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# Course:CPSC522/Improving Prediction Accuracy of User Cognitive Abilities for User-Adaptive Narrative Visualizations

## Title

This project aims to extend a previous study on User-Adaptive Narrative Visualizations by focusing on the user modeling aspect and improving the prediction of user cognitive abilities during Information visualization processing.

Principal Author: Alireza Iranpour

## Abstract

Data visualization accompanied by narrative text is often employed to convey complex information in a format known as Magazine Style Narrative Visualization or MSNV. However, simultaneous processing of two disparate information sources, namely visual and textual information, can be somewhat challenging as attention will be split. Recent studies show that these difficulties depend on the reader’s level of cognitive abilities. As a result, predicting user cognitive skills can discover individual struggles and enable tailored support. This project would be an extension of a previous study on this subject wherein eye-tracking data were leveraged to predict user’s cognitive abilities with the ultimate goal of providing personalized support with information visualization processing.

### Related Pages

Information Visualization, User-Adaptive Narrative Visualizations[1], Eye-tracking, Machine Learning, Bayesian optimization, ANOVA, .

## Content

### Previous work[2]

#### Motivations

People with different cognitive abilities have different performance in processing different aspects of Information visualization. Hence, learning the level of each user’s cognitive abilities can greatly empower personalized support. Given the perceptual nature of processing Information visualizations, eye movement patterns can reveal a good deal about user cognitive states and, therefore, can be leveraged to predict user cognitive abilities.

Three cognitive abilities, namely reading proficiency (READP), visualization literacy (VISLIT), and verbal working memory (VERWM), have been used to identify how adaptation needs to be performed.

Each cognitive ability determines the level of proficiency in processing particular aspects of narrative visualizations. Verbal working memory indicates ability in maintaining verbal information, so people with a low level of this ability might benefit from simplification of the text. Visualization literacy, on the other hand, refers to the degree to which one can efficiently and confidently perceive data visualization, so people with low levels of this trait would appreciate visual cues that link corresponding elements between the text and the visualization. Last but not least, reading proficiency is the ability of processing labels and data points, so additional guidance would benefit those who are not proficient readers.

#### Approach

To gain insight into how eye movement patterns relate to cognitive abilities, a study was conducted with 56 subjects wherein participants were asked to read 15 MSNV documents and, after each, respond to 3 comprehension questions. The cognitive skills of each subject were also evaluated and scored in advance through a battery of well-established psychological tests. During the study, as participants read these documents, their eye movement patterns were tracked using a T120 remote eye-tracker. These raw data were then processed and represented in terms of the following features over 7 salient regions in the MSNV known as areas of interest (AOIs):

• Gaze features: fixations (rate and duration) and saccades (duration, distance, and velocity)
• Pupil and head distance features: pupil size (width and dilation velocity) and head to display distance

The results from the study including eye-tracking data and cognitive ability scores were then used as training data to train classification models that classified cognitive abilities of each user into high and low based on their eye-movement data.

Since user information can be logged from several interaction sessions, the usefulness of prior user data was examined by having the classification model make predictions at the end of each task based on the accumulated data from all the previous and the current task. This was done by producing the same features for each subject considering all eye-tracking data collected up to the current task rather than the current task alone (e.g. compute fixation rate over all previous tasks and not just over the current tasks). This accumulation of eye-tracking data resulted in 15 windows (i.e. the first window (window 1) computed features over task 1 while the last window (window 15) computed the same features over all 15 tasks). All windows had the same amount of data (examples). The data from previous tasks were simply reflected in the feature values of each window.

For each cognitive ability, the study compared the following classification algorithms against a majority-class baseline:

• Logic Regression (LR)
• Random Forest (RF)
• Support Vector Machine (SVM)

The performance of each model was evaluated in terms of accuracy using a 10 times repeated 10-fold cross validation over users (10 by 10 CV). The results are as follows:

Cognitive Ability Ranking of Classifiers Accuracy at Best Window
Window Accuracy Base
READP RF > SVM > XGB > LR > Base Task 4 0.67 0.51
VISLIT RF = SVM > XGB = Base > LR Task 3 0.66 0.53
VERWM XGB > RF > LR > SVM > Base Task 10 0.72 0.51

### Extension

#### Motivations

In the study, to avoid complexity, only 4 classification algorithms have been explored and evaluated. However, there may be other classifiers that can achieve higher accuracies. Moreover, these classifiers were assessed at their default configuration without any special feature selection or feature preprocessing. Therefore, they may not have reached and demonstrated their full potential. In fact, the main purpose of the study was to establish the feasibility of predicting cognitive abilities based on eye-tracking data. As a result, not enough effort was put into fine-tuning the employed classifiers or exploring alternative ones. In this project, we will investigate the effect of feature preprocessing and feature selection on classification performance and whether optimizing the hyper-parameters would result in improved accuracy. Particularly, we will test the following hypotheses:

• H1. Preprocessing and selection of features would improve classification accuracy
• H2. Hyper-parameter optimization would improve classification accuracy

#### Considerations

The original dataset for each window contains 278 features and 53 samples (3 of the 56 subjects were removed due to incomplete data). Due to the small number of examples in each dataset, instead of following the evaluation method used in the study (repeated 10-fold CV), we will employ a leave-one-out cross validation as it completely eliminates the randomness associated with data partitioning and more importantly would leave more samples for training in each iteration.

In addition to the classification models assessed in the study, we will also evaluate gradient boosting and nearest neighbors (non-parametric model).

As a result of bootstrapping and random feature consideration, ensemble models, namely RF, GB, and XGB, would yield different results in each run. Therefore, to strengthen the stability of our evaluation, for these models, we will repeat the leave-one-out cross validation 10 times and report the average over the 10 runs.

#### Feature selection

In each of the 53 (number of samples) iterations of the leave-one-out cross validation, all of the features are scaled and normalized based on the samples in the training portion. Subsequently, analysis of variance (ANOVA) is performed on the transformed features and those with the highest F-value score are selected. Normalization of the features is specifically intended to improve the performance of the models that use Euclidean distance as part of their algorithm and also expedite the convergence of gradient descent.

In our feature selection approach, the normalized samples inside the training portion are grouped based on their target value (low or high). Then, for each feature, the F-value is calculated as the between-groups variance divided by the within group variance. In simple words, we measure the significance of the difference between the means of the two classes for each feature. Features with more significant differences (higher F-value scores) are selected as more predictive. These features are then considered for making predictions in the evaluation phase while the rest (features that did not demonstrate a significant difference) are simply disregarded (i.e. removed). Given the relatively large number of features compared to the number of samples, feature selection (i.e. keeping only a subset of the features) was intended to mitigate overfitting and improve generalizability.

Note that this is performed in addition and prior to the inherent feature selection mechanisms in some of our models (e.g. L1 regularization in Logistic Regression)

#### Hyper-parameter optimization[3]

In order to assess the effectiveness of hyper-parameter optimization, a nested cross validation approach is adopted. In the outer loop of the structure, a leave-one-out cross validations is applied on the dataset. Within each iteration of the outer cross validation, in addition to the previously mentioned preprocessing and feature selection, the hyper-parameters of each model are optimized based on a second level of cross-validation (inner loop) on the split training portion. The optimized model is then retrained on the entire training portion and tested on the held out sample from the outer loop. The inner loop is used for tuning, and the outer loop is meant for evaluation.

The task of finding optimal hyper-parameter values is treated as a Bayesian optimization problem. Unlike the randomized search approach, Bayesian optimization uses past evaluation results to choose the next best candidate values. In our case, the objective function is the cross validation error. In the Bayesian approach, instead of directly optimizing the objective function, which is intractably expensive, a probability model also known as a surrogate function is built based on the previous evaluation results (of the objective function) to enable effective exploration of the search space while restricting poor choices. For each hyper-parameter, we use a bounded distribution of possible values from which to sample (the default values are also included in the distributions).

This approach can be outlined as the following:

1. Compute a posterior expectation of the objective function using the previously evaluated values.
The posterior is an approximation of the unknown objective function and is used to estimate the cost (cross validation error) of different candidate values that we may want to evaluate.

${\textstyle P(Objective\ Function\ |\ Data)=P(Data\ |\ Objective\ Function)\times P(Objective\ Function)}$

2. Optimize the conditional probability of values in the search space to generate the next candidate value
3. Evaluate the value with the objective function (cross validate the model using the value for the hyper-parameter)
4. Update the data and the probability model
5. Repeat

This process can be repeated until a good enough result has been found or resources are depleted. In this project, each model was optimized up to 10 iterations.

#### Cognitive abilities

First, we evaluate the accuracy of our models over the 15 windows without manipulating any of the features or the default hyper-parameter configurations. The same evaluation process is also performed on the preprocessed and selected features. Finally, for each classifier, the best window (window with the highest accuracy) is used for tuning the hyper-parameters. The results are as follows:

 Figure 1. Comparison of default models on the original data Figure 2. Comparison of default models with feature preprocessing and feature selection

In the original study, for reading proficiency, the best accuracy was achieved at the fourth window by RF (0.67). However, according to our results, XGB was able to achieve 0.7 at the same window (Figure 1). The difference in the results could be explained by the difference in the evaluation method or implementation of the classifiers. The previous study used the Caret package (Kuhn 2008) in R. In this project, however, we used the scikit-learn library in python. Although, we were able to achieve an even higher accuracy at the same window, the gradient boosting classifier (GB) attained a significantly higher accuracy at window 13 (0.82). The rest of the classifiers (non-ensemble models) demonstrated close to baseline performance on the original data. In Figure 2, we can see that preprocessing and feature selection significantly improved the performance of these models. The SVM classifier provides an accuracy of 0.65 early on in the second window. At the fifth window, SVM and LR achieved accuracies of 0.77 and 0.75, respectively. Logistic Regression has the advantage of high interpretability which enables explainablility. As expected, feature selection did not prove effective for the ensemble models.

The hyper-parameters of each classifier were tuned at the best window. The results are as follows:

Classifier Window Default Tuned
RF 4 0.68 0.68
LR 5 0.75 0.75
SVM 5 0.77 0.77
GB 11 0.82 0.68
XGB 9 0.70 0.74
KNN 7 0.68 0.62

Contrary to our expectation, tuning the hyper-parameters resulted in reduced accuracy for the majority of the models. For XGB, however, there was a noticeable improvement. In order to determine the statistical significance of this improvement, an independent samples t-test was conducted over 10 runs. The results are as follows:

 Overall Low class High class XGB p = 0.005 p = 0.065 p = 0.008

The result show evidence for a statistically significant improvement in the overall accuracy and the accuracy of the high class (p < .05). For the low class, however, we have weak evidence for a significant improvement.

##### Visual literacy
 Figure 3. Comparison of default models (VISLIT) Figure 4. Comparison of default models with feature preprocessing and feature selection

According to the study, RF was also the best classifier for visual literacy yielding an accuracy of 0.66 at window 3. However, not only did GB reach a higher accuracy at this window, but it also attained 0.72 at window 14 (Figure 3). KNN had a terrible performance on the original data, but after preprocessing and feature selection, this classifier provided an accuracy of 0.67 at window 2 (Figure 4).

The results of hyper-parameter tuning are as follows:

Classifier Window Default Tuned
RF 3 0.69 0.75
LR 2 0.67 0.67
SVM 4 0.68 0.51
GB 7 0.65 0.57
XGB 6 0.66 0.68
KNN 2 0.67 0.67

RF and XGB were the only classifiers with improved accuracy. Once again, a t-test was conducted to determine the significance of these improvements.

 Overall Low class High class RF p = 0.008 p = 0.023 p = 0.158 XGB p = 0.015 p = 0.151 p = 0.004

As we can see, the overall improvement was significant for both classifiers. For RF, the improvement was significant for the low class. As for XGB, however, it was the high class which was significantly improved.

##### Verbal working memory
 Figure 5. Comparison of default models Figure 6. Comparison of default models with feature preprocessing and feature selection

Lastly, for verbal working memory, the highest reported accuracy was 0.72 at window 10 achieved by XGB. Once again, GB proved to be the best model yielding an accuracy of 0.79 even earlier at window 7 (Figure 5). Preprocessing and feature selection was most effective for XGB and improved its accuracy to 0.7 at window 2 (Figure 6).

The results of hyper-parameter tuning are as follows:

Classifier Window Default Tuned
RF 8 0.65 0.60
LR 6 0.55 0.45
SVM 7 0.58 0.57
GB 2 0.66 0.61
XGB 2 0.70 0.70
KNN 3 0.58 0.53

For this cognitive ability, optimization of hyper-parameters did not lead to improved accuracy for any of the classifiers.

#### Discussion

According to our results, feature preprocessing and feature selection improved classification accuracy in most classifiers especially the non-ensemble models (i.e LR, SVM, KNN). As a result, we can accept our first hypothesis.

H1. Preprocessing and selection of features would improve classification accuracy (Accepted)

Optimization of hyper-parameters, on the other hand, although effective for some models, mostly resulted in lowered accuracy. Therefore, our second hypothesis is rejected.

H2. Hyper-parameter optimization would improve classification accuracy (Rejected)

Interestingly, as also demonstrated in the previous work, including data from additional tasks (later windows) appeared to have a negative effect on the prediction accuracy for some cognitive abilities. That is for some cognitive abilities (also dependent on the model), optimal results are found after accumulating eye-tracking data over few tasks. Seemingly, accumulation of further eye-tracking data, at a certain point, is more disruptive than helpful. Perhaps, other abilities start to interfere. As a result, eye movements become less indicative of the cognitive abilities in question. In this regard, in addition to investigating potential models, we also compared the results at different windows to find the optimal amount of eye-tracking data for each cognitive ability depending on the model.

#### Future work

In this project, the hyper-parameters of each model were tuned based on the window on which the model performed best with the default configuration. In that regard, It would be interesting to try the other windows as well.

## Annotated Bibliography

1. Ben Steichen, Giuseppe Carenini, Cristina Conati. User-Adaptive Information Visualization - Using Eye Gaze Data to Infer Visualization Tasks and User Cognitive Abilities
2. Under Preparation, Eye-Tracking to Predict User Cognitive Abilities and Performance for User-Adaptive Narrative Visualizations, AAAI 2020 Sept 5
3. Feurer M, Springenberg JT, Hutter F. Initializing bayesian hyperparameter optimization via meta-learning. InTwenty-Ninth AAAI Conference on Artificial Intelligence 2015 Feb 16.


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# Av for small signal analysis with BJT for unbypassed emitter and ro in place

My question is about another question on this site. On that question Z(in) for small signal analysis with BJT for unbypassed emitter and r0 in place Zin is asked for a common emitter configuration with emitter and ro is in place. My question is about voltage gain Av of the same circuit. In Boylestad and Nashelsky Av is given as below without a proof:

However, I find:

$$A_v=\frac{-R_C(\beta r_o-R_E)}{Z_b(R_E+R_C+r_o)}$$

Can you tell me what am I missing or show me how to find the correct answer?

EDIT:

Here is the circuit:

I used the following equivalent circuit for solution:

Then I defined $V_o$ as:

$$V_o=-I_c R_c$$

Then from KVL I obtained:

$$\beta I_b r_o-(I_b+I_c)R_E-R_C I_c-r_o I_c=0$$

$$I_c=\frac{(\beta r_o - R_E)}{R_E+R_C+r_o}I_b$$

Replacing $I_c$ from $V_o$:

$$V_o=-R_C \frac{(\beta r_o - R_E)}{R_E+R_C+r_o}I_b$$

and finally replacing $I_b=V_i/Z_b$:

$$A_v=\frac{-R_C(\beta r_o-R_E)}{Z_b(R_E+R_C+r_o)}$$

• If you can post your procedure to get your result, it'll be better to tell where is wrong. – diverger Nov 9 '14 at 15:06
• @diverger I included my solution. – user57823 Nov 9 '14 at 15:41

Your result has no problem, and your procedure is all right. To verify this, you can substitute the $Z_{b}$

$$Z_{b} = \beta r_{e}+[\frac{(\beta+1)+R_{c}/r_{o}}{1+(R_{C}+R_{E})/r_{o}}]R_{E}$$

into your result and into the textbook's result, and do some simplification, you'll find they give the same result. Cheers!

• Are you completely sure? Because calculations get messy after some time and I can't verify this. – user57823 Nov 11 '14 at 11:50
• Yes, it's really a tedious work, but i'm sure you are right, just be confident! – diverger Nov 11 '14 at 12:12

The HF gain ($C_1$, $C_2$ short circuited) of this circuit is much simpler if you use Norton's theorem.

$$V_{out}=I_{cc}(V_{in}) \times R_{out}$$

$$I_{cc}=-\frac{V_{in} g_m r_0}{R_e+r_0(1+g_mR_e)}$$

$$R_{out}= R_c || [R_e+r_0(1+g_mR_e)]$$

$$A_{vHF}= -\frac{g_mr_oR_c}{R_c+R_e+r_0(1+g_mR_e)}$$

NOTE: $g_m=I_c/V_t$

$R_b$ does not play any role at HF.

the singularities introduced at lower frequencies by the capacitors are:

$C_1$ introduces a zero in $s=0$ and a pole in $s=-1/[C_1(Z_b||R_b)]$

$C_2$ introduces a zero in $s=0$ and a pole in $s=-1/[C_2(R_{out}||R_c)]$

I hope it can help to check your result

format: plain


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# First Time Home Buyer (Savings Question)

1. Aug 25, 2010

My wife and I are beginning to save for our first house and I had a question about saving for a down payment. What is the best option in saving for a house? Traditional savings account, IRA, or other? I was thinking an IRA option, but I'm still learning the financial game. I know with a traditional IRA you can withdraw up to $10,000 without penalty as a first time home buyer; however, I'm unsure if there is a limit for the Roth IRA. Every place I read just stated you can withdraw from a Roth IRA without penalty for first time home buying. On a plus side, my family is in good condition right now because we live, in essence, rent free due to my military commitments. Also, we have more than enough money in savings and a decent retirement nestegg for our ages. Any help is greatly appreciated. 2. Aug 26, 2010 ### Ivan Seeking Staff Emeritus The last time I checked, I was getting about a quarter of a percent interest on my savings account. At that rate, you might as well put the money under the bed mattress. 3. Aug 26, 2010 ### Insanity Unless the rules have changed recently, the$10,000 withdraw for new homebuyers (also includes building or rebuilding a first home) is the same for both a Trad & Roth IRA. There are specific details to that rule, so find an adviser you can trust about this. Don't just withdraw $10,000 on your own. Saving accounts & CDs at banks are actually about the worst place to keep money for long term purposes with the low rate of return. Their are financial companies and representatives that will help people invest in mutual funds monthly, and there is a diversity of mutual funds to meet everyone needs. Even some for which the return is exempt from federal taxes. For a shorter term goal such as a down payment, a bond fund might be best. Better return then bank, not a volatile as equity stocks. 4. Aug 26, 2010 ### Vanadium 50 Staff Emeritus Many people are giving you advice without asking the question "When do you need the money"? Where you should put it is a strong function of when you need it. As an example, if you are putting$1000 a month away for 10 months at 0.75%, the difference between having the interest taxable and not works out to about $7. 5. Aug 26, 2010 ### BishopUser If you are looking to buy a home in the next 5 years I'd say savings account. ING gives me 1.1% and I think ally bank is giving around 1.25% atm. 5 years is way too short to try the stock market unless you create a very conservative portfolio (even then you can lose money!) in which case you might as well just use a savings acount - less hassle/paperwork etc. Since it is short term saving, I would just focus on good budgeting rather than focusing on where I can make a few extra dollars in a savings/market account. 6. Aug 27, 2010 ### Cod Thanks for the advice everyone. To answer Vanadium's question, the timeline for a down payment is a bit "up in the air". I'm not exactly sure when I'll be getting out of the military. It could be three years (earliest) or 20 years (latest). Right now I'm leaning towards 20 years since I'll be able to pull in a decent pension and only be 41 years old, which is still young enough to get a job. 7. Aug 27, 2010 ### airborne18 If you are in the military, then stick to CD's. You can get better than average rates with the military credit unions. Since the rates are so low, don't lock into long term CD's, or get CD's that allow you to change the rate once. ( rates will be going up bigtime when the economy improves. The Fed has already asserted this fact, and they control rates. ). The problem with mutual funds is that there have been so many boom and bust moments in the market that they have negative returns for the 3 and 5 year. From 9/11 to present day there has been no return for your risk, so just avoid it. You are in the military and you should not worry about where your money is. 8. Aug 28, 2010 ### Astronuc ### Staff: Mentor As Vanadium indicated it depends on one's time horizon. Money markets/saving accounts are paying rather low interest rates (~1%), and that could increase once the Fed and others start raising interest rates. Some bonds pay a little more like the 2-3% range. One could look at tax free municipal bonds. Or one could look into purchasing equities (stocks) of reputable companies. Right now GE's common stock is about$14.71 per share (close Fri, Aug 26).
EPS (ttm): 0.96
Div & Yield: 0.48 (3.30%)

Disclaimer: No endorsement is expressed or implied.

So one can get a reasonable return and the stock may appreciate. Of course it could go down in the near term if the economy softens.

One could put some money in cash investment for liquidity, some in bonds and some in equities. Really high yields usually imply riskier investments, e.g., hedge funds.


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# Ideas for high-level data structure visualization?

I want to create a (graphical) representation for a custom HL7 v2 message structure. In this scenario, I'm only concerned with the high-level message structure - the arrangement and usage of the segments of a message. The official documents (which are not available unless you pay for them, AFAIK) use a table notation like this:

In this structure, the different brackets denote optional [] and repeatable {} segments as well as alternatives <x|y|z>. Our custom structure is built upon this standard and basically appends a few segments and declares that we'll never use certain other segments. So far, we have simply mimicked the table-like structure, but I find that this is hard to read, especially with multiple nestend levels of conditional groups. I was thinking about converting the TikZ file tree example to represent the data structure, but with all the child [missing] entries and other peculiarities, I was wondering whether there are other (better?) options.

• This is the sort of thing I've seen used - I think they are called railroad diagrams. – Thruston Dec 15 '14 at 14:16
• @Thruston: This might be an interesting idea - however, I haven't found a viable alternative to produce these diagrams yet. The syntax package causes the entire build process to fail with missing \endcsnames, and the TikZ approach is way too complicated to be maintained by anyone else. An external tool would not be a great idea for maintenance reasons either... – vwegert Dec 16 '14 at 10:32
• How about transforming it to json? Then it is easy with LuaLaTeX – Yiannis Lazarides Dec 17 '14 at 16:50
• @YiannisLazarides, can you explain how it would be easy with Lua? Or perhaps point to some other answer? – Thruston Dec 17 '14 at 22:32
• @Thruston Lua tables, are the datastructure, you can represent anything with them, similarly is json, which I prefer as it is now almost universal. Once you have the structure in a Lua Table, the formatting can then be programmed. I will post an answer tomorrow, as it is hectic today. See hl7.org/implement/standards/fhir/patient-examples.json.html – Yiannis Lazarides Dec 18 '14 at 2:50

I wholeheartily agree that the current format you are using is awkard and possibly it casts a light as to the many reasons that health software projects fail; which IMHO is due to bureaucratic complexification. Having to pay for the formats adds insult to injury.

I would recommend to use Lua. If you generating the data for typesetting programatically, I would use Lua tables directly. If they are generated from a db, either export them as XML or preferably json. Here is a simplified program to demonstrate this. No need for any railroads or boxes, you can read the data like a paragraph from a book.

  \documentclass{article}
\usepackage{fontspec,luacode, xcolor}
\newfontfamily{\arial}{Arial}
\begin{document}
\arial

\begin{luacode}
if type(tex)=='table' then local print = tex.print  end

local patientVisit = {
patientClass = "CON",
assignedPatientLocation = {
pointOfCare = "8152879"
},
visitNumber = {
idNumber = 16164813
}
}

function inspect_table (tab, offset)
offset = offset or "\\mbox{~~}"
for k, v in pairs (tab) do
local newoffset = offset .. offset
if type(v) == "table" then
print(offset..k .. " = \\{\\par ")
inspect_table(v, newoffset)
print(offset.."\\}\\par")
else
if k=="patientClass" then print(offset .. '{\\color{red}'.. k .. "} = " .. tostring(v), "\\par")
else
print(offset..k.."="..v.."\\par")
end
end
end
end

inspect_table(patientVisit)
\end{luacode}

The routine above is very simple, however it can easily be modified to color code better, sort the fields and add notes etc. As a bonus no need of any catcodes. If you interested in a json solution drop a message and I will post one.

An Example: find it at http://ddi.uni-wuppertal.de/material/materialsammlung/

package is on CTAN: http://ctan.org/pkg/schule

\documentclass{schulein}
\usepackage[utf8]{inputenc}
\usepackage{schulinf}

\begin{document}

\begin{tikzpicture}[syntaxdiagramm]
\node [] {};
\node [terminal] {public};
\node [terminal] {class};
\node [nonterminal] {Klassenname};
\node [terminal] {\{};
\node (endstart) [point] {};
\node [point, below=of endstart] {};
\node [point, xshift=-75mm] {};
\node (endAttribute) [endpoint, continue chain=going below] {};
\node (startAttribute) [point] {};
{[start chain=attribute going right]
\chainin (startAttribute);
\node [point, xshift=25mm] {};
\node [point, continue chain=going above] {};
\node [nonterminal, continue chain=going left] {Attribut};
\node [point, join,join=with endAttribute] {};
}

\node (startKonstruktor) [point] {};
\node (endKonstruktor) [endpoint] {};
{[start chain=konstruktor going right]
\chainin (startKonstruktor);
\node [point, xshift=30mm] {};
\node [point, continue chain=going below] {};
\node [nonterminal, continue chain=going left] {Konstruktor};
\node [point, join,join=with endKonstruktor] {};
}

\node (endMethode) [endpoint] {};
\node (startMethode) [point] {};
{[start chain=methode going right]
\chainin (startMethode);
\node [point, xshift=25mm] {};
\node [point, continue chain=going above] {};
\node [nonterminal, continue chain=going left] {Methode};
\node [point, join,join=with endMethode] {};
}

\node [point] {};
\node [terminal, continue chain=going left] {\}};

\end{tikzpicture}
\end{document}

• There's a lot hidden away in that German "school" package - is there any documentation in English, I wonder? – Thruston Dec 17 '14 at 22:34
• at this very moment, there is no English documentation, so you've to read the source-code. But most is documented in the pgfmanual as well – including the railroad-diagramm-construction as an example – L. Humbert Dec 18 '14 at 8:33


text:
# Tag Info

4

I do not know how this is teached but I think that you make the problem difficult changing the starting value of the index in the summation. Let me try $$y=\sum_{n=0}^{\infty}a_n x^n$$ $$y'=\sum_{n=0}^{\infty}n a_n x^{n-1}$$ $$y''=\sum_{n=0}^{\infty}n(n-1) a_n x^{n-2}$$ Rewrite the equation as $$(x^2 + 1)y'' - 6xy' + 10y =x^2y''+y''-6xy'+10y=0$$ and include ...

4

It's actually not so difficult. Using: $$\frac{1}{1-x-y}=\sum_{n,m\ge 0}{n + m\choose n}x^ny^m$$ You get $$\frac{x}{1-x-x^3}=x\cdot\frac{1}{1-x-x^3}=x\cdot \sum_{n,m\ge 0}{n + m\choose n}x^n(x^3)^m=\sum_{n,m\ge 0}{n + m\choose n}x^{n+3m+1}$$ And after a last manipulation you can get a formal power series of the classical form $\sum a_nx^n$

3

$$\frac{1}{1-x}=1+x+x^2+x^3+x^4+...$$ $$x(\frac{1}{1-(x+x^3)})=x(1+(x+x^3)+(x+x^3)^2+(x+x^3)^3+.....)$$

3

$$e^{x^2-1}=\frac1e\sum_{n=0}^\infty\frac{(x^2)^n}{n!}=\cdots$$

3

Using Stirling Numbers of the Second Kind, we have $\newcommand{\stirtwo}[2]{\left\{{#1}\atop{#2}\right\}}$ $$k^m=\sum_{j=0}^m\stirtwo{m}{j}\binom{k}{j}j!\tag{1}$$ Therefore, \begin{align} \sum_{k=0}^\infty k^mz^k &=\sum_{k=0}^\infty\sum_{j=0}^m\stirtwo{m}{j}\binom{k}{j}j!z^k\\ &=\sum_{j=0}^m\stirtwo{m}{j}j!\sum_{k=j}^\infty\binom{k}{j}z^k\\ ... 3 For the sake of simplicity, assume that  f  is entire. Then \begin{align} \forall (x,y) \in \mathbb{R}^{2}: \quad f(x + i y) & = \sum_{m = 0}^{\infty} a_{m} (x + i y)^{m} \\ & = \sum_{m = 0}^{\infty} \left[ a_{m} \sum_{n = 0}^{m} \binom{m}{n} x^{n} (i y)^{m - n} \right] \\ & = \sum_{m = 0}^{\infty} \left[ \sum_{n = 0}^{m} a_{m} i^{m ... 2 EDIT: Sorry, the previous answer was not correct. You have \Bigg|\frac{x^8-1}{3}\Bigg| <1 \Leftrightarrow |{x^8}-1|<3 \Leftrightarrow |x|<4^{1/8} =\sqrt[4]{2}. $$In these examples you typically want to work without getting rid of the absolute value, since the x in your series may also be considered as complex number; and those have no ... 2 Suppose we seek to evaluate$$\sum_{k\ge 1} k^m z^k.$$Put$$k^m = \frac{m!}{2\pi i} \int_{|w|=\epsilon} \frac{1}{w^{m+1}} \exp(kw) \; dw \\ = \frac{m!}{2\pi i} \int_{|w|=\epsilon} \frac{1}{w^{m+1}} \sum_{q=0}^k {k\choose q} (\exp(w)-1)^q \; dw \\ = \frac{m!}{2\pi i} \int_{|w|=\epsilon} \frac{1}{w^{m+1}} \sum_{q=0}^k \frac{k!}{(k-q)!} ...

2

Start from: $$\sum_{k\geq 1}z^k = \frac{z}{1-z}$$ then apply the operator $\varphi: f(z)\to z\cdot f'(z)$ to both terms $m$ times. That gives that for any $|z|<1$ we have: $$\sum_{k\geq 1}k\,z^k = \frac{z}{(1-z)^2},\quad \sum_{k\geq 1}k^2 z^k=\frac{z+z^2}{(1-z)^3}$$ and so on.

2

I think that the easiest way to handle $\frac{x}{e^x-1}$ is to write out the Taylor expansion of $e^x$, and use long division to compute the coefficients term by term. For $\frac{1}{1-\sin(x)}$, one needs to avoid points such that $\sin(x) =1$, but supposing that $|\sin(x)|<1$, we can use the geometric series and the Taylor expansion of $\sin(x) = ... 2 By ratio test $$\lim_{n\rightarrow\infty}\left|\frac{x^{n+1}n!\left(n+1\right)^{3}}{\left(n+1\right)!x^{n}n^{3}}\right|=\lim_{n\rightarrow\infty}\left|\frac{x}{n+1}\right|=0$$ so the radius of convergence is$\infty$and the interval of convergence is$\left(-\infty,\infty\right).$2 Just as in the expansion for$e^x$you can expand$e^{f(x)}$to $$e^{f(x)} \;\; =\;\; \sum_{n=0}^\infty \frac{(f(x))^n}{n!}.$$ For safety, I would just use functions$f(x)$which are differentiable. 2 I would say this is a very difficult thing to do (from which you can deduce that this is not going to be a very satisfactory answer...). Having just the power series about one point, the radius of convergence tells you the distance from the point of expansion of the first point at which the function fails to be analytic. This essentially divides into poles, ... 2 To answer both your old and your new question at the very same time, we can consider a surprising relationship between power series and recurrence relations through generating functions. As a simple example, consider representing$\frac{1}{1-x}$as a power series. In particular, we want to discover an$f_n$such that ... 2 The coefficient of$x^n$in the product $$(1+x)(1+x^2)(1+x^3)\cdot\ldots$$ gives the number of ways of writing$n$as a sum of distinct natural numbers, while the coefficient of$x^n$in the product: $$\frac{1}{1-x}\cdot\frac{1}{1-x^3}\cdot\ldots = (1+x+x^2+\ldots)(1+x^3+x^6+\ldots)\cdot\ldots$$ gives the number of ways to write$n$as a sum of different ... 1 Hint: Developing the two expressions with just enough terms to get the exact$x^7$coefficient, $$\frac1{1-x}\frac1{1-x^3}\frac1{1-x^5}\frac1{1-x^7}\\ =(1+x+x^2+x^3+x^4+x^5+x^6+x^7)(1+x^3+x^6)(1+x^5)(1+x^7)\\ =\color{green}1+\color{green}1x+\color{green}1x^2+\color{green}2x^3+\color{green}2x^4+\color{green}3x^5+\color{green}4x^6+\color{green}5x^7\cdots$$ ... 1$h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x) $If$n \le j$, since$k \le n$, the sums become$\sum_{k=1}^n \sum_{v=1}^{k} = \sum_{v=1}^{n} \sum_{k= v}^n $. If$n > j, the sums could be split into two parts like this: \begin{align} \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} ... 1 Your series is centered aboutx_0=1$. Since $$r=\lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|$$ where$a_k$are the coefficients, plugging in your formula gives $$r=\lim_{n\to\infty}\left|\frac{\frac{(-1)^nn}{2^n}}{\frac{(-1)^{n+1}(n+1)}{2^{n+1}}}\right|=\lim_{n\to\infty}\frac n{2^n}\cdot\frac{2^{n+1}}{n+1}=2\lim_{n\to\infty}\frac n{n+1}=2$$ so the ... 1 Your recursive formula should be $$a_{n+1} = \frac {1-n}{n+1}a_n$$ You just made a small sign error. Then just plug in$a_0=1$and find the next few values by hand. $$a_0 = 1 \implies a_1=1 \implies a_2 =0$$ Then we don't need to go any further because we can see by our recursive formula that every$a_n$for$n\ge 2$must equal$0$as well. So your ... 1 I don't think you need a recurrence relation to find the generating function: it's simply $$\underbrace{(x+x^2+\dots+x^6)\dots(x+x^2+\dots+x^6)}_{4\text{ times}} = (x+x^2+\dots+x^6)^4.$$ Each$x^k$term in the resulting expansion corresponds to one way to obtain$k$from four dice. 1 So you are trying to prove that your function is equal to its power series representation, and as you said to do this you will use Taylors theorem and inequality.$\mathbf{Thereom:}$If$f(x)=T_n(x)+R_n(x) , $where$T_n$is the$n^{th}$degree taylor polynomial of$f$at$a$and$\lim_{n\to \infty} R_n(x)=0$for$|x-a| \lt R$, then$f$is equal to the ... 1 The Maple command dsolve((x^2+1)*(diff(y(x), x, x))-6*x*(diff(y(x), x))+10*y(x)) produces a polynomial solution $$y \left( x \right) ={\it \_C1}\, \left( -5\,{x}^{2}+1 \right) +{\it \_C2}\, \left( {x}^{5}+10\,{x}^{3}-15\,x \right) .$$ 1 I like this problem! I will have to assign this sometime. Apply the root test. The denominator satisfies$\lim_{n \to \infty} \sqrt[n]{2n+1} = 1$(use L'Hopitals). The limit in the numerator simplifies to$\lim_{n \to \infty} |x-2|^{n} ,$which is 0 when$|x-2|<1.$Therefore the preliminary interval of converge is$1<x<3.$The series is ... 1 $$\sum_{n=0}^d x^n= \frac{x^{d+1}-1}{x-1}$$ So we have: $$\sum_{m=1}^{\log_2 N} 2^m=\sum_{m=0}^{\log_2 N} 2^m-2^0=\sum_{m=0}^{ \log_2 N } 2^m-1 = \frac{2^{\lfloor \log_2N \rfloor +1}-1}{2-1}-1 \leq \frac{N \cdot 2-1}{1}-1 \\=N \cdot 2-2$$ 1 For the first part, you have established that$-5$and$-3$are not part of the radius of convergence, therefore the interval is$(-5,\,3)$, and not$[-5,\,3]$(which would include$-5$and$-3$). Same logic for the second part. You've shown that the series converges for$-5 < x < -3$but diverges for$-5$and$-3\$, therefore, it is the open interval ...

1

We can write $$x+x^2+\cdots+x^6=\frac{x-x^7}{1-x}$$ and then we can expand the numerator with a binomial expansion: $$(x+x^2+\cdots+x^6)^4 = \frac{(x-x^7)^4}{(1-x)^4} = (1-x)^{-4} \sum_{k=0}^4 {4 \choose k} (-1)^k x^{6k+4}$$ $$=\left( \frac16 \sum_{n=0}^\infty (n+3)(n+2)(n+1) x^{n} \right) \left( \sum_{n=0}^4 {4 \choose n} (-1)^n x^{6n+4} \right)$$ ...

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• Effect of suction/blowing on heat-absorbing unsteady radiative Casson fluid past a semi-infinite flat plate with conjugate heating and inclined magnetic field

• # Fulltext

https://www.ias.ac.in/article/fulltext/pram/094/0127

• # Keywords

Suction; blowing; porous medium; Casson fluid; conjugate heat and mass transfer; inclined magnetic field; thermal radiation.

• # Abstract

This article studies the effect of suction/blowing, inclined magnetic field, and chemical reaction on heat-absorbing unsteady radiative Casson fluid past a semi-infinite flat plate in porous medium incorporating the oscillatory plate movement as a linear combination of ‘cosine’ and ‘sine’ functions in time. Further, the mass and heat transfer characteristics are examined under the influence of conjugate mass and heat transfer phenomena at the boundary. The governing equations of the model, viz. the energy, mass transfer, and momentum, are transformed into the non-dimensional form adopting suitable non-dimensional variables and parameters. The exact analytic solutions of the model for species concentration and fluid temperature are obtained using Laplace transform technique whilst, the solution for the fluid velocity has been obtained numerically with the help of the INVLAP routine of MATLAB. The expressions for fluid temperature, species concentration, and velocity are obtained and studied graphically for various physical parameters influencing the fluid flow model taking into account the case of both suction and blowing. Further, the solutions when the Casson fluid parameter $\alpha\rightarrow\infty$ are also obtained as special cases. Results for the skin friction coefficient, Sherwood number, and Nusselt number are numerically calculated and put in tabular form. An increment for the inclination angle of the magnetic field enhances the fluid velocity while it has a reverse effect on skin friction. Increasing the Schmidt number Sc leads to a reduction in fluid concentration and increasing the value of thermal radiation accelerates fluid temperature. This fluid flow model has several industrial applications in the field of chemical, polymer, medical sciences, etc.

• # Pramana – Journal of Physics

Volume 96, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


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Home > Confidence Interval > Confidence Standard Error

# Confidence Standard Error

## Contents

If you look closely at this formula for a confidence interval, you will notice that you need to know the standard deviation (σ) in order to estimate the mean. This may sound unrealistic, and it is. Sampling from a distribution with a large standard deviation The first data set consists of the ages of 9,732 women who completed the 2012 Cherry Blossom run, a 10-mile race held This probability is small, so the observation probably did not come from the same population as the 140 other children. check my blog

Randomised Control Trials4. SE for a proprotion(p) = sqrt [(p (1 - p)) / n] 95% CI = sample value +/- (1.96 x SE) c) What is the SE of a difference in A Concise Guide to Clinical TrialsPublished Online: 29 APR 2009Summary Bean Around The World Skip to content HomeAboutMFPH Part A ← Epidemiology - Attributable Risk (including AR% PAR +PAR%) Statistical Methods Roman letters indicate that these are sample values. http://www.healthknowledge.org.uk/e-learning/statistical-methods/practitioners/standard-error-confidence-intervals

## Standard Error And Confidence Level

These come from a distribution known as the t distribution, for which the reader is referred to Swinscow and Campbell (2002). For this purpose, she has obtained a random sample of 72 printers and 48 farm workers and calculated the mean and standard deviations, as shown in table 1. If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative

The standard error is the standard deviation of the Student t-distribution. In this scenario, the 2000 voters are a sample from all the actual voters. Lane Prerequisites Areas Under Normal Distributions, Sampling Distribution of the Mean, Introduction to Estimation, Introduction to Confidence Intervals Learning Objectives Use the inverse normal distribution calculator to find the value of Standard Error Confidence Interval Linear Regression A consequence of this is that if two or more samples are drawn from a population, then the larger they are, the more likely they are to resemble each other -

The blood pressure of 100 mmHg noted in one printer thus lies beyond the 95% limit of 97 but within the 99.73% limit of 101.5 (= 88 + (3 x 4.5)). Sem Vs Confidence Interval Journal of the Royal Statistical Society. Table 2 shows that the probability is very close to 0.0027. The true standard error of the mean, using σ = 9.27, is σ x ¯   = σ n = 9.27 16 = 2.32 {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt

Or decreasing standard error by a factor of ten requires a hundred times as many observations. Standard Error Confidence Interval Proportion A better method would be to use a chi-squared test, which is to be discussed in a later module. Thus the variation between samples depends partly on the amount of variation in the population from which they are drawn. Recall that with a normal distribution, 95% of the distribution is within 1.96 standard deviations of the mean.

## Sem Vs Confidence Interval

This estimate may be compared with the formula for the true standard deviation of the sample mean: SD x ¯   = σ n {\displaystyle {\text{SD}}_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} http://handbook.cochrane.org/chapter_7/7_7_7_2_obtaining_standard_errors_from_confidence_intervals_and.htm Correction for correlation in the sample Expected error in the mean of A for a sample of n data points with sample bias coefficient ρ. Standard Error And Confidence Level The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N. Standard Error Confidence Interval Calculator However, computing a confidence interval when σ is known is easier than when σ has to be estimated, and serves a pedagogical purpose.

Assumptions and usage Further information: Confidence interval If its sampling distribution is normally distributed, the sample mean, its standard error, and the quantiles of the normal distribution can be used to click site In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the A consequence of this is that if two or more samples are drawn from a population, then the larger they are, the more likely they are to resemble each other - The distribution of these 20,000 sample means indicate how far the mean of a sample may be from the true population mean. Standard Error Of Measurement Confidence Interval

1. Calculation of CI for mean = (mean + (1.96 x SE)) to (mean - (1.96 x SE)) b) What is the SE and of a proportion?
2. If σ is not known, the standard error is estimated using the formula s x ¯   = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} where s is the sample
3. Notice that the population standard deviation of 4.72 years for age at first marriage is about half the standard deviation of 9.27 years for the runners.
4. For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above
5. Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation".
6. ISBN 0-521-81099-X ^ Kenney, J.
7. It is important to realise that we do not have to take repeated samples in order to estimate the standard error; there is sufficient information within a single sample.

This probability is usually used expressed as a fraction of 1 rather than of 100, and written as p Standard deviations thus set limits about which probability statements can be made. To estimate the standard error of a student t-distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence The earlier sections covered estimation of statistics. news Note that this does not mean that we would expect, with 95% probability, that the mean from another sample is in this interval.

The first step is to obtain the Z value corresponding to the reported P value from a table of the standard normal distribution. Margin Of Error Confidence Interval However, it is much more efficient to use the mean +/- 2SD, unless the dataset is quite large (say >400). The following expressions can be used to calculate the upper and lower 95% confidence limits, where x ¯ {\displaystyle {\bar {x}}} is equal to the sample mean, S E {\displaystyle SE}

## Of course, T / n {\displaystyle T/n} is the sample mean x ¯ {\displaystyle {\bar {x}}} .

For each sample, calculate a 95% confidence interval. Recall that 47 subjects named the color of ink that words were written in. T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Sampling Error Confidence Interval Confidence intervals provide the key to a useful device for arguing from a sample back to the population from which it came.

For the purpose of this example, the 9,732 runners who completed the 2012 run are the entire population of interest. For an upcoming national election, 2000 voters are chosen at random and asked if they will vote for candidate A or candidate B. We can say that the probability of each of these observations occurring is 5%. More about the author Imagine taking repeated samples of the same size from the same population.


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## Basic College Mathematics (10th Edition)

$\frac{5}{6}$
$\frac{1}{2}$ + $\frac{1}{3}$ The least common multiple of 2 and 3 is 6, so first rewrite the fractions as like fractions with a denominator of 6. This is the least common denominator (LCD) of 2 and 3. Step 1 $\frac{1}{2}$ = $\frac{}{6}$ Divide 6 by 2, getting 3, multiply numerator and denominator by 3 $\frac{1}{2}$ = $\frac{1 * 3}{2 * 3}$ = $\frac{3}{6}$ $\frac{1}{3}$ = $\frac{}{6}$ Divide 6 by 3, getting 2, multiply numerator and denominator by 2 $\frac{1}{3}$ = $\frac{1 * 2}{3 * 2}$ = $\frac{2}{6}$ Now, add the like fractions $\frac{3}{6}$ and $\frac{2}{6}$ Step 2 : $\frac{3}{6}$ + $\frac{2}{6}$ = $\frac{3 + 2}{6}$ = $\frac{5}{6}$ $\frac{5}{6}$ is in the lowest term .


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Pambuccian, Victor V.

Compute Distance To:
 Author ID: pambuccian.victor Published as: Pambuccian, Victor; Pambuccian, V.; Pambuccian, Victor V.; Pambuccian, Viktor more...less Homepage: https://webapp4.asu.edu/directory/person/80568#pub_anchor External Links: MGP
 Documents Indexed: 131 Publications since 1984 Reviewing Activity: 1,095 Reviews Co-Authors: 17 Co-Authors with 23 Joint Publications 286 Co-Co-Authors
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 108 single-authored 5 Schacht, Celia 4 Struve, Rolf 2 Blass, Andreas Raphael 2 Kalhoff, Franz B. 2 Struve, Horst 1 Alama, Jesse 1 Harutyunyan, Davit 1 Havlicek, Hans 1 Hociotă, Ioana 1 Hulpke, Alexander Jörg 1 Menn, Stephen 1 Movsisyan, Yuri Movses 1 Nazaryan, Aram 1 Prażmowski, Krzysztof 1 Sakowicz, K. 1 Schnabel, Rudolf 1 Zamfirescu, Tudor I.
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 18 Journal of Geometry 13 Mathematical Logic Quarterly (MLQ) 12 Beiträge zur Algebra und Geometrie 6 Note di Matematica 5 Notre Dame Journal of Formal Logic 5 Bulletin of the Polish Academy of Sciences, Mathematics 4 Results in Mathematics 4 Studia Logica 4 Indagationes Mathematicae. New Series 4 European Mathematical Society Newsletter 3 Acta Mathematica Hungarica 3 Aequationes Mathematicae 3 Mathematical Chronicle 3 Zeszyty Naukowe. Geometria 2 Demonstratio Mathematica 2 Geometriae Dedicata 2 Publicationes Mathematicae Debrecen 2 Synthese 2 Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 2 Mathematica Pannonica 2 Elemente der Mathematik 2 Expositiones Mathematicae 2 Philosophia Mathematica. Series III 2 The Review of Symbolic Logic 1 American Mathematical Monthly 1 Journal of Mathematical Analysis and Applications 1 Rocky Mountain Journal of Mathematics 1 The Mathematical Intelligencer 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Archiv der Mathematik 1 Canadian Mathematical Bulletin 1 The Nepali Mathematical Sciences Report 1 Extracta Mathematicae 1 Forum Mathematicum 1 Glasnik Matematički. Serija III 1 Historia Mathematica 1 Georgian Mathematical Journal 1 Scientiae Mathematicae Japonicae 1 Journal of Applied Logic 1 Rendiconti del Seminario Matematico. Universitá e Politecnico di Torino 1 Annali dell’Università di Ferrara. Sezione VII. Scienze Matematiche 1 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences
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 111 Geometry (51-XX) 83 Mathematical logic and foundations (03-XX) 11 History and biography (01-XX) 7 General and overarching topics; collections (00-XX) 6 Field theory and polynomials (12-XX) 5 Convex and discrete geometry (52-XX) 3 Number theory (11-XX) 2 Differential geometry (53-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Group theory and generalizations (20-XX) 1 Real functions (26-XX) 1 General topology (54-XX) 1 Relativity and gravitational theory (83-XX)

Citations contained in zbMATH Open

96 Publications have been cited 340 times in 132 Documents Cited by Year
The axiomatics of ordered geometry: I. Ordered incidence spaces. Zbl 1227.51010
Pambuccian, Victor
2011
A logical look at characterizations of geometric transformations under mild hypotheses. Zbl 0987.51010
Pambuccian, Victor
2000
On the stepwise construction of the parallel postulate. (Zum Stufenaufbau des Parallelenaxioms.) Zbl 0815.51012
Pambuccian, Victor
1994
Axiomatizations of hyperbolic and absolute geometries. Zbl 1106.51008
Pambuccian, Victor
2006
Orthogonality as single primitive notion for metric planes. Zbl 1161.51004
Pambuccian, Victor
2007
Constructive axiomatization of non-elliptic metric planes. Zbl 1036.03051
Pambuccian, Victor
2003
Alexandrov-Zeeman type theorems expressed in terms of definability. Zbl 1141.03007
Pambuccian, Victor
2007
Book review of: D. Hilbert, David Hilbert’s lectures on the foundations of geometry, 1891–1902. Zbl 1305.00057
Pambuccian, Victor
2013
Fragments of Euclidean and hyperbolic geometry. Zbl 0995.51005
Pambuccian, Victor
2001
The simplest axiom system for plane hyperbolic geometry. Zbl 1069.03007
Pambuccian, Victor
2004
Simplicity. Zbl 0669.03005
Pambuccian, Victor
1988
Forms of the Pasch axiom in ordered geometry. Zbl 1202.51010
Pambuccian, Victor
2010
On M. T. Calapso’s characterization of the metric of an absolute plane. Zbl 1169.51017
Pambuccian, Victor; Struve, Rolf
2009
Axiomatizing geometric constructions. Zbl 1143.03007
Pambuccian, Victor
2008
The Erdős-Mordell inequality is equivalent to non-positive curvature. Zbl 1140.51009
Pambuccian, Victor
2008
Defining co-punctuality in terms of line-orthogonality in plane hyperbolic geometry. Zbl 1100.51012
Pambuccian, V.; Prazmowski, K.; Sakowicz, K.
2005
Weakly ordered plane geometry. Zbl 1205.51007
Pambuccian, Victor
2010
On the existence of a equilateral triangle in $$H$$-planes. (Zur Existenz gleichseitiger Dreiecke in $$H$$-Ebenen.) Zbl 0931.51006
Pambuccian, Victor
1998
A reverse analysis of the Sylvester-Gallai theorem. Zbl 1202.03023
Pambuccian, Victor
2009
On the equivalence of Lagrange’s axiom to the Lotschnittaxiom. Zbl 1189.51010
Pambuccian, Victor
2009
The complexity of plane hyperbolic incidence geometry is $$\forall\exists\forall\exists$$. Zbl 1066.03027
Pambuccian, Victor
2005
Ternary operations as primitive notions for constructive plane geometry. IV. Zbl 0799.03072
Pambuccian, Victor
1994
Ternary operations as primitive notions for constructive plane geometry. V. Zbl 0808.03045
Pambuccian, Victor
1994
On the axiomatics of projective and affine geometry in terms of line intersection. Zbl 1064.51001
Havlicek, Hans; Pambuccian, Victor
2004
Groups and plane geometry. Zbl 1086.03007
Pambuccian, Victor
2005
Elementary axiomatizations of projective space and of its associated Grassmann space. Zbl 1193.03022
Pambuccian, Victor
2005
Hyperbolic geometry in terms of point-reflections or of line-orthogonality. Zbl 1075.51005
Pambuccian, Victor
2004
Why are surjective lineations of the Archimedean hyperbolic plane motions? Zbl 1027.51021
Pambuccian, V.
2003
Early examples of resource-consciousness. Zbl 1057.03010
Pambuccian, Victor
2004
Construction of hyperbolic geometry from the notion of line orthogonality. (Aufbau der hyperbolischen Geometrie aus dem Geradenorthogonalitätsbegriff.) Zbl 1048.51013
Pambuccian, V.
2003
Euclidean geometry problems rephrased in terms of midpoints and point-reflections. Zbl 1082.51011
Pambuccian, Victor
2005
Negation-free and contradiction-free proof of the Steiner-Lehmus theorem. Zbl 1455.03076
Pambuccian, Victor
2018
Metric geometries in an axiomatic perspective. Zbl 1386.03005
Pambuccian, Victor; Struve, Horst; Struve, Rolf
2017
Another equivalent of the Lotschnittaxiom. Zbl 1362.51004
Pambuccian, Victor
2017
Constructive axiomatizations of plane absolute, Euclidean and hyperbolic geometry. Zbl 0971.03063
Pambuccian, Victor
2001
What is plane equiaffine geometry? Zbl 1085.51003
Pambuccian, Victor
2003
Ternary operations as primitive notions for constructive plane geometry. Zbl 0669.03029
Pambuccian, Victor
1989
Die metrisch-euklidische Geometrie als Ausgangspunkt für die geordnet- euklidische Geometrie. Zbl 0566.51018
Schnabel, Rudolf; Pambuccian, Victor
1985
Ternary operations as primitive notions for constructive plane geometry. VI. Zbl 0828.03032
Pambuccian, Victor
1995
Simple axiom systems for affine planes. Zbl 0848.03003
Pambuccian, Victor
1995
On the planarity of the equilateral, isogonal pentagon. Zbl 1017.03002
Pambuccian, Victor
2003
The sum of irreducible fractions with consecutive denominators is never an integer in $$\mathrm{PA}^-$$. Zbl 1185.03086
Pambuccian, Victor
2008
Simple axiom systems for Euclidean geometry. Zbl 0699.51010
Pambuccian, Victor
1989
Acute triangulation of a triangle in a general setting revisited. Zbl 1246.51010
Hociotă, Ioana; Pambuccian, Victor
2011
A methodologically pure proof of a convex geometry problem. Zbl 1008.52002
Pambuccian, Victor
2001
Binary relations as single primitive notions for hyperbolic three-space and the inversive plane. Zbl 1044.51012
Pambuccian, Victor
2000
An axiomatics for hyperbolic projective-metric planes in terms of lines and orthogonality. Zbl 1101.51007
Pambuccian, Victor
2004
Acute triangulation of a triangle in a general setting. Zbl 1201.51015
Pambuccian, Victor
2010
Splitting the Pasch axiom. Zbl 0854.51010
Pambuccian, Victor
1996
The Steiner-Lehmus theorem and “triangles with congruent medians are isosceles” hold in weak geometries. Zbl 1367.51008
Pambuccian, Victor; Struve, Horst; Struve, Rolf
2016
The arithmetic of the even and the odd. Zbl 1381.03041
Pambuccian, Victor
2016
Lippmann’s axiom and Lebesgue’s axiom are equivalent to the Lotschnittaxiom. Zbl 1429.51004
Pambuccian, Victor; Schacht, Celia
2019
From absolute to affine geometry in terms of point-reflections, midpoints, and collinearity. Zbl 1414.51002
Alama, Jesse; Pambuccian, Victor
2016
The simplest axiom system for plane hyperbolic geometry revisited. Zbl 1234.51011
Pambuccian, Victor
2011
Another splitting of the Pasch axiom. Zbl 1227.51011
Pambuccian, Victor
2011
Addenda et corrigenda to: “The arithmetic of the even and the odd”. Zbl 1382.03080
Menn, Stephen; Pambuccian, Victor
2016
Ternary operations as primitive notions for constructive plane geometry. III. Zbl 0799.03071
Pambuccian, Victor
1993
On the extensiveness axiom for partial affine spaces. (Über das Reichhaltigkeitsaxiom für partielle affine Räume.) Zbl 0853.51002
Pambuccian, Victor
1996
Constructive axiomatization of plane hyperbolic geometry. Zbl 1003.03052
Pambuccian, Victor
2001
On R. Baer’s generalized hyperbolic planes. Zbl 1012.51009
Pambuccian, Victor
2001
On the constructive geometry of Euclidean planes. (Zur konstruktiven Geometrie euklidischer Ebenen.) Zbl 0959.03052
Pambuccian, V.
1998
The axiomatic destiny of the theorems of Pappus and Desargues. Zbl 1454.01016
Pambuccian, Victor; Schacht, Celia
2019
The case for the irreducibility of geometry to algebra. Zbl 07511962
Pambuccian, Victor; Schacht, Celia
2022
Axiom systems implying infinity in the foundations of geometry. Zbl 1473.51012
Pambuccian, Victor; Struve, Rolf
2021
Absolute geometry proofs of two geometric inequalities of Chisini. Zbl 1366.51007
Pambuccian, Victor
2017
The ubiquitous axiom. Zbl 1490.51010
Pambuccian, Victor; Schacht, Celia
2021
The most simple axioms system in elementary geometry. (Le plus simple système d’axiomes pour la géométrie élémentaire.) Zbl 0821.51015
Pambuccian, Victor
1994
On the simplicity of an axiom system for plane Euclidean geometry. Zbl 0884.03004
Pambuccian, Victor
1997
Axiomatizations of hyperbolic geometry: A comparison based on language and quantifier type complexity. Zbl 1028.03014
Pambuccian, Victor
2002
Sphere tangency as single primitive notion for hyperbolic and Euclidean geometry. Zbl 1038.51014
Pambuccian, V.
2003
Another example of an exotic function. Zbl 0825.26007
Pambuccian, Victor
1989
Universal-existential axiom systems for geometries expressed with Pieri’s isosceles triangle as single primitive notion. Zbl 1223.51028
Pambuccian, V.
2009
A definitional view of Vogt’s variant of the Mazur-Ulam theorem. Zbl 0992.03045
Pambuccian, Victor
2001
Mohr-Mascheroni as an axiomatizability statement. Zbl 0993.51006
Pambuccian, Victor
2001
The non-planarity of $$K_{5}$$ and $$K_{3,3}$$ as axioms for plane ordered geometry. Zbl 1267.51012
Pambuccian, Victor
2012
Fragments of the plane hyperbolic geometry. (Fragmente der ebenen hyperbolischen Geometrie.) Zbl 1061.51013
Pambuccian, Victor
2004
Two notes on the axiomatics of structures with parallelism. Zbl 1162.51300
Pambuccian, Victor
2001
Positive definitions of segment congruence in terms of segment inequality. Zbl 1117.51018
Pambuccian, Victor
2006
On the Pythagorean hull of $$\mathbb Q$$. Zbl 1147.12300
Pambuccian, Victor
1990
The elementary geometry of a triangular world with hexagonal circles. Zbl 1146.51009
Pambuccian, Victor
2008
Mappings preserving the area equality of hyperbolic triangles are motions. Zbl 1213.51017
Pambuccian, Victor
2010
Corrigendum to “The complexity of plane hyperbolic incidence geometry is $$\forall \exists \forall \exists$$”. Zbl 1152.03311
Pambuccian, Victor
2008
On definitions in an infinitary language. Zbl 1021.03029
Pambuccian, Victor
2002
Ternary operations as primitive notions for constructive plane geometry. II. Zbl 0739.03033
Pambuccian, Victor
1992
Another constructive axiomatization of Euclidean planes. Zbl 0943.03052
Pambuccian, Victor
2000
A logical reading of the nonexistence of proper homomorphisms between affine spaces. Zbl 0966.51002
Pambuccian, Victor
2000
Existence of special rainbow triangles in weak geometries. Zbl 1431.51007
Pambuccian, Victor
2019
Schatunowsky’s theorem, Bonse’s inequality, and Chebyshev’s theorem in weak fragments of Peano arithmetic. Zbl 1333.03240
Pambuccian, Victor
2015
An axiomatic look at the Erdős-Trost problem. Zbl 1376.51008
Kalhoff, Franz; Pambuccian, Victor
2016
A problem in Pythagorean arithmetic. Zbl 1404.03033
Pambuccian, Victor
2018
The Green-Tao theorem on primes in arithmetical progressions in the positive cone of $$\mathbb Z [X]$$. Zbl 1293.11096
Pambuccian, Victor
2014
Paolo Pizzetti: the forgotten originator of triangle comparison geometry. Zbl 1221.01083
Pambuccian, Victor; Zamfirescu, Tudor
2011
Positive existential definability of parallelism in terms of betweenness in Archimedean ordered affine geometry. Zbl 1230.51013
Kalhoff, Franz; Pambuccian, Victor
2011
On the Euclidean geometry of the double-edged ruler. Zbl 1228.51021
Pambuccian, Victor
1999
Prolegomena to any theory of proof simplicity. Zbl 1441.03007
Pambuccian, Victor
2019
The elementary Archimedean axiom in absolute geometry. Zbl 1430.51020
Pambuccian, Victor
2019
The case for the irreducibility of geometry to algebra. Zbl 07511962
Pambuccian, Victor; Schacht, Celia
2022
Axiom systems implying infinity in the foundations of geometry. Zbl 1473.51012
Pambuccian, Victor; Struve, Rolf
2021
The ubiquitous axiom. Zbl 1490.51010
Pambuccian, Victor; Schacht, Celia
2021
Lippmann’s axiom and Lebesgue’s axiom are equivalent to the Lotschnittaxiom. Zbl 1429.51004
Pambuccian, Victor; Schacht, Celia
2019
The axiomatic destiny of the theorems of Pappus and Desargues. Zbl 1454.01016
Pambuccian, Victor; Schacht, Celia
2019
Existence of special rainbow triangles in weak geometries. Zbl 1431.51007
Pambuccian, Victor
2019
Prolegomena to any theory of proof simplicity. Zbl 1441.03007
Pambuccian, Victor
2019
The elementary Archimedean axiom in absolute geometry. Zbl 1430.51020
Pambuccian, Victor
2019
Negation-free and contradiction-free proof of the Steiner-Lehmus theorem. Zbl 1455.03076
Pambuccian, Victor
2018
A problem in Pythagorean arithmetic. Zbl 1404.03033
Pambuccian, Victor
2018
Metric geometries in an axiomatic perspective. Zbl 1386.03005
Pambuccian, Victor; Struve, Horst; Struve, Rolf
2017
Another equivalent of the Lotschnittaxiom. Zbl 1362.51004
Pambuccian, Victor
2017
Absolute geometry proofs of two geometric inequalities of Chisini. Zbl 1366.51007
Pambuccian, Victor
2017
The Steiner-Lehmus theorem and “triangles with congruent medians are isosceles” hold in weak geometries. Zbl 1367.51008
Pambuccian, Victor; Struve, Horst; Struve, Rolf
2016
The arithmetic of the even and the odd. Zbl 1381.03041
Pambuccian, Victor
2016
From absolute to affine geometry in terms of point-reflections, midpoints, and collinearity. Zbl 1414.51002
Alama, Jesse; Pambuccian, Victor
2016
Addenda et corrigenda to: “The arithmetic of the even and the odd”. Zbl 1382.03080
Menn, Stephen; Pambuccian, Victor
2016
An axiomatic look at the Erdős-Trost problem. Zbl 1376.51008
Kalhoff, Franz; Pambuccian, Victor
2016
Schatunowsky’s theorem, Bonse’s inequality, and Chebyshev’s theorem in weak fragments of Peano arithmetic. Zbl 1333.03240
Pambuccian, Victor
2015
The Green-Tao theorem on primes in arithmetical progressions in the positive cone of $$\mathbb Z [X]$$. Zbl 1293.11096
Pambuccian, Victor
2014
Book review of: D. Hilbert, David Hilbert’s lectures on the foundations of geometry, 1891–1902. Zbl 1305.00057
Pambuccian, Victor
2013
The non-planarity of $$K_{5}$$ and $$K_{3,3}$$ as axioms for plane ordered geometry. Zbl 1267.51012
Pambuccian, Victor
2012
The axiomatics of ordered geometry: I. Ordered incidence spaces. Zbl 1227.51010
Pambuccian, Victor
2011
Acute triangulation of a triangle in a general setting revisited. Zbl 1246.51010
Hociotă, Ioana; Pambuccian, Victor
2011
The simplest axiom system for plane hyperbolic geometry revisited. Zbl 1234.51011
Pambuccian, Victor
2011
Another splitting of the Pasch axiom. Zbl 1227.51011
Pambuccian, Victor
2011
Paolo Pizzetti: the forgotten originator of triangle comparison geometry. Zbl 1221.01083
Pambuccian, Victor; Zamfirescu, Tudor
2011
Positive existential definability of parallelism in terms of betweenness in Archimedean ordered affine geometry. Zbl 1230.51013
Kalhoff, Franz; Pambuccian, Victor
2011
Forms of the Pasch axiom in ordered geometry. Zbl 1202.51010
Pambuccian, Victor
2010
Weakly ordered plane geometry. Zbl 1205.51007
Pambuccian, Victor
2010
Acute triangulation of a triangle in a general setting. Zbl 1201.51015
Pambuccian, Victor
2010
Mappings preserving the area equality of hyperbolic triangles are motions. Zbl 1213.51017
Pambuccian, Victor
2010
On M. T. Calapso’s characterization of the metric of an absolute plane. Zbl 1169.51017
Pambuccian, Victor; Struve, Rolf
2009
A reverse analysis of the Sylvester-Gallai theorem. Zbl 1202.03023
Pambuccian, Victor
2009
On the equivalence of Lagrange’s axiom to the Lotschnittaxiom. Zbl 1189.51010
Pambuccian, Victor
2009
Universal-existential axiom systems for geometries expressed with Pieri’s isosceles triangle as single primitive notion. Zbl 1223.51028
Pambuccian, V.
2009
Axiomatizing geometric constructions. Zbl 1143.03007
Pambuccian, Victor
2008
The Erdős-Mordell inequality is equivalent to non-positive curvature. Zbl 1140.51009
Pambuccian, Victor
2008
The sum of irreducible fractions with consecutive denominators is never an integer in $$\mathrm{PA}^-$$. Zbl 1185.03086
Pambuccian, Victor
2008
The elementary geometry of a triangular world with hexagonal circles. Zbl 1146.51009
Pambuccian, Victor
2008
Corrigendum to “The complexity of plane hyperbolic incidence geometry is $$\forall \exists \forall \exists$$”. Zbl 1152.03311
Pambuccian, Victor
2008
Orthogonality as single primitive notion for metric planes. Zbl 1161.51004
Pambuccian, Victor
2007
Alexandrov-Zeeman type theorems expressed in terms of definability. Zbl 1141.03007
Pambuccian, Victor
2007
Axiomatizations of hyperbolic and absolute geometries. Zbl 1106.51008
Pambuccian, Victor
2006
Positive definitions of segment congruence in terms of segment inequality. Zbl 1117.51018
Pambuccian, Victor
2006
Defining co-punctuality in terms of line-orthogonality in plane hyperbolic geometry. Zbl 1100.51012
Pambuccian, V.; Prazmowski, K.; Sakowicz, K.
2005
The complexity of plane hyperbolic incidence geometry is $$\forall\exists\forall\exists$$. Zbl 1066.03027
Pambuccian, Victor
2005
Groups and plane geometry. Zbl 1086.03007
Pambuccian, Victor
2005
Elementary axiomatizations of projective space and of its associated Grassmann space. Zbl 1193.03022
Pambuccian, Victor
2005
Euclidean geometry problems rephrased in terms of midpoints and point-reflections. Zbl 1082.51011
Pambuccian, Victor
2005
The simplest axiom system for plane hyperbolic geometry. Zbl 1069.03007
Pambuccian, Victor
2004
On the axiomatics of projective and affine geometry in terms of line intersection. Zbl 1064.51001
Havlicek, Hans; Pambuccian, Victor
2004
Hyperbolic geometry in terms of point-reflections or of line-orthogonality. Zbl 1075.51005
Pambuccian, Victor
2004
Early examples of resource-consciousness. Zbl 1057.03010
Pambuccian, Victor
2004
An axiomatics for hyperbolic projective-metric planes in terms of lines and orthogonality. Zbl 1101.51007
Pambuccian, Victor
2004
Fragments of the plane hyperbolic geometry. (Fragmente der ebenen hyperbolischen Geometrie.) Zbl 1061.51013
Pambuccian, Victor
2004
Constructive axiomatization of non-elliptic metric planes. Zbl 1036.03051
Pambuccian, Victor
2003
Why are surjective lineations of the Archimedean hyperbolic plane motions? Zbl 1027.51021
Pambuccian, V.
2003
Construction of hyperbolic geometry from the notion of line orthogonality. (Aufbau der hyperbolischen Geometrie aus dem Geradenorthogonalitätsbegriff.) Zbl 1048.51013
Pambuccian, V.
2003
What is plane equiaffine geometry? Zbl 1085.51003
Pambuccian, Victor
2003
On the planarity of the equilateral, isogonal pentagon. Zbl 1017.03002
Pambuccian, Victor
2003
Sphere tangency as single primitive notion for hyperbolic and Euclidean geometry. Zbl 1038.51014
Pambuccian, V.
2003
Axiomatizations of hyperbolic geometry: A comparison based on language and quantifier type complexity. Zbl 1028.03014
Pambuccian, Victor
2002
On definitions in an infinitary language. Zbl 1021.03029
Pambuccian, Victor
2002
Fragments of Euclidean and hyperbolic geometry. Zbl 0995.51005
Pambuccian, Victor
2001
Constructive axiomatizations of plane absolute, Euclidean and hyperbolic geometry. Zbl 0971.03063
Pambuccian, Victor
2001
A methodologically pure proof of a convex geometry problem. Zbl 1008.52002
Pambuccian, Victor
2001
Constructive axiomatization of plane hyperbolic geometry. Zbl 1003.03052
Pambuccian, Victor
2001
On R. Baer’s generalized hyperbolic planes. Zbl 1012.51009
Pambuccian, Victor
2001
A definitional view of Vogt’s variant of the Mazur-Ulam theorem. Zbl 0992.03045
Pambuccian, Victor
2001
Mohr-Mascheroni as an axiomatizability statement. Zbl 0993.51006
Pambuccian, Victor
2001
Two notes on the axiomatics of structures with parallelism. Zbl 1162.51300
Pambuccian, Victor
2001
A logical look at characterizations of geometric transformations under mild hypotheses. Zbl 0987.51010
Pambuccian, Victor
2000
Binary relations as single primitive notions for hyperbolic three-space and the inversive plane. Zbl 1044.51012
Pambuccian, Victor
2000
Another constructive axiomatization of Euclidean planes. Zbl 0943.03052
Pambuccian, Victor
2000
A logical reading of the nonexistence of proper homomorphisms between affine spaces. Zbl 0966.51002
Pambuccian, Victor
2000
On the Euclidean geometry of the double-edged ruler. Zbl 1228.51021
Pambuccian, Victor
1999
On the existence of a equilateral triangle in $$H$$-planes. (Zur Existenz gleichseitiger Dreiecke in $$H$$-Ebenen.) Zbl 0931.51006
Pambuccian, Victor
1998
On the constructive geometry of Euclidean planes. (Zur konstruktiven Geometrie euklidischer Ebenen.) Zbl 0959.03052
Pambuccian, V.
1998
On the simplicity of an axiom system for plane Euclidean geometry. Zbl 0884.03004
Pambuccian, Victor
1997
Splitting the Pasch axiom. Zbl 0854.51010
Pambuccian, Victor
1996
On the extensiveness axiom for partial affine spaces. (Über das Reichhaltigkeitsaxiom für partielle affine Räume.) Zbl 0853.51002
Pambuccian, Victor
1996
Ternary operations as primitive notions for constructive plane geometry. VI. Zbl 0828.03032
Pambuccian, Victor
1995
Simple axiom systems for affine planes. Zbl 0848.03003
Pambuccian, Victor
1995
On the stepwise construction of the parallel postulate. (Zum Stufenaufbau des Parallelenaxioms.) Zbl 0815.51012
Pambuccian, Victor
1994
Ternary operations as primitive notions for constructive plane geometry. IV. Zbl 0799.03072
Pambuccian, Victor
1994
Ternary operations as primitive notions for constructive plane geometry. V. Zbl 0808.03045
Pambuccian, Victor
1994
The most simple axioms system in elementary geometry. (Le plus simple système d’axiomes pour la géométrie élémentaire.) Zbl 0821.51015
Pambuccian, Victor
1994
Ternary operations as primitive notions for constructive plane geometry. III. Zbl 0799.03071
Pambuccian, Victor
1993
Ternary operations as primitive notions for constructive plane geometry. II. Zbl 0739.03033
Pambuccian, Victor
1992
On the Pythagorean hull of $$\mathbb Q$$. Zbl 1147.12300
Pambuccian, Victor
1990
Ternary operations as primitive notions for constructive plane geometry. Zbl 0669.03029
Pambuccian, Victor
1989
Simple axiom systems for Euclidean geometry. Zbl 0699.51010
Pambuccian, Victor
1989
Another example of an exotic function. Zbl 0825.26007
Pambuccian, Victor
1989
Simplicity. Zbl 0669.03005
Pambuccian, Victor
1988
Die metrisch-euklidische Geometrie als Ausgangspunkt für die geordnet- euklidische Geometrie. Zbl 0566.51018
Schnabel, Rudolf; Pambuccian, Victor
1985
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Cited by 108 Authors

 47 Pambuccian, Victor V. 15 Struve, Rolf 8 Struve, Horst 5 Prażmowski, Krzysztof 5 Schacht, Celia 5 Székely, Gergely 5 Żynel, Mariusz 4 Havlicek, Hans 3 Madarász, Judit X. 3 Movsisyan, Yuri Movses 3 Németi, István 3 Zamani, Ali 2 Andréka, Hajnal 2 Boutry, Pierre 2 Dehghani, Mahdi 2 Giovannini, Eduardo N. 2 Kalhoff, Franz B. 2 Mandelkern, Mark 2 Narboux, Julien 2 Prażmowska, Małgorzata 2 Schreck, Pascal 2 Vesley, Richard E. 1 Agnew, Alfonso F. 1 Aiello, Marco 1 Alama, Jesse 1 Alkan, Emre 1 Alliot, Sylvain 1 Arana, Andrew 1 Babic, Joshua 1 Baldwin, John T. 1 Banjac, Bojan 1 Beeson, Michael J. 1 Bernardi, Claudio 1 Bezhanishvili, Guram 1 Bloch, Isabelle 1 Blunck, Andrea 1 Bobe, Alexandru 1 Boskoff, Wladimir-Georges 1 Braun, Gabriel 1 Brown, Elizabeth Theta 1 Bruno, Jorge L. 1 Castner, Emily 1 Changat, Manoj 1 Chmieliński, Jacek 1 Cocco, Lorenzo 1 De Risi, Vincenzo 1 Demirel, Oğuzhan 1 Deprettere, Ed F. A. 1 Gerla, Giangiacomo 1 Göral, Haydar 1 Goranko, Valentin F. 1 Gries, Charly 1 Grimson, Rafael 1 Gröger, Detlef 1 Harutyunyan, Davit 1 Hipolito, Inês 1 Hociotă, Ioana 1 Huang, Fangjian 1 Janičić, Predrag 1 Kahle, Reinhard 1 Karamzadeh, Omid Ali Shahni 1 Kharazishvili, Alexander Bezhanovich 1 Korchmáros, Gábor 1 Kozma, József 1 Kreinovich, Vladik Yakovlevich 1 Kuijpers, Bart H. M. 1 Lemaitre, Jérôme 1 List, Klaus 1 Lombard, Melinda 1 Maehara, Hiroshi 1 Makowsky, Johann-Andreas 1 Malešević, Branko J. 1 Mancosu, Paolo 1 Marchisotto, Elena Anne Corie 1 Marinković, Vesna 1 McCluskey, Aisling E. 1 Menn, Stephen 1 Meuren, Stephan 1 Miranda, Annamaria 1 Molnár, Attila 1 Moslehian, Mohammad Sal 1 Mumma, John 1 Naibo, Alberto 1 Nazari, E. M. 1 Nazaryan, Aram 1 O’Shea, Edwin 1 Othman, Walied 1 Petelczyc, Krzysztof 1 Petrović, Maja 1 Plotnitsky, Arkady 1 Narasimha-Shenoi, Prasanth G. 1 Quaresma, Pedro 1 Rainaldi, Claudio 1 Seethakuttyamma, Geetha 1 Sertbaş, Doğa Can 1 Seryozhenkov, Edouard 1 Sidoli, Nathan Camillo 1 Smadja, Ivahn 1 Sörensen, Kay 1 Steel, John R. ...and 8 more Authors
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text:
# Dirac-delta function in spherical polar coordinates

1. Oct 7, 2017

< Mentor Note -- thread moved from the Homework physics forums to the technical math forums >

Hello.I was reading recently barton's book.I reached the part corresponding to dirac-delta functions in spherical polar coordinates.
he let :$(\theta,\phi)=\Omega$ such that $f(\mathbf {r})=f(r,\Omega)$

$\int d^3r...=\int_0^{\infty}drr^2\int_0^{2π }dφ\int_0^π dθsinθ...$

define
$\int_0^{2π}dφ\int_0^πdθsinθ=\int_0^{2π}dφ\int_{-1}^{1}dcosθ=\int d\Omega$
then$δ(\mathbf {r-r^{'}})=\frac {1}{r^2} δ(r-r')δ(\Omega-\Omega^{'})$>>>(1)
where $δ(\Omega-\Omega^{'})=δ(φ-φ')δ(cosθ-cosθ')$>>>(2)

My problem is that I really didn't get how he switched from $\int_0^{2π }dφ\int_0^π dθsinθ$ into $\int_{-1}^{1}dcosθ$

same thing corresponding to relations (1) and (2), I didn't get how he obtained them?
If somebody can give me a hint for obtaining them? thanks.

Relevant equations

$\int d^3rf(\mathbf{r})δ(\mathbf{r})=f(0)$
where$δ(r)=δ(x)δ(y)δ(z)$
such that $δ(r)=1/(2π)^3\int d^3kexp(i\mathbf{k}.\mathbf{r})$

3. The attempt at a solution
I need hints to know where yo start.

Last edited by a moderator: Oct 8, 2017
2. Oct 7, 2017

### Orodruin

Staff Emeritus
That is just a regular variable substitution. Note that $d\cos(\theta) = -\sin(\theta)d\theta$. Also, the $\phi$ integral is still there.

The three-dimensional delta function should have the property
$$\int f(\vec r) \delta(\vec r - \vec r_0) dV = f(\vec r_0).$$
You know that the delta function is zero everywhere except for at $\vec r'$ so make the ansatz $\delta(\vec r - \vec r_0) = N \delta(r-r_0)\delta(\theta - \theta_0) \delta(\phi - \phi_0)$ and start computing.

3. Oct 11, 2017

I started by the relation
$\int f(\vec r)δ(\vec r- \vec r')dV=f(\vec r')$
we have$f(\vec r)=f(r,θ,φ)$
$dV=drr^2sinθdθdφ$
$\int f(\vec r)δ(\vec r -\vec r')drr^2sinθdθdφ=f(\vec r')$
Then I let$δ(\vec r -\vec r')=A_1δ(r-r')A_2δ(θ-θ')A_3δ(φ-φ')$
then I let $f(r,θ,φ)=f(r)f(θ)f(φ)$ ... I dont know if this is allowed.
Then
$\int_0^{\infty}f(r)r^2drA_1δ(r-r')\int_0^πA_2f(θ)sin(θ)dθδ(θ-θ')\int_0^{2π}A_3f(φ)δ(φ-φ')dφ=f(r',θ',φ')$
then by doing the change of variable $u=sinθdθ$ $\int_0^πA_2f(θ)sinθδ(θ-θ')dθ=\int_{-1}^{1}f(θ)dcosθδ(cosθ-cosθ')A_2$ (*)
This will lead to $A_1=1/r^2 and A_2=1 and A_3=1$
and then $δ(\vec r-\vec r')=1/r^2δ(r-r')δ(φ-φ')δ(cosθ-cosθ')$
What I'm not convinced about is relation (*),how the change of variable let $δ(θ-θ')=δ(cosθ-cosθ')$ If you can clarify it to me?

4. Oct 11, 2017

### Orodruin

Staff Emeritus
No, it is not true that $\delta(\theta - \theta') = \delta(\cos\theta - \cos\theta')$ (there is a constant factor between the two). However, you can just as well assume a factor of $\delta(\cos\theta -\cos\theta')$ instead of a factor of $\delta(\theta-\theta')$. This is in effect what you have done.

5. Oct 30, 2017

### vanhees71

One should clarify a few further things. First of all you cannot express $\delta^{(3)}(\vec{x})$ in spherical coordinates since spherical coordinates are singular along the entire polar axis and particularly at the origin.

Second you can use the general formula
$$\delta[f(x)]=\sum_{j} \frac{1}{|f'(x_j)|} \delta(x-x_j),$$
where $f$ is a function that has only 1st-order roots $x_j$.

Thus for $\vartheta, \vartheta' \in ]0,\pi[$ you have
$$\delta(\cos \vartheta-\cos \vartheta')=\frac{1}{\sin \vartheta} \delta(\vartheta-\vartheta').$$


text:
# Properties

 Label 1960.2.a.s Level $1960$ Weight $2$ Character orbit 1960.a Self dual yes Analytic conductor $15.651$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} + (\beta + 5) q^{9}+O(q^{10})$$ q + b * q^3 + q^5 + (b + 5) * q^9 $$q + \beta q^{3} + q^{5} + (\beta + 5) q^{9} + ( - \beta + 4) q^{11} + (\beta - 2) q^{13} + \beta q^{15} + ( - \beta - 2) q^{17} - 2 \beta q^{19} + 2 \beta q^{23} + q^{25} + (3 \beta + 8) q^{27} + (\beta - 2) q^{29} + 8 q^{31} + (3 \beta - 8) q^{33} - 2 q^{37} + ( - \beta + 8) q^{39} + (2 \beta - 2) q^{41} + (2 \beta - 4) q^{43} + (\beta + 5) q^{45} - 3 \beta q^{47} + ( - 3 \beta - 8) q^{51} + ( - 2 \beta + 6) q^{53} + ( - \beta + 4) q^{55} + ( - 2 \beta - 16) q^{57} - 8 q^{59} + ( - 2 \beta - 2) q^{61} + (\beta - 2) q^{65} - 4 q^{67} + (2 \beta + 16) q^{69} + 8 q^{71} + 6 q^{73} + \beta q^{75} + ( - 3 \beta + 8) q^{79} + (8 \beta + 9) q^{81} - 4 \beta q^{83} + ( - \beta - 2) q^{85} + ( - \beta + 8) q^{87} + (2 \beta - 10) q^{89} + 8 \beta q^{93} - 2 \beta q^{95} + ( - 5 \beta - 2) q^{97} + ( - 2 \beta + 12) q^{99} +O(q^{100})$$ q + b * q^3 + q^5 + (b + 5) * q^9 + (-b + 4) * q^11 + (b - 2) * q^13 + b * q^15 + (-b - 2) * q^17 - 2*b * q^19 + 2*b * q^23 + q^25 + (3*b + 8) * q^27 + (b - 2) * q^29 + 8 * q^31 + (3*b - 8) * q^33 - 2 * q^37 + (-b + 8) * q^39 + (2*b - 2) * q^41 + (2*b - 4) * q^43 + (b + 5) * q^45 - 3*b * q^47 + (-3*b - 8) * q^51 + (-2*b + 6) * q^53 + (-b + 4) * q^55 + (-2*b - 16) * q^57 - 8 * q^59 + (-2*b - 2) * q^61 + (b - 2) * q^65 - 4 * q^67 + (2*b + 16) * q^69 + 8 * q^71 + 6 * q^73 + b * q^75 + (-3*b + 8) * q^79 + (8*b + 9) * q^81 - 4*b * q^83 + (-b - 2) * q^85 + (-b + 8) * q^87 + (2*b - 10) * q^89 + 8*b * q^93 - 2*b * q^95 + (-5*b - 2) * q^97 + (-2*b + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + 11 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + 11 * q^9 $$2 q + q^{3} + 2 q^{5} + 11 q^{9} + 7 q^{11} - 3 q^{13} + q^{15} - 5 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 19 q^{27} - 3 q^{29} + 16 q^{31} - 13 q^{33} - 4 q^{37} + 15 q^{39} - 2 q^{41} - 6 q^{43} + 11 q^{45} - 3 q^{47} - 19 q^{51} + 10 q^{53} + 7 q^{55} - 34 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{65} - 8 q^{67} + 34 q^{69} + 16 q^{71} + 12 q^{73} + q^{75} + 13 q^{79} + 26 q^{81} - 4 q^{83} - 5 q^{85} + 15 q^{87} - 18 q^{89} + 8 q^{93} - 2 q^{95} - 9 q^{97} + 22 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + 11 * q^9 + 7 * q^11 - 3 * q^13 + q^15 - 5 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 + 19 * q^27 - 3 * q^29 + 16 * q^31 - 13 * q^33 - 4 * q^37 + 15 * q^39 - 2 * q^41 - 6 * q^43 + 11 * q^45 - 3 * q^47 - 19 * q^51 + 10 * q^53 + 7 * q^55 - 34 * q^57 - 16 * q^59 - 6 * q^61 - 3 * q^65 - 8 * q^67 + 34 * q^69 + 16 * q^71 + 12 * q^73 + q^75 + 13 * q^79 + 26 * q^81 - 4 * q^83 - 5 * q^85 + 15 * q^87 - 18 * q^89 + 8 * q^93 - 2 * q^95 - 9 * q^97 + 22 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.37228 3.37228
0 −2.37228 0 1.00000 0 0 0 2.62772 0
1.2 0 3.37228 0 1.00000 0 0 0 8.37228 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.a.s 2
4.b odd 2 1 3920.2.a.bt 2
5.b even 2 1 9800.2.a.bu 2
7.b odd 2 1 280.2.a.c 2
7.c even 3 2 1960.2.q.r 4
7.d odd 6 2 1960.2.q.t 4
21.c even 2 1 2520.2.a.x 2
28.d even 2 1 560.2.a.h 2
35.c odd 2 1 1400.2.a.r 2
35.f even 4 2 1400.2.g.i 4
56.e even 2 1 2240.2.a.bg 2
56.h odd 2 1 2240.2.a.bk 2
84.h odd 2 1 5040.2.a.by 2
140.c even 2 1 2800.2.a.bk 2
140.j odd 4 2 2800.2.g.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 7.b odd 2 1
560.2.a.h 2 28.d even 2 1
1400.2.a.r 2 35.c odd 2 1
1400.2.g.i 4 35.f even 4 2
1960.2.a.s 2 1.a even 1 1 trivial
1960.2.q.r 4 7.c even 3 2
1960.2.q.t 4 7.d odd 6 2
2240.2.a.bg 2 56.e even 2 1
2240.2.a.bk 2 56.h odd 2 1
2520.2.a.x 2 21.c even 2 1
2800.2.a.bk 2 140.c even 2 1
2800.2.g.r 4 140.j odd 4 2
3920.2.a.bt 2 4.b odd 2 1
5040.2.a.by 2 84.h odd 2 1
9800.2.a.bu 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1960))$$:

 $$T_{3}^{2} - T_{3} - 8$$ T3^2 - T3 - 8 $$T_{11}^{2} - 7T_{11} + 4$$ T11^2 - 7*T11 + 4 $$T_{13}^{2} + 3T_{13} - 6$$ T13^2 + 3*T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 8$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 7T + 4$$
$13$ $$T^{2} + 3T - 6$$
$17$ $$T^{2} + 5T - 2$$
$19$ $$T^{2} + 2T - 32$$
$23$ $$T^{2} - 2T - 32$$
$29$ $$T^{2} + 3T - 6$$
$31$ $$(T - 8)^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 2T - 32$$
$43$ $$T^{2} + 6T - 24$$
$47$ $$T^{2} + 3T - 72$$
$53$ $$T^{2} - 10T - 8$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} + 6T - 24$$
$67$ $$(T + 4)^{2}$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} - 13T - 32$$
$83$ $$T^{2} + 4T - 128$$
$89$ $$T^{2} + 18T + 48$$
$97$ $$T^{2} + 9T - 186$$


text:
# Integral involving the error function of log(x)

Looking for a closed form for the integral

$$\int_0^{\infty } e^{-\left(\frac{a-\log (x)}{b}\right)^2} \left(\frac{1}{2} \text{erf}\left(\frac{a-\log (x)}{b}\right)+\frac{1}{2}\right) \, \mathrm{d}x,$$ where erf is the error function $erf (z)=\frac{2}{\sqrt{\pi }}\int _0^ze^{-t^2}\mathrm{d}t.$

I've tried all manner of tricks, to no avail.

• what kind of tricks? – tired Feb 17 '15 at 15:42
• Subbing $x=e^y$ will at least do no harm. – Ron Gordon Feb 17 '15 at 15:47
• For $a=0$, one term is a Gaussian integral, and the other one is $0$, due to the parity of the integrand. – Lucian Feb 17 '15 at 16:10
• Substitution $x=e^y$ which removes the log didn't work. – Nero Feb 17 '15 at 18:09
• I'm not sure what you mean, "it didn't work." Do you mean that you get an integral that you can't evaluate? – Ron Gordon Feb 17 '15 at 19:15

Theorem. Let $a$ and $b$ be any real numbers. Then
$$\int_0^{\infty }\!\! e^{\large-\left(\frac{a-\log x}{b}\right)^2}\!\! \left(\text{erf}\left(\frac{a-\log x}{b}\right)+1\right) \mathrm{d}x=\sqrt{\pi}\:|b|\:e^{{\large {a+\frac{b^2}{4}}}}\left(1-\text{erf}\!\left(\! \frac{\sqrt{2}\:b}{4}\!\right)\! \right) \tag1$$
where $\displaystyle \text{erf}(z)=\frac{2}{\sqrt{\pi }}\int _0^ze^{\large-t^2}\mathrm{d}t.$
Proof. Let's denote the left hand side of $(1)$ by $I(a,b)$.
By the change of variable $$\displaystyle U:=\frac{a-\log x}{b}, \quad dx=-b\:e^a \:e^{-b U}dU,$$ we have \begin{align} I(a,b)&=|b|\:e^a \int_{-\infty }^{+\infty } e^{\large-U^2-bU} \left(\text{erf}\left(U\right)+1\right) \mathrm{d}U\\\\ &=|b|\:e^a \int_{-\infty }^{+\infty } e^{\large-U^2-bU} \text{erf}(U)\: \mathrm{d}U+|b|\:e^a \int_{-\infty }^{+\infty } e^{\large-U^2-bU} \mathrm{d}U \tag2 \end{align} Clearly, by the gaussian integral: $$\int_{-\infty }^{+\infty } e^{\large-U^2-bU} \mathrm{d}U =e^{\large b^2/4}\int_{-\infty }^{+\infty } e^{\large-(U+b/2)^2} \mathrm{d}U =\sqrt{\pi}\:e^{ b^2/4} .\tag3$$ Set $$f(b):=\int_{-\infty }^{+\infty } e^{\large-U^2-bU} \text{erf}(U)\: \mathrm{d}U \tag4$$ and observe that $$f(0)=\frac{\sqrt{\pi}}{4}\left[(\text{erf}(U))^2\right]_{-\infty }^{+\infty }=0. \tag5$$ Differentiating $(4)$ with respect to $b$ and performing an integration by parts gives \begin{align} f'(b)&=-\int_{-\infty }^{+\infty } U\:e^{\large-U^2-bU} \text{erf}(U)\:\mathrm{d}U\\\\ f'(b)&=\left[\frac 12 e^{\large-U^2}\left(e^{\large-bU} \text{erf}(U)\right)\right]_{-\infty }^{+\infty }-\frac 12\int_{-\infty }^{+\infty } e^{\large-U^2} \left(-b\:e^{\large-bU} \text{erf}(U)+\frac{2}{\sqrt{\pi }}e^{\large-bU}e^{\large-U^2}\right)\mathrm{d}U\\\\ f'(b)&=\frac{b}{2}f(b)-\frac{1}{\sqrt{\pi }}\int_{-\infty }^{+\infty } e^{\large-2U^2-bU}\mathrm{d}U \end{align} or $$f'(b)=\frac{b}{2}f(b)-\frac{\sqrt{2}}{2}\:e^{\large b^2/8}. \tag6$$ We classically solve the ODE $(6)$, using $(5)$, to obtain $$f(b)=-\sqrt{\pi }\:e^{\large b^2/4}\text{erf}\!\left(\! \frac{\sqrt{2}\:b}{4}\!\right) \tag7$$ then plugging $(7)$, $(4)$ and $(3)$ into $(2)$ gives $(1)$ as desired.