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(It also appears that $x$ will never be uniquely determined without an upper bound.)
-
Look up the Chinese remainder theorem. – Ｊ. Ｍ. May 8 '12 at 14:54
If you are learning from a book, surely they will discuss the CRT within a few pages of introducing such equivalences... – The Chaz 2.0 May 8 '12 at 14:55
Having ju... | {
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This is a classic example of Chinese remainder theorem. To solve it, one typically proceeds as follows. We have $$x = 2k_2 + 1 = 3k_3 + 1 = 5k_5 + 3.$$ Since $\displaystyle x = 2k_2 + 1 = 3k_3 + 1$, we have that $2k_2 = 3k_3$ i.e. $2|k_3$ and $3|k_2$, since $(2,3) = 1$. Hence, $k_3 = 2k_6$ and $k_2 = 2k_6$. Hence, we n... | {
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Note that I chose to eliminate the largest moduli first, i.e. $\rm\:x\equiv -2\ mod\ 3,5\:$ vs. $\rm\:x\equiv 1\ mod\ 2,3\:$ since that leaves the remaining modulus minimal ($= 2$ vs. $5$ above), which generally simplifies matters if we need to apply the full CRT algorithm in the final step (luckily we did not above).
... | {
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Probably the greatest difference is that the sum in the expansion need not be a finite sum.
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If two integers share every single residue, their difference is divisible by every nonzero integer and therefore zero; they are identical. Therefore the set of all residues does uniquely determine an integer. The very same ar... | {
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# Recurrence relation for Pell's equation $x^2-2y^2=1$
I am wondering how to find the recurrence relation for solutions for $$x$$ in the Pell's equation $$x^2-2y^2=1$$.
I know the formula for the general term. It is $$\frac{(3+2\sqrt2)^n+(3-2\sqrt2)^n}{2}$$ for $$x_n$$, the $$n^{th}$$ smallest solution for $$x$$. Any... | {
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Let the quadratic be $$p(x)=x^2-px+q$$, then consider $$0=A\alpha^np(\alpha)+B\beta^np(\beta)=$$$$=(A\alpha^{n+2}+B\beta^{n+2})-p(A\alpha^{n+1}+B\beta^{n+1})+q(A\alpha^{n}+B\beta^{n})=u_{n+2}-pu_{n+1}+qu_n$$from which $$u_{n+2}=pu_{n+1}-qu_n$$Where $$p=\alpha+\beta$$ and $$q=\alpha\beta$$
You simply need to identify $... | {
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$$(11,1) \; \; \; \; (37,25) \; \; \; \; (211,149) ...$$ $$(13,5) \; \; \; \; (59,41) \; \; \; \; (341,241) ...$$ $$(19,11) \; \; \; \; (101,71) \; \; \; \; (587,415) ...$$ $$(29,19) \; \; \; \; (163,115) \; \; \; \; (949,671) ...$$
If you don't mind negative values for $$x,y$$ you can combine the above four into two ... | {
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# Prove that the origin is Liapunov Stable for the given system
Consider the system $$\dot{x} = y \\ \dot{y} = -4x$$ ($\dot{x}$ means $\displaystyle \frac{dx}{dt}$ and $\dot{y}$ means $\displaystyle \frac{dy}{dt})$
I need to prove that the fixed point $\mathbf{x^{*} = 0}$ is Liapunov stable.
For reference, according... | {
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Thank you for your time and patience.
• What is the book? – CroCo May 18 at 3:58
The solution to this problem is an ellipse parameterized by the initial conditions $x_0,y_0$ that has a center at $\{0,0\}$. Therefore, for a given set of initial conditions the maximum distance a solution could have at any time from the... | {
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The norm of the solution is $$\|{\bf x}(t)\|=\sqrt{x^2(t)+y^2(t)}=\sqrt{\left(x_0\cos 2t+\frac{y_0}2\sin 2t\right)^2+\left(-2x_0\sin 2t+y_0\cos 2t\right)^2}.$$ One can use the triangle inequality: $$\sqrt{(a_1+b_1)^2+(a_2+b_2)^2}\le \sqrt{a_1^2+a_2^2}+\sqrt{b_1^2+b_2^2}$$ in order to obtain $$\|{\bf x}(t)\|\le\sqrt{x_0... | {
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# Distance between 2 planes
1. Sep 28, 2011
### cyt91
1. The problem statement, all variables and given/known data
Find the shortest distance between the 2 planes:
2x+2y-z=1 and 4x+4y-2z=5
How do we approach this problem?
I used the approach of finding the point at which the normal of one plane intersects the othe... | {
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$$= \sqrt{ \frac{4 + 4 + 1}{36} } = \sqrt{ \frac{9}{36} } = \frac{1}{2} .$$
I think the problem you may have made for yourself is that you didn't actually choose a point of intersection in either plane to build a normal line from.
A more general argument along these lines gives the perpendicular distance between two ... | {
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# Induction proof of lower bound for $\sum \sqrt n$
I'm having some trouble proving the following statement using mathematical induction:
$$\frac{1}{2}n^{\frac{3}{2}} \leq \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} + ... + \sqrt{n} ,\text{ (for all sufficiently large n)}$$
I'm sort of confused because $n^{\frac{3}{2}... | {
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$$\frac12\sqrt n\le\frac1n\sum_{k=1}^n\sqrt k\tag{1}\;.$$
The righthand side of $(1)$ is just the arithmetic mean of the square roots $\sqrt 1,\sqrt 2,\dots\sqrt n$. You’ve observed (correctly) that this mean cannot be any bigger than $\sqrt n$, the largest of the $n$ numbers, but that doesn’t mean that it must eventu... | {
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$$n\sqrt{n+1}\le n\sqrt n+\sqrt{n+1}$$ and then to $$(n-1)\sqrt{n+1}\le n\sqrt n\;.\tag{5}$$
Everything in $(5)$ is non-negative, so $(5)$ is equivalent (for positive integers $n$) to the inequality that you get by squaring it,
$$(n-1)^2(n+1)\le n^3\;.\tag{6}$$
Can you show that $(6)$ is true for all positive intege... | {
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# Tutor profile: Amadej Kristjan K.
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## Questions
### Subject:Machine Learning
TutorMe
Question:
Suppose we are trying to use a machine learning algorithm to perform a classification task, which humans can do with $$98$$% accuracy. Our... | {
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TutorMe
Question:
Prove that there are infinitely many prime numbers.
Inactive
Suppose for the sake of contradiction that there are only finitely many prime numbers. Then we can list them as follows: $$p_1,p_2,...,p_n$$. Now consider the number $$P = p_1p_2...p_n + 1$$. It is not divisible by either of the prime num... | {
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# Prove that for any integer $n, n^2+4$ is not divisible by $7$.
The question tells you to use the Division Theorem, here is my attempt:
Every integer can be expressed in the form $7q+r$ where $r$ is one of $0,1,2,3,4,5$ or $6$ and $q$ is an integer $\geq0$.
$n=7q+r$
$n^2=(7q+r)^2=49q^2+14rq+r^2$
$n^2=7(7q^2+2rq)+... | {
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Your proof is correct (fleablood's comment). Let me rephrase a bit:
$A(n):= n^2 +4$; $B(r,q) = 7 q^2 + 2rq;$
$A(n) = 7B(r,q) + (r^2 +4)$.
Fairly simple to show that:
$A(n)$ is divisible by $7 \iff$
$(r^2 +4)$ is divisible $n.$
By inspection
$(r^2 +4) , r = 0,1,2,3,4,5,6,$
is not divisible by $7$.
$\Rightarrow$... | {
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If $k = 0$ then $n^2 \equiv i^2$ which is either $0,1$ depending on the value of $i$.
If $k = 1$ then $n^2 \equiv 2\mp i + i^2$. Which is either $2$ or $4$ depending upon the value of $i$ and the sinage of $\mp$.
But that is probably way too obtuse, isn't it?
Your proof is correct. If you know congruences then we ca... | {
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Here is a proof that seems a bit roundabout, but it relates simpler methods to Fermat's Little Theorem.
First note that clearly $n^2+4$ will not be a multiple of $7$ if $n$ is one. Now for other values of $n$, raise $n^2+4$ to the power of $7-1=6$:
$(n^2+4)^6=n^{12}+(6)(4)n^{10}+(15)(4^2)n^8+...+4^6$
$\equiv n^{12}+... | {
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# How to show values of points on surface plotted with ContourPlot3D
The help for ContourPlot3D starts with this example
ContourPlot3D[x^3 + y^2 - z^2 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]
This returns a Plots of the surface $x^2 + y^2 - z^2 = 0$:
.
Now I have a function, and I would like to know how this fun... | {
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.
-
The more general question is how to interact with three dimensional graphics objects using the mouse. Tooltip does work, so there's some support. But the mouse coordinates can only be retrieved in 2D while this time you want the coordinates in 3D, on the surface. +1. – Szabolcs Mar 15 '12 at 9:52
Why not use buil... | {
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Now that we have the point on the surface, you can do with it whatever you want (calculate another functions, etc.) You can use EventHandler to just record clicks instead of tracking values dynamically.
To address your other question about how to get a number of points on the surface. One way is to use FindInstance.
... | {
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-
If you use PassEventsDown -> True, then you can use button-1 clicks while keeping the graphics rotatable (no need for button 2). Button 2 might still be more comfortable for some people. – Szabolcs Mar 15 '12 at 13:19
For the Sphere[] radius, you could use Scaled coordinates to make this independent of the plot rang... | {
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# Riemann-Stieltjes integral, integration by parts (Rudin)
Problem 17 of Chapter 6 of Rudin's Principles of Mathematical Analysis asks us to prove the following:
Suppose $\alpha$ increases monotonically on $[a,b]$, $g$ is continuous, and $g(x)=G'(x)$ for $a \leq x \leq b$. Prove that,
$$\int_a^b\alpha(x)g(x)\,dx=G(b... | {
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Theorem: Suppose $f$ and $g$ are bounded functions with no common discontinuities on the interval $[a,b]$, and the Riemann-Stieltjes integral of $f$ with respect to $g$ exists. Then the Riemann-Stieltjes integral of $g$ with respect to $f$ exists, and
$$\int_{a}^{b} g(x)df(x) = f(b)g(b)-f(a)g(a)-\int_{a}^{b} f(x)dg(x)\... | {
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### Dynamic Programming Hotel Problem | {
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dynamic programming hotel problem. Before turning to the derivation of the dynamic programming equation for the problem Vε, we introduce a notation which will be used frequently in the sequel. Dynamic Programming (DP) is a technique that solves some particular type of problems in Polynomial Time. Problem Statement. Ple... | {
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with recursion, but they don't have to be. Introduction to Operations Research. Multivariate. We can create a 2D array part[][] of size (sum/2)*(n+1). – Hotels have different costs. These methods have been used numerically to compute optimal policies, as well as analytically to determine the form of an optimal policy u... | {
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3. Yuval Tassa†, Nicolas Mansard∗ and Emo Todorov†. A recursive relation between the larger and smaller sub problems is used. Massachusetts Institute of Technology. I know that Knapsack is NP-complete while it can be solved by DP. For the various problems in area such as inventory, chemical engineering design , and con... | {
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language, such as Perl or LISP, a developer can create variables without specifying their type. def canConstruct(target,workbank,memo={}): if (target in memo): return(memo[target]) if len(target)==0. Here are just some of the benefits of using a breadcrumb trail. Dynamic programming is both a mathematical optimization ... | {
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A large part of what makes computer science hard is that it can be hard to know where to start when it comes to solving a difficult, seemingly unsurmountable problem. Adding your google account to your Android phone is a great way to get emails link various different apps to your Google account. Medium Problem Solving ... | {
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property. The analysis code may be The program analysis discussed in this dissertation is almost exclusively dynamic binary analysis (DBA) Nonetheless, in recent years these problems have been largely overcome by the advent of. Dynamically linked shared libraries are an important aspect of GNU/Linux. The first line con... | {
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9 2 10 10. Doesn't matter what kind of document it is (the example provided in the attached image will be. The heuristic restricts the size of the state space of a dynamic programming algorithm. Dynamic-Programming-Travelling-Salesman-Problem's Introduction. March 2019. Also, each of the sub-problems. There’s a reason ... | {
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having a problem with the 'note' functionality on BC. Similarly, Tile[N-1] = Tile[N-2] + Tile[N-3]. Solving these high-dimensional dynamic programming problems is exceedingly di cult due to the well-known \curse of dimensionality" (Bellman,1958, p. In interviews or contests, problems on string are really common and one... | {
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Professor Bergin, Spring 1998 Adapted from lecture notes of Kevin Salyer and from Stokey, Lucas and Prescott (1989) Outline 1) A Typical Problem 2) A Deterministic Finite Horizon Problem 2. 2021 misof: Dynamic Programming, Math 3. Coevolution and collective behavior. Dynamic programming is something every developer sho... | {
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reasons that some things can seem so tricky is that. At each stage, it ranks decisions based on the sum of the present cost and the expected future cost, assuming optimal decision making for There is a very broad variety of practical problems that can be treated by dynamic programming. It is a common pattern to combine... | {
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word recognition, IEEE Trans. In some cases, like Production Debugging, logs might be the only information you have. MapKit is free beyond your Apple Developer Program membership and doesn't have limitations on the number of API requests/day. Dynamic Programming (DP) Algorithms Culture. Dynamic Programming is a program... | {
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subproblems char-. In contrast, the dynamic programming solution to this problem runs in Θ(mn) time, where m and n are the lengths of the two sequences. Here it is You must stop at the final hotel (at distance an), which is your destination. 20; the second slice shows a couple of important examples (MRI image reconstru... | {
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algorithms do not guarantee optimality because of the hardness of the problem. In academic terms, this is called optimal substructure. This project forked from kjsce-codecell/Dynamic-Programming-Solutions. Dynamic Programming: Example Dynamic Programming Problems. It looks like you can solve this problem with dynamic p... | {
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if I have some problems related to my online registration of appointment?. The subproblems are optimized to optimize the overall solution is known as optimal substructure property. Java Programming Masterclass udemy course can be your great first stepping stone. Employees are more open to work and strive harder to reac... | {
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the unique solution of this system of nonlinear equations. Even though the problems all use the same technique, they look completely different. 1 - What is Dynamic Programming. Answer (1 of 5): Advantages 1. The L key can be used to align a handle to its position in the previous frame. And they can improve your day-to-... | {
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used in video production to bring still images to life. Title Description: We all know the Fibonacci sequence. 0 ThemeLuviana Hotel Booking theme works seamlessly within WordPress 5. Classification, Regression. A branch of mathematics studying the theory and the methods of solution of multi-step problems of optimal con... | {
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features of this kind of problem. The list of problems in each category of Dynamic. PDF Drive investigated dozens of problems and listed the biggest global issues facing the world today. Dynamic programming assumes full knowledge of the MDP It is used for planning in an MDP For prediction: Input: MDP S, A, P, R, γ and ... | {
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Bellman pioneered the systematic study of dynamic programming in the 1950s. • Linear Program (LP) is an optimization problem where. In addition to the topics. It is also a. Here, by Longest Path, we mean In general, Dynamic programming (DP) is an algorithm design technique that follows the Principle of Optimality. Get ... | {
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for understanding and solving Bayesian formulations of these problems. price discrimination, semi-parametric estimation, dynamic programming, hotel revenue management. this is a collection of tutorials for dynamic programming problems that I developed while revising the concepts. Sutton and A. 17 Amortized Analysis. Th... | {
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programming The departure point of dynamic programming (DP) method is the idea of embedding a given optimal control problem (OCP) into a family of OCP indexed by the initial data (t0 , x0 ). • The goal is to select a route to and a hotel in Perth so that the overall cost of the trip is minimized. The second line contai... | {
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Richard Bellman was facing various kinds of multistage decision problems. Athena Scientific, 1995. Job Scheduling) •Optimization problems - typically find the math, recursive function Application of Dynamic Programming to solving Dynamic Programming - Counting problems Friday, November 12, 2021 10:11 AM. Many businesse... | {
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algorithm is proposed for the P-median problem. This is the List of 100+ Dynamic Programming (DP) Problems along with different types of DP problems such as Mathematical DP, Combination DP, String DP, Tree DP, Standard DP and Advanced DP optimizations. Our customers, and their customers, are at the center of every elev... | {
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solution S for W pounds. Any recursive formula can be directly translated into recursive algorithms. It uses the bottom up approach. Problem Seeking, Fifth Edition lays out a five-step procedure that teams can follow when programming any building or series of buildings, from a small house to a hospital complex. Bellman... | {
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problems. The Traveling Salesman Problem with Hotel Selection (TSPHS) is a variant of the classic Traveling Salesman Problem. Dynamic programming was the brainchild of an American Mathematician, Richard Bellman, who described the way of solving problems where you need to find the best decisions one after another. Poste... | {
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Logic (R) Chapter 7 (A-B) - Propositional Logic. It depends on the setting up the subproblems in a way that the space for the results of all subproblems needed at one time is small enough to fit in the memory allocation for the problem. Download Ariadoss PMS for free. You could be storing up a problem for the company i... | {
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to check the availability of your hotel rooms online and book them in real-time. Before we study how to think Dynamically for a problem. WordPress 5. Hotel Booking/Central Reservation System. The main advantage is in the efficiency with which a dynamic programming language provides. By offering a. ıt is not working, an... | {
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up-to-the-minute news, breaking news, video, audio and feature stories. Programming. They converted the financial stock portfolio problem to a dynamic programming problem using some dynamic terminologies. Partner program. Dynamic Programming is just a fancy way to say 'remembering stuff to save time later'". Today, the... | {
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belongs to the NP-Hard class. effectiveness and simplicity by showing how the dynamic programming technique can be applied to several different types of problems, including matrix-chain prod-ucts, telescope scheduling, game strategies, the above-mentioned longest common subsequence problem, and the 0-1 knapsack problem... | {
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don't overlap. Dynamic Programming was invented by Richard Bellman, 1950. The problem is that we didn't compile using the -g option of gcc, which adds debugging symbols. Problem : ( Scroll to solution ). The subproblem is the following: d(i): The minimum penalty possible when travelling from the start to hotel i. confi... | {
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the algorithm that we described is a computational twin of a popu-lar alignment algorithm for sequence comparison. Example 1 input: 4 1 2 3 4 Output: 1 2 3 6 problem-solving idea: the idea of dynamic programming is similar to that of changing coins, but the combination of changing coins does not consider the order. Int... | {
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cool feature of Unreal that you might be surprised about if you are used to programming C++ in Actor iterators do not have the problem noted above, and will only return objects being used by the. The term "dynamic programming" was coined by Bellman in the 1950s. The path to the origin node is shown when selected segmen... | {
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=" on a sheet of paper. Dynamic programming used to solve problem in polynomial time which is more faster then brute force method. 9 beta Now available. Assignment 2: Optimal Policies with Dynamic Programming. → the goal is to maximize or minimize a linear objective function → over a set of feasible solutions - i. Dp w... | {
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for architects and clients¿¿—fully updated and revised Architectural programming is a team effort that requires close cooperation between architects and their clients. | {
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kmv arx rny ntf vsr nfb irw vwh fof ygx ryh fjd mfi die pzb jxy swu qxb qxo gzo | {
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Counting Distinct matrices
How many distinct (if matrix $M$ is included in count, do not include $PM$ where $P$ is permutation matrix) $3\times 3$ matrices with entries in $\{0,1\}$ are there such that each row is non-zero, distinct and such that each matrix is of real rank $2$ or $3$?
I think answer for rank $2$ is ... | {
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Here's how it goes:
• The only restriction on the first row (thinking of it as a vector in $\{0, 1\}^3$ is that it cannot be $\mathbf{0}$, so there are $2^3-1$ possibilities. Call this row $\mathbf{v}_1$.
• For the second row, it cannot be either of the $2$ multiples of $\mathbf{v}_1$, namely $\mathbf{0} = 0\mathbf{v}... | {
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and up to permutation of the rows, there are
$\dfrac{42}{6} = 7$ possibilities.
• You are getting one more than me which is strange. – Turbo Apr 10 '15 at 13:44
• Could you list rank $2$ matrices? – Turbo Apr 10 '15 at 13:47
• [111;110;001], [111;101;010], [111;011;100], [110;100;010], [101;100;001], [011;010;001] wh... | {
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# Find the range of eccentricity of an ellipse such that the distance between its foci doesn't subtend any right angle on its circumference.
What is the range of eccentricity of ellipse such that its foci don't subtend any right angle on its circumference?
I thought that the eccentricity would definitely be more than... | {
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$$(F_{+}-P)\cdot(F_{-}-P) = 0 \tag{\star}$$
That is, \begin{align} 0 &= (c - a \cos\theta )(-c-a\cos\theta) + (0 - b \sin\theta)(0-b\sin\theta) \\[4pt] &= -c^2 + a^2 \cos^2\theta + b^2\sin^2\theta \\[4pt] &= -c^2+a^2\cos^2\theta + ( a^2-c^2)(1-\cos^2\theta) \\[4pt] &= a^2 - 2 c^2 + c^2 \cos^2\theta \tag{1} \end{align}... | {
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$$0 \leq e < \frac{1}{\sqrt{2}} \tag{\star\star}$$
• The answer doesn't match, do you have any alternate solution it? – Jasmine Dec 3 '17 at 13:07
• @Jasmine: The answer doesn't match what? – Blue Dec 3 '17 at 14:19
• @Blue The Given solution, possibly. This was probably a homework question, and OP couldn't find the a... | {
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These notes and closely review Unit 1 section 9.
## Analyzing a simple iterative algorithm: findFirstMissing with binarySearch
Dr. Roche, in the Unit 1 notes, kindly analyzes our three search functions so painstakingly that he gives us exact expressions for the worst-case running time in terms of number of statements... | {
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## What happens when we replace binarySearch with linearSearch?
Let's replace binarySearch with linearSearch in findFirstMissing. The interesting thing here, is that we can't just say the call to linearSearch takes time $\Theta(n)$. We would have to prove first that it's actually possible in the context of the findFir... | {
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# “IFF” (if and only if) vs. “TFAE” (the following are equivalent)
If $P$ and $Q$ are statements,
$P \iff Q$
and
The following are equivalent:
$(\text{i}) \ P$
$(\text{ii}) \ Q$
Is there a difference between the two? I ask because formulations of certain theorems (such as Heine-Borel) use the latter, while other... | {
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-
I would never write the second one without parentheses. Also because there's also a third possible interpretation: $A\iff(B\iff C)$. As a general rule, for a nested binary operator $@$, parentheses should only be omitted iff $(A @ B) @ C$ and $A @ (B @ C)$ are equivalent. – celtschk Jul 10 '13 at 7:04
@celtschk, bic... | {
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# How often two iid variables are close?
Is there a constant $$c>0$$ such that for $$X,Y$$ two iid variables supported by $$[0,1]$$, $$\liminf_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c$$ I can prove the result if they have a density, of if they have atoms, but not in the general case.
If $$\epsilon \geqslant... | {
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Here are a few examples where the limit is infinite:
Example 1: If $$X$$ has an atom then $$\mathbb{P}[|X-Y| < \varepsilon]$$ is bounded away from 0. So the limit in question diverges like $$\varepsilon^{-1}$$.
Example 2: The Cantor ternary function is the CDF of a probability distribution on $$[0,1]$$ which is nonat... | {
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Mateusz's slick argument worked on the block diagonal, a sum of $$n$$ squares of width $$1/n$$. This covers about half of the area of the strip $$|X-Y| < \tfrac{1}{n}$$, which is why the resulting constant is half of optimal. There's probably a hands-on way to extend it, but you start using words like "convolution" and... | {
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Proof of Lemma: The Fourier transform is an isometry of the space $$L^2$$. The ($$\Leftarrow$$) direction is immediate: if $$X$$ has a density function in $$L^2$$, then certainly its Fourier transform is in $$L^2$$. For ($$\Rightarrow$$), if $$\psi$$ is in $$L^2$$ then it is the Fourier transform of some function $$f \... | {
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Suppose now that $$||\psi||_2^2 < \infty$$, i.e., $$\psi \in L^2$$. Consider the "box filter" $$B_\varepsilon$$ defined by $$B_\varepsilon(x) = (2 \varepsilon)^{-1}$$ for $$|x| < \varepsilon$$ and $$B_\varepsilon(x) = 0$$ otherwise. This has $$\hat B_\varepsilon(t) = \operatorname{sinc}(2 \pi t \varepsilon)$$. This is ... | {
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Characteristic polynomial of the matrix with zeros on the diagonal and ones elsewhere
Find the characteristic polynomial of the matrix with zeros on the diagonal and ones elsewhere.
I've been able (I believe) to guess how it looks like (by considering matrices of small orders): $(x-n+1)(x-1)^{n-1}$. I suppose I shoul... | {
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So what are the eigenvectors of $B$? It's so symmetric that it's easy to write a few down: the vector that is all $1$s has eigenvalue $n$, while any vector whose components sum to zero is an eigenvector with eigenvalue $0$. There are at least $n-1$ of these, since $e_1-e_i$ for $2 \leq i \leq n$ is a linearly independe... | {
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# Category Archives: problem-solving
## Binomial Expansion Variation
Several years ago, I posed on this ‘blog a problem I learned from Natalie Jackucyn:
For some integers A, B, and n, one term of the expansion of $(Ax+By)^n$ is $27869184x^5y^3$. What are the values of A, B, and n?
In this post, I reflect for a mom... | {
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Christopher’s approach paralleled my own. The x and y exponents from the expanded term show that n=5+3=8. Expanding a generic $(Ax+By)^8$ then gives a bit more information. From my TI-Nspire CAS,
so there are 56 ways an $x^5y^3$ term appears in this expansion before combining like terms (explained here, if needed).... | {
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EXTENSIONS & CONCLUSION
Admittedly, NB’s solution would have been complicated if the parameter was composed of something other than singleton prime factors, but it did present a fresh, alternative approach to what was becoming a comfortable problem for me. I’m curious about exploring other arrangements of the paramet... | {
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GETTING STARTED
As a simple example, my students earlier had seen the graph of $f(x)=5+2sin(x)$ as $y=sin(x)$ vertically stretched by a magnitude of 2 and then translated upward 5 units. In their return, I encouraged them to envision the function behavior dynamically instead of statically. I wanted them to see the c... | {
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For a variable amplitude, consider $y=2+1.2^{-x}*sin(x)$. The midline is $y=2$, with an “amplitude” of $1.2^{-x}$. That made a ceiling of $y=2+1.2^{-x}$ and a floor of $y=2-1.2^{-x}$, basically exponential decay curves converging on an end behavior asymptote defined by the midline.
SINUSOIDAL MIDLINES AND ENVELOPES
... | {
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WHAT HAPPENS WHEN THE PERIODS DIFFER?
Try a graph of $g(x)=cos(x)+cos(3x)$.
Using the earlier concept that any function added to a sinusoid could be considered the midline of the sinusoid, we can picture the graph of g as the graph of $y=cos(3x)$ oscillating around an oscillating midline, $y=cos(x)$:
IF you can’t se... | {
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A STUDENT WANTED MORE
In class last Friday, my students were reviewing envelope curves in advance of our final exam when one made the next logical leap and asked what would happen if both the coefficients and periods were different. When I mentioned that the exam wouldn’t go that far, she uttered a teacher’s dream pr... | {
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We didn’t go there, but recognizing that new envelopes can be found simply by rewriting sums creates an infinite number of additional envelopes. Defining these different sums with a slider lets you see an infinite spectrum of envelopes. The image below shows one. Here is the Desmos Calculator page that lets you play... | {
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INITIAL THOUGHTS
I see two big challenges here.
First, the missing location of point P is especially interesting, but is also likely to be quite vexing for many students. This led me to the first twist I found in the problem: the introduction of multiple variables and a coordinate system. Without some problem-solving... | {
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Of course, since all four triangles have the same base lengths, the given area ratios are arithmetically equivalent to corresponding height ratios. I used that to write a second equation.
$\displaystyle \frac{\Delta BCP}{\Delta CDP} = \frac{y}{12-x} = \frac{1}{3}$
Simplifying terms and clearing denominators leads to... | {
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PROBLEM VARIATIONS
Just two extensions this time. Other suggestions are welcome.
1. What’s the ratio of the area of $\Delta BCP : \Delta DAP$ at the point P that satisfies both ratios??
It’s not 1:4 as an errant student might think from an errant application of the transitive property to the given ratios. Can you s... | {
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The ratio of Women:Girls was 11:4, so the 24 girls meant each “unit” in this ratio accounted for 24/4=6 people. That gave 11*6=66 women and 66+24=90 females.
At this point, my experience working with algebraic problems tempted me to overthink the situation. I was tempted to let B represent the unknown number of boys... | {
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WHY HAVE JUST ONE SOLUTION?
Math problems involving ratios can usually be opened up to allow multiple, or even an infinite number of solutions. This leads to some interesting problem extensions if you eliminate the “24 girls” restriction. Here are a few examples and sample solutions.
What is the least number of par... | {
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What variations can you and/or your students create?
RESOLVING THE INITIAL ALGEBRA
Now to the solution variation I was initially inclined to produce. After initially determining 66 women from the given 24 girls, let B be the unknown number of boys. That gives B+24 children. It was given that adults are 4 times as ... | {
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## Unanticipated Proof Before Algebra
I was talking with one of our 5th graders, S, last week about the difference between showing a few examples of numerical computations and developing a way to know something was true no matter what numbers were chosen. I hadn’t started our conversation thinking about introducing ... | {
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on his paper and covered up all but the one’s digit. “You see,” he said, “all that matters is the units. You can make the number as big as you want and I just need to look at the last digit.” Without using this language, S was venturing into an even-odd proof via modular arithmetic.
With some more thought, he reaso... | {
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Also note that after S completed his 2s and 8s lists, he used only single digit seed numbers as the bigger starting numbers only complicated his examples. He was on a roll now.
I asked him how the “4 number” cycle was related. He noticed that the 4s used every other number in the “2 number” cycle. It was like skip ... | {
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CONCLUSION: At this point, S declared that since he had shown every possible case for evens and odds, then he had shown that any multiple of an even number was always even, and any odd multiple of an odd number was odd. And he knew this because no matter how far down the list he went, eventually any multiple had to e... | {
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Confession #2: My approach was a much longer and far less elegant solution than the identical approaches offered by a comment by “P” on my last post and the solution offered on FiveThirtyEight. Rather than just accepting the alternative solution, as too many students are wont to do, I acknowledged the more efficient ... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
My first mistake was in my calculation of the expected time if they did not choose the same initial app. The 20 numbers in blue above represent that sample space. Notice that there are 8 times where one sibling chose a 5-minute app, leaving 6 other times where one sibling chose a 4-minute app while the other chose so... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
I was initially correct in concluding there was a $\frac{1}{5}$ probability of the second app choice achieving a simultaneous finish and that this would not result in any additional total time. I missed the fact that the six non-highlighted values also did not result in additional time and that there was a $\frac{1}{5... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
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