Nomography is a branch of mathematics which studies functional dependencies through graphical representation methods, called nomograms or nomographs.
Questions tagged [nomography]
22 questions
4
votes
2 answers
Do irreducible sums span the same space?
Irreducible decompositions
Say that a decomposition $f(x,y) = \sum_i U_i(x)V_i(y)$ is irreducible if the $U_i$ are all linearly independent, as are the $V_i$. The rank of a decomposition is the number of terms in the sum.
In a previous question, I…
user326210
- 19,274
4
votes
2 answers
Nomogram for $z = x + \sqrt{x-y}$
I am attempting to make a nomogram for the equation
$$\frac{z-x}{g} - \left(\frac{z-y}{g}\right)^2 =0$$
(here $g$ is a constant). You can use the quadratic formula to solve for $z$, in which case:
$$z = \frac{g}{2} + y \,\pm \,…
user326210
- 19,274
3
votes
1 answer
Linear independence when writing a function as a sum of functions.
Consider splitting a two-variable function into a sum of products of one-variable functions like this:
$$f(x,y) = \sum_{i=1}^n g_i(x) \cdot h_i(y)$$
Such a decomposition is called irreducible if the $g_i$ are linearly independent, and the $h_i$ are…
user326210
- 19,274
3
votes
1 answer
Determinant form of quadratic equation, 3 variables, second order (nomogram)
I am looking for a determinant for a second order equation so that I can build a nomogram. The equation is simply:
$$
x^{2} +2 a x-c = 0
$$
It can also be written in another format (which is more helpful to me), but I am not sure if it can be done…
T. Calil
- 31
3
votes
1 answer
Unknown functions yield a given determinant
I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$.
This amounts to finding twelve smooth curves such that the following equations hold:
$$\det…
Dylan
- 65
2
votes
1 answer
Smith chart as a nomogram for multiplication
The Smith chart is a plot of the following Möbius transformation:
$$ \Gamma(z) = \frac{z-1}{z+1} $$
typically restricted to $\Re(z) \ge 0$ so that $|\Gamma(z)| \le 1$.
On page 36 of this slide deck, an interesting property of the Smith Chart is…
2
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0 answers
Why must the Jacobian be nonzero in this derivation?
I am reading Doerfler's Calculating Curves, and I've been puzzling over one particular derivation. I have understood most of the calculus involved, but I am still missing a step. In short:
We start with a smooth, implicitly defined function…
user326210
- 19,274
2
votes
1 answer
Is linear independence preserved by multiplication?
Say that the rank of a set of functions $\{f_1,\ldots,f_n\}$ is the size of the largest linearly independent subset.
If you have two sets of linearly independent functions $\{f_1,\ldots,f_n\}$ and $\{g_1,\ldots,g_k\}$, is it possible for the set of…
user326210
- 19,274
2
votes
0 answers
Proving linear independence of function factors
I have an algorithm for converting a two-variable function $F(x,y)$ into a sum of products of single-variable functions $F(x,y) = \sum_i g_i(x)h_i(y)$. I am attempting to determine whether (or when) the $g_i$ produced in this way are all linearly…
user326210
- 19,274
2
votes
3 answers
Nomogram for probabilities
I am trying to find a nomogram for the following equation:
$$F(a,b,c) = k(c-a)-(c-b)^2 = 0$$
where $k$ is a constant. I can rewrite it as:
$$(c^2 + kc) - ka -2bc + b^2 =0.$$
This has four or five linearly-independent terms and three variables (so…
user326210
- 19,274
1
vote
0 answers
Why does this expression look like the intercept of a line?
I have two well-behaved curves in the plane $\gamma_1,\gamma_2:\mathbb{R}\rightarrow \mathbb{R}^2$. Call their coordinates $\gamma_i = \langle f_i, g_i\rangle$.
Given any two points on these curves, I can draw the line between them. By definition,…
user326210
- 19,274
1
vote
1 answer
What does this ratio of derivatives represent, geometrically?
Suppose I have an arbitrary smooth real-valued function $F(x,y,z)$ with the property that all the first-order partials of $F$ are strictly positive everywhere.
The criterion of St-Robert says that $F$ can be written as a nomogram with three straight…
user326210
- 19,274
1
vote
0 answers
Do the zeroes of a function determine its rank?
Intro
All functions are smooth real-valued functions of real variables.
Suppose you can decompose a function $f(x,y)$ as a product as one-variable functions $f(x,y) = \sum_{i=1}^n u_i(x)\cdot v_i(y)$. The rank of $f$, if it exists, is the smallest…
user326210
- 19,274
1
vote
1 answer
If $f\cdot g$ is decomposable and $g$ is decomposable, is $f$ decomposable?
A decomposition of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a sum of products of single-value functions: $f(x,y)=\sum_{i=1}^n u_i(x)\cdot v_i(y)$. If a function has at least one decomposition, it is called decomposable; otherwise, it is…
user326210
- 19,274
1
vote
0 answers
How to eliminate variable from implicitly defined function?
I have an expression I'm trying to simplify, following along in a certain book, and I can't see how to do it. Here's what I've done so far.
I have a smooth real-valued function $F(x,y,z)$, which by the implicit function theorem, defines a function…
user326210
- 19,274