text:
# Enriching Offline Reinforcement Learning Algorithms in ReinforcementLearning.jl

This is the phase 2 technical report of the summer OSPP project Enriching Offline Reinforcement Learning Algorithms in ReinforcementLearning.jl. The report is split into the following parts: Project Information and Completed Work.

### Table of content

1. Technical Report

# Technical Report

This technical report is the second phase technical report of Project "Enriching Offline Reinforcement Learning Algorithms in ReinforcementLearning.jl" in OSPP. It includes two components: project information and completed work.

## Project Information

• Project name: Enriching Offline Reinforcement Learning Algorithms in ReinforcementLearning.jl

• Scheme Description: Recent advances in offline reinforcement learning make it possible to turn reinforcement learning into a data-driven discipline, such that many effective methods from the supervised learning field could be applied. Until now, the only offline method provided in ReinforcementLearning.jl is Behavior Cloning (BC). We'd like to have more algorithms added like Batch Constrain deep Q-Learning (BCQ), Conservative Q-Learning (CQL). It is expected to implement at least three to four modern offline RL algorithms.

• Time planning: the following is a relatively simple time table.

DateWork
Prior - June 30Preliminary research, including algorithm papers, ReinforcementLearning.jl library code, etc.
The first phase
July1 - July15Design and build the framework of offline RL.
July16 - July31Implement and experiment offline DQN and offline SAC as benchmark.
August1 - August15Write build-in documentation and technical report. Implement and experiment CRR.
The second phase
August16 - August31Implement and experiment PLAS.
September1 - September15Research, implement and experiment new SOTA offline RL algorithms.
September16 - September30Write build-in documentation and technical report. Buffer for unexpected delay.
After projectCarry on fixing issues and maintaining implemented algorithms.

## Completed Work

It summarizes all the results of the second phase.

#### Offline RL algorithms

In the second phase, we finished the offline RL algorithms: BCQ, Discrete-BCQ, BEAR(UWAC), FisherBRC. CRR and PLAS are detailed in the report of the first phase. Except for algorithms in discrete action space (such as Discrete-BCQ, Discrete-CRR), all algorithms can run on CPU and GPU, because CartesianIndex is not supported in GPU.

##### BCQ

Batch-Constrained deep Q-Learning (BCQ) uses distribution matching to constrain policy. The pseudocode of BCQ:

BCQ needs to train a VAE to sample actions based on the next state. We train VAE for a period of time, and then train VAE and BCQ learner simultaneously to ensure the quality of the generated actions. When sampling actions, we randomly samples hidden states instead of generating hidden states in PLAS. Therefore, we re-implement decode function in VAE:

function decode(rng::AbstractRNG, model::VAE, state, z=nothing; is_normalize::Bool=true)
if z === nothing
# In BCQ case. In PLAS case if z !== nothing.
z = clamp.(randn(rng, Float32, (model.latent_dims, size(state)[2:end]...)), -0.5f0, 0.5f0)
z = send_to_device(device(model), z)
end
a = model.decoder(vcat(state, z))
if is_normalize
a = tanh.(a)
end
return a
end

And it needs a perturbation network:

Base.@kwdef struct PerturbationNetwork{N}
base::N
ϕ::Float32 = 0.05f0
end

Perturbation network is used to increase the robustness of the action:

function (model::PerturbationNetwork)(state, action)
x = model.base(vcat(state, action))
x = model.ϕ * tanh.(x)
clamp.(x + action, -1.0f0, 1.0f0)
end

mutable struct BCQLearner{BA1, BA2, BC1, BC2, V} <: AbstractLearner
### Omit other parameters
policy::BA1
target_policy::BA2
qnetwork1::BC1
qnetwork2::BC2
target_qnetwork1::BC1
target_qnetwork2::BC2
vae::V
p::Int
start_step::Int
end

In BCQ, we use PerturbationNetwork: $\pi(s,a)\rightarrow a$ to model policy and use $Q(s,a)$ to model Q-network.

p is a hyper-parameter used for repeating states to obtain better Q estimation. For example, BCQ repeats states and obtains the actions and Q-values. Then it chooses an action with the highest Q-value.

function (l::BCQLearner)(env)
s = send_to_device(device(l.policy), state(env))
s = Flux.unsqueeze(s, ndims(s) + 1)
s = repeat(s, outer=(1, 1, l.p))
action = l.policy(s, decode(l.vae.model, s))
q_value = l.qnetwork1(vcat(s, action))
idx = argmax(q_value)
action[idx]
end

start_step represents the steps to train VAE alone.

Performance curve of BCQ algorithm in Pendulum (start_step=1000):

##### Discrete BCQ

Discrete BCQ is a simple offline RL algorithm in discrete action space. Its pseudocode is as follow:

The core of Discrete BCQ is calculating a mask of actions. We calculate the probability of actions in a given state and divide it by the maximum probability value. Then, we compare this value with the threshold. If this value is less than the threshold, the corresponding action will not be selected.

prob = softmax(learner.approximator.actor(s), dims=1)
mask = Float32.((prob ./ maximum(prob, dims=1)) .> learner.threshold)
new_q = q .* mask .+ (1.0f0 .- mask) .* -1f8

mutable struct BCQDLearner{Aq, At} <: AbstractLearner
### Omit other parameters
approximator::Aq
target_approximator::At
θ::Float32
threshold::Float32
end

In Discrete BCQ, we use the Actor-Critic structure. The Critic is modeled as $Q(s,\cdot)$, and the Actor is modeled as $L(s,\cdot)$ (likelihood of the state).

θ is the regularization coefficient. When calculating the loss in Discrete BCQ, it uses the likelihood of the state to regularize:

logit = AC.actor(s)
actor_loss + critic_loss + θ * mean(logit .^ 2)

Performance curve of Discrete BCQ algorithm in CartPole:

##### BEAR & UWAC

Bootstrapping error accumulation reduction (BEAR) is a policy-constraint method, which uses support constrain instead of distribution matching in BCQ. Its pseudocode is as follow:

The equation 1:

$\pi_\phi := \max_{\pi \in \triangle_{|S|}}\mathbb{E}_{s\in \mathcal{D}}\mathbb{E}_{a\sim\pi(\cdot|s)}\big[\min_{j=1,\cdots,K}\hat{Q}_j(s,a)\big] \\ s.t. \mathbb{E}_{s\in\mathcal{D}}[{\rm MMD}(\mathcal{D}(s),\pi(\cdot|s))]\leq \varepsilon$

The Actor update is the main improvement part of BEAR. We need to train a VAE to simulate sampling action from the dataset. The training of the VAE and the learner are synchronized. Then we sample action by the Actor and calculate maximum mean discrepancy (MMD) loss:

function maximum_mean_discrepancy_loss(raw_sample_action, raw_actor_action, type::Symbol, mmd_σ::Float32=10.0f0)
A, B, N = size(raw_sample_action)
diff_xx = reshape(raw_sample_action, A, B, N, 1) .- reshape(raw_sample_action, A, B, 1, N)
diff_xy = reshape(raw_sample_action, A, B, N, 1) .- reshape(raw_actor_action, A, B, 1, N)
diff_yy = reshape(raw_actor_action, A, B, N, 1) .- reshape(raw_actor_action, A, B, 1, N)
diff_xx = calculate_sample_distance(diff_xx, type, mmd_σ)
diff_xy = calculate_sample_distance(diff_xy, type, mmd_σ)
diff_yy = calculate_sample_distance(diff_yy, type, mmd_σ)
mmd_loss = sqrt.(diff_xx .+ diff_yy .- 2.0f0 .* diff_xy .+ 1.0f-6)
end

The loss of actor is:

$J=-Q_{\hat{\theta}}(s,a) + \alpha \times {\rm MMD}(\mathcal{D}(s),\pi(\cdot|s))$

Besides, lagrangian multiplier $\alpha$ needs to be updated synchronously:

$J=-Q_{\hat{\theta}}(s,a) + \alpha \times ({\rm MMD}(\mathcal{D}(s),\pi(\cdot|s))-\varepsilon)$

mutable struct BEARLearner{BA1, BA2, BC1, BC2, V, L} <: AbstractLearner
### Omit other parameters
policy::BA1
target_policy::BA2
qnetwork1::BC1
qnetwork2::BC2
target_qnetwork1::BC1
target_qnetwork2::BC2
vae::V
log_α::L
ε::Float32
p::Int
max_log_α::Float32
min_log_α::Float32
sample_num::Int
kernel_type::Symbol
mmd_σ::Float32
end

In BEAR, we use $Q(s, a)$ to model the Q-network and use a gaussian network to model the policy. log_α is a Lagrange multiplier implemented by a NeuralNetworkApproximator.

ε is used to update the Lagrangian multiplier. p is a hyper-parameter like BCQ. max_log_α and min_log_α are used to clamp the log_α. sample_num represents how many samples we sample to calculate MMD loss. mmd_σ is used to adjust the size of MMD loss. kernel_type=:laplacian/:gaussian represents what method we use to calculate MMD loss:

function calculate_sample_distance(diff, type::Symbol, mmd_σ::Float32)
if type == :gaussian
diff = diff .^ 2
elseif type == :laplacian
diff = abs.(diff)
else
error("Wrong parameter.")
end
return vec(mean(exp.(-sum(diff, dims=1) ./ (2.0f0 * mmd_σ)), dims=(3, 4)))
end

Performance curve of BEAR algorithm in Pendulum:

Uncertainty Weighted Actor-Critic (UWAC) is an improvement of BEAR. It can be implemented by adding Dropout in BEAR's Q network.

##### FisherBRC

FisherBRC is a policy-constraint offline RL algorithm, which uses Fisher distance to constrain policy. The pseudocode of FisherBRC:

Firstly, it needs to pre-train a behavior policy $\mu$ using Behavior Cloning. In official python implementation, it adds an entropy term in the negative log-likelihood of actions in a given state. Mathematical formulation:

$\mathcal{L}(\mu) = \mathbb{E}[-\log \mu(s|a) + \alpha \mathcal{H}(\mu)]$

Besides, it automatically adjusts entropy term like SAC:

$J(\alpha) = -\alpha \mathbb{E}_{a_t\sim \mu_t}[\log\mu(a_t|s_t) + \bar{\mathcal{H}}]$

Where $\bar{\mathcal{H}}$ is target entropy. But in ReinforcementLearningZoo.jl, BehaviorCloningPolicy does not contain the entropy term and does not support continuous action space. So, we define EntropyBC:

mutable struct EntropyBC{A<:NeuralNetworkApproximator}
policy::A
α::Float32
lr_alpha::Float32
target_entropy::Float32
# Logging
policy_loss::Float32
end

Users only need to set parameter policy and lr_alpha. policy usually uses a GaussianNetwork. lr_alpha is the learning rate of α, which is an entropy term. target_entropy is set to $-\dim(\mathcal{A})$, and $\mathcal{A}$ is action space.

Afterward, the FisherBRC learner is updated. When updating Actor, it adds an entropy term in Q-value loss and automatically adjusts entropy. It updates Critic by this equation:

$\min_\theta J(O_\theta + \log\mu(a|s)) + \lambda \mathbb{E}_{s\sim D, a\sim \pi_\phi(\cdot|s)}[\|\nabla_a O_\theta(s,a)\|^2]$

There are a few key concepts that need to be introduced. $J$ is the standard Q-value loss function. $O_\theta(s,a)$ is offset network:

$Q_\theta(s,a) = O_\theta(s,a) + \log\mu(a|s)$

Instead of $Q_\theta(s,a)$, $O_\theta(s,a)$ will provide a richer representation of Q-values. However, this parameterization can potentially put us back in the fully-parameterized $Q_\theta$ regime of vanilla actor critic. So it uses a gradient penalty regularizer of the form $\|\nabla_a O_\theta(s,a)\|$. The implementation is as follows:

a_policy = l.policy(l.rng, s; is_sampling=true)
q1 = l.qnetwork1(q_input) |> vec
q1_reg = mean(l.qnetwork1(vcat(s, a_policy)))
end
loss = mse(q1 .+ log_μ, y) + l.f_reg * reg  # y is target value
end

mutable struct FisherBRCLearner{BA1, BC1, BC2} <: AbstractLearner
### Omit other parameters
policy::BA1
behavior_policy::EntropyBC
qnetwork1::BC1
qnetwork2::BC2
target_qnetwork1::BC1
target_qnetwork2::BC2
α::Float32
f_reg::Float32
reward_bonus::Float32
pretrain_step::Int
lr_alpha::Float32
target_entropy::Float32
end

In FisherBRC, policy is modeled as gaussian network and Q-network is modeled as $O(s,a)$.

f_reg is the regularization parameter of $\|\nabla_a O_\theta(s,a)\|$. reward_bonus is generally set to 5, which is added in the reward to improve performance. pretrain_step is used for pre-training behavior_policy. α, lr_alpha and target_entropy are parameters used to add an entropy term and automatically adjust the entropy.

Performance curve of FisherBRC algorithm in Pendulum (pertrain_step=100):

#### Offline RL experiments

To show performance of the above algorithms, we add some built-in experiments in ReinforcementLearningExperiments.jl.

If it is an algorithm on the continuous action space, we test it in Pendulum, otherwise we test it on CartPole.

We collect the dataset by gen_JuliaRL_dataset function (source):

struct JuliaRLTransition
state
action
reward
terminal
next_state
end

function gen_JuliaRL_dataset(alg::Symbol, env::Symbol, type::AbstractString; dataset_size::Int)
dataset_ex = Experiment(
Val(:GenDataset),
Val(alg),
Val(env),
type;
dataset_size = dataset_size)

run(dataset_ex)

dataset = []
s, a, r, t = dataset_ex.policy.trajectory.traces
for i in 1:dataset_size
push!(dataset, JuliaRLTransition(s[:, i], a[i], r[i], t[i], s[:, i+1]))
end
dataset
end

Specifically, we first run the built-in data collection experiment. For CartPole environment, we use BasicDQN to collect the dataset. For Pendulum, SAC is used. Then we turn the trajectory into a vector of JuliaRLTranstion.

We can run the offline RL experiments with the following commands (list all supported commands):

run(EJuliaRL_BCQ_Pendulum(type))
run(EJuliaRL_BCQD_CartPole(type))
run(EJuliaRL_BEAR_Pendulum(type))
run(EJuliaRL_CRR_Pendulum(type))
run(EJuliaRL_CRR_CartPole(type))
run(EJuliaRL_FisherBRC_Pendulum(type))
run(EJuliaRL_PLAS_Pendulum(type))

type represents the method of collecting dataset, which can be random, medium or expert. random means dataset generated by the random agent, and medium uses an agent trained from scratch to convergence, and expert uses an agent that has converged. For example:

run(EJuliaRL_BCQ_Pendulum(medium))

## Conclusion

The implemented algorithms in this project contain most of the policy constraint methods in offline reinforcement learning (including distribution matching, support constrain, implicit constraint, behavior cloning). And these algorithms cover both discrete and continuous action spaces. Besides, we have them tested with various datasets, including random, medium, and expert.

### Corrections

If you see mistakes or want to suggest changes, please create an issue in the source repository.


text:
# Importing formatted, highlighted text from : HTML vs RTF vs. XML

We would like the ability to import and analyze ASCII documents that have been manually segmented by highlighting and changing font colors of various sections, eg:

The original line breaks, spacing and other formatting need to be preserved upon import, and in addition, need to extract the highlight or font color.

To identify a suitable format we are using MS Word for highlighting because we can then save as .rtf, .htm and .xml.

Clearly importing plain text won't contain highlight info, so on to structured formats:

So far not having much luck with MMA9's RTF:

Import[NotebookDirectory[] <> "/example-highlighting-word.rtf", "RTF"]

gives:

Notebook[{Cell[
TextData[{"I found out some info",
" on this type of acrylic..  It is called an exotic type of \
acrylic that shows a colored edge when it is cut.  Comes in different \
color types but is not a stock item at San Diego Plastics and is \
pretty expensive.", "

", "Also, the cost of ",
"a 4' x 4' sheet of 1/2\" acrylic for my discount is \$150.

", "Did you figure out the exact measurements of the two \
pieces.", "

", StyleBox["I ran some test pieces and they fit. ",
FontColor -> RGBColor[1., 0.4, 0.]],
StyleBox[
" I think we will have to get some new bolts for the larger \
pieces because the heads a", FontColor -> RGBColor[1., 0.4, 0.]],
StyleBox["re to", FontColor -> RGBColor[1., 0.4, 0.]],
StyleBox["o", FontColor -> RGBColor[1., 0.4, 0.]],
StyleBox[
" tall and I would have to make too deep of a cut to hide the \
bolt head; which would end up making a weaker attachment point.",
FontColor -> RGBColor[1., 0.4, 0.]], StyleBox["
", FontSize -> 12, FontColor -> RGBColor[1., 0.4, 0.]]}],
"Input", CharacterEncoding -> "MacintoshRoman"]},
WindowSize -> {740, 867},
WindowMargins -> {{910, Automatic}, {Automatic, 239}},
FrontEndVersion ->
"9.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (November 20, 2012)",
StyleDefinitions -> "Default.nb"]

Clearly the generated Notebook structure doesn't match the original highlighting segments.

Next we saved and then imported XML:

Import[NotebookDirectory[] <> "/example-highlighting-word.xml", "XML"]

And won't copy the output but it's lengthy, it breaks up individual sentences and the data schema is binary so would be difficult to write pattern rules to extract content and highlights.

Finally, saving and importing HTML as "Text" looks promising:

Import[NotebookDirectory[] <>
"/example-highlighting-word.htm", "Text"]

The output is less voluminous than XML (though still lengthy), but there are still Word-specific HTML tags to deal with, for example:

I found out some info on this type \
of acrylic..  It is called an exotic type of acrylic \
that shows a
colored edge when it is cut.  Comes in different color types but \
is not a
stock item at San Diego Plastics and is pretty expensive.<o:p></o:p>

Question - besides is RTF broken? - Is there documentation to handle and pattern match the MS-generated HTML tags like , and to discover potential variations in tags. For example, we'd like to recover the number of new-lines separating paragraphs (as well as indentations), but are faced with composite tags like the following that seems to represent a single newline in the original document:

 <o:p></o:p>

Any better ways to go about it - ie, other formats or Import options? (It's not necessary to use MS Word, maybe other rich text editors are more conducive to proper RTF?)

-

Seems to work fine for me. I placed a simple RTF file (created by TextEdit on my Mac) on my webspace so you can try it.

NotebookPut[
Import["http://facstaff.unca.edu/mcmcclur/temp.rtf"] /.
Cell[td_TextData, "Input", opts___] ->
Cell[td, "Text", opts], WindowTitle -> "ImportedRTF.nb"];

The one problem I did have was that the resulting cell was an Input cell, where I suppose that Text cell makes more sense. I added a bit of code to convert any cell of the form Cell[td_TextData, ___] to a Text cell. The result looks like so:

If you require further diagnosis, you might want to make your /example-highlighting-word.rtf file publicly available.

-
Great thanks, +1 - I didn't mention however that the files we're dealing with contain protected health information and are located on PC/Windows machines. We will try to use TextEdit, but did you try the same content on MS Word? –  alancalvitti Dec 10 '12 at 3:58
in addition to font color, is it possible to highlight as per MS Word? –  alancalvitti Dec 10 '12 at 18:31
@alancalvitti It is certainly possible to highlight text in a Mathematica notebook. I don't actually have MSWord, though, so it's a little hard for me know. I did quickly create a document with highlighting using Google Drive and, after export to RTF, the highlighting was lost. Does RTF even support highlighting? As I said before, you should really make your own sample file available. –  Mark McClure Dec 10 '12 at 19:21
What I meant is - is it possible to import already highlighted text (preserving the markup)? Word enables highlighting (but importing it creates the issues I described above), but your example only shows font color changes. –  alancalvitti Dec 10 '12 at 19:25
@alancalvitti Yes, that's what I understood you to mean. I'm telling you that, in spite of the fact that one can highlight text in the Mathematica Notebook, there's no way for me to answer your question without access to your "already highlighted text". –  Mark McClure Dec 10 '12 at 19:41
show 1 more comment

format: plain


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# Diy led

by Reddogracing
Tags: None
 Sci Advisor P: 4,032 I see Radio Shack have these transformers: Model: 273-1512 | Catalog #: 273-1512 Which sound like they would be suitable. $10.49 is a fair price. They also have these capacitors: Model: 272-1047 | Catalog #: 272-1047 for$2.59 They are not "LOW ESR" but may be OK to try. If they get hot, you might have to get the good ones. And they have these bridge rectifiers: Model: 276-1185 | Catalog #: 276-1185 for \$3.29 Oddly, they don't seem to have 390 ohm 1/2 watt resistors although they do have 330 ohm ones which would be OK if you couldn't get anything else. I have a flashlight with 19 LEDs in it and this can light up our whole back yard. I wonder if you have low output LEDs and might do better with higher output ones?


text:
# Confusion about $\limsup$ properties in proof of ratio test

In the proof of the ratio test in my notes, which is similar to the one here, the first step revolves around an obvious statement which is not proved. However, for whatever reason, I can't make out why it is at all obvious!

Note for our case, $$a_n\geq0$$ for all $$n$$. The comment in question is that $$\limsup_{n\to\infty} \frac{a_{n+1}}{a_n}<1$$ implies that there exists $$\epsilon>0$$ such that $$\frac{a_{n+1}}{a_n}<1-\epsilon$$ for all $$n\geq N$$, for some $$N$$.

I have also seen this post which seems to have include the same comment, but it doesn't seem to explain it in detail.

I can't figure out why this is true, but I suspect it is obvious, so any guidance would be much appreciated!

Edit: Our definition of $$\limsup$$ is $$\limsup_{n\to\infty} a_n = \lim_{N\to\infty}\sup_{n\geq N} a_n$$, or that it is the maximal limit point of $$a_n$$.

• What is the definition of $\limsup$? – Peanut Jun 27 '20 at 12:24
• @Dude Clarified in edit – Benjamin Jun 27 '20 at 12:28
• Yes. So assume on the contrary that your inequality does not hold. Is it true that $\lim sup \frac{a_{n+1}}{a_n} < 1$? No because that means that an infinite number of those ratios is getting close to something greater than or equal to $1$. – Peanut Jun 27 '20 at 12:39

This can be argued very closely from definitions, although it's just making rigorous the intuitive notion that the ratios must maintain some distance from $$1$$, as otherwise they would tend to at least $$1$$, making their $$\limsup$$ at least $$1$$.

Let $$\ell = \limsup_{n \to \infty} a_{n + 1}/a_n$$. We are given that $$\ell < 1$$, so certainly $$\tfrac 12(1 - \ell) > 0$$.

Since $$\ell = \lim_{m \to \infty} \sup_{n \ge m} a_{n + 1}/a_n$$, by the definition of the limit of a sequence (letting $$\varepsilon = \tfrac 12(1 - \ell)$$), we can find some $$N$$ such that for all $$m \ge N$$, $$\sup_{n \ge m} a_{n + 1}/a_n - \ell < \tfrac 12(1 - \ell)$$. (Here I have omitted the modulus as the sequence $$(\sup_{n \ge m} a_{n + 1}/a_n)$$ is decreasing as a sequence in $$m$$, so $$\ell \le \sup_{n \ge m} a_{n + 1}/a_n$$).

Particularly, taking $$m = N$$, we get $$\sup_{n \ge N} a_{n + 1}/a_n - \ell < \tfrac 12(1 - \ell)$$.

But then, we have $$\sup_{n \ge N} a_{n + 1}/a_n < \tfrac 12(1 + \ell) = 1 - \tfrac 12(1 - \ell)$$. Since $$\sup$$ is an upper bound, for any $$k \ge N$$, we get $$a_{k + 1}/a_k \le \sup_{n \ge N} a_{n + 1}/a_n < 1 - \tfrac 12(1 - \ell)$$.

So $$N$$ is our $$N$$, and our $$\varepsilon$$ is $$\tfrac 12(1 - \ell)$$.

You might notice that this is actually a general fact about $$\limsup$$ of a sequence with a strict upper bound. We didn't need to use positivity of $$a_n$$ for this step of the proof.

• Can one think of it this way? $\sup a_{n+1}/a_n < 1+\epsilon$ for all $\epsilon>0$, so we know that $\sup a_{n+1}/a_n < 1$ and hence $a_{n+1}/a_n < 1$, so this ratio must be less than or equal to $1-\epsilon$, for some $\epsilon>0$. – Benjamin Jun 27 '20 at 14:31
• @Benjamin, I don't follow exactly what you're saying. Do you mean $\limsup$? I don't see what the $1 + \varepsilon$ is doing. $x < 1 + \varepsilon$ for all $\varepsilon > 0$ doesn't imply that $x < 1$. It also seems like at the end you show that $a_{n + 1} / a_n < 1 - \varepsilon$ for some $\varepsilon$, but in your question you need to show that the same epsilon works for all $n$. Perhaps that's what you're arguing but it would be made clearer by saying something like "for all $n$". – Izaak van Dongen Jun 27 '20 at 14:40
• You're right, of course, this was a silly argument. I follow your proof, but I don't see what's quite so 'obvious' about it. I'm trying to also understand why the statement is 'morally right', so to speak. – Benjamin Jun 27 '20 at 15:09
• @Benjamin, The way I think of it is that the $\limsup$ of a sequence is a sort of "eventual upper bound". After a while, the sequence must quieten down and stay below any number above the $\limsup$. All this argument says is "if the eventual upper bound is strictly less than $1$, then we can pull the sequence down below $1$". Certainly if this claim were not true, then the sequence would be at least $1$ infinitely often, so it wouldn't make any sense for the eventual upper bound to be less than $1$. Maybe that helps. If not, don't worry. Analysis can be hard to build intuition about. – Izaak van Dongen Jun 27 '20 at 15:14
• This does help, thank you. – Benjamin Jun 27 '20 at 15:18

let us call the $$limsup$$ as $$p$$, now $$p<1$$ , define $$e =(1-p)/2$$ so $$1>1 - e>p$$ , if all point after some certain $$n$$ didn't lie below $$1-e$$, (let us call $$1-e$$ as $$\beta$$), we can find infinitely many point of the seq which lies above $$\beta$$ (why?). we can always create a subseq which converges to a number (in the extended reals) $$\alpha$$ greater than or equal $$\beta$$ .So it is a limit of a subseq greater than the limsup, which is a contradiction!


text:
## Ngspice to PSpice conversion

Hi All,

I have a piece of Ngspice code that I need to convert to PSpice. I feel it is a simple matter of syntax but I have so far failed to make it work.

The code models a Memristor with thresholding ability (the fourth fundamental component along with resistors, capacitors, and inductors)

The code is as follows:

.subckt memristor plus minus PARAMS: Ron=1K Roff=10K Rinit=5K alpha=0 beta=1E13 Vt=4.6
Bx 0 x I='(f1(V(plus)-V(minus))>0) && (V(x)<Roff) ? {f1(V(plus)-V(minus))}: (f1(V(plus)-V(minus))<0) && (V(x)>Ron) ? {f1(V(plus)-V(minus))}: {0}'
Cx x 0 1 IC={Rinit}
Rmem plus minus r={V(x)}
.func f1(y)={beta*y+0.5*(alpha-beta)*(abs(y+Vt)-abs(y-Vt))}
.ends

The code explanation is as follows:
- The memristive system is realized as a sub-circuit combining a behavioral resistor R, a current source ↑, and a capacitor C.
- The second line of code (Bx ...) defines the current source with the current specified through ternary functions. (A ternary function is defined in the code as a ? b : c , which means ”IF a, THEN b, ELSE c” [17].) The purpose of these functions is to limit RM between Ron and Roff.
- The third line of the code specifies the capacitor C (Cx ...) with an initial condition.
- The fourth line (Rmem ...) defines the behavioral resistor whose resistance takes the same numerical value as the voltage across the capacitor.
- The next line (.func ...) provides the function f according to Equation (not important here)

If you can help me in any way that would be great!!

Many thanks
Captainkirk
 PhysOrg.com engineering news on PhysOrg.com >> Mathematical algorithms cut train delays>> Researchers design software to detect changes in colour vision>> Trend study identifies potential for humans and technology to interact in a manufacturing environment
 I can help you. But can you be a little more clear with simple circuit diagram and simple expressions?
 Hi there, thanks for replying. I've attached the circuit diagram as image. PLEASE NOTE - I dont need help with the Model, I just need help with syntax. Thats the code I gave in my first post just needs its syntax tweaked so it is compatible with Pspice. The research paper that I'm referring to states the following for the Spice model: "SPICE model: The memristive devices with threshold that we consider are those described by I = X-1VM X ̇ = (βVM +0.5(α−β)[|VM +Vt|−|VM −Vt|]) where Vt is the threshold voltage, Ron and Roff are limiting values of the memristance RM ≡ X, and the θ-functions Note - These equations are represented in ngspice code I gave in my first post. - The important model parameters are the coefficients α and β that characterize the rate of memristance change at |VM|Vt , respectively. - These two coefficients define the slopes of the f (VM) curve below and above the threshold. When α = 0, the device state changes only if |VM| > Vt" Please let me know if this makes it clearer. I appreciate any help you can provide here! Many thanks again CaptainKirk

## Ngspice to PSpice conversion

attached file
Attached Thumbnails

 Captainkirk, I really did not try to understand the model. Actually I could not make out the ngspice syntax. I am working on it. You'll get your spice model after sometime.
 Kholdstare, see the ngspice-users forum (I am not allowed to give the link here). captainkirk is engaging at least three forums ... But is was nice just to get it up and running. Regards Holger
 Here's your PSpice macromodel as requested. To use it in your circuit just copy the "* memres ... begins" to "* memres ... ends" portion to your circuit and call the model using "X(ref) (node1) (node2) R_MEMR PARAMS: VCUT=(param) ALPHA=(param) BETA=(param)" device card. Code: * memresistor model begins .MODEL SWD D IS=1E-16 N=0.001 .FUNC GET_R1(ALPHA) {1/ALPHA} .FUNC GET_R2_R3(ALPHA,BETA) {1/ABS(BETA-ALPHA)} * ABS is given to prevent negative resistance .SUBCKT R_MEMR 1 2 * Default parameters defined below + PARAMS: VCUT=1V ALPHA=0.1M BETA=0.2M R1 1 2 {GET_R1(ALPHA)} D2 1 3 SWD V2 3 4 {VCUT} R2 4 2 {GET_R2_R3(ALPHA,BETA)} D3 5 1 SWD V3 6 5 {VCUT} R3 6 2 {GET_R2_R3(ALPHA,BETA)} .ENDS * memresistor model ends *Test ckt * The voltage source V1 1 0 0Vdc * The memresistor (new parameters are passed as arguments) XMEMR_1 1 0 R_MEMR PARAMS: VCUT=4.6V ALPHA=0 BETA=1E-3 .DC V1 -8 8 0.01 .PROBE .END The test parameters I used are Vcut = 4.6V, alpha = 0, beta = 1e-3 The test result is in the attachment. I simplified your expression as written below, I = a*V for |V| < Vt = a*V+(b-a)(V-Vt) for V>Vt = a*V+(b-a)(V+Vt) for V
 Sorry for the late. I had actually forgotten about it after your last post. Shows kinda lazy person I am. haha
 Kholdstare Thank you were much with your reply!! I can hardly call you lazy, after all you did come up with something very interesting. I'm very impressed with your model. Its is different from what I came up with but I still like the output graph that you attached! Thank you once again Kind regards, Kirk

 Similar discussions for: Ngspice to PSpice conversion Thread Forum Replies Electrical Engineering 4 Set Theory, Logic, Probability, Statistics 0 Engineering, Comp Sci, & Technology Homework 4 Electrical Engineering 1 Electrical Engineering 5


text:
# Comparing data sets¶

Last week, we discussed the comparison between two related multinomial proportions. This week, we'll step back toward the textbook and discuss the comparison between two means or two proportions.

In order, this material is covered in sections 7.2, 7.3, and 6.2 of our text.

## Textbook prices¶

I'm going to present some code to manipulate some data that all college students should be somewhat interested - textbook prices. Specifically, I've got a CSV file that contains prices from the UNCA bookstore and Amazon for more than 500 textbooks that were in use at UNCA during the Fall of 2018 semester.

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

### The data¶

Here's the data:

import pandas as pd
print(df.shape)

(518, 4)

title course bookstore_new_price amazon_max_price
0 MyAccountingLab Access Card -- for Financial A... ACCT 215 122.25 99.99
1 Financial Accounting (Loose Pgs)(w/MyAccountin... ACCT 215 263.50 214.95
2 Managerial Accounting (Loose Pgs)(w/MyAcctLab ... ACCT 216 263.25 234.33
3 Intermediate Accounting (WileyPlus Standalone ... ACCT 301 171.25 88.99
4 Intermediate Accounting (Loose Pgs)(w/ Wiley P... ACCT 301 297.00 183.31

## Comparing prices between the sellers (paired data)¶

"Common knowledge" suggest that Amazon prices might generally be lower than our bookstore's prices. If we want to compare the prices between the two vendors like this, we can simply compute the pairwise difference.

differences = df.bookstore_new_price - df.amazon_max_price
m = differences.mean()
m

18.29876447876449

I guess that computation means that the textbook price is $\$18.30$more expensive at the UNCA bookstore on average. ### A picture of the differences¶ Of course, the complete picture is more complicated than just the mean of the differences. Here's a histogram of those differences, with the mean marked by a vertical line: ### A hypothesis test¶ Now, let's explore the question: are Amazon prices genuinely cheaper the UNCA bookstore prices to a statistically significant level? To do so, let's first clearly state our null and alternative hypotheses. Let$\mu_U$denote the average price of the books at the UNCA bookstore and let$\mu_A$denote the average price of the books at Amazon. $$H_0: \mu_A = \mu_U \\ H_A: \mu_A < \mu_U$$ ### Rephrasing using differences¶ Since the data are paired, we can rephrase this by taking the pairwise difference of the data sets to get a single data set. If for each row, we take the Amazon price minus the UNCA price to get a single data set with mean$\mu$, then we could rephrase our hypotheses as $$H_0: \mu = 0 \\ H_A: \mu < 0$$ ### Computation¶ Note that we've already computed an estimate of$\mu$to be$\overline{x}=18.8$, denoted by m in our code. To examine the hypothesis, we compute a standard errror and test statistic from our set of differences. se = np.std(differences)/np.sqrt(len(differences)) se  1.669252423192934 T = (m-0)/se T  10.962251259616412 Given this huge test statistic, the$p$-value will be riduculously small. ## Comparing prices between disciplines (unpaired data)¶ What do you suppose is more expensive - the average textbook in the sciences or the average textbook in the humanities? This is a perfectly reasonable question but the data is not paired in a natural way, as in our last example. In this case, we'll compute the two means separately, take the difference, and run a hypothesis test using a combined standard error. ### Grabbing the data¶ sci_prefixes = ['ATMS', 'BIOL', 'CHEM', 'CSCI', 'MATH', 'PHYS', 'STAT'] sci_df = df[[c[:4] in sci_prefixes for c in df.course]] sci_prices = sci_df.bookstore_new_price m_sci = np.mean(sci_prices) hum_prefixes = ['CLAS', 'DRAM', 'HUM ', 'LIT ', 'PHIL', 'RELS'] hum_df = df[[c[:4] in hum_prefixes for c in df.course]] hum_prices = hum_df.bookstore_new_price m_hum = np.mean(hum_prices) [m_sci, len(sci_prices), m_hum, len(hum_prices)]  [154.0701234567901, 81, 36.0403875968992, 129] Well, the average price of a science textbook indeed looks a lot more than the average price of a humanities textbook! ### The general scenario¶ Let's place this example in the context of a general scenario. In particular, suppose that •$D_1$and$D_2$are two sets of sample observations, • with$n_1$and$n_2$elements respectively. • The mean of$D_1$is$\bar{x}_1$and the mean of$D_2$is$\bar{x}_2$• The standard deviation of$D_1$is$\sigma_1$and the standard deviation of$D_2$is$\sigma_2$### The general strategy¶ We'll analyze the difference of the two means using a hypothesis test with • Mean$\overline{x}_1 - \overline{x}_2$, • Standard error $$\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}},$$ • and we use the minimum of$n_1-1$and$n_2-1$as the degrees of freedom, if the smaller is less than 30 ### The test statistic¶ The form of the test statistic for this scenario is $$\frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$ For our problem, this boils down to: se_sci = np.std(sci_prices)/np.sqrt(len(sci_prices)) se_hum = np.std(hum_prices)/np.sqrt(len(hum_prices)) (m_sci - m_hum)/np.sqrt(se_sci**2 + se_hum**2)  10.877145243602286 Again, a very large test statistic so that we'd reject the null. ## Comparing two sample proportions¶ The ideas behind the comparison of two sample proportions is very similar to the ideas behind the comparison of two sample means. Let's examine the question in the context of a specific problem: It's widely believed that Trump's support among men is stronger than his support among women. We'll explore this question with some data. ### The data¶ According to a recent Marist's poll Trump's most recent approval rating stands at 41% but there appears to be a difference between the views of men and the views of women. Among the 712 men surveyed, 48% approve of Trump. Among the 685 women surveyed, only 36% approve of Trump. Does this data support our conjecture that Trump's support among men is higher than that among women to a 95% level of confidence? ### The hypothesis test¶ Let's first clearly state our hypotheses. Let's suppose that$p_m$represents the proportion of men who support Trump and$p_w$represent the proportion of women who support Trump. Our hypothesis test can be written •$H_0: p_m = p_w$or, put another way,$p_m-p_w = 0$•$H_A: p_m > p_w$or, put another way,$p_m-p_w > 0$. The point behind the reformulation to compare with zero is that it gives us just one number that we can apply a standard hypothesis test to. ### Applying the data¶ Now, we have measured proportions of$\hat{p}_m = 0.48$and$\hat{p}_w = 0.36$. Thus, we want to run our test with $$\hat{p} = \hat{p}_m - \hat{p}_w = 0.48 - 0.36 = 0.12.$$ We want just one standard error as well, which we get by adding the variances in the two samples. That is, $$SE = \sqrt{\frac{\hat{p}_m(1-\hat{p}_m)}{n_m} + \frac{\hat{p}_w(1-\hat{p}_w)}{n_w}} = \sqrt{\frac{0.48 \times 0.52}{712} + \frac{0.36 \times 0.64}{684}} \approx 0.026218.$$ Of course, I computed this with Python: pm = 0.48 pw = 0.36 se = np.sqrt(pm*(1-pm)/712 + pw*(1-pw)/684) se  0.026218388642629563 ### The test statistic¶ We can now compute our test statistic: $$T = \frac{\hat{p}_m - \hat{p}_w}{SE}.$$ via t = (pm-pw)/se t  4.576940316037845 With this very large test statistic, we can reject the null hypothesis and conclude with confidence that there is a difference between the way that men and women view Trump. ## Small sample sizes¶ As we move forward this semester, we will need to ask whether our sample meets the conditions to apply some specific distribution - most often, the normal distribution. In particular, when the sample size is small, we'll typically use a$t$-distribution. Let's look at such an example. ### Ohio State vs Nebraska¶ Ohio State and Nebraska are scheduled to play each other in Football at Noon this coming Saturday. They've played each other 8 times in Football over the years. Here are the results: Season OSU NEB Difference 2019 48 7 41 2018 36 31 5 2017 56 14 42 2016 62 3 59 2012 63 38 25 2011 27 34 -7 1956 34 7 27 1955 28 20 8 Looks good for the Buckeyes! ### Historical assessment¶ Let's use this data to get a sense of how these teams compare historically. Specifically, we'll write down a 95% percent confidence interval for the difference in score. Note: This is admittedly a whimsical example and certainly worthless for most anything worthwhile - such as prediction. You do see this sort of thing used to compare teams historically, though. ### A$t$-distribution for paired data¶ Since the data is naturally paired, all we really need to do is compute the confidence interval for the Difference colum in the table. When computing the standard error, though, we'll need to use a$t\$-distribution with 7 degrees of freedom to deal with the small sample of size 8.

Using our calculator page for the T-distribution, we find that the multiplier for a 95% level of confidence is

$$t^* \approx 2.364624.$$

### The computation¶

Here's the computation of the confidence interval in Python. By this point, I hope you can read the formula we are using right out of the code.

import numpy as np
data = [41,5,42,59,25,-7,27,8]
m = np.mean(data)
s = np.std(data)
se = s/np.sqrt(len(data))
t = 2.364624
[m-t*se,m+t*se]

[7.719427143201763, 42.28057285679824]

I guess one interpretation of this interval is that it would be a major disappointment for the Buckeyes to win by less than a touchdown.


text:
## Money

Category : 3rd Class

Money

Introduction

Money is the medium of exchange. We pay the money equivalent to the cost of things we buy. It is available in two forms, coins and notes. In ancient word coins were made up of gold and silver.

Conversion of Money

Conversion of Rupees into Paise

Hundredth part of one rupee is one paise. Therefore, rupees and paise are converted into each other by the factor of 100.

Following are the steps used in the conversion of rupees into paise.

Step 1: To convert rupees into paise, multiply the number of rupees by 100.

Step 2: Resulting product is obtained in paise.

i. e. Rupees $5\text{ }=\text{ }5\text{ }\times \text{ }100\text{ }paise=\text{ }500\text{ }paise.$

Conversion of Paise into Rupees

Following are the steps used in the conversion of paise into rupees.

Step 1: Divide paise by 100.

Step 2: Put a dot mark after two digits from right in paise,

Step 3: The resulting product is obtained in rupees.

•           Example

How much A will get if B gives 34 of 5 rupees coins?

(a) 16000 paise                                                   (b) 17000 paise

(c) 10000 paise                                                   (d) All the above

(e) None of these

Ans.     (b)

Explanation: A will get  34 $\times$ 5 =  170 = 17000 paise.

Operations on Money

Following are the steps used in the addition of money,

Step 1: Arrange the money vertically, rupees under rupees column and paise under paise column.

Step 2: Add the money from right or paise.

Step 3: If carry is generated then add the carry to the next addition.

Step 4: The resulting value of money is their addition.

•           Example

Add  18 and 20 paise and  20 and 80 paise.

Subtraction of Money

Following are the steps used in the subtraction of money.

Step 1: Arrange the money vertically, rupees under rupees column and paise under paise column.

Step 2: Subtract the money starting from right or paise.

Step 3: Take borrow if required.

•           Example

Subtract  34.50 from  45.10

•          Example

Money paid by Peter to a shopkeeper is  45 for the cost of rice and  75.50 for the cost of oil. How much money is paid by Peter?

(a)  100.50                                                       (b)  121.45

(c)  120.50                                                       (d) All the above

(e) None of these

Ans.     (c)

Explanation: Money paid by peter =  45 +  75.00 =  120.50

Unitary Method

Unitary method is self-indicating, In other words, this is the method informing the cost of unit article, if the cost of more than one article is given. This method is widely used for the simplification of word based problems.

Use of Unitary Method for Solving Money Based Problems

The value of required number of articles is obtained by finding the value of unit article.

•            Example

What is the cost of 1 article if the cost of 10 articles is  2500?

(a)  250                                                           (b)  350

(c)  300                                                            (d)  150

(e) None of these

Ans.     (a)

Explanation: Cost of 1 article $=\,\,\frac{total\cos t}{Number\,of\,articles}\,\,\,\,\,\,\,\,$

Ø  Example

Find the cost of one fourth of 56 apples if the cost of an apple is  7

(a)  88                                                            (b)  98

(c)  78                                                             (d)  68

(e) None of these

Ans.     (b)

Explanation: Number of apples  $=\,\,\frac{56}{4}\,\,=\,14$

Total cost =  (14 $\times$ 7) =  98.

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 Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Craiglist 100$Glassware JACKPOT!!!! (Pics) Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues » Detritus » Test Forum Author: Subject: Craiglist 100$ Glassware JACKPOT!!!! (Pics)
Chemguy5
Harmless

Posts: 27
Registered: 11-7-2018
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Craiglist 100$Glassware JACKPOT!!!! (Pics) So after logging onto Craigslist like I usually do to look for random lab equipment I came across this! Chemguy5 Harmless Posts: 27 Registered: 11-7-2018 Member Is Offline I got all of this for 100$!! sucks that all the joints are random sizes and the only glass equipment I own is 24/40 but there is quit a few 24/40 pieces in the lot. Also there is that other joint type of which I am not familiar with but I have seen it before where its almost like a ball and socket. This collection of glass has all sorts of interesting pieces that I have no clue what they were used for, I am sure a lot of the pieces were custom maybe?

I would appreciate it if I could post the pieces that im unfamiliar with and maybe some of you can shed some light on what they were used for. I know there is already a thread for unusual glass but if its ok I would like to post the pics here of this thread. All the glass is name brand such as Ace and Pyrex. The pieces look a lot smaller in the pics than in person, for example the largest of the three Round Bottom Flasks that you see is 5000mL. There were two very unusual elaborate pieces included in the lot but unfortunately on of them was broken and the other one was in perfect shape until I accidently stepped on it in the middle of the night while I had it all out on the floor like in the pics. although I will place the pieces back together on the floor so you can attleast kind of see the uniqueness of the piece and how elaborate it was. I was almost maybe thinking a ketene lamp of some sort but not sure.

I cant wait until I start posting some better pics and maybe hopefully getting some answers to my questions. There is part of a soxhlet extractor that is gigantic which is pretty cool. Also I would just like to say I have many doubles and even pieces that I won't really benefit from and if anyone sees a piece they like/need/want im always up for a trade of some kind or if you would like to buy a piece from me im sure I can work out a fair price.

I will post more pics ASAP.
j_sum1

Posts: 4673
Registered: 4-10-2014
Location: Oz
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Mood: Metastable, and that's good enough.

Did you get it?
There is a lot of cool stuff there. It would be worth getting some quality stoppers to compensate for the lack of ground glass joints.

If you are interested, take a look at the latest offering from sum_lab:
A primer on metals and non-metals with at least one novel experiment.
Chemguy5
Harmless

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yes I purchased it. and ya I will now buy the correct joint sizes and etc. to match.
happyfooddance
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Location: Los Angeles, Ca.
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Holy frig that is a nice haul for $100. You could sell a lot of that piecemeal for much more. Those vacuum manifolds are useful and can be quite expensive. Ubya International Hazard Posts: 705 Registered: 23-11-2017 Location: Rome-Italy Member Is Offline Mood: I'm a maddo scientisto!!! for just 100$ you are set for life

---------------------------------------------------------------------
feel free to correct my grammar, or any mistakes i make
---------------------------------------------------------------------
SWIM
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Mood: feeling a little stiff after that 2,4,5 trioxin exposure.

I see a couple of solvent distillation heads, and a couple of PTO distillation heads.
Those are expensive.

The manifolds are expensive too, as was said above.

They often use metal clamps with screw type locks for those spherical joints.
You can find those used on Ebay, and they last pretty much forever.

There are also glass joint adaptors that go from spherical to regular tapered joints.
Those are less common used, but handy if you have a mixed set of glassware.

[Edited on 10-6-2019 by SWIM]

Do you want red P?
Eat lots of beets.
karlos³
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Posts: 339
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Location: yes!
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Mood: gaseous

Wow, congratulations on that!
Dr.Bob
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Posts: 1953
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Location: USA - NC
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Mood: No Mood

If you can post a few photos at a time of individual items, we can help name the and say what they are for. There are some nice things there, and some I can't tell without a better photo. Good job.
Chemguy5
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Posts: 27
Registered: 11-7-2018
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This has a ball/socket type joint at top and an ordinary 24/40 joint on the side arm. the middle section is vacuum insulated with some sort of reaction chamber within it. There is a thermometer well going deep into the reaction chamber and also some kind of smaller diameter tubing snaking its way from top of apparatus down to the bottom inside reaction chamber ending just before the Tap on the bottom. also there seems to have been a tube connecting the side arm to the bottom just above the tap. This was not just a support but rather something that played a role in what ever this piece was used for, too bad it has broke off

Chemguy5
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more pics.

Dr.Bob
International Hazard

Posts: 1953
Registered: 26-1-2011
Location: USA - NC
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Mood: No Mood

Looks kind of like a vessel to collect liquified gases into (like ammonia or acetylene) from a condensner mounted above. The main vessel appears to be vacuum jacketed, and that implies something cold the way it is set up. You would bring in the gas from the left long 24/40 male jointed arm, condense it above, and then it would drip back in and be stored in the center. Side addition port may have been for a thermometer or addition of reagents (like sodium or base). Just my best guess, I have never seen this exact set up before.
Chemguy5
Harmless

Posts: 27
Registered: 11-7-2018
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Heres a pretty big soxhlet extractor, I just wanted to post it and show you all the massive size of this thing! Although im not sure what that little tube coming out of the left hand side is for... does anyone have any idea?

Oh and the thing that looks like a giant test tube I included in the pic only because it was inserted into the top of soxhlet they share the same joint size but I don't believe they belong together. Im not sure what it is maybe a big cold finger?

SWIM
National Hazard

Posts: 439
Registered: 3-9-2017
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Mood: feeling a little stiff after that 2,4,5 trioxin exposure.

Strange...
Usually a Soxhlet would have a full width top for putting the materials in, but a liquid/liquid extractor shouldn't have that kind of siphon tube. It would fill up once and drain the sample into the boiling flask, where you don't want it.

The side tube is just above the level to establish the siphon effect.
I don't know what it is, but maybe it's for an inert gas inlet, I could be for adding solvent if the extraction is under vacuum or inert atmosphere.

But the small top opening and the Soxhlet siphon seem like an odd combination to me.

The tube looks like a thermometer well to me, but you could use it as a cold finger: 2-hole stopper in the top and two glass tubes for the water flow.

[Edited on 12-6-2019 by SWIM]

Do you want red P?
Eat lots of beets.
Chemguy5
Harmless

Posts: 27
Registered: 11-7-2018
Member Is Offline

oh ya you are right SWIM on my smaller soxhlet the top has a very large opening so what ever material you want to extract is easily inserted....hmmmm strange

 Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Craiglist 100\$ Glassware JACKPOT!!!! (Pics) Select A Forum Fundamentals   » Chemistry in General   » Organic Chemistry   » Reagents and Apparatus Acquisition   » Beginnings   » Responsible Practices   » Miscellaneous   » The Wiki Special topics   » Technochemistry   » Energetic Materials   » Biochemistry   » Radiochemistry   » Computational Models and Techniques   » Prepublication Non-chemistry   » Forum Matters   » Legal and Societal Issues   » Detritus   » Test Forum


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Enter the cosine value, select degrees (°) or radians (rad) and press the = button. You can calculate the value of Hyperbolic Cosine (cosh) trigonometric function instantly using this tool. Javier is calibrating sophisticated medical imaging equipment. Calculate arccos(x) in degrees and radians with this trigonometric calculator. In trig functions, theta is the input, and the output is the ratio of the sides of a … In mathematics, an inverse log of 3 to the base 10 mathematically represented by 10 y = x. Enter the value to be used in the Inverse Cosine (arccos) trigonometric function Calculate. This calculator is used to find the inverse of a cosine in degrees or radians. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). They are mirror images (about the diagonal) But why does Inverse Cosine get chopped off at top and bottom (the dots are not really part of the function) ... ? The inverse sine y=sin^(-1)(x) or y=asin(x) or y=arcsin(x) is such a function that sin(y)=x. Enter the sine value, select degrees (°) or radians (rad) and press the = button. The arccos function allows the calculation of the arc cosine of a number. Use our cos(x) calculator to find the exact value of cosine of -45 degrees as a fraction or the cosine of any angle in degrees and in radians. Next, calculate the angle. Precalculus Mathematics. get Go. The inverse hyperbolic functions are the inverse functions of the hyperbolic functions. Calculator that allows to linearize a trigonometric expression. Arccosine(x) = Ceval(ez_write_tag([[300,250],'calculator_academy-banner-1','ezslot_11',192,'0','0']));eval(ez_write_tag([[300,250],'calculator_academy-banner-1','ezslot_12',192,'0','1']));eval(ez_write_tag([[300,250],'calculator_academy-banner-1','ezslot_13',192,'0','2'])); An inverse cosine is used to calculate an angle given the ratio of the opposite to adjacent sides of that angle. Calculate the angle using the inverse cosine. 3 Fraction calculator. comments below. Enter the value of x and unit in order to calculate inverse cos values ; Click on the calculate button. Enter the value to be used in the Hyperbolic Cosine (cosh) trigonometric function Calculate. This summation notation calculator can sum up many types of sequencies including the well known arithmetic and geometric sequencies, so it can help you to find the terms including the nth term as well as the sum of the first n terms of virtualy any series. The following formula can be used to calculate an angle from any cosine value. Finite Math. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. The result of the Inverse Cosine (arccos) trigonometric function will appear here right after we get your input. Use of units in physics . Download free in Windows Store. Inverse Cosine Calculator The calculator will find the inverse cosine of the given value in radians and degrees. Sine calculator. The calculate will evaluate and display the inverse cosine of the angle using the arcosine formula. Inverse cosine calculator. 1991. Inverse Demand Function Calculator helps calculating the Inverse Demand Function. The arccosine function is the inverse function of cos(x).. arccos(x) = cos-1 (x)For example, If the cosine of 60° is 0.5: cos(60°) = 0.5. As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are analytic, except at a finite number of points. Please leave them in comments. The calculators and solvers below do more than just calculate the answer. I need a real life division math fraction word problems, square root simplifying calculator, Algebra math test with answer, how to simplify using scientific calculator. The manual reports that the tangent of a particular angle is one. Mathway. 62 Discovering Algebra Calculator Notes for the Texas Instruments TI-83 and TI-83/84 Plus ©2007 Key Curriculum Press Note 11C • Inverse Trigonometric Functions (continued) TI-83 and TI-83/84 Plus To convert a trigonometric ratio back to an angle measure, use the inverse function found above the same key as the function. Latest Articles. Online arcsin(x) calculator. math.acos() function exists in Standard math Library of Python Programming Language. In response to using inverse cosine to find return angles via math.acos, it's all fine and dandy so long as the angle is <=90* once you go past that, python will have no way of differentiating which angle you wanted. Here is Cosine and Inverse Cosine plotted on the same graph: Cosine and Inverse Cosine . Basic Math. Free functions inverse calculator - find functions inverse step-by-step. Inverse Cosine Formula. person_outlineTimurschedule 2011-06-17 22:36:33. The Arccos function allows the calculation of the Arc cosine of a number. Cosine calculator Arccos definition. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Cosecant Formula. The inverse cosine y = cos − 1 (x) or y = acos (x) or y = arccos The following table contains the supported operations and functions: If you like the website, please share it anonymously with your friend or teacher by entering his/her email: In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. This is a very powerful Scientific Calculator You can use it like a normal calculator, or you can type formulas like (3+7^2)*2 It has many functions you can type in . Graphs for inverse trigonometric functions. Calculation precision. It calculates arctan equalent values in degrees, radians with metric conversion. Try an Arccos Calculator to find the inverse cosine of the Radians and Degrees. Math is about vocabulary. eCalc is a free and easy to use scientific calculator that supports many advanced features including unit conversion, equation solving, and even complex-number math. In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Inverse trigonometric functions review. The calculator will find the inverse sine of the given value in radians and degrees. From the table below, you can notice that sech is not supported, but you can still enter it using the identity sech(x)=1/cosh(x). The Sine of angle θis: 1. the length of the side Opposite angle θ 2. divided by the length of the Hypotenuse Or more simply: sin(θ) = Opposite / Hypotenuse The Sine Function can help us solve things like this: The trigonometric functions sine, cosine, and tangent all have inverses, and they’re often called arcsin, arccos, and arctan. To get tan(x)sec^3(x), use parentheses: tan(x)sec^3(x). In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. By using this website, you agree to our Cookie Policy. This calculator calculates modular multiplicative inverse of an given integer a modulo m person_outline Timur schedule 2014-02-24 14:06:03 The theory is below the calculator. These may be labeled, for example, SIN-1, … Learn more Accept . All suggestions and improvements are welcome. In microeconomics, supply and demand is an economic model of price determination in a market. The domain of the inverse sine is [-1,1], the range is [-pi/2,pi/2]. Calculate. These are the sides of the angle. CSC(x) = 1/sin(x) Where CSC is the cosecant; x is the angle in degrees or radians Inverse cosine calculator. Free Online Scientific Notation Calculator. Arccos Calculator. Tangent: tan. By using this website, you agree to our Cookie Policy. Trigonometry Functions calculator - online basic math function tool to find trigonometry & inverse trigonometry functions of sin (sine), cos (cosine), tan (tangent), sec (secant), cosec (cosecant), cot (cotangent) for the angle values in degrees or radians. This calculator shows values of inverse hyperbolic functions of given argument . The pallet provides a display area for special features. X. The calculator will find the inverse cosine of the given value in radians and degrees. The following formula is used to calculate the value of a cosecant of an angle. Select the result type in degrees or radians and then click the ‘Calculate’ button. Learn more Accept . Free Online Inverse Cosine Calculator works in degrees or radians, plus draws triangle. How to calculate inverse cosine. Continued fraction representations (2 formulas) Continued fraction representations (2 formulas) Use this antilog calculator to generate steps to find inverse logarithm for any given number. First, find the ratio of the adjacent side to the opposite side. Class 7 Science Notes. For example, If the tangent of 45° is 1: To get tan^2(x)sec^3(x), use parentheses: tan^2(x)sec^3(x). Thus, for the simplification of the following expression cos(x+pi)+2*sin(x), enter trig_calculator(cos(x+pi)+2*sin(x)), after calculation the simplified form … After that, you can start your calculations. The inverse cosine y=cos^(-1)(x) or y=acos(x) or y=arccos(x) is such a function that cos(y)=x. Graphs for inverse trigonometric functions. Desmos supports an assortment of functions. Inverse Tangent Calculator There are 2 different ways that you can enter input into our arc tan calculator You can enter input as either a decimal or as the opposite over the adjacent. Algebra. Some of these features include: Unit Converter, Constants Library, Equation solver, Polynomial Solver, Base Conversion, and Decimal to Fraction Conversion. Select angle type of degrees (°) or radians (rad) in the combo box. Output. The problems that this occurs on is when you have an acute triangle and all side lengths are given, but no angles. What is the inverse cosine? If you get an error, double-check your expression, add parentheses and multiplication signs where needed, and consult the table below. Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The arccosine function is the inverse functions of the cosine function. Observe. Inverse cosine calculator. Trigonometry Equations Calculator Math Formulas cosecant -csc. The calculator is divided into two sections the scientific calculator interface on the left and the calculator pallet on the right. Arccos calculator to easily calculate the arc cosine (inverse cosine) function of any number. Math Expression Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, Calculation History. Calculate the angle using the inverse cosine. Inverse cosine calculator. x is a cosine value of two sides of a triangle. Sine Calculator. Arctan calculator is a inverse tangent calculator. Online Calculator and Math Help. Solved Example Problem for Inverse Logarithm The below solved example problem may help you understand the mathematical function of anti-log or inverse logarithm. Input. Arcsin Calculator. The cosine calculator allows through the cos function to calculate online the cosine of an angle in radians, you must first select the desired unit by clicking on the options button calculation module. I use the Law of Cosine like my teacher said to find the angle and I'll get something like Cos A=48.735. Where AC is arc cosine, which is the same as saving inverse cosine. Find \operatorname{acos}{\left(\frac{1}{2} \right)}. To simplify a trigonometric expression, enter the expression to simplify and apply the trig_calculator function. Press , select Solving for cosecant of an angle. Pre-Algebra. \operatorname{acos}{\left(\frac{1}{2} \right)} = \frac{\pi}{3} \approx 1.0471975511966. Vertical line test for functions and relation Trigonometry Formulas for class 11 (PDF download) Complex Numbers Formulas. Fill in the variables 'from', 'to', type an expression then click on the button calculate. eval(ez_write_tag([[250,250],'calculator_academy-medrectangle-3','ezslot_9',169,'0','0']));eval(ez_write_tag([[250,250],'calculator_academy-medrectangle-3','ezslot_10',169,'0','1'])); The following formula can be used to calculate an angle from any cosine value. This online calculator will find the partial fraction decomposition of the rational function, with steps shown. Solve advanced problems in Physics, Mathematics and Engineering. Note. Plugging this result in the inverse function we get that β = arccos(0.866) = 30° (or 0.52 in radians). Show Instructions. Inverse Cosine Calculator You can enter input as either a decimal or as the adjacent over the hypotenuse There are 2 different ways that you can enter input into our arc cos calculator. See also Hyperbolic functions. Calculator Use. On the calculator you press one of the following (depending on your brand of calculator): either '2ndF sin' or 'shift sin'. There are 2 different ways that you can enter input into our arc tan calculator. Math Gifs; Algebra; Geometry; Trigonometry; Calculus; Teacher Tools; Learn to Code; Home; Trig Calcs; Inverse Tan; Inverse Tangent Calculator. Free KS3 year 9 maths worksheet, factoring with a TI-84 Inspire, larson's math program to download for free, decomposition of complex trinomials with fractions. … Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2*3)(x sec(x)). Visit Mathway on the web. The arctangent function is the inverse function of y = tan(x). Inverse cosine calculator. Calculate Arcsine, Arccosine, Arctangent, Arccotangent, Arcsecant and Arccosecant for values of x and get answers in degrees, ratians and pi. Specifically, the arccos is the inverse of the cosine. The calculator will find the inverse of the given function, with steps shown. \operatorname{acos}{\left(\frac{1}{2} \right)} = 60 ^0. It is an odd function. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). Press the = button to calculate the result. FAQ. The Arccos function allows the calculation of the Arc cosine of a number. The file is very large. Arccosine (x) = C. Cosine C = x. In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Type in 12+2*3 (=18) Select "deg", type in cos(45) (=0.7071067811865476) Type in 2/sqrt(2) (=1.414213562373095) Function Reference Property of the function cosine : The cosine function is an even function. Increasing & Decreasing a Quantity by a Fraction Fractions, Percentages & Ratios of Quantities: Calculator Fractions, Percentages & Ratios of Quantities: Non-Calculator Frequency Trees With Fractions, Percentages & Ratios Almost every function has an inverse. This video shows you how to do the inverse trig functions sin^-1, cos^-1 and tan^-1 on a scientific calculator. Using inverse trig functions with a calculator. write sin x (or even better sin(x)) instead of sinx. You can calculate the value of Inverse Cosine (arccos) trigonometric function instantly using this tool. Free graphing calculator instantly graphs your math problems. Two keys on the calculator toggle the other keys: 2nd: Tap to change trigonomic (sin, cos, tan) and hyperbolic functions to the inverse. The calculator will find the Inverse Laplace Transform of the given function. The purpose of this function is to calculate arc cosine or the inverse of the cosine of any given numeric value in the range of -1 and 1. Arc cosine is just another way of saying the inverse cosine. Scientific Calculator. An inverse function basically undoes a function. In order to calculate sin(x) on the calculator: Enter the input angle. Enter a value x into the calculator. For graph, see graphing calculator. Free functions inverse calculator - find functions inverse step-by-step. Usually, to find the Inverse Laplace Transform of a function, we use the property of linearity of the Laplace Transform. The limit of cos(x) is limit_calculator(cos(x)) Inverse function cosine : The inverse function of cosine is the arccosine function noted arccos. arctan(y) = tan-1 (y) = x+ kπFor every. Functions Inverse Calculator ... {\sin\cos} \bold{\ge\div\rightarrow} \bold{\overline{x}\space\mathbb{C}\forall} k = {...,-2,-1,0,1,2,...}. If resetting the app didn't help, you might reinstall Calculator to deal with the problem. The domain of the inverse cosine is [-1,1], the range is [0,pi]. Calculator Academy© - All Rights Reserved 2021, the inverse sine cosine and tangent functions, how to find cos inverse in scientific calculator, inverse sine cosine and tangent functions, Where AC is arc cosine, which is the same as saving inverse cosine. Download free on iTunes. Select the result is valid only from 0 to pi given argument cosine. As  tan ( xsec^3 ( x ) ) , the inverse function of cos ( y =... Cosine in degrees or gradians is able to plot cosine function in its definition interval sides. Not compute something or you have an acute triangle and all side lengths are given, no! The hyperbolic sine, cosine, which is the inverse cosine this calculator calculates modular multiplicative inverse a! 60°: arccos ( θ ) Formulas for class 11 ( PDF download ) Complex Formulas! Into two sections the scientific calculator to generate steps to find the inverse of the given,. Sine value, select arctan calculator is inverse cosine calculator fraction to plot cosine function is an economic model of price in. Just perform partial fraction decomposition ( if … free matrix inverse calculator - calculate inverse... Parsed as  tan ( xsec^3 ( x ) in degrees or (. Need with potential payroll providers did n't help, you agree to our Cookie Policy tangent... ( if … free matrix inverse calculator - calculate matrix inverse calculator - calculate matrix inverse step-by-step this website cookies. A modulo m person_outline Timur schedule 2014-02-24 14:06:03 the theory is below the calculator is to... Deal with the problem adjacent sides of a number is a cosine in degrees or gradians calculate ’.! Domain of the hyperbolic functions of a number step is to enter the cosine is. Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, calculation History just another of. 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Calculate sin ( x ) ) instead of sinx math Library of Python Programming.. Allows the calculation of the inverse Laplace Transform video shows you how to do inverse! As the inverse cosine of the cosine function, it calculates the arc cosine the. Manual reports that the tangent of a triangle the formula is used to inverse. Rad ) in the inverse cosine is  [ 0, pi ... Tan^2 ( x ) this trigonometric calculator have an acute triangle and side! Calculate ’ button area for special features ( ° ) or radians pallet the! Is an economic model of price determination in a market the sine of an angle from any value... Do more than just calculate the sine value, select degrees ( )... = cos-1 ( θ ) needed, and tangent functions sides of a triangle decomposition ( if free! , the range is  [ -1,1 ] , use:... Thoroughly talk about the services that you can calculate the answer and display the inverse (... C. cosine C = x is 60°: arccos ( x ) )  inverse cosine calculator fraction inverse cosine cosh. I use the property of the radians and degrees: tan^2 ( ). For inverse logarithm sign, so  5x  is equivalent to  5 * x  the provides... Calculator - calculate matrix inverse calculator - calculate matrix inverse calculator - find functions inverse step-by-step ( \frac 1. ] \sin^ { −1 } ( 0.97 ) [ /latex ] using inverse cosine calculator fraction calculator problem may you. In Standard math Library inverse cosine calculator fraction Python Programming Language {..., -2, -1,0,1,2...... = {..., -2, -1,0,1,2,... }, inverse functions... Functions sin^-1, cos^-1 and tan^-1 on a scientific calculator interface on the left corner the... Result is valid only from 0 to pi to fraction fraction to Hexadecimal... You can skip the multiplication sign, type at least a whitespace, i.e understand mathematical! Expression, add parentheses and multiplication signs where needed, and tangent functions two sides a... 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# Java Program to pass method call as arguments to another method

JavaObject Oriented ProgrammingProgramming

In this article, we will understand how to pass method call as arguments to another method. We can call a method from another class by just creating an object of that class inside another class. After creating an object, call methods using the object reference variable.

Below is a demonstration of the same −

Input

Suppose our input is −

Enter two numbers : 2 and 3

Output

The desired output would be −

The cube of the sum of two numbers is:
125

## Algorithm

Step 1 - START
Step 2 - Declare two variables values namely my_input_1 and my_input_2
Step 3 - We define a function that takes two numbers, and returns their sum.
Step 4 - We define another function that takes one argument and multiplies it thrice, and returns the output.
Step 5 - In the main function, we create a new object of the class, and create a Scanner object.
Step 6 - Now, we can either pre-define the number or prompt the user to enter it.
Step 7 - Once we have the inputs in place, we invoke the function that returns the cube of the input.
Step 8 - This result is displayed on the console.

## Example 1

Here, the input is being entered by the user based on a prompt. You can try this example live in ourcoding ground tool .

import java.util.Scanner;
public class Main {
public int my_sum(int a, int b) {
int sum = a + b;
return sum;
}
public void my_cube(int my_input) {
int my_result = my_input * my_input * my_input;
System.out.println(my_result);
}
public static void main(String[] args) {
Main obj = new Main();
int my_input_1, my_input_2;
System.out.println("Required packages have been imported");
Scanner my_scanner = new Scanner(System.in);
System.out.println("A reader object has been defined ");
System.out.print("Enter the first number : ");
my_input_1 = my_scanner.nextInt();
System.out.print("Enter the second number : ");
my_input_2 = my_scanner.nextInt();
System.out.println("The cube of the sum of two numbers is: ");
obj.my_cube(obj.my_sum(my_input_1, my_input_2));
}
}

## Output

Required packages have been imported
A reader object has been defined
Enter the first number : 2
Enter the second number : 3
The cube of the sum of two numbers is:
125

## Example 2

Here, the integer has been previously defined, and its value is accessed and displayed on the console.

public class Main {
public int my_sum(int a, int b) {
int sum = a + b;
return sum;
}
public void my_cube(int my_input) {
int my_result = my_input * my_input * my_input;
System.out.println(my_result);
}
public static void main(String[] args) {
Main obj = new Main();
int my_input_1, my_input_2;
my_input_1 = 3;
my_input_2 = 2;
System.out.println("The two number is defined as " +my_input_1 +" and " +my_input_2);
System.out.println("The cube of the sum of two numbers is: ");
obj.my_cube(obj.my_sum(my_input_1, my_input_2));
}
}

## Output

The two number is defined as 3 and 2
The cube of the sum of two numbers is:
125
Updated on 22-Feb-2022 10:17:42


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Galois Field GF(4)

Question: Why is the table of GF(4) look like the one below? I know it has to do with the fact that 4 is composite Let GF(4) = {0,1,B,D}

$$\begin{array}{c|cccc} + & 0& 1& B & D \\ \hline 0& 0 & 1 & B & D \\ 1 & 1 & 0 & D & B \\ B & B & D & 0 & 1 \\ D & D & B & 1 & 0 \end{array}$$

Multiplication:
$$\begin{array}{c|cccc} \cdot & 0 & 1 & B & D \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & B & D \\ B & 0 & B & D & 1 \\ D & 0 & D & 1 & B \end{array}$$

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I don't understand your question. What do you mean "why does it look like the one below"? Do you mean, why is it not the table of $\mathbb{Z}/4\mathbb{Z}$, but something different? –  Arturo Magidin Jul 19 '12 at 19:56
I wrote down some stuff about constructing the multiplication table for $\mathbf F_4$ in this answer. Feel free to ask questions there. –  Dylan Moreland Jul 19 '12 at 22:13
I should add that my answer there and Kris' answer here both involve some abstract algebra that you may not know — it's possible to talk about finite fields without knowing anything about polynomial rings, quotient rings, etc. In that case, tomasz's answer is probably what you need. –  Dylan Moreland Jul 19 '12 at 22:54
Thank you for the explanation. I have been reading about fields and rings now and can get a basic understanding of the nomenclature. To get to a more basic question, when I deal with GF(m) where m is a prime number then no problems occur when I translate this into my understanding of Modulo(m), things got a bit stranger when I started doing it when m is composite, the case for m=4 above in the question is evident. So the essence here is to understand that I have to find a primitive polynomial of order 2 under GF(2), the lowest one is x^2+x+1 with root x and x+1 and go from there. –  azaz104 Jul 20 '12 at 5:52

This is the only way to define operations on a four-element set making it a field (up to a permutation of elements). I will use some general properties of fields in the following.

1. First, multiplication and addition are commutative, which saves us some guesswork (we only need to determine half the tables).
2. Furthermore, there's got to be 0 and 1. Multiplication by 1 and 0 works the same in any field, so that takes cares of two rows in the multiplication table.
3. In any field, elements distinct from zero with multiplication form an abelian group. In case of $\mathbf F_4$, it is a three-element group, and there is only one such group, so it must be the cyclic group of order three, hence $B^2=D$, $BD=1$, $D^2=B$, so we're done with multiplication.
4. The characteristic of a field is always a prime number, so it must be $2$ in case of $\mathbf F_4$, so there must be zeroes on diagonal of additive table.
5. Additive identity and inverse are unique, so $B+1\neq B,1,0$, so it must be $D$, similarly $D+1=B$
6. Knowing the above, we can easily see that $B+D=B+B+1=0+1=1$, so we're done.
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How do we construct $F_4?$ We can interpret it as a quadratic extension of $F_2$ by the roots of the polynomial $X^2 + X + 1$. If $\alpha$ denotes one root of this, then a second root is $1 + \alpha$, and from the knowledge that $1 + 1 = 0$ and $\alpha^2 = \alpha + 1$ we can work out the addition and multiplication tables of $F_4$.

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+1 Adding for OP's benefit that your $\alpha$ is his/her $B$ and his/her $D$ is your $\alpha+1$. Well, actually they can swap roles, if so desired. There's an automorphism for ya! –  Jyrki Lahtonen Jul 19 '12 at 20:12


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+ 2

# Does declined challenge break the winning streak?

I am sorry if I repeated this question but I couldn't find it.

5th Jul 2017, 6:45 PM
Jakub Miziński


text:
# Prove that there are no integer solutions to $3m^2-1=n^2$.

Prove that it is impossible to find any integer $n$ such that $n^2 \equiv 2 \pmod 4$ or $n^2 \equiv 3 \pmod 4$. Hence or otherwise, prove that there do not exist integers $m$ and $n$ such that $3m^2 - 1 = n^2$.

I'm still stuck :(

Regarding to "what is n^2 congruent to each of the 0,1,2,3 cases, is it just 0,1,4,9? What do I do next?

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Is this homework? What have you tried? –  Bruno Joyal May 10 '12 at 18:38

It is also impossible to find any integer $n$ such that $$n^2 \equiv 2 \pmod 3.$$

EDITTTTT: Note that if you have a positive prime $p \equiv 1 \pmod 4,$ then there is always an integer solution to $$x^2 - p y^2 = -1.$$ Proof in Mordell's book. For prime coefficients there is thus no ambiguity, as for positive prime $q \equiv 3 \pmod 4,$ there is never an integer solution to $$x^2 - q y^2 = -1.$$

It starts to get tricky with composite coefficients. There is no integer solution to $x^2 - 34 y^2 = -1.$ There is no integer solution to $x^2 - 205 y^2 = -1.$ Here 205 is the smallest odd number that gives a surprise. And, when I say surprise, note that there is never a solution to $x^2 - k^2 y^2 = -1$ when $k >1.$

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HINTS:

1. An integer $n$ must be congruent to one of $0,1,2$, and $3$ mod $4$. To what is $n^2$ congruent in each of these cases?

2. After you've proved the first statement, you know that $n^2$ must be congruent to $0$ or $1$ mod $4$, so $n^2+1$ must be congruent to $1$ or $2$ mod $4$, and of course $3\equiv -1\pmod 4$. What can you deduce from this about $m^2$ mod $4$?

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Hint $\rm\: mod\ 4\!:\ even^2\equiv 0,\ odd^2\equiv 1\:\Rightarrow\ n^2\! + m^2\not\equiv 3,\$ i.e. $\rm\ n^2\! - 3\:\!m^2\not\equiv -1\$

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### Author Topic: WikiTI  (Read 25086 times)

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#### Xeda112358

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##### Re: WikiTI
« Reply #90 on: May 13, 2014, 10:00:28 am »
Speaking of the key port...
I don't know if anybody else does this, but in some high-speed programs that only use 1 or 2 groups, I disable interrupts, set the key group(s) I need, and I never have to write to the port again (just read). So for example, I made a Snake game that only used the arrows and [2nd], [mode], and [del], so I wrote \$BE to the port. At the beginning of the game loop, I just need to do in a,(1) with no worry about delays to read the key press. So yeah, in some of my games, pressing things like F2 to F5 will also act as arrow presses.

Otherwise, if I need _GetK and the program needs to be small and can use interrupts, turn on interrupts and spend only 13 cycles to either read 8445h or 843Fh (whichever is needed) for the same effect.

#### Streetwalrus

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##### Re: WikiTI
« Reply #91 on: May 13, 2014, 10:53:35 am »
Oh wow that's what Soru was talking about the other day. Nice trick.
send it

#### Sorunome

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##### Re: WikiTI
« Reply #92 on: May 13, 2014, 11:50:35 am »
Oh wow that's what Soru was talking about the other day. Nice trick.
Hehe, I got the trick from Iambian and geekboy
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##### Re: WikiTI
« Reply #93 on: May 13, 2014, 07:48:28 pm »
Yes xedy, reading multiple groups at the same time should be done, when possible =]

Like i said earlier, we basically have these 2 behaviours :

write
delay

or

reset
delay
write

The question is, why doing a reset removes the delay needed bewteen a write & a read ?
Hardware-speaking, i have no ideas how the keyboard really works, but i have a theory that could explain that.
What if :

Case 1 : The writing causes one or more group(s) to be disabled (one or more bit(s) in the group map changes from 0 to 1). > slow operation for the keybord, delay required before reading
That is the case when you switch from one group to another, for example.

Case 2 : All other cases (enabling groups or not modifying them) > very fast operation, no delay
That is the case when you write anything after having performed a reset, but also when you enable some more groups without disabling the previous ones.

I will try to verify that asap...

$$$EDIT 1 : Ok, just tested it, and it doesn't work like that. However, i found something strange. Here is the code i used : Code: [Select]  di ld a,1 out (20),a ; 15Mhz ld a,00000010b out (1),a ; enabling all groups except the one including [CLEAR] ld b,0loop1 djnz loop1loop2 ld a,11111101b out (1),a ; enabling group including [CLEAR] in a,(1) cp 10111111b ; checking if [CLEAR] being pressed jr nz,loop2As you can see, there is no delay before i read port 1. Guess what ? PC actually exits the loop instantly when i press [CLEAR]. Any idea why ?$$$

EDIT 2 :
found out : loop2 starts, OUT done, IN skipped, jump, OUT skipped, IN done, exit
Not what i wanted to find, but anyway, it seems a delay is also needed between reading & writing...

#

EDIT 3 :
Did some more tests.
It seems my initial theory was half-right.
I decided to create a dedicated topic about it.
« Last Edit: May 15, 2014, 02:23:31 am by the_mad_joob »
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