What is Kernel: Definition and 211 Discussions

The Linux kernel is a free and open-source, monolithic, modular, multitasking, Unix-like operating system kernel. It was conceived and created in 1991 by Linus Torvalds for his i386-based PC, and it was soon adopted as the kernel for the GNU operating system, which was created as a free replacement for UNIX. Since then, it has spawned a plethora of operating system distributions, commonly also called Linux.
Linux is deployed on a wide variety of computing systems, such as embedded devices, mobile devices (including its use in the Android operating system), personal computers, servers, mainframes, and supercomputers. It can be tailored for specific architectures and for several usage scenarios using a family of simple commands (that is, without the need of manually editing its source code before compilation); privileged users can also fine-tune kernel parameters at runtime. Most of the Linux kernel code is written using the GNU extensions of GCC to the standard C programming language and with the use of architecture specific instructions (ISA). This produces a highly optimized executable (vmlinux) with respect to utilization of memory space and task execution times.Day-to-day development discussions take place on the Linux kernel mailing list (LKML). Changes are tracked using the version control system git, which was created by Torvalds as a bespoke replacement for BitKeeper. Linux as a whole is released under the GNU General Public License version 2 (GPLv2), but it also contains several files under other compatible licenses, and an ad hoc exemption for the user space API header files (UAPI).

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  1. jack action

    Fixing Linux kernel not found

    I have this remote server where I loaded the ISP-provided OS, namely Ubuntu 22.04. The lsb_release -d shows "Ubuntu 22.04.4 LTS" and uname -r shows "5.2.0". My problem arose when there seemed to be missing modules for kernel 5.2.0 in /lib/modules/5.2.0 for my needs. There is also no...
  2. Daniel Petka

    Can the Least Squares Method be expressed as a convolution?

    I started by converting the LSM from sum to integral form: $$f(x_c) = \sum_i[S(x_i)-F(x_i,;a,b,...)]^2 to f(x_c) = \int( S(x) - F(x-x_c)^2 dx$$ Since we are not interested in the other parameters (like offset), I assumed that they are fitted correctly and thus ignored them, turning...
  3. Vanadium 50

    Troubleshooting Linux Kernel Issues with Older CPUs

    I am not sure what the next step is: Linux kernel 4.18.0-425.3.1.el8.x86_64 work s fine. 4.18.0-425.10.1.el8_7.x86_64 works on two of my five machines. On the other two, I get the screen messages "CPU Stall" but nothing obvious in the log files. The troubled machines have older CPUs (Haswell)...
  4. B

    Find a Ring with a Kernel - Image Processing

    Does anyone know of a Kernel I can use to find a ring in a image ?
  5. P

    A How Does the Non-Equality of Kernels Imply Their Sum Equals the Vector Space?

    This is actually a solved exercise from a Brazilian book on Linear Algebra. The author presented the following solution: The kernel and image theorem tells us that dimension ##\dim V=n=\dim\ker\left(F\right)+\dim \text{im}\left(F\right)=\dim\ker\left(G\right)+\dim\text{im}\left(G\right)##. As...
  6. S

    I How to calculate expectation and variance of kernel density estimator?

    This is a question from a mathematical statistics textbook, used at the first and most basic mathematical statistics course for undergraduate students. This exercise follows the chapter on nonparametric inference. An attempt at a solution is given. Any help is appreciated. Exercise: Suppose...
  7. O

    I Need some help with trace calculation in an index theorem

    I hate to create a thread for a step in a calculation, by I don't know what else to do. I'm having a lot of trouble reproducing E. Weinberg's index calculation (found here https://inspirehep.net/literature/7539) that gives the dimension of the moduli space generated by BPS solutions in the...
  8. PainterGuy

    Gaussian kernel in image processing

    I was reading the following webpage on Gaussian kernel but couldn't understand few details: https://www.imageeprocessing.com/2014/04/gaussian-filter-without-using-matlab.html . Would really appreciate it if you could guide me. Thanks in advance! Here you can find the high-res screenshot of the...
  9. LCSphysicist

    Homomorphism, Kernel and Coset

    I did the first step, that is, show that f is a homomorphism. Now i need to find the kernel K of f. But i am a little confused how to find it. Seeing the image, can we say the kernel is {0,4}?
  10. K

    Linear algebra, find a basis for the quotient space

    Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...
  11. M

    MHB Kernel of Linear Map: Show $\ker \phi$ Equation

    Hey! :o Let $1\leq n,m\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $(b_1, \ldots , b_n)$ a basis of $V$. Let $W:=\mathbb{R}^m$ and let $\phi:V\rightarrow W$ be a linear map. Show that $$\ker \phi =\left \{\sum_{i=1}^n\lambda_ib_i\mid \begin{pmatrix}\lambda_1\\ \vdots \\ \lambda_n\end{pmatrix}\in...
  12. V

    A Can the Fejér Kernel Be Approximated by Polynomials?

    Hello, I'm currently studying the Fejér kernel, which has the form of . I want to know whether there are some methods to approximate this function into polynomials. Thanks a lot for the help!
  13. J

    A Reproducing Kernel Hilbert Spaces

    I have been reading a lot about Reproducing Kernel Hilbert Spaces mainly because of their application in machine learning. I do not have a formal background in topology, took linear algebra as an undergrad but mainly have encountered things such as, inner product, norm, vector space...
  14. karush

    MHB What Is the Basis for the Kernel of the Differential Operator \(D^4-2D^3-3D^2\)?

    ok I am new to this basis of kernel and tried to understand some other posts on this but they were not 101 enough Find the basis for kernel of the differential operator $D:C^\infty\rightarrow C^\infty$, $D^4-2D^3-3D^2$ this can be factored into $D^2(D-3)(D+1)$
  15. T

    Forensic analysis using Memory Dump

    I'm in my second year in college and I've taken an Operating Systems course that has a project component. I've been assigned Memory Forensics as my project topic. On approaching the professor I was told that I need to attempt to attack the Linux Kernel ( I'm guessing that means I need to write...
  16. M

    MHB Is the Kernel of Trace Equal to the $F$-Subspace of $V_\sigma$?

    Hey! :o Let $K/F$ be a finite Galois extension and let $G= \operatorname{Gal}(K/F)$. For each $\sigma\in G$ we define $V_{\sigma}=\{\sigma (b)-b:b\in K\}$. Show that $V_{\sigma}$ is $F$-subspace of $\ker \operatorname{Tr}_{K/F}$. Show that $K/\mathcal{F} (\langle \sigma \rangle) \cong...
  17. M

    MHB Kernel of Linear Map $\theta$ in $\mathbb{F}_{q^n}$

    Hey! :o Let $q$ be a power of a prime and $n\in \mathbb{N}$. We symbolize with $Tr$ the map of the trace from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$, i.e. $Tr:\mathbb{F}_{q^n}\rightarrow \mathbb{F}_q$, $\displaystyle{Tr(a)=\sum_{j=0}^{n-1}a^{q^j}}$. I want to calculate the dimension of the image...
  18. Mr Davis 97

    I Can Multiplicative Cosets Form a Group for a Subring in a Ring Homomorphism?

    We know that kernel of a homomorphism consists of all the elements that map to the additive identity, 0. Here is my naive question: Why don't we define the kernel as all of the elements that map to the multiplicative identity, 1? Why isn't there a name for the set of all elements that map to the...
  19. A

    Show injectivity, surjectivity and kernel of groups

    Homework Statement I am translating so bear with me. We have two group homomorphisms: α : G → G' β : G' → G Let β(α(x)) = x ∀x ∈ G Show that 1)β is a surjection 2)α an injection 3) ker(β) = ker(α ο β) (Here ο is the composition of functions.) Homework Equations This is from a...
  20. Mr Davis 97

    I Enumerating the cosets of a kernel

    Suppose that we have, for purposes of example, the homomorphism ##\pi : \mathbb{R}^2 \to \mathbb{R}## such that ##\pi((x,y)) = x+y##. We see that ##\ker(\pi) = \{(x,y)\in \mathbb{R}^2 \mid x+y=0\}##. How can we enumerate all of the cosets of the kernel? My thought was that of course as we range...
  21. Mr Davis 97

    Finding the kernel of an action

    Homework Statement Let ##\mathbb{F}_3## denote the field with 3 elements and let ##V = \mathbb{F}_3^2##. Let ##\alpha, \beta, \gamma, \delta## be the four one-dimensional subspaces of ##V## spanned by ##(1,0), (0,1), (1,1)## and ##(1,-1)## respectively. Let ##\operatorname{GL}_2...
  22. M

    MHB How Do We Find the Kernel and Image Bases Using Matrix C?

    Hey! :o Let $A\in \mathbb{C}^{2\times 2}$ and $L_A:\mathbb{C}^{2\times 2}\rightarrow \mathbb{C}^{2\times 2}, \ X\mapsto A\cdot X$. We consider the matrix \begin{equation*}A=\begin{pmatrix}-1 & 2 \\ 2 & -4\end{pmatrix}\end{equation*} and the basis \begin{equation*}B=\left \{\begin{pmatrix}1...
  23. U

    Do These Functions Qualify as Group Homomorphisms?

    Homework Statement Are these functions homomorphisms, determine the kernel and image, and identify the quotient group up to isomorphism? C^∗ is the group of non-zero complex numbers under multiplication, and C is the group of all complex numbers under addition. Homework Equations φ1 : C−→C...
  24. M

    Writing a module to a linux kernel

    I was wondering about "writing a module to a linux kernel." This question haven't asked yet. Would you please explain why linuxers write such modules to a linux kernel? What is the reason? Thank you.
  25. pairofstrings

    Operating system concepts in an app by Linux System calls

    Before I begin learning what System Calls, Kernel, and Operating System is, I want to confirm that Operating System concepts like Multi-threading, Concurrency, Parallelism, Scheduling, Memory Management, Process Management, Network Management, Device Drivers can be implemented by using Linux...
  26. i_hate_math

    Heat Kernel at t=0: Dirac Delta Intuition

    Homework Statement Show that k(x,0)=δ(x). Where k(x,t) is the heat kernel and δ(x) is the Dirac Delta at x=0. Homework Equations k(x,t) = (1/Sqrt[4*π*D*t])*Exp[-x^2/(4*D*t)] The Attempt at a Solution I am just clueless from the beginning. I am guessing this is got to do with convolution...
  27. W

    I Finding the Kernel of a Matrix Map

    Hi All, I am trying to see if there is a "nice" ( relatively straightforward) way of finding the solution/kernel of the map : ##f(A)=A^n -Id ## , where A is an ## k \times k ## matrix and ##n## is a positive integer. Question: what is the kernel of this map? Cranking out matrix coefficients...
  28. Rodrigo Schmidt

    Conclusion about the dimension of C°(R)?

    [mentor note: thread moved from Linear Algebra to here hence no homework template] So, i was doing a Linear Algebra exercise on my book, and thought about this. We have a linear map A:E→E, where E=C°(ℝ), the vector space of all continuous functions. Let's suppose that Aƒ= x∫0 ƒ(t)dt. By the...
  29. Jianphys17

    I Fredholm integral equation with separable kernel

    Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...
  30. Adgorn

    Proof regarding the image and kernel of a normal operator

    Homework Statement Show that if T is normal, then T and T* have the same kernel and the same image. Homework Equations N/A The Attempt at a Solution At first I tried proving that Ker T ⊆ Ker T* and Ker T* ⊆ Ker, thus proving Ker T = Ker T* and doing the same thing with I am T, but could not...
  31. SetepenSeth

    Linear Algebra - Kernel and range of T

    Homework Statement Let ##T:M_2 \to M_2## a linear transformation defined by ##T \begin{bmatrix} a&b\\ c&d \end{bmatrix} = \begin{bmatrix} a&0\\ 0&d \end{bmatrix}## Describe ##ker(T)## and ##range(T)##, and find their basis. Homework Equations For a linear transformation ##T:V\to W##...
  32. vktsn0303

    A Understand Convolution, Singularity, Kernel, etc: Math Reading Guide

    I'm reading a book on vortex methods and I came across the above mentioned terms, however, I don't understand what they mean in mathematical terms. The book seems to be quite valuable with its content and therefore I would like to understand what the author is trying to say using the above...
  33. Adgorn

    Proving an image and annihilator of a kernel are equal

    Homework Statement Suppose T:V→U is linear and V has finite dimension. Prove that I am Tt = (Ker T)0 Homework Equations dim(W)+dim(W0)=dim(V) where W is a subspace of V and V has finite dimension. The Attempt at a Solution I first proved I am Tt ⊆ (Ker T)0. Let u be an arbitrary element of...
  34. jamalkoiyess

    I Does this theorem need that Ker{F}=0?

    I have encountered this theorem in Serge Lang's linear algebra: Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W. In the proof he starts with C1F(v1) +...
  35. ElPimiento

    I How is it that the SPH cubic spline kernel in normalized?

    Hi, (This is more of a math question but I thought Astronomy people would be more familiar with the equation and how it's used) So in Monaghan 1992 (http://adsabs.harvard.edu/abs/1992ARA&A..30..543M, bottom of pg 554) a cubic spline in three dimensions is defined. I tried to integrate it (using...
  36. Matejxx1

    Find the basis of a kernel and the dimension of the image

    Homework Statement Let ##n>1\in\, \mathbb{N}##. A map ##A:\mathbb{R}_{n}[x]\to\mathbb{R}_{n}[x]## is given with the rule ##(Ap)(x)=(x^n+1)p(1)+p^{'''}(x)## a)Proof that this map is linear b)Find some basis of the kernel b)Find the dimension of the image Homework Equations ##\mathbb{R}_{n}[x]##...
  37. D

    I How Do You Derive the Kernel of the Laplace Equation with Boundary Conditions?

    Hi! Everyone. I encounter some trouble in deriving the kernel of Laplace equation with prescribed boundary conditions. Given the following preposition: $$T(x, y) = \int_{-\infty}^{\infty}dx'\frac{y/\pi}{(x-x')^{2}+y^2}F(x')...[1]$$ satisfies the Laplace equation for ##x\in(-\infty, \infty)##...
  38. R

    MHB Proving disjoint of Kernel and Image of a linear mapping

    I am working on a problem that goes like this: Show that $Ker (F) \cap I am (F) = \{0\}$ if $F: W \rightarrow W$ is linear and if $F^4 = F.$ I have the solution but there is one step which I need help: (the delineation is mine) (1) Suppose that there exists $x$, such that $x \in Ker(F) \cap...
  39. M

    [Linear Algebra] Kernel and range

    Homework Statement Let P2 be the vector space of all polynomials of a degree at most 2 with real coefficients. Let T: P2→ℝ be the functioned defined by: ##T(p(t)) = p(2) - p(1)## a) Find a non-zero element of the Kernel of T. (I think I figured this one out, but I'm not too sure). b) Find a...
  40. M

    Linear Tranformation: Find the kernel of T

    Homework Statement Let T: P4--->P3 be a linear transformation given by T(p)=p'. What is the kernel of T? Homework EquationsThe Attempt at a Solution T(a0+a1+a2x2+a3x3+a4x4)=a1+2a2x+3a3x2+4a4x3 Ker(T)= { T(p)=0} so, a1+2a2x+3a3x2+4a4x3=0 then a1=2a2x+3a3x2+4a4x3 Ker(T)= { (-2,1,0,0)...
  41. RJLiberator

    What Are the Kernel and Image of a Group Homomorphism φ: ℝ^x -> ℝ^x?

    Homework Statement φ is a homomorphism of groups. φ: ℝ^x -> ℝ^x, where φ(α) = α^4, for all α ∈ ℝ^x. Note that ℝ^x is a group under multiplication. Describe ker(φ) and Im(φ). Homework EquationsThe Attempt at a Solution This is another one of those problems that has me scratching my head due...
  42. S

    Linear Transformation (Image, Kernel, Basis, Dimension)

    Mod note: Moved from Precalc section 1. Homework Statement Given l : IR3 → IR3 , l(x1, x2, x3) = (x1 + 2x2 + 3x3, 4x1 + 5x2 + 6x3, x1 + x2 + x3), find Ker(l), Im(l), their bases and dimensions. My language in explaining my steps is a little sloppy, but I'm trying to understand the process and...
  43. S

    Solving Linear Transformation: Image, Kernel & Vectors

    Homework Statement Given the linear transformation l : R2 → R2 , l(x, y) = (2x − 2y, −x + y), write the matrix associated to l with respect to the standard basis of R2 , find Kerl, I am l, its bases and dimensions. Find all vectors of R2 that are mapped to (4, −2). Homework Equations Ax=0...
  44. A

    MHB Kernel properties and feature maps

    I am preparing myself for maths exam and I am really struggling with kernels. I have following six kernels and I need to prove that each of them is valid and derive feature map. 1) K(x,y) = g(x)g(y), g:R^d -> R With this one I know it is valid but I don't know how to prove it. Also is g(x) a...
  45. I

    Understanding proof for theorem about dimension of kernel

    So the theorem says: Suppose that ##U## and ##V## are finite dimensional vector spaces, and that ##T:U\to V##, ##S: V \to W##. Then ##\text{dim Ker }ST \le \text{dim Ker }S + \text{dim Ker }T##. Proof: Set ##U_0 = \text{Ker }ST## and ##V_0 = \text{Ker }S##. ##U_0## and ##V_0## are subspaces of...
  46. E

    Finding the Kernel of a transformation

    Homework Statement Let T:[R[/3]→[R[/3] so that when u=[R][/3] and v=(1,2,1), then T(u)=u×v a) Show that T is a linear transformation. b) Find T((3,0,2)) c) Find a basis for Ker( T ). Give a geometric description of Ker( T ). Homework Equations Properties of a linear transformation: i) T(u+v)=...
  47. E

    Kernel Estimation: Questions about Bandwidths & R Functions

    Hi there! I'm new in the technique of Kernel Estimation, so it could be that the following questions are really elementary. There is something I don't understand about the bandwidths. Using R I have two functions to perform the estimate: kde2d from MASS bkde2D from KernelSmooth Here are my...
  48. caffeinemachine

    MHB How to Express Elements in the Kernel of the Symmetrizing Map?

    Let $V$ be a finite dimensional vector space over a field of characteristic $0$ and let $sym:\bigotimes^k V\to \bigotimes^k V$ be the map defined as $$ sym(\alpha)=\frac{1}{r!}\sum_{\sigma\in S_k}{^\sigma}\alpha $$ where $S_k$ is the permutation group on $k$ letters and ${^\sigma}\alpha$...
  49. ShayanJ

    Integral kernel in vacuum wave functional

    I'm trying to solve the exercise below in a book I'm reading. I inverted equation 1.3 to get ## \phi_{\mathbf k}(t)=\int \frac{e^{-i \mathbf k \cdot \mathbf x}}{(2\pi)^{\frac 3 2}} \phi(\mathbf x,t) d^3 \mathbf x ##. Then I put it in I to get: ## I=\int \int d^3 \mathbf x d^3 \mathbf y...
  50. C

    MHB Kernel on linear transformation proof

    hi guys :D im having trouble with this proof, any hints? let V be a vector space over a field F and let T1, T2: V--->V be linear transformations prove that