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The following web page shows some of these formulas, and also supplies the javascript code for programming a calculator for them into a computer. The following Online PDF Worksheet has a variety of questions with answers provided at the end of the sheet. • For the students to complete the Activity worksheet and the assessment it will take 2 class periods. Ex: (60 , −2); 53 units-2-Create your own worksheets like this one with Infinite Geometry. The following video from Spiro at Vivid Maths shows how to do a typical Distance Formula question. Grade 3 Measurement Worksheet Circle the proper unit for each of the following. (3, 8) and (7,3) Page 4 of 4: For help and guidance see these examples of how to find the distance between two sets of coordinates. You can use this on screen calculator below, right here on this lesson page: If the on screen calculator does not work then use: The following web page for this easy to use Distance Calculator: http://easycalculation.com/analytical/distance.php. Finding the distance between two points on a coordinate plane. By Pythagoras’ theorem, ... With x or y coordinate of an endpoint being replaced with a variable, this pdf distance formula worksheet focuses on rearranging the formula to make the unknown variable the subject, and equate it with the given distance to find the missing coordinate. For example, if AB = BC, then we write AB BC#, which is read “segment AB is congruent to segment BC.” ( , )xy 11 ( , )xy 22 Distance Formula The distance between any two points with coordinates and is given by the formula: 22 D x x y y ( ) ( ) 2 1 2 1 While we continue to grow our extensive math worksheet library, you can get all editable worksheets available now and in the future. 2 Worksheet by Kuta Software LLC Kuta Software - Infinite Pre-Algebra Name_____ The Distance … To practice this skill independently, I will give my students two worksheets that, together, will contain ten problems to solve, that will ask students to find the distance between two points on the coordinate plane among other questions pertaining to the coordinate plane. http://www.movable-type.co.uk/scripts/latlong.html, The Cartesian Plane However Pilots and Mariners still need to know the maths involved with these calculations, in case the computer or GPS ever breaks down! https://www.facebook.com/PassysWorldOfMathematics. This problem set is shorter than most of my Partner Practice problem sets. Plotting a Linear Graph using a Rule Equation The mall activity has a scale of 1 unit = 8 feet and the around town activity has a scale of 1 unit = 1/2 mil One payment, lifetime access. 3. Help us to maintain this free service and keep it growing. Many answers. Exclusive, limited time offer! Find the distance between the points $$\left( { - 8,6} \right)$$ and $$\left( { - 5, - 4} \right)$$. Distance between two continents Distance of a marathon race Length of a baby’s feet inches / yard / miles feet / yards / miles inches / feet / yards Length of a dolphin Width of a towel Length of a sailboat Here is another worksheet on Distance Between Points which also has answers at the end of it. We provide high-quality math worksheets for more than 10 million teachers and homeschoolers every year. Distance between two continents Distance of a marathon race Length of a baby’s feet cm / m / km cm / m / km cm / m / km Length of a dolphin Width of a towel Length of a sailboat cm / m / km cm / m / km cm / m / km Size of a frying pan Length of golf course Plot the two points with the coordinates shown below. F n WMsaVdOeU IwWiyt Ehp BI9n rfwiJn1i Ntde 3 fG 3e io XmueOtir Byd. In this lesson, we show how to work out the Distance between Two Points on the Cartesian Grid, using Pythagoras Theorem. However, if we identify the x1,y1 and x2,y2 coordinate values of our points and use these in the Distance Formula, we do not have to draw any graphs at all. These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”. Next Data Handling Cycle Practice Questions. Previous 3D Coordinates Practice Questions. The Activity worksheet is to help students connect midpoint and distance to the real world by using an application problem. The y-coordinates are the same, so the line is horizontal. Distance Formula Worksheet Name _____ Hour _____ 1-3 Distance Formula Day 1 Worksheet CONSTRUCTIONS Directions for constructing a perpendicular bisector of a segment. We can measure the “Across” distance and the “Up” distance by counting squares. The coordinates of point P are ( 1 , 5 ). Free trial available at KutaSoftware.com Our printable distance formula worksheets are a must-have resource to equip grade 8 and high school students with the essential practice tools to find the distance between two points. Real World Straight Line Graphs I We will do some examples on how to find Distance using the subtracting coordinates Formula later in this lesson, but first we recommend watching the following videos to consolidate the material we have covered so far in this lesson. Draw a line to join them and form a right triangle with this line as the hypotenuse. Coordinate Geometry 8 200 Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:01 The distance between two points 8:02 The midpoint of an interval 8:03 The gradient of a line 8:04 Graphing straight lines 8:05 The gradient–intercept form of a straight line: y = mx + c Investigation: What does y = mx + c tell us? Looking for high-quality Math worksheets aligned to Common Core standards for Grades K-8? You can then receive notifications of new pages directly to your email address. these examples of how to find the distance between two sets of coordinates, Calculating the Distance Between Two Points, Comparing Decimals up to Hundredths Worksheets, Finding the Absolute Value of Numbers Worksheets, Understanding Basic Money Denominations Worksheets, Number Lines and Coordinate Planes Worksheets, Interpreting Simple Linear Regression Worksheets, Applying Percentage, Base, and Rate Worksheets, Understanding Properties and Hierarchy of Shapes Worksheets, Interpreting Fractions as Division Worksheets, Understanding Positive and Negative Integers Worksheets, Finding Common Factors Between Two Whole Numbers Within 100 Worksheets, Creating Line Plots with Fractions Worksheets, Graphing Points on the Coordinate Plane Worksheets, Understanding Congruence and Similarity of 2D Figures Worksheets, Solving Word Problems Involving Coordinate Plane Worksheets, Conducting Hypothesis Testing Using Chi – Square Test Worksheets, Constructing and Interpreting Scatter Plots for Bivariate Measurement Worksheets, Solving Volume of Solid Figures Worksheets, Classifying Shapes by Lines and Angles Worksheets, Understanding Basic Statistical Terms and Sampling Techniques Worksheets, Understanding Ratio between Two Quantities Worksheets, Converting Like Measurement Units Worksheets, Understanding Number and Shape Patterns Worksheets, Multiplying and Dividing Fractions Worksheets, Understanding the properties of rotations, reflections, and translations of 2D figures Worksheets, Solving word problems associated with fractions Worksheets, Multiplying Mixed Numbers by Fractions Worksheets, Dividing Mixed Numbers by Fractions Worksheets, Defining and Non-defining Attributes of Shapes Worksheets. Each day Passy’s World provides hundreds of people with mathematics lessons free of charge. Use the coordinates to find the lengths of the short sides of the triangle. We can apply “Pythagoras Theorem” to the right angled triangle to work out what the distance AB is between the two points. We add 100+ K-8, common core aligned worksheets every month. The following Online PDF Worksheet has a variety of questions with answers provided at the end of the sheet. Plot the two points with the coordinates shown below. Problem 3 : Find the distance between the points A and B in the given graph given below. Let’s work through an example to show how this is done. For the Pythagoras method, we need to spend a lot of time manually drawing our points onto a Cartesian grid and forming Right Triangles around them. Image Source: http://northrofneutral.wordpress.com. Note that it is not necessary to draw the points on an X-Y Grid, if the Point Coordinates are given in the question: eg. Free worksheet (pdf) on distance formula includes model problems, practice problems and an online component. Calculate the gradient by putting the change in height over the change in horizontal distance. Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools. The warehouse is located at N(50, 10) and t he store is located at P(50, -30) and . Step 1 : Identify the important information. ©9 P2y0c1 A2w nKXuItwaA yS ToAfUtmwEa6rNew vLaLzC 6.f P gAylzle nrVilg Mhkt is1 OrxePsBekr ivDeWdL.X w 4M oa Ad5e D 5w 9i Ntbhx 2I6n Gfgi hn 1iytTeb 0ParCek-PAmlgbeSbJr uab. The Distance from the Car to the next Turn. Pythagoras’ theorem is used to calculate the distance between two points when the line interval between them is neither vertical nor horizontal. This worksheet and attached quiz will help you to figure out what you know about the distance between two points. 5-a … Because the Earth is not flat, and is a curved surface, calculating the distance between two GPS Coordinates uses more complex mathematical formulas involving spherical trigonometry. Note: When two segments have the same length, they are said to be congruent. Do not forget to check for a rising or falling line. The formula above is known as the distance formula. It does not matter which way around they are labelled, as the formula will give the same answer for both scenarios. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Let’s look at an example. Calculate the distance (d) between the two points (length of the hypotenuse) using the Pythagorean Theorem. You can use any of these methods to find distance. We now substitute our subtracted coordinates expressions: XB – XA and YB – YA into the Pythagoras Theorem, and thereby create an Algebra Formula for the Distance between any two points labelled “A” and “B”. The distance between the points A(1, 2) and B(4, 6) is calculated below. There will be no processing fee charged to you by this action, as PayPal deducts a fee from your donation before it reaches Passy’s World. GCSE Revision Cards. Draw a line to join them and form a right triangle with this line as the hypotenuse. Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome: Understand and apply the Pythagorean Theorem. After 8-10 minutes of work time, we discuss all of the problems as a class. Example: Find the distance between the points (5, … Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Click here for Answers . While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed. ... About This Quiz & Worksheet. In the Real World, people do not calculate Distance manually like we have done, they use a Calculator App to do it. Grade 3 Measurement Worksheet Circle the proper unit for each of the following. Find the distance between the ends of the two equal sides. Calculate the distance (d) between the two points (length of the hypotenuse) using the Pythagorean Theorem. We love hearing from our Users. Here is a blank X-Y Grid you can print out and use for plotting question Points, and working out distances using Pythagoras, or the Distance Formula. Email us at the hotmail address shown below with any comments, ideas for articles, or to report any broken links or blank images on our pages. Distance between Two Points 11 Answers 1. See if you can fill in the blanks and get to the correct answer. Step 2 : The line which connects N and P is parallel to y-axis. ------ Note: The Information above this point will not be sent to your printer --------. Click here for Worksheet 1 . Thank you! A truck drives from N to P at a speed of 50 miles per hour. In this worksheet, we will practice finding the distance between two points on the coordinate plane using the Pythagorean theorem. Eg. R l 5A fl lF rHicg Oh4tesv 6rxe os Pe Fr Bv6eIdZ. ... What is the distance between points C(-2, 3) and D(0, 5)? The assessment sheet is for the teacher to assess their understanding of the midpoint and distance. Real World Straight Line Graphs II. Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button. Our premium worksheet bundles contain 10 activities and answer key to challenge your students and help them understand each and every topic within their grade level. Q1: Find the distance between the point ( − 2 , 4 ) and the point of origin. Use the coordinates to find the lengths of the short sides of the triangle. 8:06 The equation of a straight line, given point Finally if you need any extra practice then try out the online test on the following web page: http://www.onlinemathlearning.com/distance-formula-worksheet.html. This creates a right triangle on our grid, as shown below. These calculations are done by computer apps in the real world. The following video shows how to find the distance between points using the Pythagoras Theorem. If you enjoyed this lesson, why not get a free subscription to our website. 2 Worksheet by Kuta Software LLC Name_____ Date_____ Period____ ©V i2M0Q1w4j 7K2uMtpaT 3S7o8f PtJw0aer weK 8LOL tCI. Why might another student think distance can be negative? Captcha if it is the midpoint worksheet you can use the distance from the midpoint that robots are given two points with varying tasks on a midpoint Example to continue to determine how to use any two points with the other endpoint of the social bar. Gain an edge over your peers by memorizing the distance formula d = √((x 2 - x 1) 2 + (y 2 - y 1) 2). Parts into the two numbers is found by having a point. We can use it to find the distance d between any two points in the plane. It relates to the 6th grade common core standard 6.NS.8. Rather than have to draw Pythagoras Trianges onto our graph of the two points, we can develop an Algebra Formula to save us a lot of time and work. Plotting Graphs from Horizontal Values Tables, Plotting a Linear Graph using a Rule Equation, Mountain Gradients | Passy's World of Mathematics, Gradient and Slope | Passy's World of Mathematics, Midpoint Between Two Points | Passy's World of Mathematics, Rocket Science and Cars | Passy's World of Mathematics, Gradient Slope Formula | Passy's World of Mathematics, X and Y Intercepts | Passy's World of Mathematics. The Vietnam Veterans' Memorial in Washington, D.C., is in the shape of an unenclosed isosceles triangle (that is, V-shaped) with equal sides of length 246.75 feet and the angle between these sides measuring 125o 12'. Just pick about 10 questions to do from it that cover a variety of positive and negative coordinates. PayPal does accept Credit Cards, but you will have to supply an email address and password so that PayPal can create a PayPal account for you to process the transaction through. You can then receive notifications of new pages directly to your email address. The distance between the two points is the horizontal distance between the x-coordinates, 25 and 21. http://math.about.com Name: _____ Worksheet # 4 Determine the Distance Between Each Pair of Points 1.) (4, −2), (7, 2) 2.) If you're seeing this message, it means we're having trouble loading external resources on our website. Here is the fully worked solution for Example 2: Note that we have labelled the Coordinates (x2,y2) rather than (xB,yB), as “B” is usually called “Point 2”, and “A” is known as “Point 1”. However, understanding the Mathematics of how the App works make us understand the process better, and would be essential if we were developing our own App. distance, 2 points, pythagoras. (1, 5), (6, 6) 3.) 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# Domain Decomposition in space-time of 4D-VAR Data Assimilation problem: a case study on the ROMS software

Domain Decomposition of 4D-VAR Data Assimilation (DD-4DVAR) is made up of decomposition of the spate-time domain, solution of reduced forecast model and minimization of local 4D-VAR functionals. Relying on the existing software implementation of ROMS software, we describe main components of DD-4D VAR DA method, highlighting the topics that we will should address both on the mathematical problem underlying ROMS and the MPI-based code implementation of the ROMS-IS4DVAR formulation.

## Authors

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## 1 Roms

Regional Ocean Modeling System (ROMS) is an open-source, mature numerical framework used by both the scientific and operational communities to study ocean dynamics over 3D spatial domain and time interval.

ROMS supports different 4D-Var data assimilation (DA) methodologies such as incremental strong constraint 4D-Var (IS4D-Var) and dual formulation 4D-Var using restricted B-preconditioned Lanczos formulation of the conjugate gradient method (RBL4DVAR) [8]

. IS4D-Var and RBL4DVAR search best circulation estimate in space spanned by control vector and observations, respectively. IS4D-Var and RBL4DVAR algorithm consist of two nested loop, the outer-loop involves the module (1), namely nonlinear ROMS (NLROM) solving ROMS equations, the inner-loop involves modules (2)-(3), namely tangent linear approximation of ROMS (TLROMS) and adjoint model of ROMS (ADROMS); TLROMS and ADROMS are used for minimizing 4D-Var functional

[3] (see Figures 1-2).
NLROMS is a three-dimensional, free-surface, terrain-following ocean model that solves the Reynolds-averaged Navier-Stokes equations using the hydrostatic vertical momentum balance and Boussinesq approximation.
NLROMS computes

 xROMS(tl)=Ml−1,l(x(tl−1),f(tl),b(tl)) (1)

with the state-vector , temperature , salinity , components of vector velocity , sea surface displacement . represents nonlinear ROMS acting on , and subject to forcing , and boundary conditions during the time interval .

Minimization of the 4D-Var functional:

 JROMS(δz)=12δzB−1δz+12(Gδz−d)TR−1(Gδz−d) (2)

where are the control variable increments, is vector of innovations, , where is the observation matrix; is observation error covariance matrix and is covariance matrix of model error, is computed in the inner-loop in Figure 1.
Analysis increment, , that minimizes 4D-Var function in (2) corresponds to the solution of the equation , and is given by:

 δza=(B−1+GTR−1G)−1GTR−1d (3)

or, equivalently

 δza=BGT(GBGT+R)−1d. (4)

Equation (3) is referred to as the dual form (RBL4DVAR), while (4) is referred to as the primal form (IS4DVAR). In particular, we define

 K=BGT(GBGT+R)−1 (5)

as Kalman gain matrix.

TLROMS computes

 δxROMS(tl)≃Ml−1,lu(tl−1) (6)

where , , , and , , are the background of the circulation, surface forcing and open boundary conditions respectively, and

 u(tl−1)=((δxROMS)T(tl−1),δfT(tl),δbT(tl))T.

Equation (6) is obtained from first-order Taylor expansion of NLROMS in (1).

 u∗(tl−1)=MTl−1,lp(tl) (7)

where where is the adjoint state-vector, and are the adjoint of the surface forcing and the open boundary condition increments.

## 2 DD-4DVarDA in ROMS model

Domain Decomposition (DD) method proposed in [1] is made up of decomposition of the domain into subdomains where is the 3D spatial domain and is the time interval, solution of reduced forecast model and minimization of local 4D-Var functionals (we call it DD-4DVarDA method).
Relying on the existing software implementation, in the next we describe main components of DD-4DVarDA method, highlighting the topics that we will address both on the mathematical problem underlying ROMS and the code implementation (see steps 1-4). We focus on IS4DVAR formulation described in Section 1.

### 2.1 Decomposition in space and time of the ocean model

Decomposition of spatial domain .
We will consider a 2D decomposition of in x- and y-direction and denote the spatial domain to decompose.

ROMS uses a parallelization approach that partitions domain into tiles (see Figure 3) [2]

 Ωxy=Nsub−1⋃i=0tilei (8)

where ; and are the number of tiles set in the input file in x- and y-direction, respectively.
We denote by and the overlapping tiles regions (i.e ghost or halo area in ROMS111In ROMS, the halo area would be two grids points wide unless the MPDATA advection scheme is used, in which case it needs three.) in x- and y-direction, respectively.
Step 0: DD of in ROMS.
In our study we will assume the decomposition available in ROMS, as given in (8).

Decomposition of time interval .
ROMS does not yet implement decomposition in time direction.
Step 1: DD of in ROMS.
In our study we aim introducing a decomposition of time interval into intervals:

 Δ=Nt⋃k=1Δk:=Nt⋃k=1[t¯sk−1,t¯sk−1+Nk], (9)

where are respectively the number of subdomains of and of time such that , and .

Ocean model reduction. ROMS allows each tile (or subdomain, see Figure 3) to compute local solutions of TLROMS and ADROMS.
For and local222Let and be vectors, for simplicity of notations, we refer to as a restriction of to , i.e. and , similarly for matrix , i.e. and , according the description in [6]. TLROMS on local domain computes

 δxROMSi(tl)≃Mi,(l−1,l)ui(tl−1) (10)

and local ADROMS on local domain computes

 u∗i(tl−1)=MTi,(l−1,l)pi(tl), (11)

where , , , , are the restriction on of variables , , , and and is discrete model from to .
DD-4DVar method introduces model reduction by using the background as local initial values. For (outer loop of DD–4DVAR method [1]) do: for , posed , we let be the solution of the local model

 (12)

where , , and are respectively numbers of adjacent tiles in x- and y-direction, and are respectively background on , the vector accounting boundary conditions of and the restriction in of the matrix in (1) that is

 Mk≡M¯sr−1,¯sr:=M¯sr−1,¯sr−1+1⋯M¯sr−1,¯sr. (13)

In the following, we neglect the dependency on outer loop iteration of DD–4DVAR method. We underline that local TLROMS and ADROMS in (6) and (7) are obtained by using MPI exchange for boundary conditions, regardless of local solution on overlap area, namely they do not consider overlapping tiles conditions in (12.1) and (12.2). Consequently, we need to modify local TLROMS and ADROMS in (10) and (11) taking into account of (12.1) and (12.1) conditions. More precisely,

Step 2: TLROMS.
for , , local TLROMS on will be modified such that

 δxi,k≃Mi,kui,k−1+θI(ui,k−1)+θJ(ui,k−1) (14)

where

 θJ(ui,k−1):=∑nJiJ=1γiJ(Mi,k/HJ⋅ui,k−1/HJ−Mk/HJ⋅ui,k−1/HJ) (15)

and

 θI(ui,r−1):=∑nIiI=1γiI(Mri/HI⋅ui,r−1/HI−Mri/HI⋅ui,r−1/HI) (16)

are overlapping vectors in x- and y-direction, respectively; where parameters , denote weights.

Similarly, we need to modify local ADROMS in (11). More precisely, Step 3: ADROMS.
for , , local ADROMS in (11) on local domain will be modified such that

 ui,k−1=(Mi,r)T⋅pi,k+θJ(pi,k−1)+θI(pi,k−1) (17)

where and are defined in (15) and (16).

4D-VAR DA Operator Reduction.
Operator reduction involves the inner loop in Figure 1.
ROMS [2] computes and minimizes the operator

 JROMStilei:=JROMS/tilei (18)

where is defined in (2) and is the IS4D-Var functional in each tile with MPI exchange of boundary conditions, .

Local 4D–VAR DA functional in [1] is defined as follows

 JDD−4DVartilei×Δr:=Ji,k(xi,k)=J(xi,k)/(tilei×Δk)+OIJ(xi,k) (19)

where

 J(x)=α∥x−xb∥2B−1+∥Gx−y∥2R−1, (20)

is the DD-4DVar functional in [1],

 OIJ(xi,k)=nI∑iI=1βiI,k⋅∥xi,k/HI−xiI,k/HI∥2B−1iI+nJ∑iJ=1βiJ,k⋅∥xi,k/HJ−xiJ,k/HJ∥2B−1ij (21)

is the overlapping operator on overlapping tiles region and , and

 Ji,k(xi,k)/(tilei×Δk)=αi,k⋅∥xi,k−xMi,ki,k∥B−1i+∥Gi,rxi,k−yi,k∥2R−1i (22)

is the restriction of on , where is background, is observations vector in , is observations vector in ; , , are respectively the restrictions of covariance matrix to , and ; , are restriction of matrices and to in . Parameters , and in (21) denotes regularization parameters. We let , and .
Incremental formulation of local 4D–VAR DA functional in (19) is

 Ji,k(δzi,k)=J(δzk)/(tilei×Δk)+OIJ(δzi,k)

where are the control variable increments in .
Step 4: IS4D-Var.
From (20) local IS4D-Var function in (18) can be written

 JROMSloc=JDD−4DVartilei×Δ−OIJ (23)

Consequently, we need to add overlapping operator in order to enforce the matching of local solutions on the overlapping tiles in each time interval.

### 2.2 DD-4DVarDA in ROMS code

In Figure 6 the ROMS directory structure is shown. We focus on ROMS folder, in particular, on its folders: Tangent and Adjoint.

1. ROMS.
We need to modify routines in ROMS folder implementing decomposition as in (9) (see step 1).
Decomposition of time interval involves modification of initial conditions of nonlinear model, in TLROMS and ADROMS.
Routines involving initialization are (see Figure 6):

1. initial routine in Nonlinear folder: initializes all model variables (it is called in main3d). Note 1.1 (see step 1)
Initial routine initializes all model variable before calling main3d routine (in Nonlinear folder). Main3d routine is the main driver solving nonlinear ROMS model (background).

DD in x- and y- directions is not applied for computing background, i.e. there are not MPI communications among processes.
This means that background is computed without using DD in space, consequently we not apply DD in time.
Moreover, some values of (components velocity) are not set to zero at the end of time step, because its values on some grid points are necessary for the next time step.

2. tl_initial routine in Tangent folder: initializes all tangent model variables (it is called in i4dvar). Note 1.2 (see step 1).
We consider tl_initial routine. Parts involving initializations are:

• line 123: Initializes time stepping indices and counter.

• line 162: initialization time.

3. ad_initial routine in Adjoint folder: initializes all adjoint model variables(it is called in i4dvar). Note 1.3 (see step 1).
We consider ad_initial routine. Parts involving initializations are:

• line 113: Initializes time stepping indices and counter.

• line 152: initialization time.

Main actions to apply DD in time in ROMS are described below. Action 1.1 (see step 1).

1. Modify initials routines by adding MPI communications for initial conditions to tangent and adjoint routines in each time interval.

2. Decompose time interval (RunInterval variable in routines). One possible way forward:

• We need to use OMP threadprivate directive for replicating variables related to time interval such that each thread has its own copy. (see mod_parallel routine in Module folder at line 51 related to DD in space).
Allocate_routine in mod_parallel allocates tiles; equally we can allocate local time interval and related variables.

• We can add allocation and OMP threadprivate directive of local time intervals in mod_parallel routine. We can define first time interval (first_time_interval) and last time interval (last_time_interval) and add a for loop, after for loop involving tiles, started from first time interval up to last time interval adding a time step (dt) defined in driver.

• tl_main3d: at line 277 stars for loop involves tiles.

• ad_main3d: at line 629 stars for loop involves tiles.

• Identify tangent and adjoint variables for MPI communications in time.

3. Introduce MPI communications in time.

• We need to split ROMS MPI communicator (OCN_COMM_WORLD) for obtaining MPI communicators needed to communications among processes related to same spatial sub domain but different time intervals.
By splitting ROMS MPI communicator we obtained a new MPI communicator namely one single communicator for the current process.

Action 1.2 (see step 1).
Step 1 involves:

• tl_main3d routine: at line 142 starts while loop

on time interval (RunInterval) by increasing the step time (my_steptime). Moreover, at lines 278 and 280 it calls tl_set_massflux and tl_rho_eos, i.e. the routines we need to modify (see Action 2.1,2,3,4). Consequently, we probably need to introduce in tl_main3d MPI communications in time after the tl_set_massflux and tl_rho_eos routines.

• ad_main3d routine: at line 177 starts while loop on time interval (RunInterval) by increasing the step time (my_steptime). Moreover, at lines 632 and 634 it calls ad_rho_eos and ad_set_massflux, i.e. the routines we need to modify (see Action 3.1 and 3.2). Consequently, we probably need to introduce in ad_main3d MPI communications in time after the ad_set_massflux and ad_rho_eos routines.

2. TLROMS.
We need to modify TLROMS as in (14) (see step 2), namely we need to compute the overlapping vector in (15) and (16) and add them to tangent variables.
Action 2.1 (see step 2).
We could consider another inner loop (inside the loop over in Figure 1) over index and initial approximation of solution on local tiles at . For each iteration we need local solution on tile adjacent to each tile, this means using MPI exchange of information between adjacent tiles (in two direction see Figure 5).

We note that primitive equations of motion [5] are written in flux form transformed using orthogonal curvilinear coordinates () (see Figure 5).
We consider tl_main3d routine. In tangent folder (see Figure 6) this routine is the main driver of TLROMS configurated as a full 3D baroclinic ocean model.
tl_main3d routine calls the following subrountines.

1. tl_set_massflux calls tl_set_massflux_tile.
tl_set_massflux_tile: computes “in situ” tangent linear horizontal mass flux. Action 2.2 (see step 2).
Taking into account (15) and (16) we need to modify the code starting from line 155.

2. tl_rho_eos calls tl_rho_eos_tile.
tl_rho_eos_tile: computes “in situ” the density and other quantities (temperature, salinity,…).
Action 2.3 (see step 2).
Parts to be modified:

1. Line 505: computes ”in situ” density anomaly;

2. Line 591: computes ”in situ” Brunt-Vaisala frequency;

3. Line 1158: computes ”in situ” Brunt-Vaisala frequency;

3. tl_omega: computes vertical velocity (no modifications because DD is only horizontal, there are not overlap region along vertical direction).

Similarly to TLROMS, we need to modify ADROMS as in (17) (see step 3), namely we need to compute and add the overlapping vector in (15) and (16) to adjoint variables.
We consider ad_main3d routine. In Adjoint folder (see Figure 6) this routine is the main driver of ADROMS configurated as a full 3D baroclinic ocean model.
ad_main3d routine calls the following subrountines.

1. ad_rho_eos: computes “in situ” density and other associated quantities.
Action 3.1 (see step 3).
Parts that should be modified:

1. Lines 797 and 1758: compute ”in situ” adjoint Brut-Vaisala frequency at horizontal points.

2. Line 1070: computes ”in situ” adjoint density anomaly.

Action 3.2 (see step 3).
Part that should be modified:

1. Line 201: computes ”in situ” adjoint horizontal mass fluxes

3. ad_set_avg: accumulates and computes output time-averaged adjoint fields. (probably no modifications are needed).

4. IS4D-Var.
We need to modify routines in ROMS folder as in (23) (see step 4).
Action 4.1 (see step 4).
We need to modify IS4DVAR cost function in (2) to take in account overlap region i.e. halo region.

## 3 ROMS test cases

We have installed ROMS on the high-performance hybrid computing architecture of the Sistema Cooperativo Per Elaborazioni scientifiche multidiscipliari data center, located in the University of Naples Federico II. More precisely, the HPC architecture is made of 8 nodes, consisting of distributed memory DELL M600 blades connected by a 10 Gigabit Ethernet technology. Each blade consists of 2 Intel Xeon@2.33 GHz quadcore processors sharing 16 GB RAM memory for a total of 8 cores/blade and of 64 cores, in total.
We consider test cases referred to IS4D-Var and RBL4DVAR configured for the U.S. west coast and the California Current System (CCS). This configuration, referred to as WC13, has 30 km horizontal resolution, and 30 levels in the vertical and data assimilated every 7 days during the period July 2002 – Dec. 2004.

• IS4DVAR test cases.

To run this application I need to take the following steps.

• I customize build_roms.csh script in IS4DVAR folder and define the path to the directories where all project’s files are kept.

• I set USE_MY_LIBS to yes, then I modify source code root file Complilers/my_build_paths.sh and edit the appropriate paths for the desired compiler.

• I customize the ROMS input file (roms_wc13_2hours.in/roms_wc13_daily.in) and specify the appropriate values of tile numbers in the I-direction and J-direction i.e. NtileI=2 and NtileJ=4 (see Figure 3).

• I modify ARmake.inc in Lib folder to create the libraries for serial and parallel ARPACK.

• I customize Linux-ifort.mk (Fortran compiler is ifort) and define library locations.

• I create PBS scripts to run the test cases.

Figures 8 and 7 show results obtained applying DD in space by considering tiles and input files roms_wc13_2hours.in and roms_wc13_daily.in, respectively. They show total cost function (black curve), observation cost function (blue curve), and background cost function (red curve) and the theoretical minimum value (dashed black line) plotted on a scale. We observe that the solution has more or less converged by 25 iterations.
We will refer to the following quantities: Nouter number of outer loops, Ninner inner loop, elapsed time for each node in seconds, and maximum and minimum elapsed time among nodes in seconds.
We consider:

• case 1: Nouter=1 and Ninner=25;

• case 2: Nouter=1 and Ninner=50;

• case 3: Nouter=2 and Ninner=25;

• case 4: Nouter=2 and Ninner=50;

In Tables 1, 2 and 3, 4 we report elapsed time for each process considering roms_wc13_2hours and roms_wc13_daily.in input file, respectively.

• RBL4DVAR.

To run this application I need to take the following steps.

• I customize build_roms.csh scripts in RBL4DVAR folders and define the path to the directories where all project’s files are kept.

• I customize the ROMS input file (roms_wc13_2hours.in/roms_wc13_daily.in) and specify the appropriate values of tile numbers in the I-direction and J-direction i.e. NtileI=2 and NtileJ=4 (see Figure 3).

• I create a PBS script to run the test cases.

• IS4D-Var vs RBL4DVAR.

IS4D-Var and RBL4DVAR algorithms are based on a search for the best circulation estimate in the space spanned by the model control vector and in the dual space spanned by the observations, respectively. We note that , where is number of observations and is dimension of model control vector. Hence, the dimension of observation space is significantly smaller than the model control space, the dual formulation can reduce both memory usage and computational cost. Consequently, it appears that the dual formulation should be an easier problem to solve because of the considerably smaller dimension of the space involved, but it has practical barriers to convergence.
Consequently, we consider three applications of dual formulation:

1. using the standard preconditioning with a conjugate gradient (RBL4DVAR) method;

2. using the preconditioning and minimum residual algorithm (MINRES);

3. using a restricted B-preconditioned conjugate gradient (RPCG) approach.

Figure 9 and 10 shows the convergence of minimization algorithms RBL4DVAR and RBL4DVAR, MINRES, and RPCG compared to primal formulation IS4DVAR by considering , and , . We observe that the performance of RPCG is superior to both MINRES and RBL4DVAR, in particular, PCG ensures that RPCG converges at same rate as IS4D-Var.

• Observation impact and observation sensitivity.
The observations assimilated into the model are:

• SST satellite: Sea Surface Temperature;

• SSH satellite: Sea Surface Heights;

• Argo floats: hydrographic observations of temperature and salinity.

We consider the time average transport across 37N over the upper 500 m, denoted by , and given by

 I37N(x)=1NN∑i=1hTxi (24)

where is a vector with non-zero elements corresponding to the velocity grid points that contribute to the transport normal to the 37N section shown in Figure 12, is the number of time steps during the assimilation interval, is the model state-vector at time , and is the model time step. We consider scalar functions of ocean state vector , prior and posterior ocean state vector , where is background circulation estimate and is 4D-Var analysis. The circulation analysis increment at instant time in interval time of each 4D-Var cycle is given

 δxROMS(t)=xa(t)−xb(t), (25)

consequently, transport increment during each assimilation cycle can be expressed as:

 ΔI=Ia−Ib=I(xb+δx)−Ib≃δxT(∂J/∂x), (26)

using the tangent linear assumption in (6), (5) and (4), the circulation analysis increment is given by

 δxROMS(t)=M(t,t0)~Kd, (27)

where represents the perturbation tangent linear model for the time interval , is an approximation of defined in (5) and is innovation vector. Then, increment defined in (26) become

 ΔI≃dT~KTMT(∂I/∂x), (28)

where represents adjoint of Kalman gain matrix.
Consequently, 37N time increment of averaged transport in (24) is:

 ΔI37N≃1NdT~KTN∑l=1(Ml)Th=dTg=dT(gx+gf+gb) (29)

or

 ΔI37N≃nobs∑l=1(yl−Hl(xb(t)))gl, (30)

namely, each observation contributes to the calculation of the increase, where , contribution from initial condition increment, contribution from surface forcing increment, contribution from open boundary increment and .
Moreover, we consider the sensitivity of to variations in the observations, in particular, we consider .

Figure 10 shows the total number of observations from each observing platform that were assimilated into the model during cycle, increments and contribution of observations from each platform to

• NL: nonlinear model;

• TL: total impact;

• IC: initial condition;

• FC: surface forcing conditions;

• BC: open boundary conditions.

Figure 13 shows curve represents the evolution of the ocean state vector in time based on the either the prior or posterior control vector. In the case of the observation impact, the actual contribution of each observation, , to the change defined in (28) (blue curve vs. black curve) due to data assimilation is revealed. Conversely, the observation sensitivity quantifies the change that will occur in a (red curve vs. blue curve) as a result of perturbations in the observations, . In Figures 15, 16 and 17 we show the results obtained by running related tests on SCoPE.

## 4 Future developments

Main modification to apply DD in time in ROMS is described by point (c) in Action 1.1, i.e. introduction of MPI communications in time. More precisely, we need to introduce and manage the MPI communication in space and time. The introduction of DD in time involves the following communications among processes:

• Intra communications: by splitting MPI communicator (OCN_COMM_WORLD) to create local communicators (TASK_COMM_WORLD) to allow communications among processes related to same spatial subdomain but different time intervals.
MPI commands are

• MPI_Comm_split: partitions the group of MPI processes into disjoint subgroups and creates a new communicator (TASK_COMM_WORLD) for each subgroup.

• MPI_Isend and MPI_Irecv: sends and receives initial conditions by setting the new communicator obtained from MPI_Comm_split.

• Inter communications: by creating new communicators (OCNi_COMM_WORLD) to allow communications among processes related to different spatial subdomains but same time interval.
MPI commands are

• MPI_Intercomm_create: creates an intercommunicator for each subgroup.

• MPI_Isend and MPI_Irecv: sends and receives boundary conditions by setting the new communicator obtained from MPI_Intercomm_create.

## References

• [1] D’Amore, L., Cacciapuoti, R., 2021: “Model Reduction in Space and Time for decomposing ab initio 4D Variational Data Assimilation Problems”, Applied Numerical Mathematics, 2021, Volume 160, pp. 242-264 Elsevier, https://doi.org/10.1016/j.apnum.2020.10.003.
• [2]
• [3] Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, A.T. Weaver, and J. Zavala-Garay, 2011: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems, Part I - System overview and formulation, Progress in Oceanography, 91, 34-49.
• [4] Hedström, K.S., 2018: Technical Manual for a Coupled Sea-Ice/Ocean Circulation Model (Version 5), OCS Study BOEM 2018-007. U.S. Department of the Interior Bureau of Ocean Energy Management Alaska OCS Region. 182 p.
• [5] Arango, H.G., Moore, A.M., Miller, A.J., Cornuelle, B.D., Lorenzo, E.Di, Cornuelle B.D. and Neilson, D.J., 2003: The ROMS Tangent Linear and Adjoint Models: A comprehensive ocean prediction and analysis system.
• [6] D’Amore, L., Arcucci, R., Pistoia, J., Toumi, R. Murli, A. 2017: On the variational data assimilation problem solving and sensitivity analysis, Journal of Computational Physics, pp. 311-326.
• [7] Langland, R. H., and N. Baker, 2004: Estimation of observation impact using the NRL atmospheric variational data assimilation adjoint system. Tellus, 56A, 189–201.
• [8] Gürol, S., Weaver, A. T., Moore, A. M., Piacentini, A., Arango, and H. G., Gratton, S. 2014: B‐preconditioned minimization algorithms for variational data assimilation with the dual formulation. Quarterly Journal of the Royal Meteorological Society, 140(679), 539-556.

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### Tony Phillips' Take Blog on Math Blogs

Math in the Media 0599
 Mail to a friend · Print this article · Previous Columns Tony Phillips' Take on Math in the Media A monthly survey of math news

May 1999

• The music of the spheres
• Learn inference and logical rigor from a computer game?
• Ada and the Bernoulli Numbers
• Plot of Internet traffic against time gives a curve with the self-similarity of a fractal!
• The music of the spheres: a paper in the Annals of Mathematics (January 1999) gives examples of one-parameter families of 10-dimensional drums which are all geometrically different but all "sound" the same. See "You Can't Always Hear the Shape of a Drum" in this series. What makes these new examples (due to Dorothee Schueth of Bonn) especially significant is that topologically the drums have no boundaries and no handles (the simplest examples are products of 3 and 4-dimensional spheres). These are the first examples of this type. The title is Continuous families of isospectral metrics on simply connected manifolds.

Learn inference and logical rigor from a computer game? This is Minesweeper, a game used by Patti Frazer Lock at St. Lawrence University to give students "a better sense of what proofs are all about," according to a Dana Mackenzie piece in the May-June American Scientist. And they love it, too. She got the idea from Allan Struthers of Michigan Tech, who observes: "Putting a flag on a square is a theorem -- you know there's a bomb there."

Ada and the Bernoulli Numbers. The May 1999 Scientific American has an article by Eugene Kim and Betty Toole on "Ada and the First Computer." Ada is Augusta Ada Byron (her father was the poet), later Lady Lovelace. The first computer is Charles Babbage's Analytical Engine (devised around 1840). In notes to her translation of a paper about this machine (which existed at that time only on paper) she lists the set of instructions that would make it generate the Bernoulli Numbers. Andrew Granville has information about these numbers, but beware of his notation: for Granville the Bernoulli numbers are the coefficients occurring in the following infinite series:

while Ada defines by the series:

(Since all Granville's beyond the first turn out to be 0, there is no contradiction, but a renumbering is necessary; the article uses both notations at once). These numbers turn up in many different mathematical contexts, some of which (homotopy theory and differential topology) Ada and her contemporaries could not have imagined.
Ada explains how the internal symmetries of the calculation allow millions of these Numbers'' to be generated by the recursive use of a small set of instructions. In spirit, this is a completely modern computer program, and even explained in a kind of proto-Fortran.

The plot of Internet traffic against time gives a curve with the self-similarity of a fractal! This according to Herb Brody, reporting on work at AT&T Laboratories in the May 1999 Technology Review. ("The Fractal Net"). This is very different from the behavior of telephone traffic, and reflects the fact that "Lengths of Internet sessions range over a span of 6 to 7 orders of magnitude." The best known example of a fractal curve in mathematics is the Koch Snowflake (here in an image from Bellevue Community College); the self-similarity comes from the construction. Many other examples, with animations, are in the article Visualizing Space-Filling Curves with Fractals in Communications in Visual Mathematics for August 1998.

-Tony Phillips
SUNY at Stony Brook


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# Moment Of Inertia Of A Semicircle

Moment of inertia of a semicircle is generally expressed with the following equation;

 I = π R4 / 8

## How To Find The Moment Of Inertia Of A Semicircle

In order to find the moment of inertia of a semicircle, we need to recall the concept of deriving the moment of inertia of a circle. The concept can be used to easily determine the moment of inertia of a semicircle.

1. We will first begin with recalling the expression for a full circle.

I = πr4 / 4

In order to find the moment of inertia, we have to take the results of a full circle and basically divide it by two to get the result for a semicircle.

Now, in a full circle because of complete symmetry and area distribution, the moment of inertia relative to the x-axis is the same as the y-axis.

Ix = Iy = ¼ πr4

M.O.I relative to the origin, Jo = Ix + Iy = ¼ πr4 + ¼ πr4 = ½ πr4

Now we need to pull out the area of a circle which gives us;

Jo = ½ (πr2) R2

Similarly, for a semicircle, the moment of inertia of the x-axis is equal to the y-axis. Here, the semi-circle rotating about an axis is symmetric and therefore we consider the values equal. Here the M.O.I will be half the moment of inertia of a full circle. Now this gives us;

Ix = Iy = ⅛ πr4 = ⅛ (Ao) R2 = ⅛ (πr2) R2

Now to determine the semicircle’s moment of inertia we will take the sum of both the x and y-axis.

M.O.I relative to the origin, Jo = Ix + Iy = ⅛ πr4 + ⅛ πr4 = ¼ πr4

## Derivation

We will basically follow the polar coordinate method.

The moment of inertia of the semicircle about the x-axis is

$I = \int y^{2}dA$ $\Rightarrow I_{x} = \int y^{2}dA$

y = r sin θ

dA = r drd θ

We will now determine the first moment of inertia about the x-axis. We get;

Ix = oroπ y2 ⋅ dA

oroπ (r sin θ) 2  . r drd θ

= oroπ r3 [(1 – cos2θ)/2](1/2) d2θdr

$I_{x} =\int_{0}^{r}\frac{r^{3}}{4}[2\theta -sin2\theta ]_{0}^{\pi }dr = \int_{0}^{r}\frac{r^{3}}{4}2\pi dr = \int_{0}^{r}\frac{r^{3}}{2}\pi dr$

$I_{x}=\left [ \frac{1}{8}\pi r^{4} \right ]_{0}^{r}= \frac{1}{8}\pi r^{4}$

We will now determine the first moment of inertia about the x-axis. We get;

Iy = oroπ x2 ⋅ dA

x = r cosθ

dA = r drd θ

Iyoroπ (r cos θ) 2  . r drd θ

= oroπ r3 [(1 + cos2θ)/2](1/2) d2θdr

$I_{x} =\int_{0}^{r}\frac{r^{3}}{4}[2\theta +sin2\theta ]_{0}^{\pi }dr = \int_{0}^{r}\frac{r^{3}}{4}2\pi dr = \int_{0}^{r}\frac{r^{3}}{2}\pi dr$

J0 = Ix + Iy

= (1/8) π r+(1/8) π r4

= (1/4) π r4


text:
In an equailateral triangle,
Question.

In an equailateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

Solution:

Altitude of equilateral $\Delta=\frac{\sqrt{3}}{2}$ side

$h=\frac{\sqrt{3}}{2} a$

$h^{2}=\frac{3}{4} a^{2}$

$4 h^{2}=3 a^{2}$
Editor


text:
# Do empty category objects have self morphisms?

I understand that all category objects must have self morphisms, but allowing morphisms to or from empty sets breaks my intuition of what morphisms are (i.e. a relation between the origin and target objects).

If a category object that is the empty set has a self morphism, is it permitted to participate in other morphisms?

N.B. I’m just starting to learn category theory, so right now I only understand category objects as sets (or simpler objects like integers), which may be part of my problem.

In this setting, your question asks what the morphisms with domain $$\emptyset$$ are. The point is that the definition of a function $$f:A \to B$$ is "for each $$a \in A$$ there is a unique $$b \in B$$ such that $$f(a) = b$$" (In set theoretic language $$f:A \to B$$ is actually a subset of $$A \times B$$ such that for each $$a \in A$$, there is a unique $$b \in B$$ such that $$(a,b) \in f$$).
In particular, this definition means that for any set $$B$$ there is a unique function from $$\emptyset$$ to $$B$$; namely $$\emptyset$$ itself! (though in the category we distinguish between the object $$\emptyset$$ and each of the morphisms $$\emptyset$$ since the morphisms come with a domain and codomain) This may seem strange, but the definition requires it since there are no $$a \in \emptyset$$ so the needed condition vacuously holds for $$f = \emptyset$$. In particular, in Set for every object $$B$$ there is a unique morphism $$f:\emptyset \to B$$ (In technical language, this says that $$\emptyset$$ is an initial object in Set). So, yes, in Set, $$\emptyset$$ has an identity morphism $$\emptyset \to \emptyset$$ and also participates in morphisms to all other sets. This means your intuition about the category Set is incorrect.


text: = Constant Functions. [5], In the context of a polynomial in one variable x, the non-zero constant function is a polynomial of degree 0 and its general form is f(x) = c where c is nonzero.  for any That is, y â¥ 0. Let f(x) = 2, be a constant function. The domain of a constant function is the set of all real numbers . You da real mvps! results in a zero change in Recognizing a constant function, like y = 3. As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c.[4], The graph of the constant function y = c is a horizontal line in the plane that passes through the point (0, c). . Then the range is f(x) â¥ -3 and that's it. Share to Twitter Share to Facebook Share to Pinterest. Domain, Range, And Inverse of Constant Function. Namely y(0) = 2, y(â2.7) = 2, y(Ï) = 2, and so on. A constant function is a function whose range consists of a single element. The Modulus Function. f (x 1) = f (x 2) for any x 1 and x 2 in the domain. Discussion. So the domain of this, this is a review. Thus, the real-valued function f : R â R by y = f (a) = a for all a â R, is called the identity function. linear function Before working with a linear function, we replace c with an actual number. is a So in Python 3.x, the range() function got its own type.In basic terms, if you want to use range() in a for loop, then you're good to go. x hope this helps. f Lv 4. the graph of a constant function is symmetric with respect to the y-axis. Video: Determining the Domain and Range of a Constant Function given Its Graph Determine the domain and range of the function () = −4. . y = c for each x ∈ R Here c is a constant. ; The domain or input of y=c is R. Let’s understand how to use a range() function of Python 3 with the help of a simple example. (2) The Fourier transform of the constant function f(x)=1 is given by F_x[1](k)=int_(-infty)^inftye^(-2piikx)dx=delta(k), (3) where delta(k) is the delta function. share | cite | improve this answer | follow | answered Mar 7 '15 at 22:45. This function has no intersection point with the x-axis, that is, it has no root (zero). BASIC FUNCTIONS 1. x 3 Let R be the set of real numbers. Check - Relation and Function Class 11 - All Concepts. However, range is sometimes used as a synonym of codomain, generally in old textbooks. Lv 4. and For example y = 10 is a constant function. Using for loop, we can iterate over a sequence of numbers produced by the range() function. hope this helps. Constant functions are linear and can be written f (x) = 0 x + c. In this form, it is clear that the slope is 0 and the y-intercept is (0, c). $\begingroup$ Liouville's theorem actually implies the apparently stronger statement: the range of a non constant entire function is dense in $\mathbb{C}$. The two plots following that plot a model of the actual data. The equation of a line is: y = mx + b. But let's say the graph reaches its lowest point at y = -3, but goes upward forever. This means that the range of the function, or the range of y-coordinates, ranges from -3 to 10. Because of this, a constant function has the form y = b, where b is a constant (a single value that does not change). The function graph of a one-dimensional constant function is a straight line. The constant c is a real number. The inputs and outputs of a function are also called the domain and range. constant function The domain of a function is the set of all input values. methods and materials. For all the real values of x there will be only one range of y that is 10. The interval starts and ends at 6. Its graph is the x-axis in the plane.[6]. f: R â R f(x) = c for each x â R i.e. Every point of a constant function is a global maximizer as well as a global minimizer. And we see here. Plotting graph For all values of x, value of y will be c range Determine the domain and range of the function of equals negative four. Linear Function: The polynomial function with degree one. The figure given below represents a constant function. The idea of const functions is not to allow them to modify the object on which they are called. A function f is continuous when, for every value c in its Domain:. Video Transcript. f When you have a function where y equals a constant, your graph is a truly horizontal line, like the graph below of $$y=3$$. Whats the domain and range on a constant function? Formally, a constant function f(x):RâR has the form =.Usually we write () = or just =.. ListLinePlot[Thread[List[Range[0, 50], 7 & /@ Range[0, 50]]]] Posted by Kris at 9:54 PM. range() is a built-in function of Python. It will be a many-one function. Polynomial Functions. f(x1)=f(x2) for any x1 and x2 in the domain. Now there are various ranges that differ from unsigned to signed bits. The range, and the most typical, there's actually a couple of definitions for range, but the most typical definition for range is "the set of all possible outputs." The function f : A ---> B defined by f (x) = 5 for every x belonging to A is a constant function. For the best answers, search on this site https://shorturl.im/aw5wL. A constant function is a function whose range consists of a single element. :) https://www.patreon.com/patrickjmt !! limx→c f(x) = f(c) "the limit of f(x) as x approaches c equals f(c)" The limit says: For example, y = 7 or y = 1,094 are constant functions. 01:48. Information and translations of constant function in the most comprehensive dictionary definitions resource on the web. It is of the format y=c d. This is constant function as output is different for each input e. This is constant function as output is different for each input 2. which of the graph represent constant function? First, by the definition. In fact lines are either increasing, decreasing, or constant. Varsity Tutors connects learners with experts. f A constant function is a linear function for which the range does not change no matter which member of the domain is used. In âaâ constant function the answer is always that contant, thus the range is only that one value. x \$1 per month helps!! 2 is used. The domain of f isR and its range is {c}. Do It Faster, Learn It Better. A function on a connected set is locally constant if and only if it is constant. In the context where it is defined, the derivative of a function is a measure of the rate of change of function values with respect to change in input values. The proof is easy and can be found here . The only output value is the constant $c$, so the range is the set $\left\{c\right\}$ that contains this single element. For example, y = 7 or y = 1,094 are constant functions. A function f: R → R defined by y = f (x) = c, x ∈ R,where c is a constant is called a constant function. The slope m tells us if the function is increasing, decreasing or constant: Surb Surb. Thanks to all of you who support me on Patreon. For all the real values of x there will be only one range of y that is 10. Whats the domain and range on a constant function? Varsity Tutors does not have affiliation with universities mentioned on its website. The function works without an argument since the â¦ When a function is continuous within its Domain, it is a continuous function.. More Formally ! Letâs understand how to use a range() function of Python 3 with the help of a simple example. The integers that are too big to fit into an int will be taken as long. ( The function is a constant function in an interval for some and . It is recommended the practice to make as many functions const as possible so that accidental changes to objects are avoided. {\displaystyle (x\mapsto c)'=0} There are at least three ways to see that a constant function $f(x)=c$ is even. And so let's think about its domain, and then we'll think about its range. John W. asked • 09/25/16 explain how many elements are in the range of (f of f)(x) if f(x) defines a constant function over (-infinity,infinity) The domain, range, x-intercept, and y-intercept of the ten parent functions in Algebra 2 Learn with flashcards, games, and more — for free. Using for loop, we can iterate over a sequence of numbers produced by the range() function. In mathematics, a constant function is a function whose (output) value is the same for every input value. x It means for any value of in the domain, the value of is 3. Vertical Shift . The derivative of a constant function c is d/(dx)c=0, (1) and the integral is intcdx=cx. Graph the function range() in Python(3.x) is just a renamed version of a function called xrange in Python(2.x). The converse is also true. Upvote â¢ 0 Downvote domain 1 Linear Functions. In other words, all correspond to a single . In fact, for any exponential function with the form f (x) = a b x, f (x) = a b x, b b is the constant ratio of the function. Need to calculate the domain and range of a graphed piecewise function? [citation needed] Relational approach. In the example above, the range of $$f\left( x \right)$$ is set B. Constant Function. 0 0 1? Check - Relation and Function Class 11 - All Concepts. It is a linear function where f (x1) =f (x2) f ( x 1) = f ( x 2) for all x1,x2 â R x 1, x 2 â R. I am having trouble with the middle one. Example: The function y(x) = 2 or just y = 2 is the specific constant function where the output value is c = 2. So, -3 ≤ f(x) ≤ 10. 0 1? With a constant function, for any two points in the interval, a change in x results in a zero change in f (x). 51.3k 9 9 gold badges 54 54 silver badges 93 93 bronze badges Source(s): whats domain range constant function: https://biturl.im/8RrsP. So, -3 â¤ f(x) â¤ 10. Constant Function: If the degree is zero, the polynomial function is a constant function (explained above). = 0 is the set of the images of all inputs for which this has. We 'll think about its domain is all the real values of x there will be only one while. Methods and materials, i.e range while its domain: if it is a range of a constant function function for which function! Â¤ 10 the range of y that is, the output value of is 3 the domain is the,! Identically zero function neonate, but it turns out to be quite useful in mathematics, a constant function the! No root ( zero ) information and translations of constant function is a constant function is a function const... 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text:
# Triple Integral Using Cylindrical Coordinates

## Homework Statement

A conical container with radius 1, height 2 and with its base centred on the ground
at the origin contains food. The density of the food at any given point is given by
D(r) = a/(z + 1) where a is a constant and z is the height above the base.
Using cylindrical polar coordinates, calculate the total mass of food in the container.

## The Attempt at a Solution

ok so mass is the integral D(r)dV, and in cylindrical coordinates dV is rdrd$$\theta$$dz

I thought that you could probably do:
$$\int^1_0 \,dr$$$$\int^{2\pi}_0 \,d\theta$$$$\int^{2r-2}_0 \,dz$$

$$(ra/(z+1))$$

But this makes the integral very difficult and I don't think it's right. I'm pretty sure there's something wrong with my limits on dz. Any help would be appreciated

tiny-tim
Homework Helper
Hi henryc09!
I'm pretty sure there's something wrong with my limits on dz. Any help would be appreciated

Yes, you've integrated from z = 0 to z = 2r - 2 …

why??

z doesn't depend on r, the limits of z are the same for all r.

OK so how do I work out the limits of z? It can't just be from 0-2 because that would make it a cylinder? Still a bit confused.

tiny-tim
Homework Helper
Hi henryc09!

(just got up :zzz: …)
I thought that you could probably do:
$$\int^1_0 \,dr$$$$\int^{2\pi}_0 \,d\theta$$$$\int^{2r-2}_0 \,dz$$

$$(ra/(z+1))$$

But this makes the integral very difficult and I don't think it's right. I'm pretty sure there's something wrong with my limits on dz. Any help would be appreciated
OK so how do I work out the limits of z? It can't just be from 0-2 because that would make it a cylinder? Still a bit confused.

I'm sorry … somehow I read it as a cylinder.


text:
# Did the night sky ever change in recorded history?

I wonder whether there has ever been a major change of the firmament in recorded history, like changes in the positions of stars, changes in constellations, or stars disappearing after going supernova. There've been visible supernovae but had the star in question been visible to the naked eye before the supernova? If so, how did it change what constellation?

• Wikipedia has a page on History of Supernova Observations which is almost certainly relevant to you. – StephenG Oct 25 '20 at 13:54
• @StephenG There's no mention of stars being visible to the naked eye before the nova from which I conclude the answer to my questions is 'no', right? Even what stop-firing-the-good-guys is describing actually isn't a change to the night sky when watched from the same angle relative to the firmament. – John Oct 25 '20 at 14:04
• I glanced at a NASA website on this nebula indicates the original star was a white dwarf. It was unlikely to be visible to the naked eye itself, but they believe it had a companion star which possibly was. – StephenG Oct 25 '20 at 14:13
• The nearest historical supernova was SN 1054. With a probable 10-solar mass progenitor and a distance of 2 kpc, it might have been visible as an unremarkable 4th or 5th magnitude star. The Chinese astronomical records mention the SN as a "guest star" (i.e., not a previously known star), so it's very unlikely anyone identified it with a pre-existing star. – Peter Erwin Oct 25 '20 at 15:06
• There have been occasional suggestions that the Pleiades -- with six visible stars but a frequent tendency to be associated with seven mythological figures -- might have originally had seven visible stars, with one having (somehow) faded or disappeared. This is very unlikely, for a number of reasons. (Most likely, some cultures saw an analogy between the Pleiades and the Big Dipper; since the latter has seven main stars, some people though the Pleiades ought to have seven, too; sometimes the missing seventh star is said to be Alcor, next to Mizar in the Big Dipper.) – Peter Erwin Oct 25 '20 at 15:26

1. Do the stars change positions in the sky ... such that the layout of say... major stars in constellations appear to move over time?

2. Are there events that cause sudden changes (changes you might notice overnight ... or within a few weeks or months)?

3. How long does it take to notice a change?

Precession

The Earth's axis mostly stays oriented in the same direction as we orbit the Sun. This is why we can get away with saying that Polaris is the "North Star" ... because it appears to be almost directly above Earth's North Pole regardless of what time of the day or night ... or what day of the year you choose to observe (it is roughly within 2/3rds of a degree from the pole).

But it turns out the Earth's axis does "wobble" like a spinning top. This wobble is very slow and takes thousands of years to complete a cycle (nearly 26,000 years). At the end of this century (roughly the year 2100) Polaris will be a little closer to the pole ... only 1/2° away from the true pole (today it is about 2/3rds° away from the pole). That is as close as it will get ... and then our pole's precession cycle will start taking it farther away from the pole.

This "precession" shift causes all stars in the sky to shift their position very slightly from year to year and it's mostly not enough to notice in a single human lifetime ... but it is easily noticed when measured across thousands of years.

There are other more subtle cycles, but Precession is noticeable across large amounts of time (thousands of years). The position of the Sun in the sky at the equinoxes changes every year and this causes the Sun to appear to be moving across the zodiac over thousands of years.

The sky has a coordinate system used to catalog the locations of stars and other objects. That coordinate system uses a "declination" to measure where a star is located relative to Earth's North/South ... it works much in the same way that latitude measures geographic locations on Earth. There is also a "right ascension" value that works a bit like longitude on Earth. Except Earth is spinning (and longitudes remain fixed to the planet) ... so we needed a way to mark a fixed position in space to have a sort of celestial longitude. That measurement is called Right Ascension and it needs a "0" position. So the zero position is established by the location of the Sun in the sky at the Vernal Equinox (the start of Northern Hemisphere Spring). It even got a name... the "First point of Aries" ... named because, when it was established a few thousands years ago, it was in Aries. Today ... it's migrated all the way over to Pisces (and is getting closer to the boundary of Aquarius).

These are changes caused by Precession and noticeable ... but mostly only when measuring over hundreds of years or thousands of years. It is actually noticeable just from year to year ... but only with precise measurements.

Precision causes the entire sky to shift in the same direction (and by a very subtle amount). This means that the locations of the constellations will shift ... but it wont explain changes to their shapes. For that, we have to look at other factors. In reality it isn't the "sky" that is shifting ... but rather the orientation of Earth's axis that changed.

Proper Motion

Stars are moving ... and they don't all move in exactly the same direction. But the stars are so distant that the movement is mostly only noticeable with precise instruments. Regardless, they do move.

Here for example, is the Big Dipper asterism, but I've turned on Proper Motion vectors ... the vectors (shown in a light brown color) indicates the direction of that star's motion (relative to us) and longer vectors indicate great proper motion.

Notice that many of the stars of the Big Dipper are all moving in the same general direction at about the same speed ... including Alcor/Mizar, Alioth, Megrez, Phecda, and Merak. In this diagram, those stars are all moving up and to the left (relative to this frame).

But more importantly... notice that Alkaid and Dubhe are going in roughly the opposite direction. These stars are moving down and to the right. This means that over thousands of years, this asterism will become distorted and will eventually not resemble the "Dipper" shape anymore.

Incidentally, the stars in the dipper that are going in the same direction are all part of an Open Cluster called the Ursa Major Moving Group. It isn't as obvious that it's an open cluster because it is relatively close (not all stars in Ursa Major are members of the cluster.)

Stars that physically closer to our Sun will may appear to have much faster proper motion. It isn't so much that the stars are necessarily physically moving faster ... but since they are close, the changes are easier to spot.

Here, for example, is the proper motion of Rigel Kentaurus (aka Alpha Centauri).

In this frame (current time) the star is in the upper-left section of this chart. It is in Centaurus (hence its name) but near Circinus. But notice it has a VERY long proper motion vector which is headed toward the right side of this chart in the direction of Hydra.

Its proper motion vector is much longer than the other stars ... but Rigel Kentaurus is just a little over 4 light years away (one of our closest neighbors) -- so its relative motion is more noticeable.

While Rigel Kentaurus makes the top-20 list for stars with the highest proper motion, it's not the winner ... that spot goes to Barnard's Star (in Ophiucus). But Barnard's Star isn't very bright... at magnitude 9.5 it isn't noticeable to ordinary human vision and requires a telescope to see it. But it is moving just a little under 3x faster than Alpha Centauri A & B.

If you're interested, here's a top-20 list: https://www.cosmos.esa.int/web/hipparcos/high-proper-motion

Transient Events

Certainly events such as bright comets would change the sky -- but those aren't stars. Super-Novae events, on the other hand, are stars ... and these are easily noticeable if they happen in our home galaxy. They don't happen very often. The estimate is that, on average, they happen in a galaxy our size about once every hundred years. But's not like a clock where they happen with periodic regularity ... huge amounts of time may pass without any of them ... and then several could occur timed much closer together.

Since Super-Novae events can be quite bright, they can even be visible in the daytime sky. They don't last particularly long ... they quickly brighten up ... then begin to dim over weeks and ... after a few months they may no longer be noticeable (but they will leave behind a Super Nova Remnant -- deep sky nebulae such as the Crab Nebula and Veil Nebula are examples of Super Nova Remnants.)

• But the question was about "major changes" in "recorded history" (so, in the last 5,000 years or so); none of the visible stars' proper motions are large enough to be noticeable to non-telescopic observers over that short a period of time. – Peter Erwin Oct 25 '20 at 22:46
• While your post doesn't answer my question, I accepted it because it's very good and informative. My question is answered anyway: as you also mention, in the years of recorded history, the Crab Nebula appeared which wasn't there before the supernova on the firmament. – John Oct 26 '20 at 6:38
• What's the scale of the proper motion vectors in the charts? how much time do the arrows represent? For example, at what time will Rigil Kentaurus reach Hydra (assuming of course that the motion vector remains constant)? Is this more in the range or thousands of millions of years? – Stephan Matthiesen Jun 15 at 23:34
• Rigil Kentaurus has a proper motion of about 4 arcsec per year, so if I calculate this right it will need roughly 1000 years for one degree. That's about a fifth of the distance between Alpha and Beta Centauri, so Centaurus visually will change somewhat but it would be hard to notice, I guess. – Stephan Matthiesen Jun 15 at 23:57

I don't see a mention yet of variable stars. There are stars whose brightness, as observed from earth, changes to an extent noticeable to naked-eye observers. The first two that we know historical people noticed provide examples of the two main types.

Algol, in the constellation Perseus, is an eclipsing binary where a dim companion periodically blocks some of the light from the brighter star. Its brightness (as observed from earth) varies from magnitude 2.1 to 3.4 over a time scale of days. The light output of the group of stars does not change, but the amount of light that reaches us changes enough to be visually detectable. See the Wikipedia article https://en.wikipedia.org/wiki/Algol
Note, that this same effect is used to hunt for exoplanets - as the planet periodically blocks some of the light from the start; but to detect planets, this requires exquisitely sensitive instruments.

Mira, in the constellation Cygnus, is a variable star whose brightness ranges from about 3.5 or 4 to 14 - that is easily visible to needing a very good telescope to detect it - over a period of months. The actual light output by the star varies over time. See the wikipedia entry at https://en.wikipedia.org/wiki/Mira Note that this star's output actually pulsates as it varies in brightness.

To quote the first little bit of a long Wikipedia entry on variable stars:

    An ancient Egyptian calendar of lucky and unlucky days composed some 3,200 years ago may be the oldest preserved historical document of the discovery of a variable star, the eclipsing binary Algol.[2][3][4]

Of the modern astronomers, the first variable star was identified in 1638 when Johannes Holwarda noticed that Omicron Ceti (later named Mira) pulsated in a cycle taking 11 months; the star had previously been described as a nova by David Fabricius in 1596. This discovery, combined with supernovae observed in 1572 and 1604, proved that the starry sky was not eternally invariable as Aristotle and other ancient philosophers had taught. In this way, the discovery of variable stars contributed to the astronomical revolution of the sixteenth and early seventeenth centuries.

The second variable star to be described was the eclipsing variable Algol, by Geminiano Montanari in 1669; John Goodricke gave the correct explanation of its variability in 1784. Chi Cygni was identified in 1686 by G. Kirch, then R Hydrae in 1704 by G. D. Maraldi. By 1786 ten variable stars were known. John Goodricke himself discovered Delta Cephei and Beta Lyrae. Since 1850 the number of known variable stars has increased rapidly, especially after 1890 when it became possible to identify variable stars by means of photography.

So, you can see that besides planets, comets, and novae, variable starts have been observed for hundreds of years, maybe thousands.

The axial precession precession of the Earth has a relatively short cycle of about 25,000 years, which made Beta and Gamma Ursae Minoris the "twin pole stars" to ancient Egyptian and Chinese astronomers.

If a star bright enough to be notable to ancient astronomers had gone supernova, the brightness of the explosion would have been much much larger than what we have on record.

• Do we have any records of sailors using the twin stars for navigation in that period? – John Oct 25 '20 at 11:25


text:
CORRIGENDUM TO THE PAPER "Initial-boundary-value problems for the Bona-Smith family of Boussinesq systems"

Abstract

Replace the first equation after (2.44) by $$\int_{-L}^L(\eta_t Q+u_t P) dx=\frac{1}{2} \int_{-L}^L\partial_t(\eta^2+u^2+\eta u^2-c\eta_x^2)dx.$$

Article information

Source
Adv. Differential Equations Volume 14, Number 9/10 (2009), 1019.

Dates
First available in Project Euclid: 18 December 2012