What is Functional: Definition and 414 Discussions

In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that map values to other values, rather than a sequence of imperative statements which update the running state of the program.
In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local identifiers), passed as arguments, and returned from other functions, just as any other data type can. This allows programs to be written in a declarative and composable style, where small functions are combined in a modular manner.
Functional programming is sometimes treated as synonymous with purely functional programming, a subset of functional programming which treats all functions as deterministic mathematical functions, or pure functions. When a pure function is called with some given arguments, it will always return the same result, and cannot be affected by any mutable state or other side effects. This is in contrast with impure procedures, common in imperative programming, which can have side effects (such as modifying the program's state or taking input from a user). Proponents of purely functional programming claim that by restricting side effects, programs can have fewer bugs, be easier to debug and test, and be more suited to formal verification.Functional programming has its roots in academia, evolving from the lambda calculus, a formal system of computation based only on functions. Functional programming has historically been less popular than imperative programming, but many functional languages are seeing use today in industry and education, including Common Lisp, Scheme, Clojure, Wolfram Language, Racket, Erlang, Elixir, OCaml, Haskell, and F#. Functional programming is also key to some languages that have found success in specific domains, like JavaScript in the Web, R in statistics, J, K and Q in financial analysis, and XQuery/XSLT for XML. Domain-specific declarative languages like SQL and Lex/Yacc use some elements of functional programming, such as not allowing mutable values. In addition, many other programming languages support programming in a functional style or have implemented features from functional programming, such as C++11, Kotlin, Perl, PHP, Python, Go, Rust, Raku, Scala, and Java (since Java 8).

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  1. C

    Abstract Linear Algebra, Linear Functional

    Homework Statement problem didn't state, but I assume let V be a vector space: V = C^3 and scalar is C Homework Equations Define a non-zero linear functional T on C^3 such that T ((1, 1, 1)) = T ((1, 1, −1)) = 0 The Attempt at a Solution So let X1 = (1, 1, 1); X2 = (1, 1, -1); It...
  2. B

    Need help solving functional analysis problem

    hello everyone! I had a stuck in solving problem for a week now, so need help. Please help! the problem is as follows.In a closed interval I=[0,\pi], the 2-times continuously differentiable function \phi(x) and \psi(x) meet the following conditions (they're ranged in \mathbb{R}). \psi...
  3. M

    Functional Analysis or group representations?

    I have to choose a total of 12 modules for my 3rd year. I've everything decided except four of them. I want to eventually do research either General Relativity, quantum mechanics, string theory, something like that. I'm torn between Group Representations, with one of Practical numerical...
  4. J

    Where can I get a functional PSPice?

    I tried a special edition that came with an electronics book and it had lots of problems. Then I downloaded a 9.1 student version that I found online but when I try to open a new sim profile it doesn't allow me to put any name I want or import. And the Capture Student version that I have doesn't...
  5. J

    MHB Functional Equation: A, B, C Answers

    http://latex.codecogs.com/gif.latex?\hspace{-20}$%20A%20function%20$f:\mathbb{N}%20\rightarrow%20\mathbb{N}$%20and%20satisfies%20$f(ab)%20=%20f(a)+f(b)$.\\%20Where%20$a$%20and%20$b$%20are%20Coprime%20Natural%20no.\\%20and%20$f(c+d)%20=%20f(c)+f(d)\forall$%20prime%20no.%20$c$%20and%20$d$.%20Then\\...
  6. A

    Topology, functional analysis, and group theory

    What is the relationship between topology, functional analysis, and group theory? All three seem to overlap, and I can't quite see how to distinguish them / what they're each for.
  7. O

    MHB Is $f$ a function or a functional?

    Hello everyone! I'm a bit confused about referring to a mapping as function or functional, for example: $f(x_1, x_2, x_3) = x_1+2x_2 ^2+3x_3 ^3$. $f$ takes vector $\textbf{x}=[x_1 \; x_2 \; x_3]$ and maps it to a scalar. Now, is $f$ a function or a functional? Thanks!
  8. F

    MHB OOP vs functional programming paradigm

    I was reading wikipedia about the short comings of the object oriented programing paradigm and a prof atCarnegie Mellon University states ""This semester Dan Licata and I are co-teaching a new course on functional programming for first-year prospective CS majors... Object-oriented programming...
  9. Z

    Kolmogorov & Fomin's Elements of Theory: Real Analysis or Lebesque?

    I'm looking for a Real Analysis book to start with, besides Spivak. On Amazon, one of the reviewers said it was good as a subsequent book for learning Functional Analysis/Lebesque Integration, while another said it was a good introduction to Real Analysis. For those of you that have read it...
  10. A

    Applications of Functional Analysis ?

    Hi guys, I am new to the forum. I have done a bit of reading on functional analysis lately.So I was wondering whether Functional Analysis can be related to physics in any way and what are the applications of that in physics?
  11. C

    Finding a Sup of a Functional?

    Hi guys, I need some help please! Consider the following expression: \left[1-\int_{x}^{1}F(\rho(\xi))f(\xi)d\xi\right]^{n-1} where F:[0,1]\rightarrow [0,1] is a continuously differentiable function with F'=f, x∈[0,1], and n>2. Suppose that \rho belongs to the set of continuous and...
  12. B

    Functional structure of Surface Heat

    I am facing the following interesting question. A closed room\hall contains several identical machines in it, they are fed by an electrical cable. The machines can be turned on or off. When a machine is turned on, it consumes electrical energy and as a by product generates heat. The heat...
  13. Einj

    Grassmann variables and functional derivatives

    Hi all! I'm sorry if this question has been already asked in another post... I'm studying the path integrals formalism in QED. I'm dealing with the functional generator for fermionic fields. My question is: The generating functional is: $$Z_0=e^{-i\int{d^4xd^4y \bar{J}(x)S(x-y)J(y)}}$$...
  14. J

    MHB Functional Equation: Solving for f(2012)

    If $f(x+y) = f(xy)$ and $\displaystyle f\left(-\frac{1}{2}\right) = -\frac{1}{2}$. Then $f(2012) = $
  15. N

    What can I say about this simple functional equation?

    f(x) = e^{x} f(-x) with f(x) > 0 Is there anything I can say about the general shape of this function (defined on the real axis)? For example the formula gives the derivative of f in zero in terms of f(0) (which is okay assuming I'm only interested in f up to a multiplicative constant).
  16. T

    Berezin's correspondance of (anti-)symmetric function with functional

    Hi all, I have recently been reading the book ``The Method of Second Quantization'' by Felix Berezin but I got trapped on just page 4, where the concept of generating functionals is introduced. It seems to be assigning each (anti-) symmetric function of N variables with a functional of a...
  17. A

    Functional groups of organic compounds

    Homework Statement find the functional groups in the following compound: Homework Equations C8H16O4The Attempt at a Solution I know there is an ether but there is also something else. What is it? I have tried to find a group that is in the compound, but I have had no luck. It seems to be only...
  18. J

    MHB Functional Equation: Find f(2012)

    If $f$ is a Real valued function on the set of real no. such that for any real $a$ and $b$ and $f(af(b)) = ab$. Then $f(2012) = $
  19. D

    Worried about Eng Masters using Density Functional Theory and Career Flexibility

    I just finished my undergrad in Chem Eng and am very interested in energy field ( I was thinking of doing something with battery and storage systems as it is closest to my field). I was recently offered a masters in MSE for possibly working on battery materials using Density Functional Theory...
  20. C

    Two Functional Analysis Questions

    Homework Statement 1. Given an operator H , and a sequence \{ H_n \} _{n\geq 1 } in an arbitrary Hilbert Space , such that both H and H_n are self-adjoint . How can I prove that if ||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0 and if H has an isolated eigenvalue \lambda of multiplicity...
  21. J

    What is the Range of a Differentiable Function with a Given Derivative at x=2?

    Homework Statement A function from R-->R is differentiable and follows f( (x+y)/3 ) =( 2 + f(x) + f(y) ) / 3 Derivative of f(x) at x=2 is 2 Find the range of f ( |x| ) Homework Equations The Attempt at a Solution Well the questions asks me the range of f( |x| ). But i don't...
  22. M

    What is a (linear) functional?

    Hey all, I have been reading up on Green Functions and I stumbled upon the term "linear functional". I know the properties of the linear operator, but i can't really grasp what a functional does. In my notes it says that it indicates a linear function whose domain is a function space, and...
  23. D

    Partial or Regular Derivative for Functional Derivative Stationary Value of 0?

    Hi, I have a question about a functional derivative. When determining the condition that the functional derivative have a stationary value of 0, do I use a partial derivative or a regular derivative? I would really appreciate the help. Thank you! David
  24. T

    "Understanding Functional Moieties in Gold Nanoparticles

    Homework Statement i am doing project on gold nanoparticles i was reading a per regarding synthesis of gold nanoparticles. then i came across a statement "GNPs bearing functional moieties, which are anchored with thiol-linkers, in their monolayers" i didn't understand the word functional...
  25. C

    What is the functional integral theorem for polynomial times Gaussian integrals?

    I've been watching Sidney Coleman's QFT lectures (http://www.physics.harvard.edu/about/Phys253.html). I've gotten up to his discussion of functional integration, and I have some questions. He starts out by discussing a finite-dimensional integral of a Gaussian function: \int{\frac{d^n...
  26. T

    Functional differentiability: Frechet, but not Hadamard?

    I have a question regarding functional differentiablility. I understand that Frechet differntiability of a functional T with respect to a norm \rho_1 implies Hadamard differentiability of the functional T with respect to the same norm. However, it is no surprise that there would be cases...
  27. H

    Calculation of a particular functional (path) integral

    Hello! On p.424 in the second edition of Altland & Simons, they compute an average as part of a renormalization group analysis of a model for dissipative quantum tunneling. I'd like to use their result in another situation, but I would be much happier if I understood how they derived it. I have...
  28. R

    Thermal physics. Entropy's functional dependecy

    Homework Statement I'm reading Kittel&kroemer's Thermal physics. How can I know Entropy's functional dependency? Author assume entropy's functional dependecy without explanations and derive some equaltities. So I can't follow it. N the number of particles. U Energy of the system. V. volume of...
  29. Whovian

    C/C++ C++ functional programming problem

    As a basic exercise in C++ functional programming, I wrote the following code: #include <iostream> #include <string> using namespace std; template <class T> void Wib (T& a,T& b) { T temp = a; a = b; b = temp; }; int main() { string A = "World!"; string B = "Hello, "; Wib <string>...
  30. B

    Total derivative of integral seen as a functional, how?

    To be specific, with total derivative I mean the linear map that best approximates a given function f at a given point. For f:ℝ\toℝ we have D(f,x_0):ℝ\toℝ, i.e. D(f,x_0)(h) \in ℝ. Often it is also denoted as just \delta f. Now in physics, in particular in the area of the Lagrangian, I find...
  31. A

    Difference between functional and function?

    I know function is just a subset of functional but physical example helps to understand this difference.."Any physical situation" thanks in advance
  32. alexmahone

    MHB Prove $f(x)=Cx$ for All $x$: Functional Equation

    Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$. I have shown that $f(x)=Cx$ for all rational numbers. How do I use the continuity of $f$ to show it is true for all $x$?
  33. I

    Functional Analysis & Others

    I'm looking for a rigorous introduction to functional analysis in the style of Apostol. I've looked at Introductory Functional Analysis with Applications by Kreyszig, but I find it slightly too conversational. I know that Rudin has a Functional Analysis book, but it seems to be out of print...
  34. M

    A functional depending upon x and y'(x)

    Homework Statement S[y] = \int21dx ln(1 + xny'), y(1) = 1, y(2) = 21-n where n > 1 is a constant integer, and y is a continuously differentiable function for 1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2 and ε a constant. Let ∆ = S[y + εh] − S[y]. Show that...
  35. M

    Extend the functional by continuity (Functional analysis)

    Homework Statement Let E be a dense linear subspace of a normed vector space X, and let Y be a Banach space. Suppose T0 \in £(E, Y) is a bounded linear operator from E to Y. Show that T0 can be extended to T\in £(E, Y) (by continuity) without increasing its norm. The Attempt at a Solution...
  36. B

    Functional Analysis: Open normed subspace

    Is an open normed subspace Y (subset of X) primarily defined as a set {y in X : Norm(y) < r}? Where r is some real (positive) number. I know the open ball definitions and such... but it seems like this definition is saying, an open normed space, is essentially an open ball which satisfies...
  37. E

    Help maximizing this functional please (it's simple, i think)

    the functional is \int_0^b{\frac{f(b)-f(x)}{b-x}}, . Another thing is it has defined endpoints. f(0)=0 and f(b) \in (0,10) and also f'(x)>0. oh and b \in (0,\infty) The euler-lagrange equation didn't give me anything helpful (not sure why..). One thing that is clear when looking at...
  38. E

    Mastering Functional Derivatives in Quantum Field Theory

    Alright, so I feel kind of dumb...but: I have been working on some QFT material, specifically derivation of Feynman rules for some more simple models ( \phi^{4} for example), and I have been seriously failing with functional derivatives. Every time I try to use the definition I mess up...
  39. 5

    Implicit partial differentiation functional det[]

    Hello! I was wondering how I could find the following derivatives from the given function using Jacobian determinants. f(u,v) = 0 u = lx + my + nz v = x^{2} + y^{2} + z^{2} \frac{∂z}{∂x} = ? (I believe y is constant, but the problem does not specify) \frac{∂z}{∂y} = ? (I...
  40. B

    Functional analysis with worked examples

    Folks Are there any introductory functional analysis books which show calculus examples to illustrate the different axioms? thanks
  41. B

    What is the difference between l subscript infinity and l superscript infinity?

    Folks, Is there a difference between l subscript infinity and l superscript infinity. I believe the latter is the space if bounded sequences? thanks
  42. E

    Is DFT Still the Go-To Method for Electronic Structure Calculations?

    Hi,all Is there someone's research related to DFT? I'm an undergraduate trying to get into it. I hope I could get some help hear if I have any question about that! Thanks a lot,Euphemia
  43. D

    Applied Functional Analysis by Zeidler

    "Applied Functional Analysis" by Zeidler In my book, "Applied Functional Analysis" by Zeidler, there's a question in the first chapter which, unless I got my concept of density wrong, I can't seem to see true : Let X=C[a,b] be the space of continuous functions on [a,b] with maximum norm. Then...
  44. S

    Extremum Function of a Functional

    Homework Statement J(f)=\int 2xf−f′2+3f2f′dx f(0)=0,f(1)=−1. Homework Equations Ff-\frac{d}{dx}Ff'=0 The Attempt at a Solution Ff=2x+6f f'' Ff'=-2f' + 6f2 Plugging in, I get: 2x+6f f''- [itex]\frac{d}{dx} (-2f' + 6f2) 2x+6f f''-12f f'-2f''=0 Which doesn't look...
  45. N

    Dual spaces-Existence of linear functional

    Homework Statement Let V be a finite dimensional vector field over F. Let T:V→V Let c be a scalar and suppose there is v in V such that T(v)=cv, then show there exists a non-zero linear functional f on V such that Tt(f)=cf. Tt denotes T transpose. Homework Equations Tt(f)=f°T...
  46. B

    Introductory to Functional Analysis

    Folks, I am starting a module in functional analysis undergrad level. I have been suggested introductory functional analysis by Kreyszig, but in instead of buying another expensive book is there a good online source like a pdf on in this topic that I could avail of? Any help will be...
  47. W

    Exploring Functional Powers: Seeking Information on f^n(x)

    I was reading on wikipedia about "functional powers", but I can't seem to find anything on it outside of this one section. I was wondering if there's any way to show anything for f^n(x). This is more of a general plea for more information on the topic than a specific question. Oh and here's...
  48. V

    Chemistry Hexagonal Packing Factor - Functional Molecules

    Homework Statement the sulphur atoms in the self-assembled monolayers are ~ 4.99Å apart, and that they form a hexagonal close-pack structure, estimate the number of functional molecules/cm2 of the substrate Homework Equations 1 angstrom = 1.0 × 10^{-10} metres Area of hexagon =...
  49. C

    Functional Derivatives/Euler-Lagrange

    Homework Statement Hi, I'm working on research and I hit a roadblock with something that should be very simple but I can't solve it because it gets so messy. If anyone can let me know how to do this, it would be greatly appreciated. I have a functional T: T = \int_{\lambda_{1}}^{\lambda_{2}}...
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