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Functional Analysis

Functional analysis is a branch of mathematical analysis. It is used to explain the workings of a complex system.  It was formed by the study of vector spaces with the limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense. Fourier transform as transformations defining continuous, unitary operators between function spaces. This analysis is useful for the study of differential and integral equations.

The basic idea of functional analysis is that the system is viewed as computing a function. It assumes that such processing can be explained by decomposing this complex function into a set of simpler functions that are computed by an organized system of sub processors. There are three stages in the methodology that defines functional analysis.  The first stage is where the to-be-explained function is defined. The second stage is where the analysis is performed. The to-be-explained function is decomposed into an organized set of simpler functions. This analysis can precede recursively by decomposing some of the subfunctions into sub-subfunctions. The third stage is where the analysis is stopped by subsuming the bottom level of functions. The operation of each of these operations is explained by appealing to natural laws.

In modern mathematics, functional analysis is the subject seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces. An important aspect of functional analysis is the extension of the theory of measure, integration and probability to infinite dimensional spaces, also known as infinite dimensional analysis. 

Parametric & Rectangular Equations, Slope & Concavity

Hello, I am in a fast-paced Calculus course where I must learn new concepts each week; I find it challenging to grasp the concepts while remaining on-pace and I am experiencing great difficulty. I am really finding it difficult to grasp the concept of Parametric & Rectangular Equations, Slope & Concavity, and Tangent Lines.

Need Assistance Understanding Taylor and Maclaurin Series

Hello, I am in a fast-paced Calculus course where I must learn new concepts each week; I find it challenging to grasp the concepts while remaining on-pace and I am experiencing great difficulty. I have a few weeks before my semester is over, and thankfully, I have a passing grade. I am really finding it difficult to grasp

Convergent or divergent series

Please assist with the following problems I am having a hard time solving. Please see attached. Determine if the following series are convergent or divergent Find the values of x for which and tell if the series converges or diverges when given the series Find power series for the following function

Convergent or divergent series

Use the ratio a_n/a_(n+1) to determine if the following sequences are monotonic increasing, decreasing or neither. Use the derivative test to determine if the following sequences are monotonic increasing, decreasing or neither. Use the nth term test to stae if the series converges, diverges, or cannot tell.

definite integral and area

Let B(x) represent the area bounded by the graph and the horizontal axis and vertical lines at t=0 and t=x for the graph shown below. Evaluate B(x) for x = 1 and 5. For f(x)=1/(x+1) find Find the areas of the rectangles: Rewrite as a definite integral. Do not evaluate. Represent the area bo

calculus questions in differentiation and integration

Use implicit differentiation to find dy/dx. xy+x+y=x^2 y^2 Given y = f(u) and u = g(x), find dy/dx = f'(g(x))g'(x). y=u(u-1),u=x^2+x Find dy/dx y=(4x-5)(4x^3-x^2+1) Find the derivative of the function "y" shown below. y=(x^2-8x+3)/(√x) One airplane is approaching an airport from the north at 163 km/hr

Calculus Help

Evaluate the integral shown below (try substitution) ∫▒(x dx)/(7x^2+3)^5 Evaluate the integral shown below (apply a property of logarithms first.) ∫▒(In x^6)/x dx Use the Fundamental Theorem of Calculus to find the derivative shown below. d/dx ∫_0^(x^5)▒sin⁡〖t dt〗 For the function below, s

Calculus Help

Let B(x) represent the area bounded by the graph and the horizontal axis and vertical lines at t=0 and t=x for the graph below. Evaluate B(x) for x = 1, 2, 3, 4, and 5. f(x)=x^2,g(x)=3x,and h(x)=2/x. Evaluate each sum. ∑_(i=0)^3▒〖f(1+i)〗 sketch the function and find the smallest possible value and the l

First and Second Derivative tests as appropriate

Find the derivative of the function shown below. y=(x+1)^2 (x^2+1)^(-3) Find dy/dt y=5t(2t+3)^3 Use implicit differentiation to find dy/dx x^3+3x^2 y+y^3=8 Find the derivative of y with respect to x, t, or 0 as appropriate. y=x^5 In x-1/3 x^3 A piece of land is shaped like a right triangle. Two peopl

Calculating Limit of Expressions

You have been asked to determine the least expensive route for a telephone cable that connects Andersonville with Beantown (see figure below) If it costs $5000 per mile to lay the cable on land and $8000 per mile to lay the cable across the river (with the cost of the cable included), find the least expensive route. Fi

Define Bounded Area Formula

Define A(x) to be the area bounded by the t-axis, the line y = 2t and a vertical line at t = x. (a) Find a formula for A(x). (b) Determine A'(x) The figure below shows the graph of the derivative of a continuous function f . (a) List the critical numbers of f . (b) What values of x result in a local maximum? (c) What va

Finding Critical Point Functions

find all of the critical points and local maximums and minimums of each function f(x)=〖2x〗^2-12x+7 find all critical points and local extremes of each function on the given intervals. f(x)=X^2-3x+5 on the entire real number line. find all critical points and local extremes of each function on the given interv

Calculus questions on differentiation

A circle of radius 3 inches is inside a square with 12-inch sides (see figure below). How fast is the area between the circle and square changing if the radius is increasing at 4 inches per minute and the sides are increasing at 2 inches per minute? Find dy/dx in two way (a) by differentiating implicitly and (b) by

Calculus for Parallel Lines of Graphs

solve If f(x) =7x + 11 and g(x) =5x- 1, find f(g(x)). What is f(g(2))? Find the average rate of change of the function shown below over the given interval. y=-3x^2 - x, [5, 6] Find the limit, if it exists. 〖lim┬(x→4) (x^2+4x-32)/(x^2-16) 〗⁡〖〖^〗〗 Find an equation for the tangent t

Trigonometry questions of identity

1. Express in equivalent form: log x + log 1 - 6 log (y + 4) 2. Points (4, -7) and (6, 3) are endpoints of the diameter of a circle./ (a) What is the exact length of the diameter? (b) What is the center of the circle? (c) What is the equation of the circle? 3. Given y = 9 sin(8x - p), state the (a) period (b) phas


1. Complete the square in order to put the equation into standard form. Identify the center and the radius or explain why the equation does not represent a circle. 2. Find the standard equation of the circle with endpoints of a diameter (-3,7) and (1,5). 3. Find an equation of a parabola satisfying the given: Focus (0

Solver solution for minimal shipping costs

SE Appliances manufactures refrigerators in Richmond, Charlotte, and Atlanta. Refrigerators then must be shipped to meet demand in Washington, New York, and Miami. The table below lists the shipping costs, supply, and demand information. How many units should be shipped from each plant to each retail store in order to minimiz

Polynomial Zero's

I'm have trouble understanding the following problem please assist so I may complete my assignment. 1. Find the real zeros of the function f(x) = -(x + 1)^3 (x - 2)^2 and their corresponding multiplicities. Use the information along with a sign chart (diagram) and then the end behavior to provide a rough sketch of the grap

Disspative Kepler's Potential in One Dimension

Question 1: (i) Write down the equation of motion for the point particle of mass m moving in the Kepler potential U (x) = -A /x + B /x2. Neglect the dissipation. (ii) Introduce a dissipative term in the equation of motion assuming that the viscous force acting on a particle is proportional to the particle velocity. Writ

Linearizing system of ODEs, Minimizing a functional

Question 3: A dynamical system is governed by two equations x4 y 2x, yy 8 y 12x. (a) Show that the critical points of this system are P(0, 0) and Q(1, 2). (b) Using linearation of the system in the neighbourhood of each critical point, determine the

Equation of motion is liquid

Question 1: Consider the equation of motion of a very light spherical solid particle in the creaping flow regime when the Reynolds number Re >> 1 a) Neglecting Bosinesq-Basset drag force find a solution to the equation of particle motion: (r + 1/2)(d^2*z/d*t^2) = (1 - r)g - (3v/a^2)(dz/dt) (1.1) for two limiting cases

Lagrangian, Hamilton and Variations with Constraints

1. The ground-state energy of a quantum particle of mass m in a pillbox (right-circular cylinder) is given by the following equation (see attachment). Find the ratio of R to H that will minimize the energy for a fixed volume. 2. A particle, mass m, is on a frictionless horizontal surface. It is constrained to move so that th

Implicit Function Theorem and Theorem of Lagrange Multipliers

I don't get how they can take the derivative of g_1 and g_2 with respect to x_1 and x_2 when they are defined as h_1=h_1 (x_3,x_4,...,x_n )=x_1 and h_2=h_2 (x_3,x_4,...,x_n )=x_2 I need a mathematical justification for how this can be written simply as (21) when x_1 and x_2 are defined as functions from other variables

Performance evaluation: standard vs. actual practices

It is important to have performance measures to evaluate managers as they control resources and invest in assets for the company. Describe how you could use different variances (actual to standard) to evaluate performance. Additionally, there are non-financial performance measures that can be used. Are there any that you think w

Decision Tree - Project Bidding

I need assistance setting up a decision tree in excel. Your employer wants you to bid on a project with an important client. The revenue from this project is important for the company's bottom-line this year. Estimates show that if you bid $4 million, there is a 30% chance you'll win the bid. If you bid $5 million, there is

Linear Regression Decision Model: Appraisers

I need assistance doing linear regression decision model. It needs to be done in Excel. The data you need is in the excel file. a.If the team of appraisers want to use a simple linear regression decision model (one X, one Y) based on either X1 or X2, which one of these independent variables do you recommend they use? Why?

Solving several questions on complex numbers

These questions involve trigonometry, complex numbers, polar equations, and parametric equations. See below for the problems I require assistance with, and refer to the attachments to find details regarding those problems. Show all work in a Word document. p. 357 #61, p. 358 #81 p 359, #115 p. 364 #27 p

Exponential Function V(t)

A fax machine is purchased for $5,800. Its value each year is about 80% of the value of the preceding year. So after t years the value, in dollars, of the fax machine, V(t), is given by the exponential function V(t) = 5800 (0.8)t. a. Give a sketch of the graph of the function V(t). Your graph can be a "rough draft".