content stringlengths 86 994k | meta stringlengths 288 619 |
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Consolidate your pupils' understanding of fractions and decimals with this Year 6 Fractions as Division Homework resource, consisting of two varied fluency questions and one reasoning and
problem-solving question, helping them to see the link between fractions and decimals.
Complete with an answer sheet, this clearly presented Fractions as Division Homework worksheet is ideal for extra practise at home or in the classroom. | {"url":"https://classroomsecrets.co.uk/resource/year-6-fractions-as-division-homework-2","timestamp":"2024-11-02T02:42:45Z","content_type":"text/html","content_length":"576290","record_id":"<urn:uuid:bab58cd3-7e1c-4dbb-b2b5-ba5be2f58940>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00581.warc.gz"} |
Homework 2: SDGB 7840 solved
Submit two files through Blackboard: (a) .Rmd R Markdown file with answers and code
and (b) Word document of knitted R Markdown file. Your file should be named as follows:
“HW[X]-[Full Name]-[Class Time]” and include those details in the body of your file.
Complete your work individually and comment your code for full credit. For an example of
how to format your homework see the files posted with Lecture 1 on Blackboard. Show all
of your code in the knitted Word document.
1. Read the posted article, “Bordeaux wine vintage quality and weather,” by Ashenfelter,
Ashmore, and LaLonde (CHANCE, 1995). Three regression models are considered in
this article. Answer the following questions:
(a) What is a wine “vintage”?
(b) What is the response variable for the three models described in this paper?
Now, download the data in “wine.txt”. This is some of the data the authors used
to fit their models. The columns are: vintage (VINT), log of average vintage price
relative to 1961 (LPRICE2), rainfall in the months preceding the vintage in mL
(WRAIN), average temperature over the growing season in ◦C (DEGREES), rainfall in September and August in mL (HRAIN), and age of wine in years (TIME SV).
Note: the average temperature in September is not available in our data set so we
cannot fit the third regression model from the paper.
(c) Which values of LPRICE2 are missing and, according to the article, why have they
been omitted?
(d) Make a scatterplot matrix of the variables (explanatory and response) included in
the models. Describe what you see.
(e) Fit the two regression models from the paper. Which is the best regression model?
Justify your answer and include relevant output (let α = 0.05). Did you choose the
same model as the authors?
(f) What is the sample size for your models?
(g) Write out the regression equation of the model you chose in part (e). Remember to
include the units of measurement. Interpret the partial slopes and the y-intercept.
Does the y-intercept have a practical interpretation?
(h) Make a table with the following statistics for both models: SSE, RMSE, PRESS,
and RMSEjackknife. Compare the relevant statistics. Based on this information,
would you change your answer to part (e)? Justify your answers.
(i) Could we use these regression models to predict quality for wines produced in 2005?
Justify your answer.
2. We will model the prestige level of occupations using variables such as education and
income levels. This data was collected in 1971 by Statistics Canada (the Canadian
equivalent of the U.S. Census Bureau or the National Bureau of Statistics of China)1
The data is in the file “prestige.dat” and the variables are described below:
variable description
prestige (y) Pineo-Porter prestige score for occupation, from a social survey
conducted in the mid-1960s
education average education of occupational incumbents, years, in 1971
income average income of incumbents, dollars, in 1971
women percentage of incumbents who are women
census Canadian Census occupational code
type type of occupation: “bc”=blue collar,
“prof”= professional/managerial/technical,
“wc”=white collar
(a) Do some internet research and write a short paragraph in your own words about
how the Pineo-Porter prestige score is computed. Include the reference(s) you used.
Do you think this score is a reliable measure? Justify your answer.
(b) Create a scatterplot matrix of all the quantitative variables. Use a different symbol
for each profession type: no type (pch=3), “bc” (pch=6), “prof” (pch=8), and “wc”
(pch=0) when making your plot. For the remainder of this question, we will use the
explanatory variables: income, education, and type. Does restricting our regression
to only these variables make sense given your exploratory analysis? Justify your
1Source: Canada (1971) Census of Canada. Vol. 3, Part 6. Statistics Canada; 19-1–19-21.
(c) Which professions are missing “type”? Since the other variables for these observations are available, we could group them together as a fourth professional category
to include them in the analysis. Is this advisable or should we remove them from
our data set? Justify your answer.
(d) Visually, does there seem to be an interaction between type and education and/or
type and income? Justify your answer.
(e) Fit a model to predict prestige using: income, education, type, and any interaction
terms based on your answer to part (d). Evaluate the model and include relevant
output. Use your answer to part (c) to determine which observations to use in your
(f) Create a histogram of income and a second histogram of log(income) (i.e., natural
logarithm). How does the distribution change?
(g) Fit the model in (e) but this time use log(income) (i.e., natural logarithm) instead
of income. Evaluate the model and provide the relevant output.
(h) Is the model in (e) or (g) better? Justify your answer. Why can’t we use a partial
F-test here? | {"url":"https://codeshive.com/questions-and-answers/homework-2-sdgb-7840-solved/","timestamp":"2024-11-14T07:24:27Z","content_type":"text/html","content_length":"103896","record_id":"<urn:uuid:62768deb-2482-4fe3-9747-310ad6d5cc86>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00461.warc.gz"} |
On unimodality of independence polynomials of some well-covered trees
The stability number α(G) of the graph G is the size of a maximum stable set of G. If s[k] denotes the number of stable sets of cardinality k in graph G, then I(G; x) = Σ[k=0]^α(G) s[k]x^k is the
independence polynomial of G (I. Gutman and F. Harary 1983). In 1990, Y.O. Hamidoune proved that for any claw-free graph G (a graph having no induced subgraph isomorphic to K[1,3]), I(G; x) is
unimodal, i.e., there exists some k ∈ {0, 1, ..., α(G)} such that s[0] ≤ s[1] ≤ ... ≤ s[k-1] ≤ s[k] ≥ s[k+1] ≥ ... ≥ s[α(G)]. Y. Alavi, P.J. Malde, A.J. Schwenk, and P. Erdös (1987) asked whether for
trees the independence polynomial is unimodal. J. I. Brown, K. Dilcher and R.J. Nowakowski (2000) conjectured that I(G; x) is unimodal for any well-covered graph G (a graph whose all maximal
independent sets have the same size). Michael and Traves (2002) showed that this conjecture is true for well-covered graphs with α(G) ≤ 3, and provided counterexamples for α(G) ∈ {4, 5, 6, 7}. In
this paper we show that the independence polynomial of any well-covered spider is unimodal and locate its mode, where a spider is a tree having at most one vertex of degree at least three. In
addition, we extend some graph transformations, first introduced in [14], respecting independence polynomials. They allow us to reduce several types of well-covered trees to claw-free graphs, and,
consequently, to prove that their independence polynomials are unimodal.
Original language English
Title of host publication Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Editors Cristian S. Calude, Michael J. Dinneen, Vincent Vajnovszki
Pages 237-256
Number of pages 20
State Published - 2003
Externally published Yes
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 2731
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Dive into the research topics of 'On unimodality of independence polynomials of some well-covered trees'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/on-unimodality-of-independence-polynomials-of-some-well-covered-t-3","timestamp":"2024-11-04T17:25:54Z","content_type":"text/html","content_length":"57630","record_id":"<urn:uuid:0e8989be-9e69-4706-a5f8-4f3af3913427>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00328.warc.gz"} |
Average Quiz For Sbi PO | Sbi Clerk | RRB JE & NTPC | FCI | CWC | RBI & Other Exams - GovernmentAdda
Home Quantitative Aptitude Quiz Average Quiz Average Quiz For Sbi PO | Sbi Clerk | RRB JE &...
Average Quiz For Sbi PO | Sbi Clerk | RRB JE & NTPC | FCI | CWC | RBI & Other Exams
1.In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 runs?
2.A grocer has a sale of Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500?
3.The average of 20 numbers is zero. Of them, How many of them may be greater than zero, at the most?
4.The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less
than the average age of the whole team. Find out the average age of the team.
23 years
20 years
24 years
21 years
5.The average monthly income of A and B is Rs. 5050. The average monthly income of B and C is Rs. 6250 and the average monthly income of A and C is Rs. 5200. What is the monthly income of A?
6.A car owner buys diesel at Rs.7.50, Rs. 8 and Rs. 8.50 per litre for three successive years. What approximately is the average cost per litre of diesel if he spends Rs. 4000 each year?
Rs. 8
Rs. 7.98
Rs. 6.2
Rs. 8.1
7.In Kiran’s opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Kiran and he thinks that Kiran’s weight is greater than 60 kg but less than 70 kg. His
mother’s view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Kiran?
70 kg
69 kg
61 kg
67 kg
8.The average weight of 16 boys in a class is 50.25 kg and that of the remaining 8 boys is 45.15 kg. Find the average weights of all the boys in the class.
9.A library has an average of 510 visitors on Sundays and 240 on other days. What is the average number of visitors per day in a month of 30 days beginning with a Sunday?
10.A student’s mark was wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by 1/2 .What is the number of students in the class?
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(PDF) Improved Boosting Algorithms by Pre-Pruning and Associative Rule Mining on Decision Trees for Predicting Obstructive Sleep Apnea
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RESEARCH ARTICLE Adv. Sci. Lett. 23(11), 11593–11598, 2017
11593 Adv. Sci. Lett. Vol. 23, No.11, 2018 doi:10.1166/asl.2017.10335
Copyright © 2017 American Scientific Publishers Advanced Science Letters
All rights reserved Vol.23, No. 11, 11593–11598, 2017
Printed in the United States of America
Improved Boosting Algorithms by Pre-
Pruning and Associative Rule Mining on Decision
Trees for predicting Obstructive Sleep Apnea
Doreen Ying Ying Sim1*, Chee Siong Teh1, Ahmad Izuanuddin Ismail2
1Faculty of Cognitive Sciences and Human Development, Universiti Malaysia Sarawak, Kuching, Sarawak, Malaysia
2Respiratory Medicine Unit, Department of Respiratory Medicine, UiTM Medical Specialist Centre, Faculty of Medicine,
Universiti Teknologi MARA, Selangor, Malaysia
An improvedBoosting algorithm, named as Boosted PARM-DT, was developed bypre-pruning techniques and Associative
Rule Mining (ARM) on decision trees built from the clinical datasets** collected for Obstructive Sleep Apnea (OSA). The
Pruned-Associative-Rule-Mined Decision Trees (PARM-DT) developed by adopting pre-pruning techniques on tree depth,
minimum leaf and/or parent node size observations and maximum number of tree splits, based on Apriori and/or Adaptive
Apriori (AA) frameworks, is boosted to achieve better predictive accuracies. The improved algorithms were implemented in
OSA dataset and UCI online databases for comparisons. Better predictive accuracies were achieved in all the applied
datasets/databases when comparing the classical algorithm, i.e. Boosted DT, with the improved one, i.e. Boosted PARM-DT.
Keywords: pre-pruning techniques, Associative Rule Mining, Apriori, Adaptive Apriori (AA), Boosted PARM-DT
Obstructive Sleep Apnea (OSA), like some other
diseases and medical illnesses, usually has an attribute or
a set of attributes which can perfectly or almost perfectly
confirm the medical diagnosis2-4. This attribute or set of
attributes, however, usually has low support threshold(s).
Since boosting algorithms, such as AdaBoost, are white-
box methods, and this research has raw data**collection
on OSA patients’ records**with the characteristics of it
which is fully known and understood, using pre-pruning
techniques, Associative Rule Mining (ARM) and Apriori
/Adaptive Apriori framework is a great advantage.
Sleep apnea affects both adults and children which
can result in as many as around 30 breathless episodes per
night. Untreated sleep apnea can cause death during sleep
or can incur serious health problems such as diabetes,
hypertension, stroke, and other cardiovascular diseases2-4.
If a person has Obstructive Sleep Apnea (OSA), his or her
tongue and throat muscles maybecome so relaxed and
floppy during sleep that those muscles can cause a
narrowing or even complete blockage of the airway(s)2-5.
Narrowing or complete blockage of the airway(s) can be
caused by the cephalometric anatomical abnormalities or
*Email: dsdoreenyy@gmail.com **see Acknowledgments
morphological defects (in this case, we concentrated just
on retrognathia, micrognathia and posterior pillar
webbing) and/or other anatomical defects such as throat
and/or tongue muscles flow back due to poor blood
circulation incurring muscle floppiness and relaxation2,4,5.
Table 1 shows the minimum support and minimum
confidence thresholds as per stated i.e. (1) bilateral Tonsils’
Size or TS (size ranges from 0 to 4, i.e. normal case to the
worst case); (2) crowding of oropharynx, i.e. MP
(Mallampati score ranges from 1 to 4); (3) Neck
Circumference or NC (greaterthan or equal to 40cm); (4)
Epworth Sleepiness Scale or ESS (i.e. ESS ranges from 0 to
24); (5) Morbid Obesity or MO(BMI greater than or equal
to 40); (6) Posterior Pillar Webbing or PPW; (7)
Retrognathia / Retro-positioned maxilla or RN(over-slung
or jutting lower jaw); and (8) Micrognathia / receding lower
jaw or receding chin or MN(short mentohyoid distance or
inferiorly displaced hyoid bone).
This paper is organized as follows: Section 2 deploys
ARM and pre-pruning techniques on decision trees of
OSA dataset; in Section 3, improved algorithms, i.e.
Boosted PARM-DT, are implemented. Experimental
results of comparing the algorithms proposed with
classical approaches are shown and analyzed in Section 4.
Conclusions and discussions are summarized in Section 5.
Adv. Sci. Lett. 23(11), 11593–11598, 2017 RESEARCH ARTICLE
2. ASSOCIATIVE-RULE MINING AND PRE-
Apriori lives on a uniform minimum support and
commonly used as a basic pruning strategy9-12although
most of the datasets collected for medical research
purposes, the minimum support and minimum confidence
are user-specified from the research findings1. Interesting
patterns often occur at various levels of support8-10. The
improved algorithm proposed here is mainly based on
Adaptive Apriori, a non-uniform minimum support.
Only pre-pruning techniques are used as part of the
improved algorithms, i.e. Boosted PARM-DT, in this
research because all datasets and databases applied have
well-known characteristics. The decision trees developed
for OSA dataset** and UCI online databases are pre-
pruned by the following four ways based on Apriori and
Adaptive Aprioriproperties: (1)pre-pruning of decision
trees depends on a stopping criterion set to control the
tree depth and this is done by merging the leaves, or
working out the parent node size observations (see Fig.1);
(2) “pre-pruned” by halting its construction early, i.e.
working out the item or itemset having the largest
confident threshold but the lowest support threshold as
the minimum leaf node size observations for the decision
trees developed, i.e. setting tree depth controller(s) to
minimize entropy impurity (Eq.17,8); (3) Since pruning is
the inverse of splitting1,9,11, “pre-pruning” approaches are
done by deciding not to further split the tree, i.e. set
maximum number of tree splitsstarting from a number
of 4 (i.e. number of splits starting from a perfect binary
tree of Level 2) (Eq.37-9,11-12); (4)partition the subset of
tuples ata given node so that leaf node has minimum
impurity (Eq.17,8). Pre-pruning of decision trees is done
in a top-down fashion, i.e. from root (see Eq.27,8) to
branch node and then to leaf node(s). In Eq.1, decision
trees are grown until each leaf node has the lowest
impurity7,8,10. The P(Ѡj) is the fraction of patterns at
node N in category Ѡj 8,11-12.
Entropy impurity =
i (N) =
i (Nroot) =
0.1)(log)( 2
Gini impurity =
i (N) =
ji ipPP )(1
)()( 2
Post-pruning is usually applied for datasets which its
characteristics are not well-known6-8,10. Since the
characteristics of OSA dataset and the applied databases
are well-known, post-pruning technique is not applied in
this research. So, only pre-pruning techniques, Apriori,
and AA, are applied to develop Boosted PARM-DT.
THROUGHOUT): If Cis the set of confident rules plus
the default rule, this research considers two rules r and r’
in C. In this definition, r is ranked higher than r’, i.e.
denoted in this research as r >R r’, if any of the following
conditions or criteria holds:
Criteria 1: conf (r) > conf (r’)
Criteria 2: conf (r) = conf (r’), but sup (r) > sup (r’)
Criteria 3: sup (r) = sup (r’), but size (r) < size (r’)
Refer to Table 1 for OSA dataset, certain features
such as associating relationships between Micrognathia
(MN) and Retrognathia (RN), this research adopts
Criteria 2 in Definition 1 above. This is because MN can
be considered in the higher ranking than RN when taken
their low minimum support and perfect minimum
confidence thresholds into account. In other words, MN
can be considered more general than RN, i.e. rMN >G rRN.
DEFINITION 21,8-11 (ASSOCIATION RULE): An
association rule is an implication of the form X → Y,
where X
I, Y
I , X
Y = 0.
The optimization criterion for the tree splitting criteria
taken on decision of trees is the default setting, i.e. Gini’s
diversity index (or ‘gdi’). To grow decision trees in order to
fit the characteristics of OSA dataset, pre-pruning after
ARM based on Apriori and/or AAhave to be analyzed.
conf(X → Y, D) =
In Eq.4, support-driven and confidence-driven
pruning of the pre-pruning techniques on decision trees
by using ARM to work out the following: (1)Minimum
leaf- and (2) Minimum parent-node size observations; (3)
Maximum number of tree splits. It is based on the
association based classification which uses the minimum
support thresholds derived from the attribute(s) having
the highest confidence threshold(s) so as to prune over-
fitting rules. Pruning of over-fitting rules suffers from the
dilemma that rules of high support tend to have low
confidence7,11-12. However, prediction often depends on
high confidence1,8-12. High confidence features, as shown
in Table 1, usually have low support thresholds11-12.
Since OSA dataset has 2 attributes of RN and MN
having 100% confident thresholds but low support
thresholds of respectively 8/200 (i.e. 0.04), and 12/200
(i.e. 0.06), it is good to decide on using one of these
attributes (having k-itemset) to be the minimum leaf node
size observations or ‘MinLeafSize’ while growing the
decision trees. If using classical Boosting approach, i.e.
Boosted DT, no pruning technique or any ARM will be
used. So, in Boosted DT, the default setting will be
applied, i.e. the default values of the tree depth
controllers as follows:
(1) n -1 for the maximum number of decision splits
or ‘MaxNumSplits’, (n= training sample size) -
the maximum number of tree splits is size(X, 1)-
1, i.e. the number of training data minus 1;
(2) 1 for the minimum number of leaf node
observations or ‘MinLeafSize’;
(3) 10 for the minimum number of parent or branch
node observations or ‘MinParentSize’.
RESEARCH ARTICLE Adv. Sci. Lett. 23(11), 11593–11598, 2017
Table.1. Minimum Support and Minimum
Confidence thresholds for each OSA variable in 200
patients’ records to be input to PARM-DT.
3. Neck Circum-
ference (NC)
5. Morbid Obesity
6. Posterior Pillar
Webbing (PPW)
0.40 (79/200)
0.91 (72/79)
0.909 (30/33)
For ‘MinLeafSize’, the attribute having the highest
confident threshold and lowest support threshold is taken
as the highest priority. For minimum parent node size
observations or ‘MinParentSize’, the (k-1) itemset
having one-level higher ofminimum support threshold
(but should be larger than 2 times the ‘MinLeafSize’)
should be set for ‘MinParentSize’2,3,9,12.
3. BOOSTED PARM-DT ALGORITHM BASED ON
Decision trees for OSAdatasets and other applied
UCI databases were developed based on Apriori and/or
AA frameworks, and these trees were boosted by
GentleBoost (for two-class datasets) or AdaBoostM2 (for
non-two-class datasets), 200 iterations, 15-fold cross
validations, by using MATLAB(R2016a) software.
DEFINITION 37,8. (MONOTONICITY): If X is a
subset of Y, Sup(x) must not exceed Sup(Y). That is, for
all X, Y € J: (X is a subset of Y) → f(X) ≤ f(Y).
DEFINITION 41. (MCF PRINCIPLE): If there are
choices, the rule of the highest rank has the top priority,
and a specific rule that does not have higher rank than all
general rules is never used. That specific rule will be
deemed redundant and will be pruned.
In OSA dataset, Definitions 3 and 4 above were
adopted so as to prune the redundant rules and to perform
certain pre-pruning techniques before the decision tree is
fully grown. Reasons that the minimum leaf sizechosen
for OSA dataset is 12 rather than 8 are because although
both RN and MN attributes are 100% confident attributes,
when we adopt the MCF Principle, MN has higher
minimum support threshold (i.e. 12) than RN in incurring
the OSA positive. In Table 1, MN has a support threshold
of 12 while RN only has a support threshold of 8. Setting
the minimum leaf size of 12 is to control the optimal tree
depth and to avoid over-fitting rules (since by default, the
number of splitting forn level of trees is n -1 splits).
Downward closure property, i.e. any subset of a
frequent item-set is also a frequent item-set1,6,8 ----------
when applying this property on the datasets applied, we
are going from general to specific rules, and this property
can only be applicable when there is an improvement in
the minimum confidence thresholdswhen implementing
to a certain attribute or certain set of attributes. As shown
Table 1, Fig. 1 and Fig.2, an example can be PPW and
MN OSA positive, the minconf for this item-set is
0.909, but MN OSA positive, the minconf for this item
is 1.00, so upward closure property is not applicable, but
downward closure property is applicable. The improved
boosting algorithms, i.e. Boosted PARM-DT, in narrative
form, and in pseudo-codes, are respectively as below:
Q: Is the dataset having attribute or set of attributes
with significantly high or perfect confident threshold?
Yes No Boosted DT
Improved Algorithms: Boosted PARM-DT:
OSA dataset (or UCI online databases)
Further Assoc. Rule Mining (ARM) on Decision Trees
Decision Trees (DT) Development and Completion
Boosted PARM-DT, or Boosted PARM Decision Trees
Boosted PARM-DT (in pseudo-codes):
1. Input variables: a set of OSA data (or data from
the applied UCI databases) with labels {(x1,
y1), …, (xN, yN)} where xi є X, yi є Y = {-1, +1};
the initial setting is the minimal parent node
observation of δ, i.e. δminparent; stepwise increase
of δ, i.e. δminparent+1; for tree leaves, the initial
setting is the minimal leaf node observation ζ, i.e.
ζ minleaf; ; the stepwise increase of ζ, i.e. ζ minleaf+1;
for tree splits,the initial setting of tree splitting
is 4 (i.e. Simple Tree), the maximum is ϗMaxSplit,
the stepwise increase of ϗ, i.e. ϗ initial+1.
2. Initialize:
The weights of the training OSA dataset,
= 1/N, for all i = 1, 2, … , N.
3. Do for i = 1, 2, …, N (where N=no. of iterations)
Do while (ζ >= ζ minleaf)
Support-confidence, Apriori and AA to work out
MinLeafSize, MinParentSize and MaxNumSplits.
Pre-pruning Techniques by controlling the tree
depth, maximumnumber of splits and categorical
predictor(s), surrogate splits and/or setting tree
stopping criteria (since all are white-box methods)
No post-pruning technique is used since the
characteristics of OSA dataset and the applied
databases are all well-known. Boosting is done
after the pre-pruning technique and ARM on DT.
Adv. Sci. Lett. 23(11), 11593–11598, 2017 RESEARCH ARTICLE
(a) Train Associative-Rule-Mined Decision
Tree or ARM-DT component classifier,
ht, on the weighted training OSA dataset,
(where t = number of weak classifiers)
(b) Calculate the training error of ht : εt =
, yi ≠ ht (xi),
(c) If εt > 0.5, go directly to (g);
Else, ζ minleaf = ζ minleaf+1, then proceed
normally to (d);
(d) Set the weight of ARM-DT component
classifier : ht = αt =
(e) Update weights of OSA training samples:
where Ct is the normalization constant, and
(f) Go directly to Step 4 for output;
Do while (δ >= δminparent)
(g) Train ARM-DT component classifier, ht,
on the weighted training OSA data,
(h) Calculate the training error of ht : εt =
, yi ≠ ht (xi),
(i) If εt > 0.5, go directly to (j),
Else, δminparent = δminparent+1, then proceed
normally to (d);
Do while (ϗ initial >= 4 ANDϗ initial <= ϗMaxSplit)
(j) Train ARM DT component classifier,
ht, on the weighted training OSA dataset,
(k) Calculate the training error of
ht : εt =
, yi ≠ ht (xi),
(l) If εt > 0.5, halt the loop;
Else, ϗ initial = ϗ initial+1, proceed to (d);
4. Output: the largest weighted classifier from the
associatively mined decision trees will be chosen,
f (x) = sign
END (Algorithms completed)
Apriori, Adaptive Apriori frameworks and
downward closure properties of Boosted PARM-DTon
OSA dataset are shown in Fig. 1 and Fig.2. For pre-
pruning techniques, a setting of the minimum number of
parent nodes and/or leaf nodes was done after applying
ARM on OSA dataset. These are that the minimum
number of parent nodes in between 67 to 74, and the
minimum number of leaf nodes are in between 12 to 20,
which reveals the highest prediction accuracies. These
findings are derived from the support-confidence
thresholds of the attributes, and the minimum child and
branch node size observations in the OSA dataset.
Fig.1. Support-based and confidence-based pruning of
the pre-pruning on trees developed for OSA dataset to
derive MinLeafSize and MinParentSize based on Apriori.
Fig.2. Pre-pruning techniques on decision trees for
OSA dataset based on Adaptive Apriori framework.
Table.2. Three Pre-Pruning Techniques i.e. with
MinLeafSize, MinParentSize, and MaxNumSplits
implemented after ARM as parts of Boosted PARM-DT,
to OSA dataset and another eight UCI online databases.
OSA dataset/UCI
online databases
Comparisons of Boosted PARM-DT, with Boosted
DT, were conducted in all databases and OSA dataset in
Table 3. Fig.3 then shows that Boosted PARM-DT on
AA framework produces the best ROC curve forOSA
RESEARCH ARTICLE Adv. Sci. Lett. 23(11), 11593–11598, 2017
data if comparing Boosted PARM-DT on Apriori
framework, or with the classical Boosted DT algorithms.
Table.3. Predictive Accuracies of Boosted DT and
Boosted PARM-DT to OSA dataset (based on AA and
Apriori) and all online databases based on AA only
OSA dataset/UCI
online databases
from UCI
2. Breast Cancer
a significant at P <0.05; b significant at P <0.001; c significant at P <0.0001
Table.4. Results on the Scientific Significance of the
Improvements in Predictive Accuracies shown in Table 3
2. Breast Cancer
95% confidence interval and p-values at one-tail t-test; “N-1” Chi-squared test
Fig.3. Receiver Operating Characteristics (ROC) curves
for OSA dataset based on 3 algorithms applied: green
and blue curvy lines respectively indicates the ROC
curves after applying each Boosted PARM-DTbased on
AA and Apriori framework, while light grey curvy line
indicates that of normal algorithm, i.e. Boosted DT.
Both Table 3 and Table 4 above show the scientific
reliabilities and statistics significance of applying the
improved algorithms to the datasets as accessed by
analyzing the improvements achieved through one-tailed
t-tests, p-values and 95% confidence interval. Boosted
PARM-DT, after pre-pruning techniques, augmented by
Apriori and/or AAand ARM, the predictive accuracies
are better than using the classical Boosting algorithms.
Boosted PARM-DTis a refined white-box method that
can give better predictive accuracies to all databases
applied. The most prominent characteristic of OSA
dataset is that it has two 100% or perfect confident
attributes, RN and MN, although having low support
thresholds, it is very conducive to apply the Apriori and
Adaptive Apriori before getting ARM to ‘cast on’ the
relationships for the pre-pruning on decision trees to take
place. By adoptingApriorito derive minimum support
threshold of MN to be MinLeafSize and AA with pushed
minimum support of MO to be MinParentSize for
developing decision trees for OSA dataset, Boosted
PARM-DT algorithms can prune overfitting rules which
can help medical doctors to make more accurate OSA
clinical diagnoses.
**To obtain the raw data of OSA patients’ records,
formal Research and Ethics Committee Approval on
medical ground was acquired from Universiti Teknologi
MARA (Reference: 600-RMI (5/1/6)). This research is fully
supported by Fundamental Research Grant Scheme (FRGS),
UNIMAS (Reference: FRGS/ICT02(01)/1077/2013(23)).
[1] R. Agrawal, T. Imielinski, A. Swami. Mining association rules
between sets of items in large databases. Proceeding of the 1993
ACM SIGMOD International Conference on Management of
Data, Washington, D. C., USA: ACM (1993) 207-216.
[2] D. Y. Y.Sim, C. S. Teh, P. K. Banerjee. Prediction Model by
using Bayesian and Cognition-Driven Techniques: A Study in
the Context of Obstructive Sleep Apnea. Proceeding of the 9th
International Conference on Cognitive Science, Malaysia,
Procedia - Social and Behavioral Sciences, 97 (2013) 528-537.
[3] D. Y. Y. Sim, C. S. Teh, A. I. Izuanuddin. Adaptive Apriori and
Weighted Association Rule Mining on Visual Inspected
Variables for Predicting Obstructive Sleep Apnea (OSA),
Australian Journal of Intelligent Information Processing
Systems, 14(2) (2014) 39-45.
[4] P. C. Deegan, W. T. McNicholas. Predictive Value of
Clinical Features for the Obstructive Sleep Apnea
Syndrome, European Respiratory Journal, ERS Journals Ltd.,
UK., 9:1 (1996) 117-124.
[5] T. I. Morgenthaler, R. N. Aurora, T. Brown. Practice
parameters for the use of auto-titrating continuous positive
airway pressure devices for titrating pressures and treating adult
patients with obstructive sleep apnea syndrome: an update for
2007, Sleep, 31 (2007) 141-147.
[6] A. K. Das. Mining rare item sets using both top down and
bottom up approach, Internationl Journal of Computer Science
and Information Technologies, 7(3) (2016) 1607-1614.
[7] J. Han, M. Kamber, J. Pei. Data Mining Concepts and
Techniques (3rd ed.), Elsevier, Morgan Kaufmann, USA
(2012) 17-27, 248-273, 461-488.
Adv. Sci. Lett. 23(11), 11593–11598, 2017 RESEARCH ARTICLE
[8] J. Han, J. Pei, Y. Yin. Mining frequent patterns without
candidate generation. Proceeding of the 2000 ACM SIGMOD
International Conference on Data Mining, New York, NY,
USA: CM Press, (2000) 1-12.
[9] K. Wang, Y. He, J. Han. Pushing support constraints into
association rules mining, IEEE Transactions on Knowledge and
Data Engineering, 15(3) (2003) 642-658.
[10] K. Wang, S. Zhou, S. Liew. Building hierarchical classifiers
using class proximity. Proceeding of the 25th International
Conference on Very Large Data Bases, San Francisco, CA,
USA: Morgan Kaufmann, (1999) 363-374.
[11] S. K. Pal, P. Mitra. Pattern Recognition Algorithms for Data
Mining, Chapman & Hall, Florida, USA: CRC Press LLC,
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Check into Frequent Itemset Mining, International Journal of
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Engineering, 2(7) (2012) 43-48. | {"url":"https://www.researchgate.net/publication/322097462_Improved_Boosting_Algorithms_by_Pre-Pruning_and_Associative_Rule_Mining_on_Decision_Trees_for_Predicting_Obstructive_Sleep_Apnea","timestamp":"2024-11-08T02:54:28Z","content_type":"text/html","content_length":"676902","record_id":"<urn:uuid:fd5abcf1-40b9-4e25-8f4f-5e3b4959de33>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00559.warc.gz"} |
Decimal to Octal
Free Decimal To Octal Converter - Convert Decimal To Octal Online
The decimal to octal converter is a free online tool that allows you to calculate octal values from decimal numbers. The converted value is displayed in octal digits.
What is an octal number?
An octal number is a base-8 number consisting of eight digits between 0 and 7. Octal numbers are often used in computers and electronics because they provide a simple way to represent binary values.
The octal numeral system is used in digital electronics and computer programming because it provides a more concise way of expressing large binary numbers than the decimal system.
What is a decimal number?
Decimal numbers are numbers with a base 10 number system . In other words, the place value of each digit in a decimal number is ten times the value of the previous digit. For example, the number 1234
can be represented as 1 x 10^3 + 2 x 10^2 + 3 x 10^1 + 4 x 10^0.
The octal numeral system is used in digital electronics and computing because it provides a simple way to convert binary numbers into a human-readable form. Octal numbers consist of eight digits,
from 0 through 7. In octal numbers, each digit represents three bits. Converting from binary to octal is as easy as grouping binary digits into sets of three, starting on the right side of the binary
number.The first group is 000, the second 001, the third 010, and so on until you reach 111. So the octal equivalent of 11011000 is 343.
To convert a decimal number to an octal number, divide the decimal number by 8 and write the remainder. Continue dividing until you get to 0; write all the remainders in reverse order to get your
The conversion process
The conversion process is simple and straightforward. Simply enter a decimal number into the converter and click the "Convert" button. The converted octal value is displayed in the designated area.
The formula for converting decimals to octals
To convert a decimal number to the equivalent octal value, divide the number by 8 and keep track of the remainder. Continue dividing the resulting number by 8 until no more division is possible. The
remainders of each division, read from bottom to top, give you the octal value.
For example, let's take the decimal number 253. If we divide 253 by 8, we get a quotient of 31 and a remainder of 5. If we divide 31 by 8, we get a quotient of 3 and a remainder of 7. If we divide 3
by 8, we get a quotient of 0 and a remainder of 3. Since there's nothing left to divide, we'll stop here. The remainders, read from bottom to top, give us the octal equivalent of 253, which is 375.
An example of converting a decimal to an octal number
To convert a decimal number to its octal equivalent, you must first identify the place value of each digit in the number.
For example, let's take the decimal number 42. The place values of the digits in this number are 40, 20, 10 and 1.
To convert this number into an octal number, all we need to do is divide each digit by its corresponding place value and write off the remainder.
So 42 divided by 40 is 1 with a remainder of 2. 2 divided by 20 is 0 with a remainder of 2. 2 divided by 10 is 0 with a remainder of 2. And finally, 2 divided by 1 is 2 with a remainder of 0.
Therefore, the octal equivalent of 42 is 20002.
How to convert large numbers
To convert large numbers from decimal to octal, first divide the number by 8 and write the remainder. Continue dividing the number by 8 and write the remainders until you reach 0. Then read the
remainders from bottom to top to get the octal equivalent of the original number. Let's take the number 1356 as an example:
1356 ÷ 8 = 169 with a remainder of 4
169 ÷ 8 = 21 with a remainder of 1
21 ÷ 8 = 2 with a remainder of 5
2 ÷ 8 = 0 with a remainder of 2
Therefore , the octal equivalent of 1356 is 2502.
What is a decimal to octal converter?
A decimal to octal converter is a free online tool that can convert decimal numbers to octal values. Octal is a base 8 numbering system, which uses the digits 0-7. The converted value is displayed in
octal digits.
How does a decimal to octal converter work?
A decimal to octal converter works by taking a decimal number and converting it to an octal number. The process is relatively simple and only requires a few steps.
First, the decimal number is divided by 8. The remainder of this division is the first digit of the octal number.
Then the decimal number is divided by 8 again. The remainder of this division is the second digit of the octal number.
This process is repeated until the decimal number has been divided by 8 so many times that there are no digits left in the octal number.
How do you use a decimal to octal converter?
This is a free converter that allows you to calculate octal values from decimal numbers. The converted value is displayed in octal digits.
To use this converter, simply enter a decimal value in the input field and click the "Convert" button. The converter then displays the equivalent octal value in the output field.
What are the benefits of using a decimal to octal converter?
There are many benefits to using a decimal to octal converter. One advantage is that it can help you convert decimal numbers to octal numbers quickly and easily. This can be useful if you need to
work with octal values but don't have a calculator or other tool that can easily do the conversion.
Another advantage of using a decimal to octal converter is that it can help you check your work when converting decimal numbers to octal numbers. This can be especially useful if you are new to octal
values or if you are working with large numbers. By checking your work with a converter, you can make sure you've accurately converted the decimal number to an octal number.
Finally, using a decimal to octal converter can help you save time converting decimal numbers to octal numbers. If you're working with a large number of decimals, converting them all manually can
take some time. A converter can automate this process and save you time in the long run.
Why use a decimal to octal converter?
If you need to convert a decimal number to its octal equivalent, you can use a decimal to octal converter, such as the one provided on this website. This converter is free to use and can be very
helpful in calculating octal values from decimal numbers. Simply enter the decimal number you want to convert to the converter and click the "Convert" button. The converter will then display the
equivalent octal value for your decimal number.
When you should not use a decimal to octal converter
There are a few cases where you should not use a decimal to octal converter. First, if the number you are converting has a fraction component, the converter will simply truncate the fraction and
convert the integer part. This may lead to an incorrect answer, so it is best to avoid the converter in this case.
Second, if the number you are converting is very large, the converter may not be able to accurately display it in octal form. In this case, again, it's best to find another method to do the
Finally, if you are working with negative numbers, the converter will simply output their positive counterparts.So, for example, if you enter -8 into the converter, it outputs 10. This is because
negative numbers are not used in octal notation.
Keep these cases in mind when deciding whether or not to use a decimal to octal converter - in most situations this will give you an accurate result, but there are some exceptions where it's best to
find another method of conversion .
This decimal to octal converter is a free and easy-to-use tool that helps you calculate octal values from decimal numbers. Just enter a decimal number in the box and click "Convert" to see the
equivalent octal value. Whether you're a student who needs to convert a decimal number for an assignment or a professional working with binary code, this converter will come in handy.
David Miller
CEO / Co-Founder
Our mission is to provide 100% free online tools useful for different situations. Whether you need to work with text, images, numbers or web tools, we've got you covered. We are committed to
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Common Core Assessment Analysis: Sixth Grade Statistical Questions
Each state that has adopted Common Core State Standards will select an assessment consortium to assess how children are progressing in school. The two options most widely chosen at this point are:
Smarter Balanced and PARCC. Each assessment consortium has provided practice test questions and today we will review one of these questions and discuss its potential impact on classroom instruction.
Below is an assessment question from Smarter Balanced grade 6. Smarter Balanced uses two types of assessment questions: Selected Response and Constructed Response.
• Selected Response requires the students to select one or more correct answers.
• Constructed Response requires the students to create their own answer.
Sixth Grade Statistical Questions
Common Core Standards Assessed
6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical
question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
This item assesses one sixth-grade standard. To correctly answer this item, students must be able to differentiate between a statistical question (one that expects variability in the data) and a
non-statistical question (one that has no variability in the data).
From Smarter Balanced Scoring Guide
For this item, a full-credit response (1 point) includes:
□ “Variability in Data” next to “How many pets does each 6th grader have?” and “How old are the animals at the zoo?”
□ “No Variability in Data” next to “How old is the athlete?”, “How many 6th graders attend our school?” and “How many baseball cards does the boy have?”
For full credit, students must correctly identify all questions which have Variability in Data and all questions which have No Variability in Data. No partial credit is given on this particular item.
What do we learn from this item?
This question is assessing if the students understand that a question that anticipates variability refers to a question that results in multiple quantities.
Interestingly, two options in this item are related to age. Students without thorough understanding of the concept of statistical questions may be tempted to answer these items the same way because
of similar superficial mobile casino elements.
“How old is the athlete?” is a question referring to a single individual so it will only have one answer.
“How old are the animals at the zoo?” refers to multiple subjects, so variation in the answers is expected.
How is this concept assessed in other grades?
This particular concept is not directly assessed in other grades. However, thorough understanding of statistical questions gives students a foundational understanding of how data is collected. This
understanding will lead to better success on questions related to data sets assessed in future grades.
In this 7^th-grade assessment question students must understand that the data collected in each data set answers the statistical question “How many push-ups can the students in Mr. Axt’s class do?”
Knowing what question the data set answers allows students to better interpret the data in context.
Suggestions for classroom instruction
As this is a purely concept based question, students must be taught what a statistical question is, and how to differentiate a statistical question from a non-statistical question. Defining the
content vocabulary of statistical question, as well as providing both examples and non-examples (such as the ones provided in the standard) can be beneficial in developing the concept.
Author: DataWORKS Curriculum
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APS March Meeting 2013
Bulletin of the American Physical Society
APS March Meeting 2013
Volume 58, Number 1
Monday–Friday, March 18–22, 2013; Baltimore, Maryland
Session Y24: Focus Session: Advances in Fermionic Simulatons Hide Abstracts
Sponsoring Units: DCOMP
Room: 326
Friday, Y24.00001: Determinantal Quantum Monte Carlo simulations of fermions in optical lattices
March Invited Speaker: Thereza Paiva
2013 The ability to cool fermions in optical lattices to ultra cold temperatures has led to an interdisciplinary area of research, that has attracted a lot of attention in recent years. An
8:00AM interesting development in this area is the possibility to realize models for strongly correlated fermions in the laboratory, such as the fermionic Hubbard Model. Determinantal Quantum Monte
- Carlo simulations have proven to be an important tool in the study of fermionic atoms. Nonetheless, it is important to compare the results and efficiency of different methods. Here
8:36AM comparisons with Numerical Linked Cluster Expansion and Dynamical Mean Field Theory data for double occupation and short range correlations, both relevant to current optical lattice
experiments, will be presented and discussed. Another topic relevant in the context of optical lattice experiments is the study of metal insulator transitions. Indeed, the Mott insulating
phase has been realized and observed in two-flavor mixtures of fermionic atoms loaded on optical lattices, being characterized both by the double occupation and the compressibility. An
interesting point that has been addressed in the literature over the years is whether the same fermion-fermion interaction, responsible for the Mott insulating state, could drive an
insulating system metallic. Here we show that, when fermions are loaded in optical lattices with spatially varying interactions a correlation induced Mott insulator to metal transition can
take place. The spatial modulation of the interactions was recently demonstrated and opens the possibility for the experimental realization of such exotic phases. [Preview Abstract]
Friday, Y24.00002: Quantum Monte Carlo Calculations of Entanglement
March Norm Tubman, Jeremy McMinis
2013 Spatial entanglement properties have become increasingly important in physics which includes studies in diverse fields such as condensed matter physics, astrophysics, and quantum computation.
8:36AM One of the important outstanding problems in the field of entanglement is to understand the effect of many body interactions. Recent advances in quantum Monte Carlo have facilitated such
- studies over a range of Hamiltonians that were previously inaccessible by other techniques. We apply these techniques to interacting molecular and condensed matter systems and discuss the
8:48AM effect interactions have on entanglement properties. [Preview Abstract]
Friday, Y24.00003: Excited state calculations in solids by auxiliary-field quantum Monte Carlo
March Fengjie Ma, Shiwei Zhang, Henry Krakauer
2013 We present an approach for ab initio many-body calculations of excited states in solids. Using auxiliary-field quantum Monte Carlo \footnote{S.~\ Zhang and H.~\ Krakauer, Phys. Rev. Lett. {\
8:48AM bf 90}, 136401 (2003)}, we introduce an orthogonalization constraint with virtual orbitals to prevent collapse of the stochastic Slater determinants in the imaginary-time propagation. Trial
- wave functions from density-functional calculations are used for the constraints, and detailed band structures can be calculated. Results for standard semiconductors are in good agreement
9:00AM with GW calculations and with experiment. For the challenging ZnO, we obtain a fundamental band gap of 3.30(16) eV, consistent within the range of experimental measurements \footnote{V.~\
Srikant and D.~\ R.~\ Clarke, J. Appl. Phys. 83, 5447 (1998); S.~\ Tsoi, X.~\ Lu, A.~\ K.~\ Ramdas, H.~\ Alawadhi, M.~\ Grimsditch, M.~\ Cardona, and R.~\ Lauck, Phys. Rev. B 74, 165203
(2006); H.~\ Alawadhi, S.~\ Tsoi, X.~\ Lu, A.~\ K.~\ Ramdas, M.~\ Grimsditch, M.~\ Cardona, and R.~\ Lauck, Phys. Rev. B {\bf 75}, 205207 (2007)}. Applications to other systems are currently
underway. [Preview Abstract]
Friday, Y24.00004: Bold Diagrammatic Monte Carlo for Fermionic and Fermionized Systems
March Invited Speaker: Boris Svistunov
2013 In three different fermionic cases---repulsive Hubbard model, resonant fermions, and fermionized spins-1/2 (on triangular lattice)---we observe the phenomenon of sign blessing: Feynman
9:00AM diagrammatic series features finite convergence radius despite factorial growth of the number of diagrams with diagram order. Bold diagrammatic Monte Carlo technique allows us to sample
- millions of skeleton Feynman diagrams. With the universal fermionization trick we can fermionize essentially any (bosonic, spin, mixed, etc.) lattice system. The combination of fermionization
9:36AM and Bold diagrammatic Monte Carlo yields a universal first-principle approach to strongly correlated lattice systems, provided the sign blessing is a generic fermionic phenomenon. [Preview
Friday, Y24.00005: Path Integral Quantum Monte Carlo Benchmarks for Molecules and Plasmas
March John Shumway
2013 Path integral quantum Monte Carlo is used to simulate hot dense plasmas and other systems where quantum and thermal fluctuations are important. The fixed node approximation---ubiquitous in ab
9:36AM initio ground state Quantum Monte Carlo---is more complicated at finite temperatures, with many unanswered questions. In this talk I discuss the current state of fermionic path integral
- quantum Monte Carlo, with an emphasis on molecular systems where good benchmark data exists. We look at two ways of formulating the fixed node constraint and strategies for constructing
9:48AM finite-temperature nodal surfaces. We compare different the free energies of different nodal choices by sampling an ensemble of nodal models within a Monte Carlo simulation. We also present
data on imaginary-time correlation fluctuations, which can be surprisingly accurate for molecular vibrations and polarizabilty. [Preview Abstract]
Friday, Y24.00006: Quantum Monte Carlo simulations of complex Hamiltonians
March Valery Rousseau, Kalani Hettiarachchilage, Ka-Ming Tam, Juana Moreno, Mark Jarrell
2013 In the last two decades there have been tremendous advances in boson Quantum Monte Carlo methods, which allow for solving more and more complex Hamiltonians. In particular, it is now possible
9:48AM to simulate Hamiltonians that include terms that couple an arbitrary number of sites and/or particles, such as six-site ring-exchange terms. These ring-exchange interactions are crucial for
- the study of quantum fluctuations on highly frustrated systems. We illustrate how the Stochastic Green Function algorithm with Global Space-Time Update can easily simulate such complex
10:00AM systems, and present some results for a highly non-trivial model of bosons in a pyrochlore crystal with six-site ring-exchange terms. [Preview Abstract]
Friday, Y24.00007: Quasi-adiabatic Quantum Monte Carlo algorithm for non-equilibrium quantum phase transitions
March Cheng-Wei Liu, Anders W. Sandvik, Anatoli Polkovnikov
2013 We investigate a new quantum Monte Carlo algorithm for studying static and dynamic properties of quantum phase transitions. The method, called the quasi-adiabatic quantum Monte Carlo
10:00AM algorithm, is based on evolution with a changing Hamiltonian to derive information pertinent to a quantum quench according to an arbitrary protocol. We demonstrate the method with results for
- 1D and 2D transverse-field Ising models, showing finite-size and finite-velocity scaling according to a generalization of the Kibble-Zurek mechanism. We explore ways to extract critical
10:12AM points and critical exponents to high precision. [Preview Abstract]
Friday, Y24.00008: Ground state phases in the half-filled staggered $\pi$-flux Hubbard model on square lattices
March Chia-Chen Chang, Richard T. Scalettar
2013 Ground state phase diagram of the half-filled staggered $\pi$-flux Hubbard model on a square lattice are studied by means of constrained-path quantum Monte Carlo method. Charge and spin
10:12AM excitation gaps and magnetic order are calculated as a function of interaction strength $U/t$. Within our numerical scheme, it is found that the ground state phase is a semi-metal at $U/t <
- 5.6$, and a Mott insulator with long-range antiferromagnetic order at $U/t > 6.6$. In the window $5.6 < U/t < 6.6$, the system is an insulator in which both magnetic and dimer orders are
10:24AM absent. Spin excitation in the intermediate phase appears to be gapless, and the measured equal-time spin-spin correlation function shows a power-law dependence of relative distance. Our data
suggests that the paramagnetic insulating intermediate phase might be a possible place to look for the putative algebraic spin liquid. [Preview Abstract]
Friday, Y24.00009: Momentum-dependent pseudogaps in the half-filled two-dimensional Hubbard model
March Nils Bluemer, Daniel Rost, Elena Gorelik, Fakher Assaad
2013 We compute unbiased spectral functions of the two-dimensional Hubbard model by extrapolating Green functions, obtained from determinantal quantum Monte Carlo simulations, to the thermodynamic
10:24AM and continuous time limits. Our results clearly resolve the pseudogap at weak to intermediate coupling, originating from a momentum selective opening of the charge gap. A characteristic
- pseudogap temperature $T^*$, determined consistently from the spectra and from the momentum dependence of the imaginary-time Green functions, is found to match the dynamical mean-field
10:36AM critical temperature, below which antiferromagnetic fluctuations become dominant. Our results identify a regime where pseudogap physics is within reach of experiments with cold fermions on
optical lattices.\\[2ex] D. Rost, E. V. Gorelik, F. Assaad, N. Bl\"umer, Phys. Rev. B {\bf 86}, 155109 (2012). [Preview Abstract]
Friday, Y24.00010: Series Expansion for the Green's Function of the Infinite-U Hubbard Model
March Ehsan Khatami, Edward Perepelitsky, B. Sriram Shastry, Marcos Rigol
2013 We implement computationally a strong-coupling expansion for the dynamical single-particle Green's function of the infinite-U Hubbard model up to the eighth order in the hopping, within the
10:36AM formalism introduced by Metzner [1]. We obtain analytical expressions for the finite Matsubara frequency Green's functions and the Dyson self energy in the momentum space at all densities in
- the thermodynamic limit. The results match those obtained up to the fourth order by means of another method devised by us. Furthermore, we employ Pade approximations and various numerical
10:48AM re-summation techniques to extend the region of convergence to lower temperatures.\\[4pt] Ref. [1]: W. Metzner, Phys. Rev. B 43, 8549 (1991). [Preview Abstract]
22, Y24.00011: ABSTRACT WITHDRAWN
10:48AM [Preview Abstract]
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That Nim Flashbacks - Internet burnout
• Replacing “Programmers:” with “Program:” is more accurate.
Tower of Hanoi is actually easy to write program for. Executing it on the other hand…
□ It’d be a trick if you didn’t already know the answer. Or at least, it would be for me. It’s also hard to actually visualise.
☆ I didn’t know the answer either, but usually you can compose solution from solutions of smaller problems.
solution(0): There are no disks. Nothing to do. solution(n): Let’s see if I can use solution(n-1) here. I’ll use solution(n-1) to move all but last disk A->B, just need to rename the
pins. Then move the largest disk A->C. Then use solution(n-1) to move disks B->C by renaming the pins. There we go, we have a stack based solution running in exponential time.
It’s one of the easiest problem in algorithm design, but running the solution by hand would give you a PTSD.
○ Good for you. I think I’d figure it out eventually, but it would certainly take me a while.
I’d probably be trying a number of approaches, including the recursive one. Renaming pegs is a critical piece that you’d have to realise you can do, and you can’t be sure you have a
correct inductive solution unless you actually walk through the first few solutions from the base instance. | {"url":"https://group.lt/post/1839090/3685144","timestamp":"2024-11-08T12:14:44Z","content_type":"text/html","content_length":"73713","record_id":"<urn:uuid:45769b59-1896-4f60-a337-5d3014f0d3db>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00169.warc.gz"} |
Connecticut Summer School 16 | Connecticut Summer School in Number Theory
Connecticut Summer School 16
All talks will be at the Pharmacy/Biology Building (PBB 129 and 131), and the coffee breaks will be outside of PBB 129. A campus map pointing to PBB can be found here (Google labels the building as
the “School of Pharmacy”).
Goals of the Summer School
The organizers of the summer school hope that the students attending this event will learn fundamental ideas in contemporary number theory and have a sense of some directions of current research. For
undergraduates, the summer school will expose them to topics not available in a typical college curriculum and we will encourage applications from students at institutions where advanced topics in
number theory are not ordinarily taught. The school will provide a chance for participants to meet fellow students, as well as faculty, interested in number theory.
Expected Background of Students
• Undergraduate Students: a semester each of elementary number theory, abstract algebra, and complex analysis.
• Graduate Students: a year of abstract algebra. Suggested, but not required: a semester of algebraic number theory and familiarity with p-adic numbers.
Structure of the Summer School
The summer school will take place at the Storrs campus of the University of Connecticut. Activities will be designed at two levels, targeting advanced undergraduate and beginning graduate students.
Lectures will be scheduled so that a student could attend almost all lectures if desired, choosing according to their background and interests. The daily schedule in the summer school will be as
shown in the following table.
• On the first day only, we will all meet at 8:50 at PBB 131 for some welcoming remarks.
Time PBB 129 PBB 131
8:15 – 9 Breakfast
9 – 9:50 Lecture A1 Lecture B1
9:50 – 10:10 Coffee Break
10:10 – 11 Exercises Lecture B2
11:10 – 12 Plenary Lecture (PBB 131)
12 – 2 Lunch
2 – 2:50 Magma/Sage Tutorials (PBB 131)
3 – 3:50 Exercises Lecture A2
3:50 – 4:10 Coffee Break
4:10 – 5 Exercises Lecture B3
5 – 7 Dinner
After 7 Projects/Latex/Discussion (PBB 131)
On the last day of the summer school, students will present the progress on their projects in the morning, prior to the beginning of the research conference in the afternoon.
Lecture series
Each day’s events at the summer school is as follows. The videos for the lectures can be found at the UConn Math YouTube Channel.
• Plenary Lectures: Each day will have a plenary talk, where a number theorist will give an overview (accessible to advanced undergraduates and beginning graduate students) of a current trend in
number theory. Titles of the lectures and speakers:
• Lecture A1: “Introduction to Modular Forms,” by Keith Conrad. Topics will include Eisenstein series and q-expansions, applications to sums of squares and zeta-values, Hecke operators, eigenforms,
and the L-function of a modular form.
• Lecture A2: “Introduction to Elliptic Curves,” by Álvaro Lozano-Robledo. This will be an overview of the theory of elliptic curves, discussing the Mordell-Weil theorem, how to compute the torsion
subgroup of an elliptic curve, the 2-descent algorithm, and what is currently known about rank and torsion subgroups of elliptic curves.
• Lecture B1: “Computational methods for modular and Shimura curves,” by John Voight. The classical method of modular symbols on modular curves is introduced to compute the action of the Hecke
algebra and corresponding spaces of modular forms. Generalizations to Shimura curves will then be discussed using Dirichlet modular symbols.
• Lecture B2: “Introduction to the local-global principle,” by Liang Xiao. The plan is starting with an introduction to Q[p], then introducing Hilbert symbols and the local-global principle for
quaternion algebras and central simple algebras, and ending with examples of the failure of the local-global principle.
□ Xiao’s Project: Project on computing the U[p]-eigenvalues of families of overconvergent automorphic forms.
• Lecture B3: “Gauss sums and the Weil Conjectures,” by Bin Zhao. The topics include will Gauss sums, Jacobi sums, and Weil’s original argument for diagonal hypersurfaces when he raised his
conjectures. Further developments towards the Langlands program and the modularity theorem will be mentioned at the end.
• Exercise sessions: Each day will have a period set aside for students to work on exercises together, led by senior graduate students.
• Magma and Sage tutorials, by Harris Daniels. Both Magma and Sage are extremely useful computer algebra packages for doing research in number theory. The goal for these sessions is to give an
introduction to both packages so that students can solve proposed computational exercises from the lecture courses and projects of the summer school.
• Evening Latex tutorial: (LaTeX intro files) The plan is to cover the following topics:
□ Basic use and setup of LaTeX.
□ Commutative diagrams.
□ Preparing talks with the Beamer package.
□ An introduction to graphics packages.
• Evening project and discussion session: The students will be grouped into different projects to discuss and work collaboratively. Instructors and graduate student mentors are available to assist
the students. The projects will consist of open-ended problems or more involved exercises, with computational aspects to them, that are related to the given lecture series and possibly leading to
some current research topics.
Lecture Series: day by day
• Lecture A1: “Introduction to Modular Forms,” by Keith Conrad. Videos
□ Lecture 1: Definition of modular forms, Eisenstein series, and q-expansions.
□ Lecture 2: The q-expansion of Eisenstein series, fundamental domains, and modular forms for a subgroup.
□ Lecture 3: Modular forms and sums of four squares, computing dimensions of spaces of modular forms.
□ Lecture 4: Computing dimensions of spaces of modular forms (continued), application to zeta-values, Hecke operators and the L-function for the discriminant modular form of weight 12.
• Lecture A2: “Introduction to Elliptic Curves,” by Álvaro Lozano-Robledo. Videos
□ Lecture 1: What is an elliptic curve? Curves by degree and genus. The addition law. Curves over finite fields.
□ Lecture 2: Torsion points on elliptic curves, the rank, and Z-linear independence of points.
□ Lecture 3: Calculating the Mordell-Weil, Selmer, and Sha groups.
□ Lecture 4: The modular form and L-function of an elliptic curve, Taniyama-Shimura-Weil, and BSD.
• Lecture B1: “Computational methods for modular and Shimura curves,” by John Voight. Videos
□ Lecture 1: Modular curves, fundamental domains, Farey symbols.
□ Lecture 2: Homology of modular curves via modular symbols.
□ Lecture 3: Examples, applications, and higher weight.
□ Lecture 4: Shimura curves, Dirichlet symbols, and examples.
• Lecture B2: “Introduction to the local-global principle,” by Liang Xiao. Videos
□ Lecture 1: Quaternion algebras and Q[p].
□ Lecture 2: Hilbert symbols, basic properties, and their relation to quaternion algebras.
□ Lecture 3: Product formula for Hilbert symbols, local-global principles for quaternion algebras.
□ Lecture 4: Central simple algebras, Brauer groups, and failure of local-global principles.
• Lecture B3: “Gauss sums and the Weil Conjectures,” by Bin Zhao. Videos
□ Lecture 1: Gauss sum, counting numbers of solutions of equations in finite fields, zeta functions of curves over finite fields, the Riemann hypothesis/Weil conjecture for curves over finite
□ Lecture 2: Divisors, Riemann-Roch Theorem for smooth projective curves, rationality of zeta functions of curves over finite fields;
□ Lecture 3: Intersection theory on surfaces, Weil’s proof of Riemann hypothesis;
□ Lecture 4: Generalization of Weil conjecture, modularity and Langlands programme.
• Magma and Sage Tutorials, by Harris Daniels.
□ Lecture 1: The basic syntax and working with elliptic curves in Sage.
□ Lecture 2: Congruence subgroups, modular forms and Farey symbols in Sage.
□ Lecture 3: An overview of Magma.
□ Lecture 4: A tour of the LMFDB database. | {"url":"https://ctnt-summer.math.uconn.edu/schedules-and-abstracts-ctnt2016/","timestamp":"2024-11-13T09:42:44Z","content_type":"text/html","content_length":"71031","record_id":"<urn:uuid:f44874bd-8f0a-40fa-8eb4-9b5768dedaa2>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00095.warc.gz"} |
Operational Amplifier Basics - Electronics-Lab.com
Operational Amplifier Basics
• Boris Poupet
• 13 min
• 2.426 Views
• 0 Comments
This tutorial is an introduction to the Operational Amplifiers, also known as op-amps. The fundamental goal of op-amps is to amplify a voltage difference and it is the reason why we also describe
them as differential amplifiers.
Op-amps have been invented the exact same year as the transistors (1947) and they were originally designed with vacuum tubes in order to perform basic mathematical operations. Mass-production only
started in the ’50s when the op-amps were heavy, not reliable, and cost much. Not before the late ’60s were produced large amounts of transistor-based op-amps available for just a few $.
Nowadays, op-amps are one of the most used electronic components, their cost is only of a few cents of $, and thanks to their interesting properties they are used for many applications.
In the first section, we will present in detail the architecture and definitions surrounding op-amps. Moreover, we briefly discuss the internal circuitry of op-amps.
The second section focuses on the concept of the ideal op-amp which is a model describing the functioning of a perfect op-amp.
Real op-amps will be discussed in the third section where we will look into the differences that must be considered.
An op-amp is usually represented as a triangle with 5 pins from which 4 are inputs and one is the output.
fig 1: Representation and pin configuration of an op-amp
The output is labeled V[out ]and it is the pin where the output voltage is collected. V[+] and V[–] are respectively the non-inverting and inverting inputs. V[S+] and V[S-] are respectively the
positive power supply and negative power supply rails.
We can note that in most of the op-amps representations, the power supply voltages and pins are not represented in order to simplify the drawing. Most of the time, the power configuration is just
assumed or not relevant to perform calculations on a specific op-amp.
The internal circuitry of op-amps generally consists of a succession of bipolar or field-effect transistors and other passive components that are assembled in three distinct stages as shown in Figure
fig 2: Simplified internal circuitry of an op-amp
The goal of the differential stage is to pre-amplify the differential signal V[+]-V[– ]. The special configuration used to realize this process is called a transistor long-tailed pair circuit or
differential pair. Moreover, this configuration provides a high input impedance.
The amplification stage is usually a high gain class A amplifier, the capacitor is used to assures the frequency compensation. Note that many amplification stages can be interconnected in order to
provide a higher amplification output.
Finally, the buffer stage provides no amplification (unitary gain) but has a low output impedance and, therefore, provides high output currents. It is also used in order to adapt the impedances and
protect against short-circuits.
Open-loop gain
A few major characteristics can be associated with op-amps and we will dictate their electronic behavior here. The first one is the open-loop gain (A[OL]), it is a factor that represents the
amplification applied to the input differential voltage:
eq 1: Definition of the open-loop gain
The term “open-loop” refers to the fact that no feedback is applied from the output to the inverting input of the op-amp. We will come back to that notion later on in the tutorial, however, in order
to get an idea now of this concept, we show in Figure 3 the distinction made between open-loop and closed-loop op-amps:
fig 3: Representation of an open-loop and closed-loop configuration
Input and output impedances
The input impedance Z[in] represents the ratio V[in]/I[in] with V[in]=V[+]-V[–] and I[in] being the input current. Similarly, we can also define an output impedance Z[out] which represents the ratio
V[out]/I[out ]with V[out]=A[OL].V[in ]and I[out] being the output current.
Figure 4 below shows a representation of an op-amp that takes into account these impedances:
fig 4: Equivalent representation of an op-amp showing the open-loop gain, input, and output impedances
Op-amps can be used in DC but also in the AC regime, such as for example for the amplification of audio signals. For this reason, one of the important characteristics of op-amps is their bandwidth
(B). This means that the gain (A[OL]) is dependent on the input frequency.
The bandwidth is measured in Hertz (Hz) and represents the range of frequencies that an op-amp can amplify efficiently. More precisely, the frequencies for which the gain is higher than -3 dB are
included in the bandwidth. The limit frequencies for which the gain is exactly equal to -3 dB are called cutoff frequencies and often labeled f[-3dB].
Op-amps behave actually as first-order low-pass filters, this means that the gain can be approximated as a constant from the DC regime up until its cutoff frequency. For higher frequencies, a loss of
-20 dB/decade is observed as shown in Figure 5:
fig 5: Op-amp frequency diagram
To get more detail about this topic, we recommend reading the tutorial about Bode diagrams.
Offset voltage
The offset voltage V[off] can be read at the output terminal when no input is applied to the amplifier. For example, if an op-amp has an offset voltage of 1 V, it means that the output voltage will
constantly be shifted of +1 V, even when no input signal is applied.
fig 6: Illustration of the offset voltage
Ideal op-amp model
This model describes an idealized op-amp that is free of any parasitic phenomena. It is of course not possible to build such an op-amp with ideal characteristics but only approach it.
The ideal op-amp model consists of idealizing its main characteristics previously presented in the presentation section:
• Infinite open-loop gain (A[OL]=+∞)
• Infinite input impedance (Z[in]=+∞)
• Zero output impedance (Z[out]=0)
• Infinite bandwidth (B=+∞)
• Zero offset voltage (V[off]=0)
This set of idealized characteristics highlights the fact that an ideal op-amp does not disturb the amplified signal. An ideal op-amp is usually represented with a sign “∞” within the triangle shape.
One very important property is that in an open-loop configuration, the output of an ideal op-amp can only take two values called the saturation voltages (V[sat]). If the differential input V[in] is
positive (reciprocally negative), the output is +V[sat ](reciprocally -V[sat]).
fig 7: V[out]=f(V[in]) characteristic of an ideal op-amp in open-loop
The value of |V[sat]| is slightly lower than the absolute value of the supply V[S].
In the following subsections, we will see two different modes that can be adopted for an ideal op-amp depending on which input the feedback is applied.
Saturated mode
In this mode, feedback is applied to the non-inverting input (+) of the op-amp. This means that any increase in the output voltage will increase the differential input. This kind of configuration is
also known as a comparator and represented in Figure 8:
fig 8: Ideal op-amp in saturated mode (positive feedback)
Linear mode
If instead the feedback is applied to the inverting input (-) of the op-amp, the function of the amplifier is completely different.
fig 9: Ideal op-amp in linear mode (negative feedback)
In this configuration, any increase of the output voltage tends to decrease the differential input and therefore, also tends to maintain a differential input close to zero.
The relation between the input and output voltages is given by Equation 2:
eq 2: Transfer function of the op-amp presented in Figure 7
In a closed-loop configuration with negative feedback, the characteristic V[out]=f(V[in]) is therefore linear according to Equation 2 up until -V[sat] and +V[sat] where a plateau emerges.
fig 10: V[out]=f(V[in]) characteristic of an ideal op-amp in closed-loop
Real op-amps
Op-amps that can be found in real electronic circuits have limited and non-ideal characteristics:
• Finite open-loop gain typically ranges from 10^5 to 10^6
• Finite input impedance: 10^5 up to 10^12 Ω
• Non-zero output impedance: 50 to 200 Ω
• Finite bandwidth
• Non-zero offset voltage: 1μV up to 50 mV
The gain of real op-amps depends moreover on the frequency with a variation that can be described as a first order low-pass frequency. Another important information is that the product gain-bandwidth
of op-amps is constant, this implies that “slow” op-amps can have higher gains and “fast” op-amps tend to have a lower gain.
The input impedance is not purely resistive as a parallel capacitor of a few pF modelizes the low-pass filter behavior of the op-amp and tends to reduce the impedance when the frequency increases.
We have presented the basics of operational amplifiers in this introductory tutorial. Op-amps are integrated circuits that are powered with two supply inputs and which goal is to amplify the
differential input voltage.
We have briefly presented their internal circuitry and shown that at least three stages are necessary to perform amplification.
Many characteristics can define an op-amp, however, five in particular are extremely important and are presented in detail in the first section. Moreover, we explain that two configurations can be
adopted leading to different behaviors: the open-loop or closed-loop.
The ideal op-amp model is detailed in a second section where its idealized characteristics and behavior are summarized.
Finally, we highlight the differences between this ideal model and real op-amps that can be found in many modern circuitry. The most important consequences of these differences are the finite gain
and bandwidth which limits the amplification and frequency abilities.
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Testing AngularJS with JSDom
Last week we took a look how to utilize JSDom to test JQuery code. Now let’s take a peek how we can reach the same conclusion with AngularJS code. If you went through the linked post most of the
stuff here is probably more or less the same. Regardless of that there might be few angular specific bits here that are of use on top of that.
We’ll take a look at the code bit by bit. First imports that we need to attach to our test file, lets call that test.test.js:
1 import {expect} from 'chai';
2 import jsdom from 'jsdom';
3 import fs from 'fs';
Very slim imports, the only things really needed on test side are JSdom itself, expect from chai to assert our tests and filesystem handlers from node.js. The slimness is because we will be using fs
to load the other files.
The next few lines of our file will initialize our dom implementation, jsdom. First we’ll create a virtual console for debugging purposes, if we at some point need to console.log from within our
system under test we’ll forward those messages to the console in our test environment. Then we’ll create the actual jsdom itself and attach our virtual console to it.
1 const virtualConsole = jsdom.createVirtualConsole().sendTo(console);
2 const window = jsdom.jsdom(null, {virtualConsole: virtualConsole}).defaultView;
Now let’s take a look at what we are loading in:
1 function appendScript(script, cb) {
2 const scriptEl = window.document.createElement('script');
3 scriptEl.innerHTML = script;
4 window.document.body.appendChild(scriptEl);
5 if (cb) cb();
6 }
8 function appendCommonScripts() {
9 const scripts = [
10 './lib/jquery-2.1.3.min.js',
11 './lib/lodash.min.js',
12 './lib/angular.js',
13 './app/common/angular/services.js'
14 ];
15 scripts.forEach(function (it) {
16 appendScript(fs.readFileSync(it, 'utf-8'));
17 });
18 }
Cool, that’s a bunch of files. First all the libraries that we are using jquery, lodash and angular and then an internal library that contains few of the services our system under test is using.
Since we have initialized our jsdom we have a global window element available that we can use to attach our scripts into. We’ll inline these guys into the DOM implementation and let jsdom handle
loading of them.
Next thing we can do is to jump into our test closure. We again use mocha to run the test so we’ll start with familiar call describe function:
1 describe('List page', () => {
2 let $;
3 const html = fs.readFileSync('./templates/sutTemplate.html', 'utf8');
4 const requireJSFile = fs.readFileSync('./app/sut/list.js', 'utf8');
6 before(()=> {
7 appendCommonScripts();
8 appendScript(requireJSFile, function () {
9 window.$('body').append(html);
10 $ = window.$;
11 mockApiService(window);
12 });
14 });
15 /**
16 Actual test cases, see below
17 **/
18 });
First we load two more files with the help of fs, our HTML template, the same one we use in our application code (without JS imports) and our system under test. These will be attached to the DOM in
our before function. In our before we also have a call to append all the libraries we previously defined. After that we bind Jquery dollar to a local element for ease of access and make a mysterious
call to mock an API service. Next let’s take a look at implementation of that function:
1 function mockApiService(window) {
2 window.ourOwnServices.service("apiService", function () {
3 this.get = function (url) {
4 window.getCalled = true;
5 return {
6 success: function (callback) {
7 window.successCalled = true;
8 callback([{
9 id: 1,
10 name: 'Test Name',
11 }]);
12 }
13 }
14 }
15 });
16 }
The function itself is fairly simple. Wht we do in here is just simply override a globally exposed ourOwnServices Angular service module. In that service module we target one specific thing,
apiService and more specifically only it’s get method. This is because we know that our actual application module is injected with OurOwnServices and we want to mock that out. We also know that only
apiService.get method is used in our system under test. Our mock implementation of the get method assign few global variables to window scope. These are later used in the test to assert that these
functions are called.
The code in our services.js would look something like this:
1 var ourOwnServices = angular.module('OurOwnServices', []);
2 ourOwnServices.service("apiService", ["$http", function($http) {
3 /* snip */
4 }]);
Note that we don’t even bother injecting angular $http to our mocked service, we know on instantiation time that we will be only returning a predefined object.
The last piece of the puzzle is our actual test cases. These guys are very simple for this implementation:
1 it('should make ajax request', () => {
2 expect(window.getCalled).to.be.ok;
3 expect(window.successCalled).to.be.ok;
4 });
6 it('should display correct name on first column of first row', () => {
7 const firstColumnText = $('table tr:first-child td:first-child').html();
8 expect(firstColumnText).to.be.equal('Test Name');
9 });
The first test asserts that our service method has been called by our system under test. The second checks that our system under test has replaced angular template placeholder with the correct value
in our list view. The meaningful line on our template HTML is similar to:
1 <!-- snip -->
2 <tbody>
3 <tr ng-repeat="elem in elements">
4 <td></td>
5 <!-- snip -->
And our angular code that makes the call to our API is simply
1 $scope.getElements = function() {
2 apiService.get(url)
3 .success(function(elems){
4 $scope.elements = elems;
5 });
6 };
For more thorough testing you would be probably using a proper mocking library like sinon and asserting that proper arguments are passed to mocked functions etc. For this simple demonstration though
a hardcoded manually mocked method is sufficient. | {"url":"https://jussi.hallila.com/2016/09/12/testing-angularjs-with-jsdom.html","timestamp":"2024-11-10T07:29:09Z","content_type":"text/html","content_length":"33629","record_id":"<urn:uuid:dd21f738-b1a2-416a-97b9-f86e07e0186c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00787.warc.gz"} |
Impedance to Reflection Coefficient Calculator
The formula to calculate the Reflection Coefficient (Γ) is:
\[ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} \]
• \( \Gamma \) is the Reflection Coefficient
• \( Z_L \) is the Load Impedance (Ω)
• \( Z_0 \) is the Characteristic Impedance (Ω)
What is an Impedance to Reflection Coefficient Calculator?
An Impedance to Reflection Coefficient Calculator is a tool used to determine the reflection coefficient, load impedance, or characteristic impedance in electrical circuits. The reflection
coefficient is a measure of how much of a signal is reflected back at an impedance discontinuity in a transmission line. This calculator is particularly useful in RF engineering, telecommunications,
and other fields where impedance matching is critical for efficient signal transmission.
Let's say the load impedance (Z[L]) is 75 Ω, and the characteristic impedance (Z[0]) is 50 Ω. Using the formula:
\[ \Gamma = \frac{75 - 50}{75 + 50} = \frac{25}{125} = 0.2 \]
So, the Reflection Coefficient (Γ) is 0.2.
BIT1024 Calculator© - All Rights Reserved 2024 | {"url":"https://waycalculator.com/tool/Impedance-to-Reflection-Coefficient-Calculator.php","timestamp":"2024-11-10T19:09:36Z","content_type":"text/html","content_length":"4624","record_id":"<urn:uuid:cdd1bfe4-1c92-4d95-882e-c68deeb7ba3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00863.warc.gz"} |
Alexander Millar (Stockholm)
2:00pm Friday February 25, 2022
Axion-photon conversion in neutron star magnetospheres
Axion dark matter can resonantly convert to photons in the magnetosphere of neutron stars, possibly giving rise to radio signals observable on Earth. This method for the indirect detection of axion
dark matter has recently received significant attention in the literature, which I will review in this talk. The calculation of the radio signal is complicated by a number of effects; most
importantly, the gravitational infall of the axions onto the neutron star accelerates them to semi-relativistic speed, and the neutron star magnetosphere is highly anisotropic. Both of these factors
complicate the calculation of the conversion of axions to photons. In this work, we present the first fully three-dimensional calculation of the axion-photon conversion in highly magnetised
anisotropic media. Depending on the axion trajectory, this calculation leads to orders-of-magnitude differences in the conversion compared to the simplified one-dimensional calculation used so far in
the literature, altering the directionality of the produced photons. | {"url":"https://particletheory.science.unimelb.edu.au/2022/02/16/alexander-millar-stockholm/","timestamp":"2024-11-09T11:26:13Z","content_type":"text/html","content_length":"30027","record_id":"<urn:uuid:9f1966c1-647c-49b7-9956-9192a258ae89>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00486.warc.gz"} |
Solving Equations By Combining Like Terms Worksheet - Equations Worksheets
Solving Equations By Combining Like Terms Worksheet
Solving Equations By Combining Like Terms Worksheet – The objective of Expressions and Equations Worksheets is to aid your child to learn more efficiently and effectively. They include interactive
activities as well as problems that are based on order of operations. With these worksheets, children are able to grasp simple and complex concepts in a brief amount of amount of time. These PDFs are
free to download, and can be utilized by your child to practice math equations. These are helpful for students in the 5th to 8th grades.
Download Free Solving Equations By Combining Like Terms Worksheet
These worksheets can be utilized by students from the 5th-8th grades. These two-step word puzzles are made with fractions and decimals. Each worksheet contains ten problems. They are available at any
site that is online or print. These worksheets can be used to test the practice of rearranging equations. In addition to practicing restructuring equations, they can also help your student understand
the principles of equality as well as inverse operations.
The worksheets are intended for fifth and eight graders. They are great for students who have difficulty learning to compute percentages. There are three types of questions you can choose from. You
have the choice to solve single-step questions that include whole numbers or decimal numbers, or to employ words-based methods to solve fractions and decimals. Each page is comprised of ten
equations. These worksheets for Equations can be used by students from the 5th through 8th grades.
These worksheets are a great resource for practicing fraction calculations and other concepts related to algebra. You can pick from kinds of math problems using these worksheets. You can choose the
one that is word-based, numerical or a mixture of both. It is vital to pick the right type of problem because each one will be different. There are ten challenges on each page, meaning they’re
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These worksheets aid students in understanding the relationships between numbers and variables. The worksheets give students practice in solving polynomial equations or solving equations, as well as
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Haosui Duanmu
I employ techniques from Mathematical Logic and Nonstandard Analysis to shed new light on fundamental problems in Statistics and Probability.
I am a Postdoctoral Researcher at the University of California, Berkeley, working with Robert Anderson in the Department of Economics. I received my doctorate in Statistics at the University of
Toronto, under the supervision of Jeffrey Rosenthal (Stats, Math), William Weiss (Math), and Daniel Roy (Stats, CS).
Many of my friends call me Kevin.
• Statistics: decision theory, frequentist–Bayesian interface, matching priors
• Probability: Markov chains, hitting/mixing times
• Nonstandard Analysis
• PhD, Statistics, 2017
University of Toronto
• MSc, Statistics, 2012
University of Toronto
• BSc, Mathematics, 2011
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A296099 - OEIS
The partial XOR sums are given by
This sequence is a "binary" variant of
This sequence has connections with
; graphically, both sequences have similar fractal features; in the scatterplot of the current sequence, the rays emerging from the origin correspond to the numerous terms a(n) that are multiples of
(PARI) s = 0; x = 0; for (n=1, 65, for (k=1, oo, if (!bittest(s, k) && (xx=bitxor(x, k))%n==0, x = xx; s += 2^k; print1 (k ", "); break))) | {"url":"https://oeis.org/A296099","timestamp":"2024-11-04T02:48:09Z","content_type":"text/html","content_length":"14124","record_id":"<urn:uuid:f455ff58-d629-41e4-bf29-a28f3526e459>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00137.warc.gz"} |
INTRODUCTIONAnalysis of model approximations of medium typesScientific HypothesisMATERIAL AND METHODOLOGYSamplesChemicalsInstrumentsLaboratory MethodsDesсription of the ExperimentStatistical analysisRESULTS AND DISCUSSIONSystem of realization of energy potentialsThermodynamic analysis of energy resources. Methods for determining the temperature field of a low potential energy sourceCONCLUSIONAcknowledgments:Funds:Conflict of Interest:Ethical Statement:
The system of redistribution and transformations of energy resources of environments is considered. Theoretical and practical experience of thermodynamics specialists has allowed, among other things,
to generalize the relationship between air heat capacity and temperature. Molar heat capacity at р = const, kJ / (kmol • K) is represented by the dependencies in the actual value in the interval
0-1000 °С:
c μp = 28,7558 + 0,0057208t
and on average
c ¯ μp | 0 t = 28,827 + 0,002708t
Numerical values of heat capacities are given in
Table 1, from which follows the possibility of certain comparisons.
Heat capacity of air.
Heat capacity
Temperature, °C Molar, kJ / (kmol • K) Mass, kJ / (kg • K) Volume, kJ / (m^3 • K)
C[μp] C[μv] c ¯ μp | 0 t c ¯ μv | 0 t c ¯ p | 0 t c ¯ v | 0 t c p | 0 t c v | 0 t
0 29.073 20.758 29.073 20.758 1.0036 0.7164 1.2971 0.9261
100 29.266 20.951 29.152 20.833 1.0061 0.7193 1.3004 0.9295
200 29.676 21.361 29.299 20.984 1.0115 0.7243 1.3071 0.9362
300 30266 21.951 29.521 21.206 1.0191 0.7319 1.3172 0.9462
Estimation of energy potential in 1 m^3 of air at a temperature of 303 K at a pressure of 1 bar, is:
Q p o t = c p υ T = 1.2971 1.0 303 = 393.0213 k J
Cooling this volume of air to 272 K means heat transfer Q to the receiver. At the same time, we note that the same volume of air at -30 °C or 243 K has a potential of 315.1953 kJ. Thus, technological
opportunities for the redistribution of energy potentials open up prospects for their transformation and use, due to the second law of thermodynamics.
Air is a mixture of gases. The composition of the air is not constant and varies depending on the area, region, and even the number of people around you. Air consists of about 78% nitrogen and 21%
oxygen, the rest are impurities of various compounds. Its chemical formula is O2. Under normal conditions (temperature 0 °C, pressure 101.3 kPa) oxygen is in a gaseous state, with no mass of taste,
odor, slightly heavier than air.
The flow temperature at the pump inlet and outlet was determined, after which the temperature value was averaged. Several thermocouples were used to measure the flow temperature in the middle of the
pipeline. The data of the primary temperature transducers were fed to the ADC, converted into a digital signal, and entered into the PC. Data registration interval - 10 s. Measurements were performed
with chromel-kopel thermocouples, measuring device brand Expert (manufacturer Ukraine).
At regular intervals, the surface temperature t was measured. The surface temperature of the module was determined by a thermocouple embossed into the surface. At regular intervals, the air pressure
was measured. Measurements were performed with a manovacuummeter OBMV1-1006f (manufacturer Ukraine). The value of pressure in the tables of properties of water vapor was determined by its
The determination of ambient air parameters was done by using a psychrometer brand VIT-2 (manufacturer Ukraine). There was a registration of air parameters, namely its temperature, humidity, moisture
content, enthalpy. The measurement of these parameters was performed using the ADC module "Arduino" and a computer.
A digital humidity sensor SHT 10 was used to determine the parameters of the air coming from the pump. This allowed determining with high accuracy the characteristics of the air directly in the flow
during the experiments. The program interface of the ADC converter is made in such a way that the change of air parameters from time to time is reflected in the form of graphical dependencies in the
online mode. The use of such a scheme with the use of frequency control of the pump drive allowed to quickly intervening in the course of the experiment that simplifies its planning.
The fluid flow simulation was performed using ANSYS CFD general-purpose computer software from ANSYS, Inc.
Software systems allow modeling and calculation of liquids and gases, heat and mass transfer processes, reacting flows. ANSYS CFD is fully integrated into the ANSYS Workbench environment, which is
the basis of engineering modeling, it integrates all ANSYS tools and software.
The ANSYS Workbench environment provides general access to such tools as: communication with CAD-complexes, construction, and modification of geometry and calculation grid. The software package is
widely used for modeling processes that take place in pumps, fans, compressors, gas, and hydro turbines, etc. The ANSYS CFD-Post postprocessor, which is part of the software package, can be used to
create high-quality animations, illustrations, and graphics (Bezbakh et al., 2009,; Basok, and Bazeev, 2017; Nochnichenko, and Yakhno, 2018).
Theoretical modeling of conditions of creation and reproduction of processes of redistribution of energy potentials of environment and recovery of secondary resources based on provisions and laws of
technical thermodynamics is used.
The research of the process was carried out according to the Box-Benken plan, which allows obtaining the maximum amount of objective information about the influence of factors with the help of the
smallest number of experiments.
Processing of the obtained experimental data set was performed according to well-known methods and statistical processing methods to obtain an empirical mathematical model:
Q ke = f Q ( x 1 ; x 2 ; x 3 )
using methods of correlation and regression analysis of the approximating function, which characterizes the influence of factors and their interaction on the optimization parameter, ie productivity.
Based on the constructed regression equation, the contribution of each independent variable to the variation of the studied dependent variable is determined, ie the influence of factors on the
performance indicator.
The experimental data set was processed using the Statistica-12 software package. The coefficients of the regression or approximating function equation, under the condition of orthogonality and
symmetry of the plan-matrix of the planned factorial experiment, were determined according to the standard method according to known dependences.
The obtained results were statistically processed using the standard Microsoft Office software package.
Theoretical and experimental research was performed fbased on the Problem Research Laboratory of the National University of Food Technologies. The study consisted of three parts. The first was to
assess the energy potential of ambient air as the most accessible environment.
This part of the research is the basis for the creation of a modern air heat pump ( a patent of Ukraine 17167) with the implementation of phenomenological studies relating to the redistribution of
heat fluxes based on the second law of thermodynamics using a closed circulating air circuit.
The second part of the study concerns the thermodynamic features of achieving phase transitions of water as the main filler of food industry environments with the development of mathematical
formalizations and estimation of energy consumption in the processes of heating the liquid phase and entropy transformations.
The third part is devoted to the implementation of phase transitions with the assessment of the energy potentials of the secondary pair. This creates a basis for the implementation of regenerative
processes by increasing the pressures and temperatures in the compensation processes and subsequent condensation to obtain a liquid phase of H2O in closed energy circuits.
In this article, we examine the first and second parts of the research, which substantiates our further direction of theoretically directed mathematical analysis.
Historically, humanity has received such opportunities after the creation of heat engines. In 1852, Lord Kelvin proposed new use of the heating machine for space heating. Kelvin called such a machine
a heat pump, the task of which was to cool the cold outside air and transfer the received thermal energy at a higher temperature in the room. This unnatural process of heat transfer from a cold to a
heated environment was carried out through the consumption of mechanical work. Each unit of mechanical work, brought to the ideal heat pump, before getting into the heated room "captured" 5 – 8 units
of the heat of the outside air. Therefore, 427 k Gm of work at the inlet to the heat pump was converted into 6 – 9 kcal of heat at the outlet. The same kilocalorie formed by burning a certain amount
of fuel is not supplemented by anything and remains one kilocalorie.
Burning some amount of fuel directly, bring in a heated room 1 kcal of heat. If the same amount of fuel will be burned in a heat engine, then only about 20%, equivalent to 85 kGm, will convert into
mechanical work. If these 85 kGm are brought to the heat pump, it will provide 6 times more heat to the room, ie 6 • 85 = 510 kGm or 1,2 kcal. These ratios indicate the feasibility of using the
primary energy potentials of the fuel in the circuit "heat engine - heat pump".
Thus, it is expedient to pay attention to equivalence in these thermodynamic transformations. Thus, the value of the coefficient of performance of heat engines is difficult or impossible to increase
by 20%. In the heat pump due to the generated mechanical energy, heat returns to the level of lost.
The general structure and principle of operation of the model of the laboratory installation and the "Air Heat Pump" are shown in Figure 1 and Figure 2. The laboratory complex allows obtaining the
dependence of the parameters of the heat pump installation on the temperature regimes of the heat supply and air conditioning system (Figure. 1 a, Figure 1b). In Figure 1a – the software module is
given, and in Figure 1b – a general view of the laboratory experimental setup. The research aimed to establish the optimal parameters of the developed heat pump (Figure 2), which consists of a
compressor 1 with blades 2, guides 3, partition 4, expander 5, heat exchangers 6 and 7, and a drive motor 8. It works as follows. The drive motor 8 provides rotational movement of the rotor of the
compressor 1 and due to the interaction of the airflow with the blades 2 and the guide devices, 3 is its compression with increasing temperature. This compressed air enters the zone of separation of
the internal volume of the pump and enters the heat exchanger 7, through which the external air flow. The latter absorbs thermal energy by cooling the compressed air. The cooled air is supplied to
the expander 5, in which, expanding to a given final value of pressure, gives its energy. In this process, its temperature drops sharply to a temperature well below ambient temperature. Due to this,
when in contact in the heat exchanger 6 with the flow of external air is heat transfer from the latter to the air circulating in a closed circuit. In the future, the cycle is repeated, and the cooled
outside air is sent to further technological needs.
software module (a) and General view of laboratory experimental setup (b).
Air heat pump.
Pre-rough control of the compressor shaft speed of the compressor of the appropriate diameter was set using commands from the motor control panel of the control multisystem control and reading device
Altivar 71 using Power Suite software version 2.3.0. The technical capabilities of this device and software allow to smoothly change the speed of the motor shaft of the prototype laboratory
installation in the range from 0 to 1300 rpm. The numerical value of the motor shaft speed (error within ± 1.5%) was recorded using a sensor type E40S6-10Z4-6L-5, which is connected simultaneously to
the rotor of the motor and the multisystem device.
Numerical data of energy costs and torque on the shaft of the electric drive depending on the load at a particular time of the experiment is displayed in the form of tabular data and graphical
dependences on the PC monitor.
It is worth recalling that a heat pump is an installation that converts low-potential natural thermal energy or heat from secondary low-temperature energy resources into the energy of higher
temperature potential, which is already suitable for practical use. The transformations take place in the reverse thermodynamic cycle, and the transfer of energy from the lower temperature level to a
higher one is performed due to a certain amount of mechanical (electrical) energy, which is externally supplied to the heat pump compressor and its design.
The algorithm for conducting experimental studies of the air heat pump, formalized in the form of a the structural model scheme is shown in Figure 3. It involves determining the functional patterns
of influence of individual input variables and their impact on the output value, or optimization parameter. To verify the adequacy of theoretical research (theoretical model) of productivity Q[k],
experimental studies of the model of the laboratory installation, which is shown in Figure 1 and Figure 2. To obtain an empirical regression equation characterizing the change in productivity Q[k]
depending on the parameters of the compressor rotor, implemented a planned three-factor experiment such as PFE 3^3.
Scheme of the model of the planned experiment of the PFE 3^3 type.
The total number of experiments e N one repetition was determined by the formula:
N e = P k
P – the number of levels of variation of the variable input factor;
k – the number of active variables of input factors in the experiment.
The experiments were performed in triplicate. The asymmetric plan-matrix of the planned three-factor PFE 3^3-type Box-Benkin experiment for three factors and three levels of factor variation had a
total number of experiments equal to 27.
The independent variables were: the speed of the rotor of the compressor n[k], which was encoded by the index x[1], ie n[k]; the diameter of the rotor of the compressor D[k], which is encoded by the
index x[2], ie D[k]; blade pitch T[1], encoded by the index x[3], ie T[1].
Structural model of a planned three-factor (Krutov, 1989) PFE-type experiment 3^3 shown in Figure 3.
Thus, to study the performance of Q[ke], an approximate mathematical model in the form of a functional dependence was chosen.
Q ke = f Q ( x 1 ; x 2 ; x 3 ) .
When compiling the plan-matrix of experiments, coded notations of upper (+1), lower (-1), and zero (0) levels of variation by factors were introduced (Krutov, 1989), ie a three-factor experiment was
performed at three levels of variation by input factors or a planned PFE 3^3 type experiment was implemented.
Surface response changes in productivity in the form of functionality.
The results of coding of variable input factors, the upper and lower level of variation of each factor, and the interval of its variation are given in Table. 2.
The results of coding factors and levels of their variation.
Factors Marking Interval of variation Levels of variation natural / coded
natural coded
Speed of rotation, n[k]¢, X[1] x[1] 100 100/-1 200/0 300/+1
Diameter D[k], m X[2] x[2] 0.04 0.12/-1 0.16/0 0.2/+1
Step T[1], m X[3] x[3] 0.03 0.05/-1 0.08/0 0.11/+1
Because during the experiments independent variable input factors, ie n[k] are inhomogeneous, ie they all have different physical units and different orders of arithmetic numerical values of units,
they were led to a single system of calculations by switching from the entered notation coded values to real (natural) values. Made a randomized plan matrix of the planned three-factor experiment
type PFE 3^3.
After estimating the statistical significance of the coefficients of the regression equation and checking the adequacy of the mathematical model of the logarithmic function, we obtained a regression
equation that characterizes the functional change in productivity in natural quantities (Krutov, 1989).
Q k e = 0.81 + 0.61 l n ( n k ) + 1.33 l n ( D k ) + 0.31 l n ( T 1 )
With the probability level p = 0.95 and the value of the t-alpha criterion equal to 2.053, the following statistics were obtained: coefficient of multiple determination D = 0.893; multiple
correlation coefficient R = 0.945; standard deviation of the estimate s = 0,150; Fisher's F-test is 64,212. The coefficient D is significant with the probability level P = 1.00000. The regression
equation (4) characterizes the change in the performance of the air heat pump depending on the design and kinematic parameters within the following limits of change of input factors: speed nk (from
100 to 300 rpm); diameter D[k] (from 0.12 to 0.2 m); blade pitch T1 (from 0.05 to 0.11 m). The functional change in productivity depending on the change in Qke factors is directly proportional - with
increasing speed, diameter and pitch, the value of productivity also increases.
According to the regression equation (1), the response surface of the functional change in the productivity of Qke in the form of a functional is constructed:
Q k e = f Q ( n k ; D k ) ( F i g u r e 4 , a ) ; Q k e = f Q ( n k ; T 1 ) ( F i g u r e 4 , b ) ;
The dominant factors that have a significant functional impact on the increase in productivity Qke are the speed nk and diameter D[k], which is characteristic of the graphical interpretation of the
response surface, and this is the regulation of energy potential, ie temperature.
Figure 5 represents a diagram of the change in the productivity of the Qke heat pump, based on the obtained average results of experimental studies with three repetitions of each numbered factor
field experiment according to a randomized plan matrix of the planned experiment type PFE 3^3.
The diagram of change of productivity Q[ke] of the heat pump. Note: a, b, с – T1 = 0.05; 0.08 and 0.11 m.
Based on the graph-analytical analysis (Figure 5) it can be stated that the nature of the functional change in the productivity Q[ke] of the heat pump. It is obtained for the limit values of the
corresponding points of the a three-factor experiment type PFE 3^3 compositional plan is quite adequate model Q[ke] =f[Q] (x[1]; x[2]; x[3])= Q[ke] =f[Q] (n[k]; D[k]; T[1]), Figure 4, Figure5, which
is also characteristic of the dependence, which is shown in Figure 6.
Dependence of change in productivity Qe as a functional:а – Q[k] = f[Qk] (D) , 1, 2, 3 – in accordance, n[k]= 100; 200; 300 rpm; b – Q[k]= f[Qk] (n [k]), 1, 2, 3 – in accordance, T[1] =0,05; 0,08;
0,11 m.
The discrepancy between the experimental values of the performance Q[ke] of the pump obtained according to the regression equation (4) and the experimental values of the performance Q[k] (graphical
dependences of Figure 7) is in the range of 5 – 10%.
Assessment of the general state of processes in energy effects.
This feature of energy transformations is based on the second law of thermodynamics with an indication of the need to use compensation systems by increasing the temperatures and pressures of energy
sources in closed circuits.
An important advantage of the heat pump is that it implements "reverse" processes in the modes of heating and cooling of the premises as an ideal air conditioner.
The technical implementation of heat pumps and refrigeration machines based on the Carnot reverse cycle, which is the only achievement of humanity in the implementation of the principle of energy
redistribution in existing parallel systems.
Returning to condition (3) we obtain an estimate of the heat flux dissipated from the cooling medium:
Q ′ = c p v ′ ( T ( ∏ ) − T ( K ) ) = c p v ′ ( t ( ∏ ) − t ( K ) ) , kW ,
v– volumetric flow of the gas phase supplied to the evaporator as part of the heat pump, m^3 / s;
T ( n ) , T ( K ) , t ( ∏ ) and t ( k ) – initial and final absolute temperatures and temperatures in degrees Celsius.
Similarly, it is possible to determine the energy potentials of the liquid phases of lakes, rivers, seas and oceans, which are largely used.
The study (Nikitin and Krylov, 2012) estimates the ratio of redistribution of energy flows, according to which from 60 to 70% of energy costs related to the circulating circuits.
Understanding this situation in a significant number of cases (Mandryka, 2017) prompted attempts to use this energy component in favor of energy resources. It should be noted that the kinematic
parameters of the gas-liquid medium are estimated to be approximately stable. Their transfer to the regimes characteristic of transitional processes should be assessed as a promising direction of
intensification of energy resources.
Another direction (Sniegkin et al., 2008) concerns the use of secondary energy resources that accompany most industrial technologies. Their energy potentials elate to the solid and liquid phases of
the input raw materials, to which the potentials of the vapor phases and gases are added in the transformation processes. The latter applies to a significant number of technological processes and
individual complexes. Solving the problems of recovery of secondary energy resources is most appropriate to solve in parallel-synchronized flows. This largely applies to thermal systems in which
phase transitions are carried out due to the relative ease of regeneration in them.
In cases of asynchronous situations, there is a need to use energy-saving storage devices. However, the positive results for parallel system designs are quite achievable even in mechanical systems in
which transients are generated. In this case, in addition to energy effects, it is possible to regulate the movement of machines with restrictions on total dynamic loads.
In the general list of processes that take place in food, chemical, microbiological and other technologies, there are mechanical, hydraulic, aerodynamic, thermal interactions or various combinations
of them (Figure 7). Manifestations of such interactions are in series and parallel systems with corresponding intensive and extensive parameters, based on driving factors, aero- and hydrodynamic
states of environments, heat, and mass transfer surfaces, means of increasing energy potentials, energy loss compensators, and so on. The technical organization of technologies in general and at the
level of individual processes require the interaction of material, energy, and information flows, the task of which is to achieve appropriate technological effects related to heating-cooling,
evaporation-condensation, formation of concentrated media, gas-saturated systems, aeration, etc. At the same time, efforts to minimize energy costs and limit dissipative losses to the environment at
a certain level of opportunity in the implementation of these provisions remain fundamental. Assessing the general condition, we turn to the example of the features of only one component of
convective heat transfer.
During convection, heat is transferred during the mixing of cold and warm layers of liquids or gases, and therefore this process is inextricably linked with the mechanical motion of liquid and gas
flows. Their theoretical basis relates to the relevant sections of hydro- and aerodynamics, but the level of complexity, even for simple cases in mathematical formalizations, is so significant in
combination with thermal processes that it has led to limitations in the relevant scientific interests. However, it is convection in heat transfer mechanisms in heating systems, technological
devices, electric drives, brakes, compressors, refrigeration units, etc. in terms of significance that led to the development and solution of applied problems.
Most of them concern the determination of heat transfer coefficients, which may depend on the thermal conductivity of the media, viscosity, density, heat capacity, kinematic parameters, and geometry
of the media volumes. The effects of all these parameters are combined by the phenomenon of the boundary layer. The imaginary shirt creates the main barrier to heat transfer, which barriers that most
effectively overcome in the modes of phase transitions of boiling and condensation due to the activation of heat transfer coefficients. An additional positive effect of the phase transition concerns
the production of a coolant with a thermodynamic parameter supplemented by the heat of vaporization.
Phase transitions open additional possibilities of transformation of parameters of pressure and temperatures that allow overcome natural prohibitions which features are formulated by the second law
of thermodynamics. In the classical definition, this is achieved by supplementing the closed or partially closed circuits with a compensatory process of mechanical compression or the introduction of
additional thermal potential with increasing pressure and temperature of the coolant circuit. As a result, such transformations in the reversed Carnot cycle, it is possible to transfer heat from less
heated media and bodies to more heated ones, and the purpose of such transformations may relate to the problem of cooling (heating) the local zone. In the first case there is a use of the
refrigerating machine, and in the second - the heat pump.
However, in addition to information on the creation of Lord Kelvin air heat pump with the transformation of energy potentials of air flows due to the relationship between pressure and temperature, we
note the consequences of continuing attempts to create their modern structures, which we consider in this paper.
This applies to the development of the patent of Ukraine 17167 "Air heat pump" (Figure 2).
The creation of initial energy potentials in such systems is carried out using primary energy sources. The latter in most cases relate to the resources of the generated vapor phase, electricity,
energy of hydraulic systems, compressed air systems, or chemical energy of incoming raw materials. The presence of the latter is a constant factor in any technology, according to which the energy
potential of recycled raw materials should be preserved as much as possible. However, the appropriate set of energy-material transformations provide by the influences of external energy flows due to
which the set temperatures of technological processing of environments are reached. It can be carried out without achieving the modes of phase transitions or with their implementation. Consider the
transformation of the energy potentials of airflows due to the relationships between the design parameters of the pump and the process.
The potential of devices for increase of energy efficiency and intensification of processes of a mode of phase transitions and generation of steam, gas phase, or steam-gas mixes is high enough.
Development of new designs of heat exchangers, evaporators, the definition of rational modes of their operation is possible only based on the data received at comprehensive researches of the
processes proceeding in devices.
It is expedient to refer to the peculiarities of the cycles of refrigeration units or heat pumps from the point of view of creating analogies for systems of industrial devices in which there are
modes of phase transitions and generation of steam, gas phase, or steam-gas mixtures. The existence of a closed circuit in the refrigeration cycle involves the combination of an evaporator as a steam
phase generator (Figure 8), a compressor, a condenser, and a choke operating in synchronized parallel modes with the appropriate thermodynamic parameters.
System of realization of a refrigerating cycle: 1 – the evaporator; 2 – compressor; 3 – capacitor; 4 – throttle; 5 – fan.
In the closed circulation circuit A of the refrigerant, phase transitions occur due to the supply of heat flow q0 from the cooling zone with circuit B and the removal of heat flow qk from the
condenser in circuit C. Depending on the technological tasks, circuits B and C can be closed or open.
This leads to the conclusion that the total energy balance of circuit A is supplemented by the energy consumption ℓ[K] of compressor 2, which meets the condition:
q K = q o + l K ,
and the whole system in balance calculations must take into account power consumption in the circuits B and C. The decision of technical problems is achieved in one of the circuits B or C, or both
simultaneously. It is important that the arrangement of energy-material connections of circuits B and C following the evaporator and condenser can be realized due to convective air flows of the
medium, which are formed in response to the existence of a gravitational field. This solution is present in the installation of most domestic and industrial refrigeration systems and systems.
Cooling and heating zones can exist as local, but in cases where they are open, it means that they are interconnected through the environment, and technical systems of refrigeration systems, heat
pumps, and air conditioners act as programmable energy redistributors. Important is the ratio of the potentials of the synthesized energy flows in the direction from zone B to zone C and the
potential of the compensation processes, which can be up to 5 – 10 units. This means the possibility of creating powerful energy-intensive systems using compensatory processes, the limited structure
of which is the ultimate negative result of the impact on the ecosystem. The last statement is because the synthesized heat fluxes at the end of technological processes are dissipated with the
equalization of temperatures following the law of the most probable state. To prevent further negative effects on the environment the power supply of the electric drive of the compressor-compensator
deserves attention. The use of modern systems for the transformation of world energy into electricity in such cases would practically solve the problems of energy security in a maximum way.
A scientifically based analysis of energy processes is essential for an active energy-saving policy. Modern laws of thermodynamics (Annex 49 summary report IEA ECBCS. –Fraunhofer IBR. -2011) include
the study of the properties of energy in its transformations in two approaches to efficient use: energy and exergy. Such approaches involve the use of two thermodynamic characteristics of energy -
quantity and quality: quantity - in energy, both - in exergy.
Thus, the authors (Kudelya, and Dubovskyi, 2020) consider the possibility of obtaining work when the characteristics of the system (pressure, temperature, velocity, chemical composition, and
potential energy of the system) differ from the characteristics of the state (parameters) of the environment. This possibility is completely lost when the system and the environment are in balance
and calm concerning each other.
The magnitude of the work, as a quantitative measure of energy quality, is included in the equation of energy balance (First law of thermodynamics), and the condition of convertibility S gene ≥0 - in
the equation of entropy balance (Second law of thermodynamics). Therefore, the authors (Ebrahimi, 2012) propose the use of low-potential heat in optimizing the generation of electricity taking into
account the depth temperatures and heat exchange of wells with side rocks. In this case, the Rankine cycle was chosen as the theoretical basis for the calculation of a heat engine with distributed
The calculation of authors (Domschkea, 2017) based on the total energy consumption for main gas transportation for a multi-threaded network, which includes compressor stations, heat exchange areas
with the environment, bridges, and branches
Modeling and optimization of gas flow through the pipeline network by the authors (Domschkea, 2017) used a hierarchical model based on one-dimensional isothermal Euler equations of fluid dynamics.
Therefore, in the works (Liu, et al., 2017; Orga, et al., 2017; Pouladi, 2017) based on the analysis of the topological structure of the heating network, its hydraulic and thermodynamic parameters, a
method of counting the heat flow of the network was developed. Taking into account the external heat inflow in the sections of pipelines minimizes the total heat consumption and improves the
management of technological processes.
Based on the above, the physical state of the system is determined by the values of two variables out of three, namely pressure, volume, and temperature.
It is known that the physical state of the system is determined by the values of two variables out of three, namely pressure, volume, and temperature. There is a functional connection between these
three parameters. In what follows, we will consider the pressure p and the volume v as independent variables, and then we will display this relationship in the form:
T = f ( p, v )
The set of values of p and v determines the position of a point on the plane p-v. Each such point corresponds to a certain value of temperature T (Figure 9).
Dependence of p = p (v) in thermodynamic transformations parameters.
dT = ∂ f ∂ p dp + ∂ f ∂ v dv
is a complete differential. By changing the state of the system from the parameters at point A to the parameters of point B, the temperature at point B can be determined in the form:
T B = f ( p B , v B ) .
Determining the values of the work performed by the system as a result of changing its state during the transition of parameters from point A to point B and, considering the process inverse, we
reflect the dependence:
W A-B = ∫ v A v B p dv
The graphical interpretation of the given integral is the plane under the transition curve on the p-v diagram. Since the transition from point A to point B can be done with different trajectories, it
means that these areas will be different. Their area is to some extent determined by the design parameters and speed of the drive. It follows that the value of W depends on not only the coordinates
of points A and B, but also on the selected transition trajectories. It is logical to assume that the amount of heat perceived in this transition of the system also depends on that, but the
difference between the amount of perceived heat Q and energy W does not depend on the shape of the transition trajectory. The conclusion about the constancy of the difference Q – W, which corresponds
only to the state of the system at points A and B, indicates a change in the internal energy u:
Δ u A → B = ( Q − W ) A → B = F ( p B , v B ) − F ( p A ,V A ) .
In another form, expression (11) has the form:
du = dQ − dW = ∂ u ∂ p dp + ∂ u ∂ v dv .
For the case of a closed trajectory from point A we obtain a curvilinear integral from du, then we have:
∫ v A v A du = u A - u A = 0
The written curvilinear integral is called the circulation and is denoted by a symbol
∮ ( l ) ,
ℓ – closed curve
Bringing the heat flux Q to the medium means corresponding changes in the value of entropy. The latter is determined only by the variables that characterize the physical state of the system, and in
the transition from point A to point B changes in entropy do not depend on its trajectory.
Δ s A → B ∫ A B dQ 3B T ,
dQ[B] – the amount of heat passing through the system boundaries during the reverse process.
In the elementary process we have:
ds= dQ 3B T ; dQ 3B =T ds
Replacement of values dQ i dW provided (12) leads to the form
du = T ds - p dv,
in which there are point functions and complete differentials.
Integration (16) leads to the value of u as a function of the variables s and v in the form:
u = f u ( s, v ) ,
or, disclosing condition (17), write:
du= ( ∂u ∂s ) v ds+ ( ∂u ∂v ) s d v ′
and by comparison with condition (16) we obtain:
( ∂ u ∂ s ) v = T i ( ∂ u ∂ v ) s = − p
If condition (19) is known for a mass of any homogeneous liquid, then the parameters T, p and u can be calculated for any physical state of the medium, which is determined by the independent
variables s and v. Therefore, the performance of the heat pump sets the value of the internal energy u of the process.
The paper (Oosterkamp, Ytrehus, and Galtunget, 2016) is aimed at analyzing the influence of extreme temperature conditions on heat transfer between a low-potential energy source of soil massifs
and an underground pipeline. One- and two-dimensional models were used to calculate thermal conductivity. It is shown that the accuracy of soil temperature forecasting deteriorates when using air
temperatures to assess the marginal condition of the soil surface. It is proposed to take into account the effect of heat accumulation, as well as the actual temperature of the gas in the pipeline
for more accurate efficiency of heat transfer prediction.
Therefore, the calculation of the temperature field of a low-potential energy source in the zone of influence of the compressor rotor diameter and the blade pitch reduced to solving the equation of
nonstationary thermal conductivity.
In a cylindrical coordinate system characteristic of a heat pump, the equation has the form (Lykov, 1967):
Therefore, the calculation of the temperature field of a lowpotential energy source in the zone of influence of the compressor rotor diameter and the blade pitch reduced to solving the equation of
nonstationary thermal conductivity. In a cylindrical coordinate system characteristic of a heat pump, the equation has the form (Lykov, 1967):
∂ t ∂ τ = a ( ∂ 2 t ∂ r 2 + 1 r ∂ t ∂ r + 1 r 2 ∂ 2 t ∂ θ 2 + ∂ 2 t ∂ z 2 )
t – the ambient temperature, °C;
τ – hour, s;
a – thermal conductivity, m^2/s;
r is the radial coordinate, m;
θ is the polar angle (the angle between the radius vector r and the x axis).
This is a three-dimensional problem, but given the shape and length of the rotor relative to the radius of impact, as well as the heterogeneity of the blades, it can be reduced to two-dimensional
with a sufficient degree of accuracy. For this problem statement, taking into account the symmetry of the temperature field, it is proposed to make a simplified solution, provided that τ>; 0 до rр< r
< rк (Rudenko, 2012):
∂ t ∂ τ = a ( ∂ 2 t ∂ r 2 + 1 r ∂ t ∂ r )
rр – average rotor radius, m;
r[к] – radius of the contour of influence, m.
Simplifying the task by switching from a three-dimensional to a two-dimensional model eliminates heat flow along the rotor axis. However, the heat flow in the vertical direction, despite its small
orders, must be taken into account due to its continuity in time, even when the heat pump is stopped. To solve the problem, we introduce an amendment that compensates for bulk sources and heat fluxes
(environment). Then equation (2.2) will take the following form (Rudenko, 2012; Sniegkin et al., 2008):
∂ t ∂ τ = a ( ∂ 2 t ∂ x 2 + ∂ 2 t ∂ r 2 + 1 r ∂ t ∂ r ) + g v c
t – ambient temperature, °C;
r – time, s;
a – thermal conductivity, m^2 /s;
r – is the radial coordinate, m;
qv-sources and heat runoff due to heat fluxes of the environment and heat release through the surface, W / m^3;
c – heat capacity, J / (m^3 • °C).
The paper (Biletsky, 2013) presents the concept of calculating the mode parameters of gas transportation through the network of gas wells, and the authors (Romaniuk, et al., 2019) reveal the features
of calculating electricity losses in systems. The heat exchange between the transported gas and the external environment at each of the sections of the pipeline network is not taken into account. The
problem of cork and hydrate formation in the pipeline is not considered. The calculation of electricity losses does not fully reflect the effects of the external environment, heat fluxes.
Therefore, the set of factors: background temperature of low potential energy source (t[fon], °C), ambient temperature, heat runoff (qv, W), thermophysical characteristics, the intensity of incident
solar radiation, are the basic data in the calculations. The set of factors: background temperature of low potential energy source (t[fon], °C), ambient temperature, heat runoff (qv, W),
thermophysical characteristics, the intensity of incident solar radiation, are the basic data in the calculations.
Execution of the technical and economic analysis is reduced to the definition of the temperature of the heat carrier selected from the heat pump that in turn, is defined by the created temperature.
The second feature is the need to assess the operating conditions of the heat pump in the worst, in terms of the coefficient of thermal transformation, conditions. This corresponds to the time period
of completion of the heating (cooling) cycle.
The known similarity criteria do not fully reflect the studied phenomena, in connection with which the following dimensionless complexes are proposed in the research work: relative heat flux (Q),
modified dimensionless temperature (Θ), and criterion Fo.
The temperature field of a low-potential energy source is described by a dimensionless function (Rudenko, 2012) with three dimensionless parameters:
f = ( Fo, Θ, Q ) ,
Fo –Fourier criterion,
Θ – dimensionless temperature,
Q – relative heat flux.
Considering the active load on the rotor with blades, which perturbs the temperature field factor, we propose to introduce the relative heat flux (Q), which is accepted in the research work to
calculate the formula:
Q = g n a / g f o n
q [na] – specific heat flux per unit area of the rotating rotor with blades, heat load on the pump in a certain period of operation, W / m^2;
q[fon] – background heat flux (specific, per unit area of the environment), W / m^2
The operating parameters of the heat pump depend on the difference between the generated temperature and the ambient temperature.
The Fo criterion was formed by a known formula (Lykov, 1967). The determining size is proposed to take the radius of the rotor:
F 0 = a τ r p 2 ,
а - thermal conductivity coefficient equal to а =λ/сρ m^2/s;
λ - thermal conductivity, W / m ˚С;
ρ – air density, kg / m^3;
с[p] - heat capacity of air, J / (kg × °С);
τ - characteristic time of change of external conditions, s; rр - characteristic body size (rotor radius), m.
The complex nature of the mutual influence of the defining parameters does not allow to formalize an unambiguous solution, in connection with which the traditional approach to the type of criterion
equation as a static dependence is used. The generalizing equation for single-stream operation can be described as a regression:
θ = K 1 F θ 2 Q + k 2 Q F 0 .... + k n F 0 Q
k[1,2 ... n] – determining factors.
The results of the calculations are well approximated by second-order polynomials. Applying the methods of statistical processing, the criterion equation is obtained:
θ = − 5 10 − 9 Q F 0 2 + 2 10 − 8 F 0 Q + 0.0003 Q + 5.1
The results of the calculation of the temperature at the outlet of the partition as a function of the defining parameters in the dimensionless form are presented in Figure 10 (Saprykina, 2016). This
allows you to predict the temperature change of a low-potential energy source operating in the cyclic (seasonal) mode without reversing the heat flow.
The results of the calculation of the function from the defining parameters. | {"url":"https://potravinarstvo.com/dokumenty/xml/vol15/PSJFS-15-1-680/PSJFS-15-1-680.xml","timestamp":"2024-11-09T13:19:59Z","content_type":"application/xml","content_length":"224270","record_id":"<urn:uuid:c6b54147-d6bb-4627-a47b-e183597ab95d>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00513.warc.gz"} |
Continuum Limit of Lipschitz Learning on Graphs
Roith T, Bungert L (2022)
Publication Type: Journal article
Publication year: 2022
DOI: 10.1007/s10208-022-09557-9
Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying
continuum limits, for example, of differential operators. A popular strategy here is p-Laplacian learning, which poses a smoothness condition on the sought inference function on the set of unlabeled
data. For p< ∞ continuum limits of this approach were studied using tools from Γ-convergence. For the case p= ∞, which is referred to as Lipschitz learning, continuum limits of the related infinity
Laplacian equation were studied using the concept of viscosity solutions. In this work, we prove continuum limits of Lipschitz learning using Γ-convergence. In particular, we define a sequence of
functionals which approximate the largest local Lipschitz constant of a graph function and prove Γ-convergence in the L^∞-topology to the supremum norm of the gradient as the graph becomes denser.
Furthermore, we show compactness of the functionals which implies convergence of minimizers. In our analysis we allow a varying set of labeled data which converges to a general closed set in the
Hausdorff distance. We apply our results to nonlinear ground states, i.e., minimizers with constrained L^p-norm, and, as a by-product, prove convergence of graph distance functions to geodesic
distance functions.
Authors with CRIS profile
Involved external institutions
How to cite
Roith, T., & Bungert, L. (2022). Continuum Limit of Lipschitz Learning on Graphs. Foundations of Computational Mathematics. https://doi.org/10.1007/s10208-022-09557-9
Roith, Tim, and Leon Bungert. "Continuum Limit of Lipschitz Learning on Graphs." Foundations of Computational Mathematics (2022).
BibTeX: Download | {"url":"https://cris.fau.de/publications/268924956/","timestamp":"2024-11-04T02:59:26Z","content_type":"text/html","content_length":"10065","record_id":"<urn:uuid:1983df50-3b7a-4232-b49f-79ec2545fe15>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00039.warc.gz"} |
Book Notes: Algorithm to Live By
I want to start sharing raw notes of interesting books I read.
I highly recommend this book, a lot of interesting insights. How I take notes might not make sense to most people. I take notes of keywords and interesting headlines. I'd deep dive into the keyword
if I need to go back and learn more about it.
My favorite chapters are the first (Optimal Stopping) and the last (Game Theory).
Optimal Stopping
• Multi-armed bandit problem
• A/B testing.
• Linearithmic (n log n).
• Bucket sort.
• Sort is prophylaxis to search.
• Noisy comparison.
• Eviction algo: FIFO, LRU.
• Shortest task first.
• Single-machine scheduling.
• Due date - maximum lateness.
• Earliest due date.
• Moore's algorithm.
• The sum of completion time, and shortest processing time.
• Weighted completion time.
• DoS attack.
• Priority inversion - priority inheritance, precedent constraint.
• Intractable.
• Pre-emptive and uncertainty.
• Thrashing.
• Interrupt coalescing.
Bayes' Rule
• Multiply prior probabilities.
• Predict future probabilities.
• Lifespans.
• Kapernican principle.
• Richer prior information - better prediction.
• Gaussian distribution/bell curve.
• Power law distribution / scale-free distribution.
• Good instinct when each distribution applies.
• Erlang distribution - predict a constant time longer.
• Prediction rule: multiplicative, average, additive.
• Small data is big data in disguise.
• Protect good priors.
• Moral algebra.
• Deliberately thinking less.
• Try to explain the past and predict the future.
• Idolatry of data.
• Incentive structure in business - the company builds whatever the CEO decides to build.
• Detecting overfitting: cross-validation.
• Involve less data.
• Regularization.- penalty for complexity.
• The lasso.
• Early stopping.
• When to stop? Broad strokes vs fine lines.
• Use judgment.
• Traveling salesman problem.
• Minimum spanning tree as starting point to find an optimal solution.
• Solve a simplified version of the problem to find a starting point for a solution to the real problem.
• Continuous relaxation.
• Lagrangian relaxation, put a cost to breaking constraints in discreet problems - softening hard constraints.
• Good approximate answer faster.
• Need to know when, in what way, and to what extent.
• Sampling.
• Monte Carlo method.
• How to find prime numbers - cryptography.
• False positive - witness against primality.
• Miller-Rabin Test.
• Polynomial identity testing.
• Veil of ignorance.
• Bloom filter - check URLs for malicious sites - uniqueness witness check.
• Greedy, hill-climbing algorithms, shutdown hill-climbing, metropolis algorithm - avoid local maxima.
• Annealing - how quickly or slowly a material cooled.
• Simulated annealing.
• TCP.
• Packet switching.
• Two generals problem.
• Triple-handshake.
• ACKs package.
• Set period of non-responsiveness.
• Exponential back-off.
• Aloha-net.
• Flow control & Congestion avoidance.
• Full/slow meta-communicate how fast packages should be sent.
• AIMD.
• Ant colony has this too.
• Any employees tend to rise to their level of incompetence.
• Up or out policy.
• Manage capacity.
• Dynamic hierarchy in the company.
• Back channels - ACKs and feedback/nods.
• Buffer bloat - ack packages get stuck in the buffer.
• Tail drop - packages get deleted.
• Explicit congestion notification ECN.
• Consider time as a first-class citizen.
Game Theory
• Recursion.
• Anticipate others' opinions.
• Leveling. Play 1 level above your opponent. Use game theory to break recursion.
• Equilibrium.
• John Nash - Nobel prize, Nash Equilibrium - always exists in 2 players' games.
• Truth - math, complexity - computer science.
• Algorithmic game theory - find Nash equilibrium.
• The prisoners' dilemma.
• Dominant strategy.
• The price of anarchy - maximize the outcome for an individual.
• Selfish routing in network - low price of anarchy.
• Anarchy is 4/3 worse than the ideal cooperative solution.
• Unlimited vacation game theory.
• Mechanism design / reverse game theory - worsening the unsatisfactory equilibrium. Example: stock market closed.
• Cooperation - emotions.
• Auctions.
□ Sealed bid first price auction.
□ Dutch auction / descending auction.
□ English / ascending auction.
□ Vickrey auction - sealed bid second price - strategy-proof.
• Revelation principle.
Conclusion: Computational kindness. | {"url":"https://www.amudi.org/book-notes-algorithm-to-live-by/","timestamp":"2024-11-05T09:05:36Z","content_type":"text/html","content_length":"20478","record_id":"<urn:uuid:58d4e8db-d3e3-4b54-860e-1e041b36c3c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00573.warc.gz"} |
Least Common Multiple
The least common multiple of two numbers and is denoted or and can be obtained by finding the Prime factorization of each
where the s are all Prime Factors of and , and if does not occur in one factorization, then the corresponding exponent is 0. The least common multiple is then
Let be a common multiple of and so that
Write and , where and are Relatively Prime by definition of the Greatest Common Divisor. Then , and from the Division Lemma (given that is Divisible by and ), we have is Divisible by , so
The smallest is given by ,
The LCM is Idempotent
and satisfies the Absorption Law
It is also true that
See also Greatest Common Divisor, Mangoldt Function, Relatively Prime
Guy, R. K. ``Density of a Sequence with L.C.M. of Each Pair Less than .'' §E2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 200-201, 1994.
© 1996-9 Eric W. Weisstein | {"url":"http://drhuang.com/science/mathematics/math%20word/math/l/l137.htm","timestamp":"2024-11-14T22:07:34Z","content_type":"text/html","content_length":"14924","record_id":"<urn:uuid:4d791268-1863-4982-8e62-38e0b0d6c055>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00614.warc.gz"} |
Overflow arithmetic vs saturating arithmetic vs regular arithmetic
Working on optimizing search for a byte in a buffer for [stdlib] Add to Unsafe[Mutable]RawBufferPointer implementation of _custom[Last]IndexOfEquatableElement return to blog post Finding Bytes in
Arrays. Now I bookmark it here to save search time when I will need it next time.
Working on above mentioned PR I came across &+ and alike operations. I wrongly call these operations saturating operations (saturating arithmetic). Wiki: Saturation arithmetic. Saturation arithmetic
clamps result to minimum and maximum values.
Typically, general-purpose microprocessors do not implement integer arithmetic operations using saturation arithmetic; instead, they use the easier-to-implement modular arithmetic, in which
values exceeding the maximum value “wrap around” to the minimum value, like the hours on a clock passing from 12 to 1.
This is exactly what Swift implements - Overflow Operators. | {"url":"https://valeriyvan.com/2023/03/03/TIL.html","timestamp":"2024-11-07T00:03:38Z","content_type":"text/html","content_length":"8545","record_id":"<urn:uuid:7e4f6ea0-80e9-4e13-91e2-453a56eef65a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00587.warc.gz"} |
Telephoto zoom lenses
Applet: Andrew Adams
Text: Marc Levoy
If long focal length lenses were built using a single thin lens, with object and image distances given by the Gaussian lens formula, then a 250mm lens focused on a subject 1 meter away would need to
be placed 333mm (13 inches) from the sensor. But you can buy a Tamron 18-250mm zoom lens that, even when extended to 250mm focal length, measures only 6 inches long. How is this possible? The secret
lies in a clever arrangement of convex and concave lenses that together are called a telephoto lens. Since many telephoto lenses also let you change their focal length, including the Tamron product
just mentioned, it's worth folding that functionality into our explanation. Formally, the Tamron is called a telephoto zoom lens.
How a zoom lens works
Start by clicking on the "Close-up Filter" check box. A close-up filter is a weak convex lens you can attach to the end of any lens, by screwing it into the filter threads. Its purpose is to shorten
the object distance, for example to turn a regular lens into a macro lens without the need to buy a separate macro lens. We're going to use a close-up filter here so that the in-focus plane (where
the blue and red bundles of rays individually converge to two points in object space) fits inside our applet frame. For the rest of this discussion, try to pretend that this filter doesn't exist; it
just makes the visualization easier to understand.
Now click on the "Equivalent Thin Lens" check box. A thin green lens should appear. Imagine that your long focal length lens consists solely of this lens. For the moment, ignore the two lenses to its
right. The red bundle of rays start from the in-focus plane on the left edge of the applet, diverge for a while, pass through the green lens (remember that we're ignoring the close-up filter), then
bend and follow the green lines, reconverging at the red circle on the sensor (vertical gray bar). The blue bundle of rays does the same thing, converging at the blue circle. Since the red and blue
circles lie at the two ends of the sensor, the angle subtended by the central rays of the red and blue bundles where they strike the green lens represents the field of view. To complete our analysis,
the object distance is the distance from the green lens to the left side of the applet, and the image distance is the distance from the green lens to the sensor. The focal length of the green lens is
neither of these distances, but is related to them through the Gaussian lens formula.
Try moving the focal length slider. This changes the focal length of the green lens. Note that it gets thicker and thinner as you do this, reflecting what would be required to actually change the
focal length of a single-lens system like this. As the focal length increases, the field of view (angle between the red and blue bundles) decreases, as you would expect. You can also move the sensor
size slider to change the field of view. This arrangement is called a zoom lens. Here's where it gets interesting. Notice that as you adjust the focal length, the applet keeps the in-focus object
plane and the sensor stationary. We do this by solving a system of two simultaneous equations: (1) the Gaussian lens formula, with the focal length fixed at the value you set using the slider, (2)
the sum of object distance and image distance must equal the distance from the left edge of the applet to the sensor, which is fixed by the design of the applet. This arrangement, where the optics
stays focused at the same object distance (a.k.a. subject distance) while you change the focal length, is called an optically compensated zoom lens.
How a telephoto zoom lens works
One problem with this design is that the green lens is far from the sensor. If built this way, it would yield a physically long lens, as explained in the introduction. Another problem, of course, is
that there's no way to make glass lenses change shape (get thicker and thinner) once they've been fabricated. To address both problems, we move to a multi-element design. Unclick the "Equivalent Thin
Lens" check box. Now the red and blue bundles continue spreading out as they pass the place where the green lens was, strike the convex lens, bend inwards towards the optical axis (central horizontal
line), strike the concave lens, and bend outwards again, converging to the sensor at the same points struck by the green rays. In other words, these two optical arrangements - the green lens alone or
the convex-concave lens combination - have the same effective focal length. As a result, they make the same picture. Why would you prefer the second arrangement over the first? Look how much closer
the convex-concave lens combination is to the sensor than the green lens was. This is a more compact design. It's called a telephoto lens.
Try changing the focal length. The two lenses move, and the field of view changes. So it's a telephoto zoom lens. But the in-focus object plane and sensor also remain stationary. So it's a
optically compensated telephoto zoom lens. It's interesting to see how the two lenses move; they don't move together. Explaining how we compute their motion is beyond the scope of this applet; we
do it using ray transfer matrices. Briefly, any system of thin lenses and air gaps can be modeled as a 2 x 2 matrix describing how that system bends and shifts rays of light. By constructing and
equating the matrices for an ideal thin lens and a telephoto zoom lens system, we can derive equations that make one system optically equivalent to the other. In a commerical lens these motions
are encoded into curved slots in the sides of the lens barrel, as suggested by the patent application drawing at left.
Finally, try moving the "Focus" slider. Now the location of the in-focus plane changes in object space; it is no longer fixed at the left edge of the applet. Look how the two lenses move; this time
they do move together. More slots in the lens barrel. By the way, this is not the only possible design for a telephoto zoom lens. In fact most commercial lenses have many more lens elements. However,
our applet gives the basics, and to our knowledge you can't make a simpler arrangement than the one we've shown here.
Questions or Comments? Please e-mail us.
© 2010; Marc Levoy
Last update: February 29, 2012 10:59:48 PM
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Nearest neighbour graph of datapoints sampled from the swiss roll.
Manifold learning
Elodie Maignant, Xavier Pennec, Alain Trouvé
In this work, we review popular manifold learning methods relying on the notion of neighbour graph (Laplacian Eigenmaps, Isomap, LLE, t-SNE) with a particular focus on Locally Linear Embedding (LLE).
This includes:
– A deeper analysis of the different methods [hal-03909427].
– Comparative studies on symptomatic and shape analysis examples.
– The generalisation of the methods to manifold-valued data.
Surface visualization of the permutation type. For each regions, we considered the rate at which the optimal graph matching permutes a regions with any other region (i.e. not a self permutation)
[left column], with a region that is not a 1st degree neighbour [middle column], and with a region that is not a 1st or 2nd degree neighbour [right column]. Warmer and colder colours indicate a
higher and lower permutation rates, respectively. Gray indicates no permutation. The three rows correspond to a brain parcellation with 100, 300, and 1000 regions. Our results suggest that the
optimal graph alignment of regions should be performed before further analysis but optimal permutations are in the great majority local within 1st or second degree neighbours, which drastically
simplifies the algorithmic search.
Graph Alignment Exploiting the Spatial Organisation Improves the Similarity of Brain Networks
Anna Calissano, Samuel Deslauriers-Gauthier, Theodore Papadopoulo, Xavier Pennec.
Can the structural and functional properties of a specific brain region of an atlas be assumed to be the same across subjects? In [hal-03910761] we addressed this question by looking at the network
representation of the brain, with nodes corresponding to brain regions and edges to their structural relationships. We perform graph matching on a set of control patients and on parcellations of
different granularity to measure the connectivity misalignment between regions. The graph matching is unsupervised and reveals interesting insight on local misalignment of brain regions
across subjects as shown in the figure.
Visualization of the Currents Space of Graphs for $K_3$.
The Currents Space of Graphs
James Benn, Anna Calissano, Stephen Marsland, Xavier Pennec.
We defined in [hal-03910825] a novel embedding space for binary undirected graphs, the Currents Spaces of Graphs. We represent simple graphs on n vertices by 1-forms over a complete graph K_n. It is
shown that these 1-forms lie on a hypersphere in the Hilbert space L^2 (\Omega^1_{K_n}) of all 1-forms and the round metric induces a natural distance metric on graphs. The metric itself reduces to a
global spherical Jaccard-type metric which can be computed from edge-list data alone. The structure of the graph space embedding for three vertices is illustrated in the figure.
Lastly, we describe the global structure of the 1-forms representing graphs and how these can be exploited for the statistical analysis of a sample of graphs.
The Measurement and Analysis of Shape: An Application of hydrodynamics and probability theory
James Benn, Stephen Marsland.
A de Rham p-current can be viewed as a map (the current map) between the set of embeddings of a closed p-dimensional manifold into an ambient n-manifold and the set of linear functionals on
differential p-forms. We demonstrate that, for suitably chosen Sobolev topologies on both the space of embeddings and the space of p-forms, the current map is continuously differentiable, with an
image that consists of bounded linear functionals on p-forms. Using the Riesz representation theorem we prove that each p-current can be represented by a unique co-exact differential form that has a
particular interpretation depending on p.
Embeddings of a manifold can be thought of as shapes with a prescribed topology. Our analysis of the current map provides us with representations of shapes that can be used for the measurement and
statistical analysis of collections of shapes. We consider two special cases of our general analysis and prove that: (1) if p=n-1 then closed, embedded, co-dimension one surfaces are naturally
represented by probability distributions on the ambient manifold; and (2) if p=1 then closed, embedded, one-dimensional curves are naturally represented by fluid flows on the ambient manifold. In
each case we outline some statistical applications using an \dot{H}^{1} and L^{2} metric, respectively. [hal-03556752]
Left: A realization of a phylogenetic tree generated from Brownian motions on the sphere. The green ‘+’ is the root node. The 4 green stars are leaf nodes. The two pink stars are inner nodes. Right:
Histogram of the geodesic distance between the true root r and the estimated root \hat{r}, for 1000 simulated trees on the sphere. For reference, the geodesic distance from the north pole (the true
root) to a point on the equator is \pi/2 \approx 1.57. The histogram shows that the error made by our method in estimating the root-node is low and well behaved in this setting.
Tangent phylogenetic PCA
Morten Pedersen, Stefan Sommer, Xavier Pennec.
Phylogenetic PCA (p-PCA) is a well-known version of PCA for observations that are leaf nodes of a phylogenetic tree. The method works on Euclidean data, but in evolutionary biology there is a need
for applying it to data on manifolds, particularly shape-trees. We provide in [hal-03842847] a generalization of p-PCA to data lying on Riemannian manifolds, called Tangent p-PCA. The figure
illustrates step 1 of our method, consisting of estimating the unknown root node of the phylogenetic tree.
Sphere dataset (black points) interpolated using Gaussian Process Regression (cyan) and Wrapped Gaussian Process Regression (blue).
Lower and upper bounds of the sectional curvature of the Mixed-Power-Euclidean metrics.
Geometries of covariance and correlation matrices
Yann Thanwerdas, Xavier Pennec.
In [hal-03414887], we use the principles of deformed metrics and balanced metrics to:
1. introduce Mixed-Power-Euclidean metrics which encompass the Euclidean, affine-invariant, log-Euclidean and BKM metrics;
2. relate the MPE metrics to the (u,v)-divergences of Information Geometry;
3. compute the curvature (see bounds on the figure).
In [tel-03698752], we further:
1. introduce convenient metrics on correlation matrices of full rank;
2. characterize the Bures-Wasserstein geodesics on covariance matrices. | {"url":"http://gstats.inria.fr/projects/","timestamp":"2024-11-05T00:04:18Z","content_type":"text/html","content_length":"69571","record_id":"<urn:uuid:82020447-7764-4b4c-a462-eac811f275ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00032.warc.gz"} |
Isosceles Triangle | Lexique de mathématique
Isosceles Triangle
• An isosceles triangle has at least one line of symmetry. If it has more than one, then it is an equilateral triangle.
• A non-equilateral isosceles triangle has two congruent angles, which are opposite its congruent sides.If its three angles are congruent, then it is equilateral.
In the representation of an isosceles triangle, the congruent sides are often marked with hash marks to make them easier to identify and discuss. | {"url":"https://lexique.netmath.ca/en/isosceles-triangle/","timestamp":"2024-11-05T03:16:33Z","content_type":"text/html","content_length":"64455","record_id":"<urn:uuid:0bd4ef89-a84e-4f80-9a16-4ee037db2f31>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00399.warc.gz"} |
What is a Model?
419 Models visible to you, out of a total of 666
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG upon sequential adition of purified enzymes. If the Mathematica notebook is downloaded and the data file
is downloaded in the same directory, then the notebook can be evaluated, and the figure in the manuscript for the progress curves will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG upon sequential adition of purified enzymes. If the Mathematica notebook is downloaded and the data file
is downloaded in the same directory, then the notebook can be evaluated, and the figure in the manuscript for the progress curves will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG upon sequential adition of purified enzymes. If the Mathematica notebook is downloaded and the data file
is downloaded in the same directory, then the notebook can be evaluated, and the figure in the manuscript for the progress curves will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG upon sequential adition of purified enzymes. If the Mathematica notebook is downloaded and the data file
is downloaded in the same directory, then the notebook can be evaluated, and the figure in the manuscript for the progress curves will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG upon sequential adition of purified enzymes. If the Mathematica notebook is downloaded and the data file
is downloaded in the same directory, then the notebook can be evaluated, and the figure in the manuscript for the progress curves will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG in cell free extract. If the Mathematica notebook is downloaded and the data file is downloaded in the
same directory, then the notebook can be evaluated, and the figure in the manuscript for the cell free extract with added Mn, but no NAD rec, will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG. If the Mathematica notebook is downloaded and the data file is downloaded in the same directory, then
the notebook can be evaluated, and the figure in the manuscript for cascade 12 will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG in cell free extract. If the Mathematica notebook is downloaded and the data file is downloaded in the
same directory, then the notebook can be evaluated, and the figure in the manuscript for the cell free extract with no added Mn, but with NAD rec, will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Steady state model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG. Protein levels need to be adapted to CFE levels, see SED-ML scripts.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Ordinary differential equations (ODE)
Model format: SBML
Environment: JWS Online
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG, with NAD recycling. If the Mathematica notebook is downloaded and the data file is downloaded in the
same directory, then the notebook can be evaluated, and the figure in the manuscript for cascade 13 will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG. If the Mathematica notebook is downloaded and the data file is downloaded in the same directory, then
the notebook can be evaluated, and the figure in the manuscript for cascade 10 will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG, using old enzymes, with optimal protein distribution. If the Mathematica notebook is downloaded and the
data file is downloaded in the same directory, then the notebook can be evaluated, and the figure in the manuscript for cascade 16 will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Ordinary differential equations (ODE)
Model format: SBML
Environment: JWS Online
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG in cell free extract. If the Mathematica notebook is downloaded and the data file is downloaded in the
same directory, then the notebook can be evaluated, and the figure in the manuscript for the cell free extract with added Mn and NAD rec will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG. Protein levels need to be adapted to CFE levels, see SED-ML scripts
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Ordinary differential equations (ODE)
Model format: SBML
Environment: JWS Online
Model for the Caulobacter crescentus α-ketoglutarate semialdehyde dehydrogenase, describing the initial rate kinetics for substrate dependence and product inhibition. If the Mathematica notebook is
downloaded and the data file for the XAD kinetics is downloaded in the same directory, then the notebook can be evaluated. The model in the notebook will then be parameterised and the figures in the
manuscript for KGSADH will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus Weimberg pathway, describing the conversion of Xyl to KG, with sequential addition of purified enzymes.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Ordinary differential equations (ODE)
Model format: SBML
Environment: JWS Online
Model for the Caulobacter crescentus xylose dehydrogenase, describing the initial rate kinetics including substrate dependence and product inhibition. If the Mathematica notebook is downloaded and
the data file for the XDH kinetics is downloaded in the same directory, then the notebook can be evaluated. The model in the notebook will then be parameterised and the figures in the manuscript for
XDH will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus xylonolactonase, describing the initial rate kinetics and substrate dependence. If the Mathematica notebook is downloaded and the data file for the XLA kinetics
is downloaded in the same directory, then the notebook can be evaluated. The model in the notebook will then be parameterised and the figures in the manuscript for XLA will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus xylonate dehydratase, describing the initial rate kinetics for substrate dependence. If the Mathematica notebook is downloaded and the data file for the XAD
kinetics is downloaded in the same directory, then the notebook can be evaluated. The model in the notebook will then be parameterised and the figures in the manuscript for XAD will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Model for the Caulobacter crescentus 2-keto-3-deoxy-D-xylonate dehydratase, describing the initial rate kinetics for substrate dependence and product inhibition. If the Mathematica notebook is
downloaded and the data file for the XAD kinetics is downloaded in the same directory, then the notebook can be evaluated. The model in the notebook will then be parameterised and the figures in the
manuscript for KDXD will be reproduced.
Creator: Jacky Snoep
Submitter: Jacky Snoep
Model type: Algebraic equations
Model format: Mathematica
Environment: Mathematica
Framework Model for Arabidopsis vegetative growth, version 2 (FMv2), as described in Chew et al. bioRxiv 2017 (https://doi.org/10.1101/105437; please see linked Article file).
The FMv2 model record on FAIRDOMHub has the following versions, which represent the same FMv2 model: Version 1 is an archive of the github repository of MATLAB code for the Framework Model v2,
downloaded from https://github.com/danielseaton/frameworkmodel on 06/02/17. This version was not licensed for further use and was ...
From published files, Uriel Urquiza created SBML models with all 8 parameter sets published, and versions of F2014.1 to simulate multiple clock mutants, using SloppyCell
F2014.1.2 SBML file including Stepfunction imported back into Copasi v4.8
Creators: Andrew Millar, Karl Fogelmark, Carl Troein
Submitter: Andrew Millar
Model type: Ordinary differential equations (ODE)
Model format: Copasi
Environment: Copasi
F2014.1.1 becomes the published version, with SBML file originally created from SloppyCell by Uriel Urquiza - see separate file. then Andrew Millar converted into SBML L2V4 in Copasi and added ISSF
for light input, using SBSI Stepfunction editor (see Adams et al. 2011 J Biol Rhythms).
Creators: Andrew Millar, Karl Fogelmark, Carl Troein
Submitter: Andrew Millar
Model type: Ordinary differential equations (ODE)
Model format: SBML
Environment: Not specified
Simplified model file for PLaSMo accession ID PLM_71, version 2 (use simplified if your software cannot read the file, e.g. Sloppy Cell)
Originally submitted model file for PLaSMo accession ID PLM_71, version 2
Simplified model file for PLaSMo accession ID PLM_71, version 1 (use simplified if your software cannot read the file, e.g. Sloppy Cell)
Originally submitted model file for PLaSMo accession ID PLM_71, version 1
Originally submitted model file for PLaSMo accession ID PLM_1041, version 1 | {"url":"https://fairdomhub.org/models?page=5","timestamp":"2024-11-03T12:13:18Z","content_type":"text/html","content_length":"267321","record_id":"<urn:uuid:1649e42d-8d94-4102-9e07-f650e18284bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00622.warc.gz"} |
This course includes the study of first order differential equations, higher order linear
differential equations, Laplace transforms, numerical methods, boundary value and initial value
problems, qualitative analysis of solutions, and applications of differential in solving engineering problems.
Mathematics I
Learning Outcomes
Differential equations will provide students with the needed working knowledge of advanced mathematical concepts and an awareness of their relationship to complex problems. Students wishing to major
in the sciences or engineering are required to study differential equations. It provides a solid foundation for further study in mathematics, the sciences, and
Differential equation; basic concepts and ideas; geometrical interpretation of first and second order differential equations (D.E),Separable equations, Reducible to Separable form, Exact D. E,
integrated factors, Linear first order differential equations, Bernoulli’s differential equation. Families of curves, orthogonal trajectories and applications of differential equations of first order
to relevant engineering systems. Homogeneous linear differential equations of second order, homogeneous equations with constant coefficients, the general solutions, Initial and boundary value
problems, D- operator, complementary functions and particular integrals. Real, complex and repeated roots of characteristics equations. Cauchy equation, non-homogeneous linear equations. Applications
of higher order linear differential equations. Ordinary and regular points and corresponding series solutions. Concept of Sequence and Series
Text Book
Erwin Kreyszig, “Advanced Engineering Mathematics 10th Edition”, John Wiley & Son
Recommended Books:
1. C.R. Wylie, “Advanced Engineering Mathematics 6th Edition”, McGraw- Hill Education
2. Erwin Kreyszig, “Advanced Engineering Mathematics 10th Edition”, John Wiley & Son
Sessional:20 %(Assignments 5% , Quiz 5% ,Class Attendance/Class Participation/Presentation 10%)
Mid Term Paper:30%
Final Term: 50 %
Key Dates anf Time of Class Meeting
Friday 11:00 AM to 1:30 PM
Commencement of Classes March 02, 2020
Mid Term Examination April 27 to May 04 , 2020
Final Term Examination June 22-26, 2020
Declaration of Result July 3, 2020 | {"url":"https://lms.su.edu.pk/course/838","timestamp":"2024-11-11T21:10:09Z","content_type":"text/html","content_length":"68051","record_id":"<urn:uuid:0ef25269-550e-429c-933b-139a4dd3a3b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00085.warc.gz"} |
Bayesian Estimation of Non-Gaussian Stochastic Volatility Models
Bayesian Estimation of Non-Gaussian Stochastic Volatility Models ()
Keywords:Non-Gaussian Distribution; Stochastic Volatility; Laplace Density; Fat Tails; Kullback Leiber Divengence; Bayesian Analysis; MCMC Algorithm
1. Introduction
The Stochastic Volatility models have been widely used to model a changing variance of time series Financial data [1,2]. These models usually assume Gaussian distribution for asset returns
conditional on the latent volatility. However, it has been pointed out in many empirical studies that daily asset returns have heavier tails than those of normal distribution. To account for heavy
tails observed in returns series, [3] proposes a SV model with student-t-errors. This density, although considered as the most popular basic model to account for heavier tailed returns, has been
found insufficient to express the tail fatness of returns.
[4] fitted a student-t-distribution and a Generalized Error Distribution (GED) as well as a normal distribution to the error distribution in the SV model by using the simulated maximum likelihood
method developed by [5,6]. [7] considered a mixture of normal distribution as the error distribution in the SV model. He used a bayesian method via MCMC technique to estimate the model’s parameters.
According to Bayes factors, he found that the t-distribution fits the Tokyo Index Return better than the normal, the GED and the normal mixture. However the mixture of normal distributions gives a
better fit to the Yen/Dollar exchange rate than other models.
This survey of literature proves that we can’t affirm absolutely that one distribution is better than another one. The selection of a density should be based on other parameters. In our work, we
consider a general model of nonGaussian centered error distribution. We prove that the efficiency of a specification of SV model depends on the dispersion of the data base. In fact, we find that when
the data base is very dispersed, the Gaussian specification behaves better than the non-Gaussian one. On the contrary, if the data base presents a little dispersion measure, the non-Gaussian centered
error specification will behave better than the Gaussian one. For this reason, we propose a general SV model where the diffusion of the stock return follows a non-Gaussian distribution.
Since it is not easy to derive the exact likelihood function in the framework of SV models, many methods are proposed in the literature to estimate these models. The four major approaches are: 1) the
Bayesian Markov Chain Monte Carlo (MCMC) technique suggested by [8], 2) the efficient method of moments EMM proposed by [9], 3) the Monte Carlo likelihood MCL method developed by [10], and 4) the
efficient importance sampling EIS method of [11]. In this work, we consider the MCMC method for the estimation of the model’s parameters.
The rest of the paper is organized as follow: in the second section, we present, in a comparative setting, the usual Gaussian and the general non-Gaussian SV models. Bayesian parameter estimator’s
and the MCMC algorithm are described in the third section. The fourth section develops an application of a non-Gaussian centered error density. In particular, we consider the Laplace density as an
example of non-Gaussian distribution for the data base that we have studied. We conclude in the last section.
2. Stochastic Volatility Model with Gaussian/Non-Gaussian Noise
As any nature field, Finance has adopted a simple model, developed over the years, that attempts to describe the behaviour of random time fluctuation in the prices of stocks observed in the markets.
This model assumes that the fluctuation of the stock prices follow a log-normal probability distribution function. The simple log-normal assumption would predict a Gaussian distribution for the
returns with variance growing linearly with the time lag. What is actually found is that the probability distribution for high frequency data usually deviates from normality presenting heavy tails.
In this section, we present the classical Gaussian SV model and we introduce a non-Gaussian centered error distribution as an extension to the SV models.
2.1. Gaussian Stochastic Volatility Model
The log Stochastic Volatility model is composed of a latent volatility equation and an observed return equation
According to the Euler discretization schema, we get
In order to consider a more general case of SV model, we propose in the next section, a non-Gaussian centered error distribution for
2.2. Non-Gaussian Stochastic Volatility Model
We consider a Stochastic Volatility model with a non-Gaussian noise where the return
The innovation term in the return equation [3], Generalized Error Distribution [4], Mixture of Normal distribution [7], [12], Laplace distribution, Uniform distribution....Among these nonGaussian
centered error density, we have chosen the Laplace one for being applied to SV model. We estimate the model’s parameters and we apply it to tha CAC 40 index returns data.
3. Bayesian Estimation of Non-Gaussian Stochastic Volatility Model
A long standing difficulty for applications based on SV models was that the models were hard to estimate efficiently due to the latency of the volatility state variable. The task is to carry out
inference based on a sequence of returns [13] uses the method of moments to calibrate the discrete time SV models. [14] improves the inference as they exploit the generalized method of moments
procedure (GMM). The Kalman filter was used by [15].
Recently, simulation based on inference was developed and applied to SV models. Two approaches were brought forward. The first was the application of Markov Chain Monte Carlo (MCMC) technique;
([8,16-18]). The second was the development of indirect inference or the so-colled Efficient Method of Moments; ([19-21]).
In our paper, we have chosen the bayesian MCMC approach for the estimation of the parameter’s and the volatility vector.
For the non-Gaussian SV model, we define the parameter set
We can obtain the joint distribution
[17] assume conjugate priors for the parameters
With these posterior densities, the Gibbs sampler is applied and we get a Markov Chain for each parameter and thus the parameter’s estimator. The only difficult step arises in updating the volatility
states. According to [8] the full joint posterior for the volatility is
as a function of
The simulation of the posterior density of the parameters requires the application of the Gibbs Sampler. However, we prove explicit expression for the simulated parameters. In fact, after some simple
calculations applied to the posterior density, we find the following expression for the iterated parameters by the Bayesian method in the
For the second parameter
For the parameter
We have considered some statistical model distributions characterized by different density for the noise terms. For each density listed in the second section (Student, Laplace, Uniform), we can
formulate one particular specification for SV model. We have conducted a Chi-deux test to select (among these three densities) the appropriate error distribution to our data base. The results
indicate that the Laplace density is the suitable distribution compared to the student and the uniform ones. In the second step, we will take the Laplace SV model such as a particular case of
non-Gaussian SV model to be compared to the standard Gaussian distribution model that is the Normal one. The next section will present the application study.
3.1. Application
In this section, we consider one particular case of non-Gaussian SV model that is the Laplace one. In fact, the application of the Chi-deux test has proved that this model is the more appropriate to
our data base.
The study of this density is very interesting. In fact, Laplace density has been often used for modeling phenomena with heavier than normal tails for growth rates of diverse processes such as annual
gross domestic product [22], stock prices [23], interest or foreign currency exchange rates [24,25], and other processes [26]. However, the Laplace density is not yet explored in the context of
Stochastic Volatility model.
The Laplace SV model is defined as:
The innovation term in the return equation
The probability density function of a Laplace density is expressed as:
In order to prove the efficiency of the Laplace model, we conduct a simulation analysis.
Simulation Analysis
This section illustrates our estimation procedure using the simulated data. We generate 1000 observations from the Stochastic Volatility Laplace (SVL) model given by equation (3.12), with true
We draw 6000 posterior samples of MCMC run. We discard the initial Table 1 gives the parameter’s estimates for posterior means and standard deviation. Comparison of the Mean Squared Errors calculated
with the Normal Stochastic Volatility model are larger than those calculated with the Laplace Stochastic Volatility model. This result proves that the Laplace Stochastic Volatility model better fits
the model’s parameters and state variables.
The second step in our simulation study consists of testing the hypothesis that the choice of a specification is based on the dispersion measure of the data base. So, we simulate data bases with
different variances. Results prove that when the data base is characterized by a little dispersion measure, the Laplace SV model performs a better specification for the parameters. In fact, the
calculation of the Mean Squared Errors for the Laplace and the Normal model prove that this measure is greater for the Gaussian model. So, we should consider the estimators deduced from the Laplace
(non-Gaussian) specification.
On the contrary, when the data base is very dispersed, it has been proved that the Gaussian SV model offers more precise estimation for the parameters.
From this simulation study, we can conclude that the selection of a model specification depends on data base characteristics. If we reject the normality assumption for a data base, we can accept it
for another one.
Table 1. Simulation results for the Laplace and the Normal models.
Notes: 1. This table provides a summary of the simulation results for the Laplace and the Normal model. 1000 observations were simulated off the true parameters. We report in the third and the fourth
column the average (mean(1)) and the standard deviation (Stdev.(1)) of each parameter calculated with 6000 simulated path. After discarding a burn in period of 2000 iterations, we compute the mean
(mean(2)), the standard deviation (Stdev.(2)) and the mean squared error for each parameter. Results are presented in the three last columns. We present the confidence interval between brackets. 2.
MSE for a parameter
3.2. Empirical Application
After approving our methodology using simulated data, we apply our MCMC estimation method to daily stock returns data. On our work, we focus on the study of the French stock market index: the CAC40
index returns. The sample size is 5240 observations. The log difference returns are computed as
Table 2 summarizes the descriptive statistics of the returns data. The series reveal negative skewness proving the asymmetry of the return distribution. The Kurtosis equal 24.86. The series present a
large Kurtosis
Figure 1 shows the data histogram, the Normal and the Laplace one. It appears very clear that the distribution of our database is very close to the Laplace distribution, especially in the tails. In
fact, when the Normal density ignores the queue observations, Laplace histogram represents these points so, the later density is more representative to our data then the former one.
In Table 3, we present the KL divergence that calculates the distance between the true (empirical) and the estimated distribution (Laplace, Normal). When the value of this criterion is large, the
estimated distribution differs significatively from the true distribution. On the contrary, when this measure is small, we affirm that the estimated distribution is similar to the true one. For our
data base, the KL divergence equals 0.0656 between data density and Laplace density. It equals 1.1653 between data density and Normal density. This proves that the non-Gaussian specification is more
accurate to the data considered than the standard Gaussian specification.
3.3. Estimation Results
In the last section, we have rejected the Gaussian assumption for the return series of the CAC40 index.
Table 2. Summary statistics for daily return data on the CAC40 from January 2, 1987 to November 30, 2007.
Note: This table provides summary statistics for daily return data on the CAC40 French Stoch Exchange index from January 2, 1987 to November 30, 2007.
Figure 1. Database, Laplace, student and normal histogram. Note: The histograms of this figure are obtained with Microsoft excel program. For the first histogram, that represents the database
density, we have classified our observation by class, we have computed the frequency in each class and the probability of each observation, and then we have represented the histogram. For the
Laplace, Student and Normal density, we have generated. With Matlab, random vectors that follow each distribution with parameters (mean, variance) of the true database. Then, we compute the frequency
and the probability of each observation and we represent the histogram with Microsoft excel.
Therefore, we have proven that the Laplace density is more consistent to our data base. In this section, we will apply the Laplace stochastic volatility model, that we have introduced in Equation
(12), to the analysis of the CAC40 index returns. The number of MCMC iterations is 10000 and the initial 2000 samples are discarded. Table 4 reports the estimation results: the posterior means,
standard deviations, and the Mean Squared Errors for CAC40 data.
The estimates of the volatility parameters Table 4 presents, also, the MSE calculated for each parameter for the Laplace and the Normal model. It seems clear that the Laplace model generates the
little errors, except for the
In order to test the ability of the Laplace model to predict future returns, we perform an out of sample analysis. We consider 4000 observations for the inference of parameters. We simulate (5240 -
4000) artificial observations from the Laplace model and the Normal one. We compare each simulated vector of returns with the remainder observations.
In Table 5, we present the MSE found between the true observation vector and the returns generated with the Normal model in the third column. The second column shows the MSE calculated between true
observations and returns vectors generated with the Laplace model. It seems clear that the Laplace model predict returns more accurately than the Normal one. In fact, the MSE obtained with the
Laplace model is smaller than that obtained with the Normal model. The Laplace SV model generates a Mean Squared Error less than the errors generated
Note: This table summarizes the Kullback Leiber Divergence calculated between the true density and the estimated density: non-Gaussian (Laplace) in the second row, Gaussian (Normal) in the last row.
Table 4. Parameter estimates for the CAC40 index return data.
Note: Parameter estimates for the CAC40 index data from January 2, 1987 to November 30, 2007. For each parameter, we report the mean of the posterior density, the standard deviation of the posterior
in parentheses and the MSE. Estimation for the Laplace SV model are presented in the second column. Results for the Normal model are given in the third column. The last row gives the MSE calculated
for the whole model. The first number represents the MSE between estimated returns through Laplace model and the observed returns. The second number represents the MSE for estimated returns with the
normal model and observed returns.
Table 5. Mean squared errors for the out of sample analysis.
Notes:1. In this table we present the out of sample analysis results. We take the first 4000 empirical observations to infer the parameter estimates for the Laplace SV model and the Normal SV model.
We generate (5240 - 4000) artificial observations with the two model and we calculate the MSE between observed remainder returns and the simulated vector of return with Laplace model and with Normal
model. 2. MSE for returns vector is calculated with the following formula:
by the Normal SV model. This result indicates that the Laplace model (non-Gaussian) is able to predict returns better than the Normal one or the standard Gaussian one.
4. Conclusions
In this paper, we have considered the inference of SV model with non-Gaussian noise. By applying a Chi-deux test, we have chosen the suitable non-Gaussian distribution error for the data base
considered in our study among different non-Gaussian distribution that has been considered in last studies (such as: Student, Uniform, Mixture of Normal,
We have performed MCMC technique for the stochastic volatility model when returns follow a Laplace distribution allowing for an important characteristics of returns dynamic: Heavy tails or
An application to daily CAC40 index returns over the years (1987-2007) illustrates the ability of the Laplace model to deal with heavy tails better than the Log-Normal model. An out of sample
analysis proves that the Laplace model better predicts future returns. These results have been reached according to the calculation of the Mean Squared Error calculated between estimated parameters
and true parameters. | {"url":"https://scirp.org/journal/paperinformation?paperid=43033","timestamp":"2024-11-12T09:29:38Z","content_type":"application/xhtml+xml","content_length":"138526","record_id":"<urn:uuid:b01ba0ad-c777-47d5-bc5b-78500612e057>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00010.warc.gz"} |
A.4 – Transportation and Accessibility | The Geography of Transport Systems
A.4 – Transportation and Accessibility
Author: Dr. Jean-Paul Rodrigue
Accessibility is a key element in transport geography and geography in general since it is a direct expression of mobility either in terms of people, freight, or information.
1. Defining Accessibility
Mobility is a choice made by users and is, therefore, a way to evaluate the impacts of infrastructure investment and related transport policies on regional development. Well-developed and efficient
transportation systems offer high accessibility levels, while less-developed ones have lower levels of accessibility. Thus, accessibility is linked with an array of economic and social opportunities,
but congestion can also have a negative impact on mobility.
Accessibility is the measure of the capacity of a location to be reached from, or to be reached by, different locations. Therefore, the capacity and the arrangement of transport infrastructure
are key elements in the determination of accessibility.
All locations are not equal because some are more accessible than others, which implies inequalities. Thus, accessibility is a proxy for spatial inequalities and remains fundamental since only a
small subset of an area is the most accessible. The notion of accessibility relies on two core concepts:
• The first is location, where the relativity of space is estimated in relation to transport infrastructures since they offer the means to support mobility. Each location has a set of referential
attributes, such as its population or level of economic activity.
• The second is distance, which is derived from the physical separation between locations. Distance can only exist when there is a possibility to link two locations through transportation. It
expresses the friction of distance, and the location with the least friction relative to others is likely to be the most accessible. The friction of distance is commonly expressed in units such
as in kilometers or in time, but variables such as cost or energy spent can also be used.
There are two spatial categories applicable to accessibility problems, which are interdependent:
• The first type is topological accessibility, which is related to measuring accessibility in a system of nodes and paths (a transportation network). It is assumed that accessibility is a
measurable attribute significant only to specific elements of a transportation system, such as terminals (airports, ports, or subway stations).
• The second type is contiguous accessibility, which involves measuring accessibility over a surface. Under such conditions, accessibility is a cumulative measure of the attributes of every
location over a predefined distance, as space is considered contiguous. It is also referred to as isochrone accessibility.
Last, accessibility is a good indicator of the underlying spatial structure since it takes into consideration location as well as the inequality conferred by distance to other locations.
Relationship between Distance and Opportunities
Topological and Contiguous Accessibility
Accessibility and Spatial Structure
Global Accessibility: Time to the Nearest Large City
2. Connectivity and Total Accessibility
The most basic accessibility measure involves network connectivity, where a network is represented as a connectivity matrix (C1), which expresses the connectivity of each node with its adjacent
nodes. The number of columns and rows in this matrix is equal to the number of nodes in the network, and a value of 1 is given for each cell where this is a connected pair and a value of 0 for each
cell where there is an unconnected pair. Simple networks and their connectivity matrices are rare. Thus, the matrix becomes exponentially more complex with the number of nodes. The summation of this
matrix provides a very basic measure of accessibility, also known as the degree of a node:
$\large C1 = \displaystyle\sum_{j}^{n} C_{ij}$
• C1 = degree of a node.
• Cij = connectivity between node i and node j (either 1 or 0).
• n = number of nodes.
The connectivity matrix does not consider all the possible indirect paths between nodes. Under such circumstances, two nodes could have the same degree but may have different accessibilities. To
consider this attribute, the Total accessibility matrix (T) is used to calculate the total number of paths in a network, including direct and indirect paths. Its calculation involves the following
$\large T = \displaystyle\sum_{k=1}^{D} Ck ewline$
$\large C1 = \displaystyle\sum_{j}^{n} C_{ij} ewline$
$\large Ck = \displaystyle\sum_{i}^{n} \displaystyle\sum_{j}^{n} c_{ij}^{1} \: \times \: c_{ji}^{k-1} (\forall k eq1)$
• D = the diameter of the network.
Thus, total accessibility would be a more comprehensive accessibility measure than network connectivity.
Creation of a Connectivity Matrix with a Link Table
Simple Connectivity Matrix
More Complex Connectivity Matrix
Total Accessibility Matrix T
3. The Shimbel Index and the Valued Graph
The main focus of measuring accessibility does not necessarily involve measuring the total number of paths between locations but rather the shortest paths between them. Even if several paths between
two locations exist, the shortest one is likely to be selected. In congested networks, the shortest path may change according to the current traffic level in each segment. Consequently, the Shimbel
index calculates the minimum number of paths necessary to connect one node with all the nodes in a defined network. The Shimbel accessibility matrix, also known as the D-Matrix, includes each
possible node pair with the shortest path.
The Shimbel index and its D-Matrix fail to consider that a topological link between two nodes may involve variable distances. Thus, it can be expanded to include the notion of distance, where value
is attributed to each link in the network. The valued graph matrix, or L-Matrix, represents such an attempt. It is very similar to the Shimbel accessibility matrix. The only difference is that
instead of showing the minimal path in each cell, it provides a minimal distance between each node of the network.
Shimbel Distance Matrix (D-Matrix)
Valued Graph Matrix (L-Matrix)
4. Geographic and Potential Accessibility
From the accessibility measure developed so far, it is possible to derive two simple and highly practical measures, defined as geographic and potential accessibility. Geographic accessibility
considers that the accessibility of a location is the summation of all distances between other locations divided by the number of locations. The lower its value, the more accessible a location is.
$\large A(G) = \displaystyle\sum_{i}^{n} \displaystyle\sum_{j}^{n} \frac {d_{ij}}{n} ewline$
$\large d_{ij} = L$
• A(G) = geographical accessibility matrix.
• dij = shortest path distance between location i and j.
• n = number of locations.
• L = valued graph matrix.
This measure (A(G)) is an adaptation of the Shimbel Index and the Valued Graph, where the most accessible place has the lowest summation of distances. Locations can be nodes in a network or cells in
a spatial matrix.
Potential accessibility is a more complex measure than geographic accessibility since it simultaneously includes the concept of distance weighted by the attributes of a location. All locations are
not equal, and thus, some are more important than others. Potential accessibility can be measured as follows:
$\large A(P) = \displaystyle\sum_{i}^{n} P_{i}+ \displaystyle\sum_{j}^{n} \frac {P_{j}}{d_{ij}}$
• A(P) = potential accessibility matrix.
• dij = friction of distance between place i and j (derived from valued graph matrix).
• Pj = attributes of place j, such as population, retailing surface, parking space, etc.
• n = number of locations.
The potential accessibility matrix is not transposable since locations do not have the same attributes, which brings the underlying notions of emissiveness and attractiveness:
• Emissiveness is the capacity to leave a location, the sum of the values of a row in the A(P) matrix.
• Attractiveness is the capacity to reach a location, the sum of the values of a column in the A(P) matrix.
Geographic Accessibility
Potential Accessibility
Although accessibility can be solved using a spreadsheet (or manually for simpler problems), Geographic Information Systems have proven to be a very useful and flexible tool to measure accessibility,
notably over a surface simplified as a matrix (raster representation). This can be done by generating a distance grid for each place and then summing all the grids to form the total summation of
distances (Shimbel) grid. The cell having the lowest value is thus the most accessible location.
Related Topics
• BTS (2001) Special Issue on Methodological Issues in Accessibility, Journal of Transportation and Statistics, Vol. 4, No. 2/3, Bureau of Transportation Statistics, Sept/Dec.
• Burns, L.D. (1979) Transportation, Temporal, and Spatial Components of Accessibility. Lexington, MA: Lexington Books.
• El-Geneidy, A.M., and D.M. Levinson (2006) Access to Destinations: Development of Accessibility Measures. Retrieved from the University of Minnesota Digital Conservancy, https://hdl.handle.net/ | {"url":"https://transportgeography.org/contents/methods/transportation-accessibility/","timestamp":"2024-11-01T19:34:00Z","content_type":"text/html","content_length":"160563","record_id":"<urn:uuid:27959fc7-dac1-48df-9613-e6176f492f9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00882.warc.gz"} |
Is Feminist Epistemology Different Than Male Epistemology?
Posted inPhilosophy
Is Feminist Epistemology Different Than Male Epistemology?
Euclid gave us a gloriously simple proof that there are an infinite number of primes. A prime number, of course, is a positive number that can be evenly divided only by itself or one.
Here’s Euclid’s proof. Don’t worry if you can’t follow along; it’s only important that you understand that the statement, “There are an infinite number of primes” is true given the information
provided in the proof.
Assume there are only a finite number of primes; order them from smallest to largest. Multiply them all together and then add to that product one. For ease, call that product-plus-one, P.
P is clearly larger than the largest prime we know of, because P is the product of the largest prime and all the primes smaller than it. But also, if we take all the primes we know and divide any of
them into P, we will have a remainder of one.
That means that the prime that divides evenly into P must be larger than the prime we thought was the largest. And since you can keep doing this procedure each time you discover a new “largest”
prime, the number of primes must be infinite.
Euclid’s is not the only proof that the number of primes are infinite, but it’s the simplest among all the proofs I know of.
But suppose you find another proof easier to comprehend than Euclid’s; perhaps Dirichlet’s demonstration. And let’s also imagine that your cousin has discovered an entirely new proof.
One day, the three of us meet. We all agree on the truth of the statement “There are an infinite number of primes”, but each of us believe this because of different evidence. We discuss the
differences in the evidence, but are unable to come to the conclusion that our three sets of evidence are equivalent.
By “equivalent”, I mean something like we mean when we say a sentence has identical meaning in a different language. It could be that my evidence is “The ball is blue” and that yours is “L’objet
circulaire est en bleu“, but since I don’t know French and you don’t know English, we cannot decide whether the sentences are equivalent, that they directly translate.
But even though I don’t understand your evidence, I might agree with its truth; that is, I might accept its internal coherence. Or I may simply accept it for the sake of the argument. Or again, I may
not accept its truth at all. Finally, I could understand it as well as I understand my own evidence, but I simply prefer mine to yours.
Thus far, we are on solid ground. But let’s move towards the beach and consider your cousin’s discovery. Your cousin does not prefer either of our arguments, and claims that the third offering allows
deeper understanding of truth of the statement “There are an infinite number of primes”.
Now, we might accept—and here finally arrives the crucial pronoun—her evidence as true, yet feel our evidence is easier to comprehend, or more readily adaptable to new arguments.
Or we might not be convinced of the soundness of her evidence, yet we cannot offer outside proof that her evidence is unsound. Finally, we might claim that her evidence is in fact unsound. Yet we all
agree that “There are an infinite number of primes” is true.
Because of their inherent, incontrovertible genetic differences, women think differently than men. This claim is the basis of “feminist epistemology.” Note very carefully that “differently” in no way
implies “superior to” or “inferior to.”
Let’s accept the feminist claim as true (I think it is true, but that’s irrelevant). But in doing so, to what have we acceded? Merely that some people think differently than others? And that,
thinking differently, people might, as our example attests, weigh evidence asymmetrically, even though they agree on a truth of a proposition?
None of this in the least controversial. But what if your cousin offers evidence which she claims proves “There are an infinite number of primes” is false? Further, she insists that her way of
thinking allows her to understand her evidence in ways that you, being male, cannot.
You might try to prove her evidence is unsound with respect to exterior information, but she might counter those arguments with similar ones about how that exterior information is viewed differently
by females.
All evidence, and all argument, form a linked web, the strands of which eventually hang on the single thread of the a priori, the unproved and unprovable truths from our intuitions. So, she may claim
that, being female, her intuition about axioms is just different.
She might be right. That is, there is no way to prove she is wrong. The only fundamental counter we have that feminist epistemology is no different than male, or even sentient, epistemology is our
belief (provided by our intuitions) that there is only one set of base truths; that every statement is either true or it is false (or nonsensical), but that it cannot be both.
14 Comments
It is evident to anyone with eyes to see that men and women are wired differently, and that is by design. But, as you said, truth is truth, and we know that women and men can either accept truth
or reject it. I find the “feminist” notion that “men do not have the ability to understand me” to be a bit silly.
The difference in “wiring” between the sexes, it seems to me, shows itself quite a bit in the generally more emotional mindset of a woman and the more analytical mindset of a man. Men and women
approach problems differently. Of course, each person, regardless of sex can approach any given situation any number of ways. I’m not talking rocket science here, obviously.
Men and women are counterparts intended to get together in pairs to form a complete union–complete once God is involved, that is. Each person brings features to the relationship unique to his
The marriage of a man and a woman, in a “perfect” world, is a very supportive and efficient thing, indeed. I know this is kind of a rabbit trail away from your article, but it’s just a few
I like your aritcle Mr. Briggs.
Beautiful explanation! WXRGina states a common plaint: That men don’t understand women. Looking at the problem in the aggregate, it is impossible to untangle.
But bring it down to the most fundamental matter of arriving at a logical conclusion based upon a shared truth or conflicting truths, and the problem can be solved. Or not. Maybe you discover an
entirely new question: Whose truth can stand the test of logic–but then epistemological differences rears its head again…
This is like looking in a mirror with a mirror behind you…
Ahhhh! My head hurts! Thanks for kickstarting my brain, Mr. Briggs.
Is epistemology now a synonym for hormones?
‘E piste m’ology? Well, since we’re piste-ing around, my wife saying “He doesn’t understand me” shows a deeper understanding she has for me than I for her.
“The difference in “wiring†between the sexes”
The wiring differences goes deeper then the balance between rational and emotion sides of the brain.
My daughter did some work at brown(unpublished) on visual perception.
They had originally started out trying to determine if people were biased to horizontal or vertical.
So they set up a quite simple experiment drawing checkerboards with vertical and horizontal lines of varying widths and distances from each other and asked a simple question.
Are the vertical lines closer or farther apart or the same as the horizontal lines?
The answers the females gave(the original test subjects, her friends) were in almost total agreement, so they thought they had determined a human bias towards vertical. Being fairly intelligent
she realized her sample group may have similar biases. So she sent me the test. I had a horizontal bias. I’m color blind, so maybe my vision impairment was the reason for a different bias. Then
they randomly sampled the boys on campus, they all had a horizontal bias. Then they randomly sampled the girls, vertical bias.
Of course a study that concludes the differences between male and female extend as far as visual perception will never see the light of day.
Harrywr2, I have horizontal bias myself. It’s extremely acute in the morning. My wife usually gets up before I do. Obvious vertical bias on her part.
Whatever could visual vertical/horizontal bias mean, actually? A tendency toward seeing all lines as vertical/horizontal? Seeing vertical/horizontal patterns in otherwise patternless features?
It’s not clear that even if a visual difference exists that it indicates an actual thinking difference.
“Note very carefully that “differently†in no way implies “superior to†or “inferior to.†” Would that be better as “Note that “differently†need not imply “superior to†or
“inferior to.†“?
dearieme, Brigg’s has fallen into that pc- neologism: different but somehow equal. It think “independent but otherwise equal” is a better concept.
OT: I finally discovered how to change the Foxfire spell checker language. Why it doesn’t use the Firefox language setting is beyond me. For whatever reason, it had concluded I live in Hong Kong.
Obviously, the now inescapable (!) Kantian discursively constructed gender reality is more than linguistic. The Heraclitean version of historicism rejects totalized meta-narratives. Meaning
holism entails dis-unified Marxist standpoint theory which excludes certain possibilities, i.e. social and psychological phenomena in which gender is implicated.
I could go on, but my old lady just made me lunch and she gets whiney if I don’t eat it hot.
Let me play Devils advocate.
1. It doesn’t follow from the assumption of different epistemologies that there is no commonality. ie Feminists might agree with all Mathematics but disagree on say Biology or more likely
2. Think back to Plato’s cave. There is absolute truth and there is perception of truth. That is to say there is an absolute truth but as humans we may be unable to comprehend all truths. We are
in the cave and whilst we can get closer to the entrance we can never get out. Thus our theories about the world are only simplified models of reality. Our models might be “good enough” to be
better than no model but they will never be reality. [That’s my post half a bottle of wine précis of Plato so be forgiving]
If we accept that then by inference we might have two people approaching the cave mouth and coming up with competing models. [eg. light is a wave vs light is a particle.] We might think that men
and women, being physiologically different, are more likely to come up with different models of reality (than say two members of the same sex). Assume further that it is impossible to reconcile
the two models – thus we have two epistemologies.
I think this is bollocks BTW but I doubt any feminist would openly disagree with Euclid. They might waffle around the possibility – perhaps point out the long history of exclusion preventing
Feminist Maths from gaining a footing.
I don’t think I wrote the second point sufficiently clearly. Let me try again:
“there is an absolute truth but as limited humans we may be unable to fully comprehend it.”
Epistemologists slaloming through ontologies hazard the trap of uncontrolled equivocation. Just ask the Yrmo.
A woman buys her husband a pair of ties for his birthday. They go out for a celebration dinner that evening. When he gets ready the husband puts on one of the ties. When he presents himself for
approval before they leave she says, “What’s wrong with the other one?”
Most women I’ve told this joke to tell me it’s not funny, it’s a sensible question. Most men just laugh.
If a man is alone in a forest with no woman present, and he says something, is he still wrong?
“Putting the ‘pissed’ in epistemology”.
(That’s the UK ‘pissed’ not the American one). | {"url":"https://www.wmbriggs.com/post/2443/","timestamp":"2024-11-09T12:51:23Z","content_type":"text/html","content_length":"157308","record_id":"<urn:uuid:c67b26ae-736c-4477-b64e-ac696c8a27a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00280.warc.gz"} |
Challenges Concerning Symbolic Computations on Grids
Challenges Concerning Symbolic Computations on Grids
Main Article Content
Challenges concerning symbolic computations on grids
Symbolic and algebraic computations are currently ones of fastest growing areas of scientific computing. For a long time, the numerical approach to computational solution of mathematical problems had
an advantage of being capable of solving a substantially larger set of problems than the other approach, the symbolic one. Only recently the symbolic approach gained more recognition as a viable tool
for solving large-scale problems from physics, engineering or economics, reasoning, robotics or life sciences. Developments in symbolic computing were lagging relative to numerical computing, mainly
due to the inadequacy of available computational resources, most importantly computer memory, but also processor power. Continuous growth in the capabilities of computer hardware led naturally to an
increasing interest in symbolic calculations and resulted, among others things, in development of sophisticated Computer Algebra Systems (CASs).
CASs allow users to study computational problems on the basis of their mathematical formulations and to focus on the problems themselves instead of spending time transforming the problems into forms
that are numerically solvable. While their major purpose is to manipulate formulas symbolically, many systems have substantially extended their capabilities, offering nowadays functionalities like
graphics allowing a comprehensive approach to problem solving. While, typically, CAS systems are utilized in an interactive mode, in order to solve large problems they can be also used in a batch
mode and programmed using languages that are close to common mathematical notation.
As CASs become capable of solving large problems, they follow the course of development that has already been taken by numerical software: from sequential computers to parallel machines to
distributed computing and finally to the grid. It is particularly the grid that has the highest potential as a discovery accelerator. Currently, its widespread adoption is still impeded by a number
of problems, one of which is difficulty of developing and implementing grid-enabled programs. That it is also the case for grid-enabled symbolic computations.
There are several classes of symbolic and algebraic algorithms that can perform better in parallel and distributing computing environments. For example for multiprecision integer arithmetic, that
appears among others in factorizations, were developed already twenty years ago systolic algorithms and implementations on massive parallel processors, and more recently, on the Internet. Another
class that utilize significant amount of computational resources is related to the implementations of polynomial arithmetic: knowledge based algorithms such as symbolic differentiation, factorization
of polynomials, greatest common divisor, or, more complicated, Groebner base computations. For example, in the latest case, the size of the computation and the irregular data structures make the
parallel or distributed implementation not only an attractive option for improving the algorithm performance, but also a challenge for the computational environment. A third class of algorithms that
can benefit from multiple resources in parallel and distributed environments is concerning the exact solvers of large systems of equations.
The main reason driving the development of parallel and distributed algorithms for symbolic computations is the ability to solve problems that are memory bound, i.e. that cannot fit into memory of a
single computer. An argument for this statement relies on the observation that the input size of a symbolic or algebraic computation can be small, but the memory used in the intermediate stages of
the computation may grow considerably.
Modern CASs increase their utility not only through new symbolic capabilities, but also expending their applicability using visualization or numerical modules and becoming more than only specific
computational kernels. They are real problem solving environments based on interfaces to a significant number of computational engines. In this context it appears also the need to address the ability
to reduce the wall-clock time by using parallel or distributed computing environment. A simple example is the case of rendering the images for a simulation animation.
Several approaches can be identified in the historical evolution of parallel and distributed CASs: developing versions for shared memory architectures, developing computer algebra hardware, adding
facilities for communication and cooperation between existing CASs, or building distributed systems for distributed memory parallel machines or even across Internet.
Developing completely new parallel or distributed systems, although efficient, in most cases is rather difficult. Only a few parallel or distributed algorithms within such a system are fully
implemented and tested. Still there are several successful special libraries and systems falling in this category: ParSac-2 system, the parallel version of SAC-2, Paclib system, the parallel
extension of Saclib, FLATS based on special hardware, STAR/MPI, the parallel version of GAP, ParForm, the parallel version of Form, Cabal, MuPAD, or the recent Givaro, for parallel computing
environments, FoxBox or DSC, for distributed computing environments.
An alternative approach to build parallel and distributed CASs is to add the new value, the parallelism or the distribution, to an existing system. The number of parallel and distributed versions of
most popular CASs is impressive and it can be explained by the different requirements or targeted architectures. For example, for Maple there are several implementations on parallel machines, like
the one for Intel Paragon or ||Maple||, and several implementations on networks of workstations, like Distributed Maple or PVMaple. For Mathematica there is a Parallel Computing Toolkit, a
Distributed Mathematica and a gridMathematica (for dedicated clusters). Matlab that provides a Symbolic Math Toolbox based on a Maple kernel has more than twenty different parallel or distributed
versions: DP-Toolbox, MPITB/PVMTB, MultiMatlab, Matlab Parallelization Toolkit, ParMatlab, PMI, MatlabMPI, MATmarks, Matlab*p, Conlab, Otter and others.
More recent web-enabled systems were proved to be efficient in number theory for finding large prime numbers, factoring large numbers, or finding collisions on known encryption algorithms. Online
systems for complicated symbolic computations were also built: e.g. OGB for Groebner basis computations. A framework for description and provision of web-based mathematical services was recently
designed within the Monet project and a symbolic solver wrapper was build to provide an environment that encapsulates CASs and expose their functionalities through symbolic services (Maple and Axiom
were chosen as computing engines). Another platform is MapleNet build on client-server architecture: the server manages concurrent Maple instances launched to server client requests for mathematical
computations. WebMathematica is a similar system that offers access to Mathematica applications through a web browser.
Grid-oriented projects that involve CASs were only recent initiated. The well-known NetSolve system was one of the earliest grid system developed. Version 2 released in 2003 introduces GridSolve for
interoperability with the grid based on agent technologies. APIs are available for Mathematica, Octave and Matlab. The Genss project (Grid Enabled Numerical and Symbolic Services) follows the ideas
of the Monet project and intends also to combine grid computing and mathematical web services using a common agent-based framework. Several projects are porting Matlab on grids: from small ones, like
Matlab*g, to very complex ones, like Geodise. Maple2g and MathGridLink are two different approaches for grid-enabled version of Maple and Mathematica. Simple to use front-end were recently build in
projects like Gemlca and Websolve to deploy legacy code applications as grid services and to allows the submission of computational requests.
The vision of grid computing is that of a simple and low cost access to computing resources without artificial barriers of physical location or ownership. Unfortunately, none of the above mentioned
grid-enabled CAS is responding simultaneously to some elementary requirements of a possible implementation of this vision: deploy grid symbolic services, access within CAS to available grid services,
and couple different grid symbolic services.
Moreover a number of major obstacles remain to be addressed. Amongst the most important are mechanisms for adapting to dynamic changes in either computations or systems. This is especially important
for symbolic computations, which may be highly irregular in terms of data and general computational demands. Such demands received until now relatively little attention from the research community.
In the context of a growing interest in symbolic computations, powerful computer algebra systems are required for complex applications. Freshly started projects shows that porting a CAS to a current
distributed environment like a grid is not a trivial task not only from technological point of view but also from algorithmic point of view. Already existing tools are allowing experimental work to
be initiated, but a long way is still to be cross until real-world problems will be solved using symbolic computations on grids.
Dana Petcu,
Western University of Timisoara | {"url":"https://scpe.org/index.php/scpe/article/view/330","timestamp":"2024-11-06T05:22:00Z","content_type":"text/html","content_length":"33741","record_id":"<urn:uuid:ef917764-a6c4-4041-91ad-4077fb007a92>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00498.warc.gz"} |
Approximation of the Quadratic Knapsack Problem
We study the approximability of the classical quadratic knapsack problem (QKP) on special graph classes. In this case the quadratic terms of the objective function are not given for each pair of
knapsack items. Instead an edge weighted graph G = (V,E) whose vertices represent the knapsack items induces a quadratic profit p_ij for the items i and j whenever they are adjacent in G (i.e (i,j)
are in E). We show that the problem permits an FPTAS on graphs of bounded treewidth and a PTAS on planar graphs and more generally on H-minor free graphs. This result is shown by adopting a technique
of Demaine et al. (2005). We also show strong NP-hardness of QKP on graphs that are 3-book embeddable, a natural graph class that is related to planar graphs. In addition we will argue that the
problem is likely to have a bad approximability behaviour on all graph classes that include the complete graph or contain large cliques. These hardness of approximation results under certain
complexity assumptions carry over from the densest k-subgraph problem. | {"url":"https://optimization-online.org/2013/12/4167/","timestamp":"2024-11-03T17:02:24Z","content_type":"text/html","content_length":"84784","record_id":"<urn:uuid:eff9f744-7287-4251-881c-4f922d7746d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00344.warc.gz"} |
KSDA - Graduate Center series
Kolchin Seminar in Differential Algebra
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Academic year 2010–2011
Last updated on January 31, 2020.
Other years:
2005–2006 2006–2007 2007–2008 2008–2009 2009–2010 2011–2012 2012–2013 2013–2014 2014–2015 2015–2016 2016–2017 2017–2018 2018–2019 2019–Fall
In Fall, 2010, we explored new areas in the interactions between differential algebra and related fields, such as the Hopf-algebraic approach to Galois Theory in the setting where the field of
constants is not necessarily algebraically closed (or differentially closed in the parametric case). We continued with topics in algebraic geometry, representation theory, computational complexity,
differential geometry, model theory and number theory. Professor Camilo Sanabria of Bronx Community College gave a series of lectures on foliations and holonomy. In October, KSDA members helped
organized The Fourth International Workshop on Differential Algebra and Related Topics (DART IV) October 27–30, 2010, Beijing, China. In Spring, 2011 we welcomed Professor Raymond Hoobler from The
Graduate Center and The City College of CUNY, who has done an increasingly insightful review on differential schemes. He delivered a series of exploratory lectures on the subject in honor of the late
Prof. Jerry Kovacic. We also welcomed Meghan Anderson, a model theorist just finishing her doctorate advised by Professor Tom Scanlon, from the University of California, Berkeley.
Friday,August 27, 2010 at 10:15 a.m. Room 5382
Richard Churchill, The Graduate Center, and Hunter College, CUNY
A Geometric Approach to Classical Galois Theory
We reformulate classical Galois theory and differential Galois theory in geometric forms, which closely parallel one another. In particular, the standard matrix representations of the associated
Galois groups become strikingly similar. The talk is at an introductory level — no prior knowledge of differential Galois theory is assumed.
For lecture notes, please click here.
Friday, September 3, 2010 at 10:15 a.m. Room 5382
Noson S. Yanofsky, The Graduate Center, and Brooklyn College, CUNY
Galois Theory of Algorithms
Many different programs are the implementation of the same algorithm. This gives us a surjective map from the collection of programs to the collection of algorithms. Similarly, there are many
different algorithms that implement the same computable function. This gives us a surjective map from the collection of algorithms to the collection of computable functions. Algorithms are
intermediate between programs and functions:
Programs --> Algorithms --> Functions.
We investigate the many possible intermediate structures by looking at the group of automorphisms of programs that preserve functionality. The fundamental theorem of Galois theory says that the
subgroup lattice of this group is isomorphic to the dual lattice of intermediate types of algorithms. Along the way, we formalize the intuition that one program can be substituted for another if
they are the same algorithms.
For lecture notes, please click here.
Tuesday, September 14, 2010 at 2:00 p.m. Room 5382
Alexander Levin, The Catholic University of America
Some Invariants of Difference Field Extensions
In this talk we consider some characteristics of a difference field extension, which do not depend on the system of its difference generators. We start with the discussion of invariants carried
by dimension polynomials of finitely generated difference and inversive difference field extensions (such invariants include, in particular, difference transcendental degree, difference type and
typical difference transcendental degree). Then we consider transformations of the basic set of translations of an inversive difference field and show that if L/K is an inversive difference field
extension with the basic set σ = {α[1],..., α[n]} and d is the difference type of L/K (d ≤ n), then there is a "natural" transformation of σ into a set τ = {β[1],..., β[n]} such that L is an
algebraic extension of a finitely generated difference overfield of K with respect to the basic set {β[1],..., β[d]}. The last part of the talk is devoted to algebraic (in the sense of the
classical field theory) difference field extensions. We are going to concentrate on ordinary case and discuss the concept of limit degree introduced and studied by R. Cohn, as well as the notion
of distant degree introduced in a recent work by Z. Chatzidakis and E. Hrushovski.
Friday, September 24, 2010, 10:00 a.m. Room 5382
Informal Morning Session with
Bernard Malgrange, Université Joseph Fourier – Grenoble.
On September 24, 2010, Anand Pillay from University of Leeds gave a 90-minute talk at the CUNY Logic Seminar, from 2:00–3:30 pm at the Graduate Center in Room 6417. The title of the talk is
Model-theoretic Approaches to Galois Theories: a Survey.
Please click here for the abstract and other information.
Friday, October 1, 2010, 11:00 a.m. Room 5382
Informal Morning Session with
Bernard Malgrange, Université Joseph Fourier – Grenoble.
Friday,October 1st, 2010 at 2:00 p.m. Room 5382
Camilo Sanabria, Bronx Community College, CUNY
Foliations and Holonomy
Foliations are ubiquitous in B. Malgrange's approach to non-linear differential equations. In this talk I will recall the definition of foliation, the concept of holonomy and some properties of
foliations of co-dimension 1. The talk will have a geometric approach, so the objects studied, as well as the examples, will be modeled over the real numbers.
Friday,October 8, 2010 at 10:15 a.m.Room 5382
No meeting. See 2:00 p.m. Session.
Friday, October 8, 2010 at 2:00 p.m. Room 5382
Camilo Sanabria, Bronx Community College, CUNY
Foliations and Groupoids
This is a continuation of the talk from last week. I will recall the concepts of groupoid and Lie groupoid and I will explain how these concepts are used in the study of foliations. If time
allows I will also talk about orbifolds and their relation to foliations with compact leaves. The talk will have a geometric approach, so the objects studied, as well as the examples, will be
modeled over the real numbers.
Friday, October 15, 2010 at 10:15 a.m. Room 5382
Alexey Ovchinnikov, Queens College, CUNY
Tannakian Categories and Algebraic Groupoids—Preliminaries
We will discuss Hopf algebroids in the framework of Tannakian categories and look at basic examples.
Friday, October 15, 2010 at 2:00 p.m. Room 5382
Camilo Sanabria, Bronx Community College, CUNY
Foliations and Groupoids
This is a continuation of the talk from last week.
Saturday, October 16, 2010 at 10:15 a.m.
Room 920, Hunter College, East Building
Direction: Please be advised that Hunter College has a ``card-swipe'' security system. Attendees coming to the seminar on 10/16 will have to enter the campus via the West Building on the southwest
corner of 68th and Lexington Ave,(probably) show some kind of id and/or sign in, go up to the third floor, take the bridge over Lexington Ave to get into the East Building, and then take the elevator
to the ninth floor to get to the room.
Moshe Kamensky, University of Notre Dame
Tannakian Categories
I will give a survey on Deligne's paper Catégories Tannakiennes. Among the main results of the paper are the statement that any two fibre functors on a Tannakian category are locally isomorphic;
the construction of the fundamental group of a Tannakian category; the existence of fibre functors in characteristic zero; and an alternative construction of Picard-Vessiot extensions and the
Galois group of a linear differential equation.
I plan to explain in some detail the statement of the results, and then go into some of the proofs.
Friday, October 22, 2010 at 10:15 a.m. Room 5382
Anton Leykin, University of Illinois at Chicago
Multiplier ideals via computational D-modules theory
After an introduction to computational methods in D-modules theory, we will provide an overview of the new algorithms for generalized Bernstein-Sato polynomials for an arbitrary variety. These
lead to algorithms for singularity theory invariants: log canonical thresholds, jumping coefficients, and multiplier ideals. (Based on joint work with Christine Berkesch.)
Friday, October 29, 2010
No seminar.
Fourth International Workshop on Differential Algebra and Related Topics (DART IV) October 27–30, 2010, Beijing, China.
Friday, November 5, 2010 at 10:15 a.m. Room 5382
James Freitag, University of Illinois at Chicago
Definability of Rank for Differential Varieties
We will work over an ordinary differentially closed field of characteristic zero. Given a family of differential algebraic varieties parameterized by points in affine space, we will consider the
subfamily with a common coefficient and leading term in their Kolchin polynomials. We will prove that this is a constructable condition in the Kolchin topology using geometry and model theory. We
will show that this essentially comes from the Zariski topology (or definability of Morley rank in strongly minimal theories). Then we will discuss the barriers to a similar theorem for partial
differential fields.
Friday, November 12, 2010 at 10:15 a.m. Room 5382
Alexey Ovchinnikov, Queens College, CUNY
Title: Differential representations of SL(2)
W. Sit's characterization of differential algebraic subgroups of SL(2) will be presented next week. In this talk, we will be discussing the representation theory of SL(2) including some
unexpected examples.
Friday, November 19, 2010 at 10:15 a.m. Room 5382
William Sit, City College of New York, CUNY
Differential Algebraic Subgroups of SL(2), Part I
A differential algebraic subgroup of SL(2) is a subgroup whose elements, when viewed as a quadruple in affine space, satisfy and are defined by a system of partial differential equations. A
classification of all such subgroups up to conjugation over a ground field, which is a partial differential field, was completed in 1972. In this two-part talk, we review this classification and
outline the method used to obtain it. Part I provides the motivation for the problem, a classification of the algebraic subgroups of SL(2), some useful results on linear partial differential
equations, and examples of differential algebraic subgroups.
Reference: William Sit, Differential algebraic subgroups of SL(2) and strong normality in simple extensions, Amer. J. Math., 97 (3) (1975), pp. 627–698.
For lecture slides, please click here
Friday, November 26, 2010
No meeting. Thanksgiving.
Friday, December 3, 2010 at 10:15 a.m. Room 5382
Ravi Srinivasan, Rutgers University
Hopf Algebraic approach to Picard-Vessiot Theory
This talk is based on a paper by M. Takeuchi. We will use Hopf algebras to formalize the notion of a Picard-Vessiot extension and to characterize PV extensions as a minimal splitting field of a
linear differential equation. We will also establish a Galois correspondence between Hopf ideals and intermediate differential subfields.
Friday, December 10, 2010 at 10:15 a.m. Room 5382
Andrey Minchenko, University of Western Ontario
Differential representations of SL(2)
In order to describe the linear representations of a group, it is sufficient to find all of its indecomposable representations. It is known that indecomposable algebraic representations of G=SL
(2) correspond to irreducible subrepresentations of G in the ring R of polynomials in two variables x and y. Given a derivation ' on the ground field, R extends to a G-representation R' by adding
variables x',y',x'',y'',etc. We will investigate indecomposable subrepresentations of R' and discuss their relation to description of all differential representations of G.
Friday, December 17, 2010 at 10:15 a.m. Room 5382
William Sit, City College of New York, CUNY
Differential Algebraic Subgroups of SL(2), Part II
The Zariski closures in SL(2) of differential algebraic subgroups of SL(2) are algebraic subgroups of SL(2). In Part II, we discuss "lifting" the classification of the algebraic subgroups to
obtain a classification for the differential case. If time permits, we will discuss some applications with examples of strongly normal extensions and their differential Galois groups.
For lecture notes, please click here.
Friday, January 28, 2011 at 10:15 a.m. Room 5382
Ravi Srinivasan, Rutgers University, Newark
Hopf algebraic approach to Picard-Vessiot Theory
This talk is a continuation of my talk from December 3rd 2010. We will formalize the notion of a Picard-Vessiot extension using Hopf algebras. I will give several examples and discuss briefly the
Galois correspondence between Hopf ideals and intermediate differential subfields. I will also give a quick overview on the materials from my last lecture.
Friday, February 4, 2011 at 10:15 a.m. Room 5382
Carlos Arreche, The Graduate Center, CUNY
Differential Galois theory in arbitrary characteristic for modules with iterative connection
About a decade ago, Matzat and van der Put described a Picard-Vessiot theory for iterative differential fields in arbitrary characteristic generalizing the classical theory in characteristic
zero, but their Galois correspondence was shown to be incomplete. Recently, Maurischat (arXiv:0712.3748) described a Galois theory for modules with iterative connection which generalizes that of
Matzat and van der Put and gives a complete Galois correspondence which is equivalent to Takeuchi's in this setting. I will motivate and describe Maurischat's work and relate it to the approaches
of Matzat-van der Put and Takeuchi.
Friday, February 11, 2011
No meeting. President Lincoln’s Birthday.
Friday, February 18, 2011 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center and The City College, CUNY
A Grothendieck approach to differential Azumaya algebras
Gothendieck introduced connections on a sheaf on a scheme $X$ over $S$ by considering the first order neighborhood of the diagonal map $X\rightarrow X\times_{S}X$. I will explain this definition
and connect it to the usual definition. Then I will show that an Azumaya algebra $\Lambda$ on an affine scheme $Spec(A)$ satisfies Grothendieck's definition by calculating the Hochschild
cohomology of $\Lambda$. Time permitting, I will then connect it to my talk last spring using the $\delta$-flat topology to interpret the differential Brauer group of a differential ring $A$.
For lecture notes, please click here.
Friday, February 25, 2011 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center and The City College, CUNY
Differential Schemes
I will begin by summarizing Kovacic's work and the proper sheafification procedure. I will then explain when faithfully flat descent holds for differential schemes and interpret this result in
terms of differential principal homogeneous spaces for a differential group. This provides the connection between Kolchin's constrained cohomology and the $\Delta$-flat cohomology. Given
sufficient time, I will also discuss varying the partial differential structure using adjoint functors.
Friday, March 4, 2011 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center and The City College, CUNY
Differential Cohomology
I will begin by discussing constrained extensions and show that differential principal homogeneous spaces always have points in differentially closed fields. Using this I will show that $\
Delta$-flat cohomology extends Kolchin's constrained cohomology to differential schemes and show that if the coefficients are algebraic groups, then $\Delta$-flat (= constrained) cohomology
classes split by passing to the algebraic closure of a differential field are precisely those coming from the (non differential) Galois cohomology.
Friday, March 11, 2011 at 10:15 a.m. Room 5382
Dmitry Trushin, Moscow State University, Moscow
A non-standard geometric approach to differential and difference equations
I will present a non-standard geometric approach to differential and difference equations. I will show that there are four natural classes of rings playing the role of universal domains
containing all necessary solutions. These rings are: differentially closed fields of characteristic zero and quasifields of prime characteristic in the differential case, and difference closed
fields and pseudofields in the difference case.
Saturday, March 12, 2011 at 1:00 a.m. Room R6/113, CCNY
Dmitry Trushin, Moscow State University, Moscow
A non-standard geometric approach to differential and difference equations
This is an informal continuation of Friday's talk, concentrating on difference equations.
Location: City College of New York, North Academic Center, 6th Floor, Room 113 (green side).
Friday, March 18, 2011 at 10:15 a.m. Room 5382
Michael Wibmer, RWTH, Aachen
A Chevalley theorem for difference equations
By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on ``extension of specializations'' or
``lifting of prime ideals''. We present a difference analog of this theorem. The approach is based on the philosophy that occasionally one needs to pass to higher powers of $\sigma$, where $\
sigma$ is the endomorphism defining the difference structure. In other words, we consider difference pseudo fields (which are finite direct products of fields) rather than difference fields. We
also prove a result on compatibility of pseudo fields and present some applications of the main theorem, e.g. constrained extension and uniqueness of $\sigma$-Picard-Vessiot rings for linear
differential equations with a difference parameter.
Saturday, March 19, 2011 at 1:00 a.m. Room 920, East Building, Hunter College
Michael Wibmer, RWTH, Aachen
A Chevalley theorem for difference equations
This is an informal continuation of Friday's talk.
Location: Room 920, Hunter College, East Building.
Friday, March 25, 2010, 10:15 a.m. Room 5382
Dmitry Trushin, Moscow State University, Moscow
A non-standard geometric approach to differential and difference equations
A continuation of the talk from March 11.
Friday, April 1, 2011 at 10:15 a.m. Room 5382
Brainstorming session.
Friday, April 8, 2011 at 10:15 a.m. Room 5382
Meghan Anderson, University of California, Berkeley
Solutions to Linear Equations in Valued D-fields
A model complete theory of valued D-fields was developed by Scanlon in his 1997 thesis. In this theory, valued fields are endowed with linear operator D which specializes to a derivative in the
residue field, but which in the valued field obeys a twisted Leibniz rule and is interdefinable with a valuation preserving automorphism. The theory has good model theoretic properties, notably
quantifier elimination, which should allow for some analysis of the upstairs difference field in terms of the downstairs differential structure. However, it also presents its own challenges, even
in the relatively simple setting of solution spaces to linear equations, some of which I will discuss.
Friday, April 15, 2011 at 10:15 a.m. Room 5382
Meghan Anderson, University of California, Berkeley
Solutions to Linear Equations in Valued D-fields
This is a continuation of the talk from last week. A model complete theory of valued D-fields was developed by Scanlon in his 1997 thesis. In this theory, valued fields are endowed with linear
operator D which specializes to a derivative in the residue field, but which in the valued field obeys a twisted Leibniz rule and is interdefinable with a valuation preserving automorphism. The
theory has good model theoretic properties, notably quantifier elimination, which should allow for some analysis of the upstairs difference field in terms of the downstairs differential
structure. However, it also presents its own challenges, even in the relatively simple setting of solution spaces to linear equations, some of which I will discuss.
Friday, April 29, 2011 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center and The City College, CUNY
DiffSpec Redux
They say that three times is a charm and so it appears. A Max $\Delta$ ring is a $\Delta$ ring in which all maximal ideals are $\Delta$ ideals. I will give a straightforward definition of the
structure sheaf of a Max $\Delta$ ring using the usual definition with inverting $f$ to get sections over $D(f)$ to define the $\Delta$ structure sheaf. Any $\Delta$ ring can be made into a Max $
\Delta$ ring by inverting all differential units.
I will show that there are no non-zero differential zeros in a Max $\Delta$ ring and, even better, that the tensor product of two Max $\Delta$ rings over a third Max $\Delta$ ring is a Max $\
Delta$ ring. This makes many of the standard tools from algebraic geometry available for differential algebraic geometry. If there is enough time, I will suggest an application to differential
group scheme extensions.
For lecture notes, please click here.
Friday, May 6, 2011 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center and The City College, CUNY
Projective Delta schemes
I will outline the procedure for defining projective differential schemes in a form similar to affine delta schemes. An effort will be made to describe Kolchin's results in this case. A number of
interesting questions for future work will be posed.
Other Academic Years
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Created by the KSDA Organizing Committee
Hosted by Please submit web page problems to
William Sit <wsit@ccny.cuny.edu> | {"url":"https://ksda.ccny.cuny.edu/gradcenter2010.html","timestamp":"2024-11-09T16:35:39Z","content_type":"text/html","content_length":"32042","record_id":"<urn:uuid:f8030efc-d3ca-4d57-a0ef-607e63e09d41>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00681.warc.gz"} |
How to control RGB LED color spectrum with a single potentiometer
So far, I've gotten it to fade from red to green, but what I'm trying to do is to fade from red to green to blue.
If anyone can help, you will be greatly appreciated.
int R = 11;
int G = 10;
int B = 9;
int Pot = A0;
int RVal;
int GVal;
int BVal;
void setup()
pinMode(R, OUTPUT);
pinMode(G, OUTPUT);
pinMode(B, OUTPUT);
void loop()
RVal = analogRead(Pot);
RVal = map( RVal, 0, 1023, 255,0);
GVal = analogRead(Pot);
GVal = map( GVal, 0, 1023, 255, 0);
//BVal = analogRead(Pot);
//BVal = map( BVal, 0, 1023, 255, 0);
analogWrite(R, RVal);
analogWrite(G, 255-GVal);
//analogWrite(B, RVal);
Did you expect the pot value to have changed significantly?
It is a lot easier to select colors using the HSV color scheme, than RGB.
See this post, among others: Arduino RGB LED HSV “Color Wheel” – eduardofv
Divide the pot range into four parts:
0->255: Green fades from 0 to 255 while RED stays at 255.
255->512: Red fades from 255 to 0 while Green stays at 255.
512->786: Blue fades from 0 to 255 while Green stays at 255.
768->1023: Green fades from 255 to 0 while Blue stays at 255.
int potval = analogRead(Pot);
if (potval < 255)
analogWrtite(RedPin, 255);
analogWrite(GreenPin, potval);
analogWrite(BluePin, 0);
else if (potval < 512)
analogWrite(RedPin, 512 - potval);
analogWrite(GreenPin, 255);
analogWrite(BluePin, 0);
This will be tricky...
I think you'll have to figure-out what you want to do first... What do you want to see at minimum? In the middle, at maximum? Etc.?
You'll probably need some if-statements to handle different ranges of the pot differently.
With the 10-bit ADC you can only get 1023 different color combinations.
As you may know, 0-255 can is represented by 8-bits. So a single 24-bit variable (which requires a type-long) can represent any possible color & brightness combination.
Then you do bit manipulation to "extract" the groups of 8-bits for the 3 different colors.
With 24-bits you can count to (you might want to double-check that) (1) so you could simply map 0-1023 to 0- 16,777,216. But of course, you're only going to get 1023 different colors with most values
being skipped-over.
And with this "simple" approach, you're going to get discontinuities... When a group of bytes counts 255+1, the number rolls-over and 255 suddenly becomes zero.
(1) This becomes easier if you can think and program in hexadecimal. With 8-bits you can count to FF in hex. Each byte is represented by exactly 2 hex digits. With 2-bytes we can count to FFFF and
with 24-bits we can count to FFFFFF. With RGB values each group of 2 hex digits represents a color so in hex we can "see" each color without any math.
This topic was automatically closed 180 days after the last reply. New replies are no longer allowed. | {"url":"https://forum.arduino.cc/t/how-to-control-rgb-led-color-spectrum-with-a-single-potentiometer/998734","timestamp":"2024-11-05T21:50:45Z","content_type":"text/html","content_length":"36034","record_id":"<urn:uuid:34509064-9963-4ce5-b453-281af323283b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00274.warc.gz"} |
White to play and mate in two moves
Where would you start when faced with this diagram?
If you’re a player, rather than a problemist, you’ll probably look first at the checks. If you’re a problemist, the checks will probably be the last thing you’ll look at! Why? Because composed
positions are supposed to be difficult and to be elegant, and the key move – white’s first move – is usually an unexpected one.
A problem-habitué would notice first of all that bishop in the corner, blocked by the rook. So he’d think of moving the g2 rook, but of course that gives stalemate. So he says to himself that the
problem must be based on Black moving the knight and then the white rook giving a discovered mate. That doesn’t solve it, but it’s major progress. Suppose the black knight moves, what mates have I
got? There’s a mate for every one. OK, so that means that if it were Black to move, I know what to do. All I need is to begin with a waiting move by White – one that doesn’t disturb anything. The
solver looks at every possible move – how about 1.Ka6? Oops! Black goes 1...Sc5! and that’s check to the white king, so White can’t play the 2.d5 he wanted to. 1.c4? Nope – 1...Sc3: I need that pawn
to stay on c3 so if black captures it I can play 2.Rc2 pinning. Must be 1.Rhg6?, then. That seems to do the job. Just check it one last time... dammit, if he goes 1...Sxf6! I can’t play the rook from
g2 to g7 to guard d7. Wait... I could’ve guarded d7 with the other rook. Ah-hah! 1.Rh7 does it. I didn’t need that rook and knight battery pointing at the black king after all – it fooled me into not
trying the right key move earlier. So it’s solved.
By the way, a top solver would have noticed that 1.Rhg6 Sxf6 would let White have multiple mating moves (here as many as 11 of them) – if he overlooked that d7 wouldn’t be guarded – and that is
considered really inelegant, so he would have automatically rejected 1.Rhg6 as a candidate solution.
That problem was composed by Comins Mansfield, Britain’s first ever Grandmaster (he got his title for his composing); it was published in the Morning Post in 1933.
Another aspect of the problem is that it shows a complete knight wheel – Black’s knight moves to the maximum possible number of squares (8) in the solution, and each one is met by a different white
reply. This problem is a splendidly efficient demonstration of a mate in two with a knight wheel – there are lots of such problems, but it’s very hard to compose one with as few pieces as Mansfield
managed here.
(This was first published in The British Correspondence Chess Association magazine ‘Correspondence Chess’ in 2010. | {"url":"https://www.theproblemist.org/solve.pl?type=sv_how","timestamp":"2024-11-06T13:32:38Z","content_type":"text/html","content_length":"22959","record_id":"<urn:uuid:9befc1af-a5f8-4fe2-9510-3c4c1da9b0b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00867.warc.gz"} |
MHD Ekman layer on a porous plate
An exact solution of the steady three-dimensional Navier-Stokes equations is obtained for the case of flow past a porous plate at zero incidence in a rotating frame of reference by using similarity
analysis. The behavior of the MHD Ekman layer on a flat plate, subjected to suction and blowing, is studied. It is shown that the Ekman-layer thickness is inversely proportional to suction and
directly proportional to blowing for a given Taylor number and magnetic parameter. The Ekman-layer thickness is found to be inversely proportional to both the Taylor number and the magnetic parameter
either under suction or blowing.
Nuovo Cimento B Serie
Pub Date:
October 1975
□ Ekman Layer;
□ Magnetohydrodynamic Flow;
□ Porous Boundary Layer Control;
□ Porous Plates;
□ Rotating Fluids;
□ Blowing;
□ Earth Planetary Structure;
□ Magnetic Effects;
□ Navier-Stokes Equation;
□ Suction;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1975NCimB..29..296M/abstract","timestamp":"2024-11-07T22:09:38Z","content_type":"text/html","content_length":"36430","record_id":"<urn:uuid:6826d76f-8652-4c6b-a04b-65fd5dcff620>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00353.warc.gz"} |
ACM Other ConferencesA (5/3+ε)-Approximation for Tricolored Non-Crossing Euclidean TSP
In the Tricolored Euclidean Traveling Salesperson problem, we are given k = 3 sets of points in the plane and are looking for disjoint tours, each covering one of the sets. Arora (1998) famously gave
a PTAS based on "patching" for the case k = 1 and, recently, Dross et al. (2023) generalized this result to k = 2. Our contribution is a (5/3+ε)-approximation algorithm for k = 3 that further
generalizes Arora’s approach. It is believed that patching is generally no longer possible for more than two tours. We circumvent this issue by either applying a conditional patching scheme for three
tours or using an alternative approach based on a weighted solution for k = 2. | {"url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.15/metadata/acm-xml","timestamp":"2024-11-09T08:14:53Z","content_type":"application/xml","content_length":"15619","record_id":"<urn:uuid:dfd16c8f-f241-48de-9971-2a761be39293>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00674.warc.gz"} |
RD Sharma Class 12 Solutions Chapter 11 Differentiation Ex 11.8
RD Sharma Class 12 Solutions Chapter 11 Differentiation Ex 11.8 are part of RD Sharma Class 12 Solutions. Here we have given RD Sharma Class 12 Solutions Chapter 11 Differentiation Ex 11.8 Here you
can get free RD Sharma Solutions for Class 12 Maths of Chapter 11 Differentiation Exercise 11.8. All RD Sharma Book Solutions are given here exercise wise for Differentiation. RD … [Read more...]
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Understanding Mathematical Functions: Is An Absolute Value Function On
Mathematical functions play a crucial role in various fields, from engineering to economics and even in daily life. These functions help us understand and represent relationships between different
quantities or variables. One important aspect of functions is whether they are one-to-one or not. A one-to-one function is a function where each element in the domain maps to exactly one element in
the range, and no two elements in the domain map to the same element in the range. Today, we'll delve into the concept of absolute value functions and explore whether they are one-to-one.
Key Takeaways
• Mathematical functions are crucial in various fields and help represent relationships between quantities or variables.
• A one-to-one function maps each element in the domain to exactly one element in the range, with no two elements in the domain mapping to the same element in the range.
• Absolute value functions are explored to determine if they are one-to-one, involving graphical representation and algebraic methods.
• Understanding one-to-one functions in absolute value functions has implications in mathematical analysis and real-life applications.
• The one-to-one property affects the behavior of the absolute value function and is important to understand in mathematics.
Understanding Absolute Value Functions
An absolute value function is a mathematical function that returns the absolute value of a number, which is its distance from zero on the number line. Absolute value functions are represented using
the notation |x|.
When dealing with real numbers, the absolute value of a number is always non-negative. For example, the absolute value of -5 is 5, and the absolute value of 3 is also 3.
Definition of absolute value function
• Absolute value function definition: The absolute value of a number x, denoted as |x|, is defined as follows:
□ If x is greater than or equal to 0, then |x| = x.
□ If x is less than 0, then |x| = -x.
Graphical representation of absolute value function
• Graph of the absolute value function: The graph of the absolute value function is a V-shaped graph, with its vertex at the origin (0,0). It has a slope of 1 for x > 0 and a slope of -1 for x < 0.
• Key characteristics of the graph: The graph of |x| reflects the distance of x from 0, without considering the direction. This results in a symmetrical graph about the y-axis.
Characteristics of absolute value function
• Domain and Range: The domain of the absolute value function is all real numbers. The range is also all real numbers, but the output is always non-negative.
• One-to-One Function: An absolute value function is not a one-to-one function because it fails the horizontal line test. A horizontal line intersects the graph of an absolute value function at two
points, indicating that it is not one-to-one.
Understanding Mathematical Functions: Is an absolute value function one-to-one
Mathematical functions are essential in understanding relationships between variables and their outputs. One important aspect of functions is determining if they are one-to-one, which plays a crucial
role in various mathematical concepts and applications.
A. Definition of one-to-one function
A one-to-one function, also known as an injective function, is a function in which each element in the domain maps to a unique element in the range. In other words, no two distinct elements in the
domain map to the same element in the range.
B. Criteria for determining if a function is one-to-one
• Horizontal Line Test: One way to determine if a function is one-to-one is by using the horizontal line test. If any horizontal line intersects the graph of the function at most once, then the
function is one-to-one.
• Algebraic Approach: Another method is to use algebraic techniques to analyze the function. For a function f(x) to be one-to-one, if two different inputs x1 and x2 lead to the same output f(x1) =
f(x2), then the function is not one-to-one.
C. Importance of one-to-one functions in mathematics
One-to-one functions are important in various mathematical concepts such as inverse functions, logarithms, and solving equations. Inverse functions, for example, rely on the property of one-to-one
functions to ensure that each input in the range corresponds to a unique output in the domain. Logarithms, on the other hand, are based on the inverse relationship of exponential functions, which are
Furthermore, one-to-one functions are essential in solving equations, especially when it comes to finding unique solutions for different variables. They help in ensuring that each input has only one
corresponding output, making it easier to analyze and solve mathematical problems.
Understanding Mathematical Functions: Is an absolute value function one-to-one
In the realm of mathematical functions, one important property to consider is whether a function is one-to-one, also known as injective. In this post, we will delve into the absolute value function
and analyze whether it possesses this property.
Testing the absolute value function for one-to-one property
Before we dive into the analysis, it is crucial to understand the concept of a one-to-one function. A function f is said to be one-to-one if no two different inputs produce the same output, in other
words, for any two distinct inputs x1 and x2, f(x1) does not equal f(x2).
Using algebraic methods to analyze the absolute value function
One way to test whether the absolute value function is one-to-one is by using algebraic methods. We can examine the equation f(x) = |x| and evaluate its behavior for different input values. By
testing various pairs of input values and observing the corresponding outputs, we can determine whether the function satisfies the one-to-one property.
Graphical representation to determine if the absolute value function is one-to-one
Another approach to analyzing the one-to-one property of the absolute value function is by examining its graphical representation. By plotting the function on a coordinate plane, we can visually
inspect whether the function passes the horizontal line test. If every horizontal line intersects the graph at most once, then the function is one-to-one.
Understanding Mathematical Functions: Is an Absolute Value Function One-to-One?
In mathematics, functions are a fundamental concept that describes the relationship between input and output values. One important type of function is the absolute value function, which is denoted as
|x| and returns the magnitude of a real number without considering its sign.
A. Explanation of the properties of the absolute value function
The absolute value function is defined as follows:
• |x| = x if x is greater than or equal to 0
• |x| = -x if x is less than 0
This means that the absolute value of a non-negative number is the number itself, while the absolute value of a negative number is its positive counterpart.
B. Determining if the absolute value function satisfies the criteria for being one-to-one
A function is considered one-to-one if each element of the domain maps to a unique element in the range. In other words, no two different inputs can produce the same output.
1. Using the horizontal line test
To determine if the absolute value function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is
not one-to-one. In the case of the absolute value function, it fails the horizontal line test because a horizontal line at y = 0 intersects the graph at two points, indicating that multiple inputs
map to the same output.
2. Analyzing the slope of the function
Another way to determine if a function is one-to-one is to analyze its slope. For the absolute value function, the slope changes abruptly at x = 0, as the function transitions from a slope of 1 to a
slope of -1. This sudden change in slope indicates that the function is not one-to-one, as different inputs produce the same output.
Implications of One-to-One Property in Absolute Value Functions
The one-to-one property in absolute value functions has significant implications in mathematical analysis, real-life applications, and the behavior of the function.
A. Advantages of one-to-one property in mathematical analysis
• Uniqueness: One-to-one functions ensure that each input corresponds to a unique output, allowing for straightforward analysis and interpretation of the function.
• Solvability: In mathematical equations involving absolute value functions, the one-to-one property helps in finding unique solutions, reducing ambiguity and simplifying the process of solving
• Consistency: One-to-one property ensures that the function preserves the order and relationships between input and output values, leading to consistent and predictable behavior.
B. Real-life applications of understanding one-to-one functions in absolute value functions
• Distance and direction: In real-world scenarios such as navigation and physics, absolute value functions represent distance and direction, where understanding the one-to-one property is crucial
for accurate measurements and calculations.
• Optimization problems: Applications in economics, engineering, and optimization rely on one-to-one functions to identify optimal solutions and make informed decisions based on unique
relationships between variables.
• Biomedical analysis: In medical research and analysis, absolute value functions with one-to-one property are used to model relationships between variables, leading to insights and advancements in
healthcare and pharmaceuticals.
C. How the one-to-one property affects the behavior of the absolute value function
The one-to-one property influences the behavior of the absolute value function in several ways:
• Injective nature: The one-to-one property makes the absolute value function an injective function, ensuring that distinct inputs correspond to distinct outputs, leading to a consistent and
predictable mapping.
• Reflection symmetry: Understanding the one-to-one property helps in visualizing the reflection symmetry of the absolute value function, where the function's graph reflects across the y-axis due
to the unique mapping of inputs and outputs.
• Strict monotonicity: The one-to-one property ensures that the absolute value function exhibits strict monotonicity, where the function's values either consistently increase or decrease,
reflecting the unique relationships between inputs and outputs.
Understanding one-to-one functions in mathematics is crucial for analyzing relationships between inputs and outputs. It helps us determine whether a function has a unique inverse and provides
valuable insight into the behavior of mathematical expressions.
Final thoughts on the one-to-one property of the absolute value function:
• The absolute value function is not one-to-one because it fails the horizontal line test, meaning that there are multiple inputs that result in the same output.
• Despite not being one-to-one, the absolute value function still plays a significant role in many mathematical applications and is valuable for solving equations and inequalities.
Overall, a deep understanding of mathematical functions, including whether they are one-to-one, enhances our ability to analyze and interpret mathematical models, ultimately strengthening our
problem-solving skills.
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An intuitive explanation of third-order surface behavior
We present a novel parameterization-independent exposition of the third-order geometric behavior of a surface point. Unlike existing algebraic expositions, our work produces an intuitive explanation
of third-order shape, analogous to the principal curvatures and directions that describe second-order shape. We extract four parameters that provide a quick and concise understanding of the
third-order surface behavior at any given point. Our shape parameters are useful for easily characterizing different third-order surface shapes without having to use tensor algebra. Our approach
generalizes to higher orders, allowing us to extract similarly intuitive parameters that fully describe fourth- and higher-order surface behavior.
Pushkar Joshi and Carlo H. Séquin. "An intuitive explanation of third-order surface behavior". Computer Aided Geometric Design, 27(2):150–161, February 2010. | {"url":"http://graphics.berkeley.edu/papers/Joshi-ESB-2010-02/","timestamp":"2024-11-15T02:41:28Z","content_type":"text/html","content_length":"6684","record_id":"<urn:uuid:6dc32eb4-5819-4df6-aa2d-32356acda19e>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00058.warc.gz"} |
The Facts of Factor Patterns
About a week ago, I went through pointing and clicking your way to a factor analysis. At the time, I suggested rotating the factors. Now we’re going to interpret the rotated factor pattern. Let me
recap, briefly. Agresti and Finlay (p.532) put it way better than me when they said:
Factor analysis is a multivariate statistical technique used for …
1. Revealing patterns of interrelationships among variables
2. Detecting clusters of variables, each of which contains variables that are strongly intercorrelated …
3. Reducing a large number of variables to a smaller number of statistically uncorrelated variables, the factors of factor analysis.
All of which is well and good but once you have your factors, what do they mean? How do you interpret them?
Important point one: The correlation of a variable with a factor is called the loading.
Important point two: To ease interpretation we’d really like to have “simple structure”, that is, where variables load close to 1.0 on one factor and close to zero on the others. I mean, really, if
you think about it, if your items load equally on all factors it’s going to be pretty hard to interpret.
Let’s take a look at my example from the 500 Family Study, which you have probably forgotten already. To make it easier to interpret, I copied the factor pattern output into a spreadsheet and sorted
by the loadings on the first, second and third factor. You can see that almost all of the items relating to discussion loaded on the first factor. So, I could say that factor 1 is “Communication with
parents”. The second factor seems to be mostly about rules, punishment and placing limits, such as punishments or reward for grades, curfew and time out with friends. The discussion questions that
load more on this factor than the first are on discussion of breaking rules and discussion of curfew. The third factor is all of the items related to decision-making, with the exception of family
purchases, which didn’t really load on any of the three factors.
Notice a few things— Just like correlations, loadings can be positive or negative. How late your curfew is loads negatively on the Rules Factor. That is, families that have stricter rules have an
earlier curfew. How often parents limit time out with your friends loads positively on the Rules Factor. Although it’s not ideal, variables can load on more than one factor. As noted, the discussion
of breaking rules item loads both on the Communication Factor and the Rules Factor. Variables can not load on any factor at all, like the decision on family purchases. My guess is that most parents
decide most purchases without consulting their adolescent children.
The really useful result of factor analysis is that it allows you to take your 42 items, discard one as not really fitting and distill the others down into three factors. Instead of using 41
individual items to predict your outcome of interest, say delinquent behavior, you can use three. It’s almost certain that those three factors will be far more reliable than any individual item, and
your results will be far easier to explain as well, say, “Students who have more communication with their parents, moderate rules and moderate input on decision-making have the lowest rate of
delinquent behavior and highest academic achievement.”
Not sure if that is true or not but with these factors we are now in a good position to test that. I just need a couple more measures, of delinquent behavior and academic achievement, and I can test
my hypotheses. I expect there will be a linear relationship with communication (negative for delinquency and positive for academics) and a curvilinear relationship with the other two measures
(inverse for delinquency).
I guess that will be my next thing to do when I have some spare time. Or, you can wander on over to ICPSR.org and download the 500 Family Study data yourself. | {"url":"https://www.thejuliagroup.com/blog/the-facts-of-factor-patterns/","timestamp":"2024-11-07T10:03:38Z","content_type":"text/html","content_length":"81723","record_id":"<urn:uuid:e394d1b9-fda6-47e0-9441-56b021766234>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00053.warc.gz"} |
Drive test bench for DC and BLDC motors
Special service for correct selection and equipment of drive technology
On the JBW GmbH motor test bench, characteristic curves for current, voltage, rotation speed, torque and temperature are read in and processed in order to evaluate the load capacity and functionality
of a DC motor or BLDC motor.
Together with the measurement data of the application-specific load characteristics, which are recorded on the prototype via a digital storage oscilloscope, a DC or BLDC geared motor optimized for
the application can ultimately be put together.
Brake types on the test bench
• Magnetic particle brake from 0.5-50 Nm up to 1,000 rpm
• Hysteresis brake from 0-6 Nm to 10,000 rpm
Measuring channels
• Current
• Voltage
• Rotation speed
• Torque
• Temperature
• Characteristic curve recording
• Reverse torque (static and dynamic)
• Temperature test
• Endurance test
• Freely programmable test cycle
What do you see on a motor characteristic curve?
Here, the X-axis shows the torque, the right Y-axis the rotation speed (blue) and the left Y-axis the current consumption (red).
Every DC geared motor on the JBW drive test bench is increasingly loaded with a brake and therefore slows down (the blue rotation speed line decreases). In order to continue turning, an electric
motor draws more current (the red line increases). Using the characteristic curve created in this way, it is easy to see how the respective motor behaves at which torques. In other words, how fast it
is and how much current it consumes in each case.
Why are the motors not always braked to the end, i.e. until rotation speed 0 is reached and the torque is at its highest?
Not every gear can withstand so much torque. With particularly large ratios, there is even a risk of destroying the gear. Every gear has its own nominal and maximum torque.
How do you determine the rated torque of a motor?
The nominal torque is not a fixed point for electric motors, but rather a range. The nominal torque range is approx. 1/3 of the maximum torque. It is of course possible to operate the motors up to
the defined maximum torque for a short time.
What happens to the characteristic curve when the voltage is increased or reduced?
If the voltage is increased or reduced, the rotation speed curve moves up or down almost proportionally. This means, for example, that if the voltage is only 12V instead of 24V, the rotation speed
drops by 50%. | {"url":"https://www.elektromotore.eu/en/drive-test-bench","timestamp":"2024-11-11T22:51:08Z","content_type":"text/html","content_length":"68388","record_id":"<urn:uuid:bdf21dc1-a2e4-4df6-b4a0-2e80d0aea2f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00264.warc.gz"} |
Exploring teachers’ conceptions of representations in mathematics through the lens of positive deliberative interaction
AOSIS Openjournalshttp://www.openjournals.netinfo@openjournals.nethttp://www.pythagoras.org.za1012-23462223-7895This article reports on an exploration of teachers’ views on the meaning of
mathematical representations in a democratic South Africa. We explored teachers’ conceptions of ‘mathematical representations’ as a means to promote dialogue and negotiation. These conceptions helped
us to gauge how these teachers viewed representations in mathematics. Semi-structured questionnaires were administered to 76 high school mathematics teachers who were registered for an upgrading
mathematics education qualification at a South African university. Common themes in teacher conceptions of representations were investigated as part of an inductive analysis of the written responses,
which were considered in terms of practices that support dialogue and negotiation. Findings suggest that these conceptions are in line with progressive notions of classroom interactions such as the
inquiry cooperation model. Furthermore, the findings suggest that teachers can support the development of classroom environments that promote democratic values.
Deonarain Brijlall1
1Department of
Mathematics Education, University of KwaZulu-Natal, South Africa
Sarah Bansilal1
1Department of Mathematics Education, University of KwaZulu-Natal, South Africa
Deborah Moore-Russo2
2Graduate School
of Education, University at Buffalo, The State University of New York, United States
Sarah Bansilal
bansilals@ukzn.ac.za
8 Zeeman Place, Malvern 4093, South Africa
16533210.4102/
pythagoras.v33i2.165
16 Mar. 201215 Oct. 201204 Dec. 2012Brijlall, D., Bansilal, S., & Moore-Russo, D. (2012). Exploring teachers’ conceptions of representations in mathematics through the lens of
positive deliberative interaction. Pythagoras, 33(2), Art. #165, 8 pages. http://dx.doi.org/10.4102/pythagoras.v33i2.165
© 2012.The Authors.
Licensee: AOSIS OpenJournals. This work
is licensed under
the
Creative Commons
Attribution License.Introduction
What are the specific elements of a mathematics classroom that allow it to be characterised as democratic or as a classroom that seeks to prepare
all children for life in a democratic society? Are there connections between democracy and mathematics classrooms? Skovsmose (1998) asserted that mathematics education could be related to a
discussion of democracy in terms of citizenship, mathematical archaeology, mathemacy and deliberative interaction. He illustrated how these four aspects, which concern classroom practice in
mathematics education, also concern democracy. In this article we delve into one of the above aspects, namely deliberative interaction, which Skovsmose views as possible when ‘an interaction in the
classroom which supports dialogue and negotiation’ is developed (p. 200). Figure 1 illustrates the components inherent in Skovsmose’s notion of deliberative interaction.
Deliberative interaction
excludes a view of mathematics as an unchanging body of knowledge, which a teacher transmits to learners. Such a view presupposes that mathematical tasks have only one correct answer and often only
one correct, or one preferred, method to arrive at that answer. This view sets the classroom as an autocracy in which the teacher serves as the sole authority. Alrø and Skovsmose (1996) used the
phrase classroom absolutism to refer to the type of communication between the teachers and learners that is structured by assumptions that (1) school mathematics can be organised around mathematics
activities with unique answers, and (2) the teacher’s task is to ensure that mathematical errors are removed from the classroom. In trying to identify the role of representations when teaching and
learning mathematics within such a paradigm, we extend the notion of classroom absolutism to include two further assumptions, (1) that mathematical learning can be organised around classroom activity
with one possible representation for a mathematical notion or task, and (2) it is the duty of the teacher to ensure that other representations are eradicated from mathematical learning.
In the
democratic micro-society of a mathematics classroom it is imperative for the teacher to move away from such classroom absolutism because learners should be afforded different ways to express
themselves. We are not implying that teachers accept any and all responses to mathematical tasks as final answers. Our message is this: the teacher should (1) be aware of the different mathematical
representations that can be used to achieve mathematically acceptable arguments, and (2) be willing to work with learners’ developing mathematical ideas and personal mathematical representations to
facilitate a clearer understanding of mathematics and the way it is conventionally represented.
Educational environments that discourage classroom absolutism often have a prevailing view of
mathematics as a process rather than a product. Mathematics is much more than the production of answers; it is the process of determining how to quantify, model, et cetera a situation. Rather than
producing an equation or a table or a graph, educational environments that discourage classroom absolutism should emphasise that different representations of the same mathematical concept are
possible and that doing mathematics is often the process of determining what is asked or needed and the affordances and limitations of any mathematical representation that could be used in the
answer. Such classrooms exemplify the inquiry cooperation model (Skovsmose, 1998, p. 200), which refers to a ‘pattern of communication where the student and teacher meet in a shared process of coming
to understand each other’ whilst learning about mathematics. Mathematical representations, as vehicles of communication, play a central role in such classrooms. In the mathematics education
community, the concept of mathematical representation has been based on different theoretical perspectives (English, 1997; Goldin, 1998; Presmeg, 1997). We adopt the widely used definition that a
representation is a configuration that can represent something else (Goldin, 2002).
The representations used to communicate ideas, including those involving mathematical concepts, are socially
embedded and culturally created (Greeno, 1997). Therefore, the manner in and extent to which representations mediate mathematical understanding depend as much on the individuals engaged in the task
as they do on the task itself. The use of multiple mathematical representations and the fostering of an environment that facilitates and values various representations provide a space where learners
can engage with substantial mathematics and develop the tools to become citizens who are productive and active, two qualities of democratic mathematics education (Ellis & Malloy, 2007).
Rationale
and research questions
The most recent reform in curriculum and assessment policy in South Africa aims at producing learners who are able to communicate effectively using visual, symbolic and/or
language skills in various modes (Department of Basic Education, 2011, p. 2). This is expected to occur throughout their learning where opportunity for representation arises. Teachers need to engage
in meaningful discourse with their learners so as to better recognise and appreciate the learners’ use and understanding of specific representations. Such shared exchanges result in a process in
which two groups come to understand the other’s viewpoints as well as the discursive resources and mathematical representations employed to communicate those viewpoints. In this way, deliberative
interaction is exemplified. However, to build on and from the representations of learners, teachers must have both a deep understanding of the different representations (including the affordances and
drawbacks of each) and the flexibility to use the representation that is most appropriate for the mathematical situation and the learners. The issue under investigation in this study is whether
teachers have deep, flexible understandings of mathematical representation that enable them to create democratic environments in their mathematics classrooms. This concern will inform mathematics
teacher educators and the relevant educational department authorities whether these teachers are prepared for changing educational policies (Department of Basic Education, 2011). Accordingly, this
study was designed to explore teachers’ understanding of mathematical representations. For this study, we formulated the following research questions:
What do teachers in a democratic South Africa
believe is meant by the expressions ‘mathematical representations’ and ‘representation in mathematics’?
How do teachers view representations in mathematics?
We ask these questions because we believe
that the use of a variety of mathematical representations for differing purposes is a powerful tool for teachers to foster deliberative interaction in the micro-society of the classroom. The model
presented in Figure 2 is an attempt to show the role of mathematical representations in creating democratic classroom environments.
Teachers need to be able to make flexible use of representations
before they are able to create an environment that allows learners the freedom to use developing representations. Thus, teachers’ representational fluency impacts on their ability to foster
deliberative interaction. In this study, ‘representational fluency’ means that an individual has an abundance of mathematical representations at their disposal for use when reasoning and
communicating in the mathematics classroom. In classrooms where deliberative interaction is embraced, learners’ communication and representational fluency are encouraged and developed through shared
negotiation between and amongst learners and the teacher. Development of representational fluency in learners will better prepare them to interpret mathematical tasks, share their mathematical ideas,
and interpret the mathematical communication of others. Hence, representational fluency can contribute positively to developing active citizens, by giving them a sense of freedom of expression, which
is a concern of democracy, within the mathematics classroom.
With these qualities we hope that the pupils become active citizens, taking ownership of their learning and thus becoming responsible
citizens. Responsibility is a prerequisite for upholding democracy. These connections are indicated in the model in Figure 2, which shows the links between the use of mathematical representations and
the development of a democratic society.
Classroom communication
Mathematical representations can facilitate dialogue between teachers and their learners, if teachers choose not to conform to
classroom absolutism. Of course, classroom communication may be constrained by the pre-described assigned roles of the teacher and the learner (Skovsmose, 1998). However, if the teacher accepts a
shared, negotiated dispensation with the learners then the mathematical representations used become ideal entities to promote interactive dialogue. Vithal (1999) argued that the learners in her study
who used drawings and graphs to represent their expenses and interviews as part of their project, demonstrated that the classroom ‘could serve as the arena for acting out a democratic life’ (p.
29).
It is reasonable to suppose that the development of learners’ understanding of mathematical ideas and their capacity to use representations to communicate and reason about ideas are influenced
by the nature of their teachers’ conceptions of mathematical representations. Teachers need to believe that representations can be used as tools to understand mathematical concepts and solve problems
but also as modes of communicating about these problems and concepts (Roth & McGinn, 1998). In the sciences, Ochs, Jacoby and Gonsales (1994) studied the work of a group of physicists to display how
professionals use representations to create a shared world of understanding. In mathematics education, Moore-Russo and Viglietti (2012) investigated teachers in collaborative problem-solving
situations and found that even when presented with the same task, individuals within the groups use various resources to communicate and reason mathematically, often adopting and adapting the
representations used by their group members. For teachers to value such situations, they need to foster a democratic classroom environment that departs from classroom absolutism.
Stenhagen (2011)
and Allen (2011) have suggested that teachers, teacher educators and curriculum designers place emphasis on teacher beliefs and philosophy in classroom instruction. This article explores teacher
beliefs about mathematical representations, with the aim to discover the beliefs teachers have about the practice of teaching in general and the use of representations in particular. The findings
should provide insight into the deliberative interactions in mathematics classrooms. Teachers using representations in a way that creates deliberative interactions build possibilities for the
classroom to serve as an opportunity for learners to become members of a democratic micro-society and, in doing so, preparing to be active citizens in a democratic society.
Methodology
This study
was qualitative in nature. It has been argued that interpretive researchers use mainly qualitative research methods in order to gain a more in-depth understanding of the participants’ perceptions of
the phenomenon (Cohen, Manion & Morrison, 2007; Henning, 2004). This ties in with our method of inquiry since we intended to find out what teachers in a democratic South Africa believe is meant by
the expressions ‘mathematical representations’ and ‘representation in mathematics’. The research instrument used was an open-ended questionnaire. By allowing for free responses, the instrument
allowed the research team to elicit the opinions of the teachers without influencing them to provide the answers they felt might please us. A non-probability sampling strategy was used. This is in
line with the study because qualitative researchers do not count generalisation as their primary aim but instead seek to represent a particular group (Cohen et al., 2007; Maree & Pieterse, 2007).
The
study participants were 76 teachers from historically disadvantaged schools, pursuing an Advanced Certificate in Education, specialising in high school mathematics teaching in Grades 10−12, at a
South African university. All had successfully completed the first semester course on Differential Calculus.
The questionnaire
The semi-structured questionnaire, with twelve items, was administered
to the 76 participants in the second semester of their study. For this article we consider only the teachers’ responses to the first two items of the questionnaire, namely:
Item 1: What does the
phrase ‘mathematical representation’ mean to you?
Item 2: What comes to mind when someone talks about ‘representation in mathematics’?
The teachers’ responses to the two items were analysed for
emerging themes through a general inductive analysis. Using theoretical memoing (Glaser, 1998), the research team members individually classified the teachers’ responses, then collaboratively
developed categories based on their memos. These initial categories established the themes described in Table 1.
The research team used the 12 categories to individually revisit the data to ensure
that their constant comparison method constituted a saturation of categories. Using a teacher’s response to an item as the unit of analysis, two of the team members independently coded all teachers’
responses. Working independently, the two coded each response as providing evidence, or not, for each of the 12 themes shown in Table 2. The 76 teachers’ responses to the two items provided 152 units
of analysis. The overall inter-coder agreement for the teachers’ responses was 0.95; the related Cohen’s kappa value was 0.80, above the 0.60 that is accepted to represent good agreement (Altman,
1991; Landis & Koch, 1977). After inter-coder agreement was determined, all disparities in assigned codes initially given to the responses were treated in the following manner: each disparity was
identified and then two members of the research team discussed coding until a consensus was reached for each response. The consensus codings were used for all subsequent data analysis.
Once the data
set was completely coded, the research team discussed what they saw emerging from the data and collapsed the initial categories into broader themes. The team members then individually revisited the
data once more to verify that the themes made sense of the data (Thomas, 2006). Finally, the whole team finalised the descriptions of the 12 themes that were used for data coding.
Issues of ethics
and trustworthiness
Ethical clearance was obtained from the university research office for the collection of the data. To comply with the terms of the university research policy, consent to
participate in the study was provided by all the participants.
In qualitative research, reliability and validity are conceptualised as trustworthiness criteria (Golafshani, 2003). To eliminate bias
and increase researcher truthfulness, triangulation in this study was achieved via independent coding and with agreement being reached by consensus. In addition, the researchers sought convergence of
different responses to form common themes from the categories.
FindingsAfter themes were identified and the data set was coded, the research team generated descriptive statistics to complete the
analysis of the data. The first consideration was which categories were most frequently evidenced in the teachers’ responses. Information regarding the 12 identified themes that were evidenced in the
teachers’ responses is summarised in Table 2 in order from most to least common themes.
Note that the columns in Table 2 provide information for each item as well as cumulative information on both
items. In order to read Table 2, consider the first row: 32 teachers’ responses to Item 1, 52 teachers’ responses to Item 2, and 59 teachers’ responses to only one of Item 1 or Item 2 were coded as
evidencing the Examples theme. The data for the Examples theme is illustrated in Figure 3.
During the coding process, it was apparent that many teachers’ responses provided evidence that the
teachers’ beliefs regarding representations addressed many of the 12 identified themes. For this reason, details regarding the number of themes noted in each teacher’s responses to the two items are
provided in Table 3.
In order to read Table 3, consider the first row: 2 teachers’ responses to Item 1, 3 teachers’ responses to Item 2, and 0 teachers’ responses to only one of Item 1 or Item 2
were coded as evidencing 0 themes. The final column shows how teachers responded to both items.
Analysis and discussion of data
In this discussion, note that we use the exact responses of the
teachers, without editing for language or clarity. The notation T1 is used to denote the first teacher in the list, and T76 denotes the last teacher. We will primarily emphasise the themes that are
most pertinent to promoting positive interactions and a democratic environment in the classroom.
The theme Examples was most commonly noted: almost 80% of the teachers gave examples of
representations in their responses. In considering the Examples theme, it is noteworthy that 84 responses (32 to Item 1 and 52 to Item 2) provided examples of representations although only 59
teachers used examples on one item only. This means that a significant proportion (84 − 59 = 25) of the teachers used examples in their responses to both items. Some teachers mentioned only examples
of a single representation, for example, T6 wrote ‘writing mathematics in graphical form’ for Item 1 and ‘graphs’ for Item 2. On the other hand, many other teachers listed a variety of examples of
representations, such as T31:
here mathematical knowledge is represented using verbal, pictures, symbols and manipulatives.
The Representation theme was second most common, addressed by 54% of the
teachers. In this theme, teachers described what representations are and how they are used, especially in response to Item 1. A typical example is T14:
All representations are important based on the
concept in which you are dealing with.
The teachers in this sample displayed knowledge of numerous types of representations (as seen in the responses coded under the Examples theme) as well as a
belief that mathematical ideas can be represented in different ways. This Variety theme was the third most common, with 40% of the teachers showing evidence of it in their responses. The Variety
theme was applied to teachers’ responses that explained that there are many different ways to represent mathematical concepts, ideas or relationships. Such responses show that the teachers strongly
believe that there are different ways to represent mathematical concepts. This is evidence that these teachers do not subscribe to an absolutist view of classroom communication. One example of the
Variety theme was displayed by teacher T17:
Using a variety of ways to capture concepts and relationships. Being able to develop, share and preserve thoughts in mathematics.
What was encouraging, in
terms of promoting democracy, was that this teacher perceived mathematical representation as ‘using a variety of ways’. This abundance of alternatives is crucial to create a democratic mini-society
(classroom) since these ‘variety of ways’ establish a ‘sharing’ of mathematical thoughts, which is a positive contribution to a democratic classroom. In this response, teacher T17 does not specify if
the teacher or learner initiates this variety of ways of sharing. This could imply that either the teacher or the learner could employ a variety of ways; sharing is thus regarded as a two-way
phenomenon, with importance placed on both key players in the classroom. In this case, the teacher would have no dominance, but be regarded as an equal to the learner in the classroom.
The
Communication theme was noted by 38% of the teachers. Whilst mathematical concepts are important, representations are the vehicles through which these concepts are shared with others. As evidenced by
responses that fall under the Communication theme, 21 teachers saw representations as things that convey or express mathematical information. An example of a response in this theme is T31’s response
to Item 1:
Learners can use the representations themselves to communicate their understanding of the (mathematical) concepts to the rest of the class or in smaller groups.
Here the perception of T31
is that mathematical representation is learner driven. This is in keeping with the principles of the South African school curriculum (Department of Education, 2003). T26 put it another way:
The way
you conveying the knowledge of maths to one another.
We interpreted this as being related to communication since the knowledge is being conveyed to others. This comment also reveals that the teacher
does not see the communication as being one-sided; rather it is communication with ‘one another’. This view of communication of mathematics ideas as being both from and to the teacher is aligned to
the inquiry cooperation model (Skovsmose, 1998) because representations are being used as a form of communication where the learner and teacher meet in a shared process of coming to understand each
other, whilst learning about mathematics.
Twenty-six per cent of the teachers had responses that were coded as evidence of the Aid for Understanding theme. This reveals that the teachers see
representations as facilitating the understanding of mathematical concepts or relationships. T3 expressed the view that mathematical representation means:
[a] way of delivering and presenting the
concepts such that the concept is very understandable to learn, encouraging learners to participate willingly and stay in every learner’s mind to his life-long period.
It is notable that T3 included
the need for mathematical concepts to be made understandable to learners. This provides evidence that this individual is a caring and accountable teacher who wants to make learning accessible for
all. This trait is valuable in acknowledging the purpose and function of effective schooling as desired by any democratic society. T3 specifically uses the words ‘to participate willingly’ and
mentions ‘every’ learner. Two aspects emanate implicitly from T3’s response, (1) participation by all learners and (2) freedom of expression. The first aspect evokes the concept of participatory
democracy, which requires that all individuals be afforded the opportunity to take part in the decisions that affect their lives (Devenish, 2005). The second aspect alludes to free will, as evidence
by T3’s use of the word ‘willingly’. Freedom of expression is entrenched in the Bill of Rights within the South African Constitution and is fundamental to liberal democracy (Devenish, 2005). Freedom
of expression is indispensable in establishing mathematical truth in proof or problem solving, and it is a means of fulfilment of human personality since mathematics is a human activity (Department
of Basic Education, 2011, p. 8).
With the recent emphasis worldwide on the need for links between mathematics education and real-life situations, it is no surprise that 24% of teachers identified the
role of representations in portraying Real Life situations. The linking the learning of mathematics to real life is of paramount importance. This is highlighted in the curriculum and assessment
policy statements (Department of Basic Education, 2011), which state in the first specific aim that real-life situations should be incorporated into all sections whenever appropriate. Such linking
will prevent the classroom from being a micro-society in which only mathematical abstractions prevail. T51 places emphasis on real life in his response to Item 1:
Depending on real life situation.
One problem might require a graph to solve (a mathematics task), another may require a table, while some may require a flow chart.
T51 indicated that the type of mathematical representation employed
is dependent on the real-life situation to which it applies. This shows that this teacher places greater emphasis on the need for contextualisation than on the particular mathematical representation.
This could mean that the teacher places the context first in making the choice of which mathematical representation to use to foster the learning of a particular mathematical concept.
In some of the
most common responses, teachers mentioned that representations are used for Problem Solving (17%) and discussed the Tools (12%) that they use to create mathematical representations.
Teacher T61 was
one who associated mathematical representations with problem solving:
Simplify problems by interpreting, analysis using sketches or mind maps.
She also perceived mathematical representation as a
means to simplify the problem situation. Her aim to make mathematics problem solving more understandable and, hence, more accessible to her learners indicates her respect for them.
Across the two
items, nine teachers associated representations with the Tools (equipment or resources) used to create them. For example, T21 wrote:
… being able to use different approach in sketching, use of
computer, …
whilst T46 wrote:
Mathematical representations refer to visual images which are ordinarily associated with pictures in books and drawings on a overhead projector.
These responses suggest
that these teachers see classroom resources and tools as an advantage in trying to present various forms of mathematical representations of mathematical concepts and ideas. They seem to want
everybody to have access to tools and resources, a privilege that most, if not all, of the teachers from historically disadvantaged backgrounds in this study were denied.
The remaining themes were
Flexibility, Visualisation, Differentiation & Selection, and Interrelation.
From the results, it is clear that many of these high school teachers have a rich idea of the roles played by
representation in mathematics. Table 3 presents the 12 themes that were discerned in the teachers’ responses. From these data we know that (28 + 17 + 5 + 1 =) 51 of the 76 teachers thought of
representations in multiple ways since their responses evidenced three or more themes. These results demonstrate that the majority of the teachers had a rich, broad understanding of representations,
as opposed to a narrow or limited understanding, and their roles as would be associated with an absolutist view of mathematics.
ConclusionDespite the teachers being previously disadvantaged, with
access to few resources and a varying quality of initial teacher preparation, their views on mathematical representation provide evidence of their willingness to embrace a democratic approach to
teaching mathematics. The responses have revealed that many of the teachers see representations as being interrelated and the need to move between representations showed a fluid, dynamic and flexible
understanding of mathematics, once more aligned to a democratic classroom.
This abundance of alternatives offered by mathematical representations is crucial to creating a democratic classroom
environment since this ‘variety of ways’ establishes a ‘sharing’ of mathematical thoughts thus allowing for contributions by both learners and the teacher. The choice of mathematical representation
available for classroom activity encourages free will in expression of the relevant mathematical idea. This aspect alludes to an individual’s freedom of expression regarding mathematical concepts
using mathematical representations.
The use of mathematical representations caters for greater learner involvement and participation during classroom activities, which enhances participatory
democracy. The responses of these teachers displayed that mathematical representations are potentially a means of encouraging a form of classroom interaction that promotes dialogue and negotiation in
a democratic South Africa.
We are encouraged by the teachers’ flexible and open-minded approach to the use of representations in the mathematics classroom. We believe that with the display of
mathematics teachers’ knowledge of various kinds of representations, and the various ways in which representations can be used in their classrooms, will enhance their teaching practices. Their
responses suggest that they see the learning of mathematics as a shared process and not a one-way transmission of a product from the teacher to the learner. The findings from this study also suggest
that the teachers want to engage in the inquiry cooperation model (Skovsmose, 1998), rather than following the absolutist tradition, and are keen to use a variety of representations to facilitate
understanding of mathematics processes. The findings also showed that the teachers believed that learners and teachers could use representations as a tool for communication and were positive about
freedom of expression in their classroom. All of the abovementioned findings augur well for the creation of deliberative interactions by these teachers in their classrooms, which we believe will
support the creation of a democratic environment by enhancing the development of active citizens.
More specifically, the data suggest that the teachers believe that mathematical representations can
(1) be used to reason and preserve thought in mathematics classrooms, and (2) be used as a tool for sharing thoughts and communicating ideas related to mathematical tasks. Moreover, the study
suggests that teachers believe that representational fluency (1) creates opportunities for willing participation by both learners and teachers during mathematics classroom interactions, and (2) aids
individuals as they express mathematical ideas freely. These views and beliefs on mathematical representations all facilitate communication, freedom of expression, negotiation and shared meaning, and
understanding, which are vital attributes of deliberative interaction, as displayed in Figure 2. These observed attributes are envisaged to prepare active citizens in the mathematics classrooms, thus
addressing some of the concerns of democracy.
We are mindful however that the study is based on the teachers’ reports of their views of mathematical representations and not on their actual classroom
practice, which may not be aligned with these positive reports. Further study should continue in this line of research to determine whether teachers’ apparently democratic leanings towards
mathematical representations and their uses translate into democratic classroom practices and the facilitation of democratic learning environments.
Acknowledgements
We acknowledge that the
collaboration on this research was aided by funding from a grant entitled Enhancing Secondary Mathematics Teacher Education from the United States Agency for International Development, administered
through the non-governmental organisation Higher Education for Development.
Competing interests
We declare that we have no financial or personal relationship(s) that may have inappropriately
influenced us in writing this article.
Authors’ contribution
The idea to work in this field of representations in mathematics education was encouraged by D.M-R. (State University of New York) and
D.B. (University of KwaZulu-Natal) promoted the conceptual framework of democracy and mathematics education for the research design. The creation and implementation of the research instruments were
done collaboratively by D.B., S.B. (University of KwaZulu-Natal) and D.M-R. Data collection was carried out by D.B. and S.B. The analysis of data was led by D.M-R. and worked on collaboratively with
D.B. and S.B. D.B. wrote the manuscript and it was refined by D.M-R. and S.B.
1.Allen, K. (2011). Mathematics as thinking − A response to “democracy and school math”. Democracy & Education, 19(2),
1–7. Available from http://democracyeducationjournal.org/home/vol19/iss2/10/2.Alrø, H., & Skovsmose, O. (1996). On the right track. For the Learning of Mathematics, 16(1), 2–29. Available from http:/
/www.jstor.org/stable/402481913.Altman, D.G. (1991). Practical statistics for medical research. London: Chapman and Hall.4.Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education.
(6th edn.). London: Routledge.5.Department of Basic Education. (2011). Curriculum and assessment policy statement. Mathematics Grades 10–12. Pretoria: DBE. Available from http://www.education.gov.za/
LinkClick.aspx?fileticket=QPqC7QbX75w%3d&tabid=420&mid=1216 6.Department of Education. (2003). Revised national curriculum statement for Grades 10–12 (General) Mathematics. Pretoria: DOE.7.Devenish,
G.E. (2005). The South African constitution. Durban: LexisNexis Butterworths. PMCid:12877628.Ellis, M.W., & Malloy, E.M. (2007, November). Preparing teachers for democratic mathematics education.
Paper presented at the Mathematics Education in a Global Community conference, Charlotte, NC. Available from http://edweb.csus.edu/projects/camte/monograph1.pdf 9.English, L.D. (1997). Mathematical
reasoning: Analogies, metaphors and images. Mahwah, NJ: Lawrence Erlbaum Associates.10.Glaser, B.G. (1998). Doing grounded theory: Issues and discussions. Mill Valley, CA: Sociology
Press.11.Golafshani, N. (2003). Understanding reliability and validity in qualitative research. The Qualitative Report, 8(4), 597−607. 12.10.1016/S0364-0213(99)80056-113.Goldin, G.A. (2002).
Representation in mathematical learning and problem solving. In L. English (Ed.), Handbook of international research in mathematics education (pp. 197−218). Mahwah, NJ: Lawrence Erlbaum
Associates.14.10.3102/0013189X02600100515.Henning, E. (2004). Finding your way in qualitative research. Pretoria: Van Schaik. PMid:84357116.10.2307/252931017.Maree, K., & Pieterse, J. (2007).
Sampling. In K. Maree (Ed.), First steps in research (pp. 172−180). Pretoria: Van Schaik.18.10.1016/j.jmathb.2011.12.00119.10.1353/con.1994.000320.10.1016/S0732-3123(99)80059-521.10.3102/
0034654306800103522.10.1007/s11858-998-0010-623.Stenhagen, K. (2011). Democracy and school math: Teacher beliefs, practice tensions and the problem of empirical research on educational aims.
Democracy & Education, 19(2). Available from http://democracyeducationjournal.org/cgi/viewcontent.cgi?article=1015&context=home 24.10.1177/1098214005283748 25.10.1007/s11858-999-0005-y | {"url":"https://pythagoras.org.za/index.php/pythagoras/article/view/165/253","timestamp":"2024-11-04T18:14:46Z","content_type":"application/xml","content_length":"41276","record_id":"<urn:uuid:3e2a4207-56bc-4888-80cc-1093dd67b262>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00280.warc.gz"} |
Unit 3 quick check | Quizalize
Feel free to use or edit a copy
includes Teacher and Student dashboards
Measure skills
from any curriculum
Tag the questions with any skills you have. Your dashboard will track each student's mastery of each skill.
With a free account, teachers can
• edit the questions
• save a copy for later
• start a class game
• automatically assign follow-up activities based on students’ scores
• assign as homework
• share a link with colleagues
• print as a bubble sheet
• Greater Than
Equal To
Less than
• Less Than
Greater Than
Equal To
• 2/5 * 2/7
2/1 * 2/7
4/1 * 5/14
4/5 * 10/14
• Q7
What is the main difference between multiplying fractions and adding/subtracting fractions?
Adding and Subtracting you need to find the least common denominator.
You add/ subtract vs multiplying
There isn't a difference
• Q8
When should you simplify fractions while multiplying?
Whenever you can!
What's simplifying?
• Q9
When we see a mixed number when multiplying fractions, what do we have to do?
Change it to an improper fraction
Multiply the fractions and whole numbers separately | {"url":"https://resources.quizalize.com/view/quiz/unit-3-quick-check-0e6f6e3a-7da6-4d20-b7d4-eca239906e42","timestamp":"2024-11-09T06:21:03Z","content_type":"text/html","content_length":"81727","record_id":"<urn:uuid:062be06f-e497-4d17-988f-43944292c774>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00357.warc.gz"} |
Binary Tree Archives - Sam's Blogs
In this tutorial, we will learn how to convert a given binary tree to a doubly linked list using Java programming language. Binary trees are hierarchical data structures where each node can have at
most two children, referred to as the left child and the right child. On the other hand, a doubly linked list is a linear data structure in which each node has a reference to both its previous and
next node.
To convert a binary tree to a doubly linked list, we can use the following algorithm:
1. Start by creating an empty doubly linked list.
2. Traverse the binary tree using any tree traversal algorithm.
3. For each node encountered during the traversal, perform the following steps:
1. Set the left pointer of the current node to the last node in the doubly linked list.
2. If the last node is not null, set its right pointer to the current node.
3. Set the current node as the last node in the doubly linked list.
4. Set the right pointer of the current node to the next node in the traversal (if any).
4. After traversing the entire binary tree, the doubly linked list will be formed.
Here is the Java implementation of the algorithm:
class Node {
int data;
Node left, right;
public Node(int item) {
data = item;
left = right = null;
class BinaryTreeToDoublyLinkedList {
Node root;
Node lastNode;
public void convertToDoublyLinkedList(Node node) {
if (node == null)
if (lastNode == null) {
root = node;
} else {
node.left = lastNode;
lastNode.right = node;
lastNode = node;
The above implementation uses a class named Node to represent each node in the binary tree. The convertToDoublyLinkedList method is responsible for converting the binary tree to a doubly linked list.
It uses a recursive approach to traverse the binary tree and performs the necessary pointer manipulations to form the doubly linked list.
In this tutorial, we have learned how to convert a given binary tree to a doubly linked list using Java. The algorithm involves traversing the binary tree and manipulating the pointers of each node
to form the doubly linked list. This can be a useful technique in certain scenarios where a doubly linked list is required for efficient operations. Feel free to explore further and apply this
concept to solve related problems. | {"url":"https://samsblog.in/project_tag/binary-tree/","timestamp":"2024-11-04T21:57:08Z","content_type":"text/html","content_length":"159611","record_id":"<urn:uuid:53da78d4-2985-40f3-8635-0f4ac3bb8627>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00794.warc.gz"} |
Height Calculator
About Height Calculator
• A height calculator is a simple tool used to estimate a person's adult height based on their current age, gender, and parental heights. This calculator utilizes mathematical formulas and
statistical data to make the prediction. By inputting the required information, such as age, gender, and the heights of both parents, the height calculator provides an estimate of the
individual's projected height once they reach adulthood.
• It's important to note that height calculators provide only rough estimates and cannot accurately predict an individual's final height. Genetic factors, nutrition, and overall health play
significant roles in determining adult height. Therefore, the height calculator should be used for informational and entertainment purposes and not relied upon as a precise measurement.
Frequently Asked Questions (FAQ)
• What is a height calculator?
• A height calculator is a tool that estimates a person's height based on various factors such as gender, age, parental heights, and other relevant measurements.
• How can I estimate my child's height?
• To estimate your child's height, you can use the mid-parental height method, which involves adding the heights of both parents together, dividing by 2, and then adjusting for gender-specific
growth patterns.
• Is a height calculator an accurate tool to estimate my child's height?
• A height calculator based on parents' height provides a rough estimation of a child's potential adult height, but individual variations can lead to some inaccuracy.
• Which parent determines height?
• Both parents contribute to a child's height through their genetic makeup, and various genes from both mother and father influence the final height outcome. | {"url":"http://he.symbolab.com/calculator/other/height","timestamp":"2024-11-03T06:42:47Z","content_type":"text/html","content_length":"173940","record_id":"<urn:uuid:c2e41707-ea1f-40fd-90f8-94961f3484ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00428.warc.gz"} |
Solved: To angles whose sides are opposite rays are called , Mathematics
1. Two angles whose sides are opposite rays are called _____ angles. Two coplanar angles with a common side, a common vertex, and no common interior points are called ____ angles.
A. Adjacent; vertical
B. adjacent; complementary
C. vertical; adjacent
D. vertical; supplementary
Solution Preview :
Prepared by a verified Expert
Mathematics: To angles whose sides are opposite rays are called
Reference No:- TGS01119811
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How Much is 1 Million Coins on Tiktok? - GEGCalculators
How Much is 1 Million Coins on Tiktok?
How Much is 1 Million Coins on Tiktok?
1 million TikTok Coins are worth approximately $7,000 USD. The exact value may vary slightly based on the exchange rates and any promotions or discounts offered by TikTok at the time of purchase.
TikTok Coins to Currency Converter
How much is 1 million coins on TikTok? The price of 1 million TikTok coins can vary depending on regional pricing and promotions, but it typically costs around $1,250 to $1,500 USD.
How much is 29,999 coins on TikTok? The cost of 29,999 TikTok coins can also vary, but it’s generally around $37 to $45 USD.
How much is 2,000 TikTok coins in money? 2,000 TikTok coins are typically priced at approximately $2.50 to $3 USD.
How much is $1 in TikTok coins? For around $1 USD, you can usually purchase approximately 800 to 1,000 TikTok coins.
How much are coins on TikTok in the UK? The pricing for TikTok coins in the UK is generally similar to the pricing in the United States, but it may vary slightly due to currency exchange rates and
regional promotions.
How much does TikTok pay? TikTok does not directly pay users for creating and sharing content. However, popular creators can earn money through brand partnerships, sponsored content, and live stream
gifts from viewers.
What’s the highest gift on TikTok? The highest gift on TikTok is often referred to as the “Diamond Castle” or “Diamond Gift,” and it can be worth thousands of dollars. It is typically sent by viewers
to their favorite creators during live streams as a way to show appreciation.
How many TikTok points is a pound? The exchange rate for TikTok points to pounds may vary, but as of my last knowledge update in September 2021, there wasn’t a fixed exchange rate. It’s best to check
TikTok’s official information or your local currency exchange rates for the most accurate conversion.
How much are TikTok roses worth? The cost of TikTok roses can vary, but they are typically priced between $0.50 and $1 USD each.
How much is a rose on TikTok in the UK? The pricing for TikTok roses in the UK is generally similar to the pricing in the United States, but it may vary slightly due to currency exchange rates and
regional promotions.
How much is a galaxy worth on TikTok? The price of a galaxy gift on TikTok can vary, but it is often one of the more expensive gifts and can cost several hundred to over a thousand dollars.
Why are TikTok coins so expensive? The pricing of TikTok coins is set by TikTok and may appear expensive because it’s a way for the platform to generate revenue while allowing users to support their
favorite creators. The cost can also vary based on the region and any ongoing promotions or discounts.
How much is 7,000 coins on TikTok? The cost of 7,000 TikTok coins typically ranges from approximately $8.75 to $10.50 USD.
How much is a TikTok gift worth? The value of TikTok gifts varies depending on the type of gift and the number of coins it costs. Some gifts can be as low as a few coins, while others, like the
Diamond Castle, can be worth thousands of dollars.
How much money is 1,000 coins on TikTok Live? The value of 1,000 TikTok coins on TikTok Live would depend on the specific gifts or interactions you use them for during a live stream. The actual
monetary value can vary depending on the gifts purchased with those coins.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
Leave a Comment | {"url":"https://gegcalculators.com/how-much-is-1-million-coins-on-tiktok/","timestamp":"2024-11-05T00:24:14Z","content_type":"text/html","content_length":"172451","record_id":"<urn:uuid:dd7113d1-2f27-46d6-ba70-3039cc3f5715>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00399.warc.gz"} |
First derivative with multiple indices
General questions (367)
Is there any simple way to have multiple indices on a single derivative? If I define a derivative in the usual way
then \delta{\mu\nu} is interpreted as a second derivative. I would like \delta{\mu\nu} to represent a single derivative (thus satisfying the product rule) but with multiple indices. This is important
for the implementation of derivatives with respect to tensors for example.
Well, I actually need something like $\delta B^{\mu\nu}/{\delta A^{\rho\sigma}}$. At the moment I have to use 'vary' to compute $\delta B^{\mu\nu}$ and then set $\delta A^{\rho\sigma} \to 1$ at the
end to extract $\delta B^{\mu\nu}/{\delta A^{\rho\sigma}}$. But then if I have a more complicated expression such as $\delta B^{\mu\nu}/{\delta A^{\rho\sigma}} + B^{\mu\nu}B_{\rho\sigma}$, I cannot
simply substitute whatever expression I have for $B^{\mu\nu}$ into it as the derivatives aren't automatically calculated. So I calculate the derivative separately and substitute in the expression.
The problem with that is that I still have to define $\delta{\mu\nu} (= \delta/\delta A^{\mu\nu})$ as a derivative so that Cadabra gets the indices right in the substitutions. Since $\delta\{\mu\nu}$
is a second derivative, my calculation would clash if I perform an operation such as product_rule, thus I don't use $\delta_{\mu\nu}$ for anything other than a dummy variable that gets replaced by
the derivative I obtained with vary. I was just wondering if there is any easier/cleaner route to this and it seems that a derivative with multiple indices would solve my problem. | {"url":"https://cadabra.science/qa/1690/first-derivative-with-multiple-indices?show=1692","timestamp":"2024-11-11T01:50:52Z","content_type":"text/html","content_length":"17560","record_id":"<urn:uuid:90f30b1c-8f4f-458b-aadc-d52aa40652a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00102.warc.gz"} |
Non-Euclidean Geometry
The idea of geometry was developed by Euclid around 300 BC, when he wrote his famous book about geometry, called The Elements. In the book, he starts with 5 main postulates, or assumptions, and from
these, he derives all of the other theorems of geometry. The postulates are as follows:
Illustration of Fifth Postulate
1. Given two points, there is a straight line that joins them.
2. A straight line segment can be prolonged indefinitely.
3. A circle can be constructed when a point for its centre and a distance for its radius are given.
4. All right angles are equal.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on
which the angles are less than the two right angles. [1]
The fifth postulate is clearly more complicated than the other four, and, over the years, many mathematicians were upset by this fact, believing that the fifth postulate should, in some way, be
possible to derive from the first four. However, in attempting to do this, they just ended up coming up with several equivalent postulates. A few are as follows:
1. Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
2. To each triangle, there exists a similar triangle of arbitrary magnitude.
3. The sum of the angles of a triangle is equal to two right angles (180 degrees). [2]
Thousands of years after Euclid introduced the problem, nobody had yet come up with a proof of the fifth postulate; instead they had just come up with many postulates that were equivalent. By the
mid-nineteenth century, mathematicians (Gauss, Bolyai, Lobachevsky, Reimann and Klein, to name just a few) began to explore alternative geometries, where this fifth postulate was not true.
Constant Negative Curvature
Euclidean geometry assumes that there is a unique parallel line passing through a specific point; any other line will cross the original line at some point. However, you could imagine a geometry
where there are many lines through a given point that never pass through the original line. This type of geometry is called hyperbolic geometry.
Constant Positive Curvature
Conversely, a geometry could exist where it is impossible to draw a line that never passes through another line; this type of geometry is called elliptical. Hyperbolic geometry is a mathematical
description of space of negative curvature; elliptical geometry describes space of positive curvature. 2-dimensional space of constant positive curvature is mathematically equivalent to the surface
of a sphere in 3 dimensions; the representation of 2-dimensional space of constant negative curvature in 3-dimensional space could be imagined as looking something like a saddle. Because there are
only three options (multiple lines that never cross the original, one line that never crosses the original, and no lines that never cross the original), there are only three basic types of geometry
possible (hyperbolic, Euclidean, and elliptical, respectively); all other types are combinations of these three. The CurvedLand applet models an example of elliptical geometry; in the applet, you can
see that any line you draw will eventually cross all other lines that you draw.
Curved space was simply a mathematical idea until Einstein developed his general theory of relativity in 1915 [3]. This theory posited that, instead of being a force, gravity was the result of the
curvature of space and time. Finally, the idea of curved space, or non-Euclidean geometry, had a real-world application. Although general relativity's predictions only differed from the classical
model of gravity by a small amount for most observable situations, it accounted for some unexplained inconsistencies perfectly; for example, a small deviation in Mercury's orbit was not explained by
the classical model of gravity, but was perfectly explained by general relativity. Another consequence of general relativity, called gravitational lensing, was observed in 1919 [3], shortly after the
theory was proposed; this effect involves the bending of light from a distant star as it passes by a massive object, and is discussed in more detail in this article about gravitational lensing
published by NASA, and in this video on YouTube, produced as part of the Cosmic Cinema series by the Max Planck Institute of Astrophysics. These observations support the validity of general
relativity, and the actual existence of curved space-time, giving tangible reasons why we should try to better understand non-Euclidean geometry.
1. Euclidean Geometry. (2009). Encyclopedia Britannica. Retrieved November 2, 2009 from http://www.britannica.com/EBchecked/topic/194901/Euclidean-geometry
2. O'Connor, J.J. & Robertson, E.F. (1996). Non-Euclidean geometry. Retrieved November 3, 2009 from http://www.gap-system.org/~history/HistTopics/Non-Euclidean_geometry.html
3. Lightman, A. (2005). Relativity and the Cosmos. Einstein's Big Idea Homepage. NOVA, PBS. Retrieved November 7, 2009 from http://www.pbs.org/wgbh/nova/einstein/relativity/
Close Window
Copyright (C) 2010 Stephanie Erickson, Gary Felder
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Introduction to Ultrasonic Scattering
What is/are Ultrasonic Scattering?
Ultrasonic Scattering
The ultrasonic scattering effects become stronger for higher kerogen contents with smaller rounding coefficients. ^[1] The proposed grating has the grating period in micron scale, which is shorter
than the wavelength of the incident ultrasound leading to an ultrasonic scattering. ^[2] Although the understanding of the nonlinear ultrasonic scattering at closed cracks is essential for the
practical application of nonlinear ultrasonic phased array, it has yet to be elucidated because of the lack of experimental techniques. ^[3] This work evaluates the ultrasonic scattering attenuation
of structures with complex scatterer distributions via experimental and simulation studies. ^[4] To obtain a scattering wavefield, a unique ultrasonic scattering hardware was developed, and signal
processing schemes were suggested. ^[5] Ultrasonic scattering method is a promising technique to evaluate the particle size distribution and/or the elastic properties of particle suspended in liquid.
^[6] The ultrasonic scattering matrix contains the far-field scattering coefficients of a defect for all measurable incident/scattering angles. ^[7] First, considering the large particle size and
low-density contrast characteristics of the hydrate-water dispersion, the influence of multiple scattering among particles cannot be ignored apart from the scattering attenuation caused by each
particle, so the ultrasonic scattering attenuation mechanism considering multiple scattering effects is established to solve the attenuation prediction problem of the hydrate-water dispersion. ^[8] A
mode-converted (longitudinal-to-transverse, L-T) ultrasonic scattering technique was applied to evaluate the variation of microstructural anisotropy in a railroad wheel sample. ^[9] For the latter,
PVA cryogel (PVA-c) was used as the TMM, which was made from a solution of PVA (10% by weight), distilled water, and glass spheres for ultrasonic scattering (0. ^[10] The use of thirty realizations
for each grain-size distribution allows the variation of the ultrasonic scattering to be quantified. ^[11] New concept of fine dust measurement method is suggested based on ultrasonic scattering. ^
[12] Taking advantage of the fact that ultrasonic scattering is orders of magnitude weaker than optical scattering per unit path length, PAT beats this limit and provides deep penetration at high
ultrasonic resolution and high optical contrast by sensing molecules. ^[13] The binary mixture model for ultrasonic scattering from trabecular bone was applied to predict the variations of the
ultrasound parameters with the bone volume fraction (BV/TV) and the trabecular thickness (Tb. ^[14] Spatial resolution of the method is derived by efficiency of ultrasonic scattering at objects,
sensitivity of an ultrasonic radiation-reception matrix and the number of waveguides and the distance between them. ^[15] On the purpose of improving imaging sensitivity and SNR, an ultrasonic
scattering model is developed which takes into account the interaction between the incident ultrasonic fields and the damage, then the reflectivity of the damage surface can be obtained. ^[16] This
study contributes to the understanding of ultrasonic scattering from cells undergoing cell death toward the monitoring of cancer therapy. ^[17] In this study, an ultrasonic NDE system is used for
acquiring ultrasonic scattering signals. ^[18] Ultrasonic scattering in polycrystalline media is directly tied to microstructural features. ^[19] The mode-converted ultrasonic scattering method is
utilized to characterize the structural anisotropy of a phantom mimicking trabecular bone, fabricated using metal additive manufacturing from a high resolution CT image of trabecular horse bone. ^
[20] Ultrasonic scattering method is a promising technique to evaluate the particle size distribution and/or the elastic properties of particle suspended in liquid. ^[1] The mode-converted ultrasonic
scattering method is utilized to characterize the structural anisotropy of a phantom mimicking trabecular bone, fabricated using metal additive manufacturing from a high resolution CT image of
trabecular horse bone. ^[2] This work evaluates the ultrasonic scattering attenuation of structures with complex scatterer distributions via experimental and simulation studies. ^[1] First,
considering the large particle size and low-density contrast characteristics of the hydrate-water dispersion, the influence of multiple scattering among particles cannot be ignored apart from the
scattering attenuation caused by each particle, so the ultrasonic scattering attenuation mechanism considering multiple scattering effects is established to solve the attenuation prediction problem
of the hydrate-water dispersion. ^[2] | {"url":"https://academic-accelerator.com/Manuscript-Generator/Ultrasonic-Scattering","timestamp":"2024-11-03T17:17:52Z","content_type":"text/html","content_length":"475552","record_id":"<urn:uuid:94d999b3-1097-4830-973f-a54063c7a3c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00145.warc.gz"} |
Write the following in expanded fractional form : d) 0.00016-Turito
Are you sure you want to logout?
Write the following in expanded fractional form :
d) 0.00016
Write the numbers by adding the value of digits
The correct answer is: 1/100000
Complete step by step solution:
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/Maths--qd596a254","timestamp":"2024-11-09T00:19:55Z","content_type":"application/xhtml+xml","content_length":"789972","record_id":"<urn:uuid:1601140e-53c1-4704-a261-a6ac33a41d51>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00085.warc.gz"} |
Concentration Measures
Herfindahl {DescTools} R Documentation
Concentration Measures
Computes the concentration within a vector according to the specified concentration measure.
Herfindahl(x, n = rep(1, length(x)), parameter = 1, na.rm = FALSE)
Rosenbluth(x, n = rep(1, length(x)), na.rm = FALSE)
x a vector containing non-negative elements
n a vector of frequencies (weights), must be same length as x.
parameter parameter of the concentration measure (if set to NULL the default parameter of the respective measure is used)
na.rm logical. Should missing values be removed? Defaults to FALSE.
the value of the concentration measure
The same measure is usually known as the Simpson index in ecology, and as the Herfindahl index or the Herfindahl-Hirschman index (HHI) in economics.
These functions were previously published as conc() in the ineq package and have been integrated here without logical changes. NA and weights support were added.
Achim Zeileis <achim.zeileis@r-project.org>
Cowell, F. A. (2000) Measurement of Inequality, in Atkinson, A. B., Bourguignon, F. Handbook of Income Distribution. (Eds) Amsterdam
Cowell, F. A. (1995) Measuring Inequality. Prentice Hall/Harvester Wheatshef
Hall, M., Tidemann, N. (1967) Measures of Concentration, JASA 62, 162-168.
See Also
See Gini, Atkinson and ineq() for additional inequality measures
# generate vector (of sales)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
# compute Herfindahl coefficient with parameter 1
# compute coefficient of Hall/Tiedemann/Rosenbluth
# Some more examples
version 0.99.55 | {"url":"https://search.r-project.org/CRAN/refmans/DescTools/html/Herfindahl.html","timestamp":"2024-11-01T19:10:49Z","content_type":"text/html","content_length":"4036","record_id":"<urn:uuid:88f0c48d-8fa0-4095-8036-0e3b24fd0020>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00388.warc.gz"} |
Hubble's Law of Cosmic Expansion - The Ever Expanding Universe - Physics In My View
Hubble’s Law of Cosmic Expansion – The Ever Expanding Universe
First of all, you should know that Hubble’s Law Of Cosmic Expansion is the most celebrated Paper in the history of physical science. Yup, even more than the physics of black holes.
In fact, if Hubble’s expanding universe theory has not been discovered then we might have been still living in the dilemma that whether our universe is static or expanding or even our universe is
What Is Hubble’s Law Of Cosmic Expansion?
An artistic representation of the “Metric Expansion” of the Universe/Credit: Wikimedia Commons
Hubble’s expansion law states that “Objects Observed In Extra-Galactic Space (Deep Space) Are Shifting Away Or Receding Away From Earth.
In other words, According to Hubble’s law of cosmic expansion, all the galaxies are moving away from our own. A phenomenon in today’s world is known as Cosmic Recession.
Not to mention, Hubble’s law solidified the physics of the expanding universe. Meaning, we are living in the ever-expanding universe.
This expanding universe theory is also known as Cosmological Red-Shift; only and only if the galaxy is receding away from planet earth.
In fact, you can also say that the cosmological redshift was the phenomenon observed by Edwin Hubble which was used to explain the expansion of the universe
On the other hand, if the galaxy is approaching the earth, it’s called Cosmological Blue-Shift. Take a note here. Cosmological redshift is used to measure the physics of the expanding universe.
Therefore, it is not truly a Doppler effect.
Historical Facts About Hubble’s Law
If you don’t know, it was Albert Einstein’s General Theory Of Relativity’s Papers from which the notion of the ever-expanding universe arrived.
As a matter of fact, almost a decade before Edwin Hubble’s declaration of Hubble’s Expansion Equations; in 1922, a Soviet Russian Mathematician and Physicist Alexander Friedmann had derived
Friedmann’s Equations from Einstein’s Field Equation.
Alexander Friedmann showed that the universe might be expanding, directly contradicting the Classical View of the Universe that we are living in a Static Universe or Newtonian Universe.
Moreover, five years after Alexander Friedmann’s Friedmann’s Equations and two years before Edwin Hubble’s Hubble’s Law; a Belgian Astronomer Georges Lemaître proposed The Expanding Universe Theory.
In fact, he also proposed a hypothesis of the Primeval Atom or the Cosmic Egg. A hypothesis, which in today’s world is famously known as the big bang theory of the universe.
Mathematical Representation Of Hubble’s Expansion Equation
Hubble’s law of cosmic expansion/Credit: Wikimedia Commons
The mathematical representation of Hubble’s Law Of Cosmic Expansion is stated as:
V = H × d
V = Galaxy’s recessional velocity
H = Hubble Constant or Hubble Parameter
d = Galaxy’s distance from the one which it’s being compared
Hubble Constant is constant only in space but changes over time; therefore, the currently accepted value is 70 kilometers/second per megaparsec
Hubble’s Law And Einstein’s Blunder
While developing his General Theory Of Relativity; Einstein assumed the universe to be static and dynamically stable. Meaning, the universe is neither expanding nor contracting.
In fact, he was actually using the same assumption Sir Issac Newton did for his newton’s law of universal gravitation. An assumption that we know as a Static Universe (Steady-State Theory).
But, when the experimental observations of General Relativity were compared with the theoretical prediction; both the results (experimental and theoretical) were different.
In other words, Einstein theoretically assumed the universe to be a static one. On the contrary, the experimental observation showed that the universe is rather expanding or it will contract
(observations like bending of light by large masses or the precession of the orbit of mercury).
Therefore, in order to counteract the observational results, Albert Einstein had to modify his General Theory Of Relativity by introducing the Cosmological Constant.
A constant that is used to produce a static solution to counter the effect of gravity, simply to produce a perfectly Static and Flat Universe.
However, after the discovery of Hubble’s Law Of Cosmic Expansion, Einstein had to abandon his work on Cosmological Constant.
Albert Einstein himself called his assumption of the Cosmological Constant; the biggest blunder.
Well, what do you think? Was the hypothesis of cosmological really the biggest blunder? Or, maybe we just don’t know how to effectively utilize it….!!! I mean, there was a time when people believed
in the Aristotelian universe.
But, in today’s world, it comes in the list of one of the obsolete cosmological models of the universe. What I want to say is there is always a possibility for new laws in the world of physical
science. What do you think? Is Hubble’s law of cosmic expansion is really untouchable?
That’s it for this post. If you like this article, share it if you like it, like it if you share it. You can also find us on Mix, Twitter, Pinterest, and Facebook.
14 thoughts on “Hubble’s Law of Cosmic Expansion – The Ever Expanding Universe”
1. I’ve never actually heard of this before! But you learn something new every day and this was my new thing learned for today.
2. This is all so interesting! I was actually really into Physics in high school and was the VP of our Physics club, I wish I kept up on it more as an adult.
3. Now I can see why my head wasn;t egg-shaped and full of existential information that I would barely ever use.
4. This is such an interesting article to read. Very informative and detailed about hubbles law of cosmic expansion. I really love to learn more in Physics and I have learn a lot now.
5. The universe is a continuous mystery which often leaves us wandering about the possibilities. May I ask what`s your website niche?
❀ Grace ❀
6. I’m sorry to disappoint the author, and please the astronomers, but the Hubble Constant is a truly fixed constant, as demonstrated by the following equation:- 2 X a mega parsec X C, divided by Pi
to the power of 21 = 70.98047. For this equation, a parsec is the standard 3.26 light years. This equation comes from ‘The Principle of Astrogeometry (Kindle Books). As the Hubble Constant is
‘fixed’ in value, its reciprocal of 13.778 billion light years is also ‘fixed’, and so cannot be the age of the universe, but is the Hubble horizon distance ONLY. This proves the random ‘big
bang’ is a fake hypothesis, based on the erroneous Friedmann hypothesis. It’s certain you will not understand this equation, but if you do, a proper scientific counter argument will be welcome. A
Professor at Imperial College in Kensington, London described this Hubble equation as ‘elegant’, David Hine.
7. All the contents you mentioned in post is too good and can be very useful. I will keep it in mind, thanks for sharing the information keep updating, looking forward for more posts.Thanks marriage
certificate attestation in Nigeria
8. thankyou for sharing you experience.
9. well, i would say you never say no. you can continue now.
10. well, i will say if you cant use then may be your children can. i mean if they are into physics or something.
11. thankyou so much for your valuable feed back.
12. ahmm !! in my view its physics. i mean i try to cover whole but its not possible though.
13. well, brother i got your view point. may be i wont understand the mathematics you wrote, but i definately understood the physics behind it. on the other hand, i am partially agreeing with you. i
mean i wont say that big bang is fake. i would say that there are so many big bangs that had happened before. and on the other hand, there will be so many big bangs that will happen in the future
too. just because we cant see something that dosent mean it does not exist.
14. thankyou so much for you precious reply. hope you will see more from my blog.
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product/algerbra Related topics: pre-alegbra for dummies
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Author Message
JolijC Posted: Thursday 12th of Feb 15:35
Hello, I have been trying to solve equations related to product/algerbra but I don’t seem to be getting anywhere with it . Does any one know about pointers that might help me?
From: Ohio
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ameich Posted: Friday 13th of Feb 15:29
You don’t need to ask anybody to solve any sample questions for you; in fact all you need is Algebrator. I’ve tried quite a few such algebra simulation software but Algebrator is
a lot better than most of them. It’ll solve all the questions that you have and it’ll even explain each and every step involved in reaching that answer. You can work out as many
examples as you would like to, and unlike us human beings, it would never say, Oh! I’ve had enough for the day! ;) Even I had some problems in solving questions on decimals and
radicals, but this software really helped me get over those.
From: Prague, Czech
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Sdefom Koopmansshab Posted: Sunday 15th of Feb 13:58
Algebrator is the perfect math tool to help you with assignments . It covers everything you need to know about evaluating formulas in an easy and comprehensive manner . Math had
never been easy for me to grasp but this software made it very easy to understand . The logical and step-by–step method to problem solving is really an advantage and soon you will
discover that you love solving problems.
From: Woudenberg,
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Badtj Posted: Tuesday 17th of Feb 09:58
Hey! That sounds alright. So where did you go to get the program ?
From: Belgium
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ZaleviL Posted: Wednesday 18th of Feb 13:24
You can get the program here https://softmath.com/links-to-algebra.html.
From: floating in
the light, never
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Dxi_Sysdech Posted: Thursday 19th of Feb 16:58
I remember having problems with graphing, geometry and like denominators. Algebrator is a really great piece of algebra software. I have used it through several algebra classes -
Pre Algebra, Remedial Algebra and Algebra 2. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended.
From: Right here,
can't you see me?
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Find the sum of the arithmetic series $5+11+17+\
Hint: An arithmetic series is the sum of a sequence $\left\{a_{k}\right\}, k=1,2, \ldots,$ in which each term is computed from the previous one by adding (or subtracting) a constant d. Therefore, for
$k>1$, $a_{k}=a_{k-1}+d=a_{k-2}+2 d=\ldots=a_{1}+d(k-1)$
The sum of the sequence of the first n terms is then given by:
${{S}_{n}}\text{ }\equiv \sum\limits_{k=1}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{n}{\left[ {{a}_{1}}+(k-1)d \right]}$
$\mathrm{Sn}=\mathrm{n}(\mathrm{a} 1+\mathrm{an}) 2$
Complete step-by-step answer:
An arithmetic sequence is a sequence where the difference d between successive terms is constant. An arithmetic series is the sum of the terms of an arithmetic sequence. The nth partial sum of an
arithmetic sequence can be calculated using the first and last terms as follows:
$\mathrm{Sn}=\mathrm{n/2}(\mathrm{a_1} +\mathrm{a_n}) $
An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference. A
geometric sequence is a sequence with the ratio between two consecutive terms
Given series is $5+11+17+\ldots+95$.
Thus, $a=5, d=11-5=6, l=95$
Therefore, $\mathrm{S}_{\mathrm{n}}=\dfrac{\mathrm{n}}{2}(\mathrm{a}+\mathrm{l})$
Note: Using the sum identity
$\sum_{k=1}^{n} k=\dfrac{1}{2} n(n+1)$ then gives $S_{n}=n a_{1}+\dfrac{1}{2} d n(n-1)=\dfrac{1}{2} n\left[2 a_{1}+d(n-1)\right]$
Note, however, that $a_{1}+a_{n}=a_{1}+\left[a_{1}+d(n-1)\right]=2 a_{1}+d(n-1)$
So $S_{n}=\dfrac{1}{2} n\left(a_{1}+a_{n}\right)$ or n times the arithmetic mean of the first and last terms. | {"url":"https://www.vedantu.com/question-answer/find-the-sum-of-the-arithmetic-series-class-11-maths-cbse-5fb3442ab7fb205f4fdbf888","timestamp":"2024-11-14T17:55:55Z","content_type":"text/html","content_length":"161294","record_id":"<urn:uuid:39574fd4-2b06-4242-a020-595efdcaf7b3>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00366.warc.gz"} |
What is 4 and 3/4 as a decimal?
What is 4 and 3/4 as a decimal?
So the answer is that 4 3/4 as a decimal is 4.75.
How do you change a fraction to a decimal on a Casio FX 991ES?
Press the S<>D key, (located right above “DEL”). This will shift the answer from fraction to decimal.
How do you change a decimal to a fraction on a Casio FX 9750gii?
To turn the unit off, press the yellow L key, then O key. The x key is a toggle key that will change answers or entered numbers back and forth from decimal to fraction form. The d key operates like
the back arrow on a web browser; it will take you back one screen each time you select it.
How much is Casio FX cg50?
CASIO PRIZM FX-CG50 Color Graphing Calculator
List Price: $118.99 Details
You Save: $39.99 (34%)
How to convert fractions to decimal in Casio?
Fraction To Decimal – Casio CFX/AFX/FX/Prizm – Universal Casio Forum. 1. First turn on the fx-115ES. 2. Press SHIFT. 3. Press MODE SETUP. 4. Then press 2 for LineIO. Now you can have it default
display decimals instead of fractions. 3. Press MODE SETUP.
How to switch from fractions to decimals in lineio?
1. First turn on the fx-115ES 2. Press SHIFT 3. Press MODE SETUP 4. Then press 2 for LineIO Now you can have it default display decimals instead of fractions. Can you tell me how to switch it back to
fractions now?
Is there any way to change the default to display decimal rather than fraction?
Okay, I have a Casio fx-115 ES. The answer always displays as a fraction. I almost always need to use decimal. I know you can use the S<=>D button to convert, but this is becoming very annoying for
me. Is there any way I can change the default to display decimal rather than fraction? Thanks for any help. Hello Mitoca and welcome to the UCF! 1.
How to display decimals instead of fractions in UCF?
Hello Mitoca and welcome to the UCF! 1. First turn on the fx-115ES 2. Press SHIFT 3. Press MODE SETUP 4. Then press 2 for LineIO Now you can have it default display decimals instead of fractions.
Thank you so much! I knew there had to be simple way to do it. Appreciate it greatly! Hello Mitoca and welcome to the UCF! 1. First turn on the fx-115ES
What is 4 and 3/4 as a decimal? So the answer is that 4 3/4 as a decimal is 4.75. How do you change a fraction to a decimal on a Casio FX 991ES? Press the S<>D key, (located right above “DEL”).
This will shift the answer from fraction to decimal. How do you change… | {"url":"https://www.ohare-airport.org/what-is-4-and-34-as-a-decimal/","timestamp":"2024-11-06T17:52:03Z","content_type":"text/html","content_length":"34911","record_id":"<urn:uuid:fb47e075-0996-4856-987d-7cdf1379eeac>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00689.warc.gz"} |
RIES - Find Algebraic Equations, Given Their Solution
First page . . . Back to page 3 . . . Forward to page 5 . . . Last page (page 6)
Some fairly serious maths has been done partly with the aid of RIES. For example, in 2014 December is was asked if there is a closed form for
SIGMAk=1..inf [ -1(k+1) ζ(2k) / 2(2k-1) ] = ζ(2)/2 - ζ(4)/23 + ζ(6)/25 - ...
where ζ() is the Riemann Zeta function. The series converges quickly and ries was used to find a guess for the answer:
The answer is π/2 coth(π/2) - 1 = π/(eπ-1) + π/2 - 1, and can be proven reasonably easily using the series sum definition of the Zeta function, an identity (based on the calculus of residues) for the
cotangent, and the simple identity coth x = i cot(ix).
This is ostensibly the primary purpose of ries: using some approximate measurement or calculation as the basis of a "wild guess" at an exact solution.
Here is an example: xkcd 356 presents an "infinite grid of resistors" problem that looks like it should have a closed-form solution, but with no particularly obvious way to find it.
Matthew Beckler performed circuit simulations of a number of grids ranging in size from the minimal set of 7 resistors up to 80 thousand. As the size of the simulated grid grows, the answer gets
closer to what it should be for an infinite grid. His last 4 answers were: ..., 0.7743, 0.7735, 0.7733, 0.7733. One might guess that the limit, and true answer to the resistor problem, is somewhere
around 0.7732 or 0.7733. Let's tell ries to search around 0.77325, with a maximum error of 0.0001:
Thus the simplest "wild guess" answer seems to be 4/π-1/2 = 0.773239... ohms. As others eventually showed, that is indeed the answer.
However this was a pretty lucky guess. We could have guessed that the answer is within 0.001 of the final measurement 0.7733, and ries would have told us the answer was √π-1 = 0.772453... To someone
making a wild guess, this appears at least as believable as 4/π-1/2, which illustrates why wild guessing with ries isn't actually all that useful.
Because ries uses derivatives to report how far an inequality deviates from an (exact) equality, you can use it to iterate Newton's Method. Suppose you know that the cube root of 3 starts with the
digits 1.442, and want to find more digits. ries 1.442 yields the result:
x^3 = 3 for X = T + 0.00024957 {51}
ries 1.44224957 (other answers not shown) x^3 = 3 for x = T + 3.07408e-10 {51}
Then add 3.07408×10-10 (notice you have to add a zero):
ries 1.442249570307408 (other answers not shown) x^3 = 3 for x = T + 2.22045e-16 {51} (and so on...)
ries uses about 16 digits in its internal calculations, so this is about as precise as you can get. The actual value of the cube root of 3 (to 25 digits) is 1.4422495703074083823216383... Although
this specific example is a case of extreme computational overkill, the same method can be used for things that cannot be computed directly, such as the value of x for which xx=10.
If you're like most people who took Trigonometry in high school, you remember that there was something called "trigonometric identities", but nothing more. Maybe you remember that sine could be
turned into cosine somehow, but that's about it.
With ries you can rediscover all the identities, and probably a few more you never knew existed. For this example we'll use a profile called "rad.ries":
Using a calculator (or a command like ries -prad.ries 1 --eval-expression 1S) you can find out the sine of 1 radian, which is 0.841470984807897. Naturally, if you give ries this number, it will tell
We want to discover other ways to get the same value without using the sine function. To do this we simply tell ries that sin() is not allowed using the -NS option:
So ries has told us that our number x (which we know to be the sine of 1) divided by the cosine of 1 is equal to the tangent of 1. Therefore sin(1)/cos(1) = tan(1). Going back to your calculator, you
can verify that sin(x)/cos(x) = tan(x) for lots of values of x. This is our first trigonometric identity.
Let's find another one: while still using the target value 0.841470984807897, exclude the tangent function too:
Move things around a bit with a little algebra, and this tells us that sin(1)2 + cos(1)2 = 1, which generalises into another identity: sin2x+cos2x=1. (The square of a trig function is usually written
"sin2x" rather than "sin(x)2" or "(sin(x))2" because the latter is a bit cluttered and the other might be confused with "sin(x2)"). | {"url":"https://www.mrob.com/pub/ries/index-4.html","timestamp":"2024-11-04T15:17:47Z","content_type":"text/html","content_length":"13864","record_id":"<urn:uuid:ff3cee10-d2ca-4cbb-b204-10fe7a5f19ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00751.warc.gz"} |
param.matching use matching (1, default) or not (0)
param.ordering desired symmetric reordering
"amd" (default) Approximate Minimum Degree
"metisn" Metis Multilevel Nested Dissection by nodes
"metise" Metis Multilevel Nested Dissection by edges
"mmd" Minimum Degree
"rcm" Reverse Cuthill-McKee
"amf" Approximate Minimum fill
"indset" Independent sets
"pq" ddPQ strategy from ARMS
param.droptol drop tolerance for the LU factors.
By default, 1e-2 is chosen. Here you can overwrite the default values.
param.droptolS drop tolerance for the approximate Schur complement.
By default, 0.1*param.droptol is chosen and recommended. Here you can overwrite the default values.
param.droptolc threshold for dropping small entries from the constraint part, if present (as indicated by negative entries in options.ind)
default: 0
param.condest Norm of the inverse triangular factors.
by default, 5 is chosen. Here you can overwrite the default values. As a rule of thumb, small CONDEST will allow more entries to be dropped (which may accelerate the
computation and save memory) but at the same time, more levels will be necessary (which in turn may slow down the computation and increase the memory). Typically, values
between 5 and 100 make sense. CONDEST=5 will make ILUPACK behave like AMG and select many coarse grid nodes. If you have a PDE-based problem, this might be the right
choice. Otherwise, CONDEST=100 will safeguard the ILU computation and prevent the norm of the inverse triangular factors from becoming too large.
param.restol Residual tolerance for the iterative solver.
The built-in iterative solver (CG/SQMR/restarted GMRES by default) will use this tolerance to terminate whenever the backward error (resp. relative energy norm) is less
than this threshold. By default, eps^(3/4)~1e-12 is chosen for double precision, eps^(3/4)~1e-6 is chosen for single precision.
param.elbow elbow space for the ILU.
Here please pass an estimate how much memory you are willing to spend. ILUPACK will try to keep the ILU inside the range you passed. The elbow space is a real number
measuring the number of nonzeros of the ILU relative to the fill of the original matrix. By default, 10 is chosen. Note however, if your estimate is too small, ILUPACK will
adapt elbow and overwrite this parameter. As long as enough memory is available, the ILU will be successfully computed.
param.lfil Maximum number of nonzeros per column in L (resp. per row in U).
By default n+1 is chosen, i.e. this option is disabled. You can limit the amount of memory by using some smaller value, e.g. A.ia[A.nc]-1 is the fill of A and ELBOW*(A.ia
[A.nc]-1.0)/A.nc would restrict the maximum number of fill to the average number of nonzeros of A per column (or per row) times the ELBOW. Note however that this parameter
cuts off the fill in L and U by brute force. It recommended NOT to use it.
param.lfilS Maximum number of nonzeros per row in S (approximate Schur complement).
By default n+1 is chosen, i.e. this option is disabled. You can limit the amount of memory by using some smaller value, e.g. A.ia[A.nc]-1 is the fill of A and ELBOW*(A.ia
[A.nc]-1.0)/A.nc would restrict the maximum number of fill to the average number of nonzeros of A per column (or per row) times the ELBOW. Note however that this parameter
cuts off the fill in S by brute force. It recommended NOT to use it.
param.typetv Type of test vector.
for some PDE-based problems it might be sensible to ensure that the ILU is exact when being applied to some given test vector. By default this option is disabled ("none").
If you want to use this feature you can either use "static" to pass a fixed test vector to the ILU. Or you can use any other string. In this case, using reverse
communication principle, on every level you need to pass a test vector to the ILU. The ILU passes to you the current coarse grid system and an initial guess for the test
vector. You have to return your own test vector. On entry to the first level, this initial guess is simply the test vector you prescribed. On any subsequent level, this
will be your old test vector restricted to the coarse grid.
param.tv Test vector.
If you decide to pass a test vector, then pass the associated pointer. ILUPACK will make its own copy inside AMGfactor, and you can release the memory if you like. In
PDE-based applications, a typical guess is the vector with all ones.
param.amg type of algebraic multilevel method
"ilu" multilevel ILU
"amli" on each coarse grid, an inner iteration is used based on flexible iterative solvers (e.g. fGMRES) to solve the inner coarse grid system, preconditioned by the
associated ILU. Note that this requires to maintain all coarse grid systems and increases the amount of memory.
"mg" full multigrid with pre- and post smoothing, V-cycle, W-cycle or flexible cycle is chosen. Essentially, the multilevel ILU is used to define interpolation and
restriction operators as well as the coarse grid systems, while the other components are set up as in the usual multigrid framework. Not that the flexible cycle
does not pre-select the number of coarse grid solves a priori (e.g. (1 for V-cycle, 2 for W-cycle), but on on each coarse grid, an inner iteration is used based
on flexible solvers to solve the inner coarse grid system, preconditioned by the associated full multigrid solver. Note that this type of multigrid
preconditioning requires to maintain all coarse grid systems and increases the amount of memory.
param.npresmoothing Number of pre-smoothing steps.
If classical multigrid is selected (param.amg="mg";), then here you can set the number of pre-smoothing steps. default: 1
param.npostsmoothing Number of post-smoothing steps.
If classical multigrid is selected (param.amg="mg";), then here you can set the number of post-smoothing steps. default: 1
param.ncoarse Number of coarse grid solves.
Except for multilevel ILU (i.e. param.amg="amli"; or param.amg="mg";), here you define how often the coarse grid solve is performed. By default, only one coarse grid solve
is used (V-cycle). The choice param.ncoarse=2; would correspond to a W-cycle. Note however, if a negative value is passed, a flexible solver is invoked, i.e. the number of
coarse grid solves varies from one grid to another and from one step to the next one.
param.presmoother Type of pre-smoother.
If full multigrid is used (param.amg="mg";), then here you can choose between built-in smoothers or your own hand-made smoother.
(a) "gsf"(default) Gauss-Seidel forward
(b) "gsb" Gauss-Seidel backward
(c) "j" (damped) Jacobi
(d) "ilu" ILU on the fine grid system
(e) any other string that does not match (a)-(d) will cause AMGsolver to use reverse communication principle in order to let you provide your own smoother. In that case
ILUPACK will give you the matrix, the right hand side and an initial solution (typically 0). You have to override the initial solution
param.postsmoother Type of post-smoother.
If full multigrid is used (param.amg="mg";), then here you can choose between built-in smoothers or your own hand-made smoother.
(a) "gsf" Gauss-Seidel forward
(b) "gsb"(default) Gauss-Seidel backward
(c) "j" (damped) Jacobi
(d) "ilu" ILU on the fine grid system
(e) any other string that does not match (a)-(d) will cause AMGsolver to use reverse communication principle in order to let you provide your own smoother. In that case
ILUPACK will give you the matrix, the right hand side and an initial solution (typically 0). You have to override the initial solution
param.FCpart Pre-selection of coarse grid nodes.
In some PDE-based applications it might be useful to select some coarse grid nodes in advance. Essentially this strategy uses a Ruge-Stueben-like heuristic strategy. If a
test vector is available, the coarsening strategy is applied to the matrix, which is diagonally scaled from the right with the test vector.
(a) "none"(default) leave the coarsening process to ILUPACK, inverse-based strategy will construc a coarse grid on its own.
(a) "yes" Some nodes are pre-selected as coarse grid nodes, ILUPACK might add some further nodes.
param.typecoarse Type of coarse grid system.
By default the coarse grid system S is computed from A and the ILU in
/ L11 0 \ / D11 0 \ / U11 U12 \
typical ILU manner, i.e. if A ~ | | | | | |, then S
\ L21 I / \ 0 S / \ 0 I /
is defined via S:= A22-L21*D11*U12. Alternatively one could compute W21 ~ L21*L11^{-1}, Z12 ~ U11^{-1}*U12
/ -Z12 \
and define S via S:= [-W21 I]*A* | | . This would refer to an AMG-like
\ I /
strategy to compute a coarse grid system. available are
(a) "ilu"(default) ILU-type coarse grid system
(a) "amg" AMG-type coarse grid system
param.nrestart Number of steps before GMRES is restarted.
The iterative solver uses restarted GMRES (resp. fGMRES). By default, 30 steps are computed, before the method is restarted. Note that a smaller number reduces the memory,
while a larger number can improve the convergence.
param.mixedprecision require the computation of the preconditioner in single precision
param.contraction contraction factor < 1 of the residual for inner flexible solver when AMLI or classical multigrid is used and options.ncoarse<0 (flexible coarse grid solver)
param.coarsereduce If different from zero, then the L21 and the U12 block are discarded solving with L,U is done implicitly via L11,U11 and A21 (resp. A12). If set to zero, then L21, U12 are
default: 1.
param.decoupleconstraints This allows for saddle point type problems to explictly decouple the connections between the constraint part and the free part. Applied on every level, this allows for
smaller coarse grid matrices. If set to zero, then the additional decoupling is not applied.
default: 1.
Level 1
Initial system, reordered
Initial system here: no permutation
(no initial preprocessing)
Initial Level, reordered after inverse-based ILU has been applied
Level 2
Level 2 Level 2, reordered
Initial system (Reverse Cuthill-McKee,
regular reordering)
Level 2, reordered after inverse-based ILU has been applied
Level 3
Level 3 Level 3, reordered
Initial system (Reverse Cuthill-McKee,
regular reordering)
Level 3, reordered after inverse-based ILU has been applied
Level 4
Level 4 Level 4, reordered
Initial system (Reverse Cuthill-McKee,
regular reordering)
Level 4, reordered after inverse-based ILU has been applied
Level 4
Level 4, reordered again. here ddPQ
(switched to final pivoting)
Level 4, reordered after inverse-based ILU has been applied | {"url":"http://ilupack.tu-bs.de/doc/ilupack.html","timestamp":"2024-11-02T14:26:47Z","content_type":"text/html","content_length":"63177","record_id":"<urn:uuid:469fa2e9-cb60-440c-adf8-a6744c071b82>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00134.warc.gz"} |
Best Feet to Inches Conversion Calculator (ft to in) with a fraction - convertmastertoolFeet to Inches Calculator with Fraction - Convert Feet to Inches Online
Feet to Inches with Fraction
Welcome to the "Feet to Inches Converter with fraction"
How to Convert Feet to Inches
To convert a measurement from feet to inches, you can utilize a straightforward formula. Multiply the length in feet by the conversion ratio of 12 inches per foot. This is because there are 12 inches
in one foot.
The conversion formula can be expressed as follows:
inches = feet × 12
By applying this formula, you can determine the length in inches by multiplying the length in feet by 12.
The foot is a unit of length used for measurement, and it has various characteristics and uses. Here is some information about the foot:
• The foot is equivalent to 12 inches or 1/3 of a yard. This means that there are 12 inches in one foot and three feet in one yard.
• In terms of the metric system, since the international yard is defined as exactly 0.9144 meters, one foot is equal to 0.3048 meters.
• The foot is primarily used as a unit of length in the United States customary and imperial systems of measurement.
• The abbreviation for feet is “ft.” For example, you can represent one foot as 1 ft.
• An alternative way to denote feet is by using the prime symbol (′), but it is common to use a single-quote (‘) instead for simplicity. Thus, 1 ft can be written as 1’ as well.
• When measuring in feet, a standard 12-inch ruler or a tape measure is typically used, although there are other measuring devices available for this purpose.
• The term “linear feet” is sometimes used to refer to measurements expressed in feet. It simply indicates a measurement of length in feet.
• If you need to perform calculations involving feet and other units like inches, centimeters, or meters, you may find our “inches -to-feet with fraction” calculator useful.
• An inch is equal to 1/12 of a foot or 1/36 of a yard. In terms of the metric system, with the international yard defined as precisely 0.9144 meters, one inch is equivalent to 2.54 centimeters.
• The inch is predominantly used as a unit of length in the United States customary and imperial systems of measurement.
• The abbreviation for inches is “in.” For instance, you can represent one inch as 1 in.
• Alternatively, inches can be denoted using the double-prime symbol (″). However, it is common to substitute it with a double-quote (“) for simplicity. Thus, 1 inch can be expressed as 1″.
• A quarter has a diameter of approximately .955 inches, just slightly smaller than 1 inch.
• The standard ruler is 12 inches long and serves as a widely utilized tool for measuring length in inches. Additionally, a tape measure, ranging from 6 feet to 35 feet in length, is frequently
employed for inch measurements. Other devices employed for measuring in inches include scales, calipers, measuring wheels, micrometers, yardsticks, and even lasers.
• If you wish to delve deeper into the concept of inches and its usage in measuring length, you can explore further information.
To measure length accurately in inches, it is recommended to utilize a ruler or tape measure, which can be obtained from local retailers or home centers. Ensure that you select the appropriate type
of measurement device—imperial, metric, or a combination—to suit your specific requirements.
An inch is a unit of length commonly used for measurement, and it possesses several characteristics and applications. Here is a breakdown of information about the inch:
Link this Tool to Own Website
Link this Tool to Your Website
Note: To link this tool to your website, copy the HTML code above and paste it into your website's HTML editor. The button option will also include the necessary CSS automatically. | {"url":"https://convertmastertool.com/feet-to-inches-with-fraction/","timestamp":"2024-11-03T17:21:42Z","content_type":"text/html","content_length":"623065","record_id":"<urn:uuid:eb1ce935-b2d5-4460-a1cd-d732f2582d9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00617.warc.gz"} |
Overview: INTPOINT Procedure
The INTPOINT procedure solves the Network Program with Side Constraints (NPSC) problem (defined in the section Mathematical Description of NPSC) and the more general Linear Programming (LP) problem
(defined in the section Mathematical Description of LP). NPSC and LP models can be used to describe a wide variety of real-world applications ranging from production, inventory, and distribution
problems to financial applications.
Whether your problem is NPSC or LP, PROC INTPOINT uses the same optimization algorithm, the interior point algorithm. This algorithm is outlined in the section The Interior Point Algorithm.
While many of your problems may best be formulated as LP problems, there may be other instances when your problems are better formulated as NPSC problems. The section Network Models describes typical
models that have a network component and suggests reasons why NPSC may be preferable to LP. The section Getting Started: NPSC Problems outlines how you supply data of any NPSC problem to PROC
INTPOINT and call the procedure. After it reads the NPSC data, PROC INTPOINT converts the problem into an equivalent LP problem, performs interior point optimization, then converts the solution it
finds back into a form you can use as the optimum to the original NPSC model.
If your model is an LP problem, the way you supply the data to PROC INTPOINT and run the procedure is described in the section Getting Started: LP Problems.
You can also solve LP problems by using the OPTLP procedure. The OPTLP procedure requires a linear program to be specified by using a SAS data set that adheres to the MPS format, a widely accepted
format in the optimization community. You can use the MPSOUT= option in the INTPOINT procedure to convert typical PROC INTPOINT format data sets into MPS-format SAS data sets.
The remainder of this chapter is organized as follows: | {"url":"http://support.sas.com/documentation/cdl/en/ormpug/63352/HTML/default/ormpug_intpoint_sect001.htm","timestamp":"2024-11-11T23:25:56Z","content_type":"application/xhtml+xml","content_length":"12680","record_id":"<urn:uuid:123601a6-1eeb-4643-9504-192a08a934cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00661.warc.gz"} |
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ELASTICITY | Relation among elastic constants | Thermal Stress | Torsion of a cylinder - akritinfo.com
ELASTICITY | Relation among elastic constants | Thermal Stress | Torsion of a cylinder
Relation among elastic constants : ( Y, σ, k, n, x)
Young’s modulus, bulk modulus and Rigidity modulus of an elastic solid are all together are known as elastic constants. When a deforming force is acting on a solid continuously, it results in the
change in its original dimension. In such cases, we can use the relation between elastic constants to understand the magnitude of deformation.
1. Relation among Y, σ, k –
2. Relation among Y, σ, n –
3. Relation among Y, σ, x –
4. Relation among k, n, σ –
5. Relation among Y, k, n –
6. Relation among x, n, σ –
7. Relation among n, k, x –
*** IF NECESSARY, I WILL UPLOAD THE PROOF OF THE ABOVE RELATIONS.
Limiting value of Poisson’s ratio (σ) –
The Poisson’s ratio of a stable, isotropic and linear elastic material is greater than −1 or less than 0.5 for the requirement for Young’s modulus, the shear modulus and bulk modulus to have positive
values. Most materials have Poisson’s ratio have the values between 0 and 0.5.
We know, Y = 3k (1 – 2σ)
Since Y and k are +ve, hence
1 – 2σ ≥ 0
– 2σ ≥ -1
σ ≤ 1/2
Again, Y = 2n(1+σ)
Since Y and n are +ve, hence
1 + σ ≥ 0
σ ≥ -1
Considering both cases, we may write
-1 ≤ σ ≤ 1/2
Note :
“σ is negative” which means a longitudinal extension is associated with a lateral extension, which is practically impossible.
Practical or acceptable limit of σ is
0 ≤ σ ≤ 1/2
A solid cylinder is extended longitudinally such that its volume remains constant. Find its Poisson’s ratio.
Let us consider a solid cylinder of length l and radius r is extended longitudinally by an amount dl and its radius is decreases by dr, then
Poisson’s ratio
Volume of the cylinder,
Thermal Stress
In mechanics and thermodynamics, thermal stress is mechanical stress created by any change in temperature of a material. These stresses can lead to fracturing or plastic deformation depending on the
other variables of heating, which include material types and constraints.^ Temperature gradients, thermal expansion or contraction and thermal shocks are things that can lead to thermal stress. This
type of stress is highly dependent on the thermal expansion coefficient which varies from material to material. In general, the greater the temperature change, the higher the level of stress that can
occur. Thermal shock can result from a rapid change in temperature, resulting in cracking or shattering.
If a rod is fixed at its one end and is subjected to temperature change across it, then a stress will be developed
within it, which is called thermal stress.
Let us consider a rod of length L and its temperature is increased by t, then its length is increased by l.
Hence longitudinal strain = l/L
If α be the linear coefficient of thermal expansion of the material of the rod, then
Thermal stress = Yαt
A steel rod of length 5 meters fixed rigidly between two supports. The coefficient of linear expansion of steel is 12 x 10^-6 / °C . Calculate the stress is the rod for an increase of temperature of
40°C Given Y = 2 x 10^11 N/m².
Thermal stress
Rupturing tensile stress
Stress rupture is the sudden and complete failure of a material under stress. During testing, the sample is held at a specific load level and temperature for a previously determined amount of time.
In stress rupture testing, loads can be applied by tensile bending, flexural, bi-axial, or hydro-static methods.
If a thin rod is rotated uniformly with an angular velocity, then a stress is developed within the rod. Then the stress is known as tensile stress.
Let a thin uniform rod of length l and mass m rotates uniformly with an average velocity w.
Tension at A, at a distance x from rotating axis
= centripetal force due to the portion AB
So, Tensile stress at a distance x
Torsion of a cylinder
Torsion is actually the twisting of an object due to an externally applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to it’s
Let us consider (solid) of radius a and length l· Due to an external torque cylinder is twisted by an angle Φ.
Let us consider an inner cylinder of radius r (r<a). Due to twisting Φ, let a point A on the inner cylinder is displaced to A’ which produces a shearing strain θ.
From figure,
So, shearing angle, θ = rΦ/l ———– 1
Due to elastic property, a restoring torque is produced which tends to prevent the twisting of the cylinder.
To calculate resisting tongue let us take a coaxial cylindrical shell of radius r and thickness dr.
If F be the tangential force for elementary portion – dx of the shell, then
tangential stress = F/dr.dx ————– 2
Hence rigidity modules (n)
Moment of force on that elementary portion
Restoring torque over the whole perimeter of the elementary shell,
Summing up all such elementary shells, torque over cylinder ;
Note :
called torsional rigidity.
c = τ/Φ i.e. torsional couple per unit twist is called torsional rigidity.
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Hay's Bridge - Construction, Equation, Phasor Diagram & Advantages
Hay's bridge is the electrical circuit used for the measurement of self-inductance. It is an alternating current bridge similar to Maxwell's bridge with small modifications.
The main difference between Hay's bridge and Maxwell's bridge is that Hay's bridge employs a resistance in series with the standard capacitor, whereas Maxwell's bridge uses a resistor in parallel
with the standard capacitor.
Construction of Hay’s Bridge :
The Hay's bridge is a modified form of Maxwell's inductance-capacitance bridge. It measures inductance by comparing it with a standard variable capacitance. The circuit diagram of Hay's bridge is
shown below.
It consists of an inductor with an inductance L[1] and internal resistance R[1] in arm AB and non-inductive standard resistances R[2] and R[3] in arms AD and BC respectively, and a known variable
standard capacitance C[4] in series with known non-inductive variable standard resistance R[4] in arm CD.
Operation and Theory of Hay’s Bridge :
The bridge can be balanced by adjusting the values of R[4] and C[4]. From the above figure,
Under balanced condition, we have, Equating real and imaginary terms on both sides, we get, Substituting equation 2 in 1, we get, Substituting equation 3 in 2, we get, Now, the quality factor of an
inductor is given by, Substituting equation 5 in 3, we have, For high Q coils i.e., Q > 10, 1/Q^2 is almost negligible. Hence the above equation reduces to,
L[1] = R[2] R[3] C[4]
From the above equations, we can say that for high Q coils the expression for L[1] is free from the frequency term. For low Q coils, 1/Q^2 cannot be neglected and hence to find L[1], the frequency of
source is to be accurately known. Therefore, the bridge suits only for the measurements of inductance of high Q coils.
Phasor Diagram of Hay’s Bridge :
The below shows the phasor diagram of the bridge under balanced conditions. By taking inductor current I[1] as the reference phasor. It is the current of arm AB, then the voltage drop across R[1]
will I[1] R[1] which will be in-phase with I[1]. Similarly, the voltage drop across L[1] will be I[1] ωL[1] which leads the current I[1] with 90°. Now the total voltage drop V[1] of arm AB will be
the sum of voltage drops across R[1] and L[1].
When the bridge is balanced, B and D will be at the same potential and there will be a null-deflection i.e., V[1] = V[2] and V[3] = V[4], also I[1] = I[3] and I[2] = I[4]. Therefore, the phasor V[2]
lies along with V[1] with equal magnitude, and the voltage drop I[2] R[2] (in AD arm) and current I[2] will be in-phase with V[2].
Also, when the bridge is balanced I[4] lies along with I[2] and I[3] with I[1]. Similarly, the voltage drop along the BC arm will be I[3] R[3] and will be in-phase with I[3]. Thus I[3] R[3] lies
along phasor I[3] which is nothing but V[3], and V[4] will be equal to V[3] under balanced condition.
Now the voltage drop in arm CD is the sum of voltage drops across capacitor and resistor i.e., I[4] R[4] + I[4]/ωC[4]. Due to capacitance, the drop I[4]/ωC[4] lags behind the I[4] by 90° and I[4] R
[4] lies along with I[4]. Therefore, the resultant of drop I[4] R[4] and I[4]/ωC[4] will be V[4] (also equal to V[3]). The resultant of V[1] and V[3] will be the V[s] since V[s] will be equal to (V
[1] + V[3]) or (V[2] + V[4]).
Advantages of Hay's Bridge :
• The expression obtained for the Q-factor of the coil using Hay's bridge is not a complicated one.
• From the above expression, it can be seen that the resistance R[4] is inversely proportional to the Q-factor. Lower the resistance higher the Q-factor. Thus for high Q coils, the value of
resistance R[4] should be quite small. Hence, the bridge requires the resistance of low value.
• Hay's bridge is suitable for coils whose quality factor is greater than 10 (Q > 10). Also, it gives a simple expression for unknown inductance for high Q coils.
Disadvantages of Hay's Bridge :
The major drawback of Hay's bridge is that it cannot be used for measuring coils having a Q-factor less than 10. We have,
For lower values of Q(<10) the term (1/Q)^2 in the above expression cannot be neglected. Hence this bridge cannot be used for coils that have a Q-factor less than 10. For such coils, Maxwell's bridge
can be used.
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in Northern Cape
Mathematics tutors in Northern Cape
Personalized Tutoring Near You Mathematics lessons for online or at home learning in Northern Cape
Mathematics tutors in Northern Cape near you
Almario N
Monument Heights
Almario N
Monument Heights, Kimberley
I, Almario Shelwynne Noah, obtained merit certificates in this subject at high school level and ranked fifth place in the prestigious Amesa Math competition in 2008. I also obtained a distinction in
high school Mathematics at the end of my Matric(grade 12) year and I am endorsed in Mathematics at University level. I also obtained a certificate in tutoring from the University of the Free State,
hence , I am qualified to tutor.
Teaches: Linear Algebra, Math, Algebra, Calculus, Mathematics, Pure Maths, CSS, HTML, Computer Science, Computer Programming
Available for Mathematics lessons in Northern Cape
Mohamed E
Moghul Park, Kimberley
I have yet to have a student who's marks did not increase after our lessons together. Maths is one of my favourite subjects, every problem is a puzzle seeking solution. My passion for this is one of
my key strengths when it comes to mathematics. I have tutored Maths for 3 years from Primary school to University level.
Teaches: English as a foreign Language, Statistics, Trigonometry, General Maths & Science, Mathematics Literacy, Linear Algebra, Algebra, Mathematics, Pure Maths, Chemistry, Biochemistry, Molecular
and Cellular Biology, Human Biology, Microbiology, Biotechnology
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Subjects related to Mathematics in Northern Cape
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Ordinal Numbers Транскрипция - OrdinalNumbers.com
Ordinal Numbers Транскрипция
Ordinal Numbers Транскрипция – You can enumerate unlimited sets using ordinal numbers. They can also be utilized to generalize ordinal numbers.But before you use these numbers, you need to understand
what they exist and how they function.
The ordinal number is among of the most fundamental concepts in math. It is a number that indicates the position of an object in an array of objects. Typically, ordinal numbers range from one and
twenty. While ordinal numbers can serve various purposes, they are most commonly used to indicate the sequence in which items are placed in a list.
To represent ordinal numbers you can make use of charts, numbers, or words. They can also be used to describe how a set of pieces are arranged.
The majority of ordinal numbers fall within one of two categories. Ordinal numbers that are infinite are represented by lowercase Greek letters, whereas finite ones are represented with Arabic
A well-organized collection should contain at least one ordinal in accordance to the axiom. The first student in the class, for example is the one who receives the highest grade. The contest’s winner
is the student who got the best score.
Combinational ordinal amounts
Multidigit numbers are also known as compound ordinal number. They are generated by the process of having an ordinal number multiplied by the number of its last digit. They are most commonly used to
rank and date. They don’t provide a unique ending for each number, as with cardinal numbers.
Ordinal numerals are used to indicate the order of elements within collections. The names of the elements in the collection are also indicated using the numbers. The two types of normal numbers are
regular and flexible.
Prefixing a cardinal numeral with the suffix -u results in regular ordinals. The number is then typed into words. A hyphen then added to it. It is also possible to use other suffixes. For instance,
“-nd” is for numbers that end with 2 and “-th” is for numbers ending with 9 or 4.
Suppletive ordinals are made by prefixing words using -u, -e or the suffix -ie. This suffix is used to count and is more extensive than the normal.
ordinal limit
Limit ordinal values that do not reach zero are ordinal figures. Limit ordinal numbers have a disadvantage: they do not have a limit on the number of elements. They can be constructed by joining sets
that have no element that is larger than.
Transfinite recursion definitions also employ limited ordinal numbers. Every infinite cardinal number, based on the von Neumann model can also be considered an ordinal limit.
An ordinal number that has limits is equal to the sum all of the ordinals below. Limit ordinal amounts can be enumerated using mathematics however they also be expressed as natural numbers or a
The ordinal numbers for arranging the data are employed. They offer a rationale for an object’s numerical location. They are often employed in set theory and the arithmetic. Despite sharing a similar
structure, they are not classified as natural numbers.
In the von Neumann model, a well-ordered collection is used. Let’s suppose that fy is a subfunction in an g’ function that is defined as a single function is the situation. If fy’s subfunction is (i
I, II) and g is in line with the requirements that g is a limit ordinal.
A limit ordinal of the type Church-Kleene is also known as the Church-Kleene ordinal. The Church-Kleene oral defines a limit as a properly arranged collection of the smaller ordinals. It also has an
ordinal with a nonzero value.
Stories with examples of ordinal numbers
Ordinal numbers can be used to establish the order of things between objects or entities. They are crucial to organize, count, and ranking motives. They can be used to show the position of objects in
addition to giving the order of things.
The ordinal number is typically identified by the letter “th”. In some instances the letter “nd” is substituted. The titles of books usually contain ordinal numbers.
Although ordinal numbers are typically used in lists however, they are also written in words. They may also be stated in terms of numbers or acronyms. In general, numbers are simpler to comprehend
than the cardinal numbers.
There are three kinds of ordinal number. These numbers are able to be learned through games, practice, and other activities. You can increase your math skills by understanding more about the basics
of them. Try utilizing a coloring exercise as an easy and enjoyable approach to improving. You can check your progress with a handy coloring sheet.
Gallery of Ordinal Numbers Транскрипция
Ordinal Numbers Online Worksheet For Grade 5
Ordinal Numbers Online Exercise For 4
English Room ORDINAL NUMBERS
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CS代写 CS 70 Discrete Mathematics and Probability Theory Fall 2021 – cscodehelp代写
CS 70 Discrete Mathematics and Probability Theory Fall 2021
1 Counting Cartesian Products
For two sets A and B, define the cartesian product as A×B = {(a,b) : a ∈ A,b ∈ B}.
(a) Given two countable sets A and B, prove that A × B is countable. (b) Given a finite number of countable sets A1,A2,…,An, prove that
is countable.
2 Counting Functions
A1 ×A2 ×···×An
Are the following sets countable or uncountable? Prove your claims.
(a) The set of all functions f from N to N such that f is non-decreasing. That is, f (x) ≤ f (y) whenever
CS 70, Fall 2021, DIS 7A 1
(b) The set of all functions f from N to N such that f is non-increasing. That is, f (x) ≥ f (y) whenever x≤y.
3 Undecided?
Let us think of a computer as a machine which can be in any of n states {s1,…,sn}. The state of a 10 bit computer might for instance be specified by a bit string of length 10, making for a total of
210 states that this computer could be in at any given point in time. An algorithm A then is a list of k instructions (i0,i2,…,ik−1), where each il is a function of a state c that returns another
state u and a number j. Executing A (x) means computing
(c1, j1) = i0(x), (c2, j2) = ij1(c1), (c3, j3) = ij2(c2), … until jl ≥ k for some l, at which point the algorithm halts and returns cl−1.
(a) How many iterations can an algorithm of k instructions perform on an n-state machine (at most) without repeating any computation?
CS 70, Fall 2021, DIS 7A 2
(b) Show that if the algorithm is still running after 2n2k2 iterations, it will loop forever.
(c) Give an algorithm that decides whether an algorithm A halts on input x or not. Does your contruction
contradict the undecidability of the halting problem?
4 Code Reachability
Consider triplets (M,x,L) where
M is a Java program
x is some input
L is an integer
and the question of: if we execute M(x), do we ever hit line L? Prove this problem is undecidable.
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Cool Math Stuff
You might have noticed that in this blog, I am very persistent in providing proofs for anything I say. This is because I remember wondering in math class why everything was true, and I think that
information should be provided.
Most mathematicians are the same way, where they value proofs of theorems, and will never accept something just cause it seems to work a couple times. In fact, one of the main jobs of a mathematician
is to take unproved conjectures and prove them.
One of these unproven conjectures is the
Riemann Hypothesis
, which I have talked about on this blog before. This is a problem where mathematicians have been working for decades to find a proof.
Godfrey Hardy was a British mathematician who valued proofs. He was born February 7, 1877, and had a huge contribution to the fields of number theory and mathematical analysis. He once said to a
"If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof."
You might recognize Hardy as the mentor of Srinivasa Ramanujan, who I posted about a few months ago. When Hardy was asked what his biggest contribution to mathematics was, he stated without
hesitation that it was his discovery of Ramanujan.
Hardy was an atheist, but liked to play games with a God-like being, which him and other mathematicians like to call the Supreme Fascist, or SF. One of these stories was one that I enjoyed, and
wanted to share.
When he was on a boat ride from Scandinavia home to England, the water started to get very turbulent. So, he sent a postcard to a colleague back home saying that he had proved the Riemann Hypothesis.
Knowing the importance of the problem, he didn't think the SF would let him die with everyone thinking he had a proof, so he felt that his safety was then guaranteed.
In school, multiplication is taught by giving students a specific method to use (which they call the "traditional" method), and you have to solve problems with it. By only providing one of over a
dozen multiplication methods, this eliminates the creativity, and quite possibly the fun, in multiplication. I'm not going to dwell on these issues; you can read my Capstone Research Paper for that.
I have discussed a few other methods of multiplication on this blog, one of which is the Criss-Cross Method. Again, I won't explain it here, but this method does lay the foundation for the new method
I am going to explain today.
Rather than taking you through the steps, I will post a video tutorial on how to do it (those are usually more fun anyways). If you are wondering why it works, I encourage you to refer back to the
Criss-Cross Method and notice the similarities. Try a few problems with the Criss-Cross Method, and then this method, and see how the numbers that appear throughout the problem seem to be identical.
When you watch this video, I would strongly suggest doing the examples along with the video. It can be a very useful technique for problems whose digits are of a reasonable size.
I think this is a really cool method of multiplication, and even easier to teach than the traditional method. Play around with it, and you will likely end up using it in the future.
For some more intellectual readers, it might be fun to try multiplying numbers with bigger digits using the quinary or senary number systems (base five or base six). This would make less dots in each
section. I think it might be a fun exercise.
Answers to June Problem of the Week:
*For the answers to the creation of a line of best fit, it will be written as an inequality. There is no way to get the exact answer by hand, so just make sure your answer is in the interval that is
a = 7^1/[3]
p = 14
m = ^118988/[6925]
n = ^394/[277]
y = 15.76
z = 12.08
g = 8
b = 112
q = 44
x = 33
h = 105
p = 270
t = 19
n = 92925
u = 92.925
q = 135
a = 2165.2875
d = 21.652875
g = 15
j = 46.4625
k = 23.23125
r = 27
s = 45
2 < m < 3.5
45 < b < 50
175 < x < 275
s = 107
t = 365
p = 905
q = 4096
q[0] = 4096
q[1] = 2048
q[2] = 1024
q[3] = 512
q[4] = 256
q[5] = 128
q[6] = 64
q[7] = 32
q[8] = 16
q[9] = 8
q[10] = 4
q[11] = 2
q[12] = 1
q[13] = ^1/[2]
q[14] = ^1/[4]
q[15] = ^1/[8]
x[1] = -452.5
x[2] = -2
x[3] = 4
y = 0
a = 226.25
b = -224.25
c = -454.5
t = 2635773.77
m = 34
x = 79.53%
Today is the final day of the problem of the week. In all three problems, your goal is to find the value of x. Good luck!
Find the mean of the following list of numbers:
x = ____
The data set below represents the test scores of nine students based on the amount of absences they had.
x y
f - f[1 ]- f[2] - f[3] - f[4] ceiling(n ÷ f - r)
floor(f[4 ]- f[2 ]- r) floor(8r)
floor(f[2 ]÷ f[3]) ceiling(2f[4 ]- r)
^3√(f[1 ]- f[4]) f ÷ 2
ceiling(f[2 ]- f[3 ]- r) 2f[2]
√(f[3]) floor(f[1 ]- r)
floor(n ÷ 1000) floor(n ÷ 100)
f[1 ]- f[4] f[1]
f[4 ]- f[2] f[4]
Find the line of best fit for this data set. It should be of the form y = mx + b.
m = ____
b = ____
Now, find the predicted score of a student with ten absences.
x = ____
Take the following matrix that represents the payoffs in a mathematical game.
A B
1 b, w - 6 s, -t
2 -b, a -a, c
3 -g, -d -f, b
After eliminating any dominated strategies, use mixed strategy Nash equilibria to determine the percent of the time that strategy 1 should be played.
x = ____
The answers to the problem will be up in a month. I will post the answers to June’s problem of the week with tomorrow's post.
Today is day four of the problem of the week. Good luck!
Take the following parallelogram.
Assume that this parallelogram has an area of j. Find the measure of the base b and the perimeter q.
b = ____
q = ____
Take the following ellipse:
Assume that this ellipse has an area of n. Find the measure of the short radius r. Round to the nearest tenth.
r = ____
Take the following two triangles:
Determine the length of w. Round to the nearest integer.
w = ____
Hint: Look for triangle congruence or proportionality.
Today is day three of the problem of the week. Before I begin the problems, I would like to explain one thing, which will be used throughout the medium and hard problems. You might not have learned
it in school, but the concept is very simple.
A floor function is the largest integer less than or equal to a given number. For example, the floor function of 6.7 is 6. The floor function of 4.973 is 4. The floor function of π is 3. Basically,
if you round the number down to the nearest whole, you will have its floor function.
Similarly, a ceiling function is the smallest integer greater than or equal to a given number. So, the ceiling function of 6.7 is 7. The ceiling function of 4.973 is 5. The ceiling function of π is
4. While rounding down yields the floor function, rounding up gives the ceiling function.
Normally, this is denoted by a special sort of bracket. Since I don’t know how to type these brackets into the computer, I will use the following notation:
floor(x) = the floor function of x
ceiling(x) = the ceiling function of x
This is an easy way to eliminate fractions and decimals from numbers to make the problems slightly easier and more realistic.
Good luck!
What is the Least Common Multiple of f, g, and h? Use the letter j to denote the answer.
j = ____
What is the number of dots in a regular (f[4] - f[2])-gon array whose sides are of length (f[1] - f[3])?
n = ____
Find the explicit formula for the following sequence:
v, t, s, floor((g + f)/10), (g - 60)/2, d - 2, ...
The formula should be of the form ax^2 + bx + c. So, write your answer in terms of the value of coefficients a, b, and c.
a = ____
b = ____
c = ____
Today is day two of July’s problem of the week! For the medium and hard problems, today is when you will start to use variables. For the easy problem, plug in the variables from yesterday for today.
Using the answers from yesterday, solve the following problems for f, g, and h. Remember to use the order of operations.
f = 2(a) + a ÷ 10 - 2
g = m^2 + (m - 2)(m - 7)
h = p^2 ÷ 18 - (p ÷ 30)^3
f = ____
g = ____
h = ____
The graphs of the four lines from yesterday should form a quadrilateral on your graph paper. Ignore the rest of the lines, and just analyze each segment forming the quadrilateral.
First, find the length of the each of the four segments, and use their names (ex: f[1]) for the variables equal to them. So, f[1] = the length of the segment that was originally the graph of f[1].
f[1] = ____
f[2] = ____
f[3] = ____
f[4] = ____
Then, find the perimeter of this quadrilateral. Call this perimeter f.
f = ____
First, plot points P and Q on the graph’s x-intercepts and point R on the graph’s positive y-intercept. Then, find all six measurements within triangle PQR. Use d to denote the measure of angle P, f
to denote the measure of angle Q, g to denote the measure of angle R, s to denote the measure of segment PQ, t to denote the measure of segment PR, and v to denote the measure of segment QR. Round
all angle measures to the nearest degree.
d = ____
f = ____
g = ____
s = ____
t = ____
v = ____
Today begins the second of three 2013 problems of the week. If you remember from last month, there are three different problems: easy, medium, and hard. The easy problem is at a level where you will
be fine with just a good understanding of foundational mathematics (around a sixth grade level). The medium problem requires an understanding of Algebra I, but nothing beyond that. The hard problem
requires Geometry, Algebra II, and depending on the rigor of the classes, some content from a Trigonometry or Precalculus class might be needed as well. To find your level, try looking at the
categories of my blog posts. Basic posts correspond well with the easy problem, intermediate posts correspond with the medium, and advanced correspond with the hard.
Each of the problems is broken into five parts, and I give each part on the next day of the week. On Friday, you will have to solve for x, which is the final answer. The answer to the problem gets
posted after a month. So, remember to save your results from each day to plug into the next day’s problem.
Last month’s easy problem had a lot of heavy arithmetic in it, which probably made it complicated to solve. However, this problem will be much less strenuous, and focuses more on the procedures than
the arithmetic.
Good luck!
Take the following right triangle:
You are going to determine three things about this triangle. First, find the area, which will be denoted by the letter a.
a = ____
Now, find the perimeter, which will be p.
p = ____
Lastly, find the measure of the missing angle up top, which we’ll call m.
m = ____
Normally I would start with some triangle computations, but this week, I chose to do some function graphing on Monday, and then use triangles and trigonometry on Tuesday.
Take out some graph paper and draw in your x and y axes. Then, graph the following four functions, who we’ll call f[1], f[2], f[3], and f[4].
f[1]) y = 48 - ^4/[3]x
f[2]) 10x - 24y - 360
f[3]) 3(x - 20) = -4y
f[4]) 5(y - 12) = 6(x + 15)
You won’t need to solve for any variables today, but keep the piece of paper that you graphed it on. This graph will be used in tomorrow’s problem.
Similar to the medium problem, today will be focused on graphing and tomorrow will incorporate the trigonometry.
Take out a piece of graph paper (separate from the medium problem if you are doing both) and graph the following equation:
16y^2 = 3(48 - 3x^2)
Tomorrow’s problem will be using this graph, so don’t lose it.
A common question people wonder is "What do mathematicians do?" People know that scientists make advancements in science and engineers make advancements in engineering, but what about math?
Similarly, mathematicians make advancements in math.
This is often an odd concept to people, since math seems like it is all figured out. How could it be taught so definitively if there are still things to figure out? But, there are tons of conjectures
out there that people are trying to prove.
One of them is called the Twin Prime Conjecture. You are probably aware that a prime number is a number that is only divisible by one and itself. There are an infinite number of prime numbers (click
here to see a proof). Twin primes are two prime numbers that differ by two. For instance, 3 and 5 are twin primes because they are both prime and differ by two.
We know that there are an infinite number of primes, but are there an infinite number of twin primes? This is one of the things that has yet to be figured out, and the Twin Prime Conjecture is asking
this question.
Recently, a big jump was made in trying to prove this conjecture. So, I will post the article from News Scientist that describes the latest advancements on it.
At some point during your high school math experience, whether it be during an Algebra II, Geometry, Trigonometry, or Precalculus class, you probably were presented with the Law of Sines and the Law
of Cosines. You found that the sine and cosine ratio could only solve right triangles, and that you needed something more to solve non-right triangles. So, two formulas were slapped onto your
textbook, and you had to apply them.
Today, I am just going to focus on the Law of Cosines, and save the Law of Sines for a future post. This formula is one where you probably wondered why it worked. You might be asked to prove
something using this formula, but how can you comfortably do that without being sure of the formula in the first place?
First, I need to lay a little foundation. Take a right triangle:
First off, we should know the Pythagorean Theorem. Click here for an explanation and proof of it.
Let's say we are dealing with the bottom left angle. The side opposite to this angle has a measure of 4, and will be referred to as the "opposite" side. The longest side has a measure of 5, and will
be referred to as the "hypotenuse." The remaining side has a measure of 3, and will be referred to as the "adjacent" side, since it is adjacent to the angle.
Now, let's make sure we are on the same page with terminology. The sine of this angle is the opposite side over the hypotenuse. 4/5 is 0.8, so the sine of that angle is 0.8. The cosine of this angle
is the adjacent side over the hypotenuse. 3/5 is 0.6, so the cosine of that angle is 0.6. Tangents will not be needed in this post, but it is the opposite over the adjacent. These ratios are commonly
remembered by the SOH CAH TOA acronym.
There is a pretty cool identity found in the sine and cosine ratios. Take the two ratios that we just found for the 3-4-5 triangle:
sin(x) = 0.8
cos(x) = 0.6
Now, plug those into the following expression:
sin^2(x) + cos^2(x)
0.8^2 + 0.6^2
0.64 + 0.36
Interestingly, this sum always turns out to be one. By using those ratios and the Pythagorean Theorem, you can prove that for all angles, the square of the sine plus the square of the cosine is one.
With this information, we can prove the Law of Cosines. The formula slightly resembles the Pythagorean Theorem, so we will try to keep that in mind when proving it.
Each of the derived measurements are just rewritten forms of the sine and cosine ratios. For example, the cosine of C is that adjacent side over b, so multiplying both sides by b yields the measure
of the adjacent side: b • cosC.
Let's look at the right triangle with hypotenuse c, and solve the Pythagorean Theorem. Since the Law of Cosines is similar to the Pythagorean Theorem, this might give us a start.
[a - (b • cosC)]^2 + (b • sinC)^2 = c^2
a^2 - 2abcosC + b^2cos^2C + b^2sin^2C = c^2
a^2 + b^2cos^2C + b^2sin^2C - 2abcosC = c^2
a^2 + b^2(cos^2C + sin^2C) - 2abcosC = c^2
Wait a minute - what is inside the parentheses? We have cos^2C + sin^2C. But we proved earlier that that is always equal to one. So, we can substitute one in for that sum, and see where that takes
a^2 + b^2(cos^2C + sin^2C) - 2abcosC = c^2
a^2 + b^2(1) - 2abcosC = c^2
a^2 + b^2 - 2abcosC = c^2
And we end up with the Law of Cosines: a^2 + b^2 - 2abcosC = c^2. I think that this proof is a pretty cool one, considering that it brings so many other neat identities into play. | {"url":"https://coolmathstuff123.blogspot.com/2013/07/","timestamp":"2024-11-03T22:02:32Z","content_type":"text/html","content_length":"169903","record_id":"<urn:uuid:fa2f2602-f6a5-4773-a3d4-46d884671ac5>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00634.warc.gz"} |
YTF 11
Nejc Ceplak (Queen Mary, University of London)
The aim of the talk is to present the construction of a new family of smooth horizonless solutions of supergravity that have the same charges as the supersymmetric D1-D5-P black hole. We will begin
with a brief review of the Fuzzball proposal for black holes, which states that at the length scale of the horizon a new, fuzzy, phase takes over, allowing outside observers to distinguish between
different microstates of the black hole. We will then focus on the three charge supersymmetric D1-D5-P black hole and review some of its microstate geometries. We then present a method of obtaining a
new family of solutions using supersymmetry generators. The motivation behind this construction is coming from the dual CFT multiplet structure, where these fermionic generators are used to create
new linearly independent states in the theory. On the gravity side the geometries dual to these new states are generated by the Killing spinors of AdS$_3 \times S^3 \times T^4$. Hence we present the
explicit form of these spinors and use them to construct new solutions to the supergravity equations. Finally we present these new solutions and show that they are simpler than the ones previously
known with having a fewer number of excited fields. | {"url":"https://conference.ippp.dur.ac.uk/event/748/contributions/4309/","timestamp":"2024-11-02T08:44:07Z","content_type":"text/html","content_length":"103753","record_id":"<urn:uuid:dc60bf2c-e6b2-4853-b44c-c52f5765bbca>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00779.warc.gz"} |
Policy Impacts Library | Tax Deduction for Postsecondary Tuition, Single Filers at Phase Start
The tax deduction in tuition and fees (DTF) was implemented in 2001 as part of the Economic Growth and Tax Relief Reconciliation Act. Under this program, households could deduct tuition and fees paid
for undergraduate or graduate education from gross income without needing to itemize deductions. Households are eligible to use the DTF based on adjusted gross income (AGI) net of all other
above-the-line deductions. Beginning in 2004, the maximum deductions followed a tier system in which joint (single) filers with eligible income of less than $130,000 ($65,000) were eligible for a
$4,000 maximum deduction, while households with eligible incomes of $130,000-$160,000 ($65,000-$80,000) were eligible for a $2,000 maximum deduction. Households above $160,000 ($80,000) were
ineligible for the deduction. The change in tax liability is based on whether the household claimed DTF scaled by the marginal tax rate at their income level.
Hoxby and Bulman (2016) exploit income eligibility thresholds to estimate the impact of the above-the-line deduction of tuition and fees on enrollment. They use a regression discontinuity design but
leave out observations right around the threshold to account for the possibility of manipulation in the running variable. This MVPF estimate considers the MVPF implied by the discontinuity faced by
single filers with incomes near $65,000.
Hendren and Sprung-Keyser (2020) take the causal estimates from Hoxby and Bulman (2016) and project the impact of the tuition deduction on lifetime earnings and tax revenue. They utilize estimates
from Zimmerman (2014) on the impact attendance of college on earnings and assume that the returns to college are constant in percentage terms over the lifecycle. | {"url":"https://policyimpacts.org/policy-impacts-library/tax-deduction-for-postsecondary-tuition-single-filers-at-phase-start/","timestamp":"2024-11-14T19:58:44Z","content_type":"text/html","content_length":"40693","record_id":"<urn:uuid:5d17b302-4942-4f3b-9ac7-6df5a3ccdb01>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00861.warc.gz"} |
Accounting Rate of Return Calculator | Calculate Accounting Rate of Return
What is Accounting Rate of Return?
The ARR (Accounting Rate of Return), also known as the Average Rate of Return or the Return on Investment (ROI), is a financial metric used to evaluate the profitability of an investment or project.
The ARR is expressed as a percentage and represents the return earned on the investment on an annualized basis. It provides a simple measure for comparing the profitability of different investment
opportunities or projects. However, it has some limitations, such as not considering the time value of money or the timing of cash flows, which may make it less suitable for evaluating projects with
uneven cash flows over time.
How to Calculate Accounting Rate of Return?
Accounting Rate of Return calculator uses Accounting Rate of Return = (Average Annual Profit/Initial Investment)*100 to calculate the Accounting Rate of Return, The Accounting Rate of Return formula
is defined as a financial metric used to evaluate the profitability of an investment or project. It calculates the average annual profit or return generated by an investment relative to its initial
cost. Accounting Rate of Return is denoted by ARR symbol.
How to calculate Accounting Rate of Return using this online calculator? To use this online calculator for Accounting Rate of Return, enter Average Annual Profit (AP) & Initial Investment (Initial
Invt) and hit the calculate button. Here is how the Accounting Rate of Return calculation can be explained with given input values -> 35 = (700/2000)*100. | {"url":"https://www.calculatoratoz.com/en/accounting-rate-of-return-calculator/Calc-43454","timestamp":"2024-11-12T17:11:29Z","content_type":"application/xhtml+xml","content_length":"108026","record_id":"<urn:uuid:8f3c76cd-4647-45a1-a356-238d760141ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00524.warc.gz"} |
Wilk Coefficient Applied
`"WCAW" = w * 500/ (a+bx+cx^2+dx^3+ex^4+fx^5) `
Enter a value for all fields
The Wilks Coefficient calculator computes the comparative weight lifted between weight lifters based on the weight of the lifter (w), the weight lifted (x) and the gender of the lifter, where the
weight is adjusted by the Wilks Coefficient formula.
INSTRUCTIONS: Choose units and enter the following:
• (w) Weight Lifted
• (x) Weigh of Person
• (g) Gender of Person
Wilks Coefficient Adjusted Weight (WCAW): The calculator returns the adjusted weight in kilograms. However, this can be automatically converted to other weight units (e.g. pounds) via the pull-down
The Math / Science
The formula for the Wilk's Coefficient applied to a weight lifted is:
` WCAW = 500/ (a+bx+cx^2+dx^3+ex^4+fx^5) * w`
• WCAW = Wilks Coefficient Adjusted Weight
• x = body weight of the lifter
• w = weight lifted by the weight lifter
• a,b,c,d,e,f = f(gender)
For men:
a = −216.0475144
b = 16.2606339
c = −0.002388645
d = −0.00113732
e = 7.01863E−06
f = −1.291E−08
For women:
a = 594.31747775582
b = −27.23842536447
c = 0.82112226871
d = −0.00930733913
e = 4.731582E−05
f = −9.054E−08
Enhance your vCalc experience with a free account
Sign Up Now!
Sorry, JavaScript must be enabled.
Change your browser options, then try again. | {"url":"https://www.vcalc.com/wiki/wilks-coefficient-applied","timestamp":"2024-11-06T15:34:26Z","content_type":"text/html","content_length":"51304","record_id":"<urn:uuid:6033cc56-f320-48a6-a844-1f098c9ce842>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00499.warc.gz"} |
Oberseminar Nonlinear Dynamics
Oct 25, SFB 647
Andrey Muravnik The Cauchy problem for parabolic differential-difference equations: integral representations of solutions and their long-time behavior
(Peoples' Friendship University Moscow)
Nov 01, Parabolic equations (including singular ones) containing translation (generalized translation) operators acting with respect to spatial variables are considered. Integral representations
2011 of their classical solutions are found and asymptotic closeness (stabilization) theorems are proved for their solutions.
It turns out that there are principally new effects of the long-time behavior of the above solutions caused by the non-local nature of the equation. Moreover, those effects hold even in
the case where only low-order terms of the equation are non-local.
Nov 08, SFB 647
Carlo Laing Fronts and bumps in spatially extended Kuramoto networks
(Massey University New Zealand)
Nov 22, We consider moving fronts and stationary “bumps” in networks of non-locally coupled phase oscillators. Fronts connect regions of high local synchrony with regions of complete asynchrony,
2011 while bumps consist of spatially-localised regions of partially-synchronous oscillators surrounded by complete asynchrony. Using the Ott-Antonsen ansatz we derive non-local differential
equations which describe the network dynamics in the continuum limit. Front and bump solutions of these equations are studied by either “freezing” them in a travelling coordinate frame or
analysing them as homoclinic or heteroclinic orbits. Numerical continuation is used to determine parameter regions in which such solutions exist and are stable.
Nov 29, SFB 647
Dec 06, Svetlana Gurevich Destabilization of localized structures induced by delayed feedback
2011 (Westfälische Wilhelms-Universität Münster)
Dec 13, Thomas Wagenknecht Homoclinic snaking: different ways to kill the snakes
2011 (University of Leeds)
SFB 647
Jan 10, SFB 647
Eugen Zhang Efficient Morse Decomposition of Vector Fields
Jan 24, (Oregon State University)
2012 Traditional vector field topology relies on the ability to accurately compute trajectories, which is difficult to achieve due to noise and error. Morse decomposition addresses this issue.
However, computing Morse decomposition given a simulation data set can be challenging due to the complexity in both the flows and the underlying domains. In this talk I will discuss how to
effectively compute Morse decomposition in a hierarchical fashion. The results have been applied to a number of simulation data sets.
Jan 31, SFB 647
Daria Apushkinskaya Two-Phase Parabolic Obstacle Problems: L^∞-estimates for Derivatives of Solutions
(Saarland University)
Consider the two-phase parabolic obstacle problem with non-trivial Dirichlet condition
Feb 06,
2012 Δu − ∂[t]u = λ^+χ[{u>0}] − λ^−χ[{u<0}] in Q=Ω×(0;T),
Monday u = φ on ∂[p]Q.
Free Here T<+∞, Ω ⊂ R^n is a given domain, ∂[p]Q denotes the parabolic boundary of Q, and λ^± are non-negative constants satisfying λ^++λ^−>0. The problem arises as limiting case in the model
University of temperature control through the interior.
In this talk we discuss the L^∞-estimates for the second-order space derivatives D^2u near the parabolic boundary ∂[p]Q. Observe that the case of general Dirichlet data cannot be reduced
to zero ones due to non-linearity and discontinuity at u=0 of the right-hand side of the first equation.
The talk is based on works in collaboration with Nina Uraltseva.
Free University, Institute of Mathematics, 14195 Berlin, Arnimallee 3 (rear building), room 130
Feb 07, Nina Uraltseva Two-Phase Parabolic Obstacle Problem: Regularity Properties of the Free Boundary
2012 (St.Petersburg State University)
17:15 In this talk we describe the methods, developed in the last decade, for studying the regularity of the free boundary in the vicinity of branch points. These methods are based on the use of
Free various monotonicity formulas, blow-up technique and some observations of geometric nature.
University Free University, Institute of Mathematics, 14195 Berlin, Arnimallee 6, room 031
Feb 14, NN tba
Tea and coffee will be served at 2:45 p.m. on the ground floor.
Guests are always welcome ! | {"url":"http://dynamics.mi.fu-berlin.de/lectures/oberseminar/11WS-oberseminar.php?q_xhtml=0","timestamp":"2024-11-10T18:32:28Z","content_type":"text/html","content_length":"14301","record_id":"<urn:uuid:06cc996e-2e9e-4ef1-9874-fc23a2ccfa01>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00116.warc.gz"} |
Digital Logic Design n Basics Combinational Circuits Sequential
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Digital Logic Design n Basics Combinational Circuits Sequential Circuits Adapted from the slides prepared by S. Dandamudi for the book, Fundamentals of Computer Organization and Design.
Introduction to Digital Logic Basics n Hardware consists of a few simple building blocks Ø n These are called logic gates n AND, OR, NOT, … n NAND, NOR, XOR, … Logic gates are built using transistors
n n n NOT gate can be implemented by a single transistor AND gate requires 3 transistors Transistors are the fundamental devices n n n Pentium consists of 3 million transistors Compaq Alpha consists
of 9 million transistors Now we can build chips with more than 100 million transistors
Basic Concepts n Simple gates Ø Ø Ø n Functionality can be expressed by a truth table Ø n AND OR NOT A truth table lists output for each possible input combination Precedence Ø Ø NOT > AND > OR F=
AB+AB = (A (B)) + ((A) B)
Basic Concepts (cont. ) n Additional useful gates Ø Ø Ø n n NAND NOR XOR NAND = AND + NOT NOR = OR + NOT XOR implements exclusive -OR function NAND and NOR gates require only 2 transistors Ø AND and
OR need 3 transistors!
Basic Concepts (cont. ) n Number of functions Ø Ø With N logical variables, we can define N 2 2 functions Some of them are useful n AND, NOR, XOR, … Some are not useful: n Output is always 1 n Output
is always 0 “Number of functions” definition is useful in proving completeness property
Basic Concepts (cont. ) n Complete sets Ø A set of gates is complete n If we can implement any logical function using only the type of gates in the set n Ø Ø You can uses as many gates as you want
Some example complete sets n {AND, OR, NOT} Not a minimal complete set n {AND, NOT} n {OR, NOT} n {NAND} n {NOR} Minimal complete set n A complete set with no redundant elements.
Basic Concepts (cont. ) n Proving NAND gate is universal
Basic Concepts (cont. ) n Proving NOR gate is universal
Logic Chips (cont. ) n Integration levels Ø Ø SSI (small scale integration) n Introduced in late 1960 s n 1 -10 gates (previous examples) MSI (medium scale integration) n Introduced in late 1960 s n
10 -100 gates LSI (large scale integration) n Introduced in early 1970 s n 100 -10, 000 gates VLSI (very large scale integration) n Introduced in late 1970 s n More than 10, 000 gates
Logic Functions n Logical functions can be expressed in several ways: Ø Ø Ø n Truth table Logical expressions Graphical form Example: Ø Majority function n Output is one whenever majority of inputs
is 1 n We use 3 -input majority function
Logic Functions (cont. ) 3 -input majority function A B C F 0 0 1 1 0 1 0 1 0 0 0 1 1 1 n Logical expression form F=AB+BC+AC
Logical Equivalence n All three circuits implement F = A B function
Logical Equivalence (cont. ) n Proving logical equivalence of two circuits Ø Ø Derive the logical expression for the output of each circuit Show that these two expressions are equivalent n Two ways:
n n You can use the truth table method n For every combination of inputs, if both expressions yield the same output, they are equivalent n Good for logical expressions with small number of variables
You can also use algebraic manipulation n Need Boolean identities
Logical Equivalence (cont. ) n Derivation of logical expression from a circuit Ø Trace from the input to output n Write down intermediate logical expressions along the path
Logical Equivalence (cont. ) n Proving logical equivalence: Truth table method A 0 0 1 1 B 0 1 F 1 = A B 0 0 0 1 F 3 = (A + B) 0 0 0 1
Boolean Algebra (cont. ) n Proving logical equivalence: Boolean algebra method Ø To prove that two logical functions F 1 and F 2 are equivalent n Start with one function and apply Boolean laws to
derive the other function n Needs intuition as to which laws should be applied and when n Sometimes it may be convenient to reduce both functions to the same expression Example: F 1= A B and F 3 are
equivalent n Ø Practice helps
Logic Circuit Design Process n A simple logic design process involves Ø Ø Ø Problem specification Truth table derivation Derivation of logical expression Simplification of logical expression
Deriving Logical Expressions n Derivation of logical expressions from truth tables Ø Ø n SOP form Ø Ø n sum-of-products (SOP) form product-of-sums (POS) form Write an AND term for each input
combination that produces a 1 output n Write the variable if its value is 1; complement otherwise OR the AND terms to get the final expression POS form Ø Dual of the SOP form
Deriving Logical Expressions (cont. ) n A 0 0 1 1 3 -input majority function B 0 0 1 1 C 0 1 0 1 F 0 0 0 1 1 1 n n SOP logical expression Four product terms Ø Because there are 4 rows with a 1 output
Deriving Logical Expressions (cont. ) n A 0 0 1 1 3 -input majority function B 0 0 1 1 C 0 1 0 1 F 0 0 0 1 1 1 n n POS logical expression Four sum terms Ø Because there are 4 rows with a 0 output F =
(A + B + C)
Logical Expression Simplification Ø Algebraic manipulation n Use Boolean laws to simplify the expression n n Difficult to use Don’t know if you have the simplified form
Algebraic Manipulation n Majority function example Added extra ABC+ABC+ABC = ABC+ABC+ABC+ABC n We can now simplify this expression as BC+AC+AB n A difficult method to use for complex expressions
Implementation Using NAND Gates n Using NAND gates Ø Get an equivalent expression AB+CD=AB+CD Ø Using de Morgan’s law AB+CD=AB. CD Ø Can be generalized n Majority function A B + B C + AC = A B. BC.
AC Idea: NAND Gates: Sum-of-Products, NOR Gates: Product-of-Sums
Implementation Using NAND Gates (cont. ) n Majority function
Introduction to Combinational Circuits n Combinational circuits n n Combinational circuits provide a higher level of abstraction Ø Ø n Output depends only on the current inputs Help in reducing
design complexity Reduce chip count We look at some useful combinational circuits
Multiplexers n Multiplexer Ø Ø Ø n 2 n data inputs n selection inputs a single output Selection input determines the input that should be connected to the output 4 -data input MUX
Multiplexers (cont. ) 4 -data input MUX implementation
Multiplexers (cont. ) MUX implementations
Multiplexers (cont. ) Example chip: 8 -to-1 MUX
Multiplexers (cont. ) Efficient implementation: Majority function
Demultiplexers Demultiplexer (De. MUX)
Decoders n Decoder selects one-out-of-N inputs
Decoders (cont. ) Logic function implementation (Full Adder)
Comparator n Used to implement comparison operators (= , > , < , , )
Comparator (cont. ) A=B: Ox = Ix (x=A<B, A=B, & A>B) 4 -bit magnitude comparator chip
Comparator (cont. ) Serial construction of an 8 -bit comparator
1 -bit Comparator x>y CMP x=y x<y x x y y x>y x=y x<y
8 -bit comparator xn>yn xn=yn x>y CMP xn<yn x=y x<y x y
Adders n Half-adder Ø Ø n Adds two bits n Produces a sum and carry Problem: Cannot use it to build larger inputs Full-adder Ø Ø Adds three 1 -bit values n Like half-adder, produces a sum and carry
Allows building N-bit adders n Simple technique n n Connect Cout of one adder to Cin of the next These are called ripple-carry adders
Adders (cont. ) A 16 -bit ripple-carry adder
Adders (cont. ) n Ripple-carry adders can be slow Ø n Delay proportional to number of bits Carry lookahead adders Ø Ø Eliminate the delay of ripple-carry adders Carry-ins are generated independently
n C 0 = A 0 B 0 n C 1 = A 0 B 0 A 1 + A 0 B 1 + A 1 B 1 . . . Requires complex circuits Usually, a combination carry lookahead and ripple-carry techniques are used n Ø Ø
1 -bit Arithmetic and Logic Unit Preliminary ALU design 2’s complement Required 1 is added via Cin
1 -bit Arithmetic and Logic Unit (cont. ) Final design
Arithmetic and Logic Unit (cont. ) 16 -bit ALU
Arithmetic and Logic Unit (cont’d) 4 -bit ALU | {"url":"https://slidetodoc.com/digital-logic-design-n-basics-combinational-circuits-sequential/","timestamp":"2024-11-08T05:02:59Z","content_type":"text/html","content_length":"190862","record_id":"<urn:uuid:28388167-4267-4772-a39d-d399683b0e04>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00586.warc.gz"} |
Lifeboat Foundation News Blog: Existential Risks - Page 125
by Otto E. Rössler, Faculty of Science, University of Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
Abstract: An unfamiliar result in special relativity is presented: non-conservation of rest mass. It implies as a corollary a resolution of the Ehrenfest paradox. The new result is inherited by
general relativity. It changes the properties of black holes. (June 21, 2012)
Rest mass is conserved in special relativity in the absence of acceleration. Under this condition, the well-known relativistic increase of total mass with speed is entirely due to the momentum part
of the total-mass formula, so rest mass stays invariant as is well known. However, the presence of acceleration changes the picture. Two cases in point are the constant-acceleration rocketship of
Einstein’s equivalence principle of 1907, and the rotating disk of Einstein’s friend Ehrenfest 5 years later.
First the Einstein rocket:
Continue reading “Rest-mass Nonconservation in Special Relativity’s Equivalence principle and Ehrenfest Disk (Minipaper)” »
No scientist on the planet claims to be able to prove my “Telemach theorem” wrong (you find it by adding the second keyword “African”). Only anonymous bloggers express malice against it. The
anonymous writers’ attitude is a logical consequence of the fact that CERN and Europe openly continue in defiance of my (and not only mine) results. This allegiance shown is no wonder: most everyone
is ready to defend their own trusted government. And is it not unlikely indeed that a revered multinational organization like CERN should make a terminal blunder of this magnitude?
In the remaining half year of operation of CERN’s nuclear collider, before the planned 75-percent up-scaling scheduled to take two years’ time, the cumulative yield of artificial BLACK HOLES will
grow by a factor of about 4 if everything works out optimal. So the cumulative risk to the planet will be quintupled during the next 6 months. This is all uncontested.
Of course, most everyone is sure that I have to be wrong with my published proof of danger: That black holes, (i) arise more readily than originally hoped-for by CERN, (ii) are undetectable to CERN’s
detectors and (iii) will, with the slowest specimen generated, eat the earth inside out after a refractory period of a few years. “This is bound to be ridiculous!” is a natural response.
Mr. Ben Rattray has enabled the planet to learn about the huge danger incurred by the currently running – and till the end of 2012 three times more black holes-spouting – LHC experiment. This despite
the fact that CERN’s detectors cannot detect their most anticipated products and the fact that they grow exponentially inside earth once one of them gets stuck inside. In that case, only a few years
separate us from earth being a 2-cm black hole.
Please, ask around whether anyone can name a physicist who contradicts the published proof (Telemach theorem: http://www.scribd.com/doc/82752272/Rossler-s-Telemach-paper ). This physicist is
automatically the most important living physicist today. Finding him and learning about the strength of his argument is the only aim of the present appeal to every citizen of the world. To help in
dismantling the danger before it has risen by a factor of three.
Thank you. He or she who can contradict me most is my best friend. And yours. Let us search for this human being.
“A Constantly Receding Mass at Constant Distance Has a Lower Rest-mass and Charge”
Otto E. Rossler, University of Tubingen, Auf der Morgenstelle 14, 72076 Tubingen, Germany
This “extended gravitational redshift theorem” (EGRT) is unfortunately new even though it is true as far as anyone can tell up until now. The physics community is currently betting the planet on
claiming that this result were not true. It would be gracious if a single physicist stood up saying why he thinks the theorem is not true. (For J.O.R.)
Hawking could save us all if he spoke up.
A pope knealt before Hawking to make him re-confirm the big bang. I kneel before Hawking to make him re-confirm Hawking radiation.
Then we are all safe.
… theorem and still do so on CERN’s website ( http://public.web.cern.ch/public/en/lhc/Safety-en.html ):
Dear colleagues, please, try and dismantle my much simpler, and hence both more powerful and more easy-to-disprove if false, “Telemach theorem” ( http://www.scribd.com/doc/82752272/
Rossler-s-Telemach-paper ).
The latter again proves the likely pan-biocidal nature of the currently running LHC experiment. For it shows that black holes have radically new properties: They are stable, almost frictionless at
first, undetectable by CERN’s detectors, and exponentially growing inside matter – thus forming a perfect slow bomb for planet earth. The theorem waits to be dismantled for 2 years (the former does
so for 5 years).
I grant you, my esteemed 11 colleagues, 11 days to deliver – either on the CERN website of 2008, revised, or in case CERN denies you access, on this blog. If none of you manages to deliver a
counter-proof to Telemach during this time, I shall accuse all of you of actively supporting the worst terrorist act of history, presently in progress. Acting in good faith – as you no doubt will
pledge – offers no excuse as you were alerted in time. And please, do forgive me that I did not give you the occasion to revoke your testimony earlier.
Continue reading “I herewith Challenge my 11 CERN-supporting Colleagues who in 2008 Defamed my Gothic-R …” »
The reason for the current planet-wide abandonment of major progress lies in the re-acquired belief in clairvoyance – of which anonymous peer review is a symptom. Einstein would ridicule the latter
as a “dogma-generating superstition.” While in the early 17th century, the innovators were burnt on the stakes, to date the censors choose instead to burn themselves along with their children and
The expansion theory got disproved in 1929 by Hubble’s friend Zwicky. A remaining gap was closed in 1943 by Chandrasekhar, but the two apparently never met. The final cornerstone is the discovery of
a “second statistical mechanics” besides Thermodynamics, called Cryodynamics. It can be used to break the decades-old impasse of hot fusion and hence solve earth’s energy problems.
Continue reading “Big Bang gone, Gravitational Waves gone, Hawking Radiation gone: The Dolphins Confront CERN” »
On a casual read of the appraised work of Duncan R. Lorimer on Binary and Millisecond Pulsars (2005) last week, I noted the reference to the lack of pulsars with P < 1.5 ms. It cites a mere
suggestion that this is due to gravitational wave emission from R-mode instabilities, but one has not offered a solid reason for such absence from our Universe. As the surface magnetic field strength
of such would be lower (B ∝ (P ˙P )^(1÷2)) than other pulsars, one could equally suggest that the lack of sub millisecond pulsars is due to their weaker magnetic fields allowing CR impacts resulting
in stable MBH capture… Therefore if one could interpret that the 10^8 G field strength adopted by G&M is an approximate cut-off point where MBH are likely to be captured by neutron stars, then one
would perhaps have some phenomenological evidence that MBH capture results in the destruction of neutron stars into black holes. One should note that more typical values of observed neutron stars
calculate a 10^12 G field, so that is a 10^4 difference from the borderline-existence cases used in the G&M analysis (and so much less likely to capture). That is not to say that MBH would equate to
a certain danger for capture in a planet such as Earth where the density of matter is much lower — and accretion rates much more likely to be lower than radiation rates — an understanding that is
backed up by the ‘safety assurance’ in observational evidence of white dwarf longevity. However, it does take us back to question — regardless of the frequently mentioned theorem here on Lifeboat
that states Hawking Radiation should be impossible — Hawking Radiation as an unobserved theoretical phenomenon may not be anywhere near as effective as derived in theoretical analysis regardless of
this. This oft mentioned concern of ‘what if Hawking is wrong’ of course is endorsed by a detailed G&M analysis which set about proving safety in the scenario that Hawking Radiation was ineffective
at evaporating such phenomenon. Though doubts about the neutron star safety assurance immediately makes one question how reliable are the safety assurances of white dwarf longevity – and my belief
has been that the white dwarf safety assurance seems highly rational (as derived in a few short pages in the G&M paper and not particularly challenged except for the hypothesis that they may have
over-estimated TeV-scale MBH size which could reduce their likelihood of capture). It is quite difficult to imagine a body as dense as a white dwarf not capturing any such hypothetical stable MBH
over their lifetime from CR exposure – which validates the G&M position that accretion rates therein must be vastly outweighed by radiation rates, so the even lower accretion rates on a planet such
as Earth would be even less of a concern. However, given the gravity of the analysis, those various assumptions on which it is based perhaps deserves greater scrutiny, underscored by a concern made
recently that 20% of the mass/energy in current LHC collisions are unaccounted for. Pulsars are often considered one of the most accurate references in the Universe due to their regularity and
predictability. How ironic if those pulsars which are absent from the Universe also provided a significant measurement. Binary and Millisecond Pulsars, D.R. Lorimer: http://arxiv.org/pdf/astro-ph/
Hawking radiation is dead ever since the Telemach result and its precursors surfaced on the web. No one ever defended Hawking including his own heroic voice.
The same holds true for CERN’s detectors. They are blind to its most touted anticipated success – black hole production – by virtue of the said theorem. Again not a single word of defense.
This is why a court asked CERN and the world for a safety conference on January 27, 2011.
The press cannot continue shielding the world, and Lifeboat must be relieved from its having to carry the burden of informing an otherwise lifeboat-less planet, singlehandedly.
Russia’s hastily convened international conference in St. Petersburg next month is being billed as a last-ditch effort at superpower cooperation in defense of Earth against dangers from space.
But it cannot be overlooked that this conference comes in response to the highly controversial NATO anti-ballistic missile deployments in Eastern Europe. These seriously destabilizing, nuclear
defenses are pretexted as a defense against a non-nuclear Iran. In reality, the western moves of anti-missile systems into Poland and Romania create a de facto nuclear first-strike capability for
NATO, and they vacate a series of Anti-Ballistic Missile Treaties with the Russians that go back forty years.
Deeply distrustful of these new US and NATO nuclear first-strike capabilities, the Russians announced they will not attend NATO’s planned deterrence summit in Chicago this month. Instead, they are
testing Western intentions with a proposal for cooperative project for near-space mapping, surveillance, and defense against Earth-crossing asteroids and other dangerous space objects.
The Russians have invited NATO members as well as forward-thinking space powers to a conference in June in Petrograd. The agenda: Planetary defense against incursions by objects from space. It would
be a way of making cooperative plowshares from the space technologies of hair-trigger nuclear terror (2 minutes warning, or less, in the case of the Eastern European ABMs).
It’s an offer the US and other space powers should accept. | {"url":"https://spanish.lifeboat.com/blog/category/existential-risks/page/125","timestamp":"2024-11-02T14:46:09Z","content_type":"application/xhtml+xml","content_length":"147458","record_id":"<urn:uuid:3b549763-5a03-4da1-acdb-08d9fc3b78d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00718.warc.gz"} |
Neutron Velocity Calculator - Calculator Doc
Neutron Velocity Calculator
The Neutron Velocity Calculator is a tool designed to help you calculate the speed at which neutrons travel based on their energy levels. Neutron velocity plays an essential role in nuclear physics,
research, and reactor operations, where controlling the speed of neutrons can affect the reaction rate and overall system performance.
The formula for calculating neutron velocity is:
Vn = 1.383 * 10^6 * √E / 100
• Vn is the neutron velocity in meters per second (m/s),
• E is the neutron energy in electron volts (eV),
• 1.383 * 10^6 is a constant used to convert energy to velocity.
How to use
1. Enter the neutron energy (E) in electron volts (eV).
2. Click “Calculate” to compute the neutron velocity (Vn) in meters per second.
3. The result will appear in the provided field.
Let’s say the neutron energy is 25 eV. Using the formula:
Vn = 1.383 * 10^6 * √25 / 100
Vn = 1.383 * 10^6 * 5 / 100
Vn ≈ 69150 m/s
The neutron velocity is approximately 69,150 meters per second.
1. What is neutron velocity?
Neutron velocity refers to the speed at which neutrons move based on their kinetic energy, typically measured in meters per second (m/s).
2. Why is neutron velocity important in nuclear physics?
Neutron velocity is crucial for controlling reactions in nuclear reactors, as it affects reaction rates and the efficiency of the reactor.
3. What unit is used for neutron energy?
Neutron energy is measured in electron volts (eV), a unit of energy commonly used in particle physics.
4. How does neutron energy affect velocity?
Neutron velocity increases with higher energy levels, as they gain more kinetic energy and move faster.
5. Can this formula be used for thermal neutrons?
Yes, the formula can be applied to calculate the velocity of thermal neutrons, which typically have lower energy levels.
6. What is the significance of the constant 1.383 * 10^6 in the formula?
The constant is used to convert the energy input in eV to velocity in meters per second.
7. What is the typical velocity of a neutron in a reactor?
The velocity of a neutron in a reactor depends on its energy, but thermal neutrons typically move at speeds around 2,200 meters per second.
8. Can neutron velocity be negative?
No, velocity represents speed and is always a positive value. A negative energy input would be invalid for this calculation.
9. What happens if the neutron energy is zero?
If the neutron energy is zero, the neutron would have no velocity, meaning it is at rest.
10. Is neutron velocity affected by external factors?
Neutron velocity is primarily influenced by its energy, but factors like the medium in which it travels can impact its speed.
11. How can neutron velocity be measured in experiments?
Neutron velocity is often measured using detectors that track the time of flight, which calculates how long it takes for neutrons to travel a specific distance.
12. What is the relationship between neutron velocity and nuclear fission?
Neutron velocity is important in nuclear fission as it determines whether neutrons can sustain a chain reaction by causing additional fission events.
13. Are fast neutrons slower than thermal neutrons?
No, fast neutrons have higher energy and move faster than thermal neutrons, which are slowed down to facilitate certain reactions.
14. What is the difference between neutron velocity and speed?
Velocity is a vector quantity that includes both speed and direction, while speed is a scalar quantity referring only to how fast an object moves.
15. Can neutron velocity be controlled in a reactor?
Yes, neutron velocity can be controlled using moderators, which slow down fast neutrons to thermal speeds, enhancing the efficiency of reactions.
16. What is the effect of neutron velocity on cross-sections in reactions?
The probability of certain nuclear reactions depends on neutron velocity, as cross-sections (reaction probabilities) can vary with speed.
17. What is the range of neutron velocities in a reactor?
Neutron velocities in reactors can range from a few hundred meters per second for thermal neutrons to several million meters per second for fast neutrons.
18. What is the velocity of a neutron at room temperature?
At room temperature, thermal neutrons have a velocity of approximately 2,200 meters per second.
19. Can neutron velocity be increased without increasing energy?
Neutron velocity is directly proportional to energy, so increasing velocity requires increasing the neutron’s energy.
20. How do moderators affect neutron velocity?
Moderators slow down fast neutrons by allowing them to collide with lighter atoms, reducing their velocity and facilitating certain reactions.
Neutron velocity is a fundamental aspect of nuclear physics that influences reaction rates and the behavior of neutrons in various systems. By using the Neutron Velocity Calculator, you can easily
determine the velocity of neutrons based on their energy. Understanding neutron velocity helps in the design and operation of nuclear reactors and other systems involving neutron interactions. | {"url":"https://calculatordoc.com/neutron-velocity-calculator/","timestamp":"2024-11-06T08:18:26Z","content_type":"text/html","content_length":"87417","record_id":"<urn:uuid:563fb46c-7b0f-49d7-b794-c25da5520045>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00633.warc.gz"} |
Analysis and comparison of sleeper parameters and the influence on track stiffness and performance
- Request for pdf of the paper -
In the last years, plastic railway ties have made their introduction. Amongst its characteristics, plastic ties have a good damping and a high design freedom. If used in the proper way, plastic ties
can give improvements in the track. They should not be regarded as a substitute for wood or concrete, but use should be made of their own characteristics. Existing sleeper requirements are however
applicable for wood or concrete and can hardly be seen as functional requirements suitable for the development of plastic ties.
The desired track stiffness is the first parameter to define in setting requirements. A good compromise between bending stresses in the rail versus noise and vibration seems to be a target track
stiffness of 50 kN/mm. When making a comparison between the different sleeper materials, the target track stiffness can be reached with plastic ties, where concrete tends to be on the stiffer side
and wood shows more variations. Knowing the track stiffness gives the possibility to calculate the distribution of forces over the ties. Especially at irregularities in the track, such as bridges or
viaducts, forces on ties can become high. Special attention to sleeper stiffness parameters should be given at those locations, as well to bending stiffness as to compression stiffness.
The sleeper stiffness parameters are input in calculating the system stiffness. Effects of sleeper bending stiffness on track stiffness, railhead stability and ballast contact stresses are discussed.
For a 2600 mm sleeper, a 150-250 kNm2 bending stiffness seems appropriate, where for a 2400 mm sleeper the minimum bending stiffness should be higher. The sleeper stiffness also has effects on the
strength requirements, as has the sleeper length. Where it is clear that every situation will be different, calculations have been done to give a mean value as an example. Every specific situation
can be calculated accordingly.
Railway Ties have in the past been made from wood, concrete and steel. These materials have good properties, but also have their downsides. Wood has been used since the first railway track was put in
place. Concrete ties are being used more than wooden ties nowadays. Concrete is however a stiff material, consequently dynamic forces and vibrations are high, which causes, for example, high wear and
degradation to the ballast. Wooden ties are therefore still being used in a lot of applications where concrete is too rigid a material. However, when not treated with creosotes, the lifespan of a
wooden sleeper is quite limited, giving high replacement costs. Within the European union creosotes will soon be banned, which are now used to give the wooden ties an acceptable life span. Also
availability issues, especially for longer bearers, increases the desirability of an alternative. Tropical hardwood can do without creosotes, but its environmental implications and availability do
not make it a viable alternative for large scale application. In the last years, recycled plastic ties have made their introduction (see Figure 1).
Figure 1: Plastic railroad ties in track.
Plastic ties are a good alternative that can give solutions for specific problems in the track. Plastic is however a material with other characteristics than wood or concrete. It should not be
regarded as a substitute for wood or concrete, but its unique characteristics should be made use of.
To make a suitable plastic rail road tie, the material choice to get to a pricewise compatible tie should be one of the bulk plastics, most likely Polyethylene or Polypropylene. These materials have
a bending stiffness and a thermal expansion coefficient, that makes them unsuitable to be used as is. Solving these issues can be done by either reinforcing them, for example with glass fibers,
creating a composite tie, or by embedding reinforcing elements, such as steel or glass fiber bars, creating a hybrid railroad tie. Figure 2 shows an example of a hybrid sleeper. In this case there
are 4 reinforcing metal bars in the corners of the sleeper.
Figure 2: Steel reinforced KLP-S® tie.
Ties can be made by either extrusion or injection moulding technologies. In extrusion the sleeper is formed continuously by pressing heated plastic through a die. The shape of the sleeper is
therefore uniform in longitudinal direction, except for any mechanical treatments that are done afterwards. With injection moulding, the heated plastic is pressed into a mould, after which the
material is cooled. The mould can have any desired shape and so can the sleeper.
The pros and cons of plastics:
• Plastic can be shaped in any desired shape. This is primarily the case for injection moulded ties. Optimization is possible, which for example can lead to (see Figure 3):
• Reduction of material use.
• The ballast is partly on top of the sleeper, thereby increasing the vertical stability of the sleeper.
• The change in width (and the profiled underside) of the sleeper increases the lateral stability of the sleeper.
Figure 3: Optimized railway tie shape.
• Plastic is a material with time dependent stiffness properties. That means that, static test outcomes should not be used one on one to predict dynamic behaviour. Testing should be done at the
appropriate speed of utilization, thus measuring dynamic material properties, as is being done with railpads. There is a lot known about the time related behaviour of plastics, so interpretation
of static tests is possible when material properties are available.
• Plastic materials have a high damping. This results in good performance in the area of sound or vibration reduction. Measurements on a steel girder bridge showed a 3-5 dB noise reduction after
replacing wooden sleepers for plastic ties of the type as in Figure 2, see Figure 4 (Movares, 2010).
Figure 4: Sound measurements on a Steel girder bridge near Raalte in The Netherlands.
• The thermal expansion of plastics is too large to use as is in a sleeper; normally in the range of 15·10-5 to 20·10-5 °C^-1. Adding glass fibers can bring the expansion rate down with about a
factor 2 at maximum. A more effective solution is for example to use steel inserts, which brings down the expansion rate to the level of steel or concrete, around 1,2·10-5 °C^-1.This will exclude
all problems, especially on bridges where the ties do not have temperature shielding by the ballast.
• Plastics are highly resistant against degradation from weather influences. This will in general give an advantage against wooden ties. Specific area’s where wood cannot dry properly are very
suitable for the use of plastic railroad ties. See for example Figure 5.
Figure 5: Switch and ties built into the pavement.
• Plastic materials have a high flexibility. This is a disadvantage in creating the desired bending stiffness. Adding glass fibers or reinforcements or adjusting the sleeper height is needed to get
the proper bending characteristics. The flexibility is an advantage in the compression of the sleeper. This compressive flexibility gives a good distribution of the wheel loading over multiple
rail ties, and also high dynamic forces will be distributed more easily. The high flexibility also gives a high local pressure to the ballast under the ties. In the case of a composite sleeper,
the stiffness of the sleeper is more or less the same in all directions (in flow direction of the plastic somewhat higher). Creating a high bending stiffness will therefore result in unwillingly
creating a high compressive stiffness. In the case of a hybrid sleeper, the properties in axial and lateral direction can be decoupled. A higher bending stiffness can be reached (using lower
sleeper heights and less material) where the compression stiffness can be optimized independently, creating the more optimal solution. Because a more ductile plastic material can now be chosen
and the deformation that this kind of material can experience before break can be much higher, an unbreakable sleeper can be made in a hybrid construction.
• Plastic has a good chemical resistance. Concrete sometimes experiences problems in this area on industrial tracks.
• Plastics have high rebound resilience. See Figure 6.
Figure 6: No indentation under baseplate after 133 million tons of load (30 tons axle loads).
• Plastics can be drilled and milled like wood. In concrete every bolt hole (dowel) has to be pre- casted in the factory. For example the replacement of a switch in concrete would require measuring
the complete switch, where plastic railroad ties can be easily fitted in track.
Figure 7: Switch with plastic ties.
• Plastic rail road ties are normally made of 100% recycled plastic. That gives 80-160 tonnes of high value recycling for a kilometer of track. After its lifetime, the railroad tie can be regrinded
and the material can be used again for the next generation of ties. Inserts can be removed and reused.
Figure 8: Connectable switch sleeper with constant bending characteristics over its length.
• Plastic railway ties can be designed for a specific problem. For example when very long ties are needed for switches, a connectable sleeper can solve transportation problems of the switch, see
Figure 8. Another solution can be seen in Figure 9, where a bridge sleeper is shown. Ties that are used on steel girder bridges have to be measured sleeper by sleeper to compensate for the
tolerances in the steel girders. The sleeper in this picture can be adjusted to the right height and angle by mounting insertion blocks of the right dimensions. The insertion blocks are fixed by
the screw spikes.
Figure 9: Bridge sleeper with insertion blocks.
• The expected service life of plastic railroad ties is long. Although the experience in this field is limited, there is a good track record with similar products. Table 1 gives an estimation of
the expected service life for the different sleeper materials.
Table 1: Expected service life of sleepers for UIC class 4 track (11 MGT/yr) (Prorail, 2014)
Difficulty in the development of plastic rail ties is that wood has been used for over 150 years in the track. We all know that it works, but the real mechanical requirements are not very clear. If
you look into a standard for wooden ties, you will find requirements for the dimensions, the allowable warpage and the amount of knots, but you will not find strength or stiffness requirements. The
chosen type of wood makes sure these characteristics are incorporated. It would however be too simple to regard the properties of wood or concrete as requirements for the development of plastic rail
road ties. The fact that these materials possess certain mechanical characteristics does not mean it is necessary for its function. Doing this would give some serious mistakes:
• The requirements might become much higher than necessary. The costs of the sleeper would therefore also be much higher than necessary.
• Some requirements have effect on each other. For example for strength you could require as a minimum the values that you experience in wood. For stiffness you do the same. But then you neglect
the fact that when you have a stiffer sleeper, the forces on the sleeper will be higher and the strength requirement might not be high enough. The relations that exist in wood between the
different properties are impossible to copy.
• Requiring exactly the same range as a wooden sleeper would give the problem that wood has a huge spread in its properties and that the wood properties change during the lifetime due to decay.
• Some properties that exist in wood/concrete, might not be the best properties.
It will be clear that this route is not the best way forward. At the same time it describes the current problem in developing plastic railties. Functional requirements for ties, that exist
independently from the materials used, should be developed.
To determine the necessary sleeper properties, the first priority is to determine the required system stiffness. The deflection of the rail on a train passage has to be within certain limits. If the
deflection is too high, the bending moments in the rail become too high and fatigue in the rail can become an issue. Arema advises a deflection of 3,2 - 6,35 mm (AREMA, 2006). If the deflection is
too low, impact loads on the ballast and sleeper become higher, which enhances degradation of the track and leads to more maintenance. Also ground borne noise and vibration increases with a stiff
rail construction. Riessberger advises a minimum deflection of 2 mm for this reason (Riessberger, 2014). The system stiffness also determines how much of the wheel load is transferred to one sleeper
and is therefore necessary information as input for the strength analysis. The stiffer the system is, the higher the load on one sleeper is and therefore the higher the strength requirements should
The track stiffness k is defined as the relation between the wheel load Q and the deflection δ directly under the wheel.
The wheel load is determined by the axle load and the dynamic amplification factor fd. When taking a maximum axle load of 22,5 tons and an fd of 2, the minimum track stiffness should be 35 kN/mm, to
comply with the maximum deflection as stated by Arema. For determination of the maximum track stiffness according to Riessberger, not the maximum permitted loading is of interest, but a mean expected
loading. Taking as an estimate 2/3 of the maximum allowed loading and f[d], the maximum track stiffness should be 50 kN/mm. An appropriate target track stiffness would therefore be 50 kN/mm.
As derived by Zimmermann in 1888, we can calculate the relation between the wheel load Q and the load F that is applied on the sleeper (Esveld, 2007) with Equation 2 in Table 2. As shown in Figure
10, with a target track stiffness of 50 kN/mm, we can expect that 28-37% of the wheel load is transferred to the sleeper directly under the wheel, depending on the rail profile (NP46, UIC54 and UIC60
are analyzed).
This distribution is required as input for the strength calculations.
Figure 10: Sleeper load variations, c.t.c. distance of ties 600mm.
In analyzing the system stiffness, there are three stages of load distribution, see Figure 11.
Figure 11: Load distribution on sleeper.
1. The distribution of forces caused by the wheel over the different railway ties. This is analyzed considering the rail profile as a beam on a resilient support as determined by Zimmermann in 1888,
see Equation 2. The track modulus u of the system is the main stiffness parameter of the system and has to be calculated. This is done by determining the foundation stiffness of baseplate (K[P]),
sleeper (K[S]) and railpad (K[RP]) with help of Equation 3.
2. The distribution of the force through the baseplate over the sleeper is determined by the bending stiffness of the baseplate and the compression of the sleeper. This can also be done considering
the baseplate as a beam on a resilient support (the resilient support being the sleeper). Assuming an appropriate baseplate has been chosen, a simplified assumption is often used, which assumes
that the baseplate has an evenly distributed support, and that the force distribution through the sleeper occurs under a 45 degree angle (see Equation 4). In the width direction the spreading of
the force will be limited by the width of the sleeper.
Part of the system stiffness is determined by the dynamic stiffness of the rail pad K[RP], which is added in Equation 3.
3. The distribution of the support load under the sleeper caused by the bending stiffness of the sleeper EIS and the resilient support of the ballast. This distribution has been analyzed by Hetenyi
in a similar way as the Zimmermann formula (Hetenyi, 1946), see Equation 5. The bedding modulus C of the ballast is a major variable here and can be expected to vary between 0,04 and 0,16 N/mm^3
(Manalo, 2010). The bedding modulus is the spring constant (N/mm) of the underground, defined per mm^2 surface, giving units of N/mm^3.
Table 2: Equations for stiffness calculation. (van Belkom, Railway sleeper design manual, 2014)
The derived equations have been used to make a rough comparison of the different construction materials for railties with respect to the expected track stiffness. A 1435 mm gauge track with sleepers
spaced at 600 mm and a UIC54 rail profile has been used for the evaluation. The input parameters have been specified in Table 3.
Table 3: Assumed properties for stiffness comparison (source material data plastic: (van Belkom, Material calculation data, 2013), wood: (Green, Winandy, & Kretschmann, 1999), concrete: NEN- EN
In Figure 12 the effects of the ballast and sleeper stiffness on the track stiffness can be seen. The two lines represent the upper (P95) and lower limit (P5) of the sleeper properties that can be
expected. The target track stiffness value of 50 kN/mm is highlighted in the graph. What can be seen is that the most important effect is caused by the bedding modulus of the ballast and subgrade.
Whatever you do, sleeper properties can never compensate the variations in bedding modulus. It can further be seen that:
• Concrete ties tend to create a stiff structure, usually stiffer than the target track stiffness.
• Wood ties give a rather broad range of possible track stiffnesses, due to the large variety in possible mechanical properties.
• Plastic ties give the possibility to aim for the target track stiffness, provided bedding modulus is appropriate.
Figure 12: Bedding modulus - Track stiffness relation for some sleeper materials, sleeper properties according to Table 3.
When looking at the force distribution over the railway ties, the support modulus of the total construction and the rail stiffness define how much of the wheel force is distributed to the underlying
sleeper, and how much to adjacent ties. For the case of the mean plastic sleeper (see Table 3) on a mean bedding modulus of 0,1 N/mm^3, Figure 13 gives the distribution graph. The wheel is here on
top of sleeper 0. About 32% of the wheel force, including dynamic effects, will be distributed to this sleeper. Up to 3 ties to the left and the right (about 2 meter), the influence of the load can
be seen.
Figure 13: Force distribution over ties for mean plastic sleeper of Table 3.
Due to variations in the bedding modulus and in the sleeper properties, the force distribution, and thus the maximum force on one sleeper can vary. Table 4 shows the in the first column the most
flexible sleeper on the most flexible bedding. This gives the lowest sleeper load. The second column shows the most rigid sleeper on the most rigid bedding, giving higher sleeper loads. The highest
sleeper loads however can occur when a more rigid sleeper from the tolerance range is placed amidst more flexible railties, which can be seen in the most right column. The wood ties seem to have a
higher possible maximum load than concrete, which would in the first glance be strange. The concrete sleeper in this analysis however does have a more flexible rail pad (Table 3), but also the wider
stiffness range of the wooden ties causes this effect.
Table 4: Load on sleeper as % of wheel load, for sleepers according to Table 3.
The analysis in Table 4 assumes the bedding modulus doesn’t change from sleeper to sleeper, which in general won’t happen. This is however different when the track runs over a bridge, a viaduct or
any other obstacle causing a disruption of the bedding stiffness. Figure 14 shows the distribution of forces over railties on a bridge. Sleeper 0 is the first sleeper on the bridge, the bridge
extends to the right. The track sleepers are concrete, bridge ties are Azobé, all according to Table 3, now considering a mean sleeper and bedding stiffness.
Figure 14: Force distribution over ties on edge of bridge. Sleeper 0 is the first sleeper on the bridge, the bridge extends to the right. Track ties are Concrete, bridge sleepers are Azobé, all
according to Table 3.
Because of the missing flexibility of the ballast and subgrade on the bridge, we can see here that the force on the railroad ties can be much higher than in track, specifically for the first sleeper
on the bridge. These situations should be avoided. Sleeper compressive flexibility on a bridge should be much higher than in track to compensate for the missing flexibility of the ballast. A hybrid
plastic sleeper can have a much higher compressive flexibility than wood or concrete and would be a good choice for a solution.
This analysis was just one example. For each case specifically the optimal compressive sleeper stiffness should be defined to get a continuous track stiffness. The analysis hasn’t considered any
dynamic effects yet or settlement of the track before and after the bridge, giving a height difference between bridge and ballasted track. Settlement is not prevented by a continuous track stiffness,
but it does have a positive effect. Height adjusting means of the ties on the bridge would be an additional aid to compensate any track settlements.
Also the situation in a switch requires special consideration. Since the ties in a switch are much longer and the distribution of forces over the ties is not done by 2 rails, but by 4, the load on
one sleeper is less than in regular track. Also the track stiffness in a switch is higher because of this. Rail road ties with a higher flexibility should therefore be used in a switch to create a
continuous track stiffness.
The basic thought on sleeper bending stiffness seems to be that the higher the sleeper bending stiffness is, the better the sleeper performance. The system performance has been analyzed for a change
in sleeper bending stiffness to create a better picture of these effects.
Effects of sleeper bending stiffness on track stiffness
When analyzing the plastic sleeper of Table 3, and evaluating possible different bending values for the stiffness, keeping the compressive stiffness constant as mentioned in the table, the
calculation model is as depicted in Figure 15A, being the beam on a resilient support, as previously used for the track stiffness evaluation.
Figure 15: Calculation models.
The outcome of this calculation can be seen in Figure 16. A maximum (0,16 N/mm^3), minimum (0,04N/mm^3) and a mean value (0,10N/mm^3) for the bedding modulus has been used.
Figure 16: Track stiffness as a function of sleeper stiffness, for 3 different bedding moduli of the ballast (N/mm^3), properties according to Table 3.
What can be observed is that a low bedding modulus prevents the system of achieving the target stiffness and there is nothing that even the stiffest sleeper can do to prevent that. Targeting at a
mean bedding modulus, the sleeper bending stiffness should be in the area of 150-250 kNm^2.
Effects of sleeper bending stiffness on rail head stability
Limiting the lateral displacement of the railhead when a train passes is an important function of the system and of the sleeper in particular. While this regards safety, it is not a serviceability
limit state (SLS), as was the case for the track stiffness, but an ultimate limit state (ULS). The calculation model is therefore not done regarding mean values, but extreme values. The calculation
model according to Figure 15B is therefore applicable, which describes a worst case support situation with regard to the possible rotation of the rail seat area. The support is a uniform distributed
load, which can be expected for a deteriorated track. The horizontal deflection δ[H] of the railhead can then be given as (van Belkom, Railway sleeper design manual, 2014):
Since the load situation is now a ULS calculation, worst case loads and support situations are analyzed. All applicable safety factors according to ISO 13230-6:2014 are incorporated. Doing this
calculation for the plastic sleeper of Table 3, gives the outcome as can be seen in Figure 17. An axle load of 22,5 ton is taken and a speed of 140 km/h. Be aware that the loads are the extreme
loads, mean loads will be a factor 3 lower. The minimum bedding modulus of 0,04 N/mm^3 has been used.
Figure 17: Maximum (extreme) horizontal deflection of railhead as a function of sleeper stiffness, for 3 different sleeper lengths (L[S]), properties according to Table 3, C=0,04 N/mm3.
Figure 17 shows the horizontal rail deflection for extreme loading on a straight track, for sleeper lengths of 2400, 2600 and 2700 mm. What can be seen is that the higher the sleeper bending
stiffness, the lower the deflection. The sleeper length has a lot of influence: a sleeper length of 2600 mm gives a more stable railhead than a 2400 mm sleeper. The optimum for this analysis lies
somewhere around 2730 mm. At that length the loading does not have any effect on the bending of the sleeper (for this particular load case).
Railties that are even longer will give an inward bending when loaded. When taking for example a 3 mm deflection as a maximum, the consequential minimum sleeper bending stiffness for a 2600 mm
sleeper will be around 120 kNm^2, whereas for a 2400 mm sleeper this should be 300 kNm^2. This analysis only considers straight track. For the situation in curves, additional analysis would be
Effects of sleeper bending stiffness on ballast contact stresses
If the contact stresses in the ballast are high the ballast will degrade more rapidly, with higher maintenance costs as a result. The stress in the ballast is best kept below 0,5 MPa to prevent this
(Esveld, 2007). The highest contact stress between sleeper and ballast will occur under the rail seat, as can be seen in Figure 11. It is obvious that the stiffer the sleeper is, the more evenly
distributed the stresses will be.
Therefore the sleeper bending stiffness is of interest in determining ballast stresses. Performing a static analysis however gives only a partial answer. Aspects that cannot be seen from this
analysis are:
• The dynamic effects: the more rigid the system is, the higher the dynamic impulses will be.
• The effective area of the ballast – sleeper contact: If the sleeper is made of a very rigid material, the contact area of the ballast will be very small, creating very high stresses on sleeper
and ballast. When the sleeper is softer, the point of highest stress will be moved a layer downwards, where the ballast interacts with itself. Because this is a lower layer, the force is already
more distributed, lowering stresses.
Dynamic analyses or field tests should be done to asses this subject. It is known from praxis that concrete ties have more problems with ballast degradation than wooden ties. Since concrete sties are
stiffer than wooden ties, it can be concluded that a high system stiffness and high contact stiffness are the main contributors to ballast wear. Since plastic railroad ties have a comparable system
and contact stiffness as wood, degradation of ballast is also expected to be comparable. If the bending stiffness is much lower, localizing the contact area can increase the stresses. The stresses at
the rail seat can be calculated with help of (Esveld, 2007):
We can calculate the force on the sleeper F with help of Equation 2 and the foundation stiffness of the sleeper K[S] with help of Equation 5. Performing this calculation with the parameters of Table
3 gives the graph according to Figure 18, and shows the stresses in the ballast as a function of the bending stiffness of the sleeper. As can be seen the sleeper bending stiffness should be kept over
100 kNm^2.
Figure 18: Stress in ballast at rail seat as a function of sleeper bending stiffness for 3 different bedding moduli (N/mm^3).
The distribution factors that have been calculated with help of the sleeper stiffness can now be used for calculating the sleeper strength requirements. Strength analyses of the sleeper can be done
according to the calculation model of Figure 15. The maximum bending moment in the centre of the sleeper will occur in load case B, which in praxis occurs in a deteriorated ballast bed. The maximum
centre bending moment M[C] is defined in Table 5. In the area under the rail seat, the maximum bending moment Ma will occur after tamping, which load case situation is depicted in load case C of
Figure 15.
Table 5: Equations for strength calculation
When for example taking the parameters of Table 6, this gives a required design centre bending moment of 9,8 kNm for a 2600 mm sleeper and 14,3 kNm for a 2400 mm sleeper. For the required design
bending moment at the rail seat area this value is 15,7 kNm for a 2600 mm sleeper and 11,4 kNm for a 2400 mm sleeper.
Table 6: Assumed properties for strength analysis.
While these figures are given as an example, the method of calculation can be applied for any specific situation. Doing such an analysis gives more insight in the requirements than copying wooden
sleeper properties. It shows that the system stiffness has a distinct influence on the strength requirements. It also shows that requirements cannot be set without taking the sleeper length into
Plastic rail ties can offer solutions in the track at positions where concrete ties are too stiff. In particular this is true for bridges, viaducts, and other places in the track where the ballast
stiffness has an increased value. Also switches induce an increase in track stiffness that should be compensated.
More in general, to create a proper track stiffness from a point of view of wear and vibration plastic railties can provide a good solution. Adaptability at site, for example for switches and the
one-on-one exchange with wooden ties for partial renewal of wooden track, is possible with plastic rail ties. The design freedom of plastic gives the opportunity to create optimal solutions for
specific problems.
Initiating the use of plastic railroad ties creates the necessity to have proper functional requirements for plastic railties, or better, for railties in general. The stiffness of the system plays an
important role in this analysis. It should be looked at, not only from the point of view of determining proper track stiffness, but also to form the basis for the strength calculations. Sleeper
bending stiffness requirements should not only incorporate a minimum value, but should define a range, as is the same for compressive stiffness.
Also to define strength requirements, the system stiffness as well as sleeper length should be known variables.
a : distance between end of sleeper and centre of rail (mm)
c : distance between centre of sleeper and centre of rail (mm)
c[V ]: coefficient of variation (%)
C : bedding modulus (N/mm^3)
E[C ]: E-modulus of sleeper in compression (N/mm^2)
EI[R]: Product of Young’s modulus E and moment of inertia I of rail (N/mm^2)
E[S ]: Young’s modulus of sleeper in bending (Nmm^2)
EI[S]: Product of Young’s modulus E and moment of inertia I of sleeper (Nmm^2)
f[d] : dynamic amplification factor (-)
F : applied force on sleeper (N)
h[R ]: height of rail (mm)
h[S] : height of sleeper (mm)
k : track stiffness (N/mm)
K[P ]: foundation stiffness caused by bending of baseplate & compression of sleeper (N/mm)
K[RP]: dynamic stiffness of rail pad (N/mm)
K[S ]: foundation stiffness caused by bending of sleeper & compression of ballast /subgrade (N/mm)
L[P] : length of baseplate (mm)
L[S ]: length of sleeper (mm)
M[a ]: maximum bending moment at rail seat (Nmm)
M[c ]: maximum bending moment at centre of sleeper (Nmm)
Q : wheel load = axle load/2 (N)
S : center to center distance of ties (mm)
t[P ]: thickness of baseplate (mm)
t[RP]: thickness of railpad (mm)
u : track modulus (N/mm2)
u[B] : support modulus of ballast/subgrade (N/mm2)
w[B] : width of sleeper at bottom in contact with ballast (mm)
w[P] : width of baseplate (mm)
w[S] : mean width of sleeper (mm)
δ : deflection (mm)
δ[H] : horizontal deflection of railhead (mm)
λ : characteristic of the rail (1/mm)
λ[S ]: characteristic of the sleeper (1/mm)
Ϭ[B ]: stress in ballast at rail seat (N/mm^2)
AREMA. (2006). Manual for railway Engineering. In Volume 4, Chaper 16, part 10.
Esveld, C. (2007). CT3041-Constructief ontwerp van spoorwegen. Delft: TUDelft.
Green, W., Winandy, J., & Kretschmann, D. (1999). Wood Handbook - Wood as an engineering material
- Chapter 4, mechanical properties of wood. Forest Products Laboratory, US department of Agriculture.
Hetenyi, M. (1946). Beams on elastic foundation. University of Michigan.
Manalo. (2010). Fibrecomposite sandwich beam. University of Southern Queensland. Movares. (2010). Geluidproductie spoorbrug Laag Zuthem. Utrecht.
Prorail. (2014). Levensduurverwachtingspoor en wissels ten behoeve van vervangingsplannen BID00020-V001.
Riessberger, K. (2014). Presentation on Rail technology conference, 18-20 March 2014. Dusseldorf: University of Graz.
van Belkom, A. (2013). Materialcalculationdata. Sneek: Lankhorst Engineered Products.
van Belkom, A. (2014). Railwaysleeperdesignmanual.Sneek: Lankhorst Engineered Products. Young, W. C. (1989). Roark's formulas for stress & strain. McGraw-Hill.
Aran van Belkom
Lankhorst Engineered Products
Prinsengracht 2
8607AD Sneek
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Anderson localization in optical lattices with correlated disorder
arXiv:1510.05121v1 [cond-mat.quant-gas] 17 Oct 2015
E. Fratini and S. Pilati1 1
The Abdus Salam International Centre for Theoretical Physics, 34151 Trieste, Italy We study the Anderson localization of atomic gases exposed to simple-cubic optical lattices with a superimposed
disordered speckle pattern. The two mobility edges in the first band and the corresponding critical filling factors are determined as a function of the disorder strength, ranging from vanishing
disorder up to the critical disorder intensity where the two mobility edges merge and the whole band becomes localized. Our theoretical analysis is based both on continuous-space models which take
into account the details of the spatial correlation of the speckle pattern, and also on a simplified tight-binding model with an uncorrelated distribution of the on-site energies. The mobility edges
are computed via the analysis of the energy-level statistics, and we determine the universal value of the ratio between consecutive level spacings at the mobility edge. We analyze the role of the
spatial correlation of the disorder, and we also discuss a qualitative comparison with available experimental data for interacting atomic Fermi gases measured in the moderate interaction regime.
PACS numbers: 03.75.-b, 67.85.-d,05.60.Gg
Anderson localization, namely the complete suppres-sion of wave diffusuppres-sion due to sufficiently strong disor-der [1], is one of the most important and intriguing phe-nomena studied in condensed
matter physics [2, 3]. Mak-ing reliable predictions for the critical disorder strength required to induce complete localization is a major the-oretical challenge. In the theory of solid-state
systems, studies that aim at a quantitative comparison between theory and experiments, and thus employ realistic mod-els taking into account the details of a specific material, have appeared only
recently [4].
Following the first experimental observations of Ander-son localization in quantum matter waves [5–9], ultra-cold atomic gases have emerged as the ideal setup to investigate the effects due to
disorder in quantum sys-tems [10, 11]. Feshbach resonances provide experimen-talists with a knob to turn off the interatomic scattering, allowing them to disentangle the effects due to disorder from
those due to interactions. Furthermore, using the optical speckle fields produced by shining coherent light through a diffusive plate, they can introduce disorder in a controlled manner, and even
manipulate the structure of its spatial correlations [12]; this kind of control is not possible in solid-state devices. Techniques to accurately measure the mobility edge, namely the energy threshold
which separates the localized states from the extended states, have also been implemented [13].
Several previous theoretical studies on Anderson local-ization have disclosed the fundamental role played by the disorder correlations. In low dimensional systems, the characteristics of the
correlations determine the pres-ence or abspres-ence of an effective mobility edge [14–19]. In three dimensions, varying the correlation structure dras-tically changes the localization length and the
transport properties [20, 21]. In two recent studies, the mobility edge of ultracold atoms in the presence of isotropic and anisotropic optical speckle patterns has been precisely
FIG. 1: (color online) Cross section of the three-dimensional intensity profiles of a simple-cubic optical lattice with a super-imposed blue-detuned isotropic optical speckle pattern. The optical
lattice intensity is V0 = 4Er, the disorder strength is
Vdis= 1.3Er. The speckles patterns in the two panels have
different correlation lengths: σ = d/π in panel (a) and σ = d in panel (b). The color scale represents the potential intensity in units of recoil energy Er.
determined [22, 23], highlighting again the importance of taking into account the details of the disorder correla-tions. However, the experimental configuration which re-sembles more closely the
behavior of electrons in solids is the one where the atoms are exposed to the deep periodic potential due to an optical lattice with, additionally, the disorder due to a superimposed optical speckle
pattern (see intensity profiles in Fig. 1). This configuration with both an optical lattice and a speckle field has been imple-mented in experiments performed with Bose and Fermi gases [24–26], so
far considering interacting atoms.
In this Article, we investigate the Anderson localiza-tion of noninteracting atomic gases in a simple-cubic op-tical lattice plus an isotropic blue-detuned opop-tical speckle field. The first two
mobility edges and the corresponding critical filling factors are determined as a function of the
disorder strength (see Fig. 2). Our computational proce-dure is based on the analysis of the energy-level statistics familiar from quantum-chaos theory [27] and on the de-termination of the universal
critical adjacent-gap ratio.
We employ both continuous-space models which de-scribe the spatial correlation of an isotropic speckle pat-tern, and also an uncorrelated discrete-lattice model de-rived within a tight-binding
scheme. This allows us to measure the important effect of changing the disorder correlations length, and to shed light on the inadequacy of the simple tight-binding approximation in the strong
disorder regime. Our (unbiased) results are important as a guide for future experiments performed with noninter-acting atoms in disordered optical lattices, and also as a stringent benchmark for
(inevitably approximate) theo-retical calculations of the properties of disordered inter-acting fermions based on realistic models of disorder. The rest of the Article is organized as follows: in
Sec-tion II we define our model Hamiltonians, describing the details of the optical speckle patterns; we explain our theoretical formalism and analyze the universality of the critical adjacent-gap
ratio; furthermore, we pro-vide benchmarks of our predictions with previous results for tight-binding models with box and with exponential disorder-intensity distributions. In Section III our
pre-dictions for the mobility edges and the critical filling fac-tors are reported, with an analysis on the role played by the correlation length and on the validity of the tight-binding
approximation. We also discuss the compari-son with a recent transport experiment performed with atomic Fermi gases in the regime of moderate interaction strength [26]. Section IV summarizes the main
findings of this Article and reports our conclusions.
We consider noninteracting atoms exposed to a simple-cubic optical lattice with a superimposed optical speckle pattern. The single-particle Hamiltonian which describes the system is:
ˆ H = −~
2m∆ + V (r), (1)
where ~ is the reduced Planck’s constant, m is the atomic mass, and the external potential V (r) = VL(r) + VS(r)
is the sum of the simple-cubic optical lattice VL(r =
(x, y, z)) = V0P[ι]sin2(πι/d) (here, ι = x, y, z, d is the
lattice periodicity, and V0is the optical lattice intensity),
and the disordered potential VS(r) which represents the
isotropic optical speckle pattern. This intensity-based sum corresponds to the incoherent superposition of the optical-lattice and optical-speckle fields. In the follow-ing, it will be convenient to
express V0 in units of the
recoil energy Er = ~2/(2md2). The size L of the
three-dimensional box is chosen to be a multiple of d, consis-tently with the use of periodic boundary conditions.
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 0 5 10 15 20 Ns / Ω V[dis]/E[r] V[dis]d /t σ = d/π σ = d TB[ELSS] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 2 0 5 10 15 0 5 10 15 20 (E c -E 0 )/E r (E c -E 0 )/t V
[dis]/E[r] V[dis]d /t TB[ELSS] TB[TMM] σ = d/π σ = d
FIG. 2: (color online) Phase diagrams of an atomic gas ex-posed to three-dimensional simple-cubic optical lattices with a superimposed blue-detuned isotropic disordered speckle pat-tern: (a) First
two mobility edges Ec as a function of the
disorder strength Vdis/Er (or Vdisd/t for the discrete-lattice
model, in the top axis). Empty symbols correspond to the first mobility edge Ec1, full symbols to the second mobility
edge Ec2(see text). The energies are measured with respect
to the bottom of the first band of the clean system E0. The
red rhombi and the green circles correspond to the continuos-space Hamiltonian (1) with correlation lengths σ = d/π and σ = d, respectively. The blue squares correspond to our re-sults for the
tight-binding model with exponential on-site en-ergy distribution, obtained via analysis of the enen-ergy-level spacings statistic (TBELSS). The results obtained in Ref. [28]
using the transfer-matrix method (TBTMM) are represented
by pink crosses. The optical lattice intensity is V0 = 4Er,
and the corresponding hopping energy t ∼= 0.0855Er is used
to compare the continuous-space data (bottom-left axes) with the discrete-lattice data (top-right axes). (b) Critical filling factors Ns/Ω as a function of the disorder strength,
corre-sponding to the mobility edges represented in panel (a). Ns
is the number of states below the mobility edge, Ω the adi-mensional volume.
Disordered speckle patterns are realized in cold-atom experiments by shining lasers through diffusive plates, and then focusing the diffused light onto the atomic
0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 5 5.5 6 6.5 7 <r> E/E[r] L/d=9 L/d=7 L/d=5 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.4 0.5 0.6 0.7 0.8 0.9 1 <r> E/E[σ] L=11πσ L=15πσ L=21.25πσ
FIG. 3: (color online) Ensemble-average adjacent-gap ratio hri as a function of the energy E for the continuous-space Hamil-tonian (1). Left panel: simple-cubic optical lattice with intensity V0 =
4Er plus an isotropic optical speckle pattern with
correlation length σ = d/π and intensity Vdis= Er. Right panel: optical speckle field with intensity Vdis= Eσ (without an
optical lattice, namely V0= 0). The three datasets correspond to different system sizes. The horizontal cyan solid line indicates
the value for the Wigner-Dyson distribution hriWD, the dashed magenta line the one for the Poisson distribution hriP. The
dash-dot black line indicates the universal critical adjacent-gap ratio hriC, and the light-gray bar represents its error bar. The
energy units are the recoil energy Er and the correlation energy Eσ (see text).
cloud [10, 11]. Fully developed speckle fields are char-acterized by an exponential distribution of the local in-tensities [29]. In the case of a blue-detuned optical field, the atoms experience a
repulsive potential with the local-intensity distribution: Pbd(V ) = exp (−V /Vdis) /Vdis, if
the local intensity is V > 0, and Pbd(V ) = 0 otherwise.
The (global) intensity parameter Vdisfixes both the
spa-tial average of the disordered potenspa-tial Vdis= hVS(r)i and
also its standard deviation: V2
dis =VS(r) 2[ − hV]
For sufficiently large systems, spatial averages coincide with averages over disorder realizations.
The spatial correlations of the speckle pattern depend on the details of the illumination on the plate and of the optical setup used for focusing. We consider the ide-alized case of isotropic spatial
correlations described by the following two-point correlation function [22]: Γ(r = |r|) = hVS(r′+ r)VS(r′)i /Vdis2 − 1 = [sin(r/σ)/(r/σ)]
(averaging over the position of the first point r′ [is ]
as-sumed). The parameter σ determines the length scale of the spatial correlations and, therefore, the typical size of the speckle grains. The full width at half maximum of the correlation function Γ
(r) (defined by the condition Γ(ℓc/2) = Γ(0)/2) is ℓc ∼= 0.89πσ, while the first zero is
at rz = πσ. To generate this isotropic speckle pattern
we employ the numerical recipe described in Ref. [23]. For further details on speckle pattern generation, see Refs. [22, 30, 31].
We determine the positions of the mobility edges by analyzing the statistical distribution of the spacings between consecutive energy levels. The spectrum is obtained via exact diagonalization of the
matrix represented in momentum space, using the PLASMA library [32] for large-scale linear algebra computations on multi-core architectures. Special care is taken in analyzing the convergence of the
results with the basis-set size. Further details on the numerical procedure can be found in Ref. [23].
The mobility edges can be identified as the energy thresholds where the level-spacing distribution trans-forms from the Wigner-Dyson distribution characteristic of chaotic systems in the ergodic
phase, to the Poisson distribution characteristic of localized systems, or vice versa [27]. To distinguish the Wigner-Dyson and the Poisson distributions, it is convenient to consider the parameter r
= min {δn, δn−1} / max {δn, δn−1}, where
δn = En+1− En is the spacing between the (n + 1)th
and the nth energy levels, ordered for ascending energy values [33]. Its average over disorder realizations (later on referred to as adjacent-gap ratio) is known to be hriWD ≃ 0.5307 for the
Wigner-Dyson distribution and
hri[P]≃ 0.38629 for the Poisson distribution [34].
While in an infinite system hri would change abruptly at the mobility edge Ec, in finite systems one observes a
smooth crossover from hriPto hriWD, or vice versa. The
critical point can be determined from the crossing of the curves representing hri versus energy E corresponding to different system sizes L. We fit the data using the scaling function hri = g(E − Ec)
L1/ν (universal up to
a rescaling of the argument) [3] , where ν is the critical exponent of the correlation length. We Taylor expand the function g[x] up to second order and obtain Ec from
the best-fit analysis.
0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 -7.8 -7.6 -7.4 -7.2 <r> E/t L=24 L=28 L=32 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 <r> E/t L=24 L=28 L=32
FIG. 4: (color online) Ensemble-average adjacent-gap ratio hri as a function of the energy E for the tight-binding Hamilto-nian (2). Left panel: three dimensional Anderson model with box disorder
distribution with intensity Vd
dis= 5t. Right panel:
three dimensional Anderson model with the exponential distribution with intensity Vd
dis= 7t. The three datasets correspond to
different system sizes. The horizontal cyan solid line indicates the value for the Wigner-Dyson distribution hriWD, the dashed
magenta line the one for the Poisson distribution hriP. The dash-dot black line indicates the universal critical adjacent-gap
ratio hriC, and the light-gray bar represents its error bar. The energy unit is the hopping energy t.
was previously employed in Ref. [23] for speckle pat-terns without optical lattices, requires several datasets corresponding to different system sizes with very small statistical error bars. A less
computationally expensive procedure is obtained by exploiting the universal properties of the critical point. Indeed, the level-spacing distribution at the critical point differs both from the
Wigner-Dyson and from the Poisson distributions [35, 36]; it is expected to be system-size independent and universal, meaning that it does not depend on the details of the disorder. This implies a
universal value of the critical adjacent-gap ratio, which we denote as hriC, different from hriWD and from hriP.
We verified this universality by performing the finite-size scaling analysis for various models, determining hri[C] as the value of the scaling function at vanishing argument g[0]. In Fig. 3 we
report the finite-size scaling analysis for a simple-cubic optical lattice with a superimposed speckle pattern, and also for a speckle pattern without the optical lattice (data from Ref. [23]). The
critical adjacent-gap ratios hri[C] of the two models (for the dis-ordered optical lattice we consider the first two mobility edges) agree within statistical error bar. Furthermore, we verified that
hri[C] does not depend on the disorder strength Vdis, and that a compatible value of hri[C] is
obtained also for red-detuned optical speckle fields, which have the same spatial correlations of blue-detuned speckle fields Γ(r) defined above, but the opposite local-intensity distribution Prd(V )
= Pbd(−V ).
A further verification of the universality of the critical adjacent-gap ratio hri[C] can be obtained by considering single-band models in a tight-binding scheme. The
corre-sponding discrete-lattice Hamiltonian can be written in Dirac notation as:
ˆ Hd= −t X hi,ji |ii hj| +X i Vi|ii hi| , (2)
where the indices i, j = 1, . . . , L label the sites of the cubic discrete lattice of adimensional volume Ω = L3[,]
t is the hopping energy, and the brackets hi, ji indi-cate nearest neighbor sites. The on-site energies Vi
are chosen according to a random probability distri-bution. The most commonly adopted choice in the theory of Anderson localization is the box distribution Pb(Vi) = θ(
Vi− Vdisd
)/(2V[dis]d). The parameter V[dis]d determines the disorder strength. We also consider the exponential distribution Pe(Vi) = exp Vi/Vdisd /V
d dis,
analogous to the exponential distribution Pbd(V )
de-scribed above for blue-detuned speckle patterns in the continuous-space Hamiltonian. This discrete-lattice model with the exponential on-site energy distribution is relevant to describe deep
optical lattices with super-imposed weak and uncorrelated speckle patterns, as explained more in detail in the Section III.
The finite-size scaling analyses for these two lattice models (box and exponential distributions) are shown in Fig. 4. The spectrum is obtained via exact diago-nalization of the matrix representing
the Hamiltonian
Hd, defined on the three dimensional lattice. The
universality of the critical adjacent-gap ratio is, again, confirmed within statistical uncertainty.
The average of the critical adjacent-gap ratios of the various models described above, including both the continuous-space models with correlated speckle patterns
and the uncorrelated tight-binding models, is hri[C] = 0.513 ± 0.05; the error bar represents the standard devia-tion of the populadevia-tion. This predicdevia-tion provides us with a computationally
convenient criterion to locate the transi-tion, consisting in identifying the mobility edge Ecas the
energy threshold at which the adjacent-gap ratio crosses the critical value hri[C]; the standard deviation of hri[C]will be used to define the error bar on Ec. By applying this
criterion to the isotropic speckle pattern (without optical lattice) analyzed in Fig. 3, we obtain Ec = 0.562(10)Eσ
(Eσ = ~2/mσ2 is the correlation energy), in agreement
with the transfer matrix theory of Ref. [22], which pre-dicts Ec = 0.570(7)Eσ. We further confirm the validity of
this criterion by reproducing the complete phase diagram of the discrete-lattice model with box disorder distribu-tion (typically refereed to as Anderson model), making comparison with older results
obtained using transfer-matrix theory [37] and multi-fractal analysis [38], as well as with the recent data from Ref. [39] obtained using the typical medium dynamical cluster approximation; see Fig.
5. Furthermore, in the case of the exponential disor-der distribution, our results perfectly agree with the very recent transfer-matrix theory from Ref. [28] (see Fig. 2). It is worth specifying that
our prediction for the uni-versal critical adjacent-gap ratio hri[C] applies to a cu-bic box with periodic boundary conditions. In fact, it has been predicted that the critical energy-level
distri-bution, and so possibly the corresponding value of hri[C], depends on the box shape [40] and on the boundary con-ditions [41, 42].
The continuous-space Hamiltonian (1) accurately describes atomic gases exposed to optical lattices with superimposed optical speckle patterns, for any optical lattice intensity V0 and disorder
strength Vdis. In
particular, it takes into account the spatial correlations of the optical speckle pattern. In order to make com-parison with recent experimental data, we consider the intermediate optical lattice
intensity V0 = 4Er, and we
determine the lowest two mobility edges as a function of the disorder strength Vdis, up to the critical value
where the two mobility edges merge and the whole band becomes localized.
We consider two isotropic speckle patterns with corre-lation lengths σ = d/π and σ = d. We recall that the first zero of the spatial correlation function Γ(r) (see def-inition in Section II) is at rz
= σπ. Beyond this distance
the speckle-field intensities are almost uncorrelated. The intensity profiles of the total potential V (r) correspond-ing to these two correlation lengths are shown in Fig. (1). The deformation of
the regular structure of the simple-cubic optical lattice due to the speckle pattern is evi-dent in both cases. In the first case the intensity values in nearest-neighbor wells of the optical lattice
are only
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Ec /t V[dis]d /t TMM MFA TMDCA ELSS
FIG. 5: (color online) Mobility edge Ec as a function of
the disorder strength Vd
dis for the three dimensional
Ander-son model with box disorder distribution. Our data com-puted via the analysis of the energy-level spacings statistics (ELSS, red diamonds) are compared with previous results ob-tained via
transfer-matrix method (TMM, green circles, from Ref. [37]), via multi-fractal analysis (MFA, blue squares, from Ref. [38]), and via typical medium dynamical cluster approx-imation (TMDCA, black
crosses, from Ref. [39]). The energy unit is the hopping energy t.
weakly correlated, while in the second case the correla-tions extend to a few lattice sites.
The phase diagram obtained using the methods presented in Section II, namely the analysis of the energy-level spac-ing statistics and the universality of the critical adjacent-gap ratio, is presented
in Fig. 2. The empty symbols in-dicate the lowest mobility edge Ec1, where the orbitals
transform from localized (for energies E < Ec1) to
ex-tended (for E > Ec1), while the solid symbols indicate
the second mobility edge Ec2 > Ec1, where the orbitals
transform from extended (for E < Ec2) to localized (for
E > Ec2). Other mobility edges are located at
signifi-cantly higher energies, outside the energy range investi-gated in this Article.
The data reported in Fig. 2 shed light on the fun-damental role played by the spatial correlations of the disorder pattern. The critical disorder strength beyond which the first band is fully
localized strongly depends on the correlation length. Indeed, for the short correlation length σ = d/π, full localization occurs already at the disorder strength Vdis ≃ 1.32Er, while for the longer
correlation length σ = d full localization occurs only at the much stronger disorder intensity Vdis ≃ 1.95Er.
This indicates that the disorder is more effective in inhibiting particle diffusion if the correlation length is short compared to the lattice spacing; also, it implies that, in order to
quantitatively describe experiments performed with noninteracting atomic gases exposed to disordered optical lattices, it is necessary to take into
0 50 100 0 0.5 1 1.5 2 2.5 3 3.5 4 dos (E-E[0])/E[r] σ = d/π, V[dis]=1.3E[r] σ = d,V[dis]=1.3E[r] TB, V[dis]=1.3E[r] σ = d/π, V[dis]=0.2E[r]
FIG. 6: (color online) Density of states dos (in arbitrary units) as a function of the energy E measured from the bottom of first band of the clean system E0. The energy unit is the
recoil energy Er. The continuous red, dashed green, and the
double-dash black curves correspond to the continuos-space model (1) with different correlation lengths σ and disorder intensities Vdis. The dotted blue curve corresponds to the
tight-binding (TB) model.
account the details of the optical speckle pattern. In Fig. 2, we report the two critical filling factors (defined as the number of eigenstates Nsper adimensional volume
Ω = (L/d)3 [with energy E < E]
c1 and E < Ec2) as a
function of the disorder strength. The role of the spatial correlations is again manifest. Both the mobility-edge data and the critical filling-factors data display a strong asymmetry around the band
center; this originates from the asymmetry of the exponential intensity distribution of the optical speckle pattern Pbd(V ).
Most theoretical studies of atomic gases exposed to clean optical lattices are based on single-band tight-binding Hamiltonians analogous to the one defined in Eq. (2). The conventional procedure to
map optical lattice systems to tight-binding models is based on the computation of the maximally localized Wannier function from the band-structure analysis of the periodic system. For sufficiently
deep optical lattices V0 ≫ Er,
the effect of higher Bloch bands and of hopping pro-cesses between non-adjacent Wannier orbitals can be ignored, leading to single-band tight-binding models in the discrete-lattice form defined by
Eq. (2). At the optical lattice intensity addressed in this Article, namely V0 = 4Er, the deep-lattice condition is marginally
fulfilled, with a next-nearest neighbor hopping energy |t2| ∼= 6.1 · 10−3Er, which is only one order of magnitude
smaller than the nearest neighbor hopping energy t ∼= 0.0855Er.
In the presence of additional disordered optical fields, the conventional mapping procedure [43, 44] based on band-structure calculation cannot be applied. A more
generic approach, valid also in the presence of weak optical speckle patterns with intensity Vdis ≪ V0, has
been developed in Ref. [45]; this method allows one to construct an orthonormal basis of localized Wannier-like orbitals which describes the correct low-energy properties of weakly disordered
optical-lattice systems. In the corresponding discrete-lattice Hamiltonian, the on-site energies Vi have, with good approximation,
the exponential distribution Pe(Vi), with a disorder
intensity Vd
dis ≃ Vdis, essentially coinciding with the
intensity of the optical speckle field Vdis. The on-site
energies on nearby lattice sites have significative cor-relations which depends on the details of the optical speckle pattern. Also, the nearest-neighbor hopping energies have an (asymmetric) random
distribution, characterized by strong correlations with the difference between the on-site energies of the corresponding lattice sites. In first approximation, one might neglect the hopping-energy
fluctuations and the on-site energy correlations, and retain only the exponential on-site energy distribution. This approximate model of optical lattices with superimposed speckle patterns - which
leads (in the noninteracting case) to the tight-binding Hamiltonian (2) with the on-site energy distribution Pe(Vi) - has been adopted in Ref. [46] to describe a
recent transport experiment performed with interacting ultracold atoms [46]. In this experiment, a drifting force was applied by introducing a magnetic-field gradient for a short interval of time;
after this impulse, the confining potential was switched off, and the velocity of the center of mass of the atomic cloud was measured by absorption imaging and band mapping after a time of flight;
the measurement was repeated with different intensities of the optical speckle field . Also, various optical lattice intensities were considered, ranging from V0 = 4Er to
V0 = 7Er. The authors of Ref. [46] considered mainly
the case of the deep optical lattice V0 ≃ 7Er, where
the Hubbard interaction energy of two opposite-spin fermions on the same lattice site is large: U ≃ 9t. They argue that in this strongly interacting regime the details of the correlations of the
hopping and of the on-site energies are not relevant, since transport is dominated by effective quasi-particles (not the original particles which are obviously relevant in the noninteracting case),
which experience correlated hopping and interaction processes even in the simplified model. They indeed found satisfactory agreement between the computed center-of-mass velocities and the
experimental data. Our findings indicate that in the absence of interac-tions the details of the speckle pattern are, instead, important. The mobility edges of the uncorrelated tight-binding model
(2) (with the exponential on-site energy distribution) are shown in Fig. 2, together with the results for the continuous-space model (1). To make comparison between the two models, the energies in
the lattice model have to be converted using the hopping energy t ∼= 0.0855Ercorresponding to the optical lattice
that certain qualitative features of the phase diagram are captured also by the tight-binding model. However, while at very weak disorder Vdis≈ 0.2Erthe
continuous-space and the discrete-lattice models quantitatively agree, important discrepancies appear at strong disor-der. In particular, the critical disorder strength where the whole band is
localized in the discrete-lattice model, namely Vd
dis ≃ 12t (corresponding to Vdis ≃ 0.95Er),
significantly underestimates the results obtained with the more accurate correlated continuous-space models. In principle, the details of the speckle pattern could be included also in a
discrete-lattice Hamiltonian, following the numerical procedure of Ref. [45]. This approach has been adopted in Ref. [47] to investigate an interacting Anderson-Hubbard model with correlated speckle
fields. However, the dynamical mean-field theory employed in Ref. [47] does not correctly describe the Anderson localization in the noninteracting limit, probably due to the assumed Bethe-lattice
structure. More recently, the dynamical mean-field theory has been improved using the typical medium dynamical cluster approximation [39], allowing researchers to give more accurate predictions for
the localization transition in the (uncorrelated) Anderson model with box distribution; the data from Ref. [39] are reported in Fig. 5.
Nevertheless, it should be emphasized that the numerical technique of Ref. [45] converges only as long as there is a well defined gap between the first and the second band. As shown in Fig. 6, in our
optical lattice the gap is well defined only for very weak disorder, while it is substantially filled when the intensity of the optical speckle field approaches the strength required to localize the
whole band, making that numerical technique inapplicable.
Experimental data for noninteracting atomic gases in disordered optical lattices are not available. However, in the experiment of Ref. [26] (described above), which was performed with interacting
atoms, the optical-lattice in-tensity was tuned down to V0= 4Er, corresponding to a
relatively small Hubbard interaction parameter, namely U ≃ 2.3t. It is then reasonable to discuss the comparison of these latter results with our theoretical predictions. It should be taken into
account that the optical speckle pattern employed in the experiment is anisotropic, with an axial correlation length approximately 5 times larger than the radial correlation length, and that its
spatial correlations decays as a Gaussian function. However, the propagation axis of the optical speckle field is dis-aligned with respect to the optical lattice axes; this is expected to strongly
reduce the role of the correlation anisotropy. If we consider the geometrically-averaged correlation length, we obtain a Gaussian correlation func-tion with similar full width at half maximum as our
speckle pattern with σ = d (within ≈ 15%). Further-more, in the experiment the density is inhomogeneous due to the confinement (with approximately 0.3 − 0.7 particles per lattice well in the trap
center, per spin
com-ponent) and the energy distribution is not precisely char-acterized. In Fig. 7 we plot the center-of-mass velocities vc.m. measured in the experiment, as a function of the
disorder strength. The critical point where vc.m.
van-ishes has been interpreted in Ref. [26] as the average disorder strength required to localize the whole band, since all extended states are expected contribute to trans-port. We indeed observe
that vc.m. reaches negligible
values (compatible with the experimental resolution) in the regime where we predict full localization to occur, depending on the details of the optical speckle pattern. Clearly, a quantitative
comparison with the experimen-tal data would require a precise characterization of the experimental atomic density and of the energy distribu-tion. This would also allow us to clarify the potential
role played by states in higher-energy bands. Never-theless, this qualitative agreement between experimen-tal data and theoretical predictions is encouraging, and should stimulate further
experimental efforts aiming at observing Anderson localization in noninteracting atomic gases in disordered optical lattices. All details of the op-tical speckle pattern could be included in our
theoreop-tical formalism. 0.01 0.1 1 0 0.5 1 1.5 2 0 5 10 15 20 TB
=d vc.m. (mm/sec) V
d /t
FIG. 7: (color online) Experimental data from Ref. [26]: cen-ter of mass velocity vc.m.of the atomic cloud (black squares)
as a function of the disorder strength Vdis/Er (or V d dis/t for
the tight-binding model, in the top axis). The vertical lines represent our predictions for the critical disorder strength where the whole band becomes localized in the continuos-space Hamiltonian
(1) (dashed red and dotted green lines) and in the uncorrelated tight-binding model (2) with expo-nential disorder distribution (dot-dash blue line). The gray band represents the experimental
resolution in detecting a vanishing velocity.
In summary, we have investigated the Anderson localization of noninteracting atomic gases in disordered
optical lattices. We considered both continuous-space models which describe the effect of a simple-cubic opti-cal lattice with a superimposed isotropic blue-detuned optical speckle field, taking into
account the spatial correlations of the disorder, and also an uncorrelated discrete-lattice Hamiltonian in a tight-binding scheme. Our predictions for the mobility edges and for the critical filling
factors indicate that the details of the speckle pattern play an important role; the critical disorder intensity where the whole band becomes localized strongly depends on the disorder correlation
length. The tight-binding model with an uncorrelated (exponential) disorder distribution significantly underestimates this critical disorder strength.
Our theoretical formalism is based on the analysis of the energy-level statistics familiar from random-matrix and quantum-chaos theories and on the determination of the universal critical
adjacent-gap ratio. The prediction for this universal value will be useful also in future studies of Anderson localization in different models belonging to the same universality class.
We have shown that the findings of a recent trans-port experiment performed with an atomic gas in the moderate interaction regime [26] are qualitatively consistent with our predictions; this
encouraging com-parison should stimulate further experimental efforts to accurately measure the critical point of the Anderson transition in noninteracting atomic gases exposed to controlled and well
characterized disordered fields. Such experiments would allow us to quantitatively benchmark sophisticated theories for Anderson localization based on realistic models which take into account all
details of the disorder. This would be beneficial for the field of ul-tracold atoms, and likely beyond, possibly including the research on disordered materials, on randomized optical fibers [48], and
on disordered photonic crystals [49].
We acknowledge fruitful discussions with Brian De-Marco, Vito Scarola, Estelle Maeva Inack and Giuliano Orso. Brian DeMarco is also acknowledge for providing us the data from Ref. [26]
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K. W. Koch, and A. Mafi, Optics letters 37, 2304 (2012). [49] M. Segev, Y. Silberberg, and D. N. Christodoulides, | {"url":"https://123dok.org/document/6qm3l44y-anderson-localization-in-optical-lattices-with-correlated-disorder.html","timestamp":"2024-11-09T22:03:34Z","content_type":"text/html","content_length":"191831","record_id":"<urn:uuid:384241e5-bf65-4b7d-a319-5afa32c2fc9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00202.warc.gz"} |
An Animal Contest 4 P1 - Dasher's Digits
Submit solution
Points: 5 (partial)
Time limit: 2.0s
Memory limit: 256M
Dasher the reindeer has a string with characters, where characters are 0's and the rest are uppercase Latin letters. Each 0 is assigned a "cheer value" . The cheer values are where corresponds to the
cheer value for the first 0 in , the cheer value for the second 0, and so on.
Dasher wants to get rid of all the 0's in his string, so he performs the following algorithm while there are still 0's present:
• If the frontmost character is 0, subtract that 0's cheer value by . If this cheer value becomes , remove this 0. Otherwise, move it, along with its cheer value, to the back of the string.
• Otherwise, take the frontmost character and move it to the back of the string.
For strings that are extensive in length, performing the algorithm manually is a tedious process. Dasher requires your assistance to output the string after all 0's have been removed through the
usage of the mentioned algorithm.
contains at least one non-0 character.
Subtask 1 [20%]
Subtask 2 [30%]
Subtask 3 [50%]
No additional constraints.
Input Specification
The first line will contain two space-separated integers and .
The second line will contain , which has at least one non-0 character, and may contain leading 0's.
The next line will contain space-separated integers .
Output Specification
Output the string after all 0's are removed with the algorithm described.
Sample Input
Sample Output
The following shows the results of performing the algorithm manually to obtain our answer:
US0AMOG → S0AMOGU → 0AMOGUS → AMOGUS0 → MOGUS0A → OGUS0AM → GUS0AMO → US0AMOG → S0AMOGU → 0AMOGUS → AMOGUS
• Bro why am I getting tle on a linear time solution | {"url":"https://dmoj.ca/problem/aac4p1","timestamp":"2024-11-12T12:01:22Z","content_type":"text/html","content_length":"37596","record_id":"<urn:uuid:c106590d-cd7f-4016-ac39-696966401964>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00369.warc.gz"} |
Dual Moving Average Reversal Trading Strategy
1. Dual Moving Average Reversal Trading Strategy
Dual Moving Average Reversal Trading Strategy
, Date: 2023-12-04 16:39:13
This is a reversal trading strategy based on dual moving average indicators. By calculating two groups of moving averages with different parameter settings and judging the price trend according to
their directional changes, trading signals can be generated by setting the sensitivity parameter for directional changes.
The core indicator of this strategy is the dual moving average. The strategy allows selecting the type (SMA, EMA, etc.), length and price source (close price, typical price, etc.) of the moving
average. After calculating two groups of moving averages, their directions are determined by defining the reaction parameter. A buy signal is generated when the fast line crosses above the slow line,
and a sell signal is generated when it crosses below. The reaction parameter is used to adjust the sensitivity to identify turning points.
In addition, the strategy also sets the conditions to determine the change of direction and continued rise/fall to avoid generating wrong signals. And it visualizes the rise and fall of prices in
different colors. When prices continue to rise, the movavg line is displayed in green, and in red when prices fall.
Advantage Analysis
The dual movavg strategy combines fast and slow lines with different parameter settings, which can effectively filter the noise in the trading market and identify stronger trends. Compared with a
single movavg strategy, it reduces wrong signals and allows entering the market when the trend is more distinct, thereby obtaining a higher win rate.
The reaction parameter allows the strategy to be flexible and adaptable to different cycles and varieties. The strategic process is intuitive and simple, easy to understand and optimize.
Risk Analysis
The biggest risk of this strategy is missing the turning point and losing money or taking a reverse position. This relates to the reaction parameter setting. If the reaction is too small, wrong
signals are prone to occur. If the reaction is too large, it may miss better entry points.
Another risk is the inability to effectively control losses. When prices fluctuate violently, it cannot quickly stop losses, leading to enlarged losses. This requires the use of stop-loss strategies
to control risks.
Optimization Directions
The main optimization directions of this strategy focus on selecting reaction parameters, types and lengths of moving averages. Increasing reaction appropriately can reduce wrong signals. Moving
average parameters can be tested according to different cycles and varieties to select the best combination for generating signals.
In addition, confirming trading signals with other auxiliary indicators such as RSI and KD is also an optimization idea. Or use machine learning methods to automatically optimize parameters.
Overall, this strategy is relatively simple and practical. By filtering with dual moving averages and generating trading signals, it can effectively identify trend reversals and is a typical
trend-following strategy. After optimizing the parameter portfolio, its ability to capture trends and hold positions against the market will be improved. Using it with stop loss and position
management mechanisms works better.
start: 2023-11-03 00:00:00
end: 2023-12-03 00:00:00
period: 1h
basePeriod: 15m
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
strategy(shorttitle="MA_color strategy", title="Moving Average Color", overlay=true)
// === INPUTS
ma_type = input(defval="HullMA", title="MA Type: ", options=["SMA", "EMA", "WMA", "VWMA", "SMMA", "DEMA", "TEMA", "HullMA", "ZEMA", "TMA", "SSMA"])
ma_len = input(defval=32, title="MA Lenght", minval=1)
ma_src = input(close, title="MA Source")
reaction = input(defval=2, title="MA Reaction", minval=1)
// SuperSmoother filter
// © 2013 John F. Ehlers
variant_supersmoother(src,len) =>
a1 = exp(-1.414*3.14159 / len)
b1 = 2*a1*cos(1.414*3.14159 / len)
c2 = b1
c3 = (-a1)*a1
c1 = 1 - c2 - c3
v9 = 0.0
v9 := c1*(src + nz(src[1])) / 2 + c2*nz(v9[1]) + c3*nz(v9[2])
variant_smoothed(src,len) =>
v5 = 0.0
v5 := na(v5[1]) ? sma(src, len) : (v5[1] * (len - 1) + src) / len
variant_zerolagema(src,len) =>
ema1 = ema(src, len)
ema2 = ema(ema1, len)
v10 = ema1+(ema1-ema2)
variant_doubleema(src,len) =>
v2 = ema(src, len)
v6 = 2 * v2 - ema(v2, len)
variant_tripleema(src,len) =>
v2 = ema(src, len)
v7 = 3 * (v2 - ema(v2, len)) + ema(ema(v2, len), len)
variant(type, src, len) =>
type=="EMA" ? ema(src,len) :
type=="WMA" ? wma(src,len):
type=="VWMA" ? vwma(src,len) :
type=="SMMA" ? variant_smoothed(src,len) :
type=="DEMA" ? variant_doubleema(src,len):
type=="TEMA" ? variant_tripleema(src,len):
type=="HullMA"? wma(2 * wma(src, len / 2) - wma(src, len), round(sqrt(len))) :
type=="SSMA" ? variant_supersmoother(src,len) :
type=="ZEMA" ? variant_zerolagema(src,len) :
type=="TMA" ? sma(sma(src,len),len) : sma(src,len)
// === Moving Average
ma_series = variant(ma_type,ma_src,ma_len)
direction = 0
direction := rising(ma_series,reaction) ? 1 : falling(ma_series,reaction) ? -1 : nz(direction[1])
change_direction= change(direction,1)
change_direction1= change(direction,1)
pcol = direction>0 ? lime : direction<0 ? red : na
plot(ma_series, color=pcol,style=line,join=true,linewidth=3,transp=10,title="MA PLOT")
/////// Alerts ///////
alertcondition(change_direction,title="Change Direction MA",message="Change Direction MA")
longCondition = direction>0
shortCondition = direction<0
if (longCondition)
strategy.entry("BUY", strategy.long)
if (shortCondition)
strategy.entry("SELL", strategy.short) | {"url":"https://www.fmz.com/strategy/434196","timestamp":"2024-11-03T20:33:07Z","content_type":"text/html","content_length":"14699","record_id":"<urn:uuid:368588f9-1ef7-4e0a-bb3c-38ceefcd7100>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00665.warc.gz"} |
How to calculate increaseHow to calculate increase 🚩 how to calculate payroll 🚩 Budgeting.
Set the time interval. The increased growthand allows us to estimate changes in time. Therefore, a necessary starting point and last time point recorded in the present. Suppose we need to compare the
increase in wages by July 2005 since July 2004, i.e. during the year.
Define indicators at the beginning and end. Let in July 2004, the salary received at his hands, was equal to 15 thousand rubles. In July 2005, the salary had a value of 18 thousand rubles.
Subtract from the final figure initial. Subtract from 18 thousand to 15 thousand rubles, we get 3 thousand rubles.
Divide the resulting value by the initial figure. Divide 3 thousand to 15 thousand rubles, produced of 0.2.
Multiply the resulting value by 100%. 0,2 multiply by 100, we get 20%. Thus, for the year the increase in wages amounted to 20%. They also say that "wages rose 20%.
Growth can be negative. If at the end of the period the salary is 14 thousand rubles, at the 3rd step, subtract from 14 thousand to 15 thousand rubles, we get -1 thousand.
Then at the 4th step you divide -1 thousand to 15 thousand rubles, we get about -0,07. And at the 5th step, multiply this by 100, we get -7%. Have negative growth, i.e. wages for the period under
review decreased by approximately 7%.
Useful advice
After computing the test to avoid errors. If on the calculator to 15,000 add 20%, the calculator will show 18000. Hence, the growth are identified correctly. | {"url":"https://eng.kakprosto.ru/how-11647-how-to-calculate-increase","timestamp":"2024-11-10T07:58:20Z","content_type":"text/html","content_length":"29504","record_id":"<urn:uuid:995de3f5-111f-412f-9931-16399e718487>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00461.warc.gz"} |
The Restricted Arc-Width of a Graph
The Restricted Arc-Width of a Graph
An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a
representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of
Barát and Hajnal on this subject and develop a generalization we call restricted arc-width. Our main results revolve around using this to bound arc-width from below and to examine the effect of
several graph operations on arc-width. In particular, we completely describe the effect of disjoint unions and wedge sums while providing tight bounds on the effect of cones. | {"url":"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1r41","timestamp":"2024-11-02T04:50:43Z","content_type":"text/html","content_length":"13748","record_id":"<urn:uuid:725dc5a3-79b2-4823-9efa-1481b27aedc7>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00517.warc.gz"} |
Jordan Form - (Abstract Linear Algebra I) - Vocab, Definition, Explanations | Fiveable
Jordan Form
from class:
Abstract Linear Algebra I
Jordan Form is a canonical representation of a linear operator (or matrix) that reveals its structure through Jordan blocks, which represent eigenvalues and their algebraic and geometric
multiplicities. It simplifies the analysis of linear transformations and helps in understanding the properties of matrices related to eigenvalues, diagonalization, and matrix functions.
congrats on reading the definition of Jordan Form. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Jordan Form consists of Jordan blocks, where each block corresponds to an eigenvalue and its size indicates the chain of generalized eigenvectors.
2. Not all matrices can be diagonalized; those that can be expressed in Jordan Form have at least one generalized eigenvector for every eigenvalue.
3. The Jordan Form is unique up to the order of the Jordan blocks, which means that if two matrices have the same Jordan Form, they can differ by the arrangement of those blocks.
4. Computing the Jordan Form involves finding the eigenvalues, constructing the Jordan chains, and organizing them into blocks for the final matrix representation.
5. The existence of a Jordan Form can help determine the stability and behavior of dynamic systems described by differential equations.
Review Questions
• How does the Jordan Form relate to eigenvalues and what implications does it have for understanding a matrix's properties?
□ The Jordan Form directly relates to eigenvalues by organizing them into blocks that indicate both their algebraic and geometric multiplicities. This arrangement reveals important information
about the structure of the matrix, including its potential for diagonalization. When a matrix cannot be fully diagonalized, the Jordan Form provides insight into its generalized eigenvectors
and helps understand how solutions to linear equations may behave.
• Discuss the conditions under which a matrix can be expressed in Jordan Form and how this affects its diagonalizability.
□ A matrix can be expressed in Jordan Form when it has enough generalized eigenvectors to correspond with each eigenvalue. If a matrix has fewer linearly independent eigenvectors than its
algebraic multiplicity, it cannot be fully diagonalized but can still be represented in Jordan Form. This indicates that while some aspects of its behavior can be simplified, it retains
additional complexities that need to be addressed when analyzing its applications.
• Evaluate the significance of Jordan Form in advanced applications such as differential equations and dynamic systems.
□ The significance of Jordan Form in applications like differential equations lies in its ability to simplify the analysis of systems with repeated eigenvalues or non-diagonalizable matrices.
By representing these matrices in Jordan Form, one can easily find solutions to differential equations that describe dynamic systems, identifying stability and behavior over time. Moreover,
understanding how to manipulate matrices in their Jordan Form can lead to more efficient computational methods when dealing with complex systems in engineering or physics.
"Jordan Form" also found in:
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/abstract-linear-algebra-i/jordan-form","timestamp":"2024-11-09T13:51:49Z","content_type":"text/html","content_length":"148762","record_id":"<urn:uuid:4be1184e-c997-4717-acd1-3c57a9f5ba6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00815.warc.gz"} |
Slope of the x vs. t Graph, Velocity
Here is an x vs. t, or position vs. time, graph.
Position (x) is vertical.
Time (t) is horizontal.
Initially, when t = 0 s, the object is at x = 0 m.
From then on as time passes the object moves away from the origin of the position (x) number line.
Let's look at two points on this graph....
Examine the first point, (t[1], x[1]).
When t[1] = 5 s, then x[1] = 15 m.
Examine the second point, (t[2], x[2]).
When t[2] = 15 s, then x[2] = 45 m.
Let's find the slope of this graph using those two points...
Using those two points, here is the rise and the run of the slope of this x vs. t graph.
Here, the rise is the difference of the position coordinates, or x[2] - x[1], as in:
rise = x[2] - x[1]
rise = 45 m - 15 m
rise = 30 m
Here, the run is the difference of the time coordinates, or t[2] - t[1], as in:
run = t[2] - t[1]
run = 15 s - 5 s
run = 10 s
The slope of this graph is a change in position divided by a change in time, as in:
slope = rise / run
slope = 30 m / 10 s
slope = 3 m/s
This slope is the velocity of the object, since velocity is defined as the change in position divided by the change in time. So....
Velocity = 3 m/s.
The slope of the x vs. t graph is the velocity of the object. | {"url":"http://zonalandeducation.com/mstm/physics/mechanics/kinematics/slopesAndAreas/slopeOfxvst/slopeOfxvst.html","timestamp":"2024-11-08T14:26:50Z","content_type":"text/html","content_length":"9142","record_id":"<urn:uuid:f2a59e41-e8c2-43d2-b988-e8df18a0b556>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00809.warc.gz"} |
Shortest Distance In A Plane LeetCode Solution - Leetcode Solution - TO THE INNOVATION
Shortest Distance in a Plane LeetCode Solution
/ Leetcode Solution / LeetCode Solution, Medium, MySql, SQL
Last updated on October 9th, 2024 at 10:11 pm
This Leetcode problem Shortest Distance in a Plane LeetCode Solution is done in SQL.
Level of Question
Shortest Distance in a Plane LeetCode Solution
Table of Contents
Problem Statement
Table point_2d holds the coordinates (x,y) of some unique points (more than two) in a plane.
Write a query to find the shortest distance between these points rounded to 2 decimals.
The shortest distance is 1.00 from point (-1,-1) to (-1,2). So the output should be:
1. Shortest Distance in a Plane LeetCode Solution MySQL
) as shortest
a.x = b.x
and a.y = b.y,
power(a.x - b.x, 2) + power(a.y - b.y, 2)
) as dist
point_2d as a,
point_2d as b
) as d;
Shortest Unsorted Continuous Subarray LeetCode Solution | {"url":"https://totheinnovation.com/shortest-distance-in-a-plane-leetcode-solution/","timestamp":"2024-11-02T11:24:18Z","content_type":"text/html","content_length":"194779","record_id":"<urn:uuid:53bfa6dc-b52c-4fe5-a094-d008c5f135e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00449.warc.gz"} |
21 October 2018 Archives
Here’s a thought: what’s the minimum number of votes your party would need to attract in order to be able to secure a majority of seats in the House of Commons and form a government? Let’s try to
work it out.
The 2017 general election reportedly enjoyed a 68.8% turnout. If we assume for simplicity’s sake that each constituency had the same turnout and that votes for candidates other than yours are
equally-divided amongst your opposition, that means that the number of votes you need to attract in a given constituency is:
68.8% × the size of its electorate ÷ the number of candidates (rounded up)
For example, if there was a constituency of 1,000 people, 688 (68.8%) would have voted. If there were 3 candidates in that constituency you’d need 688 ÷ 3 = 229⅓, which rounds up to 230 (because you
need the plurality of the ballots) to vote for your candidate in order to secure the seat. If there are only 2, you need 345 of them.
It would later turn out that Barry and Linda Johnson of 14 West Street had both indented to vote for the other candidate but got confused and voted for your candidate instead. In response, 89% of the
nation blame the pair of them for throwing the election.
The minimum number of votes you’d need would therefore be this number for each of the smallest 326 constituencies (326 is the fewest number of seats you can hold in the 650-seat House of Commons and
guarantee a strict majority; in reality, a minority government can sometimes form a government but let’s not get into that right now). Constituencies vary significantly in size, from only 21,769
registered voters in Na h-Eileanan an Iar (the Western Isles of Scotland, an SNP/Labour marginal) to 110,697 in the Isle of Wight (which flip-flops between the Conservatives and the Liberals), but
each is awarded exactly one seat, so if we’re talking about the minimum number of votes you need we can take the smallest 326.
Win these constituencies and no others and you control the Commons, even though they’ve tiny populations. In other news, I think this is how we end up with a SNP/Plaid coalition government.
By my calculation, with a voter turnout of 68.8% and assuming two parties field candidates, one can win a general election with only 7,375,016 votes; that’s 15.76% of the electorate (or 11.23% of the
total population). That’s right: you could win a general election with the support of a little over 1 in 10 of the population, so long as it’s the right 1 in 10.
I used a spreadsheet and everything; that’s how you know you can trust me. And you can download it, below, and try it for yourself.
I’ll leave you to decide how you feel about that. In the meantime, here’s my working (and you can tweak the turnout and number-of-parties fields to see how that affects things). My data comes from
the following Wikipedia/Wikidata sources: [1], [2], [3], [4], [5] mostly because the Office of National Statistics’ search engine is terrible.
JACOB PRETENDS TO BE STRAIGHT… | The Trueman Show! Ep 13
This is a repost promoting content originally published elsewhere. See more things Dan's reposted.
The Fratocrats at their funniest. | {"url":"https://danq.me/2018/10/21","timestamp":"2024-11-02T18:46:56Z","content_type":"text/html","content_length":"54955","record_id":"<urn:uuid:e12f8091-47d9-4992-a4ed-f3e5c747fb72>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00558.warc.gz"} |
Speed Distance and Time - Aptitude - dyclassroom | Have fun learning :-)
Important notes and formula
Relation between speed, distance and time
If a car travels D distance in time T then we say the speed of the car is S and we express the speed in terms of distance and time as follows.
Finding Distance and Time
Distance = Speed x Time
Unit of speed
Speed is expressed as distance per unit time. Most commonly used unit for speed are
meter per second (m/s)
kilometer per hour (km/hr)
Convert km/hr to m/s
We know 1 km = 1000 m and 1 hour = 3600 seconds.
Convert m/s to km/hr
We know 1000 m = 1 km and 3600 seconds = 1 hour.
Speed Ratio
If the speed of two trains are in the ratio X:Y then time taken by them to cover the same distance is Y:X
Travelling same distance but at a different speed
If a car covers a certain distance at a speed X km/hr and then covers an equal distance at Y km/hr then the average speed of the car for the complete journey is | {"url":"https://dyclassroom.com/aptitude/speed-distance-and-time","timestamp":"2024-11-11T01:04:18Z","content_type":"text/html","content_length":"35011","record_id":"<urn:uuid:969c8571-d2a9-4cf4-98b5-46e08d0908d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00634.warc.gz"} |
Single Instruction, Multiple Data (SIMD) in .NET
Single Instruction, Multiple Data (SIMD) in .NET
Photo by Antão Almada
What is SIMD?
SIMD stands for “Single Instruction, Multiple Data”. It is a type of parallel processing technique that allows a single instruction to be executed on multiple data elements simultaneously. SIMD
enables efficient and high-performance execution of repetitive operations on large sets of data, such as vector and matrix computations.
In SIMD processing, data is divided into smaller elements, often called vectors or lanes. These vectors contain multiple data items that can be processed in parallel. The SIMD processor executes a
single instruction on all the data elements in a vector simultaneously, performing the same operation on each element concurrently.
SIMD instructions are typically supported by specialized hardware or instruction sets found in modern CPUs. These instructions are designed to perform arithmetic, logical, and other operations on
vectors efficiently. SIMD instructions are commonly used in multimedia applications, scientific simulations, image and signal processing, and other computationally intensive tasks.
SIMD in .NET
SIMD can be used in .NET through the System.Numerics and System.Runtime.Intrinsics namespaces.
In .NET Core 1.0 and later versions, you can use the System.Numerics.Vector<T> class. This class provides SIMD support for a wide range of data types, including integers and floating-point numbers.
You can perform SIMD operations using Vector<T> to efficiently process large sets of data in parallel. For example, you can create Vector<T> instances, perform arithmetic or logical operations on
them, and access the individual elements of the vector using familiar array-like syntax.
Starting from .NET Core 3.0 and later versions, the System.Runtime.Intrinsics namespace provides access to lower-level SIMD capabilities. The Vector128 and Vector256 structures in this namespace
represent SIMD vector types for specific hardware instruction sets, such as SSE (Streaming SIMD Extensions) or AVX (Advanced Vector Extensions). These types allow you to perform more fine-grained
control over SIMD operations and take advantage of the full capabilities of the underlying hardware.
Optimizing the sum of the elements in a collection
As explained in my previous article, .NET 7 allows the development of a method that calculates the sum of a collection of any numeric type as follow:
public static class MyExtensions
public static T Sum<T>(this IEnumerable<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
var sum = T.AdditiveIdentity;
foreach (var value in source)
sum += value;
return sum;
This is an extension method so it can be used as follow:
var source = Enumerable.Range(0, 100).ToArray();
This simply creates an array with 100 elements ranging from 0 to 99. It writes on the console the sum of all the element of the array.
You can see it working in SharpLab.
Sum of a span of elements
Data to be used in SIMD has to be in the form of a vector. The layout of the data in an enumerable is unknown so the “vectorization” requires copying the data into the vector. For this reason it’s
not advantageous to use SIMD on enumerables.
A Span<T> represents a contiguous region of arbitrary memory. This makes it possible to convert to a vector without copies, making it possible to take advantage of the SIMD performance optimizations.
So, lets provide an override of the Sum() method that takes a ReadOnlySpan<T> as a parameter as we are not going to mutate the collection while calculating the sum.
Unfortunately the compiler does not automatically call this new override when the collection is an array. It’s a good idea to provide one more override that takes an array as parameter. It simply
cast the array to a span (no copies) and call the other override.
If an array is cast to IEnumerable<T>, the first method will be called and SIMD will not be used. It’s a good idea to check inside this method if the collection is actually an array and call the new
override so that SIMD is used on this collection.
public static class MyExtensions
public static T Sum<T>(this IEnumerable<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
// check if the enumerable is an array
if (source.GetType() == typeof(T[]))
return Sum(Unsafe.As<T[]>(source));
var sum = T.AdditiveIdentity;
foreach (var value in source)
sum += value;
return sum;
// overload that takes an array
public static T Sum<T>(this T[] source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
=> Sum<T>(source.AsSpan());
// overload that takes a span
public static T Sum<T>(this ReadOnlySpan<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
var sum = T.AdditiveIdentity;
foreach (ref readonly var value in source) // use ref to avoid value-type copies
sum += value;
return sum;
You can see it working in SharpLab.
Now we can finally add the SIMD optimizations to the Sum() override dedicated to ReadOnlySpan<T> and have the guarantee that it’s used on every case that it may be useful.
Optimisations using System.Numerics.Vector<T>
The collection must be “vectorized” so that SIMD can be used. In our case, where the type of the collection elements is not known, the use of System.Numerics.Vector<T> makes it much easier to
understand the code than using the lower-level System.Runtime.Intrinsics API:
public static class MyExtensions
public static T Sum<T>(this IEnumerable<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
if (source.GetType() == typeof(T[]))
return Sum(Unsafe.As<T[]>(source));
var sum = T.AdditiveIdentity;
foreach (var value in source)
sum += value;
return sum;
public static T Sum<T>(this T[] source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
=> Sum<T>(source.AsSpan());
public static T Sum<T>(this ReadOnlySpan<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
var sum = T.AdditiveIdentity;
// check if SIMD is available and can be used
if(Vector.IsHardwareAccelerated && Vector<T>.IsSupported && source.Length > Vector<T>.Count)
var sumVector = Vector<T>.Zero; // initialize to zeros
// cast the span to a span of vectors
var vectors = MemoryMarshal.Cast<T, Vector<T>>(source);
// add each vector to the sum vector
foreach (ref readonly var vector in vectors)
sumVector += vector;
// get the sum of all elements of the vector
sum = Vector.Sum(sumVector);
// find what elements of the source were left out
var remainder = source.Length % Vector<T>.Count;
source = source[^remainder..];
// sum all elements not handled by SIMD
foreach (ref readonly var value in source)
sum += value;
return sum;
You can see it working in SharpLab.
The code added to the third method will only be execute if hardware acceleration (SIMD) is provided by the hardware device and if the type T is supported. The JIT compiler will actually remove all
this extra code when any of the two conditions is false. Meaning that there’s no performance penalty when not used.
The size of Vector<T> may vary depending on the hardware device but it should only be used if the source is larger than the vector. Otherwise, the code defaults to the usual foreach loop.
To calculate the sum, we have to create sumVector that is a Vector<T> with all the elements initialized to zero.
The method MemoryMarshal.Cast<T, Vector<T>> provides an efficient way, without copies, of converting the source ReadOnlySpan<T> into a ReadOnlySpan<Vector<T>>. We can now use a foreach loop to
iterate through the span of vectors. On each step of the loop, the elements of the vector are added to the elements of sumVector. This means, the first element of the vector is added to the first
element of sumVector, the second element of the vector is added to the second element of sumVector, and so on.
Once the loop ends, each element of sumVector contains a partial sum of the array elements. We need to call Vector.Sum() that sums all the elements of sumVector, resulting in the total sum of the
array elements processed.
NOTE: This portion of the code does not check for overflows or deal with NaN and infinite. I you know how to do it, please let me know in the comments.
We now only have to handle the case where there are elements of source that were left out because they were not enough to fill up one last Vector<T>. To do it efficiently, without copies, we can
slice the source, leaving only these last elements. The span resulting from the slice will then be handled by the usual foreach loop, adding to the current sum value.
Optimizing the sum of the List<T> elements
List<T> is a type provided by .NET that is very commonly used. The advantage over arrays is that data can be inserted and appended. Internally it uses an array that grows as needed.
List<T> is an enumerable type so we can use the Sum() we’ve just implemented:
var source = new List<int>(Enumerable.Range(0, 100));
Although List<T> wraps an array, the first Sum() method will be used which is much slower than the third one.
.NET 5 introduced a new method CollectionsMarshal.AsSpan<T>(List<T>). It returns the List<T> internal array as a Span<T>. This means we can use the much more efficient third method to calculate the
sum of the elements of a List<T>. We just need to change the code to the following:
public static class MyExtension
public static T Sum<T>(this IEnumerable<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
if (source.GetType() == typeof(T[]))
return Sum(Unsafe.As<T[]>(source));
// check if the enumerable is a list
if (source.GetType() == typeof(List<T>))
return Sum(Unsafe.As<List<T>>(source));
var sum = T.AdditiveIdentity;
foreach (var value in source)
sum += value;
return sum;
public static T Sum<T>(this T[] source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
=> Sum<T>(source.AsSpan());
// override that takes a list
public static T Sum<T>(this List<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
=> Sum<T>(CollectionsMarshal.AsSpan(source));
public static T Sum<T>(this ReadOnlySpan<T> source)
where T : struct, IAdditionOperators<T, T, T>, IAdditiveIdentity<T, T>
var sum = T.AdditiveIdentity;
if(Vector.IsHardwareAccelerated && Vector<T>.IsSupported && source.Length > Vector<T>.Count)
var sumVector = Vector<T>.Zero;
var vectors = MemoryMarshal.Cast<T, Vector<T>>(source);
foreach (ref readonly var vector in vectors)
sumVector += vector;
sum = Vector.Sum(sumVector);
var remainder = source.Length % Vector<T>.Count;
source = source[^remainder..];
foreach (ref readonly var value in source)
sum += value;
return sum;
You can see it working in SharpLab.
This code allows SIMD to be used on a List<T> even when cast to IEnumerable<T>.
Lets now benchmark it against the basic implementation of Sum(IEnumerable<T>) without any of the optimizations introduced.
The benchmark compares the following scenarios:
1. A List<float> with 10 and 10,000 items,
2. .NET 7 and .NET 8,
3. With no SIMD support (Scalar), only Vector128 support (Vector128) and with Vector256 support (Vector256).
The use of SIMD, together with the iteration of List<T> as span, totals in performance boosts of:
• 14x faster for 10 items and 6x faster for 10,000 items when hardware acceleration is not available (Scalar jobs).
• 17x faster for 10 items and 27x faster for 10,000 items when Vector128 is available (Vector128 jobs).
• 19x faster for 10 items and 54x faster for 10,000 items when Vector256 is available (Vector256 jobs).
There is a 4x performance improvement when hardware acceleration is not available (Scalar jobs) just by upgrading from .NET 7 to .NET 8. This is an unrelated gain that you get for free by simply
The use of SIMD can radically improve performance of arithmetic intensive operations on large amounts of data.
NOTE: This implementation of Sum() should only be used when it’s guaranteed that the sum will not overflow and that the collection does not contain elements that are NaN or infinite. Use LINQ
when these are not guaranteed.
The Sum() is just one example where the use of vectorization can be used. I hope you found this article helpful in understanding the concepts so that you can apply to different scenarios in your
For documentation on the more advanced System.Runtime.Intrinsics namespace, check the “Introduction to vectorization with Vector128 and Vector256”. | {"url":"https://antao-almada.medium.com/single-instruction-multiple-data-simd-in-net-393b8cf9a90?source=user_profile_page---------5-------------fd53efd266b6---------------","timestamp":"2024-11-13T01:10:43Z","content_type":"text/html","content_length":"159145","record_id":"<urn:uuid:c1213f16-495c-4aac-b8e5-92a2cb9668cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00603.warc.gz"} |
Milliliters (mL) in an Ounce (oz) for the NAPLEX Exam
Pharmacists play a crucial role in patient care by ensuring accurate medication dispensing. A key skill in pharmacy practice is […]
Pharmacists play a crucial role in patient care by ensuring accurate medication dispensing. A key skill in pharmacy practice is mastering dosage calculations, especially when converting between units
like milliliters (mL) and ounces (oz). For students preparing for the North American Pharmacist Licensure Examination (NAPLEX), understanding such conversions is essential.
In this article, we’ll explore the mL-to-oz conversion, why it matters in pharmacy, and how to confidently tackle these calculations. By the end, you’ll have a solid understanding of how to handle
these conversions for NAPLEX success.
Why Are Conversions Important in Pharmacy?
Pharmacy is a field that demands precision. Small errors in dosage can lead to ineffective treatment or dangerous outcomes for patients. This is why pharmacists must be adept at converting
measurements, especially when working with different medication forms (liquids, tablets, capsules, etc.).
Liquid medications, in particular, often require conversion between milliliters and ounces. Whether dispensing oral solutions, injectable drugs, or IV fluids, understanding these conversions is
For the NAPLEX, these conversions are likely to appear in calculation-based questions, making it crucial to master the basic math behind them.
The Basics: What is an Ounce?
An ounce (oz) is a unit of volume commonly used in the United States, especially for liquid measurements. When discussing fluid ounces in the pharmacy context, we typically refer to the US fluid
ounce, which is different from the weight-based ounce used for solid items. One fluid ounce is equivalent to approximately 29.5735 milliliters (mL).
This may seem like an odd number, but it’s important to memorize this conversion for pharmacy calculations.
How Many mL in an Ounce?
The conversion between ounces and milliliters is a fundamental pharmacy calculation:
1 ounce (oz) = 29.5735 milliliters (mL)
This value is key when converting from ounces to milliliters or vice versa. Since medications are often dispensed in milliliters but may be prescribed in ounces, understanding how to convert between
the two is critical.
Here’s a simple way to remember this:
1 oz ≈ 30 mL (rounded for convenience)
This rounded value is often sufficient for everyday use in pharmacies unless extreme precision is required. However, for the NAPLEX exam, you should know the exact conversion and be prepared to use
it when calculating precise doses.
Practical Application of mL to Ounce Conversions in Pharmacy
To illustrate the importance of this conversion in real-life pharmacy scenarios, let’s look at some examples that could appear on the NAPLEX.
Example 1: Converting from Ounces to Milliliters
Imagine a patient has been prescribed a medication with a dosage of 2 fluid ounces. The medication is available as a liquid suspension, and you need to dispense the correct amount in milliliters.
Using the conversion formula:
\text{Volume in mL} = 2 \, \text{oz} \times 29.5735 \, \text{mL/oz}
\text{Volume in mL} = 59.147 \, \text{mL}
Therefore, you would dispense approximately 59.15 mL of the medication.
Example 2: Converting from Milliliters to Ounces
Suppose a patient has been prescribed 120 mL of a liquid medication. How many ounces is this?
Using the inverse of the conversion formula:
\text{Volume in oz} = \frac{120 \, \text{mL}}{29.5735 \, \text{mL/oz}}
\text{Volume in oz} = 4.06 \, \text{oz}
So, the patient would receive roughly 4.06 ounces of the medication.
Handling Complex NAPLEX Questions
The NAPLEX may present these conversions in a more complex format, often integrating other variables like concentration, weight-based dosing, or IV flow rates. Let’s look at a more advanced example.
Example 3: Adjusting for Concentration
A liquid medication has a concentration of 10 mg/mL, and the doctor prescribes a 2 oz dose. How many milligrams of the drug will the patient receive?
First, convert 2 oz to milliliters:
2 \, \text{oz} \times 29.5735 \, \text{mL/oz} = 59.147 \, \text{mL}
Next, calculate the amount of drug in 59.15 mL:
59.15 \, \text{mL} \times 10 \, \text{mg/mL} = 591.5 \, \text{mg}
Thus, the patient will receive 591.5 mg of the medication.
This type of multi-step question is common on the NAPLEX, so it’s important to be comfortable with performing these calculations quickly and accurately.
Tips for Mastering mL and Ounce Conversions for the NAPLEX
1. Memorize Key Conversions: While approximations like 1 oz ≈ 30 mL can be useful in daily practice, for the NAPLEX, it’s important to know the exact value (1 oz = 29.5735 mL).
2. Practice, Practice, Practice: The more you practice conversions, the more second nature they will become. Use practice questions and past NAPLEX exams to test yourself under time constraints.
3. Watch for Units: One common mistake is confusing different measurement units. Always double-check whether you are dealing with fluid ounces, milliliters, or another unit.
4. Break Down Complex Problems: Many NAPLEX questions involve multiple steps. Break down each problem into manageable parts, and focus on one conversion or calculation at a time.
5. Use Dimensional Analysis: Dimensional analysis is a helpful tool for converting between units. It allows you to set up the problem so that the units cancel out, ensuring your calculations are
Converting between milliliters and ounces is a fundamental skill for pharmacists, especially when preparing for the NAPLEX. By understanding and mastering these conversions, you’ll be well-prepared
to tackle dosage calculations and provide accurate medication doses for your patients. Remember to practice regularly and become comfortable with applying these conversions in both simple and complex
scenarios. The ability to confidently and accurately perform these calculations will serve you well throughout your pharmacy career.
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Harold Walsby: Dedication to The Paradox Principle and Modular Systems Generally
Harold Walsby Design Research Project
Paper No. 1
January 1967
Communicating mathematical ideas is a problem even among mathematicians. Many leading mathematicians are distressed over a style of mathematical writing that has become commonplace in the last
decade or two. Mathematical papers are compressed to the limit, until all intuitive ideas are squeezed out. As a result, one mathematician complains, ‘Most papers are read only three times – once
by the author, once by the editor, and once by the reviewer.’
Some mathematicians today fear that their subject is becoming almost a pure exercise in manipulating symbols. It is true that skill in symbolism can cover up and embellish trivial ideas. ‘There
is less to this than meets the eye,’ is the comment that one leading mathematician applies to impressive-looking papers that are low on ideas. While he disparages only the content of the papers,
other mathematicians are concerned about the soaring abstraction of their subject. They argue that while mathematics has always derived its most fertile inspirations from the physical world, most
creative mathematicians today are getting completely out of touch ‘with physical reality.”
The New World of Mathematics, G.A.W. Boehm.
If we marvel at the patience and the courage of the pioneers, we must also marvel at their persistent blindness in missing the easier ways through the wilderness and over the mountains. What
human perversity made them turn east to perish in the desert, when by going west they could have marched straight through to ease and plenty? …
The very crudities of the first attack on a significant problem… are more illuminating than all the pretty elegance of the standard texts which has been won at the cost of perhaps centuries of
finicky polishing.
Mathematics, E.T. Bell.
To Independent Thinking
This paper contains the beginnings of a revolution. Whether or when that revolution develops depends much on others. It rests a lot on those who are independent enough to see a new valid viewpoint
beyond that of the main current, the orthodox. However, if the content of this essay is revolutionary, so is its form. It is no brittle blossom of today’s fashionable school of ultra-formalistic
mathematics. Rather, it is a revolt against that overspecialistic trend. Its approach is holistic rather than specialistic, unitive rather than separative, since this is vital to its reconciliation
of form with content. “Content” or meaning – temporarily ignored and submerged in the present obsession with empty “form” – belongs to that “inward eye,” the source of our intuitions, without which
there would be no mathematics. Moreover, in the weary human struggle for better things, it is the evolution of content which furnishes the great theoretical and inspirational challenges, not only of
our time but of all time.
continue reading The Paradox Principle by Harold Walsby (1967):
Dedication | Aristotle’s Principle | The Role of Logic | Do Self-Contradictions Exist? | Three Types of Contradictions | Meaningful Self-Contradictions | Infinity and Self-Contradictions | Models for
Self-Contradiction | The Paradox Principle and Applications | Appendix | {"url":"https://www.gwiep.net/harold-walsby-dedication/","timestamp":"2024-11-06T05:32:02Z","content_type":"text/html","content_length":"35136","record_id":"<urn:uuid:27f667bf-b099-44ee-9704-90ff1ab36a97>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00010.warc.gz"} |
Earth's oblateness
1 Introduction
The rotating Earth is oblate, that is, it is slightly ‘flat’ in the North Pole–South Pole direction, compared to the slightly ‘bulging’ Equator. This is the result of the hydrostatic balance between
the dominant gravitational force, which wants to pull the Earth into a spherically symmetric configuration, and the centrifugal force due to Earth's rotation, which wants to expel mass away from the
rotating axis but in the end only manages to modify the Earth into an slightly oblate body.
Quantitatively, this oblateness is about 1 part in 300, which is very close (but see below) the ratio of the centrifugal force on the equator to the gravitational force. This is by far the Earth's
largest deviation from a spherically symmetric body. There are certain thermodynamic, but secondary, processes that cause departures from the rotational-hydrostatic equilibrium. Sustained by the
internal heat engine and manifested as external gravity anomalies reflecting lateral heterogeneity of internal mass distribution, these deviations are in general no more than parts per million in
relative terms.
Yet all these deviations are not static or constant; they change with time. The rotation of the Earth is itself changing over geological time, and the aforementioned mass heterogeneities also vary on
timescales upwards from millions of years. On more human timescales, there operates a myriad of dynamic processes that involve mass redistribution in or on the Earth, from tides to atmosphere–ocean
circulations, to internal phenomena like earthquakes, post-glacial rebound and core flows. For the Earth, these changes are typically on the order of parts per billion at the largest [9,24]. The
present article is a story about the oblateness in particular and how and why it changes with time, where we examine the geophysical implications.
2 The Earth's oblateness parameters and their inter-relationships
As long as the Earth is a 3-D body, we shall use the word oblateness to describe its off-spherical shape. Traditionally, the term ‘flatness’, or ‘ellipticity’, has been used; these names are
imprecise because the Earth, of course, is not ‘flat’, and it is a 2-D geometric object only when we try to draw it on paper.
There are several parameters in use to describe the oblateness; each one has its significance depending on the application in question. For simplicity let's for the moment assume the Earth is axially
symmetric, or a body of revolution and so essentially a 2-D body, which is a good approximation.
By equating the general expression of the spherical harmonic expansion of the external gravitational potential field V with that of the mass distribution of a body, one concludes that the spherical
harmonic coefficients, or Stokes coefficients, of V are simply normalized multi-poles of the density function of the body [4,5]. When specialized, the degree-2 Stokes coefficients are related to the
body's inertia tensor elements through a set of equations known as generalized MacCullagh's formulas. In particular, the degree-2 zonal (order 0) Stokes coefficient is given by:
Symbol namesake in honor of Sir Harold Jeffreys [19] and called the oblateness coefficient for the Earth, $J2$ has the physical meaning of the difference of the axial or polar (greatest) moment of
inertia C with the equatorial (least) moment of inertia A, normalized by $Ma2$ where M and a are the Earth's mass and mean Equatorial axis, respectively. Its corresponding term in the harmonic
expansion of V is the dominant term next to the ‘monopole’ term representing the total mass [4].
We can express $J2$ in the following form:
$J2=[C/Ma2][(C−A)/C]≡ηH$ (2)
where $η≡C/Ma2$ is a fundamental functional of the Earth's internal structure, and $H≡(C−A)/C$ is called the dynamic oblateness, which can be determined from the observation of the astronomical
precession of the Earth [27]. The Earth's η can be readily determined by knowing the values of $J2$ and H. However, this has not been feasible for other planets because their H's are generally
The ‘geopotential’ field is V, modified by the centrifugal potential, i.e. $V−12a2ω2$, where ω is the angular speed of Earth's rotation. If one approximates the equipotential surface, known as the
geoid, to an oblate spheroid of revolution, then one obtains the geoid oblateness for the Earth which can be given by the Clairaut's first relationship (for a review, see [27]):
where c is the mean axial or polar axis of the (ellipsoidal) Earth, and $m≡a3ω2/GM$ is the ratio of the centrifugal force $aω2$ on the equator to gravity $GM/a2$ (where G is the gravitational
constant). Eq. (3) is based on the first-order theory, which is sufficient for the present purpose for the Earth (for high-order formulas see, e.g., [23]). For planets with much larger m (for
example, Jupiter, see below), the second-order effects become rather significant [29].
Two more concepts of oblateness can be defined at this point: suppose the Earth is under rotational-thermo-hydrostatic equilibrium. A hypothetical hydrostatic geoid oblateness $fH$ representing an
idealized Earth can be defined; to first order $fH$ can be found by [18]:
$fH=(5/2)m/[1+(25/4)(1−1.5η)2]$ (4)
Under such equilibrium, the Earth's geometric surface would conform to and coincide with the geoid, so the geometric oblateness, similarly defines as Eq. (3), is simply equal to f. That is why f is
often referred to as defining the ‘figure’ of the Earth. In reality, due to its heterogeneities, the Earth's geometric oblateness (as properly defined or approximated) would depart slightly from f.
Furthermore, the Earth's true f also departs slightly from $fH$, where the departure signifies interesting geophysical dynamics sustaining non-equilibrium. The lateral heterogeneity and
non-equilibrium configuration of the Earth also manifest themselves in the (relatively small) difference between the two equatorial principal moments of inertia A and B. In the above we have assumed
that the Earth is axially symmetric where $A=B$, which would be the case if the Earth were an otherwise spherically symmetric body subject only to an axial rotation. The real Earth, of course, is not
so (we will further discuss this below.) In fact, the strict definition of $J2$ is $[C−(A+B)/2]/Ma2$, to which Eq. (2) is only an approximation, or valid only for axial-symmetric Earth.
Now let us examine the numerical values. m is known to be $3.46775×10−3$, or 1/288.371, close to 1 part in 300 or 1/300. According to Eq. (3), half of it contributes directly to the geoid oblateness
f. For f, the remaining contribution comes from $32J2$, which, of course, shares the same dynamic origin as m, i.e. Earth's rotation. $J2$ is measured from satellite geodesy (see below) to a high
accuracy, $1.082626×10−3$, about one third of 1/300. So, the two terms in (3) contribute almost the same amount to f, i.e., 50% each, and f itself becomes close to 1/300, at $f=3.35281×10−3$ or 1/
298.257. Finally, the dynamic oblateness H in Eq. (2) is observed to be $3.27379×10−3$ or 1/305.456, again close to 1/300.
Are these matchings in values just fortuitous? From dynamical considerations, one can rightly ‘guess’ that all parameters should be on the order of m, which they indeed are. However, upon closer
examination as follows, they do not necessarily have to have such similar values, so in a sense the latter is fortuitous.
For a reasonable Earth configuration, we should have $H≈f$ because of its moderate sensitivity to the internal density profile, although it is well recognized that H would be somewhat less than f
because of the smaller oblateness of the interior layers (due to smaller centrifugal force) and the higher density toward the center of the Earth (and hence proportionally lower importance in
contributing to the moment of inertia) [27]. Putting this condition into Eqs. (2) and (3), we see the following: the two terms would contribute near-equal shares to f in Eq. (3) and hence all the
values would be close to m or 1/300, only if $η=1/3$.
The interesting, but certainly not out of the ordinary, fact is that, knowing $J2$ and H in Eq. (2), the Earth in reality has $η=0.33069$, indeed almost exactly 1/3! This of course does not have to
be the case, but one does expect a η value somewhat less than 0.4, that of a uniform-density sphere, for a ‘reasonable’ centrally-heavy, terrestrial planet body such as the Earth.
Based on the PREM Earth model (Preliminary Reference Earth Model [15]) derived from seismological data, the Earth should have an estimated hydrostatic $fH$ of 1/299.66 [29], about 0.5% smaller than
the observed. This corresponds to a hydrostatic $J2$ of $1.0722×10−3$, about 1% smaller than the observed. On the other hand, Liu and Chao [21] formulated the relation between A, B and the two Stokes
coefficients of degree 2 and order 2. Using the gravity-observed values for the latter, they get $B−A=7.260×10−6Ma2$, amounting to 69.4-m difference in the equivalent geoidal semi-major axis and
semi-minor axis on the Equator. Although only $∼1/150$ that of $C−A$, this amount is comparable to the non-hydrostatic portion in $C−A$, as pointed out by [17]. They concluded that the
non-hydrostatic portion of the three principal moments of inertia A, B, C only describes a triaxial body and appears to have no preference in orientation. As far as the aforementioned excess
oblateness over the hydrostatic value is concerned, this does not favor the notion that this excess oblateness is a remnant, lagging ‘memory’ of the past, as the Earth slows down due to the tidal
3 Comparative planetology
For a contrast, let us compare the Earth with the giant planet Jupiter. Jupiter has a faster rotation and a much larger mass, and hence larger radius and gravity. Its $m=0.0892=1/11.2$. We can expect
that the geoid oblateness f and the dynamic oblateness H to be similar to m, but not necessarily very close in value. The observed $J2=0.01469$. Adopting second-order formulas [29], which are more
accurate than Eq. (2), $f=0.0649=1/15.4$. Assuming rotational-hydrostatic equilibrium, $η=0.254$ (cf. Eq. (4)), indicating that, not surprisingly, Jupiter is somewhat more centrally-heavy than the
Earth. The derived $H=(1/η)J2=0.0578=1/17.3$. We further expect that the geometric oblateness is the same as f, except possibly for some small departures from rotational-hydrostatic equilibrium.
In another extreme example, let us consider a non-rotating, uniform-density body not under hydrostatic equilibrium (hence the shape sustained by its internal material strength), such as an asteroid.
Then there exists an analytical, but complex, relationship between the spherical harmonic coefficients of gravity and geometrical shape [7]. For the present discussion, let us further assume a
special case where the body is a slightly oblate spheroid. Then, letting $m=0$ in Eq. (2), we have the geoid oblateness $f=32J2$. Since the body is not under hydrostatic equilibrium, the geometric
oblateness is not equal to f; rather, according to Eq. (9) of [7], it equals $53f$. Finally, the dynamic oblateness $H=(1/η)J2$, when $η=25$ (for a uniform spherical body), equals $53f$, the same as
the geometric oblateness, as expected.
4 Consequences of oblateness
We live in the Earth's gravity field, controlled by the dominant monopole term $GM/a2$. We hardly notice any consequence of the oblateness of the Earth (or for that matter the rotational centrifugal
force). However, dynamically, the Earth's oblateness is an essential element in our livelihood – it stabilizes our Earth's rotation. The Earth is ‘bombarded’ all the time by countless geophysical
agents exerting external torques as well as internal torques or mass transports that exchange angular momenta. Yet its rotation axis hardly changes relative to the Earth-fixed geography. This is not
true if it were spherically symmetric: then the crawl of a bug or a firing of a canon, for instance, would completely ‘tumble’ the Earth relative to the (spatially stationary) rotation axis [16,22].
On the other hand, it is well known from classical mechanics that the rotation of a body about its principal axis of the greatest moment of inertia (C) is a stable one. What prevents large shifts of
the rotation axis from happening is the extra oblateness in the form of $C−A$ with the associated extra angular momentum, which is to be overcome by any geophysical agent that tries to shift the
Earth's rotational pole positions. Since the oblateness itself is a consequence of the rotation in the first place, it can be stated that the rotating planet is self-stabilized.
A corollary of the above, but on a less dramatic scale, the Earth's dynamic oblateness under the tidal torques exerted by the Moon and Sun gives rise to the astronomical precession of the Earth's
rotation axis in space, and hence is the deciding factor for the precessional period. That in fact is how the dynamic oblateness H is determined. On the same token, H acts as the restoring factor
that prescribes the free wobble, known as the Chandler wobble, of the Earth's polar motion. The period of the Chandler wobble would be $1/H$ days if the Earth were a rigid body, but was found to be
significantly lengthened by the Earth's non-rigidity, or finite elasticity [22].
As stated, the geometrical shape of the Earth largely conforms to the oblate geoid. Therefore, the mean equatorial radius and the mean polar radius of the geoid differ by as much as $a−c=fc∼21 km$.
In particular, the global sea level follows closely this oblate geoid, only undulating on top of the geoid geographically no more than 200 m peak-to-peak and temporally less than 10 m or so. The land
topography undulates up to $∼10 km$, but largely supported isostatically.
As such, the oblateness also affects various geophysical quantities. For example, in the space geodesy enterprise using near-Earth satellites, the oblateness term resides in all Earth surface
geometry that locates the geodetic observatories and altimetric targets. Similarly, the oblateness prevails in the external gravity field that significantly affects the satellite orbits from which
geodetic measurements are made. On the Earth surface, together with the centrifugal force field, the oblateness gives the surface gravity a slight latitudinal dependence which is actually the largest
term in the surface gravity anomaly on the global scale. In another example, the Earth's elastic free oscillation modes (often excited by large earthquakes) see splitting in their otherwise
degenerate characteristic periods due to Earth's oblateness and rotation, completely analogous in the atomic world to the Stark splitting and Zeeman splitting, respectively, as such splitting is
determined by the symmetry properties common to different dynamic systems [2].
5 Historical Notes
Sir Isaac Newton, based on his law of gravitation and force laws, was the first to realize that the Earth under rotational equilibrium should possess a non-vanishing oblateness. The value of $1/f$
favored by him and given in the Principia was 230. Cassini subsequently came up with a negative value, −95, presumably owing to certain systematic errors. The value had evolved [19], since the Peru/
Lapland expedition in the 1740s from a value between 179 and 266, to 301, 295, 297.0, and finally in the early 1950s to Sir Harold Jeffreys' $297.1±0.4$ which is within 0.4% of the modern value
Then came the space age, ushered in by the launch of USSR's Sputnik I spacecraft in October 1957. A month later Sputnik II was launched, and within a few weeks, by monitoring the nodal precession of
its orbit in space, our knowledge of $J2$ grew almost an order of magnitude, to about 0.1% of the modern value. This measurement was arguably one of the very first scientific triumphs of space
Today, after nearly half a century of precise orbit determination of dozens of geodesy-quality satellite orbits around the Earth, the Earth's global gravity field has been solved to harmonic degrees
as high as 120, among which the average $J2$ coefficient has been determined to the accuracy of seven significant figures ($1.082627×10−3$) [20].
Since the 1980s, thanks to the advent of the technique of satellite laser ranging [1], tiny temporal variations around the average value of $J2$ began to be noted. The variation occurs in the last
digit of the above-quoted number and beyond, typically no more than one part in a billion! This will be discussed next.
6 How and why does Earth's $J2$ change?
Mass transports in the atmosphere–hydrosphere–cryosphere–solid Earth–core system (the ‘Earth system’) occur on all temporal and spatial scales for a host of geophysical and climatic reasons [9,24].
According to Newton's gravitational law, such mass transport will cause the gravity field to change with time, producing time-variable gravity signals.
Increasingly refined models for the Earth's static gravity field in terms of spherical harmonic components have been determined by means of decades of precise orbit tracking data of many geodetic
satellites. On top of that, low-degree components of Earth's time-variable gravity have been clearly observed by the space geodetic technique of satellite laser ranging (SLR) [1]. Although tiny in
relative terms (no more than 1 part per billion), these variations signify global-scale mass redistribution in the Earth system.
In particular, the lowest-degree zonal harmonic is Earth's oblateness coefficient $J2$, whose temporal variation was the first to be detected among all gravity components. A ‘secular’ decrease in
$J2$ (over the observed quarter century) was first identified from the SLR satellite nodal precession acceleration. Its main excitation source has since been attributed to the post-glacial rebound
(PGR) of the solid Earth [26,30], and for additional secondary causes [3]. Subsequently, many studies reported strong seasonal as well as weaker non-seasonal signals, primarily in $J2$, but of late
also in the next-lowest harmonics [13] and geocenter. The prominent seasonal $J2$ signals (with primarily annual amplitude $∼3×10−10$) have been correlated with mass transports in the atmosphere,
oceans, and land hydrology [8,11,25].
Such was the case until around the turn of the century beginning in 1998, when the SLR data began to reveal that Earth's $J2$ had suddenly deviated significantly from the PGR secular decreasing trend
(at about $−2.8×10−11 yr−1$). This ‘1998 anomaly’ embarked on a reverse, increasing trend over the following years, before quieting back down to the ‘normal’ decreasing trend. This was reported by
[10,12]. Fig. 1a shows an updated time series of the SLR-observed $J2$, using SLR data from up to nine satellites, with more satellites becoming available with time [12]. Note that the relevant
18.6-yr lunar-driven ocean tide amplitude was set to the value recovered in a 21-yr comprehensive solution for the secular zonal rates, low-degree static and annual terms, and the 18.6-yr and the
much smaller 9.3-yr lunar tides. Fig. 1(b) is the same time series but after the removal of (i) the atmospheric contribution calculated according to the global NCEP reanalysis data assuming an
inverted-barometer effect, and (ii) the least-squares fit of the remaining seasonal signals, which are attributable to (the poorly known) seasonal mass redistribution in the oceans and land
hydrology. The PGR slope (the solid line) and the 1998 anomaly are clearly evident.
Fig. 1
A number of possible causes for the 1998 anomaly was speculated by Cox and Chao [12], including oceanic water mass redistribution, melting of polar ice sheets and high-latitude glaciers, global sea
level rise, and material flow in the fluid core. Dickey et al. [14] emphasized and demonstrated the importance of the melting of high-latitude glaciers. Chao et al. [10] report an oceanographic event
that took place in the extratropic North + South Pacific basins that was found to match remarkably well with the time evolution of the $J2$ anomaly; the phenomenon appears to be part of the Pacific
Decadal Oscillation immediately following the episode of the 1997–1998 El Niño.
The difficulty in identifying the definite cause(s) in the above w.r.t. $J2$ stems from the extremely low geographical resolution of the zonal harmonic function in question, namely the degree-2
Legendre function. Thus, a positive $J2$ anomaly only tells us that a net transport of mass from higher latitudes to lower latitudes (across the nodal latitude of the degree-2 Legendre function,
namely ±35.3°) has occurred, in either or both Northern and Southern Hemispheres. For example, an equivalent of as much as 3000 km^3 of water melted from Greenland and spread into the oceans would be
needed to produce the first half of the $J2$ anomaly where the relative change is $+7×10−11$ per year; but we have no way of telling without other ancillary evidences or observations. On the other
hand, the space gravity mission of GRACE (launched in March 2002, with an expected lifetime of over 5 yr), using the satellite-to-satellite tracking (SST) technique, is yielding gravity information
at much higher geographical resolution than the SLR-based information. For example, GRACE is able to detect centimeter level water-height-equivalent mass changes over an area of about 1000 km across
from month to month [28]. However, considering the relatively weak sensitivity of the SST to the longest-wavelength (lowest-degree) gravity components, GRACE's utility in measuring the variation of
$J2$ in particular awaits to be seen. The same applies to future gravity missions of GOCE (using gravity gradiometer) and other SST measurements in ‘follow-on’ gravity missions under planning.
7 Relationship between Earth's rotation and $J2$ change
As stated, the Earth's oblateness arises from its rotation; the rotational-hydrostatic relationship, to first order, is given in Eq. (4), where the oblateness is proportional to m, which is in turn
proportional to $ω2$. Therefore,
For example, the Earth's secular spin-down due to the tidal braking would lead to a secular ‘rounding’ of the Earth (barring possible temporal retardation under viscosity), thus decreasing $J2$.
Numerically, at the tidal-braking rate of $ω˙=−6.5×10−22 rads−2$, that decreasing rate of $J2$ is about $−6.1×10−13 yr−1$, contributing only 2% of the observed decreasing rate of $J2$ (see above).
On the other hand, any change in $J2$ will cause ω to change, as dictated by the conservation of angular momentum for the Earth. For instance, a decreasing $J2$ means a faster spin (analogous to a
spinning skater pulling arms closer to the body), and vice versa. That effect can be shown to be [6]:
where the coefficient 2.01 is evaluated from Earth parameters. For example, the decrease in $J2$, at the rate of $−6.1×10−13 yr−1$ due to the tidal braking of ω given above, will in turn feedback to
cause ω to increase, but only by as little as $2.8×10−24 rads−2$, or $∼−0.053 μs$ in the equivalent length-of-day per year. That is completely negligible in today's measurement. On the other hand,
the observed $J2$ rate of change $−2.8×10−11 yr−1$ (see above) presumably speeds up the Earth rotation by $−2.4 μs$ in the equivalent length-of-day per year, which is still negligible.
8 Epilogue
Although numerically small, the oblateness is a fundamental property of the Earth under stable rotation. Its existence and cause, its dynamical and geometrical consequences, its values and departures
from idealized models, and its temporal evolution due to mass transports in the Earth system are all fascinating topics in geophysics, which reveal insights towards the understanding of the structure
and dynamical behavior of the Earth. The measurement and monitoring of the Earth's oblateness have been a triumph as well as a scientific target of the modern space geodesy. As one sees deeper and
finer into the Earth's oblateness, there is little doubt that the Earth will surprise and further fascinate us with a continuing story unfolding with time.
This paper is completed under the support of the NASA Solid Earth program. I am grateful to Christopher Cox for providing Fig. 1. | {"url":"https://comptes-rendus.academie-sciences.fr/geoscience/articles/en/10.1016/j.crte.2006.09.014/","timestamp":"2024-11-08T20:12:05Z","content_type":"text/html","content_length":"109237","record_id":"<urn:uuid:3ea5b020-0cd6-40d4-996e-5b44c0bba74b>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00696.warc.gz"} |
Q and Bandwidth of a Resonant Circuit - Electrical Engineering Textbooks
Q and Bandwidth of a Resonant Circuit
The Q, quality factor, of a resonant circuit is a measure of the “goodness” or quality of a resonant circuit. A higher value for this figure of merit corresponds to a more narrow bandwith, which is
desirable in many applications. More formally, Q is the ratio of power stored to power dissipated in the circuit reactance and resistance, respectively:
Q = P<sub>stored</sub>/P<sub>dissipated</sub> = I<sup>2</sup>X/I<sup>2</sup>R
Q = X/R
where: X = Capacitive or Inductive reactance at resonance
R = Series resistance.
This formula is applicable to series resonant circuits, and also parallel resonant circuits if the resistance is in series with the inductor. This is the case in practical applications, as we are
mostly concerned with the resistance of the inductor limiting the Q. Note: Some text may show X and R interchanged in the “Q” formula for a parallel resonant circuit. This is correct for a large
value of R in parallel with C and L. Our formula is correct for a small R in series with L.
A practical application of “Q” is that voltage across L or C in a series resonant circuit is Q times total applied voltage. In a parallel resonant circuit, current through L or C is Q times the total
applied current.
Series Resonant Circuits
A series resonant circuit looks like a resistance at the resonant frequency. (Figure below) Since the definition of resonance is X[L]=X[C], the reactive components cancel, leaving only the resistance
to contribute to the impedance. The impedance is also at a minimum at resonance. (Figure below) Below the resonant frequency, the series resonant circuit looks capacitive since the impedance of the
capacitor increases to a value greater than the decreasing inducitve reactance, leaving a net capacitive value. Above resonance, the inductive reactance increases, capacitive reactance decreases,
leaving a net inductive component.
At resonance the series resonant circuit appears purely resistive. Below resonance it looks capacitive. Above resonance it appears inductive.
Current is maximum at resonance, impedance at a minumum. Current is set by the value of the resistance. Above or below resonance, impedance increases.
Impedance is at a minumum at resonance in a series resonant circuit.
The resonant current peak may be changed by varying the series resistor, which changes the Q. (Figure below) This also affects the broadness of the curve. A low resistance, high Q circuit has a
narrow bandwidth, as compared to a high resistance, low Q circuit. Bandwidth in terms of Q and resonant frequency:
BW = f<sub>c</sub>/Q
Where f<sub>c</sub> = resonant frquency
Q = quality factor
A high Q resonant circuit has a narrow bandwidth as compared to a low Q
Bandwidth is measured between the 0.707 current amplitude points. The 0.707 current points correspond to the half power points since P = I^2R, (0.707)^2 = (0.5). (Figure below).
Bandwidth, Δf is measured between the 70.7% amplitude points of series resonant circuit.
BW = Δf = f<sub>h</sub>-f<sub>l</sub> = f<sub>c</sub>/Q
Where f<sub>h</sub> = high band edge, f<sub>l</sub> = low band edge
f<sub>l</sub> = f<sub>c</sub> - Δf/2
f<sub>h</sub> = f<sub>c</sub> + Δf/2
Where f<sub>c</sub> = center frequency (resonant frequency)
In Figure above, the 100% current point is 50 mA. The 70.7% level is 0.707(50 mA)=35.4 mA. The upper and lower band edges read from the curve are 291 Hz for f[l] and 355 Hz for f[h]. The bandwidth is
64 Hz, and the half power points are ± 32 Hz of the center resonant frequency:
BW = Δf = f<sub>h</sub>-f<sub>l</sub> = 355-291 = 64
f<sub>l</sub> = f<sub>c</sub> - Δf/2 = 323-32 = 291
f<sub>h</sub> = f<sub>c</sub> + Δf/2 = 323+32 = 355
Q = f<sub>c</sub>/BW = (323 Hz)/(64 Hz) = 5
Parallel Resonant Circuits
A parallel resonant circuit is resistive at the resonant frequency. (Figure below) At resonance X[L]=X[C], the reactive components cancel. The impedance is maximum at resonance. (Figure below) Below
the resonant frequency, the parallel resonant circuit looks inductive since the impedance of the inductor is lower, drawing the larger proportion of current. Above resonance, the capacitive reactance
decreases, drawing the larger current, thus, taking on a capacitive characteristic.
A parallel resonant circuit is resistive at resonance, inductive below resonance, capacitive above resonance.
Impedance is maximum at resonance in a parallel resonant circuit, but decreases above or below resonance. Voltage is at a peak at resonance since voltage is proportional to impedance (E=IZ). (Figure
Parallel resonant circuit: Impedance peaks at resonance.
A low Q due to a high resistance in series with the inductor produces a low peak on a broad response curve for a parallel resonant circuit. (Figure below) conversely, a high Q is due to a low
resistance in series with the inductor. This produces a higher peak in the narrower response curve. The high Q is achieved by winding the inductor with larger diameter (smaller gague), lower
resistance wire.
Parallel resonant response varies with Q.
The bandwidth of the parallel resonant response curve is measured between the half power points. This corresponds to the 70.7% voltage points since power is proportional to E^2. ((0.707)^2=0.50)
Since voltage is proportional to impedance, we may use the impedance curve. (Figure below).
Bandwidth, Δf is measured between the 70.7% impedance points of a parallel resonant circuit.
In Figure above, the 100% impedance point is 500 Ω. The 70.7% level is 0.707(500)=354 Ω. The upper and lower band edges read from the curve are 281 Hz for f[l] and 343 Hz for f[h]. The bandwidth is
62 Hz, and the half power points are ± 31 Hz of the center resonant frequency:
BW = Δf = f<sub>h</sub>-f<sub>l</sub> = 343-281 = 62
f<sub>l</sub> = f<sub>c</sub> - Δf/2 = 312-31 = 281
f<sub>h</sub> = f<sub>c</sub> + Δf/2 = 312+31 = 343
Q = f<sub>c</sub>/BW = (312 Hz)/(62 Hz) = 5
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under the terms and conditions of the CC BY License.
See the Design Science License (Appendix 3) for details regarding copying and distribution.
Revised July 25, 2007
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Worksheet on Fixed Effects
This worksheet will try to help us understand a bit more fixed-effects from a data-approach.
*First, we'll open a dataset
webuse nlswork
* Get yourself familiar with the data. These data contains information on individual's wages and education.
* Let's collapse some variables, so the data just looks like averages per year
collapse wks_work hours tenure ttl_exp, by(year)
* Now let's look at the data, specially for hours and years
list year hours
* This tell us the average hours per year. Now let's explore fixed-effects
* First, we'll run a regression of hours on year fixed effects
reg hours i.year
* Explore each of coefficient of the year, and let's try to interpret them.
* Now let's figure out what's the base year that STATA picked
reg hours i.year, base
* In my stata it seems that base year is 68, if its not in your STATA, then let's set it
fvset base 68 year
reg hours i.year, base
* So now that 68 is the base, you'll notice the coefficient on that year is 0, b* ut that means then that the constant - in this regression - should be the aver* age hours for the year 68. Check that that's the case!
* Now that you've done that, let's move on to the coefficient on 69: .597126
* This means that the year 69 is .597126 "away" from the base year 68.
* Let's test that let's add up the constant to that coefficient
fvset base 68 year
reg hours i.year, base
display _b[_cons]+_b[69.year]
. 37.947197
sum hours if year==69
Variable | Obs Mean Std. dev. Min Max
hours | 1 37.9472 . 37.9472 37.9472
* This works! This means that the coefficient on 69 is capturing the deviation from year 68, in year 69.
* Do the same for the rest of the coefficients.
* Now let's plot this coefficients to see if we can detect a pattern.
* install coefplot if you don't have it installed
* you can take out the scheme(plotplainblind) if you want.
reg hours i.year, base
coefplot, keep(*.year) vertical recast(connected) xlabel(,angle(45)) ylabel(,angle(horizontal)) scheme(plotplainblind)
* SO this tell us how hours worked has evolved over the years, and it seems that there is a big "bump" in year 69 (relative to 68) and then decreased. Let's now plot the raw data and see how it looks.
twoway (connected hours year, sort), scheme(plotplainblind)
You can notice how the fixed effects are capturing the averages changes from a baseline.
Now let’s practice a different approach of understanding FE with “individual” fixed effects.
webuse nlswork
* Let's declare the panel
xtset id year
estimates clear
eststo: reg hours i.year
coefplot est1 , keep(*.year) vertical recast(connected) xlabel(,angle(45)) ylabel(,angle(horizontal)) scheme(plotplainblind)
This tell us that hours worked has decreases since 69, after a bump from 68 to 69. Now we want to compare this trend to how the trend would look if we were to include individual level fe. The idea is
that individual level FE would be capturing time-invariant characteristics of the individual (such as race, birth year, etc), and therefore could be capturing bias or selection into the patterns we
see over time. So if the FE matter we would see something different from the line above. If they don’t then we would see the lines being the same. Let’ see if they do or don’t
webuse nlswork, clear
xtset id year
estimates clear
eststo: reg hours i.year
eststo: xtreg hours i.year, fe
coefplot est1 est2, keep(*.year) vertical recast(connected) xlabel(,angle(45)) ylabel(,angle(horizontal)) scheme(plotplainblind) legend(order(1 "Year FE" 2 "Individual FE"))
Not surprisingly, they matter; we can interpret the line from “est 2” as the changes over time in hours worked once we’ve accounted for individual time-invariant characteristics. Why is it different
from the line with just year fe? (est1). This could be for many reasons. For example, it could imply that the composition of who is reporting the hours worked may change in a way that correlates with
individual-level characteristics. Look at the drop in hours worked from 77 to 83. It is more pronounced once we account for individual-level FE than year FE. This means that within individuals, we
see large changes in hours relative to just looking at the “Year FE” line. Why is this?
Imagine that “who” is in our data changes over time. Say, for example, that over time there are fewer white individuals, and white individuals work fewer hours on average than other races, then this
could be confounding our relationship over time because as white individuals “leave” the sample, the average hours worked will be higher as - on average- these individuals work fewer hours, but if we
look at changes “within” individuals, we may actually notice that on average individuals are working fewer hours. So the compositional effect may be pushing things “upward” as we see in est1 vs. est2
for years 77,78, etc.
Let’s see if this is right. First, let’s see how hours worked varies by race:
webuse nlswork
collapse hours, by(year race)
twoway (connected hours year if race==1, sort) (connected hours year if race==2, sort) (connected hours year if race==3, sort), scheme(plotplainblind) legend(order(1 "White" 2 "Black" 3 "Other"))
Ok, this does seem to point at the fact that white individuals work, on average fewer hours than black or “other race” individuals. Now let’s see how many of them are in our sample over time.
webuse nlswork, clear
tab race, gen(racedummy)
collapse racedummy*, by(year)
twoway (connected racedummy1 year, sort) (connected racedummy2 year, sort) (connected racedummy3 year, sort), scheme(plotplainblind) legend(order(1 "White" 2 "Black" 3 "Other"))
Ok maybe there are some different patterns, but hard to see. We can do better by using year FE to see how composition changes over time. We are interested in knowing if there are “less” white
individuals over time, so we can do the following
webuse nlswork, clear
tab race, gen(racedummy)
reg racedummy1 i.year
coefplot, keep(*.year) vertical recast(connected) xlabel(,angle(45)) ylabel(,angle(horizontal)) scheme(plotplainblind)
This graph shows our point much more clearly. Over the years 75, 75, etc we see the composition changing and fewer whites. This means that hours worked should go up in the year-fe model. You can
follow the rest of the patterns.
If this theory were true, if we were to control for “race,” then the line with “year FE + race dummies” should be “closer” to the “individual FE” line than then just “year FE” line. Let’s test that
webuse nlswork, clear
xtset id year
estimates clear
eststo: reg hours i.year
eststo: xtreg hours i.year, fe
eststo: xtreg hours i.year i.race
coefplot est1 est2 est3, keep(*.year) vertical recast(connected) xlabel(,angle(45)) ylabel(,angle(horizontal)) scheme(plotplainblind)
This seems to pan out. The line with individual FE “est2” is on the bottom. Adding race has made us closer to the individual FE model, but not there quite yet. That means that individual-level FE is
still capturing other potential confounders, but controlling for race (est3) gets us closer to this line. | {"url":"https://sebastiantellotrillo.com/classes/econthaki/worksheet-on-fixed-effects","timestamp":"2024-11-10T12:25:05Z","content_type":"text/html","content_length":"212996","record_id":"<urn:uuid:b0e47f7b-d7a9-437f-9629-dc1adeafe846>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00222.warc.gz"} |
, And The United States Junior High School And Learn Mathematics Curriculum Standard Comparative Study
Posted on:2004-10-24 Degree:Master Type:Thesis
Country:China Candidate:C M Liu Full Text:PDF
GTID:2207360092986905 Subject:Curriculum and pedagogy
By making a comparative study of mathematics curriculum standards of junior secondary school between Chinese and American, the thesis rethinks some problems and put forward some suggestions, so as to
offer some references for our country's junior secondary school mathematics curriculum reform, In research method, the paper uses literature, comparison and case.Firstly, the thesis reviews the
evolutions since the 1950s of the reforms of basic education's mathematics curriculum of China and America in chronological order.Secondly, the thesis horizontally compares and analyzes the
differences, advantages and disadvantages between the two country's mathematics curriculum syllabus from three fields, namely, "Space and Shape", "Number and Algebra" and "Data analysis and
Probability" taking the content standards of junior secondary school of two country's mathematics curriculum standards for example, and then put forward some suggestions and measures for our
country's mathematics curriculum reform.At last, based on constructing the mathematics curriculum system of our country's basic education with Chinese characteristic, the paper rethinks some problems
and put forward some suggestions for our country's mathematics curriculum reform.The thesis' fruits are centered to two aspects, namely, practical level and theoretical level. In practical level, the
paper puts forward some proposes, such as "using dynamic geometry software", "extending curriculum's extent", "integrating visual geometry", "using pattern in algebraic teaching", "popularizing shape
calculator", "turning teacher's teaching concept", and so on. In theoretical level, the paper puts forward "one base, two bases, four emphases" for our country's mathematics curriculum reform.
Namely, the reform should be based on dialectical materialism, carrying forward the fine traditions of "two bases" and "basic ability", paying attention to students' emotion and discovering process,
integration of technology, the contact between mathematics study and life a, pluralistic evaluate.
Keywords/Search Tags: mathematics curriculum reform, content standards, integration of technology, "two basis", mathematics teacher | {"url":"https://www.globethesis.com/?t=2207360092986905","timestamp":"2024-11-07T12:13:12Z","content_type":"application/xhtml+xml","content_length":"8444","record_id":"<urn:uuid:be207a72-1e2c-422c-8615-44a9c204bdc2>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00664.warc.gz"} |
College Math
Easy Ways To Get Correct College Maths Homework Answers
Math homework can trouble most of us. When we are dealing with college level math, it is safe to assume that the math level will be advanced and not basic. There are some real complete tricks that
you will be looking at and some of these tricks need to be prepared for what lies ahead. In order to know what is remaining on the table, there are also a few advanced math concepts that you will
have to digest well. The real catch is in determining your own merit in the subject.
There are people who believe math is becoming an increasingly difficult language to tackle and there are those that believe that they are better off without it in the first place. But once you are
faced with the subject at say college level, there is really limited scope for you. That makes it essential to delve deep in the subject and get a thorough understanding at first. Here are some
tricks that will help you get the correct answers to problems.
Know that college level math is different
There is a difference between the math we do in school and the math we do in college. There are people tat believe college level math is only an extension of the math problems we solve at school. One
of the first issues with the problem is that there are no real threats that you will need to address here. This is again a very welcome concept that you can explore.
But college and university math is a different level altogether. Mostly, students who do not realize this are the ones that suffer the most from it. This is why you should have a clear understanding
of the subject you are leaning toward.
Can online help work for you?
When solving homework on math, a growing population is looking to have the answers checked at the local level. There is a need to make the most of available resources and you will be delighted to
know that most of these resources are available online. But even here, you will have to be very cautious if you wish to get the best out of the work you do.
It is in the favor of the student to solve the math problems first. You may then consult an online dictionary and debate and then look into the whole issue with a different angle altogether. This
will help you a lot.
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The Evolution of Signature Algorithms | Curity
The Evolution of Signature Algorithms
Public key cryptography deals with finding mathematical operations for a key pair that are easy to calculate with part of the key pair but hard to reverse without. Such an operation is called the
trapdoor function.
Signature Algorithms use the private key from the key pair and a message in some form as an input to the trapdoor function. The outcome is the signature. Normally, the message passes a hashing
function before it gets signed. The signature is easy to calculate using the private key and easy to validate with the public key. Thus everybody with the public key can validate a signed message,
verify that it comes from the expected sender (verify the message authenticity) and verify that it was not tampered with (verify the message integrity).
Because of the underlying mathematical operation, it is practically infeasible to create a valid signature without the private key. Further, the private key cannot be recalculated from the public key
or any other public information of the system, such as the message, hash value, or signature. However, if used in the wrong context, it is possible to retrieve the private key from signatures or even
create valid signatures without it despite the trapdoor function. To avoid such pitfalls and security incidents, it's essential to familiarize yourself with the different algorithms.
Not many people have had such a significant impact on modern life as Rivest, Shamir, and Adleman. In 1977, these researchers published the description of a public-key algorithm that became known as
the RSA algorithm. Even now, about 45 years later, RSA is still widely used.
It's impressive that RSA is still considered relatively safe after so many years. Yet, as great as RSA is, it has its limitations. Over the years, researchers, cryptographers, and attackers have
found vulnerabilities and impracticalities in the algorithm (mainly concerning RSA encryption and not the signature). Therefore, it's best practice not to use textbook RSA but rather
RSASSA-PKCS1-v1_5 (RS256, RS384, or RS512) or RSASSA-PSS (PS256, PS384, or PS512) for your signatures. These are the versions of RSA that we use in the Curity Identity Server.
The underlying trapdoor function in RSA relies on the factoring problem. There is no efficient algorithm known that breaks an integer in its prime factors. Finding the prime factors becomes
especially hard when the integer is composed of two large prime numbers (as is the case in all variants of RSA). However, with processing power getting stronger and cheaper, attacks on the factoring
problem are becoming more concerning.
To mitigate the risk of cracked keys, we simply increase the size of the RSA key. This way, it becomes harder and harder to guess the prime factors of the public key, which would further reveal the
corresponding private key. However, that strategy comes with a drawback and cost: considering that RSA signatures have the same size as the key, a system using large keys consumes more storage for
the keys. It also consumes greater processing power for calculations and bandwidth for transporting the messages.
Practically speaking, we cannot continue increasing the key size indefinitely. This is especially relevant for systems like small Internet of Things devices with limited resources. They simply do not
have the capacity to increase the keys. At some point, the processing power will catch up, and RSA keys might be broken before they are invalidated. That is where elliptic curves come into play.
Elliptic Curves
With Elliptic Curve Cryptography (ECC), researchers have found algorithms for signing and encryption that work with small keys and are hard to break. The algorithms are based on a problem called the
Elliptic Curve Discrete Logarithm Problem (ECDLP). The trapdoor function in ECDSA is more secure than in RSA, but it makes the algorithm slower.
There is a range of different curves, that is, named sets of defined parameters and formulas. P-256 is one example, Curve25519 another. The security of elliptic curves depends on the characteristics,
the parameters, and the formulas of the definition. However, more often, security relies on the implementation details. For example, algorithms for digital signatures based on elliptic curves (ECDSA)
require a random, single-use nonce. Consequently, the nonce becomes a weak point. A bad random generator or the reuse of a nonce eventually allows an attacker to calculate the private key — a problem
that previously affected YubiKeys and Sony's PlayStation 3.
ECDSA is complex, and it's difficult to correctly implement elliptic curves. If done wrong, elliptic curves may result in a security risk rather than a benefit. For example, the ECDSA implementation
in Java had a serious flaw that allowed an attacker to easily forge signatures. In addition, vulnerabilities for side-channel attacks are a known problem in ECDSA. So make sure you select the right
curve and a secure implementation when using ECDSA. If in doubt, choose EdDSA if you can.
EdDSA is an evolved version of the elliptic curves Curve25519 and Curve448. A smart combination of parameters and formulas eliminates two main problems with (other) elliptic curve signatures: the
requirement for a random nonce and the vulnerability for side-channel attacks. Actually, the latter depends on the implementation and is theoretically possible.
Side-channel attacks are a category of attacks that don't target vulnerabilities of the algorithm itself but of its implementation by analyzing system characteristics such as power consumption,
electromagnetic leaks in caches or memory, or timing information. EdDSA uses complete formulas, which means that the rules apply to all points on the elliptic curve. As a result, no edge cases need
expensive parameter validation and exception handling. Consequently, EdDSA is easier to implement and less prone to side-channel attacks compared to other elliptic curve signature algorithms.
Nowadays, there are even complete formulas for Weierstrass curves, meaning that it's now easier to implement elliptic curve algorithms. However, the problem with the nonce in ECDSA remains. EdDSA is
not dependent on a random number generator for the nonce and is, therefore, more secure. In addition, EdDSA was designed with high performance in mind. Since the keys are small and the operations are
fast, EdDSA saves time, money, and resources. As a result, EdDSA is a green computing alternative that reduces the environmental impact of cryptography.
Although the industry now has this progressive technology, some software providers have been slow to adopt it. When switching to EdDSA, you may experience limited support from languages, libraries,
and frameworks. However, the Curity Identity Server does not contain such an obstacle. Not only does the Curity Identity Server support EdDSA out of the box, but Curity also provides resources to
help you along the way. The right choice should be easy. Choose EdDSA and make a difference for the security and future of your system!
If you want to learn more about signature algorithms and why EdDSA is the optimal choice for securing tokens, watch our webinar - An Engineer's Guide to Signature Algorithms and EdDSA. It is
available on-demand. | {"url":"https://curity.io/blog/the-evolution-of-signature-algorithms/","timestamp":"2024-11-14T08:15:54Z","content_type":"text/html","content_length":"601302","record_id":"<urn:uuid:abb562dc-4918-49c8-9fe4-4729efffa6a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00542.warc.gz"} |
Library Stdlib.Numbers.NatInt.NZDomain
In this file, we investigate the shape of domains satisfying the NZDomainSig interface. In particular, we define a translation from Peano numbers nat into NZ.
Relationship between points thanks to succ and pred.
For any two points, one is an iterated successor of the other.
Generalized version of pred_succ when iterating
From a given point, all others are iterated successors or iterated predecessors.
In particular, all points are either iterated successors of 0 or iterated predecessors of 0 (or both).
First case: let's assume such an initial point exists (i.e. S isn't surjective)...
... then we have unicity of this initial point.
... then all other points are descendant of it.
NB : We would like to have
pred n == n
for the initial element, but nothing forces that. For instance we can have -3 as initial point, and P(-3) = 2. A bit odd indeed, but legal according to
. We can hence have
n == (P^k) m
exists k', m == (S^k') n
We need decidability of
(or classical reasoning) for this:
Second case : let's suppose now S surjective, i.e. no initial point.
To summarize:
S is always injective, P is always surjective (thanks to
I) If S is not surjective, we have an initial point, which is unique. This bottom is below zero: we have N shifted (or not) to the left. P cannot be injective: P init = P (S (P init)). (P init) can
be arbitrary.
II) If S is surjective, we have
forall n, S (P n) = n
, S and P are bijective and reciprocal.
IIa) if
exists k<>O, 0 == S^k 0
, then we have a cyclic structure Z/nZ IIb) otherwise, we have Z
An alternative induction principle using S and P.
It is weaker than
. For instance it cannot prove that we can go from one point by many
S or
, but only by many
mixed with many
. Think of a model with two copies of N:
0, 1=S 0, 2=S 1, ... 0', 1'=S 0', 2'=S 1', ...
and P 0 = 0' and P 0' = 0.
We now focus on the translation from nat into NZ. First, relationship with 0, succ, pred.
Since P 0 can be anything in NZ (either -1, 0, or even other numbers, we cannot state previous lemma for n=O.
If we require in addition a strict order on NZ, we can prove that ofnat is injective, and hence that NZ is infinite (i.e. we ban Z/nZ models)
For basic operations, we can prove correspondence with their counterpart in nat. | {"url":"https://coq.inria.fr/doc/master/stdlib/Stdlib.Numbers.NatInt.NZDomain.html","timestamp":"2024-11-14T04:23:14Z","content_type":"application/xhtml+xml","content_length":"81557","record_id":"<urn:uuid:f8456680-f892-467c-838c-7a2ecc4b2cff>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00513.warc.gz"} |
Addition and Subtraction to 10 in Kindergarten Digital Lessons Mega Bundle | TeachBuySell
Addition and Subtraction to 10 in Kindergarten Digital Lessons Mega Bundle
Resources in this bundle (4)
This mega bundle resource is perfect for teaching Kindergarten students about beginning addition and subtraction within 10. Students will engage in explicit PowerPoint lessons that are interactive
and fun! Perfect to use in Math centres and whole class teaching! Includes 40 assessment opportunities throughout the PowerPoints.
What do you get?
4 Digital PowerPoint Lessons outlined below:
1. Addition within 1-5
• 150 interactive PowerPoint slides that focus on addition to 5.
• 11 learning intentions:
1. I can combine two groups of objects to model addition to 5.
2. I can use a number line to solve addition problems to 5.
3. I can use fingers to solve addition problems to 5.
4. I can use manipulatives to solve addition problems to 5.
5. I can count on to solve addition problems to 5.
6. I can draw pictures to solve addition problems to 5.
7. I can use a five frame to solve addition problems to 5.
8. I can use dice to solve addition problems to 5.
9. I can use a domino to solve addition problems to 5.
10. I can decompose numbers by taking a whole number and breaking it into two smaller parts to solve addition problems to 5.
11. I can choose a strategy to solve addition stories to 5.
2. Addition within 6-10
• 150 interactive PowerPoint slides that focus on addition to 10.
• 11 learning intentions:
1. I can combine two groups of objects to model addition to 10.
2. I can use a number line to solve addition problems to 10.
3. I can use fingers to solve addition problems to 10.
4. I can use manipulatives to solve addition problems to 10.
5. I can count on to solve addition problems to 10.
6. I can draw pictures to solve addition problems to 10.
7. I can use a five frame to solve addition problems to 10.
8. I can use dice to solve addition problems to 10.
9. I can use a domino to solve addition problems to 10.
10. I can decompose numbers by taking a whole number and breaking it into two smaller parts to solve addition problems to 10.
11. I can choose a strategy to solve addition stories to 10.
3. Subtraction within 1-5
• 130 interactive PowerPoint slides that focus on addition to 5.
• 9 learning intentions:
1. I can use a number line to solve subtraction problems to 5.
2. I can use fingers to solve subtraction problems to 5.
3. I can use manipulatives to solve subtraction problems to 5.
4. I can count back to solve subtraction problems to 5.
5. I can draw pictures to solve subtraction problems to 5.
6. I can use a five frame to solve subtraction problems to 5.
7. I can solve subtraction stories to 5.
8. I can match the subtraction sentence to the correct answer.
9. I can match the correct answer to the subtraction sentence.
4. Subtraction within 6-10
• 130 interactive PowerPoint slides that focus on addition to 10.
• 9 learning intentions:
1. I can use a number line to solve subtraction problems to 10.
2. I can use fingers to solve subtraction problems to 10.
3. I can use manipulatives to solve subtraction problems to 10.
4. I can count back to solve subtraction problems to 10.
5. I can draw pictures to solve subtraction problems to 10.
6. I can use a ten frame to solve subtraction problems to 10.
7. I can solve subtraction stories to 10.
8. I can match the subtraction sentence to the correct answer.
9. I can match the correct answer to the subtraction sentence.
• Ready to download and teach today in your classroom!
• No prep student centred activities.
• Activities that can be used as a whole class, math centre, small group and individual task or assessment.
• Fun and engaging lessons that you and your students will look forward to!
Why do you need it?
• Let's face it, teaching can be so draining and we're so time poor already! So take back your weekends, school nights, holidays and free time and let me prepare engaging, fun, no prep, ready to
teach lessons that you can use in your classroom today!
How can you use the PowerPoints?
• Whole class Math lessons
• Daily number sense
• Math groups
• Independent Math group activity
• Math warm up activities
• Assessment activities
• Brain break activities
• Can be used on Seesaw or Google Classroom/Slides
Please reach out if you have any questions! I'm so happy you're here!
Miss Mazie
Curriculum alignment details
This resource is intended for the following use:
Australian Curriculum
Content Descriptors:
Not specified
Further context or application:
Not specified
Addition and Subtraction to 10 in Kindergarten Digital Lessons Mega Bundle
Ratings & reviews
Failed to retrieve the reviews. Please try again. | {"url":"https://teachbuysell.com.au/l/66376e3e-76cd-4288-80df-a890be38a0aa","timestamp":"2024-11-08T08:37:52Z","content_type":"text/html","content_length":"541586","record_id":"<urn:uuid:575bcb54-73d9-4844-ac8a-2cf899f41dd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00088.warc.gz"} |
CS111 Assignment 5 solved
## Overview
For this assignment, you will complete searching/sorting tasks and efficiency analysis. No code is to be written for this assignment.
Write your answers in the file assign5.txt.
## Problem 1
1. Trace selection sort on the following array of letters (sort into alphabetical order):
M U E J R Q X B
After each pass (outer loop iteration) of selection sort, show the contents of the array and the number of letter-to-letter comparisons
performed on that pass (an exact number, not big-O).
2. Trace insertion sort on the following array of letters (sort into alphabetical order):
M U E J R Q X B
After each pass (outer loop iteration) of insertion sort, show the contents of the array and the number of letter-to-letter comparisons
performed on that pass (an exact number, not big-O).
## Problem 2
For each problems segment given below, do the following:
1. Create an algorithm to solve the problem
2. Identify the factors that would influence the running time, and which can be known before the algorithm or code is executed.
Assign names (such as n) to each factor.
3. Identify the operations that must be counted. You need not count every statement separately. If a group of statements always
executes together, treat the group as a single unit. If a method is called, and you do not know the running time of that method,
count it as a single operation.
4. Count the operations performed by the algorithm or code. Express the count as a function of the factors you identified in Step 2.
If the count cannot be expressed as a simple function of those factors, define the bounds that can be placed on the count: the best
case (lower bound) and worst case (upper bound).
5. Determine what the Best Case Inputs are, and the Worst Case Inputs are, and the efficiency of your implementation
6. Transform your count formula into big-O notation by:
– Taking the efficiency with worst case input,
– Dropping insignificant terms.
– Dropping constant coefficients.
Do Problem 2 for each of these scenarios.
a. Determine if 2 arrays contain the same elements
b. Counting total number characters that have a duplicate within a string (i.e. “gigi the gato” would result in 7 (g x 3 + i x 2 + t x 2)
c. Finding an empty row in a 2-D array where empty is defined as an element with a 0 entry. | {"url":"https://codeshive.com/questions-and-answers/cs111-assignment-5-solved/","timestamp":"2024-11-05T03:33:58Z","content_type":"text/html","content_length":"99439","record_id":"<urn:uuid:7666c337-7ba0-48c2-9693-d88de9212c15>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00364.warc.gz"} |