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# 14  Graphics for regression models

In Chapter 2 of the ModernDive text introduces what they call the “five named graphs”: scatterplots, linegraphs, histograms, boxplots, barplots. Each of these graphical modalities—except maybe boxplots—are familiar to many quantitatively literate readers. That’s a good reason for using them in presentations to a general audience.

In the rest of this course, we will focus not on such general presentations, but on the contemporary statistical thinking skills needed to extract valid information from data. Our main analytic technique for this will be regression modeling. The graphics we will generate will often be used to display results from regression models. This means we need to use graphic modalities that are a good match for regression models.

The central feature of a regression model is that there is one response variable and one or more explanatory variables. The graphics frame we use will always have the response variable on the vertical axis. The horizontal axis will always be one of the explanatory variables; the choice depends on our interest.

To illustrate, consider the familiar settings of height, age, and sex. The data plotted in Figure 14.1 come from surveys conducted in 2009-2012. (That turns out to be important. See Lesson 30.) Figure 14.1 places the height variable on the vertical axis. The horizontal axis is age, and color specifies sex.

Code
ForMod <- NHANES::NHANES |>
select(Height, Age, Gender) |>
filter(Age >=15) |>
rename(Sex = Gender)
Mod2 <-  lm(Height ~ splines::ns(Age,3)*Sex, data = ForMod)
model_plot(Mod2, x = Age, color=Sex,
interval="confidence", show_data = FALSE) |>
gf_jitter(Height ~ Age, data=ForMod |> filter(Age >= 15), alpha=0.05, inherit=FALSE, color=~ Sex) |>
gf_labs(title="Height vs age and sex")

The point plot is the base layer of Figure 14.1. A model of height as a function of age and sex has been layered on top, so that you can see how the model relates to the data. The layer showing the model is more of a ribbon than a line plot. You will see the significance of this in later Lessons.

At their best, models show patterns in data that are not readily seen by eye. Figure 14.2 gives an example. Only the variables age and sex are plotted, age is the explanatory variable and sex is the response variable.

Code
Small <- NHANES::NHANES %>%
select(Height, Age, Gender, Poverty, HealthGen, Depressed) %>%
filter(Age >= 15) |>
rename(Sex = Gender) |>
mutate(Sex01 = zero_one(Sex, one="female"))
Mod3 <- glm(Sex01 ~ splines::ns(Age,4), data = Small, family="binomial")
model_plot(Mod3, x=Age, show_data=FALSE, interval="confidence") |>
gf_jitter(Sex01 ~ Age, color=~Sex, data=Small, inherit=FALSE,
height = 0.1, width=1, size=0, alpha=0.5) +
scale_y_continuous(breaks=c(0, .5, 1), labels=c("male (0)", 0.5, "female (1)")) +
ylab("Proportion of females") +
guides(color="none")

The graphics style of Figure 14.2 is important. Often, we will work with categorical response variables that have two levels: yes-or-no, alive-or-dead, success-or-failure, promoted-or-not, and so on. Such “binomial” variables (bi=two, nomial=name) can always be encoded as zero/one quantitative variables. In Figure 14.2, the raw data is shown as a jittered point plot. Each dot in the lower band of the graph corresponds to a male; the points in the upper band are for females. In the middle, the fitted model of sex versus age is plotted.

Because sex has been re-coded as a zero-one numerical value, we can use regression modeling techniques, a powerful tool. The output from the model can be interpreted as a probability (or, what is the same thing: a proportion). The blue-colored model snaking across the ages gives the proportion of females at each age. For 20-year olds, that proportion is about 50%. Starting about age 60, however, the proportion of females starts increasing. At age 80 (and above), females make up about 60% of the data points.

Underlying the use of regression modeling of binomial response variable is the transformation of the variable into a zero-one format. In Figure 14.2, this was accomplished by adding a new column (mutate() does this), like this:

data_frame %>% mutate(sex01 = zero_one(sex, one="female"))

## Violins for density

Histograms and box-and-whisker plots are often seen in presentations. They were invented to facilitate computation and pencil-and-ruler drawing, which, with the advent of computer graphics, are no longer important constraints on graphics. The violin plot is a modern alternative to both histograms (which are too visually busy and highlight random fluctuations) and box-and-whisker plots (which are unnecessarily chunky and suitable only for single-peaked distribution).

When the explanatory variable is categorical, violins make it straightforward to compare distributions and are easy to draw with the geom_violin() function.

Code
Small %>% select(HealthGen, Poverty) %>% na.omit() %>%
ggplot(aes(x=HealthGen, y=Poverty)) +
geom_jitter(size=.1, width=0.15, alpha=0.3) +
geom_violin(fill="blue", alpha=0.5, color=NA) +
xlab("Self-assessed health condition") +
ylab("Income (multiple of poverty level)")

You can think of the violin as a sideways histogram drawn with no bars and with mirror-image symmetry.

## The Null model

Occasionally, we will have reason to build models without any explanatory variables. Or, more precisely, models with only one explanatory variable whose value is the same in all rows, signified by 1. In terms of a tilde expression, such models look like income ~ 1. The left-hand side of the tilde expression is, as always, the response variable.

Graphically, the lack of an explanatory variable means that the horizontal axis is used only to provide space for jittering, violins, and other graphical annotations that extend horizontally. Tick marks on the horizontal axis are meaningless. Figure 14.4 shows the same data as Figure 14.3, but without being broken down by health condition.

Code
Stats <- df_stats(Poverty ~ 1, Small, quantile(c(0.33, 0.66)))
Small %>% select(Poverty) %>%
ggplot(aes(x=1, y=Poverty)) +
geom_jitter(size=.1, width=0.1, alpha=0.3) +
#  geom_boxplot(aes(x=1.5), width=0.1, color="green", fill=NA, alpha=1) +
geom_violin(fill="blue", alpha=0.3, color=NA) +
geom_errorbar(aes(ymin=33%, ymax=66%), y=NA,
color="magenta", linewidth=1, width=0.05,
data = Stats) +
xlab("") +
ylab("Income (multiple of poverty level)")

The violin plot shows a clear bimodal distribution, which we know from Figure 14.3 comes from the different distributions for people in different health conditions. Just for illustration, the $$\color{magenta}{\text{magenta}}$$ line segment, drawn as an “error bar,” shows the middle third of the distribution.


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# How do you Find two rational and irrational nos. between 2.1 and 2.11?

Wiki User

2017-04-17 10:27:35

For two rational numbers select any terminating or repeating decimal number which starts with 2.10 and for Irrational Numbers you require a non-terminating, non-repeating decimal which also starts with 2.10.

Wiki User

2017-04-21 12:36:50
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## hio

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Wiki User

2017-05-05 18:33:26

To the smaller of the two starting numbers, add a rational number thatis sufficiently small (to get a rational number), or add an irrational number (to get an irrational number. You can construct irrational numbers of any size, by dividing them by a rational number. For example, if you add 2.1 + pi, the result is outside the bounds specified; but instead of adding pi, you can add pi/10, pi/100, etc.

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• # question_answer Which of the following statements is correct about equivalent weight of $KMn{{O}_{4}}$ [MP PET 1994] A) It is one third of its molecular weight in alkaline medium B) It is one fifth of its molecular weight in alkaline medium C) It is equal to its molecular weight in acidic medium D) It is one third of its molecular weight in acidic medium

Correct Answer: A

Solution :

It is one third of its molecular weight in alkaline medium because it gives 3 nascent oxygen in alkaline medium. $2KMn{{O}_{4}}+{{H}_{2}}O\,\xrightarrow{+3{{e}^{-}}}\,2KOH+2Mn{{O}_{2}}+3[O]$ \ Equivalent weight = $\frac{M}{3}$

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## Results (1-50 of at least 1000)

Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim arith $\mathbb{Q}$ self-dual $\operatorname{Arg}(\epsilon)$ $r$ First zero Origin
2-1280-5.3-c0-0-1 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 5.3 $$0.0 0 0.0881 0 1.77841 Artin representation 2.1280.8t17.d.b Modular form 1280.1.p.b.513.1 2-1280-5.3-c0-0-0 0.799 0.638 2 2^{8} \cdot 5 5.3$$ $0.0$ $0$ $0.0881$ $0$ $1.37751$ Artin representation 2.1280.8t17.c.b Modular form 1280.1.p.a.513.1
2-1280-5.2-c0-0-1 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 5.2 $$0.0 0 -0.0881 0 1.20109 Artin representation 2.1280.8t17.d.a Modular form 1280.1.p.b.257.1 2-1280-5.2-c0-0-0 0.799 0.638 2 2^{8} \cdot 5 5.2$$ $0.0$ $0$ $-0.0881$ $0$ $0.983393$ Artin representation 2.1280.8t17.c.a Modular form 1280.1.p.a.257.1
2-1280-40.37-c0-0-1 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 40.37 $$0.0 0 0.286 0 1.86337 Artin representation 2.1280.8t17.a.a Modular form 1280.1.m.a.897.1 2-1280-40.37-c0-0-0 0.799 0.638 2 2^{8} \cdot 5 40.37$$ $0.0$ $0$ $0.0368$ $0$ $1.59728$ Artin representation 2.1280.8t17.b.a Modular form 1280.1.m.b.897.1
2-1280-40.19-c0-0-1 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 40.19 $$0.0 0 0.125 0 1.64330 Artin representation 2.1280.8t11.b.b Modular form 1280.1.e.a.639.1 2-1280-40.19-c0-0-0 0.799 0.638 2 2^{8} \cdot 5 40.19$$ $0.0$ $0$ $-0.125$ $0$ $1.12985$ Artin representation 2.1280.8t11.b.a Modular form 1280.1.e.a.639.2
2-1280-40.13-c0-0-1 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 40.13 $$0.0 0 -0.0368 0 1.37436 Artin representation 2.1280.8t17.b.b Modular form 1280.1.m.b.1153.1 2-1280-40.13-c0-0-0 0.799 0.638 2 2^{8} \cdot 5 40.13$$ $0.0$ $0$ $-0.286$ $0$ $0.525413$ Artin representation 2.1280.8t17.a.b Modular form 1280.1.m.a.1153.1
2-1280-20.19-c0-0-5 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 20.19 $$0.0 0 0.250 0 1.94926 Artin representation 2.1280.8t11.a.b Modular form 1280.1.h.b.1279.1 2-1280-20.19-c0-0-4 0.799 0.638 2 2^{8} \cdot 5 20.19$$ $0.0$ $0$ $0$ $0$ $1.72793$ Artin representation 2.1280.8t6.d.a Modular form 1280.1.h.c.1279.2
2-1280-20.19-c0-0-3 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 20.19 $$0.0 0 0 0 1.65679 Artin representation 2.1280.8t6.c.a Modular form 1280.1.h.a.1279.2 2-1280-20.19-c0-0-2 0.799 0.638 2 2^{8} \cdot 5 20.19$$ $0.0$ $0$ $0$ $0$ $1.26909$ Artin representation 2.1280.8t6.d.b Modular form 1280.1.h.c.1279.1
2-1280-20.19-c0-0-1 $0.799$ $0.638$ $2$ $2^{8} \cdot 5$ 20.19 $$0.0 0 -0.250 0 0.757739 Artin representation 2.1280.8t11.a.a Modular form 1280.1.h.b.1279.2 2-1280-20.19-c0-0-0 0.799 0.638 2 2^{8} \cdot 5 20.19$$ $0.0$ $0$ $0$ $0$ $0.715177$ Artin representation 2.1280.8t6.c.b Modular form 1280.1.h.a.1279.1
2-1280-1.1-c1-0-23 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 1.62671 Modular form 1280.2.a.m.1.1 2-1280-1.1-c1-0-24 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.66452$ Modular form 1280.2.a.p.1.2
2-1280-1.1-c1-0-25 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 1.80824 Modular form 1280.2.a.i.1.1 2-1280-1.1-c1-0-26 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0.5$ $1$ $2.01582$ Modular form 1280.2.a.d.1.2
2-1280-1.1-c1-0-27 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 2.01795 Modular form 1280.2.a.e.1.2 2-1280-1.1-c1-0-28 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0.5$ $1$ $2.04343$ Modular form 1280.2.a.a.1.2
2-1280-1.1-c1-0-29 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 2.05913 Modular form 1280.2.a.c.1.2 2-1280-1.1-c1-0-3 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $0.697567$ Modular form 1280.2.a.n.1.1
2-1280-1.1-c1-0-30 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 2.17471 Modular form 1280.2.a.i.1.2 2-1280-1.1-c1-0-31 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0.5$ $1$ $2.33456$ Modular form 1280.2.a.m.1.2
2-1280-1.1-c1-0-4 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0 0 0.847474 Modular form 1280.2.a.f.1.1 2-1280-1.1-c1-0-5 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $0.847842$ Modular form 1280.2.a.b.1.1
2-1280-1.1-c1-0-6 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0 0 0.852935 Modular form 1280.2.a.k.1.1 2-1280-1.1-c1-0-7 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $0.939645$ Modular form 1280.2.a.l.1.1
2-1280-1.1-c1-0-8 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0 0 0.995301 Modular form 1280.2.a.p.1.1 2-1280-1.1-c1-0-9 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.00543$ Modular form 1280.2.a.f.1.2
2-1280-20.3-c1-0-18 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 20.3 $$1.0 1 0.0493 0 0.934081 Modular form 1280.2.n.r.1023.2 2-1280-1.1-c1-0-1 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $0.547532$ Modular form 1280.2.a.g.1.1
2-1280-1.1-c1-0-10 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0 0 1.00700 Modular form 1280.2.a.b.1.2 2-1280-1.1-c1-0-11 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.07419$ Modular form 1280.2.a.o.1.1
2-1280-1.1-c1-0-12 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0 0 1.23737 Modular form 1280.2.a.k.1.2 2-1280-1.1-c1-0-13 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.28485$ Modular form 1280.2.a.g.1.2
2-1280-1.1-c1-0-14 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 1.34206 Modular form 1280.2.a.a.1.1 2-1280-1.1-c1-0-15 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.37191$ Modular form 1280.2.a.n.1.2
2-1280-1.1-c1-0-16 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 1.38644 Modular form 1280.2.a.c.1.1 2-1280-1.1-c1-0-17 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.41539$ Modular form 1280.2.a.o.1.2
2-1280-1.1-c1-0-18 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0 0 1.44674 Modular form 1280.2.a.l.1.2 2-1280-1.1-c1-0-19 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.45358$ Modular form 1280.2.a.j.1.2
2-1280-1.1-c1-0-2 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0 0 0.641103 Modular form 1280.2.a.j.1.1 2-1280-1.1-c1-0-20 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0$ $0$ $1.46548$ Modular form 1280.2.a.h.1.2
2-1280-1.1-c1-0-21 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 1.1 $$1.0 1 0.5 1 1.49867 Modular form 1280.2.a.d.1.1 2-1280-1.1-c1-0-22 3.19 10.2 2 2^{8} \cdot 5 1.1$$ $1.0$ $1$ $0.5$ $1$ $1.58507$ Modular form 1280.2.a.e.1.1
2-1280-20.7-c1-0-41 $3.19$ $10.2$ $2$ $2^{8} \cdot 5$ 20.7 $$1.0 1 -0.411 1 1.98380 Modular form 1280.2.n.d.767.1 2-1280-20.7-c1-0-42 3.19 10.2 2 2^{8} \cdot 5 20.7$$ $1.0$ $1$ $-0.315$ $0$ $2.13458$ Modular form 1280.2.n.s.767.1


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# Erlang tool -- recon

recon 是ferd 大神 释出的一个 用于生产环境诊断Erlang 问题的一个工具, 不仅仅是对Erlang stdlib 接口的封装, 还有memory fragmentation 相关的函数.

## CPU 统计相关

The reduction count has a direct link to function calls in Erlang, and a high count is usually the synonym of a high amount of CPU usage.

What’s interesting with this function is to try it while a system is already rather busy, with a relatively short interval. Repeat it many times, and you should hopefully see a pattern emerge where the same processes (or the same kind of processes) tend to always come up on top.

Using the code locations and current functions being run, you should be able to identify what kind of code hogs all your schedulers.

1> recon:proc_window(reductions, 3, 500).

## Memory Leaks

memory leaks 主要是 refc binary, 这一点主要是和binary 的内存结构有关, 之前写的一篇blog  有提到这个.

Once you’ve established you’ve got a binary memory leak using recon:bin_leak(Max) , it should be simple enough to look at the top processes and see what they are and what kind of work they do.
Generally, refc binaries memory leaks can be solved in a few different ways, depending on the source:

• call garbage collection manually at given intervals (icky, but somewhat efficient);

• stop using binaries (often not desirable);

• use binary:copy/1-210 if keeping only a small fragment (usually less than 64 bytes) of a larger binary;

• move work that involves larger binaries to temporary one-off processes that will die when they’re done (a lesser form of manual GC!);

• or add hibernation calls when appropriate (possibly the cleanest solution for inactive processes).

The first two options are frankly not agreeable and should not be attempted before all else failed. The last three options are usually the best ones to be used.

## 总结

recon 是个实际操作性很强的工具, 没有实际的使用案例, 很难说得清楚它的妙用.

posted on 2015-04-01 21:35  _00  阅读(1269)  评论(1编辑  收藏


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# Solution: 2020-15 The number of cycles of fixed lengths in random permutations

Let $$m_0=n$$. For each $$i\geq 0$$, choose a number $$x_i$$ in $$\{1,\dots, m_i\}$$ uniformly at random and let $$m_{i+1}= m_i – x_i$$. This gives a random vector $$\mathbf{x}=(x_1,x_2, \dots)$$. For each $$1\leq k\leq n$$, let $$X_k$$ be the number of occurrences of $$k$$ in the vector $$\mathbf{x}$$.

For each $$1\leq k\leq n$$, let $$Y_k$$ be the number of cycles of length $$k$$ in a permutation of $$\{1,\dots, n\}$$ chosen uniformly at random. Prove that $$X_k$$ and $$Y_k$$ have the same distribution.

The best solution was submitted by 이준호 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-15.

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# 8. Entropy Source¶

Each BSP must provide an implementation of the getentropy() system call. This system call was introduced by OpenBSD and is also available in glibc since version 2.25. This system call is used by the Newlib provided ARC4RANDOM(3) functions, which in turn are used by various cryptographic functions.

Warning

A good entropy source is critical for (nearly) all cryptographic applications. The default implementation based on the CPU counter is not suitable for such applications.

The getentropy() implementation must fill the specified memory region of the given size with random numbers and return 0 on success. A non-zero return may cause the INTERNAL_ERROR_ARC4RANDOM_GETENTROPY_FAIL internal error by one of the ARC4RANDOM(3) functions.

In general, for embedded systems it is not easy to get some real entropy. Normally, that can only be reached with some extra hardware support. Some microcontrollers integrate a true random number generator or something similar for cryptographic applications. That is the preferred source of entropy for most BSPs. For example the atsam BSP uses the TRNG for its entropy source.

There is also a quite limited default implementation based on the CPU counter. Due to the fact that it is a time based source, the values provided by getentropy() are quite predictable. This implementation is not appropriate for any cryptographic applications but it is good enough for some basic tasks. Use it only if you do not have any strong requirements on the entropy and if there is no better source.


text:
# passing present viewmodel to included page. [closed]

nsharma
917 1 11

Hello,here is a weird situation, and guess i have tried everything what i think should have worked.

look below the code first ,in this i have taken a groupbox and a div in vlayout,In div their is a include component,I want to pass the present viewModel object(vm) to the viewmodel of this included page.

    <window mode="modal" popup="true" width="1000px" height="635px"
closable="true"
apply="org.zkoss.bind.BindComposer" border="normal"
position="center,top"
viewModel="@id('vm') @init('xxx.ViewModel')">
<vlayout>
<groupbox width="100%" mold="3d" height="280px"
closable="false">
<caption>
............
</groupbox>
<div>
<include   src="@bind(vm.path)"
</include>
</div>
</vlayout>

</window>
</zk>

included page viewmodel code:

   @AfterCompose
public void afterCompose(@ExecutionParam("viewModelArg") ViewModel viewModel  ){}

initially somehow it was working ,but as i have changed some code inside my project(don't have any idea about the change,but its not relative as much i know to this code),Now it has stopped working.and gives a null value to viewModel.

The situation is that ,we can have only string type value and for non string-typed their is no clear documentation about how to retrieve the value in included page viewmodel.I get a NULL everytime. Thanks in advance.

delete retag edit


text:
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# On Legendrian dual surfaces of a spacelike curve in the $3$-dimensional lightcone

## Handan YildirimUniversity of Istanbul

In this talk which is based on the joint work with Kentaro Saji given in [3], taking into account the Legendrian dualities in [2] which are extensions of the Legendrian dualities in [1], we first introduce new extended Legendrian dualities for the 3-dimensional pseudo-spheres of various radii in Lorentz-Minkowski 4-space. Secondly, by connecting all of these Legendrian dualities continuously, we construct Legendrian dual surfaces (lying in these 3-dimensional pseudo-spheres) of a spacelike curve in the 3-dimensional lightcone. Finally, we investigate the singularities of these surfaces and show the dualities of the singularities of a certain class of such a surface in the 3-dimensional lightcone.
[1] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Moscow Mathematical Journal, 9 (2009), 325-357.
[2] S. Izumiya, H. Yildirim, Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space, Topology and its Applications, 159(2012), 509-518.
[3] K. Saji, H. Yildirim, Legendrian dual surfaces of a spacelike curve in the 3-dimensional lightcone, Journal of Geometry and Physics, 104593, https://doi.org/10.1016/j.geomphys.2022.104593

# On the canonical contact structure of the space of null geodesics of a spacetime: the role of Engel geometry in dimension 3

The space of null geodesics of a spacetime (a Lorentzian manifold with a choice of future) sometimes has the structure of a smooth manifold. When this is the case, it comes equipped with a canonical contact structure. I will introduce the theory for a countable number of metrics on the product $S^2\times S^1$. Motivated by these examples, I will comment on how Engel geometry can be used to describe the manifold of null geodesics, by considering the Cartan deprolongation of the Lorentz prolongation of the spacetime. This allows us to characterize the 3-contact manifolds which are spaces of null geodesics, and to retrieve the spacetime they come from. This is joint work with R. Rubio.

# Critical embeddings for the first eigenvalue of the Laplacian

## Eduardo LongaUniversidade de Sao Paulo

The eigenvalues of the Laplace-Beltrami operator on a closed Riemannian manifold are very natural geometric invariants. Although in many problems the Riemannian structure is kept fixed, the eigenvalues can be seen as functionals in the space of metrics. This is the suitable setting for the calculus of variations. In this vein, El Soufi and Ilias have characterised the metrics which are critical for the first eigenvalue among all metrics of fixed volume and among all metrics of fixed volume in a conformal class. In the talk, I will prove a similar characterisation for some critical metrics which are induced by embeddings into a fixed Riemannian manifold.

# Minimal surfaces as an interdisciplinary topic

## Hao ChenUniversity of Göttingen

All my works on minimal surfaces has been motivated or inspired by natural sciences, including material sciences, bio-membranes, fluid dynamics, etc. I will give an informal talk (since I’m not natural scientist) about how minimal surface theory could benefit from interdisciplinary interactions.

## Hao ChenUniversity of Göttingen

Traizet’s node opening technique has been very powerful to construct minimal surfaces. In fact, it was first applied to glue saddle towers into minimal surfaces. But for technical reasons, the construction has much room to improve. I will talk about the ongoing project that addresses to various technical problems in the gluing construction. In particular, careful treatment of Dehn twist has revealed very subtle interactions between saddle towers.

# Triply periodic minimal surfaces

## Hao ChenUniversity of Göttingen

Triply periodic minimal surfaces (TPMSs) are minimal surfaces in flat 3-tori. I will review recent discoveries of new examples of TPMSs and outline future steps towards an eventual complete classification of TPMSs of genus 3.

# Geometric Modelling of nonlinear dynamics processes by fractal approximation methods

## Olha ZalevskaNational Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”

Small variations of nonlinear dynamics by means of geometric modeling are investigated. Visualization of the dynamic process is reproduced using three-dimensional cellular automata. The dependences of the transition positions of the dynamic system between the chaotic and stable states are established. Theoretical aspects of geometric fractal approximation of the dynamic process, criteria for establishing the stable state of the system, geometric fractal derivatives and integration are considered.

# Hyperbolic domains in real Euclidean spaces

## Franc ForstneričUniversity of Ljubljana

In a recent joint work with David Kalaj (2021), we introduced a new Finsler pseudometric on any domain in the real Euclidean space $\mathbb R^n$ for $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with the Kobayashi pseudometric on complex manifolds. This "minimal pseudometric" describes the maximal rate of growth of hyperbolic conformal minimal surfaces in a given domain. On the unit ball, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric. I will discuss sufficient geometric conditions for a domain to be (complete) hyperbolic, meaning that its minimal pseudometric is a (complete) metric.

# The mathematical theory of globally hyperbolic supersymmetric configurations in supergravity

I will give a pedagogical introduction to the incipient mathematical theory of globally hyperbolic supersymmetric configurations in four-dimensional Lorentzian supergravity. First, I will introduce the basics of supergravity in four dimensions as well as the notion of globally supersymmetric configuration as a solution to a system of first-order spinorial differential equations, that is, the supergravity spinorial equations. Then, I will introduce the theory of spinorial polyforms associated with bundles of irreducible real Clifford modules, which provides a convenient geometric framework to study the global geometric and topological properties of the solutions to the supergravity spinorial equations. Then, I will consider the evolution problem for globally hyperbolic supersymmetric configurations, focusing on the constraint equations, their moduli of solutions, and the construction of explicit evolution flows, which we reformulate as the supergravity flow equations for a coupled family of functions and global co-frames on a Cauchy hypersurface. This will lead us to explore in detail the case of (possibly Einstein) globally hyperbolic Lorentzian four-manifolds equipped with a parallel or Killing spinor, obtaining several results about the differentiable topology and geometry of such manifolds. Finally, I will mention several open problems and open directions for future research.

# Sharp differential inequalities for the isoperimetric profile in spaces with Ricci lower bounds

## Gioacchino AntonelliScuola Normale Superiore di Pisa

In this talk I will discuss sharp differential inequalities for the isoperimetric profile function in spaces with Ricci bounded from below, and with volumes of unit balls uniformly bounded from below. After that, I will highlight some of the consequences of such inequalities for the isoperimetric problem. After a short introduction about the notion of perimeter in the metric measure setting, I will pass to the motivation and statement of the sharp differential inequalities on Riemannian manifolds. Hence, I will discuss the proof, which builds on a non smooth generalized existence theorem for the isoperimetric problem (after Ritoré-Rosales, and Nardulli), and on a non smooth sharp Laplacian comparison theorem for the distance function from isoperimetric boundaries (after Mondino-Semola). At the end I will discuss how to use such differential inequalities to study the behaviour of the isoperimetric profile for small volumes. This talk is based on some results that recently appeared in a work in collaboration with E. Pasqualetto, M. Pozzetta, and D. Semola. Some of the tools and ideas exploited for the proofs come from other works in collaboration with E. Bruè, M. Fogagnolo, and S. Nardulli.

# Una demostración del teorema de Gauss-Bonnet evitando triangulaciones

El objetivo de la charla es explorar la geometría compleja de superficies (orientables y compactas) para obtener una demostración del teorema de Gauss-Bonnet de forma intrínseca. Este tipo de abordaje es una buena oportunidad para introducir estructuras geométricas que se aplican en una multitud de áreas fuera de la Geometría Diferencial Clásica (e.g. Geometrías Compleja y Simpléctica); además, no se necesita ninguna familiaridad con la topología de las variedades (algo esencial al utilizar triangulaciones), solamente hace falta una buena noción de cálculo con varias variables.

Sala de Conferencias (IMAG)

»

# XXI Geometrical Seminar

»
The Geometrical Seminar started its activities more than 30 years ago under the name Yugoslav Geometrical Seminar. The aim of these meetings is to bring together mathematicians and physicists interested in geometry and its applications, to give lectures on new results, exchange ideas, problems and conjectures.
The 21st will be held in June 2021.

»

»

»

# Workshop on Surface Theory —UY60—

## Tokio (Japón) & Virtual

»
Congreso en honor de Umehara y Yamada por su 60 cumpleaños


text:
# The income I of a computer analyst varies directly as the number of hours h worked. If the analyst earns $384 for working 8 h, how much will the analyst earn for working 22 h?$

Question

The income I of a computer analyst varies directly as the number of hours h worked. If the analyst earns $384 for working 8 h, how much will the analyst earn for working 22 h?$


text:
# zbMATH — the first resource for mathematics

## Namikawa, Yoshinori

Compute Distance To:
 Author ID: namikawa.yoshinori Published as: Namikawa, Yoshinori; Namikawa, Y.
 Documents Indexed: 34 Publications since 1991
all top 5

#### Co-Authors

 29 single-authored 2 Lehn, Manfred 2 Sorger, Christoph 1 Fu, Baohua 1 Kawamata, Yujiro 1 Steenbrink, Joseph H. M. 1 van Straten, Duco
all top 5

#### Serials

 3 Compositio Mathematica 3 Inventiones Mathematicae 3 Journal für die Reine und Angewandte Mathematik 3 Mathematische Annalen 2 Duke Mathematical Journal 2 Journal of Mathematics of Kyoto University 2 Topology 2 Journal of Algebraic Geometry 2 Kyoto Journal of Mathematics 1 Advances in Mathematics 1 Annales de l’Institut Fourier 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 Sūgaku 1 International Journal of Mathematics 1 Journal of Mathematical Sciences. University of Tokyo
all top 5

#### Fields

 31 Algebraic geometry (14-XX) 17 Several complex variables and analytic spaces (32-XX) 10 Differential geometry (53-XX) 5 Nonassociative rings and algebras (17-XX) 2 Category theory; homological algebra (18-XX) 2 Manifolds and cell complexes (57-XX) 1 Topological groups, Lie groups (22-XX) 1 Global analysis, analysis on manifolds (58-XX)

#### Citations contained in zbMATH Open

29 Publications have been cited 366 times in 244 Documents Cited by Year
Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Zbl 0848.14004
Kawamata, Yujiro; Namikawa, Yoshinori
1994
Deformation theory of singular symplectic $$n$$-folds. Zbl 0989.53055
Namikawa, Yoshinori
2001
Smoothing Fano 3-folds. Zbl 0906.14019
Namikawa, Yoshinori
1997
Extension of 2-forms and symplectic varieties. Zbl 0996.53050
Namikawa, Yoshinori
2001
Global smoothing of Calabi-Yau threefolds. Zbl 0861.14036
Namikawa, Yoshinori; Steenbrink, J. H. M.
1995
Flops and Poisson deformations of symplectic varieties. Zbl 1148.14008
Namikawa, Yoshinori
2008
Counter-example to global Torelli problem for irreducible symplectic manifolds. Zbl 1028.53081
Namikawa, Yoshinori
2002
On deformations of $$\mathbb Q$$-factorial symplectic varieties. Zbl 1122.14029
Namikawa, Yoshinori
2006
Mukai flops and derived categories. Zbl 1033.18008
Namikawa, Yoshinori
2003
Poisson deformations of affine symplectic varieties. Zbl 1208.14028
Namikawa, Yoshinori
2011
Birational geometry of symplectic resolutions of nilpotent orbits. Zbl 1117.14018
Namikawa, Yoshinori
2006
On deformations of Calabi-Yau 3-folds with terminal singularities. Zbl 0813.14004
Namikawa, Yoshinori
1994
Uniqueness of crepant resolutions and symplectic singularities. Zbl 1063.14018
Fu, Baohua; Namikawa, Yoshinori
2004
Mukai flops and derived categories. II. Zbl 1086.14011
Namikawa, Yoshinori
2004
Projectivity criterion of Moishezon spaces and density of projective symplectic varieties. Zbl 1055.32015
Namikawa, Yoshinori
2002
A characterization of nilpotent orbit closures among symplectic singularities. Zbl 1423.53103
Namikawa, Yoshinori
2018
Induced nilpotent orbits and birational geometry. Zbl 1173.22011
Namikawa, Yoshinori
2009
Poisson deformations of affine symplectic varieties. II. Zbl 1211.14040
Namikawa, Yoshinori
2010
Stratified local moduli of Calabi-Yau threefolds. Zbl 1072.14523
Namikawa, Yoshinori
2002
Deformation theory of Calabi-Yau threefolds and certain invariants of singularities. Zbl 0957.14029
Namikawa, Yoshinori
1997
Slodowy slices and universal Poisson deformations. Zbl 1234.14004
Lehn, Manfred; Namikawa, Yoshinori; Sorger, Christoph
2012
Poisson deformations and birational geometry. Zbl 1334.14021
Namikawa, Yoshinori
2015
Equivalence of symplectic singularities. Zbl 1277.32029
Namikawa, Yoshinori
2013
Birational geometry and deformations of nilpotent orbits. Zbl 1140.14004
Namikawa, Yoshinori
2008
On symplectic hypersurfaces. Zbl 1376.32034
Lehn, Manfred; Namikawa, Yoshinori; Sorger, Christoph; van Straten, Duco
2016
On the birational structure of certain Calabi-Yau threefolds. Zbl 0736.14018
Namikawa, Yoshinori
1991
On the structure of homogeneous symplectic varieties of complete intersection. Zbl 1285.14051
Namikawa, Yoshinori
2013
Fundamental groups of symplectic singularities. Zbl 1392.32013
Namikawa, Yoshinori
2017
A finiteness theorem on symplectic singularities. Zbl 1400.14030
Namikawa, Yoshinori
2016
A characterization of nilpotent orbit closures among symplectic singularities. Zbl 1423.53103
Namikawa, Yoshinori
2018
Fundamental groups of symplectic singularities. Zbl 1392.32013
Namikawa, Yoshinori
2017
On symplectic hypersurfaces. Zbl 1376.32034
Lehn, Manfred; Namikawa, Yoshinori; Sorger, Christoph; van Straten, Duco
2016
A finiteness theorem on symplectic singularities. Zbl 1400.14030
Namikawa, Yoshinori
2016
Poisson deformations and birational geometry. Zbl 1334.14021
Namikawa, Yoshinori
2015
Equivalence of symplectic singularities. Zbl 1277.32029
Namikawa, Yoshinori
2013
On the structure of homogeneous symplectic varieties of complete intersection. Zbl 1285.14051
Namikawa, Yoshinori
2013
Slodowy slices and universal Poisson deformations. Zbl 1234.14004
Lehn, Manfred; Namikawa, Yoshinori; Sorger, Christoph
2012
Poisson deformations of affine symplectic varieties. Zbl 1208.14028
Namikawa, Yoshinori
2011
Poisson deformations of affine symplectic varieties. II. Zbl 1211.14040
Namikawa, Yoshinori
2010
Induced nilpotent orbits and birational geometry. Zbl 1173.22011
Namikawa, Yoshinori
2009
Flops and Poisson deformations of symplectic varieties. Zbl 1148.14008
Namikawa, Yoshinori
2008
Birational geometry and deformations of nilpotent orbits. Zbl 1140.14004
Namikawa, Yoshinori
2008
On deformations of $$\mathbb Q$$-factorial symplectic varieties. Zbl 1122.14029
Namikawa, Yoshinori
2006
Birational geometry of symplectic resolutions of nilpotent orbits. Zbl 1117.14018
Namikawa, Yoshinori
2006
Uniqueness of crepant resolutions and symplectic singularities. Zbl 1063.14018
Fu, Baohua; Namikawa, Yoshinori
2004
Mukai flops and derived categories. II. Zbl 1086.14011
Namikawa, Yoshinori
2004
Mukai flops and derived categories. Zbl 1033.18008
Namikawa, Yoshinori
2003
Counter-example to global Torelli problem for irreducible symplectic manifolds. Zbl 1028.53081
Namikawa, Yoshinori
2002
Projectivity criterion of Moishezon spaces and density of projective symplectic varieties. Zbl 1055.32015
Namikawa, Yoshinori
2002
Stratified local moduli of Calabi-Yau threefolds. Zbl 1072.14523
Namikawa, Yoshinori
2002
Deformation theory of singular symplectic $$n$$-folds. Zbl 0989.53055
Namikawa, Yoshinori
2001
Extension of 2-forms and symplectic varieties. Zbl 0996.53050
Namikawa, Yoshinori
2001
Smoothing Fano 3-folds. Zbl 0906.14019
Namikawa, Yoshinori
1997
Deformation theory of Calabi-Yau threefolds and certain invariants of singularities. Zbl 0957.14029
Namikawa, Yoshinori
1997
Global smoothing of Calabi-Yau threefolds. Zbl 0861.14036
Namikawa, Yoshinori; Steenbrink, J. H. M.
1995
Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Zbl 0848.14004
Kawamata, Yujiro; Namikawa, Yoshinori
1994
On deformations of Calabi-Yau 3-folds with terminal singularities. Zbl 0813.14004
Namikawa, Yoshinori
1994
On the birational structure of certain Calabi-Yau threefolds. Zbl 0736.14018
Namikawa, Yoshinori
1991
all top 5

#### Cited by 255 Authors

 14 Namikawa, Yoshinori 9 Fu, Baohua 7 Hanany, Amihay 6 Bellamy, Gwyn 5 Markman, Eyal 5 Mongardi, Giovanni 5 Schedler, Travis 4 Cautis, Sabin 4 Kaloghiros, Anne-Sophie 4 Kamnitzer, Joel 4 Kapustka, Michał 4 Lee, Namhoon 4 Losev, Ivan V. 4 Meachan, Ciaran 4 Morrison, David R. 4 Prokhorov, Yuriĭ Gennad’evich 4 Rossi, Michele 4 Siebert, Bernd 4 Steenbrink, Joseph H. M. 3 Greb, Daniel 3 Harder, Andrew 3 Hassett, Brendan 3 Kapustka, Grzegorz 3 Kebekus, Stefan 3 Lehn, Christian 3 Licata, Anthony M. 3 Mekareeya, Noppadol 3 Menet, Grégoire 3 Nakayama, Chikara 3 Peternell, Thomas Martin 3 Sano, Taro 3 Tschinkel, Yuri 3 Tziolas, Nikolaos 3 Wiśńiewski, Jarosław Antoni 2 Addington, Nicolas M. 2 Ambrosio, Filippo 2 Andreatta, Marco 2 Bakker, Benjamin 2 Boissière, Samuel 2 Cabrera, Santiago 2 Claudon, Benoît 2 Donovan, Will 2 Doran, Charles F. 2 Druel, Stéphane 2 Etingof, Pavel Il’ich 2 Filip, Matej 2 Fu, Lie 2 Fujisawa, Taro 2 Graf, Patrick 2 Grassi, Antonella 2 Gross, Mark 2 Hara, Wahei 2 Jahnke, Priska 2 Kaledin, Dmitry B. 2 Kalveks, Rudolph 2 Karmazyn, Joseph 2 Karzhemanov, Ilya 2 Kato, Fumiharu 2 Kato, Kazuya 2 Keel, Sean 2 Lai, Ching-Jui 2 Lehn, Manfred 2 Leung, Naichung Conan 2 Lin, Hsueh-Yung 2 Liu, Kefeng 2 Manivel, Laurent 2 Matsushita, Daisuke 2 Petracci, Andrea 2 Proudfoot, Nicholas J. 2 Radloff, Ivo 2 Ran, Ziv 2 Schröer, Stefan 2 Sorger, Christoph 2 Terpereau, Ronan 2 Tevelev, Jenia 2 Thompson, Alan Matthew 2 Tian, Zhiyu 2 Uehara, Hokuto 2 Verbitsky, Misha 2 Wang, Sz-Sheng 1 Achmed-Zade, Ismail 1 Ahmadinezhad, Hamid 1 Amerik, Ekaterina Yu. 1 Anderson, Lara B. 1 Anno, Rina 1 Arakawa, Tomoyuki 1 Arras, Philipp 1 Bayer, Arend 1 Becker, Tanja 1 Beem, Christopher 1 Bondal, Alexey I. 1 Buczyńska, Weronika 1 Căldăraru, Andrei 1 Campana, Frédéric 1 Cao, Yalong 1 Casagrande, Cinzia 1 Cascini, Paolo 1 Chacaltana, Oscar 1 Cheltsov, Ivan Anatol’evich 1 Chen, Meng ...and 155 more Authors
all top 5

#### Cited in 76 Serials

 19 Advances in Mathematics 16 Mathematische Zeitschrift 13 Journal of High Energy Physics 12 Duke Mathematical Journal 12 Journal of Algebraic Geometry 9 Compositio Mathematica 9 Mathematische Annalen 9 International Journal of Mathematics 7 Inventiones Mathematicae 5 Journal of Algebra 5 Journal für die Reine und Angewandte Mathematik 5 Manuscripta Mathematica 5 Journal de Mathématiques Pures et Appliquées. Neuvième Série 4 Annales de l’Institut Fourier 4 Geometriae Dedicata 4 Selecta Mathematica. New Series 4 Central European Journal of Mathematics 4 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 4 Kyoto Journal of Mathematics 3 Communications in Algebra 3 Nuclear Physics. B 3 Journal of Geometry and Physics 3 Journal of Pure and Applied Algebra 3 Proceedings of the American Mathematical Society 3 Transformation Groups 3 Geometry & Topology 3 Journal of the Institute of Mathematics of Jussieu 3 Science China. Mathematics 2 Communications in Mathematical Physics 2 Mathematical Notes 2 Publications Mathématiques 2 Journal of the London Mathematical Society. Second Series 2 Journal of the Mathematical Society of Japan 2 Mathematische Nachrichten 2 Nagoya Mathematical Journal 2 Proceedings of the Edinburgh Mathematical Society. Series II 2 Rendiconti del Seminario Matematico della Università di Padova 2 Tohoku Mathematical Journal. Second Series 2 Transactions of the American Mathematical Society 2 Journal of the European Mathematical Society (JEMS) 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 International Journal of Modern Physics A 1 Russian Mathematical Surveys 1 Glasgow Mathematical Journal 1 Hokkaido Mathematical Journal 1 Journal of Differential Geometry 1 Journal of the Korean Mathematical Society 1 Journal of Number Theory 1 Kodai Mathematical Journal 1 Michigan Mathematical Journal 1 Osaka Journal of Mathematics 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 Topology and its Applications 1 Sugaku Expositions 1 Differential Geometry and its Applications 1 Geometric and Functional Analysis. GAFA 1 Expositiones Mathematicae 1 Annales Mathématiques Blaise Pascal 1 Izvestiya: Mathematics 1 Mathematical Physics, Analysis and Geometry 1 Revista Matemática Complutense 1 Annals of Mathematics. Second Series 1 Acta Mathematica Sinica. English Series 1 Bulletin of the Brazilian Mathematical Society. New Series 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 International Journal of Geometric Methods in Modern Physics 1 Bulletin of the Institute of Mathematics. Academia Sinica. New Series 1 Proceedings of the Steklov Institute of Mathematics 1 Annali dell’Università di Ferrara. Sezione VII. Scienze Matematiche 1 Frontiers of Mathematics in China 1 Electronic Research Announcements in Mathematical Sciences 1 Forum of Mathematics, Pi 1 Forum of Mathematics, Sigma 1 Research in the Mathematical Sciences 1 Journal de l’École Polytechnique – Mathématiques 1 Transactions of the American Mathematical Society. Series B
all top 5

#### Cited in 20 Fields

 208 Algebraic geometry (14-XX) 61 Several complex variables and analytic spaces (32-XX) 49 Differential geometry (53-XX) 27 Nonassociative rings and algebras (17-XX) 21 Quantum theory (81-XX) 13 Category theory; homological algebra (18-XX) 12 Group theory and generalizations (20-XX) 10 Associative rings and algebras (16-XX) 5 Number theory (11-XX) 5 Commutative algebra (13-XX) 5 Topological groups, Lie groups (22-XX) 5 Global analysis, analysis on manifolds (58-XX) 3 Convex and discrete geometry (52-XX) 3 Manifolds and cell complexes (57-XX) 3 Relativity and gravitational theory (83-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 $$K$$-theory (19-XX) 1 Partial differential equations (35-XX) 1 Algebraic topology (55-XX) 1 Biology and other natural sciences (92-XX)


text:
## Saturday, January 21, 2012

### Vector fun

In Marsden & Tromba's Vector Calculus, I found the following problem. It involves time-derivatives of the position vector r(t), for which we're using the dot notation of the physicists (and Newton).

In the first panel, we have Newton's second law, then the statement that the force is minus the gradient of the gravitational potential V, followed by a calculation of the total energy in the system. The problem is to compute d/dt of the energy. (hint)

The answer is given in the last panel, below. According to the book, it's a "simple calculation."

The first step is to compute d/dt of the kinetic energy. We use the formula from above, plus a trick to convert the squared term back to the dot product of dr/dt with itself. Then we use the chain rule, and finally, the definition of the force in terms of the gradient of the potential.

Calculation of d/dt of the potential energy puzzled me for quite a while, though it really shouldn't have. My solution was to work backward from the answer, as shown.

We put the two results together, and notice that they cancel. Surprise!


text:
# Choice of trace for irreducible polynomial

say i have irreducible polynomial of $x^2 + \lambda x + 1$ for the field of $GF(2^4)$.
what are the possible value for $\lambda$


text:
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 Please use this identifier to cite or link to this item: http://hdl.handle.net/1807/24844

 Title: A Mass Spectrometry Approach to Ligand Identification for Orphan Fly and Human Nuclear Receptors Authors: Pardee, Keith Ian Advisor: Edwards, Aled M. Department: Molecular and Medical Genetics Keywords: nuclear receptororphanE75Rev-erbhemenitric oxidecrystal structureDHR3DHR96cholesterolmetabolismligandDrosophilahumanmass spectrometry Issue Date: 1-Sep-2010 Abstract: The nuclear receptor superfamily is responsible for regulating the expression of genes involved in development, reproduction and metabolism. These transcription factors control the expression of their target genes through the binding of small molecule regulators to their ligand binding domains. Classical nuclear receptors include the steroid receptors, which bind endocrine hormones and have been important targets of pharmaceutical intervention. However, approximately one half of the human nuclear receptors remain orphans and are without known cognate ligands. Focusing on the Drosophila orthologues of these orphan receptors, this project used mass spectrometry to identify the chemical diversity associated with the receptors following expression in recombinant systems. In a genome-wide screen of Drosophila nuclear receptors, this approach identified co-purifying molecules with a number of receptors. The physiological relevance of these putative ligand/receptor pairs was determined through biochemical analysis, in vivo characterization and structure determination. Ligand(s) or the ligand state was identified for the Drosophila receptors: DHR3, DHR96, E75, Ftz-f1 and USP. Of these, three were validated through the efforts of this project, and independent groups confirmed the remaining two. The most significant findings were the discoveries that the fly nuclear receptor E75 is regulated by heme, gas and redox, and that there is a similar regulatory scheme in the human orthologues, Reverbα and β. Furthermore, crystallization of the heme-bound Rev-erbβ ligand binding domain was also achieved, and this provided key insights into the mechanism of ligand regulation for the Rev-erbs. This project highlighted the role of nuclear receptors in metabolic surveillance. The ligands/signals identified in association with these receptors include: cholesterol, dehydrocholesterol, heme, NO, CO, redox and phospholipids. Unlike the classical steroid hormones, these are not dedicated signaling molecules, but instead are key substrates or products of metabolism. In the context of nuclear receptor signaling, I hypothesize that these metabolites serve as metabolic indicators/signals in the regulation of development and metabolism. Furthermore, four of these Drosophila receptors comprise the ecdysone-response pathway in the developing fly. Taken together, this suggests that both the metabolic state of the organism and steroid hormones drive nuclear receptor regulation of development. URI: http://hdl.handle.net/1807/24844 Appears in Collections: DoctoralDepartment of Molecular Genetics - Doctoral theses

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text:
# Recap GLMs

Generalization of linear models that allow to choose freely

2. which data generating distribution is assumed (data distribution).

We covered two models:

• Log link and Gaussian distribution (ozone~temp)
• Logit link and binomial distribution (mortality~concentration)

In R, GLMs are constructed using the glm() function.

# The glm Function

Generalized linear models can be fitted in R using the glm function, which is similar to the lm function for fitting linear models. The arguments to a glm call are as follows

glm(formula, family = gaussian, data, weights, subset, na.action, start = NULL, etastart, mustart, offset, control = glm.control(...), model = TRUE, method = ”glm.fit”, x = FALSE, y = TRUE, contrasts = NULL, ...)

The family argument takes (the name of) a family function which specifies

• the variance function

The exponential family functions available in R are

• binomial(link = "logit")
• gaussian(link = "identity")
• Gamma(link = "inverse")
• inverse.gaussian(link = "1/mu2")
• poisson(link = "log")

# Today’s session

GLMs to analyze:

• Binary response data with logit link and binomial distribution (known as logistic regression)
• Count data with log link and Poisson distribution

# Logistic regression

We today use a dataset where the response can only take two values. The data indicates whether trees were infested by a bark beetle or not, depending on the temperature anomaly (z-transformed temperatures with values > 0 indicating hotter than average):

dat <- read.csv("Data/infestation.csv")
head(dat)
##   infested temp_anomaly
## 1        1   -1.0785729
## 2        1    1.2728951
## 3        1    1.8699789
## 4        1    0.6327722
## 5        0    0.3342014
## 6        1    0.8407184
ggplot(dat, aes(x = temp_anomaly, y = infested)) +
geom_point()

Similar problems include: dead/alive, presence/absence, conversion/conservation, etc.

We want to model the probability of infestation given a certain temperature anomaly. The logit link seems well suited as it is bounded between 0 and 1 and alows for a quick transition between both extremes:

The regression thus can be written as:

$$Pr(infestation = 1) = logit^{−1}(\beta_0 + \beta_1 * temp)) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 * temp)}}$$

In our case the data is binomial distributed. Hence, we can relate the probability of infestation modelled through the logit link to our data using a binomial distribution with one trial, which translates into the Bernoulli distribution (a special case of the binomial distribution with only two outcomes):

$$infestation \sim Bernoulli(logit^{−1}(\beta_0 + \beta_1 * temp)))$$

This model is called binomial logistic regression, there are also multinomial (three or more outcomes) and ordinal logistic regression (ordinal outcomes).

Fitting a logistic regression model to our data:

infested.bin <- glm(infested ~ temp_anomaly, data = dat, family = binomial(link = "logit"))

And plotting the regression line:

prediction <- data.frame(temp_anomaly = seq(-2, 2, length.out = 100))
prediction$upper_link <- prediction$fitted_link + 2 * pred$se.fit prediction$link <- plogis(prediction$fitted_link) prediction$lower <- plogis(prediction$lower_link) prediction$upper <- plogis(prediction$upper_link) And add them to the plot: ggplot() + geom_point(data = dat, aes(x = temp_anomaly, y = infested)) + geom_ribbon(data = prediction, aes(x = temp_anomaly, ymin = lower, ymax = upper), alpha = 0.3) + geom_line(data = prediction, aes(x = temp_anomaly, y = fitted)) ## effects package Alternatively, we can use the effect() function, which does the transformation automatically: library(effects) efct <- effect("temp_anomaly", mod = infested.bin, xlevels = list(temp_anomaly = seq(-2, 2, length.out = 100))) efct <- as.data.frame(efct) ggplot() + geom_point(data = dat, aes(x = temp_anomaly, y = infested)) + geom_ribbon(data = efct, aes(x = temp_anomaly, ymin = lower, ymax = upper), alpha = 0.3) + geom_line(data = efct, aes(x = temp_anomaly, y = fit)) ## dose.p() We can estimate the temperature anomaly value at which the infestation probability is 50% using the dose.p() from the MASS package. library(MASS) temp_anomaly_p50 <- dose.p(infested.bin, p=0.5) temp_anomaly_p50 ## Dose SE ## p = 0.5: -0.3744093 0.2372779 ## Interpretation It is not straightforward to interpret coefficients in logistic regression, because the change in y given a one unit change in x is non-linear (though linear on the link scale!). However, there are - besides plotting - some ways of making the regression coefficients interpretable. The divide by 4 rule by Gelman and Hill (2007) takes the fact that the the logistic curve is steepest at its center (which is zero given scaled predictors). As a rough estimate, we can take logistic regression coefficients and divide them by 4 to get an upper bound of the predictive difference corresponding to a unit difference in x. This upper bound is a good approximation near the midpoint of the logistic curve, where probabilities are close to 0.5. Gelmann and Hill (2007): Data Analysis Using Regression and Multilevel/Hierarchical Models (page 82). In the our example, the regression coefficient was 2.18. Dividing 2.18 by 4 yields 0.54. We can thus say that with an increase in temperature anomaly by one standard deviation, there is an expected increase in infestation probability by no more than 54%. summary(infested.bin) ## ## Call: ## glm(formula = infested ~ temp_anomaly, family = binomial(link = "logit"), ## data = dat) ## ## Deviance Residuals: ## Min 1Q Median 3Q Max ## -1.8638 -0.6022 0.1966 0.4441 1.8595 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) 0.8154 0.5766 1.414 0.15734 ## temp_anomaly 2.1778 0.7809 2.789 0.00529 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for binomial family taken to be 1) ## ## Null deviance: 40.381 on 29 degrees of freedom ## Residual deviance: 23.488 on 28 degrees of freedom ## AIC: 27.488 ## ## Number of Fisher Scoring iterations: 5 ## Odds ratios Another way to interpret logistic regression coefficients is in terms of odds ratios. If a success has the probability $$p$$ and failure the probability of $$(1 − p)$$, then $$p/(1 − p)$$ is called the odds. Thus, the odds of success are defined as the ratio of the probability of success over the probability of failure. An odds of 1 is equivalent to a probability of 0.5, that is equally likely outcomes (1:1). An odds of 0.5 is equivalent to a probability of 0.33, indicating there is a 1:2 chance of success. An odds of 2 is equivalent to a probability of 0.66, indicating a 2:1 chance of success. • The ratio of two odds is called an odds ratio: $$\frac{(p1/(1 − p1))}{(p2/(1 − p2))}$$. • The odds ratio indicates the preference of outcome A over outcome B. Example: • We observed a plant species in 6 out of 10 plots in landscape A and in 4 out of 10 plots in landscape B. • This gives us probabilities of 0.6 and 0.4. The odds ratio of observing the plant in landscape A over observing it in landscape B is (0.6/0.4) / (0.4/0.6) = 2.25. • Hence, there is a 2.25 times higher chance of observing the plant in landscape A compared to landscape B. • The logistic regression coefficients give the change in log odds (recall the link function ln(p/1-p)). • When exponentiated, logistic regression coefficients can be interpreted as odds ratios: $$OR = \frac{odds(x + 1)}{odds(x)} = \frac{e^{\beta_0 + \beta_1 * (x + 1)}}{\beta_0 + \beta_1 * x} = e^{\beta_1}$$ • For continuous predictors, the coefficients give the change in the log odds of the outcome for a one unit increase in the predictor variable. • For categorical predictors, the coefficient estimate describes the change in the log odds for each level compared to the base level. ## Investation example Infestation example: exp(coef(infested.bin)[2]) ## temp_anomaly ## 8.826474 The odds ratio indicates that with a one SD change in temperature anomaly there is a 8.83 times higher chance of infestation over no infestation. # Poisson regression The following data set shows bird counts over a topographic relief gradient: dat <- read.table("Data/bird_counts.txt", sep = ";", header = TRUE) head(dat) ## relief Y ## 1 0.576 129 ## 2 0.576 166 ## 3 0.576 137 ## 4 0.576 105 ## 5 0.576 155 ## 6 0.576 178 ggplot(dat, aes(x = relief, y = Y)) + geom_point() ggplot(dat, aes(x = relief, y = Y)) + scale_y_continuous(trans='log') + geom_point() The data is bounded to positive values, a log-link is thus recommended: $$counts_{expected} = log^{-1}(\beta_0 + \beta_1 * relief)$$ As the data can only take natural numbers (0, 1, 2, …), we need a discrete data distrubution. For count models, the Poisson distribution is a good choice, yielding the final model of: $$counts \sim Poisson(log^{-1}(\beta_0 + \beta_1 * relief))$$ bird.poisson <- glm(Y ~ relief, data = dat, family = poisson(link = "log")) summary(bird.poisson)  ## ## Call: ## glm(formula = Y ~ relief, family = poisson(link = "log"), data = dat) ## ## Deviance Residuals: ## Min 1Q Median 3Q Max ## -11.1980 -2.1407 -0.0021 2.1986 10.7204 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) 4.665888 0.021577 216.24 <2e-16 *** ## relief 0.339559 0.008382 40.51 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for poisson family taken to be 1) ## ## Null deviance: 2921.1 on 60 degrees of freedom ## Residual deviance: 1299.6 on 59 degrees of freedom ## AIC: 1739.7 ## ## Number of Fisher Scoring iterations: 4 ## Dispersion • Dispersion quantifies, how much higher (or lower) the observed variance is relative to the expected variance of the model. • For the Poisson distribution the expected value is equal to the variance, i.e. the variance increases 1:1 with the mean value. Hence, dispersion = 1. • In othe words, when we fit a Poisson model, we expect the variance to increase with the mean value by a factor of 1. • If the variance decreases at a lower rate than the data are underdispered • If the variance decreases at a higher rate than the data are overdispersed. ## Consequences of overdispersion With correct mean model we have consistent estimates of the parameters but overdispersion leads to: • to underestimation of the standard errors of the parameter estimates. • selection of overly complex models. ## Overdispersion To get a rough estimate on overdispersion, divide the residual deviance by the residual degree’s of freedom. The ratio should be below 1. Values much larger (> 2) indicate overdispersion (or underdispersion if less than 0.6). ## Residual deviance: 1299.6 on 59 degrees of freedom summary(bird.poisson)$deviance / summary(bird.poisson)\$df.residual
## [1] 22.02709

Our model appears to be overdispered, indicating:

2. the variation of Y is greater than predicted by the model

Remedy:

2. Change optimization (quasi-likelihood)

# quasi-Poisson model

For a quasi-poisson regression, the variance is assumed to be a linear function of the mean; for negative binomial regression, a quadratic function.

bird.quasip <- glm(Y ~ relief, data = dat, family = quasipoisson)
summary(bird.quasip)
##
## Call:
## glm(formula = Y ~ relief, family = quasipoisson, data = dat)
##
## Deviance Residuals:
##      Min        1Q    Median        3Q       Max
## -11.1980   -2.1407   -0.0021    2.1986   10.7204
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  4.66589    0.09953  46.878  < 2e-16 ***
## relief       0.33956    0.03866   8.782 2.64e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for quasipoisson family taken to be 21.27853)
##
##     Null deviance: 2921.1  on 60  degrees of freedom
## Residual deviance: 1299.6  on 59  degrees of freedom
## AIC: NA
##
## Number of Fisher Scoring iterations: 4

# Parameter estimates

The coefficients of the Poisson model can be interpreted easily, as the coefficients are multiplicative when backtransformed to the response scale. That is, with a one unit increase in x, the count is multiplied by the exponent of the coefficient. For our example, we expect a 1.4 times (or 40%) increase in mean bird counts with a one unit increase in relief. You can use confint() to compute confidence intervals for the parameters.

# Poisson
coeffs <- exp(coef(bird.poisson))
coeffs_ci <- exp(confint(bird.poisson))
cbind(coeffs, coeffs_ci)
##                 coeffs      2.5 %     97.5 %
## (Intercept) 106.259866 101.846790 110.836066
## relief        1.404328   1.381446   1.427589
# quasi-Poisson
coeffs <- exp(coef(bird.quasip))
coeffs_ci <- exp(confint(bird.quasip))
cbind(coeffs, coeffs_ci)
##                 coeffs     2.5 %     97.5 %
## (Intercept) 106.259866 87.187307 128.808562
## relief        1.404328  1.301813   1.514927

# $$R^2_{pseudo}$$

Extending the Linear Model with R, Julian J. Faraway (p. 59):

$$R^2_{pseudo} = 1-\frac{Residual\ deviance}{Null\ deviance}$$

##     Null deviance: 2921.1  on 60  degrees of freedom
## Residual deviance: 1299.6  on 59  degrees of freedom

1 - 1299.6/2921.1
## [1] 0.5550991

# Fit model with effects package

efct <- effect("relief", mod = bird.quasip, xlevels = list(relief = seq(0, 4, length.out = 100)))
efct <- as.data.frame(efct)

p <- ggplot() +
geom_point(data = dat, aes(x = relief, y = Y)) +
geom_ribbon(data = efct, aes(x = relief, ymin = lower, ymax = upper), alpha = 0.3) +
geom_line(data = efct, aes(x = relief, y = fit))
print(p)

# Plot fitted values with ggplot

library(gridExtra)
plot1 <- ggplot(dat, aes(x = relief, y = Y)) +
geom_point() +
geom_smooth(method = "glm", method.args=list(family = "poisson"), col = "red")

plot2 <- ggplot(dat, aes(x = relief, y = Y)) +
geom_point() +
geom_smooth(method = "glm", method.args=list(family = "quasipoisson"), col = "red")

grid.arrange(plot1, plot2, ncol = 2)
## geom_smooth() using formula 'y ~ x'
## geom_smooth() using formula 'y ~ x'


text:
# Homegeneous Ensembles

MLJ.EnsembleModelFunction
EnsembleModel(atom=nothing,
atomic_weights=Float64[],
bagging_fraction=0.8,
n=100,
rng=GLOBAL_RNG,
acceleration=default_resource(),
out_of_bag_measure=[])

Create a model for training an ensemble of n learners, with optional bagging, each with associated model atom. Ensembling is useful if fit!(machine(atom, data...)) does not create identical models on repeated calls (ie, is a stochastic model, such as a decision tree with randomized node selection criteria), or if bagging_fraction is set to a value less than 1.0, or both. The constructor fails if no atom is specified.

Only atomic models supporting targets with scitype AbstractVector{<:Finite} (univariate classifiers) or AbstractVector{<:Continuous} (univariate regressors) are supported.

If rng is an integer, then MersenneTwister(rng) is the random number generator used for bagging. Otherwise some AbstractRNG object is expected.

The atomic predictions are weighted according to the vector atomic_weights (to allow for external optimization) except in the case that atom is a Deterministic classifier. Uniform atomic weights are used if weight has zero length.

The ensemble model is Deterministic or Probabilistic, according to the corresponding supertype of atom. In the case of deterministic classifiers (target_scitype(atom) <: Abstract{<:Finite}), the predictions are majority votes, and for regressors (target_scitype(atom)<: AbstractVector{<:Continuous}) they are ordinary averages. Probabilistic predictions are obtained by averaging the atomic probability distribution/mass functions; in particular, for regressors, the ensemble prediction on each input pattern has the type MixtureModel{VF,VS,D} from the Distributions.jl package, where D is the type of predicted distribution for atom.

The acceleration keyword argument is used to specify the compute resource (a subtype of ComputationalResources.AbstractResource) that will be used to accelerate/parallelize ensemble fitting.

If a single measure or non-empty vector of measures is specified by out_of_bag_measure, then out-of-bag estimates of performance are written to the trainig report (call report on the trained machine wrapping the ensemble model).

Important: If sample weights w (as opposed to atomic weights) are specified when constructing a machine for the ensemble model, as in mach = machine(ensemble_model, X, y, w), then w is used by any measures specified in out_of_bag_measure that support sample weights.

source


text:
• ### Recommendations of the LHC Dark Matter Working Group: Comparing LHC searches for heavy mediators of dark matter production in visible and invisible decay channels(1703.05703)

March 17, 2017 hep-ph, hep-ex, astro-ph.CO
Weakly-coupled TeV-scale particles may mediate the interactions between normal matter and dark matter. If so, the LHC would produce dark matter through these mediators, leading to the familiar "mono-X" search signatures, but the mediators would also produce signals without missing momentum via the same vertices involved in their production. This document from the LHC Dark Matter Working Group suggests how to compare searches for these two types of signals in case of vector and axial-vector mediators, based on a workshop that took place on September 19/20, 2016 and subsequent discussions. These suggestions include how to extend the spin-1 mediated simplified models already in widespread use to include lepton couplings. This document also provides analytic calculations of the relic density in the simplified models and reports an issue that arose when ATLAS and CMS first began to use preliminary numerical calculations of the dark matter relic density in these models.
• This document outlines a set of simplified models for dark matter and its interactions with Standard Model particles. It is intended to summarize the main characteristics that these simplified models have when applied to dark matter searches at the LHC, and to provide a number of useful expressions for reference. The list of models includes both s-channel and t-channel scenarios. For s-channel, spin-0 and spin-1 mediation is discussed, and also realizations where the Higgs particle provides a portal between the dark and visible sectors. The guiding principles underpinning the proposed simplified models are spelled out, and some suggestions for implementation are presented.
• ### Recommendations on presenting LHC searches for missing transverse energy signals using simplified $s$-channel models of dark matter(1603.04156)

March 14, 2016 hep-ph, hep-ex
This document summarises the proposal of the LHC Dark Matter Working Group on how to present LHC results on $s$-channel simplified dark matter models and to compare them to direct (indirect) detection experiments.
• This document is the final report of the ATLAS-CMS Dark Matter Forum, a forum organized by the ATLAS and CMS collaborations with the participation of experts on theories of Dark Matter, to select a minimal basis set of dark matter simplified models that should support the design of the early LHC Run-2 searches. A prioritized, compact set of benchmark models is proposed, accompanied by studies of the parameter space of these models and a repository of generator implementations. This report also addresses how to apply the Effective Field Theory formalism for collider searches and present the results of such interpretations.


text:
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If your products and/or services use map data that is made accessible using this service, you shall provide end-users of your products and/or services with notice, in a reasonably conspicuous manner (e.g., in the End-User Terms and on every visualized map), of the supplier notices and related terms applicable to such map data by

1. referencing the notices and related terms set here, and
2. including the following copyright notice: © 20XX HERE, XXX, where 20XX is the current year, and XXX is the label text defined for a specific zoom level range and geographical area (see below).

The HERE Raster Tile API provides the copyright endpoint that you can use to retrieve the available copyright providers of the map. To request the list of copyright providers available for a map, use the copyright endpoint.

When you make a copyright request, the response is similar to the example below.

{ "copyrights":
{
"in": [
{
"boundingBoxes": [
{
"east": -89.0225,
"north": 17.8168,
"south": 16.5943,
"west": -91.4256
},
{
"east": -88.2215,
"north": 16.1868,
"south": 14.9642,
"west": -89.0225
},
{
"east": -89.0225,
"north": 16.5943,
"south": 14.1492,
"west": -90.6245
},
{
"east": -89.8235,
"north": 14.1492,
"south": 13.7417,
"west": -91.8261
},
{
"east": -90.6245,
"north": 16.1868,
"south": 14.1492,
"west": -92.2266
},
{
"east": -90.6245,
"north": 16.5943,
"south": 16.473,
"west": -90.6686
},
{
"east": -89.6796,
"north": 14.1492,
"south": 14.0259,
"west": -89.8235
}
],
"alt": "Aprobado por el INSTITUTO GEOGRAFICO NACIONAL Resolucion del IGN No 186-2011",
"label": "IGN Guatemala",
"maxLevel": 20,
"minLevel": 10
}
],
"sat": [
{
"boundingBoxes": [
{
"east": -180,
"north": 90,
"south": -90,
"west": 180
}
],
"label": "Airbus",
"maxLevel": 20,
"minLevel": 14
}
]
},
"resources":{
"base": {
"styles": {"explore.day":["in"],"explore.satellite.day":["in","sat"]
}
}
}
}

For example, if you are using the resource base and the style explore.day you must use the copyright block with key in, as stated by .resources.base.styles."explore.day". If the bounding box {"east": -90.6245,"north": 16.5943,"south": 16.473,"west": -90.6686} is included in your visible viewport and the zoom level is between 10 and 20, then the copyright statement that needs to be clearly visible on the map in 2022 is © 2022 HERE, IGN Guatemala.

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The copyright information associated with the different styles, zoom level ranges, and geographical areas is shown. Variant copyrights are defined by the bounding boxes and the relevant zoom level range. A style may use more than one data source. In that case, all the copyright blocks defined by .resources.base.styles.\$STYLE_NAME must be used. For each copyright block, only those whose bounding boxes are visible in the viewport must be shown.

A copyright block is defined for each source of data (e.g. in, sat, and so on) and contains the following elements:

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The bounding boxes may override or add to the copyright label depending on whether the map view is contained completely within one of the bounding boxes or overlaps partially one or more of them.

When visualizing an area, all of the copyright providers that have bounding boxes overlapping to that viewport need to be visualized on top of the map.

For example, when rendering an area over Guatemala, provided the copyright information is similar to the one displayed above, the image could be similar to this one.


text:
# Electromagnetism problem

When going over my notes on the lorentz electromagnetic force in my physics 3 electromagnetism class got a bit puzzled by one statement that seems just to be taken for granted. Could anyone help me and tell me why x1i =r1 -y1j on page 5 on http://www.physics.gla.ac.uk/~dland/ELMAG305/Elmag305txt1.pdf [Broken] where frame 1 is the stationary frame and 2 the frame moving with velocity v to the right of frame 1

If its just by using pythagorus wouldnt the above quantities (x,y and r) be to the second power. i just cant get my head around this and realise its probably something really straight forward. any help would be much appreciated.

Last edited by a moderator:

Integral
Staff Emeritus
Gold Member
the vector r = xi + yj by definition, see the figure on the previous page.

Using the Pythograean theorem you get the magnetude of r.

$$\mid \hat {r} \mid = \sqrt { x^2 + y^2 }$$

You are dealing with the VECTOR r. See the difference?

Last edited:
haha yeah i do now and realise how obvious that was, thanks

Another quick question, why on page 6 of the slide does vy1k = v x r?


text:
## Roxanne Tellier – Living in the FaceBook Bubble

Posted in Opinion with tags , , , , , , , , , , , , , , , on September 20, 2015 by segarini

Dang! It always happens … I’m all gung ho about something I want to write about, and I get some of the research done, and then … life intervenes. Or, more accurately, social media. I’m looking at you, Facebook!

## JAIMIE VERNON – THE PUNKS ARE (STILL) ALRIGHT

Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on May 30, 2015 by segarini

Smashing Pumpkins leader William “Don’t Call Me Billy” Corgan bleated something about loss of innocence at the end of the 1970s with the band’s only true Pop song called “1979” in the apex of the 1990s version of a counter-culture movement (c/f Grunge, Cobain, Seattle, et al).


text:
+0

# Help asap

0
419
1

Suppose that $f(x)=\frac{1}{2x+b}$. For what value of $b$ does $f^{-1}(x)=\frac{1-2x}{2x}$?

Mar 21, 2018

#1
+22152
+1

Suppose that $f(x)=\frac{1}{2x+b}$. For what value of $b$ does $f^{-1}(x)=\frac{1-2x}{2x}$?

$$\begin{array}{|rcll|} \hline f(x) &=& \dfrac{1}{2x+b} \\ \hline y &=& \dfrac{1}{2x+b} \\\\ \dfrac{1}{y} &=& 2x+b \\\\ 2x &=& \dfrac{1}{y}-b \\\\ x &=& \dfrac{\dfrac{1}{y}-b}{2} \quad & | \quad x \leftrightarrow y \\\\ y &=& \dfrac{\dfrac{1}{x}-b}{2} \\\\ f^{-1}(x) &=& \dfrac{\dfrac{1}{x}-b}{2} \\ \hline f^{-1}(x) = \dfrac{\dfrac{1}{x}-b}{2} &=& \dfrac{1-2x}{2x} \\\\ \dfrac{\dfrac{1}{x}-b}{2} &=& \dfrac{1-2x}{2x} \\\\ \dfrac{1}{x}-b &=& \dfrac{1-2x}{x} \\\\ \dfrac{1}{x}-b &=& \dfrac{1}{x}-2 \\\\ -b &=& -2 \\\\ \mathbf{b} & \mathbf{=}& \mathbf{2} \\ \hline \end{array}$$

Mar 21, 2018


text:
• 15
• 15
• 11
• 9
• 10

how to do this ? Give me some ideas

This topic is 3416 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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Something bugs me for some time and i want to know how could i create PHP script that does this: Every game (i'm talking about browser based games) have HP (health points). Let's say that the max HP is 10. From some reasons, the user's HP drops to 5, so he has only 5 more left. Now, my question is how could i make a script that every minute that passes, his HP to increase by 1 point. So if 2 minutes pass, his HP increases from 5 to 7, but it doesn't go beyond 10 (that is maxHP). I have the fallowing idea: using CronJob, make a script like this to execute itself every 1 minute: * extract all user's HP that is < then maxHP * add 1 to the previous value * update the database with the new HP value Doesn't this technique stress the database, considering that i make every minute hundred of queries if i have hundred of members in the game ? How could i find another solution ? Something based on the user, like everytime he refreshes his profile to know what was his last activity time and what is the current time and to make a difference and see how many minutes have passed and add to his HP the amount of points, but not to go over the maxHP. But how do i save his last activity time and the time he makes a refresh (refresh means he made another activity) and see how much points to add ? (PS: in case someone didn't understand what i meant, i'm talking about a regeneration rate of 1 point/minute)

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update PLAYER set HP = HP + 1 where HP < MAXHP

1 query, not very burdensome. And ideally it wouldn't be a cron job, it'd just be a process (or part of a process) that sat there in a game loop doing updates.

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ya, i know the query, but i want to know if i should do this every minute for the whole game, considering i have hundreds of players => hundreds of queries per minute for the SQL server, OR a beter solution, to upgrade a specific account only when the user is making an action. Now, how do i know when a user is doing something ? (refresh, navigate, fight with other player, etc). How do i record the last activity time and the current time (considering the user is doing something right now)

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Update it passively, i.e.: figure out how much HP was recovered in the time since the last time the program accessed the player's stats. i.e:

int Player::GetHP(){	my_hp = my_hp + floor(now() - last_time); // where now() returns seconds elapsed since program start	if ( my_hp > my_max_hp )		my_hp = my_max_hp;	last_time = now();	return my_hp;}

Something like that, but only in whatever language you're using... :|

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well, i thought at something like this: to have a field in the user's table and keep the last time his HP have been decreased. This can happen only in some small and precise circumstencies like battle. So a code in the pages responsible for this to update that time.
Another code in some pages like profile, battle, etc that extract that time and compares it with the server's current time and makes a difference. Let's say the HP decreasing was at 10:00 and now the time is 10:05.
A code that does 10:05-10:00=5 => curent HP + 5 and updates the new value

PS: i don't need the code, just ideas

[Edited by - adicrst on November 15, 2008 12:58:58 AM]


text:
# On Cluster Algebras Arising from Unpunctured Surfaces

@article{Schiffler2007OnCA,
title={On Cluster Algebras Arising from Unpunctured Surfaces},
author={Ralf Schiffler and Hugh Thomas},
journal={International Mathematics Research Notices},
year={2007},
volume={2009},
pages={3160-3189}
}
• Published 2007
• Mathematics
• International Mathematics Research Notices
We study cluster algebras that are associated to unpunctured surfaces, with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the… Expand

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Quantum cluster algebras from unpunctured triangulated surfaces: arbitrary coefficients and quantization
We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebrasExpand
Unistructurality of cluster algebras of type A
Abstract It is conjectured by Assem, Schiffler and Shramchenko in [3] that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the clusterExpand
On cluster algebras from unpunctured surfaces with one marked point
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We extend the construction of canonical bases for cluster algebras from unpunctured surfaces to the case where the number of marked points is one, and we show that the cluster algebra is equal to theExpand
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We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts inExpand
Cluster algebras of unpunctured surfaces and snake graphs
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We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in theseExpand
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LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebrasExpand

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American Institute of Mathematical Sciences

doi: 10.3934/nhm.2021009

Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

 Imperial College London, Department of Mathematics, London, UK

Received  December 2020 Revised  March 2021 Published  April 2021

In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geqslant 3$. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $(\Phi, \mathcal{R})$. The point process $\Phi$ generating the centres of the holes is either a Poisson point process or the lattice $\mathbb{Z}^d$; the marks $\mathcal{R}$ generating the radii are unbounded i.i.d random variables having finite $(d-2+\beta)$-moment, for $\beta > 0$. We study the rate of convergence to the homogenized solution in terms of the parameter $\beta$. We stress that, for low values of $\beta$, the balls generating the holes may overlap with overwhelming probability.

Citation: Arianna Giunti. Convergence rates for the homogenization of the Poisson problem in randomly perforated domains. Networks & Heterogeneous Media, doi: 10.3934/nhm.2021009
References:

show all references

References:
The construction of $K_{ \varepsilon, z}$ from the cube $Q_{k, \varepsilon}$. The dashed grey area corresponds to the set $K_{ \varepsilon, z}$, while $Q_{ \varepsilon, z}$ is the square bounded by the thick black line. The green dots are the points of $\Phi^ \varepsilon_\delta$ that fall inside the set $Q_{k-1, z}$ (here bounded by the dashed blue line). The red dots are the points that are outside of $Q_{k, z}$ but whose associated cube intersects $\partial Q_{k, z}$. The black dots are the points that are in $Q_{k, z} \backslash Q_{k-1, z}$. Note that the cubes associated to the black and red dots are typically smaller than the ones associated to the green dots due to the cut-off $\tilde R_{ \varepsilon, z}$
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# Show that $g(r) = (a - be^{-rt})^{+}$ is neither convex nor concave in $r$ when $b > 0$.

How can I show that $$g(r) = (a - be^{-rt})^{+}$$ is not necessarily convex nor concave in $$r$$ when $$b > 0$$?

By the way,

$$x^{+} = \begin{cases} 0 & \text{ if } x < 0 \\[.7em] x & \text{ if } x \geq 0 \end{cases}$$

I tried plotting in Desmos to get an understanding of the function. It looks really strange at some points. You can also get a plot by entering

$$f\left(r\right)=\left\{r\ge-\frac{1}{t}\ln\left(\frac{b}{a}\right):0,\left(a-be^{-rt}\right)\right\}$$

Now, I don't know a great way to explain it. A result that the book gives is $$g(r) = (ae^{-rt} - b)^{+}$$ is, for $$a > 0$$, decreasing and convex in $$r$$. So I can use this result, I guess, but I don't see much relation between the two functions.

Any help would be appreciated.

• You are being asked to disprove something. That usually calls for finding a specific counterexample -- in this case, find a set of parameters $a$, $b$, and $t$ so that the function is not convex, and another set so that it is not concave. Unless you were specifically asked to do so, most likely there is no need to provide a theoretical explanation. – LarrySnyder610 May 6 at 0:31

First assume that $$t>0$$. Observe the following: $$g=0$$ for $$r$$ less then some number $$r_0$$. [ $$\lim_{r \to -\infty} (a-be^{-rt})=a-b\lim_{r \to -\infty} e^{-rt}=a-b\infty=-\infty$$]. (Hence $$g'(r)=0$$ for $$r. For $$r$$ sufficiently large $$g'(r)=bte^{-rt} >0$$. For a function which is convex or concave the derivative is monotonic. It follows that $$g'$$ is increasing. Hence $$g'' \geq 0$$. But $$g''(r)=-bt^{2}e^{-rt} <0$$ for $$r$$ sufficiently large. Hence $$g$$ is neither concave nor convex.
The case $$t<0$$ is similar.
• Sorry for the late reply, but I don't quite see why there necessarily exists an index $r_{0}$ for which $r < r_{0}$ implies $g = 0$. Can you please clarify? – user666729 May 10 at 5:10
• @gallileo22 I am assuming that $t >0$. $e^{-rt} \to \infty$ as $r \to -\infty$ so $a-be^{-rt} <0$ for $r$ less than some $r_0$. Hence $g(r)=0$ for $r$ less than $r_0$. – Kavi Rama Murthy May 10 at 5:29
• I thought $g'(r) = bte^{-rt}$, not $g'(r) = be^{-rt}$? Also, $g''(r) = -t^{2}be^{-rt}$? – user666729 May 10 at 5:49
• I'm trying to understand the $t < 0$ case. Can you help me? Here's what I've got – user666729 May 10 at 11:55