text:
# Alice and Bob Game

Alice and Bob just invented a new game. The rule of the new game is quite simple. At the beginning of the game, they write down N random positive integers, then they take turns (Alice ﬁrst) to either:

1. Decrease a number by one.
2. Erase any two numbers and write down their sum.

Whenever a number is decreased to 0, it will be erased automatically. The game ends when all numbers are ﬁnally erased, and the one who cannot play in his(her) turn loses the game.

Here's the problem: Who will win the game if both use the best strategy?

-
This appears to be a question from a programming puzzle; see here. – Douglas S. Stones Nov 4 '12 at 6:01
Moreover, your question here is identical to this; your question here is identical to this. None of which give any attribution or display any attempt to answer the question. – Douglas S. Stones Nov 4 '12 at 6:16
@DouglasS.Stones These are archived questions. They are not part of any live tournaments. Hence there is no harm in asking any of these for an optimal solution. – Adwait Kumar Nov 4 '12 at 6:29
So, why not just say where it's from when you ask the question? (And, ideally, where you're stuck in solving the problem.) It helps whoever might answer the question. – Douglas S. Stones Nov 4 '12 at 7:55
The concept of "random positive integer" is not defined: there is no uniform probability distribution on the positive integers. – Carl Mummert Nov 4 '12 at 12:47

The complete solution to this game is harder than it looks, due to complications when there are several numbers $1$ present; I claim the following is a complete list of the "Bob" games, those that can be won by the second player to move. To justify, I will indicate for each case a strategy for Bob, countering any move by Alice by another move leading to a simpler "Bob" game.

I will write game position as partitions, weakly decreasing sequences of nonnegative integers (order clearly does not matter for the game). Entries present a number of times are indicated by exponents in parentheses, so $(3,1^{(4)})$ designates $(3,1,1,1,1)$. Moves are of type "decrease" (type 1 in the question) or "merge" (type 2); a decrease from $1$ to $0$ will be called a removal.

Bob-games are:

• $(0)$ and $(2)$
• $(a_1,\ldots,a_n,1^{(2k)})$ where $k\geq0$, $n\geq1$, $a_n\geq2$, $(a_1,\ldots,a_n)\neq(2)$, and $a_1+\cdots+a_n+n-1$ is even. Strategy: counter a removal of one of the numbers $1$ by another such removal; a merge of a $1$ and an $a_i$ by another merge of a $1$ into $a_i$; a merge of two entries $1$ by a merge of the resulting $2$ into one of the $a_i$; a decrease of an $a_i$ from $2$ to $1$ by a merge of the resulting $1$ into another $a_j$; any other decrease of an $a_i$ or a merge of an $a_i$ and $a_j$ by the merge of two entries $1$ if possible ($k\geq1$) or else merge an $a_i$ and $a_j$ if possible ($n\geq2$), or else decrease the unique remaining number making it even.
• (to be continued...)

Note that the minimal possibilities for $(a_1,\ldots,a_n)$ here are $(4)$, $(3,2)$, and $(2,2,2)$. Anything that can be moved into a Bob-game is an Alice-game; this applies to any $(a_1,\ldots,a_n,1^{(2k+1)})$ where $k\geq0$, $n\geq1$, $a_n\geq2$, $(a_1,\ldots,a_n)\neq(2)$ (either remove or merge a $1$ so as to leave $a_1+\cdots+a_n+n-1$ even), and to any $(a_1,\ldots,a_n,1^{(2k)})$ where $k\geq0$, $n\geq1$, $a_n\geq2$, and $a_1+\cdots+a_n+n-1$ odd (either merge two of the $a_i$ or two entries $1$, or if there was just an odd singleton, decrease it). All cases $(3,1^{(l)})$ and $(2,2,1^{(l)})$ are covered by this, in a manner depending on the parity of $l$. It remains to classify the configurations $(2,1^{(l)})$ and $(1^{(l)})$. Moving outside this remaining collection always gives some Alice-game $(3,1^{(l)})$ or $(2,2,1^{(l)})$, which are losing moves that can be ignored. Then we complete our list of Bob-games with:

• $(1^{(3k)})$ and $(2,1^{(3k)})$ with $k>0$. Bob wins game $(1,1,1)$ by moving to $(2)$ in all cases. Similarly he wins other games $(1^{(3k)})$ by moving to $(2,1^{(3k-3)})$ in all cases. Finally Bob wins $(2,1^{(3k)})$ by moving to $(1^{(3k)})$ (unless Alice merges, but this also loses as we already saw).
-

Here is an almost complete solution.

If there is only one integer, then clearly it depends on the parity of the number to determine who wins. If the number is odd then Alice will win. If the number is even then Bob will win.

If there are two integers, then it depends on the parity of the sum of the two numbers $\Sigma$. There is one combine and $\Sigma$ reductions so the game will be won by Alice if $\Sigma + 1$ is odd. This will be true as long as there is not an opportunity to remove one of the integers (by reducing it to zero) before there is a chance to combine. It is sufficient to require that none of the integers are $1$.

Let us now prove this strategy by induction.

Suppose the game is played with $N$ positive integers greater than $1$. Call the sum of the $N$ integers $\Sigma$. Then Alice has a winning strategy if and only if $\Sigma + N - 1$ is odd.

Proof: This is clearly true for $N = 1$. Suppose that the statement holds for $N$ and consider $N+1$. If $\Sigma + N$ is odd, then Alice can choose to combine two integers on her first turn. This reduces to the case of $N$ numbers with $\Sigma + N - 1$ even. Since Bob is now starting, from the induction hypothesis, he loses.

Conversely suppose that $\Sigma + N$ is even. If Alice chooses to reduce a number by $1$, then Bob chooses to combine two numbers. Then we have $N$ numbers with $\Sigma + N - 1$ being even. It is now Alice's turn and she loses. Alternatively if Alice chooses to combine two numbers, then by the induction hypothesis $\Sigma + N$ is odd with Bob starting. Therefore Bob wins.

This leaves the problem of when the $N$ numbers contain $1$s.

-


text:
# Simple example of product not preserving coequaliser in $\mathbf{Top}$

In the category of topological spaces ($$\mathbf{Top}$$), products do not always preserve colimits. If they did then $$\mathrm{Hom}_\mathbf{Top}(-\times X,S)$$ would be representable and hence $$\mathbf{Top}$$ would be Cartesian closed (which it isn't). I think that products do preserve coproducts, so it must be that there's some coequaliser which products don't preserve. I'm trying to understand why this is in more concrete terms, but I've struggled to find a simple example that I can examine in detail.

What are some simple spaces $$A$$, $$B$$ and $$X$$ and maps $$f,g:A\to B$$ in $$\mathbf{Top}$$ such that the product of $$X$$ with the coequaliser is different from the coequaliser of the products?

The same question for the category of locales is here.

• A different angle of attack to the underlying question of why Top is not cartesian closed: if it were, then the underlying set of $Y^X$ would have to be $Hom_{Top}(1, Y^X) \simeq Hom_{Top}(X, Y)$, the set of continuous functions. And given a net $f : I \to Y^X$, $f_i \to f_0$ if and only if the map $I \cup \{ \infty \} \to Y^X$ is continuous, if and only if the map $(I \cup \{ \infty \}) \times X \to Y$, which would allow us to define what the closure operator on $Y^X$ would have to be. Then, come up with an example where this isn't idempotent. – Daniel Schepler Oct 24 '18 at 18:14
• @DanielSchepler Thanks, but I'm actually interested in the case of coequalisers in particular. I just mentioned Cartesian closedness to explain how I knew there must be an example of one that doesn't commute with products. – Oscar Cunningham Oct 24 '18 at 19:10
• Hmm, according to ncatlab.org/nlab/show/exponential+law+for+spaces it must be that if $X$ is not a core-compact space, for example $X = \mathbb{Q}$, then the functor $X \times - : Top \to Top$ does not preserve coequalizers. That doesn't really give the full proof but it might at least be somewhere to start. I think I also saw somewhere that the Sierpinski space $(\{ 0, 1 \}, \{ \emptyset, \{ 1 \}, \{ 0, 1 \} \})$ might be involved in a counterexample. – Daniel Schepler Oct 24 '18 at 21:44

## 1 Answer

Consider $$\mathbb{Z}$$, $$\mathbb{Q}$$ and $$\mathbb{R}$$ with their usual topologies. Let $$i:\mathbb{Z}\hookrightarrow\mathbb{R}$$ be the usual inclusion, and define $$j :\mathbb{Z}\to\mathbb{R}$$ by $$j(n) = i(n+1)$$. Our example of failed preservation will be that the canonical map $$\mathrm{coeq}(i\times\mathbb{Q},j\times\mathbb{Q})\to\mathrm{coeq}(i,j)\times\mathbb{Q}$$ is not a homeomorphism.

Since the forgetful functor $$\mathbf{Top}\to\mathbf{Set}$$ preserves both limits and colimits, the underlying function of this map is indeed a bijection. The underlying set of both spaces is the quotient of $$\mathbb{R}\times\mathbb{Q}$$ by the equivalence relation that relates $$(r,q)$$ to $$(r',q)$$ whenever both $$r$$ and $$r'$$ are integers. The reason this bijection is not a homeomorphism is that there are open sets in $$\mathrm{coeq}(i\times\mathbb{Q},j\times\mathbb{Q})$$ whose images are not open in $$\mathrm{coeq}(i,j)\times\mathbb{Q}$$.

To construct such an open set, consider the graphs of two continuous functions $$f,g:\mathbb{R}\to\mathbb{R}$$ with the following properties:

• Both $$f(x)$$ and $$g(x)$$ are strictly positive for all $$x$$, but tend to $$0$$ as $$x$$ tends to $$+\infty$$ and $$-\infty$$.

• We have $$f(x)=g(x)$$ iff $$x$$ is an integer, and in this case $$f(x)$$ and $$g(x)$$ are irrational.

For example we could take $$f(x)=\frac{\pi+\sin(\pi x)}{1+x^2}$$ and $$g(x)=\frac{\pi-\sin(\pi x)}{1+x^2}$$. Now define $$U$$ to be the image under the quotient map of the subset of $$\mathbb{R}\times\mathbb{Q}$$ containing the points $$(r,q)$$ for which $$q$$ is either less than both $$f(r)$$ and $$g(r)$$ or greater than both $$f(r)$$ and $$g(r)$$.

Then $$U$$ is open in $$\mathrm{coeq}(i\times\mathbb{Q},j\times\mathbb{Q})$$ since its preimage under the quotient map is open in $$\mathbb{R}\times\mathbb{Q}$$. But it is not open in $$\mathrm{coeq}(i,j)\times\mathbb{Q}$$ since every neighbourhood of $$0$$ in $$\mathrm{coeq}(i,j)$$ contains arbitrarily large nonintegers and hence every open rectangle around $$(0,0)$$ in $$\mathrm{coeq}(i,j)\times\mathbb{Q}$$ meets the area between the graphs of $$f$$ and $$g$$.

format: plain


text:
# Pulsed Nd:YAG 레이저 조사와 불소화합물 도포가 치아 내산성에 미치는 영향

• An-Hee Lee (Dept. of Oral Medicine, College of Dentistry, Chonnam National University) ;
• Woo-Cheon Kee (Dept. of Oral Medicine, College of Dentistry, Chonnam National University)
• Published : 1995.06.01

#### Abstract

This study was designed to determine the most effective concentration of fluoride and levels of laser irradiation for the remineralization of decayed teeth. After irradiation with a pulsed Nd:YAG laser and the topical application of fluoride, phosphate and fluoride concentration in enamel were measured. And then the changes on surface enamel using an scanning electron microscope were observed. Samples by extraction healthy, permanent, mandibular bovine teeth with no caries were obtained. Among them 371 healthy samples were selected and artificial carious lesions were made. 20 samples were assigned to each experimental group. After irradiation with a pulsed Nd:YAG laser with total energy densities of 10J/$\textrm{cm}^2$, 20J/$\textrm{cm}^2$ for each group. On the teeth, 2% NaF, 1.9% NH4F, 1.6% TiF4 Elmex gel(amine fluoride) and 1.23% APF gel were applied. After pH circulatory procedures, concentrations of fluoride with and Ionalyzer (Orion Research, Model 901, USA) and phosphates with an Uv/V is spectrophotometer (Uvikon 860, Kontrom Co, Switzerland) were measured. By etching the teeth in layers and measuring fluoride concentrations, a profile of fluoride penetration according to depth could be developed. And also the changes on the surface of the enamel using an electron scanning microscope were observed. The comparative analysis yielded the following results : 1. Phosphate concentration was low in all groups compared with the control group except for teeth treated Elmex gel, irradiated with 10J/$\textrm{cm}^2$ and 30J/$\textrm{cm}^2$ energy densities. Teeth treated with APF gel and 30J/$\textrm{cm}^2$ irradiation gad the lowest phosphate concentration. 2. Among all groups, fluoride concentrations in tooth enamel were highest in APF gel and NaF groups irradiated at 30J/$\textrm{cm}^2$. The APF gel group had the highest fluoride concentrations across all energy densities. 3. In the APF gel group, and the NaF group, the greater the energy density of the laser, the higher the fluoride concentrations in the enamel. 4. In all groups, the concentration of fluoride in the enamel by depth tended to decrease with depth. 5. Under the scanning electron microscope, under the condition of irradiation with 20J/$\textrm{cm}^2$, enamel crack was detected for the first time. In the NH4F group, spherical deposits were found on the surface of the enamel, and in the TiF4 group the surface of enamel was covered with an irregular, thin membranous mass in places. In the APF gel and NaF groups irradiated with 10J/$\textrm{cm}^2$, spherical and irregular particles covered the teeth. When these groups were irradiated at 20J/cm2, they were covered with amorphous crystals. These results suggest that one could obtain more effective anticariogenic effects without damage to teeth when less than 20J/$\textrm{cm}^2$ energy densities and APF gel are used.


text:
## The Annals of Applied Probability

### A Variational Method for Estimating the Parameters of MRF from Complete or Incomplete Data

#### Abstract

We introduce a new method (to be referred to as the variational method, VM) for estimating the parameters of Gibbs distributions with random variables ("spins") taking values in a Euclidean space $\mathbb{R}^n, n \geq 1$, from complete or degraded data. The method applies also to the case of iid random variables governed by exponential families, and appears to be new even in this case. For complete data, the VM is computationally more efficient than, and as reliable as, the maximum pseudo-likelihood method. For incomplete data, the VM leads to an estimation procedure reminiscent of, but simpler than, the EM algorithm. In the former case, we show that under natural assumptions a certain form of the variational estimators is strongly consistent and asymptotically normal. We also present numerical experiments that demonstrate the computational efficiency and accuracy of the variational estimators.

#### Article information

Source
Ann. Appl. Probab., Volume 3, Number 1 (1993), 103-136.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177005510

Digital Object Identifier
doi:10.1214/aoap/1177005510

Mathematical Reviews number (MathSciNet)
MR1202518

Zentralblatt MATH identifier
0771.60091

JSTOR


text:
Paper ID: 659 Title: Simultaneous Model Selection and Optimization through Parameter-free Stochastic Learning
Current Reviews

Submitted by Assigned_Reviewer_19

Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The paper studies the interesting question on online (stochastic) gradient descent
in the unconstrained setting (sometime referred to non-strongly convexity or without
explicit regularization). In the earlier work [33], it was proved that the last
iterate of stochastic gradient descent with least-square loss in the unconstrained
setting actually converges with explicit convergence rate by appropriately choosing
the step sizes (or by a stopping early rule), which, however, needs to know the
smoothness of the regression function.

The paper proposed a kernel-based stochastic gradient descent algorithm without the
need to perform model selection, which requires the loss function and its gradient
are both Lipschitz. The proposed algorithm is mainly motivated by the recent
studies [15,16] which involves a data-dependent regularization. The main technical
proof for Theorem 1 involves elegant convex-analysis estimations, although the main
idea is similar to the standard telescope techniques for proving regret bounds in
online learning. The results of theorem 2 are also very interesting and are new
compared to the existing literature.

1. Can you prove the convergence of the last iterate of the proposed algorithm,

2. line 45: arbitrary -> arbitrarily

3. In line 153-159, you may wish to emphasize the results hold true for the least
square loss.

4. line 538-539: the claim that theorem 3 improves the results in [34] seems a bit
unfair, since the results hold for different loss functions and are based on the
assumptions on the smoothness of different target functions.

5. line 653-654: the second to last inequality is wrong

6. lines 678 and 710: z -> \kappa_t

7. The proof of lemma 7 does not need lemma 6. For the upper bound of $x$, one can
directly get it by applying the basic inequality: a (x+b)^alpha \le alpha (x+b) +
(1-alpha) a^{1-alpha}.

In summary, this is a very nice piece of work. The paper is also well written and
the related work and discussion are adequate.

Submitted by Assigned_Reviewer_22

Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
Added post rebuttal: After considering the rebuttal and taking another look at the paper, I may have been a bit harsh in my criticism. I still think the paper relies a lot on techniques from previous papers, but the connections between them are indeed novel. I thank the author\s for the clarifications.

---------------------------------------------

This paper proposes a parameter-free algorithm for stochastic gradient descent of linear and kernel predictors. In particular, the step size is automatically determined and does not rely on the (initially unknown) norm of the competitor, nor on a doubling trick. The algorithm builds on recently developed online convex optimization algorithms with similar behavior, with the contributions being:
- Convergence bounds depending on the magnitude of the subgradients rather than the time horizon. In particular, for smooth losses, this leads to bounds which depend on the risk of the competitor, and interpolate between O(T^{-1/2}) and O(T^{-2/3}) rates.
- Convergence to the risk of the "Bayes-optimal" function (w.r.t. the loss function used), assuming this function lies in a function space with an appropriate spectral decay.
- Convergence bounds under a Tsybakov noise condition.
- A few experiments showing the competitiveness of the proposed algorithm and cross-validated SVM.

The question of how to avoid cross-validation and parameter tuning is certainly an interesting one and has practical implications. Moreover, the algorithm is simple and easy to implement. On the flip side, there has already been a string of recent works on online learning methods with such behavior, which can be readily applied to stochastic and batch problems (e.g. [14,16,25] which the paper cites). The proposed algorithm also builds on those methods, so the conceptual novelty is perhaps not as strong as the abstract and introduction seem to suggest. As far as I understand, the main improvement over them consists of the refined bound depending on the subgradient magnitudes, which for smooth losses lead to bounds depending on the loss of the competitor. This is an interesting contribution, although the bounds are probably suboptimal (e.g. when the loss is zero, we only get O(T^{-2/3}) rates, but with standard algorithms can get O(T^{-1}). A more minor issue is that it isn't clear if the refined bound and algorithmic change leads to anything practically noticeable (some experiments would be useful in that regard).

The results regarding convergence to the Bayes-optimal and Tsybakov noise condition are nice and technically non-trivial, but unless I missed something, appear to consist mostly of plugging the new bound into existing techniqus. Overall, I would say that the contributions are solid but a bit incremental in light of previous work.

In terms of presentation, the paper is mostly well-written, but I felt it was unnecessarily complicated at times. For example:
- In section 2, the definitions of the function spaces are very abstract, and would probably be hard to grasp for non-experts. Some specific examples would be very useful.
- Theorem 1 could be written in a much simpler way by choosing a specific value for "a", e.g. a=2.25L, and replacing \phi(x) by the resulting constant. I failed to see what additional insight is gained by keeping "a" as free parameter.

----------------------------
- The setting focuses on classification losses, which are a function of y*prediction. Can similar results be shown for regression or more general losses?

- Line 46: "While in finite hypothesis spaces simple optimal strategies are known, in infinite dimensional spaces..." - should "finite" be "finite dimensional"?

- Line 82: Since you focus on scalar-input loss functions, perhaps it's best to present things w.r.t. scalars rather than vectors, and use "sub-derivative" instead of "sub-gradient". For example, I was confused at first by algorithm 3, which talks about s_t as subgradients but are in fact used as scalars.

- Line 125: "By the Mercer's theorem"

- LIne 203: "We will present such algorithm"

- Line 315: "that is low optimal \ell-risk" -> "that has low optimal \ell-risk" (?)

- Line 408: "run few" -> "run a few"
The paper proposes a parameter-free algorithm for stochastic learning of linear and kernel predictors. This is an interesting direction, but the contribution is a bit incremental in light of recent such works.

Submitted by Assigned_Reviewer_26

Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
This paper proposes a new online algorithm to learn supervised kernel machine. The required computation per iteration is quite simple, and the method has an adaptivity on the "complexity" of the true function in a sense that depending on the complexity, the learning rate adaptively achieves the minimax optimal rate.

I think the paper gives an interesting method. The analysis is also novel. Although the method is executed in an online manner, it shows adaptivity. This is a nice point.

A simple explanation of intuition of the algorithm might be helpful to the readers.

I am wondering the following point.
1. The method is not constructed to utilize the strong convexity of the loss function. Because of this, the learning rate is not depending on b, the decaying rate of the eigen-values. The local Rademacher complexity argument utilizes the strong convexity, and thus is able to improve the convergence rate depending on b. The dependence on beta is already interesting. But, I guess, if the method could involve b, the analysis is more interesting.

2. It is stated that the computational complexity is same as the plain SGD. However, the method requires to construct g_t and compute \|g_t\| at each iteration which should cost at least O(t). The current description is not clear about the actual computational cost. (I admit that the kernel method can't avoid O(T^2) computation. I am not criticizing that, but asking to clarify.)

===
After rebuttal:
The authors claim that the computation per-iteration is O(1) because it is done incrementally from \|g_{t-1}\| and from \langle \g_{t-1}, x_t \rangle1}, x_t \rangle that is computed during the prediction step.
I see that the computation for the prediction is not included in the evaluation of the computational cost. My point was that the computation of \langle \g_{t-1}, x_t \rangle1}, x_t \rangle actually costs O(t), thus the cost per iteration should be O(t). I think it is better that this point is clearly written.
A nice theoretical paper which proposes a new online kernel method. The method seems practical and has a nice statistical convergence property.
Author Feedback
Author Feedback
Q1:Author rebuttal: Please respond to any concerns raised in the reviews. There are no constraints on how you want to argue your case, except for the fact that your text should be limited to a maximum of 6000 characters. Note however, that reviewers and area chairs are busy and may not read long vague rebuttals. It is in your own interest to be concise and to the point.
We thank the Reviewers for their very detailed and useful comments that will be useful in improving the paper.
We also thank Reviewer 19 and 26 for recognizing the potential high impact that we strongly believe this work has, from both a theoretical and practical point of view.

Reviewer 22: "The results regarding convergence to the Bayes-optimal and Tsybakov noise condition are nice and technically non-trivial, but unless I missed something, appear to consist mostly of plugging the new bound into existing techniques."

We respectfully but strongly disagree with Reviewer 22 on this point. Theorem 2 is our main result and the "existing techniques" the Reviewer mentions are actually first introduced in this very paper!
Specifically, the results in Theorem 2 are (mainly) proven thanks to the connection between optimal online learning regret bounds in unconstrained settings and optimal parameter-free SGD, sketched in Section 3 and in particular in the simple equation (4). The Reviewer seems to have missed the fact that this connection is shown for the *first time* in this paper. In fact, despite the simplicity of (4), no other paper contains even a hint at this solution. Indeed, the very opposite is true: As discussed in lines 371-399, the seminal paper by Steinwart et al. [24] strongly argued that regularizers of the form ||h||^q provides *no theoretical advantage* over the usual ||h||^2, apparently closing this research topic and making the result of Theorem 1 and the weaker results in [14,16,25] apparently useless. Here we prove the contrary.
Moreover, as also noted by Reviewer 19, we strongly believe that the result of Theorem 2 are "new" and "very interesting" with respect to the large body of literature on this topic.
The comments of the Reviewer made us realize that we did not stress enough this point, an oversight we will fix.

Reviewer 26: "Clarify complexity of the algorithm"

The calculation of \|g_t\| is just O(1). In fact, it should be done incrementally from \|g_{t-1}\| and from \langle \g_{t-1}, x_t \rangle that is computed during the prediction step. Hence, the overall computational complexity per iteration is exactly the same of SGD, with and without kernels. We will clarify it in the camera-ready version.

Reviewer 26: "how to utilize the strong convexity of the loss function?"

This is indeed a very good point and one of the important directions for future work. However, as far as we know, there are no SGD algorithms able to achieve an optimal dependency on “b”, not even with an oracle choice of the learning rate. We will add some comments addressing this in the discussion section.

Reviewer 19: “Can you prove the convergence of the last iterate of the proposed algorithm?”

Unfortunately no. The convergence of the last iterate seems currently out of reach with the analysis we used. For the same reason, even the convergence in norm K when beta>=1/2 seems difficult to achieve. We might have to switch to the analysis in [33] or find a way to fuse our approach with their approach.


text:
# which ozone sensor excel

### What's best for measuring ozone, electrochemical sensors ...

· Ozone (O3) is an important gas in industrial applications, medical or viral sterilization, ground-level pollution monitoring, and atmospheric studies. The best-known method for ozone measurement has been to use an analyzer based on the ultraviolet absorption principle. Despite significant cost reductions in recent years an ozone analyzer remains an expensive piece of …

### Aeroqual Series 500 Monitor (S-500) with Ozone Sensor Head

The Aeroqual Series 500 lets you log data for the ozone you measure, making your record-keeping easy. It can also interface with your external control systems, making it a simple solution for those needing alarms or exhaust fans controlled for safety purposes. Buying additional sensor heads lets you replace several of your existing monitors with a single Series 500 - saving you money!

### Ozone Meter | PCE Instruments

Ozone Meter AQ-500 is a highly accurate instrument for measuring ozone concentrations in the air. The Ozone Meter can be equipped with four different heads that vary in their measurement range. - High accuracy and functionality - Interchangeable sensor heads! - Battery operation - Sensor head OZU included in delivery (Ozone ppm) - Alarm function

### Ozone Sensor - Spec Sensors

Analog SDK-O3 Quickly integrate our Ozone sensors into your final design $Add to cart; Analog Gas Sensor Module for Ozone: ULPSM-O3. ULPSM-O3 converts the Ozone sensor’s linear current signal output to a linear voltage signal, while maintaining the sensor at its ideal biased operation settings.$ Add to cart; 3SP_O3_20 C Package 110-407 SPEC Sensor Operation Overview SPEC AN ...

### Ambient Ozone Monitors - 2B TecHnologies

The Model 306 Ozone Calibration Source is a highly portable ozone transfer standard that may be used to calibrate any ozone monitor or sensor. Ozone is generated at any chosen concentration in the range 0-1000 ppb in ozone-scrubbed ambient air at a volumetric flow rate of 3 L/min. No external air source is required. The instrument may be programmed to output a series of up to 10 ozone concentration …

### 190 questions with answers in OZONE | Science topic

· MQ131 ozone gas sensor has high sensitivity to ozone, and also has sensitivity to strong oxide such as Cl2, NO2 and etc. It responses oppositely to organic interference gases . my sensor ...

### Real-Time Ozone Sensor Based on Selective Oxidation of ...

sensors Article Real-Time Ozone Sensor Based on Selective Oxidation of Methylene Blue in Mesoporous Silica Films Christelle Ghazaly 1,2, Marc Hébrant 2, Eddy Langlois 1, Blandine Castel 1 ...

### Ozone sensor and sampler -

Download Citation | Ozone sensor and sampler | The Environmental Information Systems Project of NTT Energy and Environment Systems Laboratories involves research and development on systems for ...

### Dissolved Ozone Measurements with EC Sensors: Installation ...

Basically, an EC sensor for ozone receives a voltage between two electrodes immersed in an electrolyte current is generated, that is proportional to the ozone concentration that diffuses through the membrane separating the sensor electrolyte and the measured sample. EC …

### Sensor for Carbon Dioxide Gas (CO2) Oxygen (O2) Ozone (O3 ...

Sensor for Carbon Dioxide Gas (CO2) Oxygen (O2) Ozone (O3) Alcohol (C2H5OH) TVOC ETO Methane (CH4) Propane (C3H8) Butane (C4H10) Hydrogen (H2) CO H2S NH3 HCL SO2 NO2 CL2. Welcome to Winsensors. We offer Customized Sensor, Detector, Reader, Monitor, Module, Control Board Design and Manufacture Service. Free Shipping Worldwide for ALL products. Please check our gradually updated Excel …

### O3 Gas Monitoring / Automatic Switch On ... - Win Sensors

This control kit help you monitor your ozone concentration in the room / chamber, power on Ozone Generator. Automatically when ozone concentration dropped to certain low PPM (figure pre-set by you, say 15ppm) and. automatically power off generator when ozone concentration reaches certain high PPM (pre-set by you, say 85ppm, maximum 100ppm).

### Gas Sensors Ozone -

Ozone is a triatomic form of oxygen, meaning that the molecule, ozone, is made up of 3 oxygen atoms. Unlike oxygen, ozone is a visible gas with a light blue color at very high concentrations. At low concentrations ozone is colorless. Ozone also has a distinct odor that is noticeable at very low levels and may be associated with a “clean” smell, at higher levels ozone has a very pungent odor. Ozone is an unstable gas that readily decomposes to oxygen naturally. Ozone …

### Industrial Ozone Monitors - 2B TecHnologies

The Model 306 Ozone Calibration Source is a highly portable ozone transfer standard that may be used to calibrate any ozone monitor or sensor. Ozone is generated at any chosen concentration in the range 0-1000 ppb in ozone-scrubbed ambient air at a volumetric flow rate of 3 L/min. No external air source is required. The instrument may be programmed to output a series of up to 10 ozone concentration …

### PortaSens II - Analytical Technology

SENSOR CALIBRATION ... Microsoft Excel®. The application operates on Microsoft Windows ® 98,XP, 2000, and Windows 7 & 8. For more information, see C16Log on page 41. Power is provided from one primary (non-rechargeable), D-cell battery, and switches automatically to a secondary (rechargeable) battery when the primary is removed, or its power drops below an acceptable level1. 1 The internal ...

### Chem 191: Plotting Ozone Data in Microsoft Excel – Al ...

Chem 191: Plotting Ozone Data in Microsoft Excel \vspace{} Instructions: Complete this worksheet using the ozone data you collected last week. Grading: This assignments counts for 15 points toward your lab grade. Submitting: Before you leave today, turn in an electronic copy of your Excel file to Dropbox with the “Excel - Ozone Lab” Link. Make sure you include answers to the questions ...

### O3 Gas Monitoring / Automatic Switch On ... - Win Sensors

what you will get from this purchase: – One installed 0 -100ppm Ozone O3 Sensor – One Automatic Power On / Off Control Board – Screen displayed remotely with cable (default 30cm cable, length can be adjusted, free of charge) – Sensor installed remotedly via cable (default 1 metre cable, length can be adjusted, free […]

### air-Q - Luftanalysator (11 Sensoren), 389,00

Der air-Q mit 11 Sensoren - mit air-Q Smartphone App für iOS / Android und vielen smarten Funktionen. 389,00 €. inkl. 19% USt. , versandfreie Lieferung. Normalpreis: 449,00 €. verfügbar.

### Ozone measurement - Sensors | Sens2B | Sensor Technology

Sens2B - Sensor to Business is the international measurement technology and sensor suppliers directory / sensors portal. Sens2B does not manufacture or sell any sensors or measuring devices. You find in the Sens2B database sensor manufacturers, sensor suppliers and distributors (distribution & representation companies) of all kind of sensors, transducers, transmitters, measurement instruments ...

### Ozone O3 Gas Sense Module Electrochemical Sensor Air ...

Ozone O3 Gas Sensor Electrochemical Sensor Detection Module 0 – 5PPM Range: 0 – 5PPM High quality accurate measurement We can Custom Made Other Sensors / Readers / Tailor Made Functions or Other gas Detection Range upon discussion! Performance Characteristics| Nominal Range 0~5 ppm Maximum Overload 20 ppm Sensitivity(20℃) ± μA/ppm Response Time (T90) ≤ 60 s Zero …

### OZONE MONITOR

Thank you for choosing an Airmaster ozone monitor. The product is a high accuracy ozone measuring device with a state of the art electrochemical sensor. The device can be used for measuring ambient ozone levels or as a controller for an ozone generator to maintain a certain ozone level as well as a safety switch for leakage control of ozone systems. Einführung - DE Vielen Dank für Ihr ...

### MY NASA DATA Lesson

Current Ozone Updates Importing Data into Excel Directions for importing data into graphing calculator Graph of ozone proles Microset Using linear regression functionality on graphing calculator Ozone hole versus normal prole Background: Over the past several decades, the levels of ozone in the atmosphere have received a lot of media and government attention. Major focus has centered on the ...

### Portable Ozone Monitors, Sensors and Detectors | Measure O3

Our portable ozone monitors can measure up to 5 different ozone ranges, depending on your requirements. All our ozone sensors are made by Aeroqual using unique Gas Sensitive Semiconductor (GSS) technology. Aeroqual’s GSS technology allows ppb level detection of ozone with high accuracy and stability. Our ozone sensors are best-in-class for measuring ozone at low concentrations, …

### O3 Gas Monitoring / Automatic Switch Control ... - Win Sensors

– One installed high quality 0 – 5ppm Ozone O3 Sensor – One Automatic Power On / Off Control Board – Screen displayed remotely with cable (by default 30cm cable, length can be adjusted upon request, free of charge) – Sensor installed remotedly via cable (by default 1 metre cable, length can be adjusted upon request, free of charge) – Control botton can be installed remotedly via ...

### ESP32 Air Quality Monitor |

1 × Espressif Systems ESP32. 1 × CJMCU-811 CCS811 Air quality sensor. 1 × 5mm Red, yellow and green LEDs. 1 × TP4056 lithium battery protection and charging circuit Generic China. This model includes several protection features. 1 × 5V micro SD reader Generic China.

### Global Ozone Sensor Market 2021 by Manufacturers, Regions ...

Chapter 3, the Ozone Sensor competitive situation, sales, revenue and global market share of top manufacturers are analyzed emphatically by landscape contrast. Chapter 4, the Ozone Sensor breakdown data are shown at the regional level, to show the sales, revenue and growth by regions, from 2016 to 2026. Chapter 5 and 6, to segment the sales by type and application, with sales market share …

### O3 Gas Monitoring / Automatic Switch On ... - Win Sensors

O3 Gas Monitoring / Automatic Switch On / Off Control Board with Ozone Sensor \$ O3 Gas Monitoring / Automatic Switch On / Off Control Board with Ozone Sensor quantity. Add to cart. SKU: 582977020051 Categories: O3 Ozone, Ozone Generator. Description Description. what you will get from this purchase: – One installed 0 -5ppm high quality Ozone O3 Sensor,sensor type: SEMEA 7E4-O -5 ...

### Ozone and NO2 sensors | Details |

The ozone sensor voltage changes by mV for each 20ppb, while the NO2 sensor voltage changes by mV for each 20ppb change. Looking at the NO2 sensor. The output range is 0-3V. The minimum number of ADC steps needs to be 3000 mV/ mV = 13464 steps. It is quick to determine that a 12 bit ADC with 4096 steps is not capable of delivering this resolution.

Professional gas sensing solution supplier with over 30-year gas sensor manufacturing experience


text:
# Matroids over Partial Hyperstructures

Guest Post by Matt Baker.

In this post I’d like to explain a new generalization of matroids developed in this recent paper with Nathan Bowler (referred to henceforth as [BB]). We show that by working over certain algebraic structures which we call tracts, linear subspaces, matroids, valuated matroids, oriented matroids, and matroids over partial fields all become special cases of a single general concept. Actually, there are at least two natural notions of matroids over tracts, which we call weak and strong matroids, but in many cases of interest (such as all the ones mentioned above) the two notions coincide.

Two important special cases of tracts are hyperfields and partial fields. We assume familiarity with the theory of partial fields (see for example this post and its sequels by Stefan van Zwam).

Hyperfields

Roughly speaking, a hyperfield is an algebraic structure satisfying similar axioms to a field, but where addition is allowed to be multi-valued. Like fields, hyperfields come equipped with an additive identity element called zero and a negation map $x \mapsto -x$. However, one does not require that the hypersum of $x$ and $-x$ is zero, only that zero is contained in the hypersum. Before giving a precise axiomatic definition, let us give some motivating examples.

1. (Fields) Any field $K$ is in particular a hyperfield.
2. (The Krasner hyperfield) The Krasner hyperfield ${\mathbb K}$ records the arithmetic of “zero” versus “nonzero”. Specifically, define ${\mathbb K} = \{ 0,1 \}$ together with the binary operation $0 \odot 0 = 0, 0 \odot 1 = 0, 1 \odot 1 = 1$ and the binary hyperoperation $0 \boxplus 0 = \{ 0 \}, 0 \boxplus 1 = \{ 1 \}, 1 \boxplus 1 = \{ 0,1 \}$. If we think of $1$ as representing “nonzero”, the hyperaddition rules just say that zero plus zero is zero and zero plus something nonzero is always nonzero, but the sum of two things which are nonzero could be either zero or nonzero. The negation operator is the identity map, since negative zero is zero and the negative of something nonzero is again nonzero.
3. (The hyperfield of signs) The hyperfield of signs ${\mathbb S}$ records the arithmetic of “zero”, “positive”, and “negative”, represented by the symbols $0, 1, -1$, respectively. The product $x \odot y$ is defined in the obvious way for $x,y \in {\mathbb S} := \{ 0, 1, -1 \}$. Addition is also defined in the “obvious” way except we have $1 \boxplus -1 = \{ 0, 1, -1 \}$ since the sum of a positive number and a negative number could be either zero, positive, or negative. The negation operator takes $0$ to itself and interchanges $1$ and $-1$.
4. (The tropical hyperfield) The tropical hyperfield ${\mathbb T}$ records the arithmetic of valuations. More precisely, if $v : K \to {\mathbb T} := {\mathbb R} \cup \{ \infty \}$ is the valuation map on a valued field $K$ with value group ${\mathbb R}$, the hypersum $a \boxplus b$ consists of all possible values of $v(\alpha+\beta)$ where $\alpha,\beta$ are elements of $K$ with $v(\alpha)=a$ and $v(\beta)=b$. (Note that the axioms for a valuation tell us that $v(\alpha \cdot\beta) = a + b$.) Concretely, this means that we should define $a \odot b := a + b$ (with the evident conventions when one of $a,b$ equals $\infty$), and we define $a \boxplus b := {\rm min}(a,b)$ if $a \neq b$ and $a \boxplus a := \{ z \in {\mathbb T} \; : \; z \geq a \}$. The multiplicative identity element is $0$ and the additive identity element is $\infty$. The negation operator is the identity map, since the unique value of $b$ such that $\infty \in a \boxplus b$ is $b = a$.

The above examples all have an additional property not shared by all hyperfields: they are doubly distributive (see below for the definition). Here are two examples of hyperfields which are not doubly distributive:

5. (The triangle hyperfield) Let ${\mathbb V}$ be the set ${\mathbb R}_{\geq 0}$ of nonnegative real numbers with the usual multiplication and the hyperaddition rule $a \boxplus b := \{ c \in {\mathbb R}_{\geq 0} \; : \; |a-b| \leq c \leq a+b \}.$ In other words, $a \boxplus b$ is the set of all real numbers $c$ such that there exists a (possibly degenerate) Euclidean triangle with side lengths $a, b, c$. Then ${\mathbb V}$ is a hyperfield.

6. (The phase hyperfield) The phase hyperfield ${\mathbb P}$ records the arithmetic of phases of complex numbers. If $\pi : {\mathbb C} \to {\mathbb P} := S^1 \cup \{ 0 \}$ is the map taking 0 to 0 and $z \in {\mathbb C}^*$ to $z/|z|$, and if $a,b \in {\mathbb P}$, the hypersum $a \boxplus b$ consists of all possible values of $\pi(\alpha + \beta)$ where $\alpha,\beta$ are elements of ${\mathbb C}$ with $\pi(\alpha)=a$ and $\pi(\beta)=b$. More precisely, multiplication in ${\mathbb P}$ is defined as usual, and the hyperaddition law is defined for $x,y \neq 0$ by setting $x \boxplus -x := \{ 0, x, -x \}$ and $x \boxplus y := \{ \frac{\alpha x + \beta y}{\| \alpha x + \beta y \|} \; | \; \alpha, \beta \in {\mathbb R}_{>0} \}$ otherwise.
Motivated by Proposition 1, if $F$ is a tract we define a strong $F$-matroid (or strong matroid over $F$) of rank $r$ on $E$ to be a projective equivalence class of Grassmann-Plücker functions $\varphi : E^r \to F$. Thus strong matroids over fields are the same as linear subspaces, and strong matroids over the Krasner hyperfield are the same as matroids in the usual sense. (By a matroid over a partial hyperfield $F$, we mean a matroid over the corresponding tract.) One can also show that strong matroids over a partial field $P$ are the same as matroids representable over $P$ in the usual sense.

Definition of hyperrings and hyperfields

A hyperoperation on a set $S$ is a map $\boxplus$ from $S \times S$ to the collection of non-empty subsets of $S$.

If $A,B$ are non-empty subsets of $S$, we define $A \boxplus B := \bigcup_{a \in A, b \in B} (a \boxplus b)$ and we say that $\boxplus$ is associative if $a \boxplus (b \boxplus c) = (a \boxplus b) \boxplus c$ for all $a,b,c \in S$.

A (commutative) hyperring is a set $R$ together with:

1. A commutative and associative binary operation $\odot$
2. A commutative and associative binary hyperoperation $\boxplus$
3. Distinguished elements $0,1 \in R$

such that:

1. $(R, \odot, 1)$ is a commutative monoid.
2. $0 \odot x = x \odot 0 = 0$ and $0 \boxplus x = x \boxplus 0 = \{ x \}$ for all $x \in R.$
3. For every $x \in R$ there is a unique element of $R$ (denoted $-x$) such that $0 \in x\boxplus -x.$
4. $a \odot (x \boxplus y) = (a \odot x) \boxplus (a \odot y)$ for all $a,x,y \in R.$

A hyperring $R$ is called a hyperdomain if $xy=0$ in $R$ implies that $x=0$ or $y=0$.

A hyperring $R$ is called doubly distributive if it satisfies $(a \boxplus b)\odot (c \boxplus d) = (a\odot c) \boxplus (a \odot d) \boxplus (b \odot c) \boxplus (b\odot d)$ for all $a,b,c,d \in R$.

A hyperring $F$ is called a hyperfield if $0 \neq 1$ and every non-zero element of $F$ has a multiplicative inverse.

Partial hyperfields

A partial hyperfield is a pair $(G,R)$, where $G$ is a subgroup of the group of units of a hyperdomain $R$ such that $-1 \in R$ and $G$ generates $R$ as a hyperring.

We set $P := G \cup \{ 0 \}$, and denote the partial hyperfield $(G,R)$ simply by $P$ when no confusion is likely to arise.

Partial hyperfields generalize both hyperfields and partial fields in a natural way.  When $R$ is a ring, $P$ is just a partial field, and when $G = R \backslash \{ 0 \}$ is a group, $P$ is just a hyperfield.

A partial hyperfield is called doubly distributive if the ambient hyperring $R$ is doubly distributive.

Tracts

Partial hyperfields are special cases of still more general objects called tracts, which appear to be a natural setting for matroid theory.

We define a tract to be an abelian group $G$ (written multiplicatively), together with a subset $N_G$ of the group semiring ${\mathbb N}[G]$ satisfying:

(T0) The zero element of ${\mathbb N}[G]$ belongs to $N_G$.
(T1) The identity element 1 of $G$ is not in $N_G$.
(T2) There is a unique element $\epsilon$ of $G$ with $1 + \epsilon \in N_G$.
(T3) $N_G$ is closed under the natural action of $G$ on ${\mathbb N}[G]$.

We set $F := G \cup \{ 0 \} \subset {\mathbb N}[G]$, and refer to the tract $(G,N_G)$ simply as $F$ when no confusion is likely to arise.  We will also sometimes write $F^\times$ instead of $G$, and $-1$ instead of $\epsilon$.

One thinks of $N_G$ as those linear combinations of elements of $G$ which “sum to zero” (the $N$ in $N_G$ stands for “null”).

Tracts from partial hyperfields

We can associate a tract to a partial hyperfield $(G,R)$ by declaring that a formal sum $\sum a_i g_i \in {\mathbb N}[G]$ belongs to $N_G$ if and only if $0 \in \boxplus a_i g_i$ in $R$.

We note, for the experts, that one can associate a tract to any fuzzy ring in the sense of Dress and Dress-Wenzel, and that if $P$ is a doubly distributive partial hyperfield there is an associated fuzzy ring whose realization as a tract coincides with the realization of $P$ itself as a tract.

Grassmann-Plücker functions over tracts

Now let $F$ be a tract.  A function $\varphi : E^r \to F$ is called a Grassmann-Plücker function if it is nonzero, alternating (meaning that $\varphi(x_1,\ldots,x_i, \ldots, x_j, \ldots, x_r)= -\varphi(x_1,\ldots,x_j, \ldots, x_i, \ldots, x_r)$ and $\varphi(x_1,\ldots, x_r) = 0$ if $x_i = x_j$ for some $i \neq j$), and it satisfies the following generalization of the classical Grassmann-Plücker relations:

(GP) For any two subsets $X := \{ x_1,\ldots,x_{r+1} \}$ and $Y := \{ y_1,\ldots,y_{r-1} \}$ of $E$,
$\sum_{k=1}^{r+1} (-1)^k \varphi(x_1,x_2,\ldots,\hat{x}_k,\ldots,x_{r+1}) \cdot \varphi(x_k,y_1,\ldots,y_{r-1}) \in N_G.$

If $F=K$ is a field then a projective equivalence class of Grassmann-Plücker functions $\varphi : E^r \to K$ is the same thing as an $r$-dimensional subspace of $K^m$.  This is essentially just the assertion that the usual Grassmannian variety is cut out by the Plücker equations.

On the other hand:

Proposition 1: If $F = {\mathbb K}$ is the Krasner hyperfield, a projective equivalence class of Grassmann-Plücker functions $\varphi : E^r \to {\mathbb K}$ is the same thing as a rank $r$ matroid on $E$.

Indeed, if $\varphi : E^r \to {\mathbb K}$ is a nonzero alternating function and ${\mathbf B}_\varphi$ denotes the set of $r$-element subsets $\{ e_1,\ldots, e_r \}$ of $E$ such that $\varphi(e_1,\ldots,e_r) \neq 0$, it is easy to see that ${\mathbf B} := {\mathbf B}_\varphi$ satisfies the Grassmann-Plücker relations (GP) if and only if

(BE) Given $B,B’ \in {\mathbf B}$ and $b \in B \backslash B’$, there exists $b’ \in B’ \backslash B$ such that both $(B \cup \{ b’ \}) \backslash \{ b \}$ and $(B’ \cup \{ b \}) \backslash \{ b’ \}$ are in ${\mathbf B}$.

But condition (BE) is well-known to be equivalent to the usual Basis Exchange property for matroids!  In other words, ${\mathbf B}_\varphi$ is the set of bases of a rank $r$ matroid $M_{\varphi}$ on $E$.  Conversely, given such a matroid $M$, we can define $\varphi_M : E^r \to {\mathbb K}$ by setting $\varphi_M(e_1,\ldots,e_r) = 1$ if $\{ e_1,\ldots,e_r \}$ is a basis of $M$ and $\varphi_M(e_1,\ldots,e_r) = 0$ if not.  By the exchange property (BE), the function $\varphi_M(e_1,\ldots,e_r)$ will satisfy (GP).

Motivated by Proposition 1, if $F$ is a tract we define a strong $F$-matroid (or strong matroid over $F$) of rank $r$ on $E$ to be a projective equivalence class of Grassmann-Plücker functions $\varphi : E^r \to F$.  Thus strong matroids over fields are the same as linear subspaces, and strong matroids over the Krasner hyperfield are the same as matroids in the usual sense.   (By a matroid over a partial hyperfield $F$, we mean a matroid over the corresponding tract.)  One can also show that strong matroids over a partial field $P$ are the same as matroids representable over $P$ in the usual sense.

We have the following additional interesting examples of (strong) matroids over tracts:

Proposition 2: If $F = {\mathbb S}$ is the hyperfield of signs, a matroid over ${\mathbb S}$ is the same thing as an oriented matroid.

Proposition 3: If $F = {\mathbb T}$ is the tropical hyperfield, a matroid over ${\mathbb T}$ is the same thing as a valuated matroid.

Proposition 4: If $F = {\mathbb U}_0$ is the regular partial field $(\{ \pm1 \}, {\mathbb Z})$, a matroid over ${\mathbb U}_0$ is the same thing as a regular matroid.

Weak matroids over tracts

It is also of interest to consider objects which satisfy a weaker version of (GP), where we consider only the three-term Grassmann-Plücker relations. Specifically, a weak $F$-matroid is a projective equivalence class of nonzero alternating functions $\varphi : E^r \to F$ such that (a) ${\mathbf B}_\varphi$ is the set of bases for a (classical) matroid on $E$, and (b) $\varphi$ satisfies (GP) for all $X,Y$ with $|X \backslash Y| = 3$.

It will turn out that in Propositions 1-4 above, strong and weak $F$-matroids are the same.

Circuit axioms for matroids over tracts

Let ${\mathcal C}$ be a collection of pairwise incomparable nonempty subsets of $E$. We say that $C_1,C_2 \in {\mathcal C}$ form a modular pair in ${\mathcal C}$ if $C_1 \neq C_2$ and $C_1 \cup C_2$ does not properly contain a union of two distinct elements of ${\mathcal C}$.

If $F$ is a tract and $X \in F^m$, we write $\underline{X}$ for the support of $X$, i.e., the set of $i \in \{ 1,\ldots,m \}$ such that $X_i \neq 0$. If ${\mathcal C} \subset F^m$ and $X,Y \in {\mathcal C}$, we say that $X,Y$ are a modular pair in ${\mathcal C}$ if $\underline{X},\underline{Y}$ are a modular pair in $\underline{\mathcal C} := \{ \underline{X} \; : \; X \in {\mathcal C} \}.$

Theorem 1: Let $F$ be a tract and let $E = \{ 1,\ldots,m \}$. There is a natural bijection between weak $F$-matroids on $E$ and collections ${\mathcal C} \subset F^m$ satisfying:

(C0) $0 \not\in {\mathcal C}$.
(C1) If $X \in {\mathcal C}$ and $\alpha \in F^\times$, then $\alpha \cdot X \in {\mathcal C}$.
(C2) [Incomparability] If $X,Y \in {\mathcal C}$ and $\underline{X} \subseteq \underline{Y}$, then there exists $\alpha \in F^\times$ such that $X = \alpha \cdot Y$.
(C3) [Modular Elimination] If $X,Y \in {\mathcal C}$ are a modular pair in ${\mathcal C}$ and $e \in E$ is such that $X_e= -Y_e \neq 0$, there exists $Z \in {\mathcal C}$ such that $Z_e=0$ and $X_f + Y_f – Z_f \in N_G$ for all $f \in E$.

We call ${\mathcal C}$ the set of $F$-circuits of the weak $F$-matroid $M$.

In [BB], there is also a stronger version of the circuit elimination axiom (C3) which gives a cryptomorphic characterization of strong $F$-matroids in terms of circuits. Let’s say that a family of atomic elements of a lattice is modular if the height of their join in the lattice is the same as the size of the family. If $\mathcal C$ is a subset of $F^m$, a modular family of elements of $\mathcal C$ is one such that the supports give a modular family of elements in the lattice of unions of supports of elements of $\mathcal C$.

Then there is a natural bijection between strong $F$-matroids on $E$ and collections ${\mathcal C} \subset F^m$ satisfying (C0),(C1),(C2), and the following stronger version of the modular elimination axiom (C3):

[Strong modular elimination] Suppose $X_1,\ldots,X_k$ and $X$ are $F$-circuits of $M$ which together form a modular family of size $k+1$ such that $\underline X \not \subseteq \bigcup_{1 \leq i \leq k} \underline X_i$, and for $1 \leq i \leq k$ let $e_i \in (X \cap X_i) \setminus \bigcup_{\substack{1 \leq j \leq k \\ j \neq i}} X_j$ be such that $X(e_i) = -X_i(e_i) \neq 0$. Then there is an $F$-circuit $Z$ such that $Z(e_i) = 0$ for $1 \leq i \leq k$ and $X_1(f) + \cdots + X_k(f) + X(f) – Z(f) \in N_G$ for every $f \in E$.

Duality

There is a duality theory for matroids over tracts which generalizes the known duality theories for matroids, oriented matroids, valuated matroids, etc., and which corresponds to orthogonal complementation for matroids over fields.

Let $F$ be a tract. The inner product of two vectors $X,Y \in F^m$ is $X \cdot Y := \sum_{i=1}^m X_i \cdot Y_i.$ We call $X$ and $Y$ orthogonal, denoted $X \perp Y$, if $X \cdot Y \in N_G$. If $S \subset F^m$, we denote by $S^\perp$ the orthogonal complement of $S$, i.e., the set of all $X \in F^m$ such that $X \perp Y$ for all $Y \in S$.

If $M$ is a (weak or strong) $F$-matroid on $E$ whose collection of circuits is denoted ${\mathcal C}$, then $\underline{M} := \{ \underline{X} \; : \; X \in {\mathcal C} \}$ is the collection of circuits of a matroid in the usual sense on $E$, which we call the underlying matroid of $M$.

Theorem 2: Let $F$ be a tract, and let $M$ be a (weak, resp. strong) $F$-matroid of rank $r$ on $E=\{ 1,\ldots,m \}$ with $F$-circuit set ${\mathcal C}$ and Grassmann-Plücker function $\varphi.$
Then there is a (weak, resp. strong) $F$-matroid $M^*$ of rank $m-r$ on $E$, called the dual $F$-matroid of $M$, with the following properties:

1. The $F$-circuits of $M^*$ are the elements of ${\mathcal C}^\perp$ of minimal non-empty support.

2. $M^*$ has the Grassmann-Plücker function $\varphi^*(x_1,\ldots,x_{m-r}) = {\rm sign}(x_1,\ldots,x_{m-r},x_1′,\ldots,x_r’) \varphi(x_1′,\ldots,x_r’),$ where $x_1′,\ldots,x_r’$ is any ordering of $E \backslash \{ x_1,\ldots,x_{m-r} \}.$

3. The underlying matroid of $M^*$ is the dual of the underlying matroid of $M$.

4. $M^{**} = M$.

The deepest part of Theorem 2 is the fact that the elements of ${\mathcal C}^\perp$ of minimal non-empty support satisfy the circuit elimination axiom (C3) (or its strong variant).

Dual pair axioms for matroids over hyperfields

One can give another cryptomorphic characterization of $F$-matroids using the notion of dual pairs. It is perhaps the simplest of all descriptions of matroids over tracts, but it presupposes that one already knows what a (usual) matroid is.

Let $M$ be a (classical) matroid on $E$. We call a subset ${\mathcal C}$ of $F^m$ an $F$-signature of $M$ if it is closed under multiplication by nonzero elements of $F$ and $\underline{\mathcal C} := \{ \underline{X} \; : \; X \in {\mathcal C} \}$ is the set of circuits of $M$.

We call $({\mathcal C},{\mathcal D})$ a dual pair of $F$-signatures of $M$ if:

(DP1) ${\mathcal C}$ is an $F$-signature of $M$.
(DP2) ${\mathcal D}$ is an $F$-signature of the dual matroid $M^*$.
(DP3) ${\mathcal C} \perp {\mathcal D}$, meaning that $X \perp Y$ for all $X \in {\mathcal C}$ and $Y \in {\mathcal D}$.

Theorem 3: Let $F$ be a tract and let $E = \{ 1,\ldots,m \}$. There is a natural bijection between strong $F$-matroids on $E$ and matroids $M$ on $E$ together with a dual pair of $F$-signatures of $M$.

An analogous theorem holds for weak $F$-matroids if we only require that (DP3) holds when $|\underline{X} \cap \underline{Y}| \leq 3$.

Vectors, perfection, and double distributivity

Given a tract $F$ and a strong $F$-matroid $M$ on $E$ with $F$-circuit set $\mathcal C$ and $F$-cocircuit set $\mathcal C^*$, a vector of $M$ is an element of $F^m$ which is orthogonal to everything in $\mathcal C^*$. Similarly a covector of $M$ is an element of $F^m$ which is orthogonal to everything in $\mathcal C$. We say that $F$ is perfect if, for any strong matroid $M$ over $F$, all vectors are orthogonal to all covectors.

Theorem 4: Let $F$ be a tract.

1. If $F$ is perfect, then the notions of weak and strong $F$-matroids coincide.

2. Every doubly distributive partial hyperfield is perfect.

As a consequence, we see that the notions of weak and strong $F$-matroids coincide for doubly distributive partial hyperfields $F$. This holds, in particular, when $F=P$ is a partial field or when $F$ is the Krasner hyperfield, the tropical hyperfield, or the hyperfield of signs.

There are examples in [BB] which show that weak and strong $F$-matroids do not coincide when $F$ is either the triangle hyperfield or the phase hyperfield.

Some directions for future research

There are many things one would like to know about matroids over tracts which aren’t yet well-understood. Here are some concrete problems which come to mind:

1. Does the theory developed in [BB] also work for tracts in which multiplication is not assumed to be commutative? It would be nice, for example, if one could fold the theory of representations of matroids over skew partial fields (as developed in this paper by Pendavingh and van Zwam) into our framework.
2. Laura Anderson has developed a theory of vectors and covectors for matroids over hyperfields. It would be interesting to resolve some of the conjectures in her paper, for example the Elimination Conjecture 7.2 or Conjecture 6.4 concerning flats.
3. Can one extend the Lift Theorem from this paper by Pendavingh and van Zwam to matroids over tracts?
4. Can one give an “algebraic” characterization of perfect tracts?

We can make Problem #4 a bit more precise. For any perfect tract, the matroids which are strongly representable over the tract are the same as the weakly representable ones, so they are determined by the set $N_G^{(3)}$ of elements of $N_G$ which are sums of at most 3 terms. On the other hand, any set of 3-term relations in $N_G$ can be extended to a perfect tract by just including everything of size at least 4 in $N_G$. The problem is that this includes a lot of extra stuff in $N_G$ which might not be needed for strong representability. This motivates looking at the smallest tract extending $N_G^{(3)}$ which could possibly be perfect, namely the set of all combinations which appear as sums in the GP relations (or the orthogonality relations) for weak matroids over $N_G^{(3)}$. Could it be that if this tract is perfect then it must satisfy certain algebraic conditions which are also sufficient to guarantee perfection?


text:
# Trigonometry Examples

The sine of an angle is equal to the ratio of the opposite side to the hypotenuse.
Substitute the name of each side into the definition of the sine function.
Set up the equation to solve for the opposite side, in this case .
Substitute the values of each variable into the formula for sine.
Multiply by .


text:
# Tag Info

19

<x,y|x^2=y^3=1>≅PSL_2(Z) and this isomorphism identifies G with $PSL_2/T^7=1$ (where $T:z\mapsto z+1$). Result is the symmetry group of the tiling of the hyperbolic plane. From this description one can see that G is infinite (e.g. because there are infinitely many triangles in the tiling and G acts on them transitively).

12

Grigory has already answered your particular question. However, I wanted to point out that your question "How do you prove that a group specified by a presentation is infinite?" has no good answer in general. Indeed, in general the question of whether a group presentation defines the trivial group is undecidable.

11

$F_2$ acts on a certain tiling of the hyperbolic plane. It looks sort of like this: The above tiling is acted on by the modular group $\Gamma \cong \text{PSL}_2(\mathbb{Z})$, which naturally sits as a subgroup inside of the full group $\text{PSL}_2(\mathbb{R})$ of isometries of the hyperbolic plane. Abstractly, this group is the free product $C_2 \ast ... 8 As I said in the comments above, this has a positive solution due to a recent preprint of Lars Louder, which can be found here. The paper proves that surface groups have a "single Nielsen equivalence class of generating$2g$-tuples". I'll explain what this means, and then I will explain why this solves the problem. (In the comments below, Lars Louder has ... 8 The answer to your question is yes: See Bekka–de la Harpe–Valette, Kazhdan's property$(T)$, page 69 (the standard reference on property (T), freely available on Bekka's homepage): The proof is not very difficult, and it is given in a clear fashion in the book, so it doesn't make much sense to reproduce it here. Note also ... 7 This might not be the best argument, but it seems to work. First, no nontrivial finite, connected, vertex-transitive graph has a cutpoint (nontrivial here means something like cardinality at least$3$to avoid having to define cutpoint carefully). If it did, then every vertex would be a cutpoint, so you could inductively build an arbitrarily long finite ... 7 This counterexample seems to work for$\frac{1}{2}$, and beyond: $$x_g:=Cr^{\ell(g)}\qquad\forall g\in F_2$$ for the right choices of constants $$C>0\qquad\mbox{and}\qquad 0\leq r<\frac{1}{\sqrt{3}}.$$ Here$\ell(g)$denotes the word length of the element$g$, i.e. the minimal number of letters needed in the alphabet ... 7 More generally, any group$G$defined by a finite presentation with more generators than relations is infinite - in fact$G/[G,G]$is infinite. That follows from the proof of the fundamental theorem of abelian groups. You can prove it directly, by showing that there is a nontrivial epimorphism$\phi$onto${\mathbb Z}$. Let$\phi:G \to {\mathbb Z}$be any ... 7 First consider an unpainted$2\times 2\times 2$Rubik's cube: What is the "symmetry group"? Before we can discuss the symmetry group of this cube (or any Rubik's cube), we must be clear about what operations are allowed. There are three pertinent questions: Do rigid rotations of the cube count as "symmetries"?(If no, we must somehow exclude them.) Are ... 7 I'm an outsider, so maybe what I'm saying is silly. I would just think about the Tits alternative application since that really gives all the intuition. To apply it---and we'll just look at the complex case---you should produce two matrices$A$and$B$that have the following properties.$A$has a dominant eigenvalue$\lambda$(i.e., the eigenspace of ... 6 Let$z = x^2 = y^3$. This element clearly commutes with$x$and$y$. Therefore$z$lies in the center and$\langle z \rangle$is a normal subgroup. We have: $$G / \langle z \rangle = \langle x, y \mid x^2 = y^3 = 1\rangle.$$ Now the abelianization of this quotient group is$\Bbb Z_2 \times \Bbb Z_3$, which is clearly non-trivial. 6 You need to modify the definition of$\Psi$: put $$\Psi(t^na^xt^{-n})=xk^n$$ (instead of$\Psi(t^na^xt^{-n})=xk^{-n}$). The homomorphic property should follow easily along the lines of the computations given in the original question. Alternatively, you could replace$t$by$t^{-1}$in the definition of the Baumslag-Solitar group, in which case the$\Psi$... 6 You can do this using metric currents in the sense of Ambrosio-Kirchheim. This is a rather new development of geometric measure theory, triggered by Gromov and really worked out only in the last decade. I should warn you that this is rather technical stuff and nothing for the faint-hearted. Urs Lang has a set of nice lecture notes, where you can find most ... 5 If you think of$F$as acting by left multiplication on the vertices of$\Delta$, then the natural interpretation of the quotient graph is as a graph whose vertices are the distinct cosets$Gg$of$G$in$F$, with an edge (labelled$x$) from$Gg \to Ggx$for each$x \in \{a,a^{-1},b,b^{-1}\}$. So, if you ignore the labels, it is still a an infinite ... 5 If$G,H$are groups then$H$is a double cover of$G$iff$H$has a normal subgroup$K$of order 2 such that$K \leq [H,H] \cap Z(H)$and$H/K \cong G$. In other words,$H$has an element$z$that (1) commutes with every element of$H$, (2) can be written as a product of commutators$z=\left(h_1^{-1} h_2^{-1} h_1 h_2\right)\cdots\left(h_{2n-1}^{-1} ...

5

Remark: For those unacquainted with Property $(T)$, the standard reference is the freely available book by Bekka-de la Harpe-Valette. Since groups with property $(T)$ are finitely generated, we can assume that the rank of the free group $F$ is finite. If a group $G$ has property $(T)$ then so does every quotient(1), recall also my answer to your ...

5

Inspiration: in the free product $H=C_2*C_2=\langle a,b|a^2,b^2\rangle$ the element $ab$ has infinite order. And furthermore the only nontrivial coset of $\langle ab\rangle$ is $b\langle ab\rangle$ (every word in $H$ is either a product of $ab$s or it is $b$ followed by a product of $ab$s), so $\langle ab\rangle$ has index two. The given information in this ...

5

The amalgamation only identifies two isomorphic subgroups, it doesn't perform any further quotienting. So the group you are amalgamating over has to be isomorphic to a subgroup of both $A$ and $B$. Thus your first statement is correct. You cannot form an amalgamated free product of $F_2$ and a finite group with non-trivial amalgamation, because $F_2$ ...

5

Take $G = S_{3}$, $H = U = \langle (12) \rangle = \{ 1, (12) \}$. As a system of right coset representatives, take $R = \{ 1, (123), (132) \}$. Now choose $u = (12) \in U$, and $s = (123) \ne (132) = s'$. We have $$u s H = (12) (123) H = (13) H = \{ (13), (132) \} = (132) H = s' H.$$ Barring mistakes.

5

Yes, your interpretation is correct. There are no further relations since there are no loops in the Cayley graph except those deducible from the relations $a^2 = b^2 = e$. The group $\langle a, b \mid a^2, b^2 \rangle$ is known as the infinite dihedral group.

5

Gromov in his original 1987 book (Section 3.1) wrote a classification for arbitrary isometric group actions on hyperbolic spaces (with no further assumption) into 5 main classes. It goes at follows (the terminology is borrowed from here) 1: bounded: orbits are bounded 2: horocyclic: orbits are unbounded, $G$ acts with no hyperbolic isometry (hence there's ...

4

(Of course the trivial group acts freely on all spheres! Let's assume $G$ is finite and nontrivial.) Since $G$ is finite, the action is certainly properly discontinuous, and since the action is given to be free, it follows that the quotient $q: X \rightarrow X/G$ is a finite covering map. Recalling that in any $n$-sheeted covering map $q: X \rightarrow Y$ ...

4

If this was the trivial group, then it would follow that for any group $G$ and $x,y\in G$ with $x^2=y^3$ it follows that $x=y=1$. However that is not true, for example consider elements of order $2$ and $3$ in the symmetric group on $3$ elements. (This example also shows that $x$ and $y$ need not commute, so they do not commute in your group.) Similarly, ...

4

Firstly, $H \leq \operatorname{comm}_G(H)$ always, so if $H$ is a non-trivial subgroup, its commensurator is non-trivial. A natural question to ask, then, is how much bigger than $H$ its commensurator in $G$ is. The basic situations of $H \leq G$ where $\operatorname{comm}_G(H)$ is much bigger than $G$, but smaller than $G$, is when $G$ is a Lie group and ...

4

1) $G = \bigoplus_{i=1}^{\infty} \mathbb{Z}/2\mathbb{Z}$ is residually finite but not Hopfian. It is clearly not Hopfian, and any non-identity element $g$ must have $g_i \neq 0$ for at least one $i$. Then $\pi_i(G) \cong \mathbb{Z}/2\mathbb{Z}$ is a finite quotient in which the image of $g$ is nontrivial. 2) An infinitely generated free group is ...

4

We call them Cayley graphs (it doesn't seem to be usual to say 'Cayley digraph'), and they are interesting. I am doing research on semigroups, and quite often draw the Cayley graph of a semigroup to get an idea of what it's like. I don't know how much background on semigroups you have, but one reason Cayley graphs for semigroups are interesting is that the ...

4

If you want to have a group theory approach, then you are right. These graphs are not interesting! However if you have a semigroup background and you know about automaton and semiautomaton, you can find these graphs very interesting (see Kilp and Knauer's book). So the second answer is not very correct and fair since the identity and inverse elements are not ...

4

Yes if you assume a group has a normal subgroup $Z$ of finite index which is infinite cyclic then it's much easier. If $Z$ is central, then a classical theorem shows that the derived subgroup is finite. So modding out by the derived subgroup, you get an abelian group, and eventually get that the group has a homomorphism onto $\mathbf{Z}$ (with finite kernel, ...

4

Word Metric: There is something called the word metric for groups, which is a key tool in geometric group theory. Consider a group given by some generators, so $G=\langle X\rangle$. Then you can draw the Cayley graph of $G$ with respect to these generators. Take two elements $g, h\in G$, then define $d(g, h)$ to be the minimum distance in the Cayley graph ...

4

From linear algebra we know that every invertible matrix can be obtained from the identity matrix by a sequence of elementary row operations. The corresponding elementary matrices therefore generate the general linear group. Now you can find pre-images in ${\mathrm Aut}(F_{n})$ by considering Nielsen transformations, which match up fairly directly with the ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
Question 85

# If the amount received at 10% per annum Compound interest after 3 yrs is Rs 19965, then what will have been the principal (in Rs) amount?

Solution

P$$(1+\frac{10}{100})^{3}$$ = 19965

P$$(1+\frac{1}{10})^{3}$$ = 19965

P$$(\frac{11}{10})^{3}$$ = 19965

P$$(\frac{1331}{1000})$$ = 19965

P = 15000

So the answer is option B.


text:
# Calculating standard error after a log-transform

Consider a random set of numbers that are normally distributed:

x <- rnorm(n=1000, mean=10)

We'd like to know the mean and the standard error on the mean so we do the following:

se <- function(x) { sd(x)/sqrt(length(x)) }
mean(x) # something near 10.0 units
se(x)   # something near 0.03 units

Great!

However, let's assume we don't necessarily know that our original distribution follows a normal distribution. We log-transform the data and perform the same standard error calculation.

z <- log(x, base=10)
mean(z) # something near 1 log units
se(z)   # something near 0.001 log units

Cool, but now we need to back-transform to get our answer in units NOT log units.

10^mean(z) # something near 10.0 units
10^se(z)   # something near 1.00 units

My question: Why, for a normal distribution, does the standard error differ depending on whether it was calculated from the distribution itself or if it was transformed, calculated, and back-transformed? Note: the means came out the same regardless of the transformation.

EDIT #1: Ultimately, I am interested in calculating a mean and confidence intervals for non-normally distributed data, so if you can give some guidance on how to calculate 95% CI's on transformed data including how to back-transform to their native units, I would appreciate it!
END EDIT #1

EDIT #2: I tried using the quantile function to get the 95% confidence intervals:

quantile(x, probs = c(0.05, 0.95))     # around [8.3, 11.6]
10^quantile(z, probs = c(0.05, 0.95))  # around [8.3, 11.6]

So, that converged on the same answer, which is good. However, using this method doesn't provide the exact same interval using non-normal data with "small" sample sizes:

t <- rlnorm(10)
mean(t)                            # around 1.46 units
10^mean(log(t, base=10))           # around 0.92 units
quantile(t, probs = c(0.05, 0.95))                     # around [0.211, 4.79]
10^(quantile(log(t, base=10), probs = c(0.05, 0.95)))  # around [0.209, 4.28]

Which method would be considered "more correct". I assume one would pick the most conservative estimate?

As an example, would you report this result for the non-normal data (t) as having a mean of 0.92 units with a 95% confidence interval of [0.211, 4.79]?
END EDIT #2

• SE is SD divided by square root of N. Not just N. – Penguin_Knight Nov 11 '14 at 10:13
• Thanks! I fixed that problem. The issue I am having remains though. – baffled Nov 12 '14 at 4:11

Your main problem with the initial calculation is there's no good reason why $e^{\text{sd}(\log(Y))}$ should be like $\text{sd}(Y)$. It's generally quite different.

In some situations, you can compute a rough approximation of $\text{sd}(Y)$ from $\text{sd}(\log(Y))$ via Taylor expansion.

$$\text{Var}(g(X))\approx \left(g'(\mu_X)\right)^2\sigma^2_X\,.$$

If we consider $X$ to be the random variable on the log scale, here, $g(X)=\exp(X)$

If $\text{Var}(\exp(X))\approx \exp(\mu_X)^2\sigma_X^2$

then $\text{sd}(\exp(X))\approx \exp(\mu_X)\sigma_X$

These notions carry across to sampling distributions.

This tends to work reasonably well if the standard deviation is really small compared to the mean, as in your example.

> mean(y)
[1] 10
> sd(y)
[1] 0.03
> lm=mean(log(y))
> ls=sd(log(y))
> exp(lm)*ls
[1] 0.0300104

If you want to transform a CI for a parameter, that works by transforming the endpoints.

If you're trying to transform back to obtain point estimate and interval for the mean on the original (unlogged) scale, you will also want to unbias the estimate of the mean (see the above link): $E(\exp(X))\approx \exp(\mu_X)\cdot (1+\sigma_X^2/2)$, so a (very) rough large sample interval for the mean might be $(c.\exp(L),c.\exp(U))$, where $L,U$ are the upper and lower limits of a log-scale interval, and $c$ is some consistent estimate of $1+\sigma_X^2/2$.

If your data are approximately normal on the log scale, you may want to treat it as a problem of producing an interval for a lognormal mean.

• Thanks Glen_b. I never learned that in statistics class. – baffled Nov 12 '14 at 14:09
• I don't have the reputation to comment, but just in case another curious soul happens across this post, from looking at the Taylor expansion link on Wikipedia, the correct estimate for the mean should be $$\begin{eqnarray*}\text{E}[f(X)] &\approx& f(\mu_X)+\frac{f^{\prime\prime}(\mu_X)}{2}\sigma_X^2\\ &=& \exp(\mu_X)\left(1 +\frac{\sigma_X^2}{2}\right) \end{eqnarray*}$$ Otherwise as if $\exp(\mu_x)\gg\sigma_X^2$, you might underestimate $\text{E}[\exp(X)]$ – deasmhumnha Dec 19 '16 at 18:13
• Thanks @Dezmond. Yes, that's correct. I'll add in a correction to my answer, that part of it near the end is quite mangled. – Glen_b Dec 19 '16 at 22:48

It sounds like you effectively want the geometric standard error, akin to the geometric mean exp(mean(log(x))).

While it might seem reasonable to compute that as:

exp(sd(log(x)/sqrt(n-1)))

You and others have already pointed out that that isn't correct for a few reasons. Instead, use:

exp(mean(log(x))) * (sd(log(x))/sqrt(n-1))

Which is the geometric mean multiplied by the log-standard error. This should approximate the "natural" standard error pretty well.


text:
## anonymous 3 years ago Find the coordinates of the point where the vector line $$\vec{r} = (6 + \frac{7}{9}d)\vec{i} + (3 + \frac{4}{9}d)\vec{j} + (2 - \frac{4}{9}d)\vec{k}$$ intersects the plane 3x - 7y + z = 5. I'd like an explanation or better yet, a guide through it rather than just a plain answer.

1. anonymous

holy good luck with that!

2. anonymous

-_-, thanks lol

3. anonymous

you know that: x=6+7/9d, y=3+4/9d, and z=2-4/9d ?

4. anonymous

Yes. I actually think I might have an answer since I asked this yesterday. Could you check it?

5. anonymous

I sence De Moivre's Theorum comming on here =)

6. anonymous

d = -54/11, so (24/11, 9/11, 46/11) Is this right?

7. anonymous

In your question you've mentioned i..... Is that i as in i, or i as in the Square root of -1???

8. anonymous

i as in unit vector $$\vec{i}$$

9. anonymous

How so?

10. anonymous

Of course. Thank you :)

11. anonymous

that 7/9 or what, cant see the problem clearly

12. anonymous

7/9

13. anonymous

maybe I'm wrong in sth

14. anonymous

d= -54/11

15. anonymous

Here's my work if it helps... $\vec{r} = (6 + \frac{7}{9}d)\vec{i} + (3 + \frac{4}{9}d)\vec{j} + (2 - \frac{4}{9}d)\vec{k}$$3(6 + \frac79d) - 7(3 + \frac49d) + (2 - \frac49d) = 5$$18 + \frac{21}{9}d - 21 - \frac{28}{9}d + 2 - \frac49d = 5$$-\frac{11}9d = 6$$d = 6 \times -\frac9{11}$$d = -\frac{54}{11}$

16. anonymous

lol, that's right, sr

17. anonymous

lol ok, thanks :)

18. anonymous

So just plug in to find coordinate. Is this cal 3?


text:
# Using Fruitbat¶

Here is the guide of how to use the fruitbat package.

## Frb Class¶

Most calculations in fruitbat are centred around the class Frb. Hence you will need to define a fruitbat.Frb object for each FRB. The minimum required to define a Frb is the observed dispersion measure (DM).

>>> import fruitbat
>>> frb = fruitbat.Frb(635.1)

It’s also possible to ‘name’ the FRB using the keyword name, which can be useful when calculating redshifts of many FRBs at once.

>>> import fruitbat
>>> frb = fruitbat.Frb(635.1, name="FRB190229")

To see a full list of possible parameters visit the API documentation for Frb.

### Redshift Estimation¶

To estimate the redshift of the FRB, use the method calc_redshift(). Unless otherwise specified this assumes that the entire DM contribution is due to the IGM. To see how to account for the Milky Way and host galaxy contributions to the DM see the sections on Galactic Dispersion Measure and Host Galaxy Dispersion Measure respectively.

>>> frb = fruitbat.Frb(635.1)
>>> frb.calc_redshift()
<Quantity 0.63199287>

The calc_redshift() also has keywords to select alternative IGM models and cosmologies when estimating the redshift of the FRB (Default method and cosmology are Inoue2004 and Planck2018 respectively).

>>> frb.calc_redshift(method="Zhang2018")
<Quantity 0.70986024>

>>> frb.calc_redshift(method="Ioka2003", cosmology="Planck13")
<Quantity 0.52776778>

Currently avaliable methods in fruitbat include: Ioka2003, Inoue2004, Zhang2018. Currently avaliable cosmologies include: WMAP5, WMAP7, WMAP9, Planck13, Planck15, Planck18, EAGLE. It should be of note that EAGLE uses the Planck13 cosmology but is listed here for convenience.

#### Galactic Dispersion Measure¶

When estimating the redshift of an FRB, it is necessary to account for the Milky Way’s contribution to the observed DM. Depending on the location on the sky, this contribution can be as low as $$~30\ \rm{pc\ cm^{-3}}$$ out of the disk of the Milky Way and exceeding $$1000\ \rm{pc\ cm^{-3}}$$ through the disk. Without accuretly accounting for this contribution, redshift estimates of FRB would be significantly higher than their physical redshift.

Within fruitbat there are three main ways to account for the galactic contribution: calc_dm_galaxy(), dm_galaxy or dm_excess.

##### Method 1: calc_dm_galaxy()¶

The first and easiest way to account for the galactic contribution is to provide the sky coordinates of the FRB when instantiating the object, then call calc_dm_galaxy(). The calc_dm_galaxy() method of Frb estimates the total DM contribution due to the Milky Way along the line of sight of the FRB using the YMW16 galactic free electron model.

>>> frb = fruitbat.Frb(635.1, gl="35.1", gb="12.5")
>>> frb.calc_dm_galaxy()
<Quantity 114.27922821 pc / cm3>

>>> frb = fruitbat.Frb(635.1, raj="18:10:34.8668", decj="7:33:35.9289")
>>> frb.calc_dm_galaxy()
<Quantity 114.27922821 pc / cm3>

The sky coordinates can be in either ICRS or Galactic units. The calc_dm_galaxy() method will calculate the dm_excess by subtracting the estimated dm_galaxy from the observed DM. After calculating dm_galaxy, calling calc_redshift() will automatically use the calculated dm_excess to estimate the redshift.

>>> frb.calc_redshift()
<Quantity 0.52244866>

##### Method 2: dm_galaxy¶

The second method to account for the galactic dispersion meausre is to provide a value of dm_galaxy. This value could be calculated from another galactic dispersion measure model such at the NE2001 model, which has not been implemented in fruitbat yet.

>>> frb = fruitbat.Frb(635.1, dm_galaxy=114.28)
>>> frb.calc_redshift()
<Quantity 0.52244791>

##### Method 3: dm_excess¶

The third and final method is to directly specify the dm_excess. This provides the value that will be used for the redshift calculation directly.

>>> frb = fruitbat.Frb(635.1, dm_excess=520.82)
>>> frb.calc_redshift()
<Quantity 0.52244791>

#### Host Galaxy Dispersion Measure¶

It’s common to assume that the host galaxy contributes nothing to the observed dispersion measure, however this is unlikely to be true. Use the parameter dm_host_est to provide an estimate of the contribution to the observed dispersion measure due to the FRB host and set specify 'subtract_host=True' in the method calc_redshift().

>>> frb = fruitbat.Frb(635.1, gl="35.1", gb="12.5", dm_host_est=64.1)
>>> frb.calc_dm_galaxy()
>>> frb.calc_redshift()
<Quantity 0.52244866>
>>> frb.calc_redshift(subtract_host=True)
<Quantity 0.46077303>

### Calculating Distances¶

Other than redshift, fruitbat has two other distance functions however both require the redshift to be calculated first. These distance functions are calc_comoving_distance() and calc_luminosity_distance()

#### Comoving Distance¶

To calculate the comoving distance use calc_comoving_distance() after calculating the redshift.

$d_C(z) = \frac{c}{H_0} \int_0^z \frac{dz'}{\sqrt{\Omega_r (1+z)^4 + \Omega_m (1+z)^3 + \Omega_{\Lambda}}}$
>>> frb = fruitbat.Frb(635.1, gl="35.1", gb="12.5")
>>> frb.calc_dm_galaxy()
>>> frb.calc_redshift()
>>> frb.calc_comoving_distance()
<Quantity 2020.29768846 Mpc>

#### Luminosity Distance¶

To calculate the luminosity distance use calc_luminosity_distance() after calculating the redshift.

$d_L(z) = (1 + z) d_C(z)$
>>> frb = fruitbat.Frb(635.1, gl="35.1", gb="12.5")
>>> frb.calc_dm_galaxy()
>>> frb.calc_redshift()
>>> frb.calc_luminosity_distance()
<Quantity 3075.79950018 Mpc>

### Calculating Energy¶

The method calc_energy() estimates the upper limit to the isotropic energy of the FRB. To use the method calc_energy(), the FRB requires the following properties: fluence and obs_freq_central. fluence can be calculated by providing peak_flux and width.

>>> frb.fluence = 2.0
>>> frb.obs_freq_central = 1600
>>> frb.calc_energy()
<Quantity 2.37921847e+40 erg>

### Calculating Luminosity¶

The method calc_luminosity() estimates the upper limit to the isotropic peak luminosity of the FRB. To use the method calc_luminosity(), the FRB requires the following properties: peak_flux and obs_freq_central.

>>> frb.peak_flux = 0.1
>>> frb.obs_freq_central = 1600
>>> frb.calc_luminosity()
<Quantity 1.81111898e+42 erg / s>

## Custom Lookup Tables¶

### Custom Methods¶

>>> def simple_dm(z):
dm = 1200 * z
return dm
>>> fruitbat.available_methods()
['Ioka2003', 'Inoue2004', 'Zhang2018', 'simple_dm']

### Custom Cosmologies¶

>>> params = {"H0": 72.4, "Om0": 0.26}
>>> new_cosmology = fruitbat.cosmologies.create_cosmology(parameters=params)
>>> fruitbat.available_cosmologies()
['WMAP5', 'WMAP7', 'WMAP9', 'Planck13', 'Planck15', 'Planck18', 'EAGLE', 'new_cosmology']

### Custom Table¶

To create a custom lookup table you first need to define a custom method and add that method to the list of avaliable methods using add_method(). Then you can use create() to generate a lookup table of that method.

>>> def simple_dm(z):
dm = 1200 * z
return dm


text:
IFC 4.3.0.1 (IFC4X3) development

# 6.4.3.4 IfcInventory

## 6.4.3.4.1 Semantic definition

An inventory is a list of items within an enterprise.

Various types of inventory can be included. These are identified by the range of values within the inventory type enumeration which includes space, asset, and furniture. User defined inventories can also be defined for lists of particular types of element such as may be required in operating and maintenance instructions. Such inventories should be constrained to contain a list of elements of a restricted type.

There are a number of actors that can be associated with an inventory, each actor having a role. Actors within the scope of the project are indicated using the IfcRelAssignsToActor relationship in which case roles should be defined through the IfcActorRole class; otherwise principal actors are identified as attributes of the class. In the existence of both, direct attributes take precedence.

There are a number of costs that can be associated with an inventory, each cost having a role. These are specified through the CurrentValue and OriginalValue attributes.

## 6.4.3.4.4 Property sets

• Pset_Risk
• RiskName
• RiskType
• NatureOfRisk
• RiskAssessmentMethodology
• UnmitigatedRiskLikelihood
• UnmitigatedRiskConsequence
• UnmitigatedRiskSignificance
• MitigationPlanned
• MitigatedRiskLikelihood
• MitigatedRiskConsequence
• MitigatedRiskSignificance
• MitigationProposed
• AssociatedProduct
• AssociatedActivity
• AssociatedLocation

## 6.4.3.4.6 Formal representation

ENTITY IfcInventory
SUBTYPE OF (IfcGroup);
PredefinedType : OPTIONAL IfcInventoryTypeEnum;
Jurisdiction : OPTIONAL IfcActorSelect;
ResponsiblePersons : OPTIONAL SET [1:?] OF IfcPerson;
LastUpdateDate : OPTIONAL IfcDate;
CurrentValue : OPTIONAL IfcCostValue;
OriginalValue : OPTIONAL IfcCostValue;
END_ENTITY;


text:
# Article

Methods of Experimental Investigation and Measurements
1971. V. 9. № 5.
Stefanov S.R., Trokhan A.M.
Article reference:
Stefanov S.R., Trokhan A.M. , High Temp., 1971. V. 9. № 5. P. 0


text:
# Estimating the rental rate of capital from data

Take the classic optimization problem of the neo-classical firm:

$$\begin{array}{*2{>{\displaystyle}r}} \mbox{maximize (over K, L)} & f(K, L) - RK - WL \end{array}$$

The first order condition equates the marginal product of capital with the rental rate of capital $R$.

Which raises the question...

### How do macroeconomists typically estimate a time series for the rental rate of capital using U.S. data?

(Disclaimer: I don't do macro, but my curiosity has been sparked.)

One approach is to use financial market data to get $R_t$, use another first order condition that the rental rate of capital equals the nominal interest rate plus depreciation (i.e. Hall and Jorgensen). Backing a rental rate out of financial market data is not obvious though! In financial markets, prices vary based upon risk and time.

• time dimension: Long-term rates are typically higher than short-term rates. When macro-economists and macro-models talk about the rental rate of capital, what's the time frame?
• risk dimension: Eg. callable bonds have high yields than non-callable bonds. Debt and equity may have different expected returns based upon risk.
• inflation dimension: A fixed nominal rate is a stochastic real rate depending on realized inflation, and the expected real rate is the nominal rate minus inflation expectations. The real rental rate would add back inflation expectations.

A completely different direction is taken by Casey Mulligan where he sticks entirely with Bureau of Economic Analysis (BEA) data.

I don't follow this literature, and I don't have a sense of the range of approaches that are considered sensible in modern, empirical macro.

• Really late but why not have $MPK=r$? – EconJohn Feb 8 '18 at 21:41
• @EconJohn Could you expand what you mean? – Matthew Gunn Feb 8 '18 at 22:05
• Well, considering classic marginal productivity theory of wages which states the relationship of $\text{MPL}=\text{w}$, a simple extention can be drawn to capital rental rate where $\text{MPK}=\text{r}$. – EconJohn Feb 9 '18 at 3:51
• @EconJohn So then to measure marginal productivity of capital, you have to go through an exercise like this? – Matthew Gunn Feb 9 '18 at 4:26
• I'd say so. This paper is excellent, however they don't relate MPK to the price of capital- I'm suggesting (in a lack of an actual rental data) that solving where $\text{MPK=r}$ is not a bad idea. the formula of having $MPK=\alpha \frac{Y}{K}$ seems appealing. – EconJohn Feb 9 '18 at 17:55

Time series on rental price of capital can be estimated using $$r=\frac{P_k}{P}(i-inf+\delta)$$ here, $$P_k$$ is the price of capital goods (price index for capital goods), $$P$$ is a deflator, $$i$$ is nominal interest rate, $$inf$$ is an inflation rate and $$\delta$$ is depreciation rate of physical capital stock.