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Symmetry & critical points for a model shallow neural network
Using methods based on the analysis of real analytic functions, symmetry and equivariant bifurcation theory, we obtain sharp results on families of critical points of spurious minima that occur in
optimization problems associated with fitting two-layer ReLU networks with k hidden neurons. The main mathematical result proved is to obtain power series representations of families of critical
points of spurious minima in terms of 1/k (coefficients independent of k). We also give a path based formulation that naturally connects the critical points with critical points of an associated
linear, but highly singular, optimization problem. These critical points closely approximate the critical points in the original problem. The mathematical theory is used to derive results on the
original problem in neural nets. For example, precise estimates for several quantities that show that not all spurious minima are alike. In particular, we show that while the loss function at certain
types of spurious minima decays to zero like k^−1, in other cases the loss converges to a strictly positive constant.
Bibliographical note
Publisher Copyright:
© 2021
• Critical points
• Power series representation
• ReLU activation
• Spurious minima
• Student–teacher network
• Symmetry breaking
Dive into the research topics of 'Symmetry & critical points for a model shallow neural network'. Together they form a unique fingerprint.
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Asynchronous and Parallel Proof Processing
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Asynchronous and Parallel Proof Processing¶
Enrico Tassi
This chapter explains how proofs can be asynchronously processed by Coq. This feature improves the reactivity of the system when used in interactive mode via CoqIDE. In addition, it allows Coq to
take advantage of parallel hardware when used as a batch compiler by decoupling the checking of statements and definitions from the construction and checking of proofs objects.
This feature is designed to help dealing with huge libraries of theorems characterized by long proofs. In the current state, it may not be beneficial on small sets of short files.
This feature has some technical limitations that may make it unsuitable for some use cases.
For example, in interactive mode, some errors coming from the kernel of Coq are signaled late. The type of errors belonging to this category are universe inconsistencies.
At the time of writing, only opaque proofs (ending with Qed or Admitted) can be processed asynchronously.
Finally, asynchronous processing is disabled when running CoqIDE in Windows. The current implementation of the feature is not stable on Windows. It can be enabled, as described below at Interactive
mode, though doing so is not recommended.
Proof annotations¶
To process a proof asynchronously Coq needs to know the precise statement of the theorem without looking at the proof. This requires some annotations if the theorem is proved inside a Section (see
Section Section mechanism).
When a section ends, Coq looks at the proof object to decide which section variables are actually used and hence have to be quantified in the statement of the theorem. To avoid making the
construction of proofs mandatory when ending a section, one can start each proof with the Proof using command (Section Entering and leaving proof editing mode) that declares which section variables
the theorem uses.
The presence of Proof using is needed to process proofs asynchronously in interactive mode.
It is not strictly mandatory in batch mode if it is not the first time the file is compiled and if the file itself did not change. When the proof does not begin with Proof using, the system records
in an auxiliary file, produced along with the .vo file, the list of section variables used.
Automatic suggestion of proof annotations¶
The Suggest Proof Using flag makes Coq suggest, when a Qed command is processed, a correct proof annotation. It is up to the user to modify the proof script accordingly.
Proof blocks and error resilience¶
Coq 8.6 introduced a mechanism for error resilience: in interactive mode Coq is able to completely check a document containing errors instead of bailing out at the first failure.
Two kind of errors are supported: errors occurring in vernacular commands and errors occurring in proofs.
To properly recover from a failing tactic, Coq needs to recognize the structure of the proof in order to confine the error to a sub proof. Proof block detection is performed by looking at the syntax
of the proof script (i.e. also looking at indentation). Coq comes with four kind of proof blocks, and an ML API to add new ones.
blocks are delimited by { and }, see Chapter Proof handling
blocks are atomic, i.e. just one tactic introduced by the par: goal selector
blocks end with a tactic indented less than the previous one
blocks are delimited by two equal bullet signs at the same indentation level
When a vernacular command fails the subsequent error messages may be bogus, i.e. caused by the first error. Error resilience for vernacular commands can be switched off by passing
-async-proofs-command-error-resilience off to CoqIDE.
An incorrect proof block detection can result into an incorrect error recovery and hence in bogus errors. Proof block detection cannot be precise for bullets or any other non well parenthesized proof
structure. Error resilience can be turned off or selectively activated for any set of block kind passing to CoqIDE one of the following options:
• -async-proofs-tactic-error-resilience off
• -async-proofs-tactic-error-resilience all
• -async-proofs-tactic-error-resilience blocktype*,
Valid proof block types are: “curly”, “par”, “indent”, and “bullet”.
Interactive mode¶
At the time of writing the only user interface supporting asynchronous proof processing is CoqIDE.
When CoqIDE is started, two Coq processes are created. The master one follows the user, giving feedback as soon as possible by skipping proofs, which are delegated to the worker process. The worker
process, whose state can be seen by clicking on the button in the lower right corner of the main CoqIDE window, asynchronously processes the proofs. If a proof contains an error, it is reported in
red in the label of the very same button, that can also be used to see the list of errors and jump to the corresponding line.
If a proof is processed asynchronously the corresponding Qed command is colored using a lighter color than usual. This signals that the proof has been delegated to a worker process (or will be
processed lazily if the -async-proofs lazy option is used). Once finished, the worker process will provide the proof object, but this will not be automatically checked by the kernel of the main
process. To force the kernel to check all the proof objects, one has to click the button with the gears (Fully check the document) on the top bar. Only then all the universe constraints are checked.
The number of worker processes can be increased by passing CoqIDE the -async-proofs-j n flag. Note that the memory consumption increases too, since each worker requires the same amount of memory as
the master process. Also note that increasing the number of workers may reduce the reactivity of the master process to user commands.
To disable this feature, one can pass the -async-proofs off flag to CoqIDE. Conversely, on Windows, where the feature is disabled by default, pass the -async-proofs on flag to enable it.
Proofs that are known to take little time to process are not delegated to a worker process. The threshold can be configured with -async-proofs-delegation-threshold. Default is 0.03 seconds.
Batch mode¶
When working with .vio files, do not use the -vos option at the same time, otherwise stale files might get loaded when executing a Require. Indeed, the loading of a nonempty .vos file is assigned
higher priority than the loading of a .vio file.
When Coq is used as a batch compiler by running coqc, it produces a .vo file for each .v file. A .vo file contains, among other things, theorem statements and proofs. Hence to produce a .vo Coq need
to process all the proofs of the .v file.
The asynchronous processing of proofs can decouple the generation of a compiled file (like the .vo one) that can be loaded by Require from the generation and checking of the proof objects. The -vio
flag can be passed to coqc to produce, quickly, .vio files. Alternatively, when using a Makefile produced by coq_makefile, the vio target can be used to compile all files using the -vio flag.
A .vio file can be loaded using Require exactly as a .vo file but proofs will not be available (the Print command produces an error). Moreover, some universe constraints might be missing, so
universes inconsistencies might go unnoticed. A .vio file does not contain proof objects, but proof tasks, i.e. what a worker process can transform into a proof object.
Compiling a set of files with the -vio flag allows one to work, interactively, on any file without waiting for all the proofs to be checked.
When working interactively, one can fully check all the .v files by running coqc as usual.
Alternatively one can turn each .vio into the corresponding .vo. All .vio files can be processed in parallel, hence this alternative might be faster. The command coqc -schedule-vio2vo 2 a b c can be
used to obtain a good scheduling for two workers to produce a.vo, b.vo, and c.vo. When using a Makefile produced by coq_makefile, the vio2vo target can be used for that purpose. Variable J should be
set to the number of workers, e.g. make vio2vo J=2. The only caveat is that, while the .vo files obtained from .vio files are complete (they contain all proof terms and universe constraints), the
satisfiability of all universe constraints has not been checked globally (they are checked to be consistent for every single proof). Constraints will be checked when these .vo files are (recursively)
loaded with Require.
There is an extra, possibly even faster, alternative: just check the proof tasks stored in .vio files without producing the .vo files. This is possibly faster because all the proof tasks are
independent, hence one can further partition the job to be done between workers. The coqc -schedule-vio-checking 6 a b c command can be used to obtain a good scheduling for 6 workers to check all the
proof tasks of a.vio, b.vio, and c.vio. Auxiliary files are used to predict how long a proof task will take, assuming it will take the same amount of time it took last time. When using a Makefile
produced by coq_makefile, the checkproofs target can be used to check all .vio files. Variable J should be set to the number of workers, e.g. make checkproofs J=6. As when converting .vio files to
.vo files, universe constraints are not checked to be globally consistent. Hence this compilation mode is only useful for quick regression testing and on developments not making heavy use of the Type
Limiting the number of parallel workers¶
Many Coq processes may run on the same computer, and each of them may start many additional worker processes. The coqworkmgr utility lets one limit the number of workers, globally.
The utility accepts the -j argument to specify the maximum number of workers (defaults to 2). coqworkmgr automatically starts in the background and prints an environment variable assignment like
COQWORKMGR_SOCKET=localhost:45634. The user must set this variable in all the shells from which Coq processes will be started. If one uses just one terminal running the bash shell, then export
‘coqworkmgr -j 4‘ will do the job.
After that, all Coq processes, e.g. coqide and coqc, will respect the limit, globally.
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How to Calculate Return on Investment (ROI) - Stock Native
Return on investment (ROI) is the key measure of the profit derived from any investment. It is a ratio that compares the gain or loss from an investment relative to its cost. It is useful in
evaluating the current or potential return on an investment, whether you are evaluating your stock portfolio’s performance, considering a business investment, or deciding whether to undertake a new
In business analysis, ROI and other cash flow measures—such as internal rate of return (IRR) and net present value (NPV)—are key metrics that are used to evaluate and rank the attractiveness of a
number of different investment alternatives.
Although ROI is a ratio, it is typically expressed as a percentage rather than as a ratio.
Key Takeaways
• Return on investment (ROI) is an approximate measure of an investment’s profitability.
• ROI is calculated by subtracting the initial cost of the investment from its final value, then dividing this new number by the cost of the investment, and, finally, multiplying it by 100.
• ROI has a wide range of uses. It can be used to measure the profitability of stock shares, to decide whether to purchase a business, or to evaluate the success of a real estate transaction.
• One disadvantage of ROI is that it doesn’t account for how long an investment is held.
How to Calculate Return on Investment (ROI)
ROI can be calculated using either of two methods.
First method:
begin{aligned}&text{ROI} = frac { text{Net Return on Investment} }{ text { Cost of Investment} } times 100% \end{aligned}
Second method:
begin{aligned}&text{ROI} = frac { text{FVI} – text{IVI} }{ text{Cost of Investment} } times 100% \&textbf{where:} \&text{FVI} = text{Final value of investment} \&text{IVI} = text{Initial value of
investment} \end{aligned}
Interpreting Return on Investment (ROI)
When interpreting ROI calculations, it’s important to keep a few things in mind. First, ROI is typically expressed as a percentage because it is intuitively easier to understand than a ratio. Second,
the ROI calculation includes the net return in the numerator because returns from an investment can be either positive or negative.
When ROI calculations yield a positive figure, it means that net returns are in the black (because total returns exceed total costs). But when ROI calculations yield a negative figure, it means that
the net return is in the red because total costs exceed total returns.
Finally, to calculate ROI with the highest degree of accuracy, total returns and total costs should be considered. For an apples-to-apples comparison between competing investments, annualized ROI
should be considered.
The ROI formula can be deceptively simple. It depends on an accurate accounting of costs. That’s easy in the case of stock shares, for example. But it is more complicated in other cases, such as
calculating the ROI of a business project that is under consideration.
Return on Investment (ROI) Example
Assume an investor bought 1,000 shares of the hypothetical company Worldwide Wickets Co. at $10 per share. One year later, the investor sold the shares for $12.50. The investor earned dividends of
$500 over the one-year holding period. The investor spent a total of $125 on trading commissions in order to buy and sell the shares.
The ROI for this investor can be calculated as follows:
begin{aligned}text{ROI} &= frac { ( 12.50 – 10 ) times 1000 + 500 – 125 }{ 10 times 1000 } times 100 \&= 28.75% \end{aligned}
Here is a step-by-step analysis of the calculation:
1. To calculate net returns, total returns and total costs must be considered. Total returns for a stock result from capital gains and dividends. Total costs include the initial purchase price and
any trading commissions paid.
2. In the above calculation, the gross capital gain (before commissions) from this trade is ($12.50 – $10.00) x 1,000. The $500 amount refers to the dividends received by holding the stock, while
$125 is the total commissions paid.
If you further dissect the ROI into its component parts, it is revealed that 23.75% came from capital gains and 5% came from dividends. This distinction is important because capital gains and
dividends are taxed at different rates.
begin{aligned}&text{ROI} = text{Capital Gains%} – text{Commission%} + text{Dividend Yield} \end{aligned}
begin{aligned}&text{Capital Gains} = ( 2500 div 10,000 ) times 100 = 25.00% \&text{Commissions} = ( 125 div 10,000 ) times 100 = 1.25% \&text{Dividend Yield} = ( 500 div 10,000 ) times 100 = 5.00% \&
text{ROI} = 25.00% – 1.25% + 5.00% = 28.75% \end{aligned}
A positive ROI means that net returns are positive because total returns are greater than any associated costs. A negative ROI indicates that the total costs are greater than the returns.
An Alternative Return on Investment (ROI) Calculation
If, for example, commissions were split, there is an alternative method of calculating this hypothetical investor’s ROI for the Worldwide Wickets Co. investment. Assume the following split in the
total commissions: $50 when buying the shares and $75 when selling the shares.
begin{aligned}&text{IVI} = 10,000 + 50 = 10,050 \&text{FVI} = 12,500 + 500 – 75 \&phantom{ text{FVI} } = 12,925 \&text{ROI} = frac { 12,925 – 10,050 }{ 10,050} times100 \&phantom{ text{ROI} } =
28.75% \&textbf{where:}\&text{IVI} = text{Initial value (cost) of investment} \&text{FVI} = text{Final value of investment}end{aligned}
Annualized ROI helps account for a key omission in standard ROI—namely, how long an investment was held.
Annualized Return on Investment (ROI)
The annualized ROI calculation provides a solution for one of the key limitations of the basic ROI calculation. The basic ROI calculation does not take into account the length of time that an
investment is held, also referred to as the holding period. The formula for calculating annualized ROI is as follows:
begin{aligned}&text{Annualized ROI} = big [ ( 1 + text{ROI} ) ^ {1/n} – 1 big ] times100% \&textbf{where:}\&n = text{Number of years investment is held} \end{aligned}
Assume a hypothetical investment that generated an ROI of 50% over five years. The simple annual average ROI of 10%–which was obtained by dividing ROI by the holding period of five years–is only a
rough approximation of annualized ROI. This is because it ignores the effects of compounding, which can make a significant difference over time. The longer the time period, the bigger the difference
between the approximate annual average ROI, which is calculated by dividing the ROI by the holding period in this scenario, and annualized ROI.
From the formula above,
begin{aligned}&text{Annualized ROI} = big [ ( 1 + 0.50 ) ^ {1/5 } – 1 big ] times100 = 8.45% \end{aligned}
This calculation can also be used for holding periods of less than a year by converting the holding period to a fraction of a year.
Assume an investment that generated an ROI of 10% over six months.
begin{aligned}&text{Annualized ROI} = big [ ( 1 + 0.10 ) ^ {1 / 0.5 } – 1 big ] times100 = 21% \end{aligned}
In the equation above, the numeral 0.5 years is equivalent to six months.
Comparing Investments and Annualized Returns on Investment (ROI)
Annualized ROI is especially useful when comparing returns between various investments or evaluating different investments.
Assume that an investment in stock X generated an ROI of 50% over five years, while an investment in stock Y returned 30% over three years. You can determine what the better investment was in terms
of ROI by using this equation:
begin{aligned}&text{AROI}_x = big [ ( 1 + 0.50 ) ^ { 1/5 } -1 big ] times100 = 8.45% \&text{AROI}_y = big [ (1 + 0.30 ) ^ {1/3 } – 1 big ] times100 =9.14% \&textbf{where:}\&text{AROI}_x = text
{Annualized ROI for stock X} \&text{AROI}_y = text{Annualized ROI for stock Y} \end{aligned}
According to this calculation, stock Y had a superior ROI compared to stock X.
Combining Leverage With Return on Investment (ROI)
Leverage can magnify ROI if the investment generates gains. By the same token, leverage can amplify losses if the investment proves to be a losing investment.
Assume that an investor bought 1,000 shares of the hypothetical company Worldwide Wickets Co. at $10 per share. Assume also that the investor bought these shares on a 50% margin (meaning they
invested $5,000 of their own capital and borrowed $5,000 from their brokerage firm as a margin loan).
Exactly one year later, this investor sold the shares for $12.50. The shares had earned dividends of $500 over the one-year holding period. The investor also spent a total of $125 on trading
commissions when buying and selling the shares.
The calculation must also account for the cost of buying on margin. In this example, the margin loan carried an interest rate of 9%.
When calculating the ROI on this example, there are a few important things to keep in mind. First, the interest on the margin loan ($450) should be considered in total costs. Second, the initial
investment is now $5,000 because of the leverage employed by taking the margin loan of $5,000.
begin{aligned}text{ROI} &= frac { ( 12.50 – 10 ) times 1000 + 500 – 125 – 450 }{ ( 10 times 1000 ) – ( 10 times 500 ) } times 100 \&= 48.5% \end{aligned}
Thus, even though the net dollar return was reduced by $450 on account of the margin interest, ROI is still substantially higher at 48.50% (compared with 28.75% if no leverage was employed).
As another example, consider if the share price fell to $8.00 instead of rising to $12.50. In this situation, the investor decides to take the loss and sell the full position.
Here is the calculation for ROI in this scenario:
begin{aligned}text{ROI} &= frac { big [ ( 8 – 10) times1000 big ] + 500 – 125 – 450 }{ ( 10 times 1000) – (10 times 500) } times 100 \&= – frac { 2,075 }{ 5,000} \&= -41.5% \end{aligned}
In this case, the ROI of -41.50% is much worse than an ROI of -16.25%, which would have occurred if no leverage had been employed.
The Problem of Unequal Cash Flows
When evaluating a business proposal, it’s possible that you will be contending with unequal cash flows. In this scenario, ROI may fluctuate from one year to the next.
This type of ROI calculation is more complicated because it involves using the internal rate of return (IRR) function in a spreadsheet or calculator.
Assume you are evaluating a business proposal that involves an initial investment of $100,000. (This figure is shown under the “Year 0” column in the Cash Outflow row in the following table.)
The investment will generate cash flows over the next five years; this is shown in the Cash Inflow row. The row called Net Cash Flow sums up the cash outflow and cash inflow for each year.
Using the IRR function, the calculated ROI is 8.64%.
The final column shows the total cash flows over the five-year period. Net cash flow over this five-year period is $25,000 on an initial investment of $100,000. If this $25,000 was spread out equally
over five years, the cash flow table would then look like this:
In this case, the IRR is now only 5.00%.
The substantial difference in the IRR between these two scenarios—despite the initial investment and total net cash flows being the same in both cases—has to do with the timing of the cash inflows.
In the first case, substantially larger cash inflows are received in the first four years. Considering the time value of money, these larger inflows in the earlier years have a positive impact on
Advantages of Return on Investment (ROI)
The biggest benefit of ROI is that it is a relatively uncomplicated metric. It is easy to calculate and intuitively easy to understand.
Due to its simplicity, ROI has become a standard, universal measure of profitability. As a measurement, it is not likely to be misunderstood or misinterpreted because it has the same connotations in
every context.
Disadvantages of Return on Investment (ROI)
There are some disadvantages to the ROI measurement. First, it does not take into account the holding period of an investment, which can be an issue when comparing investment alternatives.
For example, assume investment X generates an ROI of 25%, while investment Y produces an ROI of 15%. One cannot assume that X is the superior investment unless the time frame of each investment is
also known. It’s possible that the 25% ROI from investment X was generated over a period of five years, while the 15% ROI from investment Y was generated in only one year.
Calculating annualized ROI can overcome this hurdle when comparing investment choices.
No Risk Adjustment
A second disadvantage of ROI is that it does not adjust for risk.
Investment returns have a direct correlation with risk: the higher the potential returns, the greater the possible risk. This can be observed firsthand in the stock market, where small-cap stocks are
likely to have higher returns than large-cap stocks but also are likely to have significantly greater risks.
An investor who is targeting a portfolio return of 12%, for example, would have to assume a substantially higher degree of risk than an investor whose goal is a return of 4%. If that investor hones
in on the ROI number without also evaluating the associated risk, the eventual outcome may be very different from the expected result.
Some Costs May Be Omitted
ROI figures can be inflated if all possible costs are not included in the calculation. This can happen deliberately or inadvertently.
For example, in evaluating the ROI on a piece of real estate, all associated expenses should be considered. These include mortgage interest, property taxes, and insurance. They also include
maintenance costs, which can be unpredictable.
These expenses can subtract from the expected ROI. Without including all of them in the calculation, the ROI figure may be grossly overstated.
Some Issues May Be Ignored
Finally, like many profitability metrics, ROI considers only financial gains when evaluating the returns on an investment. It does not consider ancillary benefits, such as social or environmental
A relatively new ROI metric, known as social return on investment (SROI), helps to quantify some of these benefits for investors.
How to Calculate ROI in Excel
What Is ROI?
Return on investment, or ROI, is a straightforward measurement of the bottom line. How much profit (or loss) did an investment make after considering its costs?
ROI is used for a wide range of business and investing decisions. It can be used to calculate the actual returns on an investment, to project the potential return on a new investment, or to compare
the potential returns on a number of investment alternatives.
For example, if a business owner is considering expanding into a new product line, the ROI formula can be used to chart out its costs and estimate its potential returns. If an entrepreneur is
evaluating a new project, an ROI calculation can help determine if the likely return is worth the expense. If an investor is evaluating past or future stock purchases, the ROI formula is a quick
indicator of real or potential stock performance.
How Is Return on Investment (ROI) Used?
ROI is a straightforward method of calculating the return on an investment. It can be used to measure profit or loss on a current investment or to evaluate the potential profit or loss of an
investment that you are considering making.
Keep in mind that ROI omits a key factor: the length of time that it took to earn that profit (or make that loss). Obviously, a stock that makes a 10% return in one year is preferable to a stock that
makes a 10% return in four years.
For this reason, the formula for annualized return on investment may be a better choice than the basic formula for return on investment. (Both are shown above.)
How Do You Calculate ROI for Real Estate?
The return on investment (ROI) formula remains the same whether you’re evaluating the performance of a single stock or considering the potential profit of a real estate investment. (See formula
Some investments are more complicated to evaluate than others, though, particularly when it comes to costs. A ROI on a real estate investment must include all of the potential costs that may be
involved, including such matters as maintenance, repairs, insurance, and lost rental income.
The Bottom Line
Return on investment (ROI) is a simple and intuitive metric of the profitability of an investment.
There are some limitations to this metric, including the facts that it does not consider the holding period of an investment and is not adjusted for risk.
Despite these limitations, ROI is a key metric used by business analysts to evaluate and rank investment alternatives.
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Aristotelian Logic No More?
I was recently reading about Aristotelian Logic at http://www.archaeonia.comwhich read…
“The heart of Aristotle’s logic is the syllogism, the classic example of which is as follows: All men are mortal; Socrates is a man; therefore, Socrates is mortal. The syllogistic form of logical
argumentation dominated logic for 2,000 years.”
(bold theirs)
“Dominated” - past tense?? I understand there are things like fuzzy logic and so on, but as far as I know, syllogism is still logic today, right?
It’s not that the Aristotelian syllogism (aka “modus ponens”) is no longer valid. Rather, it doesn’t dominate the study of logic.
I’d be very interested to hear anyone try to refute the logical example about Socrates - fuzzy logic? Let’s have some…
p1 all men are mortal
p2 socrates is a man
.: socrates is mortal
refuted in the first premise… all men (who have died) are mortal
while socrates lives he doesn’t fit the first premise
fuzzy logic? eastern or robotic?
Not eastern nor robotic, but the example you gave is a fitting one I’d say.
The term “mortal” means will die. It does not mean, “is (already) dead”. So Socrates (like you, I hope!) is mortal even if he were not dead. (Just as someone can be a smoker even if he is not smoking
at the moment.)
So. while Socrates lives he does “fit” the first premise.
But, in any case, the issue is irrelevant to Aristotelian logic, which is about the validity of the the argument, and not about whether or not the premises of the argument happen to be true or false.
Consider the following syllogism:
All dogs are reptiles.
All kangaroos are dogs
Therefore, all kangaroos are reptiles.
The above syllogism is a valid syllogism since the premises follow from the conclusion, which means that if the premises were true, then the conclusion would have to be true.
In fact, both the premises, and the conclusion are false. That is irrelevant.
Refuting syllogisms doesn’t seem much like Fuzzy Logic’s kind of thing. You’d be more likely, recasting it in FL to get an argument like:
All men are mortal 25% of the time
Socrates is a man 75% of the time
Therefore, Socrates is mortal 81% of the time.
Logic humor – arr-arr! Bite the parrot.
this “will die” is the problem…
how does one “know” with absolute certainity the future before it happens?
It doesn’t matter how we know or even if we know. The fact is, that’s the definition of “mortal”. Do we REALLY know if all men are mortal? No. But even if you assume that premise one is false, it
doesn’t change the fact that the aregument is valid because the conclusion follows from the premise. “How we know” that any particular premise is true or not, is not relevant to the question at hand.
About this thread in general…
Even if other forms of logic exist now, it still seems to me that these sorts of logical structures (Aristotelian if you will) are still the predominating, most common and standard form of logic out
there. So, I’d still say it “dominates” the field of logic. Maybe I’m out of tune with the field though.
Aristotelian logic was unable to manage relations. Infact most of the euclidean demonstrations could not be reproduced yhrough syllogisms. With Frege through the use of multiple place predicates and
variables we could finally obtain the formalization of those dimonstrations.
Anyway imi: don’t confuse an epistemological problem with a logical one. All logic has to do is to go from one sentence to another conservating the truth.
The premises of a syllogism need not be known to be true. In fact, they need not even be true. Neither of these has anything to do with the validity of the syllogism. As I pointed out earlier.
In a syllogism (or, for that matter, any other argument) if the premises are true and the argument is valid, then the conclusion must be true.
But notice: the above does not say or imply that the premises need to be known to be true, nor even that the premises need to be true. And, for that matter, it does not say or imply that the
conclusion needs to be known to be true, nor even that it needs to be true. Logic concerns only what follows from what, and not what is true or false, or known to be true or false.
That is why I have no interest in “Fuzzy Logic”.
You mean that if we do not know for certain that all men are mortal, it follows that it is false that all men are mortal? How did that happen?
Very interesting - thanks! I will have to learn more about Frege and multiple place predicates & variables.
It should perhaps be pointed out that the syllogism is the ancestral form of the conditional branching structure used in computer programming languages, the If . . . then . . . else branching
structure: If [set of conditions A] is true, then do B (i.e., B is true).
this isn’t “fuzzy logic”
yes. otherwise you are simply guessing about an empirical event that is not in view- mortality… when one guesses about empirical events that are not in view one makes a leap of induction…
Guess what I just came up with after reading this thread about FL (borrowed from BP):
All logic is fuzzy.
So go home and have a nice cup of relaxo everybody… and imp, try to take this again if you haven’t already: life is here to be farted on afterall.
That “leap” shows only that you do not know for certain that it is true. Not that you know it is false, or even that it is false.
in logic it is true or it is not true
if it is not true then logically speaking it is false
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Codeforces Round #721 Palindrome Game (easy/hard) Solution (Java/C++)
1: Consider the simplest case: the string is palindrome, and the length of string is even. For example, 000000. For this case, Bob must win. See operations below:
A:100000; B:100001; A:110001; B:110011; A:111011; B: reverse; A: 111111.
So that: Every time, after Alice changed 0 to 1. Bob also changes the corresponding 0 to 1 to keep the string palindrome. Until last two 0, at that time, Bob do one reverse operation.
But there is one exception: if the string is palindrome, and the length is even, but it is all 1s. Then the result is draw.
2: Now consider the case that: the string is palindrome, but the length is odd. Based on case 1, we find that the second player will win, and will win 2 points. So, if the center is 0, then Alice
will change on this firstly, and get win.
3: Then consider the case that string is not palindrome. For this case Alice will be happy. Let us consider there are more than 2 pairs not palindrome. For example, 00000111.
A:reverse; B:10000111; A:reverse...
Based one first cases, the winner at most win 2 points. So, before the string become palindrome, Alice continually reverses the string, and get an advantage of at least 2 points. So, Alice will win.
4: Then let us consider the case that have 1 pair not palindrome. For this case, it same to the case 2. Alice will take this firstly and get win.
But one exception: if there is only one 0 exclude this no palindrome pos. For example, 001, and exclude no palindrome pos: 0. For this case, If the first step of Alice is reverse, then Bob will
change to 101. Then Alice must change to 111. If the first step of Alice is 011 or 101, then Bob will change to 111. So, draw.
5: Now the last case is: there are exactly 2 pairs not palindrome. For example, 000011. For this case:
A:reverse; B:100011;
Now, A:110011. And Alice is second player, will win.
Submission #116877845 - Codeforces
Codeforces. Programming competitions and contests, programming community
Submission #117027679 - Codeforces
Codeforces. Programming competitions and contests, programming community
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Sperry Remington
Image: Joe Haupt, CC BY-SA 2.0
The Sperry Remington SSR-8 is a scientific calculator with 8 digits precision and algebraic logic. It has 15 functions, 29 keys, and a VFD (vacuum fluorescent) display. The power source is 4xAA
batteries. Note: Casio fx-10
Facts at a glance:
Feature Value
Type Scientific
Functions 15
Keys 29
Precision 8
Logic Algebraic
Display type VFD
Length 150mm
Width 95mm
Thickness 33mm
Weight 220g
Power Consumption 0.450000W
Power Source 4xAA batteries
Year introduced 1974
If you have a Sperry Remington SSR-8 that is no longer working, you can first try replacing the 4xAA batteries. If that doesn't solve the problem, then you can consider replacing the calculator with
a modern equivalent.
We suggest the following current model as a replacement:
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A Prime to Kill
A few weeks ago my friend invited me to a party in a part of town I didn’t know. Being a guy, I didn’t ask for directions and instead Google mapped it. I memorized the route and tried to remember the
street address.Let’s pretend the address was 7793 St. Michael’s street. Not wanting to write anything down, I looked at the number and tried to divide it. I find it’s easier to remember the smaller
multiples of a number than the whole number itself.
To my chagrin, 7793 was not evenly divisible by 3, 7 or 11. On the way there, I tried to divide it by larger and larger prime numbers in the hopes that I could break it down further. Thirteen, 17, 19
, 23 and 29 all failed, as did 31, 37 and 41. At this point the number was ingrained into my memory, but I didn’t want to give up: I felt like 7793 had challenged my manhood and I was letting it get
away with the upper hand.
I tried dividing 7793 by 43, 47, 53 and —this is how irrational I was getting— by 57 as well. Ha ha! Isn’t that ridiculous??^1
As I got closer and closer to the house, I found myself driving slower and slower to give me more time to do long-division in my head. It turned out that 59, 61, 67, 71, 73, 79 and 83 didn’t work
either. As I reached the house, I did one final calculation to see if 7793 was indeed prime. I tried multiplying my last prime number by itself. If 89 x 89 yielded a greater number than 7793, it
would effectively be declaring 7793 prime and thus a bigger man than me^2.
I multiplied the 9… 801
I multiplied the 80… 7120
I added them up… 7921
I shouted to the heavens. How could it be? Who lives on a prime number? I mean, really?? I was defeated. I could never show my face in Flatland ever again.
Final score (for those of you that keep track of the numbers):
7793: 1
Pixel: 0
1. For those of you non-geeks, 57 is divisible by 3 and 29. Since I’d already tried 3 and 29 individually, 3 x 29 would obviously not divide into 7793 as 57… obviously. [↩]
2. Again, translating geek, as the prime numbers I divided got bigger, the pair you would multiply them with to yield 7793 would get smaller. Thus, eventually, you would get to the largest possible
prime beyond which nothing could conceivably divide. Since the square root of 7793 is 88 < x < 89, and no smaller primes had worked, then this calculation would prove to me that the number was
prime. If this is confusing you as much as explaining it confused me, just assume it’s magic. [↩]
4 Comments
1. Dude. I’m lost. But as I said in my recent post, I have the math skills of a 5th grader.
I’m going to go with the magic route!
2. you obviously have way too much time on your hands… whoa.
3. First off Google map sucks. Secondly, have you ever seen the show “numbers”? You might like it…
4. I LOVE PRIME NUMBERS!!!
I have an unhealthy obsession with them. I like to bet on them. I like to have them as the number on my t-shirt. I like to scream in those quantities as necessary. I like to create haikus with
lines that are only primes. I like to sleep with them and cuddle them at night, even though they are an abstraction in my mind and provide no actual warmth. I wish I lived on a prime number. 🙁 My
house number is unfortunately divisible by three.
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A JuMP extension to use parameters in constraints constants.
ParameterJuMP adds new methods created on top of JuMP to use constant parameters in optimization problems.
To construct a parameter, pass Param() as the variable-type argument to @variable:
@variable(model, p == 1, Param())
@variable(model, p[i = 1:3] == i, Param())
anon = @variable(model, variable_type = Param())
It is possible to change the current value of a parameter with the function:
set_value(p::ParameterRef, new_value::Number)
Query the current value of the parameter with:
Finally, the dual function of JuMP is overloaded to return duals for parameters:
Last but not least! The parameter algebra was implemented so that is possible to:
• sum two parameters
• multiply parameters by constants
• sum parameters and variables
• sum parameters and affine expressions
All the operations related to linear constraints are implemented.
Lets use JuMP plus ParameterJuMP to solve the optimization problem:
where x is a variable and a is a constant. We can also solve it for different values of a.
# Create a JuMP model able to handle parameters
model = Model(SOME_SOLVER.Optimizer)
# Create a regular JuMP variable
@variable(model, x)
# Create a parameter fixed at 10
@variable(model, a == 10, Param())
# adds a constraint mixing variables and parameters to the model
@constraint(model, x >= a)
# solve the model
# query dual variable of the constant a
# modify the value of the parameter a to 20
set_value(a, 20)
# solve the model with the new value of the parameter
Currently ParameterJuMP works with Julia 1.x and JuMP 0.21.x
import Pkg; Pkg.add("ParameterJuMP")
Suppose we have linear programming problem of the following form
The only decision variable in the problem is . On the other hand, is a mere parameter.
Problems like this appear frequently in Stochastic optimization and in Decomposition frameworks.
In stochastic optimization it is frequent to solve that same problem for multiple values of , which are tipically scenario dependent.
In decomposition frameworks, it is useful to solve the same problem for multiple values of , but even more important is to be able to query dual values from . This dual values are computed by
applying the chain rule on the duals of the constraints.
In pure JuMP we can acomplish these tasks by creating dummy fixed variables, so that we can easily change their fixed values and query duals from fixing constraints.
One example in pure JuMP goes as follows:
# create a regular JuMP Model
model_pure = Model(SOME_SOLVER.Optimizer)
# add optimization variables
@variable(model_pure, x[1:N] >= 0)
# add dummy fixed variables
@variable(model_pure, y[1:M])
@variable(model_pure, y_fixed[1:M] == value_for_y[i])
@constraint(model_pure, fix_y[j in 1:M], y[i] == y_fixed[i])
# add constraints
@constraint(model_pure, ctr[k in 1:P],
sum(A[i,k]*x[i] for i in 1:N) == b[k] - sum(D[j,k]*y[j] for j in 1:M))
# create objective function
@objective(model_pure, Min, sum(c[i]*x[i] for i in 1:N))
# solve problem
# query dual values
y_duals = dual.(fix_y)
# modify y
set_value.(y_fixed, new_value_for_y)
# solve problem (again)
# query dual values (again)
y_duals = dual.(fix_y)
The main problem with this approach is that it creates to many dummy variables that are added without real need to the solver representation of the optimization problem. Hence solve times are
increased without real need!!!
The same example of the motivation can be written with parameters:
# create a ParameterJuMP Model
model_param = Model(SOME_SOLVER.Optimizer)
# add optimization variables
@variable(model_param, x[1:N] >= 0)
# add dummy fixed variables
@variable(model, y[i = 1:M] == value_for_y[i], Param())
# add constraints
@constraint(model_param, ctr[k in 1:P],
sum(A[i,k]*x[i] for i in 1:N) == b[k] - sum(D[j,k]*y[j] for j in 1:M))
# create objective function
@objective(model_param, Min, sum(c[i]*x[i] for i in 1:N))
# solve problem
# query dual values
y_duals = dual.(y)
# modify y
set_value.(y, new_value_for_y)
# solve problem (again)
# query dual values (again)
y_duals = dual.(y)
ParameterJuMP was developed by:
• Joaquim Dias Garcia (@joaquimg), PSR and PUC-Rio
• Benoît Legat (@blegat), UCLouvain
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Mathieu groupoid
You have probably encountered the 15-puzzle, where all but one of the squares of a 4 × 4 grid are occupied by counters. The only acceptable moves are to move a counter into an adjacent unoccupied
Sam Loyd challenged people to swap a pair of adjacent counters. It is relatively easy to show that this is impossible, as changing the parity of the permutation of the sixteen objects (fifteen
counters and one empty space) also changes the parity of the position of the empty space.
Unlike the Rubik’s cube, the set of positions in this puzzle do not form a group. Instead, they form a groupoid, since two operations can only be composed if the final position of the empty space in
the first operation is equal to the initial position of the empty space in the second operation. Groupoids have the same axioms as groups, except for totality; it is not always possible to compose
two arbitrary elements.
Conway, Elkies and Martin created a similar, albeit more interesting, puzzle. Place a counter on each of 12 of the 13 points of the projective plane of order 3. The plane has 13 lines of four points,
and the following operation is allowed:
• Select a counter, move it into the empty space, and swap the two remaining counters on that line.
The groupoid generated is known as M13, since it has connections with the Mathieu groups. In particular, the subgroup of positions fixing the empty space is isomorphic to M12. M12 itself has been
realised as the group of positions accessible with a mechanical puzzle. In fact, I know of at least three such descriptions, which are completely different:
• Twelve ball-bearings touch a central sphere in an icosahedral arrangement. M12 is generated by pairs of clockwise and anticlockwise twists (where we choose an equator orthogonal to a diameter
joining opposite spheres, and rotate one of the two hemispheres).
• The Number Planet mechanical puzzle by Oskar van Deventer and Jim Stasheff:
There are electronic implementations of M12, M24 and Co0 as puzzles, designed by Igor Kriz whom you may recognise from his work in Euclidean Ramsey theory.
0 Responses to Mathieu groupoid
1. I understand “Representation Theory” to mean “Study of the representation of group operations by matrix multiplication”. In these puzzles, the (initially atomic) group operation is split into
(admittedly, coupled) operations on local parts of the puzzle. Is there a connection to representation theory? Or is there a variant theory which one might call “representation by puzzles” or
“representation by coupled automata” theory, which is simply difficult to google because of the dominance of the usual linear algebra sense of “representation”?
□ These are called `permutation representations’, which are special cases of linear matrix representations.
This entry was posted in Uncategorized. Bookmark the permalink.
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Markov Chains
Sergey Bryl has an introductory-level post on what Markov chains are and how they work:
Using Markov chains allow us to switch from heuristic models to probabilistic ones. We can represent every customer journey (sequence of channels/touchpoints) as a chain in a directed Markov
graph where each vertex is a possible state (channel/touchpoint) and the edges represent the probability of transition between the states (including conversion.) By computing the model and
estimating transition probabilities we can attribute every channel/touchpoint.
Let’s start with a simple example of the first-order or “memory-free” Markov graph for better understanding the concept. It is called “memory-free” because the probability of reaching one state
depends only on the previous state visited.
Markov chains are great for behavior prediction and sentence formation. This is part one of a series I will eagerly anticipate. H/T R Bloggers.
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Nonlinear Differential Equations in Ordered Spaces - Köp
Research Seminars in Mathematics - Institutionen för
If the equation is homogeneous, i.e. g(x) = 0, one may rewrite and integrate: ′ =, = +, •The general form of a linear first-order ODE is 𝒂 . 𝒅 𝒅 +𝒂 . = ( ) •In this equation, if 𝑎1 =0, it is no
longer an differential equation and so 𝑎1 cannot be 0; and if 𝑎0 =0, it is a variable separated ODE and can easily be solved by integration, thus in this chapter 𝑎0 cannot be 0. 2017-06-17 · A linear
first order ordinary differential equation is that of the following form, where we consider that. y = y ( x), {\displaystyle y=y (x),} and. y {\displaystyle y} and its derivative are both of the
first degree.
Other applications are numerous, but most are solved in a similar fashion. Linear Equations of Order One Linear equation of order one is in the form $\dfrac{dy}{dx} + P(x) \, y = Q(x).$ The general
solution of equation in this form is $\displaystyle ye^{\int P\,dx} = \int Qe^{\int P\,dx}\,dx + C$ Derivation $\dfrac{dy}{dx} + Py = Q$ Use $\,e^{\int P\,dx}\,$ as integrating factor. First order
differential equations Calculator online with solution and steps. Detailed step by step solutions to your First order differential equations problems online with our math solver and calculator.
Solved exercises of First order differential equations. 2021-04-03 Linear Differential Equations of First Order Definition of Linear Equation of First Order. Method of variation of a constant.
TMV170/MMGD30 Matematisk analys - Canvas
Getting a copy is strongly recommended. If time permits, we might also consider first order nonlinear equations.
TMV170/MMGD30 Matematisk analys - Canvas
First Order Ordinary Linear Differential Equations • Ordinary Differential equations does not include partial derivatives.
The present book describes the state-of-art in the middle of the 20th century, concerning first order differential equations of known solution formulæ. 2. order of a differential equation. en
differentialekvations ordning. 3.
Trattoria venti titan
displaymath40. What is a linear first order equation?Edit · d y d x + P ( x ) y = Q ( x ) {\displaystyle {\frac {dy}{dx}}+P(x)y=Q(x)} · d y d x + P ( x ) y {\displaystyle {\frac {dy}{dx}}+P(x)y}. 26
Feb 2016 Scilab has a very important and useful in-built function ode() which can be used to evaluate an ordinary differential equation or a set of coupled In this lesson you'll learn how to solve a
first-order linear differential equation. We first define what such an equation is, and then we give the First, we will look at two examples of linear first-order differential equations with constant
coefficients that arise in physics.
Titta och ladda ner Differential equation introduction | First order differential equations | Khan Academy gratis, Differential equation introduction | First order First Order Linear Differential
Equations A first order ordinary differential equation is linear if it can be written in the form y′ + p(t) y = g(t) where p and g are arbitrary functions of t. This is called the standard or
canonical form of the first order linear equation. We’ll start by attempting to solve a couple of very simple is called a linear nonhomogeneous differential equation of first order. We consider two
methods of solving linear differential equations of first order: Using an integrating factor; Method of variation of a constant.
Södra dalarnas hockeyskola
king stockholm jobbakelius skattblue lotus massagesiemens s7-1500prestashop svenska språkswedish cricket boardjohannes backman loomis
system of ode - Distritec
displaymath40. What is a linear first order equation?Edit · d y d x + P ( x ) y = Q ( x ) {\displaystyle {\frac {dy}{dx}}+P(x)y=Q(x)} · d y d x + P ( x ) y {\displaystyle {\frac {dy}{dx}}+P(x)y}. 26
Feb 2016 Scilab has a very important and useful in-built function ode() which can be used to evaluate an ordinary differential equation or a set of coupled In this lesson you'll learn how to solve a
first-order linear differential equation. We first define what such an equation is, and then we give the First, we will look at two examples of linear first-order differential equations with constant
coefficients that arise in physics.
DIFFERENTIALEKVATION ▷ Engelsk Översättning - Exempel
ye P x dx. e P x dx. THEOREM 15.3 Solution of a First- Order Linear Differential Equation. It is one of the basic elements of DE. A proper understanding of first order linear differential equations
can make the process of learning DE smooth. First Order First Order Linear Equations. A first order linear differential equation has the following form: displaymath39. The general solution is given
If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and
*.kasandbox.org are unblocked.
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Variance on task list
I have a sheet that is a running list of tasks that gets added to weekly. I am looking to figure out how to show how early, or late, a task is completed in relation to the due date. If the task is on
time, I want to show 0.
I have been trying to use this formula
=NETWORKDAYS([Due Date]@row, [Actual Finish]@row) - 1
This works fine if the task is late. If the task is early, then the calculation is off by several days.
I tried combining several formulas based on what I have read elsewhere in the forum and now I just get #INCORRECT ARGUMENT.
=IF([Actual Finish]@row = [Due Date]@row, 0, IF([Due Date]@row < [Actual Finish]@row), =NETWORKDAYS([Due Date]@row, [Actual Finish]@row)-1, IF([Due Date]@row, [Actual Finish]@row) > 0, =NETWORKDAYS
([Due Date]@row, [Actual Finish]@row)+1)
Any suggestions?
Best Answer
• Hi @M38a1
There's a couple things to clean up here but you're on the right track!
Your first statement is correct, If this, then 0, Otherwise, this other formula.
However for your second and third IF statements, we'll want to adjust where your closing parentheses are and take away some of your = signs.
I'll break down each statement individually for your three outcomes, then we'll put them together.
First Statement - No Change:
=IF([Actual Finish]@row = [Due Date]@row, 0,
Second Statement - take out the ) and =
IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1,
Final Statement - we can jump right to the only other option you want to return
NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1
All Together:
=IF([Actual Finish]@row = [Due Date]@row, 0, IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1))
Let me know if this makes sense and works for you!
Need more help? 👀 | Help and Learning Center
こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 | Global Discussions
• Hi @M38a1
There's a couple things to clean up here but you're on the right track!
Your first statement is correct, If this, then 0, Otherwise, this other formula.
However for your second and third IF statements, we'll want to adjust where your closing parentheses are and take away some of your = signs.
I'll break down each statement individually for your three outcomes, then we'll put them together.
First Statement - No Change:
=IF([Actual Finish]@row = [Due Date]@row, 0,
Second Statement - take out the ) and =
IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1,
Final Statement - we can jump right to the only other option you want to return
NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1
All Together:
=IF([Actual Finish]@row = [Due Date]@row, 0, IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1))
Let me know if this makes sense and works for you!
Need more help? 👀 | Help and Learning Center
こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 | Global Discussions
• Thank you Genevieve! Your answer works brilliantly. And thank you for taking the time to break it out into individual statements so I can better understand the process for the next set of
formulas that I am working on.
Best regards,
• No problem, Eric! I'm glad I could help. 🙂
Need more help? 👀 | Help and Learning Center
こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 | Global Discussions
Help Article Resources
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t-Test, Chi-Square, ANOVA, Regression, Correlation... (2024)
This tutorial is about z-standardization (z-transformation). We will discuss what the z-score is, how z-standardization works, and what the standard normal distribution is. In addition, the z-score
table is discussed and what it's used for.
What is z-standardization?
Z-standardization is a statistical procedure used to make data points from different datasets comparable. In this procedure, each data point is converted into a z-score. A z-score indicates how many
standard deviations a data point is from the mean of the dataset.
Example of z-standardization
Suppose you are a doctor and want to examine the blood pressure of your patients. For this purpose, you measured the blood pressure of a sample of 40 patients. From the measured data, you can now
naturally calculate the average, i.e., the value that the 40 patients have on average.
Now one of the patients asks you how high his blood pressure is compared to the others. You tell him that his blood pressure is 10mmHg above average. Now the question arises, whether 10mmHg is a lot
or a little.
If the other patients cluster very closely around the mean, then 10mmHg is a lot in relation to the spread, but if the other patients spread very widely around the mean, then 10mmHg might not be that
The standard deviation tells us how much the data is spread out. If the data are close to the mean, we have a small standard deviation; if they are widely spread, we have a large standard deviation.
Let's say we get a standard deviation of 20 mmHg for our data. This means that on average, the patients deviate by 20 from the mean.
The z-score now tells us how far a person is from the mean in units of standard deviation. So, a person who deviates one standard deviation from the mean has a z-score of 1. A person who is twice as
far from the mean has a z-score of 2. And a person who is three standard deviations from the mean has a z-score of 3.
Accordingly, a person who deviates by minus one standard deviation has a z-score of -1, a person who deviates by minus two standard deviations has a z-score of -2, and a person who deviates by minus
three standard deviations has a z-score of -3.
And if a person has exactly the value of the mean, then they deviate by zero standard deviations from the mean and receive a score of zero.
Thus, the z-score indicates how many standard deviations a measurement is from the mean. As mentioned, the standard deviation is just a measure of the dispersion of the patients' blood pressure
around the mean.
In short, the z-score helps us understand how exceptional or normal a particular measurement is compared to the overall average.
Calculating the z-score
How do we calculate the z-score? We want to convert the original data, in our case the blood pressure, into z-scores, i.e., perform a z-standradiszation.
Here we see the formula for z-standardization. Here, z is of course the z-value we want to calculate, x is the observed value, in our case the blood pressure of the person in question, μ is the mean
value of the sample, in our case the mean value of all 40 patients, and σ is the standard deviation of the sample, i.e. the standard deviation of our 40 patients.
Caution: μ and σ are actually the mean and standard deviation of the population, but in our case we only have a sample. However, under certain conditions, which we will discuss later, we can estimate
the mean and standard deviation using the sample.
Let's assume that the 40 patients in our example have a mean value of 130 and a standard deviation of 20. If we use both values, we get for z: x-130 divided by 20
Now we can use the blood pressure of each individual patient for x and calculate the z value. Let's just do this for the first patient. Let's say this patient has a blood pressure of 97, then we
simply enter 97 for x and get a z-value of -1.65.
This person therefore deviates from the mean by -1.65 standard deviations. We can now do this for all patients.
Regardless of the unit of the initial data, we now have an overview in which we can see how far a person deviates from the mean in units of the standard deviation.
Now, of course, we only have a sample that comes from a specific population. But if the data is normally distributed and the sample size is greater than 30, then we can use the z-value to say what
percentage of patients have a blood pressure lower than 110, for example, and what percentage have a blood pressure higher than 110.
But how does this work? If the initial data is normally distributed, we obtain a so-called standard normal distribution through z-standardization.
The standard normal distribution is a specific type of normal distribution with a mean value of 0 and a standard deviation of 1.
The special feature is that any normal distribution, regardless of its mean or standard deviation, can be converted into a standard normal distribution.
Since we now have a standardized distribution, all we really need is a table that tells us what percentage of the values are below this value for as many z-values as possible .
And you can find such a table in almost every statistics book or here: Table of the z-distribution. Now, of course, the question is how to read this table?
If, for example, we have a z-value of -2, then we can read a value of 0.0228 from this table.
This means that 2.28% of the values are smaller than a z-value of -2. As the sum is always 100% or 1, 97.72% of the values are greater.
And with a z-value of zero, we are exactly in the middle and get a value of 0.5. Therefore 50% of the values are smaller than a z-value of 0 and 50% of the values are greater than 0. As the normal
distribution is symmetrical, we can read off the probabilities for positive z-values exactly.
If we have a z-value of 1, we only need to search for -1. However, we must note that in this case we get a value that tells us what percentage of the values are greater than the z-value. So with a
z-value of 1, 15.81% of the values are larger and 84.14% of the values are smaller.
But what if, for example, we want to read a z-value of -1.81 in the table? We need the other columns for this. We can read a z-value of -1.81 at -1.8 and at 0.01.
Now let's look at the example about blood pressure again. For example, if we want to know what percentage of patients have a blood pressure below 123, we can use z-standardization to convert a blood
pressure of 123 into a z-value, in this case we get a z-value of -0.35.
Now we can take the table with the z-distributions and search for a z-value of -0.35. Here we have a value of 0.3632. This means that 36.32 percent of the values are smaller than a z-value of -0.35
and 63.68 percent are larger.
Compare different data sets with the z-score
However, there is another important application for z-standardization. The z-standardization can help to make values measured in different ways comparable. Here is an example.
Suppose we have two classes, class A and class B, who have written a different test in mathematics.
The tests are designed differently, have a different level of difficulty and a different maximum score.
In order to be able to compare the performance of the pupils in the two classes fairly, we can apply the z-standardization.
The average score or mean score for class A was 70 points with a standard deviation of 10 points. The average score for the test in class B was 140 points with a standard deviation of 20 points.
We now want to compare the performance of Max from class A, who scored 80 points, with the performance of Emma from class B, who scored 160 points.
To do this, we simply calculate the z-value of Max and Emma. We enter 80 once for x and get a z-value of 1. Then we enter 160 for x and also get a z-value of 1.
The z-values of Max and Emma are therefore the same. This means that both students performed equally well in terms of average performance and dispersion in their respective classes. Both are exactly
one standard deviation above the mean of their class.
But what about the assumptions? Can we simply calculate a z-standardization and use the table of the standard normal distribution?
The z-standardization itself, i.e. the conversion of the data points into z-values using this formula, is essentially not subject to any strict conditions. It can be carried out independently of the
data distribution.
However, if we use the resulting z-values in the context of the standard normal distribution for statistical analyses (e.g. for hypothesis tests or confidence intervals), certain assumptions must be
The z-distribution assumes that the underlying population is normally distributed and that the mean (μ) and standard deviation (σ) of the population are known.
However, as you never have the entire population in practice and the mean value and standard deviation are usually not known, this requirement is of course often not met. Fortunately, however, there
is an alternative assumption.
Although the z-distribution is defined for normally distributed populations, the central limit theorem can be applied to large samples. This theorem states that the distribution of the sample
approaches a normal distribution if the sample size is greater than 30. Therefore, if the sample is larger than 30, the standard normal distribution can be used as an approximation and the mean and
standard deviation can be estimated using the sample.
When the standard deviation is estimated from the sample, s is usually written instead of σ and x dash instead of mu for the mean.
The z-standardization should not be confused with the z-test or the t-test. If you want to know what the t-test is, please watch the following video.
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Chapter 1: Trigonometry
1.1 Trig Functions
For any right triangle, there are specific mathematical relationships between the angles and the lengths of the sides.
A right triangle, labeled with an angle and corresponding sides
In the figure above, the sides “opposite” and “adjacent” are relative to the angle that is labeled as 𝜃 (Greek lowercase theta). The hypotenuse is the longest side. It is also the side opposite from
the right angle.
The relationships are called trigonometric (or trig) functions, and are
\sin(\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} \tag{1.1} \\ \\
\cos(\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} \tag{1.2} \\ \\
\tan(\theta) &= \frac{\text{opposite}}{\text{adjacent}} \tag{1.3}
These are all mathematical functions with input and output values. For the trig functions, the input values are lengths of the sides, and the output is given by dividing one length by the other. You
read the functions as “sine of theta,” “cosine of theta,” and “tangent of theta.” (Sometimes when saying this out loud, the “of” gets dropped, e.g., “sine theta.”)
You may find using a mnemonic, a phrase where the first letter of each word in the phrase corresponds to the first letter in each word of the definitions above, helps you remember these. You can use
whatever mnemonic works for you; my personal favorite is Some Old Hippie Caught Another Hippie Tripping On Acid. (The S in Some stands for the s in sine, the O in Old stands for the o in opposite,
and so on.)
It is more common that you will use variables to represent the lengths of the sides. Consider the triangle in the figure below, where the side opposite from the angle \(\theta\) is labeled as side \
(a\), the side adjacent to the angle is side \(b\), and the hypotenuse is side \(c\). The trig functions would then be:
\[ \sin(\theta) = \frac{a}{c} \qquad \cos(\theta) = \frac{b}{c} \qquad \tan(\theta) = \frac{a}{b} \]
Right triangle labeled with variables
Example 1.1
Find sin(𝜃), cos(𝜃), and tan(𝜃) for the diagram below.
First, identify the hypotenuse. Then, identify each side’s relationship to the angle. The longest side is the hypotenuse, so side with a length of 17 units is the hypotenuse. The side opposite from
the angle 𝜃 has a length of 8 units, and the side adjacent to the angle 𝜃 has a length of 15 units.
\sin(\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} & \cos(\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} & \tan(\theta) &= \frac{\text{opposite}}{\text{adjacent}} \\
&= \frac{8}{17} & &= \frac{15}{17} & &= \frac{8}{15} \\
&= 0.471 & &= 0.882 & &= 0.533
In physics class, you’ll almost always give your answer in decimal form, rather than as an exact fraction. It is common practice to round to two or three decimal places. You’ll learn more about
rounding in chapter 3.
Practice 1.1
Practice 1.2
Practice 1.3
Practice 1.4
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User-defined Operand: Modulo operator | Zemax Community
Hi all,
I’m not going to create a repository for this, but if you quickly need a modulo operator in your Merit Function, there you have it. Hx and Hy are the operand numbers (integer) that will be used as
the numerator and denominator respectively.
The relevant part of the code is self-explanatory (hopefully):
int numerator = (int)TheApplication.OperandArgument1;
int denominator = (int)TheApplication.OperandArgument2;
double numerator_value = TheSystem.MFE.GetOperandAt(numerator).Value;
double denominator_value = TheSystem.MFE.GetOperandAt(denominator).Value;
operandResultsl0] = numerator_value % denominator_value;
To install, download the ZIP file, extract it, and move UDOC12.exe to your
folder. Then, in the MFE you can do:
The rest of the division of 5.00 by 2.00 is 1.00.
In case you are wondering why I made this. I needed to have a linear polarization without caring about the orientation. I noticed that using CODA with data=110 (phase difference), some linear
polarization angles were not achievable somehow and if I tried known configurations that would produce a linear polarization at those unachievable angles, CODA with data=110 would report 2pi. So, to
make my original Merit function work, I’m just making sure that CODA with data=110 modulo 2pi is always zero :)
Hope this helps. Take care,
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3rd QGG Intensive Lectures: Spinfoam path integrals for Quantum Gravity
July 26 (Wed) at 10:00 - July 28 (Fri) at 19:00, 2023 (JST)
□ Etera Livine (Research Director CNRS, Ecole Normale Supérieure de Lyon, France)
Puttarak Jai-akson
At the crossroads of several approaches to quantum gravity, Spinfoams propose a discrete path integral for quantum general relativity built from topological field theory. With the spectrum of
geometric operators directly read from the representation theory of the local symmetry group, they can be interpreted as a quantized version of Regge calculus and can be understood as implementing
the dynamics of quantum states geometry in loop quantum gravity. I will explain the basics of the formalism, the motivations, the mathematical framework and the main tools. In three space-time
dimensions, the spinfoam quantization of 3d gravity is given by the Turaev-Viro topological invariant, which is intimately related to the quantization of Chern-Simons theory. I will explain in
particular how the spinfoam amplitudes solve the Wheeler-de Witt equation, implement the invariance under 3d diffeomorphisms (despite being formulated in a discretized space-time) and lead to a
quasi-local version of holography. In four space-time dimensions, general relativity can be formulated as an almost-topological theory and I will explain how the existing spinfoam models introduce a
sea of topological defects to re-create the gravitational degrees of freedom from a topological path integral. Finally, I will show how spinfoams are naturally defined in terms of group field theory,
which are generalized tensor models, and the prospects that this opens. I will conclude with the main challenges and open lines of research of the field.
July 26
10:00 - 10:15 Registration and reception
10:15 - 11:45 Lecture 1
11:45 - 13:30 Lunch & coffee break
13:30 - 15:00 Lecture 2
15:00 - 16:00 Coffee break
16:00 - 17:00 Lecture 3
17:10 - 18:30 Short talk session
July 27
10:00 - 11:45 Lecture 4
11:45 - 13:30 Lunch & coffee break
13:30 - 15:00 Lecture 5
15:00 - 16:00 Coffee break
16:00 - 17:00 Lecture 6
17:30 - 20:00 Banquet
July 28
10:00 - 11:45 Lecture 7
11:45 - 13:30 Lunch & coffee break
13:30 - 15:00 Lecture 8
15:00 - 16:00 Coffee break
16:00 - 17:30 Lecture 9 & Closing
This is a closed event for scientists. Non-scientists are not allowed to attend. All scientists, including those outside RIKEN, are welcome to attend, please register using the registration form.
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What's the best meal for a cold, overcast, damp day?
(34,544 posts) Sat Oct 13, 2018, 01:16 PM Oct 2018
What's the best meal for a cold, overcast, damp day?
I vote for a bowl of chili.
101 replies
1. or a chili topped baked potato....
Sat Oct 13, 2018, 01:18 PM
Oct 2018
I've been low carbing too long-- I LONG for a potato!
Sat Oct 13, 2018, 01:31 PM
Oct 2018
Gotta pick up some sour cream with that!
Sat Oct 13, 2018, 02:15 PM
Oct 2018
2. or Posole...or Tomato Bisque. Or Potato Leek. Meal in a bowl for sure!
Sat Oct 13, 2018, 01:19 PM
Oct 2018
7. I have heard of posole but don't know what it is
Sat Oct 13, 2018, 01:30 PM
Oct 2018
Can you describe it for me?
13. recipe below - I use chicken thighs. easiest in the world to make in a slow cooker
Sat Oct 13, 2018, 01:33 PM
Oct 2018
Sat Oct 13, 2018, 01:35 PM
Oct 2018
I am definitely going to try that! Looks like a crowd pleaser! And I have 3 grown sons who might like that.
86. Looks great, but use olive oil instead of canola. Canola is usually GMO and is inflammatory even
Sun Oct 14, 2018, 01:57 AM
Oct 2018
when it s organic. If it s GMO it will be contaminated with glyphosphate.
24. If it's Posole, I'm coming to dinner!
Sat Oct 13, 2018, 02:04 PM
Oct 2018
I love posole and one day I'm going to make at home.
26. check the recipe link I added above - it is SO easy and superb!
Sat Oct 13, 2018, 02:11 PM
Oct 2018
94. I make posole too. Have found several web articles.... the thing is....
Mon Oct 15, 2018, 01:02 PM
Oct 2018
... apparently it comes from an ancient south American culture where the meat was human...
Sat Oct 13, 2018, 01:21 PM
Oct 2018
Sat Oct 13, 2018, 01:30 PM
Oct 2018
Sat Oct 13, 2018, 01:26 PM
Oct 2018
Sat Oct 13, 2018, 01:31 PM
Oct 2018
is another winner! With cabbage!
27. I made a beef, root vegetables, barley soup the other day from a beautiful Irish cookbook.
Sat Oct 13, 2018, 02:12 PM
Oct 2018
with cabbage!
It was delicious and I have several servings to go in the freezer.
Sat Oct 13, 2018, 03:24 PM
Oct 2018
Makes almost anything better!
5. Yum! That looks good! Looks homemade too!
Sat Oct 13, 2018, 01:27 PM
Oct 2018
One of my favorites is Lima Bean soup (made with frozen Lima Beans). Slow simmered with onions, carrots and real meaty, smoked ham hocks.
Cook until the meat falls off the bone. Toss a couple whole cloves in there too! Cornbread works really well with this!
Baked apple for dessert!
6. Corn bread and baked apple ...
Sat Oct 13, 2018, 01:29 PM
Oct 2018
Sat Oct 13, 2018, 01:31 PM
Oct 2018
So few cold, rainy days here it is the only time I make it. They got it so seldom my kids think it's a special meal.
12. Sounds good right about now!
Sat Oct 13, 2018, 01:32 PM
Oct 2018
14. Just finished the leftovers from my father-in-law's
Sat Oct 13, 2018, 01:34 PM
Oct 2018
beef vegetable soup while I am making some split yellow pea soup.
I'll make a caldo de res later this week.
Oh and a new Ramen place is opening in Oak Lawn! First one south of Hyde Park.
Jesus, c'mon South Side. Plenty of good Ramen places on the North Side and North West burbs.
All of the above...
Sat Oct 13, 2018, 01:36 PM
Oct 2018
Pea soup and corn bread .....
38. I love yellow pea soup!
Sat Oct 13, 2018, 02:29 PM
Oct 2018
39. Gonna make some croutons, too
Sat Oct 13, 2018, 02:33 PM
Oct 2018
Will eat with some cayenne and a dollop of sour creme.
17. I'm making Italian Wedding Bean Soup
Sat Oct 13, 2018, 01:36 PM
Oct 2018
Sat Oct 13, 2018, 01:48 PM
Oct 2018
I had a lot of greens in my garden last year and made wedding soup all winter long. This year, not so much.
Italian food is really popular around here. Almost all restaurants offer some version of Italian wedding soup.
23. I'm in the middle of fried chicken and mashed potatoes
Sat Oct 13, 2018, 02:02 PM
Oct 2018
so I've never had a restaurant version. I'll bet it's divine.
19. Tomato soup and a grilled cheese sandwich...
Sat Oct 13, 2018, 01:52 PM
Oct 2018
21. A favorite from my childhood!
Sat Oct 13, 2018, 02:01 PM
Oct 2018
I like to dip the sangwich in the soup!
43. Tomato soup is best homemade
Sat Oct 13, 2018, 02:39 PM
Oct 2018
The canned versions are too sweet and flat. Too thin, too. Also, goes so well with grilled cheese sandwiches made with cheddar and good bread. The panini press does a good job of building an outside
crunch while melting the cheddar.
74. I can't eat Campbell's tomato soup
Sat Oct 13, 2018, 06:51 PM
Oct 2018
since I started making my own.
Sun Oct 14, 2018, 06:29 PM
Oct 2018
and the other a processed food product, so I am in agreement with you.
Sat Oct 13, 2018, 01:56 PM
Oct 2018
Chili or a really good home made curry are especially tasty on a cold, overcast, damp day.
22. Can't beat spice plus warmth!
Sat Oct 13, 2018, 02:01 PM
Oct 2018
Sat Oct 13, 2018, 02:08 PM
Oct 2018
Sat Oct 13, 2018, 02:12 PM
Oct 2018
56. We usually do tamales, with chilli as a side!
Sat Oct 13, 2018, 03:39 PM
Oct 2018
63. Back in my childhood we only had frozen Red's tamales...
Sat Oct 13, 2018, 04:30 PM
Oct 2018
and they were basically just a heaping amount of seductively spiced chili wrapped in mesa. They were a surprisingly good gut-bomb, but don't hold a candle to the tamales made by our local women and
sold out the backdoor (to Hell with health codes) of many businesses in town. One quickly learns to get you order in early because they go fast!
28. Chicken and dumplings n/t
Sat Oct 13, 2018, 02:12 PM
Oct 2018
30. Talk about comfort food!
Sat Oct 13, 2018, 02:13 PM
Oct 2018
33. my fav comfort food....
Sat Oct 13, 2018, 02:17 PM
Oct 2018
32. a mug of tomato soup with cheese melted on top
Sat Oct 13, 2018, 02:16 PM
Oct 2018
Sat Oct 13, 2018, 02:18 PM
Oct 2018
I've got a big pot o'red waiting in the kitchen, but I'm on strict orders to leave it alone until all the guests arrive. How long must I suffer?!?
41. That must be torturous, all right!
Sat Oct 13, 2018, 02:37 PM
Oct 2018
35. I made a pasta bean soup & a gr turkey chili is on the crockpot
Sat Oct 13, 2018, 02:19 PM
Oct 2018
I'm watching my cholesterol hence the turkey. Hot
garlic drop biscuits at dinner time. Stress cooking means made ahead freezer meals
Sat Oct 13, 2018, 02:41 PM
Oct 2018
Turkey chili is good, too!
36. And Beef Stew is good on a chilly grey day.
Sat Oct 13, 2018, 02:19 PM
Oct 2018
42. Absolutely! With a splash of red wine!
Sat Oct 13, 2018, 02:38 PM
Oct 2018
51. Two splashes of red wine. One for the pot, ... nt
Sat Oct 13, 2018, 03:16 PM
Oct 2018
73. my aunt makes a stew w/ red wine & fr OJ, add a little of other stuff.
Sat Oct 13, 2018, 05:54 PM
Oct 2018
i do a little coffee, any leftover coke, bit of soy sauce, worcestershire sauce. make enough for almost a week. gets better.
37. Great minds baby! Great minds!
Sat Oct 13, 2018, 02:21 PM
Oct 2018
That was my answer as soon as I saw your post!
Sat Oct 13, 2018, 02:39 PM
Oct 2018
When I saw your subject line, I just knew that was you, Glam!
40. Butter Beans with pork neck and a pan of cornbread.
Sat Oct 13, 2018, 02:34 PM
Oct 2018
46. Good home cooking Southern style!
Sat Oct 13, 2018, 02:41 PM
Oct 2018
47. Chili Yes! But Homemade Homegrown Squash-Potato-Leek-uncured-Sausage-and-Cheese Soup
Sat Oct 13, 2018, 02:47 PM
Oct 2018
48. Sounds like a bowl full of delicious!
Sat Oct 13, 2018, 02:52 PM
Oct 2018
49. Rich and thick cheesy cauliflower soup, nutmeg and a teeny pinch of hot pepper flakes...
Sat Oct 13, 2018, 02:55 PM
Oct 2018
9 grain bread croutons and salad greens. Just yum.
50. What time are you serving dinner?
Sat Oct 13, 2018, 03:14 PM
Oct 2018
58. I live just east of Cleveland.......nt
Sat Oct 13, 2018, 03:51 PM
Oct 2018
67. A fellow Northeasterner!
Sat Oct 13, 2018, 05:16 PM
Oct 2018
Sat Oct 13, 2018, 03:25 PM
Oct 2018
54. You've been watching too many horror films!
Sat Oct 13, 2018, 03:36 PM
Oct 2018
57. That and Quebecois Pea Soup is what we call Christmas Eve Dinnah
Sat Oct 13, 2018, 03:48 PM
Oct 2018
61. Like this Quebecois Pea Soup?
Sat Oct 13, 2018, 04:00 PM
Oct 2018
Last edited Tue Oct 16, 2018, 09:27 AM - Edit history (1)
Just didn't want people having to search for the recipe.
I'm not French but both of my 1st husband's parents and families were. My MIL taught me how to make meat pie and other good stuff. She passed away years ago but my kids still stay in touch with their
aunt, uncle and cousins. I remarried so I only see them once in a blue moon.
Tue Oct 16, 2018, 07:52 AM
Oct 2018
We have meat pie at Thanksgiving, Christmas and every time we want meat pie! Instead of ketchup, we smear it with gravy. We also make meat stuffing for our turkey. Every family makes it differently
and the best tasting pie is always yours!
59. tortilla soup - on the lighter side but....
Sat Oct 13, 2018, 03:52 PM
Oct 2018
very warming with the right amount of heat...........
Sat Oct 13, 2018, 08:25 PM
Oct 2018
I'll vote for tortilla soup on a cold day, or even any day for that matter!
Sat Oct 13, 2018, 03:56 PM
Oct 2018
One time I was in a restaurant with future son-in-law and he ordered the "Five-Alarm Chili". While we were waiting, there was an extremely loud "BANG!" (turns out a car ran into the kitchen wall). I
turned to him and with a straight face said "Your chili is ready!"
Sat Oct 13, 2018, 05:14 PM
Oct 2018
64. Split pea soup with ham.
Sat Oct 13, 2018, 04:43 PM
Oct 2018
Chicken noodle soup.
Go to a thai restaurant and get the cocoanut soup that comes with a fire under it.
65. My 1st thought also upon reading your headline
Sat Oct 13, 2018, 04:45 PM
Oct 2018
Chili, with sour cream, onion & a handful of fritos
mmm mmm mmm
69. Don't forget the shredded Cheddar!
Sat Oct 13, 2018, 05:17 PM
Oct 2018
Sat Oct 13, 2018, 05:46 PM
Oct 2018
68. Well - if you're Dutch, like my parents - you
Sat Oct 13, 2018, 05:17 PM
Oct 2018
make Stamppot . Kind of a shepherd s pie, all in one pot mixed up - mashed potatoes, bacon, onions, diced up fried pork chops or smoked sausage, and well-rinsed sauerkraut. Use ALL the bacon grease
and the pork chop butter to stir in with some heavy cream, salt, pepper and nutmeg.
Sometimes kale instead of sauerkraut.
Heaven. The best memories of snowy Indiana winter nights, snug and warm around the dinner table.
71. What a lovely memory!
Sat Oct 13, 2018, 05:20 PM
Oct 2018
That sounds very filling!
80. How about Sucade Ladden?
Sat Oct 13, 2018, 08:31 PM
Oct 2018
We go to a restaurant in Aruba and they have a delicious entree called Sucade Lappen. Of course we eat it while the temperature is in the seventies and when we were talking to some Dutch women where
we were staying they laughed because to them it is a dish for cold weather. Don't care, it tastes so good.
75. Split pea soup, with lentil soup a close second.
Sat Oct 13, 2018, 07:58 PM
Oct 2018
Vegetarian Scotch broth if I m feeling like spending a bit of time in the kitchen. But in a hurry a good old can of Anderson s Split Pea Soup works every time.
91. That's what I'm having for dinner!
Sun Oct 14, 2018, 08:27 PM
Oct 2018
And with a dollop of sour cream
It's amazing how full I am once done-- makes a perfect dinner
76. Large bowl of homemade soup...
Sat Oct 13, 2018, 08:01 PM
Oct 2018
and a big ole' chunk of chewy bread with butter.
Sat Oct 13, 2018, 08:03 PM
Oct 2018
Beef Stew . . .
Bouillabaisse . . . .
78. Grilled cheese and tomato soup!
Sat Oct 13, 2018, 08:04 PM
Oct 2018
81. Mac 'n cheese that is baked long enough...
Sat Oct 13, 2018, 08:39 PM
Oct 2018
to get crispy on top. The sharper the cheddar the better.
Sat Oct 13, 2018, 08:44 PM
Oct 2018
Sat Oct 13, 2018, 08:44 PM
Oct 2018
84. Mac & cheese with broccoli florets in it and a bowl of tomato soup.
Sat Oct 13, 2018, 08:45 PM
Oct 2018
87. I like making beef chili, Thai green curry w chicken and a coconut base, curry with an almond butter
Sun Oct 14, 2018, 02:00 AM
Oct 2018
Base, and just made a delicious New Mexican green chili stew with chicken and hatch chilis.
85. Beef Stew or Matzoh Ball Soup
Sat Oct 13, 2018, 11:00 PM
Oct 2018
88. Pumpkin Spice Chili !!!
Sun Oct 14, 2018, 01:22 PM
Oct 2018
89. Chunky potato soup and a cheese Frenchie.
Sun Oct 14, 2018, 01:32 PM
Oct 2018
Both homemade. And Frenchies can be made ahead of time and frozen.
Mega yummmmmm!
92. Cioppino, around the bay area it's a tradition. nothing like a big pot bubbling on the stove,
Sun Oct 14, 2018, 09:29 PM
Oct 2018
food, the aroma of tomatoes, garlic, fresh fish and shell fish. My family tradition is to make enough for a couple of days. You can also freeze it with out disastrous results. I took over cioppino
duties from my father.
"This Cioppino recipe is undeniably a San Francisco original, made famous in the 1850s by Genoese immigrant Giuseppe Bazzuro at his eponymous restaurant. Derived from the traditional ciuppin which
means "little soup" in the Genoese dialect the dish was originally a puree of cooked vegetables and leftover fish scraps. Over the years, Bay area chefs transformed it into a luxurious stew using
local delicacies such as dungeness crab, as in this version from the city's legendary Tadich Grill."
This recipe is similar to the one we work with:
93. Prime rib, twice bake potatoe, French onion soup and asparagus.
Sun Oct 14, 2018, 10:24 PM
Oct 2018
Tue Oct 16, 2018, 07:11 AM
Oct 2018
97. Home made chicken pot pie
Tue Oct 16, 2018, 12:14 PM
Oct 2018
I made one on Sunday, just finished it today. I'm making chili and a baked potato today.
Everything in these posts sounds so good!
Tue Oct 16, 2018, 01:18 PM
Oct 2018
99. Great minds think alike!
Tue Oct 16, 2018, 03:43 PM
Oct 2018
100. I'm doing a beef stew today.
Tue Oct 16, 2018, 04:34 PM
Oct 2018
Small red potatoes, carrots, red bell pepper, some wine in the broth. Because it's cold/overcast/damp here.
Wed Oct 17, 2018, 01:49 PM
Oct 2018
red lentils, sweet potatoes, black beans, carrots, corn, red peppers, turnips, parsnips and lots of onion. Must have smoked paprika.
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Mehta Research Group - Nonlinear Filtering
Controllability and observability are foundational concepts in dynamical systems and control theory. The dual nature of these concepts is of fundamental importance for both analysis and design of
Duality is coeval with the origin of modern control theory: The original duality principle appears in the seminal (1961) paper of Kalman-Bucy, where the problem of minimum variance estimation is
shown to be dual to a linear quadratic (LQ) optimal control problem. Notably, duality explains why, with the time arrow reversed, the covariance update equation of the Kalman filter is the same as
the differential Riccati equation (DRE) of optimal control.
Overview of research
For sixty years since Kalman and Bucy's seminal work, generalization of Kalman-Bucy duality to nonlinear stochastic systems (hidden Markov models) was believed to be impossible.
Not any more! Because we did the impossible. The foundational paper introduced a dual optimal control problem for nonlinear filtering. The mathematics of the paper is wild -- the constraint for
the optimal control problem is a backward stochastic differential equation (BSDE). A BSDE is a particular form of stochastic dynamical system where the arrow of time runs backwards. Just like it
did in that beautiful movie Tenet!
It is shown in our paper that the BSDE-based formulation is an exact extension of the original Kalman-Bucy duality -- in the sense that the dual optimal control problem has the same minimum variance
structure for both linear and nonlinear filtering problems. For this paper, Jin Won Kim won the Best Student Paper Award at the IEEE Conference on Decision and Control (CDC) 2019 from a competitive
field of 65 world-wide nominations for this award.
Since the foundational paper, we have written three more papers on the topic of duality:
In an MTNS paper, observability of an HMM is defined in dual terms: as controllability of the BSDE. It is shown that (i) the resulting characterization is equivalent to the definition of
observability for HMMs; and (ii) the BSDE is a dual to the Zakai equation of nonlinear filtering.
The CDC paper is the first of the two papers on the subject of nonlinear filter stability (asymptotic forgetting of the initial condition). A key contribution of the paper is the notion of
conditional Poincare inequality (PI) which is shown to yield filter stability. Using the dual methods, we are able to derive all the prior results where explicit convergence rates are obtained.
The CDC paper is the second of the two papers on the subject of nonlinear filter stability. The contributions of this paper are two-fold: (i) a definition is introduced for the stabilizability of
the BSDE; and (ii) shown to be necessary and sufficient for filter stability. The development of the paper -- introduction of the dual system, stabilizability definition, and its use the in the
filter stability analysis -- is entirely parallel to the Kalman filter stability theory.
What is the Lagrangian for nonlinear filtering? ISS Informal Systems Seminar / Séminaire informel de théorie des systèmes, McGill University, April 2, 2021.
Slides from 58th IEEE Conference of Decision and Control, Nice, France, Dec 11-13 2019. Winner of the Best Student Paper Award.
J. W. Kim, P. G. Mehta and S. P. Meyn, "What is the Lagrangian for nonlinear filtering?" in 2019 IEEE Conference on Decision and Control (CDC), Dec 2019, pp. 1607-1614.
J. W. Kim and P. G. Mehta, "A dual characterization of observability for stochastic systems," in 24th International Symposium on Mathematical Theory of Networks and Systems (MTNS), 2021.
J. W. Kim, P. G. Mehta and S. P. Meyn, "The conditional Poincaré inequality for filter stability," in 2021 IEEE Conference on Decision and Control (CDC), Dec 2021.
J. W. Kim, P. G. Mehta and S. P. Meyn, "A dual characterization of the stability of the Wonham filter," in 2021 IEEE Conference on Decision and Control (CDC), Dec 2021.
J. W. Kim and P. G. Mehta, "A dynamic programming formulation for the nonlinear filter," in 7th Indian Control Conference, 2021.
Some useful papers on filter stability
P. Chigansky, R. Liptser, and R. Van Handel, “Intrinsic methods in filter stability,” Handbook of Nonlinear Filtering, 2009.
D. L. Ocone and E. Pardoux, “Asymptotic stability of the optimal filter with respect to its initial condition,” SIAM Journal on Control and Optimization, vol. 34, no. 1, pp. 226–243, 1996.
P. Baxendale, P. Chigansky, and R. Liptser, “Asymptotic stability of the Wonham filter: Ergodic and nonergodic signals,” SIAM Journal on Control and Optimization, vol. 43, no. 2, pp. 643–669, 2004.
R. van Handel, “Nonlinear filtering and systems theory,” in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, 2010.
R. van Handel, “Observability and nonlinear filtering,” Probability theory and related fields, vol. 145, no. 1-2, pp. 35–74, 2009.
A. Budhiraja, “Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter,” in Annales de l’IHP Probabilit´es et statistiques, vol. 39, 2003, pp. 919–941.
R. Atar and O. Zeitouni, “Exponential stability for nonlinear filtering,” in Annales de l’Institut Henri Poincare (B) Probability and Statistics, Elsevier, vol. 33, 1997, pp. 697–725.
C. McDonald and S. Yüksel, “Converse results on filter stability criteria and stochastic non-linear observability,” arXiv preprint, arXiv:1812.01772, 2020.
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Lesson H: Range-only SLAM
This lesson is a summary of all the lessons of this tutorial. We will apply the concepts of constraints and interval analysis on a concrete Simultaneous Localization and Mapping (SLAM) problem, and
see how an online SLAM can be solved.
This exercise comes from IAMOOC: another MOOC related to Interval Analysis with applications to parameter estimation and robot localization. It provides complementary concepts and may be of interest
to you. https://www.ensta-bretagne.fr/jaulin/iamooc.html
This lesson is an adaptation of the Exercise 11 of IAMOOC. The difference is that we will now consider a continuous-time state equation.
Consider a robot moving in an unknown environment and described by the state \(\mathbf{x}=(x_1,x_2,x_3)^\intercal\), with \(x_1,x_2\) its 2d position and \(x_3\) its heading. The evolution of \(\
mathbf{x}\) over time is represented by the trajectory \(\mathbf{x}(\cdot):[t_0,t_f]\rightarrow\mathbb{R}^3\), with \(t_0=0\) and \(t_f=15\).
The motion of the robot is described by the state equation \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},u)\) with:
\[\begin{split}\mathbf{f}(\mathbf{x},u)=\left( \begin{array}{c} 10\cos(x_3) \\ 10\sin(x_3) \\ u + n_u \end{array}\right),\end{split}\]
where \(u\) is the desired rotational speed (input of the system) and \(n_u\) is a noise.
The desired input \(u(t)\) is chosen as:
\[u(t) = 3\sin^2(t)+\frac{t}{100}.\]
Contrary to the previous lesson, we assume that we know the initial state \(\mathbf{x}_0=(0,0,2)^\intercal\). This is common in SLAM problems.
We also assume that the heading is continuously measured from \(t_0\) to \(t_f\) (for instance by using a compass) with a small error:
\[x_3^m(t) = x_3^*(t) + n_{x_3}(t),\]
where \(x_3^*(t)\) represents the actual but unknown heading of the robot.
At any time, we consider that the errors \(n_u(t)\) and \(n_{x_3}(t)\) are bounded by \([-0.03,0.03]\).
The term simulation often refers to the integration of one dynamical system from a known initial condition. The system we are dealing with is \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},u)\) and its
initial condition is \(\mathbf{x}_0\). We will first compute the trajectory \(\mathbf{x}^*(\cdot)\) solution of this system, without uncertainties, and call it the truth.
Of course, the computation of \(\mathbf{x}^*(\cdot)\) will not be reliable: the result will depend on the integration timestep and the \(\delta\) parameter used to represent the trajectory. We will
only use the result for visualization.
H.1. Simulate the system. We will use \(\delta\) = dt = \(0.05\) for implementing the trajectories.
The simulation can be done with a classical temporal loop and an Euler integration method. With Codac, we can also compute the system at the trajectory level (applying operators on entire
trajectories), without temporal loop. For this, we will define the input of the system as a trajectory, and apply operations on it (from function \(\mathbf{f}\)) and integrations.
The following code computes \(\mathbf{x}^*(\cdot)\):
# ...
# Initial pose x0=(0,0,2)
x0 = (0,0,2)
# System input
u = Trajectory(tdomain, TFunction("3*(sin(t)^2)+t/100"), dt)
# Actual trajectories (state + derivative)
v_truth = TrajectoryVector(3)
x_truth = TrajectoryVector(3)
v_truth[2] = u
x_truth[2] = v_truth[2].primitive() + x0[2]
v_truth[0] = 10*cos(x_truth[2])
v_truth[1] = 10*sin(x_truth[2])
x_truth[0] = v_truth[0].primitive() + x0[0]
x_truth[1] = v_truth[1].primitive() + x0[1]
// ...
// Initial pose x0=(0,0,2)
Vector x0({0,0,2});
// System input
Trajectory u(tdomain, TFunction("3*(sin(t)^2)+t/100"), dt);
// Actual trajectories (state + derivative)
TrajectoryVector v_truth(3);
TrajectoryVector x_truth(3);
v_truth[2] = u;
x_truth[2] = v_truth[2].primitive() + x0[2];
v_truth[0] = 10*cos(x_truth[2]);
v_truth[1] = 10*sin(x_truth[2]);
x_truth[0] = v_truth[0].primitive() + x0[0];
x_truth[1] = v_truth[1].primitive() + x0[1];
Create a new project with this simulation.
Add a noise on \(u(\cdot)\) as mentioned in the presentation of the problem, and display the result.
We will now enclose the trajectory \(\mathbf{x}^*(\cdot)\) in a tube. For the moment, we do not take into account measurements from the environment. This is what we call deadreckoning: we estimate
the positions of the robot only from proprioceptive data, coming from the input \(u(\cdot)\) and heading measurements.
H.2. As we did for the computation of \(\mathbf{x}^*(\cdot)\), estimate the feasible state trajectories in a tube, according to the uncertainties on \(u(\cdot)\) and \(x_3(\cdot)\). We will assume
that the initial state \(\mathbf{x}_0\) is well known.
The functions cos, primitive(), etc., can be used on tubes as we did for Trajectory objects. This will propagate the uncertainties during the computations.
We will also use \(\delta\) = dt = \(0.05\) for the implementation of the tubes.
You should obtain a result similar to:
Note that if you obtain a tube \([\mathbf{x}](\cdot)\) that encloses accurately the actual trajectory \(\mathbf{x}^*(\cdot)\) without uncertainties, then you did not correctly propagate information
from the input tube \([u](\cdot)\).
We could use a Contractor Network for this deadreckoning estimation, but the use of simple operators on tubes is also fine, because we do not have observations or complex constraints to consider. If
fact, for deadreckoning, we are dealing with a causal system where information propagates in one direction from \(u(\cdot)\) to \(\mathbf{x}(\cdot)\):
The use of a CN (or more generally, contractors) is relevant when we do not know how to propagate the information on sets (when the above graphic reveals loops) and when complex constraints have to
be treated. This is typically the case when one has to consider observations on the sets, as we do in SLAM.
The environment is made of 4 landmarks. Their coordinates are given in the following table:
\(j\) Landmark \(\mathbf{b}_j\)
\(0\) \((6,12)^\intercal\)
\(1\) \((-2,-5)^\intercal\)
\(2\) \((-3,20)^\intercal\)
\(3\) \((3,4)^\intercal\)
Each \(t=2\delta\), the robot is able to measure the distance to one of these landmarks (taken randomly), with an accuracy of \(\pm0.03\). The robot does not know the landmarks coordinates (the M of
SLAM is for Mapping), but it knows which landmark \(\mathbf{b}_j\) is being observed (the landmarks are identified).
We will use a constraint propagation approach to solve the problem.
H.3. First, define the variables of the problem.
H.4. List the involved constraints and the potential decompositions to perform. This may introduce intermediate variables. Note that all the constraints describing this SLAM have been seen in the
previous lessons.
H.5. Define the initial domains of the variables:
• domains for intermediate variables will be set to infinite sets.
• other domains may be initialized from measurements or to infinite sets when nothing is known, as it is the case for the position of the landmarks.
H.6. Using a Contractor Network, improve the localization of the robot while simultaneously estimating the position of the landmarks by enclosing them into boxes.
You should obtain a result similar to:
These computations were made offline, assuming that all the data were collected before running the solver.
We could also use this approach online and make the solver run during the evolution of the robot. For this, we will use the .contract_during(ctc_dt) method instead of .contract(). This way, we will
let the solver contract as much as possible the domains during a given amount of time ctc_dt. Remaining contractions will be done during the next call to .contract_during(). This allows to spread
over time the resolution.
Hence, for real-time SLAM, we can use the following temporal loop:
import time # used for time.sleep
dt = 0.05
iteration_dt = 0.2 # elapsed animation time between each dt
tdomain = Interval(0,15) # [t0,tf]
# ...
# Create tubes defined over [t0,tf]
# Add already known constraints, such as motion equations
t = tdomain.lb()
prev_t_obs = t
while t < tdomain.ub(): # run the simulation from t0 to tf
if t-prev_t_obs > 2*dt: # new observation each 2*dt
# Creating new observation to a random landmark
# Adding related observation constraints to the network
# Updated last iteration time
prev_t_obs = t
contraction_dt = cn.contract_during(iteration_dt)
if iteration_dt>contraction_dt: # pause for the animation
time.sleep(iteration_dt-contraction_dt) # iteration delay
# Display the current slice [x](t)
#include <unistd.h> // used for usleep
// ...
double dt = 0.05;
double iteration_dt = 0.2; // elapsed animation time between each dt
Interval tdomain(0,15); // [t0,tf]
// ...
// Create tubes defined over [t0,tf]
// Add already known constraints, such as motion equations
double prev_t_obs = tdomain.lb();
for(double t = tdomain.lb() ; t < tdomain.ub() ; t+=dt)
if(t - prev_t_obs > 2*dt) // new observation each 2*dt
// Creating new observation to a random landmark
// Adding related observation constraints to the network
// Updated last iteration time
prev_t_obs = t;
double contraction_dt = cn.contract_during(iteration_dt);
usleep(max(0.,iteration_dt-contraction_dt)*1e6); // pause for the animation
// Display the current slice [x](t)
H.7. (optional) Transform the code of question H.6 to make it work online with boxes \([\mathbf{x}]\) contracted in realtime.
You should obtain an animation that looks like this:
On the above figure, we can notice that the contracted boxes \([\mathbf{x}]\) obtained during the online SLAM are sometimes larger than the blue tube computed offline as post-processing. The reasons
• at \(t\), the CN online may not have dealt with all the contractors: some contractions remain to be done. They will be processed afterwards, and the current box \([\mathbf{x}](t)\) does not
enclose optimally the set of feasible positions;
• at \(t\), the online SLAM does not take benefit from future measurements, while the offline SLAM was able to propagate all information forward and backward in time.
The tutorial ends here!
We do hope it provided you an interesting overview of what Constraint Programming methods can bring to mobile robotics. We are looking forward your feedbacks!
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Physics Cup – 2018, Problem 3
This problem required skills of using small parameters dealing with seemingly divergent integrals. At the very end, all the accepted solutions followed the rout sketched by the hints. However,
originally, there was one solution – provided by Nozomi Sakura – which was exact, but went mathematically well beyond the IPhO Syllabus, and was therefore rejected. Another reason for the rejection
was that the exact answer was expressed in terms of modified Bessel of the second kind; finding its asymptotic behaviour for large permeability was not a trivial task and had been executed
incorrectly. The solution was based on textbook exercises, (W.R. Symthe, Static and Dynamic Electricity). I feel that finding this textbook with hints for solving the problem exactly deserves a
credit, so he receives 0.4 of the bonus. I am not linking here his exact solution, because aforementioned reasons (anyone interested in the exact solution is encouraged to read the Symthe’s
textbook). Instead, here is asymptotic solution for large permeability. Keep in mind, though, that this solution has mostly formulas and explanations are minimal; as a matter of fact, the other
solutions linked below are explained much more carefully.
The rest of the bonus is distributed between those solutions which were submitted within first five weeks:
Thomas Bergamaschi;
Navneel Singhal;
Dolteanu Stefan;
Yunus Emre Parmaksiz.
And here are the results. Number of fully correct solutions: 9.
name school country Pr 2: solved; score
Nozomi Sakura Hiroo Gakuen High School Japan 2 Apr 9:22 3.869
Thomas Bergamaschi Colegio Etapa Valinhos-Brazil Brazil 2 Apr May 14:07 2.740
Navneel Singhal ALLEN Kota India 21 Apr 9:23 2.490
Dolteanu Stefan International Computer HS Bucharest Romania 29 Apr 12:20 2.264
Yunus Emre Parmaksiz Bahcesehir HSScT Turkey 30 April 8:11 2.058
Tóbiás Marozsák Óbudai Árpád Gimnázium Hungary 17 May 19:20 1.464
Mustafa Tugtekin Bahcesehir HSScT Turkey 7 May 5:40 1.449
Muhammad Farhan Husain Kharisma Bangsa High School Indonesia 18 May 4:00 1.331
Peter Elek DRK Dóczy Gimnázium Hungary 18 May 17:36 0.774
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THE UNITED REPUBLIC OF TANZANIA NATIONAL EXAMINATIONS COUNCIL
CERTIFICATE OF SECONDARY EDUCATION EXAMINATION
041 BASIC MATHEMATICS
(For Both School and Private Candidates)
Time: 3 Hours Tuesday, 31^stOctober 2017 a.m.
1. This paper consists of sections A and B with a total of sixteen (16) questions.
2. Answer all questions in sections A and four (4) questions from section B. Each question in section A carries six (06) marks while each question in section B carries ten (10) marks.
3. All necessary working and answers for each question done must be shown clearly.
4. Mathematical tables may be used.
5. Calculators, cellular phones and any unauthorised materials are not allowed in the examination room.
6. Write your Examination Number on every page of your answer booklet(s).
SECTION A (60 Marks)
Answer all questions in this section.
1. (a) Round off:
(i) 9.67 to ones,
(ii) 0.205 to one decimal place
(iii) 0.0197 to two decimal places.
Hence estimate the value of
View Ans
(b) Simplify the expressions:
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(c) Express 0.3636… in the form of ^a[b] where a and b are integers and b ? 0.
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(ii) log[3] 10 + log[3] 8.1
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(b) If nlog5125 =[ ] log264 , find the value of n.
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3(a) Factorize the following expressions:
(i) 16y^2 +xy -15x^2
(ii) 4 - (3x - 1)^2
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(b) At Moiva’s graduation ceremony 45 people drank Pepsi-Cola, 80 drank Coca-Cola and 35 drank both Pepsi-Cola and Coca-Cola. By using a Venn diagram, found out how many people were at the ceremony
if each person drank Pepsi-Cola or Coca-Cola.
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4. (a) Given the three vectors a = 4i + 6j, b = 4i + 10j and c = 2i + 4j determine the magnitude of their resultant.
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(b) Camilla walks 5 km northeast, then 3 km due east and afterwards 2 km due south. Represent these displacements together with the resultant displacement graphically using the scale 1 unit = 1 km.
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(c) Show that triangle ABC is right-angled where A = (-2,-1), B = (2,1) and C = (1,3).
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5. (a) In the figure below, AB = 10 cm, AX = 6 cm, CX = 8 cm and AB is parallel to DC.
(i) Show whether triangles AXB and CXD are similar or not.
(ii) Find the length of CD.
(iii) Find the ratio of the areas of triangles AXB and CXD.
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(b) Using a ruler and compass, construct an angle of 90°.
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6. (a) In the preparation of fanta orange drink, a bottling filling machine can fill 1,500 bottles in 45 minutes. How many bottles will it fill in 4
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(b) If X varies directly as Y and inversely as W, find the values of a and b in the table below.
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7. A computer is advertised in a shop as having a list price of sh. 2,500,000 plus value added tax (VAT) of 20%. The sales manager offers a discount of 25% before adding the VAT. Calculate:
(a) The list price including VAT.
(b) The amount of discount before VAT is added.
(c) The reduced final price of the computer.
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8. (a) If the sum of n terms of a geometric progression with first term 1 and common ratio
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(b) How many integers are there between 14 and 1,000 which are divisible by 17?
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9. In the figure below, AE = 20 m, EB = 20?2m and DAE = 45°.
(a) The length: DE, AD and AB.
(b) The area of triangle ABE, leaving the answer in surd form.
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10. (a) Solve the equation 4x^2 - 32x + 12 = 0 by using the quadratic formula.
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(b) Anna is 6 years younger than her brother Jerry. If the product of their ages is 135, find how old is Anna and Jerry.
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SECTION B (40 Marks)
11. Zelda wants to buy oranges and mangoes for her children. The oranges are sold at sh. 150 each and mangoes at sh. 200 each. She must buy at least two of each kind of fruit but her shopping bag
cannot hold more than 10 fruits. If the owner of the shop makes a profit of sh. 40 on each orange and sh. 60 on each mango, determine how many fruits of each kind Zelda must buy for the shop owner to
realise maximum profit.
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12. The heights of 50 plants recorded by a certain researcher are given below:
(a) Copy and complete this tally table for the data given above.
Height (cm) Tally Frequency
Use this table to:
(b) Draw a histogram for the height of the plants.
(c) Find the mean height of the plants (do not use the assumed mean method).
(d) Find the median of the heights of the plants.
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13. In the figure below, BC is a diameter of the circle, O is the centre of the circle and side CD of the cyclic quadrilateral ABCD is produced to E.
(a) With reasons, name the right angles in this figure.
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(b) Show that ADE = ABC.
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(c) If <ADE = 60 and <CAD = 25, find:
(i) the value of <ABD ,
(ii) the lengths AB and BD given that CB = 10cm.
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14. (a) What is a trial balance and what is its main purpose.
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(b) On January 1^st 2015 Semolina Women Group started a business with a capital in cash of
January 2 Purchased goods for cash 1,400,000/=
3 Sold goods for cash 1,000,000/=
6 Purchased goods for cash 600,000/=
15 Paid rent for cash 220,000/=
26 Paid wages for cash 220,000/=
15 Sold goods for cash 620,000/= Prepare:
(i) The cash account and balance it.
(ii) The Trial Balance.
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15. (a) Find the inverse and identity matrix of A = .
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15.(b) The triangle OAB has vertices at O(0, 0), A(2, 1) and B(-1, 3). If the triangle is enlarged by
Find the vertices of the triangle
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(c) Draw on the same x - y plane triangle OAB and the images after being:
(i) enlarged
(ii) translated
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16(a) A function f is defined as follows:
(i) Draw a pictorial diagram of f(x)
(ii) Find Domain And Range
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(b) Given that
find f^-1(4)
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(c) In a yard there are 500 vehicles, of which 160 are cars, 130 are vans and the remaining are lorries. If every vehicle has an equal chance to leave, find the probability of:
(i) A van leaving first,
(ii) A lorry leaving first,
(iii) A car leaving second if either a lorry or van had left first.
View Ans
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Sampling and reconstruction
Sampling and reconstruction are cornerstones of modern digital audio processing. We take a closer look at these processes and the limitations they impose.
Sampling is the act of converting a continuously varying signal to discrete samples suitable for digital manipulation. Reconstruction is the reverse, converting a sampled signal back to its
continuous form, which can then drive a speaker allowing us to hear it. Intuitively, the idea may appear impossible. How can a finite set of samples possibly capture the infinite variations of a
continuously changing signal? What about the signal segments between the sample points?
The sampling theorem
The questions posed above are resolved by the sampling theorem. Attributed variously to Claude Shannon, Harry Nyquist, E. T. Whittaker, and others, this theorem sets out conditions permitting a
continuous signal to be sampled and subsequently reconstructed:
1. The signal must be bandlimited.
2. The sample rate must be higher than twice the bandwidth of the signal.
If these conditions are met, the sampled values uniquely and exactly describe the full continuous waveform. There is no approximation. Increasing the sample rate cannot improve anything since there
is no error to begin with.
For a signal with bandwidth \(B\), the minimum sample rate, \(2B\), required for an accurate capture is commonly referred to as the Nyquist rate. Conversely, if the sample rate is \(f_s\), the
maximum allowed signal bandwidth, \(f_s/2\), is known as the Nyquist frequency.
Sampling a sine wave
Suppose we sample an 8 kHz sine wave at the commonly used rate of 48 kHz. The conditions of the sampling theorem are met since the sample rate is well above twice the maximum (and in this case only)
frequency of the signal. Figure 1 shows one period of this sine wave with the sample points marked as red circles.
If the signal frequency is too high, in this case above 24 kHz, a phenomenon known as aliasing occurs. In figure 2 we see a 40 kHz sine wave (green) together with the same 8 kHz signal as above
(dashed blue). Notice that the sample points end up in exactly the same locations for both waveforms. Also notice that 40 kHz is precisely 8 kHz below the sample rate.
Given only the sample data, it would be impossible to tell which of these two waveforms was the source. A similar aliasing situation occurs for again for a signal frequency 8 kHz above the sample
rate, that is 56 kHz, as can be seen in figure 3.
In general, for every valid signal frequency, there exist an infinite number of alias frequencies in symmetrical pairs around every multiple of the sample rate.
Anti-aliasing filters
Due to the aliasing effect illustrated above, it is important that the sampled signal is properly band limited. If it is not, any frequencies above the Nyquist frequency will alias into the lower
range and distort the capture. If we can’t be certain about the signal bandwidth, we must precede the sampling stage with an analogue low-pass filter. Since the purpose of this filter is to remove
alias frequencies, it is commonly called an anti-alias filter.
Ideally, the anti-alias filter would cut out everything from the Nyquist frequency and up, leaving the lower frequencies untouched. A perfect low-pass filter like this is, unfortunately, impossible
to construct in practice. The solution is to set the sample rate, not to precisely twice the highest frequency of interest, but somewhat higher, providing some margin between the top of the target
band and the point where aliasing sets in. This allows the anti-alias filter a transition band wherein its response gradually goes from passing frequencies below to blocking those above.
The generally accepted upper limit for human hearing is 20 kHz. A sampled audio system thus needs a sample rate of at least 40 kHz. With a little margin added for the anti-alias filter, we arrive at
the common sample rates of 44.1 kHz and 48 kHz. Those exact frequencies were chosen for technical reasons unrelated to the sampling process.
If we accept some aliasing distortion above 20 kHz, the width of the transition band can be doubled. This is possible since the aliases are mirrored around the Nyquist frequency, so for a 48 kHz
sample rate, a 28 kHz signal component is aliased to 20 kHz.
Even when permitting aliasing in the transition band, an anti-alias filter suitable for a 44.1 kHz or 48 kHz sample rate can be a challenge to design. This task is simplified by sampling at a much
higher rate followed by a digital decimation stage since a digital low-pass filter can readily be made very steep without adversely affecting the pass band or requiring high-precision components.
Oversampling, as this technique is called, additionally permits the use of a less accurate A/D conversion stage while maintaining the same signal to noise ratio in the audio band. In its simplest
form, each doubling of the sample rate gains one effective bit of resolution, and noise shaping can improve this further.
For audio purposes, sampling would be mostly useless without a means of converting the signal back to its analogue form. After all, our ears do not accept digital inputs.
Mathematically, a sampled signal can be viewed as a sequence of impulses, one for each sample, with heights corresponding to the sample values. This is illustrated in figure 4.
That doesn’t look much like a sine wave. However, computing the Fourier transform yields the spectrum in figure 5 below.
Below the Nyquist frequency, 24 kHz, everything looks good with a single 8 kHz tone, exactly as desired. Above 24 kHz, things are not looking so good. There are additional tones at 40 kHz, 56 kHz,
and so on around every multiple of the sample rate, and effect called imaging. For every actual frequency in the signal, this crude reconstruction has generated a multitude of image frequencies. As
the reader may have noticed, these additional frequencies coincide with the alias frequencies we encountered during the sampling process.
Frequency imaging aside, an impulse based D/A converter isn’t practical. Such fast switching while producing an accurate voltage level is not easily achieved. A more reasonable approach is to hold
the output voltage constant for the duration of each sample. This gives us the waveform displayed in figure 6.
This method is called a zero-order hold. The curve it produces looks a little more like a sine wave, though it still has some way to go. Figure 7 shows the spectrum.
As we can see, this method also produces the same image frequencies. Their level drops a little as the frequency increases, though not by much. Clearly, something must be done.
Anti-imaging filters
A solution to the problem of image frequencies is to simply remove them using an analogue low-pass filter, unsurprisingly referred to as an anti-imaging filter. If we remove everything above the
Nyquist frequency, 24 kHz, only the originally sampled signal remains.
As with the anti-aliasing filter earlier, a perfect low-pass filter is impossible to construct. We do, however, still have the margin between the limit of hearing, 20 kHz, and the Nyquist frequency
within which to work. Of course, that rather small margin still presents the same challenge.
Oversampling (again)
Once again, oversampling comes to the rescue. If we increase the sample rate by inserting one or more zeros after each sample, we obtain a digital version of the impulse sequence we looked at
previously. The image frequencies in its spectrum can now be removed using a digital low-pass filter, which as already noted, is much easier to implement.
Having done a digital oversampling of the signal, we can then pass it to the same zero-order hold D/A converter as before. The output from this process using a 2x oversampling can be seen in figure
While there are still steps, they are smaller, and the reconstruction follows the desired curve much more closely. In figure 9 we see that also the spectrum has been improved
The first pair of images, around 48 kHz, is gone, as are those for all odd multiples of the sample rate. The digital oversampling took care of that. To get rid of the remainder, a much more
reasonable analogue filter can be used. The higher the oversampled rate, the simpler the analogue anti-imaging filter can be. In practice, an oversampling factor of 8x is common, placing the first
images around 384 kHz.
Final words
Sampling captures a continuous signal up to a maximum frequency, and the reconstruction process does the reverse, turning discrete samples back into a continuous waveform. There is a lot of symmetry
between the two processes. Both rely on low-pass filters to function correctly, which presents some challenges. Likewise, digital filtering techniques operating at a higher sample rate greatly
simplify this task.
7 Replies to “Sampling and reconstruction”
1. It’s awesome that this article talked about the solution to getting around setting up a low pass filter. I appreciate you helping me learn more about these filters and how to work with them. I
will have to look into a way to work with anti alias filters in the future.
2. This makes sense in the studio… but once the digital media has been created, what good does oversampling in a DAC do? If the audio is encoded at 44.1khz, no additional information of a higher
resolution can be obtained… you’d simply be sampling the same data point multiple times, no?
Even with the 8x oversampling example above, I still don’t understand how a smooth sine wave can be generated from a series of square steps – even if it’s 8 times as many. the example is 8khz,
but consider sampling 20khz. Even with oversampling that’s not very many samples… is the analog output stage actually tracking the sampled/oversampled zero-hold output, i.e. attempting to adjust
and hold it’s voltage in steps? Are high frequency waveforms rounded out simply by virtue of analog components not being able to swing their voltages instantaneously, such that the next step up
or down in voltage is gradual and thus smoothed out? That still wouldn’t be accurate, at least not for all frequencies.
This is great information but I’m still missing something, maybe it’s the last step. How does the digital stepped representation become uniformly gradual and produce an ACTUAL sine wave such that
if we were to zoom in closely the ramp up/down would be smooth and at the proper angle?
1. ^^^ This question I still have btw! I get how the sampled data accurately represents the analog waveform… but how we turn that sampled data INTO an analog waveform from a digitized zero-hold
series of steps?
2. The stepped output is made smooth using an analogue low-pass filter. The benefit of digital interpolation is that the smaller steps it yields allow a simpler analogue filter to be used. Since
the analogue filter isn’t perfect, some residual of the unwanted higher frequencies will remain. Again, the digital interpolation helps by moving these artefacts to higher frequencies where
they are less harmful and the analogue filter is also more effective.
3. Since storage capacity is steadily increasing, why isn’t all audio recorded and delivered (either through streaming or download) at the highest possible bit-depth/resolution and sampling rate?
If I understood this article, that would allow for cheaper DACs since you wouldn’t need additional oversampling required by the analogue filters?
A side effect of this allows you to remove/not use dithering, reducing noise floor and increase dynamic range. I have never tried a really high-end audio setup, so I don’t know if it’s
appreciable enough to warrant the increased storage space and probably more expensive recording equipment?
I’m also thinking from a preservation standpoint. Perhaps in the future all DACs regardless of price can easily reconstruct 24/192 for instance, then it would be a shame if the recording wasn’t
available in high resolution (given that it is actually appreciable, as stated above) just because we wanted to save a bit of storage space or money today.
I would like to see some graphs in the “Anti-aliasing filters” – section:
1) Ideal vs. practical low-pass filter
2) Something to visualize the two last sentences in that section. What do you mean by “so for a 48 kHz sample rate, a 28 kHz signal component is aliased to 20 kHz.”? I assume it is the same as
Fig.1-3 are showing, but it’s hard to mentally visualize.
Anyways, read all your articles. Not sure I understood everything, but they were still insightfull and interesting. Great work!
4. Hello, I have an IFI Zen v2 (DSD1793) currently running with the non-MQA firmware. I am curious to know which upsampling/downsampling rate (using SOX in JRiver) is optimal to feed the DAC? Would
it be better to use 192kHz or 384kHz? Please help.
1. The answer depends on what you mean by optimal. Regardless, however, I would stick to 192 kHz or below as then the DAC chip will do a further 8x upsampling. With 384 kHz input, this is
bypassed. Looking at the audible range only (up to 20 kHz), the DAC chip tends to perform slightly better at lower sample rates, though the difference is too small to be audible.
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Target flowmeters, also known as drag force flowmeters, insert a target (drag element), usually a flat disc or a sphere with an extension rod, into the flow field. They then measure the drag force on
the inserted target and convert it to the flow velocity.
One major advantage of the target flowmeter over other flowmeters is, with a sphere drag element, a proper strain gage layout, and well thought-out mathematical formulas, a target flowmeter is
capable of measuring sporadic and multi-directional flows.
Further Information
The key to the success of a target flowmeter is the measurement of the drag force.
The drag force F[d] is given by the drag equation of incompressible flow:
where V is flow velocity, A is the projected area of the target, and C[d] is the drag coefficient to be determined experimentally based on the flow conditions and the geometry of the drag element.
For flat plate and sphere, the drag coefficients typically are:
Please note that the 1.28 drag coefficient is for flat plates that are perpendicular to the flow direction and the drag coefficient for spheres is related to the Reynolds Number.
For a given design, A and C[d] are constant. If the density of the liquid F[d] is solely a function of V^2.
With strains at certain points measured by strain gages, the drag force can be calculated by a beam-bending formula (cantilever beam with an end load). The flow velocity can consequently be obtained.
Suppose the strain gage is attached at the front and/or back surface(s) of the extension rod (
where x direction, M is the bending moment, y is the coordinate perpendicular to the rod's longitudinal (x) direction, E is Young's (elastic) modulus of the rod material, and I is the rod's cross
section's area moment of inertia. (Please refer to the beam theory for further details. The area moment of inertia can be found in Area of the Mathematics section.)
The bending moment M of a cantilever beam with length L under a concentrated end load P is
where x is measured from the fixed end. In this case, L is the length of the extension rod, and P is the drag force F[d].
The strain becomes
and the drag force can be expressed in terms of strain:
Plug the above formula into the drag equation we will have the following expression:
This formula for V is valid under the following assumptions:
1. The cross section of the extension rod is much smaller than the diameter of the drag element.
2. The drag element is much smaller than the inside diameter of the pipe.
Otherwise, further calibration is needed to determine the correction factors.
Common Specifications
Common specifications for commercially available target flowmeters are listed below:
Fluid Phase: Score Phase Condition
Gas Clean
Liquid Clean
Steam Saturated
Liquid Corrosive
Line Size: Inline models: 15 ~ 150 mm (0.5 ~ 6 inch)
Insertion models: 100 ~ 1500 mm (4 ~ 60 inch)
Turndown Ratio: 15 : 1
Pros and Cons
- Low initial set up cost
- Can be used in abrasive, contaminated, or corrosive fluid flow
- Can be made to measure flow velocity that is sporadic or multidirectional with sphere drag element designs
- Pressure drop is inevitable due to the rod and the drag element
- Less popular than it was before
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The Prisoner's Dilema
Imagine you and another person commit a crime together. Futhermore, imagine it was the perfect crime – completely foolproof – except it wasn’t – because the cops are pretty sure you two did it. In
fact, they know it was you two; the only thing stopping them from locking both of you up in jail for a very, very long time right now is this stupid burden of proof reminding them their evidence
isn’t completely airtight ^1. They could do some footwork and get more proof, but a confession seems easier. They have enough evidence to temporarily arrest you and your crime mate, and question you
in jail for a couple nights.
The first night, you two are separated. One cop takes you to an interrogation room, and offers you a deal. “Look. I know you did this. You know you did this. Everyone here knows you did this. We’re
gonna keep you in here for a couple days, then we’re gonna get the rest of the evidence we need and you’ll spend the rest of you life behind bars. Or, we can work together. Getting evidence is
annoying. I already know you and your partner did this, so I’d rather just put one of you away now. Confess. Rat on your partner, and we’ll give you a reduced sentence.” At the same time, another cop
took your crime mate to a different room, and gave him the same speech. You both have the choice giving up your partner in crime to help yourself, or staying quiet. Let’s say if you both rat each
other out, you both get 2 life sentences in jail; if you both keep quiet, there’s only enough evidence found for you both to get 1 life sentence; and if you rat him out, but he doesn’t rat you out,
you get no jail time and he gets 3 life sentences ^2. What would you do? Would you turn on your partner, or keep quiet? This is an example of what’s called a Prisoner’s Dilema.
The scenario above might seem a little contrived, and like it’s not something work considering, but Prisoner’s Dilemas show up in other places too. The main thing that makes a situation a Prisoner’s
Dilema is that you can screw someone over, or help them at your expense. Another example of this is working on a partnered project ^3. You can put in a lot of work (at the expense of having to go
through the trouble of putting in work), or be lazy and let your partner do everything themselves. Another example is performance-enhancing drugs in sports. There are two competing athletes. Each
athlete can take drugs (making them better than an undrugged opponent, but giving them health and legal risks) or compete all natural.
What do you do?
Let’s return to our original example of a Prisoner’s Dilema, and see if we can figure out what we should do. There are two people, you an your partner. You can each (D)efect from your partner by
turning on him, or (C)ooperate with him by staying quiet. Depending on what the two of you choose, you’ll be in jail for some number \(y\) of life sentences, and you want to minimize this number.
Personally, I’ve often heard that people should strive for greatness, so I think we should tweak this so you want to maximize something instead^4. That being said, minimizing \(y\) is the same as
maximizing \(3-y\), so we’ll do that. All of this can be nicely represented using the matrix^5 below
\[\begin{array}{l | c | r |} & C & D\\ \hline C & 2,2 & 0,3 \\ \hline D & 3,0 & 1,1 \\ \hline \end{array}\]
The above says, for example, that if the first (row) player^6 defects while the second (column) player player cooperates, then the first player gets 0 life sentences (3-0=0) and the second gets 3
life sentences (3-3=0). As another example, it says if both player cooperate, then they both get 1 (3-1=2) life sentence.
If you look at this matrix for a while, it might become clear what you should do. No matter what your partner does, it is always better for you to defect than to cooperate^7, so you should rat out
your partner^8. However, this matrix is symmetric, and so by the same reasoning, your partner should rat you out too. Thus, it makes sense to predict that in this situation, you’ll both defect and
spend 2 life sentences in jail.
Wait, what?
Hopefully, the argument above makes sense. It’s always in your best interest to defect, and so you should always defect. You and your partner will both want to do what’s in your individual best
interests, so you’ll both defect. Logically, this makes perfect sense, but it feels wrong. You could both cooperate instead and both spend less time in jail. This is where the dilema lies: defecting
is clearly better, but it would be nice if you both cooperated. The issue with you both cooperating though is that you don’t communicate with each other beforehand, and so if you cooperate, you don’t
know that your partner will as well. But let’s say that’s not the case. Let’s say you and your partner expected the cops to do something like this once you knew they were on to you, and so you agreed
beforehand to both cooperate. That’s it. Problem solved, right? Wrong. If you both swore to cooperate, and you believed wholehartedly that your partner was going to cooperate, then it would be in
your best interest to defect! Why go to jail for 1 life sentence, when you could go for none instead? It doesn’t stop there, though. Your partner can use this same logic to decide that he should go
back on his word and defect instead too, so even if you agree ahead of time to cooperate, it still seems reasonable to expect that you will both defect.
So maybe there’s no escaping the both of you defecting. Maybe that’s just what’s meant to happen. Maybe, but maybe not. Despite all these valid arguments against it, it can still feel like you and
your partner should just both cooperate. As we have modelled things now, there isn’t really a way of interpreting this so that we can safely arrive at this conclusion, so maybe we’re modelling this
situation incorrectly.
There’s only 1 interrogation room
In our current model, you and your partner essentially move simultaneously. The cops separate you, offer both of you deals, and then you respond before knowing what the other did. This might not be
completely realistic. Maybe the cops took you away before taking your partner away and so you were able to somehow communicate with him what you did. Maybe there’s only 1 interrogation room, so you
had to go one at a time and the other was able to figure out what the first one did. Maybe you didn’t commit a crime and there’s no police involved and you’re actually in a different prisoner’s
dilema where it makes more sense to think about you two moving one at a time (like drugs in sports)^9. It doen’t matter the reason. Let’s just say you decide to defect or cooperate first, your
partner sees what you did, then he decides to defect or cooperate. If he sees that you cooperated without a doubt, that might be enough for him to cooperate as well. Our model now looks like this
The poorly made tree above describes the situation where 1 is interrorgated first, decides to defect or cooperate, 2 sees what 1 did, then 2 is interrorgated and decides to either defect or
cooperate. Let’s say you cooperate. Unfortunately, evening knowing that, 2 still won’t cooperate with you; he will defect. It doesn’t matter than 2 knows your loyal. To him, its a choice between 1
life sentence and 0 life sentences. He’s gonna choose 0.
This whole defect thing is seeming pretty convincing, but let’s not give up yet. There’s still more ways we can think about this ^10.
We’ll meet again
Recall from the aside that there are other examples of prisoner’s dilemas than this somewhat literal scenario. If you both defect (or both cooperate), you’ll both end up in jail. Certainly, you will
interact with each other again, and in doing so, you will likely end up in other prisoner’s dilemas in the future^11 albeit less literal ones. With this in mind, maybe if you both cooperate now, you
will be willing to continue cooperating in the future since its better than both of you defecting. This could be what we need to give justification to our intuition.
Let’s say the two of you will experience \(k\) prisoner’s dilemas in total (no matter what choice you make in any of them). At each dilema, you both aquire some payoff described according to the
numbers in our original payoff matrix^12. At the end of the \(k\) dilemas, your payoff is the sum of all the payoffs you received during the individual dilemas, and this is the quantity you want to
maximize. For your convience, the payoff matrix is reproduced below.
\[\begin{array}{l | c | r |} & C & D\\ \hline C & 2,2 & 0,3 \\ \hline D & 3,0 & 1,1 \\ \hline \end{array}\]
Choose \(k\) arbitarily. Let’s say you’ve both agreed to always cooperate. Can we reasonably expect that this is what we’ll do, or will we have some incentive to deviate from this strategy. It may be
hard to think about this by imagining all \(k\) dilemas at once, so let’s say you are on the \(k^\text{th}\) dilema. You’ve come accross \(k-1\) dilema situations with your partner, and every single
time, you both cooperated. Things are going fairly well. This is your last dilema together; clearly, you should cooperate again, right? Wrong. It’s the same argument as always. If you are so sure
your partner will cooperate, then it’s in your best instance to defect!
But wait, that was just for the last dilema. Maybe you too can still cooperate up until then; play nice in the beginning to gain your partner’s trust, only to backstab him in the end. If only this
could work. Your partner can reason the same way and so you will both conclude that the other one will turn on you and defect in the end. Once you realize there’s no point in cooperating the first \
(k-1\) times (since it won’t convince your partner to cooperate the \(k^\text{th}\) time), your only choice left is to defect the \((k-1)^\text{th}\) time, and cooperate at most \(k-2\) times.
Unfortunately, this line of reasoning can be repeated again and again, resulting in both of you defecting every single time, and this is true no matter what value of \(k\) you pick. So it seems even
if you meet again, you really can’t be convinced it’s worth while to cooperate with this guy.
When is the last time, though?
The argument above all stemmed from one fact. No matter how much you cooperate, when you finally reach the end of it all, you’re gonna want to defect. This argument contains one perculiarity though;
it implicitly assumes that you know when your last dilema is. I don’t know about you, but I can’t predict the future. When I interact with people, I can’t say for sure if it’s the last time we’ll
interact or not. Certainly, it seems strange to use a model where people have these incredible powers of foresight. So, what model do we use now?
Our previous observation that future interactions can cause you to want to cooperate still seems reasonable to me. Last time, it went wrong because eventually you knew you had no more future
interactions, and then everything unwound from there, but like we said, in the real world you could always potentially have more interactions down the line. To model this, instead of saying you face
\(k\) dilemas in total, you and your partner will face infinitely many dilemas together. We’re trying to keep things realistic here, and this fails on at least two accounts. 1.) You never interact
with anyone inifintely many times in the real world. 2.) When you interact with someone, you may not know how many more times you will, but you’re probably more confident that it will happen (at
least) once more than that it will happen (at least) 10,000 times more. To address these, we will introduce a discount factor \(\delta\)^13 where \(0<\delta<1\). This number can be thought of a
measure of how patient you are, how likely it is that your current dilema is not the last one, etc.
Now let’s explicitly say what our latest model is. You and your partner play infinitely many prisoner’s dilemas, and at each one, you have the same possible actions (defect or cooperate) and the same
payoffs (given by the matrix somewhere above). For each dilema, you play it like normal (moving simultaneously), and receive some payoff. Previously, your goal was the maximize the sum of all your
payoffs. Here, your goal is the maximize the discounted sum of your payoffs. Letting \(u_t\) be your payoff from the \(t^\text{th}\) dilema, the quantity you are trying to maximize is
\[\begin{align*} \sum_{t=0}^\infty\delta^tu_t \end{align*}\]
We can’t use the same argument from before to say that you will both defect every time since that argument depended on having some last interaction to start (and argue backwards) from. So far, so
good. Let’s say you both agreed beforehand to always cooperate. Do you still have incentive to start defecting? The answer is\(\dots\) of course you do. Think about it. You both agreed to always
cooperate, so you can be confident your partner will always cooperate. If that is the case, you should always defect instead, and you will get a strictly higher payoff (independent of the value of \
(\delta\)). Your partner can use this reasoning too, and so it is still reasonable to expect both of you to always defect.
That argument makes sense, but it really only makes sense if you expect your partner to cooperate no matter what. That isn’t a very likely thing. So, let’s not give up; let’s change our strategy.
Instead of agreeing to always cooperate, you and your partner agree to always cooperate with the stipulation that if one of you ever defects, then you both start defecting from their on. Like before,
if you both follow this strategy, you will both always cooperate. However, now if one of you deviates, he won’t be forever rewarded with strictly higher payoffs. He’ll have exactly one instance where
his payoff is higher than before (3 instead of 2), and infinitely many where its lower than before (1 instead of 2). Is that it? Have we finally found justification for our intuition? I mean, there’s
no way one instance of higher payoff is worth infinitely many instances of lower payoff, right? right? Well\(\dots\)
Infinite is certainly bigger than 1, but we’re not maximizing the outright sum of individual payoffs, but the discounted sum of payoffs. Certainly, if \(\delta=0\), we would still want to defect in
the original dilema since it would be the only one giving us any payoff^14. \(\delta\) can’t be 0, but it can be close to zero. You would imagine that for small enough \(\delta\), this reasoning
still (essentially) holds, and we conclude that you should in fact still defect. It’s possible that this is true for all \(\delta\) (or at least for fairly large \(\delta\), like 0.95) and that would
mean that, unless you were very patient or very certain you will keep on seeing this guy, you should still defect.
The only way to know for sure if this model gives satisfying results is to figure out what for which values of \(\delta\) you should always cooperate. Certainly, if someone defects somewhere along
the way, then you should always defect from there on out. No matter what \(\delta\) is, once someone has defected, you expect your partner to always defect, and so it makes no sense for you to
decrease your payoff by cooperating. Thus, the only thing we need to consider in our analysis is if it is ever a good idea to defect in the first place. Fix any \(T\), and imagine you defect for the
first time at time period \(T\). Let’s compare what happens when you always conform to your agreed upon strategy and what happens when you deviate at time \(T\) by defecting.
\[\begin{matrix} \text{conform:} & (C, C) & (C,C) & \dots & (C, C) & (C, C) & (C, C) & (C, C) & \dots\\ \text{deviate:} & (C, C) & (C,C) & \dots & (D, C) & (D, D) & (D, D) & (D, D) & \dots \end
If you conform, you both always cooperate. If you deviate at \(T\), then you both cooperate for a while, then you defect while your partner cooperates, then you both defect forever. You will only
want to deviate if it increases your payoff, so let’s look at the payoffs from coforming and from devating. If you conform, your payoff is
\[\begin{align*} \sum_{t=0}^\infty2\delta^t=\frac2{1-\delta} \end{align*}\]
However, if you deviate, your payoff is
\[\begin{align*} \sum_{t=0}^{T-1}2\delta^t + 3\delta^T + \sum_{t=T+1}^\infty\delta^t =\frac{1-2\delta}{1-\delta}+\frac2{1-\delta} \end{align*}\]
You only want to deviate if your payoff is higher, so you will only deviate when \(\delta\) satisfies the following.
\[\begin{align*} \frac{1-2\delta}{1-\delta}>0 \iff 1-2\delta>0 \iff \delta<\frac12 \end{align*}\]
Which means you will want to always cooperate if \(\delta\ge\frac12\)^15! Finally, true justification for our intuition. This essentially says that if your at least as patient as average ^16, then
you want to cooperate, which makes sense. You wouldn’t really expect a short-sided person to want to cooperate, but you would expect someone willing to wait for future benefit to.
Some Thoughts
The Prisoner’s Dilema – and more generally, the type(s) of reasoning we touched upon here – come from a field known as Game Theory. Basically, game theorists study how rational beings should behave
in different situations. There’s way more to game theory than just this prisoner’s dilema, and many more types of models of games than I introduced here.
This quarter I am taking ECON 180: Honors Game Theory. Before starting the class, I knew very little about game theory. I had heard of the subject (and it seemed interesting), and I had watched a few
online lectures of a game theory course at Yale. From what little I had seen, I had the idea that game theorists studied only simple models (like the first one we had here), and that they only
searched for Nash Equilibrium^17, which didn’t always match up with intuition or accurately predict human behavior. Don’t get me wrong. I expected that there was more to this, and wanted to learn the
deeper theory behind it all; I was just not completely convinced the theory was very deep. Certainly, I did not expect to come across even just the different types of models (informally) used in this
post. Through this class, I have found that game theory is a much deeper field than I expected, and have even entertained the thought of taking more classes in it, and possibly making it the focus of
my studies. I’ve discovered that Nash equilibria aren’t the only equilibria^18, that this field has more history than I thought^19, that almost any situation is a game^20, etc.
I was reminded of this yesterday during class. A little bit of context: I was not feeling like going to class. I was super tired, I had just come off a break and was overall less motivated to get
work done, it’s an almost 2 hour long class, I wanted some food, etc. Honestly, the main thing that caused me to go was that I needed to turn in a problem set. But class started, the professor had
written on the board what we would be talking about that day, and my attitude instantly changed. We were going to be talking about learning and evolution. At the beginning of this quarter, he had
mentioned that we would cover this towards the end of the quarter, and I had been looking forward to it ever since.
For a little bit of context, my current future goals lie in AI. I hope to one day some sort of research in artificial intelligence, and specifically, machine learning, so the topic vibed with me. Not
only that, but throughout the course, many times we’ve gone over something that reminded me of reinforcement learning. The idea of players trying to take actions to maximize a payoff functions often
had parallels with the idea of an agent trying to take actions to maximize future reward in an environment. Every time I was reminded of this seeming connection, I would make a mental^21 note to do
some investigating of it on my own, to write some programs where 2 RL agents repeatedly play a game and see if their strategies converge to nash equilibrium, for example. However, I never went
through with these plans; they always just lied dormant on my mind and on my paper. Sitting in class yesterday, finally talking about learning in game theory, I kept on thinking about what I had
gained from this class, what I wanted from this class that I forgot to pursue, and what, if any, insights I can gain from exploring parallels between game theory and RL.
There’s no lesson I’m trying to convey with this. It’s just something on my mind.
1. Note: I know basically nothing about the legal system and crime and justice and how it all (attempts to) work together. Just go along with me here, and pretend everything I’m saying makes sense
and is as it should be. ↩
2. Here, these are life sentences with the possibility for parole after a few decades (depending on how many you get) of good behavior. ↩
3. I think this is a Prisoner’s Dilema, but I haven’t put enough thought into it yet to confirm that. I think it will be more clear if I just lied or not after the next section ↩
4. And this is my blog, so we do whatever I want ↩
5. table? ↩
6. From here on, I’ll use player to refer to you or your crime mate. You can be player 1 and he’ll be player 2. ↩
7. Defecting strictly dominates cooperating ↩
8. What has he every done for you anyways, other than help commit a crime that might land you in jail for the rest of your life? ↩
9. maybe its maybelline ↩
10. At this point, having spent longer than I would like to admit making the above (admittedly not very high quality) graphic, I decided to call it a night, and continue writing tomorrow. This is the
first time I have split writing a single post between two days. I’m curious to see if I remember to include everything I wanted to after spending so much time away from it. ↩
11. This is still possibly true even if one you defects while the other cooperates. ↩
12. Previously, we interpreted these numbers in terms of life sentences. That now only makes sense for the first dilema. Future dilema’s you two face may have nothing to do with jail time.
Nevertheless, your payoffs can be assigned these same values because what matters is the relationship between the numbers, and not their specific values. ↩
13. delta ↩
14. here, we’re using the convention that 0^0=1 ↩
15. and you and your partner agreed upon this specific strategy, either explicitly or implicitly, beforhand. ↩
16. assumint patience is, for example, distributed uniformly from 0 to 1 ↩
17. Situations in simultaneous games where everyone is doing the best thing given their opponent’s action. One example is, perhaps unsurprisingly, both players defecting in the prisoner’s dilema. ↩
18. In fact, there are too many of them. In class so far, we’ve talked about nash equilibria, subgame perfect equilibria, bayesian nash equilibria, weak perfect bayesian equilibria, and sequential
equilibria. These aren’t even all of them. Not even close, I think. ↩
19. There were people doing game theory in at least the 1800s. I always assumed it was something that got start in the early or mid 1900s. ↩
20. I’ve started a habit of relating things in my daily life to games from class ↩
21. and sometimes physical ↩
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Arrange the 4 pieces below so they fit perfectly inside the triangle.
Here's a hint if you need it:
Now rearrange the pieces into the same triangle using the new outline shown in the 2nd triangle:
What is happening here? The triangle didn't get bigger and the pieces stayed the same size. Yet, we have an empty square in the triangle!
The exploration is called Curry's Paradox. A paradox is a seemingly contradictory situation. Here, we have the same 4 shapes that seem to have different area based on how we arrange them. That
doesn't seem to fit with our understanding of area. Can you explain what is happening here? Mathematician Haskell Curry created this paradoxical situation.
Let's start making sense of this by finding the area of the triangle and the shapes.
We've found the issue! The triangle has an area of 32.5 while the four pieces have an area of 32. That must mean when arranged in this manner, the 4 pieces don't quite fill up the triangle:
And when arranged like this, the 4 pieces plus the extra square have an area of 33. So they must be extending beyond the triangle slightly:
The blue triangle and the red triangle do NOT make a straight line. To see this, drag the blue triangle on top of the red triangle and you'll see the longest sides (the hypotenuse) don't line up
exactly. Zoom in as needed.
Here is a close-up:
The blue triangle goes up 2 units for every 5 units over, while the red triangles goes up 3 units for every 8 units over. $2/5$ does not equal $5/8$. These triangles have different steepness.
Further, $2/5 > 5/8$, which confirms that the hypotenuse of the blue triangle is steeper.
The large black triangle goes up 5 units for every 13 units over:
• $2/5 = 0.4$
• $3/8 = 0.375$
• $5/13 \approx 0.385$
So, the steepness of the black triangle is in between the steepness of the blue and red triangle.
Below are two pictures of VERY exaggerated versions of this situation:
In this picture, the 4 shapes don't fill up the triangle.
In this picture, the 4 shapes overflow outside of the triangle.
Can you make up a paradox of your own?
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Maharashtra Board Class 6 Maths Solutions Chapter 17 Geometrical Constructions Practice Set 40
Maharashtra State Board Class 6 Maths Solutions Chapter 17 Geometrical Constructions Practice Set 40
Question 1.
Draw line l. Take point P anywhere outside the line. Using a set square draw a line PQ perpendicular to line l.
Step 1:
Step 2:
line PQ ⊥ line l.
Question 2.
Draw line AB. Take point M anywhere outside the line. Using a compass and ruler, draw a line MN perpendicular to line AB.
Step 1:
Step 2:
Step 3:
line MN ⊥ line AB.
Question 3.
Draw a line segment AB of length 5.5 cm. Bisect it using a compass and ruler.
Step 1:
Step 2:
line MN is the perpendicular bisector of seg AB.
Question 4.
Take point R on line XY. Draw a perpendicular to XY at R, using a set square.
Step 1:
Step 2:
line TR ⊥ line XY.
Maharashtra Board Class 6 Maths Chapter 17 Geometrical Constructions Practice Set 40 Questions and Activities
Question 1.
In the above construction, why must the distance in the compass be kept constant? (Textbook pg. no. 90)
The point N is at equal distance from points P and Q.
If we change the distance of the compass while drawing arcs from points P and Q, we will not get a point which is at equal distance from P and Q. Hence, the distance in the compass must be kept
Question 2.
The Perpendicular Bisector. (Textbook pg. no. 90)
1. A wooden ‘yoke’ is used for pulling a bullock cart. How is the position of the yoke determined?
2. To do that, a rope is used to measure equal distances from the spine/midline of the bullock cart. Which geometrical property is used here?
3. Find out from the craftsmen or from other experienced persons, why this is done.
1. For the bullock cart to be pulled in the correct direction by the yoke, its Centre O should be equidistant from the either sides of the cart.
2. The property of perpendicular bisector is used to make the point equidistant from both the ends
3. A rope is used just like a compass to get equal distances from the spine/midline of bullock cart.
Question 3.
Take a rectangular sheet of paper. Fold the paper so that the lower edge of the paper falls on its top edge, and fold it over again from right to left. Observe the two folds that have formed on the .
paper. Verify that each fold is a perpendicular bisector of the other. Then measure the following distances. (Textbook pg. no. 91)
i. l(XP)
ii. l(XA)
iii. l(XB)
iv. l(YP)
v. l(YA)
You will observe that l(XP) = l(YP), l(XA) = l(YA) and l(XB) = l(YB)
Therefore we can conclude that all points on the vertical fold (perpendicular bisector) are equidistant from the endpoints of the horizontal fold.
[Note: Students should attempt this activity on their own.]
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How to Round to Specific Number in Excel
Jun 29, 2020 ยท 2 minute read
you can use “MROUND” command to round your digits to specific numbers and also round the time to specific time.
Here we are going to explain them with some example;
How to Round to Specific Digit in Excel;
How to round number to first digit.
If you have decimal digit and you want to round them to the digit without any decimals you can use this formula.
In this formula
• “C5” is the cell name which we want to round their number.
• “1” means that you want to round your number to the nearest number to the first digit.
How to round number to 0.5:
If you want you can round decimal to “0.5”
For that you can follow these steps;
How to round number to 10:
You can also round number to ten, for that you can follow this formula;
How to round number to 20:
you can also round number to other digits like “20” for that, for exapmle you can use
For number “20”.
How to round number to 100:
For rounding number to one hundred you can use this formula;
How to Round Time in Excel;
we can round time in excel cells to here we will explain it by two example.
In the column “C”, we have different times and in column “D” we want to round them to 15 and 30 minitue.
• For rounding time to 15 minitue write this formula.
• In this formula “C5” is the address of the cell which you want to round the time on it.
• Specify this formula to other cells.
• Select the cells in the coulmn “D”.
• Click on the “Home” tab.
• In the Number part click on the arrow sign on the right bottom corner side or click on the drop down list.
• Click on the “Number” tab and choose “Time”.
• Choose your desired type for showing time.
• Click on the “OK”.
You can see time on your cells were rounded to “0:15”.
• If you want to round time to 30 minitue you can use this formula.
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Expressions are the building blocks of complex behavior in any programming language. They are what allow you to combine simple terms to perform meaningful operations, and eventually output a single
Expressions Expressions are the building blocks of complex behavior in any programming language.
Expression Basics Each expression is composed of one or more terms, perhaps joined together by operators or procedure calls.
Operators Operators are what connect various terms of an expression and supply the meaning of each.
Arithmetic Operators Arithmetic operators provide the basic mathematical operations on numbers.
Bitwise Operators Bitwise operators use integers and operate on the bits that represent them.
Comparison Operators Comparisons do just that, compare.
Logical Operators Logical operators perform boolean logic on the values.
Assignment Operators Assignment operators are used to put a value into something, be it a variable or database.
Operator Precedence Precedence determines the order of evaluation for all operators.
Expression Basics
Each expression is composed of one or more terms, perhaps joined together by operators or procedure calls. Almost any place that calls for a value, you can use an expression to supply that value.
Combine constants, variables, function calls, database lookups, search results, anything. As long as the types are consistent (or compatible) you’ve got a valid expression.
Operators are what connect various terms of an expression and supply the meaning of each. They compare or subtract, whatever needs to happen. They fall into one of several categories depending on
their function. Let’s look at a breakdown of each of these categories.
Arithmetic Operators
Arithmetic operators provide the basic mathematical operations on numbers. If you’re looking to add or subtract, it’ll be in this group.
+ Integer addition
- Integer subtraction
/ Integer division
% Integer modulus (remainder)
* Integer multiplication
Bitwise Operators
Bitwise operators use integers and operate on the bits that represent them. The result of bitwise operations is an integer (whose bits are the result of the operation).
| Bitwise OR
^ Bitwise XOR
& Bitwise AND
~ Unary bitwise negation
<< Bitshift left
>> Bitshift right
Comparison Operators
Comparisons do just that, compare. For integers, this is a numerical comparison, and for strings, this is an alphabetical comparison (as in an alphabetization (A-Z) of ascii values). Comparison
operations return a boolean value (0 for false, 1 for true) in an integer.
= Equality
!= Inequality
> Greater than
< Less than
>= Greater than or equal
<= Less than or equal
~= Wildcard comparison (strings only)
Logical Operators
Logical operators perform boolean logic on the values. They take boolean values on each side (or if normal values are used, the following rules are used to evaluate the truth value of a term: 1) if
it’s an integer, non-zero is true, 0 is false, 2) if it’s a string, a length greater than 0 is true, otherwise false, 3) if it’s a vector, a size greater than 0 is true, otherwise false).
or Logical OR
and Logical AND
! Logical negation (non-zero -> 0 or 0 -> 1)
Assignment Operators
Assignment operators are used to put a value into something, be it a variable or database. The assignment operator is the ‘=’ character. However, it can be combined with various other operators to
provide a shortcut in writing an assignment. For example: ‘+’ can be combined with ‘=’ to form ‘+=’. When using an operator in this way, the assignment “x += 5” is treated like “x = x + 5” for all
practical purposes.
The operators ‘++’ and ‘--’ can be used to add or substract a 1. Using ‘a++’ is identical to ‘a += 1’, and ‘a--’ corresponds to ‘a -= 1’.
Operator Precedence
Precedence determines the order of evaluation for all operators. Lower precedence operations are performed after higher precedence ones. The following listing provides a listing of the precedence for
all operators in increasing precedence.
*Operator* *Description*
= Assignment
?: Conditional operator (condition ? val1 : val2)
or Logical OR
and Logical AND
| Bitwise OR
^ Bitwise XOR
& Bitwise AND
=,!=,<,>,<=,>=,~= Comparisons
<<,>> Bitwise shift
+,- Addition and subtraction
*,/,% Multiplication, division, modulus
!,~,- Unary operations (logical, bitwise,
and mathematical negation)
.,(),[] 'Dot' connector, parentheses, indexing
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Stochastic processes on manifolds – The Dan MacKinlay stable of variably-well-consider’d enterprises
Stochastic processes on manifolds
March 1, 2021 — March 1, 2021
Hilbert space
how do science
kernel tricks
machine learning
signal processing
stochastic processes
time series
1 References
Adler, Robert J. 2010. The Geometry of Random Fields.
Adler, Robert J., and Taylor. 2007.
Random Fields and Geometry
. Springer Monographs in Mathematics 115.
Bhattacharya, and Bhattacharya. 2012.
Nonparametric Inference on Manifolds: With Applications to Shape Spaces
. Institute of Mathematical Statistics Monographs.
Borovitskiy, Terenin, Mostowsky, et al. 2020.
“Matérn Gaussian Processes on Riemannian Manifolds.” arXiv:2006.10160 [Cs, Stat]
Calandra, Peters, Rasmussen, et al. 2016.
“Manifold Gaussian Processes for Regression.”
2016 International Joint Conference on Neural Networks (IJCNN)
Manton. 2013.
“A Primer on Stochastic Differential Geometry for Signal Processing.” IEEE Journal of Selected Topics in Signal Processing
Yaglom. 1961.
“Second-Order Homogeneous Random Fields.” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory
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The effect of convergent-divergent riblets on laminar wall-bounded flows
Convergent-divergent (C-D) riblets are a type of bio-inspired surface pattern, and have begun to receive research attention in recent years, due to their potential in skin friction reduction and flow
separation control. In this thesis, the effect of C-D riblets on the secondary flow, flow separation and drag characteristics in laminar wall-bounded flows is studied via numerical simulations.
Firstly, a systematic investigation of the effect of riblet height, wavelength and yaw angle on the secondary flow in a laminar boundary layer developing over a C-D riblet section is undertaken.
Large scale secondary flow is observed in cross-stream planes which displays downward/upward motions over the diverging/converging lines. The exact structure of the secondary flow depends on the
relative size of riblet height and wavelength to the local boundary layer thickness, and three different patterns are observed. With the increase of wavelength, the average strength of the secondary
flow per unit area exhibits a peak around a ratio between wavelength and local boundary layer thickness of 1. As the yaw angle increases, the strength of the secondary flow reaches to the peak value
at a yaw angle of 45deg. Secondly, the effects of C-D riblets on momentum transfer enhancement and the extent of flow separation zone are examined by applying a section of C-D riblets upstream of a
backward-facing rounded ramp in a laminar channel flow. In comparison with the baseline case, flow separation is delayed and the reattachment occurs earlier, leading to a smaller separation zone
around the diverging line. The opposite phenomena occur around the converging line. A minimum riblet height of 3.75% of the channel height is required to produce a net reduction in the separation
zone. As riblet spacing s increases with fixed riblet height h, a maximum strength of the secondary flow and a maximum reduction in the separation zone are obtained at s/h=4. Thirdly, the effect of
C-D riblets on drag characteristics is studied by proposing an exact expression for the drag coefficient in laminar channel flows with wall roughness, whereby the drag is decomposed into
contributions from different components of the velocity gradient tensor in the flow field. Furthermore, the triple decomposition technique is used to identify the contribution to drag production from
the mean velocity field, the riblet- and wavelength-scale dispersive flow field. The normalized drag increment starts to rise when the Reynolds number is large enough to enable the secondary flow to
alter the streamwise velocity across the span. While the normalized drag increment is predominantly caused by the mean and small-scale dispersive velocity at low Reynolds number, the contribution
from the large-scale dispersive velocity field increases rapidly with the Reynolds number and gradually becomes dominant. Among C-D riblets with rectangular, triangular and sinusoidal cross-sectional
shapes, the triangular riblet pattern is found to produce a secondary motion with a similar strength with less drag penalty. Finally, a theoretical derivation is presented to prove that drag
reduction cannot be achieved by applying wall roughness structures onto the smooth inner walls of streamwise-periodic steady incompressible laminar channel/pipe flows at the same volume flow rate. It
is shown that wall roughness produces a higher drag due to two factors: a) wall roughness induces other non-zero velocity gradient terms in addition to the wall-normal/radial gradient of streamwise
velocity that exist in a smooth channel/pipe flow; b) the profile of streamwise velocity in the wall-normal/radial direction deviates from the parabolic profile that produces the minimum kinetic
energy loss at the same volume flow rate.
Date of Award 31 Dec 2021
Original language English
Awarding Institution • The University of Manchester
Supervisor Timothy Craft (Supervisor) & Shan Zhong (Supervisor)
• Convergent-divergent riblets; laminar flow; Numerical simulations; Secondary flow; Flow separation control; Drag decomposition
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GCF of 24 and 30
On this page we will define the GCF of 24 and 30, teach you the different ways of calculating the GCF of 24 and 30, and show you what you can use the GCF of 24 and 30 for.
What is the GCF of 24 and 30?
GCF is the abbreviation for Greatest Common Factor. Therefore, the GCF of 24 and 30 is the same as the Greatest Common Factor of 24 and 30. The GCF of 24 and 30 is the largest positive integer by
which both 24 and 30 can be divided. Furthermore, both 24 and 30 have a set of factors and the GCF is the greatest factor that 24 and 30 have in common.
Compare factors to get GCF of 24 and 30
Per definition above, to find the GCF of 24 and 30, you can compare the
factors of 24
with the
factors of 30
to see which factor is the greatest. When we did that, we found that the Greatest Common Factor (GCF) of 24 and 30 is 6.
Use LCM to get GCF of 24 and 30
The Least Common Multiple (LCM) of 24 and 30 is 120. You can find the GCF of 24 and 30 by dividing the product of 24 and 30 by the LCM of 24 and 30. Here is the formula and math:
Product of 24 and 30 = GCF
LCM of 24 and 30
Use computer spreadsheet to get GCF of 24 and 30
If you have a computer, you can also use a spreadsheet in Excel or Numbers to calculate the GCF of 24 and 30. You want to type =gcf(24, 30) into a cell to get the answer.
gcf(24, 30) = 6 Use the GCF of 24 and 30 to simplify a fraction
The GCF of 24 and 30 can be used for many things. You can, for example, simplify a fraction by dividing the numerator and denominator by the GCF like this:
Use GCF of 24 and 30 to simplify a ratio
Similarly, you can use the GCF of 24 and 30 to simplify a ratio by dividing each part of the ratio by the GCF like this:
= 24 : 30
= (24 ÷ 6) : (30 ÷ 6)
= 4 : 5
Use the GCF of 24 and 30 to find the LCM of 24 and 30
Since using the Least Common Multiple (LCM) is one of the ways to find the GCF of 24 and 30, you can use the GCF of 24 and 30 to find the LCM of 24 and 30. The LCM of 24 and 30 can, for example, be
used to add and subtract fractions with denominators of 24 and 30. The LCM of 24 and 30 is the product of 24 and 30 divided by the GCF of 24 and 30. Here is the math:
Product of 24 and 30 = LCM
GCF of 24 and 30
That is all there is to it! We hope this page accomplished its goal of defining the GCF of 24 and 30 by showing you how to calculate the GCF, examples of its uses, and how it relates to LCM.
GCF Calculator
Use the GCF Calculator to solve a problem similar to the one explained on this page.
GCF of 24 and 31
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1. Introduction
Image inpainting refers to the process of completing missing entries or restoring damaged regions of an image. It is a typical ill-posed inverse problem, generally solved by exploiting the image
priors [
], such as smoothness, sparsity, and low rankness. In recent years, tensor analysis including tensor low-rank decomposition and tensor completion, has attracted increasing attention [
]. A color image itself is an order-3 tensor, or it can be used to construct a high order (greater than 3) tensor, then the image inpainting problem becomes a tensor completion problem. A tensor is
more challenging to analyze than a matrix due to the complicated nature of higher-order arrays [
]. We can constrain the low tensor rank to recover the missing pixels. The effectiveness relies on the tensor rank. The lower the tensor rank is, the better the recovery results are. Thus, finding
ways to decrease the tensor rank is essential in the tensor completion problem. Unlike matrix rank, the definition of tensor rank is not unique, and relates to the tensor decomposition scheme.
Low tensor-rank completion methods can be categorized according to the tensor decomposition frameworks they use [
]. The traditional tensor decomposition tools include CANDECOMP/PARAFAC (CP), and Tucker decomposition [
]. The recently proposed decomposition frameworks include tensor singular value decomposition (t-SVD) [
], tensor train (TT) decomposition [
], tensor tree (TTR) decomposition [
] etc. As we know, CP rank is hard to estimate. Tucker rank is multi-rank, whose elements are the ranks of mode-n matrices which are highly unbalanced. TT rank is also multi-rank, whose elements are
the ranks of TT matrices. For a high-order tensor, the most TT matrices are more balanced than the mode-n matrices. Since the matrix rank minimization is only efficient when the matrix is balanced,
TT decomposition is more suitable for describing global information of high-order tensors than Tucker decomposition. T-SVD defines the tubal rank of the high order tensor, which can be easily
estimated according to a fast Fourier-based method. The tubal rank has been shown more efficient than the matrix-rank and Tucker multi-rank in video applications [
]. TTR rank is essentially equivalent to Tucker multi-rank.
Many popular tensor-completion methods have applied the traditional CP or Tucker decomposition on color image inpainting. Some recent works exploited the sparse Tucker core tensor and nonnegative
Tucker factor matrices for image restoration [
]. Some works constrained the low rankness of the mode-n matrix caused by decomposition of a color image for inpainting [
]. Since low-tensor-rank constraint cannot fully capture the local smooth and global sparsity priors of tensors, some works combine Tucker and total variation (TV). The SPCTV (smooth PARAFAC tensor
completion & total variation) method [
] used the PD (PARAFAC decomposition, a derivation of Tucker decomposition) framework, and constrained the TV (total variation) on every factor matrix of PD respectively. Some works combined the
constraints of the low rankness of every mode-n matrix and the TV regularization on every mode-n matrix for color image inpainting [
]. Some works proposed data restoration methods based on Bayesian tensor completion [
The afore-mentioned methods all take the color image as an order-3 tensor directly and haven’t deeply explored the potential low-rank prior to a color image. Since TT decomposition is efficient for
higher-order tensors, the TMac-TTKA method [
] first used the Ket augmentation (KA) scheme to permute the image to a high order data, then proposed the optimal models by enforcing low TT rankness. The KA scheme is proven to be efficient for
improving the accuracy of color image/video inpainting and dynamic MR image reconstruction in TT rank based completion methods [
]. As far as we know, the KA scheme is the only one used to permute data into a high order data.
This paper aims to deeply explore the potential low-rank structure of the image and to find an efficient way to apply the SVD, t-SVD, and TT decomposition in the image inpainting problems. The
contributions of our work are summarized as follows:
• First, we developed a novel rearrangement named as quarter augmentation (QA) scheme for permuting the image into three flexible forms of data. The first flexible QA scheme can permute an image
into an unfolding matrix (with a low matrix rank structure). The second and the third flexible QA schemes can permute the color image into a balanced 3-order form of data (with low tubal rank
structure) and a higher-order form of data (with low TT rank structure) respectively. Since those developed schemes are designed to exploit the internal structure similarity of the original data
as much as possible, the rearranged data has the corresponding kind of low-rank structure.
• Second, based on the above QA scheme, we developed three image inpainting models that exploit the unfolding matrix rank, tensor tubal rank, and TT multi-rank of the rearranged data respectively
for solving the image inpainting problem.
• Lastly, three efficient ADMM algorithms were developed for solving the above three models. Compared with numerous close image inpainting methods, the experimental results demonstrated the
superior performance of our methods.
The remainder of this paper is organized as follows. In section II, we give the related work. In section III, we mainly introduce the proposed methods. Section IV the experimental results and
analyses. The conclusion is given in section V.
2. Related work
In this section, we briefly introduce the KA scheme, the t-SVD decomposition, and tensor train decomposition. Notations and definitions are summarized in
Table 1
2.1. Ket Augmentation
The Ket Augmentation (KA) scheme was originally introduced by Latorre in [
] for casting a grayscale image into the real ket state of a Hilbert space. Bengua etc. [
] used KA to reshape a low-order tensor e.g. a color image to a higher-order tensor and proved that KA is efficient in improving the accuracy of the recovered image in TT-based completion.
Figure 1
shows the operation of KA for an 8 × 8 matrix [
]. By the KA scheme, the 8 × 8 matrix can be turned into a 3-order tensor of size 4 × 4 ×4. As well, the KA scheme can turn a 3-order tensor size of
$x N × y N × N 3$
into an
$N + 1$
-order tensor with a size of
$x y × x y × : : : × x y × N 3$
2.2. T-SVD Decomposition
Definition 1
t-product [
]. For
$A ∈ R n 1 × n 2 × n 3$
$B ∈ R n 2 × n 4 × n 3$
, the t-product
$A ∗ B = C$
is a tensor of size
$n 1 × n 4 × n 3$
$C ( i , j , : )$
is given by
$∑ k = 1 n 2 A ( i , k , : ) ∘ B ( k , j , : )$
denotes the circular convolution between the two vectors, and
$i = 1 , 2 , ⋯ , n 1$
$j = 1 , 2 , ⋯ , n 4$
The t-SVD of
$A ∈ R n 1 × n 2 × n 3$
is given by
are orthogonal tensors of size
$n 1 × n 1 × n 3$
$n 2 × n 2 × n 3$
is a rectangular f-diagonal tensor of size
$n 1 × n 2 × n 3$
and * denote t-product [
denotes tensor transpose defined in [
Figure 2
depicts the t-SVD of an order-3 tensor [
]. Tensor rank defined in t-SVD is tensor tubal rank, which is the number of nonzero singular tubes in
. [
] proposed the fast Fourier-based method to calculate the tubal rank, and used tensor nuclear norm (TNN) as the convex relaxation of the tensor tubal rank.
$A T N N = b l o c k d i a g ( A ¯ ) *$
$A ¯ = f f t ( A , [ ] , 3 )$
is the tensor obtained by applying the 1D FFT along the third dimension of
denotes nuclear norm, and
$b l o c k d i a g ( A ¯ ) = A ¯ ( 1 ) A ¯ ( 2 ) ⋱ A ¯ ( I 3 )$
2.3. Tensor Train Decomposition
$A ( i 1 , i 2 , ⋯ i n , ⋯ i N ) = U 1 ( : , i 1 , : ) U 2 ( : , i 2 , : ) ⋯ U n ( : , i n , : ) ⋯ U N ( : , i N , : )$
Given a tensor
$A ∈ R I 1 × I 2 × ⋯ I N$
, tensor train (TT) decomposition [
] can decompose it to
order-3 tensors
$U n ∈ R S n × I n × ⋯ S n + 1$
$n = 1 , ⋯ , N$
. The tensor rank defined in TT decomposition is a multi-rank i.e.
$( S 1 , S 2 , ⋯ , S N + 1 )$
, which is combined with the second-dimensional size of each
$U n$
. The details of TT decomposition are shown in the following formula and
Figure 3
The widely used way to find TT rank is to estimate the rank of each TT matrix [
] as the element of
$( S 1 , S 2 , ⋯ , S N + 1 )$
. The TT matrix
$A [ n ]$
$n = 1 , ⋯ , N − 1$
) with rank
$S n$
is the mode-
$1 , 2 , ⋯ , n$
matricization of the tensor with the size of
$m × h$
, where
$m = ∏ l = 1 n I l$
$h = ∏ l = n + 1 N I l$
3. Methods
3.1. Quarter Augmentation
To deeply explore the more efficient low-rank structure of an image, we develop a novel rearrangement scheme named as quarter augmentation (QA) scheme to turn a color image into other forms of data.
The QA schemes can maintain the internal similarity of the original image in the rearranged data.
The basic QA scheme: For example, as shown in
Figure 4
(a), M is a 2D matrix (
$8 × 8$
). We first extract the entries of M every other row and column to get four smaller matrices. Each smaller matrix with a size of
$4 × 4$
. Then we place these four smaller matrices along the third dimension in a designed order. Lastly, a 3D tensor of size
$4 × 4 × 4$
is obtained from the
$8 × 8$
matrix M without changing the total number of entries. The entries in the four smaller matrices are labeled as the MATLAB notation
$( : , : , 1 )$
$( : , : , 2 )$
$( : , : , 3 )$
$( : , : , 4 )$
respectively. If M is smooth (most images satisfy), the four smaller matrices are similar in structure due to the adjacent entries.
Applying the basic QA scheme on the single Lena image, the Lena image can be divided into 4 smaller Lena images, and as shown in
Figure 4
(b) the four smaller Lena images are similar to each other. In
Figure 4
(c), the pixel values curves of the four smaller images have overlapped into one curve. We can say that the similarity of local image structure is mainly maintained by the basic QA scheme.
Under this basic QA scheme, three flexible QA schemes are proposed for permuting the image into three flexible forms of data. The three flexible QA schemes can enhance the low-rankness for an image
by matrix SVD, tensor train decomposition, and tensor-SVD respectively. Then, by exploiting the flexible QA schemes, three low-rank constrained methods which use the TT rank, tubal rank, and matrix
rank as constrained priors respectively are exploited for image inpainting.
The three flexible QA schemes and methods are described in detail in the following three sections.
3.2. Method 1: The Low Unfolding Matrix Rank-Based Method
The unfolding method is widely used to permute the order-3 video or dynamic magnetic resonance images into an unfolding matrix, and then exploit the low rankness of this matrix for data
reconstruction [
]. The unfolding matrix has a low-rank structure because of the similarity of every slightly changed slice along the time dimension.
We try to dig out the potential low unfolding-matrix rankness of a color image by a flexible QA scheme, and we call this scheme the first flexible QA scheme.
Take a 256×256×3 Lena image as an example, as shown in
Figure 5
(a), we first permute the image into the 3-order tensor size of 32×32×192 by the basic QA scheme, then unfold the similar slices of this 3-order. Lastly, the balanced
unfolding matrix size of 1024×192 is obtained. Since the slices (32×32) in the 3-order tensor are similar, the unfolding matrix is low rank, as shown in
Figure 5
(b). In practice, the size of the designed unfolding matrix should be balanced such that the minimization of the unfolding matrix rank is efficient.
We exploit the low unfolding matrix rank in image inpainting and give the low unfolding matrix-rank-based model as follows.
$min X Μ ( Φ 1 X ) * subject to X ( i , j ) = Y ( i , j ) , ∀ ( i , j ) ∈ Ω$
where $X$ denotes the image to be recovered, $Φ 1$ denotes the operator of permuting the image into a suitable 3-order tensor by multiple basic QA schemes. $Μ$ denotes the operator of the unfolding
process, which unfolding every slice along the third dimension of the 3-order tensor $Φ 1 X$. $Ω$ is the position without painting, $Y$ is the painted image with damaged entries at the positions $Ω
To reduce the computational complexity, in the model (1), the following SVD-free approach [
] is exploited to constrain the low rankness of the unfolding matrix
$Μ ( Φ 1 X )$
instead of the nuclear norm.
$min U V H = Μ ( Φ 1 X ) 1 2 U F 2 + V F 2 = Μ ( Φ 1 X ) *$
Besides, since total variation (TV) has been proved as an effective constraint of smooth prior [
], incorporate model (1) with 2D TV to exploit the local smooth priors of visual image data. Then, the image inpainting model (1) turns to the following.
$min U , V , X 1 2 U F 2 + V F 2 + β 2 X T V subject to X ( i , j ) = Y ( i , j ) , ∀ ( i , j ) ∈ Ω , Μ Φ 1 X = U V H$
where $β$ is the regularization parameter.
We conduct the algorithm by alternating direction method of multipliers (ADMM) for solving the low unfolding matrix rank and TV-based image inpainting model (3). Firstly, introduce an auxiliary
$Z = D X$
, where
is the finite difference operator, and then rewrite (3) as the unconstrained convex optimization problem (4).
$min U , V , Λ , L , Z , X τ Ω ( X ) + 1 2 U F 2 + V F 2 + ρ 1 2 Μ Φ 1 X − U V H + Λ F 2 + β 2 Z 1 + β ρ 2 2 D X − Z + L F 2$
where $τ Ω ( X )$ denotes the indicator function:
are the Lagrangian multipliers for variables
$U V H$
respectively. The regularization parameter
is used to balance the low rankness and sparsity constraints (i.e. TV), the penalty parameters
$ρ 1 > 0$
$ρ 2 > 0$
generally affect the convergence of the algorithm. By applying ADMM, each sub-problem is performed at each iteration
as follows:
$X t = arg min X τ Ω ( X ) + ρ 1 2 Μ Φ 1 X − U t − 1 V ( t − 1 ) H + Λ t − 1 F 2$
$U t = arg min U U F 2 + ρ 1 2 Μ Φ 1 X t − U V ( t − 1 ) H + Λ t − 1 F 2$
$V t = arg min V V F 2 + ρ 1 2 Μ Φ 1 X t − U t V H + Λ t − 1 F 2$
$Z t = arg min Z Z 1 + ρ 2 2 D X t − Z + L t − 1 F 2$
$Λ t = Λ t − 1 + Μ Φ 1 X t − U t V ( t ) H$
$L t = L t − 1 + D X t − Z t$
The initial
can be determined by solving the following optimization problem using the LMaFit method [
$min U , V , X U V H − Μ Φ 1 X F 2 s u b j e c t t o X ( i , j ) = Y ( i , j ) , ∀ ( i , j ) ∈ Ω$
The whole algorithm for solving the model (3) is shown in
Table 2
3.3. Method 2: The Low Tubal-Rank-Based Method
Tensor-SVD decomposition has been efficiently used in the video image completion and dynamic MR image reconstruction problem [
]. Since the color image is highly unbalanced in the size of three dimensions, which is not suitable for the low tubal rank constraint, we exploit the second flexible QA scheme to deeply dig out the
potential low tubal-rank prior information.
Considering that tubal rank minimizations are more efficient for the balanced tensor [
], we first turn the unbalanced image into the balanced order-3 data by the second flexible QA scheme.
Take the color image size of 256×256×3 as an example, as shown in
Figure 6
(a), we can obtain the order-4 tensor size of 128×128×4×3 by the basic QA schemes, and then multiplying the basic QA schemes we can obtain the order-4 tensor size of 64×64×4×4×3. Lastly, we reshape
the order-4 tensor into the balanced order-3 tensor size of 64×64×48. Here, the context of ‘balanced’ is that the size changes from the unbalanced 256×256×3 to the more balanced size of 64×64×48. In
practice, the size of the designed order-3 tensor should be as balanced as possible. We call the above the second flexible QA scheme.
Figure 6
(b), we show the low tubal rankness of the balanced order-3 data (with the size of
$n 1 × n 2 × n 3 = 64 × 64 × 48$
here) by plotting
$δ j$
which is defined as follows.
$δ j = 1 n 3 ∑ i = 1 n 3 T ( i , i , j ) , i = 1 , 2 , ⋯ , min ( n 1 , n 2 )$
Then, TNN is used to enforce the tensor tubal rank in the image inpainting model as follows.
$min X Φ 2 X T N N s u b j e c t t o X ( i , j ) = Y ( i , j ) , ∀ ( i , j ) ∈ Ω$
$Φ 2$
denotes the operator of permuting the color image into a more ‘balanced’ order-3 tensor by the second flexible QA scheme. Combining the low tubal rank and sparsity, we introduce auxiliary variables
$B = Φ 2 X$
, and
$Z = D X$
, then rewrite (12) as the following unconstrained convex optimization problem.
$min X τ Ω ( X ) + ∑ i I 3 Z ¯ ( i ) * + ρ 2 Φ 2 X − Z + Λ F 2 + β 2 Z 1 + β ρ 2 2 D X − Z + L F 2$
$I 3$
is the third size of the 3-order tensor
$Φ 2 X$
. We conduct the algorithm by ADMM for solving model (13) as shown in
Table 3
3.4. Method 3: The Low TT-Rank-Based Method
TT decomposition works better on higher-order tensors than Tucker decomposition. To fulfill TT decomposition efficiently, we first exploit the third flexible QA scheme to permute the 3-order image
into a higher-order tensor. Based on the basic QA scheme, high-order tensors can be obtained flexibly.
The third flexible QA scheme is shown below. We take a
$16 × 16$
matrix as an example, as shown in
Figure 7
(a). We first turn a 2-order matrix into a 3D tensor via the basic QA scheme and then repeat the basic QA to obtain the final 4D tensor with the size of
$4 × 4 × 4 × 4$
. The entry comes from the i
smaller matrix of the first basic QA scheme, and the j
smaller matrix of the second basic QA scheme is labeled as the MATLAB notation
$( : , : , i , j )$
. By analogy, the third flexible QA scheme can permute a matrix with the size of
$4 P × 4 Q$
to order-
$min { P , Q }$
tensor with the size of
$4 × 4 × ⋯ × 4 ︸ min { P , Q }$
. An RGB image with the size of
$4 P × 4 Q × 3$
can be permuted into an order-
$min { P , Q } + 1$
tensor with the size of
$4 × 4 × ⋯ × 4 × 3 ︸ 1 + min { P , Q }$
. The third flexible QA scheme should ensure that the designed tensor has a higher order.
We permute the Lena image into a high-order tensor via the third flexible QA scheme, and then obtain the TT matrices of the augmented tensor. We name those TT matrices as QA-TT matrices, and their
singular values are shown in
Figure 7
(b), which demonstrates the low TT rankness of the rearranged tensor.
Then, we enforce the low TT rankness to improve the inpainting accuracy. The third model is as follows.
$min X ∑ n = 1 N α n T n Φ 3 X * s u b j e c t t o X ( i , j ) = Y ( i , j ) , ∀ ( i , j ) ∈ Ω$
$Φ 3$
stands for the third flexible QA used to permute image
into a high-dimensional tensor. We name the tensor obtained by the third flexible QA scheme as a QA tensor.
$T n$
is the operator that converts a tensor into the
TT matrix,
$n = 1 , 2 , ⋯ , N$
. The order of QA tensor is
. The inverse operators corresponding to
$T n$
$Φ − 1$
$T n - 1$
respectively. The weight
$α n$
is given by:
$α n = θ n ∑ n = 1 N − 1 θ n with θ n = min ( ∏ l = 1 n I l , ∏ l = k + 1 N I l )$
where $I 1 × I 2 × ⋯ × I N$ is the size of the QA tensor.
Combining the low TT rank and sparsity constraints, we introduce auxiliary variables
$Z = D X$
$U n V n H = T n Φ 3 X$
, rewrite (14) as the following unconstrained convex optimization problem, for all
$n = 1 , ⋯ , N − 1$
$min U n , V n , Λ n , L , Z , X τ Ω ( X ) + 1 2 ∑ n = 1 N − 1 α n ( U n F 2 + V n F 2 ) + β 2 Z 1 + ρ 1 2 ∑ n = 1 N − 1 α n T n Φ 3 X − U n V n H + Λ n F 2 ) + β ρ 2 2 D X − Z + L F 2$
By applying ADMM, each sub-problem is performed at each iteration
. Lastly, we obtain
$X * = ∑ n = 1 N − 1 α n X n ∗$
, where
$X n ∗$
represents the optimal solution of the nth subproblem. The whole algorithm for solving the model (16) is shown in
Table 4
4. Experimental Results and Analyses
In this section, we conduct the above methods 1-3 for solving image inpainting problems. For simplicity, we denote methods 1-3 which only exploit low unfolding matrix rank, low tensor tubal rank, and
low tensor train rankness as UfoldingLR, TTLR, and tSVDLR methods respectively. The methods that enhance the low rank and total variation constraints simultaneously are denoted as UnfoldingLRTV,
tSVDLRTV, and TTLRTV methods respectively. We denote the low matrix-rank completion method which is solved by the model (17) and ADMM algorithm as the MatrixLR method.
$min X r a n k ( X ) s u b j e c t t o X ( i , j ) = Y ( i , j ) , ∀ ( i , j ) ∈ Ω$
We denote the method that only exploits sparsity in the gradient domain and is solved by the ADMM algorithm as the TV method. Besides, we conduct the following numerous close methods for comparison,
some of their codes are available online.
STDC: the method exploited the images into three factor matrices and one core tensor for image inpainting [
HaLRTC: the method constrained the low rankness of the three mode-n matrices caused by decomposition of a color image for inpainting and which was solved by the ADMM [
: the smooth PARAFAC tensor completion and total variation method [
], which used the PD (PARAFAC decomposition, a derivation of Tucker decomposition) framework and constrained the TV on every factor matrix of PD respectively.
LRTV: the methods combined the constraints of the low rankness of every mode-n matrix and the TV regularization on every mode-n matrix for color image inpainting [
FBCP: the inpainting methods based on Bayesian tensor completion [
All simulations were carried out on Windows 10 and MATLAB R2019a running on a PC with an Intel Core i7 CPU 2.8GHz and 16GB of memory. For a fair comparison, every method is conducted with its optimal
parameters to ensure every method has the best performance. The reconstruction quality is quantified using the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM)
]. The original color images (from the standard image database) and missing patterns used in the experiments are shown in
Figure 8
We set the maximum number of iterations $t max = 100$ and convergence condition $η t o l = 10 - 6$ in all our methods (UnfoldingLRTV, tSVDLRTV, and TTLRTV). The pixel range of all the images is
normalized to 0-1. In UnfoldingLR, tSVDLR, and TTLR methods, we set $ρ 1 =$ 0.04, 0.002, and 0.6 respectively. In UnfoldingLRTV, Tsvdlrtv, and TTLRTV methods, we set the parameter set $( ρ 1 , β , ρ
2 )$ as (0.4, 0.004, 2), (0.6, 0.07, 0.1), and (0.7, 0.03, 0.1) respectively.
4.1. Analyses of the Three Flexible QA Schemes
Next, we call the first, second, and third flexible QA schemes QA scheme briefly. The PSNRs (dB)/SSIMs of the UnfoldingLR, tSVDLR, and TTLR methods with and without the QA scheme are shown in
Table 5
. The red numerical values correspond to the worst results. We can see that, without the QA scheme, Lena and Airplane cannot be recovered. The UnfoldingLR, tSVDLR, and TTLR methods with the QA scheme
have better numerical results than those without the QA scheme. In the low matrix-rank completion method (i.e. MatrixLR), no QA scheme is applied, i.e. the color image is dealt with as three-channel
matrices directly.
Due to the support of the QA scheme, the low tensor-rank based methods (TTLR, tSVDLR, and UnfoldingLR) with the QA scheme provide better results than the traditional low matrix-rank completion method
(i.e. MatrixLR method). So, the QA scheme is successful to be used as the first step to deeply explore the low tensor rank prior to an image.
The KA scheme and the third flexible QA scheme both can rearrange an image into a high-order tensor. However, our QA scheme is different from the KA scheme used in [
]. The KA scheme maintains the local block similarity of the image, while the third flexible QA scheme uses adjacent pixels to maintain the global similarity of the image. We conduct the comparison
of KA and the third flexible QA scheme under the corresponding TMac-TTKA [
] and TTLR methods. As shown in
Figure 9
, the small blocks are obvious in the recovered images by the TMac-TTKA method. The images recovered by the TTLR method preserve more details and without the obvious blocks.
4.2. Analyses of the Methods Exploiting Both Low Rankness and Sparsity
In this section, we analyze the recovery results of the methods both exploiting low rankness and sparsity.
Figure 10
Figure 11
Figure 12
show the visual comparisons of the eleven methods for recovering the House, Lena, and Baboon images respectively.
Table 6
shows the PSNR (dB)/SSIM results of the nine methods for recovering different color images under different missing patterns.
Figure 13
depicts the PSNR curves of the inpainting results of the different methods, the missing ratio ranges from 10% to 70% under a random missing pattern.
As shown in
Figure 10
Figure 11
Figure 12
Figure 13
Table 6
, compared to the numerous close STDC, HaLRTC, FBCP, TMac-TTKA, SPCTV, and LRTV methods, the UnfoldingLRTV, tSVDLRTV, and TTLRTV methods have the super performance on both visual and quantity
results. The SPCTV and LRTV methods also enhance the low rankness and sparsity simultaneously, but the results are worse than our methods.
Table 7
shows the PSNR (dB)/SSIM results of the eight methods: MatrixLR method only constrains the low matrix rank; TV method only exploits the TV prior; The UnfoldingLR, tSVDLR, and TTLR methods only
constrain the low unfolding matrix rank, low tubal rank and low TT rank respectively; The UnfoldingLRTV, tSVDLRTV, and TTLRTV methods combine both sparsity and low tensor rankness. As shown in
Table 7
, the combination of sparsity and low tensor rankness constraints can yield better inpainting results than enforcing sparsity or low rankness alone. TTLR method is more efficient than the MatrixLR,
and TV methods. The results of the tSVDLR method and TTLR method are comparable. The UnfoldingLR method provides the best results among the TTLR, tSVDLR, TuckerLR, MatrixLR, and TV methods.
UnfoldingLRTV, tSVDLRTV, and TTLRTV methods have improved numerical results than the corresponding UnfoldingLR, tSVDLR, and TTLR methods, which demonstrates that TV prior is efficient in improving
the accuracy of low-rank based inpainting methods.
The visual and numerical PSNR (dB)/SSIM comparisons of our methods for recovering the pepper image under 80% random missing patterns are shown in
Figure 14
. In the first row of
Figure 14
, the methods only exploit low-rank constraints. As shown in the color box, there are small blocky errors in the recovered image, these are caused by the QA scheme. This phenomenon can be solved by
combining the constraints of low rank and sparsity (TV), as shown in the second row of
Figure 14
All in all, due to the support of the QA scheme and the efficient TV prior, the low tensor-rank based methods (UnfoldingLRTV, tSVDLRTV, and TTLRTV) are superior to other close low tensor-rank based
methods. The UnfoldingLRTV method provides the best results among all the methods conducted in this paper.
4.3. Analyses of TTLR and TTLRTV Methods
In this section, we mainly focus on the analyses of the TT based methods (i.e. TTLR and TTLRTV) in detail. Since TT rank is multi-rank, how does every TT matrix rank affect the final result? We
answer this question with the below experimental results.
We conducted the experiments on recovering House, Lena, and Airplane images with a size of 256×256×3. The random missing patterns have four missing ratios: 10%, 30%, 50%, and 70% respectively. We
label the 8 TT matrices as
k=1, 2, …, 8
. Then the PSNR (dB) results of
$X n *$
(the optimal solution of the n
subproblem which exploits the n
TT matrix rank) in TTLR and TTLRTV methods are shown in
Figure 15
Figure 15
, we can see that, the PSNR (dB) results of each subproblem is steeply different in the TTLR method, which demonstrates that each TT matrix rank contributes different PSNR result. Since there are no
rules to find which TT matrix rank meets the best PSNR result, we should combine each solution of
$X n *$
to obtain the final
$X *$
, i.e.
$X * = ∑ n = 1 N − 1 α n X n *$
. Comparing the PSNR curves of TTLR and TTLRTV method in
Figure 15
, the PSNR (dB) results of each subproblem is slightly different in the TTLRTV method which demonstrates that the combination of TT and TV can make the PSNR more balanced among all
4.4. Runtime and Complexity Analysis
From a high-dimensional curse perspective, converting an image to a higher-order tensor can result in increased complexity, which inevitably leads to a longer runtime. We compare our methods
(UnfoldingLRTV, tSVDLRTV, and TTLRTV) with the traditional MatrixLR method and the close STDC, HaLRTC, FBCP, SPCQV, and LRTV methods in running time, as shown in
Table 8
UnfoldingLRTV methods: In the first step, the QA scheme is used to decompose a single image into several small graphs. Because of the similarity of these small graphs, the QA tensor can be reduced to
a matrix with a low-rank structure in an unfolding way. Ignoring TV constraints, the unfoldingLRTV method only needs to solve the low-rank matrix completion problem of an unfolding matrix, so the
running time is similar to the traditional MatrixLR method, and the accuracy is higher than the traditional MatrixLR method.
The tSVDLRTV methods: Since the color image is highly unbalanced in the size of three dimensions, which is not suitable for the low tubal rank constraint, we use the QA scheme to rearrange an image
into a third-order tensor with a more balanced size of every dimension. Then we use TNN to constrain the low tubal rank of the rearranged tensor, due to the fast Fourier scheme, it is necessary to
perform a low-rank matrix constraint on each frontal slice after the third-dimensional Fourier transform. At this time, the SVD decomposition process will increase the time consumption.
TTLRTV methods: TT multi-rank is the combination of the rank of each TT matrix. The TTLRTV method essentially completes the same data amount N-1 times, where N is the order of the QA tensor. So,
although the TITRTV method is effective, it is necessarily more computationally expensive than the low matrix-rank completion method.
In summary, among the three methods, the UnfoldingLRTV method achieves the best performance both in accuracy and runtime; The TTLRTV method reaches better accuracy, but it is time-consuming; The
tSVDLRTV method has moderate performance both in runtime.
All in all, the above three methods can deeply exploit the potential low-rank prior of an image and have been successfully used for image inpainting problems, which demonstrates that the three
flexible QA schemes are perfect ways to explore the low-rank prior of an image.
5. Conclusions
To effectively explore the potential of low tensor rank prior to an image, we first exploited a rearrangement scheme (QA) for permuting the color image (3-order) into three flexible rearrangement
forms (with more efficient low tensor rank structure). Based on the scheme, three optimization models by exploiting the low unfolding matrix rank, low tensor tubal rank, and low TT multi-rank were
proposed to improve the accuracy in image inpainting. Combined with TV constraints, we developed efficient ADMM algorithms for solving those three optimization models. The experimental results
demonstrate that our low tensor-rank-based methods are effective for image inpainting, and are superior to the low matrix-rank completion method and numerous close methods. The low tensor rank
constraint is effective for image inpainting, which is mainly due to the support of the QA scheme.
Author Contributions
Conceptualization, S.M.; methodology, S.M.; investigation, S.M. and Y.F.; resources, S.M. and S.F.; writing—original draft preparation, S.M. and S.F.; writing—review and editing, S.F., W.Y. and Y.F.;
supervision, L.L. and W.Y.; funding acquisition, S.M. and L.L. All authors have read and agreed to the published version of the manuscript.
This work was funded by the National Key Laboratory of Science and Technology on Space Microwave, No. HTKJ2021KL504012; Supported by the Science and Technology Innovation Cultivation Fund of Space
Engineering University, No. KJCX-2021-17; Supported by the Information Security Laboratory of National Defense Research and Experiment, No.2020XXAQ02.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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1 The context of ‘balanced’ is that the size changes from the unbalanced 256×3 to the more balanced size of 1024×192.
Figure 1. Example of KA for an 8 × 8 matrix. The order-2 M can be rearranged to a higher-order tensor T (order = 3) without changing the total number of entries.
Figure 4. (a) Examples of the basic QA. By the basic QA scheme, the matrix M size of 8✕8 can be turned into an order-3 tensor with a size of 4✕4✕4. (b) By the basic QA, the Lena image can be divided
into 4 small Lena images. (c) The four-pixel values curves of the four smaller Lena images have overlapped into one curve.
Figure 5. (a) The first flexible QA scheme to obtain the unfolding matrix. Take the Lena RGB image as an example, we first permute the image size of 256✕256✕3 to order-3 tensor with the size of
32×32×192 by the basic QA scheme. Then this order-3 tensor is reshaped into the unfolding matrix of size 1024✕92. (b) The singular values of this unfolding matrix.
Figure 6. (a) The second flexible QA scheme to permute the image into a balanced order-3 tensor. Take the Lena image size of 256×256×3 as an example, we obtain the balanced order-3 tensor size of
64×64×48 by multiple QA schemes. This balanced order-3 tensor is more suitable for the t-SVD decomposition than the original image size of 256×256×3. (b) The low tubal rankness of the balanced
order-3 tensor.
Figure 7. (a) Examples of the third flexible QA scheme. By the third flexible QA, the matrix size of 16×16 can be permuted into an order-4 tensor size of 4×4×4×4. (b) Singular values of TT matrices.
We permute the order-3 Lena RGB image size of 256×256×3 into an 8-order tensor with the size of 4×4×4×4×4×4×4×4×3 by the third flexible QA scheme. Then eight different TT matrices are obtained from
this higher-order tensor. We labeled those TT matrices as k=1, 2, …, 8.
Figure 9.
Comparison of KA and QA scheme under the corresponding TMac-TTKA method [
] and our TTLR method respectively. The first row lists the painted images with a random missing pattern and the missing ratio is 80%. The second row lists the recovered images by TMac-TTKA method.
The last row lists the recovered images by LRTT method.
Figure 13. The PSNR curves of the inpainting results of the six images solved by different methods. The missing ratio ranges from10% to 70% under random missing pattern.
Figure 14. The visual and numerical PSNR (dB)/SSIM comparisons of our methods for recovering the pepper image under 80% random missing patterns. In the first row, the methods only exploit low-rank
constraints. As shown in the color box, there are small blocky errors in the repaired image, this is caused by the QA scheme. This phenomenon can be solved by combining the constraints of low rank
and sparsity (TV) as shown in the second row.
Figure 15. PSNR (dB) results contributed by each TT matrix in TTLR method and TTLRTV method. We permute the image size of 256×256×3 to an order-9 tensor by the QA scheme. Then we labeled the TT
matrices of this order-9 tensor as k=1, 2, …, 8. We use the random missing patterns with four missing ratios: 10%, 30%, 50%, and 70% respectively. The tested color images for the PSNR curves in (a)-
(c) are House, Lena and Airplane images respectively.
Symbols Notations and definitions
fiber A vector defined by fixing every index but one of a tensor.
slice A matrix defined by fixing all but two indices of a tensor.
$A ( : , : , The $k t h$ frontal slice of a 3-order tensor $A$.
k )$
$A ( n )$ Mode-n matrix, the result of unfolding tensor $A$ by reshaping its mode-n fibers to the columns of $A ( n )$.
f-diagonal Order-3 tensor $A$ is called f-diagonal if each frontal slice $A ( : , : , k )$ is a diagonal matrix [10].
orthogonal Tensor $A$ with the size of $n × n × n 3$ is called orthogonal tensor if $A ∗ A H = I$, where $I$ stands for identity tensor if the first frontal slice $I ( 1 )$ is the $n × n$ identity
tensor matrix and all other frontal slices $I ( k )$ ($k = 1 , 2 , ⋯ , n 3$) are zero.
Input: $Y , Ω , ρ 1 , β , ρ 2$, maximum number of iteration $t max$, convergence condition $η t o l$.
Initialization: initial $U ( 0 )$, $V ( 0 )$ by solving the matrix completion problem (11), $Λ ( 0 )$, $L ( 0 )$, $Z ( 0 )$, t=0.
While $t < t max$ and $η < η max$do
The first flexible QA scheme: Turn an image into an order-N tensor $Φ 1 X$, then unfold it.
Solve (5)-(10) for $X *$, where * represents the optimal solution.
Update $η t + 1 = X n t + 1 ( : ) − X n t ( : ) F X n t ( : ) F$, $t = t + 1$.
End while
Output: $X *$.
Input: $Y , Ω , ρ 1 , β , ρ 2$, the maximum number of iteration $t max$, convergence condition $η t o l$.
Initialization: $Λ ( 0 )$, $L ( 0 )$, $B ( 0 )$, $Z ( 0 )$, t=0.
While $t < t max$ and $η < η max$do
QA scheme: Turn an image into the balanced order-3 tensor $Φ 2 X$.
Update $X t = arg min X τ Ω ( X ) + ρ 2 Φ 2 X − B t - 1 + Λ t - 1 F 2$
Update $B ¯ ( i ) t = arg min Z ¯ ( i ) B ¯ ( i ) * + ρ 2 Φ 2 X t − B + Λ t − 1 F 2 , i = 1 , ⋯ , I 3$
Update $Z t = arg min Z ¯ ( i ) Z 1 + ρ 2 D X t − Z + L t − 1 F 2$
Update $Λ t = Λ t − 1 + Φ 2 X t − Z t$, $L t = L t − 1 + D X t − Z t$
Update $η t + 1 = X n t + 1 ( : ) − X n t ( : ) F X n t ( : ) F$, $t = t + 1$.
End while
Output: $X *$.
Input: $Y , Ω , β , ρ 1 , ρ 2$, the maximum number of iteration $t max$, convergence condition $η t o l$.
Initialization: $U n ( 0 )$, $V n ( 0 )$ by the LMaFit method [43]; $Λ n ( 0 )$, $L n ( 0 )$, $Z ( 0 )$.
For n=1 to N-1 do
While $t < t max$ and $η < η max$do
QA scheme: permute image to order-N tensor $Φ 3 X$.
Update $X n t = arg min X τ Ω ( X ) + ρ 1 2 T n Φ 3 X − U n t − 1 V n ( t − 1 ) H + Λ n t − 1 F 2$
Update $U n t = arg min U n U n F 2 + ρ 1 2 T n Φ 3 X t − U n V n ( t − 1 ) H + Λ n t − 1 F 2$
Update $V n t = arg min V n V n F 2 + ρ 1 2 T n Φ 3 X t − U n t V n H + Λ n t − 1 F 2$
Update $Z t = arg min Z Z 1 + ρ 2 2 D X t − Z + L t − 1 F 2$
Update $Λ n t = Λ n t − 1 + T n Φ 3 X t − U n t V n ( t ) H$, $L t = L t − 1 + D X t − Z t$
Update $η t + 1 = X n t + 1 ( : ) − X n t ( : ) F X n t ( : ) F$, $t = t + 1$.
End while
End for
Output: $X * = ∑ n = 1 N − 1 α n X n *$.
PSNR (dB)/SSIM of different color images under different missing patterns
Methods House Lena Airplane Boats
Random 50% Lines Random line Random 80%
MatrixLR 9.38/0.8970 13.34/0.5850 7.118/0.1308 19.18/0.5680
Without Rearrangement TTLR 28.61/0.871 13.34/0.585 7.11/0.130 19.25/0.519
tSVDLR 32.30/0.932 13.34/0.585 7.11/0.130 21.60/0.707
UnfoldingLR 7.83/0.093 13.34/0.585 7.11/0.130 6.32/0.102
TTLR 30.21/0.9251 31.79/0.9559 25.77/0.8796 21.44/0.7144
With Rearrangement tSVDLR 29.79/0.8989 31.20/0.9561 18.91/0.8386 21.34/0.6879
UnfoldingLR 32.58/0.9416 33.45/0.9771 28.75/0.9464 23.46/0.8139
PSNR (dB)/SSIM of different color images under different missing patterns
No. Methods House Peppers Lena Airplane Baboon Boats
Random 50% Text Lines Random line Blocks Random 80%
1 STDC 32.04/0.9300 33.61/0.9813 28.56/0.8995 23.49/0.7756 27.01/0.9293 21.88/0.7340
2 HaLRTC 32.07/0.9423 25.84/0.9496 13.34/0.5850 19.94/0.6334 28.04/0.9397 20.56/0.6858
Other methods 3 FBCP 26.41/0.8701 NAN 14.56/0.5242 10.25/0.1954 18.71/0.5546 20.91/0.6947
4 TMac-TTKA 23.18/0.8113 29.47/0.9681 29.93/0.9462 20.82/0.7521 28.04/0.9429 8.83/0.1229
5 SPCTV 29.56/0.9133 23.38/0.9154 16.02/0.6107 18.58/0.6894 24.21/0.9144 20.98/0.7254
6 LRTV 30.93/0.9382 36.98/0.9945 34.07/0.9724 26.82/0.9228 27.10/0.9319 21.62/0.7541
1 TTLRTV 33.02/0.9579 37.27/0.9945 34.94/0.9823 28.82/0.9561 29.46/0.9559 22.37/0.7487
Our methods 2 tSVDLRTV 32.20/0.9550 37.49/0.9950 34.70/0.9818 28.03/0.9507 29.56/0.9574 22.86/0.8021
3 UnfoldingLRTV 35.61/0.9689 37.72/0.9952 34.87/0.9821 29.55/0.9639 29.59/0.9556 25.43/0.8863
PSNR (dB)/SSIM of different color images under different missing patterns
No. Methods House Peppers Lena Airplane Baboon Boats
Random 50% Text Lines Random line Blocks Random 80%
1 MatrixLR 9.38/0.8970 33.23/0.9814 13.34/0.5850 7.118/0.1308 27.62/0.9343 19.18/0.5680
2 TV 29.70/0.8816 34.14/0.9913 29.21/0.9107 22.85/0.8463 23.18/0.9066 20.32/0.6103
3 TTLR 30.21/0.9251 34.86/0.9892 31.79/0.9559 25.77/0.8796 25.42/0.9239 21.44/0.7144
4 tSVDLR 29.79/0.8989 33.86/0.9840 31.20/0.9561 18.91/0.8386 28.03/0.9373 21.34/0.6879
5 UnfoldingLR 32.58/0.9416 36.86/0.9938 33.45/0.9771 28.75/0.9464 22.22/0.9238 23.46/0.8139
6 TTLRTV 33.02/0.9579 37.27/0.9945 34.94/0.9823 28.82/0.9561 29.46/0.9559 22.37/0.7487
7 tSVDLRTV 32.20/0.9550 37.49/0.9950 34.70/0.9818 28.03/0.9507 29.56/0.9574 22.86/0.8021
8 UnfoldingLRTV 35.61/0.9689 37.72/0.9952 34.87/0.9821 29.55/0.9639 29.59/0.9556 25.43/0.8863
Runtime (s)
Methods House Lena Airplane Boats
Random 50% Lines Random lines Random 80%
MratrixLR 4.95 0.17 0.16 5.01
STDC 5.43 5.13 5.17 5.16
HaLRTC 8.00 0.88 0.84 6.84
FBCP 188.32 86.45 132.09 219.33
SPCTV 19.25 16.37 16.03 17.69
LRTV 19.08 20.17 21.04 21.05
TTLRTV 145.5 143.2 142.6 142.3
tSVDLRTV 15.23 15.07 15.17 15.14
UnfoldingLRTV 9.49 8.53 8.69 8.72
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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
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3/n - How do I in FP... trying several input for the same function
In the previous entry of this series, I wrote about error recovery in a functional way. But sometimes, we want to try several inputs until we retrieve (optionally) a result. This is the case if you
have a function which compute a location from a string (think geo-search), and which optionally returns a result, if a place matched. We may try this function on several input strings and only care
in the first result. In this article, we will see how to do this in a functional way.
Problem setup
Given a function def computeLocation(str: String): Option[Location] which tries to infer a location from a string and a list of input strings sorted by order of relevance (meaning that the first
element may result in a more interesting location than the second and so on), let's retrieve the location (optionally). To sum it up:
// The computeLocation function:
def computeLocation(s: String): Option[Location] = // the implementation
// this call may generate a long List
val inputs: List[String] = createInputsFromForm(aForm)
How implement val result: Option[Location] = ??? ?
From scratch
The first solution is to call the computeLocation function for every string in the input list:
This solution is the most simple ever. Let's break it into pieces to analyze it.
Let's start from the end. As we care only about the first meaningful result (which may not exist), we call headOption.
Now, back to the beginning of the line, we are calling flatMap on a List. The proper signature of flatMap in Scala standard library is slightly different from the one we are used to:
def flatMap[B](f: A => IterableOnce[B]): List[B] = //...
In the standard library, flatMap expects a function producing an IterableOnce. Fortunately Option implements this trait. The definition commonly used for flatMap is more like this one:
// Given that F is a type with one parameter
def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
// For List this results in:
def flatMap[A, B](fa: List[A])(f: A => List[B]): List[B]
In the standard library, the trick with IterableOnce spares us from writing something like:
.flatMap{ s =>
computeLocation(s).map(l => List(l)).getOrElse(Nil)
In this code, we are manually transforming an Option into a List.
What's wrong with this implementation ? If you provide a dummy implementation for both computeLocation and createInputsFromForm, you will see that computeLocation is called for every single item in
the list. If we consider that computing the location is a costly operation, it is sad to waste resource computing, while we only care about the first meaningful answer. How to make the flatMap
sequence stop when we hit a result or the end of the list ?
Recursion by hand
If we break our problem in terms of imperative programming, we will easily come up with a while statement (in pseudo code):
while no result and list has elements left
result = computeLocation with next element in the list
Let's encode this with recursion:
def recur(next: List[String]): Option[Location] = {
next match {
case Nil => None
case head :: tail =>
val r = computeLocation(head)
if (r.isDefined) r else recur(tail)
What we do is just manually traverse the list recursively until computeLocation gives a meaningful result. This is strictly the same as the imperative while version, but in FP, we like to look
further. We could for instance abstract over the computeLocation and take as an extra parameter the function returning an Option:
def recur(next: List[String],
f: String => Option[Location]): Option[Location] = {
next match {
case Nil => None
case head :: tail =>
val r = f(head)
if (r.isDefined) r else recur(tail, f)
recur(inputs, computeLocation)
Looking at this code, do we really have to deal with String and Location ? We do not use any of the properties of these types in recur. Let's make them parameters.
def recur[A, B](next: List[A], f: A => Option[B]): Option[B] = {
next match {
case Nil => None
case head :: tail =>
val r = f(head)
if (r.isDefined) r else recur(tail, f)
recur(inputs, computeLocation)
Following this reasoning we could go even further and add some mechanism to also deal with something else than Option and List. In fact, cats has such functions to hide recursion from you. It is
called tailRecM but as it deserves an entire post for itself, I won't talk about it here.
Back with the Scala standard library...
In the standard library, we can also find the collectFirst function which, well do pretty much what we need !
inputs.collectFirst { str =>
computeLocation(str) match {
case Some(r) => r
Well, the careful reader would have noticed that the function we pass to collectFirst is not total: it does not produce an output for every input.
As computeLocation is total, then it is a bit sad to write a non-total version of it... The standard library also provides a function to transform a total function returning Option to a partial one:
def unlift[T, R](f: T => Option[R]): PartialFunction[T, R]. So we can now write:
If you are using cats, you will find collectFirstSome which will make this possible:
What if computeLocation perform other effects ?
Let's change the problem a bit now. What if the computeLocation function performs some network calls, or read a database to find a result ? These things are known to fail at one point or another. How
to deal with that ?
Without adding too much complexity, let's consider that computeLocation has a more expressive output: Either[Throwable, Option[Location]]. This type means that computeLocation may fail with a
Throwable or give a result if it was able to compute one (the absence of result being completely normal). The complete signature is now:
def computeLocation(rawStr: String): Either[Throwable, Option[Location]]
As you can see, the Option result is now boxed into Either. This makes both collectFirst and collectFirstSome impossible to use, as the type do not match now. In FP, this kind of boxing appears
frequently, so that authors of functional libraries like cats already wrote functions to deal with this. In cats, you will often see functions whose name ends with a capital M. In our case, it is
easy to find collectFirstSomeM which has this signature:
def collectFirstSomeM[G[_], B](f: A => G[Option[B]])
(implicit F: Foldable[F], G: Monad[G]): G[Option[B]]
As you can see, this definition is abstract. Note that this function is part of a class parametrized on F, where F is itself a type with a "hole" : F[_]. In our example, this F will be Either
[Throwable,_] (ie Either with only one type parameter applied). To make it easier, here is a version (pseudo scala) where abstract types are replaced with the one we use in our example (without the
implicit part):
def collectFirstSomeM[Either[Throwable,_], Location](f: String => IO[Option[Location]]): Either[Throwable,Option[Location]]
Look how computeLocation has exactly the expected signature for the f parameter ! What about the implicits parameters now ? In this case, we can read the definition as: "the collectFirstSomeM method
is available if and only if for the F type we can implement the trait Foldable and for the G type the Monad trait". In our case, with List and Either it is already done for us in cats so we can do:
import cats.implicits._
val inputs: List[String] = computeInputs(param)
val result: Either[Throwable, Option[Location]] = inputs.collectFirstSomeM(computeLocation)
Yet, if you write a simple sample for this, like :
def computeLocation(rawStr: String): Either[Throwable, Option[Location]] = {
println(s"Computing with $rawStr") // to see something
// Fake implementation
if (rawStr.length == 1) Left(new IllegalStateException("wrong state"))
else if (rawStr.length % 4 == 0) Right(Some(Location(2.0, 4.0, "Plop")))
else Right(None)
val inputs = List("a", "impair?", "plop", "other")
import cats.implicits._
You will notice that the output will be:
Computing with a
Left(java.lang.IllegalStateException: wrong state)
The execution is stopping as soon as a Left is returned ! This is perfectly normal as Either is a fail fast structure. And the requirement for Monad should have caught our attention: Monad express
sequence, the next operation using the result of the previous one as input parameter.
So, as the caller of computeLocation in our situation, we have two options:
• consider a Left as a blocker, and indeed, keep it,
• try to recover from it by composing computeLocation with a function handling the situation with care.
Here is an example of the second choice, being ignoring errors (you probably don't want to do that, and perform a selective recovery, composing with things we saw earlier):
s => computeLocation(s).recoverWith(_ => Either.right(None))
In this article, we saw how to deal with something simple: calling a function for elements in a list and keeping the first meaningful result (if any). Even if it was simple, what I would like to
highlight is how this solution is composable. Every type in the design models an intent:
• Option models that the result may be irrelevant or present
• Either models that a computation may fail
• Foldable express that we want to build a single value from an initial structure (in our case, we fold the list to a single result)
• Monad express that we compute in sequence: indeed, there is no parallelism involved in our solution, we try the input strings one after the other.
Yet, the work here is not over. Next time, we will see how we can adapt this example to make it less "sequential".
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Factors and multiples: Year 4: Planning tool
Year levels
Expected level of development
Australian Curriculum Mathematics V9: AC9M4N05
Numeracy Progression: Number and place value: P7, Multiplicative strategies: P7
At this level, students solve problems involving multiplying or dividing natural numbers by multiples and powers of 10 without a calculator, using the multiplicative relationship between the place
value of digits.
Use relevant materials such as place value charts, place value expanders and virtual manipulatives to explore the effect of multiplying by 10. Make explicit that multiplying by 10 moves the digits
one place to the right. Discuss the use of zero.
Take the opportunity to introduce digital tools to explore the effect of multiplying or dividing numbers by 10 first, then multiples of 10, for example, 100 and 1,000. A spreadsheet can be a useful
tool to automate the process using a few simple formulas. A possible starting point is to have a spreadsheet for students to use that has a formula set up to multiply by 10. Use this to show them how
to create their own spreadsheet to explore emerging patterns, for example, when dividing by 10, 100 or 1000.
Teaching and learning summary:
• Use relevant materials to explore the effect of multiplying by 10.
• Introduce digital tools to explore the effect of multiplying or dividing numbers by multiples of 10.
• describe patterns of multiplying or dividing numbers by multiples of 10
• use efficient computational strategies to solve problems that include multiplying or dividing numbers by multiples of 10.
Some students may:
• believe you should you just add a zero when multiplying by 10. This approach is limited in its usefulness and can cause problems when dealing with decimal numbers. Provide challenges to this line
of thinking and present a decimal number. For example, 3.5 x 10 isn’t 3.50, because simply inserting a zero on the end gives exactly the same value.
The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate
and extend learning done at school or for home schooling.
Learning intention
• We are learning to develop understanding of multiplying and dividing natural numbers by multiples of 10.
Why are we learning about this?
• This builds number sense and helps us find efficient ways to solve problems.
What to do
1. Create a series of cards that match answers to the numbers in the grid.
2. Each equation should be written with a multiple of 10, either multiplication or division. For example, an answer for 300 could be 3,000 ÷ 10 or 30 x 10.
3. Create 30 cards and play bingo with a friend.
4. Whoever is first to get a row and then fill the whole card is the winner.
│300│9,000 │4 │90 │
│30 │7 │50 │500│
│0.5│60 │600│100│
│1 │1,000 │9 │700│
5. Create a series of cards that match answers to the numbers in the grid above.
Each equation should be written with a multiple of 10, either multiplication or division.
For example, an answer for 300 could be 3000 ÷ 10 or 30 x 10.
Success criteria
I can:
• use my knowledge of multiples of 10 when multiplying and dividing numbers.
Please note: This site contains links to websites not controlled by the Australian Government or ESA. More information here.
Teaching strategies
A collection of evidence-based teaching strategies applicable to this topic. Note we have not included an exhaustive list and acknowledge that some strategies such as differentiation apply to all
topics. The selected teaching strategies are suggested as particularly relevant, however you may decide to include other strategies as well.
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MP Board Class 7th Maths Solutions Chapter 7 Congruence of Triangles Ex 7.1
Question 1.
Complete the following statements:
(a) Two line segments are congruent if ___ .
(b) Among two congruent angles, one has a measure of 70°; the measure of the other angle is ___.
(c) When we write ∠A = ∠B, we actually mean ___.
(a) They have the same length.
(b) 70°
(c) m∠A = m∠B
Question 2.
Give any two real-life examples for congruent shapes.
(i) Sheets of same letter pad.
(ii) Biscuits in the same packet.
Question 3.
If ∆ABC ≅ ∆FED under the correspondence ABC ⟷ FED, write all the corresponding congruent parts of the triangles.
If ∆ABC ≅ ∆FED, then the corresponding angles and sides will be equal to each other.
Question 4.
If ∆DEF ⟷ ∆BCA, write the part(s) of ∆BCA that correspond to
(i) ∠E
(ii) \(\overline{E F}\)
(iii) ∠F
(iv) \(\overline{D F}\)
(i) ∠C
(ii) \(\overline{C A}\)
(iii) ∠A
(iv) \(\overline{B A}\)
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Visualize the mean, median, IQR, and MAD
You can move the data points around in the distribution below and see how the distribution of data impacts the mean, median, interquartile range (IQR), and mean absolute deviation (MAD) of the data.
Note that the MAD is actually only calculated as half the length of the vectors shown.
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Understanding Mathematical Functions: How To Write A Function Notation
Introduction to Mathematical Functions and Function Notation
Mathematical functions play a crucial role in a wide range of fields, including mathematics, physics, engineering, and economics. Functions help us describe and analyze relationships between
variables, making them essential tools for problem-solving and modeling real-world phenomena.
Overview of the importance of understanding mathematical functions in various fields of study
Understanding mathematical functions is essential in various fields of study because they allow us to represent and analyze relationships between different quantities. For example, in physics,
functions are used to describe the motion of particles, while in economics, functions help us model supply and demand curves. By understanding functions, we can make predictions, analyze trends, and
solve complex problems.
Introduction to function notation as a method to express relationships between variables
Function notation is a method used to express relationships between variables in mathematics. It is a way of representing a function using symbols and mathematical expressions. Function notation
allows us to define a function, name it, and use it in equations and calculations.
Brief history of function notation and its significance in simplifying complex mathematical concepts
Function notation has a long history in mathematics, dating back to the work of mathematicians such as Gottfried Wilhelm Leibniz and Leonhard Euler. The use of function notation has significantly
contributed to simplifying complex mathematical concepts by providing a standardized way to represent functions and their relationships with variables. By using function notation, mathematicians and
scientists can communicate ideas more effectively and work with functions in a more organized and efficient manner.
Key Takeaways
• Function notation is a way to represent mathematical relationships.
• Functions have input and output values.
• Writing a function in notation helps simplify complex expressions.
• Function notation uses f(x) to represent a function of x.
• Understanding function notation is essential in higher level math.
Fundamentals of Function Notation
Function notation is a crucial concept in mathematics that allows us to represent relationships between variables in a concise and organized manner. By using function notation, we can easily define
and work with mathematical functions. Let's delve into the key components of function notation:
A Definition of function notation and its components (eg, f(x))
Function notation is a way of representing a function using symbols and variables. The most common form of function notation is f(x), where f represents the function and x is the input variable. The
expression f(x) is read as 'f of x' and indicates that the function f operates on the input x.
Distinguishing between the input (independent variable) and output (dependent variable)
It is essential to understand the distinction between the input and output variables in function notation. The input variable, often denoted as x, is the independent variable that we can manipulate
or change. On the other hand, the output variable, represented by f(x), is the dependent variable that is determined by the function's rule or formula.
• Input (Independent Variable): The variable that is controlled or chosen by the experimenter.
• Output (Dependent Variable): The variable that is influenced by changes in the input variable.
Explanation of the domain and range within the context of function notation
In the context of function notation, the domain refers to the set of all possible input values for the function. It represents the valid inputs that the function can operate on. On the other hand,
the range is the set of all possible output values that the function can produce based on the given inputs.
Understanding the domain and range of a function is crucial for determining the behavior and limitations of the function. The domain restricts the possible inputs, while the range specifies the
possible outputs that the function can generate.
Writing Basic Function Notations
Function notation is a way to represent a mathematical function using symbols and variables. It helps us understand how one quantity depends on another and allows us to perform operations on
functions. Here is a step-by-step guide on how to write function notations from simple equations:
A Step-by-step guide on writing function notations from simple equations
• Step 1: Identify the input and output variables in the equation. The input variable is usually denoted by x, while the output variable is denoted by y.
• Step 2: Write the function notation using the input and output variables. For example, if the equation is y = 2x + 3, the function notation would be f(x) = 2x + 3.
• Step 3: Use the function notation to represent the relationship between the input and output variables. In this case, f(x) represents the output y as a function of the input x.
Examples of converting common mathematical expressions into function notations
Let's look at some examples of converting common mathematical expressions into function notations:
• Example 1: If the equation is y = x^2, the function notation would be f(x) = x^2.
• Example 2: For the equation y = 3x - 5, the function notation would be f(x) = 3x - 5.
• Example 3: If the equation is y = sin(x), the function notation would be f(x) = sin(x).
Common mistakes to avoid when writing function notations for the first time
When writing function notations for the first time, it's important to avoid common mistakes that can lead to confusion. Here are some mistakes to watch out for:
• Mistake 1: Mixing up input and output variables. Make sure to correctly identify which variable represents the input and which represents the output.
• Mistake 2: Forgetting to use function notation. Always remember to use f(x) or another appropriate notation to represent the function.
• Mistake 3: Not specifying the function domain. It's important to define the domain of the function to avoid ambiguity.
Advanced Function Notations
As we delve deeper into the realm of mathematical functions, we encounter more complex notations that involve multiple variables. Understanding these advanced function notations is crucial for
tackling higher-level mathematics such as calculus and algebra. Let's explore some examples and strategies for interpreting these intricate notations.
Introduction to more complex function notations involving multiple variables
When dealing with functions that involve multiple variables, the notation becomes more sophisticated. Instead of a simple f(x) notation, we might see functions written as f(x, y) or even f(x, y, z).
Each variable represents a different input that affects the output of the function. For example, in a function f(x, y) = x + y, both x and y contribute to the final result.
Examples of function notations in higher mathematics
In higher mathematics, such as calculus and algebra, complex function notations are commonly used to represent intricate relationships between variables. For instance, in calculus, you might come
across functions involving derivatives and integrals, denoted by symbols like f'(x) and ∫f(x)dx. These notations convey important information about the behavior of the function and its derivatives.
• Example 1: In calculus, the chain rule is often represented using function notation as (f(g(x)))' = f'(g(x)) * g'(x), where f and g are functions of x.
• Example 2: In algebra, matrices are commonly used to represent linear transformations, with functions written as f(A) = A^2 - 2A + I, where A is a matrix.
Strategies for understanding and interpreting complex function notations
When faced with complex function notations, it's essential to break them down into smaller components and analyze each part separately. Here are some strategies to help you make sense of intricate
function notations:
• Identify the variables: Determine the variables involved in the function and understand how each one contributes to the output.
• Look for patterns: Search for recurring patterns or structures within the notation that can provide insights into the function's behavior.
• Consult resources: Utilize textbooks, online resources, or consult with peers or instructors to gain a deeper understanding of complex function notations.
• Practice solving problems: Work through practice problems that involve complex function notations to improve your proficiency in interpreting them.
Applications of Function Notation in Real-world Scenarios
Illustration of how function notation is used in sciences (eg, physics, chemistry)
In the field of sciences, function notation plays a crucial role in representing relationships between variables. For instance, in physics, a function may describe the motion of an object in terms of
time. This function could be denoted as f(t), where f represents the function and t represents time. By using function notation, scientists can easily analyze and predict the behavior of physical
Exploration of function notation in economics and social sciences
In economics and social sciences, function notation is used to model various relationships and phenomena. For example, in economics, a production function may be denoted as Q(K,L), where Q represents
the quantity of output, K represents capital, and L represents labor. This notation helps economists understand how changes in inputs affect output levels.
Practical examples demonstrating the utility of function notation in technology and engineering
Function notation is widely used in technology and engineering to describe complex systems and processes. For instance, in electrical engineering, a transfer function may be denoted as H(s), where H
represents the transfer function and s represents the Laplace variable. This notation allows engineers to analyze the behavior of electrical circuits and design efficient systems.
6 Troubleshooting Common Issues with Function Notation
Function notation can sometimes be tricky to work with, leading to common errors and misunderstandings. In this section, we will discuss some of the most frequent issues that arise when dealing with
function notation and provide tips for resolving them.
A Identifying and resolving frequent errors in writing and interpreting function notations
• Missing parentheses: One common error in function notation is forgetting to include parentheses when writing a function. This can lead to confusion about the order of operations and the input
value of the function.
• Incorrect variable names: Another common mistake is using the wrong variable name in a function notation. It is important to use the correct variable to ensure the function is defined properly.
• Confusion between function notation and algebraic expressions: Sometimes, students may mix up function notation with algebraic expressions, leading to errors in interpretation. It is essential to
understand the difference between the two concepts.
B Tips for verifying the accuracy of a function notation
• Substitute values: One way to verify the accuracy of a function notation is to substitute different values for the input variable and check if the output matches the expected result.
• Check for consistency: Make sure that the function notation is consistent with the definition of the function and follows the correct mathematical rules.
• Use graphing tools: Graphing the function can also help in verifying the accuracy of the function notation. This visual representation can provide insights into the behavior of the function.
C Strategies for simplifying complex function notations for easier understanding
• Break it down: If you encounter a complex function notation, try breaking it down into smaller parts and analyzing each component separately. This can help in understanding the overall function
• Use examples: Work through examples of different function notations to gain a better understanding of how they work. Practice is key to mastering complex concepts.
• Seek help: If you are struggling with a particular function notation, don't hesitate to seek help from a teacher, tutor, or online resources. Sometimes, a fresh perspective can make all the
Conclusion & Best Practices in Function Notation
In conclusion, understanding mathematical functions and mastering function notation is essential for success in various fields such as mathematics, physics, engineering, and computer science. By
grasping the key points covered in this blog post, individuals can enhance their problem-solving skills and analytical thinking abilities.
A Recap of the key points covered and their importance in mastering function notation
• Definition of Function Notation: Function notation is a way to represent functions using symbols and variables. It helps in simplifying complex mathematical expressions and making them easier to
work with.
• Importance of Function Notation: Function notation allows us to define, evaluate, and manipulate functions efficiently. It provides a standardized way to communicate mathematical ideas and
• Understanding Function Composition: Function composition involves combining two or more functions to create a new function. It is a fundamental concept in mathematics and plays a crucial role in
solving real-world problems.
Best practices for writing and working with function notations, including continuous learning and application
• Consistent Notation: Use clear and consistent notation when writing functions to avoid confusion. Follow standard conventions and guidelines for function notation.
• Practice and Application: Regular practice and application of function notation in solving problems can help improve your understanding and proficiency. Work on a variety of problems to enhance
your skills.
• Continuous Learning: Stay updated with new developments in function notation and related concepts. Engage in continuous learning through courses, books, and online resources to deepen your
Encouragement for further exploration of mathematical functions beyond basic function notation
As you continue your journey in mathematics, I encourage you to explore advanced topics in mathematical functions beyond basic function notation. Dive into topics such as trigonometric functions,
exponential functions, logarithmic functions, and more. These concepts have wide-ranging applications in various fields and can broaden your understanding of mathematical functions.
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Daniel Bernoulli - (College Physics II – Mechanics, Sound, Oscillations, and Waves) - Vocab, Definition, Explanations | Fiveable
Daniel Bernoulli
from class:
College Physics II – Mechanics, Sound, Oscillations, and Waves
Daniel Bernoulli was a Swiss mathematician and physicist who made significant contributions to the field of fluid dynamics. He is best known for his work on the relationship between pressure,
velocity, and elevation in flowing fluids, which is now known as Bernoulli's Equation.
congrats on reading the definition of Daniel Bernoulli. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Daniel Bernoulli's work on fluid dynamics was published in his 1738 book, 'Hydrodynamica,' which laid the foundations for the modern understanding of fluid behavior.
2. Bernoulli's Equation states that the sum of the pressure, the kinetic energy per unit volume, and the gravitational potential energy per unit volume of a fluid is constant along a streamline.
3. Bernoulli's Equation is used to explain many phenomena, such as the lift generated by airplane wings, the flow of air over a curved surface, and the operation of carburetors in internal
combustion engines.
4. Bernoulli's Principle, which is derived from Bernoulli's Equation, is used to explain the behavior of fluids in motion, such as the decrease in pressure in a constricted section of a pipe or the
lift generated by airplane wings.
5. Bernoulli's work on fluid dynamics has had a significant impact on various fields, including aerodynamics, hydraulics, and meteorology, and his contributions have been essential in the
development of modern engineering and science.
Review Questions
• Explain the relationship between pressure, velocity, and elevation in flowing fluids as described by Bernoulli's Equation.
□ According to Bernoulli's Equation, the sum of the pressure, the kinetic energy per unit volume, and the gravitational potential energy per unit volume of a fluid is constant along a
streamline. This means that as the velocity of a fluid increases, the pressure within the fluid decreases, and vice versa. Additionally, as the elevation of a fluid increases, the
gravitational potential energy per unit volume increases, which leads to a decrease in the pressure and velocity of the fluid. This relationship is fundamental to understanding the behavior
of fluids in motion and has numerous applications in fields such as aerodynamics and hydraulics.
• Describe how Bernoulli's Principle, which is derived from Bernoulli's Equation, is used to explain the lift generated by airplane wings.
□ Bernoulli's Principle states that as the speed of a fluid increases, the pressure within the fluid decreases. This principle is used to explain the lift generated by airplane wings. The shape
of an airplane wing is designed to create a difference in the airflow over the top and bottom surfaces of the wing. The air traveling over the top of the wing has a longer distance to cover,
which means it must move faster to keep up with the air traveling along the bottom of the wing. According to Bernoulli's Principle, the faster-moving air over the top of the wing has a lower
pressure, while the slower-moving air along the bottom of the wing has a higher pressure. This difference in pressure creates a net upward force, which is the lift that allows the airplane to
• Analyze the significance of Daniel Bernoulli's contributions to the field of fluid dynamics and their impact on various scientific and engineering disciplines.
□ Daniel Bernoulli's work on fluid dynamics, particularly his formulation of Bernoulli's Equation and Bernoulli's Principle, has had a profound and far-reaching impact on numerous scientific
and engineering disciplines. His work laid the foundations for the modern understanding of fluid behavior, which has been essential in the development of fields such as aerodynamics,
hydraulics, and meteorology. Bernoulli's Equation and Principle have been used to explain and predict a wide range of phenomena, from the lift generated by airplane wings to the operation of
carburetors in internal combustion engines. Bernoulli's contributions have been instrumental in the design and optimization of various engineering systems, from aircraft to plumbing and
irrigation systems. Moreover, his work has had a significant impact on our understanding of the natural world, with applications in fields such as oceanography and atmospheric science. The
enduring legacy of Daniel Bernoulli's work in fluid dynamics continues to shape our understanding of the physical world and drive innovation in various scientific and engineering disciplines.
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by Lazur - uploaded on August 22, 2016, 2:53 am
A cube with 1m edge in an ortographic projection where x, and y axises are aligned to 22° and 40° from the horizontal. (In 1:1 scale @90 dpi.)
To top that, the image is cloned and transformed so that each clone has a face in the shape of a square. Such affine transformation is handy if you want to map a pattern to the faces of the cube,
although this image goes the opposite direction with the cloning for the aesthetic.
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[Solved] Construct a confidence interval for p-P2 | SolutionInn
Answered step by step
Verified Expert Solution
Construct a confidence interval for p-P2 at the given level of confidence. x =26, n =229, x2 = 31, n = 302, 95% confidence
Construct a confidence interval for p-P2 at the given level of confidence. x =26, n =229, x2 = 31, n = 302, 95% confidence The researchers are % confident the difference between the two population
proportions, P - P2, is between (Use ascending order. Type an integer or decimal rounded to three decimal places as needed.) and
There are 3 Steps involved in it
Step: 1
Calculate the sample proportions p 1 x 1 n 1 26 229 hatp1 fracx1n1 frac26229 p1n1x122926 p 2 x 2 n 2 ...
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The use of static variables position and side within the find_index function is problematic for a couple of reasons. First, it introduces statefulness to the function, which can lead to unexpected
behavior if the function is called multiple times, as the position and side will retain their values from previous calls. This makes the function non-reentrant and not thread-safe, which can cause
issues in a multi-threaded environment. Second, the logic of the binary search is flawed because it relies on these static variables to track the position and side, which is not a standard approach
and can lead to incorrect results.
To resolve these issues, remove the static keyword from the position and side variables and refactor the function to pass these variables as parameters if necessary. Additionally, consider
implementing the binary search algorithm in a more conventional and stateless manner, which typically does not require tracking the side of the search.
long unsigned int middle_piece = floor(arr.size() / 2); int mid_calc = arr.size() % middle_piece == 0 ? (arr.size() - (middle_piece - 1)) : (arr.size() - middle_piece); cout << mid_calc << endl;
position = position == 0 ? ( side ? position + middle_piece : position - middle_piece ) : ( side ? position + mid_calc : position - mid_calc );
The calculation of middle_piece and mid_calc is incorrect and does not follow the standard binary search algorithm. The variable middle_piece is intended to represent the middle index of the array,
but the calculation uses floor(arr.size() / 2) which is unnecessary because integer division will naturally truncate the decimal part. The mid_calc calculation is also incorrect and does not serve a
clear purpose in the context of a binary search. The ternary operator’s condition arr.size() % middle_piece == 0 does not make sense in this context and could lead to division by zero if middle_piece
is zero.
To correct this, middle_piece should be calculated as arr.size() / 2, and the entire mid_calc logic should be removed. Instead, the binary search should be implemented with a clear and correct
calculation of the middle index within each recursive call, and the algorithm should not modify the position based on the side variable. The standard binary search algorithm divides the array into
halves by comparing the middle element with the target item and recursing into the appropriate half without the need for additional calculations or tracking the side of the search.
position = position == 0 ? ( side ? position + middle_piece : position - middle_piece ) : ( side ? position + (middle_piece + 1) : position - (middle_piece + 1) );
The ternary operator logic for updating the position variable is incorrect and can lead to out-of-bounds access. When side is false and position is 0, subtracting middle_piece or middle_piece + 1
will result in a negative index, which is invalid for a vector. This can cause undefined behavior when accessing arr[position].
To resolve this issue, the logic for updating position should be revised to ensure that it remains within the valid range of indices for the vector. Additionally, consider using a more standard
approach to binary search without modifying the position variable in this manner, as it complicates the logic and can lead to errors.
The code is creating a new vector other_half and copying elements from arr into it in each recursive call. This is inefficient as it involves unnecessary copying of vector elements, which can be
expensive for large vectors.
Instead of copying elements into a new vector, consider passing the index range to the recursive function find_index to work on the subarray directly. This will avoid the overhead of copying and will
be more efficient in terms of both time and space complexity.
if (arr.size() == 1) { if (arr[m] == s_item) { cout << "Array cointain " << s_item << " on position: "<< s + (m-s) << endl; } else { if (s_item > arr[m]) { b_temp(arr, s_item, m + 1, s + 2*(m-s)); }
else { b_temp(arr, s_item, e - (m+e)/2, m - 1); } } }
The condition if (arr.size() == 1) is misleading and incorrect for the purpose of a binary search algorithm. Binary search does not require the array size to be 1 to find an element. Instead, it
should check if the start index s is less than or equal to the end index e to continue the search, and use the middle element m to compare with the search item s_item. If s_item is equal to arr[m],
the item is found. Otherwise, the search should continue in the left or right half of the array depending on whether s_item is less than or greater than arr[m]. The recursive calls in lines 21 and 23
also seem incorrect as they do not correctly adjust the search boundaries based on the standard binary search algorithm. The recommended solution is to remove the if (arr.size() == 1) condition and
correctly implement the binary search logic with proper recursive calls adjusting the search boundaries.
if (s_item > arr[m]) { b_temp(arr, s_item, m + 1, s + 2*(m-s)); } else { b_temp(arr, s_item, e - (m+e)/2, m - 1);
The recursive calls to b_temp in lines 20-23 use incorrect logic for adjusting the search boundaries. In a standard binary search, if the search item is greater than the middle element, the search
should continue in the right half of the array, which means updating the start index s to m + 1. Conversely, if the search item is less than the middle element, the search should continue in the left
half, updating the end index e to m - 1. The calculations s + 2*(m-s) and e - (m+e)/2 do not correctly adjust the search boundaries and can lead to incorrect behavior or infinite recursion. The
recommended solution is to correctly update the search boundaries for the recursive calls to accurately reflect the binary search algorithm: use m + 1 as the new start index when searching the right
half and m - 1 as the new end index when searching the left half.
The condition f(m) === 0 might never be true due to the floating-point arithmetic precision issues inherent in JavaScript. This could potentially cause the function to miss the exact root if it
exists. Instead of checking for strict equality to zero, consider using a tolerance value to determine if f(m) is close enough to zero to be considered as the root. For example, you could use
Math.abs(f(m)) < some_small_value as the condition.
while (Math.abs(b - a) > tol && iterations < 100) {
The loop condition checks if the absolute difference between b and a is greater than tol and if the number of iterations is less than 100. However, relying on a fixed number of iterations (in this
case, 100) as a fallback mechanism to prevent infinite loops might not be the best approach for all cases. It’s better to allow the user to specify the maximum number of iterations as a parameter to
the function. This way, the user can adjust the precision and performance trade-offs according to their specific needs.
function bisect(f, a, b, tol) { let iterations = 0; while (Math.abs(b - a) > tol && iterations < 100) { const m = (a + b) / 2; if (f(m) === 0) { return m; } else if (f(m) * f(a) < 0) { b = m; } else
{ a = m; } iterations++; } return (a + b) / 2;
The bisection method implemented in the bisect function lacks a mechanism to handle cases where the function f does not change signs over the interval [a, b]. This is a fundamental assumption for the
bisection method to work correctly. If f(a) and f(b) have the same sign, it means that there might not be a root in the interval, or there are an even number of roots, which the current
implementation does not account for.
Recommendation: Before entering the while loop, check if f(a) and f(b) have opposite signs. If they do not, either return an error or a specific value indicating that the method cannot proceed. For
if (f(a) * f(b) >= 0) { console.error('Function does not change signs over the interval. Bisection method cannot proceed.'); return null; // or any other indication of failure }
//raty malejace let raty = []; for (let i = 0; i < ilosc_rat; i++) { raty[i] = pozost * oprocen + rata_kap; if (i == 0) raty[i] += prowizja; pozost -= rata_kap; }
The loop for calculating decreasing installments (raty) does not account for the change in interest due to the decreasing principal amount. The interest component (pozost * oprocen) is calculated
based on the remaining principal (pozost), which decreases with each installment. However, the calculation of raty[i] adds the same principal installment (rata_kap) every time, which is correct, but
the interest calculation should ideally be recalculated after each iteration to reflect the decreasing principal.
Recommendation: Move the calculation of the principal installment (rata_kap) inside the loop to ensure that the interest is calculated based on the updated remaining principal. However, since
rata_kap is constant and correct as per the logic for equal principal payments, the recommendation is to clarify the intent if the interest recalculated per iteration was the intended behavior or if
the misunderstanding stems from the variable naming and expected behavior of decreasing installments:
// If the intent was to have a fixed principal payment and recalculate interest on the remaining balance: for (let i = 0; i < ilosc_rat; i++) { raty[i] = pozost * oprocen + rata_kap; if (i == 0) raty
[i] += prowizja; pozost -= rata_kap; }
Ensure the logic aligns with the intended financial model, as the current implementation suggests a fixed principal payment model rather than a decreasing installment model where both principal and
interest components decrease.
b_temp(arr, s_item, m + 1, s + 2*(m-s)); } if (s_item < arr[m]) { b_temp(arr, s_item, e - (m+e)/2, m - 1);
The recursive calls in the binary search implementation have incorrect parameters for adjusting the search range, which can lead to infinite recursion or incorrect behavior. Specifically, the
calculation for the new start and end indices in the recursive calls does not correctly narrow down the search range according to the binary search algorithm.
To fix this, ensure that the recursive calls correctly adjust the search range. The first call should set the new start index to m + 1 and keep the end index as e when the search item is greater than
the middle element. The second call should set the new end index to m - 1 and keep the start index as s when the search item is less than the middle element. This adjustment ensures that the search
range is correctly narrowed down in each step of the recursion, adhering to the binary search algorithm.
The condition if (arr.size() == 1) is misleading and incorrect for the purpose of a binary search algorithm. This condition seems to be intended to check if the search has narrowed down to a single
element, but it incorrectly checks the size of the entire array instead of the current search range. This will cause the function to behave incorrectly for arrays with more than one element.
To fix this issue, remove the condition if (arr.size() == 1) and its enclosing braces. The binary search logic should not depend on the size of the entire array but on the indices s (start) and e
(end) to determine if the search space has been narrowed down to a single element or if it needs to continue dividing the search space. The corrected code should directly proceed with comparing the
middle element with the search item and adjusting the search range accordingly without this incorrect condition.
while (num > 0) { let lastDigit = num % 10; num = (num - lastDigit) / 10; if (lastDigit == digit){ console.log(`The digit ${digit} is in the number.`); break; }else{ console.log ("no") }
The current implementation of the loop and conditional logging within the while loop can lead to excessive and potentially misleading output. Specifically, the loop will print “no” for every digit in
the number that is not equal to the target digit, which could be confusing for users or developers trying to understand the output. Additionally, if the target digit is found, the message indicating
its presence is logged, but the loop also breaks immediately, which is efficient but could be handled more cleanly.
Recommendation: To improve the clarity and efficiency of this code, consider accumulating the result in a boolean variable and logging the output once after the loop completes. This approach reduces
unnecessary logging and makes the code’s intent clearer.
let found = false; while (num > 0) { let lastDigit = num % 10; num = (num - lastDigit) / 10; if (lastDigit == digit) { found = true; break; } } console.log(found ? `The digit ${digit} is in the
number.` : "The digit is not in the number.");
The condition m * n - i * mid >= k - 1 in the binary search logic is potentially incorrect. This condition checks if the remaining area after cutting i * mid pieces is at least k-1, which does not
directly correlate to having no more than k pieces. Recommended Solution: Revise the condition to directly check against the number of pieces rather than the remaining area. For example, ensure that
the number of pieces formed by the cuts does not exceed k.
m * n - i * mid >= k - 1
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Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system
The chemotaxis–Navier–Stokes system
is considered under homogeneous boundary conditions of Neumann type for
, and of Dirichlet type for
, in a bounded convex domain
with smooth boundary, where
, and where
are nonnegative with
. Problems of this type have been used to describe the mutual interaction of populations of swimming aerobic bacteria with the surrounding fluid. Up to now, however, global existence results seem to
be available only for certain simplified variants such as e.g. the two-dimensional analogue of (⋆), or the associated chemotaxis–Stokes system obtained on neglecting the nonlinear convective term in
the fluid equation.
The present work gives an affirmative answer to the question of global solvability for (⋆) in the following sense: Under mild assumptions on the initial data, and under modest structural assumptions
on f and χ, inter alia allowing for the prototypical case when
$f(s)=sfor all s≥0andχ≡const.,$
the corresponding initial–boundary value problem is shown to possess a globally defined weak solution.
This solution is obtained as the limit of smooth solutions to suitably regularized problems, where appropriate compactness properties are derived on the basis of a priori estimates gained from an
energy-type inequality for (⋆) which in an apparently novel manner combines the standard $L2$ dissipation property of the fluid evolution with a quasi-dissipative structure associated with the
chemotaxis subsystem in (⋆).
author = {Winkler, Michael},
title = {Global weak solutions in a three-dimensional {chemotaxis{\textendash}Navier{\textendash}Stokes} system},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1329--1352},
publisher = {Elsevier},
volume = {33},
number = {5},
year = {2016},
doi = {10.1016/j.anihpc.2015.05.002},
zbl = {1351.35239},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/}
TY - JOUR
AU - Winkler, Michael
TI - Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2016
SP - 1329
EP - 1352
VL - 33
IS - 5
PB - Elsevier
UR - http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/
DO - 10.1016/j.anihpc.2015.05.002
LA - en
ID - AIHPC_2016__33_5_1329_0
ER -
%0 Journal Article
%A Winkler, Michael
%T Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 1329-1352
%V 33
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/
%R 10.1016/j.anihpc.2015.05.002
%G en
%F AIHPC_2016__33_5_1329_0
Winkler, Michael. Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 5, pp. 1329-1352. doi : 10.1016/j.anihpc.2015.05.002. http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/
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[30] Y. Tao, Boundedness in a Keller–Segel–Stokes system modeling the process of coral fertilization, preprint.
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[32] Global existence and boundedness in a Keller–Segel–Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., Ser. A, Volume 32 (2012) no. 5, pp. 1901–1914 | Zbl
[33] Locally bounded global solutions in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 1, pp. 157–178 |
Numdam | Zbl
[34] Navier–Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2, North-Holland, Amsterdam, 1977 | Zbl
[35] Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, Volume 102 (2005), pp. 2277–2282 | DOI | Zbl
[36] Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., Volume 12 (2014), pp. 545–563 | DOI | Zbl
[37] The Navier–Stokes equations – a neverending challenge?, Jahresber. Dtsch. Math.-Ver., Volume 101 (1999), pp. 1–25 | Zbl
[38] Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differ. Equ., Volume 248 (2010), pp. 2889–2905 | DOI | Zbl
[39] Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., Volume 37 (2012), pp. 319–351 | DOI | Zbl
[40] Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., Volume 100 (2013), pp. 748–767 | arXiv | DOI | Zbl
[41] Stabilization in a two-dimensional chemotaxis–Navier–Stokes system, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 2, pp. 455–487 | DOI | Zbl
[42] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, preprint.
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published by the International Education Software, IES
written by Ichiro Kobayashi, Katsuhiko Sato, and Shigeru Tsuyuki
Available Languages: English, Japanese
Manipula Math with JAVA is an interactive math education site that uses java applets (JAVA) to illustrate mathematical concepts. Each concept is has a short explanation with graphics. Users can
adjust variables. This site will be most helpful to those seeking illustrations of concepts to which they have already been introduced.
Subjects Levels Resource Types
- Instructional Material
= Activity
= Instructor Guide/Manual
- High School = Interactive Simulation
Other Sciences
- Middle School = Lecture/Presentation
- Mathematics
- Lower Undergraduate = Tutorial
- Audio/Visual
= Image/Image Set
= Movie/Animation
Intended Users Formats Ratings
- application/java
- Learners - image/gif
- Educators - image/jpeg
- text/html
Access Rights:
Free access
Has a copyright or other licensing restriction.
Mathematic Applets, calculus, geometry, trigonometry, vectors
Record Creator:
Date Metadata Instance was created June 21, 2003 by Matthew Meizlish
Record Updated:
August 25, 2016 by Lyle Barbato
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<a href="https://www.compadre.org/portal/items/detail.cfm?ID=124">Kobayashi, I, K. Sato, and S. Tsuyuki. Manipula Math with JAVA. Chiyoda ku, Tokyo: International Education Software, IES.</a>
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Kobayashi, Ichiro, Katsuhiko Sato, and Shigeru Tsuyuki. Manipula Math with JAVA. Chiyoda ku, Tokyo: International Education Software, IES. 14 Nov. 2024 <http://www.ies-math.com/math/java/>.
@misc{ Author = "Ichiro Kobayashi and Katsuhiko Sato and Shigeru Tsuyuki", Title = {Manipula Math with JAVA}, Publisher = {International Education Software, IES}, Volume = {2024}, Number = {14
November 2024}, Year = {} }
%A Ichiro Kobayashi %A Katsuhiko Sato %A Shigeru Tsuyuki %T Manipula Math with JAVA %I International Education Software, IES %C Chiyoda ku, Tokyo %U http://www.ies-math.com/math/java/ %O application/
%0 Electronic Source %A Kobayashi, Ichiro %A Sato, Katsuhiko %A Tsuyuki, Shigeru %T Manipula Math with JAVA %I International Education Software, IES %V 2024 %N 14 November 2024 %9 application/java %U
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Purrier Series (Meow) and Making Images SpeakPurrier Series (Meow) and Making Images Speak - Bilim Ne Güzel Lan
Purrier Series (Meow) and Making Images Speak
Fourier series can be explained as expressing a repetitive curve as sum of sine curves. Since “summation of sine waves” interpretation shows how many of waves are there at each frequency, it is
widely used in engineering, physics, and mathematics. The main idea in this interpretation is that sine and cosine functions are mutually orthogonal, like vectors which are perpendicular to each
\(\displaystyle{\frac{1}{\pi}\int \limits_{-\pi}^{\pi} {\color{black}\sin({\color{red}n} x)}{\color{black} \sin({\color{blue}m} x)} dx = \begin{cases} 1, & {\color{red}n} = {\color{blue}m} \\ 0, & {\
color{red}n} \neq {\color{blue}m}\end{cases}}\)
\(\displaystyle{\frac{1}{\pi}\int \limits_{-\pi}^{\pi} {\color{black}\cos({\color{red}n} x)}{\color{black} \cos({\color{blue}m} x)} dx = \begin{cases} 1, & {\color{red}n} = {\color{blue}m} \\ 0, & {\
color{red}n} \neq {\color{blue}m}\end{cases}}\)
\(\displaystyle{\frac{1}{\pi}\int \limits_{-\pi}^{\pi} {\color{red}\sin({\color{red}n} x)}{\color{blue} \cos({\color{blue}m} x)} dx =0}\)
Reminder; if two unit vectors are orthogonal, result is the same.
\({\bf \hat{{\color{red}e}}}_{\color{red}i} \cdot {\bf \hat{{\color{blue}e}}}_{\color{blue}j} = \begin{cases} 1, & {\color{red}i} ={\color{blue} j} \\ 0,& {\color{red}i} \neq {\color{blue}j} \end
Joseph Fourier
Although it was first discovered by Euler, Fourier realized and demonstrated its importance in analysis of waves; therefore it is known by his name. It is defined as (well, yes it has many forms, but
Fourier defined it as):
\(\displaystyle{f(x) = \frac{a_0}{2} + \sum \limits_{n = 1}^N \left[ a_n {\color{red} \sin(n x)} + b_n {\color{blue}\cos(nx)}\right]},\)
\(f(x)\) is a periodic function for \(x \in [-\pi, \pi]\) of which we want to find the representation. \(a_0\) is a constant,
\(a_0 = \displaystyle{\frac{1}{\pi} \int \limits_{-\pi}^{\pi} f(x) dx }\)
\(a_n\) and \(b_n\) defined as:
\(a_n = \displaystyle{\frac{1}{\pi} \int \limits_{-\pi}^{\pi} f(x) {\color{blue} \cos(nx) }dx },\)
\(b_n = \displaystyle{\frac{1}{\pi} \int \limits_{-\pi}^{\pi} f(x) {\color{red}\sin(nx) }dx }.\)
I would like to show Fourier transformations of some functions, because it is basically fun (ᵔᴥᵔ)
Our first function is a signal which is switching between +1 and -1:
A simple signal switching between +1 and -1. Now, we try to interpret this as sum of sines. The more terms we add, the merrier.
Now we are trying to interpret it as a Fourier series. Each sine wave has a certain frequency. We have to find how many of them are there for each frequency. I am writing the resultant Fourier series
\(f(x) = \displaystyle{ \sum \limits_{n = 1,3,5,7…}^{\infty} \left(\frac{4}{n \pi}\right)\sin\left(\frac{n \pi x}{L}\right)}\)
\(L\), which is the period of \(f(x)\), is \(2\pi\) for this example. In the animations below, I will show the Fourier series for n values of \(n = 1,3,5,7,9,11,13,15\).
How was that? Isn’t it marvelous? (｡◕‿◕｡) One can think of a sine wave as the distance covered by the shadow of a ball which turns around a circle. For this “square signal” wave, adding more terms
means turning circles with smaller radius around the former circle with higher speeds. Frequency of these turning circles is increasing in odd numbers (leaving the math to you). As we add more
circles (more n’s), we get closer to the function that we started with (green lines). Also, you might notice the curves that I added transparently in the background. These curves are known as the
cycloid curves! (I mentioned them in a previous article [It’s in Turkish by the way])
I can show how it approaches to the function as we add more terms in this way : We can add terms up to \(n = 240\)!
Allons-y! Now we take a “saw-tooth-like” signal and expand it to a Fourier series.
“Saw tooth” signal.
When we expand it to a Fourier series, we obtain the following relation:
\(f(x) = \displaystyle{\frac{1}{2}} -\displaystyle{ \sum \limits_{n = 1}^{\infty}\left( \frac{1}{n \pi}\right)\sin\left(\frac{n \pi x}{L}\right)}\)
Again, the more terms we add, the closer we get to this function. In animations below, I presented cases for \(n = 1,2,3,4,5,6,7,8\).
The transparent curve is, indeed, a cycloid curve!
Finally, I will expand a triangle-shaped wave to a Fourier series, but it is not as funny as the previous ones. The “triangular wave” is this:
Triangular Wave.
I derived its Fourier transformation as follows:
\(f(x) = \displaystyle{ \sum \limits_{n = 1,3,5,7,…}^{\infty} \left(\frac{8}{\pi^2} \frac{(-1)^{\frac{(n-1)}{2}}}{n^2}\right)\sin\left(\frac{n \pi x}{L}\right)}\)
As we add more terms, circles rapidly get so small that they don’t look nice as the previous ones. Anyway, I am adding animations for the first three terms as \(n = 1,3,5\) :
So, where do we use this wisdom? Basically, everywhere! This Fourier analysis thing is like four operations in engineering or science. For example, you have a signal like a noise. You want to analyze
it. Suddenly, the Purrier Transform jumps into your lap, and says (or meows) how many waves are there for each frequency.
Purr and Fourier analysis.
In the figure above, you obtain the Fourier transform of a sound, and it says that the proportion of a wave with specific frequency (1 Hz) is very high with respect to others in the composition of
this sound.T
here are a lot of charming applications of this in image and audio processing. I would like to show you, here, an interesting application.
In this work, sound can be extracted from a video by using a very sophisticated Fourier analysis.It means that, you can catch all your neighbor’s gossips about you just by silently videotaping her
flower in the living room, which is… also a little bit creepy.
Article and animations: Bilgecan Dede
Special thanks for translation: Gurkan Sonmez (Unfortunately there are no articles of him yet)
The original article in Turkish: Fuyye Serisi
The following two tabs change content below.
18 thoughts on “Purrier Series (Meow) and Making Images Speak”
• 11 December 2018 at 20:44
Music and Sound Design Implications seem likely from your work.
Synthesis from additive sine waves is nothing new (Additive Synthesis and in particular Fast Fourier Transform FFT) but I haven’t seen anyone that has tried your technique to recreate sound. I
think the future of sound design and things like samplers (For example piano or violin played by a keyboard that play backs samples of music) are soon to be a thing of the past once additive
synthesis is developed more. I feel like we are still stuck in 2004 in music production with additive synthesis. Would love to know if anyone is looking at this work to be used with audio
• 12 December 2018 at 02:41
Merhaba! Smarter Everyday brought me hear (and the link in one of your replies). Well done! Aferin.
In a certain sense, the attempt of Ptolemy of describing the movement of the planets through epicycles, is a kind of Fourier decomposition of their real movements, don’t you agree?
• 14 December 2018 at 05:49
wow that’s neat! you should make an applet so people can mess up with it. tsk
• 16 December 2018 at 09:53
Bazı periyodik fonksiyonları rahat bir şekilde fourer serisine açabiyoruz.örneğin ekg dalgalarınıda fourer serisi şeklinde açabirmiyiz yada ekg dalgaları hangi fonksiyonlardan oluşur?
• 8 January 2019 at 19:17
Excellent work,
I will keep in touch.
It has been valuable to me.
Gerardo DFS
• 17 April 2019 at 10:18
I’m absolutely blown away by this article!
I have read a lot about Fourier series. It has always been a magical thing and finally you have shown me the trick. I love it!
Thank you very much. Is there any chance that you send me the matematica notebooks used to do this? Or the link if you have published it on the wolfram demonstration projects page.
Thank you for your brilliant work!
• 1 June 2019 at 23:42
An astounding and brilliant piece of work. This is the way Fourier should be taught. I’ll be diving into my mathematica to recreate.
• 11 October 2019 at 01:13
beautiful work!!
• 27 December 2019 at 23:32
Artykuł godny polecenia. Gratulacje!!!
Piękno matematyki w czystej formie.
• 10 January 2020 at 05:56
You are a genius. Thank you. Salutes you from Argentina.
• 4 April 2020 at 15:30
How can I resolve Gibbs phenomenon? You cannot just add up term by term because it is an infinite serie, not summation, as you can see that summation of continous function to get a discountinous
fuction like square wave fuction, so overshoot and undershoot occur at jump point?
• 18 April 2020 at 01:53
This is amazing you blow my mind !
If I understood correctly you did the animations on Mathematica ?
If it the case would you be willing to share the files ? I’m an engineering student and I would love playing around with Matematica and those animations.
Thanks !
• 24 January 2021 at 02:00
I always disliked Fourier because, for the most part, I did not understand its importance. I YouTubed it, which led me to Smarter Every Day, which led me to this article. Also, I never understood
how sinusoidal waves could create square and saw-tooth waves, but now I know. I must say the video and this article helped me understand its importance. Creating images from pure sinusoidal
signals is mindblowing.
Thanks! Regards from Pakistan!
• 10 March 2021 at 23:03
Wonderful! Congrats and thank you very, very much. Beautiful work.
• 16 April 2021 at 11:34
Wow, wonderful work
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Re: [libreoffice-users] VLOOKUP for Numbers
I too hadn't realized the value of the last parameter being 1 (meaning it takes the next number below). I had trouble getting the formula to work until I realized I had pasted the array from the
email and some of the numbers were formatted as text, others not, so I was getting almost all #N/A. Then I fixed it and it works. Thanks Brian. Excellent little exercise for me to learn more about
On 3/26/13 12:48 AM, Brian Barker wrote:
At 21:17 25/03/2013 -0700, Jason C. Wells wrote:
OK. Take two. Since VLOOKUP for numbers with an inequality is not meaningful, ...
It will do what you require, I think.
... what I precisely hope to do is this. I hope that the preformatted columns survive the mailer software.
Give the following columns:
A B C D
--- --- --- ---
1.1 eqn 3 0.10
1.3 6 0.12
1.5 10 0.15
1.7 18 0.18
2 30 0.21
2.2 50 0.25
2.7 80 0.30
3.2 120 0.35
3.7 180 0.40
Compare cells in column A such that:
0 > A1 <= C1
C1 > A1 <= C2
C2 > A1 <= C3
I'm hoping you mean:
0 < A1 <= C1
C1 < A1 <= C2
C2 < A1 <= C3
is true. Then return the value of D for the appropriate row. Repeat for all values of A. When done, the length of column B will be equal to the length of column A. B will contain values of
column D. Both C and A will always be sorted. (Although it would be nice to be able to do this with A being random numbers.)
It seems like I need an array equation, but the array A is not equal in size to the array in C. Simple string matching of VLOOKUP is inadequate to the task at hand.
You are still talking of strings with respect to VLOOKUP(), but it will cope with your numbers too! And it will do your job ...
You have missed the important value zero from your column C, so please insert it in C1, moving the other values down. *Do not* move the values in column D, so that the first value, 0.10, is now
against zero in column C, 0.12 is against 3, and so on. (It seems that you don't need the maximum value 180 now in C10, in fact.)
In B1, enter:
and fill down column B.
VoilĂ !
This even works with column A values not being sorted, as you have asked. All that is required is that column C should be sorted - and that is no problem, of course.
I trust this helps.
Brian Barker
Carl Paulsen
8 Hamilton Street
Dover, NH 03820
(603) 749-2310
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Function Plotter
The calculator plots entered functions on the single coordinate plane.
The user inputs a function of x in the form of a mathematical expression and specifies the start and end boundaries and the number of points to plot. The calculator then plots all entered functions
on the same graph, making it easy to see their intersections and compare their shapes and behaviors.
Function Plotter can be useful for a variety of purposes, such as studying and analyzing mathematical functions, teaching mathematical concepts, and solving mathematical problems. It is especially
useful for students of mathematics, physics, engineering, and computer science who need to visualize and understand the behavior of mathematical functions.
The function mathematical expression syntax is the same as here: One-variable function graph
Similar calculators
PLANETCALC, Function Plotter
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TOC | Previous | Next | Index
18.5 Vector Views (.NET, C#, CSharp, VB, Visual Basic, F#)
Methods such as Row(), Column(), and Diagonal() return vector views of the data referenced by a general matrix. NMath does not generally provide such methods for structured sparse matrix types,
because of the limitations on which elements in the matrix are modifiable.
The exception is the banded matrix types which provide a Diagonal() member function that returns a vector view of a diagonal of a matrix. If no diagonal is specified, a vector view of the main
diagonal is returned. For example, this code increments every element along the main diagonal:
Code Example – C# matrix
var A = new FloatBandMatrix( 5, 5, 0, 0 );
Code Example – VB matrix
Dim A As New FloatBandMatrix( 5, 5, 0, 0 )
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What are Recall and Precision? How do they differ?
December 31, 2020
What are Recall and Precision? How do they differ?
Honorable Madras High Court said that-“1000 Culprits Can Escape but, One Innocent Should Not be Punished “. More or less legal systems around the world follow this principle.
The above motto shows that accuracy is not always the metric that is been sought out for. If you were to design a model that could assist the court in making decisions, then that model should also
incorporate this dictum. The primary goal of your model would be to ensure that no innocent be predicted as guilty. Similarly there are many business scenarios where the end goal is not accuracy. For
such cases we need metrics like precision & recall and confusion matrix forms the basis of calculating these metrics.
Further, in case of imbalanced datasets accuracy value can be sometimes misleading. Let me illustrate my point with an example, let’s say you have to build a model that has to predict if a person is
Covid positive or not based on a given set of traits. If the training data is unbalanced with overwhelming number of healthy people and only a few Covid cases, then your model would incorporate this
bias. If the the test data you use has 1000 rows in which there are 950 healthy(Covid negative) cases and 50 Covid positive cases. Now, if your model predicts 990 healthy cases (Covid negative) and
10 Covid positive cases, then your model horribly fails in its objective.
If out of 990 cases, that your model predicted negative, 940 are actually negative and of the 10 it predicted positive, 7 are actually Covid positive – then the accuracy of your model would be 94.7%.
But this accuracy number is meaningless. What you need to know if of the corona positive cases, how many were actually predicted positive i.e. True Positive, and how many were wrongly predicted as
corona positive i.e. False Positive. Similarly you would want to get the True Negative and False Negative values for the healthy cases.
In cases like this Confusion matrix based metrics like Recall and Precision come in handy.
RecallĀ measures the number of True Positives out of the Total number of actual positive cases.
This metric is used in cases where you’re willing to compromise on accuracy of predicting negative cases but not on accuracy with which positive cases are predicted. In our Covid model example,
recall is the metric that you would be most interested in. Even if the model predicts healthy people as Covid positive they physicians can rectify that error. But if Covid positive people are
classified as healthy, they would leave the process at screening stage itself and thus:
• Would be deprived of the treatment
• Won’t be quarantined and could infect others
Precision measures the number of True Positives out of the Total number of positive cases predicted by the model.
Precision is used to check the reliability of the model. If the precision of our Covid model were low, we wouldn’t rely too much on the results on the model and would:
• Build a new model with higher precision. This step would be used if the recall of the model is also low
• Include other checks in the screening process that further filter out the model results. This step would be taken if the model has high precision.
Thus recall is used to measure if the model is suited to the overall/business objective and precision is used to measure the reliability of the model results.
by : Monis Khan
Quick Summary:
Honorable Madras High Court said that-“1000 Culprits Can Escape but, One Innocent Should Not be Punished “. More or less legal systems around the world follow this principle. The above motto shows
that accuracy is not always the metric that is been sought out for. If you were to design a model that could assist […]
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Cs 321: Lecture notes 2 Programs and functions
for i=1:n
coor(i,:) = coor(i,:)-gc;
By clicking on the right top bottom it is possible to rotate the protein chain in 3D. It is not that easy to understand properties of the structure from the 3D view. For example, can you identify how
many helices are in the structure?
It is useful to have other representations (less straightforward) of the structure that are simpler to analyze. One interesting representation (in 2D) is the contact matrix. If the distance between a
pair of amino acids Otherwise it is set to zero
A short MATLAB script that prepares a contact map is below:
n = size(coor,1);
for i=1:n
cont(i,i) = 0;
for j=i+1:n
dist = norm(coor(i,:)-coor(j,:));
if (dist<=7)
We repeat the same exercise for the protein myoglobin (1mbco). Can you suggest a fingerprint for a helix? Beta sheet? Can you identify sharp turns? Explain the off-diagonal elements and what
structural features they correspond to.
Optional Exercise
We define the distance between two structures as the norm of the vector differences.
The distance between protein structures with coordinate vectors coor1(:,1:3) and coor2(:,1-3) is norm(coor1-coor2). In this exercise we compute the distances between different representations of the
same protein. Your report should include the programs, plots and explanations of the results as requested below.
Extract the coordinates of the protein 1LAP from the protein data bank
Read the CA coordinates of this protein. Plot the chain of the CA-s and the contact matrix. Assign secondary structures. You may also use the plot3 facility.
Instead of the CA-s it is possible to read the N atoms. Modify the code to read the N-s instead of the CA-s. Be careful not to lose any N! Stored the coordinates in a different array from the CA
array. Compute the distance between the N vector representation of the protein chain and the CA vector of the protein. Repeat the process after placing the geometric centers of the two protein
presentations at the origin. Any changes after the shifts of the geometric centers?
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Efficient Long-Distance Heat Transport by Microwave Photons
Efficient Long-Distance Heat Transport by Microwave Photons
Research team behind the original discovery (Reference [11] ) from left to right: Tuomo Tanttu, Joonas Goveenius, Mikko Möttönen, Matti Partanen, and Miika Mäkelä (Missing from the figure: Kuan Yen
Tan and Russell Lake). Photo Credit: Vilja Pursiainen/Kaskas Media. Authors: Matti Partanen and Mikko Möttönen Affiliation: QCD Labs, Department of Applied Physics, Aalto University, Finland.
Link to the Quantum Computing and Devices (QCD) Group >>
Quantum computers are predicted to vastly speed up the computation for certain problems of great practical interest [1]. One of the most promising architectures for quantum computing is based on
superconducting quantum bits [2], or qubits, which are the key ingredients in circuit quantum electrodynamics [3]. In such systems, the control of heat at the quantum level is extremely important,
and remote cooling may turn out to be a viable option.
In one dimension, heat transport may be described by individual heat conduction channels -- each corresponding to a certain quantized profile of the heat carriers in the transverse direction.
Importantly, the maximum heat power flowing in a single channel between bodies at given temperatures is fundamentally limited by quantum mechanics [4,5]. This quantum limit has previously been
observed for phonons [6], sub-wavelength photons [7,8], and electrons [9]. Among these, the longest distance of roughly 50 μm [7,8] was recorded in the photonic channel [10]. Such short distance may
be undesirable in cooling quantum devices which are sensitive to spurious dissipation.
In our recent work [11], we observe quantum-limited heat conduction by microwave photons flying in a superconducting transmission line of length 20 cm and 1 m. Thus we were able to extend the maximum
distance 10,000 fold compared with the previous experiments.
Figure 1: (click on the figure to view with higher resolution) Sample structure and measurement scheme. The electron temperature of the right resistor is controlled with an external voltage while the
temperatures of both resistors are measured. Microwave photons transport heat through the spiraling transmission line.
Our sample is shown in Figure 1. The heat is transferred between two normal-metal resistors functioning as black-body radiators to the transmission line [10,12]. To be able to fabricate the whole
sample on a single relatively small chip, the transmission line has a double spiral structure. We have measured such spiraling transmission lines without resistors and confirmed that photons travel
along the line; they do not jump through vacuum from one end to the other. Thus for heat transport, the distance should be measured along the line.
We measure the electron temperatures of both normal-metal resistors while we change the temperature of one of them [13]. The obtained temperature data agrees well with our thermal model, according to
which the heat conduction is very close to the quantum limit.
In contrast to subwavelength distances employed in References [7,8], we need to match the resistance of the normal-metal parts to the characteristic impedance of the transmission line to reach the
quantum limit. Furthermore, the transmission line itself has to be so weakly dissipative that almost no photons are absorbed even over distances of about a meter. However, we managed to develop
nanofabrication techniques which enabled us to satisfy these conditions well. In fact, the losses in the transmission line are so weak they allow a further increment of the distance by several orders
of magnitude.
We consider that long-distance heat transport through transmission lines may be a useful tool for certain future applications in the quickly developing field of quantum technology. If the coupling of
a quantum device to a low-temperature transmission line can be well controlled in situ, the device may be accurately initialized without disturbing its coherence properties when the coupling is
turned off [14]. Furthermore, the implementation of such in-situ-tunable environments opens an interesting avenue for the study of the detailed dynamics of open quantum systems and quantum
fluctuation relations [15].
Acknowledgements: We thank M. Meschke, J. P. Pekola, D. S. Golubev, J. Kokkala, M. Kaivola and J. C. Cuevas for useful discussions, and L. Grönberg, E. Mykkänen, and A. Kemppinen for technical
assistance. We acknowledge the provision of facilities and technical support by Aalto University at Micronova Nanofabrication Centre. We also acknowledge funding by the European Research Council
under Starting Independent Researcher Grant No. 278117 (SINGLEOUT), the Academy of Finland through its Centres of Excellence Program (project nos 251748 and 284621) and grants (nos 138903, 135794,
265675, 272806 and 276528), the Emil Aaltonen Foundation, the Jenny and Antti Wihuri Foundation, and the Finnish Cultural Foundation.
References: [1] T.D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, J.L. O'Brien, “Quantum computers”, Nature, 464, 45 (2010). Abstract. [2] J. Kelly, R. Barends, A.G. Fowler, A. Megrant, E.
Jeffrey, T.C. White, D. Sank, J.Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P.J.J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A.N.
Cleland, John M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit”, Nature, 519, 66 (2015). Abstract. [3] A. Wallraff, D.I. Schuster, A. Blais, L.
Frunzio, R.-S. Huang, J. Majer, S. Kumar, S.M. Girvin, R.J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum Electrodynamics”, Nature, 431, 162 (2004).
Abstract. [4] J.B. Pendry, “Quantum limits to the flow of information and entropy”, Journal of Physics A: Mathematical and General, 16, 2161 (1983). Abstract. [5] Luis G. C. Rego, George Kirczenow,
“Fractional exclusion statistics and the universal quantum of thermal conductance: A unifying approach”, Physical Review B, 59, 13080 (1999). Abstract. [6] K. Schwab, E.A. Henriksen, J.M.Worlock,
M.L. Roukes, “Measurement of the quantum of thermal conductance”, Nature, 404, 974 (2000). Abstract. [7] Matthias Meschke, Wiebke Guichard, Jukka P. Pekola, “Single-mode heat conduction by photons”,
Nature, 444, 187 (2006). Abstract. [8] Andrey V. Timofeev, Meri Helle, Matthias Meschke, Mikko Möttönen, Jukka P. Pekola, “Electronic refrigeration at the quantum limit”, Physical Review Letters,
102, 200801 (2009). Abstract. [9] S. Jezouin, F.D. Parmentier, A. Anthore, U. Gennser, A. Cavanna, Y. Jin, and F. Pierre, “Quantum limit of heat flow across a single electronic channel”, Science,
342, 601 (2013). Abstract. [10] D.R. Schmidt, R.J. Schoelkopf, A.N. Cleland, “Photon-mediated thermal relaxation of electrons in nanostructures”, Physical Review Letters, 93, 045901 (2004). Abstract.
[11] Matti Partanen, Kuan Yen Tan, Joonas Govenius, Russell E. Lake, Miika K. Mäkelä, Tuomo Tanttu, Mikko Möttönen, “Quantum-limited heat conduction over macroscopic distances”, Nature Physics,
Advance online publication, DOI:10.1038/nphys3642 (2016). Abstract. [12] L.M.A. Pascal, H. Courtois, F.W.J. Hekking, “Circuit approach to photonic heat transport”, Physical Review B, 83, 125113
(2011). Abstract. [13] Francesco Giazotto, Tero T. Heikkilä, Arttu Luukanen, Alexander M. Savin, Jukka P. Pekola, “Opportunities for mesoscopics in thermometry and refrigeration: Physics and
applications”, Reviews of Modern Physics, 78, 217 (2006). Abstract. [14] P. J. Jones, J.A.M. Huhtamäki, J. Salmilehto, K.Y. Tan, M. Möttönen, “Tunable electromagnetic environment for superconducting
quantum bits”, Scientific Reports, 3, 1987 (2013). Abstract. [15] Jukka P. Pekola, “Towards quantum thermodynamics in electronic circuits”, Nature Physics, 11, 118 (2015). Abstract.
Labels: Nanotechnology 10, Quantum Computation and Communication 14
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Computer Methods in Applied Mechanics and Engineering
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Which point lies on the graph of f(x) (x 4) 2? - Answers
How do you graph a line in slope intercept for?
1) You write the equation in slope-intercept form, if it isn't in that form already. 2) An easy way to graph it is to start with the y-intercept. For example, if the intercept is +5, you graph the
point (0, 5). Then you add an additional point, according to the slope. For example, if the slope is 1/2, you go 2 units to the right, and one up, and graph a point there.
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Project 3-D
Project 3-D quiver plot on axesm-based map
quiver3m(lat,lon,z,dlat,dlon,dz) plots a 3-D quiver plot on the current axesm-based map. The quiver plot has arrows with directional components dlat, dlon, and dz at the geographic coordinates
specified by lat and lon with altitude z. For example, the first arrow originates from the point lat(1), lon(1), and z(1), extends in the direction of the latitude axis according to dlat(1), extends
in the direction of the longitude axis according to dlon(1), and extends in the direction of the z-axis according to dz(1). By default, the function scales the arrow lengths so that they do not
quiver3m(lat,lon,z,dlat,dlon,dz,scale) adjusts the length of arrows:
• When scale is a positive number, the function automatically adjusts the lengths of arrows so they do not overlap, then stretches them by a factor of scale. For example, a scale of 2 doubles the
length of arrows, and a scale of 0.5 halves the length of arrows.
• When scale is "off" or 0, such as quiver3m(lat,lon,z,dlat,dlon,dz,"off"), the function does not perform automatic scaling.
quiver3m(lat,lon,z,dlat,dlon,dz,LineSpec) specifies the line style, marker, and color. Markers appear at the points specified by lat, lon, and z. If you specify a marker using LineSpec, the function
does not display arrowheads.
quiver3m(lat,lon,z,dlat,dlon,dz,LineSpec,scale,"filled") adjusts the lengths of the arrows and fills the markers and fills the markers specified by LineSpec.
h = quiver3m(___) returns the quiver plot, using any combination of input arguments from the previous syntaxes.
Project 3-D Quiver Plot on Map
Load sample data that represents air currents into the workspace, and select a subset of the data. Specify lon and dz by scaling the original data.
load wind
lat = y(13:19,13:19,1);
lon = -x(13:19,13:19,1);
z = z(13:19,13:19,1);
dlat = v(13:19,13:19,1);
dlon = u(13:19,13:19,1);
dz = 500*w(13:19,13:19,1);
Create a map and display a 3-D quiver plot of the subset you selected. The matrices lat, lon, and z represent the location of the base of each arrow, and dlat, dlon, and dz represent the directional
components of each arrow. By default, the quiver3m function shortens the arrows so they do not overlap.
axesm miller
axis off
Input Arguments
lat — Latitude coordinates of bases of arrows
scalar | vector | matrix
Latitude coordinates of the bases of the arrows, specified as a scalar, a vector, or a matrix.
The sizes of lat, lon, z, dlat, dlon, and dz must match.
Specify this argument using units that match the AngleUnits property of the axesm-based map.
Data Types: double
lon — Longitude coordinates of bases of arrows
scalar | vector | matrix
Longitude coordinates of the bases of the arrows, specified as a scalar, a vector, or a matrix.
The sizes of lat, lon, z, dlat, dlon, and dz must match.
Specify this argument using units that match the AngleUnits property of the axesm-based map.
Data Types: double
z — Altitudes of bases of arrows
scalar | vector | matrix
Altitudes of the bases of the arrows, specified as a scalar, a vector, or a matrix.
The sizes of lat, lon, z, dlat, dlon, and dz must match.
When the MapProjection property of the axesm-based map is 'globe', z is referenced to the ellipsoid.
The units of z must match the units of dz.
Data Types: double
dlat — Latitude components of arrows
scalar | vector | matrix
Latitude components of the arrows, specified as a scalar, vector, or matrix.
The sizes of lat, lon, z, dlat, dlon, and dz must match.
Specify this argument using units that match the AngleUnits property of the axesm-based map.
Data Types: single | double
dlon — Longitude components of arrows
scalar | vector | matrix
Longitude components of the arrows, specified as a scalar, vector, or matrix.
The sizes of lat, lon, z, dlat, dlon, and dz must match.
Specify this argument using units that match the AngleUnits property of the axesm-based map.
Data Types: single | double
dz — Altitude components of arrows
scalar | vector | matrix
Altitude components of the arrows, specified as a scalar, vector, or matrix.
The sizes of lat, lon, z, dlat, dlon, and dz must match.
When the MapProjection property of the axesm-based map is 'globe', dz is referenced to the ellipsoid.
The units of dz must match the units of z.
Data Types: single | double
LineSpec — Line style, marker, and color
character vector | string scalar
Line style, marker, and color, specified as a character vector or string scalar containing symbols. The symbols can appear in any order. You do not need to specify all three characteristics (line
style, marker, and color).
If you specify a marker using LineSpec, then the quiver3m function does not display arrowheads.
Example: "--or" is a red dashed line with circle markers
Line Style Description Resulting Line
"-" Solid line
"--" Dashed line
":" Dotted line
"-." Dash-dotted line
Marker Description Resulting Marker
"o" Circle
"+" Plus sign
"*" Asterisk
"." Point
"x" Cross
"_" Horizontal line
"|" Vertical line
"square" Square
"diamond" Diamond
"^" Upward-pointing triangle
"v" Downward-pointing triangle
">" Right-pointing triangle
"<" Left-pointing triangle
"pentagram" Pentagram
"hexagram" Hexagram
Color Name Short Name RGB Triplet Appearance
"red" "r" [1 0 0]
"green" "g" [0 1 0]
"blue" "b" [0 0 1]
"cyan" "c" [0 1 1]
"magenta" "m" [1 0 1]
"yellow" "y" [1 1 0]
"black" "k" [0 0 0]
"white" "w" [1 1 1]
scale — Arrow scaling factor
nonnegative number | "off"
Arrow scaling factor, specified as a nonnegative number or "off". By default, the quiver3m function automatically scales the arrows so they do not overlap. The quiver3m function applies the specified
scaling factor after it automatically scales the arrows.
To turn off automatic scaling, specify scale as "off" or 0. When you specify either of these values, the function plots the arrows from (lat, lon, z) to (lat+dlat, lon+dlon, z+dz).
Output Arguments
h — Quiver plot
vector of Line objects
Quiver plot, returned as a vector of Line objects.
Version History
Introduced before R2006a
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Real fuel consumption Auto ABC
Real fuel consumption
We have aggregated user feedback on the car's actual fuel consumption over a daily driving cycle from a number of sources and compared it with the manufacturer's fuel consumption figures. On average,
manufacturers' fuel consumption figures are 15-20% lower than actual consumption. See more fuel economy data and fuel efficiency scores by manufacturers or click on the manufacturers to see how the
real fuel consumption of specific models differs from the manufacturer's claims.
Alfa Romeo Average real petrol consumption difference: +18%Average real diesel consumption difference: +15%
Audi Average real petrol consumption difference: +23%Average real diesel consumption difference: +23%
BMW Average real petrol consumption difference: +25%Average real diesel consumption difference: +29%
Chevrolet Average real petrol consumption difference: +14%Average real diesel consumption difference: +25%
Chrysler Average real petrol consumption difference: +3%Average real diesel consumption difference: +8%
Citroen Average real petrol consumption difference: +18%Average real diesel consumption difference: +17%
Dacia Average real petrol consumption difference: +15%Average real diesel consumption difference: +28%
Dodge Average real petrol consumption difference: +10%Average real diesel consumption difference: +14%
Fiat Average real petrol consumption difference: +12%Average real diesel consumption difference: +14%
Ford Average real petrol consumption difference: +15%Average real diesel consumption difference: +18%
Honda Average real petrol consumption difference: +7%Average real diesel consumption difference: +16%
Hyundai Average real petrol consumption difference: +16%Average real diesel consumption difference: +20%
Infiniti Average real petrol consumption difference: +30%Average real diesel consumption difference: +33%
Isuzu Average real diesel consumption difference: +10%
Jaguar Average real petrol consumption difference: +20%Average real diesel consumption difference: +28%
Jeep Average real petrol consumption difference: +14%Average real diesel consumption difference: +19%
Kia Average real petrol consumption difference: +14%Average real diesel consumption difference: +25%
Land Rover Average real petrol consumption difference: +21%Average real diesel consumption difference: +23%
Lexus Average real petrol consumption difference: +55%
Mazda Average real petrol consumption difference: +9%Average real diesel consumption difference: +16%
Mercedes Average real petrol consumption difference: +20%Average real diesel consumption difference: +21%
Mini Average real petrol consumption difference: +30%Average real diesel consumption difference: +33%
Mitsubishi Average real petrol consumption difference: +9%Average real diesel consumption difference: +12%
Nissan Average real petrol consumption difference: +13%Average real diesel consumption difference: +17%
Opel Average real petrol consumption difference: +16%Average real diesel consumption difference: +17%
Peugeot Average real petrol consumption difference: +19%Average real diesel consumption difference: +16%
Porsche Average real petrol consumption difference: +25%Average real diesel consumption difference: +28%
Renault Average real petrol consumption difference: +14%Average real diesel consumption difference: +20%
Rover Average real petrol consumption difference: +7%Average real diesel consumption difference: +13%
SAAB Average real petrol consumption difference: minor
Seat Average real petrol consumption difference: +18%Average real diesel consumption difference: +20%
Skoda Average real petrol consumption difference: +18%Average real diesel consumption difference: +18%
Smart Average real petrol consumption difference: +28%Average real diesel consumption difference: +26%
Subaru Average real petrol consumption difference: +12%Average real diesel consumption difference: +12%
Suzuki Average real petrol consumption difference: +15%Average real diesel consumption difference: +7%
Toyota Average real petrol consumption difference: +12%Average real diesel consumption difference: +14%
Volkswagen Average real petrol consumption difference: +12%Average real diesel consumption difference: +15%
Volvo Average real petrol consumption difference: +11%Average real diesel consumption difference: +16%
Read more about fuel economy: Petrol or diesel - which is more favourable?
Will a diesel car always be the most economical choice compared to its petrol counterpart? What else should be considered when choosing a car based on engine type?
The fuel consumption reported by users does not reflect the experience of all users, may be selective (depending on the driving conditions, driving style, technical condition of the car and other
circumstances) and should therefore not be taken as a representative indicator.
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Portal through Mathematics: Journey to Advanced Thinkingsearch
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Portal through Mathematics: Journey to Advanced Thinking
Translated by Robert G. Burns
MAA Press: An Imprint of the American Mathematical Society
This title is not currently available
Click above image for expanded view
Portal through Mathematics: Journey to Advanced Thinking
Translated by Robert G. Burns
MAA Press: An Imprint of the American Mathematical Society
This title is not currently available
• Anneli Lax New Mathematical Library
Volume: 47; 2017; 304 pp
Reprinted edition available: NML/52
Portal through Mathematics is a collection of puzzles and problems mostly on topics relating to secondary mathematics. The problems and topics are fresh and interesting and frequently surprising.
One example: the puzzle that asks how much length must be added to a belt around the Earth's equator to raise it one foot has probably achieved old chestnut status. Ivanov, after explaining the
surprising answer to this question, goes a step further and asks, if you grabbed that too long belt at some point and raised it as high as possible, how high would that be? The answer to that is
more surprising than the classic puzzle's answer. The book is organized into 29 themes, each a topic from algebra, geometry or calculus and each launched from an opening puzzle or problem. There
are excursions into number theory, solid geometry, physics and combinatorics. Always there is an emphasis on surprise and delight. And every theme begins at a level approachable with minimal
background requirements. With well over 250 puzzles and problems, there is something here sure to appeal to everyone.
Portal through Mathematics will be useful for prospective secondary teachers of mathematics and may be used (as a supplementary resource) in university courses in algebra, geometry, calculus, and
discrete mathematics. It can also be used for professional development for teachers looking for inspiration. However, the intended audience is much broader. Every fan of mathematics will find
enjoyment in it.
□ Cover
□ Half Title Page
□ Copyright
□ Title Page
□ Contributors
□ Anneli Lax New Mathematical Library
□ Contents
□ Foreword
□ Preface for anAmerican Readership
□ Author's Preface
□ Part I Surprising and Easy
□ 1 Surprising right triangles
□ 2 Surprisingly short solutions of geometric problems
□ 3 A natural assertion with a surprising proof
□ 4 Surprising answers
□ 5 A surprising connection between three sequences
□ Part II Algebra, Calculus, and Geometry: problems
□ 6 Five problems and a function
□ 7 Five solutions of a routine problem
□ 8 Equations of the form f(x, y) = g(x, y) and their generalizations
□ 9 The generalized version of Viete's formula
□ 10 Multiple roots of polynomials
□ 11 Non-routine applications of the derivative
□ 12 Complex numbers, polynomials, and trigonometry
□ 13 Complex numbers and geometry
□ 14 Areas of triangles and quadrilaterals
□ 15 Constructions in solid geometry
□ 16 Inequalities
□ 17 Diophantine equations
□ 18 Combinatorial tales
□ 19 Integrals
□ Part III Algebra, Calculus, and Geometry: theory (a little way beyond high school mathematics)
□ 20 Functional equations of elementary functions
□ 21 Sequences given by recurrence relations
□ 22 The "golden ratio" or solving equations of the form f( x) = x
□ 23 Convex functions: inequalities and approximations
□ 24 Taylor's formula, Euler's formula, and a combinatorial problem
□ 25 Derivatives of vector-functions
□ 26 Polynomials and trigonometric relations
□ 27 Areas and volumes as functions of co-ordinates
□ 28 Values of trigonometric functions and sequences satisfying certain recurrence relation
□ 29 Do there exist further "numbers" beyond complex numbers?
□ Solutions of the supplementary problems
□ Index
• Book Details
• Table of Contents
• Additional Material
Volume: 47; 2017; 304 pp
Reprinted edition available: NML/52
Portal through Mathematics is a collection of puzzles and problems mostly on topics relating to secondary mathematics. The problems and topics are fresh and interesting and frequently surprising. One
example: the puzzle that asks how much length must be added to a belt around the Earth's equator to raise it one foot has probably achieved old chestnut status. Ivanov, after explaining the
surprising answer to this question, goes a step further and asks, if you grabbed that too long belt at some point and raised it as high as possible, how high would that be? The answer to that is more
surprising than the classic puzzle's answer. The book is organized into 29 themes, each a topic from algebra, geometry or calculus and each launched from an opening puzzle or problem. There are
excursions into number theory, solid geometry, physics and combinatorics. Always there is an emphasis on surprise and delight. And every theme begins at a level approachable with minimal background
requirements. With well over 250 puzzles and problems, there is something here sure to appeal to everyone.
Portal through Mathematics will be useful for prospective secondary teachers of mathematics and may be used (as a supplementary resource) in university courses in algebra, geometry, calculus, and
discrete mathematics. It can also be used for professional development for teachers looking for inspiration. However, the intended audience is much broader. Every fan of mathematics will find
enjoyment in it.
• Cover
• Half Title Page
• Copyright
• Title Page
• Contributors
• Anneli Lax New Mathematical Library
• Contents
• Foreword
• Preface for anAmerican Readership
• Author's Preface
• Part I Surprising and Easy
• 1 Surprising right triangles
• 2 Surprisingly short solutions of geometric problems
• 3 A natural assertion with a surprising proof
• 4 Surprising answers
• 5 A surprising connection between three sequences
• Part II Algebra, Calculus, and Geometry: problems
• 6 Five problems and a function
• 7 Five solutions of a routine problem
• 8 Equations of the form f(x, y) = g(x, y) and their generalizations
• 9 The generalized version of Viete's formula
• 10 Multiple roots of polynomials
• 11 Non-routine applications of the derivative
• 12 Complex numbers, polynomials, and trigonometry
• 13 Complex numbers and geometry
• 14 Areas of triangles and quadrilaterals
• 15 Constructions in solid geometry
• 16 Inequalities
• 17 Diophantine equations
• 18 Combinatorial tales
• 19 Integrals
• Part III Algebra, Calculus, and Geometry: theory (a little way beyond high school mathematics)
• 20 Functional equations of elementary functions
• 21 Sequences given by recurrence relations
• 22 The "golden ratio" or solving equations of the form f( x) = x
• 23 Convex functions: inequalities and approximations
• 24 Taylor's formula, Euler's formula, and a combinatorial problem
• 25 Derivatives of vector-functions
• 26 Polynomials and trigonometric relations
• 27 Areas and volumes as functions of co-ordinates
• 28 Values of trigonometric functions and sequences satisfying certain recurrence relation
• 29 Do there exist further "numbers" beyond complex numbers?
• Solutions of the supplementary problems
• Index
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How to Copy Variable to Another Graph In Tensorflow?
To copy a variable from one graph to another in TensorFlow, you can use the assign method or tf.Variable.assign method. This allows you to update the value of the variable in the target graph by
assigning the value of the variable from the source graph. By doing so, you can effectively copy the variable from one graph to another.
How to copy a variable to another graph in TensorFlow?
You can copy a variable from one graph to another in TensorFlow by using the tf.assign function. Here's an example code snippet that shows how to do this:
1 import tensorflow as tf
3 # Create a variable in the first graph
4 graph1 = tf.Graph()
5 with graph1.as_default():
6 var1 = tf.Variable(3.0)
8 # Create a second graph
9 graph2 = tf.Graph()
10 with graph2.as_default():
11 # Create a placeholder that will be assigned the value of var1
12 placeholder = tf.placeholder(tf.float32)
14 # Assign the value of var1 to the placeholder
15 assign_op = tf.assign(placeholder, var1)
17 with tf.Session(graph=graph1) as sess1:
18 sess1.run(tf.global_variables_initializer())
19 value_to_copy = sess1.run(var1)
21 with tf.Session(graph=graph2) as sess2:
22 sess2.run(assign_op, feed_dict={placeholder: value_to_copy})
In this code snippet, we first create a variable var1 in the first graph and then use the tf.assign function to copy its value into a placeholder in the second graph. We then run the assignment
operation in the second graph, feeding the value of var1 from the first graph as input to the placeholder. Finally, the value of var1 is successfully copied from the first graph to the second graph.
What is the process of copying a variable to another graph in TensorFlow?
To copy a variable to another graph in TensorFlow, you need to follow these steps:
1. Define the variable in the original graph that you want to copy to another graph.
2. Create a tf.train.Saver() object in the original graph to save the variable.
3. Initialize a new graph and define a placeholder for the variable.
4. Restore the variable from the original graph using the tf.train.Saver() object.
5. Use the restored variable in the new graph for further computations.
Here is an example code snippet that demonstrates copying a variable to another graph in TensorFlow:
1 import tensorflow as tf
3 # Original graph
4 original_graph = tf.Graph()
5 with original_graph.as_default():
6 original_variable = tf.Variable(5.0, name='original_variable')
7 saver = tf.train.Saver()
9 with tf.Session() as sess:
10 sess.run(tf.global_variables_initializer())
11 saver.save(sess, 'original_model/model.ckpt')
13 # New graph
14 new_graph = tf.Graph()
15 with new_graph.as_default():
16 new_variable_placeholder = tf.placeholder(dtype=tf.float32)
17 saver = tf.train.import_meta_graph('original_model/model.ckpt.meta')
19 with tf.Session() as sess:
20 saver.restore(sess, 'original_model/model.ckpt')
21 copied_variable = new_variable_placeholder.assign(original_variable)
23 print(sess.run(copied_variable, feed_dict={new_variable_placeholder: 10.0}))
In this example, we first define a variable original_variable in the original graph and save it using tf.train.Saver(). We then create a new graph and import the saved variable using
tf.train.import_meta_graph() and restore it using Saver.restore(). Finally, we copy the variable to the new graph using new_variable_placeholder.assign(original_variable) and run the operation in a
What is the outcome of cloning a variable between two graphs in TensorFlow?
When a variable is cloned between two graphs in TensorFlow, a new copy of the variable is created in the second graph which is independent of the original variable in the first graph. This means that
any changes made to the cloned variable in the second graph will not affect the original variable in the first graph, and vice versa. Each variable will retain its own state and values, and can be
updated independently in each graph.
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5 awesome division activities with manipulatives - Number Dyslexia
5 awesome division activities with manipulatives
Last Updated on July 16, 2022 by Editorial Team
One mantra that always work well for early learning is to make it interactive. The more interactive you make a topic, the stronger lasting impression it will make in the brain. The future contents
related to the topic can be worked out in an easy flow, if only the fundamentals are strong. There are ample ways of making things interactive. The prominent practice used in this respect is
manipulatives, specially for early learning.
The focus of this post is on division practice. We brought to you a list of 5 awesome division activities to do with manipulatives. Although, this directly improves the kid’s understanding of
division, but it also highlights certain other important areas of the topics as well.
Teachers and Parents can include these activities for their child in the routine. The requisite of the items in the activity is given with link but you can also use any relevant alternative if there
is already stuff lying around in your house. Further, the best way to make a strong impression is to follow a repetitive approach with intervals. Keeping the routine works the best.
1. Math Tiles
Math tiles manipulatives that we used for learning algebra can be used for this purpose as well. It follows the similar approach for division.
To do this activity, first we will take multiple of two different colored tiles. Let’s say we have two multiples of tiles with blue and yellow color. One should be considered as dividend and other
one as divisor.
Draw a line in-between them for better understanding.
Arrange the tiles as per the question, such that every divisor tile row have equal numbers of dividend tiles. Suppose we have to find the resultant when 24 is divided by 6. The tiles will be arranged
like this.
The number of tiles in each row or number of columns we get after the arrangement is our quotient. Since there is no dividend tile left for balance, it means 24 is divisible by 6.
Let’s us take the example of 37 divided by 6.
Here, One divdend tile is left for balancing. This means 37 is not divisible by 6 and 1 acts as a remainder.
2. Eggtray & Beads
Eggtray carton and beads manipulative follow a similar conceptual approach, it just that we put the beads (dividend) into the trays of egg carton (divisor), instead of equating them as we do in the
tiles activity.
Let’s divide 28 by 4. Here, we will take 4 trough area of the egg tray and fill it with beads equally. After arranging, every trough will have 7 beads with no bead left to balance. This means the
quotient is 7 with 0 as remainder.
Now if we divided by 30 by 4, we will left with 2 extra beads to balance. Hence, 30 is not divisible by 4 with 2 as a remainder.
3. Popsicle Puzzle
Paper popsicle can be used as manipulative for a fun division game. Follow these steps :
• First cut a paper in the shape of a popsicle.
• Split the paper popsicle in two parts.
• Write the division question on upper part and answer on the lower part.
• Make couple more similar popsicles with different division questions
• Shuffle them and ask students to clip the correct match.
4. Counters on Frame
Activities like ten frame manipulatives can be modified a little bit for division practice. Inspired from the egg tray and bead activity, counters, as dividend, are arranged on the spots of the frame
(divisor). Anything, such as coins, dices and cards, can be used as a counter for this activity.
Again, number of counters balanced equally on each spots is the quotient and number of left counters is the remainder. Good thing about this activity is that there is no limit to the divisor as you
can draw as much frames as you want. So, this gives you ample of opportunities to practice for division
5. Wreck the Fort
This activity is pretty interactive and playful. Kids will love practicing division through this. It includes building fort and destroying it with a ball. Do note that you can use anything as a
manipulative to be used for fort building. It can be anything, like boxes, plastic glasses and etc.
Players will only get as many trials as the divisor value to destroy the fort. Blocks of the fort must be arranged in groups of equal numbers representing the quotient.
With each trial, student must focus on one group and try to completely wipe it out first. The playful nature of this activity really makes kid love doing it. The repetitive approach makes kids really
hooked to it with long lasting impression. Teachers are recommended to explain the nature of the game and idea behind each move.
An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and
dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having just a simple math
fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’,
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Length Width Height - Formula, Examples | Length vs Width (2024)
The length, width and height are the dimensions of a geometrical figure that depict how long, wide and high a figure is. While length is the longest side of a figure, width is the shorter side and
height is the vertical dimension of the figure. Let us learn more about the length width height of figures.
1. What is Length Width Height?
2. Length vs Width
3. Length Width Height of a Box
4. FAQs on Length Width Height
What is Length Width Height?
Length, width, and height are the tools that are used to find the dimensions of an object. When we refer to two-dimensional shapes (2D shapes), we use the length and width, whereas when we refer to
three-dimensional shapes (3D shapes) we use the height along with the length and width. Let us understand the three terms now.
• Length: Length is used to measure the distance between two points. Length is the longest dimension of a figure and it shows how long the given object or figure is. It is expressed in linear units
like meters, centimeters, inches, and so on.
• Width: Width is the shorter distance of an object or a figure and it shows how broad or wide the given figure is. Width is also expressed in linear units like meters, centimeters, inches, and so
• Height or Depth: The height of an object refers to its depth or the third vertical dimension of the object and it shows how high or deep an object is. The height or depth of an object is
expressed in linear units like meters, centimeters, inches, and so on.
It should be noted that length, width, height and depth are words that are derived from the words long, wide, high, and deep, respectively. Hence they express the dimensions of an object. Observe the
figure given below to see the length width and height of a cuboid.
Length vs Width
The difference between the length and width of a figure is that length signifies the longer side and width signifies the shorter side of a figure.
The length shows how long the figure is and the width shows how broad or wide it is. The width is also referred to as breadth. For example, if the two sides of a rectangle are given as 8 cm and 3 cm,
we can easily identify that the length of the rectangle is 8 cm and the width of the rectangle is 3 cm. Observe the rectangle given below to see the difference between the length and width of a
Length x Width x Height
The length width and height are usually used together to find the volume of a geometrical figure like a rectangular prism which is also known as a cuboid. When we multiply the length, width, and
height of a cuboid, we get its volume. This means, Length x Width x Height = Volume of Cuboid. In other words, the capacity or volume of a cuboid or any rectangular box can be measured if we multiply
these three dimensions together. Let us understand this with an example.
Example: Find the volume of a cuboid if its length is 8 units, width is 4 units, and height is 3 units.
Solution: The volume of a cuboid can be calculated using the formula,
Volume of Cuboid = Length x Width x Height
After substituting the values we get, Volume of Cuboid = 8 × 4 × 3 = 96 units^3
Length Width Height of a Box
The length, width and height of a box can be easily identified because we know that length is the longest side, width is the shorter side and height is the vertical dimension of the box. Observe the
figure given below which shows the length, width and height of a box.
These dimensions are always expressed in the order where the length comes first, followed by the width and then the height. This means if the dimensions of a box are to be measured they are expressed
in the order of length, width and height. For example, 15'' × 10'' × 3'' means 15'' is the length of the box, 10'' is the width of the box and 3'' is the height of the box.
☛Related Articles
• Measurement
• Length Conversion
• Measuring Length
FAQs on Length Width Height
What is Length Width Height?
Length, width, height, and depth are words that are derived from the adjectives long, wide, high, and deep, respectively. Hence, they express the dimensions of an object. While length shows how long
the given object is, the width shows how broad it is and height shows how high it is. All these are expressed in linear units like centimeters, meters, inches, and so on.
What is the Formula for Length Width and Height?
When the length, width, and height of a cuboid are multiplied together, it gives the volume of the cuboid. The formula for the volume of a cuboid is, Volume of Cuboid = Length × Width × Height.
What is the Difference Between Length Width Height?
The length, width, height of an object are the different dimensions expressed in linear units. While length is the longest side of a shape, width is the shorter side, and height is the vertical
dimension or depth of the shape.
What is the Order of Length Width Height?
When the dimensions of a geometrical figure are written, they are written in the order in which the length comes first, followed by the width and then the height. For example, if the dimensions of a
cuboid are to be expressed, it will be written as Length × Width × Height, that is, 7 × 4 × 3, where 7 represents the length, 4 represents the width and 3 represents the height of the cuboid.
How to Calculate Cubic meter from Length Width Height?
Cubic meter is the unit that expresses the volume of a cuboid. Therefore, in order to find the volume of a cuboid in cubic meters the length, width and height are multiplied together. It should be
noted that all the length, width and height should have the same units (meters) so that the volume is expressed in cubic meters. For example, if the length of a cuboid is 10 m, its width is 6 m and
height is 3 m, let us find its volume in cubic meters. Volume of a cuboid = length × width × height. After substituting the values, we get, Volume of a cuboid = 10 × 6 × 3 = 180 cubic meters.
What is the Length Breadth and Height of a Cuboid?
The length of a cuboid is the longest side when the cuboid is placed horizontally. The width is the shorter side of the cuboid. The height is the vertical dimension of the cuboid.
How to Find the Length Width and Height when Volume is Given?
The formula that is used to find the volume of a cuboid is, Volume of Cuboid = Length × Width × Height. Therefore, if any one dimension is missing, it can be calculated by using this formula by
substituting the other given values. For example, let us find the height of a cuboid if the volume of a cuboid is given as 144 cubic cm, the length = 12 cm, width = 2 cm. Since the height of the
cuboid is not known, let us substitute the rest of the dimensions in the formula. Volume of Cuboid = Length × Width × Height. After substituting the known values we get, 144 = 12 × 2 × Height. After
solving this we get the height as, 6 cm.
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Maximum Bipartite Matching
Kuhn's Algorithm for Maximum Bipartite Matching¶
You are given a bipartite graph $G$ containing $n$ vertices and $m$ edges. Find the maximum matching, i.e., select as many edges as possible so that no selected edge shares a vertex with any other
selected edge.
Algorithm Description¶
Required Definitions¶
• A matching $M$ is a set of pairwise non-adjacent edges of a graph (in other words, no more than one edge from the set should be incident to any vertex of the graph $M$). The cardinality of a
matching is the number of edges in it. All those vertices that have an adjacent edge from the matching (i.e., which have degree exactly one in the subgraph formed by $M$) are called saturated by
this matching.
• A maximal matching is a matching $M$ of a graph $G$ that is not a subset of any other matching.
• A maximum matching (also known as maximum-cardinality matching) is a matching that contains the largest possible number of edges. Every maximum matching is a maximal matching.
• A path of length $k$ here means a simple path (i.e. not containing repeated vertices or edges) containing $k$ edges, unless specified otherwise.
• An alternating path (in a bipartite graph, with respect to some matching) is a path in which the edges alternately belong / do not belong to the matching.
• An augmenting path (in a bipartite graph, with respect to some matching) is an alternating path whose initial and final vertices are unsaturated, i.e., they do not belong in the matching.
• The symmetric difference (also known as the disjunctive union) of sets $A$ and $B$, represented by $A \oplus B$, is the set of all elements that belong to exactly one of $A$ or $B$, but not to
both. That is, $A \oplus B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$.
Berge's lemma¶
This lemma was proven by the French mathematician Claude Berge in 1957, although it already was observed by the Danish mathematician Julius Petersen in 1891 and the Hungarian mathematician Denés
Kőnig in 1931.
A matching $M$ is maximum $\Leftrightarrow$ there is no augmenting path relative to the matching $M$.
Both sides of the bi-implication will be proven by contradiction.
1. A matching $M$ is maximum $\Rightarrow$ there is no augmenting path relative to the matching $M$.
Let there be an augmenting path $P$ relative to the given maximum matching $M$. This augmenting path $P$ will necessarily be of odd length, having one more edge not in $M$ than the number of
edges it has that are also in $M$. We create a new matching $M'$ by including all edges in the original matching $M$ except those also in the $P$, and the edges in $P$ that are not in $M$. This
is a valid matching because the initial and final vertices of $P$ are unsaturated by $M$, and the rest of the vertices are saturated only by the matching $P \cap M$. This new matching $M'$ will
have one more edge than $M$, and so $M$ could not have been maximum.
Formally, given an augmenting path $P$ w.r.t. some maximum matching $M$, the matching $M' = P \oplus M$ is such that $|M'| = |M| + 1$, a contradiction.
2. A matching $M$ is maximum $\Leftarrow$ there is no augmenting path relative to the matching $M$.
Let there be a matching $M'$ of greater cardinality than $M$. We consider the symmetric difference $Q = M \oplus M'$. The subgraph $Q$ is no longer necessarily a matching. Any vertex in $Q$ has a
maximum degree of $2$, which means that all connected components in it are one of the three -
□ an isolated vertex
□ a (simple) path whose edges are alternately from $M$ and $M'$
□ a cycle of even length whose edges are alternately from $M$ and $M'$
Since $M'$ has a cardinality greater than $M$, $Q$ has more edges from $M'$ than $M$. By the Pigeonhole principle, at least one connected component will be a path having more edges from $M'$ than
$M$. Because any such path is alternating, it will have initial and final vertices unsaturated by $M$, making it an augmenting path for $M$, which contradicts the premise. $\blacksquare$
Kuhn's algorithm¶
Kuhn's algorithm is a direct application of Berge's lemma. It is essentially described as follows:
First, we take an empty matching. Then, while the algorithm is able to find an augmenting path, we update the matching by alternating it along this path and repeat the process of finding the
augmenting path. As soon as it is not possible to find such a path, we stop the process - the current matching is the maximum.
It remains to detail the way to find augmenting paths. Kuhn's algorithm simply searches for any of these paths using depth-first or breadth-first traversal. The algorithm looks through all the
vertices of the graph in turn, starting each traversal from it, trying to find an augmenting path starting at this vertex.
The algorithm is more convenient to describe if we assume that the input graph is already split into two parts (although, in fact, the algorithm can be implemented in such a way that the input graph
is not explicitly split into two parts).
The algorithm looks at all the vertices $v$ of the first part of the graph: $v = 1 \ldots n_1$. If the current vertex $v$ is already saturated with the current matching (i.e., some edge adjacent to
it has already been selected), then skip this vertex. Otherwise, the algorithm tries to saturate this vertex, for which it starts a search for an augmenting path starting from this vertex.
The search for an augmenting path is carried out using a special depth-first or breadth-first traversal (usually depth-first traversal is used for ease of implementation). Initially, the depth-first
traversal is at the current unsaturated vertex $v$ of the first part. Let's look through all edges from this vertex. Let the current edge be an edge $(v, to)$. If the vertex $to$ is not yet saturated
with matching, then we have succeeded in finding an augmenting path: it consists of a single edge $(v, to)$; in this case, we simply include this edge in the matching and stop searching for the
augmenting path from the vertex $v$. Otherwise, if $to$ is already saturated with some edge $(to, p)$, then will go along this edge: thus we will try to find an augmenting path passing through the
edges $(v, to),(to, p), \ldots$. To do this, simply go to the vertex $p$ in our traversal - now we try to find an augmenting path from this vertex.
So, this traversal, launched from the vertex $v$, will either find an augmenting path, and thereby saturate the vertex $v$, or it will not find such an augmenting path (and, therefore, this vertex
$v$ cannot be saturated).
After all the vertices $v = 1 \ldots n_1$ have been scanned, the current matching will be maximum.
Running time¶
Kuhn's algorithm can be thought of as a series of $n$ depth/breadth-first traversal runs on the entire graph. Therefore, the whole algorithm is executed in time $O(nm)$, which in the worst case is $O
However, this estimate can be improved slightly. It turns out that for Kuhn's algorithm, it is important which part of the graph is chosen as the first and which as the second. Indeed, in the
implementation described above, the depth/breadth-first traversal starts only from the vertices of the first part, so the entire algorithm is executed in time $O(n_1m)$, where $n_1$ is the number of
vertices of the first part. In the worst case, this is $O(n_1 ^ 2 n_2)$ (where $n_2$ is the number of vertices of the second part). This shows that it is more profitable when the first part contains
fewer vertices than the second. On very unbalanced graphs (when $n_1$ and $n_2$ are very different), this translates into a significant difference in runtimes.
Standard implementation¶
Let us present here an implementation of the above algorithm based on depth-first traversal and accepting a bipartite graph in the form of a graph explicitly split into two parts. This implementation
is very concise, and perhaps it should be remembered in this form.
Here $n$ is the number of vertices in the first part, $k$ - in the second part, $g[v]$ is the list of edges from the top of the first part (i.e. the list of numbers of the vertices to which these
edges lead from $v$). The vertices in both parts are numbered independently, i.e. vertices in the first part are numbered $1 \ldots n$, and those in the second are numbered $1 \ldots k$.
Then there are two auxiliary arrays: $\rm mt$ and $\rm used$. The first - $\rm mt$ - contains information about the current matching. For convenience of programming, this information is contained
only for the vertices of the second part: $\textrm{mt[} i \rm]$ - this is the number of the vertex of the first part connected by an edge with the vertex $i$ of the second part (or $-1$, if no
matching edge comes out of it). The second array is $\rm used$: the usual array of "visits" to the vertices in the depth-first traversal (it is needed just so that the depth-first traversal does not
enter the same vertex twice).
A function $\textrm{try_kuhn}$ is a depth-first traversal. It returns $\rm true$ if it was able to find an augmenting path from the vertex $v$, and it is considered that this function has already
performed the alternation of matching along the found chain.
Inside the function, all the edges outgoing from the vertex $v$ of the first part are scanned, and then the following is checked: if this edge leads to an unsaturated vertex $to$, or if this vertex
$to$ is saturated, but it is possible to find an increasing chain by recursively starting from $\textrm{mt[}to \rm ]$, then we say that we have found an augmenting path, and before returning from the
function with the result $\rm true$, we alternate the current edge: we redirect the edge adjacent to $to$ to the vertex $v$.
The main program first indicates that the current matching is empty (the list $\rm mt$ is filled with numbers $-1$). Then the vertex $v$ of the first part is searched by $\textrm{try_kuhn}$, and a
depth-first traversal is started from it, having previously zeroed the array $\rm used$.
It is worth noting that the size of the matching is easy to get as the number of calls $\textrm{try_kuhn}$ in the main program that returned the result $\rm true$. The desired maximum matching itself
is contained in the array $\rm mt$.
int n, k;
vector<vector<int>> g;
vector<int> mt;
vector<bool> used;
bool try_kuhn(int v) {
if (used[v])
return false;
used[v] = true;
for (int to : g[v]) {
if (mt[to] == -1 || try_kuhn(mt[to])) {
mt[to] = v;
return true;
return false;
int main() {
//... reading the graph ...
mt.assign(k, -1);
for (int v = 0; v < n; ++v) {
used.assign(n, false);
for (int i = 0; i < k; ++i)
if (mt[i] != -1)
printf("%d %d\n", mt[i] + 1, i + 1);
We repeat once again that Kuhn's algorithm is easy to implement in such a way that it works on graphs that are known to be bipartite, but their explicit splitting into two parts has not been given.
In this case, it will be necessary to abandon the convenient division into two parts, and store all the information for all vertices of the graph. For this, an array of lists $g$ is now specified not
only for the vertices of the first part, but for all the vertices of the graph (of course, now the vertices of both parts are numbered in a common numbering - from $1$ to $n$). Arrays $\rm mt$ and
are $\rm used$ are now also defined for the vertices of both parts, and, accordingly, they need to be kept in this state.
Improved implementation¶
Let us modify the algorithm as follows. Before the main loop of the algorithm, we will find an arbitrary matching by some simple algorithm (a simple heuristic algorithm), and only then we will
execute a loop with calls to the $\textrm{try_kuhn}()$ function, which will improve this matching. As a result, the algorithm will work noticeably faster on random graphs - because in most graphs,
you can easily find a matching of a sufficiently large size using heuristics, and then improve the found matching to the maximum using the usual Kuhn's algorithm. Thus, we will save on launching a
depth-first traversal from those vertices that we have already included using the heuristic into the current matching.
For example, you can simply iterate over all the vertices of the first part, and for each of them, find an arbitrary edge that can be added to the matching, and add it. Even such a simple heuristic
can speed up Kuhn's algorithm several times.
Please note that the main loop will have to be slightly modified. Since when calling the function $\textrm{try_kuhn}$ in the main loop, it is assumed that the current vertex is not yet included in
the matching, you need to add an appropriate check.
In the implementation, only the code in the $\textrm{main}()$ function will change:
int main() {
// ... reading the graph ...
mt.assign(k, -1);
vector<bool> used1(n, false);
for (int v = 0; v < n; ++v) {
for (int to : g[v]) {
if (mt[to] == -1) {
mt[to] = v;
used1[v] = true;
for (int v = 0; v < n; ++v) {
if (used1[v])
used.assign(n, false);
for (int i = 0; i < k; ++i)
if (mt[i] != -1)
printf("%d %d\n", mt[i] + 1, i + 1);
Another good heuristic is as follows. At each step, it will search for the vertex of the smallest degree (but not isolated), select any edge from it and add it to the matching, then remove both these
vertices with all incident edges from the graph. Such greed works very well on random graphs; in many cases it even builds the maximum matching (although there is a test case against it, on which it
will find a matching that is much smaller than the maximum).
• Kuhn's algorithm is a subroutine in the Hungarian algorithm, also known as the Kuhn-Munkres algorithm.
• Kuhn's algorithm runs in $O(nm)$ time. It is generally simple to implement, however, more efficient algorithms exist for the maximum bipartite matching problem - such as the
Hopcroft-Karp-Karzanov algorithm, which runs in $O(\sqrt{n}m)$ time.
• The minimum vertex cover problem is NP-hard for general graphs. However, Kőnig's theorem gives that, for bipartite graphs, the cardinality of the maximum matching equals the cardinality of the
minimum vertex cover. Hence, we can use maximum bipartite matching algorithms to solve the minimum vertex cover problem in polynomial time for bipartite graphs.
Practice Problems¶
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PAPR Reduction in OFDM using PTS Technique
Volume 06, Issue 07 (July 2017)
PAPR Reduction in OFDM using PTS Technique
DOI : 10.17577/IJERTV6IS070250
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Sohangjot Kaur Randhawa, Prof. Daljeet Singh Bajwa, 2017, PAPR Reduction in OFDM using PTS Technique, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 06, Issue 07 (July
2017), http://dx.doi.org/10.17577/IJERTV6IS070250
• Open Access
• Total Downloads : 163
• Authors : Sohangjot Kaur Randhawa, Prof. Daljeet Singh Bajwa
• Paper ID : IJERTV6IS070250
• Volume & Issue : Volume 06, Issue 07 (July 2017)
• DOI : http://dx.doi.org/10.17577/IJERTV6IS070250
• Published (First Online): 26-07-2017
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
PAPR Reduction in OFDM using PTS Technique
Sohangjot Kaur Randhawa
Department of ECE BBSBEC Fatehgarh Sahib,India
Prof. Daljeet Singh Bajwa
Department of ECE BBSBEC Fatehgarh Sahib,India
Abstract OFDM system is widely used in the field of trending communication. With the increasing use of the OFDM system many issues related to OFDM comes into existence. PAPR is one of the issues of
OFDM. It refers to the increment in the peak-to-average-power ratio of signals that are transmitted to the receiver. It occurs due to the superposition of many sub carriers. The quality of the
signals is tainted and the complexity is increased while performing conversion process. Hence it is mandatory to develop such an approach which can help to reduce the PAPR. There are many techniques
have been suggested for PAPR reduction, with different levels of success and complexity. Techniques like clipping, filtration, PTS etc were proposed but these techniques achieve PAPR reduction at the
bit error rate increase, data rate loss, computational complexity increase, and so on.
This study provides a review over the techniques meant for reducing the PAPR in OFDM systems. It represents various techniques along with their equations.
Keywords:- PAPR,SLM,PTS,Clipping and Filtering.
1. INTRODUCTION
Since the very genesis of man, communication has been one of the fundamental parts of the human life .Previously different techniques like gesture based communications were executed for this
reason. At present there are many ways that through which we communicate, email, internet, social networking, letters, cell phones, signs, newspapers, radio, magazines, the list goes on and on.
With the advancement in the technique of wireless communication channel the need of high rate data transmission comes into existence. To transfer the data on a high rate OFDM is used. OFDM stands
for Orthogonal Frequency Division Multiplexing. It is an effective method or technique adopted in wireless channels.
2. PAPR
PAPR stands for Peak-to-Average Power Ratio. PAPR is a exigent issue in OFDM systems. If the PAPR goes high then it leaves direct effects on the performance of the system by degrading it. The PAPR
degrades the efficiency of the transmitted signals. Low PAPR increases the efficiency of the power amplifier but high value of PAPR is totally opposite to this.
Figure1. Peak value issue in OFDM system
PAPR of a signal is measured or represented in the form of decibels as shown below. It will also express the need of resolving the problem of PAPR.
Equation for PAPR as:
PAPRdb = 10 log (max[x (t) x * (t)] / E[x (t) x * (t)])
1. Thus, PAPR defined as the ratio of maximum peak power which will be dividing by average power of OFDM signal.
In the above equation, E shows expected value. Now, PAPR for a single complex tone
X (t) = e2ft
2. Where t shows period and peak value of the signal: Max[x (t) x * (t)] = max [e2ft e -2ft] = max [e0] =1
As a result, calculate mean square value of the signal:
E[x (t) x * (t)] = E [e2ft e -2ft] =1
3. Generated output from the above equation is 0db i.e. shows the value of PAPR. Now, consider OFDM time signal which comprises of more than k complex tones (sub Carrier tones).These tones are known
as subcarriers. Therefore, the above signal representation will be like:
X (t) = ke j2 k t / T
Consequently, PAPR reduction can be possible by increasing
the probability of getting low PAPR values. PAPR should be reduced in order to enhance the lifetime of the network. In comparison with single carrier systems PAPR is quite high in multi carrier
system. Highest value of PAPR reduces the efficiency of the Power amplifier (Transmitter) [3]. PAPR affects the transmitted signal. PAPR is the problem exists in OFDM system. The input symbol stream
in IFFT should have a constant value of power spectrum. But the output of IFFT can result in a variable value and fluctuated wave or spikes. Only few of the sub carriers are allotted with energy to
transmit the data. This problem gives rise to other problems in OFDM system. So OFDM signal has a very large PAPR, which is very sensitive to non linear high power comprised amplifier. In OFDM, a
block of N symbols{X , k = 0,1,….,N -1} k , is formed with each symbol modulating one of a set of subcarriers,
{f , k = 0,1,…..,N -1} k .
1. The N subcarriers are chosen to be orthogonal, that is, f k f k = D , where Df =1 NT and T is the original time period.
2. PAPR of single tone: Consider a sin signal as having the period t. The peak value of the signal is: . The Mean square value of the signal is 5.
Given so, the PAPR of a single sine tone is, PAPR OF A MULTICARRIER SIGNAL
Let the data block of length N be
represented by a Duration of vector any system in the set X is T and represents one of the sub carriers set. As the N sub carriers selected for transmission of the signal are orthogonal to each
other, so we can have NT is the duration of the OFDM data block X. Reducing the max|x(t)| is the principle goal of PARP reduction techniques. In these cases Discrete Time signals are used. There
are many PAPR reducing techniques which can be used to handle amplitude value of x (t).
TECHNIQUES FOR PAPR REDUCTION
There are many techniques used for reducing the PAPR in OFDM. But all of these techniques are not suitable in every case. The use of PAPR reduction techniques based on the needs of the system and
various elements.
1. Signal Scrambling Techniques
1. Block Coding Techniques
2. Block Coding Scheme with Error Correction
3. Selected Mapping (SLM)
4. Partial Transmit Sequence (PTS)
5. Interleaving Technique
6. Tone Reservation (TR)
7. Tone Injection (TI)
2. Signal Distortion Techniques
1. Peak Windowing
2. Envelope Scaling
3. Peak Reduction Carrier
4. Clipping and Filtering
SLM stands Selected Mapping Technique. In this technique input data is divided into various sub blocks on the basis of given N length. It uses serial to parallel converter for converting the data
stream. When the conversion is applied to the signals then the blocks of the OFDM system are arranged in a sequence as follows:
Where u= [0, 1, 2.U], to make OFDM data blocks to be phase rotated. Therefore X (u) expressed as,
After data blocks are phase rotated, the rotated OFDM data blocks represents similar information which are unmodified OFDM data blocks, provided with known phase sequence.
According to PTS technique an N symbol input block is taken, and is divided into V disjoint sub-blocks. After that the divided sub-bocks are weighted by the phase vector sequence. The selection of
phase factor such that the PAPR of the resultant signal is minimum.
Where v = [1, 2, 3., V], Then, the sub-blocks X are transformed into time-domain partial transmit sequence x, by using IFFT which can be represented as
xm IFFT ( Xm)
It is a PAPR reduction technique which collects the interleavers for reducing the value of PAPR. It is quite different from PTS and SLM as it does not uses the set of phase sequence for PAPR
2. CLIPPING AND FILTERING
This is easiest and simplest way or technique to redue the PAPR. It is the combination of two processes i.e. clipping and filtering. Clipping is a technique in which a user defined threshold level or
clip level is defined. This threshold level is predefined and then the signals are compared by these pre-defined levels and the signal which crosses the threshold levels clipped or cancelled. The
process of clipping causes in-band or out-of-band noise in the signals and clipping is a non-linear process. This process may reduce the spectral efficiency of the signals and also increases the BER.
After applying clipping, filtering is applied to the signals which are received after clipping. Filtering is applied to remove the noise from the signals. It is applied to remove the out-of-band
distortion and spectrum growth efficiency of the signals. After applying filtering signal may rises above the clip level which is considered as the re-growth f the signals. This is the lacking point
of applying filtering after clipping. To overcome the disadvantage of filtering the process clipping and filtering should be applied alternatively.
From above literature it is concluded that the OFDM suffers from the problem of highest PAPR. In order to reduce the problem of PAPR large number of techniques has been developed. But these
techniques are not work sufficiently to reduce the effect of PAPR on OFDM. Hence still the research is under process to reduce the problem of PAPR.
1. Kavita Mhatrea, Efficient Selective Mapping PAPR Reduction Technique , International Conference on Advanced Computing Technologies and Applications , Pp 620 627, 2015
2. Suverna Sengar, , Performance improvement in OFDM system by PAPR reduction Signal & Image Processing : An International Journal (SIPIJ) Vol.3, No.2, April 2012,Pp 157-169
3. Md. Ibrahim Abdullah, Comparative Study of PAPR Reduction Techniques in OFDM Journal of Systems and Software, VOL. 1, NO. 8, ,Pp 263-269, 2011
4. Zhongpeng Wang, Reduction PAPR of OFDM Signals by Combining SLM with DCT Transform Int. J. Communications, Network and System Sciences, Vol 3, Pp 888-892, 2010
5. Reshma Elizabeth. Regi, Haris P.A, Performance of PAPR Reduction in OFDM System with Complex Hadamard Sequence using SLM and Clipping , International Journal of Engineering and Advanced
Technology (IJEAT), Volume-3, Issue-4, Pp 381-384,2014
6. V.B.MALODE, B.P.PATIL, Performance of Linear Block Coded OFDM system, International Journal of Applied Information Systems,
IJAIS Journal, Vol 3, issues-4,2012
7. K. Srinivas Rao,Peak to average power reduction in MIMO-OFDM systems using Sub-Optimal Algorithm, IJDPS, vol 3(3), Pp 261-273, 2012
8. Maan Singh, Vijay Kumar, Jan, Signal Scrambling Techniques for PAPR Reduction in OFDM Systems IJECS Vol. 2, No. 1, Pp. 311-317, 2013
9. Seung Hee Han, SEPTEMBER, Modified Selected Mapping Technique for PAPR Reduction of Coded OFDM Signal , IEEE Transactions On Broadcasting Vol. 50, No. 3, Pp.335-341, 2004
10. Peyali Choudhury, Achala Deshmukh, Comparison and analysis of PAPR reduction techniques in OFDM, IOSR- JECE Vol. 4, No. 5, Pp. 01-06, 2013
11. Leman Dewangan, Mangal Singh, Neelam Dewangan, November, A Survey of PAPR Reduction Techniques in LTE-OFDM System, IJRTE Vol. 1, No. 5, Pp. 10-13, 2013
12. R. Divya Kanti and R.V. Ch. Sekhar Rao, Systematic Comparison of Different PAPR Reduction Methods in OFDM Systems , IJECE Vol. 7, No. 1, Pp. 21-30.
13. Mr.Jobin Raj, Mr.M.Malleswaran, DCT Based Modified SLM Technique for PAPR Reduction in OFDM Transmission, IJSER Vol. 3,
No. 6, Pp. 1-6, 2012
14. Pawan Sharma, Seema Verma, Performance Analysis of Peak-to-Average Power Ratio Reduction Techniques for Wireless Communication Using OFDM Signals, IJCSI Vol. 7, No. 6,
Pp.261-268, 2010
15. Reshma Elizabeth Regi, Haris P.A, Performance of PAPR Reduction in OFDM System with Complex Hadamard Sequence using SLM and Clipping, IJEAT Vol. 3, No. 4, Pp.381-384, 2014
16. Kuang Xu, Beam forming MISO-OFDM PAPR Reduction, ijareeie, vol 5(8), Pp 6926-6931, 2016
17. Gurtej Singh Toor, Harjinder Singh, Amandeep singh Bhandari,REVIEW PAPER ON PAPR REDUCTION TECHNIQUES IN OFDM SYSTEM, IJTRE Vol. 1, No. 8,
Pp.640-642, 2014
18. Panduranga mukunthan,Modified PTS with interleaving for PAPR reduction of OFDM signals with QPSK sub blocks, ijfcc, vol 3(1), Pp 22-26, 2014
19. Karuna A Mahajan, Comparison of Signal Scrambling PAPR Reduction Techniques with Signal Distortion Techniques in OFDM Signal, ijca, 2012
20. Md. Mahmudul Hasan,An overview of PAPR Reduction Technique in oFDM System, ijca, vol 60(15), Pp 33-38, 2012
21. M.V.R Vittal,Performance enhancement of OOFDM signals using PAPR reduction Techniues and the comparison of their performance, ijca, vol 41(19), Pp 36-40, 2012
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Solution to Problem P – Partitioning a square
IPSC 2018
Solution to Problem P – Partitioning a square
The easy subproblem was easily solvable by hand. The smallest square that can be divided into four regions is a 2×2 square. One possible correct output looks as follows:
For the hard subproblem, we can start by making the following observation: We want to take a square of some size s and divide it into n equally-large regions. Obviously, that is only possible if n
divides s^2. Thus, if we find the smallest such s and successfully construct an s×s square, we can be certain that it’s optimal.
Suppose we have said s×s square. How can we divide it into n connected regions? There are many techniques that work. For example, you could choose any Hamiltonian path (i.e., a path that visits
each cell of the square once) and split it into n segments.
Probably the easiest solution in terms of implementation is the following one:
The size of each region will be r=s^2/n. It is obvious that r is also a divisor of s^2. But we can show a stronger statement: in fact, if s is as small as possible, r must be a divisor of s. This
is because for any prime p we have two possibilities:
• If the largest power of p that divides n is even, say 2k, then s divisible by p^k and r is not divisible by p.
• If the largest power of p that divides n is odd, say 2k+1, then s divisible by p^k+1 and r divisible by p.
Thus, for the construction we simply split each row of the square into s/r regions.
Example output for n=18:
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Incoherent neutron scattering from multi-element materials
^aUniversity of Delaware, Newark, DE 19716 , USA, and ^bCenter for Neutron Research, NIST, 100 Bureau Drive, Gaithersburg, MD 20899-6102, USA
^*Correspondence e-mail: charles.glinka@nist.gov
(Received 12 November 2010; accepted 3 March 2011; online 2 April 2011)
In a neutron diffraction measurement, including small-angle scattering, there is generally a featureless (i.e. Q-independent) component due to incoherent scattering. This scattering contains no
information about the atomic structure or structure on any scale. There may also be featureless scattering that arises from atomic disorder in multi-element materials. This scattering is sometimes
referred to as compositional or mixture incoherent scattering. However, this designation is misleading. A much better designation is diffuse coherent scattering. Here the differences and
distinguishing characteristics of incoherent scattering vis-à-vis diffuse coherent scattering due to atomic disorder are delineated and demonstrated experimentally.
1. Introduction
Most textbooks on neutron scattering discuss the concept of incoherent scattering in detail only for materials consisting of a single atomic species (e.g. Bacon, 1962 ; Squires, 1978 ; Roe, 2000 ).
Such discussions are adequate for introducing the concepts of nuclear spin incoherence and isotopic incoherence. Most materials, however, consist of more than one atomic species and hence the
question arises as to how to calculate the incoherent contribution to the scattering from such materials.
When considering the neutron scattering from a multi-element material, whether it be a compound, a solid or liquid solution, or a molecular solid or liquid, it is important to distinguish between the
incoherent scattering and any diffuse coherent scattering that may be present and is related to the degree of atomic disorder in the material. In the extreme case of complete atomic disorder, there
will in general be a component of the coherent scattering that is essentially Q independent, similar to the true incoherent scattering that arises from the isotopic and nuclear spin distributions.
1.1. Scattering formalism
The scattering cross section from a system of N atoms in a volume V is given in the Born approximation by
where the angle brackets denote averaging over an ensemble of equivalent systems.^1 In this expression, b[i] is the scattering length of the atom at location r[i]. If all of the atoms are of a single
atomic species (i.e. a single element), then the ensemble average can clearly be written as
because there is no correlation between an atom's location and the isotope or nuclear spin state at that location. What may be less obvious is that equation (2) also applies when there is more than
one type of atomic species in the sample. In that case, there is, in general, a correlation between a given location and the type of atom at the location. However, even though a given site may be
more likely to be populated by one type of atom than another, there is still no correlation with a particular isotope or nuclear spin state of that atomic species at that site. Hence equation (2)
remains valid even for multi-element materials.
For the same reasons the average of the product of scattering lengths can be written as
Only the first term in equation (4) contains information about the arrangement, or structure, of the atoms. This term is called the coherent scattering:
The second term in equation (4) contains no structural information and is called the incoherent scattering:
To reduce equation (4) further to a useful form for computations, one must indicate how many of each type of atom is present in V. If there are, say, m elements represented, and N[j] atoms of element
j, then the fraction of atoms of type j is
Now if all the atoms are disordered, then
Substituting equation (8) into equation (4) leads to
Notice the similarities, and differences, between equations (9) and (4) . The first and third terms of equation (9) correspond to the first and second terms of equation (4) , and represent coherent
and incoherent scattering, respectively. It is the middle term in equation (9) that is new, arising from the presence of more than one element, and, more importantly, the assumption that the atoms
are randomly distributed among the available sites. This term is Q independent and thus is sometimes referred to as compositional or mixture incoherent (Cotton, 1991 ; Brûlet et al., 2007 )
scattering. A better designation, however, is diffuse coherent scattering: diffuse because of the lack of Q dependence, and coherent because this term provides information about the structure of the
system, namely, that the atoms are disordered. It may at first seem odd to refer to disorder as a type of structure, yet that is exactly what it is. The essence of the assumption leading to equation
(9) is that there is no correlation between the type of atom occupying a given site and the type occupying any other site, which is, in fact, a strong statement about the structure of the system:
quite strong, for as we shall see, it does not apply to most real materials.
2. Specific examples
2.1. NaCl
Consider polycrystalline NaCl, a system with two atomic species (A and B), which has a face-centered-cubic crystal structure with mass density ρ = 2.165gcm^−3, molecular weight M[W] = 58.44 and N/V
= 1/v[m] = 2.231 × 10^22 moleculescm^−3 (v[m] is the molecular volume). For such a material, the coherent cross section, from equation (5) becomes
where G[hkl] = ha* + kb* + lc* is a reciprocal lattice vector, structure factor for the hkl Bragg reflection and the sum is over the atoms in the unit cell. For the NaCl rock salt structure,
The bound coherent scattering lengths, 〈b〉 = b[c], are b[c](Na) = 3.63fm and b[c](Cl) = 9.566fm (Sears, 1992 ).
The incoherent cross sections for each element are (Sears, 1992 )
(1 barn = 10^−28m^2). Hence the corresponding macroscopic cross section is
For hypothetical disordered NaCl, in which the sites in the rock salt structure are occupied at random by either Na or Cl, the first term in equation (9) gives
In addition, there is the diffuse coherent scattering [second term in equation (9) ; this diffuse scattering due to site disorder is called the Laue monotonic scattering (Warren, 1969 )],
as well as the incoherent scattering,
These two cases are summarized in Fig. 1 (where the multiplicity factors for the individual Bragg peaks have been ignored to emphasize the structure factors).
2.2. H[2]O
From equation (6) , the macroscopic incoherent scattering cross section for light water is
where n[j] is the number density of atoms of type j.
Using the values in Table 1 , and n[H] = 2 (ρN[A]/MW) = 6.69 × 10^22 atoms of H per cm^−3 (n[O] = 3.35 × 10^22 per cm^−3; N[A] is Avogadro's number) yields
This cross section is calculated using the bound scattering lengths for the nuclei. The actual cross section for water depends on the incoming neutron energy and the water temperature. The measured
incoherent scattering cross section from water is, for example, ∼5.7cm^−1 for 5meV neutrons at 290K, and ∼7.7cm^−1 for 1meV neutrons at 290K (Brookhaven National Laboratory, 1976 , hereafter
denoted BNL 325).
Element 〈b〉 (×10^−12cm) 〈b^2〉 = σ[s]/4π (barns)
Hydrogen −0.374 6.53
Oxygen 0.580 0.337
What about the other Q-independent term in equation (9) ? Should the middle term in equation (9) ^2 be added to the result obtained in equation (18) to give the `total incoherent scattering'? The
answer is no, because the assumption leading to equation (9) that any atom is equally likely to occupy any available site does not apply to a molecular liquid like water. [For equation (9) to apply
to water, all possible molecular permutations (H[2]O, HO[2], H[3] and O[3]) would have to be present in the liquid.]
Another way to obtain this result [equation (18) ] for the incoherent scattering, and one that will give additional insight when we consider H[2]O/D[2]O mixtures next, is to treat the water molecule
as the primary scattering entity. This approach is valid at low Q where the internal structure of the molecule is unresolvable (Qr[m] << 1, where r[m] is any intramolecular distance).
We start again from equation (1) and proceed as before to equation (4) ,
where N[m] is the number of molecules in volume V,
2.3. H[2]O/D[2]O mixtures (not including H/D exchange)
Since H[2]O/D[2]O mixtures are used extensively to control scattering contrast in aqueous solutions, this is an important case to consider. At low Q, we can again treat the individual water molecules
as the primary scattering entities (thereby ignoring their internal structure) as discussed in the previous section. We start again from equation (1) , written as
where N[m] is the number of molecules in the volume V, r[i] is the position (e.g. the center of mass) of molecule i and b[i] is the scattering length for the molecule.
If we assume there is no correlation between a site, r[i], and the type of molecule, H[2]O or D[2]O, at that site, then equation (25) can be developed as was done in arriving at equation (9) , i.e.
where v[m] is the volume of one molecule, and φ[H[2]][O] (φ[D[2]][O]) is the volume fraction of H[2]O (D[2]O). Equation (26) shows explicitly that the flat `background' seen at low Q from such
mixtures consists of a combination of diffuse coherent scattering (second term) and true incoherent scattering (third and fourth terms).
From the scattering lengths and cross sections tabulated by Sears (1992 ),
or, in terms of the total cross section per molecule,
Notice that equation (26) is the basis for the high-concentration labeling technique used to study the conformation of polymer chains in mixtures of protonated and perdeuterated chains in the melt
(Akcasu et al., 1980 ). Equation (26) can be extended to larger molecules by including the molecular form factor in the second term, which is another demonstration that this term represents coherent,
not incoherent, scattering.
2.4. H[2]O/D[2]O mixtures (including H/D exchange)
For water there is exchange of H and D. For this reason, the second (coherent scattering) term in equation (26) is reduced in real water, as pointed out by Arleth & Pedersen (2000 ). To show this
explicitly we begin again from equation (20):
In this case we have three types of molecules to consider: H[2]O, D[2]O, and HDO or DHO (HDO and DHO are indistinguishable in terms of their scattering lengths, hence there are three and not four
types of molecules to consider). Hence
where f[H[2]O], f[D[2]O] and f[HDO] are the fractions of H[2]O, D[2]O and HDO (or DHO) molecules in the mixture, respectively.
The number of H atoms in the mixture is N[m] is the number of molecules and volume fraction of H[2]O that is mixed with a volume fraction [2]O. Similarly
The terms and (31) , respectively; and
Substituting equations (36) , (37) and (38) into (33) gives, after some manipulation,
For comparison, equation (26) can be written as
The only difference between equations (39) and (40) is in the diffuse coherent scattering term (the last term in each), which is reduced by a factor of two when H/D exchange is included. It is
instructive to note that the `true' incoherent scattering terms in both (40) and (39) are the same, as they must be since this scattering does not depend on where the atoms are located.
The coherent [fourth term in (39) and (40) ] and incoherent [second and third terms in equations (39) and (40) ] contributions to the total low-Q scattering from H[2]O/D[2]O mixtures are plotted in
Fig. 2 .
The reduction in the diffuse coherent scattering when H/D exchange is included in the calculation begs the question, where does the diffuse scattering go? To understand this, we compare the Q
-dependent coherent scattering terms in equations (39) and (40) , which are proportional to 〈b〉^2:
without H/D exchange, and
with H/D exchange. Then
Hence the reduction in the diffuse coherent scattering due to H/D exchange is not accompanied by a corresponding increase in the intermolecular Q-dependent coherent scattering. The reduction in
diffuse coherent scattering therefore likely appears (although not shown here) in the intramolecular coherent scattering (at larger Q), which we have neglected in this treatment.
3. Demonstration experiment
To demonstrate the distinction between diffuse coherent scattering and true incoherent scattering, we have measured the small-angle neutron scattering (SANS) from titanium dioxide, TiO[2]. This
molecular material was chosen because titanium is one of only a few elements with a negative coherent scattering length. In addition the isotopic and nuclear spin incoherent scattering for both
oxygen and titanium are small compared to most elements. As a result, if there were a diffuse scattering term [the middle term in equation (9) ] in the cross section, it would dominate the measured
SANS and be easily identified by putting the scattering on an absolute scale. However, for such a term to exist in the cross section for TiO[2], the positions of the titanium and oxygen atoms must be
completely uncorrelated, which is physically not the case.
From equation (18) the incoherent macroscopic cross section for TiO[2] is
From the mass density (ρ = 4.23gcm^−3) and molecular weight (79.9atomic mass units) of TiO[2], the atomic number densities are n[Ti] = 3.19 × 10^22cm^−3 and n[O] = 2n[Ti]. The elemental
incoherent cross sections are (Sears, 1992 ) σ[incoh,Ti] = 2.87barns and σ[incoh,O] ≃ 0barns. Hence
The additional Q-independent scattering that would arise if the Ti and O atoms were completely disordered is given by the second term of equation (9) , which we call Σ[c,Laue] as in the NaCl example:
where 〈b[Ti]〉 = b[c,Ti] = −3.44fm and 〈b[O]〉 = b[c,O] = 5.80fm (Sears, 1992 ). The disordered scattering term is three times larger than the true incoherent scattering and hence should be
readily apparent from the scale of the Q-independent SANS.
For the SANS, measurements a 2mm-path-length quartz cell was filled with a coarse TiO[2] powder. The cell was weighed before and after filling to estimate the bulk density of the powder in the cell.
This was found to be 0.99gcm^−3. Hence the cross sections per unit volume given in equations (46) and (47) should be multiplied by 0.99/4.23 for comparison with the scattering from this particular
sample. The measurements were made on the 30m SANS instrument on neutron guide NG-7 at NIST using a wavelength of 6Å and a sample-to-detector distance of 1m. The particle size of tens of
micrometres does produce SANS at very low Q, which decays roughly as Q^−4. At larger Q, still in the SANS region, the scattering becomes essentially flat at a level that was put on an absolute scale
by measuring the neutrons per second incident on the sample. In addition to the scattering from the sample in its cell, scattering from the empty cell was also measured and subtracted taking into
account room background and the transmission of the sample. The resulting SANS for the TiO[2] is shown in Fig. 3 along with the scattering from a reference sample of D[2]O treated in the same way.
The Q-independent scattering is slightly lower, perhaps as a result of overestimating the bulk density of the sample, than the calculated level of incoherent scattering but far below the level
expected from the sum of incoherent plus disordered coherent scattering.
4. Discussion
The incoherent scattering discussed here, which arises solely from the lack of any correlation between an atom's location and that atom's nuclear spin state or nuclear isotope, is unique to neutron
scattering. There is no analog in X-ray scattering. Scattering that arises from atomic disorder, such a Laue monotonic scattering (Warren, 1969 ), is present in both X-ray and neutron scattering.
Because atomic disorder scattering may be nearly Q independent, like the incoherent scattering, authors of neutron scattering papers and texts have in some cases referred to this scattering as
compositional or mixture incoherent scattering. However, this designation blurs the distinction between true incoherent scattering, which has no structural information, and what is more properly
referred to as diffuse coherent scattering, which does contain structural information pertaining to the degree of atomic disorder.
The salient point is that incoherent cross sections are simply additive. Hence the macroscopic incoherent scattering cross section for a material is readily calculated from
where N is the number of atomic species in the material, n[j] is the number density of atomic species j and σ[incoh,j] is the tabulated incoherent scattering cross section for element j. For any
material with an appreciable amount of hydrogen, a good approximation for the bulk incoherent cross section is simply, Σ[incoh] ≃ n[H]σ[incoh,H], since the incoherent cross section for hydrogen is so
much larger than that for other elements.
The coherent and incoherent cross sections tabulated by Sears (1992 ) are calculated from the bound scattering lengths for nuclei. The actual cross sections depend on the incoming neutron energy and
sample temperature, especially for light elements. For cold neutrons, the tabulated cross sections are generally a lower limit. The measured incoherent scattering from hydrogen, for example, can be
considerably larger than its bound value. For example,
[from tables of scattering lengths given by Sears (1992 )] and
[measured for 1meV, or 0.9nm, neutrons at 290K (BNL 325 )].
^1Equivalent systems refer to all possible configurations of the atoms, including their nuclear isotopes and spin states, which may affect the scattering of a neutron beam incident on a sample.
^2The middle term in equation (9) integrated over dΩ for H[2]O would be n is the number of atoms per unit volume. For f[H] = 2/3, f[O] = 1/3, n = 3(3.35 × 10^22) atomscm^−3, Σ[Laue] = 0.255cm^−1.
Helpful discussions with B. Hammouda, D. Mildner, J. Barker and R. Cappelletti and correspondence with A. Brûlet are gratefully acknowledged.
Akcasu, A. Z., Summerfield, G. C., Jahshan, S. N., Han, C. C., Kim, C. Y. & Yu, H. (1980). J. Polym. Sci. Polym. Phys. 18, 863–869. CrossRef CAS Google Scholar
Arleth, L. & Pedersen, J. S. (2000). J. Appl. Cryst. 33, 650–652. Web of Science CrossRef CAS IUCr Journals Google Scholar
Bacon, G. E. (1962). Neutron Diffraction, 2nd ed., p. 53. Oxford University Press. Google Scholar
Brookhaven National Laboratory (1976). Neutron Cross Sections, Vol. II, 3rd ed., edited by D. I. Garber & R. R. Kinsey, Brookhaven National Laboratory Report 325, Upton, New York, USA. Google
Brûlet, A., Lairez, D., Lapp, A. & Cotton, J.-P. (2007). J. Appl. Cryst. 40, 165–177. Web of Science CrossRef IUCr Journals Google Scholar
Cotton, J. P. (1991). Neutron, X-ray and Light Scattering, edited by P. Lindner & T. Zemb, pp. 29–30. Amsterdam: North Holland. Google Scholar
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Sears, V. F. (1992). Neutron News, 3(3), 26–37. CrossRef Google Scholar
Squires, G. L. (1978). Introduction to the Theory of Thermal Neutron Scattering, p. 24. Mineola: Dover Publications. Google Scholar
Warren, B. E. (1969). X-ray Diffraction, p. 229. Reading: Addison-Wesley. Google Scholar
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For
more information, click here.
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A pentagon pyramid is placed on V.P with square as base on V.P the cutting plane is perpendicular to H.P and inclined to V.P and the section is cutting not more than 3 edges, the section will be
Q. A pentagon pyramid is placed on V.P with square as base on V.P the cutting plane is perpendicular to H.P and inclined to V.P and the section is cutting not more than 3 edges, the section will be
A. triangle
B. trapezium
C. irregular square
D. irregular pentagon
Answer» A. triangle
Explanation: : if a pyramid is cut by a plane perpendicular to its axis section gives the base shape or parallel to axis and also parallel to any edge of base then the section formed will be
trapezium if the section plane not parallel to edge of base then the section will be a triangle.
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Adding Polynomials: A Step-by-Step Guide
This article will guide you through the process of adding two polynomials: (5m³ + 2m² - m) + (m² + 4m - 2).
Understanding Polynomials
Polynomials are expressions that consist of variables and constants, combined using addition, subtraction, multiplication, and non-negative integer exponents. In our example, both expressions are
The Addition Process
1. Identify like terms: Look for terms that have the same variable and exponent. In this case, we have:
□ m³ terms: 5m³
□ m² terms: 2m² and m²
□ m terms: -m and 4m
□ Constant terms: -2
2. Combine like terms: Add the coefficients of each like term. Remember that if a term has no coefficient, it's understood to be 1.
□ m³ terms: 5m³
□ m² terms: 2m² + 1m² = 3m²
□ m terms: -1m + 4m = 3m
□ Constant terms: -2
3. Write the simplified polynomial: Combine the results of step 2 to get the final answer.
Therefore, (5m³ + 2m² - m) + (m² + 4m - 2) = 5m³ + 3m² + 3m - 2
Key Points to Remember
• Only like terms can be combined.
• When combining like terms, only the coefficients are added.
• The exponents of the variables remain unchanged.
By following these steps, you can confidently add any two polynomials.
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American Mathematical Society
In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the
other hand, rational newforms F of weight two for the congruence subgroups ${\Gamma _0}(\mathfrak {n})$, where n is an ideal in the ring of integers R of K. This continues work of the first author
and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series $L(F,s)$ at $s = 1$ and compare with the value of $L(E,1)$ which is predicted by
the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that $L(F,1) = 0$ whenever $E(K)$ has a point of infinite order. Several examples are given
in detail from the extensive tables computed by the authors. References
J. E. Cremona, Modular symbols, D.Phil. thesis, Oxford, 1981. P. Gérardin, J. P. Labesse, Base change problem for $GL(2)$, Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure
Math., vol. 33 (Part 2), Amer. Math. Soc., Providence, RI, 1979, pp. 115-133. P. F. Kurčanov, Cohomology of discrete groups and Dirichlet series connected with Jacquet-Langlands cusp forms, Math.
USSR Izv. 12 (1978), 543-555. J. Elstrodt, F. Grunewald, and J. Mennicke, On the group $PS{L_2}(\mathbb {Z}[i])$, J. Arith. 1980, LMS Lecture Notes, vol. 56, Cambridge Univ. Press, 1981. —,
Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 189, Springer-Verlag, Berlin and New York, 1971. E. Whitley, Modular symbols and elliptic curves over imaginary quadratic
number fields, Ph.D. thesis, Exeter University, 1990.
Additional Information
• © Copyright 1994 American Mathematical Society
• Journal: Math. Comp. 62 (1994), 407-429
• MSC: Primary 11F67; Secondary 11F66, 11G05, 11G40
• DOI: https://doi.org/10.1090/S0025-5718-1994-1185241-6
• MathSciNet review: 1185241
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Purdue algebra note
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Applying Maths in the Chemical & Biomolecular Sciences
Solutions Q1 - 14#
# import all python add-ons etc that will be needed later on
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from sympy import *
init_printing() # allows printing of SymPy results in typeset maths format
plt.rcParams.update({'font.size': 16}) # set font size for plots
Q1 answer#
The kinetic energy is zero and the total energy \(E\) is the same as the potential energy at the turning point, figure 38.
(a) By definition the energy is \(E=\int k(r-r_e)dr\) and because \(k\) and \(r_e\) are constants, these can be taken outside the integral
\[\displaystyle E=k\int rdr -kr_e\int dr=\frac{kr^2}{2}-kr_er+c\]
where the integral \(\int dr = r\). Note that only one constant \(c\) is needed even though there are two integrals.
To solve the specific problem, the definite integral with limits \(1.146r_e\) and \(r_e\) (in nm) is
\[\displaystyle E=k\int_{r_e}^{1.146r_e}(r-r_e)dr=kr_e\left[ \frac{r^2}{2}-r\right]_{r_e}^{1.146r_e} = 0.0107kr_e^2\]
Figure 38. Potential energy (blue) and kinetic energy (red) of a diatomic molecule as a vibrating harmonic oscillator. The turning points,where the kinetic energy is zero are marked with black dots.n
The total energy is \(E\).
(b) Using values for \(k\) and \(r_e\) this energy is \(0.89 \cdot 10^{-19}\) J / molecule or \(53.6\,\mathrm{ kJ \, mole^{-1}}\), which is \(\approx 12\)% of the dissociation energy - a surprisingly
large amount.
(c) The frequency in \(\mathrm{s^{-1}}\) is calculated using \(\displaystyle \nu = \frac{1}{2\pi}\sqrt{ \frac{k}{\mu} } =0.904\cdot 10^{14}\,\mathrm{s^{-1}}\) where \(\mu\) is the reduced mass \(35/
36 \times 1.667 \cdot 10^{-27}\) kg and as \(n = 1\), the total energy of this level is \(0.89 \cdot 10^{-19}\) J which is the same as calculated above.
Q2 answer#
(a) Taking logs of both sides of the Arrhenius equation produces \(\displaystyle \ln(k) = \ln(k_0) - \frac{E_a}{RT}\).
Differentiating with respect to \(T\) gives \(\displaystyle \frac{d\ln(k)}{dT}=\frac{E_a}{RT^2}\) as in the question.
(b) To integrate this differential equation, separate the equation into parts in \(k\) and in \(T\) as
\[\displaystyle \int_{k_1}^{k_2}d\ln(k) = \int_{T_1}^{T_2} \frac{E_a}{RT^2}dT\]
and integrate both sides separately. Because the term \(d\ln(k)\) integrates to \(\ln(k)\) the result is
\[\displaystyle \ln(k_2)-\ln(k_1)=-\frac{E_a}{RT}\bigg|_{T_1}^{T_2}\]
This can be simplified to \(\displaystyle \ln\left( \frac{k_2}{k_1} \right) = \frac{E_a}{R}\left(\frac{1}{T_1}-\frac{1}{T_2} \right)\) which can be further rearranged into the equation in the
Q3 answer#
Rearranging to separate variables produces \(\displaystyle \int\frac{dc}{c}=-k\int dt\).
Integrating both sides separately gives \(\ln(c)=-kt+q\) where \(q\) is a constant.
The constant must be included and is undefined until the problem is specified exactly. Because the concentration is \(c_0\) at \(t = 0\) the calculation can be continued in two ways.
First taking the result \(\ln(c) = -kt + q\), when \(t = 0,\, c = c_0\) then \(q = \ln(c_0)\) and therefore \(\ln(c) = -kt + \ln(c_0)\) or
and secondly by adding the limits in the integration initially as
\[\displaystyle \int_{c_0}^c\frac{dc}{c}=-k\int_0^t dt, \qquad \to \qquad\ln(c)\bigg|_{c_0}^c=-kt\bigg|_0^t\]
which produces the same result.
Q4 answer#
(a) Rearranging the equation and adding limits gives \(\displaystyle \int_{v_0}^v\frac{dv}{v}=-\frac{3\pi\delta\eta}{m}\int_0^t dt\)
and integrating gives
\[\displaystyle \ln(v)\bigg|_{v_0}^v=-\frac{3\pi\delta\eta}{m}t\bigg|_0^t\]
which can be rearranged to give \(v=v_0\exp\left(-\frac{3\pi\delta\eta }{m}t\right)\).
The exponential must be dimensionless therefore the ratio, \(\displaystyle \tau = \frac{m}{3\pi\delta\eta}\) represents a time. To prove that this is a time, add dimensions to the constants.
Viscosity is usually measured in centipoise which is not an SI unit. The SI units of viscosity are \(\mathrm{ kg\, m^{-1} \,s^{-1}}\) and \(1 cP = 0.001\,\mathrm{ kg\, m^{-1}\, s^{-1} }\). The
dimensions are then
\[\displaystyle \frac{m(\mathrm{kg}) }{\delta(\mathrm{m}) \eta(\mathrm{ kg\, m^{-1}\, s^{-1} }) }\equiv s\]
(b) Estimating the time means knowing the mass and size of the particles. Benzene and similar molecules may be estimated to be \(\approx 0.3\) nm in diameter, and of mass \(\approx 100\) amu where \
(1\, \mathrm{amu} = 1.66 \cdot 10^{-27}\) kg. The viscosity of water is 1 cP. The relaxation time for benzene is therefore approximately
\[\displaystyle \frac{100\times 1.66\cdot 10^{-27}}{3\pi \times 0.3\cdot 10^{-9} \times 0.001}\approx 6\cdot 10^{-14} \text{s}\]
or 60 fs, and this must be about the time between intermolecular collisions.
The calculation for the protein is just as straightforward, except that the mass of the protein is not known. One way round this is to use a typical density \(\rho\) and \(100 \,\mathrm{kg\, m^{-3}}
\) is typical, the mass is \( \rho\delta^3\). The time is therefore
\[\displaystyle \tau =\frac{\rho\delta^3 }{3\pi\eta}\approx 10^{-12}\text{s}\]
which means that the direction of the initial movement is lost within about a picosecond, which is a very short time considering the relatively large size of a protein.
Q5 answer#
(a) In this case,the answer can be guessed to be something like \(\ln(3x - 2)\) because of the reciprocal in \(x\) in the function. Differentiating this guess gives \(3/(3x-2) \) and therefore \(\
displaystyle \int\frac{dx}{3x-2}=\frac{1}{3}\ln(3x-2)\).
(b) Converting to exponentials gives
\[\displaystyle I=\int\cosh^2(x)dx=\frac{1}{4}\int\left(e^{2x}+2+e^{-2x}\right)dx =\frac{1}{8}\left(e^{2x}+4x-e^{-2x} \right)+c\]
With some manipulation this can be returned to a trig form which is \(x/2 + \sinh(2x)/4\).
(c) This can be converted to exponentials first giving \(I=\cosh^2(x)/2\)
(d) Expanding the integral gives \(\displaystyle I=\int 2\ln(2x)-\ln(1-x)dx\) which has a standard form, see 2.13.
(e) The function in the integral is odd so the integral is zero because the limits are symmetrical about zero.
(f) The integral becomes a standard one the type \(x^{n+1}/(n+1)\) by letting \(u-x-3\). Thus this is an example of using substitution to simplify and then solve the integral.
\[\displaystyle I=\int u^{1/2}du =\frac{2}{3}u^{3/2} \to \frac{2}{3}(x-3)^{3/2}\]
Adding limits produces
\[\displaystyle I=\frac{2}{3}(x-3)^{3/2}\bigg|_{-1}^1=\frac{2}{3}\left((-2)^{3/2}-(-4)^{3/2}\right)=\frac{4i}{3}\left(\sqrt{2}-4\right)\]
Q6 answer#
Converting to the exponential form gives
\[\displaystyle \int_0^L \frac{1}{4}\left( 2-e^{2iLx}-e^{-2iLx} \right)dx = \frac{L}{2} -\frac{\sin(2L^2)}{4L} \]
The sine cannot be greater than \(\pm 1\) so that when \(L\) is large, for example \(\gt 100\), this term becomes small because it is divided by \(L\) and the limit approaches \(L/2\).
Q7 answer#
Integrating to obtain the velocity gives
\[\begin{split}\displaystyle \boldsymbol v &=\int_0^t (2\sin(\omega_0 t)\boldsymbol{i}+\cos(\omega_0 t)\boldsymbol{j}+t\boldsymbol{k})\; dt = \left( -2\frac{\cos(\omega_0 t)}{\omega_0}\boldsymbol{i}+
\frac{\sin(\omega_0 t)}{\omega_0}\boldsymbol{j} +\frac{t^2}{2} \boldsymbol{k}\right)\Bigg|_0^t \\ &= \frac{2-2\cos(\omega_0 t)}{\omega_0}\boldsymbol{i}+\frac{\sin(\omega_0 t)}{\omega_0}\boldsymbol{j}
+\frac{t^2}{2}\boldsymbol{k} \end{split}\]
As a check at t = 0, the particle is stationary and the vector must be zero which it is; \(\boldsymbol v = 0\boldsymbol i + 0\boldsymbol j + 0\boldsymbol k\). The displacement vector \(\boldsymbol r
\) is obtained by a further integration since \(\boldsymbol v = d\boldsymbol r/dt\) or \(\boldsymbol r=\int\boldsymbol v dt\). Integrating the last results gives
\[\displaystyle \boldsymbol{r} = \frac{2}{\omega_0^2}[\omega_0 t-\sin(\omega_0 t)]\boldsymbol{i}-\frac{1}{\omega_0^2}[\cos(\omega_0 t)-1]\boldsymbol{j}+\frac{t^3}{6}\boldsymbol{k}\]
and at \(t=0\), \(\boldsymbol r = 0\boldsymbol i + 0\boldsymbol j + 0\boldsymbol k\).
Q8 answer#
Because of the term in \(1/V\) this is a logarithmic integration;
\[\displaystyle \int_{V_1}^{kV_1} pdV=\int_{V_1}^{kV_1}\frac{nRT}{V}dV=nRT\ln(V)\bigg|_{V_1}^{kV_1}=nRT\ln(k)\]
and in the last step \(\ln(kV_1)-\ln(V_1)=\ln(kV_1/V_1)\) was used.
Q9 answer#
(a) If \(V\) is the initial volume which is reduced to 10%, then the work done is calculated using equation 6 (\(\displaystyle \int x^n dx\)) with power \(n=-\gamma\).
\[\displaystyle w=\int_V^{V/10}pdV=-\int_V^{V/10}\frac{k}{V^\gamma}dV=-\frac{kV^{1-\gamma}}{1-\gamma}\bigg|_V^{V/10}=\frac{kV^{1-\gamma}}{1-\gamma}(1-10^{\gamma-1})\]
Substituting values in \(pV^\gamma = k\) makes \(k = 101325 \times 5.0^{1.404} = 9.707 \cdot 10^5\) J and therefore \(w = +1.96 \cdot 10^6\) J.
(b) The first law states that the work done on the gas plus the heat transferred to the gas must be equal to the change in internal energy of the gas, \(dU\) where \(dU = \bar dq + \bar dw\). As no
heat enters or leaves, the change in heat \(\bar dq = 0\), and the work done is the same as the change in internal energy. This depends only on the temperature and at constant volume \(dU = C_V dT\)
therefore, for \(n\) moles of gas, the work done is
\[w=n\int_{T_1}^{T_2}C_V dT =\frac{5nR}{2}\int_{T_1}^{T_2}dT=\frac{5nR}{2}(T_2-300)\]
Because the volume and pressure are known, \(n = 203.12\) moles and solving for \(T_2\) produces \(T_2 = 756\) K which is the temperature of the gas immediately after compression.
The bar notation \(\bar dq\) and \(\bar dw\) means that in the language of thermodynamics when integrated, they form path integrals, i.e. normal integrals to you or me, and their value depends on the
way the heat changes or the work is done. Some authors instead use \(\delta q\) and \(\delta w\) to indicate a path integral. The internal energy \(U\) is a state function and when integrated, its
value depends only on the starting and ending values not on how the internal energy was obtained by the molecules for example \(\displaystyle \int_{U_1}^{U_2} dU=U_2-U_1\)
Q10 answer#
(a) If \(V_0\) is the initial, \(V_1\) the final volume then the work done is
\[\displaystyle w=-\int_{V_0}^{V_1}pdV\quad\text{ where }\quad \displaystyle p=\frac{nRT}{V-nb}-\frac{an^2}{V^2}\]
Integrating gives
\[\displaystyle w=-\int_{V_0}^{V_1} \frac{nRT}{V-nb}-\frac{an^2}{V} dV=-nRT\ln\left(\frac{V_1-nb}{V_0-nb} \right) +an^2\left( \frac{1}{V_0}-\frac{1}{V_1} \right)\]
(b) The units of the second term need some care; they are
\[\displaystyle a \,(\mathrm{ bar \,dm^6 \,mol^{-2})\,n\,(\,mol^2)/V(\,dm^3) = bar\, dm^3} = 100 \text{ J}\]
Since pressure is force/area and \(1 \mathrm{ bar = 10^5}\) Pa, multiplying by \(1 \times \mathrm{ m^3}\) would convert this into \(10^5\) joules; (energy = force \(\times\) distance) but the volume
is in dm\(^3\) or \(10^{-3} \mathrm{m^3}\) so the conversion is 100. Substituting the constants gives the work as 18.3 kJ. The ideal gas by comparison needs more work at 22.4 kJ to compress it.
(c) If the gas were O\(_2\) then the van der Waals equation suggests that \(21.9\) kJ are needed and if H\(_2\) then \(23.6\) kJ.
The energy need to compress the Cl\(_2\) is less than that for ideal gas, that for O\(_2\) about the same, and H\(_2\) slightly greater. The difference is due to the interplay of the attractive
potential term \(a\) and the repulsive one \(b\). If the attractive forces dominate as the gas is compressed the molecules will remain closer to one another for longer than if they were hard spheres.
This reduces the effective pressure and so less work is needed. This is the case for the polarizable chlorine molecules. If repulsion dominates,then the molecules avoid one another effectively
increasing the pressure and this is the case for H\(_2\), which is weakly polarizable, compared to Cl\(_2\). The energy to compress is larger. Oxygen molecules seem to have a balance of repulsion and
attraction that makes them appear to behave as it they were ideal, but this is an accidental cancelling of two effects.
Q11 answer#
(a) The enthalpy change is
\[\displaystyle H_T^\mathrm{o}-H_{298}^\mathrm{o}=\int_{298}^T a+bT+\frac{c}{T^2}dT=a(T-298)+\frac{b}{2}\left(T^2-298^2\right)-c\left(\frac{1}{T}-\frac{1}{298} \right)\]
and entropy change
\[S_T^\mathrm{o}-S_{298}^\mathrm{o}=\int_{298}^T \frac{a}{T}+b+\frac{c}{T^3}dT=a\ln\left(\frac{T}{298}\right)+b(T-298)-\frac{c}{2}\left(\frac{1}{T^2}-\frac{1}{298^2} \right)\]
(b) Using the values for the constants produce \(H_T^\mathrm{o} - H_{298}^\mathrm{o} = 2.95\) kJ/mol and \(S_T^\mathrm{o} - S_{298}^\mathrm{o} = 9.14\) J/mol/K.
Q12 answer#
(a) The Clapeyron equation for a change of phase such as melting or evaporation is
\[\displaystyle \frac{dp}{dT}=\frac{\Delta S}{\Delta V}\]
Because \( \Delta S=\Delta H/T\) for the phase change, \(\displaystyle \frac{dp}{dT}=\frac{\Delta H}{T\Delta V}\).
For a liquid to vapour transition this is written as \(\displaystyle \frac{dp}{dT}=\frac{\Delta_{vap} H}{T\Delta_{vap} V}\). Separating terms in pressure and temperature produces
\[\displaystyle \int_{p_1}^{p_2}dp= \frac{\Delta H}{\Delta V}\int_{T_1}^{T_2} \frac{dT}{T}\]
and therefore
\[\displaystyle p_2-p_1=\frac{\Delta H}{\Delta V}\ln\left( \frac{T_2}{T_1} \right)\]
provided \(\Delta V\) remains constant with a change in pressure and temperature. This is the Clapeyron equation.
(b) The Clausius-Clapeyron equation is obtained by allowing the change in volume (per mole) on forming the vapour, to be far larger than that of the original liquid making \(\Delta_{vap}V \approx V_
{vap}\) and therefore
\[\Delta_{vap}V=RT/p\quad\text{ and }\quad\displaystyle \frac{dp}{dT}=p\frac{\Delta_{vap}H}{RT^2}\]
This is the Clausius-Clapyron equation equation that is usually written as
\[\displaystyle \int_{p_1}^{p_2}d\ln(p)= \frac{\Delta_{vap} H}{R}\int_{T_1}^{T_2} \frac{dT}{T^2}\]
produces \(\displaystyle \ln\left(\frac{p_2}{p_1} \right) = -\frac{\Delta_{vap}H}{R}\left( \frac{1}{T_2}-\frac{1}{T_1} \right) \)
Q13 answer#
The Clapeyron equation
\[\displaystyle p_2-p_1=\frac{\Delta H}{\Delta V}\ln\left( \frac{T_2}{T_1} \right)\]
is used for a solid-liquid transition. The changes in enthalpy and volume relate therefore to changes occurring in fusion.
The Clausius-Clapeyron equation describes solid - vapour and liquid - vapour changes because the final volume is far greater than the initial one, and is
\[\displaystyle \frac{dp}{dT}=p\frac{\Delta H}{RT^2}\]
where \(\Delta H\) the enthalpy change at the liquid - vapour or sublimation transition. Integrating this last equation from pressure p1 to p2 and temperature \(T_1 \to T_2\) gives
\[\displaystyle \ln\left(\frac{p_2}{p_1} \right) = -\frac{\Delta_{vap}H}{R}\left( \frac{1}{T_2}-\frac{1}{T_1} \right) \]
as discovered in the previous question. The change in the volume during fusion is
\[\displaystyle \Delta_{fus}V = m\left(\frac{1}{d_l}-\frac{1}{d_s} \right)\]
where \(m\) is the molar mass and \(d_l\) and \(d_s\) the densities of the liquid and solid. The pressure variation for the solid to liquid (melting or fusion) change is
\[\displaystyle p_2=p_1+\frac{\Delta_{fus}H}{\Delta_{fus}V}\ln\left(\frac{T_2}{T_1} \right)\]
and for evaporation and sublimation
\[\displaystyle p_2=p_1\exp\left( -\frac{\Delta_{vap}H}{R}\left(\frac{1}{T_2}-\frac{1}{T_1} \right) \right)\]
with the appropriate \(\Delta H\). This is \(\Delta_{vap}H\) for evaporation and \(\Delta_{fus}H + \Delta_{vap}H\) for sublimation. Sublimation is treated as two steps merged into one; melting and
instantaneous evaporation. One way of calculating the phase diagram is shown below.
# Algorithm 1. Solid-liquid-gas Phase Diagram. Data for benzene
R = 8.314 # J/mol/K
dens_sol = 981.0 # kg/m^3
dens_liq = 879.0 # kg/m^3
mol_mass = 78.0/1000.0 # kg/mol
DH_vap = 30.8*1000 # J/mol
DH_fus = 10.6*1000 # J/mol
p3 = 36.0/760*101325 # triple point pressure Pa
T3 = 5.5 + 273.16 # triple point temperture K
DV_fus = mol_mass*(1/dens_liq-1/dens_sol) # delta volume fusion
p_liq_vap= lambda T: p3*np.exp( (DH_vap/R)*(1/T3-1/T)) # pressure
p_sol_vap= lambda T: p3*np.exp( ((DH_fus+DH_vap)/R )*(1/T3-1/T) )
p_sol_liq= lambda T: p3 + DH_fus/DV_fus*(np.log(T)-np.log(T3))
# plot each function vs temperature. Use limits to restrict range and produce the figure.
Figure 39. Calculated phase diagram for benzene using the Clapeyron and Clausius-Clapeyron equations. Pressure is in pascal, temperature in kelvin.
Notice the form of the phase diagram. At low temperatures \(\approx 260\) K and pressures \(\approx 1000\) Pa, only benzene vapour exists. As the pressure is increased, a vertical movement in the
diagram, the vapour condenses to the solid phase. Not until the temperature reaches \(\approx 280\) K is there enough energy for the vapour to form the liquid phase when the pressure is increased.
Below this temperature, the solid sublimes. Notice also how vertical the phase transition line between solid and liquid is. In many textbooks, it is misrepresented as a somewhat sloping line. The
solid-liquid boundary will still appear as an almost vertical line if the graph is plotted with the log of the pressure up to \(10^6\) Pa.
Exercise: Now that you can produce any gas-liquid-solid phase diagram, calculate these for water and carbon dioxide using the data given below or choose some molecules of your own.
Some data is in the table
\[\begin{split}\displaystyle \begin{array}{lll} \hline &\mathrm{water} & \mathrm{carbon \,dioxide} & & \\ \hline \mathrm{molec \,weight, g/mol} & 18.01528 & 44.012 \\ \mathrm{Triple \,point, K} &
273.16 & 216.55 \\ \mathrm{Triple \,point}& 611.29 \,\mathrm{ Pa} & 5.185\,\mathrm{ bar}\\ \mathrm{H_{fus},\, kJ/mol} & 6.01 & 196.10/44.012 &\text{at triple point} \\ \mathrm{H_{vap},\, kJ/mol} &
43.990 & 571.08/44.012&\text{at triple point}\\ \mathrm{liquid \,density, g/cm^3 } & 0.99978 & 1.562\\ \mathrm{solid\, density, g/cm^3 } & 0.917 & 1.032 \\ \hline \end{array}\end{split}\]
Q14 answer#
(a) \(\displaystyle \left( \frac{\partial q}{\partial p} \right)_S =V\) integrates, at constant entropy, to become
\[\displaystyle \int_{q_0}^{q}dq=\int_{p_0}^pVdp=\int_{p_0}^p\left( \frac{C}{p} \right)^{1/\gamma}dp\]
This integration is of the \(x^{n+1}/(n + 1)\) form, equation 6, therefore,
\[\begin{split}\displaystyle q-q_0 =&\int_{p_0}^p\left( \frac{C}{p} \right)^{1/\gamma}dp \\ &=\frac{C^{1/\gamma} p^{1-1/\gamma}}{1-1/\gamma} \bigg|_{p_0}^p \\&=\frac{\gamma}{\gamma-1}C^{1/\gamma}\
left( p^{(\gamma-1)/\gamma}-p_0^{(\gamma-1)/\gamma} \right) \end{split}\]
Dividing by \(p_0\) and converting the change in enthalpy into velocity via \(m.u^2/2=q_0 -q\) gives,
\[\displaystyle -m.u^2=\frac{2\gamma}{\gamma-1}C^{1/\gamma} p_0^{(\gamma -1)/\gamma} \left( \left(\frac{p}{p_0} \right)^{(\gamma-1)/\gamma}-1 \right)\]
and finally substituting \(p_0V_0^\gamma = C\) and rearranging produces
\[\displaystyle u=\sqrt{\frac{2\gamma}{\gamma-1}\frac{p_0V_0}{m} \left( 1-\left(\frac{p}{p_0} \right)^{(\gamma-1)/\gamma} \right) }\]
which is the equation describing the speed of a gas in a rocket or jet engine at pressure \(p\). This equation can also be related to \(T_0\), the inlet temperature, via \(p_0V_0 = RT_0\).
(b) The volume of gas \(V\) is given by \(\displaystyle V=\left(\frac{p_0}{p}\right)^{1/\gamma}\) and because \(\sigma m .u=V\mu\) the gas and hence nozzle cross section \(\sigma\) varies as
\[\displaystyle \sigma= \frac{\mu}{m}\left(\frac{p_0}{p}\right)^{1/\gamma}V_0 \left( \frac{2\gamma}{\gamma-1}\frac{p_0V_0}{m} \left( 1-\left(\frac{p}{p_0} \right)^{(\gamma-1)/\gamma} \right) \right)^
The equation \(\sigma m .u = V\mu\) follows because the mass entering / sec must be the same as that leaving and this is why the gas speeds up at the throat. (\( \mu \) is the mass flow rate in kg
sec\(^{-1}\)). The cross section vs reduced pressure \(p/p_0\) is shown in figure 40 with all the constants set to one except \(\gamma\) which is 1.22.
Figure 40 Left: Universal calculated shape of gas cross-section in a jet or rocket (Laval) nozzle, vs reduced pressure \(p/p_0\). Right: Gas velocity relative to the speed of sound us. Note that the
inlet side is on the right in both figures where the pressure is high. The dashed lines show the position of the minimum nozzle width which is where the gas is at Mach 1.
(c) The minimum nozzle cross section vs pressure is the derivative of \(\sigma\) vs \(p\). This is not difficult to evaluate but messy and is easily performed by Sympy. The constants need not be
included because the derivative is set to zero at the minimum and they will cancel out. (note g is used instead of \(\gamma\) )
p0, p, g = symbols('p0, p, g')
eq = (p0/p)**(1/g)/sqrt( 1 - (p/p0)**((g - 1)/g) )
simplify(diff(eq,p) )
This equation must be zero at the minimum nozzle diameter and can be solved by factoring the bracket in the numerator to give the pressure ratio in the throat of
\[\displaystyle \frac{p}{p_0}=\left( \frac{2}{\gamma+1} \right)^{\gamma/(\gamma-1)}\]
which, somewhat surprisingly, depends only on the ratio of heat capacities \(\gamma\). The gas velocity in the throat is obtained by substituting this pressure into the equation for the speed, giving
\[\displaystyle U_s=\sqrt{\frac{2\gamma}{\gamma+1}\frac{RT_0}{m}}\]
which is the speed of sound in the gas under the prevailing conditions.
It is interesting to note, and was pointed out by Reynolds a long time ago (1886), that the rate of gas discharge depends on the cross-sectional area and not on the backing pressure. The gas cannot
move faster than the speed of sound through the nozzle, so increasing the input pressure has no effect once the gas is moving at this speed. The gas velocity is given by
\[\displaystyle u=u_s\sqrt{\frac{\gamma+1}{\gamma-1}\left( 1-\left( \frac{p}{p_0}\right)^{(\gamma-1)/\gamma} \right)}\]
which is shown in the figure.
The gas first reaches supersonic speed Mach 1 at the throttle point, called ‘choked flow’, and increases thereafter as it expands into a vacuum. In a real molecular beam experiment in the lab, and
presumably for a rocket, the exhaust gas is slowed down by the residual background gas present but, if the pressure is low, this occurs only at some distance from the nozzle; this stationary
shock-wave (relative to nozzle) is called the Mach disc.
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2. Time-Saving: Quick calculations allow for faster decision-making and more efficient project planning.
3. Cost Efficiency: Accurate material estimates help prevent over-ordering or under-ordering, potentially saving money and reducing waste.
4. Versatility: Most calculators can handle various project sizes, from small residential driveways to large commercial parking lots.
5. User-Friendly: With simple input requirements, even those with limited technical knowledge can operate an asphalt calculator effectively.
How to Use our Asphalt Calculator
Using an asphalt calculator is straightforward. Simply enter the width, length, and thickness of your paving area. The calculator will estimate the approximate tonnage of asphalt needed for your job.
Here’s a more detailed guide:
1. Measure the Length: Determine the length of the area to be paved in meters or feet.
2. Measure the Width: Calculate the width of the paving area using the same unit of measurement as the length.
3. Determine the Thickness: Decide on the desired thickness of the asphalt layer in inches or centimeters. This may vary depending on the project requirements and local regulations.
4. Input the Data: Enter these measurements into the appropriate fields in the asphalt calculator.
5. Calculate: Click the “Calculate” button to generate your estimate.
6. Review Results: The calculator will provide the estimated tonnage of asphalt required for your project.
Remember, while the asphalt calculator helps you estimate the approximate tonnage, it’s always wise to consult with a professional for final quotes and material orders.
Factors Affecting Asphalt Tonnage Calculations
While an asphalt calculator provides a good baseline estimate, several factors can influence the actual amount of material needed:
1. Compaction: The degree of compaction can affect the final thickness and density of the asphalt layer.
2. Waste Factor: It’s common to add a small percentage (usually 5-10%) to account for waste and spillage during installation.
3. Surface Irregularities: Uneven base surfaces may require additional material to achieve a level finish.
4. Mix Design: Different asphalt mixes have varying densities, which can affect the tonnage required.
5. Project Specifications: Specific requirements for load-bearing capacity or durability may necessitate adjustments to standard calculations.
Applications of the Asphalt Calculator
The asphalt calculator is a versatile tool with applications across various project types:
1. Road Construction: For estimating material needs in new road construction or resurfacing projects.
2. Parking Lot Paving: To calculate asphalt requirements for commercial or residential parking areas.
3. Driveway Installation: Homeowners and contractors can use it for accurate estimates on residential driveway projects.
4. Airport Runway Maintenance: For large-scale asphalt paving jobs in aviation facilities.
5. Recreational Facilities: To estimate material needs for tennis courts, basketball courts, or running tracks.
Tips for Accurate Asphalt Estimation
To get the most out of your asphalt calculator, consider these tips:
1. Double-Check Measurements: Accurate input is crucial for reliable output. Measure your project area carefully.
2. Consider the Base: Ensure your base is properly prepared and compacted before calculating asphalt needs.
3. Account for Edges: Don’t forget to include additional material for edges and transitions.
4. Consult Local Standards: Be aware of local regulations regarding asphalt thickness for different applications.
5. Factor in Compaction: Remember that the asphalt will compact during installation, potentially affecting your final thickness.
Limitations of Asphalt Calculators
While asphalt calculators are incredibly useful, it’s important to understand their limitations:
1. Simplified Calculations: They may not account for all variables present in real-world scenarios.
2. Standard Density Assumptions: Calculators often use a standard asphalt density, which may vary slightly from your specific mix.
3. No Consideration for Site Conditions: Factors like soil type, drainage, and climate are not typically factored into calculator estimates.
4. Lack of Professional Judgment: Calculators can’t replace the expertise of experienced professionals who can assess unique project needs.
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Morphing planar graph drawings with a polynomial number of steps
In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between
the two drawings so that the drawing remains straight-line planar at all times. Cairns's original proof required exponentially many morphing steps. We prove that there is a morph that consists of O(n
^2) steps, where each step is a linear morph that moves each vertex at constant speed along a straight line. Using a known result on compatible triangulations this implies that for a general planar
graph G and any two straight-line planar drawings of G with the same embedding, there is a morph between the two drawings that preserves straight-line planarity and consists of O(n^4) steps.
Publication series
Name Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Other 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
Country/Territory United States
City New Orleans, LA
Period 1/6/13 → 1/8/13
All Science Journal Classification (ASJC) codes
• Software
• General Mathematics
Dive into the research topics of 'Morphing planar graph drawings with a polynomial number of steps'. Together they form a unique fingerprint.
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[OS X TeX] psfrag troubles/PostScript in 10.4.4
Ross Moore ross at ics.mq.edu.au
Fri Jan 13 00:15:54 CET 2006
Hi Tom, Bruno
and others following this thread.
On 13/01/2006, at 6:57 AM, Thomas Schröder wrote:
>> Thus it seems using single-letter strings is the only way to use
>> psfrag reliably, which limits its usability severely. For this
>> reason I've given up on it. Have you tried WARMreader?
> I don't have Illustrator, so. PLus, I had psfrag working on the
> other computer.
WARMreader doesn't need the full functionality of Illustrator.
It just needs a way to mark positions in an image, and allow
one or more strings to be associated with each marked position.
This information then needs to be saved into a .bb file, using
a well-defined format, analagous to what Illustrator's MO plugin
An interested clever Mac programmer should be able to write an
application that does this. One feature would be to add visual
markers in a layer above the image, so that it is clear where
points have been marked already.
This layer does *not* get added to the graphic itself.
It's this latter point that makes WARMreader superior to the
psfrag approach, and allows it to work with arbitrary
graphics, not just .ps or .eps files.
Besides, the first front-end for WARMreader was a little
MacOS 7 application, named Zephyr, which did just what
was described above --- see the URL below --- as well as
many other (unrelated) things.
>> And yes, the absence of EPS export in KaleidaGraph is really
>> annoying, especially after all its years of existence and its
>> successive updates.
Any difficulties of this kind are quite irrelevant with
the WARMreader approach to labeling graphics.
> But there's no pdffrag to enhance PDF files.
No, and there cannot be --- at least not easily.
This is because PDF has indexed tagging of byte-positions.
Anything that worked like psfrag would have to parse
a good portion of the PDF so as to adjust all these byte-counts
after inserting your labels.
And that is after having decompressed streams, add the labels,
then recompress.
>> By contrast, printing to a PS or PDF file from KaleidaGraph
>> creates a PS or PDF graphic with the bounding box of a full A4 or
>> Letter page (depending on your setup), and is then inadequate for
>> a figure to be included in a LaTeX document.
> Oh, I set up a new paper size with which I create my graphs, so no
> problem there. It's basically A6 landscape but the height is
> smaller by 2cm or, so I can fit two graphs on one page. They look
> really nice, actually :-)
Yes. Keeping to a fixed size is a good idea when writing a book
or series of books. However, sometimes you may want to use
the same images in a different setting;
e.g. at a different size for a presentation.
With WARMreader, you can use the same underlying graphic
and .bb file. Just change the size at which the image
is shown with \includegraphics[scale=...]{....} .
Then use \Large, \Huge, etc. with the labels.
> Bye, Thomas
Hope this helps,
and maybe it will inspire someone to write a simple utility
to make WARMreader more accessible, without Illustrator.
Ross Moore ross at maths.mq.edu.au
Mathematics Department office: E7A-419
Macquarie University tel: +61 +2 9850 8955
Sydney, Australia 2109 fax: +61 +2 9850 8114
------------------------- Info --------------------------
Mac-TeX Website: http://www.esm.psu.edu/mac-tex/
& FAQ: http://latex.yauh.de/faq/
TeX FAQ: http://www.tex.ac.uk/faq
List Archive: http://tug.org/pipermail/macostex-archives/
More information about the macostex-archives mailing list
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Research Guides: MATH 1350 Mathematics for Teachers I: Library Resources
This resource guide is meant to help students enrolled in MATH 1350 classes find additional resources. All online resources listed here are freely available to any TCC student. If you prefer a print
book, you can check one out with your TCC ID card. If you have questions, please contact Ask a Librarian or your instructor.
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what is the meaning… - QuestionCove
Ask your own question, for FREE!
Mathematics 68 Online
OpenStudy (anonymous):
what is the meaning of improper integral?
Still Need Help?
Join the QuestionCove community and study together with friends!
OpenStudy (anonymous):
improper integrals are integrals that cannot be evaluated like you would evaluate a normal integral. The classic example is \[\int\limits_{1}^{\infty} 1/x^2\]
OpenStudy (anonymous):
instead you have to evaluate the improper integral by replacing the infinity with a variable, doing the integral, and then taking the limit of the result as the variable goes to infinity
OpenStudy (anonymous):
Note: not all improper integrals have an answer. Most improper integrals go to infinity. This makes sense when you think about how an integral is the sum of the area under the curve. If you add up
the area under the curve over the distance to infinity, the curve has to approach zero.
OpenStudy (anonymous):
You mean the improper integral is the sum of the area the curve, therefore..the limit is approaches to zero.doing the integral, the limits became infinity?? can you give me an example that the limit
is infinity??
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Latest Questions
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!
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eet to Arpent
Feet to Arpent Converter
β Switch toArpent to Feet Converter
How to use this Feet to Arpent Converter π €
Follow these steps to convert given length from the units of Feet to the units of Arpent.
1. Enter the input Feet value in the text field.
2. The calculator converts the given Feet into Arpent in realtime β using the conversion formula, and displays under the Arpent label. You do not need to click any button. If the input changes,
Arpent value is re-calculated, just like that.
3. You may copy the resulting Arpent value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Feet to Arpent?
The formula to convert given length from Feet to Arpent is:
Length[(Arpent)] = Length[(Feet)] / 191.99999984784384
Substitute the given value of length in feet, i.e., Length[(Feet)] in the above formula and simplify the right-hand side value. The resulting value is the length in arpent, i.e., Length[(Arpent)].
Calculation will be done after you enter a valid input.
Consider that a luxury yacht has a beam width of 60 feet.
Convert this width from feet to Arpent.
The length in feet is:
Length[(Feet)] = 60
The formula to convert length from feet to arpent is:
Length[(Arpent)] = Length[(Feet)] / 191.99999984784384
Substitute given weight Length[(Feet)] = 60 in the above formula.
Length[(Arpent)] = 60 / 191.99999984784384
Length[(Arpent)] = 0.3125
Final Answer:
Therefore, 60 ft is equal to 0.3125 arpent.
The length is 0.3125 arpent, in arpent.
Consider that a skyscraper's floor-to-ceiling height is 15 feet.
Convert this height from feet to Arpent.
The length in feet is:
Length[(Feet)] = 15
The formula to convert length from feet to arpent is:
Length[(Arpent)] = Length[(Feet)] / 191.99999984784384
Substitute given weight Length[(Feet)] = 15 in the above formula.
Length[(Arpent)] = 15 / 191.99999984784384
Length[(Arpent)] = 0.0781250000619125
Final Answer:
Therefore, 15 ft is equal to 0.0781250000619125 arpent.
The length is 0.0781250000619125 arpent, in arpent.
Feet to Arpent Conversion Table
The following table gives some of the most used conversions from Feet to Arpent.
Feet (ft) Arpent (arpent)
0 ft 0 arpent
1 ft 0.00520833334 arpent
2 ft 0.01041666667 arpent
3 ft 0.01562500001 arpent
4 ft 0.02083333335 arpent
5 ft 0.02604166669 arpent
6 ft 0.03125000002 arpent
7 ft 0.03645833336 arpent
8 ft 0.0416666667 arpent
9 ft 0.04687500004 arpent
10 ft 0.05208333337 arpent
20 ft 0.1042 arpent
50 ft 0.2604 arpent
100 ft 0.5208 arpent
1000 ft 5.2083 arpent
10000 ft 52.0833 arpent
100000 ft 520.8333 arpent
A foot (symbol: ft) is a unit of length used in the United States, the United Kingdom, and Canada. One foot is equal to 0.3048 meters.
The foot originated from various units used in ancient civilizations. Its current definition is based on the international agreement of 1959, which standardized it to exactly 0.3048 meters.
Feet are commonly used to measure height, length, and short distances. Despite the global shift to the metric system, the foot remains in use in these countries.
An arpent is a historical unit of length used primarily in French-speaking regions and in land measurement. One arpent is approximately equivalent to 192.75 feet or 58.66 meters.
The arpent was used in various regions, including France and the former French colonies, to measure land and property. Its length could vary slightly depending on the specific region and historical
Arpents were used in land surveying and agriculture, particularly in historical and regional contexts. Although less common today, the unit provides historical insight into land measurement practices
and regional variations in measurement standards.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Feet to Arpent in Length?
The formula to convert Feet to Arpent in Length is:
Feet / 191.99999984784384
2. Is this tool free or paid?
This Length conversion tool, which converts Feet to Arpent, is completely free to use.
3. How do I convert Length from Feet to Arpent?
To convert Length from Feet to Arpent, you can use the following formula:
Feet / 191.99999984784384
For example, if you have a value in Feet, you substitute that value in place of Feet in the above formula, and solve the mathematical expression to get the equivalent value in Arpent.
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You are here
Date Issued:
The area of focus for this research is the Stochastic Resource Constrained Project Scheduling Problem (SRCPSP) with Stochastic Task Insertion (STI). The STI problem is a specific form of the
SRCPSP, which may be considered to be a cross between two types of problems in the general form: the Stochastic Project Scheduling Problem, and the Resource Constrained Project Scheduling
Problem. The stochastic nature of this problem is in the occurrence/non-occurrence of tasks with deterministic duration. Researchers Selim (2002) and Grey (2007) laid the groundwork for the
research on this problem. Selim (2002) developed a set of robustness metrics and used these to evaluate two initial baseline (predictive) scheduling techniques, optimistic (0% buffer) and
pessimistic (100% buffer), where none or all of the stochastic tasks were scheduled, respectively. Grey (2007) expanded the research by developing a new partial buffering strategy for the initial
baseline predictive schedule for this problem and found the partial buffering strategy to be superior to Selim's "extreme" buffering approach. The current research continues this work by focusing
on resource aspects of the problem, new buffering approaches, and a new rescheduling method. If resource usage is important to project managers, then a set of metrics that describes changes to
the resource flow would be important to measure between the initial baseline predictive schedule and the final "as-run" schedule. Two new sets of resource metrics were constructed regarding
resource utilization and resource flow. Using these new metrics, as well as the Selim/Grey metrics, a new buffering approach was developed that used resource information to size the buffers. The
resource-sized buffers did not show to have significant improvement over Grey's 50% buffer used as a benchmark. The new resource metrics were used to validate that the 50% buffering strategy is
superior to the 0% or 100% buffering by Selim. Recognizing that partial buffers appear to be the most promising initial baseline development approach for STI problems, and understanding that
experienced project managers may be able to predict stochastic probabilities based on prior projects, the next phase of the research developed a new set of buffering strategies where buffers are
inserted that are proportional to the probability of occurrence. The results of this proportional buffering strategy were very positive, with the majority of the metrics (both robustness and
resource), except for stability metrics, improved by using the proportional buffer. Finally, it was recognized that all research thus far for the SRCPSP with STI focused solely on the development
of predictive schedules. Therefore, the final phase of this research developed a new reactive strategy that tested three different rescheduling points during schedule eventuation when a complete
rescheduling of the latter portion of the schedule would occur. The results of this new reactive technique indicate that rescheduling improves the schedule performance in only a few metrics under
very specific network characteristics (those networks with the least restrictive parameters). This research was conducted with extensive use of Base SAS v9.2 combined with SAS/OR procedures to
solve project networks, solve resource flow problems, and implement reactive scheduling heuristics. Additionally, Base SAS code was paired with Visual Basic for Applications in Excel 2003 to
implement an automated Gantt chart generator that provided visual inspection for validation of the repair heuristics. The results of this research when combined with the results of Selim and Grey
provide strong guidance for project managers regarding how to develop baseline predictive schedules and how to reschedule the project as stochastic tasks (e.g. unplanned work) do or do not occur.
Specifically, the results and recommendations are provided in a summary tabular format that describes the recommended initial baseline development approach if a project manager has a good idea of
the level and location of the stochasticity for the network, highlights two cases where rescheduling during schedule eventuation may be beneficial, and shows when buffering proportional to the
probability of occurrence is recommended, or not recommended, or the cases where the evidence is inconclusive.
Title: STOCHASTIC RESOURCE CONSTRAINED PROJECT SCHEDULING WITH STOCHASTIC TASK INSERTION PROBLEMS.
Archer, Sandra, Author
Name(s): Armacost, Robert, Committee Chair
University of Central Florida, Degree Grantor
Type of text
Date Issued: 2008
Publisher: University of Central Florida
Language(s): English
The area of focus for this research is the Stochastic Resource Constrained Project Scheduling Problem (SRCPSP) with Stochastic Task Insertion (STI). The STI problem is a specific form
of the SRCPSP, which may be considered to be a cross between two types of problems in the general form: the Stochastic Project Scheduling Problem, and the Resource Constrained Project
Scheduling Problem. The stochastic nature of this problem is in the occurrence/non-occurrence of tasks with deterministic duration. Researchers Selim (2002) and Grey (2007) laid the
groundwork for the research on this problem. Selim (2002) developed a set of robustness metrics and used these to evaluate two initial baseline (predictive) scheduling techniques,
optimistic (0% buffer) and pessimistic (100% buffer), where none or all of the stochastic tasks were scheduled, respectively. Grey (2007) expanded the research by developing a new
partial buffering strategy for the initial baseline predictive schedule for this problem and found the partial buffering strategy to be superior to Selim's "extreme" buffering
approach. The current research continues this work by focusing on resource aspects of the problem, new buffering approaches, and a new rescheduling method. If resource usage is
important to project managers, then a set of metrics that describes changes to the resource flow would be important to measure between the initial baseline predictive schedule and the
final "as-run" schedule. Two new sets of resource metrics were constructed regarding resource utilization and resource flow. Using these new metrics, as well as the Selim/Grey metrics,
a new buffering approach was developed that used resource information to size the buffers. The resource-sized buffers did not show to have significant improvement over Grey's 50%
buffer used as a benchmark. The new resource metrics were used to validate that the 50% buffering strategy is superior to the 0% or 100% buffering by Selim. Recognizing that partial
Abstract/ buffers appear to be the most promising initial baseline development approach for STI problems, and understanding that experienced project managers may be able to predict stochastic
Description: probabilities based on prior projects, the next phase of the research developed a new set of buffering strategies where buffers are inserted that are proportional to the probability of
occurrence. The results of this proportional buffering strategy were very positive, with the majority of the metrics (both robustness and resource), except for stability metrics,
improved by using the proportional buffer. Finally, it was recognized that all research thus far for the SRCPSP with STI focused solely on the development of predictive schedules.
Therefore, the final phase of this research developed a new reactive strategy that tested three different rescheduling points during schedule eventuation when a complete rescheduling
of the latter portion of the schedule would occur. The results of this new reactive technique indicate that rescheduling improves the schedule performance in only a few metrics under
very specific network characteristics (those networks with the least restrictive parameters). This research was conducted with extensive use of Base SAS v9.2 combined with SAS/OR
procedures to solve project networks, solve resource flow problems, and implement reactive scheduling heuristics. Additionally, Base SAS code was paired with Visual Basic for
Applications in Excel 2003 to implement an automated Gantt chart generator that provided visual inspection for validation of the repair heuristics. The results of this research when
combined with the results of Selim and Grey provide strong guidance for project managers regarding how to develop baseline predictive schedules and how to reschedule the project as
stochastic tasks (e.g. unplanned work) do or do not occur. Specifically, the results and recommendations are provided in a summary tabular format that describes the recommended initial
baseline development approach if a project manager has a good idea of the level and location of the stochasticity for the network, highlights two cases where rescheduling during
schedule eventuation may be beneficial, and shows when buffering proportional to the probability of occurrence is recommended, or not recommended, or the cases where the evidence is
Identifier: CFE0002491 (IID), ucf:47673 (fedora)
Note(s): Engineering and Computer Science, Department of Industrial Engineering and Management Systems
This record was generated from author submitted information.
stochastic project scheduling
resource constrained
Subject(s): task insertion
project buffers
reactive scheduling
predictive scheduling
Link to This http://purl.flvc.org/ucf/fd/CFE0002491
Restrictions public
on Access:
Host UCF
In Collections
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Clinical Lab Math at Free-Ed.Net
This course was developed to prepare and sustain your mathematical skills as a clinical laboratory technician. The emphasis is upon computations related to solutions and their concentrations.
After completing each set of lesson exercises, compare your answers with those on the solution sheet that follows the exercises. If you have answered an exercise incorrectly, check the reference
cited after the answer on the solution sheet to determine why your response was not the correct one.
Lesson 1. General Mathematics Review.
Lesson 2. Introduction to Solution Mathematics.
Lesson 3. Molar Solutions.
Lesson 4. Equivalent Solutions.
Lesson 5. Conversion of Concentration Units.
Lesson 6. Dilutions.
Lesson 7. Titration.
Lesson 8. Concentrated Acids and Bases.
Lesson 9. pH and Buffers.
Appendix A appears at the bottom of this page.
APPENDIX A REVIEW OF DIMENSIONAL ANALYSIS
1. Read the problem carefully. What is the problem asking for? Be sure the entire problem has been read and understood. This may require you to read the problem two or three times. YOU CANNOT ANSWER
THE PROBLEM IF YOU DO NOT KNOW WHAT IT IS ASKING!
2 Determine exactly what results are to be produced by the calculations.
3. Determine what principles and relationships are involved.
4. Think about possible methods to use in solving the problem.
5 Use the sample problems to help you set up and solve the problem.
6. Based on definition, determine the appropriate factors that allow you to solve for the unknown quantity.
7. Once you have selected the appropriate factors for that specific problem type, write them down on your paper.
8. Units are treated the same as numbers in any mathematical calculations.
9. Write the intermediate stages of the calculations clearly. Avoid writing one number on top of another as a method of correction. Make each digit legible. This will allow you to go back and check
your work later.
10. Mentally estimate an answer before working the problem.
11. Do the mathematics involved and check your work. Do not round off any intermediate calculations. Be extremely careful in positioning the decimal point and make certain the final answer has the
appropriate number of significant figures.
12. Cancel units. The units you have left should be an appropriate unit for what the problem asked. Example: If the problem asked for "How many grams," your final answer should be in grams. If it is
not, go back and check your work. Often, all that is required is a simple metric conversion.
13. Compare the calculated result with your estimated answer. If the two figures disagree drastically, determine which result is wrong.
14. Finally, go back and read the problem again. Did you answer the question correctly and does your answer make sense?
15. Ratio and proportion is consistent with and is the basis of dimensional analysis. 16. Example problems will serve as a reference to the various problem solving techniques.
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Simulation and Optimization of Control of Selected Phases of Gyroplane Flight
^ †
Institute of Aviation, Al. Krakowska 110/114, 02-256 Warszawa, Poland
This paper is an extended version of Stalewski, W. Simulation and Optimization of Control of Selected Phases of Gyroplane Flight. In Proceedings of the 7th International Conference on Experiments/
Process/System Modeling/Simulation/Optimization (7th IC-EpsMsO), Athens, Greece, 5–8 July 2017.
Submission received: 15 December 2017 / Revised: 22 January 2018 / Accepted: 2 February 2018 / Published: 6 February 2018
Optimization methods are increasingly used to solve problems in aeronautical engineering. Typically, optimization methods are utilized in the design of an aircraft airframe or its structure. The
presented study is focused on improvement of aircraft flight control procedures through numerical optimization. The optimization problems concern selected phases of flight of a light gyroplane—a
rotorcraft using an unpowered rotor in autorotation to develop lift and an engine-powered propeller to provide thrust. An original methodology of computational simulation of rotorcraft flight was
developed and implemented. In this approach the aircraft motion equations are solved step-by-step, simultaneously with the solution of the Unsteady Reynolds-Averaged Navier–Stokes equations, which is
conducted to assess aerodynamic forces acting on the aircraft. As a numerical optimization method, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm was adapted. The developed methodology was
applied to optimize the flight control procedures in selected stages of gyroplane flight in direct proximity to the ground, where proper control of the aircraft is critical to ensure flight safety
and performance. The results of conducted computational optimizations proved the qualitative correctness of the developed methodology. The research results can be helpful in the design of
easy-to-control gyroplanes and also in the training of pilots for this type of rotorcraft.
1. Introduction
Optimization methods are widely considered to be very effective tools that can significantly improve the performance of contemporarily designed and constructed aircraft. Typically, optimization
methods are utilized in the design of an aircraft airframe or its structure that may be optimized simultaneously using a multi-disciplinary approach [
]. Fast development of both computational methods and computer hardware offers opportunities to expand the range of applications of optimization methods. As part of this trend, the application of
modern computational methods to optimize aircraft flight control procedures is presented in this paper. The developed methodology for the optimization of flight control procedures is discussed in an
example of the flight of a light gyroplane.
A gyroplane is an aerodyne equipped with unpowered main rotor and engine-powered propeller, generating a thrust force necessary to move the aircraft forward. During gyroplane flight, the air flowing
around the rotating blades of the main rotor generates an aerodynamic reaction whose vertical component balances the aircraft weight, while the aerodynamic moment is driving the main rotor that
rotates in autorotation. However, to induce the autorotation phenomenon, the rotor should be initially pre-rotated, which is usually done by the engine driving the propeller. Before the gyroplane
loses contact with the ground, the drive of the rotor must be disconnected because this type of rotorcraft does not have any anti-torque devices.
The primary flight control means of gyroplanes are:
• longitudinal and lateral tilting of the rotor shaft to ensure a pitch and roll control;
• deflections of the rudder to ensure a yaw control.
Gyroplanes may be optionally equipped with the following secondary flight controls:
• a pre-rotator that drives the rotor to initiate its rotation,
• a changeable collective pitch of rotor blades that may be used for torque reduction in pre-rotation and is necessary to conduct so-called “jump takeoff.”
The regular takeoff of a gyroplane is similar to the typical takeoff of an airplane. The gyroplane, with pre-rotated main rotor, starts accelerating along the runway. The rotor generates more and
more lift (thrust). When the lift exceeds the weight of the aircraft, the gyroplane takes off. Gyroplanes usually need a short runway to conduct regular takeoff and mostly belong to the STOL (Short
Takeoff and Landing) category of aircraft.
In the case of so-called “jump takeoff”, the gyroplane takes off directly from the ground, without a run along the runway. To perform this maneuver, the rotor head design should allow for changing
the angle of blade collective pitch during the flight. After initial pre-rotation of the rotor, the drive is disconnected and simultaneously the higher angle of the blade collective pitch is
established. The inertia-driven rotor generates high thrust, which makes the gyroplane jump upwards. The propeller starts driving the gyroplane in the horizontal direction, which makes the horizontal
velocity grow; after some time the rotor starts rotating in autorotation, similar to the case of the regular takeoff.
All studies presented in this paper have been conducted for the gyroplane presented in
Figure 1
. The gyroplane is equipped with a teetering main rotor, three-bladed tractor-type propeller, front landing gear, and inverted V-tail that also serves as a rear landing gear. Like most gyroplanes,
the main rotor of the presented gyroplane is a simple design. It is a two-bladed, teetering rotor. Its blades have a rectangular planform, uniform spanwise distribution of airfoil, and are not
twisted. To enhance the controllability of the gyroplane, its rotor-head design enables control of the collective pitch of rotor blades.
An effective and safe takeoff of the gyroplane requires accurate and rapid deflections of flight control devices. This is especially true for the control of the rotor pitch angle, which has to be
changed optimally during the takeoff so as to enhance the autorotation effect as much as possible. Additionally, during jump takeoff, the dynamic changes of the rotor pitch angle have to be
synchronized with the dynamic changes of the collective pitch of the rotor blades.
The main idea of the presented research was to search for optimal procedures to control the gyroplane flight through the application of a numerical optimization methodology. To realize this idea, the
following three main goals of the research undertaken were established:
• To develop a computational methodology for the simulation of gyroplane flight, especially directed towards high-fidelity simulations of takeoff and ascent.
• To develop a numerical optimization methodology that would be applicable to solving problems concerning the optimization of rotorcraft flight control procedures.
• To optimize flight control procedures (i.e., functions describing time-varying settings of the flight control devices) during the classic takeoff and jump takeoff of the gyroplane, so as to
achieve measurable improvement in the aircraft performance of these flight modes.
Numerical optimization methods are widely utilized in rotorcraft engineering. Most of them are directed towards optimization of the rotorcraft crucial components such as main rotors or tail rotors.
Nowadays, such optimization problems are usually formulated in multidisciplinary form [
]. This includes more and more reliable methods in such areas as Computational Fluid Dynamics (CFD), Computational Structural Mechanics (CSM) or Flight Dynamics. On the other hand, attempts are made
to automate the design process itself, by applying numerical optimization methods. The CFD methods have reached sufficient maturity to compute very accurately helicopter rotor aerodynamic
performance. The current CFD codes’ efficiency potentially enables their use in automatic optimization chains. Such optimization strategies involving URANS (Unsteady Reynolds-Averaged Navier–Stokes)
solvers have been applied in aeronautics on fixed wing configurations [
] or via adjoint formulation on aircraft configurations [
] and have demonstrated their ability to be successfully and efficiently integrated in design cycles. Alternative approaches are based on stochastic optimization methods (e.g., Genetic Algorithm).
Among others, such an approach was applied in [
] for optimization of helicopter fuselage (with simulation of main and tail rotor influence) as well as in [
] for multidisciplinary optimization of aircraft wings. To cope with the extreme complexity of coupling advanced methods of CFD and numerical optimization, some authors utilize surrogate models of
physical phenomena, as presented in [
]. In [
] the authors describe an optimization strategy for helicopter rotor blade shape, based on the coupling of an optimization gradient method with a 3D Navier–Stokes solver.
Advanced numerical optimization methods coupled with Navier–Stokes solvers are actually used mostly for optimization of aircraft external shapes (aerodynamic design) or structure. In the case of
optimization of rotorcraft flight control procedures, simplified computational models of aerodynamics are usually used. In particular, this concerns advanced computational tools dedicated for flight
control optimization. Such tools usually utilize simple models of rotorcraft aerodynamics. The aerodynamic effects generated by crucial components of a rotorcraft, such as the main rotor or tail
rotor, are usually modeled using the lifting-line theory or are even based on databases of global aerodynamic characteristics of these rotors. For the case of the rotorcraft flight control design and
optimization tool CONDUIT, details regarding the optimization methods and aerodynamic models are discussed thoroughly in [
In comparison to previously used computational tools supporting the design and optimization of rotorcraft flight control, the approach discussed in this paper is not directly geared to industrial
applicability but rather to exploring new solutions that could in future be applied to designing flight control systems. The proposed solution distinguishes itself from most of the currently utilized
approaches by the following assumptions:
• The rotorcraft flight control optimization is based on advanced, gradient-based optimization methods coupled with advanced CFD methods used directly during the rotorcraft flight simulation for
the current determination of aerodynamic loads acting on the aircraft.
• The developed methodology is applied to optimize the gyroplane flight control procedures (while most of the applications cited in the literature are focused on the optimization of helicopter
flight control).
The research presented in the paper is aimed mainly at proving a possibility of successfully using advanced CFD models even in such complex computational problems as the optimization of aircraft
flight control procedures, during highly unsteady flight conditions. So far, such an approach has been limited by the computational efficiency of computer systems. However, it is expected that the
use of high-fidelity CFD methods can significantly improve the accuracy and reliability of the optimization results. Methods using simplified aerodynamic models are burdened with large errors,
especially in relation to the modeling of physical phenomena that play a large role during the takeoff and ascent of the gyroplane. These are, among others, the ground effect (increase of lift in
direct proximity of the ground); autorotation; aerodynamic interactions between the main rotor, propeller, fuselage and tail; and the unsteadiness of all physical phenomena. Advanced CFD models
enable taking into account such phenomena; therefore, their application should significantly improve the accuracy of modeling of gyroplane takeoff and ascent.
2. Research Methodology
The general scheme of the developed methodology of rotorcraft flight simulation is presented in
Figure 2
. The flight simulation procedure is embedded in the URANS (Unsteady Reynolds Averaged Navier–Stokes) solver ANSYS FLUENT [
]. Flow effects caused by rotating lifting surfaces are modeled by application of the developed UDF (User Defined Function) module Virtual Blade Model (VBM) [
], which is compiled and linked with the essential code of the ANSYS FLUENT software. In the VBM approach, real rotors are replaced by volume discs influencing the flow field in a similar manner to
rotating blades. Time-averaged aerodynamic effects of rotating lifting surfaces are modeled by means of artificial momentum source terms placed inside the volume-disc zones placed in regions of
activity of real rotors. Such zones, replacing the real rotor and propeller in investigated gyroplane, are shown in
Figure 3
. The momentum rates, injected from these zones into the fluid, are evaluated based on the Blade Element Theory, associating local flow parameters in rotor-disc zones with aerodynamic characteristics
of blade airfoils. Data bases of these characteristics (in general: lift and drag coefficients as functions of angle of attack, for several sets of Mach and Reynolds numbers, similar to those that
are expected on the real rotor blades) should be prepared before starting the flight simulation. The original VBM code was significantly modified and expanded by the author of this paper. Beside the
essential modules ANSYS FLUENT and VBM, the methodology presented in
Figure 2
utilizes two additional modules. The module FLIGHT-DYNAMIC gathers information on all instantaneous loads acting on the rotorcraft and solves six-degrees-of-freedom equations of rigid body motion.
The module KINEMATICS is responsible for modeling of effects of motion and changes of rotorcraft geometry, which is realized through a redefinition of the boundary conditions for the ANSYS FLUENT
solver and through deformations of computational mesh. The computational model of the gyroplane, shown in
Figure 3
, was developed so as to enable simulation of flight in proximity to the ground and tilting of the rotor. These motions are realized through appropriate deformations of computational mesh, which is
done with the use of the Dynamic Mesh technique, implemented in the ANSYS FLUENT solver. Examples of such deformations of computational mesh are presented in
Figure 4
. Additionally, the developed model enables deflecting the gyroplane control surfaces (option not used in the presented study), which is realized through application of Sliding-Mesh and
Non-Conformal-Interface techniques implemented in ANSYS FLUENT.
All URANS simulations discussed in this paper were conducted with the one-equation, Spalart–Allmaras turbulence model.
During gyroplane takeoff and ascent simulations, the flight velocity was changing within the range of 0 m/s (at the beginning of takeoff) to approx. 25 m/s. Taking into account the diameter of the
main rotor 9.4 m as the reference length, the gyroplane flight Reynolds numbers were changing within the range of 0 to approx. 16 × 10^6.
The aerodynamic characteristics databases of airfoils of blades of the gyroplane main rotor and propeller, necessary for the VBM module, were calculated using a 2D variant of ANSYS FLUENT software.
For the main rotor blade airfoils, the calculations were conducted for Mach number range 0.2 ÷ 0.82 and Reynolds number range 1 × 10^6 ÷ 5.7 × 10^6. For the propeller blade airfoils, these ranges
were: 0.2 ÷ 0.93 for Mach number and 1 × 10^6 ÷ 6.5 × 10^6 for Reynolds number.
The computational mesh was generated and validated using the ANSYS GAMBIT—a commonly used mesh generator included in earlier distributions of ANSYS CFD software packages. During the rotorcraft flight
simulations, the mesh was deformed due to: tilting of the main rotor, changeable distance between the aircraft and the ground, and (optional) deflections of control surfaces. This last type of mesh
deformation was not conducted in the flight simulations discussed in this paper. The quality of the mesh deformed during the simulations was controlled by both the internal procedures of the ANSYS
FLUENT and the dedicated procedures included within the developed in-house UDF module KINEMATICS.
The total number of cells of the computational mesh used in the CFD simulations discussed in this paper was 2,157,487. This number refers to the initial state of the mesh. During the rotorcraft
flight simulation, this number could be slightly changed due to remeshing procedures conducted with respect to the mesh surrounding the cylindrical zone modeling the main rotor. The mesh filling the
space between the aircraft and the ground was not remeshed, having a constant number of cells during the whole simulation.
During the flight control optimization process, gyroplane flight simulations were conducted simultaneously for several different flight control procedures. On a typical single-processor computer
system, realization of a single simulation of gyroplane takeoff and initial stage of ascent, with a time step equal 0.01 s, required approximately 15 h of computational time. In total, the complete
single step of the optimization process was conducted within 30 h. Increasing the time step of integration of equations of gyroplane motion could proportionally decrease the total time of
The described complex model of the gyroplane flight was used in optimization studies on flight control procedures during the gyroplane takeoff. Though the developed methodology enables the solution
of six-degrees-of-freedom rotorcraft flight dynamics, the simulations presented in the paper were conducted taking into account only force balance equations, while moment balance equations were
omitted. To reduce the computational complexity of the optimization problem, it was assumed that it would not concern directional stability and control. It was assumed that, during the simulated
flight states, the pilot will be able to provide the correct directional position and direction of the rotorcraft flight.
The method of computational simulation of gyroplane flight has been applied in optimization studies on flight control procedures. In these studies, functions describing changes in time of the main
gyroplane flight control means (i.e., tilt of the main-rotor shaft and collective pitch of rotor blades) have been parameterized. In general, they were assumed to have the form of continuous,
piecewise linear functions, defined in a domain consisting of M sections:
$Ψ M ( t ) = f i − 1 + ( f i − f i − 1 ) t − t i − 1 t i − t i − 1 , for : t i − 1 ≤ t ≤ t i , i = 1 , 2 , … , M ,$
where (t
, t
, …, t
) were knots defining the function domain while (f
, f
, …, f
) were function values in subsequent knots. In the presented approach both the function knots {t
and values {f
were expressed as linear functions of a few unknown real numbers—the design parameters. In practice, when defining a given optimization problem, small values of M are preferred so as to reduce the
number of independent design parameters, and thus the dimension of the problem. On the other hand, in every case the number M was selected so as to ensure sufficient degree of freedom of flight
control procedures. The objective function, considered as a function of unknown design parameters, was defined as the altitude that would be achieved by the gyroplane after reaching the assumed
distance from the takeoff point.
To solve the defined optimization problem, the appropriately adapted BFGS Algorithm [
] has been applied. The method BFGS, named after C.G. Broyden, R. Fletcher, D. Goldfarb and D. Shanno, belongs to quasi-Newton methods—a class of hill-climbing optimization techniques that seek a
stationary point of a given objective function (preferably twice continuously differentiable). Gradient-based methods utilize a necessary condition for optimality, saying that at an optimum point the
gradient of the objective function is a zero vector. Usually such methods do not guarantee convergence to an exact optimum, unless the objective function has a quadratic Taylor expansion near an
optimum. However, the BFGS Algorithm has proven to have good performance even in cases where the objective functions were not smooth.
In quasi-Newton methods, including the BFGS Algorithm, the Hessian matrix of second derivatives does not need to be evaluated directly. Instead, the Hessian matrix is approximated using updates
specified by gradient evaluations or approximate gradient evaluations. The latter approach has been applied in the presented optimization studies.
Like in all cases of gradient-based methods, the optimal solution has been searched for in sequential iterative steps. In each step, the components of the gradient vector (partial derivatives of the
objective function with respect to the unknown design parameters) were evaluated by means of a one-sided finite-difference approximation. In the presented approach, this required performing at least
N + 1 simulations of the gyroplane flight, where N was the number of design parameters.
In each step of the BFGS algorithm, an auxiliary one-dimensional optimization problem is solved. The problem consists of searching for the optimal movement in the newly determined search direction.
In the presented approach, the solution of this problem required conducting several additional simulations of gyroplane flight in each sequential step of the optimization process. It should be
mentioned that the BFGS method has also been developed in a variant with simple box constraints [
] and usually this variant has been applied in the presented optimization studies.
3. Optimization of Gyroplane Takeoff Control Procedures
The two examples presented below of numerical optimizations of flight control procedures pertain to two types of takeoff of the gyroplane: classic takeoff and jump takeoff. In both cases, the
optimization process aimed at maximization of the altitude reached by the gyroplane after traveling a certain distance from the takeoff place. Both optimizations were conducted for the same general
flight conditions:
• total mass of the gyroplane: 600 kg;
• maximum static thrust of the propeller: 2943 N (during takeoff, when the forward velocity of the vehicle was growing, the thrust of the propeller was changing, which was a result of modeling the
propeller effects through VBM methodology).
3.1. Classic Takeoff of the Gyroplane
The optimization of classic takeoff control procedure has been conducted with respect to time varying pitch angle of main rotor (
), which was considered the only flight control parameter. The angle of collective pitch of rotor blades (
), unchangeable during the flight, has been assumed as an additional unknown parameter. Based on these assumptions, the flight control parameters
) and
) were assumed to have the form defined by Equation (1) as piecewise linear functions Ψ
). Parameters M, {t
and {f
of these functions were related to the design parameters D
, F
and F
presented in
Figure 5
, according to the following dependencies:
ϕ[R](t) = Ψ[2](t), for: t[0] = 0, t[1] = D[1], t[2] = +∞, f[0] = 0, f[1] = F[1], f[2] = F[1]
θ[0](t) = Ψ[1](t), for: t[0] = 0, t[1] = +∞, f[0] = F[2], f[1] = F[2].
Based on the opinions of gyroplane designers and pilots, such a relatively simple parameterization should sufficiently approximate the real flight control procedures utilized during the takeoff and
ascent of gyroplane. This especially concerns the presented study, which is more about demonstration than implementation. In the discussed case, the optimization aimed at maximization of the altitude
(H) reached by the gyroplane after traveling the distance X = 200 m from the initial position, which is explained in
Figure 6
. The optimization problem was formulated mathematically as a search for the set of the design parameters D
, F
and F
maximizing the following objective function Φ:
Φ(D[1], F[1], F[2]) = H(X = 200 m),
taking into account the following constraints:
where λ
—limit of angular speed of change of ϕ
, ϕ
—maximum of rotor pitch and θ
—maximum of blade collective pitch.
The optimization problem has been solved iteratively using the BFGS algorithm, discussed in
Section 2
. In the sequential steps of the optimization process, this procedure has been improved from the point of view of maximizing the objective function Φ in Equation (4). In each iterative step, the
partial derivatives of the objective function Φ with respect to the unknown design parameters had to be approximated due to the requirements of the BFGS Algorithm. These derivatives were evaluated by
means of the one-sided finite-difference approximation. This required performing at least
+ 1 simulations of the classic takeoff of the gyroplane, where
= 3 was the number of independent variables in Equation (4). In addition, solving the auxiliary one-dimensional problem of finding the optimal movement in the newfound search direction, eight
additional classic gyroplane takeoff simulations were conducted in each optimization step.
In the presented optimization process only four iterative steps of the BFGS method have been conducted. Due to time limits, the calculations were not continued until full convergence of the BFGS
algorithm was achieved. Changes in the objective function Φ in the four sequential steps are shown in
Figure 7
. The presented results show that at a distance of 200 m, the gyroplane controlled by the optimized procedure reached an altitude about 5 m higher than the baseline. The improved final flight control
procedure is compared with the baseline procedure in
Figure 8
The final solution of the optimization over the baseline procedure differs in the blade collective pitch greater by 1.3°, the rotor pitch angle greater by 4.7° and angular speed of change of rotor
pitch greater by 14.3%.
Figure 9
compares gyroplane flight trajectories during the classic takeoff for two gyroplane flight control strategies: baseline and optimized. Compared to the baseline, the aircraft trajectory corresponding
to the optimized procedure shows a shorter run of the aircraft on the runway and faster climb, at least during arrival to the assumed altitude-control point, localized 200 m from the starting point.
As shown in
Figure 10
, during the classic takeoff and initial stage of ascent, the flight velocity (V) of the gyroplane controlled by the optimized procedure was lower (by approx. 3 m/s) than the velocity of gyroplane
controlled by baseline procedure.
Figure 11
Figure 12
Figure 13
present snapshots of velocity magnitude contours around the gyroplane, taken during the classic takeoff simulation at
= 0, 10, 17.5 s (time elapsed from the beginning of run on a runway), for two compared flight control strategies: baseline and optimized.
3.2. Jump Takeoff of the Gyroplane
Improvement of gyroplane flight control strategy during the jump takeoff was conducted based on the numerical optimization approach described in
Section 2
. In this case, the optimization was conducted with respect to the time-varying pitch angle of the main rotor (
) and the angle of collective pitch of the rotor blades (
). Based on these assumptions, the time-varying flight control parameters
) and
) were assumed to have the form defined by Equation (1) as piecewise linear functions Ψ
). Parameters M, {t
and {f
of these functions were related with the presented in
Figure 14
design parameters D
, D
, D
, D
, F
, F
and F
according to the following dependencies:
ϕ[R](t) = Ψ[3](t), for: t[0] = 0, t[1] = D[1], t[2] = D[1] + D[2] , t[3] = +∞, f[0] = 0, f[1] = 0, f[2] = F[1], f[3] = F[1]
θ[0](t) = Ψ[4](t), for: t[0] = 0, t[1] = 1, t[2] = 1 + D[3], t[3] = 1 + D[3] + D[4], t[4] = +∞, f[0] = 0, f[1] = F[2], f[2] = F[2], f[3] = F[3], f[4] = F[3].
Based on the opinions of gyroplane designers and pilots, such a relatively simple parameterization should sufficiently approximate the real flight control procedures utilized during the takeoff and
ascent of gyroplane. This especially concerns the presented research, which is more about demonstration than implementation. In the discussed case, the optimization aimed at maximization of the
altitude (H) reached by the gyroplane after traveling the distance X = 100 m from the initial position, which is explained in
Figure 15
. The optimization problem was formulated mathematically as a search for values of the design parameters D
, D
, D
, D
, F
, F
and F
maximizing the following objective function Φ:
Φ(D[1], D[2], D[3], D[4], F[1], F[2], F[3]) = H(X = 100 m),
taking into account the following constraints:
D[1] ≥ 0, D[2] ≥ 1, D[3] ≥ 0, D[4] ≥ 1,
where λ
, λ
—limits of angular speed of changes of ϕ
and θ
respectively, ϕ
—maximum of rotor pitch and θ
maximum of blade collective pitch. The defined optimization problem has been solved by application of the BFGS algorithm. At every step of the iterative process of the optimization, the gradient of
the objective function Equation (11) was determined using the one-sided finite-difference approximation. This required conducting at least
+ 1 (where
= 7) independent simulations of gyroplane jump takeoff for different sets of values of unknown design parameters D
, D
, D
, D
, F
, F
, and F
. In addition, to solve the auxiliary one-dimensional problem of finding the optimal movement in the newfound search direction, eight additional gyroplane jump takeoff simulations were performed at
each optimization step.
The optimization process consisted in gradual improvement of this strategy, so as to increase as much as possible the product of Equation (6). In the present optimization only four iterative steps of
the BFGS method have been conducted. Due to time limits, the calculations have not been continued until full convergence of the BFGS algorithm was achieved. Changes in the objective function Φ
(Equation (6)) in the sequential steps are presented in
Figure 16
. The final solution of the optimization is compared with the baseline in
Figure 17
. The presented results show that the optimized flight control strategy is characterized by nearly the same values of parameters F
and F
. This means that these two parameters might be replaced by only one in the assumed parametric model of flight control strategy (shown in
Figure 14
) and the phase of decreasing of the rotor pitch, which we assume might be omitted in this model.
Figure 18
compares gyroplane flight trajectories during the jump takeoff for two flight control strategies: baseline and optimized. It may be concluded that the optimized trajectory is growing monotonically
while the baseline trajectory has a local minimum. Additionally, for the optimized flight control strategy the result of Equation (11) is higher by approximately 5 m than for the baseline strategy.
As shown in
Figure 19
, during the jump takeoff and initial stage of ascent, the flight velocity (V) of the gyroplane controlled by optimized procedure, after 8 s the flight stabilized itself at a level of approx. 17 m/s.
In this case, to increase the flight velocity, after the jump takeoff and ascent the backward pitch of the rotor should be reduced. For the baseline flight control procedure, the flight velocity was
still growing and it reached V ≈ 25 m/s at
= 25 s.
Figure 20
Figure 21
Figure 22
present snapshots of flow field (velocity magnitude contours) around the gyroplane taken during the jump takeoff at
= 0.5, 1.5, 10 s (time elapsed from the beginning of the takeoff), for two compared flight control strategies: baseline and optimized.
4. Discussion
A methodology of computational simulation of gyroplane flight has been developed. The methodology was applied to simulate the classic takeoff of a gyroplane and the so-called “jump takeoff”—a
maneuver in which the gyroplane takes off similarly to a helicopter, without the accelerating run along a runway. For these two specific flight conditions, a numerical optimization of flight control
procedures has been conducted, using a gradient-based method, the BFGS algorithm. In this process, the assumed design parameters described the time-varying settings of gyroplane flight control: tilt
of main-rotor shaft and collective pitch of rotor blades. The optimization aimed at a determination of the flight control procedures that are optimal from the point of view of the highest possible
altitude reached by the gyroplane at an assumed distance during both classic takeoff and jump takeoff.
In both cases of optimization discussed, at the control point the gyroplane controlled according to the optimized strategy reached an altitude approximately 5 m higher than the gyroplane controlled
according to a baseline flight control procedure.
The presented computational optimizations have proven the qualitative correctness of the developed methodology. So far the obtained quantitative results have not been validated experimentally. The
main reason for this is that the gyroplane that is the subject of the investigation has not yet started its flight tests. In addition, the problem is the difficulty of obtaining accurate measurements
for the actual flight parameters. In a wind tunnel, investigations of strongly unsteady flight states such as takeoffs or landings are virtually unfeasible. On the other hand, it is difficult to
measure a number of important parameters in real test flights. Therefore, a thorough verification of the presented methodology is not possible. However, both the classic and jump takeoffs of other
gyroplanes, documented especially in video materials, indicate good convergence, both qualitative and quantitative, with the results of the research presented in this paper. This especially concerns
such gyroplane flight parameters as runway run length, flight velocity, flight altitude, duration of takeoff, or shape of trajectory. More accurate comparison of such data concerning takeoffs of
other gyroplanes and cases discussed in this paper is impossible because we would need to conduct thorough experimental studies on the gyroplane considered. The gyroplane flight parameters are
strongly influenced by such factors as aircraft weight, total area of rotor blades, rotor aerodynamic properties, moments of inertia of rotor and its blades, engine power, performance characteristics
of propeller, aerodynamic properties of fuselage, etc. All these data were taken into account in the simulations presented in this paper. Unfortunately, most of these data are not available in the
case of potential reference gyroplanes.
It is expected that in the case of alternative definitions of the optimization problem (i.e., another selection of design parameters or another definition of the objective function), the optimization
methodology developed would also confirm its effectiveness and reliability.
The research results can be helpful in designing easy-to-control gyroplanes and also in training pilots for this type of rotorcraft. However, the presented methodology seems to have a much wider
potential for future applications. These possible applications may concern not only other gyroplanes or rotorcrafts in general but also the optimization of flight control procedures for any aircraft,
e.g., taking off or landing airplanes.
The research leading to these results was co-financed by the European Regional Development Fund under the Operational Programme Innovative Economy 2007–2013, within the project “Modern Gyroplane Main
Conflicts of Interest
The author declares no conflict of interest.
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7. Imiela, M. High-Fidelity Optimization Framework for Helicopter Rotors. Aerosp. Sci. Technol. 2012, 23, 2–16. [Google Scholar] [CrossRef]
8. Johnson, C.; Barakos, G. Optimizing Aspects of Rotor Blades in Forward Flight, AIAA 2011-1194. In Proceedings of the 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and
Aerospace Exposition, Orlando, FL, USA, 4–7 January 2011. [Google Scholar]
9. Glaz, B.; Friedmann, P.P.; Liu, L. Surrogate Based Optimization of Helicopter Rotor Blades for Vibration Reduction in Forward Flight. Struct. Multidiscip. Optim. 2008, 35, 341–363. [Google
Scholar] [CrossRef]
10. Glaz, B.; Friedmann, P.P.; Liu, L. Helicopter Vibration Reduction throughout the Entire Flight Envelope Using Surrogate-Based Optimization. J. Am. Helicopter Soc. 2009, 54, 12007. [Google Scholar
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Memorandum 112203 USAATCOM Technical Report 97-A-009; National Aeronautics and Space Administration, Ames Research Center: Moffett Field, CA, USA, 1997.
13. Tischler, M.B.; Berger, T.; Ivler, C.M.; Mansur, M.H.; Cheung, K.K.; Soong, J.Y. Practical Methods for Aircraft and Rotorcraft Flight Control Design: An Optimization-Based Approach; American
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Figure 4. Cross-section of computational mesh around the gyroplane in two different stages of flight Left: a ground pre-rotation, right: a forward flight a few meters above the ground.
Figure 5. Parametric model of the gyroplane flight control procedure utilized in the optimization of the classic takeoff of the gyroplane.
Figure 7. Values of the maximized objective Φ in sequential iterative steps of the process of optimization of the classic takeoff control procedure of the gyroplane.
Figure 8. Comparison of the baseline (left) and optimized (right) classic takeoff control procedures, showing the pitch angle of main rotor (ϕ[R]), collective pitch of rotor blades (θ[0]) and
collective pitch of propeller blades (θ[P]) as functions of time (t).
Figure 9. Aircraft trajectories obtained for the baseline and optimized procedures of flight control during the classic takeoff of the gyroplane.
Figure 10. Aircraft flight velocity (V) vs. time (t) during the classic takeoff, for the baseline and optimized procedures of gyroplane flight control.
Figure 11. Comparison of velocity magnitude contours around the gyroplane during the classic takeoff, for two configurations related to baseline (left) and optimized (right) flight control
procedures. The initial time (t = 0 s) of the aircraft run on a runway.
Figure 12. Comparison of velocity magnitude contours around the gyroplane during the classic takeoff, for two configurations related to baseline (left) and optimized (right) flight control
procedures. Time elapsed from the beginning of the aircraft run: t = 10 s.
Figure 13. Comparison of velocity-magnitude contours around the gyroplane during the classic takeoff, for two configurations related to baseline (left) and optimized (right) flight control
procedures. Time elapsed from the beginning of the aircraft run: t = 17.5 s.
Figure 14. Parametric model of the gyroplane flight control procedure utilized in the optimization of jump takeoff of the gyroplane.
Figure 15. Definition of the objective (Φ) for the optimization of jump takeoff control strategy on the example of a gyroplane trajectory controlled by the baseline procedure.
Figure 16. Values of the maximized objective Φ in sequential iterative steps of the process of optimization of the jump takeoff control procedure of the gyroplane.
Figure 17. Comparison of the baseline (left) and optimized (right) jump takeoff control procedures, showing the pitch angle of main rotor (ϕ[R]), collective pitch of rotor blades (θ[0]) and
collective pitch of propeller blades (θ[P]) as functions of time (t).
Figure 18. Aircraft trajectories obtained for the baseline and optimized procedures of flight control during the jump takeoff of the gyroplane.
Figure 19. Aircraft flight velocity (V) vs. time (t) during the jump takeoff, for the baseline and optimized procedures of gyroplane flight control.
Figure 20. Comparison of velocity magnitude contours around the gyroplane during the jump takeoff, for two configurations related to baseline (left) and optimized (right) flight control procedures.
The initial time moment (t = 0.5 s) of the jump takeoff.
Figure 21. Comparison of velocity magnitude contours around the gyroplane during the jump takeoff, for two configurations related to baseline (left) and optimized (right) flight control procedures.
Time elapsed from the beginning of the jump takeoff: t = 1.5 s.
Figure 22. Comparison of velocity magnitude contours around the gyroplane during the jump takeoff, for two configurations related to baseline (left) and optimized (right) flight control procedures.
Time elapsed from the beginning of the jump takeoff: t = 10 s.
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Stalewski, W. Simulation and Optimization of Control of Selected Phases of Gyroplane Flight
. Computation 2018, 6, 16. https://doi.org/10.3390/computation6010016
AMA Style
Stalewski W. Simulation and Optimization of Control of Selected Phases of Gyroplane Flight
. Computation. 2018; 6(1):16. https://doi.org/10.3390/computation6010016
Chicago/Turabian Style
Stalewski, Wienczyslaw. 2018. "Simulation and Optimization of Control of Selected Phases of Gyroplane Flight
" Computation 6, no. 1: 16. https://doi.org/10.3390/computation6010016
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Applications of Geometric Series | Stage 5 Maths | HK Secondary S4-S5 Compulsory
There are many real life applications of geometric series and we discuss a few of these here.
Formulae are often developed for many of these applications, particularly when they occur regularly in industry.
For our purposes, it is often best to find solutions to various problems starting from first principles.
Example 1
A bank client deposits $\$1000$$1000 at the beginning of each year, and is given $7%$7% interest per year for $50$50 years. How much will accrue in the account over that time?
To answer this, we might begin by searching for a pattern by examining what happens in the first few years.
If we set $A_n$An as the amount of money accrued after $n$n years have elapsed, then we have:
Then $A_1=1000+1000\times\frac{7}{100}=1000\left(1+\frac{7}{100}\right)=1000\times\left(1.07\right)^1$A1=1000+1000×7100=1000(1+7100)=1000×(1.07)1
This means that the amount accrued after $1$1 year becomes $\$1070$$1070.
By the end of the second year, another $\$1000$$1000 has been added, with interest, but the original $\$1000$$1000 has been boosted by two interest payments.
The total amount is determined as:
By the end of the third year, the total accrual becomes:
A pattern is emerging, and so by the end of $50$50 years, the total accrued becomes:
Inside the square brackets is a geometric series with first term and common ratio both equal to $1.07$1.07.
Recalling the formula for the sum of a geometric sequence as $S_n=\frac{a\left(r^n-1\right)}{r-1}$Sn=a(rn−1)r−1 we have for this series:
Hence the amount accrued in the account will be approximately $\$434986$$434986.
Example 2
For example $1$1 above, devise a formula that a banker might use for any client wishing to deposit an amount $P$P at the beginning of ever year for $n$n years where an interest rate of $r%$r% p.a. is
applied. Use the formula to find the accrued amount of the regular annual payment of $\$1000$$1000 after $50$50 years where $8%$8% is applied each year.
From our solution to example $1$1, and calling $R=1+\frac{r}{100}$R=1+r100, we have the generalised formula given by:
$A_n$An $=$= $P\times\frac{R\left(R^n-1\right)}{R-1}$P×R(Rn−1)R−1
Hence, at $8%$8%, we have:
This means that one extra percentage in interest each year makes a difference in total accrual of approximately $\$138784$$138784.
Example 3
Show that the repeating decimal $N=0.2323232323...$N=0.2323232323... is a rational number. That is to say, the number can be put in the form $\frac{p}{q}$pq where $p$p and $q$q are integers and $q\
The repeating decimal can be written:
$N=0.23+0.0023+0.000023+...$N=0.23+0.0023+0.000023+... which is an infinite geometric series whose first term is given by $a=0.23$a=0.23 and whose common ratio is given by $r=0.001$r=0.001.
The limiting sum becomes:
$N$N $=$= $\frac{a}{1-r}$a1−r
$=$= $\frac{0.23}{1-0.001}$0.231−0.001
$=$= $\frac{\frac{23}{100}}{1-\frac{1}{100}}$231001−1100
$=$= $\frac{\frac{23}{100}}{\frac{100-1}{100}}$23100100−1100
$=$= $\frac{23}{99}$2399
The same strategy can be applied to any repeating decimal.
Example 4
The recipe for making a Koch snowflake is as follows;
1. Draw an equilateral triangle with area $A$A as shown in figure $(a)$(a) below.
2. To each of the three sides of figure $(a)$(a) add an equilateral triangle with sides of length $\frac{1}{3}$13 that of the original triangle, as shown in figure $(b)$(b). Note that the area of
each of the new triangles is $\frac{A}{9}$A9.
3. To each of the $12$12 sides of figure $(b)$(b) add an equilateral triangle with sides of length $\frac{1}{3}$13 that of the triangles formed in step $2$2. The area of each of these $12$12 new
triangles becomes $\frac{A}{9^2}$A92.
4. To each of the $48$48 sides of figure $(c)$(c) add an equilateral triangle with sides of length $\frac{1}{3}$13 of the $12$12 triangles formed in step $3$3.
5. Continue in this manner indefinitely to form what becomes the Koch snowflake.
The emerging figure has an infinite perimeter but we can show that is has a finite area as follows.
The total area $A_T$AT of the snow flake becomes:
$A_T$AT $=$= $A+3\left(\frac{A}{9}\right)+12\left(\frac{A}{9^2}\right)+48\left(\frac{A}{9^3}\right)+192\left(\frac{A}{9^4}\right)+...$A+3(A9)+12(A92)+48(A93)+192(A94)+...
$=$= $A\left[1+\left(\frac{3}{9}\right)+\left(\frac{12}{9^2}\right)+\left(\frac{48}{9^3}\right)+\left(\frac{192}{9^4}\right)+...\right]$A[1+(39)+(1292)+(4893)+(19294)+...]
The expression on the right and within the main bracket begins with the number $1$1, but all the other terms form a geometric series with first term $\frac{1}{3}$13 and common ratio $\frac{4}{9}$49
. We can see this because the numerator of each fraction is increasing by a factor of $4$4 and the denominator is increasing by a factor of $9$9.
Since $|\frac{4}{9}|$|49| is less than $1$1, the series sums to $\frac{a}{1-r}=\frac{\frac{1}{3}}{1-\frac{4}{9}}=\frac{3}{5}$a1−r=131−49=35. Adding the extra $1$1 at the beginning, we see that
the total sum within the main brackets is $\frac{8}{5}$85.
This means that Koch snowflake has a total area given by $A_T=\frac{8A}{5}$AT=8A5.
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Zariski topology
, the
Zariski topology
is a structure basic to
algebraic geometry
, especially since 1950.
In this topology, named after Oscar Zariski, the closed sets are the sets consisting of the mutual zeroes of a set of polynomials.
This definition indicates the kind of space that can be given a Zariski topology: for example, we define the Zariski topology on an n-dimensional vector space F^n over a field F, using the definition
above. That this definition yields a true topology is easily verified.
Using the Noetherian property of the ring of polynomials over F, one sees that any closed set is the set of zeroes of a finite set of equations.
The Zariski topology given to some finite-dimensional vector space doesn't depend on the specific basis chosen; for that reason it is an intrinsic structure. It is usually regarded as belonging to
the underlying affine space, since it is also invariant by translations.
One can generalise the definition of Zariski topology to projective spaces, and so to any algebraic variety as subsets of these. The general case of the Zariski topology is based on the affine scheme
and spectrum of a ring constructions, as local models.
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Important Principles For Design The Column
The column is the vertical member of the building construction. The essential purpose of the column is the given to vertical supports for the building. They transfer building loads from top to bottom
on the foundation or footing. Rules For Design The Column | Most Important Rules For Column Design |What is the Definition of Column? Basic And Important Rules For Design The Column. Important
Principles For Design The Column
What is the Definition of Column?
The column is the vertical member of the building construction. The essential purpose of the column is the given to vertical supports for the building. They transfer building loads from top to bottom
on the foundation or footing. The columns so many types according to shape and uses. Recantange shape column square shape column circular column etc.
Important Principles For Design The Column
Designing the rules for the column is very important because if we make a column without following the design rules then we will face a big problem in structures or building construction. Must use
the suitable size and shape of the column in the construction.
Now we discuss the basic important rules for the design of the column. we have some most important Princilpes for design the column is mentioned bellow.
1- The Size of the column is must be at least 225mm × 225mm or 9″ × 9″.
2- Must be slect the sequre column if the load acts at the axis of column.
3- Must be slect the Rectangle column if the load acts at the acenstials of column.
4- Must be use the length of overlaping in column must be at least 48d.
5- If we construct the sequre and Rectangle column then use the 4 numbers of verticle or longitudinal bar.Â
6- If we construct the Circular column then use the 8 numbers of verticle or longitudinal bar.
7- Must be use the diameter of the vertical bar is at least 12mm.
8- Must be use the diameter of the latera.l ties or ring is at least 8mm
9- Verticel bar or longitudinal bar is must be the duck leg with connected with the foundation.
10- The Area of Reinforcement of the column is must be at least 0.8% of the column cross section.
11- The Area of Reinforcement of the column is not more then 0.6% of the column cross section.
12-Use the Concrete grade for column is M20.
13- Use the Steel for column is +fc-500.
14- The lateral tie or Ring Using the column spcing between both ties is Not more then 100mm for Zone A.
15- The lateral tie or Ring Using the column spcing between both ties is Not more then 150mm for Zone B.
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Lesson 1 Homework Practice Integers And Graphing Answers
lesson 1 homework practice integers and graphing answers, lesson 1 homework practice integers and graphing answers page 71
Chapter 3 - Lesson 1: Integers and Graphing. Notes. Positive whole ... Practice: Write an integer for each situation. 1.) a gain)of $2 a share_+2 2.) 10 degrees below)zero_-10 ... 228 #1-5 in
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4.. Lesson 1 Homework Practice. Integers and Absolute Value. DOS. Write an integer for each situation. 1. a profit of $12. 2. ... Graph each set of integers on a number line. 5. {-5, 0,5}. 6. (-3,
-2, 1,4} ... 2 choices for answers the altitude increased.. Positive whole numbers, their opposites, and zero are called integers. To represent data ... Graph the set of integers {-4,2,-1} on a
number line.. You will be learning about Integers and Absolute Value. ... Get your Lesson 1 Homework Practice (Integers and Graphing) laying on the computer cart! When finished - place ... You can
only eat the ghost that answers the question/statement!. NS.6c. Mathematical Practices. 1, 3, 4, 5, 7. Common Core. State Standards. connectED.mcgraw-hill.com. Lesson 1 Integers and Graphing 345....
Example 1. Write an integer for each situation. a. 16 feet under the ground b. a gain of 5 hours. To graph an integer on a number line.... Chapter 5 Lesson 1 Integers and Graphing. Billi Sparacino.
Loading... Unsubscribe from Billi Sparacino .... Graph the set of integers {-1, 3, -2} on a number line.
How to Graph Charts With Integers : Math & Algebra - Duration: 1:42. eHow 3,568 views 1:42. 6th Grade ... f99c0e132e
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ANOVA Full Form: Of What Is The Full Form Of ANOVA?
In this article we are going to know ANOVA full form
ANOVA Full Form
What Is The Full Form Of ANOVA?
• The full form of ANOVA stands for Analysis Of Variance, the term frequently used in Mathematics as a part of Academic & Science over the world.
Below We Are Going To Explain The Meaning Of ANOVA,
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Find The Meaning Of ANOVA
What Is The Meaning Of ANOVA?
The Meaning Of ANOVA is a statistical technique in which the variation in a set of observations is split into distinct components.
Find The Abbreviation Of Analysis Of Variance
What Is The Abbreviation Of Analysis Of Variance?
The Abbreviation Of Analysis Of Variance Is ANOVA.
ANOVA Full formSee This Also: What Is The Full Form Of AO? Find Out The Full Form Of AO.
The meaning of Analysis of Variance (ANOVA) signifies a statistical method in which the variation in a set of interpretations is divided into discrete components.
The acronym ANOVA is used to examine the dissimilarity among group means in a sample. The famous statistician and evolutionary biologist Ronald Fisher was behind the developed of ANOVA. The
contraction ANOVA is based on the law of total variance.
The different ANOVA techniques are used for making superior decisions, and case studies are also taken up to analyze how ANOVA works.ANOVA is Abbreviation for?
• Analysis of variance
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Frequently Asked Questions:
What Is The Anova Test?
An ANOVA test is a way to find out if a survey or experiment results are significant. In other words, they assist you to work out if you would like to reject the null hypothesis or accept the
alternate hypothesis. Basically, you’re testing groups to ascertain if there is a difference between them.
What Does Anova Stand For?
Analysis of Variance
ANOVA (Analysis of Variance)
ANOVA may be a statistical technique that assesses potential differences during a scale-level variable by a nominal-level variable having 2 or more categories. This test is additionally called the
Fisher analysis of variance.
Why Would You Use Anova?
The ANOVA is used to dope out whether there are any statistically unusual differences between the means of two or more independent groups (although you incline to only see it used when there is a
minimum of three, rather than two groups).
What Is An Anova Table?
Analysis of Variance (ANOVA) is a statistical analysis to test the degree of differences between two or more groups of an experiment. The ANOVA table displays the statistics wont to test hypotheses
about the population means. The ANOVA table can be either one way or two way ANOVA table.
Where Is Anova Used?
The ANOVA is used to figure out whether there are any statistically significant differences between the means of two or more independent groups.
How Do You Solve Anova?
Steps for Using ANOVA
Step 1: Compute the Variance Between. First, the sum of squares (SS) between is computed.
Step 2: Compute the Variance Within. Again, first, compute the sum of squares within.
Step 3: Compute the Ratio of Variance Between and Variance Within. This is called the F-ratio.
When Should I Use An Anova Test?
The one-way analysis of variance (ANOVA) is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.
What Is The Full Form And Formula Of ANOVA?
Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The
systematic factors have a statistical influence on the given data set, while the random factors do not.
I Hope We Had Covered Your All Queries Regarding
ANOVA Full Form In English?
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ANOVA Full Form.
Check Out: What Is The Full Form Of ANS? Find Out The Full Form Of ANS.
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: removing multiple variables - context vector
I'm using formulas within the vector context to define difference equations - e.g., $f1 = $f2, where $f1 is the AR part and $f2 is the MA part. In DSP this is usually written
y(n) = a1*y(n-1) + ... aN*y(n-N) + b0*x(n) + ... + bM*x(n-M).
In the vector context, x, y, z are standard variables. So I remove them, add the variable n, and define dummy functions x(n), y(n). The problem is that the equation comes outyn = and not y(n) =,
The x(n) seems fine. I've tried various modifications with various errors, but nothing works. I can change y(n) to q(n) and that works, but I really need yno to agree with standard texts.
code below
DOCUMENT(); # This should be the first executable line in the problem.
"PGstandard.pl", # Standard macros for PG language
$showPartialCorrectAnswers = 1;
Context()->variables->remove(x); # only variables should be n
Context()->variables->add(n=> 'Real');
# how to remove y and z from Vector context?
# Component values here
parserFunction("q(n)" => "n"); # dummy function
parserFunction("x(n)" => "n"); # dummy function
$Na1 = random(3,6,1);
do {$Nb1 = random(3,6,1);} until ($Nb1 != $Na1);
$Nmax = 6;
for ($k =1;$k <= $Nmax;$k++) {$a1[$k] =0};
for ($k =1;$k <= $Nmax;$k++) {$b1[$k] =0};
$a1[0] = non_zero_random(-9,9,1); # a0
for ($k =1;$k <= $Na1;$k++) {$a1[$k] = random(-5,5,1)};
$b1[0] = non_zero_random(-9,9,1); # b0
for ($k =1;$k <= $Nb1;$k++) {$b1[$k] = non_zero_random(-5,5,1)};
for ($k =1;$k <= $Nmax;$k++) {$a1ans[$k] =0};
for ($k =1;$k <= $Nmax;$k++) {$b1ans[$k] =0};
$a1ans[0] = 1;
for ($k =1;$k <= $Na1;$k++) {$a1ans[$k] = -$a1[$k]/$a1[0]};
for ($k =0;$k <= $Nb1;$k++) {$b1ans[$k] = $b1[$k]/$a1[0]};
$f1 = Formula("$a1[0]*y(n) + $a1[1]*y(n-1) + $a1[2]*y(n-2) + $a1[3]*y(n-3) +
$a1[4]*y(n-4) + $a1[5]*y(n-5) + $a1[6]*y(n-6) ")->reduce;
$f2 = Formula("$b1[0]*x(n) + $b1[1]*x(n-1) + $b1[2]*x(n-2) + $b1[3]*x(n-3) +
$b1[4]*x(n-4) + $b1[5]*x(n-5) + $b1[6]*x(n-6) ")->reduce;
$Na2 = 1;
$Nb2 = random(4,6,1);
for ($k =1;$k <= $Nmax;$k++) {$a2[$k] =0};
for ($k =1;$k <= $Nmax;$k++) {$b2[$k] =0};
$b2[0] = non_zero_random(-9,9,1); # b0
for ($k =1;$k <= $Nb2;$k++) {$b2[$k] = random(-4,4,1)};
$Nmax = 6;
$f3 = Formula("y(n)")->reduce;
$f4 = Formula("$b2[0]*x(n) + $b2[1]*x(n-1) + $b2[2]*x(n-2) + $b2[3]*x(n-3) +
$b2[4]*x(n-4) + $b2[5]*x(n-5) + $b2[6]*x(n-6) ")->reduce;
$a2ans[0] = 1;
for ($k =1;$k <= $Na1;$k++) {$a2ans[$k] = -$a2[$k]};
for ($k =0;$k <= $Nb1;$k++) {$b2ans[$k] = $b1[$k]};
\{image("direct_form_2.jpg")\} $BR
This Problem is related to Problem 2.46 in the text.
The problem is to determine the coefficients for the Direct Form
II (figure above). For this problem, we will assume the maximum
delay is $Nmax. You will need to find all coefficients. There are
likely to be several zero coefficients, if the orders of the AR and
MA parts are not equal.
Give the coefficients of Direct Form II for the difference equation ,
a) \( $f1 = \) $BR
For the coefficients, enter as vectors, \( {\bf a} = <a_0, a_1, ..., a_6>\) $BR
and \( {\bf b} = <b_0, b_1, ..., b_6>\) $BR
Note that some of the coefficients may be zero$BR
\({\bf a} =\) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \} $BR
\({\bf b} = \) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \}
b) \( $f3 = $f4\)
For the coefficients, enter as vectors, \( {\bf a} = <a_0, a_1, ..., a_6>\) $BR
and \( {\bf b} = <b_0, b_1, ..., b_6>\) $BR
Note that some of the coefficients may be zero$BR
\({\bf a} =\) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \} $BR
\({\bf b} = \) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \}
$BBOLD SOLUTION $EBOLD
tmp holder
ENDDOCUMENT(); # This should be the last executable line in the problem.
One way to remove the variables from the context is
Another would be
But if you are adding new variables, the easier is to use are() rather than add(), which first removes all the existing variables and adds the new ones. So replace
Context()->variables->remove(x); # only variables should be n
Context()->variables->add(n=> 'Real');
# how to remove y and z from Vector context?
Context()->variables->are(n => 'Real');
There are several other things you probably want to change as well. First, you probably want to disable a coupe of reduction rules that reorder or regroup the terms of your expressions. You can use
to do that. This should keep your terms in the same order you gave them originally.
Next, it looks like you have used a pretty old problem as a starting point, so have some old-style code that should be updated.
and that will being you more up to date.
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How to solve difficult SSC CGL Geometry problems in a few steps 1
Submitted by Atanu Chaudhuri on Mon, 18/07/2016 - 16:22
Shape analysis, deductive reasoning and use of basic concepts result in quick solution
Geometry is no different from Algebra or Profit and loss with respect to solving a difficult SSC CGL problem in a few steps by applying Basic subject concepts together with Problem solving strategies
and techniques. In this first episode of Geometry problem solving in a few steps, we will analyze solution process of two chosen problems.
For refreshing the basic geometry concepts you may refer to the concept tutorials, Basic concepts on Geometry 1 - points, lines and triangles, Basic concepts on Geometry 2 - quadrilaterals and
polygons, and Basic concepts on Geometry 3 - Circles.
Chosen Problem 1.
In a circle with centre at $O$, the $\angle OAC = 15^0$ and $\angle OBC=50^0$ where $A$, $B$ and $C$ are the points on the circle periphery. The $\angle AOB$ is then,
a. $30^0$
b. $70^0$
c. $20^0$
d. $40^0$
Problem analysis
The two angles given belong to the two triangles $\triangle OAC$ and $\triangle OBC$. The crucial aspect of these two triangles are, in each, the pair of sides OA, OC and OB, OC are the radii of the
circle and so are equal. As a result the two triangles of interest turn out to be isosceles triangles.
This is the key information discovery based on basic Geometry concepts.
Note: wherever you encounter triangles in circles look for sides as radii. Often this will be the key information contributing towards quick elegant solution. By using this property of circles (equal
radii), often the triangles in circles can be identified as isosceles triangles and sometimes with a bit more favorable information, to even equilateral ones. We can term this as a rich geometric
concept classified under Triangle Sides as equal radii concept.
End state analysis
On analyzing the end requirement, we find the required $\angle AOB = \angle AOC - \angle BOC$, both these angles being the vertex angle of the two triangles of interest.
Last reasoning
The two triangles are then isosceles with base angles equal in each. Also for each triangle value of one of the base angles is given. So for each of the two triangles, the vertex angle can be found
out. Final step will then be just one step away.
Problem solving execution
In $\triangle AOC$, at the vertex,
$\angle AOC = 180^0 - 2\times{\angle OAC} $
$\hspace{15mm}= 180^0 - 2\times{15^0}$
$\hspace{15mm}=180^0 -30^0$
Similarly in $\triangle BOC$, at the vertex,
$\angle BOC = 180^0- 2\times{50^0} = 80^0$.
$\angle AOB = \angle AOC - \angle BOC $
$\hspace{15mm}= 150^0 - 80^0 $
$\hspace{15mm}= 70^0$
Answer: Option b: $70^0$.
Key concepts used: Identifying two target triangles from two given angles -- on end state analysis identifying the end requirement as the difference of two vertex angles of the two triangles of
interest -- identifying the triangles as isosceles -- deducing the vertex angles of the two triangles from two equal base angles -- getting the target value of the angle as a difference of two vertex
Ability to arrive at the target in stages using related geometric entities and concepts starting from the given values.
Chosen Problem 2.
Two chords $AB$ and $CD$ in a circle subtend angles $90^0$ and $60^0$ respectively at the centre $O$. If the length of the chord $AB = 4$cm then length of the chord $CD$ (in cm) is,
a. $4\sqrt{2}$
b. $2$
c. $2\sqrt{2}$
d. $\sqrt{2}$
Problem analysis
This is a two chord circle problem where two triangles with vertices at the centre of the circle are involved. The base of the two triangles are the chords.
We have to establish useful relationship between the two triangles so that the unknown chord length can be deduced from the known length of the chord of the first triangle with angle subtended as $90
^0$ at the centre.
As we see it, the first triangle is a right triangle with two equal sides as radii and the given known value of hypotenuse which is the chord $AB$. We can then easily deduce the length of the radius
by applying the most basic Pythagoras theorem.
We go over next to the second triangle with this known value of radius and then the final result is only a step away.
Problem solving execution
In right $\triangle AOB$,
$AO=OB=R$, say, and $AB=4$cm.
So by Pythagoras theorem,
$2R^2 = 4^2=16$,
Or, $R^2 = 8$,
Or, $R=2\sqrt{2}$cm.
In the second $\triangle COD$, $\angle COD=60^0$ and as $CO=OD=R$, the triangle is isosceles with equal base angles summing up to $180^0 - 60^0=120^0$. These two angles are also then $60^0$ and the $
\triangle COD$ turns out to be an equilateral one. This is a rich geometric concept under the class equilateral triangle conditions.
Rich concept of Equilateral triangle conditions
The rich concept that we have used here says,
An isosceles triangle of equal base angles and vertex angle of $60^0$ is an equilateral triangle.
This concept can be confirmed easily,
$\text{Base angle } = \frac{1}{2}(180^0 - 60^0) = 60^0$.
Thus in the equilateral $\triangle COD$,
$CD = R = 2\sqrt{2}$cm.
Ans. Option c: $2\sqrt{2}$.
Key concepts used: Two equal sides of a right triangle immediately urges us to use Pythagoras theorem and determines the length of radius $R$ -- while on the second triangle, the vertex angle of $60^
0$ in an isosceles triangle classifies the triangle as equilateral and hands us the solution without any more deduction.
A roundabout way to the solution
None can miss the first part of getting the value of radius $R=2\sqrt{2}$.
But while on the $\triangle COD$, with a little bit of imagination you can cook up the relation,
$CD = 2R\cos \text{(base angle)}$
$\hspace{8mm}=2R\cos 60^0 $
$\hspace{8mm}=R $
$\hspace{8mm}= 2\sqrt{2}$, as $\cos 60^0 = \frac{1}{2}$
This solution takes more steps as the rich concept of Equilateral triangle conditions is not used.
Nevertheless it shows different facets of the solution by the application of Many ways technique.
We have shown two elegant methods of solution each with its own spcialties; shape analysis and concept based deductive reasoning lying at the heart of the few step solutions.
In the first solution assessment of what is required with respect to what are given and use of the rich resource of equal radii as two sides of a triangle in a circle, quickly handed us the solution.
The second elegant solution saw the use of the most basic geometric concept of Pythagoras theorem to determine the first unknown of length of the radius and then by using the rich concept of
Equilateral triangle conditions identification of the second triangle as an equilateral triangle immediately provided us the solution.
In the second problem by following Many ways technique we have shown a second method of solution that missed the rich geometric concept used in the elegant solution and consequently took more steps
to the solution. Though this solution is inefficient, practice of Many ways technique enhances the power of your problem solving skill in general.
Guided help on Geometry in Suresolv
To get the best results out of the extensive range of articles of tutorials, questions and solutions on Geometry in Suresolv, follow the guide,
Suresolv Geometry Reading and Practice Guide for SSC CHSL, SSC CGL, SSC CGL Tier II and Other Competitive exams.
The guide list of articles includes ALL articles on Geometry and relevant topics in Suresolv and is up-to-date.
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When to arrive at a queue with earliness, tardiness and waiting costs
We consider a queueing facility where customers decide when to arrive. All customers have the same desired arrival time (w.l.o.g. time zero). There is one server, and the service times are
independent and exponentially distributed. The total number of customers that demand service is random, and follows the Poisson distribution. Each customer wishes to minimize the sum of three costs:
earliness, tardiness and waiting. We assume that all three costs are linear with time and are defined as follows. Earliness is the time between arrival and time zero, if there is any. Tardiness is
simply the time of entering service, if it is after time zero. Waiting time is the time from arrival until entering service. We focus on customers’ rational behavior, assuming that each customer
wants to minimize his total cost, and in particular, we seek a symmetric Nash equilibrium strategy. We show that such a strategy is mixed, unless trivialities occur. We construct a set of equations
that its solution provides the symmetric Nash equilibrium. The solution is a continuous distribution on the real line. We also compare the socially optimal solution (that is, the one that minimizes
total cost across all customers) to the overall cost resulting from the Nash equilibrium.
All Science Journal Classification (ASJC) codes
• Software
• Modelling and Simulation
• Hardware and Architecture
• Computer Networks and Communications
Dive into the research topics of 'When to arrive at a queue with earliness, tardiness and waiting costs'. Together they form a unique fingerprint.
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Learn Wrapping Up – The Last Algorithms Course You'll Want (Part 2)
Check out a free preview of the full The Last Algorithms Course You'll Want (Part 2) course
The "Wrapping Up" Lesson is part of the full, The Last Algorithms Course You'll Want (Part 2) course featured in this preview video. Here's what you'd learn in this lesson:
ThePrimeagen wraps up the course by summarizing the topics covered, such as trees, graphs, analysis of algorithms, and useful algorithms like union find path compression and B-trees. They also answer
some questions from the audience, including the difference between dynamic programming and recursion, the drawbacks of dynamic programming, and the best programming languages for implementing data
Transcript from the "Wrapping Up" Lesson
>> I could do a Q and A. Also use Vim. And if you don't understand this course and it's too hard, remember I have a first one that's for free forever. Free forever algorithms course on
frontendmasters.com. Aljean, so I'm reading the chat now.
>> Is there any specific reason you use three hash functions?
>> No, I just did Grab a set of hash functions.
>> Over sufficient over a sufficiently large period of time wouldn't we always say things are available or shouldn't there be a strategy to clear the bloom filter?
>> So that would be where you a, if you have a sufficiently large amount of data like a megabyte of ones, is a million ones, and so depending on how many index points you get per item, it should be
pretty sparse for a while, it obviously at some point.
I mean, again, these are like more practical questions which is how long do you keep something running before you reset it, is the reset worth it? Meaning that you don't go and check things up. So
you got to play with that because if you're actually using when you're using it to try to speed up a system practically, which means that there is no right answer, it depends on the problem and how
much memory.
I hope at the end of this you never wanna look at algorithms again. We just went way hard and you're just like, that's it, and you just never wanna do it again. That's why the other one's the one you
need because you want that for interviewing, this one's the one you want to stop wanting ever again.
>> Could you briefly describe the difference between dynamic programming and recursion?
>> Yeah, recursion uses a stack and calls itself and you may solve the same sub-problem multiple times, as we saw with Fibonacci, over and over and over again.
>> Are there any drawbacks to dynamic program?
>> It's hard, it's really hard.
Like really good dynamic programming problems are hard until you understand it, then it's simple. And so if you actually had like if you really did a dynamic programming problem for your job, for
somebody to look at it, they'd be like, what the hell am I even looking at?
This makes no sense. And so until they understand it, you completely don't understand it. Recursion tends to be really elegant, right? You can look at it and just totally understand everything that's
happening, and so it's nice. Skill issues, dynamic programming is the epitome of skill issues.
>> Do you think that there's like a particular area of math study, combinatorics, something like that that would lend to understanding dynamic programming problems better?
>> I don't know, I don't know if that's true. At one point that was pretty good at the old combinatorics. I can't remember any of that stuff at this point, the permutation. I could use to do the 5C4
and be able to tell you all that stuff, now I can't remember any of that stuff, it's been too long since discrete math.
But that never helped me. I never personally got help from it, discrete math didn't do a lot for me.
>> Got you.
>> I think one of the problems is I also took discrete math before I took a data structure course. And so I solving all these theoretical problems that I never actually seen before, and then all of a
sudden it was really hard to relate that.
And then when I did the data structures course, I looked back on the on the discrete math I was like, that makes perfect sense why you do that? And then it was like really valuable but that's like
such a value in the past problem.
>> Where do you work?
>> I work at Netflix, by the way. Okay, I'm almost at my ten-year anniversary, okay, and Netflix. At some point I'm not gonna work at Netflix, I won't be able to say that anymore and it'll be a sad
day. By the way, just casual flex, not a big deal, whatever.
>> What language do you think lends itself well to implementing these kinds of things?
>> Any language but Rust. Rust is a horrible language to implement data structures in, it's just awful, it's just awful. It's just the worst thing in the universe, don't do Rust and data structures,
You'll get stuck on a linked list and then you'll quit. C is great because C gives you the actual picture, but you can also use things like, you can use TypeScript, you can use Go. Go kind of sucks
cuz Go doesn't have a really strong generics and methods, and so it's just like it's not as good for this.
C, TypeScript, those are really easy ones to do.
>> I did your previous one in C Sharp.
>> Okay, C Sharp would be a good one, Java, great one to do these things in, they're very concrete and simple. OCaml, I don't know OCaml good enough to know if OCaml is great for this kind of stuff.
I assume it is because it has a really strong type system, and it's garbage collected, which means you can do all the things that Rust can't easily. All right, so there you go, that is the entirety
of the course, that's what I wanted to go through. I really wanted to emphasize the fun stuff at the beginning, the trees, the graphs, and really kinda dive a little bit into some analysis.
Hopefully, you like the Prim's algorithm analysis where we actually broke it down and went several levels down. I really liked union find path compression, that's a pretty useful algorithm just to
know because there are times where you want to be able to shortcut something but you don't wanna always be able to do it.
B-trees, super, super-duper useful. You could imagine you have you can have, there's just situations, I've ran into these range to binary trees where I want to have a binary tree that has ranges to
be able to find quick values out and spend super useful to use concepts like it or M -Way trees.
So hopefully everyone learned something. Does everyone feel pretty good? Do you feel like you've gotten smarter, faster, stronger? All right, fantastic. Well, the name,
>> [APPLAUSE]
>> Right there, is the Prime Jam. All right, thank you, appreciate that.
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Sound Doppler Shift Calculator
he Doppler Effect calculator for sound waves calculates the observed frequency and wavelength of a sound wave given the source frequency, speed of source, and speed of sound in the air.
The user can input the source frequency in Hertz (Hz), the speed of the source in meters per second (m/s), and the speed of sound in air in meters per second (m/s).
The calculator uses the formula for the Doppler effect (can be found below the calculator), which relates the observed frequency to the source frequency, speed of the source, and speed of sound in
air. The formula accounts for the change in frequency due to the relative motion of the source and the observer, resulting in either a higher or lower pitch. A negative speed of the source indicates
that it is moving away from the observer.
The Doppler Effect
Source: http://en.wikipedia.org/wiki/Doppler_effect
The Doppler effect is the change in frequency of a wave for an observer moving relative to its source. It is commonly heard when a vehicle sounding a siren or horn approaches, passes and moves away
from an observer. The received frequency is higher (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower during the moving away.
The relative changes in frequency can be explained as follows. When the source of the waves is moving toward the observer, each successive wave crest is emitted from a position closer to the observer
than the previous wave. Therefore each wave takes slightly less time to reach the observer than the previous wave. Therefore the time between the arrival of successive wave crests at the observer is
reduced, causing an increase in the frequency. While they are traveling, the distance between successive wavefronts is reduced; so the waves "bunch together". Conversely, if the source of waves is
moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the
frequency. The distance between successive wavefronts is increased, so the waves "spread out".
The Doppler effect for sound can be expressed as follows:
Frequency change
Wavelength change
For the approaching source, the speed v' should be negative; for receding source, speed v' should be positive.
v - the speed of sound in air. By default, it is equal to the speed of sound in the dry air at 20 degrees Centigrade, see Sound Speed in Gases
URL copiée dans le presse-papiers
Calculatrices similaires
PLANETCALC, Sound Doppler Shift Calculator
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Seminars — LAGP.
Room P3.10, Mathematics Building
TQFT approach to meromorphic connections I
The moduli space of flat $C^\infty$ connections on rank $n$ vector bundles on a smooth complex algebraic curve $\Sigma$ has an explicit description as the space of representations of the fundamental
group: $${\mathcal M}_B = {\rm Hom}(\pi_1(\Sigma),G)/G,\qquad G={\rm GL}_n({\mathbb C})$$ often called the character variety (or Betti moduli space). It has an algebraic Poisson structure with
symplectic leaves ${\mathcal M}_B({\bf{\mathcal C}})\subset {\mathcal M}_B$ given by fixing the conjugacy classes of monodromy around each puncture. Choosing suitable generators of $\pi_1(\Sigma)$
this becomes the quotient by $G$ of a space of matrices satisfying a relation of the form $$[A_1,B_1]\cdots[A_g,B_g]M_1\cdots M_m=1$$ where $[a,b]=aba^{-1}b^{-1}$. Thus ${\mathcal M}_B$ “looks like”
a multiplicative version of a symplectic (Marsden-Weinstein) quotient, with $G$-valued moment map $\mu$ given by the left-hand side of the relation, so that ${\mathcal M}_B=\mu^{-1}(1)/G$. This
“quasi-Hamiltonian” theory was set-up by Alekseev-Malkin-Meinrenken, and leads to the construction of the symplectic manifolds ${\mathcal M}_B({\bf \mathcal C})$ as the multiplicative symplectic
quotient of the fusion product of two basic types pieces (conjugacy classes ${\mathcal C}$ and doubles ${\rm \bf D}$).
However this story is just the tip of the iceberg: In algebraic geometry the space ${\mathcal M}_B$ just parametrises the special class of algebraic connections on rank $n$ algebraic vector bundles
on $\Sigma$ with tame/regular singular behaviour at the punctures. In this course I’ll first review the topological description of the full category of algebraic connections in terms of Stokes local
systems, and the resulting explicit presentations of the wild character varieties (many of which go back to Birkhoff 1913). In turn I’ll describe the extension of the above story, constructing the
wild character varieties as algebraic symplectic/Poisson varieties, by fusing together and reducing some new basic pieces (the fission spaces ${\mathcal A}$ and ${\mathcal B}$).
If time permits I'll discuss other topics such as
1. the fact the Drinfeld-Jimbo quantum group quantises a quite simple wild character variety,
2. the notion of wild Riemann surface and the resulting wild mapping group action on the wild character varieties,
3. the upgrading of the symplectic structure to a hyperkahler structure, yielding the link to meromorphic Higgs bundles,
4. the quasi-Hamiltonian fusion approach to complex WKB.
Some references/sources:
Quasi-Hamiltonian geometry (for compact Lie groups) was defined in:
• Alekseev, Malkin, Meinrenken, Lie group valued moment maps, math.DG/9707021
and the extension of this formalism to complex groups that we use is in:
• Alekseev, Bursztyn, Meinrenken, Pure spinors on Lie groups, arXiv:0709.1452
The wild extension is in the sequence of papers:
An earlier analytic approach to these symplectic manifolds is in P.Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137-205, and the hyperkahler upgrade is in:
• P. Boalch and O. Biquard, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179-204
A “simple as possible” description of several intrinsic approaches to Stokes data for general linear groups is in
The introduction of this paper aims to give a good guide to the Stokes data literature.
Several explicit descriptions of some of the simplest examples is in arXiv:1501.00930 (Wild Character Varieties, points on the Riemann sphere and Calabi's examples), and reviews of several different
directions related to this story are in arXiv:1305.6593, arXiv:1703.10376.
Funded under FCT projects UIDB/MAT/04459/2020 and PTDC/MAT-PUR/30234/2017.
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How do you make a cumulative graph in Excel?
How do you make a cumulative graph in Excel?
Click the “Insert” tab at the top of the window and select your desired chart type from the “Charts” section of the ribbon. Your cumulative chart will be generated and displayed on your worksheet.
What is a cumulative line graph?
Use a cumulative line chart when you have one important grouping representing an ordered set of data and one value to show, summed over time.
How do you make a cumulative bar graph on Excel?
How to Make a Stacked Area Chart in Excel
1. Enter the data in a worksheet and highlight the data.
2. Click the Insert tab and click Chart. Click Area and click Stacked Area.
What are cumulative graphs used for?
Cumulative graphs are used: When progress toward a predetermined number of behaviors should be demonstrated.
What is cumulative bar graph?
The stacked bar chart (aka stacked bar graph) extends the standard bar chart from looking at numeric values across one categorical variable to two. Each bar in a standard bar chart is divided into a
number of sub-bars stacked end to end, each one corresponding to a level of the second categorical variable.
What does cumulative mean in Excel?
Excel Cumulative Sum – easy way to calculate running total. A running total, or cumulative sum, is a sequence of partial sums of a given data set. It is used to show the summation of data as it grows
with time (updated every time a new number is added to the sequence).
What is cumulative data?
The definition of cumulative data is information gathered over a period of time. An example of cumulative data is a graph showing how a company’s sales have increased since the beginning of the year.
When would you use cumulative data?
Cumulative frequency is used to determine the number of observations that lie above (or below) a particular value in a data set. The cumulative frequency is calculated using a frequency distribution
table, which can be constructed from stem and leaf plots or directly from the data.
What is cumulative graph?
A cumulative frequency graph shows the total number of values that fall below the upper boundary of each variable. All this means is that it represents the running-total of frequencies.
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Math Colloquia - Structural stability of meandering-hyperbolic group actions
Sullivan sketched a proof of his structural stability theorem for differentiabl group actions satisfying certain expansion-hyperbolicity axioms.
We relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces.
This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds.
At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable.
We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.
This is a joint work with Michael Kapovich and Jaejeong Lee.
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1—10 of 16 matching pages
Wagon), published by Key College Press in 2000, and
A Radical Approach to Lebesgue’s Theory of Integration
, published by the Mathematical Association of America and Cambridge University Press in 2007. …
Lebesgue Constants
1.8.8 $L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left|\sin\left(n+\frac{1}{2}\right)t% \right|}{\sin\left(\frac{1}{2}t\right)}\,\mathrm{d}t,$ $n=0,1,\dots$.
1.8.9 $L_{n}\sim(4/{\pi}^{2})\ln n;$
Riemann–Lebesgue Lemma
$x,y$ real variables.
$L^{2}\left(X,\,\mathrm{d}\alpha\right)$ the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$.
§1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$
For a
–Stieltjes measure
be the space of all
–Stieltjes measurable complex-valued functions on
which are square integrable with respect to
, …The space
becomes a separable Hilbert space with inner product …
Eigenfunctions corresponding to the continuous spectrum are non-
functions. …
The well must be
deep and broad
enough to allow existence of such
discrete states. …
Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being
and forming a complete set. …
The spectrum is mixed, as in §
, the positive energy, non-
, scattering states are the subject of Chapter
. …
with an infinite set of orthonormal
eigenfunctions …
The bound state
eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§
, and the
-function normalized (non-
) in Chapter
, where the solutions appear as Whittaker functions. …
The fact that non-
eigenstates may be expressed in terms or (infinite) sums of
functions allows a reformulation of scattering theory in atomic physics wherein no non-
functions need appear. …
Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals
A more general concept of integrability of a function on a bounded or unbounded interval is
Lebesgue integrability
, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for
. …Similarly the Stieltjes integral can be generalized to a
Lebesgue–Stieltjes integral
with respect to the
Lebesgue-Stieltjes measure $\,\mathrm{d}\mu(x)$
and it is well defined for functions
which are integrable with respect to that more general measure. … …
nondecreasing on the closure
of an interval
, the measure
absolutely continuous
is continuous and there exists a
weight function $w(x)\geq 0$
, Riemann (or
) integrable on finite subintervals of
, such that …
to the maximum error of the minimax polynomial
is bounded by
, where
is the
Lebesgue constant
for Fourier series; see §
. … Moreover, the set of minimax approximations
requires the calculation and storage of
coefficients, whereas the corresponding set of Chebyshev-series approximations requires only
coefficients. …
Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.
, a function
which is absolutely
integrable on every compact subset of
) such that …More generally, for
a nondecreasing function the corresponding
–Stieltjes measure
(see §
) can be considered as a distribution: …
is the
–Stieltjes measure
corresponding to
(see §
), formula (
) is a special case of (
), (
) for that choice of
. …
More generally than (
may be replaced in (
) by
, where the measure
is the
–Stieltjes measure
corresponding to a bounded nondecreasing function
on the closure of
with an infinite number of points of increase, and such that
for all
. …
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Sympathetic Vibratory Physics | 14.31 - Preponderance Russell
"All form is generated from the One source of thinking Mind by a preponderance of the concentrative, contractive pressures of the centripetal force of thinking." Russell, The Universal One
"All form is radiated back into the One source of thinking Mind by a preponderance of the decentrative, expansive pressures of the centrifugal force of thinking." Russell, The Universal One
"All idea is registered in the little particles heretofore referred to as light units. These units of light, heat, sex, electricity, and magnetism are all male and all female. Every unit is either
preponderantly male or preponderantly female. Just so is every unit either preponderantly electric or preponderantly magnetic. Just so is every unit either preponderantly negatively or preponderantly
positively, electromagnetic. Just so is every unit preponderantly generative or preponderantly radiative. And each unit is all of these. And each unit is variable, becoming preponderantly one or
another of these in its turn, from the beginning to the end of its being." Russell, The Universal One
"All mass is both electric and magnetic.
"All mass simultaneously expresses both opposites of all effects of motion, and each opposite is cumulatively preponderant in sequence.
"All electro-magnetic mass forms into systems of units which revolve in spiral orbits both centripetally toward and centrifugally away from nucleal centers.
"All preponderantly charging systems are positive systems.
"All preponderantly discharging systems are negative systems.
"All preponderantly contracting systems are positive systems.
"All preponderantly expanding systems are negative systems.
"All systems whose spirals are preponderantly closing spirals are positive systems.
"All systems whose spirals are preponderantly opening spirals are negative systems.
"All systems of preponderantly lessening volume are positive systems.
"All systems of preponderantly increasing volume are negative systems.
"All systems of preponderantly increasing potential are positive systems.
"All systems of preponderantly lowering potential are negative systems.
"All preponderantly integrating systems are positive systems.
"All preponderantly disintegrating systems are negative systems.
"All preponderantly generating systems are positive systems.
"All preponderantly radiating systems are negative systems.
"All preponderantly heating systems are positive systems.
"All preponderantly cooling systems are negative systems." Russell, The Universal One, pages 67-68
"All form is generated from the One source of thinking Mind by a preponderance of the concentrative, contractive pressures of the centripetal force of thinking.
"All form is radiated back into the One source of thinking Mind by a preponderance of the decentrative, expansive pressures of the centrifugal force of thinking." Russell, The Universal One, Book 01
- Chapter 03 - Mind, The One Universal Substance)
Figure 14.09 - Force Contracts to Center - Energy Radiates from Center.
Taking an ordinary sine wave pattern as a SYMBOL over time of Expansion then Contraction it can be visualized how one enharmonic chord decreates while a harmonic chord creates over time. (The notes
shown are arbitrary and symbolic and do not represent any particular functional chord.)
Notice the triplet of notes (three notes sounded simultaneously) forming a chord. These REPRESENT the three currents (three notes) making up a Whole Chord or Whole Flow. Keely's use of musical
triplets then is an accurate depiction of the structure of Whole Flows composed of thirds, sixths and ninths, as functional portions, of that Flow.
See Also
See Also
Dynaspheric Force
Etheric Elements
Father-Mother Principle
Light Units
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Multiplying Complex Numbers
Question Video: Multiplying Complex Numbers Mathematics • First Year of Secondary School
What is β 7π (β 5 + 5π )?
Video Transcript
What is negative seven π multiplied by negative five plus five π ?
We have a complex number negative five plus five π and we want to multiply it by a purely imaginary number negative seven π . And we know that multiplying complex numbers is just like
multiplying algebraic expressions. Here, we can apply the distributive property for expanding brackets. We multiply each part inside the bracket by the number on the outside. Thatβ s negative seven
π multiplied by negative five which is 35π and negative seven π multiplied by five π which is negative 35π squared.
And here, we recall the fact that π is the solution to the equation π ₯ squared equals negative one such that π squared must be equal to negative one. So negative 35π squared is the same as
negative 35 multiplied by negative one which is simply 35. And since we now have a complex number which is of course a result of adding a real and a purely imaginary number, we write it as 35 plus
35π .
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Problems & Exercises
2.1 Displacement
Find the following for path A in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Find the following for path B in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Find the following for path C in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Find the following for path D in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
2.3 Time, Velocity, and Speed
(a) Calculate Earth's average speed relative to the Sun. (b) What is its average velocity over a period of one year?
A helicopter blade spins at exactly 100 revolutions per minute. Its tip is 5.00 m from the center of rotation. (a) Calculate the average speed of the blade tip in the helicopter's frame of reference.
(b) What is its average velocity over one revolution?
The North American and European continents are moving apart at a rate of about 3 cm/y. At this rate, how long will it take them to drift 500 km farther apart than they are at present?
Land west of the San Andreas fault in southern California is moving at an average velocity of about 6 cm/y northwest relative to land east of the fault. Los Angeles is west of the fault and may thus
someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is 590 km northwest, assuming the motion remains
On May 26, 1934, a streamlined, stainless steel diesel train called the Zephyr set the world's nonstop long-distance speed record for trains. Its run from Denver to Chicago took 13 hours, 4 minutes,
58 seconds, and was witnessed by more than a million people along the route. The total distance traveled was 1,633.8 km. What was its average speed in km/h and m/s?
Tidal friction is slowing the rotation of Earth. As a result, the orbit of the Moon is increasing in radius at a rate of approximately 4 cm/year. Assuming this to be a constant rate, how many years
will pass before the radius of the Moon's orbit increases by $3.84×106m3.84×106m size 12{3 "." "84" times "10" rSup { size 8{6} } `m} {}$ (1 percent)?
A student drove to the university from her home and noted that the odometer reading of her car increased by 12.0 km. The trip took 18.0 min. (a) What was her average speed? (b) If the straight-line
distance from her home to the university is 10.3 km in a direction $25.0º25.0º size 12{"25" "." 0°} {}$ south of east, what was her average velocity? (c) If she returned home by the same path 7 h 30
min after she left, what were her average speed and velocity for the entire trip?
The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord to
your feet is 1.1 m long, and the nerve impulse speed is 18 m/s, how long does it take for the nerve signal to travel this distance?
Conversations with astronauts on the lunar surface were characterized by a kind of echo in which the earthbound person's voice was so loud in the astronaut's space helmet that it was picked up by the
astronaut's microphone and transmitted back to Earth. It is reasonable to assume that the echo time equals the time necessary for the radio wave to travel from Earth to the Moon and back, that is,
neglecting any time delays in the electronic equipment. Calculate the distance from Earth to the Moon given that the echo time was 2.56 s and that radio waves travel at the speed of light $(3.00×108
m/s).(3.00×108 m/s). size 12{ \( 3 "." "00" times "10" rSup { size 8{8} } " m/s" \) } {}$
A football quarterback runs 15.0 m straight down the playing field in 2.50 s. He is then hit and pushed 3.00 m straight backward in 1.75 s. He breaks the tackle and runs straight forward another 21.0
m in 5.20 s. Calculate his average velocity (a) for each of the three intervals and (b) for the entire motion.
The planetary model of the atom pictures electrons orbiting the atomic nucleus much as planets orbit the Sun. In this model you can view hydrogen, the simplest atom, as having a single electron in a
circular orbit $1.06×10−10 m1.06×10−10 m$ in diameter. (a) If the average speed of the electron in this orbit is known to be $2.20×106 m/s,2.20×106 m/s,$ calculate the number of revolutions per
second it makes about the nucleus. (b) What is the electron's average velocity?
2.4 Acceleration
A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s. What is its acceleration?
Professional Application
Dr. John Paul Stapp was a U.S. Air Force officer who studied the effects of extreme deceleration on the human body. On December 10, 1954, Stapp rode a rocket sled, accelerating from rest to a top
speed of 282 m/s (1,015 km/h) in 5.00 s, and was brought jarringly back to rest in only 1.40 s! Calculate his (a) acceleration and (b) deceleration. Express each in multiples of $gg$$(9.80 m/s2)(9.80
m/s2)$ by taking its ratio to the acceleration of gravity.
A commuter backs her car out of her garage with an acceleration of $1.40 m/s2.1.40 m/s2. size 12{1 "." "40 m/s" rSup { size 8{2} } } {}$ (a) How long does it take her to reach a speed of 2.00 m/s?
(b) If she then brakes to a stop in 0.800 s, what is her deceleration?
Assume that an intercontinental ballistic missile goes from rest to a suborbital speed of 6.50 km/s in 60.0 s (the actual speed and time are classified). What is its average acceleration in $m/s2m/s2
size 12{"m/s" rSup { size 8{2} } } {}$ and in multiples of $gg$$(9.80 m/s2)?(9.80 m/s2)?$
2.5 Motion Equations for Constant Acceleration in One Dimension
An Olympic-class sprinter starts a race with an acceleration of $4.50 m/s2.4.50 m/s2. size 12{4 "." "50 m/s" rSup { size 8{2} } } {}$ (a) What is her speed 2.40 s later? (b) Sketch a graph of her
position vs. time for this period.
A well-thrown ball is caught in a well-padded mitt. If the deceleration of the ball is $2.10×104 m/s2,2.10×104 m/s2,$ and 1.85 ms $(1 ms=10−3 s)(1 ms=10−3 s) size 12{ \( "1 ms"="10" rSup { size 8{-3}
} " s" \) } {}$ elapses from the time the ball first touches the mitt until it stops, what was the initial velocity of the ball?
A bullet in a gun is accelerated from the firing chamber to the end of the barrel at an average rate of $6.20×105 m/s26.20×105 m/s2 size 12{6 "." "20"´"10" rSup { size 8{5} } " m/s" rSup { size 8{2}
} } {}$ for $8.10×10−4 s.8.10×10−4 s.$ What is its muzzle velocity, that is, its final velocity?
(a) A light-rail commuter train accelerates at a rate of $1.35 m/s2.1.35 m/s2. size 12{1 "." "35 m/s" rSup { size 8{2} } } {}$ How long does it take to reach its top speed of 80.0 km/h, starting from
rest? (b) The same train ordinarily decelerates at a rate of $1.65 m/s2.1.65 m/s2. size 12{1 "." "65 m/s" rSup { size 8{2} } } {}$ How long does it take to come to a stop from its top speed? (c) In
emergencies, the train can decelerate more rapidly, coming to rest from 80.0 km/h in 8.30 s. What is its emergency deceleration in $m/s2?m/s2? size 12{"m/s" rSup { size 8{2} } } {}$
While entering a freeway, a car accelerates from rest at a rate of $2.40 m/s22.40 m/s2 size 12{2 "." "40 m/s" rSup { size 8{2} } } {}$ for 12.0 s. (a) Draw a sketch of the situation. (b) List the
knowns in this problem. (c) How far does the car travel in those 12.0 s? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After
choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car's final velocity? Solve for this unknown in the
same manner as in part (c), showing all steps explicitly.
At the end of a race, a runner decelerates from a velocity of 9.00 m/s at a rate of $2.00 m/s.22.00 m/s.2 size 12{2 "." "00 m/s" rSup { size 8{2} } } {}$ (a) How far does she travel in the next 5.00
s? (b) What is her final velocity? (c) Evaluate the result. Does it make sense?
Professional Application:
Blood is accelerated from rest to 30.0 cm/s in a distance of 1.80 cm by the left ventricle of the heart. (a) Make a sketch of the situation. (b) List the knowns in this problem. (c) How long does the
acceleration take? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for
the unknown, checking your units. (d) Is the answer reasonable when compared with the time for a heartbeat?
In a slap shot, a hockey player accelerates the puck from a velocity of 8.00 m/s to 40.0 m/s in the same direction. If this shot takes $3.33×10−2 s3.33×10−2 s size 12{3 "." "33"´"10" rSup { size 8
{-2} } " s"} {}$, calculate the distance over which the puck accelerates.
A powerful motorcycle can accelerate from rest to 26.8 m/s (100 km/h) in only 3.90 s. (a) What is its average acceleration? (b) How far does it travel in that time?
Freight trains can produce only relatively small accelerations and decelerations. (a) What is the final velocity of a freight train that accelerates at a rate of $0.0500 m/s20.0500 m/s2 size 12{0 "."
"0500 m/s" rSup { size 8{2} } } {}$^ for 8.00 min, starting with an initial velocity of 4.00 m/s? (b) If the train can slow down at a rate of $0.550 m/s2,0.550 m/s2, size 12{0 "." "550 m/s" rSup {
size 8{2} } } {}$ how long will it take to come to a stop from this velocity? (c) How far will it travel in each case?
A fireworks shell is accelerated from rest to a velocity of 65.0 m/s over a distance of 0.250 m. (a) How long did the acceleration last? (b) Calculate the acceleration.
A swan on a lake gets airborne by flapping its wings and running on top of the water. (a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of
$0.350 m/s2,0.350 m/s2, size 12{0 "." "350 m/s" rSup { size 8{2} } } {}$ how far will it travel before becoming airborne? (b) How long does this take?
Professional Application:
A woodpecker's brain is specially protected from large decelerations by tendon-like attachments inside the skull. While pecking on a tree, the woodpecker's head comes to a stop from an initial
velocity of 0.600 m/s in a distance of only 2.00 mm. (a) Find the acceleration in $m/s2m/s2$^ and in multiples of $gg=9.80 m/s2gg=9.80 m/s2 size 12{g left (g=9 "." "80"" m/s" rSup { size 8{2} } right
)} {}$. (b) Calculate the stopping time. (c) The tendons cradling the brain stretch, making its stopping distance 4.50 mm (greater than the head and, hence, less deceleration of the brain). What is
the brain's deceleration, expressed in multiples of $g?g?$
An unwary football player collides with a padded goalpost while running at a velocity of 7.50 m/s and comes to a full stop after compressing the padding and his body 0.350 m. (a) What is his
deceleration? (b) How long does the collision last?
In World War II, there were several reported cases of airmen who jumped from their flaming airplanes with no parachute. Some fell about 20,000 feet (6,000 m), and some of them survived, with few
life-threatening injuries. For these lucky pilots, the tree branches and snow drifts on the ground allowed their deceleration to be relatively small. If we assume that a pilot's speed upon impact was
123 mph (54 m/s), then what was his deceleration? Assume that the trees and snow stopped him over a distance of 3.0 m.
Consider a grey squirrel falling out of a tree to the ground. (a) If we ignore air resistance in this case, only for the sake of this problem, determine a squirrel's velocity just before hitting the
ground, assuming it fell from a height of 3.0 m. (b) If the squirrel stops in a distance of 2.0 cm through bending its limbs, compare its deceleration with that of the airman in the previous problem.
An express train passes through a station. It enters with an initial velocity of 22.0 m/s and decelerates at a rate of $0.150 m/s20.150 m/s2 size 12{0 "." "150 m/s" rSup { size 8{2} } } {}$ as it
goes through. The station is 210 m long. (a) How long is the nose of the train in the station? (b) How fast is it going when the nose leaves the station? (c) If the train is 130 m long, when does the
end of the train leave the station? (d) What is the velocity of the end of the train as it leaves?
Dragsters can actually reach a top speed of 145 m/s in only 4.45 s—considerably less time than given in Example 2.10 and Example 2.11. (a) Calculate the average acceleration for such a dragster. (b)
Find the final velocity of this dragster starting from rest and accelerating at the rate found in (a) for 402 m (a quarter mile) without using any information on time. (c) Why is the final velocity
greater than that used to find the average acceleration? Hint—Consider whether the assumption of constant acceleration is valid for a dragster. If not, discuss whether the acceleration would be
greater at the beginning or end of the run and what effect that would have on the final velocity.
A bicycle racer sprints at the end of a race to clinch a victory. The racer has an initial velocity of 11.5 m/s and accelerates at the rate of $0.500 m/s20.500 m/s2 size 12{0 "." "500 m/s" rSup {
size 8{2} } } {}$ for 7.00 s. (a) What is his final velocity? (b) The racer continues at this velocity to the finish line. If he was 300 m from the finish line when he started to accelerate, how much
time did he save? (c) One other racer was 5.00 m ahead when the winner started to accelerate, but he was unable to accelerate, and traveled at 11.8 m/s until the finish line. How far ahead of him (in
meters and in seconds) did the winner finish?
In 1967, New Zealander Burt Munro set the world record for an Indian motorcycle, on the Bonneville Salt Flats in Utah, with a maximum speed of 183.58 mi/h. The one-way course was 5.00 mi long.
Acceleration rates are often described by the time it takes to reach 60.0 mi/h from rest. If this time was 4.00 s, and Burt accelerated at this rate until he reached his maximum speed, how long did
it take Burt to complete the course?
(a) A world record was set for the men's 100-m dash in the 2008 Olympic Games in Beijing by Usain Bolt of Jamaica. Bolt coasted across the finish line with a time of 9.69 s. If we assume that Bolt
accelerated for 3.00 s to reach his maximum speed, and maintained that speed for the rest of the race, calculate his maximum speed and his acceleration. (b) During the same Olympics, Bolt also set
the world record in the 200-m dash with a time of 19.30 s. Using the same assumptions as for the 100-m dash, what was his maximum speed for this race?
2.7 Falling Objects
Assume air resistance is negligible unless otherwise stated.
Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial velocity of 15.0 m/s. Take the point of release to be
$y0=0.y0=0. size 12{y rSub { size 8{0} } =0} {}$
Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down with an initial velocity of 14.0 m/s from the Verrazano
-Narrows Bridge in New York City. The roadway of this bridge is 70.0 m above the water.
A basketball referee tosses the ball straight up for the starting tip-off. At what velocity must a basketball player leave the ground to rise 1.25 m above the floor in an attempt to get the ball?
A rescue helicopter is hovering over a person whose boat has sunk. One of the rescuers throws a life preserver straight down to the victim with an initial velocity of 1.40 m/s and observes that it
takes 1.8 s to reach the water. (a) List the knowns in this problem. (b) How high above the water was the preserver released? Note that the downdraft of the helicopter reduces the effects of air
resistance on the falling life preserver, so that an acceleration equal to that of gravity is reasonable.
A dolphin in an aquatic show jumps straight up out of the water at a velocity of 13.0 m/s. (a) List the knowns in this problem. (b) How high does its body rise above the water? To solve this part,
first note that the final velocity is now a known and identify its value. Then identify the unknown, and discuss how you chose the appropriate equation to solve for it. After choosing the equation,
show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long is the dolphin in the air? Neglect any effects due to its size or orientation.
A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m/s, and her takeoff point is 1.80 m above the pool. (a) How long are her feet
in the air? (b) What is her highest point above the board? (c) What is her velocity when her feet hit the water?
(a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m/s. (b) How long would it take to
reach the ground if it is thrown straight down with the same speed?
A very strong, but inept, shot putter puts the shot straight up vertically with an initial velocity of 11.0 m/s. How long does he have to get out of the way if the shot was released at a height of
2.20 m and he is 1.80 m tall?
You throw a ball straight up with an initial velocity of 15.0 m/s. It passes a tree branch on the way up at a height of 7.00 m. How much additional time will pass before the ball passes the tree
branch on the way back down?
A kangaroo can jump over an object 2.50 m high. (a) Calculate its vertical speed when it leaves the ground. (b) How long is it in the air?
Standing at the base of one of the cliffs of Mount Arapiles in Victoria, Australia, a hiker hears a rock break loose from a height of 105 m. He cannot see the rock right away but then does, 1.50 s
later. (a) How far above the hiker is the rock when he can see it? (b) How much time does he have to move before the rock hits his head?
An object is dropped from a height of 75.0 m above ground level. (a) Determine the distance traveled during the first second. (b) Determine the final velocity at which the object hits the ground. (c)
Determine the distance traveled during the last second of motion before hitting the ground.
There is a 250-m-high cliff at Half Dome in Yosemite National Park in California. Suppose a boulder breaks loose from the top of this cliff. (a) How fast will it be going when it strikes the ground?
(b) Assuming a reaction time of 0.300 s, how long will a tourist at the bottom have to get out of the way after hearing the sound of the rock breaking loose, neglecting the height of the tourist,
which would become negligible anyway if hit? The speed of sound is 335 m/s on this day.
A ball is thrown straight up. It passes a 2.00-m-high window 7.50 m off the ground on its path up and takes 1.30 s to go past the window. What was the ball's initial velocity?
Suppose you drop a rock into a dark well and, using precision equipment, you measure the time for the sound of a splash to return. (a) Neglecting the time required for sound to travel up the well,
calculate the distance to the water if the sound returns in 2.0000 s. (b) Now calculate the distance taking into account the time for sound to travel up the well. The speed of sound is 332.00 m/s in
this well.
A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just
after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 0.0800 ms $(8.00×10−5 s).(8.00×10−5 s). size 12{ \( 8 "." "00" times
"10" rSup { size 8{ - 5} } " s" \) } {}$ (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?
A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m/s upward. For the coin, find (a) the maximum height reached, (b) its position and velocity 4.00 s after
being released, and (c) the time before it hits the ground.
A soft tennis ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.10 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity
just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 3.50 ms $(3.50×10−3 s).(3.50×10−3 s). size 12{ \( 3 "." "50"
times "10" rSup { size 8{ - 3} } " s" \) } {}$ (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?
2.8 Graphical Analysis of One-Dimensional Motion
Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.
(a) By taking the slope of the curve in Figure 2.72, verify that the velocity of the jet car is 115 m/s at $t=20 s.t=20 s. size 12{t="20"`s} {}$ (b) By taking the slope of the curve at any point in
Figure 2.73, verify that the jet car's acceleration is $5.0 m/s2.5.0 m/s2. size 12{5 "." "0 m/s" rSup { size 8{2} } } {}$
Using approximate values, calculate the slope of the curve in Figure 2.74 to verify that the velocity at $t=10.0 st=10.0 s size 12{t="10"`s} {}$ is 0.208 m/s. Assume all values are known to 3
significant figures.
Using approximate values, calculate the slope of the curve in Figure 2.74 to verify that the velocity at $t=30.0 st=30.0 s$ is 0.238 m/s. Assume all values are known to 3 significant figures.
By taking the slope of the curve in Figure 2.75, verify that the acceleration is $3.2 m/s23.2 m/s2$ at $t=10 s.t=10 s. size 12{t="10"`s} {}$
Construct the displacement graph for the subway shuttle train as shown in Figure 2.30(a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the
information on acceleration and velocity given in the examples for this figure.
(a) Take the slope of the curve in Figure 2.76 to find the jogger's velocity at $t=2.5 s.t=2.5 s. size 12{t=2 "." 5`s} {}$ (b) Repeat at 7.5 s. These values must be consistent with the graph in
Figure 2.77.
A graph of $vtvt$ is shown for a world-class track sprinter in a 100-m race (see Figure 2.79). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at $t=5 s?t=5
s?$ (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?
Figure 2.80 shows the displacement graph for a particle for 5 s. Draw the corresponding velocity and acceleration graphs.
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Determining the Displacements between Positions
Question Video: Determining the Displacements between Positions Physics • First Year of Secondary School
A speedboat passes by markers at the points A, B, and C, as shown in the diagram. Positive displacement is considered to be away from A, toward C. What is the displacement of the boat from A when it
is at C minus the displacement of the boat from C when it is at A?
Video Transcript
A speedboat passes by markers at the points A, B, and C, as shown in the diagram. Positive displacement is considered to be away from A, toward C. What is the displacement of the boat from A when it
is at C minus the displacement of the boat from C when it is at A?
In this question, we’re shown a diagram of a speedboat and three different position markers. We’re told that positive displacement is defined to be away from A toward C. In other words, positive
displacement is from right to left across the screen. We need to figure out the displacement of the boat from point A when it’s at point C minus its displacement from point C when it’s at A.
Let’s start by figuring out the boat’s displacement from point A when it’s at point C. We can see from the diagram that there is a straight-line distance of 250 meters between point C and point A.
So, the magnitude of the boat’s displacement from point A must also be 250 meters. However, since displacement is a vector quantity, we also need to take direction into account. We already know that
positive displacement is considered towards the left-hand side of the screen. Since point C is to the left of point A, the displacement of the boat from point A is also in the positive direction. So,
the displacement of the boat from point A is equal to positive 250 meters.
Now let’s find the boat’s displacement from point C when it’s at point A. Again, the magnitude of this displacement must be 250 meters. However, this displacement is now towards the right-hand side
of the screen, which is the opposite direction to before. This means the boat’s displacement is now in the negative direction. So, the boat’s displacement is equal to negative 250 meters.
Now, all we need to do is work out the difference between the two values we have found. The displacement of the boat from A when it is at C minus the displacement of the boat from C when it is at A
is equal to positive 250 meters minus negative 250 meters. This is equal to 500 meters.
So, our answer to this question is that the displacement of the boat from A when it is at C minus the displacement of the boat from C when it is at A is equal to 500 meters.
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Specialization Review: Mathematics for Machine Learning
Probably you’ve already realized that for getting started with machine learning you’ll have to understand certain areas of math. I took the Mathematics of Machine Learning specialization on
Coursera last year as the first step on that path. With this review, I’d like to help those people make a decision who are also thinking about taking it.
1. What you’ll learn?
MML is a three courses specialization taught by the Imperial College of London. It covers three areas of mathematics: linear algebra, multivariate calculus, and statistics. The latter focuses mostly
on a specific technique called principal component analysis (PCA).
1.1. Linear algebra
Linear algebra plays an important role in working with data. Properties of real-world phenomena are often represented as a vector and such samples or measurements are organized as matrices. Professor
David Dye demonstrates that through the example of estimating prices of real estate based on their characteristics, such as how many bedrooms they have, their area in square meters, etc.
Image courtesy of Jakob Scholbach
In this first course, you’ll learn what vectors are, how they span space, what matrices are, and how they can operate on, and transform vectors. You’ll understand how solving linear equations through
elimination is related to geometric interpretations of matrix transformations. The fifth and last module will introduce you to eigenvectors and eigenvalues that are useful for modeling processes that
evolve in iterative steps (or in time). You’ll implement and analyze the PageRank algorithm with the help of eigenvalues.
1.2. Multivariate calculus
A big part of machine learning is being able to fit a model to data, that fitting is done through minimizing some objective function or in other words minimizing the error. Thus, in finding the
optimal solution, calculus has an important application in attempting to find parameter values where the fitting is best.
Image courtesy of John B. Lewis
In this second course, Dr. Sam Cooper will introduce you to the basics of calculus in an intuitive manner, then you’ll move on to the multivariate case and to the Taylor series. From the second
module of the course, you’ll see linear algebra applied as a means of dealing with the multivariate case. As an application of the multivariate chain rule, you learn how neural networks work and even
implement one. In the last two modules, David Dye combines all that you’ve learned so far. You’ll see gradient descent in action, that’s the heart of ML algorithms and you’ll venture into the realm
of statistics with linear and non-linear regression.
1.3. Principal Component Analysis
PCA is a technique to reduce the dimensionality of data. Sometimes certain aspects of the data are highly correlated and eventually, you can express the dataset with fewer features.
In this last course, Professor Marc Deisenroth will teach you some basics of statistics, like means and variances. Linear algebra will show appearances as a means of projecting data points to a
lower-dimensional subspace and you’ll get your hands dirty by implementing PCA yourself in Python at the end.
2. Will you enjoy it?
Definitely, the specialization is very rewarding, it isn’t too hard, yet you’ll see that much to get the gist of what ML is about. The emphasis is mostly on building mathematical intuition and less
or even not at all about grinding through formal proofs of theorems.
Especially for a guy like me who studied these subjects in the prussian-style eastern European education system, where teaching happens mostly on theorem-proof-basis, this was very enjoyable.
3. How hard is it?
The first course is super easy, I would even say that it’s even missing certain details that I would have found useful. Nevertheless, it’s not a full undergraduate-level course. The second one is a
bit harder, but still not that much and there are lots of visualizations used as a means of demonstrating what the formulas are actually doing.
The last third course requires a more mathematical maturity and it’s also more abstract than the first two ones. It will require you to be more fluent in Python and at certain places you'll feel
you're own your own when you do your homework. You'll also need to gather some details from external sources.
4. What additional resources are useful?
4.1. Textbooks
Mathematical Methods in the Physical Sciences is one of the recommended books for the first two courses. I managed to purchase a copy on eBay at a decent price.
Mathematics for Machine Learning is a book written by Professor Deisenroth and two others. At that time when I was taking the specialization, it wasn’t yet available in print, but I used the WiP
online version. You can read the details of PCA in this book and of course, it’s also very useful for understanding linear algebra and multivariate calculus.
4.2. Courses
For the first course, Professor Gil Strang’s book Introduction to Linear Algebra is one of the recommended sources. But why read, if you can watch him teaching online? MIT’s linear algebra course
(often referred to just as “18.06”) proved to be edifying especially for understanding eigenvalues deeper in the few last modules of the first course. I think I should write a review of that as well
because I ended up watching most of the lectures.
For the last course (PCA), you need to look into other sources anyway for example to grasp why a covariance matrix is always positive definite. At this point, being aware of the fact that MIT OCW is
an intellectual goldmine, a relatively new course also from Professor Strang helped me fill in the details: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (often referred to
just as “18.065”). I watched most of the lectures here as well, but particularly for PCA lectures 4 - 7 are relevant.
5. Conclusion
Was too long to read, hah? No worries, I’ll summarize what’s the catch.
• The Mathematics for Machine Learning specialization taught by three professors from Imperial College of London is worth taking, you’ll get to know to the basics of math required to get started
with ML
• The first two courses are easier and the last one is a bit more challenging and requires you to be more proficient in Python
• As neither of the three courses is full graduate-level ones, Professor David Dye mentions this fact right at the beginning, you might feel that you want more details here and there.
• It’s worth taking it. Go ahead and do it! Don't forget to shoot a photo at Imperial when you finished and share it. ;)
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Seminar Archives
Ridgway Scott - Department of Computer Science and Department of Mathematics, Univ. of Chicago
Loredana Lanzani - University of Arkansas
The classical Cauchy integral is a fundamental object of complex
analysis whose analytic properties are intimately related to the
geometric properties of its supporting curve. In this talk I will
begin by reviewing the most relevant features of the classical Cauchy
integral. I will then move on to the (surprisingly more involved)
construction of the Cauchy integral for a hypersurface in $\mathbb
C^n$. I will conclude by presenting new results joint with E. M. Stein
concerning the regularity properties of this integral and their
relations with the geometry of the hypersurface. (Time permitting) I
will discuss applications of these results to the Szeg\H o and Bergman
projections (that is, the orthogonal projections of the Lebesgue space
$L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic
Benjamin Bakker - Courant
It is well known that the extremal rays in the cone of effective curve classes on a K3 surface are generated by rational curves $C$ for which $(C,C)=-2$; a natural question to ask is whether
there is a similar characterization for a higher-dimensional holomorphic symplectic variety $X$. The intersection form is no longer a quadratic form on curve classes, but the Beauville-Bogomolov
form on $X$ induces a canonical nondegenerate form $(\cdot,\cdot)$ on $H_2(X;\mathbb{R} )$ which coincides with the intersection form if $X$ is a K3 surface. We therefore might hope that extremal
rays of effective curves in $X$ are generated by rational curves $C$ with $(C,C)=-c$ for some positive rational number $c$. In particular, if $X$ contains a Lagrangian hyperplane $\mathbb P^n\
subset X$, the class of the line $\ell\subset\mathbb P^n$ is extremal. For $X$ deformation equivalent to the Hilbert scheme of $n$ points on a K3 surface, Hassett and Tschinkel conjecture that $
(\ell,\ell)=-\frac{n+3}{2}$; this has been verified for $n<4$. In joint work with Andrei Jorza, we prove the conjecture for $n=4$, and discuss some general properties of the ring of Hodge classes
on $X$.
Patrick Fitzsimmons - UCSD
1. I'll show that Hunt's hypothesis (H) fails for Leonard Gross' infinite dimensional Brownian motion, by exhibiting a subset of the state space of the motion that is hit exactly once for certain
starting points.
2. It is well known that a Lebesgue measurable additive function from R to R is necessarily continuous (and linear). I'll show how D. Stroock's recent proof of L. Schwartz's ``Borel graph
theorem" can be adapted to show that a ``universally Gaussian measurable" and additive map from one Banach space to another is automatically continuous (and linear).
Bob Chen - UCSD
In this talk we consider the necessary and sufficient conditions for a
formal language to be represented by an infinite word. We extend our
results to the case of partial words and prove a partial uniqueness
result as well
Brandon Rhoades - USC
Let X be a finite set, $C = <c>$ be a finite cyclic group acting on $X$, and $X(q) \in N[q]$ be a polynomial with nonnegative integer coefficients. Following Reiner, Stanton, and White, we say
that the triple $(X, C, X(q))$ exhibits the $\emph{cyclic sieving phenomenon}$ if for any integer $d>0$, the number of fixed points of $c^d$ is equal to $X(\zeta^d)$, where $\zeta$ is a primitive
$|C|^{th}$ root of unity. We explain how one can use representation theory to prove instances of the cyclic sieving phenomenon involving the action of tropical Coxeter elements on (complexes
closely related to) cluster complexes. The representation theory involves cluster monomial bases of geometric realizations of finite type cluster algebras.
Alan Reid - UT Austin
In broad terms this talk will discuss how much information about a f.g. residually finite group is carried by the collection of its finite quotients. For example a precise question in this
direction (and which has been open for many years) is: Given a f.g. residually finite group G with the same collection of finite quotients as a free group of rank n, is G isomorphic to a free
group of rank n?
Albert Chau - University of British Columbia
In this talk I will discuss nonnegatively curved compact
Kahler manifolds and their classification. An overview of past results
will be given in the cases of bisectional and orthogonal bisectional
curvature. The more recent case of quadratic orthogonal bisectional
curvature will then be discussed along with recent results. The talk
is based on joint work with L.F. Tam.
Frederick Fong - Stanford University
In this talk, I will discuss my recent works on the
collapsing behavior of the Kahler-Ricci flow. The first work studies
the Kahler-Ricci flow on $P^1$-bundles over Kahler-Einstein manifolds.
We proved that if the initial Kahler metric is constructed by the
Calabi's Ansatz and is in the suitable Kahler class, the flow must
develop Type I singularity and the singularity model is $P^1 X C^n$. It
is an extension of Song-Weinkove's work on Hirzebruch surfaces. The
second work discusses the collapsing behavior in a more general
setting without any symmetry assumption. We showed that if the
limiting Kahler class of the flow is given by a holomorphic submersion
and the Ricci curvature is uniformly bounded from above with respect
to the initial metric, then the fibers will collapse in an optimal
rate, i.e. diam $\sim (T-t)^{1/2}$. It gives a partial affirmative answer to
a conjecture stated in Song-Szekelyhidi-Weinkove's work on projective
Elisenda Grigsby - Boston College
The low-dimensional topology community has been energized in recent
years by the introduction of a wealth of so-called ``homology-type"
invariants. One associates to an object in low-dimensional topology
(e.g., a link or a 3-manifold) an abstract chain complex whose
homology is an invariant of the topological object. Such invariants
arise in two apparently different ways: ``algebraically," via the
representation theory of quantum groups and ``geometrically," via
constructions in symplectic geometry. I will discuss what is known
about the relationship between two such invariants: Khovanov homology,
an ``algebraic" invariant of links and tangles defined by Khovanov and
Heegaard-Floer homology, a ``geometric" invariant of 3-manifolds
defined by Ozsvath-Szabo. The portions of the talk describing my own
work are joint with Denis Auroux and Stephan Wehrli.
Adam Jacob - Columbia University
In this talk I will describe the limiting properties
Yang-Mills flow on a holomorphic vector bundle E, in the case where
the flow does not converge. In particular I will describe how to
determine the $L^2$ limit of the curvature endomorphism along the flow.
This proves a sharp lower bound for the Hermitian-Yang-Mills
functional and thus the Yang-Mills functional, generalizing to
arbitrary dimension a formula of Atiyah and Bott first proven on
Riemann surfaces. I will then show how to use this result to identify
the limiting bundle along the flow, which turns out to be independent
of metric and uniquely determined by the isomorphism class of $E$.
Jianfeng Lu - Courant Institute
Modern material sciences focus on studies on the microscopic scale. This calls for mathematical understanding of electronic structure and atomistic models, and also their connections to continuum
theories. In this talk, we will discuss some recent works where we develop and generalize ideas and tools from mathematical analysis of continuum theories to these microscopic models. We will
focus on macroscopic limit and microstructure pattern formation of electronic structure models.
Jason Teutsch - Johns Hopkins
Wei-Kuo Chen - University of California, Irvine
Mark Tiefenbruck - UCSD
We will re-enact the story of proving a conjecture by Anders
Claesson and Svante Linusson. Along the way, we are naturally led to a
particular bijection between matchings and permutations; however, this
bijection is somewhat flawed. We will discover a general technique to
circumvent these flaws, leading to a new bijection that completes the
proof. Time permitting, we will also use this technique to prove a
recent conjecture by Miles Jones and Jeffrey Remmel. This talk should
be accessible to graduate students of all areas of math.
Tommy Occhipinti - University of California, Irvine
The existence of elliptic curves of large rank over number
fields is an open question, but it has been known for decades that
there exist elliptic curves of arbitrarily large rank over global
function fields. In this talk we will discuss some results of Ulmer
that showcase the ubiquity of large ranks over function fields, as
well as some newer work in the area.
Ioan Bejenaru - University of Chicago
I will introduce and motivate the Schroedinger Map problem. I will review the results obtain in the field. Then I will talk about the global regularity of equivariant maps in two dimensions with
large data.
James Mckernan - MIT
Compact Riemann surfaces are naturally divided into three types; the Riemann sphere, elliptic curves
and curves of higher genus.
We will explain the conjectural analogue of this classification in higher dimensions, recent progress towards
this classification and some open problems.
Elham Izadi - University of Georgia, Athens
Given a curve (Riemann surface), one can construct an abelian variety: its Jacobian. Abelian
varieties are quotients of vector spaces by lattices. The classical Torelli theorem states that the Jacobian
determines the curve. We discuss some generalizations of this and their history.
Zhiwei Yun - MIT
We will use geometric Langlands theory to solve two problems
concerning number fields. One is Serre's question of whether there
exist motives over $\Bbb Q$ with motivic Galois group of type $E_8$ or $G_2$; the other is the concrete question of whether there are Galois extensions of $\Bbb Q$ with Galois group $E_8(p)$ or
$G_2(p)$ (the finite simple groups of Lie type), for sufficiently large primes $p$.
The answer to both questions is ``YES".
Please note the change of day for this week's colloquium.
Michael Shulman - UCSD
The new subject of "homotopy type theory" has been created by a fusion
of homotopy theory, higher category theory, and constructive type
theory. On one hand, it enables us to apply homotopical ideas in type
theory, giving new ways to deal with things like proof-irrelevance,
singleton elimination, type equivalence, and universes. On the other
hand, it gives us a formal language in which to do homotopy theory. A
proof written in this language will automatically be valid in many
different ``homotopy theories", and can also be formalized and checked
by a computer proof assistant. Taken to an extreme, the subject
offers the possibility of a new foundation for mathematics in which
the basic objects are homotopy types, rather than sets.
This is the beginning of a weekly seminar which will introduce the
subject and some of its highlights, assuming no background in either
homotopy theory or type theory. In addition to the mathematical
theory, we will learn to formalize it using the computer proof
assistant Coq.
Nicolaos Kapouleas - Brown University
In the first part of the talk I will discuss doubling
constructions. In particular I will discuss in some detail a recent
doubling construction for an equatorial two-sphere in the round
three-sphere, and also potential generalizations for self-shrinkers of
the Mean Curvature flow. In the second part of the talk I will briefly
discuss the current understanding of desingularization constructions
for minimal surfaces and self-shrinkers. In the third and final part I
will discuss open uniqueness questions for closed embedded minimal
surfaces in the round three-sphere inspired by the above
Ruochuan Liu - University of Michigan
Sara Pollock - UCSD
In this talk, we will discuss a goal-oriented adaptive method for second order semilinear PDEs. In goal-oriented methods we are concerned with approximating a given quantity of interest, a
function of the weak solution to the PDE. In linear problems, this is accomplished by defining a dual problem or formal adjoint and solving the two problems simultaneously. For the semilinear
case, we will discuss the formation of the linearized and approximate dual problems. We will then review the standard contraction framework and discuss some additional estimates used to show
convergence of the method. Finally, we introduce an appropriate notion of error to derive a strong contraction result.
Adriano Garsia - UCSD
The Shuffle Conjecture gives a Combinatorial setting to
the bi-graded Frobenius Characteristic of the Diagonal
Harmonic Module of $S_n$. We report here on the progress
in joint work with Angela Hicks in a three year effort to
prove this conjecture. Angela Hicks reduced a combinatorial side of
the problem to proving a deceptively simple property of a remarkable
family of polynomials in a single variable $x$ with coefficients
polynomials in $N[q]$. In this lecture and possibly following ones we
describe what remains to be done to resolve this decade old Algebraic
Combinatorial problem.
Brian Rider - University of Colorado, Boulder
The largest eigenvalue of a rank one perturbation of random hermitian matrix is known to exhibit a phase transition. If the perturbation is small, one sees the famous Tracy-Widom law; if the
perturbation is large, the result is simple Gaussian fluctuations. Further, there is a scaling window about a critical value of the perturbation which leads to a new one parameter family of limit
laws. The same phenomena exists for random sample covariance matrices in which one of the population eigenvalues is "spiked", or takes a value other than one. Bloemendal-Virag have shown how this
picture persists in the context of the general beta ensembles, giving new formulations of the discovered critical limit laws (among other things). Yet another route, explained here, is to go
through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to limit distributions. (Joint work
with Jose Ramirez.)
Dino Lorenzini - University of Georgia
Let $K$ be a field. Suppose that the algebraic variety is given by the
set of common solutions
to a system of polynomials in n variables with coefficients in $K$.
Given a solution $P=(a_1,\dots,a_n)$ of this system
with coordinates in the algebraic closure of $K$, we associate to it
an integer called the degree of $P$,
and defined to be the degree of the extension $K(a_1,\dots,a_n)$ over
$K$. When all coordinates $a_i$ belong to $K$,
$P$ is called a $K$-rational point, and its degree is 1. The index of
the variety is the greatest common divisor of all possible degrees of
points on $P$. It is clear that if there exists a $K$-rational point
on the variety, then the index equals 1. The converse is not true in
general. We shall discuss in this talk various properties of the index,
including how to compute it when $K$ is a complete local field using
data pertaining only to a reduction of the variety. This is joint work
with O. Gabber and Q. Liu.
Frank Sottile - TAMU
Building on work of Jordan from 1870, in 1979 Harris showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the Galois group of an associated field
extension. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. These Galois groups are difficult to determine, yet they contain subtle
geometric information.
Exploiting Harris's equivalence, Leykin and I used numerical homotopy continuation to compute Galois groups of problems involving mostly divisor Schubert classes, finding all to be the full
symmetric group. (This included one problem with 17589 solutions.) With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least
alternating Galois group.
My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds,
investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy
continuation, and require the development of new algorithms and software.
Frank Sottile - Texas A&M University
Computing, counting, or even deciding on the existence
of real solutions to a system of polynomial equations
is a very challenging problem that is important in many
applications of mathematics. There is an emerging
landscape of structure in the possible numbers of
real solutions to systems of polynomial equations.
These include fewnomial upper bounds, gaps or congruences,
and lower bounds. My talk will survey what is known
about these bounds, focussing on lower bounds---which
are existence proofs of solutions---and open problems,
including some concrete challenges.
Paul Norbury - University of Melbourne
Eynard and Orantin have recently defined invariants of any
compact Riemann surface equipped with two meromorphic functions, as a
tool for studying enumerative problems in geometry. I will give a brief
introduction to these invariants and describe a particular example that
encodes the stationary Gromov-Witten invariants of the two-sphere. This
brings new insight into the well-studied problem of the Gromov-Witten
invariants of the two-sphere. Conversely, we gain insight into the
Eynard-Orantin invariants showing that in this example they are related
to the Landau-Ginzburg model dual to Gromov-Witten invariants.
Adriano Garsia - UCSD
We give a bijection between Parking Functions and Crossed Bar Diagrams and use it to derive properties of Parking Functions that play a crucial role in our attack on the Shuffle Conjecture. This
is Bijective Combinatorics at its best, in that a simple bijective weight preserving correspondence between
two families of objects allows us to prove difficult results about one family by working with the other.
The talk will be completely self contained.
Jim Isenberg - University of Oregon
We discuss the detailed nature of the geometry of
rotationally symmetric degenerate neckpinch singularities which
develop in the course of Ricci flow.
Tonci Antunovic - UC Berkeley
Johanna Hennig - UCSD
Inspired by Prof. Jeff Rabin's talk ``What is a Supermanifold?" we will
discuss the analogous algebraic question: What is a Superalgebra? We
will start with basic definitions and examples, and then focus in
particular on Lie Superalgebras and their connections to other areas
of mathematics.
Yu-Chen Shu - National Cheng Kung Univ., Taiwan
In this talk, a coupling interface method (CIM) is proposed for solving complex interface problems, especially for Poisson-Boltzmann Equation. The coefficients, the source terms, and the
solutions may be discontinuous or singular across the interfaces. The method uses adaptive-order strategy and is extended to high dimensions through a dimension-by-dimension approach. To connect
information from each dimension, a coupled equation for the principal derivatives is derived through the jump conditions in each coordinate direction. The cross derivatives are approximated by
one-side interpolation. The method is easy to implement and flexible to integrate with other approaches. We compare our method with some existing methods. Numerical tests are carefully performed
to show the efficiency, robustness and accuracy of our method.
Arian Maleki - Rice University
Compressed sensing refers to a growing body of techniques that `undersample' high-dimensional signals and yet recover them accurately using efficient non-linear reconstruction algorithms. Instead
of sampling a signal at a rate proportional to its frequency bandwidth, such techniques use a sampling rate proportional to the `information content' of the signal. There exist several useful
theories in the literature that promise improvements over ordinary sampling rules in recovering sparse signals. However, most questions regarding the fundamental performance limits of the
recovery algorithms are widely open. Such questions are of particular interest in the applications where we need to design the parameters of the systems in advance. In this talk, I present a new
framework that settles such questions for a large class of algorithms including the famous $\ell_1$-penalized least squares (LASSO). As a special case of our result, we will derive tight bounds
on the noise sensitivity of the LASSO. Furthermore, I will explain the implications and contributions of the new framework for some applications.
This talk is based on a joint work with David Donoho, Iain Johnstone and Andrea Montanari.
Jason Bell - Simon Fraser University
In 1983, Makar-Limanov showed that the quotient division algebra of the complex Weyl algebra contains a copy of the free algebra on two generators. This results shows that, unlike in the
commutative case, noncommutative localization can behave very pathologically. Stafford and Makar-Limanov conjectured that the following general dichotomy should hold: if a division ring is not
finite-dimensional over its center (essentially commutative) then it must contain a free algebra on two generators. We show that for division algebras with uncountable centers a weaker dichotomy
holds: such a division ring must either contain a free algebra on two generators or it must be in some sense algebraic over certain division subalgebras. We use this to show that if $A$ is a
finitely generated complex domain of Gelfand-Kirillov dimension two then the conjectured dichotomy of Stafford and Makar-Limanov holds for the quotient division ring of $A$; that is, it is either
finite-dimensional over its center or it contains a free algebra on two generators. This is joint work with Dan Rogalski.
Everett W. Howe - Center for Communications Research
Abstract: I will talk about a computational problem inspired by the desire to improve the tables of curves over finite fields with many points (http://www.manypoints.org). Namely, if $q$ is a
large prime power, how does one go about producing a genus-4 curve over $\mathbb F_q$ with many points? I will discuss the background to this problem and give a number of algorithms, one of which
one expects (heuristically!) to produce a genus-4 curve whose number of points is quite close to the Weil upper bound in time $O\left(q^{3/4 + \epsilon}\right).$
Sergey Kitaev - University of Strathclyde
Mesh patterns are a generalization of vincular patterns. Mesh patterns
were introduced by Branden and Claesson to provide explicit expansions
for certain permutation statistics as, possibly infinite, linear
combinations of (classical) permutation patterns.
We introduce the notion of a boxed mesh pattern and study avoidance of
these patterns on permutations. We prove that the celebrated former
Stanley-Wilf conjecture is not true for all but eleven boxed mesh
patterns; for seven out of the eleven patterns the former conjecture is
true, while we do not know the answer for the remaining four
(length-four) patterns. Moreover, we show that an analogue of a
well-known theorem of Erdos and Szekeres does not hold for boxed mesh
patterns of lengths larger than 2. Finally, we discuss enumeration of
permutations avoiding simultaneously two or more length-three boxed mesh
patterns, where we meet generalized Catalan numbers.
This is joint work with Sergey Avgustinovich and Alexander Valyuzhenich.
Helge Ruddat - University of Mainz
Assuming the natural compactification X of a hypersurface in $(C^*)^n$ is
smooth, it can exhibit any Kodaira dimension depending on the size and
shape of the Newton polyhedron of X. In a joint work with Ludmil
Katzarkov, we give a construction for the expected mirror symmetry
partner of a complete intersection X in a toric variety which works for
any Kodaira dimension of X. The mirror dual might be reducible and is
equipped with a sheaf of vanishing cycles. We give evidence for the
duality by proving the symmetry of the Hodge numbers when X is a
hypersurface. The leading example will be the mirror of a genus two
curve. If time permits, we will explain relations to homological mirror
symmetry and the Gross-Siebert construction.
Zhiwei Wu - UC Irvine
The KdV equation is one of the most important equations in
soliton theory. It can be generalized to Gelfand-Dikii hierarchy and
there have been a lot of work related to it. In this talk, I will give
a geometric interpretation of the equations in Gelfand-Dikii hierarchy
as curve flows in $R^n$. I will also discuss Backlund transformation and
Hamiltonian structures for these curve flows.
Zhiyu Tian - Caltech
There is a close relation between some aspects of algebraic
geometry (in particular, birational geometry) and symplectic geometry via
Gromov-Witten theory. For example, Kollar made some conjectures about the
symplectic topology of rationally connected varieties, and Ruan speculated
the existence of the so-called symplectic birational geometry. A theorem of
Graber-Harris-Starr states that sections of rationally connected fibrations
over a curve always exist, which has many important consequences in the
theory of uniruled and rationally connected varieties. In this talk I will
discuss the symplectic analogues of their result and how these results
might be used to understand the conjectures of Kollar and Ruan.
"Alex Trebek"
Title says it all. ``Alex" will provide math-related clues and you will provide math-related questions. Come have some food and fun!
Sergey Kitaev - Univ Strathclyde
A poset is called (2+2)-free if it does not contain an induced subposet that is isomorphic to the union of two disjoint 2-element chains. In 1970, Fishburn proved that (2+2)-free posets are in
one-to-one correspondence with intensively studied interval orders. Recently, Bousquet-Melou et al. (M. Bousquet-Melou, A. Claesson, M. Dukes, and S. Kitaev, (2+2)-free posets, ascent sequences
and pattern avoiding permutations, J. Combin. Theory Ser. A 117 (2010) 884-909.) invented so-called ascent sequences which not only allowed to enumerate (2+2)-free posets (and thus interval
orders), but also to connect them to other combinatorial objects, namely to Stoimenow's diagrams (also called regular linearized chord diagrams which were used to study the space of Vassiliev's
knot invariants), to certain upper triangular matrices, and to certain pattern avoiding permutations (a very popular area of research these days). Several other papers appeared following the
influential work by Bousquet-Melou et al. Among other results, two conjectures, of Pudwell and Jovovic, were solved while dealing with (2+2)-free posets and ascent sequences.
In my talk, I will overview relevant results and research directions.
James Hall
Last spring, we presented Spectral Variational Integrators, a class of variational integrators that had both excellent conservation properties and exhibited geometric convergence. This talk will
present extensions to this work, including Spectral Variational Integrators on lie groups, pseudospectral variational integration of the one dimensional wave equation, and several conjectures
about the behavior of Spectral Variational Integrators based on observations of numerical examples.
Jon Voight - University of Vermont
Triangle groups, the symmetry groups of tessellations of the
hyperbolic plane by triangles, have been studied since early work of
Hecke and of Klein--the most famous triangle group being $\textrm{SL}_2(\mathbb Z).$ We
present a construction of congruence subgroups of triangle groups
(joint with Pete L. Clark) that gives rise to curves analogous to the
modular curves, and provide some applications to arithmetic. We
conclude with some computations that highlight the interesting
features of these curves.
Li-Sheng Tseng - UC Irvine
In this talk, I will introduce new cohomologies and elliptic
operators on symplectic manifolds. Their construction follows from a
simple decomposition of the exterior derivative into two first-order
symplectic differential operators, which are analogous to the
Dolbeault operators in complex geometry. These symplectic cohomologies
encode new geometrical invariants especially for non-Kahler symplectic
manifolds. This is joint work with S.-T. Yau.
Mirela Ciperiani - The University of Texas at Austin
In this talk I will report on progress on the following two
questions, the first posed by
Cassels in 1961 and the second considered by Bashmakov in
1974. The first question is
whether the elements of the Tate-Shafarevich group are
infinitely divisible when considered
as elements of the Weil-Chatelet group. The second question
concerns the intersection of
the Tate-Shafarevich group with the maximal divisible subgroup
of the Weil-Chatelet group.
This is joint work with Jakob Stix.
Todd Kemp
The standard playground for a lot of analysis is $L^{p}$ spaces. These function spaces have great global properties (in terms of their relationships with each other and inequalities that connect
them) but typically have very bad local properties (most of their constituent functions are extremely rough).
Instead, we will look at some $L^{p}$ spaces of holomorphic (aka complex analytic) functions. These spaces have extremely nice local properties: their elements are as smooth as can be, and they
moreover satisfy universal growth estimates you might not expect. By contrast, their global properties are not as nice: for example, they are not related to their dual spaces in the way one might
We'll discuss some of these dichotomies and try to give the flavor of modern research in holomorphic spaces. And we'll discover the truth about the delta function...
Aravind Asok - USC
By definition, an affine variety is a closed subvariety of some
affine space. A classical result asserts that every smooth affine variety
of dimension n is isomorphic to a closed subvariety of a $2n+1$-dimensional
affine space. Given a fixed smooth affine variety X it is natural to ask
when X can be realized as a closed subvariety of affine space of dimension
$n+d$ for $d < n+1$. In general, there are cohomological obstructions to the
existence of such embeddings, and we will discuss such obstructions in the
context of homotopy theory of varieties (no prior knowledge of this theory
will be assumed).
Fernando Rodriguez Villegas - UT Austin
In this talk I will discuss a combinatorial calculation of the polynomial that counts the number of indecomposable representations of a certain quiver and dimension vector. I will start by
introducing quivers, their representations and Kac's results and conjectures on such counting polynomials in general. The combinatorial calculation involves the reliability polynomial of
alternating graphs. I will end with the main motivation for the calculation: its relation to the geometry of character varieties.
Franklin Hardin Jones Kenter
Francesc Fit\'e - Universitat Politecnica de Catalunya
The (general) Sato-Tate Conjecture for an abelian variety A of
dimension g defined over a number field k predicts the existence of a
compact subgroup ST(A) of the unitary symplectic group USp(2g) that is
supposed to govern the limiting distribution of the normalized Euler
factors of A at the primes where it has good reduction. For the case
g=1, there are 3 possibilities for ST(A) (only 2 of which occur for
k=Q). In this talk, I will give a precise statement of the Sato-Tate
Conjecture for the case of abelian surfaces, by showing that if g=2,
then ST(A) is limited to a list of 52 possibilities, exactly 34 of
which can occur if k=Q. Moreover, I will provide a characterization of
ST(A) in terms of the Galois-module structure of the R-algebra of
endomorphisms of A defined over a Galois closure of k.
This is a joint work with K. S. Kedlaya, V. Rotger, and A. V. Sutherland
Cedric Villani - l'Institut Henri Poincare
Landau damping is relaxation without dissipation. For more than a half century it has been considered as a key phenomenon in plasma physics, and studied both in physics and mathematics, however
mainly at the linear level. In this lecture I explain about the physical and mathematical theory of Landau damping, and the recent progress by Mouhot and myself about Landau damping in the
nonlinear, close-to-equilibrium regime.
Patrick Gallagher
This introductory talk will cover several basic elements of convex
analysis, with particular attention paid to Fenchel conjugation and
infimal convolution. The coverage will begin with an introduction to
Fenchel conjugation and a consideration of its basic properties.
Examples of convex conjugate function pairs will be followed by the
introduction of infimal convolution. General properties of infimal
convolution will be considered, along with some particular properties
of the Moreau envelope case of infimal convolution.
Stephen Simpson - Penn State University
Maria Eulalia Vares - Universidade Federal do Rio de Janeiro
Manny Reyes - Bowdoin College
A Calabi-Yau algebra is a noncommutative analogue of the coordinate ring of a Calabi-Yau variety. It is well-known that if $G$ is a group acting on a Calabi-Yau algebra $A$, then the smash
product $A \#G$ remains Calabi-Yau under sufficiently good conditions. However, there are cases in which a smash product $A \# G$ may become Calabi-Yau even if $A$ is not Calabi-Yau. We will
explain how this can occur by studying the more general notion of a \emph{skew Calabi-Yau algebra}. This is joint work with D.~Rogalski and J.J.~Zhang.
Samuel Shen - SDSU
Uncertainties in the Assessment and Detection of Climate Changes
Kay Kirkpatrick - UIUC
Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous
connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll
discuss recent progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body systems, a step towards large deviations for Bose-Einstein condensation.
Jimmy Hall - UCSD
The paradox of the falling a cat is a famous example in
geometric mechanics. Simply stated, a cat in free fall can execute a
180 degree turn of its body, even though has zero angular momentum
throughout the entire maneuver. In this talk I will discuss how this
seeming paradoxical behavior can be explained through differential
geometry and holonomy, which in turn can offer insights into the
behavior of other mechanical systems. This talk is meant for a general
audience and no knowledge of differential geometry will be assumed.
Ms. Shenshen Wang - Department of Physics and Center for Theoretical Biological Physics, UCSD
Spontaneous directed motion, a hallmark of cell biology, is unusual in classical statistical physics. Here we study, using both numerical and analytical methods, organized motion in models of the
cytoskeleton in which constituents are driven by energy-consuming motors. Although systems driven by small-step motors are described by an effective temperature and are thus quiescent, at higher
order in step size, both homogeneous and inhomogeneous, flowing and oscillating behavior emerges. Motors that respond with a negative susceptibility to imposed forces lead to an apparent negative
temperature system in which beautiful structures form resembling the asters seen in cell division.
Pierre Albin - University of Illinois Urbana-Champaign
The signature operator of a Riemannian metric is an
important tool for studying topological questions with analytic
machinery. Though well-understood for smooth metrics on compact
manifolds, there are many open questions when the metric is allowed to
have singularities. I will report on joint work with Eric Leichtnam,
Rafe Mazzeo, and Paolo Piazza on the signature operator on stratified
pseudomanifolds and some of its topological applications.
Bernhard Lamel - Vienna
In recent work with Martin Kolar, a complete classification of ``ruled" infinite type hypersurfaces was found. We discuss this result as well as some work in progress with Ebenfelt and Zaitsev
for the general $1$-nonminimal case.
Mary Radcliffe - UCSD
Define a graph G by taking the vertices as $\mathbb{R}^2$ and the edges to be any pair of vertices that are distance 1 apart. The Hadwiger-Nelson Problem asks the chromatic number of this graph,
written $\chi(\mathbb{R}^2)$. It is known that either $4\leq \chi(\mathbb{R}^2)\leq 7$ or $5\leq \chi(\mathbb{R}^2)\leq 7$. We explore some approaches to solving this problem, encountering along
the way the Axiom of Choice (or lack thereof) and other infinite oddities.
Gabriel Stylianides - University of Oxford, UK
Students of all levels of education tend to have ‘justification schemes’ (Harel & Sowder, 1998) that are inconsistent with conventional validation methods. Yet, there is limited research
knowledge about how mathematics instruction can support progressions in students’ justification schemes so that they better approximate conventional validation methods. In this talk, I will draw
on findings from a four-year design experiment in an undergraduate mathematics course to present and exemplify an instructional intervention that has been successful in supporting progressions in
students’ justification schemes. The notion of ‘cognitive conflict’ featured prominently in the theoretical framework that underpinned the design of the intervention.
Albert Gilg - Coporate Research and Technologies, Siemens AG, Germany
Mathematical optimization is still dominated by deterministic models and corresponding algorithms. But many engineering and industrial optimization challenges demand for more realistic modelling
including stochastic effects. Common Monte-Carlo methods are too expensive for engineering applications. Polynomial chaos expansions have found to be an efficient mathematical approach for
several industrial applications, like turbomachinery design and production failure reduction.
Ryan Williams - Stanford University
Jiayi Wen
Competitive adsorption of counterions of multiple species to charged surfaces is studied by a size-effect included mean-field theory and Monte Carlo (MC) simulations. The mean-field electrostatic
free-energy functional of ionic concentrations, constrained by Poisson's equation, is numerically minimized by an augmented Lagrangian multiplier method. Unrestricted primitive models and
canonical ensemble MC simulations with the Metropolis criterion are used to predict the ionic distributions around a charged surface. It is found that, for a low surface charge density, the
adsorption of ions with a higher valence is preferable, agreeing with existing studies. For a highly charged surface, both of the mean-field theory and MC simulations demonstrate that the
counterions bind tightly around the charged surface, resulting in a stratification of counterions of different species. The competition between mixed entropy and electrostatic energetics leads to
a compromise that the ionic species with a higher valence-to-volume ratio has a larger probability to form the first layer of stratification. In particular, the MC simulations confirm the crucial
role of ionic valence-to-volume ratios in the competitive adsorption to charged surfaces that had been previously predicted by the mean-field theory. The charge inversion for ionic systems with
salt is predicted by the MC simulations but not by the mean-field theory. This work provides a better understanding of competitive adsorption of counterions to charged surfaces and calls for
further studies on the ionic size effect with application to large-scale biomolecular modeling. This is joint work with Shenggao Zhou, Zhenli Xu, and Bo Li.
Sorin Popa - UCLA
A famous problem of Murray and von Neumann (1943)
asks whether the II$_1$ factors $L(\Bbb F_n)$ associated with free groups
with $n$ generators, $\Bbb F_n$, are non-isomorphic for distinct $n$'s.
While this problem is still open, its ``group measure space'' version,
showing that
the II$_1$ factors $L^\infty(X)\rtimes \Bbb F_n$ arising from
free ergodic probability measure preserving actions $\Bbb
F_n\curvearrowright X$ are non-isomoprphic for $n= 2, 3, ...$,
independently of the actions, has been recently settled by Stefaan Vaes
and myself. I will comment on this, as
well as on related results by Gaboriau, Ozawa, Ioana, Peterson.
Ameera Chowdhury
This thesis makes contributions to extremal combinatorics, specifically extremal set theory questions and their analogs in other structures. Extremal set theory studies how large or small a
family of subsets of a finite set $X$ can be under various constraints. By replacing the set $X$ with another finite object, one can pose similar questions about families of other structures.
Remarkably, a question and its analogs essentially have the same answer, regardless of the object. Despite these similarities, not much is known about analogs because standard techniques do not
always apply. Our main results establish analogs of extremal set theory results for structures such as vector spaces and subsums of a finite sum. We also study intersecting families and shadows
in their classical context of sets by researching a conjecture of Frankl and F\"{u}redi."
Vyacheslav Kungurtsev
Sequential Quadratic Programming (SQP) methods are a popular and successful class of methods for minimizing a generally nonlinear function subject to nonlinear constraints. Under a standard set
of assumptions, conventional SQP methods exhibit a fast local convergence rate. However, in practice, a conventional SQP method involves solving an indefinite quadratic program (QP), which is NP
hard. As a result, approximations to the second-derivatives are often used, slowing the local convergence rate and reducing the chance that the algorithm will converge to a local minimizer
instead of a saddle point. In addition, the standard assumptions required for convergence often do not hold in practice. For such problems, regularized SQP methods, which also require
second-derivatives, have been shown to have good local convergence properties; however, there are few regularized SQP methods that exhibit convergence to a minimizer from an arbitrary initial
starting point. My thesis considers the formulation, analysis and implementation of: (i) practical methods that use exact second-derivative information but do not require the solution of an
indefinite QP, (i) a regularized SQP method with global convergence and (iii) a rigorously defined version of a conventional SQP method with features that have been observed to work in practice
for degenerate problems.
Michael Kozdron - University of Regina & MSRI
The Schramm-Loewner evolution is a one-parameter family of
random growth processes in the complex plane introduced by Oded
Schramm in 1999. In the past decade, SLE has been successfully used to
describe the scaling limits of various two-dimensional lattice models.
One of the first proofs of convergence was due to Lawler, Schramm, and
Werner who gave a precise statement that the scaling limit of
loop-erased random walk is SLE with parameter 2. However, their result
was only for curves up to reparameterization. There is reason to
believe that the scaling limit of loop-erased random walk is SLE(2)
with the very specific natural time parameterization that was recently
introduced by Lawler and Sheffield, and further studied by Lawler,
Zhou, and Rezaei. I will describe several possible choices for the
parameterization of the discrete curve that should all give the
natural time parameterization in the limit, but with the key
difference being that some of these discrete time parameterizations
are easier to analyze than the others. This talk is based on joint
work in progress with Tom Alberts and Robert Masson.
Professor Burkhard Duenweg - Phys., Max-Planck Institute, Mainz, Germany
The talk gives an introduction to the method for the calculation of electrostatic interactions put forward by Maggs, both in its Monte Carlo and in its Molecular Dynamics version. It is shown
that the latter can be viewed as a straightforward application of the Car-Parrinello approach to the coupled dynamics of charges and electromagnetic fields, which is equivalent to a
Galilei-invariant form of Maxwell theory. The talk then focuses on more recent developments, where the same idea is applied to solving the Poisson-Boltzmann equation. It is shown that the
resulting algorithm is rather simple and intrinsically stable.
Hans Wenzl
We give the basic definitions of invariants for classifying von
Neumann subfactors, as well as more recent formulations in terms of tensor categories. This will be illustrated with some examples.
Peter Winkler - Dartmouth & MSRI
We find optimal strategies for a pursuit and evasion game which, when pitted against each other, solve the problem of constructing a small area in the plane in which a unit-length line segment
can be rotated. Joint work with Y. Babichenko, Y. Peres, R. Peretz and P. Sousi.
Sara Pollock
In this talk, we will discuss goal-oriented adaptive methods for second order elliptic PDEs. In particular, we will look at linear nonsymmetric and semilinear problem classes. In goal-oriented
methods we are concerned with approximating a given quantity of interest, a function of the weak solution to the PDE. The adaptive algorithm is driven by estimating the error in both the primal
and a dual problem, which involves the quantity of interest. We will discuss the formation of an appropriate dual for each type of problem, and how the errors in the primal and dual problems
relate to the error in the goal function. Finally, we will look at the contraction framework in each instance and address the appropriate notion of error to show convergence.
STEM Education, Economics, and Equity - SDSU
Join us in several interactive, hands-on stations ranging from preschool through high school level, formal and informal math and science education. Then take part in a discussion surrounding the
implementation of hands-on learning in various environments and grade levels. Presenters include: Ricardo Nemirovsky, SDSU Center for Research in Mathematics and Science Education (CRMSE); Molly
Kelton, Doctoral Student SDSU/UCSD Mathematics and Science Education (MSED); Nan Renner, SD Natural History Museum; Sandy Silverman, SD County Office of Education, and more! (See the flyer for
details and to register.)
Nathan Ross - UC Berkeley
We discuss a new method for obtaining a local limit theorem
(LLT) from a known distributional limit theorem. The method rests on a
simple analytic inequality (essentially due to Hardy, Landau, and
Littlewood) which can be applied directly after quantifying the
smoothness of the distribution of interest. These smoothness terms are
non-trivial to handle and so we also provide new (probabilistic) tools
for this purpose. We illustrate our approach by showing LLTs with
rates for the magnetization in the Curie-Weiss model at high
temperature and for some counts in an Erdos-Renyi random graph. This
is joint work with Adrian Roellin.
Xun Jia - Department of Radiation Medicine and Applied Sciences, UCSD
Radiation therapy aims at delivering a prescribed dose to cancerous targets using high-energy radiation beams, while sparing dose to surrounding normal tissues and organs at risks. For this
purpose, a treatment plan is customized for each individual patient, where parameters in a treatment plan, e.g. beam direction and fluence, are adjusted. Such a problem is mathematically
formulated as an optimization problem and is solved with numerical algorithms. This talk will first give an introduction to the treatment plan optimization problem in radiotherapy, including
intensity-modulated radiation therapy (IMRT) and volumetric modulated arc therapy (VMAT). It will then focus on a particular problem in IMRT, beam orientation optimization (BOO), which tries to
find a solution that contains nonzero fluence map at only a small number of beam angles to achieve a dosimetric objective. We noticed that the objective of the BOO problem is equivalent to
finding a fluence map that is sparse at the beam angle level. As such, we introduce a sparsity energy into the total energy function, which takes an L2 norm of beamlet intensities within each
angle and then takes a weighted L1 norm over angles. Such an energy term favors solutions with nonvanishing fluence map at only a few beam angles. During optimization, the weighting factors in
the L1 norm are adaptively adjusted. Starting with all candidate angles, the optimization process identifies unimportant orientations gradually and removes them without largely sacrificing the
dosimetric objective. The whole process terminates when a target number of beams is achieved. The developed BOO algorithm is found to be effective for identifying important beam angles, which
leads to better plan qualities than unoptimized beam configurations.
Thomas Sinclair - UCLA
I will present some structural results for $II_1$ factors of products of
hyperbolic groups and their ergodic actions. Applications will be given to
the measure equivalence theory of such groups. This is joint work with
Ionut Chifan and Bogdan Udrea.
Efim Zelmanov
We will discuss the recent advances in the theory of profinite
groups and their verbal subgroups.
Dietmar Bisch - Vanderbilt University
The first "non-prime" Jones indices are 4, 3 + $\sqrt{5}$ and 6. All
hyperfinite subfactors with index 4 are known, and it follows from work of
Nicoara, Popa and myself that the set of subfactors with composite integer
index is wild. I'll explain some of the beautiful structures appearing
here and will make some comments about the situation of hyperfinite
subfactors with index 3 + $\sqrt{5}$.
Johanna Hennig - UCSD
Talk time starts at 3:45 PM.
Produced by MSRI
Taking the Long View examines the life of a remarkable mathematician whose classical
Chinese philosophical ideas helped him build bridges between China and the West.
Shiing-shen Chern (1911-2004) is one of the fathers of modern differential geometry. His
work at the Institute for Advanced Study and in China during and after World War II led to
his teaching at the University of Chicago in 1949. Next came Berkeley, where he created a
world-renowned center of geometry, and in 1981 cofounded the Mathematical Sciences
Research Institute. During the 1980s he brought talented Chinese scholars to the United
States and Europe. By 1986, with Chinese government support, he established a math
institute at Nankai University in Tianjin. Today it is called the Chern Institute of Mathematics.
Jon Wolfson - Michigan State University
A connected Riemannian manifold $M$ has constant vector
curvature $\epsilon$, denoted by cvc$(\epsilon)$, if every tangent
vector $v \in TM$ lies in a 2-plane with sectional curvature
$\epsilon$. By scaling the metric on $M$, we can always assume that
$\epsilon = -1, 0$, or $1$. When the sectional curvatures satisfy an
additional bound sectional curvature $\leq \epsilon$ or sectional
curvature $\geq \epsilon$, we say that $\epsilon$ is an {\it extremal}
In this talk we first motivate the definition and then describe the
moduli spaces of cvc$(\epsilon)$ metrics on three manifolds for each
case, $\epsilon = -1, 0$, or $1$, under global conditions on $M$. For
example, in the case $\epsilon = -1$ is extremal, we show, under the
assumption that $M$ has finite volume, that $M$ is isometric to a
locally homogeneous manifold. In the case that $M$ is compact,
$\epsilon = 1$ is extremal and there are no points in $M$ with all
sectional curvatures identically one, we describe the moduli space of
cvc$(1)$ metrics in terms of locally homogeneous metrics and the
solutions of linear elliptic partial differential equations. Solutions
of some nonlinear elliptic equations arise in the proof.
Chi Li - Princeton University
Abstract: In the continuity method to Kahler-Einstein problem, Tian
conjectured the Bergman kernels of solution metrics are uniformly
bounded below away from 0. I will show that Tian's partial $C^0$
estimate holds on any toric Fano manifold. This allows us to calculate
the multiplier ideal sheaf for certain toric Fano manifolds with large
symmetry. This is an corollary of my earlier study on the limit
behavior of solutions to continuity method on toric Fano manifolds.
James Ferris - UCSD
This will be a hands-on class on LaTeX typesetting on May 5 in from 10:00 AM to 11:30 AM. The material covered is appropriate for beginners and those with intermediate knowledge of the markup
language. Individual questions are welcome and will be answered as time permits.
Monica Vazirani - University of California, Davis
What is categorification? If you de-categorify Vector-Spaces, you replace isomorphism classes of objects with natural numbers (their dimensions), replace direct sum with addition of those
numbers, and replace tensor product with multiplication. To categorify is to undo this process--for instance, one might start with the ring of symmetric functions and realize it has replaced the
representation theory of the symmetric group.
In this talk, I will discuss how Khovanov-Lauda-Rouquier (KLR) algebras categorify quantum groups. I will discuss their simple modules, and in particular that they carry the structure of a
crystal graph. This is joint work with Aaron Lauda.
Li-Tien Cheng - UCSD
Cancer radiotherapy, together with chemotherapy and surgery, form the basis of modern day cancer treatment. Its treatment pro- cess generally involves directing a high energy radiation beam at an
identied cancerous growth from dierent directions and with varying beam shapes, durations, and intensities in order to kill the cancerous tissues while preserving nearby healthy ones. Volumetric
modulated arc therapy comprises a recently developed setup using a full-rotation trajectory of the beam about the patient along with a multi-leaf collimator for beam shape sculpting. We introduce
a variational model in this setup for the optimization of beam shapes and intensities while preserving certain constraints imposed by the equipment used. We apply a binary level-set strategy to
represent beam shapes and a fast sweeping technique to satisfy beam intensity variation limits. The result is a ow-based shape optimization algorithm that guarantees constraint satisfaction and
energy decrease for the generation of improved treatment plans in volumetric modulated arc therapy.
N. Romanovskiy - Russian Academy of Sciences
Mitchel T. Keller - London School of Economics and Political Science
In his 1985 monograph Interval Orders and Interval Graphs, Fishburn noted the dearth of enumerative results for interval orders and labelled semiorders, standing in contrast to the
well-understood case of interval graphs and unlabelled semiorders. (The latter are enumerated by the Catalan numbers.) Recently, work by Bousquet-Mélou et al. linked certain integer sequences
termed ascent sequences to unlabelled interval orders. This allowed for an asymptotic enumeration of unlabelled interval orders through earlier work by Zagier involving the same generating
function that enumerates ascent sequences. Building on subsequent work by Khamis, this talk develops an asymptotic enumeration of the labelled interval orders on an $n$-element set. This is joint
work with Graham Brightwell (LSE).
Yifei Lou - UCSD
We present a method to enhance the quality of a video sequence captured through a turbulent atmospheric medium. Enhancement is framed as the inference of the radiance of the distant scene,
represented as a latent image," that is assumed to be constant throughout the video. Temporal distortion is thus zero-mean and temporal averaging produces a blurred version
of the scene's radiance, that is processed via a Sobolev gradient flow to yield the latent image in a way that is reminiscent of the lucky region" method. Without enforcing prior knowledge, we
can stabilize the video sequence while preserving ne details. We also present the well-posedness theory for the stabilizing PDE and a linear stability analysis of the numerical scheme.
This is a joint work with Sung Ha Kang, Stefano Soatto and Andrea Bertozzi.
Fan Chung - UCSD
We will discuss recent developments
in the probabilistic and spectral approaches for graph limits.
In particular, we will extend the notion of quasi-randomness,
which concerns a class of equivalent properties that random graphs satisfy.
For example, we will give several necessary and sufficient conditions for a graph to be the union of two or more quasi-random graphs.
One of these characterizations involves eigenvalues and scalable eigenspaces,
that we call "graphslets", which dictate the behavior of graph limits
for both dense and sparse graphs.
Mr. Michael White - Math and CTBP, UCSD
The interplay between geometry and electrostatics contributes significantly to hydrophobic interactions of biomolecules in an aqueous solution. With an implicit solvent, such a system can be
described macroscopically by the dielectric boundary that separates the high-dielectric solvent from low-dielectric solutes. This work concerns the motion of a model cylindrical dielectric
boundary as the steepest descent of a free-energy functional that consists of both the surface and electrostatic energies. The effective dielectric boundary force is defined and an explicit
formula of the force is obtained. It is found that such a force always points from the solvent region to solute region. In the case that the interior of a cylinder is of a lower dielectric, the
motion of the dielectric boundary is driven initially by the surface force but is then driven inward quickly to the cylindrical axis by both the surface and electrostatic forces. In the case that
the interior of a cylinder is of a higher dielectric, the competition between the geometrical and electrostatic contributions leads to the existence of equilibrium boundaries that are circular
cylinders. Linear stability analysis is presented to show that such an equilibrium is only stable for a perturbation with a wavenumber larger than a critical value. Numerical simulations are
reported for both of the cases, confirming the analysis on the role of each component of the driving force. Implications of the mathematical findings to the understanding of charged molecular
systems are discussed. This is joint work with Li-Tien Cheng, Bo Li, and Shenggao Zhou.
Siu-Cheong Lau - Institute for the Physics and Mathematics of the Universe, University of Tokyo
Open Gromov-Witten invariants are essential ingredients of Lagrangian-Floer intersection theory, and they serve as quantum corrections in mirror symmetry from SYZ viewpoint. They are difficult to
compute in general due to non-trivial obstructions in the moduli. In this talk, I will illustrate by examples how to compute open Gromov-Witten invariants of toric manifolds, by relating them to
closed Gromov-Witten invariants which are better understood. This also gives an enumerative meaning of mirror maps. This is joint work with Kwokwai Chan, Naichung Leung and Hsian-Hua Tseng.
Susan Montgomery - University of Southern California
Ilya Kossovskiy - University of Western Ontario
The classical result of H.Poincare states that a local
biholomorphic mapping of an open piece of the 3-sphere in
$\mathbb{C}^2$ onto another open piece extends analytically to a global holomorphic automorphism of the sphere. A big stream of further publications was dedicated to the possibility
to extend local biholomorphic mapping between real hypersurfaces in complex space. The most general results were obtained by D.Hill, R.Shafikov and K.Verma who generalized Poincare's extension
phenomenon for the case of an essentially finite hypersurface in the preimage and a quadric in the image, and also for the case of a minimal hypersurface (in the sense of Tumanov) in the preimage
and a sphere
in the image.
In this joint work with R.Shafikov we consider the - essentially new - case where a hypersurface $M$ in the
preimage contains a complex hypersurface, i.e. where $M$ is nonminimal. We demonstrate that the above extension results fail in this case, and prove the following analytic continuation
phenomenon: a local biholomorphic mapping of $M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends
to a punctured neighborhood of the complex hypersurface $X$, lying in $M$, as a multiple-valued locally biholomorphic mapping. The extension phenomenon is based on the properties of Segre sets
introduced by Baouendi, Ebenfelt and Rothschild
near the complex hypersurface $X$. We also establish an interesting interaction between nonminimal spherical real hypersurfaces and linear differential equations with an isolated singular point.
Peter Blomgren - SDSU
We are exploring the viability of a novel approach to cardiocyte contractility assessment based on biomechanical properties of the cardiac cells, energy conservation principles, and information
content measures. We define our measure of cell contraction as being the distance between the shapes of the contracting cell, assessed by the minimum total energy of the domain deformation
(warping) of one cell shape into another. To guarantee a meaningful vis-a-vis correspondence between the two shapes, we employ both a data fidelity term and a regularization term. The data
fidelity term is based on nonlinear features of the shapes while the regularization term enforces the compatibility between the shape deformations and that of a hyper-elastic material. We tested
the proposed approach by assessing the contractile responses in isolated adult rat cardiocytes and contrasted these measurements against two different methods for contractility assessment in the
literature. Our results show good qualitative and quantitative agreements with these methods as far as frequency, pacing, and overall behavior of the contractions are concerned. We hypothesize
that the proposed methodology, once appropriately developed and customized, can provide a framework for computational cardiac cell biomechanics that can be used to integrate both theory and
experiment. For example, besides giving a good assessment of contractile response of the cardiocyte, since the excitation process of the cell is a closed system, this methodology can be employed
in an attempt to infer statistically significant model parameters for the constitutive equations of the cardiocytes.
Michael Kasa - UCSD
Toric varieties have rich connections to plane geometry, which allows questions about algebraic geometry (hard) to be reformulated into questions about combinatorics (easy). In this talk, we will
introduce toric varieties, and we will discuss several examples. This talk is intended to be fun, and should be generally accessible.
Rafael Sorkin - Perimeter Institute
Among the various ideas put forward in the search for a theory
of quantum gravity, the causal set hypothesis is distinguished
by its logical simplicity and by the fact that it incorporates
the assumption of an underlying spacetime discreteness
organically and from the very beginning. After presenting the
problem of quantum gravity in general, I will precis the
causal set programme and touch on some old and some recent
Ruth Williams
Stochastic networks are used as models for complex processing systems involving dynamic interactions subject to uncertainty. Applications arise in high-tech manufacturing, the service industry,
telecommunications, computer systems and bioengineering. The control and analysis of such networks present challenging mathematical problems. In this talk, a concrete application will be used to
illustrate a general approach to the study of stochastic processing networks based on deriving more tractable approximate models. Specifically, we will consider a model of Internet congestion
control in which processing can involve the simultaneous use of several resources (or links), a phenomenon that is not well understood. Elegant fluid and diffusion approximations will be derived
and used to study the performance of this model. A key insight from this analysis is a geometric representation of the consequences of using a "fair" policy for the sharing of resources. The talk
will conclude with a summary of the current status and description of open problems associated with approximate models for general stochastic processing networks.
Feng Xu - UC Riverside
In 1961 G.E.Wall conjectured that the number of maximal subgroups in a finite group is bounded by the order of group. In this talk I will discuss a generalization of this conjecture in the
setting of subfactors and recent progress on related problems.
Nolan Wallach - UCSD
The talk will describe old and new(er) results on the structure of modules for the Virasoro algebra. Joint work of Rocha and Goodman with the speaker will be the old work. A newer result to be is
a description of the annihilators of Verma modules (analogous to Dixmier's result for semi-simple Lie algebras) and its relationship with Small's albatross.
Lek-Heng Lim - Dept. of Statistics, University of Chicago
The human brain connectome is an ambitious project to provide a complete map of neural connectivity and a recent source of excitement in the neuroscience community. Just as the human genome is a
triumph of marrying technology (high throughput sequencers) with theory (dynamic programming for sequence alignment), the human connectome is a result of a similar union. The technology in
question is that of diffusion magnetic resonance imaging (dMRI) while the requisite theory, we shall argue, comes from three areas: PDE, harmonic analysis, and convex algebraic geometry.
The underlying mathematical model in dMRI is the Bloch-Torrey PDE but we will approach the 3-dimensional imaging problem directly. The main problems are (i) to reconstruct a homogeneous
polynomial representing a real-valued function on a sphere from dMRI data; and (ii) to analyze the homogeneous polynomial via a decomosition into a sum of powers of linear forms. We will focus on
the nonlinear approximation associated with (ii) and discuss a technique that combines (i) and (ii) for mapping neural fibers.
This is joint work with T. Schultz of MPI Tubingen.
Michael Kelly
Cancer is currently viewed as an evolutionary process. In an organ there is a population of cells that give birth, die and mutate according to population dynamics that are determined by the types
of cells under consideration. If certain cell mutations are acquired then the cells can become cancerous. In this manuscript we consider two evolutionary models that may each be viewed as a model
of cancer. One is a model of colorectal cancer. We discuss results pertaining to the time it takes to develop cancer and the location of the mutations. The other model is a general
Moran-type model. We discuss results pertaining to the rate of adaptation.
Konstantinos Spiliopoulos - Brown University & Boston University
The need to simulate rare events occurs in many application areas,
including telecommunication, finance, insurance, computational physics
and chemistry. However, virtually any simulation problem involving
rare events will have a number of mathematical and computational
challenges. As it is well known, standard Monte Carlo sampling
techniques perform very poorly in that the relative errors under a
fixed computational effort grow rapidly as the event becomes more
rare. In this talk, I will discuss large deviations, rare events and
Monte Carlo methods for systems that have multiple scales and that are
stochastically perturbed by small noise. Depending on the type of
interaction of the fast scales with the strength of the noise we get
different behavior, both for the large deviations and for the
corresponding Monte Carlo methods. Using stochastic control arguments
we identify the large deviations principle for each regime of
interaction. Furthermore, we derive a control (equivalently a change
of measure) that allows to design asymptotically efficient importance
sampling schemes for the estimation of associated rare event
probabilities and expectations of functionals of interest. Standard
Monte Carlo methods perform poorly in these kind of problems in the
small noise limit. In the presence of multiple scales one faces
additional difficulties and straightforward adaptation of importance
sampling schemes for standard small noise diffusions will not produce
efficient schemes. We resolve this issue and demonstrate the
theoretical results by examples and simulation studies. Applications
of these results in chemistry problems and in mathematical finance
will also be discussed.
Jesse Peterson - Vanderbilt University
Zuojun Guo - Genomics Institute of the Novartis Resesearch Foundation
The closely placed phosphate charges along the charged biopolymer DNA backbone leads to strong electrostatic repulsion. However, when the DNA is immersed in an aqueous solution containing
monovalent or divalent cations from added salts, the free energy of the system is lowered when counterions from the bulk condense on the backbone of the DNA. According to counterion condensation
theory, each phosphate charge is reduced by the factor z theta, where theta is the number counterions associated per phosphate charge, z is the valence of counterions. Brownian Dynamics
simulations also can be used to quantitatively describe condensation of monovalent and multivalent ions (from added salt) on the backbone of DNA.
The tumor suppressor gene p53 is responsible for maintaining the integrity of the human genome and plays a vital role in DNA repairing machinery. Loss of p53 tumor suppressor activity is a
frequent defect in ~ 50% of human cancers. MDM2 controls the stability of p53 through ubiquitation to target the tumor suppressor protein for degradation by the proteasome. Inhibition the
interactions between p53 and the E3 ubiquitin ligase MDM2/MDMX will reactivate the p53 pathway and selectively kill tumor cells. Extensive molecular dynamics simulations were used to study
hydrocarbon linker stapled alpha-helical peptides which could be potential inhibitors of p53 peptide and MDM2.
Gizem Karaali - Pomona College
Differential geometry and Lie theory have traditionally provided the mathematical framework for our most intuitive physical theory: classical mechanics. However, as is well-known, in the last
century physicists developed newer theories which incorporate different kinds of symmetries, and bold concepts like the uncertainty principle have arisen that need to be addressed mathematically.
Mathematical physicsists' response has been a constant search for methods of quantization and superization, thus allowing the integration of older techniques into these newer, broader theories.
This talk will explain one part of this story in more detail. In particular we will describe super quantum group theory, an eclectic collection of theorems and conjectures whose development is
very much still in progress, but one that promises a solution to some foundational questions in mathematical physics. The mathematical background needed is limited (I will provide the relevant
definitions), the physical background needed is none (I will, however, assume that all members of the audience were born in the twentieth century); the main prerequisite for this talk is a
curious mind which is willing to accommodate some occasional vague language.
Ionut Chifan - Vanderbilt University, University of Iowa
Craig Timmons
In this talk we will introduce a variation of the classical Turan problem of determining the maximum number of edges in an $n$-vertex graph that does not contain a fixed forbidden graph. We will
present some results and open problems. The talk is intended for a general audience and should be accessible for advanced undergraduates.
Dr. Changsun Eun - Chem & Biochem, UCSD
We performed molecular dynamics simulations to study the character of hydrophobic interaction between two nanoscale particles in water. For a systematic study of water density fluctuations
induced by the hydrophobic interaction, we prepared a graphene plate and also other model plates made of “carbon†atoms that had different interaction strength with water. We calculated the
interaction between two identical “carbon†plates immersed in water, and calculated the fluctuations in the number of water molecules in the confined space between two plates. The result
showed that fluctuations in some cases are strongly enhanced compared to the fluctuations observed next to a single plate. If the character of water fluctuations in the confined space determines
the character of hydrophobic interactions, then it is possible to conclude that the interaction between graphene plates in water is hydrophobic.
In another study, we investigated the effect of roughness on hydrophobic interaction (the rough hydrophobic surface was created by attaching non-polar headgroups to the graphene plates). Our
study demonstrated that roughness enhances hydrophobic interactions. As a result of this enhancement, we observed a dewetting transition between two rough hydrophobic surfaces, which would not
occur between the corresponding particles with smooth surfaces.
Mark Gross and Ken Intrilligator - UCSD Mathematics / Physics
On June 1st and June 8th, Ken Intrilligator and myself will run
an informal seminar on cluster algebras as entering into string
theory and some of my own work on mirror symmetry. Anyone interested
is welcome.
John D. Foley
Kac-Moody groups generalize Lie groups but are typically infinite
dimensional. This defense will quickly introduce discrete and
topological Kac-Moody groups and outline a direct comparison between complex topological Kac-Moody groups and discrete Kac-Moody groups over the algebraic closure of the field with p elements.
This result uses newly constructed homotopy decompositions for the "unipotent" factors of parabolic subgroups of a discrete Kac-Moody group in terms of unipotent algebraic groups. Additional
applications will be given and the topics of infinite Coxeter groups, BN-pairs, and root group data systems will be visited.
Cyril Houdayer - CNRS and ENS Lyon
Leszek Kolodziejczyk - Warsaw
Tom Alberts - Caltech
A multiplicative cascade is a randomization of any measure on the unit interval, constructed from an iid collection of random variables indexed by the dyadic intervals. Given an arbitrary initial
measure I will describe a method for constructing a continuous time, measure valued process whose value at each time is a cascade of the initial one. The process also has the Markov property,
namely at any given time it is a cascade of the process at any earlier time. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. I will
discuss applications of this process to models of tree polymers and one dimensional random geometry.
Joint work with Ben Rifkind (University of Toronto).
Son Duong
Embedding problem for real-algebraic hypersurfaces dates back to
1978 when Webster proved that real-algebraic hypersurfaces is embeddable into a hyperquadric of possibly higher dimension. In a recent paper joint with Peter Ebenfelt, we showed that this is not
true for the spheres case. We will exhibit an explicit example of a close, strictly pseudoconvex hypersurface and show that it is not locally holomorphically embeddable into a sphere of any
dimension whatsoever by showing that the point at infinity is an obstruction for local embedding at all point.
Nate Broaddus - Ohio State
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call
this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit
homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve
Shi (Fox) Cheng - UCSD
The main well developed numerical methods for Stochastic PDEs are
Stochastic Galerkin method and Stochastic Collocation method. The error
estimators of linear Poisson problem from those two methods corresponding
to numerical solutions, mean and second moment of numerical solution are
analyzed properly already. However, the analysis of other types of linear
and nonlinear models are still open. My talk will consider a stochastic
nonlinear Diffusion Reaction model, and analyze well-posedness of its weak
form in a new extended group of Banach spaces, additionally the
discretization of weak solution will be discussed.
Sebastian Herr - Universitat Bielefeld
In this talk I will present recent small data global well-posedness results for energy-critical nonlinear Schroedinger equations on specific compact manifolds, such as tori and spheres. Key
ingredients are certain multilinear estimates of Strichartz type as a replacement for the classical dispersive estimates which fail in this setup.
Adriano Garsia
In the recent FPSAC meeting in Nagoya, Michele D'Adderio
posed the problem of proving that a family of polynomials in N[q,t] that q,t-enumerate the convex polyominos in the nxm square are symmetric in q,t. In this talk we show how two beautiful
bijections of Michele D'Adderio and Angela Hicks combine to yield the symmetry result as well as its connection to the theory of Macdonald Polynomials and Diagonal Harmonics.
Devavrat Shah - MIT, visiting Stanford
We consider a switched (queueing) network in which there are constraints on which queues may be served simultaneously; such networks have been used to effectively model input-queued switches,
wireless networks and more recently data-centers. The scheduling policy for such a network specifies which queues to serve at any point in time, based on the current state or past history of the
system. As the main result, we shall discuss a new class of online scheduling policies that achieve optimal scaling for average queue-size for a class of switched networks including input-queued
switches. Time permitting, we shall discuss various exciting open questions in the domain of stochastic networks.
This is based on joint work with Neil Walton (Univ of Amsterdam) and Yuan Zhong (MIT).
James Pascoe - UCSD
This refers to the following problem. Given a sequence of numbers, when are they the moments
of some measure (or distribution if you like to normalize to the case that is known as probability.)
That is, given a sequence $(a_n)^{\infty}_{n=0}$ when is there a measure such that $a_n = \int x^n d\mu{x}.$
To solve this problem, we introduce some basic modern analysis, specifically Hilbert space techniques, which
were invented early in the last century to solve problems like the Hamburger moment problem.
Klaus Boehmer - Philipps-University Marburg
This lecture is an appetizer for my two books in OUP: Numerical Methods for Nonlinear Elliptic Differential Equations, A Synopsis, and Numerical Methods for Bifurcation and Center Manifolds in
Nonlinear Elliptic and Parabolic Differential Equations, 2010 and 2011. We extend for the first time the linear discretization theory of Schaback, developed for meshfree methods, to nonlinear
operator equations, relying heavily on methods of Boehmer, Vol I. There is no restriction to elliptic problems nor to symmetric numerical methods like Galerkin techniques. Trial spaces can be
arbitrary, but have to approximate the solution well, and testing can be weak or strong. We present Galerkin techniques as an example. On the downside, stability is not easy to prove for special
applications, and numerical methods have to be formulated as optimization problems. Results of this discretization theory cover error bounds and convergence rates. As an example we present the
meshless method for some nonlinear elliptic problems of order 2. Some numerical examples are added for illustration.
Franklin Kenter
On average, I only get to do this once every 7 years
Matt Tucker-Simmons - U. C. Berkeley
"Symmetric algebra" is a fancy way of saying "polynomial ring." The symmetric algebra of a k-vector space V is the enveloping commutative algebra of V in the category kVect, and can be realized
as the polynomial ring generated by any basis of V.
Quantum symmetric algebras are analogues of polynomial rings in the category of modules over the quantized universal enveloping algebra of a semisimple Lie algebra. Familiar examples include
quantum polynomial and matrix algebras as well as coordinate algebras of quantum Euclidean and symplectic vector spaces, but there are more exotic ones also. I will describe the general
construction of quantum symmetric algebras and show that they satisfy a universal mapping property analogous to the one for ordinary symmetric algebras. This requires an appropriate notion of
commutativity for algebras in Uq(g)-Mod.
I will try to illustrate the general theory with simple examples.
Asif Shakeel - Haverford College
Quantum cellular automata (QCA) are models of quantum computation of particular interest from the point of view of quantum simulation. Quantum lattice gas automata (QLGA - equivalently
partitioned quantum cellular automata) represent an interesting subclass of QCA. Prior work on QCA has investigated the relationship between these two classes of models. In the present paper we
establish necessary and sufficient conditions for unbounded, finite Quantum Cellular Automata (QCA) (finitely many active cells in a quiescent background) to be Quantum Lattice Gas Automata. We
define a local condition that classifies those QCA that are QLGA, and we show that there are QCA that are not QLGA.
Peter Love - Haverford College
Simulation of fermionic systems has been a topic of interest in quantum
simulation since Feynman's first papers on the topic. It has been known
for some time how to simulate fermionic systems and scalable proposals
for electronic structure calculations on quantum computers require some
solution to this problem. Current work makes use of the Jordan-Wigner
transformation to track phases arising from exchange anti-symmetry. For
a single term in a fermionic Hamiltonian on N modes the Jordan wigner
transformation requires an overhead of O(N) gates. In this talk I will
give an alternative to the Jordan Wigner transformation, originally
developed by Bravyi and Kitaev, which reduces this overhead to O(log N).
We give the details of this transformation for electronic structure
Hamiltonians and give the minimal basis model of the Hydrogen molecule
as an example.
Brendan Farrell - Cal Tech
The Jacobi ensemble is one of three ensembles of classical random matrix theory. It has a corresponding matrix form, so that a natural endeavor is to prove universality for the spectral
properties of the matrix form. In joint work with L. Erdõs we provide the first such result. More interestingly, this matrix form has special relevance to other areas of mathematics because its
eigenvalues describe the angles between random subspaces. We will consider random subspaces spanned by Euclidean and Fourier vectors and show how the Jacobi ensemble is related to discrete
uncertainty principles.
Jeremy Semko - UCSD
The notion of convexity of vector spaces can be generalized to dimension-free sets of matrices. A natural question that then arises is how to identify a set's matrix convex hull. We introduce
some techniques for getting a grasp on the matrix convex hull of semi-algebraic sets and look at one of these in particular: The bent TV screen. This is the set
$$ \{ (X, Y) : I - X^2 - Y^4 \succeq 0 \} $$
It is convex in the scalar case but is not matrix convex. In fact, there is no known "simple" formula for its matrix convex hull.
Markus Grassl - National University of Singapore
Polynomial invariants provide a tool to characterise quantum states
with respect to local unitary transformations. Unfortunately, the
situation becomes very complicated already for mixed states of three
qubits due to combinatorial explosion.
After an introduction to the mathematical background and general tools,
the talk will present preliminary results for mixed quantum states and
Hamiltonians for three-qubit systems.
The talk is based on joint work in progress with Robert Zeier.
Todd Kemp - UCSD
Consider two random subspaces of a finite-dimensional vector space -- i.e. two random projection matrices P and Q. What is the dimension of their intersection? This (random) integer is almost
surely equal to its minimal possible value, which corresponds to the subspaces being in general position. Many more delicate questions about the geometry of the configuration are encoded by the
principle angles between the subspaces, which are determined by the eigenvalues of the operator-valued angle matrix PQP.
The situation is much more complicated in infinite-dimensions. Even the question of whether two random projections are likely to be in general position is difficult to make sense of, let alone
answer. Nevertheless, understanding the operator-valued angle in an infinite-dimensional setting is of critical importance to the biggest open problem in free probability theory -- the so-called
``Unification Conjecture'' -- with ramifications for operator algebras, information theory, and random matrices.
In this talk, I will discuss recent and ongoing joint work with Benoit Collins, addressing the configuration of random subspaces in an infinite-dimensional context. Using a mixture of techniques
from stochastic processes, PDEs, and complex analysis, we prove the general position claim and give a complete understanding of the associated geometry. This work proves an important special case
of the Unification Conjecture, and has interesting implications for the original finite-dimensional setting as well.
Paul Roberts - University of Utah
The study of Cohen-Macaulay rings and modules has been a central topic in Commutative Algebra for many years. Among other things, they have played a major role in the investigations of the
"Homological Conjectures", a set of problems on finite projective dimension, intersection theory, and related subjects. More recently, as a result of advances in the homological conjectures and
developments in Arithmetic Geometry, a number of questions have come up about "almost" Cohen-Macaulay modules and algebras. In this talk I will give some background to these topics, discuss what
almost Cohen-Macaulay algebras are and why they are interesting, and present various recent developments and open problems in this area.
Paul Bryan - UCSD
In this talk I shall present an isoperimetric comparison
theorem for the Ricci flow on surfaces, inspired by Hamilton's
isoperimetric estimate. I will show how this can be used to prove the
Hamilton/Chow theorem, that the Ricci flow converges to a constant
curvature metric, thus for example providing a proof of the famous
uniformization theorem. This was joint work with Ben Andrews.
Craig Timmons - UCSD
Answering a question of Paul Erd\H{o}s, Antal Balog and Endre
Szemerédi proved that a finite set $A \subset \mathbb{Z}$ with many
three term arithmetic progressions must have a long arithmetic
progression. We will
discuss the proof of this result which uses the Balog-Szemer\'{e}di-Gowers
Theorem, Freiman's Theorem, and Szemeredi's Theorem on arithmetic
progressions. No previous knowledge of additive number theory will be
Adrian Ioana - UCSD
I will survey some recent progress on the classification of von
Neumann algebras arising from countable groups and their measure
preserving actions on probability spaces. This includes the finding of the
first classes of (superrigid) groups and actions that can be entirely
reconstructed from their von Neumann algebras.
Alireza Shabani - UC Berkeley
Rapid advance of quantum technologies demands novel mathematical tools for engineering complex quantum systems. Characterization of the structural and dynamical properties of large-scale quantum
devices, e.g., a quantum computer with 100 qubits, is among the current challenges. The major obstacle is the size of the Hilbert space and therefore the required experimental and computational
resources that grow exponentially with the number of the system components. Recently, compressed sensing method has been applied for efficient characterization of quantum systems. Originally
developed in classical signal processing, compressed sensing is a method to compress high-dimensional signals with a small number of measurements assuming that the signals live on a
low-dimensional manifold, and then to
reliably reconstruct them. In this presentation, I talk about the compressed sensing theory for quantum inversion problems, its first experimental realization, and the new problems motivated by
quantum applications.
[1] A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome,
M.P. Almeida, A. Fedrizzi and A. G. White,
”Efficient measurement of quantum dynamics via compressive sensing”,
Phys. Rev. Lett 106, 100401 (2011).
[2] A. Shabani, M.Mohseni, S. Lloyd, R. L. Kosut and H. Rabitz,
”Estimation of many-body quantum Hamiltonians via compressive sensing”,
Phys. Rev. A 84, 012107 (2011).
Nolan Wallach - UCSD
Practically every result that is presented in an elementary course
in number theory (i.e. Math 104 at UCSD) is used in the proof that this
joint work with D. Meyer works and gives an algorithm in the quantum
computing class analogous to P (polynomial).
Sebastian Cioaba - University of Delaware
Kirchhoff's Matrix Tree Theorem is one of the classical results in
spectral graph theory and it gives a formula for the number of
spanning trees of a graph in terms of the eigenvalues of its
In 1973, Chvatal introduced the notion of graph toughness and made two
important conjectures: 1. Any graph with sufficiently large toughness
is Hamiltonian. 2. Any graph with sufficiently large toughness is
The first conjecture is still open, but the second conjecture was
disproved by Alon who showed that there exist graphs with arbitrarily
large toughness and girth. The key to Alon's argument was determining
a close relation between the toughness of a regular graph and its
Independently and around the same time 1995, Brouwer found a slightly
better result relating the toughness of a regular graph to its
eigenvalues. In this talk, I will present some tight connections
between the eigenvalues of a connected regular graph and the maximum
number of edge-disjoint spanning trees in the graph that can be seen
as a spectral version of Nash-Williams/Tutte Theorem. I will show some
improvements of Brouwer's bound in certain ranges of toughness and
discuss another problem of Brouwer related to the toughness of graphs
attaining equality in the Hoffman ratio bound for the independence
number. This is joint work with my Ph.D. student, Wiseley Wong.
Dan Rogalski - UCSD
We discuss algebras over a field with the unusual property that all of their subalgebras are noetherian. We discuss some of the general results one can prove about such algebras. Some well-known
algebras associated to elliptic curves turn out to have this property, and we discuss these examples in detail.
Paul Horn - Harvard
A corollary of the Erd\H{o}s-Stone theorem is that, for any $0 \leq
\alpha < 1$, graphs with density greater than $\alpha$ contain an
(arbitrarily) large subgraph of density at least $\alpha+c$ for some
fixed $c = c(\alpha)$, so long as the graph itself is sufficiently
large. This phenomenon is known as a jump at $\alpha$. Erd\H{o}s
conjectured that similar statements should hold for hypergraphs, and
multigraphs where each edge can appear with multiplicity at most $q$,
for $q \geq 2$ fixed. Brown, Erd\H{o}s, and Simonovits answered this
conjecture in the affirmative for $q=2$, that is for multigraphs where
each edge can appear at most twice. R\"{o}dl answered the question in
Brett Kotschwar - ASU
We will show that smooth complete solutions to the Ricci flow of
uniformly bounded curvature are analytic in time in the interior of
their interval of existence. The analyticity is a consequence of
classical Bernstein-Bando-Shi type estimates on the temporal and
spatial derivatives of the curvature tensor, and offers an alternative proof of the unique continuation of solutions to the Ricci flow. As a further application of these estimates, we will show
that, under the above global hypotheses, about any interior space-time point (x0, t0), there exist local coordinates x on a neighborhood U of x0 in which the representation of the metric is
real-analytic in both x and t on some cylinder over U.
Brian Camley - Department of Physics and CTBP, UCSD
Biological membranes are composed of (among other things) hundreds of different lipids, which are believed to segregate into fluid rafts, which may be relevant to processes like virus assembly.
I'll talk about the spherical cow version of cells, synthetic membranes with three components (saturated and unsaturated lipids and cholesterol), which also segregate into two fluid phases.
Membranes are also particularly interesting from a physical standpoint because they have both two- and three-dimensional hydrodynamic behavior ("quasi-2D"), with many strange features, such as
diffusion coefficients of membrane rafts being effectively independent of their size. These quirks are characteristic of many interfacial fluids, and also appear in thin layers of liquid crystals
and protein films at the air-water interface. I'll show some continuum stochastic simulations of membrane domains and phase separation, discuss new ways of measuring membrane viscosity, and
suggest why some well-known dynamical scaling laws can change their exponents or even break down for phase separation in membranes. If there's time, I will also discuss how the dynamics of
protein diffusion can be altered by coupling to the lipid membrane.
Brendon Rhoades - UCSD
A sequence $(a_1,..., a_n)$ of positive integers is a parking function if its nondecreasing rearrangement $(b_1 \leq ... \leq b_n)$ satisfies $b_i < i + 1$ for all $i$. Parking functions were
introduced by Konheim and Weiss to study a hashing problem in computer science, but have since received a great deal of attention in algebraic combinatorics. We will define two new objects
attached to any (finite, real, irreducible) reflection group which generalize parking functions and deserve to be called parking spaces. We present a conjecture (proved in some cases) which
asserts a deep relationship between these constructions. This is joint work with Drew Armstrong at the University of Miami and Vic Reiner at the University of Minnesota.
Fan Chung Graham - UCSD
We will discuss some recent developments in several new directions of
spectral graph theory and mention a number of open problems.
Nolan Wallach - UCSD
This talk will be an exposition of joint work with Man Wai (Mandy)
Cheung on the effect of the Ricci flow on homogeneous metrics of
positive sectional curvature on flag varieties over the complex,
quaternions and octonians. The speaker’s 1972 paper shows that these
metrics exist only in the case of the variety of flags in the two
dimension projective space over these fields. Here are some of the
All cases can flow from strictly positive curvature to some negative
sectional curvature.
All cases can flow from positive definite Ricci curvature to
indefinite Ricci curvature
The quaternionic and octonianic cases can flow from strictly positive
sectional curvature to indefinite Ricci curvature (in the case of the
quaternions this is a result of Boehm and Wilking).
In the complex case the flow keeps the metrics of strictly positive
curvature in the metrics with positive definite Ricci curvature.
Elena Yudovina - University of Michigan
I consider a Markov process inspired by a toy model of a limit order book. "Bid" and "ask" orders arrive in time; the prices are iid uniform on [0,1]. (I'll discuss some extensions.) When a match
is possible (bid > ask), the highest bid and lowest ask leave the system. This process turns out to have surprising dynamics, with three limiting behaviours occurring with probability one. At low
prices (< 0.21...), bids eventually never leave; at high prices (>0.78...), asks eventually never leave; and in between, the system "ought to" be positive recurrent. I will show how we can derive
explicitly the limiting distribution of certain marginals for the middle prices; this makes it possible to extract the numerical values above from a 0-1 Law result.
Jacob Hughes - UCSD
Lights Out is a single player game on graph G. The game starts with a coloring of the vertices of G with two colors, 0 and 1. At each step, one vertex is toggled which switches the color of that
vertex and all of its neighbors. The game is won when all vertices have color 0. This game can be analyzed using linear algebra over a finite field, for example the number of solvable colorings
of a graph is 2 to the rank of A + I, where A is the adjacency matrix of the graph, and I is the identity.
We consider the stochastic process arising from toggling a sequence of random vertices. We demonstrate how the process can be viewed as a random walk on an associated state graph. We then find
the eigenvalues of the state graph, and use them to bound the rate of convergence and hitting times. We also provide bounds on the average number of steps until this random process reaches the
all 0 coloring that are asymptotically tight for many families of graphs.
Tom Goldstein - Post-Doctoral Fellow at the Rice University Department of Electrical Engineering
Alternating direction methods are a commonplace tool for general mathematical programming and optimization. These methods have become particularly important in the field of variational image
processing, which frequently requires the minimization of non-differentiable objectives. This paper considers accelerated (i.e., fast) variants of two common alternating direction methods: the
Alternating Direction Method of Multipliers (ADMM) and the Alternating Minimization Algorithm (AMA). The proposed acceleration is of the form first proposed by Nesterov for gradient descent
methods. In the case that the objective function is strongly convex, global convergence bounds are provided for both classical and accelerated variants of the methods. Numerical examples are
presented to demonstrating the superior performance of the fast methods.
Skip Garibaldi - Emory Univ. and CCR La Jolla
If two simple linear algebraic groups have the same F-isomorphism classes of maximal F-tori, are the two groups necessarily isomorphic? When F is a number field, it is an old question attributed
to Shimura. We describe the recent solution to this question (which relies on the notion of weak commensurability introduced by Gopal Prasad and Andrei Rapinchuk) and its connection with the
question "Can you hear the shape of a drum?" for arithmetic quotients of locally symmetric spaces.
Ioan Bejenaru - UCSD
This talk will cover some of the main problems in the field of nonlinear dispersive equations. I will discuss the stability, instability and blow-up for some simpler models such as the cubic
Nonlinear Schr\"odinger equations
Dave Penneys - University of Toronto
I will start with a basic introduction to planar algebras. I will
then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and
recent joint work with Stephen Bigelow (arXiv:1208:1564). With
Morrison, we construct a new exotic subfactor planar algebra using
Bigelow's jellyfish algorithm. With Bigelow, we determine exactly
when a planar algebra has a presentation by generators and jellyfish
Dave Penneys - University of Toronto
I will start with a basic introduction to planar algebras. I will
then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and
recent joint work with Stephen Bigelow (arXiv:1208:1564). With
Morrison, we construct a new exotic subfactor planar algebra using
Bigelow's jellyfish algorithm. With Bigelow, we determine exactly
when a planar algebra has a presentation by generators and jellyfish
Darren Creutz - Vanderbilt University
The Margulis Normal Subgroup Theorem states that any normal
subgroup of an irreducible lattice in a center-free higher-rank semisimple
group is of finite index. Stuck and Zimmer, expanding on Margulis'
approach, showed that any properly ergodic probability-preserving
ergodic action of such a lattice is essentially free.
I will present similar results: my work with Y. Shalom on normal
subgroups of lattices in products of simple locally compact groups and
normal subgroups of commensurators of lattices, and my work with J.
Peterson generalizing this result to stabilizers of ergodic
probability-preserving actions of such groups. As a consequence,
S-arithmetic lattices enjoy the same properties as the arithmetic
lattices (the Stuck-Zimmer result) as do lattices in certain product
groups. In particular, any nontrivial ergodic probability-preserving
action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is
essentially free.
The key idea in the study of normal subgroups is considering
nonsingular actions which are the extreme opposite of
measure-preserving. Somewhat surprisingly, the key idea in
understanding stabilizers of probability-preserving actions also
involves studying such actions and the bulk of our work is directed
towards properties of these contractive actions.
Alireza Salehi Golsefidy - UCSD
In this talk I will show how understanding of the possible limiting measures of translations of a measure can help us to deal with certain counting problems. Then I talk about the limiting
measures of translations of a horospherical measure. Finally I discuss how one can use this result to count the number rational points in a flag variety with respect to any line-bundle, reproving
a result of Franke-Manin-Tschinkel (anti-canonial line-bundle) and Batyrev-Tschinkel (arbitrary line-bundle). (Joint with A. Mohammadi)
Lillian Pierce - Oxford
Must the Fourier series of an $L^2$ function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative, by studying a particular type of maximal
singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson
operator. In this talk we will introduce the Carleson operator and survey several generalizations, and then describe new joint work with Po Lam Yung that introduces curved structure to the
setting of Carleson operators.
Lillian Pierce - Oxford University
Over the last hundred years, the circle method has become one of the
most important tools of analytic number theory. This talk (on joint work
with Roger Heath-Brown) will describe a new application of the circle
method to pairs of quadratic forms, via a novel two-dimensional analogue
of Kloosterman's version of the circle method. As a result, we prove
(under a mild geometric constraint) that any two quadratic forms with
integer coefficients, in 5 variables or more, simultaneously attain
prime values infinitely often.
Brandon Levin - Stanford University
I will begin with a friendly introduction to the deformation theory of
Galois representations and its role in modularity lifting, focusing on
the case of elliptic curves over Q. This will motivate the study of
local deformation rings and more specifically flat deformations. Next,
we will discuss Kisin’s resolution of the flat deformation ring at l = p
and describe conceptually the importance of local models of Shimura
varieties in analyzing its geometry. In the remaining time, we will
address the title of the talk; the additional structure we consider
could be a symplectic form, an orthogonal form, or more generally any
reductive subgroup G of $GL_N$. I will describe briefly the role that
recent advances in p-adic Hodge theory and local models of Shimura
varieties play in this situation.
Gilad Gour - University of Calgary
An important open problem in quantum information concerns with the question whether entanglement between signal states can help in sending classical information over a quantum channel. Recently,
Hastings proved that entanglement does help by finding a counter example for the long standing additivity conjecture that the minimum entropy output of a quantum channel is additive under taking
tensor products. In this talk I will show that the minimum entropy output of a quantum channel is locally additive. Hastings' counter example for the global additivity conjecture makes this
result somewhat surprising. In particular, it indicates that the non-additivity of the minimum entropy output is related to a global effect of quantum channels. I will end with a few related open
Markus Schmuck - Math and Chemical Engineering, Imperial College, London
We consider a classical continuum model which allows to describe essential electrokinetichenomena such as electro-phoresis and -osmosis. Applications and correspondingheory range from design of
micro fluidic devices, energy storage devices,emiconductors to emulating communication in biological cells by synthetic nanopores.
Based on this classical formulation, we derive effective macroscopic equationshich describe binary symmetric electrolytes in porous media. Theeterogeneous materials naturally induce corrected
transport parameters which weall "material tensors". A better understanding of the influence ofeterogeneous media on ionic transport is expected by the new formulation.he new equations provide
also an essential computational advantage by reliablyeducing the degrees of freedom required to resolve the microstructure.
The presented results are gained by asymptotic multi-scale expansions.his formal procedure is then made rigorous by the derivation of error boundsetween the exact microscopic solution and the new
upscaled macroscopic approximation.
Michael Young - Iowa State University
A \emph{mixed graph} is a graph with directed edges, called
arcs, and undirected edges. A $k$-coloring of the vertices is
\emph{proper} if colors $1,2,\ldots,k$ are assigned to each vertex such
that vertices $u$ and $v$ have different colors if $uv$ is an edge and
the color of $u$ is less than or equal to (resp. strictly less than) the
color of $v$ if $uv$ is an arc. The \emph{weak (resp. strong) chromatic
polynomial} of a mixed graph is a counting function that counts the
number of proper $k$-colorings. This talk will discuss previous work on
reciprocity theorems for other types of chromatic polynomials, and our
reciprocity theorem for weak chromatic polynomials which uses partially
ordered sets and order polynomials. This is joint work with Matthias
Beck, Daniel Blado, Joseph Crawford, and Taina Jean-Louis.
Liping Li - UC Riverside
The classical Koszul theory plays an important role in the representation theory of graded algebras. However, there are a lot of structures (algebras, categories, etc) having natural gradings
with non-semisimple degree 0 parts, to which the classical theory cannot apply. Particular examples include polynomial rings over non-semisimple algebras, extension algebras of modules, etc. In
this talk I'll introduce a generalized Koszul theory which does not demand the semisimple property. It preserves many classical results as Koszul duality and has a close relation to the classical
one. Applications of this generalized theory to extension algebras of modules and modular skew group algebras will be described.
Dionogi Benincasa - Imperial College London
David Zimmerman - UCSD
Logarithmic Sobolev inequalities (LSIs) show up in several areas of analysis; in particular, in probability. In this talk I will give some background and applications of LSIs. I will also discuss
some recent work and show how LSIs can be used to give a new proof of the classical result that the empirical law of eigenvalues of a sequence of random matrices converges weakly to its mean in
Otmar Venjakob - Univ. of Heidelberg
In non-commutative Iwasawa theory K-theoretic properties of Iwasawa algebras, i.e. completed group rings of e.g. compact p-adic Lie groups play a crucial role. Such groups arise naturally as
Galois groups attached to p-adic representations as for example on the Tate module of abelian varieties. In this talk we address in particular the question for which such groups the invariant
$SK_1$ vanishes. We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras. Also we shall try
to indicate what arithmetic consequences the vanishing of $SK_1$ has.
Mitchell Luskin - University of Minnesota and IPAM, UCLA
Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long
ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling
atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and
formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and sources of error. Our theoretical analysis and benchmark simulations have guided the
development of optimally accurate and efficient coupling methods.
Otmar Venjakob - Univ. of Heidelberg
Motivated by the question whether (some) Diophantine equations are related to special values of $\zeta$- or $L$-functions we first describe the origin of classical Iwasawa theory. Then we give a
survey on generalizations of these ideas to non-commutative Iwasawa theory, a topic which has been developed in recent years by several mathematicians, including the author.
Susan Hermiller - University of Nebraska
For finitely generated groups, growth of the elements of the group, and the series (or generating functions) associated to the growth function, have been widely studied. Recently researches have
begun to study the growth of conjugacy classes in these groups. Disconcertingly, the conjugacy growth series had been found by Rivin not to be rational for free groups with respect to a free
basis. In this talk I will introduce the notion of geodesic conjugacy growth functions and series, and discuss the effects of various group constructions on rationality of both the geodesic
conjugacy and (spherical) conjugacy languages whose growth is measured by these functions. In particular, we show that rationality of the geodesic conjugacy growth series, as well as on
regularity of the geodesic and spherical conjugacy growth series is preserved by both direct and free products. This is joint work with Laura Ciobanu.
Marc-Hubert Nicole - Institut Mathematique de Luminy
The classical Hasse invariant is defined via the determinant of the
Hasse-Witt matrix. It allows cutting out the so-called ordinary locus
within the special fiber of a modular curve: this is the affine locus
where the Hasse invariant is invertible. For more general Shimura
varieties, the ordinary locus may be empty, and the Hasse invariant is
then trivial. On the other hand, there exist for all Shimura varieties
of PEL-type so-called generalized Hasse-Witt invariants which are
vector-valued, but they are typically not robust enough to carry over
the usual applications of the classical Hasse invariant. In this talk,
we specialize to the scalar-valued cases that are most similar to the
classical invariant (joint work with W. Goldring).
Jason Morton - Penn State
I will discuss some common mathematical structures arising in information-processing networks in computer science, statistics and machine learning, and quantum information and many-body systems.
These seemingly disparate fields are connected by variations on the graphical modeling language of tensor networks, or more generally monoidal categories with various additional properties.
Tools from algebraic geometry, representation theory, and category theory have recently been applied to problems arising from such networks. Basic questions about each type of
information-processing system (such as what probability distributions or quantum states can be represented, or what word problems can be solved efficiently) quickly become interesting problems in
shared algebraic geometry, representation theory, and category theory. The result has been new insights into problems ranging from recognizing images to classifying quantum phases of matter and
interesting challenges in pure mathematics.
Sam Xing Peng - UCSD
The spectra of Erdos-Renyi random graphs have been long studied. We consider random graphs of which each edge is determined by an independent random indicator variable with the expected value not
all equal in general. We prove that the eigenvalues of the adjacent matrix and the normalized Laplacian matrix of such random graphs can be approximated by those of the `expectation graph ’.
Marc-Hubert Nicole - Institut Mathematique de Luminy
Shimura varieties are generalizations of modular curves which
are at the heart of the programs of both R. Langlands and S. Kudla. We
will focus on the geometric infrastructure built using Barsotti-Tate
groups, and we will present some arithmetic applications related to the
above programs or their p-adic variants.
Shenggou Zhou - UCSD
We apply the variational implicit solvent model (VISM), numerically minimized by the level-set method, to study hydration effects in the high-affinity binding of the B2 bicyclo[2.2.2]octane
derivative to the CB[7] molecule. For the unbounded host molecule, we find two equilibrium dielectric interfaces with two different types of initial wraps. The host cavity shows capillary
evaporation when the initial guess is a loose wrapping; while the host is completed hydrated when a tight initial guess is prescribed. In good agreement with MD simulation results, the hydrated
case is more favorable due to solvent-solute electrostatic interaction advantage. For the guest binding we find reasonable agreement to experimental binding affinities. Dielectric interfaces of
the system during the binding process are given by level-set simulations. Individual free energy contributions show that water-mediated hydrophobic interactions based on decreasing water
unfavorable concave surface upon binding and electrostatic interactions are two major driving forces for the binding process. The findings are in line with recent computer simulations and
experiment data. With further refinement the VISM could be a promising tool for an efficient calculation of molecular binding affinities.
Song Sun - Imperial College
Herbert Heyer - Univ. Tuebingen, Germany
There are two basic theorems on arithmetic properties of probability measures on Euclidean space: the Levy decomposition of infinitely divisible probability measures as convolutions of Poisson
and Gaussian measures, and the Khintchine factorization of arbitrary probability measures in terms of indecomposable measures and measures without indecomposable factors. Both theorems have been
generalized by K. R. Parthasarathy to measures on an Abelian locally compact group. Within this framework the role of Gaussian factors will be discussed. Moreover, characterizations of Gaussian
measures (in the sense of Cramer and Bernstein) will be presented whose validity depends on the structure of the underlying group.
Michael Kasa - UCSD
We discuss recent work by Gross and Siebert defining logarithmic Gromov-Witten invariants.
Daniele Venturi - Brown University
Darryl D. Holm - Imperial College, London
A $G$-strand is a map $\mathbb{R}\times\mathbb{R}\to G$ into a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. $G$-strands on
finite-dimensional groups satisfy $1+1$ space-time evolutionary equations. A large class of these equations have Lax-pair representations that show they admit soliton solutions. For example, the
$SO(3)$-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Various other examples will be discussed, including collisions of solutions
with singular support (e.g., peakons) on ${\rm Diff}(\mathbb{R})$-strands, in which ${\rm Diff}(\mathbb{R})$ is the group of diffeomorphisms of the real line $\mathbb{R}$, for which the group
product is composition of smooth invertible functions.
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Training Configuration | Axolotl Training Platform
gradient_accumulation_steps controls the number of forward and backward passes to skip before updating the model's weights. It is useful when the batch size is too large to fit into GPU memory at
once. Gradients are accumulated over the specified number of steps before performing weight updates.
micro_batch_size specifies the number of samples included in each batch sent to each GPU. It helps in determining the mini-batch size for training.
eval_batch_size sets the batch size for evaluation. It determines how many samples are processed in each evaluation step.
num_epochs defines the number of training epochs, which represent the number of times the entire dataset is processed during training.
warmup_steps specifies the number of warm-up steps for the learning rate scheduler. During the warm-up phase, the learning rate gradually increases to its full value.
learning_rate sets the initial learning rate for training. It is a critical hyperparameter that determines the step size for weight updates during optimization.
lr_quadratic_warmup is a field related to learning rate scheduling. It is used to specify a quadratic warm-up schedule for the learning rate, which can be beneficial for certain training scenarios.
logging_steps sets the frequency at which training logs are generated. It controls how often training progress is reported.
save_strategy determines when model checkpoints are saved during training. Setting it to 'no' skips checkpoint saves, while other options control the timing of saves.
save_steps specifies the frequency at which model checkpoints are saved. You can leave it empty to save at each epoch or specify a different number of steps.
eval_steps controls the frequency of model evaluation during training. It can be specified as an integer for every N steps or as a decimal for a fraction of total steps.
save_total_limit limits the maximum number of checkpoints saved at a time. Older checkpoints are deleted to keep the total number within this limit.
max_steps defines the maximum number of iterations to train for. It takes precedence over num_epochs. For example, if you set max_steps to 100, the training will stop after 100 steps, regardless of
the number of epochs.
eval_table_size specifies the approximate number of predictions sent to wandb (Weights and Biases) depending on the batch size. This field is enabled above 0 and is useful for tracking evaluation
eval_table_max_new_tokens sets the total number of tokens generated for predictions sent to wandb. It helps control the amount of data sent for monitoring.
When specified, save_safetensors indicates saving the model as safetensors, requiring the safetensors package for compatibility.
train_on_inputs determines whether to mask out or include the human's prompt from the training labels. Setting it to 'false' omits the prompt from training.
When set to 'true,' group_by_length groups data with similar sequence lengths together to minimize padding. This can help improve training efficiency but may lead to an oscillating training loss.
gradient_checkpointing controls whether to use gradient checkpointing, a technique that can reduce memory consumption during training. When enabled, it trades off computation for memory.
early_stopping_patience determines when to stop training if evaluation losses increase consecutively for a specified number of times. It helps prevent overfitting.
lr_scheduler specifies the learning rate scheduler to use during training. Options include 'one_cycle,' 'log_sweep,' or leaving it empty for cosine scheduling.
lr_scheduler_kwargs can be used to provide additional arguments to the learning rate scheduler, depending on the chosen scheduler type.
For 'one_cycle' optimizer, lr_div_factor determines the learning rate division factor during the one-cycle learning rate schedule.
For 'log_sweep' optimizer, log_sweep_min_lr sets the minimum learning rate for the logarithmic learning rate sweep.
For 'log_sweep' optimizer, log_sweep_max_lr sets the maximum learning rate for the logarithmic learning rate sweep.
optimizer specifies the optimizer to use for training. There are various optimizer options available, and the choice depends on the model and use case.
weight_decay determines the weight decay applied during optimization. It is a regularization term that prevents overfitting by penalizing large model weights.
For 'adamw' optimizer, adam_beta1 sets the beta1 hyperparameter for the Adam optimizer. It controls the exponential moving average of past gradients.
For 'adamw' optimizer, adam_beta2 sets the beta2 hyperparameter for the Adam optimizer. It controls the exponential moving average of past squared gradients.
For 'adamw' optimizer, adam_epsilon sets the epsilon value added to the denominator to prevent division by zero.
max_grad_norm specifies the maximum gradient norm value. Gradients are clipped to this value during training to prevent exploding gradients.
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Noise Figure (NF) in RF Systems - Rahsoft
Noise Figure (NF) in RF Systems
What is Noise Figure (NF) in RF Systems?
Noise Figure (NF) is a critical parameter in RF (Radio Frequency) systems that quantifies the impact of noise introduced by components, such as amplifiers, on the overall system’s signal-to-noise
ratio (SNR). In other words, NF measures how much additional noise a component adds to the signal as it passes through.
In RF systems, minimizing noise is crucial because it directly affects the quality of received signals. A lower NF indicates better performance, as it means the component adds less noise to the
incoming signal.
The Formula for Noise Figure (NF)
The Noise Figure (NF) of an RF component is typically expressed in decibels (dB) and can be calculated using the following formula:
Example of Noise Figure Calculation
Let’s consider an example to calculate the Noise Figure (NF) of an RF amplifier. Suppose we have an amplifier with the following characteristics:
Using the formula for Noise Figure (NF), we can calculate:
So, the Noise Figure (NF) of the RF amplifier is 50 dB.
Noise Figure Calculation for RF Cascaded System
In RF systems, it’s common to have multiple components cascaded together, each with its Noise Figure. To calculate the cascaded Noise Figure (NF_total) of the entire system, you can use the following
This formula takes into account the gains (G1,G2,G3,…) of the components to calculate the cumulative noise contribution of the entire system.
Let’s illustrate this with an example:
Suppose we have two RF amplifiers cascaded together, each with its Noise Figure and gain:
To calculate the cascaded Noise Figure (NF_total) of the system:
So, the cascaded Noise Figure (NF_total) of the two amplifiers is approximately 3.3 dB. This calculation accounts for the noise contributions and gains of both amplifiers in the cascaded system.
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How to Configure a Grade Point Average
••• Jupiterimages/Comstock/Getty Images
Keeping a high grade point average, or GPA, can mean the difference between graduating with or without honors. But with the confusion between number grades, letter grades and credit hours,
determining your GPA can be a little tricky. Keeping tabs on your GPA during the semester can help you maintain higher grades and keep you motivated to improve your work. With a few simple
calculations, you can determine your GPA for all of your courses.
Determining your GPA
Determine your institution's letter grade point equivalent. For instance, in many schools an A is consider a 4.0, but other schools offer 4.3 points for an A+ or 3.8 points for an A-. Copy these
point equivalents on your sheet of paper as a reference.
Determine the total number of course credits (hours) you are taking in a semester. In general, the number of credits for a course is the number of hours per week that you spend in that class.
For instructional purposes, let's say you took three 3-credit courses and one 4-credit course in a semester. You would have 13 total course credits for that semester.
4 + 3 + 3 + 3 = 13 credits
Multiply your letter grade's point equivalent (as determined by your institution, which you recorded in Step 1 by the number of course credits (hours) for that class. Record those numbers on your
paper, and use your calculator if necessary.
Continuing with our instructional example, let's say you made a 3.5 in your four-credit course, a 3.0 in one of your three-credit courses, a 4.0 in your other three-credit course and a 2.5 in
your final three-credit course.
3.5 x 4 = 14 points 3.0 x 3 = 9 points 4.0 x 3 = 12 points 2.5 x 3 = 7.5 points
Add all of your total points for each class. if necessary, use your calculator
In our example: 14 + 9 + 12 + 7.5 = 42.5 points
Divide your total points by your number of course credits. Record this number on your paper, and use your calculator if you need to.
42.5 points/13 hours = 3.27
We did it. The total GPA in this example is 3.27.
Things You'll Need
□ Course grades
□ Pencil
□ Paper
□ Calculator
□ Be sure to check your school's point equivalents for minus and plus grades, such as points for an A- or a B+. Many schools have slightly different point equivalents for these in-between
• Be sure to check your school's point equivalents for minus and plus grades, such as points for an A- or a B+. Many schools have slightly different point equivalents for these in-between grades.
About the Author
Lindsey Robinson Sanchez, from Bessemer, Ala., has written for the "Troy Messenger," "The Alabama Baptist" and "The Gainesville Times," where her work was featured on the AP wire. She has a Bachelor
of Science in journalism from the University of Florida. She writes style, beauty, fitness, travel and culture.
Photo Credits
Jupiterimages/Comstock/Getty Images
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