5.1 Defining and visualizing functions. A handout.Jan Plaza
This document introduces concepts related to functions including:
- Defining functions in terms of unique mappings between inputs and outputs
- Distinguishing between total, partial, and non-functions
- Specifying domains and ranges
- Using vertical line tests to identify functions from graphs
- Examples of functions defined by formulas or mappings
The document defines key concepts related to functions including: domain, codomain, range, one-to-one (injective) functions, increasing/decreasing functions, onto (surjective) functions, bijective functions, inverse functions, function composition, and floor and ceiling functions. It provides examples to illustrate these concepts and determine if specific functions have given properties.
5.1 Defining and visualizing functions. A handout.Jan Plaza
This document introduces concepts related to functions including:
- Defining functions in terms of unique mappings between inputs and outputs
- Distinguishing between total, partial, and non-functions
- Specifying domains and ranges
- Using vertical line tests to identify functions from graphs
- Examples of functions defined by formulas or mappings
The document defines key concepts related to functions including: domain, codomain, range, one-to-one (injective) functions, increasing/decreasing functions, onto (surjective) functions, bijective functions, inverse functions, function composition, and floor and ceiling functions. It provides examples to illustrate these concepts and determine if specific functions have given properties.
This document discusses inverse functions including:
- Verifying that two functions are inverse functions by showing their compositions result in the identity function
- Determining whether a function has an inverse function using the horizontal line test
- Finding the derivative of an inverse function using the theorem that the derivative of the inverse is the reciprocal of the derivative of the original function
To have an inverse function, a function must be one-to-one and pass the horizontal line test. If a function fails either of these, the inverse is considered a relation instead of a function. The domain of the original function becomes the range of the inverse, and the range of the original becomes the domain of the inverse. An example given is reflecting points across the x or y-axis.
This document discusses inverse functions and their derivatives. It defines inverse functions as switching the x- and y-values of a function to "undo" the original function. A function has an inverse only if it passes the horizontal line test. The derivative of an inverse function at a point equals the reciprocal of the derivative of the original function at the corresponding point.
The document discusses inverse functions. It defines inverse functions as pairs of one-to-one functions that undo each other. To find the inverse of a function, interchange the x and y variables and solve for y. The graph of an inverse function is a reflection of the original function across the line y = x. Examples are provided to illustrate how to determine if a function is one-to-one, find the inverse function, and relate the graph of a function to its inverse.
The document discusses function operations such as addition, subtraction, multiplication, and division of functions. It provides examples of performing these operations on specific functions f(x) and g(x). For addition, subtraction and multiplication, the results are simply the corresponding arithmetic operations performed on the individual functions. For division, the domain is restricted to values where g(x) is not equal to 0 to avoid division by zero.
The document discusses inverse functions, including:
- An inverse function undoes the output of the original function by relating the input and output variables.
- For a function to have an inverse, it must be one-to-one so that each output is paired with a unique input.
- To find the inverse of a function, swap the input and output variables and isolate the new output variable.
This document discusses finding the tangent line to the graph of a function f. It outlines the steps as: 1) finding the domain of f, 2) finding the derivative of f, and 3) considering cases based on whether the point of tangency (x0, f(x0)) is known or unknown. If the point is known, the slope of the tangent line is f'(x0) and the equation can be found. If unknown, additional information is needed, such as if the line is parallel/perpendicular to another line or passes through a specific point. The derivative f'(x0) and this extra information can be used to find the equation of the tangent line. Care must be taken to understand
This document contains a mathematics exam with 4 problems (Themes A, B, C, D) involving functions, derivatives, monotonicity, convexity, extrema, asymptotes and limits.
Theme A involves properties of differentiable functions, the definition of the derivative, and Rolle's theorem. Theme B analyzes the monotonicity, convexity, asymptotes and graph of a given function.
Theme C proves properties of a continuous, monotonically increasing function and finds extrema of related functions. Theme D proves properties of a power function and its relation to a given line, defines a new function, and proves monotonicity and existence of a single real root for a polynomial equation.
Lecture 20 fundamental theorem of calc - section 5.3njit-ronbrown
The document discusses indefinite integrals and introduces the notation used to represent them. It explains that an indefinite integral of a function f(x) is represented by ∫f(x) dx and denotes the general antiderivative of f(x), not a specific function. The document also provides an example of finding the indefinite integral of 10x4 - 2sec2x and explains how to check the answer.
This document discusses inverses of functions and their graphs. It provides an example of finding the inverse of a point (-4, 3) by reflecting it over the line y=x and switching the x and y values. It also gives an example function f(x)=4x+3 and explains how to write the inverse equation, draw the graph, and state the domain and range of both the function and its inverse. Finally, it provides two functions and explains how to determine algebraically if they are inverses of each other.
This document contains a mathematics exam for high school students in Greece. It is divided into 4 sections with multiple questions in each section. The questions cover topics related to functions, limits, derivatives, and integrals. Some questions ask students to prove statements, find domains of functions, determine if functions are injective or have critical points. The document is 3 pages long and aims to test students' understanding of key concepts in calculus and mathematical analysis.
Composition of functions g◦f means applying the function g to the output of f. You apply f first to get an output in its range, then apply g to that output. For example, if f maps x to y, then g◦f maps x to g(y). The composition only includes values where both functions are defined, staying within their domains and ranges. An example composition is given of functions f: A→B and g: B→C, showing how to follow the inputs and outputs through each function to determine the composition g◦f.
1. Prove that the function f(x) = x^2 if x is rational, 0 otherwise, is differentiable at 0 but discontinuous everywhere else.
2. Use the Chain Rule to find an expression for the derivative of the inverse function g(f(c)) in terms of f'(c), given that f and g are inverse bijective functions between intervals with f differentiable at c and f(c) = 0, and g differentiable at f(c).
3. Prove several rules for derivatives, including the Power Rule and derivatives of composite functions.
Continuity is the property that the limit of a function near a point is the value of the function near that point. An important consequence of continuity is the intermediate value theorem, which tells us we once weighed as much as our height.
This document summarizes a machine learning homework assignment with 4 problems:
1) Probability questions about constructing random variables.
2) Questions about Poisson generalized linear models (GLM) including log-likelihood, prediction, and regularization.
3) Questions comparing square loss and logistic loss for an outlier point.
4) Questions about batch normalization including its effect on model expressivity and gradients.
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
The document discusses the Fundamental Theorem of Calculus. It states that if f(x) is an integrable function and g(x) is an antiderivative of f(x), then the integral of f(x) from a to b is equal to g(b) - g(a). This theorem links the concepts of differentiation and integration by showing that one can find the area under a curve by subtracting the antiderivative evaluated at the bounds. It is an important foundational theorem in calculus.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This document discusses inverse functions including:
- Verifying that two functions are inverse functions by showing their compositions result in the identity function
- Determining whether a function has an inverse function using the horizontal line test
- Finding the derivative of an inverse function using the theorem that the derivative of the inverse is the reciprocal of the derivative of the original function
To have an inverse function, a function must be one-to-one and pass the horizontal line test. If a function fails either of these, the inverse is considered a relation instead of a function. The domain of the original function becomes the range of the inverse, and the range of the original becomes the domain of the inverse. An example given is reflecting points across the x or y-axis.
This document discusses inverse functions and their derivatives. It defines inverse functions as switching the x- and y-values of a function to "undo" the original function. A function has an inverse only if it passes the horizontal line test. The derivative of an inverse function at a point equals the reciprocal of the derivative of the original function at the corresponding point.
The document discusses inverse functions. It defines inverse functions as pairs of one-to-one functions that undo each other. To find the inverse of a function, interchange the x and y variables and solve for y. The graph of an inverse function is a reflection of the original function across the line y = x. Examples are provided to illustrate how to determine if a function is one-to-one, find the inverse function, and relate the graph of a function to its inverse.
The document discusses function operations such as addition, subtraction, multiplication, and division of functions. It provides examples of performing these operations on specific functions f(x) and g(x). For addition, subtraction and multiplication, the results are simply the corresponding arithmetic operations performed on the individual functions. For division, the domain is restricted to values where g(x) is not equal to 0 to avoid division by zero.
The document discusses inverse functions, including:
- An inverse function undoes the output of the original function by relating the input and output variables.
- For a function to have an inverse, it must be one-to-one so that each output is paired with a unique input.
- To find the inverse of a function, swap the input and output variables and isolate the new output variable.
This document discusses finding the tangent line to the graph of a function f. It outlines the steps as: 1) finding the domain of f, 2) finding the derivative of f, and 3) considering cases based on whether the point of tangency (x0, f(x0)) is known or unknown. If the point is known, the slope of the tangent line is f'(x0) and the equation can be found. If unknown, additional information is needed, such as if the line is parallel/perpendicular to another line or passes through a specific point. The derivative f'(x0) and this extra information can be used to find the equation of the tangent line. Care must be taken to understand
This document contains a mathematics exam with 4 problems (Themes A, B, C, D) involving functions, derivatives, monotonicity, convexity, extrema, asymptotes and limits.
Theme A involves properties of differentiable functions, the definition of the derivative, and Rolle's theorem. Theme B analyzes the monotonicity, convexity, asymptotes and graph of a given function.
Theme C proves properties of a continuous, monotonically increasing function and finds extrema of related functions. Theme D proves properties of a power function and its relation to a given line, defines a new function, and proves monotonicity and existence of a single real root for a polynomial equation.
Lecture 20 fundamental theorem of calc - section 5.3njit-ronbrown
The document discusses indefinite integrals and introduces the notation used to represent them. It explains that an indefinite integral of a function f(x) is represented by ∫f(x) dx and denotes the general antiderivative of f(x), not a specific function. The document also provides an example of finding the indefinite integral of 10x4 - 2sec2x and explains how to check the answer.
This document discusses inverses of functions and their graphs. It provides an example of finding the inverse of a point (-4, 3) by reflecting it over the line y=x and switching the x and y values. It also gives an example function f(x)=4x+3 and explains how to write the inverse equation, draw the graph, and state the domain and range of both the function and its inverse. Finally, it provides two functions and explains how to determine algebraically if they are inverses of each other.
This document contains a mathematics exam for high school students in Greece. It is divided into 4 sections with multiple questions in each section. The questions cover topics related to functions, limits, derivatives, and integrals. Some questions ask students to prove statements, find domains of functions, determine if functions are injective or have critical points. The document is 3 pages long and aims to test students' understanding of key concepts in calculus and mathematical analysis.
Composition of functions g◦f means applying the function g to the output of f. You apply f first to get an output in its range, then apply g to that output. For example, if f maps x to y, then g◦f maps x to g(y). The composition only includes values where both functions are defined, staying within their domains and ranges. An example composition is given of functions f: A→B and g: B→C, showing how to follow the inputs and outputs through each function to determine the composition g◦f.
1. Prove that the function f(x) = x^2 if x is rational, 0 otherwise, is differentiable at 0 but discontinuous everywhere else.
2. Use the Chain Rule to find an expression for the derivative of the inverse function g(f(c)) in terms of f'(c), given that f and g are inverse bijective functions between intervals with f differentiable at c and f(c) = 0, and g differentiable at f(c).
3. Prove several rules for derivatives, including the Power Rule and derivatives of composite functions.
Continuity is the property that the limit of a function near a point is the value of the function near that point. An important consequence of continuity is the intermediate value theorem, which tells us we once weighed as much as our height.
This document summarizes a machine learning homework assignment with 4 problems:
1) Probability questions about constructing random variables.
2) Questions about Poisson generalized linear models (GLM) including log-likelihood, prediction, and regularization.
3) Questions comparing square loss and logistic loss for an outlier point.
4) Questions about batch normalization including its effect on model expressivity and gradients.
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
The document discusses the Fundamental Theorem of Calculus. It states that if f(x) is an integrable function and g(x) is an antiderivative of f(x), then the integral of f(x) from a to b is equal to g(b) - g(a). This theorem links the concepts of differentiation and integration by showing that one can find the area under a curve by subtracting the antiderivative evaluated at the bounds. It is an important foundational theorem in calculus.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This document summarizes key concepts from a chapter on convex optimization, including:
1) The chapter introduces primal and dual problems, where the primal problem minimizes a convex function f over Rn and the dual problem maximizes the dual function f* over Rm.
2) Weak duality is proven, showing that the optimal value of the dual problem is always less than or equal to the primal problem.
3) For strong duality to hold (where the optimal values are equal), conditions are introduced like the function φ being proper, closed, and convex. The chapter explores when these conditions are satisfied.
4) Polyhedra and the finite basis theorem are discussed, showing that projections
1 IntroductionThese notes introduces a particular kind of HiAbbyWhyte974
1 Introduction
These notes introduces a particular kind of Hilbert space known as a reproducing kernel Hilbert space (RKHS).
We will establish connections with kernels, defined previously, and show that kernels and RKHSs are in one-to-
one correspondence. This material is largely drawn from Chapter 4 of [1], although some results are presented
in a slightly different way to ease digestion.
2 Reproducing Kernel Hilbert Spaces
Throughout these notes, we use the term “Hilbert function space over X ” to refer to a Hilbert space whose
elements are functions f : X 7→ R.
Definition 1. (Reproducing Kernel) Let F be a Hilbert function space over X . A reproducing kernel of F is a
function k : X × X 7→ R which satisfies the following two properties:
1. k (·,x) ∈F for any x ∈X.
2. (Reproducing property) For any f ∈F and any x ∈X, f (x) = 〈f,k (·,x)〉F.
Theorem 1. Let F be a Hilbert function space over X. The following are equivalent:
1. F has a reproducing kernel.
2. For any x ∈X, the function δx : F 7→ R defined by δx (f) = f (x) is continuous.
Definition 2. Let F be a Hilbert function space over X. We say F is a reproducing kernel Hilbert space
over X if it satisfies the conditions of the previous theorem.
Proof. ( (1) ⇒ (2) ) . Suppose that F has a reproducing kernel, k : X ×X 7→ R. We need to show that if
(fn)
∞
n=1 converges to f ∈F, i.e., ‖fn −f‖F → 0, then for every x ∈X , |δx (fn) −δx (f)| = |fn (x) −f (x)|→
0. Now
δx (f) = f (x) = 〈f,k (·,x)〉F = 〈 lim
n→∞
fn,k (·,x)〉F
(a)
= lim
n→∞
〈fn,k (·,x)〉F = lim
n→∞
fn (x) = lim
n→∞
δx (fn)
Note that the identity “(a)” is implied by the continuity of inner product operator of its first argument.
Before proving the reverse implication, we state a classical result from functional analysis named the Riesz
representation theorem.
Theorem 2 (Riesz representation theorem). Let F be a Hilbert space and L : F 7→ R be a linear functional
on F. Then L is continuous if and only if there exists Φ ∈F such that L (f) = 〈f, Φ〉F for any f ∈F.
1
2
( (2) ⇒ (1) ) . Let x ∈ X . It’s trivial to see that δx is a linear functional, so due to the assumed
continuity of δx, Theorem 2 ensures the existence of a Φx ∈ F such that δx (f) = f (x) = 〈f, Φx〉F. Now
define k (x2,x1) := Φx1 (x2) ∈ R. Note that k (·,x) = Φx ∈F (establishing the first property of reproducing
kernels). Also,
f (x) = 〈f, Φx〉F = 〈f,k (·,x)〉F
which shows that the reproducing property holds.
The next result establishes that kernels and reproducing kernels are the same.
Theorem 3. Let k : X ×X 7→ R. Then k is a kernel if and only if k is a reproducing kernel of some RKHS
F over X.
Proof. (⇐) . We first prove the reverse implication. Let k be the reproducing kernel of F and define Φ :
X 7→F by Φ (x) = k (·,x). Now for any x′ ∈X , consider the function fx′ = k (·,x′). Using the reproducing
property and symmetry of the inner product, we obtain
k (x,x′) = fx′ (x) = 〈fx′,k (·,x)〉F = 〈k (·,x′) ,k (·,x)〉F
= 〈Φ (x′) , Φ (x)〉F = 〈Φ (x) , Φ (x′ ...
1 IntroductionThese notes introduces a particular kind of HiMartineMccracken314
1 Introduction
These notes introduces a particular kind of Hilbert space known as a reproducing kernel Hilbert space (RKHS).
We will establish connections with kernels, defined previously, and show that kernels and RKHSs are in one-to-
one correspondence. This material is largely drawn from Chapter 4 of [1], although some results are presented
in a slightly different way to ease digestion.
2 Reproducing Kernel Hilbert Spaces
Throughout these notes, we use the term “Hilbert function space over X ” to refer to a Hilbert space whose
elements are functions f : X 7→ R.
Definition 1. (Reproducing Kernel) Let F be a Hilbert function space over X . A reproducing kernel of F is a
function k : X × X 7→ R which satisfies the following two properties:
1. k (·,x) ∈F for any x ∈X.
2. (Reproducing property) For any f ∈F and any x ∈X, f (x) = 〈f,k (·,x)〉F.
Theorem 1. Let F be a Hilbert function space over X. The following are equivalent:
1. F has a reproducing kernel.
2. For any x ∈X, the function δx : F 7→ R defined by δx (f) = f (x) is continuous.
Definition 2. Let F be a Hilbert function space over X. We say F is a reproducing kernel Hilbert space
over X if it satisfies the conditions of the previous theorem.
Proof. ( (1) ⇒ (2) ) . Suppose that F has a reproducing kernel, k : X ×X 7→ R. We need to show that if
(fn)
∞
n=1 converges to f ∈F, i.e., ‖fn −f‖F → 0, then for every x ∈X , |δx (fn) −δx (f)| = |fn (x) −f (x)|→
0. Now
δx (f) = f (x) = 〈f,k (·,x)〉F = 〈 lim
n→∞
fn,k (·,x)〉F
(a)
= lim
n→∞
〈fn,k (·,x)〉F = lim
n→∞
fn (x) = lim
n→∞
δx (fn)
Note that the identity “(a)” is implied by the continuity of inner product operator of its first argument.
Before proving the reverse implication, we state a classical result from functional analysis named the Riesz
representation theorem.
Theorem 2 (Riesz representation theorem). Let F be a Hilbert space and L : F 7→ R be a linear functional
on F. Then L is continuous if and only if there exists Φ ∈F such that L (f) = 〈f, Φ〉F for any f ∈F.
1
2
( (2) ⇒ (1) ) . Let x ∈ X . It’s trivial to see that δx is a linear functional, so due to the assumed
continuity of δx, Theorem 2 ensures the existence of a Φx ∈ F such that δx (f) = f (x) = 〈f, Φx〉F. Now
define k (x2,x1) := Φx1 (x2) ∈ R. Note that k (·,x) = Φx ∈F (establishing the first property of reproducing
kernels). Also,
f (x) = 〈f, Φx〉F = 〈f,k (·,x)〉F
which shows that the reproducing property holds.
The next result establishes that kernels and reproducing kernels are the same.
Theorem 3. Let k : X ×X 7→ R. Then k is a kernel if and only if k is a reproducing kernel of some RKHS
F over X.
Proof. (⇐) . We first prove the reverse implication. Let k be the reproducing kernel of F and define Φ :
X 7→F by Φ (x) = k (·,x). Now for any x′ ∈X , consider the function fx′ = k (·,x′). Using the reproducing
property and symmetry of the inner product, we obtain
k (x,x′) = fx′ (x) = 〈fx′,k (·,x)〉F = 〈k (·,x′) ,k (·,x)〉F
= 〈Φ (x′) , Φ (x)〉F = 〈Φ (x) , Φ (x′ ...
This document provides an introduction to the concepts of continuity and differentiability in calculus. It begins by giving two informal examples of functions that are and aren't continuous at a point to build intuition. It then provides a formal definition of continuity as the limit of a function at a point equaling the function value at that point. Several examples are worked through to demonstrate checking continuity at points and for entire functions. The document introduces the concept of limits approaching infinity to discuss the continuity of functions like 1/x. Overall, it lays the groundwork for understanding continuity and differentiability through examples and definitions.
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
A Komlo ́sTheorem for general Banach lattices of measurable functionsesasancpe
This document summarizes research on generalizing the Koml ́os theorem to Banach lattices of measurable functions. Specifically:
1) It generalizes previous results showing the Koml ́os theorem holds for Banach function spaces to more general spaces of integrable functions with respect to vector measures.
2) It proves the Fatou and Koml ́os properties are equivalent for sublattices of spaces L1(ν) with the Fatou property and for L1w(ν) spaces.
3) It provides an example of a Banach lattice of measurable functions that has the Fatou property but does not satisfy the Koml ́os theorem, showing the properties are not always equivalent
The Radon-Nikody ́m Theorem for vector measures and factorization of operator...esasancpe
This document presents notation and concepts related to Banach function spaces, including:
- Banach function spaces (B.f.s.) defined over a finite measure space
- Multiplication operators between B.f.s. and their associated multiplier spaces
- Product spaces of two B.f.s. and properties like order continuity
It also states a theorem regarding when the product space of two B.f.s. is order continuous.
On Frechet Derivatives with Application to the Inverse Function Theorem of Or...BRNSS Publication Hub
This document summarizes and reviews concepts related to Frechet derivatives. It begins by defining Frechet derivatives on Banach spaces and their properties such as differentiability of compositions of functions. It then discusses applications to ordinary differential equations, including the inverse function theorem. Higher order Frechet derivatives and partial derivatives on product spaces are also introduced. The document concludes by discussing concepts like the mean value theorem and Taylor's theorem in the context of Frechet derivatives.
This document summarizes and reviews concepts related to Frechet derivatives. It begins by defining Frechet derivatives on Banach spaces and their properties such as differentiability of compositions of functions. It then discusses applications to ordinary differential equations, including the inverse function theorem. Higher order Frechet derivatives and their properties are also introduced. The document concludes by stating results on the mean value theorem, Taylor's theorem, and Riemannian integration as they apply to Frechet derivatives.
This document summarizes an investigation into extending the Strong Maximum Principle for integral functionals involving Minkowski gauges. It begins by introducing the integral functional and definitions related to Minkowski gauges. It then discusses prior work establishing the Strong Maximum Principle under certain conditions, and extending the class of comparison functions used. The document aims to further generalize the Strong Maximum Principle by considering inf- and sup-convolutions of functions with the Minkowski gauge, which allows the principle to unify previous properties into a single extremal extension principle. It provides background, definitions, and auxiliary results to support the generalization proposed in Section 3.
The document discusses several key concepts regarding derivatives:
(1) It explains how to use the derivative to determine if a function is increasing, decreasing, or neither on an interval using the signs of the derivative.
(2) It provides theorems and rules for finding local extrema (maxima and minima) of functions using the first and second derivative tests.
(3) It also discusses absolute extrema, monotonic functions, and the Rolle's Theorem and Mean Value Theorem which relate the derivative of a function to values of the function.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
This document presents theorems and results regarding hypercyclic operators on the space Hbc(E), where E is a Banach space. Theorem 3.1 shows that the collection of functions {eφ : φ ∈ E*} forms an independently linear subset of Hbc(E). Theorem 3.2 proves that the span of {eφ : φ ∈ U} is dense in Hbc(E), where U is an open subset of E*. Theorem 3.3 demonstrates that if φ is an entire function of exponential type, then the operator φα(D) is hypercyclic on Hbc(E). The document also provides two corollaries: if E has a separable dual, then
This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.
6.2 Reflexivity, symmetry and transitivity (dynamic slides)Jan Plaza
This document introduces basic concepts of set theory including definitions of reflexive, symmetric and transitive relations. It provides examples of relations between lines that are parallel or perpendicular. It also discusses relations between people like siblings and ancestors. The document proves properties of congruence relations and discusses counting relations with certain properties over sets with a given number of elements. It addresses a claim about relations that are symmetric and transitive necessarily being reflexive, finding a counterexample.
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5.7 Function powers
1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Function powers
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of November 10, 2017
2. Definition
Let f : X −→ X. One defines recursively:
f0 = idX,
fn+1 = f ◦ fn, for any n ∈ N.
For any natural number n, fn is called the n-th function power of f or
the function power of f with the exponent n or the function power n of f .
Convention
We can drop the adjective “function” and say the n-th power of f
instead of “the n-th function power of f”.
Notes
Function powers are not defined for every f : X −→ Y ;
they are defined only if Y ⊆ X.
Power 0 is not defined for arbitrary binary relations.
Power 0 is defined for functions satisfying the condition above.
3. Informal Example
1. Let f(x) = 1.05x. This function gives the value of principal x after a year,
if deposited in a bank account that brings 5% annual interest.
Then, the power f10(x) is the total value after 10 years.
2. Let f be a function whose argument represents the atmospheric conditions, and
whose value represents the resulting atmospheric conditions 10 minutes later.
If x is an approximate current state of the atmosphere,
f24·6(x) is the state forecasted for 24 hours later.
(Chaos theory explains why
even a good approximation x of current conditions
makes fn(x), for high values of n,
a poor approximation of the actual future conditions.
So, long-term weather forecast is inherently unreliable.)
4. Proposition
Let f : X −→ X and m, n ∈ N. Then:
1. fm+n = fm ◦ fn = fn ◦ fm.
2. fm·n = (fm)n = (fn)m.
This can be proved by mathematical induction.
Exercise
Let f : X
1-1
−→ X.
1. Disprove: f1 = f2 ◦ f−1.
2. Disprove: (f0)−1 = (f−1)0.