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1

You could have a scoring system where the problem is worth $n$ points. You can guess as many options (choices) as you want. You get $1$ point for each correctly chosen correct option and $1$ point for each nonchoice of each incorrect option. Thus in your scenario you would score $1 +(n-k)$ (one correct choice of the correct option; and $n-k$ correct ...


2

More context is needed to establish what is "fair". To illustrate this, take an extreme example. Say the question is "What is $1+1$?", and $99$ ridiculous answers are provided, along with the correct one, and $50$ tries are allowed. One student needs $10$ attempts to get it right, while another needs $40$. It's hard to argue that the first student has ...


1

One humble suggestion: Read chapter 1.8.2 of the famous "Feynman Lectures on Physics." In order to introduce the concept of instantaneous speed, Feynman quotes this joke there: The cop stops the lady and says: "Lady, you were going 60 miles per hour!" She says, "That is impossible, sir, I was travelling only for seven minutes..." The conversation between ...


1

Imagine that you are in a room with Albert (A) and Bernard (B). The three of you are perfectly rational. The first thing to realise is that if B has number 18 or 19, then he automatically knows the whole date. Albert: "I know that B doesn't know." The only way A can possibly know this is if A does not have May or June (Since, if A has May or June, then (as ...


2

Case 1: The reason is because Bernard told him. If that was the case, the dialog would start with: Bernard: I don't know when her birthday is. In mathematics we never make any extra assumptions, is always understood that no information is withhold. Bernard telling something to Albert, information which is needed to solve the problem, would automatically ...


5

I want to briefly sketch how this kind of problems can be solved in a systematic way. One does not have to rely on intuition, scribbling and hand-waving. The wording of the problem is indeed somewhat imprecise. It is not clear how Albert comes to his first statement. The problem statement should be changed to the following (notice point 2): 1) Cheryl gives ...


0

Re: Remark 2 - producing something original. I think this is unnecessary and will create unneeded stress. There was a thesis option in the Masters Degree I did at Cambridge University in 1986. This was NOT expected to be original research. I wrote a summary of research on a particular question in Banach Algebra Functional Calculus. It was 77 pages. I did ...


0

You might want to consider the website of public-domain mathematics instructional materials that I have created. The materials range in difficulty from kindergarten to college, but the bulk of them are at the high school level. The address is www.public-domain-materials.com. These materials, under development for over 30 years, have been recognized by a ...


7

Cheryl sure is a good one for wasting people's time. :-) The convention in these kinds of problems is to be conservative: No one says anything that we are not told they said. In particular, Albert deduces that Bernard doesn't know Cheryl's birthday, not because Bernard told him, but because his knowledge of the month permits him to conclude that. What ...


1

monotonic for sure is to be used. Monotonic functions are those functions which are either increasing or decreasing. They are such that for each specific value of x there is a unique y(value of function) which does not repeat for any other x.


5

That would be monotonic function. Monotonic is always used in relation to the function you are talking about. http://mathworld.wolfram.com/MonotonicFunction.html Monotonic describes something this is unchanged or altered, such as the function in maths whereas Monotonous describes something lacking in variety and is usually used in reference to tone.


3

"Monotonic" or "monotone", but not "monotonous.


0

Think about this: when you were 17, was the mathematics you were doing in school very hard arithmetic problems? For me it was geometry, algebra, trigonometry, calculus: things that were of a different kind, not harder-of-the-same-kind. If higher mathematics were just multiplying bigger numbers faster, it wouldn't be very interesting.


0

The amount for all $N$ people is $\frac{34800}{N}$. The amount for $N-5$ people is $\frac{34800}{N-5}$. If the amount is increased to 1160 You know that $$\frac{34800}{N-5}=1160$$ Thus $34800=1160(N-5)$ and $$N=\frac{34800}{1160}+5=35$$ If the amount is increased by 1160 Then $\frac{34800}{N-5}=\frac{34800}{N}+1160$, and ...


2

Silverman and Tate - "Rational Points on Elliptic Curves" Anything by Keith Conrad - http://www.math.uconn.edu/~kconrad/blurbs/ Stopple - "Primer of Analytic Number Theory" Flanigan - "Complex Variables" to name a few


1

Honestly, for the most part you can even skip reading most of the book. Problems are millions of times more important. When you get a new book, don't even read it. It doesn't matter. Go directly to the problems, skipping everything in the book. Don't even glance at chapter titles. Go directly to the first problem. Obviously the problem probably won't make ...


0

I think I have a good source for you, provided you are looking for wordy yet mathematically interesting properties. The wikipedia pages for the integers from 1 to 12, entitled 'n_(number)', all have interesting mathematical facts about those numbers in the "Mathematics" section. For instance, 7: Seven is the lowest natural number that cannot be ...


0

These have some your numbers in them... What do you get when you link primes, transcendentals, $0$, unity, subtraction, and equality? $$e^{2 \pi i}-1=0$$ A root inside a root inside a root... $$2=\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}}$$ What do you get when you sum all the natural numbers? Not what you think according to the Riemann zeta ...


3

You didn't say your geographical area. In the United States, departments are typically not only willing to consider applicants without an MS--that most admitted PhD candidate have not had master's training is sort of the norm. As a consequence, the US degree process usually involve some number of years of course work (1 or 2 typically) with a ...


1

Start from begin{align} d(O, P) = \sqrt{\frac{\hat{x_1^2}}{\hat{s_{11}}} + \frac{\hat{x_2^2}}{\hat{s_{22}}}}. \end{align} You want to convert that to begin{align} d(O, P) = \sqrt{a_{11}x_1^2 + 2a_{12}x_1x_2 + a_{22}x_2^2} \end{align} and work out the values of the $a_{ij}$. To do so, I'll show that the things under the square roots are equal, that is, I'll ...


-2

Your kinda jumping in the deep end trying to teach yourself them topics, unless you have prior mathematical background. The reason your probably not getting the concepts is that maths is a novel. The concepts were thought up by mathematicians for a reason and if you understand them reasons then you can normally think the maths up without a textbook, hence do ...


1

I would go as far as saying that the problems Rota points out are too localised. The specialisation of students. We don't try to teach future mathematicians all the fine details of finite element methods (however, we should at least give them the basics at some point), they are needed by engineers. Might as well not teach future engineers the ...


2

I can sort of see what Rota is arguing for but it's kind of a fine line because his arguments should also apply to calculus: "why teach epsilon-delta methods", "why teach volume integration of ridiculous rotated surfaces" etc etc. More to the point, "why teach abstract nonsense to applied-math majors?" I would argue that, aside from giving students an ...


1

What the function you get in the end represents depends on your problem. One thing to note is that a certain equation, just expressed as a PDE, might appear in several different application problems. E.g. the heat equation can be used to model e.g. the temperature over time in an object, but can also be used to model financial things (more examples can be ...


0

There are tons of situations where the solution of a PDE discribes the behaviour over time: 1) pendulum 2) (radio active) decay 3) mass-spring-damper systems 4) inductance-capacitance-resistance systems 5) heat distribution over time 6) water level in a bucket with a leak 7) Temperature of a heated house 8) (electromagnetic) waves


0

The solution is just a function. That's what you're aiming for at the end. E.g. in the 1D wave equation, you get $u(x,t)$. You know what a function is right? For the solution to the wave equation, you plug in the position on the x axis and the time and $u(x,t)$ will tell you the amplitude of the wave at that position at that point in time. That's all there ...


0

Let's just use the heat equation as an example. The solution of the heat equation $u(x,t)$ can be used to describe the temperature at any position $x$ of a thin rod, at any time $t$. See the following picture: Suppose a rod was heated at the middle point and we measured its temperature afterwards. The temperature would be like this. And this could be ...


0

They can represent nearly anything, it depends on what you are modelling with the differential equation. In general, the solution of a differential equation is simply a function. Using the example of the heat equation, the solution $u(\mathbf{x},t)$ is the temperature at position $\mathbf{x}$ at time $t$. See for example the figures in this link.


1

If you are truly interested in Cryptography and Computer Science you can complete a CS course in college, with a focus on mathematics.Thus combining two things you love. It's entirely possible and quite a few people follow this path. Most of the courses, focus on possibilities, algorithms,advanced math and so on. Later you can pursue a career as a ...


2

Well, I think that the meaning of the algebraic operator is a just a convention based on the meaning applied historically to the minus (-) operator, according to the standard algebra, for that reason a double minus is for instance in terms of money like "a quantity which is not (-) a debt (-)" so by that reason it turns to be a possesion you have (+), but I ...


6

Why do you think we are treating them differently? We have $$++\to +\\--\to +\\+-\to -\\-+\to-,$$ which has a remarkable symmetry. And, on top of that, the image: the minus sign is a "make a 180 degree turn" and the plus sign is "don't turn". So, if you have $-(-5)$, you make a turn, then you make a turn, then walk 5 units of length. Two turns of 180 ...


-1

Perhaps I would advise to use SCaVis math program. It is easy to make plots, draw functions etc. in this program. In addition, it can teach kids a programming language (Python) at the same time, which is an industry standard.


-1

SCaVis is a free math program written in Java and runs on Mac/Windows/Linux. It has hundreds of example that makes it perfect for education.


2

And yet another approach: Splitting $A$ into a diagonal matrix $D$ and a pertubation matrix $P$. We look at the first powers of $P$: $$ P^0 = \left( \begin{matrix} 1 & 0\\ 0 & 1 \end{matrix} \right) \quad P^1 = \left( \begin{matrix} 0 & -1 \\ 0 & 0 \end{matrix} \right) \quad P^2 = \left( \begin{matrix} 0 & 0 \\ 0 & 0 ...


1

Look at the first few powers: $$\left(\begin{matrix}2&-1\\0&1 \end{matrix} \right)$$ $$A^2=\left(\begin{matrix}2&-1\\0&1 \end{matrix} \right)\left(\begin{matrix}2&-1\\0&1 \end{matrix} \right)=\left(\begin{matrix}4&-3\\0&1 \end{matrix} \right)$$ $$A^3=\left(\begin{matrix}8&-7\\0&1 \end{matrix} \right)$$ This suggests ...


7

An unorthodoxical but quick solution. Let us compute the powers by recurrence: $$ \begin{bmatrix} a_{n+1} & b_{n+1} \\ c_{n+1} & d_{n+1} \end{bmatrix}= \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} a_n & b_n \\ c_n & d_n \end{bmatrix} =\begin{bmatrix} 2a_n-c_n & 2b_n-d_n \\ c_n & d_n \end{bmatrix} ,$$ with ...


7

This is for the updated version of question where $A = \begin{bmatrix}2 & -1\\0 & 1\end{bmatrix}$. For the original version of $A$, the derivation is similar. The characteristic polynomial for matrix $A$ is $$\chi_A(\lambda) = \det\begin{bmatrix}\lambda-2 & 1\\0 & \lambda - 1\end{bmatrix} = (\lambda - 2)(\lambda - 1) = \lambda^2 - 3\lambda ...


1

To diagonalize a matrix, you first need to find the eigenvalues, solve: $$\det(A - \lambda I ) = 0$$ $$\det \begin{bmatrix} 2 - \lambda & 0 \\ 2 & 1 - \lambda \end{bmatrix} = 0$$ $$(1 - L)(2 - L) = 0$$ $$L \in \{1, 2\}$$ So the diagonal values are $1$ and $2$. Then you need to find the eigenvectors using nullspace basis: $$\begin{align} V_1 ...


0

Hint: Find a matrix $P$ such that $P^{-1}AP=D$ where $D=diag(2,1)$. Then $P^{-1}A^{50}P=D^{50}=diag(2^{50},1).$


5

You can also compute it with the following method. First note that $50 = 32 + 16 + 2$. Let $$A=\left(\begin{array}{cc}2&0\\2&1\end{array}\right).$$ Then $$A^{50} = A^{32} A^{16} A^2$$. We can compute $A^{2^n}$ easier than $A^{50}$, $$A^2 = \left( \begin{array}{cc} 4 & 0\\ 6 & 1 \end{array} \right)$$ $$A^4 = \left( \begin{array}{cc} 4 ...


5

Hint: Diagonalize the matrix you have been given and then take powers.


1

You may find the following website interesting/useful: Teaching with Original Historical Sources in Mathematics Use of original historical sources in lower and upper division university courses is discussed. Reinhard Laubenbacher and David Pengelley were inspired by William Dunham to cover "mathematical masterpieces from the past, viewed as works of art." ...


2

Let $I=[a,b]\subseteq\mathbb{R}$ and $\{f_n(x)\}_{n\in\mathbb{N}}$ the sequence of real polynomials defined by: $$ f_n(x) = \prod_{j=0}^{n}\left(1-\frac{4\,x^2}{(2j+1)^2\,\pi^2}\right). $$ We have that the sequence $\{f_n(x)\}_{n\in\mathbb{N}}$ is uniformly convergent to $\cos x$ over $I$. From the factorization of Chebyshev polynomials of the first and ...


2

To gain mathematical maturity, I think you should work on both new stuff (that you might find quite challenging) and also old stuff (which is usually easier). Good ideas feed off each other given a half a chance. When learning new stuff, it is often good to seek out ideas that unify and/or clarify what you already know. The following subjects are ...


0

Testimonials will not be a good way of deciding what type of medium you learn from. The way one person learns best may not be the way another person does. I suggest whenever starting a new subject, try out a few different materials and see what works best for you. If you like the video lectures you find for a subject -- use them. If you find a textbook ...


0

Why integrals? Why not computers? Why learn? 1: It's the framework to understand many "advanced" techniques and technology topics used in different areas of the science. This means you will be a professional on what you are doing. The one who is not just using technology but knows how it works or even can reproduce the technology. Trivial things will be ...


0

Learning materials , including video lessons, are all just support for your learning effort. What is more important is your eagerness to learn the subject. Reading books and solving exercises will be more beneficial than watching multimedia learning materials.


2

For a general subset $S\subseteq G$, you define (see e.g. Wikipedia) $$C_G(S) := \{g\in G\ |\ gs=sg\text{ for all } s\in S\},\qquad N_G(S) := \{g\in G\ |\ gS=Sg\}.$$ Taking $S=\{a\}$ to be a singleton, you see that both coincide and are given by $\{g\in G\ |\ ga=ag\}$. Note, however, that already $N_G(\langle a\rangle)\neq C_G(\langle a\rangle)=C_G(\{a\})$ ...


6

Try framing multiplication $a \times b$ as the area of a rectangle with sides of length $a$ and $b$. So a $2 \times 3$ box has area 6: A $2 \times 0$ box has area 0, as does a $0 \times 3$ box. And a $0 \times 0$ box has area…?


0

Let's say I have a school. The school has three classrooms, with two kids in each room. (Clearly an underfunded school.) Hopefully you can convince him — perhaps drawing it might help? — that there must be $3\times2=6$ kids in the school. Now, let's say that there is another school, but with no rooms and no kids in each room. (Horribly underfunded.) How ...



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