Questions on quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

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101 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on Fibonacci quadratic residue: $F(x)^2 mod F(y)$ = { if y is even: $(F(|x-y|)^2$ } { if y is odd: $-(F(|x-y|)^2$ } for ...
4
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0answers
73 views

Quadratic system of ODEs

I have a quadratic ODE system that looks like this: $\dot{x}=Ax+diag(x)Nx$ where $x \in R^n$ and $A,N \in R^{n \times n}$ and $diag(x) \in R^{n \times n}$ is a diagonal matrix in which $x$ is its ...
3
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69 views

Solving quadratic diophantine equations in two variables

I've looked at the recommended questions, but none of them seem to match my question. Consider the equation $2015 = \frac{(x+y)(x+y-1)}{2} - y + 1$. This can trivially be simplified to $4030 = x^2 + ...
3
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0answers
286 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
3
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81 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
2
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0answers
73 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
2
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0answers
28 views

Taylor expansion need help understanding.

I am at the moment reading a paper (SURF) and trying to understand what is happening here and how the things works as it does.... a non maximum supression is performed on the scale space ...
2
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0answers
154 views

Lagrange multiplier expression

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1\\ \text{subject to} & ...
2
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0answers
25 views

Extraction of quadratic terms with state-space representation

I am having trouble with transforming the dynamics of a 4DOF gyroscope to a neat state-space representation. The system has the following set of equations: $T_i + f_i(\omega, \alpha) = 0;\;i:1-4$ . ...
2
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0answers
56 views

Does $2^2=4$ imply $2=\pm \sqrt{4}$?

I read the square root property from the book, College Algebra by Raymond A Barnett and Micheal R Ziegler that, The square root property says, If $A^2=C$ then $A=\pm \sqrt{C}$ I took the ...
2
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0answers
49 views

$A(x+p)²-B(x-p)²=y$, historical/math reference

I'm trying to build a reminder of all that I found about the quadratic function over the years. Here I came across this form of quadratic equation that I did not know: A(x+p)²-B(x-p)²=y I have no ...
2
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32 views

can someone break this quad formula down for me?

Can someone explain how this person yield the stuff on the right side using quad formula?
2
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0answers
152 views

Proving an equality involving binomial coefficients and summations

Question: $$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right ...
1
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0answers
16 views

Finding point closest to origin on a hyperboloid

(1) Let A be 3x3 real symmetric matrix. The eigenvalues of $A$ are $\lambda_1 = -6, \lambda_2 = 1, \lambda_3=4$ $q(x_1,x_2,x_3) = -x_1^2 + x_2^2 -x^2_3 + 10x_1x_3 = 1$. $A$ is the matrix of $q$. I ...
1
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0answers
37 views

Differential Equations: Confocal Ellipse and Hyperbola

I am currently brushing up on Conic Sections, and I am having some problems on solving a first order quadratic differential equation. I would appreciate any help on the topic! I know that confocal ...
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68 views

Minimum Curvature Path

Let's say we are given a closed race track with a given and constant width. I am to implement an algorithm which finds both shortest path trajectory and minimum curvature trajectory for the car. I ...
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23 views

Reference about quadratic forms with discriminant 1

When I am reading Serre's $A$ $Course$ $In$ $Arithmetic$, Chapter 5, it deals with $quadratic$ $forms$ of some vector space $V$, which can be viewed as an extension of an $abelian$ $group$ $E$ of ...
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22 views

Optimization of a Quadratic on a Linear Variety

We have a linear subspace $L_j := L[s_0,s_1,...,s_j]$ and a linear variety: $x_0 + L_j := [x_0 + y : y \in L_j]$ and a standard quadratic cost function $V(x) = a + b^Tx + 0.5x^TCx, \ \ \ C^T = C ...
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50 views

Show Equivalence of Binary Quadratic Forms

I've been stuck on these two problems from my problem set for quite a while. Any help would be appreciated! 2)Suppose that $ax^2 + bxy + xy^2$ is equivalent to $Ax^2 + Bxy + Cy^2$. Show that $gcd ...
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60 views

Real world application of slanted conics (parabolae especially)

I am writing a report on slanted conics of the form $$(x-h)^2+(y-k)^2= \dfrac{d}{\sqrt h}$$ Where $(h, k)$ is the focus, and $d$ is the directrix. Are there any real world applications for slanted ...
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27 views

All solutions to Quadratic matrix polynomials

I am after two things: 1- algorithms for finding all solutions of possibly large quadratic matrix equations of the form $AX^2+BX+C=0$ 2- (if possible) software implementing the algorithms ...
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18 views

Probabilty of even number of games won/lost uses auxiliary variables for a quadratic equation. Why?

In a problem of finding the probability that an even number of games (even S) not being lost in $l$ games, I read the following explanation : "We form the equation, $x^2 - 4rx + 2r^2 = 0$, and ...
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33 views

Linear constraints in Quadratic equation

I have been going through this paper, and wish to implement the same algorithm in java. I have also managed to write equivalent code for the same, but I have not completely understood the mathematics ...
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55 views

Solving matrix equation of the form $(AX)^2+(BY)^2=D$

Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$) I originally have two ...
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0answers
49 views

Quadratic Congruence in $\mathbb Z/2^n \mathbb Z$

Given the congruence $ax^2+bx+c \equiv 0 \pmod {2^n}$, how precisely does one go about finding its roots? I'm comfortable with quadratic congruence mod n with n odd, but 2's lack of a multiplicative ...
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88 views

Showing DO NOT exist GCD of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$.

Showing DO NOT exist gcd of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$. I tried it. Suppose $d$ is GCD of $6$ and $2+2 \sqrt(-5)$. then there exist $x,y \in \Bbb ...
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0answers
46 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
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22 views

Variable quadratic functions

I have some doubts in proving the following:- C is the curve $$y = \frac 1 {k+1} [2x^2 + (k + 7)x + 4]$$, where k is a real number not equal to -1. Show that C always passes through two fixed points ...
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21 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
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65 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
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0answers
83 views

Quadric equation in $\mathbb{Z}/n\mathbb{Z}$?

I would like to ask for some help about the following problem. Given is a polynomial $f(x)=ax^{2}+bx+c$ in $\mathbb{Z}/n\mathbb{Z}$, we know that this quadric equation $f(x)=0$ has exactly 8 ...
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0answers
92 views

Can the following quadratic equation be solved for M without iterating over possible values

I have developed this equation for a piece of software I am writing. Not being very mathematically minded I am stumped at how I can solve for M without iterating over all possible values for M. The ...
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0answers
41 views

Calculating $(x_1^{x_2})(x_2^{x_1})$ in a quadratic equation

Supposing $S=x_1+x_2$ and $P=x_1x_2$ where $x_1$ and $x_2$ are the roots of the quadratic equation $ax^2+bx+c=0$ (it's clear the equation is equivalent to $x^2-Sx+P=0$ where $S=-b/a$ and $P=c/a$), how ...
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0answers
70 views

Having trouble with finding a Quadratic Expression

I am having trouble trying to work out a quadratic expression for uni, being a external student its hard to find help. My problem is Perimeter = 1000m Part 1 Solve L in terms of w in regard to the ...
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16 views

Find a set of linear equations whose solution is the same as the minimum of a given quadratic objective function

Given an objective function $||x_1 a_1 - x_2 a_2 - b||$, where $a_1, a_2$ and $b$ are 3-dimensional vectors, how can I find the two linear equations for the $x_1$ and $x_2$ whose solution will find ...
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0answers
12 views

Solving a quadratic vector/tensor equation arising from connected Markov chains

I have a discrete-time finite-state aperiodic irreducible Markov chain, which is composed of $m$ identical component sub-chains. With probability $1-\mu$, in each time step each of these chains ...
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34 views

Let $a,b,c$ be the sides of triangle. No two of them are equal and $\lambda\in\Re$…

Problem : Let $a,b,c$ be the sides of triangle. No two of them are equal and $\lambda\in\Re$. If the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real, then find the range of ...
0
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0answers
22 views

Find Next Position and Velocity from Instantaneous Values

To find the position of an object at a given point in time: $y_0 + v_0t - \frac{32t^2}{2} = y_t$. And to find the object's speed at a given point in time: $v_0 - 32t = v_t$ So say I give the ...
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0answers
24 views

Convex quadratic problem solver gives different answers?!!

I'm not a mathematics girl but I'm pretty sure that the variance of a vector X should be a convex quadratic problem. my objective function is as follows: arg min var(sum(L) + X*L) x>0 vector X is ...
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17 views

Second-order quadratic model with bias term

I have 3 points in the 3-d space and I would like to estimate the parameters of a second-order quadratic model with a bias term $z=f(x,y)=ax^2+bxy+cy^2+dx+ey=\theta^TQ\theta+\eta^T\theta$ where the ...
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56 views

Confusion regarding dF/dx=0, F=constant

I thought I found a theorem Given a curve in the $(y,x)$ plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of $F$ along this curve satisfies relation $g = ...
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27 views

Solving the quadratic optimization problem with quadratic inequality constraint

I have a quadratic optimization problem which which both objective function and constraint are convex. As the problem is very big, I used decomposition technique and divide the problem to smaller ones ...
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0answers
12 views

Solution to Equation involving Volatility

The following question will have little context, though, it is not relevant. To summarise though, I am trying to find solutions $u$ and $d$ to the following equation given that $d = \frac{1}{u}$: ...
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0answers
48 views

Line intersection with Sphere

I'm trying to get a formula for calculating intersection points of a line with a sphere (3d space). I've been following this one: Wiki Line-sphere intersection But I'm 99% sure that this one is ...
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0answers
37 views

The property of roots of quadratic equation

I have a problem with two tasks: Given a quadratic equation $ax ^ 2 + bx + c = 0 $ (roots can be complex or real), $a, b, c \in Q$. Prove that ${x_1} ^ m + {x_2} ^ n \in Q$. We have a trinomial ...
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22 views

Alternative method for y-vertex calculation

So, I've been wondering the following: If you can determine the x coordinate of the vertex of a quadratic function by averaging the x coordinates of both roots, would it be possible to determine the ...
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15 views

Determine the multi-dimensional relationship given the data

I have a dependent variable - A and 3 independent variables, H,V and N I have a data for all the variables and dependency relationship is based on my operational knowledge. I'd like to know what ...
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17 views

Constrained Motion Study

I'm working on a motion study for a disk moving within a mechanical enclosure and I'm having trouble reducing my equations. The system can be defined as 4 circles which are bound inside each other. ...
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0answers
21 views

Quadratic Polynomial Formula [$ax^2+b_1x+c$ vs. $ax^2+2b_2x+c$]

I thought the quadratic formula was $ax^2+b_1x+c$. However, in my linear algebra book when they deal with $x^TKx$, they use the formula $ax^2+2b_2x+c$. I understand that $b_1$ = $2b_2$, but what is ...
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30 views

Calculating speed of an object between time intervals using quadratic function

I'm actually getting stuck with a quite difficult part of a 1D math problem using quadratic functions. An object is propelled with an initial speed, has its altitude given after t seconds by the ...