What is Cubic: Definition and 427 Discussions

CUBIC (abbreviation for “clear, unobstructed brain/body imaging cocktails and computational analysis) is a histology method that allows tissues to be transparent (process called “tissue clearing”). As a result it makes investigation of large biological samples with microscopy easier and faster.
The method was published in 2014 by Etsuo A. Susaki and Hiroki R. Ueda, primarily for use in neurobiology research of brains from model organisms like rodents or small primates. But in upcoming years there were other works published, using CUBIC method on other tissues like lymph nodes or mammary glands. CUBIC can be also combined with CLARITY-based tissue clearing methods.

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  1. I

    Find a cubic function that has horizontal tangents at two given points

    Homework Statement Find a cubic function ax^3 + bx^2 + cx + d whose graph has horizontal tangents at the points (-2, 6) and (2. 0).Homework Equations No idea.The Attempt at a Solution I realize that the trick in here somewhere is to work "backwards". I completed a problem earlier that asked me...
  2. M

    How can I get help with solving cubic equations for my homework?

    Homework Statement I was just wondering if anyone could give me a quick lesson on solving cubic equations(Ax^3+Bx^2+Cx+D=0), or a good place to go to, if anyone can help thanks alot Homework Equations The Attempt at a Solution
  3. O

    Why are there 3 roots to a cubic equation?

    I find complex numbers very fascinating. But i don't understand something. Why does a cubic equation have 3 answers instead of 6? I know that there are 3 cube roots of a complex number, and the imaginary part of the complex number can be either positive or negative, so there should be 6...
  4. B

    Cubic Lattice Atom Diameter Calculation

    Homework Statement Barium metal crystallizes in a body-centred cubic lattice. The density of the metal is 3.50gcm^-3. Calculate the radius(in pm) of a barium atom. M(Ba)=137.3g/mol , NA = 6.022x10^23/mol The Attempt at a Solution For 1 unit cell: m=2x137.3/6.022x10^23...
  5. F

    Number of electrons per cubic metre

    If you stretch a wire, increasing its length will the number of electrons per cubic metre decrease? Assuming the Cross sectional area stays the same. The number of electrons will surely stay the same, and won't increase just by you stretching it, and the number of cubic metres has increased, so...
  6. N

    Calculating States in a 3D Cubic Potential Well

    Homework Statement I am to calculate the number of states in a 3Dcubic potential well with impenetrable walls that have energy less than or equal to E Homework Equations \ E_n=\frac{\hbar^2\pi^2}{\ 2 \ m \ a^2}\ (\ {n_x}^2 + \ {n_y}^2 + \ {n_z}^2) The Attempt at a Solution We may...
  7. K

    Prove that a cubic has no rational roots

    1) Prove that the acute angle whose cosine is 1/10 cannot be trisected with straightedge and compass. ... I worked it out and at the end found out that , if I can prove that the cubic polynomial 40x3 - 30x -1 has no rational roots, then I am done. Now, is there any way to prove (e.g...
  8. M

    Finding a Cubic Polynomial with Remainders

    Homework Statement A cubic polynomial gives remainders (5x + 4) and (12x -1) when divided by x^2 - x + 2 and x^2 + x - 1 respectively. Find the polynomial. Homework Equations The Attempt at a Solution
  9. A

    Proving bc=d for Cubic Function with r=-s

    Hey I need help with this proof: Consider the cubic function: x^3 + bx^2 + cx + d = 0 . If the two solutions of the cubic function are not equal, ie r != s, but r=-s, then prove that bc=d. Thanks
  10. B

    The Cubic Formula: ax^3+bx^2+cx+d=0

    The quadratic formula is, as you are well aware: ax^2+bx+c=0 x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} What is the cubic formula? ax^3+bx^2+cx+d=0 Thanks.
  11. S

    How can you solve cubic equations in R with multiple variables?

    solve in R (x^2+2)^{1/3}+(4x^2+3x-2)^{1/3}=(3x^2+x+5)^{1/3}+(2x^2+2x-5)^{1/3}
  12. N

    What is the Method for Finding a Cubic Function with Specific Zeros?

    Homework Statement From James Stewart's Essential Calculus Early Trancendentals, p.21 #5. Find an expression for a cubic function f if f(1)=6 and f(-1)=f(0)=f(2)=0 Homework Equations Used zeros of the function. The Attempt at a Solution I understand that the values of x...
  13. S

    What is the estimated cubic function for given x and y-intercepts?

    Homework Statement X-intercepts: (-1.57,0) , (0.65, 0) , (2.83, 0) Y-intercept: 11.33 Homework Equations I've got to convert that information into an estimated cubic function. The Attempt at a Solution I tried subbing the y-intercept in; although that didn't work. 11.33...
  14. H

    Is Cubing a Permutation in Zp for Primes Equivalent to 2 Mod 3?

    [SOLVED] cubic reciprocity? I would like to prove the following conjecture: If p \equiv 2\ (mod\ 3) is a prime, then the cubing function x \mapsto x^3 is a permutation of \mathbb{Z}_p . I've tried to find a contradiction to the negation by assuming that if n \neq m\ (mod\ 3), but n^3...
  15. S

    Working out a cubic sequence

    I am completely stuck on working out cubic sequences.. i know its sumthing along the lines of an3+bn2+cn+d but i don't know what each part means as in what does a = b= c= and d=?. And how to go about working it out. I have this sequence which i would like to work out : 1,4,10,20,35
  16. A

    Cubic Bezier curve: given X,Y, solve for t

    I'm programming some commands in a graphic application, and once again my extremely limited mathematical knowledge has found me rather stumped. Here is my cubic bezier function, written in JavaScript syntax(this is not actually JavaScript, though): function getBezier(percent,p1,cp1,cp2,p2) {...
  17. T

    Cubic differenation problem

    Hi The curve C has equation y = f(x) and the point P(3,5) lies on C Given that f'(x) = 3x² - 8x + 6 The point Q also lies on C, and the tangent to C at Q is parallel to the tangent to C at P Find the x-coordinate of Q So if there parallel the gradients are equal but i...
  18. B

    How Do You Factor a Cubic Equation Like a³c - a³b + b³a - b³c + c³b - c³a?

    Factorising cubic equation Anyone here know how to factor this equation? Homework Statement a^{3}c-a^{3}b+b^{3}a-b^{3}c+c^{3}b-c^{3}a The Attempt at a Solution I tried factoring by grouping but ended up getting nowhere. If anyone can factor this equation, please tell me step by...
  19. P

    Factorising a cubic polynomial

    1. A cubic polynomial is given by (f)x = x3 + x2 - 10x + 8 1) i) Show that (x -1) is a factor of f(x) ii) Factorise f(x) fully iii) Sketch the graph 2. Factor theorem? 3. Attempt at solution i) I know that f(1) = 0, so (x - 1) is a factor of (f)x = x3 + x2 - 10x + 8...
  20. A

    What is the coordination number of an atom in a primitive cubic struct

    I don't understand this question at all. But since my Professor gave it to me I'm guessing it relates to the chapter of structures and types of solids. What is the coordination number of an atom in a primitive cubic structure? What the hell is this question asking for?
  21. D

    Calculate the lattice constant of a body-centered cubic iron

    Calculate the lattice constant of a body-centered cubic iron crystal using the molar mass of iron, the density of iron and the Avogadro number.
  22. A

    How can I find the equation of the inverse function of a cubic function?

    I am working on a paper to find the equation of the inverse function of a cubic function. The function is f(x) = x^3 + 6x^2 +12x +7 I have already graphed the function and its inverse. I have found the inflection point of (-2, -1). I found the 3 roots (1 real & 2 complex). The real...
  23. D

    Yet another dumb question (zeros of a cubic function)

    f(x)= x^3-2x^2-11x+52 ----> Find all of the zeros of the function I just need some advice/direction on how to start solving this. It is not in the textbook and my brain is just kinda farting when I look at it. I think i need to get it in an (x-a)(bx^2+cx+d) form or something, but I don't know.
  24. I

    How can I use least squares to find coefficients for a cubic polynomial?

    Hi. I know how to find the coefficients of a quadratic with a data set using least sqaures. I now would like to do with to find a polynomial to the 3rd and 4th term. Could someone help me with this? The data set I am using: (-4.5,0.7) (-3.2,2.3) (-1.4,3.8) (0.8,5) (2.5,5.5)...
  25. Q

    Cubic and Ellipse problem

    The question I am looking at states: Prove that the elliptical function x^2 + 3y^2=b is orthogonal to the cubic y=3ax^3. I'm thinking they want me to prove if the functions are orthogonal when they intersect. If I am correct that would just lead to; For the ellipse, 2x+6y(m1)=0...
  26. W

    Finding a Cubic Root: Pencil and Paper Technique

    I still remember how to extract a square root without a computer but could somebody remind me the technique to find a cubic root just with the pencil and paper?
  27. T

    Finding the Roots of Cubic

    Homework Statement Find an integer c such that the equation 4x^3 + cx - 27 = 0 has a double root. Homework Equations Ax^3+Bx^2+Cx+K = 0 Sum of Roots = -B/A Product of Roots = (-1)^n * k/a etc. The Attempt at a Solution I tried using P/Q with synthetic division to find a...
  28. R

    Roots of a Cubic Polynomial: Proving Coefficient Inequalities

    Homework Statement In the equation x^3+ax^2+bx+c=0 the coefficients a,b and c are all real. It is given that all the roots are real and greater than 1. (i) Prove that a<-3 (ii)By considering the sum of the squares of the roots,prove that a^2>2b+3 (iii)By considering the sum of the cubes of...
  29. H

    Are There Closed Form Versions for Cubic Polynomials with 3 or 4 Points?

    Here's an interesting question. I'm aware of closed forms of cubic polynomials that go through 1 or 2 specific (x,y) points. Are there closed form versions for 3 or 4 points? 1 pt: y = a(x-x_0)^3 + b(x-x_0)^2 + c(x-x_0) + y_0 2 pt: y = a(x-x_0)^2(x-x_1)\ +\ b(x-x_0)(x-x_1)^2 \ +\...
  30. H

    Cubic Spline Interpolation Tutorial

    Attached below are two cubic spline tutorials: 1. Explanation of the classic tri-diagonal cubic spline formulation. Included are 2 example problems. 2. Extension to parametric cubic splines. Included are 2 example problems . :smile:
  31. J

    Cubic Equation Cardano solution

    Homework Statement To solve the following equation 3x^3 + x^2 + 15x + 27 = 0 Homework Equations The Attempt at a Solution {(x-> 1/9{-1 - 134/(-3079+27Sqrt.(16305)^(1/2) + (-3079+27Sqrt.(16305)^(1/3)} {x-> -1/9 + 67(1+iSqrt.3)/9(-3079+27Sqrt.(16305)^(1/2) -1/18...
  32. E

    Graphical solution of cubic with real roots

    Does anyone know whether the graphical solution of cubic equations with real roots by means of intersecting a circle and a parabola or hyperbola (or just a parabola and hyperbola) is known or not? That solution has to give the equations for the circles, parabolas and hyperbolas involved and not...
  33. E

    Solving Cubic Roots with Square Roots

    Hi! Who knows: can any cubic root like \sqrt[3]{x} with x real be written as a form in which only square roots (real or complex) are involved?
  34. E

    Finding Answers to Cubic Equation Questions

    L.S. I am looking for detailed information on the subject of cubic equations. Problem is that I cannot find anything but the most basic properties of such equations. For example, if given an arbitrary cubic equation with roots x(i),where to find the equation like with roots y(i) = x(i) - x(j)...
  35. 5

    Looking for pipe flow information in cubic feet per minute.

    Does anyone have a chart of what a straight pipe flows in cfm? Like 3 inch inside diam.? 4 inch? Etc? Thanks:)
  36. R

    Some Cubic Bézier Curve Questions

    I'm using a bézier curve in a Computer Game I'm working on that has four points: p0 (start point) p1 (directional helper for p0) p2 (directional helper for p3) p3 (end point) I nabbed the drawing formula from Wikipedia and it works fine. I have some problems though. I really need...
  37. S

    Cubic equation for weak acid hydronium

    If one would make no assumtions to try to simplify, except for just assuming OH- were negligible in a solution of weak diprotic acid, H2A, we can create four equations to represent concentrations for : H2A == H + HA- HA- == H + A- Each of those has a K value for equilibrium constant...
  38. T

    Cubic+ Quad Equations - y Axis

    Hi, I was wondering in a cubic equation such as x³ - 5x = 2x - 1 can be rearranged to x³ - 7x + 1 = 0 is that 0 the y value, and +1 where the line crosses the y axis? Is - 7x the gradient or is x³ - 7x the gradient? Thankyou
  39. J

    Calculating cubic ft/ min and PSI

    Has anyone shopped for an air compressor lately? It's amazing how far retailers will go in providing exaggerated/skewd/completely misleading info on these things. Due to this, I have a question: -Is there a simple formula for working with cubic feet per minute (cfm) and PSI? Ex: If you...
  40. L

    Solving Cubic Equation: Positive Root Exists

    Homework Statement Show that for negative c (a,b,c - real) equation x^3+ax^2+bx+c=0 has at least one positive root. 2. The attempt at a solution Considering the equivalent form of the equation above for large |x|: x^3(1+\frac{a}{x}+\frac{b}{x^2}+\frac{c}{x^3})=0 we can conclude that there...
  41. S

    What is the radius of the spherical air bubble before it reaches the surface?

    Homework Statement Hi,i came across this problem in a past exam paper and i got a bit stuck on it,can anyone enlighten me please?im prob just missing something really obvious and basic... A spherical air bubble at the bottom of a lake,10m below the surface,has a radius of 1mm. The...
  42. U

    Cubic with three real irrational roots.

    Considering the case of cubic polynomials with integer coefficients and three real but irrational roots. Is it true that it's impossible that all three roots can be in the form of simple surd expressions like r+s \sqrt{n} (where r and s are rational and sqrt(n) is a surd). The argument is that...
  43. D

    How can I use the Rational Zeros Theorem to factorize cubic functions?

    Hi. I am a bit lost as to how to factorize a cubic function. Would anyone be able to help me out? Thanks, Dan.
  44. Curious3141

    Analytical solutions of cubic and quartic equations

    Even though it is uncommon to see questions asking for an analytical solution to equations of degree 3 or 4, they have been asked on the forum. It's also good to know how, in any case. Cubic : http://www.karlscalculus.org/cubic.html Quartic : http://www.karlscalculus.org/quartic.html...
  45. E

    Cubic function factored with zeroes

    Let f(x) = ax3+bx2+cx+d and let f have x-intercepts x1, x2, and x3. How can I show that the expression for f(x) becomes f(x) = a(x-x1)(x-x2)(x-x3) ?
  46. B

    Solving cubic equation with matrices

    I need to find b, c and d in a cubic equation of the form y=x3+bx2+cx+d (ie. a=1 if a is the coefficient of x3). I've been given three points (-2,-13), (-1,0), (1,2). If I had a fourth piece of data i could solve the equation I suppose, but with only three I've really struggled. Here's what...
  47. J

    Find a cubic function

    Find a cubic function f(x) = ax^3 - bx^2 + cx - d that has a local maximum value of 40 at x = 1 and a local minimum valud of -68 at x = 4 Since x = 1 and x = 4 are the max and min, respectively, then f'(x) must equal 0 at x = 1 and x = 4. Therefore f'(x) = (x-1)(x-4) = x^2 - 5x +4 Using...
  48. D

    Solving cubic or higher degree polynomials by hand

    How does one find the roots of an equation, for instance: r^4 + r^3 - 7r^2 - r + 6 = 0 ...completely by hand. Is there some type of non-lengthy process or trick to find special circumstances for easy solving? Thanks.
  49. tony873004

    3rd degree polynomial is called a cubic

    A 3rd degree polynomial is called a cubic, and a 2nd degree is a quadratic. But quad implies 4. Why is a 2nd degree called quadratic?
  50. S

    How do I construct a free cubic spline to approximate e^-x with given values?

    Construct a free cubic spline to approximate f(x) = e^ -x , by using values for x = 0 , 0.25, 0.75, 1 now i know i have to contstruct something like this ss_{j} = a_{j} + b_{j} (x - x_{j}) + c_{j} (x-x_{j})^2 + d_{j} (x- x_{j})^3 also we know from the initial conditions that a0 = f(0), but...
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