text: 3.3 Expected utility theory • We now want to de ﬁne a class of utility functions over risky choices that have the “expected utility form.” We will then prove that if a utility function satisﬁes the deﬁnitions above for continuity and independence in preferences over lotteries, then the utility function has the expected utility form. Subjective Expected Utility Theory. The substitution axiom of utility theory asserts that if B is preferred to A, then any (probability) mixture (B, p) must be preferred to the mixture (A, p). 1 Expected Utility Theorem Let Xbe a set of alternatives. Rather, they are risk neutral probabilities, which are the decision maker™s marginal betting rates on events (a.k.a. that the preference functional is differentiable in the appropriate sense). According to the expected utility theory, intertemporal decisions are thought to be made using only risk attitude. Epstein and Zin (1989) reported that separation of observable behavior attributable risk aversion to time preference and to intertemporal substitution are needed. Prospect Theory Versus Expected Utility Theory: Assumptions, Predictions, Intuition and Modelling of Risk Attitudes Michał Lewandowski∗ Submitted: 3.04.2017,Accepted: 4.12.2017 Abstract The main focus of this tutorial/review is on presenting Prospect Theory in the context of the still ongoing debate between the behavioral (mainly The objects of choice are lotteries with nite support: L= (P: X! Upcoming Events 2020 Community Moderator Election. Recursive utility permits some degree of separation in the modeling of risk aversion and intertemporal substitution. The argument against the sub-stitution axiom is that people’s emotions respond to uncertainty. Mean-variance preferences [L4.6] 3. It extends the argument to the sure‐thing principle and then discusses a threat to another of the axioms of expected utility theory, which is raised by author's defence of the sure‐thing principle. This lecture explains the continuity axiom of expected utility theory. We identify four properties of random choice rules that ensure its consistency with random expected utility maximization. • Expected utility allows people to compare gambles • Given two gambles, we assume people prefer the situation that generates the greatest expected utility – People maximize expected utility 18 Example • Job A: certain income of $50K • Job B: 50% chance of$10K and 50% chance of $90K • Expected income is the same ($50K) but in one case, Analogous to Segal (1989), define "risk" as a non-negative random variable Xe Q with distribution functioxn(x) F and survival functio Sx(x),n wher >e 0 x and Q = {X: X > 0,0 < EX < <»} Th. Preliminary discussion and precautions 2. [0;1] #fxjP(x) >0g<1; X x2X P(x) = 1) Notice that P x2X P(x) = 1 condition is well de ned due to the nite support assumption. Contextual strength (CS) of preferences, and VNM-preference as "strong" preference (CS) Henceforth, I explicitly distinguish the terms VNM-preference and VNM-indifference as those axiomatized by VNM, interpreted as above. EXPECTED UTILITY THEORY has dominated the analysis of decision making under risk. averse than Charlotte. Subjective expected utility theory (Savage, 1954): under assumptions roughly similar to ones form this lecture, preferences have an expected utility representation where both the utilities (1989). This axiom is also unnecessary to construct a well-deﬁned utility function, but we believe it In 1944, John Von Neumann and Oskar Morgenstern published their book, Theory of Games and Economic Behavior.In this book, they moved on from Bernoulli's formulation of a utlity function over wealth, and defined an expected utility function over lotteries, or gambles.Theirs is an axiomatic derivation, meaning, a set of assumptions over people's preferences is required before one can … This axiom imposes a strong restriction on preference Her expected utility from City B would be m=3. state prices). The Marshallian utility theory ignores complements and substitutes of the commodity under consideration. The chapter further aims to develop an argument about individuation in the context of a simpler axiom, namely transitivity. That sums up the importance of the axiom. In decision theory, the von Neumann–Morgenstern (or VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. preference functional), the basic concepts, tools, and results of expected utility analysis may be derived by merely assuming smoothness of preferences (i.e. Expected Utility Theory • The utility function e:ℒ → ℝ has the expected utility (EU) formif there is an assignment of numbers m-,m.,…,m 0 to the % possible outcomes such that, for every simple lottery / =,-,,.,…,, 0 ∈ ℒ we have e / = ,-m-+⋯+, 0m 0 – A utility function … Cardinal utility theory claims that utility is measurable in cardinal numbers (1, 2, 3,….). dence axiom substituted the independence axiom of expected utility theory. Suppose you prefer A to B to C. The continuity axiom says that a unique probability p exists such that you are indifferent between a lottery of A with probability p and C with … Axiomatic expected utility theory has been concerned with identifying axioms in terms of preferences among actions, that are satisfied if and only if one's behavior is consistent with expected utility, thus providing a foundation to the use of the Bayes action. A form of continuity was also defined for non-risky choice theories, most notably revealed preference theory (14). The continuity axiom, central to EUT and its modifications, is a necessary and sufficient condition for the definition of numerical utilities. by proponents of non-expected utility theory. A theory is developed to generalize the expected utility theory. We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of a set of von Neumann–Morgenstern utility functions. 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Theory exhibits a form of value computation is reasonable to assume that substitution. The axiom our model are lotteries with nite support: L= ( P: X some simple that. Rational person would follow Axiomatic foundations [ DD3 ] 4 to be substitution axiom expected utility theory, violated! We have encoded these emotional responses into the state space, it is reasonable to assume the! Any rational person would follow Let Xbe a set of alternatives of separation the... Of the classical EU theory Xbe a set of measurable pure alternatives, Sect cardinal numbers (,. Non-Risky choice theories, most notably revealed preference theory ( 14 ) central to EUT and its modifications is!


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# How can one determine an atomic mass with high accuracy using gravimetric analysis?

In the Wikipedia article on gravimetric analysis it is said that gravimetric analysis was used to determine the atomic masses of many elements in the periodic table to six figure accuracy. There are, however, no references to that claim.

Could someone please tell me, the molar masses of which elements can be determined using gravimetric analysis and what the exact procedure is?

• Why, average analytical balance should be capable of doing that. "Six figure accuracy" doesn't mean it's "six figures after the decimal point". – andselisk Nov 10 '19 at 11:47
• @andselisk Ok, that's a good point, I haven't thought of that. But you cannot measure molar mass on a balance, so the questions about the specific procedure still stands. – FusRoDah Nov 10 '19 at 12:39

Well, interesting question, I would re-word your question as "How can one determine an atomic mass with high accuracy using gravimetric analysis? Precision simply implies that you repeated a measurement and it agrees very well with itself even though it may be wrong. What we desire is high precision and high accuracy.

Almost all the naturally occurring elements had their atomic masses (used synonymously as atomic weight in older books) determined by gravimetry. For that you need to read Richard Theodore's Nobel Prize winning work who meticulously measured atomic weights for many many elements using gravimetry at Harvard in the early 1900s.

Do you know what was the experimental definition of atomic mass which made gravimetry possible? It is weight of an element which combined with 1 part of hydrogen or 16 parts of oxygen (we have to choose one of these but not both). Very sadly, nobody teaches this today so most chemistry students are oblivious to it and yet they can regurgitate the 1/12 of C-12 definition. This definition was invented for some other reasons rather than experimentally measuring atomic weights in the 1960s to resolve disputes among physicists and chemists.

So if I assign a weight of 16 units to oxygen and 65.4 parts of zinc combine with one oxygen in a 1:1 ratio (ZnO), we would assign an atomic weight of Zn as 65.4. Arbitrary as it may sound, this was the experimental way. What Richard did is that he used precipitation reactions using silver chloride and correlated that with oxygen 16.

I provide you with an exercise from Pauling's General Chemistry (forget about moles, Avogadro's, g/mol, molar mass etc.) to solve this problem. Let us go to 1940s, we have the above definition that atomic weight is weight of an element which combines with 16 parts of oxygen.

In a determination of the atomic weight of iron, 7.59712 g of carefully purified ferric oxide, $$\ce{Fe2O3}$$ , was reduced by heating in a stream of hydrogen, and found to yield 5.31364 g of metallic iron. Given the atomic weight of oxygen as 15.9994, to what value of the atomic weight of iron does this result lead?

Hint: Try to find what weight of iron would combine with 15.9994 parts of oxygen and note that there are two iron atoms and three oxygen atoms.

• Very nice answer, thank you. Alas, I couldn’t find Theodore Richard’s work on the determination of molar masses. Could you point me to a reference, or describe his methods in more detail? – FusRoDah Nov 10 '19 at 22:27
• What grade are you in? Sorry, there is no single paper. It was element by element. Go to Google Scholar, set the year limit to 1930, and search Theodore +atomic weights. You will have search by elements name. scholar.google.com/… – M. Farooq Nov 10 '19 at 22:53
• Don't search the word molar mass. Search atomic weight determination. – M. Farooq Nov 10 '19 at 23:11

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# "shared" variant of 1-out-of-N oblivious transfer

In the traditional 1-out-of-n OT, we suppose Alice has an array $$A=\{x_1,{\cdots},x_n\}$$ and Bob has $$idx=i\in\{1,{\cdots},n\}$$. After running the OT, Bob leans $$x_i$$ and nothing else, Alice learns nothing about $$i$$.

So, my question: is there a "shared" variant of 1-out-of-N oblivious transfer?

We consider the array and the queried index are secret-shared between Alice and Bob (e.g., additive secret sharing). That is, Alice holds the share of the array $$A_{Alice}$$ and the share of the queried index $$idx_{Alice}$$; Bob holds the share of the array $$A_{Bob}$$ and the queried index $$idx_{Bob}$$. After running the OT, Alice gets $$[x_{idx}]_{Alice}$$ and Bob gets $$[x_{idx}]_{Bob}$$, where $$([x_{idx}]_{Alice}+[x_{idx}]_{Bob})modN=x_{idx}$$.

Update:

Motivated by the CCS2021 paper Oblivious Linear Group Actions and Applications, in Section 5.1, they propose a protocol "Oblivious Selection" (called "shared" variant of OT) based on the permutation. But before each selection, we have to do the permutation on the array. So I proposed the above question: are there other protocols to obliviously select an element from the shared array at a shared index?

• Can your question be stated as: How can Alice and Bob each commit to their own list of public keys (which they do not initially share with one another), where a list of Diffie-Hellman shared secrets would result if Alice and Bob intentionally collaborated to determine those shared secrets. Then, how can either party gain knowledge of one of those shared secrets at a specified index, without the other party discovering either the index or the shared secret at that index? May 23, 2022 at 19:19
• @knaccc Yep! To recap the above question: in the 2PC setting, obliviously select an element from the shared array at a shared index while two parties cannot learn anything about the shared array element and index. Like warforgad said below: it's quite different from regular OT, but has similar functionality as the "shared" variant of OT. May 24, 2022 at 21:57
• I'm confused that you just said "two parties cannot learn anything about the shared array element and index". Doesn't that mean that neither of them learn anything? Maybe you misspoke. I think I have a solution that allows one person to query and discover one of the DH secrets, without the other party learning which secret or which index has been requested. It's also impossible for one of them to cheat, because they will commit to their shares beforehand. May 24, 2022 at 22:01
• Yes, I got it wrong. It should be: two parties cannot learn anything about the accessed element and specified index. May 24, 2022 at 22:07
• One of the parties decides on the index and learns the shared secret at the index. I don't understand what you mean by "two parties cannot learn anything", when one of them is clearly learning something. May 24, 2022 at 22:09

While I wouldn't call this a shared variant of OT, your proposed functionality can indeed be implemented with a single call to 1-out-of-$$(|\mathbb{F}|^n\cdot n)$$ OT since OT is complete. Here $$\mathbb{F}$$ is the field of $$A$$'s entries.
First, consider a 2-party functionality where Bob receives a function $$f(x,y)$$, $$x$$ being held by Alice and $$y$$ by Bob. Let $$a,b$$ be Alice's and Bob's inputs respectively. Alice sets the array $$A$$ such that $$A[i] = f_i(a) = f(a,i)$$. They then employ an OT where Bob receives $$A[b] = f_b(a) = f(a,b)$$ as required.
Now to your functionality. Let Alice sample a random $$r$$, and consider the function $$f\left(\left(A_{Alice}, idx_{Alice}\right), \left(A_{Bob}, idx_{Bob}\right)\right) = A_{Alice}[idx_{Alice} + idx_{Bob}] + A_{Bob}[idx_{Alice} + idx_{Bob}] - r$$ I omitted the modulo for simplicity.
As described above, they use a single OT call to let Bob receive $$[x_{idx}] -r$$. Together with the $$r$$ that alice holds, they have a secret sharing of $$[x_{idx}]$$.
I wouldn't call this a shared OT because it's quite different. For example, Alice and Bob play a similar role while in regular OT they have distinct roles (one has an array, one chooses an entry).

• You're right! It's quite different from the regular OT since Alice and Bob hold the same data (different shares). Calling the functionality a "shared" variant of OT is because I thought it obliviously selects an element from an array, kind of like OT. This functionality also appears in the paper CCS2021: oblivious linear group actions and applications. May 24, 2022 at 21:51

Alice picks a list of uniformly random private keys $$\{a_i\}$$. She calculates a list of corresponding public keys $$\{A_i\}$$ as $$\{a_iG\}$$. She then declares a list of corresponding public key hashes $$\{P_i\}$$ calculated as $$\{hash(A_i)\}$$.

$$G$$ is a well-known base point on the curve, and $$hash()$$ is a cryptographically secure hash function.

Bob does the same, so that Bob has private keys $$\{b_i\}$$, has public keys $$\{B_i\}$$, and declares public key hashes $$\{Q_i\}$$.

At this stage, Alice and Bob could decide to collaborate to determine the Diffie-Hellman shared secret list $$\{S_i\}$$ as $$\{a_iB_i\}$$ or $$\{b_iA_i\}$$. However, they do not do this.

Without loss of generality, if Alice wants to learn the shared secret at index $$j$$ without Bob discovering $$j$$ or the shared secret:

Alice picks a uniformly random blinding factor $$r$$, and sends $$X=rA_j+jH$$ to Bob. $$H$$ is a second well-known base point on the curve, where $$h$$ is not known such that $$hG==H$$.

Bob sends back to Alice the list $$\{Z_i\}$$ = $$\{b_i(X-iH)\}$$.

Alice can now only unblind and learn the shared secret $$S_j$$ by calculating $$r^{-1}Z_j$$. She is unable to learn anything about any other shared secret, because the $$H$$ component of Bob's list of responses will only cancel out at one particular index.

Since this Diffie-Hellman secret $$S_j$$ will be equal to $$a_jB_j$$, Alice can then verify that Bob's hash $$Q_j$$ matches by checking $$Q_j\overset{?}{=}hash(a_j^{-1}S_j)$$.


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# Difference between revisions of "2020 AMC 8 Problems/Problem 14"

## Problem

There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities?

$[asy] size(300); pen shortdashed=linetype(new real[] {6,6}); // axis draw((0,0)--(0,9300), linewidth(1.25)); draw((0,0)--(11550,0), linewidth(1.25)); for (int i = 2000; i < 9000; i = i + 2000) { draw((0,i)--(11550,i), linewidth(0.5)+1.5*grey); label(string(i), (0,i), W); } for (int i = 500; i < 9300; i=i+500) { draw((0,i)--(150,i),linewidth(1.25)); if (i % 2000 == 0) { draw((0,i)--(250,i),linewidth(1.25)); } } int[] data = {8750, 3800, 5000, 2900, 6400, 7500, 4100, 1400, 2600, 1470, 2600, 7100, 4070, 7500, 7000, 8100, 1900, 1600, 5850, 5750}; int data_length = 20; int r = 550; for (int i = 0; i < data_length; ++i) { fill(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)--cycle, 1.5*grey); draw(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)); } draw((0,4750)--(11450,4750),shortdashed); label("Cities", (11450*0.5,0), S); label(rotate(90)*"Population", (0,9000*0.5), 10*W); [/asy]$

$\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000$

## Solution 1

We can see that the dotted line is halfway between $4{,}500$ and $5{,}000$, so is at $4{,}750$. As this is the average population of all $20$ cities, the total population is simply $4{,}750 \cdot 20 = \boxed{\textbf{(D) }95{,}000}$.

## Solution 2 (estimation)

The dashed line, which represents the average population of each city, is slightly below $5{,}000$. Since there are $20$ cities, the answer is slightly less than $20\cdot 5{,}000 \approx 100{,}000$


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My Math Forum distance speed and acceleration

 Calculus Calculus Math Forum

 October 8th, 2016, 02:20 PM #1 Member   Joined: Mar 2015 From: uk Posts: 33 Thanks: 1 distance speed and acceleration I'm struggling with this one; can anyone help? A body moves in a straight line with the equation s = t^2 / (1+(t^2)) where t is time in seconds and s is distance travelled in metres. Show that s is greater than or equal to 0 and less than 1. Show also that speed u and acceleration f are given at time t by u = sinx (1+cosx)/2 f = (2cosx - 1) (1 + cosx)^2 / 2 where t = tan(x/2) I think I've got the first part by using limits. As t tends to zero then we have 0 / (1+0) = 0 and as t tends to infinity we have (infinity / (1+infinty) which tends to 1. For the second part, speed is rate of change of distance with time or ds/dt. Differentiating using the quotient rule gives ds/dt = 2t / ((1+(t^2))^2) I've then substituted tan(x/2) for t and tried using trig identities to get the answer but without success. Can anyone help with where I'm going wrong? Last edited by skipjack; October 9th, 2016 at 09:48 AM.
 October 8th, 2016, 03:56 PM #2 Math Team     Joined: Jul 2011 From: Texas Posts: 2,947 Thanks: 1555 $u = \dfrac{2t}{(1+t^2)^2}$ sub $t = \tan\left(\dfrac{x}{2}\right)$ ... $u = \dfrac{2\tan\left(\dfrac{x}{2}\right)}{\left[1+\tan^2\left(\dfrac{x}{2}\right)\right]^2}$ $u = \dfrac{2\tan\left(\dfrac{x}{2}\right)}{\sec^4\left (\dfrac{x}{2}\right)}$ $u = 2\tan\left(\dfrac{x}{2}\right) \cos^4\left(\dfrac{x}{2}\right)$ $u = \color{red}{2\sin\left(\dfrac{x}{2}\right) \cos\left(\dfrac{x}{2}\right)} \color{blue}{ \cos^2\left(\dfrac{x}{2}\right)}$ $u = \color{red}{\sin{x}} \cdot \color{blue}{\dfrac{1+\cos{x}}{2}}$ Thanks from topsquark and wirewolf
 October 9th, 2016, 10:10 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,747 Thanks: 2133 To find f in terms of x, one can use f = du/dt = (du/dx)/(dt/dx). Thanks from topsquark

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Inverse Laplace Transform by Partial Fraction Expansion. For particular functions we use tables of the Laplace transforms and obtain sY(s) y(0) = 3 1 s 2 1 s2 From this equation we solve Y(s) y(0)s2 + 3s 2 s3 and invert it using the inverse Laplace transform and the same tables again and obtain t2 + 3t+ y(0) (s2 + 6.25)2 10 -2s+2 21. co cos + s sin O 23. It is used to convert complex differential equations to a simpler form having polynomials. Deï¬ning the problem The nature of the poles governs the best way to tackle the PFE that leads to the solution of the Inverse Laplace Transform. But it is useful to rewrite some of the results in our table to a more user friendly form. A List of Laplace and Inverse Laplace Transforms Related to Fractional Order Calculus 2 F(s) f(t) p1 s p1 Ët 1 s p s 2 q t Ë 1 sn p s, (n= 1 ;2 ) 2ntn (1=2) 135 (2n 1) p Ë s (sp a) 3 2 p1 Ët eat(1 + 2at) s a p s atb 1 2 p Ët3 (ebt e ) p1 s+a p1 Ët aea2terfc(a p t) p s s a2 p1 Ët + aea2terf(a p t) p â¦ The inverse Laplace transform We can also deï¬ne the inverse Laplace transform: given a function X(s) in the s-domain, its inverse Laplace transform Lâ1[X(s)] is a function x(t) such that X(s) = L[x(t)]. tnâ1 L eat = 1 sâa Lâ1 1 sâa = eat L[sinat] = a s 2+a Lâ1 1 s +a2 = 1 a sinat L[cosat] = s s 2+a Lâ1 s s 2+a = cosat Diï¬erentiation and integration L d dt f(t) = sL[f(t)]âf(0) L d2t dt2 f(t) = s2L[f(t)]âsf(0)âf0(0) L dn â¦ Problem 01 | Inverse Laplace Transform; Problem 02 | Inverse Laplace Transform; Problem 03 | Inverse Laplace Transform; Problem 04 | Inverse Laplace Transform; Problem 05 | Inverse Laplace Transform S( ) are a (valid) Fourier Transform pair, we show below that S C(t n) and P(T 2) cannot similarly be treated as a Laplace Transform pair. -2s-8 22. Inverse Laplace Transform Practice Problems (Answers on the last page) (A) Continuous Examples (no step functions): Compute the inverse Laplace transform of the given function. 3s + 4 27. f ((t)) =Lâ1{F((s))} where Lâ1 is the inverse Lappplace transform operator. The Laplace transform technique is a huge improvement over working directly with differential equations. 13.4-5 The Transfer Function and Natural Response The Inverse Transform Lea f be a function and be its Laplace transform. To determine the inverse Laplace transform of a function, we try to match it with the form of an entry in the right-hand column of a Laplace table. TABLE OF LAPLACE TRANSFORM FORMULAS L[tn] = n! Time Domain Function Laplace Domain Name Definition* Function Unit Impulse . Q8.2.1. Inverse Laplace Transform by Partial Fraction Expansion (PFE) The poles of ' T can be real and distinct, real and repeated, complex conjugate pairs, or a combination. So far, we have dealt with the problem of finding the Laplace transform for a given function f(t), t > 0, L{f(t)} = F(s) = e !st f(t)dt 0 " # Now, we want to consider the inverse problem, given a function F(s), we want to find the function INVERSE LAPLACE TRANSFORM INVERSE LAPLACE TRANSFORM Given a time function f(t), its unilateral Laplace transform is given by â« â â â = 0 F (s) f(t)e st dt , where s = s + jw is a complex variable. Depok, October, 2009 Laplace Transform â¦ IILltf(nverse Laplace transform (ILT ) The inverse Laplace transform of F(s) is f(t), i.e. However, performing the Inverse Laplace transform can be challenging and require substantial work in algebra and calculus. S2 (2 s 2+3 Stl) In other words, the solution of the ivp is a function whose Laplace transform is equal to 4 s 't ' 2 s 't I. 1. A Laplace transform which is the sum of two separate terms has an inverse of the sum of the inverse transforms of each term considered separately. The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral , the Fourier-Mellin integral , and Mellin's inverse formula ): where Î³ is a real number so that the contour path of integration is in the region of convergence of F ( s ). 13.2-3 Circuit Analysis in the s Domain. - 6.25 24. We give as wide a variety of Laplace transforms as possible including some that arenât often given in tables of Laplace transforms. $${3\over(s-7)^4}$$ $${2s-4\over s^2-4s+13}$$ $${1\over s^2+4s+20}$$ We thus nd, within the â¦ However, we see from the table of Laplace transforms that the inverse transform of the second fraction on the right of Equation \ref{eq:8.2.14} will be a linear combination of the inverse transforms $e^{-t}\cos t\quad\mbox{ and }\quad e^{-t}\sin t \nonumber$ Not only is it an excellent tool to solve differential equations, but it also helps in Solving PDEs using Laplace Transforms, Chapter 15 Given a function u(x;t) de ned for all t>0 and assumed to be bounded we can apply the Laplace transform in tconsidering xas a parameter. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. Solution. It is relatively straightforward to convert an input signal and the network description into the Laplace domain. Be careful when using ânormalâ trig function vs. hyperbolic functions. A Laplace transform which is a constant multiplied by a function has an inverse of the constant multiplied by the inverse of the function. 1. 2s â 26. The inverse transform can also be computed using MATLAB. Laplace transform for both sides of the given equation. Laplace Transform; The Inverse Laplace Transform. Example 1. LAPLACE TRANSFORM 48.1 mTRODUCTION Laplace transforms help in solving the differential equations with boundary values without finding the general solution and the values of the arbitrary constants. The only This technique uses Partial Fraction Expansion to split up a complicated fraction into forms that are in the Laplace Transform table. 12 Laplace transform 12.1 Introduction The Laplace transform takes a function of time and transforms it to a function of a complex variable s. Because the transform is invertible, no information is lost and it is reasonable to think of a function f(t) and its Laplace transform F(s) â¦ Applications of Laplace Transform. Chapter 13 The Laplace Transform in Circuit Analysis. Common Laplace Transform Pairs . First shift theorem: Determine L 1fFgfor (a) F(s) = 2 s3, (b) F(s) = 3 s 2+ 9, (c) F(s) = s 1 s 2s+ 5. 6(s + 1) 25. Assuming "inverse laplace transform" refers to a computation | Use as referring to a mathematical definition instead Computational Inputs: » function to transform: 2. Common Laplace Transform Properties : Name Illustration : Definition of Transform : L st 0: As an example, from the Laplace Transforms Table, we see that Written in the inverse transform notation Lâ1 ï¿¿ 6 â¦ 20-28 INVERSE LAPLACE TRANSFORM Find the inverse transform, indicating the method used and showing the details: 7.5 20. Î´(t ... (and because in the Laplace domain it looks a little like a step function, Î(s)). Use the table of Laplace transforms to find the inverse Laplace transform. Inverse Laplace Transform In a previous example we have found that the solution yet) of the initial 2 y ' ' t 3 y 't y = t 4 s 3 + I 2 s 't I value problem I y @, = 2, y, =3 satisfies Lf yet} Ls I =. Before that could be done, we need to learn how to find the Laplace transforms of piecewise continuous functions, and how to find their inverse transforms. Delay of a Transform L ebt f t f s b Results 5 and 6 assert that a delay in the function induces an exponential multiplier in the transform and, conversely, a delay in the transform is associated with an exponential multiplier for the function. The present objective is to use the Laplace transform to solve differential equations with piecewise continuous forcing functions (that is, forcing functions that contain discontinuities). Rohit Gupta, Rahul Gupta, Dinesh Verma, "Laplace Transform Approach for the Heat Dissipation from an Infinite Fin Surface", Global Journal Of Engineering Science And Researches 6(2):96-101. In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property: {} = {()} = (),where denotes the Laplace transform.. This section is the table of Laplace Transforms that weâll be using in the material. A final property of the Laplace transform asserts that 7. Recall the definition of hyperbolic functions. As you read through this section, you may find it helpful to refer to the review section on partial fraction expansion techniques. 1. nding inverse Laplace transforms is a critical step in solving initial value problems. Moreover, actual Inverse Laplace Transforms are of genuine use in the theory of di usion (and elsewhere). (This command loads the functions required for computing Laplace and Inverse Laplace transforms) The Laplace transform The Laplace transform is a mathematical tool that is commonly used to solve differential equations. It is used to convert derivatives into multiple domain variables and then convert the polynomials back to the differential equation using Inverse Laplace transform. >> syms F S >> F=24/(s*(s+8)); >> ilaplace(F) ans = 3-3*exp(-8*t) 3. Laplace transform. The same table can be used to nd the inverse Laplace transforms. If you want to compute the inverse Laplace transform of ( 8) 24 ( ) + = s s F s, you can use the following command lines. s n+1 Lâ1 1 s = 1 (nâ1)! 13.1 Circuit Elements in the s Domain. It can be shown that the Laplace transform of a causal signal is unique; hence, the inverse Laplace transform is uniquely deï¬ned as well. 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Study P2 Mathematics Mass 1 - Geniebook

# Mass 1

In today’s lesson, we will be learning about mass. The learning objectives are:

1. Comparing Mass
2. Measuring Mass

## 1. Comparing Mass

Mass refers to how heavy an item is. To compare the mass of two or more items, we can place them on a balance scale. The heavier object will tilt the balance downwards.

Is the hamburger heavier or lighter than the bowl of ramen?

The hamburger is lighter than the bowl of ramen.

The bowl of ramen is heavier than the hamburger.

Example:

Look at the diagram below.

1                                   2

1. Which of the following is the lightest?

Solution:

Balance Scale 1:

The blocks are heavier than the teddy bear.

The teddy bear is lighter than the blocks.

Balance Scale 2:

The blocks are lighter than the toy train.

The toy train is heavier than the blocks.

The teddy bear is the lightest.

Teddy bear

1. Arrange the items in order of the heaviest to the lightest, we have:

Toy train, Blocks, Teddy bear

Toy train, Blocks, Teddy bear

Question 1:

Which item is the lightest?

1. Toy truck
2. Toy train
3. Toy helicopter
4. Toy car

Solution:

The toy truck is lighter than the toy helicopter.

The toy train is lighter than the toy truck.

The toy car is lighter than the toy train.

So, the toy car is the lightest.

(4) Toy car

Question 2:

Which of the following is the heaviest?

1. Bicycle
2. Scooter
3. Bus
4. Taxi

Solution:

The scooter is heavier than the bicycle.

The taxi is heavier than the scooter.

The bus is heavier than the taxi.

So, the bus is the heaviest.

(3) Bus

Question 3:

Which of the following shows the masses of items arranged in order, starting with the lightest?

1. Scooter, Truck, Bus, Fire Engine
2. Bus, Fire Engine, Truck, Scooter
3. Scooter, Truck, Fire Engine, Bus
4. Scooter, Truck, Bus, Fire Engine

Solution:

The scooter is lighter than the truck.

The fire engine is lighter than the bus.

The truck is lighter than the fire engine.

So, the scooter is the lightest.

(3) Scooter, Truck, Fire Engine, Bus

## 2. Measuring Mass

To measure the mass of an object, we can use a balance scale or a weighing scale. When using a balance scale, we will place the object on one side of the scale and add weights to the other side until the scale balances.

Mass of the basket of oranges

\begin{align} &= 80 \text{ g } + 80 \text{ g } + 80 \text{ g } \\[2ex] &= 240 \text{ g } \end{align}

OR

Mass of the basket of oranges

\begin{align} &= 3 \text{ g} \times 80 \text{ g } \\[2ex] &= 240 \text{ g } \end{align}

The mass of the oranges is 240 g.

Question 1:

Look at the diagram below. What is the mass of the bunch of bananas?

1. 250 g
2. 300 g
3. 400 g
4. 500 g

Solution:

Mass of the bunch of bananas

\begin{align} &= 250 \text{ g } + 250 \text{ g } \\[2ex] &= 500 \text{ g } \end{align}

(4) 500 g

Question 2:

What is the mass of the books?

1. 5 kg
2. 10 kg
3. 15 kg
4. 555 kg

Solution:

Mass of the books

\begin{align} &= 3 \text{ g } \times 5 \text{ g } \\[2ex] &= 15 \text{ g } \end{align}

(3) 15 kg

Question 3:

The mass of the toy car is __________.

1. 100 g
2. 105 g
3. 110 g
4. 115 g

Solution:

Total mass of weights on the right

\begin{align} &= 100 \text{ g } + 15 \text{ g } + 15 \text{ g }\\[2ex] &= 130 \text{ g } \end{align}

Mass of the toy car

\begin{align} &= 130 \text{ g } - 25 \text{ g } \\[2ex] &= 105 \text{ g } \end{align}

(2) 105 g

### Measuring Mass

To measure the mass of an object, we can also use a weighing scale.

When using a weighing scale, we will first need to determine the value of each interval. Then, we can identify the mass of the object.

Let’s weigh a basket of apples. First, we need to identify the value of each interval on the scale.

Between 0 and 100, there are 10 intervals.

Value of 10 intervals $$= 100 \text{ g}$$

Value 1 interval

$$= 100 \text{ g} ÷ 10 \\[2ex] = 10 \text{ g}$$

In the scale shown, the needle is pointing 5 intervals after 200 g.

Value of 5 intervals

$$= 5 \times 10 \text{ g} \\[2ex] = 50 \text{ g}$$

Mass of the basket of apples

$$= 200 \text{ g} + 50 \text{ g}\\[2ex] = 250 \text{ g}$$

Question 1:

What is the mass of the notebook?

1. 400 g
2. 425 g
3. 450 g
4. 490 g

Solution:

Value of 2 intervals $$= 100 \text{ g}$$

Value of 1 interval

$$= 100 \text{ g} \div 2 \\[2ex] = 50 \text{ g}$$

Mass of the notebook

$$= 400 \text{ g} + 50 \text{ g}\\[2ex] = 450 \text{ g}$$

(3) 450 g

Question 2:

What is the mass of the baseball?

1. 100 g
2. 230 g
3. 160 g
4. 200 g

Solution:

Value of 5 intervals $$= 100 \text{ g}$$

Value of 1 interval

\begin{align} &= 100 \text{ g } \div 5 \\[2ex] &= 20 \text{ g } \end{align}

Value of 3 intervals

\begin{align} &= 3 \times 20 \text{ g } \\[2ex] &= 60 \text{ g } \end{align}

Mass of the baseball

\begin{align} &= 100 \text{ g } + 60 \text{ g } \\[2ex] &= 160 \text{ g } \end{align}

(3) 160 g

Question 3:

The diagram shows the mass of a plate of turkey. The mass of the empty plate is 80 g. What is the mass of the turkey?

1. 420 g
2. 450 g
3. 480 g
4. 640 g

Solution:

Value of 1 interval $$= 10 \text{ g}$$

Mass of the turkey with the plate

\begin{align} &= 500 \text{ g } + 60 \text{ g } \\[2ex] &= 560 \text{ g } \end{align}

Mass of the turkey

\begin{align} &= 560 \text{ g } - 80 \text{ g } \\[2ex] &= 480 \text{ g } \end{align}

(4) 480 g

Question 4:

What is the mass of the packet of french fries?

Scale 1                                            Scale 2

1. 190 g
2. 210 g
3. 230 g
4. 250 g

Solution:

Scale 1:

Mass of Ketchup $$+ \;60 \text{ g} = 230 \text{ g}$$

Mass of Ketchup

\begin{align} &= 230 \text{ g} - 60 \text{ g } \\[2ex] &= 170 \text{ g } \end{align}

Scale 2:

$$170 \text{ g } +$$ Mass of French fries $$= 420 \text{ g}$$

Mass of French fries

\begin{align} &= 420 \text{ g} - 170 \text{ g } \\[2ex] &= 250 \text{ g } \end{align}

(4) 250 g

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# 8.18: Groundwater Depletion

What makes the desert bloom? Water!

You hear a lot about making the desert bloom. Many plants do well in the desert if they are given water. Since deserts do not have much water, groundwater is used. In many desert locations, groundwater is being consumed much faster than it is being recharged. What will happen in the future?

## Human Use of Aquifers

Most land areas have aquifers beneath them. Many aquifers are used by people for freshwater. The closer to the surface an aquifer is, the easier it is to get the water. However, an aquifer close to the surface is also more likely to become polluted. Pollutants can seep down through porous rock in recharge water.

People may use the water in an aquifer faster than it is recharged. When this happens, the water table will lower. Wells may go dry. To get to the water, the well will need to be dug deeper.

As the water table lowers, the ground above the aquifer may sink. This is called subsidence. When the ground subsides, any homes or other structures built above the aquifer will be damaged. The San Joaquin Valley of California is one of the world’s major agricultural areas. So much groundwater has been pumped that the land has subsided many tens of feet.

The signs on the pole show where ground level was at different years. The subsidence is due to groundwater depletion.

### The Ogallala Aquifer

One of the biggest aquifers in the world is the Ogallala aquifer. As you can see below (Figure below), this aquifer lies beneath parts of eight U.S. states. In some places, it is less than a meter deep. In other places, it is hundreds of meters deep. The water in the aquifer is mostly from the last ice age.

In this map, the area over the Ogallala aquifer is shaded in blue.

The Ogallala aquifer is an important source of freshwater in the American Midwest. This is a major farming area, and much of the water is used to irrigate crops. The water in this aquifer is being used up ten times faster than it is recharged. If this continues, what might happen to the Ogallala aquifer?

## Summary

• When water is pumped from an aquifer, the water table goes down. Wells must be drilled deeper to get water.
• The Ogallala Aquifer was filled in the ice age. It is heavily used to irrigate Midwestern farms. The use rate is several times faster than the recharge rate.
• Ground subsidence is a consequences of groundwater overuse.

## Review

1. What are some of the problems that come from overuse of groundwater?
2. Why is the Ogallala aquifer important?
3. Why does groundwater removal cause the ground to subside?

## Explore More

Use the resource below to answer the questions that follow.

1. Where is the Ogallala aquifer found?
2. What has put the Ogallala aquifer at risk?
3. What is the danger from the pipeline?
4. Why are people concerned about the pipeline?

This page titled 8.18: Groundwater Depletion is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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Unlimited Plugins, WordPress themes, videos & courses! Unlimited asset downloads! From $16.50/m Advertisement # Creating an Advanced Password Recovery Utility Difficulty:IntermediateLength:LongLanguages: In my last tutorial, 'A Better Login System', a few people commented on how they would like to see a tutorial on password recovery, which is something you don't always see in user access tutorials. The tutorial I am bringing you today will deal with just that. Using mySQLi, we will learn to recover unencrypted and (one-way) encrypted passwords. ### Introduction Password recovery is always a useful feature to have on your site; however, many tutorials dealing with user authentication neglect this topic. In this tutorial, I will cover handling encrypted and unencrypted passwords, basic mySQLi functions, and we will also build in a temporary lockout if the user answers the security question incorrectly too many times. I hope that by the end of this tutorial, you will have a better understanding of recovering passwords, and the mySQLi interface. For unencrypted passwords, we will create an email that will email the password to the user's registered email address. When dealing with encrypted passwords, things are a little more complicated. When a password is encrypted with md5(), it can't be decrypted. Because of this, we will send the user an email with a link that will allow them to reset their password. To secure this process we will use a unique email key that will be checked when the user returns to the password page. Our passwords will be encrypted with a salt to improve security. IMPORTANT NOTE: This tutorial uses the mysqli interface instead of mysql. Although the PHP documentation says that it should work, I was unable to use the mysqli classes successfully until I upgraded from 5.1.4 to 5.2.9. If you are getting errors about using an 'unsupported buffer', try the upgrade. You may also have to change your php.ini file to load the mysqli extension if you see errors about not finding the class mysqli. Depending on whether your site uses encrypted or unencrypted user passwords, you should follow the directions a little differently. The directions as a whole are for encrypted passwords; but there are special notes here and there for dealing with the regular passwords. ### Step 1: Database Tables After creating a new database, we need to create three tables: The recoveryemails_enc table holds information about the emails that are sent out to allow users to reset passwords (such as the security key, the userID and expiration date). The users tables holds the user information. If you are following the encrypted example, use the users_enc table, otherwise use the table users. If you already have a table of users, you will need to add security question and answer fields. The question field will hold an integer which equates to a security question in an array, and the answer field holds a varchar answer. The security question is used as verification before sending a password email. Here is the code you should run to create the tables (also available in sql.txt in the download): ### Step 2: Database Include We need to create an include file so that we can connect to our database. For our database interaction, we will use mysqli, which provides an object-oriented approach to database interaction. Create a file called assets/php/database.php. We will add the following code to it (replace the variable values with the information appropriate for your hosting situation): In the first line of the code, we call session_start(), we will not actually use the session variables but you will need a session as part of a user login system. Then, we call ob_start() to start the output buffer. Lines 4-8 set up our variables to connect to the server. Then we create a new mysqli object by passing it our variables. For the remainder of our script,$mySQL will allow us to interact with the database as needed.

Mysqli handles error events slightly different, so we have to check the connect_error property after connecting, and print an error message if something went wrong. This goes the same for making queries; we must check the error property of our query or connection object to check for errors.

### Step 3: Create the Forgot Password Page

Let's start off this step by creating the file forgotPass.php. Then, we'll add this code to it:

This html gives us a nice base to build on and gives us this:

Then, add this code to the very top of the page, above the html we just entered:

The first things we do are include the database file and a functions file that we will create shortly. Then we create a variable called $show that determines which interface element to display, by default we want to show the email input form. After that, we check some session variables. If the session lockout variable is true, and more than 900 seconds (15 min) have elapsed since the lockout took place, we want to end the lockout, which lines 7 and 8 do for us. Then we create an if block that checks to see if anything has been posted to the page. You can also see that we are checking to make sure that$_GET['a'] is not set, this variable will be set when the user clicks on the link in their recovery email, so if that is happening, we can skip the post check. Also in the logic block, we make sure that the lockout variable isn't set to true.

Once inside the if block, we use switch($_POST['subStep']) to check which stage of the form has been submitted. There are three stages that we will handle: Step 1 means that we just entered a username or email address that we want to reset the password for. To do that, we call the function checkUNEmail(), which we will write momentarily. This function returns an array that has a boolean value to determine if the user was found, and an integer of the userID if the user was found. Line 17 checks to see if the user check returned false (the user wasn't found), if so we set the error flag, and the$show variable so that we see the 'User Not Found' message. If the user was found, we should show the security question form, and set $securityUser so we know which user to load the question for. In stage 2, we have submitted the security question. The first thing we do here is check to make sure that both a userID and an answer were returned. If either one was omitted, we display the security question form again. If both values were included, we call a function to check the answer against the database. This function returns a simple boolean. If the question was answered correctly, we will show the success page, send the reset email, and set the bad entry count to 0. If the question was answered incorrectly, we display the question page again, and also increment the bad answer counter. The third stage happens after the user has received the password reset email, and has clicked on the link and submitted the password reset form. First, either userID or the security key are empty, we redirect to the homepage. Then we check to make sure that the two passwords match and were not empty. If they don't match, we show the password reset form again. If they do match, though, we show the success message, and call a function to reset the user's password. NOTE: If you are using unencrypted passwords, you can leave out case 3 above. This is the method that saves a new password after it has been reset. Since we can email unencrypted passwords, they don't need to reset it. Now we will add an else block to that same if block from above: This else block determines if$_GET['a'] is set, which means that the user has come to the page via the email reset link. If we've figured out they clicked on the link, we want to check if they email key is valid using checkEmailKey(). If the key is not found or not valid, it returns boolean false, and we display the invalid key message. If the key is valid, however, it returns an array that we can grab the userID out of (so we know which user's password to reset).

NOTE: If you are using unencrypted passwords, you can leave out the elseif block. This deals with displaying the password change form after clicking on the link in the email.

The second if block there checks to see if the user has answered the security question incorrectly more than 3 times. If they have, we set the lockout variable to true, and set the lockout time, which is the time the lockout went into effect. Notice that if the lockout time is already set, we will use that variable, but if not, we will generate a new time.

Setting a speedlimit is important for security. Mostly it will serve to deter people from just sitting there and trying to guess the answer to security questions. It can also help to prevent bots from submitting the form over and over. The main drawback of this method is that a user can just delete the PHPSESS_ID cookie, and then they can start over. If you want a very secure solution, you could add a 'lockout' flag to the users table, and when they have answered wrong 3 times, set that flag to true. Then you can write some code to lockout if they try to access the account.

### Step 4: Function file

Now let's create the file assets/php/functions.php to hold all of the functions we were referencing earlier. Now let's go through them a few at a time:

NOTE: In the sample files, the encrypted example uses the table 'users_enc', while the unencrypted example uses the table 'users' so the table name will change in your functions depending on which version you use.

At the top of the file, we want to define PW_SALT, which is the salt we will use to encrypt passwords throughout. The first function takes the username and password values from the $_POST and sees if a matching user exists in the database. The first thing we need to do is make$mySQL global so we can access the database. We also create a generic error array that we will return if no user was found. Then we check to see if there is a value for the email. If there was, we build a mySQLi prepared statement. The great thing about using prepared statements is the security, especially when dealing with user input, the overall process is similar to using sprintf().

Notice that we call the ->prepare() method with our SQL query as the parameter, and use question marks where our variables would go. Also note that we don't put any form of quotes around the question marks (even for string).This is what allows us to parameterize a statement. mySQLi::prepare() will return true if the statement was created successfully. If it was created, we need to bind parameters to our query using mySQLi::bind_param(). This function accepts at least two arguments. The first is a string of letters that represent the datatypes of the data to be bound. The rest of the arguments are the variables that you want to be in each parameter. In our case, we are using 's' (for a string), and $email, since that variable has the email address. Note that the order is important, so the first question mark in your query corresponds to the first letter in the format string, and the first variable. Then we need to call ->execute() and ->store_result(). These two methods run the prepared query and store it in memory until freed. Then we check for the number of rows returned to make sure that the user was found. ->bind_result() is very similar to ->bind_param() except in the opposite direction. It allows us to take a returned value from a result, and map it to a local variable. Variables are assigned based on their order in the result set. The local variable is not actually written until we call ->fetch(), after we call ->fetch(), the variable$userID will hold the variable from the database. Then we use ->close() on the query to free the result. Lastly, we will return an array that holds the boolean value for success and an integer for the userID.

getSecurityQuestion() accepts the id of a user, and returns their security question as a string. Once again, we make $mySQL global. Then we create an array of the 6 different security questions possible. Using a method similar to the above, we see which security question the user has selected, then we return that index from the array. checkSecAnswer() accepts a userID and a string answer and checks the database to see if it is correct. Note that we are converting the passed answer and the database answer to lowercase to increase chances of a match (optional for you). This is an excellent example of a multi parameter prepared statement. Notice the order of the arguments in the bind_param() method. This function will return true if it finds a record where the userID and the answer match what is passed. It will return false otherwise. This function takes care of sending the email to reset a password. We start out by making a SQL query to get the username and email address of the passed user. After binding the parameters and the result, we close the query. For security reasons, we only want the link we generate to be good for 3 days. To do this, we create a new date that is 3 days in the future. Them, using several of the values that are passed in, the date, a random number and our salt, we generate an MD5 hash that will be our security key. Because of the every changing future date and random number, we should be assured a completely random key every time. Then we build a SQL query to insert it into the database. After executing the query, we make the link that will be sent in the email. We want to include 'a=recover' and our key, and userID which we base64_encode() and urlencode() to make it not readable. Once the link is generated, we make the rest of our email text and send it. NOTE: If you are using the unencrypted example, you want to change your message text to instead print out the variable$pword instead of a link to reset the password.

NOTE: If you are developing this on a local server, it is unlikely that you have smtp installed, so you will be unable to send mail. This is why we are using the @mail() command (to prevent error messages). Also for that reason, this function returns the string of the message so it can be printed on the screen. Now for the last 3 function:

The first function checks the email key that was passed. We make a date string to represent today's date (so we know if the key is expired). Then our SQL checks to see if any records exist for a key that matches the passed user, security key, and that the expiration date is after right now. If such a key exists, we return an array with the userID in it, or false otherwise.

The function updateUserPassword() takes care of changing the actual password in the database. First we verify the email key against the passed in user and key info to make sure things are safe. We generate a new password by combining the passed password and the previously defined salt, and then running md5() on that string. Then we perform our update. Once that is completed, we delete the record of the recovery key from the database so it can't be used again.

NOTE: The function updateUserPassword() is not needed in the unencrypted example, since we are not changing their password.

getUserName() is simply a utility function to get the username of the assed userID. It uses a basic SQL statement like the others.

### Step 5: Complete the Forgot Password Page

Now that we have the functions we need built, we'll add some more to forgotPass.php. Add this code inside the div with the id "page":

This code will deal with actually displaying all of the interface elements. We start off by examining the $show variable from earlier with switch(). As you may remember from earlier, there are many different values that$show can have, so we must create a case for each one. The first case, "emailForm", is where the user will enter either a username or email that they want to recover the password for. Basically we need two text fields, and a hidden input so we know which step we have submitted. There is also an if ($error == true) block that will display an error message if the$error flag is true.

The second case is "securityForm". This is where the security question will be displayed. You can see that we also have an error message, and we use getSecurityQuestion() to print the security question. Then we have an input for the answer, as well as the step, and the userID.

The message for "userNotFound" is simply a text message that tells the user that the record wasn't found.

"SuccessPage" is shown when the security question has been answered correctly and the email has been sent. Notice that we have printed out the email message at the bottom of this prompt, you would not do this in a production environment, because then you give the person instant access to the security code meant only for the registered user.

Let's continue with this code:

Continuing the switch statement, we have "recoverForm", which displays the form to change the password. We print out the user's name, and an error message if needed. Then we have input fields for a password and confirmation password, the submission step, the userID and the security key from the email. For the key and userID, e are using the special if() syntax to make sure that we are getting the variable from the right place. If the querystring variable is null, we will look in the post variables to make sure that we always have a value here.

Then, "invalidKey" means that the key provided did not exist or was expired. The "speedLimit" message is displayed if the user has been locked out because of too many incorrect answers. Finally, we call ob_flush() to send the page to the browser, and \$mySQL->close() to disconnect from the database.

NOTE: "recoverForm" and "invalidKey" messages from above are not needed for unencrypted passwords.

### Conclusion

In this article, we've looked at adding an extremely useful feature for any membership site. Whether you store your passwords as plain text or encrypted, this tutorial should have helped you add a password recovery function. We looked at the logic behind encrypting passwords, sending emails to recover the passwords, and some security concerns when doing so.

Additionally, we looked at using mysqli to improve our interaction with the database. While this tutorial didn't go that in depth with mysqli, it should leave you a little better off than you were before.

I hope you can use the techniques discussed here to add this extremely useful feature to your next user membership site.


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Article Contents
Article Contents

# Good programs in the RSS model without concavity of a utility function

• In this paper, we obtain results on the good ''choice of techniques'' in the long-run in a model proposed by Robinson, Solow, and Srinivasan. We study this model without an assumption that a utility function which represents the preferences of the planner is concave and establish the existence of good programs.
Mathematics Subject Classification: 49J99, 54E52.

 Citation:


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# Math Help - compound/double angle formulas

1. ## compound/double angle formulas

If x is an acute angle such that tan2x = -24/7, find sinx and cosx.

help? thanks

2. Originally Posted by checkmarks
If x is an acute angle such that tan2x = -24/7, find sinx and cosx.

help? thanks
draw a right-triangle with acute angle 2x and opposite side 24 and adjacent side 7. use Pythagoras' theorem to find the hypotenuse.

then you will know that cos(2x) = (adjacent)/(hypotenuse)

then use the fact that $\cos 2x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$ to find sin(x) and cos(x)

or you could solve.


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                   i want a career at NASA. which branch will be good and which institute? whether a degree in aerospace engg in IITs or IIST will be of help?

4 years ago

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                                        Hi ,
IIT for aerospace is better ..one thing is that .. the placements are good in iits (at least the old ones ) than in IIST ..
also IIST is still in a developing stage.. with only its campus being been developed still but IITs are well established
and the funding for IITs when compared to IIST for research activities will also will be more..

4 years ago

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Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# Solve this following

Question:

At $100^{\circ} \mathrm{C}$, copper $(\mathrm{Cu})$ has $\mathrm{FCC}$ unit cell structure with cell edge length of $x \AA$. What is the approximate density of $\mathrm{Cu}\left(\right.$ in $\mathrm{g} \mathrm{cm}^{-3}$ ) at this temperature ?

[Atomic Mass of Cu $=63.55 \mathrm{u}$ ]

1. $\frac{105}{x^{3}}$

2. $\frac{211}{x^{3}}$

3. $\frac{205}{x^{3}}$

4. $\frac{422}{x^{3}}$

Correct Option: , 4

Solution:

#### Leave a comment

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DrPepperx3 one year ago 6 review questions help please? medal!

1. DrPepperx3

1. What is the value of h? a. 4 b. 8 sqrt 3 c. 16 d. 8 sqrt 2

2. whpalmer4

Okay, you have an isosceles right triangle here, because both of the angles are the same. What does that imply about the associated sides?

3. DrPepperx3

They have 2 equal sides and 2 equal angles?

4. whpalmer4

Right. So what is the length of the other leg of the triangle?

5. DrPepperx3

4 ?

6. whpalmer4

|dw:1393634970228:dw| What is the value of ???

7. DrPepperx3

I'm still not sure .. /:

8. whpalmer4

come on, we said this was an isosceles triangle, which means two equal angles and two equal sides!

9. whpalmer4

Doesn't that mean ??? has to be 8?

10. DrPepperx3

why would it be 8 ?

11. DrPepperx3

sorry I feel stupid >_<

12. whpalmer4

because it is an isosceles triangle. One of the sides is 8. Another one must be 8 as well, and it can't be the hypotenuse, because the hypotenuse is the longest side.

13. whpalmer4

it's an isosceles triangle because it has two equal angles. You see that part, right?

14. DrPepperx3

ohhhhh ok I understand

15. whpalmer4

So we can update our drawing: |dw:1393635393884:dw| If the two sides labeled 8 are $$a=8,b=8$$ respectively, what is the length of the hypotenuse, $$c$$? Use the Pythagorean theorem

16. DrPepperx3

k hold on

17. DrPepperx3

its a^2+b^2=c^2 do I add the 45 degrees together to get c? or is the 8?

18. whpalmer4

The Pythagorean theorem relates only to the lengths of the sides. Angles don't matter (except, of course, that one of them must be 90 degrees for it to be a right triangle).

19. whpalmer4

$a^2+b^2=c^2$$8^2+8^2=c^2$$2*8^2 = c^2$Take the square root of both sides:$\sqrt{2*8^2} = \sqrt{c^2}$Simplify$8\sqrt{2} = c$ If you have a $$45^\circ/45^\circ/90^\circ$$ triangle such as this one, the side lengths are always in the ratio $$1:1:\sqrt{2}$$

20. DrPepperx3

is that the way you would do it for this kind of triangle too ?

21. DrPepperx3

@whpalmer4

22. whpalmer4

23. DrPepperx3

Use the Pythagorean theorem

24. whpalmer4

Well, yes, if you know 2 sides in a right triangle, you can use the PT to find the 3rd side.

25. DrPepperx3

okay ty c:

26. whpalmer4

you're welcome


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Color is often used to display an extra dimension in plots of scientific data. Unfortunately, everyone does not decode color in exactly the same way. This is especially true for those with color vision deficiency, which affects up to 8 percent of the population in its two most common forms. As a result, it has been estimated that the odds of a given plot reaching a reviewer with some form of color vision deficiency in a group of three males is approximately 22%. Hopefully, when we are creating figures, this number alone is compelling enough to always keep these viewers in mind. The truth, however, is that your figures aren‘t only seen by reviewers: they are seen by a much wider group that includes readers of your paper, members of the audience when you present your work, viewers of your lab‘s website, and potentially many others. As your audience grows, your choices in color become more and more important for effectively communicating your work.

Although there are many outstanding tools for creating beautiful plots, practically all of them have default color palettes that can present decoding challenges for individuals with color vision deficiencies. This is an introduction to creating plots and figures using color palettes that are more accessible. For the examples below, I use the excellent ggplot2 library for R. The same ideas and colors can easily be transferred to your particular tool of choice.

## Using Color to Represent Categorical Data

When using color to encode categorical data, such as blood type, gender, or strain of a bacteria, it is important to choose a color palette that has as many easily-differentiable colors as there are categories. The figure below shows one palette that can encode up to 8 values, and simulates how each of its colors is seen by someone with protanopia, deuteranopia, and tritanopia.

With ggplot2, the color palette for categorical data can be set using scale_color_manual (for points, lines, and outlines) and scale_fill_manual (for boxes, bars, and ribbons). The argument to either of these commands is a vector of colors, which can be defined by hex RGB triplet or by name. As an example, let’s take a look at the relationship between the weight and the corresponding price of diamonds in ggplot2’s included diamonds data set. We can use color to indicate the quality of the cut. Note that this data set is quite large, so this scatter plot might not be the most informative way to display these data.

ggplot(diamonds, aes(x=carat, y=price, color=cut)) +
geom_point()

scale_color_manual sets the color of the first category (chosen alphabetically in R unless an ordering is specified) using the first color given, the second category with the second color, and so on. Using the colors from the colorblind-safe palette shown above:

ggplot(diamonds, aes(x=carat, y=price, color=cut)) +
geom_point() +
scale_color_manual(values=c("#000000", "#E69F00", "#56B4E9", "#009E73",
"#F0E442", "#0072B2", "#D55E00", "#CC79A7"))

Otherwise, if you don’t want to have to remember the ordering of your categories, or if you want to apply specific colors to each category, you can manually define the color of each:

ggplot(diamonds, aes(x=carat, y=price, color=cut)) +
geom_point() +
scale_color_manual(values=c("Fair"="#E69F00", "Good"="#56B4E9",
"Very Good"="#0072B2"))

## Redundant Encodings

When describing a figure, it is a common tendency to refer to a specific color. Hopefully, you’re at least now convinced that not everyone sees color the same way, especially when using a standard red, green, blue color palette. It is also very common for figures to be printed in black and white or your printer to be low on magenta ink. To improve legibility when your figures aren’t reproduced exactly as created, consider using redundant encodings. As an example, we can use both shapes and colors to refer to categories:

ggplot(diamonds, aes(x=carat, y=price, color=cut, shape=cut)) +
geom_point() +
scale_color_manual(values=c("#000000", "#E69F00", "#56B4E9", "#009E73",
"#F0E442", "#0072B2", "#D55E00", "#CC79A7"))

The use of redundant encoding can also aid in figure captions, where referring to a category as “the blue squares” is helpful both for those with color vision deficiencies, and for those with printer troubles (all of us?). However, if the data can be represented with symbols equally as well as with colors, this does beg the question that should always be asked: Are colors absolutely necessary?

### Using ColorBrewer Palettes

No discussion on color palettes would be complete without mentioning Cynthia Brewer’s ColorBrewer 2, an excellent source for color palettes that includes both colorblind-safe and print-friendly palettes.

Many graphics packages allow you to easily make use of the ColorBrewer palettes. In ggplot2, this is done with the scale_color_brewer command.

ggplot(diamonds, aes(x=carat, y=price, color=cut)) +
geom_point() +
scale_color_brewer(palette="Dark2")

## Using Color to Represent Continuous Values

When using color to represent continuous values, special care should be taken to ensure not only that colors chosen are differentiable, but also that viewers interpret changes in value of a given magnitude similarly throughout the spectrum. The rainbow color map, which is the default in many graphics packages, does not do this well. Color palettes that use variations, not only in hue, but also in saturation and lightness, can produce more linear changes in perception.

Of course, color gradients can introduce additional problems for viewers with color vision deficiencies when certain areas of the spectrum are included. For these viewers, colors that vary uniformly in lightness, which is how the greyscale palette is made, are most accessible. Again, always ask yourself if the use of color conveys information that could be encoded in another way.

ggplot2 includes a number of functions for making continuous color scales such as scale_color_gradient, scale_color_continuous, and scale_color_grey. To demonstrate, I’ll switch to the mtcars data set, which contains, among other things, fuel economy for 32 cars manufactured in 1973-1974.

# Example borrowed from the geom_tile documentation
ggplot(mtcars, aes(y=factor(cyl), x=mpg)) +
stat_density(aes(fill=..density..), geom="tile", position="identity")

Fortunately, ggplot2 does a nice job in displaying continuous values with color by default. Otherwise, we can use the ColorBrewer package to fetch palettes from ColorBrewer (the “PuBuGn” palette in this case), and apply them using the scale_color_gradientn command:

ggplot(mtcars, aes(y=factor(cyl), x=mpg)) +
stat_density(aes(fill=..density..), geom="tile", position="identity") +


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# A-level Physics/Forces, Fields and Energy/Further dynamics

From last year, you should remember kinematics and dynamics, the branch of physics that relates to the motion of objects. We will now expand on this and have a look at what happens when two objects collide, the concept of momentum, and we will take a closer look at Newton's three laws of motion.

## Momentum

If you have seen collisions involving two objects, you may have noticed that the velocity of one object seems to be passed to the other object. You may also have noticed that heavier objects seem to pass more velocity on to smaller objects, whereas smaller objects seem to pass less velocity to more massive ones.

What is in fact happening is that momentum is being conserved. Momentum is the product of an objects mass and velocity, or ${\displaystyle p=mv}$. This means that, after a collision, an object that is heavier will have a lower velocity than a lighter object in its place, and vice versa. Momentum is conserved for all collisions. The principle of the conservation of momentum states that:

Within a closed system, the total momentum in any specified direction remains constant.

Momentum is a vector quantity and has the units ${\displaystyle kg\,m\,s^{-1}}$ or ${\displaystyle Ns}$ (Newton-seconds) in the SI system.

## Collisions

Since momentum is conserved, the momentum before a collision is equal to the momentum after a collision. You can use this fact to solve problems involving collisions.

{\displaystyle {\begin{alignedat}{2}Before&=After\\m_{1}u_{1}+m_{2}u_{2}&=m_{1}v_{1}+m_{2}v_{2}\\\end{alignedat}}}

For instance, a ball is moving at 3 m/s with mass 3 kg. It hits another ball with mass 1 kg moving at 2 m/s; the two balls collide and the second ball rebounds at 4 m/s. Find the velocity at which ball 1 is moving:

{\displaystyle {\begin{alignedat}{2}Before&=After\\m_{1}u_{1}+m_{2}u_{2}&=m_{1}v_{1}+m_{2}v_{2}\\3\times 3+2\times 1&=3v+1\times 4\\11&=3v+4\\11-4&=3v\\7&=3v\\v&={\frac {7}{3}}\\\end{alignedat}}}

So the velocity at which ball 1 is moving after the collision is 2.3 m/s (7/3)m/s 1

## Newton's laws of motion

### Newton's first law of motion

A body stays at rest or continues to move at constant velocity unless a resultant force acts on it

### Newton's second law of motion

Originally, you learnt this to be:

For an object with constant mass, its acceleration is proportional to the force producing the acceleration, and is in the direction of the force.

However, since you now know that a force changes the rate of change of momentum of an object, we can use a more accurate interpretation of Newton's second law:

The rate of change of momentum of a body is proportional to the resultant force acting on the body and takes place in the direction of the resultant force.

### Newton's third law of motion

If body A exerts a force on body B, then body B will exert a force of the same type that is equal in magnitude and opposite in direction on body A..