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# How do I explain a unicorn discharging powerful electricity at a distance?

I am thinking of copying the design of an electric eel to create the horn, however as air is not a good conductor of electricity it cannot discharge any electrical shock unless there is a physical contact.

Can you help me to come up with a believable unicorn that can stun its predator from at least a few meters away, and also how it prevents electrocuting itself or its own kind?

• Why do you think you need or should explain it at all? – Mołot Apr 11 '18 at 9:31
• @Mołot Because there's no magic tag so it would probably break suspension of disbelief. – wizzwizz4 Apr 11 '18 at 11:30
• @Evorlor That is correct. Why is it: “A Unicorn” – Virusbomb Apr 11 '18 at 13:41
• So you are creating one of these? Does that look like your idea? In Monster Hunter's wikia you can find a bit about the biology of the kirin. It's not much (not enough for a full answer) but it might help, even if only for inspiration. – xDaizu Apr 11 '18 at 14:34
• The horn is actually a super-soaker attached to a toaster...? – Magic Octopus Urn Apr 11 '18 at 22:34

The unicorns are symbiotic with a jumping spider species, residing on the horn.

The horn is kept at (or can be quickly charged to) a sufficient high potential, and the jumping spiders can be directed to jump on the target while threading a thin wire of cobweb attached to the horn.

As the spider or a group of spiders land on the target the circuit is closed and the target is zapped.

Alternative to the spiders: the unicorn can blow mucus covered particles to the target, with the particles keeping a thin slimy rope of mucus attached to the horn. Zap as above.

Pick the one more suited for your story and stomach.

• Comments are not for extended discussion; this conversation has been moved to chat. – Monica Cellio Apr 17 '18 at 16:12

The solution is Lasers

Simply, much like a Narwhal* the Unicorn can shoot lasers out of it's horn. From there, it's not too much of a stretch to allow for the possibility of an Electric Unicorn to discharge it's built up charge along the laser-induced plasma channel immediately after firing it's horn-laser, in much the same manner as an Electrolaser.

*Apocryphal

• This is what I came to say. A chemically pumped laser seems like something an electric unicorn would have. Turn the unicorn into something Rhino like and you've got yourself a winner. – bob0the0mighty Apr 11 '18 at 15:29
• A chemically pumped laser seems like something an electric unicorn would have.  This is why I love this site. – Renan Apr 11 '18 at 20:48
• OMG. This idea is awesome! – ElmoVanKielmo Apr 13 '18 at 8:18

While it is true that air does not conduct electricity as nicely as water, it is also true that there is no (engineering) problem that cannot be solved by judicious application of brute force.

Lightning is a thing after all, so we know a discharge will happen in air if there is sufficient difference in electric potential.

Copy the design of the eel. This is what the wiki says about it:

In the electric eel, some 5,000 to 6,000 stacked electroplaques can make a shock up to 860 volts...

Now notice what the wiki says about its typical dimensions:

The electric eel has an elongated, cylindrical body, typically growing to about 2 m (6 ft 7 in) in length, and 20 kg (44 lb)...

If your unicorn weights as much as a draft horse... let's say one metric ton. It will have enough size and mass to have a lot more electroplaques. A back-of-the-napkin calculation says that, keeping the same volts-to-mass ratio, the unicorn will be able to produce 43,000 volts. You can buff that up - nothing is keeping the unicorn from having even more electroplaques. Let's say the unicorn is able to produce 50,000 volts. That is about as much as a typical low end Tesla coil. Now check the pictures in the link. It also has this to say about the coil:

The high electric field causes the air around the high voltage terminal to ionize and conduct electricity, allowing electricity to leak into the air in colorful corona discharges, brush discharges and streamer arcs.

A pointy part in a body is much more likely to produce a discharge than a round part or a toroid, so the horn is perfect for shooting out lightning. As for how the beast does not fry itself, it may have glass or some other very good electrical insulator material in its hooves.

• Sadly it's not that easy I believe. This article suggests that the eel would have major problems if it wasn't in water scientificamerican.com/article/how-do-electric-eels-gene . Also how do you direct your lightning at the predator? It would just hit wherever the electric field is densest, wouldn't it? – Raditz_35 Apr 11 '18 at 13:20
• @Raditz_35 you stand and wait. This doesn't have to be aimed, just like the electric eel also does not do aimed shots. The horn is going to be giving out sparks until something more conductive than air comes close, and then nature runs its course. As for the problem of being in air, if the unicorn lives in a forest, it will have plenty of things to be dispersing extra charge. Mind you, you will be able to tell where unicorns live by the amount of charred up foliage, but then again you can't make an omelet without some kitchen fires. – Renan Apr 11 '18 at 13:27
• @JohnDvorak the question says "at least a few meters away" since its first version. This method could do for two or three meters. – Renan Apr 11 '18 at 20:45
• Judicious application of brute force is a great phrase. Sounds very similar to the idea of "percussive maintenance". – brichins Apr 12 '18 at 15:06
• This is no more nor less than a biological Tesla coil. Entirely feasible. – Brian Lacy Apr 12 '18 at 21:35

The horn has a tiny hole in the end. The unicorn can pump a saline solution out through this hole - like a spitting cobra or bombardier beetle - which acts as a conductive stream from horn to target.

Additionally, this liquid contains a paralytic agent, causing "tingles" that aren't actually caused by the initial, brief, jolt of electricity.

A unicorn has tall, non-conductive hooves, which prevent them from 'grounding', and protects them from accidental shocks. Young unicorns, who have not yet learned to completely control their bioelectric-capacitors yet, are thus prone to accidentally shocking themselves while eating.

• @marcelm I do not believe that is how a circuit works. If that was the case, then how would a cloud manage to discharge to ground? As in the answer provided by L.Dutch, the other animal will be the ground connection. The reason the unicorn itself doesn't get hurt might be because the current doesn't pass through any vital organs to get to their target. – Daniel Vestøl Apr 11 '18 at 14:43
• @marcelm A circuit needs to be complete so that the closed system stays balanced. It's push/pull like a pulley system - when an electron leaves at one end of a component, another enters at the other end. What you get here - and what electric eels / torpedo rays use - is more of an electrostatic discharge - since the system is already unbalanced, you get an entirely one-way flow to level them out (the shocker & the shocked) - like an overflowing bucket. There're already too many/few electrons in one component, so they just leave. – Chronocidal Apr 11 '18 at 14:56
• Static discharges are totally different than "circuits" with a (dynamic) current flowing through them. All that is required is two bodies at different potentials - you don't need a common ground or a "complete" circuit for current to return, since the potentials of the bodies are different before vs after the discharge. The difference in potential builds up indirectly, and separately (before) from the actual discharge (ie rubbing your feet on a carpet transfers electrons between you and the ground, then later on you touch a doorknob and get shocked). – dwizum Apr 11 '18 at 15:10
• @Chronocidal That is not how electric eels work. Their shocks are not electrostatic in nature, and their electric organs are much akin to battery cells. According to this picture (from this paper) they have a positive and a negative end, with current flowing between the two, completing the circuit. – marcelm Apr 11 '18 at 17:50
• A unicorn peeing through its horn. Cringe. – DonQuiKong Apr 12 '18 at 4:09

In order to avoid electrocuting itself it could simply have tissues that are highly conductive (potentially metal) as skin.

It will act as a Faraday cage leaving all other tissue mostly unharmed.

• Also means that the unicorn will leave charred hoof prints, nice. – Clumsy cat Apr 11 '18 at 13:29
• See comments on @Chronocidal's answer. During a static discharge, there is no "return path" for current. The electrons jump from the horn to the other object. They don't "flow back" to the horn - there are no electrons moving otherwise. There is no need for a Faraday cage, and there will be no burned hoof prints. There would, however, need to be a mechanism for the unicorn to "recharge" the horn, which would (presumably) be much slower and safer than the discharge - as is the case with electric eels, giving a balloon a static charge by rubbing on your hair, etc. – dwizum Apr 11 '18 at 15:14
• @dwizum consider this response focused on this part of the OP: "also how it prevents electrocuting itself or its own kind?" this way it will be largely immune to shocking itself and other unicorns – Bomaz Apr 11 '18 at 15:22
• Yes - but if we assume the shock mechanism is static discharge, then it literally can't shock itself, because it is the thing that is charged, it is the thing that needs to shed/gain electrons. Other unicorns are also safe by default, since they're all (presumably) charged the same - it would be like forcing the north poles of two magnets together. Electrons would never jump from one charged body to another body with the same charge. – dwizum Apr 11 '18 at 15:26
• This does raise an interesting story potential: if you were an enemy of the unicorn and wanted to "defeat" the mechanism, you could charge a decoy oppositely to the unicorns and use it as a sort of lightning rod. – dwizum Apr 11 '18 at 15:28

## The sparks just travel through the air.

Why would this be a problem? No, air isn't as conductive as water. But electricity can travel through the air for several miles at sufficient voltages. Machines can generate sparks multiple meters long. Even the tiny static spark that you get from rubbing your sock on the carpet and touching a grounded object can be three or four centimeters.

But natural creatures don't generate voltages that high

You are talking about a literal unicorn. Some creative license is allowed. Electrocuting itself isn't a problem for the same reason it isn't a problem in real-world electricity-generating animals: their bodies are adapted to it as a natural function.

If you really must include some vague science, say that there are at least two separate organs, or parts to the single organ, that generates the electricity: a low-voltage power supply and a transformer. Stun guns ramp up quite low voltages (household batteries, maybe 20 volts at most) to tens of thousands of volts to make their sparks. The unicorn has a structure that, in addition to simply generating a current, transforms that current into a higher voltage, accumulates a charge over short amounts of time and releases it in bursts. (Note that most of what we see as sparks, lightning included, are bursts of accumulated charges and not continuous streams).

Hope this helps!

• You seem to proposing magic as a response for an question not tagged with "magic". – March Ho Apr 11 '18 at 19:31
• You are talking about a literal unicorn. Some creative license is allowed. Thank you! +1 – Renan Apr 11 '18 at 20:48
• Stun guns often use nothing more than a cheap 9v battery. Also, although high-voltage electricity can travel a very long distance through air, all it cares about is finding ground. When the source is up in the sky, it'll travel as far as it can until it hits some nice positive dirt, but if it's coming from a unicorn's horn, assuming the unicorn is about as tall as a boring old equine, the electricity would go straight down and hit the ground in front of the creature's feet. – forest Apr 11 '18 at 23:42
• @MarchHo When we talk about unicorns, magic is implied isn't it? – ElmoVanKielmo Apr 13 '18 at 8:27
• Holy crap, I only just thought of this now but I actually love this answer. Why? Because we don't actually know how lightning moves through air, right? Even lightning rods, while helpful, are not exact. Lightning can actually bypass a rod and strike someone nearby. Or the rest of your building. That's why buildings typically have dozens of rods. We don't know how lightning travels but guess what does? Unicorns. I can't find it right now but there's a video of seals hunting blindfolded - they use their whiskers to sense prey in the water. Unicorns use their horn to "see" their strike path. – Wayne Werner Apr 13 '18 at 14:29

By projecting water droplets in an arc towards it's target allowing the electricity to conduct through the air between the unicorn and it's aggressor.

It goes without saying that these droplets would create an arch of different colours caused by the refraction and dispersion of the sun's light. I imagine these colours would be red, orange, yellow, green, blue, indigo, and violet.

• Note that you actually have to have a laminar stream to have full electric conductivity, which is really hard to do. That being said, there's no reason that the electricity can't arc from drop to drop. – Wayne Werner Apr 13 '18 at 14:30
• @WayneWerner I'm not going to lie, I just wanted to work a rainbow in there somehow :) – Dark Hippo Apr 13 '18 at 14:52

The unicorn will still be electrocuted but to a lesser degree. Thinking of it like a circuit, the current will follow the path of least resistance more readily than the higher one. (Assuming that there is a path between the unicorn and the intended target. )

Adding resistance to the unicorn`s biology would be in the form of insulation. An example is keratin which is less electrically conductive. Insulation could be placed strategically to protect vital organs from the "recoil".

Body size may also play a role here. If the unicorn is much bigger than the target, the charge may be sufficient to stun the smaller organism but not enough to harm the unicorn itself.

The unicorn has a coat of gold (or some other metal, but probably only in the silver, platinum, etc. variety because unicorns). Each hair, particularly in its mane and tail, are basically long, sharp needles. The unicorn can somehow embed these needles into other creatures or objects, maybe by rubbing on it or flinging the needle-hairs with majestic head shakes or glorious tail flicks.

Any object that has a long, sharp, conductive antenna (like golden hair-needle-spikes) will attract the high voltage discharge from the unicorn. All the unicorn needs to do is get one of its hairs embedded into its target, then release its electrical fury.

Additionally the metallic coat can do other things like ground the unicorn and protect itself like a Faraday cage. Not only that, but its valuable pelt would be sought after by evil unicorn-hunting villains.

Air is a fantastic conductor as a plasma.

Simply turn the air in a column between the horn and the target into a plasma. Boom, instant lightning gun.

Granted... you'd have to explain why the unicorn doesn't just cook the target with the plasma canon.

The spirals of the horn form a resonant cavity with the atmosphere. The unicorn's horn is literally linked to the air and charges up. You know how lightning works.


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# Calculate the standard score (z-score) of the given sample mean: Round your answer to two decimal places_0 = 45 andI;

###### Question:

Calculate the standard score (z-score) of the given sample mean: Round your answer to two decimal places_ 0 = 45 and I; n = 50; 1 =

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##### BRISCOE COMPANY Worksheet for the Month Ended June 30, 2010 Trial Balance Account Titles Dr. Cr....
BRISCOE COMPANY Worksheet for the Month Ended June 30, 2010 Trial Balance Account Titles Dr. Cr. Cash $2,320 Accounts Receivable 2,440 Supplies 1,880 Accounts Payable$1,120 Unearned Revenue 240 Lenny Briscoe, Capital 3,600 Service Revenue 2,400 Salaries Expense 560 Miscellaneous Expense 160 \$7,360 ...
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##### I want to know how to solve this problem step by step. 31.7.7 Determine if the...
I want to know how to solve this problem step by step. 31.7.7 Determine if the columns of the matrix form a linearly independent set 13 -3 4 273 -1 28 3-19 Select the correct choice below and fill in the answer box to complete your choice 100 50 The columns are not linearly independent because the r...

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There are two lights that blink at regular intervals. When each one blinks, it turns on and then immediately back off; they don’t toggle. They are both off at time $t = 0$. The first one blinks at $t=p, 2p, 3p, \ldots$ seconds; the second one blinks at $t=q, 2q, 3q, \ldots$ seconds. Once they start, they both keep blinking forever. It is very exciting to see them blink at the same time (on the same second). But your patience is going to run out eventually, in $s$ seconds. Will they blink at same time between $t=1$ and $t=s$ (inclusive)? Write a program that can answer this question, quick, before they start blinking again!

## Input

Input consists of one line containing three space-separated integers $p$, $q$, and $s$. The bounds are $1 \le p, q \le 100$ and $1 \le s \le 10\, 000$. The first light blinks every $p$ seconds, the second every $q$ seconds. The value of $s$ represents the maximum number of seconds to consider when determining if the two lights blink at the same time.

## Output

Output yes if the two lights blink on the same second between time $1$ and time $s$, or no otherwise.

Sample Input 1 Sample Output 1
2 5 20

yes

Sample Input 2 Sample Output 2
4 3 11

no


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# The energy stored in the capacitor in the circuit shown below is zero at the instant...

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The energy stored in the capacitor in the circuit shown below is zero at the instant the switch is closed at t-0. At what time does the ideal op amp reaches saturation? 500 nF 12k Wv 5 V 4V 5 V voS 10k

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# Line

### What is a Line?

• A line can be defined as a straight set of points that extend in opposite directions
• has no ends in both directions(infinite)
• has no thickness
• it is one-dimensional

### What is a Line segment?

• A line segment is part of a line
• It has a beginning point and an ending point

What is a Ray?

• A ray is a part of a line that has one endpoint (i.e. starting point)
• Extends in one direction endlessly.

Geometry is the most practical branch of mathematics that deals with shapes and sizes of figures and their properties. A line is one-dimensional. That is, a line has length, but no width or height. A line extends forever in both directions. A line is uniquely determined by two points.

The number line is well-suited to support informal thinking strategies of students because of its inherent linearity. In contrast to blocks or counters with a “set-representation” orientation, young children naturally recognize marks on a number line as visual representations of the mental images that most people have when they learn to count and develop an understanding of number relationships. It is important to note the difference between an “open number line” (shown below) and a ruler with its predetermined markings and scale.

### An open number line:

The open line allows students to partition, or subdivide, the space as they see fit, and as they may need, given the problem context at hand. In other words, the number line above could be a starting point for any variety of number representations, two of which are shown below: the distance from zero to 1, or the distance from zero to 100. Once a second point on a number line has been identified, the number line moves from being an open number line, to a closed number line. In addition, the open number line allows for flexibility in extending counting strategies from counting by ones, for example, to counting by tens or hundreds all on the same sized open number lines.

### Closed number lines:

Ray is part of a line; has one endpoint and continues on and on in one direction  As its one end is non-terminating, length of the line  cannot be measured.

The definition of a ray depends upon the notion of between points on a line. It follows that rays exist only for geometries for which this notion exists

We can name a ray using its starting point and one other point that is
on the ray: this is ray AC or ray. Or, we can name a ray using a lowercase letter: this is ray r.

### Other Types of Lines include:

#### 1.Tangent lines

The Tangent is a straight line which just touches the curve at a given point. The normal is a straight line which is perpendicular to the tangent. To calculate the equations of these lines we shall make use of the fact that the equation of a straight line passing through the point with coordinates (x1, y1) and having gradient m is given by

$\frac{y-y_{1}}{x-x_{1}}=1$

We also make use of the fact that if two lines with gradients m1 and m2 respectively are perpendicular, then m1m2 = −1.

Example: Suppose we wish to find points on the curve y(x) given by $x^{3} – 6x^{2} + x + 3$ where the tangents are parallel to the line y = x + 5.

Solution: If the tangents have to be parallel to the line then they must have the same gradient. The standard equation for a straight line is y = mx + c, where m is the gradient. So what we gain from looking at this standard equation and comparing it with the straight line y = x + 5 is that the gradient, m, is equal to 1. Thus the gradients of the tangents we are trying to find must also have gradient 1.

We know that if we differentiate y(x) we will obtain an expression for the gradients of the tangents to y(x) and we can set this equal to 1. Differentiating, and setting this equal to 1 we find

$\frac{dy}{xy}=3x^{2}-12x+1=1$

from which

$3x^{2} – 12x = 0$

This is a quadratic equation which we can solve by factorisation

$3x^{2} – 12x = 0$

$3x (x-4) = 0$

$\Rightarrow 3x = 0 \;\; or \;\; (x-4) = 0$

$\Rightarrow x = 0 \;\; or x = 4$

Now having found these two values of x we can calculate the corresponding y coordinates. We do this from the equation of the curve: $y = x^{3} – 6x^{2} + x + 3$.

when x = 0:    y = $(0)^{3} – 6(0)^{2} + 0 + 3 = 3$

when x = 4: y = $(4)^{3} – 6(4)^{2} + 4 + 3= -25$

So the two points are (0, 3) and (4, −25)

These are the two points where the gradients of the tangent are equal to 1, and so where the tangents are parallel to the line that we started out with, i.e. y = x + 5.

#### 2. Secant lines

A line in the plane is a secant line to a circle if it meets the circle in exactly two points. It is also equivalent to the average rate of change, or simply the slope between two points. The average rate of change of a function between two points and the slope between two points are the same thing.

In the left figure above,

$0=\frac{1}{2}$ (arc A C + arc B D),

while in the right figure,

$\phi =\frac{1}{2}$ (arc R T – arc S Q),

Where arc AB denotes the angular measure of the arc AB

#### Practise This Question

The common end point where two rays meet to form an angle is called:


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Is the last digit of integer x^2 - y^2 a zero?

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Is the last digit of integer x^2 - y^2 a zero? [#permalink]  13 Jun 2010, 09:06
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Is the last digit of integer x^2 - y^2 a zero?

(1) x - y is an integer divisible by 30
(2) x + y is an integer divisible by 70

I would like to know, if the Q would have been that x and y are integers, then each statement would be sufficient to ans the Q.

Can somebody please explain why?
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Re: Gmat club test 24 [#permalink]  13 Jun 2010, 10:52
gmatcracker2010 wrote:
Is the last digit of integer x^2 - y^2 a zero?

1. x - y is an integer divisible by 30
2. x + y is an integer divisible by 70

Oa is
[Reveal] Spoiler:
c

I would like to know, if the Q would have been that x and y are integers, then each statement would be sufficient to ans the Q.

Can somebody please explain why?

The question doesn't need to say that x and y are integers...

I don't understand the OA. The question says "integer x^2-y^2", which means it is already given that x^2-y^2 is an integer.

I. x^2-y^2=integer=(x+y)(x-y) Thus, if x-y is an integer divisible by 30, then x^2-y^2=30n*(x+y), which always ends in a zero as long as x+y is an integer. And x+y has to be an integer in order for x^2-y^2 to be an integer.

Same reasoning for II - where am I wrong?
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Re: Gmat club test 24 [#permalink]  13 Jun 2010, 11:22
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gmatcracker2010 wrote:
Is the last digit of integer x^2 - y^2 a zero?

1. x - y is an integer divisible by 30
2. x + y is an integer divisible by 70

Oa is
[Reveal] Spoiler:
c

I would like to know, if the Q would have been that x and y are integers, then each statement would be sufficient to ans the Q.

Can somebody please explain why?

Given: $$x^2-y^2=(x-y)(x+y)=integer$$. Question is the units digit of this integer zero, which can be translated as is $$x^2-y^2$$ divisible by 10.

(1) $$x-y=30m$$ (where $$m$$ is an integer) --> if $$x$$ and $$y$$ are integers, then $$x+y=integer$$ and $$x^2-y^2=30m*(x+y)=30m*integer$$, which is divisible by 10 BUT if $$x=30.75$$ and $$y=0.75$$, then $$x^2-y^2=(x-y)(x+y)=30*31.5=945=integer$$, which is not divisible by 10. Not sufficient.

(2) $$x+y=70n$$ (where $$n$$ is an integer) --> if $$x$$ and $$y$$ are integers, then $$x-y=integer$$ and $$x^2-y^2=(x-y)*70n=integer*70n$$, which is divisible by 10 BUT if $$x=69.75$$ and $$y=0.25$$, then $$x^2-y^2=(x-y)(x+y)=69,5*70=4865=integer$$, which is not divisible by 10. Not sufficient.

(1)+(2) $$x^2-y^2=30m*70n=integer$$, Which is divisible by 10. Sufficient.

Answer: C.

Hope it's clear.
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m24 #11 [#permalink]  03 Jan 2011, 17:42
I got this question correct, but I would just like to understand the problem in more depth. The explanation in the test review was rather brief. Can someone please elaborate. Thanks!

Is the last digit of integer $$x^2 - y^2$$ a zero?

1. $$x - y$$is an integer divisible by 30

2. $$x + y$$ is an integer divisible by 70
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Re: m24 #11 [#permalink]  04 Jan 2011, 01:21
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Re: Is the last digit of integer x^2 - y^2 a zero? 1. x - y is [#permalink]  22 Nov 2011, 14:18
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Hello Bunuel,

Thanks very much for your explanation to this question, which by the way I came across while working through your excellent "700+ GMAT Data Sufficiency Questions With Explanations" document (wow -- they're tough!)

I had a quick question on this particular question. Firstly, I believe it is the case that x^n - y^n is always divisible by (x-y), and is divisible by (x+y) when the powers of x and y are even.

If I have not mis-stated, then would you mind pointing out the flaw in the following reasoning?
(1) "x-y is an integer divisible by 30" >> adding in the rule that "if a is a factor of b and b is a factor of c, then a is a factor of c", then: x^n - y^n is an integer divisible by (x-y), and (x-y) is an integer divisible by 30, then does it not follow that x^n - y^n is also an integer divisible by 30? and this is sufficient to say that x^n - y^n must end in zero.
(2) similar logic applies since the power of x and y are indeed even and therefore divisible by (x+y).

Would be very grateful if you could take 2 mins to point out the error in that alternative method.

Thanks very much for your contributions and for taking the time to help.
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Re: Is the last digit of integer x^2 - y^2 a zero? 1. x - y is [#permalink]  04 Jan 2012, 07:27
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I am not sure if this is the best approach to handle such problems but the following is the way I dealt with this problem.

Question: Is the last digit of (x + y)(x - y) zero?

Statement 1: x - y is an integer divisible by 30
Thus, the least possible value for (x - y) is 30. However, this does not tell us anything about (x + y). If (x + y) = 0.1, then the last digit of the product (x + y)(x - y) is not 0. Thus INSUFFICIENT.

Statement 2: x + y is an integer divisible by 70
Thus, the least possible value for (x + y) is 70. However, this does not tell us anything about (x - y). Using a similar logic as above, if (x - y) = 0.1, then the last digit of their product (x + y)(x - y) is not 0. Thus INSUFFICIENT.

Combining the two statements, we have the least possible value of (x + y) as 70 and the least possible value of (x -y) as 30. Thus, the least possible value of their product will always be divisible by 10 and thus their product will always have zero as the last digit. SUFFICIENT.

Answer: C
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Re: Is the last digit of integer x^2 - y^2 a zero? 1. x - y is [#permalink]  04 Jan 2012, 13:21
Very nice problem. No info on x, y so they could be frac, int

Rephrase: (x+y)(x-y) = ..htu, is u = 0?

1. x-y = 30, so x+y could be (53.3+23.3) = 76.6. Therefore, (x+y)(x-y) will clearly not have u = 0. On the other hand, (53+23) = 76. This will have u = 0. Insufficient.
2. x+y = 70. Same deal as 1.

Together, both x+y and x-y are explicitly mentioned. Sufficient - C.
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Re: Gmat club test 24 [#permalink]  28 May 2012, 00:23
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Bunuel wrote:
gmatcracker2010 wrote:
Is the last digit of integer x^2 - y^2 a zero?

1. x - y is an integer divisible by 30
2. x + y is an integer divisible by 70

Oa is
[Reveal] Spoiler:
c

I would like to know, if the Q would have been that x and y are integers, then each statement would be sufficient to ans the Q.

Can somebody please explain why?

Given: $$x^2-y^2=(x-y)(x+y)=integer$$. Question is the units digit of this integer zero, which can be translated as is $$x^2-y^2$$ divisible by 10.

(1) $$x-y=30m$$ (where $$m$$ is an integer) --> if $$x$$ and $$y$$ are integers, then $$x+y=integer$$ and $$x^2-y^2=30m*(x+y)=30m*integer$$, which is divisible by 10 BUT if $$x=30.75$$ and $$y=0.75$$, then $$x^2-y^2=(x-y)(x+y)=30*31.5=945=integer$$, which is not divisible by 10. Not sufficient.

(2) $$x+y=70n$$ (where $$n$$ is an integer) --> if $$x$$ and $$y$$ are integers, then $$x-y=integer$$ and $$x^2-y^2=(x-y)*70n=integer*70n$$, which is divisible by 10 BUT if $$x=69.75$$ and $$y=0.25$$, then $$x^2-y^2=(x-y)(x+y)=69,5*70=4865=integer$$, which is not divisible by 10. Not sufficient.

(1)+(2) $$x^2-y^2=30m*70n=integer$$, Which is divisible by 10. Sufficient.

Answer: C.

Hope it's clear.

Quote:
Hello Bunuel,

Thanks very much for your explanation to this question, which by the way I came across while working through your excellent "700+ GMAT Data Sufficiency Questions With Explanations" document (wow -- they're tough!)

I had a quick question on this particular question. Firstly, I believe it is the case that x^n - y^n is always divisible by (x-y), and is divisible by (x+y) when the powers of x and y are even.

If I have not mis-stated, then would you mind pointing out the flaw in the following reasoning?
(1) "x-y is an integer divisible by 30" >> adding in the rule that "if a is a factor of b and b is a factor of c, then a is a factor of c", then: x^n - y^n is an integer divisible by (x-y), and (x-y) is an integer divisible by 30, then does it not follow that x^n - y^n is also an integer divisible by 30? and this is sufficient to say that x^n - y^n must end in zero.
(2) similar logic applies since the power of x and y are indeed even and therefore divisible by (x+y).

Would be very grateful if you could take 2 mins to point out the error in that alternative method.

Thanks very much for your contributions and for taking the time to help.

Hi Bunuel,

I also have the same doubt. Please suggest how is this approach incorrect.

Thanks
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Re: Gmat club test 24 [#permalink]  28 May 2012, 02:46
kunalbh19 wrote:
If I have not mis-stated, then would you mind pointing out the flaw in the following reasoning?
(1) "x-y is an integer divisible by 30" >> adding in the rule that "if a is a factor of b and b is a factor of c, then a is a factor of c", then: x^n - y^n is an integer divisible by (x-y), and (x-y) is an integer divisible by 30, then does it not follow that x^n - y^n is also an integer divisible by 30? and this is sufficient to say that x^n - y^n must end in zero.
(2) similar logic applies since the power of x and y are indeed even and therefore divisible by (x+y).

Would be very grateful if you could take 2 mins to point out the error in that alternative method.

Thanks very much for your contributions and for taking the time to help.

So your logic is:

x-y is an integer divisible by 30
a = 30, b = x-y, c = (x-y)(x+y)
a is a factor of b, b is a factor of c, therefore a is a factor of c

But you have made an assumption here that (x+y) is an integer (i.e. that c / a = (x+y) = integer), which is not necessarily true.
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Re: Is the last digit of integer x^2 - y^2 a zero? 1. x - y is [#permalink]  28 May 2012, 04:00
Ohhh.. ok.. i think i got it now.. thanks
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Re: Is the last digit of integer x^2 - y^2 a zero? 1. x - y is [#permalink]  31 May 2012, 20:00
A tricky question.... Overlooked x+y or x-y could be in Decimals....
Must not skip this
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  01 Jun 2012, 01:01
Amazing question... Thanks

Posted from my mobile device
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  21 Sep 2013, 13:13
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  19 Oct 2013, 01:03
gmatcracker2010 wrote:
Is the last digit of integer x^2 - y^2 a zero?

(1) x - y is an integer divisible by 30
(2) x + y is an integer divisible by 70

This is how I approached the question:

Statement 1:
$$x - y =30$$
$$x = y + 30$$

Now, If any we take square any value of $$X$$ and$$Y + 30$$the last digit of their difference will be O.

Let Y = -2 => then X = 28 and $$(28)^2 - (-2)^2 = 784 - 4 = 780$$
Let Y = -1 => then X = 29 and $$(29)^2 - (1)^2 = 841 - 1 = 840$$
Let Y = 0 => then X = 30 and $$(30)^2 - (0)^2 = 900 - 0 = 900 0$$
Let Y = 1 => then X = 31 and $$(31)^2 - (1)^2 = 961 - 1 = 960$$
Let Y = 2 => then X = 32 and $$(32)^2 - (2)^2 = 1024 - 4 = 1020$$
Let Y = 3 => then X = 33 and $$(33)^2 - (3)^2 = 1089 - 9 = 1080$$
Let Y = 4 => then X = 34 and $$(34)^2 - (4)^2 = 1156 - 16 = 1140$$
....
Let Y = 25 => then X = 55 and $$(55)^2 - (25)^2 = 3025 - 625 = 2400$$
....
Let Y = 107 => then X = 137 and $$(137)^2 - (107)^2 = 18769 - 11449 = 7320$$

Therefore, Sufficient!

Statement 2
$$X + Y = 70$$
$$Y = 70 - X$$
Let X = -2 => then Y = 72 and $$(-2)^2 - (72)^2 = 4 - 5184 = - 5180$$
Let X = -0 => then Y = 70 and $$(0)^2 - (70)^2 = 0 - 4900 = - 4900$$
Let X = -1 => then Y = 69 and $$(-2)^2 - (72)^2 = 4 - 5184 = - 5180$$

Sufficient

Therefore, option D

Now can anyone kindly point out the problem in this execution?
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  19 Oct 2013, 02:19
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Expert's post
suk1234 wrote:

Statement 1:
$$x - y =30$$
$$x = y + 30$$

Now can anyone kindly point out the problem in this execution?

Firstly, if it is said that (x-y) is divisible by 30, it doesn't mean x-y = 30. It just means (x-y) is a multiple of 30.

Again, you have taken only integral values for x,y. The fact that x,y can be non-integral values also makes your solution incorrect.

Please refer to the post above by Bunuel to get the correct solution.

Hope this helps.
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  08 Dec 2013, 22:44
gmatcracker2010 wrote:
Is the last digit of integer x^2 - y^2 a zero?

(1) x - y is an integer divisible by 30
(2) x + y is an integer divisible by 70

I would like to know, if the Q would have been that x and y are integers, then each statement would be sufficient to ans the Q.

Can somebody please explain why?

-- Well.. i see there is a rule which says...
"RULE: for x^n-y^n:
is ALWAYS divisible by (x-y) .
is divisible by (x+y) when n is even. "

Now, in this case...
1) x^2 - y^2 is divisible by (x-y), which in turn is divisible by 30. so if the property holds true, x^2 - y^2 should be divisible by 30, or in 10*3. Which would imply it would end with a 0.
2) Again, going by the above rule, x^2 - y^2 should be divisible by 70. which would imply it is divisible by 10. and thus ends with 0.

I am probably misinterpreting the property , or have got it wrong. Could someone please explain?
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  08 Dec 2013, 23:24
Expert's post
gmarchanda wrote:
gmatcracker2010 wrote:
Is the last digit of integer x^2 - y^2 a zero?

(1) x - y is an integer divisible by 30
(2) x + y is an integer divisible by 70

I would like to know, if the Q would have been that x and y are integers, then each statement would be sufficient to ans the Q.

Can somebody please explain why?

-- Well.. i see there is a rule which says...
"RULE: for x^n-y^n:
is ALWAYS divisible by (x-y) .
is divisible by (x+y) when n is even. "

Now, in this case...
1) x^2 - y^2 is divisible by (x-y), which in turn is divisible by 30. so if the property holds true, x^2 - y^2 should be divisible by 30, or in 10*3. Which would imply it would end with a 0.
2) Again, going by the above rule, x^2 - y^2 should be divisible by 70. which would imply it is divisible by 10. and thus ends with 0.

I am probably misinterpreting the property , or have got it wrong. Could someone please explain?

The property holds true if x and y are integers but we are not given that.
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  10 Dec 2013, 10:16
since the first statement says that x-y is an integer divisible by 30 it means x-y can be any multiple of 30
so (x+y)*(x-y)=end with 0
we can pluggin some numbers and see
lets assume (x+y) as any integer for e.g= 2
so 2*30 = ends with 0.
now lets take a non integer 1/5*30 = 6 so doesnot end with 0
since it is a yes for one example and no for other the first statement is insufficient.

second statement says that (x+y) is divisible by 70
so (x+y)*(x-y)=end with 0
we can pluggin some numbers and see
lets assume (x-y) as any integer for e.g= 2
so 70*2 = ends with 0.
now lets take a non integer 140*1/5 = 28 so doesnot end with 0
since it is a yes for one example and no for other the second statement is insufficient.

both statements individually do not satisfy the condition.
when we combine them it is sufficient hence the answer is C
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Re: Is the last digit of integer x^2 - y^2 a zero? [#permalink]  30 Nov 2014, 14:30
IS the last digit of INTEGER (x^2 - y^2) a zero?

1. x- y is an integer divisible by 30. Lets call it 30n where n is a positive integer.

Thus x^2 - y^2 = (x + y)*(x - y) = (x + y) * 30n

Now if (x + y) is an integer, it will lead to (x^2 - y^2) having units digit of zero.

eg. x = 60, y = 30; x - y = 30; x + y = 90 => x^2 - y^2 = 2700

However if (x+y) is non-integer, this would lead to (x^2 - y^2) possibly having a non-zero units digit.

eg. x = 60.3, y = 30.3 => x - y = 30 but x + y = 90.3 => x^2 - y^2 = 2709

INSUFFICIENT

2. x + y is an integer divisible by 70. Let’s call it 70m.

This is exactly analogous to statement 1, and we will still be able to choose non-integer values of x and y that lead to non-zero units digits in x^2 - y^2

INSUFFICIENT

Both 1 and 2:

x - y = 30n and x + y = 70m

Thus x^2 - y^2 = 2100mn

m and n are positive integers, so we can be sure that x^2 - y^2 has a units digit of zero.

SUFFICIENT

Correct C.

Class GMAT MATH Claudio Hurtado, GMAT CHILE
Re: Is the last digit of integer x^2 - y^2 a zero?   [#permalink] 30 Nov 2014, 14:30
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# Worrying about logic and foundations too much [closed]

I am studying undergraduate physics and I am always interested in why we consider certain physical models instead of the others. That ultimately leads to the question why we consider one mathematical description over the other in a specific physics problem.

Unfortunately these kinds of questions now make me struggle to learn anything new. In this question I would be happy to receive some advice or some personal experience.

I do not know why but I have the philosophy that we do not understand physics because we do not understand mathematics good enough. Therefore, I took some mathematics courses to make the best of my physics education. I took, for example, set theory, which is assumed to be foundations of mathematics, but I had trouble understanding it as when we wrote something intuitively (such as index sets) then I had great trouble as I wanted for it to be defined well and not just "intuitive". I understood that even in mathematics a lot of proofs are based on our intuition probably assuming that good mathematician could make intuitive argument rigorous. I tried then to learn basics of logic to make my arguments more rigorous and then started spending too much time on understanding why we choose one definition over the other (for example, why we assume vacuous truth). In the end, no matter what subject I take, I get destroyed by the need to try to explain myself why we make definitions the way we do.

For example, the last thing I was thinking about was why do we use real numbers in the physics? For now, the most satisfactory answer would be that we measure some things using some apparatus, say, length with a ruler, and we divide it in equal parts to make it easier (say, in 10 parts). Then we measure length. But probably the object for which the length is being measured does not coincide with the marks on the ruler. So we divide each interval again in 10 parts. Then we make iteration (possibly infinite, whatever that means) until we obtain a number which would be infinite decimal by construction. And real numbers might be constructed as the set of all infinite decimals.

Of course, I could assume that there exists set of real numbers with some properties that we want and then prove all the theorems, for example, in real analysis. But then why do we need exactly these properties?

Anyways, I have several possible answers on why this is happening :

1) I am not mathematically mature enough and this is something that stops being a problem with age,

2) I am not doing mathematics right and this kind of behaviour just has to be stopped by myself because it is destructive (I spend too much time on it instead of studying things that I could apply and to do research on).

## closed as off-topic by Andrés E. Caicedo, B. Mehta, Chris Custer, user, José Carlos SantosMay 6 '18 at 19:46

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is not about mathematics, within the scope defined in the help center." – Andrés E. Caicedo, B. Mehta, Chris Custer, user, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.

• Are you able to walk ? Do you know enough physics and chemistry and physiology in order to produce a good and detailed explanation of the "mechanism" of walking ? If NOT, why you are not "destroyed" and you are still walking ? (You can try the mental experiment with different example based on digestion or other...) – Mauro ALLEGRANZA Mar 4 '18 at 16:59
• I sympathize with you, and I think it's difficult to know how to "calibrate" oneself as a mathematician. How much time should I spend grokking foundational issues and how much time should I spend learning more advanced topics? (Are some mathematicians lucky that their natural calibration happens to be highly effective for research?) One comment is that we can introduce whatever model we want, and then check experimentally how well it describes the real world. Another comment is that you can recognize that a mathematical result is neat or beautiful even if you aren't sure about its foundation. – littleO Mar 4 '18 at 17:03
• Another joke a tittle less stupid: there is no evidence that math used to design bridges and spacecrafts will fail because we are not able to settle the Continuum Hypotheses. – Mauro ALLEGRANZA Mar 4 '18 at 17:06
• Conversely, math and physics need people who worry about foundational issues. If you find yourself naturally attracted to these things, there’s absolutely nothing wrong with it, IMO. It’s not quite the same, but Borovik shares stories that Gelfand was particularly fond of going “back to the basics” to start discussions, which apparently was quite fruitful for him and his collaborators (search this book for “gelfand”, it’s the 2nd occurrence. Beware, it is a large-ish file...). – pjs36 Mar 4 '18 at 18:29
• Regarding rigor. You can make mathematics completely rigorous for yourself. I mean, there are formal deductive systems, software tools. I worried too, then I found these things, learned them, and stopped worrying. I do not use them though. Maybe you should try it. Regarding definitions. If you want to improve definitions, this is a worthy enterprise actually. Mathematics is not set in stone. Aimless worrying is not good, of course. Obviously, you like mathematics, logic, or philosophy more than physics. – beroal Mar 4 '18 at 18:33

The use of the real numbers can be understood in some sense by their construction and in another by their utility:

1. The natural numbers are just that, natural. If you want to describe the world around you, then counting is vital. Integers arise from that as you want some semblance of direction, i.e. if you give me a cat i have +1 cats, remove a cat and i have -1. The rationals then arise from the desire to partition things, e.g. I have 5 chocolate bars and six friends. Where, then, do the reals come from? This is a little more involved but if you take the rationals and then add in the limit points for every possible Cauchy sequence, you end up at the reals. There are all sorts of questions in nature where you want to find the limit of processes, so the reals end up being a very natural construction. The complex numbers come from algebraic closure but as regards real numbers, we use them because we want all of the above to hold and we also want an infinite field. There's also a historical basis, that moving from naturals->integers->rationals->reals->complexes is how human thinking has progressed.

2. Utility. Physics does not determine absolute truth (i suspect expecting it to may be the root your philosphical dilemma). Nor is it supposed to. Physics constructs models of the world to allow accurate predictions to be made, nothing more. If i want to hit a boat with my cannon, i can use physics and be pretty confident it'll provide me the correct solution. If you want to heat a house, physics allows us to design systems that will do that within constraints of power, etc. Do we obtain the absolute truth of the universe? No. Are we always infallible? No. But physics is an incredibly useful tool. Universal truths are for philosophers to debate. Note that mathematics is not, in my opinion, universal truth either. We use ZFC for reasons of history and practicality. It is not the only option open to us.

• (It is not me who downvoted your answer.) I am not sure that the OP is asking about universal truths. I guess that the OP is just seeking better truths. – beroal Mar 4 '18 at 18:22
• Thank you for your answer! I agree with what you are saying. Just a little clarification - I understand that physics is basically only about making the predictions. But as you know there are a lot of predictions that are wrong and there are a lot of phenomena for which we do not have prediction. And I just think that it might be due to the fact that in physics we do not use careful definitions and we do not specify presicely axiomatic systems we use. But thinking about this makes my progress as a student very slow. – Daniels Krimans Mar 6 '18 at 16:21

There are parts of physics like quantum field theory that still have not been made rigorous, so if you insist on rigor in everything you do that will eventually be a serious problem in physics! Of course if you can reign in this impulse enough to get that far in physics, then perhaps your drive for rigor will one day help put that subject on firmer foundations.

Keep in mind that much of physics historically was done without being completely rigorous. Newton was doing calculus using differentials, yet he made it work quite well. In my experience the people who are best at physics are very good at straddling the line between intuition and rigor, and history shows that this ability was somewhat crucial in the development of physics. Imagine if we waited until all the mathematical foundations of calculus were in place before investigating classical mechanics using calculus. Physics would have been retarded by 2-3 centuries at least.

The people who are good at using math properly without using it rigorously are usually people who would make very good mathematicians, but they don't need the rigor to understand what is going on. Most physics students (again in my experience) are just mimicking this smaller group, and while they are certainly capable of computing, they don't entirely understand what they're doing, and often they don't realize that they don't understand what they're doing. Then there is a third group of people in physics for whom this kind of thing rubs them the wrong way or is too difficult, so they often drop physics and switch to math (like me!).

In summary, if you plan to stay in physics I think it is important that you learn to think intuitively about math even when you don't have rigor, and it can be a good exercise to see how much you can infer from the physics books before you open a math book to see how it should really be done. Developing this ability may help you one day in developing your own mathematical formulation of various physics.

Of course if this is too painful or difficult for you, as it was for me, then perhaps you are better suited for math.

• Thank you for your point of view on this and for kind words of encouragement! I understand what you were saying about physics, but on the other hand I feel that nowadays theory in physics is very abstract and we do not have enough experimental techniques to check them (for example, high energy physics). Then I feel the best way to argue which model for physics would work the best is only by rigorous mathematics. Could you expand more on "but they don't need rigor to understand what is going on", please? – Daniels Krimans Mar 6 '18 at 16:18

## What is the foundation of math?

Ultimately, logic. But getting from logic to set theory is much more difficult than it seems. Mathematicians more or less agreed on some self-evident truths they then call axioms (which kinda can be translated as "worthy" from Greek). You can almost measure the level of agreement on these axioms by counting the number of different axiomatic systems.

If you want to, you can basically construct almost all mathematical objects by using the empty set $\{\}$ and the other axioms from ZFC (see here and here). There are some exceptions though (like categories and strongly inaccessible cardinals), but you can try to add axioms to add the exceptions to the mix (although they will stay incomplete).

## Why are the definitions done the way they are?

Short answer: Because they work. Longer answer: They developed this way. Set theory alone has a rich history (ever wondered where $\in$/$\varepsilon$ comes from?), a short overview of it is given on Wikipedia. The development of set theory actually led to the question "What is the foundation of math?" (see first part of answer). But back to this question: It is kind of an evolutionary process, like all of science. What works well, survives. I managed myself to convince a group of students I tutored why it's good that linear maps between vector spaces are defined they way they are (because their kernel and image are then vector spaces again).

When you read scientific papers from the frontier of mathematical knowledge (e.g. on arxiv.org) you will often see new definitions which you may see never again or slightly changed if their current form turns out to have "flaws". But the mathematical basics, like almost anything done in the undergraduate courses, stand firm since about half a century and there is nothing to indicate that this would change soon.

## Are you being too rigorous?

Well, that depends. Do you like to be rigorous?

If not, then stop. Especially as a physics, you are allowed to do some handwaving. If you are simply ignorant of conditions like even being able to exchange limit and integral, you are to worry much less in most of your day-to-day physics, as I am told. The word "ignorant" here is used without prejudice, ignorance is bliss. I would fail at studying physics because I simply can't overlook these details and would invest too much time in it.

If you do like being rigorous, you may prefer doing more math, as others suggested. If usual math still takes to many intuition from you where you don't agree, I suggest working with theorem checkers, e.g. Mizar. If that still isn't rigorous enough for you, I don't know what to tell you anymore.

I have fun writing Mizar articles especially because it's so rigorous. Proving seemingly trivial things can take quite some time, because it isn't as trivial as it seems. If you want to learn more about Mizar, I recommend the PDF "Mizar in a Nutshell" or the PDF "Writing a Mizar article in nine easy steps". The official homepage is here, you may also like the HTML-linked articles, e.g. the one about boolean properties of sets (you can click on "proof").

This is a lemma I needed recently: Let $I$ be a set and $\{X_i\}_{i\in I}$ a family of sets (i.e. a function $X$ with domain $I$ and for all $i\in I$ holds $X_i$ is a set). Let $\prod_{i\in I} X_i$ denote the product set as usual (more formally it is the set of all functions $f$ with domain $I$ such that for all $i\in I$ holds $f(i)\in X_i$). Now fix $j,k\in I$, let A, B denote non empty sets and define the set families $\{Y_i\}_{i\in I}$, $\{Z_i\}_{i\in I}$ by

$$Y_i=\left\{\begin{array}{rl} A&\text{if i=j}\\ X_i&\text{if i\neq j} \end{array}\right.\qquad Z_i=\left\{\begin{array}{rl} B&\text{if i=k}\\ X_i&\text{if i\neq k} \end{array}\right.$$

Now does $\prod_{i\in I}Y_i = \prod_{i\in I} Z_i$ imply $j=k$ and $A=B$? In fact not, as I found out, the trivial case $A=X_j, B=X_k$ is the exception (but it holds otherwise). And as "without loss of generality" as this may be, I have fun discovering these subtleties and proving them, rigorously :)


text:
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 Automorphisms of the groups $SL_3(K)$, $GL_3(K)$V. M. Petechuk 657 Prime Mal'tsev algebrasV. T. Filippov 669 Theory of spaces with dominating mixed derivativeYu. A. Brychkov 679 A criterion for a convolution to be epimorphic in arbitrary regions of the complex planeO. V. Epifanov 695 Divergence of the third derivatives of interpolational cubic splines in the metrics of $L_p$N. L. Zmatrakov 707 Expansion of analytic functions in series of rational functionsYu. F. Korobeinik 723 Expansions of functions into Dirichlet seriesN. I. Rakhimkulov 739 Class of differential operators with abnormal behavior of the spectrum of the Dirichlet problemA. I. Karol', M. Z. Solomyak 747 Rectification of measurable actions of solvable Lie groups of type $(R)$A. V. Safonov 753 Total variation of optimal control in linear systemsD. B. Silin 761 Operators generated by homeomorphisms of a circle conjugate to a rotationA. B. Antonevich, V. B. Ryvkin 773 Approximative properties of sets in Hilbert spaceV. S. Balaganskii 785 Homotopy classification of a class of continuous mappingsV. T. Dmitrienko, V. G. Zvyagin 801


text:
## Doctoral Dissertations

Dissertations that have an embargo placed on them will not be available to anyone until the embargo expires.

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

2016

May

Eyal Markman

#### Subject Categories

Algebraic Geometry

#### Abstract

Consider any rational Hodge isometry

$\psi:H^2(S_1,\QQ)\rightarrow H^2(S_2,\QQ)$ between any two K\"ahler $K3$

surfaces $S_1$ and $S_2$. We prove that the cohomology class of $\psi$ in $H^{2,2}(S_1\times S_2)$

is a polynomial in Chern classes of coherent analytic sheaves

over $S_1 \times S_2$. Consequently, the cohomology class of $\psi$ is algebraic

whenever $S_1$ and $S_2$ are algebraic.


text:
A general search for new phenomena at HERA

Aaron, F D; Alexa, C; Andreev, V; Antunovic, B; Aplin, S; Müller, K; Nowak, K; Robmann, P; Schmitz, C; Schöning, A; Straumann, U; Truöl, P (2009). A general search for new phenomena at HERA. Physics Letters B, 674(4-5):257-268.

Abstract

A model--independent search for deviations from the Standard Model prediction is performed using the full $e^\pm p$ data sample collected by the H1 experiment at HERA. All event topologies involving isolated electrons, photons, muons, neutrinos and jets with transverse momenta above 20 GeV are investigated in a single analysis. Events are assigned to exclusive classes according to their final state. A dedicated algorithm is used to search for deviations from the Standard Model in the distributions of the scalar sum of transverse momenta or the invariant mass of final state particles and to quantify their significance. Variables related to angular distributions and energy sharing between final state particles are also introduced to study the final state topologies. No significant deviation from the Standard Model expectation is observed in the phase space covered by this analysis.

A model--independent search for deviations from the Standard Model prediction is performed using the full $e^\pm p$ data sample collected by the H1 experiment at HERA. All event topologies involving isolated electrons, photons, muons, neutrinos and jets with transverse momenta above 20 GeV are investigated in a single analysis. Events are assigned to exclusive classes according to their final state. A dedicated algorithm is used to search for deviations from the Standard Model in the distributions of the scalar sum of transverse momenta or the invariant mass of final state particles and to quantify their significance. Variables related to angular distributions and energy sharing between final state particles are also introduced to study the final state topologies. No significant deviation from the Standard Model expectation is observed in the phase space covered by this analysis.

Citations

5 citations in Web of Science®
14 citations in Scopus®

Altmetrics

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Physics Institute 530 Physics English 2009 18 Nov 2009 13:42 05 Apr 2016 13:32 Elsevier 0370-2693 10.1016/j.physletb.2009.03.034 http://arxiv.org/abs/0901.0507v1
Permanent URL: http://doi.org/10.5167/uzh-23792


text:
# Programming Task 3: Robot Rally

### Reviewed October 29th, 2014

#### Guidelines

• You are working in teams of four (two sets of partners). This will require workplace-style maturity.
• These tasks will test your ability to put notions into practice within a team setting.
• Submit your teams' working code to andy at novocin.com prior to your scheduled review.
• I will run the code on real inputs using ideone.com with the C++11 environment.
• Presentation day will consist of a scheduled 45 minute meeting.
• At that meeting I will run the code.
• I will select one person from your team using a random integer, that person will explain the code and answer my questions.
• I will assign a grade that day for all of you based on the randomly selected team member's explanation.
• It is your responsibility to make sure that everyone on your team understands the concepts behind your solution.
• Coding in C++ is a soft requirement of the course, this method allows the coding aspects of the project to be spread while ensuring some practical awareness.

### The Programming tasks for this assignment:

This assignment involves guiding a robot through a treasure cave. You are going to be given an ASCII map of a $$20 \times 20$$ rectangular grid cave. Each cell in the grid is surrounded by four corner + symbols. East/west walls are marked by a |, north/south walls are marked by a -, and a space where a wall would be means an open corridor. Each cell has one of four symbols in it: .,$,s, or x. There will be only one s in the top left cell, one x in the bottom right cell, and four $ treasures scattered about the cave. All other (394) cells will have a . inside.

### A sample input:

+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|s .|.|.|.|.|. . .|. .|. . . . . . .|. .|
+ +-+ + + + +-+ +-+-+ +-+-+-+ +-+-+ +-+ +
|. . .|. .|$.|. . . . .|. . . .|. . . .| +-+-+ +-+ + +-+ +-+-+-+ +-+ + + +-+ +-+ + |. .|.|. . . . . .|. . . . .|.|.|. .|. .| + +-+ +-+ +-+ + +-+-+-+ + +-+-+-+-+ +-+ + |. . . . .|. .|. .|.|. .|.|. . .|. .|. .| + +-+-+-+ + +-+ + + + + +-+-+-+ +-+-+-+ + |.|. . . .|. .|.|.|.|.|. . .|. . .|.|. .| + +-+-+ + + +-+-+-+ +-+ +-+-+-+-+ + +-+ + |.|.|.|.|.|.|. .|. . . . .|. . .|. . . .| +-+ + + +-+ +-+ +-+ +-+-+-+-+-+ +-+-+ + + |. . . .|. .|. . . . .|. . . . . .|. .|.| +-+-+-+ + +-+-+ +-+ + + +-+ +-+ + +-+-+ + |. . .|.|.|. . .|. .|.|.|. .|. .|. . . .| + +-+-+ +-+-+-+ + + +-+-+-+-+-+-+-+ +-+ + |. . . . . .|. .|.|.|.|. . . . . . .|. .| +-+-+-+ +-+-+-+-+ +-+ +-+-+-+-+-+ + +-+ + |$ . . . .|$|.|.|.|.|.|.|. .|.|. .|.|. .| +-+-+ +-+-+ + + +-+ + + +-+ + +-+-+-+-+ + |.|.|.|.|.|.|. . .|. . . . . . . . .|. .| + + +-+ + + + +-+ +-+ + +-+-+ + +-+ +-+ + |.|. . .|. . .|.|. . .|.|. . .|.|. . . .| + +-+ +-+-+-+-+ + +-+-+ +-+-+-+-+-+-+-+ + |. . . . . . . . .|.|. .|. . . .|.|. . .| +-+-+ +-+-+-+-+-+-+ + + +-+-+ +-+ + +-+ + |.|. .|. .$ . . . . .|.|. . . . .|.|. .|
+ +-+-+ +-+ + +-+-+ + +-+-+-+ + +-+ +-+-+
|. . . . .|.|.|. .|.|.|.|.|.|.|. . . . .|
+-+-+ +-+-+ +-+-+ +-+-+ + + + + +-+ +-+-+
|. . .|. . . .|. . .|. .|. . .|.|. . . .|
+-+-+ +-+-+ + +-+-+ + +-+-+-+-+-+ + + +-+
|. . .|. . .|.|.|. . . . . . . . .|.|. .|
+ + +-+-+-+-+-+ +-+ + +-+-+ + +-+-+ +-+-+
|.|.|. .|. .|. . . .|.|. . .|.|. . . . .|
+ +-+ +-+-+ +-+-+ + + +-+ + +-+ +-+-+ +-+
|.|.|.|. . . .|. .|.|.|. .|.|.|.|. . . .|
+-+ + +-+-+-+ +-+ +-+ + +-+ + +-+ +-+ +-+
|. . . . . . . . . .|.|.|. .|. . .|. . x|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

1. Input: an ascii map of a cave, as described above. (Here are more sample files: above_maze, maze2, maze3.)

Output: the fastest route out of the cave (ignoring any treasure) (from s to x). Each line should contain one of four directions "up", "down", "left", "right".

### Sample output (For the above input. As a text file: fast_output.txt.)

down
right
right
down
down
right
right
up
right
right
right
up
right
right
right
right
down
right
right
up
right
up
right
right
right
down
right
right
down
down
down
down
down
down
down
down
down
down
down
left
left
down
down
down
down
down
right
down
down
right

Notes: I will test your code on a randomly generated cave. The caves are assumed to be connected (you can reach any cell). Explain the choices and the complexity of your code.

2. Input: A cave map, matching the above specifications.

Output: A route out of the cave which uses no more than 100 steps (your robot is running out of battery power) but captures as many treasures as possible. The first line of output should be the number of steps followed by a space followed by the amount of treasure you captured. After that the output will be of the same format as question 1 (one of four directions on each line).

### Sample output with treasure: (as a text file: money_output.txt)

72 2
down
right
right
down
down
right
right
down
left
down
down
down
down
down
left
left
left
right
right
right
up
up
up
up
up
right
up
up
right
up
down
right
right
up
right
right
right
right
down
right
right
up
right
up
right
right
right
down
right
right
down
down
down
down
down
down
down
down
down
down
down
left
left
down
down
down
down
down
right
down
down
right

Notes: There might be multiple acceptable answers to this question which escape the cave and attain the same amount of treasure. If you can choose the answer which uses less power/moves/gas I will prefer that. Again, don't blow up.


text:
1a)
TABULATE:
S/N: 1, 2 , 3, 4, 5
L(cm): 90.00, 80.00, 70.00, 60.00, 50.00
T(s): 38.00, 36.00, 34.00, 31.00, 28.00
T = t/20(sqt): 1.90, 1.80, 1.70, 1.55, 1.40
√L(cm^1/2): 9.49, 8.94, 8.37, 7.73, 7.07
(1aix)
Slope(s) = Δvertical/Δhorizontal
=1.90 – 1.55/9.49 – 7.75
= 0.35/1.74
S = 0.201
(1ax)
g = 4π²/S²
= 4 ×(3.142)²/(0.201)²
= 977.4cm/s²
(1axi)
(i) I ensured the angle of oscillation is relatively small.
(ii) I ensured that no external force is added to the system of oscillation.
(iii) I ensured the oscillation is perfect
(2a)
In a tabular form:

Under tita°:
75, 65, 55, 45, 35

Under MO(cm):
1.1, 2.0, 2.5, 3.4, 3.9

Under NO(cm):
6.2, 6.4, 6.6, 6.8, 7.2

Under H=MO/NO:
0.177, 0.313, 0.379, 0.500, .542

Under Costita:
0.2588, 0.4226, 0.5736, 0.7071, 0.8192

(2axiii)
From the graph
Slope, S =Δcostita/^ΔH
= 0.75-0.45/0.5-0.3
= 0.3/0.2
=1.5

(2axiv)
(i) I ensured both the object and the pins were in straight lines so as to avoid error due to parallax.
(ii)  I made sure there was no air interference

(2bi)
Snell's law of refraction states that the ratio of the sine of angle of incidence to the sine of the angle of refraction is a constant for a given pair of media.
I.e Sini/sinr = Π
Where Π is known as refractive index.

(2bii)
Given refractive index of glass = 1.5
i.e aΠg = 1.5(from air to glass)
SinC/sin90 = gΠa
SinC/Sinn90 = 1/aΠg
SinC/1 = 1/1.5
SinC= 0.6667
C = sin^-1(0.6667)
Critical angle for glass C = 42°

The concave mirror, mounted in its holder, is moved to and fro in front of them until a sharp image of the cross wire of the ray box is formed on the screen adjacent to the object. The distance between the mirror and the screen was measured as 30.1cm. Since the radius of the curvature r,=2fo ,then half this distance is the focal length of the mirror fo . Thus focal length was determined to be 15.05cm approximately 15cm
==================

(3a)
(ii) Vo = 2.00v

(3av)
In a tabular form
Under S/N
1, 2, 3, 4, 5

Under R(ohms):
2, 3, 4, 5, 6

Under V(v):
2.10, 2.30, 2.40, 2.50, 2.60

Under R^-1:
0.500, 0.333, 0.250, 0.200, 0.167

Under V^-1(v^-1):
0.476, 0.435, 0.417, 0.400, 0.385

3vii Slope, = Δv^-1/ΔR-1
= 0.5 - 0.355/0.6 - 0
= 0.145/0.6
S = 0.242

Intercept, C = 0.355v^-1

(3aviii)
K = S/C
K = 0.242/0.355
K = 0.68

(3ix)
(i) I ensured tight connections.
(ii) I ensured clean terminals.

(3bi)
(i) Temperature of wire
(ii) Cross sectional area of wire
(iii) Length of wire
(iv) Nature of wire

(3bii)
Draw the diagram

Effective E.m.f = 2v(parallel connection)
Effective internal resistance
= r * r/r + r = r/2ohms

Current, I , 0.8A
External resistance R = 2A
Using,
E = I(R+r)
2 = 0.8(2+r/2)
2.5 = (2+r/2)
2.5 - 2 = r/2
0.5 = r/2

r = 0.5*2 = 1ohms