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. K

    Emulating SUN3 or SUN4 kernel in cygwin

    Anyone has any idea about how to emulate a SUN3 or SUN4 kernel in cygwin under windows 7? I want to run these two softwares SUPREM IV GS and SEDAN III. The are free to download and use from http://www-tcad.stanford.edu/ It seems from the makefile that it uses sun3 / sun4 architecture and was...
  2. A

    Poisson Kernel: Examining Half Plane Limit Case

    Homework Statement Can you look at Poisson's formula for a half plane as a limit case of Poisson's formula for a disk? http://en.wikipedia.org/wiki/Poisson_kernel I can find lots of information about the Poisson kernel for a disk, but not for the half plane. I do know on can mat the unit...
  3. A

    Kernel of linear transformation

    Homework Statement I am having lots of trouble understanding how to get the kernel of linear transformations. I get that you basically set it equal to zero and solve. T: P3 → P2 given by T(p(x)) = p΄΄(x) + p΄(x) + p(0) Find ker(T) The Attempt at a Solution So P3 = ax^3 + bx^2 +...
  4. T

    Projection onto the kernel of a matrix

    If we have a matrix M with a kernel, in many cases there exists a projection operator P onto the kernel of M satisfying [P,M]=0. It seems to me that this projector does not in general need to be an orthogonal projector, but it is probably unique if it exists. My question: is there a standard...
  5. L

    Kernel and image of linear transformation

    Homework Statement For the linear transformation T: R4 --> R3 defined by TA: v -->Av find a basis for the Kernel of TA and for the Image of of TA where A is 2 4 6 2 1 3 -4 1 4 10 -2 4Homework Equations Let v = a1 b1 c1 a2 b2 c2 a3 b3 c3 a4 b4 c4 The...
  6. L

    Finding Kernel and Image of Matrix transformation

    Homework Statement Matrix A = 0 1 0 0 0 1 12 8 -1 Let E1 = a(A)(A+2I)2 Let E2 = b(A)(A-3I) For each of these, calculate the image and the kernel Homework Equations I found a(A) to be 1/25 and b(A) to be 1/25*(A-7I) Also, if I am not mistaken, I think KernelE1 =...
  7. C

    Question about Linear Operator's Image and Kernel

    Homework Statement If T:V\rightarrow V is linear, then Ker(T^2)=Ker(T) implies Im(T^2)=Im(T). Homework Equations Let T:V\rightarrow V be a linear operator such that \forall x\in V, T^2(x)=0\Rightarrow T(x)=0 (Ker(T^2)=Ker(T)). Prove that \forall x\in V, \exists u\in V\ni...
  8. T

    Linear Algebra: Kernel and Image question

    Homework Statement T : R^{3} -> R^{3} is a linear transformation. We need to prove the equivalence of the three below statements. i) R^{3} = ker(T) \oplus im(T); ii) ker(T) = ker(T^{2}); iii) im(T) = im(T^{2}). Homework Equations R^{3} = ker(T) \oplus im(T), if for all v \in...
  9. M

    Finding a basis for the Kernel of T

    Homework Statement So the question is a map T: R^2x2 ---> R^2x2 by T(A) = BAB, where B = (1 1) (1 1) so i made A = (a c) and T(A) = ((a+b) + (c+d) (a+b) + (c+d))...
  10. N

    Understanding nullspace (kernel) of a matrix

    Homework Statement Find the kernel of the matrix: http://img256.imageshack.us/img256/9015/53369959.jpg The Attempt at a Solution So I row-reduce it and get: [PLAIN][PLAIN]http://img812.imageshack.us/img812/1391/97980793.jpg The system of equations the row-reduced form equals 0. So I set...
  11. Y

    Find a basis for the kernel of the matrix

    Homework Statement find a basis of the kernel of the matrix that 1 2 0 3 5 0 0 1 4 6Homework Equations how the vectors are linearly independent and span the kernel The Attempt at a Solution Does it mean I need to samplify the 1 2 0 3 5 0 0 1...
  12. S

    Kernel and centralizer of irreducible characters

    Homework Statement Find the kernel of all irreducible characters of G, when given the character table. Find the centralizer of each irreducible character of G, when given the character table. Find Z(G) (the centralizer of G) for the same character table. Homework Equations I know that...
  13. Shackleford

    Determine is endomorphism, kernel, epimorphism, monomorphism

    I'm reasonably certain I did 1b correctly. I'm not sure about 1h. In both cases, since phi is the transformation from the multiplicative group G to the multiplicative group G, the identity is 1. http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110729_200311.jpg?t=1311988303...
  14. A

    Finding an orthonormal basis for a reproducing kernel Hilbert space.

    Hello all, I'm currently working on a problem in which I'm attempting to characterize a centered Gaussian random process \xi(x) on a manifold M given a known covariance function C(x,x') for that process. My current approach is to find a series expansion $\xi(x) = \sum_{n=1}^{\infty} X_n...
  15. W

    Interpretation of Gaussian Kernel

    Hello. In a software application I am attempting to smooth a data set by convoluting it with a discrete Gaussian kernel. Based upon information garnered online, I've been using this Mathematica command to generate the kernel: kern = Table[Exp[-k^2/100]/Sqrt[2. Pi], {k, -range, range}]; where...
  16. K

    Image of ring hom is ideal, kernel is subring.

    Homework Statement Let \phi: R \to S be a ring homomorphism from R to S. What can you say about \phi if its image \text{im}\phi is an ideal of S? What can you say about \phi if its kernel \ker \phi is a subring (w unity) of R? The Attempt at a Solution I think the second one...
  17. T

    What is the nullity and geometric description of the kernel and range of T?

    Homework Statement Use the given information to find the nullity of T and give a geometric description of the kernel and range of T. T is the projection onto the vector v = (1,2,2): T(x,y,z) = (x + 2y + 2z)/9 (1,2,2)Homework Equations Kernel of T = T(v) = 0. Nullity of T = dimension of the...
  18. H

    Invertible, kernel, and range of a Linear Transformation

    1) Let L:R3 >>>R3 be defined by L([1 0 0]) = [1 2 3], L([0 1 0]) = [0 1 1], L([0 0 1]) = [1 1 0] How to prove that L is invertible? I have the idea of one-to-one and onto, but I do not know how to apply them to this proof. 2) Find a linear transformation L:R2 >>>R3 such that {[1 -1 2], [3 1...
  19. E

    Help Using Heat Kernel and Convolutions

    My text gives the following definition for the solution of the heat equation with initial temperature distribution f is the convolution of f with the heat kernel. u(x,t)= \frac{1}{c \sqrt{2t}}e^{-x^{2}/4c^{2}t} \ast f~=~ \frac{1}{2c \sqrt{\pi t}}...
  20. J

    How to calculate the bandwidth for a Kernel density estimation

    Hello everybody! Since some time I am trying the estimate the density of a set of numbers (in my case the numbers are distances to some object from a laser scanner). As I read, that the kernel density estimation technique is a basic approach for that kind of problem. Different Kernels can...
  21. T

    Linear Algebra: Image, Kernel and Basis

    Homework Statement I've been browsing the Internet but can't find a straightforward explanation for a procedure on how to find the image and kernel of a matrix. Question: Find a basis of the image of A, and a basis of the kernel of A. \[ A = \left[ {\begin{array}{ccc} 1 & 2 & 1 \\...
  22. T

    Kernel of Matrix Homework Solution

    Homework Statement Find the kernel of \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right)\Homework Equations The Attempt at a Solution I know how to find the kernel of a matrix that has numbers in all of the columns, but I think that since this matrix has a zero...
  23. T

    Linear Algebra: find the Kernel and Image

    Homework Statement (a)Find the Kernel and Image of each of the following linear transformations. ... (iv)\varphi : V\rightarrow V given \varphi(f)=f'+f where V is the subspace of the space of smooth functions \Re\rightarrow\Re spanned by sin and cos, and f' denotes the derivative. ...
  24. L

    Semilinear Transformation, Kernel

    Hello I'm trying to proof the following: f is a semilinear transformation between the vectorspaces V \rightarrow W,c^\ast \in W^\ast , G:=ker \ c^\ast . Show that f^{-1}(G)=ker(f^T(c^\ast )) and that the f-preimage of a hyperplane of W a hyperplane of V or V as a whole is. Can you help me?
  25. A

    Find the Kernel & Image of A: $\mathbb{R}^\infty \rightarrow \mathbb{R}^\infty$

    Homework Statement Find the kernel and image of the linear function A: \mathbb{R}^\infty \rightarrow \mathbb{R}^\infty defined on the vector space (with usual operations) of sequences of real numbers x \in \mathbb{R}^\infty, x = (x_1, x_2,...) . given by A(x) = (y_1, y_2, ...) with y_k =...
  26. T

    Finding a Basis for the Kernel Space of a Matrix - Solving the RREF Method

    Homework Statement Find a basis for the kernel space of the following matrix: -1 -2 -1 2 2 -2 -4 -4 10 2 1 2 2 -5 2 -1 -2 0 -1 0 row reduce to 1 2 0 1 0 0 0 1 -3 0 0 0 0 0 1 0 0 0 0 0 Somehow read the solution as { [-2 1 0 0 0]T, [-1 0 3 1 0]T } .. I don't...
  27. S

    Linear Algebra Question regarding basis of a kernel

    Homework Statement Let V be the vector space of all 2x2 matrices over Q V= {[x1 x2] : xi \in Q} ... x3 x4 Let A = [ -1 0 ] and let C:V --> V be the linear map C(X) = XA + AX .... -1 1 Find a basis for Ker(C) and a basis for Im(C) The Attempt at a Solution I used C(X) =...
  28. H

    Kernel and Linear transformation

    Homework Statement U = [Polynomial of degree 3 such that 3p(1) = p(0)] Find the basis of U and find a linear transformation T: P3 ---> R such that U is the kernel of T.Homework Equations The Attempt at a Solution The basis part is easy. 3p(1) = p(0) 3a + 3b + 3c +d = d c= -b-a Basis ...
  29. I

    Linear algebra - kernel and image question

    Homework Statement f: K^{3} \rightarrow K^{4} is a linear transformation of vector spaces: K^{3} = \left\langle \vec{e}_{1}, \vec{e}_{2}, \vec{e}_{3} \right\rangle and K^{4} = \left\langle \vec{e}^{*}_{1}, \vec{e}^{*}_{2}, \vec{e}^{*}_{3}, \vec{e}^{*}_{4} \right \rangle as well...
  30. I

    What Can be Said About the Kernel of a Tensor Product of Linear Maps?

    Suppose f_1 is a linear map between vector spaces V_1 and U_1, and f_2 is a linear map between vector spaces V_2 and U_2 (all vector spaces over F). Then f_1 \otimes f_2 is a linear transformation from V_1 \otimes_F V_2 to U_1 \otimes_F U_2. Is there any "nice" way that we can write the kernel...
  31. P

    Derivative of this bicubic interpolation kernel

    Hello everyone, I am using the bicubic kernel described here ( http://en.wikipedia.org/wiki/Bicubic_interpolation#Bicubic_convolution_algorithm ) to interpolate my image after applying some transformations. I an using the matrix kernel described here with a = -0.5. Now, what I also...
  32. B

    Nullspace, kernel and rank

    Homework Statement for the set of vectors: v_1 = 1, -2, 0, 0, 3 v_2 = 2, -5, -3, -2, 6 v_3 = 0, 5, 15, 10, 0 v_4 = 2, 6, 18, 8, 6 (a) find a basis for the set of vectors and state the dimension of the space spanned by these vectors, what is the rank of this matrix? (b) construct a matrix whose...
  33. N

    Linear Transformations,Find basis of kernel and range

    suppose that vectors in R3 are denoted by 1*3 matrices, and define T:R4 to R3 by T9x,y,z,t)=(x-y+z+t,2x-2y+3z+4t,3x-3y+4z+5t).Find basis of kernel and range.
  34. S

    Computing Matrix, finding kernel and image

    Let T: R[x]2\rightarrow R[x]3 be defined by T(P(x))=xP(x). Compute the matrix of x with respect to bases {1,x,x2} and {1,x,x2,x3}. Find the kernel and image of T. I know how to do this when given bases without exponents, however I do not know exactly what this is saying and therefore am...
  35. O

    How Does the Propagator Relate to the Heat Kernel on a Manifold?

    Hi, I have a question about the relation between the propagator of a scalar field and the heat kernel. I'm not sure wether I should rather put this question into the math section: Given a Laplacian D on some manifold M, what I mean by heat kernel is just K(x,y;s) = \langle x | \exp(-sD)...
  36. B

    Integrating a function involved error functions and a Gaussian kernel

    I am currently facing a problem of integrating exp(-ax^2)*erf(bx+c)*erf(dx+f) with the integral boundaries 0 and infinity. I have gone through some handbooks but what I could locate is the integration of exp(-ax^2)*erf(bx)*erf(cx) from 0 to infinity which yields...
  37. C

    Finding kernel and range for polynomials transformation

    I have troubles arriving at the solution to this question: Consider the transformation T: P3-->P3 given by: T(f)=(1-x^2)f '' - 2xf ' Determine the bases for its range and kernel and nullity and rank Can anyone explain how should i go about finding the bases for its kernel and range?? i get 0...
  38. U

    Kernel of GL(n,F) acting on F^n

    Homework Statement Suppose GL(n,F) acts on F^n in the usual way. Consider the induced action on the set of all k-dimensional subspaces of F^n. What's the kernel of this action? Is it faithful The Attempt at a Solution Well, I anticipate that the kernel of this action consists of scalar...
  39. G

    Integrated kernel and spectral zeta function

    I was looking at a paper about strong-coupling expansion (N. F. Svaiter, Physica (Amsterdam) 345A, 517 (2005) ) and it claims that -\int d^d x \int d^d y (-\Delta + m^2)\delta^d(x-y) = \textbf{Tr} I + \left.\frac{d}{ds}\zeta(s)\right|_{s=0} where \zeta(s) is the spectral zeta function, and I...
  40. S

    Proof That A & A^T Have the Same Nullspace (Kernel)

    Hello, can you help me with the proof? If A is normal A^TA=AA^T then A and A^T have the same nullspace (kernel). And ||Ax||=||A^Tx|| Thank you.
  41. K

    Orthogonal Complement to the Kernel of a Linear Transformation

    Hey all, I'm trying to find an orthogonal complement (under the standard inner product) to a space, and I think I've found the result mathematically. Unfortunately, when I apply the result to a toy example it seems to fail. Assume that A \in M_{m\times n}(\mathbb R^n), y \in \mathbb R^n and...
  42. A

    Finding the Bases for kernel and range of linear transformation.

    Homework Statement How do I find the bases for both the kernel and range of this linear transformation? Let T: R4 ----> R4 be the linear transformation that takes [1101] and [1011] to [2304] and takes [1110] and [0111] to [3120] a. Find the bases for both the kernel and the range of...
  43. L

    What is the Kernel Approach in Non-Parametrised Unbinned Analysis?

    i came across the term kernel approach when reading about a non parametrised unbinned method of analysis. what does this mean? cheers
  44. T

    Rank & Kernel of A: Solving Linear Equations

    Homework Statement Let A=[{1,3,2,2},{1,1,0,-2},{0,1,1,2}] i) Find the rank ii) Viewing A as a linear map from M4x1 to M3x1, find a basis for the kernel of A and verify directly that these basis vectors are indeed linearly independent. Homework Equations None The Attempt at a Solution...
  45. T

    Matrix Image and Kernel

    Homework Statement i) Find the Image and Kernel of A = (2,1)(-4,-2) (where each bracket is a row). ii) Calculate A2 and use i) to explain your result. Homework Equations None The Attempt at a Solution So I can do everything up to the very last bit (i think anyway). i) The Kernel =...
  46. X

    Differnece between the Kernel and the Nullspace?

    What is the distinct difference between the kernel and the null space? They both have the same definition, namely, Ax=0.
  47. G

    Linear Transformation / Kernel Question

    Homework Statement L(p(t)) = t*dp/dt + t^2*p(1) If p(t) = a*t^2 + b*t + c, find a basis for the kernel of L. Homework Equations None. The Attempt at a Solution I know that L(a*t^2 + b*t + c) = 0, so that would mean that the derivative needs to be zero and p(1) needs to be zero. This...
  48. T

    Basis for kernel of linear transform

    Hey guys! I am having a major brain problem today, with this problem. L is a linear transform that maps L:P4\rightarrowP4 As such that (a1t3+a2t2+a3t+a4 = (a1-a2)t3+(a3-a4)t. I am trying to find the basis for the kernel and range. I know that the standard basis for P4 is...
  49. C

    Kernel, Range, Basis (linear algebra)

    Hey all! I am working on this and got confused. Any help at all would be much appreciated! Determine the kernel and range of the transformation T and find a basis for each: T(x,y,z)=(x,y,z) from R3 to R2. I have found the kernel to be the set {(r, -r, 0)}. Range is R2. I"m not sure how...
  50. belliott4488

    Kernel density estimation for non-I.I.D. data

    I am wondering if Kernel Density Estimation (KDE) is appropriate for some data analysis I'm working on. I have a simulated process that produces a large number N of pieces of debris, and I want to know how these objects are distributed spatially. In other words, I'd like to estimate a density...
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