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# Can the subset sum problem work with an undefined set?

To give a brief outline of my actual problem, I'm attempting to identify possible impurities in proteins. I have a theoretical mass and an experimental mass, and the difference between the two would in theory be caused by any impurities present in the proteins. Impurities could take the form of an amino acid insertion, an amino acid deletion, an amino acid substitution, or an amino acid modification (modifications can both add and subtract mass).

Every impurity can be associated to a known change in mass, so I want to use the subset sum problem to get all subsets of impurities that add up to the delta mass. My issue is that I'm working with an undefined set. I know what impurities I can have, but I don't know how many of each I can have.

Is it possible to get the subset sum problem to work with an undefined set such as this? Is there a better algorithm I could implement to solve the issue?

• I don't understand your problem. Can you formulate it using more abstract terms? – Yuval Filmus Jun 20 '18 at 21:11
• Given that you can both take away and add mass, there might be infinitely many possibilities. – Tom van der Zanden Jun 20 '18 at 21:22
• I'm confused. What is known? You know about the possible impurities. Do you know what the effect on change in mass of each possible impurity is? How accurately can you measure the total change in mass? Is measurement error an issue? Do you have data on the probability distribution of number of impurities, and number of each type? If not it would probably be useful to gather that. You probably need a prior on the space of possible impurity-combinations. – D.W. Jun 21 '18 at 3:24
• I think he has $m$ different impurities, where a single copy (molecule?) of the $i$th impurity has mass $x_i$, and is looking for a solution to a subset sum problem in which the target sum is an experimentally measured mass difference $d$, and the set contains the numbers $kx_i$ and $-kx_i$ for $1 \le i \le m$ and $k$ can take any positive integer value. That is, the set is infinite (not undefined). – j_random_hacker Jun 21 '18 at 11:45

If you only have extra impurities, then the DECOMP program described in the following paper will efficiently solve the problem -- even if you only have an approximate mass difference $d$ to work with:
To deal with "negative impurities" (components that should be present but are absent from the experimental sample), I think your best option is to try all possible "missing mass" values $a$ in a small range that includes zero, and for each such value $a$, use DECOMP to solve two problems: One that finds all possible combinations of "missing" components that solves for $a$, and one that finds all possible combinations of "extra" components that solves for $d+a$. Every pair of (missing, extra) component multisets is then a solution for the chosen $a$.
• @Draconis: Right, "subset" wasn't quite right. I should have been more precise, but I couldn't think of a consise way to express "Let the weight of one unit of the $i$-th component be $x_i$. If there exists an integer multiplier $m_i$ for each component type $i$, such that at least one multiplier is nonzero and $\sum_{i=1}^{k}m_ix_i = 0$, then there will be an infinite number of solutions". – j_random_hacker Jun 21 '18 at 16:09


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# Planting flowers of unknown colours

A gardener has a sack of indistinguishable bulbs and a flat allotment. In the spring, each bulb will produce either a red, green or blue flower. She wishes to plant some of the bulbs so that when spring comes, some two flowers of the same colour will be exactly one metre apart.

How many bulbs must she plant to achieve her goal?

• I'm assuming the garden is a 2 dimensional plane? Feb 25 '16 at 0:08
• Yes. And infinite, as gardens in puzzles tend to be :) Feb 25 '16 at 0:09
• well that ruins my tetrahedron plan Feb 25 '16 at 0:10
• If you like this puzzle like I do, I might recommend (spoilers ahead!) this wikipedia article regarding the "chromatic number of the plane" Feb 25 '16 at 7:40
• @Zerris: "In his house at R'lyeh, Cthulhu has planted a nice flower garden" Feb 26 '16 at 14:47

The problem in graph theoretic terminology is to find the minimal number of vertices required for a unit distance graph with chromatic number of $4$.

Brokes' theorem states that:

Any connected, undirected graph of degree $d$ has chromatic number $k<=d$ unless the graph is either complete or an odd cycle when $k=d+1$.

Unit distance graphs are already undirected. Furthermore any minimal solution will be connected since:

• unconnected vertices may be any colour and thus do not increase the chromatic number; and
• the chromatic number of the disjoint union of two connected graphs is equal to the maximum chromatic number of those two graphs.

All odd cycle graphs are unit distance graphs (odd-sided, regular polygons of side $1$). However odd cycles always have degree $2$ and hence, by Brokes' theorem, a chromatic number of $3$ so no odd cycle is a solution.

All even cycle graphs are unit distance graphs (even-sided, regular polygons of side $1$). However all even cycles have degree $2$ and hence a chromatic number less than or equal to $3$. In fact they have chromatic number $2$ which is easy to see: colour any vertex $colour 1$ and follow an edge to the next vetex and colour it $colour 2$, and continue following edges around the polygon alternating colours as you go until all vertices are coloured; the last vertex will be coloured $colour 2$ and hence will not conflict via the ultimate edge leading back to the first coloured vertex.

We can construct the complete graph containing $3$ vertices, $K_3$, as a unit distance graph - it is an equilateral triangle of side $1$, this also has degree $2$ and a chromatic number of $3$.

To construct a complete unit distance graph with more than $3$ vertices we would need to first find a point on the Euclidean plane that lies at a distance of $1$ from all three of the vertices of an equilateral triangle of side $1$, but there is no such point - so there are no complete unit distance graphs with more than $3$ vertices.

Therefore any solution must have degree $>= 4$ (at least one vertex must have at least four neighbours).

Any neighbours of a vertex $P$ lie on a unit circle centred at $P$.

We now not only know that the solution must be at least $5$, but that any valid instance will contain a vertex $P$ at the centre of a unit circle upon which a further four vertices $Q, R, S, T$ exist.

The only way $5$ could be the solution is if such an instance, with no additional vertices, exists with a chromatic number of $4$. We can check the few cases as follows.

If no pair of $Q, R, S, T$ are $1$ apart the chromatic number would be $2$
- $P$ being $colour 1$ and $Q, R, S, T$ being $colour 2$.

If only one pair of $Q, R, S, T$ are $1$ apart, say $Q$ and $R$, then the graph would be an equilateral triangle of side $1$ $PQR$ and two spokes $PS$ and $PT$, which has a chromatic number of $3$
- $P$ being $colour 1$, $Q$ being $colour 2$, $R$ being $colour 3$, and $S$ and $T$ being free to be either $colour 2$ or $colour 3$.

If exactly two pairs of $Q, R, S, T$ are $1$ apart then there are two cases (ignoring label order isomorphisms):

1. $Q$ and $R$ are one pair and $S$ and $T$ are the other; or
2. $Q$ and $R$ are one pair and $R$ and $S$ are the other.

Case 1 is two vertex-adjoined equilateral triangles $PQR$ and $PST$, which has a chromatic number of $3$
- $P$ being $colour 1$, $Q$ and $S$ (or $Q$ and $T$) being $colour 2$, and $R$ and $T$ (or $R$ and $S$) being $colour 3$.

Case 2 is a unit diamond $PQRS$ and a spoke $PT$ from one of the corners on the diagonal of length $1$, which has a chromatic number of $3$
- $P$ being $colour 1$, $Q$ and $S$ being $colour 2$, $R$ being $colour 3$, and $T$ being free to be either $colour 2$ or $colour 3$.

If exactly three pairs of $Q, R, S, T$ are $1$ apart the only formation is the unit triamond $QRST$ with $P$ at the only vertex of the central equilateral triangle that is not a corner of the triamond, which has a chromatic number of $3$
- $P$ being $colour 1$, $Q$ and $S$ being $colour 2$, and $R$ and $T$ being $colour 3$.

In order for $6$ to be the solution we would need to extend any of these instances by one vertex $U$ to yield a graph of chromatic number $4$ by forcing $Q, R, S, T$ to require three colours. This can only be done by placing $U$ at an unoccupied point $1$ away from two vertices already forced to be of equal colour.

If no pair or one pair of $Q, R, S, T$ are $1$ apart no two are already forced to be of the same colour so it will not work.

If two pairs of $Q, R, S, T$ are $1$ apart in case 1 we may swap the colour groups as indicated by the parenthesised options, hence no two are already actually forced to be of the same colour so it also wont work; in case 2 $Q$ and $S$ are forced to be the same colour but the only points one unit away from both are already occupied ($P$ and $R$).

If exactly three pairs of $Q, R, S, T$ are $1$ apart there are two pairs which are already forced to be of the same colour:

1. $Q$ and $S$, but the only points one unit away from both are already occupied ($P$ and $R$); and
2. $R$ and $T$, but the only points one unit away from both are already occupied ($P$ and $S$).

Thus the solution is $> 6$.

Since an example of a solution with $7$ vertices is provided in ralphmerridew's answer and is illustrated below we can now say that:

The solution is $7$.

• in this example $P$ would be $A$ and $Q, R, S, T$ would be $B, E, D, G$ (in any order) this unit distance graph is known as the Moser spindle.

ralphmerridew's solution in visual form - the three colours may be rotated, and additionally either or both of (BD) & (EG) may permute to give the same "conflict" on (CF):

If we label $A$ as the origin, with the unit vectors pointing right and down for convenience, and use $\theta$ to name the angle of rotation to transform the unit diamond* $ABCD$ to $AEFG$ then:

$C = (\sqrt{3}\cos(30), \sqrt{3}\sin(30)) = (\frac32, \sqrt{\frac34})$

$F = (\sqrt{3}\cos(30+\theta), \sqrt{3}\sin(30 + \theta))$

So for $C$ and $F$ to be one unit apart:

$\sqrt{(\frac32-\sqrt{3}\cos(30+\theta))^2+(\sqrt{3}\sin(30+\theta)-\sqrt{\frac34})^2}=1$

$\Rightarrow(\frac32-\sqrt{3}\cos(30+\theta))^2+(\sqrt{3}\sin(30+\theta)-\sqrt{\frac34})^2=1$

$\Rightarrow\frac94-\sqrt{27}\cos(30+\theta)+3\cos^2(30+\theta)+3\sin^2(30+\theta)-3\sin(30+\theta)+\frac34 = 1$

$\Rightarrow\frac94-\sqrt{27}\cos(30+\theta)+3-3\sin(30+\theta)+\frac34 = 1$

$\Rightarrow\sqrt{27}\cos(30+\theta)+3\sin(30+\theta) = 5$

$\Rightarrow\sqrt{27}(\cos(30)\cos(\theta)-\sin(30)\sin(\theta))+3(\sin(30)\cos(\theta)+\cos(30)\sin(\theta)) = 5$

$\Rightarrow\sqrt{27}(\sqrt{\frac34}\cos(\theta)-\frac12\sin(\theta))+3(\frac12\cos(\theta)+\sqrt{\frac34}\sin(\theta)) = 5$

$\Rightarrow\frac92\cos(\theta)-\sqrt{\frac{27}4}\sin(\theta)+\frac32\cos(\theta)+\sqrt{\frac{27}4}\sin(\theta) = 5$

$\Rightarrow\cos(\theta) = \frac56$

• $AEFG$ is a rotation about $A$ of $ABCD$ by $\theta=33.557$ degrees $= 0.58569$ radians (5sf)

• Thus angle $BAG = 93.557$ degrees (5sf)

*two unit equilateral triangles adjoined by one side forming a rhombus with 60 and 120 degree angles

Seven bulbs suffice. Plant them so that ABCD and AEFG are rhombuses with a 1-meter side and a sixty-degree angle at A, and such that CF is 1 meter.

• This works, but can you show that it may not happen with 6 or fewer? Feb 25 '16 at 0:58
• Note that this configuration is also the unique (up to affine transforms) configuration of 7 points such that if you pick any three then at least two of the three are 1m apart. Feb 25 '16 at 19:39

Suppose you were to plant just six bulbs. Could we fail to have two flowers the same color a meter apart? I'll show the answer is yes. (This might be overly complicated, but I couldn't do it in less than four cases)

Call two bulbs adjacent if they are planted a meter apart.

1. First note that if you plant a bulb $X$ adjacent to only two others, then we've already failed. $X$ may come up the third color, and hence in the worst case scenario it is useless to our quest for two flowers the same color.
2. Suppose we choose four points $ABCD$ on a unit rhombus with a unit diagonal. Then these four points correspond to points on a triangular lattice. The other two bulbs $E$ and $F$ must be adjacent to at least two of the points $ABCD$, but this is only possible if $E$ and $F$ are both on the triangular lattice as well. But then $E$ and $F$ are forced into one of the following four locations marked x:

...

  .   .   .   .
.   .   x   .   .
.   A   B   x
.   x   C   D   .
.   .   x   .
.   .   .   .

...

Both $E$ and $F$ are then adjacent to only two other flowers, putting us back in case 1.

1. Suppose you have an equailateral triangle $ABC$ . Any bulb adjacent to two of these puts us back in case 2, so that forces the other bulbs to be adjacent to at most one of the set $ABC$. To avoid case 1, we need $D,E,F$ to be adjacent to $A,B,C$ respectively, and we also need $DEF$ to form an equilateral triangle. However, if $ABC$ comes up red, green, blue and $DEF$ comes up green, blue, red, we're in trouble.
2. Finally, consider bulb $A$. To avoid case 1 we must have at least three bulbs adjacent to $A$. Suppose $AB$, $AC$, and $AD$ are all adjacent. To avoid a triangle (case 3), we know $BC$ are not adjacent, nor are $CD$, nor $BD$. That means $B, C, D$ are all adjacent to the last two bulbs $E$ and $F$ (to avoid case 1), and $E$ and $F$ are not adjacent to each other or $A$ (to avoid case 3). But if $A,E,F$ all come up blue and $B, C, D$ all come up green we're in trouble.

Perhaps I am not understanding the question correctly but what about this, which appears to do it with 4 bulbs: plant one bulb at the centre of a circle of 1 metre radius and then three other bulbs on the circumference of the circle. There are only three colours so one of those on the circumference will be the same colour as the one in the centre.

• Not necessarily. For example, she could end up with a red flower in the center, and two blues and one green on the circumference. Feb 26 '16 at 14:51
• Yep, that won't work, even when planting infinitely many bulbs on the prescribed circle there is an infinitesimal chance it will not cut it as there may be colour (1) in the centre and only colours (2) and (3) on the circumference such that every sequence formed of the corners of the infinitely many hexagons of 1m cords are alternating (2,3,2,3,2,3). Feb 26 '16 at 17:13

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# Researchers portal

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## Observation of electroweak production of W? with two jets in proton-proton collisions at ?s=13 TeV

Abstract: A first observation is presented for the electroweak production of a W boson, a photon, and two jets in proton-proton collisions. The W boson decays are selected by requiring one identified electron or muon and an imbalance in transverse momentum. The two jets are required to have a high dijet mass and a large separation in pseudorapidity. The measurement is based on data collected with the CMS detector at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb-1. The observed (expected) significance for this process is 4.9 (4.6) standard deviations. After combining with previously reported CMS results at 8 TeV, the observed (expected) significance is 5.3 (4.8) standard deviations. The cross section for the electroweak W?jj production in a restricted fiducial region is measured as 20.4±4.5fb and the total cross section for W? production in association with 2 jets in the same fiducial region is 108±16fb. All results are in good agreement with recent theoretical predictions. Constraints are placed on anomalous quartic gauge couplings in terms of dimension-8 effective field theory operators.

Otras publicaciones de la misma revista o congreso con autores/as de la Universidad de Cantabria

Fuente: Physics Letters B 811 (2020) 135988

Editorial: Elsevier

Fecha de publicación: 01/12/2020

Nº de páginas: 27

Tipo de publicación: Artículo de Revista

ISSN: 0370-2693,1873-2445

Url de la publicación: https://doi.org/10.1016/j.physletb.2020.135988

### Autores/as

SIRUNYAN, A. M.

MARCOS FERNANDEZ GARCIA

PEDRO JOSE FERNANDEZ MANTECA

CEDRIC GERALD PRIEELS

FRANCESCA SHUN-NING ANNAROSA RICCI-TAM


text:
Enter the initial velocity, the final velocity, and the mass into the calculator to determine the Change in Momentum.

## Change in Momentum Formula

The following equation is used to calculate the Change in Momentum.

dp = (Vf-Vi) *m
• Where dp  is the Change in Momentum (m/s*kg)
• Vi is the initial velocity (m/s)
• Vf is the final velocity (m/s)
• m is the mass (kg)

To calculate the change in momentum, multiply the change in velocity by the change in mass.

## What are the units for Change in Momentum?

The most common units for Change in Momentum are m/s*kg.

## How to Calculate Change in Momentum?

Example Problem:

The following example problem outlines the steps and information needed to calculate the Change in Momentum.

First, determine the initial velocity. In this example, the initial velocity is determined to be 60 (m/s).

Next, determine the final velocity. For this problem, the final velocity is measured to be 70 (m/s).

Next, determine the mass. In this case, the mass is found to be 10 (kg).

Finally, calculate the Change in Momentum using the formula above:

dp = (Vf-Vi) *m

Inserting the values from above and solving the equation with the imputed values gives:

dp = (70-60) *10 = 100 (m/s*kg)


text:
# When is $\| f \|_\infty$ a norm of the vector space of all continuous functions on subset S?

Let S be any subset of $\mathbb{R^n}$. Let $C_b(S)$ denote the vector space of all bounded continuous functions on S. For $f \in C(S)$, define $\| f \|_\infty = \sup_{x \in S} |f(x)|$

When is this a norm of the vector space of all continuous functions on S?

-
The presentation here is confusing. Why do you mention what the notation $C_b(S)$ means? Note that by definition of norm, $\|f\|_\infty$ must be a real number for all such $f$. When is it true that for all $f\in C(S)$, $\sup\limits_{x\in S}|f(x)|<\infty$? – Jonas Meyer Feb 6 '13 at 3:33

Like Jonas mentioned, your question seems incomplete, since you define $C_{b}(S)$, yet make no use of it. Here are some facts to perhaps answer the question you intended:

(1) For continuous functions with compact support, $||f||_{\infty}$ is in fact equivalent to the ''sup norm' as you defined in your question. Note that $||f||_{L^{\infty}(S,\mathscr{S},\mu)}:=\inf\limits_{\alpha\in\mathbb{R}}\mu\left(\{x:|f(x)|>\alpha\}\right)=0.$ In other words, it is the smallest positive real number $\alpha$ whereby the set on which $|f|(x)>\alpha$ has measure zero (we call this the essential upper bound, since as far as integration is concerned, what $f$ does on a set of measure zero can be ignored). The previous assertion now follows from the observation that compact support and continuity means $f$ achieves its (finite) optimal values on its support.

(2) When the space of continuous functions with compact support, $C_{B}(S)$ as you refer to it, is endowed with the $L^{p}$ norm ($1\leq p<\infty$), then unlike for the usual $L^{p}$ spaces, $||\cdot||$ defines an actual metric without having to first quotient out almost everywhere equal functions: this is because if two continuous functions differ at a point, then they necessarily (because of continuity) differ on an open set, and such an open set has positive measure since it also contains a k-cell. But $C_{B}(S)$ is not complete, for it is an open and dense sub(space!) within $L^{p}$, which is complete (this is, when combined with Lusin's theorem, a simple consequence of step functions also being a dense and open subspace within $L^{p}$). Recall that complete metric spaces are closed and can never be open and dense within a complete metric space, for the metric completion of such a subspace is unique upto isomorphic embedding. Anyway, the point is, with the $||\cdot||_{L^{p}}$ metric, the completion of $C_{B}(S)$ is in fact $L^{p}$ (note $p=\infty$ is excluded from this point!).

(3) Recall that the ordinary Riemann integral is defined only for continuous functions with compact support (boundedness being implied). Then (2) then implies that the Lebesgue integral is the proper construction for extending the Riemann integral! Compare this to the usual direct proof.

(4) For the case $p=\infty$, the completion of $C_{B}(S)$ is not $L^{\infty}$!. It is in fact the space of continuous functions that are in some sense "locally compactly supported." Precisely, the completion of $C_{B}(S)$ consists of those continuous functions defined on $S$ which vanish at infinity: for any $\epsilon>0$ there exists a compact set $K\subset\text{supp}f$ whereby $|f|(x)<\epsilon$ for $x\in K^{c}$. The proof of this is just a direct application of usual metric space techniques. Despite this, going back to your original question, this does not mean $|\cdot|_{\infty}$ doesn't define a metric on $C_{B}(S)$, it of course does. Just that its completion is not in relation to the $L^{\infty}$ space as it is for the case $1<p<\infty$ (note that the cases $0<p<1$ are not treated since, among other things, the $L^{p}$ norm here only defines a quasimetric).

-
I thought the question was probably trying to get at what the characterization is of subspaces of $\mathbb R^n$ such that every continuous $\mathbb R$ valued function on the subspace is bounded, i.e., pseudocompactness, which for subspaces of metric spaces is equivalent to compactness. – Jonas Meyer Feb 6 '13 at 17:29
Honestly, the question is from my textbook and i think it has a typo, since it refers to two cases of continuous functions, thanks for your response, i shall consult with my professor to see if it indeed a typo – bobdylan Feb 6 '13 at 18:36
Well there is another case, which is just the bounded continuous functions defined on an arbitrary metric space. This space is complete in and of itself with respect to the "sup norm" metric, which is just the $L^{\infty}$ norm. Note carefully though the distinction between this particular space of continuous functions and the ones mentioned above. – Taylor Martin Feb 6 '13 at 18:39


text:
Inside Story

MAGNETIC FIELD DUE TO STRAIGHT CURRENT-CARRYING CONDUCTOR

INTRODUCTION

We know that static electric charges create an electric field around itself whose magnitude depends on the value of charge and the distance of the point where we want to find the electric field.

But we also know that when these charges start moving with constant acceleration then it creates an electric field as well as a magnetic field. The produced electric field and magnetic field are perpendicular to each other. The combined motion of the electric field and the magnetic field is called electromagnetic waves. [latexpage]

The magnitude of the magnetic field is proportional to the value of the electric current passing through the wire and the distance of the point where we want to find the magnetic field and its value also depends on the angle between the direction of the current element and the distance vector of the point. See figure below:

The magnetic field is a vector quantity because it has both magnitude and direction. It is denoted by the capital letter B. Its SI unit is Tesla but sometimes Weber and gauss are also used as units of the magnetic field. $$1T=10^4G(gauss)$$ $$1T=1Wb m^{-2}$$

MAGNETIC FIELD DUE TO STRAIGHT CURRENT-CARRYING CONDUCTOR

We know the current-carrying conductor produces a magnetic field around itself. This magnetic field around the wire seems like a concentric cylinder as a whole but if we take one plane of their cross-section then it appears as a concentric circle, see figure above.

Let’s understand the behavior of the magnetic field around a straight current-carrying conductor through an experiment/activity.

MATERIALS REQUIRED

You need these materials to perform this activity.

• A cardboard
• Some length of wire
• Ammeter
• Battery
• Iron filings

PROCEDURE

• Connect the positive terminal of the battery to the positive terminal of the ammeter as shown above.
• Connect the wire to the negative terminal of the ammeter to the negative terminal of the battery.
• Insert the wire from the center of the plane cardboard before connecting it to the battery.
• Keep the middle wire straight and named it X and Y, as shown above.

Now your experimental setup is ready. Supply electric current in the wire. When electric current starts flowing through the wire then sprinkle some iron filings on the cardboard and give some little jerk to the cardboard with your fingers.

Now, what did you see on the cardboard? Did you see any patterns on the cardboard? And how this pattern is looking? You must have seen a pattern like as seen in the figure below.

When you sprinkle iron filings on the plane of the cardboard and after giving some gentle jerk, it tries to align in the direction of the magnetic field produced by the electric current and the direction of the magnetic field is perpendicular to the conductor.

The direction of the magnetic field around the straight current-carrying conductor is given by the right-hand thumb rule.

According to this rule – when you catch the wire in your fist such that your thumb points the direction of the electric current then your curled fingers denote the direction of the magnetic field around the wire. See figure below:

Alignment of iron filings shows that produced magnetic field forms a concentric circle around the wire. And this magnetic field lines in the plane of the cardboard and is perpendicular to the conductor.

 If the gaps between the concentric circles are small then it shows a strong magnetic field and if it is large then it shows a weak magnetic field.

CHARACTERISTICS OF MAGNETIC FIELD DUE TO STRAIGHT CURRENT-CARRYING CONDUCTOR

The characteristics of the magnetic field produced due to a current-carrying conductor are listed below:

• It forms concentric circles around the conductor.
• It lies in a plane and perpendicular to the conductor.
• If the direction of current flow reverses then the direction of the magnetic field is also reversed.
• The strength of the field is directly proportional to the magnitude of the current. This means a high electric current produces a strong magnetic field and a low electric current produces a weak magnetic field.
• The strength of the field at any point is inversely proportional to the distance of the point from the wire.

Watch this video for more understanding.

Stay tuned with Laws Of Nature for more useful and interesting content.


text:
# General Sobolev metrics on the manifold of all Riemannian metrics

Presented by:
Peter Michor Universität Wien, Universität Wien
Date:
Monday 13th November 2017 - 09:45 to 10:30
Venue:
INI Seminar Room 1
Abstract:
Based on collaborations with M.Bauer, M.Bruveris, P.Harms. For a compact manifold $M^m$ equipped with a smooth fixed background Riemannian metric $\hat g$ we consider the space $\operatorname{Met}_{H^s(\hat g)}(M)$ of all Riemannian metrics of Sobolev class $H^s$ for real $s>\frac m2$ with respect to $\hat g$. The $L^2$-metric on $\operatorname{Met}_{C^\infty}(M)$ was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on $\operatorname{Met}_{C^\infty}(M)$ were considered in [M.Bauer, P.Harms, and P.W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. J. Differential Geom., 94(2):187-208, 2013.] In this talk we consider variants of these Sobolev metrics which include Sobolev metrics of any positive real (not integer) order $s$. We derive the geodesic equations and show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.


text:
Lemma 30.10.3 (Artin-Rees). Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}$ be a quasi-coherent subsheaf. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Then there exists a $c \geq 0$ such that for all $n \geq c$ we have

$\mathcal{I}^{n - c}(\mathcal{I}^ c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^ n\mathcal{F} \cap \mathcal{G}$

Proof. This follows immediately from Algebra, Lemma 10.51.2 because $X$ has a finite covering by spectra of Noetherian rings. $\square$

Comment #943 by correction_bot on

In the statement of the lemma, it should be $\mathcal{I}^n\mathcal{F} \cap \mathcal{G}$ not $\mathcal{I}^n\mathcal{F}$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
Examveda

# A select group of 4 is to be formed from 8 men and 6 women in such a way that the group must have at least 1 women. In how many different ways can it be done ?

A. 364

B. 728

C. 931

D. 1001

E. None of these

$$= \left( {{}^6{C_1} \times {}^8{C_3}} \right) + \left( {{}^6{C_2} \times {}^8{C_2}} \right)$$     $$+ \left( {{}^6{C_3} \times {}^8{C_1}} \right)$$   $$+ \left( {{}^6{C_4} \times {}^8{C_0}} \right)$$
$$= \left\{ {6 \times \frac{{8 \times 7 \times 6}}{{3 \times 2 \times 1}}} \right\} +$$    $$\left( {\frac{{6 \times 5}}{{2 \times 1}} \times \frac{{8 \times 7}}{{2 \times 1}}} \right)$$   $$+ \left( {\frac{{6 \times 5 \times 4}}{{3 \times 2 \times 1}} \times 8} \right)$$    $$+ \left( {{}^6{C_2} \times 1} \right)$$
$$= \left\{ {6 \times \frac{{8 \times 7 \times 6}}{{3 \times 2 \times 1}}} \right\}$$    $$+\, 420\, +$$  $$\left( {\frac{{6 \times 5 \times 4}}{6} \times 8} \right)$$   $$+ \left( {\frac{{6 \times 5}}{{2 \times 1}} \times 1} \right)$$
$$= \left( {336 + 420 + 160 + 15} \right)$$
$$= 931$$


text:
Infoscience

Journal article

# Increased tolerance of sunflower mutant seedlings to Cd and Zn in hydroponic culture

Cadmium and zinc show similarities in their chemistry, geochemistry and environmental properties. Whereas Cd is a phytotoxic non-essential element, Zn is an essential trace element for plant growth and development. Zinc plays a fundamental role in several key cellular functions, such as protein metabolism, gene expression, chromatin structure, photosynthetic carbon metabolism and indole acetic acid metabolism (Vallee and Falchuk, 1993; Marschner, 1995; Bonnet et al., 2000; Cakmak and Braun, 2001). Zinc is also an important component of many enzymes and proteins (Broadley et al., 2007). Cadmium can readily inhibit most of the Zn-dependent processes by binding to the membrane and to enzyme active sites, thus inactivating their functions (Aravind and Prasad, 2005). However, increasing Zn concentrations can replace such wrongly bound Cd (Van Assche and Clijsters, 1990; Shaw et al., 2004). Cadmium has been shown to cause many morphological, physiological, biochemical and structural changes in plants, such as growth inhibition, reduction in photosynthesis, transpiration or water imbalance (Sanità di Toppi and Gabbrielli, 1999; Schützendübel et al., 2001; Benavides et al., 2005; Mishra et al., 2006). Cadmium and high zinc concentration affect plant growth and metabolism, but the intensity of their toxic effects depends on plant species and the way and duration of metal exposure. Plants called hyperaccumulators are able to tolerate and accumulate extraordinary levels of trace elements in their above-ground tissue without developing any toxicity symptoms (Baker and Brooks, 1989; Reeves and Baker, 2000). Their high metal accumulation capability makes them interesting for the decontamination technique called phytoextraction, which uses plants for environmental clean-up (Salt et al., 1998). However, their low biomass production strongly limits the real application of this soil decontamination strategy. The ideal plant for phytoextraction should provide both a high biomass and a high tolerance and metal accumulation (Schwitzguébel et al., 2002). High biomass producing plants with improved tolerance to trace elements and with an enhanced metal accumulation capacity should be good candidates for metal removal from contaminated area. Previous studies have shown enhanced tolerance and metal accumulation properties in transgenic tobacco plants (Pomponi et al., 2006; Gorinova et al., 2007; Wojas et al., 2008). However, the main disadvantage of genetic engineering is still public acceptance and free-land application. In vitro breeding techniques have also been successfully used to obtain tobacco lines with a considerably enhanced Cd, Zn and Cu tolerance and metal accumulation (Herzig et al., 1997; Guadagnini et al., 1999; Herzig et al., 2003). In addition, the chemical mutagen ethyl methane sulphonate has been used to develop new sunflower lines with a significantly enhanced biomass and metal uptake on a metal contaminated field (Nehnevajova et al., 2007, 2009). Although these field experiments have revealed the potential of sunflower mutants for metal phytoextraction, their tolerance to trace elements and the effect of Cd and Zn on growth and photosynthetic pigments were not studied within previous free-land experimentation. The aim of the present work was thus to assess the tolerance to Cd and excess Zn of selected sunflower mutant lines with improved biomass and metal uptake capacity, by measuring different physiological parameters. For such a purpose, sunflower seedlings of eight selected mutant lines were cultivated under hydroponic conditions in the presence and in the absence of Cd and Zn to study the effect of these trace elements on (1) growth parameters, such as root elongation, root and shoot dry mass; (2) chlorophyll content; (3) carotenoid content; (4) Cd and Zn accumulation capacity of sunflower mutants; and (5) a possible correlation between Cd and Zn accumulation in sunflower shoots and roots.

### Fulltext

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text:
# Question 621cc

Nov 28, 2015

$285.7 \text{ mL}$

#### Explanation:

Use the following rule: $\frac{{P}_{1} {V}_{1}}{T} _ 1 = \frac{{P}_{2} {V}_{2}}{T} _ 2$

(Remember that we will always use temperature in Kelvin. When we have a degree measure in Celsius, we add 273˚# to convert.)

Plug in what we know:

$\frac{720 \times 325}{303} = \frac{900 \times {V}_{2}}{333}$

${V}_{2} = 285.7 \text{ mL}$


text:
# Derivative of the Area of a Circle

by husky88
Tags: circle, derivative
 P: 79 1. The problem statement, all variables and given/known data Show that the rate of change of the area of a circle with respect to its radius is the same as the circumference of the circle. Can you suggest why? 2. Relevant equations A = $$\pi$$r$$^{2}$$ = f(r) L = 2$$\pi$$r = g(r) 3. The attempt at a solution I have showed that the derivative of f(r) is equal to g(r). But I have no idea why the area and the circumference of the circle are related in such a way. Any suggestions greatly appreciated. Thank you.


text:
Services & ResourcesWolfram ForumsMathGroup Archive

Re: Simplifying and Rearranging Expressions

• To: mathgroup at smc.vnet.net
• Subject: [mg95956] Re: Simplifying and Rearranging Expressions
• From: AES <siegman at stanford.edu>
• Date: Fri, 30 Jan 2009 05:47:50 -0500 (EST)
• Organization: Stanford University
• References: <gls1u8$hjl$1@smc.vnet.net>

In article <gls1u8$hjl$1 at smc.vnet.net>,
"David Park" <djmpark at comcast.net> wrote:

> I want to start a thread on this because I believe many MathGroup people
> will have some useful things to say.

I'll bite, because I've done a bit of thinking on this.

> A common task for Mathematica users is to obtain an expression that is in a
> particular form. For students and teachers this may often be a textbook
> form, or there may be other reasons that a particular form is desired.

Just to add a bit of specificity to this, let's consider expressions
that arise in optics and e-m theory, which generally involve a set of
physical quantities (velocity of light, propagation constant,
permeabilities, index of refraction, frequency, wavelength,
characteristics impedance, critical angle of refraction, and multiple
others) that are conventionally written as

c, k, (or beta), mu, epsilon, n, f (or omega), lambda,
eta or z_0, thetaCrit, and so on

**each of which is directly linked or coupled to (that is, can be
calculated from) several others in the same set**.

[You have to put up with a side story at this point.  The distinguished
physicist W. K. H. Panofsky, who just recently died, early in his career
co-authored with Melba Phillips a small but excellent text on classical
e-m, colloquially known as "Panofsky and Phillips", from which I and
many others studied.  This was long enough ago that cgs and mks units
were still fighting it out in the physics community.]

[P and P beautifully sidestepped this issue by, as they noted in the
Preface to their book, writing every equation in their book using an
appropriate (but generally different) subset of the above symbols, such
that every equation was valid in mks units as it stood, **and could be
instantly converted to be exactly valid in cgs units as well, simply by
replacing any factor of epsilon that appeared in any of these equations
by 1/ 4 pi **.]

> It might be thought that this should be an easy task but quite often it can
> be a very difficult task, even involving mathematical derivation and many of
> the capabilities of Mathematica. Not obtaining a specific form may be a
> matter of not knowing how to solve the problem in the first place.

It may not be just a difficult task;  in fact, **it may be an impossible

1)  As already noted above, you may want to write expressions that
contain some subset of a linked set of variables in different ways at
different points in an exposition, because these different ways are
conventional in the field, and/or make the physical meaning clearer.

For example, you may want to write the space-time variation of a phasor
wave amplitude as Exp[ I k z - I omega t] because that's neat, simple,
and conventional.

But then, in discussing a waveguide mode where a factor k d (d =
waveguide width) appears, you may want to write that factor instead in
the form 2 pi d/lambda to emphasize that it's the width in wavelengths
that's important.

But if at some point in your notebook you're going to insert any of the
dependences within this set -- e.g., k := 2 pi / lambda -- then you're
stuck with this from then on.

2)  A second point:  My experience has been that useful identities that
often arise in analyses -- for example, with suitable qualifications the
infinite integral of Exp[- a x^2 + b x] == Exp[b^2/4a] -- sometime just
won't fall out (i.e., won't be explicitly evaluated by Mathematica if a
and b are actually more complicated expressions.

3)  More generally, one very often wants to do the eventual numerical
calculations using only one or another form of dimensionless or
normalized variables, because that's numerically efficient as well as
physically and practically useful in expressing the results.

And, if at some point in an exposition you're going to convert your
analytical and expositional formulas into dimensionless formulas for
numerical calculation purposes -- **at that point you really don't care
how Mathematica arranges the resulting expression**.

> Nevertheless, even simple rearrangement can be difficult. I sometimes think
> of it as doing surgery on expressions. I believe it is generally desirable
> to use Mathematica to rearrange an expression and not retype the expression.
> Retyping is too error prone.

Last sentence is true; immediately preceding sentence may be true as
phrased -- but is a mistaken belief.  Comments on this below.

> Simplify and FullSimplify are amazingly useful but it is difficult to
> control them and obtain a precise result. One will often have to do
> additional piecemeal operations. One downside of Simplify and FullSimplify
> is that they can return different forms with different Mathematica versions.
> Then any additional operations in an old notebook may no longer work. It
> would be nice if there was a method of using these commands that would be
> more version independent.

> Various routines such as Together, Apart, Factor, TrigReduce, TrigFactor,
> TrigExpand, TrigToExp, GroebnerBasis etc., can be useful in getting a
> specific form. MapAt is very useful for doing surgery on specific parts of
> an expression. Mathematica often gets two factors that have extra minus
> signs. You can correct that by mapping Minus onto the two factors. For
> integrals in the wrong form you could cheat by trying to find the constant
> by which they differ by subtracting and simplifying, and then use that in
> the derivation.

Let's say it like it is:  It's not just "difficult" for ordinary users
to use and control many of these advanced tools:  It's basically
**impossible** for the average user to learn what some of these tools
do, because they're so complex and the results can depend so critically
on what you put into them; all you end up doing is thrashing around
endlessly, trying to get them to produce the results you want.

The more powerful they get, the less they're worth trying to learn.

> It is very useful to get Mathematica generated expressions into the form
> that one wants. I believe that this is probably a sticking point with many
> users. In general it is not a trivial topic. Others may have some good
> general ideas that I don't know about.

1)  Accept that "Retyping is...error prone" -- and more generally that
"To err is human..." -- and to the extent that you have to do any form
of retyping, do a _lot_ of checking, rechecking, testing with simple
cases, and looking to see that results are physically meaningful.

2)  Nonetheless, in general, "It is GENERALLY NOT very useful to get
Mathematica generated expressions into the form that one wants" -- at
least, not very often, and not if it involves any significant amount of
effort.  It's wasted energy, and can add its own errors, or divert one
from seeing one's own errors.

3)  Instead, if what you're doing is a complex analysis and/or
exposition, tackle the analysis portion initially with paper, pencil,
and a good soft eraser, the way God intended analysis to be done.

4)  When and if certain calculations (series expansions, etc.) get
messy, run separate Mathematica symbolic calculations in auxiliary
notebooks to carry them out.

5)  When it comes time for exposition, do the exposition using a tool
that's designed for exposition (e.g., TeX),  while doing the numerical
calculations and graphing using a tool that's good at those things --
and while doing this repeat item 1) multiple times.

> Someday someone may even write a good tutorial on it.

of Mathematica's basic capabilities?

That is:

*  TeX is (I believe) totally open source, free, highly stable, and
widely known and studied -- and it's full source code is very compact.

*  So how about building the TeX source code into Mathematica's already
immense repertoire of rules and stuff, and allowing one to include at
any point in a "text portion" (I.e., a "non-evaluation portion") of a
Mathematica notebook cell the syntax

TeX[ ---any valid TeX syntax---]

such as TeX[$\alpha = \beta / \gamma^2$] to get that bit of inline math
into a Text or Header cell, or TeX[$$\alpha = \left( \beta /over \gamma^2 \right)$$] to insert a display equation,

and just having Mathematica display the typeset box produced by TeX
using that syntax into the Mathematica notebook at that point, with the
stuff inside the [ ] brackets having no other evaluational function or
effect in Mathematica itself, except to be displayed?

Is there some reason this would be conceptually impossible?  Would it be
that difficult to accomplish? Could it at least be implemented with some
reasonable subset of TeX syntax and capabilities?

If your goal is to have Mathematica notebooks serve simultaneously as
"exposition documents" and "calculation performing documents", might
this be a lot easier than endless fighting with option-laden and
temporally unstable Mathematica expressions like "Together, Apart,
Factor, TrigReduce, TrigFactor, TrigExpand, TrigToExp, GroebnerBasis"
and all their even more arcane extensions?

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text:
# Family and Register in Musical Tones

Roy Patterson , Etienne Gaudrain, Tom Walters

 The text and figures that appear on this page were subsequently published in: Patterson RD, Gaudrain E, Walters TC (2010) The Perception of Family and Register in Musical Tones. In: Jones MR, Fay RR, Popper AN (eds), Music Perception. Springer-Verlag, New York, pp.13-50.
Figure 1 – The waveform and spectrum of a child's /a/ vowel.

This Chapter is about the sounds made by musical instruments and how we perceive those sounds. The Chapter is intended to explain the basics of musical note production, such as, why a particular instrument plays a specific range of notes; why instruments come in families; and why we hear distinctive differences between members of a given instrument family - even when they are playing the same note. The chapter focuses on the relationship between the physical variables that determine how musical instruments produce their notes (like the length, mass and tension of a string) and the acoustic variables of sounds (like intensity and frequency). The discussion reveals that there are three acoustic properties of musical sounds, as they occur in air, that are particularly useful in (a) summarizing the action of the physical properties of instruments and their limitations, on the one hand, and (b) explaining the dimensions of musical note perception, on the other hand. Accordingly, the first section of the Chapter sets out the aspects of note perception to be explained. The second section describes the three important acoustic properties of musical notes as they pertain to note perception. The third section explains the relationship between the physical variables of note production (length, mass, tension, etc) and the acoustic variables observed in the sounds. The final section presents a set of novel melodies to illustrate the interaction of the two acoustic scale variables in determining the qualities of the instrument or voice producing the musical tones.

## Instrument families, registers within families, and melodies

The discussion of note perception focuses on the sounds produced by the sustained-tone instruments of the orchestra and chorus, that is, the families of instruments referred to collectively as brass, strings, woodwinds, and voice. Table I shows four of the instruments in each of the families, ordered in terms of their register, or voice. With just a little training, most people can identify these sixteen instruments from a simple melody (van Dinther and Patterson 2006). From this perspective, the purpose of the chapter is to explain how auditory perception enables us to make these distinctions.

Figure 2 – Sixteen common instruments illustrating four registers within each of four instrument families

When members of different instrument families play the same note - say a trombone, a cello, a bassoon, and a baritone all produce their version of C3 (the C below middle C on the keyboard) - it is the pitch of the notes that is the same and it is the timbre of the notes that is different. This is the traditional distinction between the perceptual variables pitch and timbre. The instruments of a given family have similar physical shapes, they are made of similar materials, and they are excited in similar ways, so it is not surprising that the instruments of a family produce a category of notes with a distinctive, unifying, sound quality, or timbre. The categories of timbre associated with instrument families are labelled with words that describe some physical aspect of the source. So, the trumpet is a brass instrument, the clarinet is a wood-wind instrument, and the violin is a string instrument. The family aspect of timbre is largely determined by the shape of the envelope of the magnitude spectrum and the pitch is largely determined by the repetition rate of the note. So, these aspects of note perception would appear straightforward. Within a family of instruments, the different members are distinguished physically by their size and perceptually by their register: soprano, alto, tenor, baritone or bass. Register is a very important aspect of note perception, but it is not as straightforward as pitch and instrument family because there are two separate aspects of instrument size involved in determining the perceived register of an instrument.

The first aspect of instrument size associated with the perception of register is the size of the source that activates, or excites, the instrument. In the string family, for example, it is the bow pushing on an individual string that excites the instrument, and it is primarily the length, mass, and tension of the string that determine the note that is produced. Together, these physical variables determine the repetition rate of the vibration that the instrument produces. Repetition rate is one of three acoustic properties that are important in musical note perception. It is a major determinant of the perceived register of an instrument, as well as the prime determinant of the pitch of a note. Register and pitch are two different aspects of note perception, and this is one of the distinctions that the chapter is intended to explain. At this point, it is sufficient to note that the musician varies the length of the string by depressing it against the neck of the instrument to vary the repetition rate and produce a melody. The three physical variables, length, mass and tension, are used by instrument makers to set the four open-string notes on each instrument, and the repetition rates of the open-string notes play a large role in establishing the register of the instrument.

The second aspect of instrument size associated with the perception of register is the size of the resonators in the body of the instrument; together, the resonators filter the pulse stream produced by the source. In the string family, the filter is primarily the bridge and face plate in conjunction with the volume of air in the body of the instrument and the f holes in the face plate. Acoustically, the resonances of the filter give the frequency spectrum of the note a distinctive shape which is closely associated with the sound quality, or timbre, of the instrument family. Spectral shape is the second of the three acoustic properties that are important in musical note perception. The spectrum also contributes to the perception of register. Within a family, if the spectrum is plotted on a logarithmic frequency axis, the distribution of activity shifts towards the origin without changing its basic shape, as the size of the instrument increases. Thus, the position of the spectral pattern contributes to the perception of instrument register, and the position of the spectral pattern is the last of the three acoustic properties that are important in musical note perception.

In general terms, then, the purpose of this chapter is to describe how the physical variables of note generation are related to the acoustic variables of notes as sounds, and how these acoustic variables are related to the perception of melodies, the perception of instrument family, and the perception of instrument register.

## Pulse-resonance sounds and acoustic scale

The acoustic variables that are important to the discussion of music perception are conveniently illustrated with the sustained notes produced by singers, that is, sustained vowels. This also provides an opportunity to introduce the voice as a family of musical instruments. The waveform and spectrum of a synthetic /a/ vowel, like that spoken by a child, are presented in the upper and lower panels of Fig. 1, respectively. The waveform shows that a vowel is a stream of glottal pulses, each of which is accompanied by a decaying resonance that reflects the filtering of the vocal tract above the larynx. The set of vertical lines in the lower panel of Fig. 1 shows the long-term magnitude spectrum of the sound, and the dashed line connecting the tops of the vertical lines shows the spectral envelope of the vowel. In this example, the glottal pulse rate (GPR) is 200 pulses per second (pps), so the time between glottal pulses in the upper panel of Fig. 1 is 5 ms, and the spacing between the harmonics that form the fine structure of the spectrum in the lower panel is 200 Hz. The resonances in the spectral envelope are the formants of the vowel; the shape of the envelope in the spectral domain corresponds to the shape of the damped resonance in the time domain.

When children begin to speak they are about 0.85 meters tall, and as they mature their height increases by about a factor of two. The GPR of the voice decreases by about an octave as the child grows up and the vocal cords become longer and more massive. The decrease is greater than an octave for males and less than an octave for females; on average it is more than an octave but much less than two octaves. A person's vocal tract length increases in proportion to their height (Fitch and Giedd 1999; Turner et al. 2009 , their Fig. 4), and so the formant frequencies of children's vowels decrease by about an octave as they mature (Lee et al. 1999; Turner et al. 2009).

The effect of growth on the spectrum of a vowel is quite simple to characterize, provided the spectrum is plotted on a logarithmic frequency scale. In this case, the set of harmonics that define the fine structure of the spectrum simply moves, as a unit, towards the origin as the child matures into an adult, by somewhat more than one octave. In speech, the pattern of formants that defines a given vowel type remains largely unchanged as people grow up (Peterson and Barney 1952; Lee et al. 1999; Turner et al. 2009). So the shape of the spectral envelope does not change as a child grows up, the envelope just shifts towards the origin by about an octave, without changing shape.

These shifts in the fine structure and the envelope of the vowel do not change our perception of the vowel type; in the current example, the vowel remains an /a/, and does not change to an /e/, an /o/ or an /u/, as the size of the singer changes. The shifts do, however, have systematic effects on our perception of who is singing and what melody they are singing, so they are important variables in the perception of music. In acoustic terms, 'the position of the fine-structure of the spectrum on a logarithmic frequency scale' is the acoustic scale of the fine structure, and it is closely associated with the physical variables that control the source of excitation - the vocal folds - and the rate at which they vibrate. The position of the fine structure is a property of the sound as it occurs in the air (Cohen, 1993) - a property that appears in the time domain as the Glottal Pulse Rate (GPR) of the speaker. For brevity, it will be referred to as 'the scale (S) of the source (s)' and designated Ss. Similarly, in acoustic terms, 'the position of the envelope on a logarithmic frequency scale' is the acoustic scale of the set of resonances in the vocal tract that produce the formants in the spectrum; this is another property of the sound, and in this case, the most relevant physical variable is the speaker's vocal tract length (VTL). For brevity, this property will be referred to as 'the scale (S) of the filter (f)' and designated Sf. Turner et al. (2009) have recently reanalysed several large databases of spoken vowels and shown that almost all of the variability in formant frequency data that is not vowel-type information is Sf information.

When vowel type changes, e.g., from /a/ to /i/, the shape of the envelope changes, so the different vowel types are, in some sense, like different instrument families; however, the set of vowel types is like a cluster of families that is, nevertheless, distinct from the other musical families. In auditory terms, since different vowels can be sung by the same person on the same note, vowel-type is like family timbre, and since singers of with different vocal registers, e.g., alto and baritone, can sing the same vowel, vocal register is essentially the same as instrument register.

## Pulse-resonance sounds and the source-filter model of musical sounds

This section of the chapter reviews the "source-filter" model of musical sound generation as it pertains to several families of musical instruments. The sustained-tone instruments of the orchestra produce pulse-resonance sounds whose basic properties are similar to the vowel illustrated in Fig. 1; that is, a source within the instrument produces a temporally regular stream of pulses and, then, resonators in the body of the instrument filter the pulses and introduce the spectral resonances that give the note, and thus, the instrument, its distinctive timbre. In this section, the source-filter model is used to illustrate the relationship between the physical properties of the instrument, on the one hand, and the three main acoustic properties of these sounds, on the other hand.

### The source of excitation and the acoustic variable, Ss

In general terms, the source in these instruments is a highly nonlinear resonant system that produces a temporally regular stream of acoustic pulses similar to a click train. The systems themselves are quite diverse. For example, in the voice, it is the vocal folds; in brass instruments, it is the lips coupled to the mouth piece; in the woodwinds it is the reed coupled to the lips; and in string instruments, it is the bow coupled to one of the strings. Despite the diversity of mechanisms, all of the sources produce streams of very precise pulses and this virtually ensures that the waves produced by sustained-tone instruments are pulse-resonance sounds, like vowels. [In Fourier terms, the overtones of the pulse rate are locked to the pulse times both in frequency and phase up to fairly high harmonic numbers.]

The acoustic scale of the source of excitation, Ss, is the repetition rate of the wave as it occurs in the air between the instrument and the listener. Ss is determined by physical properties of the instrument, like length and mass, which are not themselves acoustic variables. Ss largely determines the pitch we hear, but Ss is not an auditory variable. It is an intervening, acoustic variable that describes a property of the sound in air, and it should be distinguished from pitch which is the auditory variable of perception. The relationship between Ss and the physical variables of the instrument will be illustrated by comparing how Ss is determined in the vocal tract and in stringed instruments.

#### The source of excitation in the human voice

The vocal folds produce glottal pulses in bursts and, although the vocal folds are rather complicated structures, the effect of the physical variables on the rate of pulses can be calculated using the expression for a tense string. The glottal pulse rate, GPR, is largely determined by the length, L, mass, M, and tension, T, of the vocal folds, and the form of the relationship is

$\mathrm{GPR} \thicksim \sqrt{\frac{T}{M 4 L}}$(1)

Two of these physical variables are determined by the size of the person - the length and mass of the vocal folds. Both of these variables increase as a child grows up, and both of these terms are in the denominator on the right-hand side of the equation, so as the child increases in height the pitch of the voice decreases. The average GPR for small children is about 260 Hz, both for males and females. For females GPR just decreases with height throughout life dropping to, on average, about 160 Hz in women. For males, GPR decreases with height until puberty at which point the vocal folds suddenly increase in mass and the GPR drops to, on average, about 120 Hz in men. So it is the length and mass of the vocal folds that primarily determine whether a person is a soprano, alto, tenor, baritone or bass.

To produce a melody within their specific register, a singer varies the tension of the vocal folds. So learning to sing in tune is largely a matter of learning to control the tension of the vocal folds, holding the tension fixed during sustained notes and changing it abruptly between notes. Tension is in the numerator of the expression, and so as you increase the tension, you increase the GPR. There is considerable overlap in the note ranges of the vocal registers; in fact, the highest note of a bass is typically a note or two above the lowest note of a soprano. Finally note that the effect of all of these variables is restrained by the fact that it is the square root of the product of the physical variables that determines the specific GPR value. So, for example, a singer has to change the tension of the voice by a factor of sixteen to produce a two octave range of notes.

In summary, for a specific individual, their size, in the form of the length and mass of their vocal folds, determines their long-term average GPR, and the Ss component of the register of their voice. Varying the GPR to produce a melody involves varying the tension of the vocal folds, and thus, it also changes Ss. So the long-term average GPR, calculated over a sequence of musical phrases determines the register of the singer's voice, and short term changes in Ss are used to produce melodies.

#### The source of excitation in the string family

The excitation mechanism in stringed instruments is the string pushed by the bow. As the musician draws the bow across a string, the string is pushed or pulled away from its resting position until the tension becomes too great and then it snaps back to its resting position, vibrating in the process. The string and bow actually produce a coupled system that includes the musician, and over the first 50 ms or so of the note, the resonant properties of the string force the system to produce a standing wave. The shape of the cycle is unusual; it is the offset of each cycle that is abrupt rather than the onset, but the result is, nevertheless, a pulse-resonance sound. Although the bow-string system is rather complicated physically (McIntyre, 1983), the relationship between the rate of pulses and the main physical variables is the same as for the vocal folds, namely,

$\mathrm{PR} \thicksim \sqrt{\frac{T}{M 4 L}}$(2)

In this case, however, T, M, and L are the Tension, Mass and Length of the string, rather than the vocal folds. The two physical variables associated with the size of the source (the length and mass of the string) are the most important variables in this family of instruments and they each have two roles to play. Consider first the pulse rates of the open-strings on these instruments, and note that both the mass and length terms are in the denominator on the right-hand side of the equation, so increases in size, be they length or mass, lead to decreases in pulse rate. For a given member of the family (violin, viola, cello or contra bass), the length of the four strings is fixed, and as the size of the instrument increases, the string length gets longer in large steps. As a result, string length plays an important role in determining the register of the instrument. The mass of the string increases with its length, so it also contributes to the perceived register. But mass also plays another important role within the set of strings on an individual family member; the mass is varied across the strings to produce different note ranges on the four strings. Finally, the musician varies the length of individual strings to get the different notes within that string's range.

Instrument makers are very adept at using mass and length to vary the pulse rate of notes within a family. If a musician shortens the lightest string on the largest instrument, the contra base, to a point near the end of the neck, the pulse rate of the note will actually be a little higher than the pulse rate of the open-string note of the heaviest string on the smallest member of the family, the violin. These are the notes just below middle C on the keyboard.

#### Excitation mechanisms of the woodwind and brass instrument families

The excitation of woodwind and brass instruments is described in terms of fluid mechanical 'valves' that momentarily close the flow of air through the instrument. The closure causes a sharp acoustic pulse which resonates in the body of the instrument. For woodwind instruments, the valve is the reed in conjunction with the lips. For brass instruments, the source is not clearly localised within the instrument. The source of energy is the stream of air produced by the player who controls the pressure with the tension of the lips. The source of excitation is pulsatile because the mouthpiece is coupled to the tube between the mouthpiece and the bell (i.e. the body of the instrument), and the tube can only resonate at certain frequiencies. Thus, the pulses originate from the lips, but the pulse-rate is effectively determined by the length of the instrument, and the length is varied by the valves or slide to control the pulse rate of the note. In any event, these two families of instruments also produce pulse-resonance sounds in which the acoustic scale of the source controls the repetition rate of the note, and thus, the Ss component of the instrument's register. The pulsive nature of the excitation generated by these systems, and the temporal regularity of the pulse stream, mean that the dominant components of the spectrum are strictly harmonic and they are phase locked (Fletcher and Rossing 1998). Fletcher (1978) provides a mathematical basis for understanding what is referred to as mode locking in musical instruments. Detailed descriptions of the mechanisms are provided in, for example, Benade (1976), Fletcher (1978), and McIntyre et al. (1983); a brief overview is provided in van Dinther and Patterson (2006).

#### Summary of the role of Ss in the perception of melody and register

Comparison of the excitation mechanisms for the different instrument families shows that the mechanisms are similar, inasmuch as they all produce regular streams of sharp pulses and the pulse rate is affected in the same way by the size of the components in the source; specifically, the pulse rate decreases as the size of the components increases. At the same time, the method whereby the pulse rate is varied to produce a melody is fundamentally different: the variable that controls pulse rate in the voice is the tension of the vocal folds, and the singer increases the tension to increase the pulse rate; whereas the variable that controls pulse rate in string instruments is string length, and the musician decreases the length to increase the pulse rate. The brass and woodwind instruments are like the strings, inasmuch as the pulse rate is varied to produce a melody varying length rather than tension; they are different from the strings inasmuch as the length in this case is tube length rather than string length.

This brief overview of excitation mechanisms is intended to illustrate that, although different instrument families employ very different mechanisms to produce a regular stream of sharp pulses, and it is important for musicians to understand something of these mechanisms in order to play their instruments properly, nevertheless, all of these instruments produce pulse-resonance sounds, and the melody information in music is a sequence of pulse-rate values that specify the momentary acoustic scale of the source of excitation. Although the relationship between the physical variables involved in instrument excitation and the repetition rate of a given note is complex, the relationship between the acoustic variable, Ss, which summarizes the action of the source, and the pitch we perceive is straightforward. Indeed, the relationship is so direct that we usually use the units of acoustic scale, Hz or pulses per second, as the units for the auditory variable, pitch.

### The filtering of the excitation pulses and the acoustic scale of the filter, Sf

Each of the pulses produced by the excitation mechanism of a sustained-tone instrument is filtered by body resonances within the instrument, and it is these body resonances that produce the resonances we observe following the pulses in the waveform. The resonances also produce the distinctive shape of the envelope in the frequency domain, and ultimately, the timbre of the instrument family. For stringed instruments, the prominent resonances are associated with the plates of the body (wood resonances), the body cavities (air resonances), and the bridge (Benade 1976). For brass and woodwind instruments, the prominent resonances are associated with the shape of the mouthpiece, which acts like a Helmholtz resonator, and the shape of the bell which determines the efficiency with which the spectral components radiate into the air (Benade and Lutgen 1988). Woodwind instruments have a tube resonance like brass instruments, but the filtering is complicated by the 'open-hole cutoff frequency' for woodwinds. The dominant resonances of speech sounds are determined by the shape of the vocal tract (Chiba and Kajiyama 1941; Fant 1960). So, just as there are many source mechanisms for generating the pulse stream, so there are many systems of body resonances to produce distinctive spectral envelopes.

Within a family, the most prominent distinction between the members of the family is the size of the body of the instrument, and the primary effect of instrument size on the perception of register is straightforward: If the size of an instrument is changed while keeping its shape the same, the result is a proportionate change in Sf, the acoustic scale of the filter mechanism in the body of the instrument. That is, if the three spatial dimensions of an instrument are increased by a factor, a, keeping the materials of the instrument the same, the natural resonances decrease in frequency by a factor of 1/a. The shape of the spectral envelope is preserved under this transformation, and so, if the spectral envelope is plotted on a log-frequency axis, the envelope shifts as a unit towards the origin, without changing shape, and the change in Sf will be the log of the relative size of the two instruments. This uniform scaling relationship is called 'the general law of similarity of acoustic systems' (Fletcher and Rossing 1998), and it is used to produce much of difference in Sf between instruments. Numerical examples illustrating how the spatial dimensions of an instrument affect its resonances are provided by van Dinther and Patterson (2006) for two specific forms of resonator, Helmholtz resonators and flat plates.

Comparison of the filter systems of the different instrument families shows that the spectral envelope is affected in the same way by changes in the size of the filter-system components; specifically, the resonant frequencies decrease as body size increases and so the spectral envelope shifts towards the origin as the sizes of the components increase. So size affects the filter system in the same way as it affects the excitation mechanism, simply because bigger things vibrate more slowly. The wood-plate and bridge resonances of the string-family filter system are complex, and they are fundamentally different from the bell and mouth piece resonances of the brass-family filter system, which are also complex. Despite the complexity of the relationship between the physical variables involved in body filtering and the shape of the resultant spectral envelope, the relationship between the acoustic properties and the perception of the notes is fairly straightforward; the shape of the envelope determines the sound quality, or family timbre, the acoustic scale of the filter, Sf, summarizes the filter component of the perception of register. In all of these instrument families, the register decreases from soprano to base as instrument size increases and the spectral envelope shifts toward the origin.

### Acoustic scale and register range

In sections 3.1 and 3.2, the relationship between the physical variables involved in the production of musical notes, and the acoustic scale of the source and filter, was presented in theoretical terms without reference to the practicalities of constructing and playing instruments. It turns out, that it is not possible to simply scale the spatial dimensions of instruments to achieve registers ranging from soprano to base in most instrument families; the base member would be too large and the soprano member to small. This section reviews the spatial scaling problem, and describes how the instrument makers produce notes with a wide range of acoustic scale without using excessively large or small instruments. The spatial scaling problem arises from the desire to simultaneously satisfy three design criteria for families of sustained-tone instruments:

The first criterion is that instruments should produce notes which are heard to have a strong musical pitch whose clarity and salience provide for effortless communication of novel melodies. This places an important constraint on the relationship between the acoustic scale variables, Ss and Sf. The instrument's filter system must resonate at frequencies corresponding to the first eight harmonics of the pulse rate of each note that the instrument is intended to play; that is, the instrument must emit significant amounts of acoustic energy in the range from the pulse rate of each note to three octaves above that pulse rate. This is necessary because the pitch of notes where the energy is carried by harmonics above about the tenth is not sufficiently salient to support accurate perception of novel melodies (Krumbholz et al. 2000; Pressnitzer et al. 2001). The second criterion is that the members of each instrument family should, together, produce notes that cover a significant portion of the musical scale, which for the keyboard encompasses about seven-octaves from about 27.5 - 3520 Hz. When combined with the first criterion, the second criterion effectively requires that the instruments of a given family have matched Ss and Sf values for all of the registers in the range from soprano to bass. This is a very demanding constraint, particularly when combined with the third criterion, which is that the instruments should be playable and portable. This last, practical constraint places limitations on the sizes of instruments which, in turn, means that the desired range of notes cannot be achieved by simply scaling instrument size in accordance with the law of acoustic similarity.

There are problems for the instrument maker at both ends of the register range. For example, in the string family, there is a limit to how short the neck can be on the violin if the contact points for the notes are to be spaced far enough apart for a musician to play the instrument accurately and quickly. And at the other end of the range, if the instrument maker attempts to scale up the soprano version of the family to provide the bass member, the instruments become too large to play and too large to carry. Hutchins (1967, 1980) described the problems encountered when you try to construct a family of eight stringed instruments covering the entire range of orchestral registers based on the properties of the violin. The double bass member of the family would have to be six times the size of the violin, if simple scaling of instrument dimensions were to be used to provide a shift of six octaves in the spectral envelope. The length of a violin is about 0.6 m, so the double bass in this hypothetical family would have to be 3.6 m tall. The lower notes on the strings of such a double bass would not be reachable for most musicians and the instrument would not be portable. So, although instrument makers scale the dimensions of instruments to achieve much of the desired change in Ss and Sf, it is not possible to use the scaling of spatial dimensions, on its own, to provide the full range of registers in each family, and at the same time, ensure that the pitch of each note is sufficiently strong to support novel melody perception.

So, how do the instrument makers construct families of instruments that produce notes with salient pitches over the full range of registers from soprano to bass, and which are, at the same time, playable and portable? The first criterion of instrument production is immutable; the instrument must produce energy in the first three octaves of the pulse rate if the note is to have a well defined pitch. The third criterion is essential; the instruments have to be playable and portable. So how do the instrument makers provide such a wide range of notes on instruments with manageable sizes? This is where the knowledge and craft of the instrument maker come to the fore. What is required is not that the soprano instruments be excessively small and the bass instruments be excessively large; what matters is that the instruments produce notes with a wide range of Ss and Sf values, and that the Ss and Sf values are coordinated throughout the range. So what the musicians have done is find ways of extending the range of Ss and Sf, beyond what is practical with spatial-dimension scaling, by adjusting other physical properties of the instruments such as the mass of the strings, the thickness of the plates or the depth of the whole instrument. They scale the physical dimensions of the family so that the largest member is portable and the smallest member is playable, and then they adjust other physical properties of the instrument to achieve the desired acoustic scale values for the source mechanism and the filter system (e.g. Schelleng 1963).

With regard to the source of excitation in the string family, the strings on the larger members are not as long as the law of acoustic similarity would require because it would make the instruments unwieldy. The instrument makers increase the linear mass of the strings (the mass per meter) by increasing their diameter and by winding metal coils around the string. The increased linear mass causes the strings to vibrate more slowly as dictated by equation 2. The instrument makers use a change in mass to obtain the lower ranges of notes on the lower strings of any given member of the family. Comparison of the top and bottom strings on the violin will be used to illustrate the point. The highest string is tuned at a pulse rate of 660 pps (E5) while the lowest one is tuned at 196 pps (G3) -- a pulse-rate ratio of 3.4. The typical linear mass for the E5 string is about 0.4 g/m, and its tension is generally around 77 N. For the G3 string, these values are 2.4 g/m and 40 N. The pulse rates of the two strings can be calculated from equation 2, and it is found to be 3.4, as required.

With regard to the filter systems in the string family, the filter systems of the larger members of the family are not as large as the law of acoustic similarity would require, because it would make the instruments too heavy and too large. The instrument makers adapt the characteristics of the instruments to preserve the sound quality while making them usable at the same time. The main resonance is driven by the cavity mode of the body which then work as a Helmoltz resonator. The volume of the instrument as well as the surface of the f holes are then the key parameters. The open strings of the cello are tuned to pulse rates three times lower than those of the violin. However, the plates of the body are only 2.1 times larger than those of the violin (Schelleng 1963), while the rib height of the cello is about four times that of the violin (Fletcher and Rossing 1998). The volume of the cello is, then, 17 times larger than that of the violin; this is equivalent to uniform spatial scaling by a factor of 2.6. To lower the body resonances to the desired values, the instrument makers vary the mass, thickness and arching of the body plates. Specifically, the body plate of the cello is made proportionally thinner than that of the violin which lowers the body resonance frequency (Molin et al. 1988).

Finally, note that that the range of notes covered by the vocal registers, from soprano to bass, is only about four octaves in total (from about C6 down to a little over C2), while the string family covers almost seven octaves (from just under C8 to just over C1. The singing teacher can help a vocalist strengthen notes at the end of their natural range, but they cannot stretch the vocal tract length or add significant mass to the vocal folds. (As an aside, when you have a cold, the phlegm adds mass to the vocal cords which is why you can hit lower notes than normal for a few days.)

### Acoustic-scale 'melodies' and the perception of pitch and timbre

The discussion of pitch and timbre in the remainder of the chapter is more readily understood when presented in terms of 'melodies' in which the acoustic scale values of the notes, Ss and Sf, vary according to the diatonic scale of Western music. These melodies and the discussion are presented on the Acoustic-scale melodies page to keep the current page under the 50k wiki limit.

## Conclusions

The discovery that acoustic scale is a basic property of sound (Cohen 1993) leads to the conclusion that the major categories of timbre (vowel type and instrument family) are determined by spectral envelope shape, and that these categories of timbre are relatively independent of both the acoustic scale of the excitation source and the acoustic scale of the resonant filter. In speech, the acoustic scale variables, Ss and Sf, largely determine the voice quality of the speaker, and thus our perception of their sex and size (e.g., Smith and Patterson, 2005). With regard to timbre, this suggests that when dealing with tonal sounds that have pronounced resonances like the vowels of speech, it would be useful to distinguish between aspects of timbre associated with the shape of the spectral envelope, on the one hand, and aspects of timbre associated with the acoustic scale variables, Ss and Sf, on the other hand. This would lead to a distinction between the 'what' and 'who' of timbre, that is, what is being said, and who is saying it. This kind of distinction would at least represent progress towards a more informed use of the term timbre.

## Acknowledgements

Research supported by the Medical Research Council (G0500221; G9900369)

## References

• Benade AH (1976) Fundamentals of Musical Acoustics. Oxford University Press. [1] [2]
• Benade AH, Lutgen SJ (1988) The saxophone spectrum. J Acoust Soc Am 83:1900-1907. [1]
• Chiba T, Kajiyama M (1941) The vowel, its nature and structure. Tokyo-Kaiseikan Pub Co., Tokyo. [1]
• Cohen L (1993) The scale representation. IEEE Trans. Sig. Proc. 41:3275-3292. [1]
• Fant G (1960) Acoustic Theory of Speech Production. Mouton De Gruyter, The Hague. [1]
• Fitch WT, Giedd J (1999) Morphology and development of the human vocal tract: A study using magnetic resonance imaging. J Acoust Soc Am 106:1511-1522. [1]
• Fletcher NH (1978) Mode locking in nonlinearly excited inharmonic musical oscillators. J Acoust Soc Am 64:1566-1569. [1] [2]
• Fletcher NH, Rossing TD (1998) The Physics of Musical Instruments. Springer, New-York. [1] [2] [3]
• Hutchins CM (1967) Founding a family of fiddles. Phys Today 20:23-37. [1] [2]
• Hutchins CM (1980) The new violin family. In: Benade AH (ed), Sound Generation in Winds, Strings, Computers. The Royal Swedish Academy of Music, pp.182-203. [1]
• Krumbholz K, Patterson RD, Pressnitzer D (2000) The lower limit of pitch as determined by rate discrimination. J Acoust Soc Am 108:1170-1180. [1]
• Lee S, Potamianos A, Narayanan S (1999) Acoustics of children's speech: developmental changes of temporal and spectral parameters. J Acoust Soc Am 105:1455-68. [1] [2]
• McIntyre ME, Schumacher RT, Woodhouse J (1983) On the oscillations of musical instruments. J Acoust Soc Am 74:1325-1345. [1]
• Molin NE, Lindgren L-E, Jansson EV (1988) Parameters of violin plates and their influence on the plate modes. J Acoust Soc Am 83:281-291. [1]
• Peterson GE, Barney HL (1952) Control Methods Used in a Study of the Vowels. J Acoust Soc Am 24:175-184. [1]
• Pressnitzer D, Patterson RD, Krumbholtz K (2001) The lower limit of melodic pitch. J Acoust Soc Am 109:2074-2084. [1]
• Schelleng JC (1963) The Violin as a Circuit. J Acoust Soc Am 35:326-338. [1] [2]
• Turner RE, Walters TC, Monaghan JJ, Patterson RD (2009) A statistical, formant-pattern model for segregating vowel type and vocal-tract length in developmental formant data. J Acoust Soc Am 125:2374-2386. [1] [2] [3] [4]
• van Dinther R, Patterson RD (2006) Perception of acoustic scale and size in musical instrument sounds. J Acoust Soc Am 120:2158-76. [1] [2] [3]


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# Is there a “jagged” real-valued function that is “smooth” in cardinalities greater than the reals?

My background: I have a bachelor's CS degree and have never taken anything beyond part of a first course in abstract algebra - no real analysis or complex analysis. I learned about higher cardinalities than $R$ in my automata course.

The inverse of addition works on integers.

The inverse of multiplication forces us to get to the rationals, but that's still the same cardinality.

The inverse of exponentiation forces us out of the rationals and into complex numbers, but we're still at integer cardinality because it's algebraic.

Consider $a^x$ is transcendental at least for $a>1$ and when given imaginary variables (at $x=0$ it will look like a sine wave, probably different period than $sin(x)$, which is associated with e^x, along the imaginary part of y axis) needs a real cardinality.

Therefore, $x^x$, the next level up from exponentiation is going to be at least real-valued in cardinality, as well. By simply increasing the "level" of my operation, I bumped my way out of the integers.

My intuition says if I keep bumping up my operation, eventually I may have a function such that if I need a cardinality greater even than the reals to smooth it out.

Is there a "jagged" real-valued function that is "smooth" in cardinalities greater than the reals?

I'd say something is "smooth" if it is not "flat" and if it always approaches being "flatter" as you "zoom in". Flat would mean minimum distance and flatter would mean approaches without overshooting. Like $sin(x) * e^x$ and $e^x$ both "approach" 0 (in the the range necessary to contain all values towards infinity strictly decreases) as x grows arbitrarily negative, even though in the first case, it "overshoots" and crosses the $x$-axis.

Just like the reals are in between the integers that fill in the gaps of some functions, I wondered if there were numbers in between the reals to fill in the gaps of some functions, but I decided to try to word it a little more precisely.

In response to "How can a function on the reals be extended to a completely different set? And how do you measure it?":

Consider all finite non-empty sequences of $1$'s and $0$'s that start with a $1$, call it $S_1$. This is all positive integers base-2.

Now consider all proper subsets of that set, call it $S_2$.

Now consider $f_1$ that takes in an element of $S_2$ and returns a string whose $i$th digit is $1$ if the base-2 representation of $i$ is in $S_2$, $0$ otherwise.

Now consider the elements of $S_3 = \{ (s_1, f_1(s_2)) | s_2 \in S_2, s_1 \in S_1 \cap \{0\} \}$. This will correspond to all non-negative real numbers. The first thing in the tuple will be the part of the number to the left of the decimal. The second thing in the tuple will be the decimal part.

Is $S_1$ a subset of $S_3$? We can map $S_1$ to $S_3$ by saying elements $s_1$ in $S_1$ correspond to the elements in $S_3$ that have $s_2$ as the empty set and have $s_1$ as the first thing in their tuple. In other words integers get mapped to the reals that have all $0$'s as their decimal portion.

Now consider all proper subsets of $S_3$, call it $S_4$.

Now consider the elements of $S_5 = \{ (s_3, s_4) | s_3 \in S_3, s_4 \in S_4 \}$. This is our greater-than-the-reals set (non-negative reals), and we say the reals are a subset of this set by saying they correspond to the elements of $S_5$ where $s_4$ is the empty set.

I can give a partial definition of measuring by assuming $s_3$ is greater than $t_3$ and that $s_4$ is a superset of $t_4$ and say their distance is $(s_3 - t_3, s_4 - t_4)$. I don't know what to do if $s_3 > s_4$ and $t_3 \not \supset t_4$ because you need some way to "borrow" like in basic arithmetic if you try to remove a set that isn't there.

• Your writing is a bit imprecise, but in your last sentence you essentially ask about there being gaps in the real numbers, and the answer is that there are not. This is due to the completeness of the real numbers. – Santeri Apr 15 '16 at 23:19
• @Santeri "In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number." Well that only works if you assume your infinite string is $R$ in cardinality. If I choose $Z$ $0$'s and $1$'s I can put those in a real number (say the binary decimals in $[0, 1)$). I don't think I can fit $2^Z=R$ 0's and 1's into a real number in the same way I can't fit $Z$ 0's and 1's into $Q$ or $Z$. I would need $2^R$ to do that. Also completeness from that article seems like it's an axiom. – Words Like Jared Apr 15 '16 at 23:50
• Simply requiring that exponentiation have an inverse doesn't necessitate moving to the real numbers. You just need to take the algebraic closure of the rationals, which is the field of algebraic numbers. This field is still countable. There is nothing inherently algebraic about the reals, and algebra will not lead you to them in a natural way. Most constructions either require an order relation or a topology. – Matt Samuel Apr 15 '16 at 23:56
• @MattSamuel I see what you're saying. Division and multiplication of integers needs only 2 integers to represent. In general, as long as any finite sequence of operations (and finite number of operations) result in a finite number of integers to represent the result (and a countably infinite input set), it will still be the integers in terms of cardinality. I guess that doesn't change my question - I was just trying to construct a candidate example, although it may be generalized inverses of addition, multiplication, exponentiation, etc. may not take me there... – Words Like Jared Apr 16 '16 at 0:09
• @Santeri I do see, now, how my writing even was more imprecise than I realized at the time. I was conflating the decimal notation of something (infinite length for irrationals) with other possible encodings, like the algebraic field works with. If I took a finite number of steps, $k$, to get to my result, I can represent those steps as a $k$ tuple which will be $Z$ in cardinality. – Words Like Jared Apr 16 '16 at 0:16

The functions of analysis have ranges that are within $\Bbb C$ (often $\Bbb R$), so you will not need a larger set to capture the range. Even if they are wildly discontinuous, the range still $\Bbb C$.

In your effort to make functions continuous by adding points, it seems you are motivated by the fact that any function on the integers can be made continuous by defining it properly on the reals between the integers. A critical difference is the topology of the sets. The integers are discrete, so you can define intervals like $(0.1,0.6)$ that do not contain any integers. You can then define the function to do what you want between the integers to make it continuous over the reals.

The reals are dense in themselves, so as long as you use the standard $\epsilon-\delta$ definition of continuity, you cannot make a discontinuous real function continuous by adding more points.

• Towards the end of my question when I went to define how to represent the extra information necessary of these numbers by taking the cross product of the reals with its powerset; if you had $(a,b)$ and $(a,c)$, with $a \in \Bbb R$ and $b,c \in 2^{\Bbb R}$ then you'd have a range that doesn't contain any real numbers in between them. This is akin to $(0,1)$ and $(0,6)$ from your example. I would assume you're right on the "standard definition" - I was just wondering if a self-consistent "extension" could be made... and I ran into the problem of not always being able to define ordinality. – Words Like Jared Apr 17 '16 at 23:54
• Now that I think about it, we could look at the countably finite sequence of digits after your decimal point as a countably infinite polynomial. Maybe the digits of your uncountably infinite number could be defined as an integral or summation or something... You would just need some way to make the larger reals not as signifiant/independent like digits not being as significant for higher order polynomial terms. – Words Like Jared Apr 17 '16 at 23:58
• You have to think about the topology of your set as well as just defining it. Certainly you can embed the reals in a set of large cardinality, but now you have to figure out how to deal with open sets so you can talk about continuity. You also want the embedding to respect the real functions we are used to. I don't think you can make the reals discrete as a subset of the larger set while maintaining the density. – Ross Millikan Apr 18 '16 at 0:02
• I had never heard of an open set before. I've never studied Topology. Could you give me a link to what you mean by "density"? – Words Like Jared Apr 18 '16 at 0:07
• The reals being dense (so are the rationals) as an order just means that between any two there is another. Another use of density is to say that the rationals are dense in the reals which means there is a rational arbitrarily close to any real. Open sets are generalizations of open intervals in the reals. In metric spaces they contain an open ball around each of their points. – Ross Millikan Apr 18 '16 at 0:19


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# Parametric form of square

What is the appropriate parametric equation of the boundary of a square? For example, the unit circle has a parametric equation $x(t)=\cos(t)$ and $y(t)=\sin(t)$.

• $$[0,2\pi) \ni t\;\;\mapsto\;\; (x(t),y(t)) = \!\left(\frac{\cos(t)}{\rho(t)},\frac{\sin(t)}{\rho(t)}\right) \quad\text{ where }\quad \rho(t) = \max\big(|\cos(t)|,| \sin(t)|\big)$$ – achille hui Oct 17 '14 at 17:09
• The above is a parametric equation of circle not a square. Thank you. – Janak Oct 18 '14 at 7:28

Lets consider a square of size $2 \times 2$ in a rectangular coordinate system, where each side is a part of the lines $x = \pm 1, y = \pm 1$. Then you can define

$$\gamma(\alpha) = \begin{cases} (1,\tan(\alpha)) & \alpha \in [-\pi/4 , \pi/4] \\ (\cot(\alpha),1) & \alpha \in [\pi / 4, 3 \pi /4 ] \\ (-1,-\tan(\alpha)) & \alpha \in [3\pi/4 , 5\pi/4] \\ (-\cot(\alpha),-1) & \alpha \in [5\pi/4, 7\pi/4] \end{cases}$$

which describes the parametric path of square shape, with the angle as parameter.

A parametrization of a straight line from $$P$$ to $$Q$$ is $$r(t)=tP + (1-t)Q$$ as $$t$$ goes from $$0$$ to $$1$$. Four of these pieces with suitable start and stop values for $$t$$ will take you round the square for $$t$$ going from $$0$$ to $$4$$.

$$f(\alpha) = \sec(\alpha- \frac \pi2 \lfloor \frac {4\alpha + \pi}{2\pi}\rfloor )$$ Where $f$ is the function that gives the distance between the center of the square and a point of the square at the $\alpha$ angle

Giving us $$\begin{cases} \cos(\alpha) f(\alpha) \frac X2 +x_0\\ sin(\alpha) f(\alpha) \frac Y2 + y_0\\ \end{cases}$$

where X and Y the length and height respectivly and $x_0$ and $y_0$ are the coordonated of the square.