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# User:Mary Mendoza/Notebook/CHEM 571 Experimental Biological Chemistry I/2012/10/09

(Difference between revisions)
 Revision as of 16:54, 23 October 2012 (view source) (→Continuation of Chemiluminescence)← Previous diff Current revision (08:56, 7 December 2012) (view source) (→Fluorescence Spectrometer Graph) (30 intermediate revisions not shown.) Line 12: Line 12: * The sample cell used for this procedure is the ignited flame. * The sample cell used for this procedure is the ignited flame. * A gold hollow cathode tube was selected for the procedure. The wavelength of the lamp was 242.8 nm. * A gold hollow cathode tube was selected for the procedure. The wavelength of the lamp was 242.8 nm. - * Two blanks were used; water and HCl. Water was chosen since it is part of the ternary mixture. The standards provided by Dr. Miller were made with HCl. In addition, HCl ionizes better and keeps the metal in solution. + * Two blanks were used; water and HCl. Water was chosen since it is part of the ternary mixture. The standards provided by [[User:Abigail E. Miller/Notebook/CHEM571 Lab Project/2012/10/08|Dr. Miller]] were made with HCl. In addition, HCl ionizes better and keeps the metal in solution. The standards were 5, 8, 10, 15, 20, 25, 30, 40 ppm Au which appeared as clear and colorless liquid solutions. + * Measurement was done by inserting a transparent, plastic tube attached to the equipment into the solution. + * Each blank, water and HCl, was measured and followed by autozero. Next, standards 1, 2, and 3 were measured. + * The flame is blue green in its normal state. Having run the blanks, the flame remained blue green during the re-measurement of each blank. + * During the measurement of the standards, the flame was observed to turn orange. The color orange is indicative of the ions present in the solution. + * The other standards and samples were run. The calibration curve is on [[User:Puja Mody/Notebook/Chem 571: Gold Nanoparticles/2012/10/09|Puja's notebook.]] + + ==Chemiluminescence of Luminol== + * The reaction of 3-aminophthalhydrazide (luminol) and hydrogen peroxide is catalyzed by the addition of horseradish peroxidase (HRP) producing 3-aminophthalate. + * The reagents originated from the HRP Assay of aminoantipyrine (AAP) laboratory period [[User:Mary Mendoza/Notebook/CHEM 571 Experimental Biological Chemistry I/2012/10/02|10/02/2012.]] For this procedure, 4-iodophenol, hydrogen peroxide, HRP, and water will be used. The AAP will be replaced in the reaction with luminol. + * A concentration of 10 mM of luminol was used. This will be further diluted down to 3 mM upon the addition of water in the cuvette. + * The following calculations were made for luminol (FW 177.16): + + + $\frac{177.16 g}{10}$ = 17. 716 g of luminol in 1000 mL of water (.1M) + + + 17.716 g × 3 = 53.148 g of luminol in 1000 mL of water (.3 M = 300 mM) + + + $\frac{53.148 g}{10}$ = 5.3148 g of luminol in 1000 mL of water (30 mM) + + + x = $\frac{2 mL * 5.3148 g}{1000}$ = .0106296 g of luminol in 2 mL of water (30 mM) + + + * The actual amount of the yellow pastel powder of luminol transferred in 2 mL of water was 0.0108 g. Luminol was insoluble in water. Hence, 0.0729 g of the white granular solid of sodium carbonate was added into solution. The solution of luminol and sodium carbonate in 2 mL water was vortexed for faster dissolution. + * To buffer the alkalinity of sodium carbonate, the amount of 0.4373 g of sodium bicarbonate was added. + * Sodium bicarbonate precipitated in the solution. Addition of 4 mL of water diluted luminol from 30 mM to 10 mM. + * A pH paper was used to estimate the pH level of the solution since the electrode of the pH meter did not fit into the falcon tube of the solution. + * The images of the pH paper during qualitative analysis of the solution are embedded below: + + + [[Image:2012-10-09 14.17.45.jpg]] + + Image 1. The navy blue color is attributed to a pH of 10 to 11. + + + [[Image:2012-10-09 14.59.55.jpg]] + + Image 2. Below each pH paper is shown the number of drops of 1 M HCl into the solution of luminol. The light green color corresponds to the pH level of 7 to 8. + + + * The lamp of LS 55 Perkin Elmer fluorescence spectrometer was turned off. The signal was set to a function of time. The wavelength was set at 425 nm. The excitation slit was placed at 15 nm while the emission slit was placed at 20 nm. The duration of data collection was settled at 300 sec. + * The reagents were transferred into the cuvette using automatic delivery micropipets. The HRP was added last for this initiates the reaction through catalysis of hydrogen peroxide. + * The collected data are shown in the next section. + + ==Fluorescence Spectrometer Graph== + [[Image:Fluor10.09.12.png]] + + No luminescence was observed for any runs. + + The concentration of reagents are shown below the table: + + {| {{table}} + | align="center" style="background:#f0f0f0;"|'''' + | align="center" style="background:#f0f0f0;"|''HRP'' + | align="center" style="background:#f0f0f0;"|''4-iodophenol'' + | align="center" style="background:#f0f0f0;"|''luminol'' + | align="center" style="background:#f0f0f0;"|''H2O2'' + | align="center" style="background:#f0f0f0;"|''H2O'' + |- + | Molarity of Stock Solutions||2.3 μM||18 mM||10 mM||1.7 mM||neutral + |- + | Volume||30 μL||54 μL||900 μL||176 μL||1.8 mL + |- + +

## Current revision

Project name Main project page
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## Atomic Absorption Spectroscopy of Au/BSA

• A rack of Au/BSA solutions was recovered and used for the assigned Atomic Absorption Spectroscopy (AAS). AAS was used to strip the molecule to its elemental state.
• The entire solution set were observed to lack fibers even at high mole ratios where formation of fibers is typical. The date of which the solutions were made is unknown.
• The equipment used for AAS is by the make of Shimadzu AA-6200 Atomic Absorption Flame Emission.
• The sample cell used for this procedure is the ignited flame.
• A gold hollow cathode tube was selected for the procedure. The wavelength of the lamp was 242.8 nm.
• Two blanks were used; water and HCl. Water was chosen since it is part of the ternary mixture. The standards provided by Dr. Miller were made with HCl. In addition, HCl ionizes better and keeps the metal in solution. The standards were 5, 8, 10, 15, 20, 25, 30, 40 ppm Au which appeared as clear and colorless liquid solutions.
• Measurement was done by inserting a transparent, plastic tube attached to the equipment into the solution.
• Each blank, water and HCl, was measured and followed by autozero. Next, standards 1, 2, and 3 were measured.
• The flame is blue green in its normal state. Having run the blanks, the flame remained blue green during the re-measurement of each blank.
• During the measurement of the standards, the flame was observed to turn orange. The color orange is indicative of the ions present in the solution.
• The other standards and samples were run. The calibration curve is on Puja's notebook.

## Chemiluminescence of Luminol

• The reaction of 3-aminophthalhydrazide (luminol) and hydrogen peroxide is catalyzed by the addition of horseradish peroxidase (HRP) producing 3-aminophthalate.
• The reagents originated from the HRP Assay of aminoantipyrine (AAP) laboratory period 10/02/2012. For this procedure, 4-iodophenol, hydrogen peroxide, HRP, and water will be used. The AAP will be replaced in the reaction with luminol.
• A concentration of 10 mM of luminol was used. This will be further diluted down to 3 mM upon the addition of water in the cuvette.
• The following calculations were made for luminol (FW 177.16):

$\frac{177.16 g}{10}$ = 17. 716 g of luminol in 1000 mL of water (.1M)

17.716 g × 3 = 53.148 g of luminol in 1000 mL of water (.3 M = 300 mM)

$\frac{53.148 g}{10}$ = 5.3148 g of luminol in 1000 mL of water (30 mM)

x = $\frac{2 mL * 5.3148 g}{1000}$ = .0106296 g of luminol in 2 mL of water (30 mM)

• The actual amount of the yellow pastel powder of luminol transferred in 2 mL of water was 0.0108 g. Luminol was insoluble in water. Hence, 0.0729 g of the white granular solid of sodium carbonate was added into solution. The solution of luminol and sodium carbonate in 2 mL water was vortexed for faster dissolution.
• To buffer the alkalinity of sodium carbonate, the amount of 0.4373 g of sodium bicarbonate was added.
• Sodium bicarbonate precipitated in the solution. Addition of 4 mL of water diluted luminol from 30 mM to 10 mM.
• A pH paper was used to estimate the pH level of the solution since the electrode of the pH meter did not fit into the falcon tube of the solution.
• The images of the pH paper during qualitative analysis of the solution are embedded below:

Image 1. The navy blue color is attributed to a pH of 10 to 11.

Image 2. Below each pH paper is shown the number of drops of 1 M HCl into the solution of luminol. The light green color corresponds to the pH level of 7 to 8.

• The lamp of LS 55 Perkin Elmer fluorescence spectrometer was turned off. The signal was set to a function of time. The wavelength was set at 425 nm. The excitation slit was placed at 15 nm while the emission slit was placed at 20 nm. The duration of data collection was settled at 300 sec.
• The reagents were transferred into the cuvette using automatic delivery micropipets. The HRP was added last for this initiates the reaction through catalysis of hydrogen peroxide.
• The collected data are shown in the next section.

## Fluorescence Spectrometer Graph

No luminescence was observed for any runs.

The concentration of reagents are shown below the table:

 ' HRP 4-iodophenol luminol H2O2 H2O Molarity of Stock Solutions 2.3 μM 18 mM 10 mM 1.7 mM neutral Volume 30 μL 54 μL 900 μL 176 μL 1.8 mL


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Browse Questions

# Mention a characteristic feature and a function of zoospores in some algae.

Can you answer this question?

format: plain


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Characterization of integral polyhedra

A rational polyhedron $$P \subseteq \mathbb{R}^n$$ is an integral polyhedron if it is the convex hull of its integer points. That is, if $$P = conv(P \cap \mathbb{Z}^n)$$.

Equivalently, $$P$$ is integral if for any $$c \in \mathbb{R}^n$$, the program $$\max\{cx : x \in P\}$$ has an integer optimizer.

Now according to the theorems, to show that $$P$$ is integral, it merely suffices to show that for any $$c \in \mathbb{Z}^n$$, the program $$\max\{cx : x \in P\}$$ has an integer optimal value (whenever the optimum is finite).

Why should having an integer optimal value imply that there is an integer optimal solution? The proofs that I read for this use duality and give me no intuition.

Is there a better explanation of why this is true, or some kind of intuition?

• Here's an intuition. Suppose that some extreme point $x$ of the polytope (that is, a vertex of $P$) is not integral. Let $c$ be any vector such that $x$ is a unique maximizer of $c x$ in $P$. (It must exist if $c$ is an extreme point.) If $c\cdot x$ is not integer, you are done, so assume otherwise. Then let $c'$ be obtained from $c$ by a random infinitesimal perturbation of every coordinate. Then (i) the probability that $c'\cdot x$ is an integer is zero, and (ii) $x$ is also a unique maximizer of $c'\cdot x$ in $P$. So you are done. May 11, 2021 at 14:51
• @NealYoung in your explanation the perturbed vector $c'$ need not be an integer vector even if $c$ is. The OP is asking about the Edmonds-Giles theorem which is not as straightforward. A simpler version is that for each $c$ rational, $\max\{c x : x \in P\}$ is attained at an integer vector. May 11, 2021 at 15:50
• @ChandraChekuri Oh right. Okay, assume $c\in \mathbb Z^n$ above, take $c'= M c$ for a very large integer $M$. Now take any coordinate $i$ such that $x_i$ is not an integer, and obtain $c''$ from $c'$ by adding 1 to $c'_i$ (and leaving all other coordinates unchanged). Then (i) $c''x$ is not an integer, and (ii) $x$ is also a unique maximizer of $c''x$ in $P$. And $c''$ is integral. (??) May 11, 2021 at 17:48
• @NealYoung Yes, that works and is the right intuition. One has to argue that for sufficiently large $M$ the $+1$ in one coordinate does not matter much as far as the cone of directions for which $x$ is the optimum solution. May 11, 2021 at 19:36
• Okay I've tried to formalize this as an answer, below. May 11, 2021 at 19:59

Here's a proof (sketch) that doesn't explicitly use duality. More precisely, it replaces duality by a seemingly weaker (and hopefully easily believable) geometric fact, in Step 3 below.

EDIT: But, per the comment, the proof applies only to polytopes, not (unbounded) polyhedra!

Let $$P\subset \mathbb R^n$$ be any polyhedron polytope such that, for all $$c\in \mathbb Z^n$$, the (optimal) value of the LP $$\max\{cx : x\in P\}$$ is an integer.

Lemma 1. $$P$$ is integral. That is, every extreme point (vertex) of $$P$$ has integer coordinates.

Proof sketch.

1. Suppose otherwise for contradiction. That is, there exists a vertex $$x'$$ of $$P$$ with a coordinate $$x'_k$$ that is not an integer.

2. Given any cost vector $$c\in \mathbb R^n$$ such that $$x'$$ is an optimal solution to the linear program $$\max\{c\cdot x : x\in P\}$$, say that $$c\cdot x$$ is maximized by $$x'$$.''

3. We will use the following, hopefully intuitive, geometric fact: the set of vectors $$c$$ such that $$c\cdot x$$ is maximized by $$x'$$ contains a ball of positive volume in $$\mathbb R^n$$. (Draw some pictures for intuition.)

4. Equivalently, there is a vector $$c'\in \mathbb R^n$$ and an $$\epsilon>0$$ such that, for all $$d\in \mathbb R^n$$, if $$\max_i |d_i - c'_i| \le \epsilon$$, then $$d\cdot x$$ is maximized by $$x'$$.

5. Now define $$d' = c'/\epsilon$$ (scaling up $$c'$$ by $$1/\epsilon$$).

6. Then, for all $$d\in \mathbb R^n$$, if $$\max_i |d_i - d'_i| \le 1$$, then $$d\cdot x$$ is maximized by $$x'$$. (Here we use Line 4 above, and that $$d\cdot x$$ is maximized by $$x'$$ iff $$(\alpha d)\cdot x$$ is, provided $$\alpha > 0$$. We take $$\alpha=1/\epsilon$$.)

7. Define $$b$$ and $$b'$$ in $$\mathbb Z^n$$ by taking $$b_i = \lfloor d'_i \rfloor$$ for all $$i$$, and obtaining $$b'$$ from $$b$$ by increasing $$b_k$$ by 1. (Recall that $$k$$ is such that $$x'_k$$ is not an integer.)

8. Then $$b\cdot x$$ is maximized by $$x'$$ (using $$\max_i |b_i-d'_i| \le 1$$), and $$b\in\mathbb Z^n$$, so (by our assumption on $$P$$) $$b\cdot x'$$ is an integer.

9. By the same reasoning, $$b'\cdot x'$$ is an integer. So $$(b'-b)\cdot x'$$ is an integer.

10. But $$(b'-b)\cdot x'$$ equals $$x'_k$$, which is not an integer. $$~~~\Box$$

• I really like this argument and I think I understand the intuition now. Thank you. But I wanted to note: your definition of an integral polyhedron isn't quite right. For example if $P = \{(x,y) : x = 1/2\}$, then all extreme points have integer coordinates, because there are none. We can say instead that $P$ is integral if every minimal face contains an integral point. But this is an edge case and I'm sure things can be made to work. Thanks again May 13, 2021 at 6:02
• Oh, good point. This is not a full answer because the proof only applies to polytopes. I wonder if it can be extended to unbounded polyhedra. May 13, 2021 at 12:12
• Edmond-Giles theorem is for unbounded polyhedra. J. Edmonds and F. R. Giles, A min-max relation for submodular functions on graphs, Ann. Discrete Math. l:B-204 (1977). There are technicalities in handling polyhedra vs polytopes. Schrijver's books show all the details. May 13, 2021 at 15:56
• Your proof works for any 'pointed' polyhedron. That is, it works when every face of the polyhedron contains a vertex. Also, if $P$ has even a single vertex, then it is pointed. May 13, 2021 at 19:09
• The general case looks substantially different! E.g. consider some rational polyhedron $$P=\{( x_1,x_2) \in \mathbb R^2 ~|~ a_1 x_1 + a_2 x_2 = b\}.$$ Assume WLOG by scaling that $a_1, a_2\in \mathbb Z$. Then this polyhedron is integral iff $b$ is an integer multiple of $\text{gcd}(a_1, a_2)$. If it isn't, then, by taking $(c_1, c_2) = (\alpha a_1, \alpha a_2)$, where $\alpha=1/\text{gcd}(a_1, a_2)$, we get an LP $\max\{cx : x\in P\}$ with non-integer optimal value. May 13, 2021 at 20:41


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# `dy/dx = 5e^(-x/2)` Use integration to find a general solution to the differential equation

The given problem: ` (dy)/(dx) = 5e^(-x/2)`  is in form of a first order ordinary differential equation. To evaluate this, we may follow the variable separable differential equation: `N(y) dy= M(x)dx`

Cross-multiply `dx` to the other side, we get:

`dy= 5e^(-x/2)dx`

In this form, we may now proceed...

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

The given problem: ` (dy)/(dx) = 5e^(-x/2)`  is in form of a first order ordinary differential equation. To evaluate this, we may follow the variable separable differential equation: `N(y) dy= M(x)dx`

Cross-multiply `dx` to the other side, we get:

`dy= 5e^(-x/2)dx`

In this form, we may now proceed to direct integration on both sides:

`int dy= int 5e^(-x/2)dx`

For the left side, we apply basic integration property: `int (dy)=y` .

For the right side, we may apply u-substitution by letting: `u = -x/2` then `du =-1/2 dx` or `-2du= dx` .

Plug-in the values: `-x/2=u` and `dx=-2du` , we get:

`int 5e^(-x/2)dx=int 5e^(u)* (-2 du)`

` =int -10e^(u)du`

Apply the basic integration property: `int c*f(x)dx= c int f(x) dx` .

`int -10e^(u) du=(-10) int e^(u) du`

Apply basic integration formula for exponential function:

`(-10)int e^(u) du= -10e^(u)+C`

Plug-in` u=-x/2` on` -10e^(u)+C` , we get:

`int 5e^(-x/2) dx=-10e^(-x/2)+C`

Combining the results from both sides, we get the general solution of differential equation as:

`y=-10e^(-x/2)+C`

Approved by eNotes Editorial Team


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## Roxanne Tellier – We All Loved You, Frank Gutch Jr – Tributes, Tales, and Tears

Posted in Opinion, Review with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on April 29, 2018 by segarini

Yesterday I went through all of the private messages I’d shared with Frank Gutch Jr, since I’d first encountered him. It was in 2013, just after I’d begun writing this weekly column, and right from that first message, it was as though we were separated at birth.

## Frank Gutch Jr: It’s Gonna Be a Blues, Blues Christmas; A Look Back at Rich McCulley; Plus Notes

Posted in Opinion, Review with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on November 14, 2017 by segarini

I’m not a blues guy, really.  The closest I come to the real blues is B.B. King, maybe— electric guitar a must.  I listened to a little Sonny Terry & Brownie McGhee during my college years and there was always John Mayall, who seemed to have the best of the up-and-comers in his band no matter what period.  I accepted Cream as blues on certain songs, which probably tells you more about my non-blues background than anything.  Loved The Blues Project and Paul Butterfield.  Didn’t like Delta Blues.  Stevie Ray Vaughn is pretty much the best blues I have heard.  That should give you an idea of where I am coming from and where I am going because I am going to review some blues albums I enjoyed this year and tell you why.  They will be short and a couple even have stories behind them.

## Frank Gutch Jr: Rain Perry, Mark Hallman (The Shopkeeper), and Congress House Studio; Spotify Once Again; and Notes Hitting the Spot

Posted in Opinion, Review with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on June 27, 2017 by segarini

You can file this one under “and I thought I knew something.”  I just watched a documentary which starts “When I was a kid, music was everything,” a statement as acute to me as author Scott Turow‘s line “It suddenly hit me how much I missed music for which I once felt a yearning as keen as hunger.”  It struck a note so deep in me that I watched  all one-hour-and-thirty-one minutes feeling a kinship with the narrator (and, as it turns out, producer of the film), almost relieved that I was not alone.

## Frank Gutch Jr: Product Placement and the Things We Sometimes Miss, Plus Notes…..

Posted in Opinion, Review with tags , , , , , , , , , , , , , , , , , , , , , , , , on December 15, 2015 by segarini

You’re sitting there watching television and all of a sudden music starts seeping into your subconsciousness and you think what the hell, because some elephant has entered the room but you can’t see it, but it dawns on you that you can hear it.  You think.

## Frank Gutch Jr: Read It Now: A Look at the Edward R. Murrow I Remember… plus Notes You Should Read…..

Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , , , , on November 26, 2013 by segarini

This will not be like any other column I will write for DBAWIS.  I will sound different and write differently and will, in all probability, mimic the style of one of the most influential people in my life outside of my immediate family.  That style may seem outdated in this world of soundbytes and visual chicanery, a world in which you have seven seconds to catch a potential reader/viewer’s attention.  It is solid and straightforward enough but would be looked upon certain pundits of pop culture as dull and outdated.  Seven seconds.  For most music programmers for the chains of radio stations gathered under the ever-growing corporate umbrellas, that is how much time you have to make your case.  It would be enough to make Edward R. Murrow‘s eyes roll back in his head, though I am pretty sure they never did.  Roll back in his head, that is.  Murrow was never surprised, or didn’t appear so onscreen.  And he was seldom caught off guard.

## Frank Gutch: Lost Sounds Montana: Mining Gold… Time For Dinner: Michael Dinner, That Is… A Night at the Bombs Away With Rich McCulley… And Notes…..

Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , on August 21, 2013 by segarini

History?  A waste of time?  I have heard that said way too many times over the years.  Admittedly, most of those years were academic years.  I loved the time I spent in school, elementary on, and even considered taking classes at the University of Washington during my tenure at Peaches Records back in the eighties (for fun, not for credit).  There is something about the smell of a campus, be it major college or small technical school, which makes my heart beat a bit faster.  Just the act of learning changes things for me and I love watching people expand their horizons— in positive ways, of course.

## Frank Gutch Jr: You’ve Got Your Heads On Backwards, Babies; Indie Labels You Should Pay More Attention To; Rich McCulley and Dan Miraldi— Bookends of the Pop Spectrum; and Whatever Else Comes To Mind…..

Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , , on August 6, 2013 by segarini

But first, a word from our sponsor.  Well, not sponsor, exactly.  In fact, just the opposite.  Well, not exactly opposite.  A few weeks ago, you might remember that I posted a few words about Sheldon Gomberg, the man behind Sweet Relief III: Pennies From Heaven, an album recorded for charity (all proceeds will go to Sweet Relief, an organization committed to helping musicians in need), and before that, I devoted a column to that organization through an interview with Sweet Relief‘s Rob Max.

## Frank Gutch Jr: Up-and-Comers for 2013, Why All the Covers, Mental Tracks, and Notes…..

Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on February 20, 2013 by segarini

Remember when I said 2012 was the best year ever in music?  2013 is even at this early date biting  2012 on the ass.  Artists are lining up with what I am sure is going to be a run and a half of outstanding and, in some cases, mindblowing offerings.  Rather than wait and write about them pretty much after the fact, let us dig in now in a sort of anticipatory way, shall we?


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# Codomains reconsidered

Date 2015-11-30

In my recent efforts to delve into the core of mathematics, I've uncovered an anomaly. Codomains are weird and there is something missing.

### The classic definition of a function

Let $A$ and $B$ be sets and $f$ a set of tuples of elements respectively in $A$ and $B$. $f:\ A \rightarrow B$ is a function if $f$ is single-valued ...

$$(x,y_{1}) \in f \wedge (x,y_{2}) \in f \Rightarrow y_{1} = y_{2}$$

... and $f$ is total:

$$\forall x \in A, \exists y \in B: (x, y) \in f$$

The domain of a function $f:\ A \rightarrow B$ is defined as the set ${ x \mid \exists y \in B: (x, y) \in f }$ and the range (or image) is defined as ${ y \mid \exists x \in a: (x, y) \in f }$.

$B$ is called the codomain of $f$.

#### Preliminary notes

Note firstly that $A$ and $B$ are included in the definition of a function $f:\ A \rightarrow B$. This also means that the equality of two functions is defined as conjunction of equalities between the respective $A$'s, $B$'s and $f$'s.

Note also that the domain of a function $f:\ A \rightarrow B$ is equal to $A$ while the range of that function is not equal to $B$.

Lastly, because $B$ is not equal to the range of the function, $B$ can be an arbitrary superset of the range. There is no connection whatsoever between $B$ and the set of tuples $f$.

This codomain $B$ is only used to define things like surjectivity or in theorems like, for example, the isomorphism theorems.

#### Problems with this definition

That was a lot of theory that any mathematician should have known by heart for years.

##### Ugly

The first problem with this definition is that it's ugly. The sets $A$ and $B$ are baked into the definition of a function while $f$ could just be a single-valued set of tuples! The domain and range definitions would stay the same because they are inherent to $f$ and can be defined irrespective of $A$ and $B$.

##### No partial functions

The second problem is that it doesn't allow for non-total functions.

Most mathematicians often have the luxury to only have to think about total functions. Ironically, people like information theorists and computer scientists often use the more abstract notion of partial functions.

To remind you, partial functions are functions where there isn't necessarily an output defined for every input. I would say $A$ is not equal to $dom(f)$ if the above definition hadn't disallowed that!

In the above definition of functions, there is no way to define non-total functions and that limits our thinking. Powerful mathematical concepts like Turing-machines can't be defined unless we allow for the more abstract notion of partial functions.

##### Asymmetric

Now, suppose you really want to define a codomain each time, then there is a third problem. The definition is now asymmetric:

$$f:\ dom(f) \rightarrow img(f) \subseteq B$$

I hope it's obvious that there is something missing here. Much nicer would be the following:

$$f:\ dom(f) \subseteq A \rightarrow img(f) \subseteq B$$

We can't do this, however, because the above definition of a function dictates that $A$ must equal $dom(f)$.

### A new definition

Now clear your mind and have a look at this new and improved definition:

Let $A$ and $B$ be sets and $f$ a set of tuples of elements respectively in $A$ and $B$. $f:\ A \rightarrow B$ is a function if $f$ is single-valued:

$$(x,y_{1}) \in f \wedge (x,y_{2}) \in f \Rightarrow y_{1} = y_{2}$$

The domain of a function $f:\ A \rightarrow B$ is defined as the set ${ x \mid \exists y \in B: (x, y) \in f }$ and the range (or image) is defined as ${ y \mid \exists x \in A: (x, y) \in f }$.

$A$ is called the corange of $f$ and $B$ is called the codomain of $f$.

Note that we can't call $A$ the coimage because that name is taken.

This new definition solves all the problems of the last definition:

$f$ could (possibly) be defined independent of a corange and codomain, though that would still require another definition of equality. This definition is not as ugly as the last one.

Because the corange $A$ is now different from the domain of $f$: $dom(f) \subseteq A$, $f$ is no longer necessarily total. This means that a function can now be partial and still be a function. This is precisely the case when $dom(f)$ is a strict subset of $A$.

The definition of a function is now symmetric:

$$f:\ dom(f) \subseteq A \rightarrow img(f) \subseteq B$$

... and as an added bonus, this new way of defining functions requires you to acknowledge the existence of partial functions.

### Feedback

While suggesting this idea to mathematicians, I have often been figuratively pelted by vegetables. However, they've never been able to tell me why it was that they couldn't be open to other definitions. After all, there is no such thing as a true definition.

I hope that, if there are any other mathematicians that feel deeply and personally offended by this, they will be able to explain to me what is mathematically (!) wrong with this new definition. As of yet, I have not been able to find any flaws.

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# Science for everyone – Pint of Science comes to Hamilton

Science took over the world last week.

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Pint of Science, a global science-for-everyone festival that spans three days in May, made its debut in Hamilton last week. The festival is run predominantly by volunteers and aims to bring science to the people through engaging, digestible public lectures – often in a bar or pub.

If this sounds familiar, it is very much like the Science on Tap event I wrote about earlier this year. While Pint of Science made its debut this year in Hamilton, it is actually a long standing event originating in just three UK cities in 2013. Pint of Science has since spread to over 300 cities across 21 countries.

For its first year in Hamilton, we were treated to three days of two simultaneous lectures at two local bars – The Phoenix on McMaster campus and West Town bar. Topics focused on astronomy, light, and neuroscience and featured researchers from McMaster and their graduate students.

Lots of fun to be had in Hamilton last week.

All of these lectures looked good, so I decided where to go based on beer list.

Welcome to The Phoenix. They claim they have the largest beer selection and patio in Hamilton.

The first day I got to see Dr. Alan Chen and his graduate student (and my roommate) Johnson Liang talk about where all the big elements come from. In short, during their average life cycle, stars will produce elements as large as Iron via nuclear fusion. To make anything larger requires a star to go through a rare life event like going supernova, which partially explains the high abundance of smaller elements in the universe, and the scarcity of heavy metals.

The following evening featured Dr. Laurel Trainor and her graduate student Andrew Chang of the Department of Psychology, Neuroscience & Behaviour at McMaster University. Their research focuses on musical and rhythmic development in infants and children, and how the study of music on the human brain can help us understand our own interpersonal communication.

Some of their research is done in McMaster’s LIVElab, one of the coolest “venues” I’ve ever heard of. I say venue, because it is a mix of concert hall and laboratory. It’s so cool that I promise to do an article on it soon. To quote the lab’s website,

The LIVE (Large Interactive Virtual Environment) Lab is a unique 106-seat Research Performance Hall designed to investigate the experience of music, dance, multimedia presentations, and human interaction.

The space includes Active Acoustic Control; Sound Recording Equipment; and measurement of Behavioral Responses (96 tablets), Movement (motion capture), Brain Responses (EEG), Muscle Tension (EMG), Heart Rate, Breathing Rate, and Sweating Responses (GSR).

The final day featured Dr. Kalai Saravanamuttu from the Department of Chemistry at McMaster University and graduate student  Kathryn Benincasa speaking about their work on light. What makes their light interesting is how they can focus it and keep it coherent over long distances.

Typically light will disperse over long distances, meaning a tight laser beam, or the light from a flashlight for that matter, will spread and become weaker as it travels. Dr. Saravanamuttu’s lab uses techniques to keep the light from dispersing, and is exploring applications for these types of long-range, coherent beams. One particularly interesting application is to give solar panels bug eyes.

Picture the honey-comb-like eye of an insect. By multiple facets over a lens allows them to focus light over nearly 180-degrees. For a bug, that means it can see everything and have a minimal blind spot. For a solar panel, this means it can absorb light from all directions at once. This requires a system of keeping light focused and coherent (check), in a geometry similar to a bug’s eye, that can coat a solar panel. By making these thin, specially designed coatings for solar panels, the researchers hope to increase the efficiency of solar panels making them more economically viable.

Imagine building this structure so the straws point outward over a full 180-degrees. Then shrink it down so you can paint it over a solar panel. You would get light from every direction focusing onto your solar panel.

The best thing about all of these lectures is that they were all completely free to attend. I have written about how important it is for scientists to keep the public informed about their work, and seeing how successful these public lecture events are suggests there is a strong desire for more of them. If you are interested in this sort of thing, keep an eye on this facebook page. There will be some very exciting news in the near future.

# Elastogranularity and how soil shapes the roots of plants, continued

This week a short article of mine was posted on Softbites, a blog “run by a group of young scientists who want to attract a wider audience to the beautiful world of soft matter.” To quote their homepage,

## “Softbites brings you digestible summaries of the latest research in soft matter.

If you have a soft spot for the science of bubbles, liquid crystals and other squishy materials you might have heard of soft matter! If you have not, this branch of physics is a fascinating interdisciplinary field studying various kinds of materials from gels to biological systems. They all share the fact that they are soft, which means they are not exactly solid nor liquid. For instance, if you poke a bit of foam, it will resist like a solid at short times, but it will flow at longer times.”

I wrote about a paper recently published in Physical Review Letters (but also available for free on the arXiv) by DJ Schunter Jr of the Douglas Holmes Group at Boston University. In this paper they take two existing, well understood systems and combine them to create something new and exciting. In particular, they take buckling of beams and combine it with granular rearrangement.

This post is a more technical article and is an addendum to my article on Softbites. I will be diving into some specifics of the analysis of the experiment, and is intended for those who have read the article on Softbites (here) and want more details. I encourage you to please go and read the Soft bites article before continuing below.

## Exploring the difference in buckling as a function of packing fraction

Recall the difference in buckling behavior for different packing fraction $\phi_{0}$, demonstrated in Figure 1:

Figure 1. An elastic beam in inserted into a rigid box filled with beads. Depending on the packing fraction,  $\phi_{0}$, of the beads, the beam exhibits one or two buckles. The curvature of these buckles are affected by $\phi_{0}$.

To quantify the difference in buckling that happens at different packing fractions, the increase in amplitude relative to wavelength of the buckles, $A_{0}/\lambda$, is plotted as a function of the length of insertion relative to initial beam length, $\sqrt{\Delta/L}$. For small values of $\Delta$, the curvature of the buckle is small and can be approximated as a triangle with height $A_{0}$ and base $\lambda = 2L_{0}$. As $\Delta$ increases, the height of the buckle increases according to $A_{0}/\lambda \propto \sqrt{\Delta/L}$. This is shown by the black dashed line in Figure 4, and experiments at both high and low $\phi_{0}$ (red and blue dots respectively) follow this relationship for small $\Delta$. As $\Delta$ increases, even in the absence of beads, the analytic solution for the beam’s buckling (shown by the blue curve) begins to deviate from this approximation. For intermediate values of $\Delta$, experiments at both high and low $\phi_{0}$ are shown to follow the analytic solution of a beam bending in the absence of beads. This suggests the beads are not yet confining the beam in any meaningful way. At even higher $\Delta$, low $\phi_{0}$ experiments continue following the behavior of a free beam but high $\phi_{0}$ begins to deviate rapidly. This deviation shows that confinement due to the beads has a strong effect on the geometry of the buckles in these cases, and the results are consistent with the numerically calculated solution for buckling in the presence of beads (red line).

Figure 2.  Increase in relative amplitude $A_{0}/\lambda$ as a function of $\sqrt{\Delta/L}$. Solutions for small buckles (dashed line), buckling in the absence of beads (blue line), and buckling accounting for beads (red line) are shown. Experimental data for small $\phi_{0}$ (blue dots) and large $\phi_{0}$ (red dots) show good agreement with each of their corresponding solutions.

## Local curvature and bead dislodging

So far we have seen that the nature of buckling is highly dependent on the bead packing fraction $\phi_{0}$. At low $\phi_{0}$, the beam buckles as if the beads were not present. At high $\phi_{0}$, the buckles become highly confined. Confining the buckles forces them to take on a higher curvature – the extra length of beam has less area to spread out in. This forces the beam into an unfavorable shape and effectively stores excess energy in the beam like compressing a spring would store energy within the coils of the spring. Like a spring, if you were to suddenly remove beads from the box (or the compression on the spring), the beam would suddenly pop into a more straight, less curved shape. What we can take away from this observation is that the higher the local curvature of the beam, the more stored energy there is within the beam, and the more the beam is pushing back on the beads in the area of high curvature. Eventually the pressure on the beads will be too great, and one will be forced to pop out of plane, on top of the rest of the beads.

The plot in Figure 3 shows maximum curvature $\kappa$ multiplied by beam thickness $h$ ($\kappa$ ~ compression of the spring, $h$ ~ stiffness of the spring) plotted against $\phi_{0}$ for higher and lower insertion lengths $\Delta/L$ (light blue and dark blue dots, respectively). The plot shows that higher $\phi_{0}$ leads to higher maximum curvature in the beam, as does larger insertion lengths. The plot also shows experiments where a bead becomes dislodged (red squares). This happens more frequently for highly curved and thick beams. The additional images in Figure 3 shows examples of bead movement for systems at various points on the plot. Longer arrows represent larger bead movement, and red circles represent a bead which dislodges from the rest of the beads. Also, notice the direction the beads move in Figure 3 (I). The arrows show how the beads would be pressing into the beam, and help explain why the buckles would move together at high packing fractions.

Figure 3. Beam curvature $\kappa$ normalized by beam thickness$h$ as a function of packing fraction $\phi_{0}$ for normalized insertion lengths $\Delta/L = 0.1$ (light blue) and $\Delta/L = 0.4$. $\kappa h$ is shown to increase for larger $\phi_{0}$. To the right of the plot are examples of bead movement (arrows) for experiments corresponding to regions within the plot. Dislocated beads are shown in red.

This is a unique experiment, and one that I found very satisfying to write about. What I like most about this work is how well quantified each observation is. In many papers, authors will present a system, show off some interesting observations and leave it at that. However, DJ Schunter Jr et al. take this neat experiment and quantitatively explain so much about it, including the spontaneous disloging of beads from the system. The experiment is not too hard to understand qualitatively, but being able to make actual predictions about a system requires a quantitative explanation. When you can predict exactly how a system will behave, it is then possible to use it for something constructive.

# The Scientist’s Guide to Vinyl // The Hipster’s Guide to Atomic Force Microscopy (Happy Record Store Day)

Anyone worth their gourmet-pink-Himalayan-rock-salt knows that vinyl is the premier way to listen to music. The snap crackle pop, the warm analogue glow – vinyl connects us to our music in a primal way akin to how a campfire connects us to the natural world.

Just kidding.

There have been many articles written about the dubious claims surrounding sound quality and vinyl. To me, there is no perceived benefits to vinyl compared to high-quality digital music. (This is constantly being written about. A quick google search gave me some good stuff, like this, this, and this)

So if sound quality isn’t a consideration, and if you are going to insist on buying music rather than streaming it, you will need to choose a physical medium. For me, this comes down to aesthetics. I like the big album art, the weightiness of the records themselves, and the cumbersome, almost ritualistic act of putting on a record that forces you to really pay attention and be involved in what you listen to. But what I like most is the beautiful simplicity of the technology.

The record player operates on the principle that sound is a pressure-wave in the air that will physically vibrate whatever it hits. This vibration is transmitted to a needle, carving little mountain ranges into the “master” record. This record can then be used as a model for large-scale reproduction.

To get the sound out, you reverse the process. This is the familiar picture of a record player – a needle is dragged across the grooves, following the topography. This movement is translated along the arm of the record player to a wire coil in a magnetic field. By moving the coil, an electric signal is created which is then amplified and sent to the speakers.

What I find incredible about this process is how intricate the topography within the grooves must be such that complicated musical pieces can be contained within it. This means the needle needs to be both sensitive to very small movements of the tip, and also be able to probe very narrow “valleys” in the record. The former is achieved by having a very flexible shaft and a device sensitive to very small fluctuations in the produced electric signal. The latter requires a needle tip that is very thin. A demonstration of this second point is shown below.

Figure from an untraceable textbook, I found this on a forum.

A stylus tip that is larger than the undulations within the groove will not be able to transmit the full information to the speakers. However, a very thin tip will be able to track in the narrowest of grooves.

Perhaps even more impressive, is the robustness of this technique. (This is a story I first heard from 99% Invisible – one of my favorite podcasts.)

In the 1950’s, beneath the brutal censorship of the Iron Curtain, the youth of Soviet Russia were just as obsessed with rock and roll as the rest of the world. However, as if to win a “you think your parents are bad?” contest, consumption of “American” media like rock and roll records was punished severely. Therefore, importing records was a very dangerous proposition and obtaining one was nearly impossible. Yet, this quarantined media still found a way to spread in the form of homemade records.

DIY record cutters were made, and nearly everything was tried as a substitute for vinyl. One of the more successful mediums was used x-ray sheets. You know, those transparent sheets you see on TV when a doctor is showing off a broken bone? The plastic used in these sheets was thin and flimsy, but allowed for reasonable reproduction of a smuggled copy of Johnny B. Good. The resulting records were very low quality, highly illegal, creepy, and ultimately, the most hipster thing imaginable.

How’s this for a Record Store Day exclusive? Bone music via Paul Heartfield, St. Louie Blues via Igor White, Image via Jozsef Hajdu, from 99PI.

Now, the enterprising hipster may be thinking to themselves, “if this technique is so sensitive to undulations, can we use this to measure the topography of other surfaces?”

With the right tip (perhaps some sort of silicon or silicon nitride, with a radius of curvature about 100x smaller than a record player’s?) and an ultra-flexible cantilever, it would seem possible to measure incredibly small surface features, no?

Congratulations, you just invented the Atomic Force Microscope!

An animation of an operating AFM. Notice the laser is deflected differently as the tip tracks along the topography of the sample. (Thanks, Carmen Lee)

At its most basic, an AFM is just a ridiculously sensitive record player. A very sharp tip at the end of a flexible cantilever is dragged along the surface of a sample. As the tip rises and falls with the topography of the sample, the cantilever bends to accommodate these elevation changes. Of course there are some differences to accommodate the differences between the two devices. For example, rather than converting the undulations into an electric signal with a coil and magnetic field, the AFM uses what I think is an even more simple and intuitive technique. A laser beam is reflected off the top of the cantilever into a detector. The angle the cantilever makes with the detector changes as it bends, meaning the laser will hit the detector at a different spot. The longer the beam travels, the more it amplifies the size of the undulations. This allows for the measurement of surface features accurate to the sub-nanometer scale.

Atomic force microscope topographical scan of a glass surface. The micro and nano-scale features of the glass can be observed, portraying the roughness of the material. The image space is (x,y,z) = (20 µm × 20 µm × 420 nm). (wiki: Chych derivative work: Materialscientist (talk))

Despite being based on very old technology, the AFM wasn’t invented until the late 1980’s. Manufacturing such fine tips is not easy, and keeping errant vibrations out of your measurements is non-trivial. In fact, in a lab I used to work in, very accurate measurements required one to wait until the frequency of buses to the University decreased. The vibration of buses driving past the building was enough to compromise the results.

However, if you are able to compensate for these vibrations, this technique is so accurate and robust that it has become the go-to technique for most surface sciences. An AFM can image the structure of biological molecules and cellular components, individual polymers, and with the proper set-up, individual atoms within a structure (!!!).

So if you chose to observe Record Store Day this year, don’t feel silly for “wasting money on a dead technology”. You can feel proud that you are actually wasting your money on a very crappy AFM.

# “Hey you’re a physicist…” tangled cables

Two of my best friends have this joke. They will spontaneously ask me a question prefaced with “Hey Adam, you’re a physicist…” The question is often ridiculous, and always out of the blue. We may go weeks without speaking only to receive a text like, “Hey Adam, you’re a physicist. Why is Neapolitan ice cream so bad?” or “Hey Adam, you’re a physicist. How much should I invest in Bitcoin? Also, what is Bitcoin?”

But sometimes they end up being interesting, and something I can answer them…

## “Hey Adam, you’re a physicist – why do my headphone cables keep getting tangled?”

The short answer is “cuz math tells us so”. I’m not going to pretend I understand said math, but I can give a very qualitative description.

Aw what the hell?! Photo from Business Insider

Researchers devised a model of a coiled string and assigned some probability to “braid moves” as the coil is agitated. Braid moves are described as a spontaneous crossing of the string from adjacent lengths of cable.  By this they mean a cross-over between to lengths of the coil that are parallel to each other. They can pause after a certain number of braid moves and decide if the string has knotted.

Schematic by Dorian M. Raymer and Douglas E. Smith from their manuscript, originally published in PNAS. The wire coils within the box, forcing free ends to lay parallel to the rest of the coil. Knots form as the free end of the wire crosses adjacent segments.

This model predicts a few interesting things. The probability that the string will knot increases rapidly at short times and approaches 100%. Also, tangling and subsequent knotting happens more readily for longer and more flexible strings.

These researchers then performed an experiment to test these predictions. One at a time, they took strings of various lengths and stiffness, coiled them, and placed them in a box. They shook the box, then observed if a knot had formed. They indeed found longer, flexible strings will knot more readily, and that knots will form within seconds of agitating the box.

So how can you stop your wires from tangling? You can instead use short, stiff rods instead of wires. Or maybe just try this.

# #ScientistsWhoSelfie and the Roll of Social Media in Science

Recently Science, a high-profile journal, published an opinion piece calling into question the effectiveness of social media – specifically Instagram – on science outreach. The piece, written by Meghan Wright used popular Instagram blogger Samantha Yammine (Science Sam) as an example of the way this specific type of “outreach” that is over-glorified and ineffective.

There has already been much discussion over the article, mostly focusing on the seemingly personal nature of the criticism. This has created the #ScientistsWhoSelfie movement to show support and appreciation for the people who donate their time to make science more inclusive via social media.  However many of the points discussed in the article have yet gone analyzed. For this, I spoke to to Liz Grey – a physicist who has recently turned to intellectual property law in Ottawa, and Carmen Lee – a current physics graduate student at McMaster University.

Meghan Wright says in “Why I don’t use Instagram…” that using social media as an outreach tool is ineffective at combating female discrimination.

…it disturbs me that these efforts are celebrated as ways to correct for the long held and deeply structured forms of discrimination and exclusion that female scientists face. I wonder whether our efforts should instead be directed toward advocating for policy changes at institutional and governmental levels.

Social media visibility may not solve deep-seeded discrimination, but there is proof that it may be helping to broaden what a “typical” scientist is perceived to be – at least in children. It won’t solve the underlying issues  like equal pay or necessarily convince people who make hiring decisions, but it can help women and minorities from feeling like outsiders. “Of course there is work that needs to be done in administration, but the number of times I’ve met people in public and been told ‘you don’t look like a scientist’ is crazy!” said Carmen. And of course women aren’t the only group that can benefit from this. “The more diverse of a social media representation of science is, the better”

While many people have differing opinions on the effectiveness of social media on correcting for deep discrimination, no one would claim it is the way of doing this. “Of course there is a danger in only communicating through social media, but if that’s how individuals want to do it, that’s ok too”, explains Liz. Expanding public perception of who scientists are will at the very least make the STEM community feel more inviting. However, making STEM appear more inclusive is really just a band-aid solution. “Band-aids are very useful and important sometimes, but bringing them into surgery is of course ridiculous”.

Wright goes on to make another interesting point:

Time spent on Instagram is time away from research, and this affects women in science more than men. That’s unfair. Let’s not celebrate that.

While this point does not seem to apply to Yammine – a graduate student hoping to work outside academia, potentially as a science communicator – this is an important point for others not pursuing this path.

“Beginning in undergrad women are chosen to be the face of diversity – you are chosen to to meet with people to show how inclusive the department is. 30 minutes here and there definitely add up over a career.” said Carmen.

Liz agreed, adding that there is a pressure to fill these roles.”I feel there is some amount of pressure to be representative of women who do science, like science, and aren’t afraid of math.” This feeling also carried over in Law. “Many lawyers that are women are still considered ‘woman-lawyers’ or ‘female lawyers'”, as if this were a different classification of lawyer. And with this differentiation in gender comes gender-based programs that cut into research. “For example, there is a Women’s Legal Mentorship program in Ottawa. I could be a member of this program but like the author, I don’t want to present myself like this or spend time away from my work. Sometimes I feel like I’m doing a disservice to my community by not being involved.”

Some of these groups, like the Women’s Legal Mentorship program are women-exclusive, but many are not. “In many cases men are making the decisions. You want to empower girls to rise above and overcome, but in some ways it may be more useful to frame it as a men’s problem.” said Liz. ” Carmen, a member of Graduate Women in Physics and Astronomy (GWIPA) said, “these issues are everyone’s problem –  GWIPA makes a point of inviting all male students to our events.” Yet male attendance continues to be abysmal. She seemed particularly amused to observe, “men who come seem to be sensitive to being the only man in a room and.. get very hand-wring-y”.

Something both Carmen and Liz emphasized was that any attempt to make STEM more welcoming to under-represented people is helpful, but it can’t just be an effort on the part of the under-represented. Everyone needs to work to make their community inclusive. At McMaster, this could mean volunteering and supporting events put on by GWIPA and WISE (Women in Science and Engineering), especially if you are male. Please consider joining and helping these groups or your local equivalent.

# Book Club: The River of Consciousness – Oliver Sacks

As the author of several books, most notably, Awakenings (1973) and The Man Who Mistook His Wife for a Hat (1985), Dr. Oliver Sacks educated his readers on the importance of treating all patients with humility, especially those with invisible illnesses plaguing the dark fissures of the brain. Dr. Sacks, a trained neurologist, had a talent for listening to his patients, learning about their symptoms, then backtracking to find the neurological sources for the experienced symptoms. No headache or bout of unsteadiness went under-analyzed by Dr. Sacks.

The River of Consciousness, published posthumously, explores the interdependent relationship between science and the human condition. Common within his prior publications, Dr. Sacks contextualizes neurological conditions by examining historical medical anthologies and past scientific innovations that may be deemed relevant to a patient’s case.

The first chapter opens with an examination of Darwin’s evolutionary studies. As Dr. Sacks notes, Darwin was primarily recognized for his research into animal evolution, but he spent most of his early years enthralled by botany and the world of plant evolution. Dr. Sacks identifies several unlikely connections between modern neurology and early studies in plant evolution. For example, Venus flytraps use electrical stimulation to capture their victims which shows similarities between the electrical activities in the brain of humans.

The weakness of our memory is also explored in The River of Consciousness; weakness which can be exploited by the strength of our mind and imaginative capabilities. Dr. Sacks shares a descriptive and vivid story of a bomb raid he recalled having lived through in England during World War II. However, once Dr. Sacks fact-checked this event with his brother, his brother clarified that both he and Dr. Sacks were at boarding school during this particular raid and that a very detailed letter sent from their other brother, who was indeed present during the raid, instilled the realistic memories into the mind of Dr. Sacks. When given enough information, our brains have the power to create memories of events that may not have happened.

Near the end of the book, Dr. Sacks remarks on individual’s consciousness being derived from past experiences, making consciousness subjective and unique to each individual. We use our memories to place ourselves in present moments, making us conscious to certain events taking place around us. We may spend an entire day with someone and be experiencing the same present events, but our consciousness during those events are uniquely our own and completely different from the other person’s consciousness.

Dr. Sacks strategically concludes his book by examining the importance of the evolution of the sciences in unison to the evolution of humans and plants (as the author notes in his opening remarks referencing Darwin’s research). New theories should not replace older, outdated theories; rather, they should be developed alongside older theories, paving the way for even newer scientific innovation. Scientific innovation is derived from reworking and disproving itself.

Dr. Oliver Sacks passed away on August 30, 2015 at the age of eighty-two as a result of cancer. He was an empathetic and diverse neurologist who treated his patients as humans and their conditions with respect. What’s more, he was able to educate the public with well-written books, journal articles, and newspaper editorials. His brilliant mind will forever be missed.

By: Sydney Myles

# Science on Tap: a Night of Drinking and Deriving with Local Physicists

“Do you know where you’re going?”

“Generally, yes… I know it’s around here somewhere…”

“How did Joey find this place?”

“I have no clue…”

On February 28 I met Connor, a fellow physics graduate student at McMaster, for dinner (which he still owes me for). As we ate I described a science outreach event that I would be attending later that night. After explaining that beer would be served, it was fairly easy to convince him to come with me. The event was being hosted at Artword Artbar, just down the street from where we were eating. Investigating their website, it seemed they were predominantly an event space, hosting poetry and jazz several nights a month.

Joey Rucska and a handful of physics graduate students had been planning this event for several months. Inspired by the very successful Astronomy on Tap event hosted by the University of Toronto, Science on Tap places short and accessible lectures by local scientists in a welcoming pub atmosphere. For the first event, the speakers included Ashley Bemis, an astrophysics graduate student, McMaster’s Radiation Sciences Program Director Dr. Fiona McNeill, and Dr. Paolo Bianchini, a post-doc in the Astronomy department. To keep things light, informal question-answer periods were sprinkled throughout the night, as well as small discussions about scientific stories that had been featured in the news recently. Additionally, a rotating set of images were displayed throughout the night and guests were challenged to guess whether the images were “micro”, “macro”, or “celestial” scale. Prizes were awarded to the top three scorers.

Connor and I arrived slightly before 7:00 pm, entering a cozy single-room dotted with round tables and chairs. A piano lurked in the corner of the room, opposite a short, dramatically-lit stage. We were immediately received by a cheerful older couple, clearly the owners of the bar, and clearly very busy preparing for the evening.

Connor (left) talking to another physics graduate student between lectures.

“Hello! Come in, come in! We have a special event tonight, are you two interested in science?” the couple asked. “You bet. I’m actually helping with the event, they have me taking pictures.” “That’s really great. Do you know how many people will come?” “I have no idea…”

The room filled steadily as we drifted closer to the official 8:00 pm start time. The space boasted a perhaps ambitious 60-person capacity and it seemed like this would become a relevant factor. As the room filled, anticipatory buzz grew out of the crowd. A several-people-deep row lined the bar as the hosts made final preparations next to the stage.

Local science enthusiasts began arriving at Artword Artbar at 7:00 pm, nearly filling the space by 8:00 pm.

At 8:00 pm, Ryan Plestid presented a short, and only slightly sarcastic look at recent science headlines. After a brief discussion with the audience, he introduced the first speaker of the evening – Ashley Bemis.

Ashley Bemis explained that good science comes out of learning from your mistakes.

Ashley led us down a winding path of scientific accidents that resulted in paradigm-shifting realizations. We learned about the first-discovered collection of Jovian moons, that the universe is expanding… and actually speeding up, and of course the many mistakes that resulted from trying to shoe-horn real astronomical observations into an Earth-centered “theory” of the Universe. The last point turned into an interesting discussion with the audience on the conflict between science and politics. In hindsight it seems ridiculous that Galileo would be put under house arrest for claiming the Earth orbits the Sun, yet you can still find examples of “controversial” science being censored today. The discussion took an introspective twist, ending with the question from the audience, “if you were Galileo, would you have the courage to go against the church?”

After a short break, Dr. Fiona McNeill, the Radiation Sciences Program Director at McMaster University, took to the stage. Dr. McNeill chose to address something that brings all Hamiltonians together – a distrust towards tap water. Specifically, Dr. McNeill addressed the fluoride that is put into tap water. Studies show that some fluoride intake is beneficial to bone health, but like most things, moderation is key. Too much fluoride has also been shown to be detrimental to bone health. So where is Hamilton’s tap water on this scale? “It’s actually pretty good!” Dr. McNeill assured us. “And it happens to be very, very clean.”

Dr. Fiona McNeill assuring the audience “Hamilton’s tap water is actually very clean.” Even still, Carmen chose beer instead.

The story changes, however, for avid tea drinkers. “I’m Scottish, and I love my cup of tea, so of course I was also worried about this” she said. Dr. McNeill’s findings show that drinking several cups of tea a day increases the uptake of fluoride in humans, and in fact can push people into the danger-zone. Her research group tested a wide variety of teas, finding small differences in brand name, but most importantly showed this effect to be greatest with black teas. However, she finished by bringing the tea-lovers in the room back from the brink. “So black tea is the worst, but there is good news – adding some form of dairy will actually neutralize the threat.”

The last speaker was Dr. Paolo Bianchini, and he was a treat.

Dr. Paolo Bianchini looking over his slides before his talk.

Paolo ascended the stage to cheers from the audience – I guess he has some fans in Hamilton. On this particular evening, Paolo showed us the entire life-cycle of a star. In short, depending on its size, a star may go “supernova” or collapse into a black hole. However, this is an extremely simplified description. Condensing millions of years into 30 minutes is not an easy task and required some ingenuity. If a picture is worth a thousand words, and emoji are the hieroglyphics of our time, then Dr. Paolo Bianchini may have found the sweet spot in conveying information.

A summary of Paolo’s talk – the life cycle of a star.

During the discussion period, Paolo addressed a common misconception about black holes. “Black holes don’t actually suck you in,” Paolo explained. “They behave like any massive body from a distance. It’s only when you get too close does it become impossible to escape.” That is to say, if the Sun were to suddenly be compressed to the size of McMaster campus, it would become a black hole, but the planets would continue orbiting the same way they always have.

After the final talk, the winners of the trivia contest were announced. Congratulations to Katie C., Annie W., and Katie P., who were among 8 people who tied for first.

The event continued for some time as guests discussed what they heard that evening with the presenters, volunteers, and other guests. As I talked to some of the attendees, a common sentiment was that this sort of event was sadly lacking in Hamilton. Case in point – trivia nights have become increasingly popular, showing there is a desire to both have a night out and be mentally engaged.

Kathleen, a health science researcher had been “looking forward to this event for weeks”. She thought that there was a real lack of mentally stimulating night-out events and is looking forward to the next one.

The response to this event was so favorable that many attendees asked if another Science on Tap is in the works. It’s likely that a follow up will be scheduled some time this summer. Follow their Facebook page and be the first to know when and where it will be. I am excited to see this event grow, and reach out to other scientific disciplines. I am grateful to Joey, Artword Artbar, and everyone else who helped organize this event. And of course, thank you to the speakers who made the night so enjoyable.

_______________________

Ashley Bemis is a PhD candidate at McMaster University studying astronomy. She received a bachelor’s of science from the University of Amherst, Massachusetts, and a master’s of science from the Max Plank Institute for Radio Astronomy in Bonn Area, Germany.

Dr. Fiona McNeill is the Radiation Sciences Program Director at McMaster University. Her research focuses on advancing radiation-based measurement techniques and uses them to study trace toxic elements in humans, as well as performing behind-the-paint measurements on famous pieces of art (the latter of which was featured in a travelling exhibition. Check out The Unvarnished Truth interactive website.)

Dr. Paolo Bianchini is a post-doctoral researcher from Milan. He received his MSc from the University of Milan, and his doctorate at the Max Planck Institute for Astronomy in Heidelberg. His work focuses on the structure and internal dynamics of stellar systems.


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# 11.5 Matrices and matrix operations

 Page 1 / 10
In this section, you will:
• Find the sum and difference of two matrices.
• Find scalar multiples of a matrix.
• Find the product of two matrices.

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. [link] shows the needs of both teams.

Wildcats Mud Cats
Goals 6 10
Balls 30 24
Jerseys 14 20

A goal costs $300; a ball costs$10; and a jersey costs \$30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

## Finding the sum and difference of two matrices

To solve a problem like the one described for the soccer teams, we can use a matrix    , which is a rectangular array of numbers. A row    in a matrix is a set of numbers that are aligned horizontally. A column    in a matrix is a set of numbers that are aligned vertically. Each number is an entry    , sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named $\text{\hspace{0.17em}}A,B,\text{}$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ are shown below.

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 7\\ 0& -5& 6\\ 7& 8& 2\end{array}\right],C=\left[\begin{array}{c}-1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}3\\ 2\\ 1\end{array}\right]$

## Describing matrices

A matrix is often referred to by its size or dimensions: indicating $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ rows and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ identified as $\text{\hspace{0.17em}}{a}_{ij},\text{}$ we look for the entry in row $\text{\hspace{0.17em}}i,\text{}$ column $\text{\hspace{0.17em}}j.\text{\hspace{0.17em}}$ In matrix $\text{\hspace{0.17em}}A\text{, \hspace{0.17em}}$ shown below, the entry in row 2, column 3 is $\text{\hspace{0.17em}}{a}_{23}.$

$A=\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right]$

A square matrix is a matrix with dimensions meaning that it has the same number of rows as columns. The $\text{\hspace{0.17em}}3×3\text{\hspace{0.17em}}$ matrix above is an example of a square matrix.

A row matrix is a matrix consisting of one row with dimensions

$\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]$

A column matrix is a matrix consisting of one column with dimensions

$\left[\begin{array}{c}{a}_{11}\\ {a}_{21}\\ {a}_{31}\end{array}\right]$

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations .

## Matrices

A matrix    is a rectangular array of numbers that is usually named by a capital letter: $\text{\hspace{0.17em}}A,B,C,\text{}$ and so on. Each entry in a matrix is referred to as $\text{\hspace{0.17em}}{a}_{ij},$ such that $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ represents the row and $\text{\hspace{0.17em}}j\text{\hspace{0.17em}}$ represents the column. Matrices are often referred to by their dimensions: $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}×\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ indicating $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ rows and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ columns.

## Finding the dimensions of the given matrix and locating entries

Given matrix $\text{\hspace{0.17em}}A:$

1. What are the dimensions of matrix $\text{\hspace{0.17em}}A?$
2. What are the entries at $\text{\hspace{0.17em}}{a}_{31}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}_{22}?$
$A=\left[\begin{array}{rrrr}\hfill 2& \hfill & \hfill 1& \hfill 0\\ \hfill 2& \hfill & \hfill 4& \hfill 7\\ \hfill 3& \hfill & \hfill 1& \hfill -2\end{array}\right]$
1. The dimensions are because there are three rows and three columns.
2. Entry $\text{\hspace{0.17em}}{a}_{31}\text{\hspace{0.17em}}$ is the number at row 3, column 1, which is 3. The entry $\text{\hspace{0.17em}}{a}_{22}\text{\hspace{0.17em}}$ is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.

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text:
# Uniformly convergence of the continuous functions sequence, $f_n$

Let $$f_n, f : D \to \mathbb{R}$$ with $$A(\neq \phi) \subset \mathbb{R}$$

Here the $$f_n$$ are the sequence of the continuous functions.

I already knew the fact "$$f_n$$ is uniformly converge to $$f$$ on $$D$$ $$\Rightarrow$$ $$f$$ is continuous on $$D$$"

But There is a lecturer's claim that

Say $$f_n$$ is continuous on $$A$$

"$$f_n$$ is uniformly converge to $$f$$ on $$A$$ $$\not\Rightarrow$$ $$f$$ is continuous on $$A$$" (statement (*))

He suggested the counter-example as like the below.

$$f_n, f : \mathbb{R} \to \mathbb{R}$$ with $$A(\neq \phi) \subset \mathbb{R}$$

Here the $$D = \mathbb{R}, A = \mathbb{Q}^c$$ and

Since the rational number set,$$\mathbb{Q}$$ is countable so $$\mathbb{Q} = \{x_1, x_2, .... \}$$

Then Defining each functions like the below

$$f_n(x) = \begin{cases}1 & \text{x \in \{ x_1, x_2, ...,x_n\}} \\ 0 & \text{o.w.}\end{cases}$$

$$f(x) = \begin{cases} 1 & \text{x \in \mathbb{Q} } \\ 0 & \text{x \in \mathbb{Q}^c} \end{cases}$$

From here, my question begins.

First question) I put in to the sequence $$x_{n+1} \in \mathbb{Q}$$ to $$\Vert f_n(x) - f(x) \Vert$$

Then $$\Vert f_n(x_{n+1}) - f(x_{n+1}) \Vert$$ = $$\Vert 0-1 \Vert = 1 \geq \epsilon(= {1 \over 2})$$

So my conclusion is this counterexample is totally wrong, since $$f_n$$ is not uniformly converge to $$f$$. What do you think about that? Is my thought wrong?

Second question) Like the lecture's thought, I guess the statement (*) is false. But I can't find any counterexample. If the lecture's counterexample is false, Please give me a counterexample.

p.s.) If the statement is true, Why does the statement(*) hold?

Thanks.

There seems to be no connection between $$D$$ and $$A$$ in $$(*)$$. So $$(*)$$ is correct. For one example take $$D=[0, \frac 12 ], A=[0,1]$$, $$f_n(x)=x^{n}, f(x)=0$$ for $$x<1$$, $$f(1)=1$$. Then $$f_n \to f$$ uniformly on $$D$$ and $$f$$ is not continuous on $$A$$.
However, the example given by the lecturer is wrong since $$f_n$$ does not converge uniformly to $$f$$ on $$D=\mathbb R$$.
Answer for the revised question: If $$f_n \to f$$ uniformly on $$A$$ and each $$f_n$$ is continuous on $$A$$ then the restriction of $$f$$ to $$A$$ is continuous. But you cannot say that $$f$$ is continuous at points of $$A$$ as in the example by your lecturer.
• Thanks for reply, Dear Mr,@Murthy. But I should have been asked "$f_n$ is uniformly converge to $f$ on $A$" in statement (*) not "$f_n$ is uniformly converge to $f$ on $D$" Sorry for typo. I edited my question. – se-hyuck yang Jan 28 '20 at 8:25
• What about the case if we add the condition $A \subset D$? Though adding more condition, his claim and counterexample are still not hold, right? – se-hyuck yang Jan 28 '20 at 14:31
• @se-hyuckyang Actually $f_n(x)=f(x)=0$ for all $x$ in $A$ in his example. So $f_n \to f$ uniformly on $A$. – Kavi Rama Murthy Jan 28 '20 at 23:15


text:
# Trying to prove soundness property of propositional logic (Hilbert)

So for starters I want to prove that propositional logic is sound, using a Hilbert system such as Łukasiewicz's.

This system comes with three axioms:

$A \to (B \to A)$

$(A \to (B \to C)) \to ((A \to B) \to (A \to C))$

$(\lnot A \to \lnot B) \to (B \to A)$

And one rule of inference (modus ponens):

$P, (P \to Q) \vdash Q$

To prove soundness I need to show that:

$A \vdash B \implies A \vDash B$

Axiom 1:

$A = \top, B = \top$:

$\top \to (\top \to \top) = \top \to \top = \top$

$A = \top, B = \bot$:

$\top \to (\bot \to \top) = \top \to \top = \top$

$A = \bot, B = \top$:

$\bot \to (\top \to \bot) = \bot \to \bot = \top$

$A = \bot, B = \bot$:

$\bot \to (\bot \to \bot) = \bot \to \top = \top$

And then similar idea for the other two axioms (which I'm not showing here for space purposes since the second axiom has $8$ cases and then the third axiom has $4$ cases again)... but all cases come out $\top$ over all possible inputs of $\top$ and $\bot$ for the propositional variables.

So each axiom is always a true statement, i.e. a tautology (given our definitions/truth tables for $\top$, $\bot$, $\to$, and $\lnot$).

Now I am a little stuck. We still need to show $A \vdash B \implies A \vDash B$, which I believe means we have to somehow show that modus ponens always allows us to derive true deductions, but then somehow using our result that we have true axioms in the form of tautologies.

You're almost there. Yes, you need to show that Modus Ponens is a valid inference scheme, but that is easy: whenever $P$ is true, and $P \rightarrow Q$ is true, then indeed it must be the case that $Q$ is true.

Ok, but how do you show that if $A \rightarrow B$ (I assume $A$ is a set of assumptions?), then $A \Rightarrow B$? Well, since $A \rightarrow B$, there is a sequence of statements, where each statement is either an element of $A$, or an instance of one of the axioms, or the result of the application of Modus Ponens to two earlier statements in the sequence, and where the last statement in the sequence is statement $B$. What you want to do, is to show, by strong induction on the steps in this sequence, that each statement in this sequence is a logical consequence of $A$

• First paragraph: Are we assuming that any formula $P$, $P \to Q$, etc, are built up from the axioms we just proved were true? If the axioms are true then as long as we keep plugging them into each other (no matter how deep), they'll always be true, so $P, P \to Q \vdash Q$ becomes $\top, \top \to \top \vdash \top$? – user525966 Sep 10 '18 at 6:47
• Second paragraph: Is what you're discussing here the "deduction theorem"? I'm not sure what's being described here exactly or what/why we need to prove it? If we've shown the axioms are true and the inference rule always yields true statements, what more are we missing exactly? – user525966 Sep 10 '18 at 6:49
• I found this proofwiki.org/wiki/Soundness_Theorem_for_Hilbert_Proof_System which suggests the axioms and modus ponens thing are all we need to show in order to prove soundness? – user525966 Sep 10 '18 at 10:04
• @user525966 First paragraph: No, the $P$'s and $Q$'s here can be anything, because they can come form the assumption set $A$. Indeed, the $P$'s and $Q$'s need not be tautologies. But what is true, is that if $P$ and $P \rightarrow Q$ are assumed to be true, then $Q$ has to be true as well, which is easily verified by a truth-table. – Bram28 Sep 10 '18 at 15:46
• @user525966 Second paragraph: No, this is not about the deduction theorem, but rather about the fact that you need to show that the last line ($B$) in your derivation $A \vdash B$ is a logical consequence of $A$. This actually requires a proof (and a strong inductive proof on the steps in the derivation is the way to go) ... it's not hard, but without it, it cannot be considered a proof. Indeed, I condiser the proof on that ProofWiki to be incomplete. – Bram28 Sep 10 '18 at 15:49

$$A \vdash B \implies A \vDash B$$

Make sure you are certain about what this actually means: "For every propositional proof $P$, for every valuation of the propositional variables, when the valuation makes the assumptions of $P$ true, then the valuation also makes the conclusion of $P$ true." Also note that $B$ is not necessarily a propositional variable, it is an expression containing multiple propositional variables, and $A$ is a set of such expressions.

So the normal manner to prove statements of the form "For every proof $A \vdash B$, stuff" is to proceed by inductive construction of proofs. So to construct a proof, you have 1 base case:

• $A \vdash A$

and you have 2 recursive cases:

• The last step of the proof is an instantiation of an axiom schema, and the inductive assumption the soundness of the rest of the proof is irrelevant
• The last step of the proof is an invocation of modus ponens

In this case, we can assume some $A \vdash X$ is sound and $A \vdash X \to B$ is sound. That is , we can assume for every valuation of propositional variables, whenever every $A$ evaluates to true, so does $X$ and $X \to B$.

We have to establish that for every valuation of propositional variables, whenever every $A$ valuates to true, so does $B$. So write down all 4 cases for $X$ true/false and $B$ true/false. Cross out the cases that contradict the inductive assumptions. If what is left is only cases where $B$ is true then you've established the second recursive case.

• So A, B, etc, are these technically formulas (in terms of "data type" to use a software analogy) – user525966 Sep 10 '18 at 20:36
• @user525966 Yes, that is a good way of putting it. And $A$ is actually a set of formulas. – DanielV Sep 11 '18 at 1:12