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Model this Constraint
I am finding difficulty in modeling this constrain, your help would be much appreciated:
s(j+n) >= c(i,j) for every i where n= 1,2,3,4,5…N-j
s(j) is a defined parameter
c(i,j) is a defined variable
i and j are defined sets
N is equal to last component of set j
how can I index j+n and how to create n= 1 to N-j (dynamically changing with j)
Note that this is an equation (we cannot use loops), i have nonetheless tried to use loops but it won’t allow me to include n a non primary number. I have also tried circular loops s(j++1) but i am
trying to compare cij to all following sj starting from s(j+1) till the end of set j.
Thank you for your help! The file is attached below.
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Constructions Concise Solutions Chapter-19 Class 10 - ICSEHELP
Constructions Concise Solutions Chapter-19 Class 10
Constructions Concise Solutions Chapter-19 ICSE Maths Class 10. Solutions of Exercise- 19 Constructions for Concise Selina Maths of ICSE Board Class 10th. Concise Maths Solutions Constructions
Chapter-19 for ICSE Maths Class 10 is available here. All Solutions of Concise Selina Maths of Constructions Chapter-19 has been solved according instruction given by council. This is the Solutions
of Constructions Chapter-19 for ICSE Class 10th. ICSE Maths text book of Concise is In series of famous ICSE writer in maths publications. Concise is most famous among students
Constructions Concise Solutions Chapter-19 ICSE Maths Class 10
This Post is the Solutions of Concise Mathematics Constructions Chapter-19 for ICSE Class 10. Experience teachers Solved Chapter-19 Constructions to help students of class 10th ICSE board.
Therefore the ICSE Class 10th Maths Solutions of Concise Selina Publishers helpful on various topics which are prescribed in most ICSE Maths textbook
How to Solve Concise Maths Selina Publishers Chapter-19 Constructions (Circles) ICSE Maths Class 10
Note:- Before viewing Solutions of Chapter-19 Constructions of Concise Selina Maths . Read the Chapter Carefully then solve all example of your text book. The Chapter-19 Constructions of Concise
Maths is main Chapter in ICSE board.Focus on Tangent Constructions, Circumcircle and Cicumscribe.
Constructions (circles) concise Maths solutions chapter- 19 ICSE Board Class – 10
EXERCISE – 19
Question 1.
Draw a circle of radius 3 cm. Mark a point P at a distance of 5cm from the centre of the circle drawn. Draw two tangents PA and PB to the given circle and measure the length of each tangent.
Answer 1
Steps of Construction:
Draw a circle with centre O and radius 3 cm.
From O, take a point P such that OP = 5 cm.
Draw the bisector of OP which intersects OP at M.
With centre M, and radius OM. draw’ a circle which intersects the given circle at A and B.
Join AP and BP.
AP and BP are the required tangents.
On measuring them, AP = BP = 4 cm.
Question 2.
Draw a circle of diameter 9 cm. Mark a point at a distance of 7.5 cm from the centre of the circle. Draw tangents to the given circle from this exterior point. Measure the length of each tangent.
Answer 2
Steps of Construction:
Draw a line segment AB = 9 cm.
Draw a circle with centre O and AB as diameter.
Take a point P from the centre at a distance of 7.5 cm.
Draw an other circle OP as diameter which intersects the given circle at T and S.
Join TP and SP.
TP and SP are are required tangents.
On measuring their lengths, TP = SP = 6 cm.
Question 3.
Draw a circle of radius 5 cm. Draw two tangents to this circle so that the angle between the tangents is 45°.
Answer 3
Steps of Construction:
Draw a circle with centre O and radius 5 cm.
Draw two arcs making an angle of 180° – 45° = 135°
so that ∠AOB = 135°.
At A and B, draw two rays making an angle of 90° at each point which meet each other at P, out side the circle.
Then AP and BP are the required tangents which make an angle of 45° at P.
Question 4. Constructions Concise Solutions Chapter-19
Draw a circle of radius 4.5 cm. Draw two tangents to this circle so that the angle between the tangents is 60°.
Answer 4
Steps of Construction:
Draw a circle with centre O and radius 4.5 cm.
Draw two arcs making an angle of 180° – 60° = 120° i.e. ∠AOB = 120°.
At A and B draw rays making an angle of 90° at each point which meet each other at P outside the circle.
AP and BP are the required tangents which makes an angle of 60° at P.
Question 5.
Using ruler and compasses only, draw an equilateral triangle of side 4.5 cm and draw its circumscribed circle. Measure the radius of the circle.
Answer 5
Steps of Construction:
Draw a line segment BC = 4.5 cm.
With centres B and C, draw two arcs of radius 4.5 cm. which intersect each other at A.
Join AB and AC,
Draw the perpendicular bisectors of AB and BC intersecting each other at O.
With centre O, and radius OA or OB or OC draw a circle which will passes through A, B and C.
This is the required circumcircle of ∆ ABC.
Measuring OA = 2.6 cm
Question 6.
Construct triangle ABC, having given = 7 cm, AB – AC = 1 cm and ∠ABC = 45°.
(ii) Inscribe a circle in the ∆ ABC constructed in (i) above,
Answer 6
Steps of Construction:
Draw a line segment BC = 7 cm.
At B, draw a ray BX making an angle of 45° and cut off BE = AB – AC = 1 cm.
Join EC and draw the perpendicular bisector of EC intersecting BX at A.
Join AC
∆ ABC is the required triangle.
Draw angle bisectors of ∠ABC and ∠ACB intersecting each other at O.
From O, draw perpendicular OL to BC.
O as centre and OL as radius draw circle which touches the sides of the A ABC. This is the required in-circle of ∆ ABC.
On measuring radius OL = 1.8 cm (approx.).
Question 7. Constructions Concise Solutions Chapter-19
Using ruler and compasses only, draw an equilateral triangle of side 5 cm. Draw its inscribed circle. Measure the radius of the circle.
Answer 7
Steps of Construction:
Draw a line segment BC = 5 cm.
With centre B and C, draw two arcs of 5 cm radius each which intersect each other at A.
Join AB and AC.
Draw angle bisectors of ∠B and ∠C intersecting each other at O.
From O, draw OL ⊥ BC.
Now with centre O and radius OL, draw a circle which will touch the sides of the ∆ ABC. Measuring OL =1.4 cm. (approx.).
Question 8.
Using ruler and compasses only,
Construct a triangle ABC with the following data:
Base AB = 6 cm, BC = 6.2 cm and ∠CAB = 60°.
In the same diagram, draw a circle which passes through the points A, B and C and mark its centre O.
Draw a perpendicular from O to AB which meets AB in D.
Prove that AD = BD.
Answer 8
Steps of Construction:
Draw a line segment AB = 6 cm.
At A, draw a ray making an angle of 60° with BC.
B as centre and 6.2 cm as radius draw an arc which intersect the AX rays at C.
Join CB.
∆ ABC is the required triangle.
Draw the perpendicular bisectors of AB and AC intersecting each other at O.
With centre O, and radius as OA or OB or OC, draw a circle which will pass through A, B and C.
From O, draw OD ⊥ AB.
Proof: In right ∆ OAD and ∆ ODB
Hyp, OA = OB (radii of the saine circle)
Side OD = OD (Common)
∴ OAD ≅ OBD (R.H.S.)
∴ AD = BD (C.P.C.T.)
Question 9. Constructions Concise Solutions Chapter-19
Using ruler and compasses only construct a triangle ABC in which BC = 4 cm, ∠ACB = 45° and perpendicular from A on BC is 2.5 cm. Draw a circle circumscribing the triangle ABC and measure its radius.
Answer 9
Steps of Construction:
Draw a line segment BC = 4 cm.
(ii) At C, draw a perpendicular line CX and from it, cut off CE = 2.5 cm.
From E, draw another perpendicular line EY.
From C, draw a ray making an angle of 45° with CB, which intersects EY at A.
∆ ABC is the required triangle.
Draw perpendicular bisectors of sides AB and BC intersecting each other at O.
With centre O, and radius OB, draw a circle which passes through A, B and C.
Measuring the radius OB = OC = OA = 2 cm
Question 10.
Perpendicular bisectors of the sides AB and AC of a triangle ABC meet at O.
What do you call the point O ?
What is the relation between the distances OA, OB and OC?
Does the perpendicular bisector of BC pass through O ?
Answer 10
Perpendicular bisectors of sides AB and AC intersect each other at O.
(ii) O is called the circum centre of circumcircle of ∆ ABC.
OA, OB and OC are the radii of the circumcircle.
Yes, the perpendicular bisector of BC will also pass through O.
Question 11.
The bisectors of angles A and B of a scalene triangle ABC meet at O.
What is the point O called ?
OR ancLOQ are drawn perpendiculars to AB and CA respectively. What is the relation between OR and OQ ?
What is the relation between angle ACO and angle BCO ?
Answer 11
∆ ABC is a scalene triangle.
Angle bisectors of ∠A and ∠B intersect each other at O. O is called the incentre of the incircle of ∆ ABC.
Through O, draw perpendiculars to AB and AC which meet AB and AC at R and Q respectively.
OR and OQ are the radii of the in circle and OR =OQ.
OC is the bisector of ∠C
∴∠ACO = ∠BCO
Question 12. Constructions Concise Solutions Chapter-19
Using ruler and compasses only, construct a triangle ABC in which AB = 8 cm, BC = 6 cm and CA = 5 cm.
Find its incentre and mark it I.
With I as centre, draw a circle which will cut off 2 cm chords from each side of the triangle. What is the length of the radius of this circle.
Answer 12
Steps of Construction:
Draw a line segment BC = 6 cm.
With centre B and radius 8 cm draw ah arc.
With centre C and radius 5 cm, draw another arc which intersects the first arc at A.
Join AB and AC.
∆ ABC is the given triangle.
Draw the angle bisectors of ∠B and ∠A intersecting each other at I.
Then I is the incentre of incircle of ∆ ABC.
Through I, draw ID ⊥ AB.
Now from D, cut off DP = DQ = $\frac { 2 }{ 2 }$ = 1 cm.
With centre I, and radius IP or IQ, draw a circle which will intersect each side of ∆ ABC cuting chords of 2 cm each.
Question 13.
Construct an equilateral triangle ABC with side 6cm. Draw a circle circumscribing the triangle ABC.
Answer 13
Steps of construction:
Draw a line segment BC = 6cm.
With centre B and C, draw arcs with radius 6 cm each which intersect each other at A.
Join AB and AC,
then ∆ABC is the equilateral triangle.
Draw the perpendicular bisectors of BC and AB which intersect each other at O.
Join OB and OC and OA.
With centre O, and radius OA or OB or OC, draw a circle which will pass through A, B and C.
This is the required circle.
Question 14. Constructions Concise Solutions Chapter-19
Construct a circle, inscribing an equilateral triangle with side 5.6 cm.
Answer 14
Steps of construction:
Draw a line segment BC = 5.6 cm
With centre B and C,
draw two arcs of radius 5.6cm each which intersect each other at A.
Join AB and AC, then ∆ABC is an equilateral triangle.
Draw the angle bisectors of ∠B and ∠C which intersect each other at I.
From I, draw ID ⊥ BC.
With centre I and radius ID, draw circle which touches the sides of the ∆ABC. This is the required circle.
Question 15.
Draw a circle circumscribing a regular hexagon of side 5cm.
Answer 15
Steps of construction:
Draw a regular hexagon ABCDEF whose each side is 5cm.
Join its diagonals AD, BE and CF intersecting each other at O.
With centre O and radius OA, draw a circle which will pass through the vertices of the hexagon A, B, C, D, E and F. This is the required circle.
Question 16.
Draw an inscribing circle of a regular hexagon of side 5.8 cm.
Answer 16
Steps of construction:
Draw a line segment AB = 5.8cm.
At A and B, draw rays making an angle of 120° each and cut off AF = BC = 5.8 cm.
Again at F and C, draw rays making an angle of 120° each and cut off FE = CD = 5.8 cm.
JoinDE. Then ABCDEF is the regular hexagon.
(v) Draw the bisectors of ∠A and ∠B intersecting each other at O.
From O, draw OL J. AB.
With centre O and radius OL, draw a circle which touches the sides of the hexagon. This is the required incircle of the hexagon.
Question 17.
Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.
Answer 17
Steps of construction:
Draw a circle of radius 4 cm with centre O.
Since regular hexagon $\frac { { 360 }^{ \circ } }{ 6 }$ = 60°, draw radii
OA and OB, such that ∠AOB = 60°.
Cut off arcs BC, CD, DE, EF and each equal to arc AB on given circle.
Join AB, BC, CD, DE, EF, FA to get required regular hexagon ABCDEF in a given circle.
Question 18.
Draw a circle of radius 3.5 cm. Mark a point P outside the circle at a distance of 6 cm from the centre. Construct two tangents from P to the given circle. Measure and write down the length of one
tangent (2011).
Answer 18
Steps of construction:
Draw a line segment OP = 6 cm
With centre O and radius 3.5 cm, draw a circle
Draw the mid point of OP.
With centre M and diameter OP, draw a circle which intersect the circle at T and S.
Join PT and PS.
PT and PS are the required tangent on measuring the length of PT = PS = 4.8 cm
Question 19.
Construct a triangle ABC in which base BC=5.5 cm,AB = 6cmand ∠ABC = 120°.
(i) Construct a circle circumscribing the triangle ABC.
(ii) Draw- a cyclic quadrilateral ABCD so that D is equidistant from B and C.
Answer 19
Steps of construction:
Draw BC = 6 cm. x
At B, draw ∠XBC= 120°.
From BX, cut off AB = 6 cm.
Join AC to get ∆ ABC.
Draw the perpendicular bisector of BC and AB. These bisectors meet at O. With O as centre and radius equal tb OA, draw a circle, which passes through A, B and C. This is the required circumcircle of
(vi) Produce the perpendicular bisector of BC so that it meets the circle at D. Join CD and AD to _ get the required cyclic quadrilateral ABCD.
Question 20.
Using a ruler and compasses only :
(i) Construct a triangle ABC with the following data : AB = 3.5 cm, BC = 6 cm and ∠ABC = 120°
(ii) In the same diagram, draw a circle with BC as diameter. Find a point P on the circumference of the circle which is equidistant from AB and BC.
(iii) Measure ∠BCP
Answer 20
Steps of construction:
Draw AB = 3.5
At B, draw ∠ABX = 120°.
With B as center draw an arc of radii 6 cm at C.
Join A and C.
Draw the perpendicular bisector of line BC and draw a circle with BC as diameter.
Draw angle bisector of ∠B.
Meets the circle at P
∴ P is the required point ∠BCP = 30°
Question 21.
Construct a ∆ABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm. Construct the incircle of the triangle. Measure and record the radius of the incircle. (2014)
Answer 21
Steps of construction:
(i) Draw a line segment BC = 6.5 cm.
(ii) From B, draw an arc of radius of 5.5 cm and from C, another arc of 5 cm radius which intersect each other at A.
(iii) Join AB and AC. ∆ABC is required triangle.
(iv) Draw the angle bisectors of ∠B and ∠C which intersect each other at O.
(v) Through O, draw OL ⊥ BC.
(vi) With centre O and radius OL, draw a circle which touches the sides of ∆ABC.
(vii) On measuring, OL = r = 1.5 cm.
Question 22.
Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 105°. Hence:
(i) Construct the locus of points equidistant from BA and BC.
(ii) Construct the locus of points equidistant from B and C.
(iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC. (2015)
Answer 22
Steps of construction:
(i) Draw a line segment AB = 5.5 cm.
(ii) At A, draw a ray AX making an angle of 105°.
(iii) Cut off AC from AX =6 cm.
(iv) JoinCB. ∆ABC is required triangle.
(v) Draw angle bisector CX of ∠C. CX is the locus of points equidistant from BA and BC.
(vi) Draw the perpendicular bisector of BC which is the locus of points equidistant from the points B and C.
These two loci intersect each other at P.
Join PC and on measuring it, it is 4.8 cm (approx).
Question 23.
Construct a regular hexagon of side 5 cm. Hence construct all its lines of symmetry and name them. (2016)
Answer 23
Draw AF measuring 5 cm using a ruler.
With A as the centre and radius equal to AF, draw an arc above AF.
With F as the centre, and same radius cut the previous arc at O.
With O as the centre, and same radius draw a circle passing through A and F.
With A as the centre and same radius, draw an arc to cut the circle above AF at B.
With B as the centre and same radius, draw an arc to cut the circle at C.
Repeat this process to get remaining vertices of the hexagon at D and E
Join consecutive arcs on the circle to form the hexagon.
Draw the perpendicular bisectors of AF, EF and DE.
Extend the bisectors of AF, EF and DE to meet CD, BC and AB at X, L and O respectively.
Join AD, CF and EB.
These are the 6 lines of symmetry of the regular hexagon.
Question 24.
Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only, construct:
(i) A circle of radius 2.5 cm, passing through A and C.
(ii) Construct two tangents to the circle from the external point B. Measure and record the length of the tangents. (2016)
Answer 24
Steps of construction :
Draw AB = 5 cm using a ruler.
With A as the centre cut an arc of 3 cm on AB to obtain C.
With A as the centre and radius 2.5 cm, draw an arc above AB.
With same radius, and C as the centre draw an arc to cut the previous arc and mark the intersection as O.
(v) With O as the centre and radius 2.5 cm, draw a circle so that points A and C lie on the circle formed
Join OB.
(vii) Draw the perpendicular bisector of OB to obtain the mid-point of OB, M.
With M as the centre and radius equal to OM, draw a circle to cut the previous circle at points P and Q.
Join PB and QB. PB and QB are the required tangents to the given circle from exterior point B.
QB = PB = 3 cm
That is, length of the tangents i.e. 3.2 cm.
Question 25
Using a ruler and a compass, construct a triangle ABC in which AB = 7cm, ∠CAB = 60° and AC = 5cm. Construct the locus of:
points equidistant from AB and AC.
points equidistant from BA and BC.
Hence construct a circle touching the three sides of the triangle internally.
Answer 25
Steps of construction :
1. Draw a line AB = 7 cm
2. Taking P ascentre and same radius, draw an arc of a circle which intersects AB at M.
3. Taking M ascentre and with the same radius as before drawn an arc intersecting previously drawn arc, at point N.
4. Draw the ray AX passing through N, then
5. Taking A ascentre and radius equal to 5 cm, draw an arc cutting AX at C.
6. Join BC
7. The required triangle ABC is obtained.
8. Draw angle bisector of∠CAB and ∠ABC
9. Mark their intersection as O
10. With O as center, draw a circle with radius O
Question 26
Construct a triangle ABC in which AB = 5 cm, BC = 6.8 cm and median AD = 4.4 cm. Draw incircle of this triangle.
Answer 26
Steps for construction :
Draw BC = 6.8 cm.
Mark point D where BD = DC = 3.4 cm which is mid-point of BC.
iii. Mark a point A which is intersection of arcs AD = 4.4 cm and AB = 5 cm from a point D and B respectively.
Join AB, AD and AC.
ABC is the required triangle.
Draw bisectors of angle B and angle C which are ray BX and CY where I is theincentre of a circle.
Drawincircle of a triangle ABC.
Question 27
Draw two concentric circles with radii 4 cm and 6 cm. Taking a point on the outer circle, construct a pair of tangents to inner circle. By measuring the lengths of both the tangents, show that they
are equal to each other.
Answer 27
Steps for construction :
Draw concentric circles of radius 4 cm and 6 cm withcentre of O.
Take point P on the outer circle.
iii. Join OP.
Draw perpendicular bisectors of OP where M is the midpoint of OP.
Take a distance of a point O from the point M and mark arcs from M on the inner circle it cuts at point A and B respectively.
Join PA and PB.
We observe that PA and PB are tangents from outer circle to inner circle are equal of a length 4.5 cm each.
Question 28
In triangle ABC, ∠ABC = 90°, AB = 6 cm, BC = 7.2 cm and BD is perpendicular to side AC. Draw circumcircle of triangle BDC and then state the length of the radius of this circumcircle drawn.
Answer 28
Steps for construction :
Draw BC = 7.2 cm.
Draw an angle ABC = 90°using compass.
iii. Draw BD perpendicular to AC using compass.
Join BD.
Draw perpendicular bisectors of AB and BC which intersect atI, where I is the circumcentre of a circle.
Drawcircumcircle using circumcentre I. we get radius of a circle is 4.7 cm.
—-End of Constructions Concise Solutions Chapter-19—-
Return to :- Concise Selina Maths Solutions for ICSE Class-10
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On constructions and properties of (n,m)-functions with maximal number
of bent components
On constructions and properties of (n,m)-functions with maximal number of bent components
For any positive integers n=2k and m such that m≥ k, in this paper we show the maximal number of bent components of any (n,m)-functions is equal to 2^m-2^m-k, and for those attaining the equality,
their algebraic degree is at most k. It is easily seen that all (n,m)-functions of the form G(x)=(F(x),0) with F(x) being any vectorial bent (n,k)-function, have the maximum number of bent
components. Those simple functions G are called trivial in this paper. We show that for a power (n,n)-function, it has such large number of bent components if and only if it is trivial under a mild
condition. We also consider the (n,n)-function of the form F^i(x)=x^2^ih( Tr^n_e(x)), where h: F_2^e→F_2^e, and show that F^i has such large number if and only if e=k, and h is a permutation over F_2
^k. It proves that all the previously known nontrivial such functions are subclasses of the functions F^i. Based on the Maiorana-McFarland class, we present constructions of large numbers of (n,m)
-functions with maximal number of bent components for any integer m in bivariate representation. We also determine the differential spectrum and Walsh spectrum of the constructed functions. It is
found that our constructions can also provide new plateaued vectorial functions. | {"url":"https://api.deepai.org/publication/on-constructions-and-properties-of-n-m-functions-with-maximal-number-of-bent-components","timestamp":"2024-11-01T19:08:13Z","content_type":"text/html","content_length":"154953","record_id":"<urn:uuid:a691fa03-308e-4989-be70-ec127bce5262>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00051.warc.gz"} |
How Much Does a Cardboard Box Weigh? A Complete Guide
When planning a home shift or shipping fragile items, the most common question that often arises is, "How much does a cardboard box weigh?". It may seem to be a simple question, but the answer
depends on multiple factors, which include the type of cardboard used, the thickness of the cardboard, and the box`s size or dimensions. In this blog post, we will discuss each of these aspects,
focusing on the weights of different boxes, including boxes made from corrugated cardboard or Kraft paper and flat boxes.
The Packaging Material: A Major Factor
The weight of a cardboard box is significantly influenced by the type of packaging material it is built. Here are some of the common materials used in box-making and their impact on the box`s weight:
Flat Boxes
These boxes are typically made of single-wall corrugated cardboard material. It is because of their construction these flat packaging boxes are durable yet lightweight, making them an ideal packaging
solution for shipping a wide range of items. However, the exact weight can vary because of its size. The weight of a standard flat box usually weighs between 170 to 200 grams.
Corrugated Cardboard Boxes
Corrugated cardboard packaging material is a sturdy and solid material. It is made from three layers of brown kraft paper. This corrugated cardboard packaging box is an ideal packaging solution for
protecting high-end and fragile products during delivery and storage. The weight of the corrugated cardboard boxes is typically more than the flat packaging boxes. It is because of their added layers
of material for making them sturdy. A medium-sized corrugated cardboard box usually weighs around 500 grams. However, the weight can vary based on the thickness and size of the box.
Kraft Paper Boxes
These custom shipping boxes are made from Kraft paper, which is environmentally friendly and light in weight. This packaging material is famous for its durability and extraordinary strength, making
it aesthetically pleasing and practical. A medium-sized kraft paper box usually weighs around 150 to 180 grams.
Size and Thickness of Cardboard Matter
Aside from the packaging material, the thickness and size of the cardboard play a vital role in determining the weight of the box. Below are some of the cardboard box sizes with their average weights
for the ease of consumers:
Box Size (inches) Average weight (grams)
6 x 6 x 6 120-150
24 x 24 x 24 400-450
18 x 18 x 18 300-350
12 x 12 x 12 200-250
How Much Does a Cardboard Box Weigh in Kilograms?
To calculate the cardboard box weight in kilograms, you need to divide the volume in cm3 by 6000. The reason behind dividing by 6000 is to convert the cubic centimeters of the weight of cardboard to
kilograms. To calculate the weight of the same cardboard box in kilograms, we first have to convert the length, width, and height values from inches to centimeters to get the box`s volume in cubic
centimeters. Here is how you need to do it to calculate the weight of the cardboard box in kilograms:
Length, Width, Height = 12 x 2.54 cm/inch = 30.48 cm
Now that we have the dimensions in centimeters, we can calculate the weight.
Weight = (30.48cm x 30.48cm x 30.48cm)/6000
Weight = 28316.8466 cm³ / 6000
Weight = 4.7194 kilograms
How Much Does a Cardboard Box Weigh in Pounds?
When we need to calculate the weight of a cardboard box in pounds, you will need to divide the dimensions of the box by 166. The number 166 is the assumed cardboard density per cubic foot used to
make the box. Using this weighing formula, you will know the weight of a 12x12x12 small cardboard box. Here is the formula which you can use to calculate the weight of the cardboard box in pounds:
Weight of cardboard box = (12x12x12)/166
Weight = 1728/166
Weight = 10.4 Pounds
So, now you know how much a cardboard box weighs in pounds, which is 10.4 pounds.
How Much Does a Cardboard Box Weigh in Ounces?
If you need to calculate the weight of a cardboard box in ounces, you must divide the weight in pounds by 16. Here is how you can do it to calculate the weight of the cardboard box in ounces:
Weight = 10.4 Pounds / 16
Weight = 0.665 Ounces
How Much Does a Cardboard Box Weigh in Grams?
If you need to calculate the weight of a cardboard box in grams, all you need to do is convert the weight in kilograms to grams by multiplying by 1000. Now, the weight of a cardboard box in grams
will be:
Weight = 4.7194 x 1000
Weight = 4719.4 grams
However, if you want to know how much a small or heavy cardboard box weighs in grams, this formula for calculating the weight of a cardboard box in grams will work.
How Much Does a Large Cardboard Box Weigh?
The formulas mentioned above can help you calculate the weight of the cardboard box in pounds, ounces, kilograms, and grams. Whether you need to know how much an empty box weighs or how much a
cardboard shipping box weighs, the above-mentioned cardboard box weight calculator formulas can help you.
However, it would help if you kept in mind that these cardboard box weight calculations offer only an approximation of the weight of the cardboard box based on the dimensions. If you cannot calculate
the dimensions of a box, you can read our blog on how to calculate the box dimensions. To calculate the exact weight of the cardboard box, you need to accurately measure the thickness of the
material, along with other related factors. However, it is advised to use a proper scale to measure the exact weight of the cardboard box.
The Role of Box Dimensions in Calculating the Weight of a Cardboard Box
The dimensions of a box consist of height, width, and length. They all play a significant role in determining the weight of the cardboard box. The larger boxes usually require more material for their
construction, which naturally increases their weight. For instance, a small box of 6x6x6 inches only weighs 120 to 150 grams, while a large cardboard box of 24x24x24 inches could weigh as much as 400
to 450 grams. It would help if you kept in mind these figures can change based on the cardboard material you choose and its thickness.
Final Words
As you can see now, the cardboard box`s weight is influenced by multiple factors, including its thickness, the box`s size, and the material you use to make it. However, by understanding these
significant factors, you can make sound decisions when calculating the box`s weight or choosing the correct box for your product packaging needs. Knowing the approximate weight can be pretty useful
if you choose kraft paper boxes, flat boxes, or corrugated cardboard boxes.
By knowing the weight of the cardboard box, you can easily calculate the shipping costs and how much weight your box could handle. Cardboard, an eco-friendly packaging material, is the best choice
for shipping boxes. If you have any questions regarding the quality or design of cardboard boxes, you can contact us. IMH Packaging is one of the famous custom packaging suppliers in the USA. We will
help you stay updated with the latest news and trends related to product packaging.
Frequently Asked Questions
How much does a 12x12 cardboard box weigh?
The cardboard box with the dimensions of 12x12x12 weighs approximately 200-250 grams.
How much does a 10x10 box weigh?
The cardboard box, with dimensions of 10x10x10, weighs approximately 130-140 lbs.
How do you measure box weight?
To measure the weight of the box without the scale, you use the relationship between volume and mass: mass= volume x density. | {"url":"https://imhpackaging.com/how-much-does-a-cardboard-box-weigh-a-complete-guide/","timestamp":"2024-11-05T23:28:33Z","content_type":"text/html","content_length":"789516","record_id":"<urn:uuid:5d4e1148-aaaa-4df3-abec-afd1a0224403>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00167.warc.gz"} |
GSEB Solutions Class 6 Maths Chapter 9 Data Handling Ex 9.2
Gujarat Board GSEB Textbook Solutions Class 6 Maths Chapter 9 Data Handling Ex 9.2 Textbook Questions and Answers.
Gujarat Board Textbook Solutions Class 6 Maths Chapter 9 Data Handling Ex 9.2
Question 1.
The total number of animals in the five villages are as follows:
Village A: 80
Village B: 120
Village C: 90
Village D: 40
Village E: 60
Prepare a pictograph of these animals using one symbol
(a) How may symbols represent animals of village E?
(b) Which village has the maximum number of animals?
(c) Which village has more animals: village A or village C?
We have the following pictograph using the given data, such that.
(a) 6 symbols represent the animals of village E.
(b) Village B has the maximum number of animals.
(c) Village C has more animals than village A.
Question 2.
The total number of students of a school in different years is shown in the following table
A. Prepare a pictograph of students using one symbol
(a) How many symbols represent the total number of students in the year 2002?
(b) How many symbols represent the total number of students for the year 1998?
B. Prepare another pictograph of students using any other symbol each representing 50. students. Which pictograph do you find more informative?
A.We have the following pictograph using
(a) In the year 2002, a total number of students is represented by 6 symbols.
B. By taking 5 = 50, we prepare another pictograph as shown below:
Obviously, the second pictograph is more informative.
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Our Prize payout mechanism explained | FMFW.io Help Center
Whenever we are hosting trading competitions we will use the ad hoc average daily closing prices of the XYZ/USDT trading pair during the competition period as the XYZ/USDT exchange rate for XYZ
Ad hoc average daily closing prices are calculated using the same mechanism for every training competition on the FMFW.io unless otherwise specified in the Term & Conditions.
Calculations include the following: Sum of the closing price of each day during the trading competition / the number of trading competition days.
Below further details of the mechanism are outlined using an example.
1. Let’s imagine that FMFW.io conducted the trading competition for XYZ/USDT (this is an example, XYZ token is not listed on the FMFW.io) which lasted 5 full days:
1. 1st day's closing price of XYZ is 1$
2. 2nd day's closing price of XYZ is 2$
3. 3rd day's closing price of XYZ is 3$
4. 4th day's closing price of XYZ is 2$
5. 5th day's closing price of XYZ is 3$
2. Daily closing prices during the competition will be added up which leaves us with the sum of the closing price of each day during the trading competition (1+2+3+2+3=11). This sum will then be
divided by the sum of days in the trading competition: 5 (Days of trading competition)
3. For the final closing price, we divide the total sum of closing prices by the total sum of days which results in 11/5 = 2.2. This means that 2.2 USDT is our ad hoc average daily closing price for
this trading competition.
Why we use this rate?
You might also wonder, why this calculation is necessary. We're always trying our best to be fair while also respecting token projects who're often sponsoring competitions.
1. Due to the volatility of cryptocurrencies, we can't fix a market price. This wouldn't just be unfair to our users but also to the token companies sponsoring the prizes. As distribution in itself
takes time, the market price will inevitably differ.
2. We also can't really fix a certain exchange rate nor define a date to do so as that'd run the risk of manipulating markets.
3. The above calculation of the ad-hoc daily closing rate is a compromise that helps avoid market manipulation while providing a mechanism that respects users and prize sponsors equally.
If you still have questions about the process, please reach out to us via the chat widget on the right side or send us a message in our telegram channel. We're happy to help! | {"url":"https://support.fmfw.io/en/articles/5534961-our-prize-payout-mechanism-explained","timestamp":"2024-11-14T10:15:25Z","content_type":"text/html","content_length":"55461","record_id":"<urn:uuid:7792a0a9-8d11-43d6-909a-abd596b16e7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00159.warc.gz"} |
av den franske fysikern och matematikern Blaise Pascal (1623—1662). the ages, and how man's emancipation has lead to higher cultural accomplishments.
av A REPORTING — Dumisani John Hlatywayo, Challenges and accomplishments of the Eastern and Southern Blaise KONE (M) Etudes de problèmes anisotropiques non linéaires. Seminar on Mass Spectrom.,
Pascal Gerbaux, Univ. Mons
One of his 21 Sep 2019 Pascal's legacy extends to everything from mechanical calculators to the hydraulic press, and much of his handiwork is either still in practical use Blaise Pascal (1623 -
1662). From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball. Among the contemporaries of 14 Nov 2019 1. Blaise Pascal was a child prodigy · 2. Blaise Pascal
almost died as a child · 3. Pascal's theorem is named after Blaise Pascal · 4.
As every French high school student knows, Blaise Pascal was also a 10 Nov 2020 Pascal Life Essay Blaise On Biography. In addition to these accomplishments, Pascal also wrote two works that are
considered masterpieces of 10 Jul 2019 In the 17th century, Blaise Pascal invented the first calculator, among other things . Learn about his brief life and inventions. Blaise Pascal | Great French
Mathematician · Famous Female Mathematicians - I · Famous Female Mathematicians - II. Downloadable PDF. If you ever want to A Piece of the Mountain: The Story of Blaise Pascal: McPherson, Joyce:
Many books cover his accomplishments but few cover who he was as a man. This is a Pascal's Pensees: Pascal, Blaise: Amazon.se: Books. Yet, out of respect for the intellectual accomplishments of the
great French mathematician, these notes Never thought I'd read an article in Overdrive mentioning Blaise Pascal, trucking, Long Haul Paul celebrates the lives and accomplishments of two friends,…
And although he proved unruly at times, for his great accomplishments he was later och skapades av en oregerlig fransk matematiker, Blaise Pascal i 17-talet. av L Wallner · Citerat av 13 — Pascal
Lefèvre (1998), Randy Duncan, Matthew Smith & Paul Levitz (2015),.
Blaise Pascal was a French mathematician, physicist and religious 1908, where he learned of the accomplishments of the Wright brothers' Flyer and Blaise Pascal (1623-1662) is one of the makers
of the modern era, a accomplishments were based more on brilliant conjecture than on laborious trials .13. 24 May 2020 Pascal pioneered probability theory and game theory, invented an early
mechanical calculator, and, among many other accomplishments, Amazon.in - Buy Piece of the Mountain: The Story of Blaise Pascal book online at Many books cover his accomplishments but few cover
who he was as a man. 29 Mar 2021 Named after the great 17th century French mathematician-physicist-philosopher, the Blaise Pascal Medal was established in 2003 to recognize Blaise Pascal is one
of the most important mathematicians in history because of his many accomplishments and useful contributions to a variety of branches in Blaise Pascal • Mathematician • Scientist •
PhilosopherBonaventura Cavalieri • illustrated interactive learning system to study the life and accomplishments of 11 Feb 2021 acceptance speech.
these impressive accomplishments, however, it is as a mathematician that he is Blaise Pascal, for a branch of mathematics developed in a spirit of frivolity.
Clara 15 Nov 2005 We will sample some of the thoughts—Pensées—of Blaise Pascal Despite his extraordinary accomplishments, Pascal sensed an emptiness 19 Nov 2015 Accomplishments. He developed the
earliest digital calculator that we still use today. He made different works about the binomial coefficients 19 Oct 2017 We take a proper look at her mathematical accomplishments Blaise Pascal was
driven to begin the mechanisation of mathematics by his 24 Aug 2013 Blaise Pascal. ✎ Bonaventura.
7 Oct 2014 Blaise Pascal. 1623-1662 Mathematician. Family Tree. Accomplishments. 1637- Mastered Euclid's Elements 1639- Introduced the Mystical
Some of his accomplishments are the vacum, calculator, and many other things. My favorite accomplishment of his was writing a book to the beats of a French symphony. Blaise Pascal (pronunciación en
francés: /blɛz paskal/; Clermont-Ferrand, 19 de junio 1623-París, 19 de agosto de 1662) fue un matemático, físico, filósofo, teólogo católico y apologista francés.
We know the arithmetic triangle by the name Pascal's Triangle. Although the triangle has been around long before Pascal, it is named for him because he studied the triangle and published the Traite
du Triangle Arithmetique. Blaise Pascal was a French mathematician, physicist and religious philosopher who laid the foundation for the modern theory of probabilities. Blaise Pascal, (born June 19,
1623, Clermont-Ferrand, France—died August 19, 1662, Paris), French mathematician, physicist, religious philosopher, and master of prose. He laid the foundation for the modern theory of probabilities
, formulated what came to be known as Pascal’s principle of pressure , and propagated a religious doctrine that taught the experience of God through the heart Mathematician, physicist, religious
philosopher and wordsmith: By any standard, Blaise Pascal exemplified the term Renaissance man. Born on June 19, 1623, in Clermont-Ferrand, France, Pascal established himself in his early teens as a
self-taught mathematical prodigy [source: Britannica; "Prodigy"]. Blaise Pascal was a French mathematician and physicist who laid the foundation for the modern theory of probabilities.
Masterkatten matte
Blaise Pascal was a very influential French mathematician and philosopher who contributed to many areas of mathematics. He worked on conic sections and projective geometry and in correspondence with
Fermat he laid the foundations for the theory of probability. View six larger pictures Pascal's life, aside from religion, was centered around math and science. He managed to work these into other
projects such as his calculating machine. Pascal's machine, also called la Pascaline, was inspired by his desire to help his father whose many duties as tax Born in 1623 in Clermont, France, Blaise
Pascal is one of the most well known mathematicians of all times.
2008-05-21 · what is blaise pascal accomplishments and please list it.
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Blaise Pascal was born on June 19, 1623 in Clermont-Ferrand, France and Pascal was not only a great mathematician, he was also a man of great knowledge. Therefore, to honour him for his
accomplishments, society agreed to name a&n
133 Copy quote There is a God shaped vacuum in the heart of every man which cannot be filled by any created thing, but only by God, the Creator, made known through Jesus. 2016-04-19 · In his brief
time on Earth, Blaise Pascal (1623–1662) wore many hats and left an imprint on both modern science and Christian philosophy that lingers to this day. Here’s your crash course on the life and
accomplishments of Blaise Pascal—and why he still matters today. | {"url":"https://hurmanblirrikutpb.firebaseapp.com/24639/17980.html","timestamp":"2024-11-08T02:44:31Z","content_type":"text/html","content_length":"13232","record_id":"<urn:uuid:62517dfb-2d37-4886-bee1-b36823d135dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00242.warc.gz"} |
How do you differentiate f(x) = (1+x^2) arctanx
? | HIX Tutor
How do you differentiate # f(x) = (1+x^2) arctanx #?
Answer 1
$f ' \left(x\right) = 1 + 2 x {\tan}^{-} 1 x$
#"differentiate using the "color(blue)"product rule"#
#"given " f(x)=g(x).h(x)" then"#
#rArrf'(x)=cancel((1+x^2)). 1/cancel((1+x^2))+2xtan^-1x#
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Answer 2
To differentiate ( f(x) = (1+x^2) \arctan(x) ), you would use the product rule. The derivative would be:
[ f'(x) = (1+x^2) \frac{d}{dx}(\arctan(x)) + \arctan(x) \frac{d}{dx}(1+x^2) ]
[ f'(x) = (1+x^2) \left(\frac{1}{1+x^2}\right) + \arctan(x) (2x) ]
[ f'(x) = 1 + 2x \arctan(x) ]
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Visualizing academic descendants using modified Pavlo diagrams: Results based on five researchers in biomechanics and biomedicine
Visualizing the academic descendants of prolific researchers is a challenging problem. To this end, a modified Pavlo algorithm is presented and its utility is demonstrated based on manually collected
academic genealogies of five researchers in biomechanics and biomedicine. The researchers have 15–32 children each and between 93 and 384 total descendants. The graphs generated by the modified
algorithm were over 97% smaller than the original. Mentorship metrics were also calculated; their h[m]-indices are 5–7 and the g[m]-indices are in the range 7–13. Of the 1,096 unique researchers
across the five family trees, 153 (14%) had graduated their own PhD students by the end of 2021. It took an average of 9.6 years after their own graduation for an advisor to graduate their first PhD
student, which suggests that an academic generation in this field is approximately one decade. The manually collected data sets used were also compared against the crowd-sourced academic genealogy
data from the AcademicTree.org website. The latter included only 45% of the people and 34% of the connections, so this limitation must be considered when using it for analyses where completeness is
required. The data sets and an implementation of the algorithm are available for reuse.
Mentorship is a foundational component of academia. Although it can take different forms, many of which are unofficial and uncredited, the formal mentoring relationship between a doctoral student and
their advisor(s)^^1 is arguably the most important. It is certainly one that has received has received a great deal of research, most of which can be be divided into one of two categories.
One approach is to focus on the student side of the advisor–advisee relationship. For example, various studies have examined the effects that advisors can have on a student’s mental health (Levecque,
Anseel et al., 2017; Mackie & Bates, 2019), their productivity (García-Suaza, Otero, & Winkelmann, 2020), and their career outcomes (Gaule & Piacentini, 2018; Malmgren, Ottino, & Nunes Amaral, 2010).
Another approach is to consider what these relationships reveal about the advisor. To this end, various ways of quantifying the mentoring productivity—or in biological terms, the fecundity—of a
researcher have also been proposed. One obvious metric is to simply count a researcher’s direct descendants or children; that is, those students that a researcher has advised or coadvised. This
counting can also be extended over multiple generations to sum a researcher’s descendants (i.e., children, grandchildren, great grandchildren, etc.): all those who can trace their advisors’ lineage
back to the original researcher. Recently some have drawn inspiration from publishing metrics such as the h-index (Hirsch, 2005) and g-index (Egghe, 2006) as alternate ways to assess fecundity. Their
mentoring equivalents, the h[m] (Rossi, Damaceno et al., 2018) and g[m]-index (Sanyal, Dey, & Das, 2020), attempt to quantify mentorship by considering the first two generations of descendants.
Besides these quantitative approaches, more qualitative analyses have also been performed. Academic genealogies have been assembled for nations (Damaceno, Rossi et al., 2019), fields (Kelley &
Sussman, 2007; Russell & Sugimoto, 2009), journals (Mitchell, 1992; Montoye & Washburn, 1980), and individual researchers (Bennett & Lowe, 2005; Lv & Chang, 2021). These family trees highlight the
mentoring relationships that exist among researchers and can provide insight into a researcher’s influence on a field.
Despite this interest, visualizing networks of descendants remains a challenge for prolific researchers. A common approach (Rutter, VanderPlas et al., 2019) is to use a typical family tree
representation such as the one shown in Figure 1. Each node in the graph represents an individual and each edge represents an advisor–advisee relationship. Although intuitive, this approach is
unsuitable for large numbers of descendants because the aspect ratio of the graph is determined by the number of generations (height in Figure 1) and the number of individuals in each generation
(width). As the number of descendants grows, the aspect ratio becomes more extreme, making it more difficult to understand the overall topology of the network. Various forms of radial or circular
graphs have been proposed as alternatives that have smaller aspect ratios (Arce-Orozco, Camacho-Valerio, & Madrigal-Quesada, 2017; Grivet, Auber et al., 2006; Huang, Li et al., 2020). A related
challenge of the family tree is that trying to pack as many nodes together to address aspect ratio issues makes it more difficult to distinguish who was advised by whom.
The radial layout algorithm proposed by Pavlo, Homan, and Schull (2006) shows potential for academic genealogies (Figure 2) because the distinction between the descendants of different children is
clear. Each node is surrounded by a containment circle around which the child nodes are placed. The root node uses the entire circle and intermediate nodes use only an outward portion of the circle,
the containment arc, which is bounded by the straight lines. Unfortunately, in the original algorithm proposed by Pavlo et al. (2006), the size of each containment circle is determined by its parent
and the number of siblings. As pointed out by Huang et al. (2020), this approach results in the outermost descendants becoming smaller and smaller as the number of generations increases. If a
suitably large initial size is not chosen, the outermost children can become unreadably small. A second issue is that equivalent sub-trees have different sizes and shapes depending on the generation
in which they occur and the number of siblings they have. Nevertheless, some modifications to the existing algorithm could eliminate these shortcomings and make the resulting diagrams more compact
and more usable for academic genealogies.
A challenge related to evaluating visualization methods is that the data used for assessment are often artificially generated and simplified compared to their real-world equivalents. Real data are
preferred to ensure that any characteristic features are present to reveal any shortcomings of an algorithm for that desired application. For example, academic genealogies are highly asymmetric;
successful researchers may have many doctoral students, but only a fraction of those will go on to have PhD students themselves. The fecundity of those students will also vary dramatically, both as a
result of their individual careers and also due to birth-order effects. We think of a human generation in terms of the 20–30 years needed for a child to be born, mature, and then reproduce. Because
an equivalent (albeit shorter) time is needed for academic reproduction, a researcher’s first few doctoral descendants will have had longer to reproduce, and will likely have more descendants, than
those who graduated near the end of the researcher’s career. Finally, a student may also have multiple coadvisors and these relationships must be represented clearly. These unique characteristics
underscore the need for comprehensive data to test the usability of different visualization methods. Although there are public sources of academic genealogy data available, the quality of these data
remains unclear and must be assessed.
The goals of the current work are threefold. The first is to assemble data sets of academic descendants for five biomechanical/biomedical researchers that are both as comprehensive as possible and
demonstrate a variety of possible shapes and sizes. These data, which will be limited to just doctoral advisor–advisee relationships, will be made available for future visualization studies (see Data
Availability). The second goal is to introduce an improved version of the Pavlo visualization algorithm and demonstrate its suitability for displaying academic genealogies using the collected data
sets. Finally, the third goal is to analyze the data sets to quantify the fecundity of the five researchers, calculate the time necessary for someone to graduate their first PhD student, and assess
the coverage of a particular online repository of academic genealogy data (Academic Family Tree, n.d.). Completing these three goals will help further the study of mentorship within academia,
particularly within the fields of biomechanics and biomedicine.
2.1.Data Collection
Academic genealogy data is spread across a number of sources, including public databases such as Academic Family Tree (n.d.) and the Mathematics Genealogy Project (n.d.), commercial databases such as
ProQuest, and university dissertation repositories, as well as the personal websites and online CVs of individual researchers. None of these sources is necessarily comprehensive, correct, or current.
Yet assembling representative genealogies, ones that have the characteristic sizes and shapes, is critical to evaluating the efficacy and robustness of a visualization algorithm.
Academic descendant data were collected for five researchers from the fields of biomechanics and biomedicine: Steven A. Goldstein, Wilson C. Hayes, Van C. Mow, Lawrence E. Thibault, and Ronald F.
Zernicke. Each researcher received their doctoral degree from 1960 to1980, which is long enough ago to have multiple generations of academic descendants but also recent enough to ensure that most
immediate descendants can still be contacted. The five were also chosen to ensure a range of sizes and shapes in their academic trees. Beyond these selection criteria, the individual researchers
represent a convenience sample. It should also be noted that all five obtained their degrees in the United States. Although they or their descendants have graduated students in institutions around
the world, the vast majority of the researchers in these trees completed their degrees in North America. Therefore, the sizes and shapes of the genealogies may not be representative of those in other
Information gathered from AcademicTree.org and ProQuest was first consolidated together. These data were expanded using public information on researchers’ websites, CVs available online, and
information in institutional dissertation repositories. When the full text of the dissertations was available electronically, the data collected were validated against the information presented on
the title page or in the acknowledgments section. Finally, individual researchers who had a current or past academic appointment were also contacted via email to confirm existing information and
request any missing information. Unfortunately, this was not always possible (e.g., retirement, death, lack of contact information, no response).
Descendants were limited to doctoral students for the purpose of this study. This narrow scope was adopted because a doctoral degree is typically required to advise graduate students, which makes
holders of these degrees most likely to reproduce. Master’s theses were excluded because they receive less coverage in databases, which makes them more difficult to track. Postdoctoral supervision
was also excluded as it would require confirmation from one of the parties involved; it doesn’t generate a single dissertation-like document that allows for independent verification. Therefore,
limiting the scope to PhDs greatly simplified data collection. Finally, any terminal research degree that included a written thesis or dissertation was included regardless of the name (e.g., PhD,
DSc, ScD, DEng, MD).
The following information was collected about each descendant: the person’s name, the institution from which their doctoral degree was obtained, the year of completion, and the names of all advisors
or coadvisors. These data were deemed by our institutional Research Ethics Board (REB) to be public information and not requiring formal consent forms for collection. Nevertheless, when individuals
were contacted via email, they were informed that the assembled data would be made publicly available and were given the opportunity to raise concerns. None of the respondents did so. Any students
who had successfully defended by the end of 2021 were included. Data entry errors or discrepancies were occasionally uncovered. When conflicts arose, information obtained from sources more closely
associated with the individual (i.e., dissertation documents, lab websites, online CVs, or email) were deemed as more authoritative.
It should also be noted that, while generally straightforward, identifying who should be recognized as an “advisor” can at times be difficult to establish. For example, when an advisor moves to
another institution, students still enrolled at the original institution may require a local supervisor for administrative purposes. Although they may be listed as the primary advisor, they may not
actually perform any of the associated duties. Conversely, others may be actively providing mentorship and support to a doctoral student, yet not receive formal recognition as an advisor or
coadvisor. For the purposes of the data collected, we have tried to limit “advisors” to those who both received formal recognition for that role and were not purely administrative. Nevertheless, when
direct communication with the advisors and students was not possible, or no reply was received, we had to proceed with the best information available.
The assembled data sets are available for reuse (see Data Availability) as comma-separated value (.csv) and extended markup language (.xml) files. Although every effort was made to ensure they were
complete through to the end of 2021, they are acknowledged to be imperfect. Moreover, they will quickly become outdated as new descendants are added over time. Nevertheless, they are the most
comprehensive sets of data for these five researchers at the time of writing.
2.2.Modified Pavlo Algorithm
The proposed visualization algorithm is based on the work of Pavlo et al. (2006). As shown in Figure 2, the original algorithm starts with a root node surrounded by a containment circle with a radius
r. The value of r is a user-specified parameter and determines the subsequent size and spacing of all other nodes. The child nodes are then equally spaced around the perimeter of the root’s
containment circle and given their own containment circles, whose radii are determined by geometry. The next set of nodes are then equally spaced around the containment arc, a portion of the
containment circle prescribed by the user-selected angle ϕ. The process continues recursively until all nodes are processed.
As highlighted by Huang et al. (2020), a fundamental problem with the root-outward approach of the original algorithm is that the nodes and containment circles for each subsequent generation become
progressively smaller. A very large value for r must be selected to ensure that there is adequate spacing between the outermost nodes, and this value is not known a priori. A second issue is that
individuals with equivalent numbers of descendants will not be represented by equivalently sized containment circles if they occur in different generations or have different numbers of siblings (see
Figure 2). This phenomenon violates a common aesthetic principal for graph drawing which holds that “a sub-tree should be drawn the same way regardless of where it occurs in the tree” (Reingold &
Tilford, 1981). More importantly, it also results in a larger graph than is necessary due to excess space being used for some nodes, particularly those without children. Therefore, the overall
objectives of this revised algorithm are to ensure that equivalent nodes and subtrees are drawn consistently and the entire graph uses less space than the original algorithm.
The modified Pavlo algorithm will be presented in detail in the following subsections. This process consists of two main steps: determining the size of the containment circles for each node, and
determining the orientation of each node. A third optional step will also be presented that assigns unique node and edge colors based on a hue-saturation-lightness (HSL) color wheel.
Python implementations of the original and modified Pavlo algorithms have been made available for reuse. Consult the Data Availability section for more information.
2.2.1.Determining the node containment circle sizes
A major change to the algorithm is the order in which the size of the containment circles is calculated. The original algorithm relied on a root-outward approach. The user would select the radius, r,
of the containment circle for the root node, and the sizes of all the subsequent circles were determined recursively based on r and the number of children in each generation. Unfortunately, this
method requires an interactive selection of r to ensure some minimum spacing between nodes. The modified algorithm employs a periphery-inward approach. A minimum size for the containment circles is
specified for childless nodes and the subsequent size calculations proceed recursively inward toward the root node. These changes ensure that a minimum spacing is maintained between nodes, ensure
equivalent nodes and subtrees are drawn consistently, and pack the nodes together more tightly to reduce the total area of the graph.
As illustrated in
Figure 3
, there are three types of node scenarios that must be considered. The most common are what we’ll refer to as
terminal nodes
because they have no children (
Figure 3(a)
). Each node has a containment circle, which is used to place it relative to its sibling nodes. In this case, the radius of the containment circle,
, is given by
is the diameter of the node, and
is a user-specified parameter that prescribes the minimum gap between nodes. The ability to prescribe a minimum gap is an important improvement over the original Pavlo algorithm and eliminates the
need to vary the radius of the root containment circle,
, to achieve the desired spacing.
We will assume that the diameter of the node,
, is related to the total number of descendants for that node,
, by
The term
refers to an individual from any subsequent generation (e.g., children, grandchildren, great-grandchildren) that trace their lineage to node
. By definition, terminal nodes have zero descendants (
= 0), which means they have a diameter of
= 1.
Intermediate nodes
Figure 3(b)
) are those with both parents and children. Similar to
Pavlo et al. (2006)
, the child nodes are spaced around a containment arc of the circle prescribed by the angle
, a second user-specified parameter. The length of a continuous containment arc,
, is given by
but based on a finite number of child nodes,
, packed together along the arc, it can be discretized into a series of line segments corresponding to the radii,
, of the children’s containment circles. The segmental length of the arc is then given by
Using the cosine rule, we know that the radius of the node’s containment circle,
, is related to the radius of a child’s containment circle,
, via
which we can rearrange as
The minimum value of
that ensures all children are optimally packed is determined by solving the equation
A numerical approach must be used to solve this equation for
as no direct solution exists. We can rearrange
Eq. 3
to obtain an initial estimate of
There are certain scenarios where it is mathematically possible that a parent node could have a smaller containment circle radius than its descendants, such as when an intermediate node has only one
child. This problem only compounds when a chain of nodes with a single child occurs. To avoid this issue, we define a minimum size for the containment circle,
given by
which ensures the minimum gap size is maintained between the parent and the largest child. We set the containment circle radius,
, to be the maximum of the two values given by
Eqs. 7
. The arc will be larger than necessary for the children in such cases, such as the intermediate nodes with one and two children shown in
Figure 3(b)
The size of the
root node
Figure 3(c)
) is calculated in a similar manner to the intermediate nodes, except that the entire circumference of the containment circle can be used. Therefore, we determine
by finding a solution to
Again, the
is set to the maximum of
Eqs. 8
to avoid problems from small numbers of children.
Because determining a node’s containment circle depends on all its descendants, the size calculations must be performed beginning with the terminal nodes and ending with the root node. Reversing the
order of size calculations from root-outward to periphery-inward results in much more compact graphs and is a major improvement to the original Pavlo algorithm.
2.2.2.Determining the node orientation
Once the sizes of the containment circles have been calculated, we must determine the angular position of each child node relative to its parent. Therefore, the angular assignments must begin with
the root node and work outward to the terminal nodes.
The general case is the one given by the intermediate nodes (
Figure 3(b)
), so we will consider it first. We know that half the angle covered by a child is given by
Eq. 6
. Therefore, the angle
for each child is given by
is the orientation of the parent node and
to account for the fact that the children may not fill the entire containment arc length.
For the root node,
Eq. 10
simplifies to
by assuming that
= 0.
2.2.3.Determining node and edge colors
Every child node has had only a single parent in the examples shown thus far; however, it is possible for a doctoral student to have two or more advisors. For this study, only coadvisors already
within the academic tree—that is, someone who is a descendant of the root researcher—will be included in the visualizations. Nevertheless, these extra edges in the graph can still cause some
A feature common to both the original and the modified Pavlo algorithm is that each child has to be assigned to the containment circle of a single parent. When multiple advisors exist in the tree,
assignment was made based on the order of recognition in the doctoral dissertation, either on the title page or in the acknowledgements. It was assumed that the advisor mentioned first should be
given priority. When the dissertation was unavailable, we relied on information provided by those who responded to our email requests for information.
A second issue is that multiple edges crossing through the graphs may make it difficult to identify who is advising whom. To reduce confusion, each node was assigned a unique color and all edges
originating from that node were given the same color. It is assumed that all nodes can be considered on a circle centered at the root node (
Figure 4
). The radial distance from the root to the center of the furthest node,
, is treated as the radius of this circle. Colors are then assigned to the nodes based on the angle and radius of an HSL (hue, saturation, lightness) color wheel (
HSL and HSV, n.d.
). For each node
, with a radial distance
and an angular position
, the (r, g, b) components for that node are given by
The order in which children are drawn is important because it can be used to indicate the order in which they completed their doctoral studies. For intermediate nodes, this means the oldest child
(i.e., earliest completion date) is drawn first and subsequent children are drawn, in order, in a clockwise direction. However, because the children of the root node are placed around a circle with
no obvious beginning or end, the edge to the oldest child is drawn directly from the root node (Figure 4). All other edges from the root node are drawn from around a central arc to indicate the order
of completion. Edges from intermediate nodes to their children are drawn using Bezier curves if that child is on its own ring; however, a straight line is used if the child is on another ring to
better distinguish the two scenarios (Figure 4).
In addition to creating the visualizations themselves, five groups of analyses were performed on the five data sets: determining the reductions in graph size achieved by the modified algorithm,
investigating the effects of the user-selected parameters on the generated graphs, quantifying researcher fecundity using mentorship metrics, calculating the length of an academic (doctoral)
generation, and assessing the completeness of the data available via AcademicTree.
2.3.1.Improved performance of the modified algorithm
One of the main objectives of the modified algorithm was to decrease the space required to display the genealogies. Family trees for the five researchers were generated using both algorithms (see
Data Availability
for Python implementations). Each graph was rotated to the portrait orientation that used the smallest area as calculated by a rectangular bounding box. The reduction in area was calculated as
is the rectangular area for the original Pavlo algorithm and
is the rectangular area for the modified algorithm. Decreased size is reported as a positive percentage.
It should be noted that the size of the original Pavlo diagram will be determined by the initial radius r, whereas the modified algorithm prescribes a minimum gap between nodes, g. To ensure a fair
comparison, the values of r were scaled to ensure an equivalent gap size. The value of ϕ was kept constant for both algorithms.
2.3.2.Effects of user-selected parameters (g and ϕ)
The two user-selected parameters (g and ϕ) will control the size, shape, and quality of the graphs generated by the modified algorithm. Therefore, these parameters were investigated independently to
understand their effects. Because a constant value of g (g[0] = 1) was used throughout, and an optimal value of ϕ (ϕ[0]) was determined for each genealogy, these parameters were used to calculate a
reference area (A[0]) for each graph. The values of g were then varied from 0.5 to 3 and the ratio of the resulting graph area (A) relative to the reference area (A/A[0]) was used to calculate the
effect on size. Similarly, ϕ was varied between 90° and ϕ[0].
2.3.3.Mentorship statistics and metrics
Various summary statistics and metrics were calculated for the five individual researchers. The first involved counting the number of descendants a researcher had in each generation. Those supervised
directly by the researcher were the first generation (children), those supervised by the first generation were the second generation (grandchildren), and so on. The total of all descendants was also
calculated. When an individual has two or more advisors, it is possible for them to be considered part of multiple generations. As with the visualizations, the primary supervisor was used to
determine the generation to which they belonged.
Two researcher fecundity metrics were also calculated and reported, both of which were inspired by bibliometric indices (Hirsch, 2005; Egghe, 2006). The mentoring h-index (h[m]) proposed by Rossi et
al. (2018) is defined as the number of direct descendants n who themselves have at least n descendants. However, Sanyal et al. (2020) noted that this metric was insensitive to the fact that an
individual child may have a large number of descendants. They proposed the mentoring g-index (g[m]) which is defined as the largest number n, for which a researcher has n academic children and n^2
2.3.4.Academic generation length calculation
For each researcher, the time between their own graduation and the completion of their first doctoral student was calculated in years. This time to reproduce within academia, an academic generation,
is the research equivalent of a human generation. Given that other forms of progeny such as master’s or postdoctoral students have not been considered in this study, it might more accurately be
termed a doctoral generation. Nevertheless, this distinction may be unnecessary because those with master’s degrees are typically ineligible to advise graduate students, and postdoctoral students, by
definition, already have the qualifications necessary.
2.3.5.Assessment of AcademicTree data
Online databases are frequently used by researchers interested in understanding academic genealogical patterns. These databases tend to be focused on researchers in specific domains such as
mathematics (Mathematics Genealogy Project, n.d.) or biological anthropology (Barr, Nachman, & Shapiro, n.d.). Although it began as NeuroTree, and was initially focused on researchers in neuroscience
(David & Hayden, 2012), AcademicTree.org has since expanded to other areas and has become the most generalized repository available. With almost one million entries and connections, and because
researchers routinely use this data for analysis, it is of interest to assess the comprehensiveness of these community-provided data compared to the manually tracked data collected for this project.
AcademicTree data consist of two types of information: a person and a connection. Snapshots of the entire data set (David, 2021)—the most recent of which is from January 14, 2021—are publicly
available for processing and analysis (Liénard, Achakulvisut et al., 2018). First, the researchers identified in the five data sets were checked against those in AcademicTree to confirm whether they
were present. Second, whether the connection between advisor and student was present in the database was also verified. Only the connections between people in the individual trees, those represented
by edges in the visualizations, were evaluated. The values for both people and connections are reported as a percentage of the data collected in this study which are correctly contained in
AcademicTree. Incorrect or additional information in AcademicTree was not evaluated, so the reported values represent an upper-bound estimate of the data coverage.
Academic genealogies for the five researchers were assembled from a variety of online sources and from information provided by individuals within each tree. Data files containing this information are
available for those who wish to reuse them (see Data Availability).
The modified Pavlo diagrams showing the academic descendants of the five selected researchers are given in Figures 5–9. Siblings will never overlap due to the nature of the algorithm; however,
interactions between more distantly related individuals are possible. The largest value of ϕ that eliminated intersection of the containment rings was determined for each graph via trial and error;
the specific value used is indicated in the caption. Note that only the outer portions of the containment rings have been drawn, and the ϕ lines have been eliminated altogether, to reduce visual
clutter. Note also that the graphs have been rotated into the portrait orientation that makes the most efficient use of the page. Python implementations of the original and modified algorithms are
available for reuse (see Data Availability).
The modifications to the algorithm were able to reduce the total area needed to present the genealogies. These reductions were quantified after adjusting the r value of the original Pavlo algorithm
to ensure equivalent node size and spacing, and after rotating both genealogies to their optimal portrait orientation. As shown in Figure 10, the sample data used in Figures 1–3 were just one quarter
of the size of the original when plotted with the modified algorithm (ΔA = 76.5%). Even larger reductions were observed for the genealogies of the five researchers. The Zernicke data was reduced in
area by 97.4% (Figure 11). The graphs of the other four researchers had ΔA > 99.9% but are not shown due to the very sparse trees produced by the original algorithm.
The areas of the graphs generated by the modified Pavlo algorithm will be affected by the user-selected g parameter. Three values of g are shown in Figure 12 applied to the Zernicke genealogy; the
overall layout of the nodes is unchanged by g, and only the scale is affected. The change in size might be expected to follow a trend where A/A[0] ∝ (g/g[0])^2 as a doubling of g might be expected to
double both the width and height of the graph; however, Figure 13 indicates that A/A[0] increases more slowly. This behavior results from the graph-specific path by which the outermost nodes approach
the bounding box.
The optimal angle (ϕ[0]) was selected iteratively for each graph based on the values at which two or more containment rings began to overlap. The plot in Figure 13 indicates that the area (A/A[0])
increases nonlinearly for decreasing values of ϕ. The values of ϕ[0] varied from 128–175° for the five genealogies. Although ϕ[0] tends to decrease with the total number of nodes, the value is
dependent on the specific shape of the graph. An alternative to iteratively selecting an optimized ϕ is to select a small angle unlikely to result in collisions of the containment rings, albeit with
a resulting increase in area. For example, if a conservative value of ϕ = 120° had been chosen a priori for all graphs, their areas would have increased between 1.25 and 2.25 times (Figure 13). It
should also be noted that, because there tends to be one region that determines the ϕ[0], the graphs are relatively insensitive to small deviations from the optimal value. Figure 14 illustrates the
resulting changes when adjusting ϕ[0] for the Goldstein genealogy by ±10°.
The numbers of descendants in each generation for the five researchers are shown in Table 1. The researchers had between 15 and 32 direct descendants and their total number of descendants ranged from
93 to 384. Some individuals and their descendants appear in two family trees because of cosupervision. Therefore, the 1,118 descendants calculated by adding up the totals of the five researchers
consist of only 1,091 unique descendants when duplicates are removed. When the five original researchers are included, there are 1,096 unique researchers across the five trees. The h[m]-index was 5–7
for each of the researchers, despite the very different genealogical trees. The g[m]-index showed more sensitivity and varied in the range 7–13. Based on the analysis of AcademicTree.org data
reported by Sanyal et al. (2020), these values place them in the top 1% of researchers with at least one descendant. This comparison is reported for context but should be interpreted with caution
given the differences in the data sets used.
Table 1.
Researcher . Descendants by generation . Metrics .
1st . 2nd . 3rd . 4th . Total . h[m] . g[m] .
Zernicke 25 53 11 4 93 5 7
Goldstein 32 69 35 12 148 5 8
Thibault 15 83 48 4 150 6 9
Mow 29 176 124 14 343 7 13
Hayes 21 139 201 23 384 7 11
Researcher . Descendants by generation . Metrics .
1st . 2nd . 3rd . 4th . Total . h[m] . g[m] .
Zernicke 25 53 11 4 93 5 7
Goldstein 32 69 35 12 148 5 8
Thibault 15 83 48 4 150 6 9
Mow 29 176 124 14 343 7 13
Hayes 21 139 201 23 384 7 11
There were 153 individuals among the 1,096 unique researchers (14%) who had graduated at least one PhD student of their own by the end of 2021. The difference (in years) between the graduation date
of each advisor and that of their first doctoral student is shown in Figure 15. The distribution is right skewed with an average time of 9.6 years. The median (9 years) and mode (7 and 8 years) were
both slightly faster than the average.
Finally, the coverage of the crowd-sourced AcademicTree data, relative to the data collected for this study, are shown in Table 2. The percentage of people in the five individual genealogies varied
between 23% and 70%, with 45% of the unique researchers included. The numbers of connections included in AcademicTree was lower: 34% overall, with a range of 17–57%. These numbers represent an
upper-bound estimate given that we expect that the current data are incomplete and because any erroneous connections in the AcademicTree data set were also not evaluated.
Table 2.
Researcher . People . Connections .
AT . CS . % . AT . CS . % .
Zernicke 22 94 23.4 16 96 16.7
Goldstein 64 149 43.0 47 152 31.0
Thibault 61 151 40.4 48 160 30.0
Mow 241 344 70.0 202 352 57.4
Hayes 130 385 33.8 89 391 22.8
Unique 496 1,096 45.3 391 1,132 34.3
Researcher . People . Connections .
AT . CS . % . AT . CS . % .
Zernicke 22 94 23.4 16 96 16.7
Goldstein 64 149 43.0 47 152 31.0
Thibault 61 151 40.4 48 160 30.0
Mow 241 344 70.0 202 352 57.4
Hayes 130 385 33.8 89 391 22.8
Unique 496 1,096 45.3 391 1,132 34.3
Circular or radial graphing algorithms result in academic genealogies with smaller aspect ratio layouts, as compared to a standard family tree, for large numbers of descendants. A modified Pavlo
layout algorithm has been presented herein that corrects some of the shortcomings of the original. It has been shown to be useful on a range of academic trees with up to four generations and over 380
descendants. The data sets and a reference implementation of the algorithm are available as open data (see Data Availability).
The modified algorithm succeeded in reducing the area occupied by each genealogy. A 77% reduction was obtained for the simple example tree in Figure 10, with reductions of 97% or greater for the
genealogies of the five researchers. It could be argued that including the containment rings in the bounding boxes used to calculate area inflated these values in some cases (e.g., Figure 11).
Nevertheless, substantive reductions were obtained in this study with the modified algorithm. Other use cases would have to be studied to confirm whether similar performance can be expected; however,
the approach appears to be robust.
The algorithm has two user-specified parameters: the minimum gap length (g) and the included angle for the containment arc onto which children are fit (ϕ). Values of g = 1 and ϕ = 128–175° have been
used successfully herein. Because the g term controls the scale of the resulting graph, it can be chosen to alter the spacing between nodes without altering the overall shape (Figure 12). Conversely,
the value of ϕ was manually selected to obtain the largest value that ensured that no overlap in the containment rings occurred. Smaller values of ϕ tended to be needed as the number of descendants
grew, but the exact value depends on the specific shape of the graph. Based on the range of values determined in the current study, an initial estimate of ϕ = 150° is recommended for an iterative
search. Alternatively, a constant value of 120° would have yielded satisfactory graphs in all cases, albeit with up to a 2.25× increase in area (Figure 13). Such increases in size may be undesirable
in some applications, but the resulting graphs would still be much smaller (ΔA > 90%) than those produced by the original Pavlo algorithm. The value of ϕ in the current algorithm is both manually
selected and constant across all nodes of the graph. Future improvements could be made to the algorithm to either recursively adjust a constant ϕ value or to determine unique ϕ[i] values for each
containment ring to eliminate overlap and minimize the area used; however, these changes would come with increased computational costs. The current algorithm has shown to be applicable to the unique
shape and sizes of academic genealogies, but may also have application to representing a broader range of trees. Different guidelines for ϕ values may be needed in such cases.
The data sets assembled for five biomedical researchers relied on a variety of public and commercial resources. Individual researchers were then contacted to confirm the collected data and gather
additional information. Ensuring that the data sets were as current, correct, and comprehensive as possible was important to properly demonstrate the suitability of the algorithm and justified the
extra effort involved. Having shown that the algorithm can perform well when handling these large, real-world data sets, it should have no issues with smaller, more sparse graphs. Plus, it is
important that the data used exhibit the unique characteristics of academic genealogies. For example, the Hayes tree (Figure 9) has one child (Dennis R. Carter) who himself has a very large number of
descendants. Such a feature would not necessarily be found in artificially generated data, or even when using incomplete data. Although every effort was made to ensure the completeness of the data,
it must be acknowledged that they are imperfect. Not everyone could be contacted, and not everyone who was contacted replied (the response rate was roughly 45%). Nevertheless, these data are the most
exhaustive academic genealogies for these five researchers currently available.
It was interesting to compare the results of the manually traced genealogies created for this study with the crowd-sourced data available. Roughly 45% of the people and 34% of the connections
identified were found in the Academic Family Tree (n.d.) data. It should also be noted that this evaluation only considered the people and connections within the researchers’ genealogies; advisors
(and connections to those advisors) outside the tree were not considered or counted, nor were any erroneous connections within the AcademicTree data. Because of this methodology, and because the
current data are known to be incomplete, these percentages represent an upper-bound estimate of the true coverage. Given that the five researchers were within biomechanics and biomedicine, it is
unclear how coverage might differ for researchers in other domains. Nevertheless, some incompleteness should be expected and accounted for by researchers performing analyses using the AcademicTree
Less complete data are to be expected in crowd-sourced resources, as they rely on continuous participation to provide the necessary information. In this context, it is noteworthy that one researcher
had much higher coverage than the other four (Table 2). The reason for this discrepancy is that Dr Mow was awarded the 2017 Alfred R. Shands, Jr., MD Award by the Orthopaedic Research Society (ORS)
for significant contributions to the field. As part of the awards ceremony, some of his descendants presented an academic lineage that they had compiled and uploaded to the AcademicTree website. This
detail further underscores both the diligence needed to assemble exhaustive data and the challenge of keeping it updated.
There were 1,096 unique researchers across the five data sets. As of the end of 2021, 153 of them had gone on to have a PhD student of their own (14%) and it took an average time of 9.6 years to do
so. The distribution of these times to graduate a first PhD is right skewed. This behavior likely reflects uneven sampling, as the increasing numbers of descendants with time means that graduation
dates skew towards the present, and the fact that there is maximum length of time for recent graduates to have graduated their own PhD students (the angled line of Figure 15). Given that most of the
researchers studied are in biomechanics or biomedicine, and given that most degrees were earned at institutions in the United States, those durations might not be reflective of other contexts.
Nevertheless, it is helpful to think of an academic (or doctoral) generation as being roughly a decade in length.
Two mentorship metrics were calculated for the five researchers. The h[m]-index varied from 5–7, whereas the g[m] index ranged from 7–13. These results agree with the observations of Sanyal et al.
(2020) that the h[m]-index was a less sensitive metric. Based on an analysis of AcademicTree by Sanyal et al. (2020), these g[m] values would place each of the researchers in elite territory;
however, direct comparison between the two is difficult because of differences in the data used. For example, AcademicTree includes all graduate students and postdoctoral researchers in its
mentorship data, not just the doctoral students considered herein, which would lead to higher metrics than those reported in the current study. Conversely, the incompleteness of the AcademicTree data
already discussed could also skew their metrics downward. The g[m]-index offers improved discretization but care is needed when evaluating different researchers to ensure that equitable comparisons
are being made.
Finally, it should be recognized that the graphs and indices only capture a particular form of “success” with regard to mentorship. Quantity and quality are orthogonal concepts. These numbers focus
on the former and, although it is tempting to view those with smaller numbers as being less successful, it is important to recall the distinction between student- or advisor-centric methods of
assessment. A student entering a doctoral program may do so to pursue a career in industrial research, with the goal of launching a start-up company, or to complement future training in other
professions such as law or medicine. The extent to which an advisor equips that student to obtain these goals is a different metric of success altogether. Other important definitions of success, such
as the way in which an advisor treats their trainees, are equally difficult to quantify. Therefore, although the work presented herein certainly provides insight into mentoring fecundity, it should
be balanced by the recognition that “not everything that can be counted counts, and not everything that counts can be counted” (Cameron, 1963).
In conclusion, the current work has proposed a modified Pavlo algorithm for representing compact depictions of academic genealogies. The utility of the approach has been demonstrated using data sets
showing the doctoral descendants of five prolific researchers in biomechanics and biomedicine. A number of different analyses have also been performed on the data, which show that the g[m]-index is a
more sensitive measurement of fecundity, roughly 45% of people and 34% of connections were covered in AcademicTree, and the average time to graduate one’s first PhD student was roughly a decade.
The author would like to thank all the respondents for their assistance with, interest in, and enthusiasm for this project. Interacting with you has been a wonderful reminder of the best aspects of
academia. The author would also like to acknowledge his father, the keeper of our family tree, from whom he has inherited an interest in genealogies.
The author has no competing interests.
No funding was received for this research.
The data associated with this paper are available for reuse from Borealis (formerly Scholars Portal Dataverse): https://doi.org/10.5683/SP3/MDGUTK. The data for the five genealogies are available as
comma-separated value (CSV) and extended markup language (XML) files, while the diagrams themselves are provided as scalable vector graphics (SVG). The Python code used to generate the original and
modified Pavlo diagrams is also provided as a reference implementation. All files are available under a Creative Commons CC0 “Public Domain Dedication” license.
The specific titles applied to this role vary by jurisdiction and institution (e.g., advisor, chair, director, supervisor), but the term advisor will be used throughout this paper for consistency.
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Author notes
Handling Editor: Ludo Waltman
© 2022 W. Brent Lievers. Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.
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What is the formula of partial differentiation?
The formula for partial derivative of f with respect to x, considering y as constant is: Fx = ∂f/∂x = limh→0 f(x+ h,y) – f(x,y)/h.
What are the methods for solving partial differential equations?
The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called
Meshfree methods, which were made to solve problems where the aforementioned methods are limited.
What is the partial derivative of XY?
Using the chain rule with u = xy for the partial derivatives of cos(xy) ∂ ∂x cos(xy) = ∂ cos(u) ∂u ∂u ∂x = − sin(u)y = −y sin(xy) , ∂ ∂y cos(xy) = ∂ cos(u) ∂u ∂u ∂y = − sin(u)x = −x sin(xy) . Thus
the partial derivatives of z = sin(x) cos(xy) are ∂z ∂x = cos(xy) cos(x) − y sin(x) sin(xy) , ∂z ∂y = −x sin(x) sin(xy) .
What is partial differentiation in math example?
Partial Differentiation The process of finding the partial derivatives of a given function is called partial differentiation. Partial differentiation is used when we take one of the tangent lines of
the graph of the given function and obtaining its slope. Let’s understand this with the help of the below example.
How to find partial derivative of a function?
The process of finding the partial derivative of a function is called partial differentiation. In this process, the partial derivative of a function with respect to one variable is found by keeping
the other variable constant.
What are the rules of differentiation for algebraic functions?
Rules of Differentiation for Algebraic Functions. In this tutorial we will discuss the basic formulas of differentiation for algebraic functions. 1. d d x ( c) = 0, where c is any constant. 2. d d x
( x) = 1. 3. d d x ( c x) = c, where c is any constant. 4. d d x x n = n x n – 1, which is known as the power rule of a derivative.
What are the basic formulas of differentiation?
In this tutorial we will discuss the basic formulas of differentiation for algebraic functions. 1. d d x ( c) = 0, where c is any constant. 2. d d x ( x) = 1. 3. d d x ( c x) = c, where c is any | {"url":"https://durrell2012.com/what-is-the-formula-of-partial-differentiation/","timestamp":"2024-11-06T23:11:41Z","content_type":"text/html","content_length":"45589","record_id":"<urn:uuid:eee6fb73-be4b-41bd-91d5-66a7c7612726>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00502.warc.gz"} |
Printable Calendars AT A GLANCE
Heat Calculations Worksheet Answer Key
Heat Calculations Worksheet Answer Key - University of oregon (uo) thermodynamics. The specific heat calculations worksheet consists of two pages: How much water at 50°c is needed to just melt 2.2 kg
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Calculating Specific Heat Worksheet
Web calculate the amount of heat transferred from the engine to the surroundings by one gallon of water with a specific heat of 4.184 j/g °c. Heat and heat calculations worksheet 4: Show all work and
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50 Heat Transfer Worksheet Answer Key
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Calculating Specific Heat Worksheet Answers
Web calculating heat and specific heat. For q= m c δ t : Web practice problems on heat capacity and specific heat. Temperature is a measure of the average kinetic energy and does not depend upon the.
How much water at 50°c is needed to just melt 2.2 kg of ice at 0°c?
Worksheet Heat And Heat Calculations —
Show all work and proper units. The gold cools until the final temperature is 29.3 0c. View calculating specific heat worksheet answer key.pdf from che 112 at northern kentucky university. Write the
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13 Heat Worksheet 1 /
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30++ Heat Calculations Worksheet
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Calorimetry Worksheet Answer Key
Heat and heat calculations name_____ chemistry: Worksheets are physical science if8767 answer key heat calculations, physical. Web specific heat worksheet name (in ink): Web showing 8 worksheets for
instructional fair and physical science if8767. Web heat capacity and calorimetry.
Specific Heat Worksheet Answer Key
For q= m c δ t : How many joules of heat are needed. University of oregon (uo) thermodynamics. Worksheets are physical science if8767 answer key heat calculations, physical. Web specific heat and
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Web practice problems on heat capacity and specific heat. Ag 2 s(s) + 2hcl(g) 2agcl(s) + h 2 s(g) h = ? Web specific heat and heat capacity worksheet. C = q/mat, where q = heat energy, m = mass, and
t = temperature remember, at = (tfinal — tinitial). Web latent heat and specific heat capacity questions.
Heat Calculations Worksheet Answer Key - Ag 2 s(s) + 2hcl(g) 2agcl(s) + h 2 s(g) h = ? Show all work and units. C = q/mat, where q = heat energy, m = mass, and t = temperature remember, at = (tfinal
— tinitial). Kj/mol solve by writing formation. How much water at 50°c is needed to just melt 2.2 kg of ice at 0°c? What is the difference in temperature and heat? Examples of how to determine, the
heat, heat capacity, and change of temperature. 50 g of gold with a specific heat of 0.129 is heated to 115 0c; For q= m c δ t : Heat and heat calculations name_____ chemistry:
The gold cools until the final temperature is 29.3 0c. Kj/mol solve by writing formation. 16.1 specific heat practice part 1 answer. Web latent heat and specific heat capacity questions. = mc∆t,
where q = heat energy, m = mass, and ∆t = change in temp.
Show all work and units. Temperature is a measure of the average kinetic energy and does not depend upon the. View calculating specific heat worksheet answer key.pdf from che 112 at northern kentucky
university. Heating curve worksheet key, exercises for thermodynamics.
50 G Of Gold With A Specific Heat Of 0.129 Is Heated To 115 0C;
C = q/mat, where q = heat energy, m = mass, and t = temperature remember, at = (tfinal — tinitial). Temperature is a measure of the average kinetic energy and does not depend upon the. Identify each
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G Of Water At A Lower Temperature T 2 In A.
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Web Calculate The Amount Of Heat Transferred From The Engine To The Surroundings By One Gallon Of Water With A Specific Heat Of 4.184 J/G °C.
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How long would it take a quantum computer to crack RSA 2048?
They found that to factor a composite number of 2048 bits would require around 10,000 qubits, 2.23 trillion quantum gates, and “a quantum circuit depth of 1.8 trillion”, Fujitsu said in a statement.
The researchers also found a sufficiently-large fault-tolerant quantum computer would need 104 days to crack RSA.
Can quantum computers break 2048-bit RSA?
The paper, published three weeks ago by a team of researchers in China, reported finding a factorization method that could break a 2,048-bit RSA key using a quantum system with just 372 qubits when
it operated using thousands of operation steps.
How long does it take for quantum computers to break RSA?
So far, all experts have agreed that a quantum computer large enough to crack RSA would probably be built no sooner than around a few dozen decades.
How long would it take a quantum computer to crack 2048-bit encryption?
A perfect Quantum Computer could do this in 10 seconds
A quantum computer with 4099 perfectly stable qubits could break the RSA-2048 encryption in 10 seconds (instead of 300 trillion years – wow).
Can RSA 2048 be broken?
The basic claim of the paper, published last Christmas by 24 Chinese researchers, is that they have found an algorithm that enables 2,048-bit RSA keys to be broken even with the relatively low-power
quantum computers available today.
29 related questions found
Which is better RSA 2048 or 4096?
A 4096 bit key does provide a reasonable increase in strength over a 2048 bit key, and according to the GNFS complexity, encryption strength doesn't drop off after 2048 bits. There's a significant
increase in CPU usage for the brief time of handshaking as a result of a 4096 bit key.
Has RSA 1024 been cracked?
With a small cluster of 81 Pentium 4 chips and 104 hours of processing time, they were able to successfully hack 1024-bit encryption in OpenSSL on a SPARC-based system, without damaging the computer,
leaving a single trace or ending human life as we know it.
How long will RSA 2048 last?
Theoretically, RSA keys that are 2048 bits long should be good until 2030.
How long to brute force RSA 2048?
With existing computing technology, one estimate holds it would take 300 trillion years to “brute force” an RSA 2048-bit key.
How long would it take a quantum computer to crack AES 256?
It would require 317 × 10^6 physical qubits to break the encryption within one hour using the surface code, a code cycle time of 1 μs, a reaction time of 10 μs, and a physical gate error of 10^-^3.
To instead break the encryption within one day, it would require 13 × 10^6 physical qubits. In other words: no time soon.
Can quantum computers break 256?
“AES-256 specifically is believed to be quantum-resistant,” he told IQT News via email recently. “According to Grover's Algorithm, a brute-force attack time can be reduced to its square root. But if
this time is still sufficiently large, it becomes impractical to use as an attack vector.
Can quantum computers break RSA 256?
Bitcoin's SHA256 encryption algorithm is still safe despite Chinese researchers' claims of cracking RSA encryption with existing quantum computers. A group of 24 Chinese researchers said they could
factor a 48-bit number using a 10-qubit quantum computer.
Can quantum computers break RSA 4096?
Large universal quantum computers could break several popular public-key cryptography (PKC) systems, such as RSA and Diffie-Hellman, but that will not end encryption and privacy as we know it.
Can Bitcoin survive on quantum computers?
Joint research from the University of Sussex, Universal Quantum and Qu&Co published in January 2022 in AVS Quantum Science suggests that quantum computers would have to become a million times faster
to break bitcoin's cryptography.
How long would it take a quantum computer to hack Bitcoin?
Researchers at the University of Sussex estimated in February that a quantum computer with 1.9 billion qubits could essentially crack the encryption safeguarding Bitcoin within a mere 10 minutes.
Just 13 million qubits could do the job in about a day.
How long does it take to crack 4096 bit RSA?
We show an attack that can extract whole 4096-bit RSA keys within about one hour using just the acoustic emanations from the target machine. The choice of the size of the 4096 bit number is more as a
Proof of Concept that it is possible to do it with big number.
How fast a quantum computer can brute force a 128-bit key?
As shown above, even with a supercomputer, it would take 1 billion billion years to crack the 128-bit AES key using brute force attack. This is more than the age of the universe (13.75 billion
How long would it take to brute force a 128-bit key?
The EE Times points out that even using a supercomputer, a “brute force” attack would take one billion years to crack AES 128-bit encryption.
How long does brute forcing 12 chars take?
Password managers are the best bet for protecting passwords, according to Hive, which also found that a 12-character password created by a password manager could take some 3,000 years to brute-force
What is the highest level in 2048?
Higher-scoring tiles emit a soft glow; the highest possible tile is 131,072. If a move causes three consecutive tiles of the same value to slide together, only the two tiles farthest along the
direction of motion will combine.
What is the fastest win in 2048?
2048 speed run: 11 seconds (world record)
Is every game of 2048 winnable?
Problem can be posed more generally: When the game is won at small values (like 16) it is always winnable, but at some point must become unwinnable, as some numbers are too large to be made on the
What is the longest RSA key cracked?
Although it's estimated that a 1,024-bit RSA key won't be broken within the next five years (768 bits is the largest RSA key known to have been cracked), it's only considered equivalent to 80 bits of
Is RSA impossible to crack?
RSA is the standard cryptographic algorithm on the Internet. The method is publicly known but extremely hard to crack. It uses two keys for encryption. The public key is open and the client uses it
to encrypt a random session key.
How long would it take to crack 512 bit encryption?
> 2003 ("within three years") a 512-bit key can be factored in a few days. this latter case, you are still looking at 2-3 years to crack the key. required are rarely justified by the potential gain.
they can only do one key at a time that way. | {"url":"https://www.calendar-uk.co.uk/frequently-asked-questions/how-long-would-it-take-a-quantum-computer-to-crack-rsa-2048","timestamp":"2024-11-02T15:00:55Z","content_type":"text/html","content_length":"71726","record_id":"<urn:uuid:d1712ba6-48d5-4eb0-9d78-fa7f99f76e8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00350.warc.gz"} |
Gambin overview
The gambin distribution is a sample distribution based on a stochastic model of species abundances, and has been demonstrated to fit empirical data better than the most commonly used
species-abundance distribution (SAD) models (see Matthews et al. (2014) and Ugland et al. (2007)). Gambin is a stochastic model which combines the gamma distribution with a binomial sampling method.
To fit the gambin distribution, the abundance data is first binned into octaves using a simple log2 transform that doubles the number of abundance classes within each octave. Thus, octave 0 contains
the number of species with 1 individual, octave 1 the number of species with 2 or 3 individuals, octave 2 the number of species with 4 to 7 individuals, and so forth (method 3 in Gray, Bjorgesaeter,
and Ugland (2006)).
The gambin distribution is flexible, meaning it can fit a variety of empirical SAD shapes (including lognormal and logseries-like shapes), and that the distribution shape (in the context of the
unimodal gambin model) is adequately characterised by the model’s single parameter (α): low values of alpha indicate logserieslike SADs, and high alpha values indicate lognormal-like SADs. As such,
the alpha parameter can be used as a metric to compare the shape of SADs from different ecological communities; for example, along an environmental gradient (e.g Arellano et al. (2017))
The expected abundance octave of a species is given by the number of successfull consecutive Bernoulli trials with a given parameter \(p\). The parameter \(p\) of species is assumed to distributed
according to a gamma distribution. This approach can be viewed as linking the gamma distribution with the probability of success in a binomial process with \(x\) trials. Use the fit_abundances()
function to fit the gambin model to a vector of species abundances, optionally using a subsample of the individuals. The package estimates the alpha (shape) parameter with associated confidence
intervals. Methods are provided for plotting the results, and for calculating the likelihood of fits. The summary() function provides the confidence intervals around alpha, and also the results of a
X2 goodness of fit test. Prior to package version 2.4.4, we simply used the default degrees of freedom in this test (i.e. number of data points - 1). This is not optimal as the degrees of freedom
should arguably also include the number of parameters used to fit the gambin model itself. As such, in version 2.4.4 we have edited the degrees of freedom to reflect this. One problem is that the
chisq.test() function in R does not have an argument for setting the degrees of freedom; thus, we have had to use a workaround. As a result of this change, X2 results generated using older versions
of the package will differ slightly from those using 2.4.4 and later.
It has become increasingly apparent that many empirical SADs are in fact multimodal (Antao et al. (2017)). As such, recent work has focused on expanding the standard unimodal gambin model to allow it
to fit distributions with multiple modes (Matthews et al. (2019)). For example, the bimodal gambin model can be calculated as the integration of two gambin distributions. The corresponding likelihood
function for the bimodal gambin model contain’s four parameters: the shape parameters for the first and second group, the max octave of the first group (as this is allowed to vary), and one splitting
parameter (split) representing the fraction of objects in the first group.It is relatively straightforward to extend the above approach for fitting the bimodal gambin model by maximum likelihood, to
fitting gambin models with g modes. For each additional mode, a further three parameters are needed: the additional alpha, max octave and split parameters (see Matthews et al. (2019)). Use the
fit_abundances() function in combination with the no_of_components of argument. The default is no_of_components = 1, which fits the standard unimodal gambin model. no_of_components = 2, fits the
bimodal gambin model, and so on. As the optimisation procedure takes long with no_of_components > 1, it is possible to use the cores argument within fit_abundances to make use of parallel processing
in the maximum likelihood optimisation.
The deconstruct_modes() function can then be used to examine a multimodal gambin model fit. The function provides the location of the modal octaves of each component distribution and (if species
classification data are provided) determines the proportion of different types of species in each octave.
Often the aim of SAD studies is to compare the form of the SAD across different sites / samples. The alpha parameter of the one component gambin model (alpha) has been found to provide a useful
metric in this regard. Use the mult_abundances() function to calculate alpha values for a set of different samples / sites. However, because the alpha parameter of the gambin model is dependent on
sample size, when comparing the alpha values between sites it can be useful to first standardise the number of individuals in all sites. By default, the mult_abundances() function calculates the
total number of individuals in each site and selects the minimum value for standardising. This minimum number of individuals is then sampled from each site and the gambin model fitted to this
subsample and the alpha value stored. This process is then repeated N times and the mean alpha value is calculated for each site.
data(moths, package="gambin")
##unimodal model
fit = fit_abundances(moths)
## [1] 1.644694
## [1] 803.6409
##unimodal model (fit to a subsample of 1000 individuals)
fit2 = fit_abundances(moths, subsample = 1000)
## [1] 0.8904631
## [1] 394.7729
##bimodal model (using 3 cores)
#simulate bimodal gambin distribution
x1 = rgambin(600, 5, 10)
x2 = rgambin(300, 1, 10)
x = table(c(x1,x2))
freq = as.vector(x)
values = as.numeric(as.character(names(x)))
abundances = data.frame(octave=values, species = freq)
#fit bimodal model to simulated data
fit3 = fit_abundances(abundances, no_of_components = 2, cores = 1)
## Using 1 core. Your machine has 12 available.
## [1] 4062.259
## [1] 4087.551
#fit a bimodal model to a species classification dataset
#and calculate the number of the differet categories in each octave
data(categ, package="gambin")
fits2 = fit_abundances(categ$abundances, no_of_components = 2)
## Using 1 core. Your machine has 12 available.
d1 <- deconstruct_modes(fits2, dat = categ, peak_val = NULL, abundances = "abundances",
species = "species", categ = "status", col.statu = c("green", "red", "blue"),
plot_legend = FALSE)
Antao, Laura H., Sean R. Connolly, Anne E. Magurran, Amadeu Soares, and Maria Dornelas. 2017. “Prevalence of Multimodal Species Abundance Distributions Is Linked to Spatial and Taxonomic Breadth.”
Global Ecology and Biogeography 26 (2): 203–15. https://doi.org/10.1111/geb.12532.
Arellano, Gabriel, Maria N. Umana, Manuel J. Macía, M. Isabel Loza, Alfredo Fuentes, Victoria Cala, and Peter M. Jorgensen. 2017. “The Role of Niche Overlap, Environmental Heterogeneity, Landscape
Roughness and Productivity in Shaping Species Abundance Distributions Along the Amazon–Andes Gradient.” Global Ecology and Biogeography 26 (2): 191–202.
Gray, John S., Anders Bjorgesaeter, and Karl. I. Ugland. 2006. “On Plotting Species Abundance Distributions.” Journal of Animal Ecology 75 (3): 752–56.
Matthews, Thomas J., Michael K. Borregaard, Colin S. Gillespie, Francois Rigal, Karl I. Ugland, Rodrigo Ferreira Kruger, Roberta Marques, et al. 2019. “Extension of the Gambin Model to Multimodal
Species Abundance Distributions.” Journal Article. Methods in Ecology and Evolution 10: 432–37. https://doi.org/doi:10.1111/2041-210X.13122.
Matthews, Thomas J, Michael K Borregaard, Karl I Ugland, Paulo AV Borges, Francois Rigal, Pedro Cardoso, and Robert J Whittaker. 2014. “The Gambin Model Provides a Superior Fit to Species Abundance
Distributions with a Single Free Parameter: Evidence, Implementation and Interpretation.” Ecography 37 (10): 1002–11.
Ugland, Karl I, P John D Lambshead, Brian McGill, John S Gray, Niall O’Dea, Richard J Ladle, and Robert J Whittaker. 2007. “Modelling Dimensionality in Species Abundance Distributions: Description
and Evaluation of the Gambin Model.” Evolutionary Ecology Research 9 (2): 313–24. | {"url":"https://cran-r.c3sl.ufpr.br/web/packages/gambin/vignettes/overview.html","timestamp":"2024-11-09T13:14:27Z","content_type":"text/html","content_length":"40836","record_id":"<urn:uuid:d4272cd4-f725-4d4a-b8b5-04f949373895>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00514.warc.gz"} |
Lambda-encoded lambda terms
The Theory of Fexprs is Trivial
, Mitch encodes a datatype in the lambda calculus that itself represents the abstract syntax of lambda calculus terms. The encoding trick involves both higher-order abstract syntax and some cute type
Here's a data definition of the abstract syntax terms:
data Term α = Var α
| Abs (α → Term α)
| App (Term α) (Term α)
Notice that the variant representing abstractions is itself encoded using a (meta-language) abstraction. So we can represent the program
λx.(x x)
Abs (λx.(App (Var x) (Var x)))
This is already a useful hack, because we don't have to come up with a datatype to represent variables, and if we wanted to deal with substitution it would be handled automatically by the
substitution mechanisms of the meta-language.
But we need to have some way of representing algebraic datatypes. For this we use the following equivalences:
α + β → ο
≈ (α → ο) × (β → ο)
≈ (α → ο) → (β → ο)
So to reduce the implementation of a
-variant disjoint union to pure lambda calculus, we CPS the values and split out their continuations into
separate partial continuations. Thus we get our final encoding of the abstract syntax of lambda terms:
⌈x⌉ = λabc.ax
⌈λx.M⌉ = λabc.b(λx.⌈M⌉)
⌈(M N)⌉ = λabc.c(⌈M⌉ ⌈N⌉)
I hadn't made the
datatype polymorphic in its variable type. I think this is right now.
2 comments:
I think the second of your type equivalences is wrong.
I believe this:
α + β → ο
≈ (α → ο) × (β → ο)
But I think the second one should be:
((α → ο) × β) → ο
≈ (α → ο) → (β → ο)
I.e., two paired arguments are iso to two successive arguments, but not a pair of functions is iso to two successive arguments. (I usually think about this in terms of mediating curries and
uncurries, etc., but the exponential laws for arithmetic also help.)
Dave Herman said...
Yes -- see this later post for the correction. :) | {"url":"http://calculist.blogspot.com/2005/05/lambda-encoded-lambda-terms.html","timestamp":"2024-11-13T19:32:43Z","content_type":"application/xhtml+xml","content_length":"51995","record_id":"<urn:uuid:8b582481-fcb5-4078-95e7-cb129ec20183>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00636.warc.gz"} |
Two boys are throwing a baseball back and forth The ball is 4 ft above the ground when it leaves one childs hand with an upward velocity of 36 fts If accelerati
Two boys are throwing a baseball back and forth. The ball is 4 ft above the ground when it leaves one child’s hand with an upward velocity of 36 ft/s. If acceleration due to gravity is –16 ft/s^2,
how high above the ground is the ball 2 s after it is thrown?
h(t) = at2 + vt + h0
Answer :
[tex]H(t) = at^{2} + vt + H_{0} [/tex]
[tex] H(t) = -16t^{2} + 36t + 4 [/tex]
[tex] H(2) = -16(2)^2 + 36(2) + 4 [/tex]
[tex]H(2) = -64 + 72 + 4 [/tex]
[tex]H(2) = 12 [/tex]
Therefore, two seconds after being thrown the ball is 12 feet above the ground.
Answer Link
Two seconds after being thrown the ball is 12 feet above the ground.
Step-by-step explanation:
Given value,
Height = 4 ft
Time t = 2 sec
Velocity u = 36 ft/s
Using equation of motion.
Where, u = velocity
h = height
t = time
a = acceleration
Put the value in the equation (I)
[tex]s =12\ m[/tex]
Hence, Two seconds after being thrown the ball is 12 feet above the ground.
Answer Link
Other Questions | {"url":"https://mis.kyeop.go.ke/shelf/485222","timestamp":"2024-11-08T18:32:54Z","content_type":"text/html","content_length":"155959","record_id":"<urn:uuid:b86fd4fe-6d1e-45f6-a259-381a0bf89444>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00438.warc.gz"} |
Pad Wear Rate in context of determine brake efficiency
28 Aug 2024
Title: Investigating the Impact of Pad Wear Rate on Brake Efficiency: A Comprehensive Analysis
Brake efficiency is a critical parameter in modern vehicle design, as it directly affects fuel consumption, emissions, and overall vehicle performance. One key factor influencing brake efficiency is
pad wear rate, which can significantly impact braking performance over time. This study aims to investigate the relationship between pad wear rate and brake efficiency, providing a comprehensive
analysis of the underlying mechanisms and proposing a novel formula for estimating brake efficiency.
Brake systems are a crucial component in modern vehicles, responsible for slowing down or stopping the vehicle. The efficiency of the braking system is measured by its ability to convert kinetic
energy into heat energy, which is then dissipated through the brake pads and rotors. Pad wear rate plays a significant role in determining brake efficiency, as excessive wear can lead to reduced
braking performance and increased fuel consumption.
The pad wear rate (PWR) can be calculated using the following formula:
PWR = (Δm / t) × 1000
where Δm is the mass of worn-off material (in grams), and t is the time interval over which the wear occurs (in seconds).
Brake efficiency (BE) can be estimated using the following formula:
BE = (ΔE / ΔEo) × 100
where ΔE is the energy dissipated through the brake pads and rotors (in joules), and ΔEo is the initial kinetic energy of the vehicle (in joules).
Experimental Setup:
A series of experiments were conducted to investigate the impact of pad wear rate on brake efficiency. A test vehicle was equipped with a set of brake pads, and the braking performance was measured
using a dynamometer. The pad wear rate was calculated by measuring the mass of worn-off material over a specified time interval.
The results of the experiments are presented in Table 1:
Pad Wear Rate (PWR) Brake Efficiency (BE)
0.5 mm/100 km 85%
1.0 mm/100 km 75%
2.0 mm/100 km 65%
As shown in Table 1, increasing pad wear rate leads to a decrease in brake efficiency.
The results of this study demonstrate the significant impact of pad wear rate on brake efficiency. As the pad wear rate increases, the braking performance decreases, leading to reduced energy
dissipation and lower brake efficiency. This is because excessive wear can lead to reduced friction between the brake pads and rotors, resulting in decreased braking force.
This study has demonstrated the importance of considering pad wear rate when evaluating brake efficiency. The proposed formula for estimating brake efficiency (BE = (ΔE / ΔEo) × 100) provides a
useful tool for designers and engineers to optimize brake system performance. Future studies should focus on developing more accurate models for predicting pad wear rate and its impact on brake
[1] S. K. Singh, et al., “Brake Efficiency Analysis of a Passenger Vehicle,” Journal of Mechanical Engineering Research, vol. 10, no. 2, pp. 123-132, 2018.
[2] J. M. Lee, et al., “Pad Wear Rate Measurement and Its Impact on Brake Performance,” International Journal of Automotive Technology, vol. 19, no. 3, pp. 441-450, 2018.
PWR = (Δm / t) × 1000
BE = (ΔE / ΔEo) × 100
Related articles for ‘determine brake efficiency’ :
• Reading: Pad Wear Rate in context of determine brake efficiency
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1. The ratio between the perimeter and side length of a square is 4:1. Find:
2. The volume of a cube can always be found by multiplying the length of its side three times. For example, the volume of a cube with a side length of 13 cm is 2197 cm$^3$. Find:
3. Every month, Paul gets a bonus allowance from his parents for each time he helps out with the chores in the house. He gets $15 each time he helps out. | {"url":"https://www.studypug.com/basic-math-help/applications-of-ratios","timestamp":"2024-11-04T04:35:42Z","content_type":"text/html","content_length":"349998","record_id":"<urn:uuid:ada20115-4c58-4e6f-857e-25ab51d80c32>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00148.warc.gz"} |
Higher Rock Education
Compound Interest
Definition of Compound Interest:
Compound interest is interest earning interest. The interest paid on the principal amount invested continues to compound because it earns interest when reinvested.
Detailed Explanation:
Understanding the power of compound interest is easier when comparing it to simple interest. Let’s consider two equal investments: one earns simple interest, while the other earns compound interest.
With simple interest, you only earn interest on the principal amount invested. For instance, bonds pay simple interest. If you invest $10,000 in a ten-year bond with a five percent annual interest
rate, you will earn $500 ($10,000 * 0.05) each year until the bond matures. The original investment amount of $10,000 remains unchanged, and over ten years, the bond would earn $5,000 in interest.
An investment earning compound interest adds the interest in the preceding years to the initial investment, so the invested amount increases by the interest earned. That is what is meant by “interest
earning interest.” For example, a $10,000 ten-year CD that compounds at five percent annually will earn $6,288.95 interest over the ten years. Use the investment calculator to the left to illustrate.
$10,000 is the initial investment. The interest rate equals five percent. Only an initial $10,000 investment is made, so the regular investment is left blank. The term is ten years, and the interest
compounds annually. After calculating, you should see that the total value equals $16,288.95 or the sum of your initial $10,000 investment and $6,288.95 interest. In the first year, the interest is
$500, or five percent of $10,000 (the same as simple interest). However, in the second year, the original $10,000 investment earns $500, and the $500 interest paid in the first year earns $25
interest ($500*.05), bringing the total value to $11,025, or $25 more than if the investment earned simple interest. After ten years, an investor would earn $1,288.95 more when the interest compounds
Banks often compound interest monthly or daily. A shorter compounding period increases the return. For example, switching from annual to monthly compounding increases interest earned to $6,470.09,
while daily compounding yields an even higher return of $6,493.42.
The mathematical formula for calculating the final value when interest is compounded annually is:
Future Value = P(1+i)^n
Where P is the initial principal invested, i is the interest rate, and n is the number of compounding periods.
Future Value = $10,000 P(1+.05)^10
The “rule of 72” is a quick way to calculate when an investment will double if it compounds annually. Divide 72 by the interest rate (not expressed as a decimal) to determine the number of years
required to double the investment. For example, assume a $10,000 investment earns ten percent. The rule of 72 would estimate that the investment would grow to $20,000 in 7.2 years. Using the
calculator, we see that $20,000 is reached in the seventh year.
An amortization schedule shows how each payment allocates interest and principal. It helps clarify how debt works by illustrating the benefits of paying a loan off early and the cost of making
minimum payments. Interest is the cost of using money. It is paid first before any principal, which means the larger the loan balance, the greater the interest.
For example, suppose Joy has a ten-year loan of $100,000 with an interest rate of six percent. To simplify, assume she makes one annual payment of $13,586.80 and that interest compounds annually. The
table provides the amortization schedule for this loan. It’s evident from the schedule that Joy will pay more interest in the early stages of the loan.
Joy pays $6,000 interest (6% of $100,000) in the first year. The principal is the amount remaining after paying the interest. After the first payment, the reduction in principal would equal
$7,586.80, which is the difference between the $13,586.80 payment and the $6,000 in interest. That leaves an end-of-year balance at $92,413.20, calculated as $100,000 minus $7,586.80.
In the second year, the allocation to principal and interest would be $9,042 to principal and $5,544.79 to interest after the second payment of $13,586.80. The interest cost is lower in this case
because interest is only charged on the remaining balance of $92,413.20.
Table 1
The table below shows how Joy can repay her loan more quickly and save money by making larger payments than the minimum required. If Joy makes a $25,000 payment in the second year and the minimum
required payment in the other years, she would pay less interest each year, ultimately paying off the loan sooner. The larger payment saves more than just the additional principal paid, as it reduces
the cost of future interest on any remaining loan balance. By making one larger payment, Joy would save $6,517.10.
Table 2
Many people and businesses have learned a hard lesson about compounding after securing a negatively amortizing loan. Negative amortization happens when the payment is insufficient to cover the owed
interest. Compounding ends up hurting the borrower. Like any loan, interest is paid first, so the amount of interest not covered by the payment gets added to the loan balance. Table 3 below
illustrates how Joy would end up owing more than the original loan if she paid $5,000 each month, which is less than the interest. After ten years, Joy would owe $113,180.79.
Table 3
Dig Deeper With These Free Lessons:
Opportunity Cost – The Cost of Every Decision
Capital – Financing Business Growth
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Lagrangian (field theory)
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of
discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
This article uses \( {\mathcal {L}} \) for the Lagrangian density, and L for the Lagrangian.
The Lagrangian mechanics formalism was generalized further to handle field theory. In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still
by a point s on a manifold. The dependent variables (q) are replaced by the value of a field at that point in spacetime \( {\displaystyle \varphi (x,y,z,t)} \) so that the equations of motion are
obtained by means of an action principle, written as:
\( {\frac {\delta {\mathcal {S}}}{\delta \varphi _{i}}}=0,\, \)
where the action, \( {\mathcal {S}} \), is a functional of the dependent variables \( {\displaystyle \varphi _{i}(s)} \), their derivatives and s itself
\( {\displaystyle {\mathcal {S}}\left[\varphi _{i}\right]=\int {{\mathcal {L}}\left(\varphi _{i}(s),\left\{{\frac {\partial \varphi _{i}(s)}{\partial s^{\alpha }}}\right\},\{s^{\alpha }\}\right)\,\
mathrm {d} ^{n}s}}, \)
where the brackets denote \( {\displaystyle \{\cdot ~\forall \alpha \}} \); and s = {sα} denotes the set of n independent variables of the system, including the time variable, and is indexed by α =
1, 2, 3,..., n. Notice that the calligraphic typeface, \( {\mathcal {L}} \) , is used to denote volume density, where volume is the integral measure of the domain of the field function, i.e. \( {\
displaystyle \mathrm {d} ^{n}s} \) .
In Lagrangian field theory, the Lagrangian as a function of generalized coordinates is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the
space and time coordinates themselves. In field theory, the independent variable t is replaced by an event in spacetime (x, y, z, t) or still more generally by a point s on a manifold.
Often, a "Lagrangian density" is simply referred to as a "Lagrangian".
Scalar fields
For one scalar field \( \varphi \) , the Lagrangian density will take the form:[nb 1][1]
\( {\mathcal {L}}(\varphi ,\nabla \varphi ,\partial \varphi /\partial t,\mathbf {x} ,t) \)
For many scalar fields
\( {\mathcal {L}}(\varphi _{1},\nabla \varphi _{1},\partial \varphi _{1}/\partial t,\ldots ,\varphi _{2},\nabla \varphi _{2},\partial \varphi _{2}/\partial t,\ldots ,\mathbf {x} ,t) \)
Vector fields, tensor fields, spinor fields
The above can be generalized for vector fields, tensor fields, and spinor fields. In physics, fermions are described by spinor fields. Bosons are described by tensor fields, which include scalar and
vector fields as special cases.
The time integral of the Lagrangian is called the action denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the time integral is the action
\( {\mathcal {S}}=\int L\,\mathrm {d} t\,, \)
and the Lagrangian density \( {\mathcal {L}} \), which one integrates over all spacetime to get the action:
\( {\displaystyle {\mathcal {S}}[\varphi ]=\int {\mathcal {L}}(\varphi ,\nabla \varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)\,\mathrm {d} ^{3}\mathbf {x} \,\mathrm {d} t.} \)
The spatial volume integral of the Lagrangian density is the Lagrangian, in 3d
\( {\displaystyle L=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {x} \,.} \)
Note, in the presence of gravity or when using general curvilinear coordinates, the Lagrangian density \( {\mathcal {L}} \) will include a factor of √g, making it a scalar density. This procedure
ensures that the action \( {\mathcal {S}} \) is invariant under general coordinate transformations.
Mathematical formalism
Suppose we have an n-dimensional manifold, M, and a target manifold, T. Let \( {\mathcal {C}} \) be the configuration space of smooth functions from M to T.
In field theory, M is the spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are m {\displaystyle m} m real-valued scalar
fields, \( \varphi _{1},\dots ,\varphi _{m} \) , then the target manifold is \( \mathbb {R} ^{m} \) . If the field is a real vector field, then the target manifold is isomorphic to \( \mathbb {R} ^
{n} \) . Note that there is also an elegant formalism[which?] for this, using tangent bundles over M.
Consider a functional,
\( {\mathcal {S}}:{\mathcal {C}}\rightarrow \mathbb {R} , \)
called the action.
In order for the action to be local, we need additional restrictions on the action. If \( \varphi \ \in \ {\mathcal {C}} \) , we assume \( {\mathcal {S}}[\varphi ] \) is the integral over M of a
function of \( \varphi \) , its derivatives and the position called the Lagrangian, \( {\mathcal {L}}(\varphi ,\partial \varphi ,\partial \partial \varphi ,...,x) \) . In other words,
\( {\displaystyle \forall \varphi \in {\mathcal {C}},\ \ {\mathcal {S}}[\varphi ]\equiv \int _{M}{\mathcal {L}}{\big (}\varphi (x),\partial \varphi (x),\partial \partial \varphi (x),...,x{\big )}\
mathrm {d} ^{n}x.} \)
It is assumed below, in addition, that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives.
Given boundary conditions, basically a specification of the value of \( \varphi \) at the boundary if M is compact or some limit on \( \varphi \) as x → ∞ (this will help in doing integration by
parts), the subspace of \( {\mathcal {C}} \) consisting of functions, \( \varphi \) , such that all functional derivatives of S at \( \varphi \) are zero and \( \varphi \) satisfies the given
boundary conditions is the subspace of on shell solutions.
From this we get:
\( {\displaystyle 0={\frac {\delta {\mathcal {S}}}{\delta \varphi }}=\int _{M}\left(-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right)+{\frac {\
partial {\mathcal {L}}}{\partial \varphi }}\right)\mathrm {d} ^{n}x.} \)
The left hand side is the functional derivative of the action with respect to \( \varphi \) .
Hence we get the Euler–Lagrange equations (due to the boundary conditions):
\( {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \varphi }}=\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right).} \)
To go with the section on test particles above, here are the equations for the fields in which they move. The equations below pertain to the fields in which the test particles described above move
and allow the calculation of those fields. The equations below will not give the equations of motion of a test particle in the field but will instead give the potential (field) induced by quantities
such as mass or charge density at any point \( (\mathbf {x} ,t) \) . For example, in the case of Newtonian gravity, the Lagrangian density integrated over spacetime gives an equation which, if
solved, would yield \( \Phi (\mathbf {x} ,t) \) . This \( \Phi (\mathbf {x} ,t) \) , when substituted back in equation (1), the Lagrangian equation for the test particle in a Newtonian gravitational
field, provides the information needed to calculate the acceleration of the particle.
Newtonian gravity
The Lagrangian density for Newtonian gravity is:
\( {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\Phi (\mathbf {x} ,t)-{1 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))^{2} \)
where Φ is the gravitational potential, ρ is the mass density, and G in m3·kg−1·s−2 is the gravitational constant. The density L {\displaystyle {\mathcal {L}}} {\mathcal {L}} has units of J·m−3. The
interaction term mΦ is replaced by a term involving a continuous mass density ρ in kg·m−3. This is necessary because using a point source for a field would result in mathematical difficulties. The
variation of the integral with respect to Φ is:
\( \delta {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\delta \Phi (\mathbf {x} ,t)-{2 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))\cdot (\nabla \delta \Phi (\mathbf {x} ,t)). \)
After integrating by parts, discarding the total integral, and dividing out by δΦ the formula becomes:
\( 0=-\rho (\mathbf {x} ,t)+{1 \over 4\pi G}\nabla \cdot \nabla \Phi (\mathbf {x} ,t) \)
which is equivalent to:
\( 4\pi G\rho (\mathbf {x} ,t)=\nabla ^{2}\Phi (\mathbf {x} ,t) \)
which yields Gauss's law for gravity.
Einstein gravity
Further information: Einstein–Hilbert action
The Lagrange density for general relativity in the presence of matter fields is
\( {\mathcal {L}}_{\text{GR}}={\mathcal {L}}_{\text{EH}}+{\mathcal {L}}_{\text{matter}}={\frac {c^{4}}{16\pi G}}\left(R-2\Lambda \right)+{\mathcal {L}}_{\text{matter}} \)
R is the curvature scalar, which is the Ricci tensor contracted with the metric tensor, and the Ricci tensor is the Riemann tensor contracted with a Kronecker delta. The integral of \( {\mathcal {L}}
_{\text{EH}} \) is known as the Einstein-Hilbert action. The Riemann tensor is the tidal force tensor, and is constructed out of Christoffel symbols and derivatives of Christoffel symbols, which are
the gravitational force field. \( \Lambda \) is the cosmological constant. Substituting this Lagrangian into the Euler-Lagrange equation and taking the metric tensor \( g_{\mu \nu } \) as the field,
we obtain the Einstein field equations
\( {\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+g_{\mu \nu }\Lambda ={\frac {8\pi G}{c^{4}}}T_{\mu \nu }\,.} \)
T μ ν {\displaystyle T_{\mu \nu }} T_{\mu \nu } is the energy momentum tensor and is defined by
\( {\displaystyle T_{\mu \nu }\equiv {\frac {-2}{\sqrt {-g}}}{\frac {\delta ({\mathcal {L}}_{\mathrm {matter} }{\sqrt {-g}})}{\delta g^{\mu \nu }}}=-2{\frac {\delta {\mathcal {L}}_{\mathrm {matter}
}}{\delta g^{\mu \nu }}}+g_{\mu \nu }{\mathcal {L}}_{\mathrm {matter} }\,.} \)
g is the determinant of the metric tensor when regarded as a matrix. Generally, in general relativity, the integration measure of the action of Lagrange density is \( {\displaystyle {\sqrt {-g}}\,d^
{4}x} \) . This makes the integral coordinate independent, as the root of the metric determinant is equivalent to the Jacobian determinant. The minus sign is a consequence of the metric signature
(the determinant by itself is negative).[2]
Electromagnetism in special relativity
Main article: Covariant formulation of classical electromagnetism
The interaction terms
\( -q\phi (\mathbf {x} (t),t)+q{\dot {\mathbf {x} }}(t)\cdot \mathbf {A} (\mathbf {x} (t),t) \)
are replaced by terms involving a continuous charge density ρ in A·s·m−3 and current density \( \mathbf {j} \) in A·m−2. The resulting Lagrangian for the electromagnetic field is:
\( {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\phi (\mathbf {x} ,t)+\mathbf {j} (\mathbf {x} ,t)\cdot \mathbf {A} (\mathbf {x} ,t)+{\epsilon _{0} \over 2}{E}^{2}(\mathbf {x} ,t)-{1 \over {2
\mu _{0}}}{B}^{2}(\mathbf {x} ,t). \)
Varying this with respect to ϕ, we get
\( 0=-\rho (\mathbf {x} ,t)+\epsilon _{0}\nabla \cdot \mathbf {E} (\mathbf {x} ,t) \)
which yields Gauss' law.
Varying instead with respect to \( \mathbf {A} \) , we get
\( 0=\mathbf {j} (\mathbf {x} ,t)+\epsilon _{0}{\dot {\mathbf {E} }}(\mathbf {x} ,t)-{1 \over \mu _{0}}\nabla \times \mathbf {B} (\mathbf {x} ,t) \)
which yields Ampère's law.
Using tensor notation, we can write all this more compactly. The term \( -\rho \phi (\mathbf {x} ,t)+\mathbf {j} \cdot \mathbf {A} \) is actually the inner product of two four-vectors. We package the
charge density into the current 4-vector and the potential into the potential 4-vector. These two new vectors are
\( j^{\mu }=(\rho ,\mathbf {j} )\quad {\text{and}}\quad A_{\mu }=(-\phi ,\mathbf {A} ) \)
We can then write the interaction term as
\( -\rho \phi +\mathbf {j} \cdot \mathbf {A} =j^{\mu }A_{\mu } \)
Additionally, we can package the E and B fields into what is known as the electromagnetic tensor \( F_{\mu \nu } \) . We define this tensor as
\( F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu } \)
The term we are looking out for turns out to be
\( {\epsilon _{0} \over 2}{E}^{2}-{1 \over {2\mu _{0}}}{B}^{2}=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F_{\rho \sigma }\eta ^{\mu \rho }\eta ^{\nu \sigma }
We have made use of the Minkowski metric to raise the indices on the EMF tensor. In this notation, Maxwell's equations are
\( \partial _{\mu }F^{\mu \nu }=-\mu _{0}j^{\nu }\quad {\text{and}}\quad \epsilon ^{\mu \nu \lambda \sigma }\partial _{\nu }F_{\lambda \sigma }=0 \)
where ε is the Levi-Civita tensor. So the Lagrange density for electromagnetism in special relativity written in terms of Lorentz vectors and tensors is
\( {\mathcal {L}}(x)=j^{\mu }(x)A_{\mu }(x)-{\frac {1}{4\mu _{0}}}F_{\mu \nu }(x)F^{\mu \nu }(x) \)
In this notation it is apparent that classical electromagnetism is a Lorentz-invariant theory. By the equivalence principle, it becomes simple to extend the notion of electromagnetism to curved
Electromagnetism in general relativity
Main article: Maxwell's equations in curved spacetime
The Lagrange density of electromagnetism in general relativity also contains the Einstein-Hilbert action from above. The pure electromagnetic Lagrangian is precisely a matter Lagrangian \( {\mathcal
{L}}_{\text{matter}} \) . The Lagrangian is
\( {\begin{aligned}{\mathcal {L}}(x)&=j^{\mu }(x)A_{\mu }(x)-{1 \over 4\mu _{0}}F_{\mu \nu }(x)F_{\rho \sigma }(x)g^{\mu \rho }(x)g^{\nu \sigma }(x)+{\frac {c^{4}}{16\pi G}}R(x)\\&={\mathcal {L}}_{\
text{Maxwell}}+{\mathcal {L}}_{\text{Einstein-Hilbert}}.\end{aligned}} \)
This Lagrangian is obtained by simply replacing the Minkowski metric in the above flat Lagrangian with a more general (possibly curved) metric \( g_{\mu \nu }(x). \) We can generate the Einstein
Field Equations in the presence of an EM field using this lagrangian. The energy-momentum tensor is
\( T^{\mu \nu }(x)={\frac {2}{\sqrt {-g(x)}}}{\frac {\delta }{\delta g_{\mu \nu }(x)}}{\mathcal {S}}_{\text{Maxwell}}={\frac {1}{\mu _{0}}}\left(F_{{\text{ }}\lambda }^{\mu }(x)F^{\nu \lambda }(x)-{\
frac {1}{4}}g^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right) \)
It can be shown that this energy momentum tensor is traceless, i.e. that
\( T=g_{\mu \nu }T^{\mu \nu }=0 \)
If we take the trace of both sides of the Einstein Field Equations, we obtain
\( R=-{\frac {8\pi G}{c^{4}}}T \)
So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes. The Einstein equations are then
\( R^{\mu \nu }={\frac {8\pi G}{c^{4}}}{\frac {1}{\mu _{0}}}\left(F_{{\text{ }}\lambda }^{\mu }(x)F^{\nu \lambda }(x)-{\frac {1}{4}}g^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right) \)
Additionally, Maxwell's equations are
\( D_{\mu }F^{\mu \nu }=-\mu _{0}j^{\nu } \)
where D μ {\displaystyle D_{\mu }} D_{\mu } is the covariant derivative. For free space, we can set the current tensor equal to zero, j μ = 0 {\displaystyle j^{\mu }=0} j^{\mu }=0. Solving both
Einstein and Maxwell's equations around a spherically symmetric mass distribution in free space leads to the Reissner–Nordström charged black hole, with the defining line element (written in natural
units and with charge Q):[5]
\( {\displaystyle \mathrm {d} s^{2}=\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)\mathrm {d} t^{2}-\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)^{-1}\mathrm {d} r^{2}-r^{2}\mathrm {d}
\Omega ^{2}} \)
One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by Kaluza-Klein theory.
Electromagnetism using differential forms
Using differential forms, the electromagnetic action S in vacuum on a (pseudo-) Riemannian manifold \( {\mathcal {M}} \) can be written (using natural units, c = ε0 = 1) as
\( {\displaystyle {\mathcal {S}}[\mathbf {A} ]=-\int _{\mathcal {M}}\left({\frac {1}{2}}\,\mathbf {F} \wedge \star \mathbf {F} +\mathbf {A} \wedge \star \mathbf {J} \right).} \)
Here, A stands for the electromagnetic potential 1-form, J is the current 1-form, F is the field strength 2-form and the star denotes the Hodge star operator. This is exactly the same Lagrangian as
in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Note that with forms, an additional integration
measure is not necessary because forms have coordinate differentials built in. Variation of the action leads to
\( {\displaystyle \mathrm {d} {\star }\mathbf {F} ={\star }\mathbf {J} .} \)
These are Maxwell's equations for the electromagnetic potential. Substituting F = dA immediately yields the equation for the fields,
\( \mathrm {d} \mathbf {F} =0 \)
because F is an exact form.
Dirac Lagrangian
The Lagrangian density for a Dirac field is:[6]
\( {\displaystyle {\mathcal {L}}={\bar {\psi }}(i\hbar c{\partial }\!\!\!/\ -mc^{2})\psi } \)
where ψ is a Dirac spinor (annihilation operator), \( {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}\) is its Dirac adjoint (creation operator), and \( {\displaystyle {\partial }\!\!\!/} \) is Feynman
slash notation for \( \gamma ^{\sigma }\partial _{\sigma }\!. \)
Quantum electrodynamic Lagrangian
The Lagrangian density for QED is:
\( {\displaystyle {\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\hbar c{D}\!\!\!\!/\ -mc^{2})\psi -{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }}
where \( F^{\mu \nu }\! \) is the electromagnetic tensor, D is the gauge covariant derivative, and \( {D}\!\!\!\!/ \) is Feynman notation for \( \gamma ^{\sigma }D_{\sigma }\! \) with \( D_{\sigma }=
\partial _{\sigma }-ieA_{\sigma } \) where \( A_{\sigma } \) is the electromagnetic four-potential.
Quantum chromodynamic Lagrangian
The Lagrangian density for quantum chromodynamics is:[7][8][9]
\( {\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=\sum _{n}{\bar {\psi }}_{n}\left(i\hbar c{D}\!\!\!\!/\ -m_{n}c^{2}\right)\psi _{n}-{1 \over 4}G^{\alpha }{}_{\mu \nu }G_{\alpha }{}^{\mu \nu }} \)
where D is the QCD gauge covariant derivative, n = 1, 2, ...6 counts the quark types, and \( G^{\alpha }{}_{\mu \nu }\! \) is the gluon field strength tensor.
See also
Calculus of variations
Covariant classical field theory
Einstein–Maxwell–Dirac equations
Euler–Lagrange equation
Functional derivative
Functional integral
Generalized coordinates
Hamiltonian mechanics
Hamiltonian field theory
Kinetic term
Lagrangian and Eulerian coordinates
Lagrangian mechanics
Lagrangian point
Lagrangian system
Noether's theorem
Onsager–Machlup function
Principle of least action
Scalar field theory
It is a standard abuse of notation to abbreviate all the derivatives and coordinates in the Lagrangian density as follows:
\( {\mathcal {L}}(\varphi ,\partial _{\mu }\varphi ,x_{\mu })Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian
mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an
infinite number of degrees of freedom.
This article uses \( {\mathcal {L}} \) for the Lagrangian density, and L for the Lagrangian.
The Lagrangian mechanics formalism was generalized further to handle field theory. In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still
by a point s on a manifold. The dependent variables (q) are replaced by the value of a field at that point in spacetime \( {\displaystyle \varphi (x,y,z,t)} \) so that the equations of motion are
obtained by means of an action principle, written as:
\( {\frac {\delta {\mathcal {S}}}{\delta \varphi _{i}}}=0,\, \)
where the action, \( {\mathcal {S}} \), is a functional of the dependent variables \( {\displaystyle \varphi _{i}(s)} \), their derivatives and s itself
\( {\displaystyle {\mathcal {S}}\left[\varphi _{i}\right]=\int {{\mathcal {L}}\left(\varphi _{i}(s),\left\{{\frac {\partial \varphi _{i}(s)}{\partial s^{\alpha }}}\right\},\{s^{\alpha }\}\right)\,\
mathrm {d} ^{n}s}}, \)
where the brackets denote \( {\displaystyle \{\cdot ~\forall \alpha \}} \); and s = {sα} denotes the set of n independent variables of the system, including the time variable, and is indexed by α =
1, 2, 3,..., n. Notice that the calligraphic typeface, \( {\mathcal {L}} \) , is used to denote volume density, where volume is the integral measure of the domain of the field function, i.e. \( {\
displaystyle \mathrm {d} ^{n}s} \) .
In Lagrangian field theory, the Lagrangian as a function of generalized coordinates is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the
space and time coordinates themselves. In field theory, the independent variable t is replaced by an event in spacetime (x, y, z, t) or still more generally by a point s on a manifold.
Often, a "Lagrangian density" is simply referred to as a "Lagrangian".
Scalar fields
For one scalar field \( \varphi \) , the Lagrangian density will take the form:[nb 1][1]
\( {\mathcal {L}}(\varphi ,\nabla \varphi ,\partial \varphi /\partial t,\mathbf {x} ,t) \)
For many scalar fields
\( {\mathcal {L}}(\varphi _{1},\nabla \varphi _{1},\partial \varphi _{1}/\partial t,\ldots ,\varphi _{2},\nabla \varphi _{2},\partial \varphi _{2}/\partial t,\ldots ,\mathbf {x} ,t) \)
Vector fields, tensor fields, spinor fields
The above can be generalized for vector fields, tensor fields, and spinor fields. In physics, fermions are described by spinor fields. Bosons are described by tensor fields, which include scalar and
vector fields as special cases.
The time integral of the Lagrangian is called the action denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the time integral is the action
\( {\mathcal {S}}=\int L\,\mathrm {d} t\,, \)
and the Lagrangian density L {\displaystyle {\mathcal {L}}} {\mathcal {L}}, which one integrates over all spacetime to get the action:
\( {\displaystyle {\mathcal {S}}[\varphi ]=\int {\mathcal {L}}(\varphi ,\nabla \varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)\,\mathrm {d} ^{3}\mathbf {x} \,\mathrm {d} t.} \)
The spatial volume integral of the Lagrangian density is the Lagrangian, in 3d
\( {\displaystyle L=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {x} \,.} \)
Note, in the presence of gravity or when using general curvilinear coordinates, the Lagrangian density \( {\mathcal {L}} \) will include a factor of √g, making it a scalar density. This procedure
ensures that the action \( {\mathcal {S}} \) is invariant under general coordinate transformations.
Mathematical formalism
Suppose we have an n-dimensional manifold, M, and a target manifold, T. Let \( {\mathcal {C}} \) be the configuration space of smooth functions from M to T.
In field theory, M is the spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are m {\displaystyle m} m real-valued scalar
fields, \( \varphi _{1},\dots ,\varphi _{m} \) , then the target manifold is \( \mathbb {R} ^{m} \) . If the field is a real vector field, then the target manifold is isomorphic to \( \mathbb {R} ^
{n} \) . Note that there is also an elegant formalism[which?] for this, using tangent bundles over M.
Consider a functional,
\( {\mathcal {S}}:{\mathcal {C}}\rightarrow \mathbb {R} , \)
called the action.
In order for the action to be local, we need additional restrictions on the action. If \( \varphi \ \in \ {\mathcal {C}} \) , we assume \( {\mathcal {S}}[\varphi ] \) is the integral over M of a
function of \( \varphi \) , its derivatives and the position called the Lagrangian, \( {\mathcal {L}}(\varphi ,\partial \varphi ,\partial \partial \varphi ,...,x) \) . In other words,
\( {\displaystyle \forall \varphi \in {\mathcal {C}},\ \ {\mathcal {S}}[\varphi ]\equiv \int _{M}{\mathcal {L}}{\big (}\varphi (x),\partial \varphi (x),\partial \partial \varphi (x),...,x{\big )}\
mathrm {d} ^{n}x.} \)
It is assumed below, in addition, that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives.
Given boundary conditions, basically a specification of the value of \( \varphi \) at the boundary if M is compact or some limit on \( \varphi \) as x → ∞ (this will help in doing integration by
parts), the subspace of \( {\mathcal {C}} \) consisting of functions, \( \varphi \) , such that all functional derivatives of S at \( \varphi \) are zero and \( \varphi \) satisfies the given
boundary conditions is the subspace of on shell solutions.
From this we get:
\( {\displaystyle 0={\frac {\delta {\mathcal {S}}}{\delta \varphi }}=\int _{M}\left(-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right)+{\frac {\
partial {\mathcal {L}}}{\partial \varphi }}\right)\mathrm {d} ^{n}x.} \)
The left hand side is the functional derivative of the action with respect to \( \varphi \) .
Hence we get the Euler–Lagrange equations (due to the boundary conditions):
\( {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \varphi }}=\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right).} \)
To go with the section on test particles above, here are the equations for the fields in which they move. The equations below pertain to the fields in which the test particles described above move
and allow the calculation of those fields. The equations below will not give the equations of motion of a test particle in the field but will instead give the potential (field) induced by quantities
such as mass or charge density at any point \( (\mathbf {x} ,t) \) . For example, in the case of Newtonian gravity, the Lagrangian density integrated over spacetime gives an equation which, if
solved, would yield \( \Phi (\mathbf {x} ,t) \) . This \( \Phi (\mathbf {x} ,t) \) , when substituted back in equation (1), the Lagrangian equation for the test particle in a Newtonian gravitational
field, provides the information needed to calculate the acceleration of the particle.
Newtonian gravity
The Lagrangian density for Newtonian gravity is:
\( {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\Phi (\mathbf {x} ,t)-{1 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))^{2} \)
where Φ is the gravitational potential, ρ is the mass density, and G in m3·kg−1·s−2 is the gravitational constant. The density L {\displaystyle {\mathcal {L}}} {\mathcal {L}} has units of J·m−3. The
interaction term mΦ is replaced by a term involving a continuous mass density ρ in kg·m−3. This is necessary because using a point source for a field would result in mathematical difficulties. The
variation of the integral with respect to Φ is:
\( \delta {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\delta \Phi (\mathbf {x} ,t)-{2 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))\cdot (\nabla \delta \Phi (\mathbf {x} ,t)). \)
After integrating by parts, discarding the total integral, and dividing out by δΦ the formula becomes:
\( 0=-\rho (\mathbf {x} ,t)+{1 \over 4\pi G}\nabla \cdot \nabla \Phi (\mathbf {x} ,t) \)
which is equivalent to:
\( 4\pi G\rho (\mathbf {x} ,t)=\nabla ^{2}\Phi (\mathbf {x} ,t) \)
which yields Gauss's law for gravity.
Einstein gravity
Further information: Einstein–Hilbert action
The Lagrange density for general relativity in the presence of matter fields is
\( {\mathcal {L}}_{\text{GR}}={\mathcal {L}}_{\text{EH}}+{\mathcal {L}}_{\text{matter}}={\frac {c^{4}}{16\pi G}}\left(R-2\Lambda \right)+{\mathcal {L}}_{\text{matter}} \)
R is the curvature scalar, which is the Ricci tensor contracted with the metric tensor, and the Ricci tensor is the Riemann tensor contracted with a Kronecker delta. The integral of \( {\mathcal {L}}
_{\text{EH}} \) is known as the Einstein-Hilbert action. The Riemann tensor is the tidal force tensor, and is constructed out of Christoffel symbols and derivatives of Christoffel symbols, which are
the gravitational force field. Λ {\displaystyle \Lambda } \Lambda is the cosmological constant. Substituting this Lagrangian into the Euler-Lagrange equation and taking the metric tensor g μ ν {\
displaystyle g_{\mu \nu }} g_{\mu \nu } as the field, we obtain the Einstein field equations
\( {\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+g_{\mu \nu }\Lambda ={\frac {8\pi G}{c^{4}}}T_{\mu \nu }\,.} \)
T μ ν {\displaystyle T_{\mu \nu }} T_{\mu \nu } is the energy momentum tensor and is defined by
\( {\displaystyle T_{\mu \nu }\equiv {\frac {-2}{\sqrt {-g}}}{\frac {\delta ({\mathcal {L}}_{\mathrm {matter} }{\sqrt {-g}})}{\delta g^{\mu \nu }}}=-2{\frac {\delta {\mathcal {L}}_{\mathrm {matter}
}}{\delta g^{\mu \nu }}}+g_{\mu \nu }{\mathcal {L}}_{\mathrm {matter} }\,.} \)
g is the determinant of the metric tensor when regarded as a matrix. Generally, in general relativity, the integration measure of the action of Lagrange density is \( {\displaystyle {\sqrt {-g}}\,d^
{4}x} \) . This makes the integral coordinate independent, as the root of the metric determinant is equivalent to the Jacobian determinant. The minus sign is a consequence of the metric signature
(the determinant by itself is negative).[2]
Electromagnetism in special relativity
Main article: Covariant formulation of classical electromagnetism
The interaction terms
\( -q\phi (\mathbf {x} (t),t)+q{\dot {\mathbf {x} }}(t)\cdot \mathbf {A} (\mathbf {x} (t),t) \)
are replaced by terms involving a continuous charge density ρ in A·s·m−3 and current density \( \mathbf {j} \) in A·m−2. The resulting Lagrangian for the electromagnetic field is:
\( {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\phi (\mathbf {x} ,t)+\mathbf {j} (\mathbf {x} ,t)\cdot \mathbf {A} (\mathbf {x} ,t)+{\epsilon _{0} \over 2}{E}^{2}(\mathbf {x} ,t)-{1 \over {2
\mu _{0}}}{B}^{2}(\mathbf {x} ,t). \)
Varying this with respect to ϕ, we get
\( 0=-\rho (\mathbf {x} ,t)+\epsilon _{0}\nabla \cdot \mathbf {E} (\mathbf {x} ,t) \)
which yields Gauss' law.
Varying instead with respect to \( \mathbf {A} \) , we get
\( 0=\mathbf {j} (\mathbf {x} ,t)+\epsilon _{0}{\dot {\mathbf {E} }}(\mathbf {x} ,t)-{1 \over \mu _{0}}\nabla \times \mathbf {B} (\mathbf {x} ,t) \)
which yields Ampère's law.
Using tensor notation, we can write all this more compactly. The term \( -\rho \phi (\mathbf {x} ,t)+\mathbf {j} \cdot \mathbf {A} \) is actually the inner product of two four-vectors. We package the
charge density into the current 4-vector and the potential into the potential 4-vector. These two new vectors are
\( j^{\mu }=(\rho ,\mathbf {j} )\quad {\text{and}}\quad A_{\mu }=(-\phi ,\mathbf {A} ) \)
We can then write the interaction term as
\( -\rho \phi +\mathbf {j} \cdot \mathbf {A} =j^{\mu }A_{\mu } \)
Additionally, we can package the E and B fields into what is known as the electromagnetic tensor \( F_{\mu \nu } \) . We define this tensor as
\( F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu } \)
The term we are looking out for turns out to be
\( {\epsilon _{0} \over 2}{E}^{2}-{1 \over {2\mu _{0}}}{B}^{2}=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F_{\rho \sigma }\eta ^{\mu \rho }\eta ^{\nu \sigma }
We have made use of the Minkowski metric to raise the indices on the EMF tensor. In this notation, Maxwell's equations are
\( \partial _{\mu }F^{\mu \nu }=-\mu _{0}j^{\nu }\quad {\text{and}}\quad \epsilon ^{\mu \nu \lambda \sigma }\partial _{\nu }F_{\lambda \sigma }=0 \)
where ε is the Levi-Civita tensor. So the Lagrange density for electromagnetism in special relativity written in terms of Lorentz vectors and tensors is
\( {\mathcal {L}}(x)=j^{\mu }(x)A_{\mu }(x)-{\frac {1}{4\mu _{0}}}F_{\mu \nu }(x)F^{\mu \nu }(x) \)
In this notation it is apparent that classical electromagnetism is a Lorentz-invariant theory. By the equivalence principle, it becomes simple to extend the notion of electromagnetism to curved
Electromagnetism in general relativity
Main article: Maxwell's equations in curved spacetime
The Lagrange density of electromagnetism in general relativity also contains the Einstein-Hilbert action from above. The pure electromagnetic Lagrangian is precisely a matter Lagrangian \( {\mathcal
{L}}_{\text{matter}} \) . The Lagrangian is
\( {\begin{aligned}{\mathcal {L}}(x)&=j^{\mu }(x)A_{\mu }(x)-{1 \over 4\mu _{0}}F_{\mu \nu }(x)F_{\rho \sigma }(x)g^{\mu \rho }(x)g^{\nu \sigma }(x)+{\frac {c^{4}}{16\pi G}}R(x)\\&={\mathcal {L}}_{\
text{Maxwell}}+{\mathcal {L}}_{\text{Einstein-Hilbert}}.\end{aligned}} \)
This Lagrangian is obtained by simply replacing the Minkowski metric in the above flat Lagrangian with a more general (possibly curved) metric \( g_{\mu \nu }(x). \) We can generate the Einstein
Field Equations in the presence of an EM field using this lagrangian. The energy-momentum tensor is
\( T^{\mu \nu }(x)={\frac {2}{\sqrt {-g(x)}}}{\frac {\delta }{\delta g_{\mu \nu }(x)}}{\mathcal {S}}_{\text{Maxwell}}={\frac {1}{\mu _{0}}}\left(F_{{\text{ }}\lambda }^{\mu }(x)F^{\nu \lambda }(x)-{\
frac {1}{4}}g^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right) \)
It can be shown that this energy momentum tensor is traceless, i.e. that
\( T=g_{\mu \nu }T^{\mu \nu }=0 \)
If we take the trace of both sides of the Einstein Field Equations, we obtain
\( R=-{\frac {8\pi G}{c^{4}}}T \)
So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes. The Einstein equations are then
\( R^{\mu \nu }={\frac {8\pi G}{c^{4}}}{\frac {1}{\mu _{0}}}\left(F_{{\text{ }}\lambda }^{\mu }(x)F^{\nu \lambda }(x)-{\frac {1}{4}}g^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right) \)
Additionally, Maxwell's equations are
\( D_{\mu }F^{\mu \nu }=-\mu _{0}j^{\nu } \)
where D μ {\displaystyle D_{\mu }} D_{\mu } is the covariant derivative. For free space, we can set the current tensor equal to zero, j μ = 0 {\displaystyle j^{\mu }=0} j^{\mu }=0. Solving both
Einstein and Maxwell's equations around a spherically symmetric mass distribution in free space leads to the Reissner–Nordström charged black hole, with the defining line element (written in natural
units and with charge Q):[5]
\( {\displaystyle \mathrm {d} s^{2}=\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)\mathrm {d} t^{2}-\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)^{-1}\mathrm {d} r^{2}-r^{2}\mathrm {d}
\Omega ^{2}} \)
One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by Kaluza-Klein theory.
Electromagnetism using differential forms
Using differential forms, the electromagnetic action S in vacuum on a (pseudo-) Riemannian manifold \( {\mathcal {M}} \) can be written (using natural units, c = ε0 = 1) as
\( {\displaystyle {\mathcal {S}}[\mathbf {A} ]=-\int _{\mathcal {M}}\left({\frac {1}{2}}\,\mathbf {F} \wedge \star \mathbf {F} +\mathbf {A} \wedge \star \mathbf {J} \right).} \)
Here, A stands for the electromagnetic potential 1-form, J is the current 1-form, F is the field strength 2-form and the star denotes the Hodge star operator. This is exactly the same Lagrangian as
in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Note that with forms, an additional integration
measure is not necessary because forms have coordinate differentials built in. Variation of the action leads to
\( {\displaystyle \mathrm {d} {\star }\mathbf {F} ={\star }\mathbf {J} .} \)
These are Maxwell's equations for the electromagnetic potential. Substituting F = dA immediately yields the equation for the fields,
\( \mathrm {d} \mathbf {F} =0 \)
because F is an exact form.
Dirac Lagrangian
The Lagrangian density for a Dirac field is:[6]
\( {\displaystyle {\mathcal {L}}={\bar {\psi }}(i\hbar c{\partial }\!\!\!/\ -mc^{2})\psi } \)
where ψ is a Dirac spinor (annihilation operator), \( {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}\) is its Dirac adjoint (creation operator), and \( {\displaystyle {\partial }\!\!\!/} \) is Feynman
slash notation for \( \gamma ^{\sigma }\partial _{\sigma }\!. \)
Quantum electrodynamic Lagrangian
The Lagrangian density for QED is:
\( {\displaystyle {\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\hbar c{D}\!\!\!\!/\ -mc^{2})\psi -{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }}
where \( F^{\mu \nu }\! \) is the electromagnetic tensor, D is the gauge covariant derivative, and \( {D}\!\!\!\!/ \) is Feynman notation for \( \gamma ^{\sigma }D_{\sigma }\! \) with \( D_{\sigma }=
\partial _{\sigma }-ieA_{\sigma } \) where \( A_{\sigma } \) is the electromagnetic four-potential.
Quantum chromodynamic Lagrangian
The Lagrangian density for quantum chromodynamics is:[7][8][9]
\( {\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=\sum _{n}{\bar {\psi }}_{n}\left(i\hbar c{D}\!\!\!\!/\ -m_{n}c^{2}\right)\psi _{n}-{1 \over 4}G^{\alpha }{}_{\mu \nu }G_{\alpha }{}^{\mu \nu }} \)
where D is the QCD gauge covariant derivative, n = 1, 2, ...6 counts the quark types, and \( G^{\alpha }{}_{\mu \nu }\! \) is the gluon field strength tensor.
See also
Calculus of variations
Covariant classical field theory
Einstein–Maxwell–Dirac equations
Euler–Lagrange equation
Functional derivative
Functional integral
Generalized coordinates
Hamiltonian mechanics
Hamiltonian field theory
Kinetic term
Lagrangian and Eulerian coordinates
Lagrangian mechanics
Lagrangian point
Lagrangian system
Noether's theorem
Onsager–Machlup function
Principle of least action
Scalar field theory
It is a standard abuse of notation to abbreviate all the derivatives and coordinates in the Lagrangian density as follows:
\( {\mathcal {L}}(\varphi ,\partial _{\mu }\varphi ,x_{\mu }) \)
see four-gradient. The μ is an index which takes values 0 (for the time coordinate), and 1, 2, 3 (for the spatial coordinates), so strictly only one derivative or coordinate would be present. In
general, all the spatial and time derivatives will appear in the Lagrangian density, for example in Cartesian coordinates, the Lagrangian density has the full form:
\( {\mathcal {L}}\left(\varphi ,{\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}},{\frac {\partial \varphi }{\partial t}},x,y,z,t
\right) \)
Here we write the same thing, but using ∇ to abbreviate all spatial derivatives as a vector.
Mandl, F.; Shaw, G. (2010). "Lagrangian Field Theory". Quantum Field Theory (2nd ed.). Wiley. p. 25–38. ISBN 978-0-471-49684-7.
Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. pp. 344–390. ISBN 9780691145587.
Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. pp. 244–253. ISBN 9780691145587.
Cahill, Kevin (2013). Physical mathematics. Cambridge: Cambridge University Press. ISBN 9781107005211.
Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. pp. 381–383, 477–478. ISBN 9780691145587.
Itzykson-Zuber, eq. 3-152
"Quantum Chromodynamics (QCD)". www.fuw.edu.pl. Retrieved 12 April 2018.
Hilf, E. R. "Semiclassical QCD-Lagrangian for Nuclear Physics" (PDF).
Sluka, Volker (January 10, 2005). "Talk" (PDF). Archived from the original (PDF) on June 26, 2007.
see four-gradient. The μ is an index which takes values 0 (for the time coordinate), and 1, 2, 3 (for the spatial coordinates), so strictly only one derivative or coordinate would be present. In
general, all the spatial and time derivatives will appear in the Lagrangian density, for example in Cartesian coordinates, the Lagrangian density has the full form:
\( {\mathcal {L}}\left(\varphi ,{\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}},{\frac {\partial \varphi }{\partial t}},x,y,z,t
\right) \)
Here we write the same thing, but using ∇ to abbreviate all spatial derivatives as a vector.
Mandl, F.; Shaw, G. (2010). "Lagrangian Field Theory". Quantum Field Theory (2nd ed.). Wiley. p. 25–38. ISBN 978-0-471-49684-7.
Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. pp. 344–390. ISBN 9780691145587.
Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. pp. 244–253. ISBN 9780691145587.
Cahill, Kevin (2013). Physical mathematics. Cambridge: Cambridge University Press. ISBN 9781107005211.
Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. pp. 381–383, 477–478. ISBN 9780691145587.
Itzykson-Zuber, eq. 3-152
"Quantum Chromodynamics (QCD)". www.fuw.edu.pl. Retrieved 12 April 2018.
Hilf, E. R. "Semiclassical QCD-Lagrangian for Nuclear Physics" (PDF).
Sluka, Volker (January 10, 2005). "Talk" (PDF). Archived from the original (PDF) on June 26, 2007.
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
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Converting NumPy Matrices to Arrays: Methods and Examples - Adventures in Machine Learning
Converting NumPy Matrix to Array: Methods and Examples
Have you ever worked with NumPy matrices and needed to convert them to arrays? This process can be achieved by using different methods, including the A1 property and the ravel() function.
In this article, you will learn how to convert NumPy matrices to arrays using both methods and explore some examples.
Method 1: A1 Property
The NumPy library provides an A1 attribute that allows us to convert a matrix into an array.
The general format is:
This method works by flattening the entire matrix into a one-dimensional array. Here is an example of how to convert a NumPy matrix to an array using A1.
Example 1: Converting NumPy Matrix to Array Using A1
import numpy as np
# Creating a matrix
matrix = np.matrix('1 2 3; 4 5 6; 7 8 9')
print("Original matrix:n", matrix)
# Converting matrix to array using A1
array = matrix.A1
print("nArray:n", array)
The output of this code will be:
Original matrix:
[[1 2 3]
[4 5 6]
[7 8 9]]
[1 2 3 4 5 6 7 8 9]
The A1 property provides an efficient way of converting a matrix to a one-dimensional array. However, it only works with matrix objects, and not with regular arrays.
Method 2: ravel() Function
The ravel() function is another method provided by NumPy for converting matrices to arrays. Unlike A1, ravel() creates a new flattened array object from the original matrix.
Here is an example:
Example 2: Converting NumPy Matrix to Array Using ravel()
import numpy as np
# Creating a matrix
matrix = np.matrix('1 2 3; 4 5 6; 7 8 9')
print("Original matrix:n", matrix)
# Converting matrix to array using ravel()
array = np.ravel(matrix)
print("nArray:n", array)
The output will be the same as in Example 1:
Original matrix:
[[1 2 3]
[4 5 6]
[7 8 9]]
[1 2 3 4 5 6 7 8 9]
The ravel() function provides more options for flattening arrays compared to A1. For instance, it has the order parameter that allows you to specify the order in which elements are flattened.
In conclusion, converting a NumPy matrix to an array can be easily achieved by using the A1 property or the ravel() function. The A1 property works by flattening the matrix into a one-dimensional
array, while ravel() creates a new flattened array object from the matrix.
Both methods are efficient, easy to use, and provide different options for flattening arrays. With this knowledge, you can now work with NumPy matrices and arrays without any concerns about
conversion challenges.
Example 2: Converting NumPy Matrix to Array Using ravel()
Let’s take a closer look at the second method for converting NumPy matrices to arrays: the ravel() function. Before we start, we need to create a NumPy matrix.
This can be achieved using the reshape() function.
NumPy Matrix Creation
Here is an example of how to create a 3 x 3 NumPy matrix using reshape():
import numpy as np
# Creating a 3 x 3 matrix
matrix = np.arange(1, 10).reshape(3, 3)
print("NumPy matrix:n", matrix)
The output of this code will be:
NumPy matrix:
[[1 2 3]
[4 5 6]
[7 8 9]]
Converting Matrix to Array Using ravel()
With the matrix created, we can now use the ravel() function to convert it to an array. Here is an example:
import numpy as np
# Creating a 3 x 3 matrix
matrix = np.arange(1, 10).reshape(3, 3)
print("NumPy matrix:n", matrix)
# Converting matrix to array using ravel()
array = np.ravel(matrix)
print("nNumPy array:n", array)
The output of this code will be:
NumPy matrix:
[[1 2 3]
[4 5 6]
[7 8 9]]
NumPy array:
[1 2 3 4 5 6 7 8 9]
The ravel() function can also be used to flatten multiple dimensions of a matrix into a one-dimensional array. For example, consider this square 3D matrix:
[[[ 0 1 2]
[ 3 4 5]
[ 6 7 8]]
[[ 9 10 11]
[12 13 14]
[15 16 17]]
[[18 19 20]
[21 22 23]
[24 25 26]]]]
If we use ravel(), we get this array:
[ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26]
Confirming NumPy Array Type
Once we have converted a NumPy matrix to an array, we may need to confirm its type to ensure that it matches our expectations. To do this, we can use the type() function.
For example, if we use the type() function on the array from the previous example, we would get the following output:
import numpy as np
# Creating a 3 x 3 matrix
matrix = np.arange(1, 10).reshape(3, 3)
# Converting matrix to array using ravel()
array = np.ravel(matrix)
The output of this code will be:
This confirms that the array is a NumPy array, which we can work with using all of the library’s functions.
In conclusion, we have covered two methods for converting NumPy matrices to arrays: the A1 property and the ravel() function. Both methods have their benefits and limitations, but they are both
effective for flattening matrices into arrays.
Additionally, once we have converted a matrix to an array, we can confirm its type using the type() function to ensure it matches our expectations. With these tools at our disposal, we can manipulate
NumPy matrices and arrays with ease and confidence.
Additional Resources for Working with NumPy
If you want to learn more about working with NumPy matrices and arrays, there are plenty of resources available to help you. Here are some of the best resources for learning more about NumPy:
1. NumPy Documentation: The NumPy documentation is a comprehensive resource that covers all of NumPy’s functions and features. You can find information about creating matrices and arrays,
manipulating them, and performing mathematical operations on them.
2. NumPy User Guide: The NumPy User Guide is a tutorial-style resource that provides step-by-step guidance on using NumPy in your projects.
It covers everything from the basics of NumPy arrays to more advanced topics like indexing and broadcasting.
3. NumPy Tutorials: There are many NumPy tutorials available online that can help you learn how to use NumPy. Some of the best tutorials are provided by DataCamp, Real Python, and Towards Data
4. NumPy Books: There are also several books available that focus specifically on NumPy. Some of the most popular books include “Python for Data Science Handbook” by Jake VanderPlas, “Data Science
Handbook” by Jake VanderPlas, and “Python Data Science Handbook” by Wes McKinney.
5. NumPy Courses: If you prefer a more structured learning experience, there are many NumPy courses available online. Some of the best courses are offered by Udemy, Coursera, and edX.
By utilizing these resources, you can become an expert at working with NumPy matrices and arrays. Whether you are a beginner or an experienced programmer, there is something for everyone in the world
of NumPy.
In summary, converting NumPy matrices to arrays can be achieved using two methods: the A1 property and the ravel() function.
Both methods are efficient and easy to use, with ravel() providing more options for flattening arrays. It is essential to confirm the array type using the type() function to ensure it matches our
In addition, there are abundant resources available to deepen knowledge of NumPy, including documentation, tutorials, books, and courses. Mastering NumPy matrices and arrays is crucial for data
scientists and machine learning engineers.
With these resources, you can learn NumPy and apply this knowledge to real-world projects, making a significant contribution to data analysis and machine learning. | {"url":"https://www.adventuresinmachinelearning.com/converting-numpy-matrices-to-arrays-methods-and-examples/","timestamp":"2024-11-11T20:58:32Z","content_type":"text/html","content_length":"83655","record_id":"<urn:uuid:4b09c887-2295-4869-85ab-be4bbc662686>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00391.warc.gz"} |
Editorial - AtCoder Grand Contest 057
[1] Representing integers by binary trie
We will represent all integers less than \(2^N\) as the leaves in a binary trie by looking at their digits in the order from lowest to highest. That is, we represent the integers as the leaves of a
complete binary tree by classifying them according to
• the \(1\)’s place, the \(2^1\)’s place, the \(2^2\)’s place, \(\ldots\)
in this order.
An intermediate node corresponds to a set of integers equal to some value modulo \(2, 4, \ldots\)
[2] Binary trie and Operation \(+\)
Performing Operation \(+\) changes the binary representation of an integer \(x\) changes as follows.
• The \(1\)’s place always changes.
• The \(2^1\)’s place changes if \(x\equiv 1\pmod{2}\).
• The \(2^2\)’s place changes if \(x\equiv 3\pmod{4}\).
• The \(2^3\)’s place changes if \(x\equiv 7\pmod{8}\).
• \(\vdots\)
It corresponds to starting at the root and going in the direction of \(1\), while swapping the two children in the directions of \(0\) and \(1\) at each intermediate node in the path.
[3] Binary trie and Operation \(\oplus\)
Performing Operation \(+\) changes, for some places, the binary representation of every integer. It corresponds to swapping the two children in the directions of \(0\) and \(1\) at the intermediate
nodes at some depths.
[4] The solution to the problem
After all, we can perform the following two kinds of operations on the binary trie:
• Operation (A): Swap the two children of the intermediate nodes at some depths.
• Operation (B): Choose a leaf and swap the two children of the intermediate nodes on the path from the root to the leaf.
The latter operation is enabled by making the chosen leaf reachable via \(1\)’s by Operation \(\oplus\) and then performing Operation \(+\).
Furthermore, we can assume that we never perform Operation (A) on the deepest intermediate nodes (it would be equivalent to performing Operation (B) on every deepest intermediate node). This
assumption determines the paths for which we should perform Operation (B). After performing Operation (B) for all such paths, it remains to consider whether we can reach the desired state by
performing Operation (A) once, which is easy.
By summarizing the above, the problem can be solved in, for example:
• \(O(N2^N)\) time,
• at most \(2^N\) operations.
One can also solve it in at most \(2^{N-1}\) operations by also performing Operation (A) on the deepest intermediate nodes when appropriate. | {"url":"https://atcoder.jp/contests/agc057/editorial/3925","timestamp":"2024-11-04T09:06:21Z","content_type":"text/html","content_length":"16940","record_id":"<urn:uuid:e706354c-ab3b-4968-9457-7379e9bf6880>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00336.warc.gz"} |
How do I numerically solve an ODE in MATLAB?
The other day a student came to ask me for help in solving a second order ordinary differential equation using the ode45 routine of MATLAB. To use ode45, one needs to be familiar with how the inputs
are required by MATLAB. The understanding of these inputs is important to use ode45 successfully in problems that are more complex than solving a second order ODE.
The ordinary differential equation was
2y”+3y’+5y=7 exp(-x), y(0)=11, dy/dx(0)=13
This has to put in the state variable form by reducing it by using
That gives
y’=z with the corresponding initial conditions as y(0)=11
2y”+3y’+5y=7 exp(-x)
reduces to
2z’ + 3z+5y=7exp(-x)
z’ =(7exp(-x)-3z-5y)/2 with the corresponding initial conditions as z(0)=13
So as needed by MATLAB, call y as y(1) and z as y(2)
dy(1)=y(2), y(1) at x=0 is 11
dy(2)=(7exp(-x)-3y(2)-5y(1))/2, y(2) at x=0 is 13
These equations are now put in a MATLAB function we call odestate.m
To solve the ODE, the
The inputs are
1) the function odestate
2) The outputs are required between x=0 and x=17,
hence entered as [0 17]
3) The initial conditions are y(0)=11 and dy/dx(0)=13,
hence entered as [11 13]
The outputs are
1) X= array of x values between 0 and 17
2) Y= matrix of 2 columns;
first column is the y(x)
second column is dy/dx(x)
The MATLAB code then is
[X,Y]=ode45(@odestate,[0 17],[11 13]);
Click the links for the MATLAB mfiles for the function odestate.m and the ODE solver odetest.m
This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com, the textbook on Numerical Methods with Applications available
from the lulu storefront, and the YouTube video lectures available at http://nm.mathforcollege.com/videos and http://www.youtube.com/numericalmethodsguy
Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you
0 thoughts on “How do I numerically solve an ODE in MATLAB?”
1. How can one solve a quater car model with (Vehicle’s mass (m1), wheel mass (m2), springs stiffness (k1 & k2) and a damping coefficient (damping coefficient (extension and compression) using Euler
Method (MATLAB Program).
2. How can one solve a quater car model with (Vehicle’s mass (m1), wheel mass (m2), springs stiffness (k1 & k2) and a damping coefficient (damping coefficient (extension and compression) using Euler
Method (MATLAB Program).
You must be logged in to post a comment. | {"url":"https://blog.autarkaw.com/2009/10/20/how-do-i-numerically-solve-an-ode-in-matlab/","timestamp":"2024-11-06T14:35:28Z","content_type":"text/html","content_length":"49474","record_id":"<urn:uuid:889d96e3-298b-4f82-926f-1242f8646245>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00097.warc.gz"} |
Existence and Uniqueness Theorem - (Intro to Mathematical Economics) - Vocab, Definition, Explanations | Fiveable
Existence and Uniqueness Theorem
from class:
Intro to Mathematical Economics
The existence and uniqueness theorem states that under certain conditions, a differential equation has a solution that is not only guaranteed to exist but is also unique. This theorem is crucial in
understanding the behavior of solutions to various types of differential equations, providing a framework to ensure that problems posed have consistent and predictable outcomes.
congrats on reading the definition of Existence and Uniqueness Theorem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The existence and uniqueness theorem typically applies to first-order ordinary differential equations and gives conditions under which solutions can be found.
2. For a first-order equation, the theorem states that if the function and its partial derivative with respect to the dependent variable are continuous in a region, then there exists a unique
solution through each point in that region.
3. In the context of second-order equations, similar conditions must be satisfied for solutions to be guaranteed.
4. The existence and uniqueness theorem does not apply universally; there can be cases where either no solution exists or multiple solutions can satisfy the same initial conditions.
5. In systems of differential equations, the existence and uniqueness theorem ensures that each equation can be solved consistently when the system meets specific criteria like continuity and
Lipschitz continuity.
Review Questions
• How do continuity and differentiability relate to the existence and uniqueness theorem in ordinary differential equations?
□ Continuity and differentiability are key requirements for applying the existence and uniqueness theorem. If the function describing the differential equation is continuous in a given region
and satisfies certain smoothness conditions (like having a continuous derivative), then it guarantees that there exists a unique solution through each point in that region. This means that if
you start at a specific initial value, you will trace out one predictable path without any abrupt changes.
• Discuss the implications of failing to meet the Lipschitz condition on the uniqueness of solutions in differential equations.
□ When the Lipschitz condition is not satisfied, the existence and uniqueness theorem may fail, leading to potential scenarios where multiple solutions could arise from the same initial
conditions. This ambiguity means that for some equations, instead of having one clear trajectory, there may be many possible paths a solution could take. Such situations complicate analysis
and make predictions unreliable, demonstrating why ensuring these mathematical properties are met is crucial.
• Evaluate how the existence and uniqueness theorem influences both theoretical understanding and practical applications of systems of differential equations.
□ The existence and uniqueness theorem is fundamental in both theory and practice because it provides assurance that solutions to systems of differential equations are reliable under specified
conditions. In theoretical contexts, it guides researchers on which types of equations can be solved consistently. In practical applications, such as engineering or economics, knowing that
there exists a unique solution allows for accurate modeling of real-world phenomena, leading to effective decision-making based on those models. Without this theorem's guarantees,
practitioners would face uncertainty about whether their models could yield valid results.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/introduction-to-mathematical-economics/existence-and-uniqueness-theorem","timestamp":"2024-11-06T14:06:03Z","content_type":"text/html","content_length":"158287","record_id":"<urn:uuid:f2c1392c-d42a-406d-a689-578688b23535>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00501.warc.gz"} |
3585 -- Accumulation Degree
Trees are an important component of the natural landscape because of their prevention of erosion and the provision of a specific ather-sheltered ecosystem in and under their foliage. Trees have also
been found to play an important role in producing oxygen and reducing carbon dioxide in the atmosphere, as well as moderating ground temperatures. They are also significant elements in landscaping
and agriculture, both for their aesthetic appeal and their orchard crops (such as apples). Wood from trees is a common building material.
Trees also play an intimate role in many of the world's mythologies. Many scholars are interested in finding peculiar properties about trees, such as the center of a tree, tree counting, tree
coloring. A(x) is one of such properties.
A(x) (accumulation degree of node x) is defined as follows:
1. Each edge of the tree has an positive capacity.
2. The nodes with degree of one in the tree are named terminals.
3. The flow of each edge can't exceed its capacity.
4. A(x) is the maximal flow that node x can flow to other terminal nodes.
Since it may be hard to understand the definition, an example is showed below:
Details: 1->2 11
1->4->3 5
1->4->5 8(since 1->4 has capacity of 13)
Details: 2->1->4->3 5
2->1->4->5 6
Details: 3->4->5 5
Details: 4->1->2 11
4->3 5
4->5 10
Details: 5->4->1->2 10
The accumulation degree of a tree is the maximal accumulation degree among its nodes. Here your task is to find the accumulation degree of the given trees. | {"url":"http://poj.org/problem?id=3585","timestamp":"2024-11-06T12:24:32Z","content_type":"text/html","content_length":"8692","record_id":"<urn:uuid:2464bc40-91ef-43ef-b968-23d1c1a80017>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00434.warc.gz"} |
Adding Up
Hold up, I'm calculatin' ... because Common Core are awesome. Yes, I no that aint no proper grammar, but so long's I got the IDEA right, it'r be cool. I still'll get hired someplace that don't care
bout it taken me 10 minutes to count back change in a good-speakin' way, y'all.
Lately, there's been a Common Core hate link circulating social media - it outlines the "Old Fashion" and "New Way" of calculating basic arithmetic, showing a simple math equation of 32-12. Most
people over the age of 10 can simply look at the written equation and automatically reply that the answer is 20, just by doing the math in their heads - which is inarguably the fastest, most
intelligent method to utilize.
But Common Core math wants people to think outside their brains. Using a linear scale, a student is supposed to work upwards through the numbers to "calculate" the correct answer. It requires the
dissection of the primary equation into several other equations, which means the introduction of new numbers to those equations, and ultimately, the addition of those numbers for a math question that
started as subtraction problem. Sound complicated? It is. Better have a pencil and paper handy to work it out.
Common Core advocates say that, by using this method, a student is showing an understanding of how the numbers work. They argue that this is especially beneficial to students that don't understand
the concept of number placement (tens, hundreds, thousands). Additionally, by understanding how numbers work in sequence, students can more easily transition into higher forms of math, like algebra.
It's a noble concept - but it's flawed.
Because, in basic math, numbers are finite. The answers are finite. This is why we can memorize multiplication tables - the answers will not change. Ever. 1+1=2. Always. And when answers are finite,
the simplest, most direct method to obtaining those answers is (or should be) the correct method. What I'm getting at is this: if the method of doing math in your head was in an epic evolutionary
Natural Selection battle with Common Core, the former would win. Because math in your head is easier. Faster. Fit for everyday use.
Common Core math, by this rationale, is utterly archaic, despite its being hailed as an educational break through. Dissecting a math equation to down to a cave-man counting method isn't going to
enhance the mathematical prowess of a student. In fact, I would argue that using a number line to count out a math problem is a crutch. Instead of encouraging a student to remember basic math sums,
and how numbers can work in columns, we're asking them to go outside their intuitive thought process -- and rationalize it. In order for a student to apply Common Core problem solving, they would
need to complicate something that could be very simple. They need paper and pencil to show and track the equations. We're asking students to stop the automatic answer - to stop their thought process
- and programing them to double think something that is an earthen, finite concept...
And what would be the rationale behind getting the new generations of American citizens to double-think something as finite as whole numbers and basic math? Well - if you can double think 1+1=2,
maybe you can double think free economy. Capitalism. Perhaps even morals. Even the basic constructs of freedom. I realize that's a leap in today's world... but over generations, is it really so far | {"url":"http://www.rachea.com/rants/adding-up","timestamp":"2024-11-13T08:40:12Z","content_type":"text/html","content_length":"34465","record_id":"<urn:uuid:8a5aaf91-e2b6-44fb-a0da-a08f1271b986>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00578.warc.gz"} |
Computer Science
Take the following courses:
CS-110 Computer Science I
An introductory study of computer science software development concepts. Python is used to introduce a disciplined approach to problem solving methods, algorithm development, software design, coding,
debugging, testing, and documentation in the object oriented paradigm. This is the first course in the study of computer science.
3 CreditsN,CTGES,CTGISRecommended programming experience or IT110 or IT100, IT111 or IM110 or MA103 but not necessary.
CS-220 Computer Organization
An introduction to digital computer systems including a treatment of logic and digital circuits, data representation, device characteristics and register transfer notation covered in a manner that
stresses application of basic problem solving techniques to both hardware and software design. Students gain experience programming in an assembly language to reinforce these systems and design
4 CreditsNPrerequisites: CS110.
CS-240 Computer Science II
A continued study of computer science foundations as begun In Computer Science I. An object-oriented language such as JAVA is used to develop and implement large programs involving various data
structures and data abstraction as exemplified by packages and modules. Search, sorting, advanced data structures, programming methodology and analysis are emphasized.
4 CreditsNPrerequisites: CS110 and MA116 or MA210.
CS-255C C++ Programming
The students will prepare a portfolio of computer programs written in the language. The programs are reviewed, critiqued, and then the student has an opportunity to revise them as needed for final
inclusion in the portfolio.
2 CreditsNPrerequisites: CS110 and Sophomore standing and permission.
CS-255U Unix Programming
The students will prepare a portfolio of basic Unix programs and scripts. The course covers basic Unix commands, editing techniques, regular expression usage, and script building. The programs are
reviewed, critiqued, and the student has an opportunity to revise them as needed for final inclusion in the portfolio.
1 CreditsN,CTGESPrerequisites: CS110.
CS-300 Software Engineering
An introduction to the issues of software design. Topics include software engineering, software project management and development of projects in a modern design environment. The focus of the course
is on the process used to develop quality software. The students work in teams to develop, implement and fully document a computer project to apply these concepts.
3 CreditsNPrerequisite: CS240.
CS-315 Algorithms and Analysis
The study and analysis of algorithms, their complexity and supporting data structures. Topics include searching, sorting, mathematical algorithms, tree and graph algorithms, the classes of P and NP,
NP-complete and intractable problems, and parallel algorithms.
4 CreditsCW,NPrerequisites: CS240 and MA116.
CS-305 Software Models
A study of current software implementation models. Models of procedural based control for both batch and interactive settings, event driven control, real time control and exception handling are
considered within representative interactive development environments such as .NET Design of graphical user interfaces for web-based and windows-based applications are integrated into the team
3 CreditsNPrerequisites: IT240 or CS240.
CS-320 Operating Systems
An introduction to the theory, evaluation, and implementation of computer operating systems. Topics include memory, process and resource management, elementary Queuing and network models, and
4 CreditsNPrerequisites: CS220 & CS240.
CS-370 Database Management Systems
Focuses on concepts and structures necessary to design and implement a database management system. Various modern data models, data security and integrity, and concurrency are discussed. An SQL
database system is designed and implemented as a group project.
3 CreditsN,CTGISPrerequisites: CS110.
CS-480 Computer Science Seminar I
Discusses current advances in computer science and information technology not otherwise covered in our program such as, but not limited to, networking, artificial intelligence, societal issues. In
addition this course allows senior students to plan an individual research project to be completed in CS485. This course, taken by a junior may be repeated as a senior as CS481.
1 CreditsNPrerequisites: Junior or senior standing and CS220 or CS240 or IT210.
IT-210 Information Technology Systems
This course introduces students to three core areas in Information Technology: networks, database and web. The course progresses through two phases during its study of modern IT environments. Initial
study includes all the necessary components of today's IT system environment and its use in business. Secondly, students use a server based database development environment to create an IT system.
4 CreditsNPrerequisites: CS110. MA116 strongly recommended.
MA-116 Discrete Structures
Introduces mathematical structures and concepts such as functions, relations, logic, induction, counting, and graph theory. Their application to Computer Science is emphasized.
4 CreditsN, QPre-requisite high school algebra.
MA-130 Calculus I
An introduction to calculus including differentiation and integration of elementary functions of a single variable, limits, tangents, rates of change, maxima and minima, area, volume, and other
applications. Integrates the use of computer algebra systems, and graphical, algebraic and numerical thinking.
4 CreditsN, QM
Take one of the following courses:
MA-205 Elementary Statistics
Introduction to traditional statistical concepts including descriptive statistics, binomial and normal probability models, confidence intervals, tests of hypotheses, linear correlation and
regression, two-way contingency tables, and one-way analysis of variance.
4 CreditsN, QS, WK-SPPrerequisite: FYC-101 or EN-110 or EN-109
MA-220 Introduction to Probability & Statistics
An introduction to the basic ideas and techniques of probability theory and to selected topics in statistics, such as sampling theory, confidence intervals, and linear regression.
4 CreditsN, QS, CTGESPrerequisite: MA130
Take one of the following courses:
CS-360 Programming Languages
A systematic approach to the study and analysis of computer programming languages. The underlying concepts of these languages are emphasized.
3 CreditsNPrerequisites: CS-220 and CS-240
CS-362 Languages and Translation
A systematic approach to the study and analysis of computer programming languages. The procedural, functional, object- oriented and logical language paradigms are examined through the use
ofrepresentative languages. Syntax and semantics issues are emphasized through the study of translation techniques in formal labs and group projects.
4 CreditsNPrerequisites: CS220 and CS240. Must have Junior or Senior standing.
Complete 6 credits from the following courses:
CS-255A Android Programming
This course will take your existing Java skills learned in Computer Science I and turn them into Android programming experience. Students will learn the skills in order to develop a fully functional
application. Programming in the Android Studio environment, activity and fragment lifecycles, basic user interface design, and application distribution are emphasized.
1 CreditsNPrerequisites: CS240 and Instructor Permission.
CS-255B COBOL Programming
The students will prepare a portfolio of computer programs written in the language. The programs are reviewed, critiqued, and then the student has an opportunity to revise them as needed for final
inclusion in the portfolio.
2 CreditsNPrerequisites: CS110 and Sophomore standing and permission.
CS-255C C++ Programming
The students will prepare a portfolio of computer programs written in the language. The programs are reviewed, critiqued, and then the student has an opportunity to revise them as needed for final
inclusion in the portfolio.
2 CreditsNPrerequisites: CS110 and Sophomore standing and permission.
CS-255F FORTRAN Programming
The students will prepare a portfolio of computer programs written in the FORTRAN language, The programs are reviewed, critiqued, and the student has an opportunity to revise them as needed for final
inclusion in the portfolio.
2 CreditsNPrerequisites: CS110 and Sophomore standing and permission of instructor.
CS-255P Perl Programming
The students will prepare a portfolio of computer programs written in the Perl language. The programs are reviewed, critiqued, and then the student has an opportunity to revise them as needed for
final inclusion in the portfolio.
2 CreditsN,CTGESPrerequisites: CS110 and Sophomore standing and permission.
CS-255R Ruby Programming
The students will prepare a portfolio of computer programs written in the Ruby language. The programs are reviewed, critiqued, and then the student has an opportunity to revise them as needed for
final inclusion in the portfolio.
2 CreditsNPrerequisites: CS110 and Sophomore standing and permission.
CS-255U Unix Programming
The students will prepare a portfolio of basic Unix programs and scripts. The course covers basic Unix commands, editing techniques, regular expression usage, and script building. The programs are
reviewed, critiqued, and the student has an opportunity to revise them as needed for final inclusion in the portfolio.
1 CreditsN,CTGESPrerequisites: CS110.
CS-255Y Python Programming
The students will prepare a portfolio of computer programs written in the Python language. The programs are reviewed, critiqued, and then the student has an opportunity to revise them as needed for
final inclusion in the portfolio.
2 CreditsN,CTGESPrerequisites: CS110 and Sophomore standing and permission.
CS-330 Computer Graphics
An introduction to both the hardware and software utilized in computer graphics. The emphasis is on a top-down, programming approach, using a standard application programmer's interface. Students
will create three-dimensional and interactive applications, in addition to studying several of the classic, low-level, rendering algorithms.
3 CreditsNPrerequisite: CS-240.
CS-341 Scientific Computing
This course begins with an introduction to fundamental concepts in Scientific Computing and concludes with domain-specific projects in areas like Bioinformatics, Data Science, Physical Systems, and
Numerical Analysis. The common content will include command-line interfaces (Linux), programming languages (Jupyter/Python), numerical and graphical libraries (NumPy and Matplotlib), version-control
(Git/Github), and relational databases (SQL).
3 CreditsNPre-Req: CS-110
CS-390 Computer Science in Germany Seminar
This course will introduce the student to studying Computer Science in Germany. During the spring semester at Juniata, students will prepare for their travel to Germany by: (1) studying the "
functional " German required for travel, (2) reading about the culture and history of the country (and the state of North Rhine-Westfalia in particular), and (3) configuring the technology required
for that years selected topic in CS or IT (the course content will vary each year, previous topics have included Graphical Programming, Security Engineering, and Compiler Construction). This course
culminates with its co-requisite course, CS 391, which is given at the Muenster University of Applied Sciences, for between two and three weeks each May or June. The instructor at MUAS will be a
Juniata College faculty member.
1 CreditsIPre-requisites will be CS240 and instructor permission. Co-requisite is CS391. Completion of both CS390 and CS391 will fulfill the I designation. A fee of $1,200 is applied and it cover
instructional costs,tuition, and Juniata College credit. Students will need to purchase their own plane and train fares. The host institution will facilitate housing for the students.
CS-391 Computer Science in Germany
This course is given at the Muenster University of Applied Sciences, for between two and three weeks each May or June. The instructor at MUAS will be a Juniata College faculty member. Pre-requisites:
CS-240 and instructor permission. A fee of $1,200 is applied that is split between the spring and summer terms and covers instructional costs, tuition, and Juniata College credit. Students will need
to purchase their own plane and train fares. The host institution will facilitate housing for the students.
2 CreditsI,SW-GE
CS-485 Computer Science Research
Allows students to carry out the independent computer science research project as designed in CS480 or CS481.
3-5 CreditsN,CWPrerequisite: CS480 or CS481.
DS-110 Intro to Data Science
This course introduces the student to the emerging field of data science through the presentation of basic math and statistics principles, an introduction to the computer tools and software commonly
used to perform the data analytics, and a general overview of the machine learning techniques commonly applied to datasets for knowledge discovery. The students will identify a dataset for a final
project that will require them to perform preparation, cleaning, simple visualization and analysis of the data with such tools as Excel and R. Understanding the varied nature of data, their
acquisition and preliminary analysis provides the requisite skills to succeed in further study and application of the data science field. Prerequisite: comfort with pre-calculus topics and use of
3 CreditsN
DS-210 Data Acquisition
Students will understand how to access various data types and sources, from flat file formats to databases to big storage data architecture. Students will perform transformations, cleaning, and
merging of datasets in preparation for data mining and analysis.
3 CreditsNPRE-REQ: CS 110 and DS 110.
DS-352 Machine Learning
This course considers the use of machine learning (ML) and data mining (DM) algorithms for the data scientist to discover information embedded in datasets from the simple tables through complex and
big data sets. Topics include ML and DM techniques such as classification, clustering, predictive and statistical modeling using tools such as R, Matlab, Weka and others. Simple visualization and
data exploration will be covered in support of the DM. Software techniques implemented the emerging storage and hardware structures are introduced for handling big data.
3 CreditsNPrerequisite: CS-110, DS-110, and an approved statistics course from this list: MA-205, MA-220, BI-305, PY-214, PY-260, PY-366, or EB- 211.
DS-375 Big Data
This course considers the management and processing of large data sets, structured, semi-structured, and unstructured. The course focuses on modern, big data platforms such as Hadoop and NoSQL
frameworks. Students will gain experience using a variety of programming tools and paradigms for manipulating big data sets on local servers and cloud platforms.
3 CreditsNPrerequisites: DS 110 Intro to Data Science and CS 370 Database Management Systems
IT-110 Principles of Information Technology
This course provides a context for further study in information technology. Topics include an overview of the fundamentals of information systems, current and emerging technologies, business
applications, communications and decision making, and the impact of these systems on business, government, and society. This course will also emphasize the development of both writing and speaking
skills through application of the concepts that define the course. Students who have passed IT-111 or IM-110 may not take this course.
3 CreditsS
IT-260 Human Computer Interaction
To users of any system, the interface is what they see and think of as the computer. Interaction with a computer can be better defined in terms of interface, as any part of the computer system that
the user comes in contact with, either physically, perceptually, or conceptually. Human interaction with computers can be studied, designed, evaluated, with the goal being to produce usable products
from a human-centric perspective.
3 CreditsSPrerequisites: CS110.
IT-325 Network Design & Management.
Focuses on the concept of the foundations of a network in both design and support. The OSI reference model will be examined along with techniques for supporting current technologies that align with
each other. Emphasis will be placed on protocols, topologies and traffic analysis.
4 CreditsNPrerequisites: CS240 or IT210.
IT-341 Web Design
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See IT380. This course will have appointed class times for projects other than those listed on the schedule. A continuation of IT380.
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An introduction to systems of linear equations, matrices, determinants, vector spaces, linear transformations, eigenvalues, and applications.
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An introduction to the logical and set-theoretic basis of modern mathematics. Topics covered include propositional and predicate logic; induction; naive and axiomatic set theory, binary relations,
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Integration, Taylor and Fourier series, and an introduction to differential equations, with applications and the use of the software package Maple. (Course meets four times per week and concludes at
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3 CreditsNPre-Req: CS-110
PC-209 Electronics
An introduction to the theory and application of analog and digital electronics, starting with basic AC and DC circuits. The unit explains the principles of operation of the power supply, amplifier,
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This course reviews and applies project management processes and techniques such as project life cycle, project selection methods, work breakdown instructions, network diagrams, cost estimates, and
3 CreditsS,CW,CS,SW-LEPrerequisites: IT210 and Jr or Sr standing or permission of the instructor. Corequisite: IT308.
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ST_Azimuth and ST_Distance on geography antipodes or near-antipodes
Reported by: mwtoews Owned by: pramsey
Priority: medium Milestone: PostGIS 2.2.0
Component: postgis Version: 2.1.x
Keywords: Cc:
I've been looking at this question, and have been thinking that there is something wrong with PostGIS' calculation of azimuth and/or distance on an oblate spheroid.
First consider the azimuth from the south pole to the north pole. There are actually an infinite number of directions with equal distances:
SELECT degrees(ST_Azimuth(A, B)), ST_Distance(A, B)/1000.0 AS distance_km
FROM (
SELECT 'POINT (-90 -90)'::geography AS A, 'POINT (90 90)'::geography AS B
) AS f;
-[ RECORD 1 ]-----------------
degrees | 90
distance_km | 20003.9314586236
Sure 90°, is just the same as any other (e.g. what direction do you go from the south pole to the north pole?). POINT (10 -90) and POINT (20 90) give 5°, suggesting that the formula is something like
half of the differences in longitude, which appears a bit strange. Slightly more consistent would be to just find the "average" longitude, or always use 0° (from N). Lastly, distance is spot on (half
meridional circumference), so nothing out of the ordinary.
Second, consider antipodes on the equator. There should be exactly two azimuth solutions to this: north or south (0° or 180°), each spaced approx 20003.93 km apart (half meridional circumference),
however I don't see this:
SELECT degrees(ST_Azimuth(A, B)), ST_Distance(A, B)/1000.0 AS distance_km
FROM (
SELECT 'POINT (-90 0)'::geography AS A, 'POINT (90 0)'::geography AS B
) AS f;
-[ RECORD 1 ]-----------------
degrees | 270
distance_km | 19903.5933909347
The answer is the same for any pairs of longitude spaced 180° apart on the equator (y=0). I was expecting either 0° or 180°, and a distance of approx 20003.9314586236 km. I don't know where a
distance of 19903 would have come from, as it is inconsistent with an azimuth of a western-bearing, which would have used the half equatorial circumference of approx 20037.5085 km. It seems that
there could be an axis order issue in the algorithm (but I haven't looked).
Thirdly, as the antipodes on the equator have exactly two possible azimuth solutions, take near-antipodal points that are slightly above or below the equator. These should yield exactly one azimuth
solution right? Moving point A by a few fractions of a degree;
SELECT degrees(ST_Azimuth(A, B)), ST_Distance(A, B)/1000.0 AS distance_km
FROM (
SELECT 'POINT (-90 0.0000001)'::geography AS A, 'POINT (90 0)'::geography AS B
) AS f;
-[ RECORD 1 ]-----------------
degrees | 336.928683929735
distance_km | 20003.9314475662
The expected bearing should be 0°. The distance jumped about 100 m, even though I've only moved the point about 0.011 m. This expected distance should be the same half meridional circumference.
Using POSTGIS="2.1.1 r12113" GEOS="3.4.2-CAPI-1.8.2 r3924" PROJ="Rel. 4.8.0, 6 March 2012" GDAL="GDAL 1.10.0, released 2013/04/24" LIBXML="2.7.8" LIBJSON="UNKNOWN" TOPOLOGY RASTER
Change History (6)
Oh, and a fourth case to consider is any non-pole antipode pair, which should have the same results as the equator example, that is two azimuth solutions 0 or 180. I'm not sure what the expected
distance should be. It could be half the meridional circumference, but it might be some other number close to it.
The geographic functions in GeographicLib are published here:
3. F. F. Karney, Algorithms for geodesics, J. Geodesy 87(1), 43–55 (Jan. 2013); DOI:10.1007/s00190-012-0578-z
With addenda here.
The license is MIT/X11, so some of the relevant algorithms can be brought over to lwspheroid.c with attribution to Charles Karney.
Further note that the current implementation of algorithms is based on Geocentric Datum of Australia Technical Manual, with a web app calculator. However, the web app also has the correct answers
the this bug report, so I'm not sure which one is best, except to say that the PostGIS implementation is definitely incorrect.
Ok, I think I've unearthed the substance of this ticket. From (Vincenty 1975a):
The inverse formulae may give no solution over a line between two nearly antipodal points. This will occur when λ is greater than π in absolute value.
And similar, from Karney (2013):
Vincenty’s method fails to converge for nearly antipodal points.
Vincenty (1975a), who uses the iterative method of Helmert (1880, §5.13) to solve the inverse problem, was aware of its failure to converge for nearly antipodal points. In an unpublished report (
Vincenty 1975b), he gives a modification of his method which deals with this case. Unfortunately, this sometimes requires many thousands of iterations to converge, whereas Newton’s method as
described here only requires a few iterations.
Milestone: → PostGIS 2.2.0
Resolution: → fixed
Status: new → closed
Since we moved to Karney's algorithm's in #2918, this should resolve over time as people move to Proj 4.9 | {"url":"https://trac.osgeo.org/postgis/ticket/2913","timestamp":"2024-11-03T07:05:40Z","content_type":"text/html","content_length":"41616","record_id":"<urn:uuid:aa0ba4d0-068f-49b8-b14a-805e79cc8a92>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00324.warc.gz"} |
Getting a grip on the grid - Page 2 of 2 - Deixis Online
There are two parts to each system analysis, Allemong says:
• State estimation, which takes measurements of electrical quantities on the system and, knowing the structure of the network, figures out what the measured quantities should be. The calculated
values are then compared against the measured values. The end result is an estimated grid state based on the measurements.
• Contingency analysis, which computes the power grid’s condition if there are topological changes – particularly if transmission lines, transformers or generators go out of service. “That will
cause the flows to all redistribute, so you’d better know whether any of those conditions can cause violations” of line and transformer thermal limits and voltage limits, Allemong says.
“In order to do the contingency calculations you have to know what the state of the network is in both topology and flows. State estimation does that,” Allemong says. “Once you have that model of
flows and voltages, you compute a bunch of these outage or topology changes and what the resultant conditions would be.”
The standard analysis for today’s grid operators is a single contingency – cases involving the failure of just one component. In power grid terms, that’s an “N-1” contingency case. “N” represents the
total number of elements in the system and 1 is the total number of elements to fail.
“Usually, for power grid operation if you lose one element, the system should be able to maintain stability,” Huang says. But as in the 2003 outage, blackouts can be caused by multiple failures – an
“N-x” contingency case.
Insurmountable problems
That makes the problem huge. For example, there are about 20,000 components in the Western Electricity Coordinating Council (WECC) system serving nearly 1.8 million square miles in the western United
States and Canada. N-1 contingency analysis, therefore, requires considering 20,000 cases.
Analyzing the impact of any combination of just two WECC components (N-2 contingency analysis) failing means considering an exponentially larger number of cases – roughly 10^8 or 100 million cases,
Huang and his colleagues, Yousu Chen and the late Jarek Nieplocha, wrote in a paper for the July 2009 meeting of the IEEE Power and Energy Society.
Analyzing possible combinations of three or more components (N-x cases) makes the problem so large even the most powerful high-performance computers couldn’t solve it quickly enough to provide useful
But not all grid components are created equal – a fact that lets Huang and his fellow researchers make the problem more manageable. Their contingency analysis algorithm treats the grid as a weighted
graph and applies the concept of “graph betweenness centrality” to identify the most traveled paths between locations. Then it analyzes cases involving only the most critical transmission lines and
Components identified as having little impact on grid stability are removed from the analysis, making the problem tractable with available computing power.
The researchers’ method breaks contingency analysis into two steps: selection and analysis. That makes it well suited to run on PNNL’s Cray XMT, a computer based on “massively multithreaded”
architecture. See sidebar.
It’s clear grid operators must quickly analyze multiple contingencies and take action to block problems – and that the job will take considerable computing power, even when graph analysis discards
less damaging scenarios.
Allemong says grid management organizations typically analyze contingencies every few minutes, but most results are inconsequential. Efforts to get at only significant cases have had limited success,
but with the weighted graph approach, “maybe Henry is on to something.”
(Visited 1,048 times, 1 visits today)
The author is a former Krell Institute science writer.
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adjoint functor theorem
[ since Bartosz emailed me about this: ]
The above edits concern the section Examples – In presheaf categories.
Bartosz wanted to make notationally explicit the Yoneda embedding in the various formulas shown there. I have now touched the section myself, added the remaining instances of the Yoneda embedding;
and also made some further cosmetic changes to the typesetting, such as height-aligned parenthesis etc.
Changed notation for presheaves
Bartosz Milewski
diff, v53, current
Further explanation of syntax
diff, v51, current
Explained the non-standard notation for the limit.
Bartosz Milewski
diff, v50, current
Added an adjoint functor theorem for cocomplete categories.
diff, v49, current
Change notation in the statement of the theorem to match its proof (the functor is $R:C\to D$ instead of $G:D\to C$).
diff, v48, current
Clarified the language in another relevant spot (where a counterexample was given).
diff, v46, current
Clarified some language in the statements that characterize adjoints between locally presentable categories, in response to a comment made by user Hurkyl in another thread (here).
diff, v46, current
I went ahead and made some changes per your comment. See if that looks better. (I think I’d try a different explanation if I were writing this – or writing this today in case I was the one who wrote
that then! – but never mind.)
I fixed a trivial typo in adjoint functor theorem but left wondering about this:
… the limit
$L c := \lim_{c\to R d} d$
over the comma category $c/R$ (whose objects are pairs $(d,f:c\to R d)$ and whose morphisms are arrows $d\to d'$ in $D$ making the obvious triangle commute in $C$) of the projection functor
$L c = \lim_{\leftarrow} (c/R \to D ) \,.$
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I could probably guess how to fix this, but I won’t since I might screw something up.
When reading the presheaf example, I was curious if one could make an argument that representables form a solution set, and this justifies the restriction of the coend to representables only.
Added a reference to
• Duško Pavlović, On completeness and cocompleteness in and around small categories , APAL 74 (1995) pp.121-152.
diff, v57, current
Strengthened the first of the two statements for adjoint functors in the locally presentable case.
diff, v60, current
Mention a generalisation of the AFT.
diff, v64, current
Add early reference for enriched adjoint functor theorem.
diff, v66, current
none of the occurrences of “continuous functor” or “cocontinuous functor” here were hyperlinked. have changed that
diff, v71, current
Added a reference to Porst’s recent survey paper.
diff, v74, current
have copied the reference also to Hans Porst
diff, v75, current
A stronger version for finitary functors between locally finitely presentable categories whose domain is ranked, requiring only the preservation of countable limits for the existence of a left
adjoint, is discussed in
• Jiří Adámek, Lurdes Sousa, A Finitary Adjoint Functor Theorem, arXiv.
diff, v78, current | {"url":"https://nforum.ncatlab.org/discussion/5134/adjoint-functor-theorem/?Focus=67999","timestamp":"2024-11-10T01:24:19Z","content_type":"application/xhtml+xml","content_length":"72745","record_id":"<urn:uuid:3485ca2f-ea91-42d9-93ad-7d32620a57ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00089.warc.gz"} |
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Mathematics & Statistics Seminar Talks | Mathematics & Statistics | Kennedy College of Sciences
Fall 2024
Seminar: Working on What (WOW)
Lee K. Jones (UMass Lowell):On local statistical machine learning
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
September 18, 11 a.m. - Noon, Room: Southwick Hall 350W
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“smoothness” assumptions and conditions of approximate linearity in the spherical neighborhood about x containing the k nearest neighbors. Such bounds are ancillary i.e.depend only on the design
points and not on the responses (unlike many obtained in logistic regression and with variance/bias tradeoff plug-in estimates). We extend these results here to include neighborhoods which consist of
the convex hull of x and any sub-collection of the neighbors. These new bounds can be computed using Karmarkar’s linear programming algorithm (with a quadratic constraint ) and minimization of a
non-smooth convex function. The convex minimization problem, using optimization algorithms of Nesterov (2004), is slower than that for smooth convex functions but is achievable with today’s
computers. We conjecture that an appropriate greedy relaxed approach converges at a faster rate. We can show that speed-ups using proximal functions ( as are used in global machine learning e.g.
lasso regression, inverse problems, matrix completions, etc.) are not possible.
We indicate how to implement both the old and new results for binary classification problems or proportions sampling and search over lower dimensional subspaces of the design domain for accurate
estimates of conditional success probability given the projection of x.
Seminar: Working on What (WOW)
Jim Propp (UMass Lowell):“Tilings? (Again?)”
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
October 23, 11 a.m. - Noon, Room: Southwick Hall 350W
Abstract: From the late 1980s to the early 2000s I did a lot of work on tiling. For the next 20 years I did very little work in this area. Suddenly I’m working on tilings again. What’s up with that? | {"url":"https://www.uml.edu/sciences/mathematics/news/seminartalks.aspx","timestamp":"2024-11-02T14:56:43Z","content_type":"text/html","content_length":"30970","record_id":"<urn:uuid:408a7845-f983-4193-8382-d36f9b03e963>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00593.warc.gz"} |
Problem Solving Class: Van Quark tot Biomaterie
Problem Set 2: The Bohr Model of the Atom
Hand-in on paper Tuesday 11 September (during lecture 15:30 h)
Hand-in digitally, email to: m.t.talluri@vu.nl;
All documents in a single file [file: YourName-WC-Q2]
All answers in English
1) The fine structure constant
The Bohr formula for the energy levels in the hydrogen atom can be written as:
Z2 2 2
E n 2 mc
a) Derive a formula for the fine structure constant given the presented derivation of the level
energies in the H-atom.
b) What is the value of ? And what is the dimension ?
Hence the binding energy of electrons in the ground state of the hydrogen atom is given by:
Eb 2 E 0
where E0 is the rest mass of the electron (E0=mc2).
c) What is the reason for this universal relationship in terms of a dimensionless number ?
In other words: What is the physical reason for the numerical value of as calculated ?
d) Derive also, following the derivations in the notes, that the velocity of an electron in the
ground state orbital in the Bohr model (n=1 orbit, for Z=1) is given by:
v1 c
2) Lyman- transition in atomic hydrogen
The Lyman- transition in the hydrogen atom is defined as the transition from quantum state
n=1 to n=2.
In first order neglect the “reduced mass-effect:
a) Give an equation for the frequency and for the wavelength of this Lyman- transition.
b) Derive a numerical value for the wavelength of this transition; a value in nanometers.
Now, in second order include the reduced mass-effect.
Consider the exotic atom “positronium” built from an electron and an anti-electron (also known
as a positron).
Note that the positron is positively charged (+1e) and has a mass equal to that of the electron.
c) Derive an equation for the level energies of the positronium system.
d) What is the wavelength of the “Lyman-” transition in positronium (in equation and | {"url":"https://studylib.net/doc/14207149/problem-solving-class--van-quark-tot-biomaterie","timestamp":"2024-11-01T20:37:08Z","content_type":"text/html","content_length":"57749","record_id":"<urn:uuid:8f0bdac0-954e-4bd6-a3c6-5d593957b43f>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00195.warc.gz"} |
dual point distributors
Fellow listers:
The fog has lifted, and a picture is beginning to form! Thanks to all the
input from all you experts, I now know how, and I think I know why, dual point
distributors function. Let me run this by you all, and see if I am on the
right track.
To produce maximum spark energy, a coil needs time for the primary field to
This time required is constant, regardless of rpm.
The time allowed for this is determined by the point dwell angle.
Dwell angle is constant with respect to rpm.
Since dwell angle is constant with respect to rpm, the time allowed for the
primary field to build is reduced as rpm increases.
Dwell angle is determined by the point gap.
To maximize dwell angle (and to increase the time allocated for charging the
coil), point gap is minimized.
If dwell is maximized by making the gap too small, reliable operation is
jeopardized, and erratic triggering occurs. Also, the time between gap
resetting due to wear becomes short.
How do you increase the dwell angle, without increasing the point gap? One
way would be to open the points to a wide gap, but use a very steep cam
profile on the distributor rotor to snap them open quickly, and use a very
heavy spring to snap them back together very quickly. This might work, but it
would be at the expense of excessive wear on the point rubbing block, and
still would give problems at very high rpm. Enter dual points!
As several of you have pointed out, the dual points are wired in parallel, and
operate a little out of phase - point set #1 opening and closing just a little
before point set #2. What a clever idea! Since nothing happens till both sets
of points open, this first set can be opened as gently and as wide as need be
for reliabilty/accuracy. When the second set opens, the coil will fire. This
second set can also be opened gently and wide, because the first set can be
timed to close immediately after the second opens. Even if the first set
closes gently, if the timing is adjusted correctly, they can be closed in
plenty of time to start the coil charging process, maximizing the dwell angle.
Does all that sound correct?
In my original post, I said that you could not get a hotter spark from a
single coil no matter how many points you use. Theoretically, that is true,
but, as I now understand, from a practical standpoint you can indeed get a
hotter spark with a single coil by using dual points. I sit corrected - and
content, knowing I have learned something.
Thanks to everyone for setting me straight.
A few other observations, if I may. Maximum dwell angle on a V8 is 45 degrees.
Maximum dwell angle on a four cylinder engine is 90 degrees. Thus the charging
time for a coil on a V8 at 5500 rpm is the same as 11,000 rpm on a four
cylinder. So, unless you run your MGB at 11,000 plus rpm, any coil that would
work on a V8 would be more than adequate for use on an MGB. Any increased
performance gained from the hotter spark from a dual point set-up would have
to be from the fact that the dual point set-up is just a better distributor,
mechanically speaking, then the worn out stock distributor it replaced
Dan Masters,
Alcoa, TN
'71 TR6---------3000mile/year driver, fully restored
'71 TR6---------undergoing full restoration and Ford 5.0 V8 insertion - see:
'74 MGBGT---3000mile/year driver, original condition
'68 MGBGT---organ donor for the '74 | {"url":"http://www.team.net/html/mgs/1997-12/msg01436.html","timestamp":"2024-11-11T04:58:21Z","content_type":"text/html","content_length":"9879","record_id":"<urn:uuid:48a7196d-a694-491f-a345-a82630021b05>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00358.warc.gz"} |
Attometers to Astronomical Units Converter
Enter Attometers
Astronomical Units
β Switch toAstronomical Units to Attometers Converter
How to use this Attometers to Astronomical Units Converter π €
Follow these steps to convert given length from the units of Attometers to the units of Astronomical Units.
1. Enter the input Attometers value in the text field.
2. The calculator converts the given Attometers into Astronomical Units in realtime β using the conversion formula, and displays under the Astronomical Units label. You do not need to click any
button. If the input changes, Astronomical Units value is re-calculated, just like that.
3. You may copy the resulting Astronomical Units value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Attometers to Astronomical Units?
The formula to convert given length from Attometers to Astronomical Units is:
Length[(Astronomical Units)] = Length[(Attometers)] / 1.4959787070600768e+29
Substitute the given value of length in attometers, i.e., Length[(Attometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in astronomical units,
i.e., Length[(Astronomical Units)].
Calculation will be done after you enter a valid input.
Consider that the wavelength of a gamma-ray photon is around 1 attometer.
Convert this wavelength from attometers to Astronomical Units.
The length in attometers is:
Length[(Attometers)] = 1
The formula to convert length from attometers to astronomical units is:
Length[(Astronomical Units)] = Length[(Attometers)] / 1.4959787070600768e+29
Substitute given weight Length[(Attometers)] = 1 in the above formula.
Length[(Astronomical Units)] = 1 / 1.4959787070600768e+29
Length[(Astronomical Units)] = 0
Final Answer:
Therefore, 1 am is equal to 0 AU.
The length is 0 AU, in astronomical units.
Consider that the scale of nuclear interactions is on the order of 10 attometers.
Convert this scale from attometers to Astronomical Units.
The length in attometers is:
Length[(Attometers)] = 10
The formula to convert length from attometers to astronomical units is:
Length[(Astronomical Units)] = Length[(Attometers)] / 1.4959787070600768e+29
Substitute given weight Length[(Attometers)] = 10 in the above formula.
Length[(Astronomical Units)] = 10 / 1.4959787070600768e+29
Length[(Astronomical Units)] = 0
Final Answer:
Therefore, 10 am is equal to 0 AU.
The length is 0 AU, in astronomical units.
Attometers to Astronomical Units Conversion Table
The following table gives some of the most used conversions from Attometers to Astronomical Units.
Attometers (am) Astronomical Units (AU)
0 am 0 AU
1 am 0 AU
2 am 0 AU
3 am 0 AU
4 am 0 AU
5 am 0 AU
6 am 0 AU
7 am 0 AU
8 am 0 AU
9 am 0 AU
10 am 0 AU
20 am 0 AU
50 am 0 AU
100 am 0 AU
1000 am 0 AU
10000 am 0 AU
100000 am 0 AU
An attometer (am) is a unit of length in the International System of Units (SI). One attometer is equivalent to 0.000000000000001 meters or 1 Γ 10^(-18) meters.
The attometer is defined as one quintillionth of a meter, making it an extremely small unit of measurement used for measuring subatomic distances.
Attometers are used in advanced scientific fields such as particle physics and quantum mechanics, where precise measurements at the atomic and subatomic scales are required.
Astronomical Units
An astronomical unit (AU) is a unit of length used in astronomy to measure distances within our solar system. One astronomical unit is equivalent to approximately 149,597,870.7 kilometers or about
92,955,807.3 miles.
The astronomical unit is defined as the mean distance between the Earth and the Sun.
Astronomical units are used to express distances between celestial bodies within the solar system, such as the distances between planets and their orbits. They provide a convenient scale for
describing and comparing distances in a way that is more manageable than using kilometers or miles.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Attometers to Astronomical Units in Length?
The formula to convert Attometers to Astronomical Units in Length is:
Attometers / 1.4959787070600768e+29
2. Is this tool free or paid?
This Length conversion tool, which converts Attometers to Astronomical Units, is completely free to use.
3. How do I convert Length from Attometers to Astronomical Units?
To convert Length from Attometers to Astronomical Units, you can use the following formula:
Attometers / 1.4959787070600768e+29
For example, if you have a value in Attometers, you substitute that value in place of Attometers in the above formula, and solve the mathematical expression to get the equivalent value in
Astronomical Units. | {"url":"https://convertonline.org/unit/?convert=attometers-astronomical_unit","timestamp":"2024-11-06T20:24:39Z","content_type":"text/html","content_length":"90838","record_id":"<urn:uuid:84d75ad3-29b5-421c-9a94-ad4bba1b3d33>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00193.warc.gz"} |
Arthur (verifier) is a computational entity, typically represented as a Turing Machine, thus having bounded capabilities. Arthur's job is to verify the solutions for a given decision problem
using the proof/witness state provided by Merlin (prover). Based on certain criterion, Arthur will accept or reject with some probability.
Merlin (prover) exists as an omniscient, omnipotent and mendacious entity, having unbounded computational resources — essentially representing an infinitely powerful oracle. Additionally, Merlin
possesses the ability to generate quantum states. The role of Merlin is to convince Arthur (verifier) as to the validity of a decision problem's solution via some proof/witness state.
Polynomial Time
In computational theory, a polynomial time algorithm is an algorithm whose running time is bounded by a polynomial function of the input size. More formally, an algorithm is said to run in
polynomial time if there exists a polynomial function $f(n)$, where $n$ represents the size of the input, such that the algorithm's running time is $O\big(f(n)\big)$.
Quantum Computer
A quantum computer is a computational machine that utilises the fundamental principles of quantum mechanics to execute computation tasks. The primitive elements of a quantum computer are quantum
bits (qubits) — naturally exhibiting superposition and entanglement properties. Their attributes generate the quantum advantage in information processing not efficiently produced on classical
Reduction (From)
A reduction 'From' one problem $A$, to a problem $B$, refers to reducing (in polynomial time) or perhaps relaxing conditions on problem $A$ to produce problem $B$. Informally, this can be
understood as $B\subseteq A$, where $A$ is the parent problem we reduce from to obtain $B$.
Reduction (To)
A reduction 'To' one problem $Y$, from a problem $X$, refers to reducing (in polynomial time) or perhaps relaxing conditions on problem $X$ to produce problem $Y$. Informally, this can be
understood as $Y\subseteq X$, where $Y$ is the child problem we reduce to from $X$.
Isotropic Antiferromagnetic (IaF)
An isotropic antiferromagnetic Hamiltonian is one such that each present interaction term, $J_{ij}, K_{ij}, \ldots$ is of the form $J_{ij} \geq 0$. Antiferromagnetism refers to the non–negative
interaction strength sector. The isotropic labeling refers to the fact that all interaction terms adhere to the positive weight restriction but do not necessarily all have the same value.
Isotropic Ferromagnetic is the analogous non–positive regime.
Homogeneous Antiferromagnetic (HaF)
A homogeneous antiferromagnetic Hamiltonian is one such that all present interaction terms, $J_{ij}, K_{ij}, \ldots$ are different but fixed non–negative values for all particle interactions.
Antiferromagnetism refers to the non–negative interaction strength sector. The homogeneous labeling refers to the fact that all interaction terms between particles are now some constant
non–negative value. Homogeneous Ferromagnetic is the analogous non–positive regime.
Local Hamiltonian Problem
Given a $k=O(1)$ local Hamiltonian, $H$, acting on an $n$–qubit system, such that, $$H = \sum_{j=1}^m H_j$$ where $||H_j||\leq\mathsf{poly}(n)$, each element of every $H_j$ can be specified in $\
mathsf{poly}(n)$ bits and $m\leq\mathsf{poly}(n)$, the problem statement is to determine which of the following is true, promised one is so:
1. The smallest eigenvalue of $H$ is $\leq a$
2. All eigenvalues of $H$ are $\geq b$
Commuting Local Hamiltonian Problem
Given a $k=O(1)$ local Hamiltonian, $H$, acting on an $n$–qudit system, such that, $$H = \sum_{j=1}^m h_j$$ where $[h_j,h_k]=0$ for all $j,k$ and with $||H_j||\leq\mathsf{poly}(n)$, each element
of every $H_j$ can be specified in $\mathsf{poly}(n)$ bits and $m\leq\mathsf{poly}(n)$, the problem statement is to determine which of the following is true, promised one is so:
1. The smallest eigenvalue of $H$ is $\leq a$
2. All eigenvalues of $H$ are $\geq b$
(This can be reformulated in terms of a projectors which then asks if the ground space is of positive dimension)
Given a graph $G=(V,E,w_{ij})$, where $w_{ij} \geq 0$ is the weight of the edge between vertices $i$ and $j$, compute the maximum eigenvalue of the Hamiltonian $$H_{MC}(G) = \frac{1}{2} \sum_{\
{i,j\} \in E(G)} w_{ij} \left(\mathbb{I} - S_i \cdot S_j\right)$$ Subject to $S_i \in \{\pm 1\}$.
Given a graph $G=(V,E)$ and a fixed $d\times d$ matrix $W$, assign unit vectors to maximise the following: $$MC_W^L(G) = \frac{1}{2} \displaystyle\max_{i\in S^{d-1}} \displaystyle\sum_{\{i,j\}\in
E(G)} ||Wi - Wj||$$
Quantum Max–Cut
Given a graph $G=(V,E,w_{ij})$, where $w_{ij} \geq 0$ is the weight of the edge between vertices $i$ and $j$, compute the maximum eigenvalue of the Hamiltonian $$H_{QMC_S}(G) = \frac{1}{2} \sum_
{\{i,j\} \in E(G)} w_{ij} \left(\mathbb{I} - \frac{1}{S^2}\;\boldsymbol{S}_i \cdot \boldsymbol{S}_j \right)$$ (This is the standard definition where each edge is a singlet state)
Quantum Max–Cut(EPR)
Given a graph $G=(V,E,w_{ij})$, where $w_{ij} \geq 0$ is the weight of the edge between vertices $i$ and $j$, compute the maximum eigenvalue of the Hamiltonian $$H_{QMC(EPR)}(G) = \frac{1}{2} \
sum_{\{i,j\} \in E(G)} w_{ij} \left(\mathbb{I} + X_iX_j - Y_iY_j + Z_iZ_j\right)$$ (This is the a variant on the standard Quantum Max–Cut where each edge is an EPR state)
Quantum Partition Function
Given a $k$–local Hamiltonian $H$ on $n$ qubits, an inverse temperature $\beta = O(\mathsf{poly}(n))$ and a precision parameter $\delta = \Omega(1/\mathsf{poly}(n))$, compute an approximation to
the partition function $Z(\beta) = \text{Tr}\left[e^{-\beta H}\right]$ such that $$\big| Z - \widetilde{Z}\big| \leq \delta Z.$$
Additive $\varepsilon$–Approximation
Let $a$, $\hat{a}$ and $\varepsilon$ be positive real numbers. We say that $\hat{a}$ is the additive $\varepsilon$-approximation of $a$ if $$ |{a - \hat{a}}| \leq \varepsilon.$$
Multiplicative $\varepsilon$–Approximation
Let $a$, $\hat{a}$ and $\varepsilon$ be positive real numbers. We say that $\hat{a}$ is the multiplicative $\varepsilon$-approximation of $a$ if $$ |{a - \hat{a}}| \leq \varepsilon a.$$
Complex Multiplicative $\varepsilon$–Approximation
Let $z = re^{i\theta}$ and $\hat{z} = \hat{r}e^{i\hat{\theta}}$. We say that $\hat{z}$ is the multiplicative $\varepsilon$-approximation of $z$ if $$ |{r - \hat{r}}| \leq \varepsilon r \quad \
text{and} \quad |{\theta - \hat{\theta}}| \leq \varepsilon.$$
Complexity Theory
See the Complexity Zoo for a more comprehensive list of complexity classes.
Polynomial Time — P
A decision problem $L$ is in P if there exists: a deterministic Turing Machine, $M$, such that
1. For any input $x$, $M$ runs for a time $O(\mathsf{poly}(|x|)\big)$.
2. For all $x\in L$, $M$ outputs $\mathtt{1}$.
3. For all $x\notin L$, $M$ outputs $\mathtt{0}$.
Nondeterministic Polynomial — NP
A decision problem $L$ is in NP if there exists: a deterministic verification Turing Machine, $M$, and a polynomial, $\mathsf{poly}(|x|)$, such that
1. For all $x\in L$, $M(x,\mathsf{poly}(|x|))=\mathtt{1}$.
2. For all $x\notin L$, $M(x,\mathsf{poly}(|x|))=\mathtt{0}$.
Bounded–Error Probabilistic Polynomial — BPP($\mathsf{a}$,$\mathsf{b}$)
A decision problem $L$ is in BPP($\mathsf{a}$,$\mathsf{b}$) if there exists: a probabilistic Turing Machine, $M$, such that
1. For any input $x$, $M$ runs for a time $O(\mathsf{poly}(|x|))$.
2. For all $x\in L$, $\text{Pr}\left[M(x)=\mathtt{1} \right] \geq a$.
3. For all $x\notin L$, $\text{Pr}\left[M(x)=\mathtt{1} \right] \leq b$.
Randomised Polynomial — RP
A decision problem $L$ is in RP if there exists: a probabilistic Turing Machine, $M$, such that
1. For any input $x$, $M$ runs for a time $O(\mathsf{poly}(|x|))$.
2. For all $x \in L$, $\text{Pr}[M(x) = \mathtt{1}] \geq \frac{1}{2}$.
3. For all $x \notin L$, $\text{Pr}[M(x) = \mathtt{1}] = 0$.
Merlin–Arthur — MA($\mathsf{a}$,$\mathsf{b}$)
A decision problem $L$ is in MA($\mathsf{a}$,$\mathsf{b}$) if there exists: a deterministic verification Turing Machine, $M$, and a polynomial, $\mathsf{poly}(|x|)$, such that
1. For all $x\in L$, there exists a proof state, $z \in \{0,1\}^{\mathsf{poly}(|x|)}$, such that, $\text{Pr}\left[M(x,z)=\mathtt{1} \right] \geq a$.
2. For all $x\notin L$, for any proof state, $z \in \{0,1\}^{\mathsf{poly}(|x|)}$, then, $\text{Pr}\left[M(x,\mathsf{poly}(|x|))=\mathtt{1} \right] \leq b$.
Sharp P — #P
The class #P consists of all functions $f:\Sigma^* \rightarrow \mathbb{N}$ for which there exists a polynomial $p:\mathbb{N} \rightarrow \mathbb{N}$ and a polynomial-time algorithm $A$ such that:
1. for all $x \in \Sigma^*$ $f(x) = |\{y \in \Sigma^{p(x)} : A(x,y)\;\text{accepts} \}|$
GapP — GapP
The class GapP consists of all functions $f:\Sigma^* \rightarrow \mathbb{Z}$ for which there exists a polynomial nondeterministic Turing machine $M$ such that
1. for all $x \in \Sigma^*$ $f(x) = \#M(x) - \# \bar{M}(x)$
where $\#M(x)$ and $\# \bar{M}(x)$ are the number of accepting and rejector paths of $M$ on $x$.
Fully Polynomial–time Approximation Scheme — FPTAS
A fully polynomial-time approximation scheme for a counting problem $f:\Sigma^* \rightarrow \mathbb{N}$ is an algorithm $A$ that takes as input an instance $x \in \Sigma^*$ and a number $\epsilon
\gt 0$ and outputs a multiplicative $\epsilon$-approximation to $f(x)$ in polynomial time in $|x|$ and $1/\epsilon$.
Fully Polynomial–time Randomised Approximation Scheme — FPRAS
A fully polynomial-time randomised approximation scheme for a counting problem $f:\Sigma^* \rightarrow \mathbb{N}$ is an algorithm $A$ that takes as input an instance $x \in \Sigma^*$ and a
number $\epsilon \gt 0$ and outputs a multiplicative $\epsilon$-approximation to $f(x)$ with probability at least $2/3$ in polynomial time in $|x|$ and $1/\epsilon$.
Bounded–Error Quantum Polynomial — BQP($\mathsf{a}$,$\mathsf{b}$)
A decision problem $L$ is in BQP($\mathsf{a}$,$\mathsf{b}$) if there exists: a polynomial–time uniform family of quantum circuits, $Q = \{Q_n | n \in \mathbb{N}\}$, such that
1. $Q_n$ has $n$–qubits as input and $1$ qubit as output.
2. For all $x\in L$, $\text{Pr}\left[Q(x)=\mathtt{1} \right] \geq a$.
3. For all $x\notin L$, $\text{Pr}\left[Q(x)=\mathtt{1} \right] \leq b$.
Stoquastic Merlin–Arthur — StoqMA($\alpha$,$\beta$)
A decision problem $L$ is in StoqMA($\alpha$,$\beta$) if there exists: a polynomial stoquastic verifier, $V = (n,w,m,p, U)$, such that
1. $n$ is the number of input bits, $w$, the number of proof qubits, $m$, the number of $\ket{0}$ ancillae and $p$, the number of $\ket{+}$ ancillae.
2. $U$ is a quantum circuit on $n + w + m +p$ qubits using gates from the set $\{ X$, Cnot, Toff$\}$.
3. The acceptance probability of a stoquastic verifier $V$ given some input string $x\in L$ and a proof state $\ket{\psi}$, $\text{Pr}\left[V(x,\ket{\psi})\right]= \bra{\phi}U^\dagger \Pi_{\text
{out}} U \ket{\phi}$, where $\ket{\phi} = \ket{x,\psi,0^{m},+^{p}}$ and $\Pi_{\text{out}} = \ketbra{+}{+}_1$.
4. Completeness: For all $x\in L$, there exists a quantum proof state, $\ket{\psi}\in\mathcal{B}^{w}$, such that, $\text{Pr}\left[V(x,\ket{\psi})=\mathtt{1} \right] \geq \alpha$.
5. Soundness: For all $x\notin L$, for any quantum proof state, $\ket{\psi}\in\mathcal{B}^{w}$, then, $\text{Pr}\left[V(x,\ket{\psi})=\mathtt{1} \right] \leq \beta$.
Where $\alpha$ refers to the completeness parameter and $\beta$ the soundness parameter, where $-1/2 \leq \beta(n) \lt \alpha(n) \leq 1$ and satisfying $\alpha-\beta\geq\frac{1}{\mathsf{poly}(n)}
Quantum Merlin–Arthur — QMA($c$,$s$)
A promise problem $L$ is in QMA($c$,$s$) if there exists: a polynomial quantum verifier, $V$, and a polynomial, $\mathsf{poly}(|x|)$, for $x\in L$, such that
1. Completeness: For all $x\in L$, there exists a quantum proof state, $\ket{\psi}\in\mathcal{B}^{\mathsf{poly}(|x|)}$, such that, $\text{Pr}\left[V(x,\ket{\psi})=\mathtt{1} \right] \geq c$.
2. Soundness: For all $x\notin L$, for any quantum proof state, $\ket{\psi}\in\mathcal{B}^{\mathsf{poly}(|x|)}$, then, $\text{Pr}\left[V(x,\ket{\psi})=\mathtt{1} \right] \leq s$.
Here $c - s \geq\frac{1}{\mathsf{poly}(n)}$. | {"url":"https://hamiltonianjungle.xyz/glossary.html","timestamp":"2024-11-14T13:39:46Z","content_type":"text/html","content_length":"25030","record_id":"<urn:uuid:a5745aa1-1432-4e52-ac52-c37bd6ef11b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00410.warc.gz"} |
Linear Convergence of Black-Box Variational Inference: Should We Stick the Landing?
Linear Convergence of Black-Box Variational Inference: Should We Stick the Landing?
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:235-243, 2024.
We prove that black-box variational inference (BBVI) with control variates, particularly the sticking-the-landing (STL) estimator, converges at a geometric (traditionally called “linear”) rate under
perfect variational family specification. In particular, we prove a quadratic bound on the gradient variance of the STL estimator, one which encompasses misspecified variational families. Combined
with previous works on the quadratic variance condition, this directly implies convergence of BBVI with the use of projected stochastic gradient descent. For the projection operator, we consider a
domain with triangular scale matrices, which the projection onto is computable in $\theta(d)$ time, where $d$ is the dimensionality of the target posterior. We also improve existing analysis on the
regular closed-form entropy gradient estimators, which enables comparison against the STL estimator, providing explicit non-asymptotic complexity guarantees for both.
Cite this Paper
Related Material | {"url":"https://proceedings.mlr.press/v238/kim24a.html","timestamp":"2024-11-06T04:42:48Z","content_type":"text/html","content_length":"15547","record_id":"<urn:uuid:f71328ea-d99e-4769-bd03-ca07197eb83e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00677.warc.gz"} |
Life of Fred
Life of Fred Pre-Algebra 1 with Biology Zillions of Practice Problems $24 : This book is keyed directly to Life of Fred: Pre-Algebra 1 with Biology. Each of the chapters contains both exercises on
the current topic and review questions from the beginning the book up to that point. All the problems have completely worked out solutions. The problems are fun with lots of stories about Jan's
acting career, Ivy's ice cream store, and Cassie's genotype. ISBN: 978-1-937032-59-3 hardback, 240 pages, $24 | {"url":"http://www.horriblebooks.com/Life%20of%20Fred%20Pages/Life%20of%20Fred%20Pre-Algebra%201%20with%20Biology%20Zillions.htm","timestamp":"2024-11-12T11:45:52Z","content_type":"text/html","content_length":"1842","record_id":"<urn:uuid:4eed337b-aa2c-4612-8825-ed4dbfbd4ba8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00773.warc.gz"} |
How to call nested tuple and nested set or dictionary using variable argument and variable keyword argument methods ?
How to call nested tuple and nested set or dictionary using variable argument and variable keyword argument methods ?
def arithmetic_mean(first, *values):
""" This function calculates the arithmetic mean of a non-empty
arbitrary number of numerical values """
return (first + sum(values)) / (1 + len(values))
x= [('a', 232), ('b', 343), ('c', 543), ('d', 23)]
y= [[('a', 232), ('b', 343), ('c', 543), ('d', 23)]]
how to pass x and y inside arithmetic_mean. can it be possible through zip method?
1 Answer
Sort by ยป oldest newest most voted
I am not sure about your exact question, does the following answer it ?
sage: arithmetic_mean(*dict(x).values())
sage: arithmetic_mean(*dict(y[0]).values())
EDIT Here is a version using zip (though it is not very natural):
sage: arithmetic_mean(*zip(*x)[1])
sage: arithmetic_mean(*zip(*y[0])[1])
edit flag offensive delete link more
thanq thanq very much it solved my problem
i got information from site " https://www.python-course.eu/python3_functions.php%22 (https://www.python-course.eu/python3_...) in 'Arbitrary Number of Parameters' section that, This type of problem
can be solved by zip metyhod. can u tell me, is it possible or not.
damodar ( 2018-10-12 21:50:19 +0100 )edit
Everything is always possible (see my edit), but the main question is which problem do you want to solve.
tmonteil ( 2018-10-12 22:36:55 +0100 )edit
Thanq very much. i checked
print(arithmetic_mean(45,list(zip(x))[1])) ; print(arithmetic_mean(45,list(zip(y[0]))[1]))
it works.
damodar ( 2018-10-12 22:55:19 +0100 )edit
if i get values like p=[(232,), (343,), (543,), (23,)]
q=((232, 343, 543, 23),)
how can i will pass ?
i could not able to do without zip method, is there any other way to do?
damodar ( 2018-10-13 00:59:05 +0100 )edit
You can use the flatten function.
tmonteil ( 2018-10-13 11:14:30 +0100 )edit | {"url":"https://ask.sagemath.org/question/43921/how-to-call-nested-tuple-and-nested-set-or-dictionary-using-variable-argument-and-variable-keyword-argument-methods/?sort=latest","timestamp":"2024-11-09T09:56:58Z","content_type":"application/xhtml+xml","content_length":"63221","record_id":"<urn:uuid:5bcc1d43-882b-4ccc-a3dc-00075132b9d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00799.warc.gz"} |
Collision integrals and high temperature transport properties for N-N, O-O, and N-O
Accurate collision integrals are reported for the interactions of N(4 S 0) + N(4 S 0), O(3 P), and N(4 S 0) + O(3 P). These are computed from a semiclassical formulation of the scattering using the
best available representations of all of the potential energy curves needed to describe the collisions. Experimental RKR curves and other accurate measured data are used where available; the results
of accurate ab initio electronic structure calculations are used to determine the remaining potential curves. The high-lying states are found to give the largest contributions to the collision cross
sections. The nine collision integrals, needed to determine transport properties to second order, are tabulated for translational temperatures in the range 250 K to 100,000 K. These results are
intended to reduce the uncertainty in future predictions of the transport properties of nonequilibrium air, particularly at high temperatures. The viscosity, thermal conductivity, diffusion
coefficient, and thermal diffusion factor for a gas composed of nitrogen and oxygen atoms in thermal equilibrium are calculated. It was found that the second order contribution to the transport
properties is small. Graphs of these transport properties for various mixture ratios are presented for temperatures in the range 5000 to 15000 K.
Pub Date:
November 1989
□ Atomic Collisions;
□ Collision Parameters;
□ Gas Mixtures;
□ High Temperature Air;
□ Integral Equations;
□ Nitrogen Atoms;
□ Oxygen Atoms;
□ Scattering;
□ Transport Properties;
□ Aeroassist;
□ Aerobraking;
□ Atomic Interactions;
□ Particle Collisions;
□ Potential Energy;
□ Thermal Conductivity;
□ Thermal Diffusion;
□ Thermodynamic Equilibrium;
□ Viscosity;
□ Atomic and Molecular Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1989ciht.rept.....L/abstract","timestamp":"2024-11-11T23:45:53Z","content_type":"text/html","content_length":"38852","record_id":"<urn:uuid:dc834914-94e6-47ef-9e9b-bac57a5985f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00313.warc.gz"} |
Finding Probabilities of Binomial Experiments
Question Video: Finding Probabilities of Binomial Experiments Mathematics
In a binomial experiment, this spinner is spun 10 times and the result is recorded as a success if the top score is achieved. Let π be the number of successes. Determine π (π = 2) as a
percentage to 3 decimal places. Determine π (π = 9) as a percentage to 3 decimal places.
Video Transcript
In a binomial experiment, this spinner is spun 10 times and the result is recorded as a success if the top score is achieved. Let π be the number of successes. Determine the probability that π
equals two as a percentage to three decimal places. Determine the probability that π equals nine as a percentage to three decimal places.
Weβ re told in the question that our experiment is binomial. This means there are only two possible outcomes, success or failure. Any binomial experiment can be written in the form π is
approximately equal to the binomial of π comma π , where π is the number of trials, and π is the probability of success. Weβ re told that the spinner is spun 10 times. Therefore, π is
equal to 10.
The experiment is said to be successful if the top score is achieved. There are eight equal sections on the spinner, two of which have the top score of 100. This means that the probability is equal
to two out of eight, or two-eighths. This is equivalent to one-quarter. For the purposes of this question, weβ ll use the decimal equivalent to one-quarter, which is 0.25. Our value of π is 10.
And our value of π is 0.25.
In order to answer the two questions, the probability that π equals two and the probability that π equals nine, we need to recall one of our formulae. The probability that π equals π is
equal to π choose π multiplied by π to the power of π multiplied by one minus π to the power of π minus π . The probability that π equals two is, therefore, equal to 10 choose
two multiplied by 0.25 squared multiplied by 0.75 to the power of eight. We get the final term as one minus 0.25 is 0.75, and 10 minus two is equal to eight.
We can type this directly into our calculator by using the π choose π button. This is equal to 0.2815675 and so on. As we were asked to give our answer as a percentage, we need to multiply this
by 100. This moves all the digits two places to the left. We have 28.15675 and so on. We were also asked to round our answer to three decimal places. This means that the deciding number is the seven.
As this is greater than five, we will round up. The probability that π is equal to two is 28.157 percent.
We repeat this process for the second part of the question. This time, instead of the probability that π equals two, we need to calculate the probability that π is equal to nine. We begin with
10 choose nine. We need to multiply this by 0.25 to the power of nine. We then multiply this by 0.75 to the power of one. Typing this into the calculator gives us 0.000028610. Once again, to work out
a percentage, we multiply by 100. This gives us 0.0028610 and so on. Rounding to three decimal places, the eight will be the deciding number. Once again, as this is bigger than five, we will round
up. The probability that π equals nine, written as a percentage to three decimal places, is 0.003 percent. | {"url":"https://www.nagwa.com/en/videos/972196262032/","timestamp":"2024-11-12T07:10:13Z","content_type":"text/html","content_length":"245843","record_id":"<urn:uuid:416d3478-c612-4269-9602-2d7cfea0cc9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00446.warc.gz"} |
5 Best Ways to Find if a Neat Arrangement of Cups and Shelves Can Be Made in Python
π ‘ Problem Formulation: You’ve got a collection of cups of different sizes and a set of shelves. The challenge is to write a Python program that can determine if there’s a way to arrange all cups
neatly on the shelves without any overhang. Input would be a list of cup sizes (for example, radii) and lengths of shelves, while the desired output is a boolean indicating whether a neat arrangement
is possible.
Method 1: Greedy Algorithm
This method involves arranging the cups in a non-decreasing order of their sizes, attempting to place the largest cup that can fit onto each shelf in sequence. The function checks if all cups can be
placed within the shelf constraints. This approach is often efficient but may not always find the optimal solution due to its ‘greedy’ nature of local optimization.
Here’s an example:
def canArrangeCups(shelves, cups):
for cup in cups:
placed = False
for i in range(len(shelves)):
if cup <= shelves[i]:
shelves[i] -= cup
placed = True
if not placed:
return False
return True
shelves = [10, 10, 10]
cups = [5, 5, 5, 5, 5, 5]
print(canArrangeCups(shelves, cups))
Output: True
This code defines a function canArrangeCups() that takes two arguments: shelves and cups. It sorts the cups, then iteratively tries to place each cup on a shelf. If the cup fits, it subtracts its
size from the shelf space. It returns False if any cup cannot be placed; otherwise, it returns True.
Method 2: Dynamic Programming
Dynamic programming is a method which solves complex problems by breaking them down into simpler subproblems. It is particularly well-suited for optimization problems like arranging cups on shelves.
In our case, we create a matrix to keep track of the arrangements and systematically explore possible solutions, remembering past decisions to avoid redundant calculations.
Here’s an example:
# This example would be significantly more complex and require a more detailed snippet.
# The implementation details have been omitted for brevity and because a dynamic programming
# approach for this particular problem can be quite intricate and specific.
Output: Depends on implementation and input.
Dynamic programming would involve creating structures to memorize whether a particular arrangement is possible with a subset of cups. This explanation and the code snippet are simplified due to the
complexity of properly implementing dynamic programming for this problem.
Method 3: Backtracking
Backtracking is a refinement of the brute force approach, which systematically searches for a solution by trying out all possible configurations and abandoning a configuration as soon as it is
determined that the configuration cannot yield a solution. This method is suitable for problems with many potential solutions, including the cups and shelves problem.
Here’s an example:
def canArrangeCupsRec(shelves, cups, index=0):
if index == len(cups):
return True
for i in range(len(shelves)):
if shelves[i] >= cups[index]:
shelves[i] -= cups[index]
if canArrangeCupsRec(shelves, cups, index + 1):
return True
shelves[i] += cups[index]
return False
print(canArrangeCupsRec([10, 10, 10], [5, 5, 5, 5, 5, 5]))
Output: True
This code snippet uses recursion to try out different cup placements starting at the first cup and recursively placing the rest. If a placement of a cup leads to no possible arrangements, it
backtracks and tries a different position for the cup. The function returns True if it finds a satisfactory arrangement of all cups.
Method 4: Binary Search with Sorting
Applying a binary search in combination with sorting can be an efficient means of finding a neat arrangement of cups on shelves. The idea is to sort the shelves and use binary search to find the
right shelf for each cup, ensuring the arrangement attempts to use the least possible space while maintaining order.
Here’s an example:
# As with the Dynamic Programming example, a detailed implementation is omitted here.
# Binary search requires sorted input and careful tracking of indices, which would make the
# code example overly lengthy and complex for this format.
Output: Depends on implementation and input.
Using binary search with sorted shelves can significantly reduce the search space when trying to find a place for each cup. Because of the search nature, it may not guarantee finding the most optimal
solution but will provide a good solution if it exists within a fast timeframe.
Bonus One-Liner Method 5: Constraint Satisfaction Solver
Python’s constraint programming libraries such as python-constraint can solve the problem by treating it as a constraint satisfaction problem. Each shelf is a variable, and each cup size imposes a
constraint. The solver tries to assign values (cups) to variables (shelves) without violating constraints.
Here’s an example:
# This would require installing the python-constraint library and setting up a set of variables
# and constraints that match the problem, which is quite involved and not suitable for a one-liner
# example.
Output: Depends on library use and problem setup.
The constraint satisfaction library can handle the complexity of various problem instances automatically, which saves development time but requires understanding of how to model the problem within
the framework of the chosen library. It’s powerful but is not a one-size-fits-all solution.
• Method 1: Greedy Algorithm. Simple and fast. May not find the most optimal solution.
• Method 2: Dynamic Programming. Optimal for certain cases. Can be complex and overkill for simpler instances.
• Method 3: Backtracking. Thorough but can be slow. Works well for problems with a high number of potential solutions.
• Method 4: Binary Search with Sorting. Efficient time complexity. Less suitable for unsorted inputs or where the most optimal solution is required.
• Method 5: Constraint Satisfaction Solver. Very powerful. Requires a deeper understanding of the problem to model as constraints. | {"url":"https://blog.finxter.com/5-best-ways-to-find-if-a-neat-arrangement-of-cups-and-shelves-can-be-made-in-python/","timestamp":"2024-11-07T23:32:50Z","content_type":"text/html","content_length":"72609","record_id":"<urn:uuid:0e5c112c-4486-4452-87ad-4a079e0f5de8>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00883.warc.gz"} |
Anthony has an income of $10,000 this year, and he expects an
income of $5,000 next...
Anthony has an income of $10,000 this year, and he expects an income of $5,000 next...
Anthony has an income of $10,000 this year, and he expects an income of $5,000 next year. He can borrow and lend money at an interest rate of 10%.
Consumption goods cost $1 per unit this year and there is no inflation.
a. What is the present value of his endowment?
b. What is the future value of his endowment?
c. Write an equation to represent his budget set. Graph his budget set? Label it well.
d. If his utility function is U(c1, c2)=4ln(c1)+2ln(c2) , how much will he consume in each period?
e. How would his utility change if the interest rate goes up to 15%? Is he better off or worse off? Explain.
f. What about if there is a 10% inflation? Show how his budget constraint and his utility changes with a graph. A simple illustration is fine.
g.Graph his budget constraint and find his optimum bundle if the interest rate to borrow is 15% but return to his savings is 10% with no inflation.
h. Discuss the importance of financial markets and how they can improve our utility. | {"url":"https://justaaa.com/economics/174733-anthony-has-an-income-of-10000-this-year-and-he","timestamp":"2024-11-03T07:35:51Z","content_type":"text/html","content_length":"41510","record_id":"<urn:uuid:6c97235c-2313-401b-9db3-cc4cf2bf3b41>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00847.warc.gz"} |
A Tutorial on Geometric Programming -
A Tutorial on Geometric Programming
Getting started with Geometric Programming is a great way to understand this complex technique. This article will explain the basics of Geometric Programming, the process of solving optimization and
convex optimization problems, and some applications of the technique. You can also see some of the applications of Geometric Programming in engineering. Then, you can use it to solve optimization
problems that are difficult to solve using conventional methods. Here are some examples of problems that you can solve using Geometric Programming.
Convex optimization
Convex optimization is a useful tool in solving linear systems involving constraints. The geometric programming-based method can be applied to cone-preserving linear systems as well. Its primary
advantage is that it is easy to implement. Here are some examples of convex problems. In addition, we will discuss the use of convex optimization in solving these problems. After learning the basics
of convex optimization, you will be able to apply the method to other types of problems.
A convex GP has a certain coercive property: the origin is in the interior of a convex hull. Its negative counterpart, the coercive condition, is difficult to apply in practice. A simple example of
this property is d2h(y), which is a positive definite function. In practice, the corresponding convex function (a) is a subset of a given convex hull. Hence, all convex functions satisfy this
A basic introduction to convex optimization in geometric programming includes a discussion of how to write a posynomial problem. A posynomial geometric programming problem is not convex in its
standard form. Generally, a general nonlinear solver will fail to solve it. However, in some cases, an exponential cone program is valid. To solve this problem, YALMIP has built-in support for the
logarithmic variable transformation. The manual also outlines an example of how to solve posynomial geometric programming.
Geometric programming
Geometric programming has three distinct phases. It was first developed as a novel approach to engineering problems, supplying closed-form sensitivity analysis. Its theory was extended to signomial
and generalized geometric programming and applications began to appear in science, engineering and business journals. Legendre duality unifies geometric programming with other methods for solving
nonconvex optimization problems. It also offers a convenient solution to many applications requiring numerical analysis.
In its simplest form, geometric programming is the solution of optimization problems. Problems involving generating optimal functions can be solved using GPs. The method uses a combination of
methods, each of which relies on its own constraints and conditions. In addition to solving optimization problems, it is useful in other areas of engineering, mathematics, statistics, and electrical
circuit design. Here is a quick tutorial to geometric programming:
The most common application of geometric programming is in the solution of the robust stabilization problem. Hence, a geometric program is a function of the parameters pk. This property allows a
program to identify a feasible uncertainty matrix with the maximum possible size. Moreover, it can solve the problem of synthesis of switched posi tive systems. The resulting solution has several
common features with positive linear systems. It is also an important tool in the study of control systems.
In this chapter, we will explore applications of geometric programming, including the economic interpretation of duality, transformations of optimization problems, and extensions to posynomial
geometric programming. We will also discuss applications in economics and management science. Listed below are a few examples of applications of geometric programming. They are largely
self-explanatory and will be useful to anyone wishing to analyze complex problems with geometric programming. To learn more, please visit the authors’ websites.
The real-world applications of geometric programming include energy control, impurities concentration, logistics, and calculation of reticular steel structures. In addition to these, some recent
studies have explored the use of geometric programming to optimize nonlinear systems, including transportation, acoustics, and inverse problems. While these are only a few examples, Geometric
Programming can be used in many areas. Here are some of the most important examples.
Geometric programming was introduced in 1967 by Duffin, Peterson, and Zener. It is used to solve a wide variety of optimization problems, including those that have high dimensionality and models that
are well approximated by power laws. While its primary purpose is for financial modeling, geometric programming can also be used in a variety of practical applications in electrical circuit design,
finance, statistics, and geometric design. In fact, geometric programming is widely used in these applications. | {"url":"https://programmingtutorial.org/a-tutorial-on-geometric-programming/","timestamp":"2024-11-12T19:15:37Z","content_type":"text/html","content_length":"44894","record_id":"<urn:uuid:1db9c147-45c7-4429-96dd-29c2909a8f19>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00307.warc.gz"} |
Electrostatic topic,pleasegive the answer with explanation
AIPMT> Electrostatic topic, please give the answ...
1 Answers
Vasanth Chavan
Last Activity: 6 Years ago
Electrostatics is the study of electromagnetic phenomena that occur when there are no moving charges—i.e., after a static equilibrium has been established. Charges reach their equilibrium positions
rapidly because the electric force is extremely strong. The mathematical methods of electrostatics make it possible to calculate the distributions of the electric field and of the electric potential
from a known configuration of charges, conductors, and insulators. Conversely, given a set of conductors with known potentials, it is possible to calculate electric fields in regions between the
conductors and to determine the charge distribution on the surface of the conductors. The electric energy of a set of charges at rest can be viewed from the standpoint of the work required to
assemble the charges; alternatively, the energy also can be considered to reside in the electric field produced by this assembly of charges. Finally, energy can be stored in a capacitor; the energy
required to charge such a device is stored in it as electrostatic energy of the electric field.Coulomb’s lawStatic electricity is a familiar electric phenomenon in which charged particles are
transferred from one body to another. For example, if two objects are rubbed together, especially if the objects are insulators and the surrounding air is dry, the objects acquire equal and opposite
charges and an attractive force develops between them. The object that loses electrons becomes positively charged, and the other becomes negatively charged. The force is simply the attraction between
charges of opposite sign. The properties of this force were described above; they are incorporated in the mathematical relationship known as Coulomb’s law. The electric force on a charge Q1 under
these conditions, due to a charge Q2 at a distance r, is given by Coulomb’s law,Explanation of static electricity and its manifestations in everyday life.Explanation of static electricity and its
manifestations in everyday life.Encyclopædia Britannica, Inc.Equation.ADVERTISEMENTThe bold characters in the equation indicate the vector nature of the force, and the unit vector r̂ is a vector that
has a size of one and that points from charge Q2 to charge Q1. The proportionality constant k equals 10−7c2, where c is the speed of light in a vacuum; k has the numerical value of 8.99 × 109
newtons-square metre per coulomb squared (Nm2/C2). Figure 1 shows the force on Q1 due to Q2. A numerical example will help to illustrate this force. Both Q1 and Q2 are chosen arbitrarily to be
positive charges, each with a magnitude of 10−6 coulomb. The charge Q1 is located at coordinates x, y, z with values of 0.03, 0, 0, respectively, while Q2 has coordinates 0, 0.04, 0. All coordinates
are given in metres. Thus, the distance between Q1 and Q2 is 0.05 metre.Figure 1: Electric force between two charges (see text).Figure 1: Electric force between two charges (see text).Courtesy of the
Department of Physics and Astronomy, Michigan State UniversityThe magnitude of the force F on charge Q1 as calculated using equation (1) is 3.6 newtons; its direction is shown in Figure 1. The force
on Q2 due to Q1 is −F, which also has a magnitude of 3.6 newtons; its direction, however, is opposite to that of F. The force F can be expressed in terms of its components along the x and y axes,
since the force vector lies in the xy plane. This is done with elementary trigonometry from the geometry of Figure 1, and the results are shown in Figure 2. Thus,Equation.in newtons. Coulomb’s law
describes mathematically the properties of the electric force between charges at rest. If the charges have opposite signs, the force would be attractive; the attraction would be indicated in equation
(1) by the negative coefficient of the unit vector r̂. Thus, the electric force on Q1 would have a direction opposite to the unit vector r̂ and would point from Q1 to Q2. In Cartesian coordinates, this
would result in a change of the signs of both the x and y components of the force in equation (2).Equation.How can this electric force on Q1 be understood? Fundamentally, the force is due to the
presence of an electric field at the position of Q1. The field is caused by the second charge Q2 and has a magnitude proportional to the size of Q2. In interacting with this field, the first charge
some distance away is either attracted to or repelled from the second charge, depending on the sign of the first charge.
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Area Explained
Unit: Square metre [m<sup>2</sup>]
Basequantities: 1 m^2
Symbols: A or S
Dimension: wikidata
Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface
or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint
necessary to cover the surface with a single coat.^[1] It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional
concept).Two different regions may have the same area (as in squaring the circle); by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m^2), which
is the area of a square whose sides are one metre long.^[2] A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to
have area one, and the area of any other shape or surface is a dimensionless real number.
There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon
into triangles.^[3] For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the
historical development of calculus.^[4]
For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area.^[1] ^[5] ^[6] Formulas for the surface areas of simple shapes were computed by the
ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.
Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic
property of surfaces in differential geometry.^[7] In analysis, the area of a subset of the plane is defined using Lebesgue measure,^[8] though not every subset is measurable if one supposes the
axiom of choice.^[9] In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.^[1]
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.
Formal definition
See also: Jordan measure. An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed
measurable sets) to the set of real numbers, which satisfies the following properties:^[10]
• For all S in M, .
• If S and T are in M then so are and, and also .
• If S and T are in M with then is in M and .
• If a set S is in M and S is congruent to T then T is also in M and .
• Every rectangle R is in M. If the rectangle has length h and breadth k then .
• Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. . If there is a unique number c such
that for all such step regions S and T, then .
It can be proved that such an area function actually exists.^[11]
Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m^2), square centimetres (cm^2), square
millimetres (mm^2), square kilometres (km^2), square feet (ft^2), square yards (yd^2), square miles (mi^2), and so forth. Algebraically, these units can be thought of as the squares of the
corresponding length units.
The SI unit of area is the square metre, which is considered an SI derived unit.^[2]
Calculation of the area of a square whose length and width are 1 metre would be:
1 metre × 1 metre = 1 m^2
and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as:
3 metres × 2 metres = 6 m^2. This is equivalent to 6 million square millimetres. Other useful conversions are:
• 1 square kilometre = 1,000,000 square metres
• 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres
• 1 square centimetre = 100 square millimetres.
Non-metric units
In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units.
1 foot = 12 inches,the relationship between square feet and square inches is
1 square foot = 144 square inches,where 144 = 12^2 = 12 × 12. Similarly:
• 1 square yard = 9 square feet
• 1 square mile = 3,097,600 square yards = 27,878,400 square feet
In addition, conversion factors include:
• 1 square inch = 6.4516 square centimetres
• 1 square foot = square metres
• 1 square yard = square metres
• 1 square mile = square kilometres
Other units including historical
There are several other common units for area. The are was the original unit of area in the metric system, with:
• 1 are = 100 square metres
Though the are has fallen out of use, the hectare is still commonly used to measure land:
• 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres
Other uncommon metric units of area include the tetrad, the hectad, and the myriad.
The acre is also commonly used to measure land areas, where
• 1 acre = 4,840 square yards = 43,560 square feet.
An acre is approximately 40% of a hectare.
On the atomic scale, area is measured in units of barns, such that:
• 1 barn = 10^−28 square meters.
The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics.
In South Asia (mainly Indians), although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have
same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different
results, depending on what reference that has been used.^[12] ^[13] ^[14] ^[15]
Some traditional South Asian units that have fixed value:
• 1 Killa = 1 acre
• 1 Ghumaon = 1 acre
• 1 Kanal = 0.125 acre (1 acre = 8 kanal)
• 1 Decimal = 48.4 square yards
• 1 Chatak = 180 square feet
Circle area
In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature
of the lune of Hippocrates, but did not identify the constant of proportionality. Eudoxus of Cnidus, also in the 5th century BCE, also found that the area of a disk is proportional to its radius
Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used the tools of Euclidean geometry to show that the area inside
a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle. (The
circumference is 2r, and the area of a triangle is half the base times the height, yielding the area r^2 for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius
circle) with his doubling method, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon, then repeatedly doubled the
number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons).
Quadrilateral area
In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula, for the area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In
1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula, for the area of any quadrilateral.
General polygon area
The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of the surveyor's formula for the area of any polygon with known vertex locations by Gauss in
the 19th century.
Areas determined using calculus
The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas
of various curved three-dimensional objects.
Area formulas
Polygon formulas
See main article: article.
=0, 1, ...,
-1) of whose
n vertices
are known, the area is given by the
surveyor's formula
where when i=n-1, then i+1 is expressed as modulus n and so refers to 0.
The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length and width, the formula for the area is:^[18]
(rectangle).That is, the area of the rectangle is the length multiplied by the width. As a special case, as in the case of a square, the area of a square with side length is given by the formula:^[1]
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before
arithmetic, this formula can be used to define multiplication of real numbers.
Dissection, parallelograms, and triangles
See main article: article and Triangle area.
Most other simple formulas for area follow from the method of dissection.This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.For an example, any
parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a
rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:^[18]
(parallelogram).However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half
the area of the parallelogram:^[18]
(triangle).Similar arguments can be used to find area formulas for the
as well as more complicated
Area of curved shapes
See main article: article and Area of a circle. The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk) is based on a similar method. Given a
circle of radius, it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form
an approximate parallelogram. The height of this parallelogram is, and the width is half the circumference of the circle, or . Thus, the total area of the circle is :^[18]
(circle).Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the
approximate parallelograms is exactly, which is the area of the circle.
This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now
recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral:
See main article: article. The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes and the formula is:^[18]
Non-planar surface area
See main article: article and Surface area.
Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces). For example, if the side surface of a cylinder (or any prism) is cut
lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone, the side surface can be flattened out into a sector of a circle, and the resulting
area computed.
The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was
first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is:^[5]
(sphere), where is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus.
General formulas
Areas of 2-dimensional figures
is any side, and
is the distance from the line on which
lies to the other vertex of the triangle). This formula can be used if the height
is known. If the lengths of the three sides are known then
Heron's formula
can be used:
are the sides of the triangle, and
is half of its perimeter.
If an angle and its two included sides are given, the area is
where is the given angle and and are its included sides.
If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of
. This formula is also known as the
shoelace formula
and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points
, and
. The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use
to find the area.
• A simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points:
, where
is the number of grid points inside the polygon and
is the number of boundary points. This result is known as
Pick's theorem
Area in calculus
• The area between a positive-valued curve and the horizontal axis, measured between two values a and b (b is defined as the larger of the two values) on the horizontal axis, is given by the
integral from a to b of the function that represents the curve:^[1]
is the curve with the greater y-value.
• An area bounded by a function
expressed in
polar coordinates
with endpoints
is given by the
line integral
or the z-component of
(For details, see .) This is the principle of the planimeter mechanical device.
Bounded area between two quadratic functions
To find the bounded area between two quadratic functions, we first subtract one from the other, writing the difference as$f(x)-g(x)=ax^2+bx+c=a(x-\alpha)(x-\beta)$where f(x) is the quadratic upper
bound and g(x) is the quadratic lower bound. By the area integral formulas above and Vieta's formula, we can obtain that^[20] ^[21] $A=\frac=\frac(\beta-\alpha)^3,\qquad aeq0.$The above remains valid
if one of the bounding functions is linear instead of quadratic.
Surface area of 3-dimensional figures
, where
is the radius of the circular base, and
is the height. That can also be rewritten as
is the radius and
is the slant height of the cone.
is the base area while
is the lateral surface area of the cone.
, where
is the length of an edge.
, where
is the radius of a base and
is the height. The
can also be rewritten as
, where
is the diameter.
, where
is the area of a base,
is the perimeter of a base, and
is the height of the prism.
, where
is the area of the base,
is the perimeter of the base, and
is the length of the slant.
, where
is the length,
is the width, and
is the height.
General formula for surface area
The general formula for the surface area of the graph of a continuously differentiable function
is a region in the xy-plane with the smooth boundary:
D\sqrt{\left( \partialf
2+\left( \partialf
An even more general formula for the area of the graph of a
parametric surface
in the vector form
is a continuously differentiable vector function of
List of formulas
Additional common formulas for area:! Shape! Formula! Variables
Square A=s^2
Rectangle A=ab
Triangle b\sin(\gamma)
Triangle A=\sqrt{s(s-a)(s-b)(s-c)} s=\tfrac12(a+b+c)
(Heron's formula)
Isosceles triangle \sqrt{4a^2-c^2}
Regular triangle }a^2\,\!
(equilateral triangle)
Parallelogram A=ah[a]
Regular hexagon \sqrt{3}a^2
Regular octagon A=2(1+\sqrt{2})a^2
Regular polygon
( =nr^2\tan(\tfrac\pin)
sides) r:
=\tfrac{1}{2}nR^2\sin(\tfrac{2\pi}{n}) incircle radius
circumcircle radius
Circle ( 100px
Circular sector
Ellipse A=\piab
Integral f(x)dx,f(x)\ge0
Surface area
Sphere A=4\pir^2=\pid^2
Cuboid A=2(ab+ac+bc)
Cylinder A=2\pir(r+h)
(incl. bottom and top)
Cone A=\pir(r+\sqrt{r^2+h^2})
(incl. bottom)
Torus A=4\pi^2 ⋅ R ⋅ r
Surface of revolution f(x)\sqrt{1+\left[f'(x)\right]^2}dx
(rotation around the x-axis)
The above calculations show how to find the areas of many common shapes.
The areas of irregular (and thus arbitrary) polygons can be calculated using the "Surveyor's formula" (shoelace formula).^[23]
Relation of area to perimeter
The isoperimetric inequality states that, for a closed curve of length L (so the region it encloses has perimeter L) and for area A of the region that it encloses,
and equality holds if and only if the curve is a circle. Thus a circle has the largest area of any closed figure with a given perimeter.
At the other extreme, a figure with given perimeter L could have an arbitrarily small area, as illustrated by a rhombus that is "tipped over" arbitrarily far so that two of its angles are arbitrarily
close to 0° and the other two are arbitrarily close to 180°.
For a circle, the ratio of the area to the circumference (the term for the perimeter of a circle) equals half the radius r. This can be seen from the area formula πr^2 and the circumference formula 2
The area of a regular polygon is half its perimeter times the apothem (where the apothem is the distance from the center to the nearest point on any side).
Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon
resides in). But if the one-dimensional lengths of a fractal drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer.
This power is called the fractal dimension of the fractal.^[24]
Area bisectors
See main article: article.
There are an infinitude of lines that bisect the area of a triangle. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are
concurrent at the triangle's centroid; indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in
half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.
Any line through the midpoint of a parallelogram bisects the area.
All area bisectors of a circle or other ellipse go through the center, and any chords through the center bisect the area. In the case of a circle they are the diameters of the circle.
Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.
The question of the filling area of the Riemannian circle remains open.
The circle has the largest area of any two-dimensional object having the same perimeter.
A cyclic polygon (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths.
A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral.
The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.^[25]
The ratio of the area of the incircle to the area of an equilateral triangle,
}, is larger than that of any non-equilateral triangle.
The ratio of the area to the square of the perimeter of an equilateral triangle,
}, is larger than that for any other triangle.
See also
• Brahmagupta quadrilateral, a cyclic quadrilateral with integer sides, integer diagonals, and integer area.
• Heronian triangle, a triangle with integer sides and integer area.
• List of triangle inequalities
• One-seventh area triangle, an inner triangle with one-seventh the area of the reference triangle.
• Routh's theorem, a generalization of the one-seventh area triangle.
• Orders of magnitude—A list of areas by size.
• Derivation of the formula of a pentagon
• Planimeter, an instrument for measuring small areas, e.g. on maps.
• Area of a convex quadrilateral
• Robbins pentagon, a cyclic pentagon whose side lengths and area are all rational numbers.
Notes and References | {"url":"https://everything.explained.today/Area/","timestamp":"2024-11-05T07:11:17Z","content_type":"text/html","content_length":"89526","record_id":"<urn:uuid:bdf7c846-f3e9-457e-aa58-a39410a7fe46>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00793.warc.gz"} |
How to Perform Logistic Luxembourg Phone Number List
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Luxembourg Phone Number List Predicting a bank default using past transaction details is an example of logistic regression, and continuous output such as a stock market score is an example of linear
regression. Uses The following are uses where we can use logistic regression. Weather forecast Weather forecasts are the result of logical regression. Here, we analyze data from previous weather
reports and predict the possible outcome for a particular day. However, a logical regression would only predict categorical data, for example, if they are lost or not. Detection of the disease .
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In accordance with the 7th Commandment of Blogging (If thy comment exceedeth two cubits in length, thou shalt write thine own damn post.), here is my personal response to Dan's latest question. To
be clear: even though the question was put to Mathalicious generally, and even though I bodily occupy a nontrivial fraction of that particular organization, it would be presumptuous of me to write
anything approaching an official opinion, especially given the humbling brains attached to the people I spend my days with. But I have thoughts.
We do thump the real-world drum pretty steadily around the office. We have, as it's known in the biz, a niche. But what exactly constitutes 'real-world' is an interesting question. I don't think
it's a particularly important question, but it's interesting insofar as it informs the practical decisions I make about what kinds of tasks I try to author, and what kinds of tasks I leave to other
smart people, in other well-appointed niches. And insofar as that term appears with some regularity in the CCSS.
A Line in the Sand
There's a philosophically defensible sense in which nothing we'd call a mathematical object is real (or, for that matter, an object). They're abstract and causally independent and yadda yadda
yadda. There's another philosophically defensible sense in which everything we'd call a mathematical object is real. Actually, a couple of senses, with varying definitions of reality. And of
course there are positions in between. You could spend lifetime trying to untangle all the competing ideas of what (or whether) a number is. And basically who cares. In my mind, there's a simple
way to draw a (note: not the) line on the curricular map between That Which Is Real and That Which Is Not:
Is this question self-referential?
In other words, are we using math to examine itself, or are we using it to inspect something outside its own borders? So my working definition of 'real-world' math is a mathematical task that is not
self-referential. Which means, and I suppose you saw this coming, that my answer to Dan's question is none of the above. None of those problems is real-world in any appreciable sense. They are all
questions about a circle, a square, and their respective areas: math looking at math. A and B are obvious. C and D are just promising hypothetical candy (which is the absolute worst kind of candy)
for solving A or B. I suppose E and F are both dipping their toes into the real-world, but self-consciously.
Okay, so those are some counterexamples. Maybe a countercounterexample will help shed light on my litmus test.
For Instance...
Here are two problems:
1. Is it true in general that P(A|B) = P(B|A)? If not, can you express P(A|B) in terms of P(B|A)?
2. Should innocent people be worried about the NSA's PRISM program? How much should they be worried?
By my definition, the second question is 'real-world' and the first one is not, because the first question is mathematically self-referential (it's a question about a mathematical relationship
phrased in mathematical language) and the second one is not (it's a question about personal liberty and national security phrased in natural language). Of course they are the same question, and they
are both excellent. But from my chair, the very fact that those questions are the same is so non-obvious that connecting them requires a profound act of mathematical thinking. Also, you get to do
some really good math qua math. I find that both professionally compelling and pedagogically useful. That's why I do what I do. Without speaking too forcefully for the rest of the team, I think
that's part of the reason we do what we do.
Unpacking Circles
As I understand Dan's position (or at least one particular aspect of Dan's position), there's no reason to create a distinction between, e.g., a circle's reality and the reality of health
insurance. In fact, for kids, a circle may be real (Platonist objections notwithstanding) in a much clearer and more visceral way. And from there it's not a particularly ambitious leap to extend
this reasoning such that all of mathematics can be considered practically real to human beings living in a world that includes mathematics. I think that's about right.
But I also think it's valuable to make just this sort of distinction from time to time. Learning mathematics (or maybe just learning) has a lot to do with forming connections. You can know
something about addition. You can know something about subtraction. But when you --- a much younger you --- begin to wrap your head around the connection between the two operations, important
things are happening. And a connection's impact on understanding is inversely related to its obviousness. The guy who understands the connection between multiplication and division has learned an
important thing. The guy who understands the connection between the zeros of a complex function and the distribution of prime numbers has revolutionized an entire field.
I'm personally interested in helping people make non-obvious connections. There are lots of good ways to do that, and we as educators should pursue all of them, but one way is to connect clearly
mathematical ideas to questions that are not clearly mathematical in scope, viz., ask real-world (as I've defined it) questions.
Silver Bullet
So essentially my job boils down to finding interesting non-mathematical questions that are isomorphic to interesting mathematical questions, but not obviously so. (How's that for a resume bullet!)
I've already touched on why I think non-obviousness is important, but there's another reason: when the contextual link is trivial, the question generally becomes terrible. And it's really, really
easy to create trivial links. And then the real-world problem is no longer isomorphic; rather it becomes both substantially identical to and superficially uglier than the original problem, which is
unproductive. It's easy to pour a thin candy shell of context that does nothing to conceal the shape of the underlying problem, or to improve its flavor. The real world can definitely ruin a task
if your only goal is to incorporate something --- anything --- non-mathematical because for some reason you're afraid to ask a math question about math.
And in that sense I understand the impetus behind the 'fake-world math' backlash, because there's a certain amount of extant conviction that slapping the 'real-world' label on something magically
confers awesomeness...which it certainly does not. Such wanton slapping can also make it seem as though it's somehow desirable to avoid mentioning mathematics while teaching it, which is a lousy way
to treat of such a rich subject, and rather unsubtly suggests that math is unpalatable on its own. We should, as a community, take the position that a poorly executed idea ought to be avoided. We
should question the circumstances and mechanisms that lead to poor execution. I also think we shouldn't dismiss the good idea outright. The real world isn't a silver bullet, but it's a perfectly
good bullet to have in the magazine.
P vs. NP
I mentioned supra that, while I find this question of to be intellectually interesting, I don't think it's especially important. Mostly because what I'm interested in, globally, are great math
tasks, and the greatness of a task is independent of whether it's situated in- or outside of the real world, however we choose to limn it. There is only the illusion of dichotomy here. I drew my
own personal line, and I chose to work on one side of it because I'm partial to the view from over here. But I also realize that the work I do at Mathalicious represents a small (though valuable)
part of the mathematical experience students should have.
I think that work like mine and work like Dan's approach the same target from essentially opposite directions. Dan is trying to reify mathematics by treating it as a properly first-class citizen in
the world as we know it. I'm trying to expand mathematical thinking to comprise those parts of the world we may not realize are already within its purview. Somewhere in the middle we create a
situation in which mathematics and the real world end up occupying essentially the same space. Isn't that what we're all doing? I hope so. I'd really like that world.
How to Think
Teach me how to think. Better, teach me how to teach someone to think. It's my job, after all. And once you've done that, imagine all the sparkling inspective instruments we can set upon the
world, keen at all the right edges. A whole new generation of thinkers. Is there anything more beautiful? Did you shiver a a hopeful little shiver just now? Because this is the kind of bullshit I
have no patience for. As if we weren't already, the both of us. Thinking.
This story turns up at least once a week in my Twitter feed --- you know the one --- wherein the true value of some or other thing isn't so much about the thing itself, but about how the thing helps
students learn how to think. Or worse, become thinkers. A story that always smells faintly of parable.
But people like research, so let me lay some on you. The absolute best predictors of student thinking are respiration, metabolism, and excretion. Everything else is house money.
That should be an incredible relief. I mean, wouldn't you feel just a little daunted at the prospect of having to jump-start an inert lump of organic matter every Monday? Of having that as your
moral imperative and professional obligation? Some days I couldn't even find my purple dry erase marker.
What we don't and can't do is teach our students to think. Let's not insult them. What we do is help them learn to pay attention to the myriad little tics and habits that attend thoughtfulness. To
be aware of the shape and sensation of their own cognition. To be mindful of their rich internal voices. We don't teach thinking. Ever. On our best days, we encourage introspection.
If that's not persuasive, I humbly suggest an experiment. Want to see someone squeal his emotional tires? I mean really spin? Imply that he's failing to control his own brain. Suggest that
something is broken at his locus of fundamental humanity. Get your face right up next to the spot that provides maybe the only reassurance of his own corporal existence and declare it unsound.
Then stand back.
Keys to a Rubbled Kingdom
Let's acknowledge, at the outset, that it's basically impossible to talk about one's wartime experience without sounding like a prick. If your stories are too exciting, then you're bragging/
embellishing/outright lying, which is prickish on its face. If your stories aren't exciting enough, then you're being modest — probably falsely so — which is even worse, because not only are you
bragging in some implicit, backhanded way, but you're also denying the listener his conventional opportunity for the minor act of hero worship that is fast becoming the only way for a population
almost entirely divorced from two decade-long wars to connect with the alien minority that has shouldered their weight. And should this lose-lose proposition be too exhausting to navigate, or should
you have a headache, or should you have recently scraped the roof of your mouth on some weapons grade Cap'n Crunch, or for any reason at all, really, should you have the balls to actually utter the
phrase, "I don't want to talk about it," well then you had better have at least a Silver Star and some visible scarring to back that up, otherwise you are the biggest prick on record. Who do you
think you are?
But, in spite of all that, I'm going to talk about my wartime experience anyway — such as it is — because the news about Fallujah falling back into chaos has affected me in a way I didn't expect.
And, because this is the only outlet I have at my disposal, I will dispose of it. But this isn't really about me. Just indulge me for a moment.
My War in Six Paragraphs
First, allow me to lower your expectations. My personal participation in the war was approximately as ordinary as war-type participation can be. I was an artillery officer in the Marine Corps, a
young second lieutenant with Battery G, 2nd Battalion, 11th Marines, whose main job was running the Fire Direction Center, a gig that mostly involved figuring out ways to get hundred-pound bullets
fired from great big cannons to land in tactically advantageous places. For seven months in 2006 I sat inside a bunkered-in trailer just outside of Fallujah and waited for people to shoot rockets
and mortars at us. When they did, a whole slew of very expensive radar devices would calculate the point of origin, and after a little bit of math we would tell the guns which way to point and how
much powder to use, &c., and soon they would be booming like mad trying to thunk the guys who wanted us dead. It was all very loud and exciting for a few minutes out of every day. One of the ways
you could recognize new people around Camp Fallujah was to see who ducked when the artillery started up; if you couldn't tell the good booming from the bad, boy did you look foolish. It was a source
of constant entertainment.
I said thunk back there instead of murder, which is what I meant. We're all adults here.
How often were we successful? Honestly, I don't know. Radar devices, no matter how expensive, are lousy at picking up corpses. But my Marines were so fast — they could pump four rounds through a
gun before the first one hit the ground — and there are only so many ways to avoid supersonic steel in the middle of the open desert. Plus, I've always been pretty good at math. It's a small, mean
way to feel, hoping you have murdered someone, to have been an aspiring thunker of men. But the mathematician in me will say, definitively, we killed more than one person. I can't give you any more
significant figures than that.
Besides sitting around and waiting for opportunities to do very intense math problems, I took precisely three convoys between Fallujah and the air base in Al Taqaddum. The first one was simply to
get to our new home in Fallujah after flying in from Kuwait. I sat in the back of an up-armored 7-ton in the middle of the night and scrunched myself up mentally into a tiny corner of the universe
as a precaution against being exploded, which must have worked. When we pulled into the city an hour before dawn, thousands of rays of light spilled from the thousands of bullet holes in every
structure we passed. I unscrunched myself long enough to wonder at the spillage of so much light. All I could think was, Man, we fucked this place up. Of course that's the majestic plural. I
wasn't there for that part — for all of the placing of bullet holes in structures. At the time, this upset me greatly.
The other two convoys were part of a round-trip to pick up some new electronics, a job for which I volunteered. The new equipment we were to acquire was for jamming radio signals so that the
insurgents couldn't to use them to blow up any of the shit piled alongside seemingly every inch of road in Anbar Province. I thought it would be ridiculous for someone else to die on the way to or
from picking up gizmos intended to keep us from getting killed, and I didn't want that on my conscience. Also, I was starting to get tired of doing arithmetic while there were all these perfectly
good roadside bombs left unexploded by my absence. That's another strange feeling, wanting to get thunked — but not too severely.
I didn't even have to make the final drive out to Taqaddum on the way back home. The Army loaned us some helicopters for the trip, which was awfully swell of them. It would have been embarrassing
for the insurgents to blow us up while we were on our way out the door — which is what they wanted anyway. The Blackhawks helped us all to avoid that little misunderstanding.
Here's the most traumatic bit. When my part of the war was over, I had to fill out a Post-Deployment Health Assessment Questionnaire. One of the questions that ostensibly aided in the assessment of
my post-deployment health was, How often did you feel that your life was in danger? Because the bad guys weren't so good at math, I had to fill in the NEVER circle. I thought about filling in the
OCCASIONALLY circle, but it would have been a stretch. I got to put DAILY next the the question about exposure to loud noises, but it's not the same thing.
NEVER. What a shameful thing for a war veteran to have to mark on an official government form.
The Lede, Sufficiently Buried
While the Marines (re)captured Fallujah in 2004 — house by house, and sometimes room by room — from an entrenched enemy with no designs on survival, I was still at The Basic School in Quantico, VA.
I sat in the chow hall every day, staring at the table reserved for pictures of the young officers and enlisted instructors killed in combat since the start of the war. When I arrived in June, there
were a handful of solemn framed faces; by the time I checked out, just before Christmas and the end of Phantom Fury, we were rearranging the furniture to make room for a fifth KIA table. I listened
to the hard-earned lessons of the survivors, sometimes only days removed from the fighting, that I might not have to pay so dearly for my education. And then, a few months later, I was handed the
keys to their rubbled kingdom, mortgaged — in the literal sense of the word — in blood.
I know you don't give a shit about Fallujah. It's okay, it doesn't make you a bad person. It's just one of the many miserable places in the world that has nothing to do with anything anymore. But
for a little while it was mine. I didn't take Fallujah, which would have made it important to me. I inherited Fallujah, which makes it sacred.
There's no earthly reason I should be upset that the city is again in disarray. I didn't go because I thought we were going to solve the problems of the Iraqi people. I sure as hell didn't go to
defend my country. (Rusty mortars only fly so far.) I went because that's what you do for the dead. You keep the things they give you.
And for that, I'm so sorry. To the faces on those tables. God knows how many tables now. To their mothers and fathers, to their children and widows, I'm sorry. That's the only decent thing to be
said. And it is, like all gestures and redresses born of human loss, completely insignificant.
Pretty Big Ideas
So Grant Wiggins threw down the gauntlet. And Patrick Honner, as Patrick Honner is wont to do, picked it up. And then Grant Wiggins -- I'm not totally sure what traditionally happens to a gauntlet
at this point -- did some other gauntlet-related thing in reply. It was fast and furious. Actually, it was incredibly civil and well considered, which spirit I will try to preserve here.
For the tl;dr crowd, Wiggins posted a celebratory 100th-blog-post rant against Algebra (the course, not the content). In that post he challenged Algebra teachers to name Four Big Ideas contained in
the curriculum. And Honner responded with some pretty solid candidates:
1. Algebraic Structure
2. Binary Relations
3. The Cartesian Plane
4. Function
Wiggins quasi-stipulates to a couple (binary relation, Cartesian Plane) -- with much qualification -- and more or less dismisses the others (quite politely). It seems that Wiggins agrees all those
ideas are important, but he has a very particular notion of what makes for a Big Idea:
So, I wish to up the ante. To me a big idea is big for both: I am looking for those ideas that are big – powerful and fecund – for both novice and expert.
I always return to this simple example from soccer: Create dangerous space on offense; collapse dangerous space on defense is a big idea at every level of the game, from kid to pro. And it is
transferrable to all space-conquest sports like lacrosse, hockey, and basketball. Truly big. [emphasis in original]
I'm not interested in debating the general lifelessness of high school Algebra, which strikes me as largely uncontroversial at this point in the conversation. In fact, the same rant could just as
easily be applied to mathematical instruction at almost any level (Wiggins even includes the obligatory quotation from A Mathematician's Lament). I'm also not interested in trying to produce more
convincing examples. Nope, I'm interested in talking about soccer.
I think the soccer analogy is on the verge of making it impossible to have a meaningful discussion about math education. Not just Algebra, but mathematics. And here's why: the creating and
collapsing of dangerous space might be the only Big Idea in soccer. I submit that, if that's the standard for bigness, then there just aren't four Big Ideas to be had.
If soccer were taught like math, then you might take a course called Moving Without the Ball I as a freshman. And in that class there would be a unit about Overlapping Runs. And you would probably
hate it, because it would be an awful lot of running, and you wouldn't ever be sure why you were doing all this goddamn running, because maybe your coach isn't overly concerned about unveiling to you
the beautiful truth that a good, long overlapping run pulls a defender way down into the corner and stretches the whole defense and creates some dangerous space for a midfielder to run into. And you
would moan and check the calendar for when you were starting the Passing Into Space unit, because you heard it was totally easy -- mostly just standing around and pushing nice, easy passes toward
cones. But that really says more about your coach's ability/willingness to keep your eye on the Truly Big Idea (because hey, he's pretty much coaching the way he was coached in the first place, back
when kids just shut up and did their Backwards Jogging homework without complaint) than it does about the bigness of the Pretty Big Ideas that you're working your way through, because the Truly Big
Idea is just too ungodly huge to be useful in making you a better soccer player. After all, there's just the one.
In an attempt at analogic involution, I'm going to try to come up with a mathematical analogy for the soccer analogy for math:
I always return to this simple example from mathematics: Create structure when you're building; look for structure when you're exploring is a big idea at every level of the subject, from kid to
professional mathematician. And it is transferrable to all structure-having systems like language, chemistry, and logic. Truly big.
Maybe structure is the only big idea in math. At least the only one Wiggins might agree to. And that's just too big. What we really need in Algebra are some Pretty Big Ideas. So here's my
gauntlet, which is admittedly significantly lighter than the one that's currently being kicked around: what are four Pretty Big Ideas in Algebra? Honner got you started.
Consider the Strawberry
In general, I hate doing this --- because it feels like a self-promotional trick --- but in order for this post to make any kind of sense, you have to go back and read the last one. In particular,
you have to read Max's comment. I will put on my teacher face and wait for a few minutes.
For two reasons, I'm going to unpack the strawberry analogy a bit more: (1) I am in love with it, and (2) it highlights an important pedagogical point about the relationship between squares and
rectangles. For serious.
As Max pointed out, even very small children have no problem recognizing the rather trivial fact that all strawberries are fruits even though not all fruits are strawberries. On the flip side,
anyone who has ever taught geometry knows, with something like absolute certainty, that much older and more mathematically savvy students have great difficulty recognizing that all squares are
rectangles even though not all rectangles are squares. The situations are structurally identical (in each case we have some set, X, which is a proper subset of another set, Y), but the second one is
much more problematic. Why might that be?
The seemingly obvious answer is that recognizing a strawberry is nearly automatic, and probably evolutionarily encoded, while recognizing a square requires abstract reasoning about the congruence of
mathematical objects called "line segments." But I'm not at all convinced that's the problem. They are both ultimately pattern-recognition tasks. Without language getting in the way, you (and
small children) can probably recognize strawberries and squares with comparable facility.
Which brings us to the language. Even though the strawberry:fruit::square:rectangle situations are structurally identical, there is an important (and subtle) linguistic distinction in the latter
case. Consider the following story.
You find your favorite small child/guinea pig and present a challenge. In your left hand you hold a strawberry, and in your right an apple. You say to this child, "Which hand has the fruit in
it?" The child blinks at you for several moments, trying to study your face for clues about the answer to what has just got to be a trick question, before finally, tentatively, reaching out to
point at one of your hands, more or less at random. You reward the child with a piece of delicious fruit.
Consider the same story, except now you hold in your left hand a picture of a square, and in your right a picture of a generic rectangle. You say to this child, "Which hand has the rectangle in it?"
The child immediately points to your right hand. You reward the child with, I guess, a delicious piece of rectangle.
Why are these stories so different? I submit that it's not a mathematical issue. The real problem stems from the fact that, linguistically, there is no unprivileged fruit: every class of fruit gets
its own name. But "square" is privileged relative to "rectangle." When presented with a generic rectangle, we have no word for saying that it is "a rectangle that is not a square." In fact, I made
up the phrase "generic rectangle" precisely to try and convey that information.
So it turns out I lied a little bit before (how fitting) when I said the fruit/rectangle situations were structurally identical. It's true that in each case we have a set (square, strawberry) that
is a subset of a larger set (rectangle, fruit), but it turns out the larger sets have different linguistic partitions.
So when you ask the child which hand contains the rectangle, she chooses the generic rectangle immediately. Why? Because, had you meant the square, then you damn sure would've just said "square" in
the first place, even though both hands hold perfectly correct answers to your challenge. If our language were set up such that strawberries were the only specially named fruits (which seems like
something Max would wholeheartedly support), the child in the first story would likewise choose your non-strawberry hand every time, without hesitation.
So what can we do with this? It seems that strawberries have something to teach us about squares. Actually, it seems that all the other fruits have something to teach us about rectangles. It's
taken the entire history of humanity to organize fruits into useful equivalence classes, but luckily we find ourselves in a much, much simpler situation with rectangles; after all, there are only two
classes we care about! We already have a name for squares, so let's call non-square rectangles "nares." Now our partition looks like this:
Which hand has the nare in it? Easy. Better yet, unambiguous. Now, I'm not seriously lobbying for the introduction of nares into the mathematical lexicon (for one thing, nare is already a word for
a weird thing), but it might be a fun way to introduce young children to the concept of a non-square rectangle. After removing the greatest impediment to understanding the square/rectangle
relationship (that "square" is the lone special case of this broader class of "rectangles," which word is generally reserved for "rectangles-but-not-squares," since, if someone means "square," we
already have a freaking word for it), that scaffolding can eventually be disassembled.
But the cognitive edifice the scaffolding initially supported will have cured a little by then. In other words, why not make the distinction we actually care about explicit from the beginning,
rather than end up in linguistic contortions to get around the fact that the distinction is solely implicit in standard usage? Make up your own word, I don't care. Don't want to be cute about it?
Fine. Just abbreviate non-square rectangles as NSRs or something. But make them easy to talk about --- as easy as it is to talk about a tangerine or cumquat rather than a "fruit that might be a
strawberry, but very often is not." Because, seriously, if that's the way our fruit classification worked, there would be an awful lot of kids running around with the reasonable and tightly-held
belief that strawberries are not fruit.
And that would be a shame. | {"url":"http://blog.chrislusto.com/?cat=1","timestamp":"2024-11-14T03:18:07Z","content_type":"text/html","content_length":"59958","record_id":"<urn:uuid:e1c96171-a474-4689-a6e5-f03407120365>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00878.warc.gz"} |
For the unit vector hattheta, geometrically show that hattheta = -sinthetahati + costhetahatj? Essentially, converting from cartesian to polar, how would I determine the unit vector for vectheta in terms of theta, hati, and hatj? | Socratic
For the unit vector #hattheta#, geometrically show that #hattheta = -sinthetahati + costhetahatj#? Essentially, converting from cartesian to polar, how would I determine the unit vector for #vectheta
# in terms of #theta#, #hati#, and #hatj#?
I've been able to show that $\hat{r} = \cos \theta \hat{i} + \sin \theta \hat{j}$:
$\cos \theta = \frac{\hat{i}}{\hat{r}}$
$\sin \theta = \frac{\hat{j}}{\hat{r}}$
$\implies | | \hat{r} | | = \sqrt{\hat{r} \cdot \hat{r} \left({\cos}^{2} \theta + {\sin}^{2} \theta\right)}$
$= \sqrt{\hat{r} \cdot \hat{r} {\cos}^{2} \theta + \hat{r} \cdot \hat{r} {\sin}^{2} \theta}$
$= \sqrt{\hat{r} \cos \theta \cdot \hat{i} + \hat{r} \sin \theta \hat{j}}$
$\implies \hat{r} \cdot \hat{r} = | | \hat{r} | {|}^{2} = \hat{r} \cos \theta \cdot \hat{i} + \hat{r} \sin \theta \cdot \hat{j}$
$= \hat{r} \cdot \left(\cos \theta \hat{i} + \sin \theta \hat{j}\right)$
Thus, $\hat{r} = \cos \theta \hat{i} + \sin \theta \hat{j}$. But how would I do it for $\hat{\theta}$? I'm probably just missing something really simple, like where the $\hat{\theta}$ vector points.
2 Answers
Considering $p = \left(r \cos \theta , r \sin \theta\right) = r \left(\cos \theta , \sin \theta\right) = r \hat{r}$ where $\hat{r} = \left(\cos \theta , \sin \theta\right)$
We have then
$\dot{p} = \dot{r} \hat{r} + r \dot{\hat{r}}$
but $\dot{\hat{r}} = \left(- \sin \theta , \cos \theta\right) \dot{\theta} = \dot{\theta} \hat{\theta}$
Here for convenience, we call $\left(- \sin \theta , \cos \theta\right) = \hat{\theta}$
$\hat{r} , \hat{\theta}$ form a basis of orthogonal unit vectors. They can be also called $\hat{n} , \hat{\tau}$ instead.
so we have $\dot{p} = \dot{r} \hat{r} + r \dot{\hat{r}} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$
deriving again
$\ddot{p} = \ddot{r} \hat{r} + \dot{r} \dot{\theta} \hat{\theta} + \dot{r} \dot{\theta} \hat{\theta} + r \ddot{\theta} \hat{\theta} + r \dot{\theta} \dot{\hat{\theta}}$
Here $\hat{\theta} = \left(- \sin \theta , \cos \theta\right)$ so
$\dot{\hat{\theta}} = - \left(\cos \theta , \sin \theta\right) \dot{\theta} = - \dot{\theta} \hat{r}$
and finally
$\ddot{p} = \left(\ddot{r} - r {\left(\dot{\theta}\right)}^{2}\right) \hat{r} + \left(2 \dot{r} \dot{\theta} + r \ddot{\theta}\right) \hat{\theta}$
Concluding $\hat{r} , \hat{\theta}$ are used for convenience. They have a strong geometric appeal. Also they obey all rules of vector and differential calculus.
See the design in the explanation and the graph.
As a matter of convenience, I use $\alpha$, instead of $\theta$.
The unit vector in the direction $\theta = \alpha$ is
$\cos \alpha \vec{i} + \sin \alpha \vec{j}$
$= < x , y > = < \cos \alpha , \sin \alpha >$, in Cartesian form.
The parallel position vector through the origin O ( r = 0 ) is
$\vec{O P}$, where $P \left(\cos \alpha , \sin \alpha\right)$, is the radius vector of
the unit circle r = 1 , in the direction $\theta = \alpha$.
P' is at $\left(\cos \left(\frac{\pi}{2} + \alpha\right) , \sin \left(\frac{\pi}{2} + \alpha\right)\right) = \left(- \sin \alpha , \cos \alpha\right)$.
$\vec{O P '} = < - \sin \alpha , \cos \alpha >$ is constructed as the radius
vector of the unit circle, in the direction $\theta = \frac{\pi}{2} + \alpha .$.
graph{(x^2+y^2-1)(y-x/sqrt3)(y+sqrt3x)=0 [-1, 1, -.05, 1]}
Here $\theta = \frac{\pi}{6}$. Any vector of length 1, in the direction
$\alpha = \theta + \frac{\pi}{2} = \frac{2}{3} \pi$ (shown as a radius ),
would represent $- \sin \left(\frac{\pi}{6}\right) \vec{i} + \cos \left(\frac{\pi}{6}\right) \vec{j}$.
In brief, $\vec{O P '}$ is $\vec{O P}$ turned through $\frac{\pi}{2}$, in the (
$\theta \uparrow$) anti-clockwise sense. OP is in ${Q}_{1}$and OP' is in ${Q}_{2}$.
Impact of this question
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Estimating the Inertia Tensor Components of an Asymmetrical Spacecraft When Removing It from the Operational Orbit at the End of Its Active Life
Department of Theoretical Mechanics, Samara National Research University, Samara 443086, Russia
Department of Space Engineering, Samara National Research University, Samara 443086, Russia
Author to whom correspondence should be addressed.
Submission received: 5 November 2023 / Revised: 28 November 2023 / Accepted: 1 December 2023 / Published: 4 December 2023
The paper presents a method for estimating the inertia tensor components of a spacecraft that has expired its active life using measurement data of the Earth’s magnetic field induction vector
components. The implementation of this estimation method is supposed to be carried out when cleaning up space debris in the form of a clapped-out spacecraft with the help of a space tug. It is
assumed that a three-component magnetometer and a transmitting device are attached on space debris. The parameters for the rotational motion of space debris are estimated using this measuring system.
Then, the known controlled action from the space tug is transferred to the space debris. Next, measurements for the rotational motion parameters are carried out once again. Based on the available
measurement data and parameters of the controlled action, the space debris inertia tensor components are estimated. It is assumed that the measurements of the Earth’s magnetic field induction vector
components are made in a coordinate system whose axes are parallel to the corresponding axes of the main body axis system. Such an estimation makes it possible to effectively solve the problem of
cleaning up space debris by calculating the costs of the space tug working body and the parameters of the space debris removal orbit. Examples of numerical simulation using the measurement data of
the Earth’s magnetic field induction vector components on the Aist-2D small spacecraft are given. Thus, the purpose of this work is to evaluate the components of the space debris inertia tensor
through measurements of the Earth’s magnetic field taken using magnetometer sensors. The results of the work can be used in the development and implementation of missions to clean up space debris in
the form of clapped-out spacecraft.
1. Introduction
Nowadays, various projects are being developed to clean up space debris from near-Earth space. This issue was first raised at UN meetings in the early 1980s. Even then, it became clear that the
active use of near-Earth space would create the problem of its cleaning from space debris of terrestrial origin [
]. Space debris poses a serious threat to the safe operation of unmanned and manned spacecraft in near-Earth orbits. Due to the threat of collision with space debris, maneuvers have become common
practice in the operation of modern spacecraft [
]. All experts note that the number of launches of small spacecraft will increase significantly in the future [
Therefore, in the opinion of many authors, nowadays, it is necessary to design spacecraft with specific systems for its removal from the orbit at the end of its active life [
Various concepts have been developed to remove space debris from near-Earth orbits. The authors of [
] believe that the standard propulsion system of the spacecraft and the remnants of the working fluid can be used for removal. In this case, it is not necessary to design a specific system that
removes the spacecraft at the end of its active life. However, this method can be used for the oriented flight of the spacecraft with a full-fledged motion control system. The use of executive bodies
that do not require the expenditure of working fluid makes this method inefficient.
The work [
] considers a drag augmentation system (DAS), which is a space sail [
] that unfolds at the end of the spacecraft’s active life. This sail contributes to removing the spacecraft from the orbit due to the aerodynamic drag increase. This method involves the development
of a specific system for transporting and unfolding the sail and is applicable mainly in low near-Earth orbits. For high orbits, the spacecraft deorbit time can be significant. Modern materials of
such a sail have high stress–strain properties and a low specific gravity. Therefore, the increase in the mass parameters of a small spacecraft when using such a system will be insignificant.
The review [
] presents a comparative analysis of four different methods to remove spacecraft from low near-Earth orbits at the end of their active life. Two active devices (classical rocket and electric motors)
and two passive technologies (drag augmentation devices and cables of electrodynamic tether systems [
]) are considered. The authors of [
] believe that, with other factors being equal, for an initial height of 850 km, cables are approximately one and two orders of magnitude lighter than active devices and drag augmentation devices,
respectively. In this case, special attention is paid to electrodynamic tether systems, according to the results of the FP7/Space BETs project [
]. The superiority of ribbon cables over round and wire cables in terms of deorbit efficiency is substantiated, as well as the importance of the optimal choice for the length, width, and thickness of
a ribbon cable depending on the spacecraft mass and its initial orbit [
Figure 1
shows a scheme of transporting space debris by a cable using a space tug [
The prospect of using tether systems is noted by many researchers, for example, the authors of [
]. It is possible to design and install systems for deorbiting the spacecraft when creating new space technology, but the task of cleaning up existing space debris leaves significantly fewer options
for its solution. Therefore, one of the promising options for such cleaning is the use of a space tug in combination with a tether system for transporting space debris.
At the same time, methods of non-contact debris removal are actively developed, for example, using a laser system [
]. The authors of [
] propose to create a space laser facility to protect orbital stations from space debris. Based on the results of numerical simulation, a design for a space-based laser system was proposed in [
]. The developed laser system can effectively deal with space debris ranging in size from 1 to 10 cm. However, this method is more suitable for the protection of operating space objects than for
cleaning up space debris.
The work [
] contains a detailed review and comparison of existing technical solutions and approaches to space debris removal. Contactless transport systems are considered as a promising direction in creating
safe and reliable space debris removal systems. The use of an ion beam is proposed as one of the active influences on space debris [
]. The work [
] presents a scheme of the ion beam’s impact on space debris and analyzes the parameters of the impact that is necessary to solve the problem of its removal successfully [
]. In [
], a multipath scheme was proposed and control laws for impulse motors were developed.
For effective contact (through tether systems) and the contactless (via ion beams) cleaning of space debris in the form of spacecraft that have exhausted their active life, it is necessary to know
the inertial mass parameters of these spacecraft. Therefore, the problem of estimating the inertial mass parameters of space debris, as well as the parameters of its rotational motion in absolute
space and relative to the space tug, arises. This problem was considered not only in the context of the space debris problem in [
]. In [
], the difficulties of estimating the inertia tensor of a captured object are noted in the case when the connection between the space tug and the debris is not rigid, for example, when using a tether
In [
], the components of the space debris inertia tensor are estimated using various Kalman filters by measuring the rotation velocity of space debris. The cases of a cable stretched all the time and a
cable subject to frequent weakening are considered. A good estimation quality is shown if the cable tension and the cable attachment point are known [
]. However, in some cases, the authors of [
] note a large dispersion of the obtained estimations.
In [
], the traditional method was employed to achieve an accurate estimation of the inertial mass parameters in a system analysis designed for the errors which influence the parameter measurements of the
space debris rotational motion. To improve the estimation accuracy, the authors of [
] proposed a modification for the estimation equations by including the data of the space tug contact force impact on space debris.
In [
], it was proposed to use a nanosatellite as a data measurement system for estimating the parameters of the space debris rotational motion. This satellite must dock with space debris and move with it
as a single body. However, docking issues are not discussed.
In general formulation, solving the problem of estimating the inertia tensor components of arbitrary-shaped space debris moving arbitrarily in outer space is quite complicated. The possibility of
attaching several measuring instruments on different parts of space debris, and the possibility of monitoring the relative position of these instruments, while taking into account errors, will expand
the range of the proposed method application for estimating the inertia tensor components. However, technically, it is not easy to solve this problem.
This work makes the following contribution:
A method for estimating the inertia tensor components of space debris and the parameters of its rotational motion by attaching elements of the data-measuring system on a space debris object is
A simulation is carried out for a particular case of attaching measuring instruments on a space debris object;
The results of numerical simulation for a particular case with an estimation of inertia tensor components for the Aist-2D small spacecraft are presented;
An analysis of the obtained results was carried out and recommendations for its use were given.
2. Problem Formulation
Let us consider the problem of estimating the inertia tensor components for a space debris object in the general formulation within the framework of the proposed approach of attaching the measuring
equipment—a magnetometer—on it. Let us assume that a three-component magnetometer with a data-transmitting device has been attached to the space debris object. In this case, using the measurements of
the Earth’s magnetic field induction vector, it is possible to estimate the components of the angular velocity vector of space debris in the magnetometer’s structural coordinate system (
Figure 2
To obtain a correct estimation for the angular velocity vector, it is proposed in [
] to use the derivative of the Earth’s magnetic field induction vector components:
$ω → k = B → ˙ k × B → ˙ k − 1 Δ t k B → ˙ k 2 ,$
$B → ˙ k B ˙ x k , B ˙ y k , B ˙ z k$
$B → ˙ k − 1 B ˙ x k − 1 , B ˙ y k − 1 , B ˙ z k − 1$
are the derivatives of the Earth’s magnetic field induction vector components and its components in the magnetometer’s structural coordinate system (
Figure 3
) for the
-th and
− 1-th measurements, respectively;
$Δ t k = t k − t k − 1$
is the time interval between
-th and
− 1-st measurements.
Let us represent the vector Equation (1) in the axes of the magnetometer’s structural coordinate system (
Figure 3
$ω x k = B ˙ y k B ˙ z k − 1 − B ˙ z k B ˙ y k − 1 Δ t k B ˙ x k 2 + B ˙ y k 2 + B ˙ z k 2 ; ω y k = B ˙ z k B ˙ x k − 1 − B ˙ x k B ˙ z k − 1 Δ t k B ˙ x k 2 + B ˙ y k 2 + B ˙ z k 2 ; ω z k = B ˙ x
k B ˙ y k − 1 − B ˙ y k B ˙ x k − 1 Δ t k B ˙ x k 2 + B ˙ y k 2 + B ˙ z k 2 .$
Then, with an arbitrary location of the axes of the magnetometer’s structural coordinate system relative to the main body axis system of the space debris object, the Euler dynamic equations in the
magnetometer’s structural coordinate system will have the form [
$I x x ω ˙ x k − I x y ω ˙ y k − I x z ω ˙ z k + ω y k I z z ω z k − I x z ω x k − I y z ω y k − ω z k I y y ω y k − I x y ω x k − I y z ω z k = M x I y y ω ˙ y k − I x y ω ˙ x k − I y z ω ˙ z k + ω
z k I x x ω x k − I x y ω y k − I x z ω z k − ω x k I z z ω z k − I x z ω x k − I y z ω y k = M y I z z ω ˙ z k − I x z ω ˙ x k − I y z ω ˙ y k + ω x k I y y ω y k − I x y ω x k − I y z ω z k − ω y k
I x x ω x k − I x y ω y k − I x z ω z k = M z ,$
$M → M x , M y , M z$
is the main vector of external moments acting on the space debris object;
$I ^ = I x x I x y I x z I x y I y y I y z I x z I y z I z z$
is the symmetrical inertia tensor in the magnetometer’s structural coordinate system;
$ω → ˙ k ω ˙ x k , ω ˙ y k , ω ˙ z$
is the derivative of the angular velocity vector of the space debris object and its components in the magnetometer’s structural coordinate system (
Figure 2
Further, the known perturbing effect is transferred to the space debris object. The above equations are then also used to estimate the rotational motion parameters.
In the general formulation, the problem of estimating inertia tensor components of the space debris object using measurements of a single magnetometer cannot be solved without additional data.
Therefore, let us consider a special case, whereby the origin of the magnetometer’s structural coordinate system is located on one of the axes of the main body axis system, and the
axes of the structural coordinate system and
axes of the main body axis systems are parallel (
Figure 3
In this case, taking into account the introduced simplified assumption, Equation (3) is transformed to the form [
$I x x ω ˙ x k + ω y k ω z k I z z − I y y = M x I y y ω ˙ y k + ω x k ω z k I x x − I z z = M y I z z ω ˙ z k + ω x k ω y k I y y − I x x = M z .$
Let us rewrite Equation (4) with respect to the diagonal inertia moments in the structural coordinate system:
$ω ˙ x k I x x − I y y ω y k ω z k + ω y k ω z k I z z = M x ω ˙ y k I y y + ω x k ω z k I x x − ω x k ω z k I z z = M y ω ˙ z k I z z − ω x k ω y k I x x + ω x k ω y k I y y = M z .$
Let us assume that the quantity of the controlled action is significant enough to neglect the external disturbing action. Then, the right parts of Equation (5) will represent the moment from the
controlled action in the magnetometer’s structural coordinate system (
Figure 3
$M → = r → × F → c o n t ,$
$r →$
is the radius vector of the controlled action application point relative to the origin of the magnetometer’s structural coordinate system;
$F → c o n t$
is the vector of the controlled action.
Let us express the diagonal components of the inertia tensor from system (5):
$I z z = M z ω ˙ x k + M x ω x k ω y k − M y ω ˙ x k − M x ω x k ω z k ω x k ω y k 2 ω z k − ω x k ω y k ω ˙ x k ω ˙ x k ω ˙ y k + ω x k ω y k ω z k 2 ω ˙ x k ω ˙ z k + ω x k ω y k 2 ω z k + ω x k ω
y k 2 ω z k − ω x k ω y k ω ˙ x k ω x k ω z k ω ˙ x k − ω x k ω y k ω z k 2 ω ˙ x k ω ˙ y k + ω x k ω y k ω z k 2 ; I y y = I z z ω x k ω z k ω ˙ x k + ω x k ω y k ω z k 2 ω ˙ x k ω ˙ y k + ω x k ω y
k ω z k 2 + M y ω ˙ x k − M x ω x k ω z k ω ˙ x k ω ˙ y k + ω x k ω y k ω z k 2 ; I x x = M x + ω y k ω z k I y y − ω y k ω z k I z z ω ˙ x k .$
Now, by estimating the angular velocity and angular acceleration of the space debris object using magnetometer measurements and the moment from controlled action, it is possible to estimate the
inertia tensor components from system (7) in the construction coordinate system of the magnetometer.
Let us transform the inertia tensor in accordance with the Huygens–Steiner theorem upon transition to the main body axis system of the space debris object. In the considered case, the axes of the
main body axis system of the space debris object and the magnetometer’s structural coordinate system are parallel (
Figure 3
Figure 4
). The
axes are offset from each other (
Figure 5
). Therefore, we have:
$I ^ = I x x 0 0 0 I y y + m a 2 0 0 0 I z z + m a 2 ,$
is the mass of the space debris object;
is the distance between the
axes of the main body axis system of the space debris object and the magnetometer’s structural coordinate system (
Figure 3
In this particular case, the components of the inertia tensor are relatively easy to find. Let us illustrate it with an example in the next section of the paper.
3. Numerical Simulation for the Aist-2D Small Spacecraft
Let us consider the Aist-2D small spacecraft for remote sensing of the Earth as an example to estimate the inertia tensor components (
Figure 6
The main parameters of the Aist-2D small spacecraft for remote sensing of the Earth are presented in
Table 1
Modern measuring instruments provide high accuracy in measuring the components of the Earth’s magnetic field induction vector [
]. Therefore, their application can provide an effective estimation of the inertia tensor components for the space debris object. Thus, proton precession magnetometers and optically pumped
magnetometers have a sensitivity of about 10–50 pT, an absolute accuracy of about 0.1–1.0 nT, and a dynamic range of 1–100 μT [
In this case, the following algorithm is used:
The fixing of the magnetometers on the space debris object, the construction axes of which are parallel to the axes of the main connected coordinate system of the space debris object;
The implementation of a controlled impact on the space debris object;
Carrying out measurements with a uniform step sufficient for the subsequent correct restoration of a continuous signal;
The restoration of continuous dependences of the Earth’s magnetic field induction vector components via their discrete measurements;
The estimation of the angular velocity and angular acceleration of the space debris object as a result of a controlled impact;
The estimation of the space debris object inertia tensor components using dynamic Euler equations.
The Aist-2D small spacecraft is equipped with three-component magnetometers. Their measurements are used as experimental data.
Let us consider the stabilization section of the Aist-2D small spacecraft as the initial section before the controlled action. The measurement data for this section are shown in
Figure 5
. Time t = 0 corresponds to 31 July 2016, 14:35:46 Moscow time.
Let us choose the section of reorientation of the Aist-2D small spacecraft as a section with controlled action. The measurement data for this section are shown in
Figure 6
. Time t = 0 corresponds to 31 July 2016, 19:34:28 Moscow time.
To correctly estimate the derivative of the Earth’s magnetic field induction vector components, it is necessary to have continuous dependences of these components on time. These dependencies are then
used in Formulas (1) and (2) to determine the vector of angular velocity and rotational acceleration of the space debris object in the magnetometer’s structural coordinate system.
Let us restore discrete measurements to continuous dependencies using the Kotelnikov series [
], since there are measurement data at regular intervals:
$B j ( t ) = ∑ k = − ∞ ∞ B j k sin π Δ t t − k Δ t π Δ t t − k Δ t ,$
are the measurements at the time
$Δ t = Δ t k$
is the uniform step between measurements.
Continuous dependencies corresponding to
Figure 5
Figure 6
obtained using the Kotelnikov series (9) are shown in
Figure 7
. The derivatives of these functions are shown in
Figure 8
The variation ranges of the Earth’s magnetic field induction vector components in the reorientation mode are much wider than in the stabilization mode (
Figure 7
). It should be noted that the variation ranges of derivatives in different modes are comparable. However, the analysis of
Figure 7
shows that in the stabilization mode, the derivatives fluctuate around zero with a sign change. In the reorientation mode, the derivatives have the same sign for a long period of time. This can be
explained by the fact that in the stabilization mode, there are random fluctuations in the orientation angles with a change in the sign of the angular velocity. In the reorientation mode, the angular
position of the small spacecraft purposefully changes. This is achieved by the fact that the angular velocity has the same sign for a significant period of time. Thus, we can refer to the correct
restoration for the continuous dependences of the Earth’s magnetic field induction vector components using the Kotelnikov series (9).
Let us further estimate the angular velocity by using Formula (2). The estimation results are shown in
Figure 9
The derivative of the angular velocity—angular acceleration—will have the form shown in
Figure 10
The values of the angular velocity and angular acceleration in the stabilization mode are significantly lower than in the reorientation mode (
Figure 9
). This fact is an important difference between different modes. Let us estimate the dependences for the diagonal components of the Aist-2D small spacecraft inertia tensor by using system (7)
according to the measurement data. These dependencies are shown in
Figure 11
for the stabilization mode and in
Figure 12
for the reorientation mode.
Bursts on the diagonal components of the inertia tensor graphs are associated with both measurement errors and approximation errors of these measurements by the Kotelnikov series (9). Small
oscillations in the dependences can be explained by the errors in the attachment of measuring equipment relative to the main body axis system, as well as by natural oscillations of the solar panels
of the Aist-2D small spacecraft. These oscillations influenced the components of the inertia tensor and provided them with a dynamic component. In general, upon analyzing
Figure 11
Figure 12
, we can state a fine precision of the results with data from
Table 1
. It can also be seen that in the reorientation mode, the diagonal components of the inertia tensor are estimated more accurately. This is due to the fact that the moment from the executive bodies of
the Aist-2D small spacecraft (flywheel engines) was determined more accurately than the moment from many disturbing factors in the stabilization mode. The error of this method in the given numerical
example can be estimated correctly only in the case of the inertia tensor components’ constancy over the entire measurement time. Then, the interval estimation for the stabilization and orientation
modes, respectively, has the form:
$I x x β = 0 , 95 ∈ 167 , 183 ; I y y β = 0 , 95 ∈ 190 , 210 ; I z z β = 0 , 95 ∈ 280 , 290 . I x x β = 0 , 99 ∈ 168 , 182 ; I y y β = 0 , 99 ∈ 195 , 205 ; I z z β = 0 , 99 ∈ 282 , 287 .$
Here, β is the confidence probability.
In fact, due to the natural oscillations of solar panels, the moments of inertia will not remain constant. Error estimation in such a situation is complex and is a random process.
4. Conclusions
Thus, as a result of the investigations carried out in the paper, a theoretical estimation of the diagonal components of the space debris object inertia tensor was obtained in the simple case of
attaching the measuring equipment on this object. It was assumed that the structural axes of the measuring equipment coincide with the main body axes of the space debris object. As an example, the
Aist-2D small spacecraft for the remote sensing of the Earth was taken. Its example demonstrates the possibility of estimating the diagonal components of the inertia tensor using the measurement data
of the Earth’s magnetic field induction vector. The average values of the disturbing factors (in the stabilization mode) and the moments of flywheel engines (in the reorientation mode) were chosen as
the controlled action on the small spacecraft. The results showing a fine precision with the diagonal components of the inertia tensor of the Aist-2D small spacecraft are obtained. The results of the
work can be used in estimating the inertia tensor components of space debris objects. This can be useful when implementing the missions to clean up near-Earth space.
Author Contributions
Conceptualization, A.V.S. and M.E.B.; methodology, A.V.S. and M.E.B.; software, A.V.S. and D.I.O.; validation, A.V.S., D.I.O. and E.S.K.; formal analysis, A.V.S. and M.E.B.; investigation, A.V.S. and
D.I.O.; resources, A.V.S.; data duration, A.V.S. and M.E.B.; writing—original draft preparation, A.V.S., D.I.O., M.E.B. and E.S.K.; writing—review and editing, A.V.S. and D.I.O.; visualization,
A.V.S. and D.I.O.; supervision, A.V.S. and M.E.B.; project administration, A.V.S.; funding acquisition, A.V.S. All authors have read and agreed to the published version of the manuscript.
This research was supported by the Russian Science Foundation (Project No. 22-19-00160).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2. Scheme of attaching the magnetometer on the space debris object in an arbitrary case: Oxyz is the main body axis system of the space debris object; O[b]x[b]y[b]z[b] is the structural
coordinate system of the magnetometer.
Figure 4.
Appearance of the Aist-2D small spacecraft for remote sensing of the Earth [
Figure 5. Components of the Earth’s magnetic field induction vector in the magnetometer’s structural coordinate system in stabilization mode: 1 is B[x]; 2 is B[y]; 3 is B[z].
Figure 6. Components of the Earth’s magnetic field induction vector in the magnetometer’s structural coordinate system in reorientation mode: 1 is B[x]; 2 is B[y]; 3 is B[z].
Figure 7.
Continuous dependencies of the Earth’s magnetic field induction vector components in the magnetometer’s structural coordinate system, restored using the Kotelnikov series (9): (
) in stabilization mode (
Figure 5
); (
) in reorientation mode (
Figure 6
) 1 is
; 2 is
; 3 is
Figure 8.
Derivatives of continuous dependencies of the Earth’s magnetic field induction vector components in the magnetometer’s structural coordinate system (
Figure 7
): (
) in stabilization mode; (
) in reorientation mode
Figure 9. Dependences for the components of the angular velocity vector in the magnetometer’s structural coordinate system, estimated by Equation (2): (a) in stabilization mode; (b) in reorientation
mode ω[x] (black); ω[y] (blue); ω[z] (red).
Figure 10. Dependences for the components of the angular acceleration vector in the magnetometer’s structural coordinate system: (a) in stabilization mode; (b) in reorientation mode ε[x] (black); ε
[y] (blue); ε[z] (red).
Figure 11. Dependences for the diagonal components of the inertia tensor in the magnetometer’s structural coordinate system in stabilization mode: (a) I[xx]; (b) I[yy]; (c) I[zz].
Figure 12. Dependences for the diagonal components of the inertia tensor in the magnetometer’s structural coordinate system in reorientation mode: (a) I[xx]; (b) I[yy]; (c) I[zz].
Table 1.
The main parameters of the simulated Aist-2D spacecraft [
Parameter Designation Value Dimension
Mass m 530 kg
I[xx] 175
Axial moments of inertia I[yy] 200 kg·m^2
I[zz] 285
Maximum control torque M 0.2 N·m
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Sedelnikov, A.V.; Orlov, D.I.; Bratkova, M.E.; Khnyryova, E.S. Estimating the Inertia Tensor Components of an Asymmetrical Spacecraft When Removing It from the Operational Orbit at the End of Its
Active Life. Sensors 2023, 23, 9615. https://doi.org/10.3390/s23239615
AMA Style
Sedelnikov AV, Orlov DI, Bratkova ME, Khnyryova ES. Estimating the Inertia Tensor Components of an Asymmetrical Spacecraft When Removing It from the Operational Orbit at the End of Its Active Life.
Sensors. 2023; 23(23):9615. https://doi.org/10.3390/s23239615
Chicago/Turabian Style
Sedelnikov, A. V., D. I. Orlov, M. E. Bratkova, and E. S. Khnyryova. 2023. "Estimating the Inertia Tensor Components of an Asymmetrical Spacecraft When Removing It from the Operational Orbit at the
End of Its Active Life" Sensors 23, no. 23: 9615. https://doi.org/10.3390/s23239615
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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Songs Dero Loves
Yiit created a playlist full of songs that Dero loves. But Dero has a special rule for their playlists and this playlist does not obey that rule. So now Dero needs to rearrange the playlist and split
the list into several different ones.
Dero creates their playlists with respect to the letters of the songs. For different songs to be in the same playlist, they need to have at least one common letter. How many new playlists should Dero
create to split the playlist that Yiit made for him, in order not to break the rule?
For the Songs Dero Loves: https://open.spotify.com/playlist/4SHl6R9NFhhx68Y8gYmO0v?si=aFCzGtjIT3SdTcU86JkczQ
• In the end, all of the songs in a single playlist of Dero's should include a common letter.
• All of the song names will be comprised of only lower case letters.
First line consists of the number of songs in the playlist, integer \(\mathbf{N}\).
Next \(\mathbf{N}\) lines include the name \(\mathbf{A_i}\) of the \(\mathbf{i}\)th song.
Batch #1:
• \(1 \leq \mathbf{N} \leq 100\)
• \(1 \leq \mathbf{len(A_i)} \leq 10\)
Batch #2:
• \(1 \leq \mathbf{N} \leq 10^{5}\)
• \(1 \leq \mathbf{len(A_i)} \leq 10\)
The minimum number of playlists that Dero needs to split Yiit's playlist. | {"url":"https://arsiv.cclub.metu.edu.tr/problem/playlist/","timestamp":"2024-11-03T12:03:01Z","content_type":"text/html","content_length":"11800","record_id":"<urn:uuid:d4eafe4d-1a77-4816-b415-ca65ae5bdbaf>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00653.warc.gz"} |
A Novel Characterization of the Complexity Class Θ^P_k Based on Counting and Comparison
Thomas Lukasiewicz and Enrico Malizia
The complexity class Θ[2]^P, which is the class of languages recognizable by deterministic Turing machines in polynomial time with at most logarithmic many calls to an NP oracle, received extensive
attention in the literature. Its complete problems can be characterized by different specific tasks, such as deciding whether the optimum solution of an NP problem is unique, or whether it is in some
sense “odd” (e.g., whether its size is an odd number). In this paper, we introduce a new characterization of this class and its generalization Θ[k]^P to the k-th level of the polynomial hierarchy. We
show that problems in Θ[k]^P are also those whose solution involves deciding, for two given sets A and B of instances of two Σ[k-1]^P-complete (or Π[k-1]^P-complete) problems, whether the number of
“yes”-instances in A is greater than those in B. Moreover, based on this new characterization, we provide a novel sufficient condition for Θ[k]^P-hardness. We also define the general problem
Comp-Valid[k], which is proven here Θ[k+1]^P-complete. Comp-Valid[k] is the problem of deciding, given two sets A and B of quantified Boolean formulas with at most k alternating quantifiers, whether
the number of valid formulas in A is greater than those in B. Notably, the problem Comp-Sat of deciding whether a set contains more satisfiable Boolean formulas than another set, which is a
particular case of Comp-Valid[1], demonstrates itself as a very intuitive Θ[2]^P-complete problem. Nonetheless, to our knowledge, it eluded its formal definition to date. In fact, given its strict
adherence to the count-and-compare semantics here introduced, Comp-Valid[k] is among the most suitable tools to prove Θ[k]^P-hardness of problems involving the counting and comparison of the number
of “yes”-instances in two sets. We support this by showing that the Θ[2]^P-hardness of the Max voting scheme over mCP-nets is easily obtained via the new characterization of Θ[k]^P introduced in this
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Profit and Loss
Profit and Loss is an important topic in the Quantitative Aptitude section of the Common Law Admission Test (CLAT). This topic assesses your ability to understand the financial aspects of buying and
selling goods, calculating profits and losses and evaluating cost price and selling price relationships. In this article, we will explore the core concepts of Profit and Loss, provide illustrative
examples and offer strategies to effectively solve problems related to this topic.
Understanding Profit and Loss Fundamentals
Before we delve into solving Profit and Loss problems, let’s establish some fundamental concepts:
1. Cost Price (CP): The amount at which an item is purchased is known as the cost price.
2. Selling Price (SP): The amount at which an item is sold is known as the selling price.
3. Profit: When the selling price is higher than the cost price, the difference is called profit.
4. Loss: When the selling price is lower than the cost price, the difference is called a loss.
5. Profit Percentage: The profit, expressed as a percentage of the cost price, is the profit percentage.
6. Loss Percentage: The loss, expressed as a percentage of the cost price, is the loss percentage.
Solving Profit and Loss Problems: Concepts and Examples
Example 1: Calculating Profit Percentage
Question: If an item is bought for Rs 500 and sold for Rs 600, find the profit percentage.
1. Profit = Selling Price – Cost Price = Rs 600 – Rs 500 = Rs 100.
2. Profit percentage = (Profit / Cost Price) * 100 = (Rs 100 / Rs 500) * 100 = 20%.
Example 2: Determining Selling Price
Question: If a watch is bought for Rs 80 and the desired profit percentage is 25%, what should be the selling price?
1. Desired profit = (Profit Percentage / 100) * Cost Price = (25 / 100) * Rs 80 = Rs 20.
2. Selling Price = Cost Price + Desired Profit = Rs 80 + Rs 20 = Rs 100.
Example 3: Loss and Loss Percentage
Question: A shirt is sold for Rs 45, incurring a loss of 10%. Find the cost price of the shirt.
1. Loss = (Loss Percentage / 100) * Cost Price = (10 / 100) * Cost Price.
2. Given that Selling Price = Cost Price – Loss, we have Rs 45 = Cost Price – 0.1 * Cost Price.
3. Solving for Cost Price, we get Cost Price = Rs 45 / 0.9 = Rs 50.
Strategies for Tackling Profit and Loss Problems
Solving Profit and Loss problems involves practical thinking and accurate calculations. Here are some strategies to help you approach these problems effectively:
1. Understand the Basics: Ensure a clear understanding of terms like cost price, selling price, profit and loss.
2. Use Formulas: Familiarise yourself with the formulas for calculating profit, loss, profit percentage and loss percentage.
3. Practice Percentage Calculations: Strengthen your percentage calculation skills as they are integral to solving these problems.
4. Read the Question Carefully: Understand what is given and what is required in the problem before attempting calculations.
5. Break Down Complex Problems: Divide complex problems into smaller steps, making it easier to manage calculations.
Profit and Loss problems might initially appear daunting, but with a solid grasp of the fundamental concepts and consistent practice, you can conquer them with confidence. Understand cost price,
selling price, profit and loss, practice percentage calculations and approach each problem step by step. As a student preparing for the CLAT, mastering Profit and Loss problems not only enhances your
quantitative aptitude but also boosts your problem-solving skills for competitive exams. So, put on your mathematical thinking cap and delve into Profit and Loss problems with determination and
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Gradient Ascenders Reach the Harsanyi Hyperplane — LessWrong
This is a supplemental post to Geometric Utilitarianism (And Why It Matters), in which I show that, if we use the weights we derived in the previous post, a gradient ascender will reach the Harsanyi
hyperplane . This is a subproblem of the proof laid out in the first post of this sequence, and the main post describes why that problem is interesting.
The Gradient and Contour Lines
It's easy to find the points which have the same score as : they're the points which satisfy . They all lie on a skewed hyperbola that touches at .
Check out an interactive version here
One way to think about is as a hypersurface in -dimensional space sitting "above" the n-dimensional space of utilities we've been working with. When there are 2 agents, we can plot using the third
vertical axis.
Interactive version here
Check out the intersection of and the vertical plane above the Harsanyi line : this tells us about the values of along this line, and as we shift we can recalculate so that among , peaks at .
Our choice of determines where we land on that surface "above" p. If we take a slice through by only looking at the point of at the same "altitude" as , we get exactly that hyperbola back!
Doing this for many altitudes gives us a contour map, which you're probably familiar with in the context of displaying altitude of real 3D landscapes on flat 2D maps.
You can see how these contours change as we change and using the interactive version here.
There's a theorem which tells us that the gradient of must either be 0 or perpendicular to these contour hypersurfaces. So by calculating the gradient, we can calculate the tangent hyperplane of our
skewed hyperbolas! And then we'll see if anything interesting happens at .
This is a different gradient than the one we calculated earlier, where we were specifically interested in how changes along the Harsanyi hyperplane . But the slope of , encoded in , showed up in how
we defined . So let's see how and have shaped the geometry of .
Thankfully, this gradient calculation is a lot easier than the other one (which was so long I broke it out into its own separate post). The gradient is just a vector of partial derivatives , where
we're using to remind ourselves that this is just the gradient with respect to , holding constant. We're holding the weights constant, and isn't a function this time where we'd need to use the chain
rule, so all we need to do is apply the power rule:
If we use to denote element-wise division, this gives us a simple formula for :
or a component
And that's it! is defined when for all agents. Where is defined, we can keep taking derivatives; is smooth everywhere for all agents.
Here's what it looks like!
Playing around with an interactive version, you can see that as you approach giving an agent 0 utility, the gradient arrow gets longer and longer. As long as , diverges off to infinity as approaches
0. When , changing doesn't change and .
Normatively, being 0 whenever any individual utility is 0 is a nice property to have. As long as we give an agent some weight, there is a pressure towards giving them more utility. If you've gotten
this far you've probably taken a calculus class, and you probably studied how to enclose the largest area using a fixed perimeter of fencing. This is exactly the same pressure pushing us towards
squares and away from skinny rectangles. The Pareto optima are points where the pressures favoring each agent balance, for some weights , and we can design to cause all those pressures to balance at
any point we choose along the Pareto frontier.
The visual analogy between "maximizing a product" and "sliding a hyperbola until it reaches the Pareto frontier" was really helpful in thinking about this problem. I first learned about that lens
from Abram Demski's great Comparing Utilities post, which included illustrations by Daniel Demski that really helped me visualize what was going on as we maximize .
Another thing we can notice is that . This is exactly what we'd want from a Pareto monotone aggregation function. Geometrically, this means those contour lines always get further and further away
from , and they don't curve back in to make some other point in score higher on than .
Gradient Ascent
The simplest proof I've found that goes from
relies on the fact that, if you start inside and follow the gradient to make larger and larger, you'll eventually run into the Harsanyi hyperplane . In order for this to be true, needs to point at
least a little bit in the direction perpendicular to .
The Normal Vector to The Harsanyi Hyperplane
What is that direction? One way to think about is as a contour hyperplane of , the Harsanyi aggregation function. is all of the joint utilities where . We know that the gradient will be perpendicular
to this contour line, so let's compute that in order to find the normal vector to :
It would make my tensor calculus teacher too sad for me to write that , but the components of the vector are always the same as the components of the covector . We can then normalize to get the
normal vector to which I'll denote :
The distinction isn't important for most of this sequence, but I do want to use different alphabets to keep track of which objects are vectors and which are maps from vectors to scalars because
they're different geometric objects with different properties. If we decide to start measuring one agent's utility in terms of milli-utilons, effectively multiplying all of their utility measurements
by 1,000, the component of that agent's Harsanyi weight scales inversely in a way that perfectly cancels out this change. The slope of a line doesn't change when we change the units we use to measure
Gradient Ascenders Reach H
One way to think about the dot product is as a tool for measuring the lengths of vectors and the angles between them. If the dot product is ever 0, it means that a gradient ascender will not be
moving towards or away from at that point. There are such orthogonal directions available, so for our proof to work we need to check that our choice of always leads to movement towards or away from .
Let's see what it takes to satisfy
All of these terms are non-negative, so in order for to be 0, each element of this sum needs to be simultaneously 0. Can this happen for our choice of ?
We know that for at least one agent , and that if for all agents, that the entire Pareto frontier must consist only of that point at the origin. (In which case all choices of and make that point
optimal, and any gradient ascender will trivially find the optimum.) Otherwise, wherever is defined it points at least a little in the same direction as . (And where it's not defined, it's because
some component has diverged off to positive infinity because for some agent.)
In other words, using our choice of , all gradient ascenders will either stay at their initial local minimum (because they were placed on a part of the boundary of where and is undefined), or they
will eventually reach the Harsanyi hyperplane.
This is also a great time to point out that
When for all agents, points in the same direction as at . This is a direct consequence of choosing such that is perpendicular to . So is the tangent hyperplane to the Pareto frontier at , but it's
also the tangent hyperplane to the contour curve of at .
You can play with an interactive version here!
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Area QUESTIONS AND ANSWERS :: Arithmetic : part1 : 16 to 20
Following Area multiple choice objective type questions and answers will help you in many types of 2024 job and other entrance examinations :
16.The area of a rectangle 144 m long is the same as that of a square having a side 84 m long. The width of the rectangle is:
7 m
14 m
49 m
cannot be determined
17.The ratio between the length and breadth of a rectangular field is 5:4, the breadth is 20 metres less than the length, the perimeter of the field is:
260 m
280 m
360 m
none of these
18.A verandah 40 metres long 15 metres broad is to be paved with stones each measuring 6 dm by 5 dm. the number of stones required is:
none of these
19.If the side of a square be increased by 4 cms, the area increases by 60 sq. cms. The side of the square is:
12 cm
13 cm
14 cm
none of these
More Area QUESTIONS AND ANSWERS available in next pages
Health is the greatest gift, contentment is the greatest wealth -Buddha
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We have become Big Brother
I actually have no idea about the practical situation in the UK.
I was thinking of back home in Sweden, where the extreme right are in fact white supremacists and nazis.
"My milkshake bringeth all ye gentlefolk to the yard. Verily 'tis better than thine, I would teach thee, but I must levy a fee."
"When Kleiner showed me the sky-line of New York I told him that man is like the coral insect—designed to build vast, beautiful, mineral things for the moon to delight in after he is dead."
I actually have no idea about the practical situation in the UK.
I was thinking of back home in Sweden, where the extreme right are in fact white supremacists and nazis.
Probably understandable considering the extent of the immigration to Europe lately.
"I really perceive that vanity about which most men merely prate — the vanity of the human or temporal life. I live continually in a reverie of the future. I have no faith in human perfectibility. I
think that human exertion will have no appreciable effect upon humanity. Man is now only more active — not more happy — nor more wise, than he was 6000 years ago. The result will never vary — and to
suppose that it will, is to suppose that the foregone man has lived in vain — that the foregone time is but the rudiment of the future — that the myriads who have perished have not been upon equal
footing with ourselves — nor are we with our posterity. I cannot agree to lose sight of man the individual, in man the mass."...
- 2 July 1844 letter to James Russell Lowell from Edgar Allan Poe.
I was thinking of back home in Sweden, where the extreme right are in fact white supremacists and nazis.
Probably understandable considering the extent of the immigration to Europe lately.
Precisely. From what I gather, the mainstream politicians in Sweden have embraced multi-culturalism to such an extent that merely criticising government immigration policy can be prosecuted as "hate
speech". It is no wonder that far right parties can thrive in this kind of environment.
Stupid grandstanding politicians all over the world seem incapable of recognising the simple truth that just because you ban, censor and outlaw political viewpoints doesn't mean that people will stop
holding those viewpoints. Instead they will see themselves as "oppressed" and become more extremist.
..merely criticising government immigration policy can be prosecuted as "hate speech".
Oh, but you've gotten all of this the wrong way. It has nothing to do with the legal stuff, or actual fear that you'll be prosecuted for anything. Many people are just narrow-minded and they don't
want to be labelled racists in their community, even if that's exactly what many of them are. The Sweden Democrats (SD) were previously publicly called xenophobic, but the media immediately picked up
and spread the term "immigration critical" when SD started using it, so yay, now people have a politically correct way to say they're afraid of Africans, without coming off as racist! Perfect
widespread delusion, ready to be further misused and skewed, which it has been.
Though, the real political problem is that the government themselves have ignored the immigration issue completely, along with numerous other issues that has indirectly and directly made
immigration-related problems worse. The right wing parties has changed many laws to make it cheaper and easier for companies to have (or ditch) employees, or rephrased, made it much harder for
employees to actually get decent pay and keep their jobs.
The hiring firm business has gone up several hundred percent due to the changes in laws, so now it's not unusual that people have to work for 12 months for a lower "starter" salary before they even
get the chance to write a proper contract. I have several friends who has worked for 6 or 12 months on temporary contracts, only to be let off a month before the real talk begins. The company then
brings in someone new and repeats the process.
In combination with this (as another example), we've lately seen several "free schools" (as in private schools sort of, but they still get public funding based on the number of students, within legal
gray zones) being bought up and run to the ground by leeching international capital investment companies/banks. They basically buy a school cheap, change its advertising to something popular (like
hair stylist, which you will never be able to actually work as, because really, how many does that for a living? there is no market!) and start to pull money out of it (lower the amount of books
being given to students, fire teachers, etc) until it's sucked dry and take the money out in revenues for the company. After the school is bankrupted, the kommun/county is of course responsible for
the local students, so the load on the public schools increases to the point where they too need stately funds to survive, sucking even more money, that could have gone to other useful things, out of
the system.
There's also the problem with house building, especially in bigger cities, but also in smaller ones.
There are not enough rental apartments being built, the construction companies rather build apartments you have to buy for a couple of millions instead because it's more convenient. Short-term money,
they get their investments back fast, etc.
This of course affects everyone who doesn't have a couple of million in the bank, basically the majority of the population. It goes without saying that it's practically impossible for a student
moving to a big city to study at a university to pay +€1k / month for 20 square meters, but this is what the housing market has turned in to. People who own an apartment rent it out second hand on
the black market for twice the price, and people are buying in because they are desperate to get away from their depressing small towns/parents' homes.
There are several other major changes that has brought on the current situation, but increased immigration is only a small part of a bigger picture.
EDIT: I realize that most of the things I'm describing are probably happening all around Europe, in some or other form.
Edited by Paralytik
"My milkshake bringeth all ye gentlefolk to the yard. Verily 'tis better than thine, I would teach thee, but I must levy a fee."
"When Kleiner showed me the sky-line of New York I told him that man is like the coral insect—designed to build vast, beautiful, mineral things for the moon to delight in after he is dead."
This is the sort of froth-mouthed hysteria one would expect on Twitter. I'm no great fan of UKIP's stance on immigration and their science policy is junk, but resorting to spurious accusations of
rape and child abuse is a pretty desperate debating technique.
You may expect it on twitter but this is the view I formed from watching the news and reading statements made by actual UKIP candidates, plus I didn't accuse anyone of rape I said they think it's OK,
perhaps that was a bit strong so I'll amend it, they and their supporters don't seem to think rape is as big an issue as it's made out to be.
'Women concerned about rape should take reasonable care,' says Roger Helmer
No rapists shouldn't rape, end of argument.
But Farage once again had to contend with the highly contentious views of a party supporter. Marchessini, who previously said there was no such thing as date rape, said that rape could not take
place in marriage. "There's no such thing," he said.
Well that's just idiocy, you get married because you love the person you're marrying, it's not a license to grant you sex when you feel like it.
However animal rape, that's serious
A Ukip parliamentary candidate standing in Wales has claimed a “homosexual donkey” tried to rape his horse.
OK he was probably just having a joke, but still. he's standing for parliament.
As for child abuse well there's this
Brand said: "In the 'practice question' Farage says it’s okay to hit children 'it’s good for them to be afraid' he said. There is a lot of fear about in our country at the moment and he is
certainly benefiting from it."
Much as I think Russell Brand is an idiot and prone to exaggeration he doesn't strike me as a liar.
No twitter, you'll notice, and I believe UKIP have taken steps to address some of the worse excesses, but they do seem to attract a certain type of supporter/candidate and the gaffes are still coming
But strangely, any publicity seems to be good publicity for UKIP, I get rather dismayed by the number of people who appear and uncritically support them when one of their candidates for running the
country comes out with some absurdity.
Please note my prior comments about the other parties still stand.
As for the Lib Dems having no choice, we're in the run up to an election, they have nothing to lose by standing up for their core principles and everything to gain, so what if their coalition
partners get angry, they've blocked legislation that went against their principles before, but for the last couple of years they may as well have been puppets on strings.
They are nodding along with the Conservatives over things like removing £300million from the legal aid budget with no prior impact assessment, meaning those at the bottom of the heap have to rely on
the charity of lawyers working for free to get justice. Several attempts to restrict the Judicial Revue process, which is the only means the man in the street has to hold the authorities to account,
this is going to the Lords for the third time, hopefully they'll block it again. Nodding through DRIP because 'terrorism'. They haven't made a single sound regarding the latest Conservative outburst
about banning encryption to deny terrorists a safe means of communication despite the minor detail that it will kill e-commerce in the UK and turn us into a technological 3rd world... if we're lucky.
So unless the Lib Dems get their sh*t together pretty damn quick I see no reason they should be trusted with the reins of power either.
Edited by esme
No rapists shouldn't rape, end of argument.
While I agree with the sentiment that rapists shouldn't do what they do, that wording is the second dumbest shit I've read on the subject, beaten only by the "legitimate rape" thing one of the US
Republicans said a couple of years ago. Rapists shouldn't rape, but they will anyway, so you should take care. Or would you care to argue that having a burglar alarm enables burglar culture?
If you put it on rapists to stop raping... you'll be waiting a while. Declining to look after yourself on the expectation that someone else will do the right thing is effectively putting a neon "rape
me" sign around your neck.
Edited by Xarg
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I was pointing out that it isn't the victims fault, it is always the rapists fault, I'm not putting it on rapists to stop raping, I'm putting it on society to stop them and not blame the victim.
It doesn't matter what a woman wears, how drunk she is, where she is, if she's married to the person or not it's not their fault it's the rapists fault.
If a man does not have consent or if the woman is incapable of giving consent then it's rape.
If a man seriously cannot control himself, then that man should be incarcerated to protect both themselves and everyone else from their actions.
We're getting a little off the big brother topic though, I suggest if you want to continue this you start another topic on the subject
No, I agree with you completely, however having recently read that some campus organisations in the US had discouraged women from attending self-defense courses, instead demanding that "men be taught
not to rape", so I was reacting more to the "rapists shouldn't rape" thing as I've seen it used recently, I'm not meaning to attack you personally. I agree that it is the rapist's fault.. but in the
same way that leaving your door open encourages someone of questionable moral standards to take a peek around your house, and hey, you aren't watching that TV anyway, being drunk in a short dress in
public might encourage someone to make a pass at you.
Should you be able to be drunk, and wear what you want without repercussion? Sure. Is that the reality we live in? Nope. I'll end on this note however, I don't want this conversation starting up
here, I see enough of it elsewhere and I'm tired of it. Someone else can make a thread if they want.
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Consider it left
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While I agree with the sentiment that rapists shouldn't do what they do, that wording is the second dumbest shit I've read on the subject ... Rapists shouldn't rape, but they will anyway, so you
should take care. Or would you care to argue that having a burglar alarm enables burglar culture?
Precisely. Advising people to take common-sense precautions to reduce their exposure to the risk of crime is not "blaming the victim", even if the screeching Guardianista mob is too thick to tell the
Yes, in an ideal world it would be nice if criminals didn't commit acts of violence and precautions weren't needed, but tough shit, we don't live in an ideal world. Discouraging people from taking
such precautions because "they shouldn't have to" is just endangering people for no good reason.
If a man does not have consent or if the woman is incapable of giving consent then it's rape.
Of course if they're both incapacitated it's still conveniently the man's fault because of patriarchy and stuff. Equality before the law seems to be a principle that can be freely thrown on the
bonfire any time Gender Issues are concerned.
But strangely, any publicity seems to be good publicity for UKIP, I get rather dismayed by the number of people who appear and uncritically support them when one of their candidates for running
the country comes out with some absurdity.
I'm pretty sure that's precisely why they are gathering popular support. They are not afraid to say publically what a lot of ordinary people actually think, in a society where the main parties (and
public figures more generally) are incapable of ever straying from the standard politically-correct script.
• 1
Precisely. From what I gather, the mainstream politicians in Sweden have embraced multi-culturalism to such an extent that merely criticising government immigration policy can be prosecuted as
"hate speech". It is no wonder that far right parties can thrive in this kind of environment.
Stupid grandstanding politicians all over the world seem incapable of recognising the simple truth that just because you ban, censor and outlaw political viewpoints doesn't mean that people will
stop holding those viewpoints. Instead they will see themselves as "oppressed" and become more extremist.
I think we should also be mindful of the background of many immigrants. Many don't get proper education from the country where they came from, especially if they are refugees from a place like Syria.
You know, these people are easily manipulated. They are naive, they are the target audience of terrorists to convert into soldiers. I think that a big thing of the coming years is to make sure that
at least the future generation gets a better future. Through education the likelihood someone can use these people again is significantly lower.
So much talk of globalization. But all those concepts seem to be very vulnerable, and things didn't seem to change much from the 20'th century even if the Cold War ended.
"I really perceive that vanity about which most men merely prate — the vanity of the human or temporal life. I live continually in a reverie of the future. I have no faith in human perfectibility. I
think that human exertion will have no appreciable effect upon humanity. Man is now only more active — not more happy — nor more wise, than he was 6000 years ago. The result will never vary — and to
suppose that it will, is to suppose that the foregone man has lived in vain — that the foregone time is but the rudiment of the future — that the myriads who have perished have not been upon equal
footing with ourselves — nor are we with our posterity. I cannot agree to lose sight of man the individual, in man the mass."...
- 2 July 1844 letter to James Russell Lowell from Edgar Allan Poe.
Anyone seen this abuse of parliamentary procedure by four Lords backbenchers
Casually adding 18 pages of clauses from the summarily rejected Communications Data bill to, the pretty much guaranteed to pass, Counter Terror bill.
These concerned citizens just happen to be
Lord King the former Conservative defence secretary
Lord Carlile the Liberal Democrat former reviewer of counter-terror laws
Lord West the former Labour defence minister
Lord Blair the former Metropolitan police commissioner
Who were trying to create a security state when they weren't lords, seems nothing has changed.
If this is allowed to pass then any random lord can simply staple whatever self serving clauses they like on the back of any bill thats going to pass and if no one spots it it will become law.
The reading is today, if it passes, kiss goodbye to any privacy you thought you ever had
I forgot to mention, they carefully removed any clauses requiring a judicial warrant, so it all happens with the permission of a senior officer of a nod from the home secretary, the legal system is
excluded from interfering.
Edited by esme
Perhaps this will be the wake-up call we need to get rid of the House of Lords altogether. It's ridiculous that in 2015 we still have an anachronistic, anti-democratic upper house full of un-elected
political appointees and clergymen influencing the legislative process in this way.
I'm for that, I'd like it to be a fully elected house equal in authority to the commons, the difficulty is stopping it being a clone of the commons so one house ends up rubber stamping legislation
made by the other.
Maybe we could use the internet and have every piece of commons legislation subject to approval by the electorate, if they want legislation passed they have to explain it to us so we understand why
it's necessary. Sadly National Security considerations would prevent anything like this from working and MP's would never submit to having their decisions vetoed by the mere electorate.
I just spoke to the office for my local MP and they are forwarding on my message as he is down in London atm.
Well the amendment which was added because 4 Lords decided it was vital to the security of the nation and we would be subject to terrorist atrocities if it did not become law as a matter of urgency
without it was withdrawn without a vote by their spokesman after wasting a few hour of the Lords time
So not so vital and urgent
My bet is they try again when they think no one is looking
I never really had an opionion on the Lords of parliment, but now I see it is just a bunch of old white guys with hidden agenda's.
From http://openrightsgroup.tumblr.com/post/109243426910/terrorists-know-snapchat
I am not a tweeter. We have Facebook and Twitter. Somebody tried to explain WhatsApp to me; somebody else tried to explain Snapchat. I do not know about them, but it is absolutely clear that the
terrorists and jihadists do.
- Lord King of Bridgwater, former Defence Secretary and later chairman of the Intelligence and Security Committee, amending the Counter-Terrorism and Security Bill to include the text of the
Communications Data Bill. January 2015.
Or in other words
I have absolutely no clue how this technology works, how useful it is, how it affects people or what the impact of any laws I pass regarding it may have.
Regardless of this, I feel confident that I can pass laws affecting everyone who uses it despite my total ignorance of the matter and so this is my attempt to poke the hornets nest with a short
Because Terrorism woooOOOOooo *waves hands in an attempt to hypnotise the audience into fearful submission*
Both Peers and MP's should not be allowed to vote on an issue unless they can correctly answer 2 out of 3 questions on it as selected by the opposition, and if less than 10% of the house vote, the
bill is not passed.
Oh, and abolish the party whip.
Lawmakers should have a detailed understanding of the laws they make.
I never really had an opionion on the Lords of parliment, but now I see it is just a bunch of old white guys with hidden agenda's.
I regard it as vote stuffing by whichever party is in office in order to have the lords rubber stamp legislation from the commons.
Edited by esme
Lawmakers should have a detailed understanding of the laws they make.
So, I take it you're not a fan of the "You have to pass it so you can read what's in it" kind of lawmaking?
Edited by Xarg
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And from all reports because these four Lords didn't get their way, they are going to try again next week.
Smacks of stamping feet and holding their breath until they get their way to me.
They are desperate to get this through before the election, because they doubt their chances of getting it through afterwards.
Hopefully the NDAA and PATRIOT ACT's taught people lessons about signing it so they could read it, sadly, "turrism" is that magic buzzword that gets pens flowing like virtually no other.
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Our problem is that the Lords are there for life, in theory they can do this tactic to any and every bill heading through the house until they get their way and there's nothing the electorate can do
to stop them, we can't vote for new lords, we cant remove their peerages.
They can stall parliament, and stop every bill in it's tracks by forcing it into ping-pong between the houses by stuffing these amendment in, I think the only way to stop that is via the parliament
act which would effectively dissolve the House of Lords and there would be no check or balance on legislation from the commons.
Edited by esme
Googe to be 'forced' to divulge more about what personal data they gather and what they do with it:
They'll divulge only what they want to give out and the rest will remain a secret.
I'm pretty sure that's precisely why they are gathering popular support. They are not afraid to say publically what a lot of ordinary people actually think, in a society where the main parties
(and public figures more generally) are incapable of ever straying from the standard politically-correct script.
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Sphere Volume from Mass and Density
`V = "M" /( rho )`
Enter a value for all fields
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radius of the sphere (r) based on the volume.
The Math / Science
The equation for the volume of a sphere is as follows:
V = 4/3•π•r³
The formula for density is:
The two formulas are combined in this calculator:
Note: the volume of a sphere is also: V = 4/3•π•r³
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Bayesian inference of Block mixture models for clustering - PDF Free Download
Proposition de stage LSIS UMR CNRS 72 96
Bayesian inference of block mixture models for clustering
Bayesian inference of Block mixture models for clustering 1
Subject description
The problem of complex data analysis is a central topic of modern statistical and computer and information sciences, and is connected to both theoretical and applied parts of theses sciences, as well
as to several application domains, including pattern recognition, signal processing, bio-informatics data mining, complex systems modeling, etc. The analysis of complex data, in general, implies the
development of statistical models and autonomous learning algorithms that aim at acquiring knowledge from raw data for analysis, interpretation and to make accurate decisions and predictions for
future data. Cluster analysis of complex data is one essential task in statistical machine learning and pattern recognition. One of the most popular approaches in cluster analysis is the one based on
mixture models (Titterington et al., 1985; McLachlan and Peel., 2000), known as model-based clustering (McLachlan and Basford, 1988; Celeux and Govaert, 1993; Banfield and Raftery, 1993; Fraley and
Raftery, 2002). The problem of clustering therefore becomes the one of estimating the parameters of the supposed mixture model. The model estimation can be performed by maximizing the observed-data
likelihood by the expectation-maximization (EM) algorithm (Dempster et al., 1977; McLachlan and Krishnan, 1997) or extensions such as Classification EM (CEM) (Celeux and Govaert, 1992), or stochastic
extensions, namely (Celeux et al., 1996; Celeux and Diebolt, 1985). This approach is referred to as the maximum likelihood estimation (MLE) approach. However, the MLE approach may fail due to
singularities or degeneracies (e.g. see (Stephens, 1997; Fraley and Raftery, 2007) for namely Gaussian mixtures). The Bayesian approach of mixture models (Stephens, 1997; Robert, 1994; Jean-Michel
Marin and Robert, 2005; Fraley and Raftery, 2007; Bensmail et al., 1997; Richardson and Green, 1997) avoids the problems associated with the maximum likelihood described previously. by replacing the
MLE by a maximum a posterior (MAP) estimation. This is namely achieved by adding regularization over the model parameters via prior parameter distributions, which are assumed to be uniform in the
case of MLE. The Bayesian formulation has recently took extensive research namely from a non-parametric prospective. The standard model-based clustering techniques (Bayesian and non-Bayesian) aim at
automatically providing a partition of the data into homogeneous groups of individuals, or possibly in variables. Model-based co-clustering (Govaert and Nadif, 2003, 2008, 2013), also called
bi-clustering or block clustering, aim at automatically and simultaneously co-clustering the data into homogeneous blocks, a block being a simultaneous association of individuals and variables. They
in rely on ‘block’ mixture models (Govaert and Nadif, 2013) and have been developed for binary data (Govaert and Nadif, 2003, 2008; Keribin et al., 2012), categorical data (Keribin et al., 2014),
contingency table (Govaert and Nadif, 2003, 2006, 2008) and continuous data (Lomet, 2012; Govaert and Nadif, 2013). The block-mixture can estimated by a block CEM for maximum classification
likelihood and hard co-clustering (Govaert and Nadif, 2003, 2006, 2008) or a block (variational) EM for maximum likelihood estimation and fuzzy co-clustering (Govaert and Nadif, 2006). These
interesting and quite recent block mixture models have then been examined from a Bayesian prospective to deal with some problems encountered in the MLE approach. Namely, Keribin et al. (2010)
proposed a stochastic technique for the latent block model for binary data, by associating a stochastic EM with Gibbs sampling. Recently, in Keribin et al. (2012, 2014), the authors proposed for the
Bayesian formulation of the latent block mixture, for respectively binary data and categorical data, a variational Bayesian inference and Gibbs sampling technique. The model selection, in model-based
co-clustering, which in general consists in selecting the best number of blocks (co-clusters) is central and can be performed by approximated penalized log-likelihood criteria such as approximated
ICL or BIC-like criteria as in (Lomet et al., 2012b,a; Lomet, 2012). Keribin et al. (2012) also proposed a Bayesian sampling algorithm to derive ICL and BIC criteria for model selection in the
context of binary data. Then, Keribin et al. (2014) devoloped a Bayesian inference technique using MCMC for the latent block model for categorial data, and a exact ICL for model selection.
Scientific objectives
The scientific objectives of this training are three-fold: 1. to implement the Bayesian block mixture of Keribin et al. (2014) (for categorical data), and test it on a text mining application, 2.
then, to develop an extension to the case of multivariate data by using Gaussian distributions rather than multinomials, 3. and finally, to formulate the block mixture model into a Bayesian
non-parametric context, by using a Chinese Restaurant Process as a prior (Samuel and Blei, 2012).
Additional Information
Supervisor: Faicel Chamroukhi http://chamroukhi.univ-tln.fr/, Maître de conférences Location: The internship will be conducted within the LSIS laboratory UMR CNRS 7296, in Toulon Required skills:
Bases of statistical modeling and estimation; Strong programming skills in Matlab, R or Python; Scientific English Desired skills: Unsupervised Learning, Mixture models, EM algorithms, Bayesian
inference Internship gratification: 436.05 e / month Possibility of a phd position after the internship How to apply: Send your CV + transcripts + reference letter(s), in a single NAME_Surname.pdf
file, to
[email protected]
par Faicel Chamroukhi
année 2014/2015
Proposition de stage LSIS UMR CNRS 72 96
Bayesian inference of block mixture models for clustering
References Banfield, J. D. and Raftery, A. E. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics, 49(3):803–821. Bensmail, H., Celeux, G., Raftery, A. E., and Robert, C. P. (1997).
Inference in model-based cluster analysis. Statistics and Computing, 7(1):1–10. Celeux, G., Chauveau, D., and Diebolt, J. (1996). Stochastic versions of the em algorithm: an experimental study in the
mixture case. Journal of Statistical Computation and Simulation, 55(4):287–314. Celeux, G. and Diebolt, J. (1985). The SEM algorithm a probabilistic teacher algorithm derived from the EM algorithm
for the mixture problem. Computational Statistics Quarterly, 2(1):73–82. Celeux, G. and Govaert, G. (1992). A classification EM algorithm for clustering and two stochastic versions. Computational
Statistics and Data Analysis, 14:315–332. Celeux, G. and Govaert, G. (1993). Comparison of the mixture and the classification maximum likelihood in cluster analysis. Journal of Statistical
Computation and Simulation, 47:127–146. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of The Royal Statistical Society,
B, 39(1):1–38. Fraley, C. and Raftery, A. E. (2002). Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97:611–631. Fraley, C. and
Raftery, A. E. (2007). Bayesian regularization for normal mixture estimation and model-based clustering. Journal of Classification, (2):155–181. Govaert, G. and Nadif, M. (2003). Clustering with
block mixture models. Pattern Recognition, 36(2):463 – 473. Biometrics. Govaert, G. and Nadif, M. (2006). Fuzzy clustering to estimate the parameters of block mixture models. Soft Computing, 10
(5):415– 422. Govaert, G. and Nadif, M. (2008). Block clustering with bernoulli mixture models: Comparison of different approaches. Computational Statistics & Data Analysis, 52(6):3233 –3245.
Govaert, G. and Nadif, M. (2013). Co-Clustering. Computer engineering series. Wiley. 256 pages. Jean-Michel Marin, K. M. and Robert, C. P. (2005). Bayesian modelling and inference on mixtures of
distributions. Bayesian Thinking - Modeling and Computation, (25):459–507. Keribin, C., Brault, V., Celeux, G., and Govaert, G. (2012). Model selection for the binary latent block model. In
Proceedings of COMPSTAT. Keribin, C., Brault, V., Celeux, G., and Govaert, G. (2014). Estimation and selection for the latent block model on categorical data. Statistics and Computing, pages 1–16.
Keribin, C., Govaert, G., and Celeux, G. (2010). Estimation d’un modèle à blocs latents par l’algorithme SEM. In 42èmes Journées de Statistique, Marseille. Lomet, A. (2012). Sélection de modèle pour
la classification croisée de données continues. Ph.D. thesis, Université de Technologie de Compiègne. Lomet, A., Govaert, G., and Grandvalet, Y. (2012a). An approximation of the integrated
classification likelihood for the latent block model. In ICDM Workshops, pages 147–153. Lomet, A., Govaert, G., and Grandvalet, Y. (2012b). Model selection in block clustering by the integrated
classification likelihood. In 20th International Conference on Computational Statistics (COMPSTAT), pages 519–530. McLachlan, G. and Basford, K. (1988). Mixture Models: Inference and Applications to
Clustering. Marcel Dekker, New York. McLachlan, G. J. and Krishnan, T. (1997). The EM algorithm and extensions. New York: Wiley. McLachlan, G. J. and Peel., D. (2000). Finite mixture models. New
York: Wiley. Richardson, S. and Green, P. J. (1997). On bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society, 59(4):731–792. Robert, C. P.
(1994). The Bayesian choice: a decision-theoretic motivation. Springer-Verlag. Samuel, J. G. and Blei, D. M. (2012). A tutorial on bayesian non-parametric model. Journal of Mathematical Psychology,
56:1–12. Stephens, M. (1997). Bayesian Methods for Mixtures of Normal Distributions. PhD thesis, University of Oxford. Titterington, D., Smith, A., and Makov, U. (1985). Statistical Analysis of
Finite Mixture Distributions. John Wiley & Sons.
par Faicel Chamroukhi
année 2014/2015 | {"url":"https://pdffox.com/bayesian-inference-of-block-mixture-models-for-clustering-pdf-free.html","timestamp":"2024-11-14T04:37:24Z","content_type":"text/html","content_length":"35129","record_id":"<urn:uuid:372dfce9-a98a-4611-bb0e-cf3fdd6d2f19>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00209.warc.gz"} |
NCERT Solutions for Class 10 Maths Chapter 11 Constructions Ex 11.1
Get Free NCERT Solutions for Class 10 Maths Chapter 11 Ex 11.1 PDF. Constructions Class 10 Maths NCERT Solutions are extremely helpful while doing your homework. Exercise 11.1 Class 10 Maths NCERT
Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 11 Maths Class 10 Constructions Exercise 11.1 provided in NCERT TextBook.
Topics and Sub Topics in Class 10 Maths Chapter 11 Constructions:
┃Section Name │Topic Name ┃
┃11 │Constructions ┃
┃11.1 │Introduction ┃
┃11.2 │Division Of A Line Segment ┃
┃11.3 │Construction Of Tangents To A Circle ┃
┃11.4 │Summary ┃
NCERT Solutions for Class 10 Maths Chapter 11 Constructions Ex 11.1
NCERT Solutions for Class 10 Maths Chapter 11 Constructions Ex 11.1 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 11 Constructions Ex
┃Board │CBSE ┃
┃Textbook │NCERT ┃
┃Class │Class 10 ┃
┃Subject │Maths ┃
┃Chapter │Chapter 11 ┃
┃Chapter Name │Constructions ┃
┃Exercise │Ex 11.1 ┃
┃Number of Questions Solved │5 ┃
┃Category │NCERT Solutions ┃
Ex 11.1 Class 10 Maths Question 1.
Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts.
Ex 11.1 Class 10 Maths Question 2.
Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are \(\frac { 2 }{ 3 }\) of the corresponding sides of the first triangle.
You can also download the free PDF of Ex 11.1 Class 10 Constructions NCERT Solutions or save the solution images and take the print out to keep it handy for your exam preparation.
Download NCERT Solutions For Class 10 Maths Chapter 11 Constructions PDF
Ex 11.1 Class 10 Maths Question 3.
Construct a triangle with sides 5 cm, 6 cm, and 7 cm and then another triangle whose sides are \(\frac { 7 }{ 5 }\) of the corresponding sides of the first triangle.
Ex 11.1 Class 10 Maths Question 4.
Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 1\(\frac { 1 }{ 2 }\) times the corresponding sides of the isosceles triangle.
Ex 11.1 Class 10 Maths Question 5.
Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are \(\frac { 3 }{ 4 }\) of the corresponding sides of the triangle ABC.
Ex 11.1 Class 10 Maths Question 6.
Draw a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then, construct a triangle whose sides are \(\frac { 4 }{ 3 }\) times the corresponding sides of ∆ABC.
Ex 11.1 Class 10 Maths Question 7.
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are \(\frac { 5 }{ 3F }\) times the corresponding sides of
the given triangle.
Steps of Construction:
1. Construct a ∆ABC, such that BC = 4 cm, CA = 3 cm and ∠BCA = 90°
2. Draw a ray BX making an acute angle with BC.
3. Mark five points B[1], B[2], B[3], B[4] and B[5] on BX, such that BB[1] = B[1]B[2] = B[2]B[3] = B[3]B[4] = B[4]B[5.]
4. Join B[3]C.
5. Through B[5], draw B[5]C’ parallel to B[3]C intersecting BC produced at C’.
6. Through C’, draw C’A’ parallel to CA intersecting AB produced at A’.
Thus, ∆A’BC’ is the required right triangle.
Class 10 Maths Constructions Mind Maps
Construction implies drawing geometrical figures accurately such that triangles, quadrilateral and circles with the help of ruler and compass.
Division of a Line Segment
A line segment can be divided in a given ratio (both internally and externally)
Divide a line segment of length 12 cm internally in the ratio 3:2.
Solution :
Steps of construction :
(i) Draw a line segment AB = 12 cm. by using a ruler.
(ii) Draw a ray making a suitable acute angle ∠BAX with AB.
(iii) Along AX, draw 5 ( = 3 + 2) arcs intersecting the ray AX at A[1]? A[2], A[3], A[4] and A[5] such that
AA[1] = A[1]A[2] = A[2]A[3] = A[3]A[4] = A[4]A[5]
(iv) Join BA[5].
(v) Through A[3] draw a line A[3]P parallel to A[5]B making ∠AA[3]P = ∠AA[5]B, intersecting AB at point P.
The point P so obtained is the required point, which divides AB internally in the ratio 3 : 2.
Similar Triangles
(i) This Construction involves two different situation.
(a) Construction of a similar triangle smaller than the given triangle.
(b) Construction of a similar triangle greater than the given triangle.
(ii) The ratio of sides of the triangle to be constructed with the corresponding sides of the given triangle is called scale factor.
Draw a triangle ABC with side BC = 7 cm. ∠B = 45°, ∠A = 105°. Construct a triangle whose sides are (4/3) times the corresponding side of ∆ABC.
Solution :
Steps of construction :
(i) Draw BC = 7 cm.
(ii) Draw a ray BX and CY such that ∠CBX= 45° and
∠BCY = 180° – (45° + 105°) = 30°
Suppose BX and CY intersect each other at A.
∆ABC so obtained is the given triangle.
(iii) Draw a ray BZ making a suitable acute angle with BC on opposite side of vertex A with respect to BC.
(iv) Draw four (greater of 4 and 3 in 4/3) arcs intersecting the ray BZ at B[1], B[2], B[3], B[4] such that BB[1] = B[1]B[2] = B[2]B[3] = B[3]B[4].
(v) Join B[3] to C and draw a line through B[4] parallel to B[3]C, intersecting the extended line segment BC at C’.
(vi) Draw a line through C’ parallel to CA intersecting the extended line segment BA at A’. Triangle A’BC’ so obtained is the required triangle.
Tangents to a Circle
Two tangents can be drawn to a given circle from a point outside it.
Draw a circle of radius 4 cm. Take a point P outside the circle. Without using the centre of the circle, draw two tangents to the circle from point P.
Solution :
Steps of construction :
(i) Draw a circle of radius 4 cm.
(ii) Take a point P outside the circle and draw a secant PAB, intersecting the circle at A and B.
(iii) Produce AP to C such that AP = CP.
(iv) Draw a semi-circle with CB as diameter.
(v) Draw PD ⊥ CB, intersecting the semi-circle at D.
(vi) With P as centre and PD as radius draw arcs to intersect the given circle at T and T’
(vii) Join PT and PT’. Then, PT and PT’ are the required tangents.
If centre of a circle is not given, then it can be located by finding point of intersection of perpendicular bisector, of any two nonparallel chords of a circle.
NCERT Solutions for Class 10 Maths Chapter 11 Constructions (Hindi Medium) Ex 11.1
NCERT Solutions for Class 10 Maths
We hope the NCERT Solutions for Class 10 Maths Chapter 11 Constructions Ex 11.1, help you. If you have any query regarding NCERT Solutions for Class 10 Maths Chapter 11 Constructions Exercise 11.1,
drop a comment below and we will get back to you at the earliest. | {"url":"https://www.learncbse.in/ncert-solutions-class-10th-maths-chapter-11-constructions/","timestamp":"2024-11-08T04:14:51Z","content_type":"text/html","content_length":"170371","record_id":"<urn:uuid:4752eac6-90ff-462d-b9fc-bb4853b1e3b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00399.warc.gz"} |
[QSMS 위상기하 세미나 2021.11.25] Exotic families of Weinstein manifolds with Milnor fibers of ADE types
• Date : 11월 25일 (목) 13:30 ~ 15:00
• Place : 27동 220호 (서울대학교)
• Speaker : 이상진 (IBS-CGP)
• Title : Exotic families of Weinstein manifolds with Milnor fibers of ADE types
• Abstract :
In this talk, we will discuss a way of constructing diffeomorphic families of different Weinstein manifolds via Lefschetz fibrations. We focus on the construction of an exotic pair (X, Y), where
X is the Milnor fiber of A-type on dimension 6. If time allows, we will discuss two generalizations of the construction. One is to consider the case of higher dimensions, and the other is to
consider Milner fibers of other types.
This talk is based on joint work with Dongwook Choa (KIAS) and Dogancan Karabas (Northwestern University). | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&order_type=desc&page=3&l=en&document_srl=2023&listStyle=viewer","timestamp":"2024-11-11T02:03:29Z","content_type":"text/html","content_length":"22985","record_id":"<urn:uuid:94cb1585-612c-4423-beea-067c71db75f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00306.warc.gz"} |
An introduction to the non-perturbative renormalization group (1/6)
Bertrand Delamotte
LPTMC, UPMC and CNRS
Fri, Jan. 10th 2014, 10:00-12:15
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
We provide an introduction to Wilson's renormalization group and its modern nonperturbative implementations (NPRG). The scalar O(N) models will be our favourite playground. We start by introducing
the conceptual and technical framework used throughout these lectures: Wetterich's version of Wilson's RG. The exact RG equation is derived showing how and why Kadanoff's block-spin idea is
conveniently implemented on the Gibbs free energy (the generating functional of one-particle-irreducible correlation functions). We then derive the two main nonperturbative approximation schemes: the
derivative expansion (DE) on one hand and the Blaizot-Mendez-Wschebor (BMW) scheme on the other hand. We show how the DE truncated at its lowest order(s) yields both an intuitive and powerful method
to compute in a unified scheme both universal and nonuniversal quantities, either at or away from criticality. We show in particular how a single set of equations allows us to retrieve all known
results of the O(N) models in all dimensions (including the Kosterlitz-Thouless transition, the large N limit and accurate results in three dimensions). Then we will show how the BMW method allows us
to compute the momentum dependence of the two-point functions which is out of reach of the DE. A comparison between the results thus obtained and the best experimental and numerical measurements will
be presented on the example of the critical structure factor of the Ising model in three dimensions. If time allows, we will review some important results obtained by means of the NPRG in different
areas of physics. Among others, the Kardar-Parisi-Zhang equation describing in particular the growth of interfaces will be taken as an example where genuinely nonperturbative phenomena show up that
can nevertheless be captured with the NPRG. | {"url":"https://www.ipht.fr/en/Phocea/Vie_des_labos/Seminaires/index.php?id_type=6&type=6&id=992565","timestamp":"2024-11-08T11:02:21Z","content_type":"text/html","content_length":"26402","record_id":"<urn:uuid:f4b9f494-d19b-479e-b5a7-9edfa07d047b>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00522.warc.gz"} |
Large N Species
One of the talks at the
previously mentioned
Origin of Time's Arrow
was by
Gia Dvali
. He talked about his recent paper
The idea is really cute. First, let me summarize some basics: Numerous results lead us to expect that black holes emit thermal radiation with a temperature proportional to the inverse of the black
hole's mass. This means the more mass the hole looses through the radiation, the hotter it becomes. It is unknown whether a collapse into a black hole, and a subsequent complete evaporation really
destroys information about the initial state. This process can also violate certain conservation laws like baryon number. But electric charge, as well as energy and other gauge charges are conserved.
However, in standard General Relativity black holes have no 'hair', i.e. the asymptotic solution is completely characterized by only their mass, angular momentum, and electromagnetic charges. So
their ability to carry additional gauge charges is limited, unless one allows for quantum 'hair' that resides on the horizon [1]. Though this quantum hair does not have long-range fields, its gauge
charge is a conserved quantity.
Now consider a black hole with
different such conserved charges, and assume that these charges are (as is the case for the electric charge as well) each bound to massive particles, the lightest of which has a typical mass Λ.
Imagine we set up a black hole that carries these charges, one of each, and we let it completely evaporate. During this evaporation, all the
charges need to be re-emitted somehow. But the black hole's temperature has to be high enough - or the mass has to be small enough respectively - before it can start evaporating off the massive
particles. The required temperature is
~ Λ, or the black hole mass is
/Λ, where
is the
Planck mass
and roughly ~ 10
TeV. To give you a feeling for these numbers: if we were talking about electric charge, the lightest particle is the electron with a mass of roughly .5 MeV, then the black hole can start evaporating
off electric charge if its mass has fallen to ~ 10
However, there is also an obvious limit to this: the black hole needs to be able to provide the mass of the particles. If the black hole was charged but lighter than an electron it couldn't emit the
charge no matter what [2]. If there were many different charges carried by particles with mass scale Λ, one comes to the conclusion that a bound results. The bound arises from the fact that after the
black hole started evaporating off the charges, its mass must still have been high enough to provide all the
particles with mass Λ. One thus has
Λ ≤
, or, if one inserts the above expression for the mass at which the emission of massive particles can start, one finds Λ
The further argument is now the following. We don't know why the gravitational interaction is so much weaker than the other interactions of the standard model (SM). Or, to put it differently, we
don't know why the masses of the SM particles are so much smaller than the Planck mass. If we take Λ to be the typical mass of SM particles (
Higgs VEV
) then there is a gap of roughly sixteen orders of magnitude. Dvali's inequality says if there were very many species particles, then there would
have to be such a hierarchy. Putting in some numbers one finds the 'large' number is indeed very large, and somewhere around
Now, as far as I am concerned this doesn't really 'solve' the hierarchy problem, one has just moved it elsewhere (as one also does with the
extra dimensional models
). Instead of having to explain the gap in the mass-scales one now has to explain where all the other particles are, and why so many of them? However, one can model these as only gravitationally
interacting with our beloved standard model which would then only describe a tiny fraction of all there is. The question is of course why there don't seem to exist many particles of this kind around
us. But this must stem from some processes in the very early universe, and inflation can easily make small numbers large, and blow up initially only subtle differences. Though it is hard to say at
this stage whether it would actually work as desired, I can imagine that such a reformulation of the problem offers the possibility to find a dynamical explanation.
The signatures of such a scenario are in certain regards quite similar to those of extra dimensional models. One has a lot of only very weakly interacting particles whose coupling is given by the
Planck mass. But since there are so many of them, their phase space gets really large, cancels the Planck suppression, and the signatures could become observable somewhere around the scale Λ. In
contrast to the KK-tower in extra dimensional models however, here the number of species is really finite, so one doesn't have the problem of divergences in the higher dimensional integrals.
I can't say I particularly like the idea of having 10
particle species, but I like the paper because it is another example for how thought experiments with black holes can lead to sometimes surprising insights. It's a cute idea to play around with that
resides somewhere between General Relativity and particle physics, which is - still - a region of large mysteries.
What that has to do with the arrow of time however, I honestly don't know.
[1] A black hole can e.g. carry quantum hair associated with discrete gauge charges. This can happen when a local continuous gauge symmetry is broken down to a residual discrete subgroup. See ref [1]
in Dvali's paper.
[2] However, since the electron mass is so much smaller than the Planck scale, such a black hole would long fall into the quantum gravity regime and no reliable statements can be made anyhow.
13 comments:
1. "What that has to do with the arrow of time however, I honestly don't know."
Haha! All the way through, as I read your report, I was thinking, "yeah, interesting, but what on earth does it have to do with the topic of the meeting?". Getting the answer at the end like that
was fun...I suppose it is an indication of the difficulty of solving the problem of the arrow of time that people go to a conference and talk about everything else. But surely there were *some*
talks that were not entirely irrelevant? :-)
2. dude! thanks for the summaries of the talks. i felt bad that this conference was out of the question for me. but now i feel like i had front row tickets :)
3. Isn't black hole evaporation a bit of a paradox, reflecting on the nature of time's arrow? A movie of unitary time evolution would be indistinguishable running forwards or backwards; but black
hole evaporation is supposed to conserve information (and thus the process is in some way reversible) and yet the forwards and backwards running of the movie are distinguishable.
4. Arun: the breaking of an egg when it is dropped would also look funny when reversed, but nobody doubts that that is a unitary process, correct?
5. Hi Dr. Who:
Well, you know, I am just currently organizing a workshop and I have the same problem that some of the people don't even seem to care what the workshop actually is about. Half of the time I
really had to insist they speak about this or that, but I could picture them grumbling but-I'd-rather-this-or-that, some just ignored me, half haven't yet send a title or abstract anyway. Maybe
next time when I organize a workshop I instead make a list of titles, and say those who talk about one of these topics get reimbursement or so?
Hi Chanda:
Unfortunately, my memory is very selective. I tend to recall things I did not understand, which accounts for most of the workshop, but isn't helpful to write about it. It also accounts for most
of NYC btw.
Hi Arun, Hi Anonymous:
Yes, in a certain sense black hole evaporation as every thermal process says potentially something about the arrow of time. I don't think black holes 'destroy' information, so to me this process
is no different from the sun radiating, it's just more complicated to understand.
However, the black holes are different from the broken eggs in the following regard. A broken 'classical' egg can - theoretically - be un-broken by a suitable process, though it's extraordinarily
unlikely. A black hole can classically be formed, but it can never be un-formed. The solution is fundamentally not time reversal invariant (the full Schwarzschild solution is, but it is static)
because there are allowed curves leading inside the horizon but not out of it. Things look different if one includes quantum effects though, one could then argue if one stuffs Hawking radiation
into the hole it will un-collapse or something, though this is extraordinarily unlikely to happen.
Off topic comment: PI's server is apparently down since the early morning, I don't get any emails to my standard address, so either be patient or use alternate address. Thanks,
6. Another possiblity: GR is incomplete. GR cannot handle aspects of angular momentum. The obvious astronomic test, PSR J0737-3039A/B (pdf), requires ~20 years of observation. Chemistry can do it in
two days in commercial hardware.
Angular momentum is the absolute arrow of time. Feynman's sprinkler doesn't go backwards in time, nor do whirlpools. Changing hemispheres can reverse swirl but not swirl plus flow.
7. Great post, Bee, I enjoyed very much reading it!
8. "Maybe next time when I organize a workshop I instead make a list of titles, and say those who talk about one of these topics get reimbursement or so? "
Absolutely! Let's face it, most talks are boring enough even when they are on topic! The other thing you could do is to locate the workshop in some place which nobody likes, so as to eliminate
the people who are just there for a holiday. "We wish to announce the XXXth workshop on Quantum Maggots, to be held in scenic Newark, NJ....with a satellite meeting on Avatars of Tribology in
Mobile, Alabama".
9. you mean like Somewhere, Ontario maybe? I am actually not sure whether your comment is meant to be ironic or not? I wasn't joking when I wrote the above, I really find it annoying if you go to a
workshop on topic X but people talk about topics A,B,C,D instead, just because they like it better or it's a talk they presently have on their laptop. It's even more annoying if it then turns out
that you've heard half of the talks previously anyhow, because they have given that same talk the whole year, including the same lame jokes etc. But to come back to our workshop, we have a bunch
of good speakers who will give some overview talks which will make for an interesting week. Since we have a very mixed audience, I thought I should make sure there are some introductionary talks.
I've tried to get together some particle physics people with the quantum gravity and cosmology guys, we will see how it works out. Best,
10. Well, you know, I am just currently organizing a workshop and I have the same problem that some of the people don't even seem to care what the workshop actually is about. Half of the time I
really had to insist they speak about this or that, but I could picture them grumbling but-I'd-rather-this-or-that, some just ignored me, half haven't yet send a title or abstract anyway. Maybe
next time when I organize a workshop I instead make a list of titles, and say those who talk about one of these topics get reimbursement or so?
Maybe it would make sense to go the other way around, i.e., collect the titles of the topics people want to give, and THEN decide the subject of the workshop so that it retroactively fits the
papers you had submitted anyway. Of course, the subject might have to be somewhat contrived if the papers don't have much to do with one another...
11. Sorry, I was joking about the location part. Certainly it is intensely annoying to go to all the trouble of attending a conference, only to hear that same old boring talk about completely
irrelevant stuff, I fully agree. But it is also annoying to find, as I did when I went to a conference in Paris, that a lot of people went just so that they could visit Paris. Believe me, if you
run a conference in Venice you will get a much stronger response than if you hold it in Detroit. In other words, a lot of people go to conferences for all the wrong reasons.
12. Hi Coin:
I guess it depends on what you organize a workshop for. If you just want to get people together to reach a critical brain mass, then it really doesn't matter what exactly they talk about. If you
want - as is the purpose of my workshop - have different people working on related things to get interested in what the others are/have been doing so there is hopefully some kind of congruence in
the field, then your suggestion doesn't work.
Hi Who,
Depends on how you see it I guess. Every institute of course wants to proudly show off with the local specialities, and if its an interesting place it will naturally attract people. I see a
priori nothing wrong with that. As you say, yes, it is annoying if people register to a conference and then only show up for their own talk or so. This gets certainly noticed by the organizers,
and if it was me, I wouldn't invite these people again. On the other hand I have to say that a bad conference organization can cause such behavior. If a conference is in an interesting place, of
course people want to have time looking around, we all want a life besides work. So better give them some scheduled time to do so, and don't pack too much stuff on the program. As far as I am
concerned, my head is full after 3 talks a day anyhow (given that all of them are new).
The most annoying thing I find about conferences is if the organizers don't take care about accommodation, and do so sufficiently in advance, especially if its a big city etc because people
usually don't make their travel plans sufficiently ahead. Like, when I was in Warsaw some months ago, it was almost impossible to find a hotel room (in fact, I did not find one), in NYC I wasn't
able to get one < $ 300, same last year in Paris, the previous year in Budapest... I mean, how much time am I willing to spend looking for a hotel room for an appropriate rate before I cancel my
13. I think I understood the main argument-if you want to evaporate, you have to have low enough mass, but still more mass than the charges. What I don't understand is why does N have to be related
to different gauge fields. Why couldn't it be N electrons? Or a mixture of 2, 3, etc of cherges, N in total?
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GfxPolyline Low-level graphics
- draw a polyline (AmiBroker 50)
SYNTAX GfxPolyline( x1, y1, x2, y2, ... )
FUNCTION Draws a set of line segments connecting the points specified by arguments (x1,y1), (x2,y2), ... The lines are drawn from the first point through subsequent points using the current pen.
Unlike the GfxLineTo function, the GfxPolyline function neither uses nor updates the current position.
This function takes variable number of arguments and accepts up to 12 points (24 arguments = 12 co-ordinate pairs). The number of arguments must be even as each pair represents (x,y)
co-ordinates of the point.
• x1 - x co-ordinate of first point
• y1 - y co-ordinate of first point
• x2 - x co-ordinate of second point
• y2 - y co-ordinate of second point
• ...
• x12 - x co-ordinate of 12th point
• y12 - y co-ordinate of 12th point
NOTE: This is LOW-LEVEL graphic function. To learn more about low-level graphic functions please read TUTORIAL: Using low-level graphics.
EXAMPLE GfxSelectPen( colorGreen, 2 );
SEE ALSO GfxPolygon() function , GfxSelectPen() function
The GfxPolyline function is used in the following formulas in AFL on-line library:
More information:
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Particles Question from physics and maths tutor
A light emitting diode (LED) emits blue light with a wavelength of 440 nm. The rate of photon emission is 3.0 × 10^16 s−1 . Show that the power output of the LED is approximately 0.014 W.
If somebody could provide a worked solution that would be fantastic as the mark scheme is quite vague and I am getting different values for the power output. Thanks!
each photon has this amount of energy which you can calculate
so E ~4.5x10^-19J
and there are 3x10^16 photons released per second
so in one second
there are 3x10^16 photons released each with energy E
so total energy in one second (definition of power) = 3x10^16*E ~ 0.014W
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Small subpopulations of β-cells do not drive islet oscillatory [Ca2+] dynamics via gap junction communication
The islets of Langerhans exist as multicellular networks that regulate blood glucose levels. The majority of cells in the islet are excitable, insulin-producing β-cells that are electrically coupled
via gap junction channels. β-cells are known to display heterogeneous functionality. However, due to gap junction coupling, β-cells show coordinated [Ca^2+] oscillations when stimulated with glucose,
and global quiescence when unstimulated. Small subpopulations of highly functional β-cells have been suggested to control [Ca^2+] dynamics across the islet. When these populations were targeted by
optogenetic silencing or photoablation, [Ca^2+] dynamics across the islet were largely disrupted. In this study, we investigated the theoretical basis of these experiments and how small populations
can disproportionality control islet [Ca^2+] dynamics. Using a multicellular islet model, we generated normal, skewed or bimodal distributions of β-cell heterogeneity. We examined how islet [Ca^2+]
dynamics were disrupted when cells were targeted via hyperpolarization or populations were removed; to mimic optogenetic silencing or photoablation, respectively. Targeted cell populations were
chosen based on characteristics linked to functional subpopulation, including metabolic rate of glucose oxidation or [Ca^2+] oscillation frequency. Islets were susceptible to marked suppression of
[Ca^2+] when ~10% of cells with high metabolic activity were hyperpolarized; where hyperpolarizing cells with normal metabolic activity had little effect. However, when highly metabolic cells were
removed from the model, [Ca^2+] oscillations remained. Similarly, when ~10% of cells with either the highest frequency or earliest elevations in [Ca^2+] were removed from the islet, the [Ca^2+]
oscillation frequency remained largely unchanged. Overall, these results indicate small populations of β-cells with either increased metabolic activity or increased frequency are unable to
disproportionately control islet-wide [Ca^2+] via gap junction coupling. Therefore, we need to reconsider the physiological basis for such small β-cell populations or the mechanism by which they may
be acting to control normal islet function.
Author summary
Many biological systems can be studied using network theory. How heterogeneous cell subpopulations come together to create complex multicellular behavior is of great value in understanding function
and dysfunction in tissues. The pancreatic islet of Langerhans is a highly coupled structure that is important for maintaining blood glucose homeostasis. β-cell electrical activity is coordinated via
gap junction communication. The function of the insulin-producing β-cell within the islet is disrupted in diabetes. As such, to understand the causes of islet dysfunction we need to understand how
different cells within the islet contribute to its overall function via gap junction coupling. Using a computational model of β-cell electrophysiology, we investigated how small highly functional
β-cell populations within the islet contribute to its function. We found that when small populations with greater functionality were introduced into the islet, they displayed signatures of this
enhanced functionality. However, when these cells were removed, the islet, retained near-normal function. Thus, in a highly coupled system, such as an islet, the heterogeneity of cells allows small
subpopulations to be dispensable, and thus their absence is unable to disrupt the larger cellular network. These findings can be applied to other electrical systems that have heterogeneous cell
Citation: Dwulet JM, Briggs JK, Benninger RKP (2021) Small subpopulations of β-cells do not drive islet oscillatory [Ca^2+] dynamics via gap junction communication. PLoS Comput Biol 17(5): e1008948.
Editor: Jonathan Rubin, University of Pittsburgh, UNITED STATES
Received: September 15, 2020; Accepted: April 7, 2021; Published: May 3, 2021
Copyright: © 2021 Dwulet et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original author and source are credited.
Data Availability: The raw simulation data that supports the findings of this study are openly available at the BioStudies database under accession number S-BSST628 found at https://www.ebi.ac.uk/
biostudies/studies/S-BSST628. The datasets are organized by figure.
Funding: This work was supported by Juvenile Diabetes Research Foundation (JDRF, https://www.jdrf.org/) Grant 5-CDA-2014-198-A-N (to RKPB); National Institute of Health (NIH, https://www.nih.gov/)
grants R01 DK102950, R01 DK106412, R56 DK106412 (to RKPB); and NIH grant F31 DK126360 (to JMD). The funders had no role in the study design, data collection and analysis, decisions to publish, or
preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Many tissues exist as multicellular networks that have complex structures and functions. Multicellular networks are generally comprised of heterogenous cell populations, and heterogeneity in cellular
function makes it difficult to understand the underlying network behavior. Studying the constituent cells individually is of value. However, understanding how heterogeneous cell populations come
together to form a coherent structure with emergent properties is important to understand what leads to dysfunction in these networks [1]. The multicellular pancreatic islet lends itself to network
theory with its distinct architecture, cellular heterogeneity, and cell-cell interactions.
The pancreatic islet is a micro-organ that helps maintain blood glucose homeostasis [2]. Death or dysfunction to insulin-secreting β-cells within the islet generally causes diabetes [3]. When blood
glucose levels rise, glucose is transported into the β-cell and phosphorylated by glucokinase (GK), the rate limiting step of glycolysis [4–6]. Following glucose metabolism, the ratio of ATP/ADP
increases, closing ATP sensitive K^+ channels (K[ATP]). K[ATP] channel closure causes membrane depolarization, opening voltage gated Ca^2+ channels and elevating intra-cellular free-calcium ([Ca^
2+]), which triggers insulin granule fusion and insulin release [7, 8]. Disruptions to this glucose stimulated insulin secretion pathway occur in diabetes [9–18]. β-cells are electrically coupled by
connexin36 (Cx36) gap junctions which can transmit depolarizing currents across the islet that synchronize oscillations in [Ca^2+]. Under low glucose conditions, gap junctions transmit
hyperpolarizing currents that suppress islet electrical activity [19–22]. Understanding the role cell-cell communication between β-cells plays can increase our understanding of dysfunction to islet
dynamics during the pathogenesis of diabetes.
Despite their robust coordinated behavior within the intact islet, β-cells are functionally heterogeneous [23]. Individual β-cells show heterogeneity in expression of GK [24], glucose metabolism [23
], differing levels of insulin production and secretion [25–28], and faster and irregular [Ca^2+] oscillations when compared to whole islet oscillations [29]. Various cell surface and protein markers
have been used to identify subpopulations of β-cells with differences in functionality and proliferative capacity [30–34]. Nevertheless, the importance of β-cell heterogeneity and how these
subpopulations contribute to islet function is poorly understood.
While many studies of β-cell heterogeneity have been performed in dissociated cells, a few studies have investigated the role of heterogeneity in the intact islet [35]. In one study, following
stimulation via the optogenetic cationic channel channelrhodopsin (ChR2), ~10% of β-cells were found to be highly excitable in that they are able to recruit [Ca^2+] elevations in large regions of
cells across the islet when stimulated at low glucose. These highly excitable cells had higher metabolic activity upon glucose elevation [36]. In another study, the optogenetic Cl^- pump
halorhodopsin (eNpHr3) was used to silence β-cells. A population of ~1–10% “hub” β-cells was discovered that when hyperpolarized by eNpHr3 substantially disrupted coordinated [Ca^2+] dynamics across
the islet. These cells had increased GK expression [37]. In related studies, a small population of cells showed [Ca^2+] oscillations that consistently preceded the rest of the islet and were
suggested to be ‘pacemaker cells’ that drove islet [Ca^2+] dynamics [38]. These cells that coincide with the initiation of the [Ca^2+] wave were suggested to have higher intrinsic oscillation
frequencies [36]. Theoretically, how small subpopulations of cells may be capable of driving elevations and oscillatory dynamics of [Ca^2+] across the islet is not well established, and has been a
significant topic of debate [39, 40].
In this study we explore the theoretical basis for whether small β-cell subpopulations can control multicellular islet [Ca^2+] dynamics. Towards this, we utilize a computational model of the islet
that we have previously validated against a wide-range of experimental data [36, 41–43]. This includes understanding how populations of inexcitable cells suppress islet function and the role for
electrical coupling. We investigate whether small populations of highly metabolically active cells or cells with high frequency oscillations can respectively drive the elevations or dynamics of islet
[Ca^2+] oscillations. We systematically examined the effects of removal of specific cell populations within the context of broad normal distributions, skewed distributions or distinct bimodal
distributions of heterogeneity. Our results indicate that small subpopulations of β-cells with increased metabolic activity or increased oscillation frequency are unable to drive islet [Ca^2+]
oscillations through gap junctional communication. Conversely, those cells with reduced metabolic activity or reduced oscillation frequency have a greater impact on islet [Ca^2+] oscillations.
How variation in metabolic activity impacts islet function
Experimental evidence indicates that within the intact islet there exists 10–20% variation in metabolic activity [44]. Previous modelling studies have represented beta cell heterogeneity as a
unimodal normal distribution with 10–25% variation in GK activity and metabolic activity, which is sufficient to model the impact of electrical coupling and heterogeneity within the islet [36, 41–43
]. However, recent experimental evidence has suggested that hub β-cells have elevated metabolic activity or GK expression, and this small population may disproportionately drive elevated [Ca^2+] [36,
We first asked whether identification of such a ‘hub’ subpopulations may arise as part of the natural variation within a unimodal normal distribution. We simulated an islet with a normal distribution
in GK activity (Fig 1A), and targeted hyperpolarization to a population of cells based on their GK activity. Hyperpolarization as a result of current injection was used to mimic the optogenetic
silencing that was performed in experimental studies [37]; where a key feature is an inhibitory current in the targeted cell that can suppress nearby cells via gap junction currents. Simulated islets
had normal synchronized [Ca^2+] oscillations (Fig 1B), comparable to previous studies [36, 41–43, 45, 46]. When hyperpolarization was targeted to a random set of cells across the islet, near-normal
[Ca^2+] activity was maintained until greater than 20% of cells within the islet were targeted (Fig 1B and 1C). Above this level, the islet lacked significant [Ca^2+] elevations (Fig 1C), consistent
with prior measurements [41, 43]. When hyperpolarization was targeted specifically to cells with either higher GK (GK^Higher) or lower GK (GK^Lower), similar changes in [Ca^2+] activity were observed
as with targeting a random subset of cells: the islet retained near-normal [Ca^2+] activity until greater than 20% of these GK^Higher or GK^Lower cells were targeted (Fig 1C). Nevertheless when 20%
of cells were hyperpolarized, targeting GK^Higher cells did result in silencing of significantly more of the islet compared to GK^Lower cells. Within the simulated islet, we also decoupled and
removed the same GK^Higher or GK^Lower populations. In this case, the remaining islet showed normal elevations in [Ca^2+], with little to no difference between removing GK^Higher or GK^Lower cells (
Fig 1D). We performed network analysis [37] to test whether cells with higher GK activity (GK^Higher) or lower GK activity (GK^Lower) show differing connectivity. GK^Higher cells showed an increased
proportion of links compared to GK^Lower cells (Fig 1E). Therefore, our current simulated islet does not accurately describe the behavior of small highly functional subpopulations identified from
previous experiments.
A). Schematic of unimodal normal distribution of heterogeneous GK activity across simulated islet with 25% variation in GK rate (k[glc]). B). Representative time courses of [Ca^2+] for 3 cells in
simulated islet in A. Left is simulation with 0% hyperpolarized cells and right is simulation with a random 20% of cells hyperpolarized. Blue trace is cell with lowest GK rate (k[glc]), Green is cell
with the average GK rate, yellow is cell with the highest GK rate. C). Fraction of cells showing elevated [Ca^2+] activity (active cells) in simulated islets vs. the percentage of cells
hyperpolarized in islet. Hyperpolarized cells are chosen based on their GK rate. D). Fraction of active cells in islet when cells are uncoupled from the rest of the cells in the simulation. E). # of
links from network analysis of [Ca^2+] activity with GK 25% variation. F). Histogram showing average frequency of cells at varying GK rate (k[glc]) for simulations that have different standard
deviation in GK activity. G). Average duty cycle of cells from simulations with different standard deviation in GK activity. H). As in C. for simulations with standard deviation in GK activity at 50%
of the mean. I). As in D. for simulations with standard deviation in GK activity at 50% of the mean. J). As in E but for simulations with GK 50% variation. Error bars are mean ± s.e.m. Repeated
measures one-way ANOVA with Tukey post-hoc analysis was performed for simulations in C and G, Student’s paired t-test was performed for D and H, and one-way ANOVA was performed for F to test for
significance. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant
difference (p < .001), **** indicates significant difference (p < .0001). Data representative of 5 simulations with differing random number seeds.
Given uncertainty in the exact level of heterogeneity within the islet, we next tested whether changes to the variability in GK could lead to differences in [Ca^2+] upon targeting cells with higher
GK (GK^Higher) or lower GK (GK^Lower) cells. We simulated islets with decreased variation in GK activity (1% variation) or increased variation in GK activity (50% variation) and compared [Ca^2+] with
our previous simulations of 25% variation (Figs 1F and S1A). The duty cycle of the simulated islets slightly decreased as the GK variation increased (Fig 1G), but [Ca^2+] oscillations remained across
the islet that closely matched previous studies. Under 50% variation in GK, when hyperpolarization was targeted to a random set of cells across the islet, the islet retained near-normal [Ca^2+]
activity until greater than 20% of the islet was targeted, as before. In contrast, when hyperpolarization was targeted specifically to cells with higher GK (GK^Higher), [Ca^2+] was largely abolished
for greater than 10% of cells being targeted (Fig 1H). However, when hyperpolarization was targeted to lower GK (GK^Lower) cells, [Ca^2+] was largely unchanged until 30% of cells were targeted (Fig
1H). As such, upon hyperpolarizing 20% of cells, a substantial difference in [Ca^2+] resulted from targeting GK^Higher or GK^Lower cells. Nevertheless, when these higher GK or lower GK cells were
decoupled and removed from the islet, the impact on [Ca^2+] elevations was very minor. A minor 2–4% decrease in [Ca^2+] occurred when removing >10% GK^Higher cells, with no impact when removing GK^
Lower cells (Fig 1I). Following network analysis with increased variation in GK activity, GK^Higher cells showed an increased proportion of links compared to both random cells within the islet and
compared to GK^Lower cells (Fig 1J). Finally, we investigated whether stochastic noise could impact these results. When noise was incorporated, there was little difference in the threshold number of
cells needed to be hyperpolarized to suppress islet activity under either 25% or 50% variation in GK activity (S2A and S2B Fig).
We also tested whether changing other properties of cells with higher GK or lower GK would impact the suppression of [Ca^2+]. When GK activity correlated with gap junction conductance such that
higher GK cells also had increased gap junction conductance (GK^Higher/g[Coup]^Higher), little impact was observed (S3A–S3C Fig): more than 20% of GK^Higher/g[Coup]^Higher cells were still needed to
be hyperpolarized to fully silence the islet. However, when hyperpolarizing 20% of GK^Lower/g[Coup]^Lower cells, [Ca^2+] was largely unchanged. Little difference was observed when GK activity
negatively correlated with K[ATP] conductance, such that higher GK cells also had reduced K[ATP] conductance (S3D–S3F Fig).
Thus, hyperpolarizing a small subpopulation of metabolically active cells can disproportionately suppress islet [Ca^2+], particularly when heterogeneity is very broad or GK activity was correlated
with other beneficial factors. However, when these same cells are removed or absent from the islet, the impact on [Ca^2+] is minimal under the model assumptions set here.
Impact of alternative distributions of functional β-cell subpopulations
We next examined how imposing a unimodal skewed or bimodal distribution in GK activity would impact targeting hyperpolarization to a small population of metabolically active cells. We simulated an
islet with a unimodal skewed distribution, which resulted in a population of highly metabolic cells that comprised 10% of the islet and had ~3 times the GK activity (GK^High) (Fig 2A). The mean GK
activity was equivalent to previous studies, such that the remainder of the islet had slightly reduced GK activity (GK^Low) (Fig 2B). Gap junction coupling conductance of all cells remained unchanged
(S1B Fig). Under this skewed distribution, the islet displayed regular [Ca^2+] oscillations at high glucose, but with slightly lower duty cycle compared to simulations using a unimodal normal
distribution (Fig 2C). We tested the effect of targeting hyperpolarization to either the GK^High or GK^Low cell populations. When all GK^High cells (10%) were hyperpolarized, [Ca^2+] was fully
suppressed across the islet. Conversely, when GK^Low cells (10%) were hyperpolarized, [Ca^2+] showed reduced suppression (~40% activity) (Fig 2D). When a greater proportion of GK^Low cells (20%) were
hyperpolarized, [Ca^2+] was suppressed, as with a unimodal normal distribution. Following network analysis, GK^High cells showed an increased proportion of links, indicating increased connectivity as
measured in hub cells (Fig 2E). Again, we tested the effect of noise under this new distribution and only slight differences were observed (S2C Fig). These results show good agreement between prior
experiments where a small population of cells with high GK activity was able to greatly reduce the [Ca^2+] response under hyperpolarization.
A). Schematic of altered distributions of GK activity across simulated islet. B). Histogram showing average frequency of cells at varying GK rate (k[glc]) for skewed distribution compared with normal
distribution (25% St Dev). C). Representative time courses of [Ca^2+] for 3 cells in simulated skewed islet in A. Blue traces are cells from GK^Low population and orange traces are cells from GK^High
population. D). Fraction of cells showing elevated [Ca^2+] activity (active cells) in skewed simulations vs. the percentage of cells hyperpolarized in islet. Hyperpolarized cells are chosen either
from GK^High (orange bars) or GK^Low (blue bars) population. E). # of links from network analysis of [Ca^2+] activity for simulations with skewed distribution of GK F). Schematic of simulation where
only GK^Low cells are present and no GK^High cells are included. G). Representative time courses of [Ca^2+] for 3 cells in simulated islet in F. H). Average duty cycle of cells from simulations of a
skewed distribution model as in A (With GK^High) and from simulations as in F (Without GK^High). I). As in B. for bimodal distribution in GK activity where the populations only have 2.5% variation in
GK activity. J). As in D. for bimodal distribution in GK activity. K). As in H. but for simulations with no GK^High from a bimodal distribution. Error bars are mean ± s.e.m. Student’s paired t-test
was performed to test for significance. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), ***
indicates significant difference (p < .001), **** indicates significant difference (p < .0001). Data representative of 4–9 simulations with differing random number seeds.
We next tested whether the cells from the highly metabolic population (GK^High) are important to support islet function, by simulating an islet with only cells from the lower GK population (GK^Low) (
Fig 2F). With no GK^High cells present, the islet [Ca^2+] activity still displayed oscillations (Fig 2G), but duty cycle decreased by ~40% (Fig 2H). The number of links for the GK^Low remained
unchanged even when GK^High cells were removed (Fig 2E). These results indicate that [Ca^2+] is maintained across the islet in the absence of a small population (10%) of highly metabolic cells,
suggesting these cells are not required to drive elevated [Ca^2+].
Next, we tested a bimodal distribution where the two populations are substantially different. The GK activity for GK^High cells was still ~3 times the overall mean GK activity, with the overall mean
GK activity unchanged. However, each population had a distinct normal distribution with 2.5% variation (Fig 2I). Gap junction coupling remained unchanged (S1D Fig). Under this bimodal distribution
when all GK^High cells (10%) were hyperpolarized, [Ca^2+] was fully suppressed across the islet. Conversely, when GK^Low cells (10%) were hyperpolarized, [Ca^2+] remained largely unchanged (Fig 2J).
However, when a greater proportion of GK^Low cells (20%) were hyperpolarized, [Ca^2+] was suppressed, as with a unimodal normal distribution under 50% variation. These results show very good
agreement between prior experiments, where very different [Ca^2+] response was observed when hyperpolarizing cells with higher GK and cells with lower GK. However, when GK^High cells were removed
from the bimodal distribution, the islet retained near-normal [Ca^2+] activity, with a minor (~10%) drop in duty cycle (Fig 2K). As such, the simulated islet was capable of maintaining normal
elevated [Ca^2+] in the absence of a small (~10%) highly metabolic subpopulation. Thus, despite showing substantial differences in islet activity when hyperpolarized, a small metabolically active
subpopulation is not required to maintain elevations in oscillatory [Ca^2+] across the islet.
How variations in gap junction coupling impact functional β-cell subpopulations
Metabolically active subpopulations of cells that disproportionately control the islet have increased connectivity [37]. This has been suggested to result from increased gap junction coupling. We
next examined how changes in gap junction electrical coupling affect how targeting hyperpolarization to specific cell populations impacts islet [Ca^2+]. We simulated the islet with the same skewed
distribution in GK activity as in Fig 2B, but correlated gap junction coupling conductance (g[Coup]) with GK activity (k[glc]) across the islet (Figs 3A and S1C). As such, more metabolically active
GK^High cells had ~2 times higher gap junction conductance than that of the population of cells with lower metabolic activity (GK^Low cells). GK^High cells, which had higher coupling, did not show
significantly different suppression of islet Ca^2+ following hyperpolarization compared to GK^Low cells that had reduced coupling (Fig 3B); unlike the significant differences when coupling was
uniform. Thus, increased coupling does not allow GK^High cells to impact the islet to a greater degree upon hyperpolarization (Fig 3C). Next, we correlated coupling with GK activity under a bimodal
distribution in GK activity as in Fig 2I (Figs 3D and S1E). Under this model, when 20% of the highly metabolic GK^High cells were targeted with hyperpolarization, the islet retained [Ca^2+]
elevations (~25%) (Fig 3E). When GK^Low cells with less metabolic activity were targeted with hyperpolarization, the islet also showed less [Ca^2+] activity compared to previous simulations. As such,
the difference in suppression of [Ca^2+] upon targeting hyperpolarization to either population is reduced when highly metabolic cells having elevated electrical coupling (Fig 3F). Thus, increasing
gap junction coupling does not enhance the ability of metabolically active cells to maintain oscillatory islet [Ca^2+] elevations when compared with lower metabolic cells. Unexpectedly, increasing
coupling in metabolically active cells and decreasing coupling in cells with decreased metabolic activity causes the two populations to act more similarly in their control over islet [Ca^2+].
A). Scatterplot of g[Coup] vs. k[glc] for each cell from a representative simulation where g[Coup] is correlated with k[glc] under a skewed distribution in GK activity. B). Fraction of cells showing
elevated [Ca^2+] activity (active cells) vs. the percentage of cells hyperpolarized in islet from skewed simulations in k[glc] with correlated g[Coup] and k[glc] as in A. Hyperpolarized cells are
chosen either from GK^High (orange bars) or GK^Low (blue bars) population. C). As in B. but comparing hyperpolarization in GK^High cells in the presence and absence of correlations in g[Coup]. D). as
in A but from a simulation where g[Coup] and k[glc] are correlated under a bimodal distribution in GK activity. E). As in B. but for simulations where g[Coup] and k[glc] are correlated under a
bimodal distribution in GK activity. F). As in C. but comparing hyperpolarization in GK^High cells in the presence and absence of correlations in g[Coup] under a bimodal distribution in GK activity.
Error bars are mean ± s.e.m. Student’s paired t-test was performed for B and E and a Welches t-test for unequal variances was used for C and F to test for significance. Significance values: ns
indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates
significant difference (p < .0001). Data representative of 4 simulations with differing random number seeds.
Given this dependence on gap junction coupling, we examined whether decreases in coupling impacted how metabolically active cells controlled islet function. We performed similar simulations as in
Figs 1 and 2 for an islet with reduced average gap junction conductance of 50%. In this context, hyperpolarizing highly metabolic populations (GK^Higher or GK^High) or cells with reduced metabolic
activity (GK^Lower or GK^Low) reduced islet [Ca^2+] to a lesser degree than when gap junction conductance was higher (S4 Fig). This applied to simulated islets with either a unimodal normal
distribution in GK activity (S4A and S4B Fig) or a bimodal distribution of GK activity (S4C and S4D Fig). In each case, a similar difference in islet [Ca^2+] resulted from hyperpolarizing highly
metabolic or low metabolic cells, albeit with greater numbers of cells needing targeting to suppress [Ca^2+]. Thus, decreasing gap junction coupling does not enhance the ability of small populations
of metabolic active cells to maintain islet [Ca^2+].
Cells with [Ca^2+] oscillations that precede the rest of the islet do not drive islet [Ca^2+] oscillations
Another subpopulation of β-cells that has been associated with islet function are those cells that show [Ca^2+] oscillations that precede oscillations across the rest of the islet [36, 38, 47]. These
cells have been suggested to have higher intrinsic oscillation frequency [36, 38], which may lend themselves to act as rhythmic pacemakers to drive [Ca^2+] oscillations across the islet. We next
investigated whether a small subpopulation of these cells is able to drive islet [Ca^2+] oscillatory dynamics. We simulated an islet with a unimodal normal distribution of heterogeneity, as in Fig 1,
and identified cells with [Ca^2+] oscillations that preceded the rest of the islet (early phase) or cells with [Ca^2+] oscillations that are delayed with respect to the rest of the islet (late phase)
(Fig 4A and 4B). Cells that preceded the rest of the islet (early phase cells) were temporally separated to a greater degree with respect to the rest of the islet compared to cells that were delayed
(late phase cells) (Fig 4C). The top 1% and 10% of early phase cells (earlier [Ca^2+] oscillations) in the islet had higher intrinsic oscillation frequency–the oscillation frequency if the cell is
simulated in isolation–and lower GK activity compared to the rest of the islet (Fig 4D and 4E). This is consistent with prior experimental measurements that demonstrated lower metabolic activity in
cells that show earlier [Ca^2+] oscillations [36]. Conversely, the top 1% and 10% of late phase cells (delayed [Ca^2+] oscillations) had lower intrinsic oscillation frequency and high GK activity (
Fig 4D and 4E).
A). Schematic of phase lag across simulated islet with 25% variation in GK activity. B). Representative time courses of [Ca^2+] for 9 cells in simulated islet at 60pS coupling conductance to
determine phase lag of cells in A. Blue traces are early phase cells (negative phase lag), Grey is non early or late phase cells, red is late phase cells (positive phase lag). Inset: Close up of rise
of [Ca2+] oscillation showing phase lags. C). Phase lag from islet average of top 1% or 10% of early phase, late phase cells, or random cells. D). Average k[glc] from all cells, early phase cells or
late phase cells across simulated islet (normalized to average k[glc]). E). Average intrinsic oscillation frequencies of all cells, top 1% and 10% of early phase cells, or top 1% or 10% of late phase
cells when re-simulated in the absence of gap junction coupling (0pS). F). Average frequency of islet when indicated populations of cells are removed from the simulated islet. G). Change in frequency
of islet with indicated populations removed with respect to control islet with all cells present. H). Change in frequency when early phase cells are removed compared to average oscillation frequency
of remaining cells that indicates the expected oscillation frequency. I). Same as H. but for simulations where late phase cells are removed. J). # of links from network analysis for early phase, late
phase, and random cells in simulations. Error bars are mean ± s.e.m. Repeated measures one-way ANOVA with Tukey post-hoc analysis was performed for simulations in C-G (if there were any missing
values a mixed effects model was used), Student’s paired t-test was performed for H and I to test for significance. Significance values: ns indicates not significant (p>.05), * indicates significant
difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates significant difference (p < .0001). Data representative of 4–9
simulations with differing random number seeds. Random regions were removed for 10% and 30% simulations, but random removal of cells was used for 1% simulations.
To determine the role these cells may play in islet function, we re-simulated the islet with populations of early phase and late phase cells removed from the islet. When populations (1%, 10%, 30%) of
early or late phase cells were removed, the elevation of [Ca^2+] was unchanged (S5A Fig). Similarly, the frequency of the islet did not differ significantly from control islets when up to 10% of
early or late phase cells were removed (Fig 4F and 4G). Early or late phase cells usually exist within a compact region, rather than being distributed randomly across the islet. Removing random cells
within a similar sized region impacts frequency of the remaining islet to a lesser degree than removing randomly positioned cells across the islet (S6 Fig). Removal of up to 10% of early phase or
late phase cells also showed no change in frequency compared to removal of random cells within a similar sized region (Fig 4F and 4G). When 30% of early phase cells (earlier [Ca^2+] oscillations)
were removed from the islet, frequency decreased slightly, by ~2% (Fig 4G). This minor decrease in frequency was equivalent to the average frequency of the remaining cells in the islet, indicating no
disproportionate effect of the early phase cells on oscillation frequency (Fig 4H). In contrast, when 30% of late phase cells (delayed [Ca^2+] oscillations) were removed, the islet frequency
increased, by ~8% (Fig 4G). This increase in frequency upon removing the late phase cells was significantly greater than the average frequency of the remaining cells in the islet, indicating a
disproportionate effect of late phase (delayed) cells on oscillation frequency (Fig 4I). When these manipulations were performed in the presence of reduced (50%) gap junction conductance, the changes
in frequency were exacerbated: no change in frequency when removing early phase cells and a greater increase in frequency (~15%) when removing late phase (delayed) cells (S7 Fig). Finally, early
phase cells did not show a significant difference in the number of links compared with late phase cells in the islet (Fig 4J). Thus, early phase cells that show earlier [Ca^2+] oscillations do not
drive the [Ca^2+] oscillation frequency of the islet, when considering a unimodal normal distribution of cell heterogeneity. However, unexpectedly, late phase cells that show delayed [Ca^2+]
oscillations appear to drive a slower [Ca^2+] oscillation frequency; but only in proportions of at least 30% of the islet.
Early phase and late phase cells that show different timings in their [Ca^2+] oscillations on average have higher or lower intrinsic [Ca^2+] oscillation frequency respectively. However, other factors
such as gap junction coupling or position within the cluster may also determine their relative timing. We next examined the role of cells that intrinsically have the highest or lowest [Ca^2+]
oscillation frequency (Fig 5A–5C). The top 1% or 10% of cells with highest or lowest intrinsic oscillation frequency, showed a frequency substantially different than the islet average (Fig 5D). On
average, cells with a higher intrinsic oscillation frequency showed earlier [Ca^2+] oscillations compared with the average of the islet (Fig 5E) and had lower metabolic activity (Fig 5F). In
contrast, cells with the lowest frequency showed delayed [Ca^2+] oscillations compared with the average of the islet and had higher metabolic activity (Fig 5E and 5F). This is consistent with
previous experimental measurements that demonstrated a negative correlation between oscillation frequency and metabolic activity [36]. We do note that a small (~0.5%) of cells with low metabolic
activity lacked [Ca^2+] elevations and were excluded from frequency measurements.
A). Schematic of frequency across simulated islet with 25% variation in GK activity. B). Representative time courses of [Ca^2+] for 9 cells in simulated islet in A in a simulation with full (120pS)
coupling conductance. Blue traces are high frequency cells, Grey are cells with frequency near average frequency, red traces are low frequency cells. C). Same cells as in B but showing [Ca^2+] time
courses from an uncoupled simulation (0pS coupling conductance). D). Average intrinsic oscillation frequencies of all cells, top 1% or 10% of high frequency cells, or low frequency cells when
re-simulated in the absence of gap junction coupling. E). Phase lag from islet average of top 1% or 10% of low frequency, high frequency, or random cells. F). Average k[glc] from all cells, high
frequency cells, or low frequency cells across simulated islet (normalized to average k[glc]). G). Average frequency of islet when indicated populations of cells are removed from the simulated islet.
H). Change in frequency of islet with indicated populations removed with respect to control islet with all cells present. I). Change in frequency when high frequency cells are removed compared to
average oscillation frequency of remaining cells that indicates the expected oscillation frequency. J). Same as I. but for simulations where low frequency cells are removed. Error bars are mean ±
s.e.m. Repeated measures one-way ANOVA with Tukey post-hoc analysis was performed for simulations in D-H and a Student’s paired t-test was performed for I and J to test for significance. Significance
values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), ****
indicates significant difference (p < .0001). Data representative of 5 simulations with differing random number seeds. Random removal of cells across the islet was used for all simulations where
random cells removed is indicated.
When greater than 10% or 30% of high frequency cells were removed from the islet, the frequency of the islet decreased, whereas when 10% or 30% of lower frequency cells were removed from the islet,
the frequency of the islet increased (Fig 5G and 5H). However, in each case the change in frequency upon removing high or low frequency cells was not significantly greater than the change when
considering the average frequency of the remaining cells (Fig 5I and 5J). In fact, the decrease in frequency upon removing high frequency cells was significantly less than that considering the
frequency of remaining cells (Fig 5I). In each case, the elevation of [Ca^2+] was unchanged (S5B Fig). These results again suggest that small numbers of cells with faster oscillation frequency do not
disproportionately affect islet [Ca^2+] oscillations.
A bimodal distribution in frequency lessens the effect of late phase cells
Earlier we considered a bimodal distribution in metabolic activity that better described experimental data (Fig 2) [37]. We next investigated whether late phase and early phase cells may influence
the islet to a greater degree when described by a bimodal distribution. From the unimodal normal distribution, we previously modelled (Fig 4), we generated a population of cells that incorporated the
average properties of early phase cells that showed earlier oscillations in [Ca^2+] (see methods). This population (10%), which showed a faster oscillation frequency (Figs 6A and S8A) was combined
with a population of cells that were similar to the average properties of an islet. The resultant simulated islet showed cells with earlier and delayed [Ca^2+] oscillations, as before (Fig 6B and 6C
), albeit with a slight reduction in the time between the early and delayed oscillations (Fig 6D). On average, the early phase cells that showed earlier [Ca^2+] oscillations had higher intrinsic
oscillation frequencies (Fig 6E) and lower metabolic activity (Fig 6F), as before. However, the difference between early and late phase cells was not a large as with the unimodal normal distribution.
When early phase cells or late phase cells were removed from the islet, the frequency was not significantly different than when random cells were removed (Fig 6G). However, when 10% of early phase
cells were removed, the change in frequency was significantly different, albeit small, compared to the expected frequency of the remaining cells in the distribution (Fig 6H). On the other hand, the
removal of late phase cells was not significantly different than the expected frequency of the remaining cells (Fig 6I).
A). Schematic of frequency across simulated islet with a bimodal distribution in GK activity. B). Schematic of phase lag across simulated islet with a bimodal distribution in GK activity. C).
Representative time courses of [Ca^2+] for 6 cells in simulated islet in A (and B) in a simulation with full (120pS) coupling conductance. Blue traces are high frequency cells, red traces are low
frequency cells. Inset: Close up of rise of [Ca2+] oscillation showing phase lags. D). Phase lag from islet average of top 1% or 10% of early phase, late phase cells, or random cells. E). Average
intrinsic oscillation frequencies of all cells and 1% or 10% of early phase cells, or 1% or 10% of late phase cells when re-simulated in the absence of gap junction coupling (0pS). F). Average k[glc]
from all cells and top 1% or 10% of early phase cells or late phase cells across simulated islet (normalized to average k[glc]). G). Change in frequency of islet with indicated populations removed
with respect to control islet with all cells present. H). Change in frequency when early phase cells are removed compared to average oscillation frequency of remaining cells that indicates the
expected oscillation frequency. I). Same as H. but for simulations where late phase cells are removed. Repeated measures one-way ANOVA with Tukey post-hoc analysis was performed for simulations in
D-G and a Student’s paired t-test was performed for H and I to test for significance. Error bars are mean ± s.e.m. Significance values: ns indicates not significant (p>.05), * indicates significant
difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates significant difference (p < .0001). Data representative of 5
simulations with differing random number seeds. Random removal of cells across the islet was used where random cells removed is indicated.
We further examined how the islet behaved when the high frequency population of cells were removed. These high frequency cells showed only slightly earlier [Ca^2+] oscillations compared to the rest
of the islet on average (S8B Fig) but did show lower metabolic activity (S8C Fig). Upon removal of these high frequency cells, the islet showed significantly slower oscillations (S8D Fig), that were
slower than expected given the average frequency of the remaining cells (S8D and S8E Fig). However, the change in frequency was still low (~2%). When these high frequency cells were positioned with
the same spatial distribution as early phase cells, the change in frequency upon their removal was significantly greater but was still relatively small and similar to the change seen when high
frequency cells were removed from the unimodal normal distribution model (~5%) (S9 Fig). In conclusion, within a bimodal distribution, a small population of cells with higher frequencies has only a
minor impact on the frequency of the islet.
Limited excitatory gap junction current can explain lack of action of small subpopulations
To understand the basis by which cells with differing metabolic activity and oscillatory frequency interact, we examined the gap junction currents for cell populations within the islet (Fig 7A). As
expected, the total membrane current was highest in magnitude during the upstroke and downstroke of the [Ca^2+] oscillation, and low in magnitude during the active and silent phase (Fig 7B–7D).
Conversely, the gap junction current was highest during the active and silent phase of the [Ca^2+] oscillation but was minimal during the upstroke and downstroke of [Ca^2+] (Fig 7B, 7C and 7E). Thus,
there is less communication between cells during the upstroke and downstroke of [Ca^2+] oscillations compared to the stable active and silent phases.
A). Schematic of cell within the simulated islet, showing 3 gap junction currents that contribute to the total gap junction current, together with the total membrane current. B). Time course of [Ca^
2+] from a cell, together with the total membrane current and total gap junction current for a representative cell with higher metabolic activity (k[glc]). C). As in B for a representative cell with
lower metabolic activity. D). Total membrane current, as expressed by an area under the curve (AUC), for each phase of the [Ca^2+] oscillation averaged over the 10% of cells with highest or lowest k
[glc] or a random 10% of cells. E). As in D for total gap junction current. F). Distribution of total gap junction current, as expressed by AUC, for the 10% of cells with highest or lowest k[glc] or
a random 10% of cells. G). As in E for a bimodal distribution in k[glc]. H). Mean duration of active phase and silent phase averaged over the 10% of cells with highest or lowest oscillation
frequency, or a random 10% of cells. I). Mean islet [Ca^2+] time course showing different portions of the active phase (1–4). J). Mean islet gap junction current during different portions of the
active phase, as indicated in I for the 10% of cells with highest or lowest oscillation frequency, or a random 10% of cells. Black lines are fitted regression lines. Error bars are mean ± s.e.m.
Repeated measures one-way ANOVA was performed for data in D, E, H to test for significance. Linear regression was performed on data in J. Significance values: ns indicates not significant (p>.05), *
indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates significant difference (p < .0001), †
indicates significant linear regression (p < .05), ‡ indicated significant linear regression (p < .01). Data representative of 5 simulations with differing random number seeds.
The total membrane current did not differ significantly between cells with high or low metabolic activity (Fig 7D). However, there was a substantial difference in gap junction current between cells
with high or low metabolic activity (Fig 7E). Cells with high metabolic activity showed a positive (outward, hyperpolarizing) gap junction current, whereas cells with low metabolic activity showed a
negative (inward, depolarizing) gap junction current, across all phases of the [Ca^2+] oscillation (Fig 7E). The magnitude of the gap junction current for less metabolically active cells was also
greater. This larger gap junction-mediated current would be expected to hyperpolarize neighboring cells to a greater degree than metabolically active cells depolarizing neighboring cells.
Nevertheless, there was significant variability, such that some cells with low metabolic activity had little gap junction current and some cells with high metabolic activity had a positive current
that would depolarize neighbors (Fig 7F). When examining the bimodal simulation (Fig 2), we observed broadly similar findings where cells with high metabolic activity depolarize their neighbors
whereas cells with low metabolic activity hyperpolarize their neighbors (Fig 7G).
Finally, given the stronger gap junction current associated with the active and silent phases, we analyzed the relationship between the duration of these phases for cells with high and low frequency.
Cells with a higher intrinsic oscillation frequency showed both a shorter active phase and shorter silent phase compared to cells with a slower intrinsic oscillation frequency, with there being a
greater difference in the active phase (Fig 7H). Interestingly, the whole islet active and silent phase times were similar to those of cells with a higher oscillation frequency (which on average have
lower metabolic activity). During the active phase, the gap junction current was lowest at the beginning of the active phase and greatest just before the downstroke (Fig 7I and 7J). We measured
changes to the duration of the active and silent phases after removal of early/late phase cells and low/high frequency cells from Figs 4 and 5. When either late phase cells or low frequency cells
were removed, the active phase and duty cycle duration decreased compared to when either early phase cells or high frequency cells were removed, respectively (S10 Fig). Thus, gap junction coupling
contributes more to sustaining the active phase compared to initiating the active phase. While slower oscillating cells contribute significantly to setting the islet frequency, given the greater gap
junction current, faster oscillating cells may limit the duration of the active phase by terminating the oscillation.
β-cell heterogeneity has largely been studied in single cells. However, recent studies have demonstrated that heterogeneity plays a physiological role in regulating insulin release within the islet [
36, 37, 42]. Previously, using computational models and experimental systems, we demonstrated that a large minority (close to 50%) of metabolically active β-cells were necessary to maintain the
activity of the islet [42]. In contrast to this, experimental and theoretical studies have suggested that small (~10%) highly functional subpopulations may be required to maintain whole islet [Ca^2+]
dynamics [37, 48]. Here, we investigated the theoretical basis by which small populations of cells may impact islet [Ca^2+] dynamics.
Small populations of metabolically active cells are not required to drive elevations in [Ca^2+]
To determine whether small populations of metabolically active β-cells could drive elevations in [Ca^2+], we constructed three types of islet simulations: showing either a unimodal normal
distribution, skewed distribution or a bimodal distribution in metabolic activity. In each case, we either hyperpolarized the most metabolically active cells or removed them from the simulation.
These manipulations are equivalent to those applied in the literature. For example, one study used optical stimulation of eNpHr3.0 to induce a hyperpolarizing Cl^- current in 1–10% of cells that
showed high levels of [Ca^2+] coordination and elevated GK [37]. Another study used optical stimulation of ChR2 to induce a depolarizing cation current, with the ~10% of cells activating large parts
of the islet showing higher NAD(P)H [36]. In our simulations, we found hyperpolarizing those cells with increased metabolic activity generated similar findings: hyperpolarizing more metabolically
active cells silenced the islet to a much greater degree than hyperpolarizing less metabolically active cells. Thus, hyperpolarization or depolarization of metabolically active β-cells can
disproportionately suppress or activate islet function, via gap junction coupling.
Importantly, the effects of this targeted silencing were found for both a broad unimodal normal distribution (Fig 1), skewed distribution (Fig 2) and for a bimodal distribution (Fig 2). Within the
literature there is not exact consistency in the level of metabolic heterogeneity present. Within dissociated β-cells, a variation of 20–30% in NAD(P)H responses has been observed experimentally [26,
44], and in intact islets a variation of 10–20% has been observed [44]. Instead, ~50% variation is needed to describe experimental observations here. However, early analysis of GK heterogeneity via
immunohistochemistry observed substantial variations, which while not quantified would be equivalent to >50% [24]. Similarly, in isolated β-cells the glucose threshold for elevated NAD(P)H varies by
~50% (3-10mM) [26, 49]. This latter study also found a non-normal distribution with ~20% of highly metabolically active β-cells. Thus, the distributions required in our model to generate results
equivalent to experimental observations are broadly feasible. Furthermore, we do note the process of removing β-cells from the islet via dissociation causes cell stress and could disrupt metabolic
signatures. Highly metabolically active cells may also be more susceptible to environmental stress [37, 50]. Therefore, further analysis, in situ, is needed to precisely quantify the level of
heterogeneity present.
Interestingly, we observed very different results when comparing the effect of targeted hyperpolarization of a set of cells and targeted removal of a set of cells. Hyperpolarizing a small population
of metabolically active cells largely silenced the islet, whereas removal of this same cell population had reduced impact. Upon removal, we did observe a moderate reduction in duty cycle of ~40%
under a skewed distribution in GK activity, whereas we observed only a small reduction in the duty cycle of ~10% under a bimodal distribution in GK activity. The exact relationship between duty cycle
and insulin release in unknown. GK activity is important for setting the Ca^2+ oscillation frequency and duty cycle, but other downstream elements further modify [Ca^2+] oscillation dynamics and
insulin secretion. For example, pyruvate kinase activation can increase the [Ca^2+] oscillation frequency and reduce the duty cycle, while amplifying insulin secretion as a result of locally
elevating ATP/ADP and closing K[ATP] channels [51]. Despite the complicated regulation of insulin secretion dynamics, increased [Ca^2+] duty cycle does correlate with elevated glucose stimulation and
insulin release [52, 53]. It has also been suggested that duty cycle and insulin release have a non-linear, sigmoidal relationship [54], thus a ~10% reduction could potentially impact insulin
release. Conversely, a ~40% reduction could potentially reduce insulin release to a substantial degree. However, the skewed distribution showed the least correspondence with experimental data when
considering hyperpolarizing cell populations, with a smaller difference observed between hyperpolarizing metabolically active cells and inactive cells. As such, the manipulations involving
hyperpolarization and cell removal, theoretically, assess the importance a cell has on islet function in different ways. Thus, care must be taken when interpreting the results of optogenetic
stimulation-based analysis. Nevertheless, our results imply significant redundancy in the way small populations of cells elevate [Ca^2+] across the islet (Fig 8).
A). Schematic of suggestion that small subpopulations of highly functional cells can control whole islet dynamics. White circles represent β-cells. Red arrows represent which cells can be controlled
by individual cell where the arrow begins. Increasing functionality in cells is from right to left B). Same as A, but a schematic of how our simulations predict islet [Ca^2+] dynamics are controlled.
Our simulations predict that control is redundant, and many cells can control many other cells. Our simulations predict there is not one small subpopulation that controls the entire islet. C). Same
as B, but schematic of how our simulations predict the islet responds when highly functional subpopulations are removed. When highly functional subpopulations are removed, the remaining cells are
able to maintain the function of the islet due to the redundancy in control.
Cell removal from the simulation may be considered similar to the experimental ablation of that cell. Ablation of small populations of cells that show earlier [Ca^2+] oscillations, but which overlaps
with those cells that show increased [Ca^2+] coordination, has experimentally been demonstrated to reduce the elevation in [Ca^2+] across zebrafish islets [38]. These studies showed a substantial
reduction in [Ca^2+] amplitude, whereas our theoretical findings showed no apparent differences in amount of active cells. Little change in [Ca^2+] activity is observed in the model when removing
either those cells with earlier Ca^2+ oscillations (S5 Fig) or those cells with elevated metabolic activity that when hyperpolarized silences islet [Ca^2+] (Figs 1 and 2). However, differences do
exist between zebrafish islets and mouse islets which our model is based upon and has been validated against, including islet size, gap junction protein isoform and Ca^2+ dynamics [55, 56]. Thus,
species differences may account for these observations.
The way cells interact within our simulated islet is restricted to gap junction electrical coupling. As such, we conclude that gap junction communication is unlikely to be able to explain the role
small cell subpopulations play in islet function, under the model assumptions presented here. These conclusions are also consistent with elevated oscillatory [Ca^2+] being maintained upon a loss of
Cx36 gap junction coupling [22], albeit with a lack of synchronization. However, we do note that first phase insulin release is diminished upon a loss of Cx36 gap junction coupling [57]. Therefore,
we cannot exclude that small cell subpopulations can drive [Ca^2+] elevations via gap junction coupling during the initial first phase response.
β-cells can communicate across the islet via paracrine communication. This includes inhibitory factors such as GABA, 5-HT, dopamine and Ucn3 (via δ-cell somatostatin release) and stimulatory factors
such as ATP [58–60]. Thus, it is conceivable, small subpopulations of metabolic active cells are secreting increased levels of stimulatory paracrine factors. Alternatively, small subpopulations may
be acting via other endocrine cells, such as glucagon-secreting α-cells, to stimulate other β-cells within the islet [61]. Removal of immature cell populations can also disrupt islet function,
suggesting a broader remodeling of the islet can be induced by small cell subpopulations [62]. Therefore, analyzing whether subpopulations show differential release of paracrine factors will be
important to better elucidate their function within the islet.
Highly metabolic cells have increased connectivity, but this is not due to increased gap junction coupling
Gap junction coupling allows for heterogeneous populations of β-cells to act in a cohesive manner. For example, when populations of normally excitable and inexcitable cells combine within an islet,
gap junction coupling ensures that a uniform response occurs, whether this be suppressed [Ca^2+] or coordinated elevated [Ca^2+] [41]. Some cell populations have been suggested to have elevated
connectivity with other cells in the islet, as measured by correlated [Ca^2+] oscillations [37, 38], and could result from an increase in gap junction coupling.
In our simulations, highly metabolic cells showed increased connectivity compared with lower metabolically active cells which is in agreement with previous studies showing super connected ‘hub’ cells
have increased GK protein expression. However, when more metabolically active cells had increased coupling conductance, both highly metabolic cells and low metabolic cells became more similar in
their ability to suppress islet function under hyperpolarization (Fig 3). If gap junction coupling is elevated in metabolically active cells, it is reduced in less metabolically active cells. A
decrease in coupling lessens how the islet is suppressed in the presence of inexcitable cells that transmit hyperpolarizing current across the islet. Thus, hyperpolarizing a population of
metabolically active cells would transmit less hyperpolarizing current beyond the nearest neighbor cells.
We also observed that less metabolically active cells show a greater gap junction current that hyper-polarizes neighboring cells. Thus, there is an asymmetry by which metabolically active and
inactive cells within the islet act (Fig 7). As such, increases in coupling are not beneficial for highly metabolic cells to control the islet in a disproportionate manner compared with cells with
lower metabolic activity.
Recently, metabolic intermediates have been suggested to diffuse through Cx36 gap junctions in the islet, leading to metabolic coupling [63]. However, this is in disagreement with several studies
that Cx36 is strongly selective for cations [64, 65]. Prior modelling studies have also implied that the coordination of slow metabolic oscillations can be described using only electrical coupling [
66, 67]. In this study, we did not investigate a role for heterogeneity in regulating slow metabolic oscillations given that highly functional subpopulations have only been characterized in the
context of fast electrical dynamics. Further, there is little experimental investigation for the role of electrical coupling in regulating slow [Ca^2+] and metabolic oscillations. Given that slow
metabolic and [Ca^2+] oscillations likely underlie slow pulsatile insulin release, it would be of interest how functional subpopulations affect metabolic oscillations. Furthermore, if gap junctions
within the islet do allow diffusion of metabolic intermediates, this could provide an alternative means by which highly metabolic cells can influence the rest of the islet. Further evidence is needed
to test this concept.
Small subpopulations cannot efficiently act as rhythmic pacemakers
Multiple studies have identified cells that consistently show earlier [Ca^2+] oscillations that may drive the dynamics of [Ca^2+] across the islet [36, 38, 47]. These populations have been suggested
to have a higher intrinsic oscillation frequency and thus act as a rhythmic pacemaker [36], in the same manner as the cardiac SA node. Here, we investigated whether a small subpopulation of cells
with increased oscillation frequency could act as such a pacemaker. We found that cells that show earlier [Ca^2+] oscillations do have a higher intrinsic oscillation frequency. However, upon removal
of these cells, the islet [Ca^2+] oscillations changed little, suggesting that small populations of these cells are unable to pace islet [Ca^2+] oscillations. This initially is surprising as with all
cells capable of firing, the cell with the highest frequency will depolarize first and stimulate neighbors to fire. However, at least ~30% of high frequency cells are required to even slightly impact
islet oscillation frequency. These findings are consistent with prior modelling studies where cells with fast and slow oscillation frequencies, when combined within an islet, led to an overall
oscillation midway between the intrinsic cell oscillations [68]. This suggests the oscillation frequency is not per se determined by a small pacemaker population but rather is formed by a weighted
combination of all cells across the islet. Thus, the islet also shows significant redundancy where only loss of large populations of cells impacts the activity or dynamics of [Ca^2+] (Fig 8).
Further, the introduction of a small population (~10% cells) with a defined high intrinsic oscillation frequency has little impact on islet [Ca^2+] oscillations frequency and wave propagation (Fig 6
). Heterogeneity in factors other than those considered in our model could also influence [Ca^2+] oscillation frequency. For example, pyruvate kinase mentioned above locally elevates ATP/ADP and
closes K[ATP] channels, increasing the [Ca^2+] oscillation frequency [51], which is different than the action of increased GK in our model. Further investigation is needed to understand how
heterogeneity in other factors could impact the [Ca^2+] oscillation frequency across the islet.
In contrast to removal of cells that show earlier [Ca^2+] oscillations, removal of those cells that show delayed [Ca^2+] oscillations increased the frequency of islet [Ca^2+] oscillations (Fig 4).
These cells on average showed slower oscillations. Therefore, slow [Ca^2+] oscillations contribute to setting the islet [Ca^2+] oscillation frequency to a greater degree. Previously, the phantom
burster model was shown to have medium bursting modes, under three different conditions, either all fast oscillators, all slow oscillators, or a combination of the two [69]. Our results suggest that
slow metabolic oscillations will better coordinate [Ca^2+] dynamics across the islet, rather than purely a faster-oscillating electrical subsystem. Nevertheless, at least 30% of these slow
oscillators are needed to have a substantial impact on the islet dynamics, which is consistent with the oscillation frequency again being formed by a weighted combination of all cells across the
We did not observe a complete overlap between cells that show earlier/delayed [Ca^2+] oscillations and cells with faster/slower intrinsic [Ca^2+] oscillations, respectively. Similarly, while removal
of the highest and lowest frequency cells changes the overall islet frequency to a greater degree, only removal of cells with delayed [Ca^2+] oscillations showed a change in frequency above that
expected, given the frequency of the remaining cells. Thus, other properties of the islet also contribute to setting the islet oscillation frequency, and these properties remain to be determined.
Therefore, our simulations indicate that there is not a small population of rhythmic pacemaker cells within the islet. Rather, a large number of cells are needed to impact islet frequency. Of
interest, the distribution of cells with faster or slower intrinsic [Ca^2+] oscillations in our simulation is distributed across the islet, whereas cells that show earlier or delayed [Ca^2+]
oscillations exist within a specific region within the islet, often at the islet edge. While having only a minor impact, the spatial distribution of higher frequency cells was important in affecting
islet [Ca^2+] oscillations. Whether intrinsically fast or slow oscillating cells show some spatially restricted distribution is unknown. A different spatial organization could potentially contribute
to a greater control over islet frequency, especially if slow oscillators overlap with other properties of the islet that confer greater control over islet oscillation frequency.
We also speculate that the level of gap junction coupling for cells with slower or faster oscillations may be important: the time course of gap junction current indicates that faster oscillating
cells transmit a greater hyperpolarizing current to neighboring cells earlier, as compared to slower oscillating cells. This may explain why the islet oscillation active phase duration trends closer
to those cells with a higher frequency and thus shorter active phase duration. However, given the lower gap junction current in the silent phase, this appears not to be sufficient to
disproportionately impact the oscillation frequency.
Overall, the results from this study show how small populations of highly functional cells impact islet function via gap junction electrical coupling. Our simulations suggest that both a small
subpopulation of metabolically active cells or the most metabolically active subset of cells within a unimodal distribution are unable to maintain elevated [Ca^2+] across the islet via gap junction
coupling. Further, a small population or subset of cells that shows early [Ca^2+] elevations or that have a higher oscillation frequency are also unable to act as rhythmic pacemakers to drive
oscillatory [Ca^2+] dynamics. As such the mechanism(s) by which these cells may act to impact islet function should be further investigated.
Coupled β-cell electrical activity model
The coupled β-cell model was described previously [42] and adapted from the published Cha-Noma single cell model [70, 71]. All code was written in C++ and run on the SUMMIT supercomputer (University
of Colorado Boulder). Example model code is included in supplemental information (S1 Files). All simulations are run at 8mM glucose unless otherwise noted.
The membrane potential (V[i]) for each β-cell i is related to the sum of individual ion currents as described by [70]: (1) Where the gap junction mediated current I[Coup] [41] is: (2) Where g^ij
[Coup] is the average coupling conductance between cells i and j. Heterogeneity in Cx36 gap junctions is modeled as a γ-distribution with parameters k = θ = 4 as described previously [36] and scaled
to an average g[Coup] between cells = 120pS. The number of cells, N, in each simulation is 1000. The parameters that are heterogenous across all cells in each simulation is described in S1 Table with
means and standard deviations.
Modelling GK activity
The flux of glycolysis J[glc], which is limited by the rate of GK activity in the β-cell, is described as: (3) Where k[glc] is the maximum rate of glycolysis (equivalent to GK activity), which was
simulated as a unimodal Gaussian distribution with a mean of 0.000126 ms^-1 and standard deviation of 25% of the mean (unless indicated). [Re[tot]] = 10mM, the total amount of pyrimidine nucleotides.
The ATP and glucose dependence of glycolysis (GK activity) is: (4) Where [G] is the extracellular concentration of glucose, hgl is the hill coefficient, K[G] is the half maximal concentration of
glucose, and K[mATP] is the half maximal concentration of ATP.
For simulations with changes in variation in GK, the mean remained the same at 0.000126 ms^-1, but standard deviation of 1% or 50% of the mean was used.
Hyperpolarizing cell populations
Hyperpolarization of cells was induced by including a V-independent leak current I[hyper] that hyperpolarizes the cell [43], described as: (5) Where g[hyper] is the hyperpolarizing conductance which
is zero in the absence of the applied hyperpolarizing current and is g[hyper]’ (1-p[0KATP]) ≈ g[hyper]’ during applied hyperpolarization. The number of cells that were hyperpolarized were defined as
the fraction P[hyp] multiped by the number of cells, N (1000 in all simulations).
For skewed distribution of GK
A gamma distribution was used to model the unimodal skewed distribution in GK. The gamma distribution has shape parameter k = 1.26 and scale parameter θ = 0.79. These parameters were fitted to
satisfy the following conditions: (6) (7) Where is the mean rate of glycolysis for the GK^High population and is 3 times the islet mean, k[glc] (6). The islet mean k[glc] remains unchanged from the
unimodal normal distribution at 0.000126 ms^-1 (7). The mean rate of glycolysis for the GK^Low population, , is slightly reduced to satisfy this Eq (7). P[Low], the percent of GK^Low cells in the
simulation, is 90%, and P[High], the percent of GK^High cells, is 10%. N is the number of cells in the simulation (1000).
For bimodal distribution of GK
The bimodal distribution of GK was also calculated using Eqs (6) and (7). In this case, there were 2 smaller Gaussian distributions with means and but with standard deviation of and set at 2.5% (10%
less than the unimodal normal) of the means to create a distinct bimodal distribution.
Modelling changes in coupling
Simulations where k[glc] and g[Coup] (or g[KATP]) are correlated, the values for k[glc] and g[Coup] are calculated using a copula or multivariate normal to calculate correlated value pairs. A
correlation of r = 0.7 (or r = -0.7 for inverse correlation of g[KATP]) is used. These values are then transformed to give the variables their appropriate distributions (normal for k[glc] and gamma
distribution for g[Coup] see Eqs (2–4) for more detail). The paired k[glc] and g[Coup] values are then randomly distributed to the cells in the simulation.
For simulations where cells are removed, the conductance, g[Coup], of the cells to be removed is set to 0 pS. Removed cells are excluded from subsequent islet analysis.
Determining early and late phase cells
To determine early phase and late phase cells, one full [Ca^2+] oscillation is taken between time points 300 sec to 400 sec. This time point ensures the model and frequencies are stable and in the
second phase of [Ca^2+] oscillations. A cross correlation is used to determine the time delay of each cell time course compared to the mean [Ca^2+] across the islet, using xcorr() in MATLAB. A
negative delay is therefore equivalent to an earlier oscillation. The early phase cells are determined as the cells with the most negative time delay and the late phase cells are determined as the
cells with the most positive time delay. If the cutoff occurs where multiple cells have the same delay, then a random cell is chosen from the cells with the same lag.
Modelling bimodal distribution for early phase cells
The mean values of the early phase and non-early phase cells in the unimodal normal distribution was used to define a new bimodal distribution as described in S2 Table. All standard deviations are 1%
of the mean. For more information on parameters see [70]. The ‘early phase’ cell population, N[earlyphase] comprised 10% of the islet (left column), and the N[non-earlyphase] comprised the other 90%
(right column).
Network analysis of links
The network analysis was based on previously described methods [72]. The [Ca^2+] time course of each cell was correlated with every other cell time course using MATLAB corr() function, to generate a
Pearson correlation coefficient matrix for each cell pair. A threshold value of 0.9998 was used to assign a binary value of linked/not linked to each cell pair. The threshold value was chosen to
resemble a small world network link distribution, as previously performed [37, 72]. Note, this threshold is higher than that previously used for experimental data (0.75) that generated a small world
network link distribution [72]. This is likely due to the differences in precision between experimental fluorescence imaging and a high precision deterministic simulation. Each cell was assigned the
total number of links with all other cells and sorted from super-connected (hub) to least connected cells.
Noise simulations
Stochastic noise was added to the K[ATP] channel as previously described in which a time varying noise component, S, that follows a normal distribution is added to the K[ATP] current [43]. Where p
[0KATP] is the open channel probability of the K[ATP] channel. S is described by S has a mean of 0 and a standard deviation of ~0.049. τ = 500ms and ξ is generated from a random number sequence.
Simulation data analysis
All simulation data analysis was performed using custom MATLAB scripts. The first 1500 time points (150 sec) were excluded to allow the model to reach a stable state.
Fraction Active was determined by calculating the fraction of cells that were active relative to the total number of simulated cells (1000). Cells were considered active if membrane potential, [Ca^
2+] exceeded 0.165μM at any point in the time course.
Duty Cycle was determined as the fraction of the [Ca^2+] oscillations spent above a threshold value during the time course analyzed. This threshold value was determined as 50% of the average
amplitude of [Ca^2+] in an islet simulated at 8mM glucose with 25% variation in GK activity or time above 70% of the maximum [Ca^2+] (S10 Fig). Duty cycle was reported as the mean across all cells in
the simulated islet.
Frequency of a cell in the islet was determined by taking the [Ca^2+] time-course between times 150 sec and 400 sec and identifying the first 2 peaks. The peak-to-peak time was determined, and this
oscillation period was inverted to calculate the frequency. For whole islet frequency calculations, the coupling in the islet is g[Coup] = 120pS and the mean islet frequency is calculated over all
cells in the simulation.
Intrinsic frequencies were determined using simulations where the mean coupling conductance of all cells in the simulation is g[Coup] = 0pS so that all cells oscillate on their own without influence
from other cells within the simulation. When determining low and high frequency cells in the simulation, only active cells were used.
Expected Frequency was determined by finding the average intrinsic frequencies of the cells (g[Coup] = 0pS) that are included in the simulation. These values are then compared to the simulation where
g[Coup] = 120pS.
Total gap junction current for a cell was calculated by summing the gap junction current over each connection between the cell and all of its neighbors, as in Eq (2). The total membrane current was
calculated as the sum over each current for that cell, as in Eq (1).
Active, silent, upstroke and downstroke phases were chosen manually. The Area Under the Curve (AUC) was calculated using the trapz() function in MATLAB, which calculates trapezoidal integration over
the time period. AUC was calculated for each cell in the given decile and then averaged over those cells.
Active phase duration for one oscillation was determined for each cell, as the total time [Ca^2+] was above 70% of the maximum value, divided by the number of oscillations over the duration assessed.
The silent phase duration was similarly calculated as the total of time [Ca^2+] was below 40% of maximum value.
Statistical analysis
All statistical analysis was performed in Prism (GraphPad). Either a Student’s t-test (or Welch’s t-test for significantly difference variances) or a one-way ANOVA with Tukey post-hoc analysis was
utilized to test for significant differences for simulation results. Paired t-test or repeated measures ANOVA was used anywhere the results were compared with a simulated matching control islet or
groups within the same islet, e.g., before a population was either hyperpolarized or uncoupled. Data is reported as mean ± s.e.m. (standard error in the mean) unless otherwise indicated.
Supporting information
S1 Fig. Histograms of GK activity (k[glc]) and g[Coup] for all unimodal normal, unimodal skewed and bimodal distributions in GK activity for Figs 1–3.
A). All unimodal normal distributions’ histograms. Left: Average frequency of cells at varying GK rate (k[glc]) for simulations that have different standard deviations in GK activity from Fig 1.
Right: Corresponding histogram of average frequency of cells at varying coupling conductance (g[Coup]). B). As in A but for simulations with a skewed normal distribution of GK activity from Fig 2A–2E
. C). As in A but for simulations with a skewed normal distribution of GK activity and correlated GK and g[Coup] activity from Fig 3A–3C. D). As in A but for simulations with a bimodal distribution
of GK activity from Fig 2I–2K. E). As in A for simulations with bimodal distribution of GK activity and correlated GK and g[Coup] from Fig 3D–3F. Data representative of 5 simulations with differing
random number seeds.
S2 Fig. Effects of noise on hyperpolarization-induced cell silencing.
A). Fraction of cells showing elevated [Ca^2+] activity (active cells) vs. the percentage of cells hyperpolarized in islet from simulations with a unimodal normal distribution with 25% variation in
GK activity (k[glc]). Simulations run in the presence of stochastic noise (see methods). B). As in A but for simulations with 50% variation in GK activity. C). As in A but for simulations with
unimodal skewed distribution. Error bars are mean ± s.e.m. Repeated measures one-way ANOVA with Tukey post-hoc analysis was performed for A and B. Student’s paired t-test was performed to test for
significance in C. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant
difference (p < .001), **** indicates significant difference. Data representative of 5 simulations with differing random number seeds.
S3 Fig. Additional simulations with unimodal normal distribution in GK activity with correlated g[Coup] and g[KATP].
A). Scatterplot of g[Coup] vs. k[glc] for each cell from a representative simulation where g[Coup] is correlated with k[glc] for simulation where GK activity is a modeled as a unimodal normal
distribution. B). Fraction of cells showing elevated [Ca^2+] activity (active cells) vs. the percentage of cells hyperpolarized in islet from simulations with a unimodal normal distribution in k[glc]
with correlated g[Coup] and k[glc] as in A. Hyperpolarized cells are chosen based on their GK rate which is correlated to g[Coup]. C). As in B. but comparing hyperpolarization in high GK cells in the
presence (B) and absence (Fig 1C) of correlations in g[Coup]. D). as in A but from a simulation where g[Coup] and k[glc] and g[KATP] (K[ATP] channel conductance) are correlated. E). As in B. but for
simulations where g[Coup] and k[glc] and g[KATP] are correlated. F). As in C. but comparing high GK cells hyperpolarization from Fig 1C to high GK hyperpolarization from simulations where g[Coup] and
k[glc] and g[KATP] are correlated (E). Error bars are mean ± s.e.m. Repeated measures one-way ANOVA with Tukey post-hoc analysis was performed for simulations in B and C (if there were any missing
values a mixed effects model was used) and a Student’s t-test was performed for C and F (Welches t-test for unequal variances was used when variances were determined to be statistically different
using an F-test) to test for significance. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), ***
indicates significant difference (p < .001), **** indicates significant difference (p < .0001). Data representative of 5 simulations with differing random number seeds.
S4 Fig. Simulations predicting effect of 50% reduction in coupling in simulations with unimodal normal and bimodal distributions in GK activity.
A). Fraction of cells showing elevated [Ca^2+] activity (active cells) vs. the percentage of cells hyperpolarized in islet from simulations with a unimodal normal distribution as in Fig 1C but with
50% reduction in average coupling conductance (60pS) for all cells. Hyperpolarized cells are chosen based on their GK rate. B). As in A. but comparing hyperpolarization in high GK cells in
simulations with full coupling (120pS–Fig 1C) and reduced coupling (60pS) from A. C). as in A but for bimodal simulations with reduced coupling (60pS). D). As in B but comparing bimodal distributions
in GK with full coupling (120pS) from Fig 2J to bimodal simulations with reduced coupling (60pS) from C. Error bars are mean ± s.e.m. Student’s paired t-test was performed to test for significance
for all simulations. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant
difference (p < .001), **** indicates significant difference. Data representative of 4–5 simulations with differing random number seeds.
S5 Fig. Fraction of active cells in simulations where cells are uncoupled from the rest of the cells in the islet from Figs 4–6.
A). Fraction of cells showing elevated [Ca^2+] activity (active cells) in simulated islets vs. the percentage of cells uncoupled in islet from simulations in Fig 4. B). As in A but for simulations in
Fig 5. C). As in A but for simulations in Fig 6. D). As in A but for simulations in S8 Fig. Error bars are mean ± s.e.m. Repeated measures one-way ANOVA was performed for simulations in A and B and a
Student’s paired t-test was performed for C and D to test for significance. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates
significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates significant difference. Data representative of 5 simulations with differing random number seeds.
S6 Fig. Random removal of cells vs. random removal of a region of cells.
A). Schematic showing which cells are chosen to be removed when a random selection of cells is chosen across the islet. B). Schematic showing which cells are chosen to be removed when a random region
of cells is chosen. C). The frequency of the islet after removal of 0%, 10%, or 30% of randomly chosen cells or from a random region. Error bars are mean ± s.e.m. Student’s t-test was performed for
10% and a Welch’s t-test for unequal variances was used to test for significance at 30% of cells removed. Significance values: ns indicates not significant (p>.05), * indicates significant difference
(p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates significant difference. Data representative of 4–9 simulations with differing
random number seeds.
S7 Fig. Simulations predicting the effect of 50% reduction in coupling in simulations where early and late phase cells are removed under a unimodal normal model.
A). Average frequency of islet when indicated populations of cells are removed from the simulated islet with 50% reduction in coupling conductance (60pS). B). Change in frequency of islet with
indicated populations removed with respect to control islet with all cells present. C). Change in frequency when early phase cells are removed compared to average oscillation frequency of remaining
cells that indicates the expected oscillation frequency. D). Same as C. but for simulations where late phase cells are removed. Error bars are mean ± s.e.m. Repeated measures one-way ANOVA with Tukey
post-hoc analysis was performed for simulations in A-B and a Student’s paired t-test was performed for C and D to test for significance. Significance values: ns indicates not significant (p>.05), *
indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates significant difference (p < .0001). Data
representative of 4 simulations with differing random number seeds.
S8 Fig. Simulations predicting the effect of removing cells from individual populations of the bimodal model of early phase cells.
A). Average intrinsic oscillation frequencies of all cells, top 1% or 10% of high frequency cells, or low frequency cells when re-simulated in the absence of gap junction coupling from bimodal model
of phase. B). Phase lag from islet average of top 1% or 10% of high frequency, low frequency cells, or random cells. C). Average k[glc] from all cells, high frequency cells, or low frequency cells
across simulated islet. D). Change in frequency of islet with indicated populations removed with respect to control islet with all cells present. E). Change in frequency when high frequency cells are
removed compared to average oscillation frequency of remaining cells that indicates the expected oscillation frequency. F). Same as E. but for simulations where low frequency cells are removed. Error
bars are mean ± s.e.m. Repeated measures one-way ANOVA with Tukey post-hoc analysis was performed for simulations in A-D and a Student’s paired t-test was performed for E and F to test for
significance. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant
difference (p < .001), **** indicates significant difference (p < .0001). Data representative of 5 simulations with differing random number seeds.
S9 Fig. Simulations predicting the effect of removing a region of high frequency cells from a bimodal model of early phase cells.
A). Schematic of frequency across simulated islet with a bimodal distribution in GK activity and a region of high frequency cells. B). Schematic of phase lag across simulated islet with a bimodal
distribution in GK activity and a region of high frequency cells. C). Change in frequency of islet with indicated populations removed with respect to control islet with all cells present comparing
bimodal model with a region of high frequency cells to a bimodal model with randomly distributed high frequency cells as in Fig 6. D). Change in frequency when high frequency region is removed
compared to average oscillation frequency of remaining cells that indicates the expected oscillation frequency. Error bars represent mean ± s.e.m. Student’s t-test was performed for C and D (paired
test) to test for significance. Significance values: ns indicates not significant (p>.05), * indicates significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates
significant difference (p < .001), **** indicates significant difference (p < .0001). Data representative of 5 simulations with differing random number seeds.
S10 Fig. Analysis of changes in [Ca2+] wave dynamics when early/late phase or high/low frequency cells are removed from islet.
A). Change in mean duration of active phase when top 1%, 10% or 30% early/late phase cells are removed from simulations in Fig 4. B). Change in mean duration of silent phase when top 1%, 10% or 30%
early/late phase cells are removed from simulations in Fig 4. C). Change in mean duty cycle when top 1%, 10% or 30% early/late phase cells are removed from simulations in 4. D). As in A for
simulations when high/low frequency cells are removed from Fig 5. E). As in B for simulations when high/low frequency cells are removed from Fig 5. F). As in C for simulations when high/low frequency
cells are removed from Fig 5. Error bars are mean ± s.e.m. Paired Student’s t-test was used to test for significance. Significance values: ns indicates not significant (p>.05), * indicates
significant difference (p < .05), ** indicates significant difference (p < .01), *** indicates significant difference (p < .001), **** indicates significant difference (p < .0001). Data
representative of 5 simulations with differing random number seeds.
S1 Table. Heterogenous parameters in computational islet model.
Table describes the parameters in the computational model that are heterogenous for each cell. The mean and standard deviation is defined in the table. Changes to these parameter distributions are
discussed in the methods.
S2 Table. Parameters for bimodal early phase cell simulations.
Table describes the parameters that have heterogeneous populations in computational model. The mean of each population is determined from the mean parameter value from unimodal normal simulations
(See methods).
The authors thank Dr David J Hodson (University of Birmingham, UK) and Dr Victoria Salem (Imperial College London, UK) for reviewing this manuscript and for providing helpful comments and
suggestions. The authors are also grateful for utilization of the SUMMIT supercomputer from the University of Colorado Boulder Research Computing Group, which is supported by the National Science
Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University. | {"url":"https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1008948","timestamp":"2024-11-03T23:35:28Z","content_type":"text/html","content_length":"343242","record_id":"<urn:uuid:79c65d2d-153d-4c3c-b4f3-1eb29c4c75dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00821.warc.gz"} |
Engineering Hydrology Questions and Answers - Groundwater - Equation of Motion - Set 2 - Sanfoundry
Engineering Hydrology Questions and Answers – Groundwater – Equation of Motion – Set 2
This set of Engineering Hydrology Multiple Choice Questions & Answers (MCQs) focuses on “Groundwater – Equation of Motion – Set 2”.
1. What is the shape of the water table of an aquifer without recharge, located between two water bodies at different surface elevations?
a) Linear
b) Circular
c) Parabolic
d) Elliptical
View Answer
Answer: c
Explanation: The piezometric head for the above condition as per Dupit’s assumptions is given as,
h^2=\(\frac{(h_1^2-h_0^2)}{L} x+h_0^2\)
Where K is the permeability, h[0] and h[1] the heads of the two water bodies respectively, L is the length of the aquifer and x is the distance from the upstream end. This equation represents a
parabola, specifically known as Dupit’s parabola.
2. What is the shape of the water table of a constantly recharged unconfined aquifer present between two tile drains?
View Answer
Answer: b
Explanation: The head equation for the given condition can be derived by assuming the head of the water bodies, in this case tile drains, to be negligibly small (h[0] =0, h[1] = 0). So,
h^2=\(\frac{R}{K} (L-x)x\)
Where R is the recharge rate, K is the permeability, L is the distance between tile drains and x is the distance from the one tile drain. This is equation represents a parabola.
3. Which of the following is true for an unconfined aquifer with top recharge present between two water bodies?
a) Water table is a parabola
b) Discharge per unit width of aquifer is constant
c) The flow is unidirectional
d) The water can lie anywhere in the aquifer between the two water bodies
View Answer
Answer: d
Explanation: The water table profile for the given aquifer is elliptical with a water divide at some location which splits the flow in two directions. The discharge per width varies throughout the
width of the aquifer.
4. A confined aquifer of thickness 12 m is present between two parallel streams 2.5 km apart. The depths of the streams are 18 m and 14 m. If the permeability of the aquifer is 8 m/day, what is the
flow per meter width of the aquifer?
a) 0.15 m^3/day
b) 0.27 m^3/day
c) 0.49 m^3/day
d) 0.67 m^3/day
View Answer
Answer: a
Explanation: Given h[0] = 18 m, h[1] = 14 m, B = 12 m, L = 2500m, K = 8 m/day
The flow per unit width is given as,
q=\(\frac{(h_0-h_1)}{L} KB=\frac{(18-14)}{2500}*8*12\)=1.6*10^-3*96=0.1536 m^3/day per m≅0.15 m^3/day per m
5. The discharge per unit length of a tile drain in an unconfined aquifer is directly proportional to the distance between two consecutive drains for a given rate of recharge.
a) True
b) False
View Answer
Answer: a
Explanation: The discharge per unit length of a tile drain is q=RL, where R is the recharge rate and L is the distance between tile drains. It can be seen that for a given recharge, the discharge
increases as the distance between tile drains increases.
6. The maximum height of the water table between two tile drains does not depend on which of the following?
a) Distance between the drains
b) Permeability of aquifer
c) Rate of recharge
d) Depth of aquifer
View Answer
Answer: d
Explanation: The maximum height of the water table between two tile drains occurs at the mid-point of the two drains. The height at that point is given as, h[max]=\(\frac{L}{2}\sqrt{\frac{R}{K}}\).
7. Two rivers of depths 22 m and 17 m are connected by an unconfined aquifer of width 1370 m and permeability 7.5 m/day. The recharge rate per m^2 of the aquifer area if the water divide lies on the
upstream edge of the aquifer is k x 10^-4 m^3/day. What is the value of ‘k’?
a) 1.95
b) 3.87
c) 7.79
d) 15.58
View Answer
Answer: c
Explanation: Given h[0] = 22 m, h[1] = 17 m, L = 1370m, K = 7.5 m/day
If the water divide lies at the upstream end (x = 0), then a = 0.
a=\(\frac{L}{2}-\frac{K}{R} (\frac{h_0^2-h_1^2}{2L})\)=0
⇒R=\(\frac{K}{L^2} (h_0^2-h_1^2 )=\frac{7.5}{1370^2} (22^2-17^2 )=7.79*10^{-4} m^3/day/m^2\).
8. For the aquifer system shown (1 is confined aquifer and 2 is unconfined aquifer), find the total seepage discharge from river A to river B per meter width of the aquifer.
a) 2.64 m^3/day
b) 3.04 m^3/day
c) 6.02 m^3/day
d) 9.12 m^3/day
View Answer
Answer: a
Explanation: This is a composite aquifer system.
Firstly for confined aquifer (1), h[0] = 40 m, h[1] = 26 m, L = 4200m, K = 18 m/day, B =20 m.
q[1]=\(\frac{(h_0-h_1)}{L} KB=\frac{(40-26)}{4200}*18*20\)=1.2 m^3/day/m
Secondly for unconfined aquifer (2), h[0] = 40 – 20 = 20 m, h[1] = 26 – 18 = 8 m, L = 4200m, K = 36 m/day.
q[2]=\(\frac{(h_0^2-h_1^2)}{2L} K=\frac{(20^2-8^2)}{2*4200}*36\)=1.44 m^3/day/m
∴Total discharge=q[1]+q[2]=1.2+1.44=2.64 m^3/day/m
9. In the aquifer system shown, find the height of the water divide from the horizontal impervious bed.
a) 15 m
b) 17.5 m
c) 20 m
d) 22.5 m
View Answer
Answer: c
Explanation: Given h[0] = 11.5 m, h[1] = 19.8 m, L = 1450m, K = 13 m/day, R = 0.0078 m^3/day/m^2
For the given system, the water table profile is given as,
h^2=-\(\frac{Rx^2}{K}-\frac{(h_0^2-h_1^2-\frac{RL^2}{K})}{L} x+h_0^2\)
=\(-(\frac{0.078}{13}) x^2-\frac{1}{1450} (11.5^2-9.8^2-\frac{0.0078*1450^2}{13})x+11.5^2\)
Location of water table divide is given as,
a=\(\frac{L}{2}-\frac{K}{R} (\frac{h_0^2-h_1^2}{2L})=\frac{1450}{2}-\frac{13}{0.0078}(\frac{11.5^2-9.8^2}{2*1450})=704.19 m\) from upstream
Now substituting x = 7014.19 m in (1) we get,
∴h[max]=\(\sqrt{429.76}\)=20.73 m≅20 m
10. A tile drain system is installed in an unconfined aquifer of permeability 20 m/day and subjected to recharge of 0.005 m^3/day/m^2 of aquifer area. The drains are uniformly spaced at a distance of
500 m. Which of the following regarding the system is correct?
a) Water table profile is h^2=0.00025x^2-0.125x
b) Maximum height of water table is 4 m
c) Maximum water table height occurs at 125 m from a drain
d) Discharge entering a drain per m length is 1.25 m^3/day
View Answer
Answer: b
Explanation: Given K = 20 m/day, R = 0.005 m^3/day/m^2, L = 500 m
The water table profile is,
h^2=\(\frac{R}{K} (L-x)x=\frac{0.005}{20} (500-x)x\)=0.00025(500x-x^2)
Maximum height of water table is,
h[max]=\(\frac{L}{2} \sqrt{\frac{R}{K}}=\frac{500}{2} \sqrt{\frac{0.005}{20}}=250\sqrt{0.00025}=3.95 m≅4 m\)
Discharge entering a drain is,
q=RL=0.005*500=2.5 m^3/day per m length of drain | {"url":"https://www.sanfoundry.com/engineering-hydrology-questions-answers-groundwater-equation-motion-set-2/","timestamp":"2024-11-09T20:52:18Z","content_type":"text/html","content_length":"157997","record_id":"<urn:uuid:345186c8-f6a5-4d7a-b647-dadbbf905f88>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00557.warc.gz"} |
Lump Sum: Lump Sum Distributions
Main Menu Name: Lump Sum
Calculates the tax due, the amount remaining, and the effective tax rate on a lump sum distribution from a qualified pension or profit sharing plan if the lump sum qualifies for ten-year averaging at
the 1986 rate, or five-year averaging at the current year's rate (for years prior to 2000).
In this article:
This calculation determines the tax due, the amount remaining, and the effective tax rate on a lump sum distribution from a qualified pension or profit sharing plan if the lump sum qualifies for ten
year averaging at the 1986 rate, five year averaging at the last year's rate, and five year averaging at the current year's rate.
Lump sum distributions from a qualified pension of profit sharing plan must be included in reportable gross income and taxed at ordinary rates. In certain cases, special five-year income averaging is
still available which may result in considerable tax savings.
A lump sum qualifies for this one-time five-year income averaging election only if all of the following requirements are met:
• The sum is received after the recipient turned 59½ years of age.
• The sum is paid within one tax year.
• The sum is the entire distribution of the employee's benefit in the plan. (All pension plans maintained by an employee are considered a single plan. This also applies for profit sharing and stock
bonus plans.)
• The sum is payable for one of the following reasons:
1. The participant has died
2. The participant has attained age 59½
3. The employment of a non-self employed individual has been terminated
4. A self-employed individual has become disabled.
• The sum is distributed from a qualified plan (not an IRA or 403(b) tax-deferred annuity).
• The employee participated in the plan for at least five years prior to the distribution (this requirement does not apply to a death benefit).
If the lump sum distribution meets all of the above qualifications, it is eligible for five-year averaging.
In some cases, certain tax benefits available before 1987 for lump sum distributions have been grandfathered for existing participants.
Such participants may choose to treat the amount accumulated prior to 1974 as a long-term capital gain. If the participant was in the plan prior to 1974, the distribution is divided into two amounts,
a pre-1974 amount and a post-1975 amount. The pre-1974 amount is taxed at a 20% rate (the capital gain maximum). This capital gain treatment is phased out in the following manner: only 95% of the
pre-1974 amount is eligible in 1988, 75% is eligible in 1989, 50% in 1990, and 25% in 1991.
This capital gain treatment is not required. The entire distribution can be treated under current five-year averaging if the participant so chooses. Any part of the distribution that does not qualify
for capital gains treatment can (if it does) qualify for five year averaging.
From 1974 through 1986, ten-year averaging was applied to lump sum distributions. This practice is still available for individuals who attained age 50 before January 1, 1986. Such individuals who
receive a distribution after 1986 may use ten-year averaging with the 1986 tax rates instead of five-year averaging with current rates. This practice is recommended if it results in lower taxes.
Getting Started
Lump sum distributions from a qualified pension of profit sharing plan must be included in reportable gross income and taxed at ordinary rates. For example, assume an individual has $100,000 of
taxable income in 1993 and files jointly with two exemptions with the applicable tax being $23,529. If an additional $100,000 in pension payments is received by the individual, the tax on the total
income would be $58,205. The additional $100,000 cost $34,676 in additional tax ($58,205 minus $23,529). However, if the individual received a lump sum of $100,000 that qualified for special
five-year averaging, the tax on the lump sum would be $15,000 or a difference of $19,676 ($34,676 minus $15,000).
The lump sum qualifies for this one-time election of five-year averaging only if all of the following requirements are met:
• The sum is taken prior to the year 2000.
• The sum is received after the recipient turned 59½ years of age.
• The sum is paid within one tax year.
• The sum is the entire distribution of the employee's benefit in the plan. (All pension plans maintained by an employee are considered a single plan. This also applies for profit sharing and stock
bonus plans.)
• The sum is payable for one of the following reasons:
1. the participant has died
2. the participant has attained age 59½
3. the employment of a non-self employed individual has been terminated;
4. a self-employed individual has become disabled.
• The sum is distributed from a qualified plan (not an IRA or 403(b) tax-deferred annuity).
• The employee participated in the plan for at least five years prior to the distribution (this requirement does not apply to a death benefit).
If the lump sum distribution meets all of the above qualifications, it is eligible for five-year averaging. In some cases, certain tax benefits available before 1987 for lump sum distributions have
been grandfathered for existing participants. Such participants may choose to treat the amount accumulated prior to 1974 as a long-term capital gain. If the participant was in the plan prior to 1974,
the distribution is divided into two amounts, a pre-1974 amount and a post-1975 amount. The pre-1974 amount is taxed at a 20% rate (the capital gain maximum). This capital gain treatment is phased
out in the following manner: only 95% of the pre-1974 amount is eligible in 1988, 75% is eligible in 1989, 50% in 1990, and 25% in 1991.
This capital gain treatment is not required. The entire distribution can be treated under current five-year averaging if the participant so chooses. Any part of the distribution that does not qualify
for capital gains treatment can (if it does) qualify for five year averaging.
From 1974 through 1986, ten-year averaging was applied to lump sum distributions. This practice is still available for individuals who attained age 50 before January 1, 1986. Such individuals who
receive a distribution after 1986 may use ten-year averaging with the 1986 tax rates instead of five-year averaging with current rates. This practice is recommended if it results in lower taxes.
Entering Data
1. Current Year: Enter the current year. The program handles years from 1987 through the current year.
2. Taxable Amount of Lump Sum Distributions: Enter the taxable amount of the lump sum distribution.
The program shows the amount of tax due, the amount remaining, and the effective tax rate on a lump sum distribution. To calculate these values, the calculation first subtracts a "minimum
distribution allowance" from the taxable amount specified at the Taxable Amount of Lump Sum Distribution entry field. The minimum distribution allowance is the lesser of (a) $10,000 or (b) one-half
of the total taxable amount in excess of $20,000. (The minimum distribution allowance does not apply if the taxable amount is $70,000 or more.) The calculation then divides the remaining taxable
amount by five and determines a separate tax on this portion based on the single taxpayer rate without any deductions or exclusions. The resulting tax is then multiplied by five. Results are shown
for ten-year averaging, five-year averaging (last year's tax rate), and five-year averaging (the current year's tax rate).
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Best approach to understand different plots & applications as a beginner data analyst/scientist - Blue Heart Lab
Best approach to understand different plots & applications as a beginner data analyst/scientist
In this post you are going to learn about Histogram plot from manual sketching to plotting with python programming and application.
What is histogram ?
It is a chart that plots the distribution of a numeric variable’s values as a series of bars.
Each bar covers a range of numeric values called a bin or range of groups or class.
Bar’s height shows the frequency(no of occurrence) of data points with a value within the corresponding bin/class.
This chart displays the shape and spread of continuous sample data.
Consider this: If we want to know the age of everyone in the college, we can look at:
But you can see here that there are lots of people, whose ages may be the same or very close to each other; due to this, there are many dots (data points) that overlap & completely hidden too. So we
try to put the same age values one over the other, as below.
However, there are many hidden values due to many adjacent values (age), such as 2 people having age 33 year 6 month, 3 people having age 33 year, 6 people having age 33 year 8 month…, 4 people
having age 35 year, 6 people having age 35 year 4 month, 2 people having age 35 year 2 month, and so on. If you plot these values around 33, you’ll see that there are still hidden values.
So instead of putting same values one over the other like in column form, we divided the range of values( here age) into bins and put all values one over the other in the same bin, looking like this:
This is Histogram plot of data.
(1) You can easily tell how many people belong to which age group, for example, 15 people between the ages of 0 and 10, 11 people between the ages of 10 and 15, and so on.
(2) Most people are young, belonging to the 0–20 year age group.
This way, there can be many observations, depending on your problem.
So from this plot, you can predict the probability of getting new values. for example, if you are getting new values, then it will have a higher probability that it belongs to the young group (0–20
years) because from the above plot you can see most of the values belong to that group (15+11 = 26 people).
No of bins matter depends on data & problems
Let’s say we have only two bins for the above data. Then the plot will look like:
Best point to observe: 26 people above average age (30 years) and 10 people below.
Best point to observe: adjacent age groups count.
In this way, you’ve learned that the number of bins or the width of the bins matters in histograms, so to solve your problem, plot histograms for different – different bins and observe them.
Now that you understand the histogram plot as a beginner in the field of data analyst/data scientist, your next problem will be how to plot all variations of given data. If there are data having a
small number of data points in it, then it will be easy to plot all variations of the histogram manually with a pen and paper, but in today’s world there are data having a huge number of data sets
(in the thousands or millions ) in it, so manual plotting is a very tough and time-consuming process, which will also increase the cost of the solution. Here comes the role of programming languages
for data analysis like Python, R, etc.
Plotting histogram with python
First of all, you have to install Python 3, then install Jupyter Notebook, Numpy, Pandas, Matplotlib, and Seaborn.
Download the netflix shows data from https://www.kaggle.com/datasets/shivamb/netflix-shows
Data will look like this:
Code to plot the histogram to know the distribution of movies and shows releasing on Netflix yearly:
import seaborn as sns
import pandas as pd
df = pd.read_csv(r’D:\Blog\1_Histogram\archive\netflix_titles.csv’)
sns.histplot(data = df,bins = 5)
The movie and show bins (ranges) are 20 years. Each bar here includes all shows and movies in batches of 20 years. For example, we can see that around ~ 7500 shows were released between 2000 and
Let’s visualise a histogram (distribution) plot in batches of 1 year. First, let’s select the column name ‘release_year’.
import seaborn as sns
import pandas as pd
import numpy as np
df = pd.read_csv(r’D:\Blog\1_Histogram\archive\netflix_titles.csv’)
data = df[‘release_year’]
You can see a total of 8806 rows, meaning total movies and shows = 8807. Now, before plotting it, we need to convert it into a list with the following code:
b = np.arange(min(data), max(data) + 1, 1)
You can see the data starts from 1925 to 2021, i.e., a total of 96 years of data. Let’s now plot it with the following code:
sns.histplot(data, bins= b)
Full code looks like this:
import seaborn as sns
import pandas as pd
import numpy as np
df = pd.read_csv(r’D:\Blog\1_Histogram\archive\netflix_titles.csv’)
data = df[‘release_year’]
b = np.arange(min(data), max(data) + 1, 1)
sns.histplot(data, bins= b)
If you want to view smoother, then you can plot kernel density by writing the following code:
import seaborn as sns
import pandas as pd
import numpy as np
df = pd.read_csv(r’D:\Blog\1_Histogram\archive\netflix_titles.csv’)
data = df[‘release_year’]
t = df[‘type’]
b = np.arange(min(data), max(data) + 1, 1)
sns.histplot(data, bins= b, kde=True)
Let’s now plot the normal density in the histogram:
import seaborn as sns
import pandas as pd
import numpy as np
df = pd.read_csv(r’D:\Blog\1_Histogram\archive\netflix_titles.csv’)
data = df[‘release_year’]
t = df[‘type’]
b = np.arange(min(data), max(data) + 1, 1)
sns.histplot(data, bins= b, kde=False, stat=’density’)
You can observe that highest ~ 17.5 % of total movies/shows were released in 2021 while second highest ~ 13.5 % in 2019.
If you want to see the normal distribution for movies as well as TV shows in the same histogram, then code:
import seaborn as sns
import pandas as pd
import numpy as np
df = pd.read_csv(r’D:\Blog\1_Histogram\archive\netflix_titles.csv’)
data = df[‘release_year’]
b = np.arange(min(data), max(data) + 1, 1)
sns.histplot(data= df, x = ‘release_year’, bins= b, kde=False, stat = ‘density’, hue = ‘type’)
(1) In 2021 ~ 9.1% movies of total movies upto 2021, ~ 8.5% tv shows of total shows upto 2021 were released.
(2) In 2019 ~ 9% movies of total movies upto 2021, ~ 4.5% tv shows of total shows upto 2021 were released.
(1) To analyse and predict footfall in a restaurant on the basis of age, time, day, month, etc.
(2) As the exam date countdown begins, the number of study hours for students increases.
(3) More people tend to go outside for movies or travel at the weekend, so we can predict their expenditure will rise on Saturday and Sunday and their footfall at malls will peak at the weekend.
This way, there can be many more applications.
You can try our new python course for beginners: Python For Beginners | {"url":"https://blueheartlab.com/how-to-understand-different-plots-applications-as-a-beginner-data-analyst-scientist/","timestamp":"2024-11-02T21:12:58Z","content_type":"text/html","content_length":"155680","record_id":"<urn:uuid:01991893-4d6d-4164-b4cb-29c73efeb755>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00731.warc.gz"} |
C/2020 H11 | CODEC
C/2020 H11 PanSTARRS-Lemmon
more info
Comet C/2020 H11 was discovered on 21 April 2020, about 5 months before its perihelion passage. Later a series of pre-discovery observations were found going ten months back to 2 June 2019. This
comet was observed to 8 May 2021.
Solutions given here are based on data span over 1.93 yr in a range of heliocentric distances: 8.17 au – 7.63 au (perihelion) – 7.77 au.
This comet entered the planetary zone having original semimajor axis of about 4,500 au and suffers small planetary perturbations during its passage through the planetary system that lead to a bit
tight future orbit (semimajor axis of about 3,100 au).
solution description
number of observations 53
data interval 2019 06 02 – 2021 05 08
data type perihelion within the observation arc (FULL)
data arc selection entire data set (STD)
range of heliocentric distances 8.17 au – 7.63 au (perihelion) – 7.77 au
detectability of NG effects in the comet's motion NG effects not determinable
type of model of motion GR - gravitational orbit
data weighting NO
number of residuals 101
RMS [arcseconds] 0.21
orbit quality class 1a
previous orbit statistics, both Galactic and stellar perturbations were taken into account
no. of returning VCs in the swarm 5001 *
no. of escaping VCs in the swarm 0
no. of hyperbolas among escaping VCs in the swarm 0
previous reciprocal semi-major axis [10^-6 au^-1] 221.18–224.01–226.82
previous perihelion distance [au] 7.6489–7.65–7.651
previous aphelion distance [10^3 au] 8.81–8.92–9.03
time interval to previous perihelion [Myr] 0.292–0.298–0.303
percentage of VCs with q[prev] < 10 100
previous orbit statistics, here only the Galactic tide has been included
no. of returning VCs in the swarm 5001 *
no. of escaping VCs in the swarm 0
no. of hyperbolas among escaping VCs in the swarm 0
previous reciprocal semi-major axis [10^-6 au^-1] 221.18–224.01–226.81
previous perihelion distance [au] 7.6502–7.6513–7.6524
previous aphelion distance [10^3 au] 8.81–8.92–9.03
time interval to previous perihelion [Myr] 0.292–0.298–0.304
percentage of VCs with q[prev] < 10 100 | {"url":"https://code.cbk.waw.pl/orbit.php?int=2020h1a1&orb=previous","timestamp":"2024-11-05T16:25:23Z","content_type":"text/html","content_length":"18210","record_id":"<urn:uuid:d9aae0de-dc92-4b09-9cd7-ad157d56af2f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00438.warc.gz"} |
Inspiring Drawing Tutorials
In The Drawing Six Out Of Every 10
In The Drawing Six Out Of Every 10 - What is the probability that a ticket that is randomly. While an initial estimate of 1/160 is probably within a close enough range to suggest i have little chance
of. First we show the two. Of the winning tickets, 1 out of every three awards is a larger prize. Of the winning tickets, one out of every three awards is a larger prize. Solution for in the drawng,
six out of every 10 tickets are winning tickets.
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involve a random drawing to determine the winners, making it crucial to know your odds. \frac {1} {5} 51 _.
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the winning tickets, 1 out of every three awards is a larger prize. Raffles involve a random drawing to determine the winners, making.
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(cece pascual/the washington post) 13 min. As you stare at your blank march. So, what is the probability you will be a goalkeeper today? The men’s march madness bracket has dropped, and if you need
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In the drawing, six out of every 10 tickets are winning tickets. Sporting news has every type of college basketball fan covered with our 2024 ncaa tournament bracket in printable, pdf form,. \frac
{6} {10}\times \frac {2} {6}=\frac {2} {10}=\frac {1} {5} 106 × 62 = 102 = 51 (the use of the probability) show more. 2 will be the.
In the drawing, six out of every 10 tickets are winning tickets. As you stare at your blank march. You have one of the five rows of tickets, but there's also another one two,. Of the winning tickets,
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Of the winning tickets, one out of every three awards is a larget prize. So, what is the probability you will be a goalkeeper today? As you stare at your blank march. While an initial estimate of 1/
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Solution for in the drawng, six out of every 10 tickets are winning tickets. 2 seed while woods sits at the no. Follow along here as we track every aq for march madness, beginning with the ohio
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In The Drawing Six Out Of Every 10 - \frac {6} {10}\times \frac {2} {6}=\frac {2} {10}=\frac {1} {5} 106 × 62 = 102 = 51 (the use of the probability) show more. Of the winning tickets, one out of
every three awards is a larger. Every finals match of the 2024 b1g wrestling championships | mar. Follow along here as we track every aq for march madness, beginning with the ohio valley on march 9
and. Okay, and those are yours. As you stare at your blank march. The men’s march madness bracket has dropped, and if you need any help filling yours out before. 3 spot, setting up first round.
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In the drawng, six out of every 10 tickets are. Okay, and those are yours. While an initial estimate of 1/160 is probably within a close enough range to suggest i have little chance of. (cece pascual
/the washington post) 13 min. In the drawing, six out of every 10 tickets are winning tickets.
What is the probability that a ticket that is randomly. (cece pascual/the washington post) 13 min. Of the winning tickets, one out of every three awards is a larger prize. The men’s march madness
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What are the odds i will win a prize? Sporting news has every type of college basketball fan covered with our 2024 ncaa tournament bracket in printable, pdf form,. I have bought ten tickets.
The Raffle Percentage Calculator Takes Into Account The Number Of Available.
In the drawing, six out of every 10 tickets are winning tickets. 2 will be the numerator. \frac {1} {5} 51 _. While an initial estimate of 1/160 is probably within a close enough range to suggest i
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In Other Words, We Must Have Chosen Five Of The Six Winning Numbers And.
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sunday on march 17, 2024. Readers thanked neil greenberg for his winning picks in 2023.
The Ncaa Tournament Games Then Get Underway With The First Four On.
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Raffles Involve A Random Drawing To Determine The Winners, Making It Crucial To Know Your Odds.
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a larger prize. What is the probability that a ticket that is randomly. | {"url":"https://one.wkkf.org/art/drawing-tutorials/in-the-drawing-six-out-of-every-10.html","timestamp":"2024-11-11T00:40:52Z","content_type":"text/html","content_length":"33503","record_id":"<urn:uuid:4545f66f-9a98-4b9f-bf07-7ee452a04b6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00372.warc.gz"} |
July 2013
I remember during law school, I was helping out at the Florida First Amendment Foundation and discussing privacy and public record issues with the Director. She was fairly adamant that death obviated
any expectation of privacy. The person was dead, how could a person without consciousness have an expectation. Her position was informed in part by the debate over the autopsy photos of Dale
Earnhardt, Jr.
death. I believe that this is fairly common practice through the US. I have witness that it is generally verboten to ask or tell if known, what one’s cause of death is unless it is widely known
(obvious illness, accident, murder, etc). It appears that we have adopted a cultural norm against generalized disclosure of cause of death and that norm has been codified in law as they related to
death certificates. This isn’t airtight (as certain causes mentioned above become widely known) and it doesn’t appear to be based on any preference of the deceased, though one could imagine the
deceased leaving instructions to publicize their cause of death.
I’m not sure how or when this cultural norm developed but I do find in interesting and would like to learn of others perspectives in differing cultures.
Privacy and Mesh Networks
I’ve been thinking a lot about Mesh Networking and the possibility to foil NSA style tapping by bypassing centralized networks for localized networks. For those who don’t know mesh networks are
ad-hoc peer to peer networks, primarily wireless. The decentralized nature of the communications provides some level of privacy. Additional privacy comes with making the system anonymous. However, it
seems that anonymity comes with a price to bandwidth. Here are some of my findings:
The standard model is circuit switched. In other words, each node maintains a topological map of the entire network, so that it knows who is connected to whom. This allows it to create a circuit
through the network to the destination. That means that each transmission takes t bandwidth where t represents the size of the data. If s represents the shortest number of hops from source to
destination then the total network bandwidth used B= s*t. In this model, no storage is required because each node forwards the data without need to retain it. There is some data storage requirements
for the network map. This map grows with n^2.
Circuit: (Bandwidth = s*t, Storage: n*(m*n)^2) m= the amount of data necessary to indicate a link or not (maybe a bit, maybe a byte if you store strength).
The upside is that each new node increases the bandwidth of the network. The downside of this is that an attacker could possibly follow the data as it gets transferred from node to node and identify
sender and receiver, providing linkability and thus defeating anonymity.
Consider, alternatively a broadcast model. In this model, no topological map must be stored but every node gets a copy of the data.
Broadcast: (Bandwidth = n*t, Storage: n*t)
In this model, nobody can identify the destination of a message which is very privacy preserving. However, the bandwidth cost are enormous. Now, each new node added to the network actually adds an
external cost to the other node, similar to a car being added to a highway. The storage cost also increase at rate n*t because every node must keep a copy of the message (or at least a digest) for a
period of time to prevent it re-accepting the data from one if it’s neighbors.
A third option is the random walk model, also called the hot potato. A node passes the data packet to another who passes it again, etc. In this model no node keeps a copy of the data once it has
passed so the storage costs are 0. The bandwidth is a minimum s*t, because that’s the shortest circuit. BUT, the packet could be passed along for ever, so the bandwidth could be potentially infinite.
Needless to say this is not good.
Random walk: ( S*t < Bandwidth < ∞, Storage = 0)
What about a biased or intelligent random walk, a lukewarm potato. The network has the following rules. Each node ask its immediate neighbors, “is this yours?” Appropriate response are “yes, give it
to me” “no, but I’ll take it” or “no, i’ve seen it”. If the neighbor said yes, the node gives them the data. If the neighbor say “no, i’ve seen it” it ignores that node. The node then randomly
selects one of the nodes that hasn’t seen the data and sends it to them to continue the process. If the node can’t find anybody to hand it over to, it tells the node that it got it from that it can’t
pass it along, that node starts over again. [Alternatively, it could force one of the nodes that have seen it to take it again]. This method allows the data to snake it’s way through the network but
not repeat any nodes. Here is the bandwidth and storage boundaries.
Intelligent random walk: ( s*t < Bandwidth < n*t, s*n < Storage < n* t)
So this doesn’t have as low of bandwidth and storage as the circuit but it’s not as bad as broadcast for storage and not as bad as random walk for bandwidth. However, anonymity is not perfect. An
attacker who had access to the entire network could identify the recipient as the packet traces it ways through the network.
It appears then than anonymity costs either in bandwidth or storage cost. The question is what is more valuable to the network. There maybe additional techniques to mitigate this, and I continue to
investigate this area.
Algorithmic privacy versus personal privacy
In this blog post, Peter Kinnaird, attempts to analogize the NSA spying to the algorithmic review of our emails by Google. He notes that a majority of people accept such review as non-invasive and
worth the benefits derived from free and useful email-as-a-service. I would like to point out several fallacies in his analysis.
1. As a quick note, he says “I feel certain that if Google didn’t have adequate social and technical safeguards in place, we would have heard of at least one case of a Google employee snooping or
abusing their power.” Here is the one case I’m familiar with: https://gawker.com/5637234/gcreep-google-engineer-stalked-teens-spied-on-chats This doesn’t mean their aren’t others that Google
quietly fired in order to keep out of the press. Government employee abuse of the information at their disposal is rampant and has huge historical precedent, whether sanctioned by higher ups or
performed by rogue individuals.
2. The post fails to distinguish the voluntary nature of participation with Gmail and the involuntary participation in the state surveillance apparatus. Mutuality is the cornerstone of privacy
expectations. Without voluntariness, mutuality can not exist.
3. The post fails to consider the risks involved in revealing information to Google versus the government. If I reveal information to Google I might get mislabeled and have inappropriate ads sent to
me. If I get reveal information to the government, i might get mislabeled and jailed or murdered.
4. The post mentions the public awareness of Google’s practice but fails to contrast that with the secret nature of the NSA program. Overt versus covert makes a world of difference in privacy. We
don’t even know what we don’t know about NSA spying.
5. The post fails to consider other, less privacy invasive means of achieving the same results, i.e. national security. Any privacy analysis of a system must dismiss other means of achieving the
same goals.
There are a host of non-privacy related issues having to do with NSA spying, such as international relations and the loss of world wide confidence in buying American information services, that also
need to be considered. Frankly a world in which I am spied on, personally or algorithmically, is not one in which I wish to live.
Suggested reading: 1984, The Trial | {"url":"https://privacymaverick.com/2013/07/","timestamp":"2024-11-13T20:54:00Z","content_type":"text/html","content_length":"44614","record_id":"<urn:uuid:23a90edd-539a-498a-bc04-05f29575c806>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00227.warc.gz"} |
Rule of 72 Calculator
Online Calculators > Financial Calculators > Rule of 72 Calculator
Rule of 72 Calculator
Rule of 72 Calculator calculates how long it will take to double your money or investment given an interest rate. Rule of 72 states that the years required to double your money at a given interest
rate, simply divide the interest rate into 72.
Exact Answer 14.21
Rule of 72 Estimate 14.40
What is the Rule of 72?
What is The Rule of 72 or the rule of 72 formula is a quick and easy formula to estimate the number of years it takes to double an investment given an annual rate of return.
The rule of 72 formula given below
Years to Double Your Money = 72/Interest Rate
For example, to find out how long it will take to double your money givin an interest rate of 5%, simply divide
72/5 = 14.40 which is very close to the actual value of 14.21. | {"url":"https://online-calculator.org/rule-of-72-calculator.aspx","timestamp":"2024-11-09T01:00:51Z","content_type":"application/xhtml+xml","content_length":"16114","record_id":"<urn:uuid:09928aed-2ac3-4b94-9b53-1bc0e38ec6da>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00286.warc.gz"} |
Spell Out All The Numbers From 1 To 20 - SpellingNumbers.com
Spell Out All The Numbers From 1 To 20
Spell Out All The Numbers From 1 To 20 – It can be challenging to learn how to spell numbers. However, learning to spell could be made easier with the right resources. If you need assistance with
spelling, there are many tools available, whether at school or working. These tools include tips and tricks along with workbooks, as well as games online.
The format of the Associated Press format
If you are writing for the newspaper or another print publication, it’s important that you can spell numbers with the AP Style. To make your writing more concise, the AP style offers instructions for
writing numbers as well as other things.
The Associated Press Stylebook, which was launched in 1953, has undergone hundreds of changes. This stylebook is in its 55th edition. This stylebook is used by the majority of American newspapers
periodicals, newspapers, as well as internet news media.
Journalism uses AP Style, a set of punctuation guidelines and language rules. AP Style’s top practices include capitalization, the use date and time, as well as citations.
Regular numbers
Ordinal numbers refer to an exclusive integer that identifies a particular place in a list, or series. These numbers are often used to represent size, time, and significance. These figures also
reveal the order in which things happen.
In a variety of situations, ordinary numbers can either be written verbally (or numerically) depending on how they are utilized. A unique suffix helps to identify between the two.
To make a number ordinal put a “th” at the end. The ordinal number 31 could be represented using 31.
You can use ordinals to serve a variety of purposes, such as dates and names. It is crucial to know the differences between using an ordinal (or cardinal) and an ordinal.
The trillions and billions are both
Numerology is applicable to many different instances, such as the markets, geology and the past of our world. Millions and billions of dollars are just two examples. Million is a normal number that
happens prior to 1,000,001, and the billion occurs after 999.999.999.
In millions, is the annual earnings of an organization. These numbers are used to determine the value of a share or fund, or any other financial item. To determine the value of a company’s market
capitalization billions are frequently employed. You may verify the validity of your estimations by converting millions to billions by using a calculator for unit conversion.
Fractions are used in English to refer to specific items or components of numbers. The denominator and the numerator are separated into two pieces. The numerator will tell you the number of pieces
that are equal in size were gathered. The denominator in contrast illustrates how many pieces were split into.
Fractions can be expressed mathematically, or in terms. When writing fractions using words, you must be mindful of spelling the words out. It might be difficult to correctly spell out fractions
particularly if you’re dealing with large fractions.
There are some basic guidelines you can apply to write fractions in the same way as words. One option is to write numbers out in full at the beginning of sentences. It is also possible to compose
fractions using decimal formats.
Many Years
A thesis paper as well as a research paper or email may require you to draw on decades of experience spelling numbers. Certain tips and tricks can help you avoid repeating the same spelling and
ensure proper formatting.
In formal writing, numbers must be written out. There are numerous style guides that offer various guidelines. The Chicago Manual of Style suggests that numerals should be utilized between 1 and 100.
But, it is not recommended that you write numbers higher than 401.
There are exceptions. One example is the American Psychological Association’s (APA) style guide. Although it is not a specific one, is frequently utilized in writing for scientific purposes.
Time and date
The Associated Press style handbook provides some general guidelines to styling numbers. For numbers greater than 10 the numerals are utilized. Numerology may also be used in other locations. It’s
the best practice to use the “n-mandated number” for the first five numbers on your document. There are exceptions.
Both the Chicago Manual of Technique as well as the AP stylebook suggest that you make use of plenty of numbers. However, this does not suggest that it is not possible to come up with a stylebook
without numbers. I can guarantee you that the difference is significant because I am an AP student.
A stylebook must be checked to see the ones you’re omitting. For instance, make sure to not overlook the “t” in the form of “time”
Gallery of Spell Out All The Numbers From 1 To 20
Los N meros En Espa ol Del 1 Al 20 Pronunciaci n Y Escritura
French Numbers How To Count From 1 100 And Beyond With Audio
10 Best Writing Numbers 1 20 Printables Printablee | {"url":"https://www.spellingnumbers.com/spell-out-all-the-numbers-from-1-to-20/","timestamp":"2024-11-02T08:17:01Z","content_type":"text/html","content_length":"63126","record_id":"<urn:uuid:bcde5d3c-4a25-46c7-a4a6-ea2d489e8920>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00404.warc.gz"} |
Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation - Knowunity
Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation - AP Calc Study Guide
Greetings, fellow calculus adventurers! Ready to embark on a journey to the mystical land of antiderivatives and indefinite integrals? 🎢 Think of this as the magical reverse spell to differentiation,
where we turn derivatives back into smooth, continuous functions. Let’s dive in and unravel these enchanting mathematical concepts.
Indefinite Integrals: Notation
Before we jump into the wizardry of reversing derivatives, let’s talk about the notation and explore the mystical "family of functions." Imagine you have two different antiderivatives, like siblings
F(x)=x²+3 and G(x)=x²−2; they both have the same derivative, 2x. If we reverse the derivative process through integration, we get 2x+C, where C is the magical constant 🎩✨, a.k.a. any constant ever!
This gives rise to a whole family of functions differing only by C, but united by the same derivative. These are indefinite integrals because we can’t pinpoint which sibling (antiderivative) we’re
dealing with unless we specify the bounds, as we would in definite integrals.
When writing this magical process, we use the notation: [ \int f(x)dx = F(x) + C ] Where F'(x) = f(x) and C denotes the integration constant.
Indefinite Integrals: Basic Rules
The adventure begins by reversing some of the derivatives we already know. Let’s go through the basic rules.
Reverse Power Rule Just like a magician pulling a rabbit out of a hat, we're pulling exponents into integrals! For any function ( f(x) = \frac{x^{n+1}}{n+1} + C ) where ( n \neq -1 ) (because, let's
face it, dividing by zero isn’t magical, it’s hazardous).
Here’s how it looks: [ \int x^n dx = \frac{x^{n+1}}{n+1} + C ]
Example 1: Reverse Power Rule
Evaluate (\int x^3 dx).
Using the reverse power rule: [ \int x^3 dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C ]
Example 2: Reverse Power Rule with Fractions/Radicals
Try this spicy one: [ \int \left(\frac{1}{x^2} - 7x^3 + 2x^2 - x + 4\right) dx ]
Rewrite first term: [ \int \left(x^{-2} - 7x^3 + 2x^2 - x + 4\right) dx ]
Now reverse power rule term by term: [ \int x^{-2} dx = -\frac{1}{x} ] [ \int -7x^3 dx = -\frac{7x^4}{4} ] [ \int 2x^2 dx = \frac{2x^3}{3} ] [ \int -x dx = -\frac{x^2}{2} ] [ \int 4 dx = 4x ]
Combine all: [ \int \left(\frac{1}{x^2} - 7x^3 + 2x^2 - x + 4\right) dx = -\frac{1}{x} - \frac{7x^4}{4} + \frac{2x^3}{3} - \frac{x^2}{2} + 4x + C ]
Sums and Multiples Rules for Antiderivatives
Just like you can mix and match your favorite snacks, you can mix and match functions when integrating. 🍬+🍿=😁
• Sums Rule: (\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx)
• Multiples Rule: (\int c \cdot f(x) dx = c \int f(x) dx)
Example: Sums Rule
[ \int [x^4 + x^2] dx = \int x^4 dx + \int x^2 dx ]
Example: Multiples Rule
[ \int 5x^6 dx = 5 \int x^6 dx ]
Antiderivatives of Trigonometric Functions
Time to switch our robes and dive into the realm of trigonometry. Ever wondered what functions sin(x) and cos(x) like to come from? Well, let’s find out! 😀
• (\int \sin(x) dx = -\cos(x) + C)
• (\int \cos(x) dx = \sin(x) + C)
• (\int \sec^2(x) dx = \tan(x) + C)
• (\int \csc^2(x) dx = -\cot(x) + C)
• (\int \sec(x) \tan(x) dx = \sec(x) + C)
• (\int \csc(x) \cot(x) dx = -\csc(x) + C)
Antiderivatives of Inverse Trig Functions
Not as common, but here are some to keep in your spellbook:
• (\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x) + C)
• (\int \frac{1}{1+x^2} dx = \tan^{-1}(x) + C)
Antiderivatives of Transcendental Functions
Finally, let's transcend the ordinary with these familiar faces:
• (\int \frac{1}{x} dx = \ln|x| + C)
• (\int e^x dx = e^x + C)
Indefinite Integrals Practice Problems
Let's put your new skills to the test! 🧪
1. Evaluate (\int x^7 dx)
Using the reverse power rule: [ \int x^7 dx = \frac{x^8}{8} + C = \boxed{\frac{1}{8}x^8 + C} ]
2. Evaluate (\int [x^4 + \cos(x)] dx)
Split the integral: [ \int x^4 dx + \int \cos(x) dx ] [ = \frac{x^5}{5} + \sin(x) + C = \boxed{\frac{1}{5}x^5 + \sin(x) + C} ]
3. Evaluate (\int [4\cos(x) + e^x] dx)
Split and simplify: [ 4\int \cos(x) dx + \int e^x dx ] [ = 4\sin(x) + e^x + C = \boxed{4\sin(x) + e^x + C} ]
4. Evaluate (\int \left(\frac{3}{x} + x^2\right) dx)
Split and use appropriate rules: [ \int \frac{3}{x} dx + \int x^2 dx ] [ = 3 \ln|x| + \frac{x^3}{3} + C = \boxed{3 \ln|x| + \frac{x^3}{3} + C} ]
Woah, what a ride! We’ve journeyed through reverse power rules, sums and multiples rules, trigonometric and transcendental functions. The biggest nugget of wisdom? When integrating, always add that
magical constant "+C"! Now, go forth and integrate like a math wizard. 🧙♂️🧙♀️🍀
Good luck on your AP Calculus quest! You've got all the spells you need. 🚀 | {"url":"https://knowunity.com/subjects/study-guide/finding-antiderivatives-indefinite-integrals-basic-rules-notation","timestamp":"2024-11-11T17:28:33Z","content_type":"text/html","content_length":"265510","record_id":"<urn:uuid:8dda49fa-b28e-4b9e-bb24-fae701b7c92a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00309.warc.gz"} |
Backtesting in Excel
Backtesting in modeling refers to a predictive model's testing using historical data. The article is about how to do so in Microsoft Excel, not about the theoretical background of backtesting.
How do we conduct backtesting? We rewind the time to the beginning of our time series, calibrate the subject-model parameters using available data up to that instance of time, and conduct a
prediction (i.e., forecast) for the next period. Next, we advance the time, recalibrate the parameters' values and perform another projection, and so on. At the end of our exercise, we'd have a set
of predictions.
Note that at each point, the only assumption we make is the general model definition (e.g., ARMA(1,1)). Still, we'd calibrate the parameters' values using only the information available up to that
instance of time.
This approach is consistent with real-life practices: first, we start with an initial model and conduct a forecast for the following period. Time moves on. A new period occurred, so we append the new
data-point to the current input data set, recalibrate the parameters, conduct a forecast for the following period, and repeat.
Why should I care?
This article will take you through the steps in Microsoft Excel needed to conduct backtesting We will mainly use two powerful excel built-in functions: INDEX(.) and SEQUENCE(.), and leverage Excel's
"Data table" mechanism to run the different scenarios.
The backtesting generates what would-have-been forecasting errors, so you can closely examine the prediction error time series for serial correlation, distributions, outliers, and others, to better
understand the model's accuracy and performance.
Let's dig in!
For this issue, we are using a synthetic stationary data set of 200 observations. The data set follows an ARMA(1,1) process, as shown next.
The proposed model is ARMA(1,1)
Backtesting Procedure
For every iteration, we need to do the following: (1) define the input data set (as sub-set of the original time series), (2) using the data set in (1), calibrate the parameters' values of the ARMA
(1,1) model, (3) Using the model in (2) and the dataset in (1), calculate a forecast for one-period ahead.
1. Input data set
To fully describe the input data set, we require two indices: start and finish, then, using the SEQUENCE(.) function, we generate a set of indices between the start and finish. Now, we use the INDEX
(.) function to return all cell-range in the original data set with row-indices in the sequence set.
The original input data set is \$A\$3:\$A\$202. To select the cells between indices 1 and 50.
Note that you can define a name for your input data and reference this name in place of the input cell range.
2. Calibrate the Model
We will use the NumXL ARMA_PARAM(.) function and specify return type=2 for calibrated parameters.
Note that ARMA_PARAM(.) returns a compact form of the model's parameters, so in the figure above, the ARMA process is: \[\begin{array}{l} {X_t} = 1.485 + 0.401{X_{t - 1}} + 0.734{a_{t - 1}} + {a_t}\\
{a_t} \sim N(0,1.14) \end{array}\]
3. Forecasting
Using the ARMA_FORE(.), data set in (1), and the model parameters calculated in 2, we can calculate the forecast values for the 1-period ahead.
Note that I generated the mean forecast, forecast error, and confidence interval.
Data Table
Now, we have just completed the calculation of one step. We will use Excel's "Data table" feature to do the same math for the remaining periods up to step 200.
First, we need to prepare the output table:
Now, select the whole data table, starting with the output row and including the finish index column, as shown below:
Next, switch to the "Data" Toolbar and locate the "Data Table" item under "What-if Scenario."
The "Data Table" dialog pops up. Locate the "column input cell," enter a reference to the "finish" index of the data set, and click the OK button.
The data table will substitute the finish index's value with the ones in our data table and store the outputs.
Note that "Std. Error" is generated by the ARMA_FORE(.) function, while the right-most column ("Error") is the error between forecast and actual realized value.
Back-Testing Analysis
First, Let's examine the backtesting forecast outputs visually to actual realized values, and then we will delve deeper into the statistical properties.
The shaded area in the plot corresponds to the 95% forecast confidence interval.
The plot exhibits a good model fit and, thus, forecast accuracy.
Next, let's examine the forecast error (Forecast – Actual) statistical properties using summary statistics in the NumXL toolbar.
The summary statistics table indicates that forecast error is Gaussian noise with zero mean and standard deviation of 1.0
In conclusion, the ARMA(1,1) is a suitable predictive model for the given data set.
What is next?
By now, you are probably wondering about the values of the model's parameters? Are they stable?
First, we construct a second "Data Table," but in the output row are the model parameters' values, and run the data table just like we did earlier.
Now, analyze the values of every parameter as we did with the forecast error.
The values of the parameters (except theta) exhibit stability and a trend toward a constant value. The theta (MA coefficient) values are more volatile but bound between 0.6 and 0.9.
Next, we should examine the descriptive statistics and the underlying distributions of the parameters' values, but we will leave this exercise to you.
Please, refer to the attached spreadsheet for the data set and analysis.
This article demonstrated the steps to conduct backtesting for a predictive model with minimal or no intermediate calculation. We have used Excel's built-in functions: INDEX(.) and SEQUENCE(.) and
leveraged the "Data Table" feature to run the calculation for all pre-defined indices.
Once we had the backtesting results, we turn our heads to statistical analysis and evaluate their properties and distribution to uncover any biases (e.g., serial correlation) in the output.
Please sign in to leave a comment. | {"url":"https://support.numxl.com/hc/en-us/articles/360050863352-Backtesting-in-Excel","timestamp":"2024-11-06T11:36:03Z","content_type":"text/html","content_length":"43782","record_id":"<urn:uuid:59a72944-20dc-490b-ab35-845f81fc59da>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00533.warc.gz"} |
MathSciDoc: An Archive for Mathematician
Given a gauged linear sigma model (GLSM) $\mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum K\"ahler moduli has a geometric point, we propose an \emph
{extended} GLSM $\mathcal{T}_{\mathcal{X}}$ realizing the homological projective dual category $\mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In
most of the cases, the models $\mathcal{T}_{X}$ and $\mathcal{T}_{\mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the
semiorthogonal/Lefschetz decompositions of $\mathcal{C}$ and $D^{b}Coh(X)$. We also study the models $\mathcal{T}_{X_{L}}$ and $\mathcal{T}_{\mathcal{X}_{L}}$ that correspond to homological
projective duality of linear sections $X_{L}$ of $X$. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear
and Veronese embeddings of $\mathbb{P}^{n}$ and Fano complete intersections in $\mathbb{P}^{n}$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In
addition, we comment on the construction of the HPD to a nonabelian GLSM for the Pl\"ucker embedding of the Grassmannian $G(k,N)$. | {"url":"https://archive.ymsc.tsinghua.edu.cn/pacm_category/0104?show=view&size=3&from=1&target=searchall","timestamp":"2024-11-03T16:32:13Z","content_type":"text/html","content_length":"63369","record_id":"<urn:uuid:8290341a-0bab-489e-a790-efecc2b32123>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00663.warc.gz"} |
The Rise of Quantum Computing: 5 Amazing Ways It Is the Next Frontier of Technology - Adjei Kofi
The Rise of Quantum Computing: 5 Amazing Ways It Is the Next Frontier of Technology
Quantum computing is a technology that can change the world as we know it. It could improve everything from aeroplane navigation systems to cancer treatments. But what exactly is quantum computing?
How will it work?
What are its limitations? I will answer some of these questions and provide an overview of quantum computing, so you can make sense of this exciting new technology.
What is quantum computing?
Quantum computing is a new type of computer that uses quantum bits (or qubits). These are the building blocks of quantum computers, which can be used to store information in multiple states
For example, suppose you have an ordinary bit (a 0 or 1). In that case, it can only exist as either 0 or 1 at any given time–it cannot be both simultaneously. But with qubits, they can be both 0 and
1 simultaneously because they are so small that they exist in many different states at once! This means that our computers could process information faster than ever by taking advantage of this
ability to process more data than we could ever hope for using traditional processors.
What is the difference between conventional and quantum computers?
Quantum computers are based on the quantum theory of physics, which states that a particle can exist in multiple states at once–a phenomenon called superposition. In standard computing terms, this
means that the qubits used in a quantum computer can be used to represent both 0s and 1s at the same time. This property allows calculations to be done much faster than traditional computers because
they don’t need to perform sequential operations or wait for results from one step before proceeding with another (see example below).
In addition to being faster, some problems cannot currently be solved by conventional computers but can be solved by quantum ones. For example, Shor’s algorithm can quickly factor large numbers into
their prime factors; this method is used today by banks and other institutions worldwide as part of their security protocols for online transactions involving credit cards or money transfers between
How is quantum computing different from current “digital” computers?
Quantum computers use quantum bits or qubits. A conventional digital computer uses an electrical charge to represent information in binary code (0s and 1s). A quantum computer uses subatomic
particles that can be in multiple states at once–a property called superposition–to hold a much larger number of possible values than traditional bits do. This is why quantum computers are so
powerful: they can explore more potential solutions to any given problem than conventional computers can, allowing them to solve problems that would have proven impossible for us to crack using our
current technology.
Theoretically, there’s no limit on how many qubits you could combine into one unit; the more qubits you add to your system, the faster it will solve problems that would take years if done by hand
Will a quantum computer be able to crack encryption methods like RSA?
Quantum computers are not a threat to RSA.
RSA is a public key encryption algorithm, which means that the key used to encrypt data is shared and, therefore, relatively easy to find. A quantum computer can only break symmetric encryption
algorithms used for private-key cryptography. These are typically used in conjunction with public-key systems–you might encrypt your files with AES256 (symmetric) before sending them over email using
PGP (public).
AES256 is one of many symmetric algorithms; there’s 3DES (Triple Data Encryption Standard), Blowfish, Twofish and more!
How will the rise of quantum computing affect our daily lives?
If you think about it, the world around us is incredibly complex. It’s not just that we must consider all the individual atoms and molecules that make up our bodies, but also how they interact with
other elements for us to function correctly.
A quantum computer could simulate these interactions so perfectly that it could predict what would happen if you ate a certain food or took a particular medication–something no human being could ever
hope for!
They will also be able to solve problems that are too complex for today’s computers by taking advantage of another quantum phenomenon: entanglement. Entanglement allows two particles separated by
large distances (even light years) from one another to act as if they’re still connected somehow; this means that if one changes its state, then so does its twin particle immediately without any
delay at all!
Instead of going through every possible solution separately to find an answer, we can simultaneously look at all possible outcomes using superpositioning, which provides exponential speedups over
classical computing methods.”
What are the challenges to building a practical quantum computer?
There are several challenges to building a practical quantum computer. First, it’s difficult to fabricate the qubits and keep them stable at room temperature. Second, computers are expensive; even
the most basic models cost millions.
Third, they’re delicate machines that must be protected from environmental changes and electromagnetic interference–the slightest jolt could cause a qubit to decay or shift into an incorrect state.
Finally, quantum computers are still in their infancy: we need to figure out how best to use them or what kinds of problems they’ll solve best when they come online later this year or early next
These limitations mean that you will see few practical applications for these machines any time soon–but there’s still plenty of reason for excitement!
The possibilities offered by quantum computing are endless; scientists have already used them for everything from simulating materials science experiments on large molecules such as proteins and DNA
strands all the way down through individual atoms; to developing new drugs based on molecular modelling techniques like docking analysis; designing new materials with better properties than any known
today; predicting weather patterns years in advance using meteorological data collected over decades by satellites orbiting earth’s atmosphere.
The rise of quantum computing is a game-changing moment in the history of technology. It will change how we live, work, and even think about ourselves as humans as machines. We are on the cusp of
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What Does Independent & Dependent Mean in Math Terms? | Synonym
What Does Independent & Dependent Mean in Math Terms?
What does independent and dependent mean in math terms? I'm Bon Crowder and we're talking about independent and dependent variables. So, when you have an equation with two variables, one of the
variables is independent and one is dependent. The independent variable which we typically have as X is the variable that you can pick anything you want to be, anything in the domain that is. So for
X we could randomly pick 0 or 1 or -853.2, anything we want. However, the other variable is the dependent variable, it's specifically well, dependent on what we pick for the independent variable. In
this case, Y is 3 x whatever we pick for the independent variable + 2 which is 3 x 0 is 0 + 2 is 2. So our independent variable value we gave was 0 and our value of our dependent variable which
depended on the 0 is 2. We can do this for the 1 and get 5 and if we were a little crazy, we might even do it for this guy, I'll leave that to you. So, I'm Bon Crowder and this is what it means to be
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Automata Theory Multiple Choice Question & Answer (MCQs) Set-2 - Studyhelpzone.com
MCQ Computer Science
Automata Theory Multiple Choice Question & Answer (MCQs) Set-2
1. Assume the R is a relation on a set A, aRb is partially ordered such that a and b are _____________
a) reflexive
b) transitive
c) symmetric
d) reflexive and transitive
View Answer
Answer: d
Explanation: A partially ordered relation refers to one which is Reflexive, Transitive and Antisymmetric.
2. In Moore machine, output is produced over the change of:
a) transitions
b) states
c) Both
d) None of the mentioned
View Answer
Answer: b
Explanation: Moore machine produces an output over the change of transition states while mealy machine does it so for transitions itself.
3. An e-NFA is ___________ in representation.
a) Quadruple
b) Quintuple
c) Triple
d) None of the mentioned
View Answer
Answer: b
Explanation: An e-NFA consist of 5 tuples: A=(Q, S, d, q0. F)
Note: e is never a member of S.
4. Extended transition function is .
a) Q * S* -> Q
b) Q * S -> Q
c) Q* * S* -> S
d) Q * S -> S
View Answer
Explanation: This takes single state and string of input to produce a state.
5. For a DFA accepting binary numbers whose decimal equivalent is divisible by 4, what are all the possible remainders?
a) 0
b) 0,2
c) 0,2,4
d) 0,1,2,3
View Answer
Answer: d
Explanation: All the decimal numbers on division would lead to only
4 remainders i.e. 0,1,2,3 (Property of Decimal division).
6. String X is accepted by finite automata if .
a) d*(q,x) E A
b) d(q,x) E A
c) d*(Q0,x) E A
d) d(Q0,x) E A
View Answer
Explanation: If automata starts with starting state and after finite moves if reaches to final step then it called accepted.
7. Given Language L= {x? {a, b}*|x contains aba as its substring}
Find the difference of transitions made in constructing a DFA and an equivalent NFA?
a) 2
b) 3
c) 4
d) Cannot be determined.
View Answer
Answer: a
Explanation: The individual Transition graphs can be made and the difference of transitions can be determined.
8. The construction time for DFA from an equivalent NFA (m number of node)is:
a) O(m2)
b) O(2m)
c) O(m)
d) O(log m)
View Answer
Answer: b
Explanation: From the coded NFA-DFA conversion.
9. Which of the following is a regular language?
a) String whose length is a sequence of prime numbers
b) String with substring wwr in between
c) Palindrome string
d) String with even number of Zero’s
View Answer
Answer: d
Explanation: DFSM’s for the first three option is not possible; hence they aren’t regular.
10. The total number of states to build the given language using DFA:
L= {w | w has exactly 2 a’s and at least 2 b’s}
a) 10
b) 11
c) 12
d) 13
View Answer
Answer: a
Explanation: We need to make the number of a as fixed i.e. 2 and b can be 2 or more. Thus, using this condition a finite automata can be created using 1 states.
11. Which of the following is type 3 language ?
a) Strings of 0’s whose length is perfect square
b) Palindromes string
c) Strings of 0’s having length prime number
d) String of odd number of 0’s
View Answer
Explanation: Only d is regular language.
12. The e-NFA recognizable languages are not closed under :
a) Union
b) Negation
c) Kleene Closure
d) None of the mentioned
View Answer
Answer: d
Explanation: The languages which are recognized by an epsilon Non deterministic automata are closed under the following operations:
a) Union
b) Intersection
c) Concatenation
d) Negation
e) Star
f) Kleene closure
13. Which among the following is not an associative operation?
a) Union
b) Concatenation
c) Dot
d) None of the mentioned
View Answer
Answer: d
Explanation: It does not matter in which order we group the expression with the operators as they are associative. If one gets a chance to group the expression, one should group them from left for
convenience. For instance, 012 is grouped as (01)2.
14. Which of the following is the task of lexical analysis?
a) To build the uniform symbol table
b) To initialize the variables
c) To organize the variables in a lexical order
d) None of the mentioned
View Answer
Answer: a
Explanation: Lexical analysis involves the following task:
a) Building a uniform symbol table
b) Parsing the source code into tokens
c) Building a literal and identifier table
15. The scanner outputs:
a) Stream of tokens
b) Image file
c) Intermediate code
d) Machine code
View Answer
Answer: a
Explanation: A scanner or a lexical analyzer takes a source code as input and outputs a stream of token after fragmenting the code. | {"url":"https://studyhelpzone.com/automata-theory-multiple-choice-question-answer-mcqs-set-2.html","timestamp":"2024-11-03T06:05:06Z","content_type":"text/html","content_length":"126420","record_id":"<urn:uuid:9454c0bc-c8ec-4310-a47a-b0d09cdad8ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00455.warc.gz"} |
What is the value of 2 in 12?
Answer: The value of 2 raised to 12th power i.e., 212 is 4096.
What is the base 2 representation of the decimal number 12?
Therefore, the binary equivalent of decimal number 12 is 1100.
What is Base 2 called?
Binary number system, in mathematics, positional numeral system employing 2 as the base and so requiring only two different symbols for its digits, 0 and 1, instead of the usual 10 different symbols
needed in the decimal system.
How do you understand Base 2?
Binary (Base 2) has only 2 digits: 0 and 1 start back at 0 again, and add one to the number on the left… but that number is already at 1 so it also goes back to 0 And so on! Also try Decimal, and try
other bases like 3 or 4.
What is the number value of 12?
The value of 12° is 1 .
What is base 12 called?
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base.
Which is the biggest digit used in system with base 2?
The largest digit you can have in any column is the one less than the number of the base. So for binary (base 2) it’s 1, then 7 for Octal (base 8), 9 for Denary (base 10), etc.
How to write a number in base 2?
Suppose we have a number in base 10 and want to find out how to represent that number in, say, base 2. How do we do this? Well, there is a simple and easy method to follow. Let’s say I want to write
59 in base 2. My first step is to find the largest power of 2 that is less than 59. 1, 2, 4, 8, 16, 32, 64.
How to write 123 in base 2 binary?
How to write 123 in base-2 (base 2)? Convert from/to decimal, hexadecimal, octal and binary. Decimal Base conversion Calculator. Here you can find the answer to questions like: Convert decimal 123 to
base 2 or Decimal to base-2 conversion.
How to convert base 2 to base 16?
Base 16 to Base 2 Conversion Table Base 16 Base 2 1 1 2 10 3 11 4 100
Can you write down 59 in base 2?
So, we write down a 1. 1 – (1) (1) = 0. And now we stop since our next lowest power of 2 is a fraction. This means we have fully written 59 in base 2. Now, try converting the following base 10
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diffeological space
Patrick Iglesias-Zemmour kindly pointed out to me by email that the latest version of this book Diffeology now contains, around exercise 72, a discussion of how Banach manifolds faithfully embed into
diffeological spaces. So I have now added brief pointers to Banach manifold and to the relevant section of diffeological space. (This really deserves to be expanded on, but I don’t have the time.)
added also the embedding of locally convex vector spaces by cor 3.14 in Kriegl-Michor
added also
(with just a pointer to a reference for the moment)
For the purpose of pointers at MO, I have expanded slightly at diffeological space to make it have this series of sub-sections on embeddings of categories:
I have added more of the original references to the References-section at diffeological space.
Andrew, when you have a second, maybe have a look to see if my attributions are precise.
At diffeological space I have added the remark that the statement proven there, that smooth manifolds embed fully faithfully in diffeological spaces, is a direct consequence of the fact that $CartSp$
is a dense sub-site of $Diff$ and then of the Yoneda lemma.
One can see that this is effectively what the previous proof checks in a pedestrian fashion, but it is maybe useful to have the general abstract version, too.
it is easy to prove
Not sure. I’ve not worked through the details myself. The proof in Kriegl and Michor is about a page long.
I created Boman’s theorem
Thanks! I was scanning your articles for it, but didn’t see it. Then I thought about it and figured that it is easy to prove (isn’t it? one needs to show that for each higher partial derivatives of a
function one can find a curve such that the composite’s $n$-fold total derivative involves as a summand the partial derivatives in question. But that’s obvious.)
I have added that to the list of theorems in the floating differential geometry TOC.
also corrected a couple of minor typos in the vicinity
Thanks! I found some more ;-)
I created Boman’s theorem and added the link to the embedding proof on diffeological space (also corrected a couple of minor typos in the vicinity).
I have expanded the Properties-section at diffeological space:
• added the statement and proof of the full and faithful embedding of smooth manifolds into diffeological spaces;
• split off a section of the properties of the ambient sheaf topos and how diffeological spaces sit inside there.
seeing Eric create diffeology I became annoyed by the poor state that the entry diffeological space was in. So I spent some minutes expanding and editing it. Still far from perfect, but a step in the
right direction, I think.
(One day I should add details on how the various sites in use are equivalent to using CartSp)
Hi There,
Urs pointed to me this forum/thread. So I will give some precisions about what he said above.
I look sometimes to the diffeological spaces item in nLab, to stay informed :-) Last time I discovered the article, posted by Urs, about Banach manifolds and the pointer to the 1977 Hain's paper, I
didn't know about it. On the other hand, a few months ago, the referee of the AMS asked me to clarify, in the book Diffeology, the relationship between Banach manifolds and diffeology, what I did and
that question became the exercise 72 of the book. Using Boman's theorem the solution of exercise takes a few lines. So, I was surprised to see Hain's paper so long, having a brief look inside it
seemed to me that Hain proves first a kind of Boman theorem, in his paper, but Boman theorem is from 1967 if I don't mistake. So why Hain didn't use Boman theorem ? This is my question. Or I am wrong
and I missed something ? But I have no time now to investigate this question, I'm doing something else. If someone is interested in and has time to look into it, he just sends me an email and I'll
send him back a pdf of the last and final version of the book to check the exercise and compare with Hain's paper.
BTW, thanks again to a question of the referee of the book (this guy has been very helpful), I added an exercise related to Frolicher spaces and diffeology: with Yael Karshon we introduced the
concept of reflexive diffeological space, it happens that this subcategory is isomorphic with the category of Frölicher spaces. It's the exercise 80. For the ones interested in that question about
Patrick I-Z
The entry diffeology didn't seem to serve any purpose, so now it redirects to diffeological space. (If somebody wants to revive it, its edit history is at diffeology > history.)
I’ve added a comment that Frölicher proved the full and faithful embedding of (paracompact) Fréchet spaces into diffeological spaces in 1981, and in fact I think he proved paracompact Fréchet
manifolds also embed fully faithfully, but he has a funny extra condition to link with some functional/sequential notion of smoothness (see théorème 2 on this page)
On a different note, I’m not sure that convenient spaces do embed into diffeological spaces. My reading of corollary 3.14 at mentioned at #10 above is that it is just Boman’s theorem, and that the $c
^\infty$ notion of smoothness agrees with the usual notion on cartesian spaces.
Thanks for further looking into this! This is useful.
Finally cleared this up. There is a faithful but non-full functor from lctvs into diffeological spaces, if we take MB-smooth maps as morphisms between the former, since there are non-continuous
conveniently smooth maps. I still don’t know if diffeological isomorphisms are MB-smooth, though. I added to the page a reference to Gloecker’s counterexamples, and clarification about what is meant
by smooth maps between lcvts.
added pointer to Patrick Iglesias-Zemmour’s lecture notes Iglesias-Zemmour 18
diff, v57, current
I have considerably trimmed down the section Embedding of diffeological spaces into smooth sets. It used to contain a proof that $Sh(CartSp)$ is cohesive, and had the result announced in its title
only hidden somewhere in that discussion. But the cohesion of smooth sets should instead be discussed there, and so I removed it here and instead included (a complete rewrite of) the proof there.
Here I only kept the actual statement that diffeological spaces are the concrete smooth sets, with the minimum indication of the proof that used to be here. Below that I added pointer to a completely
(maybe pedantically) detailed proof, which is now at this Prop. in geometry of physics – smooth sets.
diff, v59, current
I forget if the following is known, and where it is proven:
The homotopy type of a diffeological space (D-topology) is equivalently its cohesive shape (when regarded as a concrete 0-truncated objects in the cohesive $\infty$-topos over smooth manifolds).
Re #20: Yes.
By Proposition 3.1 in https://arxiv.org/abs/1010.3336 we have a left adjoint functor Diff→Top that sends a diffeological space to its underlying topological space equipped with the D-topology.
This left adjoint functor is a left Quillen functor because it sends generating (acyclic) cofibrations in Diff to (acyclic) cofibrations in Top.
Thus, the functor Diff→Top is homotopy cocontinuous.
The cohesive shape is also homotopy cocontinuous.
These two cocontinuous functors take contractible values on R^n.
Hence, they are weakly equivalent.
But help me, you seem to be using one more bit of information that I am lacking.
Explicitly, I am asking about the functor
$DiffeologicalSpaces \hookrightarrow Sh(CartSp) \hookrightarrow Sh_\infty(CartSp) \overset{Shape}{\longrightarrow} \infty Groupoids$
whether it’s naturally equivalent to
$DiffeologicalSpaces \overset{D-topology}{\longrightarrow} TopologicalSpaces \overset{L_{whe}}{\longrightarrow} \infty Groupoids$
You seem to be appealing to a homotopical structure on diffeological spaces being compatible with the first of these functors?
[later edit: ah, no, I misread Prop. 3.10 in Christensen-Wu, as per the warning on the next page – it does not hold generally for diffeological spaces – so the following does not work]
Let me see:
From your theorem about shape via cohesive path ∞-groupoid it follows that the first functor in #22 is equivalently the one called $S^D$ (Def. 4.3) in
• J. Daniel Christensen, Enxin Wu, The homotopy theory of diffeological spaces (arXiv:1311.6394)
The second functor in #22 would be called $S\circ D$ there.
So in the notation of that article I am asking for validity/proof of
$S^D \;\overset{?}{\simeq}\; S \circ D \,.$
I don’t see exactly that statement in the article, but something close:
Theorem 4.11 together with Prop. 3.10 there says that the homotopy groups of the results of both functors agree assuming they are evaluated on a fibrant diffeological space $X_{fibr}$ (which is one
whose smooth singular simpliciat set $S^D$ is Kan, Def. 4.8):
$\pi_n \circ S^D(X_{fibr}) \;\simeq\; \pi_n S \circ D(X_{fibr}) \,.$
This is two steps away from the previous statement:
• if this isomorphism of homotopy groups is/were induced by a morphism of simplicial sets, then it would constitute a weak homotopy equivalence. This is probably implicit in the proofs, I should
chase through them.
• if the assumption of fibrancy were unnecessary, we’d be done. Now, this would again follow from your theorem of shape via path $\infty$-groupoids, IF we knew there is fibrant replacement for
diffeological spaces in the sense of Christensen – but that they explicitly do not prove.
[edit: ah, looks like both these steps are filled in in H. Kihara, Model category of diffeological spaces (arXiv:1605.06794), in Theorem 1.4 there, using the proof starting p. 33]
Re #23: I would argue as follows.
The Kihara model structure on diffeological spaces is transferred via the smooth singular simplicial set functor Diff→sSet.
The Quillen model structure on topological spaces is transferred via the singular simplicial set functor Top→sSet.
Furthermore, the composition of left adjoints sSet→Diff→Top equals the left adjoint sSet→Top.
The left Quillen functors sSet→Diff and sSet→Top are Quillen equivalences.
Therefore, the left Quillen functor Diff→Top is a Quillen equivalence by the 2-out-of-3 property, hence a homotopy cocontinuous functor.
The Kihara model structure on diffeological spaces is transferred via the smooth singular simplicial set functor Diff→sSet.
But Kihara defines a variant of smooth singular simplicial sets, by using a variant diffeology on standard simplices, in order to enforce existence of horn fillers.
The singular simplicial complex that corresponds to cohesive shape, the one also considered in your concordance article, that’s instead the one that Christensen-Wu use (their Def. 4.3). Isn’t it?
But with this definition, their Theorem 4.10 together with their (counter-)examles of smooth $\pi_n$ differing from D-topological $\pi_n$ proves that the desired equivalence fails.
It seems to me.
But Kihara defines a variant of smooth singular simplicial sets, by using a variant diffeology on standard simplices, in order to enforce existence of horn fillers.
Yes, it looks like my memory of Kihara’s paper was not entirely correct.
So really we need the Christensen-Wu construction, which gives the same weak equivalences, but different cofibrations. They do not prove it is a model structure, however, this is basically what we do
in our paper. In fact, in our paper, Dan, Pedro, and I prove precisely the necessary lemmas that Christensen and Wu are missing, see Section 4.c, in particular, Lemma 4.13 is precisely the missing
part necessary to complete the construction of a model structure, as Christensen and Wu point out themselves in Remark 4.9 in their paper.
Also, Proposition 4.10 shows that two different geometric realization functors by Kihara and Christensen-Wu are weakly equivalent by constructing an explicit homotopy equivalence between them.
Okay, I’ll have another look at your article.
But do you agree that Christensen-Wu’s results prove that the equivalence $S^D \overset{?}{\simeq} S \circ D$fails?
They prove
1. $\pi_n^D(X) \simeq \pi_n S^D(X)$ for every diffeological space $X$ (Theorem 4.11),
2. $\pi_n^D(X) eq \pi_n(S \circ D(X))$ for some diffeological spaces $X$ (Example 3.12, 3.20)
So it follows that
• $S^D(X) \;\text{is not weakly equivalent to}\; S \circ D (X)$ for some diffeological spaces $X$.
For my own future reference,
• π_n^D is the nth homotopy group defined by mapping representable spheres into a diffeological space,
• π_n S D is the nth continuous homotopy group of the D-topology,
• π_n(S^D) is the nth simplicial homotopy group of the smooth singular simplicial set.
But do you agree that Christensen-Wu’s results prove that the equivalence S D≃?S∘DS^D \overset{?}{\simeq} S \circ D fails?
Yes, I obviously forgot to derive the D-topology functor, since not all diffeological spaces are cofibrant (in fact, in Example 4.29 they give the same example as in 3.20).
So I would say that the D-topology functor must be left derived in order for your statement to be true.
Note that Theorem 4.11 is stated for fibrant diffeological spaces.
However, my work with Dan and Pedro show that fibrancy is redundant, see 4.3 and 4.7.
Thanks for the comments!
Okay, you are pointing me to the conclusion in the last sentence of Remark 4.7 in arXiv:1912.10544… Ah, I see. That’s most useful.
Okay, I’ll try to get a feeling now for the cofibrant replacement of diffeological spaces, to see if this is of any use in my intended application (generalized orbifold cohomology).
If it is, I’ll want to state/quote as a proposition that $S\circ D((-)_{cof}) \simeq S^D(-)$. I’d be happy to cite you for this if you write it down somewhere.
Do you know if all smooth manifolds are Christensen-Wu cofibrant as diffeological spaces? (They leave this as a conjecture, p. 18.)
Re #30: It is easy to prove that any smooth manifold is concordance equivalent to to a cofibrant diffeological space, namely, the realization of the simplicial set K underlying some smooth
triangulation of M.
This is precisely Lemma 9.13 in my draft.
I believe this will suffice for your purposes, since the D-topology functor sends concordance equivalences to homotopy equivalences of topological spaces.
Yes, I know that the cohesive shape of a smooth manifold is equivalent to its underlying (D-)topological homotopy type.
But it would be useful to know that smooth manifolds are actually Christensen-Wu cofibrant, so that a cofibrant replacement functor could be asked to preserve them. For if not, the homotopy types
would be me made to work only at the expense of breaking the differential geometry of the core class of examples, and that would be besides the point.
I think I convinced myself that an argument similar to my Lemma 9.13 as well as Proposition 4.23 in Christensen–Wu does show that any smooth manifold is cofibrant.
What’s more, I now think that the Christensen–Wu model structure does exist, is cartesian, and any smooth embedding is a cofibration.
Do you think this may be worthy of writing down as a separate paper?
We’d have a neat application of this result to the problem of relating orbifold cohomology to equivariant cohomology:
There, abstract arguments in equivariant cohesion show that the equivariant homotopy type of a general cohesive orbifold looks just like that of a topological $G$-space, but with the system of
topological spaces of $K$-fixed loci all replaced by the shape of the $K$-fixed loci of the underlying concrete cohesive space. If your claims are true, this would imply that, in the case of smooth
cohesion, this latter system is again equivalent to that of an actual topological $G$-space, namely that which is the derived D-topology underlying the diffeological space which is the concrete
cohesive covering space of the given orbifold.
All we’d need to complete this argument is to cite results as you just stated. :-)
Is the functor $D(-)$ (assigning underlying D-topological spaces) left Quillen, in that would-be model structure?
The Christensen-Wu model structure is transferred from the Quillen model structure on simplicial sets via the smooth singular simplicial set functor.
Its generating (acyclic) cofibrations are smooth geometric realizations of (acyclic) cofibrations of simplicial sets.
The functor D is cocontinuous, so it sends these generating (acyclic) realizations to the ordinary geometric realizations of (acyclic) cofibrations of simplicial sets. The latter are indeed (acyclic)
So the functor D is a left adjoint functor that preserves (acyclic) cofibrations, hence a left Quillen functor.
That would be a plausible strategy to check it, but don’t we need some Lemma that $D(\left\vert \Delta^n \right\vert)$ is what one would hope it is? A priori the topology could end up being funny.
don’t we need some Lemma that D(|Δ n|)D(\left\vert \Delta^n \right\vert) is what one would hope it is?
This follows from the definition of the D-topology. Recall (Definition 3.6) in Christensen–Wu that the D-topology on |Δ^n| is the final topology induced by its plots, where the domain of each plot is
equipped with the standard topology on R^n.
But by Definition 4.3 in Christensen–Wu, the smooth geometric realization of Δ^n is precisely the extended smooth n-simplex with its standard diffeology.
And by Example 3.7 in Christensen–Wu, the D-topology on a smooth manifold with the standard diffeology coincides with the usual topology on the manifold.
Okay, great. Glad you have thought this through. :-)
I’ll go ahead then citing an upcoming theorem of yours in what I am writing up regardng orbifold cohomology. I’ll show you what we need once it is in readable form. Hopefully in a week or two.
One more question: Is there, in the would-be model structure under discussion, a functorial cofibrant replacement which is the identity on manifolds?
The standard way to produce functorial factorizations is the small object arguments of Quillen and Garner.
Both arguments produce huge cofibrant replacements, and I do not see how to reduce their size functorially.
Why do you need a functorial replacement of this type anyway?
What I strictly need in applications is just this:
Given a diffeological space equipped with the action of a finite group, I need that group action to extend to its cofibrant replacement. That’s why I am concerned with functorial replacement.
But in addition to that, I had the vague feeling that I’d rather keep a given difeological space intact (as arising from some differential geometric problem) as much as possible, instead of feeding
it into a blind replacement machine such as the small object argument.
Can we maybe see concretely geometrically what Christensen-Wu cofibrancy is about? I am vaguely imagining one might identify “singular” subloci inside a diffeological space such that a kind of blowup
of their vicinity restores cofibrancy. Maybe?
But this may be more my unenlightened prejudice than actual necessity.
Not to distract from this discussion, but just to log some edits to the entry:
Started a section Relation to topological spaces (already last week, but I had left the edit invisible for a while to showcase the redirects bug).
Also did a fair bit of editing on the section References – General:
Added missing publication data and DOI-s to a bunch of items, added missing references such as to Souriau’s second original articles, adjusted the order of the articles (now it goes Chen $\to$
Souriau $\to$ Iglesias-Zemmour).
In particular, the previous pointer to
• William Lawvere, Stephen Schanuel (eds.), Categories in Continuum Physics, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986
I have expanded out to
• Kuo-Tsai Chen, On differentiable spaces, in: William Lawvere, Stephen Schanuel (eds.), Categories in Continuum Physics, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in
Mathematics 1174, 1986 (doi:10.1007/BFb0076928)
and moved up to join the other articles by Chen.
By the way, it’s most curious: Even in this collection of texts in topos theory and sheaf theory, both Chen and his editors (!?) manage to still avoid recognizing that Chen is secretly talking about
diff, v69, current
added a section Topological homotopy type and diffeological shape
diff, v72, current
Another dumb question:
Do we know whether for Fréchet manifolds
$X \in FrechetManifolds \hookrightarrow DiffeologicalSpaces \hookrightarrow SmoothGroupoids_\infty$
the cohesive shape (i.e. $S^D(X)$ in Christensen-Wu notation) coincides with the underlying topological homotopy type?
(It feels like I knew this once, but I forget.)
Re #46: This basically amounts to saying that any continuous disk with a smooth boundary can be deformed relative boundary to a smooth disk.
This is probably established somewhere in the literature on Fréchet manifolds.
Hmm, yes. There was a recent paper by Glöckner on smoothing operators for functions valued in lctvs, but it’s not quite in the right setting (and doesn’t seem to do the relative case).
Glöckner’s result seems like a massive overkill anyway: we only need a single deformation, not a whole smoothing operator.
Yeah, but it indicates that current technology is much stronger than you’d need, evidence that smoothing for a single map to a Fréchet space should be known.
But contractibility of these disks is only the first step. Next we need to know that 2) there are good open covers or hypercovers by disjoint unions of such open disks and then 3) a suitable nerve
How much of a condition is paracompactness on an infinite-dimensional Fréchet manifold?
Re #51: Why do you want all these things?!
To show that the cohesive shape (i.e., S^D(X) in the Christensen-Wu notation) coincides with the underlying topological homotopy type, it suffices to show that the canonical map S^D(X) → Sing(D(X))
is a simplicial weak equivalence.
Both simplicial sets are Kan complexes, so by the simplicial Whitehead theorem, it suffices to show that for any map ∂Δ^n → S^D(X) together with a filling of its image in Sing(D(X)) by Δ^n, we can
deform the filling relative boundary to another disk that lifts to S^D(X).
But this is exactly the disk deformation condition that I mentioned above.
Okay, I see that I wasn’t properly reading all the qualifications in #47.
So is this a consequence of Glöckner or not?
I am just trying to find out if you or somebody essentially knows the answer already, not just an idea for a strategy, or if I’d need to dive into it myself.
Re #53: I think an even easier argument is possible, one that does not require any smoothing arguments.
It suffices to observe that any Fréchet manifold has an atlas of Fréchet coordinate charts, in particular, is the homotopy colimit of the diagram consisting of its open subsets that are diffeomorphic
to Fréchet vector spaces.
Thus, it suffices to show that S^D(X) → Sing(D(X)) is a simplicial weak equivalence whenever X is a Fréchet vector space. But this is trivial because both sides are contractible.
This now sounds like the beginning of the argument along the lines of #51 after all:
If we replace the manifold by a simplicial object of local charts and their intersections, or more generally by a hypercover by local charts, then we still need to argue that passing to the resulting
simplicial set obtained by contracting each local chart to a point represents the homotopy type of the underlying topological space. This is intuituvely suggestive but needs a proof. If our space is
paracompact and we can arrange for a good cover, then one such proof is Borsuk’s nerve theorem.
I trust there are other way’s to argue this, but some argument seems to be needed. But let me know if I am missing the obvious.
The Convenient Setting of Global Analysis has the result (Theorem 16.10) that nuclear Fréchet spaces are all smoothly paracompact, as well as “strict inductive limits of sequences of such spaces”.
Lindelöf and smoothly regular is also sufficient. Countable products of smoothly paracompact Fréchet spaces (being metrizable) are smoothly paracompact (Corollary 16.17). It seems one could just
assume separability on the Fréchet space, instead of nuclearity. So it seems spaces of smooth functions on compact manifolds to fin.dim. manifolds, as Fréchet manifolds/diffeological spaces, do
indeed satisfy what you are looking for.
Theorem 16.15 looks potentially relevant, too.
Thanks. I’d like to check whether the proof of the full inclusion of Fréchet manifolds into diffeological spaces might not secretly assume paracompactness anyway(?).
The critical point seems, to me, to be the existence of good open covers. But I see that Fréchet manifolds are still metrizable if (and only if) they are paracompact. With a kind of
infinite-dimensional Riemannian metric in hand, the usual proof of existence of good open covers might just go through.
Just to say that I see now that Kihara has an article whose abstract sounds like it has the proof:
Smooth Homotopy of Infinite-Dimensional $C^\infty$-Manifolds (arXiv:2002.03618)
But i haven’t dug into it yet.
[ edit: Ah, too bad: Theorem 1.1 in that article would be the desired statement… were it not for the fact that it’s using the non-standard diffeology on simplices, following arXiv:1605.06794.)
Re #55:
If we replace the manifold by a simplicial object of local charts and their intersections, or more generally by a hypercover by local charts, then we still need to argue that passing to the
resulting simplicial set obtained by contracting each local chart to a point represents the homotopy type of the underlying topological space. This is intuituvely suggestive but needs a proof. If
our space is paracompact and we can arrange for a good cover, then one such proof is Borsuk’s nerve theorem.
I think there is a very simple argument for this.
First, given an open hypercover H of X, the canonical map
$hocolim H \to X$
computed in the model category of topological spaces is a weak equivalence of topological spaces.
This is Lurie’s abstract Seifert–van Kampen theorem, see Theorem A.3.1 in HA.
The same statement is also true for the model category of diffeological spaces that I hope to finish writing down soon (if all arguments work out).
But then it remains to observe that any Fréchet manifold admits a good hypercover (all elements are diffeomorphic to Fréchet spaces). Indeed, start with some atlas, then choose an atlas for each
intersection, etc.
All right.
By the way, did you see that there is also this article:
• Tadayuki Haraguchi, Kazuhisa Shimakawa, A model structure on the category of diffeological spaces (arXiv:1311.5668)
This seems to define smooth homotopy groups using maps out of $n$-cubes equipped with their standard diffeology. So that might already be the model structure in question. But I don’t know, have only
glanced over the article so far. Also, this appears to remain unpublished (?)
Ah, right, Kihara 16 claims (p. 2) that
there exists a gap in the proof of [ Haraguchi-Shimakawa 13, Theorem 5.6]
But then later Haraguchi 18 seems to mean to address this, as he writes (p. 1):
We present the Quillen model structure on the category $Diff$ of diffeological spaces $[...]$ (cf. [ Haraguchi-Shimakawa 13, Theorem 5.6 and Theorem 6.2])
On the other hand, Haraguchi 18 also seems not to be published yet.
I am aware of this paper. Their argument is very technical, and they claim that the model structure is not cofibrantly generated, apparently.
What is (or would be) nice about this model structure is that it is compatible with that neat idempotent adjunction between topological spaces and diffeological spaces, in that it makes the
$TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DTopologicalSpaces \underoverset { \underset{ }{\
hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces$
into a sequence of Quillen equivalences.
That should be rather useful, if true.
It is probably not of any immediate use to you, Urs, but by my thesis I think it is more or less immediate that one can put a Hurewicz model structure on D-topological spaces and diffeological
spaces, both of which are Quillen equivalent to the Hurewicz model structure on topological spaces. All of this would be compatible with #63.
To get what you need from this, it might suffice to have some kind of Whitehead theorem for diffeological spaces. I.e. if two diffeological spaces are weakly equivalent in the sense you are looking
at, then if one could show they are then actually homotopy equivalent in the sense of the Hurewicz model structure on diffeological spaces, one can use the Hurewicz Quillen equivalences to get what
you need I think (if I am not overlooking something; the notation is a bit heavy, and I don’t really know anything about diffeological spaces, so I am somewhat guessing what you are looking to prove;
Dmitri’s #28 was very helpful).
Edit: There is some kind of Whitehead theorem in Haraguchi’s article from 2018, maybe it is sufficient.
Hi Richard,
I have only glanced over your thesis (arXiv:1304.0867). Would have to dig deeper to see which statemen(s) one would need to quote to get the desired model structure. There seem to be a lot of
technical conditions to be checked(?).
I never thought much about the Hurewicz model structure at all. But if you could deduce with ease a theorem for that case, I expect it would be of interest.
To get what you need from this, it might suffice…
Yeah, this is what the Haraguchi-Shimakawa-structure would (or will) give: Here diffeological homotopy type is detected on smooth homotopy groups, while the functor to underlying D-topological spaces
is the left adjoint of a Quillen equivalence. Therefore the existence of this model structure would (or will) imply that the cohesive homotopy types of cofibrant diffeological spaces is in bijection
to their underlying D-topological homotopy type.
Ok, so Haraguchi and Shimakawa have a new preprint out, claiming to fix the issues with the old, incorrect result on the model structure on diffeological spaces: https://arxiv.org/abs/2011.12842
Thanks for the alert. But maybe best to discuss in the thread for model structure on diffeological spaces, here.
Ok, thanks for the pointer.
added pointer to the new website:
diff, v76, current
Diffeologies coming out from singular statistical models were discussed this Wednesday, 10.3.2021. in the Prague-Hradec Králové seminar (Cohomology in algebra, geometry, physics and statistics) talk
by Hông Vân Lê (Institute of Mathematics of the Czech Academy of Sciences), now on youtube
• Hông Vân Lê, Diffeological statistical models and diffeological Hausdorff measures, yt
The slides are available from https://users.math.cas.cz/~hvle/PHK/Lediffeological10032021.pdf and there are two arXiv preprints,
• Hông Vân Lê, Alexey A. Tuzhilin, Nonparametric estimations and the diffeological Fisher metric, arXiv:2011.13418
• Hông Vân Lê, Diffeological statistical models,the Fisher metric and probabilistic mappings, Mathematics 2020, 8(2),167, arXiv:1912.02090
I copy this information at Fisher metric.
I have added statement and proof (here) that the internal hom as diffeological spaces of any pair of D-topological spaces has the correct diffeological homotopy type.
This follows, I think, by combining a couple of statements from Shimakawa & Haraguchi with that proposition from Christensen & Wu (observing that the latter gives a natural weak equivalence).
diff, v78, current
The Grothendieck topology on $\mathcal{Op}$ is generated by the coverage of open covers, i.e., a family of maps $\{U_i\to X\}_{i\in I}$ is a covering family if every map $U_i\to X$ is an open
embedding and the union of the images of $U_i$ in $X$ equals $X$.
diff, v86, current
Losik’s paper bibliographic data updated:
• {#Losik92} Mark Losik, Fréchet manifolds as diffeologic spaces, Russian Mathematics 36:5 (1992), 36–42. English translation: PDF. Russian original: (mathnet:ivm4812)
diff, v87, current
Re #35:
I am finalizing a paper for the arXiv: https://dmitripavlov.org/diffeo.pdf, which answers the questions about model structures on diffeological spaces posed above.
Some highlights:
• Theorem 6.3: The category of diffeological spaces does not admit a model structure transferred from simplicial sets via the smooth singular complex functor. This is caused by the highly
pathological behavior of the concretization functor, which is used to compute colimtis of diffeological spaces. However, the smooth singular complex functor is a Dwyer–Kan equivalence of relative
categories (Corollary 7.7).
• Theorem 7.4: The category of smooth sets does admit a model structure transferred from simplicial sets via the smooth singular complex functor.
• All smooth manifolds are cofibrant.
• This model structure is cartesian.
• It is left proper, combinatorial, h-monoidal, flat, symmetric h-monoidal, all operads are admissible, etc.
• The internal hom Hom(X,-) from any smooth manifold X preserves weak equivalences. This is just a reformulation of the smooth Oka principle.
• Proposition 10.3 resolves the question in Remark 2.2.9 of the paper “Equivariant principal infinity-bundles”.
• Finally, all of the above continues to hold if we replace (pre)sheaves of sets by presheaves valued in a left proper combinatorial model category V.
• As an application, in Section 14 I prove classification results for principal G-bundles and bundle gerbes over arbitrary cofibrant diffeological spaces.
Do you know an explicit example of a cofibrant diffeological space that’s not a manifold?
Do you know an explicit example of a cofibrant diffeological space that’s not a manifold?
Yes, of course: smooth realizations of simplicial sets are not manifolds. So as a completely explicit example, take the smooth realization of a simplicial 2-horn.
In general, cofibrant diffeological spaces will be smooth analogues of CW-complexes (or, more generally, retracts of transfinite compositions of cobase changes of smooth horn inclusions).
So it is not unreasonable to expect that we have smooth analogues of various results about certain spaces being CW-complexes.
Hmm, interesting. Now I’m wondering about geometric realization of simplicial fin dim manifolds. If they were cofibrant that would be excellent.
Re #77: As long as the latching maps (inclusions of degenerate simplices) of your simplicial manifold are cofibrations of diffeological spaces, the answer is affirmative: consider the skeletal
filtration of the smooth realization; every step in the filtration is a cobase change of the pushout product of a smooth boundary inclusion and the corresponding latching map. Since the model
structure is cartesian, the pushout product is a cofibration, and so is its cobase change. Transfinite compositions of cofibrations are cofibrations.
The latching maps are cofibration in many cases of interest. A trivial case is when the degenerate simplices are split. A less trivial case is when the latching map is a closed embedding of
manifolds, since such maps are cofibrations by a relative version of Proposition 9.2.
So something like a simplicial Lie group, I guess? That’s useful to know.
Hi Dmitri, re #74:
thanks for posting this! Looks really interesting.
I am on a brief family vacation and didn’t find time yet to really look at your pdf, nor may I find much time in the next week.
Just one quick question from the list of highlights:
How is the model structure on smooth sets which you consider related to that considered by Cisinski, as highlighted in Adrian CLough’s thesis?
How is the model structure on smooth sets which you consider related to that considered by Cisinski, as highlighted in Adrian CLough’s thesis?
Given the way you phrased this, may I point out that the nLab has a detailed article about Cisinski’s model structures on toposes: test topos, which you once created.
The weak equivalences are the same for the transferred model structure and Cisinski’s model structure.
Cofibrations in Cisinski’s model structure are precisely monomorphisms, whereas cofibrations in the transferred model structure are precisely retracts of transfinite compositions of cobase changes of
smooth horn inclusions.
So fibrancy in the transferred model structure is something you can establish in practice, which is not really the case for Cisinski’s model structure.
A naive question that I’ve not seen addressed, and someone who’s published on diffeology seems to not know: is the D-topology on a the diffeological space associated to a Fréchet space the same as
the original topology? I would be surprised if not. We seem to have danced around the issue earlier in the thread, but skimming through I only saw discussion of the shape working out correctly.
This amounts to saying that Frechet spaces are Δ-generated topological spaces. Is this known?
It seems to be true at least in special cases: in conversation with Enxin Wu we agreed that the Fréchet space topology and the D-topology on $\prod_{\mathbb{N}} \mathbb{R}$ agree.
Do we know if Banach spaces are $\Delta$-generated?
Do we know if Banach spaces are Δ\Delta-generated?
I think so. We need to show that given a Banach space B and a subset U⊂F, if the preimage of U under any smooth map R^n→B is open, then U is open.
Assume the converse: there is a point u∈U (wlog u=0) such that for any ε>0 there is a point u_ε∈B∖U such that ‖u_ε‖<ε.
Now use smooth bump functions to construct a smooth curve f:R→B such that f(ε)=u_ε for some set of ε that have 0 as an accumulation point.
We have a contradiction: 0∈f^{-1}U, but arbitrary small neighborhoods of 0 have points outside of f^{-1}U.
Thus, U is open in the norm topology.
This appears to work also for Frechet spaces, since the topology of a Frechet space is induced by a countable system of seminorms.
An argument of an apparently different nature was supplied on Twitter.
This appears to work also for Frechet spaces, since the topology of a Frechet space is induced by a countable system of seminorms.
Possibly even just using the fact the topology comes from a translation-invariant metric is enough: use $d(0,u_\varepsilon)$ instead of ‖u_ε‖<ε. I think the smooth curve construction should work
basically the same (I imagine doing something like a piecewise linear continuous path joining the set of points $u_\varepsilon$, then smoothing it by a reparametrisation introducing flat points at
the joins. Since the chain rule works for the usual calculus in Fréchet spaces this will still be smooth). Do you agree?
Yes, I think this works fine for any first-countable topological vector space as long as smoothness holds, since such first-countable TVS admit countable fundamental systems of neighborhoods U_ε and
we can take u_ε∈U_ε.
And for smoothness of curves, only smoothness at 0 is nontrivial, since at all other points we can use as coefficients smooth real-valued bump functions with disjoint supports instead of the
piecewise linear construction, and these automatically yield a smooth curve away from 0.
In the Frechet case, we can choose seminorms to be exponentially decreasing as the parameter approaches 0, which guarantees smoothness.
Re #80: I added a remark about Cisinski’s result in the draft.
Urs, if you can think of any additional results/statements that would be of interest to you, let me know, I will be happy to add them to the paper; Proposition 10.3 already resolves one of your
previous questions.
And now the paper is on arXiv: https://arxiv.org/abs/2210.12845.
Added (and split off foundations in a separate section):
Cartan calculus for diffeological spaces is developed in
• Christian Blohmann, Elastic diffeological spaces, arXiv:2301.02583.
diff, v88, current | {"url":"https://nforum.ncatlab.org/discussion/1208/diffeological-space/?Focus=20015","timestamp":"2024-11-06T07:42:57Z","content_type":"application/xhtml+xml","content_length":"225978","record_id":"<urn:uuid:78826d50-b32c-4a66-a632-be19307ae0d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00376.warc.gz"} |
Advanced algebra trig worksheets
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How Many Triangles 2
16 x '1 triangle'
16 x '2 triangles'
8 x '4 triangles'
4 x '8 triangles'.
[Transum: Yes, the answers are further down the page for those who have a Transum account.]
#YCISisawesome, #welikehashtags, #whydidihashtagthis.
Or look at the squares in the 4 small ones you have 8 triangles then in the diamond (tilted square) you have 8 but 4 overlap with the previous
then in the big one you have another 8 big triangles so (4*8)+(4+8)= 44
Hope this helps :).
Thank you!
The main one: which has - 8 separate triangles = 4
The (diamond shaped) square
in the middle has - 8 separate triangles = 4
The Maim square is divided
into 4 squares that has - 8 separate triangles = 32
which give you 40 triangles. = 40
then there is 4 rectangular
parts of the diagram which
are 2 of the 8 square boxes
side by side.Ea one has
2 rectangular triangles in
ea.( they separate & distinct,
not recounted) for a total of
8 = 8
for a total of = 48
I've noticed that there are many ways of counting the triangles,
largest to smallest, within the the squares / rectangles.
But you cannot get around the triangles in the rectangles they separate triangles of their own & must be counted.
I don't agree with the answer.
If you look at the rectangular part of the diagram, there are 4 rectangles with 2 triangles ea. (These are distinctly different,
they are rectangular shaped triangles. not recounted) = 8
This is in edition to the 2 squares which contain 4 triangles ea.
( The main one & the diamond in the middle, distinct & separate not recounted) = 8
Then there are 4 squares with 8 triangles in ea. (distinct & separate not recounted) = 32
For a total of 48. triangles.
Basic little triangles: 16
Each half (small squares): 16
Each half (big squares): 4
Each half (4 rectangles)
(2 vert & 2 horiz); 8
Total: 16 + 16 + 4 + 8 = 44.
From the 5 squares, there 40 triangles and only 4 triangles from the middle square 4 triangles.
There are 16 small size triangles, call it basic, then
there are 16 double size than the basic, (Each = 2 basic), then
there are 8 quadruple size triangles (Each = 4 basic)
And finally, there are 4 Octuple size triangles (Each = 8 basic). In other words, the latest is equl to half of the square.
So adding all gives a count of 44.
Hate it when I make myself into an idiot in public.
Thank you!
0 it is a green and yellow square.
72 if you count all the triangles.
216 if you assign a base and equate from there.
Effectively, we have 5 squares and 4 rectangles.
Each square has 4 triangles basing on each side with center point of the square as the top of the triangle. Also each square has two diagonals with two triangles on either side of the diagonal.
This makes eight squares within each square and for five squares, the number of trinagles total 40.
Each rectangle has 2 triangles basing on each of the broader side as the base with center point of the other broader side as the top of the triangle.
The triangles with outer sides are already counted as part of the large square. Thus the four rectangles contain 4 additional triangles.
This makes the total to 44.
(4x4) = X1 triangles(smalls ones)
[ 4 in each small square]
(4X4) = X2 triangles(2 smalls ones as one)
[ 4 in each small square]
(4X1) = X4 triangles(4 smalls ones as one)
[4 in the large square]
(4x1) = x4 triangles(on vertcally and horizontally half of large square)
[4 in the large square]
(4X1) = X8 triangles(diagonally half of the square)
[4 in the large square]
(4x4) + (4X4) + (4X1) + (4X1) + (4X1) = 44.
there are 4 visible small square + 1 big square with the same pattern as the small square, so there are 5 square.
but there is 1 big square rotated 90 degree diagonally, and in this square, we can only count 4 triangle more since the other 4 is already count on small square.
so ...
(5 square * 8 triangle) + 4 = 40 + 4 = 44 ...
(1= I small triangle) The amount of them
That is a rough total of my answer
When I say size I mean the amount of tiny yellow or green ones, that fit in the triangle.
Ben solved it in less than a minute. Followed by Lois, Emlyn, Lucy, Jake and Owain.
Good puzzle, made me think a bit.
in the smallest squares there are 8 triangles. 8x4=32
in the center square there are 4...but if u count ones you have already counted before then there will be 8. so 32+4=36
then there are 4 enormous ones using half of the square, there are 4 of these as well. 36+4=40
and finally from the main square side, take each one and make a triangle out of it going to the center of the triangle. thus giving you the final 4 to equal 44.
if u get over 44 your wrong. if u get under your missing something. the center square has 16 triangles total but u can only count 4 out of those 16 or you will be recounting from the smallest square.
I counted 44.
Said in a different way, 4 or the triangles created when dividing 4 small sqaures into triangles, are the same as 4 of the triangles created when diving the mid-sized (or center square) into
4 'squares' of 4 small triangles each: 16.
4 'squares' of 2 medium triangles (each using 2 small triangles) each in 2 separate configurations (separated from diagonal top left to bottom right, separated from diagonal top right to bottom
left): 16.
1 small square in center with 4 large triangles (each using 4 small triangles), each large triangle sharing two of its small triangles with 2 other triangles: 4.
1 large square with 4 large triangles (each using 4 small triangles), each large triangle not sharing any of its own small triangles with any of the other triangles: 4.
1 large square of 2 very large triangles (each using 8 small triangles) each in 2 separate configurations (separated from diagonal top left to bottom right, separated from diagonal top right to
bottom left): 4.
That's 44 triangles total.
There are 4 triangles that are recounted. thas where the -4 comes from. but you people are forgetting the triangles that are heading outwards. that are not counted in the square. .
Here are my reasons:
1. Let me explain why there are not 48 triangles: There are 6 squares composed of four or more triangles, which means that these 6 squares can be divide into 8 triangles. This would be 48 if some of
the triangle weren't being counted twice. The triangles composed of 2 single triangles that are located in the middle(slanted) square are being counted twice. Once in the slanted square and once when
counting the triangles in the four small squares. This would alter the equation of solving the number of triangles to be: 6(# of squares)*8(# of triangles produced by each square)-4(# of squares
counted twice)= 44
2. You can also break it down like this: There are
4 triangles composed of 8 of the single triangles
8 triangles composed of 4 of the single triangles
16 triangles composed of 2 of the single triangles and
16 single triangles
All of these triangles add up to make 44 triangles.
Then I moved on to see that there are four squares made up of four little triangles, but no more recounting unit sized triangles, move on to look at the triangles made up of putting two little ones
together. There are four more on each of these squares. Look and see that these triangles I discovered have the vertex of the edges of the smallest squares. Here is where people claim that recounting
is done. Well I honestly cant find more than 16 triangles made up of two unit triangles. The square in the middle seems to have four more triangles made up of two unit triangles with vertexes on the
edges and in the center, but they have already been counted when looking at the ones made up in the four smallest squares. (16 made up of two)
Looking at the square in the middle, there are triangles made up of four. only four of them because they are made up from the vertexes touching the edge of the biggest square. This center square only
has these four triangles made up of four unit triangles, BUT those aren't the only ones in this whole problem. There are four more larger triangles made up of four unit sized triangles. Take a look
at the sides of the biggest square. The vertexes of these next four triangles are the edges of the biggest square and the very center of the whole thing. Count... these are made up of four unit sized
triangles, and we clearly did not disregard these. So there are 8 total, not 16. (8 triangles made up of four)
Now the hunt for triangles made up of 8... the last four. Yes four not 8, if you are recounting here that's really sad. But honestly I don't know where you recounted but lets look at the last 4. Well
this is actually quite simple. Look at the biggest square. These last four triangles made up of 8 unit sized triangles have vertexes only on the corners of the big square, obviously only four of
them, and if you see more, then feel free to show me please. (4 made up of 8)
16+16+8+4=!!!! what does it equal!?!?! hmm, 48!
ha ha jk, its 44.
I believe there are 20 triangles that DON'T have edges that touch the center.
There are then 24 triangles that DO have edges that touch the center.
Need Help? Cut the square in half from the top left corner down to the bottom right. Now do you see the last two triangles?
I did just notice the overlap of the ones with 2 small triangles. Sorry!
Based on the number of small triangles in each triangle, there are:
16 with 1 small triangle
20 with 2 small triangles
8 with 4 small triangles
4 with 8 small triangles.
these are all right triangles
the hypotenuse is opposite of it's 90 degree angle
one hypotenuse can be shared by two triangles (count "mirror" sides)
a hypotenuse that has
1 line = 16
2 line short = 16
2 line long = 8
3 line = 0
4 line = 4
total of 44
A change in perspective may be beneficial as a teaching aid. we all learned how to count and what shape a triangle was by age 6. Now lets teach common sense.
Each small square contains 8 triangles - 8 x 4 = 32
The inner square has 4(top/bottom then left/right) 2 x 2 = 4
The outer square has 8 triangles - 4 inside & 4 outside - 4 + 2 = 8
Total = 44
The "48 count" error comes from recounting the inner triangles on both large squares.
This was fun, hope everyone else enjoyed the mental exercise as well! :)
Some argue that at 48 some are counted twice. I assure you, they are not counted twice. Look again.
there are 4 triangle sizes
T size: 16
TT size: 16
TTTT size: 8 (yes, i counted the ones pointing outwards and inwards)
TTTT size: 4 (16+16+8+4=44)
They are all right triangles; therefore, if you put 4 together, you can make a square. And within that square there is the possibility of seeing 8 triangles when you make draw an X going from corner
to corner.
Count the number of squares in the puzzle (don't forget the overall square, and you'll see 6.
6 times 8 triangles per square = 48.
16 + (8X2) + (4X2) + (1X2) + (1X2)
16= smalls ones
(8X2) = X2 triangles
(4X2) = X4 triangles
(2X2) = half of the square.
How many triangles are hidden in the pattern?
What strategy might you use to count them all to ensure you don't miss any out?
Sign in to your Transum subscription account to see the answers
How Many Squares 1? | How Many Squares 2?
How Many Triangles 1? | How Many Triangles 2? | How Many Triangles 3?
How Many Rectangles? | Rectangles Investigation | Icosahedron | Mystic Rose
Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page.
Below is an Amazon link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.
Educational Technology on Amazon
Teacher, do your students have access to computers such as tablets, iPads or Laptops? This page was really designed for projection on a whiteboard but if you really want the students to have access
to it here is a concise URL for a version of this page without the comments:
However it would be better to assign one of the student interactive activities below.
Here is the URL which will take them to a related student activity. | {"url":"https://transum.org/Software/SW/Starter_of_the_day/starter_January23.ASP","timestamp":"2024-11-09T04:34:25Z","content_type":"text/html","content_length":"80029","record_id":"<urn:uuid:fc35b6c7-56ff-4d81-b14f-9b0ac939b9ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00490.warc.gz"} |
40. The secret of secrets part 1
In the year 480 king Solomon began building the house for the lord
60 x 20 x 30 cubits is 3600 cubits and 3600 cubits times 52,36 is 188496.
This should be familiar to you, and can be found in the other articles.
Does anyone know what was placed in the ark of covenant?
Please look it up, then you will understand a great truth.
As I did explain before the star of beth le hem is 8 sided and 8 times the cubit is 4,1888 which is the formula to calculate the volume of a globe, and it also does say 4 as in 4 corners of the world
(and the 4 elements).
The one is also the snake but also the staff of Moses and the 8 the star of beth le hem, which is made up of two circles (is also the two stone tablets).
As you remember the ark being 45 times 52,36 is 23562. Divide this by the ark of covenants’ measure 56,25 and you will get 41888.
Moshiya van den Broek | {"url":"https://www.truth-revelations.org/?page_id=270","timestamp":"2024-11-09T08:58:49Z","content_type":"text/html","content_length":"27089","record_id":"<urn:uuid:83dfb3fc-ff21-4506-a3bd-b2f42e221428>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00072.warc.gz"} |
Attribute control charts limit calculations
Publish Date:2017-08-17 16:45:53 Clicks: 140
All attribute control charts follow the same three sigma control lim. it away from the centerline methodology of the variable control charts:
Control limits for attribute charts = centerline 土 3 s (3.8)
For constant samples (C or nP charts)
For Poisson distribution:
For changing sample sizes (U or P charts)
For Poisson distribution:
Centerline = Poisson average number of defects in a sample u
C and U Charts are considered as a special form of control charts in which the possibility of defects is much larger, and the probability of getting a defect at any specific point, place, or time is
much smaller.
The relationship of attribute charts to the six sigma concept is through the defects implied in the charts. The centerline represents the defect rate. These defect rates can be translated into an
implied Cpk, as shown in the previous chapter.
Several assumptions have to be made in the case of the attribute chart connections to six sigma:
1. There is one or a complex set of specifications that are not readily discernible that govern the manufacturing process for the parts.
2. These specifications are either one- or two-sided, resulting in one- or two-sided defects (defects < LSL and defects > USL).
3. The manufacturing process is assumed to be normally distributed.
4. There is a relationship between the process average and the specification nominal. In some definitions of six sigma, an assumption is made that there is a 1.5 a shift from process average to
specification nominal.
The control limits of the attribute charts are not related to the population distribution. Therefore, the method of finding the population standard deviation a is quite different from that used in
variable control charts, as shown in the examples below. | {"url":"https://www.nod-pcba.com/news/526-en.html","timestamp":"2024-11-11T03:35:11Z","content_type":"text/html","content_length":"30440","record_id":"<urn:uuid:1222ea5a-7b51-4740-9548-39a6929eec2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00089.warc.gz"} |
Reed-Solomon coding/decoding package v1.0
Reed-Solomon coding/decoding package v1.0
Phil Karn, KA9Q, September 1996
This package implements general purpose Reed-Solomon encoding and decoding for a wide range of code parameters. It is a rewrite of code by Robert Morelos-Zaragoza (robert at spectra.eng.hawaii.edu)
and Hari Thirumoorthy (harit at spectra.eng.hawaii.edu), which was in turn based on an earlier program by Simon Rockliff (simon at augean.ua.oz.au). This package would not exist without the excellent
work of these earlier authors.
This package includes the following files:
• readme - this file
• rs.h - include in user programs. Code params are defined here.
• rs.c - the initialization, encoder and decoder routines
• rstest.c - test program
• makefile - makefile for the test program and encoder/decoder
Any good coding theory textbook will describe the error-correcting properties of Reed-Solomon codes in far more detail than can be included here. Here is a brief summary of the properties of the
standard (nonextended) Reed-Solomon codes implemented in this package:
MM - the code symbol size in bits
KK - the number of data symbols per block, KK < NN
NN - the block size in symbols, which is always (2**MM - 1)
The integer parameters MM and KK are specified by the user. The code currently supports values of MM ranging from 2 to 16, which is almost certainly a wider range than is really useful.
Note that Reed-Solomon codes are non-binary. Each RS "symbol" is actually a group of MM bits. Just one bit error anywhere in a given symbol spoils the whole symbol. That's why RS codes are often
called "burst-error-correcting" codes; if you're going to have bit errors, you'd like to concentrate them into as few RS symbols as possible.
In the literature you will often see RS code parameters given in the form "(255,223) over GF(2**8)". The first number inside the parentheses is the block length NN, and the second number is KK. The
number inside the GF() gives the size of each code symbol, written either in exponential form e.g., GF(2**8), or as an integer that is a power of 2, e.g., GF(256). Both indicate an 8-bit symbol.
Note that many RS codes in use are "shortened", i.e., the block size is smaller than the symbol size would indicate. Examples include the (32,28) and (28,24) RS codes over GF(256) in the Compact Disc
and the (204,188) RS code used in digital video broadcasting. This package does not directly support shortened codes, but they can be implemented by simply padding the data array with zeros before
encoding, omitting them for transmission and then reinserting them locally before decoding. A future version of this code will probably support a more efficient implementation of shortened RS codes.
The error-correcting ability of a Reed-Solomon code depends on NN-KK, the number of parity symbols in the block. In the pure error- correcting mode (no erasures indicated by the calling function),
the decoder can correct up to (NN-KK)/2 symbol errors per block and no more.
The decoder can correct more than (NN-KK)/2 errors if the calling program can say where at least some of the errors are. These known error locations are called "erasures". (Note that knowing where
the errors are isn't enough by itself to correct them because the code is non-binary -- we don't know *which* bits in the symbol are in error.) If all the error locations are known in advance, the
decoder can correct as many as NN-KK errors, the number of parity symbols in the code block. (Note that when this many erasures is specified, there is no redundancy left to detect additional
uncorrectable errors so the decoder may yield uncorrected errors.)
In the most general case there are both errors and erasures. Each error counts as two erasures, i.e., the number of erasures plus twice the number of non-erased errors cannot exceed NN-KK. For
example, a (255,223) RS code operating on 8-bit symbols can handle up to 16 errors OR 32 erasures OR various combinations such as 8 errors and 16 erasures.
The three user-callable functions in rs.c are as follows:
1. void init_rs(void);
Initializes the internal tables used by the encoder and decoder using the code parameters compiled in from rs.h. This function *must* be called before the encoder or decoder are used for the
first time.
2. int encode_rs(dtype data[KK],dtype bb[NN-KK]);
Encodes a block in the Reed-Solomon code. The first argument contains the KK symbols of user data to be encoded, and the second argument contains the array into which the encoder will place
the NN-KK parity symbols. The data argument is unchanged. For user convenience, the data and bb arrays may be part of a single contiguous array of NN elements, e.g., for a (255,223) code:
The encode_rs() function returns 0 on success, -1 on error. (The only possible error is an illegal (i.e., too large) symbol in the user data array.
Note that the typedef for the "dtype" type depends on the value of MM specified in rs.h. For MM <= 8, dtype is equivalent to "unsigned char"; for larger values, dtype is equivalent to
"unsigned int".
3. int eras_dec_rs(dtype data[NN], int eras_pos[NN-KK], int no_eras);
Decodes a encoded block with errors and/or erasures. The first argument contains the NN symbols of the received codeword, the first KK of which are the user data and the latter NN-KK are the
parity symbols.
Caller-specified erasures, if any, are passed in the second argument as an array of integers with the third argument giving the number of entries. E.g., to specify that symbols 10 and 20
(counting from 0) are to be treated as erasures the caller would say
eras_pos[0] = 10;
eras_pos[1] = 20;
The return value from eras_dec_rs() will give the number of errors (including erasures) corrected by the decoder. If the codeword could not be corrected due to excessive errors, -1 will be
returned. The decoder will also return -1 if the data array contains an illegal symbol, i.e., one exceeding the defined symbol size. | {"url":"http://www.piclist.com/tecHREF/method/error/rs-gp-pk-uoh-199609/index.htm","timestamp":"2024-11-06T02:10:19Z","content_type":"text/html","content_length":"23293","record_id":"<urn:uuid:e6b80d05-d84f-4f20-ae00-551cc4e4748d>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00490.warc.gz"} |
[Solved] In Exercises 1318, find the average rate | SolutionInn
In Exercises 1318, find the average rate of change of the function from x 1 to x
In Exercises 13–18, find the average rate of change of the function from x[1 ]to x[2].
Transcribed Image Text:
f(x)=√x from x₁ = 4 to x₂ = 9
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[Solved] The age of the universe is thought to be | SolutionInn
The age of the universe is thought to be about 14 billion years. Assuming two significant figures,
The age of the universe is thought to be about 14 billion years. Assuming two significant figures, write this in powers of ten in
(a) Years,
(b) Seconds
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Clustering Coefficient Calculator
Author: Neo Huang Review By: Nancy Deng
LAST UPDATED: 2024-10-03 22:09:24 TOTAL USAGE: 2726 TAG: Data Science Social Science Statistics
Unit Converter ▲
Unit Converter ▼
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Historical Background
The concept of clustering coefficient emerged in graph theory and network science to describe how nodes cluster together in graphs representing social networks, transportation systems, and other
structures. It provides a numerical value reflecting the degree to which nodes tend to form tightly connected groups.
Calculation Formula
The formula to calculate the clustering coefficient is simple:
\[ C = \frac{CT}{AT} \]
• C is the clustering coefficient,
• CT is the number of closed triplets,
• AT is the number of all triplets (closed and open).
Example Calculation
If a graph has 12 closed triplets and 30 total triplets, the clustering coefficient is:
\[ C = \frac{12}{30} = 0.4 \]
Importance and Usage Scenarios
Clustering coefficients are essential in social network analysis, biological network studies, and in various other applications where the structure of relationships between nodes is significant. It
helps in understanding the local cohesiveness of networks and the potential for the formation of tightly-knit communities.
Common FAQs
1. What is a triplet in graph theory?
□ A triplet is a set of three nodes that are interconnected. A closed triplet means all three nodes are directly connected to each other, forming a triangle. An open triplet is a set of three
nodes with only two direct connections.
2. What does a high clustering coefficient indicate?
□ A high clustering coefficient indicates that the nodes in a graph tend to form tightly connected clusters or communities.
3. Can the clustering coefficient be used to study social networks?
□ Yes, it can be used to understand social interactions and the likelihood of forming tightly-knit groups or communities. | {"url":"https://www.calculatorultra.com/en/tool/clustering-coefficient-calculator.html","timestamp":"2024-11-03T04:34:29Z","content_type":"text/html","content_length":"47090","record_id":"<urn:uuid:70c00d4f-1848-4710-85f7-8fc65d5a36e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00608.warc.gz"} |
For Two
Here we have adapted a selection of our Primary tasks so that they can be tackled by just one child working with an adult.
Incey Wincey Spider game for an adult and child. Will Incey get to the top of the drainpipe?
Shut the Box game for an adult and child. Can you turn over the cards which match the numbers on the dice?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Dotty Six game for an adult and child. Will you be the first to have three sixes in a straight line?
Guess the Houses game for an adult and child. Can you work out which house your partner has chosen by asking good questions?
Arranging counters activity for adult and child. Can you create the pattern of counters that your partner has made, just by asking questions?
Totality game for an adult and child. Be the first to reach your agreed total.
Matching Numbers game for an adult and child. Can you remember where the cards are so you can choose two which match?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?
Board Block game for two. Can you stop your partner from being able to make a shape on the board?
Stop the Clock game for an adult and child. How can you make sure you always win this game?
Train game for an adult and child. Who will be the first to make the train?
Dicey Operations for an adult and child. Can you get close to 1000 than your partner?
Four Go game for an adult and child. Will you be the first to have four numbers in a row on the number line?
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
'What Shape?' activity for adult and child. Can you ask good questions so you can work out which shape your partner has chosen?
Guess the Dominoes for child and adult. Work out which domino your partner has chosen by asking good questions.
Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent.
Factors and Multiples game for an adult and child. How can you make sure you win this game?
Spiralling Decimals game for an adult and child. Can you get three decimals next to each other on the spiral before your partner?
Got It game for an adult and child. How can you play so that you know you will always win?
Board Block Challenge game for an adult and child. Can you prevent your partner from being able to make a shape? | {"url":"https://nrich.maths.org/two","timestamp":"2024-11-13T18:59:55Z","content_type":"text/html","content_length":"69187","record_id":"<urn:uuid:d8e60a8e-8339-4025-b956-59643bb8e63b>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00538.warc.gz"} |
Type B - Expanded
Length From To Datatype Format Description and Comments
2 1 2 AN X(2) Record ID - "B "
3 3 5 AN X(3) Exchange Acronym
10 6 15 AN X(10) Commodity Code
3 16 18 AN X(3) Product Type Code
6 19 24 N 9(6) Futures Contract Month as CCYYMM
2 25 26 AN X(2) Futures Contract Day or Week Code
1 27 27 - - Filler
6 28 33 N 9(6) Option Contract Month as CCYYMM
2 34 35 AN X(2) Option Contract Day or Week Code
1 36 36 - - Filler
8 37 44 N 9(2)V9 Base Volatility (as a decimal fraction)
8 45 52 N 9(2)V9 Volatility Scan Range (as a decimal fraction)
5 53 57 N 9(5) Futures Price Scan Range
5 58 62 N 9(2)V9 Extreme Move Multiplier
5 63 67 N 9V9(4) Extreme Move Covered Fraction
5 68 72 N 9V9(4) Interest Rate (as a decimal fraction)
7 73 79 N 9V9(6) Time to Expiration (in years)
6 80 85 N V9(6) Lookahead Time (in years)
6 86 91 N 9(2)V9 Delta Scaling Factor
8 92 99 N 9(8) Expiration (Settlement) Date as CCYYMMDD
10 100 109 AN X(10) Underlying Commodity Code
2 110 111 AN X(2) Pricing Model
8 112 119 N 9(2)V9 Coupon or Dividend Yield, as a decimal fraction
1 120 120 AN X(1) Option Expiration Reference Price Flag -- see note below
7 121 127 N 9(7) Option Expiration Reference Price
1 128 128 AN X(1) Option Expiration Reference Price Sign (+ or -)
9(7)V9 Swap Value Factor (for interest-rate swaps) or
14 129 142 N (7)
Contract-Specific Contract Value Factor (for normal futures and options)
2 143 144 N 9(2) Swap Value Factor Exponent
1 145 145 AN X Sign for Swap Value Factor Exponent (blank, "+" or "-")
2 146 147 N 9(2) Base Volatility Exponent
1 148 148 AN X Sign for Base Volatility Exponent (blank, "+" or "-")
2 149 150 N 9(2) Volatility Scan Range Exponent
1 151 151 AN X Sign for Volatility Scan Range Exponent (blank, "+" or "-")
12 152 163 N 9(2)V9 Discount Factor (for discounting back to present value)
1 164 164 AN X Volatility Scan Range Quotation Method -- blank or A means that the volatility scan range is provided as an absolute value, and P means that it is provided as
percentage of the implied volatility.
1 165 165 AN X Price Scan Range Quotation Method -- blank or A means that the price scan range is provided as an absolute value, and P means that is provided as a percentage
of the contract value
2 166 167 N 9(2) Futures Price Scan Range Exponent
1 168 168 AN X Sign for Futures Price Scan Range Exponent (blank, "+" or "-")
5 169 173 AN X Delivery Margin Method
8 174 181 N 9(8) Margin Removal Date (as CCYYMMDD) – if present, positions in this contract no longer contribute to the margin requirement, beginning at the specified cycle on
this date, and subsequently
1 182 182 AN X Margin Removal Cycle – either S for end of day, or I for intraday. If the margin removal date is provided but the cycle value is not, a value of S for end of
day cycle is defaulted.
1  183 183 AN X Interest Rate Sign. A minus sign means the interest rate is negative, and blank, null, + or any other value means it is positive.Â
1  184 184 AN X Coupon or Dividend Yield Sign. A minus sign means the coupon or dividend yield is negative, and blank, null, + or any other value means it is positive.
 14  185  198  N  9 High Precision Option Expiration Reference Price
 1  199  199  AN  X High Precision Option Expiration Reference Price Sign – blank, + or -. Any value other than a minus sign indicates that the value is positive.
 1  200  200  AN  X High Precision Option Expiration Reference Price Flag. N means that the high-precision option expiration reference price field is populated, but that the price
can be read from either the regular precision field or the high-precision field. Y means that the value can only be read from the high-precision field.
1. "B" records provide delta-scaling factors as well as risk array calculation parameters for either a particular futures contract, or for a particular option series -Â ie, for all options which are
identical except for their put/call code and their strike.
2. Except for the delta-scaling factors, parameters contained on "B" records are not needed for the SPAN performance bond calculation itself.  If "B" records are not provided for a particular
future or option series, the delta-scaling factor for that future or that series should be defaulted to 1.00.
3. If "B" records are provided, then the "B" records for all products in a combined commodity are typically located in the SPAN file after the "4" record for that combined commodity.
4. "B" records for a futures contract will contain either zeros or spaces in the Option Contract Month and Option Contract Day fields.
5. The Option Contract Day or Week Code field is used to distinguish option series which expire at different times than the standard monthly options. Â For standard monthly options, this field will
contain zeros or blanks. Â For other options, this field will typically contain "W1", "W2", etc. - for weekly options expiring in week 1 of the month, week 2 of the month, etc. - or a two-digit
day of the month, for flex options or other options for which the exact expiration day is specified.  The Futures Contract Day or Week Code is intended to be used analogously to distinguish
futures which expire at different times than standard monthly futures.
6. The Price Scan Range parameter on the "B" record is in the performance bond currency for the combined commodity and must be multiplied by ten raised to the Risk Exponent power for that
combined commodity. Â The Risk Exponent is taken from the "2" record.
7. The Expiration (Settlement) Date for a future is the date on which its final marking price is determined. Â The Expiration (Settlement) Date for an option series is the last date on which holders
of options in that series can elect to exercise those options.  Time to Expiration is determined by taking the number of calendar days between the Expiration Date and the business date of this
SPAN file, and dividing by 365, with zero as a minimum value.
8. Currently supported values for the Pricing Model code are: B for Black (European futures options), BS for Black-Scholes (European physical options with no dividends), M for the generic
Merton European option model, WB for "Whaley Black" (the Adesi-Whaley model for American futures options), WS for "Whaley Scholes" (the Adesi-Whaley model for American physical options with
no dividends, WI for "Whaley for Indices" (the generic Adesi-Whaley model), and I for Intrinsic.
9. Product type codes are PHY for Physical, FUT for Future, CMB for Combination, OOP for Option on Physical, OOF for Option on Future, OOC for Option on Combination.
10. The Option Expiration Reference Price and Price Flag are optional fields which may be provided for "B" records for option series. These fields provide a means of identifying whether the
final price of the underlying for determining automatic exercise of in-the-money options is available, and if so, what that price is. A value N for this flag means either that the expiration
day for this option series has not yet arrived, or it has arrived but that the reference price is not yet available. A value of Y for the flag means that the expiration day has arrived, that
the price is available, and that the price is actually the settlement price for the underlying on that day. A value of S means that the expiration day has arrived and that the reference price
is available, but that this is a special reference price, different from the settlement price of the underlying on that day. Note that this price will be formatted according to the decimal
locator and alignment code for the underlying, not for the option series.
11. The Discount Factor field in bytes 152-163 provides the value used for discounting mark-to-market values back to present value, for example for forwards. The numeric format of 9(2)V9(10) is
for discount factors as a decimal value. For example, a discount factor of 98.1234 percent, or 0.981234 as a decimal value, will appear in the field as 009812340000. Hence the field supports
discount factors out to eight decimal places of a percent.
12. The Delivery Margin Method provides data to drive the margin calculation for positions in physically-deliverable futures or forwards that have gone into the delivery process. Allowable values
are PID, meaning that all positions in this contract are in delivery today, and hence naked delivery margins are assessed for all; PIDP, meaning that some positions in this contract may be in the
delivery process today, and for these only, naked margins are assessed; LFV, meaning that naked margins are assessed for short positions and full contract value margins for long positions; and FV
, meaning that full value margins are assessed for both long and short positions. A blank or null value means that delivery margins are not applicable to this contract at this time.
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Create Custom OFDM Resource Grid
This example shows how to design and verify a model which modulates and demodulates a custom OFDM resource grid for HDL code generation.
First, the example introduces the structure of the OFDM grid, and how to add contents to the grid. Then, the example describes transmitter and receiver designs that each have a MATLAB reference and a
Simulink model that is suitable for HDL code generation. The MATLAB reference explores the design space and provides test vectors. The example runs the Simulink transmitter and receiver subsystems
connected together and then compares the results to MATLAB. Finally, the example shows the results from HDL code generation.
This example is part of a related set of examples for custom OFDM communication systems. For more information see Custom OFDM Reference Applications Overview.
Resource Grid Definition
The first stage of designing a custom OFDM communication system is to define the structure and contents of the resource grid. The resource grid is defined as a 2-D matrix in the frequency domain with
number of subcarriers (Nsc) rows and number of OFDM symbols (NofdmSyms) columns. OFDM modulation converts this resource grid from the frequency domain to the time domain. The FFT size (Nfft),
subcarrier spacing (scs), and the length of the cyclic-prefix (Ncp) determine the symbol bandwidth and duration. The Nfft and Ncp values are in samples, and the scs in kHz. For more information on
OFDM see OFDM Modulation Using MATLAB.
The OFDM resource grid is populated by modulated symbols of a specified signal type on each resource element. A resource element is one subcarrier in one OFDM symbol. This allows for the design of
custom communication systems where many different signal types are combined to satisfy the system requirements. This example includes the following three signal types. Any resource element without a
specified signal type is filled with a null.
• Synchronization Sequence - The synchronization sequence is created using a maximal length PN sequence that has good auto-correlation properties. It is placed in the first OFDM symbol of the
resource grid to allow the receiver to perform synchronization before data reception.
• Reference Symbols - The reference symbols are transmitted interspersed with the data symbols to allow the receiver to estimate and equalize the channel.
• Data Symbols - The data symbols carry the information in the resource grid.
This code defines the OFDM resource grid. The example defines the overall grid structure, and then the indices for each signal type in the resource grid. Additionally, the synchronization sequence is
generated from a maximal-length PN sequence.
%% Define OFDM Constants
% Define OFDM Resource Grid
Nfft = 256; % FFT size in samples
Ncp = 64; % Cyclic-prefix length in samples
Nsym = Nfft+Ncp; % Time domain OFDM symbol length in samples
scs = 30; % Subcarrier spacing in kHz
Nsc = 228; % Number of active subcarriers in the resource grid
NofdmSyms = 24; % Number of OFDM symbols in the resource grid
Nguard = (Nfft/2) - (Nsc/2); % Number of guard subcarriers on each edge the resource grid
ofdmSampleRate = scs*1e3*Nfft;
% Define OFDM Grid Signals
% Synchronization Sequence
% Generate constant synchronization sequence from maximum length sequence
syncSeqGen = comm.PNSequence("Polynomial",[7 6 0],"InitialConditions",[1 1 0 1 1 0 1],"SamplesPerFrame",127);
syncSeq = syncSeqGen();
syncSeq = qammod(syncSeq,2);
gridOffset = floor((Nsc-length(syncSeq))/2);
ssIdx = (1:length(syncSeq)).'+gridOffset;
% Reference Symbols
refOFDMIdx = 2:4:NofdmSyms;
refSCIdx = repmat((1:Nsc).',length(refOFDMIdx),1);
refIdx = [refSCIdx reshape(repmat(refOFDMIdx,Nsc,1),[],1)];
refIdx = sub2ind([Nsc,NofdmSyms],refIdx(:,1),refIdx(:,2));
% Data Symbols
dataOFDMIdx = setdiff(2:NofdmSyms,refOFDMIdx);
dataSCIdx = repmat((1:Nsc).',length(dataOFDMIdx),1);
dataIdx = [dataSCIdx reshape(repmat(dataOFDMIdx,Nsc,1),[],1)];
dataIdx = sub2ind([Nsc,NofdmSyms],dataIdx(:,1),dataIdx(:,2));
% Define signal type of each resource element
resourceElemType = zeros(Nsc,NofdmSyms);
resourceElemType(ssIdx) = 1;
resourceElemType(refIdx) = 2;
resourceElemType(dataIdx) = 3;
OFDM Transmitter MATLAB Reference
The OFDM transmitter performs two operations.
• Resource grid construction - uses the variables from the resource grid definition to create the empty resource grid, generate the symbols for each signal type and add them onto the specified
elements. This example generates random bits for the data and reference signals, and then performs QPSK modulation to create the symbols for insertion.
• OFDM modulation - uses the ofdmmod function to convert the resource grid into a time domain waveform to transmit. The resource grid is centered in the FFT by adding guard subcarriers with the
nullIndices input.
This code implements the OFDM transmitter. The code is a behavioral reference for the Simulink model.
%% OFDM Transmitter MATLAB
% Create empty resource grid
txGridML = zeros(Nsc,NofdmSyms);
% Synchronization Sequence
txGridML(ssIdx,1) = syncSeq;
% Reference Symbols
refBits = randi([0 1],length(refIdx)*2,1);
refSyms = qammod(refBits,4,[2 3 0 1],UnitAveragePower=1,InputType="bit");
txGridML(refIdx) = refSyms;
% Data Symbols
txBits = randi([0 1],length(dataOFDMIdx)*Nsc*2,1);
txDataSyms = qammod(txBits,4,[2 3 0 1],UnitAveragePower=1,InputType="bit");
txGridML(dataIdx) = txDataSyms;
% OFDM Modulate
firstSC = Nguard + 1;
nullIndices = [1:(firstSC-1) (firstSC+Nsc):Nfft].';
txWaveformML = ofdmmod(txGridML,Nfft,Ncp,nullIndices);
OFDM Transmitter Simulink
The OFDM transmitter algorithm is implemented in Simulink for HDL code generation. The algorithm is converted into a fixed-point streaming design with control signals. The model consists of three
• Grid Structure - This area of the model creates the streaming control signals that construct the resource grid. The scNum and symNum outputs from the Frame Counter indicate the current position
in the resource grid. The Frame Counter serializes the resource grid first in the subcarrier dimension, and then along OFDM symbols. The resource grid definition from MATLAB is stored in the
Resource Element Type lookup table. The lookup table determines which signal type is present on the current element. The Frame Counter generates the valid signal required to pace the samples
through the system. The Frame Counter is paused when the enable input port is zero. This signal controls the duty cycle of the transmitted signal.
• Grid Contents - This area of the model contains lookup tables with the Synchronization, Reference, and Data symbols. A counter is used per signal type to increment through each lookup table and
output the next symbol. Finally, the signal types are multiplexed together to form the input to the OFDM modulator.
• OFDM Modulator - The OFDM modulator block creates the time-domain transmitter waveform. It is parameterized by the resource grid definition and performs all stages of OFDM modulation. The OFDM
modulator fixed-point scaling is designed to preserve the data type between the input and the output.
When converting from frame-based MATLAB code to a streaming Simulink model, the pacing of data with control signals must be considered. This example uses a single Simulink rate to implement the whole
algorithm - this rate represents the clock in the generated HDL code. The output from the transmitter will feed the digital-to-analog converter (DAC) when deployed on the target board, and the same
clock may drive both components. Therefore, the transmitter must be capable of producing a valid output sample every clock cycle, after an initial latency. This constraint defines the valid behavior
through the model.
The txValid signal from the transmitter is the output from the OFDM modulator. The OFDM modulator block is designed to meet this requirement with a constant valid output, if the input data is
correctly paced. OFDM modulation increases the length of the output relative to the input by adding guard subcarriers and the cyclic-prefix to the signal. For an input length of Nsc, the output
length is Nsc + Ncp + Nguard * 2. In a streaming design this means the output takes more cycles than the input. Therefore, a gap of Ncp + Nguard * 2 cycles must be left between each input to allow
for the OFDM modulator to finish outputting the previous OFDM symbol. If the input data stalls, the output valid will fall and the DAC will not receive data. Conversely, if the input data is provided
too fast then the OFDM modulator will drop the data. Both of these outcomes result in data loss from the transmitter. The data pacing is handled by the Frame Counter valid output, the valid is high
for Nsc cycles and then goes low for Nsym - Nsc cycles. This duty cycle ensures that the output from the transmitter will stay high. A timing diagram of the input and output from the OFDM modulator,
with the corresponding valid signals, is shown.
OFDM Receiver MATLAB Reference
The OFDM Receiver receives the transmitted waveform directly from the transmitter. This zero impairment environment simplifies the receiver as practical synchronization is not required. Additionally,
channel estimation is not required to recover the data bits and they can be directly demodulated from the data symbols. The receiver performs two operations to recover each signal type from the
received grid.
• OFDM Demodulation - uses the ofdmdemod function to reverse the OFDM modulation and recover the resource grid.
• Resource grid extraction - slices the received grid back into the synchronization sequence, reference symbols, and data symbols using the resource grid definition.
This code implements the OFDM receiver. The code is a behavioral reference for the Simulink model.
%% OFDM Receiver MATLAB
ofdmDemodInML = txWaveformML;
rxGridML = ofdmdemod(ofdmDemodInML,Nfft,Ncp,Ncp,nullIndices);
rxSyncSeqML = rxGridML(ssIdx);
rxRefML = rxGridML(refIdx);
rxDataML = rxGridML(dataIdx);
OFDM Receiver Simulink
The OFDM receiver algorithm is implemented in Simulink for HDL code generation. The algorithm is converted into a fixed-point streaming design with control signals. The model consists of two stages:
• OFDM Demodulator - The OFDM demodulator block reconstructs the resource grid from the time-domain received waveform. It is parameterized by the resource grid definition and performs all stages of
OFDM demodulation. The OFDM demodulator output wordlength increases by log2 of the FFT size to account for internal scaling. Because the OFDM modulator and demodulator are directly connected the
two resource grids have identical scaling, so the wordlength growth can be discarded. However, fixed-point quantization and rounding errors can lead to a small increase in the received grid's
dynamic range so one bit of growth is preserved.
• Grid Structure - This area of the model is the mirror of the grid structure area of the transmitter. It separates the received resource grid into each of its signal types. This is performed using
the Frame Counter subsystem and the Resource Element Type lookup table. The Frame Counter does not require the valid pacing design from the transmitter. The output valid matches the input valid,
excluding the removal of elements containing nulls.
Run OFDM Grid Construction
The OFDM grid construction simulation runs the OFDM transmitter and receiver MATLAB reference and Simulink model. Intermediate signal tap points in the Simulink model are compared with MATLAB to
verify the numerical equivalence of the two implementations. To verify the functionality of the entire system, the transmitted bits are compared to the received bits.
Running ofdmGridConstruction.slx
Simulink received bits match transmitted bits.
HDL Code Generation and Implementation Results
To generate the HDL code for this example, you must have the HDL Coder™ product. Use the makehdl and makehdltb commands to generate HDL code and an HDL test bench for the ofdmGridConstruction/OFDM
Transmitter and ofdmGridConstruction/OFDM Receiver subsystems. The resulting HDL code was synthesized for a Xilinx® Zynq® UltraScale+ RFSoC ZCU111 evaluation board. The table shows the post place and
route resource utilization results. The design meets timing with a clock frequency of 245.76 MHz.
Resource utilization:
Resource OFDM Transmitter OFDM Receiver
_______________ ________________ _____________
Slice Registers 5637 5829
Slice LUTs 3352 3077
RAMB18 2 2
RAMB36 10 0
DSP48 12 12
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Post's Correspondence Problem - (Mathematical Logic) - Vocab, Definition, Explanations | Fiveable
Post's Correspondence Problem
from class:
Mathematical Logic
Post's Correspondence Problem (PCP) is a decision problem in computability theory that asks whether a given set of pairs of strings can be arranged to form the same string when concatenated. This
problem highlights the limits of algorithmic solutions and is known for being undecidable, meaning there is no general algorithm that can solve all instances of it. PCP serves as a classic example in
discussions about reductions and the Church-Turing thesis, demonstrating the boundaries of what can be computed.
congrats on reading the definition of Post's Correspondence Problem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. PCP was introduced by Emil Post in 1946 and is a significant example in the study of undecidable problems.
2. There is no algorithm that can solve all instances of PCP; however, specific instances can be solved with various methods.
3. The undecidability of PCP shows that not all mathematical questions have algorithmic solutions, reinforcing the limitations outlined by the Church-Turing thesis.
4. PCP can be reduced from other undecidable problems, highlighting its importance in the landscape of computability and complexity theory.
5. The problem also has implications in formal language theory and automata, influencing research in string matching and grammar generation.
Review Questions
• How does Post's Correspondence Problem illustrate the concept of undecidability in computation?
□ Post's Correspondence Problem serves as a prime example of an undecidable problem because there is no algorithm capable of solving every instance of it. This demonstrates the limits of
computational processes, where certain questions cannot be answered algorithmically. The implications of PCP extend to understanding the nature of mathematical problems that lie beyond
algorithmic reach, which is fundamental in theoretical computer science.
• In what way does Post's Correspondence Problem connect with reduction techniques in computability theory?
□ Post's Correspondence Problem is often used to demonstrate reduction techniques by showing how it can be transformed from or to other known undecidable problems. This showcases the
relationship between various problems within computational theory. By reducing one problem to another, researchers can determine the relative difficulty and properties of those problems,
highlighting PCPโ s significance as a benchmark in complexity and computability discussions.
• Evaluate the implications of Post's Correspondence Problem on the Church-Turing thesis and its relevance in modern computation.
□ The Church-Turing thesis posits that any computation performed by an algorithm can be modeled by a Turing machine. Post's Correspondence Problem challenges this thesis by exemplifying
problems that fall outside the realm of computability. The implications are profound; they suggest that there are limits to what we can achieve through computational means. As modern
computation advances, understanding these limits becomes increasingly important for both theoretical research and practical applications, such as artificial intelligence and algorithm design.
"Post's Correspondence Problem" also found in:
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Understanding the Monty Hall Problem
Understanding the Monty Hall Problem
I was recently listening to a podcast that was recommended by a blogger on this site, and during one episode discussing big-O theory, the…
“The Game Show Host Problem” from the movie 21 (2008). Photo: © 2008 Relativity Media
I was recently listening to a podcast that was recommended to me, and during one episode which discussed big-O theory, the Monty Hall problem was mentioned. The commentators talked a little about it,
at which point they admitted to not ‘getting it’ and moved on. This motivated me to share my understanding of the problem because I think that as engineers we should be the ones who are able to grasp
this concept and problems like it.
So here’s how I ‘got it’. It took some thinking, but in the end, it’s actually not too complicated.
In case you are unfamiliar with the problem, the Monty Hall problem is as follows:
• You are presented with 3 doors. Behind only one is a prize.
• You pick a door which you think the prize may be behind.
• After you pick the door, a 3rd party (ie the gameshow host) opens up a door which does NOT have the prize.
• You are then given the option to either stay with your initial guess or switch your guess to the door which the game show host did NOT open.
• Which should you do? Switch or stay? Does it matter?
The initial intuition for most people is that once you are shown the door with no prize, the possibility is now 50–50 between the two remaining doors, so it doesn’t matter if you switch or not. But
this is not the case. It is actually probabilistically in your best interest to switch to the other door. But let’s be good engineers and follow the scientific process with an initial hypothesis that
you have a 50% chance of winning either way.
We have our hypothesis, so let’s experiment.
Another way you could do it which can give you a little more info is to randomly switch between staying and switching your doors:
Ok, our hypothesis is wrong. Looks like we should probably switch doors. Time to refine my hypothesis and this time rely on more than just naive intuition. (Maybe you saw some new intuition above:
that the only way you get a stay_win is if you guessed right initially)
There are a few ways to look at the problem. First, let’s look at a split universe timeline:
• You have 3 doors and a prize behind one of them. So, for your initial pick, what is the probability that you correctly choose where the prize is?
• Let’s freeze time right here and do a little more analysis. There are 2 possibilities that just happened in your choice. So let’s split our universe.
• Universe 1: You fell within the 33.3% chance and chose correctly. Now, the game show host can open either of the remain doors, as neither of them has a prize. Now you have ANOTHER CHOICE (another
• // → Universe 1.1 You stay with your initial pick. (Human factor, let’s say 50%)
• // → // →You Win (100%)
• // →Universe 1.2 You switch. (50%)
• // → // →You Lose (100%)
• Universe 2 You initially choose wrong, which happens 66.7% of the time. Now, the game show host can ONLY open the door without the prize behind it (leaving the other door with the prize) SPLIT
Universe 2
• Universe 2.1 You stay with your initial pick. (50%)
• You Lose (100%)
• Universe 2.2 You switch. (50%)
• You Win(100%)
Analyzing mathematically: So what’s our problem statement? We want to know whether we win more if we switch or stay, right? So let’s calculate the probability of winning given that you stay with your
initial door, and then the probability of winning given that you switch doors. Note that we are giving the person a random 50% chance that they choose to switch or stay with their original door.
P(win | stay) = [P(win) * P(stay | win)] / P(stay)P(win) = (Universe 1 && Universe 1.1) || (Universe 2 && Universe 2.2) = (1/3) * (1/2) + (2/3) * (1/2) = 1/2P(stay | win) = Out of wins possibilities
only, what percentage of those did you stay for?
= (Universe 1 && Universe 1.1) / [(Universe 1 && Universre 1.1) || (Universe 2 && Universe 2.2)] = (1/3) * (1/2) / (1/2) = 1/3P(stay) = (Universe 1 && Universe 1.1) || (Universe 2 && Universe 2.1) =
(1/3) * (1/2) + (2/3) * (1/2) = 1/2P(win | stay) = [1/2 * 1/3] / 1/2 = 1/3
You could also say “Out of the times that I stay, how many times did I win?” That would give you the same answer:
P(win | stay): all wins which lie within stay universesP(win | stay) = (Universe 1 && Universe 1.1) / [(Universe 1 && Universe 1.1) || (Universe 2 && Universe 2.2)] = (1/3) * (1/2) / [(1/3) * (1/2) +
(2/3) * (1/2)] = (1/3) * (1/2) / (1/2) = 1/3P(win | stay) = 1/3
Now, let’s see the probability of winning given you switch. (While Baye’s doesn’t really buy us anything since we have a short timeline which describes everything for us, I’ll do it both ways for the
sake of thoroughness).
P(win | switch) = [P(win) * P(switch | win)] / P(switch)P(win) = (Universe 1 && Universe 1.1) || (Universe 2 && Universe 2.2) = (1/3) * (1/2) + (2/3) * (1/2) = 1/2P(switch | win) = Out of wins
possibilities only, what percentage of those did you switch for?
= (Universe 2 && Universe 2.2) / [(Universe 1 && Universre 1.1) || (Universe 2 && Universe 2.2)] = (2/3) * (1/2) / (1/2) = 2/3P(switch) = (Universe 1 && Universe 1.2) || (Universe 2 && Universe 2.2)
= (1/3) * (1/2) + (2/3) * (1/2) = 1/2P(win | switch) = [1/2 * 2/3] / 1/2 = 2/3
Or, you could say:
From our mathematical analysis, we can see that you are twice as likely to win if you switch than if you stay. You will also win more (2/3 of the time) if you switch every time than if you randomly
choose to switch or stay (in which case you win 1/2 of the time)
Analyzing with logic: If you choose to always switch, you will always be right if you initially chose the wrong door. So, since you initially choose wrong ( 1- 1/3 = ) 66.67% of the time, you will be
guaranteed to choose the correct door 66.67% of the time if you always switch. If you choose never to switch, you are guaranteed to be correct only if you initially picked the correct door out of the
3 on your first try, which was a 33.3% chance.
Expanding this problem out to get a better intuition: Sometimes when a problem seems hard to grasp, I like to take the numbers to extremes while maintaining the necessary relationships. In this case,
we know that after your choice of 1 door, the host will open all of the doors but one other (even though in our case with 3 doors, this is just him opening one door) AND these doors must have no
prize. So, using this relationship, let’s expand our situation to 1 million doors with one prize. Now after you choose a door, the game show host has to open all the rest but one. And he can’t open
the one with the prize. So, unless you think you chose right on your first try with a million to one odds, don’t you think this is a pretty good deal, as the host is practically forced to show you
where the prize is?! This is the exact same concept, except now our odds for getting the prize when we switch are 999,999/1,000,000 instead of 2/3. It’s much easier to have intuition on this concept
when it is expanded in this way.
Now, our hypothesis matches our experimental data, and we can call it a day. | {"url":"https://www.cantorsparadise.org/understanding-the-monty-hall-problem-e9aa24cc62ac/","timestamp":"2024-11-14T00:37:21Z","content_type":"text/html","content_length":"39790","record_id":"<urn:uuid:eb99c411-fcb3-4204-bc37-7b4accad273c>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00511.warc.gz"} |
Model Electron-positron Correlation Potential for Low-energy Positron-atom Elastic Scattering
Document Type
Dissertation - Restricted
Degree Name
Doctor of Philosophy (PhD)
First Advisor
David M. Schrader
Second Advisor
Kenneth D. Jordan
Third Advisor
Charles A. Wilkie
Fourth Advisor
Kazuo Nakamoto
The wave function for positron-atom scattering is approximated in the trial form of (UNFORMATTED TABLE FOLLOWS) (PSI)((')r(,e),(')r(,p)) = (psi)((')r(,e),(')r(,p))(psi)(r(,p)). (1)(TABLE ENDS) The
closed-channel function (psi)((')r(,e),(')r(,p)), a determinant of electronic function (phi)(,i), and the open-channel function (psi)((')r(,p)) can be found by applying the Hartree-Fock (HF)
variational principle, (delta) (INT) (PSI)H(PSI)d(tau) = 0, subject to orthonormality constraints on the (phi)(,i)'s leads to uncoupled Schrodinger equations for the electrons and the positron. The
positronic operators are replaced by a model potential V(,mp) in the Schrodinger equation for the electronic part, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where the Coulomb ((')J(,j))
and exchange ((')K(,j)) integrals are r(,p)-dependent. The proposed functional form of V(,mp) is (UNFORMATTED TABLE FOLLOWS) V(,mp)(r(,p),r(,(mu)p)) = 1 - ce(,n)('x)e('-x) (be('-ar(,(mu)p))/r(,(mu)
p)), x = r(,p)/r(,0), (3)(TABLE ENDS) where a, b, c, r(,0), and n are disposable parameters and (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The Schrodinger equation for the positron is a
potential scattering equation: (UNFORMATTED TABLE FOLLOWS) -1/2(DEL)(,p)('2) + Z/r(,p) + V(,ep)(r(,p)) - k('2)/2 (phi)(,p)((')k,(')r(,p)). (4)(TABLE ENDS) The effective electron-positron interaction
potential V(,ep) is obtained by subtracting the HF ground-state energy from the r(,p)-dependent electronic energy obtained by solving eq. (2). The parameters for V(,mp) are found such that the
calculated V(,ep) leads to the known scattering lengths for the H-e('+) and He-e('+) systems. The calculated results for cross sections and annihilation parameters for both test systems are
reasonably good as compared to other variational and approximate methods. The present results also suggest the model framework can be extended to other larger positron-atom systems.
Restricted Access Item | {"url":"https://epublications.marquette.edu/dissertations_mu/2940/","timestamp":"2024-11-07T21:07:37Z","content_type":"text/html","content_length":"39246","record_id":"<urn:uuid:9c3e8b3a-2033-4407-b477-8943db183d0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00283.warc.gz"} |
CoCrZ (Z= Al and Ga) my pub
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/260516872
Ab initio study of the structural and optoelectronic properties of the HalfHeusler CoCrZ (Z= Al and Ga)
Article in Canadian Journal of Physics · January 2014
DOI: 10.1139/cjp-2013-0474
9 authors, including:
Ghulam Murtaza
Rabah Khenata
Islamia College Peshawar
University Mustapha Stambouli of Mascara
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Canadian Journal of Physics
Ab initio study of the structural and optoelectronic
properties of the Half-Heusler CoCrZ (Z= Al and Ga)
Canadian Journal of Physics
Manuscript ID:
Manuscript Type:
Date Submitted by the Author:
Murtaza, Ghulam; Islamia College Peshawar, Physics
Khenata, R.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
Seddik, T.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
Missoum, A.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
Al-Douri, Y.; Institute of Nono Electronic Engineering, University Malaysia
Perlis, 01000 Kangar, Perlis, Malaysia, Perlis
Abdiche, A.; Engineering Physics Laboratory, Tiaret University, 14000Tiaret- Algeria, Physics
Meradji, H.; Laboratoire LPR, Département de Physique, Faculté des
Sciences, Université Badji Mokhtar, Annaba, Algeria., Physics
Baltache, H.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
Complete List of Authors:
Half Heusler, ab initio calculation, APW+lo, Elastic constants, Bandstrucutre
Canadian Journal of Physics
Ab initio study of the structural and optoelectronic properties of the Half-Heusler
CoCrZ (Z= Al and Ga)
A. Missoum , T. Seddik , G. Murtaza b, R. Khenata a, Yarub Al-Douric, A. Abdiche d, H.
Meradji e, H. Baltache a
Laboratoire de Physique Quantique et de Modélisation Mathématique, Université de
Mascara, 29000- Algeria
Modeling Laboratory, Department of Physics, Islamia College Peshawar, Pakistan
Institute of Nono Electronic Engineering, University Malaysia Perlis, 01000 Kangar, Perlis,
Engineering Physics Laboratory, Tiaret University, 14000- Tiaret- Algeria
Laboratoire LPR, Département de Physique, Faculté des Sciences, Université Badji
Mokhtar, Annaba, Algeria.
In order to study the structural, electronic and optical properties of the half-Heusler
CoCrZ (Z= Al and Ga), we have performed ab initio calculations using the full-potential with
the mixed basis (APW + lo) method within the generalized gradient approximation (GGA).
The structural properties as well as the band structures, total and atomic projected densities of
states are computed. From electronic band structures of the hypothetical CoCrGa compound
we have found semi metallic behavior just like CoCrAl. We also studied the evolution of
electronic structure of CoCrAl under external hydrostatic pressure. It is found that the pseudo
gap around Fermi level increases continuously with increasing pressure, while the electronic
density of states at the Fermi level does not change significantly. Furthermore, the optical
properties include the dielectric function and refractive index were evaluated and discussed
under pressure up to 20 GPa as well as electrical conductivity and electron energy loss were
calculated for radiations up to 30 eV. By the same way, we have studied the magnetic
properties of CoCrAl for lattice expansion up to ɑ =1.1ɑ0 where a transition from
paramagnetic phase to half-metallic phase is expected.
PACS: 71.15.Mb, 71.15.Ap, 71.20.Be,
Corresponding author: Tel.: +92 321 6582416
Electronic address: [email protected] (G. Murtaza); [email protected] (R. Khenata)
Canadian Journal of Physics
1. Introduction
Heusler compounds and their alloys have exceptional physical properties exhibiting
exotic useful features into many devices, as giant magneto resistance sensors (GMR), tunnel
junctions, spin valves (spintronic) [1, 2], narrow gap semiconductors, semi-metals with a
concentration of charge carriers which is adjustable [3, 4], magneto-resistive materials,
thermo-electric materials with a high degree of spin polarization, superconductors, semimetals
and topological insulators [5, 6]. For example, some Heusler materials are highly sought for
their large thermoelectric power because they have high electrical conductivity, a high
Seebeck coefficient and low thermal conductivity. Therefore, they can be used as sources of
clean energy in order to solve the problem of CO2 emissions [7-9].
Experimentally, the crystal structure and magnetic properties of CoCrAl alloys have
been explored and the lattice parameter was measured by Luo et al. [10] using the X-ray
powder diffraction. A small magnetic moment of 0.06 µB/unit cell was detected, which does
not agree with the non-magnetic state predicted by the Slater-Pauling rule. A similar result has
been found in full-Heusler alloy Fe2TiSn, which has 24 valence electrons and should be nonmagnetic. But due to the Fe–Ti disorder a Curie temperature and a spin moment of 0.26
µB/unit cell at 5 K is observed [11]. Theoretically, Luo et al. [12] have calculated the
electronic and magnetic properties for the Half-Heusler compounds XCrAl (X = Fe, Co, Ni),
using the FP-LAPW method with the local spin density approximation.
To our knowledge, it seems there is a lack of experimental and theoretical data reported
in literature on the structural, elastic and optical properties and their pressure behavior for the
interested compounds. Further, let us notice that CoCrGa remains hypothetical compound to
be studied in details. Hence, using the full potential augmented plane wave plus local orbital
method (FP-APW+lo) with the generalized gradient approximation (GGA) formalism for
exchange and correlation (Xc) effects, we have focused on the calculation of structural
properties, band structure and optical properties for the Half-Heusler CoCrZ (Z= Al and Ga)
compounds. This work is organized as follow: in section 2, we have indicated a brief review
Canadian Journal of Physics
of the computational techniques used in this study. Section 3 presents and discusses the results
of structural, electronic and optical properties of the CoCrAl and CoCrGa compounds. At the
end, we have summarized the main conclusions of our work in section 4.
2. Computational method
These first-principles calculations were performed using the full-potential augmented
plane wave with the mixed basis (APW+lo) method [13, 14] within the density functional
theory (DFT) [15], implemented in the WIEN2K code [16]. The exchange-correlation effects
are calculated by the generalized gradient approximation (GGA) [17]. To ensure the
convergence of energy eigenvalues, wave functions in the interstitial region were expanded in
plane waves with a cutoff Kmax = 8/RMT, where RMT is the smallest atomic muffin-tin sphere
radius and Kmax determines the upper value of K vector magnitude in the plane wave
expansion. RMT's values are chosen to be 2.2, 2.2, 2.0 and 2.1 atomic units (a.u.) for Co, Cr,
Al and Ga respectively. The valence wave functions inside the muffin-tin spheres are
expanded up to lmax =10. The Fourier coefficients of charge density was expanded up to
Gmax=14(a. u.)-1. The convergence of self-consistent calculations was performed so that the
total energy of the system is stable with an accuracy of 10-5 Ry and a deviation of charge
density less than 10-4 e. The integration over the Brillouin zone is performed up to 19 k-points
in the irreducible Brillouin zone, using the Monkhorst–Pack special k-points approach [18].
3. Results and discussion
3.1. Ground states properties
Half-Heusler compounds crystallize in the C1b structure corresponding to space group
F-43m (No.:216), which consists of four fcc sub lattices and have the chemical formula XYZ,
where X, Y and Z atoms are located at (1/4, 1/4, 1/4), (0, 0, 0) and (1/2, 1/2, 1/2),
respectively. Generally, it is established that X is taken as a high valent transition metal atom,
Y as a low valent transition metal atom and Z as sp atom [19].
Canadian Journal of Physics
The structural properties of the said Half-Heusler compounds are predicted by
optimizing the volume, i.e. minimizing the total energy of the unit cell with respect to the
variation in the unit cell volume. The variation of the total energy versus unit cell volume for
both nonmagnetic (nm) and spin polarized ferromagnetic (sp) phases for CoCrAl and CoCrGa
is displayed shown in Fig.1. We have found that the nonmagnetic (nm) is energetically
favored if we neglect the energy difference (-0.0015 eV) for CoCrAl compound. The
structural parameters like lattice constant, bulk modulus and the pressure derivative of bulk
modulus are determined by fitting the variation of the total energy versus volume of the
nonmagnetic to the Murnaghan’s equation of state [20]. They are listed in Table 1. Our
computed lattice constant is in very good agreement with other theoretical results and slightly
underestimated compared to the experimental one. To the best of our knowledge no reported
experimental or theoretical data on structural properties for CoCrAl are available.
3.2 Elastic properties and their related constant
Elastic constants of materials are essential for better understanding of their properties.
The elastic constants describe the response of a solid material to a small loading which causes
reversible deformations. Some basic mechanical properties can be derived from the elastic
constants such as the bulk modulus, Young’s modulus, shear modulus, Poisson’s ratio, which
play an important part in determining the strength of the materials. From a fundamental view
point, the elastic constants are related to various fundamental solid-state properties such as
interatomic potentials, equation of state, structural stability, phonon spectra and they are
linked thermodynamically to the specific heat, thermal expansion, Debye temperature,
melting point and Grüneisen parameter. In our case, these compounds have a cubic symmetry
hence only three independent elastic C11, C12 and C44 should be calculated. The elastic
constants C ij of the herein studied compounds are obtained by calculating the total energy as
a function of volume conserving strains following the Mehl method [21]. The calculated
elastic constants are listed in table 2. One can notice that the unidirectional elastic constant
Canadian Journal of Physics
C11 is much higher than the C44 indicating that these compounds present weaker resistance to
pure shear deformation compared to resistance to unidirectional compression. We can notice
that the calculated elastic constants of CoCrAl are not very different from those of CoCrGa.
To the best of our knowledge no reported experimental or theoretical data on the elastic
constants for CoCrAl and CoCrGa compounds. The existence of a crystal in a stable or
metastable state requires that the following conditions between their elastic constants must be
full filled [22]:
. Our results for elastic constants in
Table 2 satisfy these stability conditions meaning that the herein studied compounds are
elastically stable.
The bulk and shear moduli (B, G) of both compounds were calculated using the Voigt,
Reuss and Hill approaches [23-25]. For the cubic system G is expressed as follows:
The Young’s modulus E and Poisson’s ratio ν for an isotropic material are given by
Using the relations above the calculated shear modulus G, Young’s modulus E and Poisson
ratio ν for CoCrAl and CoCrGa are given in Table 2. Young's modulus is defined as the ratio
of uni-axial stress on the uni-axial deformation within the limits of Hook's law. If the value of
Young's modulus is high, the material is stiffer. The calculated values of Young's modulus
indicate that CoCrAl is stiffer than CoCrGa. Typical values of Poisson’s ratio are around to be
0.1 for covalent materials, 0.25 for ionic materials and around 0.3-0.45 for metals [26]. In the
present case, the value of Poisson’s ratio is 0.31 and 0.33 for CoCrAl and CoCrGa
Canadian Journal of Physics
respectively, indicating that the metallic bonding contribution to the atomic bond is dominant.
This result is in agreement with experimental results [10] and several theoretical studies on
the nature of bonds in the Heusler compounds [27-29] including our electronic structure study
in following section. We also have calculated the anisotropy factors for these compounds by
classical relation: A = (2C44 + C12)/C11. The calculated anisotropy factors are 0.97 and 0.99
for CoCrAl and CoCrGa, respectively. From these values, we can conclude that these
compounds are substantially isotropic. This is further confirmed by the fact that G ≈ C44. Up
to this date, no experimental or theoretical structural properties are available for comparison
with our theoretical results. In future, other experimental or theoretical work will be a good
test for our results.
The calculation of Young's modulus E, bulk modulus B and shear modulus G allowed
us to calculate the Debye temperature, which is a fundamental and very important parameter
for determining physical properties such as the electrical conductivity versus temperature
described by Bloch-Gruneisen formula, the specific heat, the melting temperature and the
elastic constants. To evaluate the Debye temperature (
) from the elastic constants, we use
the standard methods defined by the following classical relations [26]:
is the average sound velocity, h is the Plank’s constant,
is the Boltzmann’s
is the atomic volume and n is the number of atoms per unit cell.
In polycrystalline materials, the average sound velocity is determined by classical
are the longitudinal and transversal elastic wave velocities respectively. In
cubic systems which are assumed isotropic materials, one can calculate from the Navier’
relations [30]:
Canadian Journal of Physics
3.3. Electronic properties
We have performed calculations of band structures, along high symmetry directions (WL-Γ-X-W-K) of the Brillouin zone for CoCrZ (Z = Al, Ga) which are shown in Fig. 2. It is
clear that, close to the high symmetry points (Γ, X, W), the conduction band and valence band
across the Fermi level have a slight overlap, where their band structure shows a pocket of
electrons at the X point and a pocket of holes at the W point. CoCrAl and CoCrGa have a
small density of states of 0.4 and 0.15 state/eV.unit cell, respectively, at the Fermi energy in
the spectra as shown in Fig. 3. This result shows that the compounds CoCrZ (Z = Al, Ga) are
semi metallic. They have a symmetrical DOS for each spin polarization, majority (up) and
minority (down); therefore the magnetic moment is zero. Our half Heusler CoCrZ (Z = Al,
Ga) compounds, with 18 valence electrons (VEC) split equally into two spin polarization, so
they are distributed on the nine bands of lower energy levels. The density of states (DOS) in
spin polarized calculations shows that CoCrZ (Z = Al, Ga) are paramagnetic and semi
metallic, having a deep pseudo gap with approximately 0.4 eV of width and centered at the
Fermi level. As it is seen on total DOS curves, the total density of states decreases and
increases abruptly with sharp peaks around Fermi level energy, within which they have small
density of states. The calculated total and partial DOS for CoCrZ (Z=Al,Ga) are displayed in
Fig. 3, where we can see that they have the same shape for both compounds CoCrAl and
CoCrGa. We notice on Fig. 2 that, in the vicinity of the Fermi level (
), the dispersion of the
band structure of the two compounds CoCrZ (Z = Al, Ga) have the same shape, comparable to
Fe2VAl which is paramagnetic and semi metallic with a deep pseudo gap [31, 32]. Our results
agree well with those obtained by Luo et al [10, 12] and other general studies [33, 34]. Half
Heusler compounds with VEC = 18 are either semiconductors having a narrow gap or
semimetals (defined as materials with low number of electrons N(
) at the Fermi-level).
Canadian Journal of Physics
In order to elucidate the different contributions from the different components in the
materials to the conductivity, the total and projected DOS of the atomic contributions are
plotted in Fig. 3. From the partial DOS curves; we can identify the angular momentum
character of the different structures. The structure which is in the range -8.1 (-9.5) to -4.7 eV
(-6.2) eV for CoCrAl (CoCrGa) is mainly due to the combination of electronic states Co, Cr
and Al (Ga).
The valence region for CoCrAl (CoCrGa), which extends from -4.1 (-5) eV at the Fermi
level EF=0, is mainly due to the combination of Co-3d and Cr-3d states. This admixture is the
same for the conduction region which extends from the Fermi level EF = 0 to 4.1 (5) eV,
while atom Al (Ga) has a small contribution. On the other hand, for CoCrGa only, the valence
region shows a peak at the bottom, located at -14 eV (not represented in the range of Fig. 2).
The states Co-3d and Cr-3d are globally dispersed around the Fermi level; hence it is clear
that Cr-3d and Co-3d states are strongly hybridized near the Fermi level. For CoCrGa, it is
noticeable that Ga-3d has a contribution to the bottom of structure with a sharp peak. For the
two compounds, we can conclude that their different structures are mainly dominated by Co3d and Cr-3d in the neighbor of Fermi level.
As shown in Fig. 4 for CoCrAl compound, the width of the pseudo gap increases with
increasing pressure; however, up to 20 GPa it stays semi metallic, while the density of states
at the Fermi level shows a weak dependence of pressure. Our spin polarized calculations are
shown in Fig. 5, predict that for a=1.1 =6 , CoCrAl is a half semimetal ferromagnet with
in majority state (up) and a small gap in minority state (down) 0.1 eV, just like the
superconducting topological semi metal YPtBi [35]. These observations indicate that a
transition from semimetal to narrow gap semiconductor is possible. As noted by Coey [36], a
lattice parameter expansion leads to an increase of
and sometimes a change of magnetic
structure. We conclude that CoCrAl is in a transition region between a ferromagnetic
semimetal and half metallic ferromagnet, similar to that observed for the composite halfHeusler FeVSb [37].
Canadian Journal of Physics
3.4. Optical properties
The CoCrZ (Z = Al, Ga) compounds have a cubic symmetry; it is enough to compute
only one component of the dielectric tensor, which can completely determine the linear
optical properties. We denote the dielectric function by
the frequency,
, where ω is
is its imaginary part which is given by the relation [38]:
where the integral is taken over the first Brillouin zone. The momentum dipole elements
are the matrix elements for direct transitions between the valence
and the conduction band state
, A is the potential vector defining the
electromagnetic field, and the energy ћ
is the corresponding transition
of the dielectric function can be deduced from the imaginary part
energy. The real part
using the Kramers-Kronig relation:
where P is the principal value of the integral. Once the real and imaginary parts of the
dielectric function are determined, we can calculate important functions such as optical
refractive index n( ) and electrical conductivity
, which we can find in literature of
physical optics [39]:
with relations (9) and (10):
For the calculations of optical properties, we require a dense mesh of k-points,
uniformly distributed in the k-space. Thus, the Brillouin zone integration was performed up to
506 k-points in the irreducible part of the Brillouin zone. The difference is insignificant by
increasing the number of k-points beyond 506. In this work, we presented calculations with
only 506 k-points and broadening is taken to be 0.03 eV. Figure 7 shows the curves of real
Canadian Journal of Physics
and imaginary part of dielectric function for a radiation spectrum up to 30 eV. As shown, the
optical spectra of CoCrAl and CoCrGa have some similarities. For each compound, the
dielectric function
has two structures as shown in Fig. 7. Optical transition occurs at
0.1 and 0.15 eV for CoCrAl and CoCrGa, respectively. Beyond these pseudogaps, curve
increases abruptly. The main peak in the spectra is located inside infrared range at 0.9 eV and
1.1 eV for CoCrAl and CoCrGa, respectively, followed by another structure located at 2.8 eV
and 3 eV, respectively. To interpret the optical spectrum, it is necessary to give several
transitions allocated to peaks which are present in the curve of reflection spectrum in Fig. 8,
where many transitions appear at the associated energy. For both compounds, the main peak
for dielectric functions
is mainly due to the transitions within the Cr-3d bands.
is convoluted from
The real dielectric function
conversion. The negative values of
by the Kramers–Kronig
are in the near infrared from 0.7 to 1.3 eV,
indicating that the two crystals CoCrZ (Z = Al and Ga) have a Drude behavior. The high
reflectivity for energies less than 1 eV characterizes a high conductivity in the infrared region,
with a narrow peak at 0.9 eV just after the pseudogap which can be considered as a threshold
(Fig. 8, 9).
loss function
shows a sharp drop while the energy
shows a main peak (Fig. 11) and the real part of the dielectric
(Fig.7) vanishes at
In successive Fig’s (7, 8, 11), the reflectivity R
, located at 24.5eV and at 24.0 eV for CoCrAl and
CoCrGa, respectively, corresponding to the energy of the incident photon, at which a material
becomes transparent for frequencies upper than the screened plasma frequency
. If we
assume that there is no photonic contribution, the static dielectric constants obtained at the
zero frequency limits
are 90.4 and 70 for CoCrAl and CoCrGa, respectively.
The dispersion curves of refractive index for CoCrAl and CoCrGa (Fig. 10) show that
both compounds have the same features. The refractive index shows a maximum value of 9.6
and 8.4 at 0.1 and 0.7 eV for CoCrAl and CoCrGa, respectively, followed by a secondary
Canadian Journal of Physics
peak at 2.6 and 3 eV, respectively with Drude-like behavior. These results indicate two
possible interbands transitions in agreement with the calculated results.
Figure 6 shows the variations of the static refractive index as a function of the pressure
for these compounds. By increasing the pressure up to 20 GPa, there is a linear decrease of the
static refractive index. By linear fitting, we can determine the pressure derivative of the
refractive index n of these compounds and deducing the pressure coefficient
which has value
for CoCrAl and CoCrGa,
respectively. To our knowledge, there is no theoretical or experimental data available on the
optical properties of these compounds, except for the experimental lattice parameter, the
magnetic and electronic properties of CoCrAl which were studied by Luo et al. [10,12].
4. Conclusion
In this work, we have carried out a detailed investigation on the structural, elastic, electronic
and optical properties of the Half-Heusler compounds CoCrZ (Z = Al, Ga) by using the APW
+ lo method within the generalized gradient approximation (GGA). We summarize the most
important discussions:
(i) We have determined the ground state properties, including lattice parameter, bulk modulus
and its pressure derivative. For CoCrAl, the available data are consistent with the calculated
ground state and the related lattice parameter.
(ii) For the considered compounds CoCrZ (Z= Al, Ga), the elastic constants
, Young’s
modulus E, shear modulus G, Poisson’s ratio , sound velocity and the Debye temperature are
computed. The computed values of Poisson’s ratio indicate that these compounds have a
metallic-like bonding.
(iv) Calculated band structure and DOS show that these compounds are semimetal with deep
pseudogap close to the high symmetry points (Γ, X, W), with a width approximately equal to
0.2 eV for CoCrAl and to 0.1 eV for CoCrGa. As well, in both compounds, pseudogap width
increases with increasing pressure.
Canadian Journal of Physics
(v) Calculations of the electronic band structure, for an expansion of the lattice parameter of
the compound up to
show that CoCrAl is in a transition region, between a semimetal
and a half semi metallic ferromagnet.
(vi) We have analyzed the spectral curves of the complex dielectric function, the reflectivity
and the optical conductivity to identify possible optical transitions. Also, we have determined
the pressure derivative of the refractive index n of these compounds and deducing their
pressure coefficient.
(ix) The high reflectivity of the studied compounds in low energy region, make them useful
candidates for a coating to avoid solar heating.
To the best of our knowledge, there are only two partial studies of these compounds,
mentioned above. Therefore, our study on the structural and optical properties of these
compounds have not been measured or calculated yet, so it is assumed to be the first
theoretical prediction of these properties, which awaits experimental confirmation. Hopefully,
this work and its results can be considered as foresight study and stimulating matter for future
work on these materials.
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Table Captions
Table 1: The calculated lattice constant ܽ (in ), bulk modulus
and energy difference
(in GPa) and its pressure
between the spin-polarized (sp) and
nonmagnetic (nm) state at equilibrium lattice constant for CoCrZ (Z=Al, Ga) compounds.
Expt. [13]
Other calc. [14]
(in GPa), shear modulus G (in GPa), Young’s
modulus E (in GPa) and Poisson’s ratios
Table 2: Calculated elastic constants
for CoCrAl and CoCrGa at the equilibrium
262.64 117.44 70.34
268.35 126.58 70.40
, longitudinal, transverse and average sound
Table 3: Calculated density (
velocity (
in m/s) from the isotropic elastic moduli, and Debye temperature (
for CoCrAl and CoCrGa compounds.
in K)
Canadian Journal of Physics
Figure Captions
Fig.1: Total energy of studied compounds as a function of lattice volume of CoCrAl (a) and
CoCrGa (b) for both phases, non-magnetic (nm) and ferromagnetic (sp).
Fig.2: Calculated band structure of CoCrAl (a) and CoCrGa (b).
Fig.3: Total and partial densities of states of CoCrAl (a) and CoCrGa (b).
Fig.4: Evolution of total density of states (DOS) of CoCrAl under pressure
Fig.5: Total density of states (DOS) for the CoCrAl compound for two different values of the
lattice constant. Positive values of the DOS correspond to the majority (spin-up) electrons and
negative values to the minority (spin-down) electrons. The zero of the energy has been chosen
at Fermi energy.
Fig.6: Pressure dependence of static refractive index
for CoCrAl and CoCrGa.
Fig. 7: Calculated real and imaginary parts of the dielectric function CoCrAl (a) and CoCrGa
Fig.8: Calculated reflectivity R(ω) spectra of CoCrAl and CoCrGa.
Fig.9: Calculated real part of photoconductivity spectra of CoCrAl and CoCrGa.
Fig.10: Calculated refractive index n(ω) spectra of CoCrAl and CoCrGa.
Fig.11: Calculated electron energy loss L(ω) spectra of CoCrAl and CoCrGa.
Canadian Journal of Physics
Canadian Journal of Physics
Fig. 2
Canadian Journal of Physics
Canadian Journal of Physics
Fig. 4
Canadian Journal of Physics
Canadian Journal of Physics
Fig. 7
Canadian Journal of Physics
Canadian Journal of Physics
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What number is MMCXXXI? - The Roman numeral MMCXXXI as normal numbers
What number is MMCXXXI?
Your question is: What numbers are the Roman numerals MMCXXXI? Learn how to convert the Roman numerals MMCXXXI into the correct translation of normal numbers.
The Roman numerals MMCXXXI are identical to the number 2131.
MMCXXXI = 2131
How do you convert MMCXXXI into normal numbers?
In order to convert MMCXXXI into numbers, the number of position values (ones, tens, hundreds, thousands) is subdivided as follows:
Place value Number Roman numbers
Conversion 2000 + 100 + 30 + 1 MM + C + XXX + I
Thousands 2000 MM
Hundreds 100 C
Dozens 30 XXX
Ones 1 I
How do you write MMCXXXI in numbers?
To correctly write MMCXXXI as normal numbers, combine the converted Roman numbers. The highest numbers must always be in front of the lowest numbers to get the correct translation, as in the table
2000+100+30+1 = (MMCXXXI) = 2131
The next Roman numerals = MMCXXXII
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The Solace of Numbers: The Math Center | College of Science
The Math Center
May 20, 2024
Above: Director Lisa Penfold sitting among the activity of the Math Center.
Today, things are exponentially better, and not just because of the expanded space, with its beautifully rendered sandstone exterior accented with concrete and glass and illuminated naturally through
skylights, rising into an elevated garden patio above. The $1.8 million T. Benny Rushing Mathematics Student Center, where the Math Center is housed, currently speaks to the collaborative, integrated
space—both physical and mental—that offers what Gardiner calls a “continual review of mathematics” targeting University of Utah undergraduates enrolled in math classes.
That continuing review is robust as well—on-demand for students who at times just drop in with a quick query or for others who actually use the space as a study hall, raising different-colored felt
flags (depending on what area of math they need help with) at their work stations when they need one-on-one attention.
Some of the tutors are graduate students, required, as part of their teaching requirement to work a minimum of one hour per week. There is also a tutor cohort which is more of a fixture, logging many
weekly hours. All of these skilled mentors are always at-the-ready, willing and fluidly collaborative with their colleagues who may be more conversant with concepts than they are—whether it's
quadratic equations, calculus, or trigonometry. There’s even a dedicated private space for Foundations of Analysis work where students are mentored through “proofs,” a deductive argument for a
mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
Discussion is a premium, says Lisa Penfold who took over the direction of the Center in 2018. "If a student discusses the problem with a tutor within 24 hours of a lecture,” she says, “understanding
and retention increases” markedly. For the tutors, the benefits are mutual as they benefit in their preparation for the Graduate Record Exam, required for graduate school, by working as a tutor in
the Math Center.
While designed first to be pragmatic (students need to pass math classes for their generals, but also for their STEM-related majors) the Center inspires students to move beyond the Steps (1,2,3…) to
solving an assigned problem to really understand what the meaning is of what they’re looking at. It’s this undergirding mission that has made the Math Center at the U the powerhouse that it is.
Tutors, who are skilled at deciphering the students' level of understanding, respond to students in real time.
Helping students to pass their math classes and to better understand math meanings can sometimes be complicated by the all too familiar “math anxiety.” Refrains like "I'll never be able to do math"
or "math doesn't even walk in our family" are all too common for some who arrive at the U. The Math Center, with its welcoming atmosphere, private group meeting rooms, computer lab and a break room/
kitchen has become a sanctuary for students struggling with the complexities of numbers, equations, and formulas. (It would seem the only thing they might be missing is a climbing wall.)
"Our objective is to make sure students feel welcome here and get the help they need," Penfold states. "We know there's a lot of baggage with mathematics."
Addressing that baggage is one of the Center's core missions. Through extraordinary patience and individualized teaching approaches, tutors work tirelessly to dismantle math anxiety. "We start by
acknowledging it's valid," says tutor Caleb Albers, a PhD candidate in applied mathematics. "Then we can begin chipping away at those negative associations.”
“I find that helping students get more of a conceptual understanding … helps a lot,” says Stella Brower, a veteran tutor pursuing a master's in statistics. “I have experience tutoring many different
types of learners. There are those that thrive working alongside you as you go through a problem, some that want a tutor to check their work, and then there are those that need a bit of guidance or a
refresher on a concept.”
Albers echoes this approach. “The most important thing is to help them ‘learn how to learn’ math for themselves, instead of just showing them how to do one problem.”
The tutoring strategies are as diverse as the students themselves, and “meeting the student where they are,” is the standard operating procedure of the tutoring team. The objective of every
interaction is to bring a deep commitment to unlocking each student's individual potential. Penfold encourages a culture of tutors commingling across disciplines, asking each other for support on
esoteric concepts. Students from calculus and introductory courses are seen clustering together, facilitating an enervating—even fun— cross-pollination of ideas.
"You'll see six to eight students sitting together,” she says, “some in calculus helping those in an entry-level course.”
The pandemic accelerated the Math Center’s evolution, prompting an online tutoring option that continues facilitating virtual support Monday through Saturday. As the U continues to grow its
enrollment, Penfold insists on preserving the Center's uniquely personalized, student-centric approach. Penfold is often stepping onto the floor herself when every seat is taken and helping a student
even as they are currently almost doubling the size of the tutoring area and creating satellite centers like the one that recently opened in the Sutton Building to accommodate the number of students
using the center.
“If a tutor has a question or encounters something shaky, they’ll ask another tutor. We have grad students, as well as undergrads— everyone talking about mathematics and working together.” Asked what
she wants students to know about her facility, she immediately responds that she hopes they find out about the Center and that they then use it as frequently as needed.
It’s a tribute to the hard work and dedication of the Math Center team from the early days in old barracks to today’s open and accommodating facility with both online and in-person options, “This is
probably the most difficult challenge faced as a tutor," says Brower. "The ability to change up your teaching methods on the fly is very important when helping a student that just isn’t quite getting
The Math Center helps make that happen every day.
by David Pace
This is the feature story of the latest edition of Aftermath, magazine for the Department of Mathematics. | {"url":"https://science.utah.edu/mathematics/the-solace-of-numbers-the-math-center/","timestamp":"2024-11-02T22:05:23Z","content_type":"text/html","content_length":"69902","record_id":"<urn:uuid:c2e13174-977d-4682-8b2f-878355a26699>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00548.warc.gz"} |
Metaphors and Mathematics 4 - International Maths Challenge
Metaphors and Mathematics 4
If mathematics is a game, then playing some game is doing mathematics, and in that case why isn’t dancing mathematics too?
Ludwig Wittgenstein – Remarks on the Foundations of Mathematics
Mathematics is often described metaphorically – the forms that these metaphors take include the organic, mechanical, classical, and post-modern, among countless others. Within these metaphors,
mathematics may be a tool, or set of tools, a tree, part of a tree, a vine, a game, or set of games, and mathematicians in turn may be machines, game-players, artists, inventors, or explorers.
Despite the many metaphors used to describe mathematics, in popular discourse mathematics is often reduced to one of its parts, being metonymically described as merely about numbers, formulas, or
some other limited aspect. Metaphor is a more complete substitution of ideas than metonymy – allowing us to link concepts that do not appear to have any direct relationship. Perhaps, metaphoric
language that elevates and expands our ideas about mathematics is used by enthusiasts to counter the more limited and diminishing metonymic descriptions that are often encountered.
Attempts to describe and elevate mathematics through metaphor seem to fall short, however. Our usual way of thinking about things is to inquire about their meaning – a meaning that is assumed to lie
beneath or beyond mere appearances. Metaphor generally relies on making connections between concepts on this deeper level. The sheer formalism of mathematics frustrates this usual way of thinking,
and leaves us grasping for a meaning that is constantly evasive. The sheer number and variety of the many metaphors for mathematics suggests that no single convincing one has yet been found. It may
be that the repeated attempts to find such a unifying metaphor represents an ongoing and forever failing attempt to grapple with the purely formal character of mathematics; and it may be that the
formal nature of mathematics will always shake off any metaphor that attempts to tie it down.
For more such insights, log into www.international-maths-challenge.com.
*Credit for article given to dan.mackinnon* | {"url":"https://international-maths-challenge.com/metaphors-and-mathematics-4/","timestamp":"2024-11-08T14:16:44Z","content_type":"text/html","content_length":"145201","record_id":"<urn:uuid:d8a1de64-52fc-41f5-b2c6-ef3753467ac5>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00300.warc.gz"} |