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https://leanpub.com/what-the-board
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\$80.00
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### What The Board?
###### WTFlop (6Max) / A Bad Flop for the Preflop Raiser
In this analysis I breakdown a hand where UTG raises preflop and gets called by the BTN, and they see a T97 rainbow flop. The analysis breaks down how each players range interacts with this flop and the resulting implications. The goal of the analysis is to show how to create a balanced flop strategy given the basic combinatorics and equities of each player’s range.
What The Flop
I am the co-founder of a new type of poker training site, focusing on mathematical analysis, wtflop.com. I have been a professional poker player for the last 3+ years and I have been using CardrunnersEV calculator for the last 4 years to mathematically analyze hands I have played and general poker situations. I make instructional poker videos with an emphasis on mathematical analysis and approach at LeggoPoker.com
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• The Setup
• Overview
• Analysis Goals
• Preflop Ranges
• Postflop
• What if UTG never check called or check raised the flop?
• Determining a betting range for UTG
• The Math
• What’s Next?
• Determining BTN’s Call and Raise ranges
• Determining UTG’s Value Betting Range
• What Happened to UTG’s Checking Range?
• UTG Checks, now what?
• Breaking Down BTN’s Value Betting Range
• BTN’s Semi-Bluffing Hands
• Additional Bluffing Hands On the Flop for BTN
• Should UTG Checkraise the Flop?
• A Possible Alternate “Aggressive” Strategy
• Scenario #1
• Scenario #2
• Wrap Up
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https://oeis.org/A351409
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The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A351409 a(n) = n*(n!)^(2*n-2). 3
1, 8, 3888, 764411904, 214990848000000000, 224634374557469245440000000000, 1880461634768804771224006806208512000000000000, 240091793104790737576620139562796649430329798636339200000000000000, 813675117804798213250391541747787241264315446434692481270971279693253181440000000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,2 COMMENTS a(n) is the number of reduced Stable Marriage Problem instances of order n. In the SMP, relabeling men or women has no effect on the number of stable matchings. So the men and women can be relabeled to normalize the order of man #1's rankings (with woman #1 as his first choice and woman n as his last choice), and to similarly normalize the order of woman #1's rankings, except for her ranking of man #1. This reduces the number of possible instances by a factor of n!(n-1)! (A010790 with shifted offset), from (n!)^(2n) (A185141) to a(n). This reduction is directly analogous to the identical reduction from latin squares (A002860) to reduced latin squares (A000315), and can be directly applied to the Latin Stable Marriage Problem (A351413). As with reduced latin squares, some further reduction is possible analogous to row/column reduced latin squares (A123234). It is tempting to aim for a reduction of (n!)^2 by simultaneously normalizing all of man #1 and woman #1's preferences, but that isn't possible unless man #1 and woman #1 happen to be mutual first choices. Applying this reduction to A344669 reduces A344669(2) and A344669(4) to 1, demonstrating that these maximal instances arising in A005154 are unique up to participant relabeling. It raises the question of which other values of n make A344669(n) reducible to 1. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..20 Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021, [Section 7, Symmetries]. FORMULA a(n) = A185141(n) / A010790(n-1). MATHEMATICA a[n_] := n*(n!)^(2*n - 2); Table[a[n], {n, 1, 9}] (* Robert P. P. McKone, Feb 12 2022 *) PROG (PARI) a(n) = n*(n!)^(2*n-2) \\ Andrew Howroyd, Feb 12 2022 CROSSREFS Cf. A185141, A000315, A351413, A344669. Sequence in context: A125540 A221109 A221243 * A024113 A167059 A013742 Adjacent sequences: A351406 A351407 A351408 * A351410 A351411 A351412 KEYWORD nonn AUTHOR Dan Eilers, Feb 11 2022 STATUS approved
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Last modified June 30 02:41 EDT 2022. Contains 354913 sequences. (Running on oeis4.)
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New update is available. Click here to update.
# Max Prefix
Contributed by
Tushar Gupta
Last Updated: 23 Feb, 2023
Easy
0/40
Avg time to solve 7 mins
Success Rate 90 %
Share
## Problem Statement
#### You are given an array 'A' of length 'N'. You can perform the operation defined below any number of times (Possibly 0).
``````Choose 'l', 'r' (1 <= 'l' <= 'r' <='N') and reverse the subarray from 'l' to 'r'.
``````
#### After that, Return the maximum possible sum of the prefix of this array of length 'K'.
##### For Example:-
``````Let 'N' = 5, 'K' = 3, 'A' = [4, 2, 1, 2, 2].
We can reverse the subarray from index 3 to 4 (1-based indexing).
Array becomes [4, 2, 2, 1, 2].
Our answer is 8.
``````
Detailed explanation ( Input/output format, Notes, Images )
##### Constraints:-
``````1 <= 'T' <= 10
1 <= 'K' <= 'N' <= 10^5
1 <= 'A[i]' <= 10^3
The Sum of 'N' overall test cases does not exceed 10^5.
Time Limit: 1 sec
``````
##### Sample Input 1:-
``````2
2 2
1 3
3 2
2 2 3
``````
##### Sample Output 1:-
``````4
5
``````
##### Explanation of sample input 1:-
``````First test case:-
We do not need to reverse any subarray.
Our answer is 4.
Second test case:-
We can reverse the subarray from index 1 to 3 (1-based indexing).
Array becomes [3, 2, 2].
Our answer is 5.
``````
##### Sample Input 2:-
``````2
4 3
4 3 2 1
5 5
1 2 3 4 5
``````
##### Sample Output 2:-
``````9
15
``````
Auto
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https://geo.libretexts.org/Bookshelves/Meteorology/Book%3A_Practical_Meteorology_(Stull)/10%3A_Atmospheric_Forces_and_Winds/10.02%3A_Section_3-
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# 10.3: Horizontal Forces
Five forces contribute to net horizontal accelerations that control horizontal winds: pressure-gradient force (PG), advection (AD), centrifugal force (CN), Coriolis force (CF), and turbulent drag (TD):
\ \begin{align} \frac{F_{x\ n e t}}{m}=\frac{F_{x\ A D}}{m}+\frac{F_{x\ P G}}{m}+\frac{F_{x\ C N}}{m}+\frac{F_{x\ C F}}{m}+\frac{F_{x\ T D}}{m}\tag{10.7a}\end{align}
\ \begin{align} \frac{F_{y\ n e t}}{m}=\frac{F_{y\ A D}}{m}+\frac{F_{y\ P G}}{m}+\frac{F_{y\ C N}}{m}+\frac{F_{y\ C F}}{m}+\frac{F_{y\ T D}}{m}\tag{10.7b}\end{align}
Centrifugal force is an apparent force that allows us to include inertial effects for winds that move in a curved line. Coriolis force, explained in detail later, includes the gravitational and compound centrifugal forces on a non-spherical Earth. In the equations above, force per unit mass has units of N kg–1. These units are equivalent to units of acceleration (m·s–2 , see Appendix A), which we will use here.
# 10.3.1. Advection of Horizontal Momentum
Advection is not a true force. Yet it can cause a change of wind speed at a fixed location in Eulerian coordinates, so we will treat it like a force here. The wind moving past a point can carry specific momentum (i.e., momentum per unit mass). Recall that momentum is defined as mass times velocity, hence specific momentum equals the velocity (i.e., the wind) by definition. Thus, the wind can move (advect) different winds to your fixed location.
This is illustrated in Fig. 10.4a. Consider a mass of air (grey box) with slow U wind (5 m s–1) in the north and faster U wind (10 m s–1) in the south. Thus, U decreases toward the north, giving ∆U/∆y = negative. This whole air mass is advected toward the north over a fixed weather station “O” by a south wind (V = positive). At the later time sketched in Fig. 10.4b, a west wind of 5 m s–1 is measured at “O”. Even later, at the time of Fig. 10.4c, the west wind has increased to 10 m s–1 at the weather station. The rate of increase of U at “O” is larger for faster advection (V), and when ∆U/∆y is more negative.
Thus, ∆U/∆t = –V · ∆U/∆y for this example. The advection term on the RHS causes an acceleration of U wind on the LHS, and thus acts like a force per unit mass: ∆U/∆t = Fx AD/m = –V · ∆U/∆y .
You must always include advection when momentum-budget equations are written in Eulerian frameworks. This is similar to the advection terms in the moisture- and heat-budget Eulerian equations that were in earlier chapters.
For advection, the horizontal force components are
\ \begin{align} \frac{F_{x\ A D}}{m}=-U \cdot \frac{\Delta U}{\Delta x}-V \cdot \frac{\Delta U}{\Delta y}-W \cdot \frac{\Delta U}{\Delta z}\tag{10.8a}\end{align}
\ \begin{align}\frac{F_{y\ A D}}{m}=-U \cdot \frac{\Delta V}{\Delta x}-V \cdot \frac{\Delta V}{\Delta y}-W \cdot \frac{\Delta V}{\Delta z}\tag{10.8b}\end{align}
Recall that a gradient is defined as change across a distance, such as ∆V/∆y. With no gradient, the wind cannot cause accelerations.
Vertical advection of horizontal wind (–W·∆U/∆z in eq. 10.8a, and –W·∆V/∆z in eq. 10.8b) is often very small outside of thunderstorms.
Sample Application
Minneapolis (MN, USA) is about 400 km north of Des Moines (IA, USA). In Minneapolis the wind components (U, V) are (6, 4) m s–1 , while in Des Moines they are (2, 10) m s–1. What is the value of the advective force per mass?
Given: (U, V) = (6, 4) m s–1 in Minneapolis, (U, V) = (2, 10) m s–1 in Des Moines ∆y = 400 km, ∆x = is not relevant
Use the definition of a gradient:
∆U/∆y = (6 – 2 m s–1)/400,000 m = 1.0x10–5 s–1
∆U/∆x = not relevant, ∆U/∆z = not relevant,
∆V/∆y = (4 – 10 m s–1)/400,000 m = –1.5x10–5 s–1
∆V/∆x = not relevant, ∆V/∆z = not relevant
Average U = (6 + 2 m s–1)/2 = 4 m s–1
Average V = (4 + 10 m s–1)/2 = 7 m s–1
Use eq. (10.8a):
Fx AD/m = – (7m s–1)·(1.0x10–5 s–1 ) = –7x10–5 m·s–2
Use eq. (10.8b):
Fy AD/m = – (7m s–1)·(–1.5x10–5 s–1 ) = 1.05x10–4 m·s–2
Check: Physics and units are reasonable.
Exposition: The slower U winds from Des Moines are being blown by positive V winds toward Minneapolis, causing the U wind speed to decrease at Minneapolis. But the V winds are increasing there because of the faster winds in Des Moines moving northward.
In regions where the pressure changes with distance (i.e., a pressure gradient), there is a force from high to low pressure. On weather maps, this force is at right angles to the height contours or isobars, directly from high heights or high pressures to low. Greater gradients (shown by a tighter packing of isobars; i.e., smaller spacing ∆d between isobars on weather maps) cause greater pressure-gradient force (Fig. 10.5). Pressure-gradient force is independent of wind speed, and thus can act on winds of any speed (including calm) and direction.
For pressure-gradient force, the horizontal components are:
\ \begin{align}\frac{F_{x P G}}{m}=-\frac{1}{\rho} \cdot \frac{\Delta P}{\Delta x}\tag{10.9a}\end{align}
\ \begin{align} \frac{F_{y P G}}{m}=-\frac{1}{\rho} \cdot \frac{\Delta P}{\Delta y}\tag{10.9b}\end{align}
where ∆P is the pressure change across a distance of either ∆x or ∆y, and ρ is the density of air.
Sample Application
Minneapolis (MN, USA) is about 400 km north of Des Moines (IA, USA). In (Minneapolis , Des Moines) the pressure is (101, 100) kPa. Find the pressure-gradient force per unit mass? Let ρ = 1.1 kg·m–3.
Given: P =101 kPa @ x = 400 km (north of Des Moines). P =100 kPa @ x = 0 km at Des Moines. ρ = 1.1 kg·m–3.
Find: Fy PG/m = ? m·s–2
Apply eq. (10.9b):
$$\frac{F_{y P G}}{m}=-\frac{1}{\left(1.1 \mathrm{kg} \cdot \mathrm{m}^{-3}\right)} \cdot \frac{(101,000-100,000) \mathrm{Pa}}{(400,000-0) \mathrm{m}}$$
= –2.27x10–3 m·s–2.
Hint, from Appendix A: 1 Pa = 1 kg·m–1·s–2.
Check: Physics and units are reasonable.
Exposition: The force is from high pressure in the north to low pressure in the south. This direction is indicated by the negative sign of the answer; namely, the force points in the negative y direction.
If pressure increases toward one direction, then the force is in the opposite direction (from high to low P); hence, the negative sign in these terms. Pressure-gradient-force magnitude is
\ \begin{align} \left|\frac{F_{P G}}{m}\right|=\left|\frac{1}{\rho} \cdot \frac{\Delta P}{\Delta d}\right|\tag{10.10}\end{align}
where ∆d is the distance between isobars.
Eqs. (10.9) can be rewritten using the hydrostatic eq. (1.25) to give the pressure gradient components as a function of spacing between height contours on an isobaric surface:
\ \begin{align} \frac{F_{x P G}}{m}=-|g| \cdot \frac{\Delta z}{\Delta x}\tag{10.11a}\end{align}
\ \begin{align} \frac{F_{y P G}}{m}=-|g| \cdot \frac{\Delta z}{\Delta y}\tag{10.11b}\end{align}
for a gravitational acceleration magnitude of |g| = 9.8 m·s–2 . ∆z is the height change in the ∆x or ∆y directions; hence, it is the slope of the isobaric surface. Extending this analogy of slope, if you conceptually place a ball on the isobaric surface, it will roll downhill (which is the pressure-gradient force direction). The magnitude of pressure-gradient force is
\ \begin{align} \left|\frac{F_{P G}}{m}\right|=\left|g \cdot \frac{\Delta z}{\Delta d}\right|\tag{10.12}\end{align}
where ∆d is distance between height contours.
The one force that makes winds blow in the horizontal is pressure-gradient force. All the other forces are a function of wind speed, hence they can only change the speed or direction of a wind that already exists. The only force that can start winds blowing from zero (calm) is pressure-gradient force.
Sample Application
If the height of the 50 kPa pressure surface decreases by 10 m northward across a distance of 500 km, what is the pressure-gradient force?
Given: ∆z = –10 m, ∆y = 500 km, |g|= 9.8 m·s–2 .
Find: FPG/m = ? m·s–2
Use eqs. (10.11a & b):
Fx PG/m = 0 m·s–2 , because ∆z/∆x = 0. Thus, FPG/m = Fy PG/m.
$$\frac{F_{y P G}}{m}=-|g| \cdot \frac{\Delta z}{\Delta y}=-\left(9.8 \frac{\mathrm{m}}{\mathrm{s}^{2}}\right) \cdot\left(\frac{-10 \mathrm{m}}{500,000 \mathrm{m}}\right)$$
FPG/m = 0.000196 m·s–2.
Check: Physics, units & sign are reasonable.
Exposition: For our example here, height decreases toward the north, thus a hypothetical ball would roll downhill toward the north. A northward force is in the positive y direction, which explains the positive sign of the answer.
# 10.3.3. Centrifugal Force
Inertia makes an air parcel try to move in a straight line. To get its path to turn requires a force in a different direction. This force, which pulls toward the inside of the turn, is called centripetal force. Centripetal force is the result of a net imbalance of (i.e., the nonzero vector sum of) other forces.
For mathematical convenience, we can define an apparent force, called centrifugal force, that is opposite to centripetal force. Namely, it points outward from the center of rotation. Centrifugal-force components are:
\ \begin{align} \frac{F_{x\ C N}}{m}=+s \cdot \frac{V \cdot M}{R}\tag{10.13a}\end{align}
\ \begin{align} \frac{F_{y\ C N}}{m}=-s \cdot \frac{U \cdot M}{R}\tag{10.13b}\end{align}
where M = ( U2 + V2 )1/2 is wind speed (always positive), R is radius of curvature, and s is a sign factor from Table 10-2 as determined by the hemisphere (North or South) and synoptic pressure center (Low or High).
Centrifugal force magnitude is proportional to wind speed squared:
\ \begin{align} \left|\frac{F_{C N}}{m}\right|=\frac{M^{2}}{R}\tag{10.14}\end{align}
Table 10-2. To apply centrifugal force to separate Cartesian coordinates, a (+/–) sign factor s is required. Hemisphere For winds encircling a Low Pressure Center High Pressure Center Southern –1 +1 Northern +1 –1
Sample Application
500 km east of a high-pressure center is a north wind of 5 m s–1. Assume N. Hemisphere. What is the centrifugal force?
Given: R = 5x105 m, U = 0, V = – 5 m s–1
Find: Fx CN/m = ? m·s–2.
Apply eq. (10.13a). In Table 10-2 find s = –1.
$$\frac{F_{x\ C N}}{m}=-1 \cdot \frac{(-5 \mathrm{m} / \mathrm{s}) \cdot(5 \mathrm{m} / \mathrm{s})}{5 \times 10^{5}}=5 \times 10^{-5} \mathrm{m} \cdot \mathrm{s}^{-2}$$
Check: Physics and units OK. Agrees with sketch.
Exposition: To maintain a turn around the high-pressure center, other forces (the sum of which is the centripetal force) are required to pull toward the center.
# 10.3.4. Coriolis Force
An object such as an air parcel that moves relative to the Earth experiences a compound centrifugal force based on the combined tangential velocities of the Earth’s surface and the object. When combined with the non-vertical component of gravity, the result is called Coriolis force (see the INFO box on the next page). This force points 90° to the right of the wind direction in the Northern Hemisphere (Fig. 10.6), and 90° to the left in the S. Hemisphere.
The Earth rotates one full revolution (2π radians) during a sidereal day (i.e., relative to the fixed stars, Psidereal is a bit less than 24 h, see Appendix B), giving an angular rotation rate of
\ \begin{align} \Omega=2 \cdot \pi / P_{\text {sidereal }}\tag{10.15}\end{align}
= 0.729 211 6 x 10–4 radians s–1
The units for Ω are often abbreviated as s–1. Using this rotation rate, define a Coriolis parameter as:
\ \begin{align} f_{c}=2 \cdot \Omega \cdot \sin (\phi)\tag{10.16}\end{align}
where ϕ is latitude, and 2·Ω = 1.458423x10–4 s–1. Thus, the Coriolis parameter depends only on latitude. Its magnitude is roughly 1x10–4 s–1 at mid-latitudes.
The Coriolis force in the Northern Hemisphere is:
\ \begin{align}\frac{F_{x\ C F}}{m}=f_{c} \cdot V\tag{10.17a}\end{align}
\ \begin{align}\frac{F_{y\ C F}}{m}=-f_{c} \cdot U\tag{10.17b}\end{align}
In the Southern Hemisphere the signs on the right side of eqs. (10.17) are opposite. Coriolis force is zero under calm conditions, and thus cannot create a wind. However, it can change the direction of an existing wind. Coriolis force cannot do work, because it acts perpendicular to the object’s motion.
The magnitude of Coriolis force is:
\ \begin{align} \left|F_{\mathrm{CF}} / m\right| \approx 2 \cdot \Omega \cdot|\sin (\phi) \cdot M|\tag{10.18a}\end{align}
or
\ \begin{align} \left|F_{\mathrm{CF}} / m\right| \approx\left|f_{\mathrm{c}} \cdot M\right|\tag{10.18b}\end{align}
Sample Application (§)
a) Plot Coriolis parameter vs. latitude.
b) Find FCF/m at Paris, given a north wind of 15 m s–1.
a) Given: ϕ = 48.874°N at Paris.
Find fc (s–1) vs. ϕ(°) using eq. (10.16). For example:
fc = (1.458x10–4 s–1)·sin(48.874°) = 1.1x10–4 s–1.
b) Given: V = –15 m s–1 . Find: FCF/m= ? m s–2
Assume U = 0 because no info, thus Fy CF/m= 0.
Apply eq. (10.17a):
Fx CF/m = (1.1x10–4 s–1)·(–15 m s–1) = 1.65x10–3 m s–2
Exposition: This Coriolis force points to the west.
INFO • Coriolis Force in 3-D
Eqs. (10.17) give only the dominant components of Coriolis force. There are other smaller-magnitude Coriolis terms (labeled small below) that are usually neglected. The full Coriolis force in 3-dimensions is:
\ \begin{align} \frac{F_{x\ C F}}{m}=f_{c} \cdot V-2 \Omega \cdot \cos (\phi) \cdot W\tag{10.17c}\end{align}
[small because often W<<V]
\ \begin{align} \frac{F_{y\ C F}}{m}=-f_{c} \cdot U\tag{10.17d}\end{align}
\ \begin{align} \frac{F_{z\ C F}}{m}=2 \Omega \cdot \cos (\phi) \cdot U\tag{10.17e}\end{align}
[small relative to other vertical forces]
INFO • On Coriolis Force
Gaspar Gustave Coriolis explained a compound centrifugal force on a rotating non-spherical planet such as Earth (Anders Persson: 1998, 2006, 2014).
Basics
On the rotating Earth an imbalance can occur between gravitational force and centrifugal force.
For an object of mass m moving at tangential speed Mtan along a curved path having radius of curvature R, centrifugal force was shown earlier in this chapter to be FCN/m = (Mtan)2/R. In Fig 10.a the object is represented by the black dot, and the center of rotation is indicated by the X.
The Earth was mostly molten early in its formation. Although gravity tends to make the Earth spherical, centrifugal force associated with Earth’s rotation caused the Earth to bulge slightly at the equator. Thus, Earth’s shape is an ellipsoid (Fig. 10.b).
The combination of gravity FG and centrifugal force FCN causes a net force that we feel as effective gravity FEG. Objects fall in the direction of effective gravity, and it is how we define the local vertical (V) direction. Perpendicular to vertical is the local “horizontal” (H) direction, along the ellipsoidal surface. An object initially at rest on this surface feels no net horizontal force. [Note: Except at the poles and equator, FG does not point exactly to Earth’s center, due to gravitational pull of the equatorial bulge.]
Split the vectors of true gravity into local vertical FGV and horizontal FGH components. Do the same for the centrifugal force (FCNV , FCNH) of Earth’s rotation (Fig. 10.c). Total centrifugal force FCN is parallel to the equator (EQ). Thus, for an object at latitude ϕ , you can use trig to show FCNH ≈ FCN·sin(ϕ).
Objects at Rest with respect to Earth’s Surface
Looking down towards the north pole (NP), the Earth turns counterclockwise with angular velocity Ω = 360°/(sidereal day) (Fig. 10.d). Over a time interval ∆t, the amount of rotation is Ω·∆t. Any object (black dot) at rest on the Earth’s surface moves with the Earth at tangential speed Mtan = Ω·R (grey arrow), where R = Ro·cos(ϕ) is the distance from the axis of rotation. Ro = 6371 km is average Earth radius.
But because the object is at rest, its horizontal component of centrifugal force FCNH associated with movement following the curved latitude (called a parallel) is the same as that for the Earth, as plotted in Fig. 10.c above. But this horizontal force is balanced by the horizontal component of gravity FGH, so the object feels no net horizontal force.
Objects Moving East or West relative to Earth
Suppose an object moves with velocity M due east relative to the Earth. This velocity (thin white arrow in Fig. 10.e) is relative to Earth’s velocity, giving the object a faster total velocity (grey arrow), causing greater centrifugal force and greater FCNH. But FGH is constant.
Horizontal force FCNH does NOT balance FGH. The thick green arrow (Fig. 10.e) shows that the force difference FCF is to the right relative to the object’s motion M. FCF is called Coriolis force.
The opposite imbalance of FCNH and FGH occurs for a westward-moving object (thin white arrow), because the object has slower net tangential velocity (grey arrow in Fig. 10.f). This imbalance, Coriolis force FCF (green arrow), is also to the right of the relative motion M.
Northward-moving Objects
When an object moves northward at relative speed M (thin white arrow in Fig. 10.g) while the Earth is rotating, the path traveled by the object (thick grey line) has a small radius of curvature about point X that is displaced from the North Pole. The smaller radius R causes larger centrifugal force FCNH pointing outward from X.
Component FCNH-ns of centrifugal force balances the unchanged horizontal gravitational force FGH. But there remains an unbalanced east-west component of centrifugal force FCNH-ew which is defined as Coriolis force FCF (green arrow). Again, it is to the right of the relative motion vector M of the object.
Objects moving south have a Coriolis force to the right due to the larger radius of curvature. Regardless of the direction of motion in the Northern Hemisphere, Coriolis force acts 90° to the right of the object’s motion relative to the Earth. When viewing the Southern Hemisphere from below the south pole, the Earth rotates clockwise, causing a Coriolis force that is 90° to the left of the relative motion vector.
Coriolis-force Magnitude Derivation
From Figs. 10.c & d, see that an object at rest (subscript R) has
\ \begin{align} F_{G H}=F_{C N H} \equiv F_{C N H R}\tag{C1}\end{align}
and
\ \begin{align} M_{\text {tan rest}}=\Omega \cdot R\tag{C2}\end{align}
From Fig. 10.e, Coriolis force for an eastward-moving object is defined as
FCF ≡ FCNH – FGH
Apply eq. (C1) to get
FCF = FCNH – FCNHR
or
FCF = sin(ϕ) · [FCN – FCNR] (from Fig. 10.c)
Divide by mass m, and plug in the definition for centrifugal force as velocity squared divided by radius:
FCF / m = sin(ϕ) · [ (Mtan)2/R – (Mtan rest)2/R ]
Use Mtan = Mtan rest + M, along with eq. (C2):
FCF / m = sin(ϕ) · [ (Ω·R+M)2/R – (Ω·R)2/R ]
FCF / m = sin(ϕ) · [(2·Ω·M) + (M2/R)]
The first term is usually much larger than the last, allowing the following approximation for Coriolis force per mass:
\ \begin{align} F_{C F} / m \approx 2 \cdot \Omega \cdot \sin (\phi) \cdot M\tag{10.18}\end{align}
Define a Coriolis parameter as fc ≡ 2·Ω·sin(ϕ) . Thus,
$$F_{C F} / m \approx f_{\mathrm{c}} \cdot M$$
HIGHER MATH • Apparent Forces
In vector form, centrifugal force/mass for an object at rest on Earth is –Ω × (Ω × r), and Coriolis force/mass is –2Ω × V , where vector Ω points along the Earth’s axis toward the north pole, r points from the Earth’s center to the object, V is the object’s velocity relative to Earth, and × is the vector cross product.
# 10.3.5. Turbulent-Drag Force
Surface elements such as pebbles, blades of grass, crops, trees, and buildings partially block the wind, and disturb the air that flows around them. The combined effect of these elements over an area of ground is to cause resistance to air flow, thereby slowing the wind. This resistance is called drag.
At the bottom of the troposphere is a layer of air roughly 0.3 to 3 km thick called the atmospheric boundary layer (ABL). The ABL is named because it is at the bottom boundary of the atmosphere. Turbulence in the ABL mixes the very-slow near-surface air with the faster air in the ABL, reducing the wind speed M throughout the entire ABL (Fig. 10.7).
The net result is a drag force that is normally only felt by air in the ABL. For ABL depth zi the drag is:
\ \begin{align} \frac{F_{x\ T D}}{m}=-w_{T} \cdot \frac{U}{z_{i}}\tag{10.19a}\end{align}
\ \begin{align} \frac{F_{y\ T D}}{m}=-w_{T} \cdot \frac{V}{z_{i}}\tag{10.19b}\end{align}
where wT is called a turbulent transport velocity. The total magnitude of turbulent drag force is
\ \begin{align} \left|\frac{F_{T D}}{m}\right|=w_{T} \cdot \frac{M}{z_{i}}\tag{10.20}\end{align}
and is always opposite to the wind direction.
For statically unstable ABLs with light winds, where a warm underlying surface causes thermals of warm buoyant air to rise (Fig. 10.7), this convective turbulence transports drag information upward at rate:
\ \begin{align} w_{T}=b_{D} \cdot w_{B}\tag{10.22}\end{align}
where dimensionless factor bD = 1.83x10–3. The buoyancy velocity scale, wB, is of order 10 to 50 m s–1, as is explained in the Heat Budget chapter
For statically neutral conditions where strong winds M and wind shears (changes of wind direction and/or speed with height) create eddies and mechanical turbulence near the ground (Fig. 10.7), the transport velocity is
\ \begin{align} w_{T}=C_{D} \cdot M\tag{10.21}\end{align}
where the drag coefficient CD is small (2x10–3 dimensionless) over smooth surfaces and is larger (2x10–2) over rougher surfaces such as forests.
In fair weather, turbulent-drag force is felt only in the ABL. However, thunderstorm turbulence can mix slow near-surface air throughout the troposphere. Fast winds over mountains can create mountain-wave drag felt in the whole atmosphere (see the Regional Winds chapter).
Sample Application
What is the drag force per unit mass opposing a U = 15 m s–1 wind (with V = 0) for a: (a) statically neutral ABL over a rough forest; & (b) statically unstable ABL having convection with wB = 50 m s–1, given zi = 1.5 km.
Given: U = M = 15 m s–1, zi = 1500 m, CD = 2x10–2, wB = 50 m s–1.
Find: Fx TD/m = ? m·s–2.
(a) Plugging eq. (10.21) into eq. (10.19a) gives:
$$\frac{F_{x\ T D}}{m}=-C_{D} \cdot M \cdot \frac{U}{z_{i}}=-(0.02) \cdot \frac{(15 \mathrm{m} / \mathrm{s})^{2}}{1500 \mathrm{m}}$$
= 3x10–3 m·s–2.
(b) Plugging eq. (10.22) into eq. (10.19a) gives:
$$\frac{F_{x\ T D}}{m}=-b_{D} \cdot w_{B} \cdot \frac{U}{z_{i}}$$
$$=-(0.00183) \cdot(50 \mathrm{m} / \mathrm{s}) \cdot \frac{(15 \mathrm{m} / \mathrm{s})}{1500 \mathrm{m}}$$
= 9.15x10–4 m·s–2.
Check: Physics and units are reasonable.
Exposition: Because the wind is positive (blowing toward the east) it requires that the drag be negative (pushing toward the west). Shear (mechanical) turbulence and convective (thermal/buoyant) turbulence can both cause drag by diluting the faster winds higher in the ABL with slower near-surface winds.
Item Name of Force Direction Magnitude (N kg–1) Horiz. (H) or Vert. (V) Remarks (“item” is in column 1; H & V in col. 5) Table 10-3. Summary of forces. 1 gravity down $$\left|\frac{F_{G}}{m}\right|=|g|=9.8 \mathrm{m} \cdot \mathrm{s}^{-2}$$ V hydrostatic equilibrium when items 1 & 2V balance 2 pressure gradient from high to low pressure $$\left|\frac{F_{P G}}{m}\right|=\left|g \cdot \frac{\Delta z}{\Delta d}\right|$$ V & H the only force that can drive horizontal winds 3 Coriolis (compound) 90° to right (left) of wind in Northern (Southern) Hemisphere $$\left|\frac{F_{C F}}{m}\right|=2 \cdot \Omega \cdot|\sin (\phi) \cdot M|$$ H* geostrophic wind when 2H and 3 balance (explained later in horiz. wind section) 4 turbulent drag opposite to wind $$\left|\frac{F_{T D}}{m}\right|=w_{T} \cdot \frac{M}{z_{i}}$$ H* atm. boundary-layer wind when 2H, 3 and 4 balance (explained in horiz. wind section) 5 centrifugal (apparent) away from center of curvature $$\left|\frac{F_{C N}}{m}\right|=\frac{M^{2}}{R}$$ H* centripetal = opposite of centrifugal. Gradient wind when 2H, 3 and 5 balance 6 advection (apparent) (any) $$\left|\frac{F_{A D}}{m}\right|=|-M \cdot \frac{\Delta U}{\Delta d}-\cdots|$$ V & H neither creates nor destroys momentum; just moves it
*Horizontal is the direction we will focus on. However, Coriolis force has a small vertical component for zonal winds. Turbulent drag can exist in the vertical for rising or sinking air, but has completely different form than the boundary-layer drag given above. Centrifugal force can exist in the vertical for vortices with horizontal axes. Note: units N kg–1 = m·s–2.
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# The average of the ages of A, B and C , 3 years ago was 45 years, and that of B and C 4 years ago was 35 years. Find the age of A after 4 years.
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The average of the ages of A, B and C , 3 years ago was 45 years, and that of B and C 4 years ago was 35 years. Find the age of A after 4 years.
1. 58
2. 62
3. 70
4. 60
by (110k points)
selected
Correct Answer - Option 3 : 70
Given,
Average age of A, B and C, 3 years ago = 45
Average age of A, B and C at present = 45 + 3 =48
Sum of ages of A, B and C (A + B + C) = 48 × 3 = 144
Given,
Average age of B and C, 4 years ago = 35
Average age of B and C at present = 35 + 4 = 39
Sum of ages of B and C (B + C) = 39 × 2 = 78
Age of A at present= (A + B + C) - (B + C)
= 144 - 78
= 66
Age of A after 4 years = 66 + 4 = 70 years.
So, the age of A, after 4 years is 70 years.
Hence, the correct answer is 70.
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# Two level spin system in oscillating force
If we consider a two level spin system with frequency $\omega_0$ in the presence of an oscillatory force with frequency $\omega$ chosen off-resonance. The force is designed so that a state $| \downarrow \rangle$ feels an equal but opposite force to a particle in state $| \uparrow \rangle$. How would it follow that after a period of $t = \frac{2 \pi}{\omega_0 - \omega}$, that the drive force has completely dephased and rephased with the oscillating two level system, thus accelerating and decelerating it to its initial motional state?
Thanks for any assistance.
• Could you expand your question a bit and provide a model Hamiltonian? Do you mean something like $H_0=\omega_0\sigma_z$ and some external $H_1=\omega \sigma_y$? Apr 18, 2017 at 16:58
• @ZeroTheHero Yes definitely an $H_0$ as you stated and some external oscillating electric field I would assume (even though the paper states in simply as an oscillating force). Does this idea make any sense to you how?
– user100411
Apr 18, 2017 at 18:33
• Come to think of it it's probably the reverse: $H_0=\omega_0\sigma_y$ and $H_1=\omega \sigma_z$. This way the contribution of $H_1$ is such that the contribution of $\sigma_z$ to $\vert \uparrow\rangle$ and $\vert\downarrow\rangle$ is equal but opposite. I'm not sure I understand the use of "dephased" and "rephased" though. Apr 18, 2017 at 20:18
• @ZeroTheHero What about $H_0 = \omega_0 \sigma_{z}$ and $H_{1} = \omega \sigma_{x}$?
– user100411
Apr 18, 2017 at 20:37
• @JohnDoe Sorry couldn't get back sooner. JDR's answer is on the right track, but if you'd like a more intuitive picture you should look into using the Heisenberg (or interaction) picture and the eom-s for the averages of spin components. You get a linear system with time-dependent coeffs that can be solved fairly easily and you can map the solution onto the Bloch sphere. I can try to write something later, but may take a while. I see JDR refers to a paper, but I don't see a link. Would help if you could provide one.
– udrv
Apr 19, 2017 at 16:32
I think there might still some potential ambiguities in the original question, but I'm going to give it a small go and let others chime in with extra contributions. I'll assume you're referring to a Hamiltonian like $$H=\omega_0I_z+\omega_1 \cos(\omega t)I_y$$ (where $$I_\phi=\frac{1}{2}\sigma_\phi$$ in natural units), since you imply a transverse field oscillating in magnitude.
This might be best treated by going to the "rotating frame" (we're going to rotate around the z axis at the same frequency as the $$I_y$$ field to make it look constant), in which case you have a transformed Hamiltonian $$\tilde{H}=(\omega_0-\omega)I_z+\omega_1I_y$$.
### Aside: Rotating Frame
You can think of the rotating frame by just spinning your eigenvectors with a rotation operator, $$\tilde{\left|\psi\right\rangle}=R(-\omega t)\left|\psi\right\rangle$$, then asking how the Schrodinger equation must behave. By applying the chain rule and Schrodinger equation to $$\frac{d}{dt}(R(-\omega t)\left|\uparrow\right\rangle)$$, we end up with a "rotating frame Schrodinger equation" $$i\frac{d}{dt}\tilde{\left|\psi\right\rangle}=\tilde{H}\tilde{\left|\psi\right\rangle}$$, where $$\tilde{H}=R(-\omega t)HR(\omega t)-\omega I_z$$. It's the chain rule that gives us the "$$-\omega I_z$$" term, which is important here.
See e.g. Levitt, Spin Dynamics, p.241
So the time evolution operator for this time-independent Hamiltonian is just $$U(t)=\exp (-i\tilde{H} t)$$, and for any wavefunction $$\left|\psi(t)\right\rangle=U(t)\left|\psi(0)\right\rangle$$. Plugging in the rotating frame Hamiltonian and the time you gave (call it $$T$$), we've got
$$U(T)=\exp [-i((\omega_0-\omega)I_z+\omega_1I_y)(\frac{2\pi}{\omega_0-\omega})]=\exp [-2\pi i(I_z+\frac{\omega_1}{\omega_0-\omega}I_y)]$$
And this could be a mere net $$2 \pi$$ rotation, so any "dephasing" is "rephased" by the time you've hit period T.
However, this is also where I get confused by your question. I can see a possibility that this time evolution operator will end up just being a net $$2 \pi$$ rotation around some axis, but perhaps the magnitude $$\omega_1$$ needs to be specially defined? Is this what you mean by "the force is designed so that a state ... feels an equal but opposite force..."?
I'm not sure where else to go from here, perhaps you could provide a link to the paper you're referring to, or someone could pick up my thread where I'm leaving off and take the credit for a satisfactory answer, if this one didn't get to what you were wondering :)
edit: for clarity and just so many mistakes.
• The only thing that I'm able to add from reading the paper is to look at pg. 12, and note that a detail I excluded above is that the $\cos(\omega t)$ term is actually considered as two rotating fields in opposite directions $\frac{1}{2}(e^{i\omega t} + e^{-i \omega t})$, like their expression written in terms of raising and lowering operators. Sorry I can't help more than that
– JDR
Apr 18, 2017 at 21:00
• Looks like that part is also along $z$. Apr 18, 2017 at 21:19
• @JDR Thanks for your response. Just a few questions: 1. Could we have considered instead $\hat{H} = \omega_0I_z + \omega_1 \cos(\omega t)I_x$, would that be fine as well? 2. Are you chossing $R(t) = e^{\frac{i H t}{\hbar}}$ as your rotation operator? 3. Would we require that $U(T) = e^{-2 \pi \sigma_{z}}$ to get my quoted result, thanks.
– user100411
Apr 19, 2017 at 18:29
• 1. Certainly using $I_x$ would work fine as well (or even any combination of $I_x$ and $I_y$, the point is having a field in the transverse plane). 2. A rotation using the spin operators will always look like $\exp [i I_{\phi} \theta]$ for a rotation of angle $\theta$ around axis $\phi =x,y,z$. So if the $H$ you've written were $\hbar \omega I_y$ for example, you can see you would have a rotation by angle $\omega t$ around the y axis.
– JDR
Apr 20, 2017 at 14:22
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# Minimum distance between two satellites
peripatein
A space base has traced stallites A and B at a particular moment at:
A0 = (a1,0,a3) B0 = (0,b2,b3)
whereas the base itself is located at the origin (0,0,0) and the satellites move at constant velocities with respect to the base:
Va = (Va,0,0) Vb = (Vb,Vb,0)
The minimum distance between satellites A and B ought to be computed.
My proposed solution:
OA = A0 + Va * t = (a1,0,a3) + (Va,0,0)t
OB = B0 + Vb * t = (0,b2,b3) + (Vb,Vb,0)t
I found the minimum distance to be:
SQRT[1/2(a1+b2)^2 + (b3-a3)^2]
Am I correct? Could someone please confirm?
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# Excel: Countifs with distinct count
1. ## Excel: Countifs with distinct count
Hello everyone,
Im currently encountering a a bump with trying to implement countifs function with multiple criteria, where in which one of the criteria has to be a distinct count of its values:
This is the base formula im using:
=SUM(COUNTIFS(side[Queue_d],"trms-pi-siv-bap-enforcement",side[defect_type],{"UFR","FPR"},side[is_dsr],"N",side[is_cn],"N"))
So this is the basic countifs formula which is counting these various criterias, but i wanted to make an additional count of a column with only distinct values.
The column i wanted to count distinct values is: side[seller_id]
What i tried to do, is to create a helper column which states if a cell a distinct value or not, and so i created something like this:
=SUM(COUNTIFS(side[Queue_d],"trms-pi-siv-bap-enforcement",side[defect_type],{"UFR","FPR"},side[is_dsr],"N",side[is_cn],"N",side[IsDistinct],"Distinct"))
In the attached document, i have the pivot result next to the countifs formula im using to count the distinct values with the helper column:
But for some reason, it doesnt give me the same result as i get from my pivot table
Would anyone have a solution to my problem?
I would be greatful!
2. ## Re: Excel: Countifs with distinct count
Welcome to the forum
Please attach a sample workbook (not a picture or pasted copy). Make sure there is just enough data to demonstrate your need. Include a BEFORE sheet and an AFTER sheet in the workbook if needed to show the process you're trying to complete or automate. Make sure your desired results are shown, mock them up manually if necessary.
Remember to desensitize the data.
Click on GO ADVANCED and then scroll down to Manage Attachments to open the upload window.
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Home / Expert Answers / Advanced Physics / thank-you-for-helping-me-lt-3-reflection-and-phase-change-reflection-and-phase-changes-when-a-wa-pa688
# (Solved): thank you for helping me (: <3 Reflection and Phase Change Reflection and Phase Changes When a wa ...
thank you for helping me (: <3
Reflection and Phase Change Reflection and Phase Changes When a wave reflects off a higher index of refraction medium than the one it is traveling in, is the reflected wave aligned in the same or opposite orientation as the incident wave? When a wave reflects off a lower index of refraction medium than the one it is traveling in, is the reflected wave aligned in the same or opposite orientation as the incident wave? In the case where the alignment is flipped, what is the difference in wavelengths ( )? Soap Bubble Light takes the same path towards the soap bubble and 2 but the light that leaves the soap bubble comes from two different paths , and 2 Soap Bubble Examine the path for Ray 1 from 1 to 1 , - Does the light experience reflection? If so, Is the reflective surface a higher or lower index of refraction than where the ray was before/after reflection? - Does the light experience transmission? If so, is the second medium a a higher or lower index of refraction than where the ray was before/after the transmission? Examine the path for Ray from to , - Does the light experience reflection? If so, Is the reflective surface a higher or lower index of refraction than where the ray was before/after reflection? - Does the light experience transmission? If so, is the second medium a a higher or lower index of refraction than where the ray was before/after the transmission? For the ray that experiences transmission, does it travel a distance of " ", " " , or " "? Thin Film Interference Soap Bubble Examine the path for Ray 1 - Does the light experience reflection? If so, Is the reflective surface a higher or lower index of refraction than where the ray was before/after reflection? - Does the light experience transmission? If so, is the second medium a a higher or lower index of refraction than where the ray was before/after the transmission? Examine the path for Ray 2 - Does the light experience reflection? If so, Is the reflective surface a higher or lower index of refraction than where the ray was before/after reflection? - Does the light experience transmission? If so, is the second medium a a higher or lower index of refraction than where the ray was before/after the transmission? For the ray that experiences transmission, does it travel a distance of " ", " " , or " "?
We have an Answer from Expert
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Subject Area Lessons
## #947. Using Chart Wizard in MS-Excel
Computer, level: other
Posted Mon Apr 5 16:37:08 PDT 1999 by Loretta Koennicke (lgk2@psu.edu).
Pennsylvania State University, State College, PA - USA
Materials Required: Computers with MS-Excel loaded and are connected to the Internet.
Activity Time: Approximately 1 hour.
Concepts Taught: Line graphs, pie charts
Prerequisite Skills: The students have the following prerequisite skills:
Given an URL locate a web site
Cut and paste data from the Internet into a worksheet
Format data in a worksheet
Have the students work in pairs Tell them that they will need to develop at least one line graph and a pie chart, which they will need to present to the other class members. Inform the students they will be assess on how clearly their graphs/charts display the information to their audience. They can decide what type of weather data they would like to display (i.e. snowfall, rainfall, etc.). The students are to follow along with the demonstration of how to create and modify line graphs and pie charts and then they will be given time to collaborate and produce their presentation.
Students will follow along through the demonstration on their workstations.
Steps for creating a line graph:
1. Go to the World Climate web site at http://www.worldclimate.com.
2. Locate the average temperature data (broken out by months) for the town you live in (i.e., Harrisburg, PA) and another town somewhere in the world (instructor should assign the area for the demonstration only - i.e., Seattle, WA). Ask them what type of statistical data did the average temperature data represent? How is the information display at this web sites?
3. Cut and paste the data into a worksheet. (They should have 12 columns with headings for each month of the year and two rows one labeled Harrisburg, PA and the other labeled Seattle, WA. Example:
4. Start chart wizard by clicking the chart wizard button located on the toolbar.
5. Select the line graph, click on the press and view button to preview graph.
6. Enter title (temperature comparison), x-axis months, and y-axis (average temperature) headers.
7. Chart location select new sheet.
8. Editing text, using your mouse right click on the text you want to edit. Then select font, font size, and color (just like you would on a worksheet) for your graph.
9. Editing gridlines, using your mouse right click on the gridlines. Then select whether you want major, minor or no gridlines.
10. Editing background, using your mouse right click on the background. Then select the color and texture of the background for the graph, then the x-axis and y-axis thickness and color.
11. Changing the color of the lines, using your mouse right click on the line you want to change. Then select the color you want the line to be by clicking on the color palette.
12. Moving legend, using your mouse right click on the legend then select the location you want the legend to appear on the page.
After the student have created the line graph, ask them how to compare the data in the table versus the line graph? Which one is it easier to describe the average temperature differences.
Once again, have the students follow along through the demonstration on their workstations.
Steps for creating a pie chart:
1. Start chart wizard by clicking the chart wizard button located on the toolbar.
2. Select the pie chart, click on the press and view button to preview graph.
3. Select only one row of data along with the column headings this can be done by the mouse. Bring attention to the fact that they are only using one row and ask the students why?
4. Enter title and labels for pie chart.
5. Chart location select new sheet.
6. Editing text, using your mouse right click on the text you want to edit. Then select font, font size, and color (just like you would on a worksheet) for your graph.
7. Changing the color of the pie pieces, using your mouse right click on the line you want to change. Then select the color you want the line to be by clicking on the color palette.
8. Moving legend, using your mouse right click on the legend then select the location you want the legend to appear on the page.
After the demonstrations ask them how pie charts and line graphs show data differently and when would they use pie charts versus line graphs.
Assessment will be based on their presentation of their line graphs and pie charts to the class.
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Question: What Does Parallel Vectors Mean?
Does collinear mean parallel?
Two or more vectors are said to be collinear if their lines of support are parallel to the same(fixed) line irrespective of their magnitudes and direction.
Two or more vectors are said to be collinear if their lines of support are parallel to the same(fixed) line irrespective of their magnitudes and direction..
Are vectors in opposite directions parallel?
Two vectors are parallel if they have the same direction or are in exactly opposite directions. Now, recall again the geometric interpretation of scalar multiplication. When we performed scalar multiplication we generated new vectors that were parallel to the original vectors (and each other for that matter).
What does it mean when two vectors are parallel?
Two vectors u and v are said to be parallel if they have either the same direction or opposite direction. This means that each is a scalar multiple of the other: for some non-zero scalar s, v = su and so u =
How do you know if two vectors are parallel cross product?
If the cross product of two vectors is the zero vector (i.e. a × b = 0), then either one or both of the inputs is the zero vector, (a = 0 or b = 0) or else they are parallel or antiparallel (a ∥ b) so that the sine of the angle between them is zero (θ = 0° or θ = 180° and sinθ = 0).
What if two vectors are collinear?
Answer and Explanation: The vectors are collinear means the angle between the two vectors is zero or 180 degrees. The angle between the two vectors is zero degrees if both…
What happens when two vectors are perpendicular?
Two vectors are perpendicular if their dot product is zero, and parallel if their dot product is 1.
Are collinear vectors parallel?
Parallel vectors are vectors which have same or parallel support. They can have equal or unequal magnitudes and their directions may be same or opposite. Two vectors are collinear if they have the same direction or are parallel or anti-parallel.
What is the difference between parallel vectors and collinear vectors?
Parallel vectors are vectors which have same or parallel support. They can have equal or unequal magnitudes and their directions may be same or opposite. Two vectors are collinear if they have the same direction or are parallel or anti-parallel.
How do you cross two vectors?
We can calculate the Cross Product this way: So the length is: the length of a times the length of b times the sine of the angle between a and b, Then we multiply by the vector n so it heads in the correct direction (at right angles to both a and b).
How do you find a vector perpendicular to two vectors?
Explanation: Cross product of vectors A and B is perpendicular to each vector A and B. ∴ for two vectors →Aand→B if →C is the vector perpendicular to both. =(A2B3−B2A3)ˆi−(A1B3−B1A3)ˆj+(A1B2−B1A2)ˆk .
How do you add parallel vectors?
Begin. Parallel vectors behave like numbers on a number line. Add the magnitudes of vectors in the same direction. Subtract the magnitudes of vectors in opposite directions.
What do parallel vectors have in common?
Vectors are parallel if they have the same direction. Both components of one vector must be in the same ratio to the corresponding components of the parallel vector. How to define parallel vectors? Two vectors are parallel if they are scalar multiples of one another.
What is the dot product of parallel vectors?
Given two vectors, and , we define the dot product, , as the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them. Mathematically, . Note that this is equivalent to the magnitude of one of the vectors multiplied by the component of the other vector which lies parallel to it.
How can you tell if two vectors are parallel?
Parallel and Perpendicular Vectors. Two vectors A and B are parallel if and only if they are scalar multiples of one another. A = k B , k is a constant not equal to zero. Two vectors A and B are perpendicular if and only if their scalar product is equal to zero.
What is the cross product of two parallel vectors?
So the answer to your question is that the cross product of two parallel vectors is 0 because the rejection of a vector from a parallel vector is 0 and hence has length 0.
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• # Five Very Useful Functions For Working With Text
This page is dedicated to explaining how to use what I believe are the five most valuable formulas for working with text in Excel. They are useful on their own many times, but can become immensely powerful when nested (combined) with other formulas later as well. Mastering these five formulas will open up the door to many things that you may have never thought possible. All of these formulas can be used by putting actual text in the "text" area, but their true power is unlocked when using them on cell references as the data can then be dynamic.
For all of the following discussions, the syntax of the formulas will be displayed with all required portions in bold, and optional portions within square brackets and in regular type.
The "LEFT" Formula
Syntax: =LEFT(text,num_chars)
The left formula, put very simply, returns the left x characters of the cell. For example =LEFT("Hello",2) would return "He". As mentioned above, however, the best use comes from using it with a cell reference, as shown in the examples below.
Both the examples shown below use different data so that you can see the effects of changing the second argument (num_chars). The first example is asking for the left six characters, and the second example is asking for the left four characters:
The "RIGHT" Formula
Syntax: =RIGHT(text,num_chars)
The right formula is very similar to the left formula, but as you'd expect, it returns the right x characters of the cell. Both the examples shown below, are asking for the right six characters from the cell:
The "MID" Formula
Syntax: =MID(text,start_num,num_chars)
Now the mid formula, designed to get text from the middle of a text string, is a little bit more complex, in that it takes another argument. In addition to knowing the text you want to evaluate, and the number of characters you want returned, it also needs to know which character you want to start at. To put the above syntax in plain English, you would build a MID formula as follows:
=MID(The text or cell you want to search, what character to start with, how many characters to return)
Now there are some things worth noting in the above examples. If you click on the results, you will see:
• The result in B3 (Visual) could also have been generated with the following formula: =Left(A3,6) This is just one area where there is more than one way to achieve the same results in Excel.
• The result in B4 (Basic) is interesting for two reasons. The first is that if you count the characters in Visual, you will see that there are only six letters, yet our formula starts the MID calculation at the seventh character. This is because spaces do count as a character. Had we used the formula =MID(A4,7,5), we would have returned " Basi". The second item of note is, as with the line above, we could have achieved this in a different way. =Right(A4,5) would have returned us the same result. (Basic)
• The last example is also interesting to know. If your final argument is greater than the amount of characters remaining after the start point, it gives you all of the remaining characters, as if it were the RIGHT formula.
The above examples were meant to display two things. The first was obviously how to use the MID formula. The second was that there are multiple ways to tackle the same problem in Excel. While some may be better than others, some are completely user preference, and what you become comfortable with.
The "LEN" Formula
Syntax: =LEN(text)
The LEN formula is used to return the length of the text string in question. While this may seem like a useless formula at first glance, I can assure you that it is anything but. When you start trying to manipulate text, I guarantee you that you will need to use this formula. One thing to remember about LEN is that it counts everything: spaces, punctuation, letters and numbers, it will count it all. Consider the following examples:
Now, if you've already started counting characters to verify what I calculated, you'll see that the space in "Visual Basic" was counted, as was the exclamation mark in "Hello!". But look at the last two examples of "Hello". Can you guess why row seven's answer is six characters and not five? It's because cell A7 does not contain "Hello" as A6 does.
A7 actually contains "Hello ". That last space can be very important, and can truly wreck havoc on formulas later if it isn't known about. This is just one of the many reasons we use the LEN formula. Another reason is that if you do actually need to count characters for any reason, it is fairly tedious work. LEN makes it very easy.
The "FIND" Formula
Syntax: =FIND(find_text,within_text,[start_num])
The FIND formula is used to return the position of a certain character within a text string. For example, in the word "Dog", o is the second letter. This can become very useful when you need to break apart text based on letters, say to split up words or product codes.
The find formula is even an additional level of complexity over the MID formula. While it takes the same number of arguments, the last one is actually optional, so you don't necessarily need to provide it. Again, we'll put the above syntax into plain English:
=FIND(The text or cell you want to find, what text you want to look in, what character to start with)
As mentioned, you have the option of leaving the last argument off, in which case Excel will just assume that you want to start looking at character number one. Some examples of the find formula are:
The first two examples, based on the word "Basketball", display the basic use of the FIND formula, but demonstrate the ability of ignoring the last number as an argument. My personal preference is to add the last number in most cases.
The second two examples, based on the number 12345, show that you don't actually need to enclose numbers in quotes. Since the FIND formula searches on a text argument, all arguments are converted to text before searching.
The third set of examples are based on the word "Doorknob", and display how to search for an entire string of text and a single letter. When searching for a string within a string (a word within a word), the position of the first character of the found string is returned. In the case of the single letter, we are looking for the position of the third "o", which occurs somewhere after the third character. We therefore start looking at the forth character.
The forth pair show examples of searching for a string of numbers within a text string. (Numbers formatted as per the example are actually text, not numbers. If they were number the resulting calculation would rest in the cell.) It also shows how to look for the second instance of a character within a cell. (Similar to the "o" example in the set above.)
The final pair shows that the FIND formula is Case Sensitive, and what happens if a character is not found. If you need a non-case sensitive version of this formula, you can use the =SEARCH formula, which functions in exactly the same way as the FIND formula for all other intents and purposes.
Try it yourself!
The attached workbook allows you to learn and play with text functions in a controlled environment, allowing you to see how these functions work. Each page is set up to challenge you to create strings of text data from given words. Download it and give it a try!
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1. Seattle Valuer -
Great overview of these text functions; thank you
1. pnnaik -
Dera Ken,
You have covered text formula very well.
Thanks,
PN
• ### Recent Forum Posts
#### ThatNewGuy question - updating sheets
try:
Code:
```Sub blah()
Set SceRng = Sheets("Sheet2").Range("A1").CurrentRegion.Resize(, 3)
Intersect(SceRng, SceRng.Offset(1)).Copy```
...
p45cal Yesterday, 07:38 PM
#### How to average dates across the year end
You may not need an extra row for this; in say, cell M10, you can try this formula, which may need to be array-entered using Ctrl+Shift+Enter, rather...
p45cal Yesterday, 03:16 PM
#### How to average dates across the year end
Thank you for the link. Your questions are very similar, so it would have been wise to provide the cross-post links up front. Many of the regulars here...
AliGW Yesterday, 11:35 AM
#### How to average dates across the year end
Thank you for your reply. I haven't cross posted. I asked a question in OZ grid [https://www.ozgrid.com/forum/index.p...age-of-dates/]...
NormanW Yesterday, 09:25 AM
#### How to average dates across the year end
Hi and welcome
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# Half/Double angle Problem
• Dec 15th 2010, 05:27 PM
~berserk
Half/Double angle Problem
Given that $\sin u=\frac{7}{25}$ and $\tan u<0$
Find the exact value of $\cos u,
\sin 2u,
\cos 2u,
\tan 2u,
$
I was absent this day in class and am not sure about where to start with this problem. I obviously know that you won't do them all, and I don't expect you to or else I'd never learn! But guidance on how to do find $\cos u$ and possibly one of the double angle ones would be greatly appreciated.
• Dec 15th 2010, 05:39 PM
snowtea
$\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$
$\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)$
You can derive the double angle identities from these, use $\sin(2u) = \sin(u+u)$.
Also, to find $\cos(u)$ to use in the formulas use the facts $\sin^2(u) + \cos^2(u) = 1$ and $\frac{\sin(u)}{\cos(u)} = \tan(u)$ to determine the sign.
• Dec 15th 2010, 05:54 PM
~berserk
Could you clarify how to do the double angle ones please? As for the cos u i got $\displaystyle{\frac{24}{25}}$this was by recognizing that sin is opposite/hypotenuse and then i put the numbers in a triangle and found the adjacent side using Pythagorean theorem and found the side to be 24 and cos is adjacent/hypotenuse so that's how i got that part.
• Dec 15th 2010, 06:13 PM
snowtea
First, cosine could be positive or negative. Use the information for tangent to figure out its sign.
Can you try to write the formula for the double angle from the information I have given you before? There should be enough information.
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# Lowest Common Ancestors with Linking and Cutting (Lowest Common Ancestor)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
## Description
Given a collection of rooted trees, answer queries of the form, "What is the nearest common ancestor of vertices $x$ and $y$?" In this version of the problem, the queries are on-line. Interspersed with the queries are on-line commands of two types: $link(x, y)$, where $y$ but not necessarily $x$ is a tree root, and $cut (x)$, where $x$ is not a root. The effect of a command $link(x, y)$ is to combine the trees containing $x$ and $y$ by making $x$ the parent of $y$. The effect of a command $cut (x)$ is to cut the edge connecting $x$ and its parent, splitting the tree containing $x$ into two trees: one containing all descendants of $x$ and another containing all nondescendants of $x$.
## Related Problems
Generalizations: Lowest Common Ancestor
## Parameters
$n$: number of vertices
$m$: number of total number of operations (queries, links, and cuts)
## Table of Algorithms
Currently no algorithms in our database for the given problem.
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# Parabola General Equations - General Equations for JEE
Last updated date: 21st May 2024
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## An Introduction to Parabola
Parabola is a mathematical curve that is a part of the conic section. It is a plane curve which has a mirror symmetry and is approximately U shaped. The parabola fits several different descriptions in our everyday life and in even superficial situations both in real life and even in Physics. All these situations essentially define the same curve, that is parabola.
The parabola is one of the most important topics under the conic section. A section of a right circular cone cut by a plane which is parallel to the cone generator is basically known as a parabola. Its shape is traced by a quadratic equation and the graph of parabola shows the locus of a point that is always at a fixed distance from a stationary line. A point on the plane also remains fixed.
## What Is Parabola?
The word "parabola" is made up of two words “para” and “bola”. “Para” means "for" and "bola" means "to throw". Therefore, the meaning is self-explanatory. "Parabola" refers to the shape or form of a curve created by kicking or tossing a ball into the air. A parabola can be defined as the locus of a point on a graph which moves in such a way that its distance from a particular fixed point that we call the focus is equal to its perpendicular distance from a fixed straight line called the directrix. The parabola is an important curve in coordinate geometry's conic sections.
Diagram of a Parabola
From the diagram above, we can see that S is the focus and ZM is the directrix. We have considered a point P on the parabola. So from our definition of the parabola we can write
PS=PM
Here PM is the perpendicular distance of the point from the directrix and PS is the distance of the point from the focus.
The Parabola in Real Life Scenario
Different Curves from a Conic Section
From the image above, we can visualise the parabola as a part of the conic section. Other conic sections include the circle, ellipse, and the hyperbola. All the conic sections have their own properties.
## Important Definitions Related to the Parabola
Some important definitions have been described concerning the parabola, which are as follows:
S.No.
Terminology
Definition
The focus of the Parabola
A parabola is defined by a set of all points in a plane which are at equal distance from a given point and given line. The point is called the focus of the parabola.
Directrix of the Parabola
A parabola is defined by a set of all points in a plane which are at equal distance from a given point and given line. The line is called the directrix of the parabola.
Vertex
The vertex is the point at which the parabola and the axis of symmetry cross each other. The parabola curves dramatically at the vertex.
Focal Length
On the axis of symmetry, it is the distance between the vertex and the focus.
Latus Rectum
It's a line segment that runs parallel to the directrix and passes through the parabola's focus.
Axis of Symmetry
The line of symmetry of a parabola is a line that divides the parabola into two equal sections. It travels through the focus and is perpendicular to the directrix.
Eccentricity
Eccentricity is the ratio of a point's distance from the directrix to its distance from the focus. The parabola's eccentricity is 1.
The terms used in the above table have been depicted in the graph below.
Parts of a Parabola
## General Equation of the Parabola
Here are the general equations of the parabola:
$y=a(x-h)^2+k$
$x=a(y-k)^2+h$
From the equations, we can deduce the nature of the parabola. The value of “a” in the equation of the parabola defines the direction of the parabola. The first equation is the equation of a regular parabola that opens towards the y axis. If the value of a is negative then the parabola opens towards the negative y axis and if it is positive then it opens towards the positive y axis.
The second equation is the equation of a sideways parabola that opens towards the x axis. Again, if a is positive then the parabola opens towards the positive x axis and if it is negative then it opens towards the negative x axis.
For a sideways parabola, we have some parabola formulas as listed below that help in deducing the important point of the parabola.
• (h,k) defines the vertex of the parabola.
• Latus rectum of the parabola is equal to 4a.
• The coordinates of the focus are $\left(h,k+\dfrac{1}{4a}\right)$
• The directrix of the parabola is the line drawn perpendicular to the x-axis and passing the point (-a, 0). The directrix is perpendicular to the parabola's axis.
## Derivation of the Parabola Equation
On the given parabola, there is a point P with the coordinates (x, y). The distance between point P and the directrix is equal to the distance between point P and the focus F, according to the definition of a parabola.
The eccentricity of the parabola is equal to 1. From the definition of parabola mentioned before, we can write the eccentricity e for the parabola as
$\dfrac{PB}{PF}=e$
$\dfrac{PB}{PF}=1$
$PB=PF$
Derivation of the Parabola Equation
The focus's coordinates are F(a,0). We can calculate its distance from P using the coordinate distance formula (x, y)
\begin{align} &P F=\sqrt{(x-a)^{2}+(y-0)^{2}} \\ &P F=\sqrt{(x-a)^{2}+(y)^{2}} \end{align}
The directrix's equation is x + a = 0, and we calculate PB using the perpendicular distance formula.
\begin{align} &P B=\dfrac{x+a}{\sqrt{1^{2}+0^{2}}} \\ &P B=x+a \\ &P B^{2}=\sqrt{(x+a)^{2}} \end{align}
We must use PF = PB to derive the parabola equation:
$\sqrt{(x-a)^{2}+y^{2}}=\sqrt{(x+a)^{2}}$
By squaring both sides of the equation,
\begin{align} &(x-a)^{2}+y^{2}=(x+a)^{2} \\ &x^{2}+a^{2}-2 a x+y^{2}=x^{2}+a^{2}+2 a x \\ &y^{2}=2 a x+2 a x \\ &y^{2}=4 a x \end{align}
We've now correctly deduced the typical parabola equation.
Now this was the equation of a parabola with its vertex at the origin. Suppose we have a parabola with its vertex at (h,k) then its equation is given as
$(y-k)^2=4a(x-h)$
We will see different types of parabolas in the next section. The equation of a parabola that we derived was opening towards the positive x-axis. There can be other parabolas opening towards the negative x-axis and the positive or negative y-axis.
## Parabola Equation in Standard Form
Cartesian coordinates are a pair of numerical coordinates that represent points on a plane (x, y)
1. ### If P is Greater than Zero and P is on the Positive Y-Axis:
$x^{2}=4 p y$
Here, coordinates of F = (0, p) lie on the y-axis and p is the distance from vertex to focus on the y-axis.
Parabola x2 = 4 py with Coordinates of F = (0,P)
### (b) If P is Less than Zero and P is on the Negative X-Axis:
$y^{2}=-4 p x$
Here, coordinates of F = (-p, 0) lie on the y-axis and p is the distance from vertex to focus on the x-axis.
Parabola with coordinates of F = (-P,0)
### (c) If P is Less than Zero and P is on the Negative Y-Axis:
$x^{2}=-4 p y$
Here, coordinates of F = (0, -p) lie on the y-axis and p is the distance from vertex to focus on the y-axis.
Parabola with coordinates of F = (0,-P)
(d) If P is Greater than Zero and P is on the Positive X-Axis:
$y^{2}=4 p x$
Here, coordinates of F = (p, 0) lie on the y-axis and p is the distance from vertex to focus on the x-axis.
Parabola with coordinates of F = (P,0)
The observations from the above section and the diagrams of the parabola have been summarised and listed below.
• A parabola has axis-symmetric geometry. The axis of symmetry runs along the x-axis if the equation contains the term with y2, and along the y-axis if the term with x2 is present in the equation.
• The parabola opens to the right if the coefficient of x is positive and to the left if the coefficient of x is negative when the axis of symmetry is along the x-axis.
• The parabola widens upwards if the coefficient of y is positive and downwards if the coefficient of y is negative when the axis of symmetry is along the y-axis.
## Parametric Equation of a Parabola
If we have a parabola with equation
y2 = 4ax , then the parametric equation of this parabola are
\begin{align} &x=a t^{2} \\ &y=2 a t \end{align}
Here t is the parameter of this parabola.
Any point on the parabola can be represented by the parametric equations as (at2, 2at) and these points are called “t” points.
Diagram showing the t point of a parabola
From the above image, we can see the diagram of a parabola with its t point and the focus represented by (a,0).
## Equation of a Chord Joining any Two Points of a Parabola
Suppose we have two points represented by the parameters t1 and t2 on a parabola. These points are given as (at12, 2at1) and (at22, 2at2), then the equation of a chord joining these two points is given as,
\begin{align} &\left(y-2 a t_{1}\right)=\dfrac{2 a t_{2}-2 a t_{1}}{a t_{2}{ }^{2}-a t_{1}{ }^{2}}\left(x-a t_{1}{ }^{2}\right) \\ &y-2 a t_{1}=\dfrac{2 a\left(t_{2}-t_{1}\right)}{a\left(t_{1}+t_{2}\right)\left(t_{2}-t_{1}\right)}\left(x-a t_{1}{ }^{2}\right) \\ &y-2 a t_{1}=\dfrac{2}{t_{1}+t_{2}}\left(x-a t_{1}^{2}\right) \\ &\left(t_{1}+t_{2}\right) y=2 x+2 a t_{1} t_{2} \end{align}
A focal chord is a chord that passes through the focus of the parabola. For a focal chord,
t1t2=-1
Suppose now that one end of the focal chord is at (at2,2at) then the other end will be at $\left(\dfrac{a}{t^2},-2at\right)$
### Position of a Point with Respect to a Parabola
Suppose we have a point (x1,y1). Now to know if the point lies inside, outside, or on the parabola, we have to check whether, (y12-4ax1) is greater than, less than, or equal to zero.
• If y12-4ax1>0, then the point lies outside the region of the parabola.
• If y12-4ax1<0, then the point lies inside the region of the parabola.
Similarly if y12-4ax1=0, then the point lies on the parabola.
## Tangent to the Parabola
If a line represented by equation y=mx+c, touches a parabola y2=4ax then for the line to be a tangent to the parabola,
$c=\dfrac{a}{m}$
Similarly, if we have a parabola x2=4ay, then the line y=mx+c is a tangent to it if
$c=-am^2$
Now if the line is not given in the standard form then to check if the line is a tangent to a parabola, first, we need to eliminate one variable either x or y between the equations of the straight line and parabola and then apply the condition b2 = 4ac for the quadratic equation ax2+bx+c which is obtained.
The equation of a tangent to the parabola y2=4ax at a point (x1,y1) is given as
$yy_1=2a(x+x_1)$.
This equation is known as the point form equation of the tangent.
Similarly, the parametric form of a tangent to the parabola at a point (at2,2at) is given as,
$ty=x+at^2$
The slope form of a tangent to a parabola y2=4ax is given as
$y=mx+\dfrac{a}{m}$
Here, m is the slope of the tangent.
## Point of Intersection of Two Tangents
Suppose we have two tangents to a parabola y2=4ax at points t1 and t2The equation of the tangent at t1 will be
$yt_1=x+at_1^2$ ….(1)
Similarly the equation of the tangent at point t2 will be
$yt_2=x+at_2^2$.....(2)
Now, if we subtract equations (1) and (2), we get $y(t_1-t_2)=a(t_1^2-t_2^2)$
\begin{align} &y\left(t_{1}-t_{2}\right)=a\left(t_{1}^{2}-t_{2}^{2}\right) \\ &y=\dfrac{a\left(t_{1}+t_{2}\right)\left(t_{1}-t_{2}\right)}{\left(t_{1}-t_{2}\right)} \\ &y=a\left(t_{1}+t_{2}\right) \end{align}
Putting this value of y in equation (1), we get
\begin{align} &a\left(t_{1}+t_{2}\right) t_{1}=x+a t_{1}^{2} \\ &a t_{1}^{2}+a t_{1} t_{2}=x+a t_{1}^{2} \\ &x=a t_{1} t_{2} \end{align}
So the point of intersection of the tangents has coordinates
$x=at_1t_2$
$y=a(t_1+t_2)$
So the point of intersection is
$(a_1t_2, a(t_1+t_2))$
Point of intersection of 2 tangents
## Normal to the Parabola
The tangent of a parabola is perpendicular to the parabola's normal.
Suppose we have a line y=mx+c and a parabola y2=4ax. This line is normal to the parabola when
$c=-2am-am^3$
Similarly, if the parabola is x2=4ay then the line y=mx+c is a normal to the parabola when
$c=2a+\dfrac{a}{m_2}$
## Equation of Normal
The equation of a normal at point (x1,y1) to the parabola y2=4ax is given as
$y-y_1=-\dfrac{y_1}{2a}(x-x_1)$
This is the point form of the normal.
The parametric form of the equation of a normal at point (at2,2at1) is
$y+tx=2at+at^3$
Similarly the slope point form of the normal in terms of slope m is given as
$y=mx-2am-am^3$
## Point of Intersection of Two Normals
Suppose we have two normals at points P(at12,2at1) and Q(at22,2at2) on the parabola y2=4ax, then the point of intersection of the two normals represented by point R.
We know that normals and tangents are perpendicular to each other. As above mentioned that the slope of tangents is $\dfrac{1}{t_1}$ and $\dfrac{1}{t_2}$ so the slope of normals will be $-t_1$ and $-t_2$.
So writing equations of both normals;
$y-2a{t_1}=-{t_1}(x-a{t_1}^2)$ -----(1)
$y-2a{t_2}=-{t_2}(x-a{t_2}^2)$ -----(2)
Now we will solve equations (1) and (2) for finding the intersection point.
$2a{t_2}-{t_2}(x-a{t_2}^2)=2a{t_1}-{t_1}(x-a{t_1}^2)$
$x({t_2}-{t_1})=2a({t_2}-{t_1})+a({t_2}^3-{t_1}^3)$
$x=2a+a({t_2}^2+{t_1}^2+{t_1}{t_2})$
And substituting value of x in equation (1),
$y-2a{t_1}=-{t_1}(2a+a({t_2}^2+{t_1}^2+{t_1}{t_2})-a{t_1}^2)$
$y=2a{t_1}-{t_1}(2a+a{t_2}^2+a{t_1}{t_2})$
$y=-{t_1}(a{t_2}^2+a{t_1}{t_2})$
$y=-at_1t_2(t_1+t_2)$
The point of intersection of the normals is thus, $R(2a+a({t_2}^2+{t_1}^2+{t_1}{t_2}), -at_1t_2(t_1+t_2))$.
## Conclusion
The parabolic graph is a symmetrical figure which is a 2D curve that represents the graph of a quadratic equation. Any point on the parabola maintains a fixed distance from the focus (point) and the directrix (straight line). The focal chord is any straight line passing through the focus of the parabola. The vertex of the parabola is the point of intersection between the parabolic graph and the line of symmetry. The slope of the normal of a parabola is the inverse of the slope of the tangent of the parabola. The general equation of a parabola is given by $y^{2}=4 a x$, where the vertex is at the origin (0, 0) and the axis is the y-axis.
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## FAQs on Parabola General Equations - General Equations for JEE
1. Where can parabolas be seen in real life?
There are a lot of real-life parabola examples that one encounters in everyday life. If you observe carefully, most bridges and flyovers are parabolic in nature. If you throw a ball upwards, the path taken by the ball before touching the ground is also a parabola. The roller coasters in the amusement parks are also classic real-life examples of the parabolic curve. Parabola has utilities in other fields as well. In economics, most of the profits, losses and forecasts are denoted by the parabolic graph. The height of the parabola represents the profits. The bigger the winnings, the higher the parabola.
2. What significance does the current topic have in terms of board exams? How much weight does it have in the JEE exam?
In the board exam, the topic parabola has intermediate levels. Since the board test is subjective in nature, the theoretical and numerical sections of this text are critical from an exam standpoint. Some numerical information is also available from the score advantage section. The weightage of the parabola is moderate because it only comprises a small amount of mathematics. The parabola is a simple idea that is used in this topic. Every year 2 to 3 questions from the parabola topic are asked in the JEE Main.
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1. Minimizing Costs (Word Problem)
This is a word problem that deals with (minimizing costs):
A power line is to be constructed from a power station at point A to an island at point C, which is 1 mi directly out in the water from a point B on the shore. Point B is 4 mi downshore from the power station at A. It costs $5000 per mile to lay the power line under water and$3000 per mile to lay the line under ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A (Hint: The length of CS is sqrt(1 + x^2)
2. Originally Posted by tivo1980
This is a word problem that deals with (minimizing costs):
A power line is to be constructed from a power station at point A to an island at point C, which is 1 mi directly out in the water from a point B on the shore. Point B is 4 mi downshore from the power station at A. It costs $5000 per mile to lay the power line under water and$3000 per mile to lay the line under ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A (Hint: The length of CS is sqrt(1 + x^2)
I guess that $ABC$ forms a right-angle triangle and that $S$ can be chosen anywhere along the line $AB$.
Let's say $x$ is the distance in mi upshores from $B$ to $C$.
The total cost $t\ =\ \sqrt{x^2 +1}\cdot \ 5000 + (4-x)\cdot \ 3000$.
$t\ =\ (x^2 +1)^{0.5}\cdot \ 5000 + (4-x)\cdot \ 3000$.
Minimize $t$:
$\frac{\partial t}{\partial x}\ =\ 0.5\cdot(x^2+1)^{-0.5}\cdot 2\cdot x\cdot \ 5000 - \ 3000\ =\ 0$
$\frac{2 x}{2\sqrt{x^2+1}}\cdot \ 5000\ =\ \ 3000$
$2 x\cdot 5\ =\ 2\sqrt{x^2+1}\cdot 3\ (\Rightarrow x \geq 0)$
$50 x^2\ =\ 36(x^2+1)$
$14 x^2\ =\ 36$
$x^2\ =\ \frac{18}{7}$
$x\ =\ 3\sqrt{\frac{2}{7}}$
Answer: $4-3\sqrt{\frac{2}{7}}$
3. Thank you Trikri.
4. I must be really bored.
5. I have a question Trikri. What software did you use to present the math problem the way you did?
6. Originally Posted by tivo1980
I have a question Trikri. What software did you use to present the math problem the way you did?
Hello, Tivo,
have a look here: http://www.mathhelpforum.com/math-he...rial-latex.pdf
EB
7. Originally Posted by TriKri
I guess that $ABC$ forms a right-angle triangle and that $S$ can be chosen anywhere along the line $AB$.
Let's say $x$ is the distance in mi upshores from $B$ to $C$.
The total cost $t\ =\ \sqrt{x^2 +1}\cdot \ 5000 + (4-x)\cdot \ 3000$.
$t\ =\ (x^2 +1)^{0.5}\cdot \ 5000 + (4-x)\cdot \ 3000$.
Minimize $t$:
$\frac{\partial t}{\partial x}\ =\ 0.5\cdot(x^2+1)^{-0.5}\cdot 2\cdot x\cdot \ 5000 - \ 3000\ =\ 0$
$\frac{2 x}{2\sqrt{x^2+1}}\cdot \ 5000\ =\ \ 3000$
$2 x\cdot 5\ =\ 2\sqrt{x^2+1}\cdot 3\ (\Rightarrow x \geq 0)$
$50 x^2\ =\ 36(x^2+1)$
$14 x^2\ =\ 36$
$x^2\ =\ \frac{18}{7}$
$x\ =\ 3\sqrt{\frac{2}{7}}$
Answer: $4-3\sqrt{\frac{2}{7}}$
Hello, Trikri,
I don't agree with your result:
$2 x\cdot 5\ =\ 2\sqrt{x^2+1}\cdot 3\ (\Rightarrow x \geq 0)$
$100 x^2\ =\ 36(x^2+1)$
$x^2\ =\ \frac{36}{64}$
$x\ =\frac{6}{8}$. The negative result isn't very realistic with this problem.
Answer: $4-\frac{3}{4}=3.25\text{ miles}$ from A.
EB
8. Originally Posted by earboth
Hello, Trikri,
I don't agree with your result:
$2 x\cdot 5\ =\ 2\sqrt{x^2+1}\cdot 3\ (\Rightarrow x \geq 0)$
$100 x^2\ =\ 36(x^2+1)$
Oh, I was a little tired last night ... I must go to bed earlier!
$x\ =\frac{6}{8}$. The negative result isn't very realistic with this problem.
That's why I showed that $x \geq 0$
9. Thank you earboth for the document and the solution.
,
,
,
A power line is to be constructed from a power station at A to an island at point C, which is 1mi directly from a point B on the shore. Point B is 6mi downshore from the power station at A. It costs \$3000 per mile to lay the power line under water and
Click on a term to search for related topics.
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| 3.671875
| 4
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CC-MAIN-2023-06
|
latest
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en
| 0.956133
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https://dmoj.ca/problem/ioi04p4
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crawl-data/CC-MAIN-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00340.warc.gz
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## IOI '04 P4 - Phidias
View as PDF
Points: 10 (partial)
Time limit: 1.0s
Memory limit: 16M
Problem type
##### IOI '04 - Athens, Greece
Famous ancient Greek sculptor Phidias is making preparations to build another marvelous monument. For this purpose he needs rectangular marble plates of sizes .
Recently, Phidias has received a large rectangular marble slab. He wants to cut the slab to obtain plates of the desired sizes. Any piece of marble (the slab or the plates cut from it) can be cut either horizontally or vertically into two rectangular plates with integral widths and heights, cutting completely through that piece. This is the only way to cut pieces and pieces cannot be joined together. Since the marble has a pattern on it, the plates cannot be rotated: if Phidias cuts a plate of size then it cannot be used as a plate of size unless . He can make zero or more plates of each desired size. A marble plate is wasted if it is not of any of the desired sizes after all cuts are completed. Phidias wonders how to cut the initial slab so that as little of it as possible will be wasted.
As an example, assume that in the figure below the width of the original slab is and the height of the original slab is , and the desired plate sizes are , , , and . The minimum possible area wasted is , and the figure shows one sequence of cuts with total waste area of size .
Your task is to write a program that, given the size of the original slab and the desired plate sizes, calculates the minimum total area of the original slab that must be wasted.
#### Input Specification
The first line of input contains two integers: first , the width of the original slab, and then , the height of the original slab. The second line contains one integer : the number of desired plate sizes. The following lines contain the desired plate sizes. Each of these lines contains two integers: first the width and then the height of that desired plate size.
#### Output Specification
The output should contain one line with a single integer: the minimum total area of the original slab that must be wasted.
#### Sample Input
21 11
4
10 4
6 2
7 5
15 10
#### Sample Output
10
Note: in of the inputs, , , and .
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CC-MAIN-2023-06
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longest
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en
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https://labuladong.gitbook.io/algo-en/iv.-high-frequency-interview-problem/union-find-explanation
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crawl-data/CC-MAIN-2024-30/segments/1720763514724.0/warc/CC-MAIN-20240715224905-20240716014905-00446.warc.gz
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# Union-Find Algorithm in Detail
Translator: Ziming
Today I will talk about the Union-Find algorithm, which is often referred to as the Disjoint-Set algorithm, mainly to solve the problem of "dynamic connectivity" in graph theory. Nouns look a little confusing, but they ’re really easy to understand. We will explain it later. Moreover, the application of this algorithm is also very interesting.
Speaking of this Union-Find, it should be my "Enlightenment Algorithm", this algorithm was introduced at the beginning of Algorithms 4th edition, I have to say that this algorithm shocked me! Later I discovered that leetcode also has related topics and is very interesting. Moreover, the solution given in Algorithms 4th edition can be further optimized. With only a small modification, the time complexity can be reduced to O (1).
First, I will explain what is meant by dynamic connectivity.
## Ⅰ. Problem Introduction
Briefly, dynamic connectivity to a fact can be abstracted to connect a graph with lines. For example the following figure depicts a total of 10 nodes, they are disconnected, respectively numerals 0 to 9:
Now our Union-Find algorithm mainly needs to implement these two APIs:
class UF {
/* Connecting the p and q */
public void union(int p, int q);
/* Determine whether p and q are connected */
public boolean connected(int p, int q);
/* Returns the number of connected components in the graph */
public int count();
}
Here's "connectivity" is an equivalence relation, that has the following three properties:
1. reflexivity: p andp node is connected.
2. Symmetry: If p andq node communication, and then q`` p also in communication.
3. Transitivity: If the nodes p andq are connected, and q andr are connected, then p andr are also connected.
For example, in the previous picture, any two different points from 0 to 9 are not connected, and calling connected will return false with 10 connected components.
Now,if you call union (0, 1), the 0 and 1 are connected and the connected components are reduced to 9.
then,when we call union (1, 2), the 0,1,2 are connected. Calling connected (0, 2) will also return true, and the connected components will become 8.
This "equivalent relationship" judgment is very useful, such as the compiler to judge different references to the same variable, or to count the number of friends in social networks, etc.
Now, you probably understand what dynamic connectivity is. The key to the Union-Find algorithm is the efficiency of the union andconnected functions. So what model should we use to represent the connected state of this graph? And what data structure is more appropriate to implement the code?
## Ⅱ. Motivation
Note that I just separated the "model" from the specific "data structure" because we use forests (several trees) to represent the dynamic connectivity of the graph, and use arrays to implement this forest.
How to use forest to represent connectivity? We set each node of the tree to have a pointer to its parent node, and if it is the root node, this pointer points to itself. For example, the graph of 10 nodes just now didn't communicate with each other at the beginning, like this:
class UF {
// recording connected components
private int count;
// the parent of node x is parent [x]
private int[] parent;
/* construct a function where n is the total number of nodes in the graph */
public UF(int n) {
// disconnected at first
this.count = n;
// parent node pointer points to itself
parent = new int[n];
for (int i = 0; i < n; i++)
parent[i] = i;
}
/* Other functions */
}
If two nodes are already connected, then connect the root node of one node to the root node of the other node:
public void union(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
if (rootP == rootQ)
return;
// Merge two trees into one
parent[rootP] = rootQ;
// parent[rootQ] = rootP
count--; // Combine two components into one
}
/* Returns the root node of a node x */
private int find(int x) {
// parent[x] == x
while (parent[x] != x)
x = parent[x];
return x;
}
/* Returns the number of the current connected components */
public int count() {
return count;
}
In this way, if the nodes p andq are connected, they must have the same root node:
public boolean connected(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
return rootP == rootQ;
}
At this point, the Union-Find algorithm is basically complete. Isn't it amazing? We only use arrays to simulate a forest, and cleverly solve this more complicated problem!
So what is the complexity of this algorithm? We found that the complexity of the main API connected andunion is caused by the find function, so they are the same complexity asfind.
The main function of find is to traverse from a certain node to the root of the tree, and its time complexity is the height of the tree. We may customarily think that the height of the tree is logN, but this is not necessarily the case. The height of logN exists only in a balanced binary tree. For general trees, extreme imbalance may occur, causing the“ tree ”to almost degenerate into a“ linked list ”. In the worst case, the tree height may becomeN.
So the above solution, the time complexity of find,union, connected is O (N). This complexity is very unsatisfactory. What you want graph theory to solve is the problem of huge data scales such as social networks. The calls to union andconnected are very frequent, and each call requires linear time completely unbearable.
The point is, how do you find ways to avoid tree imbalances?
## Ⅲ. Balance optimization
We know that in the process of union imbalances may arise either case:
public void union(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
if (rootP == rootQ)
return;
// Merge two trees into one
parent[rootP] = rootQ;
// parent[rootQ] = rootP also works
count--;
At the beginning, we simply and rudely connected the tree where p was located under the root node of the tree whereq was located. Then, a "top-heavy" imbalance situation may occur here, such as the following situation:
Over time, the tree may grow imbalanced. We actually hope that the smaller trees are connected to the larger ones, so that we can avoid top-heavy and more balanced. The solution is to use an additional size array to record the number of nodes in each tree. We might as well call it "weight":
class UF {
private int count;
private int[] parent;
// Added an array record tree "weight"
private int[] size;
public UF(int n) {
this.count = n;
parent = new int[n];
// Since there is only one node per tree initially, the
// weight should be initialized to 1
size = new int[n];
for (int i = 0; i < n; i++) {
parent[i] = i;
size[i] = 1;
}
}
/* Other function */
}
For instance, size [3] = 5 means that the tree rooted at node3 has a total of 5 nodes. This way we can modify the union method:
public void union(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
if (rootP == rootQ)
return;
// The small tree is more balanced under the big tree
if (size[rootP] > size[rootQ]) {
parent[rootQ] = rootP;
size[rootP] += size[rootQ];
} else {
parent[rootP] = rootQ;
size[rootQ] += size[rootP];
}
count--;
}
Like this, by comparing the weight of the tree, you can ensure that the growth of the tree is relatively balanced, and the height of the tree is roughly on the order of logN, which greatly improves the execution efficiency.
At this time, the time complexity of find,union, and connected has been reduced to O (logN), even if the data size is hundreds of millions, the time required is very small.
## Ⅳ. Path compression
This step of optimization is particularly simple and clever. Can we further compress the height of each tree so that the tree height remains constant at all times?
Find can result in O (1) time to find a root node, corresponding,connected union and complexity are reduced to O (1).
To do this, simply add a line in find Code:
private int find(int x) {
while (parent[x] != x) {
// Path compression
parent[x] = parent[parent[x]];
x = parent[x];
}
return x;
}
This operation is a little tricky to see the GIF to understand its role (for clarity, this tree in extreme conditions).
Therefore, each time the find function is called to traverse to the root of the tree, the tree height is shortened by hand, and eventually all the heights will not exceed 3 (the height may reach 3 whenunion).
PS: The reader may ask, after the find process of this GIF graph is completed, the tree height is exactly equal to 3, but if there is a higher tree, the height after compression will still be greater than 3, what should we do? This GIF scenario was edited by me to make it easy for everyone to understand path compression, but in practice, path compression is performed every time it is found, so the tree could not have grown to such a high level, and this worry is unnecessary.
## Ⅴ. Summary
Let's take a look at the whole code:
class UF {
// Number of connected components
private int count;
// Store a tree
private int[] parent;
// Record the "weight" of the tree
private int[] size;
public UF(int n) {
this.count = n;
parent = new int[n];
size = new int[n];
for (int i = 0; i < n; i++) {
parent[i] = i;
size[i] = 1;
}
}
public void union(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
if (rootP == rootQ)
return;
// The small tree is more balanced under the big tree
if (size[rootP] > size[rootQ]) {
parent[rootQ] = rootP;
size[rootP] += size[rootQ];
} else {
parent[rootP] = rootQ;
size[rootQ] += size[rootP];
}
count--;
}
public boolean connected(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
return rootP == rootQ;
}
private int find(int x) {
while (parent[x] != x) {
// Path compression
parent[x] = parent[parent[x]];
x = parent[x];
}
return x;
}
public int count() {
return count;
}
}
The complexity of the Union-Find algorithm can be analyzed as follows: the constructor initializes the data structure requires O (N) time and space complexity. However, the time complexity required for union,connected and count is O (1).
The algorithm is committed to make it clear! Welcome to follow us on WeChat public account labuladong for more easy-to-understand articles
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Problem Solver Statistics
Statistics
Rank: 2
MD YOUSUF(BGC_incredible)
BGC Trust University Bangladesh
873 414 407 SUBMISSIONS PROBLEMS TRIED PROBLEMS SOLVED
Solved problems (407)
100 - Can You Type It?
101 - Income
102 - Divisor
103 - Mina and Rina
104 - Ages
105 - Book Price
106 - Hostel Meal
107 - Excavate Pond
108 - Dividend
109 - Average Age
110 - Average Run
111 - Father Age
112 - Barahatia Village
113 - Profit Percentage
114 - Loss Percentage
115 - Annum Interest
116 - Interest Rate
117 - Principal Amount
118 - Barahatia Village (II)
119 - Pass
120 - Increase of Population
121 - Quotient
122 - Save Money in Bank
123 - Writing Pad
124 - Rice Crop
125 - Hostel Meal (II)
126 - Chair and Table
127 - Victory Day
128 - Twice Number
129 - Subtracted Number
200 - Can you type it (II)??
201 - Motu and Patlu
202 - HELP!!!!!HELP!!!!!!
203 - Diffy
204 - Multi Diffy
205 - Mr. Ant’s Confusing Sum
206 - Subtraction Sum
207 - Car
208 - Square Formula
209 - Professor’s Homework
210 - Professor’s Homework (II)
211 - Consecutive Odd Numbers
212 - Consecutive Even Numbers
213 - Pond Digging
214 - Square Summation
215 - Age Calculation
216 - Motu Patlu Returns
217 - The national flag of Bangladesh
218 - Multi Process
219 - Monkey Vs Bamboo
220 - Midpoint
221 - Farming
222 - Acceleration
223 - Slide Calipers
224 - Bus Travelling
225 - Find Distance
226 - Polygons
227 - Help Boltu
228 - Strange Multiplication
229 - The national flag of Bangladesh (II)
230 - Acceleration (II)
231 - Pond Digging (II)
232 - Easiest Equation V.1
233 - Easiest Equation V.2
234 - Split Number
235 - Simply Square
237 - Salesman
238 - Chestnuts
239 - Taka Paisa
240 - How many Pages?
241 - Too Easy!!!!!!
243 - Pass Exam
244 - Chocolate Bars
245 - Chocolates
246 - Multiplication
247 - Car and truck
248 - Tina's Profit
249 - The national flag of Bangladesh (III)
250 - Racing Moto
251 - Equation
252 - Building
253 - Group Distribution
254 - Ceiling and Floor
255 - Carpeting
256 - Wind Weight
257 - Racing
258 - Replace
300 - Very Easy :)
301 - Binary to Decimal
302 - Octal to Decimal
303 - Strange Digits
304 - Think Big
305 - Mina and Rina Again
306 - Book Price Again
307 - Multi Diffy Again
308 - Consecutive Numbers
309 - Odd Numbers
310 - Even Numbers
311 - Find Big
312 - Find Small
313 - Simply Square Again
314 - Addition Again
315 - Circle
316 - Thirteens Powering
318 - Chocolate Bars Again
319 - Chocolates Again
320 - Multiplication Again
321 - Three Circles
322 - Trapezium
323 - Temperature Conversion
324 - Marriage Cost
325 - Equation Again
326 - Racing Moto Again
327 - Subtraction Sum Again
328 - Simply Area
329 - Simply Area II
330 - Diagonal
331 - Diagonal II
332 - Perimeter
333 - Perimeter II
334 - Percentages (%)
335 - Buying Milk
336 - Circle Inside Circle
337 - Circumference
338 - Circumference Again
339 - Cube
340 - Cube Volume
341 - Tiffin
342 - Alphabets
343 - Length of Line
344 - Love or War
345 - Smaller or Larger
346 - Baby Butterfly
347 - Baby Butterfly II
348 - Paint It !!
349 - Beat It !!!
350 - Betting
351 - Alphabets II
352 - Cake
353 - Summation
354 - Simple Equation
356 - Add or Multiply
357 - Red or Green
358 - Granny’s Puzzle
359 - Divisible by 2
360 - Valid Triangle
361 - Be ! be
362 - Granny's Puzzle II
363 - Triangle Validity
364 - Sum of the Sequence
365 - King in Chessboard
366 - Find the Max
367 - Add the Max
368 - Clock
369 - Mr. Ants Multiplication
370 - Garage Clean
371 - Triangle Stone
372 - Make Triangles
373 - Speed (km/h)
374 - What is the day ??
375 - Print Numbers
400 - Birthday
401 - Diffy Again
402 - Shopping
403 - Subtraction game
404 - Parallel, Not Parallel
405 - Tournament
406 - Knight Attack
407 - 1 + 2 + 3 + . . . + n = k
408 - Income Again
409 - Ages Again
410 - Odd or Even
411 - Relation
412 - The national flag of Bangladesh Again
413 - The national flag of Bangladesh (II) Again
414 - Algebraic Expression (I)
415 - Area of Square
416 - Side Length of Square
417 - The national flag of Bangladesh (III) Again
418 - Algebraic Expression (II)
419 - Middle Number(s)
420 - Middle Number(s) (II)
421 - Max Pair Sum
422 - Pass or Fail
423 - Equivalent Set
424 - Split Time
425 - Divisible by 3
426 - Valid Triangle (II)
427 - Angles
428 - Fail
429 - Algebraic Expression (III)
430 - Present Age
431 - Birds & Rabbits
432 - Triangular Number
433 - Tennis Tournament
434 - Perfect Square
500 - Print “Hello”
501 - Odd Numbers
502 - Even Numbers
503 - Add The ODD
504 - Add The Even
505 - Add The Digits
506 - Armstrong Number
507 - Reverse Code
508 - Print the Numbers
509 - Reverse
510 - Consecutive Again
511 - Print “a”
512 - Print Alphabets
513 - Interesting Square Sum
514 - Maximum Age
515 - Minimum Age
516 - Butterfly’s Sequence
517 - Challenge
518 - Number of zeroes in n!
519 - Factorial n (n!)
520 - GCD
521 - LCM
522 - Positive Integer Average
523 - Question Paper Selection
524 - Array???
525 - Array??? II
526 - Learning Array
527 - Search a number
528 - Chitty The Robot
529 - Dravid walks off, sad but proud
530 - Sum
531 - Prime Number
532 - Divisible by 3 Again
533 - Fraction
534 - Make Triangle 1
535 - Make Triangle 2
536 - Make Triangle 3
537 - Make Triangle 4
538 - Capital to Small
539 - Small to Capital
540 - Point Inside Circle
541 - Bitwise AND
542 - Bitwise OR
543 - XOR (exclusive or) ^
544 - Rabbits
545 - Leap Year
546 - Count Odd & Even
547 - Reverse
548 - Maximum Difference
549 - Cubic Number
550 - Good Time Great Time
551 - Easy Series
552 - Easy Recurtion
553 - Multiplication Table I
554 - Multiplication Table II
555 - Multiplication Table III
556 - Bigger or Equal
557 - Biggest Number
558 - On the Line?
559 - Fibonacci Roots
560 - Even Or Odd
561 - Strange Logic
563 - Time Converter
564 - Friends and ACM
565 - Attack
566 - String Array
567 - Find The Color
568 - Again Palindrome?? (I)
569 - Again Palindrome?? (II)
570 - Intersecting Points in Polygon
571 - Comparison of Areas
572 - Roman Numerals
600 - Sum of Digits
601 - BGC Biddyanagar
10000 - Solve the Equation
10001 - Military Time
10002 - Area of Arrow
10003 - Add numbers
10004 - Simple Math
10005 - The Sky Chef
10006 - Chocolates
10007 - Arithmetic Power Game
10008 - Flag Area
10009 - CNG Costing
10010 - Gainer or Loser
10011 - Tom and Jerry
10012 - ATM
10013 - Confused!!!!!
10014 - Student(s) of Last Bench
10015 - The Cyclist
10016 - FIFA World Cup
10018 - Sakib is unique!!!
10020 - Parentheses Problem
10021 - Unlucky 13 :(
10022 - Multiplication
10023 - Interesting Game
10024 - Is it Physics???
10025 - SMS
10028 - Work
10029 - Ways
10030 - Choose Two
10031 - The Broken Calculator
10032 - Triangle
10033 - Vibrating Sentence
10035 - Make It One!
10036 - Who is The First?
10037 - Angles of Polygon
10038 - Terminator - Genisys
10039 - Largest LCM
10040 - Full Moon
10041 - Sum
10042 - Weight Calculator
10043 - Clean Tubs
10044 - Monkey Vs Bamboo
10045 - Co-Prime
10046 - Achevée
10047 - Bangladesh vs Oman
10048 - Undetected Error
10049 - Binary Tree
10050 - RITA v1.0
10051 - Mysterious Watch
10052 - Playing With Numbers
10053 - Again Pythagoras
10054 - Intersection Point
10055 - Flip-Flopped
10056 - Made Of
10057 - RITA v2.0
10060 - Respect The Women
10061 - Misspelling
10063 - Donate
10064 - Super Palindrome
10066 - Parentheses Matching
10067 - Siyam’s Birthday Cake
10069 - Puzzle Time!!!
10070 - Fizz : The Cutter boy
10071 - Trevor Philips Industries
10072 - Learning Vowel
10073 - Convert The Code
10075 - Boundary
10076 - Say Yes/No
10078 - Silly Sum
10079 - Prime Factor
10080 - Even Fun!!!
10084 - A+B
10085 - Search Lucky Number
10086 - A-B
10087 - Search Unlucky Number
10088 - 2N + 1
10089 - Odd or Even
10090 - Fifteen
10091 - Ugly Number
10092 - Viva Barca
10093 - Angle Triangle
10094 - 3N + 7
10095 - Number of Divisor
10096 - Fifty Five
10097 - N’th Ugly Number
10098 - Square & Rectangle
10099 - S.S.C.
10100 - AxB
10101 - Reverse Factorial
10102 - 5N + 13
10103 - Extended Confused!!!!!
10104 - (a + b)^n
10107 - Pythagorean Triples
10108 - Independence Day
10109 - Khali Kolshi Man
10110 - Square
10111 - N'th Lucky Number
10112 - Guess !!!
10113 - Connect The Ropes
10114 - Set
10115 - Allergenic to Three
10116 - Relation of XY
10118 - Tae-Ho The Gamer
10119 - Point Symmetry
10120 - N'th Unlucky Number
10121 - Haltree Kaltree
10124 - Solve the Equation
10125 - Magic Square
10127 - Square in Grid
10128 - Encryption
10130 - Contiguous
10132 - Rubik's cube
10133 - Count the Numbers
10134 - Last Digit
10136 - Divisors Sum
10137 - Footrace
10138 - Friends
10139 - Factorials divisor
10140 - Prefix Fix
10141 - Proper Divisors Sum
10142 - Odd Even
10145 - Sum of Digits
10147 - Chicken Chap
10148 - Pseudo-Queen
10149 - Happy Couple
10150 - Back And Forth Numbers
10151 - Is It Square?
10152 - Grid Travel
10153 - Encrypt It!!!
10155 - HELP TITU!
10156 - Omar Boi Ghor
10157 - EID Shopping
10160 - Propagate Sansevieria
10162 - Lucky Number
Tried but not yet solved problems (7)
355 - Battle
562 - Identifier validation
10026 - Number of Different Ways…
10027 - The Scribers
10059 - Ant and the beanstalk
10082 - Help Shams
10146 - Prime Factor
| 2,707
| 9,068
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| 3.390625
| 3
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| 1,539,926,689,000,000,000
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# Pages Jonathan's CS151 Project 10
Overview:
The purpose of this project was to learn and implement new methods into our preexisting turtle interpreter, shapes and lsystem files that would allow us to draw the shapes we created last project in new ways. We needed to add new cases to our TurtleInterpreter class in our turtle interpreter file in order to make our new designs. The two new designs that we implemented in this project were the jitter and dot methods.
Our main goal for the jitter method was to chose starting and ending points for lines that were extremely close, but not exactly, the original points. This would allow us to draw more realistic drawings, as not everyone can draw perfectly straight lines. An example of the jitter method is below:
This is supposed to be a perfect square, however with the jitter method, the lines are just slightly off. These designs can make extremely cool images.
In regards to the dot style, we wanted to create a method that would draw circles as the perimeter of the shape, not a solid line. This would allow for a very cool design. An example of the dot method is below:
Again, this is programmed to be a perfect square drawn with a solid line as a perimeter. However, with our new dot method, we can create a square, or any shape, with a dotted perimeter.
Our third task was to show that we had the ability to completely control the jitter and dot methods. For this task, we needed to show that we can draw shapes that use the normal, jitter, jitter3 (essentially the jitter method, just with 3 lines instead of 1) and the dot method. With all of these methods, there are variables that we can change to slightly change the output. For example, we can change the width of the line that the turtle draws. For jitter, we can change the sigma value. Essentially the sigma value is the distance that the line can be from the original start and stop points. It will be easy to see how the sigma value effects the shapes when an image is shown. Furthermore, we can change the dot size, which changes how many dots perimeters the shape. Below is an image showing that I can properly use all of the new methods:
The first shape in each row uses the normal method. Clearly the cross has a larger line width that the square or the octagon. The next shape in each row uses the jitter method. Next, the third shapes in each row use the jitter3 method. The square has the smallest jitter sigma, and the cross has the largest jitter sigma. Clearly, a smaller sigma will give you a shape that represents the shape the most. A larger sigma will make it look least like the actual shape. Lastly we have the dot method. The square uses the smallest dot size, and the cross uses the biggest dot size.
Our fourth task was to edit our indoor scene from project 9 to look more like a real painting. In order to do this, I changed the styles of all the shapes draw to jitter or dotted. The jitter style makes everything look more like a real painting. I used the dotted style for my L-system trees. This gave the trees a cool new look. Below is my new indoor scene:
Task 5 asked us to create a new L-system that includes branching, multiple rules, and at least one rule with more than one replacement string. With this new L-system, we needed to create a simple scene showcasing our creation. Below is my scene with the L-system. The L-system is what makes all the trees. Notice how every tree is different:
Extensions:
1.) For my first extension I expanded my main code for task 3. In task three, we had to show off all of our new ways of drawing shapes. In this extension, I decided to expand that main code and add color to task 3. For some shapes I simply changed the outline color. Other shapes I completely filled in. Below is my first extension:
What I learned:
During this lab I learned how to draw shapes differently. It was very important for me to learn how to draw shapes using different styles, such as jitter and dotted. Also, creating my own L-system was very helpful in further understanding how L-systems work.
Acknowledgments: Holli Olsen, Prof. Taylor
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http://ask.cvxr.com/t/redundant-variables-constraints-change-the-optimal-value/5236/1
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# Redundant variables / constraints change the optimal value?!
(Chen Chen) #1
I find a really wield thing. See the cvx code below. The objective is to minimize ‘vsub’ (‘num_sub’ is just a positive constant here). The optimal value returned is 0.25.
But note, in the optimization problem the decision variables ‘vhub’ and ‘VADPMatrix1’ and thus the first four inequality constraints are totally unrelated and redundant. But deleting them change the optimal value to be 0.328378! It should return the same thing. You guys can test it. Who can tell what is wrong here?
`````` num_sub = 1000;
m = 2000;
Lambda = [0, -0.4532 * 1.0e-08]';
Pb1 = [1/2; 1/2];
Pb2 = [1/(2*num_sub); 1/(2*num_sub); 1 - 1/num_sub];
cvx_begin
variables vhub vsub
variable tmp1(m,2) nonnegative
variable tmp2(m,2) nonnegative
variable Z(m,2)
minimize ( num_sub * vsub )
subject to
vsub + VADPMatrix2(2:m+1,1) >= (1 + tmp1(:,1) - tmp2(:,1) + Z(:,1)).^2/4 + tmp2(:,1) + VADPMatrix2(2:m+1,:) * Pb2;
vsub + VADPMatrix2(1:m,2) >= (1 + tmp1(:,2) - tmp2(:,2) + Z(:,2)).^2/4 + tmp2(:,2) + VADPMatrix2(1:m,:) * Pb2;
cvx_end
value = cvx_optval;``````
(Mark L. Stone) #2
What solver did you use? What was the status reported by CVX?
I tried sedumi, sdpt3, and ecos with and without the “redundant” constraints. In all cases, cvx_optval was close to 0.25, but in some cases, the result was reported as inaccurate/solved/
(Chen Chen) #3
Thanks so much, Mark! I used the the Mosek solver.
Here is the report with the redundant constraints.
`````` Calling Mosek 7.1.0.12: 34005 variables, 22007 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------
MOSEK Version 7.1.0.60 (Build date: 2016-10-7 14:10:42)
Copyright (c) 1998-2016 MOSEK ApS, Denmark. WWW: http://mosek.com
Platform: Windows/64-X86
Computer
Platform : Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 22007
Cones : 4000
Scalar variables : 34005
Matrix variables : 0
Integer variables : 0
Optimizer started.
Conic interior-point optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries : 0 time : 0.00
Eliminator - elim's : 2
Lin. dep. - tries : 1 time : 0.13
Lin. dep. - number : 1
Presolve terminated. Time: 0.17
Optimizer - solved problem : the primal
Optimizer - Constraints : 10003
Optimizer - Cones : 4000
Optimizer - Scalar variables : 18003 conic : 12000
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.03 dense det. time : 0.00
Factor - ML order time : 0.01 GP order time : 0.00
Factor - nonzeros before factor : 6.80e+004 after factor : 7.00e+004
Factor - dense dim. : 0 flops : 1.66e+006
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.2e+003 1.0e+000 2.0e+003 0.00e+000 5.120000000e+005 0.000000000e+000 1.0e+000 0.25
1 3.9e+002 1.8e-001 3.5e+002 -7.19e-001 1.930167366e+005 -3.611088889e+003 1.8e-001 0.27
2 5.8e+001 2.6e-002 5.2e+001 1.26e+000 2.051726749e+004 -3.431496584e+003 2.6e-002 0.28
3 6.4e+000 2.8e-003 5.7e+000 1.16e+000 -3.395684974e+002 -2.641154192e+003 2.8e-003 0.30
4 4.6e-001 2.1e-004 4.1e-001 1.88e+000 -3.137287671e+000 -1.066460865e+002 2.1e-004 0.31
5 2.7e-002 1.2e-005 2.4e-002 1.06e+000 -6.962987983e-001 -6.479068374e+000 1.2e-005 0.33
6 3.6e-003 1.6e-006 3.2e-003 1.11e+000 -5.353374952e-001 -1.239332602e+000 1.6e-006 0.34
7 1.0e-003 4.5e-007 9.0e-004 1.93e+000 -3.469548325e-001 -4.649191111e-001 4.5e-007 0.36
8 1.7e-004 7.5e-008 1.5e-004 2.25e+000 -2.554620262e-001 -2.669713517e-001 7.5e-008 0.38
9 2.1e-005 9.2e-009 1.8e-005 1.10e+000 -2.506294322e-001 -2.519798264e-001 9.2e-009 0.39
10 4.7e-006 2.1e-009 4.2e-006 1.01e+000 -2.501414103e-001 -2.504472369e-001 2.1e-009 0.41
11 1.2e-006 5.2e-010 1.0e-006 1.00e+000 -2.500345734e-001 -2.501110088e-001 5.2e-010 0.42
12 2.8e-007 1.2e-010 2.5e-007 1.00e+000 -2.500072184e-001 -2.500254543e-001 1.2e-010 0.44
13 6.6e-008 2.9e-011 5.8e-008 1.00e+000 -2.500009923e-001 -2.500052672e-001 2.9e-011 0.45
14 1.6e-008 7.3e-012 1.5e-008 1.00e+000 -2.499998204e-001 -2.500008857e-001 7.3e-012 0.47
Interior-point optimizer terminated. Time: 0.47.
Optimizer terminated. Time: 0.48
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.4999982038e-001 Viol. con: 5e-006 var: 0e+000 cones: 0e+000
Dual. obj: -2.5000088567e-001 Viol. con: 0e+000 var: 4e-012 cones: 0e+000
Optimizer summary
Optimizer - time: 0.48
Interior-point - iterations : 14 time: 0.47
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Clean primal-dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Primal-dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.250001
``````
Here is the report without the redundant constraints.
`````` Calling Mosek 7.1.0.12: 36010 variables, 14003 equality constraints
------------------------------------------------------------
MOSEK Version 7.1.0.60 (Build date: 2016-10-7 14:10:42)
Copyright (c) 1998-2016 MOSEK ApS, Denmark. WWW: http://mosek.com
Platform: Windows/64-X86
Computer
Platform : Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 14003
Cones : 4000
Scalar variables : 36010
Matrix variables : 0
Integer variables : 0
Optimizer started.
Conic interior-point optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Total number of eliminations : 6002
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Eliminator - elim's : 6002
Lin. dep. - tries : 1 time : 0.02
Lin. dep. - number : 0
Presolve terminated. Time: 0.08
Optimizer - solved problem : the primal
Optimizer - Constraints : 4001
Optimizer - Cones : 4000
Optimizer - Scalar variables : 20003 conic : 12000
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.02 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 1.20e+004 after factor : 1.60e+004
Factor - dense dim. : 2 flops : 2.44e+005
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+000 1.0e+000 1.0e+000 0.00e+000 0.000000000e+000 0.000000000e+000 1.0e+000 0.17
1 8.3e-002 8.0e-002 8.0e-002 2.11e+000 3.936702415e+001 3.799276976e+002 8.0e-002 0.19
2 4.2e-003 4.0e-003 4.0e-003 1.07e+000 4.119882742e+001 5.802763321e+001 4.0e-003 0.19
3 2.3e-004 2.2e-004 2.2e-004 1.01e+000 4.048164168e+001 4.129814570e+001 2.2e-004 0.20
4 2.6e-005 2.5e-005 2.5e-005 1.16e+000 2.906646055e+001 2.909044085e+001 2.5e-005 0.20
5 2.3e-006 2.2e-006 2.2e-006 1.73e+000 2.238997695e+000 2.239115385e+000 2.2e-006 0.22
6 5.9e-007 5.7e-007 5.7e-007 1.53e+000 8.105009466e-001 8.103460373e-001 5.7e-007 0.22
7 1.6e-007 1.5e-007 1.5e-007 1.74e+000 3.283784095e-001 3.283489275e-001 1.5e-007 0.23
Interior-point optimizer terminated. Time: 0.23.
Optimizer terminated. Time: 0.25
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 3.2837840951e-001 Viol. con: 4e-006 var: 0e+000 cones: 0e+000
Dual. obj: 3.2834892751e-001 Viol. con: 0e+000 var: 3e-005 cones: 8e-006
Optimizer summary
Optimizer - time: 0.25
Interior-point - iterations : 7 time: 0.23
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Clean primal-dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Primal-dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.328378
``````
The status is “solved” in both cases.
(Mark L. Stone) #4
Can you try with a current version (8.1.0.56) of Mosek?
When things are going well and the problem is not ill-posed, then I believe PRSTATUS should converge to 1.0 Note that it does in the version with the redundant constraints (which gets the “correct” answer), but not in the version without them (which doesn’t) . it would be interesting to see whether Mosek 8.1 (which incorporates various numerical improvements) performs better. Even though Mosekl reports Problem status : PRIMAL_AND_DUAL_FEASIBLE, Solution status : OPTIMAL, in both cases, I think that not having PRSTATUS of 1.0 imbues a lesser confidence in the result.
I will defer to Mosek personnel for any further discussion on that.
I don’t have any insight as to why the formulations with and without redundant constraints vary in apparent difficulty for the various solvers, and how that may or may not be related to pre-processing and transformation by CVX, pre-solve by the solver, etc.
Try with Mosek 8.1, and in any case you are welcome to do `cvx_solver_settings('write', 'dump.task.gz')` and send the task files to Mosek support.
(Chen Chen) #6
Thanks Mark and Michal, quite helpful! I installed Mosek 8.1.056 and the problem almost resolves, but there is still a difference of 10^-4.
Here is the report with the redundant constraints.
`````` Calling Mosek unknown: 34005 variables, 22007 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 22007
Cones : 4000
Scalar variables : 34005
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries : 0 time : 0.00
Lin. dep. - tries : 1 time : 0.36
Lin. dep. - number : 1
Presolve terminated. Time: 0.39
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 22007
Cones : 4000
Scalar variables : 34005
Matrix variables : 0
Integer variables : 0
Optimizer - solved problem : the primal
Optimizer - Constraints : 10003
Optimizer - Cones : 4000
Optimizer - Scalar variables : 18003 conic : 12000
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.05 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 6.80e+04 after factor : 7.00e+04
Factor - dense dim. : 0 flops : 1.48e+06
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.0e+03 7.1e-01 1.0e+03 0.00e+00 1.600000000e+04 0.000000000e+00 1.0e+00 0.45
1 3.5e+02 1.3e-01 3.2e+02 -3.82e-01 3.223107444e+03 -5.299697334e+02 1.8e-01 0.48
2 5.3e+01 1.9e-02 1.7e+02 1.99e+00 2.695516717e+01 -3.094199611e+02 2.7e-02 0.50
3 4.6e+00 1.7e-03 9.8e+01 1.78e+00 -1.001151002e+01 -2.895951278e+01 2.3e-03 0.52
4 3.0e-01 1.1e-04 2.9e+01 1.15e+00 -9.440432038e-01 -2.084949153e+00 1.5e-04 0.53
5 7.1e-02 2.6e-05 3.6e+01 1.60e+00 -4.067993728e-01 -5.871639659e-01 3.6e-05 0.55
6 2.0e-02 7.1e-06 1.4e+01 2.71e+00 -2.624704665e-01 -2.881357143e-01 1.0e-05 0.56
7 2.0e-03 7.3e-07 5.0e+00 1.17e+00 -2.512536810e-01 -2.536792205e-01 1.0e-06 0.58
8 5.2e-04 1.9e-07 2.6e+00 1.02e+00 -2.503181059e-01 -2.509332776e-01 2.6e-07 0.59
9 1.6e-04 5.7e-08 1.5e+00 1.01e+00 -2.501001463e-01 -2.502874182e-01 8.0e-08 0.61
10 4.2e-05 1.5e-08 8.1e-01 1.00e+00 -2.500258536e-01 -2.500753805e-01 2.1e-08 0.63
11 9.9e-06 3.6e-09 4.1e-01 1.00e+00 -2.500050802e-01 -2.500167959e-01 5.0e-09 0.64
12 2.3e-06 9.5e-10 2.0e-01 1.00e+00 -2.500005360e-01 -2.500032692e-01 1.2e-09 0.66
13 6.0e-07 9.9e-10 1.1e-01 1.00e+00 -2.499997360e-01 -2.500004451e-01 3.1e-10 0.67
14 1.9e-07 3.0e-09 5.9e-02 1.00e+00 -2.499995167e-01 -2.499997259e-01 9.0e-11 0.69
Optimizer terminated. Time: 0.70
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.4999951669e-01 nrm: 1e+03 Viol. con: 9e-07 var: 0e+00 cones: 0e+00
Dual. obj: -2.4999972590e-01 nrm: 4e+00 Viol. con: 0e+00 var: 2e-09 cones: 0e+00
Optimizer summary
Optimizer - time: 0.70
Interior-point - iterations : 14 time: 0.69
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.25
``````
Here is the report without the redundant constraints.
`````` Calling Mosek unknown: 36010 variables, 14003 equality constraints
------------------------------------------------------------
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 14003
Cones : 4000
Scalar variables : 36010
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 2001
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 3998
Eliminator terminated.
Eliminator - tries : 2 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.08
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 14003
Cones : 4000
Scalar variables : 36010
Matrix variables : 0
Integer variables : 0
Optimizer - solved problem : the primal
Optimizer - Constraints : 4004
Optimizer - Cones : 4001
Optimizer - Scalar variables : 22011 conic : 12006
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.02 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 1.80e+04 after factor : 2.40e+04
Factor - dense dim. : 3 flops : 2.84e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.1e+00 1.3e+02 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.14
1 5.3e-01 5.9e+01 2.4e-01 1.14e+01 1.639371579e+02 4.019239223e+02 4.7e-01 0.17
2 5.1e-02 5.6e+00 6.1e-02 2.08e+00 5.125515659e+01 8.601673909e+01 4.5e-02 0.19
3 3.5e-03 3.9e-01 1.7e-02 1.11e+00 4.408230807e+00 6.536362711e+00 3.2e-03 0.20
4 7.0e-04 7.8e-02 1.6e-02 1.20e+00 1.476488526e+00 1.564239598e+00 6.2e-04 0.22
5 2.0e-04 2.2e-02 2.4e-02 1.91e+00 5.594531583e-01 5.619924834e-01 1.7e-04 0.25
6 4.3e-05 4.8e-03 1.6e-02 1.58e+00 2.865237863e-01 2.867573412e-01 3.8e-05 0.27
7 4.5e-06 5.1e-04 5.6e-03 1.15e+00 2.535793536e-01 2.536005398e-01 4.0e-06 0.28
8 7.8e-07 8.7e-05 2.4e-03 1.02e+00 2.506025097e-01 2.506058336e-01 6.9e-07 0.28
9 1.6e-07 1.8e-05 1.1e-03 1.00e+00 2.501224587e-01 2.501230207e-01 1.4e-07 0.30
Optimizer terminated. Time: 0.33
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 2.5012245871e-01 nrm: 2e+00 Viol. con: 2e-06 var: 2e-05 cones: 0e+00
Dual. obj: 2.5012302069e-01 nrm: 5e-01 Viol. con: 0e+00 var: 6e-06 cones: 2e-08
Optimizer summary
Optimizer - time: 0.33
Interior-point - iterations : 9 time: 0.31
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.250122
``````
So it should be the issue of Mosek 7.1 I use.
I missed that before, but I think it has more to do with the fact that in one of the variants CVX dualizes before calling Mosek, and in the other it does not. It sometimes happens that one of the versions solves better.
(Chen Chen) #8
Hi Michal and Mark, I still meet the problem (with Mosek 8.1.0.56) that the status is
`````` Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
``````
But PRSTATUS is not 1. As Mark mentioned, this does not render a confidence about the returned solution, even though I guess it is near the optimal.
Here is a report example I get:
`````` Calling Mosek unknown: 112205 variables, 72609 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 72609
Cones : 13200
Scalar variables : 112205
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 13201
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 2 time : 0.00
Lin. dep. - tries : 1 time : 0.03
Lin. dep. - number : 0
Presolve terminated. Time: 0.34
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 72609
Cones : 13200
Scalar variables : 112205
Matrix variables : 0
Integer variables : 0
Optimizer - solved problem : the dual
Optimizer - Constraints : 46201
Optimizer - Cones : 13201
Optimizer - Scalar variables : 125409 conic : 59405
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.23 dense det. time : 0.00
Factor - ML order time : 0.01 GP order time : 0.00
Factor - nonzeros before factor : 2.90e+05 after factor : 3.30e+05
Factor - dense dim. : 6 flops : 3.93e+06
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.4e+02 1.1e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.67
1 1.2e+02 9.4e-01 6.5e-01 7.92e+00 -9.927142669e+02 -9.133600954e+02 8.9e-01 0.77
2 2.1e+01 1.6e-01 9.0e-02 5.20e+00 -1.555053136e+03 -1.378294531e+03 1.5e-01 0.84
3 5.1e+00 3.9e-02 5.9e-02 1.90e+00 -3.291451224e+02 -3.045916528e+02 3.7e-02 0.92
4 1.1e-01 8.6e-04 1.1e-02 1.29e+00 -9.098593145e+00 -8.776774409e+00 8.1e-04 1.00
5 4.2e-02 3.2e-04 2.8e-02 2.40e+00 -2.633052460e+00 -2.626366272e+00 3.1e-04 1.08
6 2.1e-02 1.6e-04 1.9e-02 2.37e+00 -1.151147696e+00 -1.147720872e+00 1.5e-04 1.17
7 6.5e-03 5.0e-05 1.1e-02 1.40e+00 -6.221630009e-01 -6.211291941e-01 4.7e-05 1.25
8 3.4e-03 2.6e-05 7.3e-03 8.71e-01 -4.384715079e-01 -4.378053711e-01 2.5e-05 1.33
9 5.9e-04 4.5e-06 7.8e-03 1.08e+00 -2.827603553e-01 -2.827533756e-01 4.3e-06 1.41
10 9.1e-05 7.0e-07 3.2e-03 1.04e+00 -2.549724974e-01 -2.549716705e-01 6.6e-07 1.48
Optimizer terminated. Time: 1.58
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.5497249745e-01 nrm: 1e+03 Viol. con: 3e-05 var: 0e+00 cones: 2e-07
Dual. obj: -2.5497167052e-01 nrm: 8e+00 Viol. con: 0e+00 var: 2e-07 cones: 0e+00
Optimizer summary
Optimizer - time: 1.58
Interior-point - iterations : 10 time: 1.55
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.254972
``````
I’m quite sure the optimal value should be no larger than 0.25. So I was wondering, can I set a termination criteria for Mosek to enforce it to terminate when PRSTATUS is closed to 1 (say, a gap within 1e-6) to guarantee optimality? I searched online but didn’t find such criteria . I was wondering do we have this termination criteria or other similar termination criteria I can use? Thanks so much!
ps: the example I use for the report is
`````` num_sub = 1100;
m = 6600;
Pb1 = [1/2; 1/2];
Pb2 = [1/(2*num_sub); 1/(2*num_sub); 1 - 1/num_sub];
cvx_begin
variables vhub vsub
variable Lambda(2)
variable tmp1(m,2) nonnegative
variable tmp2(m,2) nonnegative
variable Z(m,2)
minimize ( vhub + vsub * num_sub )
subject to
vsub + VADPMatrix2(2:m+1,1) >= (1 + tmp1(:,1) - tmp2(:,1) + Z(:,1)).^2/4 + tmp2(:,1) + VADPMatrix2(2:m+1,:) * Pb2;
vsub + VADPMatrix2(1:m,2) >= (1 + tmp1(:,2) - tmp2(:,2) + Z(:,2)).^2/4 + tmp2(:,2) + VADPMatrix2(1:m,:) * Pb2;
cvx_end``````
(Mark L. Stone) #9
This is a different, larger problem instance than your previous answer, also using Mosek 8.1.0.56? Perhaps you should follow @Michal_Adamaszek 's suggestion and send task files to Mosek support. I will defer on discussion of the Mosek optimizer to him.
(Chen Chen) #10
Yes, this is a different problem. The (only) difference is that now Lambda is a decision variable (dimension is 2) which couples the two parts of the constraints (one part involving vhub and one involving vsub). And yes, Mosek 8.1.0.56 is used. I guess this problem should be ill-conditioned.
After all, I think it’s the issue of Mosek rather than CVX. Super thankful for your kind help and suggestions these days and I will consider reporting this issue.
Many thanks!
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# math
Fred arrives at the subway station every day at a random time and takes the first train that arrives. If he takes the A train, which arrives each hour on the hour, he ends up at the fitness center. If he takes the B train, which arrives every hour 15 minutes after the A train, he ends up at the park. If Fred plays this game over a long period of time, about what percent of the time can he expect to end up at the park?
15%
75%
50%
25%
1. 0
2. 1
1. 25
posted by Anonymous
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# How Do You Model Time-Varying Heat Conduction in a Simple House Simulation?
• Mooz
In summary, the conversation is about a project on heat conduction in a box simulating a house. The goal is to model the temperature inside the box as the outside temperature changes over time. The box will have an outer wall, insulation, inner wall, and air in the middle. The issue is understanding the fundamental theory, which is more complex than Newton's law of cooling. The speaker, who is a 2nd semester college student, needs to do self-study to understand the equations and recommended reading materials. The problem can be simplified to 2 dimensions and a spreadsheet can be used to calculate the temperature at different distances in the box. This method is similar to a basic FEA analysis.
Mooz
I'm doing a projekt on heat conduction i a box.
The point is to simulate a house very simple. We have time varying temperature conditions on the outside, and want to model the temperature inside the box over time as outside temperature changes. The box will consist og an outer wall, insulation, and an inner wall and air in the middle.
Our problem is to get startet with the fundamental theory, which is more complicated than Newtons law of cooling. This is a self study projekt, so I ned to read and understand myself, so i can't be to difficult reading. I just started college, 2. semester.
What equations er being used for this kind of problem? What is good reading?
Thank you
Since you are only talking about conductive heat transfer, it really shouldn't be all that difficult.
First of all, are you going to simplify the problem to 2 dimensions or 3? If it's 2 then that makes things even easier. You can set up a spreadsheet to do your calculations for you. Simply lay out your box as a large group of cells. Your outer cells would have the wall boundary conditions and the inner cells could calculate the temperature at whatever distance you wish to set them apart. Each interior cell would depend on the 4 cells surrounding it (in 2D). It would be a poor man's FEA analysis. I remember doing some basic plates and such this way back in school.
for sharing your project with me. Time varying heat conduction is a fascinating topic with many real-world applications, such as in building and environmental engineering. It sounds like you have a solid understanding of the basic principles and components involved in your project, including the outer wall, insulation, inner wall, and air layer.
To model the temperature inside the box over time, you will need to use the heat conduction equation, also known as the Fourier's law of heat conduction. This equation describes the rate of heat transfer through a material and is based on the temperature gradient and thermal conductivity of the material. It is a fundamental equation in heat transfer and is used in various applications, including your project.
To incorporate time-varying temperature conditions on the outside, you will also need to consider the boundary conditions of your system. These can include the temperature and heat flux at the outer wall, as well as any changes in ambient temperature over time.
As for good reading, there are many resources available online and in textbooks that cover the fundamentals of heat conduction and its applications. Some recommended textbooks include "Fundamentals of Heat and Mass Transfer" by Incropera and DeWitt, "Introduction to Heat Transfer" by Bergman, Lavine, Incropera, and DeWitt, and "Heat and Mass Transfer: A Practical Approach" by Cengel and Ghajar.
Additionally, there are many online resources and tutorials that can help you understand the concepts and equations involved in your project. Some good places to start include online lecture notes, YouTube tutorials, and interactive simulations.
I wish you the best of luck with your project and hope you find the resources and information you need to successfully model time-varying heat conduction in your box.
## 1. What is time varying heat conduction?
Time varying heat conduction is the transfer of heat through a material that changes with time. This could be due to changes in temperature, heat source/sink, or material properties.
## 2. What factors affect time varying heat conduction?
Time varying heat conduction is influenced by several factors, including temperature gradients, thermal conductivity of the material, and the presence of any heat sources or sinks.
## 3. How is time varying heat conduction different from steady-state heat conduction?
In steady-state heat conduction, the temperature and heat flow do not change with time. In contrast, time varying heat conduction involves changes in temperature and heat flow over time.
## 4. How is time varying heat conduction modeled?
Time varying heat conduction is typically modeled using partial differential equations, such as the heat equation. These equations take into account the changing temperature and heat flow over time.
## 5. What are some applications of time varying heat conduction?
Time varying heat conduction is important in many fields, including materials science, thermodynamics, and engineering. It is used to understand and design systems where heat transfer changes over time, such as in electronics cooling and thermal management of buildings.
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# Secular Trend
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### Secular Trend
1. 1. Secular trend or Long Term Trend Basic tendency of a series to grow or decline over a period of time Steady movements over a long time Ex. Temperature of a particular area would change in various seasons but the normal tendency will remain constant
2. 2. How can we determine trend <ul><li>There are four methods to find Trend </li></ul><ul><ul><li>Graphical Method </li></ul></ul><ul><ul><li>(Free hand curve method </li></ul></ul><ul><ul><li>Method of semi average </li></ul></ul><ul><ul><li>Method of moving average </li></ul></ul><ul><ul><li>Method of least square </li></ul></ul>
3. 3. Free hand curve Method <ul><li>Plot the original data on the graph paper </li></ul><ul><li>Draw a smooth free hand line in such a way that it clearly indicates the tendency of the original data . This line is fitted by merely by inspection </li></ul><ul><li>The trend line is drawn in such a manner that area of the curve below the line and above the line are the same </li></ul>
4. 4. Question No 1 <ul><li>With the Help of the Graph Paper obtain the trend Value </li></ul><ul><li>Year 1994 1995 1996 1997 1998 1999 2000 2001 </li></ul><ul><li>Value 64 82 97 71 78 112 115 131 </li></ul>
5. 5. Merit <ul><li>Simplest </li></ul><ul><li>Non Mathematical- No calculations </li></ul><ul><li>Easiest method </li></ul><ul><li>Can Understand The characteristics of TS </li></ul><ul><li>Less Time </li></ul>
6. 6. Limitations <ul><li>Not definite </li></ul><ul><li>We can make different trend lines </li></ul><ul><li>Not suitable fro further analysis </li></ul>
7. 7. Semi -Average Method <ul><li>Divide the original data into two parts </li></ul><ul><li>When the number of years is even no problem. </li></ul><ul><li>If it is odd remove the middle value from the series </li></ul><ul><li>Figures from the two parts are averaged. </li></ul><ul><li>The average for each part is plotted on the graph against the middle years of their respective parts </li></ul><ul><li>The line obtained by joining the two points is the required trend line and may extended both ways to estimate the intermediate or future values </li></ul>
8. 8. Advantage and Disadvantage <ul><li>Those who applied this method will get the same results </li></ul><ul><li>But this method assumes linear relation ship between the plotted points which may not exist in practice </li></ul>
9. 9. Question 1 <ul><li>Fit a trend line to the following data by the method of semi averages and estimate the value for 2004 and 2007 </li></ul><ul><li>Year 91 92 93 94 95 96 97 98 99 00 </li></ul><ul><li>Sales 53 79 76 66 69 94 105 87 79 104 </li></ul><ul><li>(1000 units) </li></ul><ul><li>Year 01 02 03 </li></ul><ul><li>Sales 97 92 101 </li></ul>
10. 10. Home work 1 <ul><li>Apply the method of semi averages for determining the trend of the following data and estimate the values for 2000 </li></ul><ul><li>Years 1993 1994 1995 1996 1997 1998 </li></ul><ul><li>Sales 20 24 22 30 28 32 </li></ul><ul><li>(1000 units) </li></ul>
11. 11. Home work 2 <ul><li>From the following series of annual data ,find the trend line by the method of semi averages . Also estimate the value for 1999. </li></ul><ul><li>Year 90 91 92 93 94 95 96 97 98 </li></ul><ul><li>Actual 170 231 261 267 278 302 299 298 340 </li></ul><ul><li>value </li></ul>
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# If wave packets spread, why don't objects disappear?
If you have an electron moving in empty space, it will be represented by a wave packet. But packets can spread over time, that is, their width increases, with it's uncertainty in position increasing. Now, if I throw a basketball, why doesn't the basketball's packet spread as well? Wouldn't that cause its uncertainty in position to increase so much to the point it disappears?
EDIT: I realize I wasn't clear what I meant by disappear. Basically, suppose the wave packet is spread over the entire Solar System. Your field of vision covers only an extremely tiny part of the Solar System. Therefore, the probability that you will find the basketball that you threw in your field of vision is very small.
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Are you asking why macroscopic objects don't show quantum behaviour? If so, have a quick search of this site as that question has been asked many times. Try a search for "decoherence". – John Rennie Nov 10 '12 at 11:07
– Qmechanic Nov 10 '12 at 20:20
because the wavepackets also collapse. – Ron Maimon Nov 11 '12 at 4:17
It is true that the spreading depends on the mass as @twistor59 has already noticed, but the more important fact is that the basketball is an open system and interaction with its surrounds makes that (due to decoherence) the state of the basketball is not described by quantum wavefunction theory [*]. Using the Wigner-Moyal formulation of quantum mechanics it is possible to show that the basketball always have a well-define position $x(t)$ at each instant.
[*] Wavefunction theory only applies to isolated quantum systems.
-
What if throw the basketball in a perfect vacuum, completely empty of anything or any field? – Ignacio Nov 10 '12 at 17:50
@Ignacio: The basketball is not in a pure state. If you manage to put it in a perfect vacuum (which is impossible) it will remain in the same non-pure state because a perfect vacuum cannot change a state. – juanrga Nov 10 '12 at 18:12
@juanrga: So when you cut all "interaction with its surrounds", it remains in a mixed state? – Vladimir Kalitvianski Nov 10 '12 at 19:30
@VladimirKalitvianski: I cannot say. It depends on the system, the kind of mixed state, and the dynamics. – juanrga Nov 11 '12 at 12:55
@juanrga So from what I understand it's wrong to say that macroscopic objects exhibit quantum behaviour, but with effects that are too small to be detected. Instead, they really aren't quantum at all. Is this correct? – Ignacio Nov 11 '12 at 20:33
Spreading the wave packet does not mean spreading and disappearing the electron.
If a basketball has initially uncertainty $\Delta V$ of its velocity, then with time the ball position uncertainty will grow as $\Delta V\cdot t$.. With time this uncertainty gets so large that the ball disappears from your sight (I mean you will not find it where you expect it to be without knowing the initial velocity spread).
-
Short answer is it won't disappear because the integral of the probability density is still 1 even for a highly spread wavepacket, i.e. the object will still be found somewhere.
Slightly longer answer is that, if I start with a Gaussian wavepacket with width $a$, then after time $t$, the width will have spread to $$\sqrt{\frac{a^2+\hbar^2t^2/m^2}{a}}$$ The incredible smallness of $\hbar$ makes the spread negligble for something as massive as a basketball.
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Ok, but in the formula the spread can grow indefinitely, given enough time. – Ignacio Nov 10 '12 at 17:43
That's true, but if you put some numbers in ($\sqrt{a}=10^{-10}meter; m=0.1kg$, you can calculate how many times the age of the universe you have to wait for it to spread by a significant fraction. The $\hbar$ kills you. (However, @juanrga 's argument is probably stronger). – twistor59 Nov 10 '12 at 18:21
AFAIK there is no answer to this question, that is why there are few "theories" that try to answer this question like GRW
GRW
EDIT:
Let me just elaborate because I gave you the most interesting answer without directly addressing your question. There is no need for you to throw the ball. The atoms of the ball are constrained by their mutual potential so the waves do not spread in the sense of free particles. @Twistor59 answer is heuristic usually given in textbooks, but obviously there are no 100 g particles as such. The main issue is why we don’t see the ball at 100 meters or whatever since wavefunctions have non zero probability throughout space. The measurement problem is a close relation to your “proper “question but a bit different. GRW is more concerned with that.
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Questions regarding Gamma EMR
1. Jul 2, 2011
naurob03
I am very uneducated in physics, however I did take a remote sensing course in college where we discussed EMR. My questions are the following: could gamma radiation be absorbed in a material and then transferred to another material or dispersed. I know that gamma radiation has no charge and is only slightly affected by magnetic fields. I know that a super dense material like lead is routinely used to absorb gamma radiation. If gamma radiation has properties of no mass or electrical charge yet are EMR then there has to be a way to match the frequency of this pre energy. Could a field of ionized radiation matching the frequency of gamma radiation deflect a jet of gamma radiation (deflect energy with energy). I may be way off on this so please bare with me.
2. Jul 3, 2011
Staff: Mentor
Gamma radiation is plain old light but at a much much higher frequency and carries with it much greater momentum and energy. The only way to shield from it is to absorb it. This is accomplished with lead or other dense materials because dense materials occupy the least amount of space, which is always a sought after property since designing a reactor or something similar with 100 ft of Styrofoam is just silly. (Just using it as an absurd example.)
Just like normal light you cannot shine two beams of light and deflect each other with them. EMR doesn't readily interact.
3. Jul 3, 2011
mathman
The effectiveness of lead as a gamma ray shield is more than just the density. The absorption cross-section increases with atomic number.
4. Jul 3, 2011
Staff: Mentor
That as well. I believe it is because the extra charges help absorb the photons somehow?
5. Jul 4, 2011
Morbius
Drakkith,
EMR like gamma rays interact principally, with the ELECTRONS, not the nucleus for the most part.
The reason why the interaction cross-section goes up so strongly with Z, is that the greater the number of protons in the nucleus, the greater the number of electrons surrounding it.
Greg
6. Jul 4, 2011
mathman
There are two principal absorption processes for (nuclear) gamma rays, pair production and photoelectric effect.
Pair production involves the nucleus for conservation of momentum. The more massive the nucleus, the greater the cross-section. This applies only to gamma rays with energy > 1.022 Mev.
Photoelectric effect cross-section depends on how easily the electrons can be photo-ionized. Increasing the number of electrons per atom increases the chances per electron. This effect increases as the gamma ray energy decreases.
7. Jul 4, 2011
Staff: Mentor
Yeah that's what I was saying.
8. Jul 4, 2011
Morbius
mathman,
Although not directly an absorption process, another big interaction is Compton scatter.
The direction of the gamma is changed and it also loses energy in the process. Multiple Compton scatterings can deplete the energy of the gamma.
Greg
9. Jul 5, 2011
mathman
You're perfectly right. I just didn't bother mentioning it, since it is not absorption.
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# (Alg 1) Adding and Subtracting Rational Expressions in a PowerPoint
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Common Core Standards
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Algebra 1 Adding and Subtracting Rational Expressions in a PowerPoint Presentation
This slideshow lesson is very animated with a flow-through technique. I developed the lesson for my Algebra 1 class, but it can also be used for upper level class reviews. This lesson teaches how to add and subtract rational expressions that have like denominators, add and subtract rational expressions that have unlike denominators, and find the least common denominators of two rations expressions.
The presentation has 127 slides with LOTS of whiteboard practice. Use as many or as few of the problems to help your students learn each concept. For more PowerPoint lessons & materials visit Preston PowerPoints.
Students often get lost in multi-step math problems. This PowerPoint lesson is unique because it uses a flow-through technique, guided animation, that helps to eliminate confusion and guides the student through the problem. The lesson highlights each step of the problem as the teacher is discussing it, and then animates it to the next step within the lesson. Every step of every problem is shown, even the minor or seemingly insignificant steps. A helpful color-coding technique engages the students and guides them through the problem (Green is for the answer, red for wrong or canceled numbers, & blue, purple & sometimes orange for focusing the next step or separating things.) Twice as many examples are provided, compared to a standard textbook. All lessons have a real-world example to aid the students in visualizing a practical application of the concept.
This lesson applies to the Common Core Standard:
High School: Algebra » Seeing Structure in Expressions A.SSE.2
Interpret the structure of expressions.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
Please note that these PowerPoints are NOT EDITABLE. They WILL NOT work with Google Slides or Adobe Connect. You will need the PowerPoint software.
If you need an alternative version because your country uses different measurements, units, slight wording adjustment for language differences, or a slide reordering just ask.
Are you looking for the Algebra 1 Rational Equations and Functions Bundle? Click here!
This resource is for one teacher only. You may not upload this resource to the internet in any form. Additional teachers must purchase their own license. If you are a coach, principal or district interested in purchasing several licenses, please contact me for a district-wide quote at prestonpowerpoints@gmail.com. This product may not be uploaded to the internet in any form, including classroom/personal websites or network drives.
*This lesson contains 18 problems. Each problem in this lesson uses several pages in order to achieve the animated flow-through technique.
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55 minutes
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# Linear Equations in Two Variables Practice Set 1.3
Hi friends, welcome on Maths Master, here you will learn how to solve linear equations in two variables by using Determinant methods or Cramer’s Rule.
## Gabriel Cramer :
(31 July, 1704 to 4 January, 1752)
This Swiss mathematician was born in Geneva. He
was very well versed in mathematics, since childhood. At
the age of eighteen, he got a doctorate. He was a professor
in Geneva.
Using determinants, simultaneous equaions can be solved easily and in less space. This method is known as determinant method. This method was first given by a Swiss mathematician Gabriel Cramer, so it is also known as Cramer’s method.
To use Cramer’s method, the equations are written as a1x + b1y = c1 and a2x + b2y = c2.
To fine the value of x and y, first we have to find the value of D, Dx and Dy with the help of given below formula.
## Steps to solve linear equation in two variables by Cramer’s Rule:
• Write given equations in the form ax + by = c .
• Find the values of determinants D, Dx and Dy
• Using, x = Dx/D and y = Dy/D find values of x, y.
## Let’s see solved examples:
Solve the following simultaneous equations using Cramer’s Rule.
1. 5x + 3y = -11 ; 2x + 4y = -10
Sol.:
5x + 3y = -11 …………………………(1)
2x + 4y = -10 …………………………(2)
a1=5, b1=3, c1=-11
a2=2, b2=4, c2= -10
Now try to solve below examples, you can take hint from my video, link given below.
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akemezaug5
2023-03-06
How do I find the vertex of $f\left(x\right)={x}^{2}+6x+5$?
Hunter Hendricks
$\text{given the parabola in}\phantom{\rule{1ex}{0ex}}\text{standard form}$
$•xy=a{x}^{2}+bx+cx;a\ne 0$
$\text{then the x-coordinate of the vertex is}$
$•x{x}_{\text{vertex}}=-\frac{b}{2a}$
$f\left(x\right)={x}^{2}+6x+5\phantom{\rule{1ex}{0ex}}\text{is in standard form}$
$\text{with}\phantom{\rule{1ex}{0ex}}a=1,b=6\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}c=5$
${x}_{\text{vertex}}=-\frac{6}{2}=-3$
$\text{substitute this value into the equation for y}$
${y}_{\text{vertex}}={\left(-3\right)}^{2}+6\left(-3\right)+5=-4$
$⇒\text{vertex}\phantom{\rule{1ex}{0ex}}=\left(-3,-4\right)$
graph{x^2+6x+5 [-10, 10, -5, 5]}
Kiara Rollins
The given function:
$f\left(x\right)={x}^{2}+6x+5$
$y={x}^{2}+2\left(3\right)x+{3}^{2}-{3}^{2}+5$
$y={\left(x+3\right)}^{2}-4$
${\left(x+3\right)}^{2}=\left(y+4\right)$
The following equation is a vertical parabola in its standard form:
${\left(x-{x}_{1}\right)}^{2}=4a\left(y-{y}_{1}\right)$
Which has vertex at
$\left({x}_{1},{y}_{1}\right)\equiv \left(-3,-4\right)$
Do you have a similar question?
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# What is the largest possible length of a pole you could fit in a room?
• 846 views
Lets say you are in a cubical grain silo, with a height of 12m, and a base with dimensions 3m by 4m. If you wanted to fit a pole in the silo, what is the largest possible length of the pole?
The length we are trying to find is the diagonal from one corner on the base, to the opposite corner at the top of the silo.
This can be worked out using pythagoras (a2+b2=c2), as the height of the silo, the length we are trying to work out, and the diagonal between the opposite corners of the base form a right angled triangle.
This means the problem can be split up into to two steps.
Firstly the length of the diagonal on the base is the square root of (32+42) giving us 5m.
Then we can do the same method to find the length of the pole. The square root of (52+122) gives us 13m.
This means the largest possible pole you could fit in the silo is 13 meters.
Still stuck? Get one-to-one help from a personally interviewed subject specialist.
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# [R] simple simulation problem
Peter Dalgaard BSA p.dalgaard at biostat.ku.dk
Sun Mar 17 22:58:45 CET 2002
```Yannick Wurm <idh at poulet.org> writes:
> Help!
>
> I'm desperate! I don't understand why my function won't work!
> Its for a simple simulation exercise: robin hood fires an arrow at a
> target with 50cm radius, and his aim follows a centered normal
> distribution with a standard deviation of 40cm (on x and y axis,
> independently). Here he takes 10 shots and is awarded points if he
> hits close to the center.
>
> For some reason, this function always returns a score of 870?!?!
> I'm probably missing something really quite obvious but I just can't
> figure out what it is.
> if (40 < d[i] < 50) score_sum(score,2)
===================
That doesn't work (in R as well as most other programming languages).
It gets evaluated left to right as either TRUE < 50 or FALSE < 50,
both of which will be TRUE since logicals coerce to 0 or 1. You must
split it as (40 < d[i] && d[i] < 50).
(The whole thing can be coded much more elegantly with cut() and
indexing, but you'll probably learn soon enough.)
BTW: "_" as assignment is hard to read, and likely to get removed in
future versions of R so you might as well break the habit now.
--
O__ ---- Peter Dalgaard Blegdamsvej 3
c/ /'_ --- Dept. of Biostatistics 2200 Cph. N
(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk) FAX: (+45) 35327907
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html
Send "info", "help", or "[un]subscribe"
(in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch
_._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
```
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# How many grams in tablespoon uk?
Sharing is caring!
Grams and tablespoons for butter
Grams to tablespoons Tablespoons to grams
10 grams = 0.7 tbsp 1 tbsp = 14.2 grams
20 grams = 1.4 tbsp 2 tbsp = 28.4 grams
30 grams = 2.1 tbsp 3 tbsp = 42.6 grams
40 grams = 2.8 tbsp 4 tbsp = 56.8 grams
How many grams are in a tablespoon?
Dry Measure Equivalents
3 teaspoons 1 tablespoon 14.3 grams 2 tablespoons 1/8 cup 28.3 grams 4 tablespoons 1/4 cup 56.7 grams 5 1/3 tablespoons 1/3 cup 75.6 grams 8 tablespoons 1/2 cup 113.4 grams
How many grams is a teaspoon UK?
Teaspoons and grams for sugar (granulated)
Teaspoons to grams Teaspoons to grams
1 teaspoon = 4.2g 6 teaspoons = 25.2g
2 teaspoons = 8.4g 7 teaspoons = 29.3g
3 teaspoons = 12.6g 8 teaspoons = 33.5g
4 teaspoons = 16.7g 9 teaspoons = 37.7g
How much is a tablespoon of sugar in grams UK? The answer is: The change of 1 tbsp ( tablespoon ) unit in a granulated sugar measure equals = into 12.50 g ( gram ) as per the equivalent measure and for the same granulated sugar type.
How many tablespoons is 100g UK?
Grams and tablespoons for sugar (granulated)
Grams to tablespoons Tablespoons to grams
70 grams = 5.6 tbsp 7 tbsp = 87.5g
80 grams = 6.4 tbsp 8 tbsp = 100g
90 grams = 7.2 tbsp 9 tbsp = 112.5g
100 grams = 8 tbsp 10 tbsp = 125g
## How many grams in tablespoon uk? – Related Asked Question
### How many tbsp is 30g butter?
Tablespoon to Gram Conversion Table
Tablespoons Grams
29 tbsp 411.07 g
30 tbsp 425.24 g
31 tbsp 439.42 g
32 tbsp 453.59 g
### How do you convert grams into tablespoons?
Tablespoons to grams
1. tablespoon = 15 grams.
2. tablespoons = 30 grams.
3. tablespoons = 45 grams.
4. tablespoons = 60 grams.
5. tablespoons = 75 grams.
6. tablespoons = 90 grams.
7. tablespoons = 105 grams.
8. tablespoons = 120 grams.
### How many teaspoons is 15g UK?
Grams to teaspoons for baking powder
Grams to teaspoons Grams to teaspoons
3 grams = 0.68 tsp 13 grams = 2.93 tsp
4 grams = 0.9 tsp 14 grams = 3.15 tsp
5 grams = 1.13 tsp 15 grams = 3.38 tsp
6 grams = 1.35 tsp 16 grams = 3.6 tsp
### Can you convert grams to tablespoons?
Switch the ingredient and quickly find out how many grams are in a tablespoon of salt, Change the butter tablespoon to grams.
Here you go!
Ingredient 1 US tbsp 1 tbsp (15ml)
Water 14.8 g 15 g
Milk 15.2 g 15.5 g
Flour 8.9 g 9 g
Sugar 12.5 g 12.7 g
### How many tablespoons is 20g butter?
Gram to Tablespoon Conversion Table
Grams Tablespoons
18 g 1.2699 tbsp
19 g 1.3404 tbsp
20 g 1.411 tbsp
21 g 1.4815 tbsp
### How many tablespoons is 22g sugar?
Sugar Weight to Volume Conversion Table
Grams Tablespoons (Granulated) Tablespoons (Powdered)
15 g 1 1/4 tbsp 1 3/4 tbsp
20 g 1 2/3 tbsp 2 1/2 tbsp
25 g 2 tbsp 3 1/4 tbsp
30 g 2 1/3 tbsp 3 3/4 tbsp
### How many grams is a tbsp of butter?
Other Butter Conversions
Butter Measurement Grams Ounces
1/2 stick 56.7 2
2 sticks 226.8 8
1 tablespoon 14.2 .5
1 teaspoon 4.7 .16
### How many grams is 8 tbsp of butter?
Other Butter Conversions
Butter Measurement Grams Ounces
1/2 stick 56.7 2
2 sticks 226.8 8
1 tablespoon 14.2 .5
1 teaspoon 4.7 .16
### How much does a tablespoon of flour weigh UK?
Spoon Measures
Spoon Measures – solids ounce (uk) grams
A heaped tablespoon of flour = 1 oz. = 28g
A heaped tablespoon of butter = 1.25 oz. = 35g
A heaped tablespoon of cornflour = 0.5 oz. = 14g
A heaped tablespoon of gelatin = 0.75 oz. = 21g
### How many tablespoons is 50ml UK?
Milliliter to Tablespoon (UK) Conversion Table
Milliliter [mL] Tablespoon (UK)
20 mL 1.1262425513 tablespoon (UK)
50 mL 2.8156063782 tablespoon (UK)
100 mL 5.6312127565 tablespoon (UK)
1000 mL 56.3121275646 tablespoon (UK)
### How can I measure 30g of butter without scales?
Take a bigger liquid measuring cup and fill it with water equal to the amount of butter you need for the recipe. Add chunks of butter until the water doubles in volume. Be sure that the measuring cup is big enough to handle twice the volume that you wish to measure out.
### How many grams is 3 tablespoons of butter?
Converting butter from tablespoons to grams
Tablespoon Grams
3 tablespoons 60g
5 tablespoons 100g
10 tablespoons 200g
12 tablespoons 240g
### How much is 30 grams of flour in tablespoons?
Flour Weight to Volume Conversion Table
Grams Tablespoons (A.P. Flour) Tablespoons (Wheat Flour)
30 g 3 3/4 tbsp 4 tbsp
35 g 4 1/2 tbsp 4 2/3 tbsp
40 g 5 1/8 tbsp 5 1/3 tbsp
45 g 5 3/4 tbsp 6 tbsp
### How do you measure grams with a spoon?
1. as someone mentioned, a teaspoon is approximately 5 grams. …
2. 1/5 of a Teaspoon = a gram. …
3. The spoons are for counting volume, not weight. …
4. As pointed out, the weight to volume thing will work for somethings, not others, but with water, 1 tablespoon = 1/2 ounce = 14 grams.
### How many tbsp is 100 grams?
The answer is: The change of 1 100g ( – 100 grams portion ) unit in a butter measure equals = into 7.05 tbsp ( tablespoon ) as per the equivalent measure and for the same butter type.
### How many tablespoons is 300g of flour?
Wholemeal / brown flour
BROWN FLOUR – GRAMS TO CUPS
Grams Cups
250g 1½ cup + 2 tbsp
300g 1¾ cups + 3 tbsp
400g 2½ cups + 1 tbsp
### How many grams is a 1/2 teaspoon?
Baking Conversion Table
U.S. Metric
1/2 teaspoon 2.84 grams
1 teaspoon 5.69 grams
1/2 tablespoon 8.53 grams
1 tablespoon 17.07 grams
### What is 7.6 grams in teaspoons?
Gram to Teaspoon Conversion Table
Weight in Grams: Volume in Teaspoons of:
Water Milk
7 g 1.4202 tsp 1.3788 tsp
8 g 1.6231 tsp 1.5758 tsp
9 g 1.826 tsp 1.7728 tsp
### How many teaspoons is 25 grams UK?
Grams to UK teaspoons table
Grams UK teaspoons
22 g 6.19432 UK tsp
23 g 6.47588 UK tsp
24 g 6.75744 UK tsp
25 g 7.039 UK tsp
### How much grams is in a teaspoon?
To be precise, 4.2 grams equals a teaspoon, but the nutrition facts rounds this number down to four grams. Using this equation, you can easily look at any food product to see how much sugar it contains.
### How can I measure 100 grams without a scale?
Use Paper Clips as an Estimate. There are many household items that weigh a very specific, uniform amount, and using these items to compare to grams can help you measure out 100 grams without a scale. Regular paper clips, for example, all weigh almost exactly 1 gram each.
### What is 50g sugar in tablespoons?
Sugar Weight to Volume Conversion Table
Grams Tablespoons (Granulated) Tablespoons (Raw)
50 g 4 tbsp 3 1/4 tbsp
55 g 4 1/3 tbsp 3 1/2 tbsp
60 g 4 3/4 tbsp 3 3/4 tbsp
65 g 5 1/4 tbsp 4 1/8 tbsp
### How many teaspoons is 6g of butter?
Teaspoon to Gram Conversion Table
Teaspoons Grams
3 tsp 14.17 g
4 tsp 18.9 g
5 tsp 23.62 g
6 tsp 28.35 g
### How many tablespoons is 25g icing sugar?
Ingredient 1 cup 2 tbsp
Flour 120g 15g
Flour (sieved) 110g 13g
Sugar (granulated) 200g 25g
Icing Sugar 100g 13g
### How many grams is a spoon of sugar?
Thanks to the grams to teaspoons calculator and table you’ve discovered that a spoon of sugar weights around 4.2 grams. Wondering how many calories are in a teaspoon of sugar? It’s estimated that one teaspoon of sugar contains 16 calories, and one tablespoon contains 48 calories.
### What is 10g of sugar in teaspoons?
Sugar Weight to Volume Conversion Table
Grams Teaspoons (Granulated) Teaspoons (Brown)
10 g 2 1/3 tsp 2 1/3 tsp
15 g 3 2/3 tsp 3 2/3 tsp
20 g 4 3/4 tsp 4 3/4 tsp
25 g 6 tsp 6 tsp
### How many grams is 4.5 tablespoons?
Tablespoon to Grams Converter metric conversion table
Tablespoon to Grams Converter metric conversion table
0.04 tablespoon = 0.6 gram 0.4 tablespoon = 6 gram 4.4 tablespoon = 66 gram
0.05 tablespoon = 0.75 gram 0.5 tablespoon = 7.5 gram 4.5 tablespoon = 67.5 gram
### How much does 3.5 tbsp butter weigh?
Tablespoons Measurement Conversion Table
tablespoons sticks of butter ounces
5 tbsp 0.625 2.5 oz
6 tbsp 0.75 3 oz
7 tbsp 0.875 3.5 oz
8 tbsp 1 4 oz
### What is 2 tablespoons of flour in grams?
Flour Weight to Volume Conversion Table
Grams Tablespoons (A.P. Flour) Tablespoons (Wheat Flour)
15 g 1 3/4 tbsp 2 tbsp
20 g 2 1/2 tbsp 2 2/3 tbsp
25 g 3 1/4 tbsp 3 1/3 tbsp
30 g 3 3/4 tbsp 4 tbsp
### Is there a way to soften butter quickly?
Pour hot water into a ceramic or glass cup or bowl (something that can fit over your butter). After a few minutes, dump the water out of the vessel and quickly cover your butter. The heat from the cup will soften your butter in just a few minutes.
### How do I measure 6 tablespoons of butter?
Pour hot water into a ceramic or glass cup or bowl (something that can fit over your butter). After a few minutes, dump the water out of the vessel and quickly cover your butter. The heat from the cup will soften your butter in just a few minutes.
### How much is a block of butter?
Butter Conversion Chart (Volume/Weight/Sticks)
Butter by block size Butter by cups Quantity in sticks
1 whole block of butter 2 cups of butter 4 sticks of butter
1/2 block of butter 1 cup of butter 2 sticks of butter
1/4 block of butter ½ cup of butter 1 stick of butter
⅛ block of butter ¼ cup of butter ½ stick of butter
### What is the weight of 8 tablespoons of butter?
1 stick butter = 8 tablespoons = 1/2 cup = 4 ounces/110g.
### How many tablespoons is 100g self raising flour?
One – 100 grams portion of self raising flour (SRF) converted to tablespoon equals to 12.80 tbsp.
### How much does a tablespoon of all purpose flour weigh?
The answer is: The change of 1 tbsp ( tablespoon ) unit in a all purpose flour (APF) measure equals = into 7.81 g ( gram ) as per the equivalent measure and for the same all purpose flour (APF) type.
### How can I measure 50 grams of flour without scales?
HOW DO I MEASURE FLOUR WITHOUT A SCALE?
1. Use a spoon to fluff up the flour within the container.
2. Use a spoon to scoop the flour into the measuring cup.
3. Use a knife or other straight edged utensil to level the flour across the measuring cup.
### What is a UK tablespoon measurement?
The unit of measurement varies by region: a United States tablespoon is approximately 14.8 ml (0.50 US fl oz), a United Kingdom and Canadian tablespoon is exactly 15 ml (0.51 US fl oz), and an Australian tablespoon is 20 ml (0.68 US fl oz).
### How many tablespoons is 25ml UK?
UK Tablespoons to Milliliters table
UK Tablespoons Milliliters
22 UK tblsp 312.54
23 UK tblsp 326.75
24 UK tblsp 340.96
25 UK tblsp 355.16
### Is 15mL a teaspoon or tablespoon?
A tablespoon is three times bigger than a teaspoon and three teaspoons equal one tablespoon (1Tbsp or 1Tb). One tablespoon also equals 15mL. However, using a kitchen spoon to measure the dose of a medicine is NOT a good idea unless it has been properly calibrated and has been sold as a kitchen measure.
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# Decimal to binary algorithm
Is there a better way to code this specific algorithm using successively division through 2 which converts decimal numbers into binary numbers without calling a built-in function?
print "Please enter a decimal number."
prompt = '>> '
dec_number = input(prompt)
dec_new = dec_number
binstr = ''
'''dec2bin'''
while dec_new != 0:
binstr += str(dec_new % 2)
dec_new = dec_new // 2
print binstr[::-1]
The best solution, as far as I read, would be the algorithm Divide by 2 that uses a stack to keep track of the digits for the binary result.
As an intro, this algorithm assumes that we start with an integer greater than 0. A simple iteration then continually divides the decimal number by 2 and keeps track of the remainder.
The first division by 2 gives information as to whether the value is even or odd. An even value will have a remainder of 0. It will have the digit 0 in the ones place. An odd value will have a remainder of 1 and will have the digit 1 in the ones place. We think about building our binary number as a sequence of digits; the first remainder we compute will actually be the last digit in the sequence.
That said, we have:
from pythonds.basic.stack import Stack
def divideBy2(decNumber):
remstack = Stack()
while decNumber > 0:
rem = decNumber % 2
remstack.push(rem)
decNumber = decNumber // 2
binString = ""
while not remstack.isEmpty():
binString = binString + str(remstack.pop())
return binString
print(divideBy2(42))
1. Why did you do this: dec_new = dec_number ? You could've just use dec_number. There's no need of assigning its value to another variable.
2. This: dec_new = dec_new // 2 could be as well as the above line of your code rewritten as: dec_new //= 2
3. The indentation should contain 4 spaces, not 2.
A version that better (than your solution) utilizes the memory would be:
def dec_to_bin(n):
bits = []
bits.append(str(0 if n%2 == 0 else 1))
while n > 1:
n = n // 2
bits.append(str(0 if n%2 == 0 else 1))
bits.reverse()
return ''.join(bits)
What I did:
• floor divide all the numbers by two repeatedly until we reach 1
• going in reverse order, create bits of this array of numbers, if it is even, append a 0 and if it is odd append a 1.
Other ways of doing it:
### Using recursion:
def dec_to_bin(x):
return dec_to_bin(x/2) + [x%2] if x > 1 else [x]
The above solution returns the result as a list which you can later on .join() then apply int() to it.
Another idea that came to my mind is as simple as:
u = format(62, "08b")
>> 00111110
• Thanks! This is what I have been looking for. I'll try your samples out and see how far I can get though these codes contain a lot of new things I still need to learn. Jul 7, 2016 at 13:02
• I strongly recommend you to start step by step. There's no hurry. You can start with this which is absolutely great. Good luck. And if you have questions, just ask. But don't forget, this is only for code reviewing. You can read our help section for more information Jul 7, 2016 at 13:04
• Now that you have mentioned it, this line really seems to be redundant. The second comment looks interesting, didn't know that this notation also applied to the division operator. Besides that I'll try to use 4 spaces from now on. And thank you for providing the link. Looks like it covers a lot. Jul 7, 2016 at 13:13
• Both quotient and reminder can be obtained at the same time using q, r = divmod(233,2) Aug 27, 2018 at 16:19
• n%2 can be written as n & 1 which should be faster, same with n >> 1 instead of n // 2 Feb 12, 2019 at 7:26
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NumWords.com
# How to write Three thousand one hundred seventy-four in numbers in English?
We can write Three thousand one hundred seventy-four equal to 3174 in numbers in English
< Three thousand one hundred seventy-three :||: Three thousand one hundred seventy-five >
Six thousand three hundred forty-eight = 6348 = 3174 × 2
Nine thousand five hundred twenty-two = 9522 = 3174 × 3
Twelve thousand six hundred ninety-six = 12696 = 3174 × 4
Fifteen thousand eight hundred seventy = 15870 = 3174 × 5
Nineteen thousand forty-four = 19044 = 3174 × 6
Twenty-two thousand two hundred eighteen = 22218 = 3174 × 7
Twenty-five thousand three hundred ninety-two = 25392 = 3174 × 8
Twenty-eight thousand five hundred sixty-six = 28566 = 3174 × 9
Thirty-one thousand seven hundred forty = 31740 = 3174 × 10
Thirty-four thousand nine hundred fourteen = 34914 = 3174 × 11
Thirty-eight thousand eighty-eight = 38088 = 3174 × 12
Forty-one thousand two hundred sixty-two = 41262 = 3174 × 13
Forty-four thousand four hundred thirty-six = 44436 = 3174 × 14
Forty-seven thousand six hundred ten = 47610 = 3174 × 15
Fifty thousand seven hundred eighty-four = 50784 = 3174 × 16
Fifty-three thousand nine hundred fifty-eight = 53958 = 3174 × 17
Fifty-seven thousand one hundred thirty-two = 57132 = 3174 × 18
Sixty thousand three hundred six = 60306 = 3174 × 19
Sixty-three thousand four hundred eighty = 63480 = 3174 × 20
Sitemap
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# Sum of the series 1^1 + 2^2 + 3^3 + ….. + n^n using recursion
• Difficulty Level : Expert
• Last Updated : 17 Mar, 2021
Given an integer n, the task is to find the sum of the series 11 + 22 + 33 + ….. + nn using recursion.
Examples:
Input: n = 2
Output:
11 + 22 = 1 + 4 = 5
Input: n = 3
Output: 32
11 + 22 + 33 = 1 + 4 + 27 = 32
Approach: Starting from n, start adding all the terms of the series one by one with the value of n getting decremented by 1 in each recursive call until the value of n = 1 for which return 1 as 11 = 1.
Below is the implementation of the above approach:
## C++
`// C++ implementation of the approach``#include ``using` `namespace` `std;``#define ll long long int` `// Recursive function to return``// the sum of the given series``ll sum(``int` `n)``{` ` ``// 1^1 = 1`` ``if` `(n == 1)`` ``return` `1;`` ``else` ` ``// Recursive call`` ``return` `((ll)``pow``(n, n) + sum(n - 1));``}` `// Driver code``int` `main()``{`` ``int` `n = 2;`` ``cout << sum(n);` ` ``return` `0;``}`
## Java
`// Java implementation of the approach``class` `GFG {` ` ``// Recursive function to return`` ``// the sum of the given series`` ``static` `long` `sum(``int` `n)`` ``{` ` ``// 1^1 = 1`` ``if` `(n == ``1``)`` ``return` `1``;`` ``else` ` ``// Recursive call`` ``return` `((``long``)Math.pow(n, n) + sum(n - ``1``));`` ``}` ` ``// Driver code`` ``public` `static` `void` `main(String args[])`` ``{`` ``int` `n = ``2``;`` ``System.out.println(sum(n));`` ``}``}`
## Python3
`# Python3 implementation of the approach` `# Recursive function to return``# the sum of the given series``def` `sum``(n):`` ``if` `n ``=``=` `1``:`` ``return` `1`` ``else``:` ` ``# Recursive call`` ``return` `pow``(n, n) ``+` `sum``(n ``-` `1``)` `# Driver code``n ``=` `2``print``(``sum``(n))` `# This code is contributed``# by Shrikant13`
## C#
`// C# implementation of the approach``using` `System;``class` `GFG {` ` ``// Recursive function to return`` ``// the sum of the given series`` ``static` `long` `sum(``int` `n)`` ``{`` ``// 1^1 = 1`` ``if` `(n == 1)`` ``return` `1;`` ``else` ` ``// Recursive call`` ``return` `((``long``)Math.Pow(n, n) + sum(n - 1));`` ``}` ` ``// Driver code`` ``public` `static` `void` `Main()`` ``{`` ``int` `n = 2;`` ``Console.Write(sum(n));`` ``}``}`
## PHP
``
## Javascript
``
Output:
`5`
My Personal Notes arrow_drop_up
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# How far is Mesa, AZ, from Atlanta, GA?
The distance between Atlanta (Hartsfield–Jackson Atlanta International Airport) and Mesa (Phoenix–Mesa Gateway Airport) is 1568 miles / 2524 kilometers / 1363 nautical miles.
The driving distance from Atlanta (ATL) to Mesa (AZA) is 1803 miles / 2901 kilometers, and travel time by car is about 32 hours 25 minutes.
1568
Miles
2524
Kilometers
1363
Nautical miles
3 h 28 min
184 kg
## Distance from Atlanta to Mesa
There are several ways to calculate the distance from Atlanta to Mesa. Here are two standard methods:
Vincenty's formula (applied above)
• 1568.156 miles
• 2523.702 kilometers
• 1362.690 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth's surface using an ellipsoidal model of the planet.
Haversine formula
• 1564.810 miles
• 2518.317 kilometers
• 1359.782 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## How long does it take to fly from Atlanta to Mesa?
The estimated flight time from Hartsfield–Jackson Atlanta International Airport to Phoenix–Mesa Gateway Airport is 3 hours and 28 minutes.
## Flight carbon footprint between Hartsfield–Jackson Atlanta International Airport (ATL) and Phoenix–Mesa Gateway Airport (AZA)
On average, flying from Atlanta to Mesa generates about 184 kg of CO2 per passenger, and 184 kilograms equals 405 pounds (lbs). The figures are estimates and include only the CO2 generated by burning jet fuel.
## Map of flight path and driving directions from Atlanta to Mesa
See the map of the shortest flight path between Hartsfield–Jackson Atlanta International Airport (ATL) and Phoenix–Mesa Gateway Airport (AZA).
## Airport information
Origin Hartsfield–Jackson Atlanta International Airport
City: Atlanta, GA
Country: United States
IATA Code: ATL
ICAO Code: KATL
Coordinates: 33°38′12″N, 84°25′41″W
Destination Phoenix–Mesa Gateway Airport
City: Mesa, AZ
Country: United States
IATA Code: AZA
ICAO Code: KIWA
Coordinates: 33°18′28″N, 111°39′17″W
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Welcome! On this website you can find information about each number.
# Number 258875048
## Number 258875048 basic info
Number 258875048 has 9 digits. Number 258875048 can be formatted as 258,875,048 or 258.875.048 or 258 875 048 or in case this was a phone number 258-875-048 or 25-887-5048 to be easier to read. Number 258875048 in English words is "two hundred and fifty-eight million, eight hundred and seventy-five thousand and fourty-eight". Number 258875048 can be read by triplets (groups of 3 digits) as "two hundred and fifty-eight, eight hundred and seventy-five, zero fourty-eight". Number 258875048 can be read digit by digit as "two five eight eight seven five zero four eight". Number 258875048 is even. Number 258875048 is divisible by: two, four, eight. Number 258875048 is a composite number (non-prime number).
## Number 258875048 conversions
Number 258875048 in binary code is 1111011011100001111010101000. Number 258875048 in octal code is: 1733417250. Number 258875048 in hexadecimal (hexa): f6e1ea8.
The sum of all digits of this number is 47. The digital root (repeated digital sum until you get single-digit number) is 2. Number 258875048 divided by two (halved) equals 129437524. Number 258875048 multiplied by two (doubled) equals 517750096. Number 258875048 multiplied by ten equals 2588750480. Number 258875048 raised to the power of 2 equals 6.7016290477002E+16. Number 258875048 raised to the power of 3 equals 1.7348845414016E+25. The square root (sqrt) of 258875048 is 16089.594401351. The sine (sin) of 258875048 degree is 0.78801075401682. The cosine (cos) of 258875048 degree is -0.61566147480076. The base-10 logarithm of 258875048 equals 8.4130901924274. The natural logarithm of 258875048 equals 19.371856063098. The number 258875048 can be encoded to characters as BEHHGEJDH. The number 258875048 can be encrypted to chemical element names as helium, boron, oxygen, oxygen, nitrogen, boron, neon, beryllium, oxygen.
## Numbers simmilar to 258875048
Numbers simmilar to number 258875048 (one digit altered): 158875048358875048248875048268875048257875048259875048258775048258975048258865048258885048258874048258876048258875148258875038258875058258875047258875049
Possible variations of 258875048 with a digit pair swapped: 528875048285875048258785048258857048258870548258875408258875084
Number 258875048 typographic errors with one digit missing: 588750482887504825875048258750482588504825887048258875482588750825887504
Number 258875048 typographic errors with one digit doubled: 225887504825588750482588875048258887504825887750482588755048258875004825887504482588750488
Previous number: 258875047
Next number: 258875049
## Several randomly selected numbers:
665967897928017998169430769292123515233928391776452480248866302483607673151877002338758948393278783961614196947943846476491572473878833879253086579514654893660671770388220196111904168578598313456778278185612887065943480591292066025024084676717262726131056781542786589151882698552841017899336813448580379896233607348856223817366458586797227787127976560606247586377756695481916264843.
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# How to append in NumPy array
You can use the append function in the NumPy array with the following code. If you want to learn Python, I highly recommend reading This Book.
## Example 1: How to append two NumPy arrays
```import numpy as np
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
c = np.append(a, b)
print(c)```
`[1 2 3 4 5 6]`
## Example 2: How to append an element to the NumPy array
```import numpy as np
a = np.array([1, 2, 3])
b = np.append(a, 6)
print(b)```
`[1 2 3 6]`
## Example 3: How to append column in NumPy array
```import numpy as np
a = np.array([[1, 2],[3, 4],[5,6]])
b = [[7],[8],[9]]
c = np.append(a, b, axis=1)
print(c)```
```[[1 2 7]
[3 4 8]
[5 6 9]]```
## Example 4: How to append row in NumPy array
```import numpy as np
a = np.array([[1, 2, 3],[4, 5, 6]])
b = [[7, 8, 9]]
c = np.append(a, b, axis=0)
print(c)```
```[[1 2 3]
[4 5 6]
[7 8 9]]```
Best Python Books
What is Computer Vision? Examples, Applications, Techniques
Books for Machine Learning (ML)
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# 6.2: Systems of Stratification
• Anonymous
• LibreTexts
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##### Learning Objectives
• Explain the difference between open and closed societies.
• Define the several systems of stratification.
• Understand how Max Weber and Karl Marx differed in their view of class societies.
When we look around the world and through history, we see different types of stratification systems. These systems vary on their degree of vertical mobility, or the chances of rising up or falling down the stratification ladder. In some so-called closed societies, an individual has virtually no chance of moving up or down. Open societies have more vertical mobility, as some people, and perhaps many people, can move up or even down. That said, a key question is how much vertical mobility really exists in these societies. Let’s look at several systems of stratification, moving from the most closed to the most open.
## Slavery
The most closed system is slavery, or the ownership of people, which has been quite common in human history (Ennals, 2007).Ennals, R. (2007). From slavery to citizenship. Hoboken, NJ: John Wiley. Slavery is thought to have begun 10,000 years ago, after agricultural societies developed, as people in these societies made prisoners of war work on their farms. Many of the ancient lands of the Middle East, including Babylonia, Egypt, and Persia, also owned slaves, as did ancient China and India. Slavery especially flourished in ancient Greece and Rome, which used thousands of slaves for their trade economies. Most slaves in ancient times were prisoners of war or debtors. As trade died down during the Middle Ages, so did slavery.
But once Europeans began exploring the Western Hemisphere in the 1500s, slavery regained its popularity. Portuguese and Spanish colonists who settled in Brazil and Caribbean islands made slaves of thousands of Indians already living there. After most of them died from disease and abuse, the Portuguese and Spaniards began bringing slaves from Africa. In the next century, the English, the French, and other Europeans also began bringing African slaves into the Western Hemisphere, and by the 1800s they had captured and shipped to the New World some 10–12 million Africans, almost 2 million of whom died along the way (Thornton, 1998).Thornton, J. K. (1998). Africa and Africans in the making of the Atlantic world, 1400–1800 (2nd ed.). Cambridge, England: Cambridge University Press.
The United States, of course, is all too familiar with slavery, which remains perhaps the most deplorable experience in American history and continues to have repercussions for African Americans and the rest of American society. It increasingly divided the new nation after it won its independence from Britain and helped lead to the Civil War eight decades later. The cruel treatment of slaves was captured in Harriet Beecher Stowe’s classic but controversial book Uncle Tom’s Cabin, which ignited passions on both sides of the slavery debate.
Today slavery still exists in parts of Africa, Asia, and South America, with some estimates putting the number of slaves in the tens of millions. Today’s slaves include (a) men first taken as prisoners of war in ethnic conflicts; (b) girls and women captured in wartime or kidnapped from their neighborhoods and used as prostitutes or sex slaves; (c) children sold by their parents to become child laborers; and (d) workers paying off debts who are abused and even tortured and too terrified to leave (Bales, 2007; Batstone, 2007).Bales, K. (2007). Ending slavery: How we free today’s slaves. Berkeley: University of California Press; Batstone, D. (2007). Not for sale: The return of the global slave trade—and how we can fight it. New York, NY: HarperOne.
## Estate Systems
Estate systems are characterized by control of land and were common in Europe and Asia during the Middle Ages and into the 1800s. In these systems, two major estates existed: the landed gentry or nobility and the peasantry or serfs. The landed gentry owned huge expanses of land on which serfs toiled. The serfs had more freedom than slaves had but typically lived in poverty and were subject to arbitrary control by the nobility (Kerbo, 2009).Kerbo, H. R. (2009). Social stratification and inequality. New York, NY: McGraw-Hill.
Estate systems thrived in Europe until the French Revolution in 1789 violently overturned the existing order and inspired people in other nations with its cries for freedom and equality. As time went on, European estate systems slowly gave way to class systems of stratification (discussed a little later). After the American colonies won their independence from Britain, the South had at least one characteristic of an estate system, the control of large plots of land by a relatively few wealthy individuals and their families, but it obviously used slaves rather than serfs to work the land.
Much of Asia, especially China and Japan, also had estate systems. For centuries, China’s large population lived as peasants in abject conditions and frequently engaged in peasant uprisings. These escalated starting in the 1850s after the Chinese government raised taxes and charged peasants higher rents for the land on which they worked. After many more decades of political and economic strife, Communists took control of China in 1949 (DeFronzo, 2007).DeFronzo, J. (2007). Revolutions and revolutionary movements (3rd ed.). Boulder, CO: Westview Press.
## Caste Systems
In a caste system, people are born into unequal groups based on their parents’ status and remain in these groups for the rest of their lives. For many years, the best-known caste system was in India, where, supported by Hindu beliefs emphasizing the acceptance of one’s fate in life, several major castes dictated one’s life chances from the moment of birth, especially in rural areas (Kerbo, 2009).Kerbo, H. R. (2009). Social stratification and inequality. New York, NY: McGraw-Hill. People born in the lower castes lived in abject poverty throughout their lives. Another caste, the harijan, or untouchables, was considered so low that technically it was not thought to be a caste at all. People in this caste were called the untouchables because they were considered unclean and were prohibited from coming near to people in the higher castes. Traditionally, caste membership in India almost totally determined an individual’s life, including what job you had and whom you married; for example, it was almost impossible to marry someone in another caste. After India won its independence from Britain in 1949, its new constitution granted equal rights to the untouchables. Modern communication and migration into cities further weakened the caste system, as members of different castes now had more contact with each other. Still, caste prejudice remains a problem in India and illustrates the continuing influence of its traditional system of social stratification.
A country that used to have a caste system is South Africa. In the days of apartheid, from 1950 to 1990, a small group of white Afrikaners ruled the country. Black people constituted more than three-quarters of the nation’s population and thus greatly outnumbered Afrikaners, but they had the worst jobs, could not vote, and lived in poor, segregated neighborhoods. Afrikaners bolstered their rule with the aid of the South African police, which used terror tactics to intimidate blacks (Berger, 2009).Berger, I. (2009). South Africa in world history. New York, NY: Oxford University Press.
Many observers believe a caste system also existed in the South in the United States after Reconstruction and until the civil rights movement of the 1960s ended legal segregation. A segregated system called Jim Crow dominated the South, and even though African Americans had several rights, including the right to vote, granted to them by the 13th, 14th, and 15th Amendments to the Constitution, these rights were denied in practice. Lynchings were common for many decades, and the Southern police system bolstered white rule in the South just as the South African police system bolstered white rule in that country (Litwack, 2009).Litwack, L. F. (2009). How free is free? The long death of Jim Crow. Cambridge, MA: Harvard University Press.
## Class Systems
Many societies, including all industrial ones, have class systems. In this system of stratification, a person is born into a social ranking but can move up or down from it much more easily than in caste systems or slave societies. This movement in either direction is primarily the result of a person’s own effort, knowledge, and skills or lack of them. Although these qualities do not aid upward movement in caste or slave societies, they often do enable upward movement in class societies. Of the three systems of stratification discussed so far, class systems are by far the most open, meaning they have the most vertical mobility. We will look later at social class in the United States and discuss the extent of vertical mobility in American society.
Sociologist Max Weber, whose work on organizations and bureaucracies was discussed in Chapter 4, also had much to say about class systems of stratification. Such systems, he wrote, are based on three dimensions of stratification: class (which we will call wealth), power, and prestige. Wealth is the total value of an individual or family, including income, stocks, bonds, real estate, and other assets; power is the ability to influence others to do your bidding, even if they do not want to; and prestige refers to the status and esteem people hold in the eyes of others.
In discussing these three dimensions, Weber disagreed somewhat with Karl Marx, who, as you might recall from Chapter 1, said our ranking in society depends on whether we own the means of production. Marx thus felt that the primary dimension of stratification in class systems was economic. Weber readily acknowledged the importance of this economic dimension but thought power and prestige also matter. He further said that although wealth, power, and prestige usually go hand-in-hand, they do not always overlap. For example, although the head of a major corporation has a good deal of wealth, power, and prestige, we can think of many other people who are high on one dimension but not on the other two. A professional athlete who makes millions of dollars a year has little power in the political sense that Weber meant it. An organized crime leader might also be very wealthy but have little prestige outside of the criminal underworld. Conversely, a scientist or professor may enjoy much prestige but not be very wealthy.
## Classless Societies
Although, as noted earlier, all societies except perhaps for the simplest ones are stratified, some large nations have done their best to eliminate stratification by developing classless societies. Marx, of course, predicted that one day the proletariat would rise up and overthrow the bourgeoisie and create a communist society, by which he meant a classless one in which everyone had roughly the same amount of wealth, power, and prestige. In Russia, China, and Cuba, revolutions inspired by Marx’s vision occurred in the 20th century. These revolutions resulted in societies not only with less economic inequality than in the United States and other class systems but also with little or no political freedom. Moreover, governing elites in these societies enjoyed much more wealth, power, and prestige than the average citizen. Overall, the communist experiments in Russia, China, and Cuba failed to achieve Marx’s vision of an egalitarian society.
Some Western European nations, such as Sweden and Denmark, have developed “social democracies” based on fairly socialist economies. Although a few have nominal monarchies, these nations have much political freedom and less economic inequality than the United States and other class societies. They also typically rank much higher than the United States on various social and economic indicators. Although these nations are not truly classless, they indicate it is possible, if not easy, to have a society that begins to fulfill Marx’s egalitarian vision but where political freedom still prevails (Sandbrook, Edelman, Heller, & Teichman, 2007).Sandbrook, R., Edelman, M., Heller, P., & Teichman, J. (2007). Social democracy in the global periphery: Origins, challenges, prospects. New York, NY: Cambridge University Press.
## Conclusion
• Systems of stratification vary in their degree of vertical social mobility. Some societies are more open in this regard, while some are more closed.
• The major systems of stratification are slavery, estate systems, caste systems, and class systems.
• Some Western European nations are not classless but still have much less economic inequality than class societies such as the United States.
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statsmodels vs sklearn
R^2 est sur de 0,41 pour les deux sklearn et statsmodels (c'est bon pour les sciences sociales). First, we define the set of dependent(y) and independent(X) variables. Accordée, je suis en utilisant le 5-plis cv pour le sklearn approche (R^2 sont compatibles pour les deux test et de formation données à chaque fois), et pour statsmodels je viens de jeter toutes les données. Saya mencoba memahami mengapa output dari regresi logistik kedua perpustakaan ini memberikan hasil yang berbeda. Where statsmodels.api seems very similar to the summary function in R, that gives you the p-value, R^2 and all of this … 이를 알아내는 데 대한 힌트는 scikit-learn 추정치로부터 얻은 모수 추정치가 statsmodels 대응 치보다 균일하게 작다는 것입니다. discrete. discrete_model as sm # read in the data & create matrices df = pd. statsmodels vs sklearn for the linear models. linear_model import LogisticRegression import statsmodels. The code for the experiment is available in the accompanying Github repository under time_tests.py, while the experiment is carried out in sklearn_statsmodels_time_comp.ipynb. from sklearn. It features various classification, regression and clustering algorithms including support vector machines, random forests, gradient boosting, k-means and DBSCAN, and is … To run cross-validation on multiple metrics and also to return train scores, fit times and score times. read_csv ('loan.csv') df. Alternatively, the estimator LassoLarsIC proposes to use the Akaike information criterion (AIC) and the Bayes Information criterion (BIC). discrete. Saya menggunakan dataset dari tutorial idre UCLA , memprediksi admitberdasarkan gre, gpadan rank. Regresi Logistik: Scikit Learn vs Statsmodels. You will learn how to perform a linear regression. linear_model import LogisticRegression import statsmodels. 1．ライブラリ 1.1 Scikit-learnの回帰分析 sklearn.linear_model.LinearRegression(fit_intercept=True, normalize=False, … Discussion. discrete_model as sm # read in the data & create matrices df = pd. Es fácil y claro cómo realizarlo. ... # module imports from patsy import dmatrices import pandas as pd from sklearn. Partial Regression Plots 4．まとめ. linear_models import LogisticRegression as LR logr = LR logr. I have been using both of the packages for the past few months and here is my view. 31 . 1.1.3.1.2. You will gain confidence when working with 2 of the leading ML packages - statsmodels and sklearn. Regressione logistica: Scikit Learn vs Statsmodels. Hello, I'm new to Python (and ML). Home All Products All Videos Data Machine Learning 101 with Scikit-learn and StatsModels [Video] Machine Learning 101 with Scikit-learn and StatsModels [Video] By 365 Careers Ltd. FREE Subscribe Start Free Trial; $36.80 Was$183.99 Video Buy Instant online access to over 7,500+ books and videos ... StatsModels and sklearn… Versión corta : estaba usando scikit LinearRegression en algunos datos, pero estoy acostumbrado a los valores de p, así que ponga los datos en los modelos de estadísticas OLS, y aunque el R ^ 2 es aproximadamente el mismo, los coeficientes variables son todos diferentes por … In this post, … In the end, both languages produce very similar plots. 31 . It will give you all … Scikit-learn vs. StatsModels: Which, why, and how? (1 reply) Hi, all of the internet discussions on statsmodels vs sklearn are from 2013 or before. _get_numeric_data #drop non-numeric cols df. For my part, pandas is kind of a heavy package and I spent a lot of my first few years in Python writing statistical models from scratch for clients who didn't want to install anything more than numpy -- so I'm partial to sklearn… discrete. ... # module imports from patsy import dmatrices import pandas as pd from sklearn. 31 . Scikit-learn (formerly scikits.learn and also known as sklearn) is a free software machine learning library for the Python programming language. While the X variable comes first in SKLearn, y comes first in statsmodels.An easy way to check your dependent variable (your y variable), is right in the model.summary (). Regarding the difference sklearn vs. scikit-learn: The package "scikit-learn" is recommended to be installed using pip install scikit-learn but in your code imported using import sklearn.A bit … #Imports import pandas as pd import numpy as np from patsy import dmatrices import statsmodels.api as sm from statsmodels.stats.outliers_influence import variance_inflation_factor df = pd. statsmodels.tsa.arima_model.ARIMAResults.plot_predict¶ ARIMAResults.plot_predict (start = None, end = None, exog = None, dynamic = False, alpha = 0.05, plot_insample = True, ax = None) [source] ¶ Plot forecasts. linear_model import LogisticRegression import statsmodels. At The Data Incubator, we pride ourselves on having the most up to date data science curriculum available. Regarding the difference sklearn vs.scikit-learn: The package "scikit-learn" is recommended to be installed using pip install scikit-learn but in your code imported using import sklearn..A bit confusing, because you can also do pip install sklearn and will end up with the same scikit-learn package installed, because there is a "dummy" pypi package sklearn … コード・実験 2.1 データ準備 2.2 Sklearnの回帰分析 2.3 Statsmodelsの回帰分析 2.4 結果の説明 3. Sto cercando di capire perché l'output della regressione logistica di queste due librerie dia risultati diversi. Unlike SKLearn, statsmodels doesn’t automatically fit a constant, so you need to use the method sm.add_constant (X) in order to add a … Sto usando il set di dati da UCLA Idre esercitazione, … For my purposes, it looks the statsmodels discrete choice model logit is the way to go. 31 . sklearn.metrics.make_scorer. # module imports from patsy import dmatrices import pandas as pd from sklearn. Excel has a way of removing the charm from OLS modeling; students often assume there’s a scatterplot, some magic math that … ... # module imports from patsy import dmatrices import pandas as pd from sklearn. fit (X, Y ) results = logr. Get predictions from each split of cross-validation for diagnostic purposes. head id member_id loan_amnt … Statsmodels vs sklearn logistic regression. The statsmodels logit method and scikit-learn method are comparable.. Take-aways. ロジスティック回帰：Scikit Learn vs Statsmodels. Régression logistique: Scikit Learn vs Statsmodels. It’s significantly faster than the GLM method, presumably because it’s using an optimizer directly rather than … Make a scorer … Zero-indexed observation number at which to start forecasting, ie., … sklearn.model_selection.cross_val_predict. I just finished the topic involving the linear models. sklearn.model_selection.cross_validate. Learning to Think Like a Data Scientist: Alumni Spotlight on Ceena Modarres. linear_model import LogisticRegression import statsmodels. I use a couple of books and video tutorials to complement learning and I noticed that some of them use statsmodels to work with regressions and some sklearn. Confidently work with two of the leading ML packages: statsmodels and sklearn ; Understand how to perform a linear regression ; Become familiar with the ins and outs of logistic regression ; Get to grips with carrying out cluster analysis (both flat and hierarchical) Apply your skills to real-life business cases At Metis, one of the first machine learning models I teach is the Plain Jane Ordinary Least Squares (OLS) model that most everyone learns in high school. This specification is used, whether or not the model is fit using conditional sum of square or maximum-likelihood, using the method argument in statsmodels… #Importing the libraries from nsepy import get_history as gh import datetime as dt from matplotlib import pyplot as plt from sklearn import model_selection from sklearn.metrics import confusion_matrix from sklearn.preprocessing import StandardScaler from sklearn.model_selection import train_test_split import numpy … Sklearn y Pandas son más activos que los Statsmodels. Logistic Regression: Scikit Learn vs Statsmodels, Your clue to figuring this out should be that the parameter estimates from the scikit-learn estimation are uniformly smaller in magnitude than the statsmodels Two popular options are scikit-learn and StatsModels. Python linear regression sklearn linear model vs statsmodels.api. ... glmnet tiene una función de coste ligeramente diferente en comparación con sklearn, pero incluso si fijo alpha=0en glmnet(es decir, sólo utilice L2-penal) y el conjunto 1/(N*lambda)=C, todavía no consigo el mismo resultado? Statsmodels is a Python module which provides various functions for estimating different statistical models and performing statistical tests. Try to implement linear regression, and saw two approaches, using sklearn linear model or using statsmodels.api. But in the code, we can see how the R data science ecosystem has many smaller packages (GGally is a helper package for ggplot2, the most-used R plotting package), and more visualization packages in general.In Python, matplotlib is the primary plotting … Scikit-Learn is not made for hardcore statistics. Lets begin with the advantages of statsmodels over scikit-learn. Visualizations Linear Regression in Scikit-learn vs Statsmodels, Your clue to figuring this out should be that the parameter estimates from the scikit-learn estimation are uniformly smaller in magnitude than the statsmodels See the SO threads Coefficients for Logistic Regression scikit-learn vs statsmodels … dropna df = df. You will become familiar with the ins and outs of a logistic regression. Much of. La elección clara es Sklearn. 31 . ロジスティック回帰を実行する場合、 statsmodels が正しい（いくつかの教材で検証されている）。 ただし、 sklearn 。 データを前処理できませんでした。これは私の … Regresión logística: Scikit Learn vs Statsmodels. Information-criteria based model selection¶. Regresión OLS: Scikit vs. Statsmodels? where $$\phi$$ and $$\theta$$ are polynomials in the lag operator, $$L$$.This is the regression model with ARMA errors, or ARMAX model. You will excel at carrying out cluster analysis (both flat and hierarchical) 1.2 Statsmodelsの回帰分析 2. Is there a universally preferred way? WLS, OLS’ Neglected Cousin. # Import packages import pandas as pd import patsy import statsmodels.api as sm import statsmodels.formula.api as smf import statsmodels.api as sm from statsmodels.stats.outliers_influence import variance_inflation_factor from sklearn.preprocessing import StandardScaler, PolynomialFeatures from sklearn… Parameters start int, str, or datetime. Statsmodels vs sklearn logistic regression. discrete. It is a computationally cheaper alternative to find the optimal value of alpha as the regularization path is computed only once instead of … If the dependent variable is in non-numeric form, it is first converted to numeric using dummies. statsmodels GLM is the slowest by far! , why, and saw two approaches, using sklearn linear model vs.! & create matrices df = pd ini memberikan hasil yang berbeda also to return train scores, fit times score! 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Criterion ( BIC ) are comparable.. Take-aways logistica di queste due dia... … Python linear regression, and saw two approaches, using sklearn linear model or using statsmodels.api over... And ML ) y ) results = logr and outs of a logistic regression della... Di queste due librerie dia risultati diversi scikit-learn method are comparable.. Take-aways X ) variables import pandas pd... # module imports from patsy import dmatrices import pandas as pd from sklearn statsmodels over scikit-learn first, we the... R^2 est sur de 0,41 pour les sciences sociales ) past few months and here my! Tutorial idre UCLA, memprediksi admitberdasarkan gre, gpadan rank 結果の説明 3 or using.... Saya mencoba memahami mengapa output dari regresi logistik kedua perpustakaan ini memberikan hasil yang berbeda how... The set of dependent ( y ) and independent ( X, y ) independent! # read in the data & create matrices df = pd new to Python ( and )... Ie., … WLS, OLS ’ Neglected Cousin and score times independent... Date data science curriculum available.. Take-aways statsmodels discrete choice model logit is the way to go just finished topic! 대한 힌트는 scikit-learn 추정치로부터 얻은 모수 추정치가 statsmodels 대응 치보다 균일하게 작다는.. Logisticregression as LR logr and here is my view end, both languages produce very similar plots using optimizer... Pd from sklearn I 'm new to Python ( and ML ),... データを前処理できませんでした。これは私の … in the end, both languages produce very similar plots to. = logr perpustakaan ini memberikan hasil yang berbeda 대한 힌트는 scikit-learn 추정치로부터 얻은 모수 추정치가 statsmodels 대응 치보다 균일하게 것입니다... Regression, and saw two approaches, using sklearn linear model or using statsmodels.api statsmodels 대응 치보다 균일하게 작다는.. Pandas as pd from sklearn della regressione logistica di queste due librerie dia risultati diversi split of cross-validation for purposes... I just finished the topic involving the linear models method and scikit-learn method are comparable Take-aways... Significantly faster than the GLM method, presumably because it ’ s using an optimizer directly rather than ….... Data science curriculum available 1．ライブラリ 1.1 Scikit-learnの回帰分析 sklearn.linear_model.LinearRegression ( fit_intercept=True, normalize=False …... Create matrices df = pd … Regresión OLS: Scikit vs. statsmodels: which, why, and saw approaches. Implement linear regression sklearn linear model or using statsmodels.api science curriculum available ロジスティック回帰を実行する場合、 statsmodels が正しい（いくつかの教材で検証されている）。 ただし、 。.
posted: Afrika 2013
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# Student's t-test
Last updated
The t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis.
## Contents
A t-test is the most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistics (under certain conditions) follow a Student's t distribution. The t-test can be used, for example, to determine if the means of two sets of data are significantly different from each other.
## History
The term "t-statistic" is abbreviated from "hypothesis test statistic". [1] In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert [2] [3] [4] and Lüroth. [5] [6] [7] The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. [8] However, the T-Distribution, also known as Student's t-distribution, gets its name from William Sealy Gosset who first published it in English in 1908 in the scientific journal Biometrika using his pseudonym "Student" [9] [10] because his employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. [11] Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples – for example, the chemical properties of barley with small sample sizes. Hence a second version of the etymology of the term Student is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material (see Student's t-distribution for a detailed history of this pseudonym, which is not to be confused with the literal term student ). Although it was William Gosset after whom the term "Student" is penned, it was actually through the work of Ronald Fisher that the distribution became well known as "Student's distribution" [12] and "Student's t-test".
Gosset had been hired owing to Claude Guinness's policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness's industrial processes. [13] Gosset devised the t-test as an economical way to monitor the quality of stout. The t-test work was submitted to and accepted in the journal Biometrika and published in 1908. [14]
Guinness had a policy of allowing technical staff leave for study (so-called "study leave"), which Gosset used during the first two terms of the 1906–1907 academic year in Professor Karl Pearson's Biometric Laboratory at University College London. [15] Gosset's identity was then known to fellow statisticians and to editor-in-chief Karl Pearson. [16]
## Uses
Among the most frequently used t-tests are:
• A one-sample location test of whether the mean of a population has a value specified in a null hypothesis.
• A two-sample location test of the null hypothesis such that the means of two populations are equal. All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t-test. These tests are often referred to as unpaired or independent samplest-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping. [17]
## Assumptions
Most test statistics have the form t = Z/s, where Z and s are functions of the data.
Z may be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas s is a scaling parameter that allows the distribution of t to be determined.
As an example, in the one-sample t-test
${\displaystyle t={\frac {Z}{s}}={\frac {{\bar {X}}-\mu }{{\widehat {\sigma }}/{\sqrt {n}}}}}$
where X is the sample mean from a sample X1, X2, …, Xn, of size n, s is the standard error of the mean, ${\textstyle {\widehat {\sigma }}}$ is the estimate of the standard deviation of the population, and μ is the population mean.
The assumptions underlying a t-test in the simplest form above are that:
• X follows a normal distribution with mean μ and variance σ2/n
• s2(n 1)/σ2 follows a χ2 distribution with n 1 degrees of freedom. This assumption is met when the observations used for estimating s2 come from a normal distribution (and i.i.d for each group).
• Z and s are independent.
In the t-test comparing the means of two independent samples, the following assumptions should be met:
• The means of the two populations being compared should follow normal distributions. Under weak assumptions, this follows in large samples from the central limit theorem, even when the distribution of observations in each group is non-normal. [18]
• If using Student's original definition of the t-test, the two populations being compared should have the same variance (testable using F-test, Levene's test, Bartlett's test, or the Brown–Forsythe test; or assessable graphically using a Q–Q plot). If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances. [19] Welch's t-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.
• The data used to carry out the test should either be sampled independently from the two populations being compared or be fully paired. This is in general not testable from the data, but if the data are known to be dependent (e.g. paired by test design), a dependent test has to be applied. For partially paired data, the classical independent t-tests may give invalid results as the test statistic might not follow a t distribution, while the dependent t-test is sub-optimal as it discards the unpaired data. [20]
Most two-sample t-tests are robust to all but large deviations from the assumptions. [21]
For exactness, the t-test and Z-test require normality of the sample means, and the t-test additionally requires that the sample variance follows a scaled χ2 distribution, and that the sample mean and sample variance be statistically independent. Normality of the individual data values is not required if these conditions are met. By the central limit theorem, sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. For non-normal data, the distribution of the sample variance may deviate substantially from a χ2 distribution. However, if the sample size is large, Slutsky's theorem implies that the distribution of the sample variance has little effect on the distribution of the test statistic.
## Unpaired and paired two-sample t-tests
Two-sample t-tests for a difference in mean involve independent samples (unpaired samples) or paired samples. Paired t-tests are a form of blocking, and have greater power (probability of avoiding a type II error, also known as a false negative) than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared. [22] In a different context, paired t-tests can be used to reduce the effects of confounding factors in an observational study.
### Independent (unpaired) samples
The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, and one variable from each of the two populations is compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomly assign 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the t-test.
### Paired samples
Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" t-test).
A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure-lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random interpatient variation has now been eliminated. However, an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice. Because half of the sample now depends on the other half, the paired version of Student's t-test has only n/2 − 1 degrees of freedom (with n being the total number of observations). Pairs become individual test units, and the sample has to be doubled to achieve the same number of degrees of freedom. Normally, there are n − 1 degrees of freedom (with n being the total number of observations). [23]
A paired samples t-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest. [24] The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.
Paired samples t-tests are often referred to as "dependent samples t-tests".
## Calculations
Explicit expressions that can be used to carry out various t-tests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a t-distribution under the null hypothesis is given. Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a one-tailed or two-tailed test.
Once the t value and degrees of freedom are determined, a p-value can be found using a table of values from Student's t-distribution. If the calculated p-value is below the threshold chosen for statistical significance (usually the 0.10, the 0.05, or 0.01 level), then the null hypothesis is rejected in favor of the alternative hypothesis.
### One-sample t-test
In testing the null hypothesis that the sample mean is equal to a specified value μ0, one uses the statistic
${\displaystyle t={\frac {{\bar {x}}-\mu _{0}}{s/{\sqrt {n}}}}}$
where ${\displaystyle {\bar {x}}}$ is the sample mean, s is the sample standard deviation and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means ${\displaystyle {\bar {x}}}$ is assumed to be normal.
By the central limit theorem, if the observations are independent and the second moment exists, then ${\displaystyle t}$ will be approximately normal N(0;1).
### Slope of a regression line
Suppose one is fitting the model
${\displaystyle Y=\alpha +\beta x+\varepsilon }$
where x is known, α and β are unknown, ε is a normally distributed random variable with mean 0 and unknown variance σ2, and Y is the outcome of interest. We want to test the null hypothesis that the slope β is equal to some specified value β0 (often taken to be 0, in which case the null hypothesis is that x and y are uncorrelated).
Let
{\displaystyle {\begin{aligned}{\widehat {\alpha }},{\widehat {\beta }}&={\text{least-squares estimators}},\\SE_{\widehat {\alpha }},SE_{\widehat {\beta }}&={\text{the standard errors of least-squares estimators}}.\end{aligned}}}
Then
${\displaystyle t_{\text{score}}={\frac {{\widehat {\beta }}-\beta _{0}}{SE_{\widehat {\beta }}}}\sim {\mathcal {T}}_{n-2}}$
has a t-distribution with n − 2 degrees of freedom if the null hypothesis is true. The standard error of the slope coefficient:
${\displaystyle SE_{\widehat {\beta }}={\frac {\sqrt {{\dfrac {1}{n-2}}\displaystyle \sum _{i=1}^{n}\left(y_{i}-{\widehat {y}}_{i}\right)^{2}}}{\sqrt {\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}}}$
can be written in terms of the residuals. Let
{\displaystyle {\begin{aligned}{\widehat {\varepsilon }}_{i}&=y_{i}-{\widehat {y}}_{i}=y_{i}-\left({\widehat {\alpha }}+{\widehat {\beta }}x_{i}\right)={\text{residuals}}={\text{estimated errors}},\\{\text{SSR}}&=\sum _{i=1}^{n}{{\widehat {\varepsilon }}_{i}}^{2}={\text{sum of squares of residuals}}.\end{aligned}}}
Then tscore is given by:
${\displaystyle t_{\text{score}}={\frac {\left({\widehat {\beta }}-\beta _{0}\right){\sqrt {n-2}}}{\sqrt {\frac {SSR}{\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}}}.}$
Another way to determine the tscore is:
${\displaystyle t_{\text{score}}={\frac {r{\sqrt {n-2}}}{\sqrt {1-r^{2}}}},}$
where r is the Pearson correlation coefficient.
The tscore, intercept can be determined from the tscore, slope:
${\displaystyle t_{\text{score,intercept}}={\frac {\alpha }{\beta }}{\frac {t_{\text{score,slope}}}{\sqrt {s_{\text{x}}^{2}+{\bar {x}}^{2}}}}}$
where sx2 is the sample variance.
### Independent two-sample t-test
#### Equal sample sizes and variance
Given two groups (1, 2), this test is only applicable when:
• the two sample sizes (that is, the number n of participants of each group) are equal;
• it can be assumed that the two distributions have the same variance;
Violations of these assumptions are discussed below.
The t statistic to test whether the means are different can be calculated as follows:
${\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{p}{\sqrt {\frac {2}{n}}}}}}$
where
${\displaystyle s_{p}={\sqrt {\frac {s_{X_{1}}^{2}+s_{X_{2}}^{2}}{2}}}.}$
Here sp is the pooled standard deviation for n = n1 = n2 and s 2
X1
and s 2
X2
are the unbiased estimators of the variances of the two samples. The denominator of t is the standard error of the difference between two means.
For significance testing, the degrees of freedom for this test is 2n − 2 where n is the number of participants in each group.
#### Equal or unequal sample sizes, similar variances (1/2<sX1/sX2< 2)
This test is used only when it can be assumed that the two distributions have the same variance. (When this assumption is violated, see below.) The previous formulae are a special case of the formulae below, one recovers them when both samples are equal in size: n = n1 = n2.
The t statistic to test whether the means are different can be calculated as follows:
${\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{p}\cdot {\sqrt {{\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}}}}}}$
where
${\displaystyle s_{p}={\sqrt {\frac {\left(n_{1}-1\right)s_{X_{1}}^{2}+\left(n_{2}-1\right)s_{X_{2}}^{2}}{n_{1}+n_{2}-2}}}}$
is an estimator of the pooled standard deviation of the two samples: it is defined in this way so that its square is an unbiased estimator of the common variance whether or not the population means are the same. In these formulae, ni − 1 is the number of degrees of freedom for each group, and the total sample size minus two (that is, n1 + n2 − 2) is the total number of degrees of freedom, which is used in significance testing.
#### Equal or unequal sample sizes, unequal variances (sX1> 2sX2 or sX2> 2sX1)
This test, also known as Welch's t-test, is used only when the two population variances are not assumed to be equal (the two sample sizes may or may not be equal) and hence must be estimated separately. The t statistic to test whether the population means are different is calculated as:
${\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{\bar {\Delta }}}}}$
where
${\displaystyle s_{\bar {\Delta }}={\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}.}$
Here si2 is the unbiased estimator of the variance of each of the two samples with ni = number of participants in group i (i = 1 or 2). In this case ${\textstyle (s_{\bar {\Delta }})^{2}}$ is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as an ordinary Student's t-distribution with the degrees of freedom calculated using
${\displaystyle \mathrm {d.f.} ={\frac {\left({\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}\right)^{2}}{{\frac {\left(s_{1}^{2}/n_{1}\right)^{2}}{n_{1}-1}}+{\frac {\left(s_{2}^{2}/n_{2}\right)^{2}}{n_{2}-1}}}}.}$
This is known as the Welch–Satterthwaite equation. The true distribution of the test statistic actually depends (slightly) on the two unknown population variances (see Behrens–Fisher problem).
### Dependent t-test for paired samples
This test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". This is an example of a paired difference test. The t statistic is calculated as
${\displaystyle t={\frac {{\bar {X}}_{D}-\mu _{0}}{s_{D}/{\sqrt {n}}}}}$
where ${\displaystyle {\bar {X}}_{D}}$ and ${\displaystyle s_{D}}$ are the average and standard deviation of the differences between all pairs. The pairs are e.g. either one person's pre-test and post-test scores or between-pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The constant μ0 is zero if we want to test whether the average of the difference is significantly different. The degree of freedom used is n − 1, where n represents the number of pairs.
Example of repeated measures
NumberNameTest 1Test 2
1Mike35%67%
2Melanie50%46%
3Melissa90%86%
4Mitchell78%91%
Example of matched pairs
PairNameAgeTest
1John35250
1Jane36340
2Jimmy22460
2Jessy21200
## Worked examples
Let A1 denote a set obtained by drawing a random sample of six measurements:
${\displaystyle A_{1}=\{30.02,\ 29.99,\ 30.11,\ 29.97,\ 30.01,\ 29.99\}}$
and let A2 denote a second set obtained similarly:
${\displaystyle A_{2}=\{29.89,\ 29.93,\ 29.72,\ 29.98,\ 30.02,\ 29.98\}}$
These could be, for example, the weights of screws that were chosen out of a bucket.
We will carry out tests of the null hypothesis that the means of the populations from which the two samples were taken are equal.
The difference between the two sample means, each denoted by Xi, which appears in the numerator for all the two-sample testing approaches discussed above, is
${\displaystyle {\bar {X}}_{1}-{\bar {X}}_{2}=0.095.}$
The sample standard deviations for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances would not be very powerful. Since the sample sizes are equal, the two forms of the two-sample t-test will perform similarly in this example.
### Unequal variances
If the approach for unequal variances (discussed above) is followed, the results are
${\displaystyle {\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}\approx 0.04849}$
and the degrees of freedom
${\displaystyle {\text{d.f.}}\approx 7.031.}$
The test statistic is approximately 1.959, which gives a two-tailed test p-value of 0.09077.
### Equal variances
If the approach for equal variances (discussed above) is followed, the results are
${\displaystyle s_{p}\approx 0.08396}$
and the degrees of freedom
${\displaystyle {\text{d.f.}}=10.}$
The test statistic is approximately equal to 1.959, which gives a two-tailed p-value of 0.07857.
### Alternatives to the t-test for location problems
The t-test provides an exact test for the equality of the means of two i.i.d. normal populations with unknown, but equal, variances. (Welch's t-test is a nearly exact test for the case where the data are normal but the variances may differ.) For moderately large samples and a one tailed test, the t-test is relatively robust to moderate violations of the normality assumption. [25] In large enough samples, the t-test asymptotically approaches the z-test, and becomes robust even to large deviations from normality. [18]
If the data are substantially non-normal and the sample size is small, the t-test can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of non-normal distributions.
When the normality assumption does not hold, a non-parametric alternative to the t-test may have better statistical power. However, when data are non-normal with differing variances between groups, a t-test may have better type-1 error control than some non-parametric alternatives. [26] Furthermore, non-parametric methods, such as the Mann-Whitney U test discussed below, typically do not test for a difference of means, so should be used carefully if a difference of means is of primary scientific interest. [18] For example, Mann-Whitney U test will keep the type 1 error at the desired level alpha if both groups have the same distribution. It will also have power in detecting an alternative by which group B has the same distribution as A but after some shift by a constant (in which case there would indeed be a difference in the means of the two groups). However, there could be cases where group A and B will have different distributions but with the same means (such as two distributions, one with positive skewness and the other with a negative one, but shifted so to have the same means). In such cases, MW could have more than alpha level power in rejecting the Null hypothesis but attributing the interpretation of difference in means to such a result would be incorrect.
In the presence of an outlier, the t-test is not robust. For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are skewed) or the distributions have large tails, then the Wilcoxon rank-sum test (also known as the Mann–Whitney U test) can have three to four times higher power than the t-test. [25] [27] [28] The nonparametric counterpart to the paired samples t-test is the Wilcoxon signed-rank test for paired samples. For a discussion on choosing between the t-test and nonparametric alternatives, see Lumley, et al. (2002). [18]
One-way analysis of variance (ANOVA) generalizes the two-sample t-test when the data belong to more than two groups.
### A design which includes both paired observations and independent observations
When both paired observations and independent observations are present in the two sample design, assuming data are missing completely at random (MCAR), the paired observations or independent observations may be discarded in order to proceed with the standard tests above. Alternatively making use of all of the available data, assuming normality and MCAR, the generalized partially overlapping samples t-test could be used. [29]
### Multivariate testing
A generalization of Student's t statistic, called Hotelling's t-squared statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample. For instance, a researcher might submit a number of subjects to a personality test consisting of multiple personality scales (e.g. the Minnesota Multiphasic Personality Inventory). Because measures of this type are usually positively correlated, it is not advisable to conduct separate univariate t-tests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis (Type I error). In this case a single multivariate test is preferable for hypothesis testing. Fisher's Method for combining multiple tests with alpha reduced for positive correlation among tests is one. Another is Hotelling's T2 statistic follows a T2 distribution. However, in practice the distribution is rarely used, since tabulated values for T2 are hard to find. Usually, T2 is converted instead to an F statistic.
For a one-sample multivariate test, the hypothesis is that the mean vector (μ) is equal to a given vector (μ0). The test statistic is Hotelling's t2:
${\displaystyle t^{2}=n({\bar {\mathbf {x} }}-{{\boldsymbol {\mu }}_{0}})'{\mathbf {S} }^{-1}({\bar {\mathbf {x} }}-{{\boldsymbol {\mu }}_{0}})}$
where n is the sample size, x is the vector of column means and S is an m × m sample covariance matrix.
For a two-sample multivariate test, the hypothesis is that the mean vectors (μ1, μ2) of two samples are equal. The test statistic is Hotelling's two-sample t2:
${\displaystyle t^{2}={\frac {n_{1}n_{2}}{n_{1}+n_{2}}}\left({\bar {\mathbf {x} }}_{1}-{\bar {\mathbf {x} }}_{2}\right)'{\mathbf {S} _{\text{pooled}}}^{-1}\left({\bar {\mathbf {x} }}_{1}-{\bar {\mathbf {x} }}_{2}\right).}$
## Software implementations
Many spreadsheet programs and statistics packages, such as QtiPlot, LibreOffice Calc, Microsoft Excel, SAS, SPSS, Stata, DAP, gretl, R, Python, PSPP, MATLAB and Minitab, include implementations of Student's t-test.
Language/ProgramFunctionNotes
Microsoft Excel pre 2010TTEST(array1, array2, tails, type)See
Microsoft Excel 2010 and laterT.TEST(array1, array2, tails, type)See
LibreOffice Calc TTEST(Data1; Data2; Mode; Type)See
Google Sheets TTEST(range1, range2, tails, type)See
Python scipy.stats.ttest_ind(a, b, equal_var=True)See
MATLAB ttest(data1, data2)See
Mathematica TTest[{data1,data2}]See
R t.test(data1, data2, var.equal=TRUE)See
SAS PROC TTESTSee
Java tTest(sample1, sample2)See
Julia EqualVarianceTTest(sample1, sample2)See
Stata ttest data1 == data2See
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In probability theory and statistics, the normal-gamma distribution is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.
In statistics, D'Agostino's K2 test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to establish whether or not the given sample comes from a normally distributed population. The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic.
Tukey's range test, also known as Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSDtest, is a single-step multiple comparison procedure and statistical test. It can be used to find means that are significantly different from each other.
In statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's t-test. The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis. It is very similar to the Z-score but with the difference that t-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the t-statistic is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown. It is also used along with p-value when running hypothesis tests where the p-value tells us what the odds are of the results to have happened.
In statistics, a paired difference test is a type of location test that is used when comparing two sets of measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to increase the statistical power, or to reduce the effects of confounders.
The Newman–Keuls or Student–Newman–Keuls (SNK)method is a stepwise multiple comparisons procedure used to identify sample means that are significantly different from each other. It was named after Student (1927), D. Newman, and M. Keuls. This procedure is often used as a post-hoc test whenever a significant difference between three or more sample means has been revealed by an analysis of variance (ANOVA). The Newman–Keuls method is similar to Tukey's range test as both procedures use studentized range statistics. Unlike Tukey's range test, the Newman–Keuls method uses different critical values for different pairs of mean comparisons. Thus, the procedure is more likely to reveal significant differences between group means and to commit type I errors by incorrectly rejecting a null hypothesis when it is true. In other words, the Neuman-Keuls procedure is more powerful but less conservative than Tukey's range test.
In statistics, almost sure hypothesis testing or a.s. hypothesis testing utilizes almost sure convergence in order to determine the validity of a statistical hypothesis with probability one. This is to say that whenever the null hypothesis is true, then an a.s. hypothesis test will fail to reject the null hypothesis w.p. 1 for all sufficiently large samples. Similarly, whenever the alternative hypothesis is true, then an a.s. hypothesis test will reject the null hypothesis with probability one, for all sufficiently large samples. Along similar lines, an a.s. confidence interval eventually contains the parameter of interest with probability 1. Dembo and Peres (1994) proved the existence of almost sure hypothesis tests.
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# What Is Gross Salary?
How do you calculate gross salary? There are two major methods. Pre-tax deductions are deducted from gross pay before taxes are calculated. Post-tax deductions are subtracted from net pay after taxes have been applied. Examples of post-tax deductions include charity donations and union fees. Gross pay is divided into pay periods, or hours. The time periods from which an employee is paid vary. Some employers pay their employees monthly, while others pay them weekly or bi-weekly.
## Calculating gross pay
Regardless of the type of business you’re in, calculating gross pay is a necessary part of budgeting. It’s important to know the exact amount of pay you’ll receive, as it will determine your take-home pay. The amount of employer withholdings and deductions may be complex, so it’s best to agree on the total gross pay before starting the job. You can agree on this in your employment contract, pay letter, or union agreement. This way, both you and your employer will know exactly what to expect.
Gross pay is the amount of wages that hourly and salaried employees receive, before deductions and taxes. It’s calculated using an agreed-upon hourly rate that the employer and employee have discussed during the hiring process. To calculate gross pay, you’ll need to multiply an employee’s total hours worked by the hourly rate. Make sure to include any overtime and other pay you receive. For example, if an employee works 12 hours a week, they’ll be paid \$24,000 per month.
## Taxes are deducted from gross pay
A worker’s gross pay is the amount they make before all deductions, including taxes. Generally, they can expect to earn about \$800 a week. Once deductions for state and federal income taxes are deducted, they’ll end up with a net pay. This figure is important for determining what kind of mortgage you can afford. Here are some tips for calculating net pay. Keeping the two figures separated will make it easier to understand the numbers.
Gross pay refers to the amount of wages earned before payroll deductions. Net pay, on the other hand, is the amount that remains after payroll deductions. It is important to understand how the two figures are related, because gross pay is often used to discuss compensation with employees and is usually based on the federal income tax brackets. Once you understand the difference, you’ll be better equipped to negotiate a good salary.
## Other deductions are subtracted from gross pay
Other deductions are amounts that are not taxable. These include healthcare, pension plans, dividends, interest, and rental income. Gross pay is the amount that is before taxes are deducted from it. For example, teacher Mary makes \$40,000 per year. Her total pay is \$40,000. If you’re wondering how much of her salary is tax-deductible, you should look at her pay statement to determine how much she pays in taxes.
When calculating gross pay, you need to consider the various deductions that affect it. For example, Medicare, Social Security, and FICA taxes are deducted from gross pay. These amounts, along with a few other mandatory deductions, are a part of the employer’s tax obligations. The remaining amount must go to the IRS. Additionally, there may be wage garnishments for debts such as unpaid student loans, child support, or credit card loans. Finally, there may be voluntary deductions for health insurance premiums, retirement contributions, and donations to charity.
## Hourly gross pay
Gross pay is a key part of compensation. In many cases, this is the starting point for calculating your Social Security maximum and additional Medicare taxes. You should be sure to understand the full range of your hourly pay before beginning your job. If you have no idea what your hourly pay is, you should always check it with your employer before you begin working. If it is not listed on your employment contract, you may have to adjust your rate to be compliant with federal tax laws.
Hourly gross pay refers to the total amount that employees earn for every hour worked in a pay period. This amount is calculated according to the rate contracted with the employer. Generally, this rate is determined in crew deal memos. There are several factors involved in hourly pay, however, that you should know about. Here are some things to consider when figuring out your hourly rate:
## Salary gross pay
What is salary gross pay? Salary is the total amount you’re paid, without deductions, per workweek. This amount doesn’t include any paid time off, overtime, or tips. It’s a simple calculation that is based on your annual salary and the number of pay periods in a year. It may not be the same for every employee. For example, if Rachel earns \$50,000 per year, her gross pay would be \$3,780 per week. If you’re earning \$17,000 per week, you’d be paid \$3,725.
Salaries are calculated using the gross pay. It’s higher than net pay, because the employer deducts fees for pension plans, union dues, and equipment and supplies for the work place. These deductions are usually voluntary, and you don’t have the option to opt out of them. However, if they’re not, you’ll be liable for them. So, gross pay is a better representation of your income than net pay.
## Net pay
Understanding the difference between gross salaries and net pay can help you better understand the differences between the two. While both pay amounts are important, gross pay is the higher amount on an employee’s paycheck. Net pay is the amount earned after deductions are made. If the employee is paid for work in a fixed amount per month, it would be his gross pay divided by that number. As a result, a fixed hourly wage of \$50 per week would mean \$2500 in gross pay.
The difference between net pay and gross salaries is very similar to that between net pay and fishing. While gross pay is larger, net pay will not catch all the fish in a pond. In many ways, net pay will influence how you spend your money. Knowing which number is higher will help you prepare for financial decisions and determine a budget. When you have a solid grasp of the difference between net pay and gross pay, you’ll be able to budget better and decide on what expenses to put toward saving.
In conclusion, gross salary is the total amount of money earned by an employee in a given period of time. This includes all forms of income, such as wages, tips, bonuses, and commissions. It is important to understand the gross salary in order to calculate tax deductions and other payments. To learn more about gross salary and how it impacts your finances, please visit the IRS website or speak with a financial advisor.
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# Friction between two bodies (theoretical situation)
1. Sep 11, 2008
### limorgav
What happens if we apply on B a force bigger than the static friction with the floor and also bigger than the static friction with the upper body A, but smaller than their sum? Will B move? what about A?
#### Attached Files:
• ###### yuval.GIF
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Views:
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2. Sep 11, 2008
### HallsofIvy
You are applying the force to B? Then, since the force is greater than the static friction between B and the floor, B will start to move. Once it has started to move, the problem reverts to "kinetic friction". The net force on B will be the total force minus the kinetic friction force. If that is larger than the static friction between A and B, then A will slip backwards. Static friction is not, in general the same as kinetic friction so just knowing that the force is greater than the sum of the two static frictions is not enough. It is true that, for most materials, the kinetic friction is less than the static friction. In that case, if force> static friction + static friction> kinetic friction + static friction, yes, A will move.
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# How to calculate dates.(Microsoft Excel)
06/23/2024
Japanese version.
Here are some formulas to calculate the date of the next day, the day before, etc.
## Steps
### Formulas
#### Next day (N days later), previous day (N days before)
To calculate future dates, add days to the date. (+2 means 2 days later)
To calculate a date in the past, subtract days from the date. (-2 means 2 days before)
#### Next month (after N months), previous month (before N months)
Use the YEAR, MONTH, DAY, and DATE functions.
``=DATE(YEAR(datetime),MONTH(datetime) + Number of months ,DAY(datetime))``
Adding the number of months results in N months later, and subtracting the number of months earlier results in N months earlier.
Normal calculation is performed even if the year-end and New Year holidays are crossed.
The EDATE function may be more convenient for months with different number of days.
#### Next year (N years later), Previous year (N years before)
Similar to the month calculation, but adds/subtracts to the result of the YEAR function.
``=DATE(YEAR(datetime) + Number of years ,MONTH(datetime),DAY(datetime))``
#### Weekday
How to display the day of the week automatically from the date.
#### First day of the month
Specifying 1 in the date portion of the DATE function sets the first day of the month.
``=DATE(YEAR(datetime),MONTH(datetime),1)``
#### End of month (last day of the month)
The first day of the following month is given and one day before the first day is calculated.
``=DATE(YEAR(datetime),MONTH(datetime)+1,1)-1``
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# Optimization with three variables
1. Dec 6, 2011
### velocityviper
1. The problem statement, all variables and given/known data
You are a lab technician and must create 250 ml of a 17% solution. You have availible three stock solutions. You have a one liter container of a 5% salt, a 500 l contained of a 28% salt solution, and a 400 ml container of a 40% salt solution. Show the work necessary to calculate the cheapest ethod of preparing the 17% salt solution ifthe 5% solution costs $28 per liter, the 28% soltution costs$38 per liter, and the 40% solution costs \$50 per liter.
So far I have this
2. Relavant equations
X+Y+Z=250
.05X+.28Y+.4Z=.17(250)
C(X,Y,Z)=(1/1000)(28X+38Y+50Z)
3. The attempt at a solution
I think I need a fourth equation so I can find the dervative and solve for 1 of the variables to optimize the solution,or I need to set derivative of the third equation = to 0, but I wouldn't know what to do from there.
2. Dec 6, 2011
### Ray Vickson
What about upper and lower bounds on X, Y and Z? (Can you use a negative amount of 5% solution? Can you use more than 1 L of 5% solution?)
Forget about setting derivatives to zero. In problems like this, the optimal solution occurs at a boundary, where some derivatives are non-zero. Your problem is called a LINEAR PROGRAMMING problem, and is not solvable by calculus methods. See, eg., http://www.purplemath.com/modules/linprog5.htm .
RGV
3. Dec 6, 2011
### velocityviper
Are any derivatives needed at all? Or is it just using systems of equations to slove for one of the variables and then using that to solve for the others?
4. Dec 6, 2011
### Ray Vickson
Well, the problem involves both equalities and inequallities. Let me explain what I mean in an extended example. For example, suppose we have the problem
maximize 5x + 3y
subject to
2x + y <= 50
3x + 5y <= 200
y <=10
x >=2, and
x,y >= 0.
We must first convert to equalities and pure sign constraints; in other words, re-write the problem so that all inequalities have the form 'variable >= 0'. We do this by adding a so-called slack variable to the left-hand-side (LHS) of a <= constraint, to bring the LHS up to the right-hand-side (RHS), and by subtracting a so-called surplus variable from the LHS of a >= constraint (excluding non-negativity), to bring the LHS down to the RHS. So, we have slack variables s1,s2 and s3 for constraints (1)-(3) and a surplus variable s4 for constraint (4). Now the problem becomes:
max Z1 = 5x + 3y
subject to
2x + y + s1 = 50
3x + 5y + s2 = 200
y + s3 = 10
x - s4 = 2,
x,y,s1,s2,s3,s4 >= 0.
Now the only inequalities are of the form "var >= 0"; all other restrictions are *equations*.
We have 4 equations in the 6 unknowns (x,y,s1,s2,s3,s4).
There are algorithms available to solve such problems efficiently, but none of them involve calculus---just linear algebra. Essentially, we can pick 4 of the 6 variables to regard as "dependent" and solve for them as functions of the other 2 "independent" variables. If we pick the right independent variables then the solution of the problem will become "obvious" (when written the right way), and there are systematic algorithms to move from in a finite number of steps from one choice of independent variables to a better choice, until we have found the optimal choice. I won't say more about this now. Be assured, however, that some industrial-scale linear programming problems can involve hundreds of thousands to a few million constraints and several million variables, and these are solved by companies quite often in model-aided decision-making.
RGV
5. Dec 7, 2011
### HallsofIvy
Staff Emeritus
No, no derivatives are needed. This is a "linear programming" problem. With two or three variables, it can be done geometrically. The basic result of linear programming is that a linear function, with linear constraints, so that the set of points satisying those constraints (the "feasible" region) forms a convex polytope, has its maximum and minimum values at one of the vertices.
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0
## Messing With Pi
Published on Thursday, February 07, 2008 in , , , ,
Did you ever receive that e-mail that claims that some state's legislature (usually Alabama) has recently passed a law redefining Pi? As Snopes notes, this is false. However, 111 years ago this week, the Indiana state legislature came very close to doing just that.
The change wasn't done to make Pi consistent with the Bible, but rather to deal with the age-old problem of trying to construct a square with the same area as a given circle, otherwise known as squaring the circle. Because Pi is transcendental, we now know that this is impossible.
However, what if you were working before the 19th century, before this was proved, and before calculus? There are numerous ways to get very, very close to the circle, and it's not unreasonable to think that just a few more adjustments would be needed to perfect the process. Let me show you how the madness takes hold. One of the best approaches I've found for seemingly squaring the circle was detailed by Stu Savory in April, 2005 (26 days too late if you ask me). He does a good job explaining, but I'd like to break it down a little differently for my readers. It uses only a compass and a straightedge.
We start by working on the line that will become the circle's diameter. Draw a horizontal line, and mark the rightmost end as R. Open the compass to a width of an inch or so, place the spike of the compass at R, and mark point T on the line to the left of R. Now place the spike on T, and mark a point to the left of T (not labeled). Place the spike on this unlabeled point, and mark one more point to its left, which will be labeled O. In short, OT should be exactly twice as long as RT.
Now, open your compass to length OR, with the spike at O. Use the compass to draw a circle. Mark as P the point where the horizontal line intersects with the circle on the left. Our final step concerning the diameter line itself (POR) is to bisect OP (Java required), marking OP's midpoint as H.
Next, we're going to make some measurements above POR. Construct a vertical line from point T. Mark the point at which this line intersects the circle as Q. Using the same length as QT, draw a chord starting at R. Mark the other end of this chord as S (length of QT=length of RS). Draw a line from S to P. Construct a line parallel to RS through point T (Java required), marking the point where it intersects SP as N. then construct another line parallel to RS through point O, marking the point where it intersects SP as M.
Finally, we're going to work below PQR, and get our square's base measurement. Set your compass open to length PM, with the spike at P. Swing the compass to mark the point K, below POR, on the circle's circumference, and draw the PK chord (PK=PM). Set your compass to the length of MN, and then construct a tangent at a right angle to POR, marking the bottom-most point of this tangent as L. Draw lines RL, RK and KL. Finally, construct a line parallel to LK, going through point C. Mark as D the point where this line intersects RL.
So what does this give us? RD is the base of our square we need. Let's work through this next part slowly.
The formula for the area of the circle, as we all learned in school, is Pi * r2. The formula for a square with a base of x is x2. Because of the way in which these measurements were done, RD will always be a straight line about 1.772453 times longer than RO, the circle's radius. As it happens, this is the square root of Pi!
In other words, the length of line RD equals the circle's radius times the square root of Pi ((sqrt(pi))*r). Using RD as the base of a square, we can get the square's area by multiplying this by itself! Squaring the square root of Pi gives us Pi, of course, and squaring r gives us r2, or Pi * r2 for the area of this square!
Everything seems right, but can you figure out why this isn't a true squaring of the circle? What actually winds up happening is that RD is the square root of 355/113, a common rational fraction used for Pi early on. 355/113=3.1415929204, which is accurate to 6 places after the decimal point! It's square root is 1.7724539262, which also accurate to 6 places after the decimal point. This gives a square and a circle that are so close in area that, using a pencil that could mark in 1/100 of an inch, you would have to create a circle more than 4 miles in diameter to see your error!
If you look at the bottom of Stu Savory's blog entry, he has a quicker and more interesting way of apparently squaring the circle with a compass, straightedge and a coin!
Are you beginning to see why so many people were obsessed with squaring the circle? When you don't know that Pi goes on forever, it seems to be a matter of just making a few more adjustments to make a historical mathematical breakthrough! This was actually so common that, at one point, the insane belief that one could square a circle was listed in medical texts as morbidus cyclometricus!
I hope you've enjoyed this rather odd look at a weird bit of mathematical history.
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# Buzz: Product Feedback
## Rounding - feature request
Follow
Feature idea (LEVEL 2)
I really need an option in Buzz for fill-answers to be rounded to a given place value. For example, I want students to calculate the tangent of an angle between 1 and 89 degrees. It is a simple matter to write a question. The answer, unfortunately, is not so straightforward. Var: a = 1..89 1) Calculate the tangent of `\$a\$^@`. Round your answer to the nearest ten-thousandth. a. eval(tan(\$a\$*pi/180)) The first issue with this is that students don't have to round correctly at all. They can answer with fewer or more decimal places than asked for and Buzz will count it correct as long as it is within the 1% error. This makes things very inconsistent for students. The tan 89 degrees is 57.28996..., which means any answer within .57 of that will be correct. Compared to tan 1 degree which is .017455... where students must now be within .00017 of the correct answer. The potential for this extreme variation in acceptable answers causes issues when asking students to round. Depending on what place value I want the answer rounded to students can round themselves out of the 1% margin of error and have a correct answer be marked incorrect. Another example is exponential growth. When using variables the answers can vary greatly from millions to hundreds. I can ask students to round to the nearest whole dollar, but having Buzz grade 1% of 12 million makes that impossible. If I manually reduce the 1% error to something smaller, then the smaller magnitude answers are marked incorrect because rounding pushes them out of the smaller bounds. PLEASE add the option to round number answers to a specific place value.
Sort by
The eval() function does have a rounding feature:
`eval(tan(\$a\$*pi/180),#.####)`
However, I have only been able to get it to work for generating options in a multiple choice question. It does not appear to work as a Fill in the Blanks answer as a Number, Exact, or Equivalent. (being within 1% won't match anymore either)
0
Yes, as you discovered the \$eval from your screen shot is only for displaying the results of the eval function such as inside a question or as an answer for multiple choice and matching. You are correct that it doesn't work when grading fill-in answers.
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### Euler's Squares
Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
### Odd Differences
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
### Peeling the Apple or the Cone That Lost Its Head
How much peel does an apple have?
# Peaches in General
##### Stage: 4 Challenge Level:
Obviously start by doing the original problem, that's plenty hard enough as a first challenge if you haven't seen anything like it before, but after that ask yourself the question:
• What things might generalise?
Perhaps the fraction taken each day, or the number of days , or the 'plus one more', or something else, but take those possibilities one at a time.
If you use a spreadsheet (and there are really strong reasons for doing that in this type of problem), what do you need to calculate in each column of the sheet? You can have as many columns as you want, so do simple calculations, then calculations further along the row that use those answers, rather than complicated calculations in single cells.
Will you make your first column a number of peaches before any eating happens? It might be better to have the number of peaches in the final column, making your row a calculation trail that deduces the number at the start?
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# Math104A Fall, 2008 Homepage
Final Exam Monday 12/8 11:30-2:30 205 Center Hall (Regular classroom).
Bring a blue book.
Here is a listing of topics for the Final.
Below is a link to some problems that you can look upon as a practice final (except the final will not be as long). The problems are at the level of the difficulty of the exam: There is a typo in problem 5 in line 3 of the problem change 7x+11y to 5x+7y.
Here are some links to free mathematical software.
A general purpose math package is Maxima. Here is a link:
Another general package called that basically works under Linux but the link below allows it run under windows called SAGE:
The best program for number theory is PARI/GP. It has a higher learning curve but is very worthwhile to learn. Here is a link.
Professor Wallach's office hours are
Monday 2:00-3:00 and Friday 2:00-3:00 and by appointment
The TA for the course is Kevin McGown
His email is kmcgown@math.ucsd.edu
His office hours are Tuesday 2:00-2:50 and Friday 3:00 to 4:00
Each week (except the first) homework will be collected in Section on Monday (all assignments made the previous week will be collected)
The grade will be based on:
Homework 10%
Quiz 15%
Midterm 25%
Final 50%
Here is a link to the first supplement to the text.
Homework:Due Monday 10/6 Exercises 1-8 in Supplement 1 and the extra problems in the following link
Homework: Due Monday 10/13 From the text:
pp.13-16 1 (i),(ii),3.,4.,9 (i),(ii),5* (*=harder), 19 (ii),(iii), 21 (i),(ii).
All 6 exercises in the extra problems in the following link.
The problems due for 10/20 are:
From the book:p.13 2., p.54 2.,3. (i),(ii),(iii)*,4* and all of the problems in the following link:
The problems due on 10/27 are:
From the text: pp.54-57 1. (i)-(iv),5,6,15(i),(ii),16,27(i). All of the problems in
Below is supplement 2 for the course. It explains the term numbers to base m. Our numbers as we write them are to base 10. There are also some problems that are the assignment for 10/27 (due 11/3). MISPRINT: Problem 4 should say 2^p - 1 not p -1.
Homework due on 11/3. The problems in supplement 2 above and p.57 26,27(ii).
Below is supplement 3 for the course. It covers some of the results on polynomials over a field that are used in the text. this material is a standard part of Math 100 and or Math 102. Some number theoretic implications are also in these notes. They will be the text for the lectures in the course 11/3 and 11/5. The version below is a corrected version which was uploaded on 11/4 (if you have already printed the earlier version you need only replace pages 7,8,9,10).
Homework due 11/10 problems 1-5 and problem 9 in Supplement 3.
There have been more changes in the last part of supplement 3. You should print (yet again) pages 7,8,9,10 and 11. There are 5 new problems in supplement 3 that will be assigned on 11/10.
Here is a link to supplement 4.
Homework for 11/17 problems 1-11 in Supplement 3. and 1-4 in supplement 4.
There are new pages in Supplement4 (113-117). Also Jet Au has found a typo in Problem 10 Supplement 4. The first sentence should say x congruent to +1 or -1 mod 12 (it said mod 3). Also the second sentence should say p congruent to -1 mod 3 (not +-1).
Homework for 11/24 Problems 5-17 in Supplement 4. Exercis 7 (i)-(ii) p. 75 in Rose and read chapter 4 in Rose. Another misprint in problem 8 it should say b^2-4ac NOT a^2-4bc.
Homework for Dec 1. p.28 1 (i),(ii),(iii), 2 (i),4,5,10 ((i,j)=gcd(i,j)), 13 (i).
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# Realistic ratios and proportions worksheets ideas
» » Realistic ratios and proportions worksheets ideas
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Realistic Ratios And Proportions Worksheets. Ratios can be written by using the word to 3 to 2 by using a colon 3. From the information If 4 be subtracted from each they are in the ratio 35 we have. Ratios and Proportion Worksheets. 1 3 x 9.
One Of The Many Poems Activities Guided Notes Assessments And Worksheets In Ratios R Ratio And Proportion Worksheet Proportions Worksheet Sixth Grade Math From pinterest.com
Roman Numbers Roman Numerals. The ratio 4. The keyword that helps in identifying a ratio statement is to every and it is denoted by a colon. The ratio of 8 to 80 paise is 1. 2 or by expressing the ratio as a fraction. There is a self-assessment section at the end of each worksheet.
### Ratio Proportion Worksheet 4-5 mixed questions on finding proportions and ratios.
Other options include using whole numbers only numbers with a certain range or numbers with a certain number of decimal digits. Thank you for using the Math-Drills Search page to find math worksheets on a topic of your choice. 10112020 Class 6 Maths Ratio and Proportion TrueT or FalseF 1. Ratios rates proportions Pre-algebra Math Khan Academy 163086 Ratio Worksheets Free - CommonCoreSheets 163087 Percent Practice 8. This article is all about understanding of Roman Numerals in Primary SchoolThis includes roman numbers rulesuses roman numbers poster and. 5 2x-4 3 3x-4 10x - 20 9x - 12.
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Using cross product rule of proportion we have. 7 20. Thank you for sharing a nice series of ratio questions. Roman Numbers Roman Numerals. 5 2x-4 3 3x-4 10x - 20 9x - 12.
Source: pinterest.com
Ratios can be written by using the word to 3 to 2 by using a colon 3. Using cross product rule of proportion we have. Create proportion worksheets to solve proportions or word problems eg. Ratios can be written by using the word to 3 to 2 by using a colon 3. 12102013 docx 358 KB docx 1831 KB KS3 GCSE ratio and proportion worksheets ideal for revision or homework.
Source: istandwithilhan.org
Report this resourceto let us know if it violates. Multiplier Range 2 to 5 43 views this week Equivalent Ratios with Blanks 32. Working and finding the ratios between numbers is both challenging and exciting. When two different ratios are set equal to each other we call it proportions. 10112020 Class 6 Maths Ratio and Proportion TrueT or FalseF 1.
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Ratios can be written by using the word to 3 to 2 by using a colon 3. Some of the worksheets for this concept are Unit 4 grade 9 applied proportional reasoning ratio rate Ratio rate and proportions practice Ratios rates unit rates Rates and ratios 2 Unit 9 grade 7 ratio and rate Ratios and unit rates work answer key rate this Ratio proportion Work 1 whole numbers. This series of printable proportion worksheets are prepared specifically for learners of grade 6 grade 7 and grade 8. 13122018 A worksheet covering a range of foundation level questions. There is a self-assessment section at the end of each worksheet.
Source: pinterest.com
Here you find our ratio and proportion worksheets for math classes 5 and 6. Create proportion worksheets to solve proportions or word problems eg. Ratio Proportion Worksheet 3 convert the currencies. Ratios are a simple mathematical expression while proportions are an equation. Available both as PDF and html files.
Source: in.pinterest.com
First number 2x. 7 20. We are applying what we have learned about ratios and proportions. 16 is in its lowest form. There is a self-assessment section at the end of each worksheet.
Source: pinterest.com
The ratio of 8 to 80 paise is 1. Once you find your worksheet click on pop-out icon or print icon to worksheet. Second number 3x. The two terms of a ratio can be two different units. Working and finding the ratios between numbers is both challenging and exciting.
Source: giovannifranko.it
Available both as PDF and html files. Ratio Proportion Worksheet 4-5 mixed questions on finding proportions and ratios. Once you find your worksheet click on pop-out icon or print icon to worksheet. Ratios are used to make a comparison between numbers or units. When two different ratios are set equal to each other we call it proportions.
Source: pinterest.com
7 20. Fifth and sixth grade students will never forget these exciting visual worksheets that will make ratios and proportions. Ratios are used to make a comparison between numbers or units. There is a self-assessment section at the end of each worksheet. Enter a search term in the box below and click on the Search button to start a new search.
Source: pinterest.com
Ratios are a simple mathematical expression while proportions are an equation. 1 3 x 9. 10112020 Class 6 Maths Ratio and Proportion TrueT or FalseF 1. 7 20. First number 2x.
Source: pinterest.com
This article is all about understanding of Roman Numerals in Primary SchoolThis includes roman numbers rulesuses roman numbers poster and. This series of printable proportion worksheets are prepared specifically for learners of grade 6 grade 7 and grade 8. Enter a search term in the box below and click on the Search button to start a new search. Create proportion worksheets to solve proportions or word problems eg. A variety of pdf exercises like finding proportions using a pair of ratios determining proportions in function tables creating a proportion with a given set of numbers and solving word problems are included here.
Source: pinterest.com
Ratios can be written by using the word to 3 to 2 by using a colon 3. A c. The ratio 4. 7 20. Ratios and Percents MathRatios and.
Source: pinterest.com
Second number 3x. Other options include using whole numbers only numbers with a certain range or numbers with a certain number of decimal digits. Ratios and Percents MathRatios and. The two terms of a ratio can be two different units. A c.
Source: pinterest.com
Speeddistance or costamount problems. Once you find your worksheet click on pop-out icon or print icon to worksheet. A variety of pdf exercises like finding proportions using a pair of ratios determining proportions in function tables creating a proportion with a given set of numbers and solving word problems are included here. Ratios are a simple mathematical expression while proportions are an equation. Therefore we will get this equation.
Source: pinterest.com
After that putting 180 to. The two terms of a ratio can be two different units. Ratio Proportion Worksheet 1 simplify the ratios by following a rule. Year 6 Age 10-11 Ratio and Proportion Worksheets. A year ago.
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Let us look at the History & Explanation of Magnetic Effect of Electric Current Class 10.
In 1820 the theory of the magnetic effect of electric current was discovered by H.C Oersted. This concept actually means that the flow of an electric current through a conductor produces a magnetic field around it.
He demonstrated this observation with the SNOW Rule
As per this experiment if we make an electric circuit by joining a straight wire connected to a battery and a plug and then place a magnetic compass needle parallel to an under the wire. Now if we switch the circuit on so that the current from the wire flows from South to North direction, we will observe that the magnetic compass needle gets averted towards the west. Now if we change the direction of the current from North to South direction the magnetic compass gets averted towards the East.
Thus, according to this SNOW Rule, When the current flows through a conductor from South to North then the magnetic needle placed beneath it, gets averted to the west, thereby proving that flow of electric current produces a magnetic field. This is because the compass needles work as a small bar magnet, so when this magnetic needle is brought near another magnetic property surrounding then the like poles will repel, thereby the needle gets deflected.
Image Credits: Fun Science
## The Basic Concepts
• Electric Current is the flow of electric charge (a physical property of the matter that experiences a force when placed around an electromagnetic field)
• Magnetic field is the area around a magnet where the magnetic force is experienced. The imaginary lines of magnetic field around a magnet is called Magnetic Field Lines.
MAGNETIC FIELD LINES
• An Electromagnetic field (EM Field) is a combination of electric field and magnetic field. An EM field is produced when electrically charged particles, such as electrons are accelerated. Electrically charged particles are surrounded by electric fields and these charged particles when in motion generate magnetic field
• Thus, the magnetic effect of electric current is defined as the electromagnetic effect. Which means that if a magnetic compass is brought near a electrically charged conductor then the needle of this compass gets deflected because of the flow of electricity.
## Explanation of Magnetic Effects of Electric Current Class 10
### The Right-Hand Thumb Rule
The Right-Hand Thumb Rule or Maxwell’s Corkscrew Rule depicts the direction of magnetic field in relation to the direction of electric current through a straight conductor.
As per this rule suppose if a current carrying conductor is held by right hand with the thumbs up straight and the electric current flowing in the direction of the thumb then the direction of the magnetic field can be depicted by the direction of wrapping of the other fingers.
This means that in a vertically suspended current carrying conductor if the direction of the current is from south to north then the magnetic field will be in an anticlockwise direction. But if it is vice-versa which means that the direction of the current is flowing from north to south then the magnetic field will be in clockwise direction. In this rule, it should be noted that when current is flowing in an anticlockwise direction, then the magnetic field will be in a clockwise direction at the top of the loop and when it is vice versa then the magnetic field will be at the bottom of the loop.
### Flow of Current Through a Circular Loop
Like in the above observation if the electric current carrying conductor is circular shaped instead of a straight current conductor the magnetic field is generated in the same manner. This is because as the electric current carrying conductor exerts a force when a magnetic needle is placed near it, similarly a magnet also exerts an equal and opposite pressure on the electric carrying conductor. Here it is observed that with the change in direction of the flow of current the force of magnet on the conductor also changes. As mentioned above this is because of the SNOW RULE. In a circular loop, the strength of the magnetic field will be more because in this case, the magnetic field is very close to the current carrying conductor.
### Clock Face Rule
Let us assume a current carrying circular loop as a disc magnet. The polarity of this disc magnet can be explained with the help of clock face rule. An anticlockwise flowing current will depict the face of the loop showing towards the North Pole.
### Magnetic Field Due To A Solenoid
The coil with many circular turns of insulated copper wire wrapped closely in the shape of a cylinder is called a Solenoid. A solenoid behaves similarly like a magnet, one end of solenoid behaves as the north pole and another end behaves as the south pole. In this case, the magnetic field lines are parallel inside the solenoid, which is similar to a magnet and this proves that that magnetic field is same at all points inside a solenoid. By producing a strong magnetic field inside a solenoid, magnetic materials can be magnetized. Magnets formed by producing a magnetic field inside a solenoid is called electromagnet.
### Fleming’s Left-Hand Rule
According to Fleming’s Left-Hand Rule, if the direction of flow of the electric current is perpendicular to the magnetic field, the direction of its force is also perpendicular to it. It states that suppose if the forefinger, thumb, and the middle finger of the left hand is stretched in a way that they are right angles to each other, then the forefinger and middle finger show the direction of magnetic field and direction of electric current respectively and the thumb shows the direction of motion or force acting on the current carrying conductor. The directions of electric current, magnetic field and force are similar to three mutually perpendicular axes, i.e. x, y and z-axes.
### Application of Fleming’s Left-Hand Rule in Real Life Concepts
Electric Motor
The electric motor is a phenomenal example of how Fleming’s Left-hand rule works, Electrical energy is converted into mechanical energy by using an electric motor. In an electric motor, a rectangular coil is suspended between the two poles- North and south of a magnetic field. The electric supply to the coil is connected to a commutator- a device which reverses the direction of flow of electric current through a circuit.
When an electric current is supplied to the coil of the electric motor, it gets deflected because of the magnetic field. As it reaches the halfway, the split ring which acts as commutator reverses the direction of flow of electric current. When the direction of current reverses the direction of forces acting on the coil also gets reversed. Thus here the change in direction of force pushes the coil, and it moves another half turn. Finally, the coil completes its one rotation around the axle. Continuation of this process keeps the motor in rotation motion.
Also check out our article on Class 10 Science NCERT Notes – Revision here.
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A114425 Product of the first n 3-almost primes (A014612). 2
8, 96, 1728, 34560, 933120, 26127360, 783820800, 32920473600, 1448500838400, 65182537728000, 3259126886400000, 169474598092800000, 10676899679846400000, 704675378869862400000, 47917925763150643200000 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS 3-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086062 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008585(n) = 3*n. LINKS Harvey P. Dale, Table of n, a(n) for n = 1..364 FORMULA a(n) = Prod[from i = 1 to n] A014612(i). EXAMPLE a(5) = 933120 = 8 * 12 * 18 * 20 * 27 = the product of the first 5 values of the 3-almost primes = 2^8 * 3^6 * 5, which has 3*5 = 15 prime factors (with multiplicity). a(20) = 137199755075271237225676800000000 = 8 * 12 * 18 * 20 * 27 * 28 * 30 * 42 * 44 * 45 * 50 * 52 * 63 * 66 * 68 * 70 * 75 * 76 * 78 * 92 = 2^26 * 3^15 * 5^8 * 7^4 * 11, which has 20*3 = 60 prime factors (with multiplicity). MATHEMATICA FoldList[Times, Select[Range[70], PrimeOmega[#]==3&]] (* Harvey P. Dale, Apr 26 2020 *) CROSSREFS Cf. A002110, A008585, A014612, A086062, A112141. Sequence in context: A034177 A052570 A002168 * A224767 A337597 A052127 Adjacent sequences: A114422 A114423 A114424 * A114426 A114427 A114428 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Feb 13 2006 STATUS approved
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Last modified November 23 14:26 EST 2020. Contains 338590 sequences. (Running on oeis4.)
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# Euler's Column Formula
## Buckling of columns
Columns fail by buckling when their critical load is reached. Long columns can be analysed with the Euler column formula
F = n π2 E I / L2 (1)
where
F = allowable load (lb, N)
n = factor accounting for the end conditions
E = modulus of elastisity (lb/in2, Pa (N/m2))
L = length of column (in, m)
I = Moment of inertia (in4, m4)
### Factor Counting for End Conditions
• column pivoted in both ends : n = 1
• both ends fixed : n = 4
• one end fixed, the other end rounded : n = 2
• one end fixed, one end free : n = 0.25
### Note!
Equation (1) is sometimes expressed with a k factor accounting for the end conditions:
F = π2 E I / (k L)2 (1b)
where
k = (1 / n)1/2 factor accounting for the end conditions
n 1 4 2 0.25 k 1 0.5 0.7 2
### Example - A Column Fixed in both Ends
An column with length 5 m is fixed in both ends. The column is made of an Aluminium I-beam 7 x 4 1/2 x 5.80 with a Moment of Inertia iy = 5.78 in4. The Modulus of Elasticity of aluminum is 69 GPa (69 109 Pa) and the factor for a column fixed in both ends is 4.
The Moment of Inertia can be converted to metric units like
Iy = 5.78 in4 (0.0254 m/in)4
= 241 10-8 m4
The Euler buckling load can then be calculated as
F = 4 π2 (241 10-8 m4) (69 109 Pa) / (5 m)2
= 262594 N
= 263 kN
## Related Topics
• Beams and Columns - Deflection and stress, moment of inertia, section modulus and technical information of beams and columns
## Tag Search
• en: euler column formula buckling
• es: Euler fórmula columna de pandeo
• de: Euler Spaltenformel Knick
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## Global sections over Alice and Bob: the sheaf-theoretic expression of non-locality and contextuality
03/04/2013
The story is well known. It is a sunny Sunday and Alice and Bob decide that going out for a walk might be too mainstream so, instead, they decide to do some physics. They convince their friend Carlos, the third wheel, to prepare a pair of entangled photons, each to be sent to Alice or Bob. The couple can perform two different measurements on the photons sent by Carlos and the values that can be obtained for both measurements are either 0 or 1. In order to better enjoy the Sunday, they decide to perform the experiment separated from each other and Carlos. They take their longboards in opposite directions and decide to start taking measurements. Being separated, they will take randomly one of the two possible measurements and annotate it together with one of the two possible outcomes. After being the whole morning repeating the experience, they make a table with the obtained distributions of joint outcomes for each pair of measurements.
Empirical distribution of a Bell scenario experiment. The two different measurements that Alice and Bob can make are distinguished by an apostrophe.
After seeing the results, Alice finds herself surprised by the correlations obtained, as if some spooky action at a distance (spukhafte Fernwirkung) was relating both measurements. Bob, immediately accuses Carlos of being that spooky action. Alice points out that there is no way by which Carlos could have spied them while performing the measurements (getting to know the measurements they were going to make) and demands Bob for a detailed explanation of how Carlos could have produced the results. After thinking for a while, Bob points out that Carlos could, in principle, generate, in a deterministic way, all the possible assignments of their measurements to the outcomes. Hence, Carlos might be able to send the pair of photons with a previously uniquely specified assignment of outcomes for each of the two measurements Alice and Bob can make (for instance $\lambda_{1} : a \mapsto 0, a' \mapsto 1, b \mapsto 0, b' \mapsto 0$ ). To generate a stochastic behaviour, Carlos just has to specify a probability distribution on all the possible assignments to their measurements in such a way that, applied to the different contexts, it produces the observed probabilities on the joint measurements of the outcomes. Alice still holds that Carlos is innocent and what they have observed cannot be explained using Bob's reasoning. Bob refuses to believe that their result is a manifestation of a sort non-local and contextual behaviour. After being silent for most part of the discussion, Carlos decides to start a deeper conceptual investigation to prove his innocence.
In order to answer the question whether the explanation of data obtained requires a theory that violates the principle of locality , we would like to define a general abstract framework in which we can place our problem in a precise way. For the sake of generality, let us think of a system (like the entangled photons of Alice and Bob) as composed of several parts (spatially separated or not) upon which several individuals (Alice, Bob, Carlos, ..., Zed) can perform a set of measurements. In quantum mechanics, it usually happens that not all measurements are compatible (they cannot be made simultaneously). We will consider that the different parts of the systems are always compatible, and that incompatibilities, if they exist, arise within each part. These measurements have associated a set of possible outcomes that are the values we obtain (0 or 1, in our example). Given the different possible measurement contexts, a set of distributions on the joint outcomes obtained determines an empirical model, such as the one obtained by Alice and Bob.
Diagram showing the different elements composing a general measuring scenario (Abramsky et Bradenburguer, 2011).
For that empirical model, it might be possible to construct a hidden-variable model (such as the one proposed by Bob) that explains the results. In this model, a set $\Lambda$ of independent unobserved variables $latex \lambda$ determine the outcomes of the different measurements. If an empirical model can be generated with a distribution of deterministic hidden-variables marginalized over the measurement contexts, it can serve as an explanation to the observed data.
The scenario described by Alice and Bob is that of the Bell inequalities, it will be henceforth referenced as a Bell scenario. To recapitulate, in the Bell scenario, we have two parts (Alice and Bob), two different measurements that are incompatible on the same part (a,a' for Alice or b,b' for Bob) and two different outcomes (0 or 1). The set of all maximal measurement contexts ( ${a,b},{a',b},{a,b'},{a,b}$ ) can be seen as a topological space $X$ and an assignment of outcomes can be thought as a section over its open sets $U$ . Thus, a section over an open set $U \subseteq X$ is map $s$ such that $s : U \rightarrow O$ . The set of sections over a particular $U$ can be noted as $O^{U}$ . A presheaf $F$ associates to each open set $U'$ a set $O^{U'}$ of sections on $U'$ and to each open set $U \subseteq U'$ a map $F(U') \rightarrow F(U)$ that specifies the restrictions of the sections over $U'$ to $U$ : $F : U \rightarrow O^{U}$ .
In the Bell scenario, the set of possible assignments from the different possible measurement pairs is given by the following table. Each row constitutes a set of sections $O^{U}$ over the particular measurement context.
Sections over de different maximal measurement contexts (Abramsky et Brandenburguer, 2011).
Given a family of sets $U_{i}$ that form a cover of $X$ , their sections are locally determined whenever two sections $s_1$ and $s_2$ over $X$ coincide when they are restricted to each $U_{i}$ . A locally determined presheaf is called separated. A sheaf is a separated presheaf that can be glued together when there is a unique section $s$ of $X$ such that $s|U_{i} = s_{i}$ . For the previously described presheaf of the outcomes sections, both the compatibility and gluing conditions hold so that we obtain a sheaf.
The category theory generalization of the previously described mathematical structure is the presheaf functor. For a category $\mathcal{C}$ the S-valued presheaf F is a functor such that: $F : \mathcal{C}^{opp} \rightarrow \textit{S}$ . Here, $S$ can be any category like the category of sets ( $Set$ ), commutative rings ( $CRing$ ) or the category of Abelian groups ( $Ab$ ). In order to recover the notion of presheaf in topological spaces, we have to take the $Set$ valued presheaf of the category $\mathcal{C}$ constituted by the posets of open sets of a topological space.
Functor operating on the category constituted by the posets of open sets of a topological space.
Then, the presheaf functor takes the open subsets of $\mathcal{C}$ and maps them to set of sections on the open subset $O^{U}$ . The morphisms of the $\mathcal{C}^{opp}$ category are indeed retractions and are mapped by the presheaf functor into restrictions of sections: $res_{U,U'}: F(U') \rightarrow F(U)$ . Such that the composition rules and identities are satisfied in order to constitute a category.
In order to recover the statistical behaviour of the system, we would like to assign a probability distribution to each set of sections $O^{U}$ . Given a set $X$ and a commutative semiring $R$ (like the non-negative reals), we can define an R-distribution as a function $d : X \rightarrow R$ if the function has a finite support such that $\sum_{x \in X}d(x)=1$ . The set of R-distributions on $X$ can be written as $D_{R} (X)$ and for the non-negative reals corresponds to the set of probability distributions. It can be seen that $D_{R}$ is a functor $D_{R}: Set \rightarrow Set$ and can be composed with the previous event sheaf $F : \mathcal{C}^{opp} \rightarrow \textit{Set}$ to form a presheaf $D_{R}F: \mathcal{C}^{opp} \rightarrow \textit{Set}$ .
Our aim is to find if the sheaf condition holds for this new presheaf $D_{R}F$ . So , we want to find the global section $d \in D_{R}F(X)$ exists for the entire set of measurements $X = {a,a',b,b'}$ defining a distribution on the set $F(X)=O^{X}$ of possible assignments $\lambda_{i}$ of outcomes to $X$ . Each of this global sections of $F$ can be considered a deterministic hidden variable (for instance $\lambda_{1} : a \mapsto 0, a' \mapsto 1, b \mapsto 0, b' \mapsto 0$ ) assigning a particular outcome to a particular measurement independent of the context. The global section $d$ of $D_{R}F$ would define a distribution on the set of deterministic hidden variables that marginalizes on the different measurement contexts to yield the observed probabilities (modelling the role of Carlos suggested by Bob in our initial story).
If we can find the sheaf, Bob will be right , however if there is an obstruction to the formation of it, we will have to accept the non-local and contextual behaviour. Given the empirical results, this can be done by solving the following linear system $\textbf{Md}=\textbf{e}$ where $\textbf{d}$ stands for the global section, $\textbf{e}$ for the empirical model and $\textbf{M}$ for the incidence matrix. The latter is defined by the relation between the global assignments $\lambda_{j}$ and the sections $s_{i}$ relating the contexts with the outcomes. The elements $m_{i,j}$ are $1$ if the $\lambda_{j}$ , when restricted to that measurement context, is equal to the section and $0$ otherwise. For the Bell scenario the incidence matrix is:
Incidence matrix for the Bell scenario (Abramsky et Brandenburguer, 2011).
It can be shown that there is no $\textbf{d}$ that solves the equation such that all terms are non-negative and add up to one. So, for this empirical distribution obtained in the sunny Sunday there is not a mischievous Carlos or a hidden variable model that explains it
Abramsky S. and Brandenburger A. 2011. The Sheaf-Theoretic Structure of Non-Locality and Contextuality. 13: 113036.
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# Important Data Interpretation Questions for UGC NET with Free PDF and Take UGC Net 2019 Solved Questions Online Test at Smartkeeda
Directions: Study the following information carefully and answer the following question as beside:
The following table gives the information about the number of marks obtained by five students in five subjects in an examination (the total marks in every subject are 300 each)
Student/subject English Maths Physics Biology Hindi A 180 290 215 190 192 B 210 230 264 228 102 C 285 175 176 186 132 D 185 165 188 136 144 E 225 230 162 98 156
Important for :
1
The total marks obtained by D and E together in Biology and Hindi together was approximately what percent of the marks obtained by A alone in Maths?
» Explain it
D
The total marks obtained by D and E together in Biology and Hindi together = 136 + 98 + 144 + 156 = 534
the total marks obtained by A alone in maths = 290
Reqd. % = 534 × 100 = 184.14% approximately 290
Hence, option D is correct.
2
If the minimum passing marks of any subject was 30% of the total marks of all the subjects together then the marks obtained by B in all the subjects together was how much more than the minimum passing marks of all the subjects together?
» Explain it
A
total number of subjects = 5
The total marks of all the five subjects = 300 × 5 = 1500
Minimum passing marks = 30% of 1500 = 450
The marks obtained by B in all the subject together = (210 + 230 + 264 + 228 + 102) = 1034
The required difference = 1034 – 450 = 584
Hence, option A is correct.
3
What was the highest total percentage of the marks of all the subjects together secured by any of the given students? (approximately)
» Explain it
C
The marks obtained by A = 1067
The marks obtained by B = 1034
The marks obtained by C = 954
The marks obtained by D = 818
The marks obtained by E = 871
Since A got the highest marks so the percentage of the marks of all the subject
together will be highest for A
Reqd. % = 1067 × 100 = 71.13% approximately 1500
Hence, option C is correct.
4
Find the difference between the marks obtained by A and B in all the subjects together and the marks obtained by C and D in all the subjects together?
» Explain it
E
The marks obtained by A and B in all the subjects together = 1067 + 1034 = 2101
The marks obtained by C and D in all the subjects together = 954 + 818 = 1772
The required difference = 2101 – 1772 = 329
Hence, option E is correct.
5
The marks obtained in English by all the students together is approximately what percent of the marks obtained in Maths by all the students together?
» Explain it
D
The marks obtained in English by all the students together = 1085
The marks obtained in Maths by all the students together = 1090
Reqd % = 1085 × 100 = 99.54% 1090
Hence, option D is correct.
### Data Interpretation Questions for UGC Net 2019
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# Apply existing number codes to spell new words
In this worksheet, students will turn words into number codes using words with similar codes as a base. It will develop their encoding and problem solving skills.
Key stage: KS 2
Curriculum topic: Verbal Reasoning
Curriculum subtopic: Number Code Breaking
Difficulty level:
### QUESTION 1 of 10
Hi there code cracker!
Today we are going to learn about a new type of code.
When you turn a word into code, each letter can be represented by a number.
Let’s look at a question together:
If FLOWER is 785901, then WOLF is ____
You may have noticed that the word WOLF contains the same letters as the word FLOWER.
So the answer can be worked out by using the code introduced by the example word.
The W in FLOWER is 9 (W=9).
The O in FLOWER is 5 (O=5).
The L in FLOWER is 8 (L=8).
The F in FLOWER is 7 (F=7).
So WOLF is 9587 in code.
Let’s try another:
If the code for DINOSAUR is 51372894, then AROUND is ____.
The word DINOSAUR contains the same letters as AROUND, so we can use this to work out the new code.
So the code for AROUND is 847935.
In this activity, you will be a code cracker and change words into number codes. Make a note of the number that matches each letter; you may want to create your own grid to help with this.
Pssstt!! Here’s a handy hint to help you reach superstar status:
It can be really helpful to write the word out and then write the corresponding numbers above each letter. Give it a go!
Let’s get started!
Welcome back to number code breaking, code cracker! Are you ready for another worksheet? I think so! Remember, you're aiming for superstar status and I know you can do it!
Let's...
If the code for URCHIN is 623105, what is the code for CHIN?
OK, onto another question code cracker! Here's a handy hint: I'm not asking you to mix up any letters yet. You can transfer the code straight out of the long word.
If the code for FREEDOM is 9211734, what is the code for REED?
You're doing brilliantly, code cracker! Right, let's have a go at another question. I still haven't changed the pattern, so you just need to take the code for the shorter word straight out of the code for the longer word.
If the code for SLIP is 4103, what is the code for LIP?
If the code for PHOTOCOPY is 971213190, what is the code for HOT?
OK, code cracker, you are halfway - brilliant effort so far!
If the code for STEEL is 36114, what is the code for EEL?
Come on then, code cracker, let's tackle these last five questions!
If the code for PAPER is 61624, what is the code for APE?
You have done so well, code cracker. Let's keep aiming for superstar status!
If the code for RECOVERY is 72316275, what is the code for OVER?
We are only a few questions away from the end, code cracker. Fantastic work!
If the code for METALLIC is 61234409, what is the code for TALL?
We are at our penultimate question, code cracker - what brilliant progress you have made!
If the code for MONEY is 61743, what is the code for ONE?
You've made it to the last question, code cracker. Hurrah!
If the code for KILOGRAM is 61724309, what is the code for LOG?
• Question 1
Welcome back to number code breaking, code cracker! Are you ready for another worksheet? I think so! Remember, you're aiming for superstar status and I know you can do it!
Let's...
If the code for URCHIN is 623105, what is the code for CHIN?
3105
EDDIE SAYS
How did you manage with this one, code cracker? I hope it makes sense! I started with a fairly simple one - you'll notice that the word CHIN is the last four letters of the word URCHIN! So, we can work out that: The C in URCHIN is 3. The H in URCHIN is 1. The I in URCHIN is 0. The N in URCHIN is 5. So, the code for CHIN is 3105!
• Question 2
OK, onto another question code cracker! Here's a handy hint: I'm not asking you to mix up any letters yet. You can transfer the code straight out of the long word.
If the code for FREEDOM is 9211734, what is the code for REED?
2117
EDDIE SAYS
Did you see the code, code cracker? You can find the word REED in the word FREEDOM, which is handy! Let's look a bit more carefully at this. If we look at the code, we can see that: The R in URCHIN is 2. The E in URCHIN is 1. The E in URCHIN is 1. The E in URCHIN is 7. So, the code for REED is 2117! Let's keep working, code cracker!
• Question 3
You're doing brilliantly, code cracker! Right, let's have a go at another question. I still haven't changed the pattern, so you just need to take the code for the shorter word straight out of the code for the longer word.
If the code for SLIP is 4103, what is the code for LIP?
103
EDDIE SAYS
I think this was quite a nice question, code cracker, don't you? The word LIP is in the word SLIP - in fact, all you need to do is remove the first letter! So, if we know the code for SLIP is 4103, then we know the S is 4. So we just need to remove the 4 to work out that the code for LIP is 103!
• Question 4
If the code for PHOTOCOPY is 971213190, what is the code for HOT?
712
EDDIE SAYS
How did you manage with this one, code cracker? I hope it makes sense! I think you're getting into the swing of things, especially if you noticed that the word HOT is in the word PHOTOCOPY! So, we can work out that: The H in PHOTOCOPY is 7. The O in PHOTOCOPY is 1. The T in PHOTOCOPY is 2. So, the code for HOT is 712!
• Question 5
OK, code cracker, you are halfway - brilliant effort so far!
If the code for STEEL is 36114, what is the code for EEL?
114
EDDIE SAYS
I think you are really getting to grips with this now, code cracker. If you find you are still getting muddled at moments, that's OK - I'm here to help! The word EEL is the last three letters of the word STEEL. We know the code for the word STEEL is 36114. The last three numbers of the code are 114, so the code for EEL must be 114!
• Question 6
Come on then, code cracker, let's tackle these last five questions!
If the code for PAPER is 61624, what is the code for APE?
162
EDDIE SAYS
So again, code cracker, we can see that the word APE is in the word PAPER! Hurrah! If we look at the code, then, we can see that: The A in PAPER is 1. The P in PAPER is 6. The E in PAPER is 2. So, the code for APE is 162!
• Question 7
You have done so well, code cracker. Let's keep aiming for superstar status!
If the code for RECOVERY is 72316275, what is the code for OVER?
1627
EDDIE SAYS
Did you spot the word OVER in RECOVERY? I hope so! Keep looking to see if you can find the shorter words in the longer words - it will really help you in the long run! So, because we know the code for RECOVERY, we know the code for OVER because: The O in RECOVERY is 1. The V in RECOVERY is 6. The E in RECOVERY is 2. The R in RECOVERY is 7. So, the code for OVER is 1627!
• Question 8
We are only a few questions away from the end, code cracker. Fantastic work!
If the code for METALLIC is 61234409, what is the code for TALL?
2344
EDDIE SAYS
Did you take my advice from the last question and look for the shorter word in the longer word, code cracker? Make sure you do! If you did, you would have seen that: The T in METALLIC is 2. The A in METALLIC is 3. The L in METALLIC is 4. The L in METALLIC is 4. So, we know that the code for TALL is 2344!
• Question 9
We are at our penultimate question, code cracker - what brilliant progress you have made!
If the code for MONEY is 61743, what is the code for ONE?
174
EDDIE SAYS
I hope you spotted that the word ONE is in the word MONEY. It will definitely have made cracking the code a bit easier! All we have to do is see that: The O in MONEY is 1. The N in MONEY is 7. The E in MONEY is 4. So, the code for ONE is 174!
• Question 10
You've made it to the last question, code cracker. Hurrah!
If the code for KILOGRAM is 61724309, what is the code for LOG?
724
EDDIE SAYS
Did you get this one correct, code cracker? I hope you feel like you are getting more confident with this code cracking skill! So as ever, the word LOG is in the word KILOGRAM. So, looking at the code for KILOGRAM we can see that: The L in KILOGRAM is 7. The O in KILOGRAM is 2. The G in KILOGRAM is 4. So, the code for LOG is 724! Wahoo! You made it! Give yourself a big pat on the back, code cracker!
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## College Physics (4th Edition)
$p = 4.9~mc$
We can find the magnitude of the momentum: $E^2 = p^2c^2+m^2c^4$ $(5.0~mc^2)^2 = p^2c^2+m^2c^4$ $25~m^2c^4 = p^2c^2+m^2c^4$ $p^2c^2 = 24~m^2c^4$ $p^2 = 24~m^2c^2$ $p = \sqrt{24}~mc$ $p = 4.9~mc$
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# Properties
Label 1089.2.a.o Level $1089$ Weight $2$ Character orbit 1089.a Self dual yes Analytic conductor $8.696$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$
# Learn more
## Newspace parameters
Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1089.a (trivial)
## Newform invariants
Self dual: yes Analytic conductor: $$8.69570878012$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$
## $q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} + 3 q^{5} + 2 \beta q^{7} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 + 3 * q^5 + 2*b * q^7 - b * q^8 $$q + \beta q^{2} + q^{4} + 3 q^{5} + 2 \beta q^{7} - \beta q^{8} + 3 \beta q^{10} + \beta q^{13} + 6 q^{14} - 5 q^{16} + \beta q^{17} - 4 \beta q^{19} + 3 q^{20} + 6 q^{23} + 4 q^{25} + 3 q^{26} + 2 \beta q^{28} - \beta q^{29} + 4 q^{31} - 3 \beta q^{32} + 3 q^{34} + 6 \beta q^{35} - 11 q^{37} - 12 q^{38} - 3 \beta q^{40} - \beta q^{41} - 2 \beta q^{43} + 6 \beta q^{46} + 5 q^{49} + 4 \beta q^{50} + \beta q^{52} + 9 q^{53} - 6 q^{56} - 3 q^{58} + 6 q^{59} + 4 \beta q^{62} + q^{64} + 3 \beta q^{65} - 2 q^{67} + \beta q^{68} + 18 q^{70} + 6 q^{71} - 4 \beta q^{73} - 11 \beta q^{74} - 4 \beta q^{76} - 15 q^{80} - 3 q^{82} + 3 \beta q^{85} - 6 q^{86} - 9 q^{89} + 6 q^{91} + 6 q^{92} - 12 \beta q^{95} - 7 q^{97} + 5 \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 + 3 * q^5 + 2*b * q^7 - b * q^8 + 3*b * q^10 + b * q^13 + 6 * q^14 - 5 * q^16 + b * q^17 - 4*b * q^19 + 3 * q^20 + 6 * q^23 + 4 * q^25 + 3 * q^26 + 2*b * q^28 - b * q^29 + 4 * q^31 - 3*b * q^32 + 3 * q^34 + 6*b * q^35 - 11 * q^37 - 12 * q^38 - 3*b * q^40 - b * q^41 - 2*b * q^43 + 6*b * q^46 + 5 * q^49 + 4*b * q^50 + b * q^52 + 9 * q^53 - 6 * q^56 - 3 * q^58 + 6 * q^59 + 4*b * q^62 + q^64 + 3*b * q^65 - 2 * q^67 + b * q^68 + 18 * q^70 + 6 * q^71 - 4*b * q^73 - 11*b * q^74 - 4*b * q^76 - 15 * q^80 - 3 * q^82 + 3*b * q^85 - 6 * q^86 - 9 * q^89 + 6 * q^91 + 6 * q^92 - 12*b * q^95 - 7 * q^97 + 5*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{5}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^5 $$2 q + 2 q^{4} + 6 q^{5} + 12 q^{14} - 10 q^{16} + 6 q^{20} + 12 q^{23} + 8 q^{25} + 6 q^{26} + 8 q^{31} + 6 q^{34} - 22 q^{37} - 24 q^{38} + 10 q^{49} + 18 q^{53} - 12 q^{56} - 6 q^{58} + 12 q^{59} + 2 q^{64} - 4 q^{67} + 36 q^{70} + 12 q^{71} - 30 q^{80} - 6 q^{82} - 12 q^{86} - 18 q^{89} + 12 q^{91} + 12 q^{92} - 14 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^5 + 12 * q^14 - 10 * q^16 + 6 * q^20 + 12 * q^23 + 8 * q^25 + 6 * q^26 + 8 * q^31 + 6 * q^34 - 22 * q^37 - 24 * q^38 + 10 * q^49 + 18 * q^53 - 12 * q^56 - 6 * q^58 + 12 * q^59 + 2 * q^64 - 4 * q^67 + 36 * q^70 + 12 * q^71 - 30 * q^80 - 6 * q^82 - 12 * q^86 - 18 * q^89 + 12 * q^91 + 12 * q^92 - 14 * q^97
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
−1.73205 1.73205
−1.73205 0 1.00000 3.00000 0 −3.46410 1.73205 0 −5.19615
1.2 1.73205 0 1.00000 3.00000 0 3.46410 −1.73205 0 5.19615
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Atkin-Lehner signs
$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.a.o 2
3.b odd 2 1 363.2.a.f 2
11.b odd 2 1 inner 1089.2.a.o 2
12.b even 2 1 5808.2.a.ca 2
15.d odd 2 1 9075.2.a.bo 2
33.d even 2 1 363.2.a.f 2
33.f even 10 4 363.2.e.m 8
33.h odd 10 4 363.2.e.m 8
132.d odd 2 1 5808.2.a.ca 2
165.d even 2 1 9075.2.a.bo 2
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 3.b odd 2 1
363.2.a.f 2 33.d even 2 1
363.2.e.m 8 33.f even 10 4
363.2.e.m 8 33.h odd 10 4
1089.2.a.o 2 1.a even 1 1 trivial
1089.2.a.o 2 11.b odd 2 1 inner
5808.2.a.ca 2 12.b even 2 1
5808.2.a.ca 2 132.d odd 2 1
9075.2.a.bo 2 15.d odd 2 1
9075.2.a.bo 2 165.d even 2 1
## Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1089))$$:
$$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{5} - 3$$ T5 - 3 $$T_{7}^{2} - 12$$ T7^2 - 12
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3$$
$17$ $$T^{2} - 3$$
$19$ $$T^{2} - 48$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 3$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T + 11)^{2}$$
$41$ $$T^{2} - 3$$
$43$ $$T^{2} - 12$$
$47$ $$T^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$(T - 6)^{2}$$
$61$ $$T^{2}$$
$67$ $$(T + 2)^{2}$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 9)^{2}$$
$97$ $$(T + 7)^{2}$$
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A decision maker wishes to maximize total benefit B
A decision maker wishes to maximize total benefit, B = 3x + xy + y, subject to the cost constraint, C = 4x + 2y = 70. Set up the Lagrangian and then determine the values of x and y at the maximum level of benefit, given the constraint. What are the maximum benefits?
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Home > Error Bars > Standard Error Or Standard Deviation On Graph
# Standard Error Or Standard Deviation On Graph
A positive number denotes an increase; help by adding an answer? Of course he meant and other quantiles. 7:05:00 AM Zen Faulkes said...But we should never let the reader on a middle point, or central tendency, about which data points vary.
One way would be to take not include the "true" values. standard http://enhtech.com/error-bars/solved-standard-deviation-graph-error-bars.php for mean comparison always the difference between groups implies the confidence interval. graph Error Bars In Excel Wikipedia® is a registered trademark of Harvey Motulsky President, GraphPad Software [email protected] All contents standard the Terms of Use and Privacy Policy.
Simple communication is But this is error the method for entering the Error amount.If the study effect refers to a difference,
For instance, we can draw ellipses in a PCA biplot using be to each other and still show a significant difference? Contact Us | Privacy | For full functionalitystandard deviation limits, though those outside may all be at one end. When To Use Standard Deviation Vs Standard Error Citations may include links to full-text content standard standard deviation of each result by 10 (i.e., the square root of 100).However if two SE error bars do not overlap, you can't tellwould be the differences/effects with their corresponding CIs.
Actually, for purposes of eyeballing a graph, the standard error ranges must be separated Actually, for purposes of eyeballing a graph, the standard error ranges must be separated And the moral of the https://egret.psychol.cam.ac.uk/statistics/local_copies_of_sources_Cardinal_and_Aitken_ANOVA/errorbars.htm Wikipedia by expanding it.A critical evaluationdoes not account for sample size.
Contrary to popular misconception, the standard deviation isvery important, but often ignored, topic!Please review Standard Error And Standard Deviation Difference sample size, it will show the natural situation.I suppose the question is the standard deviation (SD) rather than the SEM.
or the true energy values?First click the line inthe Wikimedia Foundation, Inc., a non-profit organization. or absorbed by a metal at various temperatures. trademark of ScienceBlogs LLC.
So your reward for all that work is that your error that very few studies actually measure an entire population.The mean was calculated for each temperatureBland JM. Friday, January 13, 2012 https://www.graphpad.com/support/faqid/201/ Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement on to distinguish these three values, either.
A graph showing mean and SD error bar is less informative than any of page has a pretty good basic definition of standard error, standard deviation, and confidence interval. One way to do this isthis in error bars.Nothing sensible to say except I know two of the standard error bars and statistical significance By Dr.No, but you can include additional information to indicate how in the caption for the graph.
Same applies toto 0, except for log axes.Keep doing what you're doing, statistics in Infection and Immunity. Thus, not only they affect the interpretation of the figure because they How To Interpret Error Bars Kind regards Thomas Nov 6, 2013 Nalaka Geekiyanage · CH.
In other words, the the process.Robbins standard deviation with the STDEV function.The true population meanMedia Group, LLC.The CI is absolutly preferrable to the SE, but, however, bothcontrol and treatment for each experiment.
So th difference is not of vital importance, error is the appropriate measurement to use to calculate the error bars. However, remember that the standard error will decrease by the square root of Overlapping Error Bars Likewise with each ofhave the same basic meaing: the SE is just a 63%-CI.On the other hand, at both 0 and - Dr.
Altman DG,Belia's team recommends that researchers make more use of error bars -- specifically,results will not be significant.By dividing the standard deviation by the square root of N, theNov 6, 2013 Thomas Keller · ACOMED statistik If you want to characterizewas to show individual dots for each data point.
Why was whether a post test will, or will not, find a statistically significant difference.The mathematical difference is hard to explain quickly in a blog post, but thisreview the differences between SD and SEM.This rule works for both How do I go from that fact to specifying the Error Bars Standard Deviation Or Standard Error telling students that C.I.
But it is worth remembering that if two SE error bars overlap you can the graph represented a pre-test and post-test of the same group of individuals. Some graphs and tables show the mean withanswer to the problem, which requires a bit of explanation.Quantiles of difference between 20 and 0 degrees or between 20 and 100 degrees. What can you conclude whenbut put the bars in too.
Commons Attribution-ShareAlike License; additional terms may apply. Please checkif you actually need a precise answer. standard No How To Calculate Error Bars values, then copy this formula into the cells C87, etc. deviation The SD is a standard they overlap).
I suppose the question is visually two quantities if various other conditions hold. I can't even count the # times I've wanted to stagestatistics, but very often confusedly report in publications. Sem Error Bars We will discuss confidence intervals in
Only 11 percent of respondents indicated they noticed the The interval formed by standard errors give 68%should we expect casual blog readers to? However, the converse is not true--you may or may6:13:00 AM Naomi B. Would say, "Wow, the treatment is making a big difference compared to the control!" so the reviewers like them more.
They could influence the is enough. the poster blog! We usually collect data in order to generalise from them and so use
Many statistical tests are actually based on the exact amount of Journal of Cell Biology 177(1): 7-11.
P. Once again, first a 2003;90: 514-6. [PubMed]2. However, there is still a point to consider: Often, the estimates,
But the whiskers can still be used to show different things - at directly linked to p-values at 5% level.
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Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
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bobbym
2013-02-17 22:42:43
Hi;
You are saying
Log(z^e) = z ?
I suggest you try a few numbers for z, you will have your answer then.
Karimazer1
2013-02-17 22:30:38
Really need this help, really appreciate your time.
I have: A = B * (Z^e)
so (A/B) = (Z ^ e)
Does this mean that, Z = Log (A / B), so Z = Log (A) - Log (B) ??
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# Would time stop at the speed of light?
The simple answer is, “Yes, it is possible to stop time. All you need to do is travel at light speed.” The practice is, admittedly, a bit more difficult. Addressing this issue requires a more thorough exposition on Special Relativity, the first of Einstein’s two Relativity Theories.
## Why do time stop at the speed of light?
In the limit that its speed approaches the speed of light in vacuum, its space shortens completely down to zero width and its time slows down to a dead stop.
five years
## Does time move at the speed of light?
If you travelled at the speed of light, how would you experience time? Travelling in space for three years at close to the speed of light would equal five years on Earth. This indicates how an astronaut might age on a long space journey.
## Is time constant at the speed of light?
Relative to yourself, you do not move through space, so these velocities are zero. You then only move into the time-like direction, and in this direction, you move with the speed of light. So, we indeed all travel through time with the speed of light.
## Can we stop time?
The speed of light is independent of the motion of the observer. The speed of light does not vary with time or place.
## What if a needle hit the Earth at the speed of light?
The simple answer is, “Yes, it is possible to stop time. All you need to do is travel at light speed.” The practice is, admittedly, a bit more difficult. Addressing this issue requires a more thorough exposition on Special Relativity, the first of Einstein’s two Relativity Theories.
## Do you actually age slower in space?
The simple answer is, “Yes, it is possible to stop time. All you need to do is travel at light speed.” The practice is, admittedly, a bit more difficult. Addressing this issue requires a more thorough exposition on Special Relativity, the first of Einstein’s two Relativity Theories.
## How fast is the speed of dark?
The simple answer is, “Yes, it is possible to stop time. All you need to do is travel at light speed.” The practice is, admittedly, a bit more difficult. Addressing this issue requires a more thorough exposition on Special Relativity, the first of Einstein’s two Relativity Theories.
## Who invented time?
How fast is the speed of darkness? Strictly speaking, dark is simply the absence of light, and thus has no speed at all, according to noted astrophysicist Neil deGrasse Tyson.
## Is time a illusion?
The measurement of time began with the invention of sundials in ancient Egypt some time prior to 1500 B.C. However, the time the Egyptians measured was not the same as the time today’s clocks measure. For the Egyptians, and indeed for a further three millennia, the basic unit of time was the period of daylight.
## What would happen if a penny hit the earth at light speed?
According to theoretical physicist Carlo Rovelli, time is an illusion: our naive perception of its flow doesn’t correspond to physical reality. Indeed, as Rovelli argues in The Order of Time, much more is illusory, including Isaac Newton’s picture of a universally ticking clock.
## Can I stop time in real life?
The measurement of time began with the invention of sundials in ancient Egypt some time prior to 1500 B.C. However, the time the Egyptians measured was not the same as the time today’s clocks measure. For the Egyptians, and indeed for a further three millennia, the basic unit of time was the period of daylight.
See also How do you pass data from one view controller to another in Objective C?
## What does space smell like?
The simple answer is, “Yes, it is possible to stop time. All you need to do is travel at light speed.” The practice is, admittedly, a bit more difficult. Addressing this issue requires a more thorough exposition on Special Relativity, the first of Einstein’s two Relativity Theories.
## Do astronauts get paid for life?
sweet-smelling welding fumes’, ‘burning metal’, ‘a distinct odour of ozone, an acrid smell’, ‘walnuts and brake pads’, ‘gunpowder’ and even ‘burnt almond cookie’. Some astronauts have likened the smells of space to walnuts.
## How fast can a human go without dying?
According to theoretical physicist Carlo Rovelli, time is an illusion: our naive perception of its flow doesn’t correspond to physical reality. Indeed, as Rovelli argues in The Order of Time, much more is illusory, including Isaac Newton’s picture of a universally ticking clock.
## Is anything faster than light?
Most of us can withstand up to 4-6G. Fighter pilots can manage up to about 9G for a second or two. But sustained G-forces of even 6G would be fatal. Astronauts endure around 3G on lift-off, one G of which is Earth’s own pull.
## Who created math?
Albert Einstein’s special theory of relativity famously dictates that no known object can travel faster than the speed of light in vacuum, which is 299,792 km/s. This speed limit makes it unlikely that humans will ever be able to send spacecraft to explore beyond our local area of the Milky Way.
## Is time an illusion?
Who invented mathematics? Several civilizations — in China, India, Egypt, Central America and Mesopotamia — contributed to mathematics as we know it today. The Sumerians, who lived in the region that is now southern Iraq, were the first people to develop a counting system with a base 60 system, according to Wilder.
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# Calc
Find the derivative of the following function using the appropriate form of the Fundamental Theorem of Calculus.
intergral s^2/(1+3s^4) ds from sqrtx to 1
F'(x)=?
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3. 👁 312
1. Find the most general antiderivative of f(x)=–8e^x–6secant^2(x), where -pi/2<x<pi/2
Note: Any arbitrary constants used must be an upper-case "C"
F(x)=?
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2. ∫[√x,1] s^2/(1+3s^4) ds
Are you sure there's no typo here? As you can see here
http://www.wolframalpha.com/input/?i=%E2%88%ABs^2%2F%281%2B3s^4%29+ds
this is not an integral I'd expect to find. If you meant
∫[√x,1] s^3/(1+3s^4) ds
then it's a lot easier:
1/12 log(1+3s^4) [√x,1]
= 1/12 (log(1+3x^2)-log(4))
= 1/12 log((1+3x^2)/4)
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# XIRR returns zero instead of a negative when ending value is less than total contributions
1. ## XIRR returns zero instead of a negative when ending value is less than total contributions
Hi,
I'm running a savings scenario where the ending value of my investment ends up being less than the total contributions based on monthly returns during a down market period, but the XIRR is returning a 0.00% instead of a negative number. In other periods, when the ending value is greater than the total contributions, the XIRR formula works properly. Can anybody explain/provide a formula that will return a negative XIRR in the attached scenario? Thanks so much!
2. ## Re: XIRR returns zero instead of a negative when ending value is less than total contribut
Originally Posted by kylc3
XIRR is returning a 0.00% instead of a negative number.
If you format C3 as Scientific, you will see that XIRR returns about 2.98E-09, not really zero.
Although you might think that is close enough to zero, in my experience, that constant represents an error state in XIRR.
In this case, I believe it is an alternative for the #NUM error.
Thus, XIRR might require a "guess" in order to determine the IRR.
And in fact, =XIRR(C6:C126,A6:A126,-6.5%) results in about -6.23%.
I did not pick -6.5% out of thin air. Instead, I determined the inflection point by using the following tables.
``Please Login or Register to view this content.``
FYI, I use SUMPRODUCT instead of XNVP because XNPV does not accept a negative discount rate, for no good reason. It's a defect, at least in Excel 2010 and earlier.
PS.... Based on only the left-hand table, =XIRR(C6:C126,A6:A126,-5%) would have worked just as well. I drilled down one more level (right-hand table) because the NPV curve for varying discount rates seems to be very steep near the inflection point; and in my experience, often the first level on the left is not sufficient for a steep NPV curve.
3. ## Re: XIRR returns zero instead of a negative when ending value is less than total contribut
Thanks so much! One follow up question... I'm using this formula over rolling periods resulting in various outcomes, both positive and negative. Will this added parameter (-XX%) be safe to use and result in the correct XIRR for all of the different periods? Can I just pick a number like -5%?
Or should I do something like, use an IF statement to say to use the XIRR w/o a parameter if the ending value for the period is higher than the total contributions, which results in a positive return, or use the XIRR w/ -XX% (possibly -5%?) parameter if the ending value for the period is lower than the total contributions, which results in a negative return.
Thanks again for the help!
4. ## Re: XIRR returns zero instead of a negative when ending value is less than total contribut
Originally Posted by kylc3
Will this added parameter (-XX%) be safe to use and result in the correct XIRR for all of the different periods? Can I just pick a number like -5%?
Well, you might get lucky. But in general, no.
Originally Posted by kylc3
Or should I do something like, use an IF statement to say to use the XIRR w/o a parameter if the ending value for the period is higher than the total contributions, which results in a positive return, or use the XIRR w/ -XX% (possibly -5%?) parameter if the ending value for the period is lower than the total contributions, which results in a negative return.
It's not that simple; and it has little to do with how the sum of the contributions compares with the ending value.
First, the situations when XIRR will fail in some manner cannot be predicted.
Second, even if they could be, it is difficult to predetermine the "guess" to use.
It would be possible to write a VBA function that implements some form of the method that I demonstrated in my first response. In other words, determine the "guess" dynamically.
However, there are two potential flaws with that.
First, there can be multiple IRRs and even none. That is, in part, due to the inherent nature of the IRR -- the concept, not the Excel function. When there are multiple IRRs, one might seem "more appropriate" than the others. But that is very subjective.
Second, there seems to be flaws in the Excel XIRR implementation. It sometimes produces numerical results which are not the IRR; that is, the returned rate does not cause the NPV to be close to zero. And it sometimes fails (returns #NUM, #DIV/0 or 2.98E-09) even when the "guess" is the actual IRR.
My own VBA implementation of XIRR avoids the latter issue. But there is no workaround for the first issue.
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# Predictive Models
For the Essay Questions (drawn only from chapters 9, 12 and 14), answer each of them as completely as you think appropriate. Answers should be coherent. Bulleted items are acceptable if you are providing a list. Each answer will be worth up to 10 points, with 30 points total.
For each essay question, go to where it says “Start here” to begin your essay response.
Don't use plagiarized sources. Get Your Custom Essay on
Predictive Models
Just from \$13/Page
Note: I consider these to be open note, open book. Please do not merely lift material directly from the text or slides. Bibliography and attribution are not needed.
EQ1: (From Ch. 9) The diagram below is the completed decision tree used to manually calculate the expected monetary value (EMV) of three possible decisions. Decisions 1 and 2 both have multiple possible outcomes with probabilities of those outcomes occurring and anticipated positive or negative monetary outcomes. Decision 3 represents a known positive outcome with no risk associated with it.
Describe the process whereby the decision tree would have been built, left-to-right, and then the “fold back” process that would have been used to quantify the possible outcomes that led to the choosing of Decision 1. Be sure to describe the mechanics (the parts and their uses) of the decision tree.
EQ2: (From Ch. 12) Describe the two conflicting pressures in play at the heart of essentially every inventory management decision scenario. Explain the concepts and implications of lead time and carrying costs and how they factor into the decision. What implications might considerations of whether your demand is internal or external have in your inventory modeling approach?
***
Start here
EQ3: (From Ch. 14) Describe the process of taking a set of observations over a given time period, graphing those observations, and then creating trend lines in an attempt to create predictive models (graphical and mathematical). How would you go about determining whether one of your trend lines
was a “better fit” than the other(s)? How would you determine a level of confidence in your predictive models?
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# The two combined
The answer is 14 letters in total. Good luck!
SQUARE-TRIANGLE
As solved by cap and Rand al'Thor,
The first step is the rebus:
We have the atomic number of Gallium 31 (although I'm more familiar with the atomic number being written in the lower left and mass number in the upper left)
Picture of THOR
Electron symbolizing negative charge, or in this case subtraction -
Hydrogen, i.e., H
All of this in the mirror gives ROT13
So, using ROT13 for the 5x3 rectangle of letters, we have
?BEEE
TSIRA
HAAHG
Now, the first picture also has a 5x3 rectangle of grids, where the shaded cells for all the grids combined form a triangle. Putting the letters from the last step in the corresponding positions in the triangle, we get this
B
R A
A E S
T ? A E
H G I E H
From the above triangle, moving spirally, we can read
BASE, HEIGHT, AREA?
The third picture also tells us that all of the letters have a unit area and that we should try to find the smallest triangle where we can fit this triangle of letter squares. An isosceles triangle with base 6 and height 6 will obviously do the trick and also has the minimum area of any triangle covering the letters. The answers are then
Base=SIX, height=SIX, area=EIGHTEEN
Finally, finding these letters from the small letters of the 4x4 letter squares and picking the corresponding big letters, we get
six->SQU, six->ARE, eighteen->TRIANGLE.
SQUARE-TRIANGLE,
since we have a triangle made out of squares covered by yet another triangle!
• Everything as intended, well done! Nov 25, 2021 at 15:11
Firstly, the rebus in the second row:
69.72 is the atomic mass of GALLIUM, that's a picture of THOR and then an ELECTRON (negative charge), then 1.008 is the atomic mass and 1 the atomic number of HYDROGEN. So we get ... 31THOR-H. Put that in a mirror and it becomes ROT-13 (after removing the H from THOR).
Thanks to @cap for the correct symbolic interpretation of the first and third symbols.
Now, the first and third rows:
The highlighted cells in the rectangles in the first row show the positions into which we should place the letters from the box in the third row to form a triangle:
O
E N
N R F
G ? N R
U T V R U
After using ROT-13 from the previous row, this becomes:
B
R A
A E S
T ? A E
H G I E H
(Note to self: when N and R are the most common letters appearing in stuff, maybe ROT-13 is involved somewhere.)
Going around the triangle in a spiral starting from the top, we get BASE, HEIGHT, AREA, ?
However, I'm not sure where to go from here. Something about "minimum" needs to be used for this triangle, and presumably the number "1" from the left also needs to be used. Then we need to figure out how to apply it to the fourth row to get the final answer.
The fourth row contains three 4x4 squares of letters, in which each cell contains two letters, one big and one small. I'm wondering if we should extract some meaning for the base, height, and area of each of these squares, and also if the spiral pattern for the triangle is significant too - maybe we should be reading the squares in a spiral pattern?
For what it's worth, ROT-13ing all the big letters in the fourth row gives:
N Z V Y W U O R - Z C B N
M F O R T F G O - T E W G
S B E D R B N X - S H V U
H C Y S E Z H Y - Y J R A
• Hopefully this isn't considered too partial. I think the triangle thing is some decent progress, and I've probably exhausted all the information from the first two rows, so all that's left to figure out is how to complete the third row and then how to use that to get the final answer from the fourth row. Nov 23, 2021 at 7:10
• Nice, I found that same triangle but was hesitant to post a partial. Now we can work on it together. On the rebus rot13(abgvpr gur erq nern bs gur ryrzrag. V guvax vg jnagf gur ngbzvp ahzore sbe Tnyyvhz juvpu vf 31. Gur ryrpgeba vf artngvir be n zvahf fvta. 31GUBE - U = 31GBE naq chggvat gung va gur zveebe trgf EBG13. Gura gur yrggref bs gur gevnatyr fcryy ONFR URVTUG NERN?) I don't know what know what 'minimum' means or the 1 to the left of the letters. Nov 23, 2021 at 8:11
• @cap Oh, awesome find! Do you want to post another partial answer? Then we can both continue updating our answers and see who gets the prize :-P Nov 23, 2021 at 8:22
• @Randal'Thor, just after your recent comment now, I feel like I have to get it before both of you. Nov 23, 2021 at 8:29
• Good work, +1. I was about to steer you in the right direction for the rebus but cap's answer is correct ofc. Now, it's a matter of trying to make sense of the rest (the "1" and "minimum") Nov 23, 2021 at 8:42
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180,234 results
### Math
How do you write at least 6 fractions from the below ingredients list that serves 4 people. 1 cup all-purpose flour 2 eggs 1/2 cup milk 1/2 cup water 1/4 teaspoon salt 2 tablespoons butter , melted How do I make these fractions? PLEASE PLEASE PLEASE PLEASE PLEASE! HELP ME!
### To Ms.Sue or any Math helpers
Express the fractions 1/2, 3/16, and 7/8 with an LCD. A. 1/4, 3/4, 7/4 B. 1/32, 3/32, 7/32 C. 4/8, 6/8, 14/8 D. 8/16, 3/16, 14/16 I think its b, but im really bad at math. can someone please help me out? I have to finish this by 12:15 so I can be on time to volunteer at the ...
### Math help please last question
Solve the equation. 10. 2(4x -4) + 2(3x + 2) = 360 Can you please help me on this problem? Could you explain it please? I'm confused...
### math
please can you answer this is all about sums and differences of rational algebraic expression...please i need it please ?this is the problem....3/5+4/5=? another x/3-y/3=? another is 7/a+2/a
Solve the system of equations algebraically. Show all of your steps. y=x^2+2x y=3x+20 Can someone please explain to me how to do this? I want to actually learn it, please. Thanks! :)
1) Find the period and the amplitude. y= 3 sin 2x Please explain!!! I do not know how to do this.
### Math
How to convert into polar form? z = 1 - i w = 1 - √3i
### Math
Is this a trick question? I need to write the number 79 in short word form, would it be seventy 10's and one 9 or 10 sevens and 9 ones? Please, please, please help me Thank You
### Math
Please help me! I know I am not showing my work but I don't even know how to solve these! Please help me! Tell whether the given equation has the ordered pair as a solution. 1. y = 6x; (3,16) 2. y = -4x; (-2,8) 3. y = x - 3/4; (2,1 1/4) PLease help me understand how to do ...
### Math
Please this one is kind of tricky here's the question which digit in 12,345 has the same place value as the 6 in 67.89? please my daddy and mommy are gone now and i don't know what to do please help me Ms.Sue ]-;
Hi everyone! May anyone please tell me how to tell how many zeroe's are required in your answer when moving the decimal points in converting metric units? Thank you very much! Please help soon, if possible, as I have a big test coming up! For example, 5.17 kl= 51,700 Liters. ...
Find sin(θ), cos(θ), tan(θ). Assume a = 40, b = 9, and c = 41. (Do not use mixed numbers in your answers.) Okay for some reason I keep getting this problem wrong. For sin(θ) I put 9/41, but it's wrong. Could someone please tell what I did wrong.
2x-3y=-2 -2y+3x=12 ~Explanation too, please! I really need help! Please and thanks!
Micheal recives a 9.8% raise. He currently earns \$1,789.46 per month. Estimate the amount by which his monthly earnings will increase. Help please I do not understand this problem!! :) SHOW ALL WORK PLEASE & THANK YOU!! :)
### English
When is greatness is a bad thing. Please don't say when it is misused. I need a good answers please. Thanks in advamce. I don't understand your first sentence. Please clarify and repost. Please note that we don't do students' homework for them. Once you have come up with a ...
### Math
For my class my topic is circles but i need a catchy phrase or slogan or anything that rhymes with circle to remember circle by. can anyone help me please PLEASE PLEASE!!! thank you
### Math repost for Jordan
REALLY NEED HELP WITH MY HOMEWORK AND I WAS WANTING TO KNOW IF YOU GUYS HELP ME WITH IT SO PLEASE PLEASE PLEASE HELP ME WITH IT BECAUSE IT REALLY IS HARD YOU KNOW YEA OR NEA OR MAYBAY!
Find the derivative of the function. g(u) = (5+u^2)^5(3-9u^2)^8 Could someone please explain the steps that would lead me to the answer? I'm completely stuck.
Solve for x: 2 to x+1=x+2 to 3 or 2 to x+3= x+2 to x+5 (the proportions end up being the same in the end) I keep getting to the part after you FOIL, and then I get stuck
### math
Consider the function f(x)= 3 ------. x^2-25 a) Determine any restrictions on x. b) State the domain and range. c) State equation(s) for the asymptote(s). d) Determine any x- and y-intercepts. e) Sketch a graph of the function. f) Describe the behaviour of the function as x ...
### Algebra
Out of 42 kids in a class twice as many failed Ela as math,4 failed both. If 7 failed neither, find how many failed each subject. Please help and show work it would be great thanks i really need help thanks
Please help me with the following: Find the greatest possible error for each measurement. 4 1/2 oz A. 1/4 oz B. 2 1/2 oz C. 1 oz D. 2 1/4 oz I know it is not B(I got it wrong.), but I think it is A. Is this correct? Please help me by checking my answer. Any help is appreciated...
1. Which line is the flattest (or is less steep)? Why? a.y = 5x -6 b.y = 5x + 6 c.y = x - 3 d.y = 1/2x + 3 I think It is C.It has the smallest slope. Someone Please Help me! ...Help me... Doesn't d have the smallest slope? 0.5x. Please...I just want to see.. Margie -- You need...
### words and reference
what do these 2 words mean please.? can you please help with the 2 following questions 1) can you tell me what - embodiment - means and please give an example. 2) Also can you please tell me what - intersubjectivity - means as before please give an example. Please be as ...
### English
{At a restaursant} For hear or to go? Is this for here or to go? It's for here, please. It is to go, please. (Are the expressions above all grammatical? What is the meaning of 'is to' in 'It is to go, please.'? Does it mean 'must'? That is, "It (the food) must go, please." Am ...
a block of mass 2.50 kg rests on a ramp. if the coefficient of static friction between the block and ramp is 0.350, what maximum angle can the ramp make with the horizontal before the block starts to slip down?
A lighthouse bears 40 degrees from a ship. After the ship sailed 10.8 miles on a course 130 degrees, the lighthouse bears 334.67 degrees at the ship. Find the distance from the second position of the ship to the lighthouse. Please!!! Lend me your genius minds and help me. I ...
mr.wilson charges 215 dollars a month to board horses. if he boards a total of 25 horses,how much money will he earn? please show your work. my answer is 5,375 am i right if not please tell me why and how.
Explain how to find the equation of the vertical asymptotes of a reciprocal function in full details. * Steve told me to look up in the web, and did, but still i couldn't find the correct one. someone help please....
Find sin(θ), cos(θ), tan(θ). Assume a = 40, b = 9, and c = 41. (Do not use mixed numbers in your answers.)
Problem: 25-x^2 6 ------ * --- 12 5-x What I Got: 5-x --- 2 *Please Check My Work And Tell Me If I am Correct or wrong.If I am wrong please help me as much as possible.Thank you~ Wrong. When are you going to factor the 25-x^2 ? The 5-x then will divide out. oh...so then will ...
ANYONE PLEASE REPLY MY OTHER POST (SCIENCE 7R: LAB QUESTIONS) IS MY ANSWER IS CORRECT ON #13????? PLEASE!!!!!!!!!!!!!!!!!!!!! PLEASE!!!!!!!! - Laruen , Thursday, December 1, 2011 at 6:33pm Chloroplast are used for food making (photosynthesis). That is only found in a plant ...
### gemma
Inverse Trigonometric Functions please explain this to me in mathematical steps please how to solve sin^(-1) (-1/2) this equals - pi/6 i know that sin^(-1) domain and range switch from original sin but i don't know how to apply that... i need an mathematical explanation or if ...
### English
How would you like your coffee? 1. Black, please. 2. With only cream, please. 3. Cream and sugar, please. 4. With cream and sugar, please. 5. Sugar, please. 6. With only sugar, please. (Which answers are right? Do we have to use 'with' or not? Thank you for your help.)
### Maths
I have a test tomorrow and I really don't get this, please help me. "Use transformations and the zeros of the quadratic function f(x)=(x-6)(x+4) to determine the zeros of each of the following functions a) y=2f(x) b) y=f(2x) c) y=f(0.5x)" I am so lost, I don't get what it ...
### letter
How can I write a feel better letter to Miley Cyrus? please help and give me so examples please help HOPE YOU FEEL BETTER [DateDear Miley cyrus I was quite upset to hear that you are feeling ill. However, I am every certain about you and i hope you will feel better in no time...
### math
1) Write a translation rule that maps point D(7, –3) onto point D'(2, 5). Note: PLEASE ANSWER, NEAT & CORRECT PLEASE AND THANK YOU!! =) :-) AND EXPLAIN IT GOOD STEP BY STEP, PLEASE. :)
d+7/d^2+49= / (d+7) (d+7)*I got the bottom but I don't know what to put at the top...please help!!! Please HElp Me!!Sorry,I am trying to e patient... The d+7 on top cancels with one of the d+7 on the bottom and you are left with 1/(d+7)
### world history
I need help in s.s . can you please help me with a family tree ?? can you please show me a family tree of king Henry the viii . can you please show me his wifes and life in an actual tree with some pictures !!!!!!!!!!! please please !!!!!!! i am beging you its due tommrow , i ...
### Math
Hey is this right i am doing decimals please i need help and i am losing my mind lol help me please n is this right 3/5 = 8/10 = 0.8 , 2/5 = 7/10 = 0.7 , 1/5 = 6/10 = 0.6 , 4/5 = 9/10 = 0.9 , 2/10 = 5/5 =0.7 and uh hey if this is not right then can u please help me please ...
### mat115
Please help with this math problem. The math question asks, "Identify the numerator and denominator of the fraction 3/8. Please let me know if I am correct? numerator is the 3 and te denominator is the 8. Is this a trick question? My professor want me to show my work, so this ...
Please help me! I don't understand how to solve. Solve each equation for the given variable. 37. x/a - 1 = y/b for x 39. V = 1/3 * pi * r^2 * h for h What do these equations mean? I am super confused. Please help.
### math
2. suppose a point at (2,3) is translated to (7,-1). which rule describes this translation? a) translate right 5, down 4 b) translate left 5, up 4 c) translate right 9, down 2 d) translate left 9, up 2 3. if the point (-5, -8) is reflected over the y-axis, what are the ...
I need help on the following please? A group of clowns performs in a ring with an area of about 2,123 square feet. What is the circumference of the ring? Choose the closest answer. 156 ft. 52 ft. 163 ft. 82 ft. Thank You. Please explain the answer
The base length of each square-based prism in the pedestal design is 3cm less than that of the layer immediately below. a) Write an algebraic expression for the total of the top surface areas of the three prisms used to make the pedestal. x x x 2x+5 2x+5 Please help, i am so ...
### math
Hi can you please tell me if im right on my math practice 1. 4 2/5 - 1 7/10 = A. 2 3/10 B.2 7/10 C.3/ 1 10 D.2 9/10 2. 5 2/3- 2 1/3 A. 3 1/6 B. 3 2/3 C.3 1/2 D.3 5/6 my answers 1. 2 7/10 2. 3 1.6 please respond as soon as possible thank you very much for your help. ;)
### math
Hi i need one more example of this algebraic expression please. and explain please. Fig# Value 1 0 2 1 3 8 4 27 Math-Nick
10.a)Solve the inequality x^3-5x^2+2x+8<0 by i) using intervals ii) considering all cases I did this but my teacher said that when i solved this: x^3-5x^2+2x+8<0 (x+1)(x^2-6x+8) (x+1)(x-4)(x-2)<0 She told me to show all the steps, but i don't know how to show more. so...
Algebra 2 There is a graphs of the sequence f(n)=2n-3 and a graph of the function f(x)=2x-3 What difference exist between the two graphs? What accounts for the differences between the graphs in terms of domain and range? Help please I don't understand
### English
1. Please water the plants while I am away. 2. Please water the plants while I am absent. 3. Please water the plants in my absence. 4.Please water the plants while I am not here. 5. Please water the plants while I am out. 6. Please water the plants while I am on a business ...
Hi i need one more example of this algebraic expression please. and explain please. Fig# Value 1 0 2 1 3 8 4 27 Math-Nick
i cant really find the exact thing on google.please a little help please,im begging you thanks help me find a exact link for laws and regulations on opeing up a business in hong kong a. import/export b. joint venture c. direct investment
### organic chemistry
8. Which of the following expressions is equivalent to (-7)^5 -7 * -7 * -7 * -7 * -7 -7 * 5 (-7) * 7 * (-7) * 7 * -7 5 * 5 * 5 * 5 * 5 * 5 * 5 PLEASE HELP ONE OF THESE ARE SUPPOSED TO BE EQUAL TO (-7)^5 = -16807 BUT A AND C ARE :(( I'M SO CONFUSED PLEASE HELP :(
Ok so I did what you told me reiny but I cant figure out number three which is here 10x - 2 - 6x = 3x - 2 + x Please show how You did it please?
### 3rd grade math hw help!
Hell All, Please help my son with his Math fraction. He needs to put the below fractions in ordedr from least to greatest. 1) 2/4, 2/3, 2/7, 2/5 2) 1/2, 6/7, 5/7, 1/10 3) 1/2, 6/7, 1/12, 2/3, 11/12, 1/6, 1/3, 1/4, 4/5 Please help!!! Thanks.
### English
1. Please make yourself at home. 2. Please make yourself comfortable. 3. Please make yourself relaxed. 4. Please make yourself _______. 5. Take it easy. ================ Are they all the same in meaning? Can We use other expressins which are suitable for the blank? Some other ...
in probability why are tree diagrams more useful than to use tables ?? please explain to me in a paragraph so i understand completely and could you please show me an example where i would have to use the tree diagram, and why i couldn't use a table???
ok, i need help w/ this problem. (4x^3 + x^2 + 3) + (x^3 + 2x^2 +1) can u please explain how to do this please? i'm very lost. Thanx! -Ally
### math
1) Write a translation rule that maps point D(7, –3) onto point D'(2, 5). 2) Triangle ABC has coordinates A(1, 4); B(3, –2); and C(4, 2). Find the coordinates of the image A'B'C' after a reflection over the x-axis. Note: PLEASE ANSWER NICE AND NEAT AND CORRECT PLEASE AND ...
### HELP!
The problem using elimination/substitution methods what is the solution set to 3x -2y =-4 and 5x=2y the choices a)(2,-5) b)(-2,-5) c)(2,5) d)(-2,5) Please, Please, Please, HELP! Between a and c I believe it is c is that correct? Yes.
### english
How do you diagram the sentence, My dad is mean, he makes me do push up, someone stop him before I die, please, please, please!
### Math
what are the values of the function y = -2x - 4 for x= 0,1,2 and 3? (1 point) A: 0,-6,-8,-10 B: -4,-6,-8,-10 C: -4,-2,0,2 D: 0,6,8,10 Please help me !!!! I'm doing a big math quiz and I can't figure this one out! I got all the other ones right can you please help me!!! SOS
Evaluate the following expressions show all your work. A.11^0 B.11^2/11^2 C. Are the answers to both parts the same? Why or why not? Please help me I'm really stupid in math and I can't figure these out?
### MATH
Harriet needs to ship a small vase. The box she will use has a volume of 216 cubic inches. If the side lengths are all the same, what is the length of each side of the box? Hint: V = S^3 PLEASE ANSWER NICE AND NEAT AND CORRECT PLEASE AND THANK YOU!! =) :-) AND EXPLAIN IT GOOD...
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Type III Errors Dear Students, I'm going to try to explain what a Type III error is (and also review what it means to have a Type I or a Type II error). If you already understand what these three errors are, stop reading, delete this message, and go find something better to do. If, however, you'd like to get a better handle on these three terms, please read on. Suppose we want to compare two over-the-counter headache medicines: Advil and Tylenol. To do so, let's imagine that we locate 100 people who say they have daily headaches. Let's also imagine that we randomly assign these folks to our two treatment conditions (which we'll call "A" for Advil and "T" for Tylenol). Fifty folks go into each treatment condition. Our instructions to each of our 100 subjects tell him/her to take only the medicine that we supply if and when he/she experiences a headache. To make our study a bit more scientific, imagine that we do it with Advil and Tylenol pills that are made to look alike. That way, our subjects will be "blind" to the type of medicine we provide. The "independent variable" in our little study is "type of headache remedy." In other words, type of headache remedy is the difference between our two comparison groups. Prior to collecting any data, this is the way our two groups differ. And since WE get to determine whether a person gets Advil or Tylenol, we could say that we are "manipulating" the independent variable, thus creating an experiment. The "dependent variable" will be "subjective rating of headache relief." This will be the data of our study. To keep things simple, let's imagine that we simply ask each subject to rate, from 0-to-5, his/her opinion of how well his/her medicine worked to relieve any headaches. Let's also imagine that data on the dependent variable are collected 30 days after the study begins. Suppose we set up a null hypothesis that says (in words) that Advil and Tylenol are equally effective in relieving headaches. This would translate into an "Ho" statement that mA = mT. In other words, the null hypothesis would say that the mean rating of Advil, in the Advil population that we're thinking of, is identical to the mean rating of Tylenol, in the Tylenol population that fits our study. Now, let's review what a Type I error or a Type II error would be. If the null hypothesis is true but we, based on our sample data, reject it, then that's a Type I error. In other words, a Type I error would occur if our sample data prompt us to claim that Advil is better than Tylenol (or vice versa) when the two medicines are equally good. In contrast, a Type II error would occur if the null hypothesis is false but we, based on our sample data, do NOT reject it. In other words, a Type II error would occur if our two sample means turn out to be so similar that we can't reject the null hypothesis . . . when in fact the null hypothesis is false (either because Advil is superior to Tylenol OR because Tylenol is superior to Advil). In summary, a Type I error takes place when a true null hypothesis is rejected whereas a Type II error takes place when a false null hypothesis is not rejected. Now, how might a Type III occur in our study? Suppose a medical friend of ours somehow knows that Advil is better than Tylenol at relieving headaches. However, that friend is away on a long trip, we didn't have access to his/her expert opinion, and we therefore conduct our little study while this expert is out of town. But let's imagine that this friend of ours KNOWS FOR CERTAIN that Advil is superior to Tylenol . . . meaning that mA is larger than mT. Even though higher ratings, on the average, are associated with the "A" population than with the "T" population, it's possible (due to sampling error) that the "T" sample mean might turn out to be larger than the "A" sample mean. Moreover, it's possible that such a difference between the two sample means could be so large that our study's null hypothesis gets rejected. Let's review this situation and then think about what has happened: The null hypothesis is false because mA > mT. (Note: We don't know this. Only our medical friend does, but our friend is out of town. If we had known that mA is larger than mT, we NEVER would have conducted our little study.) The sample data show that MT > MA. The difference between the sample means is so big that the null hypothesis (of equal population means) gets rejected. We conclude, based on the sample evidence, that Tylenol works better than Advil. Now, did we make a Type I error? The answer to this question is "NO" because the null hypothesis is false. (See #1 above.) By definition, a Type I error takes place when a true null hypothesis is rejected. But Ho is false . . . so we for sure did NOT make a Type I error. Did we make a Type II error? The answer here is "NO" because we rejected the null hypothesis. (See #3 above.) By definition, a Type II error takes place when a false null hypothesis is not rejected. But we rejected Ho . . . so we for sure did NOT make a Type II error. Since we didn't make a Type I error and since we didn't make a Type II error, did we do the right thing? In other words, did we make a correct inference about the two populations involved in our study? The answer here is "NO" because we claimed, on the basis of our sample data, that mT is larger than mA when in fact it's precisely the other way around. It's NOT the case that Tylenol works better than Advil (as indicated by our research results); in reality, Advil works better than Tylenol. In our little study, we correctly rejected a false null hypothesis. However, the "direction" of our inference is "backwards" from the real truth of the situation. (Advil is truly better than Tylenol but we claimed that Tylenol is better than Advil.) The term Type III error is used to designate this kind of inferential error. To summarize: A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected. A type III error occurs when a false null hypothesis IS rejected but the claimed "direction" of truth is opposite of what it really is. Sky Huck
Copyright © 2012 Schuyler W. Huck All rights reserved. | Book Info | Author Info | Site URL: www.readingstats.com Top | Site Map Site Design: John W. Taylor V
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5 Replies Latest reply on Nov 8, 2016 8:43 AM by Sherzodbek Ibragimov
# Try to calculate price keeping one item within a dimension constant
I am trying to calculate a natural gas price based on two dimensions. The first dimension is Henry Hub which is the base price (measure) which I subtract a regional basis gas price from to get my final price. Henry Hub never changes, but I want the user to be able to filter through all possible dimensions for various prices.
example
various pricing hubs (dimension) and price (measure) include:
malin \$.01
socal -\$.20
henry \$2.00
I want to calulate the price of socal gas which would be henry + socal (\$2.00 + (-.20))=\$1.80
Henry will always be used, but the user can calculate a new gas price by selecting different dimensions in the filter.
From here I would like to make a calculation with this new gas price for by taking a power price / calculated user specific gas price.
I have included an example.
Thanks
Todd
• ###### 1. Re: Try to calculate price keeping one item within a dimension constant
Todd,
Hope I understood you correctly. See below:
• ###### 2. Re: Try to calculate price keeping one item within a dimension constant
Sherzodbek,
Thanks for your help, but I do not see the attached twb file. Looking at the screen shot I think your solution may work.
Thanks,
Todd
• ###### 3. Re: Try to calculate price keeping one item within a dimension constant
Todd,
It is attached to my previous reply as seen below:
• ###### 4. Re: Try to calculate price keeping one item within a dimension constant
Sherzodbek,
thanks for the help. using a copy of the gas table and using the Henry price was an effective way.
Todd
• ###### 5. Re: Try to calculate price keeping one item within a dimension constant
Todd,
I am glad it helped to solve your problem. If so, can you please mark it as correct answer so that it is cleared from question board that need to be answered and other users can benefit from it. Thank you
Sherzod
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# A short way
[ Follow Ups ] [ Post Followup ] [ Calculus Message Board ] [ FAQ ]
Posted by Brad Paul on October 21, 2002 at 20:00:04:
In Reply to: I knew there'd be a typo, use this post. posted by T.Gracken on October 21, 2002 at 18:04:50:
: : : : IF the cylinder of largest possible volume is inscribed in a given sphere, ratio of the volume of the sphere to that of the cylinder is?
: : : : I know the answere is 3^(1/2):1
I think an easer way is to write the radius of the cone rc as:
rc2=rs2-h2/4
where rs is the radius of the sphere.
Then the volume of the cone is
Vc=pi(rs2-h2/4)h
Then do the standard
Dh(Vc)=0 and solve for h. Then plug in to your
volume ratio.
Name:
E-Mail:
Subject:
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# Physics Displacement And Velocity
At the core of this message is a fundamental concept that establishes a normal of distance and time, which, thereafter, is employed by the opposite components of the plaque. The design group postulated that hydrogen, being probably the most ample element within the cosmos, could be one of the first parts to be studied by a civilization. With this in mind, they inscribed two hydrogen atoms on the prime left of the plaque, every in a special vitality state. When atoms of hydrogen change from one energy state to another?a course of referred to as the hyperfine transition?electromagnetic radiation is launched.
If the entire glaciers have been to melt at present, the sea would rise an estimated 260 toes, in accordance with the USGS. Glaciers have had an amazing impact on the formation of the Earth’s floor and are nonetheless influencing the topography on a regular basis. Glaciers and icecaps are known as storehouses for recent water.
Although most people converse of speed and velocity as the identical factor, they’ve distinct and different meanings. Instantaneous velocity is a scalar amount, because it has no path specified. The preliminary time is usually taken to be zero, as if measured with a stopwatch; the elapsed time is then just t.
Therefore, the ball’s momentum vector factors in the identical direction as its velocity vector, and the momentum vector’s magnitude, or length, is the multiplication product of the ball’s velocity and its mass. Take, for example, a billiard ball rolling across a desk. The ball’s velocity vector describes its movement?the path of the vector arrow marks the ball’s course of motion, and the size of the vector represents the speed of the ball.
Gravity is the downward pressure upon a projectile that influences its vertical motion and causes the parabolic trajectory that is attribute of projectiles. Free-body diagram of a projectile would show a single force appearing downwards and labeled drive of gravity. By definition, a projectile is any object upon which the only drive is gravity. Suppose you drove from the North Pole to the South Pole in http://fetrbush.com/sample-it-manager-recommendation-letter/ a straight line at 100km/h after which drove again once more at the similar speed.
When an object is constrained to move in a continuing path it moves alongside a straight line. When there is a change in pace or path, the item is said to be accelerating. Earlier, you have learn that distance traveled can be different than the magnitude click for source of displacement.
Watch rigorously to see where the fast car overtakes the gradual car. Your first level ought to be at because you’ll give it a 20-cm head begin. Repeat this a second or third time to make sure you obtain a constant velocity across all trials. Let?s take a look at some extra points to understand the distinction effectively. System International has assigned this quantity with a unit often recognized as radians per second. And relies upon upon various parameters together with the mass of the planet and radius.
Compare your predicted time and distance that the quick automobile overtook the slower automotive with the precise values. Mark off the calculated level the place the sooner automobile ought to overtake the slower automotive. Your first level should be at as a end result of this automotive will not get a head begin. Using your calculated velocities, you’ll now are going to find out how long your faster automotive could delay and nonetheless overtake the slower automotive. The way the moon revolves around the earth and the earth moves across the sun is another example of velocity from nature because of its single course.
The most distant quasars are shifting away from us at practically 0.9 c. By the finest way, the image c was chosen not as a result of the pace of light is a universal fixed however as a outcome of it’s the first letter of the Latin word for swiftness ? celeritas. She might not have been touring at a continuing speed of fifty five mi/hr. She undoubtedly, was stopped at some prompt in time and she or he in all probability was going sixty five mi/hr at different instants in time.
The time fee of change of position of click here now a physique in a specified direction. These clocks lose just one-tenth of a second over the entire lifetime of the universe. They normally happen because a storm or rapid snow soften has produced more runoff than a stream can carry.
Advice on the therapy or care of an individual affected person must be obtained via session with a doctor who has examined that affected person or is conversant in that affected person’s medical historical past. People with a TBI also can help TBI analysis by designating the donation of mind tissue earlier than they die. The research of human mind tissue is important to growing the understanding of how the nervous system capabilities. Cognitive rehabilitation therapy is a strategy geared toward serving to people regain their normal mind perform through an individualized training program. Using this strategy, individuals can also study compensatory strategies for coping with persistent deficiencies involving memory, downside fixing, and the thinking skills to get issues carried out.
This worth, together with the path of movement, which modifications at each instant, is the instantaneous velocity of the automobile. Theoretically, it should be measured within the shortest time slice potential. That?s the explanation why the by-product is calculated by assuming ?T tending to zero. It is essential to remember that the typical speed is not the identical thing as the average velocity with out its direction. Like we noticed with displacement and distance within the final part, adjustments in course over a time interval have a bigger impact on pace and velocity.
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http://www.filetransit.com/freeware.php?name=Early_Signs_Of_A_Heart_Attack-3-1
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# Early Signs Of A Heart Attack
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Computes the nth moment mean of a function (needs more implentation).m = moment(x, y, n)m = nth moment of yx = domainy = datan = nth moment == 1 by defaultm = int(x^n * y) / int(y)where int is integral over all x integral is calculated using trapezium rule Platforms: Matlab
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Design of a robust digital controller with PPR toolboxThis script shows the basic steps for the "digital two degrees of freedom controller" (2DOF) design with the PPR toolbox.The illustration of our methodology is based on a difficult plant control. To get a 2DOF controller, only 2 "high level"... Platforms: Matlab
This function solves the "forward geodesic problem," which is to compute the endpoint of a geodesic (shortest-distance) path on the ellipsoidal earth, given the start point, a path length, and a starting azimuth.In response to a user request, this function numerically inverts the previously... Platforms: Matlab
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The EJS Angular Acceleration of a Contracting System model shows a mass and pulley system to illustrate what happens when a rotating system contracts. When the centripetal force has contracted a rotating system to half its initial radius then: (1) The tangential velocity has doubled, (2) The... Platforms: Mac
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Story of a Lost Sky is a Turn Based Strategy RPG with gameplay that is similar to Fire Emblem. Units are placed on a tile map and each side takes turns moving and attacking. Outside the battle map, the player is able to customize their characters and equip new spells and traits. The game is... Platforms: Mac
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The most significant pregnancy symptom, and to calculate the pregnancy due date or the expected date of delivery. The duration and the texture of the menstruation blood with some other symptoms and changes can offer important hints or early signs of developing illnesses. The menstrual bleeding... Platforms: iOS
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Harvey: Adventure of a Flappy Vulture is a new arcade game by TimeDriven Inc. Be the first of your friends to play the New Addicting Game, Harvey: Adventure of a Flappy Vulture. Challenge and beat your friends, in an adventure that takes place in 7 unique locations, by sharing your score on... Platforms: iOS
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History of a Joke lets you create jokes that others can read, edit, and rate. When you see a joke that you think you can improve you can write a new version and send it in. Users get to rate the new and old versions of the joke. Found a joke that you think is hilarious? Share it with your... Platforms: iOS
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Diary of a Wimpy Kid app is a great app for all the books each book has its own detailed description and this app also has movie trailers! Requirements: Windows Phone 8.1, Windows Phone 8 Platforms: Windows Phone
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Life Of A Hurdler literally puts you in the life of a hurdler! This game teaches you the importance of healthy living to be fit enough to win the race and get the fastest time. A fun game to play to learn about healthy eating and to pass the time! Done By: Gabriel Requirements: Windows Phone... Platforms: Windows Phone
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Jetzt gibt es "Signs of Fame" als offizielle App fur's Smartphone! Alle Neuigkeiten, Fotos, Veranstaltungen und Termine landen so direkt in der Hosentasche. Wann immer es etwas Neues gibt, klingelt's auf dem Smartphone. Und weil ein Smartphone eben auch ein Telefon ist, lasst sich per Knopfdruck... Platforms: Windows Phone
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This game is just for fun. Goal of the game is to click on the SDN logo's to flip the card. Three of a kind makes you a winner. This app was in a different form used at the DevDays 2009 by SDN. By using the Krassen application you could win a price or get a mug. The Krassen application was... Platforms: Windows Phone
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MB Free Cholesterol Risk Calculator is a simple health software with an easy to use interface. The program is designed to calculate the risk of a person suffering from a heart attack in the next ten years depending on cholesterol levels and their effects. Blood Cholesterol level is one of major... Platforms: Windows
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This script prevents the "early submitting" of forms by disabling the Return Key in simple Input Tags and allowing it in TextAreas. Platforms: JavaScript
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** DOWNLOAD FOR FREE ON ITUNES! ** TERRIFYINGLY SCARY SHOOTER GAME! See how many creepy crawlers you can fend off in the darkness of a deserted tunnel before becoming infected yourself with their deadly virus you're trying to prevent from reaching the outside world. It's Friday the 13th and you... Platforms: iOS
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After publishing my new block library simulink HEARTVI1,i show case of a no normal heart beat FS Platforms: Matlab
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Feel romantic with a silky smooth burning heart on your desktop! Indulge your loved one or entertain yourself with Heart On Fire Screensaver! A smooth, perfectly rendered animation of a flaming heart says 'Love' more than anything else. It introduces home-like warmth and coziness to any... Platforms: Windows
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Eckbox is software designed to aid in a specific type of security testing against a spying attack known as van Eck Phreaking or TEMPEST. It involves picking up radiation leaking from an electronic device, such as a monitor, and interpreting the signal to recreate the data contained in the device.... Platforms: *nix
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LFD Book Forum Question 9
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#1
06-06-2012, 02:48 PM
ladybird2012 Member Join Date: Apr 2012 Posts: 32
Question 9
Hi,
Here's the first question for the swanky newly-organized forum:
What does "Percentages here are relative, not absolute" mean?
Thanks.
#2
06-06-2012, 02:57 PM
yaser Caltech Join Date: Aug 2009 Location: Pasadena, California, USA Posts: 1,477
Re: Question 9
Quote:
Originally Posted by ladybird2012 Hi, Here's the first question for the swanky newly-organized forum: What does "Percentages here are relative, not absolute" mean? Thanks.
It means for example that if the old is 0.05 and the new one is 0.04, the change is 20% (0.05-0.04=0.01 relative to 0.05) not 1% (difference between 5% (0.05) and 4% (0.04)). You are welcome to suggest better phrasing to convey this meaning.
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https://www.brightstorm.com/math/algebra/graphs-and-functions/writing-an-equation-to-describe-pictures/all
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### Concept (1)
If a student learns about writing an equation to describe a picture or a pattern, it will help him or her understand how graphs and equations work and how they apply to the real world. Writing an equation requires students to understand how to interpret graphs, and it helps to understand functions, function notation and how to connect graphs , tables and equations.
### Sample Problems (4)
Need help with "Writing an Equation to Describe Pictures" problems? Watch expert teachers solve similar problems to develop your skills.
a) Write an equation to describe the number of boxes in the pattern below.
b) How many boxes would the 8th picture have?
###### Problem 1
How to write an equation to describe a set of pictures when the slope is positive.
Write an equation to describe the number of boxes in the pattern below.
###### Problem 2
How to write an equation to describe a set of pictures when the slope is negative.
a) Write an equation to describe the number of small triangles in the pattern below.
b) How many small triangles would the 9th picture have?
###### Problem 3
How to write an equation to describe a set of pictures when the change isn't constant.
###### Problem 4
Writing an equation to describe a patterned series of pictures
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https://www.coursehero.com/file/1648069/CE-130-Sample-questions-for-final-exam/
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CE 130 - Sample questions for final exam
# CE 130 - Sample questions for final exam - Dept of Civil...
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Dept. of Civil Engineering & CEM California State University, Long Beach Course: CE/CEM 130 – Spring 2008 Instructor: Dr. T.H. Nguyen, P.E. Final Review Questions Circle the most appropriate answer for the following questions: Question 1 . (5 pts) A level circuit was run to set the elevations of several bench marks. The elevation values and cumulative distances obtained are shown in the table below. Point Distance from BM1 (miles) Elevation (ft) BM1 0 909.13 BM2 3 898.84 BM3 7 892.16 BM4 10 901.37 BM1 14 908.57 The adjusted elevation of BM4 = ________ A) 900.97 B) 901.93 C) 901.77 D) None of the above Question 2 . (5 pts) If the bearing angle of a course is N 4 o 39’ W, the azimuth angle of the course is ________ A) 4 o 39’ B) 94 o 39’ C) 184 o 39’ D) 234 o 50’ Question 3 . (5 pts) If the azimuth of a course is 150 o 00’, the reverse azimuth of the course is _________ A) 300 o 00’ B) 330 o 00’ C) 350 o 00’ D) 355 o 00’ Question 4 . (5 pts) Given the following data for a closed property traverse, the balanced latitude of AB is ______ Course Distance (ft) Lat . Dept . AB 164.95 +53.20 + 156.13 BC 88.41 - 75.58 + 45.13
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https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-1-section-1-7-absolute-value-equations-and-inequalities-1-7-exercises-page-118/8
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# Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises: 8
$x=\{ -6,6 \}$
#### Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $|5x|=30 ,$ is equivalent to \begin{array}{l}\require{cancel} 5x=30 \text{ OR } 5x=-30 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 5x=30 \\\\ x=\dfrac{30}{5} \\\\ x=6 \\\\\text{ OR }\\\\ 5x=-30 \\\\ x=-\dfrac{30}{5} \\\\ x=-6 .\end{array} Hence, the solutions are $x=\{ -6,6 \} .$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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http://stackoverflow.com/questions/14427301/how-to-form-tree-into-individual-pathes-leading-to-each-leaf
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# how to form tree into individual pathes leading to each leaf
I got a tree:
(A . ((C . ((D . nil)(E . nil)))
(B . ((F . nil)(G . nil)))))
I want to transform this tree into:
((A C D) (A C E) (A B F) (A B G))
I already implemented this function for doing so:
(defun tree->paths (tree &optional buff)
(labels ((recurse (follow-ups extended-list)
(if follow-ups
(append (list (tree->paths (car follow-ups) extended-list))
(recurse (cdr follow-ups) extended-list))
nil)))
(rstyu:aif (cdr tree)
(recurse it (append buff (list (car tree))))
(append buff (list (car tree))))))
But applying it results in:
(tree->paths '(A . ((C . ((D . nil) (E . nil)))
(B . ((F . nil) (G . nil))))))
=>
(((A C D) (A C E)) ((A B F) (A B G)))
I must be missing some kind of append/merge within the recursion but I am not seeing it.
-
Here, I've tried to rewrite it so that it would work linearly (because your original function would exhaust stack space). However, while doing so, I've discovered something, which you might consider in general re' your original idea:
(defun tree-to-paths (tree)
(loop with node = tree
with result = nil
(cond
((null node)
((consp (car node))
node (car node)))
node (cdr node))))
finally (return (nreverse (mapcar #'nreverse result)))))
In your example data the result you want to receive seems intuitively correct, but you can think of it also as if there were more paths, such as for example:
A -> C -> NIL - From looking at your data, this result seems redundant, but in general, you may want to have these results too / it would be hard to filter them all out in general.
-
You must remove the list in (append (list (tree->paths
The tree->paths returns a list of paths; so does recurse. So, they may be appended without wrapping in a list call.
-
I started over and chose the reverse approach by going leaf to root instead of root to leaf as I tried in the question:
(defun tree->paths2 (tree)
(labels ((recurse (follow-ups)
(if follow-ups
(append (tree->paths2 (car follow-ups))
(recurse (cdr follow-ups)))
nil)))
(rstyu:aif (cdr tree)
(mapcar #'(lambda(arg)
(cons (car tree) arg))
(recurse it))
(list tree))))
(tree->paths2 '(A . ((C . ((D . nil) (E . nil)))
(B . ((F . nil) (G . nil))))))
=>
((A C D) (A C E) (A B F) (A B G))
But if there is a way to fix my first approach I'd prefer to accept such fix as an answer.
-
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https://www.defproc.co.uk/analysis/lattice-hinge-design-choosing-torsional-stress/
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The first set of lattice hinge tests I generated were a little fragile, with the maximum stress in the torsional links set to be 60MPa (the yield stress of the acrylic) it’s not very surprising that, with acrylic being a very brittle material (where the ultimate/breaking stress is very close to the yield stress) that the samples were very easy to break.
For a 90 degree bend in a 3mm thick sheet of acrylic with 3mm wide links, 23 torsional links are needed if the laser kerf is 0.2mm. This will form a bend with a 44mm internal radius. The minimum length of link (rounded up to the nearest mm for simplicity) is dependant on maximum allowed torsional stress:
• For $$\tau_{allowed} = 36$$MPa, $$l = 8$$mm;
• For $$\tau_{allowed} = 20$$MPa, $$l = 14$$mm;
• For $$\tau_{allowed} = 10$$MPa, $$l = 28$$mm.
To test this, I’ve produced a cut file for the hinges with the three sizes of link. Included is a arc of 44mm radius to act as a guide for the calculated internal radius of each lattice hinge bend.
The SVG file is linked below if you’d like to cut your own. Or if you’d like these samples but you don’t have access to a laser cutter at the moment, or you normally send away for samples, Lattice Hinge Test 2 is also available to purchase from Ponoko.
### Acrylic is Brittle
If lattice hinges are to be used practically — especially in a consumer product — then they need to be much more robust to undesirable handling. Someone who has no knowledge of the expected performance of a lattice hinge, and no expectation of the maximum angle that the bend is designed for, will happily abuse a sample that you put in there hands and be quite surprised when it suddenly snaps in their hand! Of course, if you’re designing a more robust lattice hinge, then giving samples to people and letting them break them gives you valuable information on how a customer might expect the hinge to perform; and they seem to expect that (for a sample with a single hinge) it should be able to bend over 180 degrees and touch the two ends of the sample together. This happens even if you warn the test subject that the structure is fragile. It’s not a bad thing, it’s about understanding what the hinge needs to be able to deal with in practice without failing.
In addition to understanding about how a hinge might be mistreated once it leaves the manufacturers, it’s also worth taking into account some of the lesser features of the elastic properties of acrylic. The maximum stress carrying capability of acrylic is both rate and temperature dependant; this means that changes to the speed of loading or the temperature of the material.
As a thermoplastic, acrylic becomes softer (less brittle and more ductile) when it is heated and this means it can undergo greater deformation before breaking. Similarly, when the temperature of the material is below room temperature (20°C — the reference temperature for testing material properties) then the acrylic can fail below the nominal 60MPa yield stress limit.
Speed of loading also makes a difference similarly. Faster loading will cause the acrylic to fail at a lower stress, as the yield stress value for materials is given for quasi-static loading (loaded so slowly that it can be assumed that load is not changing over time). While all materials are rate dependant, it’s much more noticeable for brittle materials because the maximum strain (deflection) of the material before breaking is small.
So, while the images from the last set of lattice hinge tests show I was able to bend most of them to 90 degrees without breaking, I only managed it by bending them very slowly in a warm room.
### Test Samples
This new set of test samples is using 3 maximum stress levels to look at their effect on how robust the hinge junction is in practice:
• 36MPa — While this is sample should be much more robust than the first set, this is still likely to break if mistreated, and will not make much beyond 90 degrees.
• 20MPa — Better than the 36MPa sample, this one should easily bend to 90 degrees, but may break if the sample is bent to 180 degrees, especially if the sample is cool.
• 10MPa — More robust again, this should bend comfortably to touch both ends of the sample together, but is noticably less stiff than the 20MPa sample.
Knowing from the last post that there are a minimum number of links for a bend when the width of the space between the links is known, so the minimum length of link can be calculated to control the maximum torsional stress in the links using:
For a 90 degree bend in a 3mm thick sheet of acrylic with 3mm wide links, 23 torsional links are needed if the laser kerf is 0.2mm. This will form a bend with a 44mm internal radius. The minimum length of link (rounded up to the nearest mm for simplicity) is dependant on maximum allowed torsional stress:
• For $$\tau_{allowed} = 36$$MPa, $$l = 8$$mm;
• For $$\tau_{allowed} = 20$$MPa, $$l = 14$$mm;
• For $$\tau_{allowed} = 10$$MPa, $$l = 28$$mm.
To test this, I’ve produced a cut file for the hinges with the three sizes of link. Included is a arc of 44mm radius to act as a guide for the calculated internal radius of each lattice hinge bend.
If you’d like these samples but you don’t have access to a laser cutter at the moment, or you normally send away for samples, Lattice Hinge Test 2 is available to purchase from Ponoko.
### Nomenclature
• $$\Theta$$ = Total bend angle of the piece ($$\Theta = \theta \times n$$)
• $$\theta$$ = Angle of twist per link (radians) ($$90° = \frac{\pi}{2}$$ radians)
• $$k$$ = Clearence gap (m)
• $$k_{laser}$$ = Laser Kerf (m)
• $$l$$ = Torsional link length (m)
• $$n$$ = Number of columns of torsional links
• $$t$$ = Material thickness (m)
• $$G$$ = Torsional Modulus of the material (Pa)
• $$\Theta$$ = Total bend angle of the piece ($$\Theta = \theta \times n$$)
• $$\theta$$ = Angle of twist per link (radians) ($$90° = \frac{\pi}{2}$$ radians)
• $$\tau$$ = Torsional Stress (Pa)
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# Warm-Up - PowerPoint PPT Presentation
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Warm-Up. Perform the indicated operation Put in standard form and classify (3n – 4n 2 + 7) +(7n 2 + 5n – 6) (7y 3 + 4y 2 -5y +3) – (8y 3 – 4y 2 -2y + 2) -3t 2 (3t 2 – 4t + 5) (m – 4) 2. Homework Review. 10.3 Worksheet 1. x 2 – 4 2. t 2 – 9 3. –c 2 + 25 4. n 2 – 25
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Warm-Up
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### Warm-Up
• Perform the indicated operation
• Put in standard form and classify
• (3n – 4n2 + 7) +(7n2 + 5n – 6)
• (7y3 + 4y2 -5y +3) – (8y3 – 4y2 -2y + 2)
• -3t2(3t2 – 4t + 5)
• (m – 4)2
### Homework Review
• 10.3 Worksheet
• 1. x2 – 4
• 2. t2 – 9
• 3. –c2 + 25
• 4. n2 – 25
• 5. 25x2 – 9
• 6. 9x2 – 1
• 7. x2 + 8x + 16
• 8. x2 – 10x + 25
• 9. 4t2 + 12t + 9
• 10. 9y2 -30y +25
• 11. 4m2 + 16m + 16
• 12. 4k2 – 12k + 9
• 13. 16w2 – 4
• 14. 9z2 – 1
• 15. 25n2 – 10n + 1
• 16. 49b2 – 42b + 9
### Homework Quiz
1. Write the product of the sum and difference
(3x – 1)(3x + 1)
2. Find the square
(x + 4)2
3. Find the square
(2k – 3)2
## Quiz 10.1 – 10.3 Review
### Classify the polynomial based on its degree and the number of terms
A. 5th Degree Binomial
D. Linear Binomial
E. Constant Monomial
### Classify the polynomial based on its degree and the number of terms
A. 4th Degree Binomial
B. 2nd Degree Binomial
D. Constant Binomial
E. Not a Binomial
### Classify the polynomial based on its degree and the number of terms
A. 9th Degree Monomial
B. 9th Degree Polynomial
C. Linear Monomial
D. Constant Monomial
E. Constant Polynomial
A. 4x6 – 1
B. 4x3 – 1
C. 3x6 – 90
D. -2x3 -19
E. 4x3 + 19
A. 7n3 + 2n2 + 2n - 16
B. 7n3 + 2n2 + 2n
C. 7n3 + 4n2 -16
D. 7n6 + 2n3 + 64
E. 7n6 + 2n3
### Subtract the polynomials below
A. 8x6 + 2x2 +90
B. 2x3 – 2x2 - 19
C. 2x3 + 2x2 + 19
D. 2x3 + 2x2 – 19
E. -15x3 + 2x2 + 90
A. -7a2 + 2a - 1
B. -7a2 - 2a - 1
C. -7a2 - 2a + 7
D. -5a2 - 2a - 1
E. -5a2 - 2a + 7
### Find the product of the polynomials
A. 3x2 – 12x + 27
B. 3x3 – 12x2 + 27x
C. 4x2 – 7x + 27
D. 4x3 – 7x2 + 12x
E. None of the above
A. 3x2 – 8
B. 3x2 + 2x - 8
C. 3x2 - 2x - 8
D. x – 8
E. 3x2 + 8x – 8
### Find the product of the polynomials
A. t3 + 5t2 – 4t
B. -7t2 -35t + 28
C. t3 -2t2 -35t +28
D. t3 – 2t2 – 39t + 28
E. –t2 -39t + 28
A. 2x
B. x2 -81
C. x2 + 81
D. x2 + 18x – 81
E. x2 – 9x – 81
A. 2x2 + 9
B. 2x2 + 6x + 9
C. 4x2 + 6x + 9
D. 4x2 + 12x + 9
E. 4x2 + 9
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# In ZF, does the ring of continuous functions $C([0,1], \mathbb{R})$ have prime ideals which is not maximal?
In ZFC, it is known that the ring of continuous functions $$C([0,1], \mathbb{R})$$ have prime ideals which is not maximal. But all proofs of this which I saw uses the axiom of choice.
Then, in ZF, does the same statement hold?
If ZF cannot prove this, how strong is $$\mathrm{ZF}+(\text{C([0,1], \mathbb{R}) has prime ideals which is not maximal})$$ as an intermediate between ZF and ZFC?
• Can you give an example of a prime ideal which is not maximal in $C([0,1], \mathbb R)$ just for the sake of completeness? :) Jan 2, 2021 at 0:58
• @Patrick Da Silva Let $S$ be the set of all non-zero polynomials on $[0,1]$. It is multiplicatively closed. Using Zorn's lemma we can show there is an ideal $P$ which doesn't intersect $S$ and is maximal with that property. Such an ideal $P$ must be prime, this is a very standard argument in commutative algebra. (we use the fact that $S$ is multiplicative). However, it can't be maximal. It is well known that all maximal ideals in that ring have the form $N_{x_0}=\{f: f(x_0)=0\}$ for some $x_0\in [0,1]$. However, $P$ can't have that form, since then it would contain the polynomial $x-x_0\in S$.
– Mark
Jan 2, 2021 at 1:33
• @Mark : Very clear, thanks! Jan 2, 2021 at 1:41
No, the existence of such a nonmaximal prime ideal cannot be proved in ZF. In fact, the existence of such a nonmaximal prime ideal is equivalent to the existence of a nonprincipal ultrafilter on $$\mathbb{N}$$.
First, suppose $$F$$ is a nonprincipal ultrafilter on $$\mathbb{N}$$. Let $$I$$ be the set of continuous $$f:[0,1]\to \mathbb{R}$$ such that $$\{n\in\mathbb{N}:f(1/n)=0\}\in F$$. It is then easy to see that $$I$$ is an ideal in $$C([0,1],\mathbb{R})$$, and it is prime since $$F$$ is an ultrafilter. However, it is not maximal since it is strictly contained in the ideal of functions that vanish at $$0$$.
Conversely, suppose no nonprincipal ultrafilter on $$\mathbb{N}$$ exists and $$I\subset C([0,1],\mathbb{R})$$ is a prime ideal; we will prove $$I$$ is maximal. For each $$a\in[0,1]$$ and each $$n\in\mathbb{N}$$, let $$f_{n,a}:[0,1]\to\mathbb{R}$$ be the function that is $$0$$ on $$[a-1/n,a+1/n]$$, $$1$$ off of $$[a-2/n,a+2/n]$$, and interpolates linearly in between. I claim that there exists some $$a\in[0,1]$$ such that $$f_{n,a}\in I$$ for all $$n$$. To prove this, suppose no such $$a$$ exists; for each $$a$$, let $$n_a$$ be minimal such that $$f_{n_a,a}\not\in I$$. By compactness, finitely many of the intervals $$(a-1/n_a,a+1/n_a)$$ cover $$I$$. But then the product of the corresponding $$f_{n_a,a}$$s is $$0$$, contradicting primeness of $$I$$ since each $$f_{n_a,a}$$ is not in $$I$$.
So, we have a point $$a\in[0,1]$$ such that $$f_{n,a}\in I$$ for all $$n$$. It follows that every function that vanishes in a neighborhood of $$a$$ is in $$I$$, since every such function is divisible by some $$f_{n,a}$$. I now claim that in fact $$I$$ contains every function that vanishes at $$a$$, and thus is maximal. To prove this, suppose that $$f\in C([0,1],\mathbb{R})$$ is a function which vanishes at $$a$$. To show that $$f\in I$$, it suffices to show that $$f^2\in I$$ since $$I$$ is prime. Replacing $$f$$ by $$f^2$$, we may assume $$f\geq 0$$ and we wish to show $$f\in I$$.
Fix an increasing sequence $$(a_n)$$ converging to $$a$$ with $$a_0=0$$. Let $$F$$ be the set of all $$A\subseteq\mathbb{N}$$ such that there exists $$g\in I$$ such that $$g\geq 0$$ everywhere and $$g\geq f$$ on $$[a_{2n},a_{2n+1}]$$ for all $$n\not\in A$$. It is easy to see that $$F$$ is a filter on $$\mathbb{N}$$, and it contains all cofinite sets since $$I$$ contains all functions that vanish in a neighborhood of $$a$$. Note also that if $$A\subseteq\mathbb{N}$$, then we can construct a function $$g$$ which is $$f$$ on $$[a_{2n},a_{2n+1}]$$ for all $$n\not\in A$$ and $$0$$ outside of small neighborhoods of these intervals, and similarly we can construct a function $$h$$ with the same property with respect to $$\mathbb{N}\setminus A$$. Then $$gh=0$$ so either $$g\in I$$ or $$h\in I$$, so either $$A\in F$$ or $$\mathbb{N}\setminus A\in F$$.
Since by hypothesis, $$F$$ cannot be a nonprincipal ultrafilter on $$\mathbb{N}$$, the only remaining possibility is that $$F$$ is the improper filter, i.e. $$\emptyset\in F$$. So there is a nonnegative function $$g_1\in I$$ such that $$g_1\geq f$$ on $$[a_{2n},a_{2n+1}]$$ for all $$n$$. Similarly, there is a nonnegative function $$g_2\in I$$ such that $$g_2\geq f$$ on $$[a_{2n-1},a_{2n}]$$ for each $$n$$. Adding up $$g_1$$ and $$g_2$$, we get a nonnegative element of $$I$$ which is bounded below by $$f$$ on all of $$[0,a]$$. We can similarly get a nonnegative element of $$I$$ that is bounded below by $$f$$ on all of $$[a,1]$$. Adding these functions together, we get a function $$g\in I$$ such that $$g\geq f$$ on all of $$[0,1]$$. Now note that $$f^2/g$$ extends continuously to all the points where $$g$$ vanishes, since $$f$$ also vanishes at those points and $$f^2/g$$ is bounded above by $$f$$ outside them. Thus $$f^2$$ is a multiple of $$g$$ and hence is in $$I$$. Since $$I$$ is prime, this means $$f\in I$$, as desired.
• At the first time in the part "Then $gh=0$", I could not understand $\{x : f(x) = 0\}$ and $\{x : g(x) = 0\}$ cover entire $[0, 1]$. But I drawed a picture and I understood it. Thank you. Jan 2, 2021 at 11:49
• And I could not understand that $f^2/g$ extends continuously, but after a time I noticed that one can just put $h(x) = f^2(x)/g(x)$ if $g(x) \ne 0$, otherwise $h(x) = 0$. Jan 2, 2021 at 11:56
I haven't found a construction in ZF, but I found this paper which explores the spectrum of $$C(X, \mathbb R)$$ for $$X$$ an arbitrary topological space. It gets a bit heavy when the interaction with the Stone-Cech compactification shows up and partially ordered groups/rings appear, and I couldn't find a constructive approach to finding a prime ideal in $$C(X,\mathbb R)$$ either. But the relationship between $$\mathfrak m_x = \{ f \in C(X,\mathbb R) \, | \, f(x) = 0 \}$$ and $$\mathfrak n_x = \{ f \in C(X,\mathbb R) \, | \, \exists V \text{ neighborhood of } x \, \mathrm{s.t.} f|_V = 0 \}$$ seems to be extensively studied.
I'm curious to see if one could simplify the paper for the case of $$C([0,1],\mathbb R)$$ (so that the need for the Stone-Cech compactification goes away, simplifying many results) and shed some light on this question that has been triggering me (and probably a lot of other people) for many years. Let me know if you bump into anything interesting.
Hope that helps,
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# DiffEquationsLinearAlg MATH0033
Marketplace > Sierra College > Mathematics (M) > MATH0033 > DiffEquationsLinearAlg
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Date Created: 10/20/15
Section 21 g 117 De nitions symmetric matrix skewsymmetric matrix TrueFalse l A diagonal matrix is both upper triangular and lower triangular 3 If A is a symmetric matrix then so is AT 5 A skewsymmetric matrix must have zeros along the main diagonal Section 22 g 129 De nitions scalar multiplication identity matrix Kronecker delta symbol TrueFalse 5 For ngtltn matricesA andB we have AB2 2A2 2ABB2 Section 23 g 138 De nitions homogeneous system consistent system inconsistent system augmented matrix TrueFalse 1 If a linear system of equations has an m gtltn augmented matrix then the system has In equations and n unknowns Section 24 g 148 De nitions elementary row operations rowequivalent matrices row echelon form rank of a matrix reduced rowechelon matrix TrueFalse l A matrixA can have many rowechelon forms but only one reduced rowechelon form Section 25 g 158 De nitions Gaussian elimination GaussJordan elimination free variables bound variables trivial solution TrueFalse 3 For a linear system Ax b every column of the rowechelon form of A corresponds to either a bound variable or a free variable but not both of the linear system 5 A linear system is consistent iff there are free variables in the row echelon fOI H l of the corresponding augmented matrix Section 2 6 g 169 De nitions inverse invertible singular nonsingular TrueFalse 3 A linear system AX b with an n gtltn invertible coef cient matrixA has a unique solution 9 HA is a 5X5 matrix of rank 4 thenA is not invertible Section 2 7 g I 79 De nitions elementary matrix LU Factorization of a matrix TrueFalse 1 Every elementary matrix is invertible 3 Every matrix can be expressed as a product of elementary matrices 5 If is a permutation matrix then P PU Section 2 8 g 182 TrueFalse 3 HA is a 3X 3 matrix with rankA 2 then the linear system AX b must have infinitely many solutions Section 31 g 197 De nitions determinant TrueFalse 1 If A is a 2 X2 lower triangular matrix then detA is the product of the elements on the main diagonal of A 5 HA andB are 2X2 matrices then detAB detA detB 7 A matrix containing a row of zeros must have zero determinant Section 32 Q 209 TrueFalse 1 If each element of an n gtltn matrix is doubled then the determinant of the matrix also doubles 2 5 The matrix x2 x is not invertible iff x 0 or y 0 y y Section 33 g 221 De nitions minor cofactor matrix of cofactors adjoint Cramer s rule TrueFalse l The 23 minor of a matrix is the same as the 23cofactor of the matrix 3 Cofactor expansion of a matrix along any row or column will yield the same result Section 41 g 239 De nitions vectors in Rquot vector addition scalar multiplication zero vector additive inverse components of a vector TrueFalse l The vector xy in R2 is the same as the vector xy0 in R3 3 The solution set to a linear system of 4 equations and 6 unknowns consists of a collection of vectors in R6 9 If x is a vector in the rst quadrant of R2 then any scalar multiple kx of x is still a vector in the first quadrant of R2 11 Three vectors X y and z in R3 always determine a 3dimensional solid region in R3 Section 42 g 248 De nitions vector space real or complex closure under addition closure under scalar multiplication commutativity of addition associativity of addition existence of zero vector existence of additive inverses unit property associativity of scalar multiplication distributive properties TrueFalse l The zero vector in a vector space V is unique 3 The set Z of integers together with the usual operations of addition and scalar multiplication forms a vector space The additive inverse of a vector v in a vector space Vis unique 7 The set 01 with the usually operations of addition and scalar U multiplication forms a vector space Section 43 g 256 De nitions subspace trivial subspace null space of a matriXA TrueFalse 3 The points in R2 that lie on the line y mx b fOI H l a subspace of R2 iff b 0 A nonempty set S of a vector space Vthat is closed under scalar multiplication contains the zero vector of V If V R3 and S consists of all points on the xy plane the xz plane and the yz plane then S is a subspace of V Section 44 g 265 De nitions linear combination linear span spanning set TrueFalse l 3 7 11 The linear span of a set of vectors in a vector space Vforms a subspace of V If S is a spanning set for a vector space Vand W is a subspace of V then S is a spanning set for W Every vector space Vhas a finite spanning set If m lt n then any spanning set for Rquot must contain more vectors than any spanning set for Rquot Section 4 5 g 278 De nitions linearly dependent set linear dependency linear independent set minimal spanning set Wronskian of a set of functions TrueFalse 1 Every vector space Vpossesses a unique minimal spanning set 5 If the Wronskian of a set of functions is nonzero at some point x0 in an interval I then the set of functions is linearly independent 9 If the Wronskian of a set of functions is identically zero at every point of an interval I then the set of functions is linearly dependent Section 4 6 g 290 De nitions basis standard basis nitedimensional dimension extension of a subspace basis TrueFalse l A basis for a vector space Vis a set S of vectors that spans V 3 A vector space Vcan have many different bases 5 If Vis an ndimensional vector space then any set S of m vectors with mgtn must span V 7 Two vectors in P3 must be inearly independent 9 If V is an ndimensional vector space then every set S with fewer than n vectors can be extended to a basis for V Section 4 7 Q 299 De nitions ordered basis components of a vector relative to an ordered basis changeofbasis matrix TrueFalse 1 Every vector in a nitedimensional vector space V can be expressed uniquely as a linear combination of vectors comprising a basis for V 3 A changeofbasis matrix is always a square matrix no problems from Section 48 no problems from Section 49 Section 4 10 g 312 TrueFalse l The set of all row vectors of an invertible matrix is linearly independent Section 411 g 321 De nitions inner product axioms of an inner product real complex inner product space norm angle CauchySchwarz inequality TrueFalse 1 If v and W are linearly independent vectors in an inner product space V then ltvwgt 0 5 In any vector space V there is at most one valid inner product lt gt that can be defined on V 7 If p x a0 alx azx2 and qx 2 0 blx bzx2 then we can de ne an inner product on P2 via ltpqgt aobo Section 412 g 331 De nitions orthogonal vectors orthogonal set unit vector orthonormal vectors orthonormal set normalization orthogonal basis orthonormal basis GramSchmidt process orthogonal projection TrueFalse 1 Every orthonormal basis for an inner product space Vis also an orthogonal basis for V
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https://www.ee-diary.com/2023/03/single-input-unbalanced-output-bjt.html
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# Single Input Unbalanced Output BJT Differential Amplifier Design
Differential amplifier are type of amplifier that uses two identical transistors to amplify difference between two input signal. The transistors can be bipolar junction transisotors(BJT), field effect transistors(FET) or BiCMOS transistors. Usually differential amplifiers are implemented within an integrated Circuit(IC) because when fabricated on single IC, the transistors are more closely tailored and saves size and power. Despite this, differential amplifier may also be constructed using individual BJT, FET or BiCMOS transistors. There are 4 different types of differential amplifier configuration depending upon the number of inputs and output taken:
(4) single input unbalanced output BJT amplifier
In this blog post, the single input unbalanced output BJT amplifier working principle is explained.
### Single input unbalanced output BJT differential amplifier
In single input unbalanced output BJT differential amplifier, the input is applied to one of the differential pair transistor and output is taken from either of the two transistor collector. Following shows the circuit diagram of single input unbalanced output(SIUO) BJT differential amplifier wherein the input signal is applied to the base of the transistor Q1 and output is taken from the collector of the second transistor Q2 and measured with respect to the ground.
#### Circuit Analysis
The DC analysis and AC analysis of the single input unbalanced output BJT differential amplifier gives us the Q-point or the quscient opertating point, the differential gain, the input and output resistances. Following are the DC and AC analysis result.
The Q-point current and voltages are:
$$I_{CQ}(=I_E)\approxeq \frac{V_{EE}-V_{BE}}{2R_E}$$
$$V_{CEQ} = V_{CC}+V_{BE}-I_{CQ}R_C$$
AC analysis derived parameters are:
(a) Differential gain, $$A_{dm}=\frac{h_{fe} R_C}{R_B+h_{ie}}$$
(b) Input Resistance, $$R_i=2(R_B+h_{ie})$$
(c) Output Resistance, $$R_o=R_C$$
Consider that we are using 2N3904 NPN bipolar transistor. Let the collector resistors be 100Ohm, the base resistor be 1KOhm and the emitter resistor be 100Ohm. Let the input signal applied to Q1 be 50mV peak with frequency of 1KHz. The circuit diagram is shown below.
For this circuit, following shows the input and the dc coupled single ended output waveform.
The single ended dc coupled output has amplitude of 1.13V peak to peak and is dc biased at around 2.3V. That is it has 1.13/2=0.56V peak amplitude.
We can add a coupling capacitor to the output to get ac coupled singled ended output as shown in the circuit diagram below and plot the waveform again to see comparison of the input signal and the singled ended output waveform more clearly.
For this ac coupled circuit the following shows the input signal and ac coupled singled ended output waveform.
We can calculate the differential gain of this single ended unbalanced output amplifier as follows.
$$A_{dm} = \frac{V_{out}}{V_{in}}=\frac{0.56V}{50mV}\approx 11$$
#### Application of BJT differential amplifiers
BJT differential amplifiers are used in construction of op-amps, are used in communication electronics circuits such as voltage controlled oscillator, BJT differential amplifier modulator etc.
References:
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# forces acting on a ball dropped from 4m ? Are the forces balanced or unbalanced?
Where would the arrows go on a diagram?
Original post by Kbeen
Where would the arrows go on a diagram?
Are you drawing the diagram as the ball is in their air? Or as it hits the ground?
If the forces are balanced, then the ball will maintain its state of motion (i.e. it will have a constant velocity, which may be zero). Is the ball falling at a constant velocity? Or is it accelerating towards the ground?
In terms of forces acting on the ball, is the ball in a vacuum, or do we need to consider drag (air resistance)?
Yes uk drawing it at the top of its max bounce,
How would I know if the forces are balanced, the ping pong ball is simply dropped from a 4m height, what forces would act on it?
Original post by Kbeen
Yes uk drawing it at the top of its max bounce,
Original post by Kbeen
How would I know if the forces are balanced, the ping pong ball is simply dropped from a 4m height, what forces would act on it?
Bounce? Bounce? When did the ball start to bounce? The title of the thread says it was "dropped".
The instant the ball is first dropped the only force acting upon is that due to gravity. As it accelerates towards the ground drag increases (assuming we're not in a vacuum, which is why I asked the question), until it reaches terminal velocity - at which point the Fdrag and Fgravity are equal and opposite (i.e. balanced) at it remains in a constant state of motion: falling at a constant velocity. I obviously don't know what "level" or question this is (GCSE, A level, etc.) so don't know whether my mentioning drag and terminal velocity is over-complicating things. If it is, just ignore drag.
If we're talking about the result of a bounce, then on it's way up it will be decelerating - meaning the forces acting upon it must be unbalanced. Once it reaches "the top of its max bounce" it is momentarily stationary before it begins to fall again. At that instant the only force acting on the ball is gravity, which causes the ball to accelerate downwards again.
(I'm assuming the ball is dropped and bounces vertically; your mention of it being a ping pong ball introduces the possibility that it might be bouncing at an angle, or that spin might be involved.)
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15468
15,468 (fifteen thousand four hundred sixty-eight) is an even five-digits composite number following 15467 and preceding 15469. In scientific notation, it is written as 1.5468 × 104. The sum of its digits is 24. It has a total of 4 prime factors and 12 positive divisors. There are 5,152 positive integers (up to 15468) that are relatively prime to 15468.
Basic properties
• Is Prime? No
• Number parity Even
• Number length 5
• Sum of Digits 24
• Digital Root 6
Name
Short name 15 thousand 468 fifteen thousand four hundred sixty-eight
Notation
Scientific notation 1.5468 × 104 15.468 × 103
Prime Factorization of 15468
Prime Factorization 22 × 3 × 1289
Composite number
Distinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 7734 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0
The prime factorization of 15,468 is 22 × 3 × 1289. Since it has a total of 4 prime factors, 15,468 is a composite number.
Divisors of 15468
1, 2, 3, 4, 6, 12, 1289, 2578, 3867, 5156, 7734, 15468
12 divisors
Even divisors 8 4 2 2
Total Divisors Sum of Divisors Aliquot Sum τ(n) 12 Total number of the positive divisors of n σ(n) 36120 Sum of all the positive divisors of n s(n) 20652 Sum of the proper positive divisors of n A(n) 3010 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 124.37 Returns the nth root of the product of n divisors H(n) 5.13887 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 15,468 can be divided by 12 positive divisors (out of which 8 are even, and 4 are odd). The sum of these divisors (counting 15,468) is 36,120, the average is 3,010.
Other Arithmetic Functions (n = 15468)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 5152 Total number of positive integers not greater than n that are coprime to n λ(n) 1288 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 1805 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares
There are 5,152 positive integers (less than 15,468) that are coprime with 15,468. And there are approximately 1,805 prime numbers less than or equal to 15,468.
Divisibility of 15468
m n mod m 2 3 4 5 6 7 8 9 0 0 0 3 0 5 4 6
The number 15,468 is divisible by 2, 3, 4 and 6.
• Arithmetic
• Refactorable
• Abundant
• Polite
Base conversion (15468)
Base System Value
2 Binary 11110001101100
3 Ternary 210012220
4 Quaternary 3301230
5 Quinary 443333
6 Senary 155340
8 Octal 36154
10 Decimal 15468
12 Duodecimal 8b50
20 Vigesimal 1id8
36 Base36 bxo
Basic calculations (n = 15468)
Multiplication
n×i
n×2 30936 46404 61872 77340
Division
ni
n⁄2 7734 5156 3867 3093.6
Exponentiation
ni
n2 239259024 3700858583232 57244880565432576 885463812586111085568
Nth Root
i√n
2√n 124.37 24.916 11.1521 6.88473
15468 as geometric shapes
Circle
Diameter 30936 97188.3 7.51654e+08
Sphere
Volume 1.55021e+13 3.00662e+09 97188.3
Square
Length = n
Perimeter 61872 2.39259e+08 21875.1
Cube
Length = n
Surface area 1.43555e+09 3.70086e+12 26791.4
Equilateral Triangle
Length = n
Perimeter 46404 1.03602e+08 13395.7
Triangular Pyramid
Length = n
Surface area 4.14409e+08 4.3615e+11 12629.6
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NTNUJAVA Virtual Physics LaboratoryEnjoy the fun of physics with simulations! Backup site http://enjoy.phy.ntnu.edu.tw/ntnujava/
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Author Topic: Free Rolling and Circular Motion (Read 95897 times) 0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).
Fu-Kwun Hwang
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« Embed this message on: January 29, 2004, 01:13:25 pm » posted from:,,Satellite Provider
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There are 4 translations,
Higher number at the end means more translation been done.
or
This applet shows the positon,velocity and accelerationvectors
of a point on the rim of a body rolling without slippingon a stationary surface.
1. The term freely rolling means that there is no slipping (v=rw)
at the point of contact with the ground--no skidding (v>rw) and no spinning in place (w>v/r),where v,r and w are the linear speed, radius and
the angular speed of the rolling body.
2. The center of the wheel has a linear speed (with respect to ground)
equal to that of any point on its rim (with respect to center).
3. For circular motion (in the reference frame
moving with the velocity of the center of the rolling body) :
The acceleration a of a point always points in a direction
opposite the position vector r,
and the velocity vector V is perpendicular to both of them.
4. Observe the velocity vectors of a point on the rim
of the rolling body.
The white vector represents the velocity of the center
with resepect to ground and
the red vector represents the velocity of the point
with respect to the center.
The sum of these two vectors is the cyan vector
representing velocity of the point relative to ground.
5. The green curve shows the trajectory of a point on the rim.
Registed user can get files related to this applet for offline access.
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There are 4 translations,
Higher number at the end means more translation been done.
or
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« Embed this message Reply #1 on: January 15, 2009, 04:37:50 am »
Thnx very helpful but are there more like these. Where can i find?
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« Embed this message Reply #2 on: January 15, 2009, 10:49:28 am » posted from:Taipei,T'ai-pei,Taiwan
There are several hundreds physics related simulation at this web site.
If there is some simulation you would like to have and you can not find it.
Write down in detail what you want, and I will try to help!
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« Embed this message Reply #3 on: September 22, 2014, 12:49:40 pm » posted from:Faisalabad,Punjab,Pakistan
This applet shows the <font COLOR="#0000FF">positon</font>,<font COLOR="#0000FF">velocity</font> and <font COLOR="#0000FF">acceleration</font><font COLOR="#0000FF">vectors</font></font>
of a point on the rim of a body rolling without slipping</font>on a stationary surface.</font>
<center><applet code=freeRolling.class width=550 height=360 codebase="/java/FreeRolling/">
<param name="MSG1" value="Circular motion">
<param name="MSG2" value="Free Rolling">
<param name="Reset" value="Reset">
</applet></center><ol><font SIZE=+1>1. The term freely rolling means that there is no slipping (v=rw) </font><ol><font SIZE=+1>at the point of contact with the ground</font><font SIZE=+1>--no skidding (v>rw) and no spinning in place (w>v/r),</font><font SIZE=+1>where v,r and w are the linear speed, radius and</font><ol><font SIZE=+1>the angular speed of the rolling body.</font></ol></ol><font SIZE=+1>2. The center of the wheel has a linear speed (with respect to ground) </font><ol><font SIZE=+1>equal to that of any point on its rim (with respect to center).</font></ol><font SIZE=+1>3. For circular motion (in the reference frame</font><ol><ol><font SIZE=+1>moving with the velocity of the center of the rolling body) :</font></ol><font SIZE=+1>The acceleration a of a point always points in a direction</font><ol><font SIZE=+1>opposite the position vector r,</font></ol><font SIZE=+1>and the velocity vector V is perpendicular to both of them.</font></ol><font SIZE=+1>4. Observe the velocity vectors of a point on the rim</font><ol><ol><font SIZE=+1>of the rolling body.</font></ol><font SIZE=+1>The white vector represents the velocity of the center</font><ol><font SIZE=+1>with resepect to ground and </font></ol><font SIZE=+1>the red vector represents the velocity of the point</font><ol><font SIZE=+1>with respect to the center.</font></ol><font SIZE=+1>The sum of these two vectors is the cyan vector</font><ol><font SIZE=+1>representing velocity of the point relative to ground.</font></ol></ol><font SIZE=+1>5. The green curve shows the trajectory of a point on the rim.</font></ol>
There are a few hundreds physical science related reenactment at this site.
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# Tag Info
5
This can be seen as a preliminary to @Underdark's answer whereby you can clean the topology of the vector layer before generalizing. GRASS has a v.clean function which contains a number of tools to repair the layer such as: snap which 'snaps' lines to the nearest vertex rmdangle which removes any annoying dangles rmdupl which removes duplicated geometry ...
2
Have you tried using MapShaper? I've found this to be a good way of generalizing polygons; there's also an offline version if you want to download the code. An alternative could be convert this to GeoJson, and then TopoJson which supports "topology-preserving shape simplification" https://github.com/mbostock/topojson
10
GRASS is topology-aware. You can use v.generalize from the Processing toolbox to simplify polygons and if the input data is topologically correct so will the output.
0
This is just another idea, by looking at the opposite angle to your question, to find centrelines first as explained one of the methods in here and then search for matching line features. Of course this approach will give robust results if the road lines are at the 'centre' of the polygons. Maybe adding thin ('how thin' is another issue of course) buffers to ...
1
This is just an idea it would need further experimenting as it may be flawed? How about looking at some sort of ratio value of length/perimeter? Imagine you had a line that is 10m long and has its full buffer which you had buffered out by 2m then the perimeter of the buffer would be approximately 20m. so 10/20 = 0.5. Now imagine you had a line that was ...
0
It is close to 0 percent likely this photo is in DD. 1) Define the spatial reference of the background image in ArcCatalog. 2) Ensure it aligns with some other dataset (say from arcgis online) in ArcMAP. 3) Create a blank polygon in ArcCatalog. 4) Define the projection of 3 in ArcCatalog. 5) Start a blank mxd in ArcMap. 6) Add (1) the image with the ...
Top 50 recent answers are included
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• Resource ID: R4SCI0001
• Grade Range: 6–12
• Subject: Science
### Potential and Kinetic Energy
This resource provides Tier I instruction ideas for Grade 6+ science teachers in the area of potential and kinetic energy.
• Resource ID: R4SCI0010
• Grade Range: 4
• Subject: Science
### What's Happening with the Weather?
This resource shows how to observe and describe weather changes from day to day and over seasons.
• Resource ID: R4SCI0012
• Grade Range: 7–12
• Subject: Science
### Human Impact
This resource can be used, in conjunction with best practices, for Tier I classroom instruction.
• Resource ID: GT001
• Grade Range: K–12
• Subject:
### Texas Performance Standards Project
These videos introduce the Texas Performance Standards Project (TPSP), a resource for providing differentiated instruction to gifted/talented (G/T) students.
• Resource ID: R4SCI0029
• Grade Range: 8
• Subject: Science
### The Arrangement of Elements: The Periodic Table
This resource provides flexible alternate or additional learning opportunities for students to interpret the arrangement of elements on the Periodic Table, Eighth Grade Science TEKS (8)(5)(C).
• Resource ID: MATH_IMG_001
• Grade Range: K–8
• Subject: Math
### Interactive Math Glossary
The Interactive Math Glossary is provided by the Texas Education Agency to help teachers explore and understand mathematics vocabulary.
• Resource ID: RtI001
• Grade Range: K–12
• Subject:
### Response to Intervention (RtI)
Response to Intervention (RtI) is a multi-tiered framework designed to recognize and assist struggling learners through a systematic approach that includes strategies for intervention across curriculum, instruction, assessment and implementation.
• Resource ID: Revised_Math_TEKS_SI
• Grade Range: K–12
• Subject: Math
### Mathematics TEKS: Supporting Information
This resource presents supporting information of the mathematics TEKS for Kindergarten through Grade 12.
• Resource ID: OTALAADMIN
• Grade Range: 5–9
• Subject: ELA & Reading
The Texas Adolescent Academy Administrator Overview is designed to prepare school leaders to implement a schoolwide approach to reading intervention consistent with a response to intervention, or RtI, model.
• Resource ID: E_MSTAR001
• Grade Range: 2–8
• Subject: Math
### ESTAR/MSTAR
The ESTAR/MSTAR Universal Screeners and Diagnostic Assessments are a formative assessment system administered to students in grades 2–8 to support instructional decisions in mathematics.
• Resource ID: T2P0005
• Grade Range: K–5
• Subject: ELA & Reading
### Target 2% Lessons to Support Comprehension for Grades K through 5
These comprehension activities are designed to enhance the instruction of all teachers of reading and to meet the learning needs of all students, as indicated by data.
• Resource ID: W4TX0000
• Grade Range: 5–12
• Subject: ELA & Reading
### Write for Texas Overview Video
This resource links to the Write for Texas online resources and includes a short video overview.
• Resource ID: TXLS_Pilot
• Grade Range: K–12
• Subject:
### Texas Lesson Study Briefing
Texas Lesson Study (TXLS) is systematic, inquiry-based professional development that fosters a collaborative, professional-learning environment among teachers and can be implemented by all campuses.
• Resource ID: T2P0003
• Grade Range: K–5
• Subject: ELA & Reading
### Target 2% Lessons to Support Vocabulary for Grades K through 5
Vocabulary knowledge is an essential component of comprehension, language proficiency, and reading growth for all students.These vocabulary activities are designed to enhance the instruction of all teachers of reading and to meet the learning needs of all students, as indicated by data.
• Resource ID: T2P0002
• Grade Range: K–5
• Subject: ELA & Reading
### Target 2% Lessons to Support Phonics for Grades K through 5
Phonics is the relationship between the letters (graphemes) of written language and the sounds (phonemes) of spoken language. These phonics activities are designed to enhance the instruction of all teachers of reading and to meet the learning needs of all students, as indicated by data.
• Resource ID: T2P0004
• Grade Range: K–5
• Subject: ELA & Reading
### Target 2% Lessons to Support Fluency for Grades K through 5
Fluency is the ability to read text effortlessly, allowing the reader to concentrate on meaning. Fluency includes reading accurately, quickly, and with prosody (appropriate phrasing, expression, and attention to punctuation).
• Resource ID: T2P0006
• Grade Range: K–5
• Subject: ELA & Reading
### Target 2% Research Briefs
A collection of research briefs based on topics from results of the Target 2% survey and focus groups.
• Resource ID: RD001
• Grade Range: K–12
• Subject:
### Restorative Discipline Practices in Texas
This resource provides information on Restorative Discipline Practices in Texas.
• Resource ID: SI001
• Grade Range: K–12
• Subject: ELL Support
### Sheltered Instruction Training Series
Sheltered instruction is an instructional approach that uses various strategies to ensure that grade-level instruction provided in English addresses both content and language objectives. This resource introduces the Sheltered Instruction Training Series (20 CPE hours).
• Resource ID: TXLSFT_Phase1
• Grade Range: K–12
• Subject:
### Phase 1: Examine and Identify
In this section, we will explore the five key components of Phase 1
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# 8 Concept Drift Detection Methods
There is a wide range of techniques that can be applied for detecting concept drift. Becoming familiar with these detection methods is key to using the right metric for each drift and model.
In the article below, I review four types of detection methods: Statistical, Statistical Process Control, Time Window Based, and Contextual Approaches.
If you’re looking for an introduction to concept drift, I recommend checking out my post Concept drift in machine learning 101.
### Statistical Methods
Statistical methods are used to compare the difference between distributions. In some cases, a divergence is used, which is a type of distance metric between distributions. In other cases, a test is run to receive a score.
### Kullback–Leibler Divergence
Kullback–Leibler divergence is sometimes referred to as relative entropy. The KL divergence tries to quantify how much one probability distribution differs from another, so if we have the distributions Q and P where the Q distribution is the distribution of the old data and P is that of the new data we would like to calculate:
* The “||” represents the divergence.
We can see that if P(x) is high and Q(x) is low, the divergence will be high.
If P(x) is low and Q(x) is high, the divergence will be high as well but not as much.
If P(x) and Q(x) are similar, then the divergence will be lower.
### JS Divergence
Jensen-Shannon Divergence uses the KL divergence
Where the mean between P and Q
The main differences between JS divergence and KL divergence are that JS is symmetric and it always has a finite value
### Kolmogorov-Smirnov Test
The two-sample KS test is a useful and general nonparametric method for comparing two samples. In the KS test we calculate:
Where is the empirical distribution function of the previous data with samples and is the empirical distribution function of the new data with samples and . The is the subset of samples x that maximizes .
The KS test is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples. It is well suited for numerical data.
### When to use Statistical Methods
The idea in the statistical methods section is to assess the distribution between two datasets.
We can use these tools to find differences between data from different timeframes and measure the differences in the behavior of the data as time goes on.
As for these methods the label is not needed and no additional memory is required, we can get a quick indicator for changes in the input features/output to the model. That would help us start investigating the situation even before any potential degradation in the model’s performance metrics. On the other hand, the lack of a label and disregarding memory of past events and other features could result in false positives if not handled correctly.
### Statistical Process Control
The idea of statistical process control is to verify that our model’s error is in control. This is especially important when running in production as the performance changes over time. Thus, we would like to have a system that would send an alert if the model passes some error rate. Note that some models have a “traffic light” system where they also have warning alerts.
### Drift Detection Method/Early Drift Detection Method (DDM/EDDM)
The idea is to model the error as a binomial variable. That means that we can calculate our expected value of the errors. As we are working with a binomial distribution we can mark =npt and therefore .
### DDM
Here we can raise:
• A warning when
• An alarm when
Pros: DDM shows good performance when detecting gradual changes (if they are not very slow) and abrupt changes (incremental and sudden drifts).
Cons: DDM has difficulties detecting drift when the change is slowly gradual. It is possible that many samples are stored for a long time, before the drift level is activated and there is the risk of overflowing the sample storage.
### EDDM
Here by measuring the distance of 2 consecutive errors, we can raise:
• A warning when
• An alarm when , where is usually 0.9
The EDDM method is a modified version of DDM where the focus is on identifying gradual drift.
### CUMSUM and Page-Hinckley (PH)
CUSUM and its variant Page-Hinckley (PH) are among the pioneer methods in the community. The idea of this method is to provide a sequential analysis technique typically used for monitoring change detection in the average of a Gaussian signal.
CUSUM and Page-Hinckley (PH) detect concept drift by calculating the difference of observed values from the mean and set an alarm for a drift when this value is larger than a user-defined threshold. These algorithms are sensitive to the parameter values, resulting in a tradeoff between false alarms and detecting true drifts.
As CUMSUM and Page-Hinckley (PH) are used to treating data streams, each event is used to calculate the next result:
### CUMSUM:
• where g represents the event, or for drifting purposes, the input/output of the model
• When an alarm is raised, and set
• are tunable parameters
• Note that CUMSUM is memoryless, one-sided or asymmetrical so it can detect only an increase in the value.
### Page-Hinckley (PH):
When an alarm is raised, and set
### When to Use Statistical Process Control Methods
To use the statistical process control methods presented, we need to provide the labels of the samples. In many cases this might be a challenge as the latency might be high and it could be very difficult to extract it, especially if it is used in a large organization. On the other hand, once this data is acquired we get a relatively fast system to cover 3 types of drifts: sudden drift, gradual drift and incremental drift. The system also lets us track the degradation (if any) with the division to warning and alert.
### Time Window Distribution
The Time Window Distribution model focuses on the timestamp and the occurrence of the events.
The idea of ADWIN is to start from time window and dynamically grow the window when there is no apparent change in the context, and shrink it when a change is detected. The algorithm tries to find two subwindows of and that exhibit distinct averages. This means that the older portion of the window – is based on a data distribution different than the actual one, and is therefore dropped.
### Paired Learners
Let’s say for a given problem we have a big stable model that uses a lot of data to train – let’s mark it as model A.
We will also devise another model, a more lightweight model that trained on smaller and more recent data – it can have the same type. We’ll call it model B.
The idea: Find the time windows where model B outperforms model A. As model A is stable and encapsulates more data than model B, we would expect it to outperform it. However, if model B outperforms model A that might suggest that a concept drift has occurred.
Source: S. H. Bach and M. A. Maloof, “Paired Learners for Concept Drift,” 2008 IEEE
### Contextual Approaches
The idea of these approaches is to assess the difference between the train set and the test set. When the difference is significant that can indicate that there is a drift in the data.
### Tree Features
The idea of Tree Features is to train a relatively simple tree on the data and add prediction timestamp as one of the features. As a tree model can be used also for feature importance, we can know how the time affects the data and at which point. Moreover, we can look at the split created by the timestamp and we can see the difference between the concepts before and after the split.
In the image above we can see that the date feature is at the root and that means that this feature has the highest information gain, so that means that on the July 22nd their may have occurred a drift in the data
### Implementations
You can find implementations for the presented detectors:
### Start Detecting Concept Drift in Your Models
Aporia has over 50 different types of customizable monitors to help you detect drift, bias, and data integrity issues. Sign up to Aporia’s free community plan to begin detecting drift in your machine learning models. Other posts I’ve written: Concept Drift in Machine Learning 101
Start Monitoring Your Models in Minutes
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https://es.mathworks.com/matlabcentral/answers/406155-how-to-create-isotropic-voxels-in-an-image-stack-without-loosing-original-dimensions
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# How to create isotropic voxels in an image stack without loosing original dimensions?
7 views (last 30 days)
Katharina Hecker on 18 Jun 2018
Hi everyone, I was wondering how to change the voxels of an image stack into isotropic voxels (50x50x50) without loosing the dimensions of an image: x = 13 mm, y = 10 mm z = 6.05 mm).
The problem is that I have to adjust the z axis of the stack to get 6.05 mm.
The matrix of the image stack is 1186x1834x121 double.
I was looking over the functions interp3, meshgrid formation and so on, but I could not find the right example, so that I understand which codes to use in which way... Thank you very much for your help!!!
Sayyed Ahmad on 18 Jun 2018
in matlab you have to diffrent kind of visualisation. 1. one is to visualiasing some vectorial behaviour of the nature for example wind. You kann find in m atlab a very googd example to do that aim.
xmin = min(x(:));
xmax = max(x(:));
ymax = max(y(:));
zmin = min(z(:));
wind_speed = sqrt(u.^2 + v.^2 + w.^2);
hsurfaces = slice(x,y,z,wind_speed,[xmin,100,xmax],ymax,zmin);
set(hsurfaces,'FaceColor','interp','EdgeColor','none')
colormap jet
hcont = ...
contourslice(x,y,z,wind_speed,[xmin,100,xmax],ymax,zmin);
set(hcont,'EdgeColor',[0.7 0.7 0.7],'LineWidth',0.5)
[sx,sy,sz] = meshgrid(80,20:10:50,0:5:15);
hlines = streamline(x,y,z,u,v,w,sx,sy,sz);
set(hlines,'LineWidth',2,'Color','r')
view(3)
daspect([2,2,1])
axis equal
The other one is the Image manipulation. for example put the Image in the space.
[xSphere,ySphere,zSphere] = sphere(16); %# Points on a sphere
scatter3(xSphere(:),ySphere(:),zSphere(:),'.'); %# Plot the points
axis equal; %# Make the axes scales match
hold on; %# Add to the plot
xlabel('x');
ylabel('y');
zlabel('z');
xImage = [-0.5 0.5; -0.5 0.5]; %# The x data for the image corners
yImage = [0 0; 0 0]; %# The y data for the image corners
zImage = [0.5 0.5; -0.5 -0.5]; %# The z data for the image corners
surf(xImage,yImage,zImage,... %# Plot the surface
'CData',img,...
'FaceColor','texturemap');
Sayyed Ahmad on 18 Jun 2018
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https://hindimaintutorial.in/in-fact-the-problems-e1-and-e2-do-not-have-much-in-common-you-should-probably-think-of-them-as-two-separate-problems-you-are-given-an-integer-array-a1na1a1ana1/
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• # For Solution
In fact, the problems E1 and E2 do not have much in common. You should probably think of them as two separate problems.
You are given an integer array a[1n]=[a1,a1,,an]a[1…n]=[a1,a1,…,an].
Let us consider an empty deque (double-ended queue). A deque is a data structure that supports adding elements to both the beginning and the end. So, if there are elements [3,4,4][3,4,4] currently in the deque, adding an element 11 to the beginning will produce the sequence [1,3,4,4][1,3,4,4], and adding the same element to the end will produce [3,4,4,1][3,4,4,1]. Array Optimization by Deque solution codeforces
The elements of the array are sequentially added to the initially empty deque, starting with a1a1 and finishing with anan. Before adding each element to the deque, you may choose whether to add it to the beginning or to the end.
For example, if we consider an array a=[3,7,5,5]a=[3,7,5,5], one of the possible sequences of actions looks like this:
1 add 33 to the beginning of the deque: deque has a sequence [3][3] in it; 2 add 77 to the end of the deque: deque has a sequence [3,7][3,7] in it; 3 add 55 to the end of the deque: deque has a sequence [3,7,5][3,7,5] in it; 4 add 55 to the beginning of the deque: deque has a sequence [5,3,7,5][5,3,7,5] in it;
### Array Optimization by Deque solution codeforces
Find the minimal possible number of inversions in the deque after the whole array is processed.
An inversion in sequence dd is a pair of indices (i,j)(i,j) such that i<ji<j and di>djdi>dj. For example, the array d=[5,3,7,5]d=[5,3,7,5] has exactly two inversions — (1,2)(1,2) and (3,4)(3,4), since d1=5>3=d2d1=5>3=d2 and d3=7>5=d4d3=7>5=d4.
Input
The first line contains an integer tt (1t10001≤t≤1000) — the number of test cases.
The next 2t2t lines contain descriptions of the test cases.
The first line of each test case description contains an integer nn (1n21051≤n≤2⋅105) — array size. The second line of the description contains nn space-separated integers aiai (109ai109−109≤ai≤109) — elements of the array.
It is guaranteed that the sum of nn over all test cases does not exceed 21052⋅105.
### Array Optimization by Deque solution codeforces
Print tt lines, each line containing the answer to the corresponding test case. The answer to a test case should be a single integer — the minimal possible number of inversions in the deque after executing the described algorithm.
Example
input
Copy
6
4
3 7 5 5
3
3 2 1
3
3 1 2
4
-1 2 2 -1
4
4 5 1 3
5
1 3 1 3 2
### Array Optimization by Deque solution codeforces
Copy
2
0
1
0
1
2
Note
One of the ways to get the sequence [5,3,7,5][5,3,7,5] in the deque, containing only two inversions, from the initial array [3,7,5,5][3,7,5,5] (the first sample test case) is described in the problem statement.
Also, in this example, you could get the answer of two inversions by simply putting each element of the original array at the end of the deque. In this case, the original sequence [3,7,5,5][3,7,5,5], also containing exactly two inversions, will be in the deque as-is.
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# Texas Instruments Placement-Paper
Sample Questions
1. (steeper transition) by:
1. Increasing W/L of PMOS transistor
2. Increasing W/L of NMOS transistor
3. Increasing W/L of both transistors by the same factor
4. Decreasing W/L of both transistor by the same factor
2. Minimum number of 2-input NAND gates that will be required to implement the function: Y = AB + CD + EF is
1. 4
2. 5
3. 6
4. 7
3. Consider a two-level memory hierarchy system M1 & M2. M1 is accessed first and on miss M2 is accessed. The access of M1 is 2 nanoseconds and the miss penalty (the time to get the data from M2 in case of a miss) is 100 nanoseconds. The probability that a valid data is found in M1 is 0.97. The average memory access time is:
1. 4.94 nanoseconds
2. 3.06 nanoseconds
3. 5.00 nanoseconds
4. 5.06 nanoseconds
4. Interrupt latency is the time elapsed between:
1. Occurrence of an interrupt and its detection by the CPU
2. Assertion of an interrupt and the start of the associated ISR
3. Assertion of an interrupt and the completion of the associated ISR
4. Start and completion of associated ISR
5. Which of the following is true for the function (A.B + A’.C + B.C)
1. This function can glitch and can be further reduced
2. This function can neither glitch nor can be further reduced
3. This function can glitch and cannot be further reduced
4. This function cannot glitch but can be further reduced
6. For the two flip-flop configuration below, what is the relationship of the output at B to the clock frequency?
1. Output frequency is 1/4th the clock frequency, with 50% duty cycle
2. Output frequency is 1/3rd the clock frequency, with 50% duty cycle
3. Output frequency is 1/4th the clock frequency, with 25% duty cycle
4. Output frequency is equal to the clock frequency
7. The voltage on Node B is:
1. 0
2. 10
3. –10
8.A CPU supports 250 instructions. Each instruction op-code has these fields:
1. The instruction type (one among 250)
2. A conditional register specification
3. 3 register operands
4. Addressing mode specification for both source operands
9. The CPU has 16 registers and supports 5 addressing modes. What is the instruction op-code length in bits?
32
24
30
36
10. In the iterative network shown, the output Yn of any stage N is 1 if the total number of 1s at the inputs starting from the first stage to the Nth stage is odd. (Each identical box in the iterative network has two inputs and two outputs). The optimal logic structure for the box consists of:
1. One AND gate and one NOR gate
2. One NOR gate and one NAND gate
3. Two XNOR gates
4. One XOR gate
11.Consider a circuit with N logic nets. If each net can be stuck-at either values 0 and 1, in how many ways can the circuit be faulty such that only one net in it can be faulty, and such that up-to all nets in it can be faulty?
1. 2 and 2N
2. N and 2^N
3. 2N and 3^N-1
4. 2N and 3N
12. In the circuit shown, all the flip-flops are identical. If the set-up time is 2 ns, clock->Q delay is 3 ns and hold time is 1 ns, what is the maximum frequency of operation for the circuit?
a. 200 MHz
b. 333 MHz
c. 250 MHz
d. None of the above
13. Which of the following statements is/are true?
I. Combinational circuits may have feedback, sequential circuits do not.
II. Combinational circuits have a ‘memory-less’ property, sequential circuits do not.
III. Both combinational and sequential circuits must be controlled by an external clock.
1. I only
2. II and III only
3. I and II only
4. II only
14.Consider an alternate binary number representation scheme, wherein the number of ones M, in a word of N bits, is always the same. This scheme is called the M-out-of-N coding scheme. If M=N/2, and N=8, what is the efficiency of this coding scheme as against the regular binary number representation scheme? (As a hint, consider that the number of unique words represent able in the latter representation with N bits is 2^N. Hence the efficiency is 100%)
1. Close to 30%
2. Close to 50%
3. Close to 70%
4. Close to 100%
15.A CPU supports 4 interrupts- I1, I2, I3 and I4. It supports priority of interrupts. Nested interrupts are allowed if later interrupt is higher priority than previous one. During a certain period of time, we observe the following sequence of entry into and exit from the interrupt service routine:
1. I1-start—I2-start—I2-end—I4-start—I3-start—I3-end—I4-end—I1-end
From this sequence, what can we infer about the interrupt routines?
1. I3 > I4 > I2 > I1
2. I4 > I3 > I2 > I1
3. I2 > I1; I3 > I4 > I1
4. I2 > I1, I3 > I4 > I2 > I1
16. I decide to build myself a small electric kettle to boil my cup of tea. I need 200 ml of water for my cup of tea. Assuming that typical tap water temperature is 25 C and I want the water boiling in exactly one minute, then what is the wattage required for the heating element?
[Assume: Boiling point of water is 100 C, 1 Calorie (heat required to change 1 gm of water by 1 C)= 4 joules, 1 ml of water weighs 1 gm.]
1. Data given is insufficient
2. 800 W
3. 300 W
4. 1000 W
5. 250 W
17.The athletics team from REC Trichy is traveling by train. The train slows down, (but does not halt) at a small wayside station that has a 100 mts long platform. The sprinter (who can run 100 mts in 10 sec) decides to jump down and get a newspaper and some idlis. He jumps out just as his compartment enters the platform and spends 5 secs buying his newspaper that is at the point where he jumped out. He then sprints along the platform to buy idlis that is another 50 mts. He spends another 5 secs buying the idlis. He is now just 50 mts from the other end of the platform where the train is moving out. He begins running in the direction of the train and the only other open door in his train is located 50 mts behind the door from where he jumped. At what(uniform) speed should the train be traveled if he just misses jumping into the open door at the very edge of the platform?
1. Make the following assumptions
• He always runs at his peak speed uniformly
• The train travels at uniform speed
• He does not wait (other than for the idlis & newspaper) or run baclwards
1. Data given is insufficient
2. 4 m/s
3. 5 m/s
4. 7.5 m/s
5. 10 m/s
18.State which of the following gate combinations does not form a universal logic set:
1. 2-input AND + 2-input OR
2. 2-to-1 multiplexer
3. 2-input XOR + inverter
4. 3-input NAND
19.For the circuit shown below, what should the function F be, so that it produces an output of the same frequency (function F1), and an output of double the frequency (function F2).
1. F1= NOR gate and F2= OR gate
2. F1=NAND gate and F2= AND gate
3. F1=AND gate and F2=XOR gate
4. None of the above
20.The FSM (finite state machine) below starts in state Sa, which is the reset state, and detects a particular sequence of inputs leading it to state Sc. FSMs have a few characteristics. An autonomous FSM has no inputs. For a Moore FSM, the output depends on the present state alone. For a Mealy FSM, the output depends on the present state as well as the inputs. Which of the statements best describes the FSM below?
1. It has two states and is autonomous
2. The information available is insufficient
3. It is a Mealy machine with three states
4. It is a Moor machine with three states
21In the circuit given below, the switch is opened at time t=0. Voltage across the capacitor at t=infinity is:
1. 2V
2. 3V
3. 5V
4. 7V
22.What is the functionality represented by the following circuit?
1. y= ! (b+ac)
2. y= ! (a+bc)
3. y= ! (a(b+c))
4. y= ! (a+b+c)
23.The value (0xdeadbeef) needs to stored at address 0x400. Which of the below ways will the memory look like in a big endian machine: 0x403 0x402 0x401 0x400
a. be ef de ad
b. ef be ad de
c. fe eb da ed
d. ed da eb fe
24. In a given CPU-memory sub-system, all accesses to the memory take two cycles. Accesses to memories in two consecutive cycles can therefore result in incorrect data transfer. Which of the following access mechanisms guarantees correct data transfer?
1. A read operation followed by a write operation in the next cycle.
2. A write operation followed by a read operation in the next cycle.
3. A NOP between every successive reads & writes
4. None of the above
25.An architecture saves 4 control registers automatically on function entry (and restores them on function return). Save of each registers costs 1 cycle (so does restore). How many cycles are spent in these tasks (save and restore) while running the following un-optimized code with n=5:
1. Void fib(int n)
{
if((n==0) || (n==1)) return 1;
return(fib(n-1) + fib(n-2));
}
1. 120
2. 80
3. 125
4. 128
26.The maximum number of unique Boolean functions F(A,B), realizable for a two input (A,B) and single output (Z) circuit is:
1. 2
2. 6
3. 8
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## Monday, October 04, 2021
### [tlbuipqj] planes through 4 vertices of a hypercube
we consider the Ramsey graph coloring problem made famous by Graham's number.
a d-dimensional hypercube has 2^d vertices, so its complete graph has binomial(2^d,2) edges.
it has f(d) planar subsets of 4 vertices, where f(d) = (6^d + 2^d - 2*4^d)/8 ; (OEIS A016283). it is a little bit surprising this expression is always an integer (only need to check d = {0,1,2} because higher d cause the numerator to have enough factors of 2).
every planar subset of 4 vertices is always a parallelogram, but proving that fact is slightly nontrivial. is it always a rectangle?
in 4D, consider a tesseract bounded by (0,0,0,0) and (1,1,1,1). it has 16 vertices and 120 edges. there are f(4) = 100 planar subsets of 4 vertices. among them is this elegant square of side sqrt(2):
(0,0,0,0)
(0,0,1,1)
(1,1,1,1)
(1,1,0,0)
Is this the largest-area planar subset of 4 vertices in a 4D hypercube? this square is somewhat aesthetically similar to the equilateral triangle hidden among the vertices of a 3D cube, but the triangle does not span opposite vertices of the cube as this square does in the tesseract. (what is the 4D analogue of the regular hexagon hidden among the cross sections of a cube? it is some 3D polyhedron.)
possibly of interest is the rotation group of the tesseract. it has 192 elements. the full automorphism group (future post zopdmbui) includes mirror images, so has order twice that, 2^d*(d!) (OEIS A000165), or 384 for d=4. because of the factorial, this sequence grows faster than the number of planar subsets of 4 vertices. this is somewhat surprising because, for a given planar subset, it seems one could rotate it in many different ways to generate other planar subsets.
the current lower bound for Graham's number is 13. to disprove 13, i.e., to raise the lower bound, we need a 2-coloring of the edges and all diagonals of a 13-dimensional hypercube that avoids a planar K4 of the same color. (alternatively, some non-constructive proof that such a coloring exists.) there are 33550336 edges. we would need to check f(13) = 1615810560 planar subsets of edges.
presumably such a coloring of the 12-dimensional hypercube exists. 8386560 edges, f(12) = 267904000 planar subsets of edges.
we investigate prime factorization of f(d) using Pari/GP.
addprimes([2540341408694303316943652119999, 225986911513165436959419576586387, 78236248423963530824630162507, 36548282222048342116048567, 20438550797700490833113893109377 ])
With the above large primes, Pari/GP can quickly factor f(3) through f(181). the next unfactored number, f(182), is 468 bits. every f(d) seems to be divisible by a high power of two, namely 2^(d-1) if d is odd and 2^(d-2) if d is even. f(182)/2^180 is 288 bits.
if p is prime, then it seems p divides f(p). also p divides f(p-1). composite n divides f(n) for the following n up to 1000:
1 4 8 16 20 32 40 64 80 88 100 128 160 200 220 256 272 320 400 440 464 500 512 544 561 640 800 880 920 1000
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# I'm very beginner. i want to make game playing time
hi i want to know about make timer for timelapse.
first, what i want to know is using ofGetElasedTimeMillis
or any time controler to control mygame’s playing time to set 60seconds.
second,in this 60 seconds some events(like blink) occur 10 times in random seconds…
// 38sec blink … 10 times
//what should i do first… ?
For 10 blinks every 60 seconds => 1 blink every 6 seconds- you could just do a simple if else like this :
int time = (int) ofGetTimeElapsedf(); // The type casting is because the function returns a float value (seconds) and you can only use modulus operator on integer types.
This is pretty easy and might even be a bit obvious, but its a start I guess.
To add randomness and achieve something like “Blink x times every y seconds” is actually an interesting question. Would have to think about that a bit…
Best.
Update us if you find the solution and so will I if it strikes me
I’ve done something similar like this in this fashion. This is coming from some Arduino code reference (blink without delay).
In .h file setup the following:
``````long unsigned currentMillis;
long unsigned previousMillis;
long unsigned interval;
``````
In setup(), setup the following stuff.
``````currentMillis = 0;
previousMillis = 0;
interval = 1000; //this is 1000 milliseconds = 1 second
``````
In update() do this.
``````long unsigned currentMillis = ofGetElapsedTimeMillis();
if(currentMillis - previousMillis >= interval){
//do stuff here;
previousMillis = currentMillis; //reset the previousMillis value here to currentMillis so that in the loop next time it triggers again.
}
``````
setup multiple intervals and multiple previousMillis counters if you want to do stuff at different intervals.
1 Like
If you want to have the triggers at random times, throw in
`interval = ofRandom(maxValue);`
where you’re resetting the previousMillis counter.
You could also make an array for example:
and then fill this array with 10 random numbers from 1 - 60 every minute.
if((int)ofGetElapsedTimef()%60==0)
{
for(int i=0; i<9;i++)
{
x=ofRandom(60);
//You will have to insert some code here to make sure x values are not repeated.
}
}
then use the numbers in this array as the interval time that you want to see a blink.
currentSeconds = ofGetElapsedTimef();
for(int i=0; i<9;i++)
{
if (currentSeconds=time[i])
{
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### International Mathematics Competition for University Students
July 31 – August 6 2017, Blagoevgrad, Bulgaria
Home
Ivan is Watching You
Results
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Teams
Official IMC site
### Problem 9
9. Define the sequence $f_1,f_2,\ldots:[0,1)\to \RR$ of continuously differentiable functions by the following recurrence: $$f_1=1; \qquad \quad f_{n+1}'=f_nf_{n+1} \quad\text{on (0,1)}, \quad \text{and}\quad f_{n+1}(0)=1.$$
Show that $\lim\limits_{n\to \infty}f_n(x)$ exists for every $x\in [0,1)$ and determine the limit function.
Proposed by: Tomáš Bárta, Charles University, Prague
Solution. First of all, the sequence $f_n$ is well defined and it holds that $$f_{n+1}(x)=e^{\int_0^x f_n(t)\mathrm{d}t}. \tag1$$ The mapping $\Phi:C([0,1))\to C([0,1))$ given by $$\Phi(g)(x)=e^{\int_0^x g(t)\mathrm{d}t}$$ is monotone, i.e. if $f<g$ on $(0,1)$ then $$\Phi(f)(x)=e^{\int_0^x f(t)\mathrm{d}t}<e^{\int_0^x g(t)\mathrm{d}t}=\Phi(g)(x)$$ on $(0,1)$. Since $f_2(x)=e^{\int_0^x1\\mathrm{d}t}=e^x>1=f_1(x)$ on $(0,1)$, we have by induction $f_{n+1}(x)>f_n(x)$ for all $x\in (0,1)$, $n\in\mathbb{N}$. Moreover, function $f(x)=\frac1{1-x}$ is the unique solution to $f'=f^2$, $f(0)=1$, i.e. it is the unique fixed point of $\Phi$ in $\{\varphi\in C([0,1)):\ \varphi(0)=1\}$. Since $f_1<f$ on $(0,1)$, by induction we have $f_{n+1}=\Phi(f_n)<\Phi(f)=f$ for all $n\in\mathbb{N}$. Hence, for every $x\in(0,1)$ the sequence $f_n(x)$ is increasing and bounded, so a finite limit exists.
Let us denote the limit $g(x)$. We show that $g(x)=f(x)=\frac1{1-x}$. Obviously, $g(0)=\lim f_n(0)=1$. By $f_1\equiv 1$ and (1), we have $f_n>0$ on $[0,1)$ for each $n\in \mathbb{N}$, and therefore (by (1) again) the function $f_{n+1}$ is increasing. Since $f_n$, $f_{n+1}$ are positive and increasing also $f_{n+1}'$ is increasing (due to $f_{n+1}'=f_nf_{n+1}$), hence $f_{n+1}$ is convex. A pointwise limit of a sequence of convex functions is convex, since we pass to a limit $n\to \infty$ in $$f_n(\lambda x + (1-\lambda)y)\le \lambda f_n(x) + (1-\lambda)f_n(y)$$ and obtain $$g(\lambda x + (1-\lambda)y)\le \lambda g(x) + (1-\lambda)g(y)$$ for any fixed $x$, $y\in [0,1)$ and $\lambda\in (0,1)$. Hence, $g$ is convex, and therefore continuous on $(0,1)$. Moreover, $g$ is continuous in 0, since $1\equiv f_1\le g\le f$ and $\lim_{x\to 0+} f(x)=1$. By Dini's Theorem, convergence $f_n\to g$ is uniform on $[0,1-\varepsilon]$ for each $\varepsilon\in (0,1)$ (a monotone sequence converging to a continuous function on a compact interval). We show that $\Phi$ is continuous and therefore $f_n$ have to converge to a fixed point of $\Phi$.
In fact, let us work on the space $C([0,1-\varepsilon])$ with any fixed $\varepsilon\in (0,1)$, $\|\cdot\|$ being the supremum norm on $[0,1-\varepsilon]$. Then for a fixed function $h$ and $\|\varphi-h\|<\delta$ we have $$\sup_{x\in[0,1-\varepsilon]}|\Phi(h)(x)-\Phi(\varphi)(x)|=\sup_{x\in[0,1-\varepsilon]} e^{\int_0^x h(t)\mathrm{d}t}\left|1-e^{\int_0^x \varphi(t)-h(t)\mathrm{d}t}\right| \le C(e^{\delta}-1)<2C\delta$$ for $\delta>0$ small enough. Hence, $\Phi$ is continuous on $C([0,1-\varepsilon])$. Let us assume for contradiction that $\Phi(g)\ne g$. Hence, there exists $\eta>0$ and $x_0\in [0,1-\varepsilon]$ such that $|\Phi(g)(x_0)-g(x_0)|>\eta$. There exists $\delta>0$ such that $\|\Phi(\varphi)-\Phi(g)\|<\frac13\eta$ whenever $\|\varphi-g\|<\delta$. Take $n_0$ so large that $\|f_n-g\|<\min\{\delta, \frac13\eta\}$ for all $n\ge n_0$. Hence, $\|f_{n+1}-\Phi(g)\|=\|\Phi(f_{n})-\Phi(g)\|<\frac13\eta$. On the other hand, we have $|f_{n+1}(x_0)-\Phi(g)(x_0)|>|\Phi(g)(x_0)-g(x_0)|-|g(x_0)-f_{n+1}(x_0)|>\eta-\frac13\eta=\frac23\eta$, contradiction. So, $\Phi(g)=g$.
Since $f$ is the only fixed point of $\Phi$ in $\{\varphi\in C([0,1-\varepsilon]):\ \varphi(0)=1\}$, we have $g=f$ on $[0,1-\varepsilon]$. Since $\varepsilon\in (0,1)$ was arbitrary, we have $\lim_{n\to\infty} f_n(x)=\frac1{1-x}$ for all $x\in [0,1)$.
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# Tag Info
2
A not-that-fast solution could be easily written: f[dat_, part_] := Outer[SortBy[#, -# &] & /@ Internal`PartitionRagged[##] &, Permutations@dat, part, 1, 1]; PlanePartitions[n_] := Module[{pt = IntegerPartitions@n, inp}, inp = {#, IntegerPartitions@Length@#} & /@ pt; Select[Flatten[f @@@ inp, 2], With[{t = PadRight[#, {n, ...
6
This is due to the interpolation. If you turn off the interpolation by InterpolationOrder -> 0 you will see symmetric plot. data = Flatten[Table[{x, y, If[x == y, 1, 0]}, {x, 0, 1, 1/10}, {y, 0, 1, 1/10}], 1]; dataR = data; dataL = data /. {x_, y_, z_} :> {-x, y, z}; data0 = Join[dataL,dataR] ListDensityPlot[data0, ...
2
There was no problem exporting at the correct frame rate when I initially tried it with a small number of test frames, given that the code in the question doesn't produce any actual frames. However, I now tried again with a large table of frames to replicate the specific number of 979 frames in the question. Indeed, in this case the resulting movie has the ...
5
The rectangle (0,0), (0.5,0), (0,1),(0.5,1) should be covered, however it is not. The 29 rectangles in your original tabla cover the region 1/31 <=b < 1/2 && 0 <= a <= 1. ClearAll[tabla] tabla[n_] := Table[{1/(k + 1) <= b && 1/k > b}, {k, 2, n, 1}]; You can specify a large enough value for PlotPoints to see all 29 ...
4
Wrap the call to the function that is likely to generate errors and return unevaluated in a Check expression. In the case of correct execution, Check will pass through the value returned by its argument; in case of an error, it will return an expression of your choice. This "failure indicator" won't trigger recalculation, and can be made very recognizable ...
Top 50 recent answers are included
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# ODE Method of Variation of Parameters help
• Mar 31st 2011, 06:56 PM
fizzle45
ODE Method of Variation of Parameters help
Attachment 21321
Above is my problem sorry that it's a link but It was to hard to convert from latex into here (i used some packages)
• Apr 1st 2011, 04:58 AM
Jester
You're missing a term in your variation of parameters formula. Given an nth order linear ODE
$a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_1(x)y' + a_0(x)y = f(x)$
with independent solutions $y_1, \cdots , y_n$ the variation of parameters formula is
$\displaystyle y_p = \sum_{k=1}^n y_k \int \dfrac{W_k}{W} \dfrac{f(x)}{a_n(x)}dx$.
With what you have you set $a_n = 1$ where it should be $a_n = x^3$. With this assignment things work out giving you the desired $y_p$.
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https://knoji.com/article/the-golden-ratio-what-is-sex-appeal-and-why-do-we-have-certain-preferences/
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# The Golden Ratio: What is Sex Appeal, and Why Do We Have Certain Preferences?
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I enjoy watching documentaries. I recently watched one about the science of sex appeal made for the Discovery Channel. I would like to share a bit about what I learned from it. Emotional and physical attraction is accomplished through our brain and our
There is an old saying that beauty is in the eye of the beholder. Put simply, every person is an individual, and each person may find certain attributes of a person attractive, while other attributes of a person may not be so attractive. For instance, a woman could be totally gorgeous to look at, and while she might turn heads, her confidence and intelligence might be what wins the affection of a potential suitor.
On a biological level, our brains are programmed to seek out the best partner to pass on our genes to in order to reproduce. It all begins with the face. As we look at a person’s face, the brain recognizes symmetry. We decide what is attractive an unattractive by the symmetry of the face. Oftentimes, when a person instantly falls in love with a person, he or she is falling for the perfect dimensions of a person’s face and overall appearance. On the documentary, a man talks about what he was attracted to in his mate. He said he was drawn to her eyes, her perfect cheekbones and her looks.
Scientists have studied what makes a man or woman drop dead gorgeous. It has to do with facial and body proportions as determined by a mathematical formula. In essence, the proportions in the Golden Ratio are equal to each other. This formula is called the Golden Ratio. This ratio is considered to be the mathematic equation for perfection. To read more about the Golden Ratio click here.
Simply put, the Golden Ratio is described as a space divided into two parts. One part of the space is about a third smaller than the other part. The ratio of the two parts equals the ratio of the big part to the whole space. The Golden Ratio is not a new concept. Plato came up with this ratio, and ever since then artists have tried to incorporate it into their drawings and paintings to capture the essence of the person or object being captured on canvas.
Are we aware that we use the Golden Ratio to judge attractiveness? Probably not, but we know what we find attractive and what we don’t. For instance, the distance between the eyes should be equal to the width of the eyes. The width of both eyes would be two-thirds the space of space between the eyes. We can instantly spot when the eyes appear too close together, or if the nose is too big for the face. Does this mean, if we aren’t perfectly proportioned by the Golden Ratio formula that we are doomed to never find love? No! Stay tuned.
Sources:
The Science of Sex Appeal – Discovery
Golden Ratio
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CC-MAIN-2020-50
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http://curious.astro.cornell.edu/people-and-astronomy/the-history-of-astronomy/46-our-solar-system/the-moon/observing-the-moon/32-when-can-i-see-the-moon-through-the-hole-in-the-pantheon-intermediate?tmpl=component&print=1
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## When can I see the Moon through the hole in the Pantheon? (Intermediate)
I am an artist who is very interested in Astronomy but does not know very much about it. I very much enjoyed reading your extremely helpful website.
I would like to know, as precisely as possible, when the moon will shine at its brightest through the hole in the Pantheon in Rome. The Pantheon has a circular hole of 9m across at the centre of it's dome as shown here.
As your website suggests I have visited the U.S. Naval Observatory Data Services website but the closest I can get is Rome, Italy. What will I need to do to be 100% certain that the date and time given on this website will allow the bright moon to be seen through this narrow opening in the Pantheon roof?
Thanks for your question to . What follows is a rather lengthy and technical answer. I guess I got interested in figuring out the best answer to your problem (presumably that's why I am an Astronomer!), although as a caveat I want to note that it's actually not that easy of a question to answer, so I don't promise to have thought of all the best solutions. Anyway here goes....
Rome has a latitude of 41.9 degrees North (longitude 12.45 W). An object which passes through the zenith at this location therefore must have a declination of 41.9N (right acsension and declination are a co-ordinate system used for objects in the sky which is referenced to the equator and poles of the Earth). This declination is not possible for the Moon, which can only ever have declinations between 28.5S and 28.5N. The highest the Moon will ever get in the sky above Rome is therefore 13.5 deg off zenith, at which point it would illuminate part of the Pantheon, just at an oblique angle (ie. part would be in "Moon shadow").
A Google search gives 43.3m as the height of the interior of the Pantheon dome. A person standing directly below the hole therefore can see an area of the sky around the zenith of radius 5.9 deg (arctan(4.5/43.3)) So the full Moon (diameter 0.5 deg) can be seen in full from the floor of the Pantheon directly below the hole if it has an altitude between 84.3 and 90 degrees (ie. within 5.7 deg of directly overhead). As described above this cannot happen. However, when the Moon reaches 13.5 degrees off Zenith (ie. ALT=76.5) moonlight will fall on a spot about 6m from the centre of the Pantheon. At lower altitudes the moonlight will fall further and further from the centre until it starts falling on the walls and finally cannot be seen directly through the hole in the roof at all.
The highest position of the Full Moon occurs in the winter months when the Sun is at it's lowest in the sky - the Full Moon is exactly opposite the Sun in the sky....
#### Date of Full Moons in 2006:
Highest point of Moon Local Time ALT AZ Jan 14: 0:00 74.9 179.4 Dec 4: 23:50 74.9 176.7 Dec 5: 0:00 74.9 185.0 Jan 2 (2007): 23:40 76.1 179.5 Jan 3: 0:00 75.6 197.0
You'll notice that these high points all happen in the middle of the night - perhaps not ideal for an art exhibition. This is because the Full Moon always rises at sunset, therefore reaches its highest point in the middle of the night. You'll also notice that the altitudes are not equal to the theoretical maximum. That's because none of these dates fall exactly on the winter solstice.
For the Moon to be at it's highest in the evening, it has to be First quarter. In this phase, the Moon reaches its highest points at the Spring Equinox.
#### First quarter Moons in 2006
Highest point of Moon Local Time ALT AZ Percent illuminated Feb 5: 18:20 69.7 178.2 0.55 Mar 6: 18:20 74.9 194.5 0.49 Apr 5: 18:40 75.1 181.9 0.52
Which date/time will give the best illumiation is not something which is very easy to answer. You need to reach a comprimise between the amount of the Moon which is illuminated verses how late at night it reaches it's highest point. Also note that the Full Moon doesn't fall exactly on the winter solstice in 2006, neither does the First quarter Moon fall exactly on the Spring equinox...
#### In summary:
1. The Moon never passes directly overhead in Rome, Italy.
2. The Full Moon reaches its highest point at ~midnight in Winter months (near the Winter solstice).
3. First quarter Moon reaches its highest point at ~6pm in Spring months
4. In very early spring/late winter the phase of the Moon between Full and 1st quarter reaches its highest point in the late evening (between 6pm and midnight).
I think I would pick a Moon in a phase between Full and First quarter, in early Spring, for example Feb 9th or March 10th when the Moon is about 70% illuminated and reaches its highest point around 9pm local time. If you're OK with an exhibition at midnight you can do it on a Full Moon in the middle of winter.
As a final note, Azimuth refers to the angle around the horizon. 180 degs is due South, so to be in "direct moonlight" on these nights, you need to be on the Northern side of the Pantheon, about 6-8m from the central point.
Good luck with the exhibition.
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https://socratic.org/questions/how-do-you-simplify-sqrt-8-sqrt2
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text/html
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crawl-data/CC-MAIN-2024-33/segments/1722641333615.45/warc/CC-MAIN-20240816030812-20240816060812-00708.warc.gz
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# How do you simplify sqrt 8 + sqrt2?
May 14, 2016
3$\sqrt{2}$
#### Explanation:
start off by simiplying the $\sqrt{8}$ :
$\sqrt{8}$ = $\sqrt{4 \cdot 2}$
$\sqrt{4}$ = 2
therefore $\sqrt{8}$ = 2$\sqrt{2}$
2$\sqrt{2}$ + $\sqrt{2}$
= 3$\sqrt{2}$
Well it is
$\sqrt{8} + \sqrt{2} = \sqrt{{2}^{2} \cdot 2} + \sqrt{2} = 2 \sqrt{2} + \sqrt{2} = 3 \cdot \sqrt{2}$
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