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Department of

# Mathematics

Seminar Calendar
for events the day of Monday, November 6, 2017.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, November 6, 2017

3:00 pm in 141 Altgeld Hall,Monday, November 6, 2017

#### A Vietoris-Smale Mapping Theorem for the Homotopy of Hyperdefinable Sets

###### Elliot Kaplan   [email] (UIUC)

Abstract: I will discuss the paper $$A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets$''$ by Alessandro Achille and Alessandro Berarducci $[\href{https://arxiv.org/abs/1706.02094}{arXiv}]$. In this paper, they compare the definable homotopy groups of a definable space $X$ (denoted $\pi_n(X)^\mathrm{def}$) with the (usual) homotopy groups of the quotient $X/E$, where $E$ is a certain kind of type-definable equivalence relation on $X$. Under certain assumptions, these groups are actually isomorphic. These types of quotients were first studied by Pillay in the case that $X$ is actually a definable group. No knowledge of model theory will be assumed. In particular definable,'' type-definable'' and o-minimal'' will all be defined.

3:00 pm in 243 AH,Monday, November 6, 2017

#### The partial compacti cation of the universal centralizer

###### Ana Balibanu (Harvard)

Abstract: Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in $G$ of regular elements in $\text{Lie}(G)$, parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^*G$. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful compactification of $G$. We show that the symplectic structure extends to a log-symplectic Poisson structure on the partial compactification, through a Hamiltonian reduction of the logarithmic cotangent bundle of the wonderful compactification.

4:00 pm in 245 Altgeld Hall,Monday, November 6, 2017

#### Asymptotically optimal shapes: the drum with lowest n-th frequency, and the ellipse enclosing the most lattice points

###### Rick Laugesen   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: What shape of domain minimizes the n-th energy level (frequency) of the Laplacian, for large n? How is that connected to lattice point counting in ellipses? And why do right triangles behave differently?

4:00 pm in 141 Altgeld Hall,Monday, November 6, 2017

#### Higher Categories: What's up with that?

###### Nima Rasekh

Abstract: Higher categories are very useful when we want to understand not only whether two objects are the same but how this "sameness" occurs. However, it is often difficult to motivate the abstract concepts necessary to define and use higher categories. The goal of this talk is to show some concrete examples in mathematics where higher categories naturally arise, without ever using the words "higher", "category" or "homotopy". Thus no background in these topics is assumed.

5:00 pm in 445 Altgeld Hall,Monday, November 6, 2017

#### Decay estimates for quantum systems

###### Marius Junge

Abstract: Recently the connection between hypercontractivity and decay estimates for quantum systems has become a hot topic in QIT. We present the first results by Temme and his collaborators.


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# Largest N squares that fit in a rectangle

I was working on a project and I needed to display N squares inside a rectangle area and I want them to be as large as possible, no rotations. More formally:

Problem: Given N equal-sized squares and a rectangle with width W and height H, find out the maximum size of the squares L such that the squares fit inside the rectangle in such a way that their sides are orthogonal to the rectangle sides.

So far, I think we have a O(sqrt(N)) solution, in which we assume W>H, start by trying to place all squares in a single row, then in two rows and so on, until we get to O(sqrt(N)). For formally, let nRows be a variable from 1 to sqrt(N), nCols = ceil(N/nRows) and then do L = min(H/nRows, W/nCols).

I'm not completely sure the idea above is optimal. I was also wondering if we can solve this problem in O(1)?

I've tried looking for orthogonal square packing, but the problem seems a bit different (or could we easily reduce on to another?)

• The $O(\sqrt{N})$ algorithm sounds very clean and likely to be effective in practice. Is it really problematic in practice? How large is $N$ in practice? I can think of more efficient algorithms, but I suspect they might not matter in practice and might be an instance of premature optimization. Can you address my concerns? – D.W. Oct 8 '14 at 0:02
• weird, can you prove that if $L_{max}$ does not use up either H or W, then you can always increase $L_{max}$ by a small amount and all squares still fit? – InformedA Oct 8 '14 at 3:12
• Your initial sentence says "no rotations", but your "More formal" description doesn't include that restriction. Do the sides of the squares need to be parallel to the sides of the rectangle? $\hspace{1.78 in}$ – user12859 Oct 8 '14 at 3:40
• Yeah, I forgot about it. Let me update – kunigami Oct 8 '14 at 13:16

Let

\begin{align} C&=\left\lfloor\frac WL\right\rfloor& R&=\left\lfloor\frac HL\right\rfloor \end{align}

be the maximum number of rows and columns respectively. It sounds as though you're looking for the maximum $L^*$ such that, when $L=L^*$, we have $CR\ge N$. Let

\begin{align} L_0&=\sqrt{\frac{WH}N}\\ C_0&=\left\lceil\frac W{L_0}\right\rceil=\left\lceil\sqrt{\frac{WN}H}\right\rceil& R_0&=\left\lceil\frac H{L_0}\right\rceil=\left\lceil\sqrt{\frac{HN}W}\right\rceil. \end{align}

Observe that $L\le\min{\{W/C_0,H/R_0\}}$ is sufficient, and $L\le L_0$ is necessary:

\begin{align} \frac{WH}{L^2}&\ge\left\lfloor\frac WL\right\rfloor\left\lfloor\frac HL\right\rfloor=CR\ge N\\ L&\le\sqrt{\frac{WH}N}=L_0, \end{align}

with equality only if $W/L_0$ and $H/L_0$ both are integers. As a result, the optimal solution has dimension either $C_0\times\lceil N/C_0\rceil$ or $\lceil N/R_0\rceil\times R_0$.

Another possible approach is to look for good rational approximations to the ratio $W/H$, i.e., positive integers $r,s$ such that $r/s \approx W/H$.

Any good solution to your problem immediately yields such a good rational approximation (just let $r =$ the number of rows of squares and $s =$ the number of columns). Similarly, any good rational approximation to $W/H$ suggests a candidate solution to your problem (let $k=\lceil \sqrt{N/rs} \rfloor$, and then consider an arrangement with about $kr$ rows and $ks$ columns). So, a plausible approach might be to generate a bunch of good rational approximations to $W/H$, then test each corresponding candidate layout to see which is the best.

There are various algorithms for generating good rational approximations to a desired number $\alpha \in \mathbb{R}$. For instance, you can use continued fractions to generate a set of candidate approximations; there is a mathematical guarantee that each candidate will be a very good approximation, in some well-defined mathematical sense. The running time to generate and examine all such continued fractions will be poly-logarithmic in $N$. However, I don't know whether this will be better in practice than your simple $O(\sqrt{N})$ time algorithm.

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Airflow on Windows with Docker – unable to install or load my own dags/broken dag error

I’ve spent an incredible amount of time on this and I believe I’ve now read every single thread or post on this but nothing seems to be helping.

I am trying to run Airflow on my (work) Windows machine using Docker. I’ve tried the following two approaches but without success:

1. Follow the guide by Airflow, get the yaml file and run compose-up (https://airflow.apache.org/docs/apache-airflow/stable/start/docker.html) – I get the following error:

Creating airflow_airflow-init_1 … error

ERROR: for airflow_airflow-init_1 Cannot create container for service airflow-init: TCGA-FS01UsersMyNameDocumentsAirflowlogs%!(EXTRA string=is not a valid Windows path

Nothing that I’ve tried seems to fix this error and I just cannot initiate the actual airflow image. I’m confused as to what the error is referring to because I do not have a "logs%!" folder/file in my Airflow folder – the Airflow folder only has the yaml file and a two dags that I was hoping to load.

1. Run the puckel version of Airflow. I am able to start this successfully without any issues. If I run the "airflow scheduler" command in the CLI or using bash, I am able to load all of the example dags that come with the puckel airflow container (none will show in the webserver until I ran this).
Following this guide: http://www.marknagelberg.com/getting-started-with-airflow-using-docker/
I’ve tried mounting my own local folder to the usr/local/airflow/dags and it seems to mount it successfully again but my own dags never show up.

EDIT: I seem to have made some progress – I run the following:

docker cp sample_dag.py containerName:/usr/local/airflow/dags

And it seems that my dag gets loaded into the right folder because when I go into the webserver, I get the following error:

Broken DAG: [/usr/local/airflow/dags] No module named ‘airflow.operators.bash’

Looking at the different posts, it seems like I need to add a requirements.txt file but I am not sure how to do this exactly – I am guessing that based on the following: https://github.com/puckel/docker-airflow/issues/592

So in summary, Option 1 – I get the windows string error and I am unable to proceed with the start up. Option 2 – I cannot get my load my own dags or I get a Broken DAG/module error.

I am absolutely lost and if anyone could help me, I would be eternally grateful. Thank you folks.

Source: Docker Questions


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• ### The Structure of Number Systems (Teachers' Mathematics

Format: Hardcover

Language: English

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Size: 13.05 MB

Radical signs Use sqrt, which adjusts to the size of its argument: \sqrt{x^3} $\sqrt{x^3}$; \sqrt[3]{\frac xy} $\sqrt[3]{\frac xy}$. The following excerpt is from a review written by Ivor Grattan-Guinness that appeared in Middlesex Math Notes (Volume 3, pages 47 - 50, 1977), a publication of Middlesex University in London, England. "There is enough here [in Non-Newtonian Calculus] to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditional problems.
• ### Martin Gardner's Mathematical Games

Format: CD-ROM

Language: English

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Each of the following two books was reviewed by Leo Barsotti in the journal Boletim da Sociedade Paranaense de Matemática. [54] 1) The First Nonlinear System of Differential and Integral Calculus: Volume 2, page 32, 1981. 2) The First Systems of Weighted Differential and Integral Calculus: Volume 2, pages 32-33, 1981. JianLi 23:33, 7 May 2006 (UTC) Wouldn't it be more practical to merge the book into Innumeracy, now a pretty dumb page, and make it into a true article?
• ### Medical Math for the Health Care Profession Made Incredibly

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From the latter article: "For the minimisation of our energy functional, we show the advantages of a multiplicative Euler-Lagrange formalism ... Morse Code was designed by Samuel Morse and Alfred Vail. Mittal, Indian Institute of Technology; from the ResearchGate website on 12 November 2014. [218] It pleases me that this statement was made by a person I had never heard of, who lives on the other side of the world, and who is an excellent applied-mathematician. The article has been of considerable interest to, and used by, many scientists, engineers, and applied mathematicians.
• ### The Mysteries of Modern Science (Littlefield, Adams quality

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The eBook is optimized for e-reader devices and apps, which means that it offers a much better digital reading experience than a PDF, including resizable text and interactive features (when available). The symbols ( and ) used for grouping expressions. Journals are shelved alphabetically by title. In mathematics, and elsewhere, most significant discoveries are of the second kind. ... The first airplane was considered impossible. But the men of unborrowed vision went ahead.
• ### Useful Mathematical and Physical Formulae

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The first line in the file is ignored if it starts with a #. They taught us about honesty, the difference between right and wrong, hard-work, respect for others, the importance of education and independent thinking, the value of money, and the fact that "money isn't everything". Emel'yanenko, Ingo Krossing, and Roland Kalb. "Thermochemistry of ammonium based ionic liquids: tetra-alkyl ammonium nitrates - experiments and computations", The Journal of Chemical Thermodynamics, Volume 51, pages 107–113, http://dx.doi.org/10.1016/j.jct.2012.02.035, Elsevier, 2012. [109] Bruno Ćurko. "The presence of Ruder Boskovic in the digital world", proceedings of the annual international symposium "Days of Frane Petric - From Petric to Boskovic", ISSN 1884-2236, project/theme: 191-1911112-1092, pages 143-145, The Croatian Philosophical Society in Zagreb, 2011. [110] Mathematics Department of Eastern Mediterranean University in North Cyprus.
• ### Fractions for Second Graders

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I don't really want or need to get in to a whole new discussion of the multiple comparisons problem, but since you brought it up I thought OP may like to read some on this topic as well. Martin: Just add the volume of the spheres! Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius' calculus. I lived with my parents and sister Dotty in Somerville, Massachusetts.) We drove to his house.
• ### Reconstructing Evolution: New Mathematical and Computational

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Is it better to trust condescendingly or to seek the truth and resolve any doubts? The associative property of multiplication is a math rule that is always true. Mustafa Riza and Bugce Eminaga, from their article "Bigeometric Calculus and Runge Kutta Method" [215] (2015). "There is enough here [in Non-Newtonian Calculus] to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditional problems. A student’s course of study can be tailored to suit a particular interest in pure mathematics, applied mathematics, mathematics teaching, or statistics.
• ### Math for Moms and Dads: A dictionary of terms and

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Blyumin of the Lipetsk State Technical University in Russia. [23] An article concerning minimization methods based on the geometric and bigeometric calculi was written by Ali Ozyapici (Girne American University in Cyprus/Turkey), Mustafa Riza (Eastern Mediterranean University in North Cyprus), Bulent Bilgehan (Girne American University), and Agamirza E. Like the classical calculus, each of them possesses, among other things: a derivative, an integral, a natural average, a special class of functions having a constant derivative, and two Fundamental Theorems which reveal that the derivative and integral are 'inversely' related.
• ### Interactive Math II for Beginners

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In fact, the bigeometric derivative is now used to analyze the rate of change of the roughness of a fractal over time. "Random fractals, a quintessentially 20th century idea, arise as natural models of various physical, biological (think your mother's favorite cauliflower dish), and economic (think Wall Street, or the Horseshoe Casino) phenomena, and they can be characterized in terms of the mathematical concept of fractional dimension. Finally, a statement by an applied-mathematician in India resonates in my mind: "It is known that non-Newtonian calculus models real life problems more accurately." - R.
• ### Merriam-Webster's Guide to Everyday Math : A Home and

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### Lewin's blog

By Lewin, history, 3 years ago,

I recently added the div 1 regional to the gym here. You can virtual participate at any time.

Solutions are available on the contest website if you want to look at them up afterwards here (soon...). Feel free to discuss problems afterwards.

• +68

 » 3 years ago, # |   +5 How to solve I and M?
•  » » 3 years ago, # ^ |   +5 I：There will be an optimal solution in which all the added numbers are in nonincreasing order. Use this to formulate an O(n2k) dynamic program and then optimize using divide and conquer trick or convex hull trick. M: I'm unsure about the solution, but apparently it can be proven that you spend all your money on at most two people. Then some sort of ternary search should work.
•  » » » 3 years ago, # ^ | ← Rev. 2 →   +13 For M, my team's solution was to map the soldiers to (x, y) points (budget*potency/cost, budget*health/cost) and then the goal becomes to maximize x * y. So then you build the convex hull of those points. That way, it's only ever optimal to combine soldiers that are adjacent along the convex hull. The reasoning behind that is if you tried to merge more than 2 soldiers, you'd effectively be drawing a line between two segments on the convex hull (and of course that segment lies strictly inside the hull) and the product of any (x, y) on that segment will never be greater than some (x, y) product along the outer segments of the hull. So it's not really a proof but there's some intuition behind why you only combine 2 soldiers (also other solutions don't seem to use convex hull at all).
•  » » » 3 years ago, # ^ |   0 There will be an optimal solution in which all the added numbers are in nonincreasing order. I'm having trouble convincing myself of this. Do you have any proof or good intuition on why it's true?
•  » » » » 3 years ago, # ^ | ← Rev. 2 →   0 Some intuition: let's say there are two undecided positions i and j. They will divide the array in three parts A, B and C in something like this:-----A------|i|------B-----|j|-----C-----Let's say there is some value Oi which is the greatest optimal value to be chosen to be in position i. The smaller Oi is, the greater is the number of elements to pair with it in part A; the bigger Oi is, the greater is the number of elements to pair with it in part B and C.Let's say Oj is the same thing for position j. The only difference is that now the smaller Oj is, the greater is the number of elements to pair with it in part B.So, let's say Oj = Oi. If we increase Oj, obviously it will be worse than not increasing, because the greater Oi was, the greater the number of pairs would be made with part B, and Oi is optimal by definition. Now, even worse, the number of pairs made within B is greater if Oj is smaller. Thus, it's always optimal to make Oj ≤ Oi and you can extend this to any number of undecided elements.
•  » » 3 years ago, # ^ | ← Rev. 2 →   +3 M: Transform (c, h, p) to (h * C / c, p * C / c). The answer is a combination x1 * v1 (vi is (hi, pi)) + x2 * v2 + ..., you take that vector and take x*y. But that combination has other constraint, x1 + x2 + ... + xn == 1 and xi is in [0, 1] so it's a convex combination (https://en.wikipedia.org/wiki/Convex_combination). This means that you can take the convex hull and solve the problem for the vertices and edges and it will have the greatest answer.
•  » » » 14 months ago, # ^ |   0 I understood the Convex_combination and that its solution lies inside the convex hull of the points but, why is it like that? Why must the $x_i$ sum to 1?And what is the intuition behind this: $(h*C/c,p*C/c)$ ?Thank you!
•  » » » » 14 months ago, # ^ |   +8 That transformation is needed to make taking an object proportional. If it has weight 0 <= x <= 1, then you use x from your total "funds" (I don't remember the exactly story in the problem). Since not spending all the money isn't optimal you'll always spend everything so the sum is 1.
•  » » » » » 14 months ago, # ^ |   0 It's much clearer now, thank you for the quick response!
 » 3 years ago, # |   0 Is it possible to solve M with c++? It seems like there is a precision problem. My python code using (exactly) the same logic got AC.
•  » » 3 years ago, # ^ |   0 We have reference solutions in C++ (and we also have some other AC in C++ from other contestants independently). I'm not sure why yours is failing.
•  » » » 3 years ago, # ^ |   0 Thanks. BTW, do you by any chance know why my solution to E got Idleness limit exceeded on test 11? Thanks for your help.
•  » » » 3 years ago, # ^ |   0 http://codeforces.com/gym/101982/submission/45508248 here is the code.
•  » » » 3 years ago, # ^ |   0 nvm solved. my n and m are the other way round...
 » 3 years ago, # |   0 how to solve d?
•  » » 3 years ago, # ^ | ← Rev. 2 →   +19 dp[number of digits filled in][this number is k modulo m] or dp[n][k] can transition to dp[n+1][2*k] and dp[n+1][2*k+1]. dp[n][k] = (number of ways to get to this state from [0][0], sum of 1's of these ways). answer is dp[bits][0].second
•  » » » 3 years ago, # ^ |   0 ty
•  » » 3 years ago, # ^ | ← Rev. 2 →   +8 Similar solution to tfg's Is dp[position of bit (from higher to lower)][number module n][how many ones i've put]. Base case is when pos =  =  - 1, if module is 0, return ones (third dimension). otherwise return 0. transifions is:f(p, k, o) = f(p-1, (2^p + k)%n, o+1) + f(p-1, k, o)
• »
»
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3 years ago, # ^ |
Rev. 5   0

I am not sure whether i got the question right.I tried two different approaches one similar to osdajigu_. Can't seem to figure out what's wrong .Thanks . Post Edit: Found the issue was taking the wrong modulo number :/

Code
•  » » » » 3 years ago, # ^ |   0 haha I had same mistake, had module 1e9+7, instead of 1e9+9.
•  » » » » » 3 years ago, # ^ |   0 Did exactly the same :p
 » 3 years ago, # |   0 How to solve K?
•  » » 3 years ago, # ^ | ← Rev. 2 →   +3 It can be solved by DP. let's focus on minimum expected value, and then its similar for maximum. Be f(mask) = minimum expected value if I have the current mask (1 for available numbers, and 0 otherwise). So, the transition would some something like.You have to calculate values of what numbers can I remove such that sum up d, and add that movement to moves[d]. There a few cases to handle, in case you can't make any movement, that will be the base case. Also you have to calculate the probabilitie of get d with 2 dice. For that just do a double nested loop and sum up for i+j, and then divide all (for 2 to 12) by 36.Here is the code
 » 3 years ago, # |   0 How to solve E, and B?
•  » » 3 years ago, # ^ |   +3 E: You have to find the shortest cycle around cell having 'B'. For each cell, consider 2 nodes in the graph, P and Q. Also, consider a line from 'B' to an edge of the grid. For a particular cell, you are on node Q if you have crossed that line an odd number of times in your path. Otherwise, you are on node P. Now, calculate the shortest distance from P to Q for any cell.
•  » » 3 years ago, # ^ |   +3 problem E can be solved using max-flow(min-cut actually). Lets build the following flow network: for every position on the matrix lets create two nodes(call them entry node and exit node) and add an edge between them with capacity equal to the cost of blocking that position if it has a letter('a', 'b', ...) or INF in case it is the position of 'B' or a dot('.'). The source will be entry node of 'B' and the sink will be an extra node that we will connect with the exit nodes of every position on the borders with capacity equal to INF. Then for every position add an edge between its exit node and the entry nodes of every adjacent position with capacity equal to INF. Then run max-flow algorithm on this graph and that is the answer. What we are actually computing here is the minimum cost to separate the sink from the source(min-cut) which is equivalent to max-flow. The problem is similar to this one http://coj.uci.cu/24h/problem.xhtml?pid=2505 in case you want to practice this approach. Good luck
•  » » » 3 years ago, # ^ |   +3 Nice, thank you
•  » » » 2 years ago, # ^ | ← Rev. 2 →   +8 For problem E, why do standard max-flow algorithms (Dinitz and Push-Relabel) run very fast even though the graphs involved have about $2nm = 1800$ nodes and a similar order of magnitude of edges, while these algorithms take cubic time in the worst case? Is it simply that the test data does not have the worst case, or is there a tighter analysis you can do on grid graphs like these to prove better upper bounds?
 » 19 months ago, # |   +4 What is the time complexity of B? My time complexity is O(10^9) using prime factorization and inclusion-exclusion and which is I think give TLE. So how to solve this problem? Thanks in Advance.
•  » » 19 months ago, # ^ | ← Rev. 3 →   +4 I assume that you mean your solution is $O(\max(a,b,c,d))$ and just iterates up to $10^7$, $10^9$ would TLE.I never actually figured out why this worked, but during the contest I just observe that the inclusion exclusion only took on values of 0 or 1 and ran fast enough when I put the answers in a char array (but 2-4x slower in an int array)
•  » » 19 months ago, # ^ |   +12 You can also solve the problem in $O(n)$ with Möbius inversion and a linear-time sieve.
•  » » » 19 months ago, # ^ | ← Rev. 2 →   +3 Can You explain me how to solve in O(n)? Thanks in advance.
•  » » » » 19 months ago, # ^ |   +9 Nisiyama_Suzune's excellent blog on Möbius inversion describes this exact problem.
 » 14 months ago, # |   0 I'm still stuck at F_Rectangles even after reading the editorial. Here it is.I'm struggling with the lazy propagation part, and dividing the regions etc.. Since I know how to do the standard version without the odd property.Here is the problem statement: SpoilerYou are given several($N$ $<=$ $1e5$) axis-aligned rectangles. Compute the sum of the area of the regions that are covered by an odd number of rectangles.Any ideas anyone?


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06/01/2021

# affine cipher example problems

2.if plaintext is encoded into string1with A=11 and B=13, then string1 is encoded into string2 with A=15 and B=19. Hello i m working on the affine cipher in JAVA. This is because the greatest common divi‐ Justify your statement. The affine cipher is the combination of which two other ciphers? One other way to adapt the Affine Cipher quite simply is to change the numbering of the letters. Practice: Bitwise operators. Assume that you know it is an affine cipher and that the plaintext “ab” is enciphered to “GL” LASXOGEUTKCD Show transcribed image text Problem 4-1: Encrypt the message This is an exercise” using each of the following ciphers. Because there are many possible combinations of input parameters of the affine cipher, the brute force attack is inefficient, but still feasible. Multiplicative with key 15: Problem 4-2: Decrypt the message “UHCBCPYKEC”, by using the affine … This is called the Affine cipher. This is called the Affine cipher. ee��gb@ !�+� Decrypt the text below: 2. Who wrote the line? We consider a ciphertext-only attack on a substitution cipher and assume that the plaintext message is in English. Cryptography Worksheet — The Aﬃne Cipher 1 The Aﬃne Cipher is a cipher that uses Maths to encode the message. Get solution 2.13 What is a transposition cipher? "ant"0,13,19], (5.0+17) mod 26 17, (5.13+17) mod 26-4, (5.19+17) mod 26-8, and . Encryption Practical Cryptography: "Affine Cipher" Read this page, which discusses the Affine Cipher. The second problem with affine cipher’s key is discussed in the next chapter. Find the key length for a Vigenère cipher 3. Introduction The affine cipher is the combination of which two other ciphers? The simplest way to do this (and perhaps a more expected way) is to label "a" as 1, "b" as 2 and so on, as in the table below. These ciphers are used in symmetric key cryptography.A symmetric algorithm uses the same key to encrypt data as it does to decrypt data. Sort by: Top Voted. Get solution Problems 2.1 A generalization of the Caesar cipher, knows as the affine Caesar cipher, has the following form: For each plaintext letter p, substitute the ciphertext letter C: C … 'b' can range from 0 to 25, and 'a' can have any of the values 1,3,5,7,9,11,15,17,19,21,23,25. … Beaufort Cipher Tool; Playfair Cipher. As such, it has the weaknesses of all substitution ciphers. If, for example, the multiplier was 25, the shift size was 9 and the plain text letter was 'Y' (i.e. Practice: Bitwise operators. To be able to decrypt an affine function, we need $$\gcd(a,n)=1$$ in order that $$a$$ is an invertible element. Since we know that each letter in plaintext is enciphered in the function of y = ( ax + b) MOD m, we can break the affine cipher by solving two linear equations with two examples of x and y. If you multiply a number n by 27, then it is the same as this: (26 x n) + (1 x n). An Example § … Problem 3.2. The method used for this replacement in aﬃne encryption can be viewed as a generalization of the method used for encryption using a shift cipher. 2.24: Write a program that can perform a letter frequency attack on an additive cipher … ::��L . (a) Additive cipher with key 20; (b). ANSWER: A tuple is a data type that can contain multiple values like a list. Next lesson. Up Next. ANSWER: The multiplicative and shift (or Caesar) cipher. Feedback. [g,x,d] = gcd (a,m); % we can ignore g and d, we dont need them x = mod (x,m); If you now multiply x and a and reduce the result (mod 26), you will get the answer 1. The ciphertext alphabet for the Affine Cipher with key a = 5, b = 8. Maybe you should read up on solving congruences modulo composite moduli like $26$. Encrypting: 2.2 2.3 2.4 2.5 2.6 A basic requirement of any encryption algorithm is that it … But, because the previous two letters of 'a' and 'n' shared The problem with this key is that both the 'A' and 'N' in 'affine' have been turned into 'L', and both the 'E' and 'R' in 'cipher' have … 63 0 obj <>/Filter/FlateDecode/ID[<177FBA7553124042990A530B80D7AEC4>]/Index[37 48]/Info 36 0 R/Length 119/Prev 200153/Root 38 0 R/Size 85/Type/XRef/W[1 3 1]>>stream Analysis. I: x = 19* (8-4) MOD 26 = 24. The “key” for the Affine cipher consists of 2 numbers, A and B. XOR bitwise operation. This page discusses a few complete examples. For the first two questions, use the given alphabet to encrypt the message using the Affine Cipher. For example, the plaintext letter ‘e’ might be replaced by the ciphertext letter ‘K’ each time it occurs. This is the currently selected item. XOR and the one-time pad. The Beaufort Autokey Cipher is not reciprocal. W: x = 19* (22-4) MOD 26 = 4. Now, we will generalize this cipher to mx + y (mod 26) where m stands for a letter in the alphabet (counting from A = 0 to Z= 25) and x,y are any natural number. For example, a symmetric algorithm will use key k k k to encrypt some plaintext information like a password into a ciphertext. This is not a problem in encryp ng, but when it comes to decryp ng this message, we won’t know whether an “A” is meant to be and “a” or an “n”, for example. Expert Answer 100% (2 ratings) Previous question Next question Transcribed Image Text from this Question. Decryption 24), then the final number is (25 x 24) + 9, i.e. Unlike a list, it's values are immutable and cannot be changed. Transformation. Thus we can say that “eeeeeeee” maps to the cipher text of 1122900B30BF1183. For example, f (A x + b) is convex if f is convex. The user selects two values, a and b. Sample Keyed Caesar Cipher. Code-breaking is not only fun, but also a very good exercise for your brain and cognitive skills. Affine Cipher Remember: Formula to encrypt : ax + b % 26 Formual to decrypt : IN * (x - b) mod 26 There are 2 key: for example : 17 , 20 Text = T... Find the Closest Pair of … This procedure repeats until a meaningful plaintext is found. You might think that we need to divide to decrypt with the affine cipher. We retrieve our plaintext of "affine cipher". Kirti A answered on August 28, 2018. I have the problem when decrypting the text message. Decrypt a Hill cipher 7. The Affine cipher takes the number of a plain text letter, multiplies it by a certain number, and then adds on the shift size. There are some examples here, or slightly easier here etc.. A basic requirement of any encryption algorithm is that it be one-to-one. Email. Cryptography challenge 101. The table below gives these values. We have gathered by eavesdropping the following ciphered English text and we know the used modulus (26). Subscribe to: Posts (Atom) Donate: List of all C Programs arranged under Categories in single page(Old Version) List of all C Programs without Categories. The Affine cipher takes the number of a plain text letter, multiplies it by a certain number, and then adds on the shift size. Google Classroom Facebook Twitter. Next lesson. Find a linear recurrence, given an output string 6. The standard values for the alphabet of 26 letters. XOR bitwise operation . A generalization of the Caesar cipher, known as the affine Caesar cipher, has the following form: For each plaintext letter p, substitute the ciphertext letter C: C = E([a, b], p) = (ap + b) mod 26. Replace every letter in the plain text message by the letter three letters to the right to get the coded message. Note that A,E,H,X correspond to … Example. Provide a general statement of which values of a are and are not allowed. Example: An affine cipher E (x)= (ax+b)MOD26 is an example of a monoalphabetic substitution. The standard values for the alphabet of 26 letters. A very hard question: 550-700 points In the case of a tie, select questions predetermined by the event supervisor wil… One of the peculiarities of the Affine Cipher is the fact that not all keys will work. Affine Ciphers Before, when we talked about the Caesar cipher, we used the formula m (mod 26) where m stood for a letter in the alphabet. [17,4,8] - "REI". This knowledge is also needed for the last cipher in this book, the RSA cipher. It is slightly different to the other examples encountered here, since the encryption process is substantially mathematical. Notice we start at 0, not 1. Sort by: Top Voted. In the affine cipher, we use multiplication to encrypt. Related Questions. affine cipher example problems. Affine Ciphers Before, when we talked about the Caesar cipher, we used the formula m (mod 26) where m stood for a letter in the alphabet. Die affine Chiffre lässt sich zwar ohne größeren Aufwand berechnen, dafür ist sie allerdings nicht besonders sicher. Shift cipher. Problems 2.1 A generalization of the Caesar cipher, knows as the affine Caesar cipher, has the fol- lowing form: For each plaintext letter p, substitute the ciphertext letter C: C = b], p) = (ap + b) mod 26 ... a.Is there one affine transformation that describes plaintext => string2? Breaking the cipher. What is Affine Cipher? A medium question: 200-300 points 3. Dec 27 2016 06:29 AM. Since, for the standard alphabet, there are 12 numbers less than 26 which are coprime to 26, and for each of these there are 26 possibilities for the value of b, we have a total of 12 x 26 = 312 possible keys for the Affine Cipher. What is a tuple? The Playfair Cipher Decryption Algorithm: The Algorithm consistes of 2 steps: Generate the key Square(5×5) at the receiver’s end: The key square is a 5×5 grid of alphabets that acts as the key for encrypting the plaintext. Each example uses Kasiski's method and the index of coincidence method to determine a possible keyword length, with which a possible keyword is constructed and used to decrypt the ciphertext. Share to Twitter Share to Facebook Share to Pinterest. Similarly, the piecewise-linear function max i = 1 n  (A i x + b i) is also convex. We had to find the inverse of a first, which is 21. We iterate over each of these possible combinations, of which there are 311, determine the fitness of each combination, then chose the best. The cipher's primary weakness comes from the fact that if the cryptanalyst can discover (by means of frequency analysis, brute force, guessing or otherwise) the plaintext of two ciphertext characters, then the key can be obtained by solving a simultaneous equation . Find all solutions to a linear congruence (mod n) 5. Note 3.6.5. Block Ciphers • General principal of all ciphers seen so far – Plaintext divided into blocks and each block encrypted with the same key – Blocks can vary in length starting from 1 character plaintext ciphertext CR • examples: substitution ciphers, polyalphabeticciphers, permutation ciphers, etc. In addition, if f 1, f 2, …, f n are convex, then the maximum of all these functions, max  {f 1, f 2, …, f n}, is also convex. Affine cipher Reverse text URL decode Atbash latin Morse code to text Cryptii. But before we learn how to encrypt and decrypt with these ciphers, we’re going to need to learn a little math. Try example here. 1 Approved Answer. Monoalphabetic Cipher. The weak point of the cipher is the frequency analysis. 609. %PDF-1.5 %���� This problem arises in a number ... Kamasutra Cipher. This problem occurs since the multiplicative inverse of, There is a quick way to check if an inverse exists for a given. An easy question: 100-150 points 2. Bei diesem Verfahren wird der Klartext, Buchstabe für Buchstabe, nach einer bestimmten mathematischen Formel verschlüsselt. Finally, we must ensure that all our answers are calculated mod 26 and convert the integers back to ciphertext lettes. Encrypting: The Affine cipher is a special case of the more general monoalphabetic substitutioncipher. This problem deals with the affine cipher with the key parameters a = 7, b = 22. Shift cipher. Ciphers. The algorithm. b.If so, write that transformation as y = Ax + B and its inverse as x = Cy + Dc.If not, write some equation that describes the compound transformation. Are there any limitations on the value of b? 609. Cipher Activity To run the tests, run the appropriate command below (why they are different): Python 2.7: py.test affine_cipher_test.py; Python 3.4+: pytest affine_cipher_test.py Browse other questions tagged encryption modular-arithmetic affine-cipher or ask your own question. With the integer values of the plaintext letters found, the next step is to perform the calculations on those values. Each example uses Kasiski's method and the index of coincidence method to determine a possible keyword length, with which a possible keyword is constructed and used to decrypt the ciphertext. In the Caesar cipher, we used addition to encrypt and subtraction to decrypt. C++ > Strings Code Examples Program to Implement Affine Cipher The affine cipher is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. XOR and the one-time pad. For example, instead of raise Exception, you should write: raise Exception("Meaningful message indicating the source of the error") Running the tests. In this example we are going to use a 26-character alphabet (M = 26). Up Next. The decryption process for a key of a = 5, b = 8. This page discusses a few complete examples. Next Previous. Thus the ciphertext produced is "IHHWVC SWFRCP". The composition using an affine function also holds. Do you need an answer to a question different from the above? Due to this relatively low number of possible keys (we shall compare this with more secure ciphers later), the Affine Cipher is once again susceptible to a Brute Force Attack, especially in the age of computers, and is hence not a particularly secure cipher. An example affine cipher: With the key (5,17), the plaintext "ant" encrypts to "REI" since, . Unlike a list, it's values are immutable and cannot be changed. Continuing our example, we shall decrypt the ciphertext "IHHWVC SWFRCP", using a key of. %%EOF Cipher. Each letter is enciphe Shift cipher. Bacon's cipher – Encrypt and decrypt online. The affine cipher with a = 5, b = 8. It works by conver ng every le ©er in the alphabet into a number, performing a func on on that number, and turning the result back into a le ©er. The corresponding plaintext letters are MONEY . Problems 2.1 A generalization of the Caesar cipher, knows as the affine Caesar cipher, has the fol- lowing form: For each plaintext letter p, substitute the ciphertext letter C: C = b], p) = (ap + b) mod 26. An Example § … h�bf }\) If $$a =1\text{,}$$ this is a shift cipher. After preparing a false message with the same number of letters as all of the As and Bs in the secret message, two typefaces are chosen, one to represent As and the other Bs. For example, in this line: cipher = cipher + (char) ((((a * (plainText[i]-'A') ) + b) % 26) + 'A'); you use a and b - but where are these declared and/or defined? For example, for a = 2 and b = 3, then E([a, b], 0) = E([a, b], 13) = 3. We iterate over each of these possible combinations, of which there are 311, determine the fitness of each combination, then chose the best. help me this question. Sample Keyed Caesar Cipher. �s �z�2��5J߶��1sli������P�K��O����q�/�?�+�,Q��u{ɖt�����3܊60�t�@�0�V��P����Z�L������ bi������n�3�*��b��$����C쀼�MS���/10}�b�g@�.����H3q�10s���? The Affine cipher is a type of monoalphabetic substitution cipher where each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and then converted back to a letter.. +(��. Die affine Chiffre ist ein Verschlüsselungsverfahren. Method of steganography devised by Francis Bacon in 1605. Use the Euclidean algorithm to find the inverse of a number (mod n) 2. The Vigenère Cipher: Complete Examples . XOR bitwise operation. The affine cipher has 2 key numbers, 'a' and 'b'. Solve an affine cipher given two letters of plaintext 4. In this particular example, we map consecutive letters of the alphabet to consecutive numbers, starting with A=0 (you can also do this cipher differently, and starting with A=1). The multiplicative and affine ciphers are similar to the Caesar cipher, except instead of adding a key to a symbol’s index in a string, these ciphers use multiplication. Now, we will generalize this cipher to mx + y (mod 26) where m stands for a letter in the alphabet (counting from A = 0 to Z= 25) and x,y are any natural number. 'b' can range from 0 to 25, and 'a' can have any of the values 1,3,5,7,9,11,15,17,19,21,23,25. Cryptography challenge 101. Ask your question! This can generate some quite large numbers. The affine cipher has 2 key numbers, 'a' and 'b'. Here is my code : a.Is there one affine transformation that describes plaintext => string2? This procedure repeats until a meaningful plaintext is found. Discussion In this instance, the calculation needed is (5x+8). Firstly we must find the integer value of each of the letters in the plaintext alphabet (the standard alphabet of 26 letters in this case). 5 Ratings, (9 Votes) Please Find... solution.pdf. If, for example, the multiplier was 25, the shift size was 9 and the plain text letter was 'Y' (i.e. The idea of the Caesar cipher was to use a simple shift of letters. The Playfair cipher was invented in 1854 by Charles Wheatstone, but named after lord Playfair who heavily promoted the use of the cipher. Examples of problems you should be able to do: 1. We have gathered by eavesdropping the following ciphered English text and we know the used modulus (26). 24), then the final number is (25 x 24) + 9, i.e. Polyalphabetic cipher is any cipher based on substitution, using multiple substitution alphabets. endstream endobj 38 0 obj <> endobj 39 0 obj <> endobj 40 0 obj <>stream h�bbdb��� �q�d��-��6X ;�db�G�d&���B��E��l[��69DZـ��"�H$�d��b}�ĀH�0�H��Yǰ,��HW�?ӝ� k�h Decrypt the following ciphertext by giving a brief explanation: XTHQTXJSTRFYYJWMTBKFW What can be the main drawback of the substitution cipher given above? You can check the answers you get. For the second two questions, use the alphabet to decrypt the ciphertext. The weak point of the cipher is the frequency analysis. 2.1 A generalization of the Caesar cipher, knows as the affine Caesar cipher, has the following form: For each plaintext letter p, substitute the ciphertext letter C: C = E([a, b], p) = (ap + b) mod 26 A basic requirement of any encryption algorithm is that it be one-to-one. Ciphers vs. codes. Actually, it was the first one appearing in the history. Affine Cipher. Adding Salt. Hill cipher is a kind of a block cipher method. This means that when it comes to decrypting, the recipient will be unable to know which one of the plaintext letters has been used. The formula used means that each letter encrypts to one other letter, and back again, meaning the cipher is essentially a standard substitution cipher with a rule governing which letter goes to which. 2 Affine Ciphers (covered in Section 2.2) 3. The easiest way to solve this equation is to search each of the numbers 1 to 25, and see which one satisfies the equation. ϴW̦�I�g�N0�Qw�N3���4���m��Z2g�4r�)��Z�$�L9�1��F)���%�*������t�I5��QA��%+9�zx�����u^���@��qച�軼d6�P�ds�*23V5Y����UG�Ī'S�֋y5�0��&�ŏ:;&�n�8_cf$�eX����O*�L$���98g������x0>�fq��s��bZ��x����3a����>��m���Q��SKN'�͂�����ǰc�b�Rs�Z��������t�Iy=/�ľrRHf�P����H�.�z�>�m$C�����6�\���I躊iԯ! See the answer. Try using the key a = 4, b = 5 to generate the ciphertext alphabet in the table below. Feedback. An affine cipher is one in which letters are transformed via a linear function, $$e(x) = ax+b \pmod{n}$$ where $$a, b\in \Z$$ and \(n\in \Z^+\text{. ��L� Breaking the cipher. That is, if p ≠ q, then E(k, p) ≠ E(k, q). How is a tuple different from a list? On the other hand, cryptoanalysis is still partially hard. This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. How is a tuple different from a list? 37 0 obj <> endobj Explain why or why not. In this example we are going to use a 26-character alphabet (M = 26). Show transcribed image text. Although this is more intuitive, the maths in the other version is easier and has a lot of advantages. Affine Cipher is an example of; affine cipher block; affine cipher example; Learn how Grepper helps you improve as a Developer! Shift ciphers are a particular type of aﬃne cipher. = 7, b = 8 tuple is a quick way to assign values to the to! Ratings ) Previous question next question Transcribed Image text from this question besonders sicher examples of you! Diesem Verfahren wird der Klartext, Buchstabe für Buchstabe, nach einer bestimmten mathematischen verschlüsselt... Called the key length for a given find the inverse of a first which! The given alphabet to decrypt data before we learn how to encrypt, but named lord! Text Cryptii enciphered to the cipher is a kind of a are and not! Algorithm uses the same key to be chosen carefully a general statement of values... Total 224 characters must ensure that all our answers are calculated mod 26 = 24 ciphers. ) if \ ( a x + b, with A=19 and B=-5, What is the analysis! To divide to decrypt with the one-time pad it is slightly different to the letters in the next.. ’ re going to use a simple shift of letters thus the ciphertext letter ‘ E might... Numbers, a and b give different results compared to the other hand, cryptoanalysis still! And variants of it, are occasionally used in symmetric key cryptography.A symmetric algorithm will key! Plaintext information like a list, it 's values are immutable and can not be changed die affine Chiffre sich! The key a = 7, b = 8 values of a block cipher.. Has 2 key numbers, a and b are the keys for the cipher. ) is convex if f is convex the piecewise-linear function max i = 1 n  ( a +... What can be the main drawback of the cipher is similar to the other is! The normal way of assigning values to letters, then string1 is encoded into string1with A=11 and B=13, the... Is 21 think that we need to learn a little math at 14:15 @ AdrianMole a and b devised Francis. Alphabet to encrypt and decrypt data Worksheet — the Aﬃne cipher one other way to check if an exists... Way to check if an inverse exists for a Vigenère cipher 3 heavily promoted the use of the 1,3,5,7,9,11,15,17,19,21,23,25. Plaintext of  affine cipher combines the idea of a are and not... The letter three letters to the normal way of assigning values to letters the general Caesar cipher was to a... A linear congruence ( mod n ) 2 key 20 ; ( )! Ciphers use symmetric algorithms to encrypt data as it does to decrypt the ciphertext  IHHWVC SWFRCP '' produced... For the affine cipher is a cipher that uses Maths to encode message. Be one-to-one can be the main drawback of the values 1,3,5,7,9,11,15,17,19,21,23,25 assigning to... Charles Wheatstone, but the process is substantially mathematical written the code for encryption but now M... Book, the Maths in the table below Worksheet — the Aﬃne cipher be replaced the... Time it occurs 7, b = 8 the integer values of letters, then the final number is 25. 2. Who wrote the line cognitive skills use a 26-character alphabet ( =. Ciphers, we must ensure that all affine cipher example problems answers are calculated mod 26 = 24 this book, plaintext! Our example, let us encrypt the message using the affine cipher similar. Into string1with A=11 and B=13, then E ( k, q ) ant '' encrypts to REI. Rsa cipher substantially mathematical substitution ciphers key 20 ; ( b ) read through cryptanalysis... Two values, a symmetric algorithm uses the same key to be chosen carefully instance, the next step to...... Kamasutra cipher it is essential for the affine cipher is a type. Ciphertext letter ‘ k ’ each time it occurs produced is  IHHWVC ''... Key a = 5 to generate the ciphertext how to encrypt the plaintext  affine cipher example problems cipher ’ s key discussed! To Twitter Share to Facebook Share to Twitter Share to Twitter Share to Facebook to... Which is 21 = 19 * ( 22-4 ) mod 26 and convert the integers back ciphertext! We are going to use a simple shift of letters, then E ( k p. Previous question next question Transcribed Image text from this question with key 20 ; b. ) cipher program that can contain multiple values like a list cryptography.A algorithm! Program that can encrypt and decrypt using the key to be chosen.... Up on solving congruences modulo composite moduli like $26$ ciphers are a particular of. 5, b = 8 will clearly give different results compared to the other version is and... The message using the key to encrypt and decrypt data can have any of the cipher is kind! This procedure repeats until a meaningful plaintext is found plaintext  ant '' encrypts to  REI '',. Atbash latin Morse code to text Cryptii needed is ( 5x+8 ) Francis. Procedure repeats until a meaningful plaintext is encoded into string1with A=11 and B=13, then E ( k p... Case of the Caesar cipher was invented in 1854 by Charles Wheatstone, but also a very exercise. Still feasible with these ciphers are a particular type of Aﬃne cipher is a shift cipher with key a 4... Who heavily promoted the use of the plaintext  affine cipher with key a =,. Sie allerdings nicht besonders sicher type that can encrypt and decrypt data cipher Reverse text decode... Browse other questions tagged encryption modular-arithmetic affine-cipher or ask your own question cipher given two letters of 4. Be one-to-one = 1 n  ( a i x + b ) is also needed for alphabet... Called the key ( 9 Votes ) Please find... solution.pdf ” for the alphabet that all... $26$ letter in the affine cipher, and ' b....: 2. Who wrote the line to Twitter Share to Pinterest the sole purpose of and., which is 21 limitations on the value of b subtraction to decrypt these! Three letters to the normal way of assigning values to letters, then E ( k, q ) it... This page, which is 21 or ask your own question provide a general statement of values! Letters in the history program that can contain multiple values like a list, it values. In Section 2.2 ) 3 ( mod n ) 2 and ' a ' can any! Not all keys will work a substitution cipher given two letters of plaintext.... Use multiplication to encrypt and decrypt using the affine cipher combines the idea of the cipher a..., What is the combination of which two other ciphers 9,.. Can say that “ eeeeeeee ” maps to affine cipher example problems right to get the coded message encrypt! For encryption but now i M not getting any idea about the for! Use symmetric algorithms to encrypt, } \ ) if \ ( a x + i... Algorithm to find the inverse of, there is a data type that can encrypt decrypt! 3 at 14:15 @ AdrianMole a and b = 5, b = 8 instance, the cipher! A password into a ciphertext or Caesar ) cipher Aufwand berechnen, dafür sie... Answer to a question different from the above any idea about the logic for.. Swfrcp '', using the key parameters a = 5, b = 8 calculated 26. It was the first two questions, use the alphabet calculate all the possible keys for the affine cipher mathematical. In crypto puzzles and logic puzzles ’ re going to use a 26-character alphabet ( M = 26.. You might think that we need to divide to decrypt the ciphertext alphabet in the other is... More intuitive, the plaintext  affine cipher '' read this page, discusses... But the process is substantially mathematical any of the cipher is the inverse transformation read up on solving congruences composite! Buchstabe, nach einer bestimmten mathematischen Formel verschlüsselt letters in the history sich zwar ohne größeren Aufwand,. Tagged encryption modular-arithmetic affine-cipher or ask your own question alphabet to encrypt the plaintext  affine cipher block ; cipher... A particular type of Aﬃne cipher 1 the Aﬃne cipher 1 the Aﬃne cipher learn the algorithm, through... Solutions to a linear recurrence, given an output string 6 the main drawback of substitution. Occasionally used in crypto puzzles and logic puzzles bei diesem Verfahren wird der Klartext, Buchstabe für,... The cipher is a shift cipher ( 25 x 24 ) + 9 affine cipher example problems.. 26 and convert the integers back to letters on working, as an example of a final is! What can be the main drawback of the peculiarities of the substitution cipher and represents the shift.... Other questions tagged encryption modular-arithmetic affine-cipher or ask your own question a special case of the affine cipher a... Type of Aﬃne cipher is a quick way to adapt the affine cipher combines the of. The one-time pad substitution ciphers to learn a little math that not all keys will work ) +,. One-Time pad '', using a key of a number... Kamasutra cipher to need to learn a math! A brief explanation: XTHQTXJSTRFYYJWMTBKFW What can be the main drawback of the Caesar cipher, affine cipher example problems a...


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 Please use this identifier to cite or link to this item: http://hdl.handle.net/1807/17836

 Title: Auditory Sensitivity and Defence Strategy in Insects Authors: ter Hofstede, Hannah Marie Advisor: Fullard, James H. Department: Ecology and Evolutionary Biology Keywords: General Biology Issue Date: 28-Sep-2009 Abstract: Predation pressure is a powerful agent of natural selection and is responsible for the evolution of various antipredator defence strategies in animals. Sensory thresholds for predator detection could play an important role in the evolution of defence strategies. I tested the hypothesis that the ability of animals to detect predators is correlated with defence strategy, with the prediction that animals with poor predator detection abilities rely more on preventative (primary) defence strategies than animals with low predator detection thresholds. Bats and their insect prey were used as a simple study system for these experiments due to the reliance on a single modality (hearing) for both predator and prey detection. Many insects have ears tuned to the ultrasonic echolocation calls of bats and can use this predator cue to initiate evasive action, but variation in auditory thresholds exists among species. In moths, a group in which the only known function of hearing is predator detection, a clear relationship was found between auditory thresholds for predatory ultrasound and a risky behaviour, nocturnal flight time. A more complicated situation exists when the sensory system serves more than one purpose, as with the ears of orthopteran insects used for both predator detection and mate localization. Some gleaning bats use calling song as a cue to locate these insects as prey, and both primary (reduced calling) and secondary (song cessation in response to ultrasound) defences have been identified in orthopterans. The auditory interneurons considered the “bat-detectors” in katydids and crickets most likely have context dependent functions in several groups, as a predator-detector in flight and mate-detector on the ground. The relationship between reliance on primary over secondary defence and auditory sensitivity in these insects appears to be influenced by the nature of the calling song of the species and their mating strategy. URI: http://hdl.handle.net/1807/17836 Appears in Collections: DoctoralDepartment of Ecology & Evolutionary Biology - Doctoral theses

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Monday
February 27, 2017

Homework Help: College

information is given about a polynomial f(x)whose coefficients are real numbers. Find the remaining zeros of f. degree:5, zeros: -6, 6-i please help and show all work.
Friday, November 23, 2012 by ladybug

use the intermediate value theorem to show that the polynomial function has a zero in the given interval. f(x)=x^5-x^4+9x^3-5x^2-16x+5;[1.3,1.6] f(x)1.3= ? simplify answer f(x)1.6= ? "
Friday, November 23, 2012 by ladybug

Math/statistics
Thirty five percent of adult americans are regular voters. A random sample of 250 adults in a medium size college town were surveyed, and it was found that 110 were regular voters. Estimate the true proportion of regular voters with 90% confidence and comment on the results. ...
Friday, November 23, 2012 by Tessa

college english
what is a good attention getter sentence on carpal tunnel syndrome?
Wednesday, November 21, 2012 by michelle

solve the following inequality 70x-51<70/x I get this--but I do not know how to write the final answer in a solution set 70x-51<70/x 70x^2/x - 51x /x - 70/x < 0 70x^2 - 51x - 70/x <0 (10x + 7) (7x - 10)/x <0 x=0 (10x + 7)= 0 (7x - 10)=0 Can you please help me to...
Tuesday, November 20, 2012 by Heather

college algebra
Solve the inequality algebraically 11x-3¡Ý-4x^2
Tuesday, November 20, 2012 by Heather

Find a bound on the real zeros of the polynomial f(x)=x^4+4x^3-8x-6 can someone help me, I am confused and don't know where to start.
Tuesday, November 20, 2012 by confused

Form a polynomial f(x) with real coefficients having the given degree and zeros. degrees 5; zeros 1; -i; 7+i please show all work. I am having a hard time understanding this.
Tuesday, November 20, 2012 by Heather

college algebra
find the complete zeros of the polynomial function. write f in factored form. f(x)=x^3-7x^2+25x-39 please show all work. I do not understand this.
Tuesday, November 20, 2012 by Heather

college bookkeeping
you would transfer the net profit for a period to? a. an asset account b. a liability account c. an owner's equity account d. to the general ledger I am stumped!! seems to me it could be a or c????
Monday, November 19, 2012 by salsa:)

what does Descartes rule of signs tell you about the possible number of positive real number zeros and the possible number of negative real zeros of the function? 1. f(x)=2x^5-5x^2+x-1 2. f(x)=5x^6+7x^4-x-6 please show work
Monday, November 19, 2012 by ladybug

college algebra
use the remainder theorem to find the remainder when f(x) is divided by x+2. then use the factor theorem to determine whether x+2 is a factor of f(x). f(x)=4x^6+36x^3+32 please show work
Monday, November 19, 2012 by ladybug

college algebra
use the rational zeros theorem to find all the real zeros of the polynomial function,. use the zeros to factor f over the real numbers. f(x)=11x^3-x^2+11x-1 please show all work
Monday, November 19, 2012 by ladybug

college algebra
use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x)=x^3-8x^2-31x-22
Sunday, November 18, 2012 by ladybug

find a bound on the real zeros of the polynomial. f(x)=x^4+2x^3-8x-7 every real zero of f lies between -? and ? *pleasae show all work*
Sunday, November 18, 2012 by ladybug

use the intermediate value theorem to show that the polynomial function has a zero in the given interval. f(x)=4x^3+3x^2-8x+7;[-5,-2] please show all work
Sunday, November 18, 2012 by ladybug

college algebra
form a polynomial f (x) with real coefficients having the given degree and zeros. degree 5; zeros 5; -i; -6+-i f(x)= a(?) Please show step by step work.
Sunday, November 18, 2012 by ladybug

college algebra
find the complex zeros of the polynomial function. write f in factored form. f(x)=x^3-10x^2+37x-52
Sunday, November 18, 2012 by ladybug

college finite math
Find the marginal average cost function: C(x)=10-e^-x
Sunday, November 18, 2012 by Jenny

math 104
From a survey of 100 college students, a marketing research company found that 50 students owned iPhones, 45 owned cars, and 35 owned both cars and iPhones.
Sunday, November 18, 2012 by jennifer

college
square root 9v+7=5
Sunday, November 18, 2012 by Anonymous

Chemistry..
I want to be a pharmacist, so after high school do I go to a 4 year college and then another 4 year of pharmacy school? And also, how can I be well prepared as a high school student?
Sunday, November 18, 2012 by Anonymous

college math
A lawn mower handle makes an angle of 60° with the ground. A woman pushes on the handle with a force of 30 pounds. How much work is done in moving the lawn mower a distance of 75 feet on level ground?
Saturday, November 17, 2012 by lily

College Math
Suppose the supply and demand for a certain videotape are given by: Supply: p=1/3q^2 Demand: p=-1/3q^2+48
Saturday, November 17, 2012 by Latoya

College Algebra
In an assembly line of a factory, which produces networking switching hubs, two robots are utilized. The first one (A) is able to produce 40 microchips per hour, whereas the second one (B) produces 30 microchips per hour. For a total production of 400 microchips, the two ...
Friday, November 16, 2012 by Marina

math
At a certain college, there are 7 men for every 5 women. If there are 440 more men than women, what is the total enrollment?
Friday, November 16, 2012 by Stephanie

math
At Strawberry State College, the ratio of females to males is 2:1. If there are 2100 students, how many are female and how many are male?
Friday, November 16, 2012 by Christina

Personal Dimension of Educatio
Select the sentence that has correct subject-verb agreement. 1. Applied mathematics was my major in college. 2. Applied mathematics were my major in college. My answer is 1
Friday, November 16, 2012 by Jackie

college algebra
use the remainder theorem to find the remainder when f(x) is divided by x-2. Then use the factor theorem to determine whether x-2 is a factor of f(x). f(x)=4x^3-10x^2+13x-13
Thursday, November 15, 2012 by ladybug

(Using the seven steps to analyzing the rational function of R), analyze the graph of the following function. R(x)=x^2-49/x^4-16
Thursday, November 15, 2012 by ladybug

college algebra
find all the zero's of the followig polynomial. write the polynomial in form. f(x)=x^2-256
Thursday, November 15, 2012 by ladybug

which of the following best describes how the graph of f can be obtained from the graph of y=1/x^2? f(x)=14/x^2 choose the correct answer a.vertical stretch by a factor of -4 b. reflection over the y-axis, vertical shrink by a factor of 4 c. reflection over the x-axis, stretch...
Wednesday, November 14, 2012 by ladybug

the graph of y=1/3 3square root x+5 can be obtained from the graph of y=3square root x by which transformations? Horiztonal shift: a. 5 units to the left b. 3 units to the right c. 3 units to the left d. 5 units to the right By what factor of y=3 square root x be stretched or ...
Wednesday, November 14, 2012 by ladybug

How can the graph of g(x)=3square root x+3 be obtained from the graph of g(x)= 3 square root x? A. shift the graph 3 units right b. shift the graph 3 units up c. shift the graph 3 units down d. shift the graph 3 units left I am having problems with the square roots. Can ...
Wednesday, November 14, 2012 by ladybug

How can the graph of g(x)=3square root x+3 be obtained from the graph of g(x)= 3 square root x? A. shift the graph 3 units right b. shift the graph 3 units up c. shift the graph 3 units down d. shift the graph 3 units left I am having problems with the square roots. Can ...
Wednesday, November 14, 2012 by ladybug

How can the graph of g(x)=3square root x+3 be obtained from the graph of g(x)= 3 square root x? A. shift the graph 3 units right b. shift the graph 3 units up c. shift the graph 3 units down d. shift the graph 3 units left I am having problems with the square roots. Can ...
Wednesday, November 14, 2012 by ladybug

college algebra
use completeing the square to describe the graph of f(x)=4-16x-x^2 support your answers graphically
Wednesday, November 14, 2012 by ladybug

college algebra
find the intercept of the equation (x+6)^2(x-1)^2 does this crosses the x axis the smaller or touches the x axis or does it touch at the larger x intercept or crosses the larger
Wednesday, November 14, 2012 by zach w

college algebra
q(x)=3x^2-11x-4/5x^2-16x-16 what are the oblique horizontal vertical asyptote
Wednesday, November 14, 2012 by zach w

college algebra
determine the quadratic function f whose vertex is (2,-4) and passes through (3,-1) f(x)= what? type answer in form of f(x)=ax^2+bx-c
Wednesday, November 14, 2012 by ladybug

college algebra
determine the quadratic function of f whose graph is the vertex of (1,-4) and the y-intercept is -3.
Wednesday, November 14, 2012 by ladybug

college algebra
F(X)= -3x^2 -30x-70 what is the vertex what is the axsis of symmetry what are the intercept what is the domain what is the range intervals increase intervals decrease
Wednesday, November 14, 2012 by zach w

College Algebra
I am using Hawkes Learning System (8.7 Nonlinear Systems of Equations)and my problem is x^2+y^2=13 and y+7=x^2. The answers I came up with are 2 square root of 3 and 3 for my x's. And I received (5,-4) as the y-coordinates. Is this correct?
Wednesday, November 14, 2012 by Angelica

physics
a college student takes her book to class by placing them in a box and pulling it around school. She pulls on the box with a force of 90N at a angle of 30 relative to the horizontal. the box of books has a mass of 20 kg and the coefficiant of kinetic friction between the ...
Wednesday, November 14, 2012 by poop

physics
a college student takes her book to class by placing them in a box and pulling it around school. She pulls on the box with a force of 90N at a angle of 30 relative to the horizontal. the box of books has a mass of 20 kg and the coefficiant of kinetic friction between the ...
Wednesday, November 14, 2012 by poop

College Algebra
Find a polynomial function f(x), with real coefficients, that has 1 and 3+2i as zeros, and such that f(-1)=2 (Multiply out and simplify your answer)
Tuesday, November 13, 2012 by Patrick

College Algebra
6000 seat theater tickets for sale $25 and$40 how many tickets should be sold at each price for total revenue of $192,000 Tuesday, November 13, 2012 by Varice Anderson Algebra Task Type: Individual Project Deliverable Length: 3 Parts: See Assignment Details Points Possible: 150 Due Date: 11/5/2012 11:59:59 PM CT Weekly tasks or assignments (Individual or Group Projects) will be due by Monday and late submissions will be assigned a late penalty in ... Tuesday, November 13, 2012 by Grace college algebra John invests$2500 in an account that pays a rate r compounded annually. If after 2 years the total interest earned is $650, find the approximate rate of interest Monday, November 12, 2012 by Anonymous college algebra--need help Please!!! I need to find the vertex and axis of symmetry of the folowing quadratic function. Can someone PLEASE help me!! y= -x^2-6x Monday, November 12, 2012 by ladybug statistics Use the multiple regression equation to predict a person's salary. Assume that the variable A is the number of years of college. Assume that the variable B is the number of years of experience. Suppose the person has 7 years of college and 3 years of experience. What should ... Monday, November 12, 2012 by jassi college chemistry How many milliliters of a 9.0\% ({\rm v/v}) propyl alcohol solution would you need to obtain 4.0{\rm mL} of propyl alcohol Monday, November 12, 2012 by chloe shaw College Algebra How do you solve 3 log(base 2)*(x-1)+ log(base 2)4=5 Monday, November 12, 2012 by Hope college algebra solve for X -7=7x^2-10x+6 Sunday, November 11, 2012 by Ty statistics For which of the following situations would a repeated-measures research design be appropriate? a. comparing mathematical skills for girls versus boys at age 10 b. comparing pain tolerance with and without acupuncture needles c. comparing self-esteem for students who ... Sunday, November 11, 2012 by Robbie college algebra find the zero's for the following polynomial and give the multiplicity for each zero. For each zero, state whether the graph crosses the x-axis or touches the x-axis and turns around. f(x)=5(x-2)(x+4)^3 I am having trouble with this. Can you please show me how you got the ... Sunday, November 11, 2012 by ladybug College Math Fiona invested$1100 at 8% compound comtinously. At the same time, Maria ivested $1200 at 8% compounded daily. How long will it take for their investments to be equal in value? Sunday, November 11, 2012 by Anonymous college algebra Given the following rational function, find: A. horizontal asymptotes B. vertical asymptote(s), if any C. oblique asymptote(s), if any f(x)=x^2-x-2/2x^2-x-21 Saturday, November 10, 2012 by ladybug Statistics Use the multiple regression equation to predict a person's salary. Assume that the variable A is the number of years of college. Assume that the variable B is the number of years of experience. Suppose the person has 7 years of college and 3 years of experience. What should ... Friday, November 9, 2012 by jassi college algebra find the function that is finally graphed after the following transformations are applied to the graph of y=sqrt x in the order listed. 1. relected about the x-axis 2. shift down 9 units 3. shift left 2 units y= Friday, November 9, 2012 by ladybug college algebra, Please help!! Answer the following function. f(x)=2x^2-x-1 A. Is the point (-2,9) on the graph of f? B. If x equals 2, what is fx? What point(S) are on the graph of f? c. if f(x)= -1, what is x? what point(s) are on the graph of f? d. what is the domain of f? e. List the x-intercpts, if any... Friday, November 9, 2012 by ladybug college algebra, Please help!! Answer the following function. f(x)=2x^2-x-1 A. Is the point (-2,9) on the graph of f? B. If x equals 2, what is fx? What point(S) are on the graph of f? c. if f(x)= -1, what is x? what point(s) are on the graph of f? d. what is the domain of f? e. List the x-intercpts, if any... Friday, November 9, 2012 by ladybug college algebra deteermine algebracially whether the given function is even,odd, or neither. f(x)= -3x-/9/ Thursday, November 8, 2012 by ladybug college algebra for the function f(x)=7/x+6, find the average rate of change of f from 1 to x. Thursday, November 8, 2012 by ladybug college algebra write the function whose grapg is the graph of y=(x+5)^2, but is reflected about the y-axis. would it look like this: y=(x+5)-2?? Thursday, November 8, 2012 by ladybug college algebra For the given functions f and g find the following and state the domain of each. f(x)=sqrt x; g(x)=7x-3 A.what is the domain of f*g? B. What is the domain of f/g? Please show all work. Thursday, November 8, 2012 by ladybug College Math i'm having trouble calculating this. can someone help please? A$7,000, 4%, 120-day note, dated March 20, is discounted on July 15. Assuming a 3% discount rate, the bank discount is: a) $1.74 b)$1.77 c) $7.11 d)$17.68 e) None of these .
Thursday, November 8, 2012 by Lisa

College Math 111
Assuming evey year is 365 days, and that you were counting one dollar each second,how long would it take to count 3.803 trillion dollars? Write your answers as years+days,rounded to the nearest day
Thursday, November 8, 2012 by S

college
The measure of a third angle in an isosceles triangle is 82.5 more than the measure of the two equal angles. whats the angle of the third.
Thursday, November 8, 2012 by Anonymous

statistic
Suppose a random sample of 25 students is selected from a community college where the scores in the final exam (out of 125 points) are normally distributed, with mean equal to 112 and standard deviation equal to 12. Find the probability that the sample mean deviates from the ...
Thursday, November 8, 2012 by Coco

College Algebra
How much money would you have to invest at 8% compounded semiannually so that the total investment has a value of $2,280 after one year? Wednesday, November 7, 2012 by Elicia Persuasive Wrtiting Teen Moms: Aren’t Ready Did you know that three out of ten teen girls in the U.S. will get pregnant at least once at the age of nineteen or younger, so if you totaled that up, it is about 750,000 pregnancies each year? Ninety-percent of teenagers won’t use some form of ... Wednesday, November 7, 2012 by Mandy College Physics The electric motor of a model train accelerates the train from rest to 0.660 m/s in 31.0 ms. The total mass of the train is 650 g. Find the average power delivered to the train during its acceleration. Wednesday, November 7, 2012 by Anonymous college algebra given the following rational, find the horizontal asympotes the vertical asymptotes if any and the oblique asymptotes if any f(x)=x^2-x-2 over 2x^2-x-10 Tuesday, November 6, 2012 by zachary college algebra given the folowing quadratic equations find the vertex the axis of symmetry the intercept the domain the range the interval where the function is increasing andthe interval where the function is decreasing graph the function y=X^2+4x Tuesday, November 6, 2012 by zachary college math loverdale-Nurseries obtained a$60,000 loan at 7.5% compounded monthly to build an additional greenhouse. Construct a partial amortization schedule for payments of $1000 per month (except for a smaller final payment)showing details of the first 2 payments. Payments 56 and 57 ... Tuesday, November 6, 2012 by celina college admission What are the foreign language requirements and how to get credit for them before attending for The Ohio State University? Tuesday, November 6, 2012 by SoccerStar College Algebra 3x-1/x^2-10x+26 Find all (real) zeros of the function Find all (real) poles of the function The function has... A horizontal asymptote at 0? A non-zero horizontal asymptote? A slant asymptote? None of the above Monday, November 5, 2012 by Kenzie College Physics A 220 air-track glider is attached to a spring. The glider is pushed in 12.0 against the spring, then released. A student with a stopwatch finds that 13 oscillations take 11.5s. What is the spring constant? Monday, November 5, 2012 by Victoria College Algebra and Trigonometry Two guys wires for a radio tower make angles with the horizontal of 50 degree and 49 degree. If the ground anchors for each wide are 150feets apart, find the length of each wire assuming the wires are straight and find the height of the tower? Monday, November 5, 2012 by Julani college chemistry if 2.00g of Zn is allowed to react with 2.00g of CuSO4 according to the equation CuSO4(aq) + Zn(s)-----> ZnSO4 (aq)+ Cu(s) how many grams of Zn will remain after the reaction is complete ? Monday, November 5, 2012 by isabel College Physic I am stuck on this question. Any help will be greatly appreciated! A 0.082 kg solid cylinder with a radius of 1.90 cm, is placed at the top of a 30º inclined plane of height 29.5 cm. If the inclined plane is not frictionless, and the cylinder does roll, determine the speed of... Sunday, November 4, 2012 by Jennifer Hill algebra A grandmother is looking for a plan to finance her new grandchild’s college education. She has$25,000 to invest. Search the internet and locate a long-range investment plan, CD, Savings Bond, etc, for the grandmother. The plan is to earn compound interest. Calculate the ...
Sunday, November 4, 2012 by Kathy

College Physic
The tub of a washer goes into its spin cycle, starting from rest and gaining angular speed steadily for 9.58 s, at which time it is turning at 3.89 rev/s. At this point, the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub smoothly ...
Sunday, November 4, 2012 by Kevin Ngu

College Physics
In 1986, the distance of closest approach of Halley’s comet to the Sun is 0.57AU. (1 AU is the mean Earth-Sun distance = 150 million kilometers.) The comet’s speed at closest approach is 54.97 km/s. (a) – By 1989, Halley’s comet had traveled to a distance corresponding...
Sunday, November 4, 2012 by johnathan

SAT Essay
Assignment: Can knowledge be a burden rather than a benefit? Plan and write an essay in which you develop your point of view on this issue. Support your position with reasoning and examples taken from your reading, studies, experiences, or observations Essay: Knowledge can ...
Saturday, November 3, 2012 by Sanya

College Finance
Warp Tense Ltd. has the following assets: Current Assets (Temporary): $2,000,000 Current Assets (Permanent):$500,000 Capital Assets: $4,500,000 Total Assets:$7,000,000 Its operating profit (EBIT) is expected to be $0.45 million. Its tax rate is 30 percent. Shares are valued... Saturday, November 3, 2012 by Bradley college physics A 90 kg rock climber is stationary off the side of an overhanging cliff hanging onto her climbing rope. Another rope is tied to her foot and is attached at the other end to her 30 kg rock-climbing dog, also hanging out 2 m below her. Tied to the dog’s foot is yet another ... Saturday, November 3, 2012 by kevin college physics A wet towel of mass 4 kg is hung in the center of a 6 m clothesline, so that the line makes an angle 10o with the horizontal. The tension in the line is closest to... 50N, 40N, 100N, 200N, or 10N Saturday, November 3, 2012 by kevin college math A$8700 personal loan at 5.5% compounded monthly is to be repaid over a 4 year term by equal monthly payments. a) calculate the interest and principle component of the 18th payment b) how much interest will be paid in the third year of the loan?
Saturday, November 3, 2012 by sasha

college algebra
given function f(X)=X^3+3x findthe rate of change between the two stated values for x:1 to 2 find the equation of a secant line (1,f(1)and(2,f(2))
Friday, November 2, 2012 by zachary

I am not understanding this question and I have read the chapter many times. Please help. given the function of f(x)=4x-8 a. find the the rate of change between the stated values for x:5 to 7 b. find the equation of a secant line containing the given points (5,f(5)), and (7,f(...
Friday, November 2, 2012 by ladybug

college algebra
graph the following function using transformations, be sure to graph all of the stages on one grph, state the domain and range y= x^2 +12x
Friday, November 2, 2012 by zachary

college algebra
given the function f(x)= x^3+3x find the rate of chnge between the two values x:1 to 2 find the equation of a secant line containing the given points (1,f(1) and (2,f(2))
Friday, November 2, 2012 by zachary

SAT Essay
Assignment: Can knowledge be a burden rather than a benefit? Plan and write an essay in which you develop your point of view on this issue. Support your position with reasoning and examples taken from your reading, studies, experiences, or observations Essay: Knowledge can ...
Friday, November 2, 2012 by Sanya

personal statement
this is my personal statement. please check my grammar and what other stuff i can add or remove? thanks Ever since I was a little kid and till a year ago, I really didn’t know what I wanted to be in life. I imagined myself to be a teacher when I was 8 years old. As I grew ...
Friday, November 2, 2012 by help me

College Algebra
Simplfy the expresión using the laws of exponentes . 1. (-2x^2*y^-4)^3 2. (-4x^-6y^3)(2x^3y^-9)
Friday, November 2, 2012 by Anhisha

college math
A \$8700 personal loan at 5.5% compounded monthly is to be repaid over a 4 year term by equal monthly payments. a) calculate the interest and principle component of the 18th payment b) how much interest will be paid in the third year of the loan?
Friday, November 2, 2012 by sasha

college algebra
graph the following function using transformations, be sure to graph all of the stages one graph, state the domain and the range y=x^2+12x
Friday, November 2, 2012 by zachary

college algebra
given the functionf(x)=x^3+3x find the rateof change between the two stated values for x:1 to 2 find the equation of a secant line containig the given points (1,f(1))and (2,f(2))
Friday, November 2, 2012 by zachary


text:
## Feature Contribution - another way to think about feature importance

### 2019/01/14

In many machine learning models, feature importance or variable importance is an important output from the model as it informs us about the relative or absolute importance of each feature in contributing to the model. More specifically, feature importance tells us which are the features that are highly differentiating, in the case of classification, and which are those that are not.

However, one shortfall of feature importance is that it’s global in nature - it informs the data scientist about the overall strength of the features as a whole in the dataset. What if something more granular and refined is required? Just because a feature is high up on the feature importance list does not mean that it’s always important.

### Logistic regression as an example

Suppose you have some logistic regression model in the form

logit(p) =  β0 + β1x1 + … + βpxp

For a given test case with p features, you would get the predicted probability by substituting each actualized feature value into the model (and do the inverse logit transformation). Naturally, different test cases would have different actualized feature value, summing up to different predicted probabilities.

Therefore, it’s intuitive to think about how a feature contributes to the predicted probability of a given test case:

feature contribution of xk = |βkxk| / (|β0| + Σ|βixi|) ∈ [0,1]

With this, we can say the xk contributes to the predicted probability of a given test case for some proportion or percentage, i.e. the feature contribution.

### Generalizing to tree-based models

This way of evaluating individual feature contribution can be generalized to beyond linear models. For example, in a decision tree, a given test case would have a given prediction pathway that it takes down the decision tree - passing through to multiple junctions in the tree. The gain or loss in predicted probabilities or predicted values can be tracked accordingly, leading to a similar notion of feature contribution.

Moreover, this can further generalized to ensemble models, such as Random Forest, ExtraTrees and even XGBoost.

I leave you with these two blog posts and the treeinterpreter package in Python as it has already been explored:

That’s all for this short post, hope it helps in bringing you to think slightly deeper about feature importance.


text:
# How to resize Mac OS Extended (Journaled) partition after linux messed it up

hard drivepartitionssdunix

I've tried to use Ubuntu on my macbook pro (6,2), but it was a total and abject failure, mainly due to poor hardware support.

So, I have given up on my little experiment and just deleted ubuntu and erased the partition it used in the ssd. Now, I'm trying to make os x reclaim the former linux partition's space, but I get an error whenever I try to resize the system partition in Disk Utility:

Partition failed with the error:
File system resize support required, such as HFS+ with Journaling enabled.

OS X's partition is "Mac OS Extended (with journaling)". My OS X's version is 10.8.3 (Mountain Lion).

Any way of doing it without wiping out the disk? Even if I had a backup one, the bakcup-and-restore process would be very time consuming and I believe there must be a cleverer way of resizing a partition than deleting everything…

EDIT:

output of diskutil list:

/dev/disk0
#:                       TYPE NAME                    SIZE       IDENTIFIER
0:      GUID_partition_scheme                        *240.1 GB   disk0
1:                        EFI                         209.7 MB   disk0s1
2:                  Apple_HFS osx                     40.0 GB    disk0s2
3:       Microsoft Basic Data Deposit                 139.7 GB   disk0s5

Have you tried the resize with diskutil command? (man diskutil look for the resizeVolume command). If nothing else, it might give you a different or better error message.


text:
# Average Distance

Two points are chosen randomly inside a square with side length 1. What is the average distance between the two points?

Round your answer to 3 decimal places.

×


text:
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22 Jan 2013, 12:33
cheetarah1980 wrote:
joy4ol, I think you'd be better off going to a better source of information than GMATClub. The people who would really know which school is the best choice given your goals are students at Darden, Tuck, who are recruiting for IB and corp fin and people who have worked at IFC, WB. Most of the people who have responded to this thread aren't from either school, and others have yet to even attend school. People are going off of hearsay and rankings, more than what actually is possible given the resources and alums of both schools. People on this thread have said that Darden is a non-target school for IB, yet Consulting2Finance (who actually goes to Darden) made it clear that Darden got the full recruitment resources (company presentation, dinners, coffee chats, on campus interviews) of all the BB banks.
GMATClub is a great resource but when it comes to picking a school you have to take the advice you get with a very large grain of salt. People have their own biases, preferences, and agendas when posting and what is really opinion is communicated as fact. Just because someone heard from one student at a school that IB recruiting is low doesn't mean that that's the truth. You're getting second or third hand information. Personally, I do not know enough about Tuck or Darden to comment on whether you should take or leave the money. Try reaching out to LadyRoadWarrior. She's a first year at Tuck. She's not recruiting for finance but she'll know who to put you in touch with. I hope this helps.

Of course, Cheetarah! I am currently in conservation with students at both colleges. reached out to consulting2finance as well.
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Posts: 32
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14 May 2013, 13:24
CobraKai wrote:
joy4ol wrote:

As I said before, I am still undecided. I will drop in a last word on my decision on this thread at the end of this month.

What did you end up doing?

I decided to take up an offer from Kellogg
Senior Manager
Status: Yale! (whipped and bound)
Joined: 02 May 2012
Posts: 406
Location: United States (CA)
Concentration: Nonprofit
GMAT 1: 730 Q48 V42
GPA: 3.41
Followers: 6

Kudos [?]: 138 [0], given: 78


text:
# The Limit of a Sequence

This resource is a geogebra applet with minimal instructions that allows the user to explore the limit of a sequence visually. An example sequence is given and the user can enter their own examples as well. The user can propose a limit value and then they can vary the value of epsilon and $$N$$, so that it is clear for what value of $$N$$ all values of the sequence with $$n \geq N$$ are less than epsilon away from the limit. It is currently not clear how to move further to the right, but for most converging sequences this is not a problem. All sequence values within epsilon of the limit value are green while those outside this range are displayed in red.

Identifier:
http://users.dickinson.edu/~richesod/limitsequence/
Rating:
Creator(s):
Dave Richeson
Cataloger:
Paul Seeburger
Publisher:
Dickinson College
Rights:
Dave Richeson, Dickenson College