Minimization and maximization refresher. from Wikipedia. The minimization will be performed by a gradient descent algorithm, whose task is to parse the cost function output until it finds the lowest minimum point. To find the profit maximization levels, other approaches can be taken as well. In place of dJ/dTheta-j you will. This means that, in the AC equation, q + 2 are the average variable costs and 100/q are the average fixed costs. Likely, many corporate leaders believe. 1-Input the number. How is the above interpreted? The rm wants to minimize its costs (w 1x 1 + w 2x 2) of producing y units of output. This video explains how to find the average cost function and find the minimum average cost given the total cost function. Yes, even despite having so much support from ml-class … they practically implement everything and just leave the cost and gradient functions up to you. In this article, I will be going through the basic mathematics behind K-Means Algorithm. 50 which is the amount the firm has reduced their loss by producing instead of shutting down. economic order quantity (eoq) model The economic order quantity (EOQ) is the order quantity that minimizes total holding and ordering costs for the year. The formula is useful for deriving total costs for budgeting purposes, or to identify the approximate profit or loss levels likely to be achieved at certain sales volumes. So the terminology I'm going to use is that the loss function is applied to just a single training example like so. If a firm has a production function Q=F(K,L) (that is, the quantity of output (Q) is some function of capital (K) and labor (L)), then if 2Q V/(π r^2) Find r when the slope of the area is zero:. 3) The profit a business makes is equal to the revenue it takes in minus what it spends as costs. so the function is concave up, so x = 18 is the absolute minimum. Your business should be doing the same. If Minimize is given an expression containing approximate numbers, it automatically calls NMinimize. Minimize [{f, cons}, x ∈ reg] is effectively equivalent to Minimize [{f, cons ∧ x ∈ reg}, x]. In ML, cost functions are used to estimate how badly models are performing. For example, when determining optimal cooling protocols, we ultimately only care to minimize ice-related cell death in the tissue,without regard to the state outside of the tissue. I would like to minimize the cost of a function and i have these variables and restrictions. minimize (). The optimal cost is $150. Objectives: To maximize or minimize a two-variable function. output quantity. For some types of costs, the relationship is in direct proportion; for other types, there is a direct trade-off. To delineate CVX specifications from surrounding Matlab code, they are preceded with the statement cvx_begin and followed with the statement cvx_end. Clarification of Answer by livioflores-ga on 28 May 2006 20:01 PDT Hi!! Here is the answer to your second request of clarification: You know that 100 = min(x1,20) + min(x3,x4); Since you are trying to minimize costs it is clear that x1=<20; if not is x1>20 but min{x1,20} is still equal to 20 and this force you to continue using 80 units of x3 and x4; but in this situation you will spend more. Thanks readers for the pointing out the confusing diagram. 1 guitars and 48. Cost complementary exits in a multiproduct cost function when a. Now we're ready to optimize. To do that, we make a function that gives us the wrongness of a particular set of thetas against our training data. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. return_all, available for some methods, gives you the parameter vector vs generation, but not the cost function. In most cases, when you see a decorated function, the decorator is a factory function that takes a function as argument and returns a new function that includes the old function inside the closure. Summary: The goal of the diet problem is to select a set of foods that will satisfy a set of daily nutritional requirement at minimum cost. Cost categories. 01 dollars to manufacture x Xbox 360s in a day. (1) Solve for the cost-minimizing input combination: (2) Depict the optimum in the diagram to the right. To obtain the cost function, add fixed cost and variable cost together. 3) Do not exhaust all system memory. Amid COVID-19, physicians, architects, and consultants are talking change in design — to be ready for next. Examples least-squares minimize kAx−bk2 2 • analytical solution x⋆ = A†b (A† is pseudo-inverse) • can add linear constraints, e. There are two parameters (coefficients) in our cost function we can control: weight $$m$$ and bias $$b$$. So we finally have cost as a function of x. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. prefer parameters that minimize the execution time. 3 is to be constructed in the shape of a rectangular box with a square base and an open top. - Answered by a verified Math Tutor or Teacher We use cookies to give you the best possible experience on our website. Return the absolute value. 1541765 To link to this article: https://doi. The available techniques to determine soil moisture content have practical limitations owing to their high cost, dependence on labor, and time consumption. Cost Function A company finds that it costs a total of to produce units of a new product. Rowe Price, troll, uspto. [email protected] Gradient Descent is THE most used learning algorithm in Machine Learning and this post will show you almost everything you need to know about it. the cost function itself!. Then why to use the. This value may be the expected net present value of a project or a forest property; or it may be the cost of a project; it could also be the amount of wood produced, the expected number of visitor-days at a park, the number of endangered species that will be. Eliminate Storage Complexity And Minimize Costs Enterprise IT is under unremitting pressure to reduce capital and operating expenses, driving them to virtualize infrastructure to improve hardware utilization and scalability and advance toward enhanced operational efficiency and flexibility. This elementary framew ork is the basis for a broad v ariety of mac hine learning. The math problem is: A large bin for holding heavy material must be in the shape of a box with an open top and a square base. However, this benefit comes at the cost of high computational complexity. The marketing manager should be interested in revealing the complexities of an individual buyer, the dynamics of consumer behavior and should also try to. #N#function J = computeCost ( X, y, theta) #N#%COMPUTECOST Compute cost for linear regression. I want that " t and T must be greater than zero(not equal to zero) , t < T and C > 0. 15-2P = 15-2(3)= 15-6=9-6+5P=-6+5(3)=-6+15=9. If you produce a certain amount and let's say you bring in, I don't know,$10,000 of revenue and it costs you $5,000 to produce those shoes, you'll have$5,000 in profit. This study proposes a new framework to minimize the cost function of multi-objective optimization problems by using NSGA-II in economic environments. To delineate CVX specifications from surrounding Matlab code, they are preceded with the statement cvx_begin and followed with the statement cvx_end. Since C0(x) = 30 − 253000 x2, then C. LP problems seek to maximize or minimize some quantity (usually profit or cost). the objective function (maximize/minimize) and. The goal of any Machine Learning model is to minimize the Cost Function. So in training your logistic regression model, we're going to try to find parameters W and B that minimize the overall costs function J written at the bottom. How to minimise the cost function? Our goal is to move from the mountain in the top right corner (high cost) to the dark blue sea in the bottom left (low cost). Minimize operating costs and improve energy performance Data centers have to face continually increasing cost constraints. Gradient Descent basically just does what we were doing by hand — change the. From an external point of view, it is difficult to ascertain which are the alternative considered. It costs $10 to store one set for a year. The inventory cost problem, however, is something that comes up in real-life manufacturing scenarios all the time - how can I minimize my operating costs? In fact, the problem we see here today is a simplified version of a problem I covered in a DETC conference paper that I published a few years back. Take a deep breath. The point was more to introduce the reader to a specific method, not to the cost function specifically. You can use Pythagoras to compute S in terms of U in terms of S: U^2 = 500 2 + (4000 - S) 2 U = sqrt( 500 2 + (4000 - S) 2) Thus C(S) = S + 5 sqrt( 500 2. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. **TL;DR**: Rearranging the terms in Maximum Mean Discrepancy yields a much better loss function for the discriminator of Generative Adversarial Nets. To determine the optimal amount of inputs (L and K), we solve this minimization constraint using the Lagrange multiplier method:. Inventory cost problems come up in real-life manufacturing scenarios all the time - how can I minimize my operating costs? Hot Bod Jacuzzi & Spa Company is launching a new hot tub - the Neverleak Massage-o-matic DeLux. Write a Cost Function. Thus, the C function represents the. The cost functions implemented in MIPAV: Correlation ratio. The use of closures and factory functions is the most common and powerful use for inner functions. Anthony Vu Patent, Patents "ask the patent attorney series", "The American Invents Act", aia, collateral estoppel, cost, inter partes review, issue preclusion, litigation, patent, patent litigation, price, secure Axcess, T. 2 Minimize 2 x 2 1 + 2 sub ject to x 1 + 2 =1, if w ec hange the righ t hand side from 1 to 1: 05 (i. How can equations and inequalities help a business maximize profit or minimize costs? Unanswered Questions. for one-variable real functions: limits, integrals, roots This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games. (c) Test the C. Then it is going to become impossible to properly minimize or maximize the Cost Function. be used to minimize costs and what is the cost of producing that amount of chicks? f. cost of shipping + c21x21 + c22x22 + c23x23 from a plant + c31x31 + c32x32 + c33x33 to the ware house) Supply constraints. That is, the firm must choose a specific point on the q Cost Functions come directly from the production function and prices. In practice, these attractions are balanced in order to maintain a gap between the shaft (rotor) and static parts (stator). This firm minimizes its cost of producing any given output y if it chooses the pair (z 1, z 2) of inputs to solve the problem min z 1,z 2 w 1 z 1 + w 2 z 2 subject to y = F (z 1, z 2), where w 1 and w 2 are the input prices. (A) The Cost Function The cost-minimizing choice of inputs depended on two essential sets of parameters: the given output level (Y) and the given factor prices (r and w). Write an expression for the Cost in terms of only the width (w). Authors: Gaël Varoquaux. 25 lines (16 sloc) 791 Bytes. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. There are two parameters (coefficients) in our cost function we can control: weight $$m$$ and bias $$b$$. 20, it cost$6. Operations > Time-Cost. The math problem is: A large bin for holding heavy material must be in the shape of a box with an open top and a square base. Indeed, it is the most powerful method available to reduce product cost, improve quality, and simultaneously reduce development interval. The minimization will be performed by a gradient descent algorithm, whose task is to parse the cost function output until it finds the lowest minimum point. Material indices Introduction The performance, p, Each function has an associated material index. 2x1 1 x2 1 x3 1 x4 x1, x2, x3, x4 $0. Organizations are relying on cloud to maintain business-critical processes, but the journey is not always seamless: you may be grappling with cloud governance and how to keep control over security, costs, risks. So our cost as a function of x is going to be 20x squared 36 times 5. What is the Malayalam name of tukmaria or sabja seed or falooda seed. Likely, many corporate leaders believe. In the case we are going to see, we'll try to find the best input arguments to obtain the minimum value of a real function, called in this case, cost function. Minimize costs The logistics market is characterized by higher standards for air pollution and noise as well as increasing toll fees, personnel costs and fuel prices. Output is produced according to the following process 2 1 2 1 K L = Firm Output (I chose the same function as above to simplify things). 1-Input the number. It's called the cost function, which is kind of a crappy name in this context. 20 to increase production from 49 to 50 units of output. " The problem also listed these following multiple choice answers: a) 30,000 b) 300 c) 3,000 d) 30 e) None of these Now, we have the correct answer, what we need is the actual way to do this problem. The gradient descent algorithm in a nutshell. To do this, take the derivative of C(x), set it equal to zero, and solve for x. The diet problem constraints typically regulate the number of calories and the. The optimization continues as the cost function response improves iteration by iteration. optimize for black-box optimization: we do not rely on the. Put simply, a cost function is a measure of how wrong the model is in terms of its ability to estimate the relationship between X and y. Cost Minimization: Short Run • Let us go back to the two-inputs case, with only one of them variable in the short run. We have contributed on a local. Thus you know that the cost is C(S,U) = S + 5U. 9 drums to minimize his costs. Well, your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x. C CL(q) combination of inputs that minimize the cost of producing each. Fundamental theorem of linear programming If the optimal (maximum or minimum) value of the objective function in a. Reynolds Consumer Products Inc. The aim of the linear regression is to find a line similar to the blue line in the plot above that fits the given set of training example best. This website uses cookies to ensure you get the best experience. Solution: We would like to find a function that describes this situation. 4 (GP) : minimize f (x) s. "I tried a lot but I am not getting the values of t and T as mentioned above " ". The objective function J = f(x) is augmented by the constraint equations through a set of non-negative multiplicative Lagrange multipliers, λ j ≥0. We often design algorithms for GP by building a local quadratic model of f (·)atagivenpointx =¯x. 4x + 150 t?o model the unit cost in dollars for producing x stabilizer bars. INTRODUCTION Linear programming is a mathematical technique used to find the best possible solution in allocating limited resources (constraints) to achieve maximum profit or minimum cost by modelling linear relationships. nan with np. The goal of any Machine Learning model is to minimize the Cost Function. These functions can be seen as covering functions which have many applications in di erent optimization prob-lems: Set Cover functions, Edge Cut functions in graphs, etc. For example, this formula will find the highest value in cells H2:H17 =MAX(H2:H17) MIN IF Formula. Mathematical optimization: finding minima of functions¶. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. How much food should be used to minimize costs and what is the total cost? 2. Thus the function huber can be used anywhere a traditional convex function can be used, in constraints or objective functions, in accordance with the DCP ruleset. Minimize the cost to split a number Given an integer N ≥ 2 , you can split the number as a sum of k integers i. Sourcing, procurement and vendor management leaders should use this research to navigate GDPR requirements and Microsoft’s licensing to avoid legal and financial risks. The cost functions implemented in MIPAV: Correlation ratio. The transaction cost function is not di erentiable at the kink points and is piecewise continuous. Human Resource Management Functions. You will notice that as in the case of the factor demand functions, there is a. 1 guitars and 48. That is, the quantity you want to maximize or minimize is called the objective function. The Dual of the Minimum Cost Flow Problem:. C CL(q) combination of inputs that minimize the cost of producing each. Then why to use the. That's incredible but understandable when you start adding up all the "standard" wedding costs. So it's going to be plus 180 times, let's see, x times x to the negative 2, 180x to the negative x to the negative 1 power. The specific goal is to approximate a single valued function of one variable in terms of a sequence of linear segments. To minimize energy content, use the above criteria for. The two interesting exceptions to this rule are:. How to Minimize Legal Liabilities and Risks Information throughout this subsection applies primarily to external consultants. 4 — Logistic Regression | Cost Function — [ Machine Learning | Andrew Ng] - Duration: 11:26. What is the Objective Function? The objective of a linear programming problem will be to maximize or to minimize some numerical value. Assume that is costs Microsoft approximately C x x x 2 14,400 550 0. Hi , I am using FMINCON to minimize my cost function which is a product of elements of a matrix. An optimization problem seeks to minimize a loss function. To do that, we make a function that gives us the wrongness of a particular set of thetas against our training data. An extra large server costs you$0. Minimize F x y 22 with xy 2 10. using linear algebra) and must be searched for by an optimization algorithm. Find more Statistics & Data Analysis widgets in Wolfram|Alpha. HRM is the systematic planning and control of a network of fundamental organizational processes affecting and involving all organization members (French, 2004, p. However, this benefit comes at the cost of high computational complexity. For what number of bars is the unit cost at its mimimum? What is the unit cost at that level of production? Haven't got a clue what this problem is asking of me. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal. How much are closing costs? These are the fees paid that help facilitate the sale of a home typically total 2% to 7% of the home's purchase price. applied optimization calc 1. If the material for the sides costs 15¢/in. In this article, I will be going through the basic mathematics behind K-Means Algorithm. For multi-objective improvements, the most generally used developmental algorithms such as NSGA-II, SPEA2 and PESA-II can be utilized. A cost function is a MATLAB ® function that evaluates your design requirements using design variable values. In this paper, we have applied some meta scheduling methods to a model of CIM that is referred to as an automated flow shop, where backward scheduling should be used to realize a JIT's theory. Consider the same open-top box, which is to have volume $$216in. Assume we are given a dataset as plotted by the 'x' marks in the plot above. This is the radius which will minimize the surface area and thus the cost of materials. costs into account. Artificial Intelligence - All in One 87,390 views 11:26. Chap 7: Short-Run Cost Function 2. Cost categories. 6 - Linear Programming. The material that will be used for three sides costs 30 per linear foot, and the material that will be used for the fourth side costs 15 per linear foot. So our cost as a function of x is going to be 20x squared 36 times 5. Decide what the variables are and what the constants are, draw a diagram if appropriate, understand clearly what it is that is to be maximized or minimized. 5 Q 2 v What is the marginal revenue function?. The Dual of the Minimum Cost Flow Problem:. Raw Blame History. 4 million will be recognized as a component of. Deﬁne a MATLAB function to evaluate −f(x) given x. In my opinion, the #1 cost to avoid is the “retaker” cost. I want that " t and T must be greater than zero(not equal to zero) , t < T and C > 0. Dear Sir Can you please help me to minimize the following cost function with maple 10. This function is known as the cost function and will be of considerable interest to us. Does the production function exhibit increasing, decreasing, or constant returns to scale? How can you tell? b. Given a function defined by a set of parameters, gradient descent starts with an initial set of parameter values and iteratively moves toward a set of parameter values that minimize the function. h(θ) is the the prediction from your regression model. Let's take a more in depth look at the cost function and see how it works. Examples least-squares minimize kAx−bk2 2 • analytical solution x⋆ = A†b (A† is pseudo-inverse) • can add linear constraints, e. Users who have contributed to this file. This study proposes a new framework to minimize the cost function of multi-objective optimization problems by using NSGA-II in economic environments. The available techniques to determine soil moisture content have practical limitations owing to their high cost, dependence on labor, and time consumption. (d) Find the minimum value of the marginal cost. A cost function is a MATLAB ® function that evaluates your design requirements using design variable values. Ultimately, to minimize our cost, we need to find the point with the lowest z value. You can also optimize the objective function without any loss function, e. Obtain the minimum using fmin=fminsearch(fun,x0) Maximization 1. One common application of calculus is calculating the minimum or maximum value of a function. With so many options to choose from, the best iPhone XR case can be elusive. Currently, minimize lacks the ability to do this. To do: Try the following example: Given: Q = L 1/2 K 1/2 PL = 4, PK = 1 Goal: Produce Qo = 16 units as cheaply as possible. (Remember, the average cost, (6 pts. For example, if the marginal cost of producing the 50th product is 6. If the firm ordered the item, then the setup cost is simply the order cost from Module 5. Cost Function - Intuition I11:09. They would like to offer some combination of milk, beans, and oranges. So, if you employ tactics to reduce costs in all discrete functions from manufacturing through delivery, you'll have a lower total landed cost, right? Theoretically, yes. Now, to minimize marginal cost. If the brewery produces sweet stout alone, the cost function is: CS(q2) = 8q2. Homework Statement Mary Jane grows herbs in her attic. Previous work. The Cost Function If lattes and cake (or labor and capital) have unit prices of pL and pK, respec-tively, then the total cost of purchasing L units of one and K units of the other is C(L,K) = pLL+pKK. I think relative price of L & K is (Cost of Labour Per Hour)/(Cost of Rent Per Hour), but I don't know the price of Rent Per Hour. pdf), Text File (. Next time I will not draw mspaint but actually plot it out. The problem is that officers work 8 hour shifts, yet the demand comes in 4 hour chunks. 5 Actionable Tips to Reduce Operational Costs Regardless of what the circumstances are for your business, it is always a priority to find ways to reduce operational costs. k) 0 is a (nonnegative) function for which (x k;x k) = 0;then the following function deﬁnes a majorizer for : ˚ k(x) , (x)+ (x;x k): (4. Generally speaking, Least-Squares Method has two categories, linear and non-linear. The available techniques to determine soil moisture content have practical limitations owing to their high cost, dependence on labor, and time consumption. There can be significant cost savings when a business function is outsourced. Find the level of production which will minimize the average cost per item. 02xSquared - 3. Instead, it is allowable to use a cost flow assumption that varies from actual usage. Cost & Time and Also Minimum Project Duration Using Alternative Method 405 coordinates of the normal and crash points: Cost slope = (crash cost-normal cost)/ (normal duration crash duration) As the activity duration is reduced, there is an increase in direct cost. As the magnitues of the fitting parameters increase, there will be an increasing penalty on the cost function. The extent of risk and liability in your work depends on the nature of your services. Hi , I am using FMINCON to minimize my cost function which is a product of elements of a matrix. Of course, since time is money in any manufacturing process, what this really means is that looking into ways of reducing cycle time in your injection molding process can have a major impact on. What are loss functions? And how do they work in machine learning algorithms? Find out in this article. In the case we are going to see, we'll try to find the best input arguments to obtain the minimum value of a real function, called in this case, cost function. Fundamental theorem of linear programming If the optimal (maximum or minimum) value of the objective function in a. Common benzodiazepines used for GAD include alprazolam, clonazepam, diazepam, and lorazepam. For what number of bars is the unit cost at its mimimum? What is the unit cost at that level of production? Haven't got a clue what this problem is asking of me. Actual costs refer to real transactions, wherease opportunity costs refer to the alternative taken into consideration by decision makers who might want to choose the line of activity which minimise the costs. Average cost is minimized when average cost = marginal cost is another saying that isn’t quite true; in this case, the correct statement is: Average Cost has critical points when Average Cost and Marginal Cost are equal. LP problems seek to maximize or minimize some quantity (usually profit or cost). Find the dimensions that will minimize the cost of the box's construction. Therefore the total cost is: C(x) = 10y +15(2x+y) = 30x+25y. We need to decide which sub-contractor to use for a critical activity. Hence we want to minimize the can's surface area. We refer to this property as the objective function of an LP problem. In this case, the function huber will contain a special Matlab object that represents the function call in constraints and objectives. An objective function is either a loss function or its negative (in specific domains, variously called. It's important to limit your number of serverless functions to avoid having a massive charge when a lot of work is presented at the same time. The average cost of producing one output is reduced when the output of another product is increased If the wage rate is 5 and the price of capital is 2, then in order to minimize costs the firm should use a. 5 kg is the same linear function for a mass change of 2000 kg. For example, companies often want to minimize production costs or maximize revenue. The objective of the purchasing function is to obtain proper material and services when needed at the lowest obtainable cost. 1, Major functions of an institution) on the basis of modified total direct costs (MTDC), consisting of all salaries and wages, fringe benefits, materials and supplies, services, travel, and up to the. txt) or read online for free. Minimize cost and maximize quality of function in four variables. We are the prime contractor and there is a penalty in our contract with the main client for every day we deliver late. In the following example I will minimize an arbitrary function [texi]J[texi], then in the next chapter I'll apply it to the original house pricing task. Loss functions are actually at the heart of these techniques that we regularly use. (1) Solve for the cost-minimizing input combination:. The aim of the linear regression is to find a line similar to the blue line in the plot above that fits the given set of training example best. 5 Q 2 v What is the marginal revenue function?. The major objective of a typi-cal firm is to maximize dollar profits in the long run. Variables and functions should be declared in the minimum scope from which all references to the identifier are still possible. Let's take a more in depth look at the cost function and see how it works. This is where we look back at equation (1) and solve for h in terms of w. We advise on the largest and most complex legal challenges facing the world’s most important companies. the cost function itself!. Set big goals, insist on a cultural shift, and model from the top. The objective function is the function to be minimized or maximized. Use a computer to maximize the objective function subject to the constraints where 38. Thus, applications of HRM theory differ from personnel management in their dismissal of prescriptive “one best way” models of practice as diverse. Minimising Cost function. Overall, closures have affected 25% of pork production and 10% of beef production in the U. Objective function. An optimization problem is one where you have to make the best decision (choose the best investments, minimize your company's costs, find the class schedule with the fewest morning classes, or so on). That is h = 50 3w2 (3) Plugging the value for h from (3) above into equation (2) yields C = 60w2 + 48w. 2, what should the dimensions of the cup be to minimize the construction cost?. Yes, but not by playing it safe. Actually, the objective function is the function (e. Minimizing any function means finding the deepest valley in that function. A cost function is defined as: …a function that maps an event or values of one or more variables onto a real number intuitively representing some “cost” associated with the event. Examples and exercises on the cost function for a firm with two variable inputs Example: a production function with fixed proportions Consider the fixed proportions production function F (z 1, z 2) = min{z 1, z 2} (one worker and one machine produce one unit of output). Variable costs are such cost which vary directly with change in output. Recall in the calculus of one variable, if y = f(x) is defined on a set S, then there is a relative maximum value at x0 if f(x0) ≥ f(x) for all x in S near x0, and there is a relative. If a firm has a production function Q=F(K,L) (that is, the quantity of output (Q) is some function of capital (K) and labor (L)), then if 2Q V/(π r^2) Find r when the slope of the area is zero:. His next-door neighbor agrees to pay for half of the fence that borders her property; Sam will pay the rest of the cost. The nonlinearity in this form generates from the absolute value function. In most examples/tutorial I followed, the cost function used was somewhat arbitrary. Coming up with a cost function for optimization for a complex control system Hot Network Questions Is there a word or phrase for one mistaken belief leading to a web of false ones?. However, this benefit comes at the cost of high computational complexity. : residuals) between our model and our data points. One common application of calculus is calculating the minimum or maximum value of a function. Take Exam Only When You are Ready. the production function and the cost function; the only difference is whether we hold production constant or cost constant. [email protected] I would like to use the goal seek function to minimize the value in a certain cell (total cost) by changing another the value in another cell (shipment size). Then why to use the. This firm minimizes its cost of producing any given output y if it chooses the pair (z 1, z 2) of inputs to solve the problem min z 1,z 2 w 1 z 1 + w 2 z 2 subject to y = F (z 1, z 2), where w 1 and w 2 are the input prices. The Cost Function If lattes and cake (or labor and capital) have unit prices of pL and pK, respec-tively, then the total cost of purchasing L units of one and K units of the other is C(L,K) = pLL+pKK. Find the dimensions that will minimize cost. Employee compensation costs, office space expenses and other costs associated with providing a workspace or manufacturing setup are eliminated and free up resources for other purposes. We often design algorithms for GP by building a local quadratic model of f (·)atagivenpointx =¯x. (d) Find the minimum value of the marginal cost. Cost function is the sum of losses from each data point calculated with loss function. Solving for the minimum 0 points minimize f, (z) = 20 + z2-cos(2TZ) Given the cost function f. S, according to Bloomberg News. To minimize energy content, use the above criteria for. c<=2 n=1-1000 0<=p<=0. It's called the cost function, which is kind of a crappy name in this context. For example, if the marginal cost of producing the 50th product is 6. Minimising Cost function. You’ve been ordered to reduce your department’s costs by 10%, 20%, or 30%. It is obvious that if we changed relative factor. 5\text{ x }10^6 \text{ ft}^2) in an a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. Having drawn the picture, the next step is to write an equation for the quantity we want to optimize. profits and minimize costs by using cost-revenue-profit functions. **TL;DR**: Rearranging the terms in Maximum Mean Discrepancy yields a much better loss function for the discriminator of Generative Adversarial Nets. 188, and the slope was not significantly different from 0. By using this website, you agree to our Cookie Policy. “Despite the challenges of the COVID-19. Take the derivative of the Cost with respect to width. Average Cost Per Unit Formula. Many of these materials are high in quality and low in cost. Find the Average Cost Function and Minimize the Average Cost. Outsourcing can also make your firm more attractive to investors, since you're able to pump more capital directly into revenue-producing activities. That’s the main point of any model, to minize error, to perform. How many Xboxes should be manufactured in order to minimize average cost? What is the resulting average cost of an Xbox? Give your answer to the nearest dollar. In this case, the objective is to minimize the total cost per day which is given by z= 0:6x 1 + 0:35x 2 (the value of the objective function is often denoted by z). Cost Function8:12. Indeed, it is the most powerful method available to reduce product cost, improve quality, and simultaneously reduce development interval. If this sounds a lot, here are my 5 suggestions to reduce the CPA exam cost: 1. It is the heart that makes it beat! There is a loss function, which expresses how much the estimate has missed the mark for an individual observation. The use of closures and factory functions is the most common and powerful use for inner functions. The cost function: E(Cost)=E(F-LS) 3 F is for Finished goods L is for Lambda S is for Sales After expanding the function, what assumption minimized this function with respect to F? The Attempt at a Solution F 3-3F 2 LS+3F(LS) 2-(LS) 3 I know that I need to identify the terms that include both sales and inventories. f ( x) = x 4 − 8 x 2 + 5. h(θ) is the the prediction from your regression model. output, marginal cost, average cost, price, and profit at the average-cost minimizing activity level profit-maximizing or loss-minimizing output Calculus Cost/Graph cost function Optimal capital structure to minimize cost of capital Output, Profit, Fixed Costs and Perfect Competition Finding Optimal Output Level etc. Example 4 If the total revenue and total cost functions are TR = 30Q – 5Q 2 and TC = 15 + 12Q - 0. A company producing goods wants to minimize the average cost of production. Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. The global six-year average cost of a data breach is 3. Employee compensation costs, office space expenses and other costs associated with providing a workspace or manufacturing setup are eliminated and free up resources for other purposes. Yes, even despite having so much support from ml-class … they practically implement everything and just leave the cost and gradient functions up to you to implement. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. Mathematical optimization: finding minima of functions¶. Keep in mind that, the cost function is used to monitor the. To open Outlook again after it has been hidden on the system tray and disappeared from the taskbar, double-click the Outlook system tray icon. Thus the function huber can be used anywhere a traditional convex function can be used, in constraints or objective functions, in accordance with the DCP ruleset. Minimize the total physical effort & thus the cost of moving goods into & out of storage. Budgeting for your company’s training needs does not mean using surplus money when you have it. Diesel particulate filters (DPF) are devices that physically capture diesel particulates to prevent their release to the atmosphere. Authors: Gaël Varoquaux. There are many factors to consider when selecting components and board-level solutions for a real-time embedded system. 1 Where f is the number of facilities. Deﬁne a MATLAB function to evaluate −f(x) given x. , a function that takes a scalar as input) is needed. Good parameters means that the function can produce the best possible outcomes, namely the smallest ones, because small values mean less errors. Cij = transportation cost per unit of shipping from plant Pi to the Warehouse Wj. Lagrange multiplier methods involve the modiﬁcation of the objective function through the addition of terms that describe the constraints. Mathematical optimization: finding minima of functions¶ Authors: Gaël Varoquaux. Min & Max of Functions - MATLAB Minimization 1. Gradient descent is a more generic algorithm, used not only in linear regression problems and cost functions. Along with source pipeline, candidate quality, long-term retention, and other key performance metrics, tracking your cost-per-hire will help you to understand the performance of your recruiting initiatives better and minimize your expenses across the board. Univariate function minimizers (minimize_scalar)¶ Often only the minimum of an univariate function (i. A company producing goods wants to minimize the average cost of production. 15-2P = 15-2(3)= 15-6=9-6+5P=-6+5(3)=-6+15=9. My constriants are also in the form of matix. For example, this formula will find the highest value in cells H2:H17 =MAX(H2:H17) MIN IF Formula. 2% over budget). Then, minimize that slack variable until the slack is null or negative. We have contributed on a local. Minimize an objective function whose values are given by executing a file. Real-time embedded systems require. Grab a coffee. The cost function: E(Cost)=E(F-LS) 3 F is for Finished goods L is for Lambda S is for Sales After expanding the function, what assumption minimized this function with respect to F? The Attempt at a Solution F 3-3F 2 LS+3F(LS) 2-(LS) 3 I know that I need to identify the terms that include both sales and inventories. Profit is simply the Total revenue minus the costs incurred. Gradient descent is simply used to find the values of a function's parameters (coefficients) that minimize a cost function as far as possible. How To: Calculate and use regression functions in statistical analysis How To: Write a logarithm as a sum or difference of logarithms How To: Perform a quadratic regression with a calculator How To: Calculate r-squared or coefficient of determination in statistics. Of course, since time is money in any manufacturing process, what this really means is that looking into ways of reducing cycle time in your injection molding process can have a major impact on. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view. If you produce a certain amount and let's say you bring in, I don't know, 10,000 of revenue and it costs you 5,000 to produce those shoes, you'll have 5,000 in profit. However, by substituting for , the problem can be transformed into a linear problem. An optimization problem seeks to minimize a loss function. This is the personal website of a data scientist and machine learning enthusiast with a big passion for Python and open source. 1080/09715010. costs into account. If Minimize is given an expression containing approximate numbers, it automatically calls NMinimize. the firm hires labor, and the cost is the wage rate that must be paid for the labor services Total cost (TC) is the full cost of producing any given level of output, and it is divided into two parts: • Total fixed cost. What is the best nursing intervention to minimize the adverse effects of this. The production process can often be described with a set of linear inequalities called constraints. Simplex Algorithm Calculator is an online application on the simplex algorithm and two phase method. Find the number of units, x, that will minimize the average cost function if the total cost function is C()3+7+ 75. It's important to limit your number of serverless functions to avoid having a massive charge when a lot of work is presented at the same time. The following figure (right) shows a plot of a sample cost function for a selection of transformation parameters. When you optimize or estimate model parameters, you provide the saved cost function as an input to sdo. It's a cost function because the errors are "costs", the less errors your model give, the better your model is. 2) A business’ costs include the fixed cost of 5000 as well as the variable cost of 40 per bike. One common application of calculus is calculating the minimum or maximum value of a function. Derive Draper Dan's cost function (a) in terms of input prices and output and (b) when the price of cloth, w 1, is 3/metre and the wage rate w 2 is 10 per hour. When x = 18, y = 9. Otherwise, they must be considered separately. In this article, I will be going through the basic mathematics behind K-Means Algorithm. Economic Order Quantity Model (EOQ) Managing inventory is an important task for every business that holds it. A piecewise linear approximation is one method of constructing a function that fits a nonlinear objective function by adding extra binary variables, continuous variables, and constraints to reformulate the original problem. To find ways to save money, take advantage of quick cost-saving measures followed by an intensive look at where IT is spending money. What happens when the learning rate is too small? Too large? Using the best learning rate that you found, run gradient descent until convergence to find 1. Using the quadratic formula or a calculator, we find the solutions are. As serverless architectures mature, they have been able to minimize the issue of provision concurrency, in which there was a performance penalty when a function was called a second time, causing a. It is the heart that makes it beat! There is a loss function, which expresses how much the estimate has missed the mark for an individual observation. If this sounds a lot, here are my 5 suggestions to reduce the CPA exam cost: 1. 37 e) The minimal average cost. Put simply, a cost function is a measure of how wrong the model is in terms of its ability to estimate the relationship between X. Minimize The Use Of Color In Wireframes. Find the level of production which will minimize the average cost per item. So it's going to be plus 180 times, let's see, x times x to the negative 2, 180x to the negative x to the negative 1 power. Question 107995: Minimizing Cost, A company uses the formula C(x)=0. lute extrema of the function y = 2x,一2x2-16x + 1 on [-2,3]. Given a function defined by a set of parameters, gradient descent starts with an initial set of parameter values and iteratively moves toward a set of parameter values that minimize the function. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. Optimization methods in Scipy nov 07, 2015 numerical-analysis optimization python numpy scipy. A midwife usually offers a variety of options and seeks to eliminate or minimize unnecessary interventions. Once you have installed CVX (see Installation), you can start using it by entering a CVX specification into a Matlab script or function, or directly from the command prompt. Chapter 7: The Cost of Production. Minimizing Inventory Costs. Instead, it is allowable to use a cost flow assumption that varies from actual usage. Then again, Octave provides tools for learning where you essentially just run a function, tell it where to find the cost and gradient function and give it some data. Midwives: Benefits of Having a Midwife. Most optimization problems have a single objective function, if they do not, they can often be reformulated so that they do. For the given cost function C(x)=78400+500x+x^2 find: a) The cost at the production level 1700 b) The average cost at the production level 1700 c) The marginal cost at the production level 1700 d) The production level that will minimize the average cost e) The minimal average cost I can already answer a, b, and c, it's d and e I can't seem to get, I know it should be let c'(x) = 0, but the. In this context, the function is called cost function, or objective function, or energy. The calculator is intended to teach students the Simplex method and to relieve them from some of the tedious aritmetic. We want to minimize the cost of the materials subject to the constraint that the volume must be 50ft 3. In machine learning, we use gradient descent to update the parameters of our model. The cost function used is shown aboveWe want to find parameters Ɵ which minimize J(Ɵ) To do so we can use one of the algorithms already described such as; Gradient descent; Advanced optimization algorithmsTo minimize a cost function we just write code which computes the following J(Ɵ) i. Obviously, a conservative deflection limit can be specified to minimize deflection, assuming design and construction is then performed correctly. I want that " t and T must be greater than zero(not equal to zero) , t < T and C > 0. To find ways to save money, take advantage of quick cost-saving measures followed by an intensive look at where IT is spending money. We discuss the application of linear regression to housing price prediction, present the notion of a cost function, and introduce the gradient descent method for learning. It is a minimization problem. The optimization continues as the cost function response improves iteration by iteration. Then it is going to become impossible to properly minimize or maximize the Cost Function. The Total Cost of Ownership (TCO) of your IP is staggering. 20, it cost 6. 01 dollars to manufacture x Xbox 360s in a day. To demonstrate the minimization function, consider the problem of minimizing the Rosenbrock function of the NN variables −. Take Exam Only When You are Ready. Cost complementary exits in a multiproduct cost function when a. As it stands, though, it has two variables, so we need to use the constraint equation. The slope of iso cost line = PL/Pk. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. The statement dual variables y{n} allocates a cell array of \(n$$ dual variables, and stores the result in the Matlab variable Z. For example, a random sample of a population of young offenders is generated by selecting names from a list to interview. The cost function is just a mathematical formula that gives the total cost to produce a certain number of units. Mathematical optimization: finding minima of functions¶. Custom & Stock Plastic Packaging Solutions. 1-Input the number. 2x1 1 x2 1 x3 1 x4 x1, x2, x3, x4 $0. Return the arc cosine. stiff, light beam in bending –minimize ρ/E1/2 •e. 8 trillion annually in aggregate general and administrative (G&A) expenses. Labor Union Vs. Lagrange multiplier methods involve the modiﬁcation of the objective function through the addition of terms that describe the constraints. Next time I will not draw mspaint but actually plot it out. h(θ) is the the prediction from your regression model. Minimize the total physical effort & thus the cost of moving goods into & out of storage. Material indices Introduction The performance, p, Each function has an associated material index. I recently had to implement this from scratch, during the CS231 course offered by Stanford on visual recognition. Example 4 If the total revenue and total cost functions are TR = 30Q – 5Q 2 and TC = 15 + 12Q - 0. Note that w 1, w 2, and y are given in this. This will give the quantity (q) that maximizes profits, assuming of course that the firm has already taken steps to minimize costs. An isoquant and possible isocost line are shown in the following figure. Advantages of Outsourcing Cost Savings. Good parameters means that the function can produce the best possible outcomes, namely the smallest ones, because small values mean less errors. In machine learning, we use gradient descent to update the parameters of our model. The purposes of a human resources department and a labor union are decidedly different. One common application of calculus is calculating the minimum or maximum value of a function. Model Representation8:10. Budgeting for your company’s training needs does not mean using surplus money when you have it. How many players should be produced to minimize the marginal cost? and (b). The cost function equation is expressed as C(x)= FC + V(x), where C equals total production cost, FC is total fixed costs, V is variable cost and x is the number of units. Suppose the marginal cost C(in dollars) to produce x thousand mp3 players is given by the function C(x)=x^2-100x+7600. So we have written the cost as a function of two variable, height and width. cost of shipping + c21x21 + c22x22 + c23x23 from a plant + c31x31 + c32x32 + c33x33 to the ware house) Supply constraints. Example: A retail appliance store sells 2500 TV sets per year. To determine the optimal amount of inputs (L and K), we solve this minimization constraint using the Lagrange multiplier method:. 01x 2 +120 ) dollars where x represents the number of units produced. It is possible to attach a more substantial penalty to the predictions that are located above or below the expected results (some cost functions do so, e. Instead, it is allowable to use a cost flow assumption that varies from actual usage. Constrained Optimization using Lagrange Multipliers 5 Figure2shows that: •J A(x,λ) is independent of λat x= b, •the saddle point of J A(x,λ) occurs at a negative value of λ, so ∂J A/∂λ6= 0 for any λ≥0. Your business should be doing the same. Gradient descent is an optimization algorithm used to find the values of parameters (coefficients) of a function (f) that minimizes a cost function (cost). Ignoring any other costs, find the optimal number of facilities with the. Note as well that the cost for each side is just the area of that side times the appropriate cost. Mathematical optimization is the selection of the best input in a function to compute the required value. Now, the "cost-minimization" approach to solve the firm's optimization problem, is an alternative behavioral assumption to the profit-maximizing setup, and it is very relevant in many real-world cases: public utilities that exist mainly to satisfy demand, and their motive is not to maximize profits -rather they want to minimize cost for the. To maximize the revenue function To minimize the cost function To maximize the proﬁt function. Calculus Optimization Problem: What dimensions minimize the cost of a garden fence? Sam wants to build a garden fence to protect a rectangular 400 square-foot planting area. 25 lines (16 sloc) 791 Bytes. A low-cost provider is a powerful competitive approach in markets where many buyers are price sensitive. The problem is formulated as a linear program where the objective is to minimize cost and the constraints are to satisfy the specified nutritional requirements. find the location of the minimum of fr, z. That's incredible but understandable when you start adding up all the "standard" wedding costs. x11 + x12 + x13 = S1 x21 + x22. Firms can change all their inputs, both labor and capital, in the. Optimization- What is the Minimum or Maximum? 3. Hi , I am using FMINCON to minimize my cost function which is a product of elements of a matrix. For example, if the marginal cost of producing the 50th product is$6. In this case, the objective is to minimize the total cost per day which is given by z= 0:6x 1 + 0:35x 2 (the value of the objective function is often denoted by z). 4x + 150 t?o model the unit cost in dollars for producing x stabilizer bars. The average cost of producing one output is reduced when the output of another product is increased If the wage rate is $5 and the price of capital is$2, then in order to minimize costs the firm should use a. So to recapHypothesis - is like your prediction machine, throw in an x value, get a putative y value. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. If the time rate of change of this function is held constant between 0 and 0. A low-cost provider is a powerful competitive approach in markets where many buyers are price sensitive. To do this, take the derivative of C(x), set it equal to zero, and solve for x. Outsourcing can also make your firm more attractive to investors, since you're able to pump more capital directly into revenue-producing activities. The optimization continues as the cost function response improves iteration by iteration. IT continues to be a focal point for cost reduction in organizations, but cutting costs blindly can cause serious damage to IT and the business. In the general cost function problem, there is a function g: R+!R+ given, and the goal of the scheduler is to minimize P i2[n] w ig(F i). If x engines are made, then the unit cost is given by the function C(x)=x^2-560x+94,717. To minimize the deviation, the problem is formulated in a basic form as: as the objective function, and linear constraints are. 8 trillion annually in aggregate general and administrative (G&A) expenses. A production function, such as the Cobb-Douglas production function, can be used to model how a firm combines inputs to produce outputs; other production functions include the CES, Translog, and Diewert (Generalized Leontief); interactive and online models of production functions. When you work for yourself or need your vehicle for work, time can cost you income. We need to decide which sub-contractor to use for a critical activity. More labor and less capital c. Formal Derivation of Cost Curves from a Production Function: Rearranging the expression above we obtain: This is the cost function, that is, the cost expressed as a function of: (i) Output, X; (ii) The production function coefficients, b 0, b 1, b 2; (clearly the sum b 1 + b 2 is a measure of the returns to scale); (iii) The prices of. So the terminology I'm going to use is that the loss function is applied to just a single training example like so. Formware was purpose-built to model high penetration renewables at the system level and determine how all types of storage enable cost-effective renewable energy integration. We discuss the application of linear regression to housing price prediction, present the notion of a cost function, and introduce the gradient descent method for learning. output, marginal cost, average cost, price, and profit at the average-cost minimizing activity level profit-maximizing or loss-minimizing output Calculus Cost/Graph cost function Optimal capital structure to minimize cost of capital Output, Profit, Fixed Costs and Perfect Competition Finding Optimal Output Level etc. Minimize the potential for bias in the selection of the sample through random sampling. The average wedding costs \$30,000. The final detailed cost estimate contains the material, labor and in-directs data for controlling project costs. The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. 1-Input the number. The function f is called, variously, an objective function, a loss function or cost function (minimization), a utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. Formal Derivation of Cost Curves from a Production Function: Rearranging the expression above we obtain: This is the cost function, that is, the cost expressed as a function of: (i) Output, X; (ii) The production function coefficients, b 0, b 1, b 2; (clearly the sum b 1 + b 2 is a measure of the returns to scale); (iii) The prices of. costs into account. A nurse is caring for a postsurgical patient who has small tortuous veins and had a difficult IV insertion. Return the arc sine. 1, Major functions of an institution) on the basis of modified total direct costs (MTDC), consisting of all salaries and wages, fringe benefits, materials and supplies, services, travel, and up to the. Artificial Intelligence - All in One 87,390 views 11:26. In this blog post, you will learn how to implement gradient descent on a linear classifier with a Softmax cross-entropy loss function. To do that, we make a function that gives us the wrongness of a particular set of thetas against our training data. Assume that is costs Microsoft approximately C x x x 2 14,400 550 0. Linear regression predicts a real-valued output based on an input value. One common application of calculus is calculating the minimum or maximum value of a function. The cost functions implemented in MIPAV: Correlation ratio. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. There are many costs that occur because of inventory that need to be minimized, while still providing enough inventory to operate without losing customer business. But we would like to rewrite the cost as the function of only one variable (probably width). More capital and less labor b. Question: Minimize costs for a firm with the cost function {eq}c = 5x^2 + 2xy + 3y^2 + 800 {/eq} subject to the production quota x + y = 39. Common benzodiazepines used for GAD include alprazolam, clonazepam, diazepam, and lorazepam. Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. Examples: Input : Tower heights h[] = {1, 2, 3} Costs of operations cost[] = {10, 100, 1000} Output : 120 The heights can be equalized by either "Removing one block from 3 and adding one in 1" or "Adding two blocks in 1 and adding one in 2". In this article, I will be going through the basic mathematics behind K-Means Algorithm. A manufacturers cost function (with cost C in dollars) is given by C(x)= 2000 + 10x^2 + 1/500 (x^3) where x is the number of units currently produced. The AC equation is obtained by dividing the TC equation by q. Thus, the C function represents the minimum cost necessary to produce output q with fixed input prices. com To create your new password, just click the link in the email we sent you. Gradient descent is simply used to find the values of a function's parameters (coefficients) that minimize a cost function as far as possible. Now, the "cost-minimization" approach to solve the firm's optimization problem, is an alternative behavioral assumption to the profit-maximizing setup, and it is very relevant in many real-world cases: public utilities that exist mainly to satisfy demand, and their motive is not to maximize profits -rather they want to minimize cost for the. Using given information about the Volume, express the height (h) as a function of the width (w). Objective-function.
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# Squarefree polynomials over finite fields

I'm trying to figure out how many squarefree polynomials there are of a fixed degree over $\mathbb{F}_2$ specifically (and in general, over any finite field). Looking at some low-degree examples seems to suggest that half of the polynomials of any given degree are squarefree, but I'm not sure how to prove this, or whether the pattern continues at all. I'm considering the possibility of using the formal derivative, and the fact that a polynomial is relatively prime to its formal derivative iff it is squarefree, but I don't see how to proceed with this. So is there a known formula?

• Aren't there are 32 quintics in F_2[x]? – Will Dana Dec 22 '11 at 20:33
• Sure. Sorry! And to boot I neglected to count those with irreducible quartic factors in a comment that is best forgotten ;-) – Jyrki Lahtonen Dec 22 '11 at 20:36

Recall that $$M(n, q) = \frac{1}{n} \sum_{d | n} \mu(d) q^{n/d}$$

is the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$. The statement that there are $q^n$ monic polynomials of degree $n$ over $\mathbb{F}_q$ can then be written as the generating function identity $$\zeta(t) = \prod_{n \ge 1} \frac{1}{(1 - t^n)^{M(n, q)}} = \sum_{n \ge 0} q^n t^n = \frac{1}{1 - qt}$$

which is known as the cyclotomic identity and is the analogue for $\mathbb{F}_q[t]$ of the Euler product of the Riemann zeta function. If we instead want to count the number $s_n$ of squarefree monic polynomials of degree $n$ over $\mathbb{F}_q$, we want to work out the generating function $$\sum s_n t^n = \prod_{n \ge 1} (1 + t^n)^{M(n, q)}.$$

But by inspection this is just $$\frac{\zeta(t)}{\zeta(t^2)} = \frac{1 - qt^2}{1 - qt} = 1 + \sum_{n \ge 1} (q^n - q^{n-1}) t^n.$$

(The generating function identity $\zeta(t) = \zeta(t^2) \sum s_n t^n$ merely expresses the fact that every monic polynomial can be uniquely factored into its largest square factor and its squarefree part.)

Hence for $n \ge 1$ there are $q^n - q^{n-1} = \left( 1 - \frac{1}{q} \right) q^n$ monic squarefree polynomials of degree $n$ over $\mathbb{F}_q$. Jordan Ellenberg wrote a great blog post over at Quomodocumque explaining how this is related to the braid group and the analogous question about squarefree integers here.

(Note that you don't actually have to know the closed form of $M(n, q)$ for the above argument to work; I included it for the sake of concreteness.)

• Amazing! Thank you very much. – Will Dana Dec 22 '11 at 20:32
• Could you care explaining the closed form of $M(n,q)$? I'm not really seeing it. And I think the $\mu(n)$ in the sum is actually a $\mu(d)$? – Patrick Da Silva Dec 22 '11 at 20:54
• @Patrick: as I said, it doesn't actually matter for this problem, but it falls out of the cyclotomic identity after Mobius inversion; more concretely, counting the number of elements of $\mathbb{F}_{q^n}$ according to the degree of their minimal polynomial over $\mathbb{F}_q$ gives $q^n = \sum_{d | n} d M(d, q)$ and Mobius inversion gives the result. See also en.wikipedia.org/wiki/Necklace_polynomial . – Qiaochu Yuan Dec 22 '11 at 20:58
• Hm. I didn't even think about working it out. Thanks. The sum wasn't useful for this problem but I liked the sum so I asked =P – Patrick Da Silva Dec 22 '11 at 21:00
• How did you go to "Hence for n≥1 (n>1 I think) there are q^n−q^n−1=(1−1q)q^n monic squarefree polynomials of degree n ..." from the previous line? – user33646 Jun 13 '12 at 23:07

I am five years late, but I had the same question. While Qiaochu's solution is neat and introduces interesting tools like zeta functions, the answer $q^d-q^{d-1}$ somehow seems teasingly combinatorial. Like Ofir pointed out, it almost seems magical.

I have an attempt at a more intuitive combinatorial solution without using zeta functions and cyclotomic identities. Please do check it out and tell me if there is any mistake. At some level, it might be a restatement of Qiaochu's proof, but in simpler language.

So we want to count the number of square free monic polynomials in $\mathbb{F}_q[X]$ of degree $d$. Denote this number by $S_{q}(d)$. The only important fact we shall use is that every monic polynomial $f(X) \in \mathbb{F}_q[X]$ can be $uniquely$ expressed as $$f(X)=r(X)^2.s(X)$$ where $r(X),s(X)$ are monic, and $s(X)$ is square-free. This result is folklore, but quite intuitive. The uniqueness here is essential to our counting.

Now we can build monic polynomials of degree $d$ by picking an arbitrary monic polynomial of degree $k$ for $0 \leq k \leq \lfloor d/2 \rfloor$ and a square-free polynomial of degree $d-2k$ and multiplying the square of the former with the latter (the uniqueness of the expression comes into play here in counting the possibilities). At this juncture, we would have to take cases based on whether $d$ is even or odd, and count accordingly. I'll explore the case of $d$ being even (the other case is conceptually the same).

Since $d$ is even, fix an even $k$ at most $d$. Then we have a recurrence $$S_{q}(d) + S_{q}(d-2) q^{1} + S_{q}(d-4)q^{2}+ \dots + q^{\frac{d}{2}}=\sum \limits_{k=0}^{d/2} S_{q}(d-2k) q^k = q^d$$ Note that $S_{q}(0)=1$ (that is, the only square-free monic polynomial of degree $0$ is the constant polynomial $1$). A simple induction can now be used to construct $S_{q}(d)$ from $S_{q}(d-2),S_{q}(d-4),\dots,1$. In fact, one can observe that with the induction hypothesis that $S_{q}(d-2k)=q^{d-2k}-q^{d-2k-1}$ for every $0 \leq k \leq d/2$ every summand in the above recurrence is of the form $q^{d-k}-q^{d-k-1}$ and so the sum above is telescopic and simplifies to $$S_{q}(d) + q^{d-1}=q^d$$

• @QiaochuYuan I have posted a simpler combinatorial proof. Please do look at it and tell me if there's any mistake. Thanks. – BharatRam Jul 29 '16 at 13:09


Different limits for the alternating harmonic series?

Show that the series $$\sum_{n=1}^{\infty} \dfrac{(-1)^n}{n}$$ is not absolutely convergent. Show that by permuting the terms of the series one can obtain series with different limits.

I am able to show that this is not absolutely convergent. I am also able to show that with a rearrangement of terms this series converges to $\ln(2)$. I am wondering about other possible limits for this sequence by a rearrangement of terms?

Also this is not a homework problem. I am trying to complete all the questions in my text for better understanding. Any help and comments are appreciated. Thank you.

• See here. – David Mitra Sep 29 '13 at 20:51
• To change the limit, you will need to move infinitely many of the terms. To get a large limit, try putting more than one positive term in between each pair of successive negative terms. – Trevor Wilson Sep 29 '13 at 20:52
• In addition to @DavidMitra's remark, you might try showing, for an arbitrary conditionally converging series and $-\infty\leq a\leq b\leq\infty$, that there exists a rearrangement such that $\liminf S_n=a$, $\limsup S_n=b$. – Jonathan Y. Sep 29 '13 at 21:01

A theorem of Riemann says you can permute the terms and get any limit if you want, if the series is convergent and not absolutely convergent. I encourage you to try to prove it. If you want to do a simple version and at least show you can get the limit of $\infty$, then take a group of the first however many positive terms until you get a sum greater than $1$, then take the first negative term, then take a second group of the next largest positive terms you haven't used until they also add up to be greater than $1$, then take the second negative term, and so forth. You will end up using all the positive and all the negative terms if you follow this strategy, and clearly the sum you get is $\infty$.

As others have mentioned, there's a theorem that says in fact you can rearrange it to any limit - even plus or minus infinity. However, if you just want a single different limit, try $$\frac12 + \frac14 - \frac11 + \frac16 + \frac18 - \frac13 + \frac1{10} + \frac1{12} - \frac15 + \cdots$$

Which you should be able to show approaches $-\frac12 \ln 2$.

Note that the series you give ($-1 + \frac12 - \frac13 + \cdots$) actually converges to $- \ln 2$ not $\ln 2$, so perhaps you meant to say you can show it rearranges to $-\ln 2$.

One of the simplest ways I have rearranged series is the following: if $n=2k+1$ for some $k,$ switch $n$ with $4k+2,$ and vice versa. Leave all other $n$ fixed. The new sum looks like

$$-\frac{1}{2}+1-\frac{1}{6}-\frac{1}{4}-\frac{1}{10}+\frac{1}{3}-\frac{1}{14}-\frac{1}{8}...$$

Try to show that this sum converges to $-\frac{3}{4}\log(2).$

One possible way is to show that the new sum has the same limit as the sum

$$\sum_{n=0}^\infty -\frac{1}{8n+2}+\frac{1}{2n+1}-\frac{1}{8n+6}-\frac{1}{4n+4},$$ if finding this limit is easier.


# Simple Number Theory

• May 16th 2010, 07:06 PM
FlacidCelery
Simple Number Theory
Positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N?

Note I have not yet taken a Number Theory course.

I think I have found the solution using a bit of reasoning and some luck. N=60? I figured N could not be smaller than the gcd of 30 and 72, and could not be greater than their product. I also found a pattern for (30N)/72. Inputting 10, 15, 20, 30 for N gave a result of 25/6, 25/4, 25/3, 25/2, respectively. I figured that this converged to 25/1, which would then be my solution. N=60 indeed yields 25/1.

However, I feel that this is closer to luck than anything else, and also it is not very elegant. Can someone show me another way of doing this, perhaps something more elegant?

-F
• May 16th 2010, 08:04 PM
roninpro

After noting that $\gcd(30,72)=6\leq N\leq 30\cdot 72=2160$, I would probably list all of my possibilities at this point: 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 36, 40, 45, 48, 54, 60, 72, 80, 90, 108, 120, 135, 144, 180, 216, 240, 270, 360, 432, 540, 720, 1080, 2160.

Now, 30 divides $72N$. Since $30=2\cdot 3\cdot 5$ and $72=2^3\cdot 3^2$, we must conclude that $N$ is a multiple of 5. This leaves 10, 15, 20, 30, 40, 45, 60, 80, 90, 120, 135, 180, 240, 270, 360, 540, 720, 1080, 2160.

On the other hand, 72 divides $30N$. By similar reasoning as before, $N$ must be a multiple of $2^2\cdot 3=12$. Eliminating the bad possibilities leaves 60 as the smallest number satisfying all three conditions.
• May 16th 2010, 08:30 PM
NowIsForever
8|72 so for 8 to divide 30N, 4|N. 5|30, thus 5|N, since 5∤72. 3|N since 9|72 and 9∤30. Thus N must be at minimum 2²·3·5 = 60.
• May 16th 2010, 09:49 PM
simplependulum
Consider the three numbers

$ab ~,~ bc ~,~ ca$

the set satisfies the requirement because

$(ab)(bc) = b^2 (ca)$

$(bc)(ca) = c^2 (ab)$

$(ca)(ab) = a^2 (bc)$

Let $ab = 30$ , $bc = 72$ so $N = ca$ .

To minimize $N = ca$ , we have to maximize $b$ so obviously what we are looking for is the greatest common divisor of $30$ and $72$ which is $6$

Therefore $N = 5(12) = 60$
• May 16th 2010, 10:04 PM
Soroban
Hello, FlacidCelery!

Quote:

Positive integers 30, 72 and $N$ have the property
that the product of any two of them is divisible by the third.
What is the smallest possible value of $N$ ?

We have: . $\begin{array}{ccccc}A &=&30 &=& 2\cdot3\cdot5 \\ B &=& 72 &=& 2^3\cdot3^2 \end{array}$

$A$ times $N$ is divisible by $B$: . $\frac{2\cdot3\cdot5\cdot N}{2^3\cdot3^2}$ is an integer.
This reduces to: . $\frac{5N}{12}$ . . . Hence. $N$ is a multiple of 12.

$B$ times $N$ is divisible by $A$: . $\frac{2^3\cdot3^2\cdot N}{2\cdot3\cdot5}$ is an integer.
This reduces to: . $\frac{2^2\cdot3\cdot N}{5}$ . . . Hence, $N$ is a multiple of 5.

The least number which is a multiple of 12 and a multiple of 5 is: . $N \:=\:60$

• May 17th 2010, 03:12 AM
FlacidCelery
Quote:

Originally Posted by Soroban
Hello, FlacidCelery!

We have: . $\begin{array}{ccccc}A &=&30 &=& 2\cdot3\cdot5 \\ B &=& 72 &=& 2^3\cdot3^2 \end{array}$

$A$ times $N$ is divisible by $B$: . $\frac{2\cdot3\cdot5\cdot N}{2^3\cdot3^2}$ is an integer.
This reduces to: . $\frac{5N}{12}$ . . . Hence. $N$ is a multiple of 12.

$B$ times $N$ is divisible by $A$: . $\frac{2^3\cdot3^2\cdot N}{2\cdot3\cdot5}$ is an integer.
This reduces to: . $\frac{2^2\cdot3\cdot N}{5}$ . . . Hence, $N$ is a multiple of 5.

The least number which is a multiple of 12 and a multiple of 5 is: . $N \:=\:60$

I really like this solution, it seems the more obvious to me. Thanks a lot everyone.


# What are the possible eigenvalues of a linear transformation $T$ satifying $T = T^2$ [duplicate]

This question already has an answer here:

Let $T$ be a linear transformation $T$ such that $T\colon V \to V$. Also, let $T = T^2$. What are the possible eigenvalues of $T$?

I am not sure if the answer is only $1$, or $0$ and $1$.

It holds that $T = T^2$, thus $T(T(x)) = T(x)$. Let's call $T(x) = v$, so $T(v) = v$. which means that $\lambda=1$. But I am not sure about this, while I have seen a solution that says that $0$ is possible as well.

Thanks in advance !

## marked as duplicate by Najib Idrissi, Martin R, Claude Leibovici, Community♦May 20 '15 at 8:48

Let $v\neq 0$ be an eigenvector of $T$ with eigenvalue $\lambda$, so $Tv=\lambda v$. Using $T=T^2$ we have $$Tv = T^2 v = T(Tv) = T(\lambda v) = \lambda(Tv) = \lambda^2 v.$$ Hence, $\lambda v = \lambda^2 v$. Since $v\neq 0$ we conclude $\lambda = \lambda^2$. The only solutions to this equation are $0$ and $1$.

Think of this as follows:

$$T^2=T\implies T(T-I)=0$$

Thus, $\;T\;$ is a root of $\;x(x-1)\;$ and thus the characteristic polynomial of $\;T\;$ can only have $\;0\;$ or $\;1\;$ as its roots, and thus these precisely are the only possible eigenvalues of $\;T\;$ .

So you were half right...:)

Another side remark: You say that you are not sure if 1, or both 0 and 1 can be eigenvalues.

In some cases, it is worthwhile to think of specific examples and see what they can tell us. So what are some examples of matrices $T$ that satisfy $T^2 = T$?

Well, the identity is certainly one, and its eigenvalues are all 1.

However, another such matrix is the zero matrix! It also trivially satisfies $\mathbf{0}^2 = \mathbf{0}$. Its eigenvalues are all zero, so zero can certainly be an eigenvalue as well.

Anyhow, this of course just tells you that both 0 and 1 are possible eigenvalues of such a matrix, but not that they are the only possible eigenvalues. For that, the other answers provide a full solution.


Is there a difference between instantaneous speed and the magnitude of instantaneous velocity?

Consider a particle that moves around the coordinate grid. After $$t$$ seconds, it has the position $$S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, .$$ The particle traces a quarter arc of length $$\pi/2$$ around the unit circle. This means that the average speed of the particle is $$\frac{\text{distance travelled along the arc of the circle}}{\text{time}}=\frac{\pi/2}{\pi/2} = 1 \, .$$ However, since the motion of the particle is circular, the distance travelled is not the same as the displacement. The displacement of the particle would be $$\sqrt{2}$$, and so the average velocity would be $$\frac{\text{straight line distance from initial position}}{\text{time}} = \frac{\sqrt{2}}{\pi/2} = \frac{2\sqrt{2}}{\pi} \text{ at angle of \frac{3}{4}\pi with the positive x-axis} \, .$$ Here is the part I don't quite understand: over an interval, the average speed of the particle is different from the magnitude of its velocity. In the above example, the former is $$1$$, whereas the latter is $$\frac{2\sqrt{2}}{\pi}$$. However, the magnitude of the instantaneous velocity of the particle is the same as the instantaneous speed: here, they are both equal to $$1$$. We can mathematically prove this by considering the following limit $$|S'(t)| = \lim_{h \to 0}\frac{|S(t+h)-S(t)|}{|h|}=\lim_{h \to 0}\frac{\sqrt{\left(\sin(t+h)-\sin t \right)^2+\left( \cos(t+h)-\cos t\right)^2}}{|h|} \, ,$$ which turns out to be equal to $$1$$. Hence, the magnitude of the instantaneous velocity is $$1$$. And clearly, the instantaneous speed of the particle is $$\lim_{h \to 0}\frac{h}{h} = 1 \, ,$$ since the distance travelled along the arc between $$S(t+h)$$ and $$S(t)$$ is simply $$h$$ units. However, will this always be the case? Is the magnitude of the instantaneous velocity of a particle always equal to its instantaneous speed?

• To my knowledge the definition of speed is that it is the magnitude of the velocity, $v := |\vec v|$. Feb 8 at 18:16
• What you are discovering here has nothing to do with physics but how any smooth and continuous function can appear linear when seen on a small enough scale. Feb 8 at 18:37
• @Triatticus Thanks, that makes sense. Is there a precise way of formulating this mathematically? And if so, is there a way of proving this statement as a theorem?
– Joe
Feb 8 at 21:52

By definition, $$\left|\text{instantaneous velocity}\right| = \text{instantaneous speed}.$$
However, \begin{aligned} \left|\text{average velocity}\right| &= \left|\frac{\text{displacement (i.e., change in position)}}{\text{time elapsed}}\right|\\ &= \frac{\left|\text{displacement (i.e., change in position)}\right|}{\text{time elapsed}}\\ &\leq \frac{\text{distance travelled}}{\text{time elapsed}}\\ &= \text{average speed}. \end{aligned}