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. Solve the problem involving the cubic function

The problem and solution are posted... no. 8 I may need insight on common difference ... In my lines i have, Let the roots be ##(b), (b-1)## and ##(b+1)##. Then, ##x^3-3bx^2+3cx-d = a(x-b(x-b+1)(x-b-1)## ##x^3-3bx^2+3cx-d= a(x^3-3bx^2+3b^2x-x-b^3+b)## ##a=1##. Let...
2. Inflection Point Calculation: Reduction of Cubic with Second Derivative Method

(i) I take the second derivative of Y: Y'' = 6X + 2A. Y'' = 0 when X = -A/3. Moreover, as Y'' is linear it changes sign at this X. Thus, it is the point of inflection. (iii) After the substitution, the term x^2 appears twice: one, from X^3 as -3(x^2)(A/3), and another from AX^2 as Ax^2. They...
3. Understanding Cubic Factorization: Solving for Roots with a and -2a

This is part of a longer exercise I struggled with. I checked the solutions manual, and there was a bit where they performed the following steps: $$x^3=3a^2x-2a^3 \\$$ $$(x-a)^2(x+2a)=0$$ And then concluded that the roots were ##a## and ##-2a##, which is clear. What I can't work out is how...
4. B Air tank PSI and cubic feet calculation help please

If I have a air tank that is 1 cubic feet with the pressure gauge at 0 and I pump in air to 5000 psi how many cubic feet of a 100 psi would be in that tank? Any help would be appreciated
5. Exploring Tetrahedral Stacking in Diamond Cubic Arrangements

In terms of tetrahedral stacking, as occurs in diamond cubic, what I'm describing would be a system with additional tetrahedra in the empty cubes of that figure, but with their vertices on the opposite corners of the cubes that contain them to the "regular" diamond cubic arrangement. Due to the...
6. I Can an inverse function of a special cubic function be found?

Like this one: ##f(x)=3x^3 -18x^2 +36x##.Is there any way to find the inverse of this function without using the general solution to cubic equation?
7. Stationary states infinite cubic well

For a state to be stationary it must be time independent. Naively, I tried to find the values of c where I don't have any time dependency. ##e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##...
8. I Looking for a trajectory integrator that also supports cubic potential

The context: I created an educational resource, a set of interactive diagrams that allow the user to see how Hamilton's stationary action arrrives at the true trajectory. There is a diagram for each of the following three cases: - Uniform force, hence the potential increases linear with...

10. B Just for fun: Cubic Graph Plot

Desmos.com is a great online graphing utility which I'm sure is familiar to many PF users. I wanted to experiment with the Newton-Raphson method using it so chose solution of cubic graphs as an example. The graph shows a variable cubic on which all turning points and intercepts are calculated...
11. A Cubic and monolayer difference

What is the difference for example between cubic boron phosphide and monolayer boron phosphide?
12. B Why Is a Cubic Polynomial Called 'Third Degree'?

Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.
13. Why are there only two roots of this cubic polynomial?

Hi, I was trying to find roots of the following cubic polynomial and there are only two roots. I believe there should be three roots. Could you please guide me why there are only two roots? If you say that the "1" repeats itself as a root then I'd say the same could be said of "0.9". Thank...
14. MHB Positive Integer Triples $(a,b,c)$ Satisfying $a^3+b^3+c^3=(abc)^2$

Find all triples $(a,\,b,\,c)$ of positive integers such that $a^3+b^3+c^3=(abc)^2$.
15. MHB Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations

If $\alpha,\,\beta,\,\gamma$ are the roots of the equation $x^3+ax+1=0$, where $a$ is a positive real number and $\dfrac{\alpha}{\beta},\,\dfrac{\beta}{\gamma},\,\dfrac{\gamma}{\alpha}$ be the roots of the equation $x^3+bx^2+cx-1=0$, find the minimum value of $\dfrac{|b|+|c|}{a}$.
16. MHB Find Cubic Function & Polynomial of Degree 3

Find a polynomial of degree 3 with real coefficients such that each of its roots is equal to the square of one root of the polynomial $P(x)=x^3+9x^2+9x+9$.
17. MHB Cubic splines: Uniqueness

Hey! 😊 Show that the interpolation exercise for cubic splines with $s(x_0), s(x_1), , \ldots , s(x_m)$ at the points $x_0<x_1<\ldots <x_m$, together with one of $s'(x_0)$ or $s''(x_0)$ and $s'(x_m)$ or $s''(x_m)$ has exactly one solution. Could you give me a hint how we could show that? Do...
18. Find the cubic equation given the roots

##\sum ∝=3## ##\sum ∝β=0## ##∝βγ=-4## ##\sum2 ∝=6## ##\sum 2∝.2β##=4##\sum ∝β=0## ##2∝.2β.2γ=-32## we then end up with ##x^3-6x^2+0x+32=0## ##x^3-6x^2+32=0## i am looking for alternative methods ...
19. MHB Real Roots of Cubic Equation: $x^3+a^3x^2+b^3x+c^3=0$

An equation $x^3+ax^2+bx+c=0$ has three (but not necessarily distinct) real roots $t,\,u,\,v$. For what values of $a,\,b,\,c$ are the numbers $t^3,\,u^3,\,v^3$ roots of an equation $x^3+a^3x^2+b^3x+c^3=0$?
20. MHB Roots of Cubic Equation: Finding $x_1,\,x_2$, and $x_3$

The roots $x_1,\,x_2$ and $x_3$ of the equation $x^3+ax+a=0$ where $a$ is a non-zero real number, satisfy $\dfrac{x_1^2}{x_2}+\dfrac{x_2^2}{x_3}+\dfrac{x_3^2}{x_1}=-8$. Find $x_1,\,x_2$ and $x_3$.
21. MHB Prove No Integers Solve $ax^3+bx^2+cx+d=1$ for x=19,2 for x=62

Prove that there are no integers $a,\,b,\,c$ and $d$ such that the polynomial $ax^3+bx^2+cx+d$ equals 1 at $x=19$ and 2 at $x=62$.
22. Solution of a depressed cubic equation

Hi, I was trying to solve the equation x³-2x-5=0. I solved it online and the real solution is x≈2.0946. The following is an excerpt from the Wikipedia article on cubic equation. The cubic equation, x³-2x-5=0, is a depressed cubic and in this case 4p³+27q²>0. But the real root of equation...
23. I Does this make a cubic out of a quadratic system?

Suppose the system of equations (coming from invariance of the wave equation) : $$B=-vE\\A^2-B^2/c^2=1\\E^2-c^2D^2=1\\AD=EB/c^2\\B=vA\\AE-BD=1$$ If one adds a lightspeed movement like $$A=a+f\\B=b-cf\\D=d+h\\E=e-ch$$ Then solving equ 1 for f gives $$f=(b+ve-vch)/c$$ Equ 4 for h implies...
24. MHB Understanding Cubic Equation Formula for Polynomials of Degree Three

Hi, Can someone please help me in understanding few parameters of cubic equation formula for solving polynomial of degree three. I attached the formula in the screenshot. My questions are: (1) what is ". " dot in the end of the formula and what does it mean? (2) I want to use it only for real...
25. MHB Inequality of cubic and exponential functions

Prove that $3^n\ge(n+3)^3$ for any natural number $n\ge6$.
26. MATLAB Creating the Electric Octupole Tensor of a cubic electric octupole

I created an array, where the first three entries of each column are the x,y, and z coordinates. The last entry in each column is the charge. I called this array PCQ. l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 l/2 l/2 l/2 l/2 -l/2 -l/2 -l/2 -l/2 q -q q -q q...
27. M

I Cubic Algebraic Solution: Understanding the Approximation in Equations 4 and 5

Hi PF! Here equations 4 and 5 imply $$a^3=a_0^3+3 a_0 \Sigma \implies a_0 \approx a\left( 1- \frac{1}{a^2}\Sigma \right).$$ Can someone explain how this approximation is found? Edit: I place in calculus thread because there must be some series expansion going on or a neglecting of terms.
28. Solving a cubic equation

##x^3y+4x-32=0## is there a particular method for solving this? i know that ##x=2## and ##y=3##
29. How to solve this cubic equation?

Summary: Cubic equation How to solve for t ?
30. Cubic graph containing a 1-factor

I understand how to show a given graph does/does not contain a 1-factor but I'm not sure how to show existence (or the lack thereof). Please advise.
31. 7.2 cubic metres of Nitrogen compressed, will last 30mins-1hr of usage

So I am currently looking to work on some projects with my brother. I have some questions regarding the usage of nitrogen. We are looking at a 7.2 cubic metre cylinder of Nitrogen Pure compressed and wondering at what rate of pressure will equal roughly 30-60mins of use? Nitrogen Bottle here...
32. I The solution to a cubic equation

The equation ## x^3-x+.1=0 ##, has 3 real roots that can be quickly approximated as follows: Writing the equation ## x=.1+x^3 ##, iterative methods quickly indicate that there is a root ## r_1 ## near ## x=+.1 ##, and more accurately ## r_1 \approx + .101 ##. ## \\## Doing an approximate...
33. MHB Find the cubic equation that has -1 and 2i as roots

Answer is given, but no explanation or logic for it. From HiSet free practice test
34. I If I had a cubic metre of solid osmium, a perfect cube....

If i shon a red laser across the surface of the osmium cube 5mm above the solid perfect 1000mm cube, by how many degrees would it be deflected?
35. MHB Solving Cubic Root Algebra: Don't Know the Steps

I don't understand this. a is not suppose to be -1; this is the only rule in the equation The answer is the second picture, I just don't know the steps that lead to that answer.
36. MHB Solving Cubic Equation: x^3 - kx + (k + 11) = 0

Hi, this question was in a year 11 extension maths textbook in the enrichment section. I have the answer as k>17 and k<-11 because I graphed it on GeoGebra. The Graph can be found here: https://ggbm.at/xpegwwtq. While I know the answers I would like to know how to work it out using algebra...
37. MHB Is f in the vector space of cubic spline functions?

Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...
38. MHB Least squares method : approximation of a cubic polynomial

Hey! :o I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method. So we are looking for a cubic polynomial $p(x)$...
39. Matlab Plotting Points and a Cubic Polynomial that passes through them

Homework Statement We were given a tutorial to complete which I did complete. Now the question is: By modifying the appropriate lines in your script file, find the values of a, b, c, and d so that the cubic polynomial  y = ax3 + bx2 + cx + d  passes through the (x, y) pairs (-1, 3), (0, 8)...
40. I Quadratic, cubic, quartic, quintic equations

Hello, In general, any equation is a statement of equality between two expressions. In the 2D case, equations generally involve the two variables ##x## and ##y## or either variable alone if we require the other variable to be equal to zero. The most general quadratic equation should be the...
41. Difference in cubic spline formula

Homework Statement Hi, this is more of a review question and I'm just looking at solutions of natural cubic spline equations and some will give the cubic spline as: 1. s(x) = a + bx + cx^2 + dx^3 on Wolfram while other pages will give: 2. s(x) = a + b(x - t) + c(x-t)^2 + d(x-t)^3 where...
42. I Center of inversion of cubic diamond structure?

I'm having hard time finding the center of inversion of cubic diamond structure. At first I thought (2,2,2) would be the center of inversion, but (1,1,3), (3,3,3), (1,3,1), (3,1,1) (i.e. four atoms inside the cube) are not centrosymmetric about (2,2,2).
43. 240 Mosquitos can Fit into 1 Cubic Centimeter

and survive for 24 hours. That would be 4.16 cubic millimeters per mosquito! These findings are due to testing of packing methods for shipping mosquitos around for disease prevention (like malaria which is mosquito transmitted.
44. Cubic Feet of Argon required to fill a tank under vacuum

I have a tank that is 250 Cubic Feet. The tank is under vacuum (1x10-5 Torr).. which is actually 1.93368e-7 psia. How much Argon, in cubic feet, does it take to back-fill to atmosphere or 14.7 psia? Ambient temperature in tank is 90 degrees.
45. How do I factor this cubic polynomial?

Homework Statement -x^3 - 6x^2 -12x -8 Homework EquationsThe Attempt at a Solution I don't know, I just know the roots are -2 with multiplicity 3.
46. MHB Find a,b,c,d given max and min in cubic function

f(x) = ax^3 + bx^2 + cx + d min(1, -4) max(-2, 1) find a,b,c,d
47. A Relative magnetization and a Face Centered Cubic lattice

In case of simple cubic lattice relative magnetization is given by \sigma=1-\frac{1}{S}\frac{v}{(2\pi)^3}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\mbox{d} k_x\mbox{d} k_y\mbox{d}k_z(\mbox{e}^{\frac{E(\vec{k})}{kT}}-1)^{-1}...

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49. How do we solve this cubic equation

Homework Statement solve ## x^2/y + y^2/x=9, 1/x+1/y=3/4##[/B]Homework EquationsThe Attempt at a Solution ##9x^3+9y^3=81xy## ## 4x+4y-3xy=0## ##x^3+y^3-9xy=0, 12x+12y-9xy=0## ##x^3-12x+y^3-12y=0## [/B]
50. Space-centric cubic metal

<<moderator: moved from a technical forum, no template>> Hi ... I have a question about a job in the material science courseWe give the hall petch type that denotes the dependence of the strength on the average grain diameter d, also has a value close to 1mpa. I have to show in a diagram σ-d ^...