumerate $1^2 \mod 53$, $2^2 \mod 53$, ..., $26^2 \mod 53$. – Robert Israel Apr 14 at 15:52 If you're looking for a proof that the true $$(3^{100}+2\|53)=+1$$, here is my approach: Multiply be $$3^4$$, a known square, so: $$(3^{100}+2\|53)=(3^{104}+3\|53)$$ where $$2×3^4=162\equiv 3\bmod 53$$. The exponent on $$3$$ in the large term is now a multiple of $$52$$ forcing $$3^{104}\equiv 1$$ By Fermat's Little Theorem. Thereby $$(3^{100}+2\|53)=(1+3\|53)=(2^2\|53)=+1$$ but you found that a prime number for $$(3^{100}+2)$$ should have given the Legendre symbol $$-1$$. As an old hit song says, this is how it is when doves cry. Even though the above Legendre symbol is $$+1$$, what causes $$X^2-53=0$$ to have no solutions in $$\mathbb{Z}/n\mathbb{Z}$$ is a prime factor $$p$$ of $$n=3^{100}+2$$ for which $$(53\|p)=-1$$. The above proof does not identify any such factors, but the factor $$37121$$ quoted by others has this property. • Right. Had the OP replied to my comment I was going to note $\large \!\bmod 53\!:\,3^{\large 4} n\equiv 2^{\large 2}\,$ so $\,\large\,n\equiv (2/9)^{\large 2},\,$ essentially the same as above (but doesn't require any knowledge of Legendre symbols). – Bill Dubuque Apr 14 at 19:49	2019-08-22 09:02:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7396745681762695, "perplexity": 237.00920055209403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317037.24/warc/CC-MAIN-20190822084513-20190822110513-00523.warc.gz"}
https://myassignments-help.com/2022/10/17/hypothesis-testing-dai-xie-math214/	# 统计代写\|假设检验代写hypothesis testing代考\|MATH214 ## 统计代写\|假设检验代写hypothesis testing代考\|Testing one sample mean when the variance is unknown: P-value We have studied hypothesis testing for one sample mean when the population variance is unknown and the sample size is small using the critical value (traditional) procedure for a t-test. The $P$-value procedure for one sample mean using a t-test will be illustrated employing the same examples presented in Chapter 3 , t-Test for one-sample mean, and solved using the critical value procedure. The three situations of hypothesis testing are covered and the $P$-values are calculated. Example 6.10: The concentration of trace metal iron in the air: Example $3.4$ is reproduced “A researcher at an environmental section wishes to verify the claim that the mean concentration of trace metal iron (Fe) on coarse particulate matter $\left(\mathrm{PM}_{10}\right)$ in the air is $0.36\left(\mu \mathrm{g} / \mathrm{m}^3\right)$ during the summer season of an equatorial urban coastal location. Twelve samples were selected, and the concentration of iron was measured. The collected data showed that the mean concentration of iron is $0.44$ and the standard deviation is $0.34$. A significance level of $\alpha=0.01$ is chosen to test the claim. Assume that the population is normally distributed.” The five steps for conducting hypothesis testing employing the $P$-value procedure can be used to test the hypothesis regarding the mean concentration of trace metal iron on coarse particulate matter in the air during the summer season of an equatorial urban coastal location. The results of the $P$-value procedure will be compared with the critical value (traditional) procedure. Step 1: Specify the null and alternative hypotheses The two hypotheses regarding the mean concentration of iron on coarse particulate matter in the air are presented in Eq. (6.10). $$H_0: \mu=0.36 \text { vs } H_1: \mu \neq 0.36$$ Step 2: Select the significance level $(\alpha)$ for the study The level of significance is chosen to be $0.01$. The $\mathrm{t}$ critical values for a twotailed test with $\alpha=0.01$ and $d . f=12-1=11$, are $t_{\left(\frac{a}{2}, d . f\right)}=t_{\left(\frac{0.01}{2}, 12-1\right)}=\mp 3.106$. ## 统计代写\|假设检验代写hypothesis testing代考\|Testing one sample proportion: P-value Hypothesis testing for one sample proportion using the critical value (traditional) procedure was studied in Chapter 4 , Z-test for one sample proportion. The $P$-value procedure for one sample proportion using a Z-test will be illustrated employing examples to cover various situations of hypothesis testing. Example 6.13: Mineral water: Example $4.4$ is reproduced “A research group at a university wishes to verify the claim announced by a specific factory for mineral water which says that $98 \%$ of people are satisfied with the quality of mineral water. The research group selected 700 people randomly and record their responses, they found that 26 were not satisfied. Use the sample information to make a decision whether to support the claim or not. A significance level of $\alpha=0.01$ is chosen to test the claim. Assume that the population is normally distributed.” The five steps for conducting hypothesis testing employing the $P$-value procédure can be used to test the hypothesis regarding the proportion of people who feel satisfied with the mineral water produced by the factory. The results of the $P$-value procedure will be compared with the critical value (traditional) procedure. Step 1: Specify the null and alternative hypotheses The two hypotheses regarding the proportion of people who feel satisfied with the mineral water produced by the factory are presented in Eq. (6.13) $$H_0: p=0.98 \text { vs } H_1: p \neq 0.98$$ Step 2: Select the significance level $(\alpha)$ for the study The level of significance is chosen to be $0.01$. The $\mathrm{Z}$ critical values for a twotailed test with $\alpha=0.01$ are $\pm 2.58$. The two procedures will be used to solve this problem; namely the critical value and $P$-value procedures. Step 3: Use the sample information to calculate the test statistic value The test statistic value for the $\mathrm{Z}$-test is used to make a decision regarding the proportion of people who feel satisfied with the mineral water produced by the factory. The test statistic value using the Z-test formula was calculated to be $-3.78$. Step 4: Calculate the $P$-value and specify the critical and noncritical regions for the study The rejection and nonrejection regions using the critical values for testing the proportion of people who feel satisfied with the mineral water produced by the factory are presented in Fig. $6.13$ for the standard normal curve (shaded area: blue represents the $P$-value and orange represents the significance level). # 假设检验代考 ## 统计代写\|假设检验代写假设检验代考\|当方差未知时检验一个样本均值:p值 $$H_0: \mu=0.36 \text { vs } H_1: \mu \neq 0.36$$ ## 统计代写\|假设检验代写假设检验代考\|检验一个样本比例:p值 $$H_0: p=0.98 \text { vs } H_1: p \neq 0.98$$ $\mathrm{Z}$ -test的测试统计值用于决定对工厂生产的矿泉水感到满意的人的比例。使用z检验公式计算的检验统计值为$-3.78$ . myassignments-help数学代考价格说明 1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。 2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。 3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。 Math作业代写、数学代写常见问题 myassignments-help擅长领域包含但不是全部:	2022-12-07 11:20:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7505130171775818, "perplexity": 710.6559156841764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711151.22/warc/CC-MAIN-20221207085208-20221207115208-00840.warc.gz"}
https://mothur.org/wiki/Thetan	# Thetan The thetan calculator returns the community Jaccard index of Smith (aka the Θn of Yue) describing the dissimilarity between the structure of two communities. This calculator can be used in the summary.shared and collect.shared commands. $D_{\Theta_N}=1-\frac{\left(\sum_{i=1}^{S_{12}} a_i\right)\left(\sum_{i=1}^{S_{12}} b_i\right)}{\left(\sum_{i=1}^{S_{12}} a_i\right)+\left(\sum_{i=1}^{S_{12}} b_i\right)-\left(\sum_{i=1}^{S_{12}} a_i\right)\left(\sum_{i=1}^{S_{12}} b_i\right)}$ where, $S_{12}$ = the number of shared OTUs between communities A and B $a_i$ = the relative abundance of OTU i in community A $b_i$ = the relative abundance of OTU i in community B Open the file 98_lt_phylip_amazon.fn.sabund generated using the Amazonian dataset with the following commands: mothur > read.dist(phylip=98_lt_phylip_amazon.dist, cutoff=0.10) mothur > cluster() The 98_lt_phylip_amazon.fn.shared file will contain the following two lines: 0.10 forest 55 1 1 1 1 1 1 3 3 2 2 1 1 3 2 1 1 1 1 2 1 1 2 5 1 1 1 1 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.10 pasture 55 0 0 0 1 1 0 1 0 0 5 0 0 0 0 0 2 0 0 0 3 0 0 2 1 0 1 0 0 0 0 0 0 1 2 1 1 1 1 1 7 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 This indicates that the label for the OTU definition was 0.10. The first line is from the forest sample and the second is from the pasture sample. There are a total of 55 OTUs between the two communities. Writing the data in a more presentable manner we see: index forest (A) pasture (B) ai bi 1 1 0 2 1 0 3 1 0 4 1 1 0.0204 0.0204 5 1 0 6 1 0 7 3 1 0.0612 0.0204 8 3 0 9 2 0 10 2 5 0.0408 0.1020 11 1 0 12 1 0 13 3 0 14 2 0 15 1 0 16 1 3 0.0204 0.0612 17 1 0 18 1 0 19 2 0 20 1 3 0.0204 0.0612 21 1 0 22 2 0 23 5 2 0.1020 0.0408 24 1 1 0.0204 0.0204 25 1 0 26 1 1 0.0204 0.0204 27 1 0 28 2 0 29 1 0 30 1 0 31 1 0 32 1 0 33 1 1 0.0204 0.0204 34 0 2 35 0 1 36 0 1 37 0 1 38 0 1 39 0 1 40 0 7 41 0 1 42 0 1 43 0 2 44 0 1 45 0 1 46 0 1 47 0 1 48 0 1 49 0 1 50 0 1 51 0 1 52 0 1 53 0 2 54 0 1 55 0 1 Total 49 49 0.3265 0.3673 Using these sums to evaluate C we get: $C_{\Theta_N}=1-\frac{\left(0.3265\right)\left(0.3673\right)}{\left(0.3265\right)+\left(0.3673\right)-\left(0.3265\right)\left(0.3673\right)}=0.7910$ Running... mothur > summary.shared(calc=thetan) ...and opening 98_lt_phylip_amazon.fn.shared.summary gives: label comparison ThetaN 0.10 forest pasture 0.791001 These are the same values that we found above for a cutoff of 0.10.	2019-07-23 09:02:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43093422055244446, "perplexity": 380.56653223312856}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529175.83/warc/CC-MAIN-20190723085031-20190723111031-00105.warc.gz"}
http://math.stackexchange.com/questions/129480/linear-algebra-inner-product-spaces-question	# Linear Algebra - Inner Product Spaces Question So far I have figured out everything except the angle alpha of f,g. What I tried was I drew this triangle: and found side length c by doing dist(f,g) = \|\|f-g\|\| = sqrt() and I ended up with c = sqrt(54) Then I used the law of cosines to find the angle of alpha and got 14.14 degrees but it was the wrong answer. What is wrong with my approach? - I'm still getting $\sqrt{54}$, $\int_{0}^{1}((5x^2-3)-(6x+3))*((5x^2-3)-(6x+3)) dx = 54$ then square root is $\sqrt{54}$... – StickFigs Apr 9 '12 at 2:06 You’re right: I miscopied one of the functions. The problem is in your calculation of $\alpha$. I get $\cos\alpha=\frac{11}{4\sqrt{39}}\approx 0.44$, $\alpha\approx 1.1148$ radians (or about 63.87°). – Brian M. Scott Apr 9 '12 at 2:14 Those answers are incorrect, see my other comment below. – StickFigs Apr 9 '12 at 2:55 I apologize: I seem to be brain-dead tonight. I dropped a minus sign: $\cos\alpha=\frac{-11/2}{2\sqrt{39}}=-\frac{11}{4\sqrt{39}}$. By the way, I’d expect it to want the angle in radians: that’s a nearly universal convention in mathematics. Out of curiosity, is this WeBWorK? – Brian M. Scott Apr 9 '12 at 3:11 $\frac{-11}{4\sqrt{39}}$ was the correct answer, and yes it is. – StickFigs Apr 9 '12 at 15:46	2013-05-22 15:50:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9135053157806396, "perplexity": 585.8519963730586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368701943764/warc/CC-MAIN-20130516105903-00059-ip-10-60-113-184.ec2.internal.warc.gz"}
http://ncatlab.org/michaelshulman/show/truncation+in+an+exact+2-category	# Michael Shulman truncation in an exact 2-category In a suitably exact 2-category, we can construct truncations as quotients of suitable congruences. # (-1)-truncation This case is easy and just like for 1-categories. ###### Theorem In any regular 2-category $K$ the inclusion $Sub(1) \hookrightarrow K$ of the subterminal objects has a left adjoint called the support or (-1)-truncation. ###### Proof Define the support $supp(A) = A_{\le -1}$ of an object $A$ to be the image of the unique morphism $A\to 1$. That, is $A\to supp(A) \to 1$ is an eso-ff factorization. Since $supp(A)\to 1$ is ff, $supp(A)$ is subterminal, and since esos are orthogonal to ffs, it is a reflection into $Sub(1)$. # (0,1)-truncation Perhaps surprisingly, the next easiest case is the posetal reflection. ###### Theorem In any (1,2)-exact 2-category $K$ the inclusion $pos(K) \hookrightarrow K$ of the posetal objects has a left adjoint called the (0,1)-truncation. ###### Proof Given $A$, define $A_1$ to be the (ff) image of $A^{\mathbf{2}} \to A\times A$. Since esos are stable under pullback, $A_1\;\rightrightarrows\; A$ is a homwise-discrete category, and it clearly has a functor from $ker(A)$, so it is a (1,2)-congruence. Let $A\to P$ be its quotient. By the classification of congruences, $P$ is posetal. And if we have any $f:A\to Q$ where $Q$ is posetal, then we have an induced functor $ker(A) \to ker(f)$. But $Q$ is posetal, so $ker(f)$ is a (1,2)-congruence, and thus $ker(A) \to ker(f)$ factors through a functor $A_1\to ker(f)$. This then equips $f$ with an action by the (1,2)-congruence $A_1\;\rightrightarrows\; A$, so that it descends to a map $P\to Q$. It is easy to check that 2-cells also descend, so $P$ is a reflection of $A$ into $pos(K)$. This is actually a special case of the (eso+full,faithful) factorization system?, since an object $A$ is posetal iff $A\to 1$ is faithful. The proof is also an evident specialization of that. # 0-truncation The discrete reflection, on the other hand, requires some additional structure. ###### Theorem In any 1-exact and countably-coherent 2-category $K$, the inclusion $disc(K) \hookrightarrow K$ of the discrete objects has a left adjoint called the 0-truncation or discretization. ###### Proof Given $A$, define $A_1$ to be the equivalence relation generated by the image of $A^{\mathbf{2}} \to A\times A$; this can be constructed with countable unions in the usual way. Then $A_1\;\rightrightarrows\; A$ is a 1-congruence, and as in the posetal case we can show that its quotient is a discrete reflection of $A$. There are other sufficient conditions on $K$ for the discretization to exist; see for instance classifying cosieve. We can also derive it if we have groupoid reflections, since the discretization is the groupoid reflection of the posetal reflection. # (1,0)-truncation The groupoid reflection is the hardest and also requires infinitary structure. Note that the 2-pretopos $FinCat$ does not admit groupoid reflections (the groupoid reflection of the “walking parallel pair of arrows” is $B Z$). ###### Theorem In any (2,1)-exact and countably-extensive 2-category $K$, the inclusion $gpd(K) \hookrightarrow K$ of the groupoidal objects has a left adjoint called the (1,0)-truncation. Revised on May 29, 2012 22:04:00 by Andrew Stacey? (129.241.15.200)	2014-11-23 02:18:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 45, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9674659371376038, "perplexity": 705.535805895552}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400378956.26/warc/CC-MAIN-20141119123258-00020-ip-10-235-23-156.ec2.internal.warc.gz"}
https://socratic.org/questions/5915bf1f7c014963066f444a	Question #f444a May 12, 2017 No, they cancel out. Explanation: Valency is the combining power of an element. Let's use $C {a}^{2 +}$ and ${O}^{2 -}$ Valency of $C {a}^{2 +}$= 2 Valency of ${O}^{2 -}$= 2 When they combine, they cancel out into $C a O$. (Don't put this on the exam, but you can think of the formula looking like $C {a}_{2} {O}_{2}$ before they cancel out) Let's use one more example: ${K}^{+}$ and ${O}^{2 -}$ Valency of ${K}^{+}$= 1 Valency of ${O}^{2 -}$= 2 When they combine, we basically switch the valencies so that the 2 goes by the $K$ and the 1 goes by the $O$ So ${K}_{2} {O}_{1}$ but the 1 is understood so we don't write that in the final formula which is ${K}_{2} O$	2019-08-18 17:16:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 14, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6941059827804565, "perplexity": 1391.9331859283395}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313987.32/warc/CC-MAIN-20190818165510-20190818191510-00470.warc.gz"}
https://cstheory.stackexchange.com/questions/12071/validity-implies-np-p	# Validity implies NP=#P? [closed] 1. Valid progams for NP imply every solution is a valid answer. 2. NP not equals #P implies not all solutions are answers. 3. Therefore, Validity implies NP=#P. NP is the problem class for finding single verifiable solutions. #P is the related problem class of counting solutions. If the proof is invalid, where is the flaw? By my reasoning, the proof is a three boolean variable three clause 2cnf expression, one of the smallest possible uniquely solvable boolean formulas, requiring three inferences to resolve. My best counting benchmark (4cnf 4 coloring, degree 6 graph) took eleven weeks: C4D6N66c.cnf + #P 472,406,068,323,174 retros 76865745357 infers 66385 billion Send to pehoushek1 at gmail for single file C++ program, bob, for #sat, dimacs forms. The three thousand line bob program can solve millions of small formulas in a single run, but can be exponential on large formulas. bob also solves sat, unsat, and qbfs, in roughly the same order of magnitude of time as #P, computing nearly two trillion inferences per day. My main publication in the general area is Introduction to Qspace (Satisfiability 2002), containing a short proof of the theorem #P=#Q: the number of satisfying assignments to a boolean formula equals the number of valid quantifications of the formula. bob uses #P=#Q to solve qbfs, indicating coNP=NP=#P=#Q=PSpace=Exp. Garey and Johnson is the main reference. ## closed as not a real question by Tsuyoshi Ito, Lev Reyzin♦, KavehJul 23 '12 at 18:19 It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. • Welcome to cstheory, a Q&A site for research-level questions in theoretical computer science (TCS). Your question does not appear to be a research-level question in TCS. Please see the FAQ for more information on what is meant by this and suggestions for sites that might welcome your question. Finally, if your question is closed for being out of scope, and you believe you can edit the question to make it a research-level question, please feel free to do so. Closing is not permanent and questions can be reopened, check the FAQ for more information. – Kaveh Jul 20 '12 at 6:47 • Please also see the site policy on crank-friendly topics. Note that $\mathsf{PH} \subseteq \mathsf{P^{\#P}}$ so if what you say is true then $\mathsf{PH} \subseteq \mathsf{P^{NP}}$, i.e. $\mathsf{PH}$ collapses (which is a major open problem in complexity theory). – Kaveh Jul 20 '12 at 6:50 You have some fundamental misunderstanding of what a language $L$ being in $\mathsf{NP} \cap \mathsf{coNP}$ means. You'd need to show that there exist two machines: $M$ which is an $\mathsf{NP}$ machine, and $M'$, which is a $\mathsf{coNP}$ machine, such that $L$ is decided by $M$, and $L$ is also decided by $M'$. Each one should be able to solve the problem on its own, without being given the other one as an oracle, like you do. Your construction on the other hand requires a linear number of calls to both $\mathsf{NP}$ and $\mathsf{coNP}$. In other words you have an alternating Turing machine with linear number of alternations. It's easy to see that such a machine can solve TQBF and therefore any problem in $\mathsf{PSPACE}$. Since the functional equivalent $\mathsf{FPSPACE}$ of $\mathsf{PSPACE}$ contains $\mathsf{\#P}$ and $\mathsf{FPSPACE}$ reduces easily (in the Turing reduction sense) to $\mathsf{PSPACE}$, your statement is not at all surprising. In fact your machine is more powerful than needed for solving $\mathsf{\#P}$ problems. • FPSPACE contains #P. Conflating decision and functional complexity classes is a type error which people frequently gloss over, but which is especially important to get right in your answer since the OP got it wrong (asking about #P being contained in NP and co-NP) in his question. – Huck Bennett Jul 19 '12 at 22:29 • @Huck Thanks, fixed. In any case though, confusing functional and decision complexity classes is by far not the biggest problem with OP's reasoning. If it were, then there would be an easy fix :). – Sasho Nikolov Jul 20 '12 at 1:13 • A "linear number of calls to both $NP$ and $coNP$" does not require that it's a $PSPACE$ machine. (Could mean that the simulation is only in $P^{NP}$, and $\# P$ is certainly not known to be in $P^{NP}$. Still, I can't follow much of what is said above.) – Ryan Williams Jul 20 '12 at 1:59 • @Ryan Good point. On a first read I thought he may be using an alternating machine with linear number of alternations. On a second, maybe he's only claiming to make $O(n)$ calls to the NP machine... hard to tell what's going on! – Sasho Nikolov Jul 20 '12 at 6:23 After further review, if the given construction is valid, then Certificates for coNP are possible, so coNP is in NP, or else the relatively simple construction is invalid. Imagine, certificates for unsatisfiable problems! The size increase would be O(MN) on M k-clauses and N variables. The certificate can be obtained for an unsatisfiable formula by partitioning 2^N as #rootsat on the logical complement of the original unsatisfiable formula. I am not saying that the certificate is easy to construct, just that it Exists, for unsatisfiable formulas, among shallow binary trees where leafs are all labelled with #sat and #unsat. Language relativization does not actually seem to apply to the tree construction. So, solving #P would still require binary search, which means O(N) calls to determine an N bit number. But the conclusion is entirely in NP, and so is easily checked once found! One certificate labels the precise number of solutions X in O(M) trees related to the original formula; and a second certificate labels 2^N - X as the number of solutions in O(M) trees related to the logical complement of the orignal formula. The size of both certificates is O(MN). Checking the certificates of the trees is easy, just adding of the #sat labels on each leaf of each tree. Every tree has #rootsat as the total number of satisfying assignments. So the conclusion now is that #P is entirely in NP, by using two easy to check certificates, one on the formula, and one on its complement. coNP is also in NP, with an easily checkable certificate on the complementary formula. These certificates may be hard to construct, but they Exist. The construction deserves alot of attention to details, of course. If the construction is invalid, then please ignore my rambling.	2019-09-17 15:10:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.538826048374176, "perplexity": 927.0926624116662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573080.8/warc/CC-MAIN-20190917141045-20190917163045-00132.warc.gz"}
https://math.stackexchange.com/questions/1220817/proof-that-for-two-converging-sequences-and-two-corresponding-unbound-sets-the	# Proof that for two converging sequences and two corresponding unbound sets - the limit is equal. Ok so I know I've seen this before phrased differently but I can't quite put my finger on the solution. The question goes as follows: Let $$a_n, b_n$$ be two converging sequences. Also, let $$\{n: n\in\Bbb{N}, a_n \leq b_n\}$$ and $$\{n: n\in\Bbb{N}, b_n \leq a_n\}$$ be two unbounded sets. Prove that $$\lim\limits_{n\to \infty}a_n = \lim\limits_{n\to \infty}b_n$$ I'm completely drawing a blank. I tried several approaches I know, such as Cantor's intersection theorem (which doesn't seem to fit) or other properties of converging sequences I've learned, but the question - as posed - doesn't provide enough detail (that I can see) to use any of those as far as I can tell. Can anyone shed some light on this for me? Thanks! Edit (possible solution?) After some page-turning I found a similar proof in my professor's slides and adapted it to this question. Let me know if you think it's valid: First we'll show that if $$a_n \leq b_n$$ for every $$n$$, then $$\lim\limits_{n\to \infty}a_n \leq \lim\limits_{n\to \infty}b_n$$: Assuming by contradiction that if if $$a_n \leq b_n$$ for every $$n$$, then $$\lim\limits_{n\to \infty}a_n > \lim\limits_{n\to \infty}b_n$$, there then exists an $$n$$ in $$a_n$$ for which $$a_n>b_n$$, in contradiction to the fact that $$a_n \leq b_n$$ for every $$n$$, and therefor if $$a_n \leq b_n$$ for every $$n$$, then $$\lim\limits_{n\to \infty}a_n \leq \lim\limits_{n\to \infty}b_n$$. We use the same process to prove the opposite direction, and since we have two infinite sets of indices for which $$a_n\geq b_n$$ and $$b_n \geq a_n$$, therefor $$\lim\limits_{n\to \infty}a_n \leq \lim\limits_{n\to \infty}b_n$$ and $$\lim\limits_{n\to \infty}b_n \leq \lim\limits_{n\to \infty}a_n$$ - which means $$\lim\limits_{n\to \infty}a_n = \lim\limits_{n\to \infty}b_n$$ - Q.E.D. • What do you know about the subsequences of a convergent sequence? – Daniel Fischer Apr 5 '15 at 8:47 • Technically we haven't covered that topic yet, but I know from previous courses that all sub-sequences of a convergent sequence have the same sub-limit, except I can't use that because we haven't even touched the term 'subsequence' yet in class. – Elad Avron Apr 5 '15 at 8:49 • It just occured to me that the mentioned sets are not of members in the sequences but of INDICES. Could this help me in any way? I can't seem to think how. – Elad Avron Apr 5 '15 at 8:50 • But if you could use subsequences, would you see which subsequences you could take to reach the desired conclusion? If so, you can translate that idea into a proof not mentioning subsequences. – Daniel Fischer Apr 5 '15 at 8:51 • As you emphasised above: indices of subsequences. For subsets of $\mathbb{N}$, "unbounded" and "infinite" are equivalent, so if one of the two sets were bounded, we wouldn't have the corresponding subsequence. – Daniel Fischer Apr 5 '15 at 8:57 Let the limits be $a$ and $b.$ Fix $\epsilon > 0.$ There exist $N_{\epsilon}$ such that for $n>N_{\epsilon}$ $$\|a_n- a \| < \epsilon, \; \|b_n-b\| < \epsilon.$$ or alternatively $$a-\epsilon < a_n < a+\epsilon, \; b-\epsilon < b_n < b+\epsilon.$$ There is some such $n$ with $a_n < b_n.$ So $$a-\epsilon < b + \epsilon.$$ Similarly, $$b-\epsilon < a + \epsilon.$$ So $$\|a-b \| < 2\epsilon.$$ But $\epsilon$ was arbitrary, so $a=b.$	2021-03-06 05:58:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9445034265518188, "perplexity": 176.4752665507493}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178374391.90/warc/CC-MAIN-20210306035529-20210306065529-00497.warc.gz"}
https://trac.fact-project.org/changeset/8777	# Changeset 8777 Ignore: Timestamp: Dec 7, 2007, 11:23:42 PM (13 years ago) Message: * empty log message * Location: trunk/Dwarf/Documents/ApplicationDFG Files: 2 edited ### Legend: Unmodified r8774 \maketitle %\tableofcontents \newpage x \thispagestyle{empty} \cleardoublepage \newpage \section[1]{General Information (Allgemeine Angaben)} {44221 Dortmund }&\multicolumn{2}{l\|}{58285 Gevelsberg}\\ {Germany }&\multicolumn{2}{l\|}{Germany }\\[0.5ex] {\parbox[t]{1.5cm}{Phone:}+49\,(231)\,755-3550}&\multicolumn{2}{l\|}{\parbox[t]{1.5cm}{Phone:}+49\,(931)\, }\\ {\parbox[t]{1.5cm}{Phone:}+49\,(231)\,755-3550}&\multicolumn{2}{l\|}{\parbox[t]{1.5cm}{Phone:}+49\,(173)\,284\,79\,10}\\ {\parbox[t]{1.5cm}{Fax:}+49\,(231)\,755-4547}&\multicolumn{2}{l\|}{~}\\\hline\hline \multicolumn{3}{\|c\|}{{\bf email}: wolfgang.rhode@udo.edu}\\\hline {97074 W"urzburg }&\multicolumn{2}{l\|}{97299 Zell am Main }\\ {Germany }&\multicolumn{2}{l\|}{Germany }\\[0.5ex] {\parbox[t]{1.5cm}{Phone:}+49\,(931)\,888-5031}&\multicolumn{2}{l\|}{\parbox[t]{1.5cm}{Phone:} }\\ {\parbox[t]{1.5cm}{Phone:}+49\,(931)\,888-5031}&\multicolumn{2}{l\|}{\parbox[t]{1.5cm}{Phone: +49\,(931)\,404\,81\,90} }\\ {\parbox[t]{1.5cm}{Fax:}+49\,(931)\,888-4603}&\multicolumn{2}{l\|}{~}\\\hline\hline \multicolumn{3}{\|c\|}{{\bf email}: mannhein@astro.uni-wuerzbueg.de}\\\hline \multicolumn{3}{\|c\|}{{\bf email}: mannheim@astro.uni-wuerzbueg.de}\\\hline \end{tabular} \originalTeX \newpage %\paragraph{1.2 Topic}~\\ \subsection[1.2]{Topic} \paragraph{1.2 Topic}~\\ %\subsection[1.2]{Topic} Long-term VHE $\gamma$-ray monitoring of bright blazars with a dedicated Cherenkov telescope %\paragraph{1.2 Thema}~\\ \subsection[1.2]{Thema} \paragraph{1.2 Thema}~\\ %\subsection[1.2]{Thema} Langzeitbeobachtung von hellen VHE $\gamma$-Blazaren mit einem dedizierten Cherenkov Teleskop %\paragraph{1.3 Discipline and field of work (Fachgebiet und Arbeitsrichtung)}~\\ \subsection[1.3]{Discipline and field of work (Fachgebiet und Arbeitsrichtung)} \paragraph{1.3 Discipline and field of work (Fachgebiet und Arbeitsrichtung)}~\\ %\subsection[1.3]{Discipline and field of work (Fachgebiet und Arbeitsrichtung)} Astronomy and Astrophysics, Particle Astrophysics %\paragraph{\bf 1.4 Scheduled duration in total (Voraussichtliche Gesamtdauer)}~\\ \subsection[1.4]{Scheduled duration in total (Voraussichtliche Gesamtdauer)} \paragraph{\bf 1.4 Scheduled duration in total (Voraussichtliche Gesamtdauer)}~\\ %\subsection[1.4]{Scheduled duration in total (Voraussichtliche Gesamtdauer)} After successful completion of the three-year work plan developed in this proposal, we will ask for an extension of the project for another of supermassive binary black holes. %\paragraph{\bf 1.5 Application period (Antragszeitraum)}~\\ \subsection[1.5]{Application period (Antragszeitraum)} \paragraph{\bf 1.5 Application period (Antragszeitraum)}~\\ %\subsection[1.5]{Application period (Antragszeitraum)} 3\,years. The work on the project will begin immediately after the funding. funding. \newpage %\paragraph{\bf 1.6 Summary}~\\ \subsection[1.6]{Summary} We propose to set up a robotic imaging air Cherenkov telescope with low cost, but high performance design for remote operation. The goal is to \paragraph{\bf 1.6 Summary}~\\ %\subsection[1.6]{Summary} We propose to set up a robotic imaging air-Cherenkov telescope with low cost, but a high performance design for remote operation. The goal is to dedicate this gamma-ray telescope to long-term monitoring observations of nearby, bright blazars at very high energies. We will (i) search for origin, and (iii) correlate the data with corresponding data from the neutrino observatory IceCube to search for evidence of hadronic emission processes. The observations will also trigger follow-up emission processes. The observations will furthermore trigger follow-up observations of flares with higher sensitivity telescopes such as MAGIC, VERITAS, and H.E.S.S.\ Joint observations with the Whipple MAGIC, VERITAS and H.E.S.S.\ Joint observations with the Whipple monitoring telescope will start a future 24\,h-monitoring of selected sources with a distributed network of robotic telescopes. The telescope design is based on a full technological upgrade of one of the former design is based on a complete technological upgrade of one of the former telescopes of the HEGRA collaboration (CT3) still located at the Observatorio Roque de los Muchachos on the Canarian Island La Palma \germanTeX %\paragraph{\bf 1.6 Zusammenfassung}~\\ \subsection[1.6]{Zusammenfassung} \paragraph{\bf 1.6 Zusammenfassung}~\\ %\subsection[1.6]{Zusammenfassung} {\bf Unser Vorhaben besteht darin, ein robotisches Luft-Cherenkov-Teleskop mit geringen Kosten aber hoher Leistung fernsteuerbar in Betrieb zu nehmen. Das Ziel ist es, dieses gamma-ray Teleskop ganz der nehmen. Das Ziel ist es, dieses Gammastrahlen Teleskop ganz der Langzeitbeobachtung von nahen, hellen Blazaren bei sehr hohen Energien zu widmen. Wir werden (i) nach Modulationen der Blazar-Emission durch Statistik von gamma-Ausbr"uchen und deren physikalischen Ursprung untersuchen und (iii) die Daten mit entsprechenden Daten von dem Neutrino-Telskop IceCube korrelieren, um Nachweise f"ur hadronische Neutrino-Teleskop IceCube korrelieren, um Nachweise f"ur hadronische Emissionsprozesse zu finden. Die Beobachtungen werden zus"atzlich Nachfolgebeobachtungen von gamma-Ausbr"uchen mit h"ohersensitiven Teleskopen wie MAGIC, VERITAS und H.E.S.S.\ triggern. Auf einander Teleskopen wie MAGIC, VERITAS und H.E.S.S.\ triggern. Aufeinander abgestimmte Beobachtungen zusammen mit dem Whipple Teleskop werden der Auftakt zu einer zuk"unftigen 24-Stunden-Beobachtung von selektierten \newpage \section[2]{Stand der Forschung, eigene Vorarbeiten\\(Science case, preliminary work by proposer)} \section[2]{Science case, preliminary work by proposer\\(Stand der Forschung, eigene Vorarbeiten)} \subsection[2.1]{Science case (Stand der Forschung)} Since the termination of the HEGRA observations, the succeeding experiments MAGIC and H.E.S.S. have impressively extended the physical scope of gamma ray astronomy detecting tens of formerly unknown gamma ray sources and analyzing their energy spectra, morphology, and experiments MAGIC and H.E.S.S.\ have impressively extended the physical scope of gamma-ray astronomy detecting tens of formerly unknown gamma-ray sources and analyzing their energy spectra, morphology and temporal behavior. This became possible by lowering the energy threshold from 700\,GeV to less than 100\,GeV and increasing at the same time the sensitivity by a factor of five. A diversity of astrophysical source types such as pulsar wind nebulae, supernova remnants, microquasars, pulsars, radio galaxies, clusters of galaxies, gamma ray bursts and blazars have been studied with these telescopes. micro-quasars, pulsars, radio galaxies, clusters of galaxies, Gamma-Ray Bursts and blazars have been studied with these telescopes. The main class of extragalactic, very high energy gamma-rays sources long-wavelength radio waves to multi-TeV gamma-rays. In addition, blazars are characterized by rapid variability, high degrees of polarization, and super-luminal motion of knots in their polarization, and superluminal motion of knots in their high-resolution radio images. The observed behavior can readily be explained assuming relativistic bulk motion and in situ particle acceleration, e.g. at shock waves, leading to synchrotron acceleration, e.g.\ at shock waves, leading to synchrotron (radio-to-x-ray) and self-Compton (gamma-ray) emission \citep{Blandford}. Additionally, inverse Compton scattering of external photons may play a role in producing the observed gamma rays \citep{Dermer,Begelman}. role in producing the observed gamma-rays \citep{Dermer,Begelman}. Variability may hold the key to understanding the details of the emission processes and the source geometry, and the development of time-dependent models is currently on the agenda of model builders emission processes and the source geometry. The development of time-dependent models is currently under investigation worldwide. ambient matter, will quickly dominate the momentum flow of the jet. This {\em baryon pollution} has been suggested to solve the energy transport problem in gamma ray bursts, and is probably present in transport problem in Gamma-Ray Bursts and is probably present in blazar jets as well, even if they originate as pair jets in a black hole ergosphere \citep{Meszaros}. Protons and ions accelerated in the jets of blazars can reach extremely high energies before energy losses jets of blazars can reach extremely high energies, before energy losses become important \citep{Mannheim:1993}. Escaping particles contribute to the observed flux of ultrahigh energy cosmic rays in a major way. Blazars and their unbeamed hosts, the radio galaxies, are thus the prime candidates for origin of ultrahigh energy cosmic rays \citep{Rachen}, and this can be investigated with the IceCube and AUGER \citep{Rachen}. This can be investigated with the IceCube and AUGER experiments. Recent results of the AUGER experiment show a significant anisotropy of the highest energy cosmic rays and point at either nearby with magnetic confinement \citep{Mannheim:1995}. Short variability time scales can result from dynamical changes of the emission zone, running e.g. through an inhomogeneous environment. e.g.\ through an inhomogeneous environment. The contemporaneous spectral energy distributions for hadronic and model \citep{Mannheim:1999}. These properties allow conclusions about the accelerated particles. Noteworthy, even for nearby blazars the spectrum must be corrected for attenuation of the gamma rays due to the spectrum must be corrected for attenuation of the gamma-rays due to pair production in collisions with low-energy photons from the extragalactic background radiation field \citep{Kneiske}. generally showing the largest amplitudes and the shortest time scales at the highest energies. Recently, a doubling time scale of two minutes has been observed in a flare of Mrk\,501 with the MAGIC telescope \citep{Albert:501}. A giant flare of PKS\,2155-304 discovered by H.E.S.S.\ \citep{Aharonian:2007pks} has shown similarly short doubling time scales and a flux of up to 16 times the flux of the Crab Nebula. Indications for TeV flares without evidence for an accompanying x-ray flare, coined orphan flares, have been observed, questioning the has been observed in a flare of Mrk\,501 with the MAGIC telescope \citep{Albert:501}. A giant flare of PKS\,2155-304 discovered by H.E.S.S.\ \citep{Aharonian:2007pks} has shown similarly short doubling time scales and a flux of up to 16 times the flux of the Crab Nebula. Indications for TeV flares without evidence for an accompanying x-ray flare, coined orphan flares, have been observed, questioning the synchrotron-self-Compton mechanism being responsible for the gamma-rays. Model ramifications involving several emission components, external seed photons, or hadronically induced emission may solve the problem \citep{Blazejowski}. Certainly, the database for contemporaneous multi-wavelength observations is still far from proving the synchrotron-self-Compton model. problem \citep{Blazejowski}. Certainly, the database for contemporaneous multi-wavelength observations is still far from proving the synchrotron-self-Compton model. Generally, observations of flares are prompted by optical or x-ray wave luminosity is spectacularly high, even long before final coalescence and the frequencies are favorable for the detectors under consideration (LISA). Detection of gravitational waves relies on exact consideration (LISA). The detection of gravitational waves relies on exact templates to filter out the signals and the templates can be computed from astrophysical constraints on the orbits and masses of the black Mrk\,501 during a phase of high activity in 1997 was reported by HEGRA \citep{Kranich}, and was later confirmed including x-ray and Teleacope Array data \citep{Osone}. The observations can be explained in Telescope Array data \citep{Osone}. The observations can be explained in a supermassive black hole binary scenario \citep{Rieger:2000}. Indications for helical trajectories and periodic modulation of optical exposure simultaneous to the VHE observations, and this is a new qualitative step for blazar research. For the same reasons, the VERITAS Collaboration keeps the former Whipple telescope alive, albeit its collaboration keeps the former Whipple telescope alive, albeit its performance seems to have strongly degraded. It is obvious that the large Cherenkov telescopes such as MAGIC, H.E.S.S.\ or VERITAS are mainly Assuming conservatively the performance of a single HEGRA-type telescope, long-term monitoring of at least the following known blazars is possible: Mrk\,421, Mrk\,501, 1ES\,2344+514, 1ES\,1959+650, H\,1426+428, PKS\,2155-304. We emphasize that DWARF will run as a telescope, long-term monitor\-ing of at least the following known blazars is possible: Mrk\,421, Mrk\,501, 1ES\,2344+514, 1ES\,1959+650, H\,1426+428, PKS\,2155-304. We emphasize, that DWARF will run as a facility dedicated to these targets only, providing a maximum observation time for the program. Utilizing recent developments, such of Amanda, IceCube, HEGRA and MAGIC the proposing groups contribute the necessary knowledge and experience to build and operate a small imaging air Cherenkov telescope. air-Cherenkov telescope. \paragraph{Hardware} development departure of the faculty. The ultra fast drive system of the MAGIC telscopes, suitable for fast The ultra fast drive system of the MAGIC telescopes, suitable for fast repositioning in case of Gamma-Ray Bursts, has been developed, commissioned and programmed by the W\"{u}rzburg group Mirror structures made of plastic material have been developed as Winston Cones for balloon flight experiments previously by the group of Winston cones for balloon flight experiments previously by the group of Wolfgang Dr\"{o}ge. W\"{u}rzburg has also participated in the development of a HPD test bench, which has been setup in Munich and W\"{u}rzburg. With flexible and modular enough to easily process DWARF data \citep{Bretz:2005paris,Riegel:2005icrc,Bretz:2005mars}. A method for absolute light calibration of the PMs based on Muon images has been absolute light calibration of the PMs based on Muon images, especially important for long-term monitoring, has been adapted and further improved for the MAGIC telescope \citep{Meyer:Diploma,Goebel:2005}. Both, data analysis and Monte Carlo developed to be powerful and as robust as possible to be best suited for automatic processing \citep{Dorner:2005paris}. Experience with large amount of data (up to 15\,TB/month) has been gained over five years now. The datacenter is equipped with a professional multi-stage large amount of data (up to 8\,TB/month) has been gained since 2004. The datacenter is equipped with a professional multi-stage (hierarchical) storage system. Two operators are paid by the physics faculty. Currently efforts in W\"{u}rzburg and Dortmund are ongoing to turn the old inflexible Monte Carlo programs, used by the MAGIC collaboration, into modular packages which allows easy simulation of turn the old, inflexible Monte Carlo programs, used by the MAGIC collaboration, into modular packages allowing for easy simulation of other setups. Experience with Monte Carlo simulations, especially CORSIKA, is contributed by the Dortmund group, which has actively model \citep{Haffke:Dipl,Schroeder:PhD} for the local atmosphere of La Palma. Furthermore the group has developed high precision Monte Carlos for Lepton propagation in different media \citep{hepph0407075}. An energy unfolding method and program has been adapted for IceCube and for Lepton propagation in different media %\citep{hepph0407075}. An \citep{xxx}. An energy unfolding method and program has been adapted for IceCube and MAGIC data analysis \citep{Curtef:CM,Muenich:ICRC}. \paragraph{Phenomenology} Both groups further have experience with source models and theoretical computations of gamma ray and neutrino spectra expected from blazars. Both groups have experience with source models and theoretical computations of gamma-ray and neutrino spectra expected from blazars. The relation between the two messengers is a prime focus of interest. Experience with corresponding multi-messenger data analyses involving MAGIC and IceCube data is available in the Dortmund group. Research activities are also related with relativistic particle acceleration \citep{Meli} and gamma ray attenuation \citep{Kneiske}. The W\"{u}rzburg \citep{Meli} and gamma-ray attenuation \citep{Kneiske}. The W\"{u}rzburg group has organized and carried out multi-wavelength observations of bright blazars involving MAGIC, Suzaku, the IRAM telescopes, and the optical KVA telescope \citep{Ruegamer}. Signatures of supermassive bright blazars involving MAGIC, Suzaku, the IRAM telescopes and the KVA optical telescope \citep{Ruegamer}. Signatures of supermassive black hole binaries, which are most relevant also for gravitational wave detectors, are investigated jointly with the German LISA consortium (Burkart, Elbracht ongoing research, funded by DLR). Secondary gamma rays due to dark matter annihilation events are \mbox{Secondary} gamma-rays due to dark matter annihilation events are investigated both from their particle physics and astrophysics aspects. Another main focus of research is on models of radiation and particle The aim of the project is to put the former CT3 of the HEGRA collaboration on the Roque de los Muchachos back into operation - with an enlarged mirror surface, a new camera with higher quantum efficiency, and new fast data acquisition system, under the name of DWARF. The energy threshold will be lowered, and the sensitivity of DWARF will be greatly improved compared to HEGRA CT3 (see figure~\ref{sensitivity}). Commissioning and the first year of data taking should be carried out within the three years of the requested funding period. collaboration on the Roque de los Muchachos back into operation. It will be setup, under the name DWARF, with an enlarged mirror surface (fig.~\ref{DWARF}), a new camera with higher quantum efficiency and new fast data acquisition system. The energy threshold will be lowered, and the sensitivity of DWARF will be greatly improved compared to HEGRA CT3 (see fig.~\ref{sensitivity}). Commissioning and the first year of data taking should be carried out within the three years of the requested funding period. \begin{figure}[ht] \begin{center} \includegraphics[width=0.495\textwidth,angle=0,clip]{CT3.eps} \includegraphics[width=0.495\textwidth,angle=0,clip]{DWARF.eps} \caption{Left: The old HEGRA CT3 telescope as operated within the HEGRA Sytem. Right: A photomontage how the revised CT3 telescope could look like with more and hexagonal mirrors.} \includegraphics[width=0.496\textwidth,angle=0,clip]{CT3.eps} \includegraphics[width=0.496\textwidth,angle=0,clip]{DWARF.eps} \caption{The old CT3 telescope as operated within the HEGRA System (left) and a photomontage of the revised CT3 telescope with more and hexagonal mirrors (right).} \label{CT3} \label{DWARF} The telescope will be operated robotically to reduce costs and man power demands. Furthermore, we seek to obtain know-how for the power demands. Furthermore, we seek to obtain know-how for the operation of future networks of robotic Cherenkov telescopes (e.g. a monitoring array around the globe or CTA) or telescopes at inaccessible sites. From the experience with the construction and operation of MAGIC or HEGRA, the proposing groups consider the planned focused approach (small number of experienced scientists) as optimal for achieving the project goals. The available automatic analysis package developed by the W\"{u}rzburg group for MAGIC is modular and flexible, and can thus be used with minor changes for the DWARF project. monitoring array around the globe or CTA) or telescopes at sited difficult to access. From the experience with the construction and operation of MAGIC or HEGRA, the proposing groups consider the planned focused approach (small number of experienced scientists) as optimal for achieving the project goals. The available automatic analysis package developed by the W\"{u}rzburg group for MAGIC is modular and flexible, and can thus be used with minor changes for the DWARF project. \begin{figure}[htb] \includegraphics[width=0.7\textwidth,angle=0,clip]{visibility.eps} \caption{Source visibility in hours per night versus month of the year for a maximum observation zenith angle of 65$^\circ$. Shown are all sources which we want to monitor including the CrabNebula necessary for calibration and quality assurance. } considering a maximum observation zenith angle of 65$^\circ$ for all sources which we want to monitor including the Crab Nebula, necessary for calibration and quality assurance.} \label{visibility} \end{center} The scientific focus of the project will be on the long-term monitoring of bright, nearby VHE emitting blazars. At least one of the proposed targets will be visible any time of the year (see figure~\ref{visibility}). For calibration purposes, some time will be scheduled for observations of the Crab nebula. The blazar observations will allow of bright, nearby VHE emitting blazars. At least one of the proposed targets will be visible any time of the year (see fig.~\ref{visibility}). For calibration purposes, some time will be scheduled for observations of the Crab Nebula.\\ The blazar observations will allow \begin{itemize} \item to determine the duty cycle, the baseline emission, and the power \item to determine the baseline emission, the duty cycle and the power spectrum of flux variations. \item to cooperate with the Whipple monitoring telescope for an \item to prompt Target-of-Opportunity (ToO) observations with MAGIC in the case of flares increasing time resolution. Corresponding ToO proposals to H.E.S.S.\ and Veritas are in preparation. ToO proposals to H.E.S.S.\ and VERITAS are in preparation. \item to observe simultaneously with MAGIC which will provide an extended bandwidth from below 100\,GeV to multi-TeV energies. \item to obtain multi-frequency observations together with the Mets\"{a}hovi Radio Observatory and the optical Tuorla Observatory. The measurements will be correlated with INTEGRAL and GLAST results, when available. x-ray monitoring using the SWIFT and Suzaku facilities will be proposed. Mets\"{a}hovi Radio Observatory and the optical telescopes of the Tuorla Observatory. The measurements will be correlated with INTEGRAL and GLAST results, when available. X-ray monitoring using the SWIFT and Suzaku facilities will be proposed. \end{itemize} jets. We plan to interpret the data with models currently developed in the context of the Research Training Group {\em Theoretical Astrophysics} in W\"{u}rzburg (Graduiertenkolleg, GK\,1147), including Astrophysics} in W\"{u}rzburg (Graduiertenkolleg, GK\,1147), including particle-in-cell and hybrid MHD models. \item the black hole mass and accretion rate fitting the data with emission models. Results will be compared with estimates of the black hole mass from the Magorrian relation. emission models. Results will be compared with estimates of the black hole mass from the Magorrian relation. \item the flux of relativistic protons (ions) by correlating the rate of neutrinos detected with the neutrino telescope IceCube and the rate of gamma ray photons detected with DWARF, and thus the rate of escaping of gamma-ray photons detected with DWARF, and thus the rate of escaping cosmic rays. \item the orbital modulation owing to a supermassive binary black hole. period of one year, the following steps are necessary: The work schedule assumes that the work will begin in January 2008, The work schedule assumes, that the work will begin in January 2008, immediately after funding. Later funding would accordingly shift the schedule. Each year is divided into quarters (see figure~\ref{schedule}). schedule. Each year is divided into quarters (see fig.~\ref{schedule}). \begin{figure}[htb] \begin{center} \includegraphics[angle=0,clip]{schedule.eps} \includegraphics[width=\textwidth,angle=0,clip]{schedule.eps} % \caption{Left: The old HEGRA CT3 telescope as operated within the % HEGRA Sytem. Right: A photomontage how the revised CT3 telescope \paragraph{Software} \begin{itemize} \item MC adaption (Do/W\"{u}): Due to the large similarities with the MAGIC telescope, within half a year new Monte Carlo code can be programmed using parts of the existing MAGIC Monte Carlo code. For tests and cross-checks another period of six months is necessary. \item Analysis adaption (W\"{u}): The modular concept of the Magic Analysis and Reconstruction Software (MARS) allows a very fast adaption of the telescope setup, camera and data acquisition properties within half a year. \item Adaption Drive software (W\"{u}): Since the new drive electronics will be based on the design of the MAGIC II drive system the control software can be reused unchanged. The integration into the new slow control system will take about half a year. It has to be finished at the time of arrival of the drive system components in 2009/1. \item Slow control/DAQ (Do): A new data acquisition and slow control system for camera and auxiliary systems has to be developed. Based on experiences with the AMANDA DAQ, the Domino DAQ developed for MAGIC II will be adapted and the slow control integrated within three quarters of a year. Commissioning will take place with the full system in 2009/3. \item MC adaption (Do/W\"{u}): Due to the large similarities with the MAGIC telescope, within half a year new Monte Carlo code can be programmed using parts of the existing MAGIC Monte Carlo code. For tests and cross-checks another period of six months is necessary. \item Analysis adaption (W\"{u}): The modular concept of the Magic Analysis and Reconstruction Software (MARS) allows a very fast adaption of the telescope setup, camera and data acquisition properties within half a year. \item Adaption Drive software (W\"{u}): Since the new drive electronics will be based on the design of the MAGIC~II drive system the control software can be reused unchanged. The integration into the new slow control system will take about half a year. It has to be finished at the time of arrival of the drive system components in 2009/1. \item Slow control/DAQ (Do): A new data acquisition and slow control system for camera and auxiliary systems has to be developed. Based on experiences with the AMANDA DAQ, the Domino DAQ developed for MAGIC~II will be adapted and the slow control integrated within three quarters of a year. Commissioning will take place with the full system in 2009/3. \end{itemize} \paragraph{Mirrors (W\"{u})} First prototypes for the mirrors are already available. After testing (six months), the production will start in summer 2008 and shipment will be finished before the full system assembly 2009/2. \paragraph{Drive (W\"{u})} After a planning phase of half a year to simplify the MAGIC II drive system for a smaller telescope (together with the delivering company), ordering, production and shipment should be finished in 2009/1. The MAGIC I and II drive systems have been planned and implemented successfully by the Wuerzburg group. \paragraph{Auxiliary (W\"{u})} Before the final setup in 2009/1, all auxiliary systems (weather station, computers, etc.) will have been specified, ordered and shipped. \paragraph{Camera (Do)} The camera has to be ready six month after the shipment of the other mechanical parts of the telescope. For this purpose camera tests have to take place in 2009/2, which requires the assembly of the camera within six months before. By now, a PM test bench which allows to finish planning and ordering of the camera parts and PMs until summer 2008, before the construction begins, is set up in Dortmund. In addition to the manpower permanently provided by Dortmund for production and commissioning, two engineers will participate in the construction phase. \paragraph{Full System (Do/W\"{u})} The full system will be assembled after delivering of all parts in the beginning of spring 2009. Start of the commissioning is planned four months later. First light is expected in autumn 2009. This would allow an immediate full system test with a well measured, strong and steady source (CrabNebula). After the commissioning phase will have been finished in spring 2010, full robotic operation will be provided. \paragraph{Mirrors (W\"{u})} First prototypes for the mirrors are already available. After testing (six months), the production will start in summer 2008, and the shipment will be finished before the full system assembly 2009/2. \paragraph{Drive (W\"{u})} After a planning phase of half a year to simplify the MAGIC~II drive system for a smaller telescope (together with the delivering company), ordering, production and shipment should be finished in 2009/1. The MAGIC~I and~II drive systems have been planned and implemented successfully by the W\"{u}rzburg group. \paragraph{Auxiliary (W\"{u})} Before the final setup in 2009/1, all auxiliary systems (weather station, computers, etc.) will have been specified, ordered and shipped. \paragraph{Camera (Do)} The camera has to be ready six month after the shipment of the other mechanical parts of the telescope. For this purpose camera tests have to take place in 2009/2, which requires the assembly of the camera within six months before. By now, a PM test bench is set up in Dortmund, which allows to finish planning and ordering of parts of the camera, including the PMs, until summer 2008, before the construction begins. In addition to the manpower permanently provided by Dortmund for production and commissioning, two engineers will participate in the construction phase. \paragraph{Full System (Do/W\"{u})} The full system will be assembled after the delivery of all parts in the beginning of spring 2009. Start of the commissioning is planned four months later. First light is expected in autumn 2009. This would allow an immediate full system test with a well measured, strong and steady source (Crab Nebula). After the commissioning phase will have been finished in spring 2010, complete robotic operation will be provided. Based on the experience with setting up the MAGIC telescope we estimate \section[4]{Funds requested (Beantragte Mittel)} We request funding for a total of three years. Summarizing, the expenses for the telescope are dominated by the camera and data acquisition. Summarizing, the expenses for the telescope are dominated by the camera and data acquisition. We request funding for a total of three years. %The financial volume for the complete hardware inclusive %transport amounts to {\bf 372.985,-\,\euro}. For this period, we request funding for two postdocs and two PhD students, one in Dortmund and one in W\"{u}rzburg each. The staff members shall fulfill the tasks given in the work schedule above. To cover these tasks completely, one additional PhD and a various number of Diploma students will complete the working group. students, one in Dortmund and one in W\"{u}rzburg each (3\,x\,TV-L13).The staff members shall fulfill the tasks given in the work schedule above. To cover these tasks completely, one additional PhD and a various number of Diploma students will complete the working group. Suitable candidates interested in these positions are Dr.\ Thomas At the Observatorio Roque de los Muchachos (ORM), at the MAGIC site, the mount of the former HEGRA telescope CT3 now owned by the MAGIC collaboration is still operational. One hut for electronics close to collaboration is still serviceable. One hut for electronics close to the telescope is available. Additional space is available in the MAGIC counting house. The MAGIC Memorandum of Understanding allows for To achieve the planned sensitivity and threshold (figure~\ref{sensitivity}) the following components have to be bought. (fig.~\ref{sensitivity}), the following components have to be bought. To obtain reliable results as fast as possible well known components have been chosen. \citep{Juan:2000,MAGICsensi,Vassiliev:1999} and the expectation for DWARF, with both a PMT- and a GAPD-camera. It is based on the sensitivity of HEGRA~CT1, scaled by the improvements mentioned in the text. GAPD-camera, scaled from the sensitivity of HEGRA~CT1 by the improvements mentioned in the text. } \label{sensitivity} } \end{figure} \clearpage {\bf Camera}\dotfill 207.550,-\,\euro\\[-3ex] {\bf Camera}\dotfill 206.450,-\,\euro\\[-3ex] \begin{quote} To setup a camera with 313 pixels the following components are needed:\\ \parbox[t]{1em}{~}\begin{minipage}[t]{0.6\textwidth} Photomultiplier Tube EMI\,9083B\hfill 220,-\,\euro\\ Active voltage divider ({\bf !!!!})\hfill 80,-\,\euro\\ High voltage support and control\hfill {\bf 300,-}\,\euro\\ Active voltage divider (EMI)\hfill 80,-\,\euro\\ High voltage support and control\hfill 300,-\,\euro\\ Preamplifier\hfill 50,-\,\euro\\ Spare parts (overall)\hfill 3000,-\,\euro\\ criterion. To keep the systematic errors small, a good background estimation is mandatory. The only possibility for a synchronous determination of the background is the determination from the night-sky determination of the background is the measurement from the night-sky observed in the same field-of-view with the same instrument. To achieve this, the observed position is moved out of the camera center which allows the estimation of the background from positions symmetric with respect to the camera center (so called wobble-mode). This observation mode increases the sensitivity by a factor of $\sqrt{2}$, because spending observation time for dedicated background observations becomes obsolete, i.e.\ observation time for the source is doubled. This ensures in addition a better time coverage of the observed sources. respect to the camera center (so called Wobble mode). This observation mode increases the sensitivity by a factor of $\sqrt{2}$, because spending observation time for dedicated background observations becomes obsolete, i.e.\ observation time for the source is doubled. This ensures in addition a better time coverage of the observed sources.\\ A further increase in sensitivity can be achieved by better background statistics from not only one but several independent positions for the background estimation in the camera \citep{Lessard:2001}. For wobble mode observations allowing for this, the source position should be shifted background estimation in the camera \citep{Lessard:2001}. To allow for this the source position in Wobble mode should be shifted $0.6^\circ-0.7^\circ$ out of the camera center. %} A camera completely containing shower images of events in the energy A camera completely containing the shower images of events in the energy region of 1\,TeV-10\,TeV should have a diameter in the order of 5$^\circ$. To decrease the dependence of the measurements on the camera \includegraphics[width=0.495\textwidth,angle=0,clip]{cam271.eps} \includegraphics[width=0.495\textwidth,angle=0,clip]{cam313.eps} \caption{Left: Schematic picture of the 271 pixel CT-3 camera with a field of view of 4.6$^\circ$. \caption{Left: Schematic picture of the 271 pixel CT3 camera with a field of view of 4.6$^\circ$. Right: Schematic picture of the 313 pixel camera for DWARF with a field of view of 5$^\circ$.} \label{camCT3} \end{figure} Therefor a camera with 313 pixel camera (see figure~\ref{camDWARF}) is Therefore a camera with 313 pixel camera (see fig.~\ref{camDWARF}) is chosen. The camera will be built based on the experience with HEGRA and MAGIC. 19\,mm diameter Photomultiplier Tubes (PM, EMI\,9083\,KFLA-UD) MAGIC. 19\,mm diameter Photomultiplier Tubes (PM, EMI\,9083B/KFLA-UD) will be bought, similar to the HEGRA type (EMI\,9083\,KFLA). They have a 25\% improved quantum efficiency (see figure~\ref{qe}) and ensure a a quantum efficiency improved by 25\% (see fig.~\ref{qe}) and ensure a granularity which is enough to guarantee good results even below the energy threshold (flux peak energy). Each individual pixel has to be cameras. {\bf At ETH~Z\"{u}rich currently test measurements are ongoing to prove the At ETH~Z\"{u}rich currently test measurements are ongoing to prove the ability, i.e.\ stability, aging, quantum efficiency, etc., of using Geiger-mode APDs (GAPD) as photon detector in the camera of a Cherenkov telescope. The advantages are extremely high quantum efficiency ($>$50\%), easier gain stabilization and simplified application compared to classical PMs. If these test measurements are successfully finished until 8/2008 we consider to use GAPDs in favor of classical PMs. The design of such a camera would take place at University Dortmund in close collaboration with the experts from ETH. Construction would also take place at the electronics workshop of Dortmund.} Geiger-mode APDs (GAPD) as photon detectors in the camera of a Cherenkov telescope. The advantages are an extremely high quantum efficiency ($>$50\%), easier gain stabilization and simplified application compared to classical PMs. If these test measurements are successfully finished until 8/2008, we consider to use GAPDs in favor of classical PMs. The design of such a camera would take place at University Dortmund in close collaboration with the experts from ETH. The construction would also take place at the electronics workshop of Dortmund. \end{quote}\vspace{3ex} {\bf Camera support}\dotfill 204.000,-\,\euro\\[-3ex] {\bf Camera support}\dotfill 7.500,-\,\euro\\[-3ex] \begin{quote} For this setup the camera holding has to be redesigned. (1500,-\,\euro) The camera chassis must be water tight and will be equipped with an automatic lid protecting the PMs at day-time. For further protection, a automatic lid, protecting the PMs at daytime. For further protection, a plexi-glass window will be installed in front of the camera. By coating this window with an anti-reflex layer of magnesium-fluoride, a gain in transmission of {\bf 5\%} is expected. Each PM will be equipped with a light-guide (Winston Cone) as developed by UC Davis and successfully in operation in the MAGIC camera. (3000,-\,\euro\ for all winston cones). The transmission of 5\% is expected. Each PM will be equipped with a light-guide (Winston cone) as developed by UC Davis and successfully in operation in the MAGIC camera. (3000,-\,\euro\ for all Winston cones). The current design will be improved by using a high reflectivity aluminized Mylar mirror-foil, coated with a dialectical layer ($Si\,O_2$ planned. In total a gain of {\bf $\sim$15\%} in light-collection efficiency compared to the old CT3 system can be acheived. In total a gain of $\sim$15\% in light-collection efficiency compared to the old CT3 system can be achieved. \end{quote}\vspace{3ex} %\parbox[t]{1em}{~}\parbox[t]{0.955\textwidth}{ For the data acquisition system a hardware readout based on an analog ring buffer (Domino\ II/III), currently developed for the MAGIC\ II ring buffer (Domino\ II/IV), currently developed for the MAGIC~II readout, will be used \citep{Barcelo}. This technology allows to sample the pulses with high frequencies and readout several channels with a single Flash-ADC resulting in low costs. The low power consumption will allow to include the digitization near the signal source which makes the transfer of the analog signal obsolete. The advantage is less pick-up noise and less signal dispersion. By high sampling rates allow to include the digitization near the signal source making the transfer of the analog signal obsolete. This results in less pick-up noise and reduces the signal dispersion. By high sampling rates (1.2\,GHz), additional information about the pulse shape can be obtained. This increases the over-all sensitivity further, because the short integration time allows for almost perfect suppression of noise due to night-sky background photons. The estimated trigger- (readout-) rate of the telescope is below 100\,Hz (HEGRA: $<$10\,Hz) which allows to use a low-cost industrial solution for readout of the system like USB\,2.0. due to night-sky background photons. The estimated trigger-, i.e.\ readout-rate of the telescope is below 100\,Hz (HEGRA: $<$10\,Hz) which allows to use a low-cost industrial solution for readout of the system, like USB\,2.0. %\parbox[t]{1em}{~}\parbox[t]{0.955\textwidth}{ Current results obtained with the new 2\,GHz FADC system in the MAGIC data acquisition show that for a single telescope a sensitivity data acquisition show, that for a single telescope a sensitivity improvement of 40\% with a fast FADC system is achievable \citep{Tescaro:2007}. As for the HEGRA telescopes a simple multiplicity trigger is Like for the HEGRA telescopes a simple multiplicity trigger is sufficient, but also a simple neighbor-logic could be programmed (both cases $\sim$100,-\,\euro/channel). Additional data reduction and preprocessing within the readout chain is provided. Assuming conservatively a readout rate of 30\,Hz the storage provided. Assuming conservatively a readout rate of 30\,Hz, the storage space needed will be less than 250\,GB/month or 3\,TB/year. This amount of data can easily be stored and processed by the W\"{u}rzburg %\parbox[t]{1em}{~}\parbox[t]{0.955\textwidth}{ \begin{quote} The existing mirrors are replaced by new plastic mirrors which are currently developed by Wolfgang Dr\"{o}ge's group. The cheap and light-weight material has been formerly used for Winston cones in balloon experiments. The mirrors are copied from a master coated with a The existing mirrors will be replaced by new plastic mirrors currently developed by Wolfgang Dr\"{o}ge's group. The cheap and light-weight material has been formerly used for Winston cones in balloon experiments. The mirrors are copied from a master and coated with a reflecting and a protective material. Tests have given promising results. By a change of the mirror geometry, the mirror area can be increased from 8.5\,m$^2$ to 13\,m$^2$ (see picture~\ref{CT3} and montage~\ref{DWARF}); this includes an increase of $\sim$10$\%$ per montage~\ref{DWARF}). This includes an increase of $\sim$10$\%$ per mirror by using a hexagonal layout instead of a round one. A further increase of the mirror area would require a reconstruction of parts of In both cases the mirrors can be coated with the same high reflectivity aluminized Mylar mirror-foil, and a dialectical layer of SiO2 as for the Winston Cones. By this, a gain in reflectivity of $\sim10\%$ is achieved, see figure~\ref{reflectivity} \citep{Fraunhofer}. Both solutions would require the same expenses. aluminized Mylar mirror-foil and a dialectical layer of $SiO_2$ as for the Winston cones. By this, a gain in reflectivity of $\sim10\%$ is achieved, see fig.~\ref{reflectivity} \citep{Fraunhofer}. Both solutions would require the same expenses. To keep track of the alignment, reflectivity and optical quality of the adjustment system, as developed by ETH~Z\"{u}rich and successfully operated on the MAGIC telescope, is intended. \begin{figure}[p] \centering{ \includegraphics[width=0.57\textwidth]{cherenkov.eps} \includegraphics[width=0.57\textwidth]{reflectivity.eps} \includegraphics[width=0.57\textwidth]{qe.eps} \caption{Top to bottom: The cherenkov spectrum as observed by a telescope located at 2000\,m above sea level. The mirror's reflectivity of a 300\,nm thick aluminum layer with a protection layer of 10\,nm and 100\,nm thickness respectively. For comparison the reflectivity of HEGRA CT1's mirrors \citep{Kestel:2000} are shown. The bottom plot depicts the quantum efficiency of the prefered PMs (EMI) together with the predecessor used in CT1. A proper coating \citep{Paneque:2004} will further enhance its effciency. An even better increase would be the usage of Geiger-mode APDs.} \label{cherenkov} \label{reflectivity} \label{qe} } \end{figure} %The system %{\bf For a diameter mirror of less than 2.4\,m, the delay between an %parabolic (isochronus) and a spherical mirror shape at the edge is well %parabolic (isochronous) and a spherical mirror shape at the edge is well %below 1ns (see figure). Thus for a sampling rate of 1.2\,GHz parabolic %individual mirrors are not needed. Due to their small size the \end{quote}\vspace{3ex} {\bf Calibration System}\dotfill 6.650,-\,\euro+IPR?\\[-3ex] {\bf Calibration System}\dotfill 9.650,-\,\euro\\[-3ex] \begin{quote} Components\\ \parbox[t]{1em}{~}\begin{minipage}[t]{0.6\textwidth} Absolute light calibration\hfill 2.000,-\,\euro\\ Individual pixel rate control\hfill ???,-\,\euro\\ Individual pixel rate control\hfill 3.000,-\,\euro\\ Weather station\hfill 500,-\,\euro\\ GPS clock\hfill 1.500,-\,\euro\\ %\parbox[t]{1em}{~}\parbox[t]{0.955\textwidth}{ For the absolute light calibration (gain-calibration) of the PMs a calibration box as successfully used in the MAGIC telescope will be calibration box, as successfully used in the MAGIC telescope, will be produced. To ensure a homogeneous acceptance of the camera, essential for wobble-mode observations, the trigger rate of the individual pixels Wobble mode observations, the trigger rate of the individual pixels will be measured and controlled. To correct for axis misalignments and possible deformations of the structure (e.g.\ bending of camera holding masts), a pointing correction algorithm as used in the MAGIC tracking system will be applied. It is For a correction of axis misalignments and possible deformations of the structure (e.g.\ bending of camera holding masts) a pointing correction algorithm will be applied, as used in the MAGIC tracking system. It is calibrated by measurements of the reflection of bright guide stars on the camera surface and ensures a pointing accuracy well below the pixel diameter. Therefore a high sensitive low-cost video camera, as for MAGIC\ I and~II, ({\bf 300,-\,\euro\ camera, 600,-\,\euro\ optics, 300,-\,\euro\ housing, 250,-\,\euro\ Frame grabber}) will be installed. MAGIC\ I and~II, (300,-\,\euro\ camera, 600,-\,\euro\ optics, 300,-\,\euro\ housing, 250,-\,\euro\ frame grabber) will be installed. A second identical CCD camera for online monitoring (starguider) will be bought. A GPS clock is necessary for an accurate tracking. The weather station For an accurate tracking a GPS clock is necessary. The weather station helps judging the data quality. %}\\[2ex] \end{quote}\vspace{3ex} {\bf Computing}\dotfill 12.000,-\,\euro\\[-3ex] \end{quote}\vspace{3ex} %%%%%%%%%%%%%% PLOTS HERE???? %%%%%%%%%%%%%%%%%%%%%%%%%% {\bf Mount and Drive}\dotfill 17.500,-\,\euro\\[-3ex] \begin{quote} %\parbox[t]{1em}{~}\parbox[t]{0.955\textwidth}{ The present mount is used. Only a smaller investment for safety, corrosion protection, cable ducts, etc. is needed (7.500,-\,\euro). For the movement, motors, shaft encoders and control electronics in the order of 10.000,-\,\euro\ have to be bought. The costs have been estimated with the experience from building the MAGIC drive systems. The DWARF drive system should allow for relatively fast repositioning for three corrosion protection, cable ducts, etc. is needed (7.500,-\,\euro). Motors, shaft encoders and control electronics in the order of 10.000,-\,\euro\ have to be bought. The costs have been estimated with the experience from building the MAGIC drive systems. The DWARF drive system should allow for relatively fast repositioning for three reasons: (i)~Fast movement might be mandatory for future ToO observations. (ii)~Wobble-mode observations will be done changing the wobble-position continuously (each 20\,min) for symmetry reasons. (iii)~To ensure good time coverage of more than one source visible at the same time, the observed source will be changed in constant time intervals ($\sim$20\,min). Therefore three 150\,Watt servo motors are intended to be bought. A observations. (ii)~Wobble mode observations will be done changing the Wobble-position continuously (each 20\,min) for symmetry reasons. (iii)~To ensure good time coverage of more than one source visible at the same time, the observed source will be changed in constant time intervals. For the drive system three 150\,Watt servo motors are intended to be bought. A micro-controller based motion control unit (Siemens SPS L\,20) similar to the one of the current MAGIC~II drive system will be used. For communication with the readout-system, a standard ethernet connection communication with the readout-system, a standard Ethernet connection based on the TCP/IP- and UDP-protocol will be setup. %}\\[2ex] telescope position at the time of sunrise. A fence for protection in case of robotic movement will be For protection in case of robotic movement a fence will be installed.%}\\[2ex] \end{quote}\vspace{3ex} %\parbox[t]{1em}{~}\parbox[t]{0.955\textwidth}{ For remote, robotic operation a variety of remote controllable electronic components such as ethernet controlled sockets and switches will be components such as Ethernet controlled sockets and switches will be bought. Monitoring equipment, for example different kind of sensors, is also mandatory.%}\\[2ex] \hspace{0.66\textwidth}\hrulefill\\[0.5ex] \hspace{0.66\textwidth}\hspace{0.5ex}\hfill Sum 4.2:\hfill{\bf 342.235,-\,\euro}\hfill\hspace{0pt}\\[-1ex] 340.635,-\,\euro}\hfill\hspace{0pt}\\[-1ex] \hspace{0.66\textwidth}\hrulefill\\[-1.9ex] \hspace{0.66\textwidth}\hrulefill\\ \begin{figure}[p] \centering{ \includegraphics[width=0.57\textwidth]{cherenkov.eps} \includegraphics[width=0.57\textwidth]{reflectivity.eps} \includegraphics[width=0.57\textwidth]{qe.eps} \caption{Top to bottom: The Cherenkov spectrum as observed by a telescope located at 2000\,m above sea level. The mirror's reflectivity of a 300\,nm thick aluminum layer with a protection layer of 10\,nm and 100\,nm thickness respectively. For comparison the reflectivity of HEGRA CT1's mirrors \citep{Kestel:2000} are shown. The bottom plot depicts the quantum efficiency of the preferred PMs (EMI) together with the predecessor used in CT1. A proper coating \citep{Paneque:2004} will further enhance its efficiency. An even better increase would be the usage of Geiger-mode APDs.} \label{cherenkov} \label{reflectivity} \label{qe} } \end{figure} \subsection[4.3]{Consumables (Verbrauchsmaterial)} % \parbox[t]{1em}{~}\begin{minipage}[t]{0.9\textwidth} 10 LTO\,4 tapes (8\,TB)\dotfill 750,-\,\euro\\ Consumables (overalls) tools and materials\dotfill 10.000,-\,\euro Consumables (overalls): tools and materials\dotfill 10.000,-\,\euro % \end{minipage}\\[-0.5ex] \end{quote} \hspace{0.66\textwidth}\hrulefill\\ \subsection[4.4]{Reisen (Travel expenses)} \subsection[4.4]{Travel expenses (Reisen)} The large amount of travel funding is required due to the very close cooperation between Dortmund and W\"{u}rzburg and the work demands on \subsection[4.5]{Publikationskosten (Publication costs)} \subsection[4.5]{Publication costs (Publikationskosten)} Will be covered by the proposing institutes. \setlength{\itemsep}{0pt} \setlength{\parsep}{0pt} \item Prof.\ Dr.\ Dr.\ Wolfgang Rhode (Grundausttattung) \item Prof.\ Dr.\ Dr.\ Wolfgang Rhode (Grundauststattung) \item Dr.\ Tanja Kneiske (Postdoc (Ph"anomenologie), DFG-Forschungsstipendium) \item Dr.\ Julia Becker (Postdoc (Ph"anomenologie), Drittmittel) \item Dipl.-Phys.\ Kirsten M"unich (Doktorand (IceCube), Drittmittel) \item Dipl.-Phys.\ Jens Dreyer (Doktorand (IceCube), Grundausttattung) \item Dipl.-Phys.\ Jens Dreyer (Doktorand (IceCube), Grundauststattung) \item M.Sc.\ Valentin Curtef (Doktorand (MAGIC), Grundausstattung) \item cand.\ phys.\ Michael Backes (Diplomand (MAGIC), zum F"orderbeginn diplomiert) \originalTeX \subsection[5.2]{Co-operation with other scientists\\(Zusammenarbeit mit \subsection[5.2]{Cooperation with other scientists\\(Zusammenarbeit mit anderen Wissenschaftlern)} Both applying groups co-operate with the international MAGIC-Collaboration and the institutes represented therein. (W\"{u}rzburg Both applying groups cooperate with the international MAGIC collaboration and the institutes represented therein. (W\"{u}rzburg funded by the BMBF, Dortmund by means of appointment for the moment). The group in Dortmund is involved in the IceCube experiment (BMBF funding) and maintains close contacts to the collaboration partners. Moreover on the field of phenomenology there do exist good working contacts to the groups of Prof.~Dr.~Reinhard Schlickeiser, Ruhr-Universit\"{a}t Bochum and Prof.~Dr.~Peter Biermann, MPIfR Bonn. There are furthermore intense contacts to Prof.~Dr.~Francis Halzen, Madison, Wisconsin. Moreover on the field of phenomenology good working contacts exist to the groups of Prof.~Dr.~Reinhard Schlickeiser, Ruhr-Universit\"{a}t Bochum and Prof.~Dr.~Peter Biermann, MPIfR Bonn. There are furthermore intense contacts to Prof.~Dr.~Francis Halzen, Madison, Wisconsin. The telescope design will be worked out in close cooperation with the Prof.~Dr.~Eckart Lorenz (ETH~Z\"{u}rich). They will provide help in design studies, construction and software development. The DAQ design will be contributed by the group of Prof.~Dr.~Riccardo Paoletti (Università di contributed by the group of Prof.~Dr.~Riccardo Paoletti (Universit\`{a} di Siena and INFN sez.\ di Pisa, Italy). The group of the newly appointed {\em Lehrstuhl f\"{u}r Physik und Ihre The group of the newly appointed {\em Lehrstuhl f\"{u}r Physik und ihre Didaktik} (Prof.~Dr.~Thomas Trefzger) has expressed their interest to join the project. They bring in a laboratory for photo-sensor testing, The work on DWARF will take place at the ORM on the Spanish island La Palma. It will be performed in close collaboration with the MAGIC-Collaboration. MAGIC collaboration. \subsection[5.4]{Scientific equipment available (Apparative storage as well as for data analysis are available. The faculty of physics at the University of Dortmund has modern The faculty of physics at the University Dortmund has modern equipped mechanical and electrical workshops including a department for development of electronics at its command. The chair of astroparticle \end{minipage}\hfill~ \thispagestyle{empty} \newpage x \thispagestyle{empty} \newpage \paragraph{8 List of appendices (Verzeichnis der Anlagen)} \item Letter of Support from the IceCube collaboration \item Letter of Support from KVA optical telescope \item Email with offer from EMI for the PMs \end{itemize} \newpage %\section{References} (References of our groups are marked by an asterix ) x \thispagestyle{empty} \newpage %(References of our groups are marked by an asterix ) \bibliography{application} \bibliographystyle{plainnat}	2021-01-25 16:35:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6186259388923645, "perplexity": 5177.251643576785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703587074.70/warc/CC-MAIN-20210125154534-20210125184534-00415.warc.gz"}
https://tex.stackexchange.com/questions/166389/what-is-the-secret-to-use-fonts/166392#166392	What is the secret to use fonts? Background Relatively new LaTeX user going mad here. All I want to do is use the Day Ramon S font, but it seems nigh on impossible to do so. Question What is the secret to using fonts in (La)TeX? Is there any rhyme, reason, or rules of thumb to follow? Or do I need to just keep trying random combinations of the patterns I have found until something works? note Although all I really want to do is use Day Roman S, I have intentionally kept the question focused on fonts in general as there doesn't seem to be any bullet-proof advice on working with fonts. I see a lot of solutions for specific problems, and even fonts in general, but as can be seen in the next section, it's all a jumble. note 2 Yes, I have RTFM for the Day Roman S font in question, it is installed, and has been reinstalled several times. It's still a pain to figure out a consistent and reliable way to use different fonts in a document. Previous research In trying to figure this out, I have looked at many of the samples from FontCatalogue archive that did compile and found that there seem to be a wide variety of ways of specifying fonts. I'm not sure which to use when. Sometimes you need to just load a package with the same name as the font and add \normalfont\normalfont twice (why?) before the text: \usepackage{bookman} \usepackage[T1]{fontenc} \begin{document} \normalfont\normalfont Other times you need to add \normalfont\<font-name> before the text: \usepackage{calligra} \usepackage[T1]{fontenc} \begin{document} \normalfont\calligra Still other times you need to remove ic from the font name and add family, creating \normalfont\<font-name-minus-ic>family before the text: \usepackage{egothic} \usepackage[T1]{fontenc} \begin{document} \normalfont\egothfamily And sometimes the second command after \normalfont doesn't appear to be related to the package name at all: \usepackage{yfonts} \usepackage[T1]{fontenc} \begin{document} \normalfont\swabfamily Sometimes you also need to use \renewcommand* after loading the font package and before loading fontenc: \usepackage{PTSansCaption} \renewcommand\familydefault{\sfdefault} %% Only if the base font of the document is to be sans serif \usepackage[T1]{fontenc} \begin{document} \normalfont\normalfont And sometimes you can do that without loading the font package first and after loading fontenc (all other examples are before): \usepackage[T1]{fontenc} \renewcommand\familydefault{\ttdefault} %% Only if the base font of the document is to be typewriter style \begin{document} \normalfont\normalfont Sometimes you use \renewcommand* to set other defaults such as \rmdefault (previous two examples were \familydefault: \renewcommand\rmdefault{cmfib} \usepackage[T1]{fontenc} \begin{document} \normalfont\normalfont And sometimes you have to set various defaults: \renewcommand\ttdefault{lmvtt} \renewcommand\familydefault{\ttdefault} %% Only if the base font of the document is to be typewriter style \usepackage[T1]{fontenc} \begin{document} \normalfont\normalfont \input Carrickc.fd \newcommand\initfamily{\usefont{U}{Carrickc}{xl}{n}} \begin{document} \normalfont\initfamily \fontsize{12mm}{12mm}\selectfont • Welcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. Mar 19, 2014 at 3:26 • Do you plan to change your name for every question? Then think of one: In my eyes you should ask the part “Previous research” as new question. This would also make this question here shorter and a bit more readable. Mar 19, 2014 at 3:29 • edit it if you don't like it. nothing's stopping you. Mar 19, 2014 at 3:33 • First: You misunderstood me (maybe because English is not my natural language), I wanted to say, I cannot say anything about some possible special behahiour of TL in Ubuntu. Second: In TeX.SX the manners are different (also the frequent use of SX instead of SE), see Why do people answer in comments?, but I even didn’t intend to give an answer. Mar 19, 2014 at 3:52 • By the way, I cannot think of any purpose which would be served by \normalfont\normalfont which would not be equally well served by \normalfont. Resetting the default font series, shape and family twice is no better than doing so once. Sometimes, however, \normalfont\normalsize can be helpful. (But that is not your problem here.) – cfr Mar 19, 2014 at 4:16 Here is a working (almost) MWE: \documentclass[danish]{article} \usepackage[utf8]{inputenc} \usepackage{babel} \usepackage[T1]{fontenc} \renewcommand\rmdefault{dayroms} \begin{document} \section*{Dayrom S} For mange Aar siden levede en Keiser, som holdt saa uhyre meget af smukke nye Kl\ae der, at han gav alle sine Penge ud for ret at blive pyntet. Han br\o d sig ikke om sine Soldater, br\o d sig ei om Comedie eller om at kj\o re i Skoven, uden alene for at vise sine nye Kl\ae der. Han havde en Kjole for hver Time paa Dagen, og ligesom man siger om en Konge, han er i Raadet, saa sagde man altid her: >>Keiseren er i Garderoben!<<~-- \end{document} I am almost certain this example will fail for you because I am almost certain you do not have the font installed. The font is not part of TeX Live. Three possibilities: • Ubuntu provides a package which you can install. • You install the package from CTAN manually, perhaps following the instructions in the link Speravir provided in the comments on your original question. • You install the font using the getnonfreefonts script which will automate the process for you. This is part of TeX Live but probably not packaged by Ubuntu. Two possibilities: • It is packaged by Ubuntu (despite my pessimism). • You install TeX Live from upstream using the instructions in the link Speravir provided in the comments. In this case it would be best to remove the installation you currently have to avoid future confusion. • As additional information: The example works fine in TeX Live for Windows, as well. Mar 19, 2014 at 4:14 • @Speravir Are you sure? It ought not do so unless you have installed the font either manually or using getnonfreefonts. – cfr Mar 19, 2014 at 4:17 • Oh, yes, you are right, I once have executed getnonfreefonts! Mar 19, 2014 at 4:20 • @cfr I tried that approach and still get a warning: Font shape T1/dayroms/bx/n undefined using T1/dayroms/m/n instead on input line 8. Judging from your screen-shot, which looks the same to me, you probably have the same warning. Could you comment on that? Mar 19, 2014 at 5:18 • @Martin That's because in the \section Latex looks for a bold series, which seems not to exist (dayroms is only available in one weight, and shape). Mar 19, 2014 at 7:40 There’s no “secret”; fonts in TeX systems started simple, with no choice for the user -- everyone used computer modern, and as a result, the tools for handling fonts, written using Metafont, reached a stable state in the 90s, and haven't needed much extra support. When it became possible to use Adobe Type 1 fonts, with TeX, the complications hit us, pretty hard. No longer could we just make the font sources available, and let TeX deal with compiling the fonts (and so on) behind the scenes. Which is why you have found it so hard: even if someone has created the files, you need to get them into their rightful places and to conduct some magic to make sure the system notices them. It's not exactly difficult, but finding documentation might not be simple. It seems we’re getting back to where we started, with TrueType and OpenType fonts. You need an alternative TeX engine (XeTeX or LuaTeX), but those engines are set up to find your system fonts, without much soul-searching by you. Note that I'm not solving your problems, merely showing how they arose. Your best bet is to ask about individual fonts, as and when you want to use them. imo • It isn't true that everyone used Computer Modern before type1 was a possibility. Other fonts were created using Metafont - some by Knuth, many by others. These still needed to be installed, selected etc. Of course, it was simpler because there was at least only one type of font and obviously this limited user choice. But to suggest that there was no choice at all is just wrong. fonts/source/public/ alone contains 128 sub-directories in my installation of TeX Live, and there are some under jknappen etc., too. Obviously some are newer but not all are. – cfr Mar 19, 2014 at 22:51 • @cfr i said “started ... with no choice”, which wording i chose specifically to avoid your misinterpretation. however, i've only been using tex since the 1980s, so i don't actually know (at first hand) what it was all like for users of tex-in-sail (for example). Apr 30, 2014 at 12:11	2022-08-14 19:03:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7707440853118896, "perplexity": 2444.66737234093}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572063.65/warc/CC-MAIN-20220814173832-20220814203832-00773.warc.gz"}
https://www.beatthegmat.com/which-of-the-following-expresses-the-range-of-possible-value-t300995.html	• 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to \$200 Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Reach higher with Artificial Intelligence. Guaranteed Now free for 30 days Available with Beat the GMAT members only code • Get 300+ Practice Questions Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code ## Which of the following expresses the range of possible value tagged by: Roland2rule ### Top Member Roland2rule Legendary Member Joined 30 Aug 2017 Posted: 713 messages Followed by: 4 members #### Which of the following expresses the range of possible value Thu Mar 01, 2018 12:20 pm Which of the following expresses the range of possible values for t if 0 < \|5t - 8\| < 1? A. 2/5 < t < 14/5 B. 2/5 < t < 9/5 C. 7/5 < t < 14/5 D. 7/5 < t < 9/5 E. 9/5 < t < 14/5 OA is D How do i go about setting up the correct formular to solve this question? can any expert assist me? Thank you ### Top Member Vincen Legendary Member Joined 07 Sep 2017 Posted: 601 messages Followed by: 3 members 6 Fri Mar 09, 2018 8:12 am Hello, let's apply the definition of absolute value to 0 < \|5t - 8\| < 1. Hence, $$0<\left\|5t-8\right\|<1\$$ $$-1<5t-8<1\ \ and\ \ \ \ 5t-8\ne0$$ $$7<5t<9\ \ and\ \ \ \ 5t\ne8$$ $$\frac{7}{5} < t < \frac{9}{5}\ \ and\ \ t\ne\ \frac{8}{5}.$$ Therefore, the correct option should be D. We have to be carefull here because t cannot be 8/5. ### GMAT/MBA Expert mbawisdom Senior \| Next Rank: 100 Posts Joined 16 Dec 2014 Posted: 94 messages Followed by: 3 members 2 GMAT Score: 770 Mon Mar 05, 2018 1:39 pm Roland2rule wrote: Which of the following expresses the range of possible values for t if 0 < \|5t - 8\| < 1? A. 2/5 < t < 14/5 B. 2/5 < t < 9/5 C. 7/5 < t < 14/5 D. 7/5 < t < 9/5 E. 9/5 < t < 14/5 OA is D How do i go about setting up the correct formular to solve this question? can any expert assist me? Thank you Since the absolute value of (5t - 8) is between 0 and 1 then the value of (5t - 8) must be between -1 and 1. -1 < 5t - 8 < 1 7 < 5t < 9 (add 8 to both sides) 7/5 < t < 9/5 (divide both sides by 5) _________________ Interested in GMAT tutoring? Click this pink button for a Looking for more free content? Follow me on: If you like this post please click the UPVOTE POST button below! ### Best Conversation Starters 1 lheiannie07 76 topics 2 LUANDATO 59 topics 3 ardz24 52 topics 4 AAPL 46 topics 5 M7MBA 45 topics See More Top Beat The GMAT Members... ### Most Active Experts 1 Jeff@TargetTestPrep Target Test Prep 141 posts 2 Brent@GMATPrepNow GMAT Prep Now Teacher 136 posts 3 GMATGuruNY The Princeton Review Teacher 134 posts 4 Rich.C@EMPOWERgma... EMPOWERgmat 129 posts 5 Scott@TargetTestPrep Target Test Prep 110 posts See More Top Beat The GMAT Experts	2018-03-23 13:05:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28230980038642883, "perplexity": 12294.739007689257}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648226.72/warc/CC-MAIN-20180323122312-20180323142312-00655.warc.gz"}
https://forum.azimuthproject.org/plugin/ViewComment/18558	[Anindya Bhattacharyya](https://forum.azimuthproject.org/discussion/comment/18556/#Comment_18556) > I can see how dropping axiom (d) just means the space has indistinguishable points (ie it no longer has to be T0), and dropping (f) just means it no longer has to be connected... but dropping axiom (e) seems to have stranger effects, ones that can't obviously be captured by our usual concept of topology. Well, in the very least we can think of Lawvere metric spaces as inducing [Bitopological spaces](https://en.wikipedia.org/wiki/Bitopological_space). Ordinarily, a metric space topology is defined using a metric \$$(X,d)\$$ by taking the set of [open balls](https://en.wikibooks.org/wiki/Topology/Metric_Spaces#Open_Ball) as a [topological basis](http://mathworld.wolfram.com/TopologicalBasis.html): $$B_r(p) = \\{ x \in X \mid d(x,p) < r \\} \text{ where } r \in \mathbb{R}^+$$ This would be the end of it, except in a Lawvere space \$$d(x,p) \neq d(p,x)\$$ So we can construct another topology by flipping the usual definition: $$A_r(p) = \\{ x \in X \mid d(p,x) < r \\} \text{ where } r \in \mathbb{R}^+$$ Puzzle MD1. Show these each form a topological basis. That is, show: 1. For each \$$x\$$ in \$$X\$$, there is at least one basis element \$$E\$$ containing \$$x\$$. 2. If \$$x\$$ belongs to the intersection of two basis elements \$$E_1\$$ and \$$E_2\$$, then there is a basis element \$$E_3\$$ containing \$$x\$$ such that \$$E_3\$$ subset \$$E_1\$$ intersection \$$E_2\$$. Puzzle MD2. Morphisms over enriched categories are maps \$$\phi : \mathcal{X} \to \mathcal{Y} \$$ obeying: \$$\mathcal{X}(a,b) \leq_{\mathcal{X}} \mathcal{X}(c,d) \implies \mathcal{Y}(\phi(a),\phi(b)) \leq_{\mathcal{Y}} \mathcal{Y}(\phi(c),\phi(d)) \$$ \$$\mathcal{X}(a,b) \otimes_{\mathcal{X}} \mathcal{X}(c,d) \leq_{\mathcal{X}} \mathcal{X}(e,f) \implies \mathcal{Y}(\phi(a),\phi(b)) \otimes_{\mathcal{Y}} \mathcal{Y}(\phi(c),\phi(d)) \leq_{\mathcal{Y}} \mathcal{Y}(\phi(e),\phi(f)) \$$ \$$\mathcal{X}(a,b) = I \implies \mathcal{Y}(\phi(a),\phi(b)) = I \$$ Let \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$ be Cost-enriched categories, and let \$$\phi : \mathcal{X} \to \mathcal{Y}\$$ be a morphism between them. Does \$$\phi\$$ induce a [pairwise continuous map](https://en.wikipedia.org/wiki/Bitopological_space#Continuity) over their bitopologies? Or the other way around - does a pairwise continuous map between the bitopologies of \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$ induce an enriched category morphism? ------------------- I don't know the answer to MD2...	2021-10-19 17:40:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9311879873275757, "perplexity": 2652.9251483043354}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585280.84/warc/CC-MAIN-20211019171139-20211019201139-00271.warc.gz"}
https://tex.meta.stackexchange.com/questions/6883/any-plans-to-add-on-site-compilation-to-preview-mwes-solutions/6885	# Any plans to add on-site compilation to preview MWEs/solutions? I frequently run into code posted to this site which I'd like to see in rendered form in order to quickly decide whether it fits my needs. Opening a document just to do this often feels like a hassle. Are there any plans to add the ability to, for instance, just hover the cursor over a block of code or press some button below it and have a popup appear that shows what the output looks like? • to be able to provide the rendered output would require a full latex implementation in the background of the stackexchange site. that's not likely to happen. when i've answered questions that involve examples, i've usually tried to include an image of the output (.png files are easy to add to a question or an answer), and i think that's the most reasonable thing to expect. but it is up to those who are answering questions to "do the right thing". – barbara beeton Sep 24 '16 at 18:40 • In addition to what @barbarabeeton says, the issue with a LaTeX set up here is that package versions will vary, and that means for any issues we have the problem of not being able to match people's local set ups. – Joseph Wright Sep 24 '16 at 18:43 • @barbara Could you explain, why it's not likely to happen? It doesn't seem like an impossible feat to render every block of code in a question or answer upon posting and attaching the result to that post in order to show it upon request as described above. – Casimir Sep 24 '16 at 18:44 • as @JosephWright has pointed out, not everyone has the same versions of all packages and fonts; in fact, a number of the packages mentioned in questions (and answers) aren't posted on ctan, and many fonts are commercial. there's absolutely no way to keep such a system in workable condition for such a varied user base, even if there's a tex-knowledgeable staff willing to try. – barbara beeton Sep 24 '16 at 18:50 • Not every code block can be compiled out of the box. Suppose an answer suggests to use \donotincludeinmaintoc, that will give undefined control sequence, because the package defiing the command would not be loaded. – Johannes_B Sep 24 '16 at 22:51 • When I don’t want to create a new file on my hard drive, I just paste the example into Gummi. Then I can quickly see whether it works on my system, whether the output looks as I’d hoped, and whether it’s worth saving as a file or can be discarded. (Quick testing of examples here is the only reason I have Gummi installed, since I prefer writing documents in Emacs.) – Thérèse Oct 1 '16 at 23:21 • We have discussed this a zillion times, I don't think there's any reason to discuss it again. Voting to close as a dupe of Why doesn't maths render as maths? – yo' Oct 2 '16 at 5:01 This would be very similar to requesting MathJax support, which has been declined here as we're more interested in the code rather than the output. That doesn't mean the output is not important, of course. Moreover, many of the code posted by users here contain errors which they don't know how to correct. Or, it contains customizations that are found outside of the default distribution(s). Stack Overflow implemented something similar as a feature request, although limited in language scope. The extendability of (La)TeX as well as it's different compilers would be difficult to implement easily, if at all. It's best if users can add an image of the output they see (or expect to see) as pictures are often worth a 1000 words. • Now that you mention it, the lack of MathJax support has been bugging me once or twice before as well. I wouldn't say that I'm generally more interested in the code than its output. Rather, I judge the first by the second. – Casimir Sep 27 '16 at 4:10 • @Casimir -- mathjax output often doesn't adequately represent latex input. therefore, it is of relatively little interest here. however, the output from latex is important, but it's up to the poster to provide it. – barbara beeton Sep 27 '16 at 12:22 • That may be so but during the occasions where I would have appreciated the ability to use MathJax, my intention was not to simulate rendered LaTeX code but to refer to e.g. labels like z_1, m^2, p_0^2 in a TikZ diagram or something similar. In such cases, it doesn't really matter if the MathJax output doesn't exactly reproduce the LaTeX output, it would just circumvent cluttering up posts with underscores and circumflexes. – Casimir Sep 27 '16 at 12:30 • @Casimir: It's probably unlikely to provide full MathJax support and assume people will only use it the way you suggest. People will try to post complete minimal documents containing TikZ customizations and loading external files, then questioning why MathJax doesn't show the output they see on the local system. You may only have these good intentions, but from a network point-of-view, it's all-or-nothing, and the "all" isn't really "all" in this case. – Werner Sep 27 '16 at 14:52 • @Casimir You can get simple subscripts for such uses, without using mathjax: z<sub>1</sub>, m<sup>2</sup> (in posts, not in comments, unfortunately) – David Carlisle Sep 27 '16 at 23:36 • @DavidCarlisle That's cool, thanks for the tip. I'll try that next time it comes up. – Casimir Sep 28 '16 at 4:45 • I agree with the "a picture's worth a thousand words" sentiment. Those who answer questions are often very good at giving details or responding if the asker has further questions about how to implement the solution. Also, if a command is unrecognized, a simple web search can yield a result on it -- and this leads to more user self-sufficiency, which I think is a very good thing to encourage. I've learned more here by having to figure out a few things by myself, than being given a copy/paste solution. It's really so much better in the long run. – J.D. Oct 2 '16 at 21:36 As @paul-gessler has commented, this has been asked several times before, and the community of TeX.SE is generally against such a feature, mostly because of problems with erroneous code, compatibility, or output that requires multiple files (like bibliographies). However, I am strongly in favour of this feature, if it is implemented with a separate syntax. In this way a normal code block is be shown as code, which is indeed the most sensible behaviour for this site. However, if the user wants, the output can be shown as well. For example: # \documentclass{article} # \begin{document} # Hello World # \end{document} Analogous to the syntax for block quotes. This would save time for almost every time you post an answer where you want to include a screenshot of the output. Creating a new file, copying the MWE, compiling, making the screenshot, cropping, saving, uploading sometimes takes me more time than answering the question itself (for example with answers like "use \phantom") and I do this for practically every answer. Another use case is in-line code, like "the $\sum_{\alpha\in\{-5,-10\}\cup\mathbb{Z}^+}$ is not aligned with the $\sum_\beta$". TeX.SE could support for example something analoguous to the texlive-full package in Ubuntu. This supports almost everything. If the result is different from what the OP or the answerer has on his own system, then he can choose to show a screenshot instead. If the question or answer depends on a specific engine, command line options, fonts installed, ..., then a screenshot can be used. But in the many, many cases where the setup doesn't matter, the users can benefit from a compilation feature. • just measured. Takes me 6 seconds to take a screenshot and add. Shorter on Linux. – percusse Sep 28 '16 at 13:39 • And how much time would we spend undoing edits which added the hashes to code which did not compile with SE's TeX? And what would happen when SE updated their TeX installation? Presumably existing compilations would remain, but what if I edit a minor content typo in code which compiled with the old version but no longer does? – cfr Sep 29 '16 at 1:38 • Or edits which added hashes to code blocks without duplicating them so that a post which included code no longer did? – cfr Sep 29 '16 at 1:39 • Latex-community has a button to open code blocks with Overleaf, the online editor. That is quite neat sometimes and the user can choose to do it or not. That would be a Feature i would Support on TeX.SX. – Johannes_B Oct 1 '16 at 10:45 • @Johannes_B But that is very different from rendering it inline. – cfr Oct 1 '16 at 22:33	2020-02-20 14:50:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5135742425918579, "perplexity": 1077.4314717483237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144979.91/warc/CC-MAIN-20200220131529-20200220161529-00225.warc.gz"}
http://cmstatistics.org/RegistrationsV2/CFE2019/viewSubmission.php?in=821&token=r558o58r1196rs76q8o8p8q6p5083qq3	CFE 2019: Start Registration View Submission - CMStatistics B0821 Title: Quantifying and estimating asymmetric dependence Authors: Florian Griessenberger - University Salzburg (Austria) [presenting] Wolfgang Trutschnig - University of Salzburg (Austria) Abstract: Standard dependence measures considered in the literature like Pearson correlation, Spearman rank correlation or Schweitzer and Wolff's $\sigma$ are symmetric, i.e. they assign each pair of random variables $(X,Y)$ the same dependence as they assign the pair $(Y,X)$. Since dependence structures are in general not symmetric (in contrast to independence, which is a symmetric concept), the classical dependence measures fail to detect asymmetry. The recently developed R-package qad (short for quantification of asymmetric dependence) aims at detecting asymmetries in samples. It estimates the dependence of the second variable on the first one and vice versa, and additionally quantifies the asymmetry of the underlying dependence structure. The main objectives are to sketch the idea underlying the copula-based dependence measure, to present the most relevant mathematical properties of the underlying estimator and to illustrate its capabilities by some examples.	2020-12-05 02:34:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37966179847717285, "perplexity": 1440.2773875109651}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141746033.87/warc/CC-MAIN-20201205013617-20201205043617-00399.warc.gz"}
https://www.gradesaver.com/textbooks/science/chemistry/chemistry-9th-edition/chapter-16-solubility-and-complex-ion-equilibria-exercises-page-781/25	## Chemistry 9th Edition $K_{sp} (PbBr_2) = (3.92 \times 10^{-5})$ 1. Write the $K_{sp}$ expression: $PbBr_2(s) \lt -- \gt 1Pb^{2+}(aq) + 2{Br}^{-}(aq)$ $K_{sp} = [Pb^{2+}]^ 1[{Br}^{-}]^ 2$ 2. Determine the ions concentrations: $[Pb^{2+}] = [0.0214]$ $[{Br}^{-}] = [Pb^{2+}] * 2 = 0.0428$ ** Considering a pure solution. 3. Calculate the $K_{sp}$: $K_{sp} = (0.0214)^ 1 \times (0.0428)^ 2$ $K_{sp} = (0.0214) \times (1.832 \times 10^{-3})$ $K_{sp} = (3.92 \times 10^{-5})$	2019-11-22 01:51:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9656855463981628, "perplexity": 3206.9209651604456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671106.83/warc/CC-MAIN-20191122014756-20191122042756-00318.warc.gz"}
https://phys.libretexts.org/Bookshelves/Optics/Book%3A_Geometric_Optics_(Tatum)/01%3A_Reflection_and_Refraction/1.08%3A_Differential_Form_of_Snell's_Law	Skip to main content $$\require{cancel}$$ # 1.8: Differential Form of Snell's Law Snell’s law in the form $$n \sin \theta$$ = constant is useful in calculating how a light ray is bent in travelling from one medium to another where there is a discrete change of refractive index. If there is a medium in which the refractive index is changing continuously, a differential form of Snell’s law may be useful. This is obtained simply by differentiation of $$n \sin \theta$$ = constant, to obtain the differential form of Snell’s law $\cot \theta d \theta = -\frac{dn}{n}.\label{eq:1.9.1}$ If we express this in terms of the complementary angle $$\Psi$$ (see Figure I.24), this equation takes the form $\tan\Psi d \Psi = \frac{dn}{n}.\label{eq:1.9.2}$ Let us see how this might be used. Let us suppose, for example, that we have some medium in which the refractive index diminishes linearly with the $$y$$ coordinate according to $n = n_0 - \frac{(n_0-1)y}{a}. \label{eq:1.9.3}$ This form has $$n = n_0$$ at $$y = 0$$, decreasing linearly to $$n = 1$$ at $$y = a$$. We suppose that $$n = 1$$ everywhere beyond $$y = a$$. The equation is simplified if we write $$\mu$$ for $$n −1$$ and for $$\mu_0$$ for $$n_0 −1$$ , so that $\mu = \mu_0 \left(1 - \frac{y}{a}\right),\label{eq:1.9.4}$ which has $$\mu$$ decreasing linearly from $$\mu_0$$ to zero. Let us suppose that we direct a light ray upwards from the origin in a direction making an angle $$\alpha$$ with the horizontal, and we wish to trace the ray through the medium as the refractive index continuously changes. See Figure I.24. With the refractive index changing as in Equation $$\ref{eq:1.9.4}$$, Snell’s law takes the form $tan \Psi d\Psi = -\frac{dy}{k-y}, \quad \text{where} \quad k = \frac{n_0a}{\mu_0}.\label{eq:1.9.5}$ On integration, this becomes $y = k\left(1- \frac{\cos\alpha}{\cos\Psi}\right) \quad \text{or} \quad \cos\Psi = \frac{k\cos\alpha}{k-y}.\label{eq:1.9.6}$ This is the $$(y, \Psi)$$ equation to the path taken by the light. One can see from this equation that the path becomes horizontal when $$y = k(1 − \cos\alpha$$. To find the $$(x ,y)$$ equation, we use the identities $$\cos\Psi = (1+\tan^2 \Psi)^{-1/2}$$ and $$\tan\Psi =\frac{dy}{dx}$$. Substitution of these into Equation $$\ref{eq:1.9.6}$$ gives the differential equation to the path in $$(x , y)$$ coordinates: $\left(\frac{dy}{dx}\right)^2 = \frac{(k-y)^2 -l^2}{l^2}, \quad \text{where} \quad l = k\cos\alpha. \label{eq:1.9.10}$ This can be readily integrated to give the $$(x ,y)$$ equation to the path: $x = l\ln\left[\frac{k(1+\sin\alpha)}{k-y\pm\sqrt{(k-y)^2-l^2}}\right]. \label{eq:1.9.11}$ In Figure I.25 I have taken $$a = 1$$ (in other words, I am expressing all distances in units of $$a$$), and I am taking $$n_0 = 1.5$$ (hence $$k = 3$$), and I calculate the paths for $$\alpha$$ = 15°, 30°, 45°, 60°, 75°. Figure I.25 and Equation $$\ref{eq:1.9.11}$$ show the paths of the light if the refractive index is varying from $$n_0$$ at $$y = 0$$, to 1 at $$y = a$$ as described by Equation $$\ref{eq:1.9.3}$$. But now suppose that the refractive index is varying with height according to $n = \frac{n_0a}{(n_0-1)y+a}.\label{eq:1.9.12}$ In this model, too, the refractive index goes from $$n_0$$ at $$y = 0$$, to 1 at $$y = a$$, but the variation is not linear. In fact you might like to convince yourself that it is the speed of light that is increasing linearly from $$y = 0$$ to $$y = a$$. See if you can trace the paths of the light rays in this situation. I think they are arcs of circles, and you might be able to calculate the radii and the coordinates of the centres of the circles. Here’s another one: $n = \sqrt{\frac{n^2_0a}{(n^2_0-1)y+a}}. \label{eq:1.9.13}$ Here, too, the refractive index goes from $$n_0$$ at $$y = 0$$, to 1 at $$y = a$$. You might like to try tracing the rays in this model. I think they may be arcs of cycloids. Of course these examples may seem to be very unlikely. Can you imagine a glass block of width $$a$$, made of glass whose refractive index varies continuously from 1.5 at one edge to zero at the other? Not very likely, yet there is a situation that comes to mind in which there is a continuous variation of refractive index from some basal value $$n_0$$ to zero. I am thinking of Earth’s atmosphere (or indeed the atmosphere of any planet). As light from a star travels through Earth’s atmosphere, it moves not in a straight line, but in a slight curve, so that it is deviated through a few arc minutes before it reaches the astronomer’s telescope. For a star low down near the horizon, the refraction amounts to almost half a degree. This has to be taken into account when astronomers are making precise positional measurements. And sound waves, passing through the atmosphere, are also subject to refraction via the differential form of Snell’s law. The speed of sound (and hence the refractive index) varies with the temperature of the atmosphere, and hence with height in the atmosphere. Among the many applications of this sort of theory is the path of sound waves from meteorites hurtling through the atmosphere. This is discussed in a paper published in Meteoritics and Planetary Science 34, 572-585 (1999).	2020-07-12 18:10:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9151374697685242, "perplexity": 185.2694386047367}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657139167.74/warc/CC-MAIN-20200712175843-20200712205843-00181.warc.gz"}
http://openstudy.com/updates/4de3801e4c0e8b0b83fdaad8	## sristykrishnaprasad Group Title Is it possible to count the number of flux lines around a charged particle? Surely there can't be infinite since infinite line will mean infinite field strength. What could be that finite number? 3 years ago 3 years ago 1. Owlfred Group Title Hoot! You just asked your first question! Hang tight while I find people to answer it for you. You can thank people who give you good answers by clicking the 'Good Answer' button on the right! 2. ankurchiitd Group Title Since the amount of flux in one flux line is not quantified it is always possible to have infinite number of lines. Much like it is possible to have infinite rectangles under the curve when you integrate to find its area. Comparing the density of field lines is a relative process and can be used to judge the relative strength of two fields - however it does not have much significance while talking about an isolated field. Hope this helps! Ankurch, is it possible to quantify the relative field strength if field due to one charge configuration is known and the 'relative' flux distribution due compared to the known charge configuration is mentioned for an unknown configuration? 4. ankurchiitd Group Title I am not sure I get the question completely! Please elaborate. What I gather is that in the end you are curious as to whether field density ha s unit of sorts.. Am I on the right track? Say u have two charge configurations A and B. U are given the electric field strength due to A. Also, u are given a pictorial view of field lines due to A and B. Since u said it is a relative measure, is it now possible to quantify the electric field strength at a point due to B? 6. ankurchiitd Group Title See the field lines are just a representation tool - so if the person who drew the lines drew them to be accurate when compared (the two field line configurations) then yes we can use this picture to determine field strength due to B. Can you elaborate ur view on 'accurate' drawing? In what can it be accurate? 8. ankurchiitd Group Title It should be accurate in the fact that "the field line density should correspond to the field strength at the point". 9. gokuldas_tvm Group Title I remember reading in several books that 'flux line' concept is not an accurate one. It often leads to confusing results. Don't depend on it to calculate any field strength 10. gokuldas_tvm Group Title You can however calculate flux - but not in terms of lines Ankur, it means that for a given value of field strength a known density of lines should exist. This will take us back to our starting point that it is infact possible to count the number of flux lines in a region of space. gokul, yes flux is equal charge enclosed. 13. ankurchiitd Group Title :-) No, it does not mean that for a given field strength a known density should exist. One, it is always comparative and two, it assumes that the depiction is "accurate". 14. gokuldas_tvm Group Title sristy, you can draw any number of lines for a given field over an area - even infinite. The concept of flux lines should be used only for illustration. However, to show that a field is stronger over an area, you draw more lines through it. The only problem is that you can arbitrarily chose how many you want to draw I believe we are revolving around the same point again. If known density doesnt exist then it impossible to quantify field in the case of A and B systems i mentioned above. If u say only accurate depictions work then it shld be atleast possible to count the number of lines in such 'accurate' depictions. 16. ankurchiitd Group Title @gokuldas_tvm: exactly! 17. ankurchiitd Group Title @gokuldas_tvm: any analogy of a similar depiction tool usage? 18. gokuldas_tvm Group Title @ankur: can't come up with any right now. But, i think - we need a strong description for 'flux'! 19. gokuldas_tvm Group Title sristy: Concepts of 'flux' and 'flux lines' are slightly different. 'Flux density' refers to density of 'flux', not of 'flux lines' 20. ankurchiitd Group Title @gokuldas_tvm: Physics grad? and I suppose the tvm stands for Thiruvananthpuram? 21. gokuldas_tvm Group Title @ankur: TVM does stand for trivandrum! :D I am not a physics grad, but of electronics and communication. But Maxwell's equations and vector PDEs has always been one of my favourites. 22. ankurchiitd Group Title @gokul: Cool. 23. gokuldas_tvm Group Title @Ankur: Are you from around here? 24. ankurchiitd Group Title @Gokul: Am from Delhi - an engineering physics grad. 25. gokuldas_tvm Group Title @Sristy: I have a slightly more detailed idea on flux and fields. Interested? @gokuldas_tvm : surely sir 27. gokuldas_tvm Group Title @Sristy: no need to call me sir. Anyway, imagine a 3D space, with each point denoted by x, y and z coods. Now imagine that there is a vector value at each of these points. ie, For any point you take, you get a vector. This is a 'vector field'. Examples include electric and magnetic fields. @gokuldas: So, in connection with my question how shall i appreciate ur logic? 29. gokuldas_tvm Group Title This is important for understanding flux. The next point is about that 30. gokuldas_tvm Group Title Now consider a surface within that 3D space. You can imagine that this surface will contain many points in the 3D space. And each of those points will have a vector associated with it. Now imagine accumulating all these vectors on the surface. ie, integrate the vectors on the surface. $\phi = \int\limits_{}^{}\int\limits_{}^{s} F.ds$ This value is a scalar which represents all total quantity of the vector crossing the surface. This will be a single number (a scalar). This is the flux. @gokul: apparently u are interested in Maxwell's equations. Hm? 32. gokuldas_tvm Group Title yes Can u tell me whether Maxwell's equations are linear or non-linear? 34. gokuldas_tvm Group Title I doubt that the equations themselves are linear or non-linear. Their solution could be perhaps classified as linear or non-linear Hmm point taken 36. gokuldas_tvm Group Title I am not seriously sure about that - I have never done such an analysis on Maxwell's equation. I am only going through a book on non-linear systems now. Normally, you should be able to separate out linear and non-linear systems from partial differential equations like maxwell's equations. I am a little confused about this - because most results are linear. But I have also seen non-linear behaviour called 'solitons'. I am still to make any sense of that. 37. him1618 Group Title i think Q/epsilon naught represents the nmber of flux lines 38. gokuldas_tvm Group Title @him1618: Q/E represents flux, not flux lines. Flux is the scalar product of the field and the surface, integrated over the entire surface. It is a scalar. The concept of flux lines is very misleading - because you can have them at every point on the surface. Thus you can have infinite flux lines. A lot of leading authors advice against using the 'flux line' concept, precisely because of this confusion. This includes the famed Dr. Richard Feynman. 39. him1618 Group Title @gokuldas: it represents the relative number of field lines coming out of a charged particle or going into it as well 40. gokuldas_tvm Group Title @him1618: Hmm, Now I understand why Dr. Feynman so religiously opposed the flux line concept. Let us check how many flux lines are possible through a surface. There are 2 key facts: 1. In a vector field like electric field, every point in space is associated with a 3D vector (eg: the E vector). 2. A flux line is an imaginary curve (or line) that follow the direction of these vectors from a point to the next. Now consider a surface. It has infinite number of points on it. And each one of these points has a vector. You can associate each of these vectors with a flux line. Thus you have as many flux lines as you have points - ie, infinite number of flux lines! The danger in introducing the 'flux line' concept is that it leads people into thinking that the field (say electric field, or gravity field) exists as a bunch of strings. And then you would try to count them! Dr. Feynman used to oppose this concept - saying that only the 'field' concept is real. In other words, you have to consider the field as a hazy cloud of 3D vectors. You simply cannot separate it into lines and count it.	2014-07-31 23:51:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7551525831222534, "perplexity": 812.8690661332923}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273766.32/warc/CC-MAIN-20140728011753-00336-ip-10-146-231-18.ec2.internal.warc.gz"}
https://www.preprints.org/manuscript/201703.0197/v1	Preprint Article Version 1 This version is not peer-reviewed # Co-integration and Causality Analysis of Carbon Intensity and Coal Consumption of China Version 1 : Received: 26 March 2017 / Approved: 27 March 2017 / Online: 27 March 2017 (10:33:34 CEST) How to cite: Wang, X.; Fan, Z.; Zhang, Y. Co-integration and Causality Analysis of Carbon Intensity and Coal Consumption of China. Preprints 2017, 2017030197 (doi: 10.20944/preprints201703.0197.v1). Wang, X.; Fan, Z.; Zhang, Y. Co-integration and Causality Analysis of Carbon Intensity and Coal Consumption of China. Preprints 2017, 2017030197 (doi: 10.20944/preprints201703.0197.v1). ## Abstract Co-integration and Causality was built to conduct studies on causality relation between carbon intensity and coal consumption leading to providing important basis for the transition to a low carbon economy. The EG two-step method was performed to study the relation between carbon intensity and coal consumption of China during 1990-2015 and the co-integration and Granger test was constructed to build up the co-integration and error correction models for analysis of the interaction between carbon intensity and coal consumption. The results showed that in long term there is a stable co-integration relation and a positive correlation between carbon intensity and coal consumption; whereas fluctuations exist in short term and there is a one-way Granger causality of carbon intensity with respect to coal consumption. ## Subject Areas carbon intensity; coal consumption; co-integration test; Granger causality; error correction model	2020-07-11 11:44:07	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8046064376831055, "perplexity": 8238.407890568065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655929376.49/warc/CC-MAIN-20200711095334-20200711125334-00134.warc.gz"}
http://tr.lazuk.net/ohrxpsj/latex-setmainfont.html	# Latex setmainfont Will Robertson. \documentclass{article} \usepackage{fontspec} \setmainfont{CMU Serif} \begin{document} Test test2 12345 \end{document} Scarica VerbTeX LaTeX Editor direttamente sul tuo iPhone, iPad e iPod touch. 2. Put the files (ttf, tfm, fd, map) into the proper directories in the local texmf tree. If you want to stick to LaTeX, you can generally try these steps: Create the TeX font metrics (tfm), the font description (fd) and the font mapping (map). 本文目录：题外话特殊符号 CTAN是什么字体字体族字体形状字体系列字号中文字体使用更多字体强调文字（下划线等）行距题外话转眼间回家已经几天了，毕竟近九个月没有回家，该去看望的亲戚还是要去的。 \setmainfont{Junicode} The result was: XeLaTex and Junicode font with chinese and greek characters. It allows you to create and manage LaTeX projects directly on your Android device and generate a PDF by using Verbosus (Online LaTeX and Octave Editor). \setmainfont{Arial} \setmonofont{Lucida Console} \setsansfont{Comic Sans MS} This text uses the Comic Sans MS as Sans Serif Font: Reëel, café, geïrriteerd, €, $% \setmainfont命令で、その文書の既定の欧文フォントを指定する。 \setmainjfont 命令で、その文書の既定の和文フォントを指定する。このLaTeX文書を lualatex コマンドでコンパイルすると、以下のような出力が得られます。 Serif Fonts] [Serif Fonts, Sub-Categorised] [Sans Serif Fonts] [Typewriter Fonts] [Calligraphical and Handwritten Fonts] [Uncial Fonts] [Blackletter Fonts] [Other Fonts] [Fonts with Math Support] [Fonts with OpenType or TrueType Support] [All Fonts, by category] [All Fonts, alphabetically] [About The L a T e X Font Catalogue] [Packages that provide math support] Here is a summary log about how I made XeTex work on Snow Leopard (precisely I used XeLaTex). Using the editor LyX , this can be done by checking a box under LyX > Document > Settings > […] use XeTeX . Try > \setmainfont{CMU Serif} or > \setmainfont{CMU Serif Roman} I did try that, I should have mentioned it in my previous post LaTeX associates the formatting you assigned to sections to every bit of text you mark as a section. Most people that use LaTeX to make slides seem to use Beamer, but the resulting slides are usually a bit busy, like this: LaTeX associates the formatting you assigned to sections to every bit of text you mark as a section. We can use several ways to work with Chinese in LaTeX. are explicitly defined separately (sorry, I'm sure my terminology is wrong— I'm not much of a latex whiz), so is it possible you'd have to override their fonts separately as well? github. Install XeTex. But as soon as you want to typeset Unicode text in multiple languages, you’re quickly out of luck. Features: Ruby中用来实现mixin的include和extend. How to prevent changing font used to display \sin function in math mode if I include the command \setmainfont? (As I understand, \setmainfont should not effect font used in math mode). The LuaTeX project's main objective is to provide an open and configurable variant of TeX while at the same time offering downward compatibility. With the incorporation of XeLaTeX in all major distributions of LaTeX including MikTeX, it has become easy to incorporate Devanagari and other Unicode characters in LaTeX/LyX The \substitutefont command is from the eponymous LaTeX package, which is loaded by Sphinx if needed (on Ubuntu xenial it is part of texlive-latex-extra which is a Sphinx requirement). If you did the optional step, it does need to be the same font. Ligatures and German double s. 2018-09-20 It shouldn’t be a secret that I haven’t had much time for LaTeX programming over the last few years now, and fixing up fontspec, the package that got me into this whole mess in the first place, turned out to be a good way to re-engage somewhat with my old flame. tfm. It was compiled with MiKTex 2. vpl, then vptovf to create . Now, greek worked, but still no chinese characters. Use CTeX. We can see the options of SansBuild and SansProcess in the Quickbuild dropdown box. Don't change this file in any respect. \fontspec{font display Hello, I am on Miktex 2. \setmainfont{FONTNAME} \setmainfont[Ligatures=TeX]{FONTNAME} Next, create a macro, a sort of shortcut, so that you don’t have to type out the entire complicated Greek command. Nowadays LaTeX supports the 3 smartfont technologies OpenType, AAT and Graphite directly through XeTeX, which has to be activated in one's LaTeX editor. MiKTeX brings the functionality to automatically install CTAN packages when a LaTeX package is required by \usepackage{}. e. Font implementation. My pb looks very similar to online LaTeX editor with autocompletion, highlighting and 400 math symbols. LATEX and LuaLATEX. LuaTeX and XeTeX are alternative TeX engines and both are designed to work with Unicode text. Location at CTAN. [Front Page] [Serif Fonts] [Serif Fonts, Sub- Categorised] [Sans Serif Fonts] [Typewriter Fonts] [Calligraphical and Handwritten Fonts] Sep 18, 2016 A Beginner's Guide to LATEX (on the Mac) . LuaTeX is an extended version of pdfTeX using Lua as an embedded scripting language. It turned out that even fonts which are dedicated to unicode do not yet have all possible characters implemented. The fontspec package provides an automatic and unified interface for loading fonts in LaTeX. Use LatexExpr to typeset LaTeX code that you create by hand. Pgf pictures created by plt. There is \setmainfont. This post contains examples to be tried in both, but if you want to be persuaded to choose just one, see Why choose LuaLaTeX over XeLaTeX . Simple examples for testing XeTex can be found in an older post Myanmar (Burmese) Language with XeTeX and LuaTeX . 1 for Android. This also goes in the preamble. Now changing all the sections is easy: you just change the definition of the formattingfor\sectionandallsectionswillchangeautomatically. With the GUI TeXnikCenter use the output format LaTeX=>PS=>PDF. Personally, I'd drop the font in ~/. tex - First LaTeX document LATEX Workflow in Overleaf . I like to use LaTeX to make slides for talks, largely because I prefer to write code (rather than use a mouse and menus) for control of things like colors and figure placement. \setmainfont[Ligatures=TeX]{Times Mar 25, 2014 For my dissertation (written in LaTeX), I wanted to be sure to avoid this Second, the packages fontspec and unicode-math are necessary. itrans doesn’t let you type in Tamil or Hindi (or Marathi, Sanskrit, Telugu and Gujarati) directly, rather you have to key in the ASCII transcription, then process it with itrans from the command prompt, then run (pdf)latex on the resultant file. Here is the minimal latex code. Click on “Save Project Settings”. savefig('figure. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. If you do not hold it in your list, get it from perce. With LaTeX, you do the formatting once and then focus on your content. It is usually used with the fontspec package, which provides a configurable interface for font selection, and allows complex font choices to be named and later reused. De gratis, collaborative LaTeX Editor voor je Android-toestel VerbTeX is a collaborative LaTeX Editor for your Android device. The LaTeX font encodings guide names the OML encoding TeX math italic and defines: The OML encoding contains italic Latin and Greek letters for use in mathematical formulas (typically used for variables) together with some symbols. E. Philipp Gesang 1 Introduction; 2 Setting fonts for different LaTeX elements; 3 Fonts in Overleaf \ documentclass[12pt]{article} \usepackage{fontspec} \setmainfont{Times New The bodies of LaTeX documents are set in Roman (serif) type by default, but this can be . This package also provides access to the large number of font features available with OpenType (and other) fonts, fontspec uses these features to provide the same interface in LuaTeX for loading fonts for LaTeX use: \setmainfont[Numbers=OldStyle]{Adobe Garamond Pro} Important comment #1: Before luaotfload can load fonts by name as in the example above, it must first create a database of installed fonts in the operating system and in the TeX distribution. In a typical LaTeX document, you just need to include this in your headers: It's the fontspec package that allows for \setmainfont and \setmonofont. Its LaTeX counterpart is invoked as xelatex. You may want to read the document of NFSS: LaTeX2𝜀 font selection [1], section 3; and the documented source code of LaTeX2e: The LaTeX 2𝜀 Sources [2], section 42. Praveen Krishnan, Swagatika Panda LATEX for Research Papers: L2. Latex page, line and font settings. The fontspec package provides a LaTeX interface to the modern OpenType fonts that are now fontspec error: "font-not-found" ! ! The font "persian-modern-regular" cannot be found. The reason for not using the fonts provided by the system is that Tex and Latex will give the same distinctive look no matter which platform a document was compile on, \usepackage{fontspec} \setmainfont{Asana-Math} Style examples \normalfont Numbers. This template is a general template for scientific theses. fonts/Asana-Math/ Last update of this page . 95]{Some Nice Font} – egreg – 6 years ago Please also provide information about the version of XeLaTeX and fontspec Mar 23, 2015 helloworld. Ao escrever, o escritor usa texto simples, ao invés do texto formatado encontrado em processadores de texto WYSIWYG como Microsoft Word, LibreOffice Writer e Apple Pages. How do I use my favorite TrueType font (TTF) in LaTeX? Use the command lualatex or xelatex – both of which are included in TeXLive, MikTeX, and MacTeX – and in the preamble add the following: \usepackage{fontspec} \setmainfont{Arial Unicode} % For example This also works for OpenType fonts (OTF), which is a superior format. This software is provided "as is" without warranties or conditions of any kind, either expressed or implied. Hello, I need to change the font in my latex file to Times New Roman, and I used this in the preamble to do that: \usepackage{fontspec} Times New Roman font in Latex \| Physics Forums Menu From looking at the default Rmd latex template (got there by way of LaTeX options from the R Markdown docs, it looks like title, author, etc. LaTeX (pronúncia em inglês: [ˈlɑːtɛx] ou uma abreviação de Lamport TeX) é um sistema de preparação de documentos. XeLaTeX and LuaLaTeX have full unicode support and can use any font that is installed in the operating system, making use of advanced typographic features of OpenType, AAT and Graphite. LaTeX commands generally refer to these with the shorthand rm, sf, and tt respectively. ) Under “Advanced Build Options”, from the “LaTeX engine” drop list, select XeLaTeX or LuaLaTeX. Invoking LuaLaTeX or XeLaTeX just means invoking LuaTeX or XeTeX with LaTeX format. . A full installation of MacTex will have XeTex installed. Why should you use L a T e X instead of an ordinary word processor? Here is a tentative answer comparing their typographic accuracy. Fonts can be Verify that fontspec. As a beginner, you should either start with the first lesson or, if you just want a very brief introduction, try the interactive quick start guide. . hr (PMF–MO) LATEX—Uvodiosnove 2016/17 2/46 Learn about LaTeX in short lessons with full code examples. 60 introduces fontspec-etex package, e-TeX (BaKoMa TeX) fontspec-etex uses Direct Usage of Windows Fonts in LaTeX Apr 7, 2019 Handling fonts in TEX and LATEX is a notoriously difficult task. 2 \setmainfont[Ligatures=TeX,Scale=0. Important: notați că asta merge numai pentru fonturi ce sunt deja instalate în LaTeX. ! ! See the fontspec documentation for further information. Replace FONT NAME with a font of your choice that has Greek letters. Mapping those characters to their glyphs in the fonts your document uses. XeTex is part of TexLive which is included in MacTex. There were multiple standards to map between computer codes and glyphs. Search this site. com LaTeX allows by default at most 6 levels for nesting list and quote-like environments, with at most 4 enumerated lists, and 4 bullet lists. V Commandsforaccentsandsymbols(‘encodings’) 52 1 AnewUnicode-basedencodingfromscratch 52 2 Adjustingapre-existingencoding 53 3 Summaryofcommands 55 XeLaTeX: Unicode font fallback for unsupported characters. Math fonts are quite different with text fonts. It allows you to create and manage LaTeX projects directly on your Windows device and generate a PDF by using Verbosus (Online LaTeX Editor). Dartmouth Ruzicka fonts in LaTeX. LaTeX on Mac OS X: my open source L a T Selecting XeLaTeX/LuaLaTeX as your LaTeX engine; With your project loaded in the Overleaf editor, click the Settings icon (the gear icon in the upper right, just next to your user name. The CTeX group has developed multiple document classes and packages to deal with the particular need of Chinese typesetting. Only if the document actually does contain Unicode Greek letters (in text) or Cyrillic letters, will the above default set-up cause additional requirements for the PDF build. This idea doesn’t apply just to sections, however. LaTeX printing support ¶. 1. We can use SansBuild to compile and generate pdfs directly from the . Dacă aveți un font ttf sau ceva asemănător, va trebui să-l convertiți și să-l faceți disponibil sistemului LaTeX. \usepackage{fontspec} \setmainfont{Asana-Math} Style examples \normalfont Numbers. Mar 15, 2019 The fontspec package. dn ﬁles. Use %latex_debug to get debugging output. When using a \setmainfont (. com Updated September 05, 2018 Hi Ulrike, > You are using the wrong font name. Using LaTeX will enhance both, the look of your papers and your productivity. VerbTeX Pro is the secure LaTeX Editor for your Windows device. \documentclass[a4paper,12pt,dvipdfm]{article} \usepackage{xltxtra} \defaultfontfeatures{Mapping=tex-text} \setmainfont{Times New Roman} \font\YaHei="Microsoft YaHei" at 12pt（我以前自己写的作业= =翻出来居然发现是Times New Roman）这本来是篇英语文章，但是我有reference 是中文书目。所以默认字体我使用 LaTeX is a relative old typesetting system, so when it was designed, no standards in font specification existed. PartI Gettingstarted 1 History ThispackagebeganlifeasaLATEXinterfacetoselectsystem-installedmacOSfontsinJonathan EKew’sXT EX,theﬁrstwidely V Commandsforaccentsandsymbols(‘encodings’) 52 1 AnewUnicode-basedencodingfromscratch 52 2 Adjustingapre-existingencoding 53 3 Summaryofcommands 55 LaTeX By now, most LaTeX users have probably heard of XeLaTeX , if only because it is an option in the latest versions of the standard LaTeX editors such as TeXnicCenter , WinEdt and TeXWorks . That's a bit complex in LaTeX. Join them; it only takes a minute: An online LaTeX editor that's easy to use. To properly render Myanmar fonts using LaTeX, it is necessary to use TeX typesetting engines that support Unicode such as XeTeX. \setmainfont \setsansfont \setmonofont \newfontfamily \newfontface \fontspec The first five do more than the last one, because they set up fonts for general usage in the document, whereas \fontspec directly accesses the lower level functions. LaTeX Community Announcements Community talk Comments & Wishes New Members; LaTeX Text Formatting Graphics, Figures & Tables Math & Science Fonts & Character Sets Page Layout Document Classes General; LaTeX's Friends BibTeX, biblatex and biber MakeIndex, Nomenclature, Glossaries and Acronyms Conversion Tools Viewers for PDF, PS, and DVI XeTeX Others; LaTeX Distributions How do I use TrueType fonts with LaTeX. At least in inkscape and also if I open the file with firefox or chromium not I am on Ubuntu 15. This guide shows you, that nice typesetting is easy and hassle free. For examples, any math font package is OK. \usepackage{fontspec} \setmainfont[Ligatures=TeX]{Georgia} Feb 23, 2017 Messing around with fonts in LaTeX. LATEX. BaKoMa TeX beginning from V 11. 9, Windows 7 and am trying to get (new) hebrew fonts to work with polyglossia and fontspec. Definitions \newcommand \newenvironment \newtheorem \newfont. 其它项都是LaTex自己的格式，就不做解释了关于字体，XeLaTex可以使用本系统自带字体，可以通过cmd窗口下运行fc-list>fonts. fonts, run `fc-cache ~/. 让LaTeX支持中文的另一种方法是使用德国人Werner Lemberg编写的CJK巨集包。这个巨集包不仅仅支持繁简体中文、日文、朝鲜文等东亚语言，而且它也是一个多种语言支持包，另外还支持几十种其他不同的语言。 Pregled 1 Uvod 2 Jednostavnoslovoslagarstvo 3 Strukturadokumenta 4 Klasedokumenata 5 Naredbe 6 Teoremiisličneokoline IvicaNakić nakic@math. pgf') can be embedded as raw commands in LaTeX documents. The reference to italic shape is odd: No other font encoding is specific to the font shape. We have$2006 = \sage {factor (2006)}\$. Writing unicode text: Devanagari SansBuild Now, restart Kile. Vezi secțiunea de legături externe de la sfârșitul capitolului pentru câteva resurse utile în acest sens. The method designed by LaTeX involves glyphs stored in the metafont (MF) files and hints stored in TFM (TeX Font Metric) files. Encodings. I hope this can help the one who also need to use CJK characters in LaTex under Mac OS X. The original LaTeX do not have good support for Chinese. Asana Math exists as OpenType or TrueType only. Math example. Posted on June 3, 2012 March 13, 2016 by Malaysian LaTeX User Group Rather a long time ago, I wrote about how to typeset CJK (Chinese, Japanese and Korean) in pdf L a T e X , using the CJK and ctex packages. XeTeX and LuaTeX (the latter through the luaotfload package) allows a direct interface to fonts which may be loaded by their name or filename, so no manual font installation is required. One simple solution is to use normal text mode "\textrm{}" instead of "\mathrm{}" for upright characters (it works out the same). The fonts in the resulting svg file are not the ones used in the pdf file and it looks cluttered. unicode-math –modernexpl3 packagedevelopment 21/52 Testsuite Exampletestoutput This is a generated file for the l3build validation system. Currently, it is the unofficial LaTeX template for Master, Bachelor, Diploma, and Student Theses at following institutions: University of Stuttgart, Computer Science English example; German example; German example with minted and PlantUML Latex font as ubuntu system font. Even though LaTeX provides an extensive set of fonts, you may want \documentclass[12pt]{article} \usepackage{fontspec} \setmainfont{Times New Jul 22, 2018 I've got a first release ready of a package for XeLaTeX that allows dynamic font fontspec and unicode-math were initially released pre-expl3. ‎VerbTeX is a free LaTeX Editor for your iOS device. fonts', after which I should be able to use the font using XeLaTeX (and friends). The ability to choose a multitude of font features is beyond my expertise, but I would suggest looking up some examples and seeing if this would suit your needs. Counters \addtocounter \alph \arabic \fnsymbol \newcounter \roman \setcounter \usecounter \value. If you have a long article mainly consisted of Chinese, you should choose ctexart document class LaTeX is a document preparation system used for the communication and publication of scientific documents. \usepackage{fontspec}. log文件中查看看字体对应的英文名称，换到setmainfont{}里就可以了。例子中使用的是方正舒体 The original LaTeX do not have good support for Chinese. I’d like to do this with something like Markdown. 如果在一个class中include了一个module，那么在这个module中定义的方法就变为这个class的实例方法。 12 INTRODUCTION 13 Agiledevelopmentofsciencedependsonthecontinuousexchangeofinformationbetweenresearchers 14 (Woelﬂe,Olliaro&Todd,2011). Use %slide instead to typeset slides. Part of T e X Live? Asana Math is part of T e X Live. Ask Question apart from using the LuaLaTex package fontspec and then in the preamble of in the document \setmainfont The FONTSPEC package. Licensing Information; Overview of LaTeX and Local Guide; Commands. g, \setmainfont{TeX Gyre Termes}) the resp Whether you are just getting started with LaTeX and wondering what the fuss is about, here to share the clever trick you've discovered, or need urgent help with your bibliography, welcome to the LaTeX subreddit! \section{New to \LATEX ?} Here is a thread with useful links \section{Resources} Extensions to view LaTeX on reddit: Chrome: TeX All TeXstudio - A LaTeX Editor An integrated writing environment for creating LaTeX documents Brought to you by: benibela , jsundermeyer , t_hoffmann LaTeX help 1. By default, LaTeX uses Computer Modern , a family of typefaces designed by Donald Knuth for use with TeX. 2018-09-20 \setmainfont. One of my requirements is that the document must have page numb 最近发现LaTeX真是排版的好工具呀！特别是在写外文期刊文章的时候！我下载了MiKTex进行安装，然后想写一段汉字进行输出，使用XeLaTex进行编译，结果老是报错。搞了半天也没有搞明白为什么。按理来说XeLaTex支持Unicode编码的字符。 The Beauty of LaTeX. log命令到fonts. Currently, it is the unofficial LaTeX template for Master, Bachelor, Diploma, and Student Theses at following institutions: University of Stuttgart, Computer Science English example. sty and LuaLaTeX are already available ( texlive-latex-base package on Ubuntu): The LaTeX Font Catalogue. My precise problem is that I convert a pdf file, which I compiled with xelatex,to svg using inkscape. Setting this key for example to '10' (as a string) will allow up to 10 nested levels (of all sorts). Why is \setmainfont case sensitive with XelateX but not with LuaLaTeX? Configuring LaTeX to use fonts installed through portableapp. tagged latex, xelatex, fonts, fontspec \ usepackage{fontspec} \newfontfamily\bitter[Path=fonts/Bitter/, The fontspec package enables automatic font selection for LaTeX documents typeset with XeTeX or LuaTeX. 本模板介绍的是一个学术性的LaTeX Beamer模板，优点是非常简洁，适用于多公式，多算法的应用场景，缺点是不非常灵活。你也可以随意使用和修改这个模板而不用经过我的同意，Just follow you heart!但是引用本文需要经过我的同意，联系方式见文末。 LaTeX Template for Scientific Theses . Full Text Search. LaTeX is just not made for Unicode, and you need a lot of helper packages, documentation reading, and complicated configuration in your document to get it all right. This package also provides access to Perhaps you'd be better off putting the font into your Window's font folder and omit the optional argument to \setmainfont, it should be found there by XeLaTeX and friends out-of-the-box. Selecting XeLaTeX/LuaLaTeX as your LaTeX engine; With your project loaded in the Overleaf editor, click the Settings icon (the gear icon in the upper right, just next to your user name. 10. Templates and examples of how to compose an academic curriculum vitae or resume using X Ǝ T e X and expert fonts. When using any language, LaTeX must handle two fundamental problems: Mapping the bytes of your input file into the characters of the language(s) you want to use. Oldstyle numbers. It contains serif, sans serif, and monospaced fonts, each available in several weights and optical sizes . This means that an a becomes a b, a 1 becomes a 2, etc. 让LaTeX支持中文的另一种方法是使用德国人Werner Lemberg编写的CJK巨集包。这个巨集包不仅仅支持繁简体中文、日文、朝鲜文等东亚语言，而且它也是一个多种语言支持包，另外还支持几十种其他不同的语言。 XeTeX is a TeX typesetting engine using Unicode and supporting modern font technologies XeTeX works well with both LaTeX and ConTeXt macro packages. \documentclass[a4paper, 12pt]{article} \usepackage{amssymb,amsmath,amsthm,amsfonts} \usepackage{fontspec} LaTeX Template for Scientific Theses. Αλκαίος Σουγιούλ Προσωπική Σελίδα. Menu Doing a post on typesetting Tamil and Hindi is only natural after sorting out Jawi and CJK!Just after I managed to get them working using the itrans package and the devanagari fonts, this exact question was asked on the TeX-LaTeX Stack Exchange site, to which I posted what worked for me. The basic command is. XTX and Lua TX (the latter through the luaotfload package) allows a direct interface to fonts which may be loaded by their name or filename, so no manual font installation is required. in an input cell in the notebook to get a typeset version. 我说论文，是因为就我所知，LaTeX 在公司环境下并不多见，我唯一知道的是浩方。如果不符合提问者的要求，请指出。另一点是，我给出的设置参照的是Word 的默认排版设置，因为国内很多论文的模板是假定用户在 Word 环境下完成的。 Mode of operation. If you have a long article mainly consisted of Chinese, you should choose ctexart document class With LaTeX, you do the formatting once and then focus on your content. With contributions by Khaled Hosny,. Will Robertson acknowledged it was not the best Technical answer: You are telling fontspec that whenever a bold italic font is requested, it should use upright regular Times New Roman. German example. When using the xelatex or lualatex engines, much of these problems is solved for you. Export (png, jpg, gif, svg, pdf) and save & share with note system. \setmainfont命令で、その文書の既定の欧文フォントを指定する。 \setmainjfont 命令で、その文書の既定の和文フォントを指定する。このLaTeX文書を lualatex コマンドでコンパイルすると、以下のような出力が得られます。 Whether you are just getting started with LaTeX and wondering what the fuss is about, here to share the clever trick you've discovered, or need urgent help with your bibliography, welcome to the LaTeX subreddit! \section{New to \LATEX ?} Here is a thread with useful links \section{Resources} Extensions to view LaTeX on reddit: Chrome: TeX All LaTeX error: too many fonts used in version normal while in equation mode. Cross References \label \pageref \ref. A comprehensive guide to basic and advanced features. Document Styles \flushbottom \onecolumn LaTeX学习笔记 3 \| 特殊符号、字体与行距. Typesetting an academic CV in LaTeX. Leaving it to the empty string means to obey the LaTeX default. 9 and xelatex. It is usually used with the fontspec package, which provides a configurable My impression is that the usage of \fontspec has been misunderstood from the beginning. \documentclass[a4paper, 12pt]{article} \usepackage{amssymb,amsmath,amsthm,amsfonts} \usepackage{fontspec} When I'm compiling with XeLaTeX and I use \setmainfont{CMU Serif}, the resulting characters are off by one. While the beginners course will enable you to typeset your research papers or thesis, LaTeX (pronúncia em inglês: [ˈlɑːtɛx] ou uma abreviação de Lamport TeX) é um sistema de preparação de documentos. which calls both \mathfont and fontspec's \setmainfont using the font name as XeLaTeX. de and imwharf it into TeXnikCenter by Output (Format) - define output profile (Ausgabeprofil definieren). The FONTSPEC package. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. Note that the definitions of math symbols are complex. LaTeX is free software and is distributed under the LaTeX Project Public License. It allows you to create and manage LaTeX projects directly on your iOS device and generate a PDF by using Verbosus (Online LaTeX Editor). Use latex () to typeset a Sage object. But most LaTeXers have probably not yet become XeLaTeXers. XeTeX is bundled with TeX Live, MacTeX, MiKTeX and Lyx (see the History below for dates and versions). Font types are actually a bit of a pain in Latex and so far, I have almost always been using the standard font, Knuth’s Computer Modern. Remember that you hold to upengagement the knockhs for the programs! Download VerbTeX apk 4. Font selection for X. Inthepast I need to create a large document that when printed will be a couple hundred pages. Using ttf2tfm to generate . How to Create Devanagari Documents in Lyx Using Xelatex. Thanks to all the latex gurus here (Valex et al), I have managed to create a satisfactory template (screenshots attached) It has the following features: custom title page, custom chapter page, lettrine (with faked small caps), fleuron, generous white and invisible trims to prevent vertical centering. Latex fonts installed via font packages should be automatically \usepackage{ fontspec} \usepackage{tgpagella} \setmainfont{TeX Gyre . latex setmainfont fl, wf, sm, rr, js, zv, uq, qg, cc, zi, sw, sl, wd, c0, v7, pz, eu, pt, 8z, fu, fh, ci, xq, 7k, yr, yx, ju, b9, hk, 48, je,	2019-08-21 05:40:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9076668620109558, "perplexity": 10814.794853241447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315809.69/warc/CC-MAIN-20190821043107-20190821065107-00213.warc.gz"}
https://pymc3.readthedocs.io/en/stable/notebooks/api_quickstart.html	API quickstart¶ In [1]: %matplotlib inline import numpy as np import theano.tensor as tt import pymc3 as pm import seaborn as sns import matplotlib.pyplot as plt sns.set_context('notebook') 1. Model creation¶ Models in PyMC3 are centered around the Model class. It has references to all random variables (RVs) and computes the model logp and its gradients. Usually, you would instantiate it as part of a with context: In [2]: with pm.Model() as model: # Model definition pass We discuss RVs further below but let’s create a simple model to explore the Model class. In [3]: with pm.Model() as model: mu = pm.Normal('mu', mu=0, sd=1) obs = pm.Normal('obs', mu=mu, sd=1, observed=np.random.randn(100)) In [4]: model.basic_RVs Out[4]: [mu, obs] In [5]: model.free_RVs Out[5]: [mu] In [6]: model.observed_RVs Out[6]: [obs] In [7]: model.logp({'mu': 0}) Out[7]: array(-139.7234487505281) Warning It’s worth highlighting one of the counter-intuitive design choices with logp. The API makes the logp look like an attribute, when it actually puts together a function based on the current state of the model. The current design is super maintainable, does terrible if the state stays constant, and great if the state keeps changing, for reasons of design we assume that Model isn’t static, in fact it’s best in our experience and avoids bad results. If you need to use logp in an inner loop and it needs to be static, simply use something like logp = model.logp below. You can see the caching effect with the speed up below. In [8]: %timeit model.logp({mu: 0.1}) logp = model.logp %timeit logp({mu: 0.1}) 91.8 ms ± 1.95 ms per loop (mean ± std. dev. of 7 runs, 10 loops each) 35.6 µs ± 2.01 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each) 2. Probability Distributions¶ Every probabilistic program consists of observed and unobserved Random Variables (RVs). Observed RVs are defined via likelihood distributions, while unobserved RVs are defined via prior distributions. In PyMC3, probability distributions are available from the main module space: In [9]: help(pm.Normal) Help on class Normal in module pymc3.distributions.continuous: class Normal(pymc3.distributions.distribution.Continuous) \| Univariate normal log-likelihood. \| \| .. math:: \| \| f(x \mid \mu, \tau) = \| \sqrt{\frac{\tau}{2\pi}} \| \exp\left\{ -\frac{\tau}{2} (x-\mu)^2 \right\} \| \| ======== ========================================== \| Support :math:x \in \mathbb{R} \| Mean :math:\mu \| Variance :math:\dfrac{1}{\tau} or :math:\sigma^2 \| ======== ========================================== \| \| Normal distribution can be parameterized either in terms of precision \| or standard deviation. The link between the two parametrizations is \| given by \| \| .. math:: \| \| \tau = \dfrac{1}{\sigma^2} \| \| .. plot:: \| \| import matplotlib.pyplot as plt \| import numpy as np \| import scipy.stats as st \| x = np.linspace(-5.0, 5.0, 1000) \| fig, ax = plt.subplots() \| f = lambda mu, sd : st.norm.pdf(x, loc=mu, scale=sd) \| plot_pdf = lambda a, b : ax.plot(x, f(a,b), label=r'$\mu$={0}, $\sigma$={1}'.format(a,b)) \| plot_pdf(0.0, 0.4) \| plot_pdf(0.0, 1.0) \| plot_pdf(0.0, 2.0) \| plot_pdf(-2.0, 0.4) \| plt.legend(loc='upper right', frameon=False) \| ax.set(xlim=[-5,5], ylim=[0,1.2], xlabel='x', ylabel='f(x)') \| plt.show() \| \| Parameters \| ---------- \| mu : float \| Mean. \| sd : float \| Standard deviation (sd > 0). \| tau : float \| Precision (tau > 0). \| \| Method resolution order: \| Normal \| pymc3.distributions.distribution.Continuous \| pymc3.distributions.distribution.Distribution \| builtins.object \| \| Methods defined here: \| \| __init__(self, mu=0, sd=None, tau=None, *kwargs) \| Initialize self. See help(type(self)) for accurate signature. \| \| logp(self, value) \| \| random(self, point=None, size=None, repeat=None) \| \| ---------------------------------------------------------------------- \| Methods inherited from pymc3.distributions.distribution.Distribution: \| \| __getnewargs__(self) \| \| __latex__ = _repr_latex_(self, name=None, dist=None) \| Magic method name for IPython to use for LaTeX formatting. \| \| default(self) \| \| get_test_val(self, val, defaults) \| \| getattr_value(self, val) \| \| logp_nojac(self, args, *kwargs) \| Return the logp, but do not include a jacobian term for transforms. \| \| If we use different parametrizations for the same distribution, we \| need to add the determinant of the jacobian of the transformation \| to make sure the densities still describe the same distribution. \| However, MAP estimates are not invariant with respect to the \| parametrization, we need to exclude the jacobian terms in this case. \| \| This function should be overwritten in base classes for transformed \| distributions. \| \| logp_sum(self, args, *kwargs) \| Return the sum of the logp values for the given observations. \| \| Subclasses can use this to improve the speed of logp evaluations \| if only the sum of the logp values is needed. \| \| ---------------------------------------------------------------------- \| Class methods inherited from pymc3.distributions.distribution.Distribution: \| \| dist(args, *kwargs) from builtins.type \| \| ---------------------------------------------------------------------- \| Static methods inherited from pymc3.distributions.distribution.Distribution: \| \| __new__(cls, name, args, kwargs) \| Create and return a new object. See help(type) for accurate signature. \| \| ---------------------------------------------------------------------- \| Data descriptors inherited from pymc3.distributions.distribution.Distribution: \| \| __dict__ \| dictionary for instance variables (if defined) \| \| __weakref__ \| list of weak references to the object (if defined) In the PyMC3 module, the structure for probability distributions looks like this: In [10]: dir(pm.distributions.mixture) Out[10]: ['Discrete', 'Distribution', 'Mixture', 'Normal', 'NormalMixture', '__builtins__', '__cached__', '__doc__', '__file__', '__name__', '__package__', '__spec__', 'all_discrete', 'bound', 'draw_values', 'generate_samples', 'get_tau_sd', 'get_variable_name', 'logsumexp', 'np', 'tt'] Unobserved Random Variables¶ Every unobserved RV has the following calling signature: name (str), parameter keyword arguments. Thus, a normal prior can be defined in a model context like this: In [11]: with pm.Model(): x = pm.Normal('x', mu=0, sd=1) As with the model, we can evaluate its logp: In [12]: x.logp({'x': 0}) Out[12]: array(-0.9189385332046727) Observed Random Variables¶ Observed RVs are defined just like unobserved RVs but require data to be passed into the observed keyword argument: In [13]: with pm.Model(): obs = pm.Normal('x', mu=0, sd=1, observed=np.random.randn(100)) observed supports lists, numpy.ndarray, theano and pandas data structures. Deterministic transforms¶ PyMC3 allows you to freely do algebra with RVs in all kinds of ways: In [14]: with pm.Model(): x = pm.Normal('x', mu=0, sd=1) y = pm.Gamma('y', alpha=1, beta=1) plus_2 = x + 2 summed = x + y squared = x2 sined = pm.math.sin(x) While these transformations work seamlessly, its results are not stored automatically. Thus, if you want to keep track of a transformed variable, you have to use pm.Determinstic: In [15]: with pm.Model(): x = pm.Normal('x', mu=0, sd=1) plus_2 = pm.Deterministic('x plus 2', x + 2) Note that plus_2 can be used in the identical way to above, we only tell PyMC3 to keep track of this RV for us. Automatic transforms of bounded RVs¶ In order to sample models more efficiently, PyMC3 automatically transforms bounded RVs to be unbounded. In [16]: with pm.Model() as model: x = pm.Uniform('x', lower=0, upper=1) When we look at the RVs of the model, we would expect to find x there, however: In [17]: model.free_RVs Out[17]: [x_interval__] x_interval__ represents x transformed to accept parameter values between -inf and +inf. In the case of an upper and a lower bound, a LogOdds transform is applied. Sampling in this transformed space makes it easier for the sampler. PyMC3 also keeps track of the non-transformed, bounded parameters. These are common determinstics (see above): In [18]: model.deterministics Out[18]: [x] When displaying results, PyMC3 will usually hide transformed parameters. You can pass the include_transformed=True parameter to many functions to see the transformed parameters that are used for sampling. You can also turn transforms off: In [19]: with pm.Model() as model: x = pm.Uniform('x', lower=0, upper=1, transform=None) print(model.free_RVs) [x] Lists of RVs / higher-dimensional RVs¶ Above we have seen how to create scalar RVs. In many models, you want multiple RVs. There is a tendency (mainly inherited from PyMC 2.x) to create list of RVs, like this: In [20]: with pm.Model(): x = [pm.Normal('x_{}'.format(i), mu=0, sd=1) for i in range(10)] # bad However, even though this works it is quite slow and not recommended. Instead, use the shape kwarg: In [21]: with pm.Model() as model: x = pm.Normal('x', mu=0, sd=1, shape=10) # good x is now a random vector of length 10. We can index into it or do linear algebra operations on it: In [22]: with model: y = x[0] * x[1] # full indexing is supported x.dot(x.T) # Linear algebra is supported Initialization with test_values¶ While PyMC3 tries to automatically initialize models it is sometimes helpful to define initial values for RVs. This can be done via the testval kwarg: In [23]: with pm.Model(): x = pm.Normal('x', mu=0, sd=1, shape=5) x.tag.test_value Out[23]: array([ 0., 0., 0., 0., 0.]) In [24]: with pm.Model(): x = pm.Normal('x', mu=0, sd=1, shape=5, testval=np.random.randn(5)) x.tag.test_value Out[24]: array([ 0.421665 , -1.63966614, 0.92402752, 0.14413675, -1.39721381]) This technique is quite useful to identify problems with model specification or initialization. 3. Inference¶ Once we have defined our model, we have to perform inference to approximate the posterior distribution. PyMC3 supports two broad classes of inference: sampling and variational inference. 3.1 Sampling¶ The main entry point to MCMC sampling algorithms is via the pm.sample() function. By default, this function tries to auto-assign the right sampler(s) and auto-initialize if you don’t pass anything. In [25]: with pm.Model() as model: mu = pm.Normal('mu', mu=0, sd=1) obs = pm.Normal('obs', mu=mu, sd=1, observed=np.random.randn(100)) trace = pm.sample(1000, tune=500) Auto-assigning NUTS sampler... 100%\|██████████\| 1500/1500 [00:01<00:00, 1026.78it/s] As you can see, on a continuous model, PyMC3 assigns the NUTS sampler, which is very efficient even for complex models. PyMC3 also runs variational inference (i.e. ADVI) to find good starting parameters for the sampler. Here we draw 1000 samples from the posterior and allow the sampler to adjust its parameters in an additional 500 iterations. These 500 samples are discarded by default: In [26]: len(trace) Out[26]: 1000 You can also run multiple chains in parallel using the njobs kwarg: In [27]: with pm.Model() as model: mu = pm.Normal('mu', mu=0, sd=1) obs = pm.Normal('obs', mu=mu, sd=1, observed=np.random.randn(100)) trace = pm.sample(njobs=4) Auto-assigning NUTS sampler... 100%\|██████████\| 1000/1000 [00:01<00:00, 851.68it/s] Note, that we are now drawing 2000 samples, 500 samples for 4 chains each. The 500 tuning samples are discarded by default. In [28]: trace['mu'].shape Out[28]: (2000,) In [29]: trace.nchains Out[29]: 4 In [30]: trace.get_values('mu', chains=1).shape # get values of a single chain Out[30]: (500,) PyMC3, offers a variety of other samplers, found in pm.step_methods. In [31]: list(filter(lambda x: x[0].isupper(), dir(pm.step_methods))) Out[31]: ['BinaryGibbsMetropolis', 'BinaryMetropolis', 'CategoricalGibbsMetropolis', 'CauchyProposal', 'CompoundStep', 'ElemwiseCategorical', 'EllipticalSlice', 'HamiltonianMC', 'LaplaceProposal', 'Metropolis', 'MultivariateNormalProposal', 'NUTS', 'NormalProposal', 'PoissonProposal', 'SGFS', 'SMC', 'Slice'] Commonly used step-methods besides NUTS are Metropolis and Slice. For almost all continuous models, NUTS should be preferred. There are hard-to-sample models for which NUTS will be very slow causing many users to use Metropolis instead. This practice, however, is rarely successful. NUTS is fast on simple models but can be slow if the model is very complex or it is badly initialized. In the case of a complex model that is hard for NUTS, Metropolis, while faster, will have a very low effective sample size or not converge properly at all. A better approach is to instead try to improve initialization of NUTS, or reparameterize the model. For completeness, other sampling methods can be passed to sample: In [32]: with pm.Model() as model: mu = pm.Normal('mu', mu=0, sd=1) obs = pm.Normal('obs', mu=mu, sd=1, observed=np.random.randn(100)) step = pm.Metropolis() trace = pm.sample(1000, step=step) 100%\|██████████\| 1500/1500 [00:00<00:00, 5511.71it/s] You can also assign variables to different step methods. In [33]: with pm.Model() as model: mu = pm.Normal('mu', mu=0, sd=1) sd = pm.HalfNormal('sd', sd=1) obs = pm.Normal('obs', mu=mu, sd=sd, observed=np.random.randn(100)) step1 = pm.Metropolis(vars=[mu]) step2 = pm.Slice(vars=[sd]) trace = pm.sample(10000, step=[step1, step2], njobs=4) 100%\|██████████\| 10500/10500 [00:13<00:00, 774.35it/s] 3.2 Analyze sampling results¶ The most common used plot to analyze sampling results is the so-called trace-plot: In [34]: pm.traceplot(trace); Another common metric to look at is R-hat, also known as the Gelman-Rubin statistic: In [35]: pm.gelman_rubin(trace) Out[35]: {'mu': 1.0001326715972627, 'sd': 0.99995267558629874, 'sd_log__': 0.99995398290485815} These are also part of the forestplot: In [36]: pm.forestplot(trace); Finally, for a plot of the posterior that is inspired by the book Doing Bayesian Data Analysis, you can use the: In [37]: pm.plot_posterior(trace); For high-dimensional models it becomes cumbersome to look at all parameter’s traces. When using NUTS we can look at the energy plot to assess problems of convergence: In [38]: with pm.Model() as model: x = pm.Normal('x', mu=0, sd=1, shape=100) trace = pm.sample(njobs=4) pm.energyplot(trace); Auto-assigning NUTS sampler... 100%\|██████████\| 1000/1000 [00:05<00:00, 198.29it/s] 3.3 Variational inference¶ PyMC3 supports various Variational Inference techniques. While these methods are much faster, they are often also less accurate and can lead to biased inference. The main entry point is pymc3.fit(). In [39]: with pm.Model() as model: mu = pm.Normal('mu', mu=0, sd=1) sd = pm.HalfNormal('sd', sd=1) obs = pm.Normal('obs', mu=mu, sd=sd, observed=np.random.randn(100)) approx = pm.fit() Average Loss = 150.94: 100%\|██████████\| 10000/10000 [00:03<00:00, 2988.43it/s] Finished [100%]: Average Loss = 150.93 The returned Approximation object has various capabilities, like drawing samples from the approximated posterior, which we can analyse like a regular sampling run: In [40]: approx.sample(500) Out[40]: <MultiTrace: 1 chains, 500 iterations, 3 variables> The variational submodule offers a lot of flexibility in which VI to use and follows an object oriented design. For example, full-rank ADVI estimates a full covariance matrix: In [41]: mu = pm.floatX([0., 0.]) cov = pm.floatX([[1, .5], [.5, 1.]]) with pm.Model() as model: pm.MvNormal('x', mu=mu, cov=cov, shape=2) Average Loss = 0.0068883: 100%\|██████████\| 10000/10000 [00:08<00:00, 1233.38it/s] Finished [100%]: Average Loss = 0.0065707 An equivalent expression using the object-oriented interface is: In [42]: with pm.Model() as model: pm.MvNormal('x', mu=mu, cov=cov, shape=2) Average Loss = 0.01127: 100%\|██████████\| 10000/10000 [00:08<00:00, 1225.82it/s] Finished [100%]: Average Loss = 0.011343 In [43]: plt.figure() trace = approx.sample(10000) sns.kdeplot(trace['x']) Out[43]: <matplotlib.axes._subplots.AxesSubplot at 0x11817c908> Stein Variational Gradient Descent (SVGD) uses particles to estimate the posterior: In [44]: w = pm.floatX([.2, .8]) mu = pm.floatX([-.3, .5]) sd = pm.floatX([.1, .1]) with pm.Model() as model: pm.NormalMixture('x', w=w, mu=mu, sd=sd) approx = pm.fit(method=pm.SVGD(n_particles=200, jitter=1.)) 100%\|██████████\| 10000/10000 [01:21<00:00, 123.44it/s] In [45]: plt.figure() trace = approx.sample(10000) sns.distplot(trace['x']); 4. Posterior Predictive Sampling¶ The sample_ppc() function performs prediction on hold-out data and posterior predictive checks. In [46]: data = np.random.randn(100) with pm.Model() as model: mu = pm.Normal('mu', mu=0, sd=1) sd = pm.HalfNormal('sd', sd=1) obs = pm.Normal('obs', mu=mu, sd=sd, observed=data) trace = pm.sample() Auto-assigning NUTS sampler... 97%\|█████████▋\| 968/1000 [00:01<00:00, 1050.37it/s]/Users/ferres/dev/pymc3/pymc3/step_methods/hmc/nuts.py:451: UserWarning: The acceptance probability in chain 0 does not match the target. It is 0.886449584522, but should be close to 0.8. Try to increase the number of tuning steps. % (self._chain_id, mean_accept, target_accept)) 100%\|██████████\| 1000/1000 [00:01<00:00, 949.39it/s] In [47]: with model: post_pred = pm.sample_ppc(trace, samples=500, size=len(data)) 100%\|██████████\| 500/500 [00:01<00:00, 484.68it/s] sample_ppc() returns a dict with a key for every observed node: In [48]: post_pred['obs'].shape Out[48]: (500, 100) In [49]: plt.figure() ax = sns.distplot(post_pred['obs'].mean(axis=1), label='Posterior predictive means') ax.axvline(data.mean(), color='r', ls='--', label='True mean') ax.legend() Out[49]: <matplotlib.legend.Legend at 0x11a8af940> 4.1 Predicting on hold-out data¶ In many cases you want to predict on unseen / hold-out data. This is especially relevant in Probabilistic Machine Learning and Bayesian Deep Learning. While we plan to improve the API in this regard, this can currently be achieved with a theano.shared variable. These are theano tensors whose values can be changed later. Otherwise they can be passed into PyMC3 just like any other numpy array or tensor. In [50]: import theano x = np.random.randn(100) y = x > 0 x_shared = theano.shared(x) y_shared = theano.shared(y) with pm.Model() as model: coeff = pm.Normal('x', mu=0, sd=1) logistic = pm.math.sigmoid(coeff * x_shared) pm.Bernoulli('obs', p=logistic, observed=y_shared) trace = pm.sample() Auto-assigning NUTS sampler... 100%\|██████████\| 1000/1000 [00:00<00:00, 1369.19it/s] Now assume we want to predict on unseen data. For this we have to change the values of x_shared and y_shared. Theoretically we don’t need to set y_shared as we want to predict it but it has to match the shape of x_shared. In [51]: x_shared.set_value([-1, 0, 1.]) y_shared.set_value([0, 0, 0]) # dummy values with model: post_pred = pm.sample_ppc(trace, samples=500) 100%\|██████████\| 500/500 [00:01<00:00, 463.31it/s] In [52]: post_pred['obs'].mean(axis=0) Out[52]: array([ 0.026, 0.516, 0.966])	2019-10-22 00:46:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38943761587142944, "perplexity": 13841.414414422552}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987795403.76/warc/CC-MAIN-20191022004128-20191022031628-00067.warc.gz"}
http://math.stackexchange.com/questions/266181/a-nasty-integral-of-a-rational-function	# A nasty integral of a rational function I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}.$$ Mathematica has no problem evaluating it while I haven't the slightest idea how to approach it. Of course, I would like to prove it without the use of a computer. Is this an explicit form of a special function I fail to recognize? - Where did you find this integral? – Fabian Dec 27 '12 at 23:59 Our professor gave it to us as a fun problem. – LBO Dec 28 '12 at 0:14 Maybe try a comparison test instead of trying to evaluate the integral directly. I seriously doubt your prof gave you this so you guys could spend 5 hours trying to decompose the fraction – Hawk Dec 28 '12 at 1:16 This is perhaps related to the multiple angle formulae. For instance, the coefficients of $\cos 8x = 32(4\cos^8 x - 8\cos^6 x + 5\cos^4 x - \cos^2x) + 1$ look very similar to the coefficients in the numerator. – user1551 Dec 28 '12 at 10:00 @user1551: The substitution $x = \cos t$ wouldn't work since $x \in (0,\infty)$. – LBO Dec 29 '12 at 15:21 The integral over $\mathbb{R}$ of a meromorphic function $f(z)$, $O(\|z\|^{-2})$ at infinity, non-vanishing over $\mathbb{R}$, is equal to $2\pi i$ times the sum of residues in the poles located in the complex upper-half plane. Since: $$p(y) = y^6-2y^5-2y^4+4y^3+3y^2-4y+1 = p_{+}(y)\cdot p_{-}(y),$$ $$p_{+}(y)= y^3-(i+1)y^2+(i-2)x+1,\qquad p_{-}(y)=y^3+(i-1)y^2-(2+i)y+1,$$ (I got this through a numerical calculation of the roots of $p(y)$, followed by a separation of the roots with positive and negative imaginary part, say $\zeta_1,\zeta_2,\zeta_3$ and $\bar{\zeta_1},\bar{\zeta_2},\bar{\zeta_3}$ - so $p_{+}(z)$ is just $\prod_{j=1}^3 (z-\zeta_j)$) we have: $$I = \int_{\mathbb{R}}\frac{y^2 dy}{p(y)} = 2\pi i\sum_{j=1}^{3}\operatorname{Res}_{z=\zeta_j}\left(\frac{z^2}{p_{+}(z)\cdot p_{-}(z)}\right),$$ but $p_{-}(x)-p_{+}(x)=2i(x^2-x)$, so: $$I = \int_{\mathbb{R}}\frac{y^2 dy}{p(y)} = 2\pi i\sum_{j=1}^{3}\operatorname{Res}_{z=\zeta_j}\left(\frac{z^2}{p_{+}^2(z)+2i(z^2-z)p_{+}(z)}\right).$$ By De l'Hopital theorem, and since $\zeta_j$ is a double zero of $p_{+}^2(x)$: $$\lim_{z\to\zeta_j}\frac{z^2(z-\zeta_j)}{p_{+}^2(z)+2i(z^2-z)p_{+}(z)}=\frac{\zeta_j^2}{2i(\zeta_j^2-\zeta_j)p_{+}'(\zeta_j)},$$ so: $$I = \int_{\mathbb{R}}\frac{y^2 dy}{p(y)} = \pi\sum_{j=1}^{3}\frac{\zeta_j}{(\zeta_j-1)p_{+}'(\zeta_j)}.$$ Now we compute the remainder between $(z-1)p_{+}'(z)$ and $p_{+}(z)$, in order to have: $$I = \int_{\mathbb{R}}\frac{y^2 dy}{p(y)} = \pi\sum_{j=1}^{3}\frac{\zeta_j}{-(1+i)+6\zeta_j-(2-i)\zeta_j^2}.$$ If now we take $\alpha=\frac{3+\sqrt{6-i}}{2-i}$ and $\beta=\frac{3-\sqrt{6-i}}{2-i}$ we can re-write the last line as: $$I = \frac{\pi}{(i-2)(\alpha-\beta)}\left(\sum_{j=1}^{3}\frac{\alpha}{\zeta_j-\alpha}-\sum_{j=1}^{3}\frac{\beta}{\zeta_j-\beta}\right)=-\frac{\pi}{2\sqrt{6-i}}\left(\Sigma_1-\Sigma_2\right).$$ Now $\Sigma_1$ is the sum of the reciprocal of the roots of the polynomial $p_{+}(\alpha(z+1))$, and $\Sigma_2$ is the sum of the reciprocal of the roots of the polynomial $p_{+}(\beta(z+1))$. This quantities can be computed through the coefficients of $p_{+}$, since the sum of the reciprocal of the roots of a polynomial $q(z)$ is just $-\frac{q'(0)}{q(0)}$. This gives: $$\Sigma_1 = -\alpha\frac{p_{+}'(\alpha)}{p_{+}(\alpha)},\qquad \Sigma_2 = -\beta\frac{p_{+}'(\beta)}{p_{+}(\beta)}.$$ Up to a massive amount of long but straightforward computations, we get: $$\Sigma_1 = (i-2)-\sqrt{6-i},\qquad \Sigma_2 = (i-2)+\sqrt{6-i},$$ from which $\color{red}{I=\pi}$ finally follows. I am really grateful to Jon Haussmann for the proof that $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1}dx = \frac{1}{2}\int_{\mathbb{R}}\frac{y^2 dy}{p(y)},$$ where only the second integral is treated here. IMPORTANT UPDATE: In fact, there is no need to compute the coefficients of $p_{+}(x)$ and $p_{-}(x)$ (we only need the identity $p_{-}(x)-p_{+}(x)=2i(x^2-x)$), or introduce $\alpha$ and $\beta$. Since $p_{+}(x)$ is a third-degree polynomial with roots in the upper half-plane, $$0=\int_{\mathbb{R}}\frac{dz}{p_{+}(z)}=\sum_{j=1}^{3}\frac{1}{p_{+}'(\zeta_j)}.$$ This gives: $$I = \pi\sum_{j=1}^{3}\frac{\zeta_j}{(\zeta_j-1)p_{+}'(z)} = \pi\sum_{j=1}^{3}\frac{1}{(\zeta_j-1)p_{+}'(\zeta_j)},$$ but if we decompose $\frac{1}{p_{+}(z)}$ in simple fractions, we get: $$\frac{1}{p_{+}(z)}=\sum_{j=1}^{3}\frac{1}{p_{+}'(\zeta_j)(z-\zeta_j)},$$ so the magic gives: $$I = -\frac{\pi}{p_{+}(1)}.$$ Since $p(x)=p_{+}(x)\cdot p_{-}(x)$, $p(1)=1$, $I\in\mathbb{R}^+$ and $p_{-}(1)$ is the conjugate of $p_{+}(1)$, $p_{+}(1)$ can be only $+1$ or $-1$, so $I=\pi$. - I have the strong feeling that this proof can be significantly shortened, since I do not think that $(\Sigma_1-\Sigma_2)=(i-2)(\alpha-\beta)$ is a mere coincidence. Probably the reduction to elementary functions of the roots of $p_{+}(z)$ is possible directly from: $$I=\pi\sum_{j=1}^{3}\frac{\zeta_j}{(\zeta_j-1)p_{+}'(\zeta_j)}.$$ – Jack D'Aurizio Dec 4 '13 at 14:59 In fact, the feeling was right, the only indespensable ingredient of the proof is the identity: $$p(x)=(x^3-x^2-2x+1)^2+(x^2-x)^2.$$ – Jack D'Aurizio Dec 4 '13 at 17:15 The same technique leads to: $$\frac{\pi}{10}=\int_{\mathbb{R}}\frac{x^2 dx}{x^6+x^4+4x^2+4},\qquad \frac{\pi(\sqrt{2}-1)}{4}=\int_{\mathbb{R}}\frac{x^2 dx}{x^6+x^4+x^2+1}.$$ – Jack D'Aurizio Dec 4 '13 at 19:55 Excellent job! Do you think Wolfram Alpha / Mathematica uses this method to calculate these kind of integrals? – LBO Dec 4 '13 at 22:07 Honestly, I do not know it for sure. Probably the Sigma algorithm for symbolic integration can handle a large amount of logarithm and dilogarithm identities in order to compute, by the residue theorem or whatsoever, the indefinite integral of rational functions. In fact we only need the resultant between a polynomial and its derivative, a localization algorithm for the roots of such polynomials (above/below the real axis) and proper handling of polynomial symmetric functions of the roots - all these tasks are quite expensive to be achieved by hand computations, but very CPU-like. – Jack D'Aurizio Dec 5 '13 at 9:37 Some progess: The integrand actually decomposes as $$\frac{1}{2} \left( \frac{x^2 + 2x + 1}{x^6 + 4x^5 + 3x^4 - 4x^3 - 2x^2 + 2x + 1} + \frac{x^2 - 2x + 1}{x^6 - 4x^5 + 3x^4 + 4x^3 - 2x^2 - 2x + 1} \right).$$ Note that the second term is the same as the first term, except with $-x$ instead of $x$. Thus, with some substitutions, the integral becomes $$\frac{1}{2} \int_{-\infty}^\infty \frac{y^2}{y^6 - 2y^5 - 2y^4 + 4y^3 + 3y^2 - 4y + 1} \; dy.$$ - $$\int_{-\infty}^\infty \frac{y^2}{y^6 - 2y^5 - 2y^4 + 4y^3 + 3y^2 - 4y + 1} \; dy=\pi,$$ by Wolframalpha, so the result is correct. (I guess if you have a pro version then it is possible to obtain some details.) – vesszabo Jan 2 '13 at 20:57	2016-06-26 14:04:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9400440454483032, "perplexity": 183.28124838695197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395346.6/warc/CC-MAIN-20160624154955-00057-ip-10-164-35-72.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/efficiency-prediction-minimum-versus-average.952782/	# Efficiency prediction (minimum versus average) • I ## Main Question or Discussion Point I am trying to determine the efficiency of a light source for treating plants. When using the light source, there is an equal probability of encountering two different plant species. The light source has a certain electro-optical efficiency $\eta_{\text{eo}}$. And each plant type has a different receptiveness (i.e. absorbance) to the light $\eta_{\text{a}}$. So, the total efficiency of the process will be the product of the two efficiencies: $$\eta_{\text{total}}=\eta_{\text{eo}}.\eta_{\text{a}} \tag{1}$$ Some research papers that I have read predict the efficiency of such a system to be the minimum of the two efficiencies: $$\eta_{\text{total}}=\min\left(\eta_{\text{eo}}\eta_{\text{a}_{1}},\ \eta_{\text{eo}}\eta_{\text{a}_{2}}\right). \tag{2}$$ where the subscripts denote the plant type. However, the authors did not provide any statistical justifications for that. So, what could be the reasoning for this? Is it not better to use the average value instead? i.e., $$\eta_{\text{total}}=\eta_{\text{eo}}.\left(\frac{\eta_{\text{a}_{1}}+\eta_{\text{a}_{2}}}{2}\right).$$ Any explanations would be greatly appreciated. ## Answers and Replies Related Set Theory, Logic, Probability, Statistics News on Phys.org Dale Mentor Some research papers that I have read predict the efficiency of such a system to be the minimum of the two efficiencies: Can you link to the papers? mathman	2020-04-08 13:25:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8632012605667114, "perplexity": 648.8614841332789}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371813538.73/warc/CC-MAIN-20200408104113-20200408134613-00221.warc.gz"}
https://projecteuclid.org/euclid.aos/1176350618	## The Annals of Statistics ### Testing That a Stationary Time Series is Gaussian T. W. Epps #### Abstract A class of procedures is proposed for testing the composite hypothesis that a stationary stochastic process is Gaussian. Requiring very limited prior knowledge about the structure of the process, the tests rely on quadratic forms in deviations of certain sample statistics from their population counterparts, minimized with respect to the unknown parameters. A specific test is developed, which employs differences between components of the sample and Gaussian characteristic functions, evaluated at certain points on the real line. By demonstrating that, under $H_0$, the normalized empirical characteristic function converges weakly to a continuous Gaussian process, it is shown that the test remains valid when arguments of the characteristic functions are in certain ways data dependent. #### Article information Source Ann. Statist., Volume 15, Number 4 (1987), 1683-1698. Dates First available in Project Euclid: 12 April 2007 https://projecteuclid.org/euclid.aos/1176350618 Digital Object Identifier doi:10.1214/aos/1176350618 Mathematical Reviews number (MathSciNet) MR913582 Zentralblatt MATH identifier 0644.62093 JSTOR	2019-11-15 21:08:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39890363812446594, "perplexity": 857.1619906347454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668712.57/warc/CC-MAIN-20191115195132-20191115223132-00534.warc.gz"}
http://fds.duke.edu/db/aas/math/faculty/bendich/publications/303523	Department of Mathematics Search \| Help \| Login \| \| Math @ Duke ....................... ....................... Webpage ## Publications [#303523] of Paul L Bendich Papers Published 1. Bendich, P; Edelsbrunner, H; Morozov, D; Patel, A, Homology and robustness of level and interlevel sets, Homology, Homotopy and Applications, vol. 15 no. 1 (2013), pp. 51-72 [1102.3389v1], [doi] (last updated on 2018/10/23) Abstract: Given a function $f: \Xspace \to \Rspace$ on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of $f$. In addition, we quantify the robustness of the homology classes under perturbations of $f$ using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case $\Xspace = \Rspace^3$ has ramifications in the fields of medical imaging and scientific visualization. dept@math.duke.edu ph: 919.660.2800 fax: 919.660.2821 Mathematics Department Duke University, Box 90320 Durham, NC 27708-0320	2018-10-24 04:48:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42603179812431335, "perplexity": 1560.003408330159}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583519859.77/warc/CC-MAIN-20181024042701-20181024064201-00039.warc.gz"}
https://www.clutchprep.com/physics/practice-problems/141145/1-if-the-electric-potential-at-some-point-is-large-is-the-electric-field-at-that	# Problem: If the electric potential at some point is large, is the electric field at that point also necessarily large or not? Explain your answer, and provide a counterexample if not. 1. ###### FREE Expert Solution Relationship between the electric field and electric potential: $\overline{){\mathbf{E}}{\mathbf{=}}{\mathbf{-}}\frac{\mathbf{∆}\mathbf{V}}{\mathbf{∆}\mathbf{x}}}$ Electric field depend on the change in potential and not the magnitude of the electric potential. ###### Problem Details If the electric potential at some point is large, is the electric field at that point also necessarily large or not? Explain your answer, and provide a counterexample if not. 1.	2020-05-29 13:57:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8636856079101562, "perplexity": 328.78166161455044}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347404857.23/warc/CC-MAIN-20200529121120-20200529151120-00322.warc.gz"}
https://maria-titova.com/teaching/political-economy-elections/topic6-info-aggregation-elections/	# Does One Vote Matter? • suppose there are $$2$$ candidates $$D$$ and $$R$$; $$N$$ (odd) voters; simple majority • assume that everyone else votes $$D$$ with prob. $$1/2$$ and $$R$$ with prob. $$1/2$$ • how likely is it that your vote matters? • if $$N = 3$$, the possible vote outcomes for 2 other voters are • your vote is decisive if there's a tie ( others voted DR or RD), which happens with probability $$\frac{1}{2}$$ probability $$\frac{1}{4}$$ $$\frac{1}{4}$$ $$\frac{1}{4}$$ $$\frac{1}{4}$$ # Does One Vote Matter? • suppose there are $$2$$ candidates, $$N$$ (odd) voters, simple majority • how likely is it that your vote matters? • if $$N = 10$$, then $$prob(tie) = 0.246$$ • if $$N = 1,000$$, then $$prob(tie) = 0.0252$$ • if $$N = 10,000,000$$, then $$prob(tie) = 0.025%$$, or $$1$$ in $$4000$$ # Information Aggregation in Elections • how are elections at aggregating information? • democracy vs dictatorship, when there is uncertainty • common interest setting: a JURY • $$N$$ (odd) jurors charged with finding if defendant is $$Innocent$$ or $$Guilty$$ • every juror wants to acquit the $$Innocent$$ and convict the $$Guilty$$ • issue: it is entirely unclear whether the defendant is $$Innocent$$ or $$Guilty$$ # Information and Signals • unknown state of the world: (defendant is) $$Innocent$$ or $$Guilty$$ • common prior belief: $$prob(Innocent) = \frac{1}{2}$$, $$prob(Guilty) = \frac{1}{2}$$ • each juror receives a private signal: • value of signal is either $$g$$ or $$i$$ • signal is informative: • if $$Innocent$$, signal is $$i$$ with prob. $$p>\frac{1}{2}$$ and $$g$$ with prob. $$1-p < \frac{1}{2}$$ • if $$Guilty$$, signal is $$g$$ with prob. $$p>\frac{1}{2}$$ and $$i$$ with prob. $$1-p < \frac{1}{2}$$ # Learning from Signals • if you get signal $$g$$, what is probability that (defendant is) $$Guilty$$? $$prob(Guilty \| g) = \frac{prob(Guilty) \cdot prob(g \| Guilty) }{prob(g)} = \frac{\frac{1}{2}p}{\frac{1}{2}p + \frac{1}{2}(1-p)} = p$$ • if you get signal $$i$$, what is probability that (defendant is) $$Guilty$$? $$prob(Guilty \| i) = \frac{prob(Guilty) \cdot prob(i \| Guilty) }{prob(i)} = \frac{\frac{1}{2}(1-p)}{\frac{1}{2}(1-p) + \frac{1}{2}p} = 1-p$$ # Jury • each juror receives correct signal with prob. $$p>\frac{1}{2}$$ • each juror wants to convict the $$Guilty$$ and acquit the $$Innocent$$ • if you (a juror) receive signal $$g$$, you vote to CONVICT, because $$p>\frac{1}{2}$$ • if you (a juror) receive signal $$i$$, you vote ACQUIT $$p>\frac{1}{2}$$ • note that here we assume that juror vote sincerely (according to their signal only) # Condorcet Jury Theorem • $$N$$ jurors, simple majority, signal precision $$p$$; $$Q(N,p)$$ is probability of correct decision • Condorcet Jury Theorem: for any odd $$N$$ and any $$p > \frac{1}{2}$$ • $$Q(N,p) > p$$ • $$Q(N+2,p) > Q(N,p)$$ • $$Q(N,p) \to 1$$ as $$N \to +\infty$$ # Unanimity Rule • we looked at simple majority rule, how about unanimity rule? # Best Social Choice Function • Condorcet: simple majority is most efficient because it minimizes (across all SCF) total probability of an error • he's not wrong, simple majority does minimize total prob. of error: • simple majority: $$3p^2 - 2p^3$$ • unanimity: $$\frac{1}{2}\Big[ 1-(1-p)^3 \Big] + \frac{1}{2} p^3$$ # Connection to Hypothesis Testing • we can say that the jury is testing hypothesis that defendant is $$innocent$$ • type I error: innocent person goes to jail • false positive, null hypothesis is TRUE but it is rejected • type II error: guilty person walks free • false negative, null hypothesis is FALSE but it is not rejected # Simple Majority vs Unanimity: Summary • unanimity rule has lower type I error • fewer innocent people are found guilty (6.4% vs 35.2% when $$p = 0.6$$) • majority rule has lower type II error • fewer guilty people are found innocent (35.2% vs 78.4% when $$p = 0.6$$) • majority rule has lower total probability of error • $$\frac{1}{2} \text{Type I error} + \frac{1}{2} \text{Type II error}$$ (35.2% vs 42.6% when $$p = 0.6$$) • which rule is best? • unknown state of the world ($$innocent$$ or $$guilty$$) • each juror receives private signal • each juror votes according to signal # Strategic Voting: Simple Majority • same setup as before: $$3$$ jurors, $$p>\frac{1}{2}$$, simple majority • you are the "smartest person in the room" • you know other 2 jurors are not sophisticated enough to strategize and just vote according to their signal • you receive signal $$i$$ • what is your thought process? # Strategic Voting: Unanimity • same setup as before: $$3$$ jurors, $$p>\frac{1}{2}$$, unanimity • you are the "smartest person in the room" • you know other 2 jurors are not sophisticated enough to strategize and just vote according to their signal • you receive signal $$i$$ • what is your thought process? # Strategic Voting in Juries • under majority rule, there is an equilibrium with sincere voting • if everyone else votes sincerely, you do too • our assumption that voters vote according to signal is WITHOUT loss of generality \vspace{1cm} • under unanimity rule, there is NO equilibrium with sincere voting • if everyone votes sincerely, you do NOT want to also vote sincerely • our assumption that voters vote according to signal is WITH loss of generality • technically, we did not properly solve the game and juror's behavior is a lot more complicated (they will strategize) # Connection to Auction Theory • imagine you are in auction competing with two more people for an object • object's value is unknown but similar to every bidder • you win, how do you feel about it? # Swing Voter's Curse • similarly to winner's curse in auctions, we have a swing voter's curse in juries with unanimity rule: • if you naively follow you signal and then learn that everyone else voted to convict, then you would "curse" after realizing that your signal was most likely wrong and you spoiled a perfectly good conviction	2022-12-01 11:11:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5508432984352112, "perplexity": 5320.696109114645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710808.72/warc/CC-MAIN-20221201085558-20221201115558-00421.warc.gz"}
http://aimsciences.org/article/doi/10.3934/bdia.2016017	American Institue of Mathematical Sciences 2016, 1(4): 391-401. doi: 10.3934/bdia.2016017 On identifiability of 3-tensors of multilinear rank $(1,\ L_{r},\ L_{r})$ 1 Department of Computer Science, Southern Illinois University-Carbondale, Carbondale, IL 62901, USA 2 Department of Mathematics, Lamar University, Beaumont, TX 77710, USA 3 Department of Mathematics, Southern Illinois University-Carbondale, Carbondale, IL 62901, USA * Corresponding author: Qiang Cheng, Mingqing Xiao. This research is supported in part by NSF-DMS 1419028 and NSF-IIS 1218712 of United States * Corresponding author: Qiang Cheng, Mingqing Xiao. This research is supported in part by NSF-DMS 1419028 and NSF-IIS 1218712 of United States Revised May 2017 Published May 2017 In this paper, we study a specific big data model via multilinear rank tensor decompositions. The model approximates to a given tensor by the sum of multilinear rank $(1, \ L_{r}, \ L_{r})$ terms. And we characterize the identifiability property of this model from a geometric point of view. Our main results consists of exact identifiability and generic identifiability. The arguments of generic identifiability relies on the exact identifiability, which is in particular closely related to the well-known "trisecant lemma" in the context of algebraic geometry (see Proposition 2.6 in [1]). This connection discussed in this paper demonstrates a clear geometric picture of this model. Citation: Ming Yang, Dunren Che, Wen Liu, Zhao Kang, Chong Peng, Mingqing Xiao, Qiang Cheng. On identifiability of 3-tensors of multilinear rank $(1,\ L_{r},\ L_{r})$. Big Data & Information Analytics, 2016, 1 (4) : 391-401. doi: 10.3934/bdia.2016017 References: [1] L. Chiantini, C. Ciliberto, Weakly defective varieties, Transactions of the American Mathematical Society, 354 (2002), 151-178. doi: 10.1090/S0002-9947-01-02810-0. [2] L. Chiantini, G. Ottaviani, On generic identifiability of 3-tensors of small rank, SIAM Journal on Matrix Analysis and Applications, 33 (2012), 1018-1037. doi: 10.1137/110829180. [3] L. Chiantini, G. Ottaviani, N. Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1265-1287. doi: 10.1137/140961389. [4] A. Cichocki, D. Mandic, L. De Lathauwer, G. Zhou, Q. Zhao, C. Caiafa, H. Phan, Tensor decompositions for signal processing applications: From two-way to multiway component analysis, Signal Processing Magazine, IEEE, 32 (2015), 145-163. doi: 10.1109/MSP.2013.2297439. [5] P. Comon, Tensor decompositions, State of the art and applications, J. G. McWhirter and I. K. Proudler. Mathematics in Signal Processing V, Clarendon Press, Oxford, 71 (2002), 1-24. [6] L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅰ: Lemmas for partitioned matrices, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1022-1032. doi: 10.1137/060661685. [7] L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅱ: Definitions and uniqueness, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1033-1066. doi: 10.1137/070690729. [8] L. DeLathauwer, D. Nion, Decompositions of a higher-order tensor in block terms-part Ⅲ: Alternating least squares algorithms, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1067-1083. doi: 10.1137/070690730. [9] I. Domanov, L. DeLathauwer, Generic uniqueness of a structured matrix factorization and applications in blind source separation, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 701-711. doi: 10.1109/JSTSP.2016.2526971. [10] I. Domanov, L. De Lathauwer, Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL, SIAM Journal on Matrix Analysis and Applications, 36 (2015), 1567-1589. doi: 10.1137/140970276. [11] J. Draisma, J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Mathematical Journal, 163 (2014), 35-63. doi: 10.1215/00127094-2405170. [12] J. D. Hauenstein, A. J. Sommese, Witness sets of projections, Applied Mathematics and Computation, Elsevier, 217 (2010), 3349-3354. doi: 10.1016/j.amc.2010.08.067. [13] J. D. Hauenstein, A. J. Sommese, Membership tests for images of algebraic sets by linear projections, Applied Mathematics and Computation, Elsevier, 219 (2013), 6809-6818. doi: 10.1016/j.amc.2012.12.060. [14] R. Ke, W. Li, M. Xiao, Characterization of extreme points of multi-stochastic tensors, Computational Methods in Applied Mathematics, 16 (2016), 459-474. doi: 10.1515/cmam-2016-0005. [15] T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500. doi: 10.1137/07070111X. [16] J. M. Landsberg, Tensors: Geometry and Applications, AMS, Providence, Rhode Island, USA, 2012. [17] M. W. Mahoney, L.-H. Lim, G. E. Carlsson, Algorithmic and statistical challenges in modern largescale data analysis are the focus of MMDS 2008, ACM SIGKDD Explorations Newsletter, 10 (2008), 57-60. doi: 10.1145/1540276.1540294. [18] F. Malgouyres and J. Landsberg, Stable recovery of the factors from a deep matrix product and application to convolutional network, preprint, arXiv: 1703. 08044. [19] Y. Matsushima (E. Kobayashi, translator), Differentiable Manifolds, Marcel Dekker, Inc. , North-Holland Publishing Co. , North Miami Beach, FL, U. S. A. , 1972. [20] E. E. Papalexakis, C. Faloutsos, N. D. Sidiropoulos, Tensors for data mining and data fusion: Models, applications, and scalable algorithms, ACM Transactions on Intelligent Systems and Technology (TIST), 8 (2017), 1-44. doi: 10.1145/2915921. show all references References: [1] L. Chiantini, C. Ciliberto, Weakly defective varieties, Transactions of the American Mathematical Society, 354 (2002), 151-178. doi: 10.1090/S0002-9947-01-02810-0. [2] L. Chiantini, G. Ottaviani, On generic identifiability of 3-tensors of small rank, SIAM Journal on Matrix Analysis and Applications, 33 (2012), 1018-1037. doi: 10.1137/110829180. [3] L. Chiantini, G. Ottaviani, N. Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1265-1287. doi: 10.1137/140961389. [4] A. Cichocki, D. Mandic, L. De Lathauwer, G. Zhou, Q. Zhao, C. Caiafa, H. Phan, Tensor decompositions for signal processing applications: From two-way to multiway component analysis, Signal Processing Magazine, IEEE, 32 (2015), 145-163. doi: 10.1109/MSP.2013.2297439. [5] P. Comon, Tensor decompositions, State of the art and applications, J. G. McWhirter and I. K. Proudler. Mathematics in Signal Processing V, Clarendon Press, Oxford, 71 (2002), 1-24. [6] L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅰ: Lemmas for partitioned matrices, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1022-1032. doi: 10.1137/060661685. [7] L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅱ: Definitions and uniqueness, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1033-1066. doi: 10.1137/070690729. [8] L. DeLathauwer, D. Nion, Decompositions of a higher-order tensor in block terms-part Ⅲ: Alternating least squares algorithms, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1067-1083. doi: 10.1137/070690730. [9] I. Domanov, L. DeLathauwer, Generic uniqueness of a structured matrix factorization and applications in blind source separation, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 701-711. doi: 10.1109/JSTSP.2016.2526971. [10] I. Domanov, L. De Lathauwer, Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL, SIAM Journal on Matrix Analysis and Applications, 36 (2015), 1567-1589. doi: 10.1137/140970276. [11] J. Draisma, J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Mathematical Journal, 163 (2014), 35-63. doi: 10.1215/00127094-2405170. [12] J. D. Hauenstein, A. J. Sommese, Witness sets of projections, Applied Mathematics and Computation, Elsevier, 217 (2010), 3349-3354. doi: 10.1016/j.amc.2010.08.067. [13] J. D. Hauenstein, A. J. Sommese, Membership tests for images of algebraic sets by linear projections, Applied Mathematics and Computation, Elsevier, 219 (2013), 6809-6818. doi: 10.1016/j.amc.2012.12.060. [14] R. Ke, W. Li, M. Xiao, Characterization of extreme points of multi-stochastic tensors, Computational Methods in Applied Mathematics, 16 (2016), 459-474. doi: 10.1515/cmam-2016-0005. [15] T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500. doi: 10.1137/07070111X. [16] J. M. Landsberg, Tensors: Geometry and Applications, AMS, Providence, Rhode Island, USA, 2012. [17] M. W. Mahoney, L.-H. Lim, G. E. Carlsson, Algorithmic and statistical challenges in modern largescale data analysis are the focus of MMDS 2008, ACM SIGKDD Explorations Newsletter, 10 (2008), 57-60. doi: 10.1145/1540276.1540294. [18] F. Malgouyres and J. Landsberg, Stable recovery of the factors from a deep matrix product and application to convolutional network, preprint, arXiv: 1703. 08044. [19] Y. Matsushima (E. Kobayashi, translator), Differentiable Manifolds, Marcel Dekker, Inc. , North-Holland Publishing Co. , North Miami Beach, FL, U. S. A. , 1972. [20] E. E. Papalexakis, C. Faloutsos, N. D. Sidiropoulos, Tensors for data mining and data fusion: Models, applications, and scalable algorithms, ACM Transactions on Intelligent Systems and Technology (TIST), 8 (2017), 1-44. doi: 10.1145/2915921. [1] H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50 [2] Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 [3] Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2/3 ) : 139-161. doi: 10.3934/bdia.2016001 [4] Gabriele Link, Jean-Claude Picaud. Ergodic geometry for non-elementary rank one manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6257-6284. doi: 10.3934/dcds.2016072 [5] Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 [6] Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018 [7] Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014 [8] Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741 [9] Mostafa Karimi, Noor Akma Ibrahim, Mohd Rizam Abu Bakar, Jayanthi Arasan. Rank-based inference for the accelerated failure time model in the presence of interval censored data. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 107-112. doi: 10.3934/naco.2017007 [10] Nick Cercone, F'IEEE. What's the big deal about big data?. Big Data & Information Analytics, 2016, 1 (1) : 31-79. doi: 10.3934/bdia.2016.1.31 [11] Roman Vodička, Vladislav Mantič. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. doi: 10.3934/dcdss.2017079 [12] Mariantonia Cotronei, Tomas Sauer. Full rank filters and polynomial reproduction. Communications on Pure & Applied Analysis, 2007, 6 (3) : 667-687. doi: 10.3934/cpaa.2007.6.667 [13] Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239 [14] Michael Blank. Finite rank approximations of expanding maps with neutral singularities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 749-762. doi: 10.3934/dcds.2008.21.749 [15] John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019 [16] Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841 [17] Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081 [18] Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773 [19] Pankaj Sharma, David Baglee, Jaime Campos, Erkki Jantunen. Big Data Collection and Analysis for Manufacturing Organisations. Big Data & Information Analytics, 2017, 2 (2) : 127-139. doi: 10.3934/bdia.2017002 [20] Enrico Capobianco. Born to be big: Data, graphs, and their entangled complexity. Big Data & Information Analytics, 2016, 1 (2/3 ) : 163-169. doi: 10.3934/bdia.2016002 Impact Factor:	2018-01-23 23:23:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5871363878250122, "perplexity": 4056.364767595209}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084892802.73/warc/CC-MAIN-20180123231023-20180124011023-00152.warc.gz"}