What is rank: Definition and 307 Discussions

Receptor activator of nuclear factor κ B (RANK), also known as TRANCE receptor or TNFRSF11A, is a member of the tumor necrosis factor receptor (TNFR) molecular sub-family. RANK is the receptor for RANK-Ligand (RANKL) and part of the RANK/RANKL/OPG signaling pathway that regulates osteoclast differentiation and activation. It is associated with bone remodeling and repair, immune cell function, lymph node development, thermal regulation, and mammary gland development. Osteoprotegerin (OPG) is a decoy receptor for RANKL, and regulates the stimulation of the RANK signaling pathway by competing for RANKL. The cytoplasmic domain of RANK binds TRAFs 1, 2, 3, 5, and 6 which transmit signals to downstream targets such as NF-κB and JNK.
RANK is constitutively expressed in skeletal muscle, thymus, liver, colon, small intestine, adrenal gland, osteoclast, mammary gland epithelial cells, prostate, vascular cell, and pancreas. Most commonly, activation of NF-κB is mediated by RANKL, but over-expression of RANK alone is sufficient to activate the NF-κB pathway.RANKL (receptor activator for nuclear factor κ B ligand) is found on the surface of stromal cells, osteoblasts, and T cells. Mutations affecting RANK have been associated with infantile malignant osteopetrosis in humans, mice and cats.

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  1. Pilou115

    Finding the inverse of 4th rank elasticity tensor

    I'm desperately trying to understand how to get from 2.7.11 to 2.7.16 and cannot find any reference online on how to find the inverse of an elastic tangent modulus (fourth_order tensor). Can someone help me or give me a reference I can check where they do a similar thing? I would really...
  2. F

    Intro to Linear Algebra - Nullspace of Rank 1 Matrix

    The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
  3. Euge

    POTW Comparing Rank and Trace of a Matrix

    Let ##M## be a nonzero complex ##n\times n##-matrix. Prove $$\operatorname{rank}M \ge |\operatorname{trace} M|^2/\operatorname{trace}(M^\dagger M)$$ What is a necessary and sufficient condition for equality?
  4. chwala

    Understanding the Wilcoxon-signed Rank test

    Reference; https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test#:~:text=The Wilcoxon signed-rank test,-sample Student's t-test. I have managed to go through the literature (it is pretty straight forward). In general Wilcoxon-rank method applies to data with unequal variances otherwise...
  5. chwala

    Calculate the rank correlation coefficient of the given problem

    Find the problem and solution here; I am refreshing on this topic of Correlation. The steps are pretty much clear..my question is on the given formula ##\textbf{R}##. Is it a generally and widely accepted formula or is it some form of improvised formula approach for repeated entries/data? How...
  6. C

    A Gradient of higher rank tensor

    How to write following equation in index notation? $$\nabla \cdot \left( \mathbf{e} : \nabla_{s} \mathbf{u} \right)$$ where ##e## is a third rank tensor, ##u## is a vector, ##\nabla_{s}## is the symmetric part of the gradient operator, : is the double dot product. The way I approached is...
  7. LukasMont

    Proving A Must Be of Rank 2: The 2x2 Matrix Dilemma

    My trouble is being to show A must be of rank 2. Any ideas?
  8. PainterGuy

    Find the rank of this 3x3 matrix

    Hi, I was trying to find the rank of following matrix. I formed the following system and it seems like all three columns are linearly independent and hence the rank is 3. But the answer says the rank is '2'. Where am I going wrong? Thanks, in advance!
  9. M

    MHB Rank of composition of linear maps

    Hey! :giggle: Question 1: Let $C$ be a $\mathbb{R}$-vector space, $1\leq n\in \mathbb{N}$ and let $\phi_1, \ldots , \phi_n:V\rightarrow V$ be linear maps. I have shown by induction that $\phi_1\circ \ldots \circ \phi_n$ is then also a linear map. I want to show now by induction that if $V$ is...
  10. George Keeling

    I Tensor rank: One number or two?

    When I started learning about tensors the tensor rank was drilled into me. "A tensor rank ##\left(m,n\right)## has ##m## up indices and ##n## down indices." So a rank (1,1) tensor is written ##A_\nu^\mu,A_{\ \ \nu}^\mu## or is that ##A_\nu^{\ \ \ \mu}##? Tensor coefficients change when the...
  11. V

    I Convert 2x2 Matrix to 1x1 Tensor

    If I have a matrix representing a 2nd order tensor (2 2) and I want to convert this matrix from M$$\textsuperscript{ab}$$ to $$M\textsubscript{b}\textsuperscript{a}$$ what do I do? I'm given the matrix elements for the 2x2 tensor. When applying the metric tensor to this matrix I understand...
  12. Jason Bennett

    Levi-Civita symbol and its effect on anti-symmetric rank two tensors

    I am trying to understand the following: $$ \epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp} $$ Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices...
  13. T

    A Question about the irreducible representation of a rank 2 tensor under SO(3)

    When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...
  14. S

    I Finding the Error in Computing Spherical Tensor of Rank 0 Using General Formula

    This should be a trivial question. I am trying to compute the spherical tensor ##T_0^{(0)} = \frac{(U_1 V_{-1} + U_{-1} V_1 - U_0 V_0)}{3}## using the general formula (Sakurai 3.11.27), but what I get is: $$ T_0^{(0)} = \sum_{q_1=-1}^1 \sum_{q_2=-1}^1 \langle 1,1;q_1,q_2|1,1;0,q\rangle...
  15. D

    Ranking Current Densities: How Do Different Factors Affect Current Flow?

    Homework Statement: Rank in order, from the largest to the smallest, the current densities in these four wires, that have different radius and different electric conductivity Homework Equations: current density resistivity and resistance i guess ja=I/πr2 jb=2I/πr2...
  16. D

    Rank the faces in order of decreasing water pressure

    I assumed that the same magnitude of force acts on all sides of the box. Since A had the smallest area, I ranked P(A) as having the largest pressure, followed by P(B) having the second largest and P(top) and P(bottom) having the same pressure at third largest each. However, the ranking I...
  17. H

    MHB Linear Algebra Rank of a Matrix Problem

    Let A be a n x n matrix with complex elements. Prove that the a(k) array, with k ∈ ℕ, where a(k) = rank(A^(k + 1)) - rank(A^k), is monotonically increasing. Thank you! :)
  18. Math Amateur

    I Space of Alternating Tensors of Rank r.... (Browder, 12.22)

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra ... ... I need some help in order to fully understand the proof of Theorem 12.22 on page 276 ... ...The relevant text reads as follows: In the above...
  19. M

    MHB Check the statements about a 4x5 matrix with rank 2.

    Hey! :o Let $A$ be a $4\times 5$ matrix with rank $2$ and let $U$ be the corresponding row echelon form matrix. I want to check if the following statements are true or not. If $B$ is a $5\times 5$ invertible matrix, at least two of the columns of $B$ are not in the nulity of $A$. There...
  20. karush

    MHB Null Space of A: Find Rank & Dim.

    Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...
  21. Math Amateur

    MHB Rank One Tensors .... Fortney Appendix A, Section A2 ....

    I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Appendix A : Introduction to Tensors ...I need help to understand some statements/equations by Fortney concerning rank one tensors ... Those remarks...
  22. A

    I Contravariant derivative of tensor of rank 1

    If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk
  23. J

    Rank the four objects from fastest to slowest

    Homework Statement Rank the four objects (1kg solid sphere, 1kg hollow sphere, 2kg solid sphere and 1kg hoop) from fastest down the ramp to slowest. (Please see the attached screenshot for more details.) Homework Equations KE_rot = 1/2Iw^2 (where omega = w) The Attempt at a Solution Since we...
  24. isukatphysics69

    Rank charge density of spheres

    Homework Statement Sphere 1 has net positive charge Sphere 2 has net negative charge Sphere 3 has net positive charge The ranking of net charge magnitudes are SPHERE 3 > SPHERE 2 > SPHERE 1 All spheres are conductors Sphere 2 is moved away from Sphere 1 and toward Sphere 3 so that 2 and 3...
  25. isukatphysics69

    Rank charges in spheres (Physics 2)

    Homework Statement Homework Equations None (conceptual) The Attempt at a Solution My logic here is this, Sphere 3 has a net positive charge so it is repelling the positives in sphere 2 and attracting the negatives in sphere 2. This means that D has negative charge and C has positive charge...
  26. TickleTackleTock

    The Maximum Rank of a Matrix B Given AB=0 and A is a Full Rank Matrix

    Homework Statement Suppose that AB = 0, where A is a 3 x 7 full rank matrix and B is 7 x 53. What is the highest possible rank of matrix B. Homework EquationsThe Attempt at a Solution Since each column of B is in the null space of A, the rank of B is at most 4. I don't understand why it is...
  27. isukatphysics69

    Ranking the Work on Each System: Which Case Has the Most Positive Work?

    Homework Statement Homework EquationsThe Attempt at a Solution Work = W = ΔKE Case A: ΔKE = WTENSION - WSPRING - WGRAVITY Case B: ΔKE = WGRAVITY - WTENSION Case C: ΔKE - ΔPESPRING = WTENSION - WGRAVITY Case D: ΔKE + ΔPEGRAVITY = -WTENSION Case E: WSYSTEM = 0 Velocity and tension for...
  28. Marcus95

    Time Derivative of Rank 2 Tensor Determinant

    Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...
  29. isukatphysics69

    Calculating Rank Pulley Forces with Newton's Equations

    1. Homework Statement In picture Homework Equations f=ma The Attempt at a Solution i can use Newtons equations here, but i am confused about the frictional force.. so it doesn't specify whether it is a frictional or unfrictional surface.. is there something I'm missing here? does friction not...
  30. bornofflame

    Rank forces according to magnitude: Newton's 2nd/3rd laws

    Homework Statement Two crates, A and B, are in an elevator as shown. The mass of crate A is greater than the mass of crate B. a. The elevator moves downward at constant speed. ... iii. Rank the forces on the crate according to magnitude, from largest to smallest. Explain your reasoning...
  31. T

    Proving a statement about the rank of transformations

    Homework Statement How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##? Homework Equations Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...
  32. isukatphysics69

    Rank the rate of speed changing of object?

    Homework Statement Rank the rate at which the speed of each object is changing, greatest first. Ignore the sign; rank the magnitude or absolute value only. Homework Equations The red vector is the acceleration[/B] The Attempt at a Solution I have A>F>D>E=C and the program is telling me i am...
  33. isukatphysics69

    Rank the vertical component of velocities at the landing points

    Homework Statement Rank the vertical component of the velocity of each projectile at they moment they return to the ground, greatest first. Homework Equations n/a The Attempt at a Solution This is a conceptual homework, so there are no numbers involved. But looking at the picture i was...
  34. isukatphysics69

    Rank velocities of projectiles at landing?

    Homework Statement I will attatch the picture. Can someone please help me UNDERSTAND how to rank the velocity vectors at the landing zone?? I am having a really hard time with physics >=[ i don't know how to just look at this graph and rank the velocities. Homework EquationsThe Attempt at a...
  35. isukatphysics69

    Rank the vertical component of velocity?

    Homework Statement Rank the vertical component of the initial velocity of each projectile, greatest first.Homework Equations n/a The Attempt at a Solution I took the sine of the angles, the angles are ranked B>A>C>D you can see this visually as well as looking at the maximums. why isn't this...
  36. T

    I Geometric intuition of a rank formula

    I am trying to understand the geometric intuition of the above equation. ##\rho(\tau)## represents the rank of the linear transformation ##\tau## and likewise for ##\rho(\tau\sigma)##. ##Im(\sigma)## means the image of the linear transformation ##\sigma## and lastly, ##K(\tau)## is the kernel of...
  37. SchwarzF

    Programs High overall Rank school with a low physics rank

    I am currently study physics at a top 20 us school with, however, a pretty weak physics department. I am in my sophomore year and almost finished all the major course. Till now, I am still not very into doing REU in my home school, because I feel I should learn more graduate level physics and...
  38. T

    Wilcoxon Sign Rank Test rejection region

    Homework Statement Two computer software packages are being considered for use in the inventory control department of a small manufacturing firm. The firm has selected 12 different computing task that are typical of the kinds of jobs. The results are shown in the table below. At the 0.05 level...
  39. T

    Distinguish between sign test and Wilcoxon signed rank test

    Homework Statement I know that for both method are used to test the 2 group sample for a non-normally distributed population ... But , i am not sure the difference between them . Can someone explain the difference between them ? When to use sign test and wilcoxon signed rank test ? Homework...
  40. L

    Rank Electronegativity Pairings

    Homework Statement Br2, KBr, and HBr Homework Equations none The Attempt at a Solution I understand that Br2 would be least electronegative because they both equally share the electrons, but i don't understand why KBr is more electronegative than HBr. This question was on my quiz, and i...
  41. SchroedingersLion

    A Product of 3rd rank tensor with squared vector

    Greetings, can somebody show me how to calculate such a term? P= X E² where X is a third order tensor and E and P are 3 dimensional vectors. Since the result is supposed to be a vector, the square over E is not meant to be the scalar product. But the tensor product of E with itself yields a...
  42. C

    A Understanding Rank of a Matrix: Important Theorem and Demonstration

    It is the demonstration of an important theorem I do not succeed in understanding. "A matrix has rank k if - and only if - it has k rows - and k columns - linearly independent, whilst each one of the remaining rows - and columns - is a linear combination of the k preceding ones". Let's suppose...
  43. R

    Rank the cases from greatest to smallest in order of magnitude

    Homework Statement http://imgur.com/PUrHBaa Question 13 in the middle of the page. Each case in the figure shows an example of force vectors exerted on an object. These forces are all of the same magnitude F_o. Assume the forces lie in the plane of the paper. Rank the cases from greatest to...
  44. P

    Graph theory, rank and other characteristics

    Homework Statement hello, I have this graph and i have to figure out these so to speak characteristics. 1) find incidence matrix 2) arrange peak according to rank and layers 3)draw new arranged graph 4) find new connection matricies of peaks and arcs Homework EquationsThe Attempt at a...
  45. arpon

    I Is Second rank tensor always tensor product of two vectors?

    Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...
  46. Mr Davis 97

    Showing that the 0 matrix is the only one with rank = 0

    Homework Statement Prove that if rank(A) = 0, then A = 0. Homework EquationsThe Attempt at a Solution This seems like a very easy problem, but I just want to make sure I am not missing anything. rank(A) = dim(Im(A)) = 0, so Im(A) = {0}. Thus, A is by definition the zero matrix. My only...
  47. Ali Zain

    Rank in order, from largest to smallest, the resistance (eq)

    Homework Statement There is a figure, I'll try my best to draw/describe. 1. All three resisters are in parallel ___R____ !___R____! !___R____ ! 2. 2 resisters are parallel and one in series, after the parallel (ignore the dots) ___R___ ... _____R___ !___R___ ! 3. 2...
  48. H

    Determine rank of T and whether it is an isomorphism

    Homework Statement T((x_0, x_1, x_2)) = (0, x_0, x_1, x_2) Homework Equations None The Attempt at a Solution I'm getting hung up on definitions. My book says that T is an is isomorphism if T is linear and invertible. But it goes on to say that for T of finite dimension, T can only be an...
  49. M

    MHB Inequality of the rank matrices

    Hey! :o Let $\mathbb{K}$ be a fiels and $A\in \mathbb{K}^{p\times q}$ and $B\in \mathbb{K}^{q\times r}$. I want to show that $\text{Rank}(AB)\leq \text{Rank}(A)$ and $\text{Rank}(AB)\leq \text{Rank}(B)$. We have that every column of $AB$ is a linear combination of the columns of $A$, or not...