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Home > Confidence Interval > Standard Error 95 Confidence Test

# Standard Error 95 Confidence Test

## Contents

One way of assessing optimality is by the length of the interval, so that a rule for constructing a confidence interval is judged better than another if it leads to intervals The use of Z or t again depends on whether the sample sizes are large (n1 > 30 and n2 > 30) or small. Prentice Hall International, New Jersey. These formulas assume equal variability in the two populations (i.e., the population variances are equal, or σ12= σ22), meaning that the outcome is equally variable in each of the comparison populations. this contact form

The standard error estimated using the sample standard deviation is 2.56. For 90% confidence intervals divide by 3.29 rather than 3.92; for 99% confidence intervals divide by 5.15. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). http://onlinestatbook.com/2/estimation/mean.html

## Confidence Interval For Mean Formula

Suppose we wanted to calculate a 95% confidence interval forμ. Then T = X ¯ − μ S / n {\displaystyle T={\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}} has a Student's t-distribution with n − 1 degrees of freedom.[27] Note that the distribution Admittedly, such a misinterpretation is encouraged by the word 'confidence'." A 95% confidence interval does not mean that 95% of the sample data lie within the interval.

Otherwise, use the second equation. Note that it is no longer possible to say that the (observed) interval (u(x),v(x)) has probability γ to contain the parameter θ. If there is no difference between the population means, then the difference will be zero (i.e., (1-2).= 0). 95 Confidence Interval Z Score The Annals of Mathematical Statistics.

They report that, in a sample of 400 patients, the new drug lowers cholesterol by an average of 20 units (mg/dL). 95 Confidence Interval Calculator When you need to be sure you've computed an accurate interval then use the online calculators (which we use). Thus this event has probability either 0 or1. http://stattrek.com/estimation/margin-of-error.aspx?Tutorial=AP From the same data one may calculate a 90% confidence interval, which in this case might be 37% to 43%.

A larger sample size normally will lead to a better estimate of the population parameter. 95% Confidence Interval This value is dependent on the confidence level (C) for the test and degrees of freedom. doi:10.2307/2340569. London: SAGE. ^ Zar, J.H. (1984) Biostatistical Analysis.

## 95 Confidence Interval Calculator

Journal of the Royal Statistical Society. The standard deviation of the age for the 16 runners is 10.23, which is somewhat greater than the true population standard deviation σ = 9.27 years. Confidence Interval For Mean Formula And unfortunately one does not know in which of the cases this happens. How To Calculate 95 Confidence Interval In Excel If n1 > 30 and n2 > 30, we can use the z-table: Use Z table for standard normal distribution If n1 < 30 or n2 < 30, use the

G. (1981) "In defence of the Neyman-Pearson theory of confidence intervals", Philosophy of Science, 48 (2), 269–280. http://comunidadwindows.org/confidence-interval/standard-error-for-99-confidence-level.php Later sections will present the standard error of other statistics, such as the standard error of a proportion, the standard error of the difference of two means, the standard error of One way to answer this question focuses on the population standard deviation. Confidence and prediction bands are often used as part of the graphical presentation of results of a regression analysis. Confidence Interval Formula T Test

The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean. www.dummies.com. Descriptive statistics This is closely related to the method of moments for estimation. navigate here Consider an additional random variable Y which may or may not be statistically dependent on the random sample X.

Notice that the population standard deviation of 4.72 years for age at first marriage is about half the standard deviation of 9.27 years for the runners. Confidence Interval Table Gurland and Tripathi (1971)[6] provide a correction and equation for this effect. This means that if we repeatedly compute the mean (M) from a sample, and create an interval ranging from M - 23.52 to M + 23.52, this interval will contain the

## Established rules for standard procedures might be justified or explained via several of these routes.

Vol 2: Inference and Relationship, Griffin, London. doi:10.3758/s13423-015-0947-8. Despite the small difference in equations for the standard deviation and the standard error, this small difference changes the meaning of what is being reported from a description of the variation Confidence Interval Example From the formula, it is clear that the width of the interval is controlled by two factors: As N increases, the interval gets narrower from the $$\sqrt{N}$$ term.

ISBN 0-521-05165-7 Keeping, E.S. (1962) Introduction to Statistical Inference. Identify a sample statistic. Just as the random variable X notionally corresponds to other possible realizations of x from the same population or from the same version of reality, the parameters (θ,ϕ) indicate that we http://comunidadwindows.org/confidence-interval/standard-error-standard-deviation-95-confidence-interval.php If we were to perform an upper, one-tailed test, the critical value would be t1-α,ν = 1.6527, and we would still reject the null hypothesis.

D. The critical value is the t statistic having 99 degrees of freedom and a cumulative probability equal to 0.995. They are very different if informative prior information is included in the Bayesian analysis; and may be very different for some parts of the space of possible data even if the If 40 out of 50 reported their intent to repurchase, you can use the Adjusted Wald technique to find your confidence interval:Find the average by adding all the 1's and dividing

One type of sample mean is the mean of an indicator variable, which takes on the value 1 for true and the value 0 for false. The mean of all possible sample means is equal to the population mean. Perspect Clin Res. 3 (3): 113–116. Software Confidence limits for the mean and one-sample t-tests are available in just about all general purpose statistical software programs.

Oxford University Press. 62 (1): 155–161. ^ Pratt, J. Confidence intervals are commonly reported in tables or graphs along with point estimates of the same parameters, to show the reliability of the estimates. JSTOR187185 Neyman, J. (1937) "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability" Philosophical Transactions of the Royal Society of London A, 236, 333–380. (Seminal work.) pp 43–45 ^ Cox D.R., Hinkley D.V. (1974) Theoretical Statistics, Chapman & Hall, p214, 225, 233 ^ [1] ^ [2] ^ Hoekstra, R., R.


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# Finding velocity in terms of variables

1. Jan 29, 2007

### stinlin

1. The problem statement, all variables and given/known data
(See attached for labels) After taking off, a helicopter climbs in a straight line at a constant angle beta. Its flight is tracked by radar from point A. Determine the speed of the helicopter in terms of d, beta, theta, and theta dot.

d is the distance from the radar to the initial position of the copter, theta is the angle between the horizontal and the copter, v is the velocity, and beta is the angle the copter moves at.

2. Relevant equations

v = (r dot) e radius + (r * (theta dot))e theta

e radius is the radial component and e theta is the angular component...This is done in polar form. So for those who know, it's the velocity equation in polar form.

3. The attempt at a solution

Well, it's asking for the speed, so I know I have to take the radial and angular components, square them, add them, and take the square root (or so I assume). I was also able to split the velocity into vsin(beta) and vcos(beta). Otherwise, I'm lost.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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7.6 KB
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# What is the identity of a 2nd-row element that has three valence electrons?

Feb 13, 2017

$\text{Boron.........}$

#### Explanation:

Boron, $Z = 5$, has 3 valence electrons: $2 {s}^{2} 2 {p}^{1}$. Carbon, $Z = 6$, has 4 valence electrons: $2 {s}^{2} 2 {p}^{2}$. Nitrogen, $Z = 7$, has 5 valence electrons: $2 {s}^{2} 2 {p}^{3}$. And fluorine, $Z = 7$, has 7 valence electrons: $2 {s}^{2} 2 {p}^{5}$. Of course, these valence electrons are active in the geometry and structure of the compounds of these species.


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# Drawing Ray diagrams

When drawing ray diagrams for concave mirrors, I was advised to:

1. draw a ray that is parallel to the principle axis, the reflected ray will pass through the focus.

2. Draw a ray through the center (C), the reflected ray will be same as the incident ray but opposite in direction.

I understand the reasoning behind point 2. as the incident angle will equal to zero. However, I don't understand why every single incoming parallel ray will pass through this certain point called the focus. Why not choose any other point. Is there any proof as to why this happen or was this just experimentally found?

Main question: why do parallel rays pass through the focus?

• The two rules are a reasonable approximation of what happens when light parallel to the mirror axis shines on a parabolic mirror. This can be found both experimentally, as well as mathematically, if one applies the laws of reflection to a parabolic surface. Unless I am mistaken, you are having some doubts about the universality of these rules? The good, and bad news is, that you are right. Reflection on real mirrors is much more complicated than that, and leads to many optical errors in devices like telescopes. Having said that, this approximation is still useful for many simple problems. – CuriousOne Aug 20 '14 at 5:35

You've probably heard of a curve called the parabola, and you probably interpret this as meaning it's a function something like $y = x^2$. However there is another way to define the parabola. If you draw a line (called the directrix) and then choose a point (called the focus) not on that line then the set of points that are an equal distance from the directrix and the focus form a parabola.


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# Zeros of polynomials as a finite union of manifolds

Consider a polynomial in $d$ variables, $p:\mathbb{R}^{d}\rightarrow\mathbb{R}$. Denote by $\mathcal{C}$ its set of zeros, i.e. $$\mathcal{C}=\{x\in\mathbb{R}^{d}\ |\ p(x)=0\}.$$

Q. Is it possible to find finitely many (not necessarily disjoint) manifolds $M_{1},\dots, M_{n}\subset\mathbb{R}^{d}$, with possibly different dimensions, such that $$\mathcal{C}=\bigcup_{k=1}^{n}M_{k}?$$

My question arises in the context of degenerate real matrices, namely, the case where $\mathcal{C}$ consists on the set of symmetric real matrices with at least one repeated eigenvalue (in this context, $p(x)$ is the discriminant of the symmetric real matrix $x$). Any suggestion on how to approach this problem or a reference related to the problem will be greatly appreciated.

## 1 Answer

The answer is yes.

Your set $\mathcal{C}$ is by definition a (semi)algebraic subset of $\mathbb{R}^d$, and any such a subset has a Whitney stratification with finitely many (semi)algebraic strata.

See A. Dimca, Singularities and Topology of Hypersurfaces, ZBL07535700, Chapter 1, Corollary 1.12 for more details.

Edit. It is also true that any closed analytic (or subanalytc) set of an analytic manifold admits a Whitney stratification whose strata are analytic submanifolds. In the complex setting, this is called a complex Whitney stratification. See

M. Tib$\check{\rm u}$ar, Polynomials and vanishing cycles, Cambridge Tracts in Mathematics 170 (2007), ZBL1126.32026, page 219 and the references given therein.

• Note however that the wikipedia reference that you provide contains a big mistake. If $X$ is a singular variety, the collection $Sing(X), Sing(Sing(X)),...$ is not a Whitney stratification in general. Apr 27, 2017 at 18:16
• @Libli: you are right. This does not define in general a Whitney stratification, but some refinement will, see J. L. Verdier, Stratifications de Whitney et theoreme de Bertini-Sard, Invent. Math. 36 (1976), 295-312. Thank you for the remark. Apr 27, 2017 at 18:32
• Please, what happens if we replace $\mathbb{R}$ by $\mathbb{C}$? Apr 28, 2017 at 9:34
• Whitney stratification still exist in the complex setting. See my edit to the answer. Apr 28, 2017 at 9:50


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# Principles of Mathematics

> Principles of Mathematics
Course ID
MPM2D
Level
Semester
Continuous
Method
In-Person, Online
Credit Value
1.0

Principles of Mathematics course enables students to broaden their understanding of relationships and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and abstract reasoning. Students will explore quadratic relations and their applications; solve and apply linear systems; verify properties of geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles. Students will reason mathematically and communicate their thinking as they solve multi-step problems.

A gateway into any profession and one of most important positions in an Organization

• MPM1D

#### Course structure and modules

Overall Expectations:  By the end of this course, students will:
• determine the basic properties of quadratic relations;
• relate transformations of the graph of y=x2 to the algebraic representation y=a(x– h)2+ k;
• solve quadratic equations and interpret the solutions with respect to the corresponding relations;
• solve problems involving quadratic relations
• Analytic Geometry

Overall Expectations:  By the end of this course, students will:

• model and solve problems involving the intersection of two straight lines;
• solve problems using analytic geometry involving properties of lines and line segments;
• solve quadratic equations and interpret the solutions with respect to the corresponding relations;
• Trigonometry

Overall Expectations:  By the end of this course, students will:

• determine the basic properties of quadratic relations;
• relate transformations of the graph of y=x2 to the algebraic representation y=a(x– h)2+ k;
• verify geometric properties of triangles and quadrilaterals, using analytic geometry.


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## Notes

### Code

Here's a Python implementation of the algorithm we developed:

1. Consider a regular $$n$$-gon inscribed in a circle of radius 1: Show that its perimeter is given by $$2n \sin\left(\frac{360^\circ}{2n}\right).$$
2. Recall that $$\Twopi$$ is defined to be the circumference of a unit circle, and $$\pi$$ is defined to be half this quantity. Compute approximate values of $$\Twopi$$ and $$\pi$$ (to 4 decimal points, say).


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Lemma 30.26.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes which are locally of finite type over $S$.

1. If $Y$ is separated over $S$ and $Z \subset X$ is a closed subset proper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$.

2. If $f$ is universally closed and $Z \subset X$ is a closed subset proper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$.

3. If $f$ is proper and $Z \subset Y$ is a closed subset proper over $S$, then $f^{-1}(Z)$ is a closed subset of $X$ proper over $S$.

Proof. Proof of (1). Assume $Y$ is separated over $S$ and $Z \subset X$ is a closed subset proper over $S$. Endow $Z$ with the reduced induced closed subscheme structure and apply Morphisms, Lemma 29.41.10 to $Z \to Y$ over $S$ to conclude.

Proof of (2). Assume $f$ is universally closed and $Z \subset X$ is a closed subset proper over $S$. Endow $Z$ and $Z' = f(Z)$ with their reduced induced closed subscheme structures. We obtain an induced morphism $Z \to Z'$. Denote $Z'' = f^{-1}(Z')$ the scheme theoretic inverse image (Schemes, Definition 26.17.7). Then $Z'' \to Z'$ is universally closed as a base change of $f$ (Morphisms, Lemma 29.41.5). Hence $Z \to Z'$ is universally closed as a composition of the closed immersion $Z \to Z''$ and $Z'' \to Z'$ (Morphisms, Lemmas 29.41.6 and 29.41.4). We conclude that $Z' \to S$ is separated by Morphisms, Lemma 29.41.11. Since $Z \to S$ is quasi-compact and $Z \to Z'$ is surjective we see that $Z' \to S$ is quasi-compact. Since $Z' \to S$ is the composition of $Z' \to Y$ and $Y \to S$ we see that $Z' \to S$ is locally of finite type (Morphisms, Lemmas 29.15.5 and 29.15.3). Finally, since $Z \to S$ is universally closed, we see that the same thing is true for $Z' \to S$ by Morphisms, Lemma 29.41.9. This finishes the proof.

Proof of (3). Assume $f$ is proper and $Z \subset Y$ is a closed subset proper over $S$. Endow $Z$ with the reduced induced closed subscheme structure. Denote $Z' = f^{-1}(Z)$ the scheme theoretic inverse image (Schemes, Definition 26.17.7). Then $Z' \to Z$ is proper as a base change of $f$ (Morphisms, Lemma 29.41.5). Whence $Z' \to S$ is proper as the composition of $Z' \to Z$ and $Z \to S$ (Morphisms, Lemma 29.41.4). This finishes the proof. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


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# How does an AC circuit get anything done? [duplicate]

if alternating current switches directions back and forth periodically it seems like to me that current would go forward a bit and then almost immediately back track the exact same distance, if this is the case then how does AC ever get to where we want it to? Could I be misconceiving AC in some way?

• Possible duplicate of Alternating current – Aaron Stevens Nov 21 at 4:45
• Movement of charge is movement of charge. It usually doesn't matter in which direction – Aaron Stevens Nov 21 at 4:47
• Consider that pistons move only a small distance back and forth in an automobile engine and do so very quickly. How can such a system move the vehicle so far? – JimmyJames Nov 21 at 15:29

The wording of your question suggests that you might have a fundamental misconception about how electricity works (quite a common one, unfortunately, because introductory courses in electronics often get it wrong). In an electric circuit, electric currents do not directly transfer energy. Rather, energy is transferred through the electromagnetic fields in the space surrounding the wires. The only purpose of the wires and other components is to guide the fields to the right places.

This is quite counterintuitive, since electromagnetic fields are invisible and we are used to thinking of electricity as something that flows inside wires. However, the conclusion is easy to demonstrate. The directional energy flux in an electromagnetic field (the Poynting vector) is equal to $$\mathbf{S} = \mathbf{E} \times \mathbf{H}$$ where E and H are the electric field and the magnetic auxiliary field (in simple cases, $$\mathbf{H} = \mathbf{B}/\mu$$).

In the simple circuit drawn below, the electric field points downward (since the voltage is positive in the upper wire). Meanwhile, the magnetic field circles around the current-carrying wires; thus, it points into the screen inside the circuit. Using the right-hand rule for cross products, we can see that the Poynting vector points to the right. Thus, power flows from the battery to the load. One can show that this energy flux is sufficient to explain all of the power delivered to the load.

What if we replace the battery with an AC voltage source? Now when the voltage becomes negative, both the electric field and the current (and therefore the magnetic field) reverse their direction. These changes cancel each other out and the Poynting vector still points to the right. Therefore, power continues to flow from the voltage source to the load – even though every electron in the circuit stays in nearly the same place.

• Err... What about electrons, though? Isn't that a transfer of energy? – Andrew Nov 21 at 17:37
• But there is an electromagnetic field inside the wires... why don't you say anything about that energy? – thermomagnetic condensed boson Nov 21 at 19:08
• @Andrew the electrons move slightly counterclockwise around the circuit when the voltage is positive, and slightly clockwise when the voltage is negative. No electrons are transferred from the source to the load – if that happened, charge would build up on the load side of the circuit, and the voltage on that side would quickly become large enough to prevent more current from flowing. – Thorondor Nov 21 at 22:27
• @thermomagneticcondensedboson Because there is virtually no electric field inside the wires, since they are made of a conductive material. – Thorondor Nov 21 at 22:28
• @thermomagneticcondensedboson In Ohm's law $J = \sigma E$, $\sigma$ is the conductivity, which is very high (for simple problems like this one, effectively infinite) in a metal wire. Therefore, to allow a finite current density, the electric field must vanish. – Thorondor Nov 22 at 6:46

Imagine you have a crosscut saw with two handles, one on each end and a person holding each handle. The saw rests on a log to be cut. One person pulls their end of the saw and the teeth cut into the log. Then the other person pulls their end and the blade comes back, cutting further into the log.

The point is that the saw can cut the log even though it only goes back and forth, retracing its path over and over. In the same sense, AC power can perform useful work on the "pull" part of the AC cycle as well as the "push" part.

• This is an odd analogy, the saw does work because it consistently moves in the direction that it's cutting (generally downward) if it only moved back and forth, it would achieve nothing. – DBS Nov 21 at 15:26
• @DBS That's not really how saws work. They shave off the top surface of the wood as they go across it. One a slight amount of downward force is required. – JimmyJames Nov 21 at 15:32
• @DBS The saw moves downward by gravity because the wood that it cuts through is no longer there. – Tashus Nov 21 at 15:32
• In this analogy, Direct-Current is more like a bandsaw, which is a belt with teeth that spins continuously in one direction. While AC is more like a saw with double-sided teeth, both directions apply cutting effects and an oscillating motion gets the best out of it. – Ruadhan2300 Nov 21 at 16:07
• @DBS The problem with analogies is that they can be taken too literally; read it this way - given that there exists a mechanism that makes the saw interact with the wood at the contact point (the aforementioned slight downward force, but we'll "black box" that), the back & forth periodic motion is a viable option to do useful work, with the two people being analogous to a battery. (P.S. Imagine extending the ends of the saw to form a circuit with the power source (maybe a single person or an engine) being far away.) – Filip Milovanović Nov 21 at 16:09

If alternating current switches directions back and forth periodically it seems like to me that current would go forward a bit and then almost immediately back track the exact same distance, if this is the case then how does AC ever get to where we want it to?

See this microscopic view of current, and understand that the electrons do not travel fast within the conductor, but have a drift velocity. This drift velocity , according to the classical electrodynamics equations, is what generates the current measured in the lab. In a DC circuits the electrons move slowly towards the potential , in an AC current they move back and forth, but the macroscopically generated current has a direction from high potential to low potential.

The reason that AC has prevailed in the transmission of electric power is practical considerations. When DC is transmitted one has to generate AC at home to get motors running, for example.

• The Drude's model is known not to be an accurate description of reality. The speed of the electrons that conduct is about the order of the Fermi speed, i.e. about 1% of light speed in metals. – thermomagnetic condensed boson Nov 21 at 19:28
• @thermomagneticcondensedboson Of course a model that does not include quantum mechanics cannot be the "real" one, as main stream physics is based on quantum mechanics being the underlying frame of classical. But it is useful, as the planetary model of the atom was useful (and still is) when physicists first found out that it described the data in the correct direction. – anna v Nov 22 at 5:15

and then almost immediately back track the exact same distance

Current travels, very approximately, one foot per nanosecond. Mains 50Hz cycles have a period of 20,000,000 nanoseconds. It's very far from "almost immediately".

• This doesn't really address the fundamental misunderstanding that is confusing the asker. The distance traveled by the current each half period is not really material to the way the work is performed, and it does indeed travel the same distance in the opposite direction during the second half of each period, regardless of the distance traveled. – Tashus Nov 21 at 15:39
• The asker thinks that 1/60th of a second is "almost immediate", but for an electric field moving at (nearly) the speed of light, that's enough time to travel across a continent, or zoom through the circuits in your device many thousands of times. The confusion is (in part) due to not understanding the vast difference between the speed of humans vs the speed of electricity. – Foo Bar Nov 21 at 16:02

Alternating current transmits powered by varying electric fields which makes the electron oscillate this oscillating energy transmitted through the wire in the form of electric fields and the energy is transmitted where it needs to be transmitted.

For example in a electric heater the alternating current oscillates electron in the filament which produces heat. But in dc heater there is a continuous flow of electron which generates heat due to resistance.

• But DC current can cause heating, transfer power, etc. too. This answer isn't helpful, as it basically just says AC works because it works. – Aaron Stevens Nov 21 at 5:01
• The edit didn't really help alleviate @AaronStevens concerns, since it's the resistance that results in heat in an AC heater also. If anything all you've done is made it less clear. – Trotski94 Nov 21 at 12:59


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Lesson 2, Topic 2
In Progress

# Experimental Verification of Ohm’s Law

Lesson Progress
0% Complete

Aim of Experiment: To verify ohm’s law.

Apparatus: Ammeter, voltmeter, rheostat, standard resistor, cell, connection wires and switch.

Set-up Diagram: The connection diagram for verification of Ohm’s law is shown below.

Procedure:

A conductor of resistance R ohms is connected in series with an ammeter A, rheostat (variable resistance) and a battery B through key K. To measure the potential difference across the resistance, voltmeter V is connected across the ends of the resistor.

Press Key K to close the switch so that current flows in the circuit.

Read and record the values of current that flow in the circuit and voltage across the resistor using an ammeter and voltmeter respectively.

After relapsing the key K adjust the rheostat to a new position so as to change the resistance of the circuit. Close the switch by pressing Key K again. Record the new voltmeter and ammeter readings.

Repeat the process at least three more times. Read and record the corresponding values of I and V recorded by the ammeter and voltmeter.

Results

### Graph of Voltage V Against Current I:

The different values of Voltages V that were obtained are plotted on the vertical axis against different values of current I on the horizontal axis. A straight line graph which passes through the origin is obtained as shown below.

From the observations, it will be found that the ratio of voltmeter reading and ammeter reading remains the same i.e.

$$\frac{V_1}{I_1} = \frac{V_2}{I_2} = \frac{V_3}{I_3} = \frac{V_4}{I_4} = \frac{V_5}{I_5}$$

Constant = R, the resistance of the conductor.

The slope of the graph represents the resistance as shown below.

Slope = $$\frac{V_2 \: -\: V_1}{I_2 \: – \: I_1} \\ = \frac{Voltage}{Current}$$

R = $$\frac{Voltage}{Current}$$

Therefore, Slope = Resistance.

error:


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# Prove that $\sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\frac{mi}{n}+\frac{1}{2}\right\rfloor$ is an even number

Let $m$, $n$ be positive odd numbers such that $\gcd(m,n)=1$. Show that

$$\sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\dfrac{mi}{n}+\dfrac{1}{2}\right\rfloor$$

is an even number, where $\lfloor{x}\rfloor$ is the largest integer not greater than $x$.

My try: $$\sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\dfrac{mi}{n}+\dfrac{1}{2}\right\rfloor=\left\lfloor\dfrac{m}{n}+\dfrac{1}{2}\right\rfloor+\left\lfloor\dfrac{2m}{n}+\dfrac{1}{2}\right\rfloor+\cdots+\left\lfloor\dfrac{m(n-1)}{2n}+\dfrac{1}{2}\right\rfloor$$

Then I can't it. Thank you for your help.

-
I've been able to show that the sum is the number of non-negative, integer solutions to $$mx+ny\lt\frac{(m-1)(n-1)}{2}$$ –  robjohn Nov 7 '13 at 13:27
Hello,what is $x,y?$ Thank you –  china math Nov 7 '13 at 13:37
non-negative integers –  robjohn Nov 7 '13 at 14:10
can you post you solution? Thank you –  china math Nov 7 '13 at 14:28
I haven't shown that this is even yet, just that it is the number of non-negative, integer solutions to that inequality. –  robjohn Nov 7 '13 at 15:53

Note $[x+\frac{1}{2}]=[2x]-[x]$

$$\sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\dfrac{mi}{n}+\dfrac{1}{2}\right\rfloor=\sum_{i=1}^{\frac{n-1}{2}}(\left\lfloor2x\right\rfloor-\lfloor x\rfloor)\\=\sum_{i\ge \frac{n-1}{2},i\equiv0\pmod{2}}\left\lfloor x \right\rfloor-\sum_{i\le \frac{n-1}{2},i\equiv1\pmod{2}}\left\lfloor x \right\rfloor\\=\sum_{i\ge \frac{n-1}{2},i\equiv0\pmod{2}}\left\lfloor x \right\rfloor+\sum_{i\le \frac{n-1}{2},i\equiv1\pmod{2}}\left\lfloor -x+1 \right\rfloor \\ \equiv 2\sum_{i\ge \frac{n-1}{2},i\equiv0\pmod{2}}\left\lfloor x \right\rfloor \pmod{2}\\ \equiv0\pmod{2}$$

I hope you don't mind me trying to clarify what you have above: \begin{align} \sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\frac{mi}{n}+\frac12\right\rfloor &=\sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\frac{2mi}{n}\right\rfloor -\sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\frac{mi}{n}\right\rfloor\\ &=\sum_{\substack{i=2\\i\text{ even}}}^{n-1}\left\lfloor\frac{mi}{n}\right\rfloor -\sum_{i=1}^{\frac{n-1}{2}}\left\lfloor\frac{mi}{n}\right\rfloor\\ &=\sum_{\substack{i\gt n/2\\i\text{ even}}}\left\lfloor\frac{mi}{n}\right\rfloor -\sum_{\substack{i\lt n/2\\i\text{ odd}}}\left\lfloor\frac{mi}{n}\right\rfloor\\ &=\sum_{\substack{i\gt n/2\\i\text{ even}}}\left\lfloor\frac{mi}{n}\right\rfloor -\sum_{\substack{i\gt n/2\\i\text{ even}}}\left\lfloor\frac{m(n-i)}{n}\right\rfloor\\ &=\sum_{\substack{i\gt n/2\\i\text{ even}}}\left\lfloor\frac{mi}{n}\right\rfloor -\sum_{\substack{i\gt n/2\\i\text{ even}}}\left\lfloor m-\frac{mi}{n}\right\rfloor\\ &=\sum_{\substack{i\gt n/2\\i\text{ even}}}\left\lfloor\frac{mi}{n}\right\rfloor -\sum_{\substack{i\gt n/2\\i\text{ even}}}\left(m-1-\left\lfloor\frac{mi}{n}\right\rfloor\right)\tag{\ast}\\ &=2\sum_{\substack{i\gt n/2\\i\text{ even}}}\left(\left\lfloor\frac{mi}{n}\right\rfloor-\frac{m-1}{2}\right) \end{align} $(\ast)$ is true as long as $\frac{mi}{n}\not\in\mathbb{Z}$.

-
In your fourth line, You seem to be using that $-\lfloor x\rfloor=\lfloor1-x\rfloor$ (but only when $x\not\in\mathbb{Z}$). However, that is only true when you have the same $x$ on both sides. Here, you don't. At least it is not clearly so since $x=mi/n$ and the $i$'s don't seem to match. Perhaps if you made the association more explicit (i.e. don't use $x$), things might be better. –  robjohn Nov 8 '13 at 18:03
With the latest edit, I think your proof is good :-) (+1) –  robjohn Nov 8 '13 at 18:53

This is not a full answer to the question, but a proof of a claim I made in a comment.

In Fig 1, hypotenuse of the half-sized triangle is the line $\frac mni+\frac12$.

$\hspace{2cm}$

The number of dots inside that triangle is the sum in question: $$\sum_{i=1}^{(n-1)/2}\left\lfloor\frac mni+\frac12\right\rfloor\tag{1}$$ If we scale that triangle to the full-sized triangle, we see that the dots inside the half-sized triangle correspond to the red dots in the full-sized triangle. Flipping the full-sized triangle from Fig 1 to Fig 2, we see that the red dots are the points with odd coordinates. Thus, the red dots represent the solutions in non-negative integers of $$m(2x+1)+n(2y+1)\lt mn\tag{2}$$ Since the left side of $(2)$ is even and the right side is odd, it is equivalent to $$m(2x+1)+n(2y+1)\lt mn+1\tag{3}$$ which is equivalent to $$mx+ny\lt\frac{(m-1)(n-1)}{2}\tag{4}$$ Therefore, the sum in $(1)$ counts the number of non-negative solutions of $(4)$.

-


text:
11,329
pages

The tridecatrux is equal to $$\{10,6,1,2\}\ \&\ 10$$ using the array of operator and BEAF.[1] The term was coined by Cookie Fonster.

## Approximations

Notation Lower Bound Upper Bound
FGH $$f_{\varphi(1,0,0,0)[6]}(10)$$ $$f_{\varphi(1,0,0,0)}(10)$$

## Sources

1. Pointless Gigantic List of Numbers - Part 4: The Higher RoyalsPointless Large Number Stuff. Archived from the original on 2014-08-25. Retrieved 2017-05-27.


text:
# Source code for gemseo.core.grammars.factory

# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.

# Contributors:
#    INITIAL AUTHORS - initial API and implementation and/or initial
#                           documentation
#        :author: Francois Gallard, Matthias De Lozzo
#    OTHER AUTHORS   - MACROSCOPIC CHANGES
"""A factory to instantiate a derived class of :class:.AbstractGrammar."""
from __future__ import division, unicode_literals

import logging
from typing import List

from gemseo.core.factory import Factory
from gemseo.core.grammars.abstract_grammar import AbstractGrammar

LOGGER = logging.getLogger(__name__)

[docs]class GrammarFactory(object):
"""A factory of :class:.AbstractGrammar."""

def __init__(self):  # type: (...) -> None
self.__factory = Factory(AbstractGrammar, ("gemseo.core.grammars",))

[docs]    def create(
self,
class_name,  # type: str
name,  # type: str
**options
):  # type: (...) -> AbstractGrammar
"""Create a grammar.

Args:
class_name: The name of a class deriving from :class:.AbstractGrammar.
name: The name to be given to the grammar.
**options: The options to be passed to the initialization.
"""
return self.__factory.create(class_name, name=name, **options)

@property
def grammars(self):  # type: (...) -> List[str]
"""The sorted names of the available grammars."""
return self.__factory.classes

[docs]    def is_available(self, class_name):  # type: (...) -> bool
"""Return whether a grammar class exists.

Args:
class_name: The name of a class deriving from :class:.AbstractGrammar.

Returns:
Whether the grammar class exists.
"""
return self.__factory.is_available(class_name)


text:
# When connection is lost and WinSCP resumes, some files are not uploaded but skipped

ronwilliams
Joined:
Posts:
1
Location:
South Africa

## When connection is lost and WinSCP resumes, some files are not uploaded but skipped

When connection is lost and WinSCP resumes, some files are not uploaded but skipped
option batch continue
option reconnecttime 30
option confirm off
open ftp://*****************/
put C:\WINDEX\ /public_html/Elements/2021
exit

martin
To generate the session log file, use /log=C:\path\to\winscp.log command-line argument. Submit the log with your post as an attachment. Note that passwords and passphrases not stored in the log. You may want to remove other data you consider sensitive though, such as host names, IP addresses, account names or file names (unless they are relevant to the problem). If you do not want to post the log publicly, you can mark the attachment as private.


text:
# Why energy level of one dimension problem is descrete?

1. Jan 13, 2012

in the book of quantum mechanics concept and application by Zettili, chapter 4 write a theorem, that is:

in one dimensional problem the energy level of a bound state system are discrete and not degenerate.

i can not prove this theorem.

can you help me to do this!

2. Jan 13, 2012

### DrDu

First there are exceptions, e.g. when you have a symmetric potential with an infinitely high barrier.
In that case you can find two degenerate bound states.
Generally, you can refrain to the theory of differential equations.
For any energy the Schroedinger equation, which is a differential equation of second order, has two solutions (which may not be normalizable), so there can be at most two degenerate bound states.
In the case of regular (not infinite) potentials, fiddling around with the Wronskian, you can show that at most one of the two solutions can be a bound state, the other solution will diverge and not be in the Hilbert space.

3. Jan 14, 2012


text:
# statistics question

• October 2nd 2011, 12:48 PM
jujumaan
statistics question
During the 2009-2010 basketball season, the number of points scored in each game by the Boston Celtics was approximately normally distributed with a mean of 99.2 points and a standard deviation of 10.5 points.

1)What is the 33rd percentile of points scored by the Celtics?

2)The mean number of points scored by the Los Angeles Lakers was 101.7. In what proportion of their game did the Celtics score more than the Lakers' mean score?
• October 2nd 2011, 01:07 PM
pickslides
Re: statistics question
Quote:

Originally Posted by jujumaan
1)What is the 33rd percentile of points scored by the Celtics?

Find 'a' where $\displaystyle P\left(Z> \frac{a-99.2}{10.5} \right)= 0.33$

Quote:

Originally Posted by jujumaan

1)What is the 33rd percentile of points scored by the Celtics?

2)The mean number of points scored by the Los Angeles Lakers was 101.7. In what proportion of their game did the Celtics score more than the Lakers' mean score?

$\displaystyle P\left(Z>\frac{101.7-99.2}{10.5} \right)= \cdots$


text:
Home

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Class 12

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Chemistry

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Chapter

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Chemical Equilibrium

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For the reaction, A(g)+2B(g)hA...

Updated On: 27-06-2022

Get Answer to any question, just click a photo and upload the photo and get the answer completely free,

Text Solution

0.295 M^(-1)0.673 M^(-1)1.178 M^(-1)2.36 M^(-1)

Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

Transcript

proportional to the reaction A + 2B 2c he was found to be 0.35 give to see that the time t is equal to zero the number of moles per a number of equilibrium reaction will quickly spread to the post of moles of a are reacted then a2x moles of devariya because I am too and it was the concentration of November 1965

we will be 1 - 1.5 - 285 to the question is placed at the concentration of equilibrium was 0.35 to 1 by 2 is equal to 0.3 is equal to equal to zero point 3528 cancer what do you have 0.35 / to come out to be 0.3 to 5 then which is molar is the concentrate is the unit for concentration minus 2 into X 0.35 to be equal to 0.4 M then after that we

the concentration of three visits quality which is equal to 0.3 so you have to find out the equilibrium constant now please write a metric square concentration of and concentration of these are you will get it is equal to 0.35 into 0.3 / 0.2.3 write the unit for humanity will be m square unit in the denominator and this is the correct answer for the question write if you look at the auction

with optional before right it is closer to apna before so therefore that will be the correct answer


text:
Consider $f$ from $[0,1]$to $[0,1]$ be continuous and non constant .

Then, is there $c\in[0,1]$ such that $f(c) =\int^1 _0 f^2(t) dt$ ?

-
What for $f$ constant, with a constant different from $0$ and $1$? – Davide Giraudo May 8 '12 at 15:21
I have edited the question. i hope it makes sense :D the question came just as a thought . – Theorem May 8 '12 at 15:24
Just take Davide's suggestion and introduce a very small bump. – David Mitra May 8 '12 at 15:26
$$f(x)=a+bx$$ where $a \in [0,1]$ and $b$ is very very small. – N. S. May 8 '12 at 15:28
Perhaps you meant to say $f$ is onto (i.e. surjective)? – AD. May 8 '12 at 15:30

Counterexample
Consider $f(x) = \frac{1}{2} (1-x) + \frac{1}{4} x = \frac{1}{4} \left(2-x\right)$. Then $$\int_0^1 f^2(x) \mathrm{d} x = \frac{1}{16} \int_0^1 \left(2-x\right)^2 \mathrm{d} x = \frac{1}{16} \int_1^2 y^2 \mathrm{d} y = \frac{1}{16} \cdot \frac{2^3-1^3}{3} = \frac{7}{48}$$ But the equation $$\frac{1}{4} \left(2-x\right) = \frac{7}{48} \quad x = 2 - \frac{7}{12} = \frac{24-7}{12} = \frac{17}{12}$$ but the solution $x= \frac{17}{12}$ lies outside $[0,1]$ interval.

-
@ Sasha, i think there is something wrong with ur calculations , can u check once. – Theorem May 8 '12 at 15:57
@Vedananda I filled in steps. – Sasha May 8 '12 at 16:52
i was missing a number in my calculation , stupid of me. thanks:) – Theorem May 8 '12 at 16:59


text:
Revised 1feb15

Absolute Geometry versus Euclidean Geometry

\textit{$\C$  2010, Prof. George K. Francis, Mathematics Department,
University of Illinois}

\begin{document}
\maketitle

\section{Introduction}

The second module of this course addresses the geometry which Euclid
of Alexandria (ca -300) created. In particular, we are concerned with
\textit{Book I} of his \texit{Elements}, which begins with his Postulates,
and ends with his proof of the Pythagorean Theorem (Propositions 47) and
its converse (Proposition 48). For the first 28 propositions, Euclid
carefully avoids using his fifth, the \textit{Parallel Postulate}.
This body of plane geometry is
known as \textit{Absolute Geometry}. Nowadays, it is also called
\textit{Neutral Geometry} because it does not takes sides on whether to
accept or reject the parallel postulate.
David Joyce, of Clark University, summarizes  Euclid's Elements  for your convenience. As you study this lesson, you might want to check these yourself.

It was Euclid's intention of proposing his Postulates, together with
informal definitions and common notions, as given, and to derive the
propositions logically from them. Euclid, however, made a number of
additional assumption without calling any particular attention to them.
Geometers corrected these deficiencies over the subsequent 2300 years.
The final formulation Euclid's axioms was made by David Hilbert (ca 1900).
Robert Campbells praphrase of

In this course we will neither follow Euclid and present his proofs in
the classical progression, nor will we study Hilbert's axiomatic system
in any great detail. The former is purely of historical interest, and
the latter is far too technical and difficult for a first course in
college geometry.
We will discuss the high points of absolute geometry, giving proofs
only as needed to appreciate the Euclid's grand invention. When we
come to the great divide between absolute and Euclidean geometry,
we will study the structure of the theorems and proofs in considerable
detail.

A brief look into the Elements between Propositions 30 and 46 will be
followed by several proofs of Pythagoras' theorem. There are many more
on the WWW. (There is even one by President Garfield.)

The module will conclude with Birkhoff's very economical set of
axioms (only 4 as against Hilbert's 21) . The difference of these two axiomatic
systems is considerable. Birkhoff's axioms are built on top of the
entire edifice of real numbers and their arithmetic. Hilbert's is not.
Indeed, numbers and their arithmetic appear as objects inside Hilbert's
geometry. In other words,  the axiomatic arithmetic of real numbers has a
model inside geometry.
The great achievement of Hilbert and his colleagues was to show that
geometry and arithmetic are equi-consistent. It is not necessary to
choose between them, we accept both arithmetic and geometry as equally
certain.

On the other hand, since Rene Descartes (ca 1600), we have based geometry
on the properties of the real numbers, in the form of analytic geometry,
a.k.a. \textit{Cartesian Geometry}. We will consider Birkhoff's
axioms, because there are fewer of them, and they already assume the
properties of the reals, and show that they have a model inside Cartesian
geometry.

\section{Definition of Absolute Geometry}
We will take the simplest
approach and consider all the definitions, theorems and proofs in Euclid's
Elements up to and including Proposition 28 as constituting \textit{absolute
geometry}. Another, more practical definition would be to say that any
theorem that is true in both Euclidean and Hyperbolic geometry is part
of absolute geometry. But how could we apply this second approach to
deciding whether a geometrical theorem is absolute or not?

For example, that an exterior angle of a triangle is greater than either
opposite interior angle (AXAT) is true Euclidean geometry, and in a
model of Non-Euclidean geometry, and use a geometrical construction package to tdo the relevant experiments. Or we could look into Euclid's Elements and
notice that AXAT is Proposition 16 in Euclid, and $16 \lte 28$.

We will use GGB for this purpose, but Hvidsten's GEX is actually more
suited for this purpope because he provides 3 models of hyperbolic
geometry, a model of spherical geometry, and 3 models of Euclidean geometry
for good measure.

Noet that in Euclidean geometry, there is a stronger version of this
theorem, which says that the exterior angle is actually the sum of the
opposite interior angles (EXAT). But this is false in nonEuclidean geometry,
and therefore EXAT is not absolute, it is strictly Euclidean. Can you
find the number for that proposition in Euclid's Elements? Is is larger than
28?

To repeat a central fact in this course, the \textit{Euclidean Exterior
Angle Theorem} (EXAT) says that the
exterior angle is the sum of the opposite interior angles. This is false
in hyperbolic geometry. Therefore, EXAT cannot be proved inside absolute
geometry. The last lesson of this course will reveal that the amount by
which EXAT fails in hyperbolic geometry, gives the area of the triangle.

So, how can you tell that a theorem is part of absolute geometry, or
requires the Parallel Postulate to be true? One way is to study Euclid's
Elements. Since $48>28$ we know that the Pythagorean Theorem (Prop. 48)
is not a part of absolute geometry.  Another is to use a construcion
pagacke  and check whether or not the theorem holds in both Euclidean
and in non-Euclidean models of geometry, then it must be and absolute theorem.

Question 1: Do you think the theorem that rectangles have four right angles is absolutely true? Experiment with a Euclidean and a hyperbolic model in GEX. Write your results into your Journal. Report a summary here.

\section{Book I of the Elements}

Euclid invites his students to accept a long list of \textit{common notions},
which are not particularly convincing. In fact, Hilbert insisted that the primitives
of an axiomatic system are technically undefined. Of course, once you interpret
them in a model, they have a meaning. But it depends on the model you use.
Until C. F. Gauss (ca 1800) everybody, and I mean everybody was convinced that
there was only one possible geometry. And even if we humans were imperfect in
formulating this ideal geometry, it was as much there as anything we know.

What is much more interesting for us, than arguing about the nature of a point
or a line, is the tremendous care Euclid took in arranging the theorem in a
logical order. This means, none of the conclusions of a theorem was used as
the hypothesis of a prior theorem. And, almost (but not quite perfectly) he
left nothing out between a prior and a posterior proposition.

\subsection{Equilateral Triangle Theorem}

Euclid's first proposition showed how to construct an equilateral triangle
on a given base AB, directly from his postulates.
By Postulate 3 we have two circles, one centered
at A and the other at B, and with the same radius.
Euclid assumed that is was obvious that
the two circles intersected at two points, C and D. In Hilbert's geometry this
was not taken for granted. It had to be proved from prior theorems. But once
we do have a point there, both dashed segments are also radii of their respective circles. Hence the triangle is equilateral.

You should remember from high school geometry, that if we now connect C and D
with a line, this becomes the perpendicular bisector of the segment AB.

Question 2: Do you suppose this is a theorem in absolute geometry, or not?
You'll need to do some constructions in a hyperbolic model of non Euclidean
geometry to decide this experimentally.

\subsection{Side-Angle-Side Congruence}

Euclid's Proposition 4 is an attempt to prove SAS as a theorem.
Unfortunately, this one Euclid really failed to prove in any sense that
would be considered rigorous.
So we'll just accept it as another axiom, as Hilbert and most geometers do
ever since.

Question 3: Where have you seen a use of SAS in this course so far?

\subsection{Pons Asinorum}

You will recall that we really only need SAS to prove that  the base angle of
an isosceles triangle are equal, Proposition 5.
Euclid, as usual, feels obliged to prove the
converse, Proposition 6.
Here, you need to be reminded just what this means. Note that every
proposition can be formulated as an if-then statement.
The proposition $H \implies C$  says that if the
\textit{hypothesis} holds then the \textit{conclusion} follows. Recall from
MA347 or its equivalent that

\begin{eqnarray*}
\mbox{the}  &    &    \\
\mbox{proposition}  & \mbox{in symbols}  &  \mbox{ reads }   \\
& H \implies C  &  \mbox{ if } H \mbox{ then } C   \\
\mbox{its converse}  & C \implies H  &  \mbox{ if } C \mbox{ then } H   \\
\mbox{contrapositive}  & \neg C \implies \neg H  &  \mbox{ if not} C \mbox{ then not } H   \\
\mbox{inverse}  & \neg H \implies \neg C  &  \mbox{ if not} H \mbox{ then not } C\\
H \implies C  & \mbox{ is equivalent to} &  \neg H \or C  \\
\neg( H \implies C)  & \mbox{ is equivalent to}&  H \and \neg C  \\
\end{eqnarray*}

You also need to recall just why a proposition is equivalent to its
contrapositive, and similarly its converse is equivalent to its inverse.
But a proposition is not necessarily equivalent to its converse.

Question 4: State the converse of Pons Asinorum.

Question 5: State the negation of Pons Asinorum.

Can figure out a proof of the converse? A counterexample to its negation?

\subsection{What's next?}

In the next lesson we shall thoroughly study Proposition 28.

\end{document}


text:
## Abstract

A set of sets is called a family. Two families $$\mathcal {A}$$ and $$\mathcal {B}$$ are said to be cross-$$t$$-intersecting if each set in $$\mathcal {A}$$ intersects each set in $$\mathcal {B}$$ in at least $$t$$ elements. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$$t$$-intersecting subfamilies of a given family. This incorporates the classical Erdös–Ko–Rado (EKR) problem. We prove a cross-$$t$$-intersection theorem for weighted subsets of a set by means of a new subfamily alteration method, and use the result to provide solutions for three natural families. For $$r \in [n] = \{1, 2,\ldots , n\}$$, let $${[n] \choose r}$$ be the family of $$r$$-element subsets of $$[n]$$, and let $${[n] \choose \leq r}$$ be the family of subsets of $$[n]$$ that have at most $$r$$ elements. Let $$\mathcal {F}_{n,r,t}$$ be the family of sets in $${[n] \choose \leq r}$$ that contain $$[t]$$. We show that if $$g : {[m] \choose \leq r} \rightarrow \mathbb {R}^+$$ and $$h : {[n] \choose \leq s} \rightarrow \mathbb {R}^+$$ are functions that obey certain conditions, $$\mathcal {A} \subseteq {[m] \choose \leq r}$$, $$\mathcal {B} \subseteq {[n] \choose \leq s}$$, and $$\mathcal {A}$$ and $$\mathcal {B}$$ are cross-$$t$$-intersecting, then

$\sum_{A \in \mathcal{A}} g(A) \sum_{B \in \mathcal{B}} h(B) \leq \sum_{C \in \mathcal{F}_{m,r,t}} g(C) \sum_{D \in \mathcal{F}_{n,s,t}} h(D),$
and equality holds if $$\mathcal {A} = \mathcal {F}_{m,r,t}$$ and $$\mathcal {B} = \mathcal {F}_{n,s,t}$$. We prove this in a more general setting and characterize the cases of equality. We use the result to show that the maximum product of sizes of two cross-$$t$$-intersecting families $$\mathcal {A} \subseteq {[m] \choose r}$$ and $$\mathcal {B} \subseteq {[n] \choose s}$$ is $${m-t \choose r-t}{n-t \choose s-t}$$ for $$\min \{m,n\} \geq n_0(r,s,t)$$, where $$n_0(r,s,t)$$ is close to best possible. We obtain analogous results for families of integer sequences and for families of multisets. The results yield generalizations for $$k \geq 2$$ cross-$$t$$-intersecting families, and EKR-type results.


text:
# Uniquation

Uniquation is a math expression search engine. I decided to create it when I noticed that there wasn’t any tool to find an information on the Web about a particular math concept.

Suppose you are a researcher who has just built a model f(x+1) = r f(x) (1-f(x)) for his domain. Now you have two options: either to explore the model or to find if this model was explored before. Of course you want to stand on the shoulders of giants, so you choose the second option. You go to Google and… become upset, because the SERP is garbage. One may argue that you should have searched for “logistic map” instead. Well, it is true, but how one would find the name to google if they only have the expression?

Here is where Uniquation shines. You can take a look on its SERP to the same request. Almost all links lead to the pages where the expression is associated with the name – logistic map. So after you find the name you can continue your research with Google.

Exploring a model isn’t the only use case for Uniquation. It is highly possible that a web page with an equation contains a solution too, so if you search for the equation you can find a solution. Moreover if a page contains an expression then Uniquation can find the page by a part of the expression. So if you’re looking for \cos (a+b) you will find \cos (a+b) = \cos a \cos b - \sin a \sin b.

## Aha! Moment

The most complicated part about the project was to invent an algorithm to normalize expressions. It matters because the same concept in math may have several forms: x^2+a=0 or 0=a+x^2 or even y^2+x=0. So I had to take into account commutativity and alpha equivalence (same formula but different variables). It is a hard problem and when I started working on the project I didn’t know how to handle it, so when I came to a solution I had very strong aha! moment.

## WolframAlpha

WolframAlpha was released when I was implementing Uniquation. First I was frightened, because a company behind it, Wolfram Research, has a lot of experience with symbolic manipulations and could have created a great math search engine. But they focused on computation rather then on search. For example if we enter \cos (a+b) they apply a lot of formulae to it including the one above, but if we enter the logistic map expression they just try to expand it and don’t provide any hint what it is, despite they know what logistic map is.

Beside that, WolframAlpha computes the result based on an internal database. But inside is by all means smaller that outside and always will be. So a solution based on outside data always will be more comprehensive.

## Fail

This project was a part of my thesis. After I graduated I had to abandon it because I didn’t know how to make money with it and I didn’t have spare time to evolve it. Nevertheless I don’t think that to opensource it now is a good idea because I’m sure that in the current state of the project I will fail to build a community.

To make a prototype fast I used very marginal technologies. In the university my language of choice was C#, so I targeted the .Net platform, but since a linux hosting is cheaper I switched to Mono, later I switched from C# to Nemerle. Nemerle to C# is what Scala is to Java but with more powerful macro system and far less popular. I did it because Nemerle has IMHO the best tool to make parsers – Nemerle.PEG.

Now I believe that it is almost impossible to find people who are interested in development of Uniquation and who are also ready to learn Nemerle. I don’t leave a hope to port Uniquation to a more mainstream technology and to opensource it.

## Future

If I manage to revive Uniquation I would start a new project on top of it – GaloisWiki. It is a math wiki-style encyclopedia made to improve Uniquation’s CAS abilities. Now Uniquation can find a math expression only if it was indexed before or it is a part of an indexed expression. It sucks because if a user searches for a specialized expression and a generic expression was added to the index they get nothing. For instance, if there is an article about Quadratic equation with a solution to ax^2+bx+c=0 in generic form then Uniquation wouldn’t find it by a specialized request like 3x^2-17x+10=0. I have an idea how it can be changed.

During the indexation all constants are replaced with free variables and additional conditions are added.

Then the structure is pushed to the index. Let us switch to a user request, e.g., 3x^2-17x+10=0. First, we convert it to the following structure.

Then

• the search proceeds and the engine returns a list of structures that match the request
• an algorithm removes structures which conditions conflict with the request
• an algorithm puts the request’s constants into the generic form

So the user gets a specialized solution that matches their request.

I think you’ve just got the idea of GaloisWiki: it is a computer algebra system based on a wiki database which looks like a math encyclopedia.


text:
# Using modified .z functions to trace, monitor and control execution¶

Every client interaction with a q server is handled by one of the p? functions you’ll find in the system namespace .z. These functions have reasonable, simple defaults that work fine right out of the box. What we’re doing here is taking advantage of the fact that they’re just functions, allowing you to overwrite them with your own custom code to show or modify what’s happening.

The utility scripts in github.com/simongarland/dotz are examples of how to do this, and these files are described in detail below.

In all of the examples the code to wrap up existing definitions looks complicated. The reason is that these are general scripts and so a combination of them could be loaded into applications with pre-existing custom .z.p? definitions. The wrapping code protects these definitions, but in a particular application you can probably simply replace or extend existing definitions rather than wrapping them.

## Take it for a spin¶

The simplest way to get a feeling for what’s going on is to try it out. Start up two q sessions, load traceusage.q into one of them – and then talk to it from the other server. Watch the output in the traceusage session.

Vanilla server session:

q)h:hopen 5001
q)h
3
q)h"2+2 3"
4 5
q)h"2 3 4+2 3"
'length
q)hclose h

traceusage server session:

..simon/dotz$q traceusage.q -p 5001 …   2008.05.09 12:38:10.367 ms:0.002003799 m+:0K pw a:localhost u: w:4 (simon;"***") 2008.05.09 12:38:10.370 ms:0.003025343 m+:0K po a:localhost u:simon w:4 4 2008.05.09 12:38:17.151 ms:0.02098095 m+:0K pg a:localhost u:simon w:4 2+2 3 *2008.05.09 12:38:25.438 (error:length) pg a:localhost u:simon w:4 2 3 4+2 3 2008.05.09 12:38:33.246 ms:0.002986053 m+:0K pc a:192.168.1.34 u:simon w:0 4  (On non-Windows OSs the error line will be in glorious 1980s style colour.) ## The gory details¶ What the individual files do, and how to use them. ### The toolkit¶ The “tools” you have to work with are the p? functions from .z: .z.po, .z.pc, .z.pw, .z.pg, .z.ps, .z.ph, .z.pp and .z.exit. Combined with the .z variables .z.a, .z.u and .z.w which are always set to the values of the client during execution of the .z.p? function. Depending on how the function is called, additional information may be provided as arguments to the .z.p? functions (user ID and password for .z.pw, browser environment for .z.ph and .z.pp). By default, execution is done using value so strings or symbol argument lists can be tested in a console. Q for Mortals 11.6 Interprocess Communication ### saveorig.q¶ This script just saves original values of things like .z.pg so you can revert to original definitions without having to restart the task. This is made a little more complicated by the way some of the default definitions aren’t materialised in the user workspace. For example by default .z.pg is just {value x} but that’s run in the q executable. The default values are created explicitly if need be. .dotz.exit.ORIG:.z.exit:@[.:;.z.exit;{;}]; .dotz.pg.ORIG:.z.pg:@[.:;.z.pg;{.:}]; .dotz.ps.ORIG:.z.ps:@[.:;.z.ps;{.:}]; …  Other functions and variables shared between multiple scripts (such as debug output level .debug.LEVEL, or the .z.a IP address <-> hostname cache .dotz.IPA) are defined here too. Although it would be simpler to embed this setup code in each script, allowing them to be standalone, one tires of the cut’n’paste forays required by every tiny change. Note After the various state variables have been defined the script saveorig.custom.q is loaded, if found, allowing you to customise the setup without needing to have a modified version of the saveorig script. Again, for production use you should rip out the unneeded definitions. ## Tracing execution¶ Tracing execution of the various callbacks is the simplest application. It can be as simple as just sprinkling 0N! statements around the functions, or as complicated as logging to an external file. As these are samples they also track the use of .z.pi – but that can get tiresome if you’re debugging from a console. In that case just zap the custom .z.pi definition with: q)\x .z.pi  The variable .usage.LEVEL can be set to control how much is output. By default (2) it displays all messages, a value of 1 will display only errors and a value of 0 will temporarily disable logging. In the examples below, the sample session is a simple hopen, get "2+2", get "2 3+3 4 5" then hclose. ### dumpusage.q¶ The simplest file of all, it just puts in 0N! to display the input and the results. It’s fine for debugging a simple conversation with a single client – but not informative enough for more complex setups. dotz$ q dumpusage.q -p 5001
…
q)"***"
simon
1b
4
4
"2+2"
4
"2 3+3 4 5"
4
4

### traceusage.q¶

Dumps formatted output to the console, and on non-Windows consoles it will colour errors and expensive calls. The definition of what’s expensive can be set by .usage.EXPENSIVE (in milliseconds).

dotz$q traceusage.q -p 5001 … q) 2008.05.13 11:42:56.290 ms:0.002003799 m+:0K pw a:localhost u: w:4 (simon;"***") 2008.05.13 11:42:56.319 ms:0.003968307 m+:0K po a:localhost u:simon w:4 4 2008.05.13 11:43:00.866 ms:0.0159911 m+:0K pg a:localhost u:simon w:4 2+2 *2008.05.13 11:43:08.818 (error:length) pg a:localhost u:simon w:4 2 3+3 4 5 2008.05.13 11:43:13.986 ms:0.004007597 m+:0K pc a:192.168.1.34 u:simon w:0 4  ### lastusage.q¶ When debugging it’s sometimes more helpful to be able to grab the last request that came in rather than just looking at a trace of what happened. This set of custom callbacks stores the last calls in namespace .last, allowing you to fetch the data and retry the request directly in your session. dotz$ q lastusage.q -p 5001
…
q).last
| ::
pw  | whenuwaxyzr!(::;2008.05.13T11:44:36.655;;4;2130706433;{[x;y]1b};simon;"";1b)
zcmd| pc
po  | whenuwaxyr!(::;2008.05.13T11:44:36.655;simon;4;2130706433;::;4;4)
pg  | whenuwaxyr!(::;2008.05.13T11:44:40.951;simon;4;2130706433;.:;"2+2";4)
pc  | whenuwaxyr!(::;2008.05.13T11:44:52.111;simon;0;-1062731486;::;4;4)
q).last.pg
| ::
when| 2008.05.13T11:44:40.951
u   | simon
w   | 4
a   | 2130706433
x   | .:
y   | "2+2"
r   | 4
q)value .last.pg.y
4

### monitorusage.q¶

If the monitoring is to be left running for a long time scrolling back through the console is not a sensible way to look for problems. This script logs all requests to a local table USAGE, allowing you to analyse the data. As the data is stored in an in-memory table it’s of course lost when you exit unless you choose to do something with .z.exit.

dotz$q monitorusage.q -p 5001 … q)USAGE date time ms mdelta zcmd ipa u w cmd ok error --------------------------------------------- q)USAGE date time ms mdelta zcmd ipa u w cmd ok error ------------------------------------------------------------------------------------------------- 2008.05.13 11:48:19.360 0.2459958 0 pi 192.168.1.34 simon 0 "USAGE" 1 2008.05.13 11:48:31.728 0.002003799 0 pw localhost 4 "(simon;\"***\")" 1 2008.05.13 11:48:31.729 0 0 po localhost simon 4 ,"4" 1 2008.05.13 11:48:36.360 0.0159911 0 pg localhost simon 4 "2+2" 1 2008.05.13 11:48:41.920 0 0 pg localhost simon 4 "2 3+3 4 5" 0 length 2008.05.13 11:48:46.512 0.003025343 0 pc 192.168.1.34 simon 0 ,"4" 1 q)  ### logusage.q and loadusage.q¶ Finally, the all-singing all-dancing version. This script logs all requests directly to an external logfile – using the same log mechanism as kdb+tick. This allows logging to be left running for days without having to worry about tables growing – and will ensure that the logging data is safe even if the session terminates unexpectedly. Use loadusage.q to load the logged data into a session as a table (same schema as that from monitorusage.q except hostname added). dotz$ q logusage.q -p 5001
…
q)'type
q)\\ / nothing to see here..
dotz$q loadusage.q … q)USAGE date time ms mdelta zcmd ipa host u w cmd ok error -------------------------------------------------------------------------------------------------------- 2008.05.13 13:01:42.694 0.002003799 0 pw 127.0.0.1 localhost 5 (simon;"***") 1 2008.05.13 13:01:42.695 0.0009822543 0 po 127.0.0.1 localhost simon 5 5 1 2008.05.13 13:01:46.198 0.0159911 0 pg 127.0.0.1 localhost simon 5 2+2 1 2008.05.13 13:01:57.350 0 0 pg 127.0.0.1 localhost simon 5 2 3+3 4 5 0 length 2008.05.13 13:02:00.901 0.004007597 0 pc 192.168.1.34 simon 0 5 1 q)q)select from USAGE where not ok date time ms mdelta zcmd ipa host u w cmd ok error -------------------------------------------------------------------------------------- 2008.05.13 11:50:55.988 0 0 pg 127.0.0.1 localhost simon 5 2 3+3 4 5 0 length q)  ## Slamming the doors¶ Another important use for modified .z.p? callbacks is to control access to a q session. Q contains some very coarse access controls settable by command line options – particularly -u or -U to enforce password-controlled access (with MD5’d passwords), -b to enforce read-only access and -T to set a maximum CPU time per single client call. The password control in -u and -U is all done in the q executable, so completely outside user control. However as soon as the initial check (if any) has been done control is passed to z.pw, which can say if a connection is to be allowed. This can be via some session internal table or function, or can go outside to something like a central single-signon server. ### blockusage.q¶ Use this script simply to block all client interaction: it just sets .z.pw to always return false, i.e. no connection is allowed for supplied user ID and password. ### controlaccess.q and loadinvalidaccess.q¶ There are so many ways to control access that this script is way too complicated for immediate use as it stands – just pick an interesting subset. It shows how to control access via • a user table – splitting users into superusers (who can do anything), powerusers (who can run ad-hoc queries, but can’t do things like shutdown the session) and defaultusers (who can only use a speciic list of pre-defined commands). • the client’s server – with name matching a set of wildcards • a list of valid commands • a command validator which parses input Invalid access attempts are logged, and the logfile can be loaded into a table and queried with loadinvalidaccess.q ## Tracking clients and servers¶ Q provides a a list of handles in use with the keys of .z.W. These clients provide more background information about what's “behind” the handles by extending .z.po and .z.pc. ### trackclients.q¶ Tracking of clients can be automated using this script, .z.po and .z.pc maintain the list automatically. By default, the table of clients just uses information provided by .z.po like .z.u and .z.w, but if .clients.INTRUSIVE is set, the server will ask the clients for more details like their q versions, number of slaves etc. Output from a session where a client did 3 hopen 5001s, and one hclose. dotz$ q trackclients.q -p 5001
…
q)CLIENTS
w| ipa       u     a          poz                     pcz
-| --------------------------------------------------------------------------
4| localhost simon 2130706433 2008.05.13T13:37:14.176
5| localhost simon 2130706433 2008.05.13T13:37:15.007
| localhost simon 2130706433 2008.05.13T13:37:15.735 2008.05.13T13:37:26.359
q)

### trackservers.q¶

Giving client applications the ability to track servers simplifies application design – no need to hardcode server+port settings, or to handle servers becoming unavailable.

Unlike trackclients.q you do have to add server records manually, although .z.pc handles them going away. You can either add a new server using a function like .servers.addnh (nh is name, hpup) or add a server record for an existing open handle using .servers.addw.

Servers can be public or private – a private server is not handed on to other users who request a list of current servers (the simplest way of setting up a new session, no central “handle server” to be maintained).

By default servers that disappear are retried regularly.

dotz\$ q trackservers.q
…
q).servers.addw hopen:welly3:2018
3
4
q)SERVERS
name     hpup            w hits private lastz
---------------------------------------------------------------
servers  :192.168.1.34:0 0 0    1       2008.05.13T13:44:10.337
taq2007  :welly3:2018    3 0    0       2008.05.13T13:45:00.761
lava2006 :welly3:2017    4 0    0       2008.05.13T13:45:05.400
q)SERVERS
name     hpup            w hits private lastz
---------------------------------------------------------------
servers  :192.168.1.34:0 0 0    1       2008.05.13T13:44:10.337
taq2007  :welly3:2018    3 0    0       2008.05.13T13:45:00.761
lava2006 :welly3:2017    4 0    0       2008.05.13T13:45:05.400
q).servers.handleforlava2006
4
q).servers.handleforlava2007
'lava2007.not.available
q)SERVERS
name     hpup            w hits private lastz
---------------------------------------------------------------
servers  :192.168.1.34:0 0 0    1       2008.05.13T13:44:10.337
taq2007  :welly3:2018    3 0    0       2008.05.13T13:45:00.761
lava2006 :welly3:2017    4 1    0       2008.05.13T13:45:44.824
q)

q).servers.addnh[taq;::5001]
3
q).servers.addnh[taq;::5001]
5
q).servers.addnh[taq;::5001]
6
q).servers.addnh[taq;::5001]
7
q)SERVERS
name     hpup            w hits private lastz
---------------------------------------------------------------
servers  :192.168.1.34:0 0 0    1       2008.05.13T13:44:10.337
taq2007  :welly3:2018    3 0    0       2008.05.13T13:45:00.761
lava2006 :welly3:2017    4 1    0       2008.05.13T13:45:44.824
taq      ::5001          3 0    0       2008.05.13T13:50:03.777
taq      ::5001          5 0    0       2008.05.13T13:50:05.513
taq      ::5001          6 0    0       2008.05.13T13:50:06.848
taq      ::5001          7 0    0       2008.05.13T13:50:07.640
q).servers.handlefortaq
3
q).servers.handlefortaq
5
q).servers.handlefortaq
6
q).servers.handlefortaq
7
q).servers.handlefortaq
3

## Running on other servers¶

The default Q IPC allows you to easily submit synchronous or asynchronous requests. Combined with a list of all available servers from trackservers.q above you can deal with most simple requests.

This script provides an extra way of dealing with a lot of data requests. It allows you to submit synchronous or asynchronous requests, locally or remotely – and collects all the results in a local table TASKS. So, for example, if you had to run a few hundred queries to be able to build a report and you had 10 server sessions available to query you’d simply submit all 100 queries and either pick up results as they drift in, or wait until all are complete.

You can additionally allocate requests to a request group to make it easy to check when a complete group has completed.

Here's an example session using two servers on 5001 and 5002. First create the server table entries.

q).servers.addnh[hh;::5001]
5
q).servers.addnh[hh;::5002]
6
q).servers.addnh[hh;::5002]
7
q)SERVERS
name    hpup            w hits private lastz
--------------------------------------------------------------
servers :192.168.1.34:0 0 0    1       2008.05.13T18:37:41.087
hh      ::5001          5 0    0       2008.05.13T18:38:59.455
hh      ::5002          6 0    0       2008.05.13T18:39:03.886
hh      ::5002          7 0    0       2008.05.13T18:39:05.390

q).tasks.rxa[.servers.handleforhh;"max til 10"]
10001
q).tasks.rxa[.servers.handleforhh;"max til 10"]
10002
10003
nr   | grp   startz                  endz                    w ipa       status   expr         result
-----| ----------------------------------------------------------------------------------------------
10001| 20001 2008.05.13T18:41:43.057 2008.05.13T18:41:43.058 5 localhost complete "max til 10" 9
10002| 20002 2008.05.13T18:41:46.138 2008.05.13T18:41:46.138 6 localhost complete "max til 10" 9
10003| 20003 2008.05.13T18:41:47.009 2008.05.13T18:41:47.010 7 localhost complete "max til 10" 9
10002| 9
10001| 9
10002| 9
10003| 9

q).tasks.rxa[.servers.handleforhh;"17+this"]
10004
,10004
10004| fail
nr   | grp   startz                  endz                    w ipa       status   expr         result
-----| ----------------------------------------------------------------------------------------------
10001| 20001 2008.05.13T18:41:43.057 2008.05.13T18:41:43.058 5 localhost complete "max til 10" 9
10002| 20002 2008.05.13T18:41:46.138 2008.05.13T18:41:46.138 6 localhost complete "max til 10" 9
10003| 20003 2008.05.13T18:41:47.009 2008.05.13T18:41:47.010 7 localhost complete "max til 10" 9
10004| 20004 2008.05.13T18:44:05.229 2008.05.13T18:44:05.229 5 localhost fail     "17+this"   "type"
q)
`

## Utilities¶

### hutil.q¶

Production usage

All these utility files should be treated as examples. For any particular case they probably have too many options and should be cut down to do just what you want. The access control script is the most obvious case - it probably has far too many options/checks going on.


text:
If given 4.5 mL of 0.05 M magnesium sulfate, how many moles of magnesium sulfate do we have?

Jul 6, 2016

We have $2.3 \times {10}^{- 4}$ moles of $M g S {O}_{4}$

Explanation:

Molarity is represented by the following equation:

We know the molarity and the volume of solution. However, the volume does not have the proper units since it is given in terms of milliliters instead of liters. We can convert 4.5 mL into liters by using the conversion factor 1000mL = 1L.

When 4.5 mL is divided by 1000 mL/L, we obtain a volume of 0.0045 L.

Now we can rearrange the equation to solve for the number of moles. This can be accomplished by multiplying by liters of solution on both sides of the equation. The liters of solution will cancel out on the right side, leaving the number of moles being equal to the molarity times volume like so:

Moles of solute = $\text{liters of solution} \times M o l a r i t y$

Moles of solute = $0.0045 L \times 0.05 M$

$2.3 \times {10}^{- 4} m o l$


text:
# Full-bridge converter rectifier kicks

I'm in the process of building an 8kW isolated DC/DC converter, full-bridge topology.

I'm seeing some interesting phenomena on the diodes. When each diode becomes reverse-biased, a voltage spike appears across the diode, before settling down to the expected DC bus voltage. These are 1800V fast diodes (320nS spec'd recovery time), and the spikes are hitting 1800V with only 350VDC on the secondary, well below my output voltage target. Increased deadtime doesn't help; the kick still appears when the diode is reverse-biased, and is just as large.

My suspicion is that the output choke is keeping the diodes forward biased during the dead time. Then when the transformer voltage starts to rise in the other half-cycle, the diode gets instantaneously reverse-biased long enough to appear as a short across the transformer winding. Then when the diode recovers, that current is cut off, causing the kick I'm seeing.

I've tried a few things. At one point, I added a flyback diode in parallel to my bridge. I used the same fast-recovery diodes as are in my bridge. This had no apparent effect on the spikes. I then tried adding a .01 uF cap in parallel to my bridge.

This reduced the spikes to a more manageable level, but the reflected impedance of that cap caused significant problems on the primary. My snubber caps have doubled in temperature!

A few possibilities present themselves:

1) I've diagnosed the problem incorrectly. I'm 95% sure I'm seeing what I think I'm seeing, but I've been wrong before.

2) Use a synchronous rectifier. I shouldn't have reverse recovery issues with that. Unfortunately, I'm not aware of any reverse-blocking JFETs in this power range, and there's no such thing as a reverse-blocking MOSFET. The only reverse-blocking IGBTs I can find in this power range have worse losses than the diodes.

EDIT: I've just realized I've been misunderstanding the nature of a synchronous rectifier. I don't need reverse-blocking FETs; the FETs will conduct drain-source.

3) Use zero-recovery diodes. Again, problems with losses and cost.

4) Snub the kicks. This looks like it would eat way too much power, on the order of 20% of my overall throughput.

5) Add saturable cores in line with the diodes. Two of the largest saturable cores I could find barely dented my kicks.

6) Use a zero-current-switching resonant topology. I have no experience in that area, but it sounds like if the current on the primary changes more smoothly, the voltage on the secondary should also change more smoothly, giving the diodes more time to recover.

Has anyone else dealt with a similar situation? If so, how did you solve it? Edit: primary-side FET datasheet here.

-
have you tried RC snubber and/or large ferrite beads which also have the effect of ~ ~100pF loading at same time as real impedance absoprtion of RF transients. –  Tony Stewart Dec 21 '12 at 22:13
Do your primary MOSFETs have fast body diodes? Are you using a hard-switching topology, or one of the ZVS variants? –  Adam Lawrence Dec 21 '12 at 22:45
Which diodes are you using? Could you link to a datasheet? –  Brian Drummond Dec 21 '12 at 22:54
I've added FET and diode datasheets. Hard switching, no ZVS. I've tried a ferrite bead, which had no apparent effect. It was picked because it was at hand, not for its specs, so its possible it's just woefully undersized... nice to know the idea is reasonable... –  Stephen Collings Dec 22 '12 at 4:13
Where would you suggest the RC snubber? –  Stephen Collings Dec 22 '12 at 4:13

Flogging the FREDs

Voltage fed converters with transformer isolation will exhibit ringing in the secondary. Ringing is caused by parasitic inductances and capacitances in the circuit, with the dominant elements will being the transformer leakage inductance ($L_ {\text {Lk}}$) and junction capacitance ( $C_j$)of the bridge diodes. The diode data sheet shows $C_j$ of 32pF. I'm going to make a naive guess at $L_ {\text {Lk}}$ of 500nH, but it will have to be measured to really know. So, an LC of 500nH and 32pF is what must be snubbed.

Spike amplitude without snubbing will be $2 n V_ {\text {in}}$, where $n$ is transformer turns ratio and the factor of 2 is what you get for a high Q resonance.

There are different types of voltage snubbers; Clamping, Energy transfer resonant, and Dissipative. The clamping and resonant types require more parts and some involvement of active switches which I think make them impractical for this case. So, I am only going to cover dissipative snubbers because they are the most simple and work well with passive switches (like diodes or synchronous rectifiers).

The form of dissipative snubber that I will cover is a series RC placed in parallel with each bridge diode.

Some facts about RC dampening snubbers:

• They are all about impedance matching. You don't get to choose the snubber resistor value $R_d$. The parasitic LC determines that for you by characteristic impedance Zo.
• You do get to choose the value of the snubber cap $C_d$. That's important since the cap value sets the snubber loss ($P_ {\text {Rd}}$)as $C_d F V^2$ . Where V is the pedestal voltage and F is switching frequency. The snubber cap must provide a low impedance at the LC resonance of the parasitics, so it needs to be several times $C_j$.

Some guidelines, and what to expect with RC dampening snubbers:

• For $L_ {\text {Lk}}$ of 500nH and $C_j$ of 32pF, Zo will be 125Ohms. So, $R_d$ would be 125 to match Zo. You may have to fine tune this a little since $C_j$ is non-linear and falls off with reverse voltage.

• Choosing the snubber cap $C_d$ : Choose $3 C_j\leq C_d\leq 10 C_j$ . Higher values in the range do provide better dampening. For example, $C_d$ of $3 C_j$ will result in a peak diode voltage of $1.5 n V_ {\text {in}}$, while $C_d$ of $10 C_j$ will result in a peak diode voltage of $1.2 n V_ {\text {in}}$.

• Dissipative snubber performance will not improve for $C_d$ values greater than $10 C_j$.

Power loss $P_ {\text {Rd}}$, with a pedestal voltage of 1250V and F of 50KHz.

• If $C_d$ is $3 C_j$ or 100pF, $P_ {\text {Rd}}$ = $C_d F V^2$ or 7.8W.
• If $C_d$ is $10 C_j$ or 330pF, $P_ {\text {Rd}}$ = $C_d F V^2$ or 25.8W.

$C_d$ of $10 C_j$ gives the best dampening with peak voltage of 1.2 time the pedestal voltage, but you can save some power with smaller snubbing caps if you can stand the higher peak voltage.

-
Great answer, best explanation of snubbing I've seen. –  Stephen Collings Jan 10 '13 at 17:14

This is a classic snubbering problem. A diode can't instantaneously go from conduction to blocking; the charge in the PN junction needs to get swept out, and an RC snubber across each diode should help this.

I used to design industrial soft starters and on the medium-voltage units we had a lot of design work around this particular aspect. It's been a long time since I've worked in this particular industry so I don't recall the snubber values, but I would probably start with 0.1uF and maybe 49 ohms and see where things start shaking out from there.

-
+1. Yes, this sounds like a "high frequency ringing at diode turn-off" problem, and a snubber is a good fix. a b. –  davidcary Dec 22 '12 at 8:36
yes but what kind of snubber –  Tony Stewart Dec 22 '12 at 14:46
@Richman I would start with a 0.1uF capacitor and a 3.9 ohm, 2-5W resistor in series with the cap across each diode. The power rating is just a guess, you would have a better idea than me. (edit to use Brian Drummond's calculated value) –  akohlsmith Dec 22 '12 at 17:21
I'll try this when I'm back in the lab Thursday. –  Stephen Collings Dec 22 '12 at 19:33
.1 uF + 2 ohms knocked down my kicks nicely. However, my snubber resistors are taking a beating. 100W resistors are reaching temp limit very quickly, and even two in series (4 ohms 200W total) still overheated. It seems a little ridiculous to be spending 10% of my power budget just on snubbing the diodes. Snubber design not being something I've spent much time on, I'm not sure if there's an obvious path forward, or if this is just the cost of doing business. Does anyone have other suggestions? –  Stephen Collings Dec 27 '12 at 18:55

60A reverse recovery current! (from the datasheet) That has to go somewhere...

Like Andrew Kohlsmith, my first thought would be an R-C snubber across EACH diode, but I'm reluctant to make that an answer unless you can find precedents at similar power. Andrew seems to have the experience to make that judgment; not having worked on industrial power, I do not!

But let's run some numbers : as your forward current will average something like 25A (8kw,350V) let's use the same value for Irm - 25A * Trr=230ns gives a ballpark stored charge of 5.75 uC, which would charge up an 0.1uf capacitor to a more manageable 57V. But 25A * 49R is a bit high (!) - this crude calculation would suggest 4 ohms (or even 2) rather than 49 as a starting point for the snubber resistor.

I repeat : I have not worked on industrial power, so that's just what the numbers say to me. I would appreciate Andrew's commentary given these numbers.

-
you need and RC snubber that snub 60A ..49R is about 1000x too big –  Tony Stewart Dec 22 '12 at 14:45
It was a guess, it's been almost 10 years since I was active in that industry. Your calculations do appear correct to me. –  akohlsmith Dec 22 '12 at 17:22


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# proof of Amdahl’s Law

Suppose an algorithm needs $n$ operations to compute the result. With 1 processor, the algorithm will take $n$ time units. With $N$ processors, the $(1-f)n$ parallelizable operations will take $\frac{(1-f)n}{N}$ time units and the remaining $fn$ non parallelizable operations will take $fn$ time units for a total running time of $fn+\frac{(1-f)n}{N}$ time units. So the speedup $S$ is $\frac{n}{fn+\frac{(1-f)n}{N}}=\frac{1}{f+\frac{1-f}{N}}$.

Title proof of Amdahl’s Law ProofOfAmdahlsLaw 2013-03-22 13:10:36 2013-03-22 13:10:36 lieven (1075) lieven (1075) 5 lieven (1075) Proof msc 68M20


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Estimating option prices using multilevel particle filters

Department of Statistics and Applied Probability, National University of Singapore, 117546, Singapore

## Abstract    Full Text(HTML)    Figure/Table    Related pages

Option valuation problems are often solved using standard Monte Carlo (MC) methods. These techniques can often be enhanced using several strategies especially when one discretizes the dynamics of the underlying asset, of which we assume follows a diffusion process. We consider the combination of two methodologies in this direction. The first is the well-known multilevel Monte Carlo (MLMC) method [7], which is known to reduce the computational effort to achieve a given level of mean square error (MSE) relative to MC in some cases. Sequential Monte Carlo (SMC) (or the particle filter (PF)) methods (e.g. [6]) have also been shown to be beneficial in many option pricing problems potentially reducing variances by large magnitudes (relative to MC) [11, 17]. We propose a multilevel particle filter (MLPF) as an alternative approach to price options. We show via numerical simulations that under suitable assumptions regarding the discretization of the SDE driven by Brownian motion the cost to obtain O$(\epsilon^2)$ MSE scales like O$(\epsilon^{-2.5})$ for our method, as compared with the standard particle filter O$(\epsilon^{-3})$.
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# References

1. Barndoff-Nielsen OE, Shephard N (2001) Non-Gaussian OU-based models and some of their uses in financial economics. J Roy Statist Soc B 63: 167–241.

2. Boyle PP (1977) Options: a Monte Carlo approach. J Fin Econ 4: 323–338.

3. Cerou F, Del Moral P, Guyader A (2011) A non-asymptotic theorem for unnormalized FeynmanKac particle models. Ann Inst Henri Poincaire 47: 629–649.

4. Del Moral P (2004) Feynman-Kac formulae: genealogical and interacting particle systems with applications, New York: Springer.

5. Del Moral P, Jasra A, Law KJH, et al. (2017) Multilevel Monte Carlo samplers for normalizing constants. ACM TOMACS 27: 20.

6. Doucet A, Johansen A (2011) A tutorial on particle filtering and smoothing: fifteen years later, In: Handbook of Nonlinear Filtering, Oxford: Oxford University Press.

7. Giles MB (2008) Multilevel Monte Carlo path simulation. Oper Res 56: 607–617.

8. Giles MB (2015 ) Multilevel Monte Carlo methods. Acta Numerica 24: 259–328.

9. Glasserman P (2013) Monte Carlo Methods in Financial Engineering, New York: Springer.

10. Glasserman P, Staum J (2001) Conditioning on one-step survival for barrier options. Oper Res 49: 923–937.

11. Jasra A, Del Moral P (2011) Sequential Monte Carlo methods for option pricing. Stoch Anal Appl 29: 292–316.

12. Jasra A, Stephens A, Doucet A, et al. (2011) Inference for Lévy-driven stochastic volatility models via adaptive sequential Monte Carlo. Scand J Statist 38: 1–22.

13. Jasra A, Doucet A (2009) Sequential Monte Carlo methods for di usion processes. Proc Roy Soc A 465: 3709–3727.

14. Jasra A, Kamatani K, Law KJH, et al. (2017) Multilevel particle filters. SIAM J Numer Anal 55: 3068–3096.

15. Jasra A, Kamatani K, Osei PP, et al. (2018) Multilevel particle filters: normalizing constant estimation. J Statist Comp 28: 47–60.

16. Sen D, Thiery A, Jasra A (2018) On coupling particle filter trajectories. J Statist Comp 28: 461–475.

17. Sen D, Jasra A, Zhou Y (2017) Some contributions to sequential Monte Carlo methods for option pricing. J Statist Comp Sim 87: 733–752.

18. Shevchenko P, Del Moral P (2016) Valuation of barrier options using sequential Monte Carlo. J Comp Fin 20: 1–29.


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# What statistical test should I use if I want prove for this hypothesis?

I have a point $x$ and a dataset $Y$ and I want to prove that $x$ cannot be getting from the dataset $Y$. More specifically, the point $x$ is a value I got from a classifier (it's not related to accuracy or probabilities, it interprets what the classifier has learned) and the data points in $Y$ were got by shuffling the labels and building the classifier model 1000 times. In other words, I'm trying to neglect the hypothesis that the point $x$ can be produced if you shuffled the labels, which points out that $x$ came out from the knowledge that the classifier has learned. I tried the t-test to prove that the $x$ is away from the mean of $Y$ but I'm not sure if this statistical test is sufficient.

• No statistical test can prove a hypothesis; it can only provide information that might cause you to change your belief as to whether the hypothesis is indeed true. Those who already believe firmly in the truth of the hypothesis get nothing from the new knowledge; as far as they are concerned, the hypothesis is true. But for those who are more skeptical initially, the results of the experiment might cause them to be less skeptical (or it might not). – Dilip Sarwate Jul 4 '18 at 19:24


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1. Oct 4, 2012

### Mute

I'm not sure if this is really a bug with the forum software as opposed to some incompatibility with my browser, but whenever I load the forums for the first time after booting up my computer, any page with latex on it will freeze my computer, sometimes for over a minute, while the latex plugin(?) loads.

Has anyone else encountered this issue, and is there a fix or way to load the math plugin more smoothly so that it doesn't freeze up everything while it loads? I would really love to be able to keep using my laptop while the browser renders the math.

I'm using Chrome and running Windows 7 when this occurs. If I boot up into Ubuntu instead, I do not encounter the same problem with Firefox.

Any suggestions?

Thanks.

2. Oct 4, 2012

### Greg Bernhardt

I run Chrome on Win7 with no problems. Do you get any messages?

3. Oct 5, 2012

### gabbagabbahey

What makes you suspect that the $\LaTeX$ is the culprit? My first guess would have been a malicious, or just poorly written script in one of the ads on the site. Which could also explain why Greg isn't having anyproblems, as I doubt he sees the same ads we do.

4. Oct 5, 2012

### PAllen

I'll add the obvious suggestion: contribute to PF, and no more ads. Better experience all around.

5. Oct 5, 2012

### Borg

Hmm. This sounds a lot like a freeze that I experience also. One freeze after I've started the computer and then it doesn't happen again. I'll have to see if it's occurring during a LaTeX load. I have Win7 w/ Firefox 3.6.16 - yes, I know it's ancient.

6. Oct 6, 2012

### lisab

Staff Emeritus
It happens with my home laptop the same way. Once it goes through one freeze-up it's fine after that. I run Chrome.

I seem to remember this being reported in the past...or am I imagining it ?

I sometimes get a message but it flashes up so quickly it's not readable.

7. Oct 6, 2012

### Mute

Not that I'm aware of. There are no pop-up messages or anything, at least. The system just hangs up for a minute or so.

The system only freezes up when I load a page with TeX on it. If I start up my laptop and load PF and I browse, say, the Career services forum first, nothing freezes. It's only once I go to a page with rendered LaTeX that it locks up. If I go to a math forum and there's a title with latex in it, everything freezes; if there are no titles with Latex in them, then the system only locks up once I go into a thread with the LaTeX.

Also, this only happens once following booting up the computer. Even if I close the browser entirely and come back it won't freeze up again.

So, for these reasons I believe it is my browser loading the math the first time that is the issue.

To be honest I don't mind (or notice) ads very much, unless they start taking up the whole screen or talking, which I don't think has happened on PF. In any event, I don't suspect it's the ads causing a problem. However, it is probably about time I contributed...

At least I'm not the only one!

Where does the message show up? I haven't noticed one before, but maybe I'm always just looking somewhere else when the system freezes. Also, I am using a laptop as well... I don't know why that would make a difference, though.

8. Oct 6, 2012

### Borg

I don't see a message either. The entire system freezes up for me such that I can't move my mouse and other displays freeze. For example, I have Zone Alarm which normally shows a display like an audio equilizer when it's processing web pages. Even that display freezes. I still don't know if this only happens when I visit PF because it hasn't happened to me since I've posted about it.

9. Oct 6, 2012

### lisab

Staff Emeritus
It's right in the middle of the screen, and it's for a split second - literally you could blink and miss it. All I can catch is this guy:

But a lot smaller!

Last edited by a moderator: May 6, 2017
10. Oct 7, 2012

### yenchin

Nope...you are not imagining it. This was reported way back when the new latex system was being tested: https://www.physicsforums.com/showpost.php?p=3316213&postcount=150

Nothing was done about it and I still continue to have the same problem everyday...so I got used to it :uhh:

11. Oct 7, 2012

### Staff: Mentor

Problem is this happens rarely, on rare configurations (it is not just a OS/browser combination, as other people using Chrome on Vista/Seven have seen no problems), and with a code we don't manage. All we can do is to forward the error reports to MathJax developers. My guess is that they have no idea what is happening for the same reasons - it is some rare glitch, very difficult to reproduce without access to the particular computer on which it happens.

Such things happen. I have sold tens of thousands copied of my dyslexia programs in the last 15 years. Sometimes people report glitches that are impossible to reproduce even on identical computers with identical OS. In once particular case my programs were working on several computers in school - all computers were bough at the same time, from the same source, in the same configuration. The only computer that the program refused to work on was the computer used by the school dyslexia and speech therapist. After reinstalling OS to no avail we just switched computers when nobody was looking

12. Oct 7, 2012

### Jonathan Scott

I've seen this several times, using Chrome, and I'm fairly sure it is something to do with loading web fonts when I haven't visited any pages with LaTeX on for some time. The machine just locks out completely (not even responding to mouse movements) for up to two minutes, but then it seems to be fine. When it first happened, I thought the system had crashed, but I'm getting used to it.

13. Oct 7, 2012

### Greg Bernhardt

I have sent the devs this thread to look at. I also see MathJax 2.1 beta is out and it appears to address a ton of issues.

14. Oct 20, 2012

### Borg

I didn't have this freeze for a while but it's occurred over the last several days whenever I went to the Random Thoughts thread. jhae2.718 has LaTeX in his signature and had a few posts there recently.

15. Oct 20, 2012

### Greg Bernhardt

16. Oct 20, 2012

### Borg

Not that I know of. This thread was the first that I had been able to connect it to Latex. Now that I know where it freezes consistantly for me, I'll be sure to check how the beta loads on Monday.

17. Oct 24, 2012

### Mute

I haven't had time to browse the forums this week to see if the new beta has changed anything, but I found that this xkcd page:

http://what-if.xkcd.com/16/

which has latex on it also freezes my system while the math loads. (I visited the page after restarting my computer and before coming to the forums, so it definitely appears to have something to do with the loading of latex)

EDIT: So I just visited the forums after turning on my computer, and the freezing still occurred upon loading this thread with LaTeX in it. (at time of loading only first post is/was there)

Last edited: Oct 25, 2012
18. Oct 26, 2012

### Greg Bernhardt

We are now using MathJax 2.1 Please let me know if you still get the freeze.

19. Oct 26, 2012

### Borg

I went to the first page of this thread and it froze. My XP work computer doesn't freeze even though I have the same ancient version of Firefox on it. Oh well.

20. Oct 26, 2012