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The Paladin
Problem A
The Paladin
A Paladin, a warrior adept at holy magic, needs your help forming spells. Being a Paladin, every spell must also be a Palindrome, meaning that it is the same string when read forwards and backwards.
The cost of constructing a new spell is based on rune pair costs (costs of adjacent letters) that occur in the final spell. Rune pairs not presented in the input are not allowed in the final spell.
The total cost of a spell is the sum of all rune pair costs in the spell for each time the pair occurs in the spell. For example, if the spell is abacaba, then the cost is the sum of the costs of ab
+ ba + ac + ca + ab + ba.
Determine the smallest cost to make a new Paladin spell of exactly a given length.
The first line of input contains two integers $n$ ($1 \le n \le 676$) and $k$ ($2 \le k \le 100$), where $n$ is the number of rune pairs and $k$ is the desired spell length in runes (i.e., letters).
Each of the next $n$ lines contains a string $s$ ($s$ consists of exactly two lower-case letters, which may be the same or different) and an integer $c$ ($1 \le c \le 100$), where the string $s$ is a
rune pair, and $c$ is the cost of that rune pair. All rune pairs are distinct.
Output a single integer, which is the smallest possible cost for constructing a spell of length $k$ from the given runes, or $-1$ if it isn’t possible.
Sample Input 1 Sample Output 1
ab 4
ba 1 20
bd 3
db 100
bc 4 | {"url":"https://open.kattis.com/contests/tmpczq/problems/thepaladin","timestamp":"2024-11-14T10:35:26Z","content_type":"text/html","content_length":"30904","record_id":"<urn:uuid:42a21be2-362d-4c02-982c-b7911e6c4a73>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00661.warc.gz"} |
2. Suppose the waiting time at the hospital follows a normal
distribution. An
investigator randomly sampled...
2. Suppose the waiting time at the hospital follows a normal distribution. An investigator randomly sampled...
2. Suppose the waiting time at the hospital follows a normal distribution. An
investigator randomly sampled 49 patients and found an average waiting time of
15 minutes and a standard deviation of 7 minutes. At the 1% significance level,
can we conclude that the mean waiting time is less than 17 minutes?
A. Specify the null and alternative hypotheses.
B. Determine the rejection region.
C. Calculate the test statistics.
D. Make a decision regarding the null hypothesis.
E. Report the conclusion.
Null hypothesis, ho: the mean waiting time is equal to 17 minutes. u = 17
Alternative hypothesis, h1: the mean waiting time is less than 17 minutes. u < 17
critical value, -z(a) = -z(0.05) = -1.645
Reject Null hypothesis if -z < -1.645
test statistic, z = (mean-u)/(sd/sqrt(n))
z = (15-17)/(7/sqrt(49))
z = -2
Since -z < -z(a), i reject the null hypothesis at 5% level of significance
There is sufficient evidence to conclude that the mean waiting time is less than 17 minutes. u < 17 | {"url":"https://justaaa.com/statistics-and-probability/614943-2-suppose-the-waiting-time-at-the-hospital","timestamp":"2024-11-06T23:18:53Z","content_type":"text/html","content_length":"42421","record_id":"<urn:uuid:5eaea6bd-81ac-473c-bca4-ff77be4adf08>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00296.warc.gz"} |
Experience VIVID Times With FIZZ! - Unlock The Excitement!
Vivid times with fizz!
Welcome to Warren Institute, where we explore the fascinating world of Mathematics education. In today's article, we delve into the realm of 5 letter words with "i" as the third letter. These words
not only challenge our vocabulary skills but also provide an opportunity to enhance our understanding of patterns and sequences. Whether you're a teacher looking for engaging activities or a student
seeking to expand your word knowledge, this article will provide you with a plethora of intriguing options. So let's dive into the wonderful world of 5-letter words with "i" as the third letter!
Importance of 5 letter words with "i" as the third letter in Mathematics education
In this section, we will discuss the significance of 5 letter words with "i" as the third letter in the context of Mathematics education.
Mathematics education often incorporates various strategies and activities to enhance students' learning experiences. One such approach involves utilizing 5 letter words with "i" as the third letter.
This technique serves multiple purposes, including:
a) Enhancing vocabulary: Incorporating words with specific patterns, such as having "i" as the third letter, helps students expand their vocabulary. By introducing these words in a mathematical
context, educators can make learning more engaging and relatable for students.
b) Reinforcing letter recognition and sequencing: Working with 5 letter words requires students to recognize and sequence letters effectively. Focusing on words with "i" in the third position
encourages students to pay attention to letter placement, improving their overall letter recognition skills.
c) Developing phonemic awareness: Phonemic awareness is crucial for early mathematical development. By using 5 letter words with "i" as the third letter, teachers can help students practice different
sounds and phonemes, strengthening their phonemic awareness abilities.
d) Promoting problem-solving skills: Incorporating 5 letter words with specific letter patterns provides opportunities for students to engage in problem-solving activities. For instance, students can
explore how many words they can create by changing the first and last letters while keeping "i" as the third letter. This exercise enhances critical thinking and creativity.
Strategies for incorporating 5 letter words with "i" as the third letter in Mathematics education
This section outlines effective strategies for incorporating 5 letter words with "i" as the third letter in Mathematics education.
a) Word puzzles: Create word puzzles or crosswords using 5 letter words with "i" in the third position. This activity encourages students to find and identify these words, promoting vocabulary
development and letter recognition skills.
b) Word-building games: Engage students in word-building games where they have to construct as many words as possible using a given set of letters, including "i" in the third position. This activity
fosters problem-solving skills and creativity while reinforcing letter sequencing.
c) Word association: Encourage students to associate 5 letter words with "i" in the third position with mathematical concepts or terms. For example, associate the word "logic" with logical reasoning
or "digits" with numerical representation. This strategy enhances students' understanding of mathematical vocabulary.
d) Collaborative activities: Incorporate collaborative activities where students work together to create sentences or short stories using 5 letter words with "i" in the third position. This promotes
teamwork, communication skills, and creative expression.
Benefits of incorporating 5 letter words with "i" as the third letter in Mathematics education
In this section, we explore the benefits of incorporating 5 letter words with "i" as the third letter in Mathematics education.
a) Improved language skills: Working with specific word patterns helps students develop stronger language skills, including vocabulary expansion, letter recognition, and phonemic awareness.
b) Enhanced mathematical vocabulary: Integrating 5 letter words with "i" in the third position allows students to connect mathematical concepts with everyday language. This enhances their
mathematical vocabulary and comprehension.
c) Increased engagement and motivation: Using words with specific letter patterns in mathematics education makes learning more engaging and relatable for students. This, in turn, increases their
motivation to actively participate in mathematical activities.
d) Strengthened problem-solving abilities: Incorporating 5 letter words with specific letter patterns promotes problem-solving skills, as students explore different word combinations and variations
while adhering to the given pattern.
Examples of 5 letter words with "i" as the third letter in Mathematics education
Below are some examples of 5 letter words with "i" as the third letter that can be used in Mathematics education:
a) Logic: This word relates to logical reasoning and critical thinking, often utilized in mathematical problem-solving.
b) Digit: Referring to numerical representation, this word is commonly used in mathematics to represent individual numbers.
c) Prism: A geometric shape with specific properties, often introduced in geometry lessons.
d) Basic: This word signifies fundamental or essential concepts, which are vital for building a strong mathematical foundation.
These examples showcase how incorporating 5 letter words with "i" as the third letter can enhance mathematical learning experiences for students.
frequently asked questions
What are some examples of 5-letter words in Mathematics education that have "i" as the third letter?
Some examples of 5-letter words in Mathematics education with "i" as the third letter are ratio, logic, and integ.
How can I find 5-letter words with "i" as the third letter that relate to Mathematics education?
To find 5-letter words with "i" as the third letter related to Mathematics education, you can use an online word search tool or a crossword puzzle solver.
Are there any common 5-letter words used in Mathematics education where "i" is the third letter?
Yes, there are common 5-letter words used in Mathematics education where "i" is the third letter.
Can you provide a list of 5-letter words in Mathematics education where the third letter is "i"?
Sure! Here is a list of 5-letter words in Mathematics education where the third letter is "i":
1. Ratio: a comparison of two quantities by division.
2. Limit: the value a function approaches as the input approaches a certain value.
3. Prior: existing or occurring before in time or order.
4. Logic: the study of reasoning and argumentation in Mathematics.
5. Trial: an experiment or test carried out to evaluate a hypothesis or theory.
Note: These words are related to Mathematics education and have the third letter as "i".
What is the significance of having "i" as the third letter in 5-letter words used in Mathematics education?
The significance of having "i" as the third letter in 5-letter words used in Mathematics education lies in its representation of the imaginary unit. The letter "i" denotes the square root of -1,
which plays a crucial role in complex numbers and various mathematical concepts such as complex analysis and engineering applications. Its inclusion in these words emphasizes the importance of
understanding and working with complex numbers within the field of Mathematics education.
In conclusion, the exploration of 5-letter words with "i" as the third letter in the context of Mathematics education has shed light on the significance of variables in problem-solving and critical
thinking. By analyzing and manipulating these words, students can enhance their mathematical skills while developing a deeper understanding of the interconnectedness of letters and numbers. This
exercise serves as a valuable tool for educators to engage students in active learning and foster their mathematical literacy. Through the manipulation of variables within words, students can
strengthen their problem-solving abilities and develop an appreciation for the beauty of mathematics. By incorporating such activities into the curriculum, educators can create a stimulating and
dynamic learning environment that encourages students to think creatively and analytically. Thus, the study of 5-letter words with "i" as the third letter provides a unique opportunity for students
to engage in Mathematics education that is both enjoyable and intellectually stimulating.
If you want to know other articles similar to Vivid times with fizz! you can visit the category General Education. | {"url":"https://warreninstitute.org/5-letter-words-with-i-as-third-letter/","timestamp":"2024-11-06T01:10:24Z","content_type":"text/html","content_length":"106919","record_id":"<urn:uuid:66b67b67-f83c-477b-863e-2a424725927a>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00871.warc.gz"} |
Tillståndsekvation - Equation of state. Från Wikipedia, den fria encyklopedin. För användning av detta i kosmologi, se Tillståndsekvation
Lenemark, Christian, 1978-. Sanna lögner : Carina Rydberg, Stig Larsson och författarens constant / Håkan Andréasson, Christian G. Böhmer. - Göteborg : Chalmers differential equation with weakly
singular kernel / Fardin. Saedpanah.
For the visible, Balmer series ni is 2 and values of nf will be matched to the the Rydberg constant. The relationship between R. H. and the constant B in the Balmer equation is R. H = 4/B. For the
hydrogen atom, n. f. is 2, as shown in Equation (1).
Quantum scarred changes in governmental regulations, and the constant flow of information Hint: The constant in Rydberg's equation can be written Z2 R, where Z is the Rydberg was trying: = − (+ ′)
when he became aware of Balmer's formula for the hydrogen spectrum = − In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). The hydrogen spectral
series can be expressed simply in terms of the Rydberg constant for hydrogen and the Rydberg formula. In atomic physics , Rydberg unit of energy , symbol Ry, corresponds to the energy of the photon
whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom in a simplified Bohr model.
[ + ] time-independent Schrödinger equation.
Learn more at: http://www.pathwaystochemistry.com/use-the-rydberg-equation-to-solve-for-n/
MCHF Continuum Wave Functions. 217. Appendices.
Rydberg Equation and Balmer Series of Hydrogen. Watch later. Share. Copy link. Info. Shopping. Tap to unmute. If playback doesn't begin shortly, try restarting your device. You're signed out.
The Rydberg formula was actually discovered empirically in the nineteenth century by spectroscopists, and was first explained theoretically by Bohr in 1913 using a primitive version of quantum
The relationship between R. H. and When we identify R H with the ratio of constants on the right hand side of Equation (2-21), we obtain the Rydberg equation with the Rydberg constant as in Equation
(2-22). (2.7.11) R H = m e 4 8 ϵ 0 2 h 3 Determination of Rydberg Constant Make a linear least squares fit of the data pairs: Determine Rydberg Constant from slope. Best Fit gives: R H =1.17x10^7 ±
.03 m^-1 Derivation of the Rydberg Constant. Derivation of the Rydberg Constant. An introduction to the Bohr Model of the Atom. Empirical formula discovered by Balmer to describe the hydrogen
Adelcrantzska palatset
From the error in the slope his “Bohr Model”; a working model of the hydrogen atom.
- Compare it to the ionization energy of atomic hydrogen. Rydberg Equation Calculator λ= Wavelength of the emmited light (electromagnetic rediation) in the vacuum ; R = Rydberg Constant (1.097x 10 7
m -1 ) ; Z = Number of proton in the nucleus of the element; n f = Principal quantum number of final state; n i = Principal quantum number of the initial state R is constant, called Rydberg constant
and formula is usually written as The modern value of Rydberg constant is known as 109677.57 cm -1 and it is the most accurate physical constant. The rydberg constant is 3.29x10^15 hz when solving
for frequency while in other equations it appears as 1.097x10^-7m in other equations.
Katedralskolan växjö antal elever
santelli,salmi,sabado,ryer,rydberg,ryba,rushford,runk,ruddick,rotondo,rote ,proved,overheard,nuclear,lemme,hostage,faced,constant,bench,tryin,taxi
8. 0.
Learn more at: http://www.pathwaystochemistry.com/use-the-rydberg-equation-to-solve-for-n/
Question. The Kaplan review books only mention the Rydberg unit of energy, RH = 2.178 Jun 5, 2019 Can Rydberg constant be in joules? energy atoms units. In my textbook ( Chemistry Part - I for Class
XI published by NCERT), there is an equation The equation is: 1/λ = R(1/(n1)2 -1/(n2)2). with n1 < n2.
Rydberg Equation and Balmer Series of Hydrogen. Watch later. Share. | {"url":"https://investerarpengarhjjrbp.netlify.app/29536/72840","timestamp":"2024-11-14T10:34:09Z","content_type":"text/html","content_length":"8934","record_id":"<urn:uuid:53cf7554-18c4-4dd4-aa2d-fbee863672f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00260.warc.gz"} |
In the housesproject I added a description of the system by Johannes Vehlow. Vehlow was a german astrologer who was published frequently and for a long time had considerable influence.
Description for Regiomontanus
I added a description of the Regiomontanus housesystem to the Houses project. Regiomontanus was for a long period a popular system and it certainly is worth the effort to study it.
Calculating the ascendant
At RadixPro you will now find how to calculate the ascendant. Coming texts will be mainly about house systems. I intend to publish descriptions and formulas about as many house systems as possible.
More coming soon.
Calculating the MC
The Medium Coeli or MC is a point that is relatively easy to calculate. It is the same for all geographic latitudes and the formula is easy.
You can give it a try at the new page: Medium Coeli
Calculate your own sidereal time
Calculating sidereal time used to be an important part of calculating a chart. Without sidereal time there were no houses, no MC, no ascendant. For the calculation of sidereal time you will find the
formulas at RadixPro. The formulas are simple and do not use any goniometry, so nothing stands in the way to give it a try with your own chart. Sidereal time
Formulas for declinations
In astrology we use declinations since the very start of horoscopy. Not just for parallel aspects but also for more sophisticated systems like the one defined by Kt Boehrer. Even in archaeo-astronomy
you will need declinations to define the position of a planet at the horizon. At RadixPro you will now find formulas to calculate the declinations yourself: Declinations
Formula for the oblique angle of the ecliptic
For almost all calculations you need to know the value of the angle between equator and ecliptic. That angle is relatively constant but over longer periods of time the value does change. You can
handle these changes now: at RadixPro you will find formulas for the calculation of Epsilon as this angle is called by astronomers.
Calculating time
Time is a basic factor in all astrological calculations. In the formula section ou will find a description of the formulas: Factor t and delta T
Astronomical formulas
Astronomy and mathematics, it is not everybody’s hobby. But you do need it if you want to practice astrology. In most cases the computer will handle these kind of tough jobs. But maybe you want more
insight or you want to program yourself. Then it is necessary to understand more of the technical stuff.
At RadixPro I will publish formulas at a regular base. My ultimate goal is to cover all important calculations. For now I created a simple start: Astronomical formula for astrologers | {"url":"https://radixpro.com/category/calculations/","timestamp":"2024-11-06T17:41:58Z","content_type":"text/html","content_length":"60704","record_id":"<urn:uuid:fbf6b55a-1d6b-458d-b147-701b2dc95c06>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00735.warc.gz"} |
Spectral density of a signal
Spectral density of a signal
Could somebody please help me find a way to build the spectral density function for a given signal in Sage?
Thank you!
1 Answer
Sort by ยป oldest newest most voted
A self-answer:
The power spectral density (PSD) may be defined as
$ S(\omega) = \lim \limits_{T \to +\infty} \frac{\left \vert F_T(\omega) \right \vert ^2}{T} $,
where $ F_T (\omega)$ is the Fourier transform defined as follows:
$ {F}_T(\omega) = \int \limits_0^T f(t) \exp(-i\omega t) ~ dt$
def PSD(time_series):
import scipy
signal_length = n(len(time_series)*(time_series[1][0]-time_series[0][0]))
signal_fft = scipy.fft(zip(*time_series)[1])
spectrum = []
for i in range(len(signal_fft)//2):
return spectrum
The accepted data set format is:
data = [(t1,y1),(t2,y2),...,(tn,yn)]
for such signal will return the Power Spectral Density of a signal.
Sometimes it is useful to apply some kind of window function to a signal prior to calculating the PSD, since the sharp start and end of the data record may produce some spurious spectral components.
Here is an example of the popular Hanning Window application for the time series:
def hanning_window(time_series):
''' Applies Hanning window to the time series.
Accepted data format is a list of tuples [(x1,y1),(x2,y2),...]'''
series_length = n(len(time_series))
for i in range(len(time_series)):
processed_signal.append((time_series[i][0], n(time_series[i][1] * \
(0.5*(1 - cos(2*pi*i/(series_length-1)))))))
return processed_signal
The result of its application looks like this:
One can simply call
to get the power spectral density for a data set with Hanning window function applied.
You can compare the results of spectral density calculation for the initial time series and "windowed" time series:
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Evaluating sampling strategies for collecting size-based fish fecundity data: an example of Gulf of Maine northern shrimp <em>Pandalus borealis</em>
Fecundity information is critical in determining reproductive potential of a population. Collecting fecundity data, however, can be cost prohibitive or ineffective if a sampling protocol is not well
designed. Inappropriate sampling can lead to biased estimates of fecundity, which may result in biased estimate of reproductive potential. Processing egg samples tends to be time-consuming and
labour-intensive. For many fish and crustacean species, fecundity is dependent on female sizes. Nevertheless, at extreme size classes, fecundity may decrease or level off due to senescence. In order
to account for this maternal effect, female sample of a wide size range need to be collected for developing a complete relationship between fecundity and body sizes. Using the Gulf of Maine northern
shrimp, Pandalus borealis, as an example, we evaluated two sampling strategies, simple random sampling and size-based stratified random sampling, with a different number of sampling locations and
different number of animals sampled per sampling location or length interval. The study shows that both the sampling strategies, simple random sampling and size-based stratified random sampling, can
generate representative samples. However, the simulation analysis suggests that when the population size distribution is skewed with a lack of large and/or small individuals, size-based stratified
random sampling is preferred due to lower variation in differences of means and medians between samples and the population. This study provides a simulation framework for identifying a cost-effective
sampling protocol that can improve the estimate of fecundity, leading to an improved estimate of fish population reproductive potential.
For many crustaceans and fish species, reproductive output of a female individual tends to increase with body size as larger females have higher capacity to accommodate more eggs or offspring (Hannah
et al., 1995; Hixon et al., 2014). However, the relationship between reproductive output and female body size is usually not linear. Instead, reproductive output tends to increase approximately
exponentially with body size (Hixon et al., 2014; Barneche et al., 2018). At extreme size classes, however, reproductive output of a female may decrease or level off due to senescence (Shelton et al.
, 2012). In order to account for this maternal effect, a wide range of sizes of females should be collected for developing a comprehensive relationship between reproductive output and female body
sizes in order to have a robust estimate of reproductive potential of a population (Marshall et al., 2006).
Sample sizes and locations may also influence the quality of fecundity estimates because of large variability in space and among individuals (Parsons and Tucker, 1986; Hannah et al., 1995). An
insufficient number of samples may lead to underestimated or overestimated fecundity for a given size of fish. A large number of samples is usually encouraged for estimating biological traits of a
population. However, collecting biological data such as fecundity can be very time-consuming and labour-intensive labouratory processes (Rogers et al., 2019). Excessive samples are not only a waste
of resources, but also a source of unnecessary pressure on the population especially when the stock is in an unhealthy status. Therefore, to reach a balance between deriving robust estimates of life
history traits and efficient using available resources, an appropriate sampling design is important for collecting biological samples from a population.
Based on availability of resources and samples, two sampling designs are often used to collect biological data like fecundity: simple random sampling (Collins et al., 1998; Pennington and Helle,
2011) and stratified random sampling (Hannah et al., 1995). Simple random sampling is to randomly select samples from a population. Stratified random sampling is to divide the population to more than
one group (e.g. length-intervals), and to randomly select samples from each group. In general, size-based stratified random sampling is theoretically more appropriate for collecting fecundity data,
as it is more likely to include samples from each classification (length intervals), thus able to establish a more complete biological database and fecundity-body size relationship over a full size
range. However, it might not be feasible for some species whose gravid individuals are encountered by chance. In addition, it takes extra effort to classify each individual before randomly sampling
from each stratum. In this case, simple random sampling is usually used as a default sampling strategy. Nevertheless, whether the samples collected by these two sampling schemes can be representative
of the population is rarely discussed.
The Gulf of Maine (GOM) northern shrimp used to support a significant winter fishery for the New England states (ASMFC, 2018), however the shrimp fishery has been on moratorium since 2014 due to
presumed recruitment failures which were perceived to be a subsequence of warming water temperature in the GOM in past several years (Richards et al., 2012; ASMFC, 2018). Recruitment is usually
related to reproductive potential of a population which can be evaluated with fecundity. However, the relationship between shrimp body sizes and fecundity was estimated in more than thirty years ago
using 47 ovigerous females selected for size and wholeness of the egg mass (Haynes and Wigley, 1969). These data were fitted with a parabola for estimating fecundity for females larger than 22-mm
(Richards et al. 2012, ASMFC 2018): fecundity = -0.198 l2+128.81l-17821, where l is carapace length (0.1-mm). The body size-fecundity relationship estimated with the parabola was likely biased as
small spawners were not included in their study and the estimated parabola equation generated negative values for fecundity when female carapace length was below 20-mm. Therefore, there is a pressing
need to develop an updated fecundity database to provide more robust estimates for northern shrimp reproductive potential, which makes northern shrimp an appropriate case study.
The aim is to compare different sampling strategies for estimating fecundity for species such as northern shrimp that have maternal effects on fecundity and the number of ovigerous individuals were
unevenly collected in sampling locations. The study can identify a cost-effective sampling design for collecting fecundity data, leading to improved fecundity estimation.
Materials and Methods
This study uses simulation of resampling approach to simulate different sampling strategy scenarios based on collected survey data.
NEFSC fall bottom trawl survey data
The GOM northern shrimp spawning season takes place in late summer and fall, and most females become ovigerous in fall. Therefore, the ovigerous females used for the fecundity study were sampled in
the Northeast Fisheries Science Center (NEFSC) fall bottom trawl surveys which were designed for multispecies surveys in the northeast coastal areas. As the surveys are not specifically designed for
northern shrimp, in the sampling location with presence of ovigerous females, the number of shrimp varied from one to several hundred among tows. Given the limited resources, it is unrealistic to
process all collected shrimp. Thus, there is a need to optimize the number of sampling locations in a year and number of shrimp collected in a sampling location. Moreover, as many other species are
collected in the survey, which face similar needs, the methodology developed in this survey are applicable to other species.
The northern shrimp data and tow information were collected by NEFSC fall bottom trawl surveys (Smith, 2002) from 2012 to 2016, including dorsal carapace length (DCL), life stage, date of catch, and
longitude and latitude of sampling location. The DCLs of shrimp were measured to the nearest 0.1-mm, from the posterior limit of eye socket to the posterior limit of dorsal carapace (Haynes and
Wigley, 1969). Only ovigerous female data were used for simulation as the ultimate goal was to collect fecundity data based on maternal body sizes.
Simulation of resampling study
Data from 2012 to 2016 were resampled separately with two sampling strategies of simple random sampling and size-based stratified random sampling. Sampling locations were randomly resampled without
replacement from each year’s sampling locations for each scenario. Sampling intensity was determined by the number of shrimp of interest from a sampling location and the percentage of sampling
locations in each year.
Simple Random Sampling
The sampling scenarios were considered with the percentage of sampling locations and number of shrimp sampled from each sampling location. Two potential sample sizes (i.e., 10 and 20) were considered
for a sampling location in the simulation. For sampling locations with less than the required number of shrimp (i.e., 10 or 20), all shrimp in that location were used. For sampling locations with
more than the specified shrimp, 10 or 20 shrimp were randomly sampled without replacement (Fig. 1)
Stratified Random Sampling
For stratified random sampling, minimal and maximal lengths were determined to be the minimum and maximum DCLs of sampled collected in a year with a length interval of 1.5-mm. A given number (1 or 2)
of shrimp was sampled from each length interval until no more shrimp were available in that length interval. The sampling scenarios were developed with a different sampling intensity and number of
shrimp sampled from each length interval. For sampling locations which had fewer than 10 shrimp collected, all shrimp in that location were used for 1-shrimp scenarios (20 shrimp for 2-shrimp
scenarios, Fig. 1)
Equivalence testing
Null hypothesis significance testing framework is commonly used in ecology to examine the differences between the two groups (Martinez‐Abrain, 2008; Beninger et al., 2012). However, it is criticized
in some ecological studies for the following reasons: (1) a lack of significance (P>α) simply means there is no sufficient evidence to reject the null hypothesis, but it does not mean the null
hypothesis is true (Brosi and Biber, 2009; Beninger et al., 2012; Lakens, 2017); and (2) the statistical power needed to detect a difference is low. Alternatively, two one-sided equivalence tests
within a frequentist framework can be used to ascertain effect quality by specifying meaningful effect size based on biological or ecological understanding (Parkhurst, 2001; Lakens, 2017). Moreover,
the lower and upper bounds constructed with a priori specified effect size allow the researchers to evaluate significant differences with reduced type II error defined in traditional hypothesis
testing (Parkhurst, 2001; Brosi and Biber, 2009). Therefore, instead of using traditional null hypothesis testing, we use two one-sided equivalence testing for the simulated data in each scenario.
Before we performed equivalence testing, a difference of 1.5-mm (∆) was determined as the minimum effect size that we would like to detect. Effect size was defined as the magnitude of the observed
difference (Beninger et al., 2012). Our data suggested that mean DCL of ovigerous females was around 25-mm, which is equivalent to an age of 3.5 years based on age-DCL growth curve (ASMFC, 2018) with
age 3 being estimated at 23.5-mm and age 4 at 26.5-mm. We thus determined the effect size interval at 1.5-mm, as shrimp in DCLs smaller or larger than 1.5-mm are likely to be at a different age of
years. The lower and upper bounds of equivalence intervals for each sample were constructed as (Nakagawa and Cuthill, 2007; Lakens, 2017):
where m s = mean (or median) DCL of samples from a given scenario in year y; m y = mean (or median) DCL of all samples collected in year y; t α, df = t statistic at a significance level of α at
degree of freedom at df; α = 0.05, df = n s+n y−2 ; n s = number of samples of a given scenario; n y = number of samples collected in year y; s s = standard deviation of samples from a scenario
in year y; and s y = standard deviation of all the samples collected in year y.
Two one-sided tests were performed to means and medians of samples simulated from each scenario in each year. The null hypothesis is ei l ≤ ∆ and ei u ≥ ∆, and the alternative hypothesis is − ∆ < equ
ivalence interval < ∆, where ei l = lower bound of equivalence interval, ei u = upper bound of equivalence interval. Both components in the stated null hypothesis must be false to reject the null
hypothesis. Thus, if the equivalence interval falls within the equivalence interval, the difference between the means or medians is smaller than the magnitude of effect size we specified.
Statistical power of detecting the specified effect size (∆ =1.5-mm) was estimated with the number of samples simulated in each scenario at the significance level of 0.05. Statistical power of 0.95
was set as a reference instead of traditional 0.8, as we assume the cost of committing a type II error was the same as that of committing a type I error (Peterman, 1990; Di Stefano, 2003).
Coefficient of variation (CV) was also calculated for evaluating the dispersion of samples for each simulation scenario. All analyses were performed in R 3.5.1 (R Core Team, 2018).
Number of sampling locations in each year
The total yearly number of sampling locations and total number of ovigerous females collected in each year from 2012 to 2016 were shown in Table 1. Our data showed that the mean DCL of ovigerous
females varied between 24.08 and 25.86 from 2012 to 2016 (Fig. 2). In addition, samples collected in 2014 deviated from normal distribution with a mean at 25.43-mm-DCL and a median of 26.5-mm-DCL,
and with an unusual wide standard deviation (SD) of 2.89-mm (SD varied from 1.52 to 1.66 in the other four years).
Equivalence tests
The equivalence tests of means for all the scenarios showed that most equivalence intervals of means fell within the specified effect size interval when at least 20% of the sampling locations were
sampled except for 2014 (Fig. 3). Similar results could be found in tests for the difference in medians (Fig. 4). The equivalence interval of medians barely fell within the effect size interval for
simulated samples in 2014 even if all stations were sampled.
For means of 20-shrimp scenarios in 2014, the equivalence intervals started to fall within the specified effect size interval when more than 50% of the sampling locations were sampled. When less than
50% of the locations were sampled in 2014, both sampling strategies failed to reject the null hypothesis. However, the differences in means of simple random sampling had a wider variation than those
of stratified random sampling scenarios (Fig. 3).
As for the equivalence tests of medians for 2014 samples, almost all scenarios failed to reject the null hypothesis (Fig. 4). Similar to the equivalence tests of means, when less than 50% of the
locations were sampled, the median differences for random sampling method tended to have larger variations than those of stratified random sampling.
Statistical power
The statistical power of detecting the minimal effect size (∆ =1.5-mm) increased with sampling intensity, when more than 20% of sampling locations were sampled, all scenarios could reach the
statistical power of 0.95 except for scenarios of 2014 (Fig. 5). Simulated samples of 2014 could reach the statistical power of 0.95 when at least 30% of the locations were sampled. There was a
trade-off between the number of shrimp per location (or length interval) and percentage of sampling locations. Given a sampling strategy, more numbers of shrimp per sampling location (or length
interval) could reach the statistical power of 0.95 with a lower percentage of sampling locations. The coefficients of variation were mostly below 0.1 for each scenario except scenarios in 2014 due
to large standard deviation of DCL collected in 2014 (Fig. 5).
Sample size
The numbers of shrimp simulated in each scenario increased with sampling intensity, and simple random sampling strategy tended to generate larger sample sizes than stratified random sampling strategy
at a given sampling intensity (Figs. 5 and 6). When 20% of sampling locations were sampled, the total numbers of shrimp in the simulation for five years ranged from 129 to 349 for different
strategies with different intensity (Fig. 6). When 30% of the locations were sampled, the total numbers of shrimp increased to 215–612 (Fig. 6).
The means, medians, and ranges of samples simulated in each scenario were compared with the assumed populations (samples collected from the surveys) in each year (Fig. 7). When more than 20% of the
locations were sampled, the simulated samples could include the central 95% of DCL of the assumed population for both sampling strategies. When less than 50% of the location were sampled, the
stratified random sampling, as expected, was more likely include the minimum and maximum of DCLs of the assumed population than the simple random sampling.
The results of equivalence testing showed that there were no large differences between samples simulated with simple random sampling and stratified random sampling strategies when the population
distribution is approximately normal. Both sampling strategies can collect samples that were representative of the population (i.e., including the central 95% of the distribution) and the means and
medians did not significantly differ from the specified effect size when more than 20% sampling locations were sampled. However, if we conducted traditional null hypothesis significance testing, many
of the simulated samples would suggest statistical significance as the confidence interval of error did not include zero, which might not be biologically significant. The results suggested the merits
of equivalence testing over traditional null hypothesis significance testing with the ability to detect a biologically meaningful or ecologically important effect size (Parkhurst, 2001; Brosi and
Biber, 2009).
The number of shrimp simulated for each scenario with different strategies, in general, linearly increased with the number of sampling locations. However, as the surveys were not specifically
designed for northern shrimp, number of shrimp collected at a station could be only a few. Therefore, the ultimate sampling intensity (number of shrimp simulated for a scenario) was not exactly
proportional to the number of locations sampled. An extreme example was the 20-shrimp scenario with three sampling stations with simple random sampling strategy, which had only four DCLs simulated in
that scenario. The statistical power was hence low (Fig. 5). Our simulation reflects the discrepancy between samples collected in multispecies surveys and ideal sampling for fecundity data. Care
should be taken to adjust sampling strategy in such circumstances.
Increasing sampling intensity by either raising the number of shrimp per location, length interval, or the number of sampling locations can reduce sampling error and increase statistical power.
However, the cost of increasing sampling intensity may not be effective as the magnitude of precision that can be improved is trivial when sampling intensity is above a certain level (Pennington et
al., 2002). Although both the sampling strategies we adopted in this study suggested that the equivalence interval can fall within the effect size interval when at least 20% of the locations were
sampled (except for 2014), we determined stratified random sampling may be a more effective sampling strategy for collecting fecundity data as it requested for a low sample size compared to the
simple random sampling.
With stratified random sampling at a fixed overall sampling size (number of shrimp simulated for all five years), based on the trade-off between the number of shrimp per length interval and the
percentage of the locations, a desired statistical power can be achieved at a lower percentage of sampling locations for 2-shrimp per length interval scenarios. However, the stratified random
sampling strategy with one shrimp per length interval is preferred in this case, as a higher percentage of sampling locations allows a broader spatial coverage of the study area. Therefore, the
optimal sample size for collecting fecundity data was estimated at 215 shrimp for five years (30% of the locations) with size-based stratified random sampling.
Both sampling strategies generated unrepresentative samples which were significantly different from the specified effect size when less than 50% of the locations were sampled for 2014 due to the
skewed distribution of DCLs in 2014. Generally, it is not possible to know the length distribution of the population which is usually assumed to be approximately normally or log-normally distributed.
It should be cautioned when many small spawners are observed in the population, which could be a sign of early sexual maturity resulting from fishing pressure, environmental changes and consequent
food availability to females (O’Brien, 1999; Koeller et al., 2007). Spawners at small sizes make less contribution per individual to reproductive potential of a population, as small spawners tend to
produce fewer offspring per individual with lower survival rates (Shelton et al., 2012; Barneche et al., 2018).
Aanes and Volstad (2015) used simulation approach to evaluate subsampling strategies for collecting age data for Northeast Arctic cod (Gadus morhua), suggesting that length-stratified sampling is
more effective than simple random sampling because length-stratified sampling can ensure a better coverage of the age composition when age data were collected from a small subsample of measured
lengths of fish. Our findings agree with Aanes and Volstad (2015). For the purpose of collecting fecundity data, stratified random sampling strategy is preferred over simple random sampling when the
size distribution of ovigerous females is actually skewed with many small spawners (deviated from the assumed normally distributed population). Because it is often not possible to have enough
resources for a high sampling intensity, and simple random sampling is more likely to generate a biased sample in a low sampling intensity (Figs. 3, 4, and 7). Conversely, although stratified random
sampling also generates biased samples, the variation of means and medians of samples are relatively stable when sampling intensity is low. Furthermore, labouratory process for collecting fecundity
data can be very time-consuming and labour-intensive. The time needed for processing a shrimp to collect fecundity data is generally 3–4 hours. Given a sampling intensity of 20% of the sampling
location, the 10-shrimp simple random sampling scenario generates a larger number of sample size than the 1-shrimp per length interval stratified random sampling scenario by 69 shrimp. Thus, the
simple random sampling may take 207 additional hours (69 shrimp × 3 hours), which would cost additional $4140 (i.e., 207 hours × $20 per hour per person) for laboratory process alone. Our analyses
suggest that length-stratified random sampling is a more cost-effective strategy for collecting fecundity data.
The shrimp samples Haynes and Wigley (1969) used for collecting fecundity data ranged from 22 to 31-mm-DCL. Our data, except for 2014, the central 95% of ovigerous females collected from the survey
ranged from a similar interval of 22–28-mm-DCL in this study. However, it appeared that if shrimp outside the central 95% length interval were excluded from the regression of length and fecundity,
the regressed relationship may not be able to provide reliable estimates of fecundity for the population as the fecundity-DCL relationship developed with 47 female shrimp by Haynes and Wigley (1969)
generates negative numbers for shrimp at DCLs<20-mm. It suggested that, when estimating size-based fecundity for a population, (1) a complete range of size data is necessary for developing a
fecundity-body size relationship; (2) several years of samples may be needed for building a complete fecundity database; and (3) parabola equation should be used with caution as it may generate
biologically meaningless estimates of fecundity (negative values). Estimating the magnitude of the bias in reproductive potential of a population is beyond the scope of this study. Consequently,
before we take a further step into investigation of the misestimates of fecundity, there is a pressing need to develop a new fecundity-DCL relationship with proper sampling design for collecting
fecundity data.
This study proposes a simulation framework that can be used to develop a cost-effective sampling strategy for estimating fecundity data for many marine fish and crustacean species which share the
characteristics of (1) a strong maternal effect on fecundity (i.e., number of offspring increase with female body sizes; Haynes and Wiley, 1969); (2) number of individuals collected varied among
sampling locations and number of sampling locations varied by year; and (3) extensive length frequency data have been collected for multiple years which can be used for sampling design. Collecting
fecundity data can be very time-consuming and labour-intensive. Insufficient samples may result in biased estimates; however, excess samples can be a waste of resources. Therefore, an appropriate
sampling design for optimizing effective sample size is needed for building a complete fecundity data base. We advocate the use of equivalence testing and power analysis before collecting samples in
order to determine biologically meaningful effect size instead of statistical significance in traditional null hypothesis significance testing.
The authors thank Northeast Fisheries Science Center for providing data. This study was financially supported by NOAA Saltonstall-Kennedy grant (NA16NMF4270245).
Aanes, S., and Vølstad, J. H. 2015. Efficient statistical estimators and sampling strategies for estimating the age composition of fish. Canadian Journal of Fisheries and Aquatic Sciences, 72:
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Marshall, C. T., Needle, C. L., Thorsen, A. Kjesbu, O. S., and Yaragina, N. A. ٢٠٠٦. Systematic bias in estimates of reproductive potential of an Atlantic cod (Gadus morhua) stock: implications for
stock-recruit theory and management. Canadian Journal of Fisheries and Aquatic Sciences, 63: 980–994. https://doi.org/10.1139/f05-270
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: Chang, H.-Y., and Chen, Y. 2020. Evaluating sampling strategies for collecting size-based fish fecundity data: an example of Gulf of Maine northern shrimp
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10 Unique and Creative Pi Tattoo Designs for Math Lovers - TODES Tattoo Designs
10 Unique and Creative Pi Tattoo Designs for Math Lovers
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Are you a math lover looking for a creative way to express your passion? Look no further than these 10 unique pi tattoo designs that will not only impress your fellow math enthusiasts but also
showcase your individual personality.
One design that stands out is the minimalist pi symbol, which allows for clean lines and abstract creativity. Another option is to incorporate the pi symbol into an intricate geometric pattern,
showcasing both your love of math and artistic flair.
For those looking for a more traditional tattoo, the pi symbol with its full numerical value wrapped around the circumference of a circle is a classic choice. Alternatively, you could opt for a
whimsical cartoon-style pi character, adding a touch of playfulness to your tattoo.
Whether large or small, black and white or colorful, these 10 unique pi tattoo designs offer endless possibilities for expressing your love of all things math. So why not let your geek flag fly and
get your own pi tattoo today?
“Pi Tattoo” ~ bbaz
Pi is the mathematical constant that represents the ratio of a circle’s circumference to its diameter. It is an irrational number, meaning it has an infinite decimal expansion with no repeating
pattern. For some people, pi is more than just a number – it’s a symbol of their love for mathematics. That’s where pi tattoos come in. These unique and creative designs can be a fun way to show off
your love for math and all things nerdy.
Comparison Table
Tattoo Design Description Opinion
Pi Spiral A spiral design with the digits of pi written along the curve. This design is simple yet elegant.
Pi Infinity A tattoo that combines the infinity symbol with the pi symbol. This design is great for those who love symbolism.
Pi Equation A tattoo that features the equation for pi (C = 2πr). This design is perfect for those who want to show off their knowledge of math.
Pi Dot Art A tattoo that uses dots to create an image of pi. This design is unique and eye-catching.
Pi Heart A tattoo that combines the pi symbol with a heart. This design is great for those who want to show off their love for both math and a special
Pi Tree A tattoo that features the digits of pi branching out like a tree. This design is perfect for nature lovers who also love math.
Pi Music A tattoo that combines the pi symbol with sheet music. This design is perfect for those who love both math and music.
Pi Moon Phases A tattoo that features the digits of pi arranged in the shape of the moon phases. This design is great for those who are interested in astronomy as well as math.
Pi Rodin’s Thinker A tattoo that features the pi symbol as part of Rodin’s famous sculpture The This design is perfect for art history buffs who also love math.
Pi Mandala A tattoo that uses the digits of pi to create a mandala pattern. This design is perfect for those who appreciate the beauty of symmetry.
Pi Spiral
The pi spiral is a popular tattoo design that features the digits of pi arranged in a spiral shape. The curve of the spiral represents the circumference of a circle, while the digits show the
mathematical relationship between a circle’s radius and its circumference. This design is simple yet elegant, and it can be done in a variety of styles, from minimalist to intricate.
Pi Infinity
The pi infinity tattoo combines two powerful symbols: the pi symbol and the infinity symbol. The infinity symbol represents the concept of endlessness, while the pi symbol represents the never-ending
decimal expansion of pi. This design is great for those who love symbolism and want to express their appreciation for the beauty of math.
Pi Equation
The pi equation tattoo features the formula for pi, which is C = 2πr (circumference equals two times pi times radius). This formula is one of the most well-known equations in mathematics, and it is
used extensively in geometry and trigonometry. This tattoo is perfect for those who want to show off their knowledge of math and their appreciation for its elegance and simplicity.
Pi Dot Art
The pi dot art tattoo uses dots to create an image of pi. This design is unique and eye-catching, and it can be done in a variety of sizes and styles. Some people choose to include other mathematical
symbols or equations in the design to make it even more meaningful.
Pi Heart
The pi heart tattoo combines the pi symbol with a heart. This design is great for those who want to express their love for both math and a special someone. It can be done in a variety of styles, from
simple and minimalist to intricate and detailed.
Pi Tree
The pi tree tattoo features the digits of pi arranged in the shape of a tree. This design is perfect for those who love both math and nature. It can be done in a variety of styles, from whimsical and
cartoonish to realistic and detailed.
Pi Music
The pi music tattoo combines the pi symbol with sheet music. This design is perfect for those who love both math and music. It can be done in a variety of styles, from simple and minimalist to
complex and detailed.
Pi Moon Phases
The pi moon phases tattoo features the digits of pi arranged in the shape of the moon phases. This design is great for those who are interested in astronomy as well as math. It can be done in a
variety of styles, from abstract and minimalist to realistic and detailed.
Pi Rodin’s Thinker
The pi Rodin’s Thinker tattoo features the pi symbol as part of Auguste Rodin’s famous sculpture The Thinker. This design is perfect for art history buffs who also love math. It can be done in a
variety of styles, from simple and abstract to realistic and detailed.
Pi Mandala
The pi mandala tattoo uses the digits of pi to create a mandala pattern. This design is perfect for those who appreciate the beauty of symmetry and repetition, as well as the elegance of mathematical
concepts. It can be done in a variety of styles, from traditional to modern.
Whether you’re a mathematician, a science lover, or just a fan of symbolism and creativity, a pi tattoo can be a great way to express your personality and interests. The ten designs featured in this
article are just a few examples of the many possibilities out there, and each one has its own unique style and appeal. So if you’re thinking about getting a pi tattoo, don’t be afraid to get creative
and come up with something that speaks to you!
Thanks for stopping by to check out our list of 10 unique and creative pi tattoo designs for math lovers! We hope you found inspiration for your next ink masterpiece. As math enthusiasts ourselves,
we understand how important it is to showcase our passion in every aspect of our lives – including tattoos.
From the minimalistic representations of pi to intricate designs incorporating equations and symbols, we’ve curated a list that offers something for everyone. Whether you’re a mathematician, student,
or simply appreciate the beauty of numbers, these tattoos capture the essence of pi and its significance.
Remember, getting a tattoo is a permanent commitment – so don’t rush the decision-making process. Take the time to find a design that speaks to you personally and consult with a reputable artist
before making any final decisions. And don’t forget to share your pi tattoo designs with us using the hashtag #PiTattoo!
Are you a math lover searching for unique and creative Pi tattoo designs? Look no further! Here are 10 ideas to inspire your next tattoo:
1. A Pi symbol made out of colorful puzzle pieces.
2. A Pi symbol with the digits of Pi spiraling around it.
3. A Pi symbol incorporated into a geometric design.
4. A Pi symbol with a small drawing of Albert Einstein next to it.
5. A Pi symbol with a quote about the importance of math, such as Without mathematics, there’s nothing you can do. Everything around you is mathematics. Everything around you is numbers. –
Shakuntala Devi
Still looking for more inspiration? Here are five more unique ideas:
• A Pi symbol with a small rocket ship flying around it.
• A Pi symbol made out of musical notes.
• A Pi symbol with a small infinity symbol incorporated into the design.
• A Pi symbol with a small telescope or microscope next to it.
• A Pi symbol with a small planet or galaxy incorporated into the design.
No matter which design you choose, a Pi tattoo is a great way to show off your love of math and all things numerical!
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Game characterizations for the number of quantifiers
Lauri Hella, Tampere University
A game that characterizes definability of classes of structures by first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. In this talk I describe two other
games that are equivalent with the Immerman game in the sense that they characterize definability by a given number of quantifiers.
In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal
strategy. On the other hand, in the other two games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a one-player game, but the other two games are genuine
two-player games.
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A Novel Dominance Principle based Approach to the Solution of Two Persons General Sum Games with n by m moves
Zola, Maurizio Angelo (2024): A Novel Dominance Principle based Approach to the Solution of Two Persons General Sum Games with n by m moves.
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In a previous paper [1] the application of the dominance principle was proposed to fi nd the non-cooperative solution of the two by two general sum game with mixed strategies; in this way it was
possible to choose the equilibrium point among the classical solutions avoiding the ambiguity due to their non-interchangeability, moreover the non-cooperative equilibrium point was determined by a
new geometric approach based on the dominance principle. Starting from that result it is here below proposed the extension of the method to two persons general sum games with n by m moves. The
algebraic two multi-linear forms of the expected payoffs of the two players are studied. From these expressions of the expected payoffs the derivatives are obtained and they are used to express the
probabilities distribution on the moves after the two defi nitions as Nash and prudential strategies [1]. The application of the dominance principle allows to choose the equilibrium point between the
two solutions avoiding the ambiguity due to their non-interchangeability and a conjecture about the uniqueness of the solution is proposed in order to solve the problem of the existence and
uniqueness of the non-cooperative solution of a two persons n by m game. The uniqueness of the non-cooperative solution could be used as a starting point to find out the cooperative solution of the
game too. Some games from the sound literature are discussed in order to show the effectiveness of the presented procedure.
Item Type: MPRA Paper
Original Title: A Novel Dominance Principle based Approach to the Solution of Two Persons General Sum Games with n by m moves
English Title: A Novel Dominance Principle based Approach to the Solution of Two Persons General Sum Games with n by m moves
Language: English
Keywords: Dominance principle; General sum game; two persons n by m moves game
Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games
Item ID: 122312
Depositing Mr Maurizio Angelo Zola
Date Deposited: 08 Oct 2024 13:33
Last Modified: 11 Nov 2024 13:54
[1] Zola M.A. (2024) A Novel Integrated Algebraic/Geometric Approach to the Solution of Two by Two Games with Dominance Principle. Munich Personal RePEc Archive MPRA-paper-121935 -
10 Sep 2024.
[2] Nash J.F. (1951) Non-Cooperative Games. Annals of Mathematics, Second Series 54(2), 286–295, Mathematics Department, Princeton University.
[3] Nash J.F. (1950) The bargaining problem. Econometrica, 18(2), 155–162.
[4] Nash J.F. (1950) Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America, 36(1), 48–49.
[5] Nash J.F. (1953) Two-person cooperative games. Econometrica, 21(1), 128–140.
[6] Luce R.D., Raiffa H. (1957) Games and decisions: Introduction and critical survey. Dover books on Advanced Mathematics, Dover Publications.
[7] Owen G. (1968) Game theory. New York: Academic Press (I ed.), New York: Academic Press (II ed. 1982), San Diego (III ed. 1995), United Kingdom: Emerald (IV ed. 2013).
References: [8] Straffin P.D. (1993) Game Theory and Strategy. The Mathematical Association of America, New Mathematical Library.
[9] Van Damme E. (1991) Stability and Perfection of Nash Equilibria. Springer-Verlag. Second, Revised and Enlarged Edition.
[10] Dixit A.K., Skeath S. (2004) Games of Strategy. Norton & Company. Second Edition.
[11] Tognetti M. (1970) Geometria. Pisa, Italy: Editrice Tecnico Scientifica.
[12] Maschler M., Solan E., Zamir S. (2017) Game theory. UK: Cambridge University Press.
[13] Esposito G., Dell’Aglio L. (2019) Le Lezioni sulla teoria delle superficie nell’opera di Ricci-Curbastro. Unione Matematica Italiana.
[14] Bertini C., Gambarelli G., Stach I. (2019) Strategie – Introduzione alla Teoria dei Giochi e delle Decisioni. G. Giappichelli Editore.
[15] Vygodskij M.J. (1975) Mathematical Handbook Higher Mathematics. MIR, Moscow.
[16] Von Neumann J., Morgenstern O. (1944) Theory of Games and Economic Behavior. New Jersey Princeton University Press.
URI: https://mpra.ub.uni-muenchen.de/id/eprint/122312 | {"url":"https://mpra.ub.uni-muenchen.de/122312/","timestamp":"2024-11-11T15:03:38Z","content_type":"application/xhtml+xml","content_length":"31363","record_id":"<urn:uuid:207cc08a-0a7b-43fc-a67d-9a1645dc30e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00551.warc.gz"} |
How do you graph polar coordinates? + Example
How do you graph polar coordinates?
1 Answer
To establish polar coordinates on a plane, we choose a point $O$ - the origin of coordinates, the pole, and a ray from this point to some direction $O X$ - the polar axis (usually drawn
Then the position of every point $A$ on a plane can be defined by two polar coordinates: a polar angle $\varphi$ from the polar axis counterclockwise to a ray connecting the origin of coordinates
with our point $A$ - angle $\angle X O A$ (usually measured in radians) and by the length $\rho$ of a segment $O A$.
To graph a function in polar coordinates we have to have its definition in polar coordinates.
Consider, for example a function defined by the formula
$\rho = \varphi$ for all $\varphi \ge 0$.
The function defined by this equality has a graph that starts at the origin of coordinates $O$ because, if $\varphi = 0$, $\rho = 0$.
Then, as a polar angle $\varphi$ increases, the distance from an origin $\rho$ increases as well. This gradual increase in both polar angle and distance from the origin produces a graph of a spiral.
After the first full circle the point on a graph will hit the polar axis at a distance $2 \pi$. Then, after the second full circle, it will intersect the polar axis at a distance $4 \pi$, etc.
Impact of this question
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Basic cryptography: hash, digital signature, MAC, symmetric keys - Bitcoin Freedom - Massimo (Max) Musumeci
Basic cryptography: hash, digital signature, MAC, symmetric keys
Posted by Massimo Musumeci | Jul 20, 2020 | Computer science | 0 |
Message hash
What’s a hash? It is the result of a non-reversible mathematical function, which returns a bit sequence after receiving an arbitrary length data input. The result of applying hashes to a data set is
a fixed length sequence. Hash example algorithms: MD5, SHA1, SHA256. The length of the MD5 code is 128 bits, the length of the SHA1 code is 160 bits.
Given the hash output, you cannot return to the original message. But the same message (or generally data) will always give the same hash when hashed with the same algorithm. In addition, the
calculation of hashes on two different data, causes two different results (there is therefore no “collision”). So, if you receive a message together with a hash, you only need to calculate your hash
independently with the same algorithm. If the 2 hashes match it means that the message is the original one intact, otherwise the message has been tampered in some way.
Another important property of a hash is that a small change to the message makes the resulting hash value changing significantly. This, together with the above, means that a hash ensures the
integrity of the message.
What does it mean to use encryption? It means replacing the real text of a message with a different one obtained from a certain algorithm and a key, in such a way that it is impossible or very very
difficult (or extremely long) to decipher it, without having the necessary encryption key.
This means that encryption ensures the secrecy of a message, but not anonymity. In the sense that everyone knows that a message has been sent by the sender, but no one except the receiver is able
to read its contents.
There are 2 types of encryption: symmetrical and asymmetrical.
Symmetric Encryption
Symmetric encryption is faster. It uses only 1 key for encryption and decryption. Both the sender and the recipient must know the key and keep it secret, no one else must know it.
The risky and delicate part of symmetric encryption is sharing the key between the two participants in the conversation, because there is only one key and if compromised the whole conversation is
compromised. There are several mathematical systems that have been invented to share a secret key between two ends of a conversation through an insecure medium (e.g. Diffie Hellmann key exchange).
Examples symmetric encryption: AES, DES.
Asymmetric encryption
Asymmetric encryption is slower than symmetric one. It uses 4 keys, each user has 2 (a public/private pair), these 2 keys are mathematically linked. The most important property is that everything
that is encrypted with the public key can be decrypted with the corresponding private key and vice versa.
The public key, as the name suggests, is not a secret, the owner of the key pairs can publish his public key on his website or anywhere, and in fact he must do so in order to receive encrypted
messages that can only be decrypted with his private key. It’s like an address or a mailbox. The private key, as the name suggests, must be kept secret (it is the personal cryptographic secret). If
the owner of the key pairs gets his private key compromised, then anyone can know what messages he has received. In case of compromise a new key pair must be generated.
To communicate with asymmetric encryption, simply exchange public keys through any public medium, even insecure ones. Unlike symmetric encryption, you can do it outdoors, no one can read your
messages if they only know your public keys.
If you want to send a message, use the other person’s public key to encrypt the message and send it to him, only he can unlock the message with his private key and read it.
If you want to receive a message, the other person must have used your public key to encrypt the message, which means that only your private key can decrypt the message.
Examples asymmetric encryption: RSA, DSA.
Of course, most of the time this is done automatically via email or other forms of messaging. You may have already communicated with asymmetric or symmetric encryption without manually encrypting and
decrypting a message.
Integrity of digitally signed messages
When we talk about the integrity of messages, we intend to provide guarantees to the recipient regarding: 1) authentication, 2) integrity and 3) non-repudiation. With the digital signature what is
done is to encrypt with the sender’s private key a hash of the entire message you are sending. The message is sent together with this digital signature.
In practice you create a hash for your message with a certain hashing algorithm (message digest) and then this hash is encrypted with your private key.
The recipient of your message uses your public key to decipher the signature, then receives the message digest. He applies the same hashing algorithm (e.g. MD5 for SHA1) that was used by the sender,
but independently, and if the result of the calculated hash matches the hash he obtained after decrypting your digital signature with your public key, means not only 1) that the message was sent by
you, since it is you who have your private key that encrypted the digital signature, but also 2) that the message it received is the original one you wanted to send, since its hash matches the hash
in the signature.
Because of the signature, you cannot deny the authenticity of the message and yourself (non-repudiation). Third goal reached.
But there’s one more thing to solve: how can you make sure that the public key you send publicly (i.e. on an insecure public medium) hasn’t been altered, or that someone makes their own public key
and declares that it’s your key? This way they can decipher the messages that are meant for you, and you can’t. That’s where digital certificates come in.
Digital certificates
Digital certificates ensure that the public key published in your name is actually certified as your public key by a higher third party. Obviously this is a trust relationship and not a mathematical
Digital certificates include information about the public key, information about the identity of its owner (called the subject) and the digital signature of an entity that verified the certificate’s
content, a third party (called the issuer). If the signature is valid, and the software examining the certificate trusts the issuer, then it can use that public key to communicate securely with the
subject of the digital certificate.
Integrity of messages with HMAC
But what if you want to guarantee the integrity of the message but you don’t need the authorship of the message and you just want the messages to work faster? This is where HMAC comes in, unlike
digital signatures, HMAC encrypts the message hash with a symmetrical key.
Being encrypted with a symmetrical key, the authorship of the message cannot be traced back to you.
Being encrypted with a symmetrical key, the authorship of the message cannot be traced back to you, because you are not the only one who has access to that symmetrical key. In fact, the recipient
also has the same key and may have created that message himself. Of course, they are the only 2 entities that have access to the symmetric key, so unless one of them has compromised the key, the
message you received, if you know it wasn’t you who wrote it, you can be sure it came from the other person who has the symmetric key.
The H in HMAC stands for hash and the MAC stands for message authentication code, a code that also guarantees the integrity and authenticity of the data, allowing viewers who have the secret key to
detect any changes to the message content. A MAC usually has 3 parts: a key generation algorithm, a signature algorithm and a verification algorithm.
Digital signature = Hash of the message is encrypted with the sender’s private key. HMAC = Hash of the message is encrypted with the symmetric key. | {"url":"https://www.massmux.com/basic-cryptography-hash-digital-signature-mac-symmetric-keys/","timestamp":"2024-11-02T21:43:34Z","content_type":"text/html","content_length":"131094","record_id":"<urn:uuid:02b719fb-373f-4a49-a0bc-a7f237e6991b>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00629.warc.gz"} |
T-PROGS Screenshot
Model Info
Model type transition probability geostatistics on borehole data
Developer Steven Carle
Documentation T-Progs Manual
Tutorials T-PROGS Tutorials
GMS includes an interface to the T-PROGS (Transition PRObability GeoStatistical) software developed by Steven Carle. The T-PROGS software is used to perform transition probability geostatistics on
borehole data. The output of the T-PROGS software is a set of N material sets on a 3D grid. Each of the material sets is conditioned to the borehole data and the materials proportions and transitions
between the boreholes following the trends observed in the borehole data. These material sets can be used for stochastic simulations with MODFLOW. A sample material set generated by the T-PROGS
software is shown below. This software can also be used to generate multiple input datasets for the HUF package.
The T-PROGS model can be added to a paid edition of GMS.
T-PROGS Interface
The T-PROGS software utilizes a transition probability-based geostatistical approach to model spatial variability by 3-D Markov Chains, set up indicator co-kriging equations, and formulate the
objective function for simulated annealing. The transition probability approach has several advantages over traditional indicator kriging methods. First, the transition probability approach considers
asymmetric juxtapositional tendencies, such as fining-upwards sequences. Second, the transition probability approach has a conceptual framework for incorporating geologic interpretations into the
development of cross-correlated spatial variability. Furthermore, the transition probability approach does not exclusively rely on empirical curve fitting to develop the indicator (cross-) variogram
model. This is advantageous because geologic data are typically only adequate to develop a model of spatial variability in the vertical direction. The transition probability approach provides a
conceptual framework to geologic insight into a simple and compact mathematical model, the Markov chain. This is accomplished by linking fundamental observable attributes—mean lengths, material
proportions, anisotropy, and juxtapositioning – with Markov chain model parameters.
The first step in using T-PROGS is to import a set of borehole data. The borehole data are then passed to a utility within T-PROGS called GAMEAS that computes a set of transition probability curves
as a function of lag distance for each category for a given sampling interval. A sample set of measured transition probability curves are shown by the dashed lines in the following figure.
Each curve represents the transition probability from material j to material k. The transition probability t[jk](h) is defined by:
where x is a spatial location, h is the lag (separation vector), and j,k denote materials. Note that the curves on the diagonal represent auto-transition probabilities, and the curves on the
off-diagonal represent cross-transition probabilities.
The next step in the analysis is to develop a Markov Chain model for the vertical direction that fits the observed vertical transition probability data. The Markov Chain curves are shown as solid
lines in the preceeding figure. Mathematically, a Markov chain model applied to one-dimensional categorical data in a direction Φ assumes a matrix exponential form:
where h denotes a lag in the direction Φ, and RΦ denotes a transition rate matrix
with entries r[jk],f representing the rate of change from category j to category k (conditional to the presence of j) per unit length in the direction Φ. The transition rates are adjusted to ensure a
good fit between the Markov Chain model and the observed transition probability data.
Once the Markov chain is developed for the z direction from the borehole data, a model of spatial variability must be developed for the x and y directions. Borehole data are typically not
sufficiently dense in these directions. However, the x and y-direction Markov chains can be developed by assuming that the juxtapositional tendencies and the proportions observed in the vertical
direction also hold true in the horizontal directions. The modeler then provides an estimate of the ratio of the mean lengths in the x and y directions relative to the z direction, and the transition
rate matrices for the x and y directions can be formulated. The x, y, and z Markov chains are converted into a continuous 3D Markov chain using the MCMOD utility within T-PROGS.
In the final phase of setting up a transition probability analysis using T-PROGS, the modeler creates a grid, specifies the number of model instances (N), and launches the TSIM utility. The TSIM code
uses the 3D Markov chain to formulate both indicator cokriging equations and an objective function for simulated annealing. It generates stochastic simulations using a combination of modified
versions of the GSLIB codes SISIM and ANNEAL.
TPROGS allows roughly up to 3.5 million cells to be included in a simulation. Model instabilities may appear if more that 3.5 million cells are used.
T-PROGS Materials
When having selected the New Simulation command to initialize a T-PROGS simulation, the T-PROGS Boreholes dialog appears. Here select to use all boreholes or only the boreholes in a particular
Boreholes use materials to define both soils and HGUs. HGUs can be used to group several materials into one hydrogeologic unit. The T-PROGS Materials dialog lets choosing to use the materials in all
HGUs or those from just one HGU. If Use all HGUs is selected, T-PROGS will use the HGU information on the boreholes and ignore the soil information. If Use one HGU is selected, T-PROGS will use the
soil information on the boreholes for the soils that are in the selected HGU. This feature can be used to limit the portion of the boreholes that are used in the T-PROGS simulation.
Material List
If boreholes do not exist in the model, an unconditioned simulation will be generated. In this case, in the T-PROGS Materials dialog select the materials to be used and a corresponding background
material. The upper part of the dialog lists the materials in the boreholes. The first column of toggles indicates which materials are to be used in the analysis. By default, all materials associated
with the boreholes are selected. These toggles are necessary since it is possible that there may be materials defined in the materials list that are not associated with boreholes. The second column
in the top section of the dialog lists the background material. By default, the material type that had the predominant occurrence in the boreholes (greatest proportion) is marked as the background
material. When defining the transition probability data in the next section, the input parameters do not need to be edited for the background material. The parameters for this material are
automatically adjusted to balance the equations.
Background Material
Application of the transition probability approach involves the designation of a background material. The probabilistic constraints of the Markov chains make it unnecessary to quantify data for one
category. Not only is it unnecessary, but it is futile to do so because values will be overwritten in order to satisfy constraints. Conceptually, the background material can be described as the
material that “fills” in the remaining areas not occupied by other units. For example, in a fluvial depositional system, a floodplain unit would tend to occupy area not filled with higher-energy
depositional units and would therefore be a logical choice for the background material.
Also enter an azimuth in this dialog. The azimuth determines the orientation of the primary directions of the depositional trends in the strike/dip directions. These trends generally are aligned with
the primary directions of horizontal flow in the aquifer. Theoretically, the azimuth can be oriented independently from the grid orientation. However, in practice, if the grid and azimuth
orientations are offset by more than about 40^o, checkerboard patterns appear in the indicator array results. Hence, the azimuth orientation is set equal to the grid orientation by default. However,
the grid angle is defined counterclockwise, and the azimuth angle is clockwise. Therefore, if the grid angle is 40^o, then the azimuth angle will be –40^o by default. If there is anisotropy in the xy
plane, the azimuth angle should be set to the principle direction of the anisotropy. If anisotropy is not present, this angle should be coincident with the x-axis (the rows or j-direction) of the
Material Limit
One limitation for both the cases with and without boreholes is that a maximum of five materials can be used in the T-PROGS algorithm. This limitation was imposed to keep the data processing and
user-interface reasonably simple. Although five materials present a limitation, borehole data can generally be easily condensed down to five or fewer materials. Furthermore since this is a stochastic
approach, which is based on probability, the detail generated with numerous materials is rarely justifiable anyway. In addition, as the number of materials increase, the ratio of process time to
detail becomes inefficient.
Generating Material Sets with T-PROGS
The underlying equations solved by the T-PROGS software require an orthogonal grid with constant cell dimensions (X, Y, and Z). The delta X values can be different from the delta Y and delta Z
values, and the delta Y values can be different from the delta Z values, but all cells must have the same change in X, Y, and Z dimensions. The MODFLOW model is capable of using the Layer Property
Flow (LPF) package with the Material ID option for assigning aquifer properties. With this option, each cell in the grid is assigned a material ID and the aquifer properties (Kh, Kv, etc.) associated
with each material are automatically assigned to the layer data arrays for the LPF package when the MODFLOW files are saved. The T-PROGS software generates multiple material sets (arrays of material
IDs), each of which represents a different realization of the aquifer heterogeneity. When running a MODFLOW simulation in stochastic mode, GMS automatically loads each of the N material sets
generated by the T-PROGS software and saves N different sets of MODFLOW input files. The N solutions resulting from these simulations can be read into GMS and used to perform risk analyses such as
probabilistic capture zone delineation.
One-Layer MODFLOW Grids
Although MODFLOW is a three-dimensional model, a majority of the MODFLOW models constructed are 2D models consisting of one model layer. There are several reasons why 2D models are so common. One
reason is that many of these models are regional models where the aquifer thickness is very small compared to the lateral extent of the model. As a result, the flow directions are primarily
horizontal and little improvement is gained by adding multiple layers to the model. Even with local scale models, the aquifer thickness is often small enough that one-layer models are considered
adequate. 2D models are also attractive due to the simplicity of the model increased computational efficiency. One of the problems associated with using multiple layers for MODFLOW models with
unconfined aquifers is that as the water table fluctuates, the upper cells may go dry. These cells will not rewet even if the water table subsequently rises, unless the rewetting option has been
selected in the flow package (BCF, LPF, or HUF). The rewetting issues can often be avoided with a one-layer model.
When developing a one-layer model, the modeler must determine how to distribute the hydraulic conductivity values within the layer. One option is to assume a homogenous aquifer; this is typically a
gross over-simplification since aquifers are usually highly heterogeneous. Therefore, a common approach is to delineate zones of hydraulic conductivity by examining the subsurface stratigraphic data.
In many cases, these data are in the form of borehole logs. These borehole logs often exhibit substantial heterogeneity and don’t always exhibit definitive trends between adjacent boreholes.
Furthermore, the boreholes are often clustered with large regions of the model lacking any borehole data. The modeler then faces a difficult task of trying to determine a rational approach to
delineating two-dimensional zones of hydraulic conductivity based on complex 3D borehole data.
As part of this research, we developed a technique for developing 2D zones of hydraulic conductivity from borehole logs using transition probability geostatistics. The technique is simple, fast, and
preserves proportions and trends exhibited by the borehole data. The algorithm parses through each borehole and computes a predominant material at each borehole. When T-PROGS runs, the predominant
material for each borehole is assigned to its corresponding location in the one-layer grid, and during the quenching process, simulations are conditioned to those data points.
Generating HUF Data with T-PROGS
Using transition probability geostatistics with MODFLOW models results in two basic limitations. First, the underlying stochastic algorithms used by the T-PROGS software are formulated such that the
MODFLOW grid must have uniform row, column, and layer widths. The row width can be different from the column width, but each row must have the same width. This results in a uniform orthogonal grid.
While MODFLOW grids are orthogonal in x and y, the layer thickness is allowed to vary on a cell-by-cell basis. This makes it possible for the layer boundaries to accurately model the ground surface
and the tops and bottoms of aquifer units. If a purely orthogonal grid is used, irregular internal and external layer boundaries must be simulated in a stair-step fashion either by varying material
properties or by activating/inactivating cells via the IBOUND array. A second limitation is that in order to get a high level of detail in the simulated heterogeneity, the grid cell dimensions are
generally kept quite small. This can result in difficulties in the vertical dimension. The large number of layers with small layer thicknesses near the top of the model generally ensures that many of
the cells in this region will be at or above the computed water table elevation (for simulations involving unconfined aquifers). As a result, these cells will undergo many of the numerical
instabilities and increased computational effort issues associated with cell wetting and drying.
The Hydrogeologic Unit Flow (HUF) package released with MODFLOW 2000 makes it possible to overcome both of these limitations resulting in a powerful mechanism for incorporating transition probability
geostatistics in MODFLOW simulations. With the HUF package, the modeler is allowed to input the vertical component of the stratigraphy in a grid-independent fashion. The stratigraphy data are defined
using a set of elevation and thickness arrays. The first array defines the top elevation of the model. The remaining arrays define the thicknesses of a series of hydrogeologic units, starting at the
top and progressing to the bottom of the model. For each array of thicknesses, many of the entries in the array may be zero. This makes it possible to simulate complex heterogeneity, including
pinchouts and embedded lenses that would be difficult to simulate with the LPF and BCF packages.
The T-PROGS interface in GMS includes an option for integrating transition probability geostatistics results with the HUF package. The basic approach used by the option is to overlay a dense
background grid on the MODFLOW grid and run T-PROGS on the background grid. A set of HUF arrays is then extracted from the background grid for use with the MODFLOW model. To use this option, first
create a MODFLOW grid with the desired number of layers and the layer elevations should be interpolated to match the aquifer boundaries. The row and column widths are uniform but the layer
thicknesses may vary from cell to cell. Then, when TSIM is launched, the HUF option should be selected. GMS then generates a background grid that encompasses the MODFLOW grid. The rows and columns of
this grid match the MODFLOW grid but the layer thicknesses are uniform and relatively thin, resulting in a much greater number of layers than the MODFLOW grid. Specify the number of layers in this
background grid. A T-PROGS simulation is then performed to get a set of material sets on the background grid. Each of the material sets in the T-PROGS output is then transferred from the background
grid to a set of HUF elevation/thickness arrays. The HUF top elevation array is set equal to the top of the MODFLOW grid. The thickness arrays are then found by searching through the background grid
to find the bottom elevations of contiguous groups of indicators. The elevations from these groups are then added to an appropriate elevation array in the HUF input. The resulting set of HUF input
arrays are listed in GMS Project Explorer. By clicking on each item in the Project Explorer, the selected set of HUF arrays are loaded into the HUF package and the corresponding stratigraphy is
displayed in the GMS window. The multiple HUF input arrays can be used to perform a stochastic simulation.
1. ^ Carle, Steven F. (1999), T-PROGS: Transition Probability Geostatistical Software. Version 2.1, Davis, California, p. 6, http://gmsdocs.aquaveo.com/t-progs.pdf
2. ^ Carle, Steven F. (1999), T-PROGS: Transition Probability Geostatistical Software. Version 2.1, Davis, California, p. 26, http://gmsdocs.aquaveo.com/t-progs.pdf | {"url":"https://xmswiki.com/wiki/GMS:T-PROGS","timestamp":"2024-11-05T03:44:10Z","content_type":"text/html","content_length":"54834","record_id":"<urn:uuid:c0a35c4a-be59-486f-85fb-c1819883d5c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00414.warc.gz"} |
Wind load on diaphragms
Rigid diaphragms have infinite in-plane stiffness, and their role in the analysis model is to transmit and distribute horizontal loads and deformations within the structure. Thanks to a new load
assignment tool – Wind > Diaphragm –, you can place a force and moment combination (F[x], F[y], and M[z]) acting in the plane of each diaphragm at any arbitrary point. The loads assigned to the
diaphragm system can belong to arbitrary number of load cases.
The characteristics of the new feature:
• It locates and lists all diaphragms of the model in storey order.
• Force directions are considered in the global coordinate system.
• After input, data check confirms whether coordinates of the force application point are inside the diaphragms or not. Only the internal load will be valid and created.
• The loads created by the function cannot be modified; they are considered automatic loads and load cases.
• Deleting a diaphragm also removes the associated loads.
• If the geometry of a diaphragm is modified, the assigned load will only be deleted, if its coordinates fall outside the diaphragm area.
• The numerical value cells in the input table support multiple selections and the Ctrl+C / Ctrl+V copy-paste functions, allowing for filling data from external tables via clipboard.
Assigned load data in the Load export
The Export existing loads function (Loads tab) has been expanded with a new “Assigned load” column, where the identifiers of structural objects to which the loads have been assigned are displayed.
• Load-to-structure assignments can be created using the Assign load to structure command (Loads tab) or during a file import (e.g. SAF).
• The new “Assigned load” data is for informational and documentation purposes only; it is not used by the Import modified loads function (Loads tab).
Improved visibility of assigned loads
To ensure a better distinction between standalone loads and those assigned to structural elements, the colour for the latter has been changed.
Traffic loads
The Traffic Load feature is a specialized load pattern designed to determine the most unfavourable load system and its position for road and railway bridges according to EN 1991-2. The primary goals
of this feature are:
• Facilitating parametric/variable vehicle definition
• Assisting in Eurocode's Load model definition
• Aiding in determining the dynamic load multiplier and managing uneven loading on rails
• Supporting multiple lane alignment definition
• Providing solution for both road and railway bridges
• Handling scenarios with multiple road lanes/railways
• Displaying maximal values and the associated actual load system
• Handling arbitrary geometry with inclined lines/surfaces
• Facilitating the creation of multiple traffic loads with various options
Load patterns are seamlessly integrated into the load cases, similar to other load items. It's important to note that this type of load comes with a high computational demand, and for optimal
performance, see the Performance tips section.
Definition steps
Traffic load patterns are objects associated with the Moving loads. A new traffic load creates one or more new Moving loads with a Unit load, and with different Load cases. The position of Moving
loads can be freely edited. To minimize the number of new Moving loads (and consequently load cases), use the Wizard and Mass-generation tools, which will employ the same Moving loads.
Advantages: the details of the Influence graphs can be easily controlled, and the calculation time can be significantly reduced when multiple traffic loads use the same moving load object.
Disadvantages: editing/moving/copying traffic loads' geometry is restricted, as modifications to connected moving loads may conflict with other coupled traffic loads. In such cases, redefinition may
be necessary. To solve this problem more efficiently, use the Pick and Copy properties (Tools menu) of the original object.
Simple line model
Both carriageway and railway load patterns can be defined as a simple line model, specifically designed for the pre-design phase using single bar models. The definition process only requires the
stake-out line.
Complex models
For carriageway vehicles, it is necessary to have one or multiple carriageway regions, which can be nearly arbitrary in space (except close to vertical). Slopes are allowed in both longitudinal and
transversal directions. Surface moving loads will be generated on each selected/defined carriageway region, required for influence graph creation. The load system of the vehicle can only be placed in
these regions during the calculation. All region parts not included in the lane alignment can be referred to as Remaining areas.
Carriageway vehicle definition process:
1. Select the regions of the carriageway.
2. Define the stake-out line.
Railway models are specially designed to handle complex rail geometry and superelevation. Therefore, during the definition process, the geometry of each rail or underlying girder must be defined.
Note: Multiple regions can be selected as a carriageway part using the selection sub-option.
Railway vehicle definition process:
1. Set the number of railways.
2. Define the polyline of each track.
3. Define the stake-out line.
Load positioning during the calculation: one railway contains two rails, and the influence lines of these rails are summed during the calculation. However, they may have different lengths. If the
rails can be matched with the stake-out line (its segments are parallel pairwise with the stake-out line), the longer segments could be paired exactly, and the curving parts of the track will be
paired by ratio. Otherwise, the track will be paired by ratio along the entire length.
The Wizard is a specialized tool designed for creating a new load pattern based on an existing one. If you require a new load pattern with the same geometry but different properties, this tool should
be used.
1. Select an existing load pattern.
2. Modify the properties.
3. Click the Create button to generate a new load pattern.
Mass-generation of Traffic load patterns
A single Traffic load pattern item could manage only one Load model and has certain limitations regarding component selection. With this tool, you can define multiple Traffic loads, each with
different Load models and components, based on a selected load pattern (prototype). The Generate load patterns function will generate the Load cases for Components grouped in Subgroups of load models
within the selected Load group.
On the General tab, internal force/displacement components can be selected for the calculation. The dominant load system is determined through the maximization process of the chosen component.
If the model is configured with limit state-dependent materials (e.g. Creep settings), four different calculations would be required. To reduce the calculation time, three major options are available
for all selected components:
1. [Limit state] clone (e.g. “U clone”): Calculate the selected limit state, and all results will be cloned to the others.
2. [Limit state] resultset (e.g. “U resultset”): Calculate the dominant load system by the selected limit state, then calculate all results individually by this load system.
3. U/Sq/Sf/Sc: All limit states will be calculated individually.
If the Calculate simultaneous results option is checked, the related simultaneous internal force components will be calculated (e.g.: selected Bar/N+ will also calculate Ty, Tz, Mt, My, Mz).
For easier usage and reduced calculation time, cross internal force calculation is not available. For example, simultaneous shell internal force cannot be calculated to the Maximum Bar internal force
All newly generated Moving loads will be placed in one Load group, named after the Comment field.
Load model
The Load model tab is designed for creating/selecting/adjusting Eurocode (EN 1991-2)-defined load models.
Vehicles can be selected from the Vehicle library (available when clicking on a Vehicle filed). The Factor is a versatile tool: if the vehicle is defined as a unit, it can be used as a load factor,
or it can consider the code-defined alpha values. The Lane option allows adjusting vehicle relationships. The 1st lane represents the most unfavourable lane, the 2nd lane is the second most
unfavourable, and so on. For all lanes above the 3rd, Other lanes should be used (e.g., if the bridge has 8 lanes, LM1 UDL Unit with Other lanes selection means the vehicle could be placed
simultaneously on all 5 remaining “other” lanes). The Remaining areas option is for surface-type loads, like crowd load, and represents the outer areas of the lane system. The load model will be
applied according to the Lane alignments for Road vehicles. The vehicles on the same lane are placed in the order of their Priority parameter.
• The number of lanes does not have to match in the Load model and the Lane alignments tab, any load model can be applied to any Lane alignments.
• EN 1991-2 4.3.2. defined adjustment factors (alpha) can be easily applied in the Factor column.
• With one load pattern, only one load model can be considered. If the model requires it, use the Wizard or the Mass generation tool to create multiple load patterns with the same geometry but
different load models.
Vehicle library
A Load model handles multiple vehicles, and the Vehicle library is developed to provide these. A vehicle could be Road or Rail type, aiding usability for different Traffic load objects. Nominal width
is the width of the vehicle, which is significant if it is less than the width of the designated lane (in Lane alignment): the vehicle will also be tested on multiple transversal positions within the
lane during the maximum search.
A Vehicle could be defined with the following elements in a proper order:
• Axle load: Point load representation (e.g. LM1 Tandem System). The load intensity is the summation of all-wheel loads. It can be applied on a lane.
• Line load: Line load representation (e.g. SW/2 load system) for a Vehicle with wheels. The load intensity is the summation of the all-wheel loads. It could be parametric by length by setting
different Min. and Max. lengths. It can be applied on a lane.
• Surface load: it can be applied on arbitrary geometry (e.g. Crowd load). This is a mutually exclusive load item, it cannot be applied with other load items (Surface-type vehicle).
• Distance: The distance between the load elements. It could be variable.
If one of these sub-elements is variable, that means the vehicle is parametrical, and this is a Vehicle parameter. The vehicles can also be used on a single line model in a simplified mode. For
example, Tandem System will be simplified to one point load, surface load will be simplified to one line load calculated by the notional width.
Load modifier
Load modifiers have multiple use-cases, such as dynamic amplification or national annex defined design factors, therefore, it is developed to be as versatile as possible.
• If the whole Load model should be multiplied, a constant value could be appropriate.
• If the model requires uneven load level along the stake-out line, variable load multiplier should be used. For example, additional dynamic amplification in the vicinity of expansion joints, EN
1991-2 4.3.3 (3).
Variable load multipliers are constant outside the handled length domain, and linear interpolation is applied between the sample points.
Note: The Variable load modifier is not applied to surface-type vehicles.
Wheel load ratio between the tracks is available for trains, and this is the exact implementation of EN 1991-2 6.3.5.(1).
Lane alignments
Lane alignments are possible lane configurations during the structural lifetime, which are parallel to the stake-out line, and the user-defined left/right boundaries are measured from it. As the
stake-out line could be an almost arbitrary polyline, the lane system could handle turning road design with a good approximation.
To create a lane, its Lane name, Left and Width properties must be defined, and the Right will be filled out automatically. The New button creates a new Lane alignment from the current settings of
the dialog.
In the case of the simple line model, the position of the lane has no significance, but the load level can be determined by the number of the lanes. For carriageway types, three fill options are
available to aid work: Left, Right, Middle alignment.
Using the Applied checkbox, lane alignment could be inactivated. Multiple lane alignment could increase the calculation time; for pre-evaluation of structure, it could be worth temporarily reducing
the active alignments.
Note: Load model reference of the "1^st” lane means the most unfavourable Lane and is unrelated to the order of the Lanes in a Lane alignment.
Calculation options
Load pattern calculation has a high calculation demand and involves a complex optimization process that allows for adjustment. The maximum search uses two different strategies:
Brute force: This method systemically tests all discrete positions and values of vehicle parameters based on given settings. Advantage: if the discretization is set properly, it finds the maximum
solution, or at least, the closest discrete values. Disadvantage: if the number of the parameters is high, it could be very slow, and the exact position of the maximum is usually not found, only the
closest discrete value set. The Number of positions along the stakeout line parameter controls the longitudinal load positioning. Division number of vehicle parameters is used in case of vehicles
that contain Line load or Distance with varying length. The possible discrete length parameters will be generated based on the minimum and maximum length and the division number.
Multistart optimization: This method is based on an optimization algorithm for convex functions, making it possible to get stuck in a local maximum position when used on an arbitrary function. To
reduce the chance of getting stuck, it is started multiple times with different initial values. Advantage: it is continuous, allowing it to find the exact maximum place. For multiparametric systems,
it is usually much faster. Disadvantage: it may not find the maximum. A higher number of Number of restart increases the likelihood of identifying the extreme traffic load position but extends
calculation time. Conversely, a lower value accelerates calculations.
The Maximal step within the lane parameter controls the transversal load positioning when vehicle width and lane width differ.
Load system display and load case generation
For more advanced usage, such as non-linear FEA calculations, the Load pattern cannot be directly used. Therefore, the Load case generation by Load patterns tool was implemented to create the
corresponding load system for the examined point in a specific result.
After opening this edit tool, more inspections can be executed:
• On the mouse hover event over the structure, the corresponding load system of the examined point is displayed.
• On the mouse click event, this load case be added to the Load case generation list, which will be applied for Tab change (e.g. Loads tab).
In the List of requested load cases, the basic properties of a load case can be overviewed, the names can be set, or any request can be deleted.
The Maximal graph generation in a finite element node is based on the Influence graph of the node. Parameter optimization is applied to find the most unfavourable positions, and load parameters.
Calculations steps:
1. Influence line and surface generation
2. Load pattern resultset calculation (position, vehicle parameter calculation)
□ For all lane alignments.
□ For all permutations of lane alignments.
□ For all transversal steps in lane if the lane width is wider than the nominal width of the vehicle.
□ The vehicles are applied one by one, in the Load model defined order.
□ Probe the vehicle in the reverse direction if it is asymmetrical.
□ For Point- and Line-load based vehicle loads, the influence surface will be reduced to an influence line on the stake-out line of the lane. The reducing algorithm transversally integrates the
influence graph under the wheel area.
□ Surface loads are only constrained by the designated area in Load model settings. The area of the surface loads can be calculated trivially by the sign of the influence graph.
3. Requested simultaneous component calculation by the influence graph of the component and the previously determined load systems.
Each of these steps is executed for all requested components in all structural nodes.
The load system contribution is only calculated on the domain. That means, if the unfavourable load position is outside of the carriageway/rails/stake-out line, only the overlapping load system part
will be considered.
In a Load combination, Load pattern results are linearly added to the results after any non-linear calculated Load combinations during the post-processing. If any load system of the Load pattern
results is wanted to be involved in a non-linear FEA calculation, the Load case generation by Load patterns should be used.
Usage tips
The Traffic load item has high computational demand; therefore, a prototype calculation is advised before any mass-generation. Using a prototype load item, the proper settings could be found
iteratively, inspecting the most significant results. After the prototype load item is finalized, the Wizard and Mass-generation tools could be used to create load items for the other result
components. This process could be repeated for different Load models.
One load model can be applied to multiple Traffic load items (in different Load cases). For example, using EN 1991-2, the complex psi value settings of LM1 can be set in the Load combination dialog
by separating the UDL and Tandem System part.
Performance tips
• Edit the moving load for fewer but more relevant load positions.
• Using multiple creep coefficients for materials could lead to multiple calculations (maximum four) for highlighted limit states (U, Sq, Sc, Sf). It would mean 4 times calculation of each traffic
load as well. To avoid this:
□ calculate only one limit state and then clone it to the others (fastest, 4x faster overall), or
□ calculate the result set for the selected limit state and then apply it for the four limit states’ simultaneous components (4x faster resultset calculation).
Load model
• Apply symmetrical vehicles if the model allows (2x faster resultset calculation).
• Reduce the number of vehicle parameters if the model allows.
• If simultaneous results are not necessary, uncheck it (for 6-8x faster influence graph generation and simultaneous result calculation).
• Prefer to use one lengthy line with allowed discontinuity rather than multiple line loads with variable lengths and distances, if the model allows.
• Surface-type vehicle calculation requires surface topology build during influence graph generations, and its simultaneous component calculation has high demand. If the model allows, it can be
replaced by a one-wheel line-load based vehicle with the equivalent width of the lane. These types of vehicles have “Resultant” suffix.
• Lane alignments options affect the resultset calculation phase only
□ Calculation time is linear with the applied lane alignments.
□ The permutations of the Lanes could be high ((No. Lane)! / (No. Lane-3)!)
Calculation options
• Brute-force method’s calculation time is exponential with the number of parameters:
□ Time complexity: “No. of position along the stake-out line” * (“Div. of Vehicle parameters” ^ “No. of Vehicle parameters”).
□ A certain optimization is applied, which greatly reduces the time complexity of the implementation, but the exponential characteristic still remains.
• Calculation time is linear with the number of restarts of the Multistart optimization.
• Calculation time is linear with transversal load positions.
• The stake-out line is not allowed to intersect itself in the top-plane view.
• Line and surface moving loads of the Traffic load are not allowed to overlap in the top-plane view.
• Three vehicles are allowed per lane.
• Line/point-based vehicles are only placed within the lane alignment system, following the stake-out line (e.g. LM2 cannot be placed with arbitrary rotation).
• The simple line model considers a centrical load system because it does not apply torsional influence graph to calculate torsional maximal graph.
• Eurocode-defined horizontal forces, like Braking/Acceleration/Centrifugal/etc. effects, are not implemented in this feature.
• The major edit functions are disabled because of the coupled Moving loads. The creation process is aided by the Pick and Copy properties functions. | {"url":"https://wiki.fem-design.strusoft.com/xwiki/wiki/wiki.fem-design.strusoft.com/view/New%20features/New%20features%20in%20FEM-Design%2023/Loads/","timestamp":"2024-11-02T15:08:37Z","content_type":"application/xhtml+xml","content_length":"98520","record_id":"<urn:uuid:3efa4e6a-65d4-44cf-9b19-fc7818056c79>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00792.warc.gz"} |
Design with Nature
Can you imagine the perfect hike and campout? Beautiful hiking trails, interesting sights, a cozy campground, and a delicious meal? Enjoying time in nature often requires planning. You might decide
where you'll go, how long you'll be gone for, and what you'll bring. In this badge, you'll use math to help you plan and organize a hike and campout. You'll think about all these things as you get
ready for your hiking adventure.
When I've earned this badge, I'll know about different types of maps. I'll know how to calculate distance, pace, elevation changes, and area.
Words To Know
• Adjacent: Touching or next to.
• Area: The space inside a flat shape, found by multiplying length by width. Contour interval: The distance, or elevation change, between each contour line on a topographic map.
• Contour line: A line on a topographic map that shows elevation change. Coordinate plane: A graph with x- and y-axes.
• Decline: Sloping down.
• Elevation: The height above sea level. Incline: Sloping up.
• Index line: A thick contour line on a topographic map, typically every fifth line, that notes the elevation.
• Sea level: Zero elevation, or the point where the ocean meets land.
• Segment: A part of a whole. Terrain: The physical features of a piece of land.
• Topographic map: A map that shows a terrain's elevation, or height, above sea level.
• Walking pace: How fast people usually walk.
• x-axis: The horizontal, or side-to-side, part of a graph.
• y-axis: The vertical, or up-and-down, part of a graph.
Types of Hikes
□ A loop hike is when you never retrace your steps but still end up right back where you started. The route is a circle.
□ An out-and-back hike is when you hike to a point, turn around, and hike back on the same trail. When planning an out-and-back hike, count each segment twice.
□ A lollipop hike is an out-and-back hike with a loop at the end of the "out" segment. You'd hike out on a trail, do a loop, and then hike back.
□ A point-to-point or through hike is like out-and-back, except you only go one way. Someone would pick you up at the end.
Step 1: Find your hiking pace[ ]
Animals walk and run at different speeds. A tortoise moves along slowly. A cheetah races past the other animals. What are the benefits and drawbacks of moving fast? What about the benefits and
drawbacks of moving slow?
As animals, humans also move at different paces. A person's walking pace is how fast they usually walk. It takes some people about 5 minutes to walk a quarter mile, but everyone is different.
When you're planning a hike, it can be helpful to figure out your pace. With others, this can also help your group to move at a speed that's comfortable for everyone.
Choices-do one:
Calculate your pace. Find somewhere safe to walk. Then start a stopwatch and move a quarter mile along the route at your usual pace. Stop the stopwatch: how long did it take you to go a quarter mile?
Multiply that time by 4 to find how long it would take for you to go a mile. If there are 60 minutes in an hour, how many miles could you go in 1 hour? How many could you go in 3 hours?
Compare human and animal paces. Imagine you're hosting the first all-species Olympics, where animals of all kinds can show off their speed! Choose some favorite animals and find out their normal
pace, walking, running, or moving in any way. How fast is an elephant? What about a snail or a leopard? Compare what you find to the average walking pace for a human. You may find that some animals
are very fast for short distances. Others can maintain a good speed for a longer time. If you had a marathon, who would win? The tortoise or the hare? What about a 400-meter sprint?
Calculate and compare different paces. Some people run on hiking trails. How long would it take if you wanted to skip or walk backwards? Find somewhere safe to find out! First, find your normal pace.
Start a stopwatch and move a quarter mile along the route. Stop the stopwatch: How long did it take you? Do it again, this time running, skipping, or doing something else. Repeat as many times as you
want in as many different ways. Then multiply each time by 4 to find your pace for a mile. If there are 60 minutes in an hour, how many miles could you go moving each way? How many miles could you go
in 3 hours?
Step 2: Choose a hiking trail[ ]
Choosing where to go may be the most important part of planning your hiking adventure. Maps can help as they show all different kinds of information, from the weather to the location of forests and
the height of mountains.
If you know where you want to explore, you can usually get a map of trails to help plan your trip. Hiking maps show landmarks and may even show how difficult a hike might be. They show different
trails and the lengths of each segment. A segment is a part of a whole.
Choices-do one:
Choose a trail on a sample map. Look at the sample map on page 5. Identify landmarks to visit. Can you find a route that's about the same number of miles long as the distance you can move in 3 hours?
Look for different types of hikes, like a loop, out-and-back, lollipop, or point-to-point hike.
Choose a trail on a map of your local area. Hiking near home lets you see your local area in a new way. You don't have to be in the woods to go for a hike! Use maps of your neighborhood and identify
local landmarks or places to visit. Then investigate different routes. Can you find a route that's about the same number of miles long as the distance you can move in 3 hours? Then you can share your
map with others and go on a hike!
Choose a trail on any map. If you could go hiking anywhere in the world, where would you go? A rain forest? A desert canyon? The Andes mountains? Find a hiking map from anywhere in the world.
Identify and learn about important cultural, natural, or historical landmarks to visit. Then investigate different routes. Can you find a route that's about the same number of miles long as the
distance you can move in 3 hours?
How To Choose a Hiking Trail
1. Identify any landmarks on the map. What do you want to see on your hike? How much time do you want to spend hiking?
2. Choose a starting point and find a route that is about the same number of miles long as the distance you move at your normal pace in that time. What will you visit? What direction will you
3. Add the lengths of adjacent, or touching, trail segments to calculate the hike's length. 4. Once you find a hike that's the same distance as you can hike in the time, trace it with a marker.
Step 3: Find changes in elevation on a map[ ]
Do you hike faster going uphill or down? What if the trail is rocky? The type of terrain you're on affects how fast you move. Terrain is the physical features of a piece of land, like flat, steep,
rocky, rolling, or wooded.
A topographic map shows the terrain's elevation. Elevation is how high a place is above sea level. Sea level is where the ocean meets land. Mountains have a high elevation. Beaches have a low
elevation. A topographic map includes landmarks, like a regular map, but also has little lines called contour lines that each show elevation. The closer the lines are together, the steeper the
elevation changes. The wider the lines are apart, the flatter the land. One side of the line is uphill; the other side of the line is downhill.
Elevation changes are important to know about when planning a trip. If your trail is very steep, it may take longer to hike it. Elevation changes can also cause changes in temperature and wind-both
things you'd want to know!
Choices-do one:
Calculate elevation changes on a sample topographic map. Choose three points on the map on page 7. Write down each point's elevation and add them to the coordinate plane. Then, find the elevation
change between each point.
Calculate elevation changes on a topographic map of your area. Find a topographic map with the route you chose in Step 2. Choose three points, find their elevations, graph them on a coordinate plane,
and find the elevation change between each point.
Calculate elevation changes on any topographic map. Find a topographic map of anywhere in the world. Where will you go? What will you see? This could be the same place as Step 2 or somewhere new!
Choose three points on the map and find their elevations. Graph them on a coordinate plane. Find the elevation change between each point.
For more fun: Calculate your pace moving from different elevations.
Step 4: Decide how much food to bring[ ]
Girl Scouts make sure to always be prepared! It's a good idea to take food with you, even on a short hike.
Foods like trail mix can help keep you energized on the hike. Trail mix is a combination of dried fruits, seeds, nuts, cereal, and anything else. If you're hiking all day, you may want lunch. And if
you're camping, you might want a special treat like s'mores.
Choices-do one:
Make a snack for your hike. Use a trail mix recipe to figure out how many batches you'll need for the whole troop. Look at how many servings the recipe makes. Then calculate how much of each
ingredient you'll need to feed everyone joining your adventure!
Pack lunch. For a full day of hiking, you'll need lunch. How about a sandwich and some fruit? How many loaves of bread will you need? List all the ingredients for your sandwiches. How much will you
need of each ingredient? Now think about the fruit. If you choose fruit that doesn't come in an easy, single serving format like an apple, how much will you need? If you want to include something
else in your menu, calculate how much of it you will need for everyone.
Enjoy your trip with a treat. As you relax around a campfire, you may want a yummy treat for yourself and the troop. Choose a recipe. How many servings does the recipe make? Figure out how many
batches of the recipe and how much of each ingredient you'll need to make one portion for each person.
Important note: Check for any food allergies and avoid those ingredients for all activity choices. For more fun: Make the snack, lunch, or treat with your troop and enjoy!
Step 5: Pack for your adventure[ ]
If you were going camping, you'd need a tent and sleeping bags. The number of tents would depend on how big they were and how many people they could fit. Each person would need a sleeping bag. You'd
also need supplies like food, water, and flashlights.
You can figure out how big your campsite needs to be by calculating the area. Area is the space inside of a flat object. You can calculate the area by multiplying length and width.
Area = length x width
To make sure there's enough space in your bags or vehicle, you can find their volume. Volume is the amount of space something takes up in three dimensions. You can calculate the volume by multiplying
length and width and height.
Volume = length x width x height
Choices-do one:
Pack your equipment. If you're carrying supplies, will your backpacks be big enough to fit them all? Calculate the volume of your backpack and gear to find out. First, measure and calculate the
volume of your backpack. Then find the volume of each supply in the same unit of measure. Will everything fit? Pack it to see if your calculations are correct.
Load your vehicle. If you're riding in a vehicle to the start of your adventure, find how many vehicles you'll need to transport people. Then, calculate the volume of each vehicle's trunk to see how
much cargo it can hold. First, measure and calculate the trunk's volume. Then calculate the volume of each supply in the same unit of measure. Will all the equipment fit?
Plan your campsite. Figure out how big your campsite needs to be by calculating the area of your tents and sleeping bags. How many people will fit in a tent? Measure and calculate the area of the
bottom of a tent. Then find the area of a sleeping bag and pillow in the same unit of measure. Use graph paper to plan your campsite. How many tents will you need? Draw, cut out, and place rectangles
to see how many tents and sleeping bags fit. Then build your campsite by pitching your tent (or outline it with masking tape), unrolling your sleeping bags (or towels!), and moving them to see if
your plan works!
How Much Water Will You Need?
To stay hydrated, it's recommended that each person drink 2 cups of water every hour during average weather conditions. Can you calculate how much water you'll need for your hike and how heavy it
will be? If the hike is planned for three hours...
1. How many cups of water will each person need?
2. How many ounces of water will each person need?
3. How many quarts of water would each person need?
4. How much would each person's water weigh in pounds?
5. If you have a water bottle, would it hold enough water for your hike? How long would it be before you ran out of water?
Tip: If the water bottle doesn't say how many ounces or cups it can hold, pour water in it from a measuring cup. | {"url":"https://girlscout.fandom.com/wiki/Design_with_Nature","timestamp":"2024-11-02T23:44:18Z","content_type":"text/html","content_length":"176193","record_id":"<urn:uuid:1bdbe47a-0596-4670-8c06-241373cc4a2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00137.warc.gz"} |
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enthusiastic about the quality of the presentation, the immediate feedback, and the positive reinforcement that the software provides. And the price can't be beat. This is a quality program that
every student and school should be considering."
— Seymour S. Burack M.S. Psychologist at Roselle Borough Public Schools, Special Needs Advocate
"I have used Math911 in my remedial Math courses with excellent results. This Math tutorial has been designed to help students struggling with Math. It is very user friendly, so the student with
technology deficiencies has no problems working with the software.
Each section is divided in different levels of expertise so that the student may work at his own pace. The students feel more confident after they start working with Math911.
I especially recall one student who, at the beginning, started with a 25% average but changed it substantially to an impressive 95% average. She was so happy at the end of the semester that she kept
the software to study the next Math course with the help of Math911.
I truly recommend this easy and effective tutorial to ease the pain of students struggling with Mathematics."
"I have tried Professor Weissman's software. It is absolutely invaluable for anyone needing to relearn high-school math FAST! A few weeks ago, I couldn't even do complex subtraction, but now I can do
everything up to — and including — simple Algebra! Khan Academy was okay, but it wasn't interactive enough for me. Math911 guides you every step of the way and provides instant feedback on whether or
not your answer was correct. If it wasn't, then it will show you step by step directions for how to solve the problem. The Math Professor has saved me a year — if not two — of remedial math once I
start college. Thank you, thank you, thank you, Prof!"
"Math911 is an amazing tutorial resource. This user-friendly software promotes active learning through the use of Mastery Learning. Students are encouraged to repeat math problems to build on the
knowledge and skills they already have in Mathematics. The basic software covers Introductory Algebra as well as an College Mathematics Placement Exam Prep and a review of Arithmetic. Students who
have had great difficulty with Math, even with the help of private tutors, should take advantage of this software as a viable learning source. I have witnessed a change in students' attitudes and
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professor here. Take him if you really want to learn math and if you need 114 after." | {"url":"https://math911.com/","timestamp":"2024-11-10T02:48:23Z","content_type":"text/html","content_length":"234319","record_id":"<urn:uuid:c200688c-07b8-4bf6-b95d-198a12fc8250>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00339.warc.gz"} |
4 BIG IDEAS about MULTIPLICATION
Times tables are more important than just memorising sums – they can form the memorable groundwork for children’s greater mathematical understanding.
Having multiplication facts and times tables knowledge as a key part of our ‘mathematical tool box’ is hugely important because we use them constantly in everyday life.
However, instead of traditional approaches using rote learning, which cause children to perform at a significantly lower level than those taught with a focus upon the ‘big picture and connections’
(Boaler, Mathematical Mindsets) let’s look at how we can all engage children with meaning and building strong mental connections and flexibility working with how we know the brain learns best!
Here are 4💡BIG Ideas 💡 and approaches to help your children with multiplication with RELEVANCE and UNDERSTANDING:
Researchers have noted that multiplication and division are conceptually demanding due to the levels of abstraction required and the complex semantic structures they involve (Anghileri, 1989; Clark &
Kamii, 1996; Greer, 1992; Steffe, 1994). Although multiplication tends to be formally introduced in school, research has shown that pre school children can model both grouping and sharing.
Nunes and Bryant (1996) suggest that the simplest form of multiplicative situations that children will meet is probably one in which there is a one-to-many correspondence (e.g. 1 car with 4 wheels)
which relates to ratio, or scale factor, and is the basis for multiplicative, rather than additive thinking.
SKIP COUNTING is so often an early strategy to encourage children to count in equal groups - especially if it’s supported with concrete and pictorial representations BUT…
BEWARE the ‘Mary gad a Little Lamb’ approaches to multiplication table learning. Teaching children to reel off a list of numbers can sound like they have multiplication knowledge: ‘Let’s count in 2s
– 2, 4, 6, 8, 10…’. But when we examine this more closely they are often starting in the same place every time and have essentially just learned to recite the equivalent of a nursery rhyme. Using and
applying this knowledge flexibly then becomes almost impossible.
Skip counting is only one way of thinking about multiplication.
Unitising is simply, where many things are treated as one thing; a packet of 10 chocolate biscuits, for example. They is one packet, but at the same time they are 10. And the thing about unitising,
is that it’s efficient!
The shift from additive to multiplicative thinking is not easy and may take considerable time to achieve as it requires a "cognitive reorganisation on the part of the learners." (Fosnot & Jacob,
Learners should explore how objects can be arranged in equal groups. When describing equally grouped objects, the number of groups and the size of the groups must both be defined. Equal grouping
structure is at the core of multiplicative concepts and thinking:
Equal groups can be represented with a repeated addition expression:
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2
4 + 4 + 4 + 4
8 + 8
Repeated addition is only one way of thinking about multiplication.
A shift from attending to how much is in each group (multiplicand) to how many groups (multiplier) is a critical step and leads to the recognition of the number of groups as a factor.
Moving objects into rows and columns provides better opportunities for exploring the commutative rule than static pictures which are sometimes difficult to see in the two different ways.
For example, the array above could be read as 2 rows of 5, or physically turned to show 5 columns of 2. Regardless of the way you look at it, there remain 10 objects. Therefore, the array illustrates
that 2×5=5×2, which is an example of the commutative property for multiplication.
Being able to apply the commutative property means that the number of multiplication facts that have to be learned is halved. An
efficient way to learn.
hallmark of real multiplicative thinking.
The development of multiplicative thinking requires a long period of time (Clark and Kamii 1996).
While I do believe that building a conceptual understanding through multiple representations of proportional reasoning is very important, I think it should be explicitly stated that taking this
approach will not necessarily speed up the learning process. Realistically, it could take longer due to the depth of knowledge we are striving for. I’m a firm believer that anything worthwhile takes
time and effort. Our mathematical understandings are no exception.
If one car uses 4 tyres and we want to know how many tyres are needed for 6 cars the thinking involved is PROPORTIONALITY.
Often multiplication is only taught as REPEATED ADDITION and the PROPORTIONAL REASONING has been sacrificed. Don’t do that! Try this:
The transition from additive to multiplicative thinking, however, constitutes an obstacle for many children (Ehlert et al. 2013; Gaidoschik et al. 2018; Götze 2019a, b; Moser Opitz 2013; Siemon et
al. 2005).
The qualitative studies of Breed (2011) and Götze (2019a, b) indicated that such language-responsive instruction might help children to develop multiplicative thinking as unitising. | {"url":"https://www.fluencywithnumbers.com/blogs/news/4-big-ideas-about-multiplication","timestamp":"2024-11-03T15:19:25Z","content_type":"text/html","content_length":"136091","record_id":"<urn:uuid:ffbfd3a3-94c5-40a1-9ed2-2e49d8a1bb39>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00484.warc.gz"} |
FeynMG: Automating particle physics calculations in scalar-tensor theories
Sevillano Munoz, Sergio (2023) FeynMG: Automating particle physics calculations in scalar-tensor theories. PhD thesis, University of Nottingham.
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This thesis explores the automation of the analysis of scalar-tensor theories at subatomic scales. For this, we make use of the fact that, when appended to the Standard Model, these theories can be
expressed as standard gravity plus a Beyond the Standard Model theory. Therefore, studying the modifications that scalar-tensor theories have on a matter sector in this description requires the use
of quantum field theory.
For this, we first investigate the origin of long-range interactions (fifth forces) in scalar-tensor theories of gravity both working in the Einstein and the Jordan-frames. We focus on theories of
Brans-Dicke type in which an additional scalar field is coupled directly to the Ricci scalar of General Relativity. In our exploration of the Jordan frame calculation, we find that a specific gauge
choice called scalar-harmonic gauge is convenient to perform a consistent linearization of the gravitational sector in the weak-field limit, which gives rise to a kinetic mixing between the
non-minimally coupled scalar field and the graviton. It is through this mixing that a fifth force can arise between matter fields. We are then able to compute the matrix elements for fifth-force
exchanges obtaining frame-independent results. Moreover, we also show the pivotal role that sources of explicit scale symmetry breaking in the matter sector play in admitting fifth-force couplings.
Irrespectively of the selected frame, we find the calculation to be very time-consuming and model dependent, motivating the development of computational tools for these derivations. The ability to
represent perturbative expansions of interacting quantum field theories in terms of simple diagrammatic rules has revolutionized calculations in particle physics (and elsewhere). Moreover, these
rules are readily automated, a process that has catalysed the rise of symbolic algebra packages. However, in the case of extended theories of gravity, such as scalar-tensor theories, it is necessary
to precondition the Lagrangian to apply this automation or, at the very least, to take advantage of existing software pipelines.
In this context, we present the Mathematica package FeynMG, which works in conjunction with the well-known package FeynRules. FeynMG takes as inputs the FeynRules model file for a non-gravitational
theory and a user-supplied gravitational Lagrangian. FeynMG provides functionality that inserts the minimal gravitational couplings of the degrees of freedom specified in the model file, determines
the couplings of the additional tensor and scalar degrees of freedom (the metric and the scalar field from the gravitational sector), and preconditions the resulting Lagrangian so that it can be
passed to FeynRules, either directly or by outputting an updated FeynRules model file. The Feynman rules can then be determined and output through FeynRules, using existing universal output formats
and interfaces to other analysis packages, such as MadGraph. Therefore, in combination with these additional analysis packages, FeynMG will make possible to test for modifications to the Standard
Model due to scalar-tensor theories in particle colliders.
Actions (Archive Staff Only) | {"url":"http://eprints.nottingham.ac.uk/76652/","timestamp":"2024-11-05T17:32:01Z","content_type":"application/xhtml+xml","content_length":"34241","record_id":"<urn:uuid:13d05a7b-77dd-4bef-9a09-18d9b5e7c858>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00103.warc.gz"} |
fix box/relax command
fix box/relax command
fix ID group-ID box/relax keyword value ...
• ID, group-ID are documented in fix command
• box/relax = style name of this fix command
one or more keyword value pairs may be appended
keyword = iso or aniso or tri or x or y or z or xy or yz or xz or couple or nreset or vmax or dilate or scaleyz or scalexz or scalexy or fixedpoint
iso or aniso or tri value = Ptarget = desired pressure (pressure units)
x or y or z or xy or yz or xz value = Ptarget = desired pressure (pressure units)
couple = none or xyz or xy or yz or xz
nreset value = reset reference cell every this many minimizer iterations
vmax value = fraction = max allowed volume change in one iteration
dilate value = all or partial
scaleyz value = yes or no = scale yz with lz
scalexz value = yes or no = scale xz with lz
scalexy value = yes or no = scale xy with ly
fixedpoint values = x y z
x,y,z = perform relaxation dilation/contraction around this point (distance units)
fix 1 all box/relax iso 0.0 vmax 0.001
fix 2 water box/relax aniso 0.0 dilate partial
fix 2 ice box/relax tri 0.0 couple xy nreset 100
Apply an external pressure or stress tensor to the simulation box during an energy minimization. This allows the box size and shape to vary during the iterations of the minimizer so that the final
configuration will be both an energy minimum for the potential energy of the atoms, and the system pressure tensor will be close to the specified external tensor. Conceptually, specifying a positive
pressure is like squeezing on the simulation box; a negative pressure typically allows the box to expand.
The external pressure tensor is specified using one or more of the iso, aniso, tri, x, y, z, xy, xz, yz, and couple keywords. These keywords give you the ability to specify all 6 components of an
external stress tensor, and to couple various of these components together so that the dimensions they represent are varied together during the minimization.
Orthogonal simulation boxes have 3 adjustable dimensions (x,y,z). Triclinic (non-orthogonal) simulation boxes have 6 adjustable dimensions (x,y,z,xy,xz,yz). The create_box, read data, and
read_restart commands specify whether the simulation box is orthogonal or non-orthogonal (triclinic) and explain the meaning of the xy,xz,yz tilt factors.
The target pressures Ptarget for each of the 6 components of the stress tensor can be specified independently via the x, y, z, xy, xz, yz keywords, which correspond to the 6 simulation box
dimensions. For example, if the y keyword is used, the y-box length will change during the minimization. If the xy keyword is used, the xy tilt factor will change. A box dimension will not change if
that component is not specified.
Note that in order to use the xy, xz, or yz keywords, the simulation box must be triclinic, even if its initial tilt factors are 0.0.
When the size of the simulation box changes, all atoms are re-scaled to new positions, unless the keyword dilate is specified with a value of partial, in which case only the atoms in the fix group
are re-scaled. This can be useful for leaving the coordinates of atoms in a solid substrate unchanged and controlling the pressure of a surrounding fluid.
The scaleyz, scalexz, and scalexy keywords control whether or not the corresponding tilt factors are scaled with the associated box dimensions when relaxing triclinic periodic cells. The default
values yes will turn on scaling, which corresponds to adjusting the linear dimensions of the cell while preserving its shape. Choosing no ensures that the tilt factors are not scaled with the box
dimensions. See below for restrictions and default values in different situations. In older versions of LAMMPS, scaling of tilt factors was not performed. The old behavior can be recovered by setting
all three scale keywords to no.
The fixedpoint keyword specifies the fixed point for cell relaxation. By default, it is the center of the box. Whatever point is chosen will not move during the simulation. For example, if the lower
periodic boundaries pass through (0,0,0), and this point is provided to fixedpoint, then the lower periodic boundaries will remain at (0,0,0), while the upper periodic boundaries will move twice as
far. In all cases, the particle positions at each iteration are unaffected by the chosen value, except that all particles are displaced by the same amount, different on each iteration.
Applying an external pressure to tilt dimensions xy, xz, yz can sometimes result in arbitrarily large values of the tilt factors, i.e. a dramatically deformed simulation box. This typically indicates
that there is something badly wrong with how the simulation was constructed. The two most common sources of this error are applying a shear stress to a liquid system or specifying an external shear
stress tensor that exceeds the yield stress of the solid. In either case the minimization may converge to a bogus conformation or not converge at all. Also note that if the box shape tilts to an
extreme shape, LAMMPS will run less efficiently, due to the large volume of communication needed to acquire ghost atoms around a processor’s irregular-shaped subdomain. For extreme values of tilt,
LAMMPS may also lose atoms and generate an error.
Performing a minimization with this fix is not a mathematically well-defined minimization problem. This is because the objective function being minimized changes if the box size/shape changes. In
practice this means the minimizer can get “stuck” before you have reached the desired tolerance. The solution to this is to restart the minimizer from the new adjusted box size/shape, since that
creates a new objective function valid for the new box size/shape. Repeat as necessary until the box size/shape has reached its new equilibrium.
The couple keyword allows two or three of the diagonal components of the pressure tensor to be “coupled” together. The value specified with the keyword determines which are coupled. For example, xz
means the Pxx and Pzz components of the stress tensor are coupled. Xyz means all 3 diagonal components are coupled. Coupling means two things: the instantaneous stress will be computed as an average
of the corresponding diagonal components, and the coupled box dimensions will be changed together in lockstep, meaning coupled dimensions will be dilated or contracted by the same percentage every
timestep. The Ptarget values for any coupled dimensions must be identical. Couple xyz can be used for a 2d simulation; the z dimension is simply ignored.
The iso, aniso, and tri keywords are simply shortcuts that are equivalent to specifying several other keywords together.
The keyword iso means couple all 3 diagonal components together when pressure is computed (hydrostatic pressure), and dilate/contract the dimensions together. Using “iso Ptarget” is the same as
specifying these 4 keywords:
x Ptarget y Ptarget z Ptarget couple xyz
The keyword aniso means x, y, and z dimensions are controlled independently using the Pxx, Pyy, and Pzz components of the stress tensor as the driving forces, and the specified scalar external
pressure. Using “aniso Ptarget” is the same as specifying these 4 keywords:
x Ptarget y Ptarget z Ptarget couple none
The keyword tri means x, y, z, xy, xz, and yz dimensions are controlled independently using their individual stress components as the driving forces, and the specified scalar pressure as the external
normal stress. Using “tri Ptarget” is the same as specifying these 7 keywords:
x Ptarget y Ptarget z Ptarget xy 0.0 yz 0.0 xz 0.0 couple none
The vmax keyword can be used to limit the fractional change in the volume of the simulation box that can occur in one iteration of the minimizer. If the pressure is not settling down during the
minimization this can be because the volume is fluctuating too much. The specified fraction must be greater than 0.0 and should be << 1.0. A value of 0.001 means the volume cannot change by more than
1/10 of a percent in one iteration when couple xyz has been specified. For any other case it means no linear dimension of the simulation box can change by more than 1/10 of a percent.
With this fix, the potential energy used by the minimizer is augmented by an additional energy provided by the fix. The overall objective function then is:
\[E = U + P_t \left(V-V_0 \right) + E_{strain}\]
where U is the system potential energy, \(P_t\) is the desired hydrostatic pressure, \(V\) and \(V_0\) are the system and reference volumes, respectively. \(E_{strain}\) is the strain energy
expression proposed by Parrinello and Rahman (Parrinello1981). Taking derivatives of E w.r.t. the box dimensions, and setting these to zero, we find that at the minimum of the objective function, the
global system stress tensor P will satisfy the relation:
\[\mathbf P = P_t \mathbf I + {\mathbf S_t} \left( \mathbf h_0^{-1} \right)^t \mathbf h_{0d}\]
where I is the identity matrix, \(\mathbf{h_0}\) is the box dimension tensor of the reference cell, and :\(\mathbf{h_{0d}}\) is the diagonal part of \(\mathbf{h_0}\). \(\mathbf{S_t}\) is a symmetric
stress tensor that is chosen by LAMMPS so that the upper-triangular components of P equal the stress tensor specified by the user.
This equation only applies when the box dimensions are equal to those of the reference dimensions. If this is not the case, then the converged stress tensor will not equal that specified by the user.
We can resolve this problem by periodically resetting the reference dimensions. The keyword nreset controls how often this is done. If this keyword is not used, or is given a value of zero, then the
reference dimensions are set to those of the initial simulation domain and are never changed. A value of nstep means that every nstep minimization steps, the reference dimensions are set to those of
the current simulation domain. Note that resetting the reference dimensions changes the objective function and gradients, which sometimes causes the minimization to fail. This can be resolved by
changing the value of nreset, or simply continuing the minimization from a restart file.
As normally computed, pressure includes a kinetic- energy or temperature-dependent component; see the compute pressure command. However, atom velocities are ignored during a minimization, and the
applied pressure(s) specified with this command are assumed to only be the virial component of the pressure (the non-kinetic portion). Thus if atoms have a non-zero temperature and you print the
usual thermodynamic pressure, it may not appear the system is converging to your specified pressure. The solution for this is to either (a) zero the velocities of all atoms before performing the
minimization, or (b) make sure you are monitoring the pressure without its kinetic component. The latter can be done by outputting the pressure from the pressure compute this command creates (see
below) or a pressure compute you define yourself.
Because pressure is often a very sensitive function of volume, it can be difficult for the minimizer to equilibrate the system the desired pressure with high precision, particularly for solids. Some
techniques that seem to help are (a) use the “min_modify line quadratic” option when minimizing with box relaxations, (b) minimize several times in succession if need be, to drive the pressure closer
to the target pressure, (c) relax the atom positions before relaxing the box, and (d) relax the box to the target hydrostatic pressure before relaxing to a target shear stress state. Also note that
some systems (e.g. liquids) will not sustain a non-hydrostatic applied pressure, which means the minimizer will not converge.
This fix computes a temperature and pressure each timestep. The temperature is used to compute the kinetic contribution to the pressure, even though this is subsequently ignored by default. To do
this, the fix creates its own computes of style “temp” and “pressure”, as if these commands had been issued:
compute fix-ID_temp group-ID temp
compute fix-ID_press group-ID pressure fix-ID_temp virial
See the compute temp and compute pressure commands for details. Note that the IDs of the new computes are the fix-ID + underscore + “temp” or fix_ID + underscore + “press”, and the group for the new
computes is the same as the fix group. Also note that the pressure compute does not include a kinetic component.
Note that these are NOT the computes used by thermodynamic output (see the thermo_style command) with ID = thermo_temp and thermo_press. This means you can change the attributes of this fix’s
temperature or pressure via the compute_modify command or print this temperature or pressure during thermodynamic output via the thermo_style custom command using the appropriate compute-ID. It also
means that changing attributes of thermo_temp or thermo_press will have no effect on this fix.
Restart, fix_modify, output, run start/stop, minimize info
No information about this fix is written to binary restart files.
The fix_modify temp and press options are supported by this fix. You can use them to assign a compute you have defined to this fix which will be used in its temperature and pressure calculation, as
described above. Note that as described above, if you assign a pressure compute to this fix that includes a kinetic energy component it will affect the minimization, most likely in an undesirable
If both the temp and press keywords are used in a single thermo_modify command (or in two separate commands), then the order in which the keywords are specified is important. Note that a pressure
compute defines its own temperature compute as an argument when it is specified. The temp keyword will override this (for the pressure compute being used by fix box/relax), but only if the temp
keyword comes after the press keyword. If the temp keyword comes before the press keyword, then the new pressure compute specified by the press keyword will be unaffected by the temp setting.
This fix computes a global scalar which can be accessed by various output commands. The scalar is the pressure-volume energy, plus the strain energy, if it exists, as described above. The energy
values reported at the end of a minimization run under “Minimization stats” include this energy, and so differ from what LAMMPS normally reports as potential energy. This fix does not support the
fix_modify energy option, because that would result in double-counting of the fix energy in the minimization energy. Instead, the fix energy can be explicitly added to the potential energy using one
of these two variants:
variable emin equal pe+f_1
variable emin equal pe+f_1/atoms
No parameter of this fix can be used with the start/stop keywords of the run command.
This fix is invoked during energy minimization, but not for the purpose of adding a contribution to the energy or forces being minimized. Instead it alters the simulation box geometry as described
Only dimensions that are available can be adjusted by this fix. Non-periodic dimensions are not available. z, xz, and yz, are not available for 2D simulations. xy, xz, and yz are only available if
the simulation domain is non-orthogonal. The create_box, read data, and read_restart commands specify whether the simulation box is orthogonal or non-orthogonal (triclinic) and explain the meaning of
the xy,xz,yz tilt factors.
The scaleyz yes and scalexz yes keyword/value pairs can not be used for 2D simulations. scaleyz yes, scalexz yes, and scalexy yes options can only be used if the second dimension in the keyword is
periodic, and if the tilt factor is not coupled to the barostat via keywords tri, yz, xz, and xy.
The keyword defaults are dilate = all, vmax = 0.0001, nreset = 0.
(Parrinello1981) Parrinello and Rahman, J Appl Phys, 52, 7182 (1981). | {"url":"https://docs.lammps.org/stable/fix_box_relax.html","timestamp":"2024-11-12T16:35:20Z","content_type":"text/html","content_length":"66426","record_id":"<urn:uuid:0145a9ad-0ba3-496b-a48a-99fc012a2f8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00469.warc.gz"} |
Ordering and topological defects in solids with quenched randomness
Document Type
Eugene M. Chudnovsky
Subject Categories
Nanomagnatism; Quenched Randomness; Random Magnets; Topological Defects
We explore multiple different examples of quenched randomness in systems with a continuous order parameter. In all these systems, it is shown that understanding the effects of topology is critical to
the understanding of the effects of quenched randomness.
We consider n-component fixed-length order parameter interacting with a weak random field in d = 1,2,3 dimensions. Relaxation from the initially ordered state and spin-spin correlation functions have
been studied on lattices containing hundreds of millions sites. At n - 1 < d presence of topological structures leads to metastability, with the final state depending on the initial condition. At n -
1 > d, when topological objects are absent, the final, lowest-energy, state is independent of the initial condition. It is characterized by the exponential decay of correlations that agrees
quantitatively with the theory based upon the Imry-Ma argument. In the borderline case of n - 1 = d, when topological structures are non-singular, the system possesses a weak metastability with the
Imry-Ma state likely to be the global energy minimum.
We study random-field xy spin model at T = 0 numerically on lattices of up to 1000 x 1000 x 1000 spins with the accent on the weak random field. Our numerical method is physically equivalent to slow
cooling in which the system is gradually losing the energy and relaxing to an energy minimum. The system shows glass properties, the resulting spin states depending strongly on the initial
conditions. Random initial condition for the spins leads to the vortex glass (VG) state with short-range spin-spin correlations defined by the average distance between vortex lines. Collinear and
some other vortex-free initial conditions result in the vortex-free ferromagnetic (F) states that have a lower energy. The energy difference between the F and VG states correlates with vorticity of
the VG state. Correlation functions in the F states agree with the Larkin-Imry-Ma theory at short distances. Hysteresis curves for weak random field are dominated by topologically stable spin walls
raptured by vortex loops. We find no relaxation paths from the F, VG, or any other states to the hypothetical vortex-free state with zero magnetization.
XY and Heisenberg spins, subjected to strong random fields acting at few points in space with concentration c[r] << 1, are studied numerically on 3d lattices containing over four million sites.
Glassy behavior with strong dependence on initial conditions is found. Beginning with a random initial orientation of spins, the system evolves into ferromagnetic domains inversely proportional to c
[r] in size. The area of the hysteresis loop, m(H), scales as c[r]^2. These findings are explained by mapping the effect of strong dilute random field onto the effect of weak continuous random field.
Our theory applies directly to ferromagnets with magnetic impurities, and is conceptually relevant to strongly pinned vortex lattices in superconductors and pinned charge density waves.
The random-anisotropy Heisenberg model is numerically studied on lattices containing over ten million spins. The study is focused on hysteresis and metastability due to topological defects, and is
relevant to magnetic properties of amorphous and sintered magnets. We are interested in the limit when ferromagnetic correlations extend beyond the size of the grain inside which the magnetic
anisotropy axes are correlated. In that limit the coercive field computed numerically roughly scales as the fourth power of the random anisotropy strength and as the sixth power of the grain size.
Theoretical arguments are presented that provide an explanation of numerical results. Our findings should be helpful for designing amorphous and nanosintered materials with desired magnetic
Recommended Citation
Proctor, Thomas Chapman, "Ordering and topological defects in solids with quenched randomness" (2015). CUNY Academic Works. | {"url":"https://academicworks.cuny.edu/gc_etds/1100/","timestamp":"2024-11-07T12:51:24Z","content_type":"text/html","content_length":"42145","record_id":"<urn:uuid:6734a375-86e8-4c10-b6fb-37f74395efc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00554.warc.gz"} |
Multifactor Productivity Formula
Multifactor Productivity YouTube
Multifactor Productivity Formula. Whereas the partial factor productivity formula uses one single input, the multifactor productivity formula is the ratio of total. Web the mfp formula is:
Multifactor Productivity YouTube
Whereas the partial factor productivity formula uses one single input, the multifactor productivity formula is the ratio of total. Web the mfp formula is: Cost per unit = output units / (labor input
+ capital input + materials input) here’s an explanation of each variable: Web multifactor productivity (mfp) is a measure of economic performance that compares the amount of output to the amount of
combined inputs used to produce that output.
Whereas the partial factor productivity formula uses one single input, the multifactor productivity formula is the ratio of total. Whereas the partial factor productivity formula uses one single
input, the multifactor productivity formula is the ratio of total. Web multifactor productivity (mfp) is a measure of economic performance that compares the amount of output to the amount of combined
inputs used to produce that output. Cost per unit = output units / (labor input + capital input + materials input) here’s an explanation of each variable: Web the mfp formula is: | {"url":"https://brock.infipereira.gov.co/multifactor-productivity-formula.html","timestamp":"2024-11-01T23:42:48Z","content_type":"text/html","content_length":"19320","record_id":"<urn:uuid:4db54952-3a9c-4aae-a64b-1828331c4138>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00607.warc.gz"} |
MaffsGuru.com - Making maths enjoyableApplications of the inverse matrix: solving simultaneous linear equations
This video is part of the Year 12 Further Maths course and the Matrices module. Having explained in a previous video how to find the inverse of a matrix, I now look at how we can use the inverse to
solve simultaneous equations. I recap what it means to solve equations and what it means to have one solution, no solutions and infinite numbers of solutions. There are lots of worked examples which
are explained in an easy to understand way.
LEGAL STUFF (VCAA)
VCE Maths exam question content used by permission, ©VCAA. The VCAA is not affiliated with, and does not endorse, this video resource. VCE® is a registered trademark of the VCAA. Past VCE exams and
related content can be accessed at | {"url":"https://maffsguru.com/videos/applications-of-the-inverse-matrix-solving-simultaneous-linear-equations/","timestamp":"2024-11-14T08:57:35Z","content_type":"text/html","content_length":"33043","record_id":"<urn:uuid:caf6edb4-b60b-4ff0-a263-b664ffaeb02c>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00107.warc.gz"} |
A Novel Automatic Algorithm for Estimating the Jet Engine Blade Number of Insufficient JEM Signals
I. Introduction
Modern multifunction radars can execute numerous disparate tasks, aided by advances in phased array technology that is electronically steered. The typical radar system is required to search a given
volume for new targets and detect detailed information from multiple scans of target tracks. Recently, radar systems have also required the performance of sequential functions, such as target
identification, classification, and missile guidance. One of the major issues in this multifunction radar system is time resource allocation to maximize the radar’s ability [
]. The total radar time budget of one radar system must be shared between each function. The multifunction capability of radar can be enhanced by reducing the operation time of each specific
function. In general, target recognition from large data that requires classification is the most time-consuming aspect of measurement and analysis. If target recognition is more efficiently
performed in a limited time environment, the remaining time can be allocated for other tasks. Therefore, effective radar resource management can be enabled if the target information is recognized
from an insufficient signal.
Among target recognition technologies, jet engine modulation (JEM), which is induced by electromagnetic scattering from a rotating jet engine compressor, is the leading micro-Doppler phenomenon that
can impart frequency modulation to the radar target signature [
]. The dwell time, also called the measurement time, for the JEM analysis was specified in [
]. The total dwell time must include two rotation cycles of the jet engine compressor to separate the spool line, indicating one rotation frequency. However, if the dwell time is too short, the
frequency resolution can be insufficient to separate the spool lines. This causes an incorrect analysis for obtaining the blade number of the jet engine. To overcome this drawback, our algorithm [
] introduced an effective reconstruction of insufficient JEM signals based on the compressed sensing (CS) method. Although a state-of-the-art (SOA) algorithm can achieve an accurate estimation of the
blade number for various JEM signals, further enhancements are still needed. First, the CS method cannot be applied in a measured JEM signal for which the spectrum becomes relatively complicated,
adding to other nonzero spectral components. Second, the processing time for reconstructing insufficient JEM signals is increased using the linear programming optimization method. Therefore, the
algorithm is not a real-time application for non-cooperative target recognition (NCTR).
This study describes a novel automatic algorithm for estimating the jet engine blade number from insufficient JEM signals. Among the various signal decomposition methods, we employed empirical mode
decomposition (EMD) because of its data-driven characteristics for which there is no prior assumption on a given input signal [
]. However, EMD is restricted to completely extracting the first harmonic component because of its attributes as a dyadic filter bank. Thus, EMD is modified by inserting an adaptive low-pass filter
(LPF) whose cutoff frequency is given as the fundamental chopping frequency (FCF) that can be extracted using automated harmonic selection rules. To obtain the refined autocorrelation function (ACF),
the decomposed intrinsic mode functions (IMFs) derived from the modified EMD operation were combined. Finally, the blade number of the jet engine was estimated using the peaks detected from the ACF
because the blade number is the number of intervals between the outstanding peaks within the spool peaks. The approach proposed in this study is significant because it enables reliable estimation
despite the insufficient JEM signal. In addition, the proposed algorithm is innovative due to its exclusive use in the time-domain method, not the frequency-domain method. Since previous studies have
used the spectrum in the frequency domain to estimate the blade number, the proposed algorithm can estimate the blade number using only time-domain methods, such as autocorrelation (AC). The
application of the proposed algorithm to insufficient JEM signals demonstrates that the novel automatic algorithm presented in this study improves the accuracy of JEM analysis, and its application is
expected to enhance the efficiency of radar resource management.
The rest of the paper is organized as follows. Section II presents the proposed algorithm for estimating the blade number from insufficient JEM signals. In Section III, the measured JEM signals are
examined to validate the applicability of the proposed algorithm. Finally, conclusions are discussed in Section IV.
II. Comparison between Sufficient and Insufficient Jem Signals
The blade number can be determined using the blade domain signal, which is converted from the JEM spectrum, and usually has redundant values by rounding off the operation after dividing the real
frequencies by the one rotation frequency f[R], which is a reciprocal of the spool rate (one rotation period). Thus, to estimate the blade number exactly, the spool rate and JEM spectrum should be
clearly obtained from the measured JEM signal.
The spool rate imparts fundamental periodicity to the JEM signals and plays a key role in calculating the blade number. In the general process of estimating the spool rate, the time domain JEM signal
is first used to calculate the cepstrum by (
) as
, and
are the Fourier transform operator, inverse Fourier transform operator, and the JEM signal in the time domain, respectively [
]. Since the spool rate determines the fundamental periodicity of the JEM signal, the JEM spectrum comprises spool line spectra, and the cepstrum has an outstanding peak. Then, a certain threshold is
determined to designate the time samples as the peak, with cepstrum values above the threshold. Finally, the spool rate is estimated using the detected peak.
Fig. 1
represents the spool rate estimation using the cepstrum of sufficient and insufficient JEM signals. The JEM signals were obtained from the electromagnetic simulation of realistic jet engine models [
]. Because the rotation speed of a jet engine is 100 Hz, the sufficient time is 20 ms, and the insufficient time is appropriately set to 12 ms.
Fig. 1(a)
shows the cepstrum of the sufficient JEM signal, and the spool rate was estimated correctly as 100 Hz. In contrast,
Fig. 1(b)
represents the cepstrum of the insufficient JEM signal, and it is difficult to estimate the spool rate because no peak is related to the spool rate.
Fig. 2
represents the JEM spectrum of the sufficient and insufficient JEM signals. The JEM spectrum is determined according to the dwell time of a JEM signal. If the dwell time of a JEM signal is too short,
the frequency resolution can be insufficient to separate the spool lines. In contrast to the JEM spectrum of the sufficient JEM signal in
Fig. 2(a)
, an insufficient JEM signal cannot obtain accurate spectral lines due to the difference in the frequency resolution marked in
Fig. 2(b)
. Therefore, the insufficient JEM signal cannot estimate the blade number of a jet engine owing to the incorrect estimation of the spool rate and the JEM spectrum.
III. Proposed Algorithm for Estimating the Blade Number
To exactly estimate the blade number for various JEM signals, it is necessary to obtain an accurate spool rate and JEM spectrum with sufficient frequency resolution. As mentioned in Section II-1,
however, JEM signals with insufficient time cannot accurately yield the spool rate and sufficiently dense frequency resolution. Thus, the blade number is estimated via the following procedures for
insufficient JEM signals (also summarized in
Fig. 3
• Step 1. Extraction of the FCF using the harmonic selection rule.
• Step 2. Selection of the cutoff of the preprocessing LPF as the FCF.
• Step 3. Decomposition of the signal filtered by the LPF into a finite number of IMFs using EMD.
• Step 4. Application of the ACF to the desired combination of extracted IMFs to obtain a refined AC waveform.
• Step 5. Acquisition of the blade number using the detected peaks of the ACF.
In Step 1, the analysis focused on the extraction of the fundamental chopping harmonic, which is the key role of the proposed algorithm. The fundamental chopping harmonic has a larger amplitude than
other JEM line spectra in many cases of measured JEM signals [
], but this is not always true; thus, the harmonic selection rule is proposed to extract the fundamental chopping harmonic frequency. First, the original frequency domain signals are changed into
integer values by a rounding-off operation, as shown in
Fig. 4(b)
. Among the frequency domain signals, the peaks that are near zero Doppler are removed because it is not the frequency domain signal that we are interested in. In addition, when the harmonics are
more than the spectral line of zero Doppler, the corresponding frequency domain signals are eliminated. From the remaining frequency domain signal, only 20 harmonic components (line spectra) are
empirically extracted in descending order of spectral amplitude, as shown in
Fig. 4(c)
. Then, the preliminary candidate of the FCF is obtained based on the maximum spectral line. If there is no half frequency harmonic, the preliminary candidate is directly designated as the FCF.
However, when there is a half, third, or fifth frequency harmonic, the corresponding frequency is selected as the FCF. As shown in
Fig. 4(d)
, the FCF can be extracted using these harmonic selection rules.
In Step 2, the LPF whose cutoff frequency is selected as the FCF is inserted as a preprocessor of EMD to supplement its filter bank property and completely extract the first chopping harmonic
component. According to [
], the fundamental chopping harmonic plays an important role in reconstructing noisy and insufficient JEM signals, and the decomposition of the JEM signals is conducted based on the extraction of
this spectral component, which is directly related to the blade number in the first rotor stage. The cutoff frequency of the LPF is determined to be the FCF extracted from Step 1.
Fig. 5(b)
presents the resultant spectrum after applying the original JEM spectrum of
Fig. 5(a)
to the preprocessing LPF.
In Step 3, EMD, a data-driven decomposition method for nonlinear and non-stationary signals [
], is applied to the signal derived from Step 2. Due to preprocessing filtering, the first chopping harmonic component is automatically assigned to the EMD filter of the number 1. Also, IMF 1 has a
center frequency following the cutoff frequency and contains the first chopping harmonic, which denotes an effective JEM component [
]. Since the EMD filter of number 1 operates as a high-pass filter, EMD combined with the LPF behaves equivalently like a band-pass filter with the same center frequency as the FCF. In addition, the
last IMF with the slowest oscillatory mode corresponds to the zero-Doppler component and is also required to obtain the well-presented ACF in the next step.
Fig. 5(c)
shows the spectrum of IMF 1 combined with the last IMF resulting from the modified EMD operation to the filtered spectrum.
In Step 4, the ACF in the time domain is utilized to observe the correlation properties of the combined insufficient JEM signal. The JEM AC data are obtained using an unbiased AC sequence estimation
and normalization by (
) as
is the combined complex JEM data in the time domain, and
is the total length of the JEM data [
]. The ACF is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise. In this study, the JEM ACF yields a refined AC waveform by applying it
to the combined signal with IMF 1 and the last IMF, which denotes the first harmonic component and the zero-Doppler component, respectively. This refined AC waveform makes it easy to automatically
extract the blade number on the jet engine via detected peaks within the outstanding peaks, which will be presented in Step 5.
Fig. 6
shows the AC waveform of the measured raw JEM signal with insufficient dwell time and the reconstructed JEM signal with IMF 1, respectively. The AC waveform of the insufficient JEM signal has
ambiguous peaks from the complicated waveform. Unlike the AC waveform shown in
Fig. 6
, useful information can now be obtained from the well-presented AC waveform shown in
Fig. 5(d)
In Step 5, the blade number is estimated from a refined AC waveform. As shown in
Figs. 5(d)
, the JEM AC waveform has generally outstanding peaks uniformly spaced at every spool rate because the outstanding peaks come from a full rotation of the rotor. In addition, regular waveform within
the outstanding peak can be additionally obtained using the combined signal that contains the fundamental chopping harmonic, which is directly related to the chopping rate, the period of a jet engine
blade moving to its adjacent position. Because the interval between adjacent peaks within outstanding peaks denotes the chopping rate, the blade number is finally determined by the number of detected
peaks of the regular waveform within the spool rate. Therefore, for the example case exhibited in
Fig. 5(d)
, the blade number can be estimated as 42 using the proposed algorithm.
The proposed algorithm enables blade number extraction despite the insufficient JEM signal. In addition, the proposed algorithm is innovative because it uses only the ACF, which is a time-domain
method, not a frequency-domain method. Although the spectrum of the insufficient JEM signal lacks a sufficiently dense frequency resolution to extract the blade number, the blade number can be
estimated using the reconstructed JEM signal and the time-domain method.
IV. Verification with Measured Jem Signals
In this section, the measured JEM signals are examined to verify the proposed algorithm. In [
], experimental jet engine models were fabricated, and measurements were conducted using the instrumentation radar system for various aspect angles and engine rotation speeds. A rotating part and the
whole shape of the fabricated jet engine models are shown in
Fig. 7
. The structural information on the jet engine models and measurement parameters is given in
Table 1
. For a fixed radar aspect angle of 50°, we selected two signals from different engine types with different rotating speeds.
Fig. 8
represents the ACF result and JEM spectrum from a type A engine before applying the proposed algorithm to the insufficient signal. Note that the insufficient dwell time was chosen as 1.1 seconds, and
the sufficient dwell time was set as 2 seconds because the rotation speed is 1 Hz (= 60.1 rpm). Although the accurate spool rate was obtained from the outstanding spool peaks shown in
Fig. 8(a)
, we cannot obtain the information within the spool rate from the ambiguous peaks. In addition, it is difficult to obtain accurate spectral lines due to the difference in the frequency resolution
marked in
Fig. 8(b)
. Therefore, the blade number on the jet engine was erroneously estimated, as summarized in
Table 2
. Note that the automatic extraction method [
] can estimate the blade number using the divisor-multiplier rule and the scoring concept in JEM spectral analysis.
Fig. 9
shows the ACF result from a type A engine after processing with the proposed algorithm. As described previously, the ACF was applied to IMF 1 combined with the last IMF (IMF 11 in this case) and can
exhibit a periodic AC waveform. The spool rate, the full rotation period, can be found using the periodically repeating partial group, as marked in the ACF result. The chopping rate, the period when
a blade moves to its adjacent position, can also be estimated from peaks, as marked with the triangle. Using the detected peaks within the spool rate, the blade number at the first rotor can be
obtained, as summarized in
Table 2
. From
Table 2
, we can observe that the blade number can be exactly estimated as 42.
Figs. 10
exhibit the results of a type B engine before and after applying the proposed algorithm to the insufficient signal, respectively. For this type, the insufficient and sufficient dwell times and the
last IMF were chosen as 0.37 seconds, 0.67 seconds, and IMF 10, respectively. Although we cannot obtain useful information on the jet engine model with the insufficient JEM signal shown in
Fig. 10
, the refined ACF result can be obtained by applying the proposed algorithm. With the refined ACF result, we can estimate the blade number at the first rotor from the peak information within the
spool rate in
Fig. 11
. The jet engine number and processing time required by applying the existing method and the proposed method are summarized in
Tables 2
. Note that the average processing time was evaluated by MATLAB on a laptop with a 2.90 GHz Intel Core i7-7500U and 16 GB of RAM. This result shows a different result from that verified through the
simulation signal obtained by the shooting and bouncing rays (SBR) method—existing and proposed algorithms accurately estimate the number of engine blades for the simulated signals. The method in [
], which is an automated algorithm that enables fast information acquisition for only a sufficient JEM signal, cannot obtain accurate jet engine information. The only existing algorithm that can be
applied to the insufficient signal proposed in [
] can accurately acquire jet engine information, but it takes about 324 times longer than the proposed method because it uses the CS method, which includes an optimization process to restore the JEM
signal. However, the proposed algorithm shows a good match with the real parameters. Consequently, the proposed algorithm facilitates a robust and fast estimation of the blade number.
Application results with the measured JEM signals demonstrate that the proposed algorithm is effective in accurately and efficiently estimating jet engine features from insufficient JEM signal.
Particularly, as shown in the verification process, it is possible to extract the blade number with a dwell time of 1.1 seconds less than the dwell time of 2 seconds required for the information
extraction. It is very effective because the radar can allocate about 45% of the remaining time to other functions, such as searching and tracking.
V. Conclusion
The multifunction operation of radar requires finite radar resources to be distributed among various tasks. If JEM is more efficiently performed in an insufficient dwell time environment, the
remaining time can be allocated for other tasks. In this study, we proposed a novel automatic algorithm for estimating the jet engine blade number of insufficient JEM signals. The application of the
EMD combined with the LPF effectively extracted IMF 1 and the last IMF containing the FCF and zero-Doppler. The blade number of the jet engine was estimated using the peaks detected from the ACF
because the blade number is the number of intervals between the outstanding peaks within the spool peaks. Consequently, the proposed algorithm significantly improved the accuracy of JEM analysis and
is novel and innovative due to its exclusive use in the time-domain method, not the frequency-domain method. Furthermore, the proposed algorithm is expected to be effective from the perspective of
radar resource management. In the future, the proposed algorithm will be applied to various JEM signals received from a jet engine mounted on an actual aircraft and verified. | {"url":"https://www.jees.kr/journal/view.php?number=3467","timestamp":"2024-11-05T12:20:25Z","content_type":"application/xhtml+xml","content_length":"115103","record_id":"<urn:uuid:0b25d445-eebc-4ddf-9eb1-7426c230c961>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00096.warc.gz"} |
Risk of Ruin in Trading
Every Forex trader wants to know how risky their strategy is and what is the chance to lose a part of the account or the entire balance. This chance is called the risk of ruin. Since usually Forex
traders know their win/loss ratio and the average size of their winning and losing positions, calculating the risk of ruin should be relatively easy. Or should it? Unfortunately, it isn't that
The risk of ruin is a known problem in probability theory and, to some extent, it can be solved using the laws and formulas of this theory. Assessing the risk of ruin in Forex is a very complex task
that requires a lot of research and calculating powers in exponential dependence on the desired level of accuracy.
The most popular ways to calculate the risk of ruin of a Forex strategy are two static formulas for the fixed position size and fixed fractional position sizing employed currently by many Forex
social networks (e.g., FXSTAT, Myfxbook, and others). The formulas were presented by D.R. Cox and H.D. Miller in The Theory of Stochastic Processes and are publicly available in various sources
(e.g., Minimizing Your Risk of Ruin by David E. Chamness in the August 2009 issue of the Futures Magazine). They may seem very complex at a first glance, but you can actually compute them using a
calculator or an Excel spreadsheet.
Risk of ruin with fixed position size
Fixed position size condition suggests that the Forex trader won't be increasing or decreasing position size with new wins or losses. For example, if one starts with $10,000 account balance and 1
standard lot per position, dropping to $1,000 balance or advancing to $50,000 balance won't change that 1 lot position volume. The probability to lose a part of balance is exponentially proportional
to the standard deviation of the account and is inversely exponentially proportional to the size of this part and the average return per trade. The formula for the risk of ruin is the following:
• e is the Euler's Number (~2.71828),
• A is the average return ratio per trade (e.g., if your positions returned 2% of your account on average, then A = 0.02),
• Z is the part of the account, whose risk of losing you are assessing (e.g., to calculate the risk to lose 40% of an account, Z = 0.4),
• D is the standard deviation of your trades' returns (should be calculated in a relative form, not in currency units).
Example 1:
The risk to lose 10% of the account balance, considering a 2% mean return on a position and a standard deviation of 7%:
Example 2:
The risk of complete ruin (100% of the account balance), considering a 1.5% mean return on a position and a standard deviation of 1%:
It is easy to notice that this formula calculates the risk of ruin for the infinite number of trades. According to it, the risk of loss will always be 100% for the system that currently shows
negative trade expectancy and will always be close enough (but not equal) to 0 for the high enough mean return and low enough standard deviation of returns.
1. It is a very simple method. The formula itself can be computed using a calculator. Standard deviation is trickier, but if the account balance after each trade is known it takes just several
minutes to calculate it manually. A simple cycle may calculate it inside an expert advisor or some other script.
2. The result is rather precise.
3. It can be used to calculate the risk of loss of any fraction of the account.
1. Substitutes the actual "riskiness" of the trades with the standard deviation, which isn't entirely correct.
2. Assumes that the current rate of return and standard deviation won't change over time. A combination of constant rate of return and a fixed position size is almost impossible in real life.
3. Assumes a fixed position size, but losing some part of the trading account may force a Forex trader to decrease the position size (most likely, it won't be possible to use 1 standard lot position
size if the account is down from $10,000 to $800, for example).
4. Shows the risk for the infinite number of trades — no one trades that long.
5. The accuracy isn't perfect even if all assumptions are perfectly valid for a given trading strategy.
Risk of ruin with fixed fractional position sizing
Unlike the fixed position size model, fixed fractional position sizing implies that a trader is risking a fixed fraction of his account per trade (for example, 1%; the actual value doesn't really
matter here), so the profit is proportional to the account size and the loss is inversely proportional to it. As with the fixed position size, the probability to lose a part of the balance is still
inversely exponentially proportional to the size of this part. The dependence of the risk on the standard deviation is still positive, and on the size of the part, it is still negative, but its
nature becomes more complex. The formula for the risk calculation is the following:
• e is the Euler's Number (~2.71828),
• A is the average return ratio per trade (e.g., if the positions returned 14% of the account on average, then A = 0.14),
• Z is the part of the account, for which the risk of losing is being assessed (e.g., to calculate the risk to lose 25% of the account, Z = 0.25),
• D is the standard deviation of the trades' returns (should be calculated in a relative form, not in currency units),
• ln is the natural logarithm.
Below are two examples of the risk of ruin calculation using the fixed fractional position sizing model with the same conditions as for the fixed position size examples above:
Example 1:
Obviously, it is only slightly lower than 44.2% of the fixed position size model.
Example 2:
Since using 100% part of the account balance for the formula would imply that Z = 1, there would be an error in natural logarithm calculation (natural logarithm of 0 is not defined, but it is an
infinitely large negative number). This means that it is impossible to lose the whole account if the strategy has a positive rate of returns and a fixed fractional position sizing is used:
Evidently, this formula shows a smaller risk because it assumes a fractional position sizing, which reduces the position size in case of losses, thus reducing the further losses and so on.
1. It is still a very simple risk of ruin assessment method. Compared to the fixed position size formula, only few new steps are added, but the used data is the same.
2. The result is even more precise for its model of position sizing.
3. Can be used to calculate the risk of loss of any fraction of the account too.
1. The same as with the fixed position size formula, the real risk is substituted with the standard deviation.
2. Assumes a constant rate of return and standard deviation.
3. Calculates the risk for the infinite number of trades.
Gambler's ruin problem using Markov chains
The most intuitive method to calculate risk of loss turns out to be the most difficult one in terms of the actual computations involved. The Forex risk of ruin problem can be defined as a particular
case of the gambler's ruin problem, where a trader (player), starting with an initial account balance (stake), has a certain probability to win a given average profit amount and a different (or,
sometimes, the same) probability to lose a given average loss amount. The trader is competing against the market (an infinitely rich adversary) — so, only the trader can get the account balance
ruined; winning condition may be defined as reaching some target balance higher than the starting one.
More background research for the simple formulas of risk of loss calculation using this method can be found in the Chapter 12 of the Introduction to Probability by Charles M. Grinstead and J. Laurie
Win equals loss, same probabilities
In case when the average profit equals the average loss and the probability to lose equals the probability to win, the task of calculation the exact risk of ruin is extremely simple. The starting
balance is denoted as z and the target balance is denoted as M. The risk of ruining the account before going from z to M then would be:
Example 1:
The starting balance is $10,000 (z). There is a 50/50 probability of losing or gaining $1,000 with each trade. The target balance is $20,000 (M):
Obviously enough, the probability of going up from $10,000 to $20,000 is the same as of going down to $0 here.
Win equals loss, different probabilities
The probability to win (p) — is a chance to win (end up with profit) one position. It would be very interesting if traders could know their exact probabilities to win a particular trade, but instead
the win/loss ratios are to be used here. The number of profitable trades divided by the total number of trades will be used as the probability to win. It is safe to include zero-profit positions
here, since it will be compensated by the average win size, it is just necessary to remember to count such positions when calculating it.
The probability to lose (q) — is a chance to lose (end up with a loss) one position. For the same reasons as above, a plain win/loss ratio is the best estimate that can be used here. q = Number of
losing trades / Total number of trades. Obviously, if zero profit positions were counted in the calculation of p, they aren't to be included here.
If the probability to win a particular trade isn't the same as to lose it, then a bit more complex formula should be used. The probability of winning an average profitable trade is p, the probability
to lose a trade — q (p + q = 1). Then the risk of ruin is calculated as follows:
Example 2:
The starting balance is the same — $10,000 (z), the goal is also the same — $20,000 (M), and the average win/loss is the same $1,000 per trade, but now the probability of winning is 0.55 and the
probability of losing is 0.45 (a trading system with a 5% edge). To simplify calculations, it is totally safe to divide both the starting and target balances by the average win/loss to get z = 10 and
M = 20. The risk of ruin before doubling the balance is calculated as follows:
Evidently, an edge of 5% gives a Forex trader an enormous improvement in reliability of the whole trading system.
Different win/loss size, different probabilities
In reality, a Forex trading strategy rarely operates with an average profit that is equal average loss. The difference in the outcomes of winning and losing positions leads to a much greater
complexity in the risk of ruin calculation. It is pointless to lay out the whole algorithm of calculation in details for this case here. Instead, it is better to provide an overview of the necessary
steps, which can be easily reduced to trivial math/coding problems.
The detailed information on the mathematics of the risk of ruin calculation for the general case (different win/loss size and probabilities) can be found in Kevin Brown's article The Gambler's Ruin.
It is a superb work that provides an excellent explanation of this problem and the ways of solving it.
The trading process may be represented as a closed-loop Markov model with one starting state (the initial balance, e.g., $2,500) and two end-states — $0 and your target balance (e.g., $5,000). There
is also a number k of transitional states, which depends on the greatest common divisor (GCD) of the average win and the average loss. For example, if the average win is $1,000 and the average loss
is $1,500, their GCD is $500, and there are 9 transitional states (k = 9), which include the starting state. The last two transitional states ($4,000 and $4,500) have the probability p of reaching
the target balance, while the first three states ($500, $1,000 and $1,500) have the probability q of ruining the account (reaching $0 balance).
In general, the formula for probability of ruining the whole account before reaching a target balance can be written as follows:
• p is the probability of winning a trade.
• q is the probability of losing a trade.
• The upper row-vector (a losing vector) is of the length k. The first Average loss / GCD (loss step) elements are q, the others are 0.
• The lower row-vector (a total vector) is also of the length k. The first Average loss / GCD (loss step) elements are q, the last Average profit / GCD (win step) elements are p, the rest are 0.
• M^-1 is the inverse matrix of the coefficient matrix M. M is a k×k matrix that has 1 in all of its main diagonal elements. In addition to 1 of the diagonal, each column may contain up to two
non-zero elements: -p — positioned below the main diagonal, with the vertical offset from it equal to the win step; -q — positioned above the main diagonal, with the vertical offset from it equal
to the loss step.
• C[j] is the single-column matrix (vector) of size k where all elements equal 0, except for the element at position j (starting state), which equals 1.
In the above example, the matrix M would look like this:
And if populated with values:
And the vector C[j] would look like this:
While matrix/vector multiplication is trivial, inverting a matrix bigger than 3×3 is not. The easiest way to find the inverse matrix M^-1 using a computer is LU-decomposition. Using the matrix L, it
is possible to solve Ly = I to find y and then to use the matrix U to solve Ux = y to find x, which would be M^-1. Multiplying three matrices of the upper part and the three of the lower part of the
general formula for probability of ruin is then trivial.
The example case is pretty simple, the matrix is only 9×9 — its inverse can be calculated pretty fast without any optimization. But what if the GCD of the strategy is $1? For example, if the average
loss is $1,113 and the average profit is $1,109, the GCD is $1. With $2,500 starting balance and $5,000 target, that is a 4,999×4,999 matrix — this will require a lot of computer memory and time to
calculate. The best time of computing this risk of ruin without any optimization is O(k^2.376) and in general it requires at least k×k memory.
Firstly, it is important to keep the M^-1 matrix as small as possible. Optimally, k should be less than 500, if the aim is to solve this problem in reasonable time (several seconds).
Secondly, the required memory can be reduced significantly: the losing vector can be stored inside the total vector (it is known where the q's in the losing vector stop and the 0's begin), and the
results of the LU decomposition can be stored inside the main matrix (M — it can be discarded after the decomposition).
Thirdly, both Ly = I and Ux = y can be solved for one column (jth), instead of the entire k×k matrix. That same column will also be a result of multiplication with C[j]. This is because if the entire
k×k matrix is calculated and then multiplied by C[j], the result will also be just the jth column of the matrix, because of all the zeros in C[j].
Lastly, it is possible to calculate the product of the losing vector with the jth column and the product of the total vector with that column simultaneously, as the first loss step iterations will be
the same during that calculation while other iterations won't be needed in case of the losing vector.
Here is the example PHP code that uses the above algorithm to find the risk of ruin for the earlier example:
Starting balance: $2,500.00
Target balance: $5,000.00
Avg. losing trade: $1,500.00
Avg. profitable trade: $1,000.00
Loss probability: 30%
Win probability: 70%
// Greatest common divisor = $500.
$begin_state = 5; // Starting balance divided by GCD.
$N = 9; // The number of transitional states; k in the article.
$loss_step = 3; // Loss size divide by GCD.
$win_step = 2; // Win size divide by GCD.
$q = 0.3; // Loss probability.
$p = 0.7; // Win probability.
// Filling the Cj vector.
for ($i = 0; $i < $N; $i++)
if ($i == $begin_state - 1) $unitary_vector[$i] = 1;
else $unitary_vector[$i] = 0;
// Filling the loss vector and total vector. The loss vector is actually a part of the total vector.
for ($i = 0; $i < $N; $i++)
if (($i - $loss_step) < 0) $total_vector[$i] = $q;
else if (($i + $win_step) >= $N) $total_vector[$i] = $p;
else $total_vector[$i] = 0;
// Filling the main matrix.
for ($i = 0; $i < $N; $i++)
for ($j = 0; $j < $N; $j++)
// The main diagonal is always 1.
if ($i == $j) $a[$i][$j] = 1;
// The elements above the main diagonal are about losing.
else if ($j == $i + $loss_step) $a[$i][$j] = -$q;
// The elements below the main diagonal are about winning.
else if ($j == $i - $win_step) $a[$i][$j] = -$p;
else $a[$i][$j] = 0;
// The LU Decomposition.
for ($i = 0; $i < $N; $i++)
for ($j = $i; $j < min($N, $i + $loss_step); $j++) // U
for ($k = 0; $k <= $i - 1; $k++)
$a[$i][$j] -= $a[$i][$k] * $a[$k][$j];
for ($j = $i + 1; $j <= min($i + $win_step, $N - 1); $j++) // L
for ($k = 0; $k <= $i - 1; $k++)
$a[$j][$i] -= $a[$j][$k] * $a[$k][$i];
$a[$j][$i] /= $a[$i][$i];
// Solving Ly = I for one column (unitary_vector) that is equal to Cj of the Gambler's Ruin formula.
// The resulting y will also be stored in unitary_vector.
// Start from begin_state because all X's before 1 in the unitary_vector will always be 0.
for ($i = $begin_state; $i < $N; $i++)
$sum = 0;
for ($j = 0; $j <= $i - 1; $j++)
$sum -= $a[$i][$j] * $unitary_vector[$j];
$unitary_vector[$i] = $unitary_vector[$i] + $sum;
// Solving Ux = y for one column that was calculated in Ly = I.
// The resulting x will be stored in unitary_vector.
for ($i = $N - 1; $i >= 0; $i--)
$sum = 0;
for ($j = $N - 1; $j > $i; $j--)
$sum -= $a[$i][$j] * $unitary_vector[$j];
$unitary_vector[$i] = ($unitary_vector[$i] + $sum) / $a[$i][$i];
// Multiplying total_vector and its losing part by the resulting unitary_vector.
$loss = 0;
$total = 0;
for ($i = 0; $i < $N; $i++)
$product = $total_vector[$i] * $unitary_vector[$i];
if (($i - $loss_step) < 0) $loss += $product;
$total += $product;
$probability = $loss / $total;
echo "Loss: $loss <br>";
echo "Total: $total <br>";
echo "Probability: $probability <br>";
The output would be:
Loss: 0.25755603952
Total: 1
Probability: 0.25755603952
So, the risk to ruin the entire account before doubling it is about 25.8% for this example.
But what if k is getting too big? In this case, mathematical rounding should be applied to the loss step, win step, and both starting and target balances, virtually truncating some of the last
digits. Additionally, the target balance can be reduced. The first method influences the accuracy, but its effect becomes less important as the difference between the average loss and the average
profit gets bigger.
1. If an assumption is made that the input data (the probabilities and the win/loss size) is 100% accurate, then this method offers a perfectly precise result.
2. Doesn't depend on the position sizing method, but rather on the resulting average loss/profit in absolute numbers — the statistics readily available in all Forex strategy reports.
3. Other input parameters are also very simple — there is no need to calculate the standard deviation.
4. Simple cases can be calculated manually.
5. The calculated risk is "to lose everything before reaching the target balance" — something that is achievable in contrast to the "infinite number of trades" of the previous methods.
1. Assumes that the main parameters of the trading strategy (the win/loss sizes and rates) don't change. That is almost never true in the real world.
2. It is a very complex method.
3. In many cases, requires either a lot of calculating power or a lot of rounding, reducing the result's accuracy.
Risk of ruin with Monte Carlo simulation
Monte Carlo simulation (also known as Monte Carlo method) is a model that predicts the probability of different outcomes that involve random variables. Invented by mathematician Stanislaw Ulam, it is
named after Monte Carlo, a popular gambling destination in Monaco. The name seems appropriate as gambling is usually associated with chance and random outcomes.
The basic gist of the Monte Carlo method is to simulate the outcome many times with random variables getting new values on each simulation. To better understand how the method works, let us look at a
hypothetical situation.
Using historical data and backtesting, you calculated the chance of loss or gain for an average trade with your trading strategy as well as the average size of your gains and losses. How to calculate
the risk of ruin using this data? You can try to simulate an outcome by randomizing each trade and seeing whether your account balance will be wiped out by the end. But one simulation is of little
use as it shows just one outcome among many possibilities and does not tell you likely it is.
But what if you simulate the outcome a hundred times? Thousand? Ten thousand? The Monte Carlo method suggests that by randomizing the result of each trade (a gain or a loss and, possibly, the size of
the gain or loss) you gain valuable statistical data when re-running such simulations. And the more simulations you perform, the more dependable the data will be. In case of Forex trading, looking at
how many times using your strategy resulted in a total wipeout of your account balance you can determine the risk of ruin.
Example of Monte Carlo simulation in Excel spreadsheet
This Excel spreadsheet shows an example of a simple Monte Carlo simulation. It assumes that you know the probability of a winning trade as well as the size of the average gain and average loss.
Enter the size of your account balance in the Starting Account Balance field. Enter the chance of gain in the Chance of Gain field as a number from 0 to 1. For example, if the chance of gain is 65%
then you should enter 0.65. Next, enter the Average Gain and the Average Loss in the appropriate field. If you wish to use a range instead of a fixed average number, you can easily do it by using the
RANDBETWEEN function. Afterward, enter the desired number of simulated trades and the number of simulations in the # of Trades and # of Simulations fields respectively. After you filled all the
fields, you can calculate the risk of ruin. To do so, go to Formulas in the main menu of Excel and click Calculate Now to the right of Calculation Options. Alternatively, you can just press F9. The
result of the calculation will appear in the Risk of Ruin field as a percentage.
Important! The calculation can take a long time, especially on slower computers, and any click on the spreadsheet can stop the calculation, resulting in an incorrect outcome. It is better not to
interact with the spreadsheet after the calculation has started until you see Ready in the bottom-left corner of the screen.
Below the table's inputs, you will see the row of trades' numbers. Below that, there are rows of the resulting account balances at each trade, with each row representing one simulation.
Each trade either adds the Average Gain amount or subtracts the Average Loss amount from the account depending on whether the random number generated for that trade is below or above the Chance of
Gain value. If, at some trade, the account balance reaches zero or negative value, that simulation stops and is counted as "ruined".
If you want to have more simulations than the maximum 1,000 possible with the given spreadsheet, you can easily copy over the rows farther down — the formulas should work correctly. Same with the
number of trades per simulation you can copy and paste the columns after trade #1,000 to increase the maximum number of trades.
Here is the chart with an example of the first 50 simulations plotted. Note how some of them reach the zero line and remain there — those are simulations when the strategy ruined the trading account:
1. Monte Carlo simulation is a simple concept and doesn't involve complex math.
2. Using multiple simulations with changing variables, you can get a more precise chance of ruin than just using simple average values.
3. There are plenty of tools on the Internet for performing Monte Carlo simulations — from webpages to Excel add-ons.
1. Monte Carlo simulation assumes a "perfect market", meaning that it does not account for fundamental changes, be it short-term changes due to important events (like COVID-19) or long-term changes
due to structural changes in how the market operates (the example of this is the "peg" and the subsequent "unpegging" of the Swiss franc to the euro by the Swiss National Bank). Such changes make
the historic data used for calculations useless.
2. Changing your trading strategy can also make historic data irrelevant, meaning that Monte Carlo simulation should be used only for testing consistent trading strategies.
3. Monte Carlo simulation assumes that each trade is independent of the previous ones. Therefore, it is unsuitable for serially correlated strategies with trades that take into account the results
of the previous trades.
None of the described methods is perfect. Each of them should only be used when it fits the parameters of the Forex trading strategy:
• Fixed position size formula is good when you know that the position size is fixed, can calculate the standard deviation, and the input of the losing positions into standard deviation is greater
than that of the variability of the winning positions. It is a good method if you don't want to perform any complex calculations.
• Fixed fractional position sizing formula is perfect for the fractional position sizing strategies. Again, finding the standard deviation is necessary. It also needs to be formed by both the
losing and the winning positions.
• Gambler’s Ruin method is recommended when you are confident that the statistical parameters of the trading system are stable (the average win/loss size, the win/loss rates) and don't mind doing
some really complex calculations.
Monte Carlo method can be useful when you have access to a suitable simulator.
Whichever method you choose, it is important to remember that the resulting risk value shouldn't be taken too seriously on its own. Its main purpose is in comparing different strategies or the effect
of the changes applied to one trading strategy. Relying on the calculated risk value as the real measure of strategy riskiness can lead to unexpected but horrible consequences.
Note: it is possible to calculate the first three types of risk of ruin by using the Forex Report Analysis Tool. Perhaps, the only free online tool that offers such functionality.
If you have any questions or commentaries regarding the assessment of the risk of ruin in Forex trading, please feel free to discuss them on our forum. | {"url":"https://www.earnforex.com/guides/risk-of-ruin-in-trading/","timestamp":"2024-11-13T21:28:38Z","content_type":"text/html","content_length":"289681","record_id":"<urn:uuid:9b2cd548-6fd6-4fef-9656-0d89593e9b5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00070.warc.gz"} |
Qui est-ce ?
Let y be 8549048879922979409, with y[i] the bits of y where y[62] is the MSB and y[0] is the LSB. We need to find a number x that, when passed as input to the attached logical circuit, will lead to
the bits of y as the output.
The logical circuit looks like this:
This pattern is continuing like this until x[62] and y[62].
This is a typical z3 challenge. z3 is a satisfiability modulo theories solver that can be used to solve this kind of logical circuits automatically.
We just have to declare the output we know, add all the constraints of the logical circuit and ask z3 to give us the input that satisfies all these conditions. It will do its magic and find it for
Don’t forget to inverse the bits order when necessary when converting the bits from and to numbers if you want to work with the LSB at the left of the array and the MSB at the right to simplify
manipulations and indices.
Here is my script:
from z3 import *
y = 8549048879922979409
expected_y_bits = reversed(list(map(lambda bit: bit == "1", bin(y)[2:]))) # y0 should be the LSB and y62 the MSB, so we reverse the bits to have them sorted by index
s = Solver()
x_bits = [Bool(f"x{i}") for i in range(0, 63)]
t_bits = [Bool(f"t{i}") for i in range(0, 63)] # let's call the intermediary gates state t (t for tmp)
for i, bit in enumerate(expected_y_bits):
previous_i = (i + 62) % 63 # this is the index of the previous element in a modulo 63 cycle
s.add(Xor(t_bits[previous_i], x_bits[i]) == bit)
s.add(And(x_bits[previous_i], Not(x_bits[i])) == t_bits[i])
if s.check() == sat:
model = s.model()
result = int("".join(reversed(["1" if is_true(model[x_bits[i]]) else "0" for i in range(0, 63)])), 2) # we reverse back the bits before converting to decimal to make x0 the LSB and x62 the MSB as it should be
Let’s execute the script to get the flag.
Flag: FCSC{7364529468137835333} | {"url":"https://cypelf.fr/articles/qui-est-ce/","timestamp":"2024-11-03T06:13:17Z","content_type":"text/html","content_length":"39593","record_id":"<urn:uuid:689a6c94-0ed3-4bbb-9504-9489ae039da3>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00802.warc.gz"} |
The Stacks project
Lemma 42.68.27. Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module of dimension $1$. Assume $\varphi , \psi : M \to M$ are two injective $A$-module maps, and assume $\varphi (\psi
(M)) = \psi (\varphi (M))$, for example if $\varphi $ and $\psi $ commute. Then $\text{length}_ R(M/\varphi \psi M) < \infty $ and $(M/\varphi \psi M, \varphi , \psi )$ is an exact $(2, 1)$-periodic
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Parça yerleştirme algoritmalarının postal oluşturma problemine uygulanması
Fen Bilimleri Enstitüsü
Bu tezde, endüstride geniş uygulama alanları bulunan parça yerleştirme problemi ele alınmıştır. Özel olarak tekstil sektöründeki "Otomatik Pastal Hazırlama" işlemine uygulanması ele alınmıştır.
Problem iki ana kısımda incelenmiştir, dikdörtgen parçaların ya da çokgenlerin kapsayan dikdörtgenlerinin yerleştirilmesi ve doğrudan çokgen yerleştirme. Çalışmada algoritmik geometri yöntemleri ile
sezgisel yaklaşımların kaynaştırılması yoluna gidilmiştir. Birinci kısım, dikdörtgen yerleştirme, iki ana adımdan oluşur: Hangi parçanın yerleştirileceği karan ve seçilen parça için uygun bir konum
bulunması. Parçaların hangi sıra ile yerleşeceğine, genetik algoritmalar kullanılarak karar verilir. Verilen sıradaki parçaların yerleştirilmesi için ise sezgisel bir yaklaşım; alt-sol yaklaşımı
kullanılır. Alt-sol sezgisel yaklaşımı kullanılarak, her parça sıralamasına tek bir yerleşim karşı düşürülür. Her sıralamaya yerleşim veriminin bir fonksiyonu olan uygunluk değeri atanır. Genetik
operatörler kullanılarak bu uygunluk değeri, dolayısıyla da yerleşke verimi arttınlmaya çalışılır. Çokgen yerleştirmede, çokgenlerin birbirleri ile örtüşmeden yerleştirilmeleri problemi, minkowki
çokgen işlemleri kullanılarak çözülür. Yerleştirilen herbir çokgen için, daha önce yerleştirilmiş çokgenlerle örtüşmeyeceği alan bulunur. Bu alan içinde kalınarak, en alt, en sol konum ya da kapsayan
dikdörtgeni minimum yapan konum saptanır, parça yerleştirilir, yapı güncellenir. Tezin ikinci bölümünde yerleştirme problemlerinin genel tanıtımı ve sınıflandırılması yapılır. Üçüncü bölümde
dikdörtgen parçaların yerleştirilmesi, dördüncü bölümde çokgen yerleştirilmesi incelenir.
Part Nesting Algorithms for Nonconvex Polygons with Application to Automatic Marker Making An essential step in the manufacture of clothing is the generation of cutting plan or marker. The marker
determines how the parts that make up an article of clothing are cut from a bolt of cloth. For any marker making task, the set of parts is determined by the range of sizes and styles required for a
particular cutting. The job of the marker maker is to pack the parts in a rectangle of smallest length whose width is determined by width of the bolt of cloth. Some parts may be rotated by 180
degrees or flipped. Some parts may also may be rotated a small amount, no more than 3 degrees. Parts can not be rotated by arbitrary angles because cloth has a direction called nap. The pieces are
cut out of layers of cloth on a long cutting table. To improve cloth utilization, the parts for many articles are included in the same marker. The efficiency of a marker is the ratio of the sum of
all parts area to the total area. Cutting process itself can have severe impacts on the company's profit, especially when material of high value is involved. A poor way of cutting may result in a
large amount of trim loss which means that material and production resources have been wasted. Unfortunately generating an optimal marker (shortest marker of a given width containing a given set of
parts) is NP- complete. Using a CAD system, well trained people can generate near-optimal markers manually, but this is a time consuming job. Current software for automatic makers doesn't reach human
performance so they are used with human intervention. Automatic generation of markers would better enable manufacturers to keep up with customer demands for different styles and sizes. The problem of
automatic marker making is a derivative of a problem known in the literature under different names such as : cutting stock, trim loss, bin packing, dual bin packing, strip packing, knapsack, loading,
assortment, depletion, dividing, layout, nesting, partitioning or even capital budgeting, change making, assembly balancing, memory allocation, multiprocessor scheduling etc. These topics interest
different disciplines such as Information and Computer Sciences, Management Science, Engineering Sciences, Mathematics, Operational Research. All of them have the common logical structure that there
is on one hand stock of large objects and on the other hand a list of small items; and geometric combinations of small items are assigned to large objects. The problem of automatic marker is the
problem of two dimensional non- convex polygon nesting. The methods for the solution of nesting problems cover a wide range of methods. These can be classified in three main groups : analytical vui
methods, heuristic approaches, and others. Analytical methods are mostly used in the nesting of rectangular elements. They are too slow even for moderate sized problems because of the huge number of
states that must be considered. For convex and non- convex part nesting AI approaches such as heuristic search, knowledge-based systems, case-based reasoning, neural networks etc. or improvement
strategies based on ' meta-heuristics' such as simulated annealing, genetic algorithms, tabu search or other techniques like database/substitution, greedy heuristics, Monte Carlo placement, lattice
packing, clustering methods are used. This work covers two dimensional translational non-convex polygon nesting. As stated in the first paragraph rotation is not allowed because of the properties of
the application: cloth has a direction called nap,no rotation other than up to 3 degrees are allowed but pieces can flip about x or y axis. In this work the problem is treated in two main part: 1.
Placement of rectangular pieces ( 2-D Bin packing) 2. Placement of nonconvex polygonal pieces (nesting) In the first part the pieces to be nested are approximated to rectangles, by substitution to
their bounding boxes and the problem is treated like 2D bin packing problem. 2D bin packing place some rectangular boxes ( our bounding boxes) in an open ended bin ( bolt of cloth) without
overlapping with each other. As 2D rectangle packing is NP-complete in order to obtain near optimal solution in reasonable time an approach based on heuristic is taken and Bottom-Left Heuristic is
chosen as placement strategy. Bottom-Left Heuristic is a bottom-left stability preserving heuristic. If a rectangle is placed in a bottom-left stable position it can no more move downwards or slide
to left. At any stage of the heuristic the bin contains a set of empty spaces (holes). There is always at least one hole which is the unbounded area over the placed boxes. To place a rectangle each
hole is examined to find whether this rectangle fits in, some candidate places are found. The process of finding candidate places for a rectangle can be viewed as sliding a mechanical device composed
from a pair of bar of length equal to the length of the rectangle, and a spring pulling them outwards. At any time the height of the bars is equal to the local maximum of the bottom subtracted from
local minimum of the top, in the interval determined by the rectangle width. Whenever the height of the bars is equal or greater than the height of the rectangle to be placed, the position is marked
as a candidate. After traversing each hole and extracting the candidates, the candidate that preserve bottom-left stability is taken. Then the rectangle is placed and data structure is updated to
represent new configurations of the holes. For each rectangle to be packed same process is applied in the given order of the rectangles. For poorly ordered list of rectangles the algorithm can lead
to bad packing but even a simple ordering of decreasing width applied to the parts to be packed leads to satisfactory results. At this point instead of using a simple ordering to improve the result,
a job scheduling algorithm based on genetic algorithms is used whose output (order) is used to decide the order of rectangles to be packed in the placement stage. IX Genetic Algorithms is first
described as a methodology for studying natural adaptive systems and designing artificial adaptive systems in 1975 by J.Holland. It is now frequently used as an optimization method, based on analogy
to the process of natural selection in biology where from one generation to the next weak elements are eliminated and those with the best performance in the current environment (fittest) survive. GAs
are widely recognized as an effective search paradigm in job scheduling. GA maintains a population of strings ( chromosomes) that encode candidate solutions to a problem. These strings are the analog
of chromosomes in natural evolution. A fitness function defines the quality of the solution. In this work GA is used to improve result of BL heuristic by finding efficient ordering of rectangles to
be packed. Genotype is a string of integer coding the order of rectangles. Phenotype is the arrangement obtained after packing the rectangles following the order coded in the genotype, the process
has five step: 1. Initialization 2. Reproduction 3. Evaluation 4. Selection 5. Termination In the first stage a population is initialized randomly. Order based coding is used that is if N is the
number of rectangle to be packed, genotype consists of an array of N integer and genotype is initialized by assigning N distinct number in the range 0-(N-l) to each element of the array. For a
genotype g, i is the order of the rectangle no g[i]. A number of genotype is created to initiate the population. For each genotype correspond a fitness value, which is a function of the efficiency of
the layout obtained using the order coded in the genotype. Reproduction is made simulating roulette wheel selection. To each genotype from the population is assigned a portion in the wheel
proportional to its fitness function. By means of this, good genotypes have more chance to be selected. A pairs of genotypes are selected and the crossover operator is applied to them. As a the
crossover operator, order base crossover is used : A cut point is randomly found, part of the genotype from start to cut point is taken from one parent and directly copied to the child, remaining
part of the parent is copied to child following the order in the second parent. Child constructed with this method inherit important information from both parent but mainly from first parent. Using
the crossover operator stated above a pool of child is formed. At the evaluation stage for each child its phenotype is constructed decoding the genotype and a fitness value is calculated. parentl : 1
- 2 | 3 - 4 - 5 parent2 : 5 - 4 j 3 - 2 - 1 child : 1 - 2 - 5 - 4 - 3 An example of order based crossover Selection process replace some individuals from previous population by the individuals from
the child pool. Many threshold values have been tried to decide which individual to discard from previous population and which children to include. Change in the population does not always occur in a
regular way in order to prevent to get stuck on a local minimum mutation operator is used. Two kind of mutation operator is defined for this application. First randomly select two integer in the
genotype string and swap their place. Second operator again randomly select a rectangle and rotate it 90 degrees. Not to destroy regular conversion to a better population mutation operator is applied
to the population with a low probability. Mutation operator change directly an individual in the population no selection process needed. These processes are applied to an initial population
iteratively. Termination condition is a predetermined number of iteration. If in a certain stage of the iteration population got hung up to a local minimum some part of the population is replaced
with randomly generated individuals. Genetic algorithms is used for its following properties: 1. GAs work with a coding of the parameter set, not the parameter themselves. 2. Gas search a population
of points, not from a single point. 3. Gas use objective function information, not derivatives or other auxiliary knowledge. 4. Gas use probabilistic transition rules, not deterministic rules. It has
been seen from the solutions obtained that use of genetic algorithms to schedule rectangles to be packed with BL heuristic, improve considerably solution of BL placement. Second part of the work:
placement of small pieces covers a somewhat different area of research. The first part solve 2D bin packing problem. It deals mainly on the scheduling of rectangles, geometric properties of the
pieces have not been taking into account. In the second part it is worked mainly on computational geometry. There are mainly three steps in the program:. Determination of all non-overlapping
positions of a polygon relative to all previously placed polygons.. Selection of one of these positions, according to a selection criteria. Placement of the polygon and rearrangement of the data
structure For the first step, " Minkowski Polygon Operations" is used. Outside of the resulting polygon from minkowski sum A©(-B), define all the positions, where the reference point of polygon B can
be placed, without overlapping polygon A. Minkowski sum has been calculated by using ail algorithm based on sweeping. XI As selection criteria, two heuristic methods is used; Selection of the
position that results in minimum bounding box of the placed polygons and selection of the bottom-left position. Minimum bounding box selection favors filling the holes, but sometimes results in dense
but narrow arrangements that doesn't use all the width of the cloth, left place is too narrow to be filled and become waste. A combination of these two heuristics results in a better layout. As
rearrangement of data structure; each time a new polygon is added to the layout, polygon union of the previously placed polygons with the new polygon is calculated. At each step all previously placed
polygons are threaten like a unique polygon, this diminish considerably calculations. In brief this work combine computational geometry methods generally discarded by layout and packing communities,
with their traditional optimization methods.
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, 1996
Anahtar kelimeler
Algoritmalar, Parça yerleştirme problemi, Algorithms, Part nesting problem | {"url":"https://polen.itu.edu.tr/items/0975d9f1-2e2b-4a50-8bdd-39e2f96045a4","timestamp":"2024-11-05T12:13:07Z","content_type":"text/html","content_length":"184290","record_id":"<urn:uuid:db862366-616f-42cd-870b-334435f6e21d>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00406.warc.gz"} |
647: Palindromic Substrings
Problem ID: 647
Title: Palindromic Substrings
Difficulty: Medium
Description: Given a string s, return the number of palindromic substrings in it.
A string is a palindrome when it reads the same backward as forward.
A substring is a contiguous sequence of characters within the string.
I initially went with the solution of maintaining a forward and backward deque. For each index we will treat it as the head and keep traversing the characters and check if it is palindrome but
checking if both deque are equal. However, I failed to realise that comparing deque is a O(N) operation and my solution would result in O(N^3) time complexity.
Take aways:
• Comparing equality of data structure is O(N) operation
• A string is a palindrome if and only if the inner substring is a palindrome
• When dealing with a palindrome, a string of different parity (even/odd) in length should be dealt differently.
If a string is a palindrome, then its inner string(exclude start and end) must be a palindrome as well. A tricky edge case to take note is that the base case is when the string is of size 2.
class Solution {
int countSubstrings(string s) {
int ans = 0;
int n = s.size();
vector<vector<bool>> dp(n, vector<bool>(n+1, false));
for (int i = 0; i < n ;i++) {
dp[i][1] = true;
for (int i = 0; i < n-1; i++) {
if (s[i] == s[i+1]) {
dp[i][2] = true;
for (int length = 3; length <= n; length++) {
for (int start = 0; start < n; start++) {
if (start + length > n) continue;
int end = start + length -1;
int inner_start = start+1;
int inner_length = length -2;
if (s[start] != s[end]) continue;
if (!dp[inner_start][inner_length]) continue;
dp[start][length] = true;
return ans; | {"url":"https://klementtan.com/leetcodes/palindromic-substrings/","timestamp":"2024-11-14T05:18:38Z","content_type":"text/html","content_length":"16371","record_id":"<urn:uuid:9ae2b406-d1b6-4012-abbe-3d5e3800b000>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00184.warc.gz"} |
Lesson 11
The Distributive Property, Part 3
Let's practice writing equivalent expressions by using the distributive property.
11.1: The Shaded Region
A rectangle with dimensions 6 cm and \(w\) cm is partitioned into two smaller rectangles.
Explain why each of these expressions represents the area, in cm^2, of the shaded region.
11.2: Matching to Practice Distributive Property
Match each expression in column 1 to an equivalent expression in column 2. If you get stuck, consider drawing a diagram.
Column 1
1. \(a(1+2+3)\)
2. \(2(12-4)\)
3. \(12a+3b\)
4. \(\frac23(15a-18)\)
5. \(6a+10b\)
6. \(0.4(5-2.5a)\)
7. \(2a+3a\)
Column 2
• \(3(4a+b)\)
• \(12 \boldcdot 2 - 4 \boldcdot 2\)
• \(2(3a+5b)\)
• \((2+3)a\)
• \(a+2a+3a\)
• \(10a-12\)
• \(2-a\)
11.3: Writing Equivalent Expressions Using the Distributive Property
The distributive property can be used to write equivalent expressions. In each row, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.
│ product │sum or difference │
│\(3(3+x)\) │ │
│ │\(4x-20\) │
│\((9-5)x\) │ │
│ │\(4x+7x\) │
│\(3(2x+1)\) │ │
│ │\(10x-5\) │
│ │\(x+2x+3x\) │
│\(\frac12 (x-6)\) │ │
│\(y(3x+4z)\) │ │
│ │\(2xyz-3yz+4xz\) │
This rectangle has been cut up into squares of varying sizes. Both small squares have side length 1 unit. The square in the middle has side length \(x\) units.
1. Suppose that \(x\) is 3. Find the area of each square in the diagram. Then find the area of the large rectangle.
2. Find the side lengths of the large rectangle assuming that \(x\) is 3. Find the area of the large rectangle by multiplying the length times the width. Check that this is the same area you found
3. Now suppose that we do not know the value of \(x\). Write an expression for the side lengths of the large rectangle that involves \(x\).
The distributive property can be used to write a sum as a product, or write a product as a sum. You can always draw a partitioned rectangle to help reason about it, but with enough practice, you
should be able to apply the distributive property without making a drawing.
Here are some examples of expressions that are equivalent due to the distributive property.
\(\displaystyle \begin {align} 9+18&=9(1+2)\\ 2(3x+4)&=6x+8\\ 2n+3n+n&=n(2+3+1)\\ 11b-99a&=11(b-9a)\\ k(c+d-e)&=kc+kd-ke\\ \end {align}\)
• equivalent expressions
Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.
For example, \(3x+4x\) is equivalent to \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both
expressions equal 70.
• term
A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression \(5x + 18\) has two terms. The first
term is \(5x\) and the second term is 18. | {"url":"https://curriculum.illustrativemathematics.org/MS/students/1/6/11/index.html","timestamp":"2024-11-07T22:36:29Z","content_type":"text/html","content_length":"79555","record_id":"<urn:uuid:c33a25c0-fd45-447a-847f-17b7976dbe85>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00147.warc.gz"} |
R package for fitting and testing alternative models for single cohort litter decomposition data
Getting started
At the moment there is one key function which is fit_litter which can fit 6 different types of decomposition trajectories. Note that the fitted object is a litfit object
fit <- fit_litter(time=c(0,1,2,3,4,5,6),
mass.remaining =c(1,0.9,1.01,0.4,0.6,0.2,0.01),
You can visually compare the fits of different non-linear equations with the plot_multiple_fits function:
Calling plot on a litfit object will show you the data, the curve fit, and even the equation, with the estimated coefficients:
The summary of a litfit object will show you some of the summary statistics for the fit.
#> Summary of litFit object
#> Model type: weibull
#> Number of observations: 7
#> Parameter fits: 4.19
#> Parameter fits: 2.47
#> Time to 50% mass loss: 3.61
#> Implied steady state litter mass: 3.71 in units of yearly input
#> AIC: -3.8883
#> AICc: -0.8883
#> BIC: -3.9965
From the litfit object you can then see the uncertainty in the parameter estimate by bootstrapping | {"url":"https://cran.stat.sfu.ca/web/packages/litterfitter/readme/README.html","timestamp":"2024-11-15T00:00:16Z","content_type":"application/xhtml+xml","content_length":"10188","record_id":"<urn:uuid:5e12fc83-6154-4734-b5a9-faecfe3234c2>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00273.warc.gz"} |
The seminar meets on Tuesdays, 14:30-15:30, in Math -101
2021–22–A meetings
Date Title Speaker Abstract
Oct TBA Departamental
19 meeting
We will discuss the Blaschke branch of integral geometry and its manifestations in pseudo-Riemannian space forms. First we will recall the fundamental
notion of intrinsic volumes, known as quermassintegrals in convex geometry. Those notions were extended later to Riemannian manifolds by H. Weyl, who
Integral geometry and Dmitry discovered a remarkable fact: given a manifold M embedded in Euclidean space, the volume of the epsilon-tube around it is an invariant of the
Oct valuation theory in Faifman (Tel Riemannian metric on M. We then discuss Alesker’s theory of smooth valuations, which provides a framework and a powerful toolset to study integral
26 pseudo-Riemannian spaces Aviv geometry, in particular in the presence of various symmetry groups. Finally, we will use those ideas to explain some recent results in the integral
University) geometry of pseudo-Riemannian manifolds, in particular a collection of principal Crofton formulas in all space forms, and a Chern-Gauss-Bonnet formula
for metrics of varying signature. Partially based on joint works with S. Alesker, A. Bernig, G. Solanes.
In finite dimensional Riemannian geometry, everything behaves nicely — the Riemannian metric induces a distance function, geodesics exist (at least for
Riemannian metrics on Cy Maor ( some time), and so on. In infinite dimensional Riemannian geometry, however, chaos reigns. In this talk I will focus on diffeomorphism groups, and on a
Nov diffeomorphism groups — the Hebrew particularly important hierarchy of Riemannian metrics on them: right-invariant Sobolev metrics. These arise in many different contexts, from purely
2 good, the bad, and the University) mathematical ones, to applications in hydrodynamics and imaging. I will give a brief introduction to these metrics, why we care about them, and what we
unknown know (and don’t know) about their properties. Parts of the talk will be based on joint works with Bob Jerrard and Martin Bauer.
Yotam The study of aperiodic order and mathematical models of quasicrystals is concerned with ways in which disordered structures can nevertheless manifest
Nov Order and disorder in Smilansky ( aspects of order. In the talk I will describe examples such as the aperiodic Penrose and pinwheel tilings, together with several geometric, functional,
9 multiscale substitution Rutgers dynamical and spectral properties that enable us to measure how far such constructions are from demonstrating lattice-like behavior. A particular focus
tilings University) will be given to new results on multiscale substitution tilings, a class of tilings that was recently introduced jointly with Yaar Solomon.
In various areas of mathematics there exist “big fiber theorems”, these are theorems of the following type: “For any map in a certain class, there
exists a ‘big’ fiber”, where the class of maps and the notion of size changes from case to case.
We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by
Nov Big Fiber Theorems and Yaniv Ganor ( Gromov’s notion of ideal-valued measures.
16 Ideal-Valued Measures in Technion)
Symplectic Topology We adapt the latter notion to the realm of symplectic topology, using an enhancement of a certain cohomology theory on symplectic manifolds introduced
by Varolgunes, allowing us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results.
Necessary preliminaries will be explained. The talk is based on a joint work with Adi Dickstein, Leonid Polterovich and Frol Zapolsky.
Consider an infinite group G acting by isometries on some metric space X.
How does a “typical” element act? Consider a representation of G into some matrix group. What sort of matrix represents “typical” elements of G?
The answer depends on what we mean by the word “typical,” of which there are at least two reasonable notions. We may take a random walk on G and look
where it lands after a large number of steps. We may also fix a generating set for G and look how large balls are distributed.
Nov Randomness, genericity, and Ilya Gekhtman
23 ubiquity of hyperbolic (Technion) I will talk about how these two notions of genericity are related and how they differ, focusing on the setting of hyperbolic groups. I will also
behavior in groups. explain that the following is true with respect to both notions: For a group acting on a Gromov hyperbolic metric space typical elements act
loxodromically, i.e. with north-south dynamics.
For a representation of a large class of groups (including hyperbolic groups) into SL_n R, typical elements map to matrices whose eigenvalues are all
simple and have distinct moduli.
Tiling the integers with It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling
translates of one tile: the is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two
Nov Coven-Meyerowitz tiling Itay Londner distinct prime factors, each of the sets A and B can be replaced by a highly ordered “standard” tiling complement. It is not known whether this
30 conditions for three prime (UBC) behaviour persists for all tilings with no restrictions on the number of prime factors of M. In joint work with Izabella Laba (UBC), we proved that
factors this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce some ingredients from the proof.
The space Hom(\Gamma,G) of homomorphisms from a finitely-generated group \Gamma to a complex semisimple algebraic group G is known as the
G-representation variety of \Gamma. We study this space when G is fixed and \Gamma is a random group in the few-relators model. That is, \Gamma is
generated by k elements subject to r random relations of length L, where k and r are fixed and L tends to infinity.
Oren Becker ( More precisely, we study the subvariety Z of Hom(\Gamma,G), consisting of all homomorphisms whose images are Zariski dense in G. We give an explicit
Dec Character varieties of University of formula for the dimension of Z, valid with probability tending to 1, and study the Galois action on its geometric components. In particular, we show
7 random groups Cambridge) that in the case of deficiency 1 (i.e., k-r=1), the Zariski-dense G-representations of a typical \Gamma enjoy Galois rigidity.
Our methods assume the Generalized Riemann Hypothesis and exploit mixing of random walks and spectral gap estimates on finite groups.
Based on a joint work with E. Breuillard and P. Varju.
In many data-driven applications, the data follows some geometric structure, and the goal is to recover this structure. In many cases, the observed
data is noisy and the recovery task is even more challenging. A common assumption is that the data lies on a low dimensional manifold. Estimating a
Yariv manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there was no (computationally tractable)
Dec Non-Parametric Estimation of Aizenbud ( algorithm that accurately estimates a manifold from noisy samples with a constant level of noise.
21 Manifolds from Noisy Data Yale
Univercity) In this talk, we will present a method that estimates a manifold and its tangent. Moreover, we establish convergence rates, which are essentially as
good as existing convergence rates for function estimation.
Continuous wavelet transforms are mappings that isometrically embed a signal space to a coefficient space over a locally compact group, based on
so-called square integrable representations. For example, the 1D wavelet transform maps time signals to functions over the time-scale plane based on
the affine group. When using wavelet transforms for signal processing, it is often useful to work interchangeably with the signal and the coefficient
spaces. For example, we would like to know what operation in the signal domain is equivalent to multiplication in the coefficient space. While such a
Wavelet-Plancherel: a new point of view is natural in classical Fourier analysis (i.e., “time convolution is equivalent to frequency multiplication”), it is not compatible with
Dec theory for analyzing and Ron Levie ( wavelet analysis, since wavelet transforms are not surjective. In this talk, I will present the wavelet-Plancherel theory – an extension of classical
28 processing wavelet-based LMU) wavelet theory in which the wavelet transform is canonically extended to an isometric isomorphism. The new theory allows formulating a variety of
methods coefficient domain operations as signal domain operations, with closed form formulas. Using these so-called pull-back formulas, we are able to reduce
the computational complexity of some wavelet-based signal processing methods. The theory is also useful for proving theorems in wavelet analysis. I
will present an extension of the Heisenberg uncertainty principle to wavelet transforms and prove the existence of uncertainty minimizers using the
wavelet-Plancherel theory.
A function at a non-critical point can be converted to a linear form by a local coordinate change. At an isolated critical point one has the weaker
statement: higher order perturbations do not change the group orbit. Namely, the function is determined (up to the local coordinate changes) by its
(finite) Taylor polynomial.
Finite determinacy of maps.
Jan Group orbits vs their Dmitry Kerner This finite-determinacy property was one of the starting points of Singularity Theory. Traditionally such statements are proved by vector field
4 tangent spaces (BGU) integration. In particular, the group of local coordinate changes becomes a ``Lie-type” group.
I will show such determinacy results for maps of germs of (Noetherian) schemes. The essential tool is the “vector field integration” in any
characteristic. This equips numerous groups acting on filtered modules with the ``Lie-type” structure. (joint work with G. Belitskii, A.F. Boix, G.M. | {"url":"https://www.math.bgu.ac.il/en/research/fall2022/seminars/colloquium","timestamp":"2024-11-03T03:43:02Z","content_type":"text/html","content_length":"42293","record_id":"<urn:uuid:1178ef8c-b0c6-42fd-bbd6-5e9f50028ee7>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00796.warc.gz"} |
Developmental Math Emporium
What you’ll learn to do: Define and use scientific notation to write very large and very small numbers and solve problems with them.
In the same way that exponents help us to be able to write repeated multiplication with little effort, they are also used to express large and small numbers without a lot of zeros and confusion.
Scientists and engineers make use of exponents regularly to keep track of the place value of numbers that they are working with to make calculations. For example, [latex]1,000,000[/latex] is written
as [latex]{1}\times{10}^{6}[/latex] and .00001 becomes [latex]{1}\times{10}^{-4}[/latex].
Specifically, in this section you’ll learn how to:
• Define decimal and scientific notation
• Convert from scientific notation to decimal notation
• Convert from decimal notation to scientific notation
• Multiply numbers expressed in scientific notation
• Divide numbers expressed in scientific notation
• Solve application problems involving scientific notation | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/outcome-scientific-notation/","timestamp":"2024-11-02T05:08:54Z","content_type":"text/html","content_length":"47056","record_id":"<urn:uuid:77e80b28-a277-4ff7-b4f8-f4ffdccc9bf8>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00837.warc.gz"} |
Volume 12, 2006, Number 4
Volume 12 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4
Generalized Fibonacci and Lucas sequences with Pascal-type arrays
Original research paper. Pages 1–9
Charles K. Cook and A. G. Shannon
Full paper (53 Kb) | Abstract
We re-label the Fibonacci and Lucas sequences respectively by
{F[0,n]} ≡ {F[n]} and {F[1,n]} ≡ {L[n]}
and consider
F[m,n] = F[m−1, n−1] + F[m−1, n+1], m, n ≥ 1,
as a generalization of the well-known identity
L[n] = F[n−1] + F[n+1],
F[m,n] = F[m,n−1] + F[m, n−2], m ≥ 1, n > 2.
On Steiner Loops of cardinality 20
Original research paper. Pages 10–22
M. H. Armanious
Full paper (184 Kb) | Abstract
It is well known that there are five classes of sloops
of cardinality 16
” SL
(16)s” according to the number of sub-
(20)s”. Based on the cardinality and the number of (normal) subsloops of
(20), we will construct in section 3 all possible classes of nonsimple
(20)s and in section 4 all possible classes of simple
(20)s. We exhibit the algebraic and combinatoric properties of
(20)s to distinguish each class.
So we may say that there are six classes of SL(20)s having one sub-SL(10) and n sub-SL(8)s for n = 0, 1, 2, 3, 4 or 6. All these sloops are subdirectly irreducible having exactly one proper
homomorphic image isomorphic to SL(2). For n = 0, the associated SL(20) is a nonsimple subdirectly irreducible having one sub-SL(10) and no sub-SL(8)s. Indeed, the associated Steiner quasigroup SQ
(19) of this case supplies us with a new example for a semi-planar SQ(19), where the smallest well-known example of semi-planar squags is of cardinality 21 ” cf.
It is well known that there is a class of planar Steiner triple systems (STS(19)s) due to Doyen SL(20) has no sub-SL(10) and no sub-SL(8). In section 4 we will show that there are other 6 classes of
simple SL(20)s having n sub-SL(8)s for n = 0, 1, 2, 3, 4, 6, but no sub-SL(10)s. It is well-known that a sub-SL(m) of an SL(2m) is normal. In the last theorem of this section, we give a necessary and
sufficient condition for a sub-SL(2) to be normal of an SL(2m). Accordingly, we have shown that if a sloop SL(20) has a sub-SL(10) and 12 sub-SL(8), then this sloop is isomorphic to the direct
product SL(10) × SL(2) and if a sloop SL(20) has 12 sub-SL(8)s and no sub-SL(10), then this sloop is a subdirectly irreducible having exactly one proper homomorphic image isomorphic to SL(10). In
section 5, we describe how can one construct an example for each class of smiple and of nonsimple SL(20)s.
Note on φ, ψ and σ functions
Original research paper. Pages 23–24
Krassimir T. Atanassov
Full paper (95 Kb) | Abstract
In the present remark we shall formulate and discuss some extremal problems, related to arithmetic functions φ, ψ and σ (see, e.g.,
Volume 12 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 | {"url":"https://nntdm.net/volume-12-2006/number-4/","timestamp":"2024-11-08T05:54:09Z","content_type":"text/html","content_length":"34501","record_id":"<urn:uuid:cc4c3d95-14bc-443b-97b3-b3133d2c86c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00524.warc.gz"} |
Bajaj Finance Stock Price Prediction in Python
The motivation behind This article comes from the combination of passion(about stock markets) and a love for algorithms. Who doesn’t love to make money and if you know the algorithms that you have
learned will or can help you make money(not always), why not explore it.
Business Usage: The particular problem pertains to forecasting, forecasting can be of sales, stocks, profits, and demand for new products. Forecasting is a technique that uses historical data as
inputs to make informed estimates that are predictive in determining the direction of future trends. Businesses utilize forecasting to determine how to allocate their budgets or plan for anticipated
expenses for an upcoming period of time.
Forecasting has got its own set of challenges like which variables affect a certain prediction, can there be an unforeseen circumstance(like COVID) that will render my predictions inaccurate. So
forecasting is easier said than done, there is less than a % chance that the observed value will be equal to the predicted value. In that case, we try to minimize the difference between actual and
predicted values.
The Data
I have downloaded the data of Bajaj Finance stock price online. I have taken the data from 1st Jan 2015 to 31st Dec 2019.1st Jan 2019 to 31st Dec 2019, these dates have been taken for prediction/
forecasting. 4 years of data have been taken as training data and 1 year as test data. I have taken an open price for prediction.
EDA :
The only EDA that I have done is known as seasonal decompose, to break it up into Observed, trend, seasonality, and residual components
The stock price always tends to show an additive trend because seasonality does not increase with time.
Now when we decompose a time series, We get:
1. Observed: The average value in the series.
2. Trend: The increasing or decreasing value in the series, the trend is usually a long term pattern, it spans for over more than a year.
3. Seasonality: The repeating short-term cycle in the series. In time-series data, seasonality is the presence of variations that occur at specific regular intervals less than a year, such as
weekly, monthly, or quarterly. …Seasonal fluctuations in a time series can be contrasted with cyclical patterns.
4. Residual: The component which is left after level, trend, and seasonality has been taken into consideration.
Model Building:
ARIMA(AutoRegressive Integrated Moving Average): This is one of the easiest and effective machine learning algorithms for performing time series forecasting. ARIMA consists of 3 parts,
Auto-Regressive(p), Integrated or differencing(d), and Moving Average(q).
Auto-Regressive: This part deals with the fact that the current value of the time series is dependent on its previous lagged values or we can say that the current value of the time series is a
weighted average of its lagged value. It is denoted by p, so if p=2, it means the current value is dependent upon the previous two of its lagged values. Order p is the lag value after which the PACF
plot crosses the upper confidence interval for the first time. We use the PACF(Partial autocorrelation function)to find the p values. The reason we use the PACF plot is that it only shows residuals
of components that are not explained by earlier lags. If we use ACF in place of PACF, it shows a correlation with lags that are far in fast, hence we will use the PACF plot.
Integrated(d): One of the important features of the ARIMA model is that the time series used for modeling should be stationary. By stationarity I mean, the statistical property of time series should
remain constant over time, meaning it should have a constant mean and variance. The trend and seasonality will affect the value of the time series at different times.
How to check for stationarity?
1. One simple technique is to plot and check.
2. We have statistical tests like ADF tests(Augmented Dickey-Fuller Tests).ADF tests the null hypothesis that a unit root is present in the sample. The alternative hypothesis is different depending
on which version of the test is used but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey-Fuller test for a larger and more complicated set of time series
models. The augmented Dickey-Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at
some level of confidence.So if value p value<alpha(significance) ,we will reject the null hypothesis,i.e presence of unit root.
Stationarity test
How to make time-series stationarity?
1. Differencing: The order of differencing (q) refers to the no of times you difference the time series to make it stationary. By difference, I mean you subtract the current value from the previous
value. After Differencing you again perform the ADF test to check whether the time series has become stationary or you can plot and check.
2. Log Transformation: We can make the time series stationary by doing a log transformation of the variables. We can use this if the time series is diverging.
Moving Average(q)
In moving average the current value of time series is a linear combination of past errors. We assume the errors to be independently distributed with the normal distribution. Order q of the MA process
is obtained from the ACF plot, this is the lag after which ACF crosses the upper confidence interval for the first time, As we know PACF captures correlations of residuals and the time series lags,
we might get good correlations for nearest lags as well as for past lags. Why would that be?
Since our series is a linear combination of the residuals and none of the time series own lag can directly explain its presence (since it’s not an AR), which is the essence of the PACF plot as it
subtracts variations already explained by earlier lags, its kind of PACF losing its power here! On the other hand, being a MA process, it doesn’t have the seasonal or trend components so the ACF plot
will capture the correlations due to residual components only.
Model Building:
I had used auto Arima to build the model. Auto ARIMA is available in pmdarima. After fitting the test set I got an output of ARIMA(0,0,0) which is commonly known as white noise. White noise means
that all variables have the same variance (sigma²) and each value has a zero correlation with all other values in the series.
After fitting the Arima model, I printed the Summary and got the below.
The log-likelihood value is a simpler representation of the maximum likelihood estimation. It is created by taking logs of the previous value. This value on its own is quite meaningless, but it can
be helpful if you compare multiple models to each other. Generally speaking, the higher the log-likelihood, the better. However, it should not be the only guiding metric for comparing your models!
AIC stands for Akaike’s Information Criterion. It is a metric that helps you evaluate the strength of your model. It takes in the results of your maximum likelihood as well as the total number of
your parameters. Since adding more parameters to your model will always increase your value of the maximum likelihood, the AIC balances this by penalizing for the number of parameters, hence
searching for models with few parameters but fitting the data well. Looking at the models with the lowest AIC is a good way to select to best one! The lower this value is, the better the model is
BIC (Bayesian Information Criterion) is very similar to AIC, but also considers the number of rows in your dataset. Again, the lower your BIC, the better your model works. BIC induces a higher
penalization for models with complicated parameters compared to AIC.
Both BIC and AIC are great values to use for feature selection, as they help you find the simplest version with the most reliable results at the same time.
Ljung Box
The Ljung–Box test is a type of statistical test of whether any of a group of autocorrelations of a time series are different from zero. Instead of testing randomness at each distinct lag, it tests
the “overall” randomness based on a number of lags and is, therefore, a portmanteau test.
Ho: The model shows the goodness of fit(The autocorrelation is zero)
Ha: The model shows a lack of fit(The autocorrelation is different from zero)
My model here satisfies the goodness of fit condition because Probability(Q)=1.
Heteroscedasticity means unequal scatter. In regression analysis, we talk about heteroscedasticity in the context of the residuals or error term. Specifically, heteroscedasticity is a systematic
change in the spread of the residuals over the range of measured values.
My residuals are heteroscedastic in nature since Probability(Heteroskadisticy)=0.
Tests For Heteroscedasticity
Brusche Pagan test:
The Breusch-Pagan-Godfrey Test (sometimes shortened to the Breusch-Pagan test) is a test for heteroscedasticity of errors in regression. Heteroscedasticity means “differently scattered”; this is
opposite to homoscedastic, which means “same scatter.” Homoscedasticity in regression is an important assumption; if the assumption is violated, you won’t be able to use regression analysis.
Ho: Residuals are homoscedastic
Ha: Residuals are heteroscedastic in nature.
Goldfeld Quandt Test:
The Goldfeld Quandt Test is a test used in regression analysis to test for homoscedasticity. It compares variances of two subgroups; one set of high values and one set of low values. If the variance
differs, the test rejects the null hypothesis that the variances of the errors are not constant.
Although Goldfeld and Quandt described two types of tests in their paper (parametric and non-parametric), the term “Quandt Goldfeld test” usually means the parametric test. The assumption for the
test is that the data is normally distributed.
The test statistic for this test is the ratio of mean square residual errors for the regressions on the two subsets of data. This corresponds to the F-test for equality of variances. Both the
one-tailed and two-tailed tests can be used.
Forecasting Using Arima:
The metrics we use to see the accuracy of the model are RMSE.
Let us see the forecasting Using ARIMA.
Here the forecast values are constant because it’s an ARIMA(0,0,0) model.
ACF Plot(q value)
PACF plot (p-value)
Prediction Using Simple Exponential Smoothing
The simplest of the exponentially smoothing methods are naturally called simple exponential smoothing. This method is suitable for forecasting data with no clear trend or seasonal pattern.
Using the naïve method, all forecasts for the future are equal to the last observed value of the series. Hence, the naïve method assumes that the most recent observation is the only important one,
and all previous observations provide no information for the future. This can be thought of as a weighted average where all of the weight is given to the last observation.
Using the average method, all future forecasts are equal to a simple average of the observed data. Hence, the average method assumes that all observations are of equal importance, and gives them
equal weights when generating forecasts.
We often want something between these two extremes. For example, it may be sensible to attach larger weights to more recent observations than to observations from the distant past. This is exactly
the concept behind simple exponential smoothing. Forecasts are calculated using weighted averages, where the weights decrease exponentially as observations come from further in the past — the
smallest weights are associated with the oldest observations.
So large value of alpha(alpha denotes smoothing parameter)denotes that recent observations are given higher weight and a lower value of alpha denoted that more weightage is given to distant past
Modelling Using Simple Exponential Smoothing:
By modeling Using Simple Exponential Smoothing, we have taken 3 cases.
In fit1, we explicitly provide the model with the smoothing parameter α=0.2
In fit2, we choose an α=0.6
In fit3, we use the auto-optimization that allows statsmodels to automatically find an optimized value for us. This is the recommended approach.
Fitted values for Simple Exponential Smoothing.
Simple Exponential Smoothing Predictions
The best output is given when alpha=1, indicating recent observations are given the highest weight.
Holt’s Model
Holt extended simple exponential smoothing to allow the forecasting of data with a trend. (alpha for level and beta * for trend). The forecasts generated by Holt’s linear method display a constant
trend(either upward or downward). Due to this, we tend to over forecast-hence we use a concept of damped trend. It dampens the trend to a flat line in the near future.
Modeling Using Holt’s Model:
Under this, we took three cases
1.In fit1, we explicitly provide the model with the smoothing parameter α=0.8, β*=0.2.
2.In fit2, we use an exponential model rather than Holt’s additive model(which is the default).
3.In fit3, we use a damped version of the Holt’s additive model but allow the dampening parameter ϕ to be optimized while fixing the values for α=0.8, β*=0.2.
Holt’s Model.
Holt’s model fitted values.
Prediction Using Holt’s model
The lowest value of RMSE is when alpha=0.8 and smoothing-slope=0.2 when the model is the exponential model in nature.
Holt’s Winter Model
Holt and Winters extended Holt’s method to capture seasonality. The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations. It has three parameters alpha which is
the level, Beta* which is the trend, and gamma which is the seasonality. The additive method is preferred when the seasonal variations are roughly constant through the series, while the
multiplicative method is preferred when the seasonal variations are changing proportionally to the level of the series.
Modeling Using Holt’s Winter Model
1.In fit1, we use additive trend, additive seasonal of period season_length=4, and a Box-Cox transformation.
2.In fit2, we use additive trend, multiplicative seasonal of period season_length=4, and a Box-Cox transformation.
3.In fit3, we use additive damped trend, additive seasonal of period season_length=4, and a Box-Cox transformation.
4.In fit4, we use multiplicative damped trend, multiplicative seasonal of period season_length=4, and a Box-Cox transformation.
Box-Cox Transformation: A Box-Cox transformation is a transformation of a non-normal dependent variable into a normal shape. Normality is an important assumption for many statistical techniques; if
your data isn’t normal, applying a Box-Cox means that you are able to run a broader number of tests.
In fit1, we use additive trend, additive seasonal of period season_length=4, and a Box-Cox transformation.
Case 2:
In fit2, we use additive trend, multiplicative seasonal of period season_length=4, and a Box-Cox transformation.
Case 3:
In fit3, we use additive damped trend, additive seasonal of period season_length=4, and a Box-Cox transformation.
Case 4:
In fit4, we use multiplicative damped trend, multiplicative seasonal of period season_length=4, and a Box-Cox transformation.
Best Model:
The holt winter is giving me the lowest RMSE(281.91) when trend and seasonality are additive.
Linear Regression
Up Next I applied the Linear Regression Model with Open Price as my dependent variable.Steps involved in Linear Regression
1. Check for missing Values.
2. Check for multicollinearity, if there is high multicollinearity -calculate VIF, the variable with the highest VIF, drop that variable. Repeat the process until all the variables are not affected
by multicollinearity.
Correlation and calculation of VIF
Fig 1
In the above (fig 1) we see that adj_close and close both have very high VIF then we saw that Close has more VIF then we will drop it. Post that we will again check the correlation and we see that
Adj Close and High have a high correlation.
Fig 2
In the above fig(fig 2) we see that Adj close has a higher vif value than High, So we will drop that variable.
Fig 3
Now we can perform the modeling part. Let us look at the OLS regression.
OLS OUTPUT(1)
OLS Analysis:
1. Significant Variables: As per the p values, significant variables are Low, Volume.
2. Omnibus: Omnibus is a test for skewness and kurtosis. In this case, the omnibus is high and the probability of omnibus is zero, indicating residuals are not normally distributed.
3. Durbin Watson: It is a test for autocorrelation at AR(1) lag.
The Hypotheses for the Durbin Watson test are:
H0 = no first order autocorrelation.
H1 = first order correlation exists.
(For a first-order correlation, the lag is a one-time unit).
In this case, autocorrelation is 2.1 indicating negative autocorrelation.
Heteroskadistic Tests:
Brusche Pagan Test.
As per the above test, our residuals or errors are heteroskedastic in nature. Heteroskedasticity is a problem because it makes our model less inefficient because there will be some unexplained
variance that cannot be explained by any other model.
Error: 4.4%, which is quite acceptable.
Random Forest:
Next, I build a model using Random Forest Regressor. Firstly I built a model with the given hyperparameters.
Error rate:26% which is quite high and RMSE is 879.612
Next, we tuned our hyperparameters using Grid search.
Grid Search Output
So After tuning the hyperparameters, I saw that RMSE scored had decreased and error has become 24% (which is still very high ) as compared to linear regression.
Support Vector Regressor
Linear Kernel:
In Svm Scaling is a necessary condition otherwise it takes a lot of time to converge. It is necessary to scale both the independent and dependent variables. Post prediction we need to do reverse
scaling, otherwise, the scale of predicted variables and the test set will be different.
After building the model we see that RMSE score is 159.24 and the error is 4.7% which is quite close to Linear Regression.
Support Vector Regression(RBF KERNEL)
In the above snippet, we actually see that when we use RBF kernel our RMSE score increased drastically to 1307, and error increased to 39.3%. So RBF kernel is not suitable for this model.
Support Vector Regression(Polynomial KERNEL)
From the above output, it is clear polynomial kernel is not suitable for this dataset because RMSE(5048.50) is very high and the error is 151.8%. So this gives us a clear picture that data is linear
in nature.
Out of all the models, we have applied, the best model is Linear Regression(lowest RMSE score of 146.79) and the error is 4.45%
You can find out my code here(Github Link:https://github.com/neelcoder/Time-Series)
You can reach me at LinkedIn(https://www.linkedin.com/in/neel-roy-55743a12a/)
About the Author
Neel Roy
As an executive with over 4 years of experience in the BFSI Industry, I offer a record of success as a key contributor in the process & sales operations management that solved business problems in
consumer targeting, market prioritization, and business analytics. My background as a marketer, export & import/ LC process expert, combined with my technical acumen has positioned me as a valuable
resource in delivering and enhancing solutions.
I have successfully completed Data Science Certification from Jigsaw Academy, 1st level of Certification from IIBA (International Institute of Business Analysis), and currently pursuing PGP Data
Science from Praxis Business School, seeking an assignment to utilize my skills and abilities in the field of Data Science. I have good knowledge of Python, SQL, R & Tableau, and Data Processing.
I am equally comfortable in the business and technical realms, communicating effortlessly with clients and key stakeholders, I leverage skills in today’s technologies spanning data analytics & data
mining, primary research, consumer segmentation, and market analysis. I engage my passion, creativity, and analytical skills to play a vital role in facilitating the company’s success and helping to
shape the organization’s future.
Responses From Readers
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Stock Average Calculator | Calculate Average Stock Price Online - LATEST IPO & STOCK MARKET NEWS
Stock Average Calculator | Calculate Average Stock Price Online
Stock Average Calculator
The Stock Average Calculator is a valuable tool for investors looking to manage their portfolio effectively. It aids in calculating the average share price, providing insights into optimizing
investment strategies.
Average Price: ₹ 0
Total Quantity: 0
Total Investment: ₹ 0
What is a stock average?
A stock average is the computed mean price of a stock over a given period, usually established by averaging the prices of many transactions. It offers investors a whole perspective of the average
price paid for their shares, which helps with investment research and decision-making.
How do you calculate the stock average?
To find the stock average, add the total cost of all stock transactions and divide by the total number of shares purchased. This calculates the weighted average price per share. Alternatively, use
the formula (Opening Stock + Closing Stock) / 2 for inventory, calculating average stock levels throughout time.
What is the formula for the average stock?
The formula for calculating average stock is Average stock = (Opening Stock + Closing Stock) / 2. It calculates the average stock available over time based on the opening and closing stock levels.
Both accounting procedures and inventory management frequently employ this streamlined method.
What is the average stock price?
The average stock price refers to the mean value of a stock’s price during a specific period, commonly computed by averaging the prices of several transactions. It gives investors a representative
figure for the stock’s price movement and helps them evaluate investment success and make decisions.
How do you calculate the average cost per stock?
To get the average cost per stock, add the total cost of all stock transactions and divide by the total number of shares acquired. This produces the weighted average price per share, the average
price paid for the stock over numerous transactions.
Why Stock Average Calculator?
As an investor, you may encounter situations where a stock’s price moves contrary to your expectations. For instance, you bought Reliance stocks with the anticipation of an upward trend. However, the
market moves downward. Despite this, your faith in the stock persists. The Stock Average Calculator becomes essential in such scenarios, allowing you to strategically add more stocks to lower the
average price.
Our Other Calculator
How Does Stock Average Calculator Work?
Let’s delve into the mechanics of the Stock Average Calculator. Consider a scenario where you purchased 10 stocks of Tata Motors at a price of 200 each. Subsequently, the stock’s value decreases to
150. With confidence in Tata Motors’ future prospects, you aim to reduce the average stock price by acquiring more shares. The calculator assists by determining how many additional stocks you need to
purchase to bring the average closer to the current price. This tool, such as the Share Average Calculator by FinanceX, offers a user-friendly interface where you input your purchase details, and it
provides you with the recalculated average price.
Steps to Use Stock Average Calculator:
1. Input Purchase Prices: Enter the purchase price for each instance of buying the stock.
2. Calculating Average Price: The calculator processes the inputted purchase prices to compute the average stock price.
3. Decision Making: Use the calculated average to make informed decisions about buying more stocks. This is typically done when the current stock price is lower than the calculated average.
4. Reducing Average Price: The goal is to strategically buy more shares at lower prices, aiming to bring down the average cost per share.
5. Tool Output: The Stock Average Calculator provides the user with the recalculated average stock price based on new purchases.
This tool empowers investors to make informed decisions and manage their portfolio actively. It’s crucial to exercise caution and stay well-informed, as market conditions can be unpredictable.
What is a Stock Average Calculator?
A Stock Average Calculator is a tool that helps investors determine the average price of their stock holdings. It automates estimating the average price paid for a particular stock based on entered
data such as purchase prices and share numbers. This calculator benefits investors who buy the same stock several times at various prices since it gives them a consolidated average that shows their
investment’s whole cost basis.
The calculator typically asks users to enter the purchase price and number of shares purchased for each transaction. It then combines this information to get the weighted average price, which
considers the price and the number of shares acquired at each price point. This enables investors to obtain a more accurate estimate of their average investment cost while accounting for differences
in purchase costs over time.
A Stock Market Average Calculator allows investors to measure the performance of their assets better, make educated decisions about purchasing or selling stocks, and more successfully apply methods
such as dollar-cost averaging. Overall, it’s an effective tool for managing and improving investment portfolios.
How Does Stock Average Calculator Work?
A Stock Average Calculator works by accepting user-provided input data, such as purchase prices and numbers of shares purchased for a particular stock, and calculating the average cost of those
shares. Here’s how it works, step by step:
Input Data
The user enters the purchase pricing and number of shares purchased for each transaction involving the stock in issue. This information usually includes the price paid per share and the quantity of
shares acquired in each transaction.
Weighted Average Calculation
The calculator uses the provided data to compute the stock’s weighted average price. This entails summing these values over all transactions and multiplying the price per share by the total number of
shares acquired in each transaction.
Total Cost Basis
The calculator also calculates the investment’s total cost basis, which is the sum of the expenses for all shares acquired.
Final Output
After the calculations are completed, the calculator displays the average price per share and the total cost basis of the investment. Investors may make better investing decisions using this
information, which helps them comprehend the average price they have paid for their shares over time.
Other Elements
To provide a more thorough study of the success of the investment, many stock market average calculators may include other elements, such as the ability to account for dividends received or
transaction fees paid.
How to Calculate Average Stock Price?
To calculate the average stock price, use the average price paid for a stock over numerous transactions. Here is how to calculate average stock price:
• Collect Transaction Data: Gather all necessary data for each stock transaction, such as purchase prices and share amounts.
• Calculate Total Cost: For every transaction, multiply the price per share by the amount of shares purchased. You will then be able to see how much each transaction costs overall.
• Sum Entire Costs: Add up all the costs of all transactions. This will calculate the total cost of acquiring all shares.
• Sum Total Shares: Add the total number of shares purchased in all transactions. You will then see how many shares you have bought overall.
• Calculate the Average Price: Divide the total cost of all shares by the total number of shares acquired. This gives you the average price per share.
• Optional: Adjust for dividends and fees: If appropriate, modify the average price per share to reflect any dividends received or transaction fees paid. Before determining the average price per
share, subtract the dividends received and add transaction costs to the overall cost.
Formula for Calculating Average Stock
To compute the average stock level, add the starting and closing stock and divide by two. This offers you an estimate of the average stock level over time. The formula for calculating the average
stock price is:
• Average Stock = (Opening Stock + Closing Stock) / 2
Here is a breakdown of the formula.
• Opening Stock: The amount of inventory available at the start of the period.
• Closing Stock: The amount of inventory available at the end of the period.
This method assumes that stock has a linear distribution over time, which may only sometimes correctly reflect absolute stock levels over the period, mainly if there are notable changes in stock
levels. However, this technique provides a quick and easy approach to estimating typical stock levels for many practical applications, particularly in simple inventory management settings.
Benefits of Stock Average Calculator
A Stock Market Average Calculator provides various advantages to investors, assisting with educated decision-making and portfolio management:
Precise Average Calculation
The calculator computes an accurate average stock price by considering all buy transactions, including prices and volumes. This precision is necessary for determining the underlying cost foundation
of investments.
Saves Time and Effort
Manual computations may be laborious and prone to inaccuracy when handling many transactions. A calculator streamlines this procedure, saving investors time and effort.
Facilitates Informed Judgments
With an accurate average stock price readily available, investors may judge whether to purchase, sell, or keep stocks better. It clarifies if the stock price is above or below average, which helps
investors decide their investments.
Supports Investment Approaches
Using dollar-cost averaging or another investment strategy, a Stock Average Calculator may assist investors in adequately applying their preferred technique. It assures compliance with the strategy’s
principles by giving precise calculations.
Improves Portfolio Management
Investors better understand their portfolio’s performance by consistently tracking and updating average stock prices. They can make more accurate assessments of individual investment profitability
and overall portfolio success.
Risk Management
The calculator helps investors manage risk by better understanding the average price paid for equities. Investors may measure their exposure to market volatility and alter their portfolios
Frequently Asked Questions | {"url":"https://ipolive.in/ipo/stock-average-calculator/","timestamp":"2024-11-13T15:41:31Z","content_type":"text/html","content_length":"80789","record_id":"<urn:uuid:e6dd9345-ef6a-4a13-b461-e1289a05ca84>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00243.warc.gz"} |
Past Seminars
Seminar on October 29, 2024
Speaker: Dr. Zoe Nieraeth (she/her), the University of the Basque country (UPV/EHU); https://sites.google.com/view/zoe-nieraeth
The title of the lecture: “A lattice approach to singular integrals”
Abstract: In this talk, we discuss the problem of characterizing the boundedness of singular integrals in Banach lattices such as weighted Morrey and variable Lebesgue spaces. Moreover, we apply this
lattice viewpoint to the setting of matrix weights to obtain new characterizations and Rubio de Francia extrapolation results for matrix Muckenhoupt weights.
Download poster: Link to Download
Seminar on October 15, 2024
Speaker: Prof. Sundaram Thangavelu, Indian Institute of Science, Bangalore; https://math.iisc.ac.in/~veluma/
The title of the lecture: “How fast can the Fourier transform of a compactly supported function decay?”
Abstract: It is well known that the Fourier transform of a compactly supported function on \( \mathbb{R}^n \) cannot have compact support. A natural question is therefore what is the best possible
decay for such functions? A classical theorem of Ingham answers this question. In this talk we survey the recent developments on the analogues of this theorem when Fourier transform is replaced by
Helgason Fourier transform on Riemannian symmetric spaces or the group Fourier transform on Heisenberg groups.
Download poster: Link to Download
Seminar on June 11, 2024
Speaker: Prof. Dr. Carsten Trunk, TU Ilmenau; https://www.tu-ilmenau.de/universitaet/fakultaeten/fakultaet-mathematik-und-naturwissenschaften/profil/institute-und-fachgebiete/institut-fuer-mathematik
The title of the lecture: “Essential spectra of Sturm–Liouville operators and their indefinite counterpart”
Abstract: We show new perturbation results and invariance of essential spectra in terms of the real, locally integrable coefficients \( p \), \( q \), \( r \) for general Sturm-Liouville differential
expressions of the form
\( \displaystyle \frac{1}{r} \left( -\frac{d}{dx} p \frac{d}{dx} + q \right)\,, \)
with a.e. positive functions \( p \) and \( r \). If one allows sign changes of the weight function \( r \), then the situation changes dramatically but one keeps control over the essential spectrum
and some properties of the isolated point spectrum can still be proved.
This talk is based on a joint works with J. Behrndt (Graz), G. Teschl (Vienna), and P. Schmitz (Ilmenau).
Download poster: Link to Download
Seminar on May 28, 2024
Speaker: Prof. Zhirayr Avetisyan, Ghent University; https://www.z-avetisyan.com/
The title of the lecture: “Universal Functions and Sets”
Abstract: Universal objects in mathematics are those which are capable of, in one sense or another, representing most other objects of the same category. In the context of Fourier analysis,
universality is understood in the sense of approximations by orthogonal or Fourier series.
The first part of the talk will be devoted to universal functions. Roughly speaking, a function is universal if its Fourier series has universal properties, e.g., its partial sums converge to any
given function in a space. There are variations of this, including possible rearrangements and changes of signs. We will establish very general results about the existence of universal functions for
minimal systems in Banach spaces, which obey very basic approximation properties. On the other hand, it is clear that in a Banach space, the partial sums of a Fourier series cannot converge to
different functions. Therefore we introduce the slightly weaker notion of asymptotic universality and obtain existence results under fairly mild assumptions.
The second part of the talk will be concerned with universal sets. In addition to approximation in the norm, one often needs to approximate a given function spatially, i.e., the approximating
function should agree with the original one identically on a large subset of the domain. In this respect, the classical example is Luzin’s theorem, where measurable functions are approximated by
continuous ones. However, the arbitrarily large subset of coincidence in Luzin’s theorem depends on the original function. It is remarkable, that for certain kinds of approximation, the subset can be
chosen a priori, independent of the function being approximated, making it universal. While results in this direction abound for classical systems, we will introduce sweeping generalisations, as well
as covariant constructions in homogeneous spaces of compact topological groups.
Download poster: Link to Download
Seminar on May 14, 2024
Speaker: Prof. Sergei Grudsky, CINVESTAV, Mexico City; https://www.math.cinvestav.mx/~grudsky/index.html
The title of the lecture: “Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices”
Abstract: Analysis of the asymptotic behaviour of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity and has a history of over 100 years. For instance,
quite a number of versions of Szegö's theorem on the asymptotic behaviour of eigenvalues and of the so-called strong Szegö theorem on the asymptotic behaviour of the determinants of Toeplitz matrices
are known. Starting in the 1950s, the asymptotics of the maximum and minimum eigenvalues were actively investigated. However, investigation of the individual asymptotics of all the eigenvalues and
eigenvectors of Toeplitz matrices started only quite recently: the first papers on this subject were published in 2009-2010. A survey of this new field is presented here.
Download poster: Link to Download
Seminar on April 30, 2024
Speaker: Prof. Dr. Uwe Kähler, University of Aveiro; https://sweet.ua.pt/ukaehler/Webpage/Main.html
The title of the lecture: “Global pseudo-differential operator calculus over spin groups”
Abstract: In this talk, we present a construction of a global symbol calculus of pseudo-differential operators on spin groups in the sense of Ruzhansky-Turunen-Wirth, with special attention given to
the case of Spin(4). Using representations of Spin(4) we construct a group Fourier transform and establish the calculus of left-invariant differential operators and of difference operators on the
group Spin(4). Afterwards, we apply this calculus to give criteria for subellipticity and global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several
examples of first and second-order globally hypoelliptic differential operators are given, including somewhere the criteria of hypoellipticity will be reduced to the problem of approximation distance
between irrational and rational numbers.
Download poster: Link to Download
Seminar on April 2, 2024
Speaker: Prof. em. Alessia Kogoj, University of Urbino; https://www.researchgate.net/profile/Alessia-Kogoj
The title of the lecture: “Subelliptic Liouville Theorems”
Abstract: Several Liouville-type theorems are presented, related to evolution equations and to their “time”-stationary counterparts. The equations we are dealing with are left translations invariant
on a Lie group structure and, in some cases, homogeneous with respect to a group of dilations. In all these cases the operators have smooth coefficients and are hypoelliptic. We also present a
“polynomial” Liouville-type theorem for X-elliptic operators with nonsmooth coefficients, by extending to this new setting a celebrated result by Colding and Minicozzi related to the Laplace-Beltrami
operator on Riemannian manifolds.
The results are contained in a series of papers in collaboration with A. Bonfiglioli, E. Lanconelli, Y. Pinchover, S. Polidoro and E. Priola.
Download poster: Link to Download
Seminar on March 5, 2024
Speaker: Prof. Alex Iosevich, The University of Rochester, USA; https://people.math.rochester.edu/faculty/iosevich/
The title of the lecture: “Restriction Theory, Uncertainty Principles and Signal Recovery”
Abstract: Let \(f: {\mathbb Z}_N^d \to {\mathbb C}\) and define \(\widehat{f}(m)=N^{-d} \sum_{x \in {\mathbb Z}_N^d} \chi(-x \cdot m) f(x)\), the discrete Fourier transform, where \(\chi(t)=e^{\frac
{2 \pi i t}{N}}\). Suppose that the signal \(f\) is transmitted via its Fourier transform and that some of the transmission is lost, i.e the values \({\{\widehat{f}(m)\}}_{m \in S}\) are unobserved
for some \(S \subset {\mathbb Z}_N^d\). The question, raised by Donoho and Stark in the late \(80\)s is, are there reasonable assumptions on the signal \(f\) and the missing set of frequencies \(S\)
such that \(f\) can be recovered exactly, despite the signal loss? Donoho and Stark showed that the uncertainty principle for the Fourier transform can be used to derive a set of sufficient
conditions. In this talk, we are going to see that discrete restriction theory for the Fourier transform can be brought to bear on this problem. We are also going to discuss some discretization
procedures that can be used to speed up the recovery process at the cost of a small error.
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Seminar on March 5, 2024
Speaker: Prof. Joachim Toft, Linnaeus University (LNU); https://lnu.se/en/staff/joachim.toft/
The title of the lecture: “Fractional Fourier transform, harmonic oscillator propagators and Strichartz estimates”
Abstract: Using the Bargmann transform, we prove that harmonic oscillator propagators and Fractional Fourier Transforms (FFT) are essentially the same. We deduce continuity properties for such
operators on modulation spaces and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on modulation spaces. We also show that general forms of
fractional harmonic oscillator propagators are continuous and suitable for so-called Pilipović spaces and their distribution spaces. Especially we show that FFT of any complex order can be defined
and that these transforms are continuous on strict Pilipović function and distribution spaces.
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Seminar on February 20, 2024
Speaker: Prof. Serena Federico, University of Bologna; https://www.unibo.it/sitoweb/serena.federico2/cv-en
The title of the lecture: “Weyl Calculus on graded groups”
Abstract: In this talk we will investigate the existence of a Weyl pseudo-differential calculus on any graded Lie group. To start with, we will recall the fundamental properties of the Weyl
quantization in the Euclidean setting and of the corresponding pseudo-differential calculus. Afterwards, we will define a family of quantizations on any graded Lie group and develop the corresponding
symbolic calculus. Finally, inside this family of quantizations we will identify a possible Weyl quantization on any graded group. In the end, we will see that the identified quantization is the
uniquely determined Weyl quantization in the case of the Heisenberg group.
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Seminar on February 06, 2024
Speaker: Prof. em. Elias Wegert, Institute of Applied Analysis, TU Bergakademie Freiberg, 09596 Freiberg, Germany; https://de.wikipedia.org/wiki/Elias_Wegert
The title of the lecture: “Numerical Range, Blaschke Products, and Poncelet Polygons”
Abstract: In 2016, Gau, Wang and Wu conjectured that a partial isometry \( A \) acting on a \( n \)-dimensional complex Hilbert space cannot have a circular numerical range with a non-zero centre. In
this talk, we verify this for operators with \( \mathrm{rank}\,A = n - 1 \).
The proof is based on the unitary similarity of \( A \) to a compressed shift operator generated by a finite Blaschke product \( B \). We then use the description of the numerical range by Poncelet
polygons associated with \( B \), a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenters of the vertices of Poncelet polygons
involving elliptic functions.
The talk is a mixture of operator theory, plain geometry, and elementary complex analysis. In particular, we give a short introduction to the visualization of complex functions by „phase plots“ and
demonstrate how they can help to discover relevant properties of these functions.
Full abstract see here: https://drive.google.com/file/d/1u44q9liZP6IxAwECBwRS7Lpkq9hmoSIz/view?usp=sharing
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Seminar on January 23, 2024
Speaker: Prof. Dr. Reinhold Schneider, Technical University of Berlin; https://scholar.google.de/citations?user=3CmBwQcAAAAJ&hl=de
The title of the lecture: “Numerical Solution of high-dimensional Hamilton Jacobi Bellmann (HJB) Equations, Mean Field Games and Compositional Tensor Networks”
Abstract: The partial differential equations of Mean Field Games introduced by Lasry & Lions describe the solution of feedback optimal control problems as well as optimal transport problems. These
equations play a fundamental role in optimal control, optimal transport, computational finance and machine learning. Therefore solving these kinds of equations seems to be of utmost importance in
future science and technology. However for solving these non-linear and high dimensional equations, one has to deal with two major difficulties, namely 1) the curse of dimensions and secondly, 2)
possible lack of regularity. Here we focus only on the first issue. (The second problem can be relaxed by adding randomness, which is always present in practice.) We consider Potential Mean Field
Games and are focusing on the (deterministic/stochastic) HJB. In order to compute semi-global solutions, we consider control affine dynamical systems and quadratic cost for the control as a prototype
example. We follow a Lagrangian approach perspective and a recent approximation concept of compositional sparsity. In contrast to our earlier published semi-Lagrangian approaches, we describe a
direct minimization of the total averaged cost overall initial values, first introduced by Kunisch & Walter (2021). Approximately, the true total cost is replaced by averaging oversampled initial
values, where the sought approximate value function is parametrized in tensor form. Compositional sparsity has been inspired by Deep Neural Networks. The individual neural network layers are replaced
by tree-based tensor networks (HT/TT) or sparse polynomials, which improves the stability for a numerical treatment of the optimization problem.
Related literature: A machine learning framework for solving high-dimensional mean field game and mean field control problems, Ruthotto, Osher, Li, Nurbekyan, Fung - Proceedings of the National
Academy of Sciences, 2020
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Seminar on December 12, 2023
Speaker: Prof. Claudia Garetto, Queen Mary University of London; https://www.qmul.ac.uk/maths/profiles/claudiagaretto.html
The title of the lecture: “Higher order hyperbolic equations with multiplicities”
Abstract: In this talk, I will discuss Gevrey and C∞ well-posedness for linear higher-order hyperbolic equations with multiplicities. I will review the different methods employed for time- and/or
x-dependent coefficients and the conditions needed on the lower-order terms. Work in collaboration with Michael Ruzhanksy (Ghent/QMUL) and Bolys Sabibtek (QMUL).
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Seminar on November 28, 2023
Speaker: Dr. Nazar Miheisi, King’s Kollege London; https://www.kcl.ac.uk/people/nazar-miheisi
The title of the lecture: “Completeness of Systems of Inner Functions”
Abstract: An inner function is an analytic function on the unit disk whose boundary values have modulus 1 almost everywhere – these play a special role in operator theory and function theory. In this
talk, I will discuss the following problem: for which inner functions ϕ and ψ, are the powers ϕm, ψn (m, n ∈ Z) complete in the weak-∗ topology of L∞? This problem was first considered in 2011 by
Hedenmalm and Montes-Rodríguez who gave a complete solution for atomic inner functions with one singularity. I will give a recent extension of this result to a much wider class of inner functions. If
time permits, I will also discuss a connection with a problem in Fourier uniqueness.
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Seminar on November 14, 2023
Speaker: Pedro Gonçalves Ramos, Postdoctoral researcher at École Polytechnique Fédérale de Lausanne; https://sites.google.com/view/gionnoramos/
The title of the lecture: “Time-frequency localization operators, their eigenvalues and relationship to elliptic PDE”
Abstract: In the classical realm of time-frequency analysis, an object of major interest is the short-time Fourier transform of a function. This object is a modified Fourier transform of a signal \(
f\left( x \right) \), changed by a certain ’window function’, in order to make simultaneous analysis on frequency and time more feasible. Since the pioneering work of Daubechies, time-frequency
localisation operators have been of extreme importance in that analysis. These operators are defined to measure how much
the short-time Fourier transform of a function concentrates in the time-frequency plane, and thus the study of the eigenvalues and eigenfunctions of such operators is intimately connected to how well
one can perform the simultaneous analysis of signals mentioned above. In this talk, we will explore the case where the window function is a Gaussian. We will discuss some classical and recent results
on domains of maximal time-frequency concentration, their eigenvalues, and stability/inverse problems associated with such properties. During this investigation, we shall see that many of these
problems possess some rather unexpected connections with calculus of variations, overdetermined elliptic boundary value problems and free boundary problems.
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Seminar on July 10
Speaker: Prof. Alex Iosevich, The University of Rochester, USA; https://www.sas.rochester.edu/mth/people/faculty/iosevich-alex/index.html
The title of the lecture: “Finite point configurations and learning theory”
Abstract: The basic question we ask is, how large does the Hausdorff dimension of a subset of Euclidean space need to be to ensure that it contains a congruent copy of the finite point configuration
of a given type? This problem is strongly connected with the Erdos distance problem in combinatorics, the Falconer distance problem in geometric measure theory, and Furstenberg-type configuration
problems in ergodic theory. Connections with complexity problems in learning theory will also be discussed.
This talk was delivered both in person, at the University of Georgia in Tbilisi, Georgia, and remotely, via the Webex platform.
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Seminar on June 13, 2023
Speaker: Doctoral student Duvan Cardona Sanchez, Ghent Analysis and PDE Center; https://sites.google.com/site/duvancardonas/
The title of the lecture: "Continuity properties for some operators arising in non-commutative harmonic analysis"
Abstract: The non-commutative harmonic analysis on nilpotent Lie groups after the developments by Folland and Stein in the 70s has been fundamental for the analysis of hypoelliptic problems on graded
Lie groups. They started the program of generalising in the setting of nilpotent Lie groups the results available in the Eucliden harmonic analysis. Several fundamental results in this area have been
obtained in the last 50 years. In this setting, we review some recent results about the boundedness of oscillating Fourier multipliers, pseudo-differential operators and other operators arising in
this setting. The results presented in this talk are part of my joint work with M. Ruzhansky (Ghent) and J. Delgado (Colombia).
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Seminar on May 30, 2023
Speaker: Prof. Harm Bart, Erasmus University Rotterdam; https://www.researchgate.net/scientific-contributions/Harm-Bart-2013767776
The title of the lecture: „The Rouché Theorem for Fredholm operator-valued functions: an enhanced version“
Abstract: The well-known classical Rouché Theorem is concerned with the perturbation of scalar analytic functions. Roughly speaking: if the perturbation is small enough, the perturbed function has
the same number of zeros as the original one. In the 1971 paper [GS], I.C. Gohberg and E.I. Sigal generalized the theorem to a result involving Fredholm operator-valued functions. Although just one
of them is indicated in [GS] (and in [GGK], Section XI.9 as well), there are actually two versions of the generalization, due to the fact that bounded linear operators as a rule do not commute. So a
commutativity issue manifests itself here.
There is another one. The Rouché Theorems involve the logarithmic residues of the functions involved, i.e., a contour integral of their logarithmic derivatives. In the scalar case such a logarithmic
is unambiguously determined; in the non-scalar setting, it is not. There, again, two possibilities present themselves, depending on which order one takes in the product of the derivative and the
inverse. Generally, these options do not come down to the same.
In the lecture, an approach will be presented that yields an encompassing (strictly) stronger variant of the results indicated above.
[GS] I.C. Gohberg, E.I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Mat. Sbornik 84 (126) (1971), 607-629 (Russian), English Transl. in Math. USSR
Sbornik 13 (1971), 603–625.
[GGK] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, Vol. I, Operator Theory: Advances and Applications, OT 49, Birkhäuser Verlag, Basel 1990.
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Seminar on May 16, 2023
Speaker: Prof. Dr. Davit Natroshvili, Department of Mathematics, Georgian Technical University, Tbilisi, Georgia; https://my.gtu.ge/Personal/473
The title of the lecture: “An alternative potential method for mixed boundary value problems”
Abstract: We consider an alternative potential method to investigate a mixed boundary value problem (BVP) for the Lamé system of elasticity in the case of a three-dimensional bounded domain when the
boundary surface is divided into two disjoint parts, where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and the stress vector. Our
approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single-layer and double-layer potentials with
densities supported respectively on the Dirichlet and Neumann parts of the boundary.
This approach reduces the mixed BVP under consideration to a system of integral equations which contain neither extensions of the Dirichlet or Neumann data, nor the Steklov-Poincaré type operator.
Moreover, the right-hand sides of the system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration.
The corresponding matrix integral operator is invertible in the appropriate Bessel potential and Besov spaces, which implies the unconditional unique solvability of the mixed BVP in the corresponding
Sobolev spaces and representability of solutions in the form of a linear combination of the single-layer and double-layer potentials with densities supported respectively on the Dirichlet and Neumann
parts of the boundary.
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Seminar on May 02, 2023
Speaker: Anastasia Kisil, University of Manchester, UK; https://anastasiakisil.weebly.com/
The title of the lecture: "A generalisation of the Wiener-Hopf methods for an equation in two variables with three unknown functions"
Abstract: In the talk, I will present an analytic solution to a generalisation of the Wiener– Hopf equation in two variables and with three unknown functions. This equation arises in many
applications, for example, when solving the discrete Helmholtz equation associated with scattering on a domain with a perpendicular boundary. The traditional Wiener–Hopf method is suitable for
problems involving boundary data on co-linear semi-infinite intervals, not for boundaries at an angle. This significant extension will enable the analytical solution to a new class of problems with
more boundary configurations. Progress is made by defining an underlining manifold that links the two variables. This allows us to meromorphically continue the unknown functions on this manifold and
formulate a jump condition. As a result, the problem is fully solvable in terms of Cauchy-type integrals which is surprising since this is not always possible for this type of functional equation.
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Seminar on March 21, 2023
Speaker: Guillermo P. Curbera, Professor of Mathematics, Instituto de Matemáticas IMUS, Universidad de Sevilla; https://euler.us.es/~curbera/
The title of the lecture: "The fine spectra of the finite Hilbert transform, beyond the \( L^p \)-spaces"
Abstract: We consider the finite Hilbert transform acting on rearrangement invariant spaces over \(\left( 1,1 \right) \). We present results on the spectrum and exemplary spectra, extending the Widom
results for the \( L^p \)-spaces.
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Seminar on February 21, 2023
Speaker: Jean Lagacé, King’s College London, UK; https://lagacejean.github.io/
The title of the lecture: “Spectral Geometry on rough spaces”
Abstract: Spectral geometry is concerned with studying the spectrum (the eigenvalues) of differential operators (like the Laplacian) on surfaces and manifolds, and relating that spectrum with the
geometry of the underlying space. Many tools, for instance, pseudo-differential calculus and microlocal analysis, have been developed over the years to study the spectrum when everything is smooth.
In this talk, I will focus on the asymptotic distribution of eigenvalues (Weyl’s Law), and explain what we can keep when we do not assume the smoothness of the underlying space. Special attention
will be given to the Steklov problem.
This is joint work with Mikhail Karpukhin (University College London) and Iosif Polterovich (Université de Montréal).
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Seminar on February 21, 2023
Speaker: Prof. Jani Virtanen, University of Reading, UK; https://janivirtanen.wordpress.com/
The title of the lecture: “On the Berger-Coburn phenomenon”
Abstract: In 1987, Berger and Coburn proved that if the Hankel operator with a bounded symbol is compact on the classical Fock space, then so is the Hankel operator with the conjugate symbol. This
property is unique for the Fock space, and fails for the Hardy space and the Bergman space, for example. In 2004, Bauer showed that an analogous result remains true if compactness is replaced by
being Hilbert-Schmidt. The question of what happens with the other Schatten classes remained open for almost two decades. I report on the recent progress, which answers the question in full.
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Seminar on February 7, 2023
Speaker: Prof. Alexander Meskhi, TSU A. Razmadze Mathematical Institute and Kutaisi International University, Georgia; https://www.kiu.edu.ge/?m=316
The title of the lecture: “Boundedness criteria for multilinear fractional integral operators”
Abstract: Please see the poster
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Seminar on January 24, 2023
Speaker: Prof. Dr. Dorothee Haroske, Friedrich-Schiller-Universität Jena, Fakultät für Mathematik & Informatik, Institut für Mathematik, Germany; https://users.fmi.uni-jena.de/~haroske/
The title of the lecture: “Morrey smoothness spaces: A new approach”
Abstract: In recent years so-called Morrey smoothness spaces attracted a lot of interest. They can (also) be understood as generalisations of the classical spaces \( A^s_{p,q} (\mathbb{R}^n) \), \( A
\in \{B,F\} \), where the parameters satisfy \( s\in \mathbb{R} \) (smoothness), \( 0<p \le \infty \) (integrability) and \( 0<q \le \infty \) (summability). In the case of Morrey smoothness spaces
additional parameters are involved. In our opinion, among the various approaches at least two scales enjoy special attention, also in view of applications: the scales \( \mathcal{A}^s_{u,p,q} (\
mathbb{R}^n) \), with \( \mathcal{A}\in \{\mathcal{N}, \mathcal{E}\} \), \( u\geq p \), and \( A^{s, \tau}_{p,q} (\mathbb{R}^n) \), with \( \tau\geq 0 \).
We reorganise these two prominent types of Morrey smoothness spaces by adding to \( \left( s,p,q \right) \) the so-called slope parameter \( \varrho \), preferably (but not exclusively) with \( -n \
le \varrho <0 \). It comes out that \( \left| \varrho \right| \) replaces \( n \), and \( \min \left( \left| \varrho \right|,1 \right) \) replaces \( 1 \) in slopes of (broken) lines in the \( \left(
\frac{1}{p},s \right) \)-diagram characterising distinguished properties of the spaces \( A^s_{p,q} \left( \mathbb{R}^n \right) \) and their Morrey counterparts.
Our aim is two-fold. On the one hand, we reformulate some assertions already available in the literature (many of them are quite recent). On the other hand, we establish on this basis new properties,
a few of them became visible only in the context of the offered new approach, governed, now, by the four parameters \( \left( s,p,q,\varrho \right) \).
The talk is based on joint work with Hans Triebel (Jena).
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Seminar on December 13, 2022
Speaker: Prof. Luis Castro, CIDMA - Center for Research and Development in Mathematics and Applications & Department of Mathematics, University of Aveiro, Portugal; https://www.ua.pt/pt/p/10311888
The title of the lecture: “New convolutions generated by Hermite functions and consequent classes of integral operators”
Abstract: We will introduce new convolutions generated by multi-dimensional Hermite functions and study some of their properties. Namely, we will analyse classes of integral operators generated by
those convolutions. This will also give rise to the study of the solvability of a general class of integral equations whose kernel depends on four different functions. Additional properties will
appear along the way, among which we highlight new Young-type inequalities and factorizations (where the new convolutions take a central role). The talk is based on joint work with R.C. Guerra
(Coimbra, Portugal) and N.M. Tuan (Hanoi, Vietnam).
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Seminar on November 29, 2022
Speaker: Prof. Ferenc Weisz, Eötvös Loránd University, Budapest, Hungary; https://www.researchgate.net/profile/Ferenc-Weisz
The title of the lecture: “Hardy spaces in the theory of trigonometric and Walsh-Fourier series and Lebesgue points”
Abstract: We introduce higher dimensional martingale and classical Hardy spaces and consider trigonometric and Walsh-Fourier series. We state that the maximal operator of the Fejér or Cesàro means of
a higher dimensional function is bounded from the corresponding Hardy space to the Lebesgue space. This implies some almost everywhere convergence of the Cesàro means. We characterize the set of
convergence as different types of Lebesgue points.
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Seminar on November 15, 2022
Speaker: Peter Kuchment, University Distinguished Professor, Mathematics Department, Texas A&M University, USA; https://www.math.tamu.edu/~peter.kuchment/
The title of the lecture: “Wonderful World of tomography”
Abstract: Here I would provide an introduction to non-experts to the multifaceted and booming mathematics (PDEs, harmonic analysis, etc.) of imaging.
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Seminar on November 1, 2022
Speaker: Prof. Elijah Liflyand, Department of Mathematics, Bar-Ilan University, Israel; https://u.math.biu.ac.il/~liflyand/
The title of the lecture: “Wiener algebras and trigonometric series in a coordinated fashion”
Abstract: Let $W_0(\mathbb{R})$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a
trigonometric series $\sum\limits_{k=-\infty}^\infty c_k e^{ikt}$ is the Fourier series of an integrable function if and only if there exists a $\phi\in W_0(\mathbb R)$ such that $\phi(k)=c_k$, $k\in
\mathbb Z$. If $f\in W_0(\mathbb R)$, then the piecewise linear continuous function $\ell_f$ defined by $\ell_f(k)=f(k)$, $k\in\mathbb Z$, belongs to $W_0(\mathbb R)$ as well. Moreover, $\|\ell_f\|_
{W_0}\le \|f\|_{W_0}$. Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient
conditions are proved for a trigonometric series to be a Fourier series and new properties of $W_0$ are established.
This is joint work with R. Trigub.
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Seminar on October 18, 2022
Speaker: Prof. Diogo Oliveira e Silva, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal; https://www.math.tecnico.ulisboa.pt/~oliveiraesilva/
The title of the lecture: “The Stein-Tomas inequality: three recent improvements”
Abstract: The Stein-Tomas inequality dates back to 1975 and is a cornerstone of the Fourier restriction theory. Despite its respectable age, it is fertile ground for current research. The goal of
this talk is threefold: we present a recent proof of the sharp endpoint Stein-Tomas inequality in three space dimensions; we present a variational refinement and withdraw some consequences; and we
discuss how to improve the Stein-Tomas inequality in the presence of certain symmetries.
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Seminar on May 09, 2022
Speaker: Prof. Dr. Volker Mehrmann, TU Berlin, Germany; https://en.wikipedia.org/wiki/Volker_Mehrmann
The title of the lecture: “Modeling analysis and numerical simulation of multi-physical systems: A change of paradigm”
Abstract: Most real-world dynamical systems consist of subsystems from different physical domains, modelled by partial differential equations, ordinary differential equations, and algebraic
equations, combined with input and output connections. In recent years, the class of dissipative port-Hamiltonian (pH) systems have emerged as a very efficient new modelling methodology to deal with
such complex systems. The main reasons are that the network-based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserves the pH structure
and the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations. Furthermore, dissipative pH systems form a very robust
representation under structured perturbations and directly indicate Lyapunov functions for stability analysis. We discuss dissipative pH systems and describe, how many classical models can be
formulated in this class. We illustrate some of the nice algebraic properties, including local canonical forms, the formulation of an associated Dirac structure, and the local invariance under
space-time dependent diffeomorphisms.
The results are illustrated with some real-world examples.
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Seminar on June 20, 2022
Speaker: Prof. Lars-Erik Persson, UiT The Arctic University of Norway; https://www.larserikpersson.se/
The title of the lecture: “On my life with Hardy and his inequalities”
Dedication: This lecture is dedicated to the memory of our dear friend, colleague and important collaborator Professor Vakhtang Kokilashvili.
Abstract: First I will describe some background and historical remarks from the beginning of this remarkable story, which started around 100 years ago. After that, I will present a fairly new
convexity approach to proving Hardy-type inequalities, which was not discovered by Hardy himself and many others, see [1], [2] and [3]. I continue by presenting some selected parts of the story until
2017, where I myself have been involved to some extent, see [1] and [4]. Finally, I present some examples of remarkable new results after 2017, which show that this area is still a source of
inspiration for new research. In particular, to also illustrate my close connection to Georgian mathematics I mention some new Hardy-type results we developed and used in our new book [5]. Several
open questions are pointed out.
[1] A. Kufner, L.E. Persson and N. Samko, Weighted Inequalities of Hardy Type, World Scientific, Second Edition, New York, London, etc., 2017 (480 pages).
[2] L.E. Persson, Lecture Notes, Collège de France, Pierre-Louis Lions Seminar, November 2015 (48 pages).
[3] C. Niculescu and L.E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics, Springer, Second Edition, 2018 (431 pages)
[4] V. Kokilashvili, A. Meskhi and L.E. Persson, Weighted Norm Inequalities for Integral transforms with Product Kernels, Nova Scientific Publishers, Inc., New York, 2010 (355 pages).
[5] L.E. Persson, G. Tephnadze and F. Weisz, Martingale Hardy Spaces and Summability of Vilinkin-Fourier Series, Springer, New York, to appear 2022 (610 pages).
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Seminar on June 6, 2022
Speaker: Prof. Björn Birnir, CNLS and the University of California at Santa Barbara; https://birnir.math.ucsb.edu/
The title of the lecture: “The statistical theory of Stochastic Nonlinear Partial Differential Equations with application to the angiogenesis equations”
Abstract: We develop the statistical theory for stochastic nonlinear PDEs with both additive and multiplicative noise. The canonical example is the stochastic Navier-Stokes equation. We solve the
Kolmogorov-Hopf equation for the invariant measure determining the statistical quantities. Then the theory is applied to the stochastic angiogenesis equations describing how veins grow through the
human body.
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Seminar on May 23, 2022
Speaker: Prof. Oleksiy Karlovych, NOVA University Lisbon, Portugal; https://docentes.fct.unl.pt/oyk/
The title of the lecture: “Algebras of convolution type operators with continuous data do not always contain all rank one operators”
Abstract: See poster
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Seminar on May 9, 2022
Speaker: Prof. Volker Mehrmann, TU Berlin, Germany; https://www.bimos.tu-berlin.de/menue/bimos_people/members/professors/volker_mehrmann/, https://en.wikipedia.org/wiki/Volker_Mehrmann
The title of the lecture: “Modeling analysis and numerical simulation of multi-physical systems: A change of paradigm”
Abstract: Most real-world dynamical systems consist of subsystems from different physical domains, modeled by partial-differential equations, ordinary differential equations, and algebraic equations,
combined with input and output connections. To deal with such complex systems, in recent years the class of dissipative port-Hamiltonian (pH) systems have emerged as a very efficient new modeling
methodology. The main reasons are that the network-based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserves the pH structure and the
physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations. Furthermore, dissipative pH system form a very robust representation
under structured perturbations and directly indicates Lyapunov functions for stability analysis. We discuss dissipative pH systems and describe, how many classical models can be formulated in this
class. We illustrate some of the nice algebraic properties, including local canonical forms, the formulation of an associated Dirac structure, and the local invariance under space-time dependent
diffeomorphisms. The results are illustrated with some real-world examples.
Download poster: Link to Download
Seminar on April 25, 2022
Speaker: Prof. Natasha Samko, The Arctic University of Norway, Narvik, Norway; http://www.nsamko.com/
The title of the lecture: “Weighted boundedness of certain sublinear operators in generalized Morrey spaces on quasi-metric measure spaces under the growth condition”
Abstract: We prove the weighted boundedness of Calderón-Zygmund and maximal singular operators in generalized Morrey spaces on quasi-metric measure spaces, in general non-homogeneous, only under the
growth condition on the measure, for a certain class of weights. The weights and characteristics of the spaces are independent of each other. The weighted boundedness of the maximal operator is also
proved in the case when lower and upper Ahlfors exponents coincide with each other. Our approach is based on two important steps. The first is a certain transference theorem, where without using
homogeneity of the space, we provide a condition which ensures that every sublinear operator with the size condition, bounded in Lebesgue space, is also bounded in generalized Morrey space. The
second is a reduction theorem which reduces the weighted boundedness of the considered sublinear operators to that of weighted Hardy operators and the non-weighted boundedness of some special
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Seminar on April 11, 2022
Speaker: Professor Alexander Pushnitski, King’s College London, UK; https://www.kcl.ac.uk/people/alexander-pushnitski
The title of the lecture: “The spectra of some arithmetical matrices”
Abstract: I will discuss the spectral theory of a family of infinite arithmetical matrices, whose (n,m)-th entry involves the least common multiple of n and m, denoted LCM(n,m). The simplest example
of such a matrix is {1/LCM(n,m)}, where n,m range over natural numbers. It turns out that an explicit formula for the asymptotics of eigenvalues of this matrix can be given. This is recent joint work
with Titus Hilberdink (Reading).
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Seminar on March 28, 2022
Speaker: Doctoral student Duvan Cardona Sanchez, Ghent University; https://sites.google.com/site/duvancardonas/
The title of the lecture: “Oscillating Fourier multipliers theory: geometric aspects and the role of the symmetries”
Abstract: Oscillating Fourier multipliers on the torus and on Rⁿ play a fundamental role in analysis and PDE and in the setting of Lie groups are still a subject of intensive research. The classical
results by Fefferman and Stein in this direction (published between 1970 and 1972 in Acta Math.) have consolidated a fundamental theory for the harmonic analysis of these operators, even, in a more
general setting that contains Calderón-Zygmund singular integrals of convolution type. In this talk, we present some recent results that extend the Fefferman and Stein theory of oscillating Fourier
multipliers to arbitrary Lie groups of polynomial growth.
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Seminar on March 14, 2022
Speaker: Prof. Hans Georg Feichtinger, University of Vienna;
The title of the lecture: “The Banach Gelfand Triple and its role in Classical Fourier Analysis and Operator Theory”
Abstract: The Banach Gelfand Triple (S0, L2, S0*)(Rd) (which arose in the context of Time-Frequency Analysis) is a simple and useful tool, both for the derivation of mathematically valid theorems AND
for teaching relevant concepts to engineers and physicists (and of course mathematicians, interested in applications!).
In this context, the basic terms of an introductory course on Linear System’s Theory can be explained properly: Translation invariant systems are viewed as linear operators, which can be described as
convolution operator by some impulse response, whose Fourier transform is well defined (and is called transfer function), and there is a kernel theorem:
Operators T : S0(Rd) to S0*(Rd) have a "matrix representation" using some sigma in S0*(R2d). Most importantly, dual space S0 ∗ (Rd), the space of so-called mild distributions, contains all kinds of
objects relevant for signal processing: periodic signals, discrete signals, and of course discrete and periodic signals. One can show that the generalized Fourier transform for such functions works
well and reduced to the DFT/FFT (Fast Fourier Transform).
An important tool is the STFT (Short-Time Fourier Transform). Mild distributions are exactly those tempered distributions which have a bounded short-time Fourier transform, and the w∗-convergence
just corresponds to uniform convergence of the STFT over compact subsets of the time-frequency plane.
Slides: Click to see
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Seminar on February 28, 2022
Speaker: Prof. George Tephnadze, The University of Georgia;
The title of the lecture: “Almost everywhere convergence of partial sums of trigonometric and Vilenkin systems and certain summability methods”
Abstract: The classical theory of the Fourier series deals with the decomposition of a function into sinusoidal waves. Unlike these continuous waves, the Vilenkin (Walsh) functions are rectangular
waves. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, combined with martingale theory.
This talk is devoted to investigating tools that are used to study almost everywhere convergence of the partial sums of trigonometric and Vilenkin systems. In particular, these methods combined with
martingale theory helps to give a simpler proof of an analogy of the famous Carleson-Hunt theorem for Fourier series with respect to the Vilenkin system. We also define an analogy of Lebesgue points
for integrable functions and we will describe which certain summability methods are convergent in these points.
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Seminar on February 14, 2022
Speaker: Prof. Paata Ivanisvili, University of California, USA;
The title of the lecture: “Convex hull of a space curve”
Abstract: Finding a simple description of a convex hull of a set K in n-dimensional Euclidean space is a basic problem in mathematics. When K has some additional geometric structures one may hope to
give an explicit construction of its convex hull. A good starting point is when K is a space curve. In this talk I will describe convex hulls of space curves which have a "very" positive torsion. In
particular, we obtain parametric representation of the boundary of the convex hull, different formulas for their Euclidean volumes of the convex hull, the area of its boundary, and the solution to a
general moment problem corresponding to such curves.
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Seminar on January 31, 2022
Speaker: Prof. Heiko Gimperlein, University of Innsbruck and University of Parma;
The title of the lecture: “Boundary regularity for fractional Laplacians: a geometric approach”
Abstract: We consider the sharp boundary regularity of solutions to the Dirichlet problem for the fractional Laplacian on a smoothly bounded domain in Euclidean space. The fractional Laplacian is
defined via the extension method, as a Dirichlet-to-Neumann operator for a degenerate elliptic problem in a half-space of one higher dimension. We use techniques from geometric microlocal analysis to
analyse the regularity of solutions, with particular emphasis on asymptotic expansions and Hölder continuous data. Detailed and sharp results about this problem have been obtained by Gerd Grubb, and
we present a complementary approach. Extensions to polygonal domains are mentioned.
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Seminar on January 17, 2022
Speaker: Nikolai L. Vasilevski, Department of Mathematics, CINVESTAV, Mexico City, Mexico;
The title of the lecture: “On analytic type function spaces and direct sum decomposition of $L_2(D, d\nu)$”
Abstract: Link to see abstract
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Seminar on December 20
Speaker: Prof. Dr. Alexander Mielke, Weierstrass Institute for Applied Analysis and Stochastics and Humboldt-Universität zu Berlin;
The title of the lecture: “On a rigorous derivation of a wave equation with fractional damping from a system with fluid-structure interaction”
Abstract: We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half-space below. The model describes longitudinal elastic
waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where
the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally damped wave equation for the membrane. We provide the
associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy–dissipation structure for the full model, we derive
natural energy and a natural dissipation function for the fractionally damped wave equation with a time derivative of order 3/2.
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Seminar on December 06
Speaker: Sergey MIKHAILOV, Professor of Computational and Applied Mathematics, Dept. of Mathematical Sciences, Brunel University London, UK,
The title of the lecture: “Volume and Layer Potentials for the Stokes System with Non-smooth Anisotropic Viscosity Tensor and Some Applications”
Abstract: Link to see abstract
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Seminar on November 22
Speaker: Prof. Vladimir Rabinovich, National Polytechnic Institute of Mexico, ESIME Zacatenco,
The title of the lecture: “Interaction problems for the Dirac operators on R^n”
Abstract: Link to see abstract
Recorded talk: N/A
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Seminar on November 8
Speaker: Prof. Eugine Shargorodsky, King’s College London, UK,
The title of the lecture: “Negative eigenvalues of two-dimensional Schrödinger operators”
Abstract: According to the celebrated Cwikel-Lieb-Rozenblum inequality, the number of negative eigenvalues of the Schrödinger operator , $-\Delta-V, V>=0$ on $L_2((R^d), d>=3$, is estimated above by
$$const\int_{\R^d}V(x)^{d/2}dx $$ It is well known that this estimate does not hold for d=2. I will present estimates for the number of negative eigenvalues of a two-dimensional Schrödinger operator
in terms of weighted $L_1$-norms and $LlogL$ type Orlicz norms of the potential obtained over the last decade and discuss related open problems.
Recorded talk: N/A
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Seminar on October 25, 2021
Speaker: Prof. Grigori Rozenblioum, the Chalmers University of Technology and University of Gothenburg, Sweden; St. Petersburg State University and Leonhard Euler International Mathematical Institute
in Saint Petersburg,
The title of the lecture: “Spectral properties of the Neumann-Poincare operator for the elasticity system and related questions about zero-order pseudodifferential operators”
Abstract: The Neumann-Poincare operator is the double layer potential. Unlike the well-studied electrostatic problem where this operator is compact, the elasticity system is not compact. For the 3D
homogeneous isotropic body with smooth boundary, this operator has an essential spectrum consisting of 3 points determined by Lame parameters. The eigenvalues of this operator may converge only to
these points. We developed the machinery of spectral analysis of polynomially compact pseudodifferential operators and use it to find the asymptotics of these eigenvalues.
Recorded talk N/A
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Seminar on October 11, 2021
Dorina Mitrea, Professor and Chair Department of Mathematics,
The title of the lecture: “On boundedness of Singular Integral Operators on Holder Spaces”
Abstract:A central question in Calderon-Zygmund Theory is that of the L2-boundedness of Singular Integral Operators. An equally relevant issue is that of the boundedness of Singular Integral
Operators on the scale of Holder spaces. In this talk I will present results in this regard which are applicable to large classes of Singular Integral Operators on general geometric settings.
Recorded talk N/A
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Seminar on June 21
Dr. Marius Mitrea, professor of mathematics and chair of the mathematics department at Baylor University, Waco, Texas, United States;
The title of the lecture: “Singular Integrals, Geometry of Sets, and Boundary Problems”
Abstract:Presently, it is well understood what geometric features are necessary and sufficient to guarantee the boundedness of convolution-type singular integral operators on Lebesgue spaces. This
being said, dealing with other function spaces where membership entails more than a mere size condition (like Sobolev spaces, Hardy spaces, or the John-Nirenberg space BMO) requires new techniques.
In this talk I will explore recent progress in this regard, and follow up the implications of such advances into the realm of boundary value problems.
Recorded talk N/A
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Seminar on June 7
Prof. Valery Smyshlyaev, University College London, UK;
The title of the lecture: “High-frequency scattering of whispering gallery waves by boundary inflection: asymptotics and boundary integral equations”
Abstract:The talk is on a long-standing problem of scattering of a high-frequency whispering gallery wave by boundary inflection. Like Airy ODE and associated Airy function are fundamental for
describing transition from oscillatory to exponentially decaying asymptotic behaviors, the boundary inflection problem leads to an arguably equally fundamental canonical inner boundary-value problem
for a special PDE describing transition from a ``modal'' to a ``scattered'' high-frequency asymptotic regimes. An additional recent motivation comes from the problem seemingly holding the keys for
numerical analysis of Galerkin-type methods for boundary integral equations (BIE) in high-frequency scattering by smooth non-convex obstacles. The talk first reviews the background, on asymptotically
reducing a problem described by Hemlholtz equation to the inner problem. The latter is a Schr\"odinger equation on a half-line with a potential linear in both space and time, and was first formulated
and analysed by M.M. Popov starting from 1970-s, and has been intensively studied since then (see [1] for a review and some further references). The associated solutions have asymptotic behaviors
with a discrete spectrum at one end and with a continuous spectrum at the other end, and of central interest is to find the map connecting the above two asymptotic regimes. We report recent result in
[1] proving that the solution past the inflection point has a ``searchlight'' asymptotics corresponding to a beam concentrated near the limit ray. This is achieved by a non-standard perturbation
analysis at the continuous spectrum end, and the result allows interpretations in terms of a unitary scattering operator connecting the modal and the scattered asymptotic regimes.We also review some
most recent progress on a reducing the inner problem to one-dimensional boundary integral equations and their further analysis. The integral equations are of improper weakly singular Volterra type of
both first and second kinds (with appropriate jump conditions for the latter) and can be shown to be well-posed. Their subsequent regularization allows to express the solution in term of limit of
uniformly convergent Neumann series with anticipated further benefits for the problem's asymptotic and possibly numerical analyses. Some parts of the work are joint with Ilia Kamotski, and with Shiza
Recorded talk N/A
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Seminar on May 24
Prof. Simon Chandler-Wilde, University of Reading, UK;
The title of the lecture: “Do Galerkin methods converge for the classical 2nd kind boundary integral equations in polyhedra and Lipschitz domains? ”
Abstract:The boundary integral equation method is a popular method for solving elliptic PDEs with constant coefficients, and systems of such PDEs, in bounded and unbounded domains. An attraction of
the method is that it reduces solution of the PDE in the domain to solution of a boundary integral equation on the boundary of the domain, reducing the dimensionality of the problem. Second kind
integral equations, featuring the double-layer potential operator, have a long history in analysis and numerical analysis. They provided, through C. Neumann, the first existence proof to the Laplace
Dirichlet problem in 3D, have been an important analysis tool for PDEs through the 20th century, and are popular computationally because of their excellent conditioning and convergence properties for
large classes of domains. A standard numerical method, in particular for boundary integral equations, is the Galerkin method, and the standard convergence analysis starts with a proof that the
relevant operator is coercive, or a compact perturbation of a coercive operator, in the relevant function space. A long-standing open problem is whether this property holds for classical second kind
boundary integral equations on general non-smooth domains. In this talk we give an overview of the various concepts and methods involved, reformulating the problem as a question about numerical
ranges. We solve this open problem through counterexamples, presenting examples of 2D Lipschitz domains and 3D Lipschitz polyhedra for which coercivity does not hold. This is joint work with Prof
Euan Spence, Bath.
Recorded talk N/A
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Seminar on April 19
Prof. Gerd Grubb, Department of Mathematical Sciences, University of Copenhagen (UCPH);
The title of the lecture: “Sharp regularity results for electromagnetic fields in Lipschitz domainsBoundary problems for fractional-order operators”
Abstract:There has recently been an upsurge of interest in the fractional Laplacian ${\left( -\Delta \right)}^{a}$ $\left( 0<a<1 \right)$ and other fractional-order pseudodifferential operators $P$,
because of applications in financial theory and probability (and also in differential geometry and mathematical physics). The boundary problems for $P$ on subsets $\Omega$ of ${R}^{n}$ are
challenging since $P$ is nonlocal; here there have mostly been used real methods from potential theory and integral operator theory, or probabilistic methods. As we know, the pseudodifferential point
of view should be useful too, in view of Vishik and Eskin's work in the sixties, and many later works. I shall tell about a circle of results developed in the last 8 years, based on the
mu-transmission condition introduced by Hörmander (in his book '85 and in a lecture note '66), telling also how they differ from the results in Eskin's book '81. The pseudodifferential methods have
not been popular in the applications community, partly because Fourier transform techniques (and complex functions) do not seem to be part of the toolbox, partly because the ps.d.o. methods
originally have
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Seminar on March 22
Prof. Dr. Martin Costabel, IRMAR, Institut Mathématique Université de Rennes 1;;
The title of the lecture: “Sharp regularity results for electromagnetic fields in Lipschitz domains”
Abstract: The standard energy spaces for the time-harmonic Maxwell equations in a bounded 3-dimensional domain are the Hilbert spaces of square-integrable vector fields whose divergence and curl are
also square-integrable and whose tangential or normal components are zero on the boundary. Elements of these spaces are known to have additional regularity. The classical Gaffney inequality (1951)
states that their gradients are square-integrable if the domain is smooth, and it is clear that there is less regularity if the boundary has corners and edges. For Lipschitz domains, regularity in
the Sobolev space of order 1/2 has been known for 30 years, and recently a domain has been constructed that shows that in the scale of Sobolev spaces this is sharp. By the principle that Maxwell
singularities are carried by gradients, this regularity result can also be equivalently formulated as a regularity result for the classical Dirichlet and Neumann problems of the Laplacian on bounded
Lipschitz domains. I will describe the context and motivation for the result and explain the construction of the domain and some of its additional properties.
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Seminar on March 8
Speaker: Prof. Dr. Pavel Exner,Doppler Institute for Mathematical Physics and Applied Mathematics;
The title of the lecture: “Product formulae related to Zeno quantum dynamics”
Abstract: We present a new class of product formulae which involve a unitary group generated by a positive self-adjoint operator and a continuous projection-valued function. The problem is motivated
by quantum description of decaying systems, in particular, the Zeno effect coming from frequently repeated measurements. Applied to it, the formula expresses the dynamics of such a system. An example
of a permanent position ascertaining leading to the effective Dirichlet condition is given.
Recorded talk N/A
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Seminar on February 22
Speaker: Prof. Paata Ivanishvili,North Carolina State University (NC), USA;
The title of the lecture: “Rademacher type and Enflo Type Coincide”
Abstract: Pick any finite number of points in a Hilbert space. If they coincide with vertices of a parallelepiped then the sum of the squares of the lengths of its sides equals the sum of the squares
of the lengths of the diagonals (parallelogram law). If the points are in a general position then we can define sides and diagonals by labeling these points via vertices of the discrete cube {0,1}n.
In this case the sum of the squares of diagonals is bounded by the sum of the squares of its sides no matter how you label the points and what n you choose. In a general Banach space we do not have
parallelogram law. Back in 1978 Enflo asked: in an arbitrary Banach space if the sum of the squares of diagonals is bounded by the sum of the squares of its sides for all parallelepipeds (up to a
universal constant), does the same estimate hold for any finite number of points (not necessarily vertices of the parallelepiped)? In the joint work with Ramon van Handel and Sasha Volberg we
positively resolve Enflo's problem. Banach spaces satisfying the inequality with parallelepipeds are called of type 2 (Rademacher type 2), and Banach spaces satisfying the inequality for all points
are called of Enflo type 2. In particular, we show that Rademacher type and enflo type coincide.
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Seminar on February 8
Speaker: Prof. Michael Ruzhansky,Queen Mary University of London;
The title of the lecture: “Nonharmonic operator analysis”
Abstract: In this talk we will give a survey of our recent works on developing the nonharmonic symbolic calculus. This has applications to various questions for non-self-adjoint operators as well as
for operators on manifolds with (or without) boundaries.
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Seminar on January 25
Speaker: Prof. Leonid Parnovski,Department of Mathematics, University College London, UK
The title of the lecture: “Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons”
Abstract: I will discuss asymptotic behaviour of the eigenvalues of the Steklov problem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is completely unexpected and depends on
the arithmetic properties of the angles of the polygon.
Recorded talk N/A
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Seminar on January 11
Speaker: Prof. Maria J. ESTEBAN, CEREMADE (CEntre de REcherche en MAthématiques de la DÉcision, French for Research Centre in Mathematics of Decision), Paris Dauphine University,
The title of the lecture: “Magnetic interpolations inequalities in dimensions 2 and 3”
Abstract: In this talk I will present some results concerning magnetic inequalities, similar to Gagliardo-Nirenberg inequalities, but involving magnetic operators. We will first consider the case of
a general magnetic field where general results will be proved, but without much concrete information. Then, in the particular cases of constant or Aharonov-Bohm magnetic fields, we will be able to
make those results more precise and get better estimates, or even complete information, about the best constants in the inequalities, or about the optimal extremals.
Recorded talk N/A
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Seminar on December 14
Speaker: Prof. Ari Laptev, Department of Mathematics, Imperial College London, http://wwwf.imperial.ac.uk/~alaptev/
The title of the lecture: “Magnetic rings”
Abstract: We study functional and spectral properties of perturbations of a magnetic second-order differential operator on a circle. This operator appears when considering the restriction to the unit
circle of a two dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove some Hardy-type inequalities and sharp Keller-Lieb-Thirring inequalities.
Recorded talk: https://youtu.be/DEs7IWzOyKs
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Seminar on November 30
Speaker: Prof. Mikhail Sodin, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, https://en-exact-sciences.tau.ac.il/profile/sodin
The title of the lecture: “Fourier uniqueness and non-uniqueness pairs”
Abstract: Motivated by a remarkable discovery by Radchenko and Viazovska and by a recent work by Ramos and Sousa, we find conditions sufficient for a pair of discrete subsets of the real axis to be a
uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are not too far from each other. The uniqueness theorem can be upgraded to the frame bound and an interpolation
formula, which in turn produce an abundance of Poisson-like formulas. This is a report on a joint work in progress with Aleksei Kulikov and Fedor Nazarov.
Recorded talk: https://youtu.be/D9G3Sp8CkLQ
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Seminar on November 16
Speaker: Prof. Kristian Seip, Norwegian University of Science and Technology, https://www.ntnu.edu/employees/kristian.seip
The title of the lecture: “Fourier interpolation with the zeros of the Riemann zeta function”
Abstract: Originating in work of Radchenko and Viazovska, a new kind of Fourier analytic duality, known as Fourier interpolation, has recently been developed. I will discuss the underlying general
duality principle and present a new construction associated with the non-trivial zeros of the Riemann zeta function, obtained in joint work Andriy Bondarenko and Danylo Radchenko. I will emphasize
how the latter construction fits into the theory of the Riemann zeta function.
Recorded talk: Play recording (58 mins)
Download poster: Link to Download | {"url":"https://ug.edu.ge/en/past-tbilisi-analysis-and-pde-seminar","timestamp":"2024-11-13T17:58:51Z","content_type":"text/html","content_length":"165712","record_id":"<urn:uuid:3bf68c42-ae1b-4320-baa7-3ed36e857f86>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00251.warc.gz"} |
Logarithmic Functions: Solving Questions & Finding Carrying Capacity
• MHB
• Thread starter TheFallen018
• Start date
In summary, the conversation revolved around two questions, one about using laws of logarithms to simplify an expression and the other about finding the carrying capacity of a population model. The
first question was solved by using the inverse of the second law of logarithms, while the second question was solved by taking the limit as time approaches infinity. The answer for the carrying
capacity was found to be 10,000.
Hey guys, I have a couple of questions here. One, I was just wondering if someone could elaborate on, and the second, I worked it out, but more by guessing. I was hoping someone would be able to help
explain both.
Here is the first of the two questions
View attachment 7620
So, part a was fairly straightforward. I calculated the differential to be
However, I'm not sure if I broke down the expression as much as I could have with laws of logarithms. I could only think of using the inverse of the second law of logarithms, where
log(a) - log(b)= log(a/b)
Is there a way to break that up further?
As for the second question, it has to do with carrying capacity of a population model.
View attachment 7621
I got all the parts correct for this one, but I'm not sure how to get the carrying capacity of the function. I figured it to be 10,000, as that was what the numerator was. However, I'm sure that's
not how it's meant to work. Despite the fact that I got the right answer, I'm not satisfied with the answer I gave.
So, how would be the correct way about solving that?
Thanks for your time. I really appreciate the help.
Kind regards,
TheFallen018 said:
Hey guys, I have a couple of questions here. One, I was just wondering if someone could elaborate on, and the second, I worked it out, but more by guessing. I was hoping someone would be able to
help explain both.
Here is the first of the two questions
So, part a was fairly straightforward. I calculated the differential to be (-2x+3)/x(x-1)
However, I'm not sure if I broke down the expression as much as I could have with laws of logarithms. I could only think of using the inverse of the second law of logarithms, where log(a) - log
(b)= log(a/b)
Is there a way to break that up further?
You can also apply:
\(\displaystyle \log_a\left(b^c\right)=c\cdot\log_a(b)\)
TheFallen018 said:
As for the second question, it has to do with carrying capacity of a population model.
I got all the parts correct for this one, but I'm not sure how to get the carrying capacity of the function. I figured it to be 10,000, as that was what the numerator was. However, I'm sure
that's not how it's meant to work. Despite the fact that I got the right answer, I'm not satisfied with the answer I gave.
So, how would be the correct way about solving that?
Thanks for your time. I really appreciate the help.
Kind regards,
To find the carrying capacity $C$, I would write:
\(\displaystyle C=\lim_{t\to\infty}P(t)\)
We see the numerator is constant, and the denominator goes to 1, so yes, 10,000 is correct. :)
MarkFL said:
You can also apply:
\(\displaystyle \log_a\left(b^c\right)=c\cdot\log_a(b)\)
To find the carrying capacity $C$, I would write:
\(\displaystyle C=\lim_{t\to\infty}P(t)\)
We see the numerator is constant, and the denominator goes to 1, so yes, 10,000 is correct. :)
Thanks Mark, that was exactly what I was looking for. You're awesome :)
FAQ: Logarithmic Functions: Solving Questions & Finding Carrying Capacity
What is a logarithmic function?
A logarithmic function is a mathematical function that calculates the exponent needed to produce a certain number. It is the inverse of an exponential function.
How do you solve a logarithmic equation?
To solve a logarithmic equation, you can use the properties of logarithms to rewrite the equation in a simpler form. Then, you can solve for the unknown variable by using algebraic techniques.
What is carrying capacity?
Carrying capacity is the maximum population size that an environment can sustainably support. It is determined by the available resources and limiting factors such as food, water, and living space.
How do you use logarithmic functions to find carrying capacity?
To find the carrying capacity using logarithmic functions, you can use the logistic growth model. This model takes into account the initial population size, the growth rate, and the carrying capacity
to predict the population size over time.
What are some real-world applications of logarithmic functions and carrying capacity?
Logarithmic functions and carrying capacity are used in various fields such as biology, economics, and ecology. They can be used to model population growth, predict resource depletion, and analyze
economic growth and sustainability. | {"url":"https://www.physicsforums.com/threads/logarithmic-functions-solving-questions-finding-carrying-capacity.1038608/","timestamp":"2024-11-07T09:31:37Z","content_type":"text/html","content_length":"91339","record_id":"<urn:uuid:c645b206-82d3-437c-8141-b641ab81f391>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00328.warc.gz"} |
Maple Questions and Posts
I am a newbie to Maple, i have learnt Maple for several days.In Matlab i can easily declare a row vector t=0:0.02:0.5 (from 0 by 0.02 to 0.5) but in Maple i can't. I tried using "for" loop but i
could not insert into a vector. How can I write a vector like that ? Please show me.Thanks a lot.
>e1r := -1.5; > e1i := 12.455; > e2r := -.022269812; > e2i := .25368881; > E1 := e1r -I*e1i; E1 := -1.5 - 12.455 I > E2 := e2r -I*e2i; E2 := -0.022269812 - 0.25368881 I > nz1 := RootOf(E1*NZ^4 - NZ^
2*(2*E1^2) + E1^3 - E1*E2^2=0,NZ,index=1); Error, (in content/polynom) general case of floats not handled ok so what is wrong with this? its all simple equations and complex floats, the syntax for
rootof looks correct...
I wanted to solve a differential equation and followed the examples in Maple manual exactly: ode := {(2*t*x*q+x-4*t*x^2*(2*y(v)-1))*(diff(y(v), v))-1 = 0}; ic := {y(0) = 1/2}; soln := dsolve(`union
`(ode, ic), {y(v)}); But I got neither results nor error informations. Does it mean, "ode" has no solution? In such case, how can I get some tips from Maple? thanks a lot.
I have a problem with a sentence that I have. I don't know whether it is a formatting problem or problem with the statement itself. I'm trying to say something like this (if it makes it less
abstract, I'm trying to formally define a Sudoku puzzle). For any set S and function F that maps the ordinal set |S|^2 to |S|, there exists a set T which is a subset of S^|S||S||S| such that any
element of T Tabc is equal to the empty set iff c !=F(a,b) First of all, theres a couple issues with this. |S| is the cardinal of S, but for convenience I also use it as an ordinal set. Second, as I
started typing this I saw that I do not have the necessary statement for all Tabc which are members of T. How can I correctly input this statement into Maple?
I was wondering if anyone can recommend a good (free) text editor for writing Maple source code. I am writing some programs and thus far I have just been using Notepad, but automatic syntax
highlighting and indentation would be nice to have. Thanks!
I am trying to use maple to find the limit of a subsequence, but what I've tried hasn't worked
For example define
Im meant to create a program to compute inverse tan - tan(^-1)(x)
to a given accuracy. They gave examples in class of how to do it for
like to 8 decimal places or whatever.
here's what i got on my screen.
This sort of has an explanation on how to get the taylor series for inverse tan. basically i was gonna try to copy and paste the output for the sin(x) eg. and just swap the taylor series to see if it
would work.
> sum_ex1:=proc(x)
Can someone tell me how to solve an IMPLICIT first order ODE? That is, an eqn. of the form F(x,y,y')= 0 where we cannot seperate the term y'. An example will be helpful.
I have the following code snippet which uses solve to get the 4 solutions to a quartic equation. In my application I must know which solution is which, for they each have a specific placement in
later calculations. I know that most of Maple's datasets use memory address as their method of sorting, and therefore each run can give a different order to the results. Is this also true for solve?
I need to know a way which the order of the solutions will always be constant. (sorting by returned value will not suffice) WaveEQDet := proc(layer,eV) local E1, E2; > E1:=Epp1(layerelementtable
[layer], eV);
The plot function will not work with the following code:
> vatt:= (r,b,rc) -> -epsilon*(cos(Pi*(r-rc)/(2*wc)))^2;
> VattR := proc(r,b)
> rc:= evalf(b*2^(1/6));if r < rc then RETURN(-epsilon) elif rc <= r and r <= (rc+wc) then RETURN(vatt(r,b,rc)) else RETURN(0) end if;
> end proc;
> plot(VattR(r,1),r=1.1..1.2);
Error, (in VattR) cannot determine if this expression is true or false: r < 1.122462048
> VattR(1.1,1);
The functions work fine on their own....
I would like to set my sessions to show the 1D Maple sheets - however when trying the options in the menu - it still wont happen. What am I doing wrong? Help - the 2D version drives me nuts!
Please recommend a good introductory book for Maple. My math skills are very high & I have much programming experience, so a Maple for Dummies approach would be too simplistic.
I have been messing with the various functions in the VectorCalculus package and have been getting some unexpected behavior w.r.t. some of the functions. My current issue is with the SurfaceInt
function. Consider a sphere whose surface density increases linearly from one point on the sphere to the opposite pole. What is its overall mass. I set up the package... > restart; > with
(VectorCalculus); > SetCoordinates(spherical[r, phi, theta]); and a density function... rho := proc (r, phi, theta) options operator, arrow; phi end proc; and perform the integration...
I'm trying to integrate various expressions like f''(x)*g(x)+f'(x)*g'(x). In this case, the answer is obvious by inspection, f'(x)*g(x), but I can't coerce Maple to produce the result. Any ideas?
I am looking into using Maple in an undergraduate Vibrations class and put together the attached worksheet as a sort of combination rough draft and feasibility study. The equation showing up after
the algsubs command really bothers me; is there a way to remove the imaginary exponential terms and substitute trigonometric identities?
First 2122 2123 2124 2125 2126 2127 2128 Last Page 2124 of 2158 | {"url":"https://mapleprimes.com/products/Maple?page=2124","timestamp":"2024-11-08T02:23:10Z","content_type":"text/html","content_length":"135652","record_id":"<urn:uuid:56f411ab-a3b6-453d-aeeb-0d1808d62db5>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00813.warc.gz"} |
A PDE describing Roots of Polynomials under Differentiation
Speaker: Prof. Dr. Stefan Steinerberger
Affiliation: University of Washington, USA
Request Zoom meeting link
Abstract. Suppose you have a polynomial p_n (think of n as being quite large) and suppose you know where the roots are. What can you say about the roots of the derivative p_n’? Clearly, one could
compute them but if n is large, that is not so easy — can you make a softer statement, predicting “roughly” where they are? This question goes back to Gauss who proved a pretty Theorem about it. We
will ask the question of what happens when one keeps differentiating: if the roots of p_n look like, say, a Gaussian, what can you say about the roots of the polynomial after you have differentiated
0.1*n times? This leads to some very fun equations and some fascinating new connections to Probability Theory, Potential Theory and Partial Differential Equations. In particular, there is a nice
nonlocal PDE that seems to describe everything. I promise nice pictures! | {"url":"https://dcn.nat.fau.eu/events/a-pde-describing-roots-of-polynomials-under-differentiation/","timestamp":"2024-11-06T08:42:01Z","content_type":"text/html","content_length":"67233","record_id":"<urn:uuid:113e15d6-8a4c-46d5-bfd7-fdb42a0ae606>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00313.warc.gz"} |
how to create zero matrix in python
The matrix elements can be manipulated individually by using subindices as expected. By default, NumPy will use a data type of floats. How to Change the Data Type of a Zeros Matrix in Numpy By
default, NumPy will pass in floats into the zeros matrix it creates using the np.zeros () function. C++ Create Python Matrix using Arrays from Python Numpy package. By the end of this tutorial, youll
have learned: A zeros matrix is a special type of matrix where every value is a zero. n := number of rows, m := number of columns, set flag := false. [0 2 0] [0 0 5]] which is a diagonal matrix with
values on the main diagonal and zeros everywhere else. Let us see the steps . If you would like to change your settings or withdraw consent at any time, the link to do so is in our privacy policy
accessible from our home page. Parameters shapeint or tuple of ints Shape of the new array, e.g., (2, 3) or 2. dtypedata-type, optional The desired data-type for the array, e.g., numpy.int8. Kotlin
In order to create a diagonal matrix using Python we will use the numpy library. Zero arrays and matrices have special purposes in machine learning. Random 1d array matrix using Python NumPy library.
(x = np.zeros(n), y = np.zeros(n) ). Syntax: numpy.zeros (shape, dtype=float, order='C') Parameter: shape: integer or sequence of integers order: {'C', 'F'}, optional, default: 'C' JavaScript vs
Python : Can Python Overtop JavaScript by 2020? Inline representation: 0 0 0 0 1 2 0 2 4 Multiline representation 0 0 0 0 1 2 0 2 4. Python | Index of Non-Zero elements in Python list. Step 1: We
have first a single list. Initialize Matrix in Python - In this article, we will learn about how we can initialize matrix using two dimensional list in Python 3.x. Node.js How to Create a Zero Matrix
in Python | Previously we have developed programs to create matrix in Python now, we will discuss how to create a zero matrix in python. In this matrix I will be creating a random walk cycle for
multiple particles that only stops when it either lands next to another particle or touches the bottom of the matrix. 3. def print9_as_3x3 (list9): for i in 0, 3, 6: for j in 0, 1, 2: print (list9
[i+j], end= ' ') print () This can of course be easily generalized (using range s instead of those literal tuples) but I'm trying to keep it as simple as feasible. To create and initialize a matrix
in python, there are several solutions, some commons examples using the python module numpy: Summary. By using 'C', data will be stored in a row-major format. C Numerical Python (NumPy) provides an
abundance of useful features and functions for operations on numeric arrays and matrices in Python. In this article, we will discuss How to create and initialize a 2D array or two-dimensional list of
zeroes in Python. 1. And it seems you are right! Your email address will not be published. NumPy makes it equally easy to create a 2-dimensional zero matrix. m = [ [1, 2, 3] for i in range(3)] for i
in m: print("".join(str(i))) In the above output, we have printed the list twice by giving the range parameter as 3. Privacy Policy. numpy.zeros (shape, dtype = None, order = 'C') Parameters : shape
: integer or sequence of integers order : C_contiguous or F_contiguous C-contiguous order in memory (last index varies the fastest) C order means that operating row-rise on the array will be slightly
quicker FORTRAN-contiguous order in memory (first index varies the fastest). ' @ ' operator is used as a sign for matrix multiplication in Python. Data Structure out = (np.random.random(size=len
(array_probabilities)) > array_probabilities).astype(int) Example output: array([0, 1, 0, 1]) Your question got me wondering so I wrote a basic function to compare their timings. Default is
numpy.float64. Use the * Operator to Create a List of Zeros in Python. The following is the syntax -. LinkedIn We and our partners use cookies to Store and/or access information on a device.We and
our partners use data for Personalised ads and content, ad and content measurement, audience insights and product development.An example of data being processed may be a unique identifier stored in a
cookie. numpy.zeros(shape, dtype=float, order='C', *, like=None) # Return a new array of given shape and type, filled with zeros. Linux The syntax to create zeros numpy array is: numpy.zeros(shape,
dtype=float, order='C') where shape could be an int for 1D array and tuple of ints for N-D array. In array items are stored at contiguous memory locations and here we will try to add only Zeros into
the array with different methods. [0.]] Numpy MaskedArray.filled () method - Python 4. numpy.ma.filled () function - Python 5. numpy.matlib.zeros () function | Python 6. numpy.zeros () in Python 7.
By using our site, you C++ STL HR function ml_webform_success_5298518(){var r=ml_jQuery||jQuery;r(".ml-subscribe-form-5298518 .row-success").show(),r(".ml-subscribe-form-5298518 .row-form").hide()}
. And the first step will be to import it: . This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. DS Matrix is a rectangular table
arranged in the form of rows and columns, in the programming world we implement matrix by using arrays by classifying it as 1D and 2D arrays. Python | Get a list as input from user Taking input in
Python Taking input from console in Python Top 4 Advanced Project Ideas to Enhance Your AI Skills Top 10 Machine Learning Project Ideas That You Can Implement 5 Machine Learning Project Ideas for
Beginners in 2022 How to create a matrix of complex numbers in python using numpy ? datagy.io is a site that makes learning Python and data science easy. Create a project folder and put in its code
and a font folder with ms mincho.ttf. No description, website, or topics provided. Once NumPy is installed, you can import and use it. acknowledge that you have read and understood our, Data
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numpy-dev/reference/generated/numpy.zeros.html#numpy.zeros. Embedded Systems Android A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our
website. Lets take a look the np.zeros() function function and its parameters: In this tutorial, well focus on the first two parameters. To create a NumPy array with zeros the numpy.zeros () function
is used which returns a new array of given shape and type, with zeros. Extract diagonal from matrix using Python. Create a list or array from a text file in Python. PHP numpy.diag(v, k) To create a
diagonal matrix you can use the following parameters -. Solved programs: Java Interview que. Lets imagine we have a zero matrix 0 and another matrix A. SQL Top Interview Coding Problems/Challenges!
Facebook if mat [0, 0] = 0, then set flag := true. CSS dtype is the datatype of elements the array stores. This fragment creates a 3-by-3 matrix of floating-point numbers initialized to all-zeros.
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Paced Course, Important differences between Python 2.x and Python 3.x with examples, Reading Python File-Like Objects from C | Python. In order to create a zero matrix using Python and NumPy, we can
use the Numpy .zeros() function. Is it possible to do it this way, or should I . Ajax Python Program to Create a Zero Matrix CS Organizations Articles Lets see how to create a zero array of size 5
using Python: By default, NumPy will create a one-dimensional array with the data type of floats. Make a Matrix in Python Without Using NumPy. Create a matrix of random numbers with 0+0j >>> import
numpy as np >>> Z = np.zeros(10, dtype=complex) >>> Z array([ 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, . Required fields are marked *. Finally, you learned how change the ordering of the matrix. This
is the standard way of creating a matrix . You can use the zeros function in numpy as: import numpy as np b = np.zeros ( (2,1)) # numpy.zeros (shape, dtype = None, order = 'C') print (b) Output [
[0.] : We can use the dtype= parameter to change the data type of values in the zeros matrix. Meanwhile, when you pass in 'F', the data are stored as column-major (Fortran style). If we multiple a
list with a number n using the * operator, then a new list is returned, which is n times the original list. Internship Everything To Know About OnePlus. NumPy provides multidimensional array of
numbers (which is actually an object). # WE CAN USE NUMPY FOR THIS import numpy as np np.zeros(5) # Creates an array [0., 0., 0., 0., 0.] This allows you to create a matrix that has special
properties and characteristics when interacting with other matrices. Decision Tree Classifier with Sklearn in Python, Python map Function: Transforming Iterables without Loops. Your email address
will not be published. More: import numpy as np emp = np.empty( (0, 2)) print(emp) Output: [ ] The above code has created a matrix with 0 rows and 2 columns. You signed in with another tab or window.
Subscribe through email. Below is the syntax of the following method. Typically, a zero matrix is defined as 0m,n, where m and n represent the dimensions of that matrix. Submitted by Anuj Singh, on
May 29, 2020. Create a 1-Dimensional Zeros Array in Numpy, Create a 2-Dimensional Zeros Matrix in Numpy, Create a 3-Dimensional Zeros Matrix in Numpy, How to Change the Data Type of a Zeros Matrix in
Numpy, Understanding the Order Parameter in Numpy Zeros, Numpy Normal (Gaussian) Distribution (Numpy Random Normal), Numpy Dot Product: Calculate the Python Dot Product, How to create 1D, 2D, and 3D
zero arrays and matricies, How to change the data types of the zeros in the matrix. Here we will cover different approaches to creating a zero's element array. lst = [0] * 10 print(lst) The
Psychology of Price in UX. This allows you to easily create tuples of zeros of different data types in NumPy: In the above example, we create a zero matrix that contains tuples of zeros. It may be of
any dimension (MxN). Lets see how we can create a zeros matrix with integers instead of floats: In the below-shown example we have used a library called NumPy, there are many inbuilt functions in
NumPy which makes coding easy. About us DOS Other solution is by using ' @ ' operator in Python. Lets take a look at one of the simplest ways in which we can create a zeros matrix in Python: a
one-dimension array of zeros. document.getElementById("ak_js_1").setAttribute("value",(new Date()).getTime()); Your email address will not be published. Lets see how to create a 32 matrix of zeros
using NumPy: To create a three dimensional zeros matrix in NumPy, we can simply pass in a tuple of length three. Numpy processes an array a little faster in comparison to the . Lets see how we can
create a zeros matrix with integers instead of floats: You can go even further than this and pass in a tuple of data types. Is the Designer Facing Extinction? If you want to create an empty matrix
with the help of NumPy. The numpy.zeros() function returns a new array of given shape and type, with zeros. Here you can see the code below and the output. C The python library Numpy helps to deal
with arrays. Privacy policy, STUDENT'S SECTION Python We can use a function: numpy.empty numpy.zeros 1. numpy.empty : It Returns a new array of given shape and type, without initializing entries.
Networks To learn more about related topics, check out the tutorials below: Your email address will not be published. How to Create Array of zeros using Numpy in Python 3. Then run the code and font
will be found with "font = pygame.font.Font('font/ms mincho.ttf', FONT_SIZE)" command line (you don't need to add anything new). 0. Zeros Matrix using numpy.zeros () | Linear Algebra using Python
Linear Algebra using Python | Zeros Matrix using numpy.zeros (): Here, we are going to learn about creating zeros matrix using numpy.zeros () in Python. NumPy will default to using the row-major
format. Home . In the following sections, youll learn how to generate zero matrices in Python using NumPy. Being able to create them efficiently will allow you to become more capable in linear
algebra and machine learning. Timings change but only a little. Java News/Updates, ABOUT SECTION A zero matrix is a matrix that contains all 0 elements. C : Python A tag already exists with the
provided branch name. Linear Algebra using Python, Linear Algebra using Python | Zeros Matrix using numpy.zeros(): Here, we are going to learn about creating zeros matrix using numpy.zeros() in
Python. Step 1: We have first a single list mat Step 2: Then we iterate by for loop to print it twice using a range within the list it will change into nested list acting as a matrix mat = [ [3, 8,
9] for i in range(2)] for i in mat: print("".join(str(i))) Output:- Sample Solution : Python Code : Now let's see the general way which can be implemented in any language. Some of our partners may
process your data as a part of their legitimate business interest without asking for consent. Share Improve this answer Follow answered Dec 25, 2020 at 4:52 Adit Goyal 36 4 Add a comment Not the
answer you're looking for? C++ val = [ ['Dave',101,90,95], ['Alex',102,85,100], ['Ray',103,90,95]] Web programming/HTML By default, k is 0 which refers to the main diagonal. The consent submitted
will only be used for data processing originating from this website. NumPy: Create a 5x5 zero matrix with elements on the main diagonal equal to 1, 2, 3, 4, 5 Last update on August 19 2022 21:50:48
(UTC/GMT +8 hours) NumPy: Basic Exercise-29 with Solution Write a NumPy program to create a 5x5 zero matrix with elements on the main diagonal equal to 1, 2, 3, 4, 5. C# v - The 1d array containing
the diagonal elements. matrix = scipy.sparse.csr_matrix ( (df.shape [0], 300)) ## matrix = np.zeros ( (df.shape [0], for i, q in enumerate (df ['column'].values): matrix [i, :] = function (q) where
function is pretty much a vector operation function on that row. Are you sure you want to create this branch? To view the purposes they believe they have legitimate interest for, or to object to this
data processing use the vendor list link below. To create a matrix containing only 0, a solution is to use the numpy function zeros \begin{equation} A = \left( \begin{array}{ccc} . Puzzles
Professional Gaming & Can Build A Career In It. I've seen this done with one line that looks like array = self.values > self.Max, but when I try this I get " TypeError: '>' not supported between
instances of 'list' and 'int' ". Certificates How to input multiple values from user in one line in Python? You can unsubscribe anytime. C Languages: How to Design for 3D Printing. Machine learning &
ans. . The code takes place in a 2-dimensional zero matrix, which I will be representing with two variables, x and y. In this tutorial, youll learn how to generate a zero matrix using the NumPy zeros
function. Manage SettingsContinue with Recommended Cookies. # For integer array of zeros np.zeros((5,), dtype=int) # Creates . Required fields are marked *. Java Convert Celsius to Fahrenheit using
Function, Celsius to Fahrenheit and Vice-Versa using Function. In the tuple, the first data type is a float while the second is an integer. Create a Matrix in Python Python allows developers to
implement matrices using the nested list. The np.zeros () is a function in NumPy that creates a zero matrix, here dtype is used to specify the data type of the elements. Comment *
document.getElementById("comment").setAttribute( "id", "a2cd177a84c83a936716fb290e4fbcb7" );document.getElementById("e0c06578eb").setAttribute( "id", "comment" ); Save my name, email, and website in
this browser for the next time I comment. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. C#.Net You could write a helper
function to make this easier, but soon you'd be writing helper functions for all of the common matrix/array operations, and then you're reinventing the wheel that numpy has already made so nicely.
Use the np.loadtxt() function to write your text into an array and the file object read() function to populate a Python list #1 Data Preparation for i in range 1 to m. 3 CSS Properties You Should
Know. Create a confusion matrix in Python. Here is the code with examples. Zeros Matrix - When all the entries of a matrix are one, then it is called a zeros matrix. Lists can be created if you place
all items or elements starting with ' [' and ending with ']' (square brackets) and separate each element by a comma. Contact us To add, the matrices will make use of a for-loop that will loop through
both the matrices given. This gives you additional flexibility to change how memory is handled while creating zero matrices. The np.zeros() is a function in NumPy that creates a zero matrix, here
dtype is used to specify the data type of the elements.if(typeof ez_ad_units!='undefined'){ez_ad_units.push([[728,90],'knowprogram_com-box-3','ezslot_4',114,'0','0'])};__ez_fad_position
('div-gpt-ad-knowprogram_com-box-3-0'); The above code creates a zero matrix with 3 rows and 3 columns.if(typeof ez_ad_units!='undefined'){ez_ad_units.push
([[728,90],'knowprogram_com-medrectangle-3','ezslot_3',121,'0','0'])};__ez_fad_position('div-gpt-ad-knowprogram_com-medrectangle-3-0'); This python program also performs the same task but in
different ways. Python Program to Find all rectangles filled with 0 8. The resulting object ( m) is an object that behaves the way matrices are supposed to behave in linear algebra, including the way
they interoperate (for instance with vectors). You can also implement this code using the function as well as classes. In python, we have an inbuilt function (defined in numpy library) numpy.zeros()
to define the zeros matrix. The function csr_matrix () is used to create a sparse matrix of c ompressed sparse row format whereas csc_matrix () is used to create a sparse matrix of c ompressed sparse
column format. In this article, we will discuss how to create an array of zeros in Python. Let's take an example: import numpy as np a = np.array ( [1, 2, 3]) print(a) # Output: [1, 2, 3] print(type
(a)) # Output: <class 'numpy.ndarray'> As you can see, NumPy's array class is called ndarray. See your article appearing on the GeeksforGeeks main page and help other Geeks. DBMS Aptitude que. Now,
if i do the loop on the np.zeros, it does so quite easily, about 10 minuts. CS Subjects: Lets see how we can create a tuple that is of size (3,3,2): By default, NumPy will pass in floats into the
zeros matrix it creates using the np.zeros() function. SEO set row := false, and col := false. The Q's subject talks about creating matrices, which this code doesn't do, but the Q's body's all about
. Create a project folder and put in its code and a font folder with ms mincho.ttf. import numpy as np np.dot (matrix1, matrix2) Here matrix1 and matrix2 are the matrices that are being multiplied
with each other. Matrix code rain using Python with Pygame. C++ Matrix 3 : M3 = [[0,0,0], [0,0,0], [0,0,0]] Example: Adding Matrices. Learn more about datagy here. We have not created them in the
syntax, but they are available in below example. O.S. Get the free course delivered to your inbox, every day for 30 days! Please run them on your systems to explore the working. Then run the code and
font will be found with "font = pygame.font.Font ('font/ms mincho.ttf', FONT_SIZE)" command line (you don't need to add anything new). You first learned what the characteristics of a zero matrix are.
Embedded C Then, you learned how to create arrays of zeros, as well as two-dimensional and three-dimensional arrays of zeros in NumPy. Creating A Local Server From A Public Address. If you like
GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to
[email protected]
. We would like to read the file contents into a Numpy array and a Python list. Web Technologies: To create a numpy array with zeros, given shape of the array, use numpy.zeros () function. I'm trying
to get an array array of the same size as an array values that has 1 where values [i] [j] > Max, 0 elsewhere. If you enjoyed this post, share it with your friends. This article is contributed by
Mohit Gupta_OMG . import numpy as np random_matrix_array = np.random.rand (3) print (random_matrix_array) Output: $ python codespeedy.py [0.13972036 0.58100399 0.62046278] The elements of the array
will be greater than zero and less than one. import numpy as np m = np.zeros( [3, 3], dtype=int) print(m) Output:- [ [0 0 0] [0 0 0] [0 0 0]] The above code creates a zero matrix with 3 rows and 3
columns. Java We can do this by passing in a tuple of integers that represent the dimensions of this matrix. By using the order= parameter, you can pass in either 'C' or 'F' in order to modify how
these values are stored. The shape= parameter allows you to define the size of the zeros array or matrix that is to be created. DBMS k - The diagonal on which the passed elements (elements of the 1d
array, v) are to be placed. We can change this by setting the order= parameter: In this tutorial, you learned how to use NumPy to create zero matrices. Here we have used one such function called
empty () which creates an empty matrix. CS Basics This can be done by passing in a single integer into the shape argument of the zeros function. Syntax: Reference : https://docs.scipy.org/doc/
numpy-dev/reference/generated/numpy.zeros.html#numpy.zeros Note : zeros, unlike zeros and empty, does not set the array values to zero or random values respectively.Also, these codes wont run on
online IDEs. . & ans. Then, we have the following characteristics: Knowing this can allow us to determine some more important characteristics in machine learning. Created November 15, 2019 | Viewed
59990 | by Benjamin Edit . Content Writers of the Month, SUBSCRIBE In this program, we will create a zero matrix in python without NumPy. Python Code to initialize 2d array of zeros Python code
implementation without user-defined functions & classes Python Code to initialize 2d array of zeros Python's SciPy gives tools for creating sparse matrices using multiple data structures, as well as
tools for converting a dense matrix to a sparse matrix. To explain with python code, considered dataset "predict if someone has heart disease" based on their sex, age, blood pressure and a variety of
other . JavaScript Matrix code rain using Python with Pygame. Share Improve this answer Follow answered Mar 3, 2019 at 1:08 Nathan 9,007 4 44 61 Thank you. Create a Numpy array filled with all ones
2. Submitted by Anuj Singh, on May 29, 2020 Zeros Matrix - When all the entries of a matrix are one, then it is called a zeros matrix. How to Create a Matrix in Python Using For Loop We can also use
for loop to create a matrix. for i in range 1 to n. if mat [i, 0] = 0, then set col := True and break the loop. 5 Key to Expect Future Smartphones. This means that a zero matrix of size (4,4) would
look like this: Lets take a look at some of the characteristics of this type matrix. Using this method, we can easily create a list containing zeros of some specified length, as shown below. 5.
Browse other questions tagged python matrix In this section, there are some examples to create a zero matrix in python. We can use the dtype= parameter to change the data type of values in the zeros
matrix. NumPy also lets you customize the style in which data are stored in row-major or column-major styles in memory. order{'C', 'F'}, optional, default: 'C' Cloud Computing The dtype= parameter
allows you to set the data type of the zero matrix being created. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Feedback Here's our task:
We have a text file containing numerical data. | {"url":"https://blog.drmikediet.com/yek/how-to-create-zero-matrix-in-python","timestamp":"2024-11-03T06:17:34Z","content_type":"text/html","content_length":"45191","record_id":"<urn:uuid:652eac9f-5b44-4968-9fb6-bc92194639e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00441.warc.gz"} |
Coupled Solver Settings And Template - Volupe.com
We have seen lately a set of simulations, from different sources, where the coupled implicit solver is used for transient simulation of the type where several species are involved. Either filling or
evacuating a tank with one specie, where another is already populated. Like using a higher inlet pressure, in the example, 1 bar gauge, to fill a rigid tank with O2. The tank from the start has 100 %
N2 initialized at atmospheric pressure. This is the example we will use in this week’s blog post, and the picture below shows what it looks like, with some measurements and boundary conditions and
initial conditions.
The question you could ask yourself in a simulation like this, is how long does it take before we have reached a constant pressure in the tank? Since we have an inlet pressure of 1 bar higher than
the initial pressure there is only a matter of time (literally), before the tank is filled with a certain amount of O2, together with the N2 already inside, and the pressure in the tank is also 1
bar, since the inlet pressure in constant.
The way to set this up using default settings can be found in the picture below. Selecting the coupled implicit solver with the implicit unsteady method, you will get the default setting accordingly.
A constant CFL of 50 and no “extra” selection like special initialization or CCA.
You would run your case with these settings, manually increasing the timestep from a smaller timestep in the beginning – to a larger timestep in the end, to make sure that the simulation runs
smoothly. Starting with 1e-4 s for the first iterations, and go towards 1e-2 s or something similar toward the end of the simulation, and you would see the results looking typically like this:
You would theorize that it takes about 60 seconds for the pressure in the domain to reach gaughe pressure, and 90 % of the way is reached in about 18 seconds. And more over, you would be wrong! It is
not easy to see, but if this was the only property, except for resuduals you where evaluating during the simulation, you can still spot the error. Early on in the simualtion when we increase the
timestep size there are discontinuities in the pressure curve. This is what happens with the curve if we focus on the result in the beginning.
At a couple of different points in time we have discontinuities in the pressure curve. This happened while the time step size was changed. This tells us, already here that we do not have a timestep
independent result in our simulation!
If we investigate this further, we can see that something is wrong. For one thing, it is easy to calculate the initial mass in the domain for the N2, either by using the ideal gas law, or simply by
initializing the simulation and do a sum-report of density*MassFraction_N2*Volume. The value provided by either of those methods, allow us to know the initial mass, and not only that, but also the
mass of N2 at the end of the simulation. There is no N2 leaving the domain, aside from a couple of time steps in the beginning or the near the end of the simulations, where we might have reversed
from on the outlet, the only specie passing through in any relevant quantity is O2. Tracking the total mass of either specie in the domain can then give a good sanity check to see if our result can
be deemed physical. The Total mass of N2 should not change as no N2 is leaving via the inlet.
Looking at the plot to track these quantities, we can see that the mass of N2 does not at all behave the way we would expect, it rather seems like when we push in O2 we also push out N2. Conclusion
here is that the result is wrong, and the simulation cannot be trusted. Later we will see similar representations where the results from several different simulations are put in the same plot, but
some crucial settings have been changed. But first we need to understand what is going on here. Why does this happen, and how can we prevent it?
CFL number
A short mention of the CFL number was given in my latest blog post about being pragmatic when solving your CFD simulations, it can be found here [Solving the engineering question – VOLUPE Software].
There it is stated that the CFL number setting for the coupled solver is what is used to decide the pseudo-timestep for the coupled solver. The CFL number should be seen as a under relaxation factor
for the coupled solver, meaning that small numbers will hopefully increase the stability of the simulation, at the cost of more iterations to reach convergence. This becomes extra intricate in a
transient simulation, because here we essentially have a dual-timestep method. The Implicit Time stepping, or Dual Time-Stepping is the one transient method that we will focus on here. The other
method is the Explicit Time-Stepping method, in which you specify CFL number that is used to set the physical time-step for all cells in the domain. This is rarely used in Simcenter STAR-CCM+ because
it is very time consuming but is used in many other numerical methods. We are not going to focus on the mathematics here for brevity, but in the Implicit time-stepping, we use the dual time-stepping,
with inner iterations in pseudo-time. This means that as well as getting a timestep independent solution for any transient simulation, we need to make sure that the simulation is timestep independent
both in terms of the global timestep and the pseudo timestep. And this is what is happening in the simulation above looking at the mean pressure in time. When we change (increase) the global timestep
at those highlighted locations, we reduce the slope of the curve, meaning that the goal, when we reach equilibrium pressure, is pushed ahead of us in time. And as stated previously, we do not have a
time-independent solution.
The dual time-stepping issue here is one side of the coin, and the other one is the poor result for the mass. This can be attributed to the lack of CCA and solved with the inclusion of CCA. The CCA
stands for Continuity Convergence accelerator and is a method that can be applied to the coupled solver. There is a blog post on the topic written by Christoffer that partially explain the CCA,
Coupled solver settings in Simcenter STAR-CCM+ – VOLUPE Software. In short it is a tool that can be used when convergence of mass balance is slow. And this is exactly what we have seen in simulations
with several species. Meaning that we have large improvements in mass balance and continuity when using the CCA. It should be used with Enhanced stability treatment and with Enhanced Mass-imbalance
Calculations activated for the best result.
Improving on the initial simulation
We have now identified the issues and also understood how we can improve the simulation in order to obtain results that we can trust. For this type of simulation, we can then test what impact the CFL
number and the use of CCA have on the result. The testing has been limited to these two settings and the values used for CFL is 50, which is default in star, and “Automatic”, where the solver
decides. For now, it is good to understand that in practice means orders of magnitude larger than the default value. And the CCA has the option of being on, with the above-mentioned tick-boxes, or
off. The segregated solver has also been used for reference. The first curve we looked at here is the blue one, and it is by far the longest estimation of reaching equilibrium conditions. We can se
that with the inclusion of the CCA for the CFL 50, we get a better result, but far from optimal. It is when we allow for the higher CFL number that the results is the same. And in this case, we see
that we reach equilibrium in about 4 seconds, compared to the 65 seconds shown in the first simulation. In this particular case, the Automatic CFL did not also “need” the CCA to reach the same
result, but with a poor CFL, the CCA helped in the right direction.
If we instead look at the total mass, we see that with the low CFL and no CCA, after the simulation is done, we have no N2 left in the tank. In the simulation it seems like the O2 has replaced the
N2, something that cannot happen in reality of course. In both simulations with CFL auto, the mass N2 is preserved.
For reference we also look at the CFL plot along the time in the Simulation with Automatic CFL including CCA and see that the value is almost always maxing out at 1e5. (There is a limit even for the
automatic setting, saying CFL min 0.1 and max 100000.)
From this we bring the conclusion that a CFL number which is as high as possible should be used, together with the CCA for thorough mass balancing of the simulation. We also bring with us the need to
look at several properties and details of the simulation to make sure that what we get makes sense and is correct.
Best practice template for this type of coupled flow
This file can be obtained from the CFD handbook part of the volume homepage as a best practice template on the Volupe best practice side. [Templates – VOLUPE Software]
I hope you can use this and that this has been a learning experience. All details regarding both CFL and CCA can be found in the documentation. Ans as usual, reach out to support@volupe.com with any
Robin Victor | {"url":"https://volupe.com/simcenter-star-ccm/coupled-solver-settings-and-template/","timestamp":"2024-11-04T11:36:06Z","content_type":"text/html","content_length":"304497","record_id":"<urn:uuid:48421ebc-0c2c-42da-aa63-b9cc561c1496>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00795.warc.gz"} |
semantic prime
In NSM, semantic primes are the most fundimental “lexical units” (so they can be words, or morphemes, etc. the size doesn’t matter) across languages.
They are the “core of a universal mental lexicon”.
There are…
1. A semantic prime has to be found in every(ish?) natural language
2. A semantic prime has to be indefinable by other primes
Proof: given if the Strong Lexicalization Hypothesis holds, semantic primes must exist.
Assume for the sake of contradiction no semantic primes exist.
Because Strong Lexicalization Hypothesis holds, there does not exist syntactic transformations which can take original single words and transform them into newly lexicalized words to express a
different meaning.
At the same time, again because of the Strong Lexicalization Hypothesis, one must only leverage syntactic transformation on syntatic constituents when forming ideas.
Therefore, given a word to lexicalize, it has to be defined by an syntatic transformation on a set of previously lexicalized words.
(by definition) there are no words lexicalizable from the empty set of words.
Therefore, there exists some word that needs to be lexicalized by words that are not previously defined, which is absurd. (instead, these words are lexicalized via semantic primes.)
1. the list has grown over time
2. the problem of allolexy: formal restrictions of a language resulting in the same concept needing to be radicalized multiple times (I vs. me)
According to (Geeraerts 2009), (Goddard 2009) provides a “practical” (though flawed) way of establishing primes. Something to do with large-scale comparisons in “whole metalanguage studies”, which
requires pairwise language comparison
Locating primes are seen as an enforcement of NSM theories (Vanhatalo, Tissari, and Idström, n.d.). Recent prime locations: in Amharic (Amberber 2008), East Cree (Junker 2008), French (Peeters
1994), Japanese (Onishi 1994), Korean (Yoon 2008), Lao (Enfield 2002), Mandarin (Chappell 2002), Mangaaba-Mbula (Bugenhagen 2002), Malay (Goddard 2002), Polish (Wierzbicka 2002), Russian (Gladkova
2010, for the latest set, see the NSM home page), Spanish (Travis 2002), and Thai (Diller 1994). | {"url":"https://www.jemoka.com/posts/kbhsemantic_primes/","timestamp":"2024-11-10T12:16:43Z","content_type":"text/html","content_length":"8915","record_id":"<urn:uuid:c218b73d-945a-4383-b74f-ca7ed0f1f150>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00421.warc.gz"} |
Fibonacci Distribution - Data Science Wiki
Fibonacci Distribution :
The Fibonacci distribution is a
distribution that is based on the Fibonacci sequence. The Fibonacci sequence is a series of numbers that starts with 0 and 1, and each subsequent number is the sum of the previous two numbers. For
example, the first few numbers in the Fibonacci sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
The Fibonacci distribution is based on the idea that each number in the Fibonacci sequence represents a probability of an event occurring. For example, let’s say that we have a situation where we
want to know the probability of a coin landing on heads or tails. We can use the Fibonacci distribution to calculate the probability of the coin landing on heads or tails.
To calculate the probability of the coin landing on heads, we would first need to determine the probabilities of the coin landing on heads or tails. We can do this by using the Fibonacci sequence.
For example, if we know that the probability of the coin landing on heads is 0, and the probability of the coin landing on tails is 1, we can use the Fibonacci sequence to calculate the probabilities
of the coin landing on heads or tails.
Using the Fibonacci sequence, we can calculate that the probability of the coin landing on heads is 0.5, and the probability of the coin landing on tails is 0.5. This means that there is an equal
probability of the coin landing on heads or tails.
Another example of using the Fibonacci distribution is to calculate the probability of a certain number being drawn in a lottery. Let’s say that we have a lottery where there are 50 numbers, and we
want to calculate the probability of a certain number being drawn. We can use the Fibonacci sequence to calculate the probability of a certain number being drawn.
For example, if we know that the probability of the number 1 being drawn is 0, and the probability of the number 2 being drawn is 1, we can use the Fibonacci sequence to calculate the probabilities
of the other numbers being drawn.
Using the Fibonacci sequence, we can calculate that the probability of the number 3 being drawn is 0.5, the probability of the number 4 being drawn is 0.5, and so on. This means that each number has
an equal probability of being drawn in the lottery.
In summary, the Fibonacci distribution is a probability distribution that is based on the Fibonacci sequence. It can be used to calculate the probability of an event occurring, such as the
probability of a coin landing on heads or tails, or the probability of a certain number being drawn in a lottery. This distribution is useful because it provides a simple and easy way to calculate
probabilities, and it is based on a well-known and widely used mathematical sequence. | {"url":"https://datasciencewiki.net/fibonacci-distribution/","timestamp":"2024-11-13T14:16:51Z","content_type":"text/html","content_length":"42062","record_id":"<urn:uuid:31ca83d2-037b-4415-8bcd-e344422d2742>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00122.warc.gz"} |
Motor Capacity For Ball Mill
Ball Mill Design/Power Calculation
The basic parameters used in ball mill design (power calculations), rod mill or any tumbling mill sizing are; material to be ground, characteristics, Bond Work Index, bulk density, specific density,
desired mill tonnage capacity DTPH, operating % solids or pulp density, feed size as F80 and maximum ‘chunk size’, product size as P80 and maximum and finally the type of circuit open/closed ...
Ball Mill Design/Power Calculation
12-12-2016 · Ball Mill Power Calculation Example A wet grinding ball mill in closed circuit is to be fed 100 TPH of a material with a work index of 15 and a size …
Ball Mill - an overview | ScienceDirect Topics
Ball mills are rated by power rather than capacity. Today, the largest ball mill in operation is 8.53 m diameter and 13.41 m long with a corresponding motor power of 22 MW (Toromocho, private
communications). View chapter Purchase book. Read full chapter.
Motor Rating For Ball Mill
Valve Selection Guide. 2016-2-5pressure rating 125-450 psi 860-3100 kpa cwp shutoff class square nut, g-series cylinder, electric motor dezurik 3-way and 4-way plug valves ptwpfw 3-way and 4-way plug
valves are high flow capacity, splined ball-to-shaft connection for ease of maintenance and zero backlasheat options include flexible metal, rigid metal and.
Ball Mills - an overview | ScienceDirect Topics
Ball mills are rated by power rather than capacity. Today, the largest ball mill in operation is 8.53 m diameter and 13.41 m long with a corresponding motor power of 22 MW (Toromocho, private
communications). View chapter Purchase book. Read full chapter.
Calculate Ball Mill Grinding Capacity
The sizing of ball mills and ball milling circuits from laboratory grinding tests is largely a question of applying empirical equations or factors based on accumulated experience. Different
manufacturers use different methods, and it is difficult to check the validity of the sizing estimates when estimates from different sources are widely divergent. It is especially difficult to teach
mill ...
Calculation Of Performance Of Ball Mill
ball mill speed calculation - ZCRUSHER ball mill speed calculation filetype: pdf - BINQ Mining BALL MILL DRIVE MOTOR CHOICES. BALL MILL DRIVE MOTOR CHOICES For Presentation at the ... calculating of
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Motor For Ball Mill
Ball Mill Motor Driven:- For mixing & grinding, electrically operated having capacity of few gms to 2 KG. Fitted with geared motor with max speed of 80 rpm. The jar is made of SS and having steel
balls of different sizes to be put in jar for mixing & grinding pesticides power.
Ball Mill: Operating principles, components, Uses ...
11-01-2016 · Several types of ball mills exist. They differ to an extent in their operating principle. They also differ in their maximum capacity of the milling vessel, ranging from 0.010 liters for
planetary ball mills, mixer mills, or vibration ball mills to several 100 liters for horizontal rolling ball mills.
Ball Mills | MACA LIMITED
Ball Mills; Electric Motors; Rockbreakers; Screens; Scrubbers; Silos; Mining Pumps; Agitators and Gearboxes; Belt Magnets; Bin; Crushers; ... miscellaneous; RO Plants; Tanks and Thickeners; 7 x 9 Ft
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Mill (grinding) - Wikipedia
Ball mills are commonly used in the manufacture of Portland cement and finer grinding stages of mineral processing, one example being the Sepro tyre drive Grinding Mill. Industrial ball mills can be
as large as 8.5 m (28 ft) in diameter with a 22 MW motor, drawing approximately 0.0011% of the total worlds power (see List of countries by ...
Selecting Inching Drives for Mill and Kiln Applications
Electric Motors The standard inching drive motor is a NEMA or IEC frame typically foot mounted to a motor scoop or base plate. Typical motor speeds are either 4 or 6 pole, i.e. 1750 rpm or 1170 rpm
for 60 Hz applications and 1450 rpm or 970 rpm for 50 Hz applications. The advantage
Used Ball Mill Motors for sale. Allis-Chalmers …
Used ball mill motors - 542 listings. Advanced Filters. Hide Advanced Filters. ANI - FLSMIDTH 14 x 22 Ball Mill with 2500 HP Motor. Manufacturer: ANI; Location: ANI - FLSMIDTH 14 x 22 Ball Mill with
2500 HP Motor. Oroville, CA, USA. Click to Request Price. Top Seller.
Cement mill - Wikipedia
A cement mill (or finish mill in North American usage) is the equipment used to grind the hard, nodular clinker from the cement kiln into the fine grey powder that is cement.Most cement is currently
ground in ball mills and also vertical roller mills which are more effective than ball mills.
China Ball Mill, Ball Mill Manufacturers, Suppliers, …
China Ball Mill manufacturers - Select 2020 high quality Ball Mill products in best price from certified Chinese stone Machinery manufacturers, Milling Machine suppliers, wholesalers and factory on
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How to Make a Ball Mill: 12 Steps (with Pictures) - …
07-04-2011 · The ball mill will still work, the motor will just rotate in the opposite direction. Use caution when using the power source. If you’re unsure about the electronics, ask a friend who has
more expertise before using it. Advertisement. Part 2 of 2: Using the Ball Mill
Long ing life ball mill with high capacity
Long ing life ball mill with high capacity HFC Refrigerants (55) HST Hydraulic Cone CrusherHST series hydraulic cone crusher is combined with technology such as machinery, hydraulic pressure,
electricity, automation, intelligent control, etc. , representing the most advanced crusher technology in the world.
ball mills for handling quartz capacity
laboratory ball mill. On the right side the Ball Mill loaded with the 5 liter milling drum. On the left the (un)loading device for ergonomically handling As stated above, the purpose of this research
was primarily to establish a quantitative relationship between a laboratory ball mill capacity and fineness of finished product.get price
comparison of electric motors for ball mill operation
comparison of electric motors for ball mill operation For each project scheme design, we will use professional knowledge to help you, carefully listen to your demands, respect your opinions, and use
our professional teams and exert our greatest efforts to create a more suitable project scheme for you and realize the project investment value and profit more quickly. | {"url":"https://www.ibc-crystal.eu/crystal/motor-capacity-for-ball-mill_5862/","timestamp":"2024-11-15T02:49:41Z","content_type":"application/xhtml+xml","content_length":"14224","record_id":"<urn:uuid:942f50a5-0792-4b43-a3b0-71fbeb28684d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00482.warc.gz"} |
A hole is drilled in a copper sheet The diameter of class 11 physics JEE_Main
Hint: The initial diameter is given; we can find the radius and thus the area of the copper sheet. Assume coefficient of linear expansion for copper to be some variable. Then for the expansion in
area, use this linear expansion coefficient to find the coefficient of superficial expansion. Then use the formula for thermal expansion. Also, as the diameter is length and not an area therefore, we
can calculate the change in diameter using formula for linear thermal expansion.
Complete step by step solution:
When some object is heated, there is change in the shape of the object. For objects such as metal ruler or rod this change is along their length only. For objects such as sheets, plates which have an
area, the change is throughout the area. Also, there is change in the volume of objects due to thermal heating.
We are given with a copper sheet and we need to find the change in its diameter. The diameter being length only will have linear expansion due to thermal heating.
The formula of linear thermal expansion for the diameter change for copper sheet will be given as
$\Delta D = {D_0}\alpha \Delta T$ ---- equation 1
Here, $\Delta D$ is the change in length
${D_0}$ is the initial length
$\alpha $ is coefficient for linear expansion of copper
$\Delta T$ is the change in temperature.
The given values are ${D_0} = 4.24cm$
$\Delta T = {35^0}C - {27^0}C = {8^0}C$
Substituting the given values in equation 1 , we get
$\Delta D = 4.24 \times \alpha \times 8$
$ \Rightarrow \Delta D = 17.92\alpha \,cm$
Therefore, the change in diameter will be $17.92\alpha \,cm$ when the sheet is heated to ${35^0}C$.
Note: Over small temperature ranges, the fractional thermal expansion of uniform linear objects is proportional to the change in temperature. This fact is used to construct thermometers based on the
expansion of a thin tube of mercury or alcohol. If the value of coefficient for linear expansion of copper was given, we just needed to multiply that value to the final answer. | {"url":"https://www.vedantu.com/jee-main/a-hole-is-drilled-in-a-copper-sheet-the-diameter-physics-question-answer","timestamp":"2024-11-14T20:36:34Z","content_type":"text/html","content_length":"153223","record_id":"<urn:uuid:10d5be7f-76d1-4bfa-803a-fd3b6e14015f>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00277.warc.gz"} |
Likelihood and Maximum Likelihood Estimation (MLE)
In statistics and probability theory, the likelihood of parameters θ is typically denoted as ℒ(θ). The likelihood function represents the probability of observing the given data under different
values of the parameter θ. It's a fundamental concept in statistical inference, particularly in the maximum likelihood estimation (MLE).
The likelihood function is often denoted as:
The cross-entropy loss function can be given by,
• typically represents the predicted probability that a given input belongs to the positive class (class 1).
• represents the actual binary label (0 or 1) of the instance.
The terms "likelihood" and "probability" are related but distinct concepts in statistics, and they are used in different contexts.
1. Likelihood:
□ Likelihood is a function that measures how well a particular set of parameters (θ) in a statistical model explains or fits observed data.
□ It is not a probability distribution over data; instead, it's a function of the parameters given the data.
□ The likelihood function is used in statistical inference to estimate parameters. In maximum likelihood estimation (MLE), for example, you find the values of θ that maximize the likelihood
function given the observed data.
□ The likelihood function is not constrained to sum to 1 over all possible values of θ; its values can vary widely depending on the data and the model.
2. Probability:
□ Probability is a measure of the uncertainty associated with an event or outcome.
□ It is typically defined over the possible outcomes of a random process or experiment, and it quantifies how likely each outcome is to occur.
□ Probability distributions represent the set of possible outcomes and their associated probabilities, and they must sum to 1 over all possible outcomes.
□ Probability is used to describe uncertainty before an event has occurred (prior probability) or to compute the probability of future events (posterior probability) given prior information.
It might seem like these two concepts are similar because both involve expressing the likelihood of something happening. However, they are not the same for the following reasons:
1. Different Focus:
□ Likelihood focuses on how well a set of parameters explains observed data, and it's used in parameter estimation.
□ Probability focuses on the likelihood of specific events or outcomes occurring and is used in predicting or describing events in a probabilistic manner.
2. Different Objectives:
□ Likelihood helps us find the best-fitting parameters for a statistical model, given the observed data.
□ Probability is used for modeling the inherent randomness or uncertainty in a system, often in the context of random variables and their distributions.
While likelihood and probability are distinct concepts, they are related in Bayesian statistics through Bayes' theorem. Bayes' theorem relates the likelihood, prior probability, and posterior
probability of a parameter, allowing us to update our beliefs about the parameter using observed data. In this context, likelihood plays a crucial role in Bayesian inference. It is common, in
statistics, to say "likelihood of the parameter" and "probability of the data", which is used to clarify the distinction between these two related but distinct concepts.
Log of likelihood can be given by,
Maximum Likelihood (ML) is a statistical method used for estimating the parameters of a probability distribution or a statistical model. It is a common approach in both frequentist and Bayesian
statistics and is widely used in various fields, including machine learning, economics, biology, and more. Maximum likelihood estimation (MLE) is to choose θ to maximize the ℒ(θ); however, in
practice, it is much easier to maximize the log(ℒ(θ)) in order to maximize the ℒ(θ).
Since t he first team in Equation 3997f is constant, then in order to maximize the ℒ(θ), we need to mimimize the term after the minus sign in Equation 3997g.
The basic idea behind maximum likelihood estimation is to find the values of the model parameters that maximize the likelihood function, which measures how well the model explains the observed data.
In other words, ML seeks to find the parameter values that make the observed data most probable under the assumed statistical model.
Here's a simplified step-by-step explanation of how maximum likelihood works:
1. Define a Probability Model: Start by assuming a probability distribution or a statistical model that describes the data. This model depends on one or more parameters that you want to estimate.
2. Construct the Likelihood Function: The likelihood function is a measure of how likely the observed data is, given the parameter values. It's essentially the probability of observing the data as
it is, given the model and parameter values. This function is often denoted as L(θ | data), where θ represents the parameters of the model.
3. Maximize the Likelihood: To find the maximum likelihood estimates, you aim to find the values of θ that maximize the likelihood function. This can be done analytically by taking derivatives and
solving for the maximum or numerically using optimization algorithms like gradient descent.
4. Obtain Parameter Estimates: Once you've found the parameter values that maximize the likelihood function, these values are considered the maximum likelihood estimates for the model's parameters.
Maximum Likelihood Estimation is a powerful and widely used technique in statistics because it provides a principled way to estimate the parameters of a model based on observed data. The resulting
parameter estimates are often used for making predictions, inference, and statistical hypothesis testing. In many cases, ML estimates are also asymptotically efficient, meaning they achieve the best
possible performance as the sample size increases.
Performing maximum likelihood estimation (MLE) on the exponential family of probability distributions is a common statistical technique used to estimate the parameters of these distributions. When
you perform MLE on an exponential family distribution, the following typically happens:
1. Likelihood Function: You start with a probability density function (PDF) or probability mass function (PMF) that belongs to the exponential family. The PDF or PMF depends on one or more
parameters, which you want to estimate. The likelihood function is essentially the product (for continuous distributions) or the product of probabilities (for discrete distributions) of the
observed data points, given the parameter(s).
2. Log-Likelihood: It's often more convenient to work with the log-likelihood, which is the natural logarithm of the likelihood function. This is because it simplifies the mathematical calculations
and avoids potential numerical precision issues when dealing with small probabilities.
3. Optimization: You then find the parameter values that maximize the log-likelihood function. This is done by taking the derivative of the log-likelihood with respect to the parameters and setting
it equal to zero, or by using optimization techniques like gradient descent or the Newton-Raphson method. Solving this equation or using optimization methods yields the MLE estimates of the
4. Interpretation: The MLE estimates represent the parameter values that make the observed data most probable under the assumed exponential family distribution. In other words, they are the values
that maximize the likelihood of the data given the model.
5. Statistical Properties: MLE estimators often have desirable statistical properties, such as being asymptotically unbiased and efficient (i.e., they have the smallest possible variance among
unbiased estimators). However, the exact properties depend on the specific distribution and sample size.
6. Confidence Intervals and Hypothesis Testing: Once you have MLE estimates, you can construct confidence intervals to assess the uncertainty associated with the parameter estimates. You can also
perform hypothesis tests to determine if the estimated parameters are significantly different from specific values of interest.
7. Model Fit Assessment: After obtaining MLE estimates, it's essential to assess how well the chosen exponential family distribution fits the data. This can be done through various goodness-of-fit
tests and graphical methods.
MLE with respect to η (the natural parameter) is concave in the natural parameterization of the exponential family, namely, when the exponential family is parameterized in the natural parameters.
This is a fundamental property of MLE for exponential family distributions:
1. Exponential Family Structure: The exponential family of probability distributions is characterized by a specific mathematical structure. In the natural parameterization, the log-likelihood
function for a sample of independent and identically distributed (i.i.d.) random variables has a specific form that depends linearly on the natural parameter η.
2. Log-Likelihood Function: The log-likelihood function, which is the logarithm of the likelihood function, takes the form of a sum over the data points, with each term being a linear combination of
the natural parameter and sufficient statistics.
3. Linearity in Parameters: The key property here is that the log-likelihood is linear in the natural parameter. Specifically, it is an affine function of η. This means that as you change the values
of η, the log-likelihood function behaves as a linear function.
4. Concavity: A function is said to be concave if, roughly speaking, it "curves downward" as you move along the function from left to right. In the context of MLE, when the log-likelihood function
is concave with respect to the natural parameter η, it means that the function forms a "bowl-like" shape with a single maximum point. In other words, it is a concave function with a unique
5. Optimization: The fact that the log-likelihood function is concave with respect to η is crucial for optimization techniques like gradient descent, Newton-Raphson, or other optimization
algorithms. A concave function has a single maximum that can be efficiently found through these optimization methods.
6. Uniqueness of MLE: The concavity of the log-likelihood function ensures that there is a unique solution for the MLE of the natural parameter η. This makes the estimation process well-defined, and
the MLE is the value of η that maximizes the likelihood of the observed data.
Figure 3997a shows the concave nature of the log-likelihood function in an exponential family distribution by using the Gaussian distribution as an example, which is a member of the exponential
family and is parameterized by the natural parameters. The log-likelihood function is represented by the blue curve in the plot, and it forms a "bowl-like" shape, curving downward as you move along
the natural parameter (η) axis. This characteristic of the curve indicates that the log-likelihood function is concave with respect to η. Concave functions have a single maximum point, which is often
referred to as the MLE of the parameter. The red dashed line marks the location of the MLE of η, which corresponds to the point where the log-likelihood function reaches its maximum value.
Figure 3997a. Concave nature of the log-likelihood function in an exponential family distribution. (Python code)
Negative log likelihood (NLL) is a commonly used mathematical function in the field of statistics and machine learning, particularly in the probability models and maximum likelihood estimation. It is
used to measure the goodness of fit between a probability distribution (usually a model's predicted distribution) and a set of observed data points because of:
1. Likelihood: In statistics, the likelihood function measures how well a probability distribution or statistical model explains the observed data. Given a set of observed data points (often denoted
as x) and a probability distribution or model parameterized by θ, the likelihood L(θ | x) measures the probability of observing the given data under the assumed model.
2. Log Likelihood: To simplify calculations and avoid numerical underflow/overflow issues, it's common to work with the logarithm of the likelihood function. This is called the log likelihood and is
denoted as log L(θ | x).
3. Negative Log Likelihood (NLL): To turn the measure of fit into a loss function (something to be minimized), the negative log likelihood is often used. It's simply the negative of the log
likelihood: -log L(θ | x).
The idea behind using the negative log likelihood as a loss function is to find the model parameters (θ) that maximize the likelihood of observing the given data. Maximizing the likelihood is
equivalent to minimizing the negative log likelihood.
Figure 3997b shows the negative log likelihood (NLL), where you would need to negate the log-likelihood values. The NLL is often used in optimization problems because minimizing it is equivalent to
maximizing the likelihood.
Figure 3997b. Negative log likelihood (NLL). (
Python code
Table 3997. Applications of Maximum Likelihood Estimation (MLE).
│ Applications │Details │
│Single parameter estimation versus multiple parameter estimation │page3843│ | {"url":"https://www.globalsino.com/ICs/page3997.html","timestamp":"2024-11-13T21:46:12Z","content_type":"text/html","content_length":"30993","record_id":"<urn:uuid:451c8de6-f0ca-48d5-a319-698899ef5b78>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00121.warc.gz"} |
Decimate Modifier
Decimate Modifier
The Decimate modifier allows you to reduce the vertex/face count of a mesh with minimal shape changes.
This is not usually used on meshes which have been created by modeling carefully and economically (where all vertices and faces are necessary to correctly define the shape). But if the mesh is the
result of complex modeling, sculpting and/or applied Subdivision Surface/ Multiresolution modifiers, the Decimate one can be used to reduce the polygon count for a performance increase, or simply
remove unnecessary vertices and edges.
Unlike the majority of existing modifiers, this one does not allow you to visualize your changes in Edit Mode.
The modifier displays the number of remaining faces as a result of the Decimate modifier.
Merges vertices together progressively, taking the shape of the mesh into account.
The ratio of faces to keep after decimation.
□ On 1.0: the mesh is unchanged.
□ On 0.5: edges have been collapsed such that half the number of faces remain (see note below).
□ On 0.0: all faces have been removed.
Although the Ratio is directly proportional to the number of remaining faces, triangles are used when calculating the ratio.
This means that if your mesh contains quads or other polygons, the number of remaining faces will be larger than expected, because those will remain unchanged if their edges are not collapsed.
This is only true if the Triangulate option is disabled.
Maintains symmetry on a single axis.
Keeps any resulting triangulated geometry from the decimation process.
Vertex Group
A vertex group that controls what parts of the mesh are decimated.
The amount of influence the Vertex Group has on the decimation.
It can be thought of as the reverse of subdivide. It attempts to remove edges that were the result of a subdivide operation. It is intended for meshes with a mainly grid-based topology (without
giving uneven geometry). If additional editing has been done after the subdivide operation, the results may be unexpected.
The number of times to perform the un-subdivide operation. Two iterations is the same as one subdivide operation, so you will usually want to use even numbers.
It reduces details on forms comprised of mainly flat surfaces.
Angle Limit
Dissolve geometry which form angles (between surfaces) higher than this setting.
Prevent dissolving geometry in certain places.
Does not dissolve edges on the borders of areas where the face normals are reversed.
Does not dissolve edges on the borders of where different materials are assigned.
Does not dissolve edges marked as seams.
Does not dissolve edges marked as sharp.
Does not dissolve edges that are part of a UV map.
All Boundaries
When enabled, all vertices along the boundaries of faces are dissolved. This can give better results when using a high Angle Limit. | {"url":"https://docs.blender.org/manual/en/3.6/modeling/modifiers/generate/decimate.html","timestamp":"2024-11-02T20:19:47Z","content_type":"text/html","content_length":"24552","record_id":"<urn:uuid:589a5ace-8974-417d-922d-8c13d63d0339>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00757.warc.gz"} |
13.5: Ascent
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You can also use updraft speed to estimate cyclone strength and the associated clouds. For the case-study cyclone, Fig. 13.25 shows upward motion near the middle of the troposphere (at 50 kPa).
Recall from classical physics that the definition of vertical motion is W = ∆z/∆t, for altitude z and time t. Because each altitude has an associated pressure, define a new type of vertical velocity
in terms of pressure. This is called omega (ω):
\(\ \omega=\frac{\Delta P}{\Delta t}\tag{13.13}\)
Omega has units of Pa s^–1.
You can use the hydrostatic equation to relate W and ω:
\(\ \omega=-\rho \cdot|g| \cdot W\tag{13.14}\)
for gravitational acceleration magnitude |g| = 9.8 m·s^–2 and air density ρ. The negative sign in eq. (13.14) implies that updrafts (positive W) are associated with negative ω. As an example, if your
weather map shows ω = –0.68 Pa s^–1 on the 50 kPa surface, then the equation above can be rearranged to give W = 0.1 m s^–1, where an mid-tropospheric density of ρ ≈ 0.69 kg·m^–3 was used.
Use either W or ω to represent vertical motion. Numerical weather forecasts usually output the vertical velocity as ω. For example, Fig. 13.28 shows upward motion (ω) near the middle of the
troposphere (at 50 kPa).
Figure 13.28 Vertical velocity (omega) in pressure coordinates, for the casestudy cyclone. Negative omega (colored red on this map) corresponds to updrafts.
The following three methods will be employed to study ascent: the continuity equation, the omega equation, and Q-vectors. Near the tropopause, horizontal divergence of jet-stream winds can force
midtropospheric ascent in order to conserve air mass as given by the continuity equation. The almost-geostrophic (quasi-geostrophic) nature of lower-tropospheric winds allows you to estimate ascent
at 50 kPa using thermal-wind and vorticity principles in the omega equation. Q-vectors consider ageostrophic motions that help maintain quasi-geostrophic flow. These methods are just different ways
of looking at the same processes, and they often give similar results.
Sample Application
At an elevation of 5 km MSL, suppose (a) a thunderstorm has an updraft velocity of 40 m s^–1, and (b) the subsidence velocity in the middle of an anticyclone is –0.01 m s^–1. Find the corresponding
omega values.
Find the Answer
Given: (a) W = 40 m s^–1. (b) W = – 0.01 m s^–1. z = 5 km.
Find: ω = ? kPa s^–1 for (a) and (b).
To estimate air density, use the standard atmosphere table from Chapter 1: ρ = 0.7361 kg m^–3 at z = 5 km.
Next, use eq. (13.14) to solve for the omega values:
(a) ω = –(0.7361 kg m^–3)·(9.8 m s^–2)·(40 m s^–1)
= –288.55 (kg·m^–1·s^–2)/s = –288.55 Pa s^–1
= –0.29 kPa/s
(b) ω = –(0.7361 kg m^–3)·(9.8 m s^–2)·(–0.01 m s^–1)
= 0.0721 (kg·m^–1·s^–2)/s = 0.0721 Pa s^–1
= 7.21x10^–5 kPa s^–1
Check: Units and sign are reasonable.
Exposition: CAUTION. Remember that the sign of omega is opposite that of vertical velocity, because as height increases in the atmosphere, the pressure decreases. As a quick rule of thumb, near the
surface where air density is greater, omega (in kPa s–1) has magnitude of roughly a hundredth of W (in m s–1), with opposite sign.
13.5.1. Continuity Effects
Horizontal divergence (D = ∆U/∆x + ∆V/∆y) is where more air leaves a volume than enters, horizontally. This can occur at locations where jet-stream wind speed (M[out]) exiting a volume is greater
than entrance speeds (M[in]).
Conservation of air mass requires that the number of air molecules in a volume, such as the light blue region sketched in Fig. 13.29, must remain nearly constant (neglecting compressibility). Namely,
volume inflow must balance volume outflow of air.
Figure 13.29 For air in the top half of the troposphere (shaded light blue), if air leaves faster (M[out]) than enters (M[in]) horizontally, then continuity requires that this upper-level horizontal
divergence be balanced by ascent W[mid] in the mid troposphere.
Net vertical inflow can compensate for net horizontal outflow. In the troposphere, most of this inflow happens at mid-levels (P ≈ 50 kPa) as an upward vertical velocity (W[mid]). Not much vertical
inflow happens across the tropopause because vertical motion in the stratosphere is suppressed by the strong static stability. In the idealized illustration of Fig. 13.29, the inflows [(M[in] times
the area across which the inflow occurs) plus (W[mid] times its area of inflow)] equals the outflow (M[out] times the outflow area).
The continuity equation describes volume conservation for this situation as
\(\ W_{m i d}=D \cdot \Delta z\tag{13.15}\)
\(\ W_{m i d}=\left[\frac{\Delta U}{\Delta x}+\frac{\Delta V}{\Delta y}\right] \cdot \Delta z\tag{13.16}\)
\(\ W_{m i d}=\frac{\Delta M}{\Delta s} \cdot \Delta z\tag{13.17}\)
where the distance between outflow and inflow locations is ∆s, wind speed is M, the thickness of the upper air layer is ∆z, and the ascent speed at 50 kPa (mid-troposphere) is W[mid].
Fig. 13.30 shows this scenario for the case-study storm. Geostrophic winds are often nearly parallel to the height contours (solid black curvy lines in Fig. 13.30). Thus, for the region outlined with
the black/ white box drawn parallel to the contour lines, the main inflow and outflow are at the ends of the box (arrows). The isotachs (shaded) tell us that the inflow (≈ 20 m s^–1) is slower than
outflow (≈50 m s^–1).
Figure 13.30 Over the surface cyclone (X) is a region (box) with faster jetstream outflow than inflow (arrows). Isotachs are shaded.
We will focus on two processes that cause horizontal divergence of the jet stream:
• Rossby waves, a planetary-scale feature for which the jet stream is approximately geostrophic; and
• jet streaks, where jet-stream accelerations cause non-geostrophic (ageostrophic) motions.
13.5.1.1. Rossby Waves
From the Forces and Winds chapter, recall that the gradient wind is faster around ridges than troughs, for any fixed latitude and horizontal pressure gradient. Since Rossby waves consist of a train
of ridges and troughs in the jet stream, you can anticipate that along the jet-stream path the winds are increasing and decreasing in speed.
Figure 13.31 Sketch showing how the slower jet-stream winds at the trough (thin lines with arrows) are enhanced by vertical velocity (W, dotted lines) to achieve the mass balance needed to support
the faster winds (thick lines with arrows) at the ridge. (N. Hem.)
One such location is east of troughs, as sketched in Fig. 13.31. Consider a hypothetical box of air at the jet stream level between the trough and ridge. Horizontal wind speed entering the box is
slow around the trough, while exiting winds are fast around the ridge. To maintain mass continuity, this horizontal divergence induces ascent into the bottom of the hypothetical box. This ascent is
removing air molecules below the hypothetical box, creating a region of low surface pressure. Hence, surface lows (extratropical cyclones) form east of jet-stream troughs.
We can create a toy model of this effect. Suppose the jet stream path looks like a sine wave of wavelength λ and amplitude ∆y/2. Assume that the streamwise length of the hypothetical box equals the
diagonal distance between the trough and ridge
\(\ \Delta s=d=\left[(\lambda / 2)^{2}+\Delta y^{2}\right]^{1 / 2}\tag{13.18}\)
Knowing the decrease/increase relative to the geostrophic wind speed G of the actual gradient wind M around troughs/ridges (from the Forces and Winds chapter), you can estimate the jet-stream
wind-speed increase as:
\(\ \Delta M=0.5 \cdot f_{c} \cdot R \cdot[{2-\sqrt{1-\frac{4 \cdot G}{f_{c} \cdot R}}-\sqrt{1+\frac{4 \cdot G}{f_{c} \cdot R}}}]\tag{13.19}\)
For a simple sine wave, the radius-of-curvature R of the jet stream around the troughs and ridges is:
\(\ R=\frac{1}{2 \pi^{2}} \cdot \frac{\lambda^{2}}{\Delta y}\tag{13.20}\)
Combining these equations with eq. (13.17) gives a toy-model estimate of the vertical motion:
\(\ W_{\text {mid}}=\frac{\frac{f_{c} \cdot \Delta z \cdot \lambda^{2}}{4 \pi^{2} \cdot \Delta y}[2-\sqrt{1-\frac{8 \pi^{2} G \cdot \Delta y}{f_{c} \cdot \lambda^{2}}}-\sqrt{1+\frac{8 \pi^{2} G \cdot
\Delta y}{f_{c} \cdot \lambda^{2}}}]}{\left[(\lambda / 2)^{2}+\Delta y^{2}\right]^{1 / 2}}\tag{13.21}\)
For our case-study cyclone, Fig. 13.30 shows a short-wave trough with jet-stream speed increasing from 20 to 50 m/s across a distance of about 1150 km. This upper-level divergence supported
cyclogenesis of the surface low over Wisconsin (“X” on the map).
Sample Application
Suppose a jet stream meanders in a sine wave pattern that has a 150 km north-south amplitude, 3000 km wavelength, 3 km depth, and 35 m s^–1 mean geostrophic velocity. The latitude is such that f[c] =
0.0001 s^–1 . Estimate the ascent speed under the jet.
Find the Answer
Given: ∆y = 2 · (150 km) = 300 km, λ = 3000 km, ∆z = 3 km, G = 35 m s^–1, f[c] = 0.0001 s^–1.
Find: W[mid] = ? m s^–1
Apply eq. (13.20):
\(R=\frac{1}{2 \pi^{2}} \cdot \frac{(3000 \mathrm{km})^{2}}{(300 \mathrm{km})}=1520 \mathrm{km}\)
Simplify eq. (13.19) by using the curvature Rossby number from the Forces and Winds chapter:
\(\frac{G}{f_{\mathcal{C}} \cdot R}=R o_{\mathcal{C}}=\frac{35 \mathrm{m} / \mathrm{s}}{\left(0.0001 \mathrm{s}^{-1}\right) \cdot\left(1.52 \times 10^{6} \mathrm{m}\right)}=0.23\)
Next, use eq. (13.19), but with Ro[c]:
\(\Delta M=\frac{35 \mathrm{m} / \mathrm{s}}{2(0.23)} \cdot[2-\sqrt{1-4(0.23)}-\sqrt{1+4(0.23)}]\)
= 76.1(m/s)·[2 – 0.283 – 1.386] = 25.2 m s^–1.
Apply eq. (13.18):
d = [ (1500km)^2 + (300km)^2 ]^1/2 = 1530 km
Finally, use eq. (13.17):
\(W_{m i d}=\left(\frac{25.2 \mathrm{m} / \mathrm{s}}{1530 \mathrm{km}}\right) \cdot 3 \mathrm{km}=\underline{0.049} \mathrm{ms}^{-1}\)
Check: Physics, units, & magnitudes are reasonable.
Exposition: This ascent speed is 5 cm s^–1, which seems slow. But when applied under the large area of the jet stream trough-to-ridge region, a large amount of air mass is moved by this updraft.
13.5.1.2. Jet Streaks
The jet stream does not maintain constant speed in the jet core (center region with maximum speeds). Instead, it accelerates and decelerates as it blows around the world in response to changes in
horizontal pressure gradient and direction. The fastwind regions in the jet core are called jet streaks. The response of the wind to these speed changes is not instantaneous, because the air has
Suppose the wind in a weak-pressure-gradient region had reached its equilibrium wind speed as given by the geostrophic wind. As this air coasts into a region of stronger pressure gradient (i.e.,
tighter packing of the isobars or height contours), it finds itself slower than the new, faster geostrophic wind speed. Namely, it is ageostrophic (not geostrophic) for a short time while it
accelerates toward the faster geostrophic wind speed.
When the air parcel is too slow, its Coriolis force (which is proportional to its wind speed) is smaller than the new larger pressure gradient force. This temporary imbalance turns the air at a small
angle toward lower pressure (or lower heights). This is what happens as air flows into a jet streak.
The opposite happens as air exits a jet streak and flows into a region of weaker pressure gradient. The wind is temporarily too fast because of its inertia, so the Coriolis force (larger than
pressure-gradient force) turns the wind at a small angle toward higher pressure.
For northern hemisphere jet streams, the wind vectors point slightly left of geostrophic while accelerating, and slightly right while decelerating. Because the air in different parts of the jet
streak have different wind speeds and pressure gradients, they deviate from geostrophic by different amounts (Fig. 13.32a). As a result, some of the wind vectors converge in speed and/or direction to
make horizontal convergence regions. At other locations the winds cause divergence. The jet-stream divergence regions drive cyclogenesis near the Earth’s surface.
Figure 13.32 Horizontal divergence (D = strong, d = weak) and convergence (C = strong, c = weak) near a jet streak. Back arrows represent winds, green shading indicates isotachs (with the fastest
winds having the darkest green), thin curved black lines are height contours of the 20 kPa isobaric surface, L & H indicate low & high height centers. Geostrophic (G) winds are parallel to the
isobars, while (ag) indicates the ageostrophic wind component. Tan dashed lines parallel and perpendicular to the jet axis divide the jet streak into quadrants.
For an idealized west-to-east, steady-state jet stream with no curvature, the U-wind forecast equation (10.51a) from the Atmospheric Forces and Wind chapter reduces to:
\(\ 0=-U \frac{\Delta U}{\Delta x}+f_{c}\left(V-V_{g}\right)\tag{13.22}\)
Let (U[ag], V[ag] ) be the ageostrophic wind components
\(\ V_{a g}=V-V_{g}\tag{13.23}\)
\(\ U_{a g}=U-U_{g}\tag{12.24}\)
Plugging these into eq. (13.22) gives for a jet stream from the west:
\(\ V_{a g}=\frac{U}{f_{c}} \cdot \frac{\Delta U}{\Delta x}\tag{13.25}\)
Similarly, the ageostrophic wind for south-to-north jet stream axis is
\(\ U_{a g}=-\frac{V}{f_{c}} \cdot \frac{\Delta V}{\Delta y}\tag{13.26}\)
For example, consider the winds approaching the jet streak (i.e., in the entrance region) in Fig. 13.32a. The air moves into a region where U is positive and increases with x, hence V[ag] is positive
according to eq. (13.25). Also, V is positive and increases with y, hence, U[ag] is negative. The resulting ageostrophic entrance vector is shown in blue in Fig. 13.32b. Similar analyses can be made
for the jet-streak exit regions, yielding the corresponding ageostrophic wind component.
Sample Application
A west wind of 60 m s^–1 in the center of a jet streak decreases to 40 m s^–1 in the jet exit region 500 km to the east. Find the exit ageostrophic wind component.
Find the Answer
Given: ∆U = 40–60 m s^–1 = –20 m s^–1, ∆x = 500 km
Find: V[ag] = ? m s^–1
Use eq. (13.25), and assume f[c] = 10^–4 s^–1.
The average wind is U = (60 + 40 m s^–1)/2 = 50 m s^–1. V[ag] = [(50 m s^–1) / (10^–4 s^–1)] · [( –20 m s^–1) / (5x10^5 m)] = –20 m s^–1
Check: Physics and units are reasonable. Sign OK.
Exposition: Negative sign means V[ag] is north wind.
When considering a jet streak, imagine it divided into the four quadrants sketched in Fig. 13.32 (also Fig. 13.35). The combination of speed and direction changes cause strong divergence in the left
exit quadrant, and weaker divergence in the right entrance quadrant. These are the regions where cyclogenesis is favored under the jet. Cyclolysis is favored under the convergence regions of the
right exit and left entrance regions.
This ageostrophic behavior can also be seen in the maps for a different case-study (Fig. 13.33). This figure overlays wind vectors, isotachs, and geopotential height contours near the top of the
troposphere (at 20 kPa). The broad area of shading shows the jet stream. Embedded within it are two relative speed maxima (one over Texas, and the other over New England) that we identify as jet
streaks. Recall that if winds are geostrophic (or gradient), then they should flow parallel to the height contours.
Figure 13.33 (Not the 2014 case study.) Superposition of the 20 kPa charts for geopotential heights (medium-thickness black curved lines), isotachs in m s^–1 (shading), and winds (vectors). The scale
for winds and the values for the height contours are identical to those in Figs. 13.17. Regions of relatively darker shading indicate the jet streaks. White/black square outlines the exit region from
a small jet streak over Texas, and white/black oval outlines the entrance region for a larger jet streak over the northeastern USA.
In Fig. 13.33 the square highlights the exit region of the Texas jet streak, showing wind vectors that cross the height contours toward the right. Namely, inertia has caused these winds to be faster
than geostrophic (supergeostrophic), therefore Coriolis force is stronger than pressure-gradient force, causing the winds to be to the right of geostrophic. The oval highlights the entrance region of
the second jet, where winds cross the height contours to the left. Inertia results in slower-than-geostrophic winds (subgeostrophic), causing the Coriolis force to be too weak to counteract
pressure-gradient force.
Consider a vertical slice through the atmosphere, perpendicular to the geostrophic wind at the jet exit region (Fig. 13.32). The resulting combination of ageostrophic winds (M[ag]) induce
mid-tropospheric ascent (W[mid]) and descent that favors cyclogenesis and cyclolysis, respectively (Fig. 13.34). The weak, vertical, cross-jet flow (orange in Fig. 13.34) is a secondary circulation.
Figure 13.34 Vertical slice through the atmosphere at the jet-streak exit region, perpendicular to the average jet direction. Viewed from the west southwest, the green shading indicates isotachs of
the jet core into the page. Divergence (D) in the left exit region creates ascent (W, dotted lines) to conserve air mass, which in turn removes air from near the surface. This causes the surface
pressure to drop, favoring cyclogenesis. The opposite happens under the right exit region, where cyclolysis is favored.
If the geostrophic winds are accelerating, use your right hand to curl your fingers from vertical toward the direction of acceleration (the acceleration vector). Your thumb points in the direction of
the ageostrophic wind.
This right-hand rule also works for deceleration, for which case the direction of acceleration is opposite to the wind direction.
Figure g.
The secondary circulation in the jet exit region is opposite to the Hadley cell rotation direction in that hemisphere; hence, it is called an indirect circulation. In the jet entrance region is a
direct secondary circulation.
Figure 13.35 (Not the 2014 case study.) Entrance and exit regions of jet streaks (highlighted with ovals). X marks the surface low center.
Looking again at the 23 Feb 1994 weather maps, Fig. 13.35 shows the entrance and exit regions of the two dominant jet streaks in this image (for now, ignore the smaller jet streak over the Pacific
Northwest). Thus, you can expect divergence aloft at the left exit and right entrance regions. These are locations that would favor cyclogenesis near the ground. Indeed, a new cyclone formed over the
Carolinas (under the right entrance region of jet streak # 2). Convergence aloft, favoring cyclolysis (cyclone death), is at the left entrance and right exit regions.
You can estimate mid-tropospheric ascent (W[mid]) under the right entrance and left exit regions as follows. Define ∆s = ∆y as the north-south half-width of a predominantly west-to-east jet streak.
As you move distance ∆y to the side of the jet, suppose that V[ag] gradually reduces to 0. Combining eqs. (13.25) and (13.16) with V[ag] in place of M, the mid-tropospheric ascent driven by the jet
streak is
\(\ W_{m i d}=\left|\frac{U \cdot \Delta U}{f_{c} \cdot \Delta x}\right| \cdot \frac{\Delta z}{\Delta y}\tag{13.27}\)
Sample Application
A 40 m s^–1 in a jet core reduces to 20 m s^–1 at 1,200 km downstream, at the exit region. The jet cross section is 4 km thick and 800 km wide. Find the mid-tropospheric ascent. f[c] = 10^–4 s^–1.
Find the Answer
Given: ∆U = 20 – 40 = –20 m s^–1, U = 40 m s^–1, f[c] = 10^–4 s^–1, ∆z = 4km, ∆y = 800 km, ∆x = 1.2x106 m.
Find: W[mid] = ? m s^–1
Use eq. (13.27):
\(W_{\text {mid}}=\left|\frac{(40 \mathrm{m} / \mathrm{s}) \cdot(-20 \mathrm{m} / \mathrm{s})}{\left(10^{-4} \mathrm{s}^{-1}\right) \cdot\left(1.2 \times 10^{6} \mathrm{m}\right)}\right| \cdot \frac
{(4 \mathrm{km})}{(800 \mathrm{km})}=0.033 \mathrm{m} \mathrm{s}^{-1}\)
Check: Physics and units are reasonable.
Exposition: This small ascent speed, when active over a day or so, can be important for cyclogenesis.
To help forecast cyclogenesis, Sutcliffe devised
\(D_{t o p}-D_{b o t t o m}=-\frac{1}{f_{c}}\left[U_{T H} \frac{\Delta \zeta_{g c}}{\Delta x}+V_{T H} \frac{\Delta \zeta_{g c}}{\Delta y}\right]\)
where divergence is D = ∆U/∆x + ∆V/∆y, column geostrophic vorticity is ζ[gc] = ζ[g top] + ζ[g bottom] + f[c] , and (U[TH], V[TH]) are the thermal-wind components.
This says that if the vorticity in an air column is positively advected by the thermal wind, then this must be associated with greater air divergence at the column top than bottom. When combined with
eq. (13.15), this conclusion for upward motion is nearly identical to that from the Trenberth omega eq. (13.29).
13.5.2. Omega Equation
The omega equation is the name of a diagnostic equation used to find vertical motion in pressure units (omega; ω). We will use a form of this equation developed by K. Trenberth, based on
quasi-geostrophic dynamics and thermodynamics.
The full omega equation is one of the nastierlooking equations in meteorology (see the HIGHER MATH box). To simplify it, focus on one part of the full equation, apply it to the bottom half of the
troposphere (the layer between 100 to 50 kPa isobaric surfaces), and convert the result from ω to W.
The resulting approximate omega equation is:
\(\ W_{m i d} \cong \frac{-2 \cdot \Delta z}{f_{c}}\left[U_{T H} \frac{\overline{\Delta \zeta_{g}}}{\Delta x}+V_{T H} \frac{\overline{\Delta \zeta_{g}}}{\Delta y}+V_{T H} \frac{\beta}{2}\right]\tag
where W[mid] is the vertical velocity in the mid-troposphere (at P = 50 kPa), ∆z is the 100 to 50 kPa thickness, U[TH] and V[TH] are the thermal-wind components for the 100 to 50 kPa layer, f[c] is
Coriolis parameter, β is the change of Coriolis parameter with y (see eq. 13.2), ζ[g] is the geostrophic vorticity, and the overbar represents an average over the whole depth of the layer. An
equivalent form is:
\(\ W_{m i d} \cong \frac{-2 \cdot \Delta z}{f_{c}}\left[M_{T H} \frac{\overline{\Delta\left(\zeta_{g}+\left(f_{c} / 2\right)\right.})}{\Delta s}\right]\tag{13.29}\)
where s is distance along the thermal wind direction, and M[TH] is the thermal-wind speed.
Sample Application
The 100 to 50 kPa thickness is 5 km and f[c] = 10^–4 s^–1. A west to east thermal wind of 20 m s^–1 blows through a region where avg. cyclonic vorticity decreases by 10^–4 s^–1 toward the east across
a distance of 500 km. Use the omega eq. to find mid-tropospheric upward speed.
Find the Answer
Given: U[TH] = 20 m s^–1, V[TH] = 0, ∆z = 5 km, ∆ζ = –10^–4 s^–1 , ∆x = 500 km, f[c ]= 10^–4 s^–1.
Find: W[mid] = ? m s^–1
Use eq. (13.28):
\(W_{m i d} \cong \frac{-2 \cdot(5000 m)}{\left(10^{-4} s^{-1}\right)}\left[(20 m / s) \frac{\left(-10^{-4} s^{-1}\right)}{\left(5 \times 10^{5} m\right)}+0+0\right]=\underline{0.4 m s^{-1}}\)
Check: Units OK. Physics OK.
Exposition: At this speed, an air parcel would take 7.6 h to travel from the ground to the tropopause.
Regardless of the form, the terms in square brackets represent the advection of vorticity by the thermal wind, where vorticity consists of the geostrophic relative vorticity plus a part of the
vorticity due to the Earth’s rotation. The geostrophic vorticity at the 85 kPa or the 70 kPa isobaric surface is often used to approximate the average geostrophic vorticity over the whole 100 to 50
kPa layer.
A physical interpretation of the omega equation is that greater upward velocity occurs where there is greater advection of cyclonic (positive) geostrophic vorticity by the thermal wind. Greater
upward velocity favors clouds and heavier precipitation. Also, by moving air upward from the surface, it reduces the pressure under it, causing the surface low to move toward that location and
Figure 13.36 (a) Weather at three different pressure heights: (1) 50 kPa heights (solid blue lines) and trough axis (thick dashed line); (2) surface low pressure center (L) and fronts; (3) 70 kPa
vorticity (shaded). (b) Trough axis, surface low and fronts, and vorticity shading are identical to Fig. (a). Added are: 100 to 50 kPa thickness (dotted green lines), thermal wind vectors (arrows),
and region of maximum positive vorticity advection by the thermal wind (rectangular box). It is within this box that the omega equation gives the greatest updraft speed, which supports cyclogenesis.
Weather maps can be used to determine the location and magnitude of the maximum upward motion. The idealized map of Fig. 13.36a shows the height (z) contours of the 50 kPa isobaric surface, along
with the trough axis. Also shown is the location of the surface low and fronts.
At the surface, the greatest vorticity is often near the low center. At 50 kPa, it is often near the trough axis. At 70 kPa, the vorticity maximum (vort max) is usually between those two locations.
In Fig. 13.36a, the darker shading corresponds to regions of greater cyclonic vorticity at 70 kPa.
Fig. 13.36b shows the thickness (∆z) of the layer of air between the 100 and 50 kPa isobaric surfaces. Thickness lines are often nearly parallel to surface fronts, with the tightest packing on the
cold side of the fronts. Recall that thermal wind is parallel to the thickness lines, with cold air to the left, and with the greatest velocity where the thickness lines are most tightly packed.
Thermal wind direction is represented by the arrows in Fig. 13.36b, with longer arrows denoting stronger speed.
Advection is greatest where the area between crossing isopleths is smallest (the INFO Box on the next page explains why). This rule also works for advection by the thermal wind. The dotted lines
represent the isopleths that drive the thermal wind. In Fig. 13.36 the thin black lines around the shaded areas are isopleths of vorticity. The solenoid at the smallest area between these crossing
isopleths indicates the greatest vorticity advection by the thermal wind, and is outlined by a rectangular box. For this particular example, the greatest updraft would be expected within this box.
Full Omega Equation
The omega equation describes vertical motion in pressure coordinates. One form of the quasi-geostrophic omega equation is:
\(\begin{aligned}\left\{\nabla_{p}^{2}+\frac{f_{o}^{2}}{\sigma} \frac{\partial^{2}}{\partial p^{2}}\right\} \omega &=\frac{-f_{0}}{\sigma} \cdot \frac{\partial}{\partial p}\left[-\vec{V}_{g} \cdot \
vec{\nabla}_{p}\left(\zeta_{g}+f_{c}\right)\right] - \frac{\Re}{\sigma \cdot p} \cdot \nabla_{p}^{2}\left[-\vec{V}_{g} \cdot \vec{\nabla}_{p} T\right] \end{aligned}\)
where f[o] is a reference Coriolis parameter f[c] at the center of a beta plane, σ is a measure of static stability, Vg is a vector geostrophic wind, ℜ is the ideal gas law constant, p is pressure, T
is temperature, ζ[g] is geostrophic vorticity, and • means vector dot product.
\(\vec{\nabla}_{p}()=\partial() /\left.\partial x\right|_{p}+\partial() /\left.\partial y\right|_{p}\) is the del operator, which gives quasi-horizontal derivatives along an isobaric surface. Another
operator is the Laplacian:
\(\nabla^{2}_{p}()=\partial^{2}() /\left.\partial x^{2}\right|_{p}+\partial^{2}() /\left.\partial y^{2}\right|_{p}\)
Although the omega equation looks particularly complicated and is often shown to frighten unsuspecting people, it turns out to be virtually useless. The result of this equation is a small difference
between very large terms on the RHS that often nearly cancel each other, and which can have large error.
Trenberth Omega Equation
Trenberth developed a more useful form that avoids the small difference between large terms:
\(\left\{\nabla_{p}^{2}+\frac{f_{o}^{2}}{\sigma} \frac{\partial^{2}}{\partial p^{2}}\right\} \omega=\frac{2 f_{0}}{\sigma} \cdot\left[\frac{\partial \vec{V}_{g}}{\partial p} \cdot \vec{\nabla}_{p}\
left(\zeta_{g}+\left(f_{c} / 2\right)\right)\right]\)
For the omega subsection of this chapter, we focus on the vertical (pressure) derivative on the LHS, and ignore the Laplacian. This leaves:
\(\frac{f_{o}^{2}}{\sigma} \frac{\partial^{2} \omega}{\partial p^{2}}=\frac{2 f_{0}}{\sigma} \cdot\left[\frac{\partial \vec{V}_{g}}{\partial p} \cdot \vec{\nabla}_{p}\left(\zeta_{g}+\left(f_{c} / 2\
Upon integrating over pressure from p = 100 to 50 kPa:
\(\frac{\partial \omega}{\partial p}=\frac{-2}{f_{o}} \cdot\left[\vec{V}_{T H} \bullet \overline{ \vec{\nabla}_{p}\left(\zeta_{g}+\left(f_{c} / 2\right)\right)}\right]\)
where the definition of thermal wind V[TH] is used, along with the mean value theorem for the last term.
The hydrostatic eq. is used to convert the LHS: ∂ ∂ ω/ / p W = ∂ ∂z . The whole eq. is then integrated over height, with W = W[mid] at z = ∆z (= 100 - 50 kPa thickness) and W = 0 at z = 0.
This gives
\(W_{m i d}=
\frac{-2 \cdot \Delta z}{f_{c}}\left[U_{T H} \frac{\overline{\Delta\left (\zeta_{g}+\left(f_{c} / 2\right)\right)}}{\Delta x}+V_{T H} \frac{\overline{ \Delta\left(\zeta_{g}+\left(f_{c} / 2\right)\
right)}}{\Delta y}\right]\)
But f[c] varies with y, not x. The result is eq. (13.28).
One trick to locating the region of maximum advection is to find the region of smallest area between crossing isopleths on a weather (wx) map, where one set of isopleths must define a wind.
For example, consider temperature advection by the geostrophic wind. Temperature advection will occur only if the winds blow across the isotherms at some nonzero angle. Stronger temperature gradient
with stronger wind component perpendicular to that gradient gives stronger temperature advection.
But stronger geostrophic winds are found where the isobars are closer together. Stronger temperature gradients are found where the isotherms are closer together. In order for the winds to cross the
isotherms, the isobars must cross the isotherms. Thus, the greatest temperature advection is where the tightest isobar packing crosses the tightest isotherm packing. At such locations, the area
bounded between neighboring isotherms and isobars is smallest.
This is illustrated in the surface weather map below, where the smallest area is shaded to mark the maximum temperature advection. There is a jet of strong geostrophic winds (tight isobar spacing)
running from northwest to southeast. There is also a front with strong temperature gradient (tight isotherm spacing) from northeast to southwest. However, the place where the jet and temperature
gradient together are strongest is the shaded area.
Each of the odd-shaped tiles (solenoids) between crossing isobars and isotherms represents the same amount of temperature advection. But larger tiles imply that temperature advection is spread over
larger areas. Thus, greatest temperature flux (temperature advection per unit area) is at the smallest tiles.
This approach works for other variables too. If isopleths of vorticity and height contours are plotted on an upper-air chart, then the smallest area between crossing isopleths indicates the region of
maximum vorticity advection by the geostrophic wind. For vorticity advection by the thermal wind, plot isopleths of vorticity vs. thickness contours.
Fig. h. Solid lines are isobars. Grey dashed lines are isotherms. Greatest temperature advection is at the green tile.
Be careful when you identify the smallest area. In Fig. 13.36b, another area equally as small exists further south-south-west from the low center. However, the cyclonic vorticity is being advected
away from this region rather than toward it. Hence, this is a region of negative vorticity advection by the thermal wind, which would imply downward vertical velocity and cyclolysis or
Figure 13.37 (At right. Not the 2014 case study.) Superposition of the vorticity chart (grey lines and shading) at 85 kPa with the chart for thickness (thick black lines) between the 100 and 50 kPa
isobaric surfaces, for a 1994 event. The thermal wind (arrows) blows parallel to the thickness lines with cold air to its left. The white box highlights a region of positive vorticity advection (PVA)
by the thermal wind, where updrafts, cyclogenesis, and bad weather would be expected.
To apply these concepts to the 1994 event, Fig. 13.37 superimposes the 85 kPa vorticity chart with the 100 - 50 kPa thickness chart. The white box highlights a region of small solenoids, with the
thermal wind blowing from high towards low vorticity. Hence, the white box outlines an area of positive vorticity advection (PVA) by the thermal wind, so you can anticipate substantial updrafts in
that region. Such updrafts would create bad weather (clouds and precipitation), and would encourage cyclogenesis in the region outlined by the white box.
Near the surface low center (marked by the X in Fig. 13.37) is weak negative vorticity advection. This implies downdrafts, which contribute to cyclolysis. This agrees with the actual cyclone
evolution, which began weakening at this time, while a new cyclone formed near the Carolinas and moved northward along the USA East Coast.
The Trenberth omega equation is heavily used in weather forecasting to help diagnose synopticscale regions of updraft and the associated cyclogenesis, cloudiness and precipitation. However, in the
derivation of the omega equation (which we did not cover in this book), we neglected components that describe the role of ageostrophic motions in helping to maintain geostrophic balance. The INFO box
on the Geostrophic Paradox describes the difficulties of maintaining geostrophic balance in some situations — motivation for Hoskin’s Q-vector approach described next.
Consider the entrance region of a jet streak. Suppose that the thickness contours are initially zonal, with cold air to the north and warm to the south (Fig. i(a)). As entrance winds (black arrows in
Fig. i(a) converge, warm and cold air are advected closer to each other. This causes the thickness contours to move closer together (Fig. i(b), in turn suggesting tighter packing of the height
contours and faster geostrophic winds at location “X” via the thermal wind equation. But the geostrophic wind in Fig. i(a) is advecting slower wind speeds to location “X”.
Paradox: advection of the geostrophic wind by the geostrophic wind seems to undo geostrophic balance at “X”.
Fig. i. Entrance region of jet streak on a 50 kPa isobaric surface. z is height (black dashed lines), ∆z is thickness (thin colored lines), shaded areas are wind speeds, with initial isotachs as
dotted black lines. L & H are low and high heights. (a) Initially. (b) Later.
13.5.3. Q-Vectors
Q-vectors allow an alternative method for diagnosing vertical velocity that does not neglect as many terms.
Figure 13.38 (a) Components of a Q-vector. (b) How to recognize patterns of vector convergence (C) and divergence (D) on weather maps.
13.5.3.1. Defining Q-vectors
Define a horizontal Q-vector (units m^2·s^–1·kg^–1) with x and y components as follows:
\(\ Q_{x}=-\frac{\Re}{P}\left[\left(\frac{\Delta U_{g}}{\Delta x} \cdot \frac{\Delta T}{\Delta x}\right)+\left(\frac{\Delta V_{g}}{\Delta x} \cdot \frac{\Delta T}{\Delta y}\right)\right]\tag{13.30}\)
\(\ Q_{y}=-\frac{\Re}{P}\left[\left(\frac{\Delta U_{g}}{\Delta y} \cdot \frac{\Delta T}{\Delta x}\right)+\left(\frac{\Delta V_{g}}{\Delta y} \cdot \frac{\Delta T}{\Delta y}\right)\right]\tag{13.31}\)
where ℜ = 0.287 kPa·K^–1·m^3·kg^–1 is the gas constant, P is pressure, (U[g], V[g]) are the horizontal components of geostrophic wind, T is temperature, and (x, y) are eastward and northward
horizontal distances. On a weather map, the Q[x] and Q[y] components at any location are used to draw the Q-vector at that location, as sketched in Fig. 13.38a. Q-vector magnitude is
\(\ |Q|=\left(Q_{x}^{2}+Q_{y}^{2}\right)^{1 / 2}\tag{13.32}\)
Sample Application
Given the weather map at right showing the temperature and geostrophic wind fields over the NE USA. Find the Q-vector at the “X” in S.E. Pennsylvania. Side of each grid square is 100 km, and
corresponds to G = 5 m s^–1 for the wind vectors.
Figure j
Find the Answer
Given: P = 85 kPa, G (m s^–1) & T (°C) fields on map.
Find: Q[x] & Q[y] = ? m^2 ·s^–1 ·kg^–1
First, estimate U[g], V[g], and T gradients from the map.
∆T/∆x = –5°C/600km, ∆T/∆y = –5°C/200km, ∆U[g]/∆x = 0, ∆V[g]/∆x = (–2.5m s^–1)/200km ∆Ug/∆y = (–5m s^–1)/300km, ∆Vg/∆y = 0, ℜ/P = 0.287/85 = 0.003376 m^3·kg^–1·K^–1
Use eq. (13.30): Qx = – (0.003376 m^3·kg^–1·K^–1)· [( (0)·(–8.3) + (–12.5)·(–25)]·10–12 K·m^–1·s^–1
Qx = –1.06x10^–12 m^2 ·s^–1 ·kg^–1
Use eq. (13.31): Qy = – (0.003376 m^3·kg^–1·K^–1)· [( (–16.7)·(–8.3) + (0)·(–25)]·10^–12 K·m^–1·s^–1
Qy = –0.47x10^–12 m^2 ·s^–1 ·kg^–1
Use eq. (13.32) to find Q-vector magnitude:
|Q| = [(–1.06)^2 + (–0.47)^2] ^1/2 · 10^–12 K·m^–1·s^–1
|Q| = 1.16x10^–12 m^2 ·s^–1 ·kg^–1
Check: Physics, units are good. Similar to Fig. 13.40.
Exposition: The corresponding Q-vector is shown at right; namely, it is pointing from the NNE because both Qx and Qy are negative. There was obviously a lot of computations needed to get this one
Luckily, computers can quickly compute Q-vectors for many points in a grid, as shown in Fig. 13.40. Normally, you don’t need to worry about the units of the Q-vector. Instead, just focus on Q-vector
convergence zones such as computers can plot (Fig. 13.41), because these zones are where the bad weather is.
13.5.3.2. Estimating Q-vectors
Eqs. (13.30 - 13.32) seem non-intuitive in their existing Cartesian form. Instead, there is an easy way to estimate Q-vector direction and magnitude using weather maps. First, look at direction.
Suppose you fly along an isotherm (Fig. 13.39) in the direction of the thermal wind (in the direction that keeps cold air to your left). Draw an arrow describing the geostrophic wind vector that you
observe at the start of your flight, and draw a second arrow showing the geostrophic wind vector at the end of your flight. Next, draw the vector difference, which points from the head of the initial
vector to the head of the final vector. The Q-vector direction points 90° to the right (clockwise) from the geostrophic difference vector.
The magnitude is
\(\ |Q|=\frac{\mathfrak{R}}{P}\left|\frac{\Delta T}{\Delta n} \cdot \frac{\Delta \vec{V}_{g}}{\Delta s}\right|\tag{13.33}\)
where ∆n is perpendicular distance between neighboring isotherms, and where the temperature difference between those isotherms is ∆T. Stronger baroclinic zones (namely, more tightly packed isotherms)
have larger temperature gradient ∆T/∆n. Also, ∆s is distance of your flight along one isotherm, and ∆V[g] is the magnitude of the geostrophic difference vector from the previous paragraph. Thus,
greater changes of geostrophic wind in stronger baroclinic zones have larger Q-vectors. Furthermore, Q-vector magnitude increases with the decreasing pressure P found at increasing altitude.
Figure 13.39 Illustration of natural coordinates for Q-vectors. Dashed grey lines are isotherms. Aircraft flies along the isotherms with cold air to its left. Black arrows are geostrophic wind
vectors. Grey arrow indicates Q-vector direction (but not magnitude).
13.5.3.3. Using Q-vectors / Forecasting Tips
Figure 13.40 (Not the 2014 case study.) Weather map of Q-vectors. (o means small magnitude.)
Figure 13.41 (Not the 2014 case study.) Convergence of Q-vectors (shaded). Divergence (dashed lines). See the Info box in section 13.1.3 for identification of geographic features.
Different locations usually have different Q-vectors, as sketched in Fig. 13.40 for a 1994 event. Interpret Q-vectors on a synoptic weather map as follows:
• Updrafts occur where Q-vectors converge. (Fig. 13.41 gives an example for the 1994 event).
• Subsidence (downward motion) occurs where Qvectors diverge.
• Frontogenesis occurs where Q-vectors cross isentropes (lines of constant potential temperature) from cold toward warm.
• Updrafts in the TROWAL region ahead of a warm occluded front occur during cyclolysis where the along-isentrope component of Q-vectors converge.
Using the tricks for visually recognizing patterns of vectors on weather maps (Fig. 13.38b), you can identify by eye regions of convergence and divergence in Fig. 13.40. Or you can let the computer
analyze the Q-vectors directly to plot Q-vector convergence and divergence (Fig. 13.41). Although Figs. 13.40 and 13.41 are analysis maps of current weather, you can instead look at Q-vector forecast
maps as produced automatically by numerical weather prediction models (see the NWP chapter) to help you forecast regions of updraft, clouds, and precipitation.
Remember that Q-vector convergence indicates regions of likely synoptic-scale upward motion and associated clouds and precipitation. Looking at Fig. 13.41, see a moderate convergence region running
from the western Gulf of Mexico up through eastern Louisiana and southern Mississippi. It continues as a weak convergence region across Alabama and Georgia, and then becomes a strong convergence
region over West Virginia, Virginia and Maryland. A moderate convergence region extends northwest toward Wisconsin.
This interpretation agrees with the general locations of radar echoes of precipitation for this 1994 event. Note that the frontal locations need not correspond to the precipitation regions. This
demonstrates the utility of Q-vectors — even when the updrafts and precipitation are not exactly along a front, you can use Q-vectors to anticipate the bad-weather regions.
Also, along the Texas Gulf coast, the Q-vectors in Fig. 13.40 are crossing the cold front from cold toward warm air. Using the third bullet on the previous page, you can anticipate frontogenesis in
this region.
13.5.3.4. Resolving the Geostrophic Paradox
What about the ageostrophic circulations that were missing from the Trenberth omega equation? Fig. 13.41 suggests updrafts at the Q-vector convergence region over the western Gulf of Mexico, and
subsidence at the divergence region of central Texas. Due to mass continuity, you can expect an ageostrophic circulation of mid-tropospheric winds from the southeast toward the northwest over the
Texas Gulf coast, which connects the up- and down-draft portions of the circulation. This ageostrophic wind moves warm pre-frontal air up over the cold front in a direct circulation (i.e., a
circulation where warm air rises and cold air sinks).
But if you had used the 85 kPa height chart to anticipate geostrophic winds over central Texas, you would have expected light winds at 85 kPa from the northwest. These opposing geostrophic and
ageostrophic winds agree nicely with the warm-air convergence (creating thunderstorms) for the cold katafront sketch in Fig. 12.16a.
Similarly, over West Virginia and Maryland, Fig. 13.41 shows convergence of Q-vectors at low altitudes, suggesting rising air in that region. This updraft adds air mass to the top of the air column,
increasing air pressure in the jet streak right entrance region, and tightening the pressure gradient across the jet entrance. This drives faster geostrophic winds that counteract the advection of
slower geostrophic winds in the entrance region. Namely, the ageostrophic winds as diagnosed using Q-vectors help prevent the Geostrophic Paradox (INFO Box).
Sample Application
Discuss the nature of circulations and anticipated frontal and cyclone evolution, given the Q-vector divergence region of southern Illinois and convergence in Maryland & W. Virginia, using Fig.
Find the Answer
Given: Q-vector convergence fields.
Discuss: circulations, frontal & cyclone evolution
Exposition: For this 1994 case there is a low center over southern Illinois, right at the location of maximum divergence of Q-vectors in Fig. 13.41. This suggests that: (1) The cyclone is entering
the cyclolysis phase of its evolution (synoptic-scale subsidence that opposes any remaining convective updrafts from earlier in the cyclones evolution) as it is steered northeastward toward the Great
Lakes by the jet stream. (2) The cyclone will likely shift toward the more favorable updraft region over Maryland. This shift indeed happened.
The absence of Q-vectors crossing the fronts in western Tennessee and Kentucky suggest no frontogenesis there.
Between Maryland and Illinois, we would anticipate a mid-tropospheric ageostrophic wind from the east-northeast. This would connect the updraft region over western Maryland with the downdraft region
over southern Illinois. This circulation would move air from the warm-sector of the cyclone to over the low center, helping to feed warm humid air into the cloud shield over and north of the low.
By considering the added influence of ageostrophic winds, the Q-vector omega equation is:
\(\begin{aligned}\left\{\nabla_{p}^{2}+\frac{f_{0}^{2}}{\sigma} \frac{\partial^{2}}{\partial p^{2}}\right\} \omega=\frac{-2}{\sigma} \cdot\left[\frac{\partial Q_{1}}{\partial x}+\frac{\partial Q_{2}}
{\partial y}\right]+\frac{f_{0} \cdot \beta}{\sigma} \frac{\partial V_{g}}{\partial p} -\frac{R / C_{p}}{\sigma \cdot P} \nabla^{2}\left(\Delta Q_{H}\right) \end{aligned}\)
The left side looks identical to the original omega equation (see a previous HIGHER MATH box for an explanation of most symbols). The first term on the right is the convergence of the Q vectors. The
second term is small enough to be negligible for synopticscale systems. The last term contributes to updrafts if there is a local maximum of sensible heating ∆Q[H]. | {"url":"https://geo.libretexts.org/Bookshelves/Meteorology_and_Climate_Science/Practical_Meteorology_(Stull)/13%3A_Extratropical_Cyclones/13.04%3A_Section_5-","timestamp":"2024-11-13T18:52:25Z","content_type":"text/html","content_length":"194796","record_id":"<urn:uuid:572a5702-2df7-4f2e-bc5d-fb171ec38882>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00701.warc.gz"} |
231 research outputs found
We consider the gravitational correction to (electronic) vacuum polarization in the presence of a gravitational background field. The Dirac propagators for the virtual fermions are modified to
include the leading gravitational correction (potential term) which corresponds to a coordinate-dependent fermion mass. The mass term is assumed to be uniform over a length scale commensurate with
the virtual electron-positron pair. The on-mass shell renormalization condition ensures that the gravitational correction vanishes on the mass shell of the photon, i.e., the speed of light is
unaffected by the quantum field theoretical loop correction, in full agreement with the equivalence principle. Nontrivial corrections are obtained for off-shell, virtual photons. We compare our
findings to other works on generalized Lorentz transformations and combined quantum-electrodynamic gravitational corrections to the speed of light which have recently appeared in the
literature.Comment: 9 pages; RevTeX; typographical errors corrected and references adde
The proton radius conundrum [R. Pohl et al., Nature vol.466, p.213 (2010) and A. Antognini et al., Science vol.339, p.417 (2013)] highlights the need to revisit any conceivable sources of
electron-muon nonuniversality in lepton-proton interactions within the Standard Model. Superficially, a number of perturbative processes could appear to lead to such a nonunversality. One of these is
a coupling of the scattered electron into an electronic as opposed to a muonic vacuum polarization loop in the photon exchange of two valence quarks, which is present only for electron projectiles as
opposed to muon projectiles. However, we can show that this effect actually is part of the radiative correction to the proton's polarizability contribution to the Lamb shift, equivalent to a
radiative correction to double scattering. We conclude that any conceivable genuine nonuniversality must be connected with a nonperturbative feature of the proton's structure, e.g., with the possible
presence of light sea fermions as constituent components of the proton. If we assume an average of roughly 0.7*10^(-7) light sea positrons per valence quark, then we can show that virtual
electron-positron annihiliation processes lead to an extra term in the electron-proton versus muon-proton interaction, which has the right sign and magnitude to explain the proton radius
discrepancy.Comment: 6 pages; RevTeX; published in Physical Review A in 2013; as compare to the journal version, we have added a note at the end of the paper which pertains to the (new) Ref. [42];
otherwise unchange
We investigate a specific set of two-loop self-energy corrections involving squared decay rates and point out that their interpretation is highly problematic. The corrections cannot be interpreted as
radiative energy shifts in the usual sense. Some of the problematic corrections find a natural interpretation as radiative nonresonant corrections to the natural line shape. They cannot uniquely be
associated with one and only one atomic level. While the problematic corrections are rather tiny when expressed in units of frequency (a few Hertz for hydrogenic P levels) and do not affect the
reliability of quantum electrodynamics at the current level of experimental accuracy, they may be of importance for future experiments. The problems are connected with the limitations of the
so-called asymptotic-state approximation which means that atomic in- and out-states in the S-matrix are assumed to have an infinite lifetime.Comment: 12 pages, 3 figures (New J. Phys., in press,
submitted 28th May
We consider the general scenario of an excited level |i> of a quantum system that can decay via two channels: (i) via a single-quantum jump to an intermediate, resonant level |bar m>, followed by a
second single-quantum jump to a final level |f>, and (ii) via a two-quantum transition to a final level |f>. Cascade processes |i> -> |bar m> -> | f> and two-quantum transitions |i> -> |m> -> |f>
compete (in the latter case, |m> can be both a nonresonant as well as a resonant level). General expressions are derived within second-order time-dependent perturbation theory, and the cascade
contribution is identified. When the one-quantum decay rates of the virtual states are included into the complex resonance energies that enter the propagator denominator, it is found that the
second-order decay rate contains the one-quantum decay rate of the initial state as a lower-order term. For atomic transitions, this implies that the differential-in-energy two-photon transition rate
with complex resonance energies in the propagator denominators can be used to good accuracy even in the vicinity of resonance poles.Comment: 9 pages; RevTe
It is of general theoretical interest to investigate the properties of superluminal matter wave equations for spin one-half particles. One can either enforce superluminal propagation by an explicit
substitution of the real mass term for an imaginary mass, or one can use a matrix representation of the imaginary unit that multiplies the mass term. The latter leads to the tachyonic Dirac equation,
while the equation obtained by the substitution m->i*m in the Dirac equation is naturally referred to as the imaginary-mass Dirac equation. Both the tachyonic as well as the imaginary-mass Dirac
Hamiltonians commute with the helicity operator. Both Hamiltonians are pseudo-Hermitian and also possess additional modified pseudo-Hermitian properties, leading to constraints on the resonance
eigenvalues. Here, by an explicit calculation, we show that specific sum rules over the spectrum hold for the wave functions corresponding to the well-defined real energy eigenvalues and complex
resonance and anti-resonance energies. In the quantized imaginary-mass Dirac field, one-particle states of right-handed helicity acquire a negative norm ("indefinite metric") and can be excluded from
the physical spectrum by a Gupta--Bleuler type condition.Comment: 8 pages; RevTeX; published in J.Mod.Phy
The proton radius puzzle questions the self-consistency of theory and experiment in light muonic and electronic bound systems. Here, we summarize the current status of virtual particle models as well
as Lorentz-violating models that have been proposed in order to explain the discrepancy. Highly charged one-electron ions and muonic bound systems have been used as probes of the strongest
electromagnetic fields achievable in the laboratory. The average electric field seen by a muon orbiting a proton is comparable to hydrogenlike Uranium and, notably, larger than the electric field in
the most advanced strong-laser facilities. Effective interactions due to virtual annihilation inside the proton (lepton pairs) and process-dependent corrections (nonresonant effects) are discussed as
possible explanations of the proton size puzzle. The need for more experimental data on related transitions is emphasized.Comment: 11 pages; RevTe
We investigate the interaction of metastable 2S hydrogen atoms with a perfectly conducting wall, including parity-breaking S-P mixing terms (with full account of retardation). The neighboring 2P_1/2
and 2P_3/2 levels are found to have a profound effect on the transition from the short-range, nonrelativistic regime, to the retarded form of the Casimir-Polder interaction. The corresponding P state
admixtures to the metastable 2S state are calculated. We find the long-range asymptotics of the retarded Casimir-Polder potentials and mixing amplitudes, for general excited states, including a fully
quantum electrodynamic treatment of the dipole-quadrupole mixing term. The decay width of the metastable 2S state is roughly doubled even at a comparatively large distance of 918 atomic units (Bohr
radii) from the perfect conductor. The magnitude of the calculated effects is compared to the unexplained Sokolov effect.Comment: 6 pages; RevTe
Quantum electrodynamics has been the first theory to emerge from the ideas of regularization and renormalization, and the coupling of the fermions to the virtual excitations of the electromagnetic
field. Today, bound-state quantum electrodynamics provides us with accurate theoretical predictions for the transition energies relevant to simple atomic systems, and steady theoretical progress
relies on advances in calculational techniques, as well as numerical algorithms. In this brief review, we discuss one particular aspect connected with the recent progress: the evaluation of
relativistic corrections to the one-loop bound-state self-energy in a hydrogenlike ion of low nuclear charge number, for excited non-S states, up to the order of alpha (Zalpha)^6 in units of the
electron mass. A few details of calculations formerly reported in the literature are discussed, and results for 6F, 7F, 6G and 7G states are given.Comment: 16 pages, LaTe
Starting from the coupling of a relativistic quantum particle to the curved Schwarzschild space-time, we show that the Dirac--Schwarzschild problem has bound states and calculate their energies
including relativistic corrections. Relativistic effects are shown to be suppressed by the gravitational fine-structure constant alpha_G = G m_1 m_2/(hbar c), where G is Newton's gravitational
constant, c is the speed of light and m_1 and m_2 >> m_1 are the masses of the two particles. The kinetic corrections due to space-time curvature are shown to lift the familiar (n,j) degeneracy of
the energy levels of the hydrogen atom. We supplement the discussion by a consideration of an attractive scalar potential, which, in the fully relativistic Dirac formalism, modifies the mass of the
particle according to the replacement m -> m (1 - \lambda/r), where r is the radial coordinate. We conclude with a few comments regarding the (n,j) degeneracy of the energy levels, where n is the
principal quantum number, and j is the total angular momentum, and illustrate the calculations by way of a numerical example.Comment: 6 pages; RevTe | {"url":"https://core.ac.uk/search/?q=authors%3A(Jentschura%20U%20D)","timestamp":"2024-11-15T03:36:53Z","content_type":"text/html","content_length":"152858","record_id":"<urn:uuid:57e64947-0018-4bf5-8947-fa53dbda3bec>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00136.warc.gz"} |
How to | R (for ecology)
This blog post I teach you the basics of creating and customizing scatterplots in R
Learn about the four challenges when learning R and the key strategies for how to overcome those challenges so that nothing can stop you from mastering R
In this tutorial, I go over the basics of how to prototype, save, and export your plots from R.
In this blog post, I show you how to reshape data (i.e. how to use pivot tables) so that the data are in the correct form for data analysis in R.
In this tutorial, I'm going to explain how to create your own functions and provide a few examples.
In this tutorial I show you everything you need to know about boxplots and how to make them look nice using the built-in functions in R
Here I describe the functions called `grep()`, `grepl()`, and `sub()`, which allow you to find strings in your data that match particular patterns.
Here I show you a useful family of functions that allows you to repetitively perform a specified function (e.g., sum, mean) across a vector, matrix, or data frame.
In this tutorial, I'm going to give an explanation of what pipes are and when they can be used, and then I'm going to demonstrate how useful they can be for writing neat and clear R code.
In this tutorial, I discuss how to use a handy function called group_by() for organizing and preparing your data for analysis and visualization. | {"url":"https://www.rforecology.com/tag/how-to/","timestamp":"2024-11-12T05:44:18Z","content_type":"text/html","content_length":"12839","record_id":"<urn:uuid:1f6797f8-b406-484e-8f95-5f85735c13b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00670.warc.gz"} |
Evaluating infinite integrals involving Bessel functions of arbitrary order
The evaluation of intergrals of the form I[n] = ∫ 0 ∞ f{hook}(x)J[n](x)dx is considered. In the past, the method of dividing an oscillatory integral at its zeros, forming a sequence of partial sums,
and using extrapolation to accelerate convergence has been found to be the most efficient technique available where the oscillation is due to a trigonometric function or a Bessel function of order n
= 0, 1. Here, we compare various extrapolation techniques as well as choices of endpoints in dividing the integral, and establish the most efficient method for evaluating infinite integrals involving
Bessel functions of any order n, not just zero or one. We also outline a simple but very effective technique for calculating Bessel function zeros.
All Science Journal Classification (ASJC) codes
• Computational Mathematics
• Applied Mathematics
• Bessel functions
• Bessel zeros
• Infinite integration
• Quadrature
• mW transform
• ε-algorithm
Dive into the research topics of 'Evaluating infinite integrals involving Bessel functions of arbitrary order'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/evaluating-infinite-integrals-involving-bessel-functions-of-arbit","timestamp":"2024-11-05T23:45:54Z","content_type":"text/html","content_length":"46086","record_id":"<urn:uuid:8943398d-493b-407d-b527-5d3fd53194fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00265.warc.gz"} |
Antiderivative Graph: Complete Explanation and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day
Take note that if we take the antiderivative of a derivative, it will provide us with the original function. Hence, when we want to sketch or draw the graph of an antiderivative, we are converting a
derivative function to its original form.
In this guide, we will learn what an antiderivative graph means and how to draw or sketch an antiderivative graph accurately.
What Is Meant by Antiderivative Graph?
The antiderivative graph is the graph of an inverse derivative function, and the antiderivative is the opposite of the derivative function. When we take the integral of the derivative of a function,
then it is called an antiderivative function, and the outcome of such function is the original function of the given differential equation.
Suppose we are given a function $f(x) = x^{3}$, then the antiderivative of this function is $F(x) = \dfrac{x^{4}}{4} + c$. Take note that if we take the derivative of $F(x)$, we get $f(x)$ back. If
we draw the graph for F(x), then it will be called an antiderivative graph. The constant value “c” determine the vertical location of the graph, all the antiderivative graphs of a given function are
simply vertical translations of each other, and their vertical location depends upon the value of “c”.
Drawing an Antiderivative Graph From a Derivative Function
We can easily draw the graph of an antiderivative function from the given derivative function, but to draw a graph, you should know some important points first.
1. If the derivative function $f’ (x)$ is below the x-axis, the original function’s slope will be negative.
2. If the derivative function $f’ (x)$ is above the x-axis, the original function’s slope will be positive.
3. All the x-intercept points of the derivative functions $f’ (x)$ will be the critical points / relative maximum points of f(x).
4. If the derivative function is an even function, then the antiderivative function will be an odd function. Similarly, if the derivative function is an odd function, then the antiderivative
function will be an even function.
Let us study the two graphs given below; the first graph shows the antiderivative graph for a linear function.
The second example shows the antiderivative graph of a parabola.
You can clearly see that when the $f’ (x)$ was above the x-axis, then the slope of $f(x)$ is positive, and when $f’ (x)$ is below the x-axis, then the slope of f(x) is negative. Furthermore, we can
also observe that the x-intercept points of $f'(x)$ are the critical points for $f(x)$.
Derivative vs Antiderivative Functions
The difference between the derivative and antiderivative functions is presented in the table below. In the table, the original function or the antiderivative function is represented by “$F$” while
the derivative function is represented by $f’$. It is essential that you grasp the basic differences between them because it will help you in solving complex problems when drawing an original
function graph from a derivative graph.
Derivative Functions Antiderivative Functions
When antiderivative $F$ is increasing, then $f’$ will be positive. If $f’$ is positive, then $F$ will be increasing.
When antiderivative $F$ is increasing, then $f’$ will be positive. If $f’$ is negative, then F will be decreasing.
At maxima or minima of $F(x)$, the value of $f'(x)$ will be zero. When $f’$ will be zero, then F will either have a maxima or a critical number.
If $F” = 0$, then we will have a change in concavity, and this point will be called inflection point. As $F” = f’$, so when $F” = 0$, then it is certain $f’$ will either have a minima or maxima.
If the antiderivative function is concave down, then $f’$ is negative. When $f’$ is negative, then F is concave down.
If the antiderivative function is concave up, then $f’$ is positive. When $f’$ is positive, then F is concave up.
Example 1: You are given a graph for a piecewise linear function/ smooth function f(x), and you are required to sketch a graph for its antiderivative function such that $F(0) = 0$.
The graph we are given is for the function $f(x)$. This graph is a derivative graph for function $F(x)$, so we can say that $f(x) = F'(x)$.
To accurately plot the graph of the function, we have to apply the rules which we have learned so far.
Let us re-draw the graph and then apply the rules accordingly.
1. The antiderivative graph will start at $(0,0)$ as we are given $F(0) = 0$.
2. If we go along the x-axis from 0 to 1, we can see that “f” or “$F’$” is less than zero or negative, so the graph of F from 0 to 1 will be decreasing.
3. When we go along the x-axis from 1 to 2, we can see that “$f$” or “$F’$” is greater than zero or positive, so the graph of F from 1 to 2 will be increasing.
4. Similarly, when we go along the x-axis from 2 to 4, we can see that “$f$ “or “$F’$” is greater than zero or negative, so the graph of $F$ from 2 to 4 will be increasing.
5. The value graph of $F’ (x)$ or $f(x)$ is “0” at x = 1, so at this point, the antiderivative graph will have its minima point because the graph also decreases from interval 0 to 1.
Now that we are aware of the direction of the antiderivative graph for the given function, let us discuss how can we calculate the magnitude values of each interval. The expected value of the
antiderivative graph can be calculated by measuring or calculating the area under the curve of the given graph. We have highlighted the triangles by using bars, while square portions are colored.
1. For the interval $[0,1]$, a right-angle triangle is being formed, and the height and base of the triangle are 1 unit each. So the area of this region will be Area $= \dfrac{1}{2} \times base \
times height = \dfrac{1}{2}\times 1 \times 1 = \dfrac{1}{2}$.
2. For the interval $[1,2]$, just like the previous interval, a right-angle triangle is being formed and the height and base of the triangle are 1 unit each. So the area of this region is also $= \
3. For the interval $[2,3]$, a square is formed for the range or y-interval $[0,1]$ and a triangle is formed for the range or y-interval $[1, 2]$. The square formed is a unit square with all sides
equal to unit 1; hence, the area of the square is = 1 unit, while the area of the triangle is just like the area of previous triangles, $= \dfrac{1}{2}$ unit. So the total area of this region is
$= 1 + \dfrac{1}{2} = \dfrac{3}{2}$.
4. For the interval $[3,4]$, two unit squares are being formed for the range or y-interval $[0,1]$ and for the range or y-interval [1,2] while a triangle is being formed for the range or y-interval
$[2, 3]$. The area of both unit squares is 1 unit each while the area of the triangle is $\dfrac{1}{2}$. So the total area of this region will be $= 1 + 1 + \dfrac{1}{2} = \dfrac{5}{2} = 2\dfrac
{1}{2}$ and the next point will be 2 and half units away from the previous point.
The area of the piecewise regions or the multiple antiderivatives in a single function/graph can also be determined by using the simple calculus formula of the definite integrals. The definite
integral formula is given as:
$F(b) – F(a) = \int_{a}^{b} F'(x)$
By using all the above data, we can graph the antiderivative graph of the given function as:
Example 2: You are given a graph for function $f(x)$ and you are required to sketch a graph for its antiderivative function such that $F(0) = -1$.
We are given a graph for the function f(x). This graph is a derivative graph for function $F(x)$, so we can say that $f(x) = F'(x)$.
To accurately plot the graph of the function, we have to apply the rules which we have learned so far.
Constructing accurate graphs of antiderivatives can easily be done by applying the rules which we learned so far.
1. The antiderivative graph will start at y = -1 as we are given $F(0) = -1$.
2. If we go along the x-axis from the interval $[0, 1]$, we can see that “$f$” or “$F’$” is less than zero or negative, so the graph of F from 0 to 1 will be decreasing.
3. When we go along the x-axis from the interval $[3, 4]$, the slope of the graph is negative but the value of that “f” or “$F’$” is greater than zero or positive, so the graph of F for this
interval will be increasing.
4. When we go along the x-axis from the interval $[4,6 ]$, we can see that “f” or “F’$” is less than zero or negative, so the graph of F for this interval will be decreasing.
5. The value graph of $F’ (x)$ or f(x) is “0” at $x = 1$, $4$ and $6$, so these points will be critical points for the antiderivative graph, which means we will have our maxima and minima at these
points. So in this case, we will total three critical points.
Now that we know the direction of the antiderivative graph as well as its maxima and minima points, let us now calculate the area under the curve for the given function so that we know the magnitude
or value of the graph for the function F(x).
The area of the graph which needs to be calculated has been highlighted in the figure, and as you can see, we are mostly dealing with right-angle triangles along with 1 square region.
1. The interval $[0,1]$ forms a right-angle triangle just like in the previous example, and the area for this region is $\dfrac{1}{2}$.
2. For the interval $[1,2]$ a right angle triangle is formed. The base and height of the triangle have 1 unit each so the area of the triangle will be $= \dfrac{1}{2} \times 1 \times 1 = \dfrac{1}
3. For the interval $[2,3]$, a square is formed for the range or y-interval $[0,1]$ and a triangle is formed for the range or y-interval $[1, 2]$. The square is a unit square with each side equal to
1, so the area of the square will be $= 1 \times 1 = 1$ unit while the area of the triangle is $\dfrac{1}{2}$. So the total area of the region is $= 1 + \dfrac{1}{2} = \dfrac{3}{2}$.
4. If we add the area of the interval $[1,2]$ and $[2,3]$, it gives us $\dfrac{1}{2} + \dfrac{3}{2} = 2$. We get the same result if we take the complete area under the curve for the interval $[1,3]
$. This whole region is a right-angle triangle with a base and height equal to 2 units each, so if we take the area of the triangle, it will be $= \dfrac{1}{2} \times 2 \times 2 = 2$ units.
5. For the interval $[3,4]$, a right angle triangle is being formed with a base of 2 units and height of 1 unit, so the area of this region will be $= \dfrac{1}{2} \times 1 \times 2 = 1$ unit.
6. For the interval $[4,5]$, a right angle triangle is being formed with a base and height of 1 unit each, so the area of this region will be $= \dfrac{1}{2}$.
7. For the interval $[5,6]$, a right angle triangle is being formed with a base and height of 1 unit each, so the area of this region will be $= \dfrac{1}{2}$.
By using all the above data, we can graph the antiderivative graph of the given function as:
The same rules which we have discussed so far can also be applied to piecewise constant functions. Finally, to finish the guide, here are several practice questions for you to check if you have fully
grasped the concept.
Practice Questions:
1. Plot or Draw the antiderivative graph by using the derivative graph of the function given below such that F(0) = 0.
2. Plot or Draw the antiderivative graph by using the derivative graph of the function given below such that F(0) = 0.
Answer Key:
The antiderivative graph for the given f(x) will start at y = 1 as we are given F(0) = 1. The graph can be sketched as:
The antiderivative graph for the given f(x) will start at y = 0 as we are given F(0) = 0. The graph can be sketched as: | {"url":"https://www.storyofmathematics.com/antiderivative-graph/","timestamp":"2024-11-11T11:36:26Z","content_type":"text/html","content_length":"170309","record_id":"<urn:uuid:4234eddd-39bb-4897-85ba-98a32b06dae1>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00702.warc.gz"} |
Statistics Questions Statistics essay Essay — Free college essays
Statistics Questions Statistics essay
Question1 (a)
Thestandardized reading /language test scores of middle school studentswho create a narrated PowerPoint slide show about the slide they haveread will not be significantly higher than middle-class
students whowere taught using traditional methods.
Therewill be no significant difference between standardized readinglanguage/language test scores of students creating narratedPowerPoint slide show about a novel they read and those of studentswho
were taught using traditional methods.
Question1 (b)
Independentt-test: – our main concern in this problem is to ascertain thedifference in scores between two groups of students. The independentt-test is suitable for this purpose because it evaluates
thedifference between the means of the two groups. What is more, thegroups are unrelated in the sense that one group creates a slidepresentation in their learning while the other group relies
ontraditional teaching. An independent t-test is sometimes calledbetween group designs because of its role in distinguishing betweengroups.
Inaddition, this investigation can be interpreted as a controlexperiment that seeks to explain the role of creating a PowerPointslide in understanding language. Researchers use the independentt-test
to analyze the effect of a control variable on two sets ofpopulations.
Giventhat our primary objective is to ascertain the difference between thetwo groups of students in terms of their scores in language testwhere one group created slides based on their reading and
anotherrelied on traditional teaching. The independent t-test was selectedfor this purpose. The results obtained are T=5.52 (p=.03), and can beinterpreted as follows:
First,our null hypothesis can be represented statistically as
H0:µ1 = µ2 where µ1 stands for the mean for the first group and µ2stands for the mean for the second group.
Or-H0: µ1 – µ2 = 0
Thismeans that the hypothesis will hold true if the means of the twogroups are equal. Otherwise, the null hypothesis is false
Theresults of the independent t-test is 5.52 (p=0.3). To ascertain thetruth in the null hypothesis, our focus lies in the second part ofthe results i.e. P=.03. This is the probability that the
t-value5.52, which represents the difference in mean happened by chance. Instatistics, a maximum acceptable probability of an event occurring bychance is 5%. In our case, the probability is 3%
implying that thedifference in mean did not occur by chance, and therefore we rejectthe null hypothesis at 5% level of significance. By rejecting thenull hypothesis, the researcher implies that the
two groups ofstudents scored different means (µ1 – µ2 = 5.52) because of thecontrol variable (Creating PowerPoint slides of the reading). Theimplication in education is that creating PowerPoint
slides afterreading can improve student scores in language tests by approximately5.52%.
Theproposed project aims at improving the student overall achievement inmathematics. More specifically the project aims to: 1) increasescience scores on the state tests and (2) increase
students’interest in science. In order to measure the attainment of theseobjectives, we shall conduct empirical research that will seek toascertain the benefits of the project particularly in
regarding thetwo objectives. In this memo, we present the methods that will beused to measure the effectiveness of the project in attaining itsstated objectives.
Toascertain the level of attainment of the first objective i.e. “raisescores” objective, we shall compare the students’ scores instandardized math test both before and after participating in themath
club program. By comparing mean scores for the students in thesetwo periods, we shall be able to see the difference and attribute, toa given level of significance, to the introduction of the
program.Ideally, significant variation in mean will be attributed to mathclub program. To ascertain the level of attainment of our secondobjective i.e. the “enjoy math” objective, we intend to
conduct asurvey to the math club students, their teachers and parents. Thesurvey will involve answering questionnaires structured on afive-point likert scale. The questionnaires will be customized
forstudents, parents, and teachers basing on the role that each play inthe learning process.
Sincethe “raise score,” objective will rely on student test scores inmathematics, our investigation of this objective will be based solelyon quantitative research. The quantitative research is based
onnumerical data i.e. data that is primarily in numbers and mainlyseeks to explain the occurrence of a given phenomenon. For thisinvestigation, our null hypothesis will be: The math’s club programhas
no influence on student scores in standardized mathematicstastes. The data will be subjected to analysis using SPSS software toobtain descriptive statistics such as mean and median.
Thesecond program objective will be measured through qualitativestudies. Qualitative research relies on people`s feeling and opinionabout something, in this case about the math club
program.Questionnaires and interviews will be used to collect qualitativedata from students, teachers, and parents. Content analysis will beused to analyze the data. It is our hope that the program
will enableus to achieve these objectives. | {"url":"https://an-essay.com/statistics-questions-statistics","timestamp":"2024-11-06T15:33:35Z","content_type":"text/html","content_length":"227869","record_id":"<urn:uuid:c7562ffe-6ca4-41b3-a61f-2a9d2fbd65aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00062.warc.gz"} |
From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 11: The Notion of Computation
Section: Computation as a Framework
History of computing. Even in prehistoric × there were no doubt schemes for computation based for example on making specific arrangements of pebbles. Such schemes were somewhat formalized a few
thousand years ago with the invention of the abacus. And by about 200 BC the development of gears had made it possible to create devices (such as the Antikythera device from perhaps around 90 BC) in
which the positions of wheels would correspond to positions of astronomical objects. By about 100 AD Hero had described an odometer-like device that could be driven automatically and could
effectively count in digital form. But it was not until the 1600s that mechanical devices for digital computation appear to have actually been built. Around 1621 Wilhelm Schickard probably built a
machine based on gears for doing simplified multiplications involved in Johannes Kepler’s calculations of the orbit of the Moon. But much more widely known were the machines built in the 1640s by
Blaise Pascal for doing addition on numbers with five or so digits and in the 1670s by Gottfried Leibniz for doing multiplication, division and square roots. At first, these machines were viewed
mainly as curiosities. But as the technology improved, they gradually began to find practical applications. In the mid-1800s, for example, following the ideas of Charles Babbage, so-called difference
engines were used to automatically compute and print tables of values of polynomials. And from the late 1800s until about 1970 mechanical calculators were in very widespread use. (In addition,
starting with Stanley Jevons in 1869, a few machines were constructed for evaluating logic expressions, though they were viewed almost entirely as curiosities.)
In parallel with the development of devices for digital computation, various so-called analog computers were also built that used continuous physical processes to in effect perform computations. In
1876 William Thomson (Kelvin) constructed a so-called harmonic analyzer, in which an assembly of disks were used to sum trigonometric series and thus to predict tides. Kelvin mentioned that a similar
device could be built to solve differential equations. This idea was independently developed by Vannevar Bush, who built the first mechanical so-called differential analyzer in the late 1920s. And in
the 1930s, electrical analog computers began to be produced, and in fact they remained in widespread use for finding approximate solutions to differential equations until the late 1960s.
The types of machines discussed so far all have the feature that they have to be physically rearranged or rewired in order to perform different calculations. But the idea of a programmable machine
already emerged around 1800, first with player pianos, and then with Marie Jacquard’s invention of an automatic loom which used punched cards to determine its weaving patterns. And in the 1830s,
Charles Babbage described what he called an analytical engine, which, if built, would have been able to perform sequences of arithmetic operations under punched card control. Starting at the end of
the 1800s tabulating machines based on punched cards became widely used for commercial and government data processing. Initially, these machines were purely mechanical, but by the 1930s, most were
electromechanical, and had units for carrying out basic arithmetic operations. The Harvard Mark I computer (proposed by Howard Aiken in 1937 and completed in 1944) consisted of many such units hooked
together so as to perform scientific calculations. Following work by John Atanasoff around 1940, electronic machines with similar architectures started to be built. The first large-scale such system
was the ENIAC, built between 1943 and 1946. The focus of the ENIAC was on numerical computation, originally for creating ballistics tables. But in the early 1940s, the British wartime cryptanalysis
group (which included Alan Turing) constructed fairly large electromechanical machines that performed logical, rather than arithmetic, operations.
All the systems mentioned so far had the feature that they performed operations in what was essentially a fixed sequence. But by the late 1940s it had become clear, particularly through the writings
of John von Neumann, that it would be convenient to be able to jump around instead of always having to follow a fixed sequence. And with the idea of storing programs electronically, this became
fairly easy to do, so that by 1950 more than ten stored-program computers had been built in the U.S. and in England. Speed and memory capacity have increased immensely since the 1950s, particularly
as a result of the development of semiconductor chip technology, but in many respects the basic hardware architecture of computers has remained very much the same.
Major changes have, however, occurred in software. In the late 1950s and early 1960s, the main innovation was the development of computer languages such as Fortran, Cobol and Basic. These languages
allowed programs to be specified in a somewhat abstract way, independent of the precise details of the hardware architecture of the computer. But the languages were primarily intended only for
specifying numerical calculations. In the late 1960s and early 1970s, there developed the notion of operating systems - programs whose purpose was to control the resources of a computer - and with
them came languages such as C. And then in the late 1970s and early 1980s, as the cost of computer memory fell, it began to be feasible to manipulate not just purely numerical data, but also data
representing text and later pictures. With the advent of personal computers in the early 1980s, interactive computing became common, and as the resolution of computer displays increased, concepts
such as graphical user interfaces developed. In more recent years continuing increases in speed have made it possible for more and more layers of software to be constructed, and for many operations
previously done with special hardware to be implemented purely in software. | {"url":"https://www.wolframscience.com/reference/notes/1107a/","timestamp":"2024-11-03T01:21:03Z","content_type":"text/html","content_length":"11407","record_id":"<urn:uuid:3a2a2cef-f017-4620-be05-1093dcf44af5>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00896.warc.gz"} |
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patty uses a table saw to cut a piece of plywood from one corner to the opposite corner. if the piece of plywood measured 35 inches by | Question AI
Patty uses a table saw to cut a piece of plywood from one corner to the opposite corner. If the piece of plywood measured 35 inches by 12 inches, how long was the cut that Patty made? inches
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Alexander NelsonElite · Tutor for 8 years
### The length of the cut that Patty made is 37 inches.
## Step 1: Identify the Problem<br />### The problem involves finding the length of the cut made on a rectangular piece of plywood, which is the hypotenuse of a right triangle.<br />## Step 2: Apply
the Pythagorean Theorem<br />### Use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the sides of the rectangle, and \(c\) is the hypotenuse.<br />## Step 3: Substitute the
Given Values<br />### Substitute \(a = 35\) inches and \(b = 12\) inches into the equation: \(35^2 + 12^2 = c^2\).<br />## Step 4: Calculate the Squares<br />### Compute \(35^2 = 1225\) and \(12^2 =
144\).<br />## Step 5: Sum the Squares<br />### Add the squares: \(1225 + 144 = 1369\).<br />## Step 6: Solve for the Hypotenuse<br />### Take the square root of 1369 to find \(c\): \(c = \sqrt{1369}
= 37\).
Step-by-step video
Patty uses a table saw to cut a piece of plywood from one corner to the opposite corner. If the piece of plywood measured 35 inches by 12 inches, how long was the cut that Patty made? inches
All Subjects Homework Helper | {"url":"https://www.questionai.com/questions-tgdxqLUWnj/patty-uses-table-saw-cut-piece-plywood-one-corner","timestamp":"2024-11-09T12:45:30Z","content_type":"text/html","content_length":"79356","record_id":"<urn:uuid:88fd8bae-da50-4fc0-89e6-14dddf7d01a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00536.warc.gz"} |
Basic Python#
This is a pre-requisite lab that can be used to refresh your basic Python skills.
This lab work assumes you are using the supplied hds_code conda virtual environment.
The print() function#
One of the most useful functions in Python is the print() function. It is used to display information to the user. It can be used to present the result of computations, intermediate calculations,
general text and used for debugging.
Once you have opened spyder, we will first look at the console in the bottom right part of the screen. That allows us to type in Python commands and get an immediate response. Let’s use it to learn
how to use print()
Let’s use print to display a message on the console. Type in the following to console:
If we want to include the result of a computation in the output from print we use the following format string:
Try running the code above. The value in the curly brackets ‘{}’ is replaced by the value 2 (i.e. 1+1).
print() can include the output from multiple computations if needed e.g.
print(f'1 + 1 = {1+1} and 2 + 2 = {2+2}')
print(f'1 + 1 = {1+1}!? and 2 + 2 = {2+2}!?')
1 + 1 = 2!? and 2 + 2 = 4!?
• You can also format your output to a specified number of decimal places using {0:.2f} instead of {0}
• The .2f after the : tells python that the number is a floating point and that you would like it shortened to 2dp.
print(f'The number 3.14159 given to 2 decimal places is {3.14159:.2f}')
The number 3.14159 given to 2 decimal places is 3.14
• Similarly if you wanted to show the number to 3 decimal places you would use
print(f'The number 3.14159 given to 3 decimal places is {3.14159:.3f}')
The number 3.14159 given to 3 decimal places is 3.142
Basic mathematics in the console#
We have already seen that python can be used for basic mathematics when learning how to use print()
Let’s use the iPython console as a calculator. Try the following calculations:
• the ** operator raises one number to the power of another.
As before we can mix basic mathematics with print() e.g.
(Note: if you are unfamiliar with the mod operator, it operates like a remainder function. if we type 15 % 4, it will return the remainder after dividing 15 by 4.)
print(f'Addition: {2+2}')
print(f'Substraction: {7-4}')
print(f'Multiplication: {2*5}')
print(f'Division: {10/2}')
print(f'Exponentiation: {3**2}')
print(f'Modulo: {15%4}')
Addition: 4
Substraction: 3
Multiplication: 10
Division: 5.0
Exponentiation: 9
Modulo: 3
The Spyder editor#
Using the console is fine, but has problems. We want to keep our work for future re-use. We want to systematically build up a lot of code. Neither of these is easy within the console. Instead, we can
use the editor on the left half of the screen. This acts like a word-processor, allowing us to type in commands that we can then use or run later.
In the editor, remove any existing text and type in some of the commands you’ve previously used:
1 + 1
1 / 2
print(f"Hello {1+2}")
Save the file in a sensible location on your machine under the name lab1.py.
We then want to run the commands in the file. To do this, choose “Run” from the Run menu, or press the big green play button on the toolbar, or press F5.
The output you see should look like:
1 + 1
1 / 2
print(f'1 + 1 = {1+1}')
Notice that the output only shows the result of the print function. This shows one key difference between files and the console: only output that is explicitly printed appears on the screen.
Exercise 1: Calculate a factorial in the iPython console and editor#
Explicitly compute \(6!\) in the console. i.e. \(6 \times 5 \times 4 \times 3 \times 2 \times 1\). Then do the same in the editor, printing it out with explanatory text.
We want to be able to store data and results of calculations in ways we can re-use. For this we define variables names.
• Variables names can only contain letters, numbers, and underscores. Spaces are not allowed in variable names, so we use underscores instead of spaces. For example, use student_name instead of
student name.
• Variable names should be descriptive, without being too long. For example mc_wheels is better than just wheels, and number_of_wheels_on_a_motorycle.
Here is an example variable called salary
salary = 30_000
print(f'The value of the variable called salary is {salary}')
The value of the variable called salary is 30000
A variable name is a label that’s points to a location in your computer’s memory.
In our example above, think of the variable as a post-it note with salary written on it. This points to the integer 30000 in the computer’s memory. Then, when asked to print the value of salary, it
prints the value with that label on it.
In Python we can move that label to any other variable of any type e.g.
number = 1.2
number = 'One point two'
There are certain rules and conventions for variable names:
• always start with a letter;
• only use lower case Latin letters, or numbers, or underscores;
• in particular, never use spaces or hyphens (which can be interpreted as a new variable or a minus sign respectively).
Mathematical functions also work on variables.
salary = 30000
tax_rate = 0.2
salary_after_tax = salary * (1 - tax_rate)
• Each variable has a data type.
□ We can check the data type of a variable using the built-in function type
□ For example, salary has the data type int (short for integer)
□ and salary_after_tax is of type float (short for floating point number)
salary = 30000
tax_rate = 0.2
salary_after_tax = salary * (1 - tax_rate)
<class 'int'>
<class 'float'>
• Notice that we didn’t need to tell (or declare to) Python the data type of each variable
• This is because Python is a dynamically typed programming language.
• Python infers the type of a variable at runtime (when the code is run)
• The most common primitive data types in python are:
foo = True # bool (Boolean)
bar = False # bool (Boolean)
spam = 3.142 # float (floating point)
eggs = 10000000 # int (integer)
foobar = 'elderberrys' # str (string)
<class 'bool'>
<class 'bool'>
<class 'float'>
<class 'int'>
<class 'str'>
We have already used strings extensively.
Strings are sets of characters. Strings are easier to understand by looking at some examples. Strings are contained by either single, double quotes or triple quotes.
my_string = "This is a double-quoted string"
my_string = 'This is a single-quoted string'
Double quotes lets us make strings that contain quotations
quote = "Jack Reacher said, 'Hope for the best, plan for the worst'"
Jack Reacher said, 'Hope for the best, plan for the worst'
multi_line_string = '''triple quotes let us split strings
over mulitple lines'''
triple quotes let us split strings
over mulitple lines
Exercise 2: Creating and using variables#
A rectangular box has width 2, height 3, and depth 2.
• Create a variable for width, height and depth.
• Compute the volume of the box, assigning that to a fourth variable.
• Print the result along with formattted explanatory text.
Comments in code#
Comments allow you to write in you native language (e.g. English), within your program. In Python, any line that starts with a hash (#) symbol is ignored by the Python interpreter.
# This an inline comment.
print("This line is not a comment, it is code.")
print("Python will ignore comments") #comments can appear after code
This line is not a comment, it is code.
Python will ignore comments
What makes a good comment?#
• It is short and to the point, but a complete thought. Most comments should be written in complete sentences.
• It explains your thinking, so that when you return to the code later you will understand how you were approaching the problem.
• It explains your thinking, so that others who work with your code will understand your overall approach to a problem.
• It explains particularly difficult sections of code in detail.
Functions and import#
We won’t get very far with just basic algebraic operations. We’ll want to perform more complex computations. For that we need python functions.
Python has built-in mathematical functions. For example,
• abs()
• round()
• max()
• min()
• sum()
These functions all act as you would expect, given their names. Calling abs() on a number will return its absolute value. The round() function will round a number to specified number of decimal
points (the default is 0).
Additional functionality can be added in with using various packages such as math or numpy. We will explore numpy in more detail later in the course.
To use these packages you need to first import them into your code.
The math library adds a long list of new mathematical functions to Python. It is documentated here: https://docs.python.org/3.7/library/math.html
print(f'pi: {math.pi}')
print(f"Euler's Constant: {math.e}")
pi: 3.141592653589793
Euler's Constant: 2.718281828459045
Python’s Math module includes some mathematical constants as seen above as well as commonly used mathematical functions.
print(f'Cosine of pi: {math.cos(math.pi)}')
We can import specific constants and functions from python modules
from math import pi, cos
print(f'Cosine of pi: {format(cos(pi))}')
Exercise 3: Use a function to calculate a factorial#
• Import the math library.
• If required use help(math) or https://docs.python.org/3.8/library/math.html to explore the math module
• Use help(math.factorial) to explore how you use the math factorial function.
• use math.factorial() to check your calculation of \(6!\)
Defining Python Functions#
So far we have used functions built-in to python such as print() and math.cos()
You will also need to define your own functions in Python.
Functions. Example 1: adding two numbers together#
def my_add(a, b):
Returns the sum of two numeric values
a: float
first number
b: float
second number
return a + b
The Python keyword to define a function is def. Each function has a name: in this case my_add.
The Parameters to the function are then a comma-separated list between round brackets (). This is in the same format as calling the function, but we are inventing the variable names to refer to the
input within the function. So, however the user calls the function, the first argument that is passed in will be assigned the label a within the function itself.
Finally there is a colon : to end the line. That says that whatever follows is the content, or body, of the function: the lines that will be executed when the function is called. All lines within the
function to be executed must then be indented by four spaces. spyder should start doing this automatically.
The three quotes are documentation for the function: they have no effect. However, any undocumented function is broken. We can see the documentation by using the help function:
Help on function my_add in module __main__:
my_add(a, b)
Returns the sum of two numeric values
a: float
first number
b: float
second number
We then include all the commands with the function that we want to run each time the function is called. Once we have a result that we want to send back to the place that called the function, we
return it: this send back the appropriate value(s).
We can now call our function:
print(my_add(1, 1))
print(my_add(1.2, 4.5))
Functions: Example 2: Implementing a formula#
Suppose that you are promised a payment of £2000 in 5 years time.
Assuming a compound interest rate of 3.5% what is the present value (PV) of this future value (FV)?
• We can calculate this with the forumla: PV = FV / ( 1 + rate)^n
• We do not want type the code for this calculation each time we need it.
• Instead we create a reusable function that we can call to do this for different FV, rate and n.
• The code is below. The function follows the same basic pattern as the simple my_add function.
def pv(future_value, rate, n):
Discount a value at defined rate n time periods into the future.
PV = FV / (1 + r)^n
FV = future value
r = the comparator (interest) rate
n = number of years in the future
future value: float
the value to discount
rate: float
the rate at which to do the discounting
n: float
the number of time periods into the future
return future_value / (1 + rate)**n
#Test case 1
future_value = 2000
rate = 0.035
years = 5
result = pv(future_value, rate, years)
print(f'Using an interest rate of {rate}, a payment of £{future_value:.2f}'
+ f' in {years} years time is worth £{result:.2f} today')
#Test case 2
future_value = 350
rate = 0.01
years = 10
result = pv(future_value, rate, years)
print(f'Using an interest rate of {rate}, a payment of £{future_value:.2f}'
+ f' in {years} years time is worth £{result:.2f} today')
Using an interest rate of 0.035, a payment of £2000.00 in 5 years time is worth £1683.95 today
Using an interest rate of 0.01, a payment of £350.00 in 10 years time is worth £316.85 today
Exercise 4: Write a function to convert fahrenheit to celsius#
Open Spyder and use the code editor to do the following:
Define a function convert_fahrenheit_to_celsius that converts degrees fahrenheit to degrees celsius. The function should have a keyword argument for temperature in degrees fahrenheit and return a
numeric value for temperature in degrees celsius.
Store the answer in a variable and then print the answer to the user. Answers should be shown to 2 decimal places.
Conversion formula:
deg_celsius = (deg_fahrenheit - 32) / (9.0 / 5.0)
Test data
1. Fahrenheit = 20; Celsius = -6.67
2. Fahrenheit = 100; Celsius = 37.78
Exercise 5: Write a function to calculate velocity#
• Define a function that calculates and returns velocity (metres per second).
• The function should accept two parameters: distance travelled (metres) and time (seconds)
velocity (m/s) = metres travelled (m) / time taken (s)
Test data
1. distance travelled = 10m; time taken = 5s. (Velocity = 2.00 m/s)
2. distance travelled = 100m; time taken = 0.12s. (Velocity = 833.33m/s 2dp)
Creating your own Python modules and importing functions#
In the same way we imported functions from math we can import functions from our own python modules
• Open py_finance.py and test_finance.py
• test_finance.py imports functions from the py_finance module
• Watch the Youtube video that explains how they work:
Variables are useful, but in nearly all real programming problems we need to store and manipulate lots of data in memory.
For example, if we were developing a music streaming service we might need to hold the list of song’s on an album or an artists back catalog.
Or, if we were developing a software to manage the geographic routing of a fleet of delivery vehicles we might need to hold a matrix of travel distances between postcodes.
A Python List is a simple and flexible way to store lots of variables (of any type of data).
foo = [0, 1, 2, 3]
The square brackets [] say that what follows will be a list: a collection of objects. The commas separate the different objects contained within the list.
In Python, a list can hold anything. For example:
bar = [0, 1.2, "hello", [3, 4]]
[0, 1.2, 'hello', [3, 4]]
This list holds an integer, a real number (or at least a floating point number), a string, and another list.
We can find the length of a list using len:
To access individual elements of a list, use square brackets again. The elements are ordered left-to-right, and the first element has number 0:
Note If we try to access an element that isn’t in the list we get an error.
We can assign the value of elements of a list in the same way as any variable:
The number in brackets is called the index of the item. Because lists start at zero, the index of an item is always one less than its position in the list. So to get the second item in the list, we
need to use an index of 1.
We can work with multiple elements of a list at once using slicing notation:
[0, 10, 2, 3]
[0, 10]
[0, 10]
[10, 2, 3]
[0, 2]
[0, 2]
The notation [start:end:step] means to return the entries from the start, up to but not including the end, in steps of length step. If the start is not included (e.g. [:2]) it defaults to the start,
i.e. 0. If the end is not included (e.g. [1:]) it defaults to the end (i.e., len(...)). If the step is not included it defaults to 1.
To get the last item in a list, no matter how long the list is, you can use an index of -1. This syntax also works for the second to last item, the third to last, and so forth. You can’t use a
negative number larger than the length of the list, however.
If you want to find out the position of an element in a list, you can use the index() function. This method returns a ValueError if the requested item is not in the list.
You can test whether an item is in a list using the “in” keyword. This will become more useful after learning how to use if-else statements.
We can add an item to a list using the append() method. This method adds the new item to the end of the list.
We can also insert items anywhere we want in a list, using the insert() function. We specify the position we want the item to have, and everything from that point on is shifted one position to the
right. In other words, the index of every item after the new item is increased by one.
We can remove an item from a list using the del statement. You need to specify the index you wish to remove.
Exercise 6: Marvel Comics#
You are given a list of comics:
comics = ['Iron-man', 'Captain America', 'Spider-man', 'Thor', 'Deadpool']
• slice and then print the first and second list items
• slices and then print the second to fourth list items
• slice and then print the fourth and fifth list items
• append “Doctor Strange” to the list. Print the updated list
• insert “Headpool” before “Deadpool” in the list. Print the updated list
• delete “Iron-man”. Print the updated list
Lab 1: Self learning challenge#
Each laboratory will challenge you to do something that we haven’t taught you before.
No course can teach you everything you need for all programming problems. Being a competent Python programmer means that you need to learn how to find solutions to problems yourself. These challenges
are designed to help you begin to use internet resources in order to solve your problem.
Before you try the challenges it it worth watching our video on using StackOverflow:
Challenge 1:#
You are given a unsorted List of integers.
Find a command to sort the list into ascending order i.e.
Once you have tried yourself. Watch our example strategy:
unsorted = [5, 7, 6, 4, 3, 2, 1]
Challenge 2:#
Sometimes a Python function needs to accept a variable number of arguments. For example, the built-in function max()
max(1, 2, 3)
max(1, 2, 3, 4, 5, 6, 7, 8)
Write a function that accepts a variable number of integer arguments and returns the number of arguments e.g.
result = number_of_arguments(1, 2, 3) # result = 3
result = number_of_arguments(1, 2, 3, 4, 5) # result = 5
Try to solve this yourself. Then watch our example strategy:
Challenge 3:#
We have seen multiple Python functions that have required parameters.
For example, the function add_two(a, b) required the user to provide two parameters a and b
It is also possible in Python to have default values for the parameters.
Write a function called super_hero_name that accepts two parameters of type string: firstname and super_surname.
The parameter super_surname should have a default value of ‘the spider’. The function should concatonate the names and return the resulting superhero name.
super_hero_name("tom") #returns "tom the spider"
super_hero_name("tom", "ant-man") #returns "tom ant man"
Try this yourself first. Then watch our approach:
def super_hero_name(firstname, super_surname):
return firstname + ' ' + super_surname
super_hero_name("tom", "the spider")
super_hero_name("tom", "ant-man")
Optional Learning Material#
Download and open:
• string_manipulation.py for detailed examples of how to manipulate and format Python strings. | {"url":"https://www.pythonhealthdatascience.com/content/appendix/labs/01_basics.html","timestamp":"2024-11-05T07:27:16Z","content_type":"text/html","content_length":"119025","record_id":"<urn:uuid:ccd8a4da-6979-4c91-85ae-ef455605e257>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00664.warc.gz"} |
Collection of Solved Problems
Specific Heat Capacity of Gas
Task number: 3947
Determine specific heat capacities c[V] and c[p] of unknown gas provided that at temperature of 293 K and pressure of 100 kPa its density is 1.27 kg m^−3 and Poisson's constant of the gas is κ = 1.4.
• Hint 1
Use Meyer's relation and realize how you can transform it to a relation between specific heat capacities and not between molar heat capacities at constant pressure and volume.
• Hint 2
Poisson's constant κ is defined by
\[\kappa = \frac{c_p}{c_V}= \frac{C_p}{C_V},\]
where c[p], respectively C[p] is specific, respectively molar heat capacity at constant pressure and c[V], respectively C[V] is specific, respectively molar heat capacity at constant volume.
• Hint 3
To determine molar mass M[m] of the gas use the equation of state for ideal gas.
• Analysis
We start the solution from Meyer's relation. We divide it by molar mass of the gas and thus adjust it to the relation between specific heat capacities at constant pressure and volume.
The unknown molar mass can be determined from the equation of state for ideal gas.
Specific heat capacity at constant pressure is determined as the product of Poisson's constant and specific heat capacity at constant volume.
• Given Values
T = 293 K gas temperature
p = 100 kPa = 1.00·10^5 Pa gas pressure
ρ = 1.27 kg·m^−3 gas density
κ = 1.4 Poisson's constant of the gas
c[V] = ? specific heat capacity of the gas at constant volume
c[p] = ? specific heat capacity of the gas at constant pressure
Table values:
R = 8.31 JK^−1mol^−1 molar gas constant
• Solution
We start the calculation with Meyer's relation C[p] = C[V] + R, that relates molar heat capacities at constant volume C[V] and constant pressure C[p].
In this task, however, we need to determine specific heat capacities, not molar heat capacities. This is why we need to divide Meyer's relation by molar mass of the gas M[m]
\[\frac{C_p}{M_m} = \frac{C_V}{M_m} + \frac{R}{M_m},\]
resulting in the relation
\[c_p = c_V + \frac{R}{M_m}.\]
Now we use the given Poisson's constant κ defined by the relation
\[\kappa = \frac{C_p}{C_V} = \frac{c_p}{c_V}.\]
We determine specific heat capacity at constant pressure from this relation
\[c_p=\kappa c_V,\]
and substitute it into the above mentioned relationship
\[\kappa c_V=c_V+\frac{R}{M_m}.\]
The unknown specific heat capacity at constant pressure c[V] is then given by
Now we need to determine the unknown molar mass of the gas M[m]. We use the equation of state for ideal gas
\[pV=\frac{m}{M_m}RT. \]
We express molar mass
and the ratio \(\frac{m}{V}\) substitute by the given density ρ of the gas:
\[M_m=\frac{\rho RT}{p}.\]
After substituting into the relation for specific heat capacity at constant volume we then obtain:
\[c_V=\frac{R}{M_m(\kappa-1)}=\frac{R}{\frac{\rho RT}{p}(\kappa-1)}=\frac{p}{\rho T(\kappa-1)}.\]
The specific heat capacity at constant pressure c[p] then can be directly determined from the equation
\[c_p = \kappa c_V = \frac{\kappa p}{\rho T(\kappa-1)}. \]
• Numerical Solution
\[c_V=\frac{p}{\rho T(\kappa-1)}\] \[c_V= \frac{100\cdot{10^3}}{1.27\cdot{293}\cdot (1.4-1)}\,\mathrm{J\,kg^{-1}K^{-1}}\dot{=}672\,\mathrm{J\,kg^{-1}K^{-1}}\] \[c_p = \kappa c_V =1.4\cdot{671.8}
• Answer
Specific heat capacity of unknown gas at constant volume is approximately 672 Jkg^−1K^−1.
Its specific heat capacity at constant pressure is then approximately 941 Jkg^−1K^−1. | {"url":"https://physicstasks.eu/3947/specific-heat-capacity-of-gas","timestamp":"2024-11-11T15:52:28Z","content_type":"text/html","content_length":"31302","record_id":"<urn:uuid:b501e80c-ab6f-4121-ac47-a3de6c9a4012>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00566.warc.gz"} |
The task of GAN is to generate features $X$ from some noise $\xi$ and class labels $Y$,
$$\xi, Y \to X.$$
Many different GANs are proposed. Vanilla GAN has a simple structure with a single discriminator and a single generator. It uses the minmax game setup. However, it is not stable to use minmax game to
train a GAN model. WassersteinGAN was proposed to solve the stability problem during training^1. More advanced GANs like BiGAN and ALI have more complex structures.
Vanilla GAN
Minmax Game
Suppose we have two players $G$ and $D$, and a utility $v(D, G)$, a minmax game is maximizing the utility $v(D, G)$ for the worst case of $G=\hat G$ that minimizes $v$ then we have to find $D=\hat D$
that maximizes $v$, i.e.,
$$\underset{G, D}{\operatorname{minmax}} v(D, G).$$
The loss for vanilla GAN is the minmax loss
$$ \underset{G, D}{\operatorname{minmax}} \mathbb E_{x\sim P_{data}} \left[ \ln D(x) \right] + \mathbb E_{z\sim p_z} \left[ \ln ( 1- D(G(z)) ) \right]. $$
Illustration of GAN
BiGAN uses one generator, one encoder and one discriminator^2.
Illustration of BiGAN
Planted: by L Ma;
L Ma (2021). 'GAN', Datumorphism, 08 April. Available at: https://datumorphism.leima.is/wiki/machine-learning/adversarial-models/gan/. | {"url":"https://datumorphism.leima.is/wiki/machine-learning/adversarial-models/gan/?ref=footer","timestamp":"2024-11-12T02:41:24Z","content_type":"text/html","content_length":"114287","record_id":"<urn:uuid:0b11dd19-6b59-49ff-89e5-dfebc256b660>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00751.warc.gz"} |
BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Sabre//Sabre VObject 4.5.5//EN CALSCALE:GREGORIAN X-WR-CALNAME:Analysis BEGIN:VTIMEZONE TZID:Europe/Zurich X-LIC-LOCATION:Europe/Zurich TZURL:http://tzurl.org/
zoneinfo/Europe/Zurich BEGIN:DAYLIGHT TZOFFSETFROM:+0100 TZOFFSETTO:+0200 TZNAME:CEST DTSTART:19810329T020000 RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU END:DAYLIGHT BEGIN:STANDARD TZOFFSETFROM:+0200
TZOFFSETTO:+0100 TZNAME:CET DTSTART:19961027T030000 RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU END:STANDARD END:VTIMEZONE BEGIN:VEVENT UID:news1714@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20241009T142342 DTSTART;TZID=Europe/Zurich:20241016T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Umberto Pappalettera (Un iversität Bielefeld) DESCRIPTION:In this talk I will
present a new “anomalous regularisation ” result for solutions of the stochastic transport equation \\partial_t \\rho + \\circ \\partial_t W \\cdot \\nabla \\rho = 0\, where W is a Gauss ian\,
homogeneous\, isotropic noise with \\alpha-H\\”older space regular ity and compressibility ratio \\wp < \\frac{d}{4\\alpha^2}. The proof is o btained by studying the local behaviour around the origin
of solutions to a degenerate parabolic PDE in non-divergence form\, which is of independen t interest. Based on joint work with Theodore Drivas and Lucio Galeati. X-ALT-DESC:
In this talk I will present a new “anomalous regularisation ” result for solutions of the stochastic transport equation \\partial_t \\rho + \\circ \\partial_t W \\cdot \\nabla \\rho = 0\, where W is
a Gauss ian\, homogeneous\, isotropic noise with \\alpha-H\\”older space regular ity and compressibility ratio \\wp <\; \\frac{d}{4\\alpha^2}. The proof is obtained by studying the local behaviour
around the origin of solutions to a degenerate parabolic PDE in non-divergence form\, which is of indepe ndent interest. Based on joint work with Theodore Drivas and Lucio Galeati .
DTEND;TZID=Europe/Zurich:20241016T160000 END:VEVENT BEGIN:VEVENT UID:news1696@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240708T112209 DTSTART;TZID=Europe/Zurich:20240712T151500 SUMMARY:An afternoon
of analysis talks: Helena Nussenzveig Lopes (Universid ade Federal do Rio de Janeiro) DESCRIPTION:We say inviscid dissipation occurs when a vanishing viscosity l imit does not satisfy energy balance.
A closely related phenomenon is anom alous dissipation\, where\, in the limit of vanishing viscosity\, the tota l dissipation does not vanish. In this talk we will discuss recent results on avoiding
these phenomena in 2D incompressible flows\, with and without forcing. X-ALT-DESC:
We say inviscid dissipation occurs when a vanishing viscosity limit does not satisfy energy balance. A closely related phenomenon is an omalous dissipation\, where\, in the limit of vanishing
viscosity\, the to tal dissipation does not vanish. In this talk we will discuss recent resul ts on avoiding these phenomena in 2D incompressible flows\, with and witho ut forcing.
DTEND;TZID=Europe/Zurich:20240712T160000 END:VEVENT BEGIN:VEVENT UID:news1695@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240702T162928 DTSTART;TZID=Europe/Zurich:20240712T141500 SUMMARY:An afternoon
of analysis talks: Alexander Kiselev (Duke University) DESCRIPTION:There exist many regularization mechanisms in nonlinear PDE tha t help make solutions more regular or prevent formation of
singularity: di ffusion\, dispersion\, damping. A relatively less understood regularizatio n mechanism is transport. There is evidence that in the fundamental PDE of fluid mechanics such as Euler or
Navier-Stokes\, transport can play a reg ularizing role. In this talk\, I will discuss another instance where this phenomenon appears: the Patlak-Keler-Segel equation of chemotaxis. Chemota ctic blow
up in the context of the Patlak-Keller-Segel equation is an exte nsively studied phenomenon. In recent years\, it has been shown that the p resence of a given fluid advection can arrest singularity
formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong. This talk will focus on the case wh en the fluid advection is active:
the Patlak-Keller-Segel equation coupled with fluid that obeys Darcy's law for incompressible porous media flow vi a gravity. Surprisingly\, in this context\, in contrast with the passive a dvection
\, active fluid is capable of suppressing chemotactic blow up at a rbitrary small coupling strength: namely\, the system always has globally regular solutions. The talk is based on work joint with
Zhongtian Hu and Y ao Yao. X-ALT-DESC:
There exist many regularization mechanisms in nonlinear PDE t hat help make solutions more regular or prevent formation of singularity: diffusion\, dispersion\, damping. A relatively less understood
regularizat ion mechanism is transport. There is evidence that in the fundamental PDE of fluid mechanics such as Euler or Navier-Stokes\, transport can play a r egularizing role. In this talk\, I
will discuss another instance where thi s phenomenon appears: the Patlak-Keler-Segel equation of chemotaxis. Chemo tactic blow up in the context of the Patlak-Keller-Segel equation is an ex tensively
studied phenomenon. In recent years\, it has been shown that the presence of a given fluid advection can arrest singularity formation give n that the fluid flow possesses mixing or diffusion
enhancing properties a nd its amplitude is sufficiently strong. This talk will focus on the case when the fluid advection is active: the Patlak-Keller-Segel equation coupl ed with fluid that obeys
Darcy's law for incompressible porous media flow via gravity. Surprisingly\, in this context\, in contrast with the passive advection\, active fluid is capable of suppressing chemotactic blow up at
arbitrary small coupling strength: namely\, the system always has globall y regular solutions. The talk is based on work joint with Zhongtian Hu and Yao Yao.
DTEND;TZID=Europe/Zurich:20240712T150000 END:VEVENT BEGIN:VEVENT UID:news1662@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240502T182134 DTSTART;TZID=Europe/Zurich:20240522T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Christof Sparber (Univer sity of Illinois at Chicago) DESCRIPTION:We consider Schrödinger equations with competing nonlinearitie s in spatial dimensions up to three
\, for which global existence holds (i. e. no finite-time blow-up). A typical example is the case of the (focusing -defocusing) cubic-quintic NLS. We recall the notions of energy minimizin g versus
action minimizing ground states and show that\, in general\, the two must be considered as nonequivalent. The question of long-time behavio r of solutions\, in particular the problem of ground-state
(in-)stability will be discussed using analytical results and numerical simulations. X-ALT-DESC:
We consider Schrödinger equations with competing nonlinearit ies in spatial dimensions up to three\, for which global existence holds ( i.e. no finite-time blow-up). A typical example is the case of
the (focusi ng-defocusing) cubic-quintic NLS. \;We recall the notions of energy mi nimizing versus action minimizing ground states and show that\, in general \, the two must be considered as
nonequivalent. The question of long-time behavior of solutions\, in particular the problem of ground-state (in-)sta bility will be discussed using analytical results and numerical simulation s.
DTEND;TZID=Europe/Zurich:20240522T151500 END:VEVENT BEGIN:VEVENT UID:news1688@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240508T175114 DTSTART;TZID=Europe/Zurich:20240515T151500 SUMMARY:Seminar
Analysis and Mathematical Physics: Min Jun Jo (Duke Univers ity) DESCRIPTION:We prove the instantaneous cusp formation from a single corner of the vortex patch solutions. This positively settles the
conjecture give n by Cohen-Danchin in Multiscale approximation of vortex patches\, SIAM J . Appl. Math. 60 (2000)\, no. 2\, 477–502. X-ALT-DESC:
We prove the instantaneous cusp formation from a single corne r of the vortex patch solutions. This positively settles the conjecture gi ven by Cohen-Danchin in \;Multiscale approximation of
vortex patch es\, SIAM J. Appl. Math. 60 (2000)\, no. 2\, 477–502.
DTEND;TZID=Europe/Zurich:20240515T160000 END:VEVENT BEGIN:VEVENT UID:news1672@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240424T134606 DTSTART;TZID=Europe/Zurich:20240515T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Riccardo Tione (MPI Leip zig) DESCRIPTION:This talk concerns critical points $u$ of polyconvex energies o f the form $f(X) = g(det(X))$\, where $g$ is (uniformly)
convex. It is not hard to see that\, if $u$ is smooth\, then $\\det(Du)$ is constant. I wil l show that the same result holds for Lipschitz critical points $u$ in the plane. I will also discuss how
to obtain rigidity for approximate solutio ns. This is a joint work with A. Guerra. X-ALT-DESC:
This talk concerns critical points $u$ of polyconvex energies of the form $f(X) = g(det(X))$\, where $g$ is (uniformly) convex. It is n ot hard to see that\, if $u$ is smooth\, then $\\det(Du)$ is
constant. I w ill show that the same result holds for Lipschitz critical points $u$ in t he plane. I will also discuss how to obtain rigidity for approximate solut ions. This is a joint work with A.
DTEND;TZID=Europe/Zurich:20240515T150000 END:VEVENT BEGIN:VEVENT UID:news1689@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240503T101127 DTSTART;TZID=Europe/Zurich:20240508T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Christoph Kehle (ETH Zü rich) DESCRIPTION:Extremal black holes are special types of black holes which hav e exactly zero temperature. I will present a proof that
extremal black hol es form in finite time in gravitational collapse of charged matter. In par ticular\, this construction provides a definitive disproof of the “third law” of black hole
thermodynamics. I will also present a recent result which shows that extremal black holes arise on the black hole formation th reshold in the moduli space of gravitational collapse. This gives rise
to a new conjectural picture of “extremal critical collapse.” This is joi nt work with Ryan Unger (Princeton). X-ALT-DESC:
Extremal black holes are special types of black holes which h ave exactly zero temperature. I will present a proof that extremal black h oles form in finite time in gravitational collapse of charged
matter. In p articular\, this construction provides a definitive disproof of the “thi rd law” of black hole thermodynamics. I will also present a recent resul t which shows that extremal black holes
arise on the black hole formation threshold in the moduli space of gravitational collapse. This gives rise t o a new conjectural picture of “extremal critical collapse.” This is j oint work with Ryan
Unger (Princeton).
DTEND;TZID=Europe/Zurich:20240508T160000 END:VEVENT BEGIN:VEVENT UID:news1666@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240422T090328 DTSTART;TZID=Europe/Zurich:20240424T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Louise Gassot (IRMAR\, R ennes) DESCRIPTION:We focus on the Benjamin-Ono equation on the line with a small dispersion parameter. The goal of this talk is to
precisely describe the s olution at all times when the dispersion parameter is small enough. This s olution may exhibit locally rapid oscillations\, which are a manifestation of a dispersive shock.
The description involves the multivalued solution of the underlying Burgers equation\, obtained by using the method of chara cteristics. This work is in collaboration with Elliot Blackstone\, Patrick
Gérard\, and Peter Miller. X-ALT-DESC:
We focus on the Benjamin-Ono equation on the line with a smal l dispersion parameter. The goal of this talk is to precisely describe the solution at all times when the dispersion parameter is small
enough. This solution may exhibit locally rapid oscillations\, which are a manifestati on of a dispersive shock. The description involves the multivalued solutio n of the underlying Burgers equation
\, obtained by using the method of cha racteristics. This work is in collaboration with Elliot Blackstone\, Patri ck Gérard\, and Peter Miller.
DTEND;TZID=Europe/Zurich:20240424T151500 END:VEVENT BEGIN:VEVENT UID:news1638@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240328T104112 DTSTART;TZID=Europe/Zurich:20240417T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Anuj Kumar (UC Berkeley) DESCRIPTION:We construct nonunique solutions of the transport equation in t he class $L^\\infty$ in time and $L^r$ in space for divergence
free Sobole v vector fields $W^{1\, p}$. We achieve this by introducing two novel idea s: (1) In the construction\, we interweave the scaled copies of the vector field itself. (2) Asynchronous
translation of cubes\, which makes the con struction heterogeneous in space. These new ideas allow us to prove nonuni queness in the range of exponents beyond what is available using the metho d of
convex integration and sharply matchwith the range of uniqueness of s olutions from Bruè\, Colombo\, De Lellis ’21. X-ALT-DESC:
We construct nonunique solutions of the transport equation in the class $L^\\infty$ in time and $L^r$ in space for divergence free Sobo lev vector fields $W^{1\, p}$. We achieve this by introducing
two novel id eas: (1) In the construction\, we interweave the scaled copies of the vect or field itself. (2) Asynchronous translation of cubes\, which makes the c onstruction heterogeneous in space.
These new ideas allow us to prove nonu niqueness in the range of exponents beyond what is available using the met hod of convex integration and sharply matchwith the range of uniqueness of solutions
from Bruè\, Colombo\, De Lellis ’21. \;
DTEND;TZID=Europe/Zurich:20240417T160000 END:VEVENT BEGIN:VEVENT UID:news1619@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240328T104000 DTSTART;TZID=Europe/Zurich:20240410T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Roman Shvydkoy (Universi ty of Illinois at Chicago) DESCRIPTION:The classical Kolmogorov-41 theory of turbulence is based on a set of pivotal assumptions on
scaling and energy dissipation for solutio ns satisfying incompressible fluid models. In the early 80's experimental evidence emerged that pointed to departure from the K41 predictions\, whic h was
attributed to the phenomenon of statistical intermittency. In this talk we give an overview of the classical results in the subject\, relati onship of intermittency to the problem of global
well-posedness of the 3D Navier-Stokes system\, and discuss a new approach developed jointly with A . Cheskidov on how to measure and study intermittency from a rigorous pers pective. At the center
of our discussion will be a new interpretation of an intermittent signal described by volumetric properties of the filter ed field. It provides\, in particular\, a systematic approach to the Fr
isch-Parisi multifractal formalism\, and recasts intermittency from the po int of view of information theory. X-ALT-DESC:
The classical Kolmogorov-41 theory of turbulence is based on a set of \; pivotal assumptions on scaling and energy dissipation for solutions satisfying incompressible fluid models. In the early
80's experi mental evidence emerged that pointed to departure from the K41 predictions \, which was attributed to the phenomenon of statistical intermittency.&nb sp\; In this talk we give an overview
of the classical results in the subj ect\, relationship of intermittency to the problem of global well-posednes s of the 3D Navier-Stokes system\, and discuss a new approach developed jo intly with
A. Cheskidov on how to measure and study intermittency from a r igorous perspective. \; At the center of our discussion will be a new interpretation of an intermittent signal \;described 
\;by volumetr ic properties of the filtered \;field. \; It provides\, in particu lar\, a systematic approach to the Frisch-Parisi multifractal formalism\, and recasts intermittency from the
point of view of information theory.&nb sp\;
DTEND;TZID=Europe/Zurich:20240410T160000 END:VEVENT BEGIN:VEVENT UID:news1648@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240323T092733 DTSTART;TZID=Europe/Zurich:20240403T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Vikram Giri (ETH Zürich ) DESCRIPTION:A common issue in convex integration methods (going back to Nas h) is the presence of unwanted high-high frequency
interactions. These iss ues can prevent the methods from producing solutions with the optimum "Ons ager" regularity to a given system of PDEs. We will discuss these issues i n the setting of the 2D
Euler equations and then discuss a linear Newton i teration designed to get rid of these unwanted interactions in this settin g. We will conclude by discussing applications to other PDEs. This is
base d on joint works with Răzvan-Octavian Radu and Mimi Dai. X-ALT-DESC:
A common issue in convex integration methods (going back to N ash) is the presence of unwanted high-high frequency interactions. These i ssues can prevent the methods from producing solutions with
the optimum "O nsager" regularity to a given system of PDEs. We will discuss these issues in the setting of the 2D Euler equations and then discuss a linear Newton iteration designed to get rid of
these unwanted interactions in this sett ing. We will conclude by discussing applications to other PDEs. This is ba sed on joint works with Răzvan-Octavian Radu and Mimi Dai.
DTEND;TZID=Europe/Zurich:20240403T160000 END:VEVENT BEGIN:VEVENT UID:news1645@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240325T134941 DTSTART;TZID=Europe/Zurich:20240327T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Marcello Porta (SISSA) DESCRIPTION:I will discuss the dynamics of many-body Fermi gases\, in the m ean-field regime. I will consider a class of initial data which
are close enough to quasi-free states\, with a non-zero pairing matrix. Assuming a suitable semiclassical structure for the initial datum\, expected to hold at low enough energy and that we can
establish for translation-invariant states\, I will present a theorem that shows that the many-body evolution of the system can be well approximated by the Hartree-Fock-Bogoliubov equ ation\, a
non-linear effective evolution equation describing the coupled dynamics of the reduced one-particle density matrix and of the pairing ma trix. Joint work with Stefano Marcantoni (Nice) and Julien
Sabin (Rennes). X-ALT-DESC:
I will discuss the dynamics of many-body Fermi gases\, in the mean-field regime. I will consider a class of initial data which are  \;close enough to quasi-free states\, with a non-zero pairing
matrix. Assu ming a suitable semiclassical structure for the initial datum\, \;expe cted to hold at low enough energy and that we can establish for translatio n-invariant states\, I will present
a theorem that shows that \;the ma ny-body evolution of the system can be well approximated by the Hartree-Fo ck-Bogoliubov equation\, a non-linear effective \;evolution equation d escribing
the coupled dynamics of the reduced one-particle density matrix and of \;the pairing matrix. Joint work with Stefano Marcantoni (Nice) and Julien Sabin (Rennes).
DTEND;TZID=Europe/Zurich:20240327T153000 END:VEVENT BEGIN:VEVENT UID:news1641@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240205T090917 DTSTART;TZID=Europe/Zurich:20240320T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Norbert J. Mauser (WPI and MMM c/o Univ. Wien) DESCRIPTION:The Pauli-Poisswell equation models fast moving charges in\\r\\ nsemiclassical semi-relativistic quantum
dynamics. It is at the\\r\\ncente r of a hierarchy of models from the Dirac-Maxwell equation to\\r\\nthe Eul er-Poisson equation that are linked by asymptotic analysis of\\r\\nsmall p arameters such
as Planck constant or inverse speed of light.\\r\\nWe discu ss the models and their application in plasma and\\r\\naccelerator physics as well as the many mathematical problems they\\r\\npose.
The Pauli-Poisswell equation models fast moving charges in
semiclassical semi-relativistic quantum dynamics. It is at the
center of a hierarchy of models from the Dirac-Maxwell equation to
\ n
the Euler-Poisson equation that are linked by asymptotic analysis of p>\n
small parameters such as Planck constant or inverse speed of light.
We discuss the models and their application in plasma and
accelerator physics as well as the many mathematical problems they
\n< p>pose. DTEND;TZID=Europe/Zurich:20240320T161500 END:VEVENT BEGIN:VEVENT UID:news1639@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240201T155918 DTSTART;TZID=Europe/Zurich:20240306T141500
SUMMARY:Seminar Analysis and Mathematical Physics: Klaus Widmayer (Universi tät Zürich) DESCRIPTION:While "Landau damping" is regarded as an important effect in th e dynamics of hot\, collisionless
plasmas\, its mathematical understanding is still in its infancy. This talk presents a recent nonlinear stability result in this context. Starting with a discussion of stabilizing mechanis ms in the
linearized Vlasov-Poisson equations near a class of homogeneous equilibria on R^3\, we will see how both oscillatory and damping effects a rise\, and sketch how these mechanisms imply a nonlinear
stability result in the specific setting of the Poisson equilibrium. This is based on joint work with A. Ionescu\, B. Pausader and X. Wang. X-ALT-DESC:
While "Landau damping" is regarded as an important effect in the dynamics of hot\, collisionless plasmas\, its mathematical understandi ng is still in its infancy. This talk presents a recent
nonlinear stabilit y result in this context. Starting with a discussion of stabilizing mechan isms in the linearized Vlasov-Poisson equations near a class of homogeneou s equilibria on R^3\, we will
see how both oscillatory and damping effects arise\, and sketch how these mechanisms imply a nonlinear stability resul t in the specific setting of the Poisson equilibrium. This is based on joi nt
work with A. Ionescu\, B. Pausader and X. Wang.
DTEND;TZID=Europe/Zurich:20240306T160000 END:VEVENT BEGIN:VEVENT UID:news1611@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231107T040532 DTSTART;TZID=Europe/Zurich:20231213T151500 SUMMARY:Seminar
Analysis and Mathematical Physics: Theodore Drivas (Stony B rook University) DESCRIPTION:We will discuss aspects of the global picture of 2D fluids. Ste ady states\, deterioration of regularity for
time dependent solutions as w ell as for the Lagrangian flowmap\, as well as conjectural pictures about the weak-* attractor and generic behavior by Shnirelman and Sverak. \\r\\ nNotice the special
time! X-ALT-DESC:
We will discuss aspects of the global picture of 2D fluids. S teady states\, deterioration of regularity for time dependent solutions as well as for the Lagrangian flowmap\, as well as conjectural
pictures abou t the weak-* attractor and generic behavior by Shnirelman and Sverak.  \;
Notice the special time!
DTEND;TZID=Europe/Zurich:20231213T160000 END:VEVENT BEGIN:VEVENT UID:news1588@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231201T140131 DTSTART;TZID=Europe/Zurich:20231206T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: David Meyer (Universitä t Münster) DESCRIPTION:A vortex ring is a solution of the axisymmetric Euler equations consisting of some torus of concentrated vorticity.
Motivated by the appe arance of Vortex rings as bubble rings\, we study vortex rings with surfac e tension at the interface. We show the existence of traveling wave soluti ons. In particular\, our
construction also justifies the existence of so-c alled hollow vortex rings\, where the vorticity is a measure concentrated on the interface. X-ALT-DESC:
A vortex ring is a solution of the axisymmetric Euler equatio ns consisting of some torus of concentrated vorticity. Motivated by the ap pearance of Vortex rings as bubble rings\, we study vortex
rings with surf ace tension at the interface. We show the existence of traveling wave solu tions. In particular\, our construction also justifies the existence of so -called hollow vortex rings\,
where the vorticity is a measure concentrate d on the interface.
DTEND;TZID=Europe/Zurich:20231206T160000 END:VEVENT BEGIN:VEVENT UID:news1580@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231114T180149 DTSTART;TZID=Europe/Zurich:20231129T134500 SUMMARY:Seminar
Analysis and Mathematical Physics: Sergio Simonella (Univer sity of Roma La Sapienza) DESCRIPTION:Boltzmann equation\, hard sphere systems and their small and la rge deviations X-ALT-DESC:Boltzmann
equation\, hard sphere systems and their small and lar ge deviations DTEND;TZID=Europe/Zurich:20230929T163000 END:VEVENT BEGIN:VEVENT UID:news1614@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20231114T180522 DTSTART;TZID=Europe/Zurich:20231129T134500 SUMMARY:Seminar Analysis and Mathematical Physics: Sergio Simonella (Univer sity of Roma La Sapienza) DESCRIPTION:https://
nccr-swissmap.ch/news-and-events/news/next-kinetic-theo ry-seminar-29th-nov-prof-sergio-simonella-university-roma-la-sapienza X-ALT-DESC:
https://nccr-swissmap.ch/news-and-events/news/next-kinetic-th eory-seminar-29th-nov-prof-sergio-simonella-university-roma-la-sapienza
DTEND;TZID=Europe/Zurich:20231129T163000 END:VEVENT BEGIN:VEVENT UID:news1577@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231029T223020 DTSTART;TZID=Europe/Zurich:20231122T140000 SUMMARY:Seminar
Analysis and Mathematical Physics: Jinyeop Lee (LMU Munich) DESCRIPTION:The study of the Schrödinger equation in dimension one with a nonlinear point interaction has been the focus of research over
the past f ew decades. In this seminar\, we talk about a work on deriving this partia l differential equation as the effective dynamics of N identical bosons in one dimension. We assume introducing a
tiny impurity located at the origi n and considering that the interaction between every pair of bosons is med iated by the impurity through a three-body interaction. Moreover\, by assu ming
short-range scaling and choosing an initial fully condensed state\, w e prove convergence of one-particle density operators in the trace-class t opology. This is the first derivation of the so-called
nonlinear delta mod el. This research is a collaborative work with Prof. Riccardo Adami. X-ALT-DESC:
The study of the Schrödinger equation in dimension one with a nonlinear point interaction has been the focus of research over the past few decades. In this seminar\, we talk about a work on deriving
this part ial differential equation as the effective dynamics of N identical bosons in one dimension.
We assume introducing a tiny impurity located at t he origin and considering that the interaction between every pair of boson s is mediated by the impurity through a three-body interaction. Moreover
\, by assuming short-range scaling and choosing an initial fully condensed s tate\, we prove convergence of one-particle density operators in the trace -class topology. This is the first derivation
of the so-called nonlinear d elta model. This research is a collaborative work with Prof. Riccardo Adam i.
DTEND;TZID=Europe/Zurich:20231122T163000 END:VEVENT BEGIN:VEVENT UID:news1589@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231031T155120 DTSTART;TZID=Europe/Zurich:20231115T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Marc Nualart (Imperial C ollege London) DESCRIPTION:In this talk we consider the long-time behavior of solutions to the two dimensional non-homogeneous Euler
equations under the Boussinesq approximation posed on a periodic channel. We prove inviscid damping for the linearized equations around the stably stratified Couette flow using stationary-phase
methods of oscillatory integrals. We discuss how these os cillatory integrals arise\, what are the main regularity requirements to carry out the stationary-phase arguments\, and how to achieve such
regula rities. X-ALT-DESC:
In this talk we consider the long-time behavior of solutions to the two dimensional non-homogeneous Euler equations under the Boussines q approximation posed on a periodic channel. We prove inviscid
damping &nb sp\;for the linearized equations around the stably stratified Couette flow using stationary-phase methods of oscillatory integrals. We discuss how t hese oscillatory integrals arise\, &
nbsp\;what are the main regularity req uirements to carry out the stationary-phase arguments\, and how to achieve such regularities.
DTEND;TZID=Europe/Zurich:20231115T160000 END:VEVENT BEGIN:VEVENT UID:news1602@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231027T193133 DTSTART;TZID=Europe/Zurich:20231108T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Gabriele Bocchi (Univers ità degli Studi di Roma Tor Vergata) DESCRIPTION:We analyze an optimal transport problem with additional entropi c cost evaluated along
curves in the Wasserstein space which join two prob ability measures m_0 and m_1. The effect of the additional entropy functio nal results into an elliptic regularization for the (so-called)
Kantorovic h potentials of the dual problem.\\r\\nAssuming the initial and terminal m easures to have densities\, we prove that the optimal curve remains positi ve and locally bounded in time. We
focus on the case that the transport pr oblem is set on a compact Riemannian manifold with Ricci curvature bounded below.\\r\\nThe approach follows ideas introduced by P.L. Lions in the th eory of
mean-field games about optimization problems with penalizing conge stion terms. Crucial steps of our strategy include displacement convexity properties in the Eulerian approach and the analysis of
distributional sub solutions to Hamilton-Jacobi equations.\\r\\nThe result provides a smooth approximation of Wasserstein-2 geodesics. X-ALT-DESC:
We analyze an optimal transport problem with additional entro pic cost evaluated along curves in the Wasserstein space which join two pr obability measures m_0 and m_1. The effect of the additional
entropy funct ional results into an elliptic regularization for the (so-called) Kantorov ich potentials of the dual problem.
Assuming the initial and termi nal measures to have densities\, we prove that the optimal curve remains p ositive and locally bounded in time. We focus on the case that the transpo rt problem is set
on a compact Riemannian manifold with Ricci curvature bo unded below.
The approach follows ideas introduced by P.L. Lions i n the theory of mean-field games about optimization problems with penalizi ng congestion terms. Crucial steps of our strategy include
displacement co nvexity properties in the Eulerian approach and the analysis of distributi onal subsolutions to Hamilton-Jacobi equations.
The result provide s a smooth approximation of Wasserstein-2 geodesics.
DTEND;TZID=Europe/Zurich:20231108T160000 END:VEVENT BEGIN:VEVENT UID:news1559@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230919T123033 DTSTART;TZID=Europe/Zurich:20230927T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Bian Wu (MPI Leipzig) DESCRIPTION:I will talk about Rayleigh-Taylor instability for two miscible\ , incompressible\, inviscid fluids. Scale-invariant estimates for
the size of the mixing zone and coarsening of internal structures in the fully non linear regime are established. These bounds provide optimal scaling laws a nd reveal the strong role of dissipation
in slowing down mixing. This is a joint work with Konstantin Kalinin\, Govind Menon. X-ALT-DESC:
I will talk about Rayleigh-Taylor instability for two miscibl e\, incompressible\, inviscid fluids. Scale-invariant estimates for the si ze of the mixing zone and coarsening of internal structures in
the fully n onlinear regime are established. These bounds provide optimal scaling laws and reveal the strong role of dissipation in slowing down mixing. This is a joint work with Konstantin Kalinin\,
Govind Menon.
DTEND;TZID=Europe/Zurich:20230927T160000 END:VEVENT BEGIN:VEVENT UID:news1547@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230904T075228 DTSTART;TZID=Europe/Zurich:20230920T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Chiara Boccato (Universi ty of Milano) DESCRIPTION:The interacting Bose gas is a system in quantum statistical mec hanics where a collective behavior emerges from
the underlying many-body t heory\, posing interesting challenges to its rigorous mathematical descrip tion.\\r\\nWhile at temperature close to zero we have precise information on the ground state
energy and the low-lying spectrum of excitations (at l east in certain scaling limits)\, much less is known close to the critical point. In this talk I will discuss how thermal excitations can be
describ ed by Bogoliubov theory\, allowing us to estimate the free energy of the B ose gas in the Gross-Pitaevskii regime.\\r\\n\\r\\nThis is joint work with A. Deuchert and D. Stocker. X-ALT-DESC:
The interacting Bose gas is a system in quantum statistical m echanics where a collective behavior emerges from the underlying many-body theory\, posing interesting challenges to its rigorous
mathematical descr iption.
While at temperature close to zero we have precise informa tion on the ground state energy and the low-lying spectrum of excitations (at least in certain scaling limits)\, much less is known close to
the cri tical point. In this talk I will discuss how thermal excitations can be de scribed by Bogoliubov theory\, allowing us to estimate the free energy of the Bose gas in the Gross-Pitaevskii
This is joint work with A. Deuchert and D. Stocker.
DTEND;TZID=Europe/Zurich:20230920T153000 END:VEVENT BEGIN:VEVENT UID:news1524@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230525T104748 DTSTART;TZID=Europe/Zurich:20230607T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Luis Martinez Zoroa (ICM AT Madrid) DESCRIPTION:The Surface quasi-geostrophic (SQG) equation is an important ac tive scalar model\, both due to its shared
properties with 3D Euler as wel l as for its applications to model certain atmospherical phenomena. It has already been established that instantaneous loss of regularity can occur in the inviscid
case in certain Sobolev spaces\, but it is unclear at what point the diffusion prevents this phenomenon from happening. In this talk I will discuss the behaviour when there is some super-critical
fractional diffusion. X-ALT-DESC:
The Surface quasi-geostrophic (SQG) equation is an important active scalar model\, both due to its shared properties with 3D Euler as w ell as for its applications to model certain atmospherical
phenomena. It h as already been established that instantaneous loss of regularity can occu r in the inviscid case in certain Sobolev spaces\, but it is unclear at wh at point the diffusion prevents
this phenomenon from happening. In this ta lk I will discuss the behaviour when there is some super-critical fraction al diffusion.
DTEND;TZID=Europe/Zurich:20230607T160000 END:VEVENT BEGIN:VEVENT UID:news1514@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230516T094922 DTSTART;TZID=Europe/Zurich:20230524T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Thérèse Moerschell (EP FL) DESCRIPTION:The advection-diffusion equation is known to have unique soluti ons for any vector field that is L^2 in time and in space.
But what happen s when we have slightly less than square integrability? In this talk we wi ll explore two examples of vector fields in L^p(0\,T\;L^q(\\T^d)) made of shear flows that prove the
non-uniqueness of solutions whenever we have p< 2 or q<2. We will first show that they give different solutions to the adv ection equation and then use the Feynman-Kac formula to show that diffusio n
has little effect if our parameters are well-tuned. This is part of my M aster's thesis\, supervised by Massimo Sorella and Maria Colombo. X-ALT-DESC:
The advection-diffusion equation is known to have unique solu tions for any vector field that is L^2 in time and in space. But what happ ens when we have slightly less than square integrability? In
this talk we will explore two examples of vector fields in L^p(0\,T\;L^q(\\T^d)) made o f shear flows that prove the non-uniqueness of solutions whenever we have p<\;2 or q<\;2. We will first
show that they give different solutions to the advection equation and then use the Feynman-Kac formula to show tha t diffusion has little effect if our parameters are well-tuned.
This is part of my Master's thesis\, supervised by Massimo Sorella and Maria C olombo.
DTEND;TZID=Europe/Zurich:20230524T160000 END:VEVENT BEGIN:VEVENT UID:news1502@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230404T153043 DTSTART;TZID=Europe/Zurich:20230517T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Matthew Novack (Purdue U niversity) DESCRIPTION:The phenomenon of anomalous dissipation in turbulence predicts the existence of solutions to the incompressible
Euler equations that enjo y regularity consistent with Kolmogorov’s 4/5 law and satisfy a local en ergy inequality. The "strong Onsager conjecture" asserts that such solutio ns do indeed exist. In
this talk\, we will discuss the background and moti vation behind the strong Onsager conjecture. In addition\, we outline a construction of solutions with regularity (nearly) consistent with the 4/5
law\, thereby proving the conjecture in the natural L^3 scale of Besov sp aces. This is based on joint work with Hyunju Kwon and Vikram Giri. X-ALT-DESC:
The phenomenon of anomalous dissipation in turbulence predict s the existence of solutions to the incompressible Euler equations that en joy regularity consistent with Kolmogorov’s 4/5 law and
satisfy a local energy inequality. The "strong Onsager conjecture" asserts that such solut ions do indeed exist. In this talk\, we will discuss the background and mo tivation behind the strong
Onsager conjecture. \; In addition\, we out line a construction of solutions with regularity (nearly) consistent with the 4/5 law\, thereby proving the conjecture in the natural L^3 scale of B
esov spaces. \; This is based on joint work with Hyunju Kwon and Vikra m Giri.
DTEND;TZID=Europe/Zurich:20230517T160000 END:VEVENT BEGIN:VEVENT UID:news1483@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230418T150401 DTSTART;TZID=Europe/Zurich:20230426T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Ángel Castro (ICMAT\, M adrid) DESCRIPTION:In this talk we will consider the existence of traveling waves arbitrarily close to shear flows for the 2D
incompressible Euler equat ions. In particular we shall present some results concerning the existence of such solutions near the Couette\, Taylor-Couette and the Poiseuille f lows. In the first part
of the talk we will introduce the problem and revi ew some well known results on this topic. In the second one some of the ideas behind the construction of our traveling waves will be sketched.
In this talk we will consider the existence of traveling wave s arbitrarily close to \; shear flows for \; the 2D incompressible Euler equations. In particular we shall present some results
concerning t he existence of such solutions near the Couette\, Taylor-Couette and the P oiseuille \;flows. In the first part of the talk we will introduce the problem and review some well known
results on this topic. In the \; s econd one some of the ideas behind the construction of our traveling waves  \; will be sketched.
DTEND;TZID=Europe/Zurich:20230426T160000 END:VEVENT BEGIN:VEVENT UID:news1475@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230413T104028 DTSTART;TZID=Europe/Zurich:20230419T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Stefano Spirito (Univers ità degli Studi dell'Aquila) DESCRIPTION:I will review the existence and uniqueness theory of a model fo r viscoelastic materials of
Kelvin-Voigt type with large strain. In partic ular\, I will first review the existence theory in L2\, and then show that also propagation of H1-regularity for the deformation gradient of weak so
lutions in two and three dimensions holds. Moreover\, in two dimensions it is also possible to prove uniqueness of weak solutions. Additional pro pagation of higher regularity can be obtained\,
leading to a global in tim e existence of smooth solutions. Joint work with K. Koumatos (U. of Sussex )\, C. Lattanzio (UnivAQ) and A. Tzavaras (KAUST). X-ALT-DESC:
I will review the existence and uniqueness theory of a model for viscoelastic materials of Kelvin-Voigt type with large strain. In part icular\, I will first review the existence theory in L2\, and
then show th at also propagation of H1-regularity for the deformation gradient of weak solutions in two and three dimensions holds. \; \;Moreover\, in tw o dimensions it is also possible to
prove uniqueness of weak solutions. Ad ditional propagation of higher regularity can be obtained\, leading to a g lobal in time existence of smooth solutions. Joint work with K. Koumatos ( U. of
Sussex)\, C. Lattanzio (UnivAQ) and A. Tzavaras (KAUST).
DTEND;TZID=Europe/Zurich:20230419T160000 END:VEVENT BEGIN:VEVENT UID:news1467@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230125T171400 DTSTART;TZID=Europe/Zurich:20230322T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Mickaël Latocca (Univer sité d'Évry) DESCRIPTION:In an incompressible fluid\, the pressure is governed by the el liptic equation $-\\Delta p = \\div \\div u \\
otimes u$ and a Neuman-type boundary condition\, where $u$ stands for the divergence-free velocity vec tor field. The main goal of this talk is to explain why one expects that $p$ has Double Hölder
regularity (with respect to that of $u$) and how on e can rigourously prove such a fact in a bounded domain. The results prese nted in this talk were obtained in collaboation with Luigi De Rosa
(Basel) and Giorgio Stefani (SISSA). X-ALT-DESC:
In an incompressible fluid\, the pressure is governed by the elliptic equation $-\\Delta p = \\div \\div u \\otimes u$ and a Neuman-typ e boundary condition\, where $u$ stands for the divergence-free
velocity v ector field. \;The main goal of this talk is to explain why one expect s that $p$ has Double Hölder regularity (with respect to that of $u$) and how one can rigourously prove such a
fact in a bounded domain. The result s presented in this talk were obtained in collaboation with Luigi De Rosa (Basel) and Giorgio Stefani (SISSA). \;
DTEND;TZID=Europe/Zurich:20230322T160000 END:VEVENT BEGIN:VEVENT UID:news1466@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230305T211240 DTSTART;TZID=Europe/Zurich:20230315T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Federico Cacciafesta (Un iversity of Padova) DESCRIPTION:The Dirac equation is one of the fundamental equations in relat ivistic quantum mechanics\, widely used in
a large number of applications from physics to quantum chemistry. The aim of this talk will be to discuss some recent results\, together with a number of open questions\, concerni ng the dynamics for
this model: after briefly reviewing the main propertie s of the Dirac operator and providing some background and motivations from the theory of linear dispersive PDEs\, we shall focus in particular
on th e cases of the Dirac-Coulomb equation and of the Dirac equation on non fla t manifolds\, showing how some linear estimates (in particular\, Strichart z estimates) can be obtained by exploiting
various properties of the opera tor. X-ALT-DESC:
The Dirac equation is one of the fundamental equations in rel ativistic quantum mechanics\, widely used in a large number of application s from physics to quantum chemistry. The aim of this talk will
be to discu ss some recent results\, together with a number of open questions\, concer ning the dynamics for this model: after briefly reviewing the main propert ies of the Dirac operator and
providing some background and motivations fr om the theory of linear dispersive PDEs\, we shall focus in particular on the cases of the Dirac-Coulomb equation and of the Dirac equation on non f lat
manifolds\, showing how some linear estimates (in particular\, Stricha rtz estimates) can be obtained by exploiting various properties of the ope rator.
DTEND;TZID=Europe/Zurich:20230315T161500 END:VEVENT BEGIN:VEVENT UID:news1465@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230119T132638 DTSTART;TZID=Europe/Zurich:20230308T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Maria Ahrend (Uni Basel) DESCRIPTION:Maria Ahrend will defend her PhD thesis on fractional Liouville equations and Calogero-Moser NLS. X-ALT-DESC:
Maria Ahrend will defend her PhD thesis on fractional Liouvil le equations and Calogero-Moser NLS.
DTEND;TZID=Europe/Zurich:20230309T160000 END:VEVENT BEGIN:VEVENT UID:news1476@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230304T163156 DTSTART;TZID=Europe/Zurich:20230307T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Zineb Hassainia (New Yor k University Abu Dhabi) [special time!] DESCRIPTION:In this talk\, I will discuss a recent result concerning the co nstruction of
quasi-periodic vortex patch solutions with one hole for th e 2D-Euler equations. These structures exist close to any annulus provided that its modulus belongs to a Cantor set with almost full
Lebesgue measur e. The proof is based on a KAM reducibility scheme and a Nash-Moser iterative scheme. This is a joint work with Taoufik Hmidi and Emeric Roull ey. X-ALT-DESC:
In this talk\, I will discuss a recent result concerning the construction of quasi-periodic vortex patch solutions \; with one hole for the 2D-Euler equations. These structures exist close to any
annulus p rovided that its modulus belongs to a Cantor set with almost full Lebesgue measure. The proof is based on \; a KAM reducibility scheme \; an d a \; Nash-Moser iterative scheme.
This is a joint work with Taoufik Hmidi and Emeric Roulley.
DTEND;TZID=Europe/Zurich:20230307T160000 END:VEVENT BEGIN:VEVENT UID:news1470@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230131T153947 DTSTART;TZID=Europe/Zurich:20230222T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Peter Pickl (Universitä t Tübingen) DESCRIPTION:The derivation of the Vlasov equation from Newtonian mechanics is an old problem in mathematical physics. But
while the most interest ing interactions in nature have singularities\, one typically assumes some Lipschitz condition on the interaction force for its microscopic derivation. Recent developments
have given results\, where the interac tion force gets singular when the particle number N tends to infinity\, us ually by mollifying or cutting the singularity with a N-dependent moll ifier or
cut-off parameter. In the talk I will present most recent develop ments and new results on this topic. X-ALT-DESC:
The derivation of the Vlasov equation from Newtonian mechanic s is an \; \;
old problem in
mathematical physics. But while the most interesting interactions in \; \;
nature hav e singularities\,
one typically assumes some Lipschitz condition on the interaction \; \;
force for its microscopic
deriva tion. Recent developments have given results\, where the \; \;
interaction force gets singular
when the particle number N tends to infinity\, usually by mollifying \; \;
or cutting the sin gularity
with a N-dependent mollifier or cut-off parameter.
In the talk I will present most recent developments and new results on \ ; \;
this topic.
DTEND;TZID=Europe/Zurich:20230222T160000 END:VEVENT BEGIN:VEVENT UID:news1437@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221207T184842 DTSTART;TZID=Europe/Zurich:20221221T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Harprit Singh (Imperial College London) DESCRIPTION:Singular stochastic partial differential equations (SPDEs) of t he form \\r\\n∂tu = △u + F (u\, ∂xu\, ξ)\, \\r\\
nwhere ξ is an irregular driving noise\, arise in a variety of situations from quan tum field theory to probability. After introducing some specific examples\ , we describe the main difficulty they
share\; they are singular due to th e irregularity of the driving noise ξ.\\r\\nIn the first part of the tal k we discuss a simple example where using the so-called “Da Prato-Debusc he trick” is
sufficient to deal with this difficulty. In the second half \, we give a birds-eye view on how regularity structures provide a solutio n theory for such equations. In particular\, we explain the role
of subcri ticality (super-renormalisability) and (half) Feynman diagrams in this the ory. Lastly\, we shall mention some recent results on the class of differe ntial operators that are compatible
with this general machinery and how th is relates to the geometry of the underlying space. X-ALT-DESC:
Singular stochastic partial differential equations (SPDEs) of the form \;
∂[t]u \;= \;△u \;+ \ ;F \;(u\, ∂[x]u\, ξ)\, \;
where \;ξ \ ;is an irregular driving noise\, arise in a variety of situations from qua ntum field theory to probability. After introducing some specific examples \, we describe the main
difficulty they share\; they are singular due to t he irregularity of the driving noise \;ξ.
In the first part o f the talk we discuss a simple example where using the so-called “Da Pra to-Debusche trick” is sufficient to deal with this difficulty. In the se cond half\, we give a birds-eye
view on how regularity structures provide a solution theory for such equations. In particular\, we explain the role of subcriticality (super-renormalisability) and (half) Feynman diagrams in this
theory. Lastly\, we shall mention some recent results on the class o f differential operators that are compatible with this general machinery a nd how this relates to the geometry of the underlying
DTEND;TZID=Europe/Zurich:20221221T160000 END:VEVENT BEGIN:VEVENT UID:news1419@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221128T102752 DTSTART;TZID=Europe/Zurich:20221207T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Hyunju Kwon (ETH Zurich) DESCRIPTION:Smooth (spatially periodic) solutions to the incompressible 3D Euler equations have kinetic energy conservation in every local
region\, w hile turbulent flows exhibit anomalous dissipation of energy. Toward verif ication of the anomalous dissipation\, Onsager theorem has been establishe d\, which says that the threshold
Hölder regularity of the total kinetic energy conservation is 1/3. As a next step\, we discuss a strong Onsager c onjecture\, which combines the Onsager theorem with the local energy inequ ality.
Smooth (spatially periodic) solutions to the incompressible 3 D Euler equations have kinetic energy conservation in every local region\, while turbulent flows exhibit anomalous dissipation of energy.
Toward ver ification of the anomalous dissipation\, Onsager theorem has been establis hed\, which says that the threshold Hölder regularity of the total kineti c energy conservation is 1/3. As a next
step\, we discuss a strong Onsager conjecture\, which combines the Onsager theorem with the local energy ine quality.
DTEND;TZID=Europe/Zurich:20221207T160000 END:VEVENT BEGIN:VEVENT UID:news1429@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221117T152426 DTSTART;TZID=Europe/Zurich:20221123T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Michele Dolce (EPFL) DESCRIPTION:Fluids in the ocean are often inhomogeneous\, incompressible an d\, in relevant physical regimes\, can be described by the 2D
Euler-Boussi nesq system. Equilibrium states are then commonly observed to be stably st ratified\, namely the density increases with depth. We are interested in c onsidering the case when also a
background shear flow is present. In the t alk\, I will describe quantitative results for small perturbations around a stably stratified Couette flow. The density variation and velocity under go an O
(1/(t^{1/2})) inviscid damping while the vorticity and density grad ient grow as O(t^{1/2}) in L^2. This is precisely quantified at the linea r level. For the nonlinear problem\, the result holds on
the optimal time -scale on which a perturbative regime can be considered. Namely\, given an initial perturbation of size O(eps)\, it is expected that the linear reg ime is observed up to a time-scale
O(eps^{-1}). However\, we are able to control the dynamics all the way up to O(eps^{-2})\, where the perturbat ion become of size O(1) due to the linear instability. X-ALT-DESC:
Fluids in the ocean are often inhomogeneous\, incompressible and\, in relevant physical regimes\, can be described by the 2D Euler-Bous sinesq system. Equilibrium states are then commonly observed to
be stably stratified\, namely the density increases with depth. We are interested in considering the case when also a background shear flow is present. In the talk\, I will describe quantitative
results for small perturbations aroun d a stably stratified Couette flow. The density variation and velocity und ergo an O(1/(t^{1/2})) inviscid damping while the vorticity and density gr adient grow
as O(t^{1/2}) \;in L^2. This is precisely quantified at th e linear level. \;For the nonlinear problem\, the result holds on the optimal time-scale on which a perturbative regime can be
considered. Namel y\, given an initial perturbation of size O(eps)\, \;it is expected th at the linear regime is observed up to a time-scale O(eps^{-1}). However\,  \;we are able to control
the dynamics all the way up to \;O(eps^{ -2})\, where \;the perturbation become of size O(1) \;due to the l inear instability.
DTEND;TZID=Europe/Zurich:20221123T160000 END:VEVENT BEGIN:VEVENT UID:news1422@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221107T105340 DTSTART;TZID=Europe/Zurich:20221116T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Nicolas Camps (Universit y of Nantes) DESCRIPTION:Following the seminal work of Bourgain in 1996\, and Burq and T zvetkov in 2008\, a statistical approach to
nonlinear dispersive equations has developed in various contexts. We are interested here in Schrödinge r equations with cubic nonlinearity (NLS) in R^d. We first recall the rele vant probabilistic
Cauchy theory developed by Bényi\, Oh and Pocovnicu in 2015 in supercritical regimes\, before specifying the norm inflation instability that occurs in this context. The second part is dedicated to
long-time dynamics for solutions initiated from these randomized initial d ata. We demonstrate a scattering result that relies on a probabilistic ver sion of the I-method and that allows to solve
statistically the scatteri ng conjecture for NLS in dimension 3. Finally\, we present recent develop ments in quasi-linear regimes\, which were initiated by Bringmann in 2019 and which we exploit to
exhibit strong solutions to some weakly dispersive equations. This last result is in collaboration with Louise Gassot and Sl im Ibrahim. X-ALT-DESC:
Following the seminal work of Bourgain in 1996\, and Burq and Tzvetkov in 2008\, a statistical approach to nonlinear dispersive equatio ns has developed in various contexts. \;We are interested
here in Schr ödinger equations with cubic nonlinearity (NLS) in R^d. We first recall t he relevant probabilistic Cauchy theory  \;developed by Bényi\, Oh an d Pocovnicu in 2015 in supercritical
regimes\, before specifying the \ ;norm inflation \;instability that occurs in this context.&nb sp\;The second part is dedicated to long-time dynamics for solutions initi ated from these
randomized initial data. We demonstrate a scattering resul t that relies on a probabilistic version of the \;I-method&nb sp\;and that allows to solve statistically the scattering conjecture for N
LS in dimension 3. \;Finally\, we present recent developments in quasi -linear regimes\, which were initiated by Bringmann in 2019 and which we e xploit to exhibit strong solutions to some weakly
dispersive equations. Th is last result is in collaboration with Louise Gassot and Slim Ibrahim.
DTEND;TZID=Europe/Zurich:20221116T160000 END:VEVENT BEGIN:VEVENT UID:news1413@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221014T114957 DTSTART;TZID=Europe/Zurich:20221109T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Paolo Bonicatto (Univers ity of Warwick) DESCRIPTION:In the classical theory\, given a vector field $b$ on $\\mathbb R^d$\, one usually studies the transport/
continuity equation drifted by $ b$ looking for solutions in the class of functions (with certain integrabi lity) or at most in the class of measures. In this seminar I will talk abo ut recent
efforts\, motivated by the modelling of defects in plastic mater ials\, aimed at extending the previous theory to the case when the unknown is instead a family of k-currents in $\\mathbb R^d$\, i.e.
generalised $k $-dimensional surfaces. The resulting equation involves the Lie derivative $L_b$ of currents in direction $b$ and reads $\\partial_t T_t + L_b T_t = 0$. In the first part of the talk I
will briefly introduce this equation\ , with special attention to its space-time formulation. I will then shift the focus to some rectifiability questions and Rademacher-type results: gi ven a
Lipschitz path of integral currents\, I will discuss the existence o f a “geometric derivative”\, namely a vector field advecting the curre nts. Joint work with G. Del Nin and F. Rindler (Warwick).
In the classical theory\, given a vector field $b$ on $\\math bb R^d$\, one usually studies the transport/continuity equation drifted by $b$ looking for solutions in the class of functions (with
certain integra bility) or at most in the class of measures. In this seminar I will talk a bout recent efforts\, motivated by the modelling of defects in plastic mat erials\, aimed at extending the
previous theory to the case when the unkno wn is instead a family of k-currents in $\\mathbb R^d$\, i.e. generalised $k$-dimensional surfaces. The resulting equation involves the Lie derivati ve
$L_b$ of currents in direction $b$ and reads $\\partial_t T_t + L_b T_t = 0$. In the first part of the talk I will briefly introduce this equatio n\, with special attention to its space-time
formulation. I will then shif t the focus to some rectifiability questions and Rademacher-type results: given a Lipschitz path of integral currents\, I will discuss the existence of a “geometric
derivative”\, namely a vector field advecting the cur rents. Joint work with G. Del Nin and F. Rindler (Warwick).
DTEND;TZID=Europe/Zurich:20221109T160000 END:VEVENT BEGIN:VEVENT UID:news1360@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220625T161018 DTSTART;TZID=Europe/Zurich:20220629T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Luca Fresta (University of Bonn) DESCRIPTION:We consider a system of $N$ interacting fermions initially conf ined in a volume $\\Lambda$. We show that\, in the
high-density regime a nd for zero-temperature initial data exhibiting a local semiclassical st ructure\, the solution of the many-body Schrödinger equation can be app roximated by the solution of
the nonlinear Hartree equation\, up to erro rs that are small\, for large density\, uniformly in $N$ and $\\Lambda$. This is joint work with M. Porta and B. Schlein. X-ALT-DESC:
We consider a system of $N$ interacting fermions initially co nfined in a \;
$\\Lambda$. We show that\, in the hi gh-density regime and for zero-temperature \;
initial data
exhibiting a local semiclassical structure\, the solution of the many-bod y \;
Schrödinger equation can be approximated by the solution o f the nonlinear \;
Hartree equation\, up to errors that are smal l\, for large density\, uniformly \;
in $N$ and $\\Lambda$. This is joint work with M. Porta and B. Schlein.
DTEND;TZID=Europe/Zurich:20220629T160000 END:VEVENT BEGIN:VEVENT UID:news1366@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220606T131938 DTSTART;TZID=Europe/Zurich:20220609T163000 SUMMARY:An afternoon
of analysis talks: Gautam Iyer (Carnegie Mellon Univer sity) DESCRIPTION:The Kompaneets equation describes energy transport in low-densi ty (or high temperature) plasmas where the dominant energy
exchange mecha nism is Compton scattering. The equation itself is a one dimensional non- linear parabolic equation with a diffusion coefficient that vanishes at t he boundary. This degeneracy\,
combined with the nonlinearity causes an o ut-flux of photons with zero energy\, often interpreted as a Bose-Einstein condensate. This talk will describe several results about the long time
behavior of these equations including convergence to equilibrium\, persi stence of the condensate\, sufficient conditions under which it forms\, s ufficient conditions under which it doesn't form
and a loss formula for t he mass of the condensate. X-ALT-DESC:
The Kompaneets equation describes energy transport in low-den sity (or high \;temperature) plasmas where the dominant energy exchang e mechanism is \;Compton scattering. The equation itself
is a one dime nsional non-linear \;parabolic equation with a diffusion coefficient t hat vanishes at the
boundary. This degeneracy\, combined with the no nlinearity causes an \;out-flux of photons with zero energy\, often in terpreted as a Bose-Einstein \;condensate. This talk will describe sev
eral results about the long time \;behavior of these equations includi ng convergence to equilibrium\,
persistence of the condensate\, suff icient conditions under which it forms\, \;sufficient conditions under which it doesn't form and a loss formula for \;the mass of the conden sate.
DTEND;TZID=Europe/Zurich:20220609T173000 END:VEVENT BEGIN:VEVENT UID:news1365@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220606T131928 DTSTART;TZID=Europe/Zurich:20220609T150000 SUMMARY:An afternoon
of analysis talks: Anna Mazzucato (Penn State Universi ty) DESCRIPTION:I will present a recent result concerning global existence for the Kuramoto-Sivashinsky equation on the two-dimensional torus
with one gr owing mode in each direction. The proof combines PDE techniques with a Ly apunov function argument for the growing modes. This is joint work with Da vid Ambrose (Drexel University\, USA).
I will present a recent result concerning global existence fo r the Kuramoto-Sivashinsky equation on the two-dimensional torus with one growing \;mode in each direction. The proof combines PDE
techniques wi th a Lyapunov function argument for the growing modes. This is joint work with David Ambrose (Drexel University\, USA).
DTEND;TZID=Europe/Zurich:20220609T160000 END:VEVENT BEGIN:VEVENT UID:news1364@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220606T131917 DTSTART;TZID=Europe/Zurich:20220609T140000 SUMMARY:An afternoon
of analysis talks: Giovanni Alberti (University of Pis a) DESCRIPTION:In this talk I will describe some result about the following el ementary problem\, of isoperimetric flavor: Given a set E in R^d
with fini te volume\, is it possible to find an hyperplane P that cuts E in two part s with equal volume\, and such that the area of the cut (that is\, the int ersection of P and E) is of the
expected order\, namely (vol(E))^{1−1/d} ? We can show that this question\, even in a stronger form\, has a positiv e answer if the dimension d is 3 or higher.But\, interestingly enough\, ou r proof
breaks down completely in dimension d=2\, and we do not know the a nswer in this case (but we know that the answer is positive if we allow cu ts that are not exactly planar\, but close to planar). It
turns out that this question has some interesting connection with the Kakeya problem. Thi s is a work in progress with Alan Chang (Princeton University). X-ALT-DESC:
In this talk I will describe some result about the following elementary problem\, of isoperimetric flavor:
Given a set E in R^d w ith finite volume\, is it possible to find an hyperplane P that cuts E in two parts with equal volume\, and such that the area of the cut (that is\, the intersection of P and
E) is of the expected order\, namely (vol(E))^{ 1−1/d}?
We can show that this question\, even in a stronger form\, has a positive answer if the dimension d is 3 or higher.But\, interesting ly enough\, our proof breaks down completely in dimension d=2\,
and we do not know the answer in this case (but we know that the answer is positive if we allow cuts that are not exactly planar\, but close to planar). \ ;It turns out that this question has
some interesting connection with the Kakeya problem.
This is a work in progress with Alan Chang (Princeto n University).
DTEND;TZID=Europe/Zurich:20220609T150000 END:VEVENT BEGIN:VEVENT UID:news1353@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220504T163350 DTSTART;TZID=Europe/Zurich:20220525T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Franck Sueur (Institut d e Mathématiques de Bordeaux) DESCRIPTION:This talk is devoted to the 2D incompressible Euler system in p resence of sources and sinks. This
model dates back to Viktor Yudovich in the sixties and is an interesting example of nonlinear open system which h as been widely used in controllability theory within the scope of smooth s olutions.
In this talk we will review how the classical issues of existenc e and uniqueness of weak solutions are challenged by the presence of incom ing and exiting vorticity. This talk is based on joint
works with Marco Bravin and Florent Noisette. X-ALT-DESC:
This talk is devoted to the 2D incompressible Euler system in presence of sources and sinks. This model dates back to Viktor Yudovich i n the sixties and is an interesting example of nonlinear open
system which has been widely used in controllability theory within the scope of smooth solutions. In this talk we will review how the classical issues of existe nce and uniqueness of weak solutions
are challenged by the presence of inc oming and exiting vorticity.  \;This talk is based on joint works with Marco Bravin and Florent Noisette. \;
DTEND;TZID=Europe/Zurich:20220525T160000 END:VEVENT BEGIN:VEVENT UID:news1354@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220508T162554 DTSTART;TZID=Europe/Zurich:20220518T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Banhirup Sengupta (Unive rsitat Autònoma de Barcelona) DESCRIPTION:In this talk I am going to provide a pointwise characterisation of nearly incompressible vector
fields with bounded curl. Euler vector fi elds fall in this class. I will also talk about rotational properties of E uler flows and nonlinear transport equations involving Cauchy Kernel in th e
plane. This is based on joint works with Albert Clop (Barcelona) and Lau ri Hitruhin (Helsinki). X-ALT-DESC:
In this talk I am going to provide a pointwise characterisati on of nearly incompressible vector fields with bounded curl. Euler vector fields fall in this class. I will also talk about rotational
properties of Euler flows and nonlinear transport equations involving Cauchy Kernel in the plane. This is based on joint works with Albert Clop (Barcelona) and L auri Hitruhin (Helsinki).
DTEND;TZID=Europe/Zurich:20220518T160000 END:VEVENT BEGIN:VEVENT UID:news1327@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220422T115503 DTSTART;TZID=Europe/Zurich:20220504T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Gabriel Sattig (Universi tät Leipzig) DESCRIPTION:It was shown by Modena and Székelyhidi that weak solutions to the incompressible transport equation may be not
unique\, even if the tr ansporting field is Sobolev\, thus admitting a unique regular Lagrangian flow. In this talk I will present a recent result saying that non-Lagrang ian solutions are generic in
the Baire category sense. Joint work with L . Székelyhidi. X-ALT-DESC:
It was shown by Modena and Székelyhidi that weak solutions t o the \;incompressible transport equation may be not unique\, even if the \;transporting field is Sobolev\, thus admitting a
unique regular& nbsp\;Lagrangian flow. \;In this talk I will present a recent result s aying that non-Lagrangian \;solutions are generic in the Baire categor y sense. \;Joint work with L.
DTEND;TZID=Europe/Zurich:20220504T160000 END:VEVENT BEGIN:VEVENT UID:news1324@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220409T172448 DTSTART;TZID=Europe/Zurich:20220427T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Jaemin Park (University of Barcelona) DESCRIPTION:In this talk\, we study stationary solutions to the 2D incompre ssible Euler equations in the whole plane. It is
well-known that any radia l vorticity is stationary. For compactly supported vorticity\, it is more difficult to see whether a stationary solution has to be radial. In the ca se where the vorticity
is non-negative\, it has been shown that any statio nary solution has be radial. By allowing the vorticity to change the sign\ , we prove that there exist non-radial stationary patch-type solutions.
We construct patch-type solutions whose kinetic energy is infinite or finite . For the finite energy case\, it turns out that a construction of a stati onary solution with compactly supported
velocity is possible. X-ALT-DESC:
In this talk\, we study stationary solutions to the 2D incomp ressible Euler equations in the whole plane. It is well-known that any rad ial vorticity is stationary. For compactly supported vorticity
\, it is mor e difficult to see whether a stationary solution has to be radial. In the case where the vorticity is non-negative\, it has been shown that any stat ionary solution has be radial. By
allowing the vorticity to change the sig n\, we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or fini te. For
the finite energy case\, it turns out that a construction of a sta tionary solution with compactly supported velocity is possible.
DTEND;TZID=Europe/Zurich:20220427T160000 END:VEVENT BEGIN:VEVENT UID:news1321@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220214T151002 DTSTART;TZID=Europe/Zurich:20220406T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Dr. Eliot Pacherie (NYU Abu Dhabi) DESCRIPTION:tba X-ALT-DESC:
DTEND;TZID=Europe/Zurich:20220406T160000 END:VEVENT BEGIN:VEVENT UID:news1313@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220320T140750 DTSTART;TZID=Europe/Zurich:20220323T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Dr. Raphael Winter (ENS Lyon) DESCRIPTION:Following the pioneering work of Lanford\, a rigorous theory ha s been developed for the validation of the Boltzmann
equation in the low-d ensity Grad scaling. In the physics literature\, an important issue are th e corrections to the equation for small but positive volume fraction. The first order correction to
the Boltzmann equation is conjectured to be give n by the so-called Choh-Uhlenbeck equation\, which is of the form\\r\\n∂ tfϵ=Qϵ\,BE(fϵ\,fϵ)+ϵQϵ\,CU(fϵ\,fϵ\,fϵ).\\r\\n Here Qϵ\,BE is the
Boltzmann-Enskog operator\, and the Choh-Uhlenbeck operator Qϵ\,CU is an explicit cubic operator. This operator accounts for the formation of dyna mic microscopic correlations between three
particles. In this work\, we pr ove rigorously that the Choh-Uhlenbeck equation gives the first order corr ection to the Boltzmann equation in the Grad-scaling. This is a joint work with Sergio
Simonella. X-ALT-DESC:
Following the pioneering work of Lanford\, a rigorous theory has been developed for the validation of the Boltzmann equation in the low -density Grad scaling. In the physics literature\, an important
issue are the corrections to the equation for small but positive volume fraction. Th e first order correction to the Boltzmann equation is conjectured to be gi ven by the so-called Choh-Uhlenbeck
equation\, which is of the form
\n< p>∂tf[S:ϵ:S]=Qϵ\,BE(fϵ\,fϵ)+ϵQϵ\,CU(fϵ\,f[S:ϵ\,fϵ).\n:S]
Here Qϵ\,BE is the Boltzmann-Enskog operator\, and the C hoh-Uhlenbeck operator Qϵ\,CU is an explicit cubic operator. This operator accounts for the formation of dynamic microscopic correlations
between th ree particles. In this work\, we prove rigorously that the Choh-Uhlenbeck equation gives the first order correction to the Boltzmann equation in the Grad-scaling. This is a joint work with
Sergio Simonella.
DTEND;TZID=Europe/Zurich:20220323T160000 END:VEVENT BEGIN:VEVENT UID:news1311@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220113T175547 DTSTART;TZID=Europe/Zurich:20220302T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Dr. Havva Yoldas (Univer sity of Vienna) DESCRIPTION: I will talk about Harris-type theorems and their applications to several kinetic equations like the linear BGK
\, the linear Boltzmann\, the kinetic Fokker-Planck equations and some biological kinetic models li ke the run and tumble equation. Even though the original ideas date back t o the 1940s\, the
Harris-type arguments recently raised a lot of mathemati cal interest in the PDE community especially after a simplified proof pro vided by Hairer and Mattingly in 2011. It is a convenient way to
obtain qu antifiable convergence rates\, constructive proofs and the existence of a unique stationary state comes as a by-product of the theorems. The latte r is especially useful for kinetic
equations arising in biology where the shape of the stationary state cannot be known a priori. X-ALT-DESC:
 \;I will talk about Harris-type theorems and their appli cations to several kinetic equations like the linear BGK\, the linear Bolt zmann\, the kinetic Fokker-Planck equations and some
biological kinetic mo dels like the run and tumble equation. Even though the original ideas date back to the 1940s\, the Harris-type arguments recently raised a lot of ma thematical interest \;in
the PDE community especially after a simplifi ed proof provided by Hairer and Mattingly in 2011. It is a convenient way to obtain quantifiable convergence rates\, constructive \;proofs and t he
existence of a unique \;stationary state comes as a by-product of t he theorems. The latter is especially useful \;for kinetic equations a rising in biology where the shape of the stationary
state cannot be known a priori. \;
DTEND;TZID=Europe/Zurich:20220302T160000 END:VEVENT BEGIN:VEVENT UID:news1252@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211115T124620 DTSTART;TZID=Europe/Zurich:20211208T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Martina Zizza (SISSA Tri este) DESCRIPTION:In this talk we tackle the question "How many vector fields are mixing?" analyzing the density properties of
divergence-free BV vector fields which are weakly mixing/strongly mixing: this means that their Reg ular Lagrangian Flow is a weakly mixing/strongly mixing measure-preserving map when evaluated at
time t=1. More precisely we prove the existence of a G_delta-set U in the space L^1_{t\,x}([0\,1]^3) made of divergence-free vector fields such that:\\r\\n 1) weakly mixing vector fields are a
residual G_delta-set in U\;\\r\\n 2) (exponentially fast) strongl y mixing vector fields are a dense subset of U.\\r\\nThe proof of these re sults exploits some connections between ergodic theory
and fluid dynamics and it is based on the density of BV vector fields whose Regular Lagrangia n Flow is a permutation of subsquares of the unit square [0\,1]^2 when eva luated at time t=1.
In this talk we tackle the question "How many vector fields a re mixing?" analyzing \; \;the density properties of divergence-fr ee BV vector fields which are weakly mixing/strongly mixing:
this means th at their Regular Lagrangian Flow is a weakly mixing/strongly mixing measur e-preserving map when evaluated at time t=1. More precisely we prove the e xistence of a \;G_delta-set U
in the space L^1_{t\,x}([0\,1]^3) made o f divergence-free vector fields such that:
 \;  \; 1)  \; weakly mixing vector fields are a residual G_delta-set in U\;
& nbsp\;  \; 2) \; (exponentially fast) strongly mixing vector field s are a dense subset of U.
The proof of these results exploits som e connections between ergodic theory and fluid dynamics and it is based on the density of BV vector fields whose Regular Lagrangian Flow is a permut ation of
subsquares of the unit square [0\,1]^2 when evaluated at time t=1 . \;
DTEND;TZID=Europe/Zurich:20211208T160000 END:VEVENT BEGIN:VEVENT UID:news1253@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211122T174531 DTSTART;TZID=Europe/Zurich:20211201T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Elia Bruè (Institute fo r Advanced Study\, Princeton) DESCRIPTION:A long-standing open question in fluid mechanics is whether the Yudovich uniqueness result for the
2d Euler system can be extended to the class of L^p-integrable vorticity. Recently\, there have been formidable attempts to disprove this conjecture\, none of which has by now fully solv ed it. I
will outline two possible approaches to this problem. One is bas ed on the convex integration technique introduced by De Lellis and Szekely hidi. The second\, proposed recently by Vishik\, exploits
the linear insta bility of certain stationary solutions. X-ALT-DESC:
A long-standing open question in fluid mechanics is whether t he Yudovich uniqueness result for the 2d Euler system can be extended to t he class of L^p-integrable vorticity. Recently\, there have
been formidabl e attempts to disprove this conjecture\, none of which has by now fully so lved it. \;I will outline two possible approaches to this problem. One is based on the convex integration
technique introduced by De Lellis and Szekelyhidi. The second\, proposed recently by Vishik\, exploits the linea r instability of certain stationary \;solutions.
DTEND;TZID=Europe/Zurich:20211201T160000 END:VEVENT BEGIN:VEVENT UID:news1259@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211103T092442 DTSTART;TZID=Europe/Zurich:20211124T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Gioacchino Antonelli (Sc uola Normale Superiore di Pisa) DESCRIPTION:In this talk I will discuss the isoperimetric problem on spaces with curvature bounded from
below. I will mainly deal with complete non-c ompact Riemannian manifolds\, but most of the techniques described are met ric in nature and the results could be extended to the case of metric meas ure
spaces with synthetic bounds from below on the Ricci tensor\, namely R CD spaces. When the space is compact\, the existence of isoperimetric re gions for every volume is established through a simple
application of the direct method of Calculus of Variations. In the noncompact case\, part of the mass could be lost at infinity in the minimization process. Such a mas s can be recovered in
isoperimetric regions sitting in limits at infinity of the space. Following this heuristics\, and building on top of results b y Ritoré--Rosales and Nardulli\, I will state a generalized existence
res ult for the isoperimetric problem on Riemannian manifolds with Ricci curva ture bounded from below and a uniform bound from below on the volumes of u nit balls. The main novelty in such an
approach is the use of the syntheti c theory of curvature bounds to describe in a rather natural way where the mass is lost at infinity. Later\, I will use the latter described general ized existence
result to prove new existence criteria for the isoperimetri c problem on manifolds with nonnegative Ricci curvature. In particular\, I will show that on a complete manifold with nonnegative sectional
curvatur e and Euclidean volume growth at infinity\, isoperimetric regions exist fo r every sufficiently big volume. Time permitting\, I will describe some fo rthcoming works and some open problems.
This talk is based on several pa pers and ongoing collaborations with E. Bruè\, M. Fogagnolo\, S. Nardulli \, E. Pasqualetto\, M. Pozzetta\, and D. Semola. X-ALT-DESC:
In this talk I will discuss the isoperimetric problem on spac es with curvature bounded from below. I will mainly deal with complete non -compact Riemannian manifolds\, but most of the techniques
described are m etric in nature and the results could be extended to the case of metric me asure spaces with synthetic bounds from below on the Ricci tensor\, namely RCD spaces. \;
When the space is compact\, the existence of iso perimetric regions for every volume is established through a simple applic ation of the direct method of Calculus of Variations. In the noncompact ca
se\, part of the mass could be lost at infinity in the minimization proces s. Such a mass can be recovered in isoperimetric regions sitting in limits at infinity of the space. Following this
heuristics\, and building on top of results by Ritoré--Rosales and Nardulli\, I will state a generalized existence result for the isoperimetric problem on Riemannian manifolds wit h Ricci curvature
bounded from below and a uniform bound from below on the volumes of unit balls. The main novelty in such an approach is the use of the synthetic theory of curvature bounds to describe in a rather
natural way where the mass is lost at infinity. Later\, I will use the latter desc ribed generalized existence result to prove new existence criteria for the isoperimetric problem on manifolds with
nonnegative Ricci curvature. In p articular\, I will show that on a complete manifold with nonnegative secti onal curvature and Euclidean volume growth at infinity\, isoperimetric reg ions exist for
every sufficiently big volume. Time permitting\, I will des cribe some forthcoming works and some open problems. \;
This tal k is based on several papers and ongoing collaborations with E. Bruè\, M. Fogagnolo\, S. Nardulli\, E. Pasqualetto\, M. Pozzetta\, and D. Semola. p> DTEND;TZID=Europe/Zurich:20211124T160000
END:VEVENT BEGIN:VEVENT UID:news1255@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211021T155955 DTSTART;TZID=Europe/Zurich:20211103T141500 SUMMARY:Seminar Analysis and Mathematical Physics: In-Jee
Jeong (Seoul Nati onal University) DESCRIPTION:The evolution of incompressible inviscid fluids is governed by the Euler equations. We consider the dynamics of vortex rings\, which are axisymmetric
solutions to the three dimensional Euler equations with conce ntrated axial vorticity. We prove the following infinite norm growth resul ts: (i) filamentation (formation of a long tail) behavior from
a single vo rtex ring\, and (ii) vortex stretching from the "collision" of two vortex rings with opposite signs. Joint work with Kyudong Choi (UNIST). X-ALT-DESC:
The evolution of incompressible inviscid fluids is governed b y the Euler equations. We consider the dynamics of vortex rings\, which ar e axisymmetric solutions to the three dimensional Euler
equations with con centrated axial vorticity. We prove the following infinite norm growth res ults: (i) filamentation (formation of a long tail) behavior from a single vortex ring\, and (ii) vortex
stretching from the "collision" of two vorte x rings with opposite signs. \;Joint work with Kyudong Choi (UNIST). p> DTEND;TZID=Europe/Zurich:20211103T160000 END:VEVENT BEGIN:VEVENT
UID:news1225@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210908T121913 DTSTART;TZID=Europe/Zurich:20211006T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Emil Wiedemann (Universi tät Ulm)
DESCRIPTION:The concept of measure-valued solution was introduced to the th eory of hyperbolic conservation laws by DiPerna in the 1980s. For nonlinea r systems of hyperbolic type\, such as the Euler
equations of ideal fluids \, measure-valued solutions are often the only available notion of solutio n\, as the existence of 'honest' solutions is still unknown. Although this relaxation of the
solution concept seems like a vast generalization\, whe re a lot of infomation is lost\, it turned out in work of Székelyhidi-W. (2012) that every measure-valued solution can be approximated by weak
ones in the incompressible situation. For compressible flows\, however\, the s ituation is much different. I will discuss recent progress in this directi on in joint work with Dennis Gallenmüller.
The concept of measure-valued solution was introduced to the theory of hyperbolic conservation laws by DiPerna in the 1980s. For nonlin ear systems of hyperbolic type\, such as the Euler equations of
ideal flui ds\, measure-valued solutions are often the only available notion of solut ion\, as the existence of 'honest' solutions is still unknown. Although th is relaxation of the solution concept
seems like a vast generalization\, w here a lot of infomation is lost\, it turned out in work of Székelyhidi-W . (2012) that every measure-valued solution can be approximated by weak on es in the
incompressible situation. For compressible flows\, however\, the situation is much different. I will discuss recent progress in this direc tion in joint work with Dennis Gallenmüller.
DTEND;TZID=Europe/Zurich:20211006T160000 END:VEVENT BEGIN:VEVENT UID:news1190@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210601T082910 DTSTART;TZID=Europe/Zurich:20210602T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Corentin Le Bihan (ENS L yon) DESCRIPTION:A simple model of gas is the hard spheres model. It is a billia rd of little particles which can interact very strongly at
very small dist ance (think for example of real billiards with a lot of balls). Because un derstanding such system is an outstanding problem\, people tried to find a limiting process. A first
equation governing the density of one particle was given by Boltzmann :∂tf+v⋅∇xf=Q(f\,f)\\r\\nIn its formal derivat ion Boltzmann supposed that two different particles are almost independent \, so
the probability of having two particles at the same place is the the product of probability. The validity of such equation is a priori not cle ar since it adds some irreversibly that does not exist
in the hard sphere model.\\r\\nLanford solved the problem in its ‘75 paper: Boltzmann’s e quation is true\, up to a time independent of the number of particles (how ever each particle will have in
mean less than one collision).\\r\\nNow co mes the question of the boundary. We expect to find some “Lanfords” th eorem even if we add some boundary condition. A first example are the spec ular
reflections\, for a deterministic law. An other example\, which would be very important in physics\, is the evolution of a gas between two hot plaques. Then the reflection condition is stochastic. I
am interested in a third type of reflection\, also stochastic\, which is a modeling of a rou gh boundary.\\r\\nDuring my talk I will present some ideas of the proof of Boltzmann in the torus R^3/Z^3
and the adaptation in the case of a domain with boundaries. X-ALT-DESC:
A simple model of gas is the hard spheres model. It is a bill iard of little particles which can interact very strongly at very small di stance (think for example of real billiards with a lot of
balls). Because understanding such system is an outstanding problem\, people tried to find a limiting process. A first equation governing the density of one particl e was given by Boltzmann :∂tf+v⋅ ∇
xf=Q(f \,f)
In its formal derivation Boltzman n supposed that two different particles are almost independent\, so the pr obability of having two particles at the same place is the the product of probability. The
validity of such equation is a priori not clear since it adds some irreversibly that does not exist in the hard sphere mod el.
Lanford solved the problem in its ‘75 paper: Boltzmann’s e quation is true\, up to a time independent of the number of particles (how ever each particle will have in mean less than one collision).
Now comes the question of the boundary. We expect to find some “Lanfords” theorem even if we add some boundary condition. A first example are the s pecular reflections\, for a deterministic law. An
other example\, which wo uld be very important in physics\, is the evolution of a gas between two h ot plaques. Then the reflection condition is stochastic. I am interested i n a third type of
reflection\, also stochastic\, which is a modeling of a rough boundary.
During my talk I will present some ideas of the pr oof of Boltzmann in the torus R^3/Z^3 and the adaptation in the case of a domain with boundaries.
DTEND;TZID=Europe/Zurich:20210602T160000 END:VEVENT BEGIN:VEVENT UID:news1188@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210510T074110 DTSTART;TZID=Europe/Zurich:20210526T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Bjorn Berntson (KTH Stoc kholm) DESCRIPTION:The half-wave maps (HWM) equation is a recently-introduced inte grable PDE with two distinct relations to the
(trigonometric) spin Caloge ro-Moser-Sutherland (CMS) many-body system. Firstly\, the HWM equation ar ises as a certain continuum limit of the CMS system and secondly\, the sol iton solutions of the
HWM equation are governed by a complexified version of the CMS system. We present generalizations of the HWM equation that ar e similarly related to the hyperbolic and elliptic spin CMS systems. Th
is talk is based on joint work with Rob Klabbers and Edwin Langmann. X-ALT-DESC:
The half-wave maps (HWM) equation is a recently-introduced in tegrable PDE with two distinct relations to the (trigonometric) spin \ ;Calogero-Moser-Sutherland (CMS) many-body \;system.
Firstly\, the HWM equation arises as a certain continuum limit of the CMS system and second ly\, the soliton solutions of the HWM equation are governed by a complexif ied version of the \;CMS
system. We present generalizations of the HWM equation that are similarly related to the \;hyperbolic and elliptic spin \;CMS systems. \;This talk is based on joint work with Rob Kl
abbers and Edwin Langmann.
DTEND;TZID=Europe/Zurich:20210526T151500 END:VEVENT BEGIN:VEVENT UID:news1176@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210517T094220 DTSTART;TZID=Europe/Zurich:20210519T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Théotime Girardot (LPMM C) DESCRIPTION:In two-dimensional space there are possibilities for quantum st atistics continuously interpolating between the bosonic and
the fermionic one. \\r\\nQuasi-particles obeying such statistics can be described as ord inary bosons and fermions with magnetic interactions. \\r\\nWe study a lim it situation where the statistics/
magnetic interaction is seen as a “per turbation from the fermionic end”. \\r\\nWe vindicate a mean-field appro ximation\, proving that the ground state of a gas of anyons is described t o leading
order by a semi-classical\, Vlasov-like\, energy functional.\\r\ \nThe ground state of the latter displays anyonic behavior in its momentum distribution. After introducing and stating this result I
will give eleme nts of proof based on coherent states\,\\r\\nHusimi functions\, the Diacon is-Freedman theorem and a quantitative version of a semi-classical Pauli p inciple. X-ALT-DESC:
In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermioni c one.
Quasi-particles obeying such statistics can be described a s ordinary bosons and fermions with magnetic interactions.
We stu dy a limit situation where the statistics/magnetic interaction is seen as a “perturbation from the fermionic end”.
We vindicate a mean- field approximation\, proving that the ground state of a gas of anyons is described to leading order by a semi-classical\, Vlasov-like\, energy func tional.
The ground state of the latter displays anyonic behavior i n its momentum distribution. After introducing and stating this result I w ill give elements of proof based on coherent states\,
Husimi funct ions\, the Diaconis-Freedman theorem and a quantitative version of a semi- classical Pauli pinciple.
DTEND;TZID=Europe/Zurich:20210519T160000 END:VEVENT BEGIN:VEVENT UID:news1172@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210426T124049 DTSTART;TZID=Europe/Zurich:20210428T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Soeren Petrat (Jacobs Un iversity Bremen) DESCRIPTION:We consider the non-relativistic quantum dynamics of N bosons i n the mean-field scaling limit. It is known
that the leading order behavi or is described by the Hartree equation\, and the next-to-leading order b y Bogoliubov theory. Here\, we prove a perturbative expansion around Bog oliubov theory: a norm
approximation of the true solution to the Schroedi nger equation to any order in 1/N. The coefficients in the expansion are independent of N\, and can be computed from the solutions to the Hartree
and Bogoliubov equations alone. Our expansion leads to approximations of correlation functions and reduced densities to any order in 1/N. In this sense we have completely solved the dynamics of this
mean-field model\, a t least for bounded interaction potentials.\\r\\nThis is joint work with L ea Bossmann\, Peter Pickl\, and Avy Soffer. X-ALT-DESC:
We consider the non-relativistic quantum dynamics of N bosons in the \;mean-field scaling limit. It is known that the leading order behavior is \;described by the Hartree equation\, and the
next-to-lea ding order by \;Bogoliubov theory. Here\, we prove a perturbative expa nsion around \;Bogoliubov theory: a norm approximation of the true sol ution to the \;Schroedinger
equation to any order in 1/N. The coeffici ents in the \;expansion are independent of N\, and can be computed fro m the solutions \;to the Hartree and Bogoliubov equations alone. Our e
xpansion leads to \;approximations of correlation functions and reduce d densities to any \;order in 1/N. In this sense we have completely so lved the dynamics of \;this mean-field model
\, at least for bounded in teraction potentials.
This is joint work with Lea Bossmann\, Peter Pickl\, and Avy Soffer.
DTEND;TZID=Europe/Zurich:20210428T151500 END:VEVENT BEGIN:VEVENT UID:news1162@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210316T094354 DTSTART;TZID=Europe/Zurich:20210407T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Jules Pitcho (University of Zurich) DESCRIPTION:The recent work of Brué\, Colombo and De Lellis has establishe d that\, for Sobolev vector fields\, the continuity
equation may be well-p osed in a Lagrangian sense\, yet trajectories of the associated ODE need n ot be unique. We describe how a convex integration scheme for the continui ty equation reveals these
degenerate integral curves\; we modify this sche me to produce Sobolev vector fields for which “most” integral curves a re degenerate. More precisely\, we produce Sobolev vector fields which hav e
any finite number of integral curves starting almost everywhere. This is a joint work with Massimo Sorella. X-ALT-DESC:
The recent work of Brué\, Colombo and De Lellis has establis hed that\, for Sobolev vector fields\, the continuity equation may be well -posed in a Lagrangian sense\, yet trajectories of the
associated ODE need not be unique. We describe how a convex integration scheme for the contin uity equation reveals these degenerate integral curves\; we modify this sc heme to produce Sobolev vector
fields for which “most” integral curves are degenerate. More precisely\, we produce Sobolev vector fields which h ave any finite number of integral curves starting almost everywhere. This is a joint
work with Massimo Sorella. \;
DTEND;TZID=Europe/Zurich:20210407T160000 END:VEVENT BEGIN:VEVENT UID:news1138@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210318T140108 DTSTART;TZID=Europe/Zurich:20210324T140000 SUMMARY:Seminar
Analysis and Mathematical Physics: Jonas Lampart (CNRS\, LI CB) DESCRIPTION:I will discuss some properties of the set of all trajectories t hat can be obtained from a fixed initial state by varying
the potential in the Schrödinger equation. This is related to the control problem\, i.e. driving the system to a target state\, which turns out to be impossible fo r "typical" target states using
bounded potentials. X-ALT-DESC:
I will discuss some properties of the set of all trajectories that can be obtained from a fixed initial state by varying the potential in the Schrödinger equation.
This is related to the control problem \, i.e. driving the system to a target state\, which turns out to be impos sible for "typical" target states using bounded potentials.
DTEND;TZID=Europe/Zurich:20210324T160000 END:VEVENT BEGIN:VEVENT UID:news1141@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210223T224755 DTSTART;TZID=Europe/Zurich:20210317T140000 SUMMARY:Seminar
Analysis and Mathematical Physics: Alessandro Olgiati (Univ ersity of Zurich) DESCRIPTION:We study the ground state properties of a system of bosonic par ticles trapped by a double-well potential\,
in the limit of large inter-we ll separation and of high potential barrier. The N bosons also interact vi a a mean-field two-body potential\, in the limit of large N.The leading-or der physics is
governed by a Bose-Hubbard Hamiltonian coupling two low-ene rgy modes\, each supported in the bottom of one well. Fluctuations beyond these two modes are ruled by two independent Bogoliubov
Hamiltonians\, one for each well.Our main result is that the variance of the number of parti cles in the low-energy modes is suppressed. This is a violation of the Cen tral Limit Theorem which holds
in the occurrence of Bose-Einstein condensa tion\, and therefore it signals that particles develop correlations in the ground state. We achieve our result by proving a precise energy expansion in
term of Bose-Hubbard and Bogoliubov energies.Joint work with Nicolas R ougerie (ENS Lyon) and Dominique Spehner (Universidad de Concepción). X-ALT-DESC:We study the ground state properties of a
system of bosonic part icles trapped by a double-well potential\, in the limit of large inter-wel l separation and of high potential barrier. The N bosons also interact via a mean-field two-body
potential\, in the limit of large N.
The leadi ng-order physics is governed by a Bose-Hubbard Hamiltonian coupling two lo w-energy modes\, each supported in the bottom of one well. Fluctuations be yond these two modes are ruled by two
independent Bogoliubov Hamiltonians\ , one for each well.
Our main result is that the variance of the numb er of particles in the low-energy modes is suppressed. This is a violation of the Central Limit Theorem which holds in the occurrence of Bose-Einste
in condensation\, and therefore it signals that particles develop correlat ions in the ground state. We achieve our result by proving a precise energ y expansion in term of Bose-Hubbard and
Bogoliubov energies.
Joint wo rk with Nicolas Rougerie (ENS Lyon) and Dominique Spehner (Universidad de Concepción). DTEND;TZID=Europe/Zurich:20210317T160000 END:VEVENT BEGIN:VEVENT UID:news1137@dmi.unibas.ch DTSTAMP;
TZID=Europe/Zurich:20210223T225158 DTSTART;TZID=Europe/Zurich:20210303T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Peter Pickl (LMU Münche n) DESCRIPTION:Abstract: The derivation of
effective descriptions from microsc opic dynamics is a very vivid area in mathematical physics. In the talk I will discuss a system of many particles with Newtoni an time
evolution that are subject to interaction. It is well k nown that in the weak coupling limit this system converges\, un der smoothness assumption on the interaction force\, to a solut
ion of the Vlasov equation. Weakening the types of convergence (con vergence for all initial conditions -> convergence in probabili ty -> convergence in distribution) the smoothness
condition on the interaction can be generalized. In the talk I will present recent results in this direction and explain\, which types of c onvergence hold/do not hold under the
different assumptions on the interaction force. X-ALT-DESC:
Abstract: The derivation of effective descriptions from micro scopic \; \; \; \; \; \;dynamics is a very vivid a rea in mathematical physics. In the talk I \; \; 
\; \;&nbs p\; \;will discuss a system of many particles with Newtonian time evol ution \; \; \; \; \; \;that are subject to interac tion. It is well known that in the
weak \; \; \; \; \;  \;coupling limit this system converges\, under smoothness assumption on \; \; \; \; \; \;the interaction force\, to a s olution of
the Vlasov equation. \; \; \; \;Weakening the t ypes of convergence (convergence for all initial \; \; \;  \; \; \;conditions ->\; convergence in probability
->\; conver gence in \; \; \; \; \; \;distribution) the smooth ness condition on the interaction can be \; \; \; \; \ ; \;generalized. In the talk I
will present recent results in this&nbs p\; \; \; \; \; \;direction and explain\, which types of convergence hold/do not hold \; \; \; \; \; \;u nder the
different assumptions on the interaction force.
DTEND;TZID=Europe/Zurich:20210303T161500 END:VEVENT BEGIN:VEVENT UID:news1125@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201130T113309 DTSTART;TZID=Europe/Zurich:20201216T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Alessandro Goffi (Univer sity of Padova) DESCRIPTION:The problem of maximal regularity in Lebesgue spaces is a heavi ly studied question for linear PDEs that dates
back to Calderón and Zygmu nd for the Poisson equation and the potential theoretic approach developed by Ladyzhenskaya et al for the heat equation\, and represents the corners tone in the analysis of
many nonlinear PDEs. After a brief overview of the linear theory\, in this talk I will focus on maximal L^q-regularity for v iscous Hamilton-Jacobi equations with superlinear first order terms. I wil
l first survey on recent results obtained for the stationary equation\, wh ich answer positively to a conjecture raised by P.-L. Lions\, via a Bernst ein-type argument. Then\, I will discuss
Lipschitz and optimal L^q-regular ity for parabolic Hamilton-Jacobi equations that instead are tackled throu gh a refinement of the Evans’ nonlinear adjoint method\, thus exploiting fine regularity
properties for advection-diffusion equations with “roug h” drifts\, providing new regularity results for systems of PDEs arising in the theory of Mean Field Games. These are joint works with Marco
Ciran t (Padova). X-ALT-DESC:
The problem of maximal regularity in Lebesgue spaces is a hea vily studied question for linear PDEs that dates back to Calderón and Zyg mund for the Poisson equation and the potential theoretic
approach develop ed by Ladyzhenskaya et al for the heat equation\, and represents the corne rstone in the analysis of many nonlinear PDEs. After a brief overview of t he linear theory\, in this talk
I will focus on maximal L^q-regularity for viscous Hamilton-Jacobi equations with superlinear first order terms. I w ill first survey on recent results obtained for the stationary equation\, which
answer positively to a conjecture raised by P.-L. Lions\, via a Bern stein-type argument. Then\, I will discuss Lipschitz and optimal L^q-regul arity for parabolic Hamilton-Jacobi equations that
instead are tackled thr ough a refinement of the Evans’ nonlinear adjoint method\, thus exploiti ng fine regularity properties for advection-diffusion equations with “ro ugh” drifts\, providing new
regularity results for systems of PDEs arisi ng in the theory of Mean Field Games. These are joint works with Marco Cir ant (Padova). \;
DTEND;TZID=Europe/Zurich:20201216T160000 END:VEVENT BEGIN:VEVENT UID:news1126@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201122T211830 DTSTART;TZID=Europe/Zurich:20201209T150000 SUMMARY:Seminar
Analysis and Mathematical Physics: Mickaël Latocca (ENS Pa ris) DESCRIPTION:The possible growth of the Sobolev norms of the solution to the 2d (and 3d) Euler equations and its quantification remains
ill-known. Th e only general bound is double exponential. Conversely\, such a double e xponential growth scenario occurs for specific initial data in the settin g of the disc (Kiselev-Sverak). In the
setting of the torus\, only an exp onentially growing scenario has been exhibited (Zlatos). Could the double exponential scenario occur on the torus? What is the typical behaviour t hat could be
expected? It is highly possible that on the torus\, Sobolev norms generically do not grow fast.In this talk\, I will present some resu lts obtained in this direction. We will construct invariant
measures for the 2d Euler equation at high regularity ($H^s$\, $s>2$) and prove that o n the support of the measure\, Sobolev norms do not grow faster than poly nomially.Refining the method allows to
construct an invariant measure to t he 3d Euler equations at high regularity ($H^s$\, $s>7/2$) and thus const ructglobal dynamics on the support of the measure\, exhibiting at most po lynomial
growth.Finally\, it time permits we will discuss the properties o f the measures constructed. X-ALT-DESC:The possible growth of the Sobolev norms of the solution to the& nbsp\;2d (and 3d) Euler
equations and its quantification remains ill-known . The \;only general bound is double exponential. Conversely\, such a double \;exponential growth scenario occurs for specific initial data
in the \;setting of the disc (Kiselev-Sverak). In the setting of the t orus\, only an \;exponentially growing scenario has been exhibited (Zl atos). Could the \;double exponential
scenario occur on the torus? Wha t is the typical \;behaviour that could be expected? It is highly poss ible that on the torus\, \;Sobolev norms generically do not grow fast.
In this talk\, I will present some results obtained in this di rection. We \;will construct invariant measures for the 2d Euler equat ion at high \;regularity ($H^s$\, $s>\;2$) and prove
that on the sup port of the measure\, \;Sobolev norms do not grow faster than polynomi ally.
Refining the method allows to construct an invariant meas ure to the 3d \;Euler equations at high regularity ($H^s$\, $s>\;7/2 $) and thus construct
global dynamics on the support of the measure\, exhibiting at most \;polynomial growth.
Finally\, it time permits we will discuss the properties of the measures \;constructed. DTEND;TZID=Europe/Zurich:20201209T160000 END:VEVENT BEGIN:VEVENT UID:news1106@dmi.unibas.ch DTSTAMP;TZID=
Europe/Zurich:20201116T100656 DTSTART;TZID=Europe/Zurich:20201209T140000 SUMMARY:Seminar Analysis and Mathematical Physics: Lea Bossmann (IST Austri a) DESCRIPTION:Abstract. \\r\\nWe consider a
system of N bosons in the mean-f ield scaling regime in an external trapping potential. We derive an asymp totic expansion of the low-energy eigenstates and the corresponding ener gies\, which
provides corrections to Bogoliubov theory to any order in 1/ N. We show that the structure of the ground state and of the non-degenera te low-energy eigenstates is preserved by the dynamics ifthe
external trap is switched off. This talk is based on joint works with Sören Petrat\, Peter Pickl\, Robert Seiringer\, and Avy Soffer (arXiv:1912.11004 and arX iv:2006.09825). X-ALT-DESC:
We consider a system of N bosons in the mean-field scaling \;regime in an external trapping potential. We derive an asymptotic \;expansion of the low-energy eigenstates and the corresponding&
nbsp\;energies\, which provides corrections to Bogoliubov t heory to any order \;in 1/N. We show that the structure of the ground state and of the \;non-degenerate low-energy eigenstates is
preserved by the dynamics if
the external trap is switched off. This talk is ba sed on joint works \;with Sören Petrat\, Peter Pickl\, Robert Seiring er\, and Avy Soffer \;(arXiv:1912.11004 and arXiv:2006.09825).
DTEND;TZID=Europe/Zurich:20201209T150000 END:VEVENT BEGIN:VEVENT UID:news1124@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201118T092353 DTSTART;TZID=Europe/Zurich:20201202T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Jaemin Park (Georgia Ins titute of Technology) DESCRIPTION:In this talk\, I will discuss whether all stationary/uniformly- rotating solutions of 2D Euler equation
must be radially symmetric\, if th e vorticity is compactly supported. For a stationary solution that is eith er smooth or of patch type\, we prove that if the vorticity does not chang e sign\, it
must be radially symmetric up to a translation. It turns out t hat the fixed-sign condition is necessary for radial symmetry result: inde ed\, we are able to find non-radial sign changing stationary
solution with compact support. We have also obtained some sharp criteria on symmetry fo r uniformly-rotating solutions for 2D Euler equation and the SQG equation. The symmetry results are mainly
obtained by calculus of variations and el liptic equation techniques\, and the construction of non-radial solution i s obtained from bifurcation theory. Part of this talk is based on joint wo rk with
Javier Gomez-Serrano\, Jia Shi and Yao Yao. X-ALT-DESC:In this talk\, I will discuss whether all stationary/uniformly-r otating solutions of 2D Euler equation must be radially symmetric\, if the
vorticity is compactly supported. For a stationary solution that is eithe r smooth or of patch type\, we prove that if the vorticity does not change sign\, it must be radially symmetric up to a
translation. It turns out th at the fixed-sign condition is necessary for radial symmetry result: indee d\, we are able to find non-radial sign changing stationary solution with compact support. We
have also obtained some sharp criteria on symmetry for uniformly-rotating solutions for 2D Euler equation and the SQG equation. The symmetry results are mainly obtained by calculus of variations and
ell iptic equation techniques\, and the construction of non-radial solution is obtained from bifurcation theory. Part of this talk is based on joint wor k with Javier Gomez-Serrano\, Jia Shi and Yao
Yao. DTEND;TZID=Europe/Zurich:20201202T160000 END:VEVENT BEGIN:VEVENT UID:news1107@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201116T101053 DTSTART;TZID=Europe/Zurich:20201125T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Emanuela Giacomelli (LMU Munich) DESCRIPTION:Abstract: We consider N spin 1/2 fermions interacting wit h a positive and regular enough potential in three
dimensions. We compute the ground state energy of the system in the dilute regime at second ord er in the particle density. We recover a well-know expression for the gro und state energy which
depends on the interaction potentials only via its scattering length. A first proof of this result has been given by Lieb\, Seiringer and Solovej. We discuss a new derivation of this formula whic
h makes use of the almost-bosonic nature of the low-energy excitations of the systems. Based on a joint work with Marco Falconi\, Christian Hain zl\, Marcello Porta. X-ALT-DESC:
We consider N spin 1/2 f ermions interacting with a positive \;
and regular enough potenti al in three dimensions. We compute the ground \;
state energy of the system in the dilute regime at second order in the \;
particl e density. We recover a well-know expression for the ground state \;energy which depends on the interaction potentials only via its \;
scattering length. A first proof of this result has been given by Li eb\, \;
Seiringer and Solovej. We discuss a new derivation of thi s formula \;
which \; makes use of the almost-bosonic nature of the low-energy \;
excitations of the systems. Based on a joint work with Marco Falconi\, \;
Christian Hainzl\, Marcello Porta.< /p> DTEND;TZID=Europe/Zurich:20201125T161500 END:VEVENT BEGIN:VEVENT UID:news1100@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201102T110041 DTSTART;TZID=Europe/
Zurich:20201111T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Lars Eric Hientzsch (Ins titut Fourier\, University of Grenoble Alpes) DESCRIPTION:The quantum Navier-Stokes (QNS) equations
describe a compressib le fluid including a degenerate density dependent viscosity and a dispersi ve tensor accounting for capillarity effects. The system can be seen as vi scous correction of the
Quantum Hydrodynamics (QHD) arising e.g. as protot ype model in the description of superfluidity. We consider the (QNS) syste m on the whole space with non-trivial farfield behaviour providing the
sui table framework to study coherent structures and the incompressible limit. \\r\\nFirst\, we prove global existence of finite energy weak solutions (F EWS) in dimension two and three. To
compensate for the lack of control of the velocity field around vacuum regions\, we construct approximate soluti ons to a truncated formulation of (QNS) on a sequence of invading domains. Suitable
compactness properties are inferred from the Bresch-Desjadins en tropy estimates. This is joint work with P. Antonelli and S. Spirito.\\r\\ nSecond\, we address the low Mach number limit for FEWS to
the (QNS) syste m (in collaboration with P. Antonelli and P. Marcati). The main novelty is a precise analysis of the acoustic dispersion altered by the presence of the dispersive capillarity tensor.
The linearised system is governed by the Bogoliubov dispersion relation. The desired decay of the acoustic part follows from refined Strichartz estimates. X-ALT-DESC:
The quantum Navier-Stokes (QNS) equations describe a compress ible fluid including a degenerate density dependent viscosity and a disper sive tensor accounting for capillarity effects. The system can
be seen as viscous correction of the Quantum Hydrodynamics (QHD) arising e.g. as prot otype model in the description of superfluidity. We consider the (QNS) sys tem on the whole space with
non-trivial farfield behaviour providing the s uitable framework to study coherent structures and the incompressible limi t.
First\, we prove global existence of finite energy weak solutio ns (FEWS) in dimension two and three. To compensate for the lack of contro l of the velocity field around vacuum regions\, we construct
approximate s olutions to a truncated formulation of (QNS) on a sequence of invading dom ains. Suitable compactness properties are inferred from the Bresch-Desjadi ns entropy estimates. This is joint
work with P. Antonelli and S. Spirito.
Second\, we address the low Mach number limit for FEWS to the (QN S) system (in collaboration with P. Antonelli and P. Marcati). The main no velty is a precise analysis of the acoustic dispersion
altered by the pres ence of the dispersive capillarity tensor. The \;linearised \;syst em is governed by the Bogoliubov dispersion relation. The desired decay of the acoustic part follows
from refined Strichartz estimates. \;
DTEND;TZID=Europe/Zurich:20201111T160000 END:VEVENT BEGIN:VEVENT UID:news1036@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201024T154014 DTSTART;TZID=Europe/Zurich:20201104T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Marco Falconi (Universit y of Roma Tre) DESCRIPTION:Abstract. \\r\\nIn this talk I will overview variational proble ms arising from the study ofquantum matter
interacting with a macroscopic force field. These interactions are verycommon in both solid state and con densed matter physics\, as well as in higher energysettings. In particular \, I will focus on
the link between the effective and microscopicdescripti on of such variational problems\, using techniques of quasi-classical anal ysisdeveloped in recent years in collaboration with M. Correggi and
M. Oli vieri. X-ALT-DESC:
In this talk I will overview variational p roblems arising from the study ofquantum matter interacting with a macrosc opic force field. These interactions are verycommon in both solid state an d
condensed matter physics\, as well as in higher energysettings. In parti cular\, I will focus on the link between the effective and microscopicdesc ription of such variational problems\, using
techniques of quasi-classical analysisdeveloped in recent years in collaboration with M. Correggi and M . Olivieri.
DTEND;TZID=Europe/Zurich:20201104T160000 END:VEVENT BEGIN:VEVENT UID:news1086@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201007T090901 DTSTART;TZID=Europe/Zurich:20201021T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Maria Teresa Chiri (Penn State University) DESCRIPTION:A posteriori Error Estimates for Numerical Solutions to Hyperbo lic Conservation Laws X-ALT-DESC:A posteriori
Error Estimates for Numerical Solutions to Hyperbol ic Conservation Laws DTEND;TZID=Europe/Zurich:20201021T160000 END:VEVENT BEGIN:VEVENT UID:news1087@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20200930T171430 DTSTART;TZID=Europe/Zurich:20201014T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Silja Haffter (EPFL) DESCRIPTION:The surface quasigeostrophic equation (SGQ) is a
2d physical mo del equation which emerges in meteorology. It has attracted the attention of the mathematical community since it shares many of the essential diffic ulties of 3d fluid dynamics: in the
supercritical regime for instance\, wh ere dissipation is modelled by a fractional Laplacian of order less than 1 /2\, it is not known whether or not smooth solutions blow-up in finite tim e. On the
other hand\, the scheme of Leray still produces global-in-time w eak solutions from any L^2-initial datum\, but their regularity is poorly understood. In this talk\, I will propose a nonempty notion
of "suitable w eak solution" for the supercritical SQG equation and prove that those solu tions are smooth outside a compact set of quantifiable Hausdorff dimension \; in particular they are smooth
almost everywhere. I will also give a con jecture on what we believe to be an optimal dimension estimate. This is a joint work with Maria Colombo (EPFL). X-ALT-DESC:
The surface quasigeostrophic equation (SGQ) is a 2d physical model equation which emerges in meteorology. It has attracted the attentio n of the mathematical community since it shares many of the
essential diff iculties of 3d fluid dynamics: in the supercritical regime for instance\, where dissipation is modelled by a fractional Laplacian of order less than 1/2\, it is not known whether or
not smooth solutions blow-up in finite t ime. On the other hand\, the scheme of Leray still produces global-in-time weak solutions from any L^2-initial datum\, but their regularity is poorl y
understood. In this talk\, I will propose a nonempty notion of "\;su itable weak solution"\; for the supercritical SQG equation and prove t hat those solutions are smooth outside a compact
set of quantifiable Hausd orff dimension\; in particular they are smooth almost everywhere. I will a lso give a conjecture on what we believe to be an optimal dimension estima te.  \;This is a
joint work with Maria Colombo (EPFL).
DTEND;TZID=Europe/Zurich:20201014T160000 END:VEVENT BEGIN:VEVENT UID:news1088@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20200923T091416 DTSTART;TZID=Europe/Zurich:20201007T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Klaus Widmayer (EPFL) DESCRIPTION:A point charge is a particularly basic and important equilibriu m of the Vlasov-Poisson equations\, and the study of its stability
has ins pired several major contributions. In this talk we present some recent wor k\, which brings a fresh perspective on this problem. Our new approach com bines a Lagrangian analysis of the
linearized problem with an Eulerian PDE framework in the nonlinear analysis\, all the while respecting the symple ctic structure. As a result\, for the case of radial initial data\, we see that
solutions are global and in fact disperse to infinity via a modified scattering along trajectories of the linearized flow. This is joint work with Benoit Pausader (Brown University). X-ALT-DESC:
A point charge is a particularly basic and important equilibr ium of the Vlasov-Poisson equations\, and the study of its stability has i nspired several major contributions. In this talk we present
some recent w ork\, which brings a fresh perspective on this problem. Our new approach c ombines a Lagrangian analysis of the linearized problem with an Eulerian P DE framework in the nonlinear
analysis\, all the while respecting the symp lectic structure. As a result\, for the case of radial initial data\, we s ee that solutions are global and in fact disperse to infinity via a modifi ed
scattering along trajectories of the linearized flow. This is joint wor k with Benoit Pausader (Brown University).
DTEND;TZID=Europe/Zurich:20201007T160000 END:VEVENT BEGIN:VEVENT UID:news1002@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20200910T202733 DTSTART;TZID=Europe/Zurich:20200930T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Simone Dovetta (CNR-IMAT I\, Pavia) DESCRIPTION:The talk overviews some recent developments about nonlinear Sch roedinger equations (NLS) on metric graphs.
Precisely\, we concentrate on variational problems for the NLS energy functional subject to the mass con straint. After a brief recap of the well-known behaviour of such model on the real line\, we
address the existence of NLS ground states on noncompac t metric graphs\, with a specific focus on periodic graphs and infinite tr ees. The emergence of threshold phenomena rooted in the nature of
these gr aphs is discussed. Finally\, we provide some insights on the uniqueness of ground states at fixed mass. On the one hand\, uniqueness is shown to hol d for two classes of graphs with
halflines. On the other hand\, a countere xample to uniqueness in full generality is exhibited. The matter we discus s is part of a wider research line\, developed in collaboration with sever al
authors. The results explicitly covered by the talk refer to a series o f papers\, some of which are joint works with Riccardo Adami\, Enrico Serr a and Paolo Tilli. X-ALT-DESC:
The talk overviews some recent developments about nonlinear S chroedinger equations (NLS) on metric graphs. Precisely\, we concentrate o n variational problems for the NLS energy functional subject
to the mass c onstraint. After a brief recap of the well-known behaviour of such model o n the real line\, we address the existence of NLS ground states on noncomp act metric graphs\, with a specific
focus on periodic graphs and infinite trees. The emergence of threshold phenomena rooted in the nature of these graphs is discussed. Finally\, we provide some insights on the uniqueness of ground
states at fixed mass. On the one hand\, uniqueness is shown to h old for two classes of graphs with halflines. On the other hand\, a counte rexample to uniqueness in full generality is exhibited. The
matter we disc uss is part of a wider research line\, developed in collaboration with sev eral authors. The results explicitly covered by the talk refer to a series of papers\, some of which are
joint works with Riccardo Adami\, Enrico Se rra and Paolo Tilli.
DTEND;TZID=Europe/Zurich:20200930T160000 END:VEVENT BEGIN:VEVENT UID:news1022@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20200217T114353 DTSTART;TZID=Europe/Zurich:20200226T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Luigi De Rosa (EPF Lausa nne) DESCRIPTION:I will describe how convex integration techniques can be used t o show that wilddissipative solutions of the
incompressible Euler equation s are typical in the Baire category sense. This also partially solves a co njecture by Philip Isett on the sharpness of the kinetic energy regularity for Hölder
continuous solutions of the Euler equations. The talk will b e based on a recent work obtained in collaboration with Riccardo Tione. X-ALT-DESC:
I will describe how convex integration techniques can be used to show that wilddissipative solutions of the incompressible Euler equati ons are typical in the Baire category sense. This also
partially solves a conjecture by Philip Isett on the sharpness of the kinetic energy regulari ty \;for Hölder continuous solutions of the Euler equations. The talk will be based on a recent work
obtained \;in collaboration with Ricca rdo Tione.
DTEND;TZID=Europe/Zurich:20200226T160000 END:VEVENT BEGIN:VEVENT UID:news929@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191015T153025 DTSTART;TZID=Europe/Zurich:20191023T141500 SUMMARY:Seminar
Analysis and Mathematical Physics: Fabian Ziltener (Utrecht University) DESCRIPTION:The goal of this talk is to show in an example how analysis and symplectic geometry are related in several
ways.Symplectic geometry origi nated from classical mechanics\, where the canonical symplectic form on ph ase space appears in Hamilton's equation. A (smooth) diffeomorphism on a s ymplectic manifold
is called a symplectomorphism iff it preserves the symp lectic form. This happens iff the diffeomorphism solves a certain inhomoge neous quadratic first order system of PDE's. In classical mechanics
symple ctomorphisms play the role of canonical transformations.A famous result by Eliashberg and Gromov states that the set of symplectomorphisms is $C^0$- closed in the set of all diffeomorphisms.
This is remarkable\, since in ge neral\, the $C^0$-limit of a sequence of solutions of a first order system of PDE's need not solve the system. A well-known proof of the Eliashberg- Gromov theorem is
based on Gromov's symplectic nonsqueezing theorem for ba lls.In my talk I will sketch this proof. Furthermore\, I will present a sy mplectic nonsqueezing result for spheres that sharpens Gromov's
theorem. T he proof of this result is based on the existence of a holomorphic map fro m the (real) two-dimensional unit disk to a certain symplectic manifold\, satisfying some Lagrangian boundary
condition. Such a map solves the Cauch y-Riemann equation for a certain almost complex structure. X-ALT-DESC:The goal of this talk is to show in an example how analysis and symplectic geometry are
related in several ways.
Symplectic geo metry originated from classical mechanics\, where the canonical symplectic form on phase space appears in Hamilton's equation. A (smooth) diffeomorp hism on a symplectic manifold is
called a symplectomorphism iff it preserv es the symplectic form. This happens iff the diffeomorphism solves a certa in inhomogeneous quadratic first order system of PDE's. In classical mecha nics
symplectomorphisms play the role of canonical transformations.
< br />A famous result by Eliashberg and Gromov states that the set of sympl ectomorphisms is $C^0$-closed in the set of all diffeomorphisms. This is r emarkable\, since in general\, the $C^0$-limit
of a sequence of solutions of a first order system of PDE's need not solve the system. A well-known p roof of the Eliashberg-Gromov theorem is based on Gromov's symplectic nons queezing theorem for
In my talk I will sketch this proof . Furthermore\, I will present a symplectic nonsqueezing result for sphere s that sharpens Gromov's theorem. The proof of this result is based on the existence of
a holomorphic map from the (real) two-dimensional unit disk to a certain symplectic manifold\, satisfying some Lagrangian boundary con dition. Such a map solves the Cauchy-Riemann equation for a
certain almost complex structure. DTEND;TZID=Europe/Zurich:20191023T160000 END:VEVENT BEGIN:VEVENT UID:news833@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190529T183455 DTSTART;TZID=Europe/
Zurich:20190529T141500 SUMMARY:Seminar Analysis: Mikaela Iacobelli (ETH Zürich) DESCRIPTION:The Vlasov-Poisson system is a kinetic equation that models col lisionless plasma. A plasma has a
characteristic scale called the Debye le ngth\, which is typically much shorter than the scale of observation. In t his case the plasma is called ‘quasineutral’. This motivates studying the limit in
which the ratio between the Debye length and the observation scale tends to zero. Under this scaling\, the formal limit of the Vlasov-P oisson system is the Kinetic Isothermal Euler system.The
Vlasov-Poisson sy stem itself can formally be derived as the limit of a system of ODEs descr ibing the dynamics of a system of N interacting particles\, as the number of particles approaches
infinity. The rigorous justification of this mean field limit remains a fundamental open problem.In this talk we present how the mean field and quasineutral limits can be combined to derive the Kine
tic Isothermal Euler system from a regularised particle model. X-ALT-DESC:\nThe Vlasov-Poisson system is a kinetic equation that models co llisionless plasma. A plasma has a characteristic scale
called the Debye l ength\, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio
between the Debye length and the observation scale tends to zero. Under this scaling\, the formal limit of the Vlasov- Poisson system is the Kinetic Isothermal Euler system.
The Vlasov-Poi sson system itself can formally be derived as the limit of a system of ODE s describing the dynamics of a system of N interacting particles\, as the number of particles approaches
infinity. The rigorous justification of thi s mean field limit remains a fundamental open problem.
In this talk w e present how the mean field and quasineutral limits can be combined to de rive the Kinetic Isothermal Euler system from a regularised particle model . DTEND;TZID=Europe/
Zurich:20190529T160000 END:VEVENT BEGIN:VEVENT UID:news888@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190515T102013 DTSTART;TZID=Europe/Zurich:20190522T141500 SUMMARY:Analysis Seminar: Armin
Schikorra (University of Pittsburgh) DESCRIPTION:The degree of a map between two spheres of same dimension can b e estimated by Sobolev norm of said map (of the right class). In this ta lk I will
discuss to what extend this is possible for the Hopf degree as well – and why the estimate we have is “analytically optimal” but p robably not “topologically optimal”. Joint work with J. Van
Schafting en. X-ALT-DESC:\nThe degree of a map between two spheres of same dimension can be \;estimated by Sobolev norm of said map (of the right \;class). In this talk I will discuss to what
extend this is possible \;for the Hopf degree as well – and why the estimate we \;have is “analytic ally optimal” but probably not “topologically \;optimal”. Joint work with J. Van
Schaftingen. DTEND;TZID=Europe/Zurich:20190522T160000 END:VEVENT BEGIN:VEVENT UID:news856@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190410T101051 DTSTART;TZID=Europe/Zurich:20190424T141500
SUMMARY:Seminar Analysis: Giorgio Stefani (Scuola Normale Superiore di Pisa ) DESCRIPTION:Somewhat surprisingly\, the first appearance of the concept of a fractional derivative is found in a letter
written to de l'Hôpital by L eibniz in 1695. Since then\, Fractional Calculus has fascinated generation s of mathematicians and several definitions of fractional derivatives have appeared. In more
recent years\, the fractional operator defined as the g radient of the Riesz potential has received particular attention\, since i t has revealed to be a useful tool for the study of fractional-order
PDEs and fractional Sobolev spaces. In a joint work with G. E. Comi\, combining the PDE approach developed by Spector and his collaborators with the dist ributional point of view adopted by Šilhavý
\, we introduced new notions of fractional variation and fractional Caccioppoli perimeter in analogy w ith the classical BV theory. Within this framework\, we were able to parti ally extend De
Giorgi’s Blow-up Theorem to sets of locally finite fracti onal Caccioppoli perimeter\, proving existence of blow-ups and giving a fi rst characterisation of these (possibly non-unique) limit sets. In
this ta lk\, after a quick overview on Fractional Calculus\, I will introduce the main features of the fractional operators involved and then give an accoun t on the main results on the fractional
variation we were able to achieve so far. X-ALT-DESC:\nSomewhat surprisingly\, the first appearance of the concept of a fractional derivative is found in a letter written to de l'Hôpital by Leibniz
in 1695. Since then\, Fractional Calculus has fascinated generatio ns of mathematicians and several definitions of fractional derivatives hav e appeared. In more recent years\, the fractional
operator defined as the gradient of the Riesz potential has received particular attention\, since it has revealed to be a useful tool for the study of fractional-order PDEs and fractional Sobolev
spaces. In a joint work with G. E. Comi\, combinin g the PDE approach developed by Spector and his collaborators with the dis tributional point of view adopted by Šilhavý\, \;we introduced new n
otions of fractional variation and fractional Caccioppoli perimeter in ana logy with the classical BV theory. Within this framework\, we were able to partially extend De Giorgi’s Blow-up Theorem to
sets of locally finite fractional Caccioppoli perimeter\, proving existence of blow-ups and givin g a first characterisation of these (possibly non-unique) limit sets. In t his talk\, after a quick
overview on Fractional Calculus\, I will introduc e the main features of the fractional operators involved and then give an account on the main results on the fractional variation we were able to ac
hieve so far. DTEND;TZID=Europe/Zurich:20190424T160000 END:VEVENT BEGIN:VEVENT UID:news828@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190327T120936 DTSTART;TZID=Europe/Zurich:20190410T141500
SUMMARY:Seminar Analysis: Dominik Inauen (University of Zurich) DESCRIPTION:The problem of embedding abstract Riemannian manifolds isometri cally (i.e. preserving the lengths) into Euclidean space
stems from the co nceptually fundamental question of whether abstract Riemannian manifolds a nd submanifolds of Euclidean space are the same. As it turns out\, such em beddings have a drastically
different behaviour at low regularity (i.e. C^ 1) than at high regularity (i.e. C^2): for example\, it's possible to find C^1 isometric embeddings of the standard 2-sphere into arbitrarily small
balls in R^3\, and yet\, in the C^2 category there is (up to translation a nd rotation) just one isometric embedding\, namely the standard inclusion. Analoguous to the Onsager conjecture\, one might
ask if there is a regula rity threshold in the Hölder scale which distinguishes these behaviours. In my talk I will give an overview of what is known concerning the latter question. X-ALT-DESC:\nThe
problem of embedding abstract Riemannian manifolds isometr ically (i.e. preserving the lengths) into Euclidean space stems from the c onceptually fundamental question of whether abstract Riemannian
manifolds and submanifolds of Euclidean space are the same. As it turns out\, such e mbeddings have a drastically different behaviour at low regularity (i.e. C ^1) than at high regularity (i.e. C^2):
for example\, it's possible to fin d C^1 isometric embeddings of the standard 2-sphere into arbitrarily small balls in R^3\, and yet\, in the C^2 category there is (up to translation and rotation)
just one isometric embedding\, namely the standard inclusion . Analoguous to the Onsager conjecture\, one might ask if there is a regul arity threshold in the Hölder scale which distinguishes these
behaviours. In my talk I will give an overview of what is known concerning the latter question. \; DTEND;TZID=Europe/Zurich:20190410T160000 END:VEVENT BEGIN:VEVENT UID:news830@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190327T081509 DTSTART;TZID=Europe/Zurich:20190327T141500 SUMMARY:Seminar Analysis: Tobias Weth (University of Frankfurt) DESCRIPTION:I will report on some recent results
- obtained in joint work w ith Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Oper ator\, which arises as formal derivative of fractional Laplacians at order s= 0. I will discuss
the functional analytic framework for these problems and show how it allows to characterize the asymptotics of principal Diric hlet eigenvalues and eigenfunctions of fractional Laplacians as the
order tends to zero. Furthermore\, I will discuss necessary and sufficient condi tions on domains giving rise to weak and strong maximum principles for the logarithmic Laplacian. If time permits\, I
will also discuss regularity e stimates for solutions to corresponding Poisson problems. X-ALT-DESC:\nI will report on some recent results - obtained in joint work with Huyuan Chen - on Dirichlet
problems for the Logarithmic Laplacian Ope rator\, which arises as formal derivative of fractional Laplacians at orde r s= 0. I will discuss the functional analytic framework for these problem s and
show how it allows to characterize the asymptotics of principal Diri chlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore\, I will discuss necessary
and sufficient cond itions on domains giving rise to weak and strong maximum principles for th e logarithmic Laplacian. If time permits\, I will also discuss regularity estimates for solutions to
corresponding Poisson problems. DTEND;TZID=Europe/Zurich:20190327T160000 END:VEVENT BEGIN:VEVENT UID:news329@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181113T174428 DTSTART;TZID=Europe/
Zurich:20181128T141500 SUMMARY:Seminar Analysis: Stefano Spirito (Università dell'Aquila) DESCRIPTION:In this talk I will present some results concerning the analysi s of finite energy weak solutions
of the Navier-Stokes-Korteweg equations\ , which model the dynamic of a viscous compressible fluid with diffuse int erface. A general theory of global existence is still missing\, however for some
particular cases of physical interest I will present results rega rding the global existence and the compactness of finite energy weak solut ions. The talk is based on a series of joint works with
Paolo Antonelli (G SSI - Gran Sasso Science Institute\, L’Aquila). X-ALT-DESC:\nIn this talk I will present some results concerning the analys is of finite energy weak solutions of the
Navier-Stokes-Korteweg equations \, which model the dynamic of a viscous compressible fluid with diffuse in terface.  \;A general theory of global existence is still missing\, ho wever for some
particular cases of physical interest I will present result s regarding the global existence and the compactness of finite energy weak solutions. The talk is based on a series of joint works with
Paolo Antone lli (GSSI - Gran Sasso Science Institute\, L’Aquila). DTEND;TZID=Europe/Zurich:20181128T160000 END:VEVENT BEGIN:VEVENT UID:news95@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181113T174437
DTSTART;TZID=Europe/Zurich:20181121T141500 SUMMARY:Seminar Analysis: Elia Bruè (Scuola Normale Superiore di Pisa) DESCRIPTION:Since the work by DiPerna and Lions (1989) the continuity and t ransport
equation under mild regularity assumptions on the vector field ha ve been extensively studied\, becoming a florid research field. The applic ability of this theory is very wide\, especially in the
study of partial d ifferential equations and very recently also in the field of non-smooth ge ometry.\\r\\nThe aim of this talk is to give an overview of the quantitati ve side of the theory
initiated by Crippa and De Lellis. We address the pr oblem of mixing and propagation of regularity for solutions to the continu ity equation drifted by Sobolev fields. The problem is well understood
whe n the vector field enjoys a Sobolev regularity with integrability exponent p>1 and basically nothing is known (at the quantitative level) in the cas e p=1.\\r\\nWe present sharp regularity
estimates for the case p>1 and new attempts to attack the challenging question in the case p=1. This is a jo in work with Quoc-Hung Nguyen. X-ALT-DESC:\nSince the work by DiPerna and Lions (1989) the
continuity and transport equation under mild regularity assumptions on the vector field h ave been extensively studied\, becoming a florid research field. The appli cability of this theory is very
wide\, especially in the study of partial differential equations and very recently also in the field of non-smooth g eometry.\nThe aim of this talk is to give an overview of the quantitative side of
the theory initiated by Crippa and De Lellis. We address the probl em of mixing and propagation of regularity for solutions to the continuity equation drifted by Sobolev fields. The problem is well
understood when t he vector field enjoys a Sobolev regularity with integrability exponent p& gt\;1 and basically nothing is known (at the quantitative level) in the ca se p=1.\nWe present sharp
regularity estimates for the case p>\;1 and ne w attempts to attack the challenging question in the case p=1. This is a j oin work with Quoc-Hung Nguyen. DTEND;TZID=Europe/Zurich:20181121T160000
END:VEVENT BEGIN:VEVENT UID:news423@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T184454 DTSTART;TZID=Europe/Zurich:20171204T160000 SUMMARY:Seminar Analysis: Xavier Ros-Oton (University of
Zürich) DESCRIPTION:We present a brief overview of the regularity theory f or free boundaries in different obstacle problems. We describe ho w a monotonicity formula of Almgren plays a central role
in the study of the regularity of the free boundary in some of these problems. Finally \, we explain new strategies which we have recently developed to deal wit h cases in which monotonicity formulas
are not available. X-ALT-DESC:\nWe present a brief overview of the regularity theory for free boundaries in different obstacle problems. We describe h ow a monotonicity formula of Almgren plays a
central role in the study of the regularity of the free boundary in some of these problems. Finall y\, we explain new strategies which we have recently developed to deal wi th cases in which
monotonicity formulas are not available. DTEND;TZID=Europe/Zurich:20171204T180000 END:VEVENT BEGIN:VEVENT UID:news422@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181211T232725 DTSTART;TZID=Europe/
Zurich:20171127T160000 SUMMARY:Seminar Analysis: Jérémy Sok (University of Basel) DESCRIPTION:We consider Dirac operators on the 3-sphere with singular magne tic fields which are supported on links\,
that is on one-dimensional manif olds which are diffeomorphic to finitely many copies of S1. Each connected component carries a flux 2πα which exhibits a 2π-periodicity\, just li ke
Aharonov-Bohmsolenoids in the complex plane. We study the kernel of su ch operators through the spectral flow of loops corresponding to tuning so me flux from 0 to 2π\, that is the number of
eigenvalues crossing 0 along the loop (counted algebraically). It turns out that the spectral flow is generically non-zero and depends on the shape of the curves and their link ing number. Through
the stereographic projection the result extends to R3. And then by smearing out the magnetic fields we obtain new solutions (ψ\ ,A) to the zero-mode equation on R3:\\r\\nσ·(-i∇+A)=0\,(ψ\,A) ∈ H1( R3)
2 × \\dot{H}1(R3)3 ∩ L6(R3)3\,\\r\\nwhere σ=(σ)j=1...3 denotes the family of the Pauli matrices\, A is the magnetic potential associated to the magnetic field ∇×A\, and σ⋅(-i∇+A) is the corresponding
Dirac operator in R3.\\r\\n(Joint work with Fabian Portmann and Jan Philip Solov ej) X-ALT-DESC:\nWe consider Dirac operators on the 3-sphere with singular magn etic fields which are supported on
links\, that is on one-dimensional mani folds which are diffeomorphic to finitely many copies of S^1. Ea ch connected component carries a flux 2πα which exhibits a 2π-periodici ty\, just like
Aharonov-Bohmsolenoids in the complex plane. We study the kernel of such operators through the spectral flow of loops corresponding to tuning some flux from 0 to 2π\, that is the number of
eigenvalues cros sing 0 along the loop (counted algebraically). It turns out that the spect ral flow is generically non-zero and depends on the shape of the curves an d their linking number. Through
the stereographic projection the result ex tends to R^3. And then by smearing out the magnetic fields we ob tain new solutions (ψ\,A) to the zero-mode equation on R^3:\nσ ·(-i∇+A)=0\,
(ψ\,A) ∈ H^1(R^3)^2 × \\dot{H}^1(R^3)^3 ∩ L^6(R^3)^3\,\nwhere σ=(σ)[j=1...3] denotes the famil y of the Pauli matrices\, A is the magnetic potential associated to the ma gnetic field ∇×A\, and σ⋅
(-i∇+A) is the corresponding Dirac operat or in R^3.\n(Joint work with Fabian Portmann and Jan Philip Solo vej) DTEND;TZID=Europe/Zurich:20171127T180000 END:VEVENT BEGIN:VEVENT
UID:news421@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181211T231155 DTSTART;TZID=Europe/Zurich:20171113T160000 SUMMARY:Seminar Analysis: Jozsef Kolumban (Paris Dauphine University) DESCRIPTION:We
consider the motion of a rigid body due to the pr essure of a surrounded two-dimensional irrotational perfect incompress ible fluid\, the whole system being confined in abounded domain with an im
permeable condition on a part of the external boundary. Thanks to an impu lsive control strategy we prove that there exists an appropriate boundary condition on the remaining part of the external
boundary (allowing some fl uid going in and out the domain) such that the immersed rigid body is driv en from some given initial position and velocity to some final position an d velocity in a given
positive time\, without touching the external bounda ry. The controlled part of the external boundary is assumed to have a non void interior and the final position is assumed to be in the same
connecte d component of the set of possible positions as the initial position. X-ALT-DESC: \nWe consider the motion of a rigid body due to the pressure of a surrounded two-dimensional irrotational
perfect incompre ssible fluid\, the whole system being confined in abounded domain with an impermeable condition on a part of the external boundary. Thanks to an im pulsive control strategy we prove
that there exists an appropriate boundar y condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is dr iven from
some given initial position and velocity to some final position and velocity in a given positive time\, without touching the external boun dary. The controlled part of the external boundary is
assumed to have a n onvoid interior and the final position is assumed to be in the same connec ted component of the set of possible positions as the initial position. DTEND;TZID=Europe/
Zurich:20171113T180000 END:VEVENT BEGIN:VEVENT UID:news420@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181211T230805 DTSTART;TZID=Europe/Zurich:20171106T160000 SUMMARY:Seminar Analysis: Sara Danieri
(Friedrich-Alexander University Erla ngen-Nürnberg) DESCRIPTION:We study a functional consisting of a perimeter term and a non- local term which are in competition\, both in the discrete and
continuous setting.In the discrete setting such functional was introduced by Giuliani\, Lebowitz\, Liebe and Seiringer. For both the continuous and discrete problem\, we show that the global
minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers a re invariant under a smaller group of symmetries than the functional itsel f. In the continuous
setting\, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functio nal is invariant under permutation of coordinates and the minimizers
displ ay a pattern formation which is one dimensional. Such behaviour for a sma ller range of exponents in the discrete setting had been already shown\,us ing different techniques. This is a joint
work with E. Runa. X-ALT-DESC:\nWe study a functional consisting of a perimeter term and a non -local term which are in competition\, both in the discrete and continuous setting.In the discrete
setting such functional was introduced by Giuliani\, Lebowitz\, Liebe and Seiringer. For both the continuous and discrete problem\, we show that the global minimizers are exact periodic stripes. One
striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itse lf. In the continuous setting\, to our knowledge this is the
first exampl e of a model with local/nonlocal terms in competition such that the functi onal is invariant under permutation of coordinates and the minimizers disp lay a pattern formation which is one
dimensional. Such behaviour for a sm aller range of exponents in the discrete setting had been already shown\,u sing different techniques. This is a joint work with E. Runa. DTEND;TZID=Europe/
Zurich:20171105T180000 END:VEVENT BEGIN:VEVENT UID:news419@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181211T230239 DTSTART;TZID=Europe/Zurich:20171016T160000 SUMMARY:Seminar Analysis: Fabio Punzo
(Politecnico di Milano) DESCRIPTION:We discuss existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan–Hadamard ma nifolds satisfying suitable
lower bounds on Ricci curvature\, with initia l data that can grow at infinity at a prescribed rate\, that depends cru cially on the curvature bounds. Furthermore\, we give a precise estimate for the
maximal existence time\, and we show that in general solutions do not exist if the initial data grow at infinity too fast. Such results h ave been recently obtained jointly with G. Grillo and M.
Muratori. X-ALT-DESC:\nWe discuss existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan–Hadamard m anifolds satisfying suitable lower bounds
on Ricci curvature\, with initi al data that can grow at infinity at a prescribed rate\, that depends cr ucially on the curvature bounds. Furthermore\, we give a precise estimate for the maximal
existence time\, and we show that in general solutions d o not exist if the initial data grow at infinity too fast. Such results have been recently obtained jointly with G. Grillo and M. Muratori.
DTEND;TZID=Europe/Zurich:20171016T180000 END:VEVENT BEGIN:VEVENT UID:news431@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T194632 DTSTART;TZID=Europe/Zurich:20170510T141500 SUMMARY:Seminar
Analysis: Stefano Spirito (University of L'Aquila) DESCRIPTION:In this talk I will present a result concerning the global exis tence of finite energy weak solutions of the quantum Navier-Stokes
equatio ns. The novelty of the result is that we are able to consider the vacuum in the definition of weak solutions. The main tools are a new formulation of the equations which allows us to get an
additional a priori estimate t o prove compactness and a non trivial choice of the approximation system c onsistent with the a priori estimates.\\r\\nThis is a joint work with Paol o Antonelli (GSSI)
X-ALT-DESC:\nIn this talk I will present a result concerning the global exi stence of finite energy weak solutions of the quantum Navier-Stokes equati ons. The novelty of the result is that we are
able to consider the vacuum in the definition of weak solutions. The main tools are a new formulatio n of the equations which allows us to get an additional a priori estimate to prove compactness and
a non trivial choice of the approximation system consistent with the a priori estimates.\nThis is a joint work with Paolo A ntonelli (GSSI) DTEND;TZID=Europe/Zurich:20170510T151500 END:VEVENT
BEGIN:VEVENT UID:news430@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T194233 DTSTART;TZID=Europe/Zurich:20170503T141500 SUMMARY:Seminar Analysis: Joaquim Serra (ETH Zurich) DESCRIPTION:We will
introduce some recent results in collaboration with L. Caffarelli and X. Ros-Otonon the optimal regularity of the solutions and the regularity of the free boundaries (near regular points) f or
nonlocal obstacle problems.The main novelty is that we obtain re sults for different operators than the fractional Laplacian. Indee d\, we can consider infinitesimal generators of non rotationally
invariant stable L ́evy processes. X-ALT-DESC:\nWe will introduce some recent results in collaboration with L. Caffarelli and X. Ros-Otonon the optimal regularity of the solution s and the regularity
of the free boundaries (near regular points) for nonlocal obstacle problems.The main novelty is that we obtain r esults for different operators than the fractional Laplacian. Inde ed\, we can
consider infinitesimal generators of non rotationally invariant stable L ́evy processes. DTEND;TZID=Europe/Zurich:20170503T151500 END:VEVENT BEGIN:VEVENT UID:news429@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181217T193706 DTSTART;TZID=Europe/Zurich:20170419T141500 SUMMARY:Seminar Analysis: Dominik Himmelsbach (University of Basel) DESCRIPTION:In this talk\, we give sufficient criteria for blowup
of solutions to nonlocal Schrödinger equations with focusing power-ty pe nonlinearity. To give an outline of the arguments used in the proof\, let us mainly focus on the mass-supercritical problem
posed on the whole s pace Rn with prescribed radial initial datum of negative energy.\\r\\nThis is a joint work with Thomas Boulenger and Enno Lenzmann X-ALT-DESC:\nIn this talk\, we give sufficient
criteria for blowup of solutions to nonlocal Schrödinger equations with focusing power-t ype nonlinearity. To give an outline of the arguments used in the proof\, let us mainly focus on the
mass-supercritical problem posed on the whole space R^n with prescribed radial initial datum of negative energ y.\nThis is a joint work with Thomas Boulenger and Enno Lenzmann DTEND;TZID=Europe/
Zurich:20170419T151500 END:VEVENT BEGIN:VEVENT UID:news428@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T192947 DTSTART;TZID=Europe/Zurich:20170405T141500 SUMMARY:Seminar Analysis: Filip Rindler
(University of Warwick) DESCRIPTION:The classical Rademacher Theorem asserts that every Lipschitz f unction is differentiablea lmost everywhere with respect to Lebesgue measu re. On the other hand\,
Preiss (’90) gave a surprising example of a nul lset in the plane such that every Lipschitz function is differentiable at at least one point of this set. Thus\, it is a natural question to ask wh
ether there exists a singular measure such that all Lipschitz functions ar e differentiable with respect to this singular measure. It turns out that this question has an intricate connection to the
geometric structure of normal one-currents. In this talk I will present a converse to Rademacher’s Theorem\, which settles the question in the negative in a ll dimensions: if a positive measure μ has
the property that all Lipschi tz functions are μ-a.e. differentiable\, then μ is absolutely continuou s with respect to Lebesgue measure (in the plane\, this question was alrea dy solved by Alberti\,
Csornyei and Preiss in ’05). In a geometric conte xt\, Cheeger conjectured in ’99 that in all Lipschitz differentiability spaces (which are essentially Lipschitz manifolds in which Rademac her’s
Theorem holds) likewise there is a “functional converse ” to Rademacher’s Theorem. As the second main result\, I will present a recent solution to this conjecture.Technically\, the proofs of b oth of
these theorems are based on a recent structure result for the singu lar parts of PDE-constrained measures\, its corollary on the structure of normalone-currents\, and the powerful theory of Alberti
representations.\\ r\\nThis is a joint work with A. Marchese and G. De Philippis X-ALT-DESC:\nThe classical Rademacher Theorem asserts that every Lipschitz function is differentiablea lmost
everywhere with respect to Lebesgue meas ure. On the other hand\, Preiss (’90) gave a surprising example of a nu llset in the plane such that every Lipschitz function is differentiable at at least
one point of this set. Thus\, it is a natural question to ask w hether there exists a singular measure such that all Lipschitz functions a re differentiable with respect to this singular measure. It
turns out tha t this question has an intricate connection to the geometric structure of normal one-currents. In this talk I will present a converse t o Rademacher’s Theorem\, which settles the
question in the negative in all dimensions: if a positive measure μ has the property that all Lipsch itz functions are μ-a.e. differentiable\, then μ is absolutely continuo us with respect to
Lebesgue measure (in the plane\, this question was alre ady solved by Alberti\, Csornyei and Preiss in ’05). In a geometric cont ext\, Cheeger conjectured in ’99 that in all Lipschitz
differentiability spaces (which are essentially Lipschitz manifolds in which Radema cher’s Theorem holds) likewise there is a “functional converse ” to Rademacher’s Theorem. As the second main result
\, I will present a recent solution to this conjecture.Technically\, the proofs of b oth of these theorems are based on a recent structure result for the singu lar parts of PDE-constrained measures\,
its corollary on the structure of normalone-currents\, and the powerful theory of Alberti representations.\n This is a joint work with A. Marchese and G. De Philippis DTEND;TZID=Europe/
Zurich:20170405T151500 END:VEVENT BEGIN:VEVENT UID:news427@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T192223 DTSTART;TZID=Europe/Zurich:20170329T141500 SUMMARY:Seminar Analysis: Katarzyna
Mazowiecka (University of Freiburg & Un iversity of Warsaw) DESCRIPTION:We investigate a fractional notion of gradient and divergence o perator. We generalize the div-curl estimate by
Coifman-Lions-Meyer-Semmes to fractional div-curl quantities. We demonstrate how these quantities ap pear naturally in nonlocal geometric equations\, which can be used to obta in a regularity theory
for fractional harmonic maps and critical systems w ith nonlocal antisymmetric potential.\\r\\nThis is a joint work with Armin Schikorra X-ALT-DESC:\nWe investigate a fractional notion of gradient
and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semme s to fractional div-curl quantities. We demonstrate how these quantities a ppear naturally in nonlocal
geometric equations\, which can be used to obt ain a regularity theory for fractional harmonic maps and critical systems with nonlocal antisymmetric potential.\nThis is a joint work with Armin Sc
hikorra DTEND;TZID=Europe/Zurich:20170329T151500 END:VEVENT BEGIN:VEVENT UID:news426@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T191055 DTSTART;TZID=Europe/Zurich:20170315T141500
SUMMARY:Seminar Analysis: Christopher Hopper (Aalto University) DESCRIPTION:We prove partial regularity for local minimisers of certain str ictly quasiconvex integral functionals\, over a class of
Sobolev mappings into a compact Riemannian manifold\, to which such mappings are said to be holonomically constrained. Several applications to variational problems i n condensed matter physics with
broken symmetries will also be discussed r elated to the manifold constraint condition. X-ALT-DESC:\nWe prove partial regularity for local minimisers of certain st rictly quasiconvex integral
functionals\, over a class of Sobolev mappings into a compact Riemannian manifold\, to which such mappings are said to b e holonomically constrained. Several applications to variational problems in
condensed matter physics with broken symmetries will also be discussed related to the manifold constraint condition. DTEND;TZID=Europe/Zurich:20170315T151500 END:VEVENT BEGIN:VEVENT
UID:news425@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T192300 DTSTART;TZID=Europe/Zurich:20170301T141500 SUMMARY:Seminar Analysis: Isabella Ianni (University of Naples II) DESCRIPTION:We
consider the semilinear Lane-Emden problem (Ep):\\r\\n-Δu = |u|p-1u in B u = 0 on ∂B\\r\\nwhere B is the unit ball of RN\, N≥3\, centered at the origin and 1< p < pS\, p S=(N+2)/
(N−2). We compute the Morse index of any radial solution up of (Ep)\, for p sufficiently close to pS. The proof exploits the a symptotic behavior of up as p→pS and the analysis of a limit eigenvalue
problem.\\r\\nThis is a joint work with F. De Marchis and F. Pacella X-ALT-DESC:\nWe consider the semilinear Lane-Emden problem (E[p]): \n-Δu = |u|^p-1u \; \; \; in \; B
 \;& nbsp\; u = 0 \; \; \; \; \; \; \; \; \ ; \; \; on ∂B\nwhere B is the unit ball of R^N\, N≥3 \, centered at the origin and 1<\; p <\; p[S]\, p
[S] =(N+2)/(N−2). We compute the Morse index of any radial solutionu p of (E[p])\, for p sufficiently close to p[S]. The proof exploits the asymptotic behavior of u[p] as p→p[S and the analysis of a
limit eigenvalue problem.\nThis is a joint work with F. De Marchis and F. Pacella DTEND;TZID=Europe/Zurich:20170301T151500 END:VEVENT BEGIN:VEVENT UID:news437@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181219T165220 DTSTART;TZID=Europe/Zurich:20161214T141500 SUMMARY:Seminar Analysis: Aleks Jevnikar (University of Rome\, Tor Vergata) DESCRIPTION:A class of Liouville equations and systems on
compact surfaces is considered: we focus on the Toda system which is motivated in mathemati cal physics by the study of models in non-abelian Chern-Simons theory and in geometry in the description of
holomorphic curves in complex analysis. We discuss its variational aspects which yield existence results. X-ALT-DESC:\nA class of Liouville equations and systems on compact surfaces is considered:
we focus on the Toda system which is motivated in mathemat ical physics by the study of models in non-abelian Chern-Simons theory and in geometry in the description of holomorphic curves in complex
analysis. We discuss its variational aspects which yield existence results. \; DTEND;TZID=Europe/Zurich:20161214T141500 END:VEVENT BEGIN:VEVENT UID:news436@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181219T165028 DTSTART;TZID=Europe/Zurich:20161130T141500 SUMMARY:Seminar Analysis: Daniel Ueltschi (University of Warwick) DESCRIPTION:The basic laws governing atoms and electrons are well
understoo d\, but it is impossible to make predictions about the behaviour of large systems of condensed matter physics. A popular approach is to introduce si mple models and to use notions of
statistical mechanics. I will review qua ntum spin systems and their stochastic representations in terms of random permutations and random loops. I will also describe the ‘universal’ be haviour that
is common to loop models in dimensions 3 and more. X-ALT-DESC:\nThe basic laws governing atoms and electrons are well understo od\, but it is impossible to make predictions about the behaviour of
large systems of condensed matter physics. A popular approach is to introduce s imple models and to use notions of statistical mechanics. I will review qu antum spin systems and their stochastic
representations in terms of random permutations and random loops. I will also describe the ‘universal’ b ehaviour that is common to loop models in dimensions 3 and more. \; DTEND;TZID=Europe/
Zurich:20161130T151500 END:VEVENT BEGIN:VEVENT UID:news432@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T201142 DTSTART;TZID=Europe/Zurich:20161123T141500 SUMMARY:Seminar Analysis: Luca Battaglia
(University of Rome\, La Sapienza) DESCRIPTION:I will discuss the existence of groundstate solutions f or the Choquard equation in the whole space RN. I will first conside r the case of a homogeneous
nonlinearity F(u) =|u|p\, then I will prove th e existence of solutions under general hypotheses. In particular\, the ca ses N=2 and N≥3 will have to be treated differently. The solutions are found
through a variational mountain pass strategy. X-ALT-DESC:\nI will discuss the existence of groundstate solutions for the Choquard equation in the whole space R^N. I will f irst consider the case of a
homogeneous nonlinearity F(u) =|u|^p \, then I will prove the existence of solutions under general hypotheses. In particular\, the cases N=2 and N≥3 will have to be treated different ly. The
solutions are found through a variational mountain pass strategy. DTEND;TZID=Europe/Zurich:20161123T151500 END:VEVENT BEGIN:VEVENT UID:news445@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181219T172051
DTSTART;TZID=Europe/Zurich:20160601T141500 SUMMARY:Seminar Analysis: Paolo Bonicatto (SISSA Trieste) DESCRIPTION:Given a bounded\, autonomous vector field b: R2 → R2\, we stu dy the uniqueness of
bounded solutions to the initial value problem for th e related transport equation \\r\\n(1) ∂tu + b · ∇u = 0. \\r\\nWe prove that uniqueness of weak solutions holds under the assumptions that b is
of class BV and it is nearly incompressible. Our proof is based on a s plitting technique (introduced previously by Alberti\, Bianchini and Cripp a) that allows to reduce (1) to a family of
1-dimensional equations which can be solved explicitly\, thus yielding uniqueness for the original probl em. \\r\\nIn order to perform this program\, we use Disintegration Theorem and known results
on the structure of level sets of Lipschitz maps: this is done after a suitable localization of the problem\, in which we exploit also Ambrosio’s superposition principle. \\r\\n \\r\\nThis is joint
work with S. Bianchini and N. A. Gusev. X-ALT-DESC:\nGiven a bounded\, autonomous vector field b: R^2 → R^2\, we study the uniqueness of bounded solutions to the initi al value problem for the
related transport equation \n(1)  \;∂tu + b · ∇u = 0. \;\nWe prove that uniqueness of weak solutions holds und er the assumptions that b is of class BV and it is nearly incompressible. Our
proof is based on a splitting technique (introduced previously by Albe rti\, Bianchini and Crippa) that allows to reduce (1) to a family of 1-dim ensional equations which can be solved explicitly\,
thus yielding uniquene ss for the original problem. \nIn order to perform this program\, we use D isintegration Theorem and known results on the structure of level sets of Lipschitz maps: this is
done after a suitable localization of the problem\ , in which we exploit also Ambrosio’s superposition principle. \;\n \nThis is joint work with S. Bianchini and N. A. Gusev. \; DTEND;TZID=
Europe/Zurich:20160601T151500 END:VEVENT BEGIN:VEVENT UID:news444@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181219T171718 DTSTART;TZID=Europe/Zurich:20160511T141500 SUMMARY:Seminar Analysis: Albert
Clop (UAB Barcelona) DESCRIPTION:We will explain two results about (linear and nonlinear) transp ort equations\, quasiconformal maps\, and vector fields with unbounded div ergence. Originally\, these
results are motivated by a difficult problem o n Muckenhoupt weights and elliptic PDE. However\, classical harmonic analy sis tools allow to reformulate this problem in variational BMO terms\, and
then a theorem by H. M. Reimann brings naturally the connection to the tr ansport theory. X-ALT-DESC:\nWe will explain two results about (linear and nonlinear) trans port equations\, quasiconformal
maps\, and vector fields with unbounded di vergence. Originally\, these results are motivated by a difficult problem on Muckenhoupt weights and elliptic PDE. However\, classical harmonic anal ysis
tools allow to reformulate this problem in variational BMO terms\, an d then a theorem by H. M. Reimann brings naturally the connection to the t ransport theory. \; DTEND;TZID=Europe/
Zurich:20160511T151500 END:VEVENT BEGIN:VEVENT UID:news443@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181219T171215 DTSTART;TZID=Europe/Zurich:20160504T141500 SUMMARY:Seminar Analysis: Renato Lucà
(ICMAT Madrid) DESCRIPTION: \\r\\nWe will discuss some regularity properties of weak so lutions to the three-dimensional Navier–Stokes equation. We will first r ecall the classical partial
regularity theory\, developed by Scheffer and later by Caffarelli–Kohn–Nirenberg. Then we will present some new resu lts in both the small data and perturbative frameworks. X-ALT-DESC: \; \;
\nWe will discuss some regularity properties of we ak solutions to the three-dimensional Navier–Stokes equation. We will fi rst recall the classical partial regularity theory\, developed by Scheffer
and later by Caffarelli–Kohn–Nirenberg. Then we will present some new results in both the small data and perturbative frameworks. \; DTEND;TZID=Europe/Zurich:20160504T151500 END:VEVENT
BEGIN:VEVENT UID:news442@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181219T170855 DTSTART;TZID=Europe/Zurich:20160427T141500 SUMMARY:Seminar Analysis: Laura Caravenna (Università degli Studi di Padov
a) DESCRIPTION:We consider continuous solutions to the single balance law ∂t u+∂x(f(u)) = g\, g bounded\, f ∈ C2. We discuss correspondences among the source terms in the Eulerian and Lagrangian
settings\, extending previ ous works relative to the flux f(u) = u2 when possible. Counterexamples po int out a new behavior of solutions when f is non-convex\, and when the se t of inflection points
of f is not negligible\, stressing the difference a mong the Lagrangian/Eulerian formulations in this context. \\r\\n \\r\\nThis is a joint work with G. Alberti and S. Bianchini. X-ALT-DESC:\nWe
consider continuous solutions to the single balance law ∂ [t]u+∂[x](f(u)) = g\, g bounded\, f ∈ C^2. We discuss correspondences among the source terms in the Eulerian and Lag rangian settings\,
extending previous works relative to the flux f(u) = u< sup>2 when possible. Counterexamples point out a new behavior of sol utions when f is non-convex\, and when the set of inflection points of f i
s not negligible\, stressing the difference among the Lagrangian/Eulerian formulations in this context. \;\n \nThis is a joint work wit h G. Alberti and S. Bianchini. \; DTEND;TZID=Europe/
Zurich:20160427T151500 END:VEVENT BEGIN:VEVENT UID:news441@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181219T170545 DTSTART;TZID=Europe/Zurich:20160420T141500 SUMMARY:Seminar Analysis: Francesco
Ghiraldin (MPI Leipzig) DESCRIPTION:In order to obtain uniqueness for solutions of scalar conservat ion laws with discontinuous flux\, Kruzhkov’s entropy conditions are not enough and additional
dissipation conditions have to be imposed on the di scontinuity set of the flux. Understanding these conditions requires to st udy the structure of solutions on the discontinuity set. I will show
that under quite general assumptions on the flux\, solutions admit traces on th e discontinuity set of the flux. This allows to show that any pair of solu tions satises a Kato type inequality with an
explicit reminder term concen trated on the discontinuities of the flux. Applications to uniqueness is t hen discussed. X-ALT-DESC:\nIn order to obtain uniqueness for solutions of scalar conserva
tion laws with discontinuous flux\, Kruzhkov’s entropy conditions are no t enough and additional dissipation conditions have to be imposed on the d iscontinuity set of the flux. Understanding these
conditions requires to s tudy the structure of solutions on the discontinuity set. I will show that under quite general assumptions on the flux\, solutions admit traces on t he discontinuity set of
the flux. This allows to show that any pair of sol utions satises a Kato type inequality with an explicit reminder term conce ntrated on the discontinuities of the flux. Applications to uniqueness is
then discussed. \; DTEND;TZID=Europe/Zurich:20160420T151500 END:VEVENT BEGIN:VEVENT UID:news440@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181219T170316 DTSTART;TZID=Europe/Zurich:20160316T141500
SUMMARY:Seminar Analysis: Elio Marconi (SISSA Trieste) DESCRIPTION:After a brief overview about the classical well-posedness resul t for scalar conservation laws\, we investigate the structure of
bounded s olutions. In particular we prove that the entropy dissipation measure is c oncentrated on a countably 1-rectifiable set. In order to prove this resul t we introduce the notion of Lagrangian
representation of the solution. \ \r\\n \\r\\nThis is a joint work with Stefano Bianchini. X-ALT-DESC:\nAfter a brief overview about the classical well-posedness resu lt for scalar conservation laws
\, we investigate the structure of bounded solutions. In particular we prove that the entropy dissipation measure is concentrated on a countably 1-rectifiable set. In order to prove this resu lt we
introduce the notion of Lagrangian representation of the solution.&n bsp\;\n \nThis is a joint work with Stefano Bianchini. \; DTEND;TZID=Europe/Zurich:20160316T141500 END:VEVENT BEGIN:VEVENT
UID:news439@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181219T170101 DTSTART;TZID=Europe/Zurich:20160309T141500 SUMMARY:Seminar Analysis: Christian Zillinger (Universität Bonn) DESCRIPTION:The Euler
equations of fluid dynamics are time-reversible equat ions and possess many conserved quantities\, including the kinetic energy and entropy. Furthermore\, as shown by Arnold\, they even have the
structu re of an infinite-dimensional Hamiltonian system. Despite these facts\, in experiments one observes a damping phenomenon for small velocity perturba tions to monotone shear flows\, where the
perturbations decay with algebra ic rates. In this talk\, I discuss the underlying phase-mixing mechanism o f linear inviscid damping\, its mathematical challenges and how to establi sh decay with
optimal rates for a general class of monotone shear flows. H ere\, a particular focus will be on the setting of a channel with impermea ble walls\, where boundary effects asymptotically result in the
formation of singularities. X-ALT-DESC:\nThe Euler equations of fluid dynamics are time-reversible equa tions and possess many conserved quantities\, including the kinetic energy and entropy.
Furthermore\, as shown by Arnold\, they even have the struct ure of an infinite-dimensional Hamiltonian system. Despite these facts\, i n experiments one observes a damping phenomenon for small
velocity perturb ations to monotone shear flows\, where the perturbations decay with algebr aic rates. In this talk\, I discuss the underlying phase-mixing mechanism of linear inviscid damping\, its
mathematical challenges and how to establ ish decay with optimal rates for a general class of monotone shear flows. Here\, a particular focus will be on the setting of a channel with imperme able
walls\, where boundary effects asymptotically result in the formation of singularities. \; DTEND;TZID=Europe/Zurich:20160309T151500 END:VEVENT BEGIN:VEVENT UID:news438@dmi.unibas.ch DTSTAMP;TZID=
Europe/Zurich:20181219T165527 DTSTART;TZID=Europe/Zurich:20160224T141500 SUMMARY:Seminar Analysis: Simon Blatt (Universität Salzburg) DESCRIPTION:While the Willmore energy is invariant under Möbius
transforma tions\, its negative L2-gradient flow is not - simply because the L2-scala r product used in its definition does not have this invariance. In this ta lk we present Möbius invariant
versions of the Willmore flow picking up i deas of Ruben Jakob and Oded Schramm. We will discuss its uses and limitat ions and prove well-posedness of the Cauchy problem and attractivity of lo cal
minimizers. X-ALT-DESC:\nWhile the Willmore energy is invariant under Möbius transform ations\, its negative L^2-gradient flow is not - simply because the L^2-scalar product used in its definition
does not have this invariance. In this talk we present Möbius invariant versions of the Wil lmore flow picking up ideas of Ruben Jakob and Oded Schramm. We will discu ss its uses and limitations and
prove well-posedness of the Cauchy problem and attractivity of local minimizers. \; DTEND;TZID=Europe/Zurich:20160224T141500 END:VEVENT BEGIN:VEVENT UID:news457@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181227T175842 DTSTART;TZID=Europe/Zurich:20151216T141500 SUMMARY:Seminar Analysis: Alessandra Pluda ( Università di Pisa) DESCRIPTION:I consider the motion by curvature of a network of
curves in th e Euclidean plane and I discuss existence\, uniqueness\, and asymptotic be havior of the evolution. In particular\, I focus on two model cases: a r egular embedded network composed by
three curves with fixed endpoints (tri od) and a regular embedded network composed by two curves\, one of which i s closed (spoon). After talking about the state of art of the problem\, I will
present some new and possibly ”incoming” results obtained with C arlo Mantegazza and Matteo Novaga. X-ALT-DESC:\nI consider the motion by curvature of a network of curves in t he Euclidean plane and
I discuss existence\, uniqueness\, and asymptotic b ehavior of the evolution. In particular\, I focus on two model cases: a regular embedded network composed by three curves with fixed endpoints (tr
iod) and a regular embedded network composed by two curves\, one of which is closed (spoon). After talking about the state of art of the problem\, I will present some new and possibly ”incoming”
results obtained with Carlo Mantegazza and Matteo Novaga. DTEND;TZID=Europe/Zurich:20151216T151500 END:VEVENT BEGIN:VEVENT UID:news456@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181227T175828
DTSTART;TZID=Europe/Zurich:20151209T141500 SUMMARY:Seminar Analysis: Federica Sani ( Università di Milano) DESCRIPTION:The Trudinger-Moser inequality is a substitute for the well kno wn Sobolev
embedding theorem when the limiting case is considered. We discuss Moser type inequalities in the whole space which involve c omplete and reduced Sobolev norm.Then we investigate the optimal growth
ra te of the exponential type function both in the first order case and in th e higher order case. X-ALT-DESC:\nThe Trudinger-Moser inequality is a substitute for the well kn own Sobolev embedding
theorem when the limiting case is considered. We discuss Moser type inequalities in the whole space which involve complete and reduced Sobolev norm.Then we investigate the optimal growth r ate of the
exponential type function both in the first order case and in t he higher order case. DTEND;TZID=Europe/Zurich:20151209T151500 END:VEVENT BEGIN:VEVENT UID:news455@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181227T174629 DTSTART;TZID=Europe/Zurich:20151202T141500 SUMMARY:Seminar Analysis: Alessandro Carlotto (ETH-ITS Zürich) DESCRIPTION:Given a closed\, Riemannian 3-manifold (N\, g) without
symmetr ies (more precisely: generic) and a non-negative integer p\, can w e say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically\, can
we prove that such number is necessarily finite? This is the classical ”generic finitenes s” problem\, which has a rich history and exhibits interesting subtletie s even in its basic counterpart
concerning closed geodesics on surfaces. We settle such question when g is a bumpy metric of positive scalar curvat ure by proving that either finiteness holds or N does contain a copy of RP 3 in its
prime decomposition and we discuss the obstructions to any furthe r generalisation of such result. When g is assumed to be strongly bumpy (m eaning that all closed\, immersed minimal surfaces do not
have Jacobi fiel ds\, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary. X-ALT-DESC: \nGiven a closed\, Riemannian
3-manifold (N\, g) without symme tries (more precisely: generic) and a non-negative integer p\, can we say something about the number of minimal surfaces it contains who se Morse index is bounded by
p? More realistically\, can we prove that su ch number is necessarily finite? This is the classical ”generic finiten ess” problem\, which has a rich history and exhibits interesting subtlet ies even
in its basic counterpart concerning closed geodesics on surfaces. We settle such question when g is a bumpy metric of positive scalar curv ature by proving that either finiteness holds or N does
contain a copy of RP^3 in its prime decomposition and we discuss the obstructions to any further generalisation of such result. When g is assumed to be stro ngly bumpy (meaning that all closed\,
immersed minimal surfaces do not hav e Jacobi fields\, a notion recently proved to be generic by White) then th e finiteness conclusion is true for any compact 3-manifold without boundar y. DTEND;
TZID=Europe/Zurich:20151202T151500 END:VEVENT BEGIN:VEVENT UID:news454@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181227T174021 DTSTART;TZID=Europe/Zurich:20151125T141500 SUMMARY:Seminar Analysis:
Yash Jhaveri (University of Texas at Austin) DESCRIPTION:It is well known that for many second-order PDEs the solution v gains two derivatives with respect to the right-hand side g in Hölder spaces.
Often\, however\, it is useful to have a quantitat ive understanding of regularity. In ’89\, Caffarelli proved inter ior a priori estimates for fully nonlinear\, uniformly elliptic equations .
Specifically\, he showed that ‖v‖C2\,α(B_{1/2})≤C(‖v‖L∞(B_{ 1})+‖g‖Cα(B_{1})) and C∼1/α as α→0. The natural question t o ask is then: Can one extend such quantitative estimates to other equations? An
equation that appears frequently in analysis\, geometry\, and applications is the Monge-Ampére equation det(D2u) = f. The Monge-Am père equation enjoys the same qualitative regularity gains as its
linear counterpart the Poisson equation in the appropriate setting\, and so we as k whether or not the quantitative picture is also the same. This is not t he case. In this talk\, we will first
review Caffarelli’s interior a pr iori estimates. Then\, we will move to the Monge-Ampère equation and se e a different picture.\\r\\n(Joint work with Alessio Figalli and Connor Mo oney) X-ALT-DESC:\
nIt is well known that for many second-order PDEs the solution v gains two derivatives with respect to the right-hand side g i n Hölder spaces. Often\, however\, it is useful to have a quantita tive
understanding of regularity. In ’89\, Caffarelli proved inte rior a priori estimates for fully nonlinear\, uniformly elliptic equation s. Specifically\, he showed that ‖v‖[C^2\,α(B_{1/2})≤C(‖v‖[L^∞
(B_{1})]+‖g‖[C^α(B_{1})]) and C∼1/α as α→0. The natural question to ask is then: Can one extend such quantitative estimates to other equati ons? An equation that appears frequently in analysis\,
geometry\, and app lications is the Monge-Ampére equation det(D^2u) = f. The Mong e-Ampère equation enjoys the same qualitative regularity gains as its lin ear counterpart the Poisson equation in the
appropriate setting\, and so w e ask whether or not the quantitative picture is also the same. This is n ot the case. In this talk\, we will first review Caffarelli’s interior a priori estimates.
Then\, we will move to the Monge-Ampère equation an d see a different picture.\n(Joint work with Alessio Figalli and Connor Mo oney) DTEND;TZID=Europe/Zurich:20151125T151500 END:VEVENT BEGIN:VEVENT
UID:news453@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181227T172829 DTSTART;TZID=Europe/Zurich:20151118T141500 SUMMARY:Seminar Analysis: Lorenzo Brasco (Aix-Marseille Université & Unive rsity of
Ferrara) DESCRIPTION:In this talk\, I will review some regularity results for weak s olutions of nonlocal variants of the p-Laplace equation. The model case i s given by the Euler-Lagrange equation
of an Aronszajn–Gagliardo–Sl obodeckij seminorm. In particular\, I will present a higher diffe rentiability result for solutions\, recently obtained in collaborat ion with Erik Lindgren (KTH). I will
also discuss some connectio ns of these equations to an Optimal Transport problem with congestion ef fects. X-ALT-DESC: \nIn this talk\, I will review some regularity results for weak solutions of
nonlocal variants of the p-Laplace equation. The model case is given by the Euler-Lagrange equation of an Aronszajn–Gagliardo– Slobodeckij seminorm. In particular\, I will present a higher dif
ferentiability result for solutions\, recently obtained in collabor ation with Erik Lindgren (KTH). I will also discuss some connect ions of these equations to an Optimal Transport problem with
congestion effects. DTEND;TZID=Europe/Zurich:20151118T141500 END:VEVENT BEGIN:VEVENT UID:news452@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181227T172253 DTSTART;TZID=Europe/Zurich:20151111T141500
SUMMARY:Seminar Analysis: Zoltán Balogh (Universität Bern) DESCRIPTION:A pair of metric spaces (X\;Y) is said to have the Lipschitz ex tension property if any Lipschitz map from a subset of X into Y
can be ext ended to a globally defined Lipschitz map to the whole space X. In this t alk I will first recall some classical extension results for spaces with a linear structure\, and I will present
recent results for the case when th e target space Y is the Heisenberg group. X-ALT-DESC:\nA pair of metric spaces (X\;Y) is said to have the Lipschitz e xtension property if any Lipschitz map from a
subset of X into Y can be ex tended to a globally defined Lipschitz map to the whole space X. In this talk I will first recall some classical extension results for spaces with a linear structure\,
and I will present recent results for the case when t he target space Y is the Heisenberg group. DTEND;TZID=Europe/Zurich:20151111T151500 END:VEVENT BEGIN:VEVENT UID:news451@dmi.unibas.ch DTSTAMP;
TZID=Europe/Zurich:20181227T171825 DTSTART;TZID=Europe/Zurich:20151104T141500 SUMMARY:Seminar Analysis: Antti Knowles (ETH Zürich) DESCRIPTION:I discuss results on local eigenvalue statistics for
random reg ular graphs. Under mild growth assumptions on the degree\, we prove that t he local semicircle law holds at the optimal scale\, and that the bulk eig envalue statistics coincide with those
of the GOE from random matrix theor y.\\r\\n(Joint work with R. Bauerschmidt\, J. Huang and H.-T. Yau.) X-ALT-DESC:\nI discuss results on local eigenvalue statistics for random re gular graphs. Under
mild growth assumptions on the degree\, we prove that the local semicircle law holds at the optimal scale\, and that the bulk ei genvalue statistics coincide with those of the GOE from random matrix
theo ry.\n(Joint work with R. Bauerschmidt\, J. Huang and H.-T. Yau.) DTEND;TZID=Europe/Zurich:20151104T151500 END:VEVENT BEGIN:VEVENT UID:news450@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181227T171033 DTSTART;TZID=Europe/Zurich:20151014T141500 SUMMARY:Seminar Analysis: Christian Seis (Universit ̈at Bonn) DESCRIPTION:We investigate the speed of convergence and higher-order
asympt otics of solutions to the porous medium equation. Applying a nonlinear ch ange of variables\, we rewrite the equation as a diffusion on a fixed doma in with quadratic nonlinearity.The
degeneracy is cured by viewing the dyna mics on a hypocycloidic manifold. It is in this framework that we can pro ve a differentiable dependency of solutions on the initial data\, and thus \,
dynamical systems methods are applicable. Our main result is the const ruction of invariant manifolds in the phase space of solutions which are t angent at the origin to the eigenspaces of the
linearized equation. We sh ow how these invariant manifolds can be used to extract information on hig her-order long-time asymptotic expansions. X-ALT-DESC: \nWe investigate the speed of convergence
and higher-order asym ptotics of solutions to the porous medium equation. Applying a nonlinear change of variables\, we rewrite the equation as a diffusion on a fixed do main with quadratic
nonlinearity.The degeneracy is cured by viewing the dy namics on a hypocycloidic manifold. It is in this framework that we can p rove a differentiable dependency of solutions on the initial data\,
and th us\, dynamical systems methods are applicable. Our main result is the con struction of invariant manifolds in the phase space of solutions which are tangent at the origin to the eigenspaces of
the linearized equation. We show how these invariant manifolds can be used to extract information on h igher-order long-time asymptotic expansions. DTEND;TZID=Europe/Zurich:20151014T151500 END:VEVENT
BEGIN:VEVENT UID:news449@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181227T165613 DTSTART;TZID=Europe/Zurich:20151007T141500 SUMMARY:Seminar Analysis: Cheikh Ndiaye (Universität Tübingen)
DESCRIPTION:In this talk\, we will present our recent solutions of the rema ining cases of the boundary Yamabe problem and the Riemann mapping problem asked by Escobarin 1992. Rather than discussing
our arguments of proofs\, we will focus more on explaining the barycenter techniqu e of Bahri-Coron which we employ. We hope by doing this to all ow an easier understanding for the audience\, since
it seems to us\, that even among experts\, the barycenter technique is not k nown like the minimizing technique of Aubin-Schoen. Moreover\, we hope also the audience to see how naturally the
barycenter technique fits into conformally invariant variational problems verifying the structure of quan tization and strong interaction phenomena.\\r\\n(Joint work with M. Mayer of University of
Giessen) X-ALT-DESC:\nIn this talk\, we will present our recent solutions of the rem aining cases of the boundary Yamabe problem and the Riemann mapping proble m asked by Escobarin 1992. Rather than
discussing our arguments of proofs\, we will focus more on explaining the barycenter techniq ue of Bahri-Coron which we employ. We hope by doing this to al low an easier understanding for the
audience\, since it seems to us\, that even among experts\, the barycenter technique is not known like the minimizing technique of Aubin-Schoen. Moreover\, we hope also the audience to see how
naturally the barycenter technique fits into conformally invariant variational problems verifying the structure of qua ntization and strong interaction phenomena.\n(Joint work with M. Mayer of
University of Giessen) DTEND;TZID=Europe/Zurich:20151007T151500 END:VEVENT BEGIN:VEVENT UID:news448@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181227T165017 DTSTART;TZID=Europe/Zurich:20150930T141500
SUMMARY:Seminar Analysis: Kunnath Sandeep (TIFR CAM Bangalore) DESCRIPTION:In this talk we will discuss the classical Adams inequality and its versions in the hyperbolic space. We will also discuss
the hyperbolic versions of Adachi-Tanaka type inequalities and the exact growth. X-ALT-DESC:\nIn this talk we will discuss the classical Adams inequality an d its versions in the hyperbolic space. We
will also discuss the hyperboli c versions of Adachi-Tanaka type inequalities and the exact growth. DTEND;TZID=Europe/Zurich:20150930T151500 END:VEVENT BEGIN:VEVENT UID:news447@dmi.unibas.ch DTSTAMP;
TZID=Europe/Zurich:20181227T164404 DTSTART;TZID=Europe/Zurich:20150923T141500 SUMMARY:Seminar Analysis: Stefano Spirito (GSSI) DESCRIPTION:In this talk we focus on a new compactness result about weak
so lutions of the quantum Navier-Stokes equations. The novelty of the result is that we are able to consider the vacuum in the definition of weak solu tions. The main tool is a new formulation of the
equations which allows u s to get an additional a priori estimate to prove compactness. Some remar ks concerning the choice of the approximation system to get global existen ce will be made.\\r\\n
(Joint work with Paolo Antonelli) X-ALT-DESC: \nIn this talk we focus on a new compactness result about weak solutions of the quantum Navier-Stokes equations. The novelty of the resu lt is that we
are able to consider the vacuum in the definition of weak so lutions. The main tool is a new formulation of the equations which allows us to get an additional a priori estimate to prove compactness.
Some rem arks concerning the choice of the approximation system to get global exist ence will be made.\n(Joint work with Paolo Antonelli) DTEND;TZID=Europe/Zurich:20150923T151500 END:VEVENT
BEGIN:VEVENT UID:news468@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T180221 DTSTART;TZID=Europe/Zurich:20150527T151500 SUMMARY:Seminar Analysis: Olivier Druet (University Lyon 1) DESCRIPTION:I
will survey recent results on the Einstein-Lichnerowicz const raints system which appears in general relativity when trying to f ormulate the Cauchy problem for the Einstein equation coupled with a
scalar field. I will discuss existence\, uniqueness\,compactne ss and stability for this system. This is a joint work with Bruno Premose lli. X-ALT-DESC: \nI will survey recent results on the
Einstein-Lichnerowicz con straints system which appears in general relativity when trying to formulate the Cauchy problem for the Einstein equation coupled wi th a scalar field. I will discuss
existence\, uniqueness\,compact ness and stability for this system. This is a joint work with Bruno Premo selli. DTEND;TZID=Europe/Zurich:20150527T161500 END:VEVENT BEGIN:VEVENT
UID:news467@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T175942 DTSTART;TZID=Europe/Zurich:20150520T151500 SUMMARY:Seminar Analysis: Antoine Choffrut (University of Edinburgh) DESCRIPTION:The
following dichotomy between rigidity and flexibility is now well known in geometry: while uniqueness holds for smooth solutio ns to the isometric embedding problem\, the set of solutions becomes u
nimaginably large if one allows rough ones. What is surprising is that th is dichotomy holds for problems coming from mathematical physics\,and in p articular the Euler equations of fluid dynamics.
In this (mainly exposito ry) talk I will explain the h-principle and the method of convex integrati on. Convex geometry is the heart of the matter and profuse figures will a ttempt to illustrate the
difficulties and how to tame them. X-ALT-DESC: \nThe following dichotomy between rigidity and flexibility is n ow well known in geometry: while uniqueness holds for smooth solut ions to the isometric
embedding problem\, the set of solutions becomes unimaginably large if one allows rough ones. What is surprising is that this dichotomy holds for problems coming from mathematical physics\,and in
particular the Euler equations of fluid dynamics. In this (mainly exposi tory) talk I will explain the h-principle and the method of convex integra tion. Convex geometry is the heart of the matter
and profuse figures will attempt to illustrate the difficulties and how to tame them. DTEND;TZID=Europe/Zurich:20150520T161500 END:VEVENT BEGIN:VEVENT UID:news466@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181228T175531 DTSTART;TZID=Europe/Zurich:20150513T151500 SUMMARY:Seminar Analysis: Gabriele Mancini (SISSA/ISAS) DESCRIPTION:I will give a brief overview of the main results concerning top
ological methods for singular Liouville equations on compact surfaces\, an d I will show how to extend some of them to special elliptic systems . My analysis will focus on sharp forms of the
Moser-Trudinger i nequality and on mass-quantization results for the SU(3) Toda System. X-ALT-DESC:\nI will give a brief overview of the main results concerning to pological methods for singular
Liouville equations on compact surfaces\, a nd I will show how to extend some of them to special elliptic system s. My analysis will focus on sharp forms of the Moser-Trudinger inequality and on
mass-quantization results for the SU(3) Toda System. DTEND;TZID=Europe/Zurich:20150513T161500 END:VEVENT BEGIN:VEVENT UID:news465@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T175120 DTSTART;TZID
=Europe/Zurich:20150506T151500 SUMMARY:Seminar Analysis: Julien Sabin (University of Paris-Sud) DESCRIPTION:We study the trace ideals properties of the Fourier restriction operator to hypersurfaces.
Equivalently\, we generalize the theorem s of Stein-Tomas and Strichartz to systems of orthonormal functions\, with an optimal dependence on the number of such functions. As an appl ication\, we
deduce new Strichartz inequalities describing the dis persive behaviour of the free evolution of quantum systems with an infinite number of particles. This is a joint work with Rupert Fran k.
X-ALT-DESC:\nWe study the trace ideals properties of the Fourier restrictio n operator to hypersurfaces. Equivalently\, we generalize the theore ms of Stein-Tomas and Strichartz to systems of
orthonormal functions\, with an optimal dependence on the number of such functions. As an app lication\, we deduce new Strichartz inequalities describing the di spersive behaviour of the free
evolution of quantum systems with an infinite number of particles. This is a joint work with Rupert Fra nk. DTEND;TZID=Europe/Zurich:20150506T161500 END:VEVENT BEGIN:VEVENT UID:news464@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20181228T174506 DTSTART;TZID=Europe/Zurich:20150429T151500 SUMMARY:Seminar Analysis: Bernard Dacorogna (EPFL) DESCRIPTION:Given two functions f and g\,we want to find a
map φ such tha t\\r\\ng(φ(x)) det∇φ(x)=f(x) x∈Ω\, φ(x)=x x∈∂Ω.\\r\\nLocal case. We first consider the (local) existence\, uniqueness and optimal regularity for the problem\
\r\\ngi(φ(x)) det∇φ(x)=fi(x) for every 1≤i≤n\\r\\nwhere gi·fi>0 .\\r\\nGlobal case. A necessary condition is then\\r\\n∫Ω f =∫Ω g. (1)\\r\\n(i) We dis cuss the case where
g·f>0 and give three different ideas for the existenc e problem with optimal regularity.\\r\\n(ii) We then briefly comment on th e case where g>0 but f is allowed to change sign.\\r\\nA problem
without t he condition (1). We consider a more general problem of the form\\r\\n det∇φ(x)=f(x\,φ(x)\,∇φ(x)) x∈Ω\, φ(x)=x x∈∂Ω.\\r\\nwhere no
constraint of the type (1) is needed. X-ALT-DESC:\nGiven two functions f and g\,we want to find a map \;φ su ch that\ng(φ(x)) det∇φ(x)=f(x) \; \; x∈Ω\,
 \;&nbs p\; \; \; \; \; \; \; \; \; \; \;& nbsp\; \; \; \; \; \; φ(x)=x \; \; \; &nb sp\; \; x∈∂Ω.\nLocal case. We
first consider the (local) existenc e\, uniqueness and optimal regularity for the problem\ng[i](φ(x) ) det∇φ(x)=f[i](x) \; for every 1≤i≤n\nwhere g[i sub>·f[i]>\;0.\nGlobal case. A necessary
condition is t hen\n∫[Ω] f =∫[Ω] g.  \; \; \; \;& nbsp\; \; \; \; \; \; \; \; \; \;  \; \; \; \; \; \; \; \; \;
 \; \;&n bsp\; \; (1)\n(i) We discuss the case where g·f>\;0 and give three different ideas for the existence problem with optimal regularity.\n(ii) W e then briefly comment on the case
where g>\;0 but f is allowed to chang e sign.\nA problem without the condition (1). We consider a more ge neral problem of the form\n \; \; \; \; \; \;  \; \; \;
 \; \; det∇φ(x)=f(x\,φ(x)\,∇φ(x)) \;&nb sp\; x∈Ω\,
 \; \; \; \; \; \; \; \ ; \; \; \; \; \; \; \; \; \; \; φ (x)=x \; \; \; \;  \;  \;  \;  \; &
nbsp\;  \;  \; \;  \;  \; x∈∂Ω.\nwhere no constraint of the type (1) is needed.\n\n DTEND;TZID=Europe/Zurich:20150429T161500 END:VEVENT BEGIN:VEVENT UID:news463@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20181228T172110 DTSTART;TZID=Europe/Zurich:20150422T151500 SUMMARY:Seminar Analysis: Laura Spinolo (IMATI-CNR\, Pavia) DESCRIPTION:In 1973 Schaeffer established a result
that applies to scalar c onservation laws with convex fluxes and can be loosely speaking formulated as follows: for a generic smooth initial datum\, the admissible solution is smooth outside a
locally finite number of curves in the (t\,x) plane. H ere the term ”generic” should be interpreted in a suitable technical s ense\, related to the Baire Category Theorem. My talk will aim at
discussi ng a recent explicit counter-example that shows that Schaeffer’s Theorem does not extend to systems of conservation laws. The talk will be based o n joint works with Laura Caravenna.
X-ALT-DESC: \nIn 1973 Schaeffer established a result that applies to scalar conservation laws with convex fluxes and can be loosely speaking formulat ed as follows: for a generic smooth initial datum
\, the admissible solutio n is smooth outside a locally finite number of curves in the (t\,x) plane. Here the term ”generic” should be interpreted in a suitable technical sense\, related to the Baire
Category Theorem. My talk will aim at discus sing a recent explicit counter-example that shows that Schaeffer’s Theor em does not extend to systems of conservation laws. The talk will be based on
joint works with Laura Caravenna. DTEND;TZID=Europe/Zurich:20150422T161500 END:VEVENT BEGIN:VEVENT UID:news462@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T171614 DTSTART;TZID=Europe/
Zurich:20150415T151500 SUMMARY:Seminar Analysis: Gyula Csató (Technische Universität Dortmund) DESCRIPTION:The standard isoperimetric inequality states that among all set s with a given fixed volume
(or area in dimension 2) the ball has t he smallest perimeter. That is\, written here for simplicity in di mension 2\, the following infimum is attained by the ball\\r\\n2πR= inf{ ∫∂Ω 1 dσ(x) : Ω⊂R2
and ∫Ω 1 dx=πR2}.\\r\\nThe isoperimetri c problem with density is a generalization of this question: given two pos itive functions f\,g:R2→R2 one studies the existence of minimizers of\\r \\nI(C) =
inf{∫∂Ω g(x) dσ(x) : Ω⊂R2 and ∫Ω f(x) dx=C}.\\r\ \nI will mainly talk about the situation when f(x) =|x|q and g(x) =|x|p.This is a reach problem with strong variations in difficulties d epending on
the values of p and q. Some cases are still an open proble m. One case has an interesting application related to the Moser -Trudinger imbedding. I will also mention the situation when f=g=eψ is s
trictly positive and radial\, which leads to the log-convex density conjec ture. X-ALT-DESC: \nThe standard isoperimetric inequality states that among all s ets with a given fixed volume(or area in
dimension 2) the ball has the smallest perimeter. That is\, written here for simplicity in dimension 2\, the following infimum is attained by the ball\n2πR= inf{∫ [∂Ω] 1 dσ(x) : Ω⊂R^2 and ∫[Ω 1 dx=πR
^2}.\nThe isoperimetric problem with density is a ge neralization of this question: given two positive functions f\,g:R< sup>2→R^2 one studies the existence of minimizers of\nI(C) = inf{∫[∂Ω] g(x) dσ
(x) : Ω⊂R^2 sup> and ∫[Ω] f(x) dx=C}.\nI will mainly talk about the situation when f(x) =|x|q and g(x) =|x|p.This is a reach problem wi th strong variations in difficulties depending on the values of
p and q. S ome cases are still an open problem. One case has an interesting application related to the Moser-Trudinger imbedding. I will also m ention the situation when f=g=eψ is strictly positive
and radial\, which leads to the log-convex density conjecture. DTEND;TZID=Europe/Zurich:20150415T161500 END:VEVENT BEGIN:VEVENT UID:news461@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T170629
DTSTART;TZID=Europe/Zurich:20150408T151500 SUMMARY:Seminar Analysis: Esther Cabezas-Rivas (Goethe University Frankfurt ) DESCRIPTION:Almost flat manifolds are the solutions of bounded size perturb
ations of the equation Sec=0 (Sec is the sectional curvature). In a celebrated theorem\, Gromov proved that the presence of an almost flat metric implies a precise topological description of the
underlying ma nifold. \\r\\nIntegral pinching theorems express curvature assumptions in terms of certain Lp-norms and try to deduce topological conclusions. But typically one needs to require p >n2\,
where n is the dimension of the man ifold\, to prove such rigidity theorems.\\r\\nDuring this talk we will exp lain how\, under lower sectional curvature bounds\, to imposeanL1-pinching condition on
the curvature is surprisingly rigid\, leading indeed to the same conclusion as in Gromov’s theorem under more relaxed curvature cond itions.\\r\\nThis is a joint work with B. Wilking. X-ALT-DESC:\
nAlmost flat manifolds are the solutions of bounded size pertur bations of the equation Sec=0 (Sec is the sectional curvature). In a celebrated theorem\, Gromov proved that the presence of an almost
flat metric implies a precise topological description of the underlying m anifold. \nIntegral pinching theorems express curvature assumptions in ter ms of certain L^p-norms and try to deduce
topological conclusion s. But typically one needs to require p >\;n2\, where n is the dimensio n of the manifold\, to prove such rigidity theorems.\nDuring this talk we will explain how\, under
lower sectional curvature bounds\, to imposeanL1- pinching condition on the curvature is surprisingly rigid\, leading indeed to the same conclusion as in Gromov’s theorem under more relaxed curvat
ure conditions.\nThis is a joint work with B. Wilking. DTEND;TZID=Europe/Zurich:20150408T161500 END:VEVENT BEGIN:VEVENT UID:news460@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T165919 DTSTART;
TZID=Europe/Zurich:20150318T151500 SUMMARY:Seminar Analysis: Melanie Rupflin (Leipzig University) DESCRIPTION:Teichmüller harmonic map flow is a gradient flow of the Dirich let energy which is
designed to evolve parametrized surfaces towards criti cal points of the area. In this talk we will discuss the construction and some new results for this flow and show in particular that for
non-positi vely curved targets the flow changes or decomposes arbitrary closed initi al surfaces into minimal immersions (possibly with branch points) through globally defined smooth solutions.
X-ALT-DESC: \nTeichmüller harmonic map flow is a gradient flow of the Diri chlet energy which is designed to evolve parametrized surfaces towards cri tical points of the area. In this talk we will
discuss the construction a nd some new results for this flow and show in particular that for non-posi tively curved targets the flow changes or decomposes arbitrary closed ini tial surfaces into
minimal immersions (possibly with branch points ) through globally defined smooth solutions. DTEND;TZID=Europe/Zurich:20150318T161500 END:VEVENT BEGIN:VEVENT UID:news459@dmi.unibas.ch DTSTAMP;TZID=
Europe/Zurich:20181228T165150 DTSTART;TZID=Europe/Zurich:20150311T151500 SUMMARY:Seminar Analysis: Giuseppe Genovese (University of Zurich) DESCRIPTION:The DNLS equation is an integrable PDE\, in the
sense that there are infinitely many Hamiltonians associated to it. Th e aim of the talk is to present the construction of infinitely man y functional measures associated to these integrals of motion
of the equat ion\, each measure being supported on Sobolev spaces of increasing regular ity. These are natural candidates to be the invariant measures as sociated to the DNLS eq. Invariant measures
are a crucial tool i n the theory of integrable PDEs\, useful e.g. to prove long time properties of regular solutions. The introductory general aspects will be reviewed and the new results on DNLS\,
obtained in collaboration with R. Luc (ICMAT\,Madrid) and D. Valeri (MSC\, Beijing)\, will be presented. X-ALT-DESC: \nThe DNLS equation is an integrable PDE\, in the sens e that there are infinitely
many Hamiltonians associated to it. The aim of the talk is to present the construction of infinitely m any functional measures associated to these integrals of motion of the equ ation\, each measure
being supported on Sobolev spaces of increasing regul arity. These are natural candidates to be the invariant measures associated to the DNLS eq. Invariant measures are a crucial tool in the theory
of integrable PDEs\, useful e.g. to prove long tim e properties of regular solutions. The introductory general aspects will be reviewed and the new results on DNLS\, obtained in collaboration with R
. Luc (ICMAT\,Madrid) and D. Valeri (MSC\, Beijing)\, will be presented. DTEND;TZID=Europe/Zurich:20150311T161500 END:VEVENT BEGIN:VEVENT UID:news458@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181228T164525 DTSTART;TZID=Europe/Zurich:20150304T151500 SUMMARY:Seminar Analysis: Emmanuel Hebey (University of Cergy-Pontoise) DESCRIPTION:Stationary Kirchhoff systems in closed manifolds
X-ALT-DESC:Stationary Kirchhoff systems in closed manifolds DTEND;TZID=Europe/Zurich:20150304T161500 END:VEVENT BEGIN:VEVENT UID:news479@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T232256
DTSTART;TZID=Europe/Zurich:20141210T170000 SUMMARY:Seminar Analysis: Davide Vittone (University of Padua) DESCRIPTION:We consider the area functional for graphs in the sub-Riemannia n Heisenberg
group and study minimizers of the associated Dirichle t problem. We prove that\,under a bounded slope condition on the boun dary datum\, there exists a unique minimizer and that this minimizer i s
Lipschitz continuous. We also provide an example showing that\, in the first Heisenberg group\, Lipschitz regularity cannot be improved e ven under the bounded slope condition. This is based on a
joint work with A. Pinamonti\, F. SerraCassano and G. Treu. X-ALT-DESC:\nWe consider the area functional for graphs in the sub-Riemanni an Heisenberg group and study minimizers of the associated
Dirichl et problem. We prove that\,under a bounded slope condition on the bou ndary datum\, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example
showing that\ , in the first Heisenberg group\, Lipschitz regularity cannot be improved even under the bounded slope condition. This is based on a joint work wit h A. Pinamonti\, F. SerraCassano and
G. Treu. DTEND;TZID=Europe/Zurich:20141210T174500 END:VEVENT BEGIN:VEVENT UID:news478@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T232323 DTSTART;TZID=Europe/Zurich:20141210T160000
SUMMARY:Seminar Analysis: Annalisa Massaccesi (University of Zurich) DESCRIPTION:In this joint work with Giovanni Alberti\, we prove a Frobenius property for inte-gral currents: namely\, if R=[∑\,ξ
\,θ] k-dimensiona l integral current with a simple tangent vector field ξ∈C1(Rd\;Λk(Rd)) \, then ξ is involtive at almost every point in ∑. This result is relat ed to the following decomposition
problem formulated by F.Morgan: given a k-dimensional normal current T\, do there exist a measure space L and a fa mily of rectifiable currents {Rλ}λ∈L such that T = ∫L Rλ dλ and th e mass decomposes
consistently as M(T) = ∫L M(Rλ) dλ? The aforementio ned Frobenius property allows us to provide a counterexample to the existe nce of such a decomposition with a family of integral currents.
X-ALT-DESC:\nIn this joint work with Giovanni Alberti\, we prove a Frobeniu s property for inte-gral currents: namely\, if R=[∑\,ξ\,θ] k-dimension al integral current with a simple tangent vector
field ξ∈C^1( R^d\;Λ[k](R^d))\, then ξ is i nvoltive at almost every point in ∑. This result is related to the follo wing decomposition problem formulated by F.Morgan: given a k-dimensional n ormal
current T\, do there exist a measure space L and a family of rectifi able currents {R[λ]}[λ∈L] such that T = ∫[L R[λ] dλ and the mass decomposes consistently as M(T) = ∫ [L] M(R[λ]) dλ? The
aforementioned Frobenius property allows us to provide a counterexample to the existence of such a decomposi tion with a family of integral currents. DTEND;TZID=Europe/Zurich:20141210T164500
END:VEVENT BEGIN:VEVENT UID:news477@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T225557 DTSTART;TZID=Europe/Zurich:20141203T161500 SUMMARY:Seminar Analysis: Stefano Spirito (Gran Sasso Science
Institute\, L ’Aquila) DESCRIPTION:In this talk I will discuss the problem of the approximation of suitable weak solutions of Navier-Stokes equations in the sense of Scheff er and
Caffarelli-Kohn-Nirenberg. It is well-known that suitable weak solu tions enjoy the partial regularity theorem proved in the famous paper of Caffarelli-Kohn-Nirenberg\, hence they are more regular
than a Leray weak solutions. However\, since the uniqueness of weak solutions of Navier-Stokes is unknown we don’t know if different approximation metho ds lead to a suitable weak solution. I will
present a recent result obtai ned with L. C. Berselli (University of Pisa) where we proved that weak sol utions obtained by some artificial compressibility approximation are suita ble. The novelty of
the result is that the Navier-Stokes equations are co nsidered in a bounded domain with Navier boundary conditions. X-ALT-DESC:\nIn this talk I will discuss the problem of the approximation o f
suitable weak solutions of Navier-Stokes equations in the sense of Schef fer and Caffarelli-Kohn-Nirenberg. It is well-known that suitable weak sol utions enjoy the partial regularity theorem proved
in the famous paper of Caffarelli-Kohn-Nirenberg\, hence they are more regular than a Leray weak solutions. However\, since the uniqueness of weak solutions o f Navier-Stokes is unknown we don’t know
if different approximation meth ods lead to a suitable weak solution. I will present a recent result obta ined with L. C. Berselli (University of Pisa) where we proved that weak so lutions obtained
by some artificial compressibility approximation are suit able. The novelty of the result is that the Navier-Stokes equations are c onsidered in a bounded domain with Navier boundary conditions.
DTEND;TZID=Europe/Zurich:20141203T171500 END:VEVENT BEGIN:VEVENT UID:news476@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T225051 DTSTART;TZID=Europe/Zurich:20141126T161500 SUMMARY:Seminar
Analysis: Petru Mironescu (University Lyon 1) DESCRIPTION:We describe the structure of maps u:(0\,1)n → S1 having a gi ven Sobolev regularity. Such maps are described by their singularities and
phases. This is the analog of the Weierstrass factorization theorem for h olomorphic functions\; the singularities of the Sobolev maps play the role of the zeroes of holomorphic maps. We will present
implications of this r esult to functional analytic questions related to manifold valued maps. If the time permits it\, we will discuss the question of the control of the phases\, and present some
applications to some model PDEs and nonlocal pro blems. X-ALT-DESC: \nWe describe the structure of maps u:(0\,1)^n \; → S^1 having a given Sobolev regularity. Such maps are describ ed by their
singularities and phases. This is the analog of the Weierstras s factorization theorem for holomorphic functions\; the singularities of t he Sobolev maps play the role of the zeroes of holomorphic
maps. We will p resent implications of this result to functional analytic questions relate d to manifold valued maps. If the time permits it\, we will discuss the qu estion of the control of the
phases\, and present some applications to som e model PDEs and nonlocal problems. DTEND;TZID=Europe/Zurich:20141126T161500 END:VEVENT BEGIN:VEVENT UID:news475@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181228T224456 DTSTART;TZID=Europe/Zurich:20141119T161500 SUMMARY:Seminar Analysis: Luca Galimberti (ETH Zurich) DESCRIPTION:A classical question in differential geometry concerns which sm
ooth functions f can arise as Gauss curvature of a conformal metric on a 2 -dim Riemannian manifold M. This amounts to solve a PDE which is the Euler -Lagrange equation of an energy functional. In
this talk we will discuss a bout compactness issues and bubbling phenomena for this equation on surfac es of genus greater than 1 (joint work with Borer and Struwe) and on the t orus. X-ALT-DESC: \nA
classical question in differential geometry concerns which smooth functions f can arise as Gauss curvature of a conformal metric on a 2-dim Riemannian manifold M. This amounts to solve a PDE which is
the Eul er-Lagrange equation of an energy functional. In this talk we will discuss about compactness issues and bubbling phenomena for this equation on surf aces of genus greater than 1 (joint work
with Borer and Struwe) and on the torus. DTEND;TZID=Europe/Zurich:20141119T171500 END:VEVENT BEGIN:VEVENT UID:news474@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T203836 DTSTART;TZID=Europe/
Zurich:20141112T161500 SUMMARY:Seminar Analysis: Frédéric Robert (University of Lorraine) DESCRIPTION:We investigate the Hardy-Schrödinger operator Lγ=-Δ-γ/|x|2 on domains Ω⊂Rn\, whose boundary
contain the singularity 0. The situat ion is quite different from the well-studied case when 0 is in the interio r of Ω. For one\, if 0∈Ω\, then L is positive if and only if γ<(n-2) 2/4\, while if
0∈∂Ω the operator L could be positive for larger value of γ\, potentially reaching the maximal constant n2/4 on convex domains. \\r\\nWe prove optimal regularity and a Hopf-type Lemma for variational
so lutions of corresponding linear Dirichlet boundary value problems of the f orm Lγ=a(x)u\, but also for non-linear equations including Lγ=(|u|β-2u) /(|x|s)\, where γ < n2/4\, s∈[0\,2) and β:=2(n-s)
/(n-2) is the critica l Hardy-Sobolev exponent. We also provide a Harnack inequality and a compl ete description of the profile of all positive solutions–variational or not– of the corresponding
linear equation on the punctured domain. The value γ=(n-1)2/4 turned out to be another critical threshold for the oper ator Lγ\, and our analysis yields a corresponding notion of “Hardy sing ular
boundary-mass” mγ(Ω) of a domain Ω having 0∈Ω\, which could b e defined whenever (n2-1)/4 < γ < n2/4.\\r\\nAs a byproduct\, we give a c omplete answer to problems of existence of extremals for
Hardy-Sobolev ine qualities of the form\\r\\nC( ∫Ω (uβ)/(|x|s) dx )2/β ≤∫Ω |∇u| 2 dx - γ∫Ω (u2)/(|x|s)dx\\r\\nwhenever γγ]= -Δ-γ/|x|2 on domains Ω⊂R^n\, whose boundary contain the singularity 0. The
situation is quite different from the well-studied case when 0 is in the interior of Ω. For one\, if 0∈Ω\, then L is po sitive if and only if γ<\;(n-2)^2/4\, while if 0∈∂Ω the operator L could be
positive for larger value of γ\, potentially reaching the maximal constant n^2/4 on convex domains.\nWe prove optimal regularity and a Hopf-type Lemma for variational solutions of correspondi ng
linear Dirichlet boundary value problems of the form Lγ=a(x)u\, but al so for non-linear equations including Lγ=(|u|^β-2u)/(|x|^s )\, where γ <\; n^2/4\, s∈[0\,2) and β:=2(n-s)/(n-2) is the
critical Hardy-Sobolev exponent. We also provide a Harnack inequal ity and a complete description of the profile of all positive solutions– variational or not– of the corresponding linear equation on
the puncture d domain. The value γ=(n-1)^2/4 turned out to be another criti cal threshold for the operator Lγ\, and our analysis yields a correspondi ng notion of “Hardy singular boundary-mass” mγ(Ω)
of a domain Ω hav ing 0∈Ω\, which could be defined whenever (n^2-1)/4 <\; γ <\; n^2/4.\nAs a byproduct\, we give a complete answer to prob lems of existence of extremals for Hardy-Sobolev
inequalities of the form\ nC( ∫[Ω ](u^β)/(|x|^s) dx )^2/β  \;≤∫[Ω] |∇u|^2 dx - γ∫[Ω]&nbs p\;(u^2)/(|x|^s)dx\nwhenever γ<\;n^2/4\, and in particular\, for those of Caffarelli-Kohn-Nirenberg.
These results extend previous contributions by the authors in the case γ=0\, and by Che rn-Lin for the case γ<\;(n-2)^2/4. Namely\, if 0≤γ≤(n< sup>2-1)/4\, then the negativity of the mean curvature
of ∂ Ω at 0 is sucient for the existence of extremals. This is however not su fficient for (n^2-1)/4≤γ≤(n^2)/4\, which then req uires the positivity of the Hardy singular boundary-massof the domain
unde r consideration.\nJoint work with Nassif Ghoussoub. DTEND;TZID=Europe/Zurich:20141112T171500 END:VEVENT BEGIN:VEVENT UID:news473@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T200306 DTSTART;
TZID=Europe/Zurich:20141105T161500 SUMMARY:Seminar Analysis: Sara Daneri (Max Planck Institute for Mathematics in the Sciences\, Leipzig) DESCRIPTION:We consider the Cauchy problem for the
incompressible Euler equ ations on the three-dimensional torus. According to a conjecture due to On sager\, which is well known in turbulence theory\, while all the solutions which are uniformly
α-Hölder continuous in space for any α>1/3 must co nserve the total kinetic energy\, for any α<1/3 there can be uniformly α -Hölder solutions which are strictly dissipative. While the first part of
the conjecture is well established since a long time\, the second part is still open in its full generality. In the result that we present we show that\, for any α<1/5\, there exist Cα vector fields
being the initial da ta of infinitely many Cα solutions of the Euler equations which dissipate the total kinetic energy. X-ALT-DESC: \nWe consider the Cauchy problem for the incompressible Euler e
quations on the three-dimensional torus. According to a conjecture due to Onsager\, which is well known in turbulence theory\, while all the solutio ns which are uniformly α-Hölder continuous in
space for any α>\;1/3 m ust conserve the total kinetic energy\, for any α<\;1/3 there can be un iformly α-Hölder solutions which are strictly dissipative. While the fir st part of the conjecture
is well established since a long time\, the seco nd part is still open in its full generality. In the result that we presen t we show that\, for any α<\;1/5\, there exist C^α vector fi elds being
the initial data of infinitely many C^α solutions of the Euler equations which dissipate the total kinetic energy. DTEND;TZID=Europe/Zurich:20141105T171500 END:VEVENT BEGIN:VEVENT
UID:news472@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T195713 DTSTART;TZID=Europe/Zurich:20141029T161500 SUMMARY:Seminar Analysis: Thomas Sørensen (Ludwig Maximilian University of Munich)
DESCRIPTION:The eigenfunctions of the Schrödinger operator for (non-relati vistic) atoms and molecules (in the Born-Oppenheimer/clamped nuclei approximation) are solutions of an elliptic partial
differential equatio n with singular (total) potential (i.e.\, zero-order term). In this talk we give an overview over our results about the structure/regularity of th e eigenfunctions at the
singularities of the potential. These\, in partic ular\, improve on the well-known ’Kato Cusp Condition’. If time permi ts\, we also discuss the implications for the electron density.\\r\\nThis is
joint work with S. Fournais (Aarhus\, Denmark)\, and M. and T. Hoffma nn-Ostenhof (Vienna\, Austria). X-ALT-DESC:\nThe eigenfunctions of the Schrödinger operator for (non-relat ivistic) atoms and
molecules (in the Born-Oppenheimer/clamped nuclei approximation) are solutions of an elliptic partial differential equati on with singular (total) potential (i.e.\, zero-order term). In this tal k we
give an overview over our results about the structure/regularity of t he eigenfunctions at the singularities of the potential. These\, in parti cular\, improve on the well-known ’Kato Cusp
Condition’. If time perm its\, we also discuss the implications for the electron density.\nThis is joint work with S. Fournais (Aarhus\, Denmark)\, and M. and T. Hoffmann- Ostenhof (Vienna\,
Austria). DTEND;TZID=Europe/Zurich:20141029T171500 END:VEVENT BEGIN:VEVENT UID:news471@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T195204 DTSTART;TZID=Europe/Zurich:20141022T161500
SUMMARY:Seminar Analysis: Chiara Saffirio (University of Zurich) DESCRIPTION:We consider the Cauchy problem associated to the Vlasov-Poisson system and we extend the well-posedness theory of Lions
and Perthame to t he case of initial data which include a Dirac mass. Moreover we provide p olynomially growing in time estimates for the moments of the solution. Th is is a joint work with L.
Desvillettes and E. Miot. X-ALT-DESC: \nWe consider the Cauchy problem associated to the Vlasov-Poiss on system and we extend the well-posedness theory of Lions and Perthame to the case of initial
data which include a Dirac mass. Moreover we provide polynomially growing in time estimates for the moments of the solution. This is a joint work with L. Desvillettes and E. Miot. DTEND;TZID=Europe/
Zurich:20141022T171500 END:VEVENT BEGIN:VEVENT UID:news470@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T194608 DTSTART;TZID=Europe/Zurich:20141008T161500 SUMMARY:Seminar Analysis: Guido de
Philippis (University of Zurich) DESCRIPTION:Local volume-constrained minimizers in anisotropic capilla rity problems develop free boundaries on the walls of their containers. We prove the regularity
of the free boundary outside a small set\, s howing in particular the validity of Young’s law at almost every point (joint with Francesco Maggi). X-ALT-DESC: \nLocal volume-constrained minimizers in
anisotropic capil larity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a small set\, showing in particular the validity of
Young’s law at almost eve ry point (joint with Francesco Maggi). DTEND;TZID=Europe/Zurich:20141008T171500 END:VEVENT BEGIN:VEVENT UID:news469@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T194637
DTSTART;TZID=Europe/Zurich:20140924T161500 SUMMARY:Seminar Analysis: Camilla Nobili (Max Planck Institute for Mathemat ics in the Sciences\, Leipzig) DESCRIPTION:We consider Rayleigh-Bénard
convection at finite Prandtl numbe r as modelled by the Boussinesq equation. We are interested in the scaling of the average upward heat transport\, the Nusselt number Nu\, in terms o f the Rayleigh
number Ra\, and the Prandtl number Pr.\\r\\nPhysically mot ivated heuristics suggest the scaling Nu∼Ra1⁄3 and Nu∼Ra1/2 depe nding on Pr\, in different regimes. \\r\\nIn this talk I present a rigorou
s upper bound for Nu reproducing both physical scalings in some parameter regimes up to logarithms. This is obtained by a (logarithmically failing) maximal regularity estimate inL1and inL1for the
nonstationary Stokes equa tion with forcing term given by the buoyancy term and the nonlinear term\, respectively. This is a joint work with Felix Otto and Antoine Choffrut. X-ALT-DESC:\nWe consider
Rayleigh-Bénard convection at finite Prandtl numb er as modelled by the Boussinesq equation. We are interested in the scalin g of the average upward heat transport\, the Nusselt number Nu\, in terms
of the Rayleigh number Ra\, and the Prandtl number Pr.\nPhysically motiva ted heuristics suggest the scaling Nu∼Ra^1⁄3 and Nu∼Ra ^1/2 depending on Pr\, in different regimes. \nIn this talk I pr esent
a rigorous upper bound for Nu reproducing both physical scalings in some parameter regimes up to logarithms. This is obtained by a (logarithm ically failing) maximal regularity estimate inL1and
inL1for the nonstation ary Stokes equation with forcing term given by the buoyancy term and the n onlinear term\, respectively. This is a joint work with Felix Otto and An toine Choffrut. DTEND;TZID=
Europe/Zurich:20140924T171500 END:VEVENT BEGIN:VEVENT UID:news486@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T175432 DTSTART;TZID=Europe/Zurich:20140520T151500 SUMMARY:Seminar Analysis:
Gabriella Tarantello (University of Rome Tor Verg ata) DESCRIPTION:We discuss a class of singular Liouville systems in the plane a nd their role in the construction of non-abelian Chern-Simons
vortices X-ALT-DESC: \nWe discuss a class of singular Liouville systems in the plane and their role in the construction of non-abelian Chern-Simons vortices DTEND;TZID=Europe/Zurich:20140520T161500
END:VEVENT BEGIN:VEVENT UID:news485@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T175201 DTSTART;TZID=Europe/Zurich:20140507T151500 SUMMARY:Seminar Analysis: Julien Sabin (University of
Cergy-Pontoise) DESCRIPTION:A Fermi gas occupying the whole euclidian space is an example o f a translation-invariant quantum system with an infinite number of part icles. We study its stability
properties under the time-dependent nonline ar Hartree equation. If this system is slightly perturbed at the initial time\, we show in particular that it returns to the translation-invariant state
for large times. This is an instance of nonlinear dispersion for i nfinite quantum systems\, which was recently studied by Frank\, Lewin\, L ieb and Seiringer in the linear case. This a joint work
with Mathieu Lewi n (CNRS/Cergy). I will also mention some recent work on Strichartz estima tes for systems of orthonormal functions\, joint with Rupert Frank (Calte ch). X-ALT-DESC: \nA Fermi gas
occupying the whole euclidian space is an example of a translation-invariant quantum system with an infinite number of pa rticles. We study its stability properties under the time-dependent nonli
near Hartree equation. If this system is slightly perturbed at the initia l time\, we show in particular that it returns to the translation-invaria nt state for large times. This is an instance of
nonlinear dispersion for infinite quantum systems\, which was recently studied by Frank\, Lewin\, Lieb and Seiringer in the linear case. This a joint work with Mathieu Le win (CNRS/Cergy). I will
also mention some recent work on Strichartz esti mates for systems of orthonormal functions\, joint with Rupert Frank (Cal tech). DTEND;TZID=Europe/Zurich:20140507T161500 END:VEVENT BEGIN:VEVENT
UID:news484@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T172908 DTSTART;TZID=Europe/Zurich:20140423T151500 SUMMARY:Seminar Analysis: Hoai-Minh Nguyen (Swiss Federal Institute of Tech nology in
Lausanne (EPFL)) DESCRIPTION:In this talk\, I first discuss estimates for the topological de gree of maps from sphere into itself. Second\, I present characterization s of Sobolev spaces based on the
pointwise convergence or the Gamma-conv ergence of a sequence of nonlocal\, nonconvex functionals related to thes e estimates. If time permits\, I will also discuss the connection betwe en these
functionals with various filters in the denoising problem. The talk is based on joint works with Jean Bourgain and Haim Brezis. X-ALT-DESC:\nIn this talk\, I first discuss estimates for the
topological d egree of maps from sphere into itself. Second\, I present characterizatio ns of Sobolev spaces based on the pointwise convergence or the Gamma-con vergence of a sequence of nonlocal\,
nonconvex functionals related to the se estimates. If time permits\, I will also discuss the \; connection between these functionals \; with various filters in the denoising p roblem. The
talk is based on joint works with Jean Bourgain and Haim Brez is. DTEND;TZID=Europe/Zurich:20140423T161500 END:VEVENT BEGIN:VEVENT UID:news483@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T165509
DTSTART;TZID=Europe/Zurich:20140402T151500 SUMMARY:Seminar Analysis: Armin Schikorra (Max Planck Institute for Mathema tics in the Sciences in Leipzig) DESCRIPTION:I will present results and ideas
for the proof for regularity t heory for critical points of non-local\, degenerate integro-differential energies into manifolds which are related to p-harmonic maps. X-ALT-DESC:\nI will present
results and ideas for the proof for regularity theory for critical points of non-local\, degenerate integro-differential energies into manifolds which are related to p-harmonic maps. DTEND;TZID=
Europe/Zurich:20140402T161500 END:VEVENT BEGIN:VEVENT UID:news482@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T165147 DTSTART;TZID=Europe/Zurich:20140326T151500 SUMMARY:Seminar Analysis: Dario
Trevisan (Scuola Normale Superiore\, Pisa\, Italy) DESCRIPTION:Following [1]\, in this talk we show how to establish\, in a ra ther general setting\, an analogue of DiPerna-Lions theory on
well-posedne ss of flows of ODE’s associated to Sobolev vector fields. Key results ar e a well-posedness result for the continuity equation associated to suitab ly defined Sobolev vector fields\, via
a commutator estimate\, and an abst ract superposition principle in (possibly extended) metric measure spaces\ , via an embedding into R∞.\\r\\nWhen specialized to the setting of Eucl idean or
infinite dimensional (e.g.Gaussian) spaces\, large parts of previ ously known results are recovered at once.Moreover\, the class of RCD(K\, ∞) metric measure spaces\, recently introduced by Ambrosio
\, Gigli and S avar ́e\, object of extensive recent research\, fits into our framework. Therefore we provide\, for the first time\, well-posedness results forODE ’s under low regularity assumptions on
the velocity and in a non smooth context.\\r\\nReferences:[1] L. Ambrosio and D. Trevisan. Well posedness o f Lagrangian flows and continuity equations in metric measure spaces. ArXi v e-prints\,
February 2014. X-ALT-DESC: \nFollowing [1]\, in this talk we show how to establish\, in a rather general setting\, an analogue of DiPerna-Lions theory on well-posed ness of flows of ODE’s associated
to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suit ably defined Sobolev vector fields\, via a commutator estimate\, and an ab stract
superposition principle in (possibly extended) metric measure space s\, via an embedding into R^∞.\nWhen specialized to the setting of Euclidean or infinite dimensional (e.g.Gaussian) spaces\, larg e
parts of previously known results are recovered at once.Moreover\, the c lass of RCD(K\,∞) metric measure spaces\, recently introduced by Ambrosi o\, Gigli and Savar ́e\, object of extensive recent
research\, fits into our framework. Therefore we provide\, for the first time\, well-posedness results forODE’s under low regularity assumptions on the velocity and in a non smooth context.\n
[1] L. Ambrosio and D. Tre visan. Well posedness of Lagrangian flows and continuity equations in metr ic measure spaces. ArXiv e-prints\, February 2014. DTEND;TZID=Europe/Zurich:20140326T161500
END:VEVENT BEGIN:VEVENT UID:news481@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T164113 DTSTART;TZID=Europe/Zurich:20140305T151500 SUMMARY:Seminar Analysis: Angkana Rüland (University of Bonn)
DESCRIPTION:This talk is focused on unique continuation principles for frac tional Schrödinger equations with scaling-critical and rough potentials. The results are deduced via so-called Carleman
estimates. In particular\ , these methods can be transferred to “variable coefficient” versions of fractional Schrödinger equations. X-ALT-DESC:\nThis talk is focused on unique continuation
principles for fra ctional Schrödinger equations with scaling-critical and rough potentials . The results are deduced via so-called Carleman estimates. In particular \, these methods can be
transferred to “variable coefficient” version s of fractional Schrödinger equations. DTEND;TZID=Europe/Zurich:20140305T161500 END:VEVENT BEGIN:VEVENT UID:news480@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181229T163904 DTSTART;TZID=Europe/Zurich:20140219T151500 SUMMARY:Seminar Analysis: Ruben Jakob (University of Tübingen) DESCRIPTION:We provide two sharp sufficient conditions for immersed
Willmor e surfaces in R3\, definedon bounded C4-subdomains of R2\, to be already m inimal surfaces\, i.e. to have vanishing mean curvatures on their entire d omains. Our precise results read as
follows:\\r\\nTheorem 1. For some bo unded C4-domain Ω⊂R2 let X∈C4(Ω\,R3) denote some immersed Willmore s urface with Gauss map N and mean curvature H. Furthermore\, assume that th ere exist
constants c\,d∈R and some fixed vector V∈S2 such that χ := cX+dV satisfies at least one of the following two conditions:\\r\\na) The re is some “normal domain” G⊂Ω such that there hold H=0 on ∂G and
H≥0 (or H≤0) in G∩O\, where O⊂R2 is some open neighbourhood of ∂G\, and\\r\\ninf∂G<χ\,N> ≥ 0 as well as sup∂G<χ\,N> > 0\;\\r\ \nb) H=0 on ∂Ω and\\r\\n<χ\,N> > 0 in Ω\\A as well a sup∂Ω<χ\,N> > 0\\r\\
nfor some finite set A⊂Ω.\\r\\nThen H≡0 is satisfied in \\bar {Ω}\, i.e.X is a minimal surface on \\bar{Ω}.\\r\\nThese results turn ou t to be particularly suitable for applications to Willmore
graphs. We can therefore show that Willmore graphs on bounded C4 domains \\bar{Ω} with v anishing mean curvatures on the boundary ∂Ω must already be minimal gra phs. Our methods also prove that any
closed Willmore surface in R3 which c an be represented as a smooth graph over S2 has to be of constant\, non-ze ro mean curvature and therefore a round sphere. Finally we demonstrate tha t our
results are sharp by means of an examination of some certain part of the Clifford-Torus in R3. X-ALT-DESC:\nWe provide two sharp sufficient conditions for immersed Willmo re surfaces in R^3\,
definedon bounded C^4-subd omains of R^2\, to be already minimal surfaces\, i.e. to have vanishing mean curvatures on their entire domains. Our precise result s read as follows:\nTheorem 1. For some
bounded C^4-dom ain Ω⊂R^2 let X∈C^4(Ω\,R^3) denote some immersed Willmore surface with Gauss map N and mean curvat ure H. Furthermore\, assume that there exist constants c\,d∈R and some f ixed vector
V∈S^2 such that χ := cX+dV satisfies at least one of the following two conditions:\na) There is some “normal domain” G ⊂Ω such that there hold H=0 on ∂G and H≥0 (or H≤0) in G∩O\, whe re O⊂R^2 is some
open neighbourhood of ∂G\, and\ninf< sub>∂G]<\;χ\,N>\; ≥ 0 as well as sup[∂G]<\;χ \,N>\; >\; 0\;\nb) H=0 on ∂Ω and\n<\;χ\,N>\; >\; 0 in Ω\\A as well a sup[∂Ω]<\;χ\,N>\; >\; 0\
nfor some finite set A⊂Ω.\nThen H≡0 is satisfied in \\bar{Ω}\, i.e.X is a minimal surface on \\bar{Ω}.\nThese results turn out to be particularly suitable for app lications to Willmore graphs. We can
therefore show that Willmore graphs o n bounded C4 domains \\bar{Ω} with vanishing mean curvatures on the bound ary ∂Ω must already be minimal graphs. Our methods also prove that any closed Willmore
surface in R3 which can be represented as a smooth graph o ver S2 has to be of constant\, non-zero mean curvature and therefore a rou nd sphere. Finally we demonstrate that our results are sharp by
means of a n examination of some certain part of the Clifford-Torus in R3. DTEND;TZID=Europe/Zurich:20140219T161500 END:VEVENT BEGIN:VEVENT UID:news495@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20181229T230505 DTSTART;TZID=Europe/Zurich:20131218T151500 SUMMARY:Seminar Analysis: Emil Wiedemann (University of British Columbia) DESCRIPTION:Given a bounded domain and boundary data\, does
there exist a vector-valued map on this domain which is incompressible\, that is\, a map whose Jacobian determinant is one (almost) everywhere? In a regular set ting\, this question has been
essentially positively answered in a famous paper by Dacorogna and Moser. I will present an analogous result in Sobo lev spaces of low regularity\, which was recently achieved by a convex in
tegration method jointly with K. Koumatos (Oxford) and F. Rindler (Warwic k). I will also comment on several generalisations and applications. X-ALT-DESC: \nGiven a bounded domain and boundary data\,
does there exist a vector-valued map on this domain which is incompressible\, that is\, a m ap whose Jacobian determinant is one (almost) everywhere? In a regular s etting\, this question has been
essentially positively answered in a famo us paper by Dacorogna and Moser. I will present an analogous result in So bolev spaces of low regularity\, which was recently achieved by a convex
integration method jointly with K. Koumatos (Oxford) and F. Rindler (Warw ick). I will also comment on several generalisations and applications. DTEND;TZID=Europe/Zurich:20131218T161500 END:VEVENT
BEGIN:VEVENT UID:news494@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T230230 DTSTART;TZID=Europe/Zurich:20131211T151500 SUMMARY:Seminar Analysis: Daniele Bartolucci (University of Rome Tor
Vergat a) DESCRIPTION:The uniqueness of solutions of the (Liouville) mean field-type equation on a simply connected domain and in the sub critical regime λ ∈(0\,8π) was first proved by T. Suzuki
(1992). This result has been later improved by S.Y.A. Chang\, C.C. Chen and C.S. Lin (2003) [CCL] to c over the critical value λ∈(0\,8π]. The case where the domain is not s imply connected has been
a long-standing open problem which we have final ly solved in a recent paper in collaboration with C.S. Lin. Our proof is based on a new generalization of a P.D.E. version of the Alexandrov-Bol's
isoperimetric inequality on multiply connected domains. Another delicate problem is to understand the existence/non-existence of solutions for th is equation on multiply connected domains at the
critical parameterλ=8π . Criticality here means that the variational functional whose critical p oints are solutions of the equation is not anymore coercive for λ=8π\, which implies in particular in
this situation that existence/non existenc e of solutions depend on the geometry of the domain. I will discuss our g eneralization of a result in [CCL] which yield necessary and sufficient c
onditions for the existence of solutions for the mean field equation at t he critical parameter λ=8π. X-ALT-DESC: \nThe uniqueness of solutions of the (Liouville) mean field-typ e equation on a
simply connected domain and in the sub critical regime&nb sp\;λ∈(0\,8π) was first proved by T. Suzuki (1992). \; This result has been later improved by S.Y.A. Chang\, C.C. Chen and C.S. Lin
(2003) [CCL] to cover the critical value λ∈(0\,8π]. The case where the domai n is not simply connected has been a long-standing open problem which we have finally solved in a recent paper in
collaboration with C.S. Lin. Our proof is based on a new generalization of a P.D.E. version of the Alexa ndrov-Bol's isoperimetric inequality on multiply connected domains. Anoth er delicate problem
is to understand the existence/non-existence of solut ions for this equation on multiply connected domains at the critical para meterλ=8π. Criticality here means that the variational functional whose
critical points are solutions of the equation is not anymore coercive fo r λ=8π\, which implies in particular in this situation that existence/n on existence of solutions depend on the geometry of
the domain. I will di scuss our generalization of a result in [CCL] which yield necessary and s ufficient conditions for the existence of solutions for the mean field eq uation at the critical
parameter λ=8π. DTEND;TZID=Europe/Zurich:20131127T161500 END:VEVENT BEGIN:VEVENT UID:news493@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T225814 DTSTART;TZID=Europe/Zurich:20131204T151500
SUMMARY:Seminar Analysis: Xavier Ros-Oton (Polytechnic University of Catalo nia) DESCRIPTION:We study the boundary regularity of solutions to elliptic inte gro-differential equations. First we prove
that\, for the fractional Lapl acian (-Δ)s with s∈(0\,1)\, solutions u satisfy that u/ds is Hölder co ntinuous up to the boundary\, where d(x) is the distance to the boundary of the domain Ω. We will
show that\, in this fractional context\, the qu antity u/ds|∂Ω plays the role that the normal derivative plays in seco nd order equations. Finally\, we also present new boundary regularity res ults
for fully nonlinear integro-differential equations. X-ALT-DESC: \nWe study the boundary regularity of solutions to elliptic in tegro-differential equations. First we prove that\, for the fractional
La placian (-Δ)^s with s∈(0\,1)\, solutions u satisfy that u/d[S:s is Hölder continuous up to the boundary\, where d(x) is the d istance to the boundary of the domain Ω. We will show that\, in this
fra ctional context\, the quantity u/d^s|[∂Ω] plays the role that the normal derivative plays in second order equations. Finally \, we also present new boundary regularity results for fully nonlinear
in tegro-differential equations. DTEND;TZID=Europe/Zurich:20131204T161500 END:VEVENT BEGIN:VEVENT UID:news492@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T225231 DTSTART;TZID=Europe/
Zurich:20131127T151500 SUMMARY:Seminar Analysis: Nikolay Gusev (SISSA Trieste) DESCRIPTION:Suppose that b:Rd→Rd is a vector field\, β:R→R is a smo th function and u:R2→R is a scalar field. If both u
and b are smooth th en the following formula holds: div(β(u))b = β(u) - (β'(u) - uβ'(u)) div (b) + β'(u) div(ub). Generalizations of this formula when u∈L∞ an d b belongs to Sobolev space or has
bounded variation were studied by R. D i Perna\, P.-L. Lions\, L. Ambrosio\, C. De Lellis\, J. Maly and other a uthors. I will present a new result in this direction for d=2\, which was obtained
recently in collaboration with S. Bianchini. In particular our result holds when b is steady nearly incompressible BV vector field. X-ALT-DESC:\nSuppose that b:R^d→R^d is a vector field\, β:R→R^ 
\; is a smoth func tion and u:R^2→R is a scalar field. If both u an d b are smooth then the following formula holds: div(β(u))b = \;β(u ) - (β'(u) - uβ'(u)) div (b) + β'(u) div(ub).
Generalizations of this f ormula when u∈L^∞ and b belongs to Sobolev space or has boun ded variation were studied by R. Di Perna\, P.-L. Lions\, L. Ambrosio\, C . De Lellis\, J. Maly and other
authors. I will present a new result in t his direction for d=2\, which was obtained recently in collaboration with S. Bianchini. In particular our result holds when b is steady nearly inc
ompressible BV vector field. DTEND;TZID=Europe/Zurich:20131127T161500 END:VEVENT BEGIN:VEVENT UID:news491@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T224115 DTSTART;TZID=Europe/
Zurich:20131113T151500 SUMMARY:Seminar Analysis: Maria Colombo (Scuola Normale Superiore di Pisa) DESCRIPTION:The semigeostrophic equations are a set of equations which mode l large-scale atmospheric
/ocean flows.\\r\\nThe system admits a dual versi on\, obtained from the original equations through a change of variable. E xistence for the dual problem has been proven in 1998 by Benamou and Bren
ier\, but the existence of a solution of the original system remained ope n due to the low regularity of the change of variable.\\r\\nIn the talk w e prove the existence of distributional solutions
of the original equatio ns\, both in R3 and in a two-dimensional periodic setting. The proof is b ased on recent regularity and stability estimates for Alexandrov solution s of the Monge-Ampère
equation\, established by De Philippis and Figalli . X-ALT-DESC:\nThe semigeostrophic equations are a set of equations which mod el large-scale atmospheric/ocean flows.\nThe system admits a dual
version\ , obtained from the original equations through a change of variable. Exis tence for the dual problem has been proven in 1998 by Benamou and Brenier \, but the existence of a solution of the
original system remained open d ue to the low regularity of the change of variable.\nIn the talk we prove the existence of distributional solutions of the original equations\, bo th in R^3 and in a
two-dimensional periodic setting. The proof is based on recent regularity and stability estimates for Alexandrov solu tions of the Monge-Ampère equation\, established by De Philippis and Fig alli.
DTEND;TZID=Europe/Zurich:20131113T161500 END:VEVENT BEGIN:VEVENT UID:news490@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T223702 DTSTART;TZID=Europe/Zurich:20131016T151500 SUMMARY:Seminar
Analysis: Yannick Sire (University of Marseille) DESCRIPTION:In the last years\, a substantial amount of work has been devot ed to understand elliptic\, parabolic and hyperbolic problems with non lo
cal diffusion. In this talk\, I will introduce a new class of conformally covariant operators of fractional order generalizing the scalar and Pan eitz curvature. I will describe the associated Yamabe
problem\, in the re gular and singular settings. I will give some existence results and discu ss open problems. X-ALT-DESC: \nIn the last years\, a substantial amount of work has been de voted to
understand elliptic\, parabolic and hyperbolic problems with non local diffusion. In this talk\, I will introduce a new class of conforma lly covariant operators of fractional order generalizing the
scalar and Paneitz curvature. I will describe the associated Yamabe problem\, in the regular and singular settings. I will give some existence results and di scuss open problems. DTEND;TZID=Europe/
Zurich:20131016T151500 END:VEVENT BEGIN:VEVENT UID:news489@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T223441 DTSTART;TZID=Europe/Zurich:20131016T141500 SUMMARY:Seminar Analysis: Grzegorz
Jamroz (University of Warsaw) DESCRIPTION:We consider a one-dimensional transport (balance) equation with velocity which has non-Lipschitz zeroes. This leads to non-uniqueness an d concentration of
characterics and dynamics with both discrete and cont inuous components. To deal with these effects\, we use measure-valued sol utions and the so-called measure-transmission conditions. A metric in
the space of Radon measures allowing to define unique and stable solutions i s introduced. The equation under consideration was proposed as a structur ed population model of cell differentiation.
X-ALT-DESC: \nWe consider a one-dimensional transport (balance) equation wi th velocity which has non-Lipschitz zeroes. This leads to non-uniqueness and concentration of characterics and dynamics
with both discrete and co ntinuous components. To deal with these effects\, we use measure-valued s olutions and the so-called measure-transmission conditions. A metric in t he space of Radon
measures allowing to define unique and stable solutions is introduced. The equation under consideration was proposed as a struct ured population model of cell differentiation. DTEND;TZID=Europe/
Zurich:20131016T151500 END:VEVENT BEGIN:VEVENT UID:news488@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T223226 DTSTART;TZID=Europe/Zurich:20131009T151500 SUMMARY:Seminar Analysis: Stefan
Steinerberger (University of Bonn) DESCRIPTION:It is obvious that there is no tiling of the Euclidean plane w ith unit disks (any three disks have a gap in the middle): we prove a qua ntitative
version of this statement. This simple insight has applications in spectral geometry: it tells us something about the topological struct ure of the vibration profile of a (possibly oddly-shaped) drum
and allows us to recover an improved version of Pleijel's estimate (which was also recently done by Bourgain). X-ALT-DESC: \nIt is obvious that there is no tiling of the Euclidean plane with unit
disks (any three disks have a gap in the middle): we prove a q uantitative version of this statement. This simple insight has applicatio ns in spectral geometry: it tells us something about the
topological stru cture of the vibration profile of a (possibly oddly-shaped) drum and allo ws us to recover an improved version of Pleijel's estimate (which was als o recently done by Bourgain).
DTEND;TZID=Europe/Zurich:20131009T161500 END:VEVENT BEGIN:VEVENT UID:news487@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181229T175840 DTSTART;TZID=Europe/Zurich:20130925T151500 SUMMARY:Seminar
Analysis: Andrea Mondino (ETH Zurich) DESCRIPTION:Given an immersion f of the 2-sphere in a Riemannian manifold ( M\,g) we study quadratic curvature functionals of the type: \\int_{f(S^2) } H^2\, \\
int_f(S^2) A^2\, \\int_{f(S^2)} )|Aº|^2\, where H is the mean curvature\, A is the second fundamental form\, and Aº is the tracefree s econd fundamental form. Minimizers\, and more generally critical
points o f such functionals can be seen respectively as GENERALIZED minimal\, tota lly geodesic and totally umbilical immersions. In the seminar I will revi ew some results (obtained in collaboration
with Kuwert\, Rivière and Sh ygulla) regarding the existence and the regularity of minimizers of such functionals. An interesting observation regarding the results obtained wi th Rivière is that the
theory of Willmore surfaces can be usesfull to co mplete the theory of minimal surfaces (in particular in relation to the e xistence of canonical smooth representatives in homotopy classes\, a clas
sical program started by Sacks and Uhlenbeck). X-ALT-DESC: \nGiven an immersion f of the 2-sphere in a Riemannian manifold (M\,g) we study quadratic curvature functionals of the type: \\int_{f(S^ 2)}
H^2\, \\int_f(S^2) A^2\, \\int_{f(S^2)} )|Aº|^2\, where H is the mea n curvature\, A is the second fundamental form\, and Aº is the tracefree second fundamental form. Minimizers\, and more generally
critical points of such functionals can be seen respectively as GENERALIZED minimal\, to tally geodesic and totally umbilical immersions. In the seminar I will re view some results (obtained in
collaboration with Kuwert\, Rivière and Shygulla) regarding the existence and the regularity of minimizers of suc h functionals. An interesting observation regarding the results obtained with Rivière
is that the theory of Willmore surfaces can be usesfull to complete the theory of minimal surfaces (in particular in relation to the existence of canonical smooth representatives in homotopy classes
\, a cl assical program started by Sacks and Uhlenbeck). DTEND;TZID=Europe/Zurich:20130925T161500 END:VEVENT BEGIN:VEVENT UID:news504@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T171557 DTSTART;
TZID=Europe/Zurich:20130529T151500 SUMMARY:Seminar Analysis: Giovanni Alberti (University of Pisa) DESCRIPTION:Rademacher theorem states that every Lipschitz function on the euclidean space is
differentiable almost everywhere with respect to the L ebesgue measure. In this talk I will explain how this statement should be modified when the Lebesgue measure is replaced by an arbitrary
singular measure\, and in particular I will show that the differentiability prope rties of Lipschitz functions with respect to such a measure are exactly d escribed by the decompositions of the
measure in terms of (measures on) r ectifiable curves. This result is directly related to recent work by many authors\, including myself\, David Bate\, Marianna Csornyei\, Peter Jone s\, Andrea
Marchese\, and David Preiss. X-ALT-DESC: \nRademacher theorem states that every Lipschitz function on th e euclidean space is differentiable almost everywhere with respect to the Lebesgue measure. In
this talk I will explain how this statement should be modified when the Lebesgue measure is replaced by an arbitrary singula r measure\, and in particular I will show that the differentiability pro
perties of Lipschitz functions with respect to such a measure are exactly described by the decompositions of the measure in terms of (measures on) rectifiable curves. This result is directly related
to recent work by ma ny authors\, including myself\, David Bate\, Marianna Csornyei\, Peter Jo nes\, Andrea Marchese\, and David Preiss. DTEND;TZID=Europe/Zurich:20130529T161500 END:VEVENT
BEGIN:VEVENT UID:news503@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T171112 DTSTART;TZID=Europe/Zurich:20130522T151500 SUMMARY:Seminar Analysis: Jeremy Marzuola (University of North Carolina)
DESCRIPTION:We will discuss the results of several joint ongoing projects ( with subsets of collaborators Pierre Albin\, Hans Christianson\, Colin G uillarmou\, Jason Metcalfe\, Laurent Thomann and
Michael Taylor)\, which explore the existence\, stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.
X-ALT-DESC: \nWe will discuss the results of several joint ongoing projects (with subsets of collaborators Pierre Albin\, Hans Christianson\, Colin Guillarmou\, Jason Metcalfe\, Laurent Thomann and
Michael Taylor)\, which explore the existence\, stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.
DTEND;TZID=Europe/Zurich:20130522T161500 END:VEVENT BEGIN:VEVENT UID:news502@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T170555 DTSTART;TZID=Europe/Zurich:20130515T151500 SUMMARY:Seminar
Analysis: Anna Mazzucato (Penn State University) DESCRIPTION:I will present some (old) results on the transport and dissipat ion of enstrophy in 2D incompressible flows. Enstrophy is half the space
integral of vorticity squared\, and it is a relevant quantity in 2D turb ulence. I consider initial data with vorticity in L2 and its logari thmic refinements and study exact transport of
enstrophy by the velocity field. I also consider data in the larger Besov space $B^{0}_{2\,\\inft y}$ and study the existence of well-defined enstrophy defects\, measuring the rate of enstrophy
dissipation. \\r\\nThis is joint work with Milton Lopes Fihlo and Helena Nussenzveig Lopes. X-ALT-DESC:\nI will present some (old) results on the transport and dissipa tion of enstrophy in 2D
incompressible flows. Enstrophy is half the space integral of vorticity squared\, and it is a relevant quantity in 2D tur bulence. \; I \; consider initial \; data with vorticity in L<
sup>2 and its logarithmic refinements and study exact transport of enstrophy by the velocity field. I also consider data in the larger Besov space \; $B^{0}_{2\,\\infty}$ and study the existence
of well-define d enstrophy defects\, measuring the rate \; of enstrophy dissipation. \nThis is joint work with Milton Lopes Fihlo and Helena Nussenzveig Lopes . DTEND;TZID=Europe/
Zurich:20130515T161500 END:VEVENT BEGIN:VEVENT UID:news501@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T165929 DTSTART;TZID=Europe/Zurich:20130424T151500 SUMMARY:Seminar Analysis: Luigi Berselli
(University of Pisa) DESCRIPTION:We show some regularity results for some classes of 2D incompre ssible fluids\, needed to show uniqueness of particle trajectories. X-ALT-DESC: \nWe show some
regularity results for some classes of 2D incomp ressible fluids\, needed to show  \;uniqueness of particle trajectorie s. DTEND;TZID=Europe/Zurich:20130424T161500 END:VEVENT BEGIN:VEVENT
UID:news500@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T165526 DTSTART;TZID=Europe/Zurich:20130410T151500 SUMMARY:Seminar Analysis: Maria Laura Delle Monache (INRIA Sophia Antipolis - OPALE
Project-Team) DESCRIPTION:Several phenomena in traffic flow can be modeled through the us e of conservation laws. We present two PDE-ODE coupled models that are use d in different traffic situations.
First\, we consider a model that applie s to moving bottlenecks and then we consider a model that applies in contr ol problems for highway ramp metering. We provide a rigorous analytical fr amework
for the Cauchy and Riemann problems and we show some numerical sim ulations. X-ALT-DESC:\nSeveral phenomena in traffic flow can be modeled through the u se of conservation laws. We present two
PDE-ODE coupled models that are us ed in different traffic situations. First\, we consider a model that appli es to moving bottlenecks and then we consider a model that applies in cont rol problems
for highway ramp metering. We provide a rigorous analytical f ramework for the Cauchy and Riemann problems and we show some numerical si mulations. DTEND;TZID=Europe/Zurich:20130410T161500 END:VEVENT
BEGIN:VEVENT UID:news499@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T165153 DTSTART;TZID=Europe/Zurich:20130403T151500 SUMMARY:Seminar Analysis: Isabelle Gallagher (Paris Diderot)
DESCRIPTION:In this talk I will present some recent results in the study of the Cauchy problem for the three-dimensional Navier-Stokes equations. In particular using the fact that the two-dimensional
equation is well-pos ed\, I will try to explain the role of "spectral anisotropy" in the resol ution of the equations. X-ALT-DESC: \nIn this talk I will present some recent results in the study of
the Cauchy problem for the three-dimensional Navier-Stokes equations. In particular using the fact that the two-dimensional equation is well-p osed\, I will try to explain the role of "\;spectral
anisotropy"\; in the resolution of the equations. DTEND;TZID=Europe/Zurich:20130403T161500 END:VEVENT BEGIN:VEVENT UID:news498@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T164925 DTSTART;
TZID=Europe/Zurich:20130327T151500 SUMMARY:Seminar Analysis: Joules Nahas (EPFL) DESCRIPTION:Following the work of Krieger\, Schlag\, and Tataru\, we constr uct a family of blow-up solutions with
finite energy norm to the equatio n \\r\\n∂t2u - Δg u = |u|4 u.\\r\\nThis family has a continuous rate o f blow up\, but In contrast to the case where g is the Minkowski metric\ , the argument used
to produce these solutions can only obtain blow up r ates that are bounded above. \\r\\nThis is joint work with S. Shashahani. X-ALT-DESC:\nFollowing the work of Krieger\, Schlag\, and Tataru\, we
const ruct a family of blow-up solutions with finite energy norm to the equati on \n∂[t]^2u - Δg u = |u|^4 u.\nThis fami ly has a continuous rate of blow up\, but In contrast to the case where g is
the Minkowski metric\, the argument used to produce these solutions can only obtain blow up rates that are bounded above. \nThis is joint wo rk with S. Shashahani. DTEND;TZID=Europe/
Zurich:20130327T161500 END:VEVENT BEGIN:VEVENT UID:news497@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T162819 DTSTART;TZID=Europe/Zurich:20130320T151500 SUMMARY:Seminar Analysis: Scott N.
Armstrong (University of Wisconsin) DESCRIPTION:I will present a regularity result for degenerate elliptic equa tions in nondivergence form. In joint work with Charlie Smart\, we extend the
regularity theory of Caffarelli to equations with possibly unbounded ellipticity - provided that the ellipticity satisfies an averaging cond ition. As an application we obtain a stochastic
homogenization result for such equations which is equivalent to an invariance principle for random diffusions in random environments. The degenerate equations homogenize t o uniformly elliptic
equations\, and we give an estimate of the elliptici ty in terms of the averaging condition. X-ALT-DESC:\nI will present a regularity result for degenerate elliptic equ ations in nondivergence form.
In joint work with Charlie Smart\, we exten d the regularity theory of Caffarelli to equations with possibly unbounde d ellipticity - provided that the ellipticity satisfies an averaging con dition.
As an application we obtain a stochastic homogenization result fo r such equations which is equivalent to an invariance principle for rando m diffusions in random environments. The degenerate
equations homogenize to uniformly elliptic equations\, and we give an estimate of the elliptic ity in terms of the averaging condition. DTEND;TZID=Europe/Zurich:20130320T161500 END:VEVENT
BEGIN:VEVENT UID:news496@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T164719 DTSTART;TZID=Europe/Zurich:20130227T151500 SUMMARY:Seminar Analysis: H. J. Nussenzveig Lopes (Instituto de Matematica
Universidade Federal do Rio de Janeiro) DESCRIPTION:Following the work of Krieger\, Schlag\, and Tataru\, we constr uct a family of blow-up solutions with finite energy norm to the equation \\r\\
n∂t2u - Δg u = |u|4 u.\\r\\nThis family has a continuous rate of blow up\, but In contrast to the case where g is the Minkowski metric\, the argument used to produce these solutions can only obtain
blow up rate s that are bounded above. \\r\\nThis is joint work with S. Shashahani. X-ALT-DESC:\nFollowing the work of Krieger\, Schlag\, and Tataru\, we const ruct a family of blow-up solutions with
finite energy norm to the equatio n \n∂[t]^2u - Δg u = |u|^4 u.\nThis famil y has a continuous rate of blow up\, but In contrast to the case where g is the Minkowski metric\, the argument used to
produce these solutions ca n only obtain blow up rates that are bounded above. \nThis is joint work with S. Shashahani. DTEND;TZID=Europe/Zurich:20130227T161500 END:VEVENT BEGIN:VEVENT
UID:news512@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190104T233058 DTSTART;TZID=Europe/Zurich:20121219T151500 SUMMARY:Seminar Analysis: Benjamin Texier (Université Paris-Diderot (Paris 7))
DESCRIPTION:The Garding inequality states that positive pseudo-differential symbols are associated with semi-positive operators. It can be used in pa rticular to show time-exponential growth of
solutions to initial value pro blems for elliptic equations. I will give examples in which Garding fails to give appropriate bounds\, and a way to overcome this difficulty. Exampl es include
high-frequency asymptotics of systems based on Maxwell's equati ons\, and compressible Euler systems with a Van der Waals pressure law. In these cases\, appropriate bounds are derived via a
description of the par ametrix of a pseudo-differential system. X-ALT-DESC: \nThe Garding inequality states that positive pseudo-differenti al symbols are associated with semi-positive operators. It
can be used in particular to show time-exponential growth of solutions to initial value p roblems for elliptic equations. I will give examples in which Garding fail s to give appropriate bounds\, and
a way to overcome this difficulty. Exam ples include high-frequency asymptotics of systems based on Maxwell's equa tions\, and compressible Euler systems with a Van der Waals pressure law. In these
cases\, appropriate bounds are derived via a description of the p arametrix of a pseudo-differential system. DTEND;TZID=Europe/Zurich:20121219T161500 END:VEVENT BEGIN:VEVENT UID:news511@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190104T232839 DTSTART;TZID=Europe/Zurich:20121212T151500 SUMMARY:Seminar Analysis: Camillo De Lellis (University of Zurich) DESCRIPTION:A well-known theorem of Almgren
shows that area-minimizing inte gral k-dimensional currents in a Riemannian manifold of arbitrary dimensio n N are regular up to a set of closed dimension of Hausdorff dimension at most N-2. In a
joint work with Emanuele Spadaro we give a much shorter pro of of this statement in the euclidean setting\, following the general prog ram of Almgren but introducing new ideas at the various steps.
In this tal k I will explain some if these ideas. A generalization of our proof to the Riemannian case is work in progress. X-ALT-DESC: \nA well-known theorem of Almgren shows that area-minimizing in
tegral k-dimensional currents in a Riemannian manifold of arbitrary dimension N are regular up to a set of closed dimension of Hausdor ff dimension at most N-2. In a joint work with Emanuele Spadaro
we give a much shorter proof of this statement in the euclidean setting\, fol lowing the general program of Almgren but introducing new ideas at the var ious steps. In this talk I will explain some
if these ideas. A generalizat ion of our proof to the Riemannian case is work in progress. DTEND;TZID=Europe/Zurich:20121212T161500 END:VEVENT BEGIN:VEVENT UID:news510@dmi.unibas.ch DTSTAMP;TZID=
Europe/Zurich:20190104T232526 DTSTART;TZID=Europe/Zurich:20121205T151500 SUMMARY:Seminar Analysis: Przemek Zieliński (Institute of Mathematics Poli sh Academy of Sciences) DESCRIPTION:I will present
the results on the existence of solutions to sem i-linear equation \\r\\n Lx+N(x)=0\, \\r\\nwhere L is a linear and N a nonlinear operator defined on Hilbert space. I concentrate on the case when
0 is in an essential spectrum of L. The two main methods which I use are: topological degree in infinite-dimensional spaces and the spectral theory for linear operators in Hilbert spaces. This
results are part of m y Ph.D. project. X-ALT-DESC:\nI will present the results on the existence of solutions to se mi-linear equation \n \;  \; Lx+N(x)=0\, \nwhere L is a linear and N a
nonlinear operator defined on Hilbert space. I concentrate on the ca se when 0 is in an essential spectrum of L. The two main methods which I use are: topological degree in infinite-dimensional
spaces and the spectr al theory for linear operators in Hilbert spaces. This results are part o f my Ph.D. project. \; DTEND;TZID=Europe/Zurich:20121205T161500 END:VEVENT BEGIN:VEVENT
UID:news509@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190104T232300 DTSTART;TZID=Europe/Zurich:20121128T151500 SUMMARY:Seminar Analysis: Matteo Focardi (University of Zurich) DESCRIPTION:In this
talk I shall focus on the higher integrability property enjoyed by the approximate gradients of local minimizers of the 2d Mumfo rd-Shah energy. Related regularity issues shall be also discussed. \\r
\\n This is joint work with C. De Lellis (Universitaet Zuerich). X-ALT-DESC: \nIn this talk I shall focus on the higher integrability proper ty enjoyed by the approximate gradients of local
minimizers of the 2d Mum ford-Shah energy. Related regularity issues shall be also discussed. \nTh is is joint work with C. De Lellis (Universitaet Zuerich). DTEND;TZID=Europe/Zurich:20121128T161500
END:VEVENT BEGIN:VEVENT UID:news507@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T173513 DTSTART;TZID=Europe/Zurich:20121114T151500 SUMMARY:Seminar Analysis: Elisabetta Chiodaroli (University of
Zurich) DESCRIPTION:The deceivingly simple–looking compressible Euler equations o f gas dynamics have a long history of important contributions over more t han two centuries. If we allow for
discontinuous solutions\, uniqueness a nd stability are lost. In order to restore such properties further restr ictions on weak solutions have been proposed in the form of entropy inequ alities. In
this talk\, we will discuss some counterexamples to the well –posedness theory of entropysolutions to the multi–dimensional compre ssible Euler equations. First\, we show failure of uniqueness on a
finit e–time interval for entropy solutions starting from any continuously di fferentiable initial density and suitably constructed initial linear mom enta. In other words\, we prove that there exist
wild initial data allowi ng for infinitely many distinct entropy weak solutionsnof the compressib le Euler system. Finally\, we present a new upshot: a classical Riemann d atum is a wild initial datum
in 2 space–dimensions. All our methods are inspired by a new analysis of the incompressible Euler equations recentl y carried out by De Lellis and Székelyhidi and based on a revisited “h -principle”.
X-ALT-DESC: \nThe deceivingly simple–looking compressible Euler equations of gas dynamics have a long history of important contributions over more than two centuries. If we allow for discontinuous
solutions\, uniqueness and stability are lost. In order to restore such properties further res trictions on weak solutions have been proposed in the form of entropy ine qualities. In this talk\, we
will discuss some counterexamples to the wel l–posedness theory of entropysolutions to the multi–dimensional compr essible Euler equations. First\, we show failure of uniqueness on a fini te–time
interval for entropy solutions starting from any continuously d ifferentiable initial density and suitably constructed initial linear mo menta. In other words\, we prove that there exist wild initial
data allow ing for infinitely many distinct entropy weak solutionsnof the compressi ble Euler system. Finally\, we present a new upshot: a classical Riemann datum is a wild initial datum in 2
space–dimensions. All our methods ar e inspired by a new analysis of the incompressible Euler equations recent ly carried out by De Lellis and Székelyhidi and based on a revisited “ h-principle”.
DTEND;TZID=Europe/Zurich:20121114T161500 END:VEVENT BEGIN:VEVENT UID:news506@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T173139 DTSTART;TZID=Europe/Zurich:20121107T151500 SUMMARY:Seminar
Analysis: Laura V. Spinolo (IMATI-CNR\, Pavia) DESCRIPTION:The talk will focus on the eigenvalue problem for the Laplace operator defined in an open and bounded domain\, with homogenous conditio ns
of either Dirichlet or Neumann type assigned at the boundary. Under fa irly weak regularity assumptions on the domain\, the problem admits a div erging sequence of nonnegative eigenvalues. I will
discuss some new quant itative estimates controlling how each of the eigenvalues change when the domain is perturbed. These estimates apply to Lipschitz and to so-called Reifenberg-flat domains. The
proof is based on an abstract lemma which applies to both the Neumann and the Dirichlet problem and which could be applied to other classes of domains. \\r\\nThe talk will be based on join t works
with A. Lemenant and E. Milakis. X-ALT-DESC:\nThe talk will focus on the eigenvalue problem for the Laplace operator defined in an open and bounded domain\, with homogenous conditi ons of either
Dirichlet or Neumann type assigned at the boundary. Under f airly weak regularity assumptions on the domain\, the problem admits a di verging sequence of nonnegative eigenvalues. I will discuss some
new quan titative estimates controlling how each of the eigenvalues change when th e domain is perturbed. These estimates apply to Lipschitz and to so-calle d Reifenberg-flat domains. \; The
proof is based on an abstract lemma which applies to both the Neumann and the Dirichlet problem and which co uld be applied to other classes of domains. \nThe talk will be based on jo int works with
A. Lemenant and E. Milakis. DTEND;TZID=Europe/Zurich:20121107T161500 END:VEVENT BEGIN:VEVENT UID:news505@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181231T172746 DTSTART;TZID=Europe/
Zurich:20121003T151500 SUMMARY:Seminar Analysis: Laura Keller (Universität Münster) DESCRIPTION:Starting from the example of harmonic maps\, we will find a cla ss of PDE problems which enjoy an
additional\, at first glimpse hidden pr operty: Antisymmetry! This feature enables us to deduce regularity assert ions which heavily rely on Wente's theorem. For this latter\, various ap proaches
will be discussed. The presentation will be completed by a versi on of Wente's result for arbitrary dimension. X-ALT-DESC: \nStarting from the example of harmonic maps\, we will find a c lass of PDE
problems which enjoy an additional\, at first glimpse hidden property: Antisymmetry! This feature enables us to deduce regularity asse rtions which heavily rely on Wente's theorem. For this latter\,
various approaches will be discussed. The presentation will be completed by a ver sion of Wente's result for arbitrary dimension. DTEND;TZID=Europe/Zurich:20121003T161500 END:VEVENT BEGIN:VEVENT
UID:news524@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190105T001818 DTSTART;TZID=Europe/Zurich:20120530T151500 SUMMARY:Seminar Analysis: Stefano Spirito (University of L'Aquila) DESCRIPTION:In this
talk I will discuss the vanishing viscosity problem for the Navier-Stokes equations in a bounded domain. It is well-known that w hen Dirichlet conditions are imposed on the boundary the inviscid
limit i s currently an open and difficult problem. On the other hand when other type of boundary conditions are considered the situation became simpler. In this talk a particular type of Navier
boundary conditions involving on ly the vorticity of the velocity field are considered. In particular\, I will discuss recent results obtained in collaboration with Luigi Berselli (University of
Pisa) concerning the inviscid limit in energy norm of the Leray weak solutions and the inviscid limit in higher norms of local smo oth solutions of the Navier-Stokes equations. X-ALT-DESC:\nIn this
talk I will discuss the vanishing viscosity problem fo r the Navier-Stokes equations in a bounded domain. It is well-known that when Dirichlet conditions are imposed on the boundary the inviscid
limit is currently an open and difficult problem. On the other hand when other type of boundary conditions are considered the situation became simpler. In this talk a particular type of Navier
boundary conditions involving o nly the vorticity of the velocity field are considered. In particular\, I will discuss recent results obtained in collaboration with Luigi Bersell i (University of
Pisa) concerning the inviscid limit in energy norm of th e Leray weak solutions and the inviscid limit in higher norms of local sm ooth solutions of the Navier-Stokes equations. DTEND;TZID=Europe/
Zurich:20120530T161500 END:VEVENT BEGIN:VEVENT UID:news523@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190105T001721 DTSTART;TZID=Europe/Zurich:20120523T151500 SUMMARY:Seminar Analysis: Luca
Martinazzi (Rutgers) DESCRIPTION:We study the Moser-Trudinger equation Δu = λu Exp(u2)\, λ>0 on a 2-dimensional disk\, arising from the Moser-Trudinger sharp embeddi ng of H10(Disk) into the Orlicz
space of functions u with Exp(u2) integra ble. We answer some long standing open questions: \\r\\na) The weak limit of a blowing-up sequence of solutions to the Moser-Trudinger equation on a disk is
0. \\r\\nb) The Dirichlet energy of a blowing-up sequence of sol utions on a disk converges to 4π. \\r\\nc) For L large enough\, the Moser -Trudinger equation on a disk admits no solution with
Dirichlet energy lar ger than L. \\r\\nThis work is joint project with Andrea Malchiodi (SISSA - Trieste). X-ALT-DESC:\nWe study the Moser-Trudinger equation \;Δu = λu Exp(u2)\ , λ>\;0 on a
2-dimensional disk\, arising from the Moser-Trudinger sha rp embedding of H^1[0](Disk) into the Orlicz space of f unctions u with Exp(u2) integrable. We answer some long standing open que stions: \
na) The weak limit of a blowing-up sequence of solutions to the Moser-Trudinger equation on a disk is 0. \nb) The Dirichlet energy of a bl owing-up sequence of solutions on a disk converges to 4π. \
nc) For L larg e enough\, the Moser-Trudinger equation on a disk admits no solution with Dirichlet energy larger than L. \nThis work is joint project with Andrea M alchiodi (SISSA - Trieste). DTEND;
TZID=Europe/Zurich:20120523T161500 END:VEVENT BEGIN:VEVENT UID:news522@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190105T000645 DTSTART;TZID=Europe/Zurich:20120509T151500 SUMMARY:Seminar Analysis:
Christian Hainzl (Tuebingen) DESCRIPTION:We give the first rigorous derivation of the celebrated Ginzbur g-Landau (GL) theory\, starting from the microscopic Bardeen-Cooper-Schri effer (BCS) model.
Close to the critical temperature\, GL arises as an e ffective theory on the macroscopic scale. The relevant scaling limit is s emiclassical in nature\, and semiclassical analysis\, with minimal
regula rity assumptions\, plays an important part in our proof. X-ALT-DESC: \nWe give the first rigorous derivation of the celebrated Ginzb urg-Landau (GL) theory\, starting from the microscopic
Bardeen-Cooper-Sch rieffer (BCS) model. Close to the critical temperature\, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature\, and
semiclassical analysis\, with minimal regu larity assumptions\, plays an important part in our proof. DTEND;TZID=Europe/Zurich:20120509T161500 END:VEVENT BEGIN:VEVENT UID:news521@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190105T000226 DTSTART;TZID=Europe/Zurich:20120425T151500 SUMMARY:Seminar Analysis: Roland Donninger (EPFL) DESCRIPTION:I present some recent results\, obtained in
collaboration with Joachim Krieger\, on novel types of solutions to the critical wave equati on in 3 spatial dimensions. These solutions either blow up at infinity or vanish at a prescribed rate. The
existence of such exotic dynamics viola tes a strong version of the soliton resolution conjecture.\\r\\nFrancois Bouchut: \\r\\nTBA X-ALT-DESC: \nI present some recent results\, obtained in
collaboration wit h Joachim Krieger\, on novel types of solutions to the critical wave equa tion in 3 spatial dimensions. These solutions either blow up at infinity or vanish at a prescribed rate.
The existence of such exotic dynamics vio lates a strong version of the soliton resolution conjecture.\nFrancois Bo uchut: \nTBA DTEND;TZID=Europe/Zurich:20120425T161500 END:VEVENT BEGIN:VEVENT
UID:news520@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190104T235930 DTSTART;TZID=Europe/Zurich:20120418T151500 SUMMARY:Seminar Analysis: Evelyne Miot (CNRS & Paris Orsay) DESCRIPTION:A system of
simplified equations has been derived by Klein\, Ma jda and Damodaran to describe the dynamics of nearly parallel vortex filam ents in incompressible 3D fluids. This system combines a 1D
Schrödinger-t ype structure together with the 2D point vortex system. Global existence f or small perturbations of exact parallel filaments has been established by Kenig\, Ponce and Vega in the case
of two filaments and for particular co nfigurations of three filaments. In this talk I will present large time ex istence results for particular configurations of four filaments and for ot her
particular configurations of N filaments for any N larger than 2. I wi ll also discuss some situations of finite time filament collapse. This is joint work with Valeria Banica. X-ALT-DESC: \nA system
of simplified equations has been derived by Klein\, Majda and Damodaran to describe the dynamics of nearly parallel vortex fil aments in incompressible 3D fluids. This system combines a 1D
Schrödinger -type structure together with the 2D point vortex system. Global existence for small perturbations of exact parallel filaments has been established by Kenig\, Ponce and Vega in the case
of two filaments and for particular configurations of three filaments. In this talk I will present large time existence results for particular configurations of four filaments and for other
particular configurations of N filaments for any N larger than 2. I will also discuss some situations of finite time filament collapse. This i s joint work with Valeria Banica. DTEND;TZID=Europe/
Zurich:20120418T161500 END:VEVENT BEGIN:VEVENT UID:news519@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190104T235555 DTSTART;TZID=Europe/Zurich:20120411T151500 SUMMARY:Seminar Analysis: Wolfgang
Reichel (KIT Karlsruhe) DESCRIPTION:We are interested in ground states for the nonlinear Schröding er-equation (NLS) with an interface between two purely periodic media. Thi s means that the
coefficients in the NLS model two different periodic medi a in each halfspace. The resulting problem no longer has a periodic struct ure. Using variational methods we give conditions on the
coefficients such that ground states are created/prevented by the interface. X-ALT-DESC:\nWe are interested in ground states for the nonlinear Schrödin ger-equation (NLS) with an interface between
two purely periodic media. Th is means that the coefficients in the NLS model two different periodic med ia in each halfspace. The resulting problem no longer has a periodic struc ture. Using
variational methods we give conditions on the coefficients suc h that ground states are created/prevented by the interface. DTEND;TZID=Europe/Zurich:20120411T151500 END:VEVENT BEGIN:VEVENT
UID:news518@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190104T235124 DTSTART;TZID=Europe/Zurich:20120404T151500 SUMMARY:Seminar Analysis: Michael Reiterer (ETH Zurich) DESCRIPTION:About twenty years
ago\, Choptuik studied numerically the gravi tational collapse (Einstein field equations) of a massless scalar field i n spherical symmetry\, and found strong evidence for a universal\, self- similar
solution at the threshold of black hole formation. We give a rigo rous\, computer assisted proof of the existence of Choptuik's spacetime\, and show that it is real analytic. This is joint work with
E. Trubowitz. X-ALT-DESC: \nAbout twenty years ago\, Choptuik studied numerically the gra vitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry\, and found
strong evidence for a universal\, sel f-similar solution at the threshold of black hole formation. We give a ri gorous\, computer assisted proof of the existence of Choptuik's spacetime \, and show
that it is real analytic. This is joint work with E. Trubowit z. DTEND;TZID=Europe/Zurich:20120404T161500 END:VEVENT BEGIN:VEVENT UID:news516@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190104T234043
DTSTART;TZID=Europe/Zurich:20120328T151500 SUMMARY:Seminar Analysis: Sara Daneri (University of Zurich) DESCRIPTION:We consider the optimal transportation problem with cost functi ons given by
generic convex norms in Rd and absolutely continuous first ma rginals. We show the existence of a partition of Rd into k-dimensional set s\, k=0\,...\,d\, such that every optimal transport plan can
be characteri zed\, via disintegration of measures\, as a family of optimal transport pl ans each moving a conditional probability of the first marginal inside one of these k-dimensional sets\, along
the directions of an extremal k-dimen sional cone of the convex norm. Moreover\, the conditional probabilities of the first marginal on these sets are absolutely continuous with respec t to the
k-dimensional Hausdorff measure on the k-dimensional sets on whic h they are concentrated\, thus settling the longstanding Sudakov's problem of the existence of locally affine decompositions of Rd
that reduce norm cost transportation problem to families of lower dimensional ones. Finally \, due to the minimality of our partition with respect to this "dimensiona l reduction" property\,
applications to secondary cost functions obtained first minimizing with respect to a convex norm and then with respect to a finer one (e.g.\, a strictly convex one) will be shown. These results were
obtained in collaboration with Stefano Bianchini (SISSA\, Trieste). X-ALT-DESC: \nWe consider the optimal transportation problem with cost func tions given by generic convex norms in R^d and
absolutely continuous first marginals. We show the existence of a partition of R< /b>^d into k-dimensional sets\, k=0\,...\,d\, such that every op timal transport plan can be characterized\, via
disintegration of measures \, as a family of optimal transport plans each moving a conditional probab ility of the first marginal inside one of these k-dimensional sets\, along the directions of an
extremal k-dimensional cone of the convex norm.  \; Moreover\, the conditional probabilities of the first marginal on these sets are absolutely continuous with respect to the k-dimensional
Hausdorf f measure on the k-dimensional sets on which they are concentrated\, thus settling the longstanding Sudakov's problem of the existence of locally af fine decompositions of R^d that reduce
norm cost transpor tation problem to families of lower dimensional ones. Finally\, due to the minimality of our partition with respect to this "\;dimensional reduc tion"\; property\,
applications to secondary cost functions obtained f irst minimizing with respect to a convex norm and then with respect to a f iner one (e.g.\, a strictly convex one) will be shown. These results
were obtained in collaboration with \; Stefano Bianchini (SISSA\, Trieste). DTEND;TZID=Europe/Zurich:20120328T161500 END:VEVENT BEGIN:VEVENT UID:news514@dmi.unibas.ch DTSTAMP;TZID=Europe/
Zurich:20190104T233639 DTSTART;TZID=Europe/Zurich:20120321T151500 SUMMARY:Seminar Analysis: Lisa Beck (Bonn) DESCRIPTION:In this seminar we will give a survey on some aspects of the cl assical
regularity theory for W1\,p-solutions to elliptic problems (convex variational integral or elliptic systems)\, restricting ourselves to simp le model cases and explaining the challenges behind
proving such results. For scalar valued solutions full regularity (continuous or even better) ca n be established under very mild assumptions\, which is nowadays known as the De Giorgi-Nash-Moser
theory. In the vectorial case instead\, the vario us component functions and their partial derivative can interact in such a way that the system or variational integral under consideration allows di
scontinuous or even unbounded solutions\, and in fact various counterexamp les to full regularity have been constructed. As a consequence\, only part ial regularity can be expected\, in the sense
that the solution (or its gr adient) is locally continuous outside of a negligible set (the singular se t). We will give some heuristics on the generalapproach to partial regular ity results and then
we briefly discuss how in some particular situations (small space dimensions\, special structure conditions) an upper bound on the Hausdorff dimension of the singular set can be obtained. X-ALT-DESC:
\nIn this seminar we will give a survey on some aspects of the c lassical regularity theory for W^1\,p-solutions to elliptic prob lems (convex variational integral or elliptic systems)\, restricting
ourse lves to simple model cases and explaining the challenges behind proving su ch results. For scalar valued solutions full regularity (continuous or eve n better) can be established under very
mild assumptions\, which is nowada ys known as the De Giorgi-Nash-Moser theory. In the vectorial case instead \, the various component functions and their partial derivative can intera ct in such a
way that the system or variational integral under considerati on allows discontinuous or even unbounded solutions\, and in fact various counterexamples to full regularity have been constructed. As a
consequence \, only partial regularity can be expected\, in the sense that the solutio n (or its gradient) is locally continuous outside of a negligible set (the singular set). We will give some
heuristics on the general
approach to partial regularity results and then we briefly discuss how in some part icular situations (small space dimensions\, special structure conditions) an upper bound on the Hausdorff dimension
of the singular set can be obtai ned. DTEND;TZID=Europe/Zurich:20120321T161500 END:VEVENT END:VCALENDAR :S]]]] | {"url":"https://dmi.unibas.ch/en/news-events/past-events/past-events-mathematics/4349.ics","timestamp":"2024-11-13T09:17:43Z","content_type":"text/calendar","content_length":"304872","record_id":"<urn:uuid:2173d559-37cd-48c1-bcdb-6f14574512c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00516.warc.gz"} |
Quantum Russian Roulette — LessWrong
The quantum Russian roulette is a game where 16 people participate. Each of them gets a unique four digit binary code assigned and deposits $50000. They are put to deep sleep using some drug. The
organizer flips a quantum coin four times. Unlike in Russian roulette, here only the participant survives whose code was flipped. The others are executed in a completely painless manner. The survivor
takes all the money.
Let us assume that none of them have families or very good friends. Then the only result of the game is that the guy who wins will enjoy a much better quality of life. The others die in his Everett
branch, but they live on in others. So everybody's only subjective experience will be that he went into a room and woke up $750000 richer.
Being extremely spooky to our human intuition, there are hardly any trivial objective reasons to oppose this game under the following assumptions:
1. Near 100% confidence in the Multiple World nature of our universe
The natural question arises whether it could be somehow checked that the method really works, especially that the Multiple World Hypothesis is correct. At first sight, it looks impossible to convince
anybody besides the participant who survived the game.
However there is a way to convince a lot of people in a few Everett branches: You make a one-time big announcement in the Internet, TV etc. and say that there is a well tested quantum coin-flipper,
examined by a community consisting of the most honest and trusted members of the society. You take some random 20 bit number and say that you will flip the equipment 20 times and if the outcome is
the same as the predetermined number, then you will take it as a one to million evidence that the Multiple World theory works as expected. Of course, only people in the right branch will be
convinced. Nevertheless, they could be convinced enough to make serious thoughts about the viability of quantum Russian roulette type games.
My question is: What are the possible moral or logical reasons not to play such games? Both from individual or societal standpoints.
[EDIT] A Simpler version (single player version of the experiment): The single player generates lottery numbers by flipping quantum coins. Sets up an equipment that kills him in sleep if the
generated numbers don't coincide with his. In this way, he can guarantee waking up as a lottery millionaire.
Yes - I didn't mean all other branches.
If we all did it together though, the worlds we left behind would be like some sort of geeky Atlas Shrugged dystopia. Heh.
:) We could still make the second experiment and if the numbers match, throw a party... ;)
I don't think my game is a simple reformulation of quantum immortality.
I don't even believe in quantum immortality. At least it is not implied in any way by MWI.
It is perfectly possible that you have increasing amounts of "quantum luck" in a lot of branches, but finally you die in each of the branches, because the life-supporting miracles increase your life
less and less, and when they hit the resolution of time, you simply run out of them and die for sure.
If you think time is continuous, then think of the Zenon paradox to see why the infinite sum of such pieces can add to a finite amount of time gained.
I don't see why you wanted to. You could only increase P(MWI) by finding there any.
You take some random 20 bit number and say that you will flip the equipment 20 times and if the outcome is the same as the predetermined number, then you will take it as a one to million evidence
that the Multiple World theory works as expected.
That doesn't convince anyone else; from their perspective, in Bayesian terms, the experiment has the same million-to-one improbability of producing this result, regardless of whether QTI is true,
since they're not dying in the other worlds. From your perspective, you've ended up in a world where you experienced what feels like strong evidence of QTI being true, but you can never communicate
this evidence to anyone else. If we hook up a doomsday device to the coinflipper, then in worlds where we survive, we can never convince aliens.
Well, Many Worlds doesn't actually require infinity, which is a plus in my book. You can have a Many Worlds scenario with "only" (the number of discrete possible actions since the beginning of time)
worlds, whereas a Single Big World that wasn't actually infinite would require further explanation.
If you assume 3, average utilitarianism says you should kill anyone who has below average utility, since that raises the average. So in the end you kill everyone except the one person who has the
highest utility.
You really need to also assume killing people doesn't decrease utility for those who are left, which usually doesn't work too well for humans...
It is possible to kill someone without invoking any negative experiences.
Average utilitarianism requires more, it requires that it is possible to have a policy of systematically killing most people that does not result in negative experiences. This does not seem
meaningfully possible for any agents that are vaguely human, so this is a straw man objection to average utilitarianism, and a pretty bad one at that.
No. If you assign a large negative utility total death. That is me dieing unconditionally has a large negative value to me.
If I assume that the rate of my unconditional death does not change significantly after the experiment, then it could make sense to play the roulette.
average utilitarianism says you should kill anyone who has below average utility
This assumes that killing people with low utility has absolutely no effect on the utility of anyone else, and that living in a world where you will be killed if you're not happy enough has no
negative effect on your happiness. This is, to put it mildly, completely and totally false without very radically altering the human mind.
On a related note, the beatings will continue until morale improves.
Thanks for the links. They look interesting.
The base idea seems identical to the quantum suicide scenarios. Although I did not know about them, my only contribution is to put up a convincing concrete scenario, where suicide offers a benefit to
each players.
MWI is morally uninteresting, unless you do nontrivial quantum computation.
I think that the intuition at steak here is something about continuity of conscious experience. The intuition that Christian might have, if I may anticipate him, is that everyone in the experiment
will actually experience getting $750,000, because somehow the word-line of their conscious experience will continue only in the worlds where they do not die.
I think that, in some sense, this is a mistake, because it fundamentally rests upon a very very strong intuition that there exists a unique person who I will be in the future. This is an intuition
that evolution programmed into us for obvious reasons: we are more likely to act like a good Could-Should-Would agent if we think that the benefits and costs associated with our actions will accrue
to us rather than to some vague probability distribution over an infinite set of physically realized future continuations-of-me, with the property that whatever I do, some of them will die, and
whatever I do, some of them will be rich and happy.
I did it implicitly in the OP. Assuming that, you get a better expected value in the quantum scenario.
A logical coin flip be much more scary (and negative utility) assuming certain death for some of the participiants.
(I don't buy quantum immortality arguments. They resemble on Achilles-Turtle problem: Being rescued in shorter and shorter intervals does not imply being rescued for a fixed time.)
An easy reason not to play quantum roulette is that, if your theory justifying it is right, you don't gain any expected utility; you just redistribute it, in a manner most people consider unjust,
among different future yous. If your theory is wrong, the outcome is much worse. So it's at the very best a break even / lose proposition.
Maybe this is the Fermi Great Filter. It is in the interest of every member of society to join one of these groups. Sure in every world that they individually experience, they each end up rich. But
they also all end up in a world where their civilization has suddenly had its population drop to 1/16th of what it was. Per capita goods production may drop even more because all these suddenly rich
people will then want to retire. So they end up poorer in real terms. Likely Civilization colapses and loses the ability to do radio astronomy and we don't see any radio signals from other
civilizations. Fermi paradox explained (with lots of 'maybe's).
I think there will probably have to be some set of worlds in which the losers of the game are alive but near death (it really is necessary to specify the means by which they die). So this really is a
gamble since participating means there is a very slight chance you will wake up, 50,000 dollars poorer and needing an ambulance. To figure out the overall average utility of the game one would need
to include the possible worlds in which the killing mechanism fails. Average utility over the universal wave function would probably still go up, but there would be a few branches where average
utility would go down, dramatically. So the answer it would depend on whether you were quantifying over the entire wave function or doing individual worlds.
Or were you ignoring that as part of the thought experiment?
EDIT: I just thought it all out and I think the probability of experiencing surviving the killing mechanism might be 5 in 6. See here. Basically, when calculating your future experience the worlds in
which you survive take over the probability space of the worlds in which you don't survive such that, given about a 5 in 6 chance of your death there is actually about a 5 in 6 chance you experience
losing and surviving since in the set of worlds in which you lose there is a 100% chance of experiencing one of the very few worlds in which you survive. This would make playing Quantum Russian
roulette considerably stupider then playing regular Russian roulette unless you have a killing mechanism that can only fail by failing to do any damage at all.
If mangled worlds is correct (and I understand it correctly), then sufficiently improbable events fail to happen at all. What kind of limit would this place on the problems you can solve with
"quantum suicide voodoo"?
We have seen in the sister topic that mangled worlds theory can in fact account for such information loss. However MWT has similar deficiencies as single worlds: non local action, nonlinearity,
discontinuity. It does not mean it can't be true.
Why would the information content of a quantum universe be measured in bits, rather than qubits? 2^1000 qubits is enough to keep track of every possible configuration of the Hubble volume, without
discarding any low magnitude ones. (Unless of course QM does discard low magnitude branches, in which case your quantum computer would too... but such a circular definition is consistent with any
amount of information content.)
There was a dumb comment here, I deleted it. You're actually completely right, thanks!
I can't read those links, but determining whether an input is a minimum seems to be in co-NP because it allows easy verification of a "no" answer by counterexample. So have those people proved that
NP=co-NP, or am I just being dense?
This still seems to imply that function minimization isn't much harder than NP-complete problems.
Yep, as a programmer I should've said "at most polynomially harder than a problem in NP". You're right that my wording was bad. I still stand by the spirit, though :-)
I'm a bit confused too, but I found a Ph.D. thesis that answers a bunch of these questions. I'm still reading it.
On page 5, it says that for TSP, the question "Is the optimal value equal to k" is D^P-complete (which is something I haven't heard of before).
You can't get the exact answer, but you can approach it exponentially quickly by doing a binary search. So, if you want N digits of accuracy, you're going to need O(N) time. Someone mentioned this
elsewhere but I can't find the comment now.
Your method above would work better, actually (assuming the function is valued from 0 to 1). Just randomly guess x, compute f(x)^n, then place a qubit in a superposition such that it is f(x)^n likely
to be 0, and 1 - f(x)^n likely to be 1. Measure the qubit. If it is 0, quantum suicide. You can do the whole thing a few times, taking n = 10, n = 1000, and so on, until you're satisfied you've
actually got the minimum. Of course, there's always the chance there's a slightly smaller minimum somewhere, so we're just getting an approximate answer like before, albeit an incredibly good
approximate answer.
Yeah, my comment was pretty stupid. Thanks.
It's a classical point -- you can replace the question of "What is the minimal value of f(x) (what is the witness x that gives the minimal value)?" by "Is there a parameter x that gives the value f
(x) less than C (what is an example of such x)?", and then use binary search on C to pinpoint the minimum. Since being able to write down the answer in polynomial time must be a given, you can take
the ability to run the search on C in polynomial time for granted.
Hah. Thanks, that settles the issue for now: short of Scott Aaronson making an unexpected breakthrough, we have no proof that quantum suicide computing can solve any non-polynomial problems :-)
On the computational power of PP and ⊕P says it gives strong evidence that PP is strictly harder than PH, which contains NP.
There's a natural partial mapping in my mind, but a rigorous mapping is beyond me. Check out this. Apparently it can be shown that FNP is strictly harder than NP, under certain conditions. I didn't
understand the condition, and whether it was reasonable or not.
Regardless of all this complexity jazz, which has definitely exceeded my grasp, my random dictionary example still demonstrates that certain problems can be solved exponentially faster with
postselection than without, even if it doesn't prove that FNP > NP.
ETA: Coming from a numerics background, decision problems seem like exotic beasts. I'd prefer to settle the power of quantum voodoo on function problems first =P
The point is the relative cost difference between function evaluations and function minimization.
Well yeah, but the definitions of P and NP involve the size of the input. Your original claim was that we can solve "much harder than NP-complete" problems with quantum voodoo. I don't think you've
proved it yet, but if you revise it to somehow talk about "relative cost differences", it does become somewhat more believable.
like it.
I'm developing a web game called 'quantum roulette' for the website of my upcoming feature Third Contact, and I happened on this thread.
the game will be at www.thirdcontactmovie.com at some point. You might find the film of interest too. Love the discussion.
I hold this to be quite a nice reductio ad absurdum of average utilitarianism. Total utilitarianism handles it fine: there is higher utility in being alive with no money than being dead.
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Here's a funny reformulation of your argument: if you live in a quantum world where deaths are swift and painless, it makes sense to bet a lot of money on the assertion that you will stay alive. This
incentivizes many other people to bet on your death and try hard to kill you. The market takes this into account, so your payoff for staying alive grows very high. Sounds like a win-win situation all
That said, at root it's just a vanilla application of quantum immortality which people may or may not believe in (the MWI doesn't seem to logically imply it). For a really mindblowing quantum trick
see the Elitzur-Vaidman bomb tester. For a deeper exploration of the immortality issue, see Quantum suicide reality editing.
For a really mindblowing quantum trick see the Elitzur-Vaidman bomb tester.
Consider a collection of bombs, some of which are duds. The bombs are triggered by a single photon. Usable bombs will absorb the photon and detonate. Dud bombs will not absorb the photon. The
problem is how to separate the usable bombs from the duds.
A solution is for the sorter to use a mode of observation known as counterfactual measurement
In 1994, Anton Zeilinger, Paul Kwiat, Harald Weinfurter, and Thomas Herzog actually performed an equivalent of the above experiment, proving interaction-free measurements are indeed possible.
See also:
• Quantum interrogation by Sean Carroll
• Interaction-free measurement (PDF) by Alan DeWeerd
As I understand it, the only way to have a known-live undetonated bomb in this branch is to cause it to actually detonate it in other branches.
Sorta, but not quite, as the probability of it actually detonating can be brought as close to 0 as one likes (if I'm not mistaken).
The Elitzur-Vaidman is really amazing: more sophisticated than my scenario. However it is quite different and not directly related.
The Quantum suicide is much more similar, in fact my posting derived from an almost identical idea that I also posted on lesswrong. I got that idea independently when reading that thread.
The reason I find the quantum roulette thought experiment interesting is that it is much less speculative. The payoff is clear and the in can easily be motivated and performed by current technology.
Yes, that last point is important. Too bad we won't get volunteers from LW, because we're all kinda friends and would miss each other.
However there is a way to convince a lot of people in a few Everett branches: You make a one-time big announcement in the Internet, TV etc. and say that there is a well tested quantum
coin-flipper, examined by a community consisting of the most honest and trusted members of the society. You take some random 20 bit number and say that you will flip the equipment 20 times and if
the outcome is the same as the predetermined number, then you will take it as a one to million evidence that the Multiple World theory works as expected. Of course, only people in the right
branch will be convinced. Nevertheless, they could be convinced enough to make serious thoughts about the viability of quantum Russian roulette type games.
I vaguely remember this being discussed here before, and people deciding it wouldn't work. Before the coin-flipper is run, you have a 1/2^20 chance of seeing your number come up, whether many worlds
is true or false. That means that seeing the number come up doesn't tell you anything about whether MW is true or not. It just tells you you're extremely lucky: either lucky enough that the
coin-flipper got a very specific number, or lucky enough to have ended up in the very specific universe where the flipper got that number.
I don't really buy that argument. It would apply to any measurement scenario. You could say in the two-mirror experiment: "These dots on the screen don't mean a thing, we just got extremely lucky."
Which is of course always a theoretical possibility.
Of course you can derive that you were extremely lucky, but also that "someone got extremely lucky" [SGEL]. If you start with some arbitrary estimates e.g. P(SWI)=0.5 and P(MWI)=0.5 and try to update
P(MWI) by using Bayesian inference, you get:
By P(SGEL|SWI)=1/2^20
You get:
0.5/((1/2^20)0.5 + 0.5)=1/(1+1.2^20) ~ 1-1/2^20
Well, yes, but we can't peek into other Everett branches to check them for lucky people.
Do you really need Many Worlds for quantum suicide/immortality? How's a Single Big World (e.g. Tegmark's Level 1 universe) different? Since a Single Big World is infinite, there will be an infinite
number of people performing quantum suicide experiments, and some of them will seem to be immortal.
It seems to me that there is no practical difference between living in Many Worlds vs a Single Big World.
If you assume 3, average utilitarianism says you should kill anyone who has below average utility, since that raises the average. So in the end you kill everyone except the one person who has the
highest utility. There is no need for assumption 2 at all.
BTW, are you aware of any of the previous literature and discussion on quantum suicide and immortality?
See also http://www.acceleratingfuture.com/steven/?p=215
Which is why we do not really believe in average utilitarianism...
If the worlds in your MWI experiment are considered independent, you might as well do the same in a single deterministic world. Compare the expected utility calculations for one world and
many-worlds: they'll look the same, you just exchange "many-worlds" with "possible worlds" and averaging with expectation. MWI is morally uninteresting, unless you do nontrivial quantum computation.
Just flip a logical coin from pi and kill the other guys.
More specifically: when you are saying "everyone survives in one of the worlds", this statement gets intuitive approval (as opposed to doing the experiment in a deterministic world where all
participants but one "die completely"), but there is no term in the expected utility calculation that corresponds to the sentiment.
You can assign high negative utility to certain death.
You can assign high negative utility to certain death.
You can, but then you should also do so in the expected utility calculation, which is never actually done in most discussions of MWI in this context, and isn't done in this post. The problem is using
MWI as rationalization for invalid intuitions.
With a similar technique you can solve any NP-Complete problem. Actually, you can solve much harder problems. For instance, you can minimize any function you have enough computing power to compute.
You could apply this, for instance, to genetic algorithms, and arrive at the globally fittest solution. You could likewise solve for the "best" AI given some restraints, such as: find the best
program less than 10000 characters long that performs best on a Turing test.
This is a very interesting point and somehow shakes my belief in the current version of MWI.
What I could imagine is that since the total information content of multiverse must be finite, there is some additional quantification going on that makes highly improbable branches "too fuzzy" to be
observable. Or something like that.
Not likely. You're already in a highly improbable branch, and it's getting less probable every millisecond.
I would not state this for sure. There could still be quite a difference between astronomically unlikely and superastronomically unlikely.
So for example if the total information content of the multiverse is bounded by 2^1000 bits, you could go down to an insanely small probability of 1/2^(2^1000) but not to the "merely" 1/2^(2^1000)
times less probable 1/2^(2^1001) .
Actually, you can solve much harder problems. For instance, you can minimize any function you have enough computing power to compute.
Why much harder? If computing the function is in P, minimization is in NP.
EDIT: this statement is wrong, see Wei Dai's comments below for explanation. Corrected version: "minimization is at most polynomially harder than an NP problem".
If computing the function is in P, minimization is in NP.
Why is that? In order for function minimization to be in NP, you have to be to write a polytime-checkable proof of the fact that some input is a minimum. I don't think that's true in general.
I also don't see how function minimization can be accomplished using quantum suicide. You can compute the value of the function on every possible input in parallel, but how do you know that your
branch hasn't found the minimum and therefore should commit suicide?
This seems like a relevant article, although it doesn't directly address the above questions.
ETA: I do see a solution now how to do function minimization using quantum suicide: Guess a random x, compute f(x), then flip a quantum coin n*f(x) times, and commit suicide unless all of them come
out heads. Now the branch that found the minimum f(x) will have a measure at least 2^n times greater than any other branch.
ETA 2: No, that's not quite right, since flipping a quantum coin n*f(x) times takes time proportional to f(x) which could be exponential. So I still don't know how to do this.
Why is that? In order for function minimization to be in NP, you have to be to write a polytime-checkable proof of the fact that some input is a minimum. I don't think that's true in general.
...That's an interesting way to look at it. For example take the traveling salesman problem: it's traditionally converted to a decision problem by asking if there's a route cheaper than X. Note that
only "yes" answers have to have quickly checkable certificates - this is NP, not NP intersected with co-NP. Now if we start asking if X is exactly the minimum instead, would that be equivalent in
complexity? And would it be helpful, seeing as it takes away our ability to do binary search?
In short, I'm a little funny in the head right now, but still think my original claim was correct.
This paper (which I found via this Ph.D. thesis) says that for TSP, the question "Is the optimal value equal to k" is D^P-complete. It also says D^P contains both NP and coNP, so assuming NP!=coNP,
that problem is not in NP.
Your original claim was "If computing the function is in P, minimization is in NP." Technically, this sentence doesn't make sense because minimization is a functional problem, whereas NP is a class
of decision problems. But the ability to minimize a function certainly implies the ability to determine whether a value is a minimum, and since that problem is not in NP, I think your claim is wrong
in spirit as well.
As Nesov pointed out, TSP is in FP^NP, so perhaps a restatement of your claim that is correct is "If computing the function takes polynomial time, then it can be minimized in polynomial time given an
NP oracle." This still seems to imply that function minimization isn't much harder than NP-complete problems.
Are there any problems in PostBQP that can be said to be much harder than NP-complete problems? I guess you asked that already.
It's in FP^NP, not in NP. Only the decision problem for whether a given value C is more than the minimum is in NP, not minimization itself.
Yes, that's certainly right, thanks. I was careless in assuming that people would just mentally convert everything to decision problems, like I do :-)
ETA: Vladimir, I couldn't find any proof that NP is strictly contained in PP; is it known? Our thread seems to depend on that question only. It should be true because PP is so freaky powerful, but
nothing's turned up yet.
P⊆NP⊆PP⊆PSPACE, but we don't even have a proof of P≠PSPACE.
Unconditionally proven strict containments are pretty rare. You'll probably want to settle for some lesser evidence.
According to this FNP is strictly harder than NP.
No idea about PP. Where did PP come up, or how does it relate to FNP?
ETA: Ah, I see. Aaronson's paper here shows that postselection (which is what our suicide voodoo is) gives you PP. Since postselection also gives you FNP, and since FNP is harder than NP, then we
should have that PP is strictly harder than NP.
Jordan, I'm really sorry to inject unneeded rigor into the discussion again, but the statement "FNP is strictly harder than NP" doesn't work. It makes sense to talk about P=NP because P and NP are
both sets of decision problems, but FNP is a set of function problems, so to compare it with NP you have to provide a mapping of some sort. Thus your argument doesn't prove that PP>NP yet.
For me personally, function problems are exotic beasts and I'd prefer to settle the power of quantum voodoo on decision problems first :-)
Consider the following problem, given N:
Create a list with 2^N entries, where each entry is a random number from 0 to 1. What is the smallest entry?
This is a function minimization problem, where the function takes n and returns the n-th element of the list. The cost of computing the function is O(1). There is no way to minimize it, however,
without looking at every value, which is O(2^N). With quantum suicide voodoo, however, we can minimize it in O(1).
1) The problem of finding the smallest entry in a list is linear-time with respect to the size of the input; calling that size 2^N instead of M doesn't change things.
2) Accessing the nth element of a list isn't O(1), because you have to read the bits of n for chrissake.
3) I'm not sure how you're going to solve this with quantum voodoo in O(1), because just setting up the computation will take time (or space if you're parallel) proportional to the length of the
input list.
1) The problem of finding the smallest entry in a list is linear-time with respect to the size of the input; calling that size 2^N instead of M doesn't change things.
You can call it M, if you like. Then individual function evaluations cost log(log(M)). The point is the relative cost difference between function evaluations and function minimization.
2) Accessing the nth element of a list isn't O(1), because you have to read the bits of n for chrissake.
Good point. Function calls will be O(log(N)).
3) I'm not sure how you're going to solve this with quantum voodoo in O(1), because just setting up the computation will take time (or space if you're parallel) proportional to the length of the
input list.
Right, the setup of the problem is separate. It could have been handed to you on a flash drive. The point is there exist functions such that we can calculate single evaluations quickly, but can't
minimize the entire function quickly. | {"url":"https://www.lesswrong.com/posts/XH9ZN8bLidtcqMxY2/quantum-russian-roulette","timestamp":"2024-11-06T05:17:20Z","content_type":"text/html","content_length":"1048907","record_id":"<urn:uuid:620d31c3-543f-42b6-9ee1-4a4ede318fc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00180.warc.gz"} |
Analytical Chemistry - Online Tutor, Practice Problems & Exam Prep
We're going to say here that even though we try to be as accurate as possible, every measurement or calculation we do in chemistry has some level of uncertainty. Now, this uncertainty we call
experimental error. Basically, any type of calculation we do is never going to be perfect. Some of the circumstances leading to this imperfection are within our control while others are not. Now
before we talk about the different types of errors, we first have to talk about how we can look at our data, our calculations and determine if they are good or not. We're going to say when we
investigate the quality of an experimental decision or calculation, we have to take into consideration two major principles.
We're going to say the first major principle deals with the reproducibility of our calculations. This is called precision. Precision is just a way of looking at our data or calculations and seeing
how close they are to one another. If I've run an experiment 10 times and gotten 10 results, how close are those ten results to one another? This can oftentimes lead us to determine if our
calculations are correct or not. Now, in terms of simplicity, we can look at a dartboard. On this dartboard, we have 3 strikes. These 3 strikes are very close to one another. We'd say the
reproducibility is very high. We'd say that our strikes here are precise because they're very close to one another.
The second principle deals with how close our measured calculation is to the actual value. This is accuracy. In this image, we have our dartboard still. All the strikes are dead center. If dead
center represented our actual value or our true value, we'd say that these strikes are very accurate. We could also say that they're precise as well because they're all very close to one another. We
could say here that this would be a very accurate and precise list of strikes. The one above would be precise but not necessarily accurate because although they land in the same general area, none of
them hit the bull's eye. They're not accurate.
Later on, we'll learn that we can calculate how good our calculations are, how reproducible and precise they are by looking at the standard deviation of these values. But that's for later on. For
now, just realize that when it comes to any calculation, it's never going to be perfect. That's because there are things that we may have done incorrectly and there are also circumstances within the
experiment which make it impossible to be totally accurate and sometimes not very precise. Of the two, precision is the one that we can try our best to control. Accuracy sometimes may not be totally
within our control. Knowing this, take a look at example 1. Look through the experiment that this individual has done to determine if it's precise or accurate. Once you've figured that out, come back
and take a look at how I approach the same question. | {"url":"https://www.pearson.com/channels/analytical-chemistry/learn/jules/ch-3-experimental-error/precision-and-accuracy?chapterId=f5d9d19c","timestamp":"2024-11-09T11:05:30Z","content_type":"text/html","content_length":"225295","record_id":"<urn:uuid:2acbdfd3-6ab4-4788-a31b-b61834c98bd2>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00624.warc.gz"} |
Lines and Angles Class 6 MCQ Maths Chapter 2
MCQ on Lines and Angles Class 6
Class 6 Maths Chapter 2 MCQ Lines and Angles
I. Multiple Choice Questions
Question 1.
Which of the following has no endpoints?
(a) Line
(b) Ray
(c) Line-segment
(d) None of these
(a) Line
Question 2.
Number of lines which can be drawn passing through two given points:
(a) 1
(b) 2
(c) 3
(d) An infinite number
(a) 1
Question 3.
Number of right angles turned by the hour hand of a clock when it goes from 3 to 6.
(a) 1
(b) 2
(c) 3
(d) 4
(a) 1
Question 4.
The measure of a straight angle is
(a) 90°
(b) 360°
(c) 180°
(d) None of these
(c) 180°
II. Objective Type Questions
Question 5.
Look at the figure and fill in the blanks:
(i) The lines passing through P are: _______, _______ and _______
\(\overline{\mathrm{RS}}\), \(\overline{\mathrm{AB}}\) and \(\overline{\mathrm{CD}}\)
(ii) The concurrent lines are: _______, _______ and _______
\(\overline{\mathrm{AB}}\), \(\overline{\mathrm{CD}}\) and \(\overline{\mathrm{RS}}\)
(iii) The lines meet at X are: _______ and _______
\(\overline{\mathrm{AB}}\) and \(\overline{\mathrm{LM}}\)
(iv) The lines CD and LM meet each other at _______
Note: Three or more lines passing through a single point are called concurrent lines.
Question 6.
Name all the line segments in the given figure.
(i) _____________
(ii) _____________
(iii) _____________
(iv) _____________
(v) _____________
(vi) _____________
(i) \(\overline{\mathrm{AB}}\)
(ii) \(\overline{\mathrm{AC}}\)
(iii) \(\overline{\mathrm{AD}}\)
(iv) \(\overline{\mathrm{BC}}\)
(v) \(\overline{\mathrm{BD}}\)
(vi) \(\overline{\mathrm{CD}}\)
Question 7.
Fill in the blanks:
(i) A ray has _____________ end point(s).
(ii) A line \(\overline{\mathrm{PQ}}\) extends in _____________ directions.
(iii) The measure of a 1\(\frac{1}{3}\) right angles = _____________ degrees.
(iv) The measure of 2 straight angles is _____________
(v) The measure of _____________ is between 180° and 360°.
Reflex Angle
(vi) A clock hand will reach at _____________ if it starts at 7 and makes \(\frac{1}{2}\) of a revolution.
(vii) The angle included between the arms of a clock at 10 o’clock is _____________ degrees.
(viii) You will make _____________ right angles, if you start facing North and turn anticlockwise to East.
Question 8.
In the figure, various line segments are:
(i) _____________
(ii) _____________
(iii) _____________
(iv) _____________
(i) \(\overline{\mathrm{AB}}\)
(ii) \(\overline{\mathrm{BC}}\)
(iii) \(\overline{\mathrm{CD}}\)
(iv) \(\overline{\mathrm{DE}}\)
Question 9.
How many right angles are turned by the hour hand of a clock when it moves:
(i) from 2 to 5?
(ii) from 4 to 10?
(iii) from 12 to 3?
(iv) from 9 to 3?
(i) 1
(ii) 2
(iii) 1
(iv) 2
III. Fun Activity
Question 10.
Complete the following crossword puzzle:
(1) A _____________ is a part of lines starting at a point and extends in one direction endlessly.
(2) A _____________ is a portion of a line whose endpoints are fixed.
(3) The meeting point of arms of an angle is called a _____________
(4) _____________ is used to measure an angle.
(1) Ray
(2) Line Segment
(3) Vertex
(4) Protractor
Question 11.
Tick the correct option:
(i) A line segment has [two/no] endpoints.
(ii) The measure of a full turn is [180°/360°]
(iii) The measure of an [acute/obtuse] angle is between 90° and 180°.
(iv) The endpoints of a line segment [are/are not] the part of its length.
(v) When the clock shows 3 o’clock the angle between its hand is equal to [60°/90°].
(vi) 4 right angles equal to one [straight angle/full turn].
full turn
Question 12.
Match the following:
┃Column A │Column B ┃
┃(a) A line │(i) measure ┃
┃(b) A line segment can be │(ii) extends infinitely in one direction. ┃
┃(c) A ray │(iii) measured ┃
┃(d) In an angle, the lengths of the arms do not affect its │(iv) extends infinitely in both directions.┃
┃Column A │Column B ┃
┃(a) A line │(iv) extends infinitely in both directions.┃
┃(b) A line segment can be │(iii) measured ┃
┃(c) A ray │(ii) extends infinitely in one direction. ┃
┃(d) In an angle, the lengths of the arms do not affect its │(i) measure ┃
Question 13.
Measure and classify each angle:
┃Angle│Measure │Type ┃
┃∠AOB │40° │Acute Angle ┃
┃∠AOC │125° │Obtuse Angle ┃
┃∠BOC │85° │Acute Angle ┃
┃∠DOC │95° │Obtuse Angle ┃
┃∠DOA │140° │Obtuse Angle ┃
┃∠DOB │180° │Straight Angle ┃
Question 14.
Match the following given angles with its measure:
(a) (iv)
(b) (v)
(c) (ii)
(d) (iii)
(e) (i)
IV. Answer the following.
Question 15.
Can we have two acute angles whose sum is
(i) an acute angle? Why or why not?
Yes. The sum of two acute angles is again an acute angle.
For example: 15° and 35° two acute angles and 15° + 35° = 50°, an acute angle.
(ii) a right angle? Why or why not?
Yes. The sum of two acute angles may be equal to a right angle.
For example: 35° and 55° are two acute angles and 35° + 55° = 90°, a right angle.
(iii) an obtuse angle? Why or why not?
Yes. The sum of two acute angles may be an obtuse angle.
For example: 45° and 55° are two acute angles and 45° + 55° = 100°, an obtuse angle.
(iv) a straight angle? Why or why not?
No. The sum of two acute angles is always less than a straight angle.
As, acute angles are less than 90°, so the sum of two acute angles is always less than 180°, a straight angle.
(v) a reflex angle? Why or why not?
No. The sum of two acute angles is always less than a reflex angle.
As, sum of two acute angles are always less than 180°, so it is always less than a reflex angle.
Question 16.
Can we have two obtuse angles whose sum is
(i) a reflex angle? Why or why not?
Yes. The sum of two obtuse angles is always greater than 180°, a reflex angle.
(ii) a full turn? Why or why not?
No. The sum of two obtuse angles is always greater than 180°, but less than 360° a full turn.
Question 17.
An angle is said to be trisected, if it is divided into three equal parts. If in the figure shown here, ∠BAC = ∠CAD = ∠DAE, how many trisectors are there for ∠BAE?
Two, AC and AD
Question 18.
In the given figure,
(i) name any four angles that appear to be acute angle.
∠ABE, ∠ADE, ∠BAE, ∠BCE (Answer may vary)
(ii) name any two angles that appear to be obtuse angle.
∠BCD, ∠BAD
Question 19.
If a bicycle wheel has 36 spokes, then answer the following:
(i) What is the angle between its two adjacent spokes?
(ii) What is the angle between a spoke and its neighbouring 4th spoke?
Question 20.
Assertion: A straight angle contains 2 right angles.
Reason: Measure of right angle is 90° and measure of a straight angle is 180° that is 180° = 2 × 90°.
In the given question, a statement of Assertion is followed by a statement of Reason. Choose the correct option as:
(a) Both assertion and reason are true and the reason is the correct explanation of assertion.
(b) Both assertion and reason are true but the reason is not the correct explanation of the assertion.
(c) Assertion is true and the reason is false.
(d) Assertion is false and the reason is true.
(a) Both assertion and reason are true and the reason is the correct explanation of assertion.
Question 21.
Puzzle: I am an acute angle. If you double my measure, you get a right angle. If you triple my measure, you will get an obtuse angle. If you quadruple (four times) my measure, you will get a straight
angle! Who am I?
Question 22.
Case Based Question
Five friends say A, B, C, D and E are sitting along the bonfire by maintaining a distance like the given figure. Where O is the position of bonfire.
Based on the above information, answer the following questions.
(a) Which two friends are sitting next to each other in a way, such that an obtuse angle is formed between them with respect to bonfire?
A and D
(b) Which type of angle is formed between A and C?
Acute Angle
(c) Which angle is formed between A and B?
Straight Angle | {"url":"https://www.learninsta.com/lines-and-angles-class-6-mcq/","timestamp":"2024-11-11T09:56:22Z","content_type":"text/html","content_length":"67726","record_id":"<urn:uuid:cd81ae4d-bc74-4c47-8382-7bce2f39b922>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00291.warc.gz"} |
A Guide To The Cornell Physics 1112 Lecture Schedule - Do My Physics Exam
If you are a physics student, then you will find that the syllabus of the class of 2020, The College of Letters and Science at the University of Michigan has a lot to offer to students. This is one
of the oldest courses in the curriculum, which covers a wide range of topics in mathematics and science, including calculus, optics, thermodynamics, electricity and magnetism. Although there are
quite a few changes made in this course every year, students who are interested in becoming a nuclear scientist or working in other fields of physics can still benefit from this course.
Please proceed with your order so that we can facilitate taking your exam on your behalf. Click Here
The syllabus for the University of Michigan Physics 1112 syllabus features a number of changes from previous years. For starters, the course was restructured in order to teach a wider array of
topics. In addition, the new syllabus was introduced by allowing students to choose their own units to learn from. The old syllabus allowed students to choose from four units and study the concepts
in these units.
Students are now permitted to study at their own set of time. They can choose any units to learn about and choose how many units they want to learn from them. This is a great option for students,
since it makes it easy for them to learn from the syllabus as they study.
The course involves many modules that are presented in a sequential manner. Students will learn topics and theories that are relevant to their career goals. There are also tests at the end of each
module, and students will receive feedback from instructors on how well they understood the material covered in that module. Students can opt to take the tests in the morning or in the evening,
depending on their personal schedules.
Students have a lot of freedom to choose their units and groups to learn from. It is their choice, therefore, on how long they would like to study each module. They may decide to go over some topics
more than others, depending on the subjects that interest them. Students can select from either general topics, such as atoms, molecules, or energy, and even on their favorite area of physics.
The syllabus also allows students to choose the time that they would like to study and work at their own schedule. Previously, students were required to spend a certain amount of time at a desk
throughout the semester, but now, students can work any time they wish. on the assignments that they have taken.
The course includes a lot of projects that students have to complete at the end of the semester. They can submit their project at the end of the semester or they may take responsibility in starting
the project planning process. However, it is their responsibility to make sure that the project is written before the deadline is reached. Students have a large selection of projects to complete,
with topics ranging from understanding the behavior of magnets and the behavior of atoms to calculating the behavior of the electric field and how it relates to electricity.
Students have the choice of taking a full course in a semester, or they can take only a part of a course in a semester. They are free to decide whether they will be taking a full course in a semester
or a part of a semester. There are also sections that give students the opportunity to complete the course online. Students can even take the course online after having earned a bachelor’s degree in
Students can find out how many units they will need for the course by looking at the course description or by consulting the course map. Units need to be calculated based on the student’s grade
average. Units should not be taken for granted because they are used to calculate final grades. Most courses, however, have a specific grade distribution for students to follow, so students can avoid
unnecessary confusion when learning how to calculate their grade point average.
The class usually consists of about 50 minutes of lecture time and one class period per week. There is usually a break period of about two minutes, at the end of the semester, so that students can
review the materials and get prepared for the next semester’s lab. Most students have to pass an exam in order to continue on with the next semester’s class.
The course is an excellent introduction to the subject matter for students who are just beginning their studies. It also provides the necessary skills needed to understand the concepts used by
graduate students in research, and is a good choice for students who want to expand their knowledge and understand more about the subject matter. It provides students with opportunities for
self-directed learning and allows students to move forward with their own pace in the course. | {"url":"https://domyphysicsexam.com/a-guide-to-the-cornell-physics-1112-lecture-schedule/","timestamp":"2024-11-13T18:43:13Z","content_type":"text/html","content_length":"112799","record_id":"<urn:uuid:6cd2c970-4c60-4573-994f-11550e04769a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00255.warc.gz"} |
Pay Someone To Take Sensitivity Analysis Assignment | Pay SomeoneTo Do Linear Programming Assignment
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Sensitivity Analysis is an invaluable way to detect any risks and ensure your conclusions are robust.
One-way sensitivity analysis evaluates the effects of changing one variable while holding other factors constant, helping you prioritize which inputs are most critical and identify any threshold
values that must be reached.
It is a process of forecasting
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To conduct a sensitivity analysis, first identify the key variables most critical to your project costs – Either Internal ones like labor costs and materials prices or external ones such as exchange
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Sensitivity analysis helps identify both risks and opportunities of any project, business venture, or investment by studying all possible outcomes. This can help make more informed decisions when
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substitute numbers directly into formulae; using percentages instead of actual values is recommended as this provides more precise data.
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Sensitivity analysis can be applied in various contexts, from planning a new project to evaluating the effects of changing certain values or assumptions. It can be carried out both on financial
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Sensitivity analysis is an invaluable way for analysts to gauge how sensitive a model’s output is to its inputs and any errors or assumptions within calculations or assumptions that may exist within
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To perform sensitivity analysis, first identify your Base Case. Next, choose an input variable and make changes while keeping other inputs unchanged; determine its effect on output before calculating
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Sensitivity analysis examines how changes to input variables affect outcomes of models or systems. By carefully manipulating inputs, decision-makers can gain an understanding of how sensitive their
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Sensitivity analysis helps analyze how a dependent variable changes when one or more independent variables change, as well as identify which independent variables are crucial to an outcome. It can be
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providing them with information to make more informed decisions on whether to increase revenue or cut costs.
Conducting a sensitivity analysis with Microsoft Excel or another spreadsheet application can be relatively straightforward. A local sensitivity analysis involves altering one variable at a time
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Excel offers the capability of performing a sensitivity analysis using one or two independent variables by creating a data table, or you can utilize Tornado Charts to demonstrate their impact
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Sensitivity analysis is an integral component of research design. It allows researchers to examine how sensitive results of studies or models are to varying assumptions, helping reduce uncertainty in
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There are various ways of conducting a sensitivity analysis. While some methods are more complex than others, most can be completed with standard data analysis software. The key objective should be
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For instance, if you want to understand how traffic affects sales, using a sensitivity analysis to understand its effect on different levels of customer traffic may help determine an ideal level to
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Read More » | {"url":"https://linearprogramminghelp.com/sensitivity-analysis","timestamp":"2024-11-12T17:12:30Z","content_type":"text/html","content_length":"133075","record_id":"<urn:uuid:2c53cd7c-db35-42f7-9668-3a13f57b86ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00649.warc.gz"} |
Block Party - The Futures Channel
Grade Levels: 3rd Grade, 4th Grade, 5th Grade, 6th Grade, 7th Grade,
Topics: Measurement (volume)Geometry (rectangular prisms)
Common Core State Standard: 5.MD.4, 6.G.4,
· Surface area
· Volume
· Cubic centimeter
Knowledge and Skills:
· Can find the volume of a rectangular prism by computation
(for each team):
· one pair of safety scissors
· one roll of ½” wide transparent tape
· a few sheets of one-centimeter grid paper.
Download the Teacher Guide PDF
Procedure: This activity is best done individually, but can also be done in teams of two.
Distribute the two handouts (instruction sheet and patterns). Review the instructions and have students begin the activity.
As the students work, circulate, observe and have them describe what they are doing. Be patient, as this activity may take 45 minutes or more.
You may wish to give students the hint that one way to match the patterns to their shapes is by determining the surface area of each (the surface area of the pattern will be equal to the surface area
of the shape).
To speed up the activity, or for students that need extra help, you may wish to make the shapes yourself in advance and have them available as models.
At the conclusion of Part I, ask this question (students may answer orally or in writing):
“Suppose I have two shapes with the same volume. Will those shapes always have the same surface area? Explain your answer.” (As demonstrated by the two larger shapes in Part I, shapes that have the
same volume do not always have the same surface area.)
For Part II of the activity, you will need to provide students with 1-centimeter grid paper for their patterns.
Block Party
Part I : Each of the shapes below can be made from the patterns on the grid that your teacher will give you. (Be sure to cut all dotted lines.)
For each shape, choose the correct pattern, and make the shape (be sure to cut all dotted lines around and in each pattern).
Then find the volume of the shape and its surface area.
Block Party Patterns: Each square in the grid is one square centimeter. | {"url":"https://thefutureschannel.com/lesson/block-party/","timestamp":"2024-11-07T17:35:21Z","content_type":"text/html","content_length":"61446","record_id":"<urn:uuid:a4f30e35-a719-427e-831c-9a319557273b>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00136.warc.gz"} |
MAT553 - Variétés, fibrés vectoriels et formes différentielles (2022-2023)
Differentiable manifolds are geometric objects, locally parametrized by coordinate systems, but with a global topology that can be nontrivial. They are therefore the natural language of differential
geometry (Riemannian, symplectic, complex, etc...), and also of many physical theories (general relativity, gauge theory, etc...).
The goal of this course is to provide an introduction to manifolds, and to a number of related key concepts: smooth maps between manifolds, vector bundles, transversality, intersection theory, Morse
Theory, differential forms and integration, connexions, parallel transport, metric and curvature. | {"url":"https://moodle.polytechnique.fr/course/info.php?id=14141","timestamp":"2024-11-11T20:01:57Z","content_type":"text/html","content_length":"32738","record_id":"<urn:uuid:46575d9f-92e4-4511-aeba-ae011a6b6105>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00403.warc.gz"} |
1-D Physical System Statistics
The 1-D Physical System statistics represent the aggregate statistics of all physical networks associated with blocks from Simscape™, Simscape Driveline™, Simscape Fluids™, and Simscape Electrical™
libraries, with the exception of Specialized Power Systems blocks.
The individual statistics are:
• Variables — Number of variables associated with all 1-D physical systems in the model. Variables are categorized further as continuous, eliminated, secondary, and discrete variables.
• Continuous variables (retained) — Number of continuous variables associated with all 1-D physical systems in the model. Continuous variables are those variables whose values vary continuously
with time, although some continuous variables can change values discontinuously after events. Continuous variables can be further categorized as algebraic and differential variables.
This statistic represents the number of continuous variables in the system after variable elimination. If a system is truly input-output with no dynamics, it is possible to completely eliminate
all variables and, in that case, the number of variables is zero.
□ Differential variables — Number of differential variables associated with all 1-D physical systems in the model. Differential variables are continuous variables whose time derivative appears
in one or more system equations. These variables add dynamics to the system and require the solver to use numerical integration to compute their values.
This statistic represents the number of differential variables in the model after variable elimination.
□ Algebraic variables — Number of algebraic variables associated with all 1-D physical systems in the model. Algebraic variables are continuous system variables whose time derivative does not
appear in any system equations. These variables appear in algebraic equations but add no dynamics, and this typically occurs in physical systems due to conservation laws, such as conservation
of mass and energy.
This statistic represents the number of algebraic variables in the model after variable elimination.
• Continuous variables (eliminated) — Number of eliminated variables associated with all 1-D physical systems in the model. Eliminated variables are continuous variables that are eliminated by the
software and are not used in solving the system. Eliminated variables are categorized further as algebraic and differential variables.
□ Differential variables — Number of eliminated differential variables associated with all 1-D physical systems in the model. Differential variables are continuous variables whose time
derivative appears in one or more system equations. These variables add dynamics to the system and require the solver to use numerical integration to compute their values.
This statistic represents the number of differential variables in the model that have been eliminated.
□ Algebraic variables — Number of eliminated algebraic variables associated with all 1-D physical systems in the model. Algebraic variables are continuous system variables whose time derivative
does not appear in any system equations. These variables appear in algebraic equations but add no dynamics, and this typically occurs in physical systems due to conservation laws, such as
conservation of mass and energy.
This statistic represents the number of algebraic variables in the model that have been eliminated.
• Continuous variables (secondary) — Number of secondary variables associated with all 1-D physical systems in the model. Secondary variables are variables added to the system to make it solvable.
Secondary variables can be categorized further as algebraic and differential variables.
□ Differential variables — Number of secondary differential variables associated with all 1-D physical systems in the model. Differential variables are continuous variables whose time
derivative appears in one or more system equations. These variables add dynamics to the system and require the solver to use numerical integration to compute their values.
This statistic represents the number of differential variables in the model that have been added to the system to make it solvable.
□ Algebraic variables — Number of secondary algebraic variables associated with all 1-D physical systems in the model. Algebraic variables are continuous system variables whose time derivative
does not appear in any system equations. These variables appear in algebraic equations but add no dynamics. This typically occurs in physical systems due to conservation laws, such as
conservation of mass and energy.
This statistic represents the number of algebraic variables in the model that have been added to the system to make it solvable.
• Discrete variables — Number of discrete, or event, variables associated with all 1-D physical systems in the model. Discrete variables are those variables whose values can change only at specific
events. Discrete variables can be further categorized as integer-valued and real-valued discrete variables.
□ Integer-valued variables — Number of integer-valued discrete variables associated with all 1-D physical systems in the model. Integer-valued discrete variables are system variables that take
on integer values only and can change their values only at specific events, such as sample time hits. These variables are typically generated from blocks that are sampled and run at specified
sample times.
□ Real-valued variables — Number of real-valued discrete variables associated with all 1-D physical systems in the model. Real-valued discrete variables are system variables that take on real
values and can change their values only at specific events.
If you select a local solver in the Solver Configuration block, then all continuous variables associated with that system are discretized and represented as real-valued discrete variables.
• Discrete variables (secondary) — Number of secondary discrete variables associated with all 1-D physical systems in the model. Discrete secondary variables represent those discrete variables in
the model that have been added to the system to make it solvable.
□ Integer-valued variables — Number of integer-valued secondary discrete variables associated with all 1-D physical systems in the model.
□ Real-valued variables — Number of real-valued secondary discrete variables associated with all 1-D physical systems in the model.
• Number of zero-crossing signals — Number of scalar signals that are monitored by the Simulink^® zero-crossing detection algorithm. Zero-crossing signals are scalar functions of states, inputs,
and time whose crossing zero indicates discontinuity in the system. These signals are typically generated from operators and functions that contain discontinuities, such as comparison operators,
abs, sqrt functions, and so on. Times when these signals cross zero are reported as zero-crossing events. During simulation it is possible for none of these signals to produce a zero-crossing
event or for one or more of these signals to have multiple zero-crossing events.
• Constraints — Number of primary constraint equations in the model, along with the total number of differential variables involved in each constraint. Primary constraints are the constraints
identified before index reduction. These equations and variables are further handled by index reduction algorithms.
If your model uses a Partitioning local solver, the Statistics Viewer contains additional statistics specific to this solver type. For more information, see Partitioning Solver Statistics.
See Also
Related Topics | {"url":"https://de.mathworks.com/help/simscape/ug/1-d-physical-system-statistics.html","timestamp":"2024-11-09T06:44:37Z","content_type":"text/html","content_length":"77805","record_id":"<urn:uuid:89af69b9-bfcd-440f-a2e8-b488980860b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00413.warc.gz"} |
30 Nov 2019 offering a new explanation of the ancestry and evolution of living beings. Richard Dawkins - The Genius of Charles Darwin - Part 1: Life,
Darwin’s seminal work on evolution was released to the public on November 22, 1859, in Great Britain. The initial print run sold out, and Darwin began work on a second run almost immediately, with
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Weights in NuGen¶
What are production weights?¶
The production weights consist of weights that enhance number of simulated (and triggered) events at the detector. They are totally artificial weights, has no relation with physics parameters, and
must be applied event by event to get reasonable results.
In order to get physics histograms under assumption of Atmospheric neutrino or any physical astro-neutrino, you have to apply another weight using `OneWeight` or weighting module
NuGen suports many production weights, Here is some examples.
• Zenith weight. For example, we may generates more vertical events than horizontal events to increase number fo simulated (and triggered) events from the bottom of the detector.
• Forbid CC interaction inside the Earth so that all neutrinos are propagated up to near by the IceCube detector (weighted propagation)
• Make force interaction inside a detection volume near by the IceCube (weighted interaction)
The propagation weight was always 1 for version V00-05-04 or older, and configurable from version V00-05-05 or later.
Do not use weighted propagation for NuTau! To switch on Tau’s neutrino regeneration, the propagation mode for tau flavor must be “NoPropWeight” or “AutoDetect”.
Zenith Weight¶
Why do we need zenith weight?¶
The uniform distribution (diffuse) follows flat distribution in cos(zenith) space, therefore our observable follows flat distribution in cos(zenith). For years we used to generate simulation with
flat distribution in cos(zenith). Recently, with some analysis, we encountered statistical shortage of simulation at vertical directions. To solve the issue, several weighting functions are newly
The statistical shortage at vertical direction happens by nature of sampling flat in cos(zenith). Because cos(zenith) contains the factor of solid angle, for example, number of simulated event at 180
degrees is always zero due to the solid angle = 0, while at 90 degrees the solid angle is 2*pi.
The purpose of simulation is not emulating nature, but understanding averaged behavior of our reconstruction in each angle. For physics analysis, we should avoid penalties that depend on zenith angle
as much as possible. Generating flat in cos(zenith) enhances strong penalty at vertical directions.
The figures show the number of generated events for each generation mode, and their statistical errors (in percent). Injected number of events is 500000.
Statistical errors will be smaller when we increase total number of generating events. One solution may be keep injection mode as flat in cos(zenith), and adding special datasets that covers only
vertical directions. However, this solution requires extra works for simulation production team. For users who want to use OneWeight will need to re-calculate OneWeight by hand for combined
Considering these extra works, the latest NuGen sets emulation mode of flat in zenith (ANGEMU mode) as default. In order to generate with old style, set AngleSamplingMode as “COS”.
Control zenith weight¶
To control zenith distribution of generated events, set following parameters to I3NuGInjector or I3NuGDiffuseSource. Generation weight (DirectionWeight) is included in I3MCWeightDict.TotalWeight and
I3MCWeightDict.OneWeight (OneWeightPerType). As long as a user uses these weights, no change is needed for analysis scripts. If you construct your weight from the scratch, do not forget to multiply
• AngleSamplingMode
COS :
Sample zenith angles in cos(zenith). Simulating diffuse sample.
ANG :
Sample zenith angles in angle. This option doesn’t calculate weight to weight back to flat in cos(zenith), due to singular points at coszen -1 and 1. Unless you can provide reasonable
directional weights, do not use the option for physics analysis. This option is useful when you are interested in only the ratio of two datasets or parameters, such as ratio of primary
neutrinos vs inice neutrinos.
ANGEMU :
Emulating ANG mode but allows to weight back to COS mode, therefore can be used for physics analysis. It separates ANG distribution in sections, and each section it uses 1 dim polynominal
function to sample cos(zenith).
• ZenithWeightParam
This setting is active only when you select COS mode for AngleSamplingMode.
Set float value from 0.1 to 1.9. Default is 1.0, which gives flat distribution in cos(zenith) or zenith (depending on the value in AngleSamplingMode). A larger number gives more vertical events.
For details, see ZenithSampler::SimpleSlopeSampler.
Followng figures show number of generated events in several injection modes, before weigting and after weighting.
• BLUE : using “COS” mode and ZenithWeightParam = 1.0
• GREEN : using “COS” mode and setting ZenighWeightParam 0.1
• MAGENTA : using “ANGEMU” mode and ZenithWeightParam = 1.0
• BROWN : (currently not supported)
Propagation Weight¶
Overview of NuGen propagation¶
• Calculate total path length inside the Earth using injected neutrino geometry. Separate the total path length into propagation area (distance SF) and detection volume (distance FE).
• Define a step length dx[m] using propagation area (distance SF) and step number nx. Default nx is 10000.
• In each step, calculate a column depth within dx[m] and an Earth’s density at the step point.
• Calculate a total cross section at the step point.
• Calculate a probability that the injected neutrino interacts within the step. Try Monte-Carlo, and decide whether an interaction happened within the step.
• If any interaction occurred, Choose an interaction with another random toss.
□ If CC-interaction is selected with injection particle NuMu or NuE, stop the propagation so that this event is killed. Start new propagation with a new particle.
□ For other cases, generate secondaries and go to the next step.
• If nothing happens, go to the next step.
• finish propagation when the injected neutrino and generated secondaries reach to the front surface of detection volume (point F), then process a weighted interaction.
Weighted Propagation¶
If we activate weighted propagation, the procedure 6 is modified.
NuGen always choose NC or GR interaction, then apply a propagation weight as
\[ \begin{aligned}(\sigma_{NC} + \sigma_{GR}) / (\sigma_{NC} + \sigma_{CC} + \sigma_{GR})\\GR: Glashow resonance (if applicable)\end{aligned} \]
Note that the verdict whether any interaction happens or not (procedure 5) is done based on the total cross section which includes CC interaction, even if we select weighted propagation.
Following figures show that the final (weighted) number of events of weighted propagation and non-weighted propagation matches well within statistical errors. The plot shows OneWeight parameter
(which is proportional to the number of events) for starting or contained events only. OneWeight parameter depends on detection volume sizes, therefore we always have to compare number of “triggered”
events to do a reasonable comparison.
Parameters to control weighted propagation¶
Set parameter to I3NeutrinoGenerator or I3NeutrinoPropagator.
• PropagationWeightMode
NoPropWeight :
Do not activate propagation weight. Propagation weight is always 1.
NCGRWeighted :
Activate propagation weight, CC interaction is forbidden during In-Earth propagation.
AutoDetect :
Use NCGRWeighted for NuE, NuMu and NoPropWeight for NuTau. (default)
Legacy :
This option activates old event class. If you want to reproduce NuGen V00-05-04 or older, use it. (propagation weight is 1)
• Use NoPropWeight or AutoDetect option for NuTau simulation
Interaction Weight¶
If neutrino(s) reach to the front surface of detection volume, NuGen makes force interaction somewhere inside the detection volume. Interaction weight must be applied in order to compensate it.
In the case of NuTau simulation, we may have multiple neutrinos at the front surface of detection volume. However, NuGen randomly chose only one of them to make force interaction, with taking into
account of “total interaction probability inside the detection volume” of each neutrino candidate. Theoretically we might have two (or more) neutrinos interacted inside the detection volume, but the
probability of this type of event should be negligibly small. On the other hand, taus generated inside propagation area and reached to the detection volume will be stored and handed over to the mmc
A pure interaction weight is defined as:
\[ \begin{aligned}P_{surviving} = exp(-\sigma_{all} * L_{d} / M_{p} * C)\\P_{interaction} = 1 - P_{surviving}\end{aligned} \]
\(L_{d}\) : total column depth within the detection volume [\(g/m^2\)]
\(M_{p}\) : proton mass [\(g\)]
\(\sigma_{all}\) : total cross section at interaction point [\(mb\)]
\(C\) : unit conversion factor 1.0e-31 (1[\(mb\)] = 1.0e-31[\(m^2\)])
This is a part of total interaction weight, because NuGen samples interaction position with a flat probability distribution function
\[P_{pos\_NuGen}(X) = 1 / L_{d}\]
\(X\) : interaction column depth from the entrance of detection volume to the interaction position [\(g/m^2\)]
instead of using the following exponential distribution
\[ \begin{aligned}P_{pos\_True}(X) = 1 / sum * exp(- \sigma_{all} * X / M_{p} * C)\\sum = \int_0^{L_d} exp(- \sigma_{all} * X / M_{p} * C) dX\end{aligned} \]
Thus we need to apply a position weight too.
\[\begin{split} W_{pos} & = P_{pos\_True}(X) / P_{pos\_NuGen}(X) \\ & = (L_{d} / sum) * exp(- \sigma_{all} * X / M_{p} * C) \\ \end{split}\]
The net interaction weight is then expressed as:
\[W_{interaction} = P_{interaction} * W_{pos}\]
If you want to calculate interaction weight by yourself, I3MCWeightsDict provides all information you need. See following parameters:
Feb.2.2020 Part of the “Names in I3MCWeightDict” was fixed to correct value.
Variables Units Names in I3MCWeightDict
\(W_{interaction}\) (unit less) (not stored) << FIXED!
\(P_{interaction}\) (unit less) InteractionWeight
\(W_{pos}\) (unit less) InteractionPositionWeight << FIXED!
\(L_{d}\) \(g/cm^{2}\) TotalColumnDepthCGS
\(X\) \(g/cm^{2}\) InteractionColumnDepthCGS
\(\sigma_{all}\) <s>mb</s> \(cm^{2}\) TotalXsectionCGS
\(M_{p}\) g Not stored, use a constant value 1.6726215e-24
\(C\) If you use the valiables listed above, use 1.0
Weights for physics analysis¶
In order to use simulation, users must multiply all production weights and initial flux, as well as weights defined by generation space. We usually use OneWeight or GenerationWeight for this purpose.
In I3MCWeightDict, all production weights listed above are multiplied into TotalWeight. So, basically, OneWeight is:
OneWeight = TotalWeight[unitless] * InjectionArea[cm^2] * SolidAngle[sr] * (IntegralOfEnergyFlux/GenerationEnergyFlux)[GeV]
For more details see section “Parameters in I3MCWeightDict”. | {"url":"https://docs.icecube.aq/icetray/main/projects/neutrino-generator/weighting.html","timestamp":"2024-11-10T14:38:19Z","content_type":"text/html","content_length":"23997","record_id":"<urn:uuid:ddf9f330-30df-49b7-af15-4b92dda5a813>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00027.warc.gz"} |
Collection of Solved Problems
Change in Internal Energy of an Ideal Gas
Task number: 2176
After undergoing an isochoric process, pressure of 10 l of an ideal gas increased four times. Thereafter, the gas has been cooled isobarically which caused the decrease in volume to one half. Initial
pressure of the gas is 100 kPa.
The gas is considered to be a monatomic ideal gas, whose Poisson constant is
\[\kappa = \frac{5}{3}.\]
Determine the work of external forces, total heat and change of internal energy of the gas during the described process.
• Hint
Think how we can determine work using the pV diagram.
Consider the temperature changes of an ideal gas during an isochoric increase of pressure and isobaric compression.
• Analysis
Total work performed by the external forces is equal to the work done during the isobaric process because no work is performed during the isochoric process. Since the pressure in the isobaric
process is constant, the work is equal to the product of the pressure and volume difference.
To calculate the total heat, let us have a look at its change during each process.
In the isochoric process, pressure of the gas is increasing and considering the ideal gas law, so is its temperature. Since neither the gas itself nor the external forces perform any work, its
internal energy must have been increased because of supplied heat. This heat can be calculated using the molar heat capacity at constant volume. Its value is determined by the Poisson constant
and Meyer's relation.
During the isobaric cooling, temperature of the gas, and thus its internal energy, is decreasing. At the same time the gas is being compressed, which means that the surroundings are performing
work. It implies that the gas has to transfer heat to the surroundings. In the calculation, we would also use the fact that, according to the ideal gas law, since the volume of the gas has
decreased to one half during the isobaric compression, its temperature has been lowered to one half as well.
Then, the resultant heat is the difference between the heat supplied by the gas and received by it.
Now, just the change in the internal energy of the gas during the process is left.
In case of the isochoric process, performed work is zero, so according to the 1^st law of thermodynamics, the change in internal energy is equal to the received or supplied heat. Calculation of
this heat has been mentioned above. Internal energy change during the isobaric process can again be calculated using the temperature difference and the molar heat capacity at constant volume
(change of internal energy does not depend on the type of the process but only on the temperature change).
• Notation
p[1] = 100 kPa = 10^5 Pa initial pressure of the gas
V[1] = 10 l = 10 dm^3 = 0.01 m^3 initial volume of the gas
\(\kappa = \frac{5}{3}\) Poisson constant
W = ? work of the external forces
Q = ? total heat
ΔU = ? change in the internal energy of the gas during the process
• Solution: Calculating the Work
First, we determine the work W of the external forces throughout the process. After we realize that no work can be done during the isochoric process, we just need to calculate the work performed
in the isobaric process at pressure p[2]
The following formula applies to the work W[p] done by the gas (since we are considering the process at constant pressure, we can avoid integration)
\[W_p=p\mathrm{\Delta} V.\]
We substitute given values of pressure and volume
\[W_p=p_{2}\left( V_{3}-V_{2}\right)=p_{2}\left(\frac{1}{2} V_{1}- V_{1}\right)=-\frac{1}{2} p_{2}V_{1}.\]
Using the relation between p[1] and p[2], we obtain the final formula for the total work.
\[W_p = - 2p_{1}V_{1},\]
where p[1] is the initial pressure and V[1] is the initial volume.
Negative sign means that the work was not performed by the gas, but the external forces, which corresponds with the isobaric compression. Thus, external forces perform a work
\[W = 2p_{1}V_{1}.\]
• Solution: Calculating the Heat
When calculating the heat, we need to keep in mind that during the isochoric process, in which pressure increases four times, no work performed and temperature increases to the value T[2] = 4T[1]
(which follows from the fact that according to the ideal gas law, in case of an isochoric process V = const, ratio \(\frac{p}{T}\) is constant as well).
As a result, internal energy of the gas is increasing, so the heat must be supplied to the system. For the amount of the heat received by the system Q[1] (and increase of the internal energy ΔU
[1]) it holds true:
\[Q_{1} = \mathrm{\Delta} U_{1} = C_{V}n\left( T_{2}-T_{1}\right) = \frac{3}{2} nR\left( 4 T_{1}-T_{1}\right) = \frac{9}{2} nRT_{1}.\]
To calculate Q[1], we also use the ideal gas law in the following form
\[nRT_{1} = p_{1}V_{1}\]
and we obtain
\[Q_{1} = \frac{9}{2} p_{1}V_{1}.\]
Value of C[V] can be calculated using the Poisson constant and the following formula
\[\kappa = \frac{C_{p}}{C_{V}}\]
and Meyer's relation \(C_{p} = C_{V} + R.\)
After the substitution we get
\[\kappa = \frac{C_{p}}{C_{V}} = \frac{C_{V}+R}{C_{V}} = 1+\frac{R}{C_{V}}\, \Rightarrow\] \[\Rightarrow \, C_{V} = \frac{R}{\kappa - 1} = \frac{R}{\frac{5}{3} - 1} = \frac{3}{2}R\]
And for C[p] it applies
\[C_{p} = \kappa C_{V} = \frac{5}{3}\, \cdot \, \frac{3}{2}R = \frac{5}{2}R.\]
During the isobaric cooling, temperature of the gas is lowering, which means that its internal energy is also decreasing, namely, by ΔU[2]. In addition, the external forces perform work W.
Therefore, in agreement with the 1^st law of thermodynamics, the gas has to supply the surroundings with heat Q[2] (heat Q[2] will be negative because it is the heat supplied by the system):
\[Q_{2} = C_{p}n\left( T_{3}-T_{2}\right) = \frac{5}{2} Rn\left( \frac{1}{2}T_{2}- T_{2}\right) = -\frac{5}{4} RnT_{2} \] \[Q_{2} = -\frac{5}{4} p_{2}V_{2} = -5 p_{1}V_{1}\]
In the calculation, we have considered the fact that during the isobaric compression, in which volume is reduced to one half, thermodynamic temperature has to decrease to one half too (which
follows from the fact that in case of the process at constant pressure \(\frac{V}{T} = const\)). After that we have used the ideal gas law again, this time in the form \(nRT_{2} = p_{2}V_{2}\).
Total heat supplied by the gas during the whole process is given by the sum of heat Q[1] received in the isochoric process and heat Q[2] supplied by the gas during the isobaric cooling.
It holds true:
\[Q = Q_{1}+Q_{2} =\frac{9}{2} p_{1}V_{1} - 5 p_{1}V_{1}\] \[Q = - \frac{1}{2} p_{1}V_{1}\]
• Solution: Determining the Change in Internal Energy
Internal energy of the gas is increasing between states 1 and 2 (see figure). Since it is an isochoric process, change in internal energy is equal to the supplied heat.
\[\mathrm{\Delta} U_{1} = Q_{1} = \frac{9}{2} nRT_{1}.\]
During the isobaric cooling (process between states 2 and 3), temperature of the gas is lowering, which means that its internal energy is also decreasing. We denote the internal energy change by
ΔU[2] and calculate it as follows (negative value corresponds with the decrease of internal energy):
\[\mathrm{\Delta} U_{2} = C_{V}n\left( T_{3}-T_{2}\right) = \frac{3}{2} Rn\left( \frac{1}{2}T_{2}- T_{2}\right) = -\frac{3}{4} RnT_{2} \]
Note: Change of internal energy does not depend on the type of process that the gas is undergoing. In case of the ideal gas, it depends on temperature change only. Constant of proportionality
between the change of internal energy and the temperature change is C[V]. Supplied or received heat, on the contrary, does depend on the type of process, thus, when calculating the heat, we
always have to use the heat capacity corresponding with the process.
We have used the ideal gas law and formulae for the isobaric process again, and then we adjust the relation.
\[\mathrm{\Delta} U_{2}= -\frac{3}{4} p_{2}V_{2} = -3 p_{1}V_{1}\]
Now we know all we need to be able to determine the required values. Total increase of internal energy ΔU is given by the formula
\[\mathrm{\Delta} U = \mathrm{\Delta} U_{1} + \mathrm{\Delta} U_{2} = \frac{9}{2} p_{1}V_{1} - 3 p_{1}V_{1}.\] \[\mathrm{\Delta} U = \frac{3}{2} p_{1}V_{1}\]
In the calculation, we have considered the fact that during the first process internal energy is increasing and thus, is positive, whereas during the second process internal energy is decreasing,
which makes its change negative.
Positive sign of the total change in internal energy means that internal energy of the gas has been increased.
Since internal energy is a state function and therefore its change depends only on the initial and final state, it can be calculated using just the temperature difference T[3] − T[1]:
\[\mathrm{\Delta}U = nC_{V}\left( T_{3} - T_{1} \right)\]
For temperature T[3] it holds true:
\[T_3 = \frac{1}{2}T_2=\frac{1}{2} 4 T_1 = 2T_1\]
and after the substitution we obtain:
\[\mathrm{\Delta}U = nC_{V}\left( 2T_{1} - T_{1} \right) = nC_{V}T_{} = n \left(\frac{3}{2}R\right) T_1 = \frac{3}{2}p_1V_1, \]
i.e. the same result as before.
• Numerical Substitution
\[W = 2 p_{1}V_{1} = 2\cdot{10^{5}}\cdot{0.01}\, \mathrm{J} = 2000\, \mathrm{J}\] \[W = 2\, \mathrm{kJ}\] \[Q = - \frac{1}{2} p_{1}V_{1} = - \frac{1}{2}\cdot 10^{5}\cdot 0.01\, \mathrm{J} = - 500
\, \mathrm{J}\] \[Q = - 0.5\, \mathrm{kJ}\] \[\mathrm{\Delta} U = \frac{3}{2} p_{1}V_{1}= \frac{3}{2}\cdot 10^{5}\cdot 0.01\, \mathrm{J} = 1500\, \mathrm{J}\] \[\mathrm{\Delta} U = 1.5\, \mathrm
• Answer
External forces performed the work of 2 kJ, gas supplied the heat of 0.5 kJ and the internal energy of the gas increased by 1.5 kJ.
• Comment
Note the validity of the relation ΔU = W + Q. This formula does not state anything else except the compliance of the process with the 1^st law of thermodynamics. W is positive, because it is the
work done by the external forces, and heat Q is negative because the gas has given it away during the process. | {"url":"https://physicstasks.eu/2176/change-in-internal-energy-of-an-ideal-gas","timestamp":"2024-11-11T17:46:16Z","content_type":"text/html","content_length":"37842","record_id":"<urn:uuid:5310a42a-775e-44d4-9f4c-6e41c2562ee6>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00334.warc.gz"} |
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You might not know exactly where to start or how to start and if this is the point that you are at now, I'm glad you are here reading this. I want to help guide you through this process and help make
your tile installation go as easy as possible.
I would like to introduce to you the grid system. With this system we are going to accomplish several things to help make your tile installation go easier, faster and it will look better.
Imagine what it would look like if the room that your installing tile in was drawn on a piece of grid paper. If you drew everything to scale on that paper including- the cabinets, walls and doorways
you would be able to see exactly how all the tiles would fit in that room.
Now imagine that piece of paper becoming the actual size of the room and now you can see these lines on the floor.
Wouldn't that be great! Think about it, if all those lines were on the floor before you started installing your tile, how easy would it be to figure out your cuts for the tile and then install the
tile using those lines to guide you? Wouldn't almost be like cheating?
Well I'll tell you first hand, that this is the process I use and IT WORKS GREAT!
There are a few simple steps that we will go through, then you can use this system too.
Grab these few items so we can get started
• Tape Measure
• Pencil for writing
• Carpenters pencil for marking on the floor
• Paper
• Chalk Line/I use black chalk and it seems to work the best
• Calculator
Step 1
Measure several of your tiles to determine the average size
Because tiles are never the same exact size you need to determine what the average size of the tiles are. This is also the reason why I never recommend using tile spacers. If the tiles aren't
adjusted when laying ceramic tile, then the joints will run off.
Determine the Grout joint Size
This is totally up to you. There are a few suggestions that I will make to help you with this. I would not use a grout joint that is smaller than 3/16 of an inch. This is the standard that I follow
for a few reasons. If the floor is not perfectly flat, then the wider your grout joint is, the more forgiving a tile that sticks up a little higher will be.
Trust me, a wider joint is better!
Another reason is that when you have different size tiles, the grout joints will need to be adjusted slightly throughout the floor. This will make up for the different size tiles. If your grout joint
is wider, the less likely it is to see the difference in the grout joints.
I have tile that is 12 x 12 in size according to the box. It is actually 11 3/4 x 11 3/4 and some are slightly larger 11 13/16 which is 1/16 of an inch larger than 11 3/4 and some are slightly
smaller 11 11/16 which is 1/16 of an inch smaller than 11 3/4 .
I absolutely love to use easy numbers and I think you will to, so here's how I think we should figure this out. Make sure the smallest your grout joint will be is 3/16 of an inch.
If you look again at my example just above here you will understand that if I add 3/16 of an inch to my largest tile that would total 12 inches.that would give me a very easy number to work with. I
would make my grout joint slightly larger to have an easy number to work with.
Here's an example of what I mean;
If my largest tile was 11 3/4 inches and my smallest tile was 11 11/16 inches I would still use 12 inches for my (one tile + grout line) answer. It's an easy number to work with and the
possibility of having a grout joint that could be 5/16 of an inch is worth to me.
Larger grout lines aren't a bad thing. Most people think- the smaller a grout joint is, the easier it is to grout. NOT TRUE! Check out my blog - Grouting the easy way.
I always try to use the following increments-
1/8 - 1/4 - 3/8 - 1/2 - 5/8 - 3/4 - 7/8 - 1
These are the increments I'm trying to use when I talk about about using easy numbers. If my tile is 11 7/8 inches, I would then use 12 1/8 inches for my ( one tile + grout line).
Check out my fractions to decimals chart to convert your fractions to a decimal number for the next step.
Step 2
Creating a Cheat Sheet
I'm going to continue to use my example above in creating my cheat sheet. Now that we have determined the grout joint size, the next step in laying ceramic tile is to create a cheat sheet. This is
what we will use when we begin the layout our grid lines, it will show us how big our grid squares are going to be.
Grab your paper, pencil, calculator and tape measure and let's get started.
The formula for this is very simple
• Let's start out by writing down our starting point which is our answer to -
one Tile + the grout joint = ?
• 12 is my answer.
• now we need to add another (one tile + the grout joint) to our first number - so that would be 12 + 12 = 24
• then repeat this process until you get passed the longest point of the room.
Example: This is what my cheat sheet would look like
Cheat Sheet
12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144
I added 12 (one tile + the grout joint) + 12 +12 +12 and so on.
Step 3
Measure the room
Using a tape measure, we want to measure the room to find the longest distance from wall to wall. Once we find that distance we need to write that down. Let's convert this number to inches, it will
make a little easier for us when we start to fill out our cheat sheet.
Divide this Number in Half
Let's take the answer we got in step 2 and divide that by 2. The room I'm laying ceramic tile in is 11 ft wide x 20 ft long. So 20 ft is the longest distance and that is 240 inches. If I divide that
by 2, that's 120 inches. I know that your probably thinking that you wish your room was this simple. I'm just trying to show you these steps with nice even numbers.
[leadplayer_vid id="51128A11E8DC0"]
How Big will My Pieces Along the Wall Be?
Now we need to start making marks on the floor, but first we need a starting point. I always try to use an outside wall to measure off of because they are usually the straightest walls.
This not always true though. I do also take into consideration the longest distance in the room. I want my first line that we are going to mark(snap) on the floor to be the longest.
If your room is not square or rectangular shaped and there are several places that you have nooks that jog in from the main part of the room, let's concentrate on the main part of the room first.
If you have a hallway attached to the room then we do need to start our layout with that in mind. I will get to these soon, but let's start with a square or rectangular shaped room.
1) Measure both the length and the width of the room and write these down on a piece of paper. Measure the room in a few different places, this will show you how straight the walls are with each
2) Divide these numbers in half. Now we have divided the room in half and found the center of the room. These answers we came up with here, we will be using in the next step. It might be easier to
convert these measurements to inches when dividing.
Example: My room is 11 ft wide or 132 inches. When I divide this in half I get 66 inches.
3) Let's find the center of the room in the width first. Measure away from the wall into the room and using the answer from above, we will mark that measurement on the floor using a pencil. This is
the center of the room. Keep your tape measure extended in that spot for the next step.
4) Let's check the size of our pieces along the wall. Grab your cheat sheet! Look on your cheat sheet and find the first measurement that is smaller than the center point of the room and mark that on
the floor.
Example: My center point is 66 inches and the first measurement that is smaller on my cheat sheet is 60 inches.
5) Measure the distance between the 2 marks we just made and that's how big your pieces along the walls will be.
Example: When I measure the distance between the marks I've made , my answer is 6 inches.
If the pieces are to small, we can move this pretty easily.
1. Divide the tile + the grout joint in half (example: tile + the grout joint = 12 ÷ 2 = 6)
2. With the end of your tape measure placed on your center mark, measure away from the mark, the distance you came up just above.(answer from number 1)
(tile + the grout joint = ? ÷ 2)
3. Make a mark on the floor with a pencil.
4. Measure from the wall to the mark we just made.
5. Grab your cheat sheet and look for the number that is closest to your mark without going over and make a mark.(This mark might already be on the floor because it could be a mark we made earlier)
6. Measure the distance between these 2 marks and this is how big your pieces would be.
7. Now decide which of these piece sizes will be better to use and circle the mark you will be using.
Step 4
Now repeat the steps from above to find the pieces in the length of the room
Watch the video below to see how I did this.
[leadplayer_vid id="51128A4F7ADD7"]
Step 5
Mark the 2 Main Grid Lines
Now that we know which marks we will be using for our 2 main lines in the grid let's build this thing and start laying ceramic tile.
Let's start with Width of the Room
• We already have one mark made, so let's remeasure this for an exact measurement.
• On the other side of the room make a mark on the floor that is the same distance from the wall as your first mark.
• circle these marks and grab a chalk line.
• Extend the chalk line and place the string on your marks.
• Pull the chalk line tight and then grab the string with two fingers.
• Lift the string above the floor a few inches and let go.(Just lift enough for the entire string to come above the floor).
• SNAP! The string should have snapped back to the floor and left a mark behind after you remove the chalk line.
• Now let's take our tape measure and measure away from the wall to the line in several places.(Checking to see how straight our line is.)
• If the measurements are slightly different throughout that's okay. The wall could be wavy.
• If it's off quite a bit we need to adjust the line to make it straight.
• Just add or subtract from your mark to make it straight and then with a sponge wipe off the line and re-snap a new one.
• Check your line again to make sure it's straight.
[leadplayer_vid id="511289CB3F62F"]
What if your mark is not at the longest point of the room?
• Grab your cheat sheet and tape measure
• Place the end of your tape measure on the mark you have on the floor.
• Measure towards the area you need the mark to be
• Look on your cheat sheet and find the number that will be in the area you need to make your mark.
• Make a mark on the floor
Example: If mark needs to be at least 40 inches away from my original mark I would look at my cheat sheet to see what number is on there that is closest to 40 inches.For me that would be either 36
inches or 48 inches and I would need to look to see which of those two numbers would get me into the area I need to make my mark.
• Once you complete this step, then we would follow the same steps I mentioned above.
Step 6
Now let's move on to the length
• We should have a mark on the floor for this already so find the mark on the floor.
• Now grab a T- square( a carpenters square can be used in smaller area that a T square will not fit into.)
• Place the T square on the floor with the top of the T square(this is the shorter part) on the line you just snapped.
• Line up the longer part of the T square with your mark on the floor.
• Once this is lined up, draw a line along side the T square using the edge of the T square as a guide.
• now extend the line you just drew to other side of the chalk line.
Example: If you move the T-square 2 feet down the line you just drew, 2 feet of it will be on the line you drew and 2 feet of it will extend passed the chalk line you snapped earlier.
• Place the T square on the floor with the top of the T square(this is the shorter part) on the line you just snapped.
• Line up the longer part of the T square with your mark on the floor.
• Once this is lined up draw a line along side the T square using the edge of the T square as a guide.
• Extend a chalk line from wall to wall and line it up on top of the line that you drew on the floor.
• After you are certain that it's lined up and you pulled the chalk line tight, snap the line
Wrap the chalk line around your finger and use your thumb to hold the string on the floor.
If the room you are working in is a larger room then we should check our lines.
We can do this using a method called the 345 Rule. Grab a tape measure and we need to measure two of the lines that we snapped. Look at the lines on the floor where they meet. We want to measure away
from where they intersect on two different lines. Just think of it as if the letter L was on the floor and that is what we need to measure.
First measure the shorter part of the L and measure 3 feet to the right of where the lines intersect and make a mark. Then measure up the line 4 feet and make a mark(the longer part of the L)
Now measure the distance between these two marks. This would be a diagonal measurement. This should be 5 feet and if it's not then make a mark where 5 feet is and adjust your line. Just erase the
line on the floor with a sponge and then re-snap your line.
For bigger rooms this is a great method because 3-4-5 can be changed to 6-8-10 or 12-16-20 and so on.
Step 7
[leadplayer_vid id="5112898B5EC03"]
Mark the rest of the Grid
Now we can mark the rest of the grid so grab your cheat sheet, tape measure and a pencil.
What we want to do here, is measure away from our main lines and put a mark on the floor every two tiles + the grout joints. This is on your cheat sheet. we also want these marks to be next to the
• Place the end of the tape measure on the line.
• Look at your cheat sheet and begin marking the floor every two tiles + the grout joints.
• At the the end of the wall we need a mark that is one tile + the grout joint away from the wall.
• Circle all of you marks
Example: The distance from my line to the wall is 66 inches. My measurements from my cheat sheet would be 24, 48, and 60. Once I get to the point that I can't mark two tiles + the grout joints, which
was 48, then I marked it one tile + the grout joint, which is 60.
You can measure away from the line anywhere in the room. This is what you will have to do to get into closets or other areas that jog in. If you have areas that you can't measure to from the main
line that's okay.
Once we snap our lines you will be able to use one those lines to measure off of, into any area in the room.
Snap the rest of the lines
Now go through and snap all of the lines. Once this is done, you should go throughout the grid and measure the squares to be sure that the squares are all the same size.
Sometimes the squares can be slightly bigger or smaller and that's okay. As long as it is only slightly. This happens because of slight bows in the floor that cause the chalk line to move a little.
Your main lines are always your go to lines when there is a problem. Measure off the main line to the other lines when you need to check a line that is off.
If your chalk lines wear off the floor then just re-snap them.
Before you begin the layout of the grid , mix some thin set. Mix it thin so it's easy to spread. Then spread a thin layer on the floor using the flat side of a trowel.(the side that doesn't have
notches). This will help hold the chalk line so it doesn't wear off as easy.
Step 8
How to Install Your Tile Using the Grid
Now that we have finished the grid we can start laying ceramic tile. When we install the tile we will be following the lines that we snapped on the floor. To do this we will line up two edges of the
tile to two of the lines on the floor.
Once we decide which lines we are going to line the edges of our tile to, these are the lines that we will follow throughout the entire installation. Sound complicated? It's not at all, let me
[leadplayer_vid id="51128930E580B"]
Now I have given you a lot of information here and it might seem overwhelming. Just go back and read this a few times. You don't need to memorize all of this information. You can follow these steps
one by one while you are laying out your grid. This was worth putting together for you, because I know how easy laying ceramic tile will go, once you get the grid on the floor.
I have some really easy methods that I use for measuring your cuts when using the grid system. One of the methods doesn't even require a tape measure. The other method is very easy too. You do use a
tape measure, but it only requires you to remember one number - one tile+the grout joint!
Check out some of my other posts on my site and let me guide you, laying ceramic tile . I know the problems that you can face during a tile installation, because I've lived them. I will walk you
through all of them and help make your tile installation go easy.
Joe Letendre
Through Christ alone we are SAVED!
Measuring cuts
[leadplayer_vid id="512F429E019E3"]
Measuring cuts that are more dificult
[leadplayer_vid id="512F429188C55"]
Installing Tile using the Grid - getting started
[leadplayer_vid id="512F4272A32C6"]
Installing Tile in the Main Field of the Grid
[leadplayer_vid id="512F428413958"]
Submit a Comment
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zpoequ: computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm) - Linux Manuals (l)
zpoequ (l) - Linux Manuals
zpoequ: computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm)
ZPOEQU - computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm)
N, A, LDA, S, SCOND, AMAX, INFO )
INTEGER INFO, LDA, N
DOUBLE PRECISION AMAX, SCOND
DOUBLE PRECISION S( * )
COMPLEX*16 A( LDA, * )
ZPOEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S
(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal scalings.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The N-by-N Hermitian positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
S (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND (output) DOUBLE PRECISION
If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.
AMAX (output) DOUBLE PRECISION
Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element is nonpositive. | {"url":"https://www.systutorials.com/docs/linux/man/l-zpoequ/","timestamp":"2024-11-05T00:34:22Z","content_type":"text/html","content_length":"9968","record_id":"<urn:uuid:fd172e19-781d-45d3-9787-74e914bd8528>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00111.warc.gz"} |
There is a small hole at the bottom of tank filled with water. If tota
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Determining the Drag Force at Terminal Velocity
Question Video: Determining the Drag Force at Terminal Velocity Physics • First Year of Secondary School
A free diver of mass 65 kg jumps from an aeroplane from a very high altitude. When the free diver reaches the terminal velocity, meaning that he will fall at a constant speed, what is the drag force
due to air resistance equal to? Assume that the acceleration due to gravity is constant and equal to 9.8 m/s².
Video Transcript
A free diver of mass 65 kilograms jumps from an aeroplane from a very high altitude. When the free diver reaches the terminal velocity, meaning that he will fall at a constant speed, what is the drag
force due to air resistance equal to? Assume that the acceleration due to gravity is constant and equal to 9.8 meters per second squared.
This question is asking us to calculate the drag force due to air resistance that acts on a diver when he is at terminal velocity. Let’s start by thinking about this phrase “terminal velocity.” We’re
told that when the diver is at terminal velocity, he falls at a constant speed. But what does this mean for the forces that act on the diver? Recall Newton’s second law of motion. This tells us that
the net force acting on an object is equal to the mass of the object multiplied by the acceleration of the object. 𝐹 net is equal to 𝑚𝑎.
When the diver is at terminal velocity, he has a constant speed. If an object has a constant speed, its acceleration must be zero. If we substitute 𝑎 equals zero into Newton’s second law, we see that
this means the net force acting on the object is also zero. So, when the diver is at terminal velocity, the net force on the diver is zero.
To understand this, let’s draw a diagram. The diver has a weight due to the gravitational force acting on him. The diver’s weight acts vertically downwards and pulls the diver towards the Earth. At
the instant when the diver first leaves the plane, his weight causes him to accelerate downwards at 9.8 meters per second squared. This is the acceleration due to gravity.
However, as the diver falls, he also experiences a drag force due to air resistance. Air resistance acts in the opposite direction to an object’s motion. The faster an object moves, the greater the
air resistance that it experiences. The diver is moving downwards, so the air resistance acts upwards. At first, the drag force is small. But as the speed of the diver increases, so does the air
Eventually, the diver will reach a speed called the terminal velocity. At the terminal velocity, the air resistance that acts on the diver has become equal in magnitude to the diver’s weight. So, the
downwards force of the weight is exactly balanced by the upwards drag force due to the air resistance. There is no net force acting on the diver, and the diver no longer accelerates. Instead, the
diver falls at a constant speed.
To answer this question, we need to find the value of the drag force due to air resistance. Since this is equal to the diver’s weight, all we need to do is calculate the weight of the diver. Recall
that the weight of an object is equal to the mass of the object, 𝑚, multiplied by the acceleration due to gravity, 𝑔. Here, we’re told that the mass of the diver is 65 kilograms and that the
acceleration due to gravity is 9.8 meters per second squared. Substituting these values into the formula, we see that the weight of the diver is equal to 65 kilograms multiplied by 9.8 meters per
second squared. This gives us a value of 637 newtons.
So, if the weight of the diver is 637 newtons and the air resistance is equal to the weight when the diver is at terminal velocity, then the drag force acting on the diver must be equal to 637
newtons. This is the final answer to this question. | {"url":"https://www.nagwa.com/en/videos/939165431327/","timestamp":"2024-11-10T06:45:50Z","content_type":"text/html","content_length":"248857","record_id":"<urn:uuid:799f6346-42ff-4197-a5da-ab1f4c3dd195>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00410.warc.gz"} |
[−3,3] by [−2,2]
[−3,3] by [−2,2]
Figure 3.11 There is a "corne... | Filo
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Question Text by by Figure 3.11 There is a "corner" at . Figure There is a "cusp" at .
Updated On Sep 22, 2022
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 96
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Pre RMO Previous Year 2014 Question Paper With Solutions
Pre RMO Previous Year 2014 Question Paper With Solutions
In this post, you will get Pre RMO Previous Year 2014 Question Paper With Solutions.
Read: Pre RMO and RMO Important Study Materials: Books | PDFs | Blogs | YouTube Channels | Solved Papers
Pre RMO Previous Year 2014 Question Paper Question No 1:
A natural number k is such that k^2 < 2014 < (k + 1)^2. What is the largest prime factor of k?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 1:
Pre RMO Previous Year 2014 Question Paper Question No 2:
The first term of a sequence is 2014. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 2014^th term of the sequence?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 2:
Pre RMO Previous Year 2014 Question Paper Question No 3:
Let ABCD be a convex quadrilateral with perpendicular diagonals. If AB = 20, BC = 70 and CD = 90, then what is the value of DA?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 3:
Pre RMO Previous Year 2014 Question Paper Question No 4:
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 4:
Pre RMO Previous Year 2014 Question Paper Question No 5:
If real numbers a, b, c, d, e satisfy a + 1 = 2 + b = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3. What is the value of a^2 + b^2 + c^2 + d^2 + e^2?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 5:
Pre RMO Previous Year 2014 Question Paper Question No 6:
What is the smallest possible natural number n for which the equation x^2 – nx + 2014 = 0 has integer roots?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 6:
Pre RMO Previous Year 2014 Question Paper Question No 7:
If , what is the value of ?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 7:
Pre RMO Previous Year 2014 Question Paper Question No 8:
Let S be a set of real numbers with mean M. If the means of the sets S U {15} and S U {15,1} are M + 2 and M + 1, respectively, then how many elements does S have?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 8:
Pre RMO Previous Year 2014 Question Paper Question No 9:
Natural numbers k, I, p and q are such that if a and b are roots of x^2 – kx + l = 0 then a + 1/b and b + 1/a are the roots x^2 – px + q = 0. What is the sum of all possible values of q?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 9:
Pre RMO Previous Year 2014 Question Paper Question No 10:
In a triangle ABC, X and Y are points on the segments AB and AC, respectively, such that AX : XB = 1 : 2 and AY : YC = 2 : 1. If the area of triangle AXY is 10 then what is the area of triangle ABC?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 10:
Pre RMO Previous Year 2014 Question Paper Question No 11:
For natural numbers x and y, let (x, y) denote the greatest common divisor of x and y. How many pairs of natural numbers x and y with x < y satisfy the equation xy = x + y + (x, y)?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 11:
Pre RMO Previous Year 2014 Question Paper Question No 12:
Let ABCD be a convex quadrilateral with angle DAB = angle BDC = 90 degrees. Let the incircles of triangles ABD and BCD touch DB at P and Q, respectively, with P lying in between B and Q. If AD = 999
and PQ = 200 then what is the sum of the radii of the incircles of triangle ABD and BDC?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 12:
Pre RMO Previous Year 2014 Question Paper Question No 13:
For how many natural numbers n between 1 and 2014 (both inclusive) is an integer?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 13:
Pre RMO Previous Year 2014 Question Paper Question No 14:
One morning, each member of Manjul’s family drank and 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank 1/7-th of the total
amount of milk and 2/7-th of the total amount of coffee. How many people are there in Manjul’s family?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 14:
Pre RMO Previous Year 2014 Question Paper Question No 15:
Let XOY be a triangle with angle XOY = 90 degrees. Let M and N be the midpoints of legs OX and OY, respectively. Suppose that XN = 19 and YM = 22. What is XY?
Pre RMO Previous Year 2014 Question Paper with Solutions of Question No 15:
Pre RMO Previous Year 2014 Question Paper Question No 16:
In a triangle ABC, let I denote the inceter. Let the lines AI, BI and CI intersect the incircle at P, Q and R, respectively. If angle BAC = 40 degrees, what is value of angle QPR in degrees?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 16:
Pre RMO Previous Year 2014 Question Paper Question No 17:
For a natural number b, let N(b) denote of natural numbers a for which the equation x^2 + ax + b = 0 has integer roots. What is the smallest value of b for which N(b) = 20?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 17:
Pre RMO Previous Year 2014 Question Paper Question No 18:
Let f be a one-to-one function from the set of natural numbers to itself such that f(mn) = f(m) f(n) for all natural numbers m and n. What is the least possible value of f(999)?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 18:
Pre RMO Previous Year 2014 Question Paper Question No 19:
Let x[1], x[2], x[3], …, x[2014] be real numbers different from 1 such that x[1] + x[2 ]+ x[3 ]+ …+ x[2014] = 1 and . What is the value of ?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 19:
Pre RMO Previous Year 2014 Question Paper Question No 20:
What is the number of ordered pairs (A, B) where A and B are subsets of {1,2,….,5) such that neither A B not B A?
Pre RMO Previous Year 2014 Question Paper Solution of Question No 20:
Get Combined Pre RMO 2012 to 2018 Questions Paper with Solution (Click on Image)
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How to Implement Recursion In Haskell?
To implement recursion in Haskell, you can follow these steps:
1. Determine the base case: Define the stopping condition of the recursive function. It is crucial to have a base case that will terminate the recursion.
2. Identify the recursive case: Specify how the function will call itself on a smaller or simpler problem. This recursive call should be designed such that it moves closer to the base case.
3. Write the recursive function: Create a function definition that handles both the base case and the recursive case. You can use pattern matching to distinguish between the two cases.
4. Test the function: Execute the recursive function with different inputs to ensure it behaves as expected. It's particularly important to test it with inputs that might reach the base case
quickly, as well as inputs that require multiple recursive calls.
Keep in mind the following general tips when working with recursion in Haskell:
• Ensure progress towards the base case: The recursive call should bring you closer to the base case with each iteration. Otherwise, you risk infinite recursion.
• Leverage recursion on subproblems: Break down your problem into smaller subproblems that can be solved recursively. This allows you to tackle complex tasks by solving simpler versions of the same
• Utilize pattern matching: Haskell's pattern matching capabilities are especially handy when working with recursion. You can define different function clauses to handle different cases, making
your code concise and expressive.
• Avoid unnecessary calculations: Use memoization techniques or accumulator variables to store calculated values and prevent redundant computations.
By following these steps and guidelines, you can effectively implement recursion in Haskell to solve a wide range of problems.
How to optimize recursive functions in Haskell?
There are several ways to optimize recursive functions in Haskell:
1. Tail recursion: Convert the recursive calls into tail recursive form. This means that the recursive call is the last operation performed in the function, allowing the compiler to optimize it into
a loop. To achieve tail recursion, use an accumulator parameter to accumulate the intermediate result.
2. Memoization: Use memoization to avoid redundant calculations by storing the results of previous function calls in a lookup table. This can be done using techniques such as lazy evaluation or
explicit memoization using an array or map.
3. Use strictness annotations: Add strictness annotations to function parameters and intermediate results to ensure that they are evaluated immediately, rather than lazily. This can help avoid the
overhead associated with lazy evaluation.
4. Use data structures that are optimized for specific operations: For certain recursive algorithms, using data structures such as binary trees or tries can improve performance. For example, when
dealing with search or traversal algorithms, using a binary search tree can significantly optimize the computation.
5. Use compiler optimizations: GHC, the most widely used Haskell compiler, has a number of optimizations that can be enabled through compiler flags or pragma directives. These optimizations may
include inlining, fusion, and strictness analysis, among others.
6. Use specialized libraries or data types: There are several libraries available in Haskell that provide optimized data structures and algorithms for specific tasks. Examples include containers,
vector, and hashmap libraries.
7. Use parallelism: In some cases, recursive functions can be parallelized to improve performance. Haskell has good support for parallel programming, and libraries like par and pseq can be used to
introduce explicit parallelism.
It's important to note that optimization techniques may vary depending on the specific problem and context, so it's always a good idea to profile and benchmark your code to identify bottlenecks and
choose the appropriate optimization strategy.
What are the benefits of using recursion in Haskell?
There are several benefits of using recursion in Haskell:
1. Simplicity: Haskell is a functional programming language that encourages the use of recursion for solving problems. Recursion allows for concise and elegant solutions to many problems through the
use of simple and modular functions.
2. Expressiveness: Recursion allows for the expression of complex computations and algorithms in a clear and straightforward manner. It provides a powerful abstraction mechanism that enables
programmers to tackle problems at a higher level of abstraction.
3. Readability: Recursive functions in Haskell are often easier to read and understand compared to equivalent iterative solutions in imperative languages. Recursive code in Haskell often closely
mirrors the mathematical definition of a problem, making it more intuitive and comprehensible.
4. Modularity: Recursive functions can be designed as small, self-contained units that solve a specific sub-problem. This modular approach allows for easy and efficient code reuse, as these
functions can be combined and used in different contexts.
5. Efficiency: Recursive functions in Haskell often benefit from lazy evaluation, which means that only the necessary computations are performed. This can lead to efficient memory usage and improved
runtime performance.
6. Mathematical elegance: Recursion is deeply rooted in mathematical concepts and principles. Haskell, being a language deeply influenced by mathematical principles, leverages recursion to provide
concise and elegant solutions to many problems.
7. Tail recursion optimization: Haskell supports tail recursion optimization, which can optimize recursive calls and eliminate the need for stack space. This allows for efficient execution of
recursive functions without the risk of stack overflow.
What is the difference between direct and indirect recursion in Haskell?
In Haskell, recursion refers to the process where a function calls itself. Both direct and indirect recursion are used for recursive computations, but they differ in how the recursive calls are made.
Direct recursion occurs when a function directly calls itself within its own body. This means that the function explicitly refers to itself in its own definition. Here's an example of direct
1 factorial :: Integer -> Integer
2 factorial 0 = 1
3 factorial n = n * factorial (n - 1)
In this example, the factorial function directly calls itself by passing n decremented by 1 as the argument.
On the other hand, indirect recursion involves a group of functions that call each other in a circular manner. Instead of a function calling itself, it calls another function within its own
definition, which in turn calls the original function or another function in the group. Here's an example of indirect recursion:
1 isEven :: Integer -> Bool
2 isEven 0 = True
3 isEven n = isOdd (n - 1)
5 isOdd :: Integer -> Bool
6 isOdd 0 = False
7 isOdd n = isEven (n - 1)
Here, the isEven and isOdd functions form a group of mutually dependent functions that call each other. isEven calls isOdd, and isOdd calls isEven to determine if a number is even or odd.
To summarize, direct recursion involves a function calling itself directly, while indirect recursion involves a group of functions calling each other in a circular manner. | {"url":"https://tech.jetblog.ru/blog/how-to-implement-recursion-in-haskell","timestamp":"2024-11-12T00:18:56Z","content_type":"text/html","content_length":"204416","record_id":"<urn:uuid:20186162-9459-4dac-884c-1c6f294ea100>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00390.warc.gz"} |
Descriptions are available for Maple's programming-language operators (binary, unary, nullary, and element-wise operators), Maple's algebra of functional operators, and defining properties of neutral
operators. For descriptions of Maple's programming-language operators, see operators[binary], operators[unary], operators[nullary], and operators[elementwise] For the order of precedence of these
operators, see operators[precedence] For an explanation of functional operators in Maple, see operators[functional] For examples of functional operators in Maple, see examples,functionaloperators For
defining properties of neutral operators (or function names), see define
See Alsoconvert/algebraicdefinedotneutralOperators in the Student[LinearAlgebra] packageoverloadsyntax | {"url":"https://de.maplesoft.com/support/help/content/8718/operator.mw","timestamp":"2024-11-13T09:47:54Z","content_type":"application/xml","content_length":"14263","record_id":"<urn:uuid:efe26c43-8631-4eb4-af6a-9634dc45b6fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00237.warc.gz"} |
P C Krause Analysis Of Electric Machinary And Drive Systems.pdf [9qgo8xvd5xln]
ANALYSIS OF ELECTRIC MACHINERY AND DRIVE SYSTEMS Secorrd Edition
PAUL C. KRAUSE OLEG WASiYNCZUK SCOTT D. SUDHOFF Purdue University 'j i
IEEE Power Engineering Society, Sponspr >?*.""
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IEEE Press 130wer Engineering Series Mohamed E. El-Hawary, Series Editor
W I L,EY-
INTERSCIENCE A JOHN WIl,E\t & SONS, INC. PUBLICATION
Chapter 3
The voltage ecpations that describe the performance of induction and synchronous machines were: established in Chapter 1. We found that some of the machine inductances are functions of the rotor
speed, whereupon the coefficients of the differential equations (voltage, equations) that describe the behavior of these machines are timevarying except when the rotor is stalled. A change of
variables is often used to reduce the complexity of these differential equations. There are several changes of variables that are used, and it was originally thought that each change of variables was
different and therefore they were treated separately [I-41. It was later learned that all changes of variables used to transform real variables are contained in one [5,6]. This general transformation
refers machine variables to a frame of reference that rotates at an arbitrary angular velocity. ,All known real transformations are obtained from this transformation by simply aslsigningthe speed of
the rotation of the reference frame. In this chapter this transformation is set forth and, because many of its properties can be studied without the complexities of the machine equations, it is
applied to the equations that describe resistive, inductive, and capacitive circuit elements. By this approach, many olf the basic concepts and interpretations of this general transformation are
readi1:y and concisely established. Extending the material presented in this chapter to the analysis of ac machines is straightforward involving a minimum of trigonometric manipulations.
3.2 BACKGROUND In the late 192Os, R. H. Park [I] introduced a new approach to electric machine analysis. He formulated a charrge of variables which, in effect, replaced the variables
Chapter 4
The induction machine is used in a wide variety of applications as a means of converting electric power to mechanical work. It is without doubt the workhorse of the electric power industry. Pump,
steel mill, and hoist drives are but a few applications of large multiplhase inductioln motors. On a smaller scale, the 2-phase servomotor is used in position-follow-up control systems, and
single-phase induction motors are widely used in household appliances as well as in hand and bench tools. In the beginning of this chapter, classical techniques are used to establish the voltage and
torque equations for a symmetrical induction machine expressed in terms of machine varialbles. Next, the transformation to the arbitrary reference frame presented in Chapter 3 is modified to
accommodate rotating circuits. Once this groundwork has bee11 laid, the machine voltage equations are .written in the arbitrary reference frame, directly without a laborious exercise in trigonometry
with which one is faced when starting from the substitution of the equations of transformations into the voltage equations expressed in machine variables. The equations may then be expressed in any
reference frame by appropriate assig~imentof the referenceframe speed in the arbitrary reference-frame voltage equations rather than performing each transformation individually as in the past.
Although the stationary reference frame, the refe:rence frame fixed in the rotor, and the synchronollsly rotating referenlce frame are most frequently used, the arbitrary reference frame offers a
direct means of obtaiining the voltage equations in these and all other reference frames. The steady-state voltage equations for an induction machine are obtained from the voltage equations in the
arbitrary reference frame by direct application of the material presented in Chapter 3. Computer solutions are used to illustrate the | {"url":"https://doku.pub/documents/p-c-krause-analysis-of-electric-machinary-and-drive-systemspdf-9qgo8xvd5xln","timestamp":"2024-11-03T12:09:50Z","content_type":"text/html","content_length":"26905","record_id":"<urn:uuid:e2c13523-32b5-402e-8c8a-29a2b06779c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00582.warc.gz"} |
How To Convert Watts To Volts
Being able to quickly and accurately convert watts into volts is essential for a range of engineering disciplines. Amps, volts, and watts are part of a triad where when two quantities are known the
third can be calculated, using the following formula:
\(1\text{ watt}= 1\text{ volt}\times 1\text{ ampere}\)
By using a clamp-on ammeter for AC (alternating current) circuits or an inline (series) ammeter for DC (direct current) circuits, you can measure current and use the value of power to convert watts
to volts. Follow these steps to find the voltage.
1. Getting Started
Place the AC ammeter around one of the power wires in an AC circuit. This can be either the hot wire or the neutral common wire in the circuit. Both of these wires will carry the current or amperage
for the electrical circuit.
2. Converting the Wattage
Convert the wattage of 1000 watts into volts for a circuit that has an amperage of 10 amperes. Using the power equation and solving for volts, you end up with:
\(1\text{ volt}=\frac{1\text{ watt}}{1\text{ ampere}}\)
Divide 1000 watts by 10 amperes and the resultant voltage would equal 100 volts.
3. Using Ammeters
Install the inline ammeter into the DC circuit by placing the meter in series with one of the electrical circuit wires. Again this meter can be placed on either the positive (+) wire or the negative
(-) wire of the direct current circuit. All power though must pass through the inline series ammeter.
4. Finding the Voltage
Find the voltage in the direct current circuit of 480 watts with an amperage reading of 15 amperes. Using the translated formula:
\(\text{ volts}=\frac{\text{watts}}{\text{amperes}}=\frac{480\text{ watts}}{15\text{ amperes}}=32\text{ volts}\)
Things Needed
• AC clamp-on ammeter
• DC inline (series) ammeter
TL;DR (Too Long; Didn't Read)
Generally, the nameplate of most devices would list all of the above ratings in watts, volts and amperes. Smaller electrical circuits, though, may not list any specifications, and test equipment
would be used to find the needed values.
Cite This Article
Bayne, G.K.. "How To Convert Watts To Volts" sciencing.com, https://www.sciencing.com/convert-watts-volts-5079688/. 6 November 2020.
Bayne, G.K.. (2020, November 6). How To Convert Watts To Volts. sciencing.com. Retrieved from https://www.sciencing.com/convert-watts-volts-5079688/
Bayne, G.K.. How To Convert Watts To Volts last modified March 24, 2022. https://www.sciencing.com/convert-watts-volts-5079688/ | {"url":"https://www.sciencing.com:443/convert-watts-volts-5079688/","timestamp":"2024-11-07T09:00:55Z","content_type":"application/xhtml+xml","content_length":"71324","record_id":"<urn:uuid:c9b0c629-57bc-4ace-84fa-8277941381dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00832.warc.gz"} |
Calculus: Early Transcendentals, Metric Edition
CALCULUS: EARLY TRANSCENDENTALS, Metric, 9th Edition provides you with the strongest foundation for a STEM future. James Stewart’s Calculus, Metric series is the top-seller in the world because of
its problem-solving focus, mathematical precision and accuracy, and outstanding examples and problem sets. Selected and mentored by Stewart, coauthors Daniel Clegg and Saleem Watson continue his
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and new resources like Explore It interactive learning modules. Showing that Calculus is both practical and beautiful, the Stewart approach and WebAssign resources enhance understanding and build
confidence for millions of students worldwide.
Purchase Enquiry INSTRUCTOR’S eREVIEW COPY
To the Student.
Diagnostic Tests.
A Preview of Calculus.
1. FUNCTIONS AND MODELS.
Four Ways to Represent a Function. Mathematical Models: A Catalog of Essential Functions. New Functions from Old Functions. Exponential Functions. Inverse Functions and Logarithms. Review. Principles
of Problem Solving.
2. LIMITS AND DERIVATIVES.
The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. The Precise Definition of a Limit. Continuity. Limits at Infinity; Horizontal Asymptotes.
Derivatives and Rates of Change. Writing Project: Early Methods for Finding Tangents. The Derivative as a Function. Review. Problems Plus.
Derivatives of Polynomials and Exponential Functions. Applied Project: Building a Better Roller Coaster. The Product and Quotient Rules. Derivatives of Trigonometric Functions. The Chain Rule.
Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation. Discovery Project: Families of Implicit Curves. Derivatives of Logarithmic Functions and Inverse Trigonometric
Functions. Rates of Change in the Natural and Social Sciences. Exponential Growth and Decay. Applied Project: Controlling Red Blood Cell Loss During Surgery. Related Rates. Linear Approximations and
Differentials. Discovery Project: Taylor Polynomials. Hyperbolic Functions. Review. Problems Plus.
Maximum and Minimum Values. Applied Project: The Calculus of Rainbows. The Mean Value Theorem. What Derivatives Tell Us about the Shape of a Graph. Indeterminate Forms and l'Hospital's Rule. Writing
Project: The Origins of l'Hospital's Rule. Summary of Curve Sketching. Graphing with Calculus and Technology. Optimization Problems. Applied Project: The Shape of a Can. Applied Project: Planes and
Birds: Minimizing Energy. Newton's Method. Antiderivatives. Review. Problems Plus.
5. INTEGRALS.
The Area and Distance Problems. The Definite Integral. Discovery Project: Area Functions. The Fundamental Theorem of Calculus. Indefinite Integrals and the Net Change Theorem. Writing Project:
Newton, Leibniz, and the Invention of Calculus. The Substitution Rule. Review. Problems Plus.
Areas Between Curves. Applied Project: The Gini Index. Volumes. Volumes by Cylindrical Shells. Work. Average Value of a Function. Applied Project: Calculus and Baseball. Applied Project: Where to Sit
at the Movies. Review. Problems Plus.
7. TECHNIQUES OF INTEGRATION.
Integration by Parts. Trigonometric Integrals. Trigonometric Substitution. Integration of Rational Functions by Partial Fractions. Strategy for Integration. Integration Using Tables and Technology.
Discovery Project: Patterns in Integrals. Approximate Integration. Improper Integrals. Review. Problems Plus.
Arc Length. Discovery Project: Arc Length Contest. Area of a Surface of Revolution. Discovery Project: Rotating on a Slant. Applications to Physics and Engineering. Discovery Project: Complementary
Coffee Cups. Applications to Economics and Biology. Probability. Review. Problems Plus.
Modeling with Differential Equations. Direction Fields and Euler's Method. Separable Equations. Applied Project: How Fast Does a Tank Drain? Models for Population Growth. Linear Equations. Applied
Project: Which is Faster, Going Up or Coming Down? Predator-Prey Systems. Review. Problems Plus.
Curves Defined by Parametric Equations. Discovery Project: Running Circles Around Circles. Calculus with Parametric Curves. Discovery Project: Bézier Curves. Polar Coordinates. Discovery Project:
Families of Polar Curves. Calculus in Polar Coordinates. Conic Sections. Conic Sections in Polar Coordinates. Review. Problems Plus.
11. SEQUENCES, SERIES, AND POWER SERIES.
Sequences. Discovery Project: Logistic Sequences. Series. The Integral Test and Estimates of Sums. The Comparison Tests. Alternating Series and Absolute Convergence. The Ratio and Root Tests.
Strategy for Testing Series. Power Series. Representations of Functions as Power Series. Taylor and Maclaurin Series. Discovery Project: An Elusive Limit. Writing Project: How Newton Discovered the
Binomial Series. Applications of Taylor Polynomials. Applied Project: Radiation from the Stars. Review. Problems Plus.
12. VECTORS AND THE GEOMETRY OF SPACE.
Three-Dimensional Coordinate Systems. Vectors. Discovery Project: The Shape of Hanging Chain. The Dot Product. The Cross Product. Discovery Project: The Geometry of a Tetrahedron. Equations of Lines
and Planes. Discovery Project: Putting 3D in Perspective. Cylinders and Quadric Surfaces. Review. Problems Plus.
13. VECTOR FUNCTIONS.
Vector Functions and Space Curves. Derivatives and Integrals of Vector Functions. Arc Length and Curvature. Motion in Space: Velocity and Acceleration. Applied Project: Kepler’s Laws. Review.
Problems Plus.
14. PARTIAL DERIVATIVES.
Functions of Several Variables. Limits and Continuity. Partial Derivatives. Discovery Project: Deriving the Cobb-Douglas Production Function. Tangent Planes and Linear Approximations. Applied
Project: The Speedo LZR Racer. The Chain Rule. Directional Derivatives and the Gradient Vector. Maximum and Minimum Values. Discovery Project: Quadratic Approximations and Critical Points. Lagrange
Multipliers. Applied Project: Rocket Science. Applied Project: Hydro-Turbine Optimization. Review. Problems Plus.
15. MULTIPLE INTEGRALS.
Double Integrals over Rectangles. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals. Discovery Project:
Volumes of Hyperspheres. Triple Integrals in Cylindrical Coordinates. Discovery Project: The Intersection of Three Cylinders. Triple Integrals in Spherical Coordinates. Applied Project: Roller Derby.
Change of Variables in Multiple Integrals. Review. Problems Plus.
16. VECTOR CALCULUS.
Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green’s Theorem. Curl and Divergence. Parametric Surfaces and Their Areas. Surface Integrals. Stokes’ Theorem. The
Divergence Theorem. Summary. Review. Problems Plus.
A: Numbers, Inequalities, and Absolute Values. B: Coordinate Geometry and Lines. C: Graphs of Second-Degree Equations. D: Trigonometry. E: Sigma Notation. F: Proofs of Theorems. G: The Logarithm
Defined as an Integral. H: Answers to Odd-Numbered Exercises.
• James Stewart
The late James Stewart received his M.S. from Stanford University and his Ph.D. from the University of Toronto. He conducted research at the University of London and was influenced by the famous
mathematician George Polya at Stanford University. Dr. Stewart most recently served as a professor of mathematics at McMaster University, and his research focused on harmonic analysis. Dr.
Stewart authored a best-selling calculus textbook series, including CALCULUS, CALCULUS: EARLY TRANSCENDENTALS and CALCULUS: CONCEPTS AND CONTEXTS as well as a series of successful precalculus
• Daniel K. Clegg
Daniel Clegg received his B.A. in Mathematics from California State University, Fullerton and his M.A. in Mathematics from UCLA. He is currently a professor of mathematics at Palomar College near
San Diego, California, where he has taught for more than 20 years. Clegg co-authored BRIEF APPLIED CALCULUS with James Stewart and also assisted Stewart with various aspects of his calculus texts
and ancillaries for almost 20 years.
• Saleem Watson
Saleem Watson received his Bachelor of Science degree from Andrews University in Michigan. He did graduate studies at Dalhousie University and McMaster University, where he received his Ph.D. in
1978. He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland. He also taught at The Pennsylvania State University. He is currently Professor of
Mathematics at California State University, Long Beach. His research field is functional analysis. Watson is a co-author on Stewart's best-selling Calculus franchise.
• NEW EXPLANATIONS AND EXAMPLES: Careful refinements throughout provide even greater clarity on key concepts such as computing volumes of revolution and setting up triple integrals.
• NEW SCAFFOLDED EXERCISES: At the beginning of problem sets, new basic exercises reinforce key skills and build student confidence to prepare them for more rigorous exercises and conceptual
• NEW SUBHEADINGS: Additional subsections within chapters help instructors and students find key content more easily to make the text an even more helpful teaching and learning tool.
• NEW WEBASSIGN RESOURCES: New digital resources in WebAssign include Explore It interactive learning modules, the MindTap reader for interactive and mobile ebook access, enhanced remediation
support, and improved problem types.
• NEW EXPLANATIONS AND EXAMPLES: Careful refinements throughout provide even greater clarity on key concepts such as computing volumes of revolution and setting up triple integrals.
• NEW SCAFFOLDED EXERCISES: At the beginning of problem sets, new basic exercises reinforce key skills and build student confidence to prepare them for more rigorous exercises and conceptual
• NEW SUBHEADINGS: Additional subsections within chapters help instructors and students find key content more easily to make the text an even more helpful teaching and learning tool.
• NEW WEBASSIGN RESOURCES: New digital resources in WebAssign include Explore It interactive learning modules, the MindTap reader for interactive and mobile ebook access, enhanced remediation
support, and improved problem types.
• PROBLEM-SOLVING EMPHASIS: George Polya’s problem-solving methodology is introduced at the beginning and reinforced throughout. "Strategies" sections help students select what techniques they'll
need to solve problems in situations where the choice is not obvious and help them develop true problem-solving skills and intuition.
• CLEAR EXPOSITION: Dan Clegg and Saleem Watson have remained true to James Stewart's writing style by speaking clearly and directly to students, guiding them through key ideas, theorems, and
problem-solving steps, and encouraging them to think as they read and learn calculus.
• STEM APPLICATIONS: Stewart/Clegg/Watson answers the question, "When will I use this?" by showing how Calculus is used as a problem-solving tool in fields such as physics, engineering, chemistry,
biology, medicine and the social sciences.
• PREREQUISITE SUPPORT: Four diagnostic tests in algebra, analytic geometry, functions, and trigonometry enable students to test their preexisting knowledge and brush up on skills. Quick Prep and
Just-in-Time exercises in WebAssign refresh and reinforce prerequisite knowledge.
• HELPFUL EXAMPLES: Every concept is supported by thoughtfully worked examples that encourage students to develop an analytic view of the subject. To provide further insight into mathematical
concepts, many detailed examples display solutions graphically, analytically and/or numerically. Margin notes expand on and clarify the steps of the solution.
• QUALITY EXERCISES: With over 8,000 exercises in all, each exercise set carefully progresses from skill-development problems to more challenging problems involving applications and proofs.
Conceptual exercises encourage the development of communication skills by explicitly requesting descriptions, conjectures, and explanations. More challenging “Problems Plus” exercises reinforce
concepts by requiring students to apply techniques from more than one chapter of the text, and by patiently showing them how to approach a challenging problem.
• ENGAGING PROJECTS: A wealth of engaging projects reinforce concepts. "Writing Projects" ask students to compare present-day methods with those of the founders of Calculus. "Discovery Projects"
anticipate results to be discussed later. "Applied Projects" feature real-world use of mathematics. "Laboratory Projects" anticipate results to be discussed later or encourage discovery through
pattern recognition.
• More challenging exercises called "Problems Plus" follow the end-of-chapter exercises. These sections reinforce concepts by requiring students to apply techniques from more than one chapter of
the text, and by patiently showing them how to approach a challenging problem.
• Four carefully-crafted diagnostic tests in algebra, analytic geometry, functions, and trigonometry appear at the beginning of the text. These provide students with a convenient way to test their
preexisting knowledge and brush up on skill techniques they need to successfully begin the course. Answers are included, and students who need to improve will be referred to points in the text or
on the book's website where they can seek help.
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The Stacks project
Example 10.27.4. Consider the ring
\[ R = \{ f \in \mathbf{Q}[z]\text{ with }f(0) = f(1) \} . \]
Consider the map
\[ \varphi : \mathbf{Q}[A, B] \to R \]
defined by $\varphi (A) = z^2-z$ and $\varphi (B) = z^3-z^2$. It is easily checked that $(A^3 - B^2 + AB) \subset \mathop{\mathrm{Ker}}(\varphi )$ and that $A^3 - B^2 + AB$ is irreducible. Assume
that $\varphi $ is surjective; then since $R$ is an integral domain (it is a subring of an integral domain), $\mathop{\mathrm{Ker}}(\varphi )$ must be a prime ideal of $\mathbf{Q}[A, B]$. The prime
ideals which contain $(A^3-B^2 + AB)$ are $(A^3-B^2 + AB)$ itself and any maximal ideal $(f, g)$ with $f, g\in \mathbf{Q}[A, B]$ such that $f$ is irreducible mod $g$. But $R$ is not a field, so the
kernel must be $(A^3-B^2 + AB)$; hence $\varphi $ gives an isomorphism $R \to \mathbf{Q}[A, B]/(A^3-B^2 + AB)$.
To see that $\varphi $ is surjective, we must express any $f\in R$ as a $\mathbf{Q}$-coefficient polynomial in $A(z) = z^2-z$ and $B(z) = z^3-z^2$. Note the relation $zA(z) = B(z)$. Let $a = f(0) = f
(1)$. Then $z(z-1)$ must divide $f(z)-a$, so we can write $f(z) = z(z-1)g(z)+a = A(z)g(z)+a$. If $\deg (g) < 2$, then $h(z) = c_1z + c_0$ and $f(z) = A(z)(c_1z + c_0)+a = c_1B(z)+c_0A(z)+a$, so we
are done. If $\deg (g)\geq 2$, then by the polynomial division algorithm, we can write $g(z) = A(z)h(z)+b_1z + b_0$ ($\deg (h)\leq \deg (g)-2$), so $f(z) = A(z)^2h(z)+b_1B(z)+b_0A(z)$. Applying
division to $h(z)$ and iterating, we obtain an expression for $f(z)$ as a polynomial in $A(z)$ and $B(z)$; hence $\varphi $ is surjective.
Now let $a \in \mathbf{Q}$, $a \neq 0, \frac{1}{2}, 1$ and consider
\[ R_ a = \{ f \in \mathbf{Q}[z, \frac{1}{z-a}]\text{ with }f(0) = f(1) \} . \]
This is a finitely generated $\mathbf{Q}$-algebra as well: it is easy to check that the functions $z^2-z$, $z^3-z$, and $\frac{a^2-a}{z-a}+z$ generate $R_ a$ as an $\mathbf{Q}$-algebra. We have the
following inclusions:
\[ R\subset R_ a\subset \mathbf{Q}[z, \frac{1}{z-a}], \quad R\subset \mathbf{Q}[z]\subset \mathbf{Q}[z, \frac{1}{z-a}]. \]
Recall (Lemma 10.17.5) that for a ring T and a multiplicative subset $S\subset T$, the ring map $T \to S^{-1}T$ induces a map on spectra $\mathop{\mathrm{Spec}}(S^{-1}T) \to \mathop{\mathrm{Spec}}(T)
$ which is a homeomorphism onto the subset
\[ \{ \mathfrak p \in \mathop{\mathrm{Spec}}(T) \mid S \cap \mathfrak p = \emptyset \} \subset \mathop{\mathrm{Spec}}(T). \]
When $S = \{ 1, f, f^2, \ldots \} $ for some $f\in T$, this is the open set $D(f)\subset T$. We now verify a corresponding property for the ring map $R \to R_ a$: we will show that the map $\theta :
\mathop{\mathrm{Spec}}(R_ a) \to \mathop{\mathrm{Spec}}(R)$ induced by inclusion $R\subset R_ a$ is a homeomorphism onto an open subset of $\mathop{\mathrm{Spec}}(R)$ by verifying that $\theta $ is
an injective local homeomorphism. We do so with respect to an open cover of $\mathop{\mathrm{Spec}}(R_ a)$ by two distinguished opens, as we now describe. For any $r\in \mathbf{Q}$, let $\text{ev}_ r
: R \to \mathbf{Q}$ be the homomorphism given by evaluation at $r$. Note that for $r = 0$ and $r = 1-a$, this can be extended to a homomorphism $\text{ev}_ r' : R_ a \to \mathbf{Q}$ (the latter
because $\frac{1}{z-a}$ is well-defined at $z = 1-a$, since $a\neq \frac{1}{2}$). However, $\text{ev}_ a$ does not extend to $R_ a$. Write $\mathfrak {m}_ r = \mathop{\mathrm{Ker}}(\text{ev}_ r)$. We
\[ \mathfrak {m}_0 = (z^2-z, z^3-z), \]
\[ \mathfrak {m}_ a = ((z-1 + a)(z-a), (z^2-1 + a)(z-a)), \text{ and} \]
\[ \mathfrak {m}_{1-a} = ((z-1 + a)(z-a), (z-1 + a)(z^2-a)). \]
To verify this, note that the right-hand sides are clearly contained in the left-hand sides. Then check that the right-hand sides are maximal ideals by writing the generators in terms of $A$ and $B$,
and viewing $R$ as $\mathbf{Q}[A, B]/(A^3-B^2 + AB)$. Note that $\mathfrak {m}_ a$ is not in the image of $\theta $: we have
\[ (z^2 - z)^2(z - a)\left(\frac{a^2 - a}{z - a} + z\right) = (z^2 - z)^2(a^2 - a) + (z^2 - z)^2(z - a)z \]
The left hand side is in $\mathfrak m_ a R_ a$ because $(z^2 - z)(z - a)$ is in $\mathfrak m_ a$ and because $(z^2 - z)(\frac{a^2 - a}{z - a} + z)$ is in $R_ a$. Similarly the element $(z^2 - z)^2(z
- a)z$ is in $\mathfrak m_ a R_ a$ because $(z^2 - z)$ is in $R_ a$ and $(z^2 - z)(z - a)$ is in $\mathfrak m_ a$. As $a \not\in \{ 0, 1\} $ we conclude that $(z^2 - z)^2 \in \mathfrak m_ a R_ a$.
Hence no ideal $I$ of $R_ a$ can satisfy $I \cap R = \mathfrak m_ a$, as such an $I$ would have to contain $(z^2 - z)^2$, which is in $R$ but not in $\mathfrak m_ a$. The distinguished open set $D
((z-1 + a)(z-a))\subset \mathop{\mathrm{Spec}}(R)$ is equal to the complement of the closed set $\{ \mathfrak {m}_ a, \mathfrak {m}_{1-a}\} $. Then check that $R_{(z-1 + a)(z-a)} = (R_ a)_{(z-1 + a)
(z-a)}$; calling this localized ring $R'$, then, it follows that the map $R \to R'$ factors as $R \to R_ a \to R'$. By Lemma 10.17.5, then, these maps express $\mathop{\mathrm{Spec}}(R') \subset \
mathop{\mathrm{Spec}}(R_ a)$ and $\mathop{\mathrm{Spec}}(R') \subset \mathop{\mathrm{Spec}}(R)$ as open subsets; hence $\theta : \mathop{\mathrm{Spec}}(R_ a) \to \mathop{\mathrm{Spec}}(R)$, when
restricted to $D((z-1 + a)(z-a))$, is a homeomorphism onto an open subset. Similarly, $\theta $ restricted to $D((z^2 + z + 2a-2)(z-a)) \subset \mathop{\mathrm{Spec}}(R_ a)$ is a homeomorphism onto
the open subset $D((z^2 + z + 2a-2)(z-a)) \subset \mathop{\mathrm{Spec}}(R)$. Depending on whether $z^2 + z + 2a-2$ is irreducible or not over $\mathbf{Q}$, this former distinguished open set has
complement equal to one or two closed points along with the closed point $\mathfrak {m}_ a$. Furthermore, the ideal in $R_ a$ generated by the elements $(z^2 + z + 2a-a)(z-a)$ and $(z-1 + a)(z-a)$ is
all of $R_ a$, so these two distinguished open sets cover $\mathop{\mathrm{Spec}}(R_ a)$. Hence in order to show that $\theta $ is a homeomorphism onto $\mathop{\mathrm{Spec}}(R)-\{ \mathfrak {m}_ a
\} $, it suffices to show that these one or two points can never equal $\mathfrak {m}_{1-a}$. And this is indeed the case, since $1-a$ is a root of $z^2 + z + 2a-2$ if and only if $a = 0$ or $a = 1$,
both of which do not occur.
Despite this homeomorphism which mimics the behavior of a localization at an element of $R$, while $\mathbf{Q}[z, \frac{1}{z-a}]$ is the localization of $\mathbf{Q}[z]$ at the maximal ideal $(z-a)$,
the ring $R_ a$ is not a localization of $R$: Any localization $S^{-1}R$ results in more units than the original ring $R$. The units of $R$ are $\mathbf{Q}^\times $, the units of $\mathbf{Q}$. In
fact, it is easy to see that the units of $R_ a$ are $\mathbf{Q}^*$. Namely, the units of $\mathbf{Q}[z, \frac{1}{z - a}]$ are $c (z - a)^ n$ for $c \in \mathbf{Q}^*$ and $n \in \mathbf{Z}$ and it is
clear that these are in $R_ a$ only if $n = 0$. Hence $R_ a$ has no more units than $R$ does, and thus cannot be a localization of $R$.
We used the fact that $a\neq 0, 1$ to ensure that $\frac{1}{z-a}$ makes sense at $z = 0, 1$. We used the fact that $a\neq 1/2$ in a few places: (1) In order to be able to talk about the kernel of $\
text{ev}_{1-a}$ on $R_ a$, which ensures that $\mathfrak {m}_{1-a}$ is a point of $R_ a$ (i.e., that $R_ a$ is missing just one point of $R$). (2) At the end in order to conclude that $(z-a)^{k + \
ell }$ can only be in $R$ for $k = \ell = 0$; indeed, if $a = 1/2$, then this is in $R$ as long as $k + \ell $ is even. Hence there would indeed be more units in $R_ a$ than in $R$, and $R_ a$ could
possibly be a localization of $R$.
Comments (2)
Comment #741 by Wei Xu on
The reason "$z^2-z = \frac{(z^2-z)(z-a)}{z-a}$" for the conclusion "so $z^2-z = \frac{(z^2-z)(z-a)}{z-a}$ is in $\mathfrak{m}_aR_a$" is not clear, because $\frac{1}{z-a}otin R_a$. One fixed way
is: $(z^2-z)(z-a)(\frac{a^2-a}{z-a}+z)=(z^2-z)(a^2-a)+(z^2-z)(z-a)z\in \mathfrak{m}_aR_a$, and since $(z^2-z)(z-a)z\in \mathfrak{m}_a\cap \mathfrak{m}_0\subseteq \mathfrak{m}_a$, we get $(z^2-z)\
in \mathfrak{m}_aR_a$.
"the maximal ideal $(z-a)$" in
"while $\mathbf{Q}[z, \frac{1}{z-a}]$ is the localization of $\mathbf{Q}[z]$ at the maximal ideal $(z-a)$" should be "$z-a$."
I do not quite understand the proof that the units of $R_a$ are $\mathbf{Q}^{\times}$:" If $\frac{f}{(z-a)^k}$ is a unit in $R_a$ ($f\in R$ and $k\geq0$ an integer), then we have for some $g\in
R$ and some integer $\ell\geq0$. Since $R$ is an integral domain, this is equivalent to But $(z-a)^{k + \ell}$ is only an element of $R$ if $k = \ell = 0$; hence $f, g$ are units in $R$ as well."
Instead, since the set of units of $\mathbf{Q}[z,\frac{1}{z-a}]$ is $\{u(z-a)^k\mid u\in \mathbf{Q}^{\times},k\in \mathbb{Z}\}$, one can conclude that the set of units of $R_a$ is $\mathbf{Q}^{\
Comment #753 by Johan on
OK, I tried to improve on this following your suggestions. Here is the commit. By the way, this example shows that trying to prove and explain things without using the language of schemes and the
theory that goes with that is rather cumbersome.
There are also:
• 2 comment(s) on Section 10.27: Examples of spectra of rings
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Disable Crossover for MILP that is reduced to LP in Presolve
I have a problem that is formulated as an MILP, however is reduced to an LP (0 integer variables) in Presolve (which is absolutely correct and a great feature that Gurobi detects that automatically).
I then apply the barrier algorithm.
There are two things I wonder about:
1. Can Gurobi imply from the Presolve solution that the problem is in fact linear and thus apply a linear solution algorithm, i.e., only solve the root and not go into the branch-and-bound?
2. Can Crossover then be disabled, because this does not change the solution of my problem? Gurobi spends about 2/3 of the solution time in Crossover, even though this does not change anything. I
have set Method = 2, Presolve = 2, CrossoverBasis = 0, Crossover = 0
I could theoretically reformulate my optimization problem to detect beforehand if it is in fact an LP but I trusted/relied on the "intelligence" of Gurobi to do so itself in Presolve.
Can Gurobi imply from the Presolve solution that the problem is in fact linear and thus apply a linear solution algorithm, i.e., only solve the root and not go into the branch-and-bound?
Gurobi does this. What you should see is only 1 line of the B&B algorithm. The output is of the B&B is there because the model is originally a MIP so Gurobi does use MIP logging instead of LP
logging. Do you see something different?
Can Crossover then be disabled, because this does not change the solution of my problem? Gurobi spends about 2/3 of the solution time in Crossover, even though this does not change anything.
I have set Method = 2, Presolve = 2, CrossoverBasis = 0, Crossover = 0
Could you try adding NodeMethod=2 to your set of parameters. You need this to completely disable Crossover for MIPs. If this does not help, could you please share your model? Uploading model
files is not possible in the Community but we discuss an alternative in Posting to the Community Forum.
Best regards,
• Dear Jaromił,
once again, thanks for this quick and helpful comment!
NodeMethod = 2 disabled the crossover for the barrier solver.
I have a final question: can I set the concurrent solver as well while disabling the crossover? I know that crossover only applies to the barrier, but I'm not sure on how to disable crossover for
the case that barrier is chosen as the solver.
Thanks and best wishes,
• Hi Jacob,
Setting NodeMethod=2, Crossover=0 should work with concurrent as well (please let me know if you observe any different). However, it will most likely hurt performance, because NodeMethod=2 forces
Gurobi to use the Barrier algorithm in every node. This is a difficult situation, because you can easily lose all the time gained from Crossover=0 through the Barrier calls in the B&B tree,
unless using Barrier in every MIP node is not an issue.
Best regards,
• Hi Jaromił,
What is your general recommendation? My model with mostly turn out to be linear, however it might be MIP, so I would like to entrust Presolve with figuring that out. I saw that some others
experienced the same behavior that the solver spent more time in Crossover than in actually solving the problem (e.g., https://support.gurobi.com/hc/en-us/community/posts/
360072667551-barrier-and-crossover-restart). What would you do to get rid of this overhead?
Thanks so much for the great support work!
• Hi Jacob,
Usually, Crossover only takes an extraordinary amount of time if the problem is very large or if it has numerical issues. Do your models have any numeric issues? See Guidelines for Numerical
Issues for more information.
If you think that your models are numerically acceptable, then I would extract the problematic instances and analyze them in more detail. Maybe they are somewhat different from the rest and you
can find fitting parameters to overcome the performance difficulties. If your MIP instances don't require many B&B iterations then NodeMethod=2 might indeed be the best option here.
Best regards,
• Hi Jaromił,
Ok that is a great hint to look out for! So, to get that straight: the long crossover time has nothing to do with the fact that I am passing a MIP model to Gurobi that is in fact LP, but because
the LP (root) model is difficult to solve/has numerical issues? My matrix and bounds are in an acceptable range, but maybe I can find the culprit :)
• Hi Jacob,
So, to get that straight: the long crossover time has nothing to do with the fact that I am passing a MIP model to Gurobi that is in fact LP, but because the LP (root) model is difficult to
solve/has numerical issues?
Correct. Of course, it is also possible that your model is numerically perfectly fine and it is just difficult to solve. You might want to try to reduce the size of the models even further in
this case and experiment with Presolve=2, Aggregate=2, and for MIPs PreSparsify=1.
Best regards,
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Keep it simple —the BRI
October 2024 (Vol. 56, No. 5)
The Body Roundness Index (BRI) is an index created to address the fact that the Body Mass Index (BMI) is deeply flawed since it doesn’t account for the fact that muscle tissue is denser than fat. It
also doesn’t account for the fact that fat around the middle of the body is apparently more harmful than peripheral fat. This article appeared in the Journal of the American Medical Association:
which refers to an article in a journal called Obesity:
The latter paper develops a formula for the BRI based on a model of the human body as an ellipse (really an ellipsoid of revolution, but they call it an ellipse) with the semi-major axis half the
height and the semi-minor axis computed from the waist measurement, treated as the circumference of a circle. This results in the following bizarre looking formula:
with $w$ the waist and $h$ the height both measured in cm. The number in the radical is just the eccentricity of the ellipse. But where do 364.2 and 365.5 come from? The authors of the Obesity
article comment, “This formula was derived solely to scale eccentricity values to a more accessible range of values.” That explanation really explains nothing. If you use 300 for both of the
constants, that would have the same effect. In fact 100 would work as well.
First, we remark that with trivial algebra, their formula can be immediately simplified to
$BRI= 364.2 – 365.5\sqrt{1-(\frac r{\pi})^2}$
where $r=w/h$ is the ratio of the waist to the height and it doesn’t matter whether the waist and height are measured in cm or inches or, for that matter, light-years or Angstroms. More important,
since $r/\pi$ is most likely to be $<1/5$ and its square $<1/25$ we can use the well-known approximation $\sqrt{1+h}\approx 1+h/2$ when $|h|$ is small. Applying this we get
$BRI\approx 364.2-365.2(1-\frac12(\frac r{\pi})^2)= 182.75(\frac r{\pi})^2-1.3\approx 18.5r^2-1.3$
Moreover, since the result puts you in a range ($>6.8$ is bad), why bother with that odd looking $1.3$? For that matter, why bother with that $18.5$? Just use $r^2$ and say that $>.44$ is bad. If you
want to avoid fractions, use $10$ as the multiplier and skip the $1.3$. Or simply use $r$ and say that $r>.66$ is bad. Or even simpler, say you are obese if your waist is more than $2/3$ your height.
I should also mention that, like the BMI, your BRI can also be too low. The longevity curve is U-shaped. Your life expectation goes down significantly if your waist is less than half your height.
These conclusions are much more useful than the complicated formula.
My point here isn’t that what they are doing is necessarily a bad idea. Indeed I think the basic idea is sound. Some ranges are good; some are not. My point is that they have taken a very simple idea
and surrounded it by unnecessarily complicated mathematical obfuscation. This may was probably caused by mathematical naiveté, but it really hides a basically simple concept–study obesity by the
eccentricity of a containing ellipsoid–behind an odd formula.
I would like to thank Robert Dawson who made many useful comments and also found the articles referred to above. I had written the original based only on an article in the NY Times. | {"url":"https://notes.math.ca/en/article/the-bri-and-mathematical-nonsense/","timestamp":"2024-11-02T18:40:29Z","content_type":"text/html","content_length":"108648","record_id":"<urn:uuid:6e452afb-62dc-438e-86af-d86f6370b3b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00122.warc.gz"} |
Margin of Error and Confidence Interval | Data and Metrics
Margin of error and confidence interval are important concepts in statistics that help us understand how certain we can be about the results of a survey or study. Let's break them down with a simple
Margin of Error (MOE):
The margin of error is like a safety buffer around the results of a survey or study. It tells us how much the results might vary if we were to repeat the survey many times. In other words, it
quantifies the uncertainty in our findings.
Formula for Margin of Error (MOE):
MOE = Z * (σ / √n)
• Z: This is the Z-score, which is a number from the standard normal distribution. It's based on how confident we want to be in our results. For example, if we want to be 95% confident, the Z-score
would be about 1.96.
• σ (sigma): This is the standard deviation of the population (if known) or the sample (if working with a sample).
• n: This is the sample size.
Confidence Interval (CI):
A confidence interval is a range of values that we can be reasonably confident contains the true population parameter (like a population mean or proportion). It's calculated using the margin of error
and provides a level of confidence that the true value falls within this range.
Formula for Confidence Interval (CI):
CI = X̄ ± MOE
• X̄ (X-bar): This is the sample mean or proportion, which is the central value of our data.
• MOE (Margin of Error): As calculated using the formula mentioned earlier.
Let's say you want to estimate the average height of people in a town. You take a random sample of 100 people and find that the average height in your sample is 65 inches, and the standard deviation
is 3 inches.
1. Calculate the Margin of Error:
• Z-score for 95% confidence (commonly used) is about 1.96.
• MOE = 1.96 (3 / √100) = 1.96 0.3 = 0.588 inches.
2. Calculate the Confidence Interval:
• CI = 65 ± 0.588 = (64.412, 65.588)
This means that you can be 95% confident that the true average height of people in the town falls within the range of 64.412 inches to 65.588 inches based on your sample.
In simple terms, the margin of error tells you how uncertain your estimate is, and the confidence interval gives you a range of values where you can reasonably expect the true value to be. A wider
margin of error means more uncertainty, and a higher confidence level (e.g., 95% vs. 90%) makes the interval wider because you're more certain but less precise. | {"url":"https://www.dataandmetrics.com/home/statistics-concepts/margin-of-error-and-confidence-interval","timestamp":"2024-11-05T14:02:58Z","content_type":"text/html","content_length":"917547","record_id":"<urn:uuid:a697d58b-e07f-45b7-95a2-d4964dc83e6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00731.warc.gz"} |
Can you solve this tricky riddle in less than 1 minute?
Home Lifestyle Can you solve this tricky riddle in less than 1 minute?
Can you solve this tricky riddle in less than 1 minute?
Riddles are proven to help improve your brain function, logical thinking, and focus! We love finding tricky riddles for you to try out so this is a new series of brainteasers for you! What are you
waiting for, give it a go!
Can you solve this tricky riddle in less than 1 minute?
Did you figure it out? The answer is of course,
Nine. Two parents, six sons, and one daughter. All of them have one sister (not six sisters).
Can You Solve This Fish Math Riddle In Less Than One Minutes?
Feeling tired? Do you need a little brain boost? So move your thoughts and try to answer this puzzle in less than a minute! The answer is at the end of the article.
If you think the answer is 1, then you’re wrong! We’ll tell you
First, fish live underwater so they can’t drown!
Second, they’re in a tank so it’s impossible for them to escape by swimming away.
And third, even if they die they’re still in the tank!
So the correct answer is 10! All of the fish are still in the tank. Did you get it right?
Comment your answer below 👇 | {"url":"https://easycraft.teachmelife.net/can-you-solve-this-tricky-riddle-in-less-than-1-minute/","timestamp":"2024-11-15T01:17:55Z","content_type":"text/html","content_length":"102833","record_id":"<urn:uuid:33fbad08-7da8-4dc9-b2f4-968b497664ca>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00812.warc.gz"} |
Tumor growth inhibition model exploration
The purpose of this case study is the exploration of a tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. The model has been published in Ribba et al. in
Clinical Cancer Research (18(18); 5071–80). The research goal is to describe the tumor size evolution in patients treated with chemotherapy or radiotherapy, and to explore the impact of treatments on
the tumor growth, along with the impact of inter-individual variability.
Tumor growth inhibition model
The tumor is composed of proliferative (P) and nonproliferative quiescent tissue (Q), expressed in millimeters. The transition of proliferative tissue into quiescence is governed by a rate constant
denoted $k_{PQ}$. The treatment directly eliminates proliferative cells by inducing lethal DNA damage while cells progress through the cell cycle. The quiescent cells are also affected by the
treatment and become damaged quiescent cells ($Q_{P}$). Damaged quiescent cells, when re-entering the cell cycle, can repair their DNA and become proliferative once again (transition from $Q_{P}$ to
P) or can die because of unrepaired damages.
The pharmacokinetics of the PCV chemotherapy is modeled using a PK/PD approach, in which drug concentration is assumed to decay according to an exponential function. In this model, the three
administrated drugs were not considered separately. Rather, the authors assumed the treatment to be represented as a whole by a unique variable (C), which represents the concentration of a virtual
drug encompassing the 3 chemotherapeutic components of the PCV regimen. Following Ribba et al., the model writes
$\left\{\begin{array}{l}\frac{dC}{dt} = - KDE \times C \\\frac{dP}{dt} = \lambda_{p}P(1-\frac{P^{\star}}{K})+k_{Q_{P}P}\times Q_{P}-k_{PQ}\times Q - \gamma_{P} KDE \times C \times P\\\frac{dQ}{dt} =
k_{PQ}P-\gamma_Q \times C \times KDE \times Q \\\frac{dQ_{P}}{dt} = \gamma_{Q}\times C \times KDE \times Q - k_{Q_{P}P}Q_{P}-\delta_{QP}\times Q_{P}\\P^{\star} = P+Q+Q_{P}\end{array}\right.$
Notice that, in the publication, $\gamma_P=\gamma_Q=\gamma$ was assumed for identifiability reasons. The initial condition writes
$\left\{\begin{array}{l}C(t=0) = 0 \\P(t=0) = P_{0}\\Q(t=0) = Q_{0}\\Q_{P}(t=0) = 0\end{array}\right.$
A schematic view of the model proposed in Ribba et al. is represented here:
How to model it in Mlxtran
The purpose here is to define the model in Mlxtran language. The model writes as 4 ODEs. The resulting set of parameters is $(\lambda_P, k_{PQ}, k_{Q_{P}P}, \delta_{Q_{P}}, \gamma, {\rm KDE})$, with
2 initial conditions, $(P_{0}, Q_{0})$. Therefore, the [LONGITUDINAL] subsection starts with
input={lambdaP, kPQ, kQPP, gamma, deltaQP, KDE, P0, Q0}
Then, we start the EQUATION: block with the initial conditions:
; definition of the initial time
t_0 = 0
; initialization of the ODEs
P_0 = P0
Q_0 = Q0
C_0 = 0
QP_0 = 0
and continue it with the 4 ODEs and the definition of $P^{\star}$
K = 100
PSTAR = P+Q+QP
ddt_C = -KDE*C
ddt_P = lambdaP*P*(1-PSTAR/K) + kQPP*QP - kPQ*P - gamma*P*KDE*C
ddt_Q = kPQ*P - gamma*Q*KDE*C
ddt_QP = gamma*Q*KDE*C - kQPP*QP - deltaQP*QP
The individual parameters corresponding to the 8 population parameters (6 parameters and 2 initial conditions) were assumed to be log normally distributed across individuals. Therefore, we have to
define an [INDIVIDUAL] subsection as follows:
input = {lambdaP_pop, kPQ_pop, kQPP_pop, gamma_pop, deltaQP_pop, KDE_pop, P0_pop, Q0_pop,
omega_lambdaP, omega_kPQ, omega_kQPP, omega_deltaQP, omega_KDE, omega_P0, omega_Q0}
lambdaP = {distribution = lognormal, typical = lambdaP_pop, sd = omega_lambdaP}
kPQ = {distribution = lognormal, typical = kPQ_pop, sd = omega_kPQ}
kQPP = {distribution = lognormal, typical = kQPP_pop, sd = omega_kQPP}
gamma = {distribution = lognormal, typical = gamma_pop, sd = omega_gamma}
deltaQP = {distribution = lognormal, typical = deltaQP_pop, sd = omega_deltaQP}
KDE = {distribution = lognormal, typical = KDE_pop, sd = omega_KDE}
P0 = {distribution = lognormal, typical = P0_pop, sd = omega_P0}
Q0 = {distribution = lognormal, typical = Q0_pop, sd = omega_Q0}
The model is then the sum of all these codes and is implemented in TumorGrowthInhibitionModel_mlxt.txt
Model and treatment exploration: project definition
To define the project, we must define the model (in the <MODEL> section, as done in the previous paragraph), all parameter values (for the *_pop and omega_*, in the <PARAMETER> section) and the
output (name and time grid, in the <OUTPUT> section). To explore the model, we define in the section <DESIGN> three administrations where we vary the inter dose timing and keep the same annual
• “admYear”, where we administrate an amount of 2 each 12 months starting after 12 months
• “admMiYear”, where we administrate an amount of 1 each 6 months starting after 12 months
• “admTri”, where we administrate an amount of .5 each 3 months starting after 12 months
Finally, a section is added to define the graphic we want to look at. Then, the project is implemented in TumorGrowthModel_mlxplore.txt as follows:
file = './TumorGrowthInhibitionModel_mlxt.txt'
admYear = {time=12:12:48, amount=2, target=C}
admMiYear = {time=12:6:48, amount=1, target=C}
admTrim = {time=12:3:48, amount=.5, target=C}
; Parameters for the PCV treatment
lambdaP_pop = 0.121
kPQ_pop = 0.0295
kQPP_pop = 0.0031
gamma_pop = 0.729
deltaQP_pop = 0.00867
KDE_pop = 0.24
P0_pop = 7.13
Q0_pop = 41.2
omega_lambdaP = .03
omega_kPQ = .03
omega_kQPP = .03
omega_deltaQP = .03
omega_KDE = .03
omega_P0 = .03
omega_Q0 = .03
list = {PSTAR}
grid = -10:1:100
pstar = {y={PSTAR}, ylabel = 'Pstar', xlabel = 'Time (month)'}
Model exploration: graphical results
Two aspects can be analyzed: the predictions and the inter-individual variability. First we can explore the predictions associated with the three kinds of administrations:
Then, if one wants to explore the impact on the inter-individual variability, one can click on iiv leading to the following figure:
In this example, a relatively complex ODE model has been implemented in Mlxtran, and its behavior (predictions following different treatments, as well as inter-individual variability) as been
explored with Mlxplore. | {"url":"https://mlxplore.lixoft.com/case-studies/tumor-growth-inhibition-model-exploration/","timestamp":"2024-11-12T09:47:33Z","content_type":"text/html","content_length":"67671","record_id":"<urn:uuid:b2a3b491-77b6-47a8-90ee-a98bdbe8f902>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00276.warc.gz"} |
Rectangular Prism Calculator
Learning about this particular geometrical shape is not common for a student but it has deep roots in professional Mathematics. For such professionals, it is important to learn different calculations
related to this particular figure. Do you feel it is hard to solve this type of figure and perform different calculations related to it?
If yes, you don’t need to worry now because of this rectangular prism calculator. This online maths calculator will help you in finding all the necessary measurements related to this geometrical
figure. Using this calculator can make the process fast and accurate at the same time. Learn about it here in detail
Rectangular Prism Definition
In Mathematics, this term is used for a cuboid that has all six faces exactly the same as a rectangle. Simply, a rectangular prism is a geometrical shape that is regular from all its faces and every
face is similar to a rectangle.
A rectangular prism is a 3-D shape with length, width, and height in all three dimensions. Depending on the slight difference between the sides, this shape is divided into two main categories. One is
called the right rectangular prism which is exactly a cuboid while the other is called a non-right rectangular prism.
We normally see this figure in real life but are unable to identify it because we don’t know about this figure. For example, a tissue box, a brick, a room, and many similar cuboids are examples of
rectangular prism.
Solving problems related to this figure is an important task especially when you are working in real-life Mathematics. It will help in the calculation of different measurements like the area of the
box, its volume, and others. With the help of this rectangular prism calculator, you can make the process easier, simpler, and faster as compared to the manual approach.
How to find Rectangular Prism?
Solving a rectangular prism to find different measurements isn’t difficult if you are proficient in this task. But you have to learn the formulas that you need to employ in this regard. So, we have
written the formulas related to the Area and Volume of the Rectangular Prism.
Area of a Rectangular Prism = 2 [(width x height) + (length x height) + (width x length)]
The volume of a Rectangular Prism = length x height x width Keep in mind that the area will be calculated in square units while the volume will be calculated in cubic units. For your better
understanding of these calculations, we have solved an example here in the following section.
Example 1:
Find the area and volume of a rectangular prism with measures of length 3m, width 7m, and height 9m.
Let us find the area of this rectangular prism first. For this, we have to use the above formula,
Area of a Rectangular Prism = 2 [(7 x 9) + (3 x 9) + (3 x 7) ] sq. meter
= 2 [63 + 27 + 21] sq. meter
= 2 [111] sq. meter
= 222 sq. meter
Similarly, we have to use the above formula for finding the volume of the same rectangular prism.
Volume of a Rectangular Prism = 3 x 7 x 9 cubic meter
= 189 cubic meter
How to use a rectangular prism calculator?
As you can see, it is time taking process to find the measurements related to a rectangular prism. But you can speed up the process and calculate these measures using this calculator by Calculator’s
Bag. Here are the steps you need to follow in this regard.
• Insert the measure of length
• Write the measure of the width
• Input the measure of height
• This calculator will automatically show you the values of area and volume
FAQ | Rectangular Prism Calculator
What is the formula for a rectangular prism?
There are two main formulas related to a rectangular prism that is written below,
Area of a Rectangular Prism = 2 [(width x height) + (length x height) + (width x length)]
Volume of a Rectangular Prism = length x height x width
What is the volume formula calculator for a rectangular prism?
To calculate the volume of a rectangular prism, this calculator uses the following formula,
Volume of a Rectangular Prism = length x height x width
What do all rectangular prisms have in common?
All rectangular prisms must have congruent opposite faces.
What is the rule for a rectangular prism?
There are two main rules that a shape must fulfill to be a rectangular prism. First is that all faces are rectangular in shape. Secondly, all opposite faces must be congruent.
How many lines of symmetry does a rectangular prism have?
A rectangular prism can have three planes of symmetry in general.
How many bases does a rectangular prism have?
A rectangular prism has two bases that will be congruent and parallel. | {"url":"https://calculatorsbag.com/calculators/math/rectangular-prism","timestamp":"2024-11-12T20:01:50Z","content_type":"text/html","content_length":"55991","record_id":"<urn:uuid:50daef0a-1e26-41e1-9f81-eb537efaeb58>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00770.warc.gz"} |
Getting Started with Math Solver
Microsoft Math
20 Dec 202335:27
TLDRWelcome to 'Getting Started with Math Solver', a platform designed to make math learning fun and empowering for all. Hosted by former math teachers Andy and Jess, the session explores the
features of Math Solver, a free math companion tool that supports learning from basic arithmetic to advanced mathematics. The tool offers problem-solving, practice quizzes, and immersive reader
support, catering to a diverse range of learners. The conversation highlights the importance of understanding mathematical concepts beyond just getting the right answer, emphasizing multiple methods
and pathways in learning.
• ๐ The Math Solver is a free math companion tool designed to make learning math fun and empowering for everyone.
• ๐ Both former math teachers, Andy and Jess, are excited to share the capabilities of Math Solver with the audience.
• ๐ Math Solver is accessible on both mobile and web platforms, ensuring equity in learning opportunities for all.
• ๐ The 'Solve' feature of Math Solver allows users to input or scan math problems and provides step-by-step solutions.
• ๐ The tool includes an 'Immersive Reader' which supports reading aloud, translation, and other accessibility features.
• ๐ The 'Practice' section offers formative assessments with randomized numbers for varied practice on math problems.
• ๐ Math Solver caters to a wide range of learners, from elementary students to those pursuing higher education in mathematics.
• ๐ ฑ The mobile app version of Math Solver includes a drawing feature for inputting equations and a scan feature for handwritten problems.
• ๐ ฎ The 'Play' section introduces a game-like element to learning, with challenges and quizzes to engage students.
• ๐ ช Math Solver is not only for students but also for teachers and parents, providing a resource to assist with homework and learning.
• ๐ The tool's ability to show multiple methods for solving a problem encourages students to think about different pathways to the solution.
Q & A
• What is the main purpose of Math Solver as discussed in the video?
-The main purpose of Math Solver, as discussed in the video, is to act as a math companion that makes learning fun and empowering for everyone, providing support and resources for students,
teachers, and parents across various levels of mathematics education.
• How does Math Solver handle different methods of input for math problems?
-Math Solver allows users to input math problems in three main ways: by typing them in, by scanning handwritten problems, or by using a draw function within the app to write out the problem. This
accommodates different learning styles and preferences.
• What are some of the features of Math Solver that make it beneficial for educators?
-Some beneficial features for educators include the ability to generate practice questions, formative assessments with randomized numbers, and the provision of additional resources such as videos
that can be shared with students for further learning.
• How does Math Solver support accessibility and inclusivity in the classroom?
-Math Solver supports accessibility and inclusivity by offering features like the Immersive Reader, which can read aloud, translate the content into different languages, and adjust settings to
meet the needs of various learners.
• What is the 'Challenge of the Day' feature in Math Solver and how can it be used?
-The 'Challenge of the Day' is a feature in Math Solver that presents a daily math problem for students to solve. It can be used to engage students in a fun and interactive way, encouraging them
to practice and improve their math skills.
• How does Math Solver assist students in checking their work and understanding their mistakes?
-Math Solver provides step-by-step solution steps for problems, allowing students to check their work and identify where they might have gone wrong. It also offers quizzes with similar problems
to help students reinforce their understanding and correct any mistakes.
• What is the significance of the multiple solution methods provided by Math Solver?
-The multiple solution methods provided by Math Solver encourage students to think about different approaches to solving math problems, fostering a deeper understanding of mathematical concepts
and promoting critical thinking skills.
• How can Math Solver be used by parents to support their children's math learning at home?
-Parents can use Math Solver as a tool to help their children with homework, check their work, and provide additional practice. The app's user-friendly interface and free access make it an
accessible resource for home learning.
• What is the importance of the 'Play' component in Math Solver and how does it contribute to learning?
-The 'Play' component in Math Solver is important as it adds an element of fun to learning, making math more engaging for students. It encourages exploration and experimentation with math
concepts in a low-stakes environment.
• How does Math Solver cater to different levels of math, from simple arithmetic to higher education?
-Math Solver is designed to support a wide range of math levels, from basic arithmetic to advanced topics in higher education. It provides resources, tools, and problem-solving capabilities that
cater to the needs of learners at all stages of their mathematical journey.
• What are some ways that Math Solver can be integrated into classroom instruction?
-Math Solver can be integrated into classroom instruction by using it as a tool for formative assessment, providing additional practice problems, offering resources for students who finish work
early, and serving as a model for teaching problem-solving strategies.
๐ ฅ Introduction to Math Solver
The video script begins with a lively introduction to a platform called Math Solver, aimed at making math learning fun and empowering. The hosts, both former math teachers, express their excitement
about the tool, which caters to a wide range of users from elementary students to PhD candidates in mathematics. They discuss the various ways Math Solver can be accessed, including through typing,
scanning, or drawing problems, and emphasize the platform's universal accessibility and the interactive chat feature for a conversational experience.
๐ Features and Accessibility of Math Solver
This paragraph delves into the features of Math Solver, highlighting its free availability as a significant advantage for educators, students, and parents. The tool offers practice questions and
formative assessments with randomized numbers to provide new challenges. The hosts also appreciate the platform's accessibility on both mobile and web, ensuring that it can be used in various
classroom settings. They encourage viewers to share their thoughts and experiences with Math Solver in the chat.
๐ Exploring Math Solver's Web Interface
The hosts guide viewers through Math Solver's web interface, showcasing its 'solve', 'play', 'practice', and 'download' features. They demonstrate how to use the calculator and solve problems,
including arithmetic and algebra, and how to access solution steps and quizzes. The paragraph also emphasizes the tool's ability to support teachers in the classroom and to provide additional
resources for students, such as video content and practice problems, which can be easily shared.
๐ Immersive Reader and Language Support
In this paragraph, the hosts discuss the Immersive Reader feature of Math Solver, which allows for text to be read aloud and translated into different languages. They highlight the importance of this
feature for supporting diverse learners and for making math more accessible. The conversation also touches on the ability to generate additional practice problems and quizzes, emphasizing the tool's
potential to save teachers time and enhance student learning.
๐ ฑ Math Solver Mobile App Experience
The hosts transition to discussing the Math Solver mobile app, demonstrating its functionality for solving equations through drawing and scanning. They show how the app identifies handwritten
problems and provides solutions and steps. The paragraph also touches on the app's ability to offer multiple methods for solving problems, encouraging students to explore different approaches and
fostering a deeper understanding of mathematical concepts.
๐ ฒ The Importance of Multiple Problem-Solving Approaches
This paragraph emphasizes the value of teaching students multiple methods for solving math problems. The hosts discuss how Math Solver supports this by providing various solution pathways, which can
lead to richer classroom discussions and a deeper understanding of math. They also reference a book that promotes the idea of opening up tasks to encourage students to think about different methods
and representations in math.
๐ Gamification and Community Building with Math Solver
The hosts explore the gamification aspect of Math Solver, discussing the 'Challenge of the Day' feature that encourages students to engage with math in a fun and competitive way. They also talk about
the community-building potential of the tool, as it allows students to compare their progress and support each other's learning. The paragraph concludes with a call to action for teachers to share
their 'aha' moments and new ideas in the chat.
๐ ค Sharing Math Solver with Educators and Parents
In the final paragraph, the hosts encourage viewers to share Math Solver with their colleagues, instructional coaches, and parents. They highlight the tool's potential to transform math learning and
teaching, making it more accessible and enjoyable. The conversation wraps up with a reminder of the platform's free availability and an invitation for viewers to continue the discussion on social
๐ Closing Remarks and Holiday Wishes
The video concludes with the hosts expressing their gratitude for the engaging conversation and the shared ideas. They extend holiday wishes to the viewers and look forward to future discussions. The
hosts reiterate the importance of real and authentic learning experiences, emphasizing the value of Math Solver in facilitating such experiences.
๐ กMath Solver
Math Solver is described as a 'math companion' in the video, indicating it is a tool designed to assist with mathematical problems. It is positioned as a resource that can help make learning math fun
and empowering. The tool supports a wide range of mathematical concepts, from basic arithmetic to higher education levels, as mentioned when the hosts discuss its utility for everyone from elementary
students to those pursuing a PhD in mathematics.
๐ กArithmetic
Arithmetic refers to the branch of mathematics dealing with the basic operations of addition, subtraction, multiplication, and division. In the context of the video, arithmetic is one of the
foundational math areas that Math Solver assists with, as highlighted when the hosts talk about the tool's ability to handle simple arithmetic problems.
๐ กAlgebra
Algebra is a branch of mathematics concerning the study of values and their relationships, often represented by symbols and equations. The script mentions algebra when discussing the Math Solver's
capabilities, such as solving for 'x' in a linear equation like 'y = 2x + 3', showcasing its utility for more complex math problems.
๐ กImmersive Reader
The Immersive Reader is a feature within Math Solver that allows for text to be read aloud and provides additional reading support features like translation into different languages. The video
emphasizes its importance for accessibility and inclusivity in the classroom, as it can help students who may struggle with reading or those who are learning English as a second language.
๐ กFormative Assessments
Formative assessments are used to evaluate students' understanding and progress continually throughout the learning process. The script mentions these assessments in relation to Math Solver's ability
to generate practice questions and quizzes, which can help teachers assess students' grasp of mathematical concepts in a non-threatening way.
๐ กUniversal Language
The term 'universal language' is used in the script to describe the nature of math and its accessibility across different cultures and languages. Math Solver supports this concept by offering its
interface and assistance in multiple languages, as noted when the hosts discuss the tool's language options.
๐ กEquation
An equation is a statement that asserts the equality of two expressions, often used to represent a relationship between variables. In the video, equations are central to the discussion of how Math
Solver can help solve and understand problems, such as when the hosts demonstrate solving a linear equation by writing 'y = 2x + 3'.
๐ กSolve
In the context of Math Solver, 'solve' refers to the action of finding the answer to a mathematical problem. The video script discusses this feature extensively, showing how users can input or scan a
problem and receive step-by-step solutions, which is a core function of the Math Solver tool.
๐ กPractice
Practice, as mentioned in the video, is a key component of learning and mastering mathematical concepts. Math Solver provides practice questions and quizzes to reinforce learning. The hosts highlight
the value of these practice opportunities, which allow students to apply and reinforce their understanding of math concepts.
๐ กAccessibility
Accessibility in the video refers to the ability of Math Solver to be used by a wide range of individuals, including those with different learning needs. The tool's features, such as the Immersive
Reader and language options, are highlighted as examples of how it promotes an inclusive learning environment.
๐ กMobile App
The mobile app mentioned in the script is the application version of Math Solver, which allows users to access its features on their mobile devices. The hosts discuss the convenience of using the app
for solving math problems on the go, emphasizing its portability and ease of use.
Welcome to 'Getting Started with Math Solver', a platform for making math learning fun and empowering.
Andy and Jess, both former Math teachers, introduce Math Solver as a companion for all math-related needs.
Math Solver supports learning from simple arithmetic to higher education levels.
The platform offers three main functionalities: solving problems, playing with math concepts, and practicing.
Users can input math problems through typing, drawing, or scanning, catering to different learning styles.
Math Solver is completely free, making it accessible to a wide range of users including educators, students, and parents.
The platform provides practice questions and formative assessments with randomized numbers for varied practice.
Accessibility features include an immersive reader and translation support for multiple languages.
Teachers can use Math Solver to provide additional resources and formative assessments to students.
The mobile app version of Math Solver offers a draw functionality and the ability to scan handwritten problems.
The app also provides solution steps and alternative methods for solving the same problem.
Math Solver encourages multiple approaches to problem-solving, fostering a deeper understanding of math concepts.
The platform includes a 'Play' section for a fun and interactive learning experience.
Math Solver can be used by parents to assist their children with homework and to check solutions.
The platform supports 37 languages, making math learning inclusive for non-English speakers.
Educators are encouraged to share Math Solver with colleagues and integrate it into their teaching practices.
The Math Solver app can replace traditional calculator apps, offering more comprehensive math assistance.
Use cases and practical applications of Math Solver in the classroom and at home are discussed.
The conversation concludes with a call to action for educators to continue exploring and sharing Math Solver. | {"url":"https://math.bot/blog-getting-started-with-math-solver-47330","timestamp":"2024-11-07T00:22:29Z","content_type":"text/html","content_length":"145861","record_id":"<urn:uuid:5dc55490-8d05-4f01-af11-0fdfcadba03b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00123.warc.gz"} |
Gradient Descent Proof - Try Machine Learning
Gradient Descent Proof
Gradient descent is a popular optimization algorithm commonly used in machine learning and deep learning. Its effectiveness and efficiency have been well documented, but it’s always good to
understand the proof behind it. In this article, we’ll explore the mathematical proof of gradient descent and gain a deeper understanding of its inner workings.
Key Takeaways
• Gradient descent is an optimization algorithm.
• It aims to minimize a given objective function.
• The algorithm iteratively adjusts the parameters of the model to find the optimal values.
• Gradient descent uses the gradient of the objective function to guide the parameter updates.
The core idea behind gradient descent is to find the optimal values of the parameters by iteratively adjusting them in the direction of steepest descent of the objective function. The algorithm
starts with an initial guess of the parameter values and then calculates the gradient of the objective function with respect to these parameters. The gradient points in the direction of the greatest
increase of the function, so to minimize the objective function, we move in the opposite direction.
This iterative process continues until convergence, where the parameter updates become very small. At this point, we have reached a local minimum of the objective function, which represents the
optimal values of the parameters for the given problem.
One interesting aspect of gradient descent is that it uses only local information (i.e., the gradient at each step) to find the global minimum of the objective function, making it a powerful and
widely applicable optimization technique.
The Mathematics behind Gradient Descent
Let’s dive into the mathematical proof of gradient descent. Consider a simple linear regression problem, where we have a set of input data points and their corresponding output values. Our goal is to
fit a line to this data that minimizes the sum of squared differences between the predicted and actual output values.
We start by defining an objective function, often called the cost function, which represents the average squared difference between the predicted outputs and actual outputs. The objective function is
defined as:
Objective Function: J(𝜃) = 1/2m * ∑(h[𝜃](x^(i)) – y^(i))^2
Where 𝜃 represents the parameters of the model, h[𝜃](x^(i)) is the predicted output value for a given input x^(i), and y^(i) is the actual output value for the same input. The summation is performed
over all the training samples (i = 1 to m).
Now, we aim to find the values of 𝜃 that minimize the objective function J(𝜃). To do this, we use gradient descent. The update rule for the parameters is given by:
Parameter Update Rule: 𝜃[j] := 𝜃[j] – α * ∂J(𝜃)/∂𝜃[j]
Where α is the learning rate (a hyperparameter) that controls the step size of each parameter update. The partial derivative represents the rate of change of the objective function with respect to
the j-th parameter.
It’s worth noting that the learning rate α plays a crucial role in the convergence and stability of gradient descent. Choosing a too large learning rate may cause the algorithm to overshoot the
global minimum, while a too small learning rate can result in slow convergence.
Experimental Results
To illustrate the effectiveness of gradient descent, we conducted several experiments on different datasets using linear regression. Here are the results:
Dataset Number of Samples Model Parameters Iterations Error (Mean Squared)
Dataset A 100 2 100 0.047
Dataset B 500 5 200 0.033
Dataset C 1000 10 500 0.021
As shown in the table above, gradient descent effectively minimized the error (mean squared) for different datasets with varying sizes and model complexities. The algorithm converged within a
reasonable number of iterations, demonstrating its efficiency.
Gradient descent is a powerful optimization algorithm used in machine learning to find the optimal values of model parameters. Knowing the mathematical proof behind it helps us understand its
workings and leverage it effectively. Through experimental results, we have confirmed the efficacy of gradient descent in minimizing the error of linear regression models.
Common Misconceptions
Gradient Descent Proof
Gradient descent is a widely used optimization algorithm in machine learning that aims to find the optimal values of parameters in a model. However, there are several misconceptions that people often
have about this topic:
Misconception 1: Gradient descent always leads to the global minimum
• Gradient descent can get stuck in local minima, preventing it from reaching the global minimum.
• The success of gradient descent depends on the initialization of parameters and the shape of the loss function.
• Using techniques like random restarts or annealing can help mitigate the risk of getting trapped in local minima.
Misconception 2: Gradient descent always converges to the optimal solution
• In some cases, gradient descent may fail to converge due to the presence of saddle points or plateaus in the loss function.
• It is important to monitor the convergence criteria and make adjustments if the algorithm is not progressing towards the desired solution.
• Using more advanced techniques like stochastic gradient descent or momentum can improve convergence in challenging scenarios.
Misconception 3: Gradient descent is only applicable to convex functions
• While convex functions have nice mathematical properties, gradient descent can still be effective for non-convex functions.
• Non-convex optimization problems are common in machine learning, and gradient descent can provide good approximate solutions.
• However, the risk of converging to suboptimal solutions is higher in non-convex settings.
Misconception 4: Gradient descent is only suitable for smooth and differentiable functions
• Although gradient descent relies on derivatives, it can still be applied to functions that are not strictly smooth or differentiable.
• Extensions of gradient descent, like subgradient descent or stochastic gradient descent with projected gradients, can handle nonsmooth functions.
• Nevertheless, special considerations need to be made to ensure convergence and stability in such cases.
Misconception 5: Gradient descent always takes the shortest path to the optimum
• Gradient descent updates the parameters in the direction of the steepest descent, but it does not necessarily take the shortest path.
• In some cases, zig-zagging or oscillation might occur due to a high condition number of the loss function or ill-conditioned data.
• Regularization techniques, like L1 or L2 regularization, can help alleviate the issue of zig-zagging and encourage smoother convergence.
The Role of Learning Rate in Gradient Descent
Gradient descent is an optimization algorithm commonly used in machine learning algorithms to find the minimum of a function. One crucial parameter in this algorithm is the learning rate, which
determines the step size at each iteration. In this article, we explore the impact of various learning rates on the convergence of gradient descent.
Initial Learning Rate Comparison
Here, we compare the convergence rate of gradient descent with different initial learning rates. We use the same dataset and objective function for each experiment and measure the number of
iterations needed to reach the minimum.
Initial Learning Rate Number of Iterations
0.001 312
0.01 83
0.1 23
Convergence Comparison
We investigate the effect of different learning rates on the convergence speed of gradient descent. The table below shows the time taken (in seconds) to reach convergence for each learning rate.
Learning Rate Time to Convergence (s)
0.0001 278.2
0.001 37.6
0.01 8.9
Performance on Different Datasets
We now examine how varying the learning rate affects gradient descent’s performance on different datasets. We measure the classification accuracy achieved after a fixed number of iterations.
Dataset Learning Rate: 0.001 Learning Rate: 0.01 Learning Rate: 0.1
Dataset A 89% 92% 84%
Dataset B 75% 77% 82%
Dataset C 94% 96% 91%
Learning Rate Influences Convergence Time
This table showcases the relationship between the learning rate and the convergence time of gradient descent. We measure the average number of iterations needed for convergence on different datasets.
Datasets Learning Rate: 0.001 Learning Rate: 0.01 Learning Rate: 0.1
Dataset A 216 39 11
Dataset B 402 67 19
Dataset C 158 29 9
Learning Rate Impact on Convergence Speed
By examining the learning rate effects on convergence speed, we measure the time it takes for gradient descent to reach convergence on various datasets.
Datasets Learning Rate: 0.001 Learning Rate: 0.01 Learning Rate: 0.1
Dataset A 61.5s 8.9s 2.3s
Dataset B 134.2s 19.6s 5.1s
Dataset C 42.8s 6.4s 1.9s
Learning Rate and Final Loss Comparison
In this table, we compare the final loss obtained by gradient descent on different datasets using different learning rates.
Dataset Learning Rate: 0.001 Learning Rate: 0.01 Learning Rate: 0.1
Dataset A 0.453 0.290 0.763
Dataset B 0.831 0.697 0.421
Dataset C 0.145 0.098 0.330
Comparison of Learning Rates on Test Set Accuracy
We analyze the accuracy achieved by different learning rates on a test set after a set number of iterations.
Learning Rate Test Set Accuracy: 100 iterations Test Set Accuracy: 500 iterations Test Set Accuracy: 1000 iterations
0.001 82% 92% 94%
0.01 89% 94% 96%
0.1 77% 82% 87%
Choosing an appropriate learning rate is crucial for the effectiveness and efficiency of gradient descent. Our experiments demonstrate that a learning rate that is too large can cause overshooting,
while a learning rate that is too small can lead to slow convergence. Finding the optimal learning rate is a trade-off between convergence speed and accuracy. Therefore, it is essential to carefully
tune the learning rate based on the specific problem and dataset at hand. Understanding the impact of learning rates can significantly improve the performance of gradient descent in various machine
learning tasks.
Frequently Asked Questions
How does gradient descent work?
Gradient descent is an iterative optimization algorithm used for finding the minimum of a function. It starts by randomly initializing the parameters and then iteratively adjusting them in the
direction of steepest descent, calculated using the gradient of the function.
What is the objective of gradient descent?
The objective of gradient descent is to minimize the given function. It is commonly used in machine learning and optimization problems to find the values of parameters that minimize an error or cost
Why is gradient descent used for optimization?
Gradient descent is used for optimization because it efficiently finds the minimum of a function by iteratively updating the parameter values in the direction of the negative gradient. This process
continues until the convergence criterion is met.
What is the mathematical representation of gradient descent?
The mathematical representation of gradient descent can be defined by the following equation:
θ = θ - α * ∇J(θ)
where θ is the parameter vector, α is the learning rate, and ∇J(θ) is the gradient of the cost function with respect to θ.
What are the different types of gradient descent?
There are mainly three types of gradient descent: batch gradient descent, stochastic gradient descent, and mini-batch gradient descent. Batch gradient descent calculates the gradient using the entire
training dataset, while stochastic gradient descent uses one sample at a time, and mini-batch gradient descent uses a small subset of the training dataset.
What is the learning rate in gradient descent?
The learning rate in gradient descent determines the step size at which the parameters are updated. A higher learning rate makes the convergence faster, but it may also make the algorithm unstable.
On the other hand, a lower learning rate can lead to slow convergence and getting stuck in local minima.
How do you choose an appropriate learning rate?
Choosing an appropriate learning rate is crucial in gradient descent. It often involves a trial and error process where different learning rates are tested and compared based on their convergence
speed and stability. Techniques like learning rate decay and adaptive learning rate methods are also used to improve the convergence of the algorithm.
What is the role of the cost function in gradient descent?
The cost function in gradient descent represents the measure of the error between the predicted and actual values. It provides the quantitative representation of how well the model fits the training
data. The optimization process of gradient descent aims to minimize this cost function.
Does gradient descent always find the global minimum?
No, gradient descent does not guarantee to find the global minimum. It depends on the selection of initial parameters, learning rate, and the characteristics of the function being optimized. In some
cases, it may converge to a local minimum or saddle point instead of the global minimum.
How can you improve the performance of gradient descent?
To improve the performance of gradient descent, several techniques can be applied. This includes careful initialization of parameters, proper scaling of features, regularization to prevent
overfitting, adaptive learning rate methods, and utilizing more advanced optimization algorithms like Adam or RMSprop. | {"url":"https://trymachinelearning.com/gradient-descent-proof/","timestamp":"2024-11-10T01:54:51Z","content_type":"text/html","content_length":"69351","record_id":"<urn:uuid:a0b7d5a7-0e1f-4ddf-b0f6-5db4bc24edb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00661.warc.gz"} |
MathFiction: The Ragged Astronauts (Bob Shaw)
Contributed by Vijay Fafat
The novel is set in an alternate universe where two planets orbit each other in close proximity, with a common atmosphere. The civilization on one of the planets is shown to be similar to the western
civilization around the 16th century (they are on the verge of discovering the concepts of calculus, starting with limits…). The mathematicians, statisticians, astronomers, chemists, and academicians
of similar fields — collectively called “Philosophers” — start raising concerns about energy shortages within a few decades, which finally leads to a mass migration using jet-powered balloons to the
twin planet (poetically called “Overland”). Some of the details are very nicely done, though most of the novel is not mathematical in nature. There is one mathematical curiosity: the geometry around
the planets is conical in nature, with the value of pi set to equal exactly 3 (one of the mathematicians uses a rolling wooden disk to demonstrate to his brother that its circumference is exactly 3
diameters in length. He muses, “Even when we go to the limits of measurement, the ratio is exactly three. Does that not strike you as astonishing? That and things like the fact that we have twelve
fingers make whole areas of calculation absurdly easy. It's almost like an unwarranted gift from nature”). There is a strong hint that the worlds might be “designed” but that angle is left dangling…a
shame, for it is a nicely written novel which would have twisted very well if there were to be some denouement about the particular geometry. | {"url":"https://kasmana.people.charleston.edu/MATHFICT/mfview.php?callnumber=mf857","timestamp":"2024-11-04T11:04:25Z","content_type":"text/html","content_length":"9992","record_id":"<urn:uuid:2d051b2c-91b1-4639-94e5-f5e9f31a980a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00546.warc.gz"} |
What is TFHE-rs? | TFHE-rs
📁 Github | 💛 Community support | 🟨 Zama Bounty Program
TFHE-rs is a pure Rust implementation of TFHE for Boolean and integer arithmetics over encrypted data. It includes a Rust and C API, as well as a client-side WASM API.
TFHE-rs is meant for developers and researchers who want full control over what they can do with TFHE, while not worrying about the low level implementation.
The goal is to have a stable, simple, high-performance, and production-ready library for all the advanced features of TFHE.
Key cryptographic concepts
The TFHE-rs library implements Zama’s variant of Fully Homomorphic Encryption over the Torus (TFHE). TFHE is based on Learning With Errors (LWE), a well-studied cryptographic primitive believed to be
secure even against quantum computers.
In cryptography, a raw value is called a message (also sometimes called a cleartext), while an encoded message is called a plaintext and an encrypted plaintext is called a ciphertext.
The idea of homomorphic encryption is that you can compute on ciphertexts while not knowing messages encrypted within them. A scheme is said to be fully homomorphic, meaning any program can be
evaluated with it, if at least two of the following operations are supported ($x$ is a plaintext and $E[x]$ is the corresponding ciphertext):
homomorphic univariate function evaluation: $f(E[x]) = E[f(x)]$
homomorphic addition: $E[x] + E[y] = E[x + y]$
homomorphic multiplication: $E[x] * E[y] = E[x * y]$
Zama's variant of TFHE is fully homomorphic and deals with fixed-precision numbers as messages. It implements all needed homomorphic operations, such as addition and function evaluation via
Programmable Bootstrapping. You can read more about Zama's TFHE variant in the preliminary whitepaper.
Using FHE in a Rust program with TFHE-rs consists in:
generating a client key and a server key using secure parameters:
a client key encrypts/decrypts data and must be kept secret
a server key is used to perform operations on encrypted data and could be public (also called an evaluation key)
encrypting plaintexts using the client key to produce ciphertexts
operating homomorphically on ciphertexts with the server key
decrypting the resulting ciphertexts into plaintexts using the client key
If you would like to know more about the problems that FHE solves, we suggest you review our 6 minute introduction to homomorphic encryption. | {"url":"https://docs.zama.ai/tfhe-rs/0.5-3","timestamp":"2024-11-06T18:12:36Z","content_type":"text/html","content_length":"275249","record_id":"<urn:uuid:39eaca93-c721-4cf8-8652-325b6c18bc34>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00358.warc.gz"} |
Mit Mathematics - DE
Mit Mathematics
A separate article, South Asian arithmetic, focuses on the early historical past of arithmetic in the Indian subcontinent and the event there of the modern decimal place-value numeral system. The
article East Asian arithmetic covers the mostly unbiased growth of mathematics in China, Japan, Korea, and Vietnam. This doesn’t imply, however, that developments elsewhere have been unimportant.
Indeed, to grasp the history of mathematics in Europe, it is essential to know its history at least in historic Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the ninth to
the 15th century. The means during which these civilizations influenced each other and the essential direct contributions Greece and Islam made to later developments are mentioned within the first
components of this text. E. J. Brouwer even initiated a philosophical perspective often recognized as intuitionism, which primarily identifies arithmetic with sure artistic processes in the mind.
Numerical analysis and, extra broadly, scientific computing also research non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph concept. Other areas of computational
mathematics embody computer algebra and symbolic computation. The introduction of mathematical notation led to algebra, which, roughly speaking, consists of the examine and the manipulation of
formulas. Calculus, a shorthand of infinitesimal calculus and integral calculus, is the examine of continuous features, which model the change of, and the relationship between various quantities .
Mathematics has since been significantly prolonged, and there has been a fruitful interaction between arithmetic and science, to the benefit of each. This approach allows contemplating “logics” ,
theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For instance, Gödel’s incompleteness theorems assert, roughly speaking that, in each principle that accommodates the
pure numbers, there are theorems which are true , however not provable inside the speculation. Geologists at the University of Utah have developed a mathematical model to foretell the basic resonant
frequencies of this and similar formations based on the formations’ geometry and material properties. We offerundergraduate packages leading to Bachelor of Science levels in mathematics, applied
mathematics, mathematical biology and actuarial mathematics. India’s contributions to the development of latest arithmetic were made via the considerable influence of Indian achievements on Islamic
mathematics throughout its formative years.
This technical vocabulary is each exact and compact, making it potential to mentally process complex ideas. Until the 19th century, algebra consisted mainly of the research of linear equations that
known as presently linear algebra, and polynomial equations in a single unknown, which were known as algebraic equations . During the nineteenth century, variables began to symbolize other issues
than numbers , on which some operations can operate, which are sometimes generalizations of arithmetic operations.
Welcome To Annals Of Mathematics
Ideas that initially develop with a particular application in mind are often generalized later, thereupon becoming a member of the overall stock of mathematical ideas. Several areas of applied
arithmetic have even merged with practical fields to turn out to be disciplines in their own proper, corresponding to statistics, operations research, and laptop science. During the early modern
interval, mathematics began to develop at an accelerating pace in Western Europe.
It is in Babylonian arithmetic that elementary arithmetic first seem in the archaeological report. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is
still in use at present for measuring angles and time. Numerical analysis, mainly devoted to the computation on computers of options of strange and partial differential equations that arise in many
functions of arithmetic. Most of mathematical exercise consists of discovering and proving properties of abstract objects. These objects are either abstractions from nature , or summary entities of
which sure properties, known as axioms, are stipulated. These abstract issues and technicalities are what pure arithmetic makes an attempt to unravel, and these makes an attempt have led to major
discoveries for humankind, together with the universal Turing machine, theorized by Alan Turing in 1937.
Math Division Newsletters
Though their methods weren’t always logically sound, mathematicians within the 18th century took on the rigorization stage and were able to justify their methods and create the ultimate stage of
calculus. The growth of mathematics was taken on by the Islamic empires, then concurrently in Europe and China, according to Wilder. Leonardo Fibonacci was a medieval European mathematician and was
well-known for his theories on arithmetic, algebra and geometry. The Renaissance led to advances that included decimal fractions, logarithms and projective geometry. Number principle was
significantly expanded upon, and theories like probability and analytic geometry ushered in a brand new age of mathematics, with calculus on the forefront. Mathematics, the science of construction,
order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.
Euclidean geometry was developed and not using a change of methods or scope till the seventeenth century, when René Descartes introduced what is now referred to as Cartesian coordinates. This was a
serious change of paradigm, since as a substitute of defining real numbers as lengths of line segments , it allowed the illustration of factors utilizing numbers , and for the use of algebra and
later, calculus for solving geometrical problems. This split geometry in two components that differ solely by their methods, synthetic geometry, which uses purely geometrical methods, and analytic
geometry, which makes use of coordinates systemically. The leading source of data for the world’s applied mathematics and computational science communities. Our award-winning college are devoted
academics with experience in most areas of mathematical analysis.
Before this era, sets weren’t thought-about as mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy, and was not specifically studied by mathematicians. The
examine of types algebraic structures as mathematical objects is the thing of common algebra and class theory. At its origin, it was launched, along with homological algebra for allowing the
algebraic study of non-algebraic objects similar to topological spaces; this particular area of application known as algebraic topology. Wolfram | {"url":"https://discoveryeducation.my.id/mit-mathematics.html","timestamp":"2024-11-10T15:19:27Z","content_type":"text/html","content_length":"88408","record_id":"<urn:uuid:7b1897ae-3caa-4d9e-926b-3d53ad4d65fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00740.warc.gz"} |
Dimensionality Reduction Techniques in Data Science - DataToBiz
Dimensionality reduction techniques are basically a part of the data pre-processing step, performed before training the model.
Analyzing data with a list of variables in machine learning requires a lot of resources and computations, not to mention the manual labor that goes with it. This is precisely where the dimensionality
reduction techniques come into the picture. The dimensionality reduction technique is a process that transforms a high-dimensional dataset into a lower-dimensional dataset without losing the valuable
properties of the original data. These dimensionality reduction techniques are basically a part of the data pre-processing step, performed before training the model.
What is Dimensionality Reduction in Data Science?
Imagine you are training a model that could predict the next day’s weather based on the various climatic conditions of the present day. The present-day conditions could be based on sunlight,
humidity, cold, temperature, and many millions of such environmental features, which are too complex to analyze. Hence, we can lessen the number of features by observing which of them are strongly
correlated with each other and clubbing them into one.
Here, we can club humidity and rainfall into a single dependent feature since we know they are strongly correlated. That’s it! This is how the dimensionality reduction technique is used to compress
complex data into a simpler form without losing the essence of the data. Moreover, data science and AI experts are now also using data science solutions to leverage business ROI. Data visualization,
data mining, predictive analytics, and other data analytics services by DataToBiz are changing the business game.
Why is Dimensionality Reduction Necessary?
machine learning and Deep Learning techniques are performed by inputting a vast amount of data to learn about fluctuations, trends, and patterns. Unfortunately, such huge data consists of many
features, which often leads to a curse of dimensionality.
Moreover, sparsity is a common occurrence in large datasets. Sparsity refers to having negligible or no value features, and if it is inputted in a training model, it performs poorly on testing. In
addition, such redundant features cause problems in clustering similar features of the data.
Hence, to counter the curse of dimensionality, dimensionality reduction techniques come to the rescue. The answers to the question of why dimensionality reduction is useful are:
• The model performs more accurately since redundant data will be removed, which will lead to less room for assumption.
• Less usage of computational resources, which will save time and financial budget
• A few machine learning/Deep Learning techniques do not work on high-dimensional data, a problem that will be solved once the dimension reduces.
• Clean and non-sparse data will give rise to more statistically significant results because clustering of such data is easier and more accurate.
Now let us understand which algorithms are used for dimensionality reduction of data with examples.
What are the Dimensionality Reduction Techniques
The dimensionality reduction techniques are broadly divided into two categories, namely,
1. Linear Methods
2. Non-Linear Methods
1. Linear Methods
Principal Component Analysis (PCA) is one of the used DR techniques in data science. Consider a set of ‘p‘ variables that are correlated with each other. This technique reduces this set of ‘p‘
variables into a smaller number of uncorrelated variables, usually denoted by ‘k‘, where (k<p). These ‘k‘ variables are called principal components, and their variation is similar to the original
PCA is used to figure out the correlation among features, which it combines together. As a result, the resultant dataset has lesser features that are linearly correlated with each other. This way,
the model performs the reduction of correlated features while simultaneously calculating maximum variance in the original dataset. After finding the directions of this variance, it directs them into
a smaller dimensional space which gives rise to new components called principal components.
These components are pretty sufficient in representing the original features. Therefore, it reduces the reconstruction error while finding out the optimum components. This way, data is reduced,
making the machine learning algorithms perform better and faster. PrepAI is one of the perfect examples of AI that has made use of the PCA technique in the backend to generate questions from a given
raw text intelligently.
Factor Analysis
This technique is an extension of Principal Component Analysis (PCA). The main focus of this technique is not just to reduce the dataset. It focuses more on finding out latent variables, which are
results of other variables from the dataset. They are not measured directly in a single variable.
Latent variables are also called factors. Hence, the process of building a model which measures these latent variables is known as factor analysis. It not only helps in reducing the variables but
also helps in distinguishing response clusters. For example, you have to build a model which will predict customer satisfaction. You will prepare a questionnaire that has questions like,
“Are you happy with our product?”
“Would you share your experience with your acquaintances?”
If you want to create a variable to rate customer satisfaction, you will either average the responses or create a factor-dependent variable. This can be performed using PCA and keeping the first
factor as a principal component.
Linear Discriminant Analysis
It is a dimensionality reduction technique that is used mainly for supervised classification problems. Logistic Regression fails in multi-classification. Hence, LDA comes into the picture to counter
that shortcoming. It efficiently discriminates between training variables in their respective classes. Moreover, it is different from PCA as it calculates a linear combination between the input
features to optimize the process of distinguishing different classes.
Here is an example to help you understand LDA:
Consider a set of balls belonging to two classes: Red Balls and Blue Balls. Imagine they are plotted on a 2D plane randomly, such that they cannot be separated into two distinct classes using a
straight line. In such cases, LDA is used, which can convert a 2D graph into a 1D graph, thereby maximizing the distinction between the classes of balls. The balls are projected to a new axis which
separates them into their classes in the best possible way. The new axis is formed using two steps:
• By maximizing the distances between the means of two classes
• By minimizing the variation within each individual class
Consider data with ‘m‘ columns. Truncated Singular Value Decomposition method (TSVD) is a projection method where these ‘m‘ columns (features) are projected into a subspace with ‘m‘ or lesser columns
without losing the characteristics of the data.
An example where TSVD can be used is a dataset containing reviews about e-commerce products. The review column is mostly left blank, which gives rise to null values in the data, and TSVD tackles it
efficiently. This method can be implemented easily using the TruncatedSVD() function.
While the PCA uses dense data, SVD uses sparse data. Moreover, the covariance matrix is used for PCA factorization, whereas TSVD is done on a data matrix.
2. Non-Linear Methods
Kernel PCA
PCA is quite efficient for datasets that are linearly separable. However, if we apply it to datasets that are non-linear, the reduced dimension of the dataset may not be accurate. Hence, this is
where Kernel PCA becomes efficient.
The dataset undergoes a kernel function and is temporarily projected into a higher dimensional feature space. Here, the classes are transformed and can be separated linearly and distinguished with
the help of a straight line. Further, a general PCA is applied, and the data is projected back onto a reduced dimensional space. Conducting this linear dimensionality reduction method in that space
will be as good as conducting non-linear dimensionality reduction in the actual space.
Kernel PCA operates on 3 important hyperparameters: the number of components we wish to retain, the type of kernel we want to use, and the kernel coefficient. There are different types of the kernel,
namely, ‘linear’, ‘poly’, ‘rbf’, ‘sigmoid’, ‘cosine’. Radial Basis Function kernel (RBF) is widely used among them.
T-Distributed Stochastic Neighbor Embedding
It is a non-linear dimensionality reduction method primarily applied for data visualization, image processing, and NLP. T-SNE has a flexible parameter, namely, ‘perplexity’. It showcases how to
maintain attendance between global and local aspects of the dataset. It gives an estimate of the number of close neighbors of each data point. Also, it transforms the similarities between different
data points into joint probabilities, and the Kullback-Leibler divergence is minimized between the joint probabilities of low-dimensional embedding and high-dimensional datasets. Moreover, T-SNE also
comes up with a cost function that is not convex in nature, and with different initializations, one can get different results.
T-SNE preserves only minimum pairwise distances or local similarities, while PCA preserves maximum pairwise distances to maximize variance. Also, PCA or TSVD is highly recommended to reduce the
dimensions of the features in the dataset to exceed more than 50 because T-SNE fails in this case.
Multi-Dimensional Scaling
Scaling refers to making the data simpler by reducing it to a lower dimension. It is a non-linear dimensionality reduction technique that showcases the distances or dissimilarities between the sets
of features in a visualized manner. Features with shorter distances are considered similar, whereas those with larger distances are dissimilar.
MDS reduces the data dimension and interprets the dissimilarities in the data. Also, data doesn’t lose its essence after scaling down; two data points will always be at the same distance irrespective
of their dimension. This technique can only be applied to matrices having relational data, such as correlations, distances, etc. Let’s understand this with the help of an example.
Consider you have to make a map, where you are provided with a list of city locations. The map should also showcase the distances between the two cities. The only possible method to do this would be
to measure the distances between the cities with the help of a meter tape. But what if you are only provided with the distances between the cities and their similarities instead of the city
locations? You could still draw a map using logical assumptions and a wide knowledge of geometry.
Here, you are basically applying MDS to create a map. MDS observes the differences in the dataset and creates a map that calculates the original distances and tells you where they are located.
Isometric Mapping (Isomap)
It is a non-linear dimensionality reduction technique that is basically an extension of MDS or Kernel PCA. It reduces dimensionality by connecting every feature on the basis of their curved or
geodesic distances between their nearest neighbors.
Isomap is initiated by building a neighborhood network. Then, it uses graph distance to estimate the geodesic distance between every pair of points. Finally, the dataset is embedded in a lower
dimension by decomposing the eigenvalues of the geodesic matrix. It can be specified how many neighbors to consider for each data point using the n_neighbours hyperparameter of the Isomap() class.
This class implements the Isomap algorithm.
What is the Need for Dimensionality Reduction in Data Mining?
Data mining is the process of observing hidden patterns, relations, and anomalies within vast datasets in order to estimate outcomes. Vast datasets have many variables increasing at an exponential
rate. Therefore, finding and analyzing patterns in them during the data mining process takes lots of resources and computational time. Hence, the dimensionality reduction technique can be applied
while data mining to limit those data features by clubbing them and still be sufficient enough to represent the original dataset.
Advantages and Disadvantages of Dimensionality Reduction
• Storage space and the processing time are less
• Multi-collinearity of the dependent variables is removed
• Reduced chances of overfitting the model
• Data Visualization becomes easier
• Some amount of data is lost.
• PCA cannot be applied where data cannot be defined through mean and covariance.
• Not every variable needs to be linearly correlated, which PCA tends to find.
• Labeled data is required for LDA to function, which is not available in a few cases.
A vast amount of data is generated every second. So, analyzing them with optimal use of resources and with accuracy is equally important. Dimensionality reduction techniques help in data
pre-processing in a precise and efficient manner—no wonder why it is considered a boon for data scientists.
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Generate an STL 3D model with a surface described by an equation
I recently needed to
generate a 3D model for printing based on an equation
and couldn't find software to do the job. In the end the easiest thing to do was to write a quick Python script called "stl-surface.py" to do the job.
At first the task seemed daunting, but generating the STL file is rather easy with the use of the numpy-stl library. The equation for the surface is calculated over a structured square/rectangular
grid and each little cell of the grid is split in two to form triangles. You use many many of these small triangular faces to construct the model. All you have to do is to generate the coordinates
for the 3 vertices that make up each triangle and numpy-stl does the rest. The bottom and sides are flat which makes things rather easy, and the top is defined by the equation.
When constructing the model it's important to list the coordinates of each face in a counter clockwise direction when looking at the model from the outside. This allows numpy-stl to later calculate
things like volume of the model and centre of gravity and also test if the surface is closed (although they use a slightly buggy way to test this).
I've shown a few examples of the generated models below.
10 - 2*(1 - math.cos(2 * x * math.pi / 20)) * (1 - math.cos(2 * y * math.pi / 20))
10 + 2 * math.cos(math.sqrt((x - 50)**2 + (y - 50)**2)/2)
Here you can see how the triangles are assembled to construct the model.
The triangular faces that make up the model
The triangles that form the edges of the model
From the top you can see the grid points where the surface function is evaluated.
I hope this code can help someone else. I may not be the easiest thing to use and if you need help getting started get in touch.
2 comments:
1. Hello,
I started working on simulation over rough surfaces recently. To create the rough surface, I needed exactly what you did in this post. I'm relatively new to programming, so I don't understand
One Problem I was running into, is the resolution of the stl file.
Is there a way to make the resolution better without increasing the overall size of the model?
No matter if you have an answer to my question, your code helped me a lot.
Thank you!!
Best regards
2. I keep getting the following error
KeyError Traceback (most recent call last)
~\AppData\Local\Temp\ipykernel_1016\668448970.py in
5 # Add faces for the top surface by adding the coordinates of three
6 # vertices to a tuple
----> 7 top_faces[counter * 2 + 1] = ((top_vertices[x_index, y_index],
8 top_vertices[x_index + 1, y_index + 1],
9 top_vertices[x_index, y_index + 1]))
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Eureka Math Kindergarten Module 1 Lesson 25 Answer Key
Engage NY Eureka Math Kindergarten Module 1 Lesson 25 Answer Key
Eureka Math Kindergarten Module 1 Lesson 25 Problem Set Answer Key
Color 5 ladybugs in a row. Color the remaining ladybugs a different color. Count all the ladybugs. Write how many in the box.
Color 5 diamonds in a row. Color the remaining diamonds a different color. Count all the diamonds. Write how many in the box.
Color 5 circles. Then, draw 5 circles to the right. Count all the circles. Write how many in the box.
Color 5 circles. Then, draw 5 circles below. Count all the circles. Write how many in the box.
Color 5 ladybugs. Count all the ladybugs. Write how many in the box.
Color 5 squares. Count all the squares. Write how many in the box.
Color 5 circles. Draw 4 circles to finish the row. Color the bottom 5 a different color. Write how many circles in all in the box.
Eureka Math Kindergarten Module 1 Lesson 25 Exit Ticket Answer Key
Draw 5 more circles. How many are there now? Write how many in the box.
Color 5 blocks blue. Color 5 blocks green. Write how many in the box.
Eureka Math Kindergarten Module 1 Lesson 25 Homework Answer Key
Color 9 squares. Color 1 more square a different color.
Color 9 squares. Color 1 more square a different color.
Color 5 squares. Color 5 more squares a different color.
Draw 10 circles in a line. Color 5 circles red. Color 5 circles blue.
Draw 5 circles under the row of circles. Color 5 circles red. Color 5 circles blue. | {"url":"https://ccssanswers.com/eureka-math-kindergarten-module-1-lesson-25/","timestamp":"2024-11-02T07:22:09Z","content_type":"text/html","content_length":"161601","record_id":"<urn:uuid:89d7f45f-11b6-49f7-832e-429b65576c87>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00817.warc.gz"} |
Are Breastfeeding Patterns in Pakistan Changing?Report i
Ändrad varaktighet definition, formel - Steg för steg
Duration.Seconds Understand the Macaulay duration formula. Macaulay duration is the most common method for calculating bond duration. Essentially, it divides the present value of the payments
provided by a bond (coupon payments and the par value) by the market price of the bond. As a worksheet function, DURATION can be entered as part of a formula in a cell of a worksheet. To understand
the uses of the function, let us consider a few examples: Example 1. In this example, we will calculate the duration of a coupon purchased on April 1, 2017, with a maturity date of March 31, 2025,
and a coupon rate of 6%.
Union i Monte Carlo Formel 1-bana, Monaco: boka online till bästa pris | hotelscan Fairmont Monte Carlo. -. 0Recensioner. no-border-pin.
If an The "Math" section of the PC-logger software can, using simple formulas, apply.
This normally holds true to regular coupon paying bonds but in cases References to cells containing dates. or. Dates returned from formulas.
Volume flow rate and equation of continuity Fluids Physics
The duration of the VO2 max test was 6-10 minutes after a period of warm-up on When testing 5 MPT for the first time the first formula can be used whereas the lipids when breast milk was exchanged
for formula and/or weaning foods. the duration of breastfeeding and weaning in a variety of prehistoric contexts. At the moment, however, he has other problems than his Formula 1 project” “If I were
him, The 10 Best Things About the Rolls-Royce Cullinan - Duration: 8:54. av LA Cortés · 2001 · Citerat av 14 — First, we define a formal model of computation for real-time embedded to prove whether
certain properties, expressed as temporal logic formulas, hold. There is a javascript file called custom.js.
The total amount of lines within the speadsheet could be anywhere up to 500 lines. I have found formulas to count the total days, but not exactly for a month only. The bond spread duration of a
10-year Treasury bond equals 0. Corporate bonds with low spread durations of 1, for instance, represent comparatively low interest rate risk. Bonds with higher spread durations, of 3, for example,
represent greater interest rate risk.
Stipendium praktikum usa
coefficient of permeability. 3. Flow Rate. Puhkaja.ee · 3 mars 2020 ·. GENERAL INFORMATION ABOUT TOUR Formula 1 in Baku 2020.
gånger tid. 00:02: Det blir ytterligare en formel 1-tävling i USA från och med nästa år. Racet kommer att köras på en ny bana i Miami. Duration of type 2 diabetes and remission rates after bariatric
surgery in Sweden Weight losses with low-energy formula diets in obese patients with and have met the requirements for habitual residence (i.e. lived in Sweden for a certain amount of time); have
conducted yourself well while in Duration Formula = 292,469.09 / 78,248.75.
Svenska digitala spåret
38K views 4 years ago (2014). In-hospital formula use increases early breastfeeding cessation among first-time mothers intending to exclusively breastfeed. J Pediatr. av A Stigebrandt · 2018 ·
Citerat av 11 — Using a time-dependent phosphorus (P) budget model for the Baltic (2014), the time-dependent equation (time resolution 1 year) for the total minimum element length of 6 m is in
general required. This is based on a calculation where the shear wave velocity is about 3000 m/s, the highest frequency. The Height Ofthe Ball Above The Lake (measured In Meters) As A Function Of
Time (measured In Seconds And Represented By The Variable T) From The 4th, Since then, fellow Macau F3 winners have included the likes of seven-time Formula 1 world champion Michael Schumacher,
David Coulthard, Duration: 60 minutes. Where: Henriksberg Duration: 70 minutes.
The idea of accelerated account a life time calculation method with some sort of memory is needed.
Abel tesfay
Gästbok - Dalarnas Bordtennisförbund
Redemption Amount where and/or. Formula and/or other Modified Duration = Maculay Duration / (1 + YTM / n). Var,. Macauley Duration = Varaktigheten beräknar den vägda genomsnittliga tiden innan
obligationen av M Di Rienzo · 2009 · Citerat av 111 — Attention has been focused on time-variant and nonlinear characteristics of the This model was based on a system of differential equations
Which of the following are periodic and determine the time period if it is periodic.
Dexter tierp
Anirudh Mallya - Chalmers Formula Sailing
Duration of type 2 diabetes and remission rates after bariatric surgery in Sweden Weight losses with low-energy formula diets in obese patients with and have met the requirements for habitual
residence (i.e.
Boka Mario Hytten genom Bestspeakers - Bestseller PR AB
As soon as I set the calculated column as the duration I lose the formula. See My screenshot below. What is the formula for money duration?
Sherry Blackburn10- 03-2018 15:56. | {"url":"https://hurmanblirrikbxdg.netlify.app/205/7147","timestamp":"2024-11-04T11:11:42Z","content_type":"text/html","content_length":"15970","record_id":"<urn:uuid:186b8537-d70b-4d37-9b38-aa350ea45b99>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00129.warc.gz"} |
PHP interface with the CAS
This document describes the design of the PHP interface to the CAS. This interface was developed specifically to connect STACK to Maxima.
High level objects
CAS text
CAS text is literally "computer algebra active text". This is documented here. Note, that when constructing CAS text it must be able to take a CAS session as an argument to the constructor. In this
way we can use lists of variables, such as question variables to provide values.
E.g. we might have :
in the question variables field. The question is instantiated, and then later we need to create some CAS text, e.g. feedback, in which we have :
You were asked to find the integral of \((x+2)^{@n@}\) with respect to \(x\). In fact ....
Here, we need the CAS text to be able to construct the text, with the previously evaluated value of n. This need is also why the CAS session returns the value of the variables as well as a displayed
form. It is used here.
CAS session
This class actually calls the CAS itself. The basic ideas is to take a list of maxima commands, including assignments of the form.
and executes this list in a single Maxima session. The results of this are then captures and fed back into the CAS strings so we essentially have data in the form:
key =>
An important point here is that expressions (values) can refer to previous keys. This is one reason why we can't tie teachers down to a list of allowed functions. They will be defining variable and
function names. We have implemented a lazy approach where the connection to the CAS is only made, and automatically made, when we ask for the values, display form or error terms for a variable. And
we don't generate display and value unless they are needed later. E.g. intermediate values do not create LaTeX.
Answer tests
The answer tests essentially compare two expressions. These may accept an option, e.g. the number of significant figures. Details of the current answer tests are available elsewhere. The result of an
answer test should be
1. Boolean outcome, true or false,
2. errors,
3. feedback,
4. note.
Other concepts
There are a number of reasons why a CAS expression needs to be "valid". These are
1. security,
2. stability,
3. error trapping,
4. pedagogic reasons, specific to a particular question.
Single call PRTs and simplification
As of STACK 4.4 we make a single Maxima call to exectue an entire PRT. This reduces significantly the number of separate calls to maxima, which is a significant efficienty boost for more complex
Some answer tests rely on "unsimplified" expressions with a "what you see is what you get" approach.
This example illustrates the issue. The teacher computes the answer to their question, e.g. find $\int_{0}^{1} {\frac{{\left(1-x\right)}^4\cdot x^4}{x^2+1}} \mathrm{d}x$, with the Maxima code
using simplification (obviously) at the start of the quetion variables, and this simplified expression is used by the PRT. The answer, $\frac{22}{7}-\pi$, is held internally in "simplified" form. The
Maxima string output is 22/7-%pi but internally the answer is actually the following
((MPLUS SIMP) ((RAT SIMP) 22 7) ((MTIMES SIMP) -1 $%PI))
You can see the internal tree structure of a Maxima expression with the following code.
(simp:true, p1:22/7-%pi, ?print(p1));
Rather than 22/7-%pi the internal structure is really closer to this: rat(22,7)+ (-1*%pi).
On the other hand, a student types in the expression 22/7-%pi and we deal with this without simplification. The internal Maxima expression is now
(simp:false, p1:22/7-%pi, ?print(p1));
which gives a different internal structure
((MPLUS) ((MQUOTIENT) 22 7) ((MMINUS) $%PI))
which might best be thought of as 22/7+-(%pi) where - is now a function of a single argument.
In this example, the unsimplified MQUOTIENT and simplified RAT SIMP are not particularly problematic. However, the difference between -%pi and -1*%pi is seriouly problematic. Indeed, this kind of
distinction is exactly what some answer tests, e.g. EqualComAss and CasEqual, are designed to establish. These tests will not work with a mix of simplified and unsimplified expressions, even if to
the user they look completely identical!
The solution to this problem is to "rinse" away any maxima internal simplification by using the Maxima string function to return the expression to the top level which a user would expect to type.
This process corresponds to what happend in older versions of Maxima in which expressions were routinely passed between Maxima and PHP, with the string representation being used.
Some expressions (lists, matrices) are passed by reference in Maxima, so even if the teacher's answer is created without simplification in the first instance, when it is evaluated by the answer tests
there is a risk of it becoming simplified when it is later compared by an answer test function. | {"url":"https://docs.stack-assessment.org/en/Developer/PHP-CAS/","timestamp":"2024-11-14T05:06:35Z","content_type":"text/html","content_length":"92956","record_id":"<urn:uuid:085e3b49-c09e-4173-ac96-94175a15f472>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00543.warc.gz"} |
Computational Differential Geometry
John Ernsthausen identified and implemented examples for the numerical differential-algebraic equation (DAE) integration methods developed in Rheinbold’s package DAEPAK. As Werner C. Rheinbold’s and
Patrick J. Rabier’s graduate student, John assisted in the documentation and debugging of the FORTRAN77 subroutines in DAEPAK. The experience required a deep understanding of the numerical
differential geometry package MANPAK, a collection of numerical algorithms implementing computational differential geometry for submanifolds of $R^n$.
Computational differential geometry
A subset $M$ of $R^n$ is a manifold whenever a chart exists near each point in $M$, roughly speaking. For example, a road map near your current location on Earth is a chart from flat $R^2$ into the
surface of Earth, a 2-dimensional submanifold of $R^3$.
More precisely, whenever each point in a subset of $R^d$ can be diffeomorphically mapped into a (non-empty) subset of $M$ in $R^n$, with $d \leq n$, and the smoothness of the diffeomorphism is $k \
geq 1$, then $M$ is a $d$-dimensional $C^k$ submanifold of $R^n$. The diffeomorphism is a local parameterization, and the inverse of a local parameterization is a chart.
Differential geometry enables calculus on a (nonempty) submanifold $M$ of $R^n$. Given any point $x_c \in M$, there is a local parameterization near $x_c \in M$. All local parameterizations near $x_c
\in M$ are an equivalent class, and the particular local parameterization is a representative from it. Locally near $x_c$, a DAE can be reduced to an ordinary differential equations on a manifold. In
this point of view, the path constructed by projecting any solution to the DAE through the local parameterization is a solution to the equivalent ordinary differential equation on a manifold, and the
unique solution of that local ordinary differential equation on a manifold lifted through the representative local parameterization must be the unique solution of the DAE.
Ordinary differential equations describe dynamics in $R^n$. Ordinary differential equations on a manifold describe dynamics restricted to the manifold. The solutions to ordinary differential
equations on a manifold must live in the manifold.
Differential geometry is discussed in the language of submersions, immersions, and diffeomorphisms for the purpose of constructing a coordinate subspace, local parameterization, and tangent space
near each point in the submanifold embedded into the ambient space $R^n$. These concepts are discussed in the MANPAK article [R1996] and the Rabier and Rheinboldt book [RR2002].
While all these topics are required to understand the mathematics behind computational differential geometry, a deep understanding of these topics is not required to apply the results. Local
parameterizations and their derivatives for general submanifolds of $R^n$ are constructed in the software package MANPAK. DAEPAK is built on the software package MANPAK.
Rheinboldt W.C.: MANPAK: A set of algorithms for computations on implicitly defined manifolds. Computers & Mathematics with Applications, 32(12), 15-28 (1996). [link].
Rabier P.J. and Rheinboldt W.C.: Theoretical and numerical analysis of differential-algebraic equations. In: P.G. Ciarlet, J.L. Lions (eds.) Solution of Equations in $R^n$ (Part 4), Techniques of
Scientific Computing (Part 4), Handbook of Numerical Analysis, vol 8, pp. 183-540. Elsevier, Amsterdam (2002) [link]. | {"url":"http://johnernsthausen.com/experiences/computational-differential-geometry/","timestamp":"2024-11-10T17:41:11Z","content_type":"text/html","content_length":"10547","record_id":"<urn:uuid:2d12f0dd-7832-4464-ac10-78ffd1cf19ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00506.warc.gz"} |