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# Shooting Game for Fun

Trigger Warning: Murder is mentioned.

Let there be $n>1$ people (players) on a plane, each having a loaded gun and each being a perfect shot (assuming that each bullet is laced with one gram of plutonium-239 to ensure that hit targets do not survive and that the bullets are not penetrative enough to hit multiple bodies). Suppose that the distances between the players are pairwise distinct. At a signal, each player shoots the player closest to him (all the actions occur simultaneously). For those who have studied relativity, I assume that all players are initially at rest with respect to a fixed inertial frame so that it makes sense to discuss the simultaneity of the actions.

What is the minimum possible number of survivors? What is the maximum possible number of survivors? Do the answers change in higher dimensions (or even in other geodesic spaces like the $d$-dimensional torus)?

Below are my speculations.

The minimum is $n\!\!\mod\!2$ (this part is trivial and independent of the geometry of the space). I think the maximum for $n\geq 5$ is $n-\left(2q+s_r\right)$, if $n=10q+r$, where $q$ and $r$ are integers such that $0\leq r<10$, with $s_0=0$, $s_1=s_2=s_3=s_4=1$, and $s_5=s_6=s_7=s_8=s_9=2$. The case $n=10q+5$ seems to be the most difficult case for me. I found a mistake in my original bound, and now the new bound is worse. At the moment, the best bound is that at least $n-2q-s_r$ can survive, where $n=9q+r$ with $q,r\in\mathbb{Z}$ such that $0\leq r<9$, and $s_0=0$, $s_1=s_2=1$, and $s_3=s_4=s_5=s_6=s_7=s_8=2$.
Since a $d$-dimensional Euclidean space can be locally embedded into a $d$-dimensional geodesic space, I don't expect the answers to change (for a given dimension $d$) if the space is not Euclidean. However, the dimension should play a huge role in this shooting game.
EDIT I: After some more thought, I realized the answers may indeed be different in the non-Euclidean case. For example, a player cannot be shot by more than five bullets in the $2$-dimensional Euclidean case, but in a $2$-dimensional hyperbolic space, it seems to be possible that somebody is gunned down by at least six players.
EDIT II: In the $3$-dimensional Euclidean case, I expect the maximum number of survivors to be around $\frac{10}{11}n$. In the $d$-dimensional Euclidean space, this number should be around $\frac{L_d-2}{L_d-1}n$, where $L_d$ is the Kissing number in $d$ dimension.

• Minimum possible: zero (n=2p, p couples sitting close to each other and far from other couples). – Evargalo Jun 16 '17 at 12:38
• @Goodra I don't see why partial answers can't be posted as answers. Please go ahead. – Batominovski Jun 16 '17 at 15:57
• Consider reframing this question so that it is not about murder. – dbx Jan 10 '18 at 16:16
• Because casually talking about shooting people with radioactive bullets is a weird thing to do, and completely unnecessary mathematically speaking? – dbx Jan 10 '18 at 16:29
• no, my argument is that math teachers should not frame their problems around killing people. obviously you're not receptive to the idea, no need to get so defensive. It was a suggestion, not a command. – dbx Jan 10 '18 at 17:00

This answer considers only the original planar version of the game.

We already know that each player of this not very jolly game is shot by at most $5$ players (see, for instance, Lemma 1 here. In the next Lemma 2 this results implies that the maximum possible number $s(n)$ of survivors is at most $4n/5$ for each $n$.

We can improve this bound as follows. We say that a player is hardly killed if he is shot by exactly $5$ players. We need two more lemmas .

Lemma 3. A victim of a hardly killed player is his killer.

Proof. The proof is similar to that of Lemma 1. Suppose to the contrary, that the player located at the point $O$ has been shot by players located at points $A_1,\dots, A_5$ and has shot a player located at a point $A_6$. Draw rays $OA_1,\dots,OA_6$. Then there exist points $A_i$ and $A_j$, $i\not =j$ such that the angle $A_iOA_j\le \pi/3$. Therefore the angle $A_iOA_j$ is not the largest angle of a triangle $A_iOA_j$. Therefore the side $A_iA_j$ is not the largest side of the triangle $A_iOA_j$. If none of $A_i$ and $A_j$ has been shot by $O$ then $|A_iA_j|>|A_iO|$ and $|A_jA_i|>|A_jO|$, a contradiction. So, say, $A_i$ has been shot by $O$. Then $|A_iO|<|A_jO|$ and $|A_iA_j|>|A_jO|$, a contradiction again. $\square$

Lemma 4. There are no pair of hardly killed players such that each player from the pair is a victim of the other.

Proof. Suppose to the contrary, that there is such a pair. Consider a set $S$ consisting of $10$ players constituted by this pair and $8$ players who have shot a player of the pair. Since for each player of the set $S$ his victim also belongs to $S$, if we remove the rest of the players then the players of $S$ would shot the same players as before. That is we obtain a set of $10$ players with only two killed, that is impossible by an answer to a problem G7 from 41st IMO shortlist. $\square$

Now denote the set of the players by $P$ and let $v:P\to P$ be a map such that $v(p)$ is a victim of the player $p$ for each $p\in P$. Let $H\subset P$ be the set of hardly killed players. Lemma 3 states that $v(v(p)=p$ for each $p\in H$, so the map $v|H$ is injective. By Lemma 3 and Lemma 4, $v(H)\subset P\setminus H$.

Assume now that there are $h$ hardly killed players. The each of the remaining killed players was shot by at most $4$ players, so it total there are at least $$\max\{2h,h+(n-5h)/4\}=\max\{2h,(n-h)/4\}\ge 2n/9$$ killed players. Thus $s(n)\le (7/9)n$.

The precise calculation of the maximum yield no better result. Namely, if $n=9q+r$ as in your question then it can be easily checked then $\max\{2h,(n-h)/4\}=2q+r/4$ and $\lceil 2q+r/4\rceil=\lceil 2n/9\rceil$. Nevertheless, this shows that your lower bound for all $r$ but $3$, $4$ is exact and in the remaining cases at most $1$ worse than then above upper bound.

• nice argument - straightforward to follow – martin Jan 11 '18 at 1:47


text: Title Local and global comparison of continuous functions (In Proceedings) IEEE Conference on Visualization Herbert Edelsbrunner, John Harer, Vijay Natarajan, Valerio Pascucci Visualization, Riemannian manifolds, smooth functions, time-varying data, comparison measure, differential forms. 2004 Austin, TX 275--280 We introduce local and global comparison measures for a collection of $k \leq d$ real-valued smooth functions on a common $d$-dimensional Riemannian manifold. For $k = d = 2$ we relate the measures to the set of critical points of one function restricted to the level sets of the other. The definition of the measures extends to piecewise linear functions for which they are easy to compute. The computation of the measures forms the centerpiece of a software tool which we use to study scientific datasets.

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# GSEB Solutions Class 6 Maths Chapter 11 Algebra Ex 11.3

Gujarat Board GSEB Textbook Solutions Class 6 Maths Chapter 11 Algebra Ex 11.3 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 6 Maths Chapter 11 Algebra Ex 11.3

Question 1.
Makeup as many expressions with numbers {no variables) as you can from three numbers 5, 7, and 8. Every number should be used not more than once. Use only addition, subtraction, and multiplication.
Hint: Three possible expressions are 5 + (8 – 7), 5 – (8 – 7), (5 x 8) + 7; make the other expressions.
Solution:
Here, we have to use the given three numbers altogether. In an expression, every number is not to be used more than once.
Thus, we can have:
(i) 5 + (7 + 8)
(ii) 7 × 5 + 8
(iii) (8 – 5) × 7
(iv) (7 – 5) × 8
(v) (5 × 7) – 8
(vi) (8 × 7) + 5
(vii) 8 – 5 + 7
(viii) (8 – 7) + 5, etc.

Question 2.
Which out of the following are expressions with numbers only?
(a) y + 3
(b) (7 × 20) – 8z
(c) 5(21 -7) + 7 × 2
(d) 5
(e) 3x
(f) 5 – 5n
(g) (7 × 20) – (5 × 10) – 45 + p
Solution:
In expressions (c) and (d) no variable is involved.
∵ (c) and (d) are expressions with numbers only.

Question 3.
Identify the operations (addition, subtraction, division, multiplication) informing the following expressions and tell how the expressions have been formed.
(a) z + 1, z – l, y + 17, y – 17
(b) 17y, $$\frac{y}{17}$$, 5z
(c) 2y + 17, 2y – 17
(d) 7m, – 7m + 3, – 7m – 3
Solution:

Question 4.
Give expressions for the following cases.
(b) 7 subtracted from p
(c) p multiplied by 7
(d) p divided by 7
(e) 7 subtracted from -m
(f) -p multiplied by 5
(g) -p divided by 5
(h) p multiplied by -5
Solution:
(a) P + 7
(b) p – 7
(c) 7p
(d) $$\frac{p}{7}$$
(e) (-m) -7
(f) 5 x (-p) or -5p
(g) $$\frac{-p}{5}$$
(h) -5p

Question 5.
Give expressions in the following cases.
(b) 11 subtracted from 2m
(c) 5 times y to which 3 is added
(d) 5 times y from which 3 is subtracted
(e) y is multiplied by -8
(f) y is multiplied by -8 and then 5 is added to the result
(g) y is multiplied by 5 and the result is subtracted from 16
(h) y is multiplied by -5 and the result is added to 16.
Solution:
(a) 2m + 11
(b) 2m – 11
(c) (5xy) + 3 = 5y + 3
(d) (5 x y) – 3 = 5y – 3
(e) y x (-8) = -8y
(f) [y x (-8)] + 5 = -8y + 5
(g) 16 – (y x 5) = 16 – 5y
(h) 16 + [y x (-5)] = 16 + (-5y)

Question 6.
(a) Form expressions using t and 4. Use not more than one number operation. Every expression must have t in it.
(b) Form expressions using y, 2, and 7. Every expression must have in it. Use only two number operations. These should be different.
Solution:
(a) We can have the following expressions:
t + 4, t – 4, 4t, $$\frac{t}{4}$$ , 4 – t…
(b) 2y + 7, 2y – 7, 7y + 2, 7y – 2…


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# Distance to Viewport View-Camera

How can I get the distance between specific points in the 3D View and the virtual camera of my viewport?

Using the distance, I could select the closest or furthest (farthest) vertex/object from my perspective view.

Answering my own question, although I still don't know the difference between perspective_matrix, window_matrix and view_matrix. I also would prefer more explanation on why we don't need to use the inverted matrix.)
This question came up while creating this answer.

Snapping an empty to the closest vertex of a cube. "Closest" meaning closest to the viewer.

First, loop through the areas or obtain a SpaceView3D by other means. The type of this area returns VIEW_3D. It's first space will be a VIEW_3D space. This space has a region_3d property, which has a perspective_matrix property.

The perspective_matrix appears to be the perspective view transform matrix. Mutliple a 3D Vector by the matrix, to get its position relativ to the camera.

Here is a working sample.

import bpy

# function returning the closest vertex of
# param 'ob' is an object in the scene with vertices
def get_closest_vertex_position(scene, ob):
# get an area of type VIEW_3D
areas = [a for a in bpy.context.screen.areas if a.type == 'VIEW_3D']
if not len(areas):
return

region3d = areas[0].spaces[0].region_3d

# get the view matrix
view_mat_inv = region3d.view_matrix

if region3d.is_perspective:
vertices = [[v, (view_mat_inv*v.co).length] for v in ob.data.vertices]
else:
vertices = [[v, -(view_mat_inv*v.co).z] for v in ob.data.vertices]

# use a lamda expression to get closest vertex
return min(vertices, key = lambda x: x[1])[0].co

• Cool. Suggest doing this via a modal operator, a timer, or in a draw callback on the 3dview space rather than with a scene_update_pre . (depsgraph_update_pre on 2.8) handler. If for instance have two windows open with different scenes will confuse the result above. – batFINGER Jul 15 '19 at 12:31
• @bat Yes, definitely. This was merely a side product of this answer. I used the modal operator there. I'll just remove the last line, so future readers won't get tempted.^^ – Leander Jul 15 '19 at 12:45
• Did you mean to type view_matrix in the descriptive paragraphs? – Robin Betts Jul 15 '19 at 13:05
• @Robin Actually, no. As stated in the question, I'm still confused about perspective_matrix, window_matrix and view_matrix. The code works view perspective_matrix, although I don't know why. If you could clarify, that would be great. – Leander Jul 15 '19 at 13:12
• Following (as far as I can see) exactly the conventions described here, viewing down -Z, view_matrix is the xform from world space to viewpoint space (the inverse of the world xform of the viewpoint).. what window_matrix is depends on whether your view is orthogonal or perspective, and in either case gives the xform from the current window-shaped frustum to a canonical view volume, a cube -1 to 1 in X,Y and Z, viewpoint at 0. The perspective_matrix gives the 1st xform followed by the 2nd. – Robin Betts Jul 16 '19 at 15:57


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# Algebra Expressions with Exponents

## Calculate values of numbers with exponents

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Algebra Expressions with Exponents

Sam is preparing to have an epic water balloon fight with his best friend Josh. He is at home making his water balloons and wonders what the total volume of water is in his 50 balloons. He knows his balloons are approximately spheres. He also learned that the volume of a sphere is approximately  where  is the radius of the sphere. How could Sam determine the total volume of water in his balloons if the radius of each balloon is approximately 2 inches?

In this concept, you will learn how to simplify exponential expressions using exponent rules.

### Simplifying Algebraic Expressions with Exponents

Sometimes you need to multiply a number or a variable by itself many times.

Here is an example.

is  multiplied by itself 5 times.

To avoid having to write out the  again and again, you can use an exponent. Whole number exponents are shorthand for repeated multiplication of a number or a variable by itself.

In this example, 5 is the exponent and  is the base. The exponent indicates how many times the base is being multiplied by itself.

When you use an exponent to write an expression you are using exponential form.  is exponential form. When you write out the expression using multiplication without an exponent you are using expanded form.  is expanded form.

Here is an example.

Write the following in expanded form:

First, notice that in this expression  is the base and 12 is the exponent.  means  is being multiplied by itself 12 times.

Next, write the expression in expanded form without an exponent.

It's much easier to write the expression in exponential form!

When you have a variable expression with more than one exponent, you can often simplify.

Here is an example.

Simplify .

First, write the expression in expanded form by expanding each part of the expression.

Next, rewrite your expression in exponential form using just one base and one exponent.

Notice that the exponent in the answer is the sum of the original two exponents. This example helps to illustrate the first rule of exponents.

Rule 1: When multiplying two powers that have the same base, you can add the exponents. In symbols, .

Let's apply this rule to the next example.

Simplify .

First, remember that when parentheses are right next to one another it means multiplication.

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents.

If you are ever not sure, remember you can always write the expression in expanded form and then convert back to exponential form.

You can also have an exponential term raised to a power.

Here is an example.

Simplify .

First, focus on the exponent of 3 and start expanding the expression.  is being multiplied by itself 3 times.

Now, you are multiplying three powers that each have the same base so you can use Rule 1 and add the exponents.

Notice that the exponent in the answer is the product of the original two exponents. This example helps to illustrate the second rule of exponents.

Rule 2: When raising a power to a power, you can multiply the exponents. In symbols, .

Let's apply this rule to the next example.

Simplify .

First, notice that this expression is a power raised to another power. There are two exponents, but only one variable base. This means you can use Rule 2 and simplify by multiplying the exponents.

Again, remember that you can always write your expression in expanded form and then convert back to exponential form if you want to check your answer.

The final exponent rule that you need to know right now has to do with an exponent of 0.

Rule 3: Any nonzero number raised to the power of 0 is equal to 1. In symbols,  as long as .

Here is an example.

Simplify .

First, notice that the exponent is 0. This means you can use Rule 3.

### Examples

#### Example 1

Earlier, you were given a problem about Sam and his upcoming epic water balloon fight.

He has 50 water balloons that each have a volume of approximately  where  is the radius of each balloon. He measured his balloons and found that each has a radius of approximately 2 inches. Now he wonders what the total volume of water is in his balloons.

First, Sam can figure out the volume of water in one water balloon. He can substitute 2 in for the  in his volume formula.

Next, he can expand the expression. 2 is being multiplied by itself 3 times.

Then, he can multiply.

So each water balloon has 33.52 cubic inches of water.

Now, since he has 50 water balloons, to find the total volume of water in his balloons he can multiply the volume of water in one balloon by 50.

The answer is that in total, Sam has 1676 cubic inches of water in his water balloons.

#### Example 2

Simplify .

First, remember that when parentheses are right next to one another it means multiplication.

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents.

#### Example 3

Write the following in exponential form:

First, notice that  is being multiplied by itself 7 times.  will be your base and 7 will be your exponent.

#### Example 4

Simplify .

First, remember that when parentheses are right next to one another it means multiplication.

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents.

#### Example 5

Simplify .

First, notice that this expression is a power raised to another power. This means you can use Rule 2 and simplify by multiplying the exponents.

### Review

Evaluate each expression.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.

Simplify each expression.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Evaluate To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Expanded Form Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Integer The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
substitute In algebra, to substitute means to replace a variable or term with a specific value.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.


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# Irrational Numbers

1. Sep 26, 2008

### anantchowdhary

How do we exactly define irrational numbers..
but id like to know about any infinite series,
if any which is used to define irrational numbers...
and how can one prove properties of basic operations for irrational numbers

Thanks

2. Sep 26, 2008

### Mentallic

An irrational number is one that the decimal places do not terminate or recur. There are no perceivable patterns.

I'm sure you already knew this and want further detail on the subject though.

3. Sep 26, 2008

### HallsofIvy

Staff Emeritus
"Irrational" numbers are simply defined as real numbers that are not rational. But I think you are really asking how real numbers are defined.

One method is the "Dedekind cut" (see "Baby Rudin"). A Dedekind cut is defined as a set of rational numbers such that:
1) It is not empty- there exist at least one rational number in the set.
2) It is not all rational numbers- there exist a least one rational number NOT in the set
3) It has no largest member.
4) if a is a rational number in the set and b< a, then b is also in the set.

You might argue that we must have the rational numbers a subset of the real numbers, and rational numbers are NOT "sets of rational numbers"! But we can identify each rational number with such a set.

For example, if a is a rational number then {x| x is rational and x< a} is a cut- called a "rational cut" and is identified with the rational number a. It is non-empty: it contains a-1. It is not all rational numbers: it does not contain a. It has no largest member: if b is any member, then (a+b)/2 is a rational number in the set and is large than b. If b is in the set and c< b, then c< b< a so c is in the set.

The set {x| x is rational and either x2< 2 or x< 0} is also a Dedekind cut. It is non-empty: it includes 0. It is not all rational numbers: it does not include 2. It is harder to show that it has no largest member, but basically, if there were a largest member, its square would have to be 2- and there is no such rational number. Finally, suppose a is in the set and c< a. If a$\le$0 then c< 0. If 0< c< a, then c2< a2< 2. This Dedekind cut would be identified with the irrational number $\sqrt{2}$.

The "Dedekind cut" definition has the nice property that it becomes very easy to prove one of the "fundamental" properties of the real numbers: that every non-empty, bounded set of real numbers has a least upper bound.

But since you ask specifically about infinite sequences here are two other ways of defining the real numbers:

Consider the collection of all increasing, bounded, sequences of rational numbers. Say that two such sequences, {an} and {bn} are "equivalent" if and only if the sequence {an- bn} converges to 0. That is an "equivalence relation" and so partitions the collection into "equivalence classes"- sets of sequences so that any two sequences in the same set are equivalent, two sequences in different sets are not. Now we identify the real numbers with these equivalence classes. Again, we want to have the rational numbers a subset of this and rational numbers aren't sequences of rational numbers! But given any rational number, a, we can identify it with the sequence of rational numbers {a, a, a, a, ...} and it is not too difficult to show that a sequence of rational numbers is "equivalent" to that sequence if and only if it converges to a.

Of course, there exist increasing, bounded, sequences of rational numbers that do NOT converge to a rational number- the "monotone convergence property" is not true for the rational numbers. In fact, the "naive" way of thinking about real numbers is that a real number is any number that can be "written as a decimal": of the form a.a1a2.... We can identify that with the sequence {a, a.a1, a.a1a2, ...} and then with the class of all sequeces equivalent to that. In particular, "$\pi$" is identified with the class of sequences containing {3, 3.1, 3.14, 3.141, 3.1415, ...} and "$\sqrt{2}$" is identified with the class of sequences containing {2, 2.1, 2.14, 2.141, 2.1414, ...}

That definition makes it easy to prove the "monotone convergence property" for the real numbers.

I promised two ways of defining real numbers in terms of sequences. The second is almost the same. Consider the collection of all Cauchy sequences. That is, the collection of all sequences {an} such that |an- am| goes to 0 as m and n go to infinity (independently). We again say that two such sequences {an} and {bn} are equivalent if and only if the sequence {an- bn} converges to 0 and identify the real numbers with the equivalence classes of such sequences as before.

This definition makes it easy to prove the "Cauchy Criterion"- that all Cauchy sequences converge.

4. Sep 26, 2008

### anantchowdhary

how do we prove the basic results for operations like multiplication and division of irrational numbers?

5. Sep 26, 2008

### Tac-Tics

You don't. You prove them for all the reals.

I am a little fuzzy on the details, but it goes something like this. Take cauchy sequences (a_n) and (b_n) representing two reals a and b. Define an operation + such that a+b is the sequence c_i = a_i + b_i. You then show that (c_n) is cauchy, and thus, c is real. Multiplication is defined similarly. The usual properties for arithmetic (distributivity, commutativity, associativity-type things) are all inherited in a straightfoward way from the underlying rational operations used.

6. Sep 27, 2008

### arildno

We must distinguish between two levels of analysis:

1. To specify a set of axioms that some (otherwise undefined) set of objects (numbers) are said to obey.

2. To CONSTRUCT, on basis of another axiom set and their associated "numbers", structures that can be proven (according to this latter, more "fundamental" set of axioms) to be obeying the set of axioms as specified under 1.

Halls' post shows you how to construct the real numbers out of the rational number set (which again can be constructed out of the natural number set and basic set theory).
In this approach, we need to prove that multiplication of the reals does, in fact, obey the rules arbitrarily set up in 1.

However, we might let the axioms under 1 be our fundamental axioms, and in that case, we do not prove that reals can be multiplied, rather, that they obey the multiplication rules are part of their definition.

The advantage of the more fundamental approach is mainly theoretical in that by proving the constructibility of the reals out of simpler number sets, we have also proven that there aren't any other logical problems lurking within the concept of the reals than those problems that might hide within the concept of the naturals.

7. Sep 27, 2008

### anantchowdhary

is there any particular infinite series whic may converge to a given irrational number...if there is could someone please provide a derivation

thanks...

8. Sep 27, 2008

### HallsofIvy

Staff Emeritus
I wrote a paper that essentially proved the properties of the real numbers from the Dedekind Cut definition but don't have a URL for it available right now.

Of course, the first thing you would do is DEFINE addition and multiplication.

For example, using the Dedekind Cut definition, you would define "x+ y" to be the set of rational numbers of the form {a+ b| a is in x, b is in y}. You would need to show that satifies the requirements for a Dedekind Cut itself. If I remember correctly, defining multiplication is quite a lot harder- you have to do the case x, y both positive first.

Using the "equivalence classes of increasing, bounded sequences of rational numbers" definition, you would select one "representative" of each of x and y (one sequence in each of the classes defining x and y), say {xn} and {yn} and then show that the sequence {xn+ yn} is also an "increasing, bounded sequence of rational numbers" and so is in some equivalence class. That equivalence class IS "x+ y". Basically, then, you could show the properties of addtion or real numbers, commutativity, associativity, additive identity, additive inverse, from the corresponding properties of rational numbers, applied to the numbers in the sequences.

9. Sep 29, 2008

### anantchowdhary

Is ther any general infinite series which lets us express an irrational number in terms of an irrational number...ALl proofs in calculus are pretty much incomplete without basic proofs of operations on irrational numbers!

10. Sep 29, 2008

### Tac-Tics

Do you mean "in terms of *rational* numbers"?

Operations on irrational numbers are fully defined and proofs of their properties are very simple once you are comfortable working with sequences of rational numbers.

11. Sep 29, 2008

### HallsofIvy

Staff Emeritus
Why do you keep asking the same question over and over again? I just explained- not a complete explanation, that would take a whole book- how we can define the real numbers and show the basic properties of them.


text:
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text:
# CAT2002-58

162 views

In the following figure, the area of the isosceles right triangle ABE is 7 sq.cm. If EC=3BE, then the area of rectangle ABCD id (insq.cm.)

1. 64
2. 82
3. 26
4. 56

1 vote
Given: 1) $\text{Area of$\triangle$ABC =7 cm$^{2}$}$

2) $EC=3BE$

We know that area of triangle =$\frac{1}{2}*B*H$

so Area of $\triangle$ ABE $= \frac{1}{2}*B*H$

$\implies 7=\frac{1}{2}*B*H$

$\implies B*H=14$

This can be also written as  $\text{ BE*AB=14 cm$^{2}$}\cdots\cdots(i)$

Now area of rectangle $ABCD= AB*BC$

$\implies AB* (4*BE)$

$\implies 4*14=56$ sq cm

Note: $(BC=BE+EC\implies BE+3BE=4BE)$

Option $(D)$ is correct.
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Answer the question based on the diagram In the diagram below Angle ABC=90$^o$ = Angle DCH = Angle DOE = Angle EHK = Angle FKL = Angle GLM = Angle LMN, AB=BC=2CH=2CD=EH=FK=2HK=4KL=2LM=MN If $X_n = (-1)^n x_{n-1} \text{ & } X_0 =x$, then $X_n$ is positive for n=even $X_n$ is positive for n=odd $X_n$ is negative for n=even None of these
Answer the question based on the diagram In the diagram below Angle ABC=90$^o$ = Angle DCH = Angle DOE = Angle EHK = Angle FKL = Angle GLM = Angle LMN, AB=BC=2CH=2CD=EH=FK=2HK=4KL=2LM=MN The ratio of the areas of the two quadrangles ABCD and DEFG is 1:2 2:1 12:7 None of these


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# (a) Find $y'$ by implicit differentiation.(b) Solve the equation explicitly for y and differentiate to get $y'$ in terms of $x.$(c) Check that your solutions to part (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a).$\frac {2}{x} - \frac {1}{y} = 4$

## a. $\frac{d y}{d x}=\frac{2 \cdot y^{2}}{x^{2}}$b. $y^{\prime}=\frac{1}{2(1-2 x)^{2}}$c. click for solution

Derivatives

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in this trouble were given any question or any question ist over explains why Don't why a seat before you parked anywhere. Stu is implicit differentiation to find Derek Hough. Why Celeste back there with a litter of disrespect, X first time would give us negative two eggs. Great. Uh, second term would give us plus born, uh, scraped. Don't forget, wise. Also functional ex Soviet have divined the eckstrom here, and that is zero since forces From this we see that then the i t x so very little boy would suspect X is equal to two over x described times. Why squared? So that is, too. Once Craig B y squared in Frankie were testifying Function What? Take that. We see that or what? Is he going to over X minus four? From this? We see that. Then why is he going to over X voice for immerse? All right, now let's take their We took that. Why? Prime is an equal to negative one times two over x runs, four for the negative second power times. They were cooked in a function which is made of two over x squared. So the insert is done two over X squared. Well, it's like by two over X words for to the second. Negative second around Now in Percy, where testicle period great is we find our egg and beat. So let's take derivative that we've won. Our baby had to one scraping by spring that import fee we find What? What is right? This is our function boy. So why don't we just plugged it in? If you do so, we see that this is too over x grade multiplied by a widespread where ah Weisberg is too over x once before made one. This is why skirt this. And from this we see that then everything should be direct, scraped, multiplied by two over X minus four to connect a second power and agency. It's the same as what be so. It means that implicit explicit parameters for depreciation give us the same results.

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# What is the potential hazard of oxidation?

Even in a laboratory reaction, oxidation reactions are a hard thing to accomplish successfully. And even in these (controlled?) conditions, you often gets tars, which students refer to as $\text{brown sh..}$.


text:
# Are safe primes $p=2^k \pm s$ with $s$ small less recommandable than others as a discrete log modulus?

I take the definition of safe prime as: a prime $p$ is safe when $(p-1)/2$ is prime.

Safe primes of appropriate size are the standard choice for the modulus of cryptosystems related to the discrete logarithm problem, such as Diffie-Hellman.

A modulus $p=2^k \pm s$ with $s$ small makes exponentiation $\bmod p$ appreciably simpler (the problem of quotient estimation in classical Euclidean division all but vanishes, and the cost of modular reductions becomes negligible compared to multiplications). This, and the desire to use a non-arbitrary, compactly stored modulus, makes it tempting to prefer such choice of $p$. Like "the modulus shall be the biggest 2048-bit safe prime, which is $p=2^{2048}-1942289$".

Are there any major reason not to do that, like a much faster algorithm for the discrete logarithm problem?

-
I wonder if there is more practical attacks on choosen safe primes. –  catpnosis Jan 14 at 16:42
@catpnosis: Short of SNFS as explained in the accepted answer, I do not know any attack specific to how safe primes are generated. If safe primes are not used and the DLP in $\operatorname{GF}(p)$ matters, Pohlig-Hellman might apply; which is why safe primes are used for (non-ECC) DH and DSA. –  fgrieu Jan 14 at 18:11

I have asked a similar question to Arjen Lenstra a few years ago: I was investigating three 2048-bit primes of low Hamming weight:

• $p_1 = 2^{2048} - 2^{1056} + 2^{736} - 2^{320} + 2^{128} + 1$
• $p_2 = 2^{2048} - 2^{1376} + 2^{992} + 2^{896} + 2^{640} - 1$
• $p_3 = 2^{2048} - 2^{2016} + 2^{1984} - 2^{1856} - 2^{1824} + 2^{1792} - 2^{1760} + 2^{1696} + 2^{1664} + 1$

Since all the exponents are multiple of $32$, this allows for fast arithmetics (especially modular reduction). Moreover, $p_1 = 2^{128} q_1 + 1$, $p_2 = 2 q_2 + 1$ and $p_3 = 2^{1664} q_3 + 1$, where $q_1$, $q_2$ and $q_3$ are prime, and the subgroup generated by $2$ has an order which is a multiple of $q_1$, $q_2$ and $q_3$, respectively. So these looked good for performance, when implementing Diffie-Hellman (especially with $p_2$, which is a "safe prime") or DSA (with $p_3$). But it was possible that discrete logarithm modulo such special primes was easier than with "normal" primes, especially if Special Number Field Sieve applied.

Lenstra himself was not sure of the answer, but he forwarded the question to someone else (I do not know who), who answered this:

I did a short search for SNFS-like polynomial, just examining small powers of 2 as zeros (using LLL) without caring about skewness.

For $p_1$ one gets:

f1(x)=309485009821345068724781056*x^8 - 75557863725914323419136*x^4
+ 2*x^3 - 37778931862957161709568*x
+ 340282366920938463463374607431768211457

with $f_1(2^{245})=p_1$.

This is much better than GNFS, but also worse than (usual) SNFS for a 2048 bit number. Perhaps one can get a better polynomial by also examining skewed polynomials. Similarly, for $p_2$ there is the following degree 9 polynomial:

f2(x)=16*x^9 - 17179869184*x^6 + 332306998946228968225951765074280448*x^4
+ x^3 - 4611686018427387904

with $f_2(2^{234})=p_2$ and for $p_3$:

f3(x)=170141183381241069254316454265464291329*x^9
- 737869762914022326280*x^8
+ 130668428917922779575948702919181067091968*x^7
+ 2722258934733682407888030063919506653192

with $f_3(2^{228})=p_3$.

Perhaps degree 9 is too big for 2048 bit number, but the degree 8 and degree 7 polynomials also seem to be better than GNFS polynomials. Moreover, by examining the numbers $p_1$, $p_2$ or $p_3$ more closely, one might find better polynomials, thus getting closer to SNFS complexity.

As reference I only know the following paper by O. Schirokauer which essentially does the constructions above.

Bottom-line: when $p$ has a special format with low weight, it seems that SNFS applies at least partially, and offers a faster DL algorithm than GNFS. Complexity-wise, when SNFS applies, you need a $2n$-bit modulus to get the same resistance than what you have with a $n$-bit random prime where only GNFS works. So, to be safe, if you want a special-format $p$ like the one you suggest, or those I investigated, you need to double the length (a 4096-bit modulus, not a 2048-bit modulus), which cancels any implementation advantage derived from the special format.

On the other hand, special-format primes $p$ appear to be fine for base fields for elliptic curves; the NIST curves like P-256 use such special primes.

-

The discrete logarithm can be attacked in a multitude of ways: Baby-step giant-step (BSGS), Pollard's Rho, Pohlig-Hellman, and the several variants of Index Calculus, the best of which currently is the Number Field Sieve.

Let $n$ be the order of the generator of our field $\mathbb{F}_p$; it is $n = p-1$. We are trying to find $x$ given $a$ and $b=a^x$ in the above field.

### Pollard's Rho and BSGS

In Baby-step giant step, we are trying to find $i$ and $j$ such that $b(a^{-m})^i = a^j$, where $m = \lceil \sqrt{n} \rceil$. Once we find such a pair, the discrete logarithm $x = i m + j$ follows, as $b(a^{-m})^i = a^j \Leftrightarrow a^{x - m i} = a^j \Leftrightarrow x \equiv mi + j \pmod{n}$.

To do so, we first compute a table of all $a^j$ for all $j$ up to $m-1$. Then we iterate through all $i$ up to $m-1$, and compare $b(a^{-m})^i$ with $a^j$. Ignoring arithmetic costs, the runtime of this method is at most $2(m-1) = O(\sqrt{n})$ (with the same space requirements).

Due to the large space requirements, BSGS is rarely used in practice. Instead, we turn to Pollard's Rho. The crux of this method is to find a colliding nontrivial pair $(i,j)$ and $(k,l)$ such that $a^ib^j \equiv a^kb^l$. It follows that $x = \frac{k-i}{l-j} \pmod{n}$, since $a^i a^{xj} = a^k a^{xl} \Leftrightarrow a^{i + xj} = a^{k + xl} \Leftrightarrow i + xj \equiv k + xl \pmod{n}$.

So Rho comes down to finding a collision quickly. This can be done with various algorithms, Floyd's being the oldest and best known. The good news is that we can try and find a collision without an enormous table; the not so good news is that the algorithm is probabilistic, although the birthday paradox tells us we should expect a collision in about $\sqrt{n}$ steps.

In any case, these attacks are no good against a safe prime, where the order large enough that $\sqrt{n}$ is computationally unfeasible.

### Pohlig-Hellman

The Pohlig-Hellman approach relies on the observation that there is an homomorphism $\phi$ from $a$ and $b$ from their group of order $n$, to the subgroup of order $p_i^{e_i}$ dividing $n$. In general, given $n = p_1^{e_1}p_2^{e_2}\ldots p_m^{e_m}$,

$$\phi_{p_i^{e_i}}(a) = a^{n/p_i^{e_i}}$$

This allows us to compute the discrete logarithm of $\phi_{p_i^{e_i}}(a)$ and $\phi_{p_i^{e_i}}(b)$, which really is the discrete logarithm of $a$ and $b$ modulo $p_i^{e_i}$. From this observation, it is a matter of computing the logarithm modulo all prime divisors of $n$ (using the methods in the previous section) and combining them together using the Chinese remainder theorem.

If $n$ has many small prime divisors, i.e., it is smooth, this method is very much faster than Rho or BSGS. In a safe prime, however, this is not the case, since the order $n$ is the product $2q$, for a very large $q$. Pohlig-Hellman doesn't help much here.

### Index Calculus

Index Calculus is the basis for the best-performing algorithms to compute discrete logarithms modulo safe primes. Suppose we know the logarithms of $2$ and $3$; finding the logarithm of $12$ is easy: $\log_a12 = 2 log_a2 + log_a3$, since $12$ factors into $2^2\times 3$.

We can generalize this method to arbitrary elements. Start by defining the factor base, i.e., all the primes up to some bound $B$. Then, find the logarithms of all the elements of the factor base (this is the tricky part). Finally, factor $b$ into the factor base, and simply add all the logarithms corresponding to the factorization you find. If $b$ does not factor completely into the factor base, multiply $b$ by some known exponent of $a$ and try again.

Finding the logarithms of all the primes up to $B$ requires some trickery. It has two major steps:

• For $k_i \in [1..n]$, find (usually by sieving) at least $\pi(B)$ elements $a^k$ that factor completely into the factor base. Store both $a^{k_i}$ and its complete factorization.
• Now we have the linear system (modulo $n$): $$\begin{eqnarray} e_{1,1} \log_a 2 &+& e_{1,2} \log_a 3 &+& \ldots &+& e_{1,{\pi(B)}} &=& {k_1} \\ e_{2,1} \log_a 2 &+& e_{2,2} \log_a 3 &+& \ldots &+& e_{2,{\pi(B)}} &=& {k_2} \\ &&&&\ldots&&&& \\ e_{{\pi(B)},1} \log_a 2 &+& e_{{\pi(B)},2} \log_a 3 &+& \ldots &+& e_{{\pi(B)},{\pi(B)}} &=& {k_{\pi(B)}} \\ \end{eqnarray}$$
• Solving the above linear system gives us the needed logarithms for the factor base.

The runtime of this method, for appropriate choice of $B$, is $\exp{((2+o(1)((\log n)^{1/2}(\log \log n)^{1/2}))}$. This is not strictly polynomial, but is a big improvement on the previous methods.

### Number field sieve

The number field sieve is currently the best algorithm for both integer factorization and discrete logarithms over finite fields. For the discrete logarithm, it is analogous to the above index calculus, with a few major modifications:

• We are working in the number fields $\mathbb{Q}[\alpha]$ and $\mathbb{Q}[\beta]$ instead of the integers; there is, however, a map from such fields to the integers under some conditions. The number fields are defined by the polynomials $f_1$ and $f_2$ of degree $d_1$ and $d_2$; there must exist an integer $m$ such that $f_1(m) = f_2(m) = 0 \pmod{p}$.
• The factor base is formed by the primes in both $\mathbb{Q}[\alpha]$ and $\mathbb{Q}[\beta]$, up to bounds $B_1$ and $B_2$.
• During sieving, we look for pairs $(x,y)$ such that $N_{f_1}(x + \alpha y)$ and $N_{f_2}(x + \beta y)$ are $B_1$ and $B_2$-smooth, respectively, where $N_{f_i}$ is given by $$N_{f_i}(x + \alpha y) = y^{d_i} f_i(x/y)$$

The speed of the number field sieve hinges on the speed of finding $(x,y)$ with smooth norms $N_{f_i}(x,y)$. In turn, the probability of $N_{f_i}(x,y)$ being smooth is linked to its size: the smaller it is, the more likely it is to be smooth. And in turn, the size of $N_{f_i}(x,y)$ is determined by $x$ and $y$ (obviously), but also by the coefficients of $f_i$! When $f_i$ has very small coefficients, the number field sieve becomes asymptotically faster, from

$$\exp ( (1.923 + o(1))( (\log n)^{1/3}(\log \log n)^{2/3})$$

to

$$\exp ( (1.526 + o(1))( (\log n)^{1/3}(\log \log n)^{2/3}).$$

When one chooses a prime $p$ very close to $2^k$, it becomes easy to find a sparse polynomial that has a root modulo $p$. In your example, $p = 2^{2048} - 1942289$, we can find the degree $8$ polynomial

$$x^8 - 1942289,$$

since $2^{2048} - 1942289 = (2^{256})^8 - 1942289$. This polynomial has very small coefficients, that render the number field sieve for the discrete logarithm in this field much faster that it would be for a random 2048-bit prime.

-
You're right, fixed. –  Samuel Neves Mar 25 '13 at 19:12
Thank you for the fantastic answer! Great stuff! –  D.W. Mar 25 '13 at 19:21
"[Is] much faster that it would be for a random 2048-bit prime" — hm, for 2048 bit $n$ I still get just 24 bit reduction from 177 to 153. (If I'm calculated $O$ correctly.) So this doesn't sounds like more practical or much faster attack. –  catpnosis Jan 14 at 16:38


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# Limit of an exponent when it goes to infinity

how this thing works ?

integration of e to the power of (-x) from 0 to infinity

sorry , not good in using the symbols ..^.^ ~

Hi
Do you know what the primitive of e^{-x} is ?

First integrate it, then you get [-e-x] from 0 to infinity.

The key is to find the limit as x approaches infinity of (-e-x).

(Hint: Put in a very large negative value for x and see what you get.)

sorry ..~ i missed out the negative ..~
now i know already ..~ tq tq ...


text:
"Never doubt that a small group of thoughtful, committed citizens can change the world. Indeed, it is the only thing that ever has."

Original article
peer-reviewed

# Employment Status and Its Association With Psychological Distress and Alcohol Consumption

### Abstract

Purpose: The objective of this analysis was to find an association between employment status, psychological distress, and alcohol consumption.

Methods: Data from the 2020 Health Information National Trends Survey (HINTS) data was used. Patient health questionnaire-4 (PHQ-4) data and an average number of drinks per week were used to assess psychological distress and drinking status.

Results: Out of the 3865 people who answered the survey in the year 2020, 1890 (59.11%) were employed in one or multiple jobs during the time of the survey. The sample included 1561 men and 2204 women with an average age of 48.4 years. More than half (58.7%) of them were Non-Hispanic White followed by Hispanic and Non-Hispanic Black at 15.73% and 10.32%, respectively. Bivariate analysis showed a significant association between employment, psychological distress (p value=0.032), and alcohol drinking (p value=0.002); 60.59% of participants reporting severe distress (PHQ-4 score of 9-12) were unemployed. Alternatively, 73.1% of the employed respondents reported no stress (PHQ-4 score of 0-2). While more than half (75.97%) of those who were unemployed consumed only 1-2 drinks per week on an average, 7.27% consumed >13 drinks per week on an average. After adjusting for covariates, the regression analysis showed a highly significant association (p value< 0.001) between unemployment and psychological distress (OR=1.55; 95% CI 1.03, 2.31), and alcohol consumption (OR=0.67; 95% CI 0.48, 0.92).

Conclusion: Unemployment is associated with outcomes like psychological distress and alcohol consumption. However, being employed was found to be more strongly associated with alcohol drinking. We do not know if the coronavirus disease 2019 (COVID-19) pandemic was a risk factor for the given outcomes.

### Introduction

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) or more commonly known as coronavirus disease 2019 (COVID-19), was declared to be a public health emergency of international concern (PHEIC) by the World Health Organization (WHO) in early 2020 [1]. The pandemic has had a significant effect on the global economy leading to a sudden change in the employment status of several people. The increase in unemployment has been a major concern for the United States since the shutdown of the country due to the on-going pandemic. A total of 20.6 million people were reported to have lost their jobs in the first couple months of the pandemic in 2020, resulting in an unemployment rate of 14.7%, which gradually increased to 14.8% in April 2020 before it started to decline to 6.7% in October 2020 [2,3]. The rate of unemployment peaked in April 2020 during which nearly 49 million Americans were unemployed [4]. The highest rates of unemployment were reported in New Jersey (10.2%), followed by Hawaii (10.2%), Nevada (10.1%), New York (8.4%), and Louisiana (8.3%) [5].

The COVID-19 pandemic has increased the concerns about mental distress and substance use. Psychological distress, according to American Psychiatry Association (APA), is a set of painful mental and physical symptoms associated with normal fluctuations of mood. It can be assessed using specific clinical tools and can identify the beginning of common mental health issues such as anxiety and depression. If left untreated, it may lead to mental health conditions like major depressive disorder, anxiety disorder, schizophrenia, somatization disorders, sleep disorders, and several chronic health conditions [6,7]. McLachlan and Gale [7] showed that psychological distress significantly increases the risk of several other chronic health conditions like cardiovascular diseases, arthritis, chronic obstructive pulmonary disease.

Several risk factors for psychological distress have been reported in the literature. Some of the common ones include stressful life events and family history of depression, female gender, any other comorbid medical conditions, unemployment, immigration, academic achievements and or bullying experience at school, subjective memory complaint (SMC) [8-11]. Anxiety and depression are more common in the general population. An ultra-brief screening scale [12], patient health questionnaire-4 (PHQ-4), a four-point Likert-type scale was developed to measure the symptoms of anxiety and depression. It is the combination of 2-item measure of core symptoms (PHQ-2) for depression and 2-item measure for anxiety (GAD-2). It is a quick and easy clinical tool to screen patients for symptoms of anxiety or depression. Patients scoring high on this tool are assessed further for a definitive clinical diagnosis.

The pandemic has seen an increase in overall alcohol consumption by individuals [13]. According to the Dietary guidelines for Americans, provided by the U.S. Department of Health and Human Services and the U.S. Department of Agriculture [14], adult men should limit alcohol intake to a maximum of two drinks a day, whereas adult women should have no more than one alcoholic drink a day to lead a healthy, balanced life. Heavy alcohol use is defined as four or more drinks daily, or more than 14 drinks weekly for adult men. However, more than three alcoholic drinks daily, or more than seven drinks per week is considered heavy alcohol use for women. The National Institute on Alcohol Abuse and Alcoholism (NIAAA) defines binge drinking as an alcohol consumption typically five or more drinks for men, or four or more drinks for women during a time period of two hours. As per the Substance Abuse and Mental Health Services Administration (SAMHSA), binge drinking is also defined as five drinks for men, and four drinks per woman during a few hours, at least monthly [15]. Binge drinking is also defined as alcohol consumption that brings blood alcohol concentration to 0.08%.

A study performed in a large public University in Ohio focused on analyzing the alcohol consumption of college students after campus closure at the onset of the COVID-19 pandemic. The study also focused on the relationship between psychological distress, social support and changes in alcohol intake. It was discovered that campus closure during this pandemic led to an increase in alcohol consumption throughout, and those with anxiety and depression had a higher increase than those without any psychological distress [16]. Another study that focused on alcohol consumption in patients with a known history of alcohol use disorder was performed in the UK. The COVID-19 pandemic was a risk factor in itself for increased alcohol use, those with a history of alcohol use disorder were at a higher risk of relapse. In this study, there was an increase in alcohol consumption in 21% of the participants [13].

While many studies have reported the association of employment with alcohol consumption, several others have reported unemployment being associated with increase in alcohol intake. Psychological distress, on the other hand, has been reported to be only associated with unemployment. Moreover, alcohol intake and psychological distress have been shown to have a bi-directional association. The aim of this study is the examine the association of employment status (exposure) with alcohol consumption (outcome) and psychological distress (outcome) together.

### Materials & Methods

#### Data collection

Data was obtained from the Health Information National Trends Survey (HINTS 5 Cycle 4) conducted in the year 2020 [17]. This national surveillance survey is conducted annually by the National Cancer Institute. The sample design for the survey consisted of two stages. An equal probability sample of addresses was selected from each sampling stratum in the first stage. The second stage consisted of one adult per sampled household. The sampling frame for Cycle 4 consisted of a single-mode mail survey using the Next Birthday Method for the respondent section. A total of 3865 respondents completed the survey questionnaire in this cycle. The completion rate (at least 80% survey completed) was 98.1% (n=3792).

No Institutional Review Board (IRB) approval was required to use a publicly available dataset.

#### Measures

The survey asked the question on employment status: Which of the following best describe your current occupational status? Employed full time, employed part time, homemaker, student, retired, disabled, unemployed less than one year, unemployed one year or more, others. The answers were re-categorized into employed (full time and part time) and unemployed.

Psychological distress was assessed using a combined score from four separate questions. The question included in the survey was: Over the past two weeks, how often have you been bothered by: little interest, hopelessness, nervousness, and worrying. The answers included not at all, several days, more than half days, nearly every day. A new ordinal variable was created to interpret the total calculated PHQ-4 score for all four symptoms. The new variable phq4_score_interpret was categorized based on total PHQ-4 score interpretation as 0-2 as none; 3-5 as mild; 6-8 as moderate; and 9-12 as severe psychological distress. Alcohol consumption was calculated using the average number of weekly drinks. A new ordinal variable was created to convert the responses into categories of 1-2, 3-5, 6-8, 9-12, and more than equal to 13.

#### Data analysis

The data was analyzed with STATA software, version 16.1 (Stata Corporation, College Station, TX, USA), using descriptive statistics and logistic regression analysis. To evaluate the association of unemployment with psychological distress and alcohol consumption, a weighted sample was used. A weighted sample better represents the general US adult population and adjusts for possible confounders in the sampling method. After controlling for potential confounders, logistic regression analysis was conducted.

### Results

Table 1 shows the descriptive statistics for the sample population (with weighted proportions). The sample included 1561 men and 2204 women with a mean age of 48.4 years. More than half (58.7%) of the sample were Non-Hispanic White followed by Hispanic and Non-Hispanic Black at 15.73% and 10.32%, respectively. Around 15% of the respondents had an annual household income of less than $20,000 and 8% did not complete high school. In addition, more than half of the respondents had an annual household income of over$50,000 (60.72%) and had completed some college or beyond (69.14%). 1890 (59.11%) of the respondents were employed in one or multiple jobs during the time of the survey.

 Variables n (weighted %) Age (Mean, SD) 48.4 (0.30) Gender Female 2204 (50.22%) Male 1561 (47.57%) Missing data 100 (2.21%) Race and Ethnicity Non-Hispanic White 2133 (58.7%) Non-Hispanic Black 596 (10.32%) Non-Hispanic Asian 161 (4.6%) Hispanic 481 (15.73%) Others 119 (10.65%) Education Less than High school 273 (8.03%) High school graduate 705 (22.51%) Some college 1081 (39.12%) College graduate or more 1663 (30.02%) Missing data 143 (0.32%) Household income Less than $20,000 624 (15.15%)$20,000 to $35,000 451 (11.46%)$35,000 to $50,000 460 (12.67%)$50,000 to $75,000 592 (18.26%)$75,000 or more 1321 (42.46%) Employment status Employed 1890 (59.11%) Unemployed 1888 (40.89%)

#### Association of unemployment and psychological distress

Bivariate analysis in Table 2 showed significant association between employment status and psychological distress (p value=0.032). 77.5% (n=1268) of the unemployed respondents reported having no distress (PHQ-4 score of 0) while 14.5% reported having moderate to severe distress. 60.59% of participants reporting severe distress (PHQ-4 score of 9-12) were unemployed.

 Psychological distress (PHQ-4 score) P value None (0-2) Mild (3-5) Moderate (6-8) Severe (9-12) Age 0.000* Gender 0.159 Male 35.34% 8.15% 2.93% 2.82% Female 33.55% 9.41% 4.76% 3.04% Race and Ethnicity 0.0082* Non-Hispanic White 44.22% 11.01% 4.99% 3.37% Non-Hispanic Black 7.99% 1.86% 0.36% 0.85% Non-Hispanic Asian 10.48% 3.47% 1.78% 0.96% Hispanic 4.39% 0.63% 0.22% 0.05% Others 1.48% 0.98% 0.52% 0.43% Education 0.2161 Less than High school 5.01% 1.18% 0.95% 0.67% High school graduate 14.46% 4.19% 1.82% 1.73% Some college 26.66% 7.97% 2.93% 1.84% College or more 22.34% 4.37% 2.34% 1.55% Household income 0.0008* Less than $20,000 7.55% 4.21% 1.73% 1.44%$20,000 to $35,000 7.63% 1.99% 1.12% 0.65%$35,000 to $50,000 8.35% 1.78% 1.57% 0.60%$50,000 to $75,000 12.66% 3.54% 0.85% 1.28%$75,000 or more 31.77% 6.43% 2.82% 2.05% Occupation 0.0320* Employed 42.52% 9.57% 4.69% 2.7% Unemployed 26.06% 17.35% 8.03% 6.04% *statistically significant

After adjusting for covariates, the logistic regression analysis showed a highly significant association of employment status with psychological distress. Alcohol consumption showed an insignificant association with psychological distress; hence, it was not included in the final model. Table 3 shows that the unemployed had 1.03-2.31 times the odds of having psychological distress compared to respondents who were employed (p value=0.033). Age was shown to be significantly associated (OR=0.97; 95% CI 0.96, 0.98; p value<0.001) with distress symptoms. Participants who reported an annual household income of more than 75,000$had lower odds (OR=0.39; 95% CI 0.24, 0.62) of having psychological distress compared to the ones with an annual household income of less than 20,000$ (p<0.001). Additionally, Non-Hispanic Blacks had lower odds (OR=0.69; 95% CI 0.48, 0.99) of psychological distress compared to Non-Hispanic Whites (p=0.046).

 Psychological distress Variables AOR (95% CI) P value Occupation Employed Reference Unemployed 1.55 (1.03, 2.31) 0.033* Age 0.97 (0.96, 0.98) 0.000* Household income Less than $20,000 Reference$20,000 to < $35,000 0.56 (0.35, 0.89) 0.015*$35,000 to < $50,000 0.52 (0.29, 0.93) 0.029*$50,000 to < $75,000 0.47 (0.29, 0.93) 0.008*$75,000 or more 0.39 (0.24, 0.62) 0.000* Race/Ethnicity Non-Hispanic White Reference Non-Hispanic Black 0.69 (0.48, 0.99) 0.046* Non-Hispanic Asian 0.99 (0.68, 1.43) 0.968 Hispanic 0.46 (0.23, 0.90) 0.024* Others 1.94 (0.89, 4.19) 0.089 *statistically significant

#### Association of unemployment and alcohol consumption

Bivariate analysis in Table 4 showed significant association between employment status and alcohol drinking (p value=0.002). While more than half (75.97%) of those who were unemployed consumed only 1-2 drinks per week on an average, 7.27% consumed >13 drinks per week on an average. Overall, out of 7.77% participants who reported heavy drinking (>13 drinks per week), more than half were employed.

 Alcohol consumption (# of drinks per week) P value 1-2 drinks 3-5 drinks 6-8 drinks 9-12 drinks >13 drinks Age 0.45 Gender 0.0001* Male 31.12% 5.43% 3.31% 2.56% 6.31% Female 38.4% 5.25% 3.1% 2.07% 2.44% Race and Ethnicity 0.0042* Non-Hispanic White 41.1% 7.89% 4.74% 3.93% 6.86% Non-Hispanic Black 8.4% 1.09% 0.48% 0.31% 0.37% Non-Hispanic Asian 12.87% 0.98% 0.84% 0.46% 1.2% Hispanic 4.02% 0.59% 0.31% 0.0% 0.16% Others 2.52% 0.31% 0.18% 0.0% 0.37% Education 0.0003* Less than High school 6.04% 0.23% 0.27% 0.22% 0.67% High school graduate 14.9% 1.72% 1.23% 0.38% 2.79% Some college 29.14% 4.11% 2.36% 1.56% 3.01% College or more 19.22% 4.67% 2.61% 2.48% 2.38% Household income 0.0039* Less than $20,000 11.07% 1.02% 0.72% 0.29% 0.08%$20,000 to $35,000 8.54% 0.68% 0.49% 0.31% 1.26%$35,000 to $50,000 8.66% 0.99% 0.62% 0.65% 1.09%$50,000 to $75,000 13.6% 2.0% 1.16% 0.54% 1.29%$75,000 or more 27.16% 5.91% 3.52% 2.91% 4.72% Occupation 0.0002* Employed 39.1% 7.5% 4.36% 3.35% 6.03% Unemployed 30.41% 3.07% 2.06% 1.25% 2.87% *statistically significant

After adjusting for covariates, the logistic regression analysis in Table 5 showed a highly significant association of unemployment and alcohol consumption (p=0.016). In addition, psychological distress was included in the final model. The unemployed respondents had 0.48-0.92 times the odds of consuming alcohol compared to the ones who were employed. Being a non-Hispanic Black (OR=0.53; 95% CI 0.32, 0.88) and women (OR=0.62; 95% CI 0.44, 0.89) had lower odds of alcohol consumption compared to non-Hispanic White and men. People with an annual household income of over 75,000$were at 1.01-2.96 times the odds of alcohol consumption compared to the ones with a household income of less than 20,000$ (p=0.044).

 Alcohol consumption (number of drinks per week) Variables AOR (%CI) P value Occupation Employed Reference Unemployed 0.66 (0.48, 0.90) 0.009* PHQ-4 score None (0-2) Reference Mild (3-5) 1.09 (0.74, 1.58) 0.664 Moderate (6-8) 0.80 (0.42, 1.53) 0.500 Severe (9-12) 1.85 (1.04, 3.31) 0.038* Gender Male Reference Female 0.61 (0.44, 0.83) 0.003* Race/Ethnicity Non-Hispanic White Reference Non-Hispanic Black 0.47 (0.31, 0.73) 0.001* Non-Hispanic Asian 0.47 (0.31, 0.70) 0.000* Hispanic 0.36 (0.17, 0.75) 0.007* Others 0.47 (0.18, 1.28) 0.138 Household income Less than $20,000 Reference$20,000 to < $35,000 1.38 (0.73, 2.60) 0.307$35,000 to < $50,000 1.37 (0.74, 2.40) 0.323$50,000 to < $75,000 1.11 (0.65, 1.87) 0.703$75,000 or more 1.70 (1.00, 2.85) 0.048* *statistically significant

### Discussion

Unemployment has been a major concern in the United States since 2020. The rate of unemployment was drastically influenced by the COVID-19 pandemic, and therefore the level of psychological distress. Using PHQ-4, the symptoms of anxiety and depression were measured. Results showed that levels of psychological distress were higher in those who were unemployed. Table 3 shows that unemployed individuals had higher odds of having psychological distress compared to those who were employed. Alcohol consumption, measured as the number of drinks per week, was shown to be higher in employed participants compared to the unemployed ones. This is contrary to other published studies in the literature that show association of unemployment with higher alcohol consumption. We cannot ascertain if the results of this study were influenced by the ongoing COVID-19 pandemic. However, several studies also showed alcohol consumption to be higher in those with higher education and higher annual income [18].

Several studies have published an association between employment status and psychological stress, or alcohol consumption. Paul and Moser [19] conducted a meta-analysis and reported an average of 34% of cases of psychological disorders among the unemployed compared to 16% in the employed individuals. In addition, the reported mental health outcomes due to unemployment were found to be more pronounced in countries with poor economic development [19]. Another study by Frasquilho et al. [20] conducted a questionnaire-based survey on 748 unemployed people to detect the association between economic and non-economic factors and psychological distress. The results showed the participant's mean distress was 46.5% higher than the national mean of Portugal. Several other studies have shown consistent results in various parts of the world [21-23]. According to a randomized household survey conducted in Montreal in 2009, employment status under 45-year-old is inversely related to the distress and depression, using coping strategies like social support and alcohol usage [24]. In a study conducted using a sample taken from Panel Study of Income Dynamics [25], they estimated the association between estimated income tax credit (EITC) and income and other outcomes of interest like smoking, alcohol intake, physiological distress, and overall health. They found that people with high income are associated with less physiological distress, more alcohol consumption, more cigarette smoking.

Anxiety, depression, and alcohol consumption are interlinked. Many individuals consume alcohol to reduce anxiety, and often, long-term alcohol abuse can lead to anxiety development [26]. Similarly, people with depression are more likely to abuse alcohol to suppress their symptoms and use drinking as a way to escape reality [27]. Therefore, it is critical to reach out to a primary care or mental health provider to reduce the worsening of these outcomes and their subsequent effect on daily life. SAMHSA provides various resources to address alcohol and mental health disorders including anxiety and depression. Campaigns like “National Prevention week” which takes place in May, encourages community involvement, shares resources, and aims to increase public awareness on substance abuse including alcohol use [28]. There are several other organizations such as Centerstone, that distribute educational resources including alcohol-related statistics from published research, fact sheets, and videos that cover the effects of heavy drinking and how to recognize drinking problems [29].

#### Study limitations

There are several limitations to this study. First, the HINTS dataset did not ask questions on alcohol consumption in the past 12 months. Because of the lack of baseline data on alcohol consumption, a change over time could not be assessed. Moreover, since this dataset is a cross-sectional survey design, it is not possible to infer causal relationship between the variables. Second, a two-way relationship between alcohol use or psychological distress and unemployment could not be established due to the study design. Such associations are best established using prospective study designs. Lastly, the survey data used for this study was conducted in 2020. There was no efficient measure to adjust the confounding effect of the ongoing COVID-19 pandemic on the outcome variables.

### Conclusions

Unemployment is associated with outcomes like psychological distress and alcohol consumption. However, being employed was found to be more strongly associated with alcohol drinking. We do not know if the COVID-19 pandemic was a risk factor for the given outcomes. Federal aid through the Coronavirus Aid, Relief, and Economic Security (CARES) Act 2020 provided compensation to workers who became unemployed due to the COVID-19 pandemic. In addition, mutual-support groups like the alcoholics anonymous (AA) program or pharmacological and psychotherapy treatment for those experiencing anxiety or depressive symptoms are effective options to seek. Psychotherapy focuses on thoughts, feelings, and issues that are happening in an individual's life. It teaches them skills to help cope with their life, change behaviors that are causing problems, and find solutions.

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Human subjects: Consent was obtained or waived by all participants in this study. Animal subjects: All authors have confirmed that this study did not involve animal subjects or tissue. Conflicts of interest: In compliance with the ICMJE uniform disclosure form, all authors declare the following: Payment/services info: All authors have declared that no financial support was received from any organization for the submitted work. Financial relationships: All authors have declared that they have no financial relationships at present or within the previous three years with any organizations that might have an interest in the submitted work. Other relationships: All authors have declared that there are no other relationships or activities that could appear to have influenced the submitted work.

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text:
# A Problem in Division of Polynomials

### Solution 1

We observe that, for $n\gt 1,$ the quotient of $n^{n-1}-1$ and $n-1$ is

$n^{n-2}+n^{n-3}+\ldots+1.$

We conceive this sum as resulting from the polynomial

$R(x)=(1+x)^{n-2}+(1+x)^{n-3}+\ldots+(1+x)+1$

when we substitute in it $x=n-1.$ In the expansion of $R(x)$ in powers of $x,$ the free term is $R(0)=n-1,$ and so

$R(x)=Q_{n-2}(x)x+(n-1),$

where $Q_{n-2}(x)$ is a polynomial of degree $n-3$ whose coefficients are integers. Now we substitute $n-1$ for $x$ and collect our conclusions:

\begin{align} n^{n-1}-1 &= (n-1)R(n-1)\\ &=(n-1)[Q_{n-2}(n-1)(n-1)+(n-1)]\\ &=(n-1)^2[Q_{n-2}(n-1)+1]. \end{align}

### Solution 2

Let $k=n-1$ with $k$ a positive integer. The property is trivially satisfied for $k=1$. For $k\gt 1$,

$(k+1)^k-1=k\left[1+(k+1)+(k+1)^2+...+(k+1)^{(k-1)}\right].$

Each term in the box bracket is congruent to $1$ modulo $k$ and there are $k$ such terms. Thus, the expression in the box bracket is divisible by $k$. Thus, $k^2|(k+1)^k-1$ or $(n-1)^2|n^{(n-1)}-1$.

### Solution 3

$\displaystyle \frac{n^{n-1}-1}{(n-1)^2}$ should result in an integer.

If it holds for $n$, it should hold for $n+1$. We replace $n$ by $n+1$, hence $\displaystyle f=\frac{(n+1)^n-1}{n^2}$ should give an integer.

We note that by the binomial theorem

\displaystyle\begin{align} (n+1)^n&=1+n^2+\frac{1}{2} (n-1) n^3\\ &\qquad\qquad+\frac{1}{6} (n-2) (n-1) n^4+\frac{1}{24} (n-3) (n-2) (n-1) n^5+\ldots \end{align}

which gives the desired result.

### Solution 4

With the change of variables $x = n-1$, we have

\displaystyle\begin{align} n^{n-1} - 1 &= (1+x) ^x -1\\ &=\left( \sum_{k=0}^x {{x} \choose{k}} x^{k} \right) - 1 & \text{Binomial Theorem}\\ &=\sum_{k=1}^x {{x} \choose{k}} x^{k} & {{x} \choose{0}} \; x^{0} = (1)(1)=1\\ &= x^2 + \sum_{k=2}^x {{x} \choose{k}} x^{k}. & {{x} \choose{1}} \; x^{1} = (x)(x)=x^2 \end{align}

Clearly, each term in the final expression is divisible by $x^2$.

This implies that $n^{n-1} -1$ is divisible by $(n-1)^2$, which is what we needed to demonstrate.

### Solution 5

\displaystyle \begin{align}n^{n-2} + \ldots + n^0 &\equiv 1^{n-2} + ... + 1^0\\ &= n-1 \equiv 0 \mod (n-1) \end{align}

so $n-1$ divides $\displaystyle n^{n-2} + \ldots + n^0 =\frac{n^{n-1}-1}{n-1}$.

### Solution 6

As in Solution 1,

$n^{n-1}-1=(n-1)(n^{n-2}+n^{n-3}+\ldots+1).$

By the long division algorithm,

\displaystyle \begin{align} n^{n-2}+n^{n-3}+\ldots+1&=(n-1)\left[\sum_{k=0}^{n-3}(n-2-k)n^k\right]-(n-1)]\\ &=(n-1)\left[\sum_{k=0}^{n-3}(n-2-k)n^k-1\right]. \end{align}

### Acknowledgment

The problem and Solution 1 came from The Stanford Mathematics Problem Book by G. Polya and J. Killpatrick (59.3). Solution 2 is by Amit Itagi; Solution 3 is by N. N. Taleb; Solution 4 is by Jim Henegan. Solution 5 came from anonymous comment.


text:
How do you simplify (2sqrt27)times(3 sqrt32)?

Mar 24, 2017

$72 \sqrt{6}$

Explanation:

$\left(\textcolor{b l u e}{2} \sqrt{\textcolor{g r e e n}{27}}\right) \times \left(\textcolor{red}{3} \sqrt{\textcolor{m a \ge n t a}{32}}\right)$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{b l u e}{2} \times \textcolor{red}{3} \times \sqrt{\textcolor{g r e e n}{{3}^{3}}} \times \sqrt{\textcolor{m a \ge n t a}{{2}^{5}}}$

$\textcolor{w h i t e}{\text{XXX}} = 6 \times \textcolor{g r e e n}{3} \sqrt{\textcolor{g r e e n}{3}} \times \textcolor{m a \ge n t a}{{2}^{2}} \sqrt{\textcolor{m a \ge n t a}{2}}$

$\textcolor{w h i t e}{\text{XXX}} = 6 \times \textcolor{g r e e n}{3} \times \textcolor{m a \ge n t a}{{2}^{2}} \times \sqrt{\textcolor{g r e e n}{3} \times \textcolor{m a \ge n t a}{2}}$

$\textcolor{w h i t e}{\text{XXX}} = 72 \sqrt{6}$

Mar 24, 2017

$72 \sqrt{6}$

Explanation:

Using the $\textcolor{b l u e}{\text{law of radicals}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\sqrt{a b} \Leftrightarrow \sqrt{a} \times \sqrt{b}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

To simplify the radicals consider the product of their factors of which one should be a $\textcolor{b l u e}{\text{perfect square}}$

$\Rightarrow \sqrt{27} = \sqrt{\textcolor{red}{9} \times 3} = \sqrt{\textcolor{red}{9}} \times \sqrt{3} = 3 \sqrt{3}$

$\Rightarrow \sqrt{32} = \sqrt{\textcolor{red}{16} \times 2} = \sqrt{\textcolor{red}{16}} \times \sqrt{2} = 4 \sqrt{2}$

$\Rightarrow 2 \sqrt{27} \times 3 \sqrt{32}$

$= 2 \times \left(3 \times \sqrt{3}\right) \times 3 \times \left(4 \times \sqrt{2}\right)$

$= \left(2 \times 3 \times 3 \times 4\right) \times \left(\sqrt{3} \times \sqrt{2}\right)$

$= 72 \sqrt{3 \times 2} = 72 \sqrt{6}$

Mar 24, 2017

$72 \sqrt{6}$

Explanation:

$\left(2 \sqrt{27}\right) \times \left(3 \sqrt{32}\right)$

$\therefore = 2 \sqrt{3 \cdot 3 \cdot 3} \times 3 \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}$

$\therefore = 2 \times 3 \sqrt{3} \times 3 \cdot 2 \cdot 2 \sqrt{2}$

$\therefore = 6 \sqrt{3} \times 12 \sqrt{2}$

$\therefore = 72 \sqrt{3} \sqrt{2}$

$\therefore = 72 \sqrt{2 \cdot 3}$

$\therefore = 72 \sqrt{6}$


text:
Fluorescence spectra of different dilutions of quinine forming a calibration set.

## Format

The data set has 6 fluorescence emission spectra measured on quinine concentrations between 0.05 mg/l and 0.30 mg/l. Each spectrum consists of 181 data points in the range of 405 nm to 495 nm.

## Details

See the vignette: vignette("flu", package = "hyperSpec")

## Author

M. Kammer and C. Beleites

## Examples

flu
#> hyperSpec object
#>    6 spectra
#>    3 data columns
#>    181 data points / spectrum

plot(flu)

plotc(flu)
#> Warning: Intensity at first wavelengh only is used.


text:
# Chebyshev inequality in terms of RMS

I'm self studying the book Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

In page 48, the author write: "It says,for example, that no more than 1/25 = 4% of the entries of a vector can exceed its RMS value by more than a factor of 5."

I need more explain about it. Especially about why the factor is 5?

According to Chebyshev's_inequality, the probability of a value to deviate more than $$k=5$$ standard deviations from the mean is at most $$1/k^2$$.

When applied to vectors in your specific case, and following the book you cited, let $$k$$ be the number of elements of the vector $$\vec{x}=(x_1,\ldots,x_n)$$ such that $$||x_i|| \geq a > 0$$.

Hence $$\|\vec{x}\|^2 = \sum x_i^2 \geq k a^2 + (n - k) \times 0$$, which means that we have $$k$$ values larger than $$a^2$$ and the others $$n-k$$ values are at least zero.

Since the root mean square value is $$\operatorname{rms}(\vec{x}) = \sqrt{\frac{\|\vec{x}\|^2}{n}}$$, it follows that $$\operatorname{rms}(\vec{x})^2 = \frac{\|\vec{x}\|^2}{n} \geq \frac {k a^2}{n}$$.

Therefore, we get the final expression that says

$$\frac {k}{n} \leq \left( \frac{\operatorname{rms}(\vec{x})}{a} \right) ^2$$

So, following the example, where $$a = 5 \operatorname{rms}(\vec{x})$$, we have that $$\frac {k}{n} \leq \left( \frac{1}{5} \right) ^2 = 4 \%$$, so, the fraction of elements of the vector larger (in absolute value) than $$5\operatorname{rms}$$ is at most $$4\%$$.

If we chose another number, say $$a = 2 \operatorname{rms}(\vec{x})$$, we would have that $$\frac {k}{n} \leq \left( \frac{1}{2} \right) ^2 = 25 \%$$.


text:
## Duke Mathematical Journal

### On the structure of the conformal Gaussian curvature equation on $\mathbb{R}^2$

#### Article information

Source
Duke Math. J. Volume 62, Number 3 (1991), 721-737.

Dates
First available in Project Euclid: 20 February 2004

http://projecteuclid.org/euclid.dmj/1077296513

Mathematical Reviews number (MathSciNet)
MR1104815

Zentralblatt MATH identifier
0733.35037

Digital Object Identifier
doi:10.1215/S0012-7094-91-06231-9

#### Citation

Cheng, Kuo-Shung; Ni, Wei-Ming. On the structure of the conformal Gaussian curvature equation on ℝ 2 . Duke Math. J. 62 (1991), no. 3, 721--737. doi:10.1215/S0012-7094-91-06231-9. http://projecteuclid.org/euclid.dmj/1077296513.

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