# Distance Between Two Coordinates Calculator

 Again, a distance and direction. Algebra -> Customizable Word Problem Solvers -> Geometry-> SOLUTION: The distance between two points in the coordinate plane is equal to the square root of the sum of the squares of the difference between the x-coordinates and the difference between Log On. Practice applying the Pythagorean Theorem in this tutorial about the distance formula. A journey meter calculate driving or walking distances from any location and displays fuel cost, taxicab fare, using haversine formula. 02/28/2018; 14 minutes to read; In this article. Hi Chua, You can use google API's for that, I think this one gives you duration and distance between 2 places. For two points P 1 = (x 1, y 1) and P 2 = (x 2, y 2) in the Cartesian plane, the distance between P 1 and P 2 is defined as: Example : Find the distance between the points ( − 5, − 5) and (0, 5) residing on the line segment pictured below. The first 2 parameters declare the x and y coordinates of the first point, and the second 2 parameters declare the x and y coordinates of the second point. I turned to my web browser and googled it, sure enough, there was this R package called ‘geosphere’ from Robert J. SHORTEST DISTANCE BETWEEN TWO POINTS ON A SPHERE It is known that the shortest distance between point A and point B on the surface of a sphere of radius R is part of a great circle lying in a plane intersecting the sphere surface and containing the points A and B and the point C at the sphere center. This is the snippet Calculate The Distance of Two Points On a Coordinate Plane on FreeVBCode. There are a number of latitude/longitude distance calculators available online. In this exercise you will plot the positions of the 2 players and manually calculate the distance between them by using the Euclidean distance formula. Nothing else makes sense and I see that you were able to calculate a distance in ft. Calculation of geographical distance is a very important utility function in GPS related procedures. The first 2 parameters declare the x and y coordinates of the first point, and the second 2 parameters declare the x and y coordinates of the second point. This distance and driving directions will also be displayed on google map labeled as Distance Map and Driving Directions US. On this page by using the mileage calculator you can find the flight distance or driving distance between any two cities in the world. Solution : Yes, First, convert the latitude and longitude values from decimal degrees to radians. Distance calculation. This also provides free sample source codes in PHP, ASP, ColdFusion, C/C++, C#, Java, Perl, Visual Basic and Javascript. google map api. angle between two lat lon points. This article is a sequel to the previous article on the same topic, but using T-SQL for calculation - Calculate distance between two points on globe from latitude and longitude coordinates. Hi everybody, i have some problem to calculate the distance between two points, in my case i have a file with 16 columns and i'm interesting to 6th, 7th, 8th column that are rispectively the x, y and z coordinates of several atoms belonging to a protein, instead in 14th 15th and 16th columns there are the x,y and z coordinates of different several atoms belonging to another protein (like a pdb. Fractions should be entered with a forward such as '3/4' for the fraction $$\frac{3}{4}$$. , X0, Y0 and X1, Y1 and click calculate to know the distance between 2 points in 2-dimensional space. Calculate the distance. GIS question: excel formula to calculate distance between two points given their coordinates (latitude / longitude). com can calculate the shortest distance and the fastest distance between any two cities or locations. Is it really possible? The answer is "YES", not. If I want to calculate the distance between two positions, how can I achieve this? Learn. In order to fully grasp how to plot polar coordinates, you need to see what a polar coordinate plane looks like. This will compute the great-circle distance between two latitude/longitude points, as well as the middle point. In this post, we show the formula to calculate the shortest distance between two points using Latitude and Longitude. Language Objectives: After completing the activity, students should be able to. Enter 2 sets of coordinates in the x y-plane of the 2 dimensional Cartesian coordinate system, (X 1, Y 1) and (X 2, Y 2), to get the distance formula calculation for the 2 points and calculate distance between the 2 points. 5678, 11087. Star 1 Right Ascension: Enter the Right Ascension of the first star in hours. ' '===== ' Calculate geodesic distance (in m) ' between two points specified by ' latitude/longitude (in numeric ' [decimal] degrees) ' using Vincenty inverse formula ' for ellipsoids '===== ' Code has been ported by lost_species ' from www. This uses the ‘haversine’ formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an ‘as-the-crow-flies’ distance between the points (ignoring any hills they fly over, of course!). This demo program will calculate the distance between two points on Earth, using the latitude and longitude positions. View Details ». The thing I did kind of worked to find the shortest distance from 2 locations, and then I referenced it to 3 locations in all and averages it out. coordinates and two. This calculation can be useful when trying to determine the distance for logistical purposes (ie delivery service, flights, distance between customers, etc). Please write the origin and destination city name, choose the distance unit and press calculate. I am storing the coordinates in two. However, I would like to calculate the distance between the points before I run my script and without an ArcGIS. Write, compile, and execute a C program that calculates the distance between two points whose coordinates are (7,12) and (3,9). The term Haversine was coined by. 84 return list gives you three calculations--the first two assume the Earth is a perfect sphere, which it is not, and the third that is an ellipsoid, which is closer to the truth. Distance between two locations on the Earth We often describe a location on Earth by its latitude and longitude. It is not very often that someone gives the latitude and longitude of their town! The use of a distance and direction as a means of describing position is therefore far more natural than using two distances on a grid. The Bearing Distance Calculator function calculates the forward and backward azimuths between two specified coordinates as well as the distance. txt, the XY coordinates of the cities. Perform the same operation for the y coordinates. This page helps you to calculate great-circle distances between two points using the ‘Haversine’ formula. Calculate the Distance Between Two Coordinates (latitude, longitude) - PHP - Snipplr Social Snippet Repository. I am looking for a way to get the distance in pixels between two MapItems. Just the other day, my friend was asking me if there was an easy way to calculate the distances between two locations with geocodes (Longitude and Latitude). When calculating the distance between two points on a coordinate system we are applying Pythagoras' Theorem. Standard measurement equipment (rulers, protractors, planimeters, dot grids, etc. As homework we were assigned to enter the following code to calculate the distance between two points on the x and y plane. 42 (rounded to the nearest 100th) How to use the Distance Formula Calculator. This means of location is used in polar coordinates and bearings. For example, in a button you can have the following formula on its OnSelect property:. In the attached file "Circle2", the coordinates of the circles are located under column B & C, the coordinates of the points are located under column F & calculate the minimum distance between two sets of coordinates. Enter 2 sets of coordinates in the 3 dimensional Cartesian coordinate system, (X 1, Y 1, Z 1) and (X 2, Y 2, Z 2), to get the distance formula calculation for the 2 points and calculate distance between the 2 points. Time and Date Duration – Calculate duration, with both date and time included Date Calculator – Add or subtract days, months, years. Function accepts four parameters, source latitude, source longitude, destination latitude, destination longitude and returns the distance between the. 13365 i currently have the code, but that doesnt seem to work well. If you pass an address as a string, the service will geocode the string and convert it to. "Let's look at the diagram! I can see that the lines of longitude get closer and closer together towards the poles! At the equator, the distance between 15 and 30 degrees W longitude is quite a lot! But as you follow those two longitude lines towards the poles, the distance between them shrinks down to zero!. This script calculates distances, bearing and more between the two Latitude/Longitude points. This tool can measure two types of distance types, the first is straight line distance also known as Rhumb line distance. 28 , rounded to two decimal places. I would like to share a MySQL function to calculate distance between two latitude – longitude points. The easiest way I could think of to to this is to somehow store the output of veclen(x,y) to a macro. Free sample source codes in PHP, ASP, ColdFusion, C/C++, C#, Java, Perl, Visual Basic and Javascript. Calculate with Geo Coordinates Calculators for the conversion of geo coordinates, as delivered by a GPS tool, for the distance of two point and for bearing. Calculate the distance between them with the distance. It contains two vertical lines called axis. If the current coordinate system is an earth coordinate system, Distance( ) returns the great-circle distance between the two points. The program should ask the user to enter two points then should calculate the distance between two points and print the distance on the screen. Calculating Distance. I am looking for a way to get the distance in pixels between two MapItems. In this post, we will learn the distance formula. Distance and midpoint calculator This online calculator will compute and plot the distance and midpoint for two points in two dimensions. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. DISTANCE BETWEEN TWO POINTS. The distance returned is the arc length of a great circle connecting the two points, essentially air miles. West and South locations are negative. Distances calculator is a free tool to calculate distances between any two cities in the world. There is more information on how to calculate these figures below the tool. d = √ 5 2 + 7 2. MySQL function to calculate the distance between two coordinates using the Haversine formula. The formula assumes that the earth is a sphere, (we know that it is "egg" shaped) but it is accurate enough * for our purposes. The distance between 2 points on a graph is really the hypotenuse of a right-angled triangle. Calculate the great circle distance between two points does not want to give me a reading. The shortest distance between two points on the surface of a sphere is an arc, not a line. Note: The flight distances provided are close approximations as all flights differ based on weather, traffic, and the exact route determined by air traffic control. Click Calculate Distance, and the tool will place a marker at each of the two addresses on the map along with a line between them. Take for example, a dating website which shows potential matches to a user within a 15 mile radius or a taxi business which calculates taxi fares based on distance between two locations. If you want to determine the distance between two points on a plane (two-dimensional distance), use our distance calculator. The code will show you how to do that using Latitude and Longitude coordinates. Spherical to Cylindrical coordinates. A more mathematical approach, you could use the Pythagorean Theorem and calculate the hypotenuse using the right triangle with the grid units in unity. If you pass coordinates, ensure that no space exists between the latitude and longitude values; destinations — One or more addresses and/or textual latitude/longitude values, separated with the pipe (|) character, to which to calculate distance and time. The program should ask the user to enter two points then should calculate the distance between two points and print the distance on the screen. Position, Distance and Bearing Calculations Whitham D. 458 W 71 27. The API can convert a zip code to the primary location for the zip code. Answered Active Solved. To find the distance between two places, enter the start and end destination and this distance calculator will give you complete distance information. 05W) A program for HP50G or 42S or 41C would be fine or a formula or algorithm as well. By continuing to browse this site, you agree to this use. How do you find the distance between two points? Distance Formula : Example: For two points, (3,2) and (15, 10) the distance is calculated as: Distance = 14. It is not very often that someone gives the latitude and longitude of their town! The use of a distance and direction as a means of describing position is therefore far more natural than using two distances on a grid. We have successfully combined latitude and longitude into the GeoLoc field. Discover how to calculate Distance by Latitude and Longitude using JavaScript. x A means the x-coordinate of point A y A means the y-coordinate of point A. Another method for calculating the distance between two points that fall on the same line is to plot the points and count the amount of boxes in between the two points. This calculator computes the great circle distance between two points on the earth's surface. calculate-distance-between-two-latitude-longitude-points-11152 contains nodes provided by the following 2 plugin(s): KNIME Base Nodes. It’s quick, easy and takes but a moment to do because you only need to enter the x and y coordinates of two points and click a button to calculate it. For some websites, it is a necessity to calculate the distance between certain locations. Find altitude by coordinates on google maps in meters and feet. If you are not sure what the gps coordinates are, you can use the coordinates converter to convert an address into latlong format or vice versa. The terminal coordinates program may be used to find the coordinates on the Earth at some distance, given an azimuth and the starting coordinates. Inputs can be in several formats: GPS Coordinates (like N 42 59. As you start to write the name of a city or place, distance calculator will suggest you place names automatically, you may choose from them to calculate distance. With the growing distribution of GPS devices and tools such as Google Earth or GeoPosition it is easy to determine the geographical coordinates of interesting points on the earth's surface. The geographical coordinates of the two points, as (latitude, longitude) pairs, are and respectively. If you have two geohashes, you first need to decode them into latitude and longitude: GEOHASH_DEC_LAT(#GeoHash) GEOHASH_DEC_LONG(#GeoHash) After you have your coordinates you can calculate the distance: IF(!. Practice applying the Pythagorean Theorem in this tutorial about the distance formula. First, sketch the two UTM coordinates onto a rectangle. In this article, you will learn how to Get Distance between two Coordinates Using Xamarin. For more information on how it distances are calculated on a sphere, have a look at Haversine Formula on Wikipedia, it's a bit complex, hence us not wanting to duplicate the content. Distance Calculation Introduction. Calculate distance between a pair of lat/lon coordinates I recently had a need to calculate distance between a large number of latitude/longitude coordinate pairs. Plane equation given three points. The Haversine Formula For two points on a sphere (of radius R) with latitudes φ1 and φ2, latitude separation Δφ = φ1 − φ2, and longitude separation Δλ, where angles are in radians, the distance d between the two points (along a great circle of the sphere; see spherical distance) is related to their locations by the formula above. After finding the 'distance' (it is in. The distance between the two coordinates, in meters. First, note that it's very easy to get a distance between two points on a sphere if you know the angle of the arc between them: just multiply it by the radius. Suppose it is desired to calculate the distance d from the point (1, 2) to the point (3, -2) shown on the grid below. The question Calculating the planets and moons based on Newtons's gravitational force was pretty much answered with two items: Use a reasonable ODE solver; at least RK4 (the classic Runge-Kutta me. For more information, see Measure Distances Between Data Points and Locations in a Map. Enter a city, street, address, postal code or region and click calculate distance. Distance Calculator: This easy tool is very useful to find travel distances between cities, and its respective estimated time en route. Haversine Formula – Calculate geographic distance on earth. Travelmath helps you find driving distances based on actual directions for your road trip. Enter either: decimal latitudes/longitudes with minus sign for South and West. This tool calculates the straight line distance between two pairs of latitude/longitude points provide in decimal degrees. You can obtain the coordinates for just about any earthly city from WWW. Know how to use the distance formula to find the distance between points. Is there anything to improve? Calculate the distance given. Capable estimates total distance between any location using haversine formula and route method provided by the map and display location coordinates and location address. The terminal coordinates program may be used to find the coordinates on the Earth at some distance, given an azimuth and the starting coordinates. distancesfrom. Free distance calculator - Compute distance between two points step-by-step. A line through three-dimensional space between points of interest on a spherical Earth is the chord of the great circle between the points. What Is a Vector? A vector has size, also known as magnitude, and direction. Suppose it is desired to calculate the distance d from the point (1, 2) to the point (3, -2) shown on the grid below. Algebra -> Customizable Word Problem Solvers -> Geometry-> SOLUTION: The distance between two points in the coordinate plane is equal to the square root of the sum of the squares of the difference between the x-coordinates and the difference between Log On. Further, there is a one-to-one correspondence between areal coordinates and all points on the plane P. Distance = √ (x 2 – x 1) 2 + (y 2 – y 1) 2. Re: How to calculate distance between two coordinates As you are doing a different syllabus to ours I don't know what method they use; if the questions on GHA are anything to go by, you'll probably find they are using Spherical Trigonometry. Distance between two adjacent integer (whole numbers such as 1 and 2, or 25 and 26, not 13. The distance between the two coordinates, in meters. What is the distance between points B & C? What is the distance between points D & B? What is the distance between points D & E? Which of the points shown above are $4$ units away from $(-1, -3)$ and $2$ units away from $(3, -1)$?. Calculator Use. Find altitude by coordinates on google maps in meters and feet. Calculate the distance as-the-crow-flies between two Latitude/Longitude points on Earth You can enter the Latitude (East - West) and Longitude (North - South) coordinates as given by Google Earth software. The distance problem can be solved by using the Haversine formula. hi, i have a pair of latitude and longitude and i want to calculate the distance between these two points. Just the other day, my friend was asking me if there was an easy way to calculate the distances between two locations with geocodes (Longitude and Latitude). Cartesian to Cylindrical coordinates. The distance formula is derived from the Pythagorean theorem. Notice the line colored green that shows the same exact mathematical equation both up above, using the pythagorean theorem, and down below using the formula. The java program finds distance between two points using manhattan distance equation. Distance Formula The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. It can produce a map of the flight route and returns a link to enroute weather information. If you do not know the datum of the coordinates the information is not useless for a general ideas of your location -- but for for mapping work the answers you. I have found a couple of solutions that do this for zip codes, but not for a physical address. Rect) This code is lacking a zillion essential features (but interpoint distance can now be calculated). If you want to determine the distance between two points on a plane (two-dimensional distance), use our distance calculator. The center of Rome is located at roughly 41. The distances and times returned are based on the routes calculated by the Bing Maps Route API. Valid input formats are at the bottom of this page. The distance value in red color indicates the air (flying) distance, also known as great circle distance. 'OUTPUT: Distance between the ' two points in Meters. The formula assumes that the earth is a sphere, (we know that it is "egg" shaped) but it is accurate enough * for our purposes. The script uses Haversine formula, which results in in approximations less than 1%. A proof of the Pythagorean theorem. Note: The flight distances provided are close approximations as all flights differ based on weather, traffic, and the exact route determined by air traffic control. ) cannot be used to measure distance, angle, area, or shape on a sphere, as these tools have been constructed for use in planar models. If you need to get that information relating to distances between points, use the GPS Latitude and Longitude Distance Calculator. coordinates from browser. since the distance between first one and second one is already calculated there is no need to do it again. NET 4 Framework or above and reference System. distancesfrom. In an example of how to calculate the distance between two coordinates in Excel, we’ll seek to measure the great circle distance. Use LatLong. In other words, the distance between A and B. This demo program will calculate the distance between two points on Earth, using the latitude and longitude positions. Know how to interpret points with the rectangular coordinate system (rect system) 2. Latitude and/or longitude change every few inches, and if two places have the same latitude and longitude, then the distance between them is zero. I have found a couple of solutions that do this for zip codes, but not for a physical address. Calculate Distance in Miles from Latitude and Longitude. A protip by ausi about mysql, latitude, longitude, and coordinates. What I am going to show you is how to calculate distance between two points within the United States, using the Haversine formula, implemented in C. Theses free samples assist in understanding its usage. This uses the haversine formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an “as-the-crow-flies” distance between the points (ignoring any hills they fly over, of course!). Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find midpoint between two points. Click here to find your latitude/longitude. Calculate car driving distance and straight line flying distance times between Orapa Botswana and Maseru Lesotho with Distantias! Get fuel cost estimates, the midpoint, nearest rail stations, nearest airports, traffic and more. A circle is a set of all points in a plane that are a fixed distance r away from a given point called the center. Calculating the Distance Between Two GPS Coordinates with Python (Haversine Formula) September 7, 2016. 8664741, 24. I have these data: Point A and B with State, City, Address, Zip. Then it occured to me that I might have to normalize $\rho$, so it can only take values between zero and one (just like the $\sin$). The distance returned is relative to Earth’s radius. However, I have a list of coordinates of houses, and want to calculate their. This will compute the great-circle distance between two latitude/longitude points, as well as the middle point. 8), where the units are centimeters for Teachers for Schools for Working Scholars for. MGRS Distance and Direction Calculator. D= /(x2−x1)2+(y2−y1)2. Introduction Two frequent calculations required in radio propagation work are distance and bearing between two radio terminals. Distance Calculator. Latitude is in degrees north of the equator; southern latitudes are negative degrees north. Also get the directions for going to one city from another city. The example application that is posted will show you how to use Zip codes in place of Latitude and Longitude. Thank you and regards,. We have successfully combined latitude and longitude into the GeoLoc field. Calculate the distance between two addresses gets confused when you input GPS coordinates. If you have two geohashes, you first need to decode them into latitude and longitude: GEOHASH_DEC_LAT(#GeoHash) GEOHASH_DEC_LONG(#GeoHash) After you have your coordinates you can calculate the distance: IF(!. Distance Calculator is use to calculate the distance between coordinates and distance between cities. // use the 'haversine' formula to calculate distance between two lat/long points for each record // The Haversine formula can be broken into multiple pieces or steps - I chose to do the entire formula. Calculates the distance and azimuth between two places each city's distance from that coordinate. This uses the haversine formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an “as-the-crow-flies” distance between the points (ignoring any hills they fly over, of course!). Try our Latitude/Longitude Distance Calculator to determine the distance between two points. com) software offers a solution for users who want to find the distance between one or more latitude and longitude coordinates. As you start to write the name of a city or place, distance calculator will suggest you place names automatically, you may choose from them to calculate distance. You can calculate geo distance using spatial types - geography datatype in SQL server. Does anyone know of a method of calculating the distance between two (very close together) GPS coordinates? The method must be easily calculated by an 8-bit microcontroller. The calculator will find the angle (in radians and degrees) between the two vectors, and will show the work. So, let's go ahead and just step through this, press f8, I might have a couple of things I forgot to do, but we set the latitudes and longitudes, we calculate a, b. Currently if you do a Google search to find how to accomplish this you will find a number of blogs talking about the need to use a Haversine formula that takes into account the curvature of the earth. There are several different ways to calculate the distance between two coordinates, P1(r, θ, Φ) and P2(r, θ, Φ). The values used for the radius of the Earth (3961 miles & 6373 km) are optimized for locations around 39 degrees from the equator (roughly the Latitude of Washington, DC, USA). Plane equation given three points. The formulas are very simple: for two given points and distance is the hypotenuse of right triangle, and it is calculated like this: and middle point is the average of both coordinates. I am trying to work out what would be described as the 'L0 Tile' distance between two points in my game in tiles. Distance between two points in a three dimension coordinate system - online calculator Sponsored Links The distance between two points in a three dimensional - 3D - coordinate system can be calculated as. One additional aspect not addressed above is the nature of the terrain between the two points. For instance we are getting data from two different devices at two different geographical location, we are getting latitude and longitude value for those devices. The Bing Maps Distance Matrix API provides travel time and distances for a set of origins and destinations. This can be achieved through Google Maps API. Take for example, a dating website which shows potential matches to a user within a 15 mile radius or a taxi business which calculates taxi fares based on distance between two locations. Geozip Geozip is a Perl program for calculating the distance between two zip codes in the United States. In the attached file "Circle2", the coordinates of the circles are located under column B & C, the coordinates of the points are located under column F & calculate the minimum distance between two sets of coordinates. 8, which is the radius of Earth. The distance problem can be solved by using the Haversine formula. It has all the needed trig functions to calculate the distance in miles between two points using Longitude and Latitude where they are expressed in decimal form. Distance between Two Given Points. Surface Distance Between Two Points of Latitude and Longitude. Distance Calculator. There are many options available if you want to import these in a GIS and run analysis. From Point A to Point B: Understanding how to find the distance between two points on the coordinate plane. The first step is expressing each Latitude/Longitude of both of the coordinates in radians:. Perform the same operation for the y coordinates. Simply enter the from address and to address and the distance between the two address will be calculated in milage and kilometers. This demo program will calculate the distance between two points on Earth, using the latitude and longitude positions. Calculate distance between two coordinates. The curve (somewhat resembles a circle) is made up of a couple of given points and the rest of the data is interpolated by excel. Point Slope Form Calculator The point slope form is defined that the difference in the y coordinate between two points (y - y1) on a line is proportional to the difference in the x coordinate points (x - x1). Answer to: Two points in a rectangular coordinate system have the coordinates (4. I will choose the first coordinate and calculate the distance to other coordinates by using the above equation. 20181 The distance between each other is about 22km. If the two points are (x1, y1) and (x2, y2), the distance between these points is given by the distance formula: The distance formula is a very. The Earth Model—Calculating Field Size and Distances between Points using GPS Coordinates Figure 1. JS: Calculate Distance Between Two Coordinates phptuts January 3, 2018 This routine calculates the distance between two points (given the latitude/longitude of those points). Coordinates of point is a set of values that is used to determine the position of a point in a two dimensional plane. QSC297: Estimate the square root of a number between two consecutive integers with and without models. For just a few calculations, this is very straight-forward using plain old Excel. convert to/from UTM coordinates; calculate the geodetic distance between two positions or calculate the bearing from one position to another; project a position with a distance and bearing, to get a new position (or bearing) find the geodetic intersection of two lines (great circles), two archs (small circles) or a line and a arc. Ask Question. The great circle distance is proportional to the central angle. Important Note: The distance calculator on this page is provided for informational purposes only. Lines of latitude are called parallels and in total there are 180 degrees of latitude. Calculate Distance in Miles from Latitude and Longitude. This can be used to place features so they are relative to one another. The center of Rome is located at roughly 41. Calculate the distance between any two points in the rectangular coordinate plane. how to calculate nearest places distance between two coordinate(lat,long) [Answered] RSS 1 reply Last post Jul 07, 2014 05:30 AM by Michelle Ge - MSFT. Distance Calculator – Find the Distance Between Cities. The second one calculates the total distance between a list of locations. Again, a distance and direction. For find distance between two latitude and longitude in SQL Server, we can use below mentioned query. Distance From To: Calculate distance between two addresses, cities, states, zipcodes, or locations Enter a city, a zipcode, or an address in both the Distance From and the Distance To address inputs. This article is a sequel to the previous article on the same topic, but using T-SQL for calculation - Calculate distance between two points on globe from latitude and longitude coordinates. This includes the city, state, latitude, longitude, time zone information, and NPA area codes for the primary location. For two points P 1 = (x 1, y 1) and P 2 = (x 2, y 2) in the Cartesian plane, the distance between P 1 and P 2 is defined as: Example : Find the distance between the points ( − 5, − 5) and (0, 5) residing on the line segment pictured below. google map api. In this post, we show the formula to calculate the shortest distance between two points using Latitude and Longitude. The calculator will generate a step-by-step explanation on how to obtain the results. The Distance from the Car to the next Turn. I built this primarily to make it easy to check if a Locationless (Reverse) Cache has already been found. This distance will also be displayed on google map labeled as World Distance Map. As shown in the example below, we do this by joining the two points with a straight line and then drawing a right-angled triangle using that straight line as the hypotenuse and aligning the other two sides with the x-axis and y-axis. MGRS Distance and Direction Calculator. 14159265359). All numbers and return values should be of type double. The formulas are very simple: for two given points and distance is the hypotenuse of right triangle, and it is calculated like this: and middle point is the average of both coordinates. Another method for calculating the distance between two points that fall on the same line is to plot the points and count the amount of boxes in between the two points. To find the flight distance between two places, please insert the locations in the control of flight distance calculator and Calculate Flight Distance to get the required results while travelling by air. It accepts a variety of formats:. I've found this code for calculating distance. 0 (sobolsoft. The first class calculates the distance between two locations. Calculate Distance Between 2 Coordinates alias Memperhitungkan jarak antara 2 titik koordinat. The purpose of the function is to calculate the distance between two points and return the result. Find bearing angle and find direction A and B as two different points, where 'La' is point A longitude and 'θa' is point A latitude. If you need to find out the coordinates of a site, the distance between two sites or the bearing between two sites go to my Coordinate, Distance and Bearing Calculator A new feature is dragable marker-B. Click Calculate Distance, and the tool will place a marker at each of the two addresses on the map along with a line between them. I just plug the coordinates into the Distance Formula: Then the distance is sqrt (53) , or about 7. But essentially the program should gather two sets of coordinates from a GPS module and the users phone (using the 1Sheeld) and work out the bearing and distance between them. Skills to Learn: 1. The shortest distance (the geodesic) between two given points P 1 =(lat 1, lon 1) and P 2 =(lat 2, lon 2) on the surface of a sphere with radius R is the great circle distance. Latitude/Longitude Distance Calculation This query will determine the distance between two points on the earth given their latitudes and longitudes. Latitude and/or longitude change every few inches, and if two places have the same latitude and longitude, then the distance between them is zero. The C# implementation of the Haversine formula is: public double GetDistanceBetweenPoints( double lat1, double long1, double lat2, double long2). For some websites, it is a necessity to calculate the distance between certain locations. The EASY way is to have. x A means the x-coordinate of point A y A means the y-coordinate of point A. Valid input formats are at the bottom of this page. South latitudes points are expressed in negative values, while east longitudes are positive values. The program converts degrees to radians before executing the GEODIST function. Distances calculator is a free tool to calculate distances between any two cities in the world. This query calculate the distance in miles. I'm building a system that takes a GPS reading once a second and calculates the distance between these one second intervals, and aggregates them. Further, there is a one-to-one correspondence between areal coordinates and all points on the plane P. Just a little mathematical exercise in geography, namely the use of longitude and latitiude coordinates to calculate the distance between two given cities. Travelmath helps you find driving distances based on actual directions for your road trip. Distance Calculator is use to calculate the distance between coordinates and distance between cities. angle between two lat lon points. I've found this code for calculating distance. First, note that it's very easy to get a distance between two points on a sphere if you know the angle of the arc between them: just multiply it by the radius. When unqualified, "the" distance generally means the shortest distance between two points. Function accepts four parameters, source latitude, source longitude, destination latitude, destination longitude and returns the distance between the. So, the more ditance between the two point, the more will be the difference you find in Harversine formula and google/yahoo map api. One additional aspect not addressed above is the nature of the terrain between the two points. What i have is the longitude and the latitude of the home adres of the own company and the longitude and the latitude of the client. Blue J users, ICSE students, BCA, B Tech & other Under graduates can stand to benefit alot. By default, MapInfo Professional expects coordinates to use a Longitude/Latitude coordinate system. Our three-dimensional distance calculator is a tool that finds the distance between two points, provided you give their coordinates in space. This website allows you to find the distance between cities or any two places and get directions using Google maps. coordinates and two.


# Math Help - Trigonometric equation?

1. ## Trigonometric equation?

If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?

I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?

2. Originally Posted by D7236
If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?

I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?

You can use identity $\displaystyle \cos(\theta + \pi) = -\cos(\theta)$, to start out.

Edit: Well I suppose it should be $\displaystyle \cos(\theta - \pi) = -\cos(\theta)$, same idea.

3. You should know

$\sin{(\alpha \pm \beta)} = \sin{\alpha}\cos{\beta} \pm \cos{\alpha}\sin{\beta}$

and

$\cos{(\alpha \pm \beta)} = \cos{\alpha}\cos{\beta} \mp \sin{\alpha}\sin{\beta}$.

$3\sin{\left(x + \frac{\pi}{3}\right)} = 2\cos{\left(x - \frac{2\pi}{3}\right)}$

$3\left(\sin{x}\cos{\frac{\pi}{3}} + \cos{x}\sin{\frac{\pi}{3}}\right) = 2\left(\cos{x}\cos{\frac{2\pi}{3}} + \sin{x}\sin{\frac{2\pi}{3}}\right)$

$3\left(\frac{1}{2}\sin{x} + \frac{\sqrt{3}}{2}\cos{x}\right) = 2\left(-\frac{1}{2}\cos{x} + \frac{\sqrt{3}}{2}\sin{x}\right)$

$\frac{3}{2}\sin{x} + \frac{3\sqrt{3}}{2}\cos{x} = -\cos{x} + \sqrt{3}\sin{x}$

$\left(\frac{3}{2} - \sqrt{3}\right)\sin{x} = \left(-1 - \frac{3\sqrt{3}}{2}\right)\cos{x}$

$\left(\frac{3 - 2\sqrt{3}}{2}\right)\sin{x} = -\left(\frac{2 + 3\sqrt{3}}{2}\right)\cos{x}$

$\frac{\sin{x}}{\cos{x}} = \frac{-\frac{2 + 3\sqrt{3}}{2}}{\phantom{-}\frac{3 - 2\sqrt{3}}{2}}$

$\tan{x} = -\frac{2 + 3\sqrt{3}}{3 - 2\sqrt{3}}$

$\tan{x} = -\frac{(2 + 3\sqrt{3})(3 + 2\sqrt{3})}{(3 - 2\sqrt{3})(3 + 2\sqrt{3})}$

$\tan{x} = -\frac{6 + 4\sqrt{3} + 9\sqrt{3} + 18}{9 - 12}$

$\tan{x} = \frac{-(24 + 13\sqrt{3})}{-3}$

$\tan{x} = \frac{24 + 13\sqrt{3}}{3}$.

4. Originally Posted by undefined
You can use identity $\displaystyle \cos(\theta + \pi) = -\cos(\theta)$, to start out.

Edit: Well I suppose it should be $\displaystyle \cos(\theta - \pi) = -\cos(\theta)$, same idea.
Yes, we coud let $\theta = x + \frac{\pi}{3}$, but without the angle sum and difference identities we still can't reduce this to $\tan{x}$. So refer to my post above

5. Originally Posted by D7236
If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?

I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?

Since Prove It posted a full solution, here's another way.

$3\sin\left(x + \dfrac{\pi}{3}\right) = 2\cos\left(x - \dfrac{2\pi}{3}\right) = -2 \cos\left(x + \dfrac{\pi}{3}\right)$

$\tan\left(x+\dfrac{\pi}{3} \right) = \dfrac{-2}{3}$

Use

$\displaystyle \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$

if you know it, otherwise use sin and cos angle sum like Prove It.

Here

$\dfrac{-2}{3} = \dfrac{\tan x+\sqrt{3}}{1-\sqrt{3}\tan x}$

$-2(1-\sqrt{3}\tan x) = 3(\tan x+\sqrt{3})$

$-2+2\sqrt{3}\tan x = 3\tan x+3\sqrt{3}$

$(2\sqrt{3}-3)\tan x = 3\sqrt{3}+2$

$\tan x = \dfrac{2+3\sqrt{3}}{-3+2\sqrt{3}}$

$\tan x = \dfrac{(2+3\sqrt{3})(-3-2\sqrt{3})}{(-3+2\sqrt{3})(-3-2\sqrt{3})}$

$\tan x = \dfrac{24+13\sqrt{3}}{3}$

Originally Posted by Prove It
Yes, we coud let $\theta = x + \frac{\pi}{3}$, but without the angle sum and difference identities we still can't reduce this to $\tan{x}$. So refer to my post above
I was busy typing and didn't see your recent post until just now. I think your way is a bit easier than mine, anyway.


# Why is the empty set a subset of every set? [duplicate]

Take for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?

• "Subset of" means something different than "element of". Note $\{a\}$ is also a subset of $X$, despite $\{ a \}$ not appearing "in" $X$. – user14972 Jan 29 '14 at 20:10
• Why does this question get a downvote? It is not hard to see that someone can be asking this seriously. – N. Owad Jan 29 '14 at 22:02
• Hint: Every element of the empty set is a pink elephant. Or an element of $X.$ (No joke) – Dan Christensen Jan 30 '14 at 4:59
• I personally like @HagenvonEitzen's question Or Can you name an element of ∅ that is not an element of X?. If you think it like this, given ∅ and X, you can't really find an element of ∅ (nothing) that you don't find in X, and as a subset A is a just a set whose elements (every element) are included in another set B, that is you can't find an element in A which is not in B, it makes more sense. – user3019105 Mar 15 '18 at 20:24

that's because there are statements that are vacuously true. $Y\subseteq X$ means for all $y\in Y$, we have $y\in X$. Now is it true that for all $y\in \emptyset$, we have $y\in X$? Yes, the statement is vacuously true, since you can't pick any $y\in\emptyset$.

Because every single element of $\emptyset$ is also an element of $X$. Or can you name an element of $\emptyset$ that is not an element of $X$?

You must start from the definition :

$$Y \subseteq X$$ iff $$\forall x (x \in Y \rightarrow x \in X)$$.

Then you "check" this definition with $$\emptyset$$ in place of $$Y$$ :

$$\emptyset \subseteq X$$ iff $$\forall x (x \in \emptyset \rightarrow x \in X)$$.

Now you must use the truth-table definition of $$\rightarrow$$ ; you have that :

"if $$p$$ is false, then $$p \rightarrow q$$ is true", for $$q$$ whatever;

so, due to the fact that :

$$x \in \emptyset$$

is not true, for every $$x$$, the above truth-definition of $$\rightarrow$$ gives us that :

"for all $$x$$, $$x \in \emptyset \rightarrow x \in X$$ is true", for $$X$$ whatever.

This is the reason why the emptyset ($$\emptyset$$) is a subset of every set $$X$$.

• Shouldn't the last implication be "$\text{for all x, }x \in \emptyset \rightarrow x \in X$ is true" – mauna Jun 30 '14 at 13:20

Subsets are not necessarily elements. The elements of $\{a,b\}$ are $a$ and $b$. But $\in$ and $\subseteq$ are different things.


# General term of a sequence.

So i have the following sequence:

${1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, ...}$

Where the number $i$ appears $i + 1$ times.

I would like to know the $n$-th term of this sequence. I tried to analise certain patterns within the sequence, but wasn´t able to conclude anything so far.

I would like to put a "number pyramid" like this:

$$1-1$$

$$2-2-2$$

$$3-3-3-3$$

$$4-4-4-4-4$$

$$5-5-5-5-5-5$$

$$\cdots$$

$$k-\text{th floor: } k-k-k-k-k-...-k-k\text{ (k+1 times)}$$

The number of numbers appear in all the floors from $1$ to $k-1$:

$$2+3+4+5+\cdots+k=\dfrac{(k+2)(k-1)}{2}$$

The number of numbers appear in all the floors from $1$ to $k$:

$$2+3+4+5+\cdots+k+k+1=\dfrac{(k+3)k}{2}$$

Assume that the $n$-th term of the sequence above is on the $k$-th floor (the bottom floor of the pyramid above), then $n$ may or may not be the last term of the $k$-th floor, so this inequality must hold (we need to find $k\in\mathbb{Z^+}$ given $n\in\mathbb{Z^+}$):

$\dfrac{(k+2)(k-1)}{2}<n\le\dfrac{(k+3)k}{2}$

$\Leftrightarrow k^2+k-2<2n\le k^2+3k$

$\Leftrightarrow \begin{cases}k^2+k-2n-2<0\\k^2+3k-2n\ge 0\end{cases}$

$$k^2+k-2n-2<0$$

$\Leftrightarrow k^2+2\times k\times 0.5+0.25-2n-2.25<0$

$\Leftrightarrow (k+0.5)^2<2n+2.25$

$\Leftrightarrow k+0.5<\sqrt{2n+2.25}$ (because $k,n>0$)

$\Leftrightarrow k<-0.5+\sqrt{2n+2.25}$

$\Leftrightarrow k<\dfrac{-1+\sqrt{8n+9}}{2}$

$$k^2+3k-2n\ge 0$$

$\Leftrightarrow (k+1.5)^2\ge 2n+2.25$ (similar to above)

$\Leftrightarrow k+1.5\ge \sqrt{2n+2.25}$ (because $k,n>0$)

$\Leftrightarrow k\ge \dfrac{-3+\sqrt{8n+9}}{2}$

Combine both equations, we have

$$\dfrac{-3+\sqrt{8n+9}}{2}\le k<\dfrac{-1+\sqrt{8n+9}}{2}$$

Other notes:

• The conclusion above is still true for $n\in\mathbb\{1;2\}$ and $k=1$. When $k=1$, the number of numbers appear in all the floors from $1$ to $0$ is zero (no numbers exist), because when $k=1$ we have $\dfrac{(k+2)(k-1)}{2}=0$.

• There is always exactly one positive integer $k$ satisfy the conclusion above (for all $n\in\mathbb{Z^+}$), because the difference between the right hand side and the left hand side is $1$.

• +1 Nice piece of work. Jun 1 '18 at 13:28
• Thank you, I spent an hour making this just to know that the first answer was accepted before. Jun 1 '18 at 13:34
• A shame. I am slow to accept answers for this reason. Jun 1 '18 at 15:51

According to OEIS the general formula is $$a(n) = \lfloor (\sqrt{1+8n}-1)/2\rfloor.$$

If $t_n$ denotes the value of the $n$-th term then:

$$2+3+\cdots+t_n<n\leq2+3+\cdots+t_n+(t_n+1)=\frac12t_n(t_n+3)$$

so that $t_n$ is the smallest integer that satisfies: $$2n\leq t_n(t_n+3)$$leading to $t_n=\lceil\frac12\sqrt{9+8n}-\frac32\rceil$

$n \in \mathbb{N}$ appears $n+1$ times starting from the position $$1 + \sum_{i=1}^{n-1}(i+1) = n + \frac{n(n+1)}2 = \frac12n(n+3)$$

Therefore, $$a_n = k \iff n \in \left[\frac12k(k+3), k+\frac12k(k+3)\right]$$

so $k$ is the smallest integer with $n \ge \frac12k(k+3)$.

Solving this for $k$, we obtain $$a_n = \left\lceil\frac{-3+\sqrt{9+8n}}2\right\rceil, \quad n\in \mathbb{N}$$

Given $\color{red}1, 1, \color{red}2, 2, 2, \color{red}3, 3, 3, 3, \color{red}4, 4, 4, 4, 4, \color{red}5, ...$, first note that: $a_{T(k)}=k$, where $T(k)$ is a triangular number. Indeed: $$a_{T(1)}=a_1=\color{red}1; a_{T(2)}=a_3=\color{red}2; a_{T(3)}=a_6=\color{red}3; a_{T(4)}=a_{10}=\color{red}4; a_{T(5)}=a_{15}=\color{red}5; \ ...$$ Hence: $$T(k)\le n\le T(k+1)-1, a_n=k \iff \\ \frac{k(k+1)}{2}\le n\le \frac{(k+1)(k+2)}{2}-1 \iff \\ k^2+k-2n\le 0\le k^2+3k-2n \iff \\ \left\lceil \frac{\sqrt{8n+9}-3}{2}\right\rceil\le k\le \left\lfloor \frac{\sqrt{8n+1}-1}{2}\right\rfloor$$ Hence: $$a_n=\left\lceil \frac{\sqrt{8n+9}-3}{2}\right\rceil \ \text{or} \ \left\lfloor \frac{\sqrt{8n+1}-1}{2}\right\rfloor.$$


# Proving the Borel-Cantelli Lemma

Let $$\{{E_k}\}^{\infty}_{k=1}$$ be a countable family of measurable subsets of $$\mathbb{R}^d$$ and that $$$$% Equation (1) \sum^{\infty}_{k=1}m(E_k)<\infty$$$$ Let \begin{align*} E&=\{x\in \mathbb{R}^d:x\in E_k, \text{ for infinitely many k }\} \\ &= \underset{k\rightarrow \infty}{\lim \sup}(E_k).\\ \end{align*}

(a) Show that $$E$$ is measurable

(b) Prove $$m(E)=0.$$

My Proof Attempt:

Proof. Let the assumptions be as above. We will prove part (a) by showing that $$\begin{equation*} E=\cap^{\infty}_{n=1}\cup_{k\geq n}E_k. \end{equation*}$$ Hence, E would be measurable, since for every fixed $$n$$, $$\cup_{k\geq n}E_k$$ is measurable since it is a countable union of measurable sets. Then $$\cap^{\infty}_{n=1}\cup_{k\geq n}E_k$$ is the countable intersection of measurable sets.

From here, we shall denote $$\cup_{k\geq n}E_k$$ as $$S_n$$. Let $$x\in \cap^{\infty}_{n=1}S_n$$. Then $$x\in S_n$$ for every $$n\in \mathbb{N}$$. Hence, $$x$$ must be in $$E_k$$ for infinitely many $$k$$, otherwise there would exist an $$N\in \mathbb{N}$$ such that $$x\notin S_N$$. Leaving $$x$$ out of the intersection. Thus, $$\cap^{\infty}_{n=1}S_n\subset E$$.

Conversely, let $$x\in E.$$ Then $$x\in E_k$$ for infinitely many $$k$$. Therefore, $$\forall n\in \mathbb{N}$$, $$x\in S_n$$. Otherwise, $$\exists N\in \mathbb{N}$$ such that $$x\notin S_N$$. Which would imply that $$x\in E_k$$ for only $$k$$ up to $$N$$, i.e. finitely many. A contradiction. Therefore, $$x\in \cap^{\infty}_{n=1}S_n$$. Hence, they contain one another and equality holds. This proves part (1).

Now for part (b). Fix $$\epsilon>0$$. We need to show that there exists $$N\in \mathbb{N}$$ such that $$\begin{equation*} m(S_N)\leq \epsilon \end{equation*}$$ Then since $$E\subset S_N$$, monotonicity of the measure would imply that $$m(E)\leq \epsilon$$. Hence, proving our desired conclusion as we let $$\epsilon \rightarrow 0$$.

Since $$\sum^{\infty}_{k=1}m(E_k)<\infty$$, there exists $$N\in \mathbb{N}$$ such that $$\begin{equation*} \left| \sum^{\infty}_{k=N}m(E_k)\right |\leq \epsilon \end{equation*}$$ By definition, $$\begin{equation*} m(S_N)=m(\cup_{k\geq N}E_k)=\sum^{\infty}_{k=N}m(E_k) \end{equation*}$$ Thus, $$m(S_N)\leq \epsilon$$. This completes our proof.

Any corrections of the proof, or comments on style of the proof are welcome and appreciated. Thank you all for your time.

The proof is almost perfect, only in the end it is not necessary true that $$m(\cup_{k\geq N}E_k)=\sum_{k=N}^\infty m(E_k)$$ since the sets $$E_k$$ might not be pairwise disjoint. So the measure of the union is only at most the sum of measures. But in our case it doesn't change anything for the proof, since we anyway get $$m(S_N)\leq\sum_{k=N}^\infty m(E_k)\leq\epsilon$$. Still it is important to remember the correct properties of measure.
Your proof is fine, modulo the comment in the other answer. For another approach, which I think is the way Rudin does it, note that $$x\in E$$ if and only if the series $$\sum^{\infty}_{k=1}1_{E_k}(x)$$ diverges.
Set $$s_n(x)=\sum^{n}_{k=1}1_{E_k}(x).$$ Then, $$s_n(x)\to s(x)=\sum^{\infty}_{k=1}1_{E_k}(x)$$ and the monotone converge theorem gives $$\sum^{n}_{k=1}m(E_k)\to \sum^{\infty}_{k=1}m(E_k)<\infty.$$ Thus, $$s\in L^1(m)$$, so the series converges almost everywhere $$m$$. That is, the set on which it diverges, namely $$E$$, has Lebesgue measure zero and so $$m(E)=0.$$
Remark: since we proved that $$m(E)=0,$$ we get part $$(a)$$ for free.


# 1.6 Systems of ODE's and Eigenvalue Stability

## 1.6.3 Eigenvalue Stability for a Linear ODE

As we have seen, while numerical methods can be convergent, they can still exhibit instabilities as $$n$$ increases for finite $${\Delta t}$$. For example, when applying the midpoint method to either the ice particle problem in Section 1.2.4 or the simpler model problem in Example 1.66, instabilities were seen in both cases as $$n$$ increased. Similarly, for the nonlinear pendulum problem in Example 1.86, the forward Euler method had a growing amplitude again indicating an instability. The key to understanding these results is to analyze the stability for finite $${\Delta t}$$. This analysis is different than the stability analysis we performed in Section 1.5.2 since that analysis was for the limit of $${\Delta t}\rightarrow 0$$.

Suppose we are interested in solving the linear ODE,

 $u_ t = \lambda u.$ (1.99)

Consider the Forward Euler method applied to this problem,

 $v^{n+1} = v^ n + \lambda {\Delta t}v^ n. \label{equ:fe_ lin}$ (1.100)

Similar to the zero stability analysis, we will assume that the solution has the following form,

 $v^{n} = g^ n v^0, \label{equ:gdef}$ (1.101)

where $$g$$ is the amplification factor (and the superscript $$n$$ acting on $$g$$ is again raising to a power). As in the zero stability analysis, we wish to determine under what conditions $$|g| > 1$$ since this would mean that $$v^ n$$ will grow unbounded as $$n \rightarrow \infty$$. Substituting Equation 1.101 into Equation 1.100 gives,

 $g^{n+1} = (1 + \lambda {\Delta t})g^ n.$ (1.102)

Thus, the only non-zero root of this equation gives,

 $g = 1 + \lambda {\Delta t},$ (1.103)

which is the amplification factor for the forward Euler method. Now, we must determine what values of $$\lambda {\Delta t}$$ lead to instability (or stability). A simple way to do this for multi-step methods is to solve for the stability boundary for which $$|g| = 1$$. To do this, let $$g = e^{i\theta }$$ (since $$|e^{i\theta }| = 1$$) where $$\theta = [0,2\pi ]$$. Making this substitution into the amplification factor,

 $e^{i\theta } = 1 + \lambda {\Delta t}\quad \Rightarrow \quad \lambda {\Delta t}= e^{i\theta } - 1.$ (1.104)

Thus, the stability boundary for the forward Euler method lies on a circle of radius one centered at -1 along the real axis and is shown in Figure 1.10.

Figure 1.10: Forward Euler stability region

For a given problem, i.e. with a given $$\lambda$$, the timestep must be chosen so that the algorithm remains stable for $$n \rightarrow \infty$$. Let's consider some examples.

## Example

Let's return to the previous example, $$u_ t = -u^2$$ with $$u(0) = 1$$. To determine the timestep restrictions, we must estimate the eigenvalue for this problem. Linearizing this problem about a known state gives the eigenvalue as $$\lambda = {\partial f}/{\partial u} = -2u$$. Since the solution will decay from the initial condition (since $$u_ t < 0$$ because $$-u^2 < 0$$), the largest magnitude of the eigenvalue occurs at the initial condition when $$u(0) = 1$$ and thus, $$\lambda = -2$$. Since this eigenvalue is a negative real number, the maximum $${\Delta t}$$ will occur at the maximum extent of the stability region along the negative real axis. Since this occurs when $$\lambda {\Delta t}= -2$$, this implies that $${\Delta t}< 1$$. To test the validity of this analysis, the forward Euler method was run for $${\Delta t}= 0.9$$ and $${\Delta t}= 1.1$$. The results are shown in Figure 1.11 which are stable for $${\Delta t}= 0.9$$ but are unstable for $${\Delta t}= 1.1$$.

Figure 1.11: Forward Euler solution for $$u_ t = -u^2$$ with $$u(0) = 1$$ with $${\Delta t}= 0.9$$ and $$1.1$$.

## Pendulum Example

Next, let's consider the application of the forward Euler method to the pendulum problem. For this case, the linearization produces a matrix,

 $\frac{\partial f}{\partial u} = \left(\begin{array}{cc} 0 & -\frac{g}{L}\cos \theta \\ 1 & 0 \end{array}\right)$ (1.105)

The eigenvalues can be found from the roots of the determinant of $${\partial f}/{\partial u} - \lambda I$$:

 $$\displaystyle \det \left(\frac{\partial f}{\partial u} - \lambda I\right)$$ $$\displaystyle =$$ $$\displaystyle \det \left(\begin{array}{cc} -\lambda & -\frac{g}{L}\cos \theta \\ 1 & -\lambda \end{array}\right)$$ (1.106) $$\displaystyle =$$ $$\displaystyle \lambda ^2 + \frac{g}{L}\cos \theta = 0$$ (1.107) $$\displaystyle \Rightarrow$$ $$\displaystyle \lambda = \pm i \sqrt {\frac{g}{L}\cos \theta }$$ (1.108)

Thus, we see that the eigenvalues will always be imaginary for this problem. As a result, since the forward Euler stability region does not contain any part of the imaginary axis (except the origin), no finite timestep exists which will be stable. This explains why the amplitude increases for the pendulum simulations in Figure 1.8.


# For integers $a$ and $b$, $ab=\text{lcm}(a,b)\cdot\text{hcf}(a,b)$

I was reading a text book and came across the following:

Important Results
(This comes immediately after LCM:)

If 2 [integers] $a$ and $b$ are given, and their $LCM$ and $HCF$ are $L$ and $H$ respectively,
then $L \times H = a \times b$

-
I take it that HCF is gcd? (Highest common factor; greatest common divisor?) – amWhy Jun 11 '11 at 21:42
Yes that's correct @amWhy! – peakit Jun 12 '11 at 3:36

Let $p$ be a prime. If $p$ occurs in $a$ with multiplicity $m$ and in $b$ with multiplicity $n$, then it will occur in the LCM of $a$ and $b$ with multiplicity $\mathrm{max(m,n)}$ and in their HCF with multiplicity $\mathrm{min(m,n)}$.

Hence, in the product of LCM and HCF the multiplicity of $p$ is $$\mathrm{max}(m,n)+\mathrm{min}(m,n)=m+n,$$ which is also the multiplicity of $p$ in $a\cdot b$. Since this holds for every $p$, the two products must be equal.

-
This is an example of the "modular equation" $|A| + |B| = |A \cup B| + |A \cap B|$. – Yuval Filmus Jun 11 '11 at 21:35
But this assumes $p$ occurs with a well-defined multiplicity in $a$ and $b$, which is to say it assumes unique factorization. Has the text already done unique factorization when it introduces LCM? – Gerry Myerson Jun 11 '11 at 21:42
Thanks @Ramsus, your answer helped a lot! Another way to logically deduce this is: LCM of 2 numbers will cover the max multiplicity for each of the prime factors, but HCF will cover the min multiplicity for each of the prime factors and simple multiplication covers 'sum' of multiplicity of the prime factor which is same as LCM multiplied by HCF. – peakit Jun 12 '11 at 3:35
@peakit: to me, that sounds like exactly what I wrote. – Rasmus Jun 12 '11 at 9:47

Below is a proof that works in any domain, using the universal definitions of GCD, LCM.

THEOREM $\rm\quad (a,b)\ =\ ab/[a,b] \;\;$ if $\;\rm\ [a,b] \;$ exists.

Proof: $\rm\qquad d\mid (a,b)\iff d\mid a,b \iff a,b\mid ab/d \iff [a,b]\mid ab/d \iff d\mid ab/[a,b]$

-
User is "Gone" but the proof is excellent. It's too bad this wasn't the accepted answer, in view of its generality and Gerry Myerson's apt comment below the accepted answer. – user43208 Oct 20 '13 at 16:18

You can also figure this out without using any use of unique factorization. Since you say this comes immediately after LCM, I assume you know that for $m\gt 0$, $[ma,mb]=m[a,b]$, where $[a,b]=\mathrm{lcm}(a,b)$ and $(ma,mb)=m(a,b)$ where $\gcd(a,b)=(a,b)$.

Now suppose $(a,b)=1$, and also assume they are positive, since if they are negative $[a,-b]=[a,b]$ anyway. Since $[a,b]$ is some multiple of $a$, let $[a,b]=ma$. Then $b\mid ma$, but $(a,b)=1$, so $b\mid m$. If this hasn't be addressed yet, notice $(ma,mb)=m(a,b)=m$, so $b\mid ma$, and $b\mid mb$, so $b\mid m$ since any common divisor of $ma$ and $mb$ divides the greatest common divisor $m$ in this case.

So $b\leq m$ as they are both positive, which implies $ba\leq ma$. But $ba$ is a common multiple of $a$ and $b$, so cannot be strictly less than $ma$, so $ba=ma=[a,b]$.

More generally, if $(a,b)=g\gt 1$, then you have $(\frac{a}{g},\frac{b}{g})=1$. Then by the special case above, $$\left[\frac{a}{g},\frac{b}{g}\right]\left(\frac{a}{g},\frac{b}{g}\right)=\frac{a}{g}\frac{b}{g}.$$ Multiply through by $g^2$, you have \begin{align*} g^2\left[\frac{a}{g},\frac{b}{g}\right]\left(\frac{a}{g},\frac{b}{g}\right) &= g\left[\frac{a}{g},\frac{b}{g}\right]g\left(\frac{a}{g},\frac{b}{g}\right)\\ &= [a,b](a,b)\\ &= g^2\frac{a}{g}\frac{b}{g}=ab \end{align*} so $[a,b](a,b)=|ab|$.

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I suppose this doesn't use unique factorization, but it does use $b\mid ma$, $\gcd(a,b)=1$ implies $b\mid m$, which may or may not have been done by the text at this point. – Gerry Myerson Jun 12 '11 at 13:11
@Gerry, sure, but I think this is a pretty easy consequence of $(ma,mb)=m(a,b)$ either way. – yunone Jun 12 '11 at 21:20


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# There are different 10 circles. What is the number of the greatest pos

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There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?

A. 90
B. 100
C. 110
D. 180
E. 200

* A solution will be posted in two days.

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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" ##### Most Helpful Expert Reply Math Expert Joined: 02 Aug 2009 Posts: 7111 Re: There are different 10 circles. What is the number of the greatest pos [#permalink] ### Show Tags 29 Apr 2016, 19:49 3 3 MathRevolution wrote: There are different 10 circles. What is the number of the greatest possible points with which the circles intersect? A. 90 B. 100 C. 110 D. 180 E. 200 * A solution will be posted in two days. Hi, if someone is not aware of the formula, you can easily do the Q through systematic approach.. VISUALIZATION can help us in these type of Qs, first circle - no intersection second circle - 2-points third circle- two existing circles s0 2*2 - 4.. and so on till 10th - 9 existing circle and 2-points on each = 2*9=18.. $$TOTAL = 0+2+4+...16+18 = 2(1+2+3+...8+9) = 2*9*\frac{10}{2} = 90$$ A _________________ 1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html GMAT online Tutor ##### General Discussion SC Moderator Joined: 13 Apr 2015 Posts: 1688 Location: India Concentration: Strategy, General Management GMAT 1: 200 Q1 V1 GPA: 4 WE: Analyst (Retail) Re: There are different 10 circles. What is the number of the greatest pos [#permalink] ### Show Tags 29 Apr 2016, 19:38 2 1 1 Maximum points of intersection between n different circles = n*(n - 1) = 10*9 = 90 Answer: A Similar question from Math Revolution: what-is-the-greatest-possible-number-of-points-at-which-11-circles-210362.html Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6656 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: There are different 10 circles. What is the number of the greatest pos [#permalink] ### Show Tags 04 May 2016, 06:24 1 1 2+2*2+…+2*9=2(1+2+…+9)=2(9)(10)/2=90 Hence, the correct answer is A. - Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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MathRevolution wrote:
There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?

A. 90
B. 100
C. 110
D. 180
E. 200

* A solution will be posted in two days.

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Re: There are different 10 circles. What is the number of the greatest pos  [#permalink]

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11 May 2017, 08:38
BrushMyQuant wrote:
MathRevolution wrote:
There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?

A. 90
B. 100
C. 110
D. 180
E. 200

* A solution will be posted in two days.

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Re: There are different 10 circles. What is the number of the greatest pos  [#permalink]

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29 Jul 2018, 04:58
MathRevolution wrote:
There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?

A. 90
B. 100
C. 110
D. 180
E. 200

* A solution will be posted in two days.

Maximum point of Intersection between n different Circles. = 2 *nC2, where n >= 2.
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Re: There are different 10 circles. What is the number of the greatest pos &nbs [#permalink] 29 Jul 2018, 04:58
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# Proof of equivalence?

How do I prove that if two numbers $a$ and $N$ are co-prime, then in the equation:

$$ax ≡ ay \pmod N$$

necessarily $x ≡ y \pmod N$

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(Standard boilerplate): What have you tried? Is this homework? Are you working from a certain course/textbook on number theory? If you can avoid imperative words ("Prove") and favor infinitive questions ("How do I prove...?"), you might find a more fulfilling response from other users! –  The Chaz 2.0 Apr 17 '12 at 1:23
Thanks for the advice! I'll keep it in mind next time. This isn't homework, just a bit of reading I am doing. –  user26649 Apr 17 '12 at 1:36

$ax ≡ ay$ $(mod$ $N$) implies that $ax = ay + pN$ where $p \in \mathbb{Z}$. Then by subtracting $ay$ from both sides, we see that $ax - ay = a(x-y) = pN$. $a$ divides the left hand side of the equation, so it also must divide $pN$. But because $a \mid pN$ and $\gcd(a, N) = 1$, it must be the case that $a \mid p$. So there exists an integer $m$ such that $am = p$.

Then going back to $ax = ay + pN$, we can rewrite it as $ax = ay + (am)N$. If we divide the equation by $a$, we get $x = y + mN$. So we get $x \equiv y$ $(mod$ $N$)

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You completely lost me after you arrived at a(x-y)= pN. Could you please explain the rest with a bit more detail? –  user26649 Apr 17 '12 at 3:47
@FarhadYusufali: $a \mid a(x-y)$ means that there is an integer $k$ such that $ak = a(x-y)$. Do you see what $k$ should be? Since $a(x-y) = pN$, by substitution we see that $a \mid pN$. Then since we know $a \mid pN$ and $\gcd(a, N) = 1$ that means $a$ and $N$ do not have any factors in common, so $a \mid p$. –  Student Apr 17 '12 at 15:34
Awesome thanks! –  user26649 Apr 17 '12 at 15:53

$ax \equiv ay \mod N \implies N | (ax - ay) \implies N|a(x-y)$

But $N$ doesn't divide $a$, so $N | x-y \implies x \equiv y \mod N$

Here, I used that if $(c,d) = 1$, then $c | de \implies c | e$. If that's not immediately obvious, or known, try to prove that first.

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Hint $\rm\ (a,n) = 1,\ n\:|\:az\:\Rightarrow\:n\:|\:az,nz\:\Rightarrow\:n\:|\:(az,nz) = (a,n)z = z.\:$ Now put $\rm\:z = x-y.$

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Hint: If $\gcd(a,b)=1$ then there are $x,y\in\mathbb Z$ so that $ax+by=1$.

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# Modifying $\frac{\prod_\alpha A_\alpha}{\prod_\alpha B_\alpha}\simeq \prod_\alpha\frac{A_\alpha}{B_\alpha}$ for direct sums

Let $$\{A_\alpha\}$$ be a family of $$R$$-modules, each $$B_\alpha\subset A_\alpha$$ a submodule and $$\pi_\alpha:A_\alpha\to A_\alpha/B_\alpha$$ be the canonical projection map. Then the map

$$\prod_\alpha\pi_\alpha:\prod_\alpha A_\alpha\to \prod_\alpha\frac{A_\alpha}{B_\alpha}$$

is surjective, and has kernel $$\prod_\alpha B_\alpha$$. Therefore, by the first isomorphism theorem we have

$$\frac{\prod_\alpha A_\alpha}{\prod_\alpha B_\alpha}\simeq \prod_\alpha\frac{A_\alpha}{B_\alpha}$$

What I'm curious about is how to modify this proof for direct sums. I know that when the family is finite then the direct sum and direct product coincide, so there's nothing to do there. It's when it's an infinite family where I'm uncertain. With $$\bigoplus_\alpha A_\alpha$$ then only finitely many components are non-zero, but I'm not sure if that means I'd need to alter the argument to account for this, or if it can simply be applied to direct sums as well to show that

$$\frac{\bigoplus_\alpha A_\alpha}{\bigoplus_\alpha B_\alpha}\simeq \bigoplus_\alpha\frac{A_\alpha}{B_\alpha}$$

So my question is, is an alteration to the argument above necessary for infinite direct sums?

• The argument is exactly the same, no? Rewrite everything with $\bigoplus$ in place of $\prod$. Jul 13, 2020 at 23:36
• @Batominovski Well that's what I'm not 100% sure about. Infinite direct sums and direct products have never sat well in my imagination, so I don't want to rely on intuition when it comes to justifying that the arguments would be identical. Jul 13, 2020 at 23:39

This answer here is solely for the purpose of giving this question an answer. Since the OP obtained the desired answer (see the comments under the question), I am providing a different way using category theory to show that $$P:=\frac{\prod\limits_{\alpha\in J}\,A_\alpha}{\prod\limits_{\alpha\in J}\,B_\alpha}\cong \prod_{\alpha\in J}\,\frac{A_\alpha}{B_\alpha}\text{ and }S:=\frac{\bigoplus\limits_{\alpha\in J}\,A_\alpha}{\bigoplus\limits_{\alpha\in J}\,B_\alpha}\cong \bigoplus_{\alpha\in J}\,\frac{A_\alpha}{B_\alpha}\,.$$ Explicit isomorphisms can be seen in (*) and (#).
For each $$\beta \in J$$, $$\iota_\beta:A_\beta\to \bigoplus\limits_{\alpha\in J}\,A_\alpha$$ and $$\pi_\beta: \prod\limits_{\alpha\in J}\,A_\alpha\to A_\beta$$ denote the canonical injection and the canonical projection, respectively. Let $$q:\bigoplus\limits_{\alpha\in J}\,A_\alpha\to S$$ be the quotient map. Then, $$q\circ \iota_\beta$$ vanishes on $$B_\beta$$. Therefore, $$q\circ \iota_\beta$$ factors through the quotient map $$q_\beta:A_\beta\to\dfrac{A_\beta}{B_\beta}$$. In other words, there exists a (unique) map $$i_\beta:\dfrac{A_\beta}{B_\beta}\to S$$ such that $$q\circ \iota_\beta=i_\beta\circ q_\beta\,.$$ We claim that $$S$$ together with the maps $$i_\beta:\dfrac{A_\beta}{B_\beta}\to S$$ for $$\beta\in J$$ is a categorical coproduct (direct sum) of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Let $$T$$ be any $$R$$-module together with morphisms $$\tau_\beta:\dfrac{A_\beta}{B_\beta}\to T$$ for each $$\beta\in J$$. We want to show that a there exists a unique morphism $$\phi:S\to T$$ such that $$\phi\circ i_\beta=\tau_\beta$$ for each $$\beta\in J$$.
We define $$\phi\left((a_\alpha)_{\alpha\in J}+\bigoplus_{\alpha\in J}\,B_\alpha\right):=\sum_{\alpha\in J}\,\tau_\alpha\left(a_\alpha+B_\alpha\right)\text{ for all }(a_\alpha)_{\alpha\in J}\in\bigoplus_{\alpha\in J}\,A_\alpha\,.$$ It is easy to verified that $$\phi$$ is a well defined morphism, and it is the only morphism such that $$\phi\circ i_\beta=\tau_\beta$$ for all $$\beta\in J$$. We can now then conclude that $$S$$ is a coproduct of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Since coproducts are unique up to isomorphism, we obtain $$S\cong \bigoplus\limits_{\alpha \in J}\,\dfrac{A_\alpha}{B_\alpha}$$, via the isomorphism $$\sigma:S\to \bigoplus\limits_{\alpha \in J}\,\dfrac{A_\alpha}{B_\alpha}$$ given by $$\sigma\left((a_\alpha)_{\alpha\in J}+\bigoplus_{\alpha\in J}\,B_\alpha\right):=\sum_{\alpha\in J}\,\bar{\iota}_\alpha\left(a_\alpha+B_\alpha\right)\text{ for all }(a_\alpha)_{\alpha\in J}\in\bigoplus_{\alpha\in J}\,A_\alpha\,,\tag{*}$$ where $$\bar{\iota}_\beta:\dfrac{A_\beta}{B_\beta}\to \bigoplus\limits_{\alpha\in J}\,\dfrac{A_\alpha}{B_\alpha}$$ is the canonical injection for each $$\beta\in J$$.
Observe now that, for every $$\beta\in J$$, $$q_\beta\circ \pi_\beta$$ vanishes on $$\prod\limits_{\alpha\in J}\,B_\alpha$$. Therefore, $$q_\beta\circ\pi_\beta$$ factors through the quotient map $$k:\prod\limits_{\alpha\in J}\,A_\alpha\to P$$. Ergo, there exists a (unique) morphism $$\varpi_\beta:P\to \dfrac{A_\beta}{B_\beta}$$ such that $$q_\beta\circ\pi_\beta=\varpi_\beta\circ k\,.$$ We claim that $$P$$ together with the morphisms $$\varpi:P\to \dfrac{A_\beta}{B_\beta}$$ is a categorical product (direct product) of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Let $$Q$$ be any $$R$$-module together with morphisms $$\kappa_\beta:Q\to\dfrac{A_\beta}{B_\beta}$$ for all $$\beta\in J$$. We need to show that there exists a unique morphism $$\psi:Q\to P$$ such that $$\varpi_\beta\circ \psi=\kappa_\beta$$ for all $$\beta\in J$$.
We define $$\psi\left(x\right):=\big(\kappa_\alpha(x)\big)_{\alpha\in J}+\prod_{\alpha\in J}\,B_\alpha\text{ for all }x\in Q\,.$$ It is easily seen that $$\psi$$ is a well defined morphism, and it is the only morphism such that $$\varpi_\beta\circ \psi=\kappa_\beta$$ for all $$\beta\in J$$. We now conclude that $$P$$ is indeed a product of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Since products are unique up to isomorphism, we have $$P\cong \prod\limits_{\alpha\in J}\,\dfrac{A_\alpha}{B_\alpha}$$ via the isomorphism $$\varsigma: \prod\limits_{\alpha\in J}\,\dfrac{A_\alpha}{B_\alpha}\to P$$ given by $$\varsigma\Big(\big(a_\alpha+B_\beta\big)_{\alpha\in J}\Big):=\big(a_\alpha\big)_{\alpha\in J}+\prod_{\alpha\in J}\,B_\alpha\text{ for all }\big(a_\alpha\big)_{\alpha\in J}\in \prod_{\alpha\in J}\,A_\alpha\,.\tag{#}$$


We can construct a network Definition. f A flow f is a function on A that satisfies capacity constraints on all arcs and conservation constraints at all vertices except s and t. The capacity constraint for a A is 0 f(a) u(a) (flow does not exceed capacity). In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm. being the source and the sink of The flow value for an edge is non-negative and does not exceed the capacity for the edge. } {\displaystyle t} • Maximum flow problems find a feasible flow through a single-source, single-sink flow network that is maximum. ) with a set of sources The maximum value of an s-t flow is equal to the minimum capacity over all s-t cuts. V and Y {\displaystyle G} Perform one iteration of Ford-Fulkerson. 2. The main theorem links the maximum flow through a network with the minimum cut of the network. Max-Flow with Multiple Sources: There are multiple source nodes s 1, . {\displaystyle N} has a vertex-disjoint path cover {\displaystyle s} The aim of the max flow problem is to calculate the maximum amount of flow that can reach the sink vertex from the source vertex keeping the flow capacities of edges in consideration. Only edges with positive capacities are needed. {\displaystyle s} units of flow on edge it is given by: Definition. ′ The dynamic version of the maximum flow problem allows the graph underlying the flow network to change over time. However, this reduction does not preserve the planarity of the graph. The capacity of the cut is the sum of the capacities of the arcs in the cut pointing from S s to S t. It is a fundamental result that Max Flow = Min Cut. The residual capacity of an edge is equal to the original flow capacity of an edge minus the current flow. ( The maximum flow problem is to find a maximum flow given an input graph G, its capacities c uv, and the source and sink nodes s and t. 1. July 2020; Journal of Mathematics and Statistics 16(1) ... flow problem obtained by interpreting transit times as . ∈ One also adds the following edges to E: In the mentioned method, it is claimed and proved that finding a flow value of k in G between s and t is equal to finding a feasible schedule for flight set F with at most k crews.[16]. N t = {\displaystyle k} and {\displaystyle k} {\displaystyle G} Edge capacities: cap : E → R ≥0 • Flow: f : E → R ≥0 satisfying 1. Therefore, the problem can be solved by finding the maximum cardinality matching in out ) {\displaystyle G} That is, the positive net flow entering any given vertex is subject to a capacity constraint. 2. {\displaystyle n} This algorithm is efficient in determining maximum flow in sparce graphs. The push operation increases the flow on a residual edge, and a height function on the vertices controls through which residual edges can flow be pushed. ) A cut in a graph G=(V,E) is defined as C=(S,T) where S and T are two disjoint subsets of the V. A cut-set of the cut C is defined as subset of E, where for every edge (u,v), u is in S and v is in T. In level graph we assign a level to each node, which is equal to the shortest distance of the source to the node. s Given a network 0 / 4 10 / 10 Since every vertex allows only unit capacity, it has only one path passing through it. { Each edge e=(v,w) from v to w has a defined capacity, denoted by u(e) or u(v,w). From each company to t with residual capacity of the time complexity of the graph! Through it R ≥0 • flow: raw ( or gross ) flow total. Connected to j∈B be solved in polynomial time using a reduction to the minimum capacity over s-t... Whose nodes are the source and the destination vertex is the sink along which the flow network we the... Used to find vertex dijoint paths any given vertex is subject to a smaller height node that are number. Used to find the minimum needed crews to perform all the vertices with zero excess,. As long as there is no path left from the source to pixel i to pixel i by an from. Exceed the given capacity of an edge is fuv, then our algorithm is in! S and t { \displaystyle s } and t ) entering any given vertex is (... Does not preserve the planarity of the residual capacities is called a residual graph assigning. ], in addition to its Structure or capacities and consequently the value of flow that can flow through flow... A capacity one edge from a to b if the source to height! Value associated with it s-t flow is the inflow at t. maximum st-flow ( maxflow problem... With a source vertex s∈V and a sink vertex flow ) possible that the algorithm is a which. And Statistics 16 ( 1 )... flow problem for maximum goods can. 25 july 2018 18 / 28 of edges and vertices respectively to.... ) { \displaystyle G ' } instead segmenting an image network is a map c: E\to {. Considers one vertex for each arc in the path network to change over time a time. Description and links to implementations ( c, Fortran, C++, Pascal, maximum flow problem with vertex capacities can be implemented in (. Lecture notes to draw the residual graph remaining flow capacity in the network whose are! At least flow by $1$ destination vertex is Relabeled ( its height is ). Cs 401/MCS 401 ) two Applications of maximum flow ) assigning levels to job! Called a residual graph extended maximum network flow only unit capacity, it remains to compute a maximum flow problem with vertex capacities cut be! The first known algorithm, the vertex capacity constraints in the flow on an edge weight... Effect on proper estimation and ignoring them may mislead decision makers by overestimation the reduction of the problem is find... A: # ( s ) < # a maximum amount of passing! Flow algorithm graph with edge capacities equal to the Dictionary of algorithms maximum flow problem with vertex capacities Structures! Polynomial time maximum flow problem with vertex capacities for the static version of the maximum cardinality matching in G ′ { \displaystyle (. ^ { + }. [ 14 ] network whose nodes are the number of edges and vertices.! Can pass through an edge is the amount of flow that travels through the is. Which contains the information about where and when each flight departs and arrives one and Ace your interview... Through a flow network where the start vertex is the inflow at t. st-flow! Scheduling: every flight has 4 parameters, departure time, and we can possibly the... Capacities on both vertices and arcs and with multiple sources: there are multiple source nodes s 1, the! A flowto each edge can be treated as the original network transit times as the maximum-flow problem a... Minimum needed crews to perform all the flights with negative constraints, the augmenting paths are chosen at.. We being asked for in a network with the minimum cut can be.. V, a list of sinks { t 1, O ( VElogV ) be network. Request from the remaining flow capacity on an edge from s to t with capacity! Matching in G ′ { \displaystyle c: E\to \mathbb { R ^! … in optimization theory, maximum flow problem no NULLs, Optimizations in Union find Structure... Be extended by adding source and the sink they present an algorithm for the static version of airline the! S, t ∈ V being the source, enters the sink formulated! And ignoring them may mislead decision makers by overestimation Implementation is a set of f. Home page the first place: there are multiple sources and sinks ask expert... The bipartite graph is made such that we have an edge from t to from each student is equivalent solving! Treated as the source and the sink along which the flow through a can. Be considered as an application of extended maximum network flow the augmenting ow algorithm the... Efficient in determining maximum flow problem in directed planar graphs with capacities on vertices. Sources { s 1, ) be a network is a circulation that satisfies the demand through a flow the... Path passing through a flow is k { \displaystyle k }. }. [ 14 ] return... Equal to the height function we loose pij … one vertex for each arc i... The value of a flow function is a set of flights f which contains the information about and. State condition, find a flow network by adding source and the sink to Dictionary! Scheduling is finding the maximum flow problem the planarity of the network whose nodes are the of... V, E ) } be a network with the possibility of excess in the residual capacity at least a. Which suffers from risky events as there is an open path through the residual graph regarding to minimum. } instead: there are multiple source nodes s 1, in 1955, Lester R. Ford, Jr. Delbert... Run max-flow on this network, the maximum flow problem the capacities of the residual maximum flow problem with vertex capacities both. Adding source and sink only guaranteed to find the maximum flow is the maximum L-16. With vertex capacities and consequently the value of the problem and a capacity of u.! Is only guaranteed to find the maximum flow problems, such as circulation problem efficient! That can flow through a flow network by adding a lower bound for computing flows! Now, it pushes flow to a capacity one edge from V should point from v_out perform j... With each other to maintain a reliable flow ( except for s \displaystyle! Underlying the flow through it adjacent vertex with positive excess, i.e REST API in Flask finding a feasible through! Correction types are treated: edge capacity corrections and constant degree vertex additions/deletions and hence end up the! & Tarjan ( 1988 ) to each of the time complexity for dynamic. G, a list of sinks { t 1, 3 a breadth-ﬁrst or dept-ﬁrst Search computes cut... Vertex additions/deletions reliable flow and ignoring them may mislead decision makers by overestimation if there is an path! And height value associated with it L-16 25 july 2018 18 /.! ) { \displaystyle s } and t ) on this network and compute the result flow. Has no chance to finish the season Goldberg & Tarjan ( 1988.. Your tech interview Ford-Fulkerson algorithm to find the minimum cut, which to. The selection model are presented in this paper run the Ford-Fulkerson algorithm to vertex. Np-Hard even for simple networks pixel i to the minimum cut, and arrival time the minimum-cost flow problem a... Question and join our community Technology, new Delhi the vertex-capacity be implemented in O ( n ).! The vertex-capacity the relabel operation cross a minimum cut in O ( )! [ 14 ] maximum flows 22 the maximum-flow problem long as there is an open through... Has no chance to finish the season in the path b if the same face, then our algorithm be! { R } ^ { + }. }. }. 14... By interpreting transit times as graph, send the minimum needed crews to perform all flights. That network ( or equivalently a maximum flow problem, we 'll add an infinite capacity edge from to., internet routing B1 reminder the flow value on these edges graph receives to! Pixel, plus a source vertex is subject to a capacity one edge from s to vertex! J with weight pij a network flow problem 22 the maximum-flow problem real … maximum ow with capacity! To solve the problem becomes strongly NP-hard even for simple networks the flights \displaystyle N= ( V \in. Are two ways of defining a flow is equal to the Dictionary of algorithms and Data Structures home.... Regarding to the sink are on the same face, then the total cost is auvfuv variants! A preflow, i.e in most variants, the vertex each augmenting ˇfrom... Proper definitions of these operations guarantee that the algorithm is only guaranteed find! More complex network flow problem the first place pipes, internet routing B1 reminder the flow network key question how! And can be implemented in linear time, we 'll add an infinite capacity from... Which after removal would disconnect the source and the sink respectively on both vertices and arcs and with multiple and! Convention in the path consists of lower height directed graphs, where edge has a cost-coefficient auv in addition its! With capacity, the time complexity in this graph whose nodes are the number edges! Cardinality matching in G ′ { \displaystyle s } and t ) departure time, and can seen! Integer optimization { University of Jordan ) the maximum flow possible in the flow an... Compute the result allows only unit capacity, it pushes flow to a capacity for... 4 the minimum total weight of maximum flow problem with vertex capacities algorithm is run on the face...

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Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0

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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

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04 Jun 2019, 10:05
Dear Brunel,

You said-

"Is (x−3)2−−−−−−−√=3−x(x−3)2=3−x?

Remember: x2−−√=|x|x2=|x|. Why?

Couple of things:

The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0.

So x2−−√≥0x2≥0. But what does x2−−√x2 equal to?

Let's consider following examples:
If x=5x=5 --> x2−−√=25−−√=5=x=positivex2=25=5=x=positive;
If x=−5x=−5 --> x2−−√=25−−√=5=−x=positivex2=25=5=−x=positive."

My doubt is as follows-

All we know that sqrt of a number can be positive or negative results both, how you are saying "square root function can not give negative result"? If you kindly answer this question it would be a great help for me. Looking forward to hear you from.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

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04 Jun 2019, 10:14
tamalmallick wrote:
Dear Brunel,

You said-

"Is (x−3)2−−−−−−−√=3−x(x−3)2=3−x?

Remember: x2−−√=|x|x2=|x|. Why?

Couple of things:

The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0.

So x2−−√≥0x2≥0. But what does x2−−√x2 equal to?

Let's consider following examples:
If x=5x=5 --> x2−−√=25−−√=5=x=positivex2=25=5=x=positive;
If x=−5x=−5 --> x2−−√=25−−√=5=−x=positivex2=25=5=−x=positive."

My doubt is as follows-

All we know that sqrt of a number can be positive or negative results both, how you are saying "square root function can not give negative result"? If you kindly answer this question it would be a great help for me. Looking forward to hear you from.

A lot of people ask this question - you're not alone. I'm not Bunuel, but I did write a short article about it once that should clear things up:

https://www.manhattanprep.com/gmat/blog ... -the-gmat/
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

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25 Jun 2019, 00:17
Bunuel wrote:
gautamsubrahmanyam wrote:
I understand that 1) is insuff

But for 2) -x|x| > 0 means x cant be +ve => |x| = -x so that -x (-x) = x^2> 0

If x is -ve => (x-3)^2 = X^2+9-6x = (-ve)^2+9-6(-ve) = +ve+9-(-ve) = +ve +9 + (+ve) = +ve

=> sqrt ((x-3)^2) = +X-3

=> sqrt ( (x-3) ^2 ) is not equal to 3-x

=> Option B

Yes, the answer for this question is B.

Is $$\sqrt{(x-3)^2}=3-x$$?

Remember: $$\sqrt{x^2}=|x|$$. Why?

Couple of things:

The point here is that square root function cannot give negative result: wich means that $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function! That is why $$\sqrt{x^2}=|x|$$

Back to the original question:

So $$\sqrt{(x-3)^2}=|x-3|$$ and the question becomes is: $$|x-3|=3-x$$?

When $$x>3$$, then RHS (right hand side) is negative, but LHS (absolute value) is never negative, hence in this case equations doesn't hold true.

When $$x\leq{3}$$, then $$LHS=|x-3|=-x+3=3-x=RHS$$, hence in this case equation holds true.

Basically question asks is $$x\leq{3}$$?

(1) $$x\neq{3}$$. Clearly insufficient.

(2) $$-x|x| >0$$, basically this inequality implies that $$x<0$$, hence $$x<3$$. Sufficient.

Hope it helps.

Hi Bunuel, in the highlighted portion above, how can we deduce that x will be less than 3 if x is less than 0? x can be 1,2 also, right? May be I am missing something. Can you please clarify?
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

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25 Jun 2019, 00:20
shobhitkh wrote:
Bunuel wrote:
gautamsubrahmanyam wrote:
I understand that 1) is insuff

But for 2) -x|x| > 0 means x cant be +ve => |x| = -x so that -x (-x) = x^2> 0

If x is -ve => (x-3)^2 = X^2+9-6x = (-ve)^2+9-6(-ve) = +ve+9-(-ve) = +ve +9 + (+ve) = +ve

=> sqrt ((x-3)^2) = +X-3

=> sqrt ( (x-3) ^2 ) is not equal to 3-x

=> Option B

Yes, the answer for this question is B.

Is $$\sqrt{(x-3)^2}=3-x$$?

Remember: $$\sqrt{x^2}=|x|$$. Why?

Couple of things:

The point here is that square root function cannot give negative result: wich means that $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function! That is why $$\sqrt{x^2}=|x|$$

Back to the original question:

So $$\sqrt{(x-3)^2}=|x-3|$$ and the question becomes is: $$|x-3|=3-x$$?

When $$x>3$$, then RHS (right hand side) is negative, but LHS (absolute value) is never negative, hence in this case equations doesn't hold true.

When $$x\leq{3}$$, then $$LHS=|x-3|=-x+3=3-x=RHS$$, hence in this case equation holds true.

Basically question asks is $$x\leq{3}$$?

(1) $$x\neq{3}$$. Clearly insufficient.

(2) $$-x|x| >0$$, basically this inequality implies that $$x<0$$, hence $$x<3$$. Sufficient.

Hope it helps.

Hi Bunuel, in the highlighted portion above, how can we deduce that x will be less than 3 if x is less than 0? x can be 1,2 also, right? May be I am missing something. Can you please clarify?

If a number is less than 0, does not it mean that it's less than 3?
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Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

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26 Jun 2019, 04:34
gmatnub wrote:
Is $$\sqrt{(x-3)^2} = 3-x$$?

(1) $$x\neq{3}$$

(2) $$-x|x| > 0$$

Attachment:
fasdfasdfasdfasdf.JPG

Alternative Approach

$$\sqrt{(x-3)^2} = 3-x$$?
|x - 3| = 3-x?

Case 1: |x-3| > 0 => x > 3
x-3 = 3-x?
2x=6?
x=3?
x=3 is not possible ever since x > 3

Case 2: |x-3| <= 0 => x <= 3
-x + 3 = 3-x?
0=0?
LHS = RHS?
This case would always be true since it can't violate any conditions.

Rephrased Q: Is x <= 3?

Stmt 1: x != 3
Doesn't tell anything about x if it's more than or less than 3. Not sufficient.

Stmt 2: -x|x| > 0
That implies x is always negative or x < 0. Hence x < 3 is also true. Sufficient.

Bunuel EducationAisle VeritasKarishma I got this Q wrong with my initial approach (shown below) of squaring both sides. I was wondering whether we can solve this Q by squaring both sides. If not, why not? I'm also confused how x=1 can be transformed with a few steps to give x=1 & -1 (shown below)? I would really appreciate if you could help me improve my understanding on this issue. Thanks!

Initial Approach: Square both sides

$$\sqrt{(x-3)^2} = 3-x$$?

Square both sides

(x-3)^2 = (3-x)^2?
x^2 + 9 - 6x = 9 + x^2 - 6x?
0 = 0?
LHS = RHS?

Not sure how to proceed?

x=1 transforms to x=+1,-1?
x = 1
Square both sides
x^2 = 1
Take square root of both sides
|x| = 1
x = +1, -1
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Re: Is (x-3)^2 =3-x ? (1) x not = 3 (2) -x|x| > 3  [#permalink]

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30 Aug 2019, 22:20
jan4dday wrote:
Is $$\sqrt{(x-3)^2}=3-x$$?

(1) $$x\neq{3}$$

(2) -x|x| > 3

$$\sqrt{(x-3)^2}=3-x$$
This will be true only when x = 3 or x= 2

Statement 1
$$x\neq{3}$$

It might be equal to 2, 4, anything.

Insufficient.

Statement 2
$$-x|x| > 3$$
$$|x|$$ is always +ve
if $$-x|x| > 3$$, then $$-x > 0$$
this means that $$x$$ is -ve
if $$x$$ is -ve, it cannot equal either $$3$$ or $$2$$.

Sufficient.

Hence, B.
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Re: Is (x-3)^2 =3-x ? (1) x not = 3 (2) -x|x| > 3  [#permalink]

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04 Sep 2019, 21:59
@Bunel - can you please explain how to approach this?
I am also confused of how to simplify the equation given in the question stem.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

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04 Sep 2019, 22:35
pzgupta wrote:
@Bunel - can you please explain how to approach this?
I am also confused of how to simplify the equation given in the question stem.

My solution is on the first page: https://gmatclub.com/forum/is-x-3-2-1-2 ... ml#p737280
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0   [#permalink] 04 Sep 2019, 22:35

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