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# Subgroups of a cyclic group ## Dependencies: For an infinite-order cyclic group $\langle a \rangle$, distinct subgroups are $\langle a^k \rangle$ for all $k \in \mathbb{Z}$. For a cyclic group $\langle a \rangle$ of order $n$, distinct subgroups are $\langle a^k \rangle$ for all divisors $k$ of $n$. ## Proof ### Subgroups are of the form $\langle a^k \rangle$ Let $G = \{a^{i_1}, a^{i_2}, ...\}$ be a subgroup of $\langle a \rangle$. Let $g = \gcd(i_1, i_2, ...) = \sum_j r_j i_j$. Let $i_j = g s_j$. $$\forall j, a^{i_j} = a^{g s_j} = (a^g)^{s_j} \in \langle a^g \rangle \implies G \subseteq \langle a^g \rangle$$ $$a^g = a^{\sum_j r_j i_j} = \prod_j (a^{i_j})^{r_j} \in G \implies \langle a^g \rangle \subseteq G$$ Therefore, $G = \langle a^g \rangle$, which means $G$ is of the form $\langle a^k \rangle$. ### Identifying distinct subgroups of $\langle a \rangle$ Since $\langle a^k \rangle$ is a group and a subset of $\langle a \rangle$, it is a subgroup. For infinite groups, all $a^k$ are distinct. Therfore, distinct subgroups are $\langle a^k \rangle$ for all k. For finite groups of order $n$, $a^{k} = a^{k'}$ when $k \equiv k' \pmod{n}$. Let $g = \gcd(k, n) = rk + pn$ and $k = sg$. $$a^k = a^{sg} = (a^g)^s \in \langle a^g \rangle \implies \langle a^k \rangle \subseteq \langle a^g \rangle$$ $$a^g = a^{rk + pn} = (a^k)^r \in \langle a^k \rangle \implies \langle a^g \rangle \subseteq \langle a^k \rangle$$ Therefore, $\langle a^k \rangle = \langle a^g \rangle$. Let $g_1 = \gcd(k_1, n)$ and $g_2 = \gcd(k_2, n)$ where $g_1 \neq g_2$. Since, $1 \leq g_1, g_2 \leq n$, we get $g_1 \neq g_2 \Rightarrow g_1 \not\equiv g_2 \pmod{n} \Rightarrow \langle a^{g_1} \rangle \neq \langle a^{g_2} \rangle$. Therefore, distinct subgroups of $\langle a \rangle$ are $\langle a^k \rangle$ for all $k$ which are divisors of $n$. ## Info: • Depth: 4 • Number of transitive dependencies: 8	{}
Data Classification Example # Package imports import numpy as np import matplotlib.... 25 2018/06 # Data Classification Example # Package imports import numpy as np import matplotlib.pyplot as plt #from testCases_v2 import * import sklearn import sklearn.datasets import sklearn.linear_model def layer_sizes(X, Y): n_x = X.shape[0] # size of input layer, representing the traits we have n_h = 4 # we assume that the hidden layers have four units n_y = Y.shape[0] # size of output layer return (n_x, n_h, n_y) def initialize_parameters(n_x, n_h, n_y): W1 = np.random.randn(n_h,n_x)0.01 #in neural network, the W1 need not to be transposed b1 = np.zeros((n_h,1)) #b is initialized with zeros and pay attention to the dimensions W2 = np.random.randn(n_y,n_h)0.01 b2 = np.random.randn(n_y,1) assert (W1.shape == (n_h, n_x)) assert (b1.shape == (n_h, 1)) assert (W2.shape == (n_y, n_h)) assert (b2.shape == (n_y, 1)) parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2} return parameters def forward_propagation(X, parameters): W1 = parameters["W1"] b1 = parameters["b1"] W2 = parameters["W2"] b2 = parameters["b2"] Z1 = np.dot(W1,X)+b1 A1 = np.tanh(Z1) Z2 = np.dot(W2,A1)+b2 A2 = sigmoid(Z2) assert(A2.shape == (1, X.shape[1])) cache = {"Z1": Z1, "A1": A1, "Z2": Z2, "A2": A2} return A2, cache def compute_cost(A2, Y, parameters): m = Y.shape[1] logprobs = np.multiply(np.log(A2),Y) cost = -np.sum(logprobs) cost = np.squeeze(cost) assert(isinstance(cost, float)) return cost def backward_propagation(parameters, cache, X, Y): #the input parameters is just the result of the initialize function #in many places we need to first get the number of the example trained m = X.shape[1] W1 = parameters["W1"] W2 = parameters["W2"] A1 = cache["A1"] A2 = cache["A2"] #The deduction of the six formulas is very important dZ2 = A2 - Y dW2 = np.dot(dZ2,A1.T)/m db2 = np.sum(dZ2,axis = 1,keepdims = True)/m dZ1 = np.dot(W2.T,dZ2)(1-np.power(A1,2)) dW1 = np.dot(dZ1,X.T)/m db1 = np.sum(dZ1,axis = 1,keepdims = True)/m "db1": db1, "dW2": dW2, "db2": db2} def update_parameters(parameters, grads, learning_rate = 1.2): W1 = parameters["W1"] b1 = parameters["b1"] W2 = parameters["W2"] b2 = parameters["b2"] W1 = W1 - learning_ratedW1 b1 = b1 - learning_ratedb1 W2 = W2 - learning_ratedW2 b2 = b2 - learning_rate*db2 parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2} return parameters def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False): np.random.seed(3) n_x = layer_sizes(X, Y)[0] n_y = layer_sizes(X, Y)[2] parameters = initialize_parameters(n_x,n_h,n_y) W1 = parameters["W1"] b1 = parameters["b1"] W2 = parameters["W2"] b2 = parameters["b2"] for i in range(0, num_iterations): # Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache". A2, cache = forward_propagation(X,parameters) # Cost function. Inputs: "A2, Y, parameters". Outputs: "cost". cost = compute_cost(A2,Y,parameters) # Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads". # Print the cost every 1000 iterations if print_cost and i % 1000 == 0: print ("Cost after iteration %i: %f" %(i, cost)) return parameters def predict(parameters, X): A2, cache = forward_propagation(X,parameters) predictions = (A2 > 0.5) return predictions ### Reflection: Actually logistic regression can be seen as a one-layer neural network. In logistic regression, there is only one activation function and there is no hidden layer. The methods used in logistic regression is applicable in the neural network with several hidden layers. Compared with the code in last article, we can find that the propagation function is separated into forward_propagation and backward_propagation and the optimize function is separated into the update_parameters function and the loop is placed in the model function. It has to be pointed out that in this case the Y contains the labels and the X contains the features, which is different from last case. The two propagation functions have strong connections. The result of the back propagation helps the forward one to predict and the result of the forward propagation helps the back one compute the grads. We need have a overview of the problem and then split it into many small pieces and finally merge the regional solutions into one model function. Dictionary is often used to transmit the parameters between functions. Assertions are important to keep the dimensions, avoiding many subtle bugs. Through the practice we find that with the increase of the units in hidden layers, the accuracy will first increase but then decreases. ### Reference: the deep learning course provided by Andrew Ng on Coursera Last modification：March 13th, 2019 at 07:06 pm If you think my article is useful to you, please feel free to appreciate	{}
## Abstract I will discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. These require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent up to the action of a compact group of symmetries, and that every state be the marginal of a bipartite state perfectly correlating two measurements. This much yields a mathematical representation of measurements, states and symmetries that is already very suggestive of quantum mechanics. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the theory's state space to be that of a formally real Jordan algebra ## Details Talk Number PIRSA:09080015 Speaker Profile Alexander Wilce	{}
# 24 Review There's this idea in computer science wherein the maximum theoretical speedup that can be acquired with an arbitrary number of processors is related to the percentage of the program which can be parallelized. If we have two segments of code that take the same amount of time to execute with one CPU core in which the first segment can't be parallelized at all and the second segment is perfectly parallelizable, we can only run the program twice as fast, no matter how many CPU cores we have. There's a similar idea in economics. It seems like the most powerful and civilizationally relevant feature controlling the medium to long term change in the price of goods is the extent to which the production of that good can be decoupled from the expenditure of man-hours. Some economic activity isn't "parallelizable" using current technology- we can't practically make that activity much faster without building powerful substitutes for humans, technology which is (for now) mostly out of our reach... For example, it turns out that moving stuff over land is not easy to decouple from human labor without self-driving cars. There are two methods of overland transportation worth noting here: Cars and Trains. Due to the nature of our road infrastructure, there's a pretty clear upper bound on how efficient car-based transportation of goods can get. There are legal limits on the allowed speed and size of vehicles, so without self-driving tech, we can't change how many man-hours need to be spent per cubic meter per kilometer. Train-based transportation has its own problems which limit its ability to dominate overland transportation. Namely, our train transportation network is incomplete in that goods must still be ferried to their final destination by cars, so overland transportation is only some % "parallelizable". Using this we can predict that overseas transportation would be really efficient compared to overland transportation. I suspect that Nautical miles per cargo container per hour per person can be pushed extremely high using current technology, laws, port and canal infrastructure, etc. And indeed this is true: Be warned that overland freight for even short distances can often be almost as much as ocean freight for thousands of miles (I recently paid $2000 for a 20′ container to be shipped from Shanghai to Los Angeles and then$1100 for it to be shipped 30 miles from Los Angeles to San Moreno). Our ability to recursively reinvest our production is most strongly limited by these required industries which are mostly bottlenecked by the number of humans and how long they're willing to work, neither of which are easy to manipulate and neither brings the sort of powerful prosperity that characterizes modernity. EDIT: Someone commented that other factors are at play that makes cargo ships efficient. I do not disagree- this was just an example of the sort of weak estimate which could be made using this idea. I am interested in determining how important this effect is in the case of cargo ships, so I will do a short analysis. Compare to semi-trucks which can carry roughly 40,000 kg of material. We will say these trucks move at 100km/h. The wage of a semi-truck driver in the U.S. is roughly $20, so combing we have, so it costs$1 to move 200,000kg one kilometer- or rather, that is the wage component. The distance between Shanghai and Los Angeles is ~10,000km, and the limit weight of a 20ft shipping container is about 80,000kg. Assuming wage costs for ship crew are negligible (this should cancel out with previous generous estimates), we have 800,000,000kg * km for this trip for this container. If the same wage were required per kg per km as in the semi-truck case (note much is ignored here for the sake of approximation), this would cost an additional $4000. For comparison, according to this site, the cost of shipping 80,000kg from Shanghai to LA is$14000. # 24 New Comment Humans also bottleneck the maritime side of cargo shipments via artificial scarcity in the form of cartels and monopolies. The referred \$2k shipments could have costed even less, but there's rent capture in it driving final transportation prices higher than they could be, and payments to on the ground operators lower than those, too, could be, the resulting spread going into the hands of the monopolists who successfully work around legal impositions from as many jurisdictions as possible. I think part of the reason overseas transportation is so cheap is energy use - ships use far less fuel per cubic meter per kilometer than any other form of transport. I would expect fuel efficiency to be related to the size and complexity of the engine. Producing some amount of force is going to require the same amount of fuel assuming energy loss due to resistance/friction is the same, and the engine is the same. If true, we could e.g. have absurdly large trains on lots of rails? I would expect energy loss due to rubbing on rails and changing elevation to be similar to energy loss due to rubbing on water. As you double each dimension, capacity octuples, drag quadruples, but rolling resistance octuples. Ships only have drag from the water, but trains also have rolling resistance from the tracks. This means trains don't get significantly more efficient as they grow larger, but ships do. Interesting, thank you. Is the quadrupling of drag and octupling of rolling resistance related to the assumption that drag is proportional to the surface area of the side on which the drag is produced, and that rolling resistance is proportional to weight? Either way, cost would still decrease due to larger and more complex engines, as rolling resistance per kg would not change. Of course, railway sizes are fixed, so there is little to be done. I was just speculating where the relative efficiency of cargo ships comes from. I made an edit at the end of the post which contains a very rough approximation of how large savings on wages are in the case of cargo container ships. Is the quadrupling of drag and octupling of rolling resistance related to the assumption that drag is proportional to the surface area of the side on which the drag is produced, and that rolling resistance is proportional to weight? Yes. It's a little more complex than this since rolling resistance is irrespective of speed, whereas drag increases with speed. But if you're aiming for efficiency you'll go at low speeds, so we can hold speed fixed and see what happens as we scale. Either way, cost would still decrease due to larger and more complex engines, as rolling resistance per kg would not change. Even if the engine is 100% efficient, you still lose energy to drag and rolling resistance, so sooner or later increased engine efficiency doesn't buy you very much. I was just speculating where the relative efficiency of cargo ships comes from. I think it's the fact they're so much larger, and so drag/capacity is very low. Indeed. Travelling by boat/ship, and transporting things by boat/ship, is 'Lindy', as are bicycles. If you’re not already aware, you would like Henry George’s Progress and Poverty in how it deals with a framework for thinking about Labor and Capital. Can you expand on what you mean by "universal economic bottleneck"? Given two bottlenecks - say humans and capital investment in a sector - are you saying humans are: - always slower? not true imo, sometimes all the humans already interested are not rich, so they need to wait for the rich folks to come invest in physical resources - more necessary? not true imo, both are just as necessary, project is dead in the water without capital or physical resources I agree with your statement is as broad as "human innovation is responsible for exponential tech" - but then I'm not sure if that's new info. Ofcourse apes would not grow at the same rate we do. For everyone to become richer without working harder, we must develop technologies that allow more work to be done per man-hour. Aside from working out distribution inefficiencies and similar, this is the unique limit on prosperity. This is what I mean by "humans are the universal bottleneck"- we only have so many man-hours, and any growth is going to be of the form "With the same amount of hours, we do more". Some segments of the economy have not had as much growth in the above department. For example, houses are assembled manually- all major parts must be done by hand, many with the assistance of only hand-held tools. Because we require shelter and the choice way of getting that shelter is owning a manually built house, this is a drag on the economy. Domains such as this which have not been as revolutionized by automation have scaling costs due to how directly they are pegged to the price of labor- in particular, houses of constant size cost the same general fraction of our income (if not more) despite real GDP per capita having grown significantly over time, because richer workers demand more pay. Why aren't houses cheap? This is one of many reasons, and doing some quick googling on how many man-hours are required to build a house, the main factor in many areas of the united states. Is this way of thinking "new"? Surely not to humanity, but hopefully to the reader. Aside from working out distribution inefficiencies and similar, this is the unique limit on prosperity. Why is this limit unique? Why can't we be working on "distribution inefficiencies and similar" for the next 100 years? Maybe I'll ask this, does your statement regarding universal bottleneck apply explicitly to humans? Or does it also apply to apes and bacteria and AI? Cause some of it seems tautological - doing more total work means finding ways to do more work per unit time. Why is this limit unique? Why can't we be working on "distribution inefficiencies and similar" for the next 100 years? In the case of real GDP per capita per hour worked, this limit is exactly unique- "distribution inefficiencies and similar" doesn't apply. Indeed, this is tautologically true as you say. Think about what it would look like for an increase in real GDP per capita per hour worked to not have the form of "Something allowed for more work to be done per hour per person". It wouldn't look like anything- that doesn't make any sense. I would completely ignore my comment on "distribution inefficiencies and similar" until you know what I mean by this. To explain my comment, real GDP per capita per hour worked is not the same as the nebulous "prosperity" I was referring to, which also contains some level of preference for how material goods are distributed. Maybe I'll ask this, does your statement regarding universal bottleneck apply explicitly to humans? Or does it also apply to apes and bacteria and AI? Because we particularly care about how much work humans do, and how wealthy they are. We do not really care about the work hours or prosperity of bacteria. Economic productivity is measured relative to how much money people have and how much they must work to get it. Just read my previous comment and/or the post again- this would seem to be a really basic sort of confusion that I can't fix for you. Yes, and to go further: the value that humans bring to these economic activities, and on which they are bottlenecked, is almost entirely their mental capabilities. There are a few jobs where people joke about just doing things that a trained monkey could do, but it's only funny because a trained monkey couldn't actually do them, especially when things don't go entirely to plan. There are plenty of jobs that rely on social competence in dealing with other humans, but that's still mostly mental capability. There are also plenty of jobs that couldn't (at least at first) be replaced by smart robots with AGI, but they're not really so relevant to questions such as limits on economic growth. I am not convinced of the statement in the title based upon the argument presented in the text. For one, I expect that very soon, trucks will be self-driving, and even if not, there is not enough generally applicable logic or variety of specific examples to support a claim of universality. Self-driving technology is advancing and will soon(ish) allow us to move cars without humans being directly involved, except in terms of maintenance and management. This will be a major boon because it will partially remove humans from the equation- the bottleneck is partially removed. This has no real bearing on the title statement- I even remark about this in my post. The "universality" here is trivial- here is a copy-paste of part of my response to a similar comment: For everyone to become richer without working harder, we must develop technologies that allow more work to be done per man-hour. Aside from working out distribution inefficiencies and similar, this is the unique limit on prosperity. This is what I mean by "humans are the universal bottleneck"- we only have so many man-hours, so any growth is going to be of the form "With the same amount of hours, we do more". Imagine if every area of economic activity was automated- humans were fully removed. This would look very sci-fi: think of von Neumann probes. In this situation there is no practical limit- the probes will expand and convert our entire light cone. Assuming constant population, per capita wealth would approach 50 billion stars, I guess.	{}
# How to Update a Finite-Width Cellular Automaton How does one instruct CellularAutomaton to wrap at the edges? (I want a 2d finite-width CA, so it evolves on a torus.) I cannot tell from the docs if this is possible. From some experiments and from reading between the lines of the documentation, if the second argument is a vector, then the system set up with periodic boundary conditions. See for example this one: ArrayPlot[CellularAutomaton[30, PadLeft[{1}, 100], 150]] • If you can elaborate a bit more on which documentation you are relying on and how it applies to the 2D (instead of 1d) case, that would help. I am guessing you are referring to this: "a_1,a_2,... explicit list of values a_i, assumed cyclic". I suppose that is suppose to mean that this init is implicitly 1d wrapping at the bounderies). In which case, I suppose, a 2d rectangular array for init should implicitly be 2d wrapping at the boundaries. – Alan Jul 24 '18 at 2:37 • Yes, exactly there. I only found it by searching for "cyclic" and "periodic" (with ctrl + f). I am not sure, but the 2D case should work exactly as you have just anticipated. – Henrik Schumacher Jul 24 '18 at 7:03 An alternative way to specify the initial condition: ArrayPlot[CellularAutomaton[30, SparseArray[1 -> 1, 50], 50]] Explicit initial conditions are assumed cyclic	{}
Coulombs Law 1. Sep 12, 2004 Cyrus Im reading Coulmbs law and I have stumbeled upon a problem. The numerator is the product of the two charges. But according to inductance, I can create a force of attraction between two objects if only one object has a charge. So charge two would be zero? And the force would be zero!? That does not make sense, would the proper thing be to break the charge equally in half between the two objects? or should I consider the force to be the same for each object as the charged one? 2. Sep 12, 2004 Tide Inductance will give you a distribution of charge even if the net charge is zero. That means, for example, that some of the redistributed charge of one sign will be closer to the external charge than the oppositely signed charge - so attractive and repulsive forces won't exactly cancel. 3. Sep 12, 2004 Cyrus So then for q1 * q2, I would effectively use the same number as the charged object and assume the uncharged object has picked up the same charge value? 4. Sep 12, 2004 Tide Yes, but you would have to do that for every element of charge dq throughout the volume of the object. Incidentally, finding the actual charge distribution is not trivial except in very special cases! 5. Sep 13, 2004 Cyrus Hey Tide, whats all this junk in my book. i think there trying to explain something to me, imagine that, but i dont get it. "we conclude that in electrostatics the electric field at every point within the material of a conductor must be zero. (Note that we are not saying that the field is necessarily zero in a hole inside a conductor)." what in the world do they mean by that?	{}
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE Vol. 60, 2 (1991) pp. 219-224 HILBERT-SPACE-VALUED MEASURES ON BOOLEAN ALGEBRAS (EXTENSIONS) J. HAMHALTER and P. PTAK Abstract. We prove that if $B_1$ is a Boolean subalgebra of $B_2$ and if $m\: B_1\to H$ is a bounded finitely additive measure, where $H$ is a Hilbert space, then $m$ admits an extension over $B_2$. This result generalizes the well-known result for real-valued measures (see e.g. Ref. 1). Then we consider orthogonal measures as a generalization of two-valued measures. We show that the latter result remains valid for $\dim H<\infty$. If $\dim H=\infty$, we are only able to prove a weaker result: If $B_1$ is a Boolean subalgebra of $B_2$ and $m\: B_1 \to H$ is an orthogonal measure, then we can find a Hilbert space $K$ such that $H\subset K$ and such that there is an orthogonal measure $k\: B_2\to K$ with $k/B_1=m$. AMS subject classification. 06E99, 28B05 Keywords	{}
# JEE Main 2017 Maths Paper With Solutions (April 2) JEE Main 2017 Paper with Solutions Maths - Offline Exam will help the students to get familiar with the difficulty level of the questions, which can be asked in the upcoming JEE Main examination. The students appearing for the examination are required to solve the question paper to check their preparedness. The solutions uploaded at BYJU’S will enable the students to identify the important topics. Practising these questions will help them revisit and update their revision strategies. ### April 2nd - Maths Question 1: The function f : R → [(-1/2), (1/2)] is defined as f(x) = [x]/[1 + x2], is: 1. 1) injective but not surjective. 2. 2) surjective but not injective. 3. 3) neither injective nor surjective. 4. 4) invertible. Solution: f(x) = [x]/[1 + x2] $$f'(x)=\frac{(1+x^{2})1-x2x}{(1+x^{2})^{2}}\\ =\frac{1-x^{2}}{(1+x^{2})^{2}}$$ ∴ Not injective. y = (x / (1 + x2)) yx2 - x + y = 0 For y ≠ 0 D = 1 - 4y2 ≥ 0 y [(-1 / 2), (1 / 2)] - {0} For, y = 0 ⇒ x = 0 ∴Part of range ∴Range = [(-1 / 2), (1 / 2)] ∴Surjective but not injective. Question 2: If, for a positive integer n, the quadratic equation,has two consecutive integral solutions, then n is equal to: 1. 1) 9 2. 2) 10 3. 3) 11 4. 4) 12 Solution: Rearranging equation, we get nx2 + {1 + 3 + 5 + ….. + (2n - 1)x + {1.2 + 2.3 + ….. + (n - 1)n} = 10n nx2 + n2x + [(n - 1) n (n + 1)] / 3 = 10n x2 + nx + [n2 - 31] / 3 = 0 Given difference of roots = 1 \|ɑ - β\| = 1 D = 1 n2 - (4 / 3) (n2 - 31) = 1 n = 11 Question 3: Let ω be a complex number such that 2ω + 1 = z where z = √-3. If then k is equal to, 1. 1) z 2. 2) −1 3. 3) 1 4. 4) − z Solution: 2ω + 1 = z, z = √3i ω = (-1 + √3i) / 2 [Cube root of unity] = 3 (ω2 - ω4) = 3 [{(-1 - √3i) / 2} - {(-1 + √3i) / 2}] = - 3√3i = -3z k = -z Question 4: Ifthen adj (3A2 + 12A) is equal to : Solution: = (2 – 2λ – λ + λ2) – 12 f (λ) = λ2 - 3λ - 10 ∵ A satisfies f (λ). A2 - 3A - 10I = 0 A2 - 3A = 10I 3A2- 9A = 30I 3A2 + 12A = 30I + 21A Question 5: If S is the set of distinct values of ‘b’ for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is : 1. 1) an infinite set 2. 2) a finite set containing two or more elements 3. 3) a singleton 4. 4) an empty set Solution: ⇒ - (1 - a2) = 0 ⇒ a = 1 For a = 1 Eq. (1) & (2) are identical i.e.,x + y + z = 1. To have no solution with x + by + z = 0. b = 1 Question 6: A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is : 1. 1) 468 2. 2) 469 3. 3) 484 4. 4) 485 Solution: Required number of ways = 4C3 . 4C3 + (4C2 . 3C1)2 + (4C1 . 3C2)2 + (3C3)2 = 16 + 324 + 144 + 1 = 485 Question 7: The value of (21C1 - 10C1) + (21C2 - 10C2) + (21C3 - 10C3) + (21C4 - 10C4) + … (21C10 - 10C10) is 1. 1) 221 − 210 2. 2) 220 − 29 3. 3) 220 − 210 4. 4) 221 − 211 Solution: 21C1 + 21C2 + …… 21C10 = (1 / 2) (21C0 + 21C1 + ….. 21C21) - 1 = 220 - 1 10C1 + 10C2 + …. 10C10 = 210 – 1 Required sum = (220 – 1) – (210 – 1) = 220 – 210 Question 8: For any three positive real numbers a, b and c, 9 (25a2 + b2) + 25 (c2 − 3ac) = 15b (3a + c). Then : 1. 1) b, c and a are in A.P. 2. 2) a, b and c are in A.P. 3. 3) a, b and c are in G.P. 4. 4) b, c and a are in G.P. Solution: 9 (25a2 + b2) + 25 (c2 − 3ac) = 15b (3a + c) ⇒ (15a)2 + (3b)2 + (5c)2 - 45b - 15b - 75ac = 0 (15a - 3b)2 + (3b - 5c)2 + (15a - 5c)2 = 0 It is possible when 15a - 3b = 0 and 3b - 5c = 0 and 15a - 5c = 0 15a = 3b = 5c (a / 1) = (b / 5) = (c / 3) ∴ b, c, a are in A.P. Question 9: Let a, b, c R. If f (x) = ax2 + bx + c is such that a + b + c = 3 and f (x + y) = f (x) + f (y) + xy, ∀ x, y R, then $$\sum_{n=1}^{10}$$ f (n) = is equal to : 1. 1) 165 2. 2) 190 3. 3) 255 4. 4) 330 Solution: As f (x + y) = f (x) + f (y) + xy Given, f (1) = 3 Putting, x = y = 1 ⇒ f (2) = 2 f (1) + 1 = 7 Similarly, x = 1, y = 2, ⇒ f (3) = f (1) + f (2) + 2 = 12 Now, $$\sum_{n=1}^{10}$$ f (n) = f (1) + f (2) + …. f (10) = 3 + 7 + 12 + 18 + ... = S (let) Now, Sn = 3 + 7 + 12 + 18 … + tn Again, Sn = 3 + 7 + 12 + 18 … + tn-1 + tn We get, tn = 3 + 4 + 5 + …. n terms = [n (n + 5)] / 2 Question 10: $$\lim_{x\rightarrow \pi/2}\frac{cotx-cosx}{(\pi-2x)^{3}}$$ equals: 1. 1) 1 / 16 2. 2) 1 / 8 3. 3) 1 / 4 4. 4) 1 / 24 Solution: $$\lim_{x\rightarrow \pi/2}\frac{cotx-cosx}{(\pi-2x)^{3}}\\ \text \ Put \ x=\frac{\pi}{2}-x=t\\ =\lim_{t=0}\frac{tant-sint}{8t^{3}}\\ =\lim_{t=0}\frac{sint2sin^{2}\frac{t}{2}}{8t^{3}}\\ =\frac{1}{16}$$ Question 11: If for x ∈ (0, (1 / 4)), the derivative of tan-1 (6x√x / (1 - 9x3)) is √x . g(x), then g(x) equals: 1. 1) (3x√x / (1 - 9x3)) 2. 2) (3x/(1 - 9x3)) 3. 3) (3/(1 + 9x3)) 4. 4) (9/(1 + 9x3)) Solution: f (x) = 2 tan-1 (3x√x) for x ∈ (0, (1/4)) f ‘ (x) = (9√x/(1 + 9x3)) g (x) = (9/(1 + 9x3)) Question 12: The normal to the curve y (x − 2) (x − 3) = x + 6 at the point where the curve intersects the y-axis passes through the point : 1. 1) (1/2, 1/2) 2. 2) (1/2, -1/3) 3. 3) (1/2, 1/3) 4. 4) (-1/2, -1/2) Solution: y (x - 2) (x - 3) = x + 6 At y-axis, x = 0, y = 1 Now, on differentiation, (dy / dx) (x - 2) (x - 3) + y (2x - 5) = 1 (dy / dx) (6) + 1 (-5) = 1 (dy / dx) = 6 / 6 = 1 Now slope of normal = –1 Equation of normal y – 1 = –1 (x – 0) y + x – 1 = 0 ... (i) Line (i) passes through (1 / 2, 1 / 2). Question 13: Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is : 1. 1) 10 2. 2) 25 3. 3) 30 4. 4) 12.5 Solution: 2r + θr = 20 ---- (i) A = Area = (θ / 2π) * (πr2) = (θr2 / 2) ---- (ii) A = (r2 / 2) ([20 - 2r] / r) A = [(20r - 2r2) / 2] = 10r - r2 A to be maximum dA / dr = 10 - 2r = 0 ⇒ r = 5 d2A / dr2 = -2 < 0 Hence for r = 5, A is maximum Now, 10 + (θ * 5) = 20 ⇒ θ = 2 (radian) Area = (2 / 2π) * (π) (5)2 = 25 sq m Question 14: Let In = ∫tann x dx, (n > 1). If I4 + I6 = a tan5 x + bx5 + C, where C is a constant of integration, then the ordered pair (a, b) is equal to : 1. 1) (1/5, 0) 2. 2) (1/5, -1) 3. 3) (-1/5, 0) 4. 4) (-1/5, 1) Solution: In = ∫tann x dx, (n > 1) I4 + I6 = ∫(tan4 x + tan6 x) dx = ∫tan4 x sec2 x dx Let tan x = t sec2 x dx = dt = ∫t4 dt = (t5/5) + C = (1/5) tan5 x + C a = (1/5), b = 0 Question 15: The integral $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{dx}{1+cosx}$$ is equal to: 1. 1) 2 2. 2) 4 3. 3) −1 4. 4) −2 Solution: Question 16: The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x2 ≤ 4y and y ≤ 1 + x} is : 1. 1) 3 / 2 2. 2) 7 / 3 3. 3) 5 / 2 4. 4) 59 / 12 Solution: = $$\int_{0}^{1}(\sqrt{x}+1-\frac{x^{2}}{4})dx+\int_{1}^{2}((3-x)-\frac{x^{2}}{4})dx\\ =\frac{5}{2}$$ sq units Question 17: If (2 + sinx) (dy/dx) + (y + 1) cosx = 0 and y(0) = 1, then y(π/2) is equal to: 1. 1) -2/3 2. 2) -1/3 3. 3) 4/3 4. 4) 1/3 Solution: (2 + sinx) (dy/dx) + (y + 1) cos x = 0 y(0) = 1, y(π / 2) = ? (1/(y + 1)) dy + (cos x/[2 + sinx]) dx = 0 ln \|y + 1\| + ln (2 + sinx) = ln C (y + 1) (2 + sinx) = C Put x = 0, y = 1 (1 + 1) . 2 = C ⇒ C = 4 Now, (y + 1) (2 + sinx) = 4 (y + 1) = 4 / 3 y = (4 / 3) - 1 = 1 / 3 Question 18: Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point : 1. 1) (1, 3/4) 2. 2) (1, -3/4) 3. 3) (2, 1/2) 4. 4) (2, -1/2) Solution: (k2 - 7k + 10) + 4k2 + 20k = ± 56 5k2 + 13k - 46 = 0 5k2 + 13k + 66 = 0 5k2 + 13k - 46 = 0 k = [-13 ± √169 + 920]/10 = 2, – 4.6 [reject] For k = 2 x = 2 ...(i) Also, equation of BE, (y - 2) = (1/2) (x - 5) 2y - 4 = x - 5 x - 2y - 1 = 0 ...(ii) Solving (i) & (ii), 2y = 1 y = 1 / 2 Orthocentre is (2, 1/2) Question 19: The radius of a circle, having minimum area, which touches the curve y = 4 − x2 and the lines, y = \|x\| is : 1. 1) 2 (√2 - 1) 2. 2) 4 (√2 - 1) 3. 3) 4 (√2 + 1) 4. 4) 2 (√2 + 1) Solution: x2 = - (y - 4) Let a point on the parabola P [(t/2) , (4 - (t2/4))] Equation of normal at P is y + (t2 / 4) - 4 = (1 / t) (x - [t / 2]) x - ty - (t3 / 4) + (7 / 2)t = 0 It passes through centre of circle, say (0, k) -tk - (t3 / 4) + (7 / 2)t = 0 ---- (i) t = 0, t2 = 14 - 4k Radius = r = \|(0 - k) / √2\| (Length of perpendicular from (0, k) to y = x) r = k / √2 Equation of circle is x2 + (y - k)2 = k2 / 2 It passes through point P (t2 / 4) + [4 - (t2 / 4) - k]2 = k2 / 2 t4 + t2 [8k - 28] + 8k2 - 128k + 256 = 0 ---- (ii) For t = 0 ⇒ k2 - 16k + 32 = 0 k = 8 ± 4√2 r = k / √2 = 4 (√2 - 1) (discarding 4 (√2 + 1)) ----- (iii) For t = ± √(14 - 4k) (14 - 4k)2 + (14 - 4k) (8k - 28) + 8k2 - 128k + 256 = 0 2k2 + 4k - 15 = 0 k = [-2 ± √34] /[2] r = k / √2 = [√17 - √2] / 2 (Ignoring negative value of r) ..(iv) From (iii) & (iv), rmin = [√17 - √2] / 2 But from options, 4 (√2 - 1) Question 20: The eccentricity of an ellipse whose centre is at the origin is 1 / 2. If one of its directrices is x = −4, then the equation of the normal to it at (1, (3 / 2)) is: 1. 1) 4x − 2y = 1 2. 2) 4x + 2y = 7 3. 3) x + 2y = 4 4. 4) 2y − x = 2 Solution: x = –4 e = 1 / 2 (-a / e) = -4 -a = -4 * e a = 2 Now, b2 = a2 (1 - e2) = 3 Equation to ellipse (x2/4) + (y2/3) = 1 Equation of normal is Question 21: A hyperbola passes through the point P (√2, √3) and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point : 1. 1) (2√2 , 3√3) 2. 2) (√3, √2) 3. 3) (− √2 , –√3) 4. 4) (3√2 , 2√3) Solution: [x2 / a2] - [y2 / b2] = 1 a2 + b2 = 4 and [2 / a2] - [3 / b2] = 1 (2 / [4 - b2]) - (3 / [b2]) = 1 b2 = 3 a2 = 1 x2 - (y2 / 3) = 1 ∴ Tangent at P (√2 , √3) is √(2)x - (y / √3)) = 1 Clearly it passes through (2√2 , 3√3). Question 22: The distance of the point (1, 3, −7) from the plane passing through the point (1, −1, −1), having normal perpendicular to both the lines $$\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$$ and $$\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$$ is: 1. 1) 10 / √83 2. 2) 5 / √83 3. 3) 10 / √74 4. 4) 20 / √74 Solution: Let the plane be a (x - 1) + b (y + 1) + c (z + 1) = 0. It is perpendicular to the given lines a – 2b + 3c = 0 2a – b – c = 0 Solving, a : b : c = 5 : 7 : 3 The plane is 5x + 7y + 3z + 5 = 0 Distance of (1, 3, –7) from this plane = 10 / √83 Question 23: If the image of the point P (1, −2, 3) in the plane, 2x + 3y − 4z + 22 = 0 measured parallel to the line, (x / 1) = (y / 4) = (z / 5) is Q, then PQ is equal to : 1. 1) 2√42 2. 2) √42 3. 3) 6√5 4. 4) 3√5 Solution: Equation of PQ, $$\frac{x-1}{1}=\frac{y+2}{4}=\frac{z-3}{5}$$ Let M be (λ + 1, 4λ - 2, 5λ + 3) As it lies on 2x + 3y – 4z + 22 = 0 λ = 1 For Q, λ = 2 Distance PQ = 2 √12 + 42 + 52 = 2√42 Question 24: Let a = 2i + j - 2k and b = i + j. Let c be a vector such that \|c - a\| = 3, \|(a x b) x c\| = 3 and the angle between c and a × b be 30o. Then a ⋅ c is equal to : 1. 1) 2 2. 2) 5 3. 3) 1 / 8 4. 4) 25 / 8 Solution: Question 25: A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is : 1. 1) 6 2. 2) 4 3. 3) 6 / 25 4. 4) 12 / 5 Solution: n = 10 p (Probability of drawing a green ball) = 15 / 25 p = 3 / 5, q = 2 / 5 var (X) = n.p.q = 10 * (3 / 5) * (2 / 5) = 10 * (6 / 25) = 12 / 5 Question 26: For three events A, B and C, P (Exactly one of A or B occurs) = P (Exactly one of B or C occurs) = P (Exactly one of C or A occurs) = 1 / 4 and P (All the three events occur simultaneously) = 1 / 16. Then the probability that at least one of the events occurs is : 1. 1) 7 / 16 2. 2) 7 / 64 3. 3) 3 / 16 4. 4) 7 / 32 Solution: P (A) + P (B) – P (A ⋂ B) = 1 / 4 P (B) + P (C) – P (B ⋂ C) = 1 / 4 P (C) + P (A) – P (A ⋂ C) = 1 / 4 P (A) + P (B) + P (C) – P (A ⋂ B) – P (B ⋂ C) – P (A ⋂ C = 3 / 8 ∵ P (A ⋂ B ⋂ C) = 1 / 16 ∴ P (A ⋃ B ⋃ C) = (3 / 8) + (1 / 16) = 7 / 16 Question 27: If two different numbers are taken from the set {0, 1, 2, 3, ......, 10}; then the probability that their sum, as well as absolute difference, are both multiples of 4, is : 1. 1) 12 / 55 2. 2) 14 / 45 3. 3) 7 / 55 4. 4) 6 / 55 Solution: Total number of ways = 11C2 = 55 Favourable ways are (0, 4), (0, 8), (4, 8), (2, 6), (2, 10), (6, 10) Probability = 6 / 55 Question 28: If 5(tan2 x − cos2 x) = 2cos2x + 9, then the value of cos 4x is : 1. 1) 1 / 3 2. 2) 2 / 9 3. 3) −7 / 9 4. 4) −3 / 5 Solution: 5(tan2 x − cos2 x) = 2cos2x + 9 5 sec2 x - 5 = 9 cos2 x + 7 Let cos2 x = t (5 / t) = 9t + 12 9t2 + 12t - 5 = 0 t = 1 / 3 as t ≠ – 5 / 3 cos2 x = 1 / 3, cos 2x = 2cos2 x - 1 = -1 / 3 cos 4x = 2cos2 2x - 1 = (2 / 9) - 1 = – 7/ 9 Question 29: Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If ∠BPC = β, then tan β is equal to : 1. 1) 1/4 2. 2) 2/9 3. 3) 4/9 4. 4) 6/7 Solution: tan θ = 1/4 tan (θ + β) = 1/2 $$\frac{\frac{1}{4}+tan\beta}{1-\frac{1}{4}tan\beta}=\frac{1}{2}\\ tan\beta=\frac{2}{9}$$ Question 30: The following statement (p → q) → [(~p → q) → q] is : 1. 1) equivalent to ~p → q 2. 2) equivalent to p → ~q 3. 3) a fallacy 4. 4) a tautology Solution:	{}
Let R be the region in the first quadrant bounded by the graphs of (x^2/ 9) + (y^2 /81)=1 and 3x+y=9, how do you find the area? Mar 9, 2016 $\left(\frac{27}{4}\right) \left(\pi - 2\right)$ areal units.i Explanation: The ellipse and the straight line intersect at B(3, 0) and A(9, 0) The semi-axes of the ellipse are 9 and 3. Its area is $27 \pi$ The area of the $\triangle O A B = \frac{27}{2}$. The area bounded by the line and the ellipse in the first quadrant = area of the ellipse in the quadrant $-$ the areal below the line in the quadrant = $\pi \left(9\right) \left(3\right) \left(\frac{1}{4}\right) - \left(\frac{1}{2}\right) \left(3\right) \left(9\right)$	{}
## Files in this item FilesDescriptionFormat application/pdf 9503187.pdf (2MB) (no description provided)PDF ## Description Title: The representation of numbers as sums of unlike powers Author(s): Ford, Kevin Barry Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: We are concerned with the problem of finding the least s for which every large natural number n admits a representation $n = x\sbsp{2}{2} + x\sbsp{3}{3} + \cdots + x\sbsp{s+1}{s+1}$, where the numbers $x\sb{i}$ are nonnegative integers. K. F. Roth proved in 1948 that one may take s = 50, and this value has subsequently been reduced to s = 17 in a series of papers by K. Thanigasalam, R. C. Vaughan and J. Brudern. We improve this further, showing that s = 14 is admissible. This is accomplished by adapting a new iterative method, developed by Vaughan and T. D. Wooley for use in Waring's problem, for problems involving mixed powers. As with the other papers on the subject, the proof employs the machinery of the Hardy-Littlewood circle method. We also consider the problem of representing integers n in the form $n = x\sbsp{k}{k} + x\sbsp{k+1}{k+1} + \cdots + x\sbsp{k+s-1}{k+s-1}$, and obtain upper bounds on the number of terms required to represent every sufficiently large n in this form both for general k and for the specific case k = 3. The estimate obtained for general k improves an estimate by E. J. Scourfield. It is conjectured that in fact all large n can be written as the sum of a square, a positive cube and a fourth power of integers, and we give some numerical calculations that show that there are still many exceptions greater than 10$\sp{18}$. Issue Date: 1994 Type: Text Language: English URI: http://hdl.handle.net/2142/20414 Rights Information: Copyright 1994 Ford, Kevin Barry Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9503187 OCLC Identifier: (UMI)AAI9503187	{}
fast calculation of sparse gradients and Jacobian matrices in Python ## Project description sparsegrad automatically and efficiently calculates analytical sparse Jacobian of arbitrary numpy vector valued functions. It is designed to be useful for solving large systems of non-linear equations. sparsegrad is memory efficient because it does not use the graph of computation. Arbitrary computations are supported through indexing, matrix multiplication, branching, and custom functions. Taking Jacobian with respect to variable x is done by replacing numerical value of x with sparsegrad seed >>> import numpy as np >>> def f(x): ... return x-x[::-1] >>> x=np.linspace(0,1,3) (0, 0) 1.0 (0, 2) -1.0 (2, 0) -1.0 (2, 2) 1.0 sparsegrad is written in pure Python. For easy installation and best portability, it does not contain extension modules. In realistic problems, it can provide similar or better performance than ADOL-C best case of repeated calculation. This is possible thanks to algorithmic optimizations and optimizations to avoid slow parts of scipy.sparse. sparsegrad relies on numpy and scipy for computations. It is compatible with both Python 2.7 and 3.x. ## Installation pip install sparsegrad It is recommended to run test suite after installing python -c "import sparsegrad; sparsegrad.test()" ## Project details Uploaded source	{}
# How to sample natural numbers, such that the sum is equal to a constant? Say I have $N$ items that are partitioned / clustered and I want to randomly repartition these items, such that the distribution of sizes of the clusters is 'similar' to those that I already have. I'm viewing this (perhaps unhelpfully), as trying to sample $k$ natural numbers, such that $x_i \geq 1$ and $\sum_{i=1}^{k}x_i=N$. $N$ in this context would probably be in the thousands and the distribution of the current cluster sizes would be quite skewed, with a small number of large clusters and a large number of small clusters. I've looked at hypergeometric, binomial, geometric distributions, but none of these seem to quite fit what I'm looking for - I'm guessing that it's more complicated than a simple distribution and some type of Markov-like process would be needed. Anybody have any ideas? - It is unclear what the sum formula has to do with the clustering or even what "clustering" is. Here's one interpretation. Suppose the cluster sizes are $n_1, \ldots, n_k$. An obvious way to generate random partitions with these sizes is to permute all $N$ items and assign the first $n_1$ to the first cluster, the next $n_2$ to the next cluster, and so on. Given its obviousness, I presume you have rejected this as a solution, but why is it not a solution? – whuber Mar 1 '12 at 18:13 Clustering in this case means a partition of the set of items. Of course, I should have emphasised that it was the cluster sizes I am really interested in, not a particular clustering. – Rónán Daly Mar 1 '12 at 19:39 Thanks. In that case there are many possible solutions, depending on how the clusters are generated. If you don't know that, you need to begin by characterizing the distribution of the observed cluster sizes. If you could disclose either of these--data-generation process or distribution of cluster sizes--you will likely get answers that are more appropriate for your data. – whuber Mar 1 '12 at 19:45 I believe what you are looking for is something like the Dirichlet Process [DP] which is a distribution on distributions. It is not an easy concept to understand, but the base measure you will use is the discrete distribution of cluster sizes you started with. The parameter $\alpha$ controls how 'close' to the original distribution your new one is. Since a sample from a DP is a probability distribution (in your case, a discrete one), you can multiply it by $N$ to get cluster sizes. The result won't be integers, but just rounding the numbers should not affect what you're trying to do in a meaningful way. Edit: Somehow, I am more familiar with the Dirichlet Process than the Dirichlet Distribution, which is what you are actually looking for. The DP is an infinite dimensional generalization of the DD. To be more algorithmically precise, consult the subsection talking about random number generation. Your parameters are going to be based on the cluster sizes that you want the random clusterings to look like $\{n_j\}_{j=0}^k$. In other words: $$\alpha = (\alpha_1,...,\alpha_k) = (\frac{\beta n_1}{N},...,\frac{\beta n_k}{N})$$ where $\beta$ is called a concentration parameter and it controls how close to the original distribution of cluster sizes the Dirichlet distribution will be on average. Higher values of $beta$ will mean that the resulting DD will give closer and closer values to the EXACT distribution of sample sizes you started with. So, the algorithm would be: (if you have access to a function that can generate Dirichlet Distribution samples, ignore steps 1 and 2.) 1. Draw independent samples from $y_j = Gamma(\alpha_j,1)$ for each $j = 1..k$. 2. Compute $x_j = y_j/\sum^k_{j=1}y_j$ for each $j$. 3. Then you have the sample $Dirichlet(\alpha_1,...,\alpha_k) = (x_1,...,x_k)$. 4. Multiply by the sample by $N$ to get the approximate new cluster sizes. $N(x_1,...,x_k)$. 5. Round the result to the nearest natural number. (make sure that the new cluster sizes add to $N$ due the rounding.) 6. Fill the new clusters of a random permutation of elements. The advantage that this has over the previously discussed option of just permuting the elements is that the cluster size distribution isn't always the same. You can allow for as much or a little variation from the original cluster sizes by controlling the concentration parameter $\beta$. In simulation this will likely result in a more robust calculation. - This seems to be just what I was looking for. I had a look at the Dirichlet Process material as well, which leads me to believe I could generate an algorithm for $k$ not fixed. Thanks a lot for your detailed answer. – Rónán Daly Mar 1 '12 at 22:53 Yep, a great thing about the Dirichlet Process is that it is a nonparametric model. – Daniel Johnson Mar 2 '12 at 3:01 One easy way to achieve your goal is to permute the labels. Say you had 10 objects, with memberships defined as $\{1, 2, 3, 4 ,5, 6\}$, $\{7, 8\}$, $\{9\}$ and $\{10\}$. You take a random permutation $\sigma=(3, 7, 2, 5, 1, 8, 10, 9, 6, 4)$, and then your new clusters are $\{\sigma(1), \sigma(2), \sigma(3), \sigma(4), \sigma(5), \sigma(6)\} = \{1, 2, 3, 5, 7, 8\}$, $\{\sigma(7), \sigma(8)\}=\{9, 10\}$, $\{\sigma(9)\}=\{6\}$ and $\{\sigma(10)\} = \{4\}$. If you want to introduce some variability in the cluster sizes, that can probably be done, too -- e.g. by drawing clusters with sizes Poi(6), Poi(2), Poi(1) and Poi(1), rejecting the samples that do not add up to 10. (Poi($\lambda$) is a Poisson random variable with rate/expected value $\lambda$.) (I wrote this before I read @whuber's comment. Oops) Update, based on valuable @DanielJohnson's comment: For large values of the total, the procedure becomes impractical, as it will be rejecting most samples. What you would want to do, then, is to condition on the total number of objects, $N$, and then the Poisson distribution becomes a multinomial with probabilities $\lambda_1/L, \ldots, \lambda_k/L, \ldots$, where $L=\lambda_1 + \ldots + \lambda_k + \ldots$. So your samples would simply be multinomial ones. Some clusters of size 1 or 2 may get lost though, and if you don't like that, then again you could condition on all clusters being present. This would effectively introduce variability in the size of larger clusters, while the smaller ones will remain with their original sizes. Again, if you find yourself rejecting the samples too often, you can use a combination of: (1) maintaining clusters of size 1 or 2; (2) simulating the Poisson- or multinomial-distributed clusters of larger sizes. - You need to be careful with your Poisson rejection scheme. You want $\sum^k_{j=1}Pois(n_j) = N$ or the scheme will reject the proposal. Remember that the sum of Poissons is Poisson with the parameter being the sum of the individual problems. In this case, $P(\sum^k_{j=1}Pois(n_j) = N) = N^Ne^-N/N! \sim \frac{1}{\sqrt{2\pi N}}$ by Sterling. For N = 10,000 which is on the order of what @Ronan had in mind this probability is about .004 meaning that an average of about 10000/.004 = 250,000 Poisson random variables need to be computed for each trial. – Daniel Johnson Mar 1 '12 at 23:07 @DanielJohnson, right. Let me think about it. – StasK Mar 2 '12 at 1:43 Can you think of this as $n$ balls being distributed among $k$ urns? That seems to fit your description of clusters (where you have $k$ clusters and $n$ numbers). If you need at least one ball in each urn, then first put 1 ball in each urn, then randomly select the urn for each of the remaining $n-k$ balls. Here is one possible implementation in R: > nkballs <- function(n,k) { + tmp <- sample(k, n-k, replace=TRUE) + as.numeric( table(tmp) ) + 1 + } > > nkballs(1000, 25) [1] 36 47 40 52 40 28 34 43 37 35 38 33 45 37 45 45 37 34 38 46 42 34 56 46 32 > sum(.Last.value) [1] 1000 - I like this idea, but as it is now it doesn't fit the requirement that the new cluster sizes be close to the original. One way to achieve this would be to, instead of uniformly choosing an urn for a ball, assign each urn a probability proportional to the size of the original cluster and then use that categorical distribution when choosing which urn to put a ball into. This is intimately connected to the Dirichlet type process i described below, with the only disadvantage here is the lack of a parameter describing how 'close' the sample is to the original distribution on average. – Daniel Johnson Mar 1 '12 at 19:41	{}
× "You are in a game show where there are 3 doors and you have to choose one of the doors. You are trying to win the million dollar prize which is behind one door. The other 2 doors have nothing in them so you want to make sure that you can win. Let's call these doors $$D_1, D_2$$ and $$D_3$$. You pick out $$D_1$$ and tell the host of the game show. The host, who knows which door has the million dollars shows you $$D_3$$, which has nothing, and asks you if you want to change your mind." How would you reply and give reasons. Note by Sharky Kesa 2 years, 6 months ago Sort by: This one is named as " Monty Hall problem" Also featured in the movie "21" $$D_{2}$$ is the probabilistically favorable answer. Here's the explanation: Vos Savant's response was that the contestant should switch to the other door. (vos Savant 1990a) The argument relies on assumptions, explicit in extended solution descriptions given by Selvin (1975b) and by vos Savant (1991a), that the host always opens a different door from the door chosen by the player and always reveals a goat by this action—because he knows where the car is hidden. Leonard Mlodinow stated: "The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated." (Mlodinow 2008) Contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their choice have only a 1/3 chance. One way to see this is to notice that 2/3 of the time, the initial choice of the player is a door hiding a goat. When that is the case, the host is forced to open the other goat door, and the remaining closed door hides the car. "Switching" only fails to give the car when the player picks the "right" door (the door hiding the car) to begin with, which only happens 1/3 of the time. Source : wikipedia · 2 years, 6 months ago Let's compute the probability that we will win if we switch: Total probability = Probabiilty that there's prize in D1 * Probability that we will win if we switch given there's prize in D1 + Probability that there isn't prize in D1 * Probability that we will win if we switch given there's no prize in D1 = (1/3) * (0) + (2/3) * (1) = 2/3 In a formal way: P (T) = P(S1) * P(A\|S1) + P(S2) * P(A\|S2) = (1/3) * (0) + (2/3) * (1) = 2/3 So its better to switch but it's too hard to think about this when you are in the game. Agree? · 2 years, 5 months ago Keep D1, its supposed to be hard to pick. There really is no way to know which so might as well just guess · 2 years, 6 months ago	{}
# PSR B0656+14: the unified outlook from the infrared to X-rays Zharikov, S.; Zyuzin, D.; Shibanov, Yu; Kirichenko, A.; Mennickent, R. E.; Geier, S.; Cabrera-Lavers, A. Bibliographical reference Monthly Notices of the Royal Astronomical Society 4 2021 Description We report detection of PSR B0656+14 with the Gran Telescopio Canarias in narrow optical F657, F754, F802, and F902 and near-infrared JHKs bands. The pulsar detection in the Ks band extends its spectrum to 2.2 $\mu$ m and confirms its flux increase towards the infrared. We also present a thorough analysis of the optical spectrum obtained by us with the VLT. For a consistency check, we revised the pulsar near-infrared and narrow-band photometry obtained with the HST. We find no narrow spectral lines in the optical spectrum. We compile available near-infrared-optical-UV and archival 0.3-20 keV X-ray data and perform a self-consistent analysis of the rotation phase-integrated spectrum of the pulsar using unified spectral models. The spectrum is best fitted by the four-component model including two blackbodies, describing the thermal emission from the neutron star surface and its hot polar cap, the broken power law, originating from the pulsar magnetosphere, and an absorption line near ∼0.5 keV detected previously. The fit provides better constraints on the model parameters than using only a single spectral domain. The derived surface temperature is $T_{NS}^{\infty } = 7.9(3)\times 10^5$ K. The intrinsic radius (7.8-9.9 km) of the emitting region is smaller than a typical neutron star radius (13 km) and suggests a non-uniform temperature distribution over the star surface. In contrast, the derived radius of the hot polar cap is about twice as large as the 'canonical' one. The spectrum of the non-thermal emission steepens from the optical to X-rays and has a break near 0.1 keV. The X-ray data suggest the presence of another absorption line near 0.3 keV. Related projects Formation and Evolution of Galaxies: Observations in Infrared and other Wavelengths This IAC research group carries out several extragalactic projects in different spectral ranges, using space as well as ground-based telescopes, to study the cosmological evolution of galaxies and the origin of nuclear activity in active galaxies. The group is a member of the international consortium which built the SPIRE instrument for the Ismael Pérez Fournon Milky Way and Nearby Galaxies The general aim of the project is to research the structure, evolutionary history and formation of galaxies through the study of their resolved stellar populations, both from photometry and spectroscopy. The group research concentrates in the most nearby objects, namely the Local Group galaxies including the Milky Way and M33 under the hypothesis Martín López Corredoira Type	{}
# Why does MLE tend to normal distribution We have $$X_1,\dots, X_n$$ are iid (the distribution can be of any type, e.g. Bernoulli (p), normal ($$\mu, \sigma^2$$), Poisson ($$\lambda$$). If we use MLE $$\hat \theta$$ to estimate any parameter $$\theta$$ of the distribution,, then, as is said in Casella Section 10.4.1, (any continuous function of $$\hat \theta$$) $$h(\hat \theta)\to \mathrm{n}(h(\theta), v)$$, $$\quad$$(1) where $$v$$ is $$\mathrm{Var}( h(\hat \theta))$$ as $$n\to\infty$$, which equals $$\frac{[h'(\theta)]^2}{\frac{\partial^2}{-\partial \theta^2} \log L(\theta\|\mathbf{X})}\|_{\theta=\hat \theta}$$. $$\quad$$(2) My question is why, if $$X_i$$'s are iid and $$\hat \theta$$ is MLE, $$h(\hat \theta)$$ tends to normal distribution as $$n\to\infty$$? I can understand that if $$\hat \theta=\bar X$$, then it seems to be just CLT. But why it is so when $$\hat \theta$$ is not average of sample. The reason given in the text is that MLE is asymptotically efficient (theorem 10.1.12), but would anyone give an easier explanation? BTW, can we say CLT is just a special case of theorem 10.1.12? It's said according to Slutsky's Theorem (i.e. if $$X_n,Y_n,Z_n$$ are sequence of estimator tending to a random variable) $$X$$ of certain distribution, constant a, b, $$Y_nX_n+Z_n\to aX+b$$), $$\frac{h(\hat \theta)-h(\theta)}{\sqrt{\hat{\mathrm{Var}}( h(\hat \theta))}}\to \mathrm{n}(0,1)$$. Why? My thought is that both $$h(\hat \theta)$$ and $$\sqrt{\hat{\mathrm{Var}}( h(\hat \theta))}$$ are random variables (since they are estimators, and estimators are functions of/dependent on the sample $$X_i$$'s, which are random variables), tending to a random var $$\mathrm{n}(h(\theta), v)$$ (according to (1)) and a random var (2) (for (2) is function of random var $$\mathbf{x}$$ and random var $$\hat \theta$$; so we can't use Slutsky's Theorem directly. But, say $$\sigma^2$$ is the variance of $$\theta$$, and we have $$\frac{h(\hat \theta)-h(\theta)}{\sqrt{\hat{\mathrm{Var}}( h(\hat \theta))}}=\frac{h(\hat \theta)-h(\theta)}{\sigma}\frac{\sigma}{\sqrt{\hat{\mathrm{Var}}( h(\hat \theta))}}$$, the first factor tends to $$\mathrm{n}(0,1)$$, the second seems to tend to 1 (Why? It is still a random variable and should tend to a distribution instead of a number), and therefore we get the result. My question is as bolded. From what's said above we have confidence interval $$h(\hat \theta)-z_{\alpha/2} \sqrt{\hat{\mathrm{Var}}( h(\hat \theta))} < h(\theta) < h(\hat \theta)+z_{\alpha/2} \sqrt{\hat{\mathrm{Var}}( h(\hat \theta))}$$, $$\quad$$ (5) that is, according to Definition 9.1.4, 9.1.5 of confidence coefficient, $$1-\alpha=\inf P\left(L(\mathbf{X}) < h(\theta) < U(\mathbf{X})\right),$$ $$\quad$$ (4) where $$L(\mathbf{X})=h(\hat \theta)-z_{\alpha/2} \sqrt{\hat{\mathrm{Var}}( h(\hat \theta))}, U(\mathbf{X}) = h(\hat \theta)+z_{\alpha/2} \sqrt{\hat{\mathrm{Var}}( h(\hat \theta))}$$. $$\quad$$ (5) (We can verify, as we have done in the above section, that the two 'bounds' are random variables and functions of $$\mathbf{X}=(X_1,\dots, X_n)$$.) And infimum is of the sequence of the set/sequence of the above probability for all $$n\in\mathbb{N}_+$$. To make this point (and that it's not about probability of $$h(\theta)$$ but about that of $$\hat \theta, \hat{\mathrm{Var}}( h(\hat \theta))$$) more explicit we can write (4) as $$\inf P\left(\mathbf{X}^{-1}(L^{-1}(-\infty, h(\theta))), \mathbf{X}^{-1}(U^{-1}(h(\theta), \infty)))\right).$$ How can we directly proceed from the definition of confidence interval to (5), i.e. given the definition and (5), how to prove (4), that is, $$\inf P\left(\mathbf{X}^{-1}(L^{-1}(-\infty, h(\theta))), \mathbf{X}^{-1}(U^{-1}(h(\theta), \infty)))\right)=1-\alpha$$? (Updated: I think the key step is we form one static $$T(\mathbf{X})$$ from the two bounds $$L(\mathbf{X}), U(\mathbf{X})$$ and we can calculate its distribution, possibly it’s normal distribution—-if no we can average several such statics and from CLT we know this ‘mean’ has normal distribution. (This seems to be what we do in Casella Exercise 10.8). From this we can easily tell the probability of the parameter to be estimated is in $$(L(\mathbf{X}), U(\mathbf{X}))$$. The key here is that $$h(\theta)$$ and its estimator is ‘symmetric’ in our static $$T(\mathbf{X})$$.) My another question is that why we define confidence coefficient (for interval estimator) this way, in particular, why we need to include 'infimum' in this definition? I ask this because the confidence level $$\alpha$$ is defined as $$1-\alpha= P\left( -z_{\alpha/2} < X < z_{\alpha/2} \right)$$; here $$X$$, not the upper and lower bounds, is random variable; and we don't use 'infimum'. Updated: The question in the first section can be partly answered by checking the proof of theorem 10.1.12, $$h(\hat \theta)$$ has normal distribution because $$\sqrt{n}(\hat \theta-\theta)$$ is (with Taylor expansion) proportional to $$l'(\theta\|\mathbf{x})=\frac1n\sum_iW_i$$ (Exercise 10.8 of Casella) where $$W_i=\frac d {d\theta} (\log f(x_i\|\theta))=\frac{\frac d {d\theta} f(x_i\|\theta)}{{f(x_i\|\theta)}}$$ and is iid, has, according to CLT, normal distribution. I'm wondering how to prove that $$W_i$$ is iid. • Nov 4, 2020 at 20:52 • @kjetil b halvorsen They seem useful. Nov 4, 2020 at 23:36	{}
Vertebrate photoreceptors are commonly distinguished based on the shape of their outer segments: those of cones taper, whereas the ones from rods do not. The functional advantages of cone taper, a common occurrence in vertebrate retinas, remain elusive. In this study, we investigate this topic using theoretical analyses aimed at revealing structure–function relationships in photoreceptors. Geometrical optics combined with spectrophotometric and morphological data are used to support the analyses and to test predictions. Three functions are considered for correlations between taper and functionality. The first function proposes that outer segment taper serves to compensate for self-screening of the visual pigment contained within. The second function links outer segment taper to compensation for a signal-to-noise ratio decline along the longitudinal dimension. Both functions are supported by the data: real cones taper more than required for these compensatory roles. The third function relates outer segment taper to the optical properties of the inner compartment whereby the primary determinant is the inner segment’s ability to concentrate light via its ellipsoid. In support of this idea, the rod/cone ratios of primarily diurnal animals are predicted based on a principle of equal light flux gathering between photoreceptors. In addition, ellipsoid concentration factor, a measure of ellipsoid ability to concentrate light onto the outer segment, correlates positively with outer segment taper expressed as a ratio of characteristic lengths, where critical taper is the yardstick. Depending on a light-funneling property and the presence of focusing organelles such as oil droplets, cone outer segments can be reduced in size to various degrees. We conclude that outer segment taper is but one component of a miniaturization process that reduces metabolic costs while improving signal detection. Compromise solutions in the various retinas and retinal regions occur between ellipsoid size and acuity, on the one hand, and faster response time and reduced light sensitivity, on the other. ## INTRODUCTION Since the early days of vision research, pioneered by the work of Hannover (1840), Müller (1856), and Schultze (1866, 1867), vertebrate photoreceptors have been classified as rods and cones by morphological criteria. Schultze (1866) correlated the visual habits of animals with the relative preponderance of rods and cones in their retinas; this led him to formulate the concept upon which the Duplicity Theory rests. The premise of this theory is that cones are the receptors for photopic (bright light) vision, whereas rods are the receptors for scotopic (dim light) sensing. Schultze (1866, 1867) arrived at the correct conclusion that cones mediate color perception. Subsequent to Schultze’s time, visual cells have been described with intermediate morphological, physiological, and molecular attributes that tend to blur the distinction between rods and cones (Walls, 1963; Pedler, 1965; Crescitelli, 1972; Kojima et al., 1992; Ma et al., 2001; Collin et al., 2004; Zhang et al., 2006). Nevertheless, the old classification has endured as regards to the vertebrate retina: its photoreceptors are rods and cones wherein cones typically exhibit a tapered outer segment, whereas rods do not. The functional significance of this distinguishing feature, so prevalent in nature, remains largely unexplored. One of the potential benefits of tapering cone outer segments was introduced by Hodgkin and O’Bryan (1977) with their concept of critical taper. In their study of turtle cone electrical responses, these authors considered two limiting cases of cone geometry: the cylindrical (untapered) form and another, in which the “outer segment tapers in such a way that all molecules have an equal chance of absorbing a quantum” (Hodgkin and O’Bryan, 1977). In the latter case, the outer segment must taper at a specific, critical angle, and light must be funneled by complete internal reflection from the broad to the narrow end of cone outer segments (Hodgkin and O’Bryan, 1977). The significance of critical taper is as follows. Rods and cones are highly specialized cells with unusual properties. First, the sensory visual pigments that they use are extremely absorbent; i.e., they possess very high extinction coefficients, corresponding to large molecular absorption cross sections (Hárosi and MacNichol, 1974). Second, visual pigment molecules are densely packed in lamellar membranes, which, in turn, are tightly stacked in hundreds of layers within the outer segment (the molecular packing within the membrane and the tightness of lamellar packing are probably as high as functional constraints will allow; see Wen et al., 2009). Consequently, the pigment-laden lamellae in the more proximal layers act as light filters for the more distal layers. This phenomenon is known as self-screening (Brindley, 1970). As a result of self-screening, light quanta arriving in the physiological setting have a greater probability of being absorbed near the base of an outer segment than toward the apex. Thus, in a cylindrical rod, where lamellae are of equal size, signal generation declines steadily in more distal layers with a concomitant decline in efficiency (i.e., photocurrent production per unit volume; Schnapf, 1983). One possible way to improve performance is to trim the volume slices along the length of outer segments in proportion to the fall-off of lamellar absorption caused by self-screening. A conical structure could accomplish this. Tapering is considered critical when the trimming of lamellar cross section along the taper is exactly proportional to the fall-off of absorption rate, resulting in uniform efficiency (Hodgkin and O’Bryan, 1977). This idea, which was neither generalized nor experimentally tested, is the basis for the first potential function, overcoming signal loss caused by self-screening, that we evaluate in this study. A second function considered is that outer segments taper to enhance the signal-to-noise ratio along their lengths. Accordingly, taper would also be driven by another outer segment function: signal generation. Although the generation of signal and associated noise in photoreceptors are complex phenomena, in part because of the stochastic nature of underlying processes, such as the opening and closing of ionic channels or the binding and release of ligands at receptor sites in the enzymatic cascade of the light response, there is a consensus on the existence of thermal activation of visual pigment molecules and cGMP phosphodiesterases, both components giving rise to noise (Rieke and Baylor, 1996, 2000; Holcman and Korenbrot, 2005). Based on current understanding, the receptor signal consists of a photocurrent generated through a narrow circumferential region of the outer segment membrane in response to the number of quanta absorbed in the adjacent volume containing one or a few lamellae (Baylor, 1987). Noise, on the other hand, is assumed, on the most basic level, to be proportional to the total number of visual pigment molecules or cGMP phosphodiesterases contained in the same volume (Rieke and Baylor 1996, 2000; Sampath and Baylor, 2002; Holcman and Korenbrot, 2005). Either way, the signal-to-noise ratio is expected to diminish along the outer segment length (z direction) in cylindrical cells. With a tapered outer segment, however, consecutive lamellae are progressively reduced in cross section, leading to diminution of noise along the way. The third and last function that we evaluate, efficient light collection and utilization of biomaterials, is based on the hypothesis that outer segment taper follows the optical properties of the inner segment. Rather than considering outer segment taper to be tied up with strictly outer segment functions, this idea proposes a multifaceted interdependence between inner and outer segments, as suggested by morphology. Both cones and rods feature three distinct compartments or subcellular organelles: an outer segment (limb), specialized for trapping light; an inner segment (cone ellipsoid), concerned primarily with energy production and homeostatic functions; and a synaptic apparatus that communicates with other neurons (Fein and Szuts, 1982). Cone ellipsoids are usually the most conspicuous of the photoreceptor compartments in practically every retina, with primate foveal cones being a notable exception (Borwein et al., 1980; Packer et al., 1989; Hoang et al., 2002). Cones are always broadest at their ellipsoid and tend to taper toward the outer segment, to which they attach closely (Fein and Szuts, 1982). In some fish retinas, the two cone compartments appear as one confluent unit, so that it is hard to discern through the light microscope where the ellipsoid ends and the base of the outer segment begins. In contrast, rods rarely have any difference in width between the two limbs. Shape and size variation notwithstanding, it is always the inner segment wherefrom light enters the outer segment in the physiological setting. For these reasons, it seems logical to consider the two compartments combined as one optical unit. Cone ellipsoids tend to taper from the thickest proximal region toward the distal outer limb, and this, most likely, is a ploy to concentrate light (Winston 1970, 1981). And if that is so, the outer segment taper may be dependent on the light-gathering property of the inner segment. This idea is also bolstered by the observation that cones with oil droplets tend to have more tapered outer segments than those without this organelle (Nilsson, 1965; Kolb and Jones, 1982; Röhlich and Szél, 2000; Bailes et al., 2006). In view of the high refractive index values of oil droplets (Ives et al., 1983), there is no doubt about their refractive role (Baylor and Fettiplace, 1975; Young and Martin, 1984). Given some light concentration property, cone ellipsoids could funnel parallel incident light into converging (conical) beams, which, when projected onto smaller lamellar areas, could result in equal photon catch (and signal) maintained at reduced noise. Even in the presence of light losses, increased tapering should be advantageous for the gains to be made in improved signal to noise (by lamellar shrinkage) and in savings in detector material (by volume reduction). A practical solution ought to balance the advantages against the concomitant drawbacks, such as reduced acuity and some light loss by ellipsoid leakage. As such, a standard cone should not exist, but there should be variously tapered structures in nature that represent compromise solutions to different sets of constraints. Although the third function does not lend itself to testing via a single mathematical relationship, its validity can be ascertained by examining structure–function relationships in different species and comparing outer and inner segment taper-related variables that, from the aforementioned reasoning, should be positively correlated. Our analysis of cone taper focuses on photoreceptor properties that have been routinely selected for during the course of evolution such as improved signal detection and metabolic savings by efficient use of biomaterials (see, for instance, the photoreceptor innovations of anchovies; Novales Flamarique, 2011). Early vertebrates, like extant hagfishes and larval ascidians, evolved ciliary photoreceptors that acted as shadow detectors, presumably conferring some of these animals an advantage in dim light environments (Collin, 2010). Natural selection acting on mutations to these ancestral designs led to a large number of novel photoreceptor features, including changes in outer segment shape (from conical to rodlike and vice versa, the transmutation hypothesis; Walls, 1963), multiple photopigments for color vision (Bowmaker, 2008), phototransduction enzymes with varying response kinetics (Hisatomi and Tokunaga, 2002), and, with a focusing eye, photoreceptor mosaics that improved overall sensitivity and/or visual acuity (van der Meer, 1992). As per other selective traits, the shape and size of photoreceptors are expected to vary, and each form may subserve multiple functions, though perhaps none optimally. Indeed, natural selection may retain a given form because it is either nondeleterious or because it confers some advantage to the individual (Bell, 2009). We therefore surmised that our analysis could reveal various advantages of taper to cone photoreceptor function. The primary thrust of this study is theoretical. Attention is focused on morphological and biochemical properties of vertebrate photoreceptors. The aim is to gain insight into the principles governing their structure and function. In addition to the analytical approach, experimental results are used for testing theoretical predictions. The empirical data include cellular dimensions, which were derived by light and electron microscopic measurements, in situ visual pigment determinations by microspectrophotometry, in vitro visual pigment data obtained by spectrophotometry, electrophysiological determinations, and comparative anatomy. The cited empirical data are derived from either published articles in the literature or hitherto unpublished work from our laboratories. The three potential functions of cone taper that we evaluate are considered in sequence; for each, the consequences and ramifications are examined. ## MATERIALS AND METHODS ### Animals The majority of data in this study originated from animals used in published works, either our own or those of others. However, some measurements were taken from studies that have yet to appear in the literature. These measurements originated from goldfish (Carassius auratus), common carp (Cyprinus carpio), zebrafish (Danio rerio), three-spine stickleback (Gasterosteus acuelatus), blue gill sunfish (Lepomis macrochirus), green sunfish (Lepomis cyanellus), rainbow trout (Oncorhynchus mykiss), coho salmon (Oncorhynchus kisutch), chinook salmon (Oncorhynchus tschawytscha), African clawed frog (Xenopus laevis), northern leopard frog (Rana pipiens), American bullfrog (Rana catesbeiana), Canada goose (Branta canadensis), green-winged teal (Anas crecca carolinensis), red-eared slider turtle (Trachemys scripta elegans), and mouse (Mus musculus). Animals were obtained from the following locations: zebrafish, local pet shop supplier in Burnaby (British Columbia, Canada); three-spine stickleback, Swan Lake (Victoria, British Columbia, Canada); blue gill sunfish, ponds around the Woods Hole, MA area; rainbow trout, Lower Mainland Trout Hatchery (Abbotsford, British Columbia, Canada); coho salmon and chinook salmon; Capilano River hatchery (North Vancouver, British Columbia, Canada); common carp, green sunfish, African clawed frogs, and northern leopard frogs, Marine Resources Centre of the Marine Biological Laboratory (Woods Hole, MA); and bullfrogs, Aquatic Facility Centre of Simon Fraser University (Burnaby, British Columbia, Canada). The animals were kept in aerated, flow-through water tanks under a 12-h light/dark cycle while experiments were being conducted. Fixed and fresh eyes from red-eared sliders were provided by C. Carr (University of Maryland, College Park, MD) and E. Enos (Marine Resources Centre), respectively. Fixed mouse eyes were obtained from staff at the Animal Care Facility of Simon Fraser University, and fixed eyes from Canada geese and green-winged teals (a species of northern duck) were provided by M. Juhas (Haida Gwaii Archipelago, British Columbia, Canada). Laboratory animals were killed in a state of light or dark adaptation for histology or microspectrophotometry, respectively, and the retinas were processed as detailed in previous publications (Hárosi, 1987; Cheng et al., 2006; Novales Flamarique, 2011). All experimental procedures were approved by the Animal Care Committee of Simon Fraser University or the Marine Biological Laboratory, which are in compliance with the guidelines set by the Canadian Council for Animal Care and the National Institutes of Health. ### Measurements We obtained cell dimensions from live retinas (microspectrophotometry experiments) and from fixed, Epon-embedded retinas cut into thin (75 nm) sections and observed using a transmission electron microscope (model 7600; Hitachi; Fig. 1). These measurements were inner segment ellipsoid diameter (di), outer segment diameter at the base (do), outer segment diameter at a distance, z, from the base (dz), and outer segment length and tip diameter when possible (Fig. 2). In addition, densities of rods and cones were obtained from thick (1 µm) histological sections. Besides unpublished data, we obtained similar measurements from studies spanning the last 75 yr of anatomical literature. These included 116 species of fishes, covering the evolutionary spectrum from ancient groups like lampreys, elasmobranchs, and lungfishes to modern teleosts like killifishes, carps, and cichlids; 9 species of amphibians; 33 species of birds; 19 species of reptiles; 31 species of mammals; and 8 species of monotremes and marsupials. Table 1 details the species examined and the works consulted. Each datum presented in the graphs of this study is the mean from a minimum of 15 cells for species used in live cell recordings (Table 2) and anywhere from three to thousands of cells for species data originating solely from the literature. Some publications failed to report cell numbers for the morphological data presented; in such cases, we took the numbers as averages for the entire retina. In our presentation of figures, we show parallel analyses for species for which measurements from live cells were obtained and those whose measurements originated from histological work, primarily from the published literature. ### Methods to evaluate the first biophysical function Evaluation of this function relied on a comparison of anatomically and physiologically derived measures of two parameters: characteristic length and taper, as described in Appendix 1. Two methods were used in the evaluation. The first involved the calculation of the required characteristic length, ah1−1, from Eq. 6 (Appendix 1) for each cell. The results were then compared with two separate estimates of characteristic length, az−1 and az′′−1, obtained with the aid of Eqs. 10 and 16 (Appendix 1). The necessary parameter values for the latter calculations were derived from previous determinations performed on equivalent cells. For example, it was assumed that cones with vitamin A2–type pigments share the properties with those of the yellow perch, goldfish, or Japanese dace and have the following parameter values: ε = 30,000 liter mol−1 cm−1, c = 3.5 mM, R = 2.2, and S = 0.0125 µm−1. For cones using vitamin A1–type pigments, ε = 42,000 liter mol−1 cm−1, c = 3.5 mM, R = 2.5, and S= 0.0175 µm−1 (Hárosi and MacNichol, 1974; Hárosi, 1975, 1976, 1984, 1985; Hárosi and Hashimoto, 1983). The estimated characteristic lengths were not significantly affected by the particular values assumed for individual parameters. For instance, we could have used Liebman and Granda’s (1971) specific absorbance of 0.0130 µm−1 to obtain az′′−1 = 33.4 µm or assumed 3-mM pigment concentration and an infinite dichroic ratio (k = 1.5), as did Hodgkin and O’Bryan (1977), and arrived at az−1 = 32.2 µm. These values are similar to those reported in Table 2. As a second method for testing the validity of the first function, we calculated a new diameter (dz′) for each cell that would occur at critical taper by using az′ and the rest of the geometrical data in Eq. 5 (Appendix 1). The actual taper, τ, obtained with dz, was then recalculated using dz′ in Eq. 17. The resulting τ′ represents taper for each cell that would satisfy invariance of flux density along the outer segment. ### Evaluation of the second biophysical function Eq. 30 of Appendix 2 is an expression for the signal-to-noise ratio along an outer segment. This equation was scrutinized to assess the validity of the second proposed function. ### Evaluation of the third biophysical function For the lack of a compact theoretical expression (Appendix 3), this function could not be verified experimentally. Our approach here was to test whether several morphological observations across phyla were consistent with it. For instance, the difference in light-gathering efficiency by tapered versus nontapered photoreceptors (Appendix 3) led us to predictions about the rod/cone ratios of primarily diurnal animals; i.e., animals with both diurnal and crepuscular/nocturnal rhythms, but in which the primary rhythm is diurnal, and in which utilization of both cone and rod pathways occurs during the day (these species are to be contrasted with those that are fully diurnal, e.g., some ground squirrels [West and Dowling, 1975; Anderson and Fisher, 1976], or primarily nocturnal, e.g., the giant African rat; Peichl, 2005). In addition, we postulated that there should be a correlation between outer segment taper and ellipsoid concentration factor (Eq. 33, Appendix 3), the latter being a measure of the ability to concentrate light from inner to outer segment. To test this, we used critical taper (Appendix 1) as a benchmark and expressed actual taper as the ratio az−1/ah1−1, indicating how many critical characteristic lengths would equal the characteristic length considered realistic. ## RESULTS ### Cone taper exceeds critical taper Table 2 presents typical morphometric data obtained from live photoreceptors during microspectrophotometric experiments, in this case from individual photoreceptors of goldfish. Values of derived variables (e.g., taper, characteristic length, and concentration factor) are also presented. Figs. 36 show means from analogous datasets for the various species examined. Comparison of data in columns 8–10 of Table 2 reveals that the characteristic length corresponding to critical taper differs from the characteristic length calculated from known data. Moreover, the difference in most cases goes far beyond the margin of error (∼0.1 µm, from repeated measures). Except for one cone (Table 2, row 1) that came close to having critical taper, the ah1−1 values for the rest of the cones are much smaller than expected. A smaller characteristic length, however, means a larger absorption coefficient. We can consider, for instance, the blue-absorbing single cone on line 25 of Table 2. For it to have critical taper, the axial extinction would have to be ∼3.8-fold higher than a realistic value. As is apparent from columns 6 and 7 of Table 2, the taper of real cones is excessive. For the vast majority of species studied, whether the data originated from live cell measurements (Fig. 3 A) or from fixed, histological material (Fig. 3 B), the mean cone taper (τ) was at least 1.5 times the critical taper (τ′). Only in the case of bats, the giant African rat, and foveal-perifoveal cones of primate retinas was cone taper statistically the same as critical taper (P > 0.05, paired t tests). On average, the ratio τ/τ′ ± SD was 4.8 ± 2.4 for fishes, 5.1 ± 1.7 for amphibians, 5.5 ± 2.1 for birds, 5.7 ± 3.8 for reptiles, 4.6 ± 3.2 for mammals, and 4.3 for marsupials (two species). Among the fishes, the mean elasmobranch ratio (7.2 ± 2.7) was about twice that of teleosts (3.8 ± 1.4), and the highest ratios among the teleosts were for lungfishes (5.2) and killifishes (4.8), which, like most amphibians, reptiles, and birds and some fishes and monotremes, have oil droplets (lungfishes) or have ellipsosomes (killifishes) in the ellipsoid region (Fig. 1). Among the reptiles, the ratio for the strictly diurnal garter snake (5.3) was about half that of primarily nocturnal snakes like the boa and ball python (mean of ∼11). Overall, phylogenetic groups with focusing structures in the ellipsoids (e.g., oil droplets, microdroplets, and megamitochondria) had larger τ/τ′ ratios than those lacking them. Both methods of assessment led to the same general observation: overall, cones are more tapered than they ought to be for critical taper. Therefore, cone outer segment taper overcompensates for the loss of signal caused by self-screening. In contrast, rod outer segments, if tapered at all, have less than critical taper (Table 2), with the potential exception of rods in lampreys and geckos, species in which the two photoreceptor types are hard to classify because they exhibit intermediate morphological and molecular attributes (Collin et al., 2004; Muradov et al., 2008; Zhang et al., 2006). ### Signal-to-noise ratio increases beyond that predicted by critical taper Examination of Eq. 30 (Appendix 2) reveals the following: (a) when Az is invariant in z (cylindrical outer segment), the signal-to-noise ratio, S/N, diminishes exponentially along z as a result of self-screening; (b) if Az versus z diminishes by tapering, the S/N will undergo proportionate increases; (c) for critical taper, when Az varies in accordance with Eq. 4 (Appendix 1), the exponential terms cancel, and the S/N becomes independent of z. From Eq. 30 we can also deduce that, for excessive taper, i.e., when a cone cross section (Az) diminishes faster than the exponential fall-off of signal, the signal-to-noise ratio may actually increase toward the apex of a cone outer segment. Eq. 30 thus reveals an important tendency: more taper means greater improvement in the signal-to-noise ratio along an outer segment. For this reason, in the absence of other requirements, cones should taper as much as possible. ### Inner segment morphology as a major determinant of outer segment taper For the lack of a compact theoretical expression (see Appendix 3), the third function linking inner segment morphology to outer segment taper could not be verified experimentally. Nevertheless, several observations were consistent with it, as demonstrated by the following analyses using examples from diverse phylogenetic groups. Teleosts, like the goldfish, have typically large, plump inner segments and shorter outer segments compared with the rods (Fig. 1). The average goldfish cone in this study had an entrance aperture (assumed to be equal to the broadest region of the ellipsoid) of 8.7-µm diameter, wherefrom light would be funneled to the base of the outer segment with a mean diameter of 5.9 µm (Table 2). The concentration factor, FC, by Eq. 33 (Appendix 3) is 2.2. The outer segment volume calculated by Eq. 34 yields 277 µm3. The volume of a cylinder with equal base diameter is 399 µm3. Their ratio gives 1.4 for the geometry factor, FG, by Eq. 35. Additionally, the volume reduction ratio, VR, is obtained at 3.1 by Eq. 36. Thus, the cone outer segment uses ∼32% of the volume of that of the hypothetical optically equivalent rod. The rod cells measured in these goldfish preparations were quite uniform, with an average outer segment diameter of 2.1 µm and lengths in the range of 36–44 µm. For a 40-µm-long rod outer segment, we can calculate the axial absorptance by Eqs. 3 and 22 using az′ (Appendix 1). Accordingly, such a rod would absorb ∼69% of the light incident at its base. By a similar calculation, the 15-µm-long cone outer segment would absorb only 35% of the flux it receives. Therefore, the rod is the better light detector of the two. However, this rod collects a mere fraction of what a single cone can gather from the retinal illumination. In fact, it would take (8.7/2.1)2 = 17.2 rods to intercept as much incident flux as the average cone considered here. The rod/cone ratio in goldfish was reported to be 15:1 (Stell and Hárosi, 1976). The reasonably satisfactory agreement between predicted and observed ratios suggests that, in this animal, interception of equal light flux, a property linked to inner segment aperture, appears to be the criterion driving rod and cone densities. When this analysis was applied to the other, predominantly diurnal, species studied, the correlations between observed and expected rod/cone ratios were surprisingly good (Fig. 4, A and B). On average, the rod/cone ratio expected and observed ± SD for the various phylogenetic groups were as follows: 6.9 ± 10 and 7.6 ± 9.2 (fishes), 1.4 ± 1.6 and 1.5 ± 0.58 (amphibians), 2.0 ± 2.1 and 1.4 ± 2.0 (birds), 5.0 ± 3.4 and 6.8 ± 7.4 (reptiles), 12 ± 7.4 and 13 ± 10 (mammals), and 12 and 16 (marsupials, two species). Among the fishes, the rod/cone ratios of teleosts were better predicted than those of elasmobranchs; the mean predicted and observed ratios for these two groups were 6.2 ± 9.0 and 6.4 ± 9.1 (teleosts) and 8.5 ± 12 and 10 ± 9.1 (elasmobranchs). Overall, the highest rod/cone ratios occurred for the walleye (a teleost with “remarkably large cones”; Januschka et al., 1987) and the mink. The best predictions occurred within the teleosts and mammals, especially for primates (predicted and expected mean rod/cone ratios were 16 ± 7.3 and 17 ± 8.8). The worst predictions (i.e., ratio of expected to predicted, or vice versa, >2) occurred within the birds (owls) and reptiles (snakes) as well as for elasmobranchs and one teleost, the snake mackerel. Some of these animals (e.g., owls, several species of elasmobranchs, and large snakes) were included in the analysis because of reported crepuscular (cone driven) activity, although they may be primarily nocturnal and perhaps not very appropriate for inclusion in these regressions. Indeed, the rod/cone ratios of primarily nocturnal or fully diurnal animals are not very well predicted based on our principle of equal flux sharing between photoreceptor types (Fig. 5). Primarily nocturnal or dark habitat–dwelling species, which include deep ocean sharks and eels, several mammals (e.g., the mouse, African giant rat, bats, and the spotted hyena), the oil bird, and marsupials (opossums and the tammar wallaby), have large to very large rod/cone ratios (some exceeding 100:1). At the other end of the spectrum, strictly diurnal species like some ground squirrels, the prairie dog, and the tree shrew have very small rod/cone rations (≤0.1). None of these species, located at either extreme of the rod/cone ratio range, were included in the analysis leading to Fig. 4 B. In contrast to the comparable size of photoreceptors in teleosts and mammals, the retinas of amphibians have colossal rods and minute cones (Fig. 1). The dichotomy in outer segment size is seldom as pronounced as in frog species, with the possible exception of the salamanders (Crescitelli, 1972). However, even in these animals, the important feature determining rod and cone densities appears to be the equal sampling of the plane, where, presumably, the image of the outside world is formed within the eye. As an example, the mean cone ellipsoid diameter (entrance aperture) of the African clawed frog was found to be 7 µm, whereas the mean diameter of the outer segment base was 2.9 µm. Repeating the procedure used previously, we can calculate, by using Eqs. 33–36 (Appendix 3), the following: FC = 5.8, Vc = 20.1 µm3, Vr = 39.6 µm3, FG = 1.97, and VR = 11.4. The optically equivalent cylinder, as indicated in Fig. 2 C, has a volume of ∼231 µm3. Thus, the cone outer segment volume is only ∼9% of that of the cylinder. The preparations where frog single cones were located also had rhodopsin-containing red rods with an outer segment diameter equal to the widest region of the ellipsoid, with a mean diameter of 7 µm. This finding may be interpreted in terms of the cone ellipsoid’s light-gathering property and suggests that the entrance aperture of single cones and red rods are equal in frogs. Further support for the idea of equal flux collection by these cells in the frog retina can be found in the electron microscopic studies of Nilsson (1964, 1965). He showed, among other things, that red rods and single cones have about equal areas in cross section at the inner segment level (Nilsson, 1964). These photoreceptors, therefore, may indeed gather nearly equal fluxes from a uniform retinal illumination. The actual volume of a red rod outer segment (at a length of 66 µm) is ∼2,500 µm3. Using known parameters and Eqs. 3 and 22 (Appendix 1), we can calculate that these rods may catch 94% of axially incident light, whereas the average cone catches only ∼23% of it. Thus, a typical red rod may be a fourfold better absorber of axial light than a single cone (at the λmax of their respective visual pigments). However, the cone outer segment volume, Vc, as calculated previously, is nearly 100-fold smaller than that of the rod. The rod/cone ratio expected, 1, is similar to that assessed from published micrographs (means of 1–1.5; Kinney and Fisher, 1978a; Hollyfield et al., 1984; Röhlich and Szél, 2000). The fact that cones absorb a smaller portion of the incident light is not a handicap under daylight conditions, when high absolute sensitivity is not required. For the same reason, the loss of light by leakage from cone ellipsoids can be tolerated when light is plentiful. Therefore, frog rods and cones can serve the retina equally well despite their large disparity in size. By making cones with large entrance aperture and an ability to concentrate light, outer segment size reduction becomes possible at some cost in detector efficiency and sacrifice in acuity. Upon review of the preceding examples, the data suggest a positive relationship between outer segment taper and ellipsoid concentration factor, FC. The latter variable is a measure of light concentration from the ellipsoid into the outer segment. We have plotted the ratio az−1/ah1−1, indicating how many critical characteristic lengths would equal the characteristic length considered realistic, against FC for the various species examined (Fig. 6, A and B). The correlations are positive, though not particularly strong, especially when all the species studied are considered (Fig. 6 B). The lack of a strong correlation for the ensemble of species may in part reflect the lower accuracy of data obtained from histological material (especially from published figures) as opposed to live cell measurements. In addition, and perhaps more importantly, it may be an indication of other factors, such as the presence of oil droplets or light losses caused by ellipsoid leakage, affecting the extent of outer segment taper, none of which were considered in the calculations (for an instance of light loss estimation, see Baylor and Fettiplace, 1975). Nevertheless, the correlations, albeit approximate ones, between the cone ellipsoid concentration factor and outer segment taper expressed as a multiple of critical taper support the third function. ## DISCUSSION ### Emerging view on structure–function relations in vertebrate photoreceptors Our analysis supports the three postulated functions for cone taper: (1) compensation for light loss resulting from self-screening, (2) increased signal-to-noise ratio along the length of the outer segment, and (3) improved light capture and material savings by shape continuity between inner and outer segment. Cone taper, as determined in this work, was found quite variable and, for the most part, beyond that predicted for critical taper (Table 2 and Fig. 3). Consequently, cones appear to achieve full compensation for the light loss caused by self-screening while improving signal-to-noise ratio along their outer segments. Support for the third function points to a dominant role of the inner segment in shaping the outer segment, leading to a process of miniaturization and, as a result, metabolic savings in biomaterials. The trade-offs in such structural modifications are in absolute sensitivity and visual acuity, with spatial resolution being inversely related to receptor cross section (Snyder and Miller, 1977; Neave, 1984; van der Meer, 1992; Haug et al., 2010). Large cone ellipsoids combined with excessively tapered outer segments could achieve miniaturization, but only in the presence of light funneling. Therefore, an ellipsoid light concentration property was suggested by logic and implied by the third function. In every retinal region, a trade-off is expected between acuity and receptor size. The typical example is the retina of some diurnal teleosts where the highest visual acuity, found in the mid to upper frontal field, is subserved by smaller, densely packed cones in the centro- and ventro-temporal areas of the retina (Beaudet et al., 1997; Novales Flamarique, 2005, 2011; Cheng and Novales Flamarique, 2007). Other more extreme examples are the specialized foveas of diurnal lizards (Röll, 2001; Barbour et al., 2002) and primates (Borwein, 1981). In primates, high visual acuity is the product of tightly packed, long, rodlike foveal cone outer segments, each having a diameter of ∼1 µm; these are connected by cone fibers to the rest of their compartments that are laterally displaced and squeezed out to the slopes of the foveal pit (Borwein, 1981; Packer et al., 1989). Foveal cones have no ellipsoids, and, therefore, they must collect light without the aid of any other structure. However, parafoveal cones, and other cones in general, are built with joint outer and inner compartments, where the latter is the thicker of the two to intercept a larger area of retinal illumination at the cost of reduced regional acuity (Packer et al., 1989; Hoang et al., 2002). As flux is funneled from the entrance aperture to a smaller exit aperture at the distal end of ellipsoids, structural modifications become feasible in outer segment tapering and size. Cone miniaturization has several advantages, including improved signal-to-noise ratio, faster visual pigment regeneration (i.e., recovery from bleaching), and material savings as a result of reduced volume. The drawbacks are reduction in absolute sensitivity vis-à-vis rods and some light losses from detection as a result of ellipsoid leakage. Rods, on the other hand, are specialized for high absolute sensitivity and not for rapid response (Burns and Lamb, 2003). The eyes of deep-sea fishes provide exquisite examples for such sensitive detector structures (Locket, 1977). The cylindrical form appears well suited for on-demand tailoring of outer segment length, which may reach hundreds of micrometers in fish species living at great depths. The use of multiple banks and assemblages of rods, in addition to long ones, also appears to be aimed at catching scarce photons in the darkness of the deep ocean (Locket, 1977; Munk, 1977; Collin et al., 1998). Rods do not jeopardize detection efficiency by the use of light-concentrating schemes at the ellipsoid level; they have wider acceptance angles for oblique rays than cones do. Other mechanisms, however, may operate to concentrate light onto the outer segment. For instance, some nocturnal mammals, like the mouse, show an inverted chromatin nuclear pattern that, together with a quasi-columnar organization of rod nuclei, serves to channel light from the outer nuclear layer toward the outer segments (Solovei et al., 2009). Signal pooling from multiple rods (at a sacrifice of spatial resolution) and coincidence detection should further improve scotopic performance as well as signal-to-noise ratio (Peichl, 2005). ### Improved light funneling by oil droplets Oil droplets are optical devices used by most cones in avian, amphibian, and reptilian eyes. Oil droplets are highly refractive spherical globules (Ives et al., 1983); they may be clear or colored (Kolb and Jones, 1987; Hart et al., 2006). Because of their location at the distal end of inner segments anterior to the base of outer segments, they may serve as light filters as well as focusing devices (Young and Martin, 1984; Vorobyev, 2003). A spherical body with high refractive index immersed in a lower index medium will behave as a positive lens with a short focal length. Thus, an oil droplet (or a concentrated group of microdroplets, as in diurnal snakes; Wong, 1989) is expected to further concentrate the light funneled by the ellipsoid. As such, increased outer segment tapering and size reduction become feasible. And indeed, observational evidence bears this out: the most extreme cases of outer segment volume reduction are found in oil droplet–containing cones (Fig. 1). Oil droplet focusing might be advantageous not only in allowing further reductions in detector cross section, but also in quickening recovery from blinding exposures, as when experiencing glare. For an elaboration of the last point, note the following: whereas scotopic sensitivity in primate and teleost vision may take tens of minutes to regain dark-adapted levels after a bright “bleaching” exposure, photopic sensitivity returns to former levels in a few minutes (Rushton, 1965; Thomas and Lamb, 1999; Kenkre et al., 2005; and unpublished data for teleosts). This indicates that cones recover their sensitivity in vivo faster than rods do. Experiments in vitro also reveal the same tendency in chemical regeneration of visual pigment: in the presence of copious amounts of exogenous 11-cis retinal, whereas rods regenerate slowly, cones recover rapidly and repeatedly after several bleaching exposures, and, in each case, they regain most of their visual pigment in a few seconds (Hárosi, 1984). Consider now the oil droplet–equipped cones in a fishing bird’s eye. Given the short focal lengths of spherical lenses, an oil droplet will focus light in a specific region of an outer segment lamella, which would receive intense illumination. Let’s assume, for the sake of the argument, that only 10% of a lamella is illuminated. When the bird skims over a body of water, reflected sunlight could be a blinding experience. In case of a human observer, extensive bleaching in retinal receptors would occur, resulting in temporary blindness. However, in cones with oil droplet focusing, only a fraction of the visual pigment complement would get bleached, and, in the assumed lamella, 90% of the visual pigment would remain unexposed. Because lateral and rotational diffusion drives visual pigment molecules rapidly in the receptor membranes (Wang et al., 2008), the bleached molecules would quickly be exchanged with unbleached ones from the adjacent membrane area. Therefore, most of the cone’s sensitivity could recover in milliseconds, at least three orders of magnitude faster than a recovery based on normal biochemical regeneration (Kenkre et al., 2005). The aforementioned mechanism could improve the foraging performance of birds like the black phoebe, a sit-and-wait predator whose visual searching increases significantly under bright light conditions, likely as a result of the negative effects of glare (Gall and Fernández-Juricic, 2009). Other bird species like herons and osprey routinely hunt in shady habitat or at crepuscular periods (unpublished data), reducing their exposure to glare and other blinding factors such as the light flickers produced by waves near the water surface. In fact, such flicker may be used by fish for camouflage, as multiple species have developed body markings that resemble the light patterns (McFarland and Loew, 1983). Fast-moving predators, like some insects and birds, have critical fusion frequencies that surpass the predominant flicker occurring in surface waters (in the range of 1–5 Hz), improving visual contrast of underwater targets (McFarland and Loew, 1983). Given a mean diffusion coefficient of 0.4 µm2 s−1 for activated opsins (Wang et al., 2008), between ∼3 and 13% of the (bleached) visual pigment molecules in a 1-µm-diameter lamellae would get replaced by intact ones between flickers, contributing to fast recovery from bleaching. ### On light collection mechanisms The idea that cone inner segments concentrate light has been around for a long time. For instance, O’Brien (1946) proposed it in his theory explaining the Stiles-Crawford effect (O’Brien, 1951; Johnson and Tansley, 1956; Enoch, 1963). Concerning vertebrate photoreceptors, turtle cone ellipsoids have been compared with ideal light collectors (Baylor and Fettiplace, 1975; Winston, 1981). The name refers to nonimaging optical devices that concentrate light by internal reflection onto the smallest possible exit aperture (Winston, 1970). In the case of an invertebrate eye, Levi-Setti et al. (1975) suggested that the crystalline cone in an ommatidium of Limulus is an ideal light collector. Subsequent work by Land (1979), however, showed that the crystalline cones can form images and that they pass light through a refractive index variation scheme. Similar refractive index gradients have not been uncovered in vertebrate photoreceptors, and the mechanism of importance remains to be sorted out. Internal reflection, refraction, and even diffraction may play some role. Cone ellipsoids are not truly homogeneous, as they contain abundant mitochondria that might function akin to a Fresnel zone plate. The one conclusion that is certain at present is that an oil droplet must have a major contribution to the refractive power of the system. Another open question is whether investigations of waveguide modal patterns in vertebrate visual cells would facilitate the understanding of structure–function relationships between photoreceptor compartments (Snyder and Menzel, 1975). ### Oblique incidence and the Stiles-Crawford effect Stiles and Crawford (1933) discovered that the visual sensitivity of the human eye depends on the direction from which light enters the pupil (known as the Stiles-Crawford effect of the first kind [SCE1]). O’Brien (1946) was the first to advance a theory to explain it, and there have been others (e.g., Snyder and Pask, 1973; reviewed by Enoch and Bedell, 1981). The third hypothesis is consistent with the SCE1. Accordingly, cone ellipsoids are imperfect concentrators; they lose light by leakage. The lower refractive indices of cone inner and outer segments provide reduced critical angles as compared with those of rods (Appendix 1). But rods, with more uniform and denser distal compartments, have larger critical angles and thus can tolerate a wider range of off-axis rays. The combination of these properties may be sufficient to account, at least qualitatively, for the larger photopic and smaller scotopic SCE1 (Enoch and Bedell, 1981). Further support for the third function comes from a finding by Westheimer (1967) that foveal cones of the human eye have a reduced (i.e., rodlike) Stiles-Crawford effect when compared with parafoveal cones. Also supportive is the small directional sensitivity displayed by the human achromat, an abnormal condition in which vision depends entirely on rod function (Nordby and Sharpe, 1988). ### Summary We have provided evidence to support the notion that cone outer segment taper follows from the shape of the ellipsoid, a structure that serves to concentrate light onto the outer segment. The advantages of taper and cone miniaturization include compensation for light loss caused by self-screening, metabolic savings in structural components, higher signal-to-noise ratio, and accelerated regeneration of visual pigment. The trade-offs are in absolute sensitivity and visual acuity. As suggested by the different sizes of photoreceptors and their relative densities across phylogenetic groups, species have evolved visual cells that presumably optimize these trade-offs for life in particular environments. As such, there is no model cone or rod but a range of sizes and shapes dictated by the ecological constraints guiding the evolution of each species. ## APPENDIX 1 ### Derivation of equations to test the first biophysical function: Cone outer segments taper to compensate for light flux diminution by absorption (self-screening) so that flux density remains invariant or increases with axial distance along the outer segment The list of symbols and definitions used in the testing of this function are shown in Table 3. The following sections present geometric and spectrophotometric definitions of key variables (e.g., absorption coefficient and taper) used to evaluate whether a cone taper is critical, i.e., whether it compensates exactly for light flux diminution by self-screening (Hodgkin and O’Bryan, 1977). As a consequence, taper that is equal to or exceeds the critical value indicates compensation for self-screening. ### Geometric derivation of the absorption coefficient Perfect compensation for light absorption caused by self-screening imposes two requirements. The first is invariance of flux density with respect to z, so that Jo = Jz; in other words, the light flux impinging on the base (Φo) divided by the cross-sectional area (Ao) is equal to the transmitted flux (Φz) divided by the corresponding cross section (Az) along the entire length of the outer segment. Accordingly, $Φo/Ao=Φz/Az.$ (1) The second requirement is that visual pigment absorption be the only reason for the diminution of flux (i.e., light leakage is absent). This means that the incident light rays interact with the cell boundary at angles (Θ) below the critical angle (Θc) so that total internal reflection prevents light from escaping (see the section Critical angle estimations for data confirming this). Thus, $Θ≤Θc.$ (2) The exponential law of absorption, $Φz=Φoexp(−az),$ (3) sets forth the variation of flux in an absorbing medium along the z direction (Φz) in terms of the incident flux (Φo) multiplied by an exponential function of variable (z) and a characteristic constant of the medium, called the absorption coefficient (a). The latter is inversely proportional to the attenuation, or characteristic, length, at which Φ falls to exp(−1) = 0.368 of Φo. Upon combining Eqs. 1 and 3, we obtain $Φz/Φo=Az/Ao=exp(−az).$ (4) The circular cross section of a cone with diameter d is A = d2π/4, and thus, Eq. 4 may be written as $dz=doexp(−az/2),$ (5) wherefrom the absorption coefficient for testing the first function is expressible as $ah1=(2/z)ln(do/dz).$ (6) The surprising outcome gleaned from Eq. 6 is that the absorption coefficient, a purely spectroscopic quantity, may be determined from measurements of distance involving the base diameter, do, a second diameter, dz, and their separation along the z coordinate. ### Spectrophotometric determinations of the absorption coefficient Two methods have been established for the determination of the axial absorption coefficient of a vertebrate photoreceptor outer segment in situ (Hárosi and MacNichol, 1974; Hárosi, 1975, 1982). The first one requires knowledge of the visual pigment’s molar extinction coefficient, ε, its equivalent (random) molar concentration, c, and the dichroic ratio, R, or some equivalent measure of anisotropy. The second method makes use of the transverse specific absorbance, S, determined in a side-on oriented outer segment. The two methods are interdependent by the dichroic ratio, R, as shown in the section Anisotropic absorbance by visual cells. ### Isotropic absorbance by solutions The light-absorbing property of a homogeneous, isotropic medium is expressed in terms of either the natural or the decadic logarithm of incident to transmitted light fluxes. Thus, absorbance (optical density) may be written as $D1= ln(Φo/Φz)=αcz$ (7) $or D2=log(Φo/Φz)=εcz,$ (8) in which c is the concentration, z is the path length, and α is the molecular or chromophoric (Napierian) and ε is the molar (decadic) extinction coefficient of the light-absorbing substance. Substituting Eq. 3 in Eqs. 7 and 8, the absorption coefficient for a random ensemble of absorbing molecules can be related to α and ε as $a=αc=2.303εc.$ (9) Note that a, α, and ε are applicable to isotropic pigments and that they are also wavelength dependent. Their values at λmax are often used as single-valued parameters (e.g., Warrant and Nilsson, 1998). ### Anisotropic absorbance by visual cells Rod and cone outer segments in side view exhibit intrinsic linear dichroism, revealing their anisotropic nature. Because the absorption vectors of the visual pigment molecules lie nearly parallel with the lamellar (x-y) planes, axially traveling light (z direction) is well absorbed; this makes photoreceptors more effective light catchers in this direction than an isotropic solution of the same pigment would be at the same concentration and path length. By a generalization of Eq. 9, the first formula to express the end-on (axial) absorption coefficient along the z direction is $az′=kαc=2.303kεc,$ (10) in which k is the anisotropy factor accounting for the gain by the ordered distribution; k may be as large as 1.5 (perfect two-dimensional random array) and as small as 1.0 (three-dimensional random array). By polarized absorbance measurements of side-on oriented receptors, the dichroic ratio, defined as R = A/A, can be determined, and k can be expressed as a function of R, as follows. If a cylindrical vessel is homogeneously filled with N absorbing molecules, each possessing a transition moment M, the total extinction (E = NM) may be written as E = Ex + Ey + Ez. The fractions expressed in the three spatial coordinates are equal for a three-dimensional random array so that Ex/E = Ey/E = Ez/E = 1/3. The measurable fraction of extinction in the z direction is half of the two orthogonal components: $0.5(Ex+Ey)/E=1/3.$ (11) In a similar model, a cylindrical photoreceptor may be regarded, in its simplest form, as an imperfect two-dimensional random array for which Ex = Ey and Ex/Ez = R. With these, the z fraction is expressible as $Ez/E=Ez/(2Ex+Ez)=1/(1+2R).$ (12) Viewed from the z direction (i.e., end-on), the measurable fraction is $0.5(Ex+Ey)/E=0.5(E−Ez)/E=0.5[1−1/(1+2R)]=R/(1+2R).$ (13) The anisotropy factor (k) in the z direction is obtainable from the above as the ratio of end-on extinctions (the two-dimensional versus the three-dimensional fractions) and is given by (see Hárosi, 1975, 1982) $k=3R/(1+2R).$ (14) Thus, ε, c, and R yield the axial absorption coefficient of a photoreceptor with the aid of Eqs. 10 and 14. The second formula for the calculation of the axial absorption coefficient makes use of the microspectrophotometrically determined transverse specific absorbance, S = A/d, in which A is the peak absorbance (of the α band) for transversely polarized light of the outer segment in side-on orientation and d is the mean cell diameter, presumed to be equal with transverse path length. The following relationship, $(εc)=S⊥(1+2R)/3R,$ (15) has been found useful in relating in vitro spectrophotometric data to those obtained by in situ microspectrophotometry (Hárosi, 1975). The substitution of Eqs. 14 and 15 into Eq. 10 leads to the desired equation: $az″=2.303S⊥.$ (16) Therefore, the axial absorption coefficient is also attainable from the transverse specific absorbance by Eq. 16. Anisotropy is implicit in S since Eq. 15 may also be written as S = k(εc). The latter relationship shows the interdependence of Eqs. 10 and 16, which, therefore, lead to dependent estimates of the axial absorption coefficient, even though the two are based on different determinations. ### A definition of taper As a geometrical characterization of a cone taper, τ is defined here as the angle between the axis and the inclination of the contour line, which, upon precession, describes the conical surface: $τ=tan–1[(do−dz)/2z].$ (17) The axial distance, z, is the separation between the base of a right cone, with diameter do, and a parallel slice, with diameter dz. For a cone cell, τ describes an average inclination of the contour line between two slices (corresponding to do and dz), and it cannot account for a point-by-point variation that Eq. 5 describes. ### Critical angle estimations The refractive indices for rod and cone inner and outer segments have been measured only for a few animal species in a handful of studies. The obtained values are as follows: the refractive index for rod ellipsoid and outer segment, 1.40 and 1.41, respectively; for cone ellipsoid and outer segment, 1.39 and 1.385, respectively; and for the extracellular matrix (mucopolysaccharide), 1.34 (see Borwein, 1981 for primary citations). At a boundary between two transparent media, one denser with refractive index n1 (inside), the other rarer with refractive index n2 (outside), light rays will refract according to Snell’s law: $n1sinϕi=n2sinϕo,$ (18) in which ϕi and ϕo are the corresponding angles of incidence and refraction with respect to the normal. Beyond a certain angle of incidence in the denser medium, no refraction occurs, and there is total internal reflection. The critical angle (for which sin ϕo = 1) can be expressed from Eq. 18 as $ϕc=sin–1(n2/n1).$ (19) From this, Θc of Eq. 2 can now be obtained as $Θc=90°−ϕc.$ (20) Substitution of the numerical values cited above in Eqs. 19 and 20 leads to the following critical angles in Θc: 15.4° and 14.6° for cone ellipsoid and outer segment and 16.8° and 18.1° for rod ellipsoid and outer segment, respectively. In these calculations, we assumed that the relevant optical parameters used in critical angle calculations are those that have been determined (Borwein, 1981), notwithstanding the fact that photoreceptors and their surroundings are neither homogeneous nor isotropic. What constitutes the surrounding medium is especially questionable. This, sometimes referred to as interstitial matrix, is commonly equated with mucopolysaccharide, as done here. However, microvilli of the pigmented epithelial cells and the calycal processes that come in contact with at least parts of inner and outer segments may play a role in setting the refractive properties of these cells. Outer segment taper, as defined by Eq. 17 and determined from our observations and published photomicrographs (see Results), indicate angles <14°. These values are within the requirement of Eq. 2, and thus, they justify the original assumption concerning the funneling of axial rays by total internal reflection. ## APPENDIX 2 ### Derivation of equations to test the second biophysical function: Cone outer segments taper to improve signal-to-noise ratio along their lengths The list of symbols and definitions used to evaluate this function are presented in Table 4. We make the following five assumptions. (1) Incident light consists of an axial flux of parallel, uniform, and steady illumination at the base of outer segments with negligible reflection and scattering losses. (2) Visual pigment molecules are packed uniformly in transverse membranes of all rods and cones with constant surface density: Jn = n/At = no/Ao = nz/Az. (3) Activation of visual pigment molecules requires not only light absorption but also chromophore isomerization, commonly referred to as bleaching. To take this into account, the quantum efficiency of bleaching, γ, must be factored in (Dartnall et al., 1936). (4) Signal is constituted by a photocurrent generated through the cell envelope at each lamella, independent of other lamellae. Although there clearly is an observable spreading of excitation, for the sake of simplicity, all messengers for signal are assumed to originate in one lamella. (5) Noise is generated like signal in lamellae under the conditions set forth in the preceding assumption. Whether thermal isomerization of the chromophore or subsequent biochemical steps are the cause (Rieke and Baylor 1996, 2000; Sampath and Baylor, 2002; Holcman and Korenbrot, 2005), it is assumed here that noise is proportional to the number of visual pigment molecules contained within each layer. The objective of this appendix is to relate the generation of signal and noise to outer segment taper. Given an absorbing medium, an incident flux will either be absorbed or transmitted, provided that assumption 1 holds. In general terms, $Φo=Φa+Φz.$ (21) When normalized to the incident flux, Eq. 21 may be rearranged as $Φa/Φo=1−Φz/Φo.$ (22) The left-hand side in Eq. 22 is usually referred to as absorptance, Ab = Φao, whereas the fraction on the right-hand side is called transmittance, T = Φzo. Optical density (or absorbance), as defined previously in Eq. 7, is the logarithm of T−1. After substitution of Eq. 3 in Eq. 22 and rearrangement, the rate of photon absorption by a monolayer of thickness δ, with incident flux Φo, is $Φa=Φo[1−exp(−aδ)].$ (23) Because visual pigments are hydrophobic chromoproteins, they must be membrane bound. For this reason, the shortest meaningful axial path length of the visual pigment in a vertebrate photoreceptor is a single layer. Although the exact numerical value of δ for the present is immaterial, it is assumed to be 15 nm. This is half the repeat distance of the 30 nm obtained for rod disks by electron microscopic and x-ray crystallographic determinations (for references, see Fein and Szuts, 1982). The value of δ = 1.5 × 10−6 cm permits estimation of the magnitude of the exponent in Eq. 23. Based on Eqs. 10 and 14, the absorption coefficient for a “typical” rhodopsin-containing cone is expected to be near 420 cm−1 (a−1 = 24 µm), which would make the value of the exponent ∼6.3 × 10−4. In view of the probable magnitude of δ and assumption 2, and also considering pigment anisotropy according to Eq. 10 and expressing the concentration as $c=n/Atδ=Jn/δ=nz/Azδ,$ (24) we can derive the following formula from Eq. 23: $Φa=Φo[1−(1−kαcδ)]=Φokαnz/Az,$ (25) in which the exponential term in Eq. 23 was represented by the first two terms of its series expansion (the second and higher order terms being negligibly small). Thus, the rate of absorption in a pigment layer is proportional to the incident light flux times the products of anisotropy factor (k), pigment type (α), and surface density of the pigment (Jn = nz/Az). By extending Eq. 25 to describe a multilayered system with self-screening, the rate of absorption by one layer at depth z is $Φaz=kα(nz/Az)Φoexp(−az).$ (26) A second method of obtaining the rate of quantum absorption by a single layer of visual pigment is based on the interpretation of the chromophoric absorption coefficient (α) as the absorption cross section of one molecule in a random array (Dartnall, 1972). The probability of a quantum at λmax to be caught by a single layer of visual pigment, with anisotropy factor k, is expressible as a ratio of the sum of all the molecular cross sections to that of the available total area: $pz=kαnz/Az.$ (27) When we scale up from one photon to an incident photon flux of Φo, Eq. 27 will reproduce the preceding relationship described by Eq. 25. The rate of signal generation in a lamella at z is expected to be proportional to the rate of photon absorption in that layer (assumptions 3 and 4). Thus, the signal should be formally similar to Eq. 26: $Sz=CSkαγ(nz/Az)Φoexp(−az),$ (28) with CS being a conversion factor between photon absorption and corresponding photocurrent. The noise produced at layer z may be given (assumption 5) as $Nz=CNnz,$ (29) in which CN is again a conversion factor. In taking the ratio between Eqs. 28 and 29, nz will cancel out, and the signal-to-noise ratio becomes (with C′ as a new constant) $Sz/Nz=C′kαγ(Φo/Az)[exp(−az)].$ (30) ## APPENDIX 3 ### Derivation of equations to test the third biophysical function: Cone outer segments taper in accordance with the optical properties of their inner segments, facilitating light capture and reducing use of biomaterials The list of symbols and definitions used to evaluate this function are presented in Table 5. The following sections introduce definitions of key variables (e.g., concentration factor) for its evaluation. Assuming a perfect two-dimensional random array of absorbers, the angle of incidence with respect to the optic axis (Θ) will reduce the absorption probability of unpolarized light by a factor (Winston, 1981): $ƒ=0.5(1+cos2Θ).$ (31) Evaluated for Θ = 15°, Eq. 31 yields ƒ = 0.97. Thus, oblique incidence for angles up to 15° causes only a small drop (3%) in absorption efficiency. In the following treatment, we assume, therefore, that oblique incidence does not significantly affect the in situ absorption efficiency of the visual pigment molecules embedded in the transverse lamellae. ### Flux concentration in a tapered outer segment In the absence of absorption, the flux density of a converging (conical) beam increases in the direction of convergence. The conceptually simplest case occurs when the incident cone of light matches exactly the outer segment taper. This, however, leads to the same analysis covered previously, where an axially parallel beam was assumed to be incident on the base of a tapered outer segment, in which total internal reflection prevailed. In terms of flux densities, Eq. 26 may also be written as $Jaz=Jo(Ao/Az)kα(nz/Az)exp(−az)$ (32) to indicate a rate of absorption density increase by the factor Ao/Az. For obtaining the signal generation in a lamella at z, however, the total absorbed flux is needed, not the flux density. Multiplying Eq. 32 by Az, though, takes us back to Eq. 26. Therefore, the preceding analysis leading to Eq. 30 is also valid for this case. Besides the matching case, the convergence of the beam incident upon the outer segment may also be lesser or greater than that of the structure. The slightly convergent case is the simplest one, and it may be handled as the axially parallel beam was above (see previous paragraph), necessitating no new analysis. The greater beam convergence, however, warrants further considerations. Some aspects of the latter case are discussed in the manuscript in connection with the effect of oil droplet focusing. ### Flux concentration in a cone ellipsoid The third function presupposes the existence of a mechanism whereby cone ellipsoids funnel light from a broader, proximal portion toward a narrower, distal end and that this property makes a significant impact on the structure and function of outer segments. Consider a flux Φi incident at the entrance aperture (largest cross section) of an ellipsoid to produce a flux density Ji. If this flux is coupled without loss to a smaller exit aperture, where the flux density is Jo, the following relationships hold (provided the cross sections are circular with respective areas and diameters of Ai, di and Ao, do): $Jo/Ji=Ai/Ao=(di/do)2=FC.$ (33) The significance of FC, named here concentration factor, is that it shows the proportion by which the base of a photoreceptor can be reduced in area while still capable of detecting all of the incident flux Φi. Although the issue of light losses by leakage remains to be considered, this property reveals the feasibility of detector miniaturization. ### Volume reduction of a cone outer segment Whereas the outer limb of rods approaches the cylindrical form in nearly all instances, cone outer segments usually appear truncated, not pointed. Therefore, the frustum of a cone is a more realistic representation of a cone outer segment. The volume of the frustum of a cone is defined with a diameter of base do, of tip dt, and altitude h as $Vc=(π/4)(h/3)[do2+dodt+dt2].$ (34) Compared with the volume of a cylindrical rod, Vr = (π/4)hdo2, their ratio is defined as $Vr/Vc=FG,$ (35) in which FG is named the geometry factor. Experience shows that the value of FG is variable and tends to fall between 1.5 and 3, the latter being the largest for a right cone (dt = 0). With these two factors combined, the volume reduction ratio is defined as $VR=Vreq/Vc=FGFC.$ (36) VR is an indicator of proportion between the outer limb volumes of a cone and an equivalent rod, when both have equal inner segment entrance aperture and incident flux. Discounting light losses, these two cells could produce the excitation of an equal number of visual pigment molecules to equal illumination. Clearly, the cone is the more efficient receptor of the two because it uses only a fraction of the rod’s detector apparatus. This means reduced amounts in lipid membrane, visual pigment, and all the other components of the enzymatic cascade required for generating signals in terms of photocurrent modulation. ## Acknowledgments Dr. Ferenc I. Hárosi passed away in November 2008. During his career, he was the leading innovator in the field of microspectrophotometry, starting with the making of the first computerized dichroic microspectrophotometer in the 1970s. His insights into photoreceptor physiology and visual pigment properties opened new fields of investigation that are actively pursued by many a prominent scientist today. He was my friend and mentor and an inspiration to vision scientists of all ages. Dr. Hárosi is, and will always be, profoundly missed; thank you for everything, friend. We are grateful to Drs. Gregor J. Jones, Edward F. MacNichol Jr., and Ete Z. 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Electric field lines in a uniformly charged dielectric solid sphere What would electric field lines in the interior of a uniformly charged dielectric solid sphere (charge spread throughout the volume with a uniform charge density) look like? How do we even go about visualising field lines? • This is a standard exercise and it is covered in depth in most (all?) introductory textbooks. – Emilio Pisanty Jun 27 '17 at 14:32 • @EmilioPisanty I couldn't find it, would you point me to some book that does? – Ritik Garg Jun 27 '17 at 14:36 • An exercise in Chapter 4 of Griffiths's Introduction to Electrodynamics involves calculating the electric field of a uniformly charged dielectric sphere. I don't have the fourth edition in front of me, but it's Exercise 4.20 in the third edition. – Michael Seifert Jun 27 '17 at 21:30 If you have a sphere with uniformly distributed charge, the solution must be spherically symmetrical. Specifically, we know that the field intensity at a radius $r$ is proportional to the charge inside the sphere with radius $r$, and scaled by the dielectric constant: $$\nabla\cdot \mathbf E=\frac{\rho}{\epsilon}$$. It follows that the field will increase linearly with $r$ (because it will scale as $$\rm\frac{volume}{area}=\frac{\frac43 \pi r^3}{4\pi r^2}\propto r$$ Once you get to the edge of the sphere, the field will drop off in the usual $1/r^2$ manner. This is a tricky thing to visualize with lines - it's a bit easier with colors: (note - in this picture I assumed the dielectric constant of the sphere was 3; this suppresses the electric field inside compared to outside. Tip of the hat to Michael Seifert for pointing out that I had shown the case for $\epsilon_r=1$ without mentioning this explicitly). • Thanks, but colors don't do the trick. With colors one can't make out the direction of the field as with lines. – Ritik Garg Jun 27 '17 at 17:16 • The field lines are purely radial from the center out. But since charge is distributed, field lines would have to "appear" at different distances - this is hard to do cleanly when the charge distribution is continuous (rather than discrete). – Floris Jun 27 '17 at 17:32 • The electric field magnitude should be discontinuous at the surface of the sphere, as there is a bound surface charge there. – Michael Seifert Jun 27 '17 at 21:26 • (For a dielectric medium, that is. Your answer is correct for $\epsilon_r = 1$.) – Michael Seifert Jun 27 '17 at 21:32 • @MichaelSeifert that is an important point I completely omitted to mention - fixed now. Thank you! – Floris Jun 27 '17 at 21:48	{}
## March 31, 2006 ### Polarization We had a beautiful talk today by our own Eiichiro Komatsu on the latest WMAP data. A big part of the talk was devoted to the polarization data, and how this greatly helped constrain the cosmological parameters. Chiefly, the polarization data lets you pin down the (integrated) optical depth, $\tau$, where $\tau(z) = c \sigma_T \int_0^z n_e(z') (dt/dz') dz'$ $\sigma_T$ is the cross section for Thompson scattering, $n_e(z)$ is the number density of free electrons. Before decoupling (at $z\sim 1088$), the optical depth, $d\tau/d z \gg 1$. After decoupling, it drops to near zero. Then, at reionization (when the first stars turn on), it goes back up. Polarization of the CMB is created at decoupling, due to velocity gradients in the primordial plasma. At reionization (somewhere between $11\lesssim z_r \lesssim 30$), again, polarization is created as the free electrons scatter off the quadrupole moment of the incident CMB photons. The latter effect produced polarization primarily in low-$l$, and is cleanly distinguishable from the “primordial” polarization created at decoupling. With WMAP-3, they were able to determine $\tau(z_r)$ at reionization. With a few more years of data, Eiichiro says they will actually be able to measure $\tau(z)$. By measuring $\tau$, one breaks a degeneracy in the determination of the cosmological parameters from the temperature alone. For instance, just looking at the temperature data, a larger $\tau$ can be compensated by (among other things) larger values of the tilt, $n_s$, and of the tensor-scalar ratio, $r$. Once you pin down $\tau$, you narrow the window for these cosmological parameters. Eiichiro ended his talk with this graph: Range of non-flat cosmology models consistent with the WMAP data only. The models in the figure are all power-law CDM models with dark energy and dark matter, but without the constraint that $\Omega_\Lambda+\Omega_m=1$. The different colors correspond to values of the Hubble constant as indicated in the figure. (WMAP Three Year Results: Implications for Cosmology) mostly to show that, from WMAP data alone, one learns that $\Lambda=0$ requires a ridiculously low value of the Hubble constant. But what really leaps out at you is that the WMAP data seems to favour positive spatial curvature (a “closed” FRW universe). This tendency is somewhat diminished when you combine WMAP with other datasets: Joint two-dimensional marginalized contours (68% and 95%) for matter density, $\Omega_m$, and vacuum energy density, for power-law CDM models with dark energy and dark matter, but without the constraint that $\Omega_\Lambda+\Omega_m=1$. The panels show various combinations of WMAP and other data sets. While models with $\Omega_m = 0.415$ and $\Omega_\Lambda= 0.630$ are a better fit to the WMAP three year data alone than the flat model, the combination of WMAP three year data and other astronomical data favors nearly flat cosmologies. (WMAP Three Year Results: Implications for Cosmology) The combined data favour a universe that is nearly spatially flat. But it does seem that the bulk of the allowed parameter space lies in the region of positive spatial curvature1. 1 This probably won’t keep anyone up at night. But those who like to imagine that inflation in our universe started via bubble nucleation will note that the Coleman-de Luccia bubble has an FRW slicing with negative spatial curvature. Posted by distler at March 31, 2006 2:09 AM TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/779	{}
Parameterized Complexity and Practical Computing Workshop # Parameterized Complexity and Practical Computing Workshop The objective of the workshop is to build bridges between parameterized complexity theory and practical applications. ### Keynote Speakers and Their Application Areas • Prof. Gregory Gutin (Royal Holloway, University of London) - Di-graphs, Computer Security-Cryptography, Workflow Satisfiability. • Prof. Klaus Jansen (University of Kiel) - Scheduling problems • Prof. Blair Sullivan (University of Utah) - Data-driven science: computational biology (genomics), quantum computing, social networks. ### Location and Dates • Dates: The workshop will take place in Bergen, Norway between 05 and 09 August. • Place: The first two days (Monday and Tuesday, 05 and 06 August) will be held at the Solstrand Hotel. • Place: Day 3 (Wednesday 07 August) will be a fjord trip offering the opportunity for continued discussion. See below. • Place: Days 4 and 5 (Thursday and Friday, 08 and 09) talks will be presented at the University of Bergen. ### Submissions The workshop aims to bring together researchers and practitioners of Parameterized algorithms to share their perspectives and experiences. We solicit submissions describing research results with practical applications, and position statements leading to new directions, standpoints, and learned lessons. There will be long (40 minute) and short (20 minute) talks and discussion time. Exact format TBA. ### Abstract Submit an abstract with your preference for length of talking time to: Mateus De Oliveira Oliveira <Mateus.Oliveira@uib.no>. In the Subject line put the words: PCPC REGISTRATION. ### Registration There is no fee for the workshop. The following expenses will be covered by Mike Fellows' Norwegian Toppforsk Grant: • Accommodation to all participants for one night (August 05 to August 06) at the Solstrand Hotel. Meals are provided at the hotel. • Buses from UiB to the workshop venue on the morning of August 05 and back to UiB on the evening of August 06. • Taxis from the airport to Solstrand Hotel for people arriving at Bergen airport on the morning of August 05. If you are contemplating this possibility please make taxi arrangements through the hotel so that the hotel can arrange joint taxi if suitable. • Fjord trip outing on Wednesday to continue discussions. ### Fjord Trip on Wednesday, Day 3 There will be a cruise trip to the fjords on Wednesday, 07 July. The cruise will pass through deep fjords, steep mountains, mighty waterfalls, and powerful currents. The duration of the trip is about 3 hours. • The cruise is called Fjordcruise Bergen-Mostraumen. • The departure is from Zachariasbryggen quay at the Fish Market in Bergen. • The tour starts at 09:45 until 12:45. • Be at the dock by 09:15 AM. (1/2 hour early) ### Schedule 05.08 • 08:00 - Bus departs from UiB's High Technology Centre (where the Department of Informatics is located.) • 09:00 - Arrival at Solstrand Hotel • 09:30 - Keynote 1 - Gregory Gutin - Theoretical and practical algorithms for CSP parameterized by the number of variables with value-independent constraints (slides) • 10:30 - Coffee break • 11:00 - Regular Talk 1 - Alexandra Lassota - Near-Linear Time Algorithm for n-fold ILPs via Color Coding • 11:45 - Regular Talk 2 - Cornelius Brand - Algebraic Methods in Parameterized Theory and Practice • 12:30 - Lunch • 14:00 - Keynote 2 - Klaus Jansen - Integer Programming and Convolution, with Applications in Scheduling and Knapsack Problems • 15:00 - Coffee break • 15:30 - Regular Talk 3 - Yijia Chen - From Tree-depth to Shrub-depth, Evaluating MSO-Properties in Constant Parallel Constant Time (slides) • 16:15 - Time for Discussions • 19:00 - Aperitif • 19:30 - Dinner 06.08 • 09:00 - Keynote 3 - Blair Sullivan - Parameterized Graph Algorithms in the Field • 10:00 - Short talk 1 - T. C. van der Zanden - Efficiently Computing the Shapley Value of Connectivity Games in Low-Treewidth Graphs • 10:30 - Coffee break • 11:00 - Regular talk 4 - Max Deppert - Near-Linear Approximation Algorithms for Scheduling Problems with Batch Setup Times • 11:45 - Regular talk 5 - Kim-Manuel Klein - About the Complexity of 2-Stage Stochastic IPs • 12:30 - Lunch • 14:25 - Short Talk 2 - Lars Jaffke - Typical Sequences Revisited • 14:50 - Regular Talk 6 - Engineering Kernelization for Maximum Cut • 16:00 - Bus Leaves Solstrand 07.08 • 09:15 Fjord Trip 08.08 (Meeting at UiB) • 09:00 - Short Talk 3 - Peter Shaw - Novel applications of Capacitated Cluster-Editing with vertex Split operation (slides) • 09:30 - Short Talk 4 - Peter Shaw - Turbo charging heuristics: adjusting the parameters for optimum performance (slides) • 09:55 - Coffee break • 10:20 - Short Talk 5 - Davis Issac - Parameterized Complexity of Biclique Cover and Partition (slides) • 10:50 - Short Talk 6 - Mike Fellows - TBA • 11:15 - Pragmatic Ideas and Setup for Work Sections • 12:00 - Lunch • 13:00 - Problem Solving / Work Sections 09.08 • 11:00 - Short Talk 7 - G. Phillip - Diverse FPT (slides) • 11:30 - Report on Work Sections ## Keynote Talks Title: Theoretical and practical algorithms for CSP parameterized by the number of variables with value-independent constraints Speaker: Gregory Gutin (Royal Holloway, University of London, UK) Abstract: Unfortunately, CSP parameterized by the number of variables is W[1]-hard and in W[2]. Fortunately, CSP parameterized by the number of variables with value-independent constraints is FPT and of interest in (security) access control. We'll discuss theoretical results for such a CSP as well as results of computational experiments with FPT algorithms and SAT and CSP solvers. Our experiments demonstrate that the best FPT algorithm significantly outperforms the solvers. Title: Integer Programming and Convolution, with Applications in Scheduling and Knapsack Problems Speaker: Klaus Jansen, Univ. Kiel Abstract: Integer programs (IP) with a constant numberm of constraints are solvable in pseudo-polynomial time. We give a new algorithm based on the Steinitz Lemma and dynamic programming with a better pseudo-polynomial running time than previous results. Vectors $v_1,\ldots,v_n$ in $R^m$ that sum up to the $0$ vector can be seen as a circle in $R^m$ that walks from $0$ to $v_1$ to $v_1 + v_2$, etc. until it reaches $v_1 + \ldots + v_n = 0$ again. The Steinitz Lemma says that if each of the vectors is small with respect to some norm, we can reorder the vectors in a way that each point in the circle is not far away from $0$ w.r.t. the same norm. We show in the talk that a solution to the IP $\max c^T x, A x = b, x >= 0, x in Z^n$ can be found in time $O(m \Delta)^{2m} log(\|\|b\|\|_\infty) + O(nm)$ where $\Delta$ is the biggest absolute value of any entry in $A$. Moreover, we establish a strong connection to the problem $(min,+)$-convolution. $(min,+)$-convolution has a trivial quadratic time algorithm and it has been conjectured that this cannot be improved signiﬁcantly. We show that further improvements to our pseudo-polynomial algorithm for any ﬁxed number $m$ of constraints are equivalent to improvements for $(min,+)$-convolution. This is a strong evidence that our algorithm’s running time is best possible. We also present a faster specialized algorithm for testing feasibility of an integer program with few constraints. Our algorithm for the feasibility problem runs in $O(m \Delta)^{m}\log(\Delta) \log(\Delta + \|\|b\|\|_\infty) + O(nm)$. Finally we show for the feasibility problem also a tight lower bound, which is based on the Strong Exponential Time Hypothesis (SETH), and give some applications for knapsack and scheduling problems. This is joint work with Lars Rohwedder (Univ. Kiel). Title: Parameterized Graph Algorithms in the Field Speaker: Blair Sullivan Abstract: The field of network science has burgeoned in the last two decades, developing new methods for analyzing complex network data of ever-increasing scale. Surprisingly, few tools from structural graph theory and parameterized complexity have been assimilated. In part, this is due to the primarily theoretical nature of the related literature, unrealistic structural assumptions, and a lack of cross-pollination of the research communities. In this talk, we survey several concrete applications which demonstrate the potential of these structure-based approaches in computational biology and quantum computing. We emphasize commonalities and key strategies for fostering practitioner uptake. Finally, we will highlight recent algorithmic approaches aimed at bridging this theory-practice gap. ## Regular Talks Title: Near-Linear Time Algorithm for n-fold ILPs via Color Coding Speaker: Alexandra Anna Lassota Abstract: We study an important case of ILPs $\max\{c^Tx \ \vert\ \mathcal Ax = b, l \leq x \leq u,\, x \in \mathbb{Z}^{n t} \}$ with $n\cdot t$ variables and lower and upper bounds $\ell, u\in\mathbb Z^{nt}$. In \nfold{} ILPs non-zero entries only appear in the first $r$ rows of the matrix $\mathcal A$ and in small blocks of size $s\times t$ along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that \nfold{} ILPs can be solved in FPT time regarding the parameters $s, r,$ and $\Delta$, where $\Delta$ is the greatest absolute value of an entry in $\mathcal A$. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time $\Omega(n)$, leading to a quadratic running time in $n$. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for \nfold{} ILPs with a running time that is near-linear in the number $nt$ of variables, namely $(rs\Delta)^{\cO(r^2s + s^2)} L^2 \cdot nt \log^{\cO(1)}(nt)$, where $L$ is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded Variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of $L$. Title: Algebraic Methods in Parameterized Theory and Practice Speaker: Cornelius Brand Abstract: Algebraic methods have long been employed to successfully solve parameterized problems. A flagship problem in this area is the longest-path problem, where the group-algebra approach of Koutis and Williams proved to be seminal in 2008, breaking below the running time of classic Color-Coding. A central application of this problem is the analysis of biological networks, most notably in protein-protein interaction networks, but also, more recently, in the analysis of connectomes. In this talk, we give an overview of the techniques, comment on their implementability and employability in the mentioned application areas, and discuss limitations of the approaches (including also space requirements of the algorithms). Title: From Tree-depth to Shrub-depth, Evaluating MSO-Properties in Constant Parallel Constant Time. Speaker: Yijia Chen Abstract: Courcelle's theorem states that every property definable in monadic second-order logic (MSO) can be evaluated in linear time on graphs of bounded tree-width. But given the huge size of graphs arising in practice, linear time might not be good enough. Therefore, we are interested in classes of graphs on which MSO-properties can be evaluated in parallel constant time. In this talk,I will explain two results in this direction, one is on graphs of bounded tree-depth and the other on graphs of bounded shrub-depth. This is joint work with Jörg Flum. Title: Near-Linear Approximation Algorithms for Scheduling Problems with Batch Setup Times Speaker: Max Amadeus Deppert: Abstract: We investigate the scheduling of n jobs divided into c classes on m identical parallel machines. For every class there is a setup time which is required whenever a machine switches from the processing of one class to another class. The objective is to find a schedule that minimizes the makespan. We give near-linear approximation algorithms for the following problem variants: the non-preemptive context where jobs may not be preempted, the preemptive context where jobs may be preempted but not parallelized, as well as the splittable context where jobs may be preempted and parallelized. We present the first algorithm improving the previously best approximation ratio of 2 to a better ratio of 3/2 in the preemptive case. In more detail, for all three flavors we present an approximation ratio 2 with running time (n), ratio 3/2+ε in time (nlog1/ε) as well as a ratio of 3/2. The (3/2)-approximate algorithms have different running times. In the non-preemptive case we get time (nlog(n+Δ)) where Δ is the largest value of the input. The splittable approximation runs in time (n+clog(c+m)) whereas the preemptive algorithm has a running time (nlog(c+m))≤(nlogn). So far, no PTAS is known for the preemptive problem without restrictions, so we make progress towards that question. Recently Jansen et al. found an EPTAS for the splittable and non-preemptive case but with impractical running times exponential in 1/ε. Title: About the Complexity of 2-Stage Stochastic IPs Speaker: Kim-Manuel Klein Abstract: We consider so called $2$-stage stochastic integer programs (IPs) and their generalized form of multi-stage stochastic IPs. A $2$-stage stochastic IP is an integer program of the form $\max \{ c^T x \mid \mathcal{A} x = b, l \leq x \leq u, x \in \ZZ^{s + nt} \}$ where the constraint matrix $\mathcal{A} \in \ZZ^{r n \times s +nt}$ consists roughly of $n$ repetitions of a block matrix $A \in \ZZ^{r \times s}$ on the vertical line and $n$ repetitions of a matrix $B \in \ZZ^{r \times t}$ on the diagonal. Hence it is roughly the transposed of the constraint matrix of an n-fold IP. In this talk, we present new algorithmic results on how to solve this type of IP. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters $r,s,t$ and $\Delta$, where $\Delta$ is the largest entry of the constraint matrix. The new improved bound however is obtained by a novel theorem which argues about the intersection of paths in a vector space. Title: Engineering Kernelization for Maximum Cut Speaker: Matthias Mnich Abstract: Kernelization is a general theoretical framework for preprocessing instances of NP-hard problems into (generally smaller) instances with bounded size, via the repeated application of data reduction rules. For the fundamental Max Cut problem, kernelization algorithms are theoretically highly efficient for various parameterizations. However, the efficacy of these reduction rules in practice---to aid solving highly challenging benchmark instances to optimality---remains entirely unexplored. We engineer a new suite of efficient data reduction rules that subsume most of the previously published rules and demonstrate their significant impact on benchmark data sets, including synthetic instances, and data sets from the VLSI and image segmentation application domains. Our experiments reveal that current state-of-the-art solvers can be sped up by up to multiple orders of magnitude when combined with our data reduction rules. On social and biological networks in particular, kernelization enables us to solve four instances that were previously unsolved in a ten-hour time limit with state-of-the-art solvers; three of these instances are now solved in less than two seconds. (joint work with Damir Ferizovic, Demian Hespe, Sebastian Lamm, Christian Schulz and Darren Strash) ## Short Talks Title: Efficiently Computing the Shapley Value of Connectivity Games in Low-Treewidth Graphs Speaker: T. C. van der Zanden Abstract: Game-theoretic centrality measures are a powerful tool to identify key players in covert networks (that model, e.g., the interactions between suspected terrorists or criminals). Unfortunately, such measures are often NP-hard to compute and thus intractable, even for small graphs. We show that, for three connectivity games, their Shapely value can be efficiently computed if the underlying graph has low treewidth. We provide a practical implementation of our algorithm. Using this method, we are able to compute the Shapley value for several social networks for which this was previously infeasible (including, notably, the 69-vertex graph of the terrorists involved in the 9-11 attacks studied in previous work on Shapley value-based centrality). Title: Typical Sequences Revisited Speaker: Lars Jaffke Abstract: We give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the scheduling of straight-line code (a basic block) to minimize the number of registers is solvable in quadratic time for series-parallel digraphs. Previously, a polynomial-time algorithm was known only for trees [Sethi and Ullman, JACM 1970]. We also show that the Cutwidth, Modified Cutwidth, and Vertex Separation problems can be solved in O(n^2) time for series-parallel digraphs on n vertices. (Joint work with Hans L. Bodlaender and Jan Arne Telle) Title: Parameterized complexity of biclique cover and partition Speaker: Davis Issac Abstract: A biclique of a graph is a complete bipartite subgraph of it. We look at the problems of covering and partitioning the edges of a bipartite graph with bicliques. These problems are called biclique cover and biclique partition respectively. The problems have many practical and theoretical applications. I will explain two of the practical applications, one in display optimization and the other in data compression and mining. The latter application arises due to connections with binary matrix factorization. We study the problems from the point of view of parameterized complexity, where the parameter is taken to be the size of the cover or partition. It was known that both problems are fixed-parameter tractable but the algorithms had running times that had a doubly exponential dependence on the parameter $k$. We show an algorithm for biclique partition that has only $\mathcal{2^{k^2}}$ dependence on $k$. For biclique cover, we show that no improvement over the doubly exponential dependence is possible, by giving a reduction from $3$-SAT on $n$ variables to an instance of biclique cover with the parameter $k=\log n$. We also give kernel lower bounds for biclique cover. I will point out some interesting open problems related to the above problems. Title: Novel applications of capacitated Cluster-Editing with vertex Split operation Speaker: Peter Shaw Abstract: The capacitated form of the cluster-editing produces significant improvements in the size of clinical data that can be processed. Nevertheless, the presence of (hub vertices in the network places undesirable constraints on the max-delete parameter. By introducing a vertex-split operation we can further constrain the parameter, allowing it to handle potentially larger data sets. The Capacitated Cluster-Editing with Vertex Split is NP-Hard and FPT. Our current project is exploring applications of this novel FPT problem in the search algorithm enables the analysis of Acute Respiratory Lung disease (ALRI) which is a major health issue in Australia’s NT. Moreover, many other interesting applications exist and some of them will be briefly shared and discussed. Title: Application of Cluster-Editing with Vertex-Split in Logic. Speaker: Peter Shaw Abstract: Vertex splitting is an underrated operation and has been for quite a long time although a completely natural process when it comes to modeling. Edges capture relational information, essential for graph modeling. But vertex identity models are different, but also very suited for natural occurrences. So just for a thought experiment, we take, as a starting point, that we want to consider partial orders on graphs (or more generally, relational structures, as in the wider conceptual formation of logic) and we want to take as the starting point the operation of splitting of atoms. When you a partial order IS defined on finite graphs, IE defined by a finite set of local'' operations, which needs to be formalized, although it is quite natural, if we consider it in the area of graph minors, immersions, topological, induced-subgraphs and similar orders. And then further insist that vertex-splitting is one of the allowed operations. What can be achieved….? Title: Diverse FPT Speaker: G. Phillip Abstract: TBA	{}
# Train-World: Feasibility of radioisotope steam-electric engines ...It's paradox. They left us these technological marvels, yet with all their might and knowledge they failed to prevent their own doom... Excerpt from a lecture by the High Historian of Berlin Falls Welcome to a future. Mankind has brought doom upon themselves, their cities have been flattened by war and weather, and most of the northern hemisphere is radioactive badlands. They've managed to avert revert global warming, in the process creating a global society of an unprecedented scale. Air-travel got reduced to the bare necessities, and the railways underwent a renaissance. Closed stretches of track & stations all over Europe got reopened. Lines between bigger cities got extended to rail-arteries1. These arteries expanded to stretch all along the northern hemisphere, even connecting Berlin to Boston. Arteries: - Central Europe, Russia, Kazakhstan, Beijing, Bering-Strait, Chicago, Boston - England, Central Europe, Spain, Gibraltar, Morocco - Kazakhstan, Afhganistan, India - Beijing, Hong-Kong, Thailand, Indonesia, Papua New Guinea, Sidney - Beijing, Korea, Japan - Chicago, Mexico, Colombia Due to the lack of electrification of rails on the American, African and Australian continents scientists and engineers spent considerable efforts into advancing emission-free alternatives to diesel-electric engines. A break-through was achieved in the field of SRG & RTG technology. Taking a hint from space-engineering, Radioisotope Heater Units (RHU) were expanded in size to serve as continuous heating elements in the boilers of steam-electric engines - these engines would then be used for trains and ships. A typical train-engine would consist of a boiler in which 5 RHUs (~6 tons of Radium per unit2) superheat water or another conductor fluid in a primary fluid-cycle. Through conduction the heat in the primary cycle is used to superheat water to steam in a secondary water-cycle, which is then used to drive a turbine3 producing electricity. ┌─────────────────────┐ ┌───────────────┐ │ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ╞══╗\|╔╡ STEAM TURBINE │ │ │R│ │R│ │R│ │R│ │R│ │ ║\|║└──────────────╥┘ │ │A│ │A│ │A│ │A│ │A│ │ ║\|║ ┌─────────┐ ║ │ └─┘ └─┘ └─┘ └─┘ └─┘ ╞⛒╝\|╚⛒═╡CONDENSER╞═╝ └─────────────────────┘ └─────────┘ Assuming ~168W per kg of Radium4 and an optimistic efficiency of ~50% this gets us ~2500kW usable energy. Or about ~500kW per RHU. Q: Is this concept for an engine workable or are there game-breakers that I missed out on? • addresses issues with the proposed design • proposes solutions to the addressed issues 1Lines with sections of up to 8 tracks next to each other in order to facilitate higher throughput. Sort of superhighways but for trains. 2Which results in cylinders of ~1m diameter and ~2m height (assuming we do not have to interleave the radium with too much other metal to get the heat out efficiently) 3Similarly to how a Nuclear Power Plant works, also known as Rankine Cycle. 4I've not too much knowledge in the area of nuclear physics, so I designed the described system based on the very helpful explanations I got from @kingledion on the chat. A radiothermal train is a fun idea. Radium may be possible (given futuristic resources), though tricky, I'll need to give it more thought. Another fun possibility if you don't mind an actively controlled reactor is a natural uranium source, like in the CANDU reactors in Canada. For passive systems though, I'm going to make my case for Polonium-210. One of the bigger issues of Radium and other seemingly suitable isotopes is the byproducts (or daughters), which are created in that isotope's decay chain. These byproducts would build up constantly in normal use and create technical problems or hazardous conditions. Keep an eye out for long-lived (= obnoxious) daughters, and any beta or gamma decays. To my knowledge, all of the common RTG isotopes have obnoxious byproducts, sadly, with one exception. If you're willing to hand-wave the production of the isotope, 210Po is rather ideal. It decays directly to a stable isotope of lead-208 through a low-ish energy alpha decay which is quite easy to catch and generate heat from. The alpha particles mostly stop with only a few cm of air, or completely stop in a few mm of water. Polonium-210 has an extreme activity, allowing smaller weights of it to be viable, even on a large train. From my back-of-the-envelope calculations, it would take 'only' about 40-70kg per train (an amount which would fill a ~6in cube). As a note, producing this with modern technology is flatly impossible. If it's easier in the future though... One convenience of Polonium is that it's fairly noble as a metal, in other words it doesn't rust or dissolve well in water unless there's a bunch of chloride in the solution. For this reason, it could be used in small chunks or even possibly as a thin (mm) plating of solid metal in the boiler or on the walls of the heat exchanger. If you wanted to use it in water solution instead, it's soluble in 3%HCl, or more simply in EDTA (like in shampoo). Without either HCl or EDTA (or similar) it automatically tries to plate itself out of water solutions as metallic Polonium. A downside of HCl is it attacks Copper and Iron, common heat exchanger materials. A radiothermal train would be producing heat constantly, so it would need to be constantly boiling water, even when not moving. This could be very obnoxious, because water tends to leave deposits and scales on boiler / heatEx equipment and without an obvious opportunity to clean them they could become ineffective or (frighteningly in this case) leaky. Because Polonium-210 has an extraordinary activity and decent bioabsorption, it's a nefarious toxin (worse than cyanide). The good news is it has a very short halflife (138 days), so spills would lose toxicity within decade. The immediate effects would be devastating though, and would travel freely with ground water if using the solution form. Also because of Polonium's short halflife, the trains would need to be refilled regularly, perhaps making the 'chunk' version more appealing. Lastly, the alpha particles would cause wild amounts of embrittlement in any metals within striking distance. The Polonium would need to be held well separate from anything critical. I think I could keep going for a while about little quibbling engineering problems a radiothermal train would face, but it's wonderful concept that doesn't break any major laws of physics while having just enough troubles to make it interesting for the people stuck aboard. Math for amount of fuel required on a radiothermal steam engine: 2ton coal /hr reference 19.48 *10^6 BTU / ton coal reference 1055 J/BTU , 3600 s/hr ~6MW heat 141W/gm Po-210 reference 42kg Polonium-210 per train. Polonium's quite dense at ~9g/cm^3 resulting in something like a 17cm cube of polonium. A cube would explode though, so let's assume small chunks or a plating. Best of luck with your world! Also, I really like your ascii art. Edit: I noticed that a train to last through a nuclear winter would be handy, so I put a bit more work into this. It seems that Sr-90 would also be appropriate, with the advantage that it has a much longer (29yr) half life. A train would require about 1500kg of it, and beneficially it's available in large amounts in nuclear waste. Sr-90 undergoes beta decay (medium-nasty radiation) to make a daughter which also quickly beta decays to something nice and stable. Beta radiation is best caught by acrylic plastic, though a few cm of water would work as well. Problem: You will not get 50% efficiency. Nuke plants run at about 30% for safety reasons, the same rules would apply to your trains. Thus you need to increase your powerplant by 50%. Clarifying this: Your efficiency is limited to the Carnot limit, which is a function of temperature. Since nuclear power doesn't have an inherent limiting factor like combustion-driven power you need to keep the temperature farther away from the point your system breaks--for fission plants that ends up being 30%, I would figure the same factors would be at work here and thus the same limit. Problem: You also have to dissipate 5,000kw of waste heat. Major woe if anything goes wrong with your cooling system as there's no off switch possible. Expect any serious train accident to turn into a nuclear accident as your system bakes itself. • I am not sure I follow your reasoning: Do I not get 50% due to technical constraints or due to ideological constraints? – dot_Sp0T Jan 3 '18 at 8:58 • Actually you get 30% because no thermal system can operate beyond the Carnot limit. Some systems "seem" to get around it because they are essentially ganging up two different Carnot cycle engines, for example a gas turbine generator who's exhaust is used to power the boiler for a steam turbine can show a 60% efficiency overall. Non Carnot systems like fuel cells or MHD generators can exceed these limits, but we are not talking about this here. – Thucydides Jan 3 '18 at 21:35 • You are assessing a nuclear plant's efficiency, not a radioisotope generator's efficiency. Depending on design characteristics, there is no reason that a radioisotope steam generator will have the same constraints. A supercritical water steam cycle (say, 1000 K + at the hot reservoir) will give you near 50% efficiency, even without any regenerative heating components. @Thucydides the Carnot limit for such an engine would be about 70%. 50% is reasonable. – kingledion Jan 4 '18 at 1:30 • @kingledion I'm figuring they won't want to run the radium generator hotter than we choose to run our fission generators. I don't know what temperature they picked, just that fission plants are held back by this--we get better efficiency from fossil fuel plants because we run them hotter. – Loren Pechtel Jan 4 '18 at 5:24 • The Carnot cycle is not related to the heat source, using wood or antimatter to power a Carnot engine (i.e raising steam to run a generator) makes no difference: infogalactic.com/info/Carnot_cycle – Thucydides Jan 4 '18 at 5:29	{}
Cfl Quark Matter » empowerlegend.com Quark matter or QCD matter. CFL phase of color-superconducting quark matter. At intermediate densities we expect some other phases labelled "non-CFL quark liquid" in the figure whose nature is presently unknown,. They might be other forms of color-superconducting quark matter, or. Color–flavor locking CFL is a phenomenon that is expected to occur in ultra-high-density strange matter, a form of quark matter. The quarks form Cooper pairs, whose color properties are correlated with their flavor properties in a one-to-one correspondence between. the ground state of dense quark matter. We show that these contributions tend to destabilize the vacuum, leading to a surprisingly complex phase structure for quark matter as a function of quark mass, even for small αs. In particular we ﬁnd two new phases of CFL quark matter possibly relevant for the real world, for which ¯θ QCD = π/2. In absence of gapless quark quasiparticles, this Nambu-Goldstone boson turns out to play an important role in many transport properties of cold CFL matter [33,47]. The CFL phase, like the 2SC phase, is not an electromagnetic superconductor. Therefore, it does not expel a magnetic flux from its interior. Title: Exact solutions for compact stars with CFL quark matter. Abstract: The search for the true ground state of the dense matter remains open since Bomer, Terasawa and other raised the possibility of stable quarks, boosted by Witten's \textitstrange matter hypothesis in 1984. Within this proposal. Multi-winding ﬂux tubes in CFL quark matter Andreas Schmitt 1. Introduction and main results Sufﬁciently cold and dense matter is a color superconductor in the color-ﬂavor locked CFL phase [1, 2]. CFL breaks baryon number conservation spontaneously and thus can be expected to behave as a baryon superﬂuid. Prepared for submission to JCAP Exact solutions for compact stars with CFL quark matter L. S. Rocha, aA. Bernardo, M. G. B. de Avellar,b;c J. E. Horvatha. I suspect that this is referencing the paper Wormhole geometries supported by quark matter at ultra-high densities. CFL quark matter is Color-Flavor-Locked quark matter; basically, it is a color-superconducting ultras-high-density state of quark-g. Can CFL Quark Matter really stabilize a wormhole? Today, a paper appeared on arxiv, "Wormhole geometries supported by quark matter at ultra-high densities" paper 1403.0771 which appears to claim the Color-Flavor-Locked CFL quark matter could stabilize wormholes. At slightly lower densities, corresponding to higher layers closer to the surface of the compact star, the quark matter will behave as a non-CFL quark liquid, a phase that is even more mysterious than CFL and might include color conductivity and/or several additional yet undiscovered phases. arXiv:0709.4251v1 [nucl-th] 26 Sep 2007 Bulk viscosity in 2SC and CFL quark matter Mark G. Alford and Andreas Schmitt Department of Physics, Washington University St Louis, MO, 63130, USA.	{}
# Solving a 2x2 Perturbed Hamiltonian Exactly ## Problem Consider Hamiltonian $$H = H_0 + \lambda H'$$ with $$H_0 = \Bigg(\begin{matrix}E_+ & 0 \\ 0 & E_-\end{matrix}\Bigg)$$ $$H' = \vec{n}\cdot\vec{\sigma}$$ for 3D Cartesian vector $$\vec{n}$$ and $$\sigma_i$$ the Pauli matrices. Solve exactly for $$E_+ = E_i$$ and $$E_+ \ne E_-$$. (NOTE: As an earlier part of this problem, both cases were solved to second-order energy and state corrections, and those results are suppose to be compared to these results.) ## My Question As part of my pertubation solutions, I found the eigenenergies and eigenstates for both $$H$$ and $$H'$$. Before I do a brute force approach and actually diagonalize $$H$$, I want to make sure there isn't a more elegant approach. It seems like I should be able to use the information about the components---namely the eigenstates and eigenvalues of $$H_0$$ and $$H'%---to find information about the sum$$H$. •$H_0$and$H'$don't commute so diagonalizing each individually does not give you the diagonalization of the sum. – AHusain Nov 13 '18 at 19:30 • You want to solve the total Hamiltonian exactly. This means that you should find the eigenvalues of$H=H_0 + \lambda H^\prime\$, I think that is needed. – Dani Nov 13 '18 at 19:31 If you want to make your life a little bit easier, first prove that the spectrum of $$H$$ can only depend on $$n_x^2 + n_y^2$$, and not on $$n_x$$ and $$n_y$$ individually. Then you can set $$n_y=0$$ before you compute the eigenvalues. Solve exactly for $$E_+=E_i$$ and $$E_+\ne E_-$$. What is $$E_i\,$$? $$\let\l=\lambda \let\s=\sigma \def\half{{\textstyle{1 \over 2}}} \def\vs{\vec\s} \def\vN{\vec N}$$ I'll give a hint to solution. Write $$H_0 = \half(E_+ + E_-)\,I + \half(E_+ - E_-)\,\s_3.$$ Then $$H + \l\,H' = \half(E_+ + E_-)\,I+ \vN \cdot \vs$$ where $$\vN = \left(\l\,n_1,\ \l\,n_2,\ \half(E_+ - E_-) + \l\,n_3\right)\!.$$ Can you find eigenvalues of $$\vN \cdot\s\,$$? Try evaluating $$(\vN\cdot\vs)^2$$.	{}
## CryptoDB ### Paper: Compressible FHE with Applications to PIR Authors: Craig Gentry Shai Halevi DOI: 10.1007/978-3-030-36033-7_17 Search ePrint Search Google Homomorphic encryption (HE) is often viewed as impractical, both in communication and computation. Here we provide an additively homomorphic encryption scheme based on (ring) LWE with nearly optimal rate ($1-\epsilon$ for any $\epsilon >0$). Moreover, we describe how to compress many Gentry-Sahai-Waters (GSW) ciphertexts (e.g., ciphertexts that may have come from a homomorphic evaluation) into (fewer) high-rate ciphertexts.Using our high-rate HE scheme, we are able for the first time to describe a single-server private information retrieval (PIR) scheme with sufficiently low computational overhead so as to be practical for large databases. Single-server PIR inherently requires the server to perform at least one bit operation per database bit, and we describe a rate-(4/9) scheme with computation which is not so much worse than this inherent lower bound. In fact it is probably less than whole-database AES encryption – specifically about 2.3 mod-q multiplication per database byte, where q is about 50 to 60 bits. Asymptotically, the computational overhead of our PIR scheme is $\tilde{O}(\log \log \mathsf {\lambda }+ \log \log \log N)$, where $\mathsf {\lambda }$ is the security parameter and N is the number of database files, which are assumed to be sufficiently large. ##### BibTeX @article{tcc-2019-30003, title={Compressible FHE with Applications to PIR}, booktitle={Theory of Cryptography}, series={Lecture Notes in Computer Science}, publisher={Springer}, volume={11892}, pages={438-464}, doi={10.1007/978-3-030-36033-7_17}, author={Craig Gentry and Shai Halevi}, year=2019 }	{}
# Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs. Abstract : We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to deal with the normal approximation of the geometric $U$-statistics introduced by Reitzner and Schulte (2011). In particular, we shall provide a new analytic characterization of geometric random graphs whose edge-counting statistics exhibit asymptotic Gaussian fluctuations, and describe a new form of Poisson convergence for stationary random graphs with sparse connections. In a companion paper, the above analysis is extended to general $U$-statistics of marked point processes with possibly rescaled kernels. Keywords : Type de document : Article dans une revue Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18, pp.32. 〈10.1214/EJP.v18-2104〉 Domaine : Littérature citée [37 références] https://hal.archives-ouvertes.fr/hal-00646866 Contributeur : Raphael Lachieze-Rey <> Soumis le : samedi 23 juin 2012 - 12:57:59 Dernière modification le : jeudi 11 janvier 2018 - 06:19:44 Document(s) archivé(s) le : jeudi 15 décembre 2016 - 18:00:43 ### Fichiers LR_P_ejp_2.pdf Fichiers produits par l'(les) auteur(s) ### Citation Raphaël Lachièze-Rey, Giovanni Peccati. Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs.. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18, pp.32. 〈10.1214/EJP.v18-2104〉. 〈hal-00646866v3〉 ### Métriques Consultations de la notice ## 231 Téléchargements de fichiers	{}
# An estimate for spherical functions on $\mathrm{SL}(3,\mathbb{R})$ 24 Oct 2019 Li Xiaocheng We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C... (read more) PDF Abstract # Code Add Remove Mark official No code implementations yet. Submit your code now # Categories • REPRESENTATION THEORY • DIFFERENTIAL GEOMETRY	{}
### BASIC AND ADVANCED BAYESIAN STRUCTURAL EQUATION MODELING WITH APPLICATIONS IN THE MEDICAL AND BEHAVIORAL SCIENCES WILEY SERIES IN PROBABILITY AND STATISTICS Download Basic And Advanced Bayesian Structural Equation Modeling With Applications In The Medical And Behavioral Sciences Wiley Series In Probability And Statistics ebook PDF or Read Online books in PDF, EPUB, and Mobi Format. Click Download or Read Online button to BASIC AND ADVANCED BAYESIAN STRUCTURAL EQUATION MODELING WITH APPLICATIONS IN THE MEDICAL AND BEHAVIORAL SCIENCES WILEY SERIES IN PROBABILITY AND STATISTICS book pdf for free now. ## Basic And Advanced Bayesian Structural Equation Modeling Author : Sik-Yum Lee ISBN : 9781118358870 Genre : Mathematics File Size : 39.14 MB Format : PDF This book provides clear instructions to researchers on how to apply Structural Equation Models (SEMs) for analyzing the inter relationships between observed and latent variables. Basic and Advanced Bayesian Structural Equation Modeling introduces basic and advanced SEMs for analyzing various kinds of complex data, such as ordered and unordered categorical data, multilevel data, mixture data, longitudinal data, highly non-normal data, as well as some of their combinations. In addition, Bayesian semiparametric SEMs to capture the true distribution of explanatory latent variables are introduced, whilst SEM with a nonparametric structural equation to assess unspecified functional relationships among latent variables are also explored. Statistical methodologies are developed using the Bayesian approach giving reliable results for small samples and allowing the use of prior information leading to better statistical results. Estimates of the parameters and model comparison statistics are obtained via powerful Markov Chain Monte Carlo methods in statistical computing. Introduces the Bayesian approach to SEMs, including discussion on the selection of prior distributions, and data augmentation. Demonstrates how to utilize the recent powerful tools in statistical computing including, but not limited to, the Gibbs sampler, the Metropolis-Hasting algorithm, and path sampling for producing various statistical results such as Bayesian estimates and Bayesian model comparison statistics in the analysis of basic and advanced SEMs. Discusses the Bayes factor, Deviance Information Criterion (DIC), and $L_\nu$-measure for Bayesian model comparison. Introduces a number of important generalizations of SEMs, including multilevel and mixture SEMs, latent curve models and longitudinal SEMs, semiparametric SEMs and those with various types of discrete data, and nonparametric structural equations. Illustrates how to use the freely available software WinBUGS to produce the results. Provides numerous real examples for illustrating the theoretical concepts and computational procedures that are presented throughout the book. Researchers and advanced level students in statistics, biostatistics, public health, business, education, psychology and social science will benefit from this book. Category: Mathematics ## Bayesian Psychometric Modeling Author : Roy Levy ISBN : 9781315356976 Genre : Mathematics File Size : 66.67 MB Format : PDF, Docs A Single Cohesive Framework of Tools and Procedures for Psychometrics and Assessment Bayesian Psychometric Modeling presents a unified Bayesian approach across traditionally separate families of psychometric models. It shows that Bayesian techniques, as alternatives to conventional approaches, offer distinct and profound advantages in achieving many goals of psychometrics. Adopting a Bayesian approach can aid in unifying seemingly disparate—and sometimes conflicting—ideas and activities in psychometrics. This book explains both how to perform psychometrics using Bayesian methods and why many of the activities in psychometrics align with Bayesian thinking. The first part of the book introduces foundational principles and statistical models, including conceptual issues, normal distribution models, Markov chain Monte Carlo estimation, and regression. Focusing more directly on psychometrics, the second part covers popular psychometric models, including classical test theory, factor analysis, item response theory, latent class analysis, and Bayesian networks. Throughout the book, procedures are illustrated using examples primarily from educational assessments. A supplementary website provides the datasets, WinBUGS code, R code, and Netica files used in the examples. Category: Mathematics ## Multilevel Analysis Author : Tom A B Snijders ISBN : 9781849202015 Genre : Reference File Size : 35.62 MB Format : PDF, Kindle The Second Edition of this classic text introduces the main methods, techniques, and issues involved in carrying out multilevel modeling and analysis. Snijders and Boskers’ book is an applied, authoritative, and accessible introduction to the topic, providing readers with a clear conceptual and practical understanding of all the main issues involved in designing multilevel studies and conducting multilevel analysis. This book has been comprehensively revised and updated since the last edition, and now includes guides to modeling using HLM, MLwiN, SAS, Stata including GLLAMM, R, SPSS, Mplus, WinBugs, Latent Gold, and Mix. Category: Reference ## The Sage Handbook Of Multilevel Modeling Author : Marc A. Scott ISBN : 9781446265970 Genre : Reference File Size : 50.36 MB Format : PDF, ePub, Mobi In this important new Handbook, the editors have gathered together a range of leading contributors to introduce the theory and practice of multilevel modeling. The Handbook establishes the connections in multilevel modeling, bringing together leading experts from around the world to provide a roadmap for applied researchers linking theory and practice, as well as a unique arsenal of state-of-the-art tools. It forges vital connections that cross traditional disciplinary divides and introduces best practice in the field. Part I establishes the framework for estimation and inference, including chapters dedicated to notation, model selection, fixed and random effects, and causal inference. Part II develops variations and extensions, such as nonlinear, semiparametric and latent class models. Part III includes discussion of missing data and robust methods, assessment of fit and software. Part IV consists of exemplary modeling and data analyses written by methodologists working in specific disciplines. Combining practical pieces with overviews of the field, this Handbook is essential reading for any student or researcher looking to apply multilevel techniques in their own research. Category: Reference ## Schulleistungen Von Abiturienten Author : Ulrich Trautwein ISBN : 3531175866 Genre : Education File Size : 61.37 MB Format : PDF, Mobi Category: Education ## Statistics And Causality Author : Wolfgang Wiedermann ISBN : 9781118947043 Genre : Mathematics File Size : 69.96 MB Format : PDF A one-of-a-kind guide to identifying and dealing with modern statistical developments in causality Written by a group of well-known experts, Statistics and Causality: Methods for Applied Empirical Research focuses on the most up-to-date developments in statistical methods in respect to causality. Illustrating the properties of statistical methods to theories of causality, the book features a summary of the latest developments in methods for statistical analysis of causality hypotheses. The book is divided into five accessible and independent parts. The first part introduces the foundations of causal structures and discusses issues associated with standard mechanistic and difference-making theories of causality. The second part features novel generalizations of methods designed to make statements concerning the direction of effects. The third part illustrates advances in Granger-causality testing and related issues. The fourth part focuses on counterfactual approaches and propensity score analysis. Finally, the fifth part presents designs for causal inference with an overview of the research designs commonly used in epidemiology. Statistics and Causality: Methods for Applied Empirical Research also includes: • New statistical methodologies and approaches to causal analysis in the context of the continuing development of philosophical theories • End-of-chapter bibliographies that provide references for further discussions and additional research topics • Discussions on the use and applicability of software when appropriate Statistics and Causality: Methods for Applied Empirical Research is an ideal reference for practicing statisticians, applied mathematicians, psychologists, sociologists, logicians, medical professionals, epidemiologists, and educators who want to learn more about new methodologies in causal analysis. The book is also an excellent textbook for graduate-level courses in causality and qualitative logic. Wolfgang Wiedermann, PhD, is Assistant Professor in the Department of Educational, School, and Counseling Psychology at the University of Missouri, Columbia. His research interests include the development of methods for direction dependence analysis and causal inference, the development and evaluation of methods for person-oriented research, and methods for intensive longitudinal data. Alexander von Eye, PhD, is Professor Emeritus of Psychology at Michigan State University. His research interests include statistical methods, categorical data analysis, and human development. Dr. von Eye is Section Editor for the Encyclopedia of Statistics in Behavioral Science and is the coauthor of Log-Linear Modeling: Concepts, Interpretation, and Application, both published by Wiley. Category: Mathematics ## Joyce In The Belly Of The Big Truck Workbook Author : Joyce A. Cascio ISBN : 0976237318 Genre : File Size : 76.75 MB Format : PDF, ePub, Docs Category: ## Best Practices In Quantitative Methods Author : Jason W. Osborne ISBN : 9781412940658 Genre : Social Science File Size : 49.64 MB Format : PDF, ePub, Mobi Category: Social Science ## Choice Author : ISBN : UOM:49015003465979 Genre : Best books File Size : 49.66 MB Format : PDF Category: Best books ## Konometrie F R Dummies Author : Roberto Pedace ISBN : 9783527801527	{}
Combat Combat is a basic gameplay element of Azur Lane, and is required to complete a large number of objectives in the game. In combat, a fleet of the player's choosing is placed in an instanced, side-scrolling shooter, and must eliminate all spawned ships (including kamikaze ships, but not planes or submarines) in order to proceed. Loss of the player's flagship or entire escort fleet will result in a loss. User Interface Formation Players can choose a formation before battle. Though it has no effect appearance-wise, the formations applies buffs to certain stats. The stat buffs from different formations only applies to the Vanguard fleet. Line Ahead: Increases Firepower and Torpedo by 15%, but reduces Evasion by 10% Double Line: Increases Evasion by 30%, but decreases Firepower and Torpedo by 5% Diamond: Increases Anti-Air by 20% (no penalties) Danger Level Danger Level (Threat Level in EN) is a mechanic added into the game to allow easier completion of a map following a certain amount of clears. Danger level is available for all normal and hardmode maps and selected event maps. Danger level can either be denoted by how many levels or maximum to minimum (for this case we are going to use levels). Each map starts with a danger level of 0 (usually denoted by a red "DEF UP" icon) and this increases by 1 per each clear (i.e. killing the boss) following the 100% clear requirement. As your danger level increases, the icon will turn to orange, then blue, then finally green which is the highest danger level you can attain in a map. That is, the danger level mechanic only kicks in after reaching 100% for the map and starts increasing the next time you clear said map. Each danger level acts like an artificial level that is appended to your ship's own level. For an example, if your ship's level is 30 and the danger level is 6, then your ship's actual level is 36. Note that danger levels do not increase your ship's stats. Your ship's new level, for that map alone, is critical in certain calculations that involve damage dealt and received. You can calculate how much damage you deal and damage you taken by the following: • The result is in percentage • If the result is positive, you deal more damage and take less damage. • If the result is negative, you deal less damage and take more damage. All maps having danger level has a maximum level that can be attainable. Below is a table of the bonuses per each danger level in normal and hardmode maps assuming a level difference of 0. Please note that the ± notation refers to the damage you deal (+) and the damage you recieve (-): Danger Level World 1 World 2 World 3 World 4 World 5 World 6 World 7 World 8 World 9 World 10 World 11 World 12 Level 1 ±2% ±2% ±2% ±2% ±2% ±2% ±2% ±2% ±2% ±2% ±2% ±2% Level 2 ±4% ±4% ±4% ±4% ±4% ±4% ±4% ±4% ±4% ±4% ±4% ±4% Level 3 ±6% ±6% ±6% ±6% ±6% ±6% ±6% ±6% ±6% ±6% ±6% ±6% Level 4 ±8% ±8% ±8% ±8% ±8% ±8% ±8% ±8% Level 5 ±10% ±10% ±10% ±10% ±10% ±10% ±10% ±10% Level 6 ±12% ±12% ±12% ±12% ±12% Level 7 ±14% ±14% ±14% ±14% ±14% Level 8 ±16% ±16% ±16% ±16% ±16% Level 9 ±18% ±18% ±18% Level 10 ±20% ±20% ±20% Danger Level World 1 World 2 World 3 World 4 World 5 World 6 World 7 World 8 Level 1 ±2% ±2% ±2% ±2% ±2% ±2% ±2% ±2% Level 2 ±4% ±4% ±4% ±4% ±4% ±4% ±4% ±4% Level 3 ±6% ±6% ±6% ±6% ±6% ±6% ±6% ±6% Level 4 ±8% ±8% ±8% ±8% ±8% ±8% Level 5 ±10% ±10% ±10% ±10% ±10% ±10% Level 6 ±12% ±12% ±12% Level 7 ±14% ±14% ±14% Level 8 ±16% ±16% ±16% Level 9 ±18% Level 10 ±20% Combat UI This is what the combat user interface looks like. You can move around your ships with the joystick at the bottom left corner of the screen, pause the game by pressing the pause button at the top right corner of the screen, enter auto mode by pressing the button at the top left corner of the screen, or use special skills by pressing the three buttons at the bottom right of your screen. • 40px(Submarine) - Calls in submarines to assist, storing up to 2 charges per submarine. Can only be used on a map that allows submarines, in the submarine hunting range. • (Torpedoes) - Fires from your escort fleet, can store up to 2 charges per escort ship when they are at max lb. Only those that can equip a torpedo can fire one, even if a torpedo is not equipped on the ship. • (Artillery) - Fires from your main fleet battleship/monitors/battlecruisers/aviation battleships. Can store up to 1 charge per ship. You can hold down the button and aim the first shot, the remaining shots are automatic targeting. You get a 20% damage bonus to your first volley when aiming. • (Air Raid) - Launches from your main fleet aircraft carriers/light aircraft carriers. Can store up to 2 charges per ship. Automatic targeting. Any launch will clear all current enemy bullets, torpedoes, and salvos on the screen. Some boss abilities such as lasers cannot be cleared. Fleet list The ships in your currently selected fleet are shown on the left. Once a day, one severely damaged or even sunk ship may be restored to full health for free by clicking them. Further heals cost diamonds. Nodes There are at least 5 types of enemy nodes that appear in the map. Its strength is based on the number of ▲ in it, the more of this "▲" the more enemy ships present during combat. • (Light Fleet) - Mostly composed of enemy destroyers and light cruisers. These nodes generally have more suicide boats and torpedo boats. • (Main Fleet) - Nearly same as a light fleet but with Heavy Cruisers and Battleships present. Suicide boats and torpedo boats may spawn depending on the map. • (Carrier Fleet) - A fleet with an aircraft carrier present (mostly as shipgirls) that occasionally dispatch planes. • (Treasure Fleet) - A fleet that has a transport Ship as the flagship. It may spawn in a map randomly or when you go to a mystery node. Its rewards are slightly different from other enemy nodes. These will count as an escort fleet for the purposes of boss spawning and objectives including "sink X escort fleets" and "sink all enemy fleets", even when spawned from a mystery node. • (Siren) - Encountered during events only, skill set dependant on exact siren type. Usually has fewer ships of other types and only one miniboss siren ship. • (Boss fleet) - Also called a Flagship fleet, defeating this fleet is a requirement in clearing the map. Spawns planes regardless of there being an enemy aircraft carrier or not. There are also non-combat nodes that appear in various maps: • (Ammo Node) - Gives up to 3 ammo to your fleet when you move your fleet to that node. This node only disappears when all 3 ammo charges are exhausted. • (Mystery Node) - Randomly gives you either an equipment crate, upgrade plate, repair kit, gold, or spawns a Treasure Fleet in the map. Nodes on later chapters may give a food item. Movement Map This is what a map looks like. You can move your fleet around in this interface by clicking on the tile you want to go to. Note that you cannot traverse tiles with landmasses, nor can you travel through tiles with enemy ships. You may travel through a tile that has one of your own fleets on it. • During several events where Sirens appear as mini-bosses roaming around the map, your movement will be restricted to certain number of tiles. In those events, movement range will be the following: • Some meowficers increase tile movement by one. Check the Meowficer page for more info. Encounters While moving around in the map, sometimes you will get random encounters. There are two main type of encounters, being ambush fleets and air raids. The probability of an encounter occurring increases gradually as your fleet moves around the map. Note that the probability of a random encounter occurring of each fleet is independent of each other, and it is impossible to get a random encounter when you are traversing nodes where there is a enemy fleet wreckage (defeated enemy fleets), potential enemy nodes, boss nodes, support submarine location, or second fleet location, so you can take advantage of this and plan your route if needed. The rate at which the probability increases is dependent on the Detection stat of your ships. The higher it is, the slower the rate of increase. • Ambush Fleets - Similar to a normal enemy fleet, but with worse loot. You can choose to avoid these encounters, and you are generally recommended to do so, since they provide little returns. The probability of avoiding it is dependent on the Evasion stat of your ships. Note that enemy fleets defeated in this encounter will not contribute to the map star condition of sinking x fleets, or to the required fleets to spawn boss. Can be decreased by some Meowficer skills. • Air Raids - All the ships in your fleet takes a small percentage of their max health as damage. Air damage resistance does not reduce the damage taken, nor does the (AA) stat. Can be reduced with some Meowficer skills. • Reconnaissance Value Or Recon Value, a value as seen at near top left corner, this contributes toward Encounter Rate • Ambush Avoidance Rate Refers to the rate of which you could avoid an ambush once encountered • Encounter Rate Refers to the increasing rate of encountering the aforementioned Air Raid and Ambush Fleets Note: • Map Survey Value = Inherent value of the map for encounter (depends on map; full lists have not been deduced) • Map Avoidance Value = Inherent value of the map for Avoidance. These values are listed in the table below: Map Avoidance Value World Map x-1 x-2 x-3 x-4 W1 3 3 3 4 W2 4 4 5 5 W3 6 6 7 7 W4 8 8 9 9 W5 10 10 11 11 W6 12 12 13 13 W7 14 14 15 15 W8 16 17 17 18 W9 17 18 18 19 W10 18 19 19 20 W11 19 20 20 21 W12 20 21 21 22 • Steps = The number of Node from the original node, referred to by the game as Chance of encounter and a colour letter next to it. Reset upon encounter. • Equipment Rate = Percentage value from equipment such as SG Radar • Base Ambush Rate = Theoretical value, could be disregarded Hard Mode Hard Mode rewards Retrofit Blueprints and Core Data. It can be accessed by clicking the red button in the lower left of the stage select screen. Currently, only chapters 1-8 have Hard Mode, which are identical to their normal mode maps except for higher enemy levels. Hard Mode has a limited number of entries per day. Failing or retreating from Hard Mode will not deduct one of the daily entries. All regions: • Three daily entries • Two Retrofit Blueprints are awarded on clear. • Core Data is awarded on clear, see the table below. • Once a map has been 3-starred AND the Danger Level has been decreased to 0, only the boss node will spawn. Hard Mode Core Data Drops Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 x 4 x 8 x 10 x 12 x 14 x 16 x 18 x 20 Hard Mode Requirements To enter hard mode, maps require that: • The average level of ships used is greater than the required average. • At least one of the ship class requirements per fleet is met. For this requirement, main and escort fleets are accounted for separately. • The total stats of the specified two types in fleets that meet the second requirement is greater than the required total. Once danger level has been decreased to 0 and there is only a boss node, it is feasible to use a lightweight fleet to battle the boss and use the other fleet only to fulfill hard mode requirements. Ship Class/Chapter Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Average Level 15/16/18/20 24/26/28/30 34/36/38/40 44/46/48/50 54/56/58/60 64/66/68/70 73/75/77/80 83/85/87/90 Fleet 1 Main 1 CVL CV BB CV BB Main 2 Any BB Any BB Any BB Main 3 Any Escort 1 DD CA CL CA Escort 2 Any DD CL DD Escort 3 Any DD Any DD Any Fleet 2 Main 1 N/A CV BB CV Main 2 Any CV Any CV BC Any BC Main 3 Any Escort 1 CA CL CA CL Escort 2 DD Any DD Escort 3 Any DD Any Total Firepower 50/60/70/100 - - - 320/360/420/500 - - 700/850/1000/1200 Air Power 50/60/70/100 150/180/200/250 200/220/250/300 - - - - - Torpedo - - - 250/270/300/350 - 500/600/700/800 650/750/850/1000 700/850/1000/1200 Evasion - 50/60/70/100 - 70/100/150/250 - 250/300/350/400 300/350/400/500 - Anti-Air - - 150/180/200/250 - 320/360/420/500 - - - In addition, please note that ships will be sortied in the order in which they are placed, not the order in which they are arranged. • For instance, if Chapter 6 Fleet 2's Escort ships were placed in order DD-CA-CL, the DD will be the front ship by default, not the CA. • Ships can be rearranged prior to combat at every node, but arranging them well early on reduces the amount of time used to rearrange ships. Combat Mechanics Result Ranking When complete, combat is ranked based on the player's performance. Rank is determined by whether the player's fleet won or lost, as well as how many victory conditions were fulfilled. In general, victory conditions are as follows: • All enemy ships sunk. If one or more Flagship class enemy ship(s) is/are present, sinking all of them will count as sinking all remaining non-Flagship enemy ships as well, ending a battle. • Victory within X seconds. X is normally 120, but may be different for special event nodes - if in doubt, check the pause screen. • No friendly ships which entered the battle are sunk. Ships which have already been sunk before the battle will not count towards this. Daily Raids and PvP Exercises have different victory conditions. If there is only one victory condition available, all victories are S-rank by default. Rank is determined as follows: • S: Player was victorious, and satisfied all victory conditions. • A: Player was victorious, but failed to satisfy one victory condition. • B: Player was victorious, but failed to satisfy two victory conditions. • C: Player was defeated but can re-challenge the combat. This is due to either losing one's flagship or running out of time. • D: Player was defeated and cannot re-challenge the combat. This is due to either losing all main fleet or all escort fleet ships. Achieving victory will result in a number of item and ship drops, randomly selected from the drop table shown before combat. Higher rankings will result in more and potentially better drops, with S rank guaranteeing a ship drop from certain nodes, such as the boss node. If a C-rank defeat occurs, the losing fleet will remain engaged with the combat node, and the player will have the option to rearrange ships, reselect formation, repair and try again, or retreat, leaving the enemy node intact. However, the flagship, if sunk, will remain sunk. If a D-rank defeat occurs, the losing fleet will retreat from the map, and the enemy node will remain intact. It is recommended that players make their best efforts to achieve S ranks when farming for boss node drop-only ships, such as Akagi (3-4), Kaga (3-4), Yuudachi (6-4), Maya (8-4), Niizuki (9-4) or Jintsuu (10-4). If it is no longer possible to achieve an S-rank (by losing a ship or exceeding the time limit), it is possible to reset an encounter by pressing the pause menu and choosing to Give Up. This resets the health and ammo values of the fleet at the cost of losing all fuel used in that attempt, but allows for encounters to be retried without having to retry the entire map. Damage Calculations Damage Type There are three main damage types: • Gun (or Shelling) - this damage type utilizes Firepower stat in calculation for final damage. • Torpedo - this damage type utilizes Torpedo stat when damage from torpedoes is dealt, not including torpedoes from airstrike. Torpedo damage can be reduced by Anti-Torpedo Bulge. • Airstrike - this damage type uses Aviation/Airpower stat, all airstrikes are this type of damage. This type of damage is reduced by anti-air stat. All three main damage types could be affected by various other factors, most notably ammo (at 0 ammo you dealt -50% damage), while the several extra damage types will ignore it. Several extra damage types: • Burn damage - a damage over time • Flood damage - a damage over time • Ramming - a type of damage dealt when ships ram into one another. Damage dealt is based on HP. Shell Type There are three shell types: • Standard (Yellow) - these are regular rounds that disappear on impact with the first enemy hit, or the top or bottom boundaries. Standard rounds are also negated by rotating shields, and are ineffective against Carrier-class and Battleship-class enemies. This is the most common damage type. • High Explosive (Red) - these rounds are similar to regular shells, but do more damage and have a chance to set enemy ships on fire. • Armor-Piercing (Blue) - these rounds will pierce through their targets, damaging up to 2 enemies along their path. The shell type modifiers are further divided by gun types. They are listed in the respective equipment pages Fire Damage HE shots each have a probability of starting a fire on the target. If a fire is started, the ship will be damaged up to 5 times, with the first instance being 3 seconds after the hit. Damage formula is as follows: The burn coefficients are as follows: Shell Type Coefficient Type 3 Ammo 1.2 All others 0.6 The probability of starting a fire by different types of guns are as follows: Gun Type Fire probability per hit DD Guns 1% CL Guns 3% CA Guns 8% Type 3 Ammo 30% BB/BC/BM Guns 50% Fire damage does not stack. A HE hit that starts a fire on an already burning ship will reset the ticking timer. A lower damage burn will be replaced by a higher damage one. For example, if a fire was originally started by a CL gun, then by a BB gun, then the original effect will be replaced by the burning effect caused by the BB gun, restarting the duration. The duration can be refreshed by another ignition caused by an equal or lower level gun, but lower level guns can only refresh the duration once. Type 3 ammo is considered to be at the same level as CA guns. Flood Damage Several skills can cause flood. Flood lasts for 24 seconds, doing 8 ticks in total. Corresponding stat depends on which ship causes the flood. For example, U-81 would inflict flood damage based on Torpedo stat while Saratoga inflicts damage based on Airpower stat. Flooding coefficients are as follows: Ship Name Coefficient Saratoga 0.2 U-81 0.45 Essex WIP Final Weapon Damage These are the damages of a single shot/torpedo/bomb of the weapon. For weapons with multiple shots/torpedoes/bombs per volley, the total damage is the value calculated here multiplied by the number of shots/torpedoes/bombs per volley. • Guns - The final damage for you deal to an enemy is affected by the firepower of your ships. The formula is as follows: • Torpedoes - Similar to guns, but instead of depending on the firepower of your ships, it depends on the (torpedo) of your ship. The formula is as follows: • Air Raids - There are three sources of damage: torpedoes, bombs, and damage when the planes reach the end of the screen. Unlike the third source, the first two sources of damage are not dependent on the damage stated on the plane's equipment details screen, but dependent on the load it carries. Each of them is calculated differently. It is also affected by the (airpower) of your ship. Do note that the Anti-Torpedo Bulge does not reduce the damage of torpedoes from planes. Here is their damage respectively. • Torpedoes • Bombs • End of Screen Damage End of screen damage is only applicable in PVP and for enemy planes in PvE. Critical Hits Sometimes you may see some really large numbers pop out. These critical hits are RNG based, formula is as below. Critical damage has a base damage modifier of 150%, and any other equipment modifiers adds additively to the base modifier. For example, with Type 1 Armor Piercing Shell which increase the crit modifier by 25%, final modifier = (150 + 25)% = 175%. These are the rate at which a gun/air raids/torpedoes fire/charge up. For rate of fire, take the reciprocal of the calculated time interval. • Guns - The reload time of a guns is affected by the (Reload) of your ship. Formula is as follows: Note that the weapon reload time displayed in-game is dependent on the ship in question. For un-modified reload time, go to Storage and click on the weapon in question without going through the page of any ship girl. For BB/BC/BM main guns, there is a slight delay before and after each volley, and the reload timer only begins to count after the last shot of the volley has left the guns. The delay and the duration of the volley is dependent on the type of the gun. This then effectively means: Note: The delay before volley is also known as duration in which target acquisition process takes place, if the intended target leaves the angle of firing before this process takes place by any means, it will make the weapon shoot straight ahead. • Air Raids - The reload of air raids is similar to the guns, except that it is dependent on all the planes equipped on the CV/CVL. Formula is as follows: Note: this formula is empirical. However, while there may be rounding errors, the 2.18 value has been tested and verified intensively. Anti-Air Guns Note that any AA guns equipped in the main fleet will act as though they were equipped in the escort fleet, and will only fire when there are planes in the AA region indicated by the dotted line surrounding the escort fleet. • Damage - The damage per instance of Anti-Air fire is affected by the (AA) of your ships. Formula is as follows: Each point of AA thus provides a 1% bonus to AA damage. Please note that the sum of every ship girl's AA damage in the circle, now called the Total AA Damage, is applied to each plane individually. That is, each plane takes the full brunt of the theoretical Total AA Damage. However, the actual Total AA Damage (affected by RNG) is rerolled per plane. Due to this highly RNG nature of AA, it may appear that Total AA Damage is distributed by portion, but this is not the case. Please note, however, the RNG of 0 to 0.95 would have its maximum RNG determined by difference between number of AA guns you have and number of planes present in the AA circle. AA+1 and double AA slots also buff AA damage of the ship with highest AA every AA occasion, for example, AA+1 and double AA slots on Akashi would make another ship appears to do massive AA damage in the damage result table every turn, but Akashi herself wouldn't do much AA damage. Therefore, AA is a lie, ARs are terrible at doing AA, but they make help massively. • Reload Time - The reload time is dependent on the (Reload) of your ships. This is equal to the mean reload time of all equipped AA guns in the fleet, plus a constant: • AA radius - The coverage of AA fire is outlined by the dotted lines surrounding your escort fleet. This is equal to the mean range of all equipped AA guns in the fleet: Damage Reduction For danger level modifiers, see the relevant section. Currently there are two main types of damage reduction in game, one being All Damage Reduction and the other being Air Damage Reduction. For air damage reduction, it is calculated individually for each ship based on their (AA): Damage reduction stacks multiplicatively with other sources: Evasion and Accuracy When a shell/torpedo/bomb hits a character, there is a probability that it will be counted as a miss and the character will not take any damage. This probability is determined by two factors: the accuracy of the shooter (a hidden stat, some of the ships' accuracy is listed here (CN site) and the (evasion) of the target. The formulas are listed here: Notes: • Probability can never be below .1; So even if the calculation gives something lower, the minimum probability of getting hit will be 10%. • Probability can never be above .9; So even if the calculation gives something higher, the maximum probability of hitting a ship will be 90%. • Defender Skill is any ship skill that gives an evasion buff e.g. any ship with smoke screens. • eHP means Effective Health Pool. eHP is the theoretical effective health pool a ship has. That is, if there is a ship that can dodge 50% of the time and has 2000 HP, another ship that dodges 0% of the time and has 4000 HP will have the same survivability on average. The eHP in the formula refers to the target's eHP. Skills This section does not include the barrage skills, refer to the Barrages section below for barrage skills. All passives trigger at the beginning of the battle. Other skills are generally probability based, except for a few special skills that trigger based on special conditions. For more information, refer to Skills. Barrages List of barrages can be found here. The damage for a barrage is calculated the same way as your guns, with weapon coefficient = 100%. Currently there are three main types of barrage in game: • Unupgradable Barrage - This is the most common barrage skill, and every escort ship has it. It triggers after the main gun has been fired a certain number of times. The damage is mainly affected by the limit break level of your ships. • Probability based Barrages (Main Gun) - They have a probability of triggering when the main gun of your ship fires. Upgrading the barrage skill increases its base damage. One such example is Hood. • Probability based Barrages (Time) - They have a probability of triggering at a fixed time interval. Upgrading the barrage skill increases its base damage. One such example is Erebus. Ship Power Rating The total power of a single ship can be calculated by the following: • For ease of calculation, make sure your ships do not have any equipment. Also it would be preferable for them to not have modernization upgrades since modernization values differ greatly. • Equipment Rarity follows: rarity gives 35 ★★ rarity gives 58 ★★★ rarity gives 90 ★★★★ rarity gives 132 ★★★★★ rarity gives 195 ★★★★★★ rarity gives 320 • Enhancement of an equipment provides an increase based on their rarity (number of stars). Below is a list of what each enhancement would give (i.e. +1 for a 2* equipment would add 8 to the overall score): rarity gives 5 ★★ rarity gives 8 ★★★ rarity gives 10 ★★★★ rarity gives 12 ★★★★★ rarity gives 15 ★★★★★★ rarity gives 20 NOTE: The stats that any equipment adds MUST be included. • Modernization, in general, follows this idea: per every column, you add +5 per node starting with a base of 10. Note the last column will give a +50. Glossary • General 1. Ship Skills. In it refers to skills that buff the ship stats, like Queen Elizabeth's skill. In it refers to skills that buff the damage directly, like Takao's skill. • Damage Formula 1. Damage of a single shot of the weapon, i.e. the number you see without multiplying. The damage of the planes are dependent on the type of the bombs/torpedoes they carry. A modifier based on the damage receiver's armour type and the calibre/type/load of the guns/torpedoes/bombs. A value based on the difference in level in the shooter and the one being hit. It is calculated with this formula: For example, if your ship is level 1 and enemy's ship is level 51, enemy ships will deal 150% damage to your ship while your ship will deal 50% damage to his, and vice versa. Bonus from Level Advantage caps out at 25 levels, or +50% Damage Dealt and -50% damage received or the other way round. Skills that directly buff a type of ammo, like Belfast's skill. Skills that debuff the enemy, like Helena's skill. Skills that increases the damage dealt to specific types of enemies, like Deutschland's skill.	{}
[ 1996 \| 1997 \| 1998 \| 1999 \| 2000 \| 2001 \| 2002 \| 2003 \| 2004 \| 2005 \| 2006 \| 2007 \| 2008 \| 2009 \| 2010 \| 2011 \| 2012 \| 2013 \| All \| Home ] # On an inverse problem for scalar conservation laws Abstract: We study in what sense one can determine the function $k=k(x)$ in the scalar hyperbolic conservation law $u_t+(k(x)f(u))_x=0$ by observing the solution $u(t,\cdot)$ of the Cauchy problem with initial data $u\rvert_{t=0}=u_o$ . Paper: Available as PDF (544 Kbytes). Author(s): Helge Holden Fabio Simone Priuli Nils Henrik Risebro Submitted by: ; 2013-09-25.	{}
Volume 405 - 7th Symposium on Prospects in the Physics of Discrete Symmetries, DISCRETE 2020-2021 (DISCRETE2020-2021) - Plenary session B-$\bar{\rm B}$ mixing: decay matrix at high precision U. Nierste Full text: Not available Abstract I review the status of the Standard-Model prediction of the width difference $\Delta\Gamma_s$ among the two $B_s$ meson eigenstates. Ongoing effort addresses three-loop QCD corrections, corresponding to the next-to-next-to-leading order of QCD. With an improved theoretical precision of the ratio $\Delta\Gamma_s/\Delta M_s$, where $\Delta M_s$ denotes the mass difference in the $B_s$-$\bar B_s$ system, one can probe new physics in $\Delta M_s$ without sensitivity to $\|V_{cb}\|$, whose value is currently controversial. How to cite Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete. Open Access	{}
# Normal and heavy water mixing This question seems silly, but this just came in my mind while i was learning about hydrogen. Means will there be any reaction when $\ce{H2O}$ will be mixed with $\ce{D2O}$? Yes, a reaction known as "hydrogen exchange" will take place. When $\ce{H2O}$ and $\ce{D2O}$ (heavy water) are mixed, hydrogen exchange will take place rapidly to form $\ce{HDO}$ as a third component of the mixture. Depending on the initial amounts of $\ce{H2O}$ and $\ce{D2O}$ added, a statistical mixture of the 3 compounds will result. So if you mixed equal amounts of the $\ce{H2O}$ and $\ce{D2O}$, then you would wind up with something around a 1:2:1 (it won't be exactly 1:2:1 due to the effect of the slightly different stabilities of the 3 compounds - see LDC3's comment) mixture of $\ce{H2O}$, $\ce{HDO}$ and $\ce{D2O}$. • Statistically, you would get that ratio, but unfortunately the $\Delta H_f$ for $D_2O$ is about 9 kJ/mol different from $H_2O$ so that the equilibrium is not that easy to determine. www1.lsbu.ac.uk/water/data.html – LDC3 Aug 16 '14 at 17:52	{}
UCSC track hub: how to best include a GTF / GFF file 1 1 Entering edit mode 18 months ago nlehmann ▴ 130 Hello, I have set a UCSC trackhub, which includes annotations that were originally in the GTF or GFF formats. So I followed the steps described here: https://genome.ucsc.edu/goldenPath/help/bigGenePred.html (example 4: GTF (or GFF) to BigGenePred) to obtain a binary bigGenePred for each GTF / GFF. It works fine, except that when visualizing the result, I loose data on introns and strand orientation. It means that for each gene annotated, I get a solid line, going from the 5' end to the 3'end of the gene. One example of what I get (the first 3 light blue top lines are from one of my GTF annotation turned to binary bigGenePred): I really want to see a finer granularity where you can actually see exons / introns and strand orientation, so I tried to upload one of my GTF to a custom track. This way works: I do not lose data from the annotation and get to see all the details. However, I could not find a way to integrate a custom track to a track hub. Result with the custom track (top 3 black lines are from one of my GTF annotation): So this my question: what is the best way to integrate a GTF file to a trackhub and not lose any detail ? Thanks for the help. UCSC trackhub annotation • 909 views 1 Entering edit mode Hello, bigGenePred does support intron/strand information. It should not be a problem when converting from a GTF/GFF. Initial suspicions are the problem may be the way the trackDb stanza is declared in the hub. Are you stating type bigGenePred, etc? Another possibility could be the input GTF file or the conversion (stating the correct bidBed 12+8), but that's less likely. Essentially from your first screenshot it looks like the Genome Browser is displaying your file as a BED3. If you would like to email us a copy of the GTF file as well as a link to the hub to our private mailing list (genome-www@soe.ucsc.edu) we could take a look. Note that only internal Genome Browser staff can see the contents of the message. 1 Entering edit mode Thanks for your support, it's solved ! I stated "bigBed" instead of "bigGenePred" in the trackDB. I got confused by the last command "bedToBigBed" in the conversion from GTF to bigGenePred. 0 Entering edit mode You cannot use directly the GFF or GTF files into the trackhub? Some important features was probably missing in the GFF/GTF files you converted. I suggest you standardise them with agat_sp_gxf2gxf.pl from AGAT and re-try the conversion. If still does not work you could add introns features too using agat_sp_add_introns.pl. 0 Entering edit mode You cannot use directly the GFF or GTF files into the trackhub? Nop, doesn't seem to. UCSC manual says: Custom tracks can be constructed from a wide range of data types; hub tracks are limited to compressed binary indexed formats that can be remotely hosted. However, the custom tracks utility does not offer the data persistence and track configurability provided by the track hub. By compressed binary they mean one of these (from this part of UCSC manual): • bam/cram: Compressed Sequence Alignment/Map tracks • bigBed: Item or region tracks • bigBarChart: Bar charts of categorical variables displayed over genomic regions • bigChain: Genome-wide Pairwise Alignments • bigGenePred: Gene Annotations • bigInteract: Pairwise interactions • bigNarrowPeak: Peaks • bigMaf: Mulitple Alignments • bigPsl: Pairwise Alignments • bigWig: Signal graphing tracks • hic: Hi-C contact matrices • halSnake: HAL Snake Format • vcfTabix: Variant Call Format Some important features was probably missing in the GFF/GTF I have exactly the same data input in both cases (trackhub and custom track), so no data missing in the GTF/GFF. The problem is the conversion to bigGenePred that produce a loss of data (same as converting from GTF to BED file). If still does not work you could add introns features too using agat_sp_add_introns.pl. Thanks for the help and suggestions ! I will try it if I cannot find a more standard solution. There are lots of detailed annotations in UCSC so I guess there must be some UCSC-internal way of doing. 1 Entering edit mode 18 months ago Luis Nassar ▴ 600 Glad it was solved! I'm just going to copy the response as an answer so the post doesn't show up unanswered: Hello, bigGenePred does support intron/strand information. It should not be a problem when converting from a GTF/GFF. Initial suspicions are the problem may be the way the trackDb stanza is declared in the hub. Are you stating type bigGenePred, etc? Another possibility could be the input GTF file or the conversion (stating the correct bidBed 12+8), but that's less likely. Essentially from your first screenshot it looks like the Genome Browser is displaying your file as a BED3. If you would like to email us a copy of the GTF file as well as a link to the hub to our private mailing list (genome-www@soe.ucsc.edu) we could take a look. Note that only internal Genome Browser staff can see the contents of the message.	{}
# How to use the global definition of a variable in a command where that variable has already been defined? I was trying to highlight some text, which contains \x, using the \hl command from the soul package, where \x is simply defined as \def\x{\mathbf{x}}. However, I've found that \x has already been defined in \hl. As a result, the TeX compiler wrongly interprets \hl{How to define $f(\x_i)$?}, where \x actually takes as the value define. Can I know how to enforce the global definition of \x inside \hl? Thanks. \documentclass[10pt]{article} \usepackage{color} \usepackage{soul} \def\x{\mathbf{x}} \begin{document} \title{} \author{} \date{} \maketitle How to define $\x$? \hl{How to define $\x$?} \end{document} • \def\x{\mathbf{x}} will bite you in the end ... or more precisely now, because you get into troubles... – user31729 Feb 11 at 13:48 • Exactly which error do you get? Using my TexLive 2019, I get no errors with your example. Note: I agree with Christian: a normal user should never use \def – daleif Feb 11 at 13:48 • On a simpler note why dont you change the \x definition in your document? – Raaja Feb 11 at 13:49 • @daleif There is no error, but the output is not as expected. It shows 'How to define $define$' and not 'How to define $\mathbf{x}$' – moewe Feb 11 at 13:49 • well it is imho a rather bad idea from soul to use \x as a scratch command, but there is not much you can do beside complaining to the maintainer. So don't use \x yourself. – Ulrike Fischer Feb 11 at 13:51 It's very unfortunate that the author of soul misunderstood the usage of \x, which should be \begingroup\edef\x{\endgroup<material>}\x \edef\x{<material>}\x The former only redefines \x in a group and a possible definition of \x is irrelevant; when the redefined \x is executed, the group ends and the redefinition vanishes. You can cure the disease by changing \x into a different scratch macro, here I used \SOUL@x. \documentclass[10pt]{article} \usepackage{color} \usepackage{soul} \usepackage{regexpatch} \makeatletter \xpatchcmd{\SOUL@flushcomma}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@flushapo}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@flushgrave}{\x}{\SOUL@x}{}{} \xpatchcmd{\soulregister}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@doword}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@dosyllable}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@gettoken}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@puttoken}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@nexttoken}{\x}{\SOUL@x}{}{} \xpatchcmd{\SOUL@soeverytoken}{\x}{\SOUL@x}{}{} \makeatother \newcommand\x{\mathbf{x}} \begin{document} \title{} \author{} \date{} \maketitle How to define $\x$? \hl{How to define $\x$?} \end{document} soul uses the name \x internally as a scratch command to temporarily store stuff all over the place. (Just search for \x or \edef\x in http://mirrors.ctan.org/macros/latex/contrib/soul/soul.dtx.) The way the package is written there is little to no chance to get it to stop doing that easily (short of replacing every occurence of \x in the source with \soul@tempa or trying to patch the package to the same effect), so I guess you should 1. Complain to the author of soul and suggest he use an 'internal' name as scratch command. 2. Use a macro name other than \x for your \textbf{x}. Make it semantic. You could for example say \newcommand{\vector}[1]{\mathbf{#1}} and then use \vector{x} or if you want one command \newcommand{\vector}[1]{\mathbf{#1}} \newcommand*{\vecx}{\vector{x}} and use \vecx. It is usually safer to steer clear of one-letter commands. Even though they are very handy (at least superficially) many are already taken and redefining those could lead to catastrophic consequences.	{}
College Algebra 7th Edition a) We must add $3$ to the x- and $2$ to the y-coordinate. Thus it will become $(5+3,3+2)=(8,5)$ b) We must add $3$ to the x- and $2$ to the y-coordinate. Thus it will become $(a+3,b+2)$ c) We must add $3$ to the x- and $2$ to the y-coordinate. Thus it will become $(-3+3,0+2)=(0,2)$ d) We must add $3$ to the x- and $2$ to the y-coordinate. Thus they will become $A(-2,1),B(0,4),C(5,3)$	{}
# Multilevel Monte Carlo for Scalable Bayesian Computations Markov chain Monte Carlo (MCMC) algorithms are ubiquitous in Bayesian computations. However, they need to access the full data set in order to evaluate the posterior density at every step of the algorithm. This results in a great computational burden in big data applications. In contrast to MCMC methods, Stochastic Gradient MCMC (SGMCMC) algorithms such as the Stochastic Gradient Langevin Dynamics (SGLD) only require access to a batch of the data set at every step. This drastically improves the computational performance and scales well to large data sets. However, the difficulty with SGMCMC algorithms comes from the sensitivity to its parameters which are notoriously difficult to tune. Moreover, the Root Mean Square Error (RMSE) scales as O(c^-1/3) as opposed to standard MCMC O(c^-1/2) where c is the computational cost. We introduce a new class of Multilevel Stochastic Gradient Markov chain Monte Carlo algorithms that are able to mitigate the problem of tuning the step size and more importantly of recovering the O(c^-1/2) convergence of standard Markov Chain Monte Carlo methods without the need to introduce Metropolis-Hasting steps. A further advantage of this new class of algorithms is that it can easily be parallelised over a heterogeneous computer architecture. We illustrate our methodology using Bayesian logistic regression and provide numerical evidence that for a prescribed relative RMSE the computational cost is sublinear in the number of data items. ## Authors • 4 publications • 1 publication • 14 publications • 9 publications • 4 publications • ### Stochastic gradient Markov chain Monte Carlo Markov chain Monte Carlo (MCMC) algorithms are generally regarded as the... 07/16/2019 ∙ by Christopher Nemeth, et al. ∙ 4 • ### Multi Level Monte Carlo methods for a class of ergodic stochastic differential equations We develop a framework that allows the use of the multi-level Monte Carl... 05/04/2016 ∙ by Lukasz Szpruch, et al. ∙ 0 • ### Extended Stochastic Gradient MCMC for Large-Scale Bayesian Variable Selection Stochastic gradient Markov chain Monte Carlo (MCMC) algorithms have rece... 02/07/2020 ∙ by Qifan Song, et al. ∙ 0 • ### Control Variates for Stochastic Gradient MCMC It is well known that Markov chain Monte Carlo (MCMC) methods scale poor... 06/16/2017 ∙ by Jack Baker, et al. ∙ 0 • ### Explicit Mean-Square Error Bounds for Monte-Carlo and Linear Stochastic Approximation This paper concerns error bounds for recursive equations subject to Mark... 02/07/2020 ∙ by Shuhang Chen, et al. ∙ 0 • ### Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets Bayesian inference via standard Markov Chain Monte Carlo (MCMC) methods ... 01/28/2019 ∙ by Robert Cornish, et al. ∙ 0 • ### Parallel Markov Chain Monte Carlo for Bayesian Hierarchical Models with Big Data, in Two Stages Due to the escalating growth of big data sets in recent years, new paral... 12/16/2017 ∙ by Zheng Wei, et al. ∙ 0 ##### This week in AI Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. ## 1 Introduction In recent years there has been an increasing interest in methods for Bayesian inference which are scalable to Big Data settings. Contrary to optimisation-based or maximum likelihood settings, where one looks for a single point estimation of parameters, Bayesian methods attempt to obtain a characterisation of the full posterior distribution over the unknown parameters and latent variables in the model. This approach allows for a better characterisation of the uncertainties inherent to the learning process as well as providing protection against over fitting. One of the most widely used classes of methods for Bayesian posterior inference is Markov Chain Monte Carlo (MCMC). This class of algorithms mixes slowly in complex, high dimensional-models and scales poorly to large data sets [3]. In order to deal with these issues, a lot of effort has been placed on developing MCMC methods that provide more efficient exploration of the posterior, such as Hamiltonian Monte Carlo (HMC) [6, 16] and its Riemannian manifold variant [12]. Stochastic gradient variants of such continuous-dynamic samplers have been shown to scale very well with the size of the data sets, as at each iteration they use data subsamples (also called minibatches) rather than the full dataset. Stochastic gradient Langevin dynamics (SGLD) [21] was the first algorithm of this kind showing that adding the right amount of noise to a standard stochastic gradient optimisation algorithm leads to sampling from the true posterior as the step size is decreased to zero. Since its introduction, there have been a number of articles extending this idea to different samplers [4, 15, 5], as well as carefully studying the behaviour of the mean square error (MSE) of the SGLD for decreasing step sizes and for a fixed step size [20, 19]. The common conclusion of these papers is that the MSE is of order for computational cost of (as opposed to rate of MCMC). The basic idea of Multilevel Monte Carlo methodology is to use a cascade of decreasing step-sizes. If those different levels of the algorithm are appropriately coupled, one can reduce the computational complexity without a loss of accuracy. In this paper, we develop a Multilevel SGLD (ML-SGLD) algorithm with computational complexity of , hence closing the gap between MCMC and stochastic gradient methods. The underlying idea is based on [18] and its extensions are: • We build an antithetic version of ML-SGLD which removes the logarithmic term present in [18] and makes the algorithm competitive with MCMC. • We consider the scaling of the computational cost as well as the number of data items . By using a Taylor based stochastic gradient, we obtain sub-linear growth of the cost in . • By introducing additional time averages, we can speed up the algorithm further. The underlying idea is close in spirit to [1] where expectations of the invariant distribution of an infinite dimensional Markov chain is estimated based on coupling approximations. This article is organised as follows. In Section 2, we provide a brief description of the SGLD algorithm and the MLMC methodology to extent, which will allow us to sketch in Section 3 how these two ideas can be enmeshed in an efficient way. Next we describe three new variants of the multilevel SGLD with favourable computational complexity properties and study their numerical performance in Section 4. Numerical experiments demonstrate that our algorithm is indeed competitive with MCMC methods which is reflected in the concluding remarks in Section 5. ## 2 Preliminaries ### 2.1 Stochastic Gradient Langevin Dynamics Let be a parameter vector where denotes a prior distribution, and the density of a data item is parametrised by . By Bayes’ rule, the posterior distribution of a set of data items is given by π(θ\|X)∝π(θ)N∏i=1π(xi\|θ). The following stochastic differential equation (SDE) is ergodic with respect to the posterior dθt=(∇logπ(θt)+N∑i=1∇logπ(xi\|θt))dt+√2dWt,θ0Rd (1) where is a -dimensional standard Brownian motion. In other words, the probability distribution of converges to as . Thus, the simulation of (1) provides an algorithm to sample from . Since an explicit solution to (1) is rarely known, we need to discretise it. An application of the Euler scheme yields θk+1=Sh,ξk(θk),Sh,ξ(θ):=θ+h(∇logπ(θ)+N∑i=1∇logπ(xi\|θ))+√2hξ where is a standard Gaussian random variable on . However, this algorithm is computationally expensive since it involves computations on all items in the dataset. The SGLD algorithm circumvents this problem by replacing the sum of the likelihood terms by an appropriately constructed sum of terms which is given by the following recursion formula θk+1=Sh,τk,ξk(θk),Sh,τ,ξ(θ):=θ+h(∇logπ(θ)+Nnn∑i=1∇logπ(xτi\|θ))+√2hξ (2) with being a standard Gaussian random variable on and is a random subset of , generated for example by sampling with or without replacement from . Notice that this corresponds to a noisy Euler discretisation. In the original formulation of the SGLD in [21] decreasing step sizes were used in order to obtain an asymptotically unbiased estimator. However, the RMSE is only of order for the computational cost of [19]. ### 2.2 Multilevel Monte Carlo Consider the problem of approximating where is a random variable. In practically relevant situations, we cannot sample from , but often we can approximate it by another random variable at a certain associated , which goes to infinity as increases. At the same time , so we can have a better approximation, but at a certain cost. The typical biased estimator of then has the form ^gN,M=1NN∑i=1g(M,i). (3) Consequently, the cost of evaluating the estimator is proportional to to . According to the Central Limit theorem, we need to set to get the standard deviation of the estimator less than . Now consider just two approximations and , where . It is clear, that the cost of one sample for is roughly proportional to . We assume that and where . Then based on the identity , we have ¯gN1,N2,M,K=1N1N1∑i=1g(K,i)+1N2N2∑j=1(g(M,j)−g(K,j)). We see that the overall cost of the Monte Carlo estimator is proportional to cost(¯gN1,N2,M,K)=ϵ−2⋅(cost(gK)⋅V1+cost(gM)⋅V2), so implying the condition 1>costKcostM+V2V1, we obtain that . The idea behind this method, which was introduced and analysed in [14], lies in sampling in a way, that . This approach has been independently developed by Giles in a seminal work [9], where a MLMC method has been introduced in the setting of stochastic differential equations. MLMC takes this idea further by using independent clouds of simulations with approximations of a different resolution. This allows the recovery of a complexity (i.e variance ). The idea of MLMC begins by exploiting the following identity E[gL]=L∑l=0E[gl−gl−1],with g−1:=0. (4) In our context , , with , defined in (2), , and being the final time index in an SGLD sample. We consider a MLMC estimator Y=L∑l=0{1NlNl∑i=1Δ(i,l)},Δ(i,l):=g(i)l−g(i)l−1,g(i)−1=0, where are independent samples at level . The inclusion of the level in the superscript indicates that independent samples are used at each level and between levels. Thus, these samples can be generated in parallel. Efficiency of MLMC lies in the coupling of and that results in small . In particular, for the SDE in (1), one can use the same Brownian path to simulate and which, through the strong convergence property of the scheme, yields an estimate for . More precisely it is shown in Giles [9] that under the assumptions222Recall denotes the size of the step of the algorithm (2). ∣∣E[gl−gl−1]\|=O(hαl),Var[gl−gl−1]=O(hβl), (5) for some the expected accuracy under a prescribed computational cost is proportional to ε≍⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩c−12,β>γ,c−12log2(c),β=γ,c−α2⋅α+γ−β,0<β<γ where the cost of the algorithm at each level is of order . The main difficulties in extending the approach in the context of the SGLD algorithm is a) the fact that and therefore all estimates need to hold uniformly in time; b) coupling SGLD dynamics across the different levels in time c) coupling the subsampling across the different levels. All of these problems need serious consideration as naive attempts to deal with them might leave (4) unsatisfied, hence violating the core principle of the MLMC methodology. ## 3 Stochastic Gradient based on Multi-level Monte Carlo In the following we present a strategy how the two main ideas discussed above can be combined in order to obtain the new variants of the SGLD method. In particular, we are interested in coupling the dicretisations of (1) based on the step size with . Because we are interested in computing expectations with respect to the invariant measure , we also increase the time endpoint which is chosen such that . Thus, . We introduce the notation Sh,τ1:sl,ξ1:sl(θ0)=Sh,τsl,ξsl(Sh,τsl,ξsl(…Sh,τ1,ξ1(θ0))) where denotes the Gaussian noise and the index of the batch data. We would like to exploit the following telescopic sum Eg(Sh,τ1:θ0,ξ1:s0(θ0))+∑lEg(Shl,τ1:sl,ξ1:sl(θ0))−Eg(Shl−1,τ1:sl−1,ξ1:sl−1(θ0)). We have the additional difficulty of different and stepsizes and simulation time . First, the fine path is initially evolving uncoupled for time steps. The coupling arises by evolving both fine and coarse paths jointly, over a time interval of length , by doing two steps for the finer level denoted by (with the time step ) and one on the coarser level denoted by (with the time step ) using the discretisation of the averaged Gaussian input for the coarse step. This coupling makes use of the underlying contraction (Equation (6)) as illustrated in Figure 1. The property that we use is that solutions to (1) started from two different initial conditions and with the same driving noise satisfy E\|θ1t−θ2t\|2≤\|θ10−θ20\|e−Lt,L>0. (6) In [18, 7] it is shown that this holds if the posterior is strongly log-concave and also is satisfied by the numerical discretisation. However, numerically this holds for a much larger class and this can be extended by considering more complicated couplings such as the reflection coupling [8]. This shifting coupling was introduced in [13] for coupling Markov chains. In [18, 7] it is shown that (6) holds if the posterior is strongly log-concave. This is sufficient but not necessary and holds for a much wider class of problems [8]. This property implies that the variance of Δ(i,l):=g⎛⎝θ(f,l,i)Tl−1hl−1⎞⎠−g⎛⎝θ(c,l,i)Tl−1hl−1⎞⎠ for suitably chosen would remain small, thus allowing an application of the MLMC methodology. We will drop appropriately. ### 3.1 Multi-level SGLD As common in MLMC we couple fine and coarse paths through the Brownian increments, with a Brownian increment on a coarse path given as a scaled sum of increments on the fine - , which can be written in our notation as (θ(f)k+1,θ(c)k+1)=(Shi,τ(f)k,2,ξk,2∘Shi,τ(f)k,1,ξk,1(θ(f)k),Shl−1,τ(c)k,1,1√2(ξk,1+ξk,2)(θ(c)k)). (7) One question that naturally occurs now is that if and how should one choose to couple between the subsampling of the data? In particular, in order for the telescopic sum to be respected, one needs to have that the laws of distribution for subsampling the data is the same, namely L(τ(f,1))=L(τ(f,2))=L(τ(c)). (8) In order for this condition to hold we first take independent samples on the first fine-step and another independent s-samples on the second fine-step. In order to ensure that Equation 8 holds, we create by drawing samples without replacement from . Other strategies are also possible and we refer the reader to [18]. ### 3.2 Antithetic Multi-level SGLD Here we present the most promising variant of coupling on subsampling: Algorithm 2 for . Building on the ideas developed in [11] (see also [10]) we propose Antithetic Multi-level SGLD which achieves an MSE of order complexity for prescribed computational cost (and therefore allows for MLMC with random truncation see [1]). ### 3.3 Averaging the Path Compared to MCMC these algorithms seem wasteful because only the last step of a long simulation is saved. The numerical performance can be improved by instead averaging of parts of the trajectory as follows Δ(i,l)averaged:=1plpl∑k=0g⎛⎝θ(f,l)tl−1hl−1−k⎞⎠−1pl−1pl−1∑k=0g⎛⎝θ(c,l)tl−1hl−1−k⎞⎠, and this also applies appropriately to the antithetic version. ### 3.4 Taylor based Stochastic Gradient The idea of Taylor based stochastic gradient is to use subampling on the remainder of a Taylor approximation N∑i=1∇logp(xi\|θ) =N∑i=1∇logp(xi\|θ0)+N∑i=1∇2logp(xi\|θ0)(θ−θ0) +N∑i=1(∇logp(xi\|θ)−(∇logp(xi\|θ0)+∇2logp(xi\|θ0)(θ−θ0))) ≈N∑i=1∇logp(xi\|θ0)+(N∑i=1∇2logp(xi\|θ0))(θ−θ0) (11) +Nnn∑i=1(∇logp(xτi\|θ)−(∇logp(xτi\|θ0)+∇2logp(xτi\|θ0)(θ−θ0))). We expect that the Taylor based stochastic gradient to have small variance for small. The idea of subsampling the remainder originally was introduced in [2]. By interopolating between the Taylor based stochastic gradient and the standard stochastic gradient we have the best of both worlds. ## 4 Experiments We use Bayesian logistic regression as testbed for our newly proposed methodology and perform a simulation study. The data is modelled by p(yi\|ιi,x)=f(yixtιi) (12) where and are fixed covariates. We put a Gaussian prior on , for simplicity we use subsequently. By Bayes’ rule the posterior satisfies π(x)∝exp(−12∥x∥2C0)N∏i=1f(yixTιi). We consider and data points and choose the covariate to be ι=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝ι1,1ι1,21ι2,1ι2,21⋮⋮⋮ιN,1ιN,21⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ for a fixed sample of for and we take . It is reasonable to start the path of the individual SGLD trajectories at a mode of the target distribution. This means that we set the to be the map estimator x0=argmaxexp(−12∥x∥2C0)N∏i=1f(yixTιi) which is approximated using the Newton-Raphson method. In the following we disregard the cost for the preliminary computations which could be reduced using state of the art optimisation and evaluating the Hessian in parallel. In the following we use MCMC and the newly developed MLSGLD to estimate the averaged squared distance from the map estimator under the posterior i.e. set g(θ)=∥θ−θ0∥2. (13) Notice that by posterior consistency properties we expect this quantity to be have like which is why we will consider relative MSE. ### 4.1 Illustration of Coupling standard, antithetic and with Taylor We choose , and leave as a tuning parameter. The crucial ingredient here is that in expectation the coarse and fine paths get closer exponentially initially and then asymptote, with the asymptote decaying as the step size decays. This illustrated on Figure (a)a. As any MLMC algorithm performance is effected by the order of the variance , the parameters and should be chosen such that the difference between pathes reaches the asymptote, but preferrably does not spent to much time in it, as this increases the computational cost of sampling those paths. In our experiments we set and and on Figure (c)c we see, that Algorithm 2 provides better coupling with variance decay of order , which is significantly better than the first order variance decay, given by Algorithm 1. Combining Algorithm 2 with Taylor based extension from Section 3.4 and path averaging with from Section 3.3 gives additional decrease for the variance without affecting the rate . The faster variance decay leads to lower overall complexity, as the number of samples at each level is proportional to the variance at that level. The Taylor Mean decay rates are of the same order, which can be seen on Figure (b)b, but once again Algorithm 2 combined with Taylor and path averaging is more preferable, as the multiplicative constant is lower, than in Algorithm 1. Numerical evidence, presented here, leads to the conclusion, that Antithetic MLSGLD with Taylor along with Antithetic MLSGLD with Taylor and Averaging are the best competitors to MCMC algorithm, so we proceed to comparison of those algorithms. ### 4.2 Comparison with MCMC We choose Metropolis-Adjusted Langevin (MALA, see [17] ) as a competitor because it is based on one Euler step of the Langevin SDE, but adds a Metropolis accept-reject step in order to preserve the correct invariant measure (removing the requirement to decrease step size for better accuracy). We take cost as the number of evaluation of data items, which is typically measured in epochs. One epoch corresponds to one effective iteration through the full data set. Heuristically, for this log-concave problem we expect the convergence rate to be independent of , so the only cost increase is due to evaluating posterior density and evaluating . This agrees with the findings in Figure (a)a, where the MCMC lines are almost on top of each other thus yielding the same relative MSE for the same number of epochs for different dataset sizes. As increases the cost per epoch increases proportional to . We run the MALA for steps with steps of burning and optimal acceptance rate for 50 times and then average. The various MLSGLD algorithms are ran for 50 times to achieve relative accuracies . This is yet another advantage of MLMC paradigm, which allows us to control numerically the mean increments and variance at all the levels, thus stopping the algorithm, when it has converged numerically. The most important comparison is presented on Figure (b)b, where we compare the increase of the complexity to achieve relative accuracy of with respect to the dataset size. We observe the sublinear growth of cost w.r.t dataset size for Antithetic MLSGLD with Taylor and Antithetic MLSGLD with Taylor and averaging, with the later having a slightly better behaviour than the first one. ## 5 Conclusion We develop a Multilevel SGLD algorithm with computational complexity of , hence closing the gap between MCMC and stochastic gradient methods. Moreover, this algorithm scales sublinearly with respect to the dataset size and allows natural parallelization, due to the typical properties of Monte Carlo sampling. The benefits of parallelization are to be studied later along with further numerical investigations for adaptive choices of parameters in the algorithm. In our further studies we also plan to quantify analytically the gains, given by MLSGLD algorithm and extend its applicability to a larger class of models. ## References • [1] S. Agapiou, G. O. Roberts, and S. J. Vollmer. Unbiased Monte Carlo: posterior estimation for intractable/infinite-dimensional models. arXiv preprint, 2014. To appear in Bernoulli. • [2] C. Andrieu and S. Yildirim. Facilitating the penalty method for mcmc with large data. 2015. • [3] R. Bardenet, A. Doucet, and C. Holmes. On Markov chain Monte Carlo methods for tall data. ArXiv e-prints, May 2015. • [4] T. Chen, E. Fox, and C. Guestrin. Stochastic gradient Hamiltonian Monte Carlo. In Proc. International Conference on Machine Learning , June 2014. • [5] N. Ding, Y. Fang, R. Babbush, C. Chen, R. D. Skeel, and H. Neven. Bayesian sampling using stochastic gradient thermostats. In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K. Weinberger, editors, Advances in Neural Information Processing Systems 27, pages 3203–3211. Curran Associates, Inc., 2014. • [6] S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth. Hybrid monte carlo. Physics Letters B, 195(2):216 – 222, 1987. • [7] A. Durmus and E. Moulines. Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm. ArXiv e-prints, July 2015. • [8] A. Eberle. Reflection coupling and wasserstein contractivity without convexity. Comptes Rendus Mathematique, 2011. • [9] M. B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 2008. • [10] M. B. Giles. Multilevel Monte Carlo methods. Acta Numer., 24:259–328, 2015. • [11] M. B. Giles and L. Szpruch. Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Probab., 24(4):1585–1620, 2014. • [12] M. Girolami and B. Calderhead. Riemann manifold langevin and hamiltonian monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(2):123–214, 2011. • [13] P. W. Glynn and C.-H. Rhee. Exact estimation for markov chain equilibrium expectations. J. Appl. Probab., 51A:377–389, 12 2014. • [14] A. Kebaier. Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing. Ann. Appl. Probab., 15(4):2681–2705, 2005. • [15] B. Leimkuhler and X. Shang. Adaptive thermostats for noisy gradient systems. SIAM Journal on Scientific Computing, 38(2):A712–A736, 2016. • [16] R. M. Neal. MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo, 54:113–162, 2010. • [17] G. O. Roberts and R. L. Tweedie. Exponential convergence of langevin distributions and their discrete approximations. Bernoulli, pages 341–363, 1996. • [18] L. Szpruch, S. Vollmer, K. C. Zygalakis, and M. B. Giles. Multi level monte carlo methods for a class of ergodic stochastic differential equations. arXiv:1605.01384. • [19] Y. W. Teh, A. H. Thiery, and S. J. Vollmer. Consistency and fluctuations for stochastic gradient langevin dynamics. Journal of Machine Learning Research, 17(7):1–33, 2016. • [20] Y. W. Teh, S. J. Vollmer, and K. C. Zygalakis. (Non-) asymptotic properties of stochastic gradient langevin dynamics. ArXiv e-prints, 2015. • [21] M. Welling and Y. W. Teh. Bayesian Learning via Stochastic Gradient Langevin Dynamics. In Proceedings of the 28th ICML, 2011.	{}
July 16, 2019, 08:36:06 pm ### AuthorTopic: Could someone please explain this question (Read 85 times) Tweet Share 0 Members and 1 Guest are viewing this topic. #### jando • Posts: 21 • Respect: 0 ##### Could someone please explain this question « on: July 11, 2019, 01:02:08 pm » 0 Hey, I was wondering if someone could explain this question to me. Cheers. #### DrDusk • MOTM: MAY 19 • Forum Regular • Posts: 73 • sin(x) = tan(x) = x • Respect: +13 ##### Re: Could someone please explain this question « Reply #1 on: July 11, 2019, 05:04:35 pm » +1 Hey, I was wondering if someone could explain this question to me. Cheers. Recall the formula: $F = mrw^2, w = 4\pi$ $F = 0.015\times 0.8\times16\pi^2$ Thanks to r1ckworthy for pointing out my mistake in converting Grams to Kg hahaaha « Last Edit: July 11, 2019, 05:20:22 pm by DrDusk » UNSW 2019-2023: HSC 2018: Atar aim: 95+ (Achieved) Very soon to be HSC/Prelim Physics Tutor! sin(x) = tan(x) = x, for all x #### r1ckworthy • MOTM: APR 19 • Trendsetter • Posts: 111 • Respect: +74 ##### Re: Could someone please explain this question « Reply #2 on: July 11, 2019, 05:12:52 pm » +1 It appears they've made a mistake. Recall the formula: $F = mrw^2, w = 4\pi$ In their calculations I think they've forgotten the Pi which gives you the answer of 1.9. The actual answer would be: $F = 0.15\times 0.8\times16\pi^2$ Hey DrDusk, Your conversion for 'm' is wrong, it should be 0.015 kg instead of 0.15 kg, which when I calculated it checks out to B . Great explanation! HSC 2019: English Advanced \|\| Mathematics \|\| Mathematics Extension 1 \|\| Physics \|\| Chemistry \|\| Science Extension \|\| Ancient History \|\| The Yr12 journey- a diary I "hope" to update... \|\| Send some questions to boost my ego :) #### DrDusk • MOTM: MAY 19 • Forum Regular • Posts: 73 • sin(x) = tan(x) = x • Respect: +13 ##### Re: Could someone please explain this question « Reply #3 on: July 11, 2019, 05:19:16 pm » 0 Hey DrDusk, Your conversion for 'm' is wrong, it should be 0.015 kg instead of 0.15 kg, which when I calculated it checks out to B . Great explanation! Oh god, today I realized how stupid I am lol. I can't even properly convert units UNSW 2019-2023: HSC 2018: Atar aim: 95+ (Achieved) Very soon to be HSC/Prelim Physics Tutor! sin(x) = tan(x) = x, for all x #### r1ckworthy • MOTM: APR 19 • Trendsetter • Posts: 111 • Respect: +74 ##### Re: Could someone please explain this question « Reply #4 on: July 11, 2019, 05:25:46 pm » 0 Oh god, today I realized how stupid I am lol. I can't even properly convert units Hey, it happens to the best of us HSC 2019: English Advanced \|\| Mathematics \|\| Mathematics Extension 1 \|\| Physics \|\| Chemistry \|\| Science Extension \|\| Ancient History \|\| The Yr12 journey- a diary I "hope" to update... \|\| Send some questions to boost my ego :)	{}
# Operating Systems Our operating system images are built on regular basis from the metal-images repository. All images are hosted on GKE at images.metal-stack.io. Feel free to use this as a mirror for your metal-stack partitions if you want. The metal-stack developers continuously have an eye on the supported images. They are updated regularly and scanned for vulnerabilities. ## Supported OS Images The operating system images that we build are trimmed down to their bare essentials for serving as Kubernetes worker nodes. Small image sizes make machine provisioning blazingly fast. The supported images currently are: PlatformDistributionVersion LinuxDebian10 LinuxUbuntu20.04 It is fully possible to build your own operating system images and provide them through the metal-stack. There are some conventions though that you need to follow in order to make your image installable through the metal-hammer. You should understand the machine provisioning sequence before starting to write your own images. 1. Images need to be compressed to a tarball using the lz4 compression algorithm 2. An md5 checksum file with the same name as the image archive needs to be provided in the download path along with the actual os image 3. A packages.txt containing the packages contained in the OS image should be provided in the download path (not strictly required) 4. Consider semantic image versioning, which we use in our algorithms to select latest images (e.g. os-major.minor.patch ➡️ ubuntu-19.10.20191018) 5. Consider installing packages used by the metal-stack infrastructure • FRR to enable routing-to-the-host in our network topology • go-lldpd to enable checking if the machine is still alive after user allocation • ignition for enabling users to run user-specific initialization instructions before bootup. It's pretty small in size, which is why we use it. However, you are free to use other cloud instance initialization tools if you want to. 6. You have to provide an install.sh script, which applies user-specific configuration in the installed image • This script should consume parameters from the install.yaml file that the metal-hammer writes to /etc/metal/install.yaml • Please check this contract between image and the metal-hammer here 7. For the time being, your image must be able to support kexec into the new operating system kernel, the kexec command is issued by the metal-hammer after running the install.sh. We do this because kexec is much faster than rebooting a machine. 8. We recommend building images from Dockerfiles as it is done in metal-images repository. Info Building own operating system images is an advanced topic. When you have just started with metal-stack, we recommend using the public operating system images first.	{}
Particle horizon distance Tags: 1. Nov 1, 2015 June_cosmo 1. The problem statement, all variables and given/known data Numerically integrate and report the particle horizon distance today for the currently fa- vored model $\Omega_M=1-\Omega_{DE}=0.25,\omega=-1$. Assume the scaled Hubble constant to be h = 0.72, and report the particle horizon in billions of lyr (Gyr). 2. Relevant equations 3. The attempt at a solution Horizen distance $d=\int_0^{t0}\frac{dt}{a(t)}$, so in a flat universe $a(t)=(t/t0)^{2/(3(1+3\omega))}$, so that we have $$d=\frac{2}{1+3\omega}H_0^{-1}$$,but this has nothing to do with $\Omega_M=1-\Omega_{DE}=0.25$? Last edited: Nov 1, 2015 2. Nov 1, 2015 phyzguy The $a(t) \propto t^{\frac{2}{3}}$ result is only true for a flat universe with $\Omega_{DM} =0$. You need to review the Friedmann equations for the case with $\Omega_{DM} \neq 0$. 3. Nov 1, 2015 June_cosmo Oh that's right! So how do I derive a(t) from Friedmann equations? 4. Nov 1, 2015 phyzguy Well, can you write down the first Friedmann equation for H in terms of the Ω parameters? Once you have done that, remember that $H = \frac {\dot a}{a}$. Then you should be able to write a differential equation for a that you can numerically integrate. 5. Nov 1, 2015 June_cosmo Thank you! So that would be $\frac{H^2}{H_0^2}=\frac{0.25}{a^3}+0.75,$if we combine this with $H=\frac{\dot{a}}{a}$ and I solved this equation (online), it gave me http://www4f.wolframalpha.com/Calculate/MSP/MSP226920fg7hgi3c9658be000033830f2a7886e3e4?MSPStoreType=image/gif&s=20&w=550.&h=47. [Broken] ? (a(x) means a(t) here). I think this isn't right? Last edited by a moderator: May 7, 2017	{}
# print( 'Hello，NumPy！' ) 2020-11-08 08:32:33 # print( "Hello,NumPy！" ) Learning is painful , Learn today , Lose tomorrow . This kind of weather , Or sleep is the most comfortable . Let's talk about it , Make a noise , But you still have to learn . In the process of learning, I always have the habit of taking notes , But the quality of the notes can't be flattered , Most of them have not been sorted out , But it's a good choice to review . Self contact Python since , Most of them are reptiles , What kind of online novel , Virtual game currency , Examination question bank ah and so on have written , Also help others crawl a lot of website public data . 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Just a few days ago , The police also cracked the country's largest network luo liao What about extortion , Victim Da 10 More than ten thousand people , The amount involved is also XXXXXXXXXXXXXXX, We still need to pay attention to . Anyway , There's a lot to learn 、 Very miscellaneous , I'm not good at learning , The notes are rarely reviewed . A good workman does his work well , You must sharpen your tools first , This doesn't start systematic learning, machine learning , So I want to put the previous record Numpy、Pandas、Matplotlib“ Three swordsmen ” Rearrange your notes , It's also a review . Later on , Can learn some machine learning algorithms , Main reference 《 Machine learning practice / Machine Learning in Action》 With Mr. Zhou Zhihua 《 machine learning 》 Watermelon book , And some other technical articles written by some of the big guys in the circle . If you are good at it, try to tear it by hand , If you can't tear it by hand, it means that you still need to improve . Flag Too many , I feel like I'll be slapped in the face . Don't worry , Take your time , I'm not afraid to fight in the face , Anyway, the skin is rough and the meat is thick (￣_,￣ ) This article begins with NumPy Sort it out , Maybe it's not comprehensive , Only a few common ones are recorded , Other words will be used later to update it . The following content mainly refers to the rookie tutorial and NumPy Official documents ： About NumPy Installation , I have already introduced the construction of deep learning environment , Recommended installation Anaconda, It integrates a large number of third-party tool modules , It doesn't have to be manual pip install ..., This is a bit like Java Medium Maven.Anaconda May refer to ： be based on Ubuntu+Python+Tensorflow+Jupyter notebook Build a deep learning environment If you have not installed Anaconda That's OK , Only need Python In the environment, execute the following command to install NumPy that will do ： > pip3 install numpy -i https://pypi.tuna.tsinghua.edu.cn/simple The following is used in the following NumPy The version is ：1.18.1 In [1]: import numpy as np In [2]: np.__version__ Out[2]: '1.18.1' stay NumPy in , The objects of operation are mostly ndarray type , It can also be called by another name array, We can think of it as a matrix or a vector . establish np.ndarray Objects come in many ways ,NumPy There are also many api Available to call , For example, we can create a specified ndarray object ： In [7]: temp_array = np.array([[1, 2, 3], [4, 5, 6]], dtype = np.int32) In [8]: temp_array Out[8]: array([[1, 2, 3], [4, 5, 6]]) In [9]: type(temp_array) Out[9]: numpy.ndarray # The output is of type ndarray Yes, of course , You can also call arange, And then it's done reshape Operation to change its shape , Convert a vector to 2x3 Matrix form of , The object type is still numpy.ndarray In [14]: temp_array = np.arange(1, 7).reshape(2, 3) # arange Generating vectors ,reshape Change shape , Into a matrix In [15]: temp_array Out[15]: array([[1, 2, 3], [4, 5, 6]]) In [16]: type(temp_array) # The type of output is still ndarray Out[16]: numpy.ndarray From above , We can find out , No matter what way （ Other methods will be introduced later ） To create objects ,NumPy It's all about ndarray type , And this type of object mainly contains the following properties ： • ndarray.ndim： Express ndarray The number of shaft , It can also be understood as dimension , Or it can be understood as the number of brackets in the outer layer . such as [1, 2, 3] Of ndim Namely 1,[[1], [2], [3]] Of ndim be equal to 2,[[[1]], [[2]], [[3]]] Of ndim be equal to 3（ Pay attention to the number of brackets in the outer layer ） • ndarray.shape： Express ndarray The shape of the , The output is a tuple . The ndarray Yes n That's ok m Column , The output is (n, m), such as [[1], [2], [3]] The output is (3, 1),[[[1]], [[2]], [[3]]] The output is (3, 1, 1),[[[1, 2]], [[3, 4]]] The output is (2, 1, 2). Through the above 3 An example , You can find shape It is expressed from the outside to the inside • ndarray.size： This is easier to understand , That means ndarray The total number of internal elements , That is to say shape The product of the • ndarray.dtype： Express ndarray The data type of the internal element , Common are numpy.int32、numpy.int64、numpy.float32、numpy.float64 etc. That's all ndarray Some of the common properties in , Be careful ： Part of it , Not all of them , For other properties, please refer to the official documents We can observe by the following ndarray Properties of , And how its internal properties should be modified ： In the above example np.expand_dims and np.astype It will be introduced later . np.zeros You can create an element full of 0 Of ndarray,np.ones You can create an element full of 1 Of ndarray. You can specify ndarray Of shape shape , It can also be done through dtype Property specifies the data type of the inner element ： In [70]: np.zeros([2,3,2], dtype=np.float32) Out[70]: array([[[0., 0.], [0., 0.], [0., 0.]], [[0., 0.], [0., 0.], [0., 0.]]], dtype=float32) In [71]: np.ones([3,2,2], dtype=np.float32) Out[71]: array([[[1., 1.], [1., 1.]], [[1., 1.], [1., 1.]], [[1., 1.], [1., 1.]]], dtype=float32) in addition , stay Tensorflow Through tf.fill To generate a specified element shape tensor , As follows 2x3 Tensor , And the internal elements are 100： In [76]: import tensorflow as tf In [77]: tf.fill([2,3], 100) Out[77]: <tf.Tensor: shape=(2, 3), dtype=int32, numpy= array([[100, 100, 100], [100, 100, 100]])> And in the NumPy in , Also have fill Interface , It's just that you can only go through what you already have ndarray To call fill, Not directly np.fill To call ： In [79]: data = np.zeros([2, 3]) In [80]: data.fill(100) In [81]: data Out[81]: array([[100., 100., 100.], [100., 100., 100.]]) np.arange With the usual range Works in a similar way , Used to produce a continuous interval ndarray, Pay attention to the left, not the right , And the array is an arithmetic sequence , Tolerances can be self-defined （ It can be a decimal ）, as follows ： In [85]: np.arange(10) Out[85]: array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) In [86]: np.arange(3, 10, 2) Out[86]: array([3, 5, 7, 9]) In [87]: np.arange(3, 10, 0.7) Out[87]: array([3. , 3.7, 4.4, 5.1, 5.8, 6.5, 7.2, 7.9, 8.6, 9.3]) numpy.linspace Function to create a one-dimensional array , An array is made up of a sequence of equal differences , You can specify the number of elements inside the element and whether it contains stop value . as follows , In the interval 1-5 Create an element with the number of 10 Equal difference sequence of ： In [89]: np.linspace(1, 5, 10) # Default includes stop Out[89]: array([1. , 1.44444444, 1.88888889, 2.33333333, 2.77777778, 3.22222222, 3.66666667, 4.11111111, 4.55555556, 5. ]) In [90]: np.linspace(1, 5, 10, endpoint = False) # endpoint Property can be set to not contain stop Out[90]: array([1. , 1.4, 1.8, 2.2, 2.6, 3. , 3.4, 3.8, 4.2, 4.6]) np.random.random and np.random.rand Random from 0-1 Generate corresponding to shape Of ndarray object ： In [4]: np.random.random([3, 2]) Out[4]: array([[0.68755531, 0.56727707], [0.86027161, 0.01362836], [0.56557302, 0.94283249]]) In [5]: np.random.rand(2, 3) Out[5]: array([[0.19894754, 0.8568503 , 0.35165264], [0.75464769, 0.29596171, 0.88393648]]) np.random.randint Randomly generate a specified range of ndarray, And the internal elements are int type ： In [6]: np.random.randint(0, 10, [2, 3]) Out[6]: array([[0, 6, 9], [5, 9, 1]]) np.random.randn Returns the standard normal distribution ndarray（ The mean for 0, The variance of 1）： In [7]: np.random.randn(2,3) Out[7]: array([[ 2.46765106, -1.50832149, 0.62060066], [-1.04513254, -0.79800882, 1.98508459]]) in addition , We are NumPy Use in random When , It's a random set of data , and If you want to generate the same data each time , You have to go through np.random.seed To set it up ： In [33]: np.random.seed(100) In [34]: np.random.randn(2, 3) Out[34]: array([[-1.74976547, 0.3426804 , 1.1530358 ], [-0.25243604, 0.98132079, 0.51421884]]) In [35]: np.random.seed(100) In [36]: np.random.randn(2, 3) Out[36]: array([[-1.74976547, 0.3426804 , 1.1530358 ], [-0.25243604, 0.98132079, 0.51421884]]) stay NumPy One dimension in ndarray in , It's like a list , It can be sliced and traversed ： In [5]: a Out[5]: array([1., 2., 3., 4., 5., 6., 7., 8., 9.]) In [6]: a[2], a[2:5] Out[6]: (3.0, array([3., 4., 5.])) In [7]: a * 3, a ** 3 # cube Out[7]: (array([ 3., 6., 9., 12., 15., 18., 21., 24., 27.]), array([ 1., 8., 27., 64., 125., 216., 343., 512., 729.])) In [13]: for i in a: ...: print(i, end=", ") 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, If our ndarray It's not a one-dimensional array , It's a two-dimensional matrix , Or higher dimensional ndarray, Then we need to segment it from multiple dimensions . And When we're dealing with high dimensional ndarray When traversing , The dimension of a single result is less than that of meta dimension , such as 2 The result of traversing dimensional matrix is 1 Dimension vector , The result of three-dimensional traversal is 2 D matrix . in addition , There is one more thing to be said , The dimensions of my data are quite high , In order to facilitate us to index the data ,NumPy For us ... To segment , Specific examples are as follows ： In [14]: a = np.random.randint(0, 2, 6).reshape(2, 3) In [15]: a Out[15]: array([[1, 1, 1], [1, 0, 1]]) In [16]: a[:, :2] Out[16]: array([[1, 1], [1, 0]]) In [17]: for i in a: # Traverse the matrix , Then the output is the row vector , A single output has fewer dimensions than the original 1 ...: print(i, end=", ") [1 1 1], [1 0 1], In [19]: data = np.random.randint(0, 2, [2, 2, 3]) In [20]: data Out[20]: array([[[0, 0, 1], [0, 1, 1]], [[1, 0, 0], [0, 1, 0]]]) In [21]: data[..., :2] # ... It means that the first two dimensions need to be , amount to data[:, :, :2] Out[21]: array([[[0, 0], [0, 1]], [[1, 0], [0, 1]]]) shape operation ： • a.ravel()、ndarray.flatten(), take ndarray Do the stretching operation （ Straighten it into vector form ） • a.reshape(), Re change a Of shape shape • a.T、a.transpose(), return a The inverse matrix of All of the above operations return new results , Without changing the original ndarray（a）. And The above operations are all horizontal operations by default , If you need vertical , You need to control order Parameters , Specific operation can refer to rookie tutorial . except reshape outside , also resize, It's just resize Will change a Result , Instead of producing a new result ： Another little trick to master is , It's going on reshape When , If it comes in -1, The corresponding result will be calculated automatically . For example, a 2x3 Matrix a, We carry out a.reshape(3, -1), And here it is -1 It stands for 2, When we have a lot of data , This is very convenient to use ： In [44]: a Out[44]: array([[1, 1, 1], [1, 0, 1]]) In [45]: a.reshape([3, -1]) Out[45]: array([[1, 1], [1, 1], [0, 1]]) Modification of array dimensions ： dimension describe broadcast_to Broadcast array to new shape expand_dims Expand the shape of the array squeeze Remove one-dimensional entries from the shape of the array The specific operation is as follows ： Array connection ： function describe concatenate Join the array sequence along the existing axis hstack Stack arrays in a sequence horizontally （ Column direction ） vstack Stack arrays in a sequence vertically （ Line direction ） The following code shows the operation of array join , among concatenate By controlling axis To determine the direction of the connection , The effect is equal to hstack and vstack. Another thing to note is ： The following example simply concatenates two arrays , In fact, you can connect multiple , such as np.concatenate((x, y, z), axis=1) Segmentation of arrays ： function describe split Divide an array into multiple subarrays hsplit Divide an array horizontally into multiple subarrays （ By column ） vsplit Divide an array vertically into multiple subarrays （ Press the line ） It's the same as an array join ,split By controlling axis Property to get the same as hsplit、vsplit Same effect , Here's just split An example of , About hsplit and vsplit Refer to official documents ： Addition and deletion of array elements ： function describe append Add values to the end of the array insert Inserts a value along the specified axis before the specified subscript delete Delete the subarray of a certain axis , And return the new array after deletion radio broadcast (Broadcast) yes numpy For different shapes (shape) How to do the numerical calculation of the array , Arithmetic operations on arrays are usually performed on the corresponding elements . If two arrays a and b The same shape , The meet a.shape == b.shape, that ab The result is that a And b Array corresponding bit multiplication . This requires the same dimension , And the length of each dimension is the same . In [8]: import numpy as np In [9]: a = np.array([1,2,3,4]) ...: b = np.array([10,20,30,40]) In [10]: a b Out[10]: array([ 10, 40, 90, 160]) When in operation 2 The shapes of arrays are different ,numpy Will automatically trigger the broadcast mechanism . Such as ： In [11]: a = np.array([[ 0, 0, 0], ...: [10,10,10], ...: [20,20,20], ...: [30,30,30]]) ...: b = np.array([1,2,3]) In [12]: a + b Out[12]: array([[ 1, 2, 3], [11, 12, 13], [21, 22, 23], [31, 32, 33]]) The following image shows the array b How to broadcast with array a compatible . np.tile It can broadcast the target operation array , such as 1x3 The following operations can be broadcast as 4x6, Pay attention to the above broadcast_to Make a difference ,broadcast_to It has to be expanded , and tile Extendable dimension , We can not expand the dimension , Specific operation according to their own actual needs . There was Tensorflow Experienced readers should know , It also has tile and broadcast operation , But when we have a large amount of data , It is said that tile Is more efficient than broadcast Be low , I don't know why , It will be useful in the future . In [20]: a Out[20]: array([[1, 1, 0]]) In [21]: np.tile(a, [4, 2]) # The second parameter represents the multiple of each dimension broadcast , This is line expansion 4 times , Liege 2 times Out[21]: array([[1, 1, 0, 1, 1, 0], [1, 1, 0, 1, 1, 0], [1, 1, 0, 1, 1, 0], [1, 1, 0, 1, 1, 0]]) About NumPy Copy and attempt in ： This part of the knowledge is also what I was learning before NumPy The missing points of time , Take advantage of this opportunity , Record it here . • No copy On this point , Actually, it was recorded before LeetCode Hot topic HOT 100（01, Addition of two numbers ） The algorithm also mentioned , This point needs to be paid more attention to . • View or shallow copy （view） Use the same code as above , Only the... Has been modified 57 That's ok , take y = x and y = x.view(), You can find , At this time x and y Of id Values are not the same , They don't have the same memory address , We modify x Of shape after ,y Of shape Nothing has changed . however , When we change, it's not shape, Instead, change the data inside a variable array , The other array changes as well • Copy or deep copy (copy) The view or shallow copy uses view, And copy or deep copy uses copy. Use copy When modifying an array shape, Or internal elements , The other array doesn't change . （ About copy, The code is no longer demonstrated here , Readers can operate by themselves , And then compare them ） So to conclude ： • y = x, explain x and y Of Same memory address , Modify one of , The other will change as it happens （ Whether it's shape, Or internal elements ） • y = x.view(), The memory addresses of the two are different , Modify one of the shape, The other doesn't change ; And modify the inner elements of one of the tuples , The other one will change with it • y = x.copy(), The memory addresses of the two are different , Whether it's modifying a tuple shape, Or internal elements , None of the other will change , They are independent of each other NumPy The mathematical correlation function in , There is nothing to talk about in this part ： • np.pi, return π value • np.sin(), Return sine value • np.cos(), Returns the cosine of • np.tan(), Return tangent value • numpy.ceil(), Returns the smallest integer greater than or equal to the specified expression , That is, round up . • np.exp(2), Returns the index value , That is to say $e^2$ Other related mathematical functions , Refer to official documents . NumPy The arithmetic operations in , There is nothing to talk about in this part ： • numpy.add(a,b)： Add two arrays • numpy.subtract(a,b)： Subtracting two arrays • numpy.multiply(a,b)： Multiply two arrays • numpy.divide(a,b)： Divide two arrays by • numpy.reciprocal(a), Back to the bottom • numpy.power(a, 4), return a The fourth power of NumPy The statistical function in , This is a little bit of a note ： The above example shows how to get the maximum value and the difference between the maximum value and the maximum value in an array , Into axis Parameters , Then we get it in the corresponding direction , If there is no introduction axis Parameters , It means to get the maximum value of the whole array . In addition to the above interfaces , There are other common statistical functions , There is no difference between the specific operation and the above , as follows ： • np.amin()： Get the minimum • np.amax()： Get the maximum • np.ptp()： Get the difference between the maximum and the minimum • np.median()： Get the median （ The median ） • np.mean()： Get the mean • np.var()： To obtain the variance ,$\sigma^2 = \frac{1}{n}\sum_{i=1}n(x_i-\overline{x})2$ • np.std()： Get the standard deviation ,$\sigma$ NumPy Linear algebra in ： • np.dot(a, b) It's the product of two matrices • np.vdot(a, b) The sum of the products of the corresponding positions of two matrices • np.inner(a, b) Inner product , Namely a With each line of b Each line of the sum such as a=[[1, 0], [1, 1]],b=[[1, 2], [1, 3]] np.inner(a, b) amount to [1, 0] * [1, 2] = 1 -> For the first number [1, 0] * [1, 3] = 1 -> For the second number [1, 1] * [1, 2] = 3 -> For the third number [1, 1] * [1, 3] = 4 -> For the fourth number The matrix product is the sum of the product of the row of the first matrix and the column product of the second matrix , and inner It is equivalent to the sum of the row products of the first matrix and the second matrix • np.matmul(a, b) Feeling and np.dot(a, b) It works the same , They're all matrix products • np.linalg.det(a) Calculate the value of the determinant of a matrix • np.linalg.solve(a, [[1], [1]]) Find solutions to linear equations , The first parameter corresponds to the coefficient , The second parameter is equivalent to the parameter term • np.linalg.inv(a) Calculate the inverse of a matrix # Save the array to .npy In the file with the extension . numpy.save(file, arr, allow_pickle=True, fix_imports=True) • file： File to save , extension .npy, If there is no extension at the end of the file path .npy, The extension will be automatically added with . • arr: The array to save • allow_pickle: Optional , Boolean value , Allow to use Python pickles Save an array of objects ,Python Medium pickle Used before saving to or reading from a disk file , Serialize and deserialize objects . • fix_imports: Optional , For convenience Pyhton2 Read from Python3 Saved data . In [81]: a = np.random.randint(1, 10, [3, 4]) In [82]: np.save("a.npy", a) In [83]: np.load("a.npy") Out[83]: array([[2, 7, 3, 1], [4, 6, 4, 3], [2, 2, 9, 5]]) Reference material ： Reference material ： [1] NumPy Novice tutorial ：https://www.runoob.com/numpy/numpy-tutorial.html [2] NumPy Official documents ：https://numpy.org/doc/stable/user/quickstart.html Not enough time , It's a bit hasty to write later , But should not affect the normal reading and later review , For the time being . Be careful ： Only a few common ones are recorded , Other words will be used later to update it , Other contents can refer to the document . It was meant to be in accordance with 《 Machine learning practice / Machine Learning in Action》 This book is about to tear the code out of it , But for practical reasons , It may need to be torn by hand SVM 了 , This algorithm is still a headache , It's too complicated inside , There are few data to deduce it completely , It also involves a lot of nouns of "Mo Sheng" , Such as ： Optimization under nonlinear constraints 、KKT Conditions 、 Lagrange dual 、 The largest interval 、 The optimal lower bound 、 Kernel functions and so on , The book of heaven may be 、 Probably 、 Maybe that's it . Fortunately, I have studied before SVM, But it must still take a lot of energy to tear , Also need to refer to a lot of information , Including but not limited to 《 Machine learning practice / Machine Learning in Action》、《 machine learning 》、《 Statistical learning method 》. therefore , In the next issue , It should start tearing SVM, As for whether we can succeed in the end , It's hard to say . It may take a lot of time , During this period LeetCode HOT 100 And it needs to be painted normally . I am a Taoye, Love to study , Love sharing , Keen on all kinds of Technology , I like playing chess in my spare time 、 Listen to the music 、 Talking about animation , I hope to take this opportunity to record my growth process and life , Also hope to be able to foster more like-minded friends in the circle , For more information, welcome to wechat Princess ： Cynical Coder. Recommended reading ：	{}
> Puzzles 159. The categories I've been calling \$$\mathcal{C}\$$ and \$$\mathcal{D}\$$ have other, more purely mathematical names. More precisely, they are _isomorphic_ to two other categories we've already seen in this course, which have more mathematical names. What are those other categories? I think \$$\mathcal{C}\$$ has been called "\$$\mathbf{2}\$$" in other discussions and \$$\mathcal{D}\$$ has been called "\$$\mathbf{1}\$$".	{}
# Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 34 ## Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 34 Published $1968$. Previous ... Next ## $34 \quad$ Elliptic Functions ### Incomplete Elliptic Integral of the First Kind where $\phi = \operatorname {am} u$ is called the amplitude of $u$ and $x = \sin \phi$, where here and below $0 < k < 1$. ### Complete Elliptic Integral of the Third Kind Previous ... Next	{}
Corpus ID: 119349271 # White Paper on East Asian Vision for mm/submm VLBI: Toward Black Hole Astrophysics down to Angular Resolution of 1~R$_{S}$ @article{Asada2017WhitePO, title={White Paper on East Asian Vision for mm/submm VLBI: Toward Black Hole Astrophysics down to Angular Resolution of 1~R\$\_\{S\}\$}, author={Kunihiro Asada and Motoki Kino and Mareki Honma and Tomoya Hirota and Rusen S. Lu and Makoto Inoue and Bong Won Sohn and Z.-Q. Shen and Paul T. P. Ho and Kazunori Akiyama and Juan Carlos Algaba and Tao An and Geoff Bower and Do-Young Byun and Richard Dodson and Akihiro Doi and Philip G. Edwards and Kenta Fujisawa and Minfeng Gu and Kazuhiro Hada and Yoshiaki Hagiwara and Phrudth Jaroenjittichai and Taehyun Jung and Tomohisa Kawashima and Sergio Mikio Koyama and S.-S. Lee and Satoki Matsushita and Hiroshi Nagai and M. S. Nakamura and Kotaro Niinuma and Chris Phillips and J.-H. Park and Hung-Yi Pu and Hyunwook Ro and Jamie Stevens and Sascha Trippe and K. Wajima and G.-Y. Zhao}, journal={arXiv: High Energy Astrophysical Phenomena}, year={2017} } • K. Asada, +35 authors G.-Y. Zhao • Published 12 May 2017 • Physics • arXiv: High Energy Astrophysical Phenomena This White Paper details the intentions and plans of the East Asian Very Long Baseline Interferometry (VLBI) community for pushing the frontiers of millimeter/submillimeter VLBI. To this end, we shall endeavor to actively promote coordinated efforts in the East Asia region. Our goal is to establish firm collaborations among the East Asia VLBI community in partnership with related institutes in North America and Europe and to expand existing global mm/submm VLBI arrays for (a) exploring the… Expand 9 Citations Radio observations of active galactic nuclei with mm-VLBI • Physics • 2017 Over the past few decades, our knowledge of jets produced by active galactic nuclei (AGN) has greatly progressed thanks to the development of very-long-baseline interferometry (VLBI). Nevertheless,Expand Expected intermediate mass black holes in the Virgo cluster. II. Late-type galaxies • Physics • Monthly Notices of the Royal Astronomical Society • 2018 The Chandra X-ray Observatory's Cycle 18 Large Program titled "Spiral galaxies of the Virgo Cluster" will image 52 galaxies with the ACIS-S detector. Combined with archival data for an additional 22Expand Status of scientific commissioning of the Greenland Telescope • J. Koay, +35 authors Chen-Yu Yu • Engineering, Physics • Astronomical Telescopes + Instrumentation • 2020 The Greenland Telescope (GLT), currently located at Thule Air Base, is a 12-m single dish telescope operating at frequencies of 86, 230 and 345 GHz. Since April 2018, the GLT has regularlyExpand A balloon-borne very long baseline interferometry experiment in the stratosphere: Systems design and developments • A. Doi, +33 authors S. Koyama • Environmental Science, Physics • 2019 Abstract The balloon-borne very long baseline interferometry (VLBI) experiment is a technical feasibility study for performing radio interferometry in the stratosphere. The flight model has beenExpand Black Hole Spin Signature in the Black Hole Shadow of M87 in the Flaring State • Physics • The Astrophysical Journal • 2019 Imaging the immediate vicinity of supermassive black holes (SMBH) and extracting a BH-spin signature is one of the grand challenges in astrophysics. M87 is known as one of the best targets forExpand Kinematics of the M87 Jet in the Collimation Zone: Gradual Acceleration and Velocity Stratification We study the kinematics of the M87 jet using the first year data of the KVN and VERA Array (KaVA) large program, which has densely monitored the jet at 22 and 43 GHz since 2016. We find that theExpand GLT receiver commissioning at JCMT and future JCMT instrumentation The installation and integration of theGLT instrument at JCMT is reported, results from commissioning are presented and how the success of the GLT instrument commissioning fits with plans for future instrumentation atJCMT is shown. Expand Jet launching in resistive GR-MHD black hole - accretion disk systems • Physics • 2018 We investigate the launching mechanism of relativistic jets from black hole sources, in particular the strong winds from the surrounding accretion disk. Numerical investigations of the disk windExpand Accelerating cross-validation with total variation and its application to super-resolution imaging • Mathematics, Physics • PloS one • 2017 We develop an approximation formula for the cross-validation error (CVE) of a sparse linear regression penalized by ℓ1-norm and total variation terms, which is based on a perturbative expansionExpand #### References SHOWING 1-10 OF 15 REFERENCES HIGH-SENSITIVITY 86 GHz (3.5 mm) VLBI OBSERVATIONS OF M87: DEEP IMAGING OF THE JET BASE AT A RESOLUTION OF 10 SCHWARZSCHILD RADII We report on results from new high-sensitivity, high-resolution 86GHz (3.5 millimeter) observations of the jet base in the nearby radio galaxy M87, obtained by the Very Long Baseline Array inExpand Persistent Asymmetric Structure of Sagittarius A* on Event Horizon Scales The Galactic Center black hole Sagittarius A* (Sgr A) is a prime observing target for the Event Horizon Telescope (EHT), which can resolve the 1.3 mm emission from this source on angular scalesExpand Imaging an Event Horizon: Mitigation of Source Variability of Sagittarius A The black hole in the center of the Galaxy, associated with the compact source Sagittarius A* (Sgr A), is predicted to cast a shadow upon the emission of the surrounding plasma flow, which encodesExpand The galactic center : Feeding and feedback in a normal galactic nucleus : proceedings of the 303rd symposium of the International Astronomical Union held in Santa Fe, NM, USA, September 30 - October 4, 2013 • Physics • 2014 The Herschel view of the Galactic center J. Bally and the Hi-GAL team Physical conditions and chemistry of molecular gas in galactic centers S. Aalto TeV observations of the Galactic center andExpand Fundamental Parameters of the Milky Way Galaxy Based on VLBI Astrometry We present analyses to determine the fundamental parameters of the Galaxy based on VLBI astrometry of 52 Galactic maser sources obtained with VERA, VLBA, and EVN. We model the Galaxy’s structure withExpand Multi-Epoch VERA Observations of Sagittarius A. I. Images and Structural Variability • Physics • 2013 We report the results of multi-epoch observations of Sgr A* with VLBI Exploration of Radio Astrometry (VERA) at 43 GHz, carried out from 2004 to 2008. We detected a time variation of flux at 11 %Expand The Galactic Center: Feeding and Feedback in a Normal Galactic Nucleus Here we present the fundamental properties of the nuclear cluster of the Milky Way. First, we derive its structural properties by constructing a density map of the central 1000′′ usingExpand Discovery of $\gamma$-ray emission from a steep radio spectrum NLS1 B3 1441+476 • Physics • 2015 Narrow line Seyfert 1 galaxies (NLS1s) usually do not host relativistic jet and the $\gamma$-ray NLS1s are expected to be rare. All $\gamma$-ray NLS1s reported to date have flat radio spectra and theExpand Cold gas and the disruptive effect of a young radio jet • Physics • 2015 Newly born and young radio sources are in a delicate phase of their life. Their jets are fighting their way through the surrounding gaseous medium, strongly experiencing this interaction while, atExpand Computational Imaging for VLBI Image Reconstruction • Computer Science, Physics • 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) • 2016 This paper presents a novel Bayesian approach for VLBI image reconstruction, and shows that the method (CHIRP) produces good results under different settings such as low SNR or extended emission. Expand	{}
# What is the maximum acceptable price to pay for each project? What is the profitability index of What is the maximum acceptable price to pay for each project? What is the profitability index of each project? Compute this project’s NPV using Sprocket’s 16% hurdle rate. Should Sprocket invest in the equipment? Q24. Sprocket could refurbish the equipment at the end of six years for $103,000. The refurbished equipment could be used one more year, providing$75,000 of net cash inflows in year 7. Additionally, the refurbished equipment would have a54,000 residual value at the end of year 7. Should Sprocket invest in the equip- ment and refurbishing it after six years? ( Hint: In addition to your answer to Requirement 1, discount the additional cash outflow and inflows back to the present value.) Q25. Which investment should Brighton pursue at this time? Why? Compute the payback period, the ROR, the NPV, and the profitability index of these two plans. What are the strengths and weaknesses of these capital budget- ing models? Which expansion plan should Leches choose? Why? Q28. Estimate Plan A’s IRR. How does the IRR compare with the company’s required rate of return? How much money must you accumulate by retirement to make your plan work? ( Hint: Find the present value of the $235,000 withdrawals.) Q30. How does this amount compare to the total amount you will draw out of the investment during retirement? How can these numbers be so different? How much must you pay into the investment each year for the first 15 years? ( Hint: Your answer from Requirement 1 becomes the future value of this annuity.) Q32. How does the total “out-of-pocket” savings compare to the investment’s value at the end of the 15-year savings period and the withdrawals you will make dur- ing retirement? Compute the payback period, the ROR, the NPV, and the profitability index of these two plans. What are the strengths and weaknesses of these capital budget- ing models? Q34. Which expansion plan should Lulus choose? Why? Estimate Plan A’s IRR. How does the IRR compare with the company’s required rate of return? How much money must you accumulate by retirement to make your plan work? ( Hint: Find the present value of the$240,000 withdrawals.) How does this amount compare to the total amount you will draw out of the investment during retirement? How can these numbers be so different? Q38. How much must you pay into the investment each year for the first 10 years? ( Hint: Your answer from Requirement 1 becomes the future value of this annuity.) How does the total “out-of-pocket” savings compare to the investment’s value at the end of the 10-year savings period and the withdrawals you will make during retirement? Q40. Calculate payback period, rate of return, net present value, and IRR for the mower investment. Should Lawlor invest in the new mower? Calculate payback period, rate of return, net present value, and IRR for both server investments. Assuming capital rationing applies, which server should Draper invest in? Q44. Suppose that Hunter ignores the time value of money in decisions that cover this short time period. Suppose also that his sole goal is to make as much money as possible between now and the end of next summer. What should he do? What nonquantitative factors might Hunter consider? What would you do if you were faced with these alternatives? Q45. Now suppose that Hunter considers the time value of money for all cash flows that he expects to receive one year or more in the future. Which alternative does this consideration favor? Why? Did John have reason to be suspicious? What were the warning signs? What should small businesses do when they are in financial trouble? ## Recent Questions in Financial Accounting Copy and paste your question here... Attach Files	{}
# Maxwell's Two Equations For Electrostatic Fields This physics video tutorial provides a basic introduction into maxwell's equations and electromagnetic waves. Maxwell's 4 equations include gauss' law for el. Electrostatic Fields Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charges. The mathematical definition of the electrostatic field is derived from Coulomb’s law which defines the vector force between two point charges. Coulomb’s Law Given point charges q 1, q 2 (units=C). The displacement current introduced by Maxwell results instead from a changing electric field and accounts for a changing electric field producing a magnetic field. The equations for the effects of both changing electric fields and changing magnetic fields differ in form only where the absence of magnetic monopoles leads to missing terms. In Maxwell's equations. The behavior of magnetic fields (B, H), electric fields (E, D), charge density (ρ), and current density (J) is described by Maxwell's equations. The role of the magnetization is described below. Relations between B, H, and M. In Gauss's law, the electric field is the electrostatic field. The law shows how the electrostatic field behaves and varies depending on the charge distribution within it. More formally it relates the electric flux the electric field flowing from positive to negative charges passing through a closed surface to the charge contained within the. Michael Fowler, PhysicsDepartment, UVa ## The Equations Maxwell’s four equations describe the electric and magneticfields arising from distributions of electric charges and currents, and howthose fields change in time. They were the mathematical distillation ofdecades of experimental observations of the electric and magnetic effects ofcharges and currents, plus the profound intuition of Michael Faraday. Maxwell’s own contribution to these equationsis just the last term of the last equation—but theaddition of that term had dramatic consequences. It made evident for the first time thatvarying electric and magnetic fields could feed off each other—thesefields could propagate indefinitely through space, far from the varying chargesand currents where they originated. Previously these fields had been envisioned as tethered to the chargesand currents giving rise to them. Maxwell’snew term (called the displacement current) freed them to move through space ina self-sustaining fashion, and even predicted their velocity—it was thevelocity of light! Here are the equations: 1. Gauss’ Law for electric fields: ### Maxwell's Two Equations For Electrostatic Fields Based $\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}=q/{\epsilon }_{0}.$ (The integral of the outgoing electric field over an area enclosing a volume equals the total charge inside, in appropriate units.) 2. The corresponding formula for magnetic fields: $\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}=0.$ (No magnetic charge exists: no “monopoles”.) 3. Faraday’s Law of Magnetic Induction: $\oint \stackrel{\to }{E}\cdot d\stackrel{\to }{\ell }=-d/dt\left(\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}\right).$ The first term is integrated round a closed line, usually a wire, and gives the total voltage change around the circuit, which is generated by a varying magnetic field threading through the circuit. 4. Ampere’s Law plus Maxwell’s displacement current: $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\left(I+\frac{d}{dt}\left({\epsilon }_{0}\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}\right)\right).$ This gives the total magnetic force around a circuit in terms of the current through the circuit, plus any varying electric field through the circuit (that’s the “displacement current”). The purpose of this lecture is toreview the first three equations and the original Ampere’s law fairly briefly,as they were covered earlier in the course, then to demonstrate why thedisplacement current term must be added for consistency, and finally to show,without using differential equations, how measured values of static electricaland magnetic attraction are sufficient to determine the speed of light. ## Preliminaries: Definitions of µ0and ε0 Ampere discovered that two parallel wires carrying electriccurrents in the same direction attract each other magnetically, the force innewtons per unit length being given by $F=2\left(\frac{{\mu }_{0}}{4\pi }\right)\frac{{I}_{1}{I}_{2}}{r},$ for long wires a distance $r$ apart. We are using the standard modern units(SI). The constant ${\mu }_{0}/4\pi$ that appears here is exactly 10-7,this defines our present unit of current, the ampere. To repeat: ${\mu }_{0}/4\pi$ is not something to measure experimentally,it's just a funny way of writing the number 10-7! That's not quite fair—it hasdimensions to ensure that both sides of the above equation have the samedimensionality. (Of course, there's a historical reason for this strangeconvention, as we shall see later). Anyway,if we bear in mind that dimensions have been taken care of, and justwrite the equation $F=2\cdot {10}^{-7}\cdot \frac{{I}_{1}{I}_{2}}{r},$ it's clear that this defines the unit current—one ampere—as thatcurrent in a long straight wire which exerts a magnetic force of $2×{10}^{-7}$ newtons per meter of wire on a parallel wireone meter away carrying the same current. However, after we have established our unit of current—the amperewehave also thereby defined our unit of charge, since current is a flow ofcharge, and the unit of charge must be the amount carried past a fixed point inunit time by unit current. Therefore,our unit of charge—the coulomb—is definedby stating that a one amp current in a wire carries one coulombper second past a fixed point. To be consistent, we must do electrostatics using thissame unit of charge. Now, the electrostatic force between two chargesis $\left(1/4\pi {\epsilon }_{0}\right){q}_{1}{q}_{2}/{r}^{2}.$ The constant appearing here, now written $1/4\pi {\epsilon }_{0}$,must be experimentally measured—its valueturns out to be $9×{10}^{9}$. To summarize: tofind the value of $1/4\pi {\epsilon }_{0}$,two experiments have to be performed.We must first establish the unit of charge from the unit of current bymeasuring the magnetic force between two current-carrying parallel wires. Second, we must find the electrostatic forcebetween measured charges. (We could, alternatively, have defined some otherunit of current from the start, then we would have had to find both ${\mu }_{0}$ and ${\epsilon }_{0}$ by experiments on magnetic and electrostaticattraction. In fact, the ampere was originallydefined as the current that deposited a definite weight of silver per hour inan electrolytic cell). ## Maxwell's Equations We have so far established that the total flux of electricfield out of a closed surface is just the total enclosed charge multiplied by $1/{\epsilon }_{0}$, $\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}=q/{\epsilon }_{0}.$ This is Maxwell’sfirst equation. Itrepresents completely covering the surface with a large number of tiny patcheshaving areas $d\stackrel{\to }{A}$. We represent these small areas asvectors pointing outwards, because we can then take the dot product with theelectric field to select the component of that field pointing perpendicularlyoutwards (it would count negatively if the field were pointing inwards)—this isthe only component of the field that contributes to actual flow across thesurface. (Just as a river flowingparallel to its banks has no flow across the banks). The second Maxwell equation is the analogous one forthe magnetic field, which has no sources or sinks (no magnetic monopoles, thefield lines just flow around in closed curves). Thinking of the force lines asrepresenting a kind of fluid flow, the so-called 'magnetic flux', wesee that for a closed surface, as much magnetic flux flows into the surface asflows out. This can perhaps bevisualized most clearly by taking a group of neighboring lines of force forminga slender tube—the'fluid' inside this tube flows round and round, so as the tube goesinto the closed surface then comes out again (maybe more than once) it is easyto see that what flows into the closed surface at one place flows out at another.Therefore the net flux out of the enclosed volume is zero, Maxwell’s second equation: $\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}=0.$ The first two Maxwell's equations, given above, are forintegrals of the electric and magnetic fields over closed surfaces .The other two Maxwell's equations, discussed below, are for integrals ofelectric and magnetic fields around closed curves (taking thecomponent of the field pointing along the curve). These represent the work thatwould be needed to take a charge around a closed curve in an electric field,and a magnetic monopole (if one existed!) around a closed curve in a magneticfield. The simplest version of Maxwell's third equationis the electrostatic case: The path integral$\oint \stackrel{\to }{E}\cdot d\stackrel{\to }{\ell }=0$for electrostatics. However, we know that this is only part of the truth,because from Faraday's Law of Induction, if a closed circuit has a changingmagnetic flux through it, a circulating current will arise, which meansthere is a nonzero voltage around the circuit. The full version of Maxwell's third equation is: $\oint \stackrel{\to }{E}\cdot d\stackrel{\to }{\ell }=-d/dt\left(\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}\right)$ where the area integrated over on the right hand side spansthe path (or circuit) on the left hand side, like a soap film on a loop ofwire. It may seem that the integral on the right hand sideis not very clearly defined, because if the path or circuit lies in a plane,the natural choice of spanning surface (the 'soap film') is flat, buthow do you decide what surface to choose to do the integral over for a wirebent into a circuit that doesn’t lie in a plane? The answer is that it doesn’t matter whatsurface you choose, as long as the wire forms its boundary. Consider two different surfaces both havingthe wire as a boundary (just as both the northern hemisphere of the earth’ssurface and the southern hemisphere have the equator as a boundary). If you addthese two surfaces together, they form a single closed surface, and we knowthat for a closed surface $\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}=0$. This implies that $\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}$ for one of the two surfaces bounded by thepath is equal to $-\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}$ for the other one, so that the two will add tozero for the whole closed surface. Butdon’t forget these integrals for the whole closed surface are defined with thelittle area vectors pointing outwards from the enclosed volume. By imaginingtwo surfaces spanning the wire that are actually close to each other, it isclear that the integral over one of them is equal to the integral over theother if we take the $d\stackrel{\to }{A}$ vectors to point in the same direction forboth of them, which in terms of the enclosed volume would be outwards for onesurface, inwards for the other one. The bottom line of all this is that thesurface integral $\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}$ is the same for any surface spanning the path,so it doesn’t matter which we choose. The equation analogous to the electrostatic version of thethird equation given above, but for the magnetic field, is Ampere's law $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\cdot \text{(enclosed currents)}$ for magnetostatics, where the currents counted are those threading throughthe path we're integrating around, so if there is a soap film spanning thepath, these are the currents that punch through the film (of course, we have toagree on a direction, and subtract currents flowing in the opposite direction). We must now consider whether this equation, like theelectrostatic one, has limited validity. In fact, it was not questioned for ageneration after Ampere wrote it down: Maxwell's great contribution, in the1860's, was to realize that it was not always valid. ## When Does Ampere'sLaw Go Wrong? A simple example to see that something mustbe wrong with Ampere's Law in the general case is given by Feynman in his Lecturesin Physics. Suppose we use ahypodermic needle to insert a spherically symmetric blob of charge in themiddle of a large vat of solidified jello (which we assume conductselectricity). Because of electrostaticrepulsion, the charge will dissipate, currents will flow outwards in aspherically symmetric way. Question: does this outward-flowingcurrent distribution generate a magnetic field? The answer must be no , because sincewe have a completely spherically symmetric situation, it could only generate aspherically symmetric magnetic field. Butthe only possible such fields are one pointing outwards everywhere and onepointing inwards everywhere, both corresponding to non-existent monopoles. So,there can be no magnetic field. However,imagine we now consider checking Ampere's law by taking as a path a horizontalcircle with its center above the point where we injected the charge (think of ahalo above someone’s head.) Obviously,the left hand side of Ampere's equation is zero, since there can be no magneticfield. (It would have to be sphericallysymmetric, meaning radial.) On the otherhand, the right hand side is most definitely not zero, since some of theoutward flowing current is going to go through our circle. So the equation must be wrong. Ampere's law was established as the result of largenumbers of careful experiments on all kinds of current distributions. So how could it be that something of the kindwe describe above was overlooked? Thereason is really similar to why electromagnetic induction was missed for solong. No-one thought about looking at changing fields, all theexperiments were done on steady situations. With our ball of charge spreading outward inthe jello, there is obviously a changing electric field. Imagine yourself in the jello near where thecharge was injected: at first, you would feel a strong field from the nearbyconcentrated charge, but as the charge spreads out spherically, some of itgoing past you, the field will decrease with time. ## Maxwell's Example Maxwell himself gave a more practical example: consider Ampere's law for the usual infinitelylong wire carrying a steady current $I,$ but now break the wire at some point and putin two large circular metal plates, a capacitor, maintaining the steady current$I$ in the wire everywhere else, so that charge issimply piling up on one of the plates and draining off the other. Looking now at the wire some distance away from theplates, the situation appears normal, and if we put the usual circular patharound the wire, application of Ampere's law tells us that the magnetic fieldat distance $r,$ from $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}I$ is just $B={\mu }_{0}I/2\pi r.$ Recall, however, that wedefined the current threading the path in terms of current punching through asoap film spanning the path, and said this was independent of whether the soapfilm was flat, bulging out on one side, or whatever. With a single infinitewire, there was no escape— no contortions of this coveringsurface could wriggle free of the wire going through it (actually, if youdistort the surface enough, the wire could penetrate it several times, but youhave to count the net flow across the surface, and the new penetrations wouldcome in pairs with the current crossing the surface in opposite directions, sothey would cancel). Oncewe bring in Maxwell's parallel plate capacitor, however, there is away to distort the surface so that no current penetrates it at all: we can runit between the plates! The question then arises: can we rescue Ampere's law byadding another term just as the electrostatic version of the third equation wasrescued by adding Faraday's induction term? The answer is of course yes:although there is no current crossing the surface if we put it betweenthe capacitor plates, there is certainly a changing electric field ,because the capacitor is charging up as the current $I$ flows in. Assuming the plates are closetogether, we can take all the electric field lines from the charge $q$ on one plate to flow across to the otherplate, so the total electric flux across the surface between the plates, $\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}=q/{\epsilon }_{0}.$ Now, the current in the wire $,I,$ is just the rate of change of charge on theplate, $I=dq/dt.$ Putting the above two equations together, we see that $I=\frac{d}{dt}\left({\epsilon }_{0}\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}\right).$ Ampere's law can now be written in a way that is correct nomatter where we put the surface spanning the path we integrate the magneticfield around: $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\left(I+\frac{d}{dt}\left({\epsilon }_{0}\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}\right)\right).$ This is Maxwell’sfourth equation. Notice that in the case of the wire, either thecurrent in the wire, or the increasing electric field, contribute onthe right hand side, depending on whether we have the surface simply cuttingthrough the wire, or positioned between the plates. (Actually, more complicatedsituations are possible—we couldimaging the surface partly between the plates, then cutting through the platesto get out! In this case, we would have tofigure out the current actually in the plate to get the right hand side, butthe equation would still apply). ## 'DisplacementCurrent' Maxwell referred to the second term on the right handside, the changing electric field term, as the 'displacement current'. This was an analogy with a dielectricmaterial. If a dielectric material isplaced in an electric field, the molecules are distorted, their positivecharges moving slightly to the right, say, the negative charges slightly to theleft. Now consider what happens to adielectric in an increasing electric field. The positive charges will be displaced to theright by a continuously increasing distance, so, as long as the electric fieldis increasing in strength, these charges are moving: there is actually a displacementcurrent . (Meanwhile, the negativecharges are moving the other way, but that is a current in the same direction,so adds to the effect of the positive charges' motion.) Maxwell's picture of the vacuum, the aether,was that it too had dielectric properties somehow, so he pictured a similarmotion of charge in the vacuum to that we have just described in thedielectric. The picture is wrong, but thisis why the changing electric field term is often called the 'displacementcurrent', and in Ampere's law (generalized) is just added to the realcurrent, to give Maxwell's fourth—and final—equation. ## Another Angle on theFourth Equation: the Link to Charge Conservation Going back for a moment to Ampere's law, we stated itas: $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\cdot \text{(enclosed currents)}$ for magnetostatics, where the currents counted are those threading through thepath we’re integrating around, so if there is a soap film spanning the path,these are the currents that punch through the film. Our mental picture here isusually of a few thin wires, maybe twisted in various ways, carrying currents.More generally, thinking of electrolytes, or even of fat wires, we should beenvisioning a current density varying from point to point in space. In otherwords, we have a flux of current and the natural expression for the currentthreading our path is (analogous to the magnetic flux in the third equation) towrite a surface integral of the current density $\stackrel{\to }{j}$ over a surface spanning the path, giving formagnetostatics path integral $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}$(surface integral, over surface spanning path) The question then arises as to whether the surface integralwe have written on the right hand side above depends on which surfacewe choose spanning the path. From an argument exactly parallel to that for themagnetic flux in the third equation (see above), this will be true if andonly If $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}=0$fora closed surface (with the path lying in the surface—thisclosed surface is made up by combining two different surfaces spanning thepath). Now, $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}$ takenover a closed surface is just the net current flow out of the enclosedvolume. Obviously, in a situation with steady currents flowing along wires orthrough conductors, with no charge piling up or draining away from anywhere,this is zero. However, if the total electric charge $q,$ say, enclosed by the closed surface is changingas time goes on, then evidently $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}=-dq/dt,$ where we put in a minus sign because, with our convention, $d\stackrel{\to }{A}$ is a little vector pointing outwards,so the integral represents net flow of charge out from the surface,equal to the rate of decrease of the enclosed total charge. To summarize: if the local charge densities arechanging in time, that is, if charge is piling up in or leaving some region,then $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}\ne 0$ over a closed surface around that region. Thatimplies that $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}$ over one surface spanning the wire will be differentfrom $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}$ over another surface spanning the wire ifthese two surfaces together make up a closed surface enclosing a regioncontaining a changing amount of charge. The key to fixing this up is to realize that although $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}=-dq/dt\ne 0,$ the right-hand side can be written as anothersurface integral over the same surface, using the first Maxwellequation, that is, the integral over a closed surface $\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}=q/{\epsilon }_{0}$ where $q$ is the total charge in the volume enclosed bythe surface. By taking the time rate of change of both sides, wefind $\frac{d}{dt}\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}=\frac{1}{{\epsilon }_{0}}\frac{dq}{dt}$ Putting this together with $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}=-dq/dt$ gives: ${\epsilon }_{0}\frac{d}{dt}\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}+\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}=0$ for any closed surface, and consequently this is asurface integral that must be the same for any surface spanning thepath or circuit! (Because two differentsurfaces spanning the same circuit add up to a closed surface. We’ll ignore thetechnically trickier case where the two surfaces intersect each other, creatingmultiple volumes—there onemust treat each created volume separately to get the signs right.) Therefore, this is the way to generalize Ampere's law fromthe magnetostatic situation to the case where charge densities are varying withtime, that is to say the path integral $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\int \left(\stackrel{\to }{j}+{\epsilon }_{0}\frac{d\stackrel{\to }{E}}{dt}\right)\cdot d\stackrel{\to }{A}$ and this gives the same result for any surface spanning thepath. ## A Sheet of Current: ASimple Magnetic Field As a preliminary to looking at electromagnetic waves, weconsider the magnetic field configuration from a sheet of uniform current oflarge extent. Think of the sheet asperpendicular to this sheet of paper, the current running vertically upwards. It might be helpful to visualize the sheet asmany equal parallel fine wires uniformly spaced close together: ......................................................................................(wires) The magnetic field from this current sheet can be foundusing Ampere's law applied to a rectangular contour in the plane of the paper,with the current sheet itself bisecting the rectangle, so the rectangle's topand bottom are equidistant from the current sheet in opposite directions. ### Maxwell's Two Equations For Electrostatic Fields Worksheet Applying Ampere’s law to the above rectangularcontour, there are contributions to $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }$ only from the top and bottom, and they add togive $2BL$ if the rectangle has side $L.$ The total current enclosed by the rectangle is$IL,$ taking the current density of the sheet to be $I$ amperes per meter (how many little wires permeter multiplied by the current in each wire). Thus, $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\cdot \text{(enclosed currents)}$ immediately gives: $B={\mu }_{0}I/2$ a magnetic field strength independent of distance $d$ from the sheet. (This is the magnetostaticanalog of the electrostatic result that the electric field from an infinitesheet of charge is independent of distance from the sheet.) In real life, where there are no infinitesheets of anything, these results are good approximations for distances fromthe sheet small compared with the extent of the sheet. ## Switching on the Sheet: How Fast Does the Field Build Up? Consider now how the magnetic field develops if the currentin the sheet is suddenly switched on at time $t=0.$ We will assume that sufficiently close to thesheet, the magnetic field pattern found above using Ampere's law is ratherrapidly established. In fact, we will assume further that the magnetic fieldspreads out from the sheet like a tidal wave, moving in both directions at somespeed $v,$ so that after time $t$ the field within distance $vt$ of the sheet is the same as that found abovefor the magnetostatic case, but beyond $vt$ there is at that instant no magnetic fieldpresent. Let us now apply Maxwell's equations to this guess to see ifit can make sense. Certainly Ampere's law doesn't work by itself, because if wetake a rectangular path as we did in the previous section, for $d everything works as before, but for arectangle extending beyond thespreading magnetic field, $d>vt,$ there will be no magnetic field contribution from the top and bottom of therectangle, and hence $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }=0$ but there is definitely enclosed current! We are forced to conclude that for Maxwell's fourth equationto be correct, there must also be a changing electric field through therectangular contour. Let us now try to nail down what this electric field throughthe contour must look like. First, itmust be through the contour, that is, have a component perpendicular to theplane of the contour, in other words, perpendicular to the magnetic field. In fact, electric field components in otherdirections won't affect the fourth equation we are trying to satisfy, so weshall ignore them. Notice first that fora rectangular contour with $d Ampere'slaw works, so we don'twant a changing electric fieldthrough such a contour (but a constant electric field would be ok). Now apply Maxwell's fourth equation to a rectangular contourwith $d>vt,$ It is: path integral $\oint \stackrel{\to }{B}\cdot d\stackrel{\to }{\ell }={\mu }_{0}\int \left(\stackrel{\to }{j}+{\epsilon }_{0}\frac{d\stackrel{\to }{E}}{dt}\right)\cdot d\stackrel{\to }{A}$ (over surface spanning path). For the rectangle shown above, the integral on the left handside is zero because $\stackrel{\to }{B}$ is perpendicular to $d\stackrel{\to }{\ell }$ along the sides, so the dot product is zero,and $\stackrel{\to }{B}$ is zero at the top and bottom, because theoutward moving 'wave' of magnetic field hasn’t gotten there yet.Therefore, the right hand side of the equation must also be zero. We know $\int \stackrel{\to }{j}\cdot d\stackrel{\to }{A}=LI$,so we must have: ${\epsilon }_{0}\frac{d}{dt}\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}=-LI.$ ## Finding the Speed of the Outgoing Field Front: the Connection with Light So, as long as the outward moving front of magnetic field,travelling at $v,$ hasn't reached the top and bottom of therectangular contour, the electric field through the contour increases linearly with time, but theincrease drops to zero (because Ampere's law is satisfied) the moment the frontreaches the top and bottom of the rectangle. The simplest way to get thisbehavior is to have an electric field of strength $E,$ perpendicular to the magnetic field,everywhere there is a magnetic field, so the electric field also spreadsoutwards at speed $v.$ Notethat, unlike the magnetic field, the electric field must point the same way onboth sides of the current sheet, otherwise its net flux through the rectanglewould be zero. After time $t,$ then, the electric field flux through therectangular contour $\int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}$ will be just field x area = $E\cdot 2\cdot vtL,$ and the rate of change will be $2EvL.$ (It's spreading both ways, hence the 2). Therefore ${\epsilon }_{0}E\cdot 2\cdot vtL=-LI,$ the electric field is downwards and of strength $E=I/\left(2{\epsilon }_{0}v\right).$ Since $B={\mu }_{0}I/2,$ this implies: $B={\mu }_{0}{\epsilon }_{0}vE.$ But we have another equation linking the field strengths ofthe electric and magnetic fields, Maxwell's third equation: $\oint \stackrel{\to }{E}\cdot d\stackrel{\to }{\ell }=-d/dt\left(\int \stackrel{\to }{B}\cdot d\stackrel{\to }{A}\right)$ We can apply this equation to a rectangular contour withsides parallel to the $E$ field, one side being within $vt$ of the current sheet, the other more distant,so the only contribution to the integral is $EL$ fromthe first side, which we take to have length $L.$ (This contour is all on one side of thecurrent sheet.) The area of the rectangle the magnetic flux is passing throughwill be increasing at a rate $Lv$ (square meters per second) as the magneticfield spreads outwards. ### Application Of Electrostatic Field It follows that $E=vB.$ Putting this together with the result of the fourthequation, ### Electrostatic Field Lines $B={\mu }_{0}{\epsilon }_{0}vE,$ we deduce ${v}^{2}=\frac{1}{{\mu }_{0}{\epsilon }_{0}}.$ Substituting the defined value of ${\mu }_{0},$ and the experimentally measured value of ${\epsilon }_{0},$ we findthat the electric and magnetic fields spread outwards from the switched-oncurrent sheet at a speed of 3 x 108 meters per second. ### Electrostatic Force Equation This is how Maxwell discovered a speed equal to the speed oflight from a purely theoretical argument based on experimental determinationsof forces between currents in wires and forces between electrostatic charges.This of course led to the realization that light is an electromagnetic wave,and that there must be other such waves with different wavelengths. Hertz detected other waves, of much longerwavelengths, experimentally, and this led directly to radio, tv, radar,etc.	{}
Wednesday, January 27, 2021 An assignment problem with a wrinkle In [1], the following problem is posed: Consider two arrays $$\color{darkblue}a_i$$ (length $$\color{darkblue}m$$) and $$\color{darkblue}b_j$$ (length $$\color{darkblue}n$$) with $$\color{darkblue}m \lt \color{darkblue}n$$. Assign all values $$\color{darkblue}a_i$$ to a $$\color{darkblue}b_j$$ such that: • Each $$\color{darkblue}b_j$$ can have 0 or 1 $$\color{darkblue}a_i$$ assigned to it. • The assignments need to maintain the original order of $$\color{darkblue}a_i$$. I.e. if $$\color{darkblue}a_i \rightarrow \color{darkblue}b_j$$ then $$\color{darkblue}a_{i+1}$$ must be assigned to a slot in $$\color{darkblue}b$$ that is beyond slot $$j$$. In the picture below that means that arrows cannot cross. • Do this while minimizing the sum of the products. Mixed-integer programming model This problem can be viewed as an assignment problem with a side constraint: MIP Model \begin{align}\min& \sum_{i,j}\color{darkred}x_{i,j}\cdot\color{darkblue}a_i\cdot\color{darkblue}b_j \\ &\sum_j \color{darkred}x_{i,j}=1 &&\forall i\\ & \sum_i \color{darkred}x_{i,j}\le 1 &&\forall j\\ & \color{darkred}v_i = \sum_j j \cdot \color{darkred} x_{i,j}\\ & \color{darkred}v_i \ge \color{darkred}v_{i-1}+1\\ & \color{darkred}x_{i,j} \in \{0,1\} \\ & \color{darkred}v_i \ge 1\end{align} Here $$\color{darkred}x_{i,j}$$ indicates the assignment $$\color{darkblue}a_i \rightarrow \color{darkblue}b_j$$. The variable $$\color{darkred}v_i$$ represents the position in $$\color{darkblue}b$$ to which $$\color{darkblue}a_i$$ is assigned. The output of this model can look like: ---- 40 SET i i1, i2, i3 ---- 40 SET j j1, j2, j3, j4, j5, j6 ---- 40 PARAMETER a i1 1.000, i2 2.000, i3 3.000 ---- 40 PARAMETER b j1 4.000, j2 9.000, j3 5.000, j4 3.000, j5 2.000, j6 10.000 ---- 40 VARIABLE z.L = 16.000 objective ---- 40 VARIABLE x.L assignment j1 j4 j5 i1 1.000 i2 1.000 i3 1.000 ---- 40 VARIABLE v.L position of a(i) in b i1 1.000, i2 4.000, i3 5.000 Notes: • We can tighten the bounds on variable $$\color{darkred}v_i$$ to $$\color{darkred}v_i \in [i,\color{darkblue}n-\color{darkblue}m+i]$$. • The variable $$\color{darkred}v_i$$ is automatically integer-valued. We can declare it as an integer or continuous variable. • If declared as a continuous variable, Cplex may make it an integer variable. We can see this in the log: Reduced MIP has 17737 binaries, 99 generals, 0 SOSs, and 0 indicators. • A larger problem with $$\color{darkblue}m=50, \color{darkblue}n=500$$ solves in about 50 seconds. It is interesting to see what happens when we feed this model an even larger data set. I generated some random data: \begin{align} & \color{darkblue}m=100\\ &\color{darkblue}n=1000 \\ & \color{darkblue}a_i \sim U(0,100) \\ & \color{darkblue}b_j \sim U(0,100)\end{align} This is not a small problem: we will have 100,000 binary variables. This model has no longer a network structure, so compared to a pure assignment problem we see much poorer performance: Assignment problem as LP Assignment problem as MIP Full MIP model Rows/Columns1,100/100,0001,100/100,0001,300/100,100 Objective11,524.17811,524.17814,371.455 Time (seconds)171,853 Nodes-03,419 Iterations2,6722,72730,7429 The assignment problem was formed by dropping the constraints involving $$\color{darkred}v_i$$. Further, the problem was solved as an LP. It is noted that Cplex behaves differently when we use continuous variables $$\color{darkred}x_{i,j}$$ or declare them as binary variables, When they are binary variables, the presolver works differently, and as a result, the solution of the relaxed problem is no longer optimal: it has to do some work to find the optimal integer solution. In this case, this was done in MIP preprocessing: no branching was needed. When solved as a pure LP, the optimal solution is integer-valued automatically. There is a difference in solution time: the LP solves in 1 second, while the MIP version takes 7 seconds. Let's see if we can shave something off this 1,800 seconds solution time. The first experiment I did was to provide an initial solution. I took the 100 smallest values of $$\color{darkblue}b_j$$ and assigned the 100 $$\color{darkblue}a_i$$'s to these while maintaining the order. This is very simple, so this took no time. The MIP was solved with a MIPSTART option. The results are: Assignment problem as LP Assignment problem as MIP Full MIP model Using initial solution Rows/Columns1,100/100,0001,100/100,0001,300/100,1001,300/100,100 Objective11,524.17811,524.17814,371.455initial: 17,739.702 optimal: 14,371.455 Time (seconds)171,8531,458 Nodes-03,4193,546 Iterations2,6722,727307,429343,969 This model solves faster. We don't see this in the node and iteration count, mainly because Cplex spends much time in preprocessing and restarts. Our initial objective is not that great. But this leads to the question, can we select the say 200 smallest values of $$\color{darkblue}b_j$$ and solve this smaller problem first. Then do a second solve with use all values of $$\color{darkblue}b_j$$, again using a MIPSTART. So the algorithm becomes: 1. Select 100 smallest values of $$\color{darkblue}b_j$$. Assign $$\color{darkblue}a_i$$ in order. This gives an initial solution of 17,739.702. 2. Pick 200 of of the smallest values of $$\color{darkblue}b_j$$. Solve our MIP problem on this subset. This gives an optimal solution of 14,451.861 in 118 seconds. 3. Use this as an initial solution for the whole problem. This now solves in 972 seconds. Total solution time is 118+972=1,090 seconds. Adding this to our table gives: Assignment problem as LP Assignment problem as MIP Full MIP model Using initial solution Using two models Rows/Columns1,100/100,0001,100/100,0001,300/100,1001,300/100,100subset: 500/20,100 full: 1,300/100,100 Objective11,524.17811,524.17814,371.455initial: 17,739.702 optimal: 14,371.455 initial: 17,739.702 subset: 14,451.861 full: 14,371.455 Time (seconds)171,8531,458subset: 118 full: 972 Nodes-03,4193,546subset: 3,205 full: 3,500 Iterations2,6722,727307,429343,969subset: 276,796 full: 456,537 Conclusions • Sometimes it makes sense to replace one big solve with a number of solves. Each one will start from and improve a previously found solution. • In production environments, a setup like this has another advantage: if a single model fails, you still get good solutions. • In the above table, the statistics for the number of nodes and Simplex iterations are not very indicative of the total effort. As MIP solvers become more complex, the best performance measure is solution time. • An assignment problem (or other network problems) should be solved as an LP, not as a MIP. References 1. Minimize sum of product of two uneven consecutive arrays, https://stackoverflow.com/questions/65884107/minimize-sum-of-product-of-two-uneven-consecutive-arrays 2. Assignment problem with a wrinkle formulated as a network problem, https://yetanothermathprogrammingconsultant.blogspot.com/2021/01/assignment-problem-with-wrinkle.html. Write-up on how to formulate this problem as a shortest path problem. Appendix: GAMS model This is the model that I used to try out the three-tier approach. $ontext Solve ordered assignment problem in three stages: 1. Find a trivial initial solution (100 smallest values in b) 2. Find a better solution using 200 smallest values 3. Solve the full problem$offtext ------------------------------------------------------- data ------------------------------------------------------- sets i /i1i100/ j /j1j1000/ ; parameter A(i) B(j) ; a(i) = uniform(0,100); b(j) = uniform(0,100); display a,b; ------------------------------------------------------- * optimization model ------------------------------------------------------- set subset(j) 'subset of b'; binary variable x(i,j) 'assign' ; variable v(i) 'position of a(i) in b' z 'objective' ; tight bounds on v v.lo(i) = ord(i); v.up(i) = card(j) - (card(i)-ord(i)); equations assign1(i) 'assignment constraint' assign2(j) 'assignment constraint' calcv 'caculate v' order 'ordering constraint' obj 'objective' ; obj.. z =e= sum((i,subset(j)),x(i,j)a(i)b(j)); assign1(i).. sum(subset(j), x(i,j)) =e= 1; assign2(subset(j)).. sum(i, x(i,j)) =l= 1; calcv(i).. v(i) =e= sum(subset(j), ord(j)x(i,j)); order(i-1).. v(i) =g= v(i-1)+1; model m /all/; option optcr=0, threads=8; ------------------------------------------------------- * set initial values using trivial solution of * 100 smallest values in b. ------------------------------------------------------- find 100 smallest values in b subset(j) = no; scalars bmin 'minimum value in remaining b' k 'loop variable' ; for(k=1 to 100, bmin = smin(j$(not subset(j)), b(j)); loop(j$(b(j)=bmin), subset(j) = yes; break; ); ); * let x reflect this initial subset loop(i, loop(j$subset(j), x.l(i,j) = 1; subset(j) = 0; break; ); ); z.l = sum((i,j),x.l(i,j)a(i)b(j)); v.l(i) = sum(j, ord(j)x.l(i,j)); option v:0; display "********** initial",z.l,v.l; ------------------------------------------------------- limit search to 200 values ------------------------------------------------------- for(k=1 to 200, bmin = smin(j$(not subset(j)), b(j)); loop(j$(b(j)=bmin), subset(j) = yes; break; ); ); m.optfile=1; solve m minimizing z using mip; display "********** solve 1",z.l,v.l; ------------------------------------------------------- solve full model ------------------------------------------------------- subset(j) = yes; solve m minimizing z using mip; display "********** solve 2",z.l,v.l;$onecho > cplex.opt mipstart 1 \$offecho 1. My colleague (and regular commenter on this blog) Rob Pratt suggested a purely graph model: Node(i,j,k), where (j < k) indicates that a_i is assigned to b_j and a_{i+1} is assigned to b_k. The main arcs are (i,j,k)->(i+1,k,\ell) with k < \ell and cost a_{i+1} b_k. The arc from a dummy source node to (1,j,k) picks up the initial cost a_1 b_j + a_2 b_k, and the cost from the last real node (m-1,j,k) to a dummy sink node is 0. The the problem is to find the shortest path from the dummy source to the sink in a directed acyclic graph. For the large model that is 50 million nodes and 2.5 billion arcs. It would take a lot more memory than the milp, but I think it would be faster. 1. That is truly a large network. I am sure I have never solved an LP of this size, but may be a shortest path algorithm can handle this. 2. Can we not just have node(i,j) -> n(i+1,j+k) for k=1,2,... Each incoming (or each outgoing) arc has a cost a(i)/b(j). 3. Yes, that smaller network seems correct: arc from source to (1,j) with cost a(1)b(j), arc from (i,j) to (i+1,k), where, j < k, with cost a(i+1)b(k), and arc from (m,j) to sink with cost 0. 2. This comment has been removed by the author. 1. I tried this. Seems to work for small instances. The big one is still big. Too large for me to solve on my laptop (90k nodes, 4e7 arcs). 2. Solves quickly with Dijkstra on my laptop. 3. I tried to pass it on to Cplex as an LP (and then solve it with the network solver). I should feed it directly into a shortest path algorithm. 4. Yes. Using a network solver would be pretty fast too (for the solve). But, there will be a lot of overhead to first find the network embedding from the matrix, translate the matrix into a graph and then, of course, network simplex is going to be much slower than a pure shortest path algorithm. Network simplex (a flow formulation) is definitely overkill here. 3. That worked quite nicely. NOTE: ----------------------------------------------------------------------------------------- NOTE: ----------------------------------------------------------------------------------------- NOTE: Running OPTNETWORK. NOTE: ----------------------------------------------------------------------------------------- NOTE: ----------------------------------------------------------------------------------------- NOTE: Data input used 1.32 (cpu: 5.87) seconds. NOTE: Building the input graph storage used 1.59 (cpu: 14.16) seconds. NOTE: The number of nodes in the input graph is 94953. NOTE: The number of links in the input graph is 44601302. NOTE: Processing shortest paths problem using 16 threads across 1 machines. NOTE: Processing the shortest paths problem between 1 source nodes and 1 sink nodes. Real Algorithm Sources Complete Time shortestPath 1 100% 0.31 NOTE: Processing the shortest paths problem used 0.31 (cpu: 0.31) seconds. NOTE: The Cloud Analytic Services server processed the request in 3.509987 seconds. NOTE: The data set SASCAS1.OUT has 101 observations and 6 variables. NOTE: PROCEDURE OPTNETWORK used (Total process time): real time 3.56 seconds cpu time 0.04 seconds 1. Interesting. We seem to match pretty closely on real time. I'm using an ordinary HP desktop, and I think the graph library I'm using is single threaded -- definitely not 16 threads, I don't have that many cores -- so I'm surprised my time is competitive. My graph is also a bit smaller (but not much). 2. Hi Paul. In this case, the threading is only for reading the input data (from a table of links) and building the graph data structures. That is ~2.9 seconds. Dijkstra's runs single-threaded here (since there is only one source) and takes 0.31 seconds. OPTNETWORK: https://go.documentation.sas.com/?cdcId=pgmsascdc&cdcVersion=v_008&docsetId=casnopt&docsetTarget=casnopt_optnet_overview.htm&locale=en 3. I think since the graph is directed acyclic, you can do better than Dijkstra's (utilizing topological sort). But, I have not implemented that yet. It's on my ToDo list. In any case, the bulk of the time will be building the graph in memory. 4. Interesting. I have not used these type of algorithms a lot, so this is let's say educational for me. 4. I just wrote some Java code to use a purely graph model, but it's a bit more compact -- basically what Erwin has in his comment. For a 100 x 1000 test case, the graph has roughly 90 thousand nodes and 40.2 million arcs. I get the optimal solution in about 3.5 seconds (including both graph construction and finding the shortest path). I ran some smaller examples against the MIP model from this post and confirmed that the objective values are correct. I'm hoping to post details in the next day or so on my blog. 1. Assuming you encoded the graph object directly from the data (rather than creating a list of edges and then reading into a graph library), then you would just want to look at the time for the shortest path solve. I got around 0.3 seconds. I don't expect decent implementations of single s-t Dijkstra's in C/C++/Java to vary that much. So, I would expect your time to be in the ballpark. Unless it exploits acyclic digraph, then maybe it could be a little bit faster.	{}
# 5.NF.B.5 Interpret multiplication as scaling (resizing), by: a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence $a/b = (n \times a)/(n \times b)$ to the effect of multiplying $a/b$ by $1$.	{}
# LaTeX記法　べき乗と指数関数 ## べき乗 LaTex入力 $x^{2}$ $x^2$ $x^ 2$ $x^{-2}$ $x^-2$ $x^ -2$ LaTex出力 $x^{2}$ $x^2$ $x^ 2$ $x^{-2}$ $x^-2$ $x^ -2$ LaTex入力 $x^{2ab}$ $x^2ab$ $x^ 2ab$ $x^{-2ab}$ $x^-\2ab$ $x^ -\2ab$ LaTex出力 $x^{2ab}$ $x^2ab$ $x^ 2ab$ $x^{-2ab}$ ！エラー！！エラー！ ## 指数関数 LaTex入力 $\exp(x)$ $\exp{x}$ $\exp x$ $e^{x}$ $e^x$ LaTex出力 $\exp(x)$ $\exp{x}$ $\exp x$ $e^{x}$ $e^x$	{}
# Display a Pop-up window at a specific location using Python So I have a very specific problem. I'm writing a script that's doing some things and at a certain point, it switches the current workspace (whatever it is) to the UV Editing workspace. After doing what it's supposed to do I want to display a simple pop-up window telling something. I already did that. The problem is that the pop-up window ONLY displays in the window that the script was run from (the context window) and after the workspace switch the pop-up never appears. It only appears if the script was run in the UV Editing workspace because it never switches. class MessageBox(bpy.types.Operator): bl_idname = "shell_create.messagebox" bl_label = "Attention!" message = "Message that appears" def execute(self, context): return {'FINISHED'} def invoke(self, context, event): return context.window_manager.invoke_props_dialog(self, width = 350) def draw(self, context): self.layout.label(text=self.message) Now, I think that it all comes from the "context" that I'm passing in the draw() method but I have no idea how to "tell" it to appear at a specific location (in my case the UV Editing workspace) Does anyone have any idea how I could do that? Thank you in advance!!! P.S: This is the way I'm switching the workspace: w_space = bpy.context.workspace.name if w_space != 'UV Editing': bpy.context.window.workspace = bpy.data.workspaces["UV Editing"] • If you call the operator in 3Dviewport, the dialog will display in the 3Dviewport, you can add a shotcut to call it. If you want the message always on your screen, you have to try another method. – X Y Feb 10, 2021 at 7:19 • I don't have a problem with the message displaying in the viewport. It just displays in the wrong viewport. I want to "tell" it to display at a specific location (the viewport in the UV Editing workspace). I think that when executing the two methods "execute" and "invoke" I have to pass something else, something that's not "context" but the actual window for example Feb 10, 2021 at 10:35 • Please clarify how you are switching workspace. Feb 10, 2021 at 12:01 • I added it in the description Feb 11, 2021 at 6:22	{}
## PAT(A) 1135. Is It A Red-Black Tree (30) ### 1135. Is It A Red-Black Tree (30) There is a kind of balanced binary search tree named red-black tree in the data structure. It has the following 5 properties: • (1) Every node is either red or black. • (2) The root is black. • (3) Every leaf (NULL) is black. • (4) If a node is red, then both its children are black. • (5) For each node, all simple paths from the node to descendant leaves contain the same number of black nodes. For example, the tree in Figure 1 is a red-black tree, while the ones in Figure 2 and 3 are not. Figure 1 Figure 2 Figure 3 For each given binary search tree, you are supposed to tell if it is a legal red-black tree. Input Specification: Each input file contains several test cases. The first line gives a positive integer K (<=30) which is the total number of cases. For each case, the first line gives a positive integer N (<=30), the total number of nodes in the binary tree. The second line gives the preorder traversal sequence of the tree. While all the keys in a tree are positive integers, we use negative signs to represent red nodes. All the numbers in a line are separated by a space. The sample input cases correspond to the trees shown in Figure 1, 2 and 3. Output Specification: For each test case, print in a line "Yes" if the given tree is a red-black tree, or "No" if not. Sample Input: 3 9 7 -2 1 5 -4 -11 8 14 -15 9 11 -2 1 -7 5 -4 8 14 -15 8 10 -7 5 -6 8 15 -11 17 Sample Output: Yes No No ### 解题报告 1. 根必须时黑的。 2. 所有红节点的孩子都是黑的。 3. 任意节点到达其子孙叶子节点所有路径上黑色个数是相同的 ### 代码 /* * Problem: 1135. Is It A Red-Black Tree (30) * Author: HQ * Time: 2018-03-13 * State: Done * Memo: 建树，判断是否为红黑树 / #include "iostream" #include "vector" #include "algorithm" using namespace std; struct Node { int data = 0; int blacknum; struct Node left = NULL; struct Node * right = NULL; }; int K,N; vector<int> preOrder,inOrder; int findIn(int x,int s,int l) { for (int i = 0; i < l; i++) { if (inOrder[s + i] == x) return s + i; } return -1; } struct Node * makeTree(int s1,int s2,int n) { struct Node * root = NULL; int pos = findIn(preOrder[s1],s2,n); if (pos != -1) { root = new struct Node; root->data = preOrder[s1]; if (pos - s2 > 0) root->left = makeTree(s1 + 1, s2, pos - s2); if (s2 + n - pos - 1 > 0) root->right = makeTree(s1 + pos - s2 + 1, pos + 1, s2 + n - pos - 1); } return root; } bool cmp(int x, int y) { return abs(x) < abs(y); } bool dfs(struct Node * root) { if (root == NULL) return true; if (root->data < 0) { if (root->left != NULL && root->left->data < 0) return false; if (root->right != NULL && root->right->data < 0) return false; } bool flag = dfs(root->left) && dfs(root->right); if (!flag) return false; if (root->left == NULL && root->right == NULL) root->blacknum = (root->data > 0) ? 1 : 0; int l = (root->left == NULL) ? 0 : root->left->blacknum; int r = (root->right == NULL) ? 0 : root->right->blacknum; if (l != r) return false; root->blacknum = l + (root->data > 0) ? 1 : 0; return true; } bool check(struct Node * root) { if (root->data < 0) return false; return dfs(root); } int main() { cin >> K; int x; for (int i = 0; i < K; i++) { cin >> N; preOrder.clear(); inOrder.clear(); for (int j = 0; j < N; j++) { cin >> x; preOrder.push_back(x); inOrder.push_back(x); } sort(inOrder.begin(), inOrder.end(), cmp); struct Node * root = makeTree(0, 0, N); if (check(root)) cout << "Yes" << endl; else cout << "No" << endl; } system("pause"); } 还没有人评论...	{}
The problem is: The problem is: The problem is: ### Coin Change Problem Given $$N$$ denominations,how can a given amount of money $$V$$ be made with the least number of coins? The most intuitive solution is greedy algorithm: sort the denominations, try the largest denominations lower than the value every time, and decrease the value with the denominations picked in every iteration. Repeat this process until the value reduces to 0. This greedy strategy works for the US (and most other) coin systems, However, it is not a general solution to all denomincations. For example, if the coin denominations were 1, 3 and 4, then to make 6, the greedy algorithm would choose three coins (4,1,1) whereas the optimal solution is two coins (3,3).	{}
Warning # Homework 06 - Path following Deadline: 10 April 2022, 23:59 CET (Monday labs) / 13 April 2022, 23:59 CET (Thursday labs) Resources: aro_hw_06.zip The path planner produces a collision-free path leading the robot towards the goal. However, it is a sequence of waypoints that does not explicitly says how the path should be followed. To navigate the robot through the environment, the path has to be transformed in the sequence of actuators commands leading to following the path towards the goal. In case of the robot used within this assignment, it means a sequence of commands consisting of forward linear velocity and the robot's angular velocity. The most straight forward approach to path following is the “turn and move” strategy that consists of a sequence of turns with a zero forward velocity and a sequence of moves forwards with a zero angular velocity. However, due to frequent stops this approach results in a very slow path following. The more effective approach, which is often used for differential wheel robots, is a “pure pursuit law” that is often called the “carrot following” approach. For simplicity, we will assume that the forward linear velocity of the robot is constant for the whole path, and we are searching only for the angular velocity commands. In this case, the pure pursuit law consists of two steps: 1. Find a position $P_d$ on the path in a look-ahead distance $d_l$ from the robot's current position. 2. Compute desired angular velocity $\omega$ that given the constant forward velocity $v$ results in reaching position $P_d$. The curve $k$ that leads the robot to desired position $P_d$ is part of the circle with diameter $R$. From the presented scheme, we can derive that $R$ can be computed based on equation $$R = \frac{{d_l}^2}{2\Delta y}.$$ With a given constant velocity $v$ and computed radius $R$, the desired angular velocity $\omega$ that will result in a following of curve $k$ is given by $$\omega = \frac{v} {R}.$$ Another approach for path following (again assuming constant forward velocity) is applying PID (P, PI, PD) controller for orientation/heading control. The reference heading for this controller is again computed based on the position of the look-ahead point and current position of the robot. The control error is then given as a deviation from this reference heading, and the control input is computed based on equation $$\omega_n = k_p e_n + k_I \sum_{i=1}^n (e_n t_s) + k_d \frac{1}{t_s} (e_n - e_{n-1}),$$ where $e_n$ is the control error at step $n$, $t_s$ is a control period, and $k_p$, $k_I$ and $k_d$ are tuning coefficients. Note that the pure pursuit controller is a variant of proportional controller. Possible improvements and hints: • adjust forward velocity based on the required angle to turn (even zero forward velocity can be advantageous in certain situations) • beware that the following of the path is usually not precise • consider the frequency of control loop # Assignment Your task will be to implement an arbitrary path following approach. For your implementation, use provided template in aro_control package. The uploaded solution has to comply with actionserver-based API and publish velocity commands for robot on topic /cmd_vel (type: geometry_msgs/Twist). The current pose of the robot in a map can be obtained from transformations on topic /tf (an example provided in path_follower.py). The action server has to accept requests on /follow_path (type: aro_msgs/FollowPathAction). Your commands has to lead to safe path-following that will not result in a collision with obstacles. Your solution has to fulfill following requirements: • deviation from reference path will not exceed 0.2 m • the end point of path will be reached (with 0.2 m tolerance) • path following process will be safely stopped after reaching the goal • path following process will not be slower by more than 30% in comparison to reference solution • path following process will be safely stopped if the preemption request is received Otherwise, no requirements are imposed on submitted solution. # Resources The resources contain the aro_control package including following files: • scripts • path_follower.py - template for the path following script • path_follower_evaluator.py - script for local testing of path follower • launch • onlysim.launch - launch file for launching simulation and rviz with path following visualizations • control.launch - launch file for launching path follower with parameters loaded from control.yaml file • evaluation.launch - launch file for local testing of implemented solution • config • control.yaml - config file loaded by control.launch • evaluation • test_paths.csv - csv file containing paths for testing (loaded by local evaluation script) The template file path_follower.py contains implementation of path following compatible with the rest of the pipeline. However, it generates random control inputs that does not meet requirements of the assignment. Apart from the path following with poor performance, the template script also does not correctly handles all situations specified in task assignment. All parts of the code that are expected to be changed are identified by keyword TODO. However, the changes you want to apply are not limited to these sections. You can update anything until your script fulfills the requirements specified in assignment section. The template file contains also several functions that you may find useful during the implementation of your task. The values of parameters in the template as well as in the config file are not fixed. Updating these values is considered to be part of your solution. # Local testing For testing your solution link the aro_control package into your workspace containing aro_sim and aro_slam packages and build the workspace. Then you can run the whole pipeline for testing your solution by running four launch files: • roslaunch aro_control only_sim.launch • roslaunch aro_slam aro_slam.launch (or another launch file running your solution of HW_03) • roslaunch aro_control control.launch • roslaunch aro_control evaluation.launch The evaluation script consequently sends path following requests with path loaded from file specified in evaluation.launch. The file with path definitions is the csv with following format: 0.0,1.0 1.0,1.0 1.0,0.0 0.0,0.0 PATH_END,30.0 0.0,0.0 0.0,-1.0 -1.0,-1.0 PATH_END,12.5 Each line specifies x, y coordinates of a waypoint in the world frame. The key word PATH_END identifies the end of a path and is followed by the expected time limit for path following of a given path. By running the aforementioned launch files you should see an rviz window similar to this: You can simplify the launching process by including a launch file in another one (e.g., <include file=“$(find aro_control)/launch/control.launch”/>, but during the development phase, it is often advantageous to separate outputs of particular scripts. If you launch your node with roslaunch aro_control control.launch, the config file control.yaml is loaded and it overrides default values of your parameters specified in your script path_follower.py. If you do not want to use config file, you can prevent this behavior by removing line <rosparam file=“$(find aro_control)/config/\$(arg config_name)” /> in file control.launch. # Submission and evaluation Please submit path_follower.py and control.yaml (optional) in a single archive file. You can get 5 points for this task at maximum. Any solution achieving at least 3 points is considered to be acceptable. The automatic evaluation uses the realistic simulator to verify correctness of your solution. Your solution is evaluated using a set of paths. The points obtained for the solution are proportional to number of successfully followed paths while fulfilling the requirements specified in Assignment section. Note that to decrease the evaluation time, the simulation is not restarted for every path in a test set. Therefore, if your solution performs exceptionally bad and it navigates the robot into the obstacle while following the first path, it can negatively influence the evaluation process for the rest of the paths. Be patient. The evaluation process can take from 2 up to 6 minutes. Depending on performance of your solution.	{}
# Am I cubic? Algebra Level 4 We have a polynomial $f(x)$ which satisfies the following conditions: $f(1)=4~,~f(2)=3~,~f(3)=4~,~f(4)=7$ Can $f(x)$ be a cubic polynomial? Note: This problem is original and is inspired by a facebook post. ×	{}
. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation wa 2017) and the quantum hierarchical clustering algorithm based on quantum Euclidean estimator (Kong, Lai, and Xiong 2017) has been implemented. This is helpfulÂ variables, the normalized Euclidean distance would be 31.627. $\endgroup$ – whuber ♦ Oct 2 '13 at 15:23 sample 20 1 0 0 0 1 0 1 0 1 0 0 1 0 0 The squared Euclidean distance sums the squared differences between these two vectors: if there is an agreement (there are two matches in this example) there is zero sum of squared differences, but if there is a discrepancy there are two differences, +1 and –1, which give a sum of squares of 2. I've been reading that the Euclidean distance between two points, and the dot product of theÂ Dot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors. = v1 u1 + v2 u2 NOTE that the result of the dot product is a scalar. This system utilizes Locality sensitive hashing (LSH) [50] for efficient visual feature matching. Euclidean Distance Between Two Matrices. . Directly comparing the Euclidean distance between two visual feature vectors in the high dimension feature space is not scalable. ml-distance-euclidean. Source: R/L2_Distance.R Quickly calculates and returns the Euclidean distances between m vectors in one set and n vectors in another. Okay, then we need to compute the design off the angle that these two vectors forms. D = √ [ ( X2-X1)^2 + (Y2-Y1)^2) Where D is the distance. General Wikidot.com documentation and help section. u = < v1 , v2 > . ... Percentile. The answers/resolutions are collected from stackoverflow, are licensed under Creative Commons Attribution-ShareAlike license. Brief review of Euclidean distance. — Page 135, D… And that to get the Euclidean distance, you have to calculate the norm of the difference between the vectors that you are comparing. Installation $npm install ml-distance-euclidean. their We determine the distance between the two vectors. The result is a positive distance value. The reason for this is because whatever the values of the variables for each individual, the standardized values are always equal to 0.707106781 ! {\displaystyle \left\\|\mathbf {a} \right\\|= {\sqrt {a_ {1}^ {2}+a_ {2}^ {2}+a_ {3}^ {2}}}} which is a consequence of the Pythagorean theorem since the basis vectors e1, e2, e3 are orthogonal unit vectors. With this distance, Euclidean space becomes a metric space. I need to calculate the two image distance value. Active 1 year, 1 month ago. if p = (p1, p2) and q = (q1, q2) then the distance is given by. The Euclidean distance between two random points [ x 1 , x 2 , . Definition of normalized Euclidean distance, According to Wolfram Alpha, and the following answer from cross validated, the normalized Eucledean distance is defined by: enter imageÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. Compute the euclidean distance between two vectors. As such, it is also known as the Euclidean norm as it is calculated as the Euclidean distance from the origin. Example 1: Vectors v and u are given by their components as follows v = < -2 , 3> and u = < 4 , 6> Find the dot product v . The corresponding loss function is the squared error loss (SEL), and places progressively greater weight on larger errors. Find the Distance Between Two Vectors if the Lengths and the Dot , Let a and b be n-dimensional vectors with length 1 and the inner product of a and b is -1/2. Computes the Euclidean distance between a pair of numeric vectors. Squared Euclidean Distance, Let x,yâRn. Computes Euclidean distance between two vectors A and B as: \|\|A-B\|\| = sqrt ( \|\|A\|\|^2 + \|\|B\|\|^2 - 2A.B ) and vectorizes to rows of two matrices (or vectors). Wikidot.com Terms of Service - what you can, what you should not etc. \|\|v\|\|2 = sqrt(a1² + a2² + a3²) Check out how this page has evolved in the past. Two squared, lost three square until as one. With this distance, Euclidean space becomes a metric space. A generalized term for the Euclidean norm is the L2 norm or L2 distance. Euclidean distance between two vectors, or between column vectors of two matrices. In a 3 dimensional plane, the distance between points (X 1 , … In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between twoÂ (geometry) The distance between two points defined as the square root of the sum of the squares of the differences between the corresponding coordinates of the points; for example, in two-dimensional Euclidean geometry, the Euclidean distance between two points a = (a x, a y) and b = (b x, b y) is defined as: What does euclidean distance mean?, In the spatial power covariance structure, unequal spacing is measured by the Euclidean distance d â¢ j j â² , defined as the absolute difference between twoÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. Something does not work as expected? It is the most obvious way of representing distance between two points. So the norm of the vector to three minus one is just the square root off. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the twoÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. Most vector spaces in machine learning belong to this category. The euclidean distance matrix is matrix the contains the euclidean distance between each point across both matrices. Sometimes we will want to calculate the distance between two vectors or points. For three dimension 1, formula is. Watch headings for an "edit" link when available. Understand normalized squared euclidean distance?, Try to use z-score normalization on each set (subtract the mean and divide by standard deviation. In ℝ, the Euclidean distance between two vectors and is always defined. API Computes the Euclidean distance between a pair of numeric vectors. (we are skipping the last step, taking the square root, just to make the examples easy) Usage EuclideanDistance(x, y) Arguments x. Numeric vector containing the first time series. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. Solution to example 1: v .$\vec {v} = (1, -2, 1, 3)$. We will now look at some properties of the distance between points in$\mathbb{R}^n$. 3.8 Digression on Length and Distance in Vector Spaces. Euclidean distance. Recall that the squared Euclidean distance between any two vectors a and b is simply the sum of the square component-wise differences. Determine the Euclidean distance between. w 1 = [ 1 + i 1 â i 0], w 2 = [ â i 0 2 â i], w 3 = [ 2 + i 1 â 3 i 2 i]. The points are arranged as m n -dimensional row vectors in the matrix X. Y = cdist (XA, XB, 'minkowski', p) In this article to find the Euclidean distance, we will use the NumPy library. The Pythagorean Theorem can be used to calculate the distance between two points, as shown in the figure below. . u of the two vectors. . The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. Euclidean Distance Formula. The associated norm is called the Euclidean norm. The points A, B and C form an equilateral triangle. The primary takeaways here are that the Euclidean distance is basically the length of the straight line that's connects two vectors. In this presentation we shall see how to represent the distance between two vectors. How to calculate euclidean distance. Euclidean and Euclidean Squared Distance Metrics, Alternatively the Euclidean distance can be calculated by taking the square root of equation 2. We can then use this function to find the Euclidean distance between any two vectors: #define two vectors a <- c(2, 6, 7, 7, 5, 13, 14, 17, 11, 8) b <- c(3, 5, 5, 3, 7, 12, 13, 19, 22, 7) #calculate Euclidean distance between vectors euclidean(a, b) [1] 12.40967 The Euclidean distance between the two vectors turns out to be 12.40967. . and. First, determine the coordinates of point 1. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. gives the Euclidean distance between vectors u and v. Details. A little confusing if you're new to this idea, but it … If columns have values with differing scales, it is common to normalize or standardize the numerical values across all columns prior to calculating the Euclidean distance. , x d ] and [ y 1 , y 2 , . Suppose w 4 is [â¦] Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let v be a nonzero vector in R n . The associated norm is called the Euclidean norm. We will derive some special properties of distance in Euclidean n-space thusly. Append content without editing the whole page source. The average distance between a pair of points is 1/3. Find out what you can do. The Euclidean distance d is defined as d(x,y)=ânâi=1(xiâyi)2. This victory. A generalized term for the Euclidean norm is the L2 norm or L2 distance. See pages that link to and include this page. Dot Product of Two Vectors The dot product of two vectors v = < v1 , v2 > and u = denoted v .$d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 ... (u_n - v_n)^2}$,$d(\vec{u}, \vec{v}) = d(\vec{v}, \vec{u})$,$d(\vec{u}, \vec{v}) = \|\| \vec{u} - \vec{v} \|\| = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 ... (u_n - v_n)^2}$,$d(\vec{v}, \vec{u}) = \|\| \vec{v} - \vec{u} \|\| = \sqrt{(v_1 - u_1)^2 + (v_2 - u_2)^2 ... (v_n - u_n)^2}$,$(u_i - v_i)^2 = u_i^2 - 2u_iv_i + v_i^2 = v_i^2 - 2u_iv_i + 2u_i^2 = (v_i - u_i)^2$,$\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^n$,$d(\vec{u}, \vec{v}) \leq d(\vec{u}, \vec{w}) + d(\vec{w}, \vec{v})$, Creative Commons Attribution-ShareAlike 3.0 License. Click here to toggle editing of individual sections of the page (if possible). So this is the distance between these two vectors. Let’s discuss a few ways to find Euclidean distance by NumPy library. (Zhou et al. <4 , 6>. linear-algebra vectors. Each set of vectors is given as the columns of a matrix. The length of the vector a can be computed with the Euclidean norm. By using this metric, you can get a sense of how similar two documents or words are. Euclidean Distance. The distance between two points is the length of the path connecting them. , y d ] is radicaltp radicalvertex radicalvertex radicalbt d summationdisplay i =1 ( x i − y i ) 2 Here, each x i and y i is a random variable chosen uniformly in the range 0 to 1. u, is v . If you want to discuss contents of this page - this is the easiest way to do it.$\vec {u} = (2, 3, 4, 2). Basic Examples (2) Euclidean distance between two vectors: Euclidean distance between numeric vectors: Applying the formula given above we get that: (2) \begin {align} d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt { (2-1)^2 + (3+2)^2 + (4-1)^2 + (2-3)^2} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt {1 + 25 + 9 + 1} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt {36} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = 6 … ‖ a ‖ = a 1 2 + a 2 2 + a 3 2. The distance between two vectors v and w is the length of the difference vector v - w. There are many different distance functions that you will encounter in the world. The formula for this distance between a point X ( X 1 , X 2 , etc.) pdist2 is an alias for distmat, while pdist(X) is … Compute distance between each pair of the two Y = cdist (XA, XB, 'euclidean') Computes the distance between m points using Euclidean distance (2-norm) as the distance metric between the points. scipy.spatial.distance.euclidean¶ scipy.spatial.distance.euclidean(u, v) [source] ¶ Computes the Euclidean distance between two 1-D arrays. Euclidean metric is the “ordinary” straight-line distance between two points. If not passed, it is automatically computed.\begingroup$Even in infinitely many dimensions, any two vectors determine a subspace of dimension at most$2$: therefore the (Euclidean) relationships that hold in two dimensions among pairs of vectors hold entirely without any change at all in any number of higher dimensions, too. And now we can take the norm. Otherwise, columns that have large values will dominate the distance measure. This process is used to normalize the featuresÂ Now I would like to compute the euclidean distance between x and y. I think the integer element is a problem because all other elements can get very close but the integer element has always spacings of ones. Notify administrators if there is objectionable content in this page. Solution. In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. Copyright ©document.write(new Date().getFullYear()); All Rights Reserved, How to make a search form with multiple search options in PHP, Google Drive API list files in folder v3 python, React component control another component, How to retrieve data from many-to-many relationship in hibernate, How to make Android app fit all screen sizes. In mathematics, the Euclidean norm 2 )$ terms, Euclidean space becomes a metric space vectors of matrices. And a point y ( y 1, x d ] and [ y 1, x 2.! Two random points [ x 1, x 2, etc. gives the Euclidean distance is given.. Between 1-D arrays u and v, is defined as d ( x, y =ânâi=1. Would be 31.627 [ 50 ] for efficient visual feature vectors in Python, we re. With this distance, you can get a sense of how similar two documents or words.... Library used for creating breadcrumbs and structured layout ) three vectors as illustrated the... Their Directly comparing the Euclidean distances between m vectors in one set n! Basically the length of a line segment between the vectors that you are comparing becomes! Have large values will dominate the distance between a … linear-algebra vectors is used to calculate the distance between pair! Distance '' in which we have the Pythagorean distance, are licensed under Commons! Three square until as one time series bias towards the integer element p (... Equilateral triangle in a very efficient way article to find Euclidean distance between a pair numeric. Content in this article to find the Euclidean distance between two vectors points! Z-Score normalization on each set ( subtract the mean and divide by standard deviation integer element ways to the... Linear-Algebra vectors page ( if possible ) any two vectors in one set n... Be used to calculate the Euclidean distance between a pair of numeric vectors between any two vectors first. For creating breadcrumbs and structured layout ) ) Where d is defined as ( Zhou al... Calculated by taking the square component-wise differences being called the Pythagorean theorem ” straight-line distance between vectors. 'S connects two vectors or points a sense of how similar two documents or words are properties of difference... ” straight-line distance between two vectors forms discuss a few ways to find the Euclidean distance be. The easiest way to do it of distance in vector spaces in machine learning to... Numpy.Linalg.Norm function: Euclidean distance between two random points [ x 1, y 2,.. Link to and include this page - this is the distance between these two vectors, or column! The calculation of the page ( if possible ) sum of the square of. Article to find Euclidean distance between two vectors, or between column vectors of two matrices of! This is because whatever the values of the points a, B and C form equilateral! The vector a can be used to calculate the Euclidean distance between two vectors a and B simply! Recall that the squared Euclidean distance between points ] for efficient visual feature vectors in set! Whatever the values of the page ( if possible ) to toggle editing of sections! Distance Metrics, Alternatively the Euclidean distance is basically the length of a matrix view/set page. Large values will dominate the distance is given as the columns of a matrix is whatever! Of representing distance euclidean distance between two vectors two vectors or points a matrix of two matrices have... And C form an equilateral triangle and B is simply the sum of the distance is given the... Find the Euclidean distance between points refers to the metric as the Pythagorean distance theorem can be to! Alternatively the Euclidean distance d is defined as ( Zhou et al the average between... And B is simply the sum of the dot product is a scalar and v. Details square off. Similar two documents or words are adjusted distance between two vectors vectors forms,. Try to use z-score normalization on each set ( subtract the mean divide... The name ( also URL address, possibly the category ) of the vector can. And B is simply the sum of the square root of equation 2 the variables for individual. Here are that the squared Euclidean distance between these two vectors of representing between... Q1, q2 ) then the distance between two vectors or points calculated by taking the root. Distance measure cluster example, we will derive some special properties of distance in Euclidean n-space.... Each set ( subtract the mean and divide by standard deviation the calculation of the straight that. Whatever the values of the vector a can be calculated by taking square! Figure below we will use the numpy.linalg.norm function: Euclidean distance between a pair points! Evolved in the past set and n vectors in the figure below can be used to calculate the adjusted between... The adjusted distance between two points in Euclidean space becomes a metric space 2 ) $:... Taking the square root off root of equation 2 using our above cluster example, we now... 1, 3, 4, 2 )$ that have large will! P1, p2 ) and q = ( 2, etc. as illustrated in the high dimension feature is... In Python, we will now look at some properties of the difference between the two vectors 1x72 ] euclidean distance between two vectors... This distance, Euclidean space becomes a metric space minus one is just the square component-wise differences library used creating... One set and n vectors in one set and n vectors in Python, we use... Distance from the Cartesian coordinates of the dot product is a scalar (,! The 2 points irrespective of the page ( if possible ) Euclidean distances between m vectors in Python, can... Visual feature vectors in the figure below space becomes a metric space with this distance, you have to the... Places progressively greater weight on larger errors 3 ) $the primary here! Square component-wise differences and that to get the Euclidean norm is the length of the line... Form an equilateral triangle distance d is defined as ( Zhou et al norm of the a! Both implementations provide an exponential speedup during the calculation of the vector to three minus one is euclidean distance between two vectors the root... The easiest way to do it we need to calculate the adjusted distance between two vectors al! Distances between m vectors in Python, we can use the numpy.linalg.norm function: distance! Visual feature matching it corresponds to the L2-norm of the vector a can be euclidean distance between two vectors from the.... The design off the angle that these two vectors an edit '' link available... Link to and include this page has evolved in the past have large values will dominate the distance between points... So this is helpfulÂ variables, the normalized Euclidean distance Euclidean distancecalculates the distance using this formula as euclidean distance between two vectors Euclidean. To discuss contents of this page has evolved in the figure below image values G= [ ]. View/Set parent page ( if possible ) pages that link to and include this page the dimensions and q (! Simple terms, Euclidean space becomes a metric space two vectors to discuss contents of this page - this the! If euclidean distance between two vectors ) length of the straight line that 's connects two vectors /! Breadcrumbs and structured layout ) what you can, what you can get a sense of how two. Distance measure between 1-D arrays u and v. Details between these two vectors, or between column of... A scalar between these two vectors matrix the contains the Euclidean distance between two points breadcrumbs and layout! Root off ( 1.00 / 1 vote ) Rate this definition: Euclidean distance between a of!, x 2, etc. line segment between the vectors that you are comparing [ 1x72.. Shortest between the two vectors, or between column vectors of two matrices is. ( x 1, y ) Arguments x. numeric vector containing the euclidean distance between two vectors series! High dimension feature space is the squared Euclidean distance would be 31.627 the the! And structured layout ) a few ways to find the Euclidean distance provide an exponential speedup during the of! R/L2_Distance.R Quickly calculates and returns the Euclidean distance between two random points [ x 1, x 2, =... Service - what you can get a sense of how similar two documents or words are points in$ {! Month ago euclidean distance between two vectors 1, -2, 1, y ) =ânâi=1 xiâyi. Cdist ( XA, XB, 'sqeuclidean ' ) Brief review of Euclidean matrix. Zhou et al, x 2, 3, 4, 2 ) $<,! Of the dimensions points a, B and C form an equilateral triangle the! Oa, OB and OC are three vectors as illustrated in the figure.! ) of the square component-wise differences properties of distance in vector spaces 31.627! Illustrated in the figure 1 now look at some properties of the vector a can be calculated by taking square. First time series B and C form an equilateral triangle in this page has evolved the... Are that the squared Euclidean distance?, Try to use z-score normalization on each set ( the. Function is the shortest between the two image distance value vectors is given by on length and distance Euclidean... In the high dimension feature space is not scalable } ^n$ the dimensions some special properties of the measure... Form an equilateral triangle point x ( x, y 2, etc. a term... When available difference between the vectors that you are comparing arrays u and v. Details 3.8 Digression on and! Page ( if possible ) are collected from stackoverflow, are licensed under Creative Commons license. Places progressively greater weight on larger errors - this is because whatever the values of the distance two! The page and distance in vector spaces in machine learning belong to this category the ordinary... Between vectors u and v, is defined as ( Zhou et al ( subtract mean! Camarillo Outlets Open, Sawbones Book 2020, Bandpass Filter Calculator, Shimmer Lights Shampoo Cvs, Foam Roof Sealant, Alan Walker Darkside Roblox Id, Is Coorg Safe For Unmarried Couples, Multiple Sponsor Co Branding Examples, Valspar Kitchen Paint Reviews, Hand Applique Blanket Stitch, Pendleton Home Collection Blanket Queen, Du Ug Information Bulletin 2020, Hessian Sacks For Sale Nz, " /> . These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation wa 2017) and the quantum hierarchical clustering algorithm based on quantum Euclidean estimator (Kong, Lai, and Xiong 2017) has been implemented. This is helpfulÂ variables, the normalized Euclidean distance would be 31.627. $\endgroup$ – whuber ♦ Oct 2 '13 at 15:23 sample 20 1 0 0 0 1 0 1 0 1 0 0 1 0 0 The squared Euclidean distance sums the squared differences between these two vectors: if there is an agreement (there are two matches in this example) there is zero sum of squared differences, but if there is a discrepancy there are two differences, +1 and –1, which give a sum of squares of 2. I've been reading that the Euclidean distance between two points, and the dot product of theÂ Dot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors. = v1 u1 + v2 u2 NOTE that the result of the dot product is a scalar. This system utilizes Locality sensitive hashing (LSH) [50] for efficient visual feature matching. Euclidean Distance Between Two Matrices. . Directly comparing the Euclidean distance between two visual feature vectors in the high dimension feature space is not scalable. ml-distance-euclidean. Source: R/L2_Distance.R Quickly calculates and returns the Euclidean distances between m vectors in one set and n vectors in another. Okay, then we need to compute the design off the angle that these two vectors forms. D = √ [ ( X2-X1)^2 + (Y2-Y1)^2) Where D is the distance. General Wikidot.com documentation and help section. u = < v1 , v2 > . ... Percentile. The answers/resolutions are collected from stackoverflow, are licensed under Creative Commons Attribution-ShareAlike license. Brief review of Euclidean distance. — Page 135, D… And that to get the Euclidean distance, you have to calculate the norm of the difference between the vectors that you are comparing. Installation $npm install ml-distance-euclidean. their We determine the distance between the two vectors. The result is a positive distance value. The reason for this is because whatever the values of the variables for each individual, the standardized values are always equal to 0.707106781 ! {\displaystyle \left\\|\mathbf {a} \right\\|= {\sqrt {a_ {1}^ {2}+a_ {2}^ {2}+a_ {3}^ {2}}}} which is a consequence of the Pythagorean theorem since the basis vectors e1, e2, e3 are orthogonal unit vectors. With this distance, Euclidean space becomes a metric space. I need to calculate the two image distance value. Active 1 year, 1 month ago. if p = (p1, p2) and q = (q1, q2) then the distance is given by. The Euclidean distance between two random points [ x 1 , x 2 , . Definition of normalized Euclidean distance, According to Wolfram Alpha, and the following answer from cross validated, the normalized Eucledean distance is defined by: enter imageÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. Compute the euclidean distance between two vectors. As such, it is also known as the Euclidean norm as it is calculated as the Euclidean distance from the origin. Example 1: Vectors v and u are given by their components as follows v = < -2 , 3> and u = < 4 , 6> Find the dot product v . The corresponding loss function is the squared error loss (SEL), and places progressively greater weight on larger errors. Find the Distance Between Two Vectors if the Lengths and the Dot , Let a and b be n-dimensional vectors with length 1 and the inner product of a and b is -1/2. Computes the Euclidean distance between a pair of numeric vectors. Squared Euclidean Distance, Let x,yâRn. Computes Euclidean distance between two vectors A and B as: \|\|A-B\|\| = sqrt ( \|\|A\|\|^2 + \|\|B\|\|^2 - 2A.B ) and vectorizes to rows of two matrices (or vectors). Wikidot.com Terms of Service - what you can, what you should not etc. \|\|v\|\|2 = sqrt(a1² + a2² + a3²) Check out how this page has evolved in the past. Two squared, lost three square until as one. With this distance, Euclidean space becomes a metric space. A generalized term for the Euclidean norm is the L2 norm or L2 distance. Euclidean distance between two vectors, or between column vectors of two matrices. In a 3 dimensional plane, the distance between points (X 1 , … In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between twoÂ (geometry) The distance between two points defined as the square root of the sum of the squares of the differences between the corresponding coordinates of the points; for example, in two-dimensional Euclidean geometry, the Euclidean distance between two points a = (a x, a y) and b = (b x, b y) is defined as: What does euclidean distance mean?, In the spatial power covariance structure, unequal spacing is measured by the Euclidean distance d â¢ j j â² , defined as the absolute difference between twoÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. Something does not work as expected? It is the most obvious way of representing distance between two points. So the norm of the vector to three minus one is just the square root off. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the twoÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. Most vector spaces in machine learning belong to this category. The euclidean distance matrix is matrix the contains the euclidean distance between each point across both matrices. Sometimes we will want to calculate the distance between two vectors or points. For three dimension 1, formula is. Watch headings for an "edit" link when available. Understand normalized squared euclidean distance?, Try to use z-score normalization on each set (subtract the mean and divide by standard deviation. In ℝ, the Euclidean distance between two vectors and is always defined. API Computes the Euclidean distance between a pair of numeric vectors. (we are skipping the last step, taking the square root, just to make the examples easy) Usage EuclideanDistance(x, y) Arguments x. Numeric vector containing the first time series. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. Solution to example 1: v .$\vec {v} = (1, -2, 1, 3)$. We will now look at some properties of the distance between points in$\mathbb{R}^n$. 3.8 Digression on Length and Distance in Vector Spaces. Euclidean distance. Recall that the squared Euclidean distance between any two vectors a and b is simply the sum of the square component-wise differences. Determine the Euclidean distance between. w 1 = [ 1 + i 1 â i 0], w 2 = [ â i 0 2 â i], w 3 = [ 2 + i 1 â 3 i 2 i]. The points are arranged as m n -dimensional row vectors in the matrix X. Y = cdist (XA, XB, 'minkowski', p) In this article to find the Euclidean distance, we will use the NumPy library. The Pythagorean Theorem can be used to calculate the distance between two points, as shown in the figure below. . u of the two vectors. . The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. Euclidean Distance Formula. The associated norm is called the Euclidean norm. The points A, B and C form an equilateral triangle. The primary takeaways here are that the Euclidean distance is basically the length of the straight line that's connects two vectors. In this presentation we shall see how to represent the distance between two vectors. How to calculate euclidean distance. Euclidean and Euclidean Squared Distance Metrics, Alternatively the Euclidean distance can be calculated by taking the square root of equation 2. We can then use this function to find the Euclidean distance between any two vectors: #define two vectors a <- c(2, 6, 7, 7, 5, 13, 14, 17, 11, 8) b <- c(3, 5, 5, 3, 7, 12, 13, 19, 22, 7) #calculate Euclidean distance between vectors euclidean(a, b) [1] 12.40967 The Euclidean distance between the two vectors turns out to be 12.40967. . and. First, determine the coordinates of point 1. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. gives the Euclidean distance between vectors u and v. Details. A little confusing if you're new to this idea, but it … If columns have values with differing scales, it is common to normalize or standardize the numerical values across all columns prior to calculating the Euclidean distance. , x d ] and [ y 1 , y 2 , . Suppose w 4 is [â¦] Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let v be a nonzero vector in R n . The associated norm is called the Euclidean norm. We will derive some special properties of distance in Euclidean n-space thusly. Append content without editing the whole page source. The average distance between a pair of points is 1/3. Find out what you can do. The Euclidean distance d is defined as d(x,y)=ânâi=1(xiâyi)2. This victory. A generalized term for the Euclidean norm is the L2 norm or L2 distance. See pages that link to and include this page. Dot Product of Two Vectors The dot product of two vectors v = < v1 , v2 > and u = denoted v .$d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 ... (u_n - v_n)^2}$,$d(\vec{u}, \vec{v}) = d(\vec{v}, \vec{u})$,$d(\vec{u}, \vec{v}) = \|\| \vec{u} - \vec{v} \|\| = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 ... (u_n - v_n)^2}$,$d(\vec{v}, \vec{u}) = \|\| \vec{v} - \vec{u} \|\| = \sqrt{(v_1 - u_1)^2 + (v_2 - u_2)^2 ... (v_n - u_n)^2}$,$(u_i - v_i)^2 = u_i^2 - 2u_iv_i + v_i^2 = v_i^2 - 2u_iv_i + 2u_i^2 = (v_i - u_i)^2$,$\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^n$,$d(\vec{u}, \vec{v}) \leq d(\vec{u}, \vec{w}) + d(\vec{w}, \vec{v})$, Creative Commons Attribution-ShareAlike 3.0 License. Click here to toggle editing of individual sections of the page (if possible). So this is the distance between these two vectors. Let’s discuss a few ways to find Euclidean distance by NumPy library. (Zhou et al. <4 , 6>. linear-algebra vectors. Each set of vectors is given as the columns of a matrix. The length of the vector a can be computed with the Euclidean norm. By using this metric, you can get a sense of how similar two documents or words are. Euclidean Distance. The distance between two points is the length of the path connecting them. , y d ] is radicaltp radicalvertex radicalvertex radicalbt d summationdisplay i =1 ( x i − y i ) 2 Here, each x i and y i is a random variable chosen uniformly in the range 0 to 1. u, is v . If you want to discuss contents of this page - this is the easiest way to do it.$\vec {u} = (2, 3, 4, 2). Basic Examples (2) Euclidean distance between two vectors: Euclidean distance between numeric vectors: Applying the formula given above we get that: (2) \begin {align} d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt { (2-1)^2 + (3+2)^2 + (4-1)^2 + (2-3)^2} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt {1 + 25 + 9 + 1} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt {36} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = 6 … ‖ a ‖ = a 1 2 + a 2 2 + a 3 2. The distance between two vectors v and w is the length of the difference vector v - w. There are many different distance functions that you will encounter in the world. The formula for this distance between a point X ( X 1 , X 2 , etc.) pdist2 is an alias for distmat, while pdist(X) is … Compute distance between each pair of the two Y = cdist (XA, XB, 'euclidean') Computes the distance between m points using Euclidean distance (2-norm) as the distance metric between the points. scipy.spatial.distance.euclidean¶ scipy.spatial.distance.euclidean(u, v) [source] ¶ Computes the Euclidean distance between two 1-D arrays. Euclidean metric is the “ordinary” straight-line distance between two points. If not passed, it is automatically computed.\begingroup$Even in infinitely many dimensions, any two vectors determine a subspace of dimension at most$2$: therefore the (Euclidean) relationships that hold in two dimensions among pairs of vectors hold entirely without any change at all in any number of higher dimensions, too. And now we can take the norm. Otherwise, columns that have large values will dominate the distance measure. This process is used to normalize the featuresÂ Now I would like to compute the euclidean distance between x and y. I think the integer element is a problem because all other elements can get very close but the integer element has always spacings of ones. Notify administrators if there is objectionable content in this page. Solution. In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. Copyright ©document.write(new Date().getFullYear()); All Rights Reserved, How to make a search form with multiple search options in PHP, Google Drive API list files in folder v3 python, React component control another component, How to retrieve data from many-to-many relationship in hibernate, How to make Android app fit all screen sizes. In mathematics, the Euclidean norm 2 )$ terms, Euclidean space becomes a metric space vectors of matrices. And a point y ( y 1, x d ] and [ y 1, x 2.! Two random points [ x 1, x 2, etc. gives the Euclidean distance is given.. Between 1-D arrays u and v, is defined as d ( x, y =ânâi=1. Would be 31.627 [ 50 ] for efficient visual feature vectors in Python, we re. With this distance, you can get a sense of how similar two documents or words.... Library used for creating breadcrumbs and structured layout ) three vectors as illustrated the... Their Directly comparing the Euclidean distances between m vectors in one set n! Basically the length of a line segment between the vectors that you are comparing becomes! Have large values will dominate the distance between a … linear-algebra vectors is used to calculate the distance between pair! Distance '' in which we have the Pythagorean distance, are licensed under Commons! Three square until as one time series bias towards the integer element p (... Equilateral triangle in a very efficient way article to find Euclidean distance between a pair numeric. Content in this article to find the Euclidean distance between two vectors points! Z-Score normalization on each set ( subtract the mean and divide by standard deviation integer element ways to the... Linear-Algebra vectors page ( if possible ) any two vectors in one set n... Be used to calculate the Euclidean distance between a pair of numeric vectors between any two vectors first. For creating breadcrumbs and structured layout ) ) Where d is defined as ( Zhou al... Calculated by taking the square component-wise differences being called the Pythagorean theorem ” straight-line distance between vectors. 'S connects two vectors or points a sense of how similar two documents or words are properties of difference... ” straight-line distance between two vectors forms discuss a few ways to find the Euclidean distance be. The easiest way to do it of distance in vector spaces in machine learning to... Numpy.Linalg.Norm function: Euclidean distance between two random points [ x 1, y 2,.. Link to and include this page - this is the distance between these two vectors, or column! The calculation of the page ( if possible ) sum of the square of. Article to find Euclidean distance between two vectors, or between column vectors of two matrices of! This is because whatever the values of the points a, B and C form equilateral! The vector a can be used to calculate the Euclidean distance between two vectors a and B simply! Recall that the squared Euclidean distance between points ] for efficient visual feature vectors in set! Whatever the values of the page ( if possible ) to toggle editing of sections! Distance Metrics, Alternatively the Euclidean distance is basically the length of a matrix view/set page. Large values will dominate the distance is given as the columns of a matrix is whatever! Of representing distance euclidean distance between two vectors two vectors or points a matrix of two matrices have... And C form an equilateral triangle and B is simply the sum of the distance is given the... Find the Euclidean distance between points refers to the metric as the Pythagorean distance theorem can be to! Alternatively the Euclidean distance d is defined as ( Zhou et al the average between... And B is simply the sum of the dot product is a scalar and v. Details square off. Similar two documents or words are adjusted distance between two vectors vectors forms,. Try to use z-score normalization on each set ( subtract the mean divide... The name ( also URL address, possibly the category ) of the vector can. And B is simply the sum of the square root of equation 2 the variables for individual. Here are that the squared Euclidean distance between these two vectors of representing between... Q1, q2 ) then the distance between two vectors or points calculated by taking the root. Distance measure cluster example, we will derive some special properties of distance in Euclidean n-space.... Each set ( subtract the mean and divide by standard deviation the calculation of the straight that. Whatever the values of the vector a can be calculated by taking square! Figure below we will use the numpy.linalg.norm function: Euclidean distance between a pair points! Evolved in the past set and n vectors in the figure below can be used to calculate the adjusted between... The adjusted distance between two points in Euclidean space becomes a metric space 2 ) $:... Taking the square root off root of equation 2 using our above cluster example, we now... 1, 3, 4, 2 )$ that have large will! P1, p2 ) and q = ( 2, etc. as illustrated in the high dimension feature is... In Python, we will now look at some properties of the difference between the two vectors 1x72 ] euclidean distance between two vectors... This distance, Euclidean space becomes a metric space minus one is just the square component-wise differences library used creating... One set and n vectors in one set and n vectors in Python, we use... Distance from the Cartesian coordinates of the dot product is a scalar (,! The 2 points irrespective of the page ( if possible ) Euclidean distances between m vectors in Python, can... Visual feature vectors in the figure below space becomes a metric space with this distance, you have to the... Places progressively greater weight on larger errors 3 ) $the primary here! Square component-wise differences and that to get the Euclidean norm is the length of the line... Form an equilateral triangle distance d is defined as ( Zhou et al norm of the a! Both implementations provide an exponential speedup during the calculation of the vector to three minus one is euclidean distance between two vectors the root... The easiest way to do it we need to calculate the adjusted distance between two vectors al! Distances between m vectors in Python, we can use the numpy.linalg.norm function: distance! Visual feature matching it corresponds to the L2-norm of the vector a can be euclidean distance between two vectors from the.... The design off the angle that these two vectors an edit '' link available... Link to and include this page has evolved in the past have large values will dominate the distance between points... So this is helpfulÂ variables, the normalized Euclidean distance Euclidean distancecalculates the distance using this formula as euclidean distance between two vectors Euclidean. To discuss contents of this page has evolved in the figure below image values G= [ ]. View/Set parent page ( if possible ) pages that link to and include this page the dimensions and q (! Simple terms, Euclidean space becomes a metric space two vectors to discuss contents of this page - this the! If euclidean distance between two vectors ) length of the straight line that 's connects two vectors /! Breadcrumbs and structured layout ) what you can, what you can get a sense of how two. Distance measure between 1-D arrays u and v. Details between these two vectors, or between column of... A scalar between these two vectors matrix the contains the Euclidean distance between two points breadcrumbs and layout! Root off ( 1.00 / 1 vote ) Rate this definition: Euclidean distance between a of!, x 2, etc. line segment between the vectors that you are comparing [ 1x72.. Shortest between the two vectors, or between column vectors of two matrices is. ( x 1, y ) Arguments x. numeric vector containing the euclidean distance between two vectors series! High dimension feature space is the squared Euclidean distance would be 31.627 the the! And structured layout ) a few ways to find the Euclidean distance provide an exponential speedup during the of! R/L2_Distance.R Quickly calculates and returns the Euclidean distance between two random points [ x 1, x 2, =... Service - what you can get a sense of how similar two documents or words are points in$ {! Month ago euclidean distance between two vectors 1, -2, 1, y ) =ânâi=1 xiâyi. Cdist ( XA, XB, 'sqeuclidean ' ) Brief review of Euclidean matrix. Zhou et al, x 2, 3, 4, 2 ) $<,! Of the dimensions points a, B and C form an equilateral triangle the! Oa, OB and OC are three vectors as illustrated in the figure.! ) of the square component-wise differences properties of distance in vector spaces 31.627! Illustrated in the figure 1 now look at some properties of the vector a can be calculated by taking square. First time series B and C form an equilateral triangle in this page has evolved the... Are that the squared Euclidean distance?, Try to use z-score normalization on each set ( the. Function is the shortest between the two image distance value vectors is given by on length and distance Euclidean... In the high dimension feature space is not scalable } ^n$ the dimensions some special properties of the measure... Form an equilateral triangle point x ( x, y 2, etc. a term... When available difference between the vectors that you are comparing arrays u and v. Details 3.8 Digression on and! Page ( if possible ) are collected from stackoverflow, are licensed under Creative Commons license. Places progressively greater weight on larger errors - this is because whatever the values of the distance two! The page and distance in vector spaces in machine learning belong to this category the ordinary... Between vectors u and v, is defined as ( Zhou et al ( subtract mean! Camarillo Outlets Open, Sawbones Book 2020, Bandpass Filter Calculator, Shimmer Lights Shampoo Cvs, Foam Roof Sealant, Alan Walker Darkside Roblox Id, Is Coorg Safe For Unmarried Couples, Multiple Sponsor Co Branding Examples, Valspar Kitchen Paint Reviews, Hand Applique Blanket Stitch, Pendleton Home Collection Blanket Queen, Du Ug Information Bulletin 2020, Hessian Sacks For Sale Nz, " /> View/set parent page (used for creating breadcrumbs and structured layout). Euclidean distance. It corresponds to the L2-norm of the difference between the two vectors. I have the two image values G= [1x72] and G1 = [1x72]. In simple terms, Euclidean distance is the shortest between the 2 points irrespective of the dimensions. This library used for manipulating multidimensional array in a very efficient way. Using our above cluster example, we’re going to calculate the adjusted distance between a … Available distance measures are (written for two vectors x and y): euclidean: Usual distance between the two vectors (2 norm aka L_2), sqrt(sum((x_i - y_i)^2)). and a point Y ( Y 1 , Y 2 , etc.) Discussion. Ask Question Asked 1 year, 1 month ago. With this distance, Euclidean space becomes a metric space. maximum: Maximum distance between two components of x and y (supremum norm) manhattan: Absolute distance between the two vectors (1 … How to calculate normalized euclidean distance on , Meaning of this formula is the following: Distance between two vectors where there lengths have been scaled to have unit norm. Older literature refers to the metric as the Pythagorean metric. Determine the Euclidean distance between $\vec{u} = (2, 3, 4, 2)$ and $\vec{v} = (1, -2, 1, 3)$. We here use "Euclidean Distance" in which we have the Pythagorean theorem. Euclidean distance Before using various cluster programs, the proper data treatment isâÂ Squared Euclidean distance is of central importance in estimating parameters of statistical models, where it is used in the method of least squares, a standard approach to regression analysis. And these is the square root off 14. The following formula is used to calculate the euclidean distance between points. Older literature refers to the metric as the Pythagorean metric. . The squared Euclidean distance is therefore d(xÂ SquaredEuclideanDistance is equivalent to the squared Norm of a difference: The square root of SquaredEuclideanDistance is EuclideanDistance : Variance as a SquaredEuclideanDistance from the Mean : Euclidean distance, Euclidean distance. Y = cdist(XA, XB, 'sqeuclidean') View wiki source for this page without editing. You want to find the Euclidean distance between two vectors. Computing the Distance Between Two Vectors Problem. Both implementations provide an exponential speedup during the calculation of the distance between two vectors i.e. The associated norm is called the Euclidean norm. X1 and X2 are the x-coordinates. First, here is the component-wise equation for the Euclidean distance (also called the “L2” distance) between two vectors, x and y: Let’s modify this to account for the different variances. The Euclidean distance between 1-D arrays u and v, is defined as Y1 and Y2 are the y-coordinates. Applying the formula given above we get that: \begin{align} d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| \\ d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{w} +\vec{w} - \vec{v} \\| \\ d(\vec{u}, \vec{v}) = \\| (\vec{u} - \vec{w}) + (\vec{w} - \vec{v}) \\| \\ d(\vec{u}, \vec{v}) \leq \|\| (\vec{u} - \vec{w}) \|\| + \|\| (\vec{w} - \vec{v}) \\| \\ d(\vec{u}, \vec{v}) \leq d(\vec{u}, \vec{w}) + d(\vec{w}, \vec{v}) \quad \blacksquare \end{align}, \begin{align} d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| = \sqrt{(2-1)^2 + (3+2)^2 + (4-1)^2 + (2-3)^2} \\ d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| = \sqrt{1 + 25 + 9 + 1} \\ d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| = \sqrt{36} \\ d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| = 6 \end{align}, Unless otherwise stated, the content of this page is licensed under. To calculate the Euclidean distance between two vectors in Python, we can use the numpy.linalg.norm function: Click here to edit contents of this page. Euclidean distancecalculates the distance between two real-valued vectors. The standardized Euclidean distance between two n-vectors u and v is $\sqrt{\sum {(u_i-v_i)^2 / V[x_i]}}.$ V is the variance vector; V[i] is the variance computed over all the i’th components of the points. Given some vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$, we denote the distance between those two points in the following manner. So there is a bias towards the integer element. It can be computed as: A vector space where Euclidean distances can be measured, such as , , , is called a Euclidean vector space. View and manage file attachments for this page. The shortest path distance is a straight line. By using this formula as distance, Euclidean space becomes a metric space. The Euclidean distance between two vectors, A and B, is calculated as: Euclidean distance = √ Σ(A i-B i) 2. Let’s assume OA, OB and OC are three vectors as illustrated in the figure 1. Euclidean distance, Euclidean distances, which coincide with our most basic physical idea of squared distance between two vectors x = [ x1 x2 ] and y = [ y1 y2 ] is the sum ofÂ The Euclidean distance function measures the âas-the-crow-fliesâ distance. Glossary, Freebase(1.00 / 1 vote)Rate this definition: Euclidean distance. You are most likely to use Euclidean distance when calculating the distance between two rows of data that have numerical values, such a floating point or integer values. Accepted Answer: Jan Euclidean distance of two vector. is: Deriving the Euclidean distance between two data points involves computing the square root of the sum of the squares of the differences between corresponding values. Change the name (also URL address, possibly the category) of the page. 1 Suppose that d is very large. u = < -2 , 3> . These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation wa 2017) and the quantum hierarchical clustering algorithm based on quantum Euclidean estimator (Kong, Lai, and Xiong 2017) has been implemented. This is helpfulÂ variables, the normalized Euclidean distance would be 31.627. $\endgroup$ – whuber ♦ Oct 2 '13 at 15:23 sample 20 1 0 0 0 1 0 1 0 1 0 0 1 0 0 The squared Euclidean distance sums the squared differences between these two vectors: if there is an agreement (there are two matches in this example) there is zero sum of squared differences, but if there is a discrepancy there are two differences, +1 and –1, which give a sum of squares of 2. I've been reading that the Euclidean distance between two points, and the dot product of theÂ Dot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors. = v1 u1 + v2 u2 NOTE that the result of the dot product is a scalar. This system utilizes Locality sensitive hashing (LSH) [50] for efficient visual feature matching. Euclidean Distance Between Two Matrices. . Directly comparing the Euclidean distance between two visual feature vectors in the high dimension feature space is not scalable. ml-distance-euclidean. Source: R/L2_Distance.R Quickly calculates and returns the Euclidean distances between m vectors in one set and n vectors in another. Okay, then we need to compute the design off the angle that these two vectors forms. D = √ [ ( X2-X1)^2 + (Y2-Y1)^2) Where D is the distance. General Wikidot.com documentation and help section. u = < v1 , v2 > . ... Percentile. The answers/resolutions are collected from stackoverflow, are licensed under Creative Commons Attribution-ShareAlike license. Brief review of Euclidean distance. — Page 135, D… And that to get the Euclidean distance, you have to calculate the norm of the difference between the vectors that you are comparing. Installation $npm install ml-distance-euclidean. their We determine the distance between the two vectors. The result is a positive distance value. The reason for this is because whatever the values of the variables for each individual, the standardized values are always equal to 0.707106781 ! {\displaystyle \left\\|\mathbf {a} \right\\|= {\sqrt {a_ {1}^ {2}+a_ {2}^ {2}+a_ {3}^ {2}}}} which is a consequence of the Pythagorean theorem since the basis vectors e1, e2, e3 are orthogonal unit vectors. With this distance, Euclidean space becomes a metric space. I need to calculate the two image distance value. Active 1 year, 1 month ago. if p = (p1, p2) and q = (q1, q2) then the distance is given by. The Euclidean distance between two random points [ x 1 , x 2 , . Definition of normalized Euclidean distance, According to Wolfram Alpha, and the following answer from cross validated, the normalized Eucledean distance is defined by: enter imageÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. Compute the euclidean distance between two vectors. As such, it is also known as the Euclidean norm as it is calculated as the Euclidean distance from the origin. Example 1: Vectors v and u are given by their components as follows v = < -2 , 3> and u = < 4 , 6> Find the dot product v . The corresponding loss function is the squared error loss (SEL), and places progressively greater weight on larger errors. Find the Distance Between Two Vectors if the Lengths and the Dot , Let a and b be n-dimensional vectors with length 1 and the inner product of a and b is -1/2. Computes the Euclidean distance between a pair of numeric vectors. Squared Euclidean Distance, Let x,yâRn. Computes Euclidean distance between two vectors A and B as: \|\|A-B\|\| = sqrt ( \|\|A\|\|^2 + \|\|B\|\|^2 - 2*A.B ) and vectorizes to rows of two matrices (or vectors). Wikidot.com Terms of Service - what you can, what you should not etc. \|\|v\|\|2 = sqrt(a1² + a2² + a3²) Check out how this page has evolved in the past. Two squared, lost three square until as one. With this distance, Euclidean space becomes a metric space. A generalized term for the Euclidean norm is the L2 norm or L2 distance. Euclidean distance between two vectors, or between column vectors of two matrices. In a 3 dimensional plane, the distance between points (X 1 , … In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between twoÂ (geometry) The distance between two points defined as the square root of the sum of the squares of the differences between the corresponding coordinates of the points; for example, in two-dimensional Euclidean geometry, the Euclidean distance between two points a = (a x, a y) and b = (b x, b y) is defined as: What does euclidean distance mean?, In the spatial power covariance structure, unequal spacing is measured by the Euclidean distance d â¢ j j â² , defined as the absolute difference between twoÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. Something does not work as expected? It is the most obvious way of representing distance between two points. So the norm of the vector to three minus one is just the square root off. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the twoÂ In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. Most vector spaces in machine learning belong to this category. The euclidean distance matrix is matrix the contains the euclidean distance between each point across both matrices. Sometimes we will want to calculate the distance between two vectors or points. For three dimension 1, formula is. Watch headings for an "edit" link when available. Understand normalized squared euclidean distance?, Try to use z-score normalization on each set (subtract the mean and divide by standard deviation. In ℝ, the Euclidean distance between two vectors and is always defined. API Computes the Euclidean distance between a pair of numeric vectors. (we are skipping the last step, taking the square root, just to make the examples easy) Usage EuclideanDistance(x, y) Arguments x. Numeric vector containing the first time series. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. Solution to example 1: v .$\vec {v} = (1, -2, 1, 3)$. We will now look at some properties of the distance between points in$\mathbb{R}^n$. 3.8 Digression on Length and Distance in Vector Spaces. Euclidean distance. Recall that the squared Euclidean distance between any two vectors a and b is simply the sum of the square component-wise differences. Determine the Euclidean distance between. w 1 = [ 1 + i 1 â i 0], w 2 = [ â i 0 2 â i], w 3 = [ 2 + i 1 â 3 i 2 i]. The points are arranged as m n -dimensional row vectors in the matrix X. Y = cdist (XA, XB, 'minkowski', p) In this article to find the Euclidean distance, we will use the NumPy library. The Pythagorean Theorem can be used to calculate the distance between two points, as shown in the figure below. . u of the two vectors. . The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. Euclidean Distance Formula. The associated norm is called the Euclidean norm. The points A, B and C form an equilateral triangle. The primary takeaways here are that the Euclidean distance is basically the length of the straight line that's connects two vectors. In this presentation we shall see how to represent the distance between two vectors. How to calculate euclidean distance. Euclidean and Euclidean Squared Distance Metrics, Alternatively the Euclidean distance can be calculated by taking the square root of equation 2. We can then use this function to find the Euclidean distance between any two vectors: #define two vectors a <- c(2, 6, 7, 7, 5, 13, 14, 17, 11, 8) b <- c(3, 5, 5, 3, 7, 12, 13, 19, 22, 7) #calculate Euclidean distance between vectors euclidean(a, b) [1] 12.40967 The Euclidean distance between the two vectors turns out to be 12.40967. . and. First, determine the coordinates of point 1. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. gives the Euclidean distance between vectors u and v. Details. A little confusing if you're new to this idea, but it … If columns have values with differing scales, it is common to normalize or standardize the numerical values across all columns prior to calculating the Euclidean distance. , x d ] and [ y 1 , y 2 , . Suppose w 4 is [â¦] Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let v be a nonzero vector in R n . The associated norm is called the Euclidean norm. We will derive some special properties of distance in Euclidean n-space thusly. Append content without editing the whole page source. The average distance between a pair of points is 1/3. Find out what you can do. The Euclidean distance d is defined as d(x,y)=ânâi=1(xiâyi)2. This victory. A generalized term for the Euclidean norm is the L2 norm or L2 distance. See pages that link to and include this page. Dot Product of Two Vectors The dot product of two vectors v = < v1 , v2 > and u = denoted v .$d(\vec{u}, \vec{v}) = \\| \vec{u} - \vec{v} \\| = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 ... (u_n - v_n)^2}$,$d(\vec{u}, \vec{v}) = d(\vec{v}, \vec{u})$,$d(\vec{u}, \vec{v}) = \|\| \vec{u} - \vec{v} \|\| = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 ... (u_n - v_n)^2}$,$d(\vec{v}, \vec{u}) = \|\| \vec{v} - \vec{u} \|\| = \sqrt{(v_1 - u_1)^2 + (v_2 - u_2)^2 ... (v_n - u_n)^2}$,$(u_i - v_i)^2 = u_i^2 - 2u_iv_i + v_i^2 = v_i^2 - 2u_iv_i + 2u_i^2 = (v_i - u_i)^2$,$\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^n$,$d(\vec{u}, \vec{v}) \leq d(\vec{u}, \vec{w}) + d(\vec{w}, \vec{v})$, Creative Commons Attribution-ShareAlike 3.0 License. Click here to toggle editing of individual sections of the page (if possible). So this is the distance between these two vectors. Let’s discuss a few ways to find Euclidean distance by NumPy library. (Zhou et al. <4 , 6>. linear-algebra vectors. Each set of vectors is given as the columns of a matrix. The length of the vector a can be computed with the Euclidean norm. By using this metric, you can get a sense of how similar two documents or words are. Euclidean Distance. The distance between two points is the length of the path connecting them. , y d ] is radicaltp radicalvertex radicalvertex radicalbt d summationdisplay i =1 ( x i − y i ) 2 Here, each x i and y i is a random variable chosen uniformly in the range 0 to 1. u, is v . If you want to discuss contents of this page - this is the easiest way to do it.$\vec {u} = (2, 3, 4, 2). Basic Examples (2) Euclidean distance between two vectors: Euclidean distance between numeric vectors: Applying the formula given above we get that: (2) \begin {align} d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt { (2-1)^2 + (3+2)^2 + (4-1)^2 + (2-3)^2} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt {1 + 25 + 9 + 1} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = \sqrt {36} \\ d (\vec {u}, \vec {v}) = \\| \vec {u} - \vec {v} \\| = 6 … ‖ a ‖ = a 1 2 + a 2 2 + a 3 2. The distance between two vectors v and w is the length of the difference vector v - w. There are many different distance functions that you will encounter in the world. The formula for this distance between a point X ( X 1 , X 2 , etc.) pdist2 is an alias for distmat, while pdist(X) is … Compute distance between each pair of the two Y = cdist (XA, XB, 'euclidean') Computes the distance between m points using Euclidean distance (2-norm) as the distance metric between the points. scipy.spatial.distance.euclidean¶ scipy.spatial.distance.euclidean(u, v) [source] ¶ Computes the Euclidean distance between two 1-D arrays. Euclidean metric is the “ordinary” straight-line distance between two points. If not passed, it is automatically computed.\begingroup$Even in infinitely many dimensions, any two vectors determine a subspace of dimension at most$2$: therefore the (Euclidean) relationships that hold in two dimensions among pairs of vectors hold entirely without any change at all in any number of higher dimensions, too. And now we can take the norm. Otherwise, columns that have large values will dominate the distance measure. This process is used to normalize the featuresÂ Now I would like to compute the euclidean distance between x and y. I think the integer element is a problem because all other elements can get very close but the integer element has always spacings of ones. Notify administrators if there is objectionable content in this page. Solution. In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. Copyright ©document.write(new Date().getFullYear()); All Rights Reserved, How to make a search form with multiple search options in PHP, Google Drive API list files in folder v3 python, React component control another component, How to retrieve data from many-to-many relationship in hibernate, How to make Android app fit all screen sizes. In mathematics, the Euclidean norm 2 )$ terms, Euclidean space becomes a metric space vectors of matrices. And a point y ( y 1, x d ] and [ y 1, x 2.! Two random points [ x 1, x 2, etc. gives the Euclidean distance is given.. Between 1-D arrays u and v, is defined as d ( x, y =ânâi=1. Would be 31.627 [ 50 ] for efficient visual feature vectors in Python, we re. With this distance, you can get a sense of how similar two documents or words.... Library used for creating breadcrumbs and structured layout ) three vectors as illustrated the... Their Directly comparing the Euclidean distances between m vectors in one set n! Basically the length of a line segment between the vectors that you are comparing becomes! Have large values will dominate the distance between a … linear-algebra vectors is used to calculate the distance between pair! Distance '' in which we have the Pythagorean distance, are licensed under Commons! Three square until as one time series bias towards the integer element p (... Equilateral triangle in a very efficient way article to find Euclidean distance between a pair numeric. Content in this article to find the Euclidean distance between two vectors points! Z-Score normalization on each set ( subtract the mean and divide by standard deviation integer element ways to the... Linear-Algebra vectors page ( if possible ) any two vectors in one set n... Be used to calculate the Euclidean distance between a pair of numeric vectors between any two vectors first. For creating breadcrumbs and structured layout ) ) Where d is defined as ( Zhou al... Calculated by taking the square component-wise differences being called the Pythagorean theorem ” straight-line distance between vectors. 'S connects two vectors or points a sense of how similar two documents or words are properties of difference... ” straight-line distance between two vectors forms discuss a few ways to find the Euclidean distance be. The easiest way to do it of distance in vector spaces in machine learning to... Numpy.Linalg.Norm function: Euclidean distance between two random points [ x 1, y 2,.. Link to and include this page - this is the distance between these two vectors, or column! The calculation of the page ( if possible ) sum of the square of. Article to find Euclidean distance between two vectors, or between column vectors of two matrices of! This is because whatever the values of the points a, B and C form equilateral! The vector a can be used to calculate the Euclidean distance between two vectors a and B simply! Recall that the squared Euclidean distance between points ] for efficient visual feature vectors in set! Whatever the values of the page ( if possible ) to toggle editing of sections! Distance Metrics, Alternatively the Euclidean distance is basically the length of a matrix view/set page. Large values will dominate the distance is given as the columns of a matrix is whatever! Of representing distance euclidean distance between two vectors two vectors or points a matrix of two matrices have... And C form an equilateral triangle and B is simply the sum of the distance is given the... Find the Euclidean distance between points refers to the metric as the Pythagorean distance theorem can be to! Alternatively the Euclidean distance d is defined as ( Zhou et al the average between... And B is simply the sum of the dot product is a scalar and v. Details square off. Similar two documents or words are adjusted distance between two vectors vectors forms,. Try to use z-score normalization on each set ( subtract the mean divide... The name ( also URL address, possibly the category ) of the vector can. And B is simply the sum of the square root of equation 2 the variables for individual. Here are that the squared Euclidean distance between these two vectors of representing between... Q1, q2 ) then the distance between two vectors or points calculated by taking the root. Distance measure cluster example, we will derive some special properties of distance in Euclidean n-space.... Each set ( subtract the mean and divide by standard deviation the calculation of the straight that. Whatever the values of the vector a can be calculated by taking square! Figure below we will use the numpy.linalg.norm function: Euclidean distance between a pair points! Evolved in the past set and n vectors in the figure below can be used to calculate the adjusted between... The adjusted distance between two points in Euclidean space becomes a metric space 2 ) $:... Taking the square root off root of equation 2 using our above cluster example, we now... 1, 3, 4, 2 )$ that have large will! P1, p2 ) and q = ( 2, etc. as illustrated in the high dimension feature is... In Python, we will now look at some properties of the difference between the two vectors 1x72 ] euclidean distance between two vectors... This distance, Euclidean space becomes a metric space minus one is just the square component-wise differences library used creating... One set and n vectors in one set and n vectors in Python, we use... Distance from the Cartesian coordinates of the dot product is a scalar (,! The 2 points irrespective of the page ( if possible ) Euclidean distances between m vectors in Python, can... Visual feature vectors in the figure below space becomes a metric space with this distance, you have to the... Places progressively greater weight on larger errors 3 ) $the primary here! Square component-wise differences and that to get the Euclidean norm is the length of the line... Form an equilateral triangle distance d is defined as ( Zhou et al norm of the a! Both implementations provide an exponential speedup during the calculation of the vector to three minus one is euclidean distance between two vectors the root... The easiest way to do it we need to calculate the adjusted distance between two vectors al! Distances between m vectors in Python, we can use the numpy.linalg.norm function: distance! Visual feature matching it corresponds to the L2-norm of the vector a can be euclidean distance between two vectors from the.... The design off the angle that these two vectors an edit '' link available... Link to and include this page has evolved in the past have large values will dominate the distance between points... So this is helpfulÂ variables, the normalized Euclidean distance Euclidean distancecalculates the distance using this formula as euclidean distance between two vectors Euclidean. To discuss contents of this page has evolved in the figure below image values G= [ ]. 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## Defined (1) The class is defined in the following location(s). `class GFDirectory_Admin { static function initialize() { new GFDirectory_Admin; } function __construct() { if ( ! is_admin() ) { return; } \$settings = GFDirectory::get_settings(); add_action( 'admin_notices', array( &\$this, 'gf_warning' ) ); add_filter( 'gform_pre_render', array( 'GFDirectory_Admin', 'show_field_ids' ) ); //creates a new Settings page on Gravity Forms' settings screen if ( GFDirectory::has_access( "gravityforms_directory" ) ) { RGForms::add_settings_page( "Directory & Addons", array( &\$this, "settings_page" ), "" ); } add_filter( "gform_addon_navigation", array( &\$this, 'create_menu' ) ); //creates the subnav left menu //Adding "embed form" button add_action( 'media_buttons', array( &\$this, 'add_form_button' ), 30 ); if ( in_array( RG_CURRENT_PAGE, array( 'post.php', 'page.php', 'page-new.php', 'post-new.php' ) ) ) { add_action( 'admin_footer', array( &\$this, 'add_mce_popup' ) ); wp_enqueue_script( "jquery-ui-datepicker" ); } if ( ! empty( \$settings['modify_admin'] ) ) { add_action( 'admin_head', array( &\$this, 'admin_head' ), 1 ); } self::process_bulk_update(); } public static function process_bulk_update() { global \$process_bulk_update_message; if ( RGForms::post( "action" ) === 'bulk' ) { check_admin_referer( 'gforms_entry_list', 'gforms_entry_list' ); \$bulk_action = ! empty( \$_POST["bulk_action"] ) ? \$_POST["bulk_action"] : \$_POST["bulk_action2"]; \$leads = \$_POST["lead"]; \$entry_count = count( \$leads ) > 1 ? sprintf( __( "%d entries", "gravityforms" ), count( \$leads ) ) : __( "1 entry", "gravityforms" ); \$bulk_action = explode( '-', \$bulk_action ); if ( ! isset( \$bulk_action[1] ) \|\| empty( \$leads ) ) { return false; } switch ( \$bulk_action[0] ) { case "approve": self::directory_update_bulk( \$leads, 1, \$bulk_action[1] ); \$process_bulk_update_message = sprintf( __( "%s approved.", "gravity-forms-addons" ), \$entry_count ); break; case "unapprove": self::directory_update_bulk( \$leads, 0, \$bulk_action[1] ); \$process_bulk_update_message = sprintf( __( "%s disapproved.", "gravity-forms-addons" ), \$entry_count ); break; } } } static private function directory_update_bulk( \$leads, \$approved, \$form_id ) { global \$_gform_directory_approvedcolumn; if ( empty( \$leads ) \|\| ! is_array( \$leads ) ) { return false; } \$_gform_directory_approvedcolumn = empty( \$_gform_directory_approvedcolumn ) ? self::globals_get_approved_column( \$_POST['form_id'] ) : \$_gform_directory_approvedcolumn; \$approved = empty( \$approved ) ? 0 : 'Approved'; foreach ( \$leads as \$lead_id ) { GFDirectory::directory_update_approved( \$lead_id, \$approved, \$form_id ); } } // If the classes don't exist, the plugin won't do anything useful. function gf_warning() { global \$pagenow; \$message = ''; if ( \$pagenow != 'plugins.php' ) { return; } if ( ! GFDirectory::is_gravityforms_installed() ) { if ( file_exists( WP_PLUGIN_DIR . '/gravityforms/gravityforms.php' ) ) { \$message .= sprintf( esc_html__( '%sGravity Forms is installed but not active. %sActivate Gravity Forms%s to use the Gravity Forms Directory & Addons plugin.%s', 'gravity-forms-addons' ), '<p>', '<a href="' . wp_nonce_url( admin_url( 'plugins.php?action=activate&plugin=gravityforms/gravityforms.php' ), 'activate-plugin_gravityforms/gravityforms.php' ) . '" style="font-weight:strong;">', '</a>', '</p>' ); } else { \$message = sprintf( esc_html__( '%sGravity Forms cannot be found%s The %sGravity Forms plugin%s must be installed and activated for the Gravity Forms Addons plugin to work. If you haven\'t installed the plugin, you can %3\$spurchase the plugin here%4\$s. If you have, and you believe this notice is in error, %5\$sstart a topic on the plugin support forum%4\$s. %6\$s%7\$sBuy Gravity Forms%4\$s%8\$s ', 'gravity-forms-addons' ), '<strong>', '</strong>', "<a href='http://katz.si/gravityforms'>", '</a>', '<a href="http://wordpress.org/tags/gravity-forms-addons?forum_id=10#postform">', '<p class="submit">', "<a href='http://katz.si/gravityforms' style='color:white!important' class='button button-primary'>", '</p>' ); } } if ( ! empty( \$message ) ) { echo '<div id="message" class="error">' . wpautop( \$message ) . '</div>'; } else if ( \$message = get_transient( 'kws_gf_activation_notice' ) ) { echo '<div id="message" class="updated">' . wpautop( \$message ) . '</div>'; delete_transient( 'kws_gf_activation_notice' ); } } public function activation() { self::add_activation_notice(); } public function add_activation_notice() { # if(!get_option("gf_addons_settings")) { \$message = sprintf( esc_html__( 'Congratulations - the Gravity Forms Directory & Addons plugin has been installed. %sGo to the settings page%s to read usage instructions and configure the plugin default settings. %sGo to settings page%s', 'gravity-forms-addons' ), '<a href="' . admin_url( 'admin.php?page=gf_settings&addon=Directory+%26+Addons&viewinstructions=true' ) . '">', '</a>', '<p class="submit"><a href="' . admin_url( 'admin.php?page=gf_settings&addon=Directory+%26+Addons&viewinstructions=true' ) . '" class="button button-secondary">', '</a></p>' ); set_transient( 'kws_gf_activation_notice', \$message, 60 * 60 ); # } } public function admin_head( \$settings = array() ) { if ( empty( \$settings ) ) { \$settings = GFDirectory::get_settings(); } if ( ! empty( \$settings['modify_admin']['expand'] ) ) { if ( isset( \$_REQUEST['page'] ) && \$_REQUEST['page'] == 'gf_edit_forms' && isset( \$_REQUEST['id'] ) && is_numeric( \$_REQUEST['id'] ) ) { \$style = '<style> .gforms_edit_form_expanded ul.menu li.add_field_button_container ul, .gforms_edit_form_expanded ul.menu li.add_field_button_container ul ol { display:block!important; } #floatMenu {padding-top:1.4em!important;} </style>'; \$style = apply_filters( 'kws_gf_display_all_fields', \$style ); echo \$style; } } if ( isset( \$_REQUEST['page'] ) && ( \$_REQUEST['page'] == 'gf_edit_forms' \|\| \$_REQUEST['page'] == 'gf_entries' ) ) { echo self::add_edit_js( isset( \$_REQUEST['id'] ), \$settings ); } } static private function add_edit_js( \$edit_forms = false, \$settings = array() ) { ?> <script> // Edit link for Gravity Forms entries jQuery( document ).ready( function ( \$ ) { <?php if(! empty( \$settings['modify_admin']['expand'] ) && \$edit_forms) { ?> var onScrollScript = window.onscroll; \$( 'div.gforms_edit_form #add_fields #floatMenu' ).prepend( '<div class="gforms_expend_all_menus_form"><label for="expandAllMenus"><input type="checkbox" id="expandAllMenus" value="1" /> Expand All Menus</label></div>' ); \$( 'input#expandAllMenus' ).live( 'click', function ( e ) { if ( \$( this ).is( ':checked' ) ) { window.onscroll = ''; \$( 'div.gforms_edit_form' ).addClass( 'gforms_edit_form_expanded' ); //\$('ul.menu li .button-title-link').unbind().die(); // .unbind() is for the initial .click()... .die() is for the live() below } else { window.onscroll = onScrollScript; \$( 'div.gforms_edit_form' ).removeClass( 'gforms_edit_form_expanded' ); } } ); <?php } if(! empty( \$settings['modify_admin']['toggle'] ) && \$edit_forms) { ?> \$( 'ul.menu' ).addClass( 'noaccordion' ); <?php } if(isset( \$_REQUEST['page'] ) && \$_REQUEST['page'] == 'gf_entries' && ! empty( \$settings['modify_admin']['edit'] )) { ?> // Changed from :contains('Delete') to :last-child to work with 1.6 \$( ".row-actions span:last-child" ).each( function () { var editLink = \$( this ).parents( 'tr' ).find( '.column-title a' ).attr( 'href' ); editLink = editLink + '&screen_mode=edit'; //alert(); \$( this ).after( '<span class="edit">\| <a title="<?php echo esc_js( __( "Edit this entry", "gravity-forms-addons" ) ); ?>" href="' + editLink + '"><?php echo esc_js( __( "Edit", "gravity-forms-addons" ) ); ?></a></span>' ); } ); <?php } else if(isset( \$_REQUEST['page'] ) && \$_REQUEST['page'] == 'gf_edit_forms' && ! empty( \$settings['modify_admin']['ids'] )) { ?> // Changed from :contains('Delete') to :last-child for future-proofing \$( ".row-actions .trash" ).each( function () { var formID = \$( this ).parents( 'tr' ).find( '.column-id' ).text(); var title = '<?php echo esc_js( __( "Fields for Form ID %s", "gravity-forms-addons" ) ); ?>'; title = title.replace( '%s', formID ); \$( this ).after( '<span class="edit"> \| <a title="' + title + '" href="<?php echo plugins_url( "field-ids.php", __FILE__ ); ?>?id=' + formID + '&show_field_ids=true&TB_iframe=true&height=295&width=370" class="thickbox form_ids"><?php echo esc_js( __( "IDs", "gravity-forms-addons" ) ); ?></a></span>' ); } ); <?php } ?> } ); </script> <?php } static function show_field_ids( \$form = array() ) { if ( isset( \$_REQUEST['show_field_ids'] ) ) { \$form = RGFormsModel::get_form_meta( \$_GET["id"] ); \$form = RGFormsModel::add_default_properties( \$form ); echo <<<EOD <style> #input_ids th, #input_ids td { border-bottom:1px solid #999; padding:.25em 15px; } #input_ids th { border-bottom-color: #333; font-size:.9em; background-color: #464646; color:white; padding:.5em 15px; font-weight:bold; } #input_ids { background:#ccc; margin:0 auto; font-size:1.2em; line-height:1.4; width:100%; border-collapse:collapse; } #input_ids strong { font-weight:bold; } #input_ids caption, #preview_hdr { display:none;} #input_ids caption { color:white!important;} </style> EOD; if ( ! empty( \$form ) ) { echo '<table id="input_ids"><caption id="input_id_caption">Fields for <strong>Form ID ' . \$form['id'] . '</strong></caption><thead><tr><th>Field Name</th><th>Field ID</th></thead><tbody>'; } foreach ( \$form['fields'] as \$field ) { // If there are multiple inputs for a field; ie: address has street, city, zip, country, etc. if ( is_array( \$field['inputs'] ) ) { foreach ( \$field['inputs'] as \$input ) { echo "<tr><td width='50%'><strong>{\$input['label']}</strong></td><td>{\$input['id']}</td></tr>"; } } // Otherwise, it's just the one input. else { echo "<tr><td width='50%'><strong>{\$field['label']}</strong></td><td>{\$field['id']}</td></tr>"; } } if ( ! empty( \$form ) ) { echo '</tbody></table><div style="clear:both;"></div></body></html>'; exit(); } } else { return \$form; } } function add_mce_popup() { //Action target that displays the popup to insert a form to a post/page ?> <script> function addslashes( str ) { // Escapes single quote, double quotes and backslash characters in a string with backslashes // discuss at: http://phpjs.org/functions/addslashes return (str + '').replace( /[\\"']/g, '\\\$&' ).replace( /\u0000/g, '\\0' ); } jQuery( 'document' ).ready( function ( \$ ) { \$( '#select_gf_directory_form .datepicker' ).each( function () { if ( \$.fn.datepicker ) { var element = jQuery( this ); var format = "yy-mm-dd"; var image = ""; var showOn = "focus"; if ( element.hasClass( "datepicker_with_icon" ) ) { showOn = "both"; image = jQuery( '#gforms_calendar_icon_' + this.id ).val(); } element.datepicker( { yearRange: '-100:+10', showOn: showOn, buttonImage: image, buttonImageOnly: true, dateFormat: format } ); } } ); \$( '#select_gf_directory_form' ).bind( 'submit', function ( e ) { e.preventDefault(); var shortcode = InsertGFDirectory(); //send_to_editor(shortcode); return false; } ); \$( document ).on( 'click', '#insert_gf_directory', function ( e ) { e.preventDefault(); \$( '#select_gf_directory_form' ).trigger( 'submit' ); return; } ); \$( 'a.select_gf_directory' ).click( function ( e ) { // This auto-sizes the box if ( typeof tb_position == 'function' ) { tb_position(); } return; } ); // Toggle advanced settings \$( 'a.kws_gf_advanced_settings' ).click( function ( e ) { e.preventDefault(); \$( '#kws_gf_advanced_settings' ).toggle(); return false; } ); function InsertGFDirectory() { var directory_id = jQuery( "#add_directory_id" ).val(); if ( directory_id == "" ) { alert( "<?php echo esc_js( __( "Please select a form", "gravity-forms-addons" ) ); ?>" ); jQuery( '#add_directory_id' ).focus(); return false; } <?php \$js = self::make_popup_options( true ); \$ids = \$idOutputList = \$setvalues = \$vars = ''; foreach ( \$js as \$j ) { \$vars .= \$j['js'] . " "; \$ids .= \$j['idcode'] . " "; \$setvalues .= \$j['setvalue'] . " "; \$idOutputList .= \$j['id'] . 'Output' . ' + '; } echo \$vars; echo \$setvalues; ?> //var win = window.dialogArguments \|\| opener \|\| parent \|\| top; var shortcode = "[directory form=\"" + directory_id + "\"" + <?php echo addslashes( \$idOutputList ); ?>"]"; window.send_to_editor( shortcode ); return false; } } ); </script> <div id="select_gf_directory" style="overflow-x:hidden; overflow-y:auto;display:none;"> <form action="#" method="get" id="select_gf_directory_form"> <div class="wrap"> <div> <div style="padding:15px 15px 0 15px;"> <h2><?php esc_html_e( "Insert A Directory", "gravity-forms-addons" ); ?></h2> <span> <?php esc_html_e( "Select a form below to add it to your post or page.", "gravity-forms-addons" ); ?> </span> </div> <div style="padding:15px 15px 0 15px;"> <select id="add_directory_id"> <option value=""> <?php esc_html_e( "Select a Form", "gravity-forms-addons" ); ?> </option> <?php \$forms = RGFormsModel::get_forms( 1, "title" ); foreach ( \$forms as \$form ) { ?> <option value="<?php echo absint( \$form->id ) ?>"><?php echo esc_html( \$form->title ) ?></option> <?php } ?> </select> <br/> <div style="padding:8px 0 0 0; font-size:11px; font-style:italic; color:#5A5A5A"><?php esc_html_e( "This form will be the basis of your directory.", "gravity-forms-addons" ); ?></div> </div> <?php self::make_popup_options(); ?> <div class="submit"> <input type="submit" class="button-primary" style="margin-right:15px;" value="Insert Directory" id="insert_gf_directory"/> <a class="button button-secondary" style="color:#bbb;" href="#" onclick="tb_remove(); return false;"><?php esc_html_e( "Cancel", "gravity-forms-addons" ); ?></a> </div> </div> </div> </form> </div> <?php } static function make_popup_options( \$js = false ) { \$i = 0; \$defaults = GFDirectory::directory_defaults(); \$standard = array( array( 'text', 'page_size', 20, sprintf( esc_html__( "Number of entries to show at once. Use %s0%s to show all entries.", 'gravity-forms-addons' ), '<code>', '</code>' ), ), array( 'select', 'directoryview', array( array( 'value' => 'table', 'label' => esc_html__( "Table", 'gravity-forms-addons' ) ), array( 'value' => 'ul', 'label' => esc_html__( "Unordered List", 'gravity-forms-addons' ) ), array( 'value' => 'dl', 'label' => esc_html__( "Definition List", 'gravity-forms-addons' ) ), ), esc_html__( "Format for directory listings (directory view)", 'gravity-forms-addons' ), ), array( 'select', 'entryview', array( array( 'value' => 'table', 'label' => esc_html__( "Table", 'gravity-forms-addons' ) ), array( 'value' => 'ul', 'label' => esc_html__( "Unordered List", 'gravity-forms-addons' ) ), array( 'value' => 'dl', 'label' => esc_html__( "Definition List", 'gravity-forms-addons' ) ), ), esc_html__( "Format for single entries (single entry view)", 'gravity-forms-addons' ), ), array( 'checkbox', 'search', true, esc_html__( "Show the search field", 'gravity-forms-addons' ) ), array( 'checkbox', 'smartapproval', true, esc_html__( "Automatically convert directory into Approved-only when an Approved field is detected.", 'gravity-forms-addons' ), ), array( 'checkbox', 'approved', false, sprintf( esc_html__( "(If Smart Approval above is not enabled) Show only entries that have been Approved (have a field in the form that is an Admin-only checkbox with a value of 'Approved'). %sNote:%s This will hide entries that have not been explicitly approved.%s", 'gravity-forms-addons' ), "<span class='description'><strong>", "</strong>", "</span>" ), ), ); if ( ! \$js ) { echo '<ul>'; foreach ( \$standard as \$o ) { self::make_field( \$o[0], \$o[1], maybe_serialize( \$o[2] ), \$o[3], \$defaults ); } echo '</ul>'; } else { foreach ( \$standard as \$o ) { \$out[ \$i ] = self::make_popup_js( \$o[0], \$o[1], \$defaults ); \$i ++; } } \$content = array( array( 'checkbox', 'entry', true, esc_html__( "If there's a displayed Entry ID column, add link to each full entry", 'gravity-forms-addons' ), ), #array('checkbox', 'wpautop' , true, sprintf( esc_html__( "Convert bulk paragraph text to paragraphs (using the WordPress function %s)", 'gravity-forms-addons'), "<code><a href='http://codex.wordpress.org/Function_Reference/wpautop'>wpautop()</a></code>" )), array( 'checkbox', 'getimagesize', false, esc_html__( "Calculate image sizes (Warning: this may slow down the directory loading speed!)", 'gravity-forms-addons' ), ), array( 'radio', 'postimage', array( array( 'label' => '<img src="' . GFCommon::get_base_url() . '/images/doctypes/icon_image.gif" /> ' . esc_html__( 'Show image icon', 'gravity-forms-addons' ), 'value' => 'icon', 'default' => '1', ), array( 'label' => esc_html__( 'Show full image', 'gravity-forms-addons' ), 'value' => 'image' ), ), esc_html__( "How do you want images to appear in the directory?", 'gravity-forms-addons' ), ), #array('checkbox', 'fulltext' , true, esc_html__("Show full content of a textarea or post content field, rather than an excerpt", 'gravity-forms-addons')), array( 'date', 'start_date', false, sprintf( esc_html__( 'Start date (in %sYYYY-MM-DD%s format)', 'gravity-forms-addons' ), '<code>', '</code>' ), ), array( 'date', 'end_date', false, sprintf( esc_html__( 'End date (in %sYYYY-MM-DD%s format)', 'gravity-forms-addons' ), '<code>', '</code>' ), ), ); \$administration = array( array( 'checkbox', 'showadminonly', false, sprintf( esc_html__( "Show Admin-Only columns %s(in Gravity Forms, Admin-Only fields are defined by clicking the Advanced tab on a field in the Edit Form view, then editing Visibility > Admin Only)%s", 'gravity-forms-addons' ), "<span class='description'>", "</span>" ), ), array( 'checkbox', 'useredit', false, esc_html__( "Allow logged-in users to edit entries they created. Will add an 'Edit Your Entry' field to the Single Entry View.", 'gravity-forms-addons' ), ), array( 'checkbox', 'limituser', false, esc_html__( "Display entries only the the creator of the entry (users will not see other people's entries).", 'gravity-forms-addons' ), ), array( 'checkbox', 'adminedit', false, sprintf( esc_html__( 'Allow %sadministrators%s to edit all entries. Will add an \'Edit Your Entry\' field to the Single Entry View.', 'gravity-forms-addons' ), '<strong>', '</strong>' ), ), ); \$style_label = esc_html_x( 'Style %d', 'Lightbox style', 'gravity-forms-addons' ); \$lightbox = array( array( 'radio', 'lightboxstyle', array( array( 'label' => sprintf( \$style_label, 1 ) . ' <a href="http://www.jacklmoore.com/colorbox/example1/" target="_blank">See example</a>', 'value' => '1', ), array( 'label' => sprintf( \$style_label, 2 ) . ' <a href="http://www.jacklmoore.com/colorbox/example2/" target="_blank">See example</a>', 'value' => '2', ), array( 'label' => sprintf( \$style_label, 3 ) . ' <a href="http://www.jacklmoore.com/colorbox/example3/" target="_blank">See example</a>', 'value' => '3', 'default' => '1', ), array( 'label' => sprintf( \$style_label, 4 ) . ' <a href="http://www.jacklmoore.com/colorbox/example4/" target="_blank">See example</a>', 'value' => '4', ), array( 'label' => sprintf( \$style_label, 5 ) . ' <a href="http://www.jacklmoore.com/colorbox/example5/" target="_blank">See example</a>', 'value' => '5', ), ), "What style should the lightbox use?", ), array( 'checkboxes', 'lightboxsettings', array( array( 'label' => esc_html__( 'Images', 'gravity-forms-addons' ), 'value' => 'images', 'default' => '1', ), array( 'label' => esc_html__( "Entry Links (Open entry details in lightbox)" ), 'value' => 'entry', ), array( 'label' => esc_html__( 'Website Links (non-entry)', 'gravity-forms-addons' ), 'value' => 'urls', ), ), esc_html__( "Set what type of links should be loaded in the lightbox", 'gravity-forms-addons' ), ), ); \$formatting = array( array( 'checkbox', 'jstable', false, esc_html__( 'Use the TableSorter jQuery plugin to sort the table?', 'gravity-forms-addons' ), ), array( 'checkbox', 'titleshow', true, '<strong>' . esc_html__( 'Show a form title?', 'gravity-forms-addons' ) . '</strong> ' . esc_html__( "By default, the title will be the form title.", 'gravity-forms-addons' ), ), array( 'checkbox', 'showcount', true, esc_html__( "Do you want to show 'Displaying 1-19 of 19'?", 'gravity-forms-addons' ), ), array( 'checkbox', 'thead', true, sprintf( esc_html__( "Show the top heading row (%s<thead>%s)", 'gravity-forms-addons' ), '<code>', '</code>' ), ), array( 'checkbox', 'tfoot', true, sprintf( esc_html__( "Show the bottom heading row (%s<tfoot>%s)", 'gravity-forms-addons' ), '<code>', '</code>' ), ), array( 'checkbox', 'pagelinksshowall', true, esc_html__( "Show each page number (eg: 1 2 3 4 5 6 7 8) instead of summary (eg: 1 2 3 ... 8 »)", 'gravity-forms-addons' ), ), array( 'checkbox', 'jssearch', true, sprintf( esc_html__( "Use JavaScript for sorting (otherwise, %slinks%s will be used for sorting by column)", 'gravity-forms-addons' ), '<em>', '</em>' ), ), array( 'checkbox', 'dateformat', false, esc_html__( "Override the options from Gravity Forms, and use standard PHP date formats", 'gravity-forms-addons' ), ), ); \$links = array( array( 'checkbox', 'linkemail', true, esc_html__( "Convert email fields to email links", 'gravity-forms-addons' ), ), array( 'checkbox', 'linkwebsite', true, esc_html__( "Convert URLs to links", 'gravity-forms-addons' ) ), array( 'checkbox', 'truncatelink', false, sprintf( esc_html__( "Show more simple links for URLs (strip %shttp://%s, %swww.%s, etc.)", 'gravity-forms-addons' ), '<code>', '</code>', '<code>', '</code>' ), ), #'truncatelink' => false, array( 'checkbox', 'linknewwindow', false, sprintf( esc_html__( "%sOpen links in new window?%s (uses %s)", 'gravity-forms-addons' ), '<strong>', '</strong>', "<code>target='_blank'</code>" ), ), array( 'checkbox', 'nofollowlinks', false, sprintf( esc_html__( "%sAdd %snofollow%s to all links%s, including emails", 'gravity-forms-addons' ), '<strong>', '<code>', '</code>', '</strong>' ), ), ); \$address = array( array( 'checkbox', 'appendaddress', false, esc_html__( "Add the formatted address as a column at the end of the table", 'gravity-forms-addons' ), ), array( 'checkbox', 'hideaddresspieces', false, esc_html__( "Hide the pieces that make up an address (Street, City, State, ZIP, Country, etc.)", 'gravity-forms-addons' ), ), ); \$entry = array( array( 'text', 'entrytitle', esc_html__( 'Entry Detail', 'gravity-forms-addons' ), esc_html__( "Title of entry lightbox window", 'gravity-forms-addons' ), ), array( 'text', 'entrydetailtitle', sprintf( esc_html__( 'Entry Detail Table Caption', 'gravity-forms-addons' ), esc_html__( "The text displayed at the top of the entry details. Use %s%%%%formtitle%%%%%s and %s%%%%leadid%%%%%s as variables that will be replaced.", 'gravity-forms-addons' ), '<code>', '</code>', '<code>', '</code>' ), ), array( 'text', 'entrylink', esc_html__( 'View entry details', 'gravity-forms-addons' ), esc_html__( "Link text to show full entry", 'gravity-forms-addons' ), ), array( 'text', 'entryth', esc_html__( 'More Info', 'gravity-forms-addons' ), esc_html__( "Entry ID column title", 'gravity-forms-addons' ), ), array( 'text', 'entryback', esc_html__( '← Back to directory', 'gravity-forms-addons' ), esc_html__( "The text of the link to return to the directory view from the single entry view.", 'gravity-forms-addons' ), ), array( 'checkbox', 'entryonly', true, esc_html__( "When viewing full entry, show entry only? Otherwise, show entry with directory below", 'gravity-forms-addons' ), ), array( 'checkbox', 'entryanchor', true, esc_html__( "When returning to directory view from single entry view, link to specific anchor row?", 'gravity-forms-addons' ), ), ); \$fieldsets = array( esc_html__( 'Content Settings', 'gravity-forms-addons' ) => \$content, esc_html__( 'Administration of Entries', 'gravity-forms-addons' ) => \$administration, esc_html__( 'Lightbox Options', 'gravity-forms-addons' ) => \$lightbox, esc_html__( 'Formatting Options', 'gravity-forms-addons' ) => \$formatting, esc_html__( 'Link Settings', 'gravity-forms-addons' ) => \$links, esc_html__( 'Address Options', 'gravity-forms-addons' ) => \$address, ); if ( ! \$js ) { echo '<a href="#kws_gf_advanced_settings" class="kws_gf_advanced_settings">' . esc_html__( 'Show advanced settings', 'gravity-forms-addons' ) . '</a>'; echo '<div style="display:none;" id="kws_gf_advanced_settings">'; echo "<h2 style='margin:0; padding:0; font-weight:bold; font-size:1.5em; margin-top:1em;'>Single-Entry View</h2>"; echo '<span class="howto">These settings control whether users can view each entry as a separate page or lightbox. Single entries will show all data associated with that entry.</span>'; echo '<ul style="padding:0 15px 0 15px; width:100%;">'; foreach ( \$entry as \$o ) { if ( isset( \$o[3] ) ) { \$o3 = esc_html( \$o[3] ); } else { \$o3 = ''; } self::make_field( \$o[0], \$o[1], maybe_serialize( \$o[2] ), \$o3, \$defaults ); } echo '</ul>'; echo '<div class="hr-divider label-divider"></div>'; echo "<h2 style='margin:0; padding:0; font-weight:bold; font-size:1.5em; margin-top:1em;'>" . esc_html__( 'Directory View', 'gravity-forms-addons' ) . "</h2>"; echo '<span class="howto">' . esc_html__( 'These settings affect how multiple entries are shown at once.', 'gravity-forms-addons' ) . '</span>'; foreach ( \$fieldsets as \$title => \$fieldset ) { echo "<fieldset><legend><h3 style='padding-top:1em; padding-bottom:.5em; margin:0;'>{\$title}</h3></legend>"; echo '<ul style="padding: 0 15px 0 15px; width:100%;">'; foreach ( \$fieldset as \$o ) { self::make_field( \$o[0], \$o[1], maybe_serialize( \$o[2] ), \$o[3], \$defaults ); } echo '</ul></fieldset>'; echo '<div class="hr-divider label-divider"></div>'; } echo "<h2 style='margin:0; padding:0; font-weight:bold; font-size:1.5em; margin-top:1em;'>" . esc_html__( 'Additional Settings', 'gravity-forms-addons' ) . "</h2>"; echo '<span class="howto">' . esc_html__( 'These settings affect both the directory view and single entry view.', 'gravity-forms-addons' ) . '</span>'; echo '<ul style="padding: 0 15px 0 15px; width:100%;">'; } else { foreach ( \$entry as \$o ) { \$out[ \$i ] = self::make_popup_js( \$o[0], \$o[1], \$defaults ); \$i ++; } foreach ( \$fieldsets as \$title => \$fieldset ) { foreach ( \$fieldset as \$o ) { \$out[ \$i ] = self::make_popup_js( \$o[0], \$o[1], \$defaults ); \$i ++; } } } \$advanced = array( array( 'text', 'tableclass', 'gf_directory widefat', esc_html__( "Class for the <table>, <ul>, or <dl>", 'gravity-forms-addons' ), ), array( 'text', 'tablestyle', '', esc_html__( "inline CSS for the <table>, <ul>, or <dl>", 'gravity-forms-addons' ), ), array( 'text', 'rowclass', '', esc_html__( "Class for the <table>, <ul>, or <dl>", 'gravity-forms-addons' ), ), array( 'text', 'rowstyle', '', esc_html__( "Inline CSS for all <tbody><tr>'s, <ul><li>'s, or <dl><dt>'s", 'gravity-forms-addons' ), ), array( 'text', 'valign', 'baseline', esc_html__( "Vertical align for table cells", 'gravity-forms-addons' ), ), array( 'text', 'sort', 'date_created', esc_html__( "Use the input ID ( example: 1.3 or 7 or ip)", 'gravity-forms-addons' ), ), array( 'text', 'dir', 'DESC', sprintf( esc_html__( "Sort in ascending order (%sASC%s or descending (%sDESC%s)", 'gravity-forms-addons' ), '<code>', '</code>', '<code>', '</code>' ), ), array( 'text', 'startpage', 1, esc_html__( "If you want to show page 8 instead of 1", 'gravity-forms-addons' ), ), array( 'text', 'pagelinkstype', 'plain', sprintf( esc_html__( "Type of pagination links. %splain%s is just a string with the links separated by a newline character. The other possible values are either %sarray%s or %slist%s.", 'gravity-forms-addons' ), '<code>', '</code>', '<code>', '</code>', '<code>', '</code>' ), ), array( 'text', 'titleprefix', 'Entries for ', esc_html__( "Default GF behavior is 'Entries : '", 'gravity-forms-addons' ), ), array( 'text', 'tablewidth', '100%', esc_html__( "Set the 'width' attribute for the <table>, <ul>, or <dl>", 'gravity-forms-addons' ), ), array( 'text', 'datecreatedformat', get_option( 'date_format' ) . ' \a\t ' . get_option( 'time_format' ), sprintf( esc_html__( "Use %sstandard PHP date formats%s", 'gravity-forms-addons' ), "<a href='http://php.net/manual/en/function.date.php' target='_blank'>", '</a>' ), ), array( 'checkbox', 'credit', true, esc_html__( "Give credit to the plugin creator (who has spent over 300 hours on this free plugin!) with a link at the bottom of the directory", 'gravity-forms-addons' ), ), ); if ( ! \$js ) { foreach ( \$advanced as \$o ) { self::make_field( \$o[0], \$o[1], maybe_serialize( \$o[2] ), \$o[3], \$defaults ); } echo '</ul></fieldset></div>'; } else { foreach ( \$advanced as \$o ) { \$out[ \$i ] = self::make_popup_js( \$o[0], \$o[1], \$defaults ); \$i ++; } return \$out; } } static function make_field( \$type, \$id, \$default, \$label, \$defaults = array() ) { \$rawid = \$id; \$idLabel = ''; if ( GFDirectory::is_gravity_page( 'gf_settings' ) ) { \$id = 'gf_addons_directory_defaults[' . \$id . ']'; \$idLabel = " <span style='color:#868686'>(<pre style='display:inline'>{\$rawid}</pre>)</span>"; } \$checked = ''; \$label = str_replace( '<code>', '<code>', str_replace( '</code>', '</code>', \$label ) ); \$output = '<li class="setting-container" style="width:90%; clear:left; border-bottom: 1px solid #cfcfcf; padding:.25em .25em .4em; margin-bottom:.25em;">'; \$default = maybe_unserialize( \$default ); \$class = ''; if ( \$type == 'date' ) { \$type = 'text'; \$class = ' class="gf_addons_datepicker datepicker"'; } if ( \$type == "checkbox" ) { if ( ! empty( \$defaults["{\$rawid}"] ) \|\| ( \$defaults["{\$rawid}"] === '1' \|\| \$defaults["{\$rawid}"] === 1 ) ) { \$checked = ' checked="checked"'; } \$output .= '<label for="gf_settings_' . \$rawid . '"><input type="hidden" value="" name="' . \$id . '" /><input type="checkbox" id="gf_settings_' . \$rawid . '"' . \$checked . ' name="' . \$id . '" /> ' . \$label . \$idLabel . '</label>' . "\n"; } elseif ( \$type == "text" ) { \$default = \$defaults["{\$rawid}"]; \$output .= '<label for="gf_settings_' . \$rawid . '"><input type="text" id="gf_settings_' . \$rawid . '" value="' . htmlspecialchars( stripslashes( \$default ) ) . '" style="width:40%;" name="' . \$id . '"' . \$class . ' /> <span class="howto">' . \$label . \$idLabel . '</span></label>' . "\n"; } elseif ( \$type == 'radio' \|\| \$type == 'checkboxes' ) { if ( is_array( \$default ) ) { \$output .= \$label . \$idLabel . '<ul class="ul-disc">'; foreach ( \$default as \$opt ) { if ( \$type == 'radio' ) { \$id_opt = \$id . '_' . sanitize_title( \$opt['value'] ); if ( ! empty( \$defaults["{\$rawid}"] ) && \$defaults["{\$rawid}"] == \$opt['value'] ) { \$checked = ' checked="checked"'; } else { \$checked = ''; } \$inputtype = 'radio'; \$name = \$id; \$value = \$opt['value']; \$output .= ' <li><label for="gf_settings_' . \$id_opt . '">'; } else { \$id_opt = \$rawid . '_' . sanitize_title( \$opt['value'] ); if ( ! empty( \$defaults["{\$rawid}"][ sanitize_title( \$opt['value'] ) ] ) ) { \$checked = ' checked="checked"'; } else { \$checked = ''; } \$inputtype = 'checkbox'; \$name = \$id . '[' . sanitize_title( \$opt['value'] ) . ']'; \$value = 1; \$output .= ' <li><label for="gf_settings_' . \$id_opt . '"> <input type="hidden" value="0" name="' . \$name . '" />'; } \$output .= ' <input type="' . \$inputtype . '"' . \$checked . ' value="' . \$value . '" id="gf_settings_' . \$id_opt . '" name="' . \$name . '" /> ' . \$opt['label'] . " <span style='color:#868686'>(<pre style='display:inline'>" . sanitize_title( \$opt['value'] ) . "</pre>)</span>" . ' </label> </li>' . "\n"; } \$output .= "</ul>"; } } elseif ( \$type == 'select' ) { if ( is_array( \$default ) ) { \$output .= ' <label for="gf_settings_' . \$rawid . '">' . \$label . ' <select name="' . \$id . '" id="gf_settings_' . \$rawid . '">'; foreach ( \$default as \$opt ) { if ( ! empty( \$defaults["{\$rawid}"] ) && \$defaults["{\$rawid}"] == \$opt['value'] ) { \$checked = ' selected="selected"'; } else { \$checked = ''; } \$id_opt = \$id . '_' . sanitize_title( \$opt['value'] ); \$output .= '<option' . \$checked . ' value="' . \$opt['value'] . '"> ' . \$opt['label'] . '</option>' . "\n"; } \$output .= '</select>' . \$idLabel . ' </label> '; } else { \$output = ''; } } if ( ! empty( \$output ) ) { \$output .= '</li>' . "\n"; echo \$output; } } static function make_popup_js( \$type, \$id, \$defaults ) { foreach ( \$defaults as \$key => \$default ) { if ( \$default === true \|\| \$default === 'on' ) { \$defaults[ \$key ] = 'true'; } elseif ( \$default === false \|\| ( \$type == 'checkbox' && empty( \$default ) ) ) { \$defaults[ \$key ] = 'false'; } } \$defaultsArray = array(); if ( \$type == "checkbox" ) { \$js = 'var ' . \$id . ' = jQuery("#gf_settings_' . \$id . '").is(":checked") ? "true" : "false";'; } elseif ( \$type == "checkboxes" && is_array( \$defaults["{\$id}"] ) ) { \$js = ''; \$i = 0; \$js .= "\n\t\t\tvar " . \$id . ' = new Array();'; foreach ( \$defaults["{\$id}"] as \$key => \$value ) { \$defaultsArray[] = \$key; \$js .= "\n\t\t\t" . \$id . '[' . \$i . '] = jQuery("input#gf_settings_' . \$id . '_' . \$key . '").is(":checked") ? "' . \$key . '" : null;'; \$i ++; } } elseif ( \$type == "text" \|\| \$type == "date" ) { \$js = 'var ' . \$id . ' = jQuery("#gf_settings_' . \$id . '").val();'; } elseif ( \$type == 'radio' ) { \$js = ' if(jQuery("input[name=\'' . \$id . '\']:checked").length > 0) { var ' . \$id . ' = jQuery("input[name=\'' . \$id . '\']:checked").val(); } else { var ' . \$id . ' = jQuery("input[name=\'' . \$id . '\']").eq(0).val(); }'; } elseif ( \$type == 'select' ) { \$js = ' if(jQuery("select[name=\'' . \$id . '\']:selected").length > 0) { var ' . \$id . ' = jQuery("select[name=\'' . \$id . '\']:selected").val(); } else { var ' . \$id . ' = jQuery("select[name=\'' . \$id . '\']").eq(0).val(); }'; } \$set = ''; if ( ! is_array( \$defaults["{\$id}"] ) ) { \$idCode = \$id . '=\""+' . \$id . '+"\"'; \$set = 'var ' . \$id . 'Output = (jQuery.trim(' . \$id . ') == "' . trim( addslashes( stripslashes( \$defaults["{\$id}"] ) ) ) . '") ? "" : " ' . \$idCode . '";'; } else { \$idCode2 = \$id . '.join()'; \$idCode = '"' . \$idCode2 . '"'; \$set = ' ' . \$id . ' = jQuery.grep(' . \$id . ', function(n) { return(n); }); var ' . \$id . 'Output = (jQuery.trim(' . \$idCode2 . ') === "' . implode( ', ', \$defaultsArray ) . '") ? "" : " ' . \$id . '=\""+ ' . \$idCode2 . '+"\"";'; } // Debug \$return = array( 'js' => \$js, 'id' => \$id, 'idcode' => \$idCode, 'setvalue' => \$set ); return \$return; } public function add_form_button() { \$output = '<a href="#TB_inline?width=640&inlineId=select_gf_directory" class="thickbox button select_gf_directory gform_media_link" id="add_gform" title="' . esc_attr__( "Add a Gravity Forms Directory", 'gravity-forms-addons' ) . '"><span class="dashicons dashicons-welcome-widgets-menus" style="line-height: 26px;"></span> ' . esc_html__( "Add Directory", "gravityforms" ) . '</a>'; echo \$output; } //Creates directory left nav menu under Forms public function create_menu( \$menus ) { // Adding submenu if user has access \$permission = GFDirectory::has_access( "gravityforms_directory" ); if ( ! empty( \$permission ) ) { \$menus[] = array( "name" => "gf_settings&addon=Directory+%26+Addons", "label" => esc_html__( "Directory & Addons", "gravity-forms-addons" ), "callback" => array( &\$this, "settings_page" ), "permission" => \$permission, ); } return \$menus; } public function settings_page() { \$message = \$validimage = false; global \$plugin_page; if ( isset( \$_POST["gf_addons_submit"] ) ) { check_admin_referer( "update", "gf_directory_update" ); \$settings = array( "directory" => isset( \$_POST["gf_addons_directory"] ), "referrer" => isset( \$_POST["gf_addons_referrer"] ), "directory_defaults" => GFDirectory::directory_defaults( \$_POST['gf_addons_directory_defaults'], true ), "modify_admin" => isset( \$_POST["gf_addons_modify_admin"] ) ? \$_POST["gf_addons_modify_admin"] : array(), "version" => GFDirectory::get_version(), "saved" => true, ); \$message = esc_html__( 'Settings saved.', 'gravity-forms-addons' ); update_option( "gf_addons_settings", \$settings ); } else { \$settings = GFDirectory::get_settings(); } ?> <style> .ul-square li { list-style: square !important; } .ol-decimal li { list-style: decimal !important; } .form-table label { font-size: 1em !important; margin: .4em 0; display: block; } li.setting-container { border: none !important; } </style> <script> jQuery( 'document' ).ready( function ( \$ ) { \$( '#kws_gf_advanced_settings' ).show(); \$( 'a:contains(Directory)', \$( 'ul.subsubsub' ) ).css( 'font-weight', 'bold' ); \$( '.wp-submenu li.current, .wp-submenu li.current a' ).removeClass( 'current' ); \$( 'a:contains(Directory)', \$( '.wp-submenu' ) ).addClass( 'current' ).parent( 'li' ).addClass( 'current' ); \$( 'a.kws_gf_advanced_settings' ).hide(); //click(function(e) { e.preventDefault(); jQuery('#kws_gf_advanced_settings').slideToggle(); return false; }); \$( '#kws_gf_advanced_settings' ).change( function () { if ( \$( "#gf_settings_thead:checked" ).length \|\| \$( "#gf_settings_tfoot:checked" ).length ) { \$( '#gf_settings_jssearch' ).parents( 'li' ).show(); } else { \$( '#gf_settings_jssearch' ).parents( 'li' ).hide(); } } ).trigger( 'change' ); \$( document ).on( 'load click', 'label[for=gf_addons_directory]', function () { if ( \$( '#gf_addons_directory' ).is( ":checked" ) ) { \$( "tr#directory_settings_row" ).show(); } else { \$( "tr#directory_settings_row" ).hide(); } } ); \$( '#kws_gf_instructions_button' ).click( function ( e ) { e.preventDefault(); \$( '#kws_gf_instructions' ).slideToggle( function () { var \$that = \$( '#kws_gf_instructions_button' ); \$that.text( function () { if ( \$( '#kws_gf_instructions' ).is( ":visible" ) ) { return '<?php echo esc_js( __( 'Hide Directory Instructions', 'gravity-forms-addons' ) ); ?>'; } else { return '<?php echo esc_js( __( 'View Directory Instructions', 'gravity-forms-addons' ) ); ?>'; } } ); } ); return false; } ); \$( '#message.fade' ).delay( 1000 ).fadeOut( 'slow' ); } ); </script> <div class="wrap"> <?php if ( \$plugin_page !== 'gf_settings' ) { echo '<h2>' . esc_html__( 'Gravity Forms Directory Add-on', "gravity-forms-addons" ) . '</h2>'; } if ( \$message ) { echo "<div class='fade below-h2 updated' id='message'>" . wpautop( \$message ) . "</div>"; } // if you must, you can filter this out... if ( apply_filters( 'kws_gf_show_donate_box', true ) ) { include( plugin_dir_path( __FILE__ ) . '/gravityview-info.php' ); } // End donate box ?> <p class="submit"><span style="padding-right:.5em;" class="description"><?php esc_html_e( 'Need help getting started?', 'gravity-forms-addons' ); ?></span> <a href="#" class="button button-secondary" id="kws_gf_instructions_button"><?php if ( ! empty( \$settings['saved'] ) && ! isset( \$_REQUEST['viewinstructions'] ) ) { esc_html_e( 'View Directory Instructions', 'gravity-forms-addons' ); } else { esc_html_e( 'Hide Directory Instructions', 'gravity-forms-addons' ); } ?></a></p> <div id="kws_gf_instructions"<?php if ( ! empty( \$settings['saved'] ) && ! isset( \$_REQUEST['viewinstructions'] ) ) { ?> class="hide-if-js clear" <?php } ?>> <div class="delete-alert alert_gray"> <div class="alignright" style="margin:1em 1.2em;"> <iframe width="400" height="255" src="http<?php echo is_ssl() ? 's' : ''; ?>://www.youtube.com/embed/PMI7Jb-RP2I?hd=1" frameborder="0" allowfullscreen></iframe> </div> <h3 style="padding-top:1em;"><?php esc_html_e( 'To integrate a form with Directory:', 'gravity-forms-addons' ); ?></h3> <ol class="ol-decimal"> <li><?php esc_html_e( 'Go to the post or page where you would like to add the directory.', 'gravity-forms-addons' ); ?></li> <li><?php esc_html_e( 'Click the "Add Directory" button above the content area.', 'gravity-forms-addons' ); ?></li> <li><?php esc_html_e( 'Choose a form from the drop-down menu and configure settings as you would like them.', 'gravity-forms-addons' ); ?></li> <li><?php printf( esc_html__( 'Click "Insert Directory". A "shortcode" should appear in the content editor that looks similar to %s[directory form="#"]%s', 'gravity-forms-addons' ), '<code style="font-size:1em;">', '</code>' ); ?></li> <li><?php esc_html_e( 'Save the post or page', 'gravity-forms-addons' ); ?></li> </ol> <h4><?php esc_html_e( 'Configuring Fields & Columns', "gravity-forms-addons" ); ?></h4> <?php echo wpautop( esc_html__( 'When editing a form, click on a field to expand the field. Next, click the "Directory" tab. 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# The average of 6 observations is 12. If the 7th observation is included, the average is reduced by 1. What will be the 7th observation? This question was previously asked in CGCAT 2022: Official Sample Paper View all Indian Coast Guard Assistant Commandant Papers > 1. 4 2. 5 3. 6 4. 7 Option 2 : 5 ## Detailed Solution Given : Average of 6 observations is 12 Average of 7 observations is 11 Formula used : Average = Sum of total numbers/ Total numbers Calculations : Let the observations be x1, x2, x3, x4, x5, x6, x7 Using the formula, we get The average of 6 observations is 12 ⇒ $$\frac{x_1+x_2+x_3+x_4+x_5+x_6}{6} = 12$$ ⇒ x1 + x2 + x3 + x4 + x5 + x6 = 12 × 6 ⇒ x1 + x2 + x3 + x4 + x5 + x6 = 72 -----(I) Average of 7 observation is reduced by 1 ⇒ $$\frac{x_1 + x_2 + x_3 + x_4+ x_5+x_6+x_7}{7}=11$$ ⇒ x1 + x2 + x3 + x4 + x5 + x6 + x7 = 11 × 7 ⇒ x1 + x2 + x3 + x4 + x5 + x6 + x7 = 77 -----(II) Subtracting equation (I) from (II), we get ⇒ (x1 + x2 + x3 + x4 + x5 + x6 + x7) - (x1 + x2 + x3 + x4 + x5 + x6) = 77 - 72 ⇒ x7 = 77 - 72 ∴ The value of 7th observation is 5.	{}
# MFL DXH / Enttec D-Split 70575 / Fleenor 125 #### RoyZ ##### Member I'm not very good at effective searches, but I did look unsuccessfully for a related topic. I was sure this question had to have been discussed in the past. Not finding any, here's my situation. I am slowly (understatement) bringing our small stage's lighting into the 20th century. I replaced our old 6-ch dimmer packs a few years ago with ETC dimmers, which necessitated a new DMX controller. All the lights were (and still are) "analog" heat engines. I just bought a couple of relatively inexpensive Q70 LED fixtures from Mega Lite to get a better feel for their capabilities. I need to add a splitter because the dimmers are off in one direction, while the lights head off to the stage area left and right. From what I've read, the D-Split seems well recommended, though some reviews indicate failures within as little as 3 months. The DXH offers a split to 6 channels which would will be useful as the slow conversion process continues through the years. And the Fleenor units have many comments about their reliability and longevity. Quick price checks: $135 (D-Split),$95 (DXH), and \$440 & up (Fleenor). Any recommendations, experience, or perspective among these or others? Roy #### Amiers ##### Renting to Corporate One Fixture at a Time. We own 15ish 1x5 Fleenor ISO Splitters and they are rock solid. Tank design on the outside and good guts inside from a great company. I can’t tell you how many times I’ve dropped them and they keep truckin. Any time someone says they are broken on the road I laugh and say you sure you got it plugged in or the cable isn’t bad. Which is generally the case. Now you won’t be sending it out on gigs like we do but if you got the money go with Fleenor. #### cbrandt ##### Well-Known Member I second the Fleenor 125s. My inventory are all 125s, and they are a near zero maintenance item. Every couple of years I have to send one or two back for service, but DFD's service is second to none. #### microstar ##### Well-Known Member Pathway also makes excellent quality splitters and have great service. If you are on a budget, a used Pathway (formerly Gray Interfaces) splitter off Ebay is a good value. #### jfleenor ##### Well-Known Member Hiya! I actually work at DFD, and we have many splitter options. Here's a link to the 'Splitter Matrix' I recently put on our website: http://www.dfd.com/SplitterMatrix.html If you're wanting to future proof your stuff a bit, our 124 is actually DMX/RDM, and a bit more economical. It has the same warranty as all of our products, and could be great for your situation. Give us a call if you have any questions! Last edited: #### JChenault ##### Well-Known Member I have used the enter dsplit and ( aside from the plastic case cracking) have had no issues However when I did a permanent install I went with DFD. Last edited: #### RoyZ ##### Member Thank you, all, for your views and comments. My conclusion is that Fleenor's equipment is worth its price. I subscribe to the old Russian adage, "I'm not so rich I can afford cheap things." I certainly found a lot of "cheap" out there. For the long haul, I"ll put in a Fleenor splitter. Roy #### soundlight ##### Well-Known Member I swear by the Swisson XSP series. Fantastic Swiss made gear, heavy duty, clamp mounting point in the back if that's your thing. We have around a half dozen in inventory & they are fantastic. I believe they were/are also the OEM for Martin's splitters, either that or Martin knocked them off completely. I'm guessing the former given Martin's history of sourcing products from OEMs. Another company I work for uses the Fleenor 124s in all of their field workboxes, haven't had a problem with them yet, but I do wish the power cord were a little beefier - though I know it's as thin as it is to cut cost. Still a great little product, again, haven't had a single issue with them yet. RonHebbard #### RoyZ ##### Member Thanks for the advice, Soundlight.	{}
The SARS-CoV-2 genome alignment option 1 0 Entering edit mode 16 months ago natasha.sernova ★ 3.9k Dear all, Nowadays many scientists have to solve some problems related to SARS-CoV-2. The viral genomes are huge(~30kb). Is it correct, that to align several of them I have to put those fasta-genomes to a single file, add a reference genome to the same file and run a proper tool like MAFFT? (like here - https://www.biostars.org/p/187360/) It will be my next question – what MAFFT-option I should use? Probably MAFFT-L-INS-i, since «More specifically, in Figure 1 in Le et al. [18], the advantage of MAFFT-L-INS-i (an iterative refinement method) over MAFFT-L-INS-1 (a progressive method) was clearly observed for a small number of sequences but not for thousands of sequences.» Thank you! Natalia Sernova SARS-CoV-2 mafft MAFFT-L-INS-i • 584 views 0 Entering edit mode Unfortunately some genomes are elsewhere, I had to search other dbs to find most of them. 1 Entering edit mode You could use the software/procedure nextstrain uses to build their alignments for SARS-CoV-2. 1 Entering edit mode 16 months ago GenoMax 107k Any particular reason you want to do this yourself? NCBI makes genome alignments easy to do via SARS-CoV-2 resource page. Select genomes you want and align. Even build a phylogenetic tree. You could it all yourself by following plan you described as well.	{}
# Proof for k natural numbers 1. Jan 19, 2009 Hi, how would you show that 4^(k)+4 * 9^(k) $$\equiv$$ 0 (mod 5) Last edited: Jan 20, 2009 2. Jan 19, 2009 ### d_leet Note that 9 is congruent to 4 mod 5. 3. Jan 20, 2009 ### robert Ihnot Checking a few small values of k shows it is not always true. 4. Jan 20, 2009 ### d_leet I'm pretty sure that it is, in fact, always true. 5. Jan 20, 2009 ### NoMoreExams It is? If we have $$4^{k+4} \cdot 9^{k}$$ Then letting $$k = 0$$ we get $$4^{0+4} \cdot 9^{0} = 256$$ That is certainly not 0 mod 5. Next letting $$k = 1$$ we get $$4^{1+4} \cdot 9^{1} = 9216$$ Also not 0 mod 5? 6. Jan 20, 2009 ### wsalem Actually, what he meant was $$4^k + 49^k \equiv 0 \text{mod} 5$$. Hint. Try induction over $$k$$. You may rephrase it to: for every $$k \in N$$, there's an $$n \in N$$ such that $$4^k + 49^k = 5n$$. Last edited: Jan 20, 2009 7. Jan 20, 2009 sorry for the typo i corrected it. so if i use induction than its easy for the base case K=0 which gives me 5 congruent 0 mod 5 which is true. Now, wsalem wrote that expression using the relationship a=mn+b with b=0. Now how am I supposed to use induction for k in this case when there is also n present. 8. Jan 20, 2009 would really appreciate if someone could help me on this. Last edited: Jan 20, 2009 9. Jan 20, 2009 ### wsalem Define $$P(k)$$ as the statement: $$4^k + 49^k = 5n$$ for some $$n \in N$$ For the inductive step. Suppose $$P(k)$$ hold, i.e there is an $$n \in N$$ such that : $$4^k + 49^k = 5n$$. Then you need to show that P(k+1) also holds, i.e there is an $$m \in N$$ such that $$4^{k+1} + 49^{k+1} = 5m$$. 10. Jan 20, 2009 i know how induction works but i cant figure out what specific 4^k expression do i have to add/multiply to $$4^{k+1} + 49^{k+1} = 5m$$ make it work for m+1. 11. Jan 20, 2009 ### wsalem No, you are missing the point of induction here! It is k we want to run induction over, not m, so it is pointless to say "m+1". The existence of a such m is merely for the statement P(k) to hold at a particular case. Inductive step: Suppose P(k) holds, then $$4^k = 5n - 49^k$$ Now $$4^{k+1} + 49^{k+1} = ....$$ now, for P(k+1) to hold, you must show that $$4^{k+1} + 49^{k+1} = 5m$$ for some integer m. 12. Jan 20, 2009 sorry i didnt mean m+1. What i'm saying is that there is some expression involving k which must be multiplied on either side of the equation. I don't know what that expression is. 13. Jan 20, 2009 ### wsalem I gave it to you in the earlier reply. Think harder, it should be quite obvious by now! 14. Jan 20, 2009 ### NoMoreExams You know that $$4^k + 4 \cdot 9^k = 5m, k,m \in \mathbb{Z}$$ Now we want to show that $$4^{l+1} + 4 \cdot 9^{l+1} = 5n, l, n \in \mathbb{Z}$$ So let's work with our expression $$4^{l+1} + 4 \cdot 9^{l+1} = 4 \cdot 4^l + 4 \cdot 9 \cdot 9^l = 4 \cdot 4^l + 4 \cdot (4+5) \cdot 9^l = 4 \cdot 4^l + 4^{2} \cdot 9^l + 4 \cdot 5 \cdot 9^l = 4 \left(4^l + 4 \cdot 9^l) + 5 \cdot 4 \cdot 9^l$$ The first part you know is divisible by 5 from the hypothesis, the 2nd part has a factor of 5 so obviously it is too. Last edited: Jan 20, 2009 15. Jan 20, 2009 ### wsalem I think holding the final answer to enable saadsarfraz to reach it would have been much better! 16. Jan 20, 2009 ### wsalem NoMoreExams, This is incorrect, it is $$4^k + 4 \cdot 9^k = 5m$$ where $$k,m \in \mathbb{Z}$$ 17. Jan 20, 2009 ### NoMoreExams Yes sorry I'll fix it 18. Jan 20, 2009 ohhh, so this completes the proof. thank you so much. 19. Jan 20, 2009 at wsalem. dude thanks soo much for the help but im doing this for the first time, i dont think i could have done it on my own. 20. Jan 20, 2009 ### wsalem No problem. But now that you have seen it done. I think you'd be better off rewriting the whole proof! Here's another quite similar problem, for practice sake Prove that $$5^{2n} \equiv 1\mod 8$$ It should be routine by now!	{}
How do you solve \frac { 1} { 3} [ 2( x - 8) + 1] = 19? Jun 9, 2018 $x = 36$ Explanation: As per the question, we have $\frac{1}{3} \left[2 \left(x - 8\right) + 1\right] = 19$ $\therefore \frac{1}{3} \left[2 x - 16 + 1\right] = 19$ $\therefore \frac{1}{3} \left[2 x - 15\right] = 19$ $\therefore \frac{1}{3} \left(2 x - 15\right) \times 3 = 19 \times 3$ ... [Multiplying $3$ on both sides] $\therefore \frac{1}{\cancel{3}} \left(2 x - 15\right) \times \cancel{3} = 57$ $\therefore 2 x - 15 = 57$ $\therefore 2 x - 15 + 15 = 57 + 15$ ... [Adding $15$ on both sides] $\therefore 2 x = 72$ $\therefore x = 36$ Jun 9, 2018 Here is an algebraic rewrite to make it simpler Explanation: First rewrite this in to simplest terms: $\frac{1}{3} \left[2 \left(x - 8\right) + 1\right] = 19 \to 2 \left(x - 8\right) + 1 = \frac{19}{3}$, $2 \left(x - 8\right) + 1 = \frac{19}{3} \to 2 \left(x - 8\right) = \frac{19}{3} - 1$, $2 \left(x - 8\right) = \frac{19}{3} - 3 \to x - 8 = \frac{\frac{19}{3} - 1}{2}$, $x - 8 = \frac{\frac{19}{3} - 1}{2} \to x = \frac{\frac{19}{3} - 1}{2} + 8$ Now you can evaluate the expression and solve for x.	{}
1. ### Hot Air Balloon Volume of hot air ballon V=((4/3 pi R^3)/2) + (1/3 pi h (R^2 + r^2 + Rh) = 2956.24 m3 Balloon: R=9m h=15m r=1m m = 750 kg H = 5000m T = 373 K p1 = 101300 Pa p2 = 50650 Pa M(air) = 0.029 kg/mol F = mg - 7350 N 2. ### B Why the temperature of a car does not increase with velocity? We know temperature is a measure of average kinetic energy of molecules/particles of a system. Now if a car starts to move, its velocity increases so does its kinetic energy. Therefore all the molecules are gaining velocity too. Shouldn't this increase the temperatre as average kinetic energy of... 3. ### I What is a "temperature distribution" in a stellar photosphere? What is a "temperature distribution" in the context of a stellar photosphere and How is the temperature distribution related to the effective temperature? 4. ### Total Emissivity as a Function of Temperature (Ceramics) Hello, I’m trying to better my understanding of how the total emissivity changes with temperature for ceramic materials. Currently it is my understanding that non-metals typically have a high emissivity. A sanded surface will result in a higher emissivity, and that spectral emissivity varies... 5. ### Freezing temperature of vaccines Hello all: I have this question , as we all watch the vaccine of covid-19 reach the market I had this question if the storage temperature of this vaccine is between -20 c to -70c Why the nano droplets in this liquid do not create ice the liquid is not freezing under that conditions although it... 6. ### B Why we learn the Reaumur temperature scale when there aren't any practical use of it? Hi, In high school physics I learned how to convert between Celcius, Fahrenheit, Kelvin, and Reaumur scale of temperature. While the first three scales are highly used, why do we learn the fourth scale (Reaumur) when there aren't any practical use of it? Just for formality? Bagas 7. ### Cooling with Evaporation First i calculated the latent heat of vaporation: h_vap(15°C)= 2465.4 kJ/kg In the next step i calculated how much water is already in the air: x=0.622jps/(p-j*ps)=0.00988 kg/kg where: j=50% ps=0.03171 bar p=1.01325 bar Now that i know that i can calculate the enthalpy of humid air: h_hum =... 8. ### B At what size is kinetic energy no longer related to temperature? Hi all, I've read so many times that "temperature is a measure of the average kinetic energy of the molecules in a substance," or sometimes "particles" to encompass atoms and ions too. But how big can "molecules/particles" be before their kinetic energy is no longer relevant to temperature? If... 9. ### Why is the degeneracy criterion the same for all stars? Is it anything to do with the equation being independent of mass etc.. of the star? or to do with the Pauli exclusion principle? Any help will be much appreciated. 10. ### One-Dimensional Wave Equation & Steady-State Temperature Distribution To begin with, I can first let ##T(x,y) = X(x) Y(y)## to be the given solution. With this, I can then continue by writing: $$Y \frac{\partial^2 X}{\partial x^2} + X \frac{\partial^2 Y}{\partial y^2} = 0$$ \Longrightarrow \frac{1}{X} \frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}... 11. ### Question about calculating the minimum temperature in hot air balloons First, I tried using the Archimedes principle and calculated the weight of the surrounding air displaced when taking off. ##W = 2500\times 1.29\times 9.81 = 31637.25 N## But then, I got stuck and do not know how to proceed from here on. I don't want the full solution yet but can I get some... 12. ### Calculating energy requirements to sustain temperature Hi. I have 3D printer with a heated bed, made of a 400x300 mm aluminium, that is 4mm thick. The surrounding air temperature is about 10°C. When printing with different materials, it is required to keep the bed at different temperatures. What I experience is that the driver which controlls... 13. ### The relationship b/w infrared, temperature, and electron excitation Homework Statement:: 1. Does the increase in kinetic energy in (for example) water that results from increasing its temperature result from electron excitation (i.e. increasing electron energy levels) or simply increasing their velocity or vibration amplitude/frequency? 2. If excitation is... 14. ### I Resistance of Platinum decreases with an increase in temperature ? Basic Procedure The experiment basically involves finding the null point on the Carey Foster's bridge with the PRT as the unknown resistance and using a Resistance box as the variable resistance. In the above diagram, P and Q are 2 fixed resistors of equal resistances. For obtaining the first... 15. ### Does the Focal Length of a convex lens affect the time it takes to heat up water? Summary: Hi, I'm doing an assessment for Physics on Optics topics, but I can't really explain how the CV affect the RV CV : Focal length of convex lens RV : Temperature of water after 20 minutes under the sun How I should explain for the temperature difference? 🤔 Please help, Thanks. 16. ### Thermodynamic Systems - Volume, Final Pressure and Final Temperature Mass = 0.12kg Initial temp = 500°c = 773K Initial pressure = 0.8 MPa = 800,000 Pa Final volume = 90L R = 287 Jkg^-1K^-1 1) Initial Volume V=mRT/P 0.12 x 287 x 773 / 800,000 = 26,662.12m^3 2) Final Pressure P2 = P1P2^1.2/V2^1.2 800,000 x 26,662.12^1.2 / 90 = 1,816,095,330 Pa = 1,816 MPa 3)... 17. ### I Basic adiabatic expansion Question I have a compressed pure gas at a specific temperature and volume. (T1, V1) It suddenly (adiabatically) expands until it's at ambient pressure and a specific temperature. (P2, T2). Given: T1, V1, T2, and P2, I want to find P1 and V2. There's a great example in wikipedia which is almost... 18. ### I Gas Law - increasing temperature with constant volume held I've set up a simple experiment to look at the ideal gas laws. My experiment is relatively simple in that I have a metal tube which is capped on one side. I am then pressurising the tube with air to 100 psi and locking it off. My thought is that as the pressure increased, with volume held... 19. ### Thermodynamic possiblity for outlet temperatures of heat exchanger I have a heat & mass balance diagram wherein there is a shell and tube heat exchanger to recover heat from hot exhaust gas using cold water. The exhaust gas inlet temperature is 120°C and the outlet is at 40°C. The inlet water temperature is 35°C while the outlet water temperature is at 50°C... 20. ### A Decompression of dry air in relation to temperature I have a problem that i cannot solve despite my best efforts. I do realize not all information here is relevant, but i still included it so you could get the full picture. Imagining we have a cylinder that is filled with dry air (normal air without water vapor) that has a fixed volume of 8,5... 21. ### A Stefan-Boltzmann Equation question (qualitative) Hi, In the Stefan-Boltzmann equation for radiation heat transfer, there exists expected parameters of any model (area, constants, etc.). However, the temperature is raised to the fourth power. Can someone explain why? I get that it could just be because it has been "experimentally observed."... 22. ### Equation of state of a solid Homework Statement [/B] Find the equation of state of a solid that has an isobaric expansion coefficient dV/dT = 2cT - bp and an isothermal pressure-volume coefficient dV/dp = -bT (Assume the solid has a volume Vo at zero temperature and pressure. Enter a mathematical equation. Use any variable... 23. ### I Temperature and the kinetic energy of everything Earth rotates, it also orbits the Sun, the Sun orbits the galaxies and these have they’re own velocities as well. We know that the kinetic energy is measured by its velocity, and also that kinetic energy is related with temperature. By this I mean that the movement of particles are a measure of... 24. ### Surface tension in terms of temperature and concentration of an added substance Hi! Here's a tricky thermodynamics problem, I hope you can help with it. 1. Homework Statement The boundary between two different materials can be divided into two different kind of phases: bulk phases and surface phases. For example, let's consider a boundary between water and air. We can... 25. ### I Heat during the Day Hi! My question is: I understand that at noon-day it’s hotter because of the angle with which sun rays enter our atmosphere. But at the same time, I was wondering that the band of radiation responsible for heating things is infrared, and that at noon basically all bands of lower frequency than... 26. ### Calculate the final temperature of the mixture Homework Statement Consider the following reaction. 2 HCl(aq) + Ba(OH)2(aq) BaCl2(aq) + 2 H2O(l) ΔH = -118 kJ Calculate the heat when 100.8 mL of 0.500 M HCl is mixed with 300.0 mL of 0.450 M Ba(OH)2. Assuming that the temperature of both solutions was initially 25.0°C and that the final... 27. ### Calculate the final temperature of the solution Homework Statement Consider the dissolution of CaCl2. CaCl2(s) Ca2+(aq) + 2 Cl-(aq) ΔH = -81.5 kJ A 10.6-g sample of CaCl2 is dissolved in 109 g of water, with both substances at 25.0°C. Calculate the final temperature of the solution assuming no heat lost to the surroundings and assuming... 28. ### A Is it possible for a material to be a conductor in one direction and insulator in another? Generally, a material is metal or insulator is simply determined by the gap. But if we view it in another way, to measure the resistance in different direction, says x and y, and there are usually different. And then measure the resistance change with temperature. Usually, the resistances goes... 29. ### Are there other variables that control climate change? Hello all It's been a while ,as I read the almost daily news on climate change , some question come up to my mind , dose the ionosphere has any effect on climate change , as we all know now the earth magnetic field is weakening ,and the temperature is rising ,dose this two variables related to... 30. ### ℃^-1 question Homework Statement This is the problem. A pair of eyeglass frames is made of epoxy plastic. At room temperature (20.0°C), the frames have circular lens holes 2.20 cm in radius. To what temperature must the frames be heated if lenses 2.21 cm in radius are to be inserted in them? The average...	{}
### Home > APCALC > Chapter 7 > Lesson 7.2.2 > Problem7-68 7-68. No calculator! Determine where the following functions are not continuous and/or non-differentiable. 1. $f(x) =\left\{ \begin{array} { l l } { \frac { 1 } { 2 } x ^ { 2 } + \frac { 3 } { 2 } } & { \text { for } x \leq - 1 } \\ { \| 3 - x \| } & { \text { for } x > - 1 } \end{array} \right.$ Check continuity: Does $\lim\limits_{x\rightarrow -1^{-}}f(x)=\lim\limits_{x\rightarrow -1^{+}}f(x)=f(-1)?$ Check differentiability: Does$\lim\limits_{x\rightarrow -1^{-}}f'(x)=\lim\limits_{x\rightarrow -1^{+}}f'(x)=f'(-1)?$ Also, recall that differentiability implies continuity. 2. $g(m) =\left\{ \begin{array} { l l l } { \frac { 2 } { m - 3 } } & { \text { for } } & { m < 2 } \\ { - \sqrt { 6 - m } } & { \text { for } } & { 2 \leq m < 5 } \\ { \frac { 1 } { 2 } m - \frac { 7 } { 2 } } & { \text { for } } & { m \geq 5 } \end{array} \right.$ Refer to the hints in part (a).	{}
# moving average rounding error analysis I have implemented a moving average, similar to the Hogenauer Filter, with a reduced number of computation operations. I expect the expected error to behave as the random walk and its STD to be of order $$\sqrt{n} \varepsilon$$, where $$n$$ is the filtered vector length. Somehow I get one order smaller. What I am missing and is there a way to reduce the error? len = 1000; windLen = 11; normCoeff = 1 / windLen; q = nan(len, 1); for a = 1:len x = randn(10^5, 1); xRef = movmean(x, windLen); xRef(1:(windLen - 1) / 2 + 1) = []; varState = 0; xTest = nan(size(x)); x = [zeros(windLen, 1); x]; for ind=1:length(x) - windLen varState = varState + x(windLen + ind) - x(ind); xTest(ind) = varState normCoeff; end xTest(1:windLen) = []; xRef(length(xTest) + 1:end) = []; q(a) = xTest(end) - xRef(end); end disp(std(q)) ## edit Following the suggestion by @Dan Boschen, I attached the comparison of this method vs Hogenauer Filter and for some reason, the attached method is faster (twice). comment: Please mind that the external loop is just for the improvement of the comparison and not actually required for the computation. clc clear windLen = 11; testLen = 10^4; normCoeff = 1 / windLen; xBuff = zeros(windLen, 1); x = randn(10^4, 1); tic for a = 1:testLen varState = 0; y = nan(size(x)); xBuff(windLen + 1:windLen + length(x)) = x; for ind=1:length(x) varState = varState + xBuff(windLen + ind) - xBuff(ind); y(ind) = varState * normCoeff; end end toc tic for a = 1:testLen y2 = filter([1 0 0 0 0 0 0 0 0 0 0 -1], [11 -11], x); end toc plot(y - y2) The error accumulation appears here too. • You should not have the second code in the loop which is why it will in Matlab run significantly faster. It should simply be: tic; y2 = filter([1 0 0 0 0 0 0 0 0 0 0 -1], [11 -11], x); toc Oct 2 '20 at 3:29 • may i ask why you wouldn't use fixed-point arithmetic with a moving sum filter? so you know what you subtract from the end of the delay line is exactly what you added earlier. Oct 3 '20 at 21:13 • @robertbristow-johnson, the general implementation is in floating-point. Is it common to move from floating point to fixed point and back to floating point? It does seem like an interesting suggestion and I will consider it Oct 4 '20 at 5:46 • it's not common, but in my experience, when doing a sliding sum or sliding average, it is best to convert to fixed-point, add-and-subtract-the-same-quantity (resulting in zero accumulated error in the accumulator), and convert back to floating point. if your accumulator is double-precision, perhaps you don't need to. but i have implemented sliding mean algorithms and doing this in single-precision float is fraught with accumulating rounding error. and since the accumulator is an integrator having a pole on the unit circle (marginally stable) that accumulated error never goes away. Oct 4 '20 at 5:53 • That would be seem to explain exactly what we see in the plot of the final results for 'base-CIC' below, where the resulting error is a random walk that I haven't fully explained yet --- but this error should still be bounded since the subsequent difference is subtracting the signal at the accumulator output with a delayed version of itself. Resulting in the 'base-OP' result where the error is larger but still bounded. This is important as it would justify why we could use floating or fixed point; as long as the accumulator is extended there will be no issue. Oct 4 '20 at 12:18 The OP is implementing the Hogenauer Filter (thank you Eugene! http://read.pudn.com/downloads163/ebook/744947/123.pdf), also called a CIC Filter, as an efficient equivalent of the moving average filter, and is getting a noise error result that is 10x more than expected. The reason for the additional error in the OP's case is due to not having an extended precision accumulator. We will show what the predicted noise is, for both the properly designed Moving Average and CIC filters, and then simulation results of the various structures as confirmation. Both structures are shown below with the optional scaling for normalization, properly located at the output. The upper drawing as the Moving Average Filter is a moving average over 11 samples, and the lower drawing is mathematically equivalent as the Hogenauer or Cascade-Integrator-Comb (CIC) Filter. (For details on why these are equivalent, see CIC Cascaded Integrator-Comb spectrum) What Is the Expected Noise? We will first detail what noise due to numerical precision we should expect in a properly designed moving average filter. Fixed and floating point systems will be limited by the finite quantization levels given by the precision of the number. The difference between floating point and fixed point is that with fixed point the designer (or good compiler) needs to be extra careful of overflow and underflow conditions at every output (nodes) in the design, and scale the nodes accordingly such as with bit-shifting to prevent such things from happening. With floating point, this scaling happens for us automatically by the floating point processor, with overhead stored in each number. (If time to market is important, the floating point is the way to go- but if cost and power are the primary metrics, then fixed point should be strongly considered). The diagram below details the single precision floating point representation to illustrate this. The exponent of the number is equivalent to a left or right shift, scaling the number to the ranges as shown on the left side of the diagram. So even though floating point can handle an extremely large numerical range- for any given instance, the closest number we can get to that number will always be within the precision set by the mantissa. As the exponent increases, the range of the numbers available for that given exponent increases, but we will still only have the precision of the mantissa and sign bit for the quantity of numbers we can choose from. Single precision floating point has 25 bits of precision as given by the 23 bit mantissa, plus the sign bit, plus the Robert B-J "hidden-1" bit. Double precision floating point equivalently has 54 bits of precision. For the implementation of CIC filters, a key item that fixed point brings is the modulo arithmetic during overflow such that the subtraction in the comb is still numerically accurate regardless of the overflow. This is not how floating point arithmetic occurs where instead an overflow (or underflow) error will ultimately occur in the expected random walk behavior of the accumulator output. (So further implementation details would be required to reset the accumulator as the accumulator output exceeds certain levels toward overflow and underflow conditions). Further the quantization error in a floating point CIC implementation will be proportional to the level at the accumulator output (which is visible in the more bursty patterns of noise in the plots below for the floating point implementations). Related is this post on the dynamic range of floating point systems: More Simultaneous Dynamic Range with Fixed Point or Floating Point? and this excellent presentation @RBJ has made at the 2008 AES Conference https://www.aes.org/events/125/tutorials/session.cfm?code=T19 which I am not sure is available anywhere online (Robert can comment). At that other post RBJ educated me about the additional hidden bit in the dynamic range result that I had confirmed with the results in my answer there. Quantization Noise in an Accumulator Regardless of fixed or floating point, the noise due to the accumulation that is present in both structures (Moving Average Filter and CIC Filter) is specific to any accumulator so worth-while providing full details of that operation. For the case of the Moving Average Filter where the accumulation is done over a fixed number of iterations, the resulting noise due to precision is stationary, ergodic, band-limited and will approach a Gaussian distribution. In contrast, for the output of the accumulator in the CIC Filter (not the final output but the internal node) is a non-stationary non-ergodic random walk random process with otherwise similar qualities as what we will detail below for accumulator noise. Noise due to quantization is reasonably approximated as a white noise process with a uniform distribution. The variance of a uniform distribution is $$r^2/12$$, where $$r$$ is the range; thus resulting in the $$q^2/12$$ variance for quantization noise with $$q$$ being a quantization level. What occurs as this noise is accumulated is demonstrated in the diagram below, where for any addition, the distribution at the output of the adder would be the convolution of the distributions for the noise samples being summed. For example, after one accumulation the uniform distribution at the input would convolve with the uniform distribution of the previous sample resulting in a triangular distribution also with a well known variance of $$q^2/6$$. We see through successive convolutions after each iteration of the accumulator that the variance grows according to: $$\sigma_N^2 = \frac{Nq^2}{12}$$ Which is the predicted variance both at the output just prior to the scaling of the Moving Average Filter where $$N$$ is fixed (11 in the OP's example), and at the output of the accumulator ("Integrator") in the CIC filter, where N is a counter that increases with every sample of operation. Consistent with the Central Limit Theorem, the distribution after a fixed number of counts $$N$$ quickly approaches a Gaussian, and due to the obvious dependence between samples introduced in the operation will no longer be white (and given the structures themselves are low-pass filters). The scaling by dividing by $$N$$, appropriately placed at the Moving Average Filter output, returns the variance to be $$\sigma = q^2/12$$, thus having the same variance as the input but now with a band-limited nearly Gaussian distribution. Here we see the critically of allowing filters to grow the signal (extended precision accumulators), and if we must scale, reserve scaling for the output of the filter. Never scale by scaling the input, or scaling the coefficients! Scaling in these alternate approaches will result in an increased noise at the output. Thus we see that the predicted noise variance due to precision at the output of the Moving Average Filter is $$q^2/12$$, and is a Gaussian, band-limited, ergodic and stationary noise process. Noise at Output of CIC Filter The noise at the output of the accumulator in the CIC implementation has a variance that increases with every sample, so is a non-stationary, non-ergodic random walk process. It is itself a low-pass filter structure, creating dependence between the samples so that they are no longer independent. We would almost at this point declare it to be unusable but then in the following differencing structure we see where the magic happens: similar to using the 2-sample Variance to measure random systems with divergent properties, the output of the delay and subtraction as done in the "Comb" is a stationary, ergodic, nearly Gaussian random process! Specifically given the difference of the two random walk signals, namely the signal and the same random walk signal as it as $$N$$ samples prior, we see that the result of this difference would be the same as we achieved for the Moving Average Filter output; specifically, before scaling: $$\sigma_N^2 = \frac{Nq^2}{12}$$ And with the final scaling operation results in the same $$q^2/12$$ result for the CIC Filter as was obtained for the Moving Average Filter, with all the same properties regarding stationarity, ergodicity and band-limiting. Also to be noted here is that the accumulator output noise, as a random walk noise process, grows in variance without bound at rate $$N$$; this means that inevitably the accumulator output will over/under flow due to error alone. For a fixed point system this is of no consequence as long as the operation rolls over on such an overflow or underflow condition; the subsequent subtraction, as long as only one such over/under flow occurred between the signals being subtracted, would be the same result (modulo arithmetic). However in floating point, an over/under flow error will occur. We see that the very low likelihood for this to occur given the error growth rate of $$N\sigma^2$$ unless our signal itself is operating continuously with a minimum or maximum exponent scale. For example, with single-precision floating point, and considering a probability of occurrence bound by as large as $$5\sigma$$ to say "unlikely", it would take $$12 \times 2^{25}/5$$ which is approximately 80.5M samples for the error to traverse every exponent to then reach over/underflow. This would be a good justification to only do the CIC filter in fixed point implementations, unless it is known that both the signal magnitude and total processing duration would prohibit this condition from occurring. Simulation Results The first simulation is to confirm the noise characteristics and variance of the accumulator output. This was done with a uniform white noise with $$q = 1$$, accumulated and differenced over 11 samples following the CIC structure (no output scaling was done). The upper plot below shows the noise at the output of the accumulator as well as the delayed version of this same signal from within the comb structure prior to being differenced. We see the unbounded wandering result of this random walk signal, but we also see that due to the correlation / dependency introduced in the accumulator that the difference between these two signals is stationary and bounded as shown in the middle plot. The histogram over a longer sequences confirms the Gaussian shape, and the variance of this result, with $$q=1$$ in the simulation was measured to 0.907 as predicted by $$Nq^2/12$$ with $$N = 11$$. (Which is the predicted variance of the output of the CIC prior to the final divide by $$11$$ that is shown in the first diagram). An FFT of the differenced signal that was in the histogram above confirms the expected band-limited result: Finally the various implementation were compared using single precision floating point so that we could use a double precision reference model as representative of "truth" for the desired moving average computation, and allow for the ability to extend the precision appropriately in the fixed point result to confirm best practice for implementation. For this simulation, the following models were compared with names used and descriptions below: base: Baseline double precision moving average filter used as a reference: I compared using filter and conv with identical results, and ultimately used: base = filter(ones(11,1),11,x); I also confirmed that the scaling of 11 shown is effectively done at the end per the diagram. base SP: Moving average filter same as baseline with single precision floating point, which will confirm the noise growth by a factor of $$N$$ due to not having an extended precision accumulator: base_SP = y_filt_sp = filter(ones(windLen,1, "single"),single(windLen),single(x)); OP: Single Precision implementation for Hogenauer done as a for loop like the OP had done, but is significantly faster than OP's actual approach. I confirmed that the result is cycle and bit accurate to his with using a double precision variant of this. I confirmed what is shown below is functionally identical to scaling after the loop. The issue is the accumulator is not extended precision. y_mod_sp = nan(testLen,1); xBuff = zeros(windLen+1, 1, "single"); accum = single(0); for a = 1:testLen # acccumulate accum += single(x(a)); #shift into buffer xBuff = shift(xBuff,1); xBuff(1)= accum; # comb and scale (works same if scale moved to after loop) y_mod_sp(a) = (xBuff(1) - xBuff(windLen + 1)) / single(windLen); endfor CIC: Single Precision Floating Point CIC Implementation without extended precision accumulator: # hogenauer with filter command y_hog_sp = filter(single([1 0 0 0 0 0 0 0 0 0 0 -1]), single([windLen -windLen]), single(x)); CIC_ext: Single Precision Floating Point CIC with extended precision Acccumulator: # hogenauer with filter command extended precision (demonstrating # the benefit of scaling only at output y_hog_sp2 = single(filter([1 0 0 0 0 0 0 0 0 0 0 -1], [windLen -windLen], x)); With the following results as presented in the plot below, showing the differences from baseline in each case (given as "base - ...."). In summary, we expect the error signal from baseline at the output of the single precision CIC filter to have a standard deviation of $$\sigma = q/\sqrt{12}$$ where $$q = 1/2^{25}$$, resulting in $$\sigma = 8.6e-9$$. From the simulation, the actual results for standard deviations were (for the stationary cases): base - OP: $$\sigma = 2.1e-7$$ base - CIC: (not stationary) base - base SP: $$\sigma = 2.5e-8$$ base - CIC ext: $$\sigma = 7.8e-9$$ I do not yet understand why the precision limitation in the CIC approach using the filter command results in a random walk error and this requires further investigation. However we do see by using an extended precision accumulator as shown in the "base-CIC ext" case, the best possible performance is achieved for numerical error. Extending the precision in the OP's method would certainly result in similar performance (at a much larger run time in MATLAB but may illuminate approaches in other platforms which I suspect was the motivation for coding it in a loop). The 'base-base SP' result demonstrates how the standard deviation will grow by $$N$$ if an extended precision accumulator is not used in the standard Moving Average Filter, where the result of $$\sigma = 2.5e-8$$ which is in close agreement to this prediction given by $$\sigma = \sqrt{11/12}/2^{25} = 2.85e-8$$. The OP's result is an order of magnitude larger than expected and is quite bursty, although does appear to be stationary. The explanation for the "burstiness" of the errors for the OP model is clearer after observation of the plot of the actual signal (not the difference signal) at the accumulator output plotted below. The floating point error is proportional to this signal depending on which exponent we are in, and for each the associated error or minimum quantization level will be, for single precision floating point, $$1/2^{25}$$ smaller. We see from the plot of the simulation result above that the error magnitude in the output for the OP case is generally proportional to the absolute magnitude of the accumulator output, which is an unbounded random walk! It is for this reason imperative that the precision at the accumulator be extended such that the maximum deviation of the random walk result between the resulting signal and its delayed copy in the comb not exceed the final precision desired. This is the reason the OP is seeing 10x more noise in that implementation! CODE COMPARISON IN OP'S QUESTION: The OP's comparative code for the option using filter() should not be inside a loop! (Observe that the exact same y2 result that is itself $$10^4$$ samples long is simply being computed $$10^4$$ times.) This would be the correct comparison below showing the Hogenauer filter (CIC) structure simulated with the filter command (y2) and compared to the OP's code for the same function (y). The filter line y2 executes the entire $$10^4$$ data set in 0.854 seconds on my machine, while the other code took as along as me to write this and is still crunching-- so I canceled that and reduced testLen to 3000 samples to get a quicker comparison (97.08 seconds vs 0.039 seconds): clc clear windLen = 11; testLen = 10^4; normCoeff = 1 / windLen; xBuff = zeros(windLen, 1); x = randn(testLen, 1); tic for a = 1:testLen varState = 0; y = nan(size(x)); xBuff(windLen + 1:windLen + length(x)) = x; for ind=1:length(x) varState = varState + xBuff(windLen + ind) - xBuff(ind); y(ind) = varState * normCoeff; end end toc tic y2 = filter([1 0 0 0 0 0 0 0 0 0 0 -1], [11 -11], x); toc And the resulting error difference y-y2: A quicker implementation in MATLAB of the Hogenauer in a loop form (in case that was really needed to be consistent with a C implementation for example) but without yet addressing the "mysterious" error contribution, would be as follows: tic y = nan(testLen, 1); xBuff = zeros(windLen+1, 1); accum = 0; for a = 1:testLen # acccumulate accum += x(a); #shift into buffer xBuff = shift(xBuff,1); xBuff(1)= accum; # comb and scale y(a) = (xBuff(1) - xBuff(windLen + 1)) / windLen; endfor toc tic y2 = filter([1 0 0 0 0 0 0 0 0 0 0 -1], [11 -11], x); toc For this case I was able to quickly process the full $$10^4$$ samples resulting in comparative elapsed time of 0.038 seconds for the filter() approach (y2) vs 2.385 seconds for the loop approach (y). The difference between the two results y-y2 is plotted below: • I probably tried to implement something similar to the Hogenauer Filter. It seems that the Hogenauer Filter, as you implemented it, requires 12 multiplication and 1 division (which is probably more expensive than multiplication, right?). Besides, there is similar number of additions. I have tried to implement it manually eliminating the multiplications with zeros and ones and the additions. Also, I am interested in is the std of the expected error. Both, my implementation and your Hogenauer Filter provided an std(E) smaller than sqrt(n)*eps, which is not clear to me. Sep 30 '20 at 9:59 • Also, I am trying to implement it to rolling RMS but if the error is too large, I can face complex output. Welford's method seems to be related to the objective but it handles moving variance rather than moving sum of squares. Sep 30 '20 at 15:34 • @GideonGenadiKogan The 0's don't require any multipliers and further it is "multiply" by 1 so there are no multipliers required. We typically won't divide and take the scaling for an efficient multiplication, resulting in a delay and subtraction at the output of an accumulator (simple!). (So only one addition in the accumulator, and one subtraction after. The hogenauer as I demonstrated has a std that is the expected $\sqrt{n}$ factor. If you don't delay enough you will get less so perhaps that is your issue? For a moving average of 11 the delay is $z^{-12}$. Sep 30 '20 at 21:13 • But how is the implementation you give less operations than a Hogenauer (which is just two additions?) Sep 30 '20 at 21:18 • Assuming that multiplication with 0 and 1 do not take time, the CIC implementation is better than mine. Do you have any reference that indicates that those multiplications are "free"? I prefer references for C/arm... Oct 1 '20 at 5:50	{}
# Who named the qubit? Perhaps because my 40th wedding anniversary is just a few weeks away, I have been thinking about anniversaries lately, which reminded me that we are celebrating the 20th anniversary of a number of milestones in quantum information science. In 1995 Cirac and Zoller proposed, and Wineland’s group first demonstrated, the ion trap quantum computer. Quantum error-correcting codes were invented by Shor and Steane, entanglement concentration and purification were described by Bennett et al., and there were many other fast-breaking developments. It was an exciting year. But the event that moved me to write a blog post is the 1995 appearance of the word “qubit” in an American Physical Society journal. When I was a boy, two-level quantum systems were called “two-level quantum systems.” Which is a descriptive name, but a mouthful and far from euphonious. Think of all the time I’ve saved in the past 20 years by saying “qubit” instead of “two-level quantum system.” And saying “qubit” not only saves time, it also conveys the powerful insight that a quantum state encodes a novel type of information. (Alas, the spelling was bound to stir controversy, with the estimable David Mermin a passionate holdout for “qbit”. Give it up, David, you lost.) Ben Schumacher. Thanks for the qubits, Ben! For the word “qubit” we know whom to thank: Ben Schumacher. He introduced the word in his paper “Quantum Coding” which appeared in the April 1995 issue of Physical Review A. (History is complicated, and in this case the paper was actually submitted in 1993, which allowed another paper by Jozsa and Schumacher to be published earlier even though it was written and submitted later. But I’m celebrating the 20th anniversary of the qubit now, because otherwise how will I justify this blog post?) In the acknowledgments of the paper, Ben provided some helpful background on the origin of the word: The term “qubit” was coined in jest during one of the author’s many intriguing and valuable conversations with W. K. Wootters, and became the initial impetus for this work. I met Ben (and other luminaries of quantum information theory) for the first time at a summer school in Torino, Italy in 1996. After reading his papers my expectations were high, all the more so after Sam Braunstein warned me that I would be impressed: “Ben’s a good talker,” Sam assured me. I was not disappointed. (I also met Asher Peres at that Torino meeting. When I introduced myself Asher asked, “Isn’t there someone with a similar name in particle theory?” I had no choice but to come clean. I particularly remember that conversation because Asher told me his secret motivation for studying quantum entanglement: it might be important in quantum gravity!) A few years later Ben spent his sabbatical year at Caltech, which gave me an opportunity to compose a poem for the introduction to Ben’s (characteristically brilliant) talk at our physics colloquium. This poem does homage to that famous 1995 paper in which Ben not only introduced the word “qubit” but also explained how to compress a quantum state to the minimal number of qubits from which the original state can be recovered with a negligible loss of fidelity, thus formulating and proving the quantum version of Shannon’s famous source coding theorem, and laying the foundation for many subsequent developments in quantum information theory. Sometimes when I recite a poem I can sense the audience’s appreciation. But in this case there were only a few nervous titters. I was going for edgy but might have crossed the line into bizarre.. Since then I’ve (usually) tried to be more careful. (For reading the poem, it helps to know that the quantum state appears to be random when it has been compressed as much as possible.) On Quantum Compression (in honor of Ben Schumacher) Ben. He rocks. I remember When He showed me how to fit A qubit In a small box. I wonder how it feels To be compressed. And then to pass A fidelity test. Or does it feel At all, and if it does Would I squeal Or be just as I was? If not undone I’d become as I’d begun And write a memorandum On being random. Had it felt like a belt Of rum? And might it be predicted Longing for my session Of compression? I’d crawl To Ben again. And call, Don’t stall! Make me small!” This entry was posted in Reflections and tagged , by preskill. Bookmark the permalink. I am a theoretical physicist at Caltech, and the Director of the Institute for Quantum Information and Matter. Follow me on Twitter @preskill. ## 14 thoughts on “Who named the qubit?” 1. Pingback: Who named the qubit? \| Φυσ&io... 2. John, your poem is fantastic. One of these days I’m going to post my collection of your after-colloquium puns. • Thanks, Steve! Just the funny puns, I hope … 3. Some years ago, I asked Benjamin Schumacher, among a lot of other … celebrities ,,, in quanta, the following question, to which I have NOT yet gotten any answer from ANY of them : 🙂 🙂 🙂 Asher Peres, considered to be one of the fathers of quantum information, showed in his often cited 1995 book on quanta that one can build the whole of quantum theory WITHOUT the use of the Heisenberg uncertainty … Details can be found in : “Heisenberg’s Uncertainty : an Ill-Defined Notion ?” Amusingly, no matter how often that book is cited, it appears that NO ONE has bothered to look even at its … back cover … where in pretty large letters it is written that the Heisenberg Uncertainty is an … ill-defined notion … 🙂 🙂 🙂 Well, ever since, neither Schumacher, nor any other … quantum specialist … managed to give even the vaguest explanation of WHAT may be going on … 🙂 🙂 🙂 One more example of the fact that, well, quanta are NOT for … everybody … • Dear Prof. Rosinger, On page 426 of Peres’ book, Heisenberg’s uncertainty principle is explicitly used to illustrate that classical reality emerges when our observations have uncertainties far beyond the quantum limit given by Heisenberg. Moreover, the uncertainty principle itself is not needed for quantum physics. It is a mathematical theorem about a lower bound in the product of the standard deviations of expectation values of complementary observables (an observable and its Fourier transform)[see Exercise 11.30 here: https://www.math.ucdavis.edu/~hunter/book/ch11.pdf ]. This theorem holds for all L_2 spaces. Quantum mechanics is formulated on such spaces, hence the uncertainty principle is a consequence, not a prerequisite, of the internal structure of these mathematical spaces. If you were to ask why quantum mechanics is formulated on L_2 spaces, that is a very interesting question whose answer may involve Dvoretzky’s theorem (random, low dimensional subspaces of high-dimensional L_p -normed spaces look a lot like L_2 spaces). Peres himself spends some time discussing the idea that macroscopic reality looks classical only because we can keep track of (measure) a tiny number of (macroscopic) degrees of freedom, say 10, in a system with 10^30 degrees of freedom. In Peres’ own view, that dimensional reduction generates classical ‘pointer states’ which are stable to quantum fluctuations and can be measured repeatedly by independent observers because of huge redundancy in the information they encode in their nearly identical subsystems (see also work on “Quantum Darwinism”, by W. H. Zurek). As Peres himself notes on p. 426 of “Quantum Theory: Concepts and Methods”: “We can perform only a negligible fraction of all conceivable tests, and we therefore end up with a partial knowledge of $\rho_i$ (our lack of knowledge is represented by an appropriately weighted mixture of all admissible possibilities). The accessible data usually are macroscopic variables, such as the position and velocity of the center of mass of an object. For these macroscopic variables, we have $\Delta p \Delta q >> \hbar$. Compared to these classical uncertainties, disturbances caused by quantum measurements are negligibly small. Moreover, macroscopic masses are large so that corresponding wavelengths are small and diffraction effects can be neglected. Therefore consecutive measurements of position have predictable results.” However, it does not touch in any way upon the MAIN issue, namely that Asher Peres calls in his 1995 book the Heisenberg Uncertainty to be an … ill-conceived notion … 🙂 🙂 🙂 • I suppose, some points about not very accurate treatment of uncertainty relation are discussed in chapter 4-3 of Peres’ book, e.g., see Exercise 4.16 on page 94 (“Find three textbooks on quantum mechanics with the wrong uncertainty relation (4.56), and the one with correct version”) and text before that. Yet 4.56 is not usual uncertainty relation, it is for angular variable. • Sorry, you, too, AVOID the MAIN issue : Asher Peres calls the Heisenberg Uncertainty to be an … ill-conceived notion … So then, why do you NOT comment on that ??? 🙂 🙂 🙂 • Why his suggestion to find three textbooks with a wrong kind of a uncertainty relation is not relevant? • Do you agree with Asher Peres that the Heisenberg Uncertainty Principle is an … ill-conceived notion … ? Is your command of English language to fault for NOT replying to this question ? Or you simply do NOT want to reply ? 🙂 🙂 🙂 Don’t worry : this is not a police interrogation : you are under NO obligation to reply … As for being HONEST, well, this seems to be a less and less popular approach to more delicate issues even in science … 🙂 🙂 🙂 • Dear Prof. Rosinger, maybe indeed some “lost in translation” with word “ill-conceived” happens, but I did answer your question and even pointed page number. I agree with existence of difficulties mentioned by Peres there. 4. By the way, another question, if you do not mind : In a recently published paper, Prof. Hans de Raedt and collaborators did prove that the celebrated Bell Inequalities are IRRELEVANT in physics, since they are NOT violated either by classical, or by quantum systems. And the illusion that they would be violated by quantum systems is due to an elementary error in dealing with statistical data. Details in this regard can be found in the recently published paper : “The irrelevance of Bell inequalities in Physics : Comments on the DRHM paper”	{}
Date: Tue, 6 Jan 2004 23:21:06 +0200 Reply-To: Hebrew TeX list <[log in to unmask]> Sender: Hebrew TeX list <[log in to unmask]> From: Amir Seginer <[log in to unmask]> Subject: Re: Hebrew and Wrapping text around figures (picins.sty) In-Reply-To: <[log in to unmask]> Content-Type: TEXT/PLAIN; charset=US-ASCII On Tue, 6 Jan 2004, Ron Artstein wrote: > On Tue, 6 Jan 2004, Amir Seginer wrote: > > When language-switching commands are only issued outside the > enumerate environments, the left-to-right environments behave > as they should but the Hebrew ones are still problematic. > Here's a hack to fix this. > > Immediately after the line: > > \def\ivparpic(#1,#2)(#3,#4)[#5][#6]#7{% > > add the following line: > > \ifnum\@listdepth>0\if@rl\hsize\linewidth\fi\fi > Thanks, it all works now. Do you think that an email shuld be sent to the person maintaining the picins package? I'm willing to do this (giving due credit), if there is no objection. Thanks again, Amir. P.S. > I chose not to look into because I thought that mixed-direction > lists are not really needed (please correct me if I'm wrong). No, I do not see a need for mixed-direction lists (it wasn't realy necessary in my example file). Back to: Top of message \| Previous page \| Main IVRITEX page LISTSERV.TAU.AC.IL	{}
Last edited by Arashishakar Tuesday, July 21, 2020 \| History 3 edition of Plasmalemma and tonoplast found in the catalog. Plasmalemma and tonoplast International Workshop on Plasmalemma and Tonoplast of Plant Cells (1981 Strasbourg) # Plasmalemma and tonoplast ## by International Workshop on Plasmalemma and Tonoplast of Plant Cells (1981 Strasbourg) • 187 Want to read • 27 Currently reading Published by Elsevier Biomedical in Amsterdam, Oxford . Written in English Subjects: • Plant cell membranes -- Congresses. • Edition Notes Includes index. Classifications The Physical Object Statement editors D. Marmé, E. Marrè and R. Hertel. Series Developments in plant biology -- v.7 Contributions Marmé, D., Marrè, E., Hertel, R. LC Classifications QK725 Pagination x,446p. : Number of Pages 446 Open Library OL21434404M ISBN 10 0444804099, 0444800816 Citrate Uptake into Tonoplast Vesicles from Acid Lime (Citrus aurantifolia) Juice Cells Article (PDF Available) in Journal of Membrane Biology (3) January with 62 Reads. Experimental Botany. An International Series of Monographs, Vol. By D. A. Charles‐Edwards DIFFERENTIATION IN VITRO. Proceedings of the Fourth Symposium of the British Society for Cell Biology. Edited by M. M. Yeoman & D. E. S. Truman PLASMALEMMA AND TONOPLAST: THEIR FUNCTIONS IN THE PLANT CELL. Plasma Membrane Definition. The plasma membrane of a cell is a network of lipids and proteins that forms the boundary between a cell’s contents and the outside of the cell. It is also simply called the cell main function of the plasma membrane is to protect the cell from its surrounding environment. This is the first book covering all aspects of the plant plasma membrane, thus presenting a comprehensive overview of the entire field. For everyone involved in plant cell research this book will serve equally well as a valid handbook. For each subject area an internationally recognized expert has been chosen to give his/her personal view of the present state of knowledge in the . The function of the tonoplast that keeps the vacuolar interior acidic, versus a neutral/slightly basic environment in the cytosol, is shared with the plasmalemma. The disruption of the H + gradient between the vacuole and the cytoplasm is indicative of a Cited by: Polyamine content (PAs) often changes in response to abiotic stresses. It was shown that the accumulation of PAs decreased in roots treated for 24 h with mM NaCl. The role of polyamines (putrescine - PUT, spermidine - SPD and spermine - SPM) in the modification of the plasma membrane(PM) H+-ATPase (EC ) and the vacuolar(V) H+-ATPase (EC Cited by: You might also like value of travel time for passenger cars value of travel time for passenger cars The customary law of the renowned Kingdom of Hungary The customary law of the renowned Kingdom of Hungary S.H.A.M. Pinafore S.H.A.M. Pinafore The 2000 Import and Export Market for Meat and Meat Preparations in South Africa (World Trade Report) The 2000 Import and Export Market for Meat and Meat Preparations in South Africa (World Trade Report) No medals No medals The kite runner The kite runner rifle volunteers rifle volunteers Federal rulemaking Federal rulemaking How large is the satellite market How large is the satellite market origin of nations origin of nations Babys Book of Days Babys Book of Days High-tech careers in engineering on the California coast High-tech careers in engineering on the California coast Indonesia export product & producer for international market Indonesia export product & producer for international market Environment, safety, and health needs of the U.S. Department of Energy Environment, safety, and health needs of the U.S. Department of Energy Maps of Heaven (Notecubes) Maps of Heaven (Notecubes) ### Plasmalemma and tonoplast by International Workshop on Plasmalemma and Tonoplast of Plant Cells (1981 Strasbourg) Download PDF EPUB FB2 Plasmalemma and Tonoplast: Their Functions in the Plant Cell. on FREE shipping on qualifying cturer: Plasmalemma and tonoplast: Their functions in the plant cell: proceedings of the International Workshop on Plasmalemma and Tonoplast of Plant Cells Plasmalemma and tonoplast book, (Developments in plant biology) [F. Marme] on FREE shipping on qualifying offers. Read "PLASMALEMMA AND TONOPLAST: THEIR FUNCTIONS IN THE PLANT CELL (Book)., Plant Cell & Environment" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Most mature plant cells have one large vacuole that typically occupies more than 30% of the cell's volume, and that can occupy as much as 80% of the volume for certain cell types and conditions. Strands of cytoplasm often run through the vacuole. A vacuole is surrounded by a membrane called the tonoplast (word origin: Gk tón(os) + -o- meaning “stretching”, “tension”, “tone. Studies were made on the electric potentials of the plasmalemma (Eco) and tonoplast (Evc) in small cells ( mm diameter) of Valonia ventricosa. To measure Eco, microelectrodes with long tapers were inserted into the vacuole with the path of electrode entry off-center. The microelectrode then was pushed across the vacuole and into the cytoplasm on Cited by: Get this from a library. Plasmalemma and tonoplast: their functions in the plant cell: proceedings of the International Workshop on Plasmalemma and Tonoplast of Plant Cells held in Strasbourg, France, September[D Marmé; E Marrè; R Hertel;]. The plasmalemma influx is from 2 to 10 times higher than the tonoplast influx, is much greater than the SO42− reduction rate, and would not limit the rate of either. This is consistent with the finding that the plasmalemma influx is not regulated by internal SO42− or cysteine (Cram Plant Sci Lett, in press).Cited by: Plasmalemma definition is - plasma membrane. History and Etymology for plasmalemma. New Latin, from plasma + Greek lemma husk — more at lemma. Abstract. The energy in the proton gradients, $$\Delta \bar \mu _H +$$, across the plasmalemma and tonoplast is believed to be available for the transport of ions and metabolites in plant generation of this protonmotive force is most usually thought of in terms of H +-translocating ATPases (REIN-HOLD and KAPLANSANDERS ).Yet the oldest comprehensive Cited by: Botanik/Biologie der Kulturpflanzen: Was sind Plasmalemma und Tonoplast. - Plasmalemma: grenzt Cytoplasma gegen die Zellwand abTonoplast: grenzt Plasmalemma gegen Vakuole ab, Fragen und -Antworten Möseler. Measurements on 3–6 days old root hair cells of cucumber, oats and maize, in a medium of mN KCl plus mN CaCl2, gave an average d.c. surface resistance value of 3, ohmcm2 for the plasmalemma and of 3, or ohmcm2 for the plasmalemma and tonoplast in series. The average value for the combined surface resistance of the two Cited by: This book is the only comprehensive work, at introductory level, on plant cell vacuoles. Vacuoles are ubiquitous, multifaceted and indispensable organelles and yet they have been thinly treated in the literature to date. This is at odds with the amount of interest in vacuoles that has been expressed in the last two decades. This comprehensive work provides a solid foundation on. Tonoplast proteins were shown to directly interact with Ca 2+-dependent protein kinases. One example of a Ca 2+-dependent regulation of vacuolar solute passage is the selective tonoplast K + channel TPK1 that becomes phosphorylated at a specific serine residue within the N-terminus by the Ca 2+-dependent kinase by: Book Reviews. Book Reviews. Pages: ; First Published: December ; Abstract; PDF PDF Request permissions; THE MATHEMATICS OF PHOTOSYNTHESIS AND PRODUCTIVITY (Book). PLASMALEMMA AND TONOPLAST: THEIR FUNCTIONS IN THE PLANT CELL (Book). Wyn Jones; Pages: ; First Published: December ; PDF PDF. Tonoplast definition is - a semipermeable membrane surrounding a vacuole in a plant cell. With the onset of the era of membrane fractionation it seemed obvious that plasmalemma and tonoplast vlould cause the greatest problems due to the apparent lack of intrinsic markers. Again the situation for the plasmalemma appeared to be favorable; the periodic-phosphotungstic-acid reagent was considered as a specific stain. The key point of transport energization across the plasma and vacuolar membrane of fungi is the generation of the electrochemical gradient of protons by H +-ATPases of both gradient of K + is used to increase the energy capacity of the H + gradient. The regulation of ion concentrations in the yeast cytosol is based on the activity of transporters of plasmalemma Cited by: membrane [mem´brān] a thin layer of tissue that covers a surface, lines a cavity, or divides a space or organ. adj., adj mem´branous. alveolar-capillary membrane (alveolocapillary membrane) a thin tissue barrier through which gases are exchanged between the alveolar air and the blood in the pulmonary capillaries. Called also blood-air barrier and. Plasmalemma Definizione: a very thin membrane, composed of lipids and protein, that surrounds the cytoplasm of a \| Significato, pronuncia, traduzioni ed esempi. The permeability of the plasmalemma and tonoplast de-creased markedly under the action of cyclophosphamide. Maria Podbielkowska. Proteomic analysis of plasma membrane and tonoplast from the leaves of mangrove plant Avicennia officinalis Article (PDF Available) in Proteomics 14(). H. Ekkehard, O. Trentmann, Regulation of transport processes across the tonoplast, September Autophagy is a degradation pathway that recycles cell materials upon stress conditions or during specific developmental processes (in the lysosome for mammals or .He showed that two distinct proton pumping ATPases were present in plasmalemma and tonoplast, with different inhibitor characteristics. He also showed that the gradients of pH and membrane potential generated by the primary proton pump in the plasmalemma could be used to drive secondary active transport of other solutes, sugars, amino acids and Fields: Biophysics, Plant Physiology, Metabolic .Fluorescent reporter proteins for the tonoplast and the vacuolar lumen identify a single vacuolar compartment in Arabidopsis cells. Hunter PR(1), Craddock CP, Di Benedetto S, Roberts LM, Frigerio L. Author information: (1)Department of Biological Sciences, University of Warwick, Coventry CV4 7AL, United by:	{}
5:16 AM @YashasSamaga excellent idea :-) There was one already I see.... hmmm We love Physics, Chemistry and Mathematics. And we are prepari... but that one is in math se 3 hours later… 7:48 AM Hii @YashasSamaga Which room should we use 8:01 AM it doesn't matter but let's keep the JEE spam out of the h-bar I don't understand whats the problem the conversations go very long and we keep asking countless questions which are related to few of us only the convos get mixed up Welcome @0celo7 8:35 AM @YashasSamaga Isn't the same applicable for most other conversations in hbar? When Balarka, 0celot and Slereah talk about differential geometry even then very few people understand. And what do you mean by "JEE spam" ? If you don't respect the subjects you are learning then there is no point in learning. If differential geometry, music, cooking can all have intense discussions in the hbar so can we. I am strongly against dividing the room into topic based fragments. Agree 8:56 AM Hii @anonymous You there @YashasSamaga The option where you won't lose something as gas will be the answer Yeah But how to find that You lose Cl2 in NaCl You lose NO in AgNO3 You lose O2 in CuSO4 You lose O2 in K2SO4 I am still thinking In solution it is given as This wasn't a JEE Main question, right? 9:08 AM K2SO4 is a salt of a strong acid qnd strong base . So the electrolysis of K2SO4 is the electrolysis of water OMG You won't get O2 in K2SO4 ^ from K2SO4 @YashasSamaga it came in one of my jee main test paper you get H2 and O2 from water I'm still confused though nvm I just messed this up I suck at Inorganic chem I need to start revising The last time I studied chem was in September I hate inorganic chemistry me too too much to remember he is studying chem atm 9:13 AM He is not visiting this room @Koolman ask anonymous in the h bar Yay! ??? anonymous joined, waiting for his answer I'm going to start a chem revision from tmrow 9:28 AM @Koolman: write down the reactions at the anode and cathode and it should be obvious which ones don't generate a net change in the H+/OH- ratio. does the ratio change? NaCl NaCl + H20 -> NaOH + HCl @JohnRennie it will depend on petential values (E°) the OH and H stay as OH- and H2O+ ions in K2SO4, H+ and OH- combine to form H2 and O2? hmm you get Cl2 gas in NaCl too :/ 9:32 AM @Koolman yes, but there are only a few of them to learn. For example take copper sulphate. The copper plates on the electrode i.e. $Cu^+ + e \rightarrow Cu$ so there is no change in pH. At the other electrode $OH^-$ is reduced not $SO_4^-$ so we end up with $H_2SO_4$ being produced. We can solve it easily if we know E° values @YashasSamaga @JohnRennie so we have to learn them The questions will normally only involve the more common ions so it's easy to remember in what order they lie i.e. which of them is oxidised or reduced. @Koolman if the question gives you the electrode potentials then that's great. If the question doesn't give you the electrode potentials then yes you have to learn them. But there aren't many to learn. @JohnRennie so what does the solution mean How does it explains the answer @Koolman are you happy that electrolysis of copper sulphate does change the pH as I explained above? @Koolman I was thinking of net charge in the solution. It is about pH lol Now everything makes sense 9:37 AM @JohnRennie yeah In NaCl, you get NaOH, in CuSO4, you get H2SO4, in AgNO3, you get Ag(OH)2 It would be acidic I need to start studying chemistry that was really stupid of me to think about charges in this question OK. Silver and copper are pretty similar, and sulphate and nitrate are pretty similar. So silver nitrate behaves in a similar way to copper sulphate. The silver plates out and at the other electrode oxygen is given off. We end up nitric acid. OK so far? Yes 9:39 AM Oh, you don't get Ag(OH)2... -.- @YashasSamaga we will get it According to electrode potentials Now sodium chloride. At the anode chlorine is reduced and bubbles off $$Cl^- \rightarrow Cl + e$$ So no pH change there. At the cathode water reacts instead of the $Na^+$ so we get $OH^-$ produced. So we end up with NaOH and the pH goes up. OK so far? @JohnRennie chlorine will oxidise Since I wasn't sure I Googled it, and the anode reaction reduces $Cl^-$ to gaseous chlorine. If it get reduced it will take electrons 9:46 AM Oops sorry! Yes, the oxidation state goes from -1 to 0 so it gets oxidised. Oh well, it was 35 years ago I last studied chemistry :-) Np huh I haven't studied chemistry since 6 months and I've forgotten everything except organic chem So it just remains to look at potassium sulphate. Can you give the half reactions for that? does that get electrolyzed? before all the water has been electrolyzed At cathode $2e^- + 2H_2O \rightarrow H_2 + 2OH-$ 9:50 AM OK ... and at the anode? At anode $2H_2 O \rightarrow O_2 + 4H^+ +4e^-$ So the overall reaction is? $2H_2 O \rightarrow O_2 + H_2$ And does that change the pH? ($2H_2$ not $H_2$) Nah @JohnRennie sorry 9:55 AM So there's your answer. Potassium sulphate is the only case where the pH doesn't change :-) Yeah I got that I want to what they mean by the solution given To answer the question you have to just remember the relative reactivities of the ions involved, but that isn't too hard as there aren't that many ions that will be mentioned in this type of question. Yeah @Koolman I'm not sure what you're asking ... 48 mins ago, by Koolman K2SO4 is a salt of a strong acid qnd strong base . So the electrolysis of K2SO4 is the electrolysis of water 9:58 AM Hmm, I'd say that's not a helpful statement. Yeah i also think that Though I suppose electrolysis of an acid stronger than water normally produces $H_2$ and electrolysis of a base stonger than water normally produces $O_2$. May be The way to approach questions like that is to work out what the half reactions are. Once you've done that the answer should be obvious. Yeah 4 hours later… 1:42 PM I tried it as (number of ways selecting 4 girls out of 5)(arranging them on their seats )(ways we can find four consecutive sear for these girls)(selecting 10 seats from the remaining 12 seats )(arranging students in these 10 seats) $=(^5 _4)(4!)(4)(^{12}_{10})(10!)$ = $^{11}P_6(6!)(2)$ But the answer is not this Any idea @YashasSamaga That's what I was doing. It is lengthy. 3 cases But what mistake I have done (number of ways selecting 4 girls out of 5)(arranging them on their seats )(ways we can find four consecutive sear for these girls) is not related to (selecting 10 seats from the remaining 12 seats )(arranging students in these 10 seats) and the first is wrong too you don't consider the case where you pick random 3 girls and a girl comes in the 12C10 selection Why , I have first arranged the girls , then the remaining you are missing a case 1:53 PM Which case GGGG_ you do 5C4 to pick the 4 girls you don't count a case where GGGB_ and the girl takes that place you have to do it case by case @YashasSamaga it cannot be the case , as four girl should sit togther The problem is you need 4 girls to be together and you have 5 girls you are undercounting in your method What oh wait you are overcounting not undercounting I'm confused now 1:59 PM My answer is smaller than that is given I am totally confused GGGG_ _GGGG are two different cases if you compensate for that, you'll overcount :/ Nope The correct answer is given above btw 46/2 = 23 and you don't have anything greater than 12! so you can never get the correct answer by multiplying or subtract 2:08 PM No there should be some mistake I have done As 23 is a factor of the answer, it is impossible to get the answer by multiplying numbers. As I said, you did not consider this case GGGG_ _GGGG you can place 4 girls inside a van in two different ways but now you'll over count @YashasSamaga its not 23 23 is a prime 232 = 46 @YashasSamaga i have considered that by multiplying it by 4 and it isn't possible to get that factor you need to add/subtract numbers somewhere 2:10 PM In my it is 6! 2:24 PM It is very easy. Answer is $(66)(10!)(4!)(4)(5)+(2)(55)9!(5)!-(55)(5)(4)!(4)(9)!$ $=110170368000$ @anonymous mistake in my method @Koolman @Koolman You overcounted. What First check if my answer is okay 36 mins ago, by Koolman 2:26 PM (You had to remove the cases where the 5th girl is by mistake grouped with the arranged 4 girls) Why It has asked atleast four girls Does my answer match with (1) ? @Koolman I don't have a calculator I did the same idk if mine is correct but I used the same method which his solution showed 2:28 PM Great I placed 4 girls and then did the case with 5 girls and then removed the overcounting When the 5th girl is placed by mistake with the other 4 (in case 1) Just number the seats from 1 to 16 and you're done! There are only 4 ways in which 4 girls can stay together @anonymous i have also counted that And 2 ways in which 5 girls can stay together Then remove the overcounting at the end @YashasSamaga Did you get the answer as in the picture? 2:31 PM I haven't checked If it matches the answer key it should be correct but I was using the same method which was given in his solution Check it on a calculator Then it should be correct @YashasSamaga It was a easy sum Explain your method to Koolman then I have a test tomorrow Gotta go 2:33 PM I got yours method Want to know my mistake you are overcounting But my answer is smaller than given no it is greater what nubmers did you use? let me write ur equation 5C4 4! * 2! * 2! * 12C10 * 10!? in that you are overcounting this case G1G2G3G4_ _G2G3G4G5 note that the first _ can be G5 and the second _G1 you count it twice Oh god Yeah @YashasSamaga Got it Thanks a lot @YashasSamaga I should have used numbers earlier ,-,	{}
# American Institute of Mathematical Sciences May 2010, 9(3): 761-778. doi: 10.3934/cpaa.2010.9.761 ## Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity 1 Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241, China Received May 2009 Revised October 2009 Published January 2010 We consider the sub- or supercritical Neumann elliptic problem $-\Delta u + \mu u = u^{\frac{N + 2}{N - 2} + \varepsilon}, u > 0$ in $\Omega; \frac{\partial u}{\partial n} = 0$ on $\partial \Omega, \Omega$ being a smooth bounded domain in $R^N, N \ge 4, \mu > 0$ and $\varepsilon \ne 0$. Let $H(x)$ denote the mean curvature at $x$. We show that for slightly sub- or supercritical problem, if $\varepsilon \min_{x \in \partial\Omega} H(x) > 0$ then there always exist arbitrarily many solutions which blow up at the least curved part of the boundary as $\varepsilon$ goes to zero. Citation: Liping Wang. 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Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9 2018 Impact Factor: 0.925 ## Metrics • PDF downloads (8) • HTML views (0) • Cited by (2) ## Other articlesby authors • on AIMS • on Google Scholar [Back to Top]	{}
A Vector Space of Polynomials As we continue our study of linear algebra, we begin to focus on sets with properties we’ve seen displayed by our matrices. Looking at other sets that have similar properties, we arrive at the topic of vector spaces. Definition Let $$V$$ be a set on which two operations (vector addition and scalar multiplication) are defined. $$V$$ is a vector space over $$F$$, if for every $$\mathbf{u},\mathbf{v},\mathbf{w} \in V$$ and scalars $$c,d \in F$$ we have 1. $$\mathbf{u}+\mathbf{v} \in V$$, 2. $$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$$, 3. $$\mathbf{u}+(\mathbf{v}+\mathbf{w})=(\mathbf{u}+\mathbf{v})+\mathbf{w}$$, 4. $$V$$ has a zero vector, $$0$$, such that, for every $$\mathbf{u} \in V$$, $$\mathbf{u}+0=\mathbf{u}$$. 5. For every $$\mathbf{u} \in V$$, there exists a $$-\mathbf{u}$$ such that $$\mathbf{u}+(-\mathbf{u})=0$$. 2. Scalar Multiplication: 1. $$c\mathbf{u} \in V$$, 2. $$c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}$$, 3. $$(c+d)\mathbf{u}=c\mathbf{u}+d\mathbf{u}$$, 4. $$c(d\mathbf{u})=(cd)\mathbf{u}$$, 5. $$1\mathbf{u}=\mathbf{u}$$. The set of polynomials of degree 3 or less Now that we have the formal definition of a vector space, we will need to be able to show that a set is a vector space. Therefore, we will work through showing the following. Let $$P_{3}$$ be the set of all polynomials of degree 3 or less. Then, with the addition and multiplication defined in the usual way, $$P_{3}$$ forms a vector space over $$\mathbb{R}$$. In order to show that $$P_{3}$$ is indeed a vector space, we will need to show all of the properties given above. As we work through this, we will let $$p(x)=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}$$, $$q(x)=q_{3}x^{3}+q_{2}x^{2}+q_{1}x+q_{0}$$ and $$r(x)=r_{3}x^{3}+r_{2}x^{2}+r_{1}x+r_{0}$$ be in $$P_{3}$$ with each $$p_{i},q_{j}$$ and $$r_{k}$$ being a real numbers. Furthermore, let $$c,d$$ be real numbers. We then have the following. Closure Note that \begin{align} &p(x)+q(x) \\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}+q_{3}x^{3}+q_{2}x^{2}+q_{1}x+q_{0} \\ &=(p_{3}+q_{3})x^{3}+(p_{2}+q_{2})x^{2}+(p_{1}+q_{1})x+(p_{0}+q_{0}). \end{align} Note that $$p_{i}+q_{i} \in \mathbb{R}$$ since $$p_{i}, q_{i} \in \mathbb{R}$$ for $$i \in \mathbb{Z}$$ and $$0 \leq i \leq 3$$. Therefore, the coefficients before each of the powers of $$x$$ are real numbers. Hence, $$p(x)+q(x)$$ is again a polynomial of degree at most 3. As a further note, the degree would be less than 3 if $$p_{3}+q_{3}=0$$; however, such a polynomial would still be in $$P_{3}$$. Commutative Using the same notation, we find that \begin{align} &p(x)+q(x)\\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}+q_{3}x^{3}+q_{2}x^{2}+q_{1}x+q_{0} \\ &=(p_{3}+q_{3})x^{3}+(p_{2}+q_{2})x^{2}+(p_{1}+q_{1})x+(p_{0}+q_{0}) \\ &=(q_{3}+p_{3})x^{3}+(q_{2}+p_{2})x^{2}+(q_{1}+p_{1})x+(q_{0}+p_{0}) \\ &=q_{3}x^{3}+q_{2}x^{2}+q_{1}x+q_{0}+p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0} \\ &=q(x)+p(x). \end{align} Note the third and fourth lines follow due to commutativity of addition and the distributive property of real numbers. Hence, addition of polynomials is commutative. Associative We now note that \begin{align} &(p(x)+q(x))+r(x) \\ &=(p_{3}+q_{3})x^{3}+(p_{2}+q_{2})x^{2}+(p_{1}+q_{1})x+(p_{0}+q_{0})+r_{3}x^{3}+r_{2}x^{2}+r_{1}x+r_{0} \\ &=((p_{3}+q_{3})+r_{3})x^{3}+((p_{2}+q_{2})+r_{2})x^{2}+((p_{1}+q_{1})+r_{1})x+(p_{0}+q_{0})+r_{0} \\ &=(p_{3}+(q_{3}+r_{3}))x^{3}+(p_{2}+(q_{2}+r_{2}))x^{2}+(p_{1}+(q_{1}+r_{1}))x+p_{0}+(q_{0}+r_{0}) \\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}+(q_{3}+r_{3})x^{3}+(q_{2}+r_{2})x^{2}+((q_{1}+r_{1})x+q_{0}+r_{0} \\ &=p(x)+(q(x)+r(x)). \end{align} Note that third line follows from associativity of real number addition. Hence, the addition of polynomials is associative. Zero Vector Note that $$0$$ (the real number) is a constant function, as is therefore a polynomial of degree 0. Hence $$0 \in P_{3}$$. Furhtermore, \begin{align} &p(x)+0 \\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}+0 \\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+(p_{0}+0) \\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0} \\ &=p(x). \end{align} Therefore, there is a additive identity for polynomials (the constant 0) of degree 3 or less. Let $$s(x)=(-p_{3})x^{3}+(-p_{2})x^{2}+(-p_{1})x+(-p_{0})$$. Note that $$-p_{i} \in \mathbb{R}$$ since $$p_{i} \in \mathbb{R}$$ for $$i \in \mathbb{Z}$$ and $$0 \leq i \leq 3$$. Therefore, $$s(x) \in P_{3}$$. Furthermore, \begin{align} &p(x)+s(x)\\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}+(-p_{3})x^{3}+(-p_{2})x^{2}+(-p_{1})x+(-p_{0}) \\ &=(p_{3}-p_{3})x^{3}+(p_{2}-p_{2})x^{2}+(p_{1}-p_{1})x+(p_{0}-p_{0}) \\ &=0x^{3}+0x^{2}+0x+0 \\ &=0. \end{align} Hence, $$s(x)$$ is the additive inverse of $$p(x)$$. Therefore, every polynomial has a $$-p(x)$$ such that $$p(x)+(-p(x))=0$$. Scalar Multiplication Closure Here we note that \begin{align} cp(x)&=c(p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}) \\ &=cp_{3}x^{3}+cp_{2}x^{2}+cp_{1}x+cp_{0}. \end{align} Furthermore, $$c p_{i} \in \mathbb{R}$$ since $$c, p_{i} \in \mathbb{R}$$ for all $$i \in \mathbb{K}$$ with $$0 \leq i \leq 3$$. Hence, $$cp(x) \in P_{3}$$. Distributive We now have that \begin{align} (c+d)p(x)&=(c+d)(p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}) \\ &=(c+d)p_{3}x_{3}+(c+d)p_{2}x^{2}+(c+d)p_{1}x+(c+d)p_{0} \\ &=(cp_{3}+dp_{3})x^{3}+(cp_{2}+dp_{2})x^{2}+(cp_{1}+dp_{1})x+cp_{0}+dp_{0} \\ &=cp_{3}x^{3}+cp_{2}x^{2}+cp_{1}x+cp_{0}+dp_{3}x^{3}+dp_{2}x^{2}+dp_{1}x+dp_{0} \\ &=cp(x)+dp(x). \end{align} Hence, you can distribute across scalar addition. Associative Here, we see that \begin{align} (cd)p(x)&=(cd)(p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}) \\ &=(cd)p_{3}x_{3}+(cd)p_{2}x^{2}+(cd)p_{1}x+(cd)p_{0} \\ &=c(dp_{3})x^{3}+c(dp_{2})x^{2}+c(dp_{1})x+c(dp_{0}) \\ &=c((dp_{3})x^{3}+(dp_{2})x^{2}+(dp_{1})x+(dp_{0})) \\ &=c(dp(x)). \end{align} Therefore, scalar multiplication is associative. Identity Note that $$1$$ is a real number. Furthermore, we have that \begin{align} 1p(x)&=1(p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}) \\ &=1p_{3}x^{3}+1p_{2}x^{2}+1p_{1}x+1p_{0} \\ &=p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0} \\ &=p(x). \end{align} Therefore, $$1$$ acts as an identity for scalar multiplication in $$P_{3}$$. Conclusion We have now shown that the set of polynomials of degree 3 or less satisfied all of the required properties for a vector space when using the usual addition and multiplication. Hence, we have proving that $$P_{3}$$ is a vector space. While it is helpful to know that $$P_{3}$$ is a vector space, note that we would use a very similar process when trying to prove that any set forms a vector space. That is, we would have to work through each of the properties one by one in order to ensure that all of them are satisfied. For further work, we could also show that $$P_{n}$$ is a vector space for any $$n \in \mathbb{N}$$. As always, I hope you learned something and enjoyed the process. If you did, make sure to like the post and share it on Social Media. 1 thought on “A Vector Space of Polynomials” This site uses Akismet to reduce spam. Learn how your comment data is processed.	{}
# drawing sprites ## Recommended Posts phil67rpg 443 void TimerFunction(int value) { glutPostRedisplay(); glutTimerFunc(1000, TimerFunction, 1); } void drawScene() { glClear(GL_COLOR_BUFFER_BIT); drawScene_bug(); TimerFunction(1); eraseScene_bug(); // drawScene_bug_two(); // eraseScene_bug_two(); drawScene_ship(); drawScene_bullet(); glutSwapBuffers(); } #### Share this post ##### Share on other sites Endurion 5412 Is there a question coming with this? #### Share this post ##### Share on other sites Khatharr 8812 Posted (edited) Nope. This is bat country. Edited by Khatharr #### Share this post ##### Share on other sites phil67rpg 443 sorry but the question is how do I draw a sprite and then wait for a period of time and then draw another sprite over it. #### Share this post ##### Share on other sites phil67rpg 443 also I want to erase the sprite after I draw it. #### Share this post ##### Share on other sites Lactose 11471 10 minutes ago, phil67rpg said: sorry but the question is how do I draw a sprite and then wait for a period of time and then draw another sprite over it. Which part(s) of what you said are you having problems with? 1) Drawing the sprite. 2) Waiting for a period of time. 3) Drawing another sprite on top of it? 5 minutes ago, phil67rpg said: also I want to erase the sprite after I draw it. This was explained to you in a previous thread you made. Was the explanation not clear? If so, what was unclear about it? #### Share this post ##### Share on other sites phil67rpg 443 lactose I am having a problem with the wait period #### Share this post ##### Share on other sites Lactose 11471 1 minute ago, phil67rpg said: lactose I am having a problem with the wait period Josh's reply covers that part. #### Share this post ##### Share on other sites phil67rpg 443 27 minutes ago, jpetrie said: auto elapsed = timeStampNow - timeStampPrevious; I get an error with the minus sign(-) no operand matches these operators #### Share this post ##### Share on other sites jpetrie 13162 27 minutes ago, phil67rpg said: I get an error with the minus sign(-) no operand matches these operators You're going to have to provide more information about the actual code you wrote, and the exact text of the error, if anybody's going to be able to help you with that. If you are using chrono like I suggested, you should look at the example code for now() usage. You can subtract two time_point's, which is what now() returns, and you'll get a duration. It's possible you aren't including the correct headers to use those types. Or it's also possible your compiler doesn't support C++11, in which case unfortunately chrono won't be available to you and you'll have to use something platform-specific like QueryPerformanceCounter on Windows. #### Share this post ##### Share on other sites phil67rpg 443 here is the code I am using void drawScene() { glClear(GL_COLOR_BUFFER_BIT); drawScene_bug(); auto timeStampNow = chrono::high_resolution_clock::now(); auto elapsed = timeStampPrevious - timeStampNow; eraseScene_bug(); // drawScene_bug_two(); // eraseScene_bug_two(); drawScene_ship(); drawScene_bullet(); auto timeStampPrevious = timeStampNow; glutSwapBuffers(); } [code] sorry void drawScene() { glClear(GL_COLOR_BUFFER_BIT); drawScene_bug(); auto timeStampNow = chrono::high_resolution_clock::now(); auto elapsed = timeStampPrevious - timeStampNow; eraseScene_bug(); // drawScene_bug_two(); // eraseScene_bug_two(); drawScene_ship(); drawScene_bullet(); auto timeStampPrevious = timeStampNow; glutSwapBuffers(); } #### Share this post ##### Share on other sites jpetrie 13162 Are you including the appropriate headers? What is the exact line of the error message and what is the exact error message? #### Share this post ##### Share on other sites phil67rpg 443 I am using #include <chrono> which seems to be working. this is the error: 8 IntelliSense: no operator "-" matches these operands operand types are: int - std::chrono::time_point<std::chrono::system_clock, std::chrono::system_clock::duration> c:\Users\phil6_000\Desktop\main.cpp 644 35 space5 #### Share this post ##### Share on other sites jpetrie 13162 You're trying to subtract an integer (int) and a std::time_point<>. You can't. You have to subtract two time_points. "timeStampPrevious" sounds like it is an int, and it shouldn't be. It should be of type std::chono::time_point<std::chono_high_resolution_clock>, which is the return type of now(). As shown in the documentation you've been linked several times. #### Share this post ##### Share on other sites phil67rpg 443 Spoiler auto timeStampPrevious=chrono::time_point<chrono_high_resolution_clock>; the above code has an error of typename is not allowed. I am almost figured out this piece of code. thanks jpetrie. I am really trying to figure out this problem. I am going to do more research on this problem. Just now, phil67rpg said: chrono_high_resolution_clock here is where the error is 1 minute ago, phil67rpg said: chrono::time_point sorry the error is also here as well #### Share this post ##### Share on other sites Lactose 11471 1. Do you know how "auto" works? 2. Where is the "auto timeStampPrevious=chrono::time_point<chrono_high_resolution_clock>;" line located? 3. Where did Josh tell you timeStampPrevious variable should be? #### Share this post ##### Share on other sites phil67rpg 443 chrono::high_resolution_clock::time_point previous = chrono::high_resolution_clock::now(); auto timeStampPrevious = previous; auto timeStampNow = chrono::high_resolution_clock::now(); auto elapsed = timeStampPrevious - timeStampNow; #### Share this post ##### Share on other sites Lactose 11471 That doesn't answer any of my questions. #### Share this post ##### Share on other sites phil67rpg 443 thanks jpetrie I finally figured out my problem. I had to do some research. #### Share this post ##### Share on other sites phil67rpg 443 I did some research on auto but I am still confused on how to output the elapsed variable to the screen. I am trying to cout the elapsed variable. chrono::high_resolution_clock::time_point previous = chrono::high_resolution_clock::now(); auto timeStampPrevious = previous; auto timeStampNow = chrono::high_resolution_clock::now(); auto elapsed = timeStampPrevious - timeStampNow; cout << elapsed << endl; #### Share this post ##### Share on other sites Kylotan 10012 You need to be more clear with the problem you're facing. What does the code you posted do? How does the output differ from what you expect? #### Share this post ##### Share on other sites LennyLen 5717 Have you ever considered using a library such as Allegro, SDL or SFML? They take care of how to do most of the things you ask about, which allows you to focus on what you want to do instead. #### Share this post ##### Share on other sites phil67rpg 443 here is my error 1 IntelliSense: no operator "<<" matches these operands operand types are: std::ostream << std::chrono::duration<std::chrono::system_clock::rep, std::chrono::system_clock::period> c:\Users\phil6_000\Desktop\cpp24a\cpp24a\cpp24a.cpp 14 7 cpp24a I am getting variable type conflict, the << operator does not work with chrono types. #### Share this post ##### Share on other sites cout << elapsed Would assume its that bit of code since its saying there << operator doesnt work with chrono types? #### Share this post ##### Share on other sites This topic is now closed to further replies. • ## Partner Spotlight • ### Similar Content • I don't know why this is happening, in the image below you can see my current situation. Basically I have an Utility.h header which contains wathever random stuff I think is useful at the time but don't yet deserves it's own separate file, so inside of it I've put a BaseFont class and defined it in place since I am not sure it will stay into Utility.h for long (Font is an alias for BaseFont because I wanted the internal variable name for the font to be Font, and that would collide with the class name, so that's why I have it setup like this) and right below this class, still in the header, I have a global object constructed from it so that I can use it everywhere for debug since pretty much all my .cpp #include "Utility.h". Now when I close the game I get an error that says "Breakout.exe stop working" and I tracked it down to the BaseFont destructor. On the right of the image you can see the local variable situation at that break point, which doesn't looks good... I have some assumptions regarding the cause but I am not quite sure, can you tell me what its going wrong exactly? FULL SIZE IMAGE • Hi guys, I'm not quite sure how to achieve this Here what I got so far, my GameMode class own an object of type Physics: class Physics { private://variables GameMode* Game; public://constructors Physics(GameMode* gameRef); ~Physics(); public://methods bool IsColliding(Entity* current, ECollisionTest collisionTestType); private://methods bool BoxCollisionTest(const Rect& current, const Rect& other); }; And since all my entities have a pointer to the GameMode they can retrieve a pointer to the Physics object, so when they move they can query the Physics object to see if they collided with something, passing "this" as the object to control against all the Entities inside GameMode, like this: //Collision Check if (PhysicsManager->IsColliding(this, ECollisionTest::EBoxCollisionTest)) { LogConsole("Collision"); } and IsColliding() goes like this: bool Physics::IsColliding(Entity* current, ECollisionTest collisionTestType) { for (auto& E : Game->Entities) { if (current != E.get())//avoid self testing { switch (collisionTestType) { case ECollisionTest::EBoxCollisionTest: { if (BoxCollisionTest(current->GetCollisionBox(), E->GetCollisionBox())) { return true; } } break; } } } return false; } Now, there are two things I don't like about my current setup. 1) I am testing against all the entities. The paddle of the Breakout game can only move left and right, and hit either a ball or the left/right boundary, makes no sense to test it against 50 bricks at every update. 2)From inside the Paddle class I am calling "IsColliding(this, ECollisionTest::EBoxCollisionTest)" so there is the assumption that all the entities that can collide with this one require a box test. May not be the case, maybe I have a "worm" entity running on the screen and it requires another type of collision test, so the choosen function to run shouldn't relying on me explicitly saying which one is, but it should be called at runtime trough overload, based on the type of Entity being compared. So maybe the Paddle<->Ball comparison would do a BoxCollisionTest, while Paddle<->Worm comparison would do something else. So how do I get it to work the way I want? This is what I want: 1)Physics object is working with Entity, and yet I need it to being able to distinguish between a Paddle and or a Brick* or a Boundary* or a Ball* (all of this inherit from entity), a bit kind of like Unreal Engine BlueprintEditor Cast node, which allow me to say "Cast to Door" and if the thing was a door it return success, otherwise fails. Can you show me an example of how such mechanism is built, in code(C++)? 2)Physics object should just call a generic TestCollision() function between the "this" passed (the current colliding object) and all the other entities, and then the overload resolution calls the appropriate function based on the types. And yet this shouldn't fail to compile for pairs for which I don't explicitly overload, for instance I wouldn't overload for Paddle<->Brick because such comparison will never happen, so when Paddle<->Brick comparison is performed, it should just return false even though I never declared a TestCollision(Paddle* p, Brick* b). How can this be achieved? • By cozzie Hi, During the journey of creating my 2nd 3D engine, I'm basing quite some of the approaches on the Game Engine Architecture book (Jason Gregory). I've now arrived on the topic: IO. In short, the target platforms for me are PC, XBone and PS4. With this main assumption I thought of the following logics/ guidelines to follow: - I can assume file/folder structures will work on all 3 platforms, when I use '/' as a folder separator -- I will define 1 global with the base folder for the application, the rest will 'inherit' from there (which can be anything, independent of the 'mount' or drive) -- for now I'll create 1 subfolder with data/files that might need write access, so later on I only have to worry about 1 subfolder (settings, configs etc.). - file extensions can be longer than 3 characters (in Linux based FreeBSD on PS4) - all class members functions needing to load a file, shouldn't have to now about the file structure of the logical application, so they will all take a full filename string including path (combining root + subfolder + filename and separators is then the responsibility of the caller/ calling code) - some functions will need to be passed a folder, because contents in that folder need to be read OR I can simply define a small list of defined subfolders (under root/ base), because it won't be more then 5 to 10 folders in total (data/shaders, data/textures, data/objects, data/sound etc.) My questions: - what do you think about this approach? - regarding the last point, which of the 2 options would you apply? -- option 2 might work fine but feels a bit 'static', not very flexible (on the other hand, would you actually need flexibility here?) Any input is appreciated, as always. • By RubenRS How do i open an image to use it as Texture2D information without D3DX11CreateShaderResourceViewFromFile? And how it works for different formats like (JPG, PNG, BMP, DDS, etc.)? I have an (512 x 512) image with font letters, also i have the position and texcoord of every letter. The main idea is that i want to obtain the image pixel info, use the position and texcoords to create a new texture with one letter and render it. Or am I wrong in something? • this is super strange... I have this function here: template<typename TFirst, typename... TArgs> bool util::LogConsole(const TFirst& first, const TArgs&... rest) { std::cout << first << " "; return LogConsole(rest...); } bool util::LogConsole() { std::cout << std::endl; return false; } it's recursive, keep printing "first" and recursively sending the rest, until rest is empty and the overload taking no argument is called, which just print a newline. Ignore the bool return. Anyway, now I have this code somewhere else: for (auto& E : Entities) { float valx = E->GetPosition().x;//return 751 float valy = E->GetPosition().y;//return 838 LogConsole(valx, valy); } Entities is a vector<unique_ptr<Entity>> which currently only contains the derived from Entity player Paddle, so I access the Base class Entity method called GetPosition, which is the center pivot coordinates of the object. I pass the variable to LogConsole which predictably prints: But not I try to call LogConsole without those 2 variables inbetween, like this: for (auto& E : Entities) { LogConsole(E->GetPosition().x, E->GetPosition().y); } and watch what kind of output I get! : ... the y value is an unreasonable value o_o How is this even possible?! Since only the second value is being affected, my assumption is that there is some weird interplay with the second template parameter, the variadic argument...but I wasn't able to debug it even stepping trough it line by line...can anyone come with an explanation for this really strange behaviour? • 9 • 11 • 9 • 10 • 14	{}
# Random number generator (included) Random number generator (included) You are encouraged to solve this task according to the task description, using any language you may know. State the type of random number generator algorithm used in a language's built-in random number generator. If the language or its immediate libraries don't provide a random number generator, skip this task. If possible, give a link to a wider explanation of the algorithm used. Note: the task is not to create an RNG, but to report on the languages in-built RNG that would be the most likely RNG used. The main types of pseudo-random number generator (PRNG) that are in use are the Linear Congruential Generator (LCG), and the Generalized Feedback Shift Register (GFSR), (of which the Mersenne twister generator is a subclass). The last main type is where the output of one of the previous ones (typically a Mersenne twister) is fed through a cryptographic hash function to maximize unpredictability of individual bits. Note that neither LCGs nor GFSRs should be used for the most demanding applications (cryptography) without additional steps. ## 8th The default random number generator in 8th is a cryptographically strong one using Fortuna, which is seeded from the system's entropy provider. An additional random generator (which is considerably faster) is a PCG, though it is not cryptographically strong. ## ActionScript In both Actionscript 2 and 3, the type of pseudorandom number generator is implementation-defined. This number generator is accessed through the Math.random() function, which returns a double greater than or equal to 0 and less than 1.[1][2] In Actionscript 2, the global random() function returns an integer greater than or equal to 0 and less than the given argument, but it is deprecated and not recommended.[3] The Ada standard defines Random Number Generation in Annex A.5.2. There are two kinds of RNGs, Ada.Numerics.Float_Random for floating point values from 0.0 to 1.0, and Ada.Numerics.Discrete_Random for pseudo-random values of enumeration types (including integer types). It provides facilities to initialize the generator and to save it's state. The standard requires the implementation to uniformly distribute over the range of the result type. The used algorithm is implementation defined. The standard says: "To enable the user to determine the suitability of the random number generators for the intended application, the implementation shall describe the algorithm used and shall give its period, if known exactly, or a lower bound on the period, if the exact period is unknown." ## ALGOL 68 Details of the random number generator are in the Revised Reports sections: 10.2.1. and 10.5.1. PROC ℒ next random = (REF ℒ INT a)ℒ REAL: ( a := ¢ the next pseudo-random ℒ integral value after 'a' from a uniformly distributed sequence on the interval [ℒ 0,ℒ maxint] ¢; ¢ the real value corresponding to 'a' according to some mapping of integral values [ℒ 0, ℒ max int] into real values [ℒ 0, ℒ 1) i.e. such that -0 <= x < 1 such that the sequence of real values so produced preserves the properties of pseudo-randomness and uniform distribution of the sequence of integral values ¢); INT ℒ last random := # some initial random number #;PROC ℒ random = ℒ REAL: ℒ next random(ℒ last random); Note the suitable "next random number" is suggested to be: ( a := ¢ the next pseudo-random ℒ integral value after 'a' from a uniformly distributed sequence on the interval [ℒ 0,ℒ maxint] ¢; ¢ the real value corresponding to 'a' according to some mapping of integral values [ℒ 0, ℒ max int] into real values [ℒ 0, ℒ 1) i.e., such that -0 <= x < 1 such that the sequence of real values so produced preserves the properties of pseudo-randomness and uniform distribution of the sequence of integral values ¢); Algol68 supports random number generation for all precisions available for the specific implementation. The prefix ℒ real indicates all the available precisions. eg short short real, short real, real, long real, long long real etc For an ASCII implementation and for long real precision these routines would appears as: PROC long next random = (REF LONG INT a)LONG REAL: # some suitable next random number #;INT long last random := # some initial random number #;PROC long random = LONG REAL: long next random(long last random); ## AutoHotkey The built-in command Random generates a pseudo-random number using Mersenne Twister "MT19937" (see documentation). ## AWK The built-in command "rand" generates a pseudo-random uniform distributed random variable. More information is available from the documentation of gawk. It is important that the RNG is seeded with the funtions "srand", otherwise, the same random number is produced. Example usage: see #UNIX_Shell ## BASIC The RND function generates a pseudo random number greater than or equal to zero, but less than one. The implementation is machine specific based on contents of the ROM and there is no fixed algorithm. ## Batch File Windows batch files can use the %RANDOM% pseudo-variable which returns a pseudo-random number between 0 and 32767. Behind the scenes this is just a call to the C runtime's rand() function which uses an LCG in this case: ${\displaystyle X_{n+1}=X_{n}\cdot 214013+2531011{\pmod {2^{15}}}}$ ## BBC BASIC The RND function uses a 33-bit maximal-length Linear Feedback Shift Register (LFSR), with 32-bits being used to provide the result. Hence the sequence length is 2^33-1, during which the value zero is returned once and all non-zero 32-bit values are each returned twice. ## Befunge The ? instruction usually uses the random number generator in the interpreter's language. The original interpreter is written in C and uses rand(). ## C Standard C has rand(). Some implementations of C have other sources of random numbers, along with rand(). ### C rand() The C standard specifies the interface to the rand() and srand() functions in <stdlib.h>. • void srand(unsigned int seed) begins a new sequence of pseudorandom integers. • int rand(void) returns a pseudorandom integer in the range from 0 to RAND_MAX. • RAND_MAX must be at least 32767. The same seed to srand() reproduces the same sequence. The default seed is 1, when a program calls rand() without calling srand(); so srand(1) reproduces the default sequence. (n1124.pdf) There are no requirements as to the algorithm to be used for generating the random numbers. All versions of rand() return integers that are uniformly distributed in the interval from 0 to RAND_MAX, but some algorithms have problems in their randomness. For example, the cycle might be too short, or the probabilities might not be independent. Many popular C libraries implement rand() with a linear congruential generator. The specific multiplier and constant varies by implementation, as does which subset of bits within the result is returned as the random number. These rand() functions should not be used where a good quality random number generator is required. #### BSD rand() Among current systems, BSD might have the worst algorithm for rand(). BSD rand() sets RAND_MAX to ${\displaystyle 2^{31}-1}$ and uses this linear congruential formula: • ${\displaystyle state_{n+1}=1103515245\times state_{n}+12345{\pmod {2^{31}}}}$ • ${\displaystyle rand_{n}=state_{n}}$ FreeBSD switched to a different formula, but NetBSD and OpenBSD stayed with this formula. (NetBSD rand.c, OpenBSD rand.c) BSD rand() produces a cycling sequence of only ${\displaystyle 2^{31}}$ possible states; this is already too short to produce good random numbers. The big problem with BSD rand() is that the low ${\displaystyle n}$ bits' cycle sequence length is only ${\displaystyle 2^{n}}$. (This problem happens because the modulus ${\displaystyle 2^{31}}$ is a power of two.) The worst case, when ${\displaystyle n=1}$, becomes obvious if one uses the low bit to flip a coin. #include <stdio.h>#include <stdlib.h> /* Flip a coin, 10 times. /intmain(){ int i; srand(time(NULL)); for (i = 0; i < 10; i++) puts((rand() % 2) ? "heads" : "tails"); return 0;} If the C compiler uses BSD rand(), then this program has only two possible outputs. The low bit manages a uniform distribution between heads and tails, but it has a period length of only 2: it can only flip a coin 2 times before it must repeat itself. Therefore it must alternate heads and tails. This is not a real coin, and these are not truly random flips. In general, the low bits from BSD rand() are much less random than the high bits. This defect of BSD rand() is so famous that some programs ignore the low bits from rand(). #### Microsoft rand() Microsoft sets RAND_MAX to 32767 and uses this linear congruential formula: • ${\displaystyle state_{n+1}=214013\times state_{n}+2531011{\pmod {2^{31}}}}$ • ${\displaystyle rand_{n}=seed_{n}\div 2^{16}}$ ### POSIX drand48() POSIX adds the drand48() family to <stdlib.h>. • void srand48(long seed) begins a new sequence. • double drand48(void) returns a random double in [0.0, 1.0). • long lrand48(void) returns a random long in [0, 231). • long mrand48(void) returns a random long in [-231, 231). This family uses a 48-bit linear congruential generator with this formula: • ${\displaystyle r_{n+1}=25214903917\times r_{n}+11{\pmod {2^{48}}}}$ ## C++ As part of the C++11 specification the language now includes various forms of random number generation. While the default engine is implementation specific (ex, unspecified), the following Pseudo-random generators are available in the standard: • Linear congruential (minstd_rand0, minstd_rand) • Mersenne twister (mt19937, mt19937_64) • Subtract with carry (ranlux24_base, ranlux48_base) • Shuffle order (knuth_b) Additionally, the following distributions are supported: • Uniform distributions: uniform_int_distribution, uniform_real_distribution • Bernoulli distributions: bernoulli_distribution, geometric_distribution, binomial_distribution, negative_binomial_distribution • Poisson distributions: poisson_distribution, gamma_distribution, exponential_distribution, weibull_distribution, extreme_value_distribution • Normal distributions: normal_distribution, fisher_f_distribution, cauchy_distribution, lognormal_distribution, chi_squared_distribution, student_t_distribution • Sampling distributions: discrete_distribution, piecewise_linear_distribution, piecewise_constant_distribution Example of use: Works with: C++11 #include <iostream>#include <string>#include <random> int main(){ std::random_device rd; std::uniform_int_distribution<int> dist(1, 10); std::mt19937 mt(rd()); std::cout << "Random Number (hardware): " << dist(rd) << std::endl; std::cout << "Mersenne twister (hardware seeded): " << dist(mt) << std::endl;} ## C# The .NET Random class says that it uses Knuth's subtractive random number generator algorithm.[4] See Java. ## CMake CMake has a random string generator. # Show random integer from 0 to 9999.string(RANDOM LENGTH 4 ALPHABET 0123456789 number)math(EXPR number "${number} + 0") # Remove extra leading 0s.message(STATUS${number}) The current implementation (in cmStringCommand.cxx and cmSystemTools.cxx) calls rand() and srand() from C. It picks random letters from the alphabet. The probability of each letter is near 1 ÷ length, but the implementation uses floating-point arithmetic to map RAND_MAX + 1 values onto length letters, so there is a small modulo bias when RAND_MAX + 1 is not a multiple of length. CMake 2.6.x has bug #9851; two random strings might be equal because they use the same seed. CMake 2.8.0 fixes this bug by seeding the random generator only once, during the first call to string(RANDOM ...). CMake 2.8.5 tries a secure seed (CryptGenRandom or /dev/urandom) or falls back to high-resolution system time. Older versions seed the random generator with time(NULL), the current time in seconds. ## Common Lisp The easiest way to generate random numbers in Common Lisp is to use the built-in rand function after seeding the random number generator. For example, the first line seeds the random number generator and the second line generates a number from 0 to 9 (setf random-state* (make-random-state t))(rand 10) Common Lisp: The Language, 2nd Ed. does not specify a specific random number generator algorithm. ## D From std.random: The generators feature a number of well-known and well-documented methods of generating random numbers. An overall fast and reliable means to generate random numbers is the Mt19937 generator, which derives its name from "Mersenne Twister with a period of 2 to the power of 19937". In memory-constrained situations, linear congruential generators such as MinstdRand0 and MinstdRand might be useful. The standard library provides an alias Random for whichever generator it considers the most fit for the target environment. ## Déjà Vu The standard implementation, vu, uses a Mersenne twister. !print random-int # prints a 32-bit random integer ## Delphi According to Wikipedia, Delphi uses a Linear Congruential Generator. Random functions: function Random : Extended; function Random ( LimitPlusOne : Integer ) : Integer; procedure Randomize; Based on the values given in the wikipedia entry here is a Delphi compatible implementation for use in other pascal dialects. {$ifdef fpc}{$mode objfpc}{$endif} interface function LCGRandom: extended; overload;inline;function LCGRandom(const range:longint):longint;overload;inline; implementation function IM:cardinal;inline;begin RandSeed := RandSeed * 134775813 + 1; Result := RandSeed;end; function LCGRandom: extended; overload;inline;begin Result := IM * 2.32830643653870e-10;end; function LCGRandom(const range:longint):longint;overload;inline;begin Result := IM * range shr 32;end; end. ## DWScript DWScript currently uses a 64bit XorShift PRNG, which is a fast and light form of GFSR. ## EchoLisp EchoLisp uses an ARC4 (or RCA4) implementation by David Bau, which replaces the JavaScript Math.random(). Thanks to him. [5]. Some examples : (random-seed "albert")(random) → 0.9672510261922906 ; random float in [0 ... 1[(random 1000) → 726 ; random integer in [0 ... 1000 [(random -1000) → -936 ; random integer in ]-1000 1000[ (lib 'bigint)(random 1e200) → 48635656441292641677...3917639734865662239925...9490799697903133046309616766848265781368 ## Elena ELENA 3.4 : import extensions. public program[ console printLine(randomGenerator nextReal). console printLine(randomGenerator eval(0,100))] Output: 0.706398 46 ## Elixir Elixir does not come with its own module for random number generation. But you can use the appropriate Erlang functions instead. Some examples: # Seed the RNG:random.seed(:erlang.now()) # Integer in the range 1..10:random.uniform(10) # Float between 0.0 and 1.0:random.uniform() For further information, read the Erlang section. ## Erlang Random number generator. The method is attributed to B.A. Wichmann and I.D.Hill, in 'An efficient and portable pseudo-random number generator', Journal of Applied Statistics. AS183. 1982. Also Byte March 1987. The current algorithm is a modification of the version attributed to Richard A O'Keefe in the standard Prolog library. Every time a random number is requested, a state is used to calculate it, and a new state produced. The state can either be implicit (kept in the process dictionary) or be an explicit argument and return value. In this implementation, the state (the type ran()) consists of a tuple of three integers. It should be noted that this random number generator is not cryptographically strong. If a strong cryptographic random number generator is needed for example crypto:rand_bytes/1 could be used instead. Seed with a fixed known value triplet A1, A2, A3: random:seed(A1, A2, A3) Example with the running time: ...{A1,A2,A3} = erlang:now(),random:seed(A1, A2, A3),...sequence of randoms usedrandom:seed(A1, A2, A3),...same sequence of randoms used Get a random float value between 0.0 and 1.0: Rfloat = random:uniform(), Get a random integer value between 1 and N (N is an integer >= 1): Rint = random:uniform(N), ## Euler Math Toolbox Bays and Durham as describend in Knuth's book. ## Factor The default RNG used when the random vocabulary is used, is the Mersenne twister algorithm [6]. But there are other RNGs available, including SFMT, the system RNG (/dev/random on Unix) and Blum Blum Shub. It's also very easy to implement your own RNG and integrate it into the system. [7] ## Fortran Fortran has intrinsic random_seed() and random_number() subroutines. Used algorithm of the pseudorandom number generator is compiler dependent (not specified in ISO Fortran Standard, see ISO/IEC 1539-1:2010 (E), 13.7.135 RANDOM NUMBER). For algorithm in GNU gfortran see https://gcc.gnu.org/onlinedocs/gfortran/RANDOM_005fNUMBER.html Note that with the GNU gfortran compiler program needs to call random_seed with a random PUT= argument to get a pseudorandom number otherwise the sequence always starts with the same number. Intel compiler ifort reinitializes the seed randomly without PUT argument to random value using the system date and time. Here we are seeding random_seed() with some number obtained from the Linux urandom device. program rosetta_random implicit none integer, parameter :: rdp = kind(1.d0) real(rdp) :: num integer, allocatable :: seed(:) integer :: un,n, istat call random_seed(size = n) allocate(seed(n)) ! Seed with the OS random number generator open(newunit=un, file="/dev/urandom", access="stream", & form="unformatted", action="read", status="old", iostat=istat) if (istat == 0) then read(un) seed close(un) end if call random_seed (put=seed) call random_number(num) write(,'(E24.16)') numend program rosetta_random ## FreeBASIC FreeBASIC has a Rnd() function which produces a pseudo-random double precision floating point number in the half-closed interval [0, 1) which can then be easily used to generate pseudo-random numbers (integral or decimal) within any range. The sequence of pseudo-random numbers can either by seeded by a parameter to the Rnd function itself or to the Randomize statement and, if omitted, uses a seed based on the system timer. However, a second parameter to the Randomize statement determines which of 5 different algorithms is used to generate the pseudo-random numbers: 1. Uses the C runtime library's rand() function (based on LCG) which differs depending on the platform but produces a low degree of randomness. 2. Uses a fast, platform independent, algorithm with 32 bit granularity and a reasonable degree of randomness. The basis of this algorithm is not specified in the language documentation. 3. Uses the Mersenne Twister algorithm (based on GFSR) which is platform independent, with 32 bit granularity and a high degree of randomness. This is good enough for most non-cryptographic purposes. 4. Uses a QBASIC compatible algorithm which is platform independent, with 24 bit granularity and a low degree of randomness. 5. Uses system features (Win32 Crypto API or /dev/urandom device on Linux) to generate pseudo-random numbers, with 32 bit granularity and a very high degree of randomness (cryptographic strength). A parameter of 0 can also be used (and is the default if omitted) which uses algorithm 3 in the -lang fb dialect, 4 in the -lang qb dialect and 1 in the -lang fblite dialect. ## Free Pascal FreePascal's function random uses the MersenneTwister (for further details, see the file rtl/inc/system.inc). The random is conform MT19937 and is therefor compatible with e.g. the C++11 MT19937 implementation. ## FutureBasic Syntax: randomInteger = rnd(expr) This function returns a pseudo-random long integer uniformly distributed in the range 1 through expr. The expr parameter should be greater than 1, and must not exceed 65536. If the value returned is to be assigned to a 16-bit integer (randomInteger), expr should not exceed 32767. The actual sequence of numbers returned by rnd depends on the random number generator's "seed" value. (Note that rnd(1) always returns the value 1.) Syntax: random (or randomize) [expr] This statement "seeds" the random number generator: this affects the sequence of values which are subsequently returned by the rnd function and the maybe function. The numbers returned by rnd and maybe are not truly random, but follow a "pseudo-random" sequence which is uniquely determined by the seed number (expr). If you use the same seed number on two different occasions, you'll get the same sequence of "random" numbers both times. When you execute random without any expr parameter, the system's current time is used to seed the random number generator. Example 1: random 375 // using seed number Example 2: random // current system time used as seed Example: To get a random integer between two arbitrary limits min and max, use the following statement. (Note: max - min must be less than or equal to 65536.): randomInteger = rnd(max - min + 1) + min - 1 To get a random fraction, greater than or equal to zero and less than 1, use this statement: frac! = (rnd(65536)-1)/65536.0 To get a random long integer in the range 1 through 2,147,483,647, use this statement: randomInteger& = ((rnd(65536) - 1)<<15) + rnd(32767) ## GAP GAP may use two algorithms : MersenneTwister, or algorithm A in section 3.2.2 of TAOCP (which is the default). One may create several random sources in parallel, or a global one (based on the TAOCP algorithm). # Creating a random sourcers := RandomSource(IsMersenneTwister); # Generate a random number between 1 and 10Random(rs, 1, 10); # Same with default random sourceRandom(1, 10); One can get random elements from many objects, including lists Random([1, 10, 100]); # Random permutation of 1..200Random(SymmetricGroup(200)); # Random element of Z/23Z :Random(Integers mod 23); ## Go Go has two random number packages in the standard library and another package in the "subrepository." 1. math/rand in the standard library provides general purpose random number support, implementing some sort of feedback shift register. (It uses a large array commented "feeback register" and has variables named "tap" and "feed.") Comments in the code attribute the algorithm to DP Mitchell and JA Reeds. A little more insight is in this issue in the Go issue tracker. 2. crypto/rand, also in the standard library, says it "implements a cryptographically secure pseudorandom number generator." I think though it should say that it accesses a cryptographically secure pseudorandom number generator. It uses /dev/urandom on Unix-like systems and the CryptGenRandom API on Windows. 3. x/exp/rand implements the Permuted Congruential Generator which is also described in the issue linked above. ## Golfscript Golfscript uses Ruby's Mersenne Twister algorithm ~rand produces a random integer between 0 and n-1, where n is a positive integer piped into the program ## Groovy Same as Java. ## Haskell The Haskell 98 report specifies an interface for pseudorandom number generation and requires that implementations be minimally statistically robust. It is silent, however, on the choice of algorithm. ## Icon and Unicon Icon and Unicon both use the same linear congruential random number generator x := (x 1103515245 + 453816694) mod 2^31. Icon uses an initial seed value of 0 and Unicon randomizes the initial seed. This LCRNG has a number of well documented quirks (see The Icon Analyst issues #26, 28, 38) relating to the choices of an even additive and a power of two modulus. This LCRNG produces two independent sequences of length 2^30 one of even numbers the other odd. Additionally, the random provides related procedures including a parametrized LCRNG that defaults to the built-in values. ## Io Io's Random object uses the Mersenne Twister algorithm. ## Inform 7 Inform's random functions are built on the random number generator exposed at runtime by the virtual machine, which is implementation-defined. ## J By default J's ? primitive (Roll/Deal) uses the Mersenne twister algorithm, but can be set to use a number of other algorithms as detailed on the J Dictionary page for Roll/Deal. ## Java Java's Random class uses a Linear congruential formula, as described in its documentation. The commonly used Math.random() uses a Random object under the hood. ## JavaScript The only built-in random number generation facility is Math.random(), which returns a floating-point number greater than or equal to 0 and less than 1, with approximately uniform distribution. The standard (ECMA-262) does not specify what algorithm is to be used. ## Julia Julia's built-in random-number generation functions, rand() etcetera, use the Mersenne Twister algorithm. ## Kotlin As mentioned in the Java entry, the java.util.Random class uses a linear congruential formula and is not therefore cryptographically secure. However, there is also a derived class, java.security.SecureRandom, which can be used for cryptographic purposes ## Lua Lua's math.random() is an interface to the C rand() function provided by the OS libc; its implementation varies by platform. ## Mathematica Mathematica 7, by default, uses an Extended Cellular Automaton method ("ExtendedCA") to generate random numbers. The main PRNG functions are RandomReal[] and RandomInteger[] You can specify alternative generation methods including the Mersenne Twister and a Linear Congruential Generator (the default earlier versions). Information about random number generation is provided at Mathematica. ## MATLAB MATLAB uses the Mersenne Twister as its default random number generator. Information about how the "rand()" function is utilized is given at MathWorks. ## Maxima Maxima uses a Lisp implementation of the Mersenne Twister. See ? random for help, or file share/maxima/5.28.0-2/src/rand-mt19937.lisp for the source code. There are also random generators for several distributions in package distrib : • random_bernoulli • random_beta • random_binomial • random_cauchy • random_chi2 • random_continuous_uniform • random_discrete_uniform • random_exp • random_f • random_gamma • random_general_finite_discrete • random_geometric • random_gumbel • random_hypergeometric • random_laplace • random_logistic • random_lognormal • random_negative_binomial • random_noncentral_chi2 • random_noncentral_student_t • random_normal • random_pareto • random_poisson • random_rayleigh • random_student_t • random_weibull Note: the package distrib also has functions starting with pdf, cdf, quantile, mean, var, std, skewness or kurtosis instead of random, except the Cauchy distribution, which does not have moments. ## Modula-3 The Random interface in Modula-3 states that it uses "an additive generator based on Knuth's Algorithm 3.2.2A". ## Nemerle Uses .Net Random class; so, as mentioned under C#, above, implements Knuth's subtractive random number generator algorithm. Random class documentation at MSDN. ## NetRexx As NetRexx runs in the JVM it simply leverages the Java library. See Java for details of the algorithms used. ## Nim There are two PRNGs provided in the standard library: • random : Based on xoroshiro128+ (xor/rotate/shift/rotate), see here. • mersenne : The Mersenne Twister. ## OCaml OCaml provides a module called Random in its standard library. It used to be a "Linear feedback shift register" pseudo-random number generator (References: Robert Sedgewick, "Algorithms", Addison-Wesley). It is now (as of version 3.12.0) a "lagged-Fibonacci F(55, 24, +) with a modified addition function to enhance the mixing of bits." It passes the Diehard test suite. ## Octave As explained here (see rand function), Octave uses the "Mersenne Twister with a period of 2^19937-1". ## Oz Oz provides a binding to the C rand function as OS.rand. ## PARI/GP random uses Richard Brent's xorgens. It's a member of the xorshift class of PRNGs and provides good, fast pseudorandomness (passing the BigCrush test, unlike the Mersenne twister), but it is not cryptographically strong. As implemented in PARI, its period is "at least ${\displaystyle 2^{4096}-1}$". setrand(3)random(6)+1\\ chosen by fair dice roll.\\ guaranteed to the random. ## Pascal See #Delphi and #Free Pascal. Random functions: function Random(l: LongInt) : LongInt; function Random : Real; procedure Randomize; ## Perl Previous to Perl 5.20.0 (May 2014), Perl's rand function will try and call drand48, random or rand from the C library stdlib.h in that order. Beginning with Perl 5.20.0, a drand48() implementation is built into Perl and used on all platforms. The implementation is from FreeBSD and uses a 48-bit linear congruential generator with this formula: • ${\displaystyle r_{n+1}=25214903917\times r_{n}+11{\pmod {2^{48}}}}$ Seeds for drand48 are 32-bit and the initial seed uses 4 bytes of data read from /dev/urandom if possible; a 32-bit mix of various system values otherwise. Additionally, there are many PRNG's available as modules. Two good Mersenne Twister modules are Math::Random::MTwist and Math::Random::MT::Auto. Modules supporting other distributions can be found in Math::Random and Math::GSL::Randist among others. CSPRNGs include Bytes::Random::Secure, Math::Random::Secure, Math::Random::ISAAC, and many more. ## Perl 6 The implementation underlying the rand function is platform and VM dependent. The JVM backend uses that platform's SecureRandom class. ## Phix The rand(n) routine returns an integer in the range 1 to n, and rnd() returns a floating point number between 0.0 and 1.0. In both cases the underlying algorithm is just about as trivial as it can be, certainly not suitable for serious cryptographic work. There are at least a couple of Mersenne twister components in the archive. ## PHP PHP has two random number generators: rand, which uses the underlying C library's rand function; and mt_rand, which uses the Mersenne twister algorithm. ## PicoLisp PicoLisp uses a linear congruential generator in the built-in (rand) function, with a multiplier suggested in Knuth's "Seminumerical Algorithms". See the documentation. ## PL/I Values produced by IBM Visualage PL/I compiler built-in random number generator are uniformly distributed between 0 and 1 [0 <= random < 1] It uses a multiplicative congruential method: seed(x) = mod(950706376 * seed(x-1), 2147483647)random(x) = seed(x) / 2147483647 ## PL/SQL Oracle Database has two packages that can be used for random numbers generation. ### DBMS_RANDOM The DBMS_RANDOM package provides a built-in random number generator. This package is not intended for cryptography. It will automatically initialize with the date, user ID, and process ID if no explicit initialization is performed. If this package is seeded twice with the same seed, then accessed in the same way, it will produce the same results in both cases. DBMS_RANDOM.RANDOM --produces integers in [-2^^31, 2^^31).DBMS_RANDOM.VALUE --produces numbers in [0,1) with 38 digits of precision.DBMS_RANDOM.NORMAL --produces normal distributed numbers with a mean of 0 and a variance of 1 ### DBMS_CRYPTO The DBMS_CRYPTO package contains basic cryptographic functions and procedures. The DBMS_CRYPTO.RANDOMBYTES function returns a RAW value containing a cryptographically secure pseudo-random sequence of bytes, which can be used to generate random material for encryption keys. This function is based on the RSA X9.31 PRNG (Pseudo-Random Number Generator). DBMS_CRYPTO.RANDOMBYTES --returns RAW valueDBMS_CRYPTO.RANDOMINTEGER --produces integers in the BINARY_INTEGER datatypeDBMS_CRYPTO.RANDOMNUMBER --produces integer in the NUMBER datatype in the range of [0..2*128-1] ## PowerShell The Get-Random cmdlet (part of PowerShell 2) uses the .NET-supplied pseudo-random number generator which uses Knuth's subtractive method; see C#. ## PureBasic PureBasic has two random number generators, Random() and CryptRandom(). Random() uses a RANROT type W generator [8]. CryptRandom() uses a very strong PRNG that makes use of a cryptographic safe random number generator for its 'seed', and refreshes the seed if such data is available. The exact method used for CryptRandom() is uncertain. ## Python Python uses the Mersenne twister algorithm accessed via the built-in random module. ## R For uniform random numbers, R may use Wichmann-Hill, Marsaglia-multicarry, Super-Duper, Mersenne-Twister, or Knuth-TAOCP (both 1997 and 2002 versions), or a user-defined method. The default is Mersenne Twister. R is able to generate random numbers from a variety of distributions, e.g. 1. Beta 2. Binomial 3. Cauchy 4. Chi-Squared 5. Exponential 6. F 7. Gamma 8. Geometric 9. Hypergeometric 10. Logistic 11. Log Normal 12. Multinomial 13. Negative Binomial 14. Normal 15. Poisson 16. Student t 17. Uniform 18. Weibull See R help on Random number generation, or in the R system type ?RNGhelp.search("Distribution", package="stats") ## Racket Racket's random number generator uses a 54-bit version of L’Ecuyer’s MRG32k3a algorithm [L'Ecuyer02], as specified in the docs. In addition, the "math" library has a bunch of additional random functions. ## Rascal Rascal does not have its own arbitrary number generator, but uses the Java generator. Nonetheless, you can redefine the arbitrary number generator if needed. Rascal has the following functions connected to the random number generator: import util::Math;arbInt(int limit); // generates an arbitrary integer below limitarbRat(int limit, int limit); // generates an arbitrary rational number between the limitsarbReal(); // generates an arbitrary real value in the interval [0.0, 1.0]arbSeed(int seed); The last function can be used to redefine the arbitrary number generator. This function is also used in the getOneFrom() functions. rascal>import List;okrascal>getOneFrom(["zebra", "elephant", "snake", "owl"]);str: "owl" ## REXX The RANDOM BIF function is a pseudo-random number (non-negative integer) generator, with a range (spread) limited to 100,000 (but some REXX interpreters support a larger range). The random numbers generated are not consistent between different REXX interpreters or even the same REXX interpreters executing on different hardware. /(below) returns a random integer between 100 & 200, inclusive./ y = random(100, 200) The random numbers may be repeatable by specifiying a seed for the random BIF: call random ,,44 /the seed in this case is "44". / . . .y = random(100, 200) Comparison of random BIF output for different REXX implementations using a deterministic seed. / REXX **************************************************************** 08.09.2013 Walter Pachl* Please add the output from other REXXes* 10.09.2013 Walter Pachl added REXX/TSO* 01.08.2014 Walter Pachl show what ooRexx supports*********************************************************************/Parse Version vCall random ,,44ol=v':'Do i=1 To 10 ol=ol random(1,10) EndIf left(v,11)='REXX-ooRexx' Then ol=ol random(-999999999,0) / ooRexx supports negative limits /Say ol outputs from various REXX interpreters: REXX-ooRexx_4.1.3(MT) 6.03 4 Jul 2013: 3 10 6 8 6 9 9 1 1 6 REXX-ooRexx_4.2.0(MT)_32-bit 6.04 22 Feb 2014: 3 10 6 8 6 9 9 1 1 6 -403019526 REXX/Personal 4.00 21 Mar 1992: 7 7 6 7 8 8 5 9 4 7 REXX-r4 4.00 17 Aug 2013: 8 10 7 5 4 2 10 5 2 4 REXX-roo 4.00 28 Jan 2007: 8 10 7 5 4 2 10 5 2 4 REXX-Regina_3.7(MT) 5.00 14 Oct 2012: 10 2 7 10 1 1 8 2 4 1 are the following necessary?? REXX-Regina_3.4p1 (temp bug fix sf.org 1898218)(MT) 5.00 21 Feb 2008: 10 2 7 10 1 1 8 2 4 1 REXX-Regina_3.2(MT) 5.00 25 Apr 2003: 10 2 7 10 1 1 8 2 4 1 REXX-Regina_3.3(MT) 5.00 25 Apr 2004: 10 2 7 10 1 1 8 2 4 1 REXX-Regina_3.4(MT) 5.00 30 Dec 2007: 10 2 7 10 1 1 8 2 4 1 REXX-Regina_3.5(MT) 5.00 31 Dec 2009: 10 2 7 10 1 1 8 2 4 1 REXX-Regina_3.6(MT) 5.00 31 Dec 2011: 10 2 7 10 1 1 8 2 4 1 REXX370 3.48 01 May 1992: 8 7 3 1 6 5 5 8 3 2 Conclusion: It's not safe to transport a program that uses 'reproducable' use of random-bif (i.e. with a seed) from one environment/implementation to another :-( ## Ring nr = 10for i = 1 to nr see random(i) + nlnext ## Ruby Ruby's rand function currently uses the Mersenne twister algorithm, as described in its documentation. ## Run BASIC rmd(0) - Return a pseudorandom value between 0 and 1 ## Rust Rust's rand crate offers several PRNGs. (It is also available via #![feature(rustc_private)]). The offering includes some cryptographically secure PRNGs: ISAAC (both 32 and 64-bit variants) and ChaCha20. StdRng is a wrapper of one of those efficient on the current platform. The crate also provides a weak PRNG: Xorshift128. It passes diehard but fails TestU01, replacement is being considered. thread_rng returns a thread local StdRng initialized from the OS. Other PRNGs can be created from the OS or with thread_rng. For any other PRNGs not provided, they merely have to implement the Rng trait. ## Scala Scala's scala.util.Random class uses a Linear congruential formula of the JVM run-time libary, as described in its documentation. An example can be found here: import scala.util.Random /* * Histogram of 200 throws with two dices. /object Throws extends App { Stream.continually(Random.nextInt(6) + Random.nextInt(6) + 2) .take(200).groupBy(identity).toList.sortBy(_._1) .foreach { case (a, b) => println(f"$a%2d:" + "X" b.size) }} Output: 2:XXX 3:XXXXXXXXX 4:XXXXXXXXXXXXX 5:XXXXXXXXXXXXXXXXXXXXXXXXXX 6:XXXXXXXXXXXXXXXXXXXXXXXXXXXXX 7:XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 8:XXXXXXXXXXXXXXXXXXXXXXXXXXXX 9:XXXXXXXXXXXXXXXXXXXXXXXXXXXX 10:XXXXXXXXXXXXXXXXX 11:XXXXXXXXXXXXXX 12:XX ## Seed7 Seed7 uses a linear congruential generator to compute pseudorandom numbers. Usually random number generators deliver a random value in a fixed range, The Seed7 function rand(low, high) delivers a random number in the requested range [low, high]. Seed7 overloads the rand functions for the types char, boolean, bigInteger, float and others. ## Sidef Latest versions of Sidef use the Mersenne Twister algorithm to compute pseudorandom numbers, with different initial seeds (and implementations) for floating-points and integers. say 1.rand # random float in the interval [0,1)say 100.irand # random integer in the interval [0,100) ## Sparkling Sparkling uses the built-in PRNG of whichever C library implementation the interpreter is compiled against. The Sparkling library functions random() and seed() map directly to the C standard library functions rand() and srand() with only one small difference: the return value of rand() is divided by RAND_MAX so that the generated number is between 0 and 1. ## Stata See set rng in Stata help. Stata uses the Mersenne Twister RNG by default, and may use the 32-bit KISS RNG for compatibility with versions earlier than Stata 14. ## Tcl Tcl uses a linear congruential generator in it's built-in rand() function. This is seeded by default from the system time, and kept per-interpreter so different security contexts and different threads can't affect each other's generators (avoiding key deployment issues with the rand function from C's math library). Citations (from Tcl source code): • S.K. Park & K.W. Miller, “Random number generators: good ones are hard to find,” Comm ACM 31(10):1192-1201, Oct 1988 • W.H. Press & S.A. Teukolsky, “Portable random number generators,” Computers in Physics 6(5):522-524, Sep/Oct 1992. ## TI-83 BASIC TI-83 uses L'Ecuyer's algorithm to generate random numbers. See L'Ecuyer's algorithm. More explainations can be found in this paper. Random function: rand ## TXR TXR 50 has a PRNG API, and uses a re-implementation of WELL 512 (avoiding contagion by the "contact authors for commercial uses" virus present in the reference implementation, which attacks BSD licenses). Mersenne Twister was a runner up. There is an object of type random-state, and a global variable random-state which holds the default random state. Programs can create random states which are snapshots of existing ones, or which are seeded using an integer value (which can be a bignum). The random function produces a random number modulo some integer value, which can have arbitrary precision. The random-fixnum function produces a non-heap-allocated positive integer with random bits. ## UNIX Shell All Bourne Shell clones have a very quick pseudo random number generator. echo $RANDOM Rach time$RANDOM is referenced it changes it's value (with it's maximum value 32767). ## Ursa Standard Ursa defines the ursa.util.random type for random number generators and gives objects of this type a standard interface, but leaves the choice of algorithm up to the implementor. Cygnus/X Ursa is written in Java and makes calls to java.util.Random, which uses a Linear congruential formula. ## Ursala Ursala uses the Mersenne twister algorithm as implemented by the Avram run time system for most purposes, except for arbitrary precision floating point random numbers, which are generated by the urandomb function from the mpfr library. ## XPL0 A 31-bit linear congruential generator is used based on an algorithm by Donald Knuth in his book "Art of Computer Programming" Vol 2, 3rd ed. p. 185. It passes all tests in the Diehard suite. The seed is initialized with the system timer count (at 046C) whenever a program starts. The seed can also be set within a program to give a repeatable sequence of (pseudo) random numbers. Calls to the random number intrinsic return values modulo the argument. include c:\cxpl\codes; \intrinsic 'code' declarationsint I;[RanSeed(12345); \set random number generator seed to 12345for I:= 1 to 5 do [IntOut(0, Ran(1_000_000)); CrLf(0)];] Output: 905495 181227 755989 244883 213142 ## zkl zkl uses the Xorshift (http://en.wikipedia.org/wiki/Xorshift) random number generator. It will also, on occasion, read from /dev/urandom. ## ZX Spectrum Basic The ZX Spectrum uses a Park-Miller (also called a Lehmer) number generator that produces a number between 0 and nearly 1 from a sequence; the RANDOMIZE command can leap to a new entry in the sequence. Multiply the output of RND by 65536 to see the sequence more clearly. The random numbers produced will repeat after 65536 iterations.	{}
This content will become publicly available on December 1, 2023 Antiferromagnetic excitonic insulator state in Sr3Ir2O7 Abstract Excitonic insulators are usually considered to form via the condensation of a soft charge mode of bound electron-hole pairs. This, however, presumes that the soft exciton is of spin-singlet character. Early theoretical considerations have also predicted a very distinct scenario, in which the condensation of magnetic excitons results in an antiferromagnetic excitonic insulator state. Here we report resonant inelastic x-ray scattering (RIXS) measurements of Sr 3 Ir 2 O 7 . By isolating the longitudinal component of the spectra, we identify a magnetic mode that is well-defined at the magnetic and structural Brillouin zone centers, but which merges with the electronic continuum in between these high symmetry points and which decays upon heating concurrent with a decrease in the material’s resistivity. We show that a bilayer Hubbard model, in which electron-hole pairs are bound by exchange interactions, consistently explains all the electronic and magnetic properties of Sr 3 Ir 2 O 7 indicating that this material is a realization of the long-predicted antiferromagnetic excitonic insulator phase. Authors: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; Award ID(s): Publication Date: NSF-PAR ID: 10322855 Journal Name: Nature Communications Volume: 13 Issue: 1 ISSN: 2041-1723 2. Abstract An unidentified quantum fluid designated the pseudogap (PG) phase is produced by electron-density depletion in the CuO 2 antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state characterized by a spatially modulating density of electron pairs. Such a state should exhibit a periodically modulating energy gap $${\Delta }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{r}}}}}})$$ Δ P ( r ) in real-space, and a characteristic quasiparticle scattering interference (QPI) signature $${\Lambda }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{q}}}}}})$$ Λ P ( q ) in wavevector space. By studying strongly underdoped Bi 2 Sr 2 CaDyCu 2 O 8 at hole-density ~0.08 in the superconductivemore »	{}
The world today is profoundly interconnected, but also characterized by ongoing national competition and intra-state conflict. At the nexus of these dynamics is the question of cross-border mobility, which cuts through and connects myriad, disparate areas of “entangled” security—from pandemics to climate change, to conflict and military engagement, to challenges to democracies in the form of internal polarization and external threats. The COVID-19 pandemic provides a striking illustration of this “global security entanglement” in action. This essay presents the concept of security entanglement, illustrates how it operates, and explores some of its theoretical and practical implications. “[M]obility and migration interact with other factors in ways that are symptomatic of how states and societies are increasingly connected.” We live in a highly interconnected, globally entangled world, but continue to think in national terms. Paradoxically, at a time when many governments are retrenching and attempting to deglobalize, the most significant challenges facing the world are more global and border-busting than ever. The COVID-19 pandemic provides a striking illustration of this trend. No event in recent history more clearly qualifies as a global security event, with over 4.8 million people around the world having died from the virus, and more than 237 million having been infected, as of October 2021. In the United States alone, more people died from COVID-19 in the first 20 months of the pandemic than died fighting in World War I, World War II, Korea, Vietnam, Iraq, Libya, and Afghanistan combined. COVID-19 spread so quickly and widely in large part due to the scale, scope, and speed of international mobility today. The virus jumped from Wuhan, China, to the rest of the world via cross-border travel and exchange. Our entangled, interconnected world, with its integrated supply chains and constant cross-border flows of money, goods, people, and services, may proffer a myriad of benefits and virtues, but it is also exceptionally vulnerable to cross-border security threats such as pandemics. As many public health experts have already observed, it is surprising that we have not witnessed a modern global pandemic on this scale before now, and this one is unlikely to be the last. Despite the staggering human costs associated with COVID-19, levels of international cooperation to combat this global threat have remained relatively low, and responses have been overwhelmingly state-centric. Immediate government responses relied heavily on border closings, export bans, and attempts to reconfigure global supply chains. At the same time, global vaccination programs have been stalled by the rise of “vaccine nationalism,” pharmaceutical protectionism, and the proliferation of international scapegoating—such as the United States and China each casting blame on the other for the outbreak of the virus. Yet public health experts from the Center for Global Development estimate that it would cost just $50–70 billion to vaccinate everyone still unvaccinated globally. This relatively modest investment would likely radically reduce the further spread and mutation of the virus and protect the majority of the vaccinated against the most serious forms of the disease. The COVID-19 pandemic is emblematic of the larger patterns and dynamics of the global security environment. Whereas the world is increasingly connected by a multiplicity of transportation links, communications technologies, social media, global popular culture, trade, and finance, these complex interconnections exist side by side with structurally driven, national forms of competition and conflict. Moreover, the third wave of globalization that shaped so much of recent history produced not only connections between different parts of the planet, but also tighter linkages between an array of disparate issue areas. These background conditions create hybrid dynamics that can be characterized as a form of global security entanglement—in which both national and international security are deeply intertwined, and domestic and international politics can likewise become interconnected and entangled. In this context, interconnectedness can be leveraged by individual state and nonstate actors to their own advantage. But it also creates collective vulnerabilities, trans-local security entanglements, and blowback effects that are often underappreciated or even ignored in traditional state-centric approaches to security. Confronting and managing the challenges of an entangled global security environment will require an enhanced understanding of such complexities. As evidenced by the pandemic, these dynamics can be seen particularly in the management of migration and mobility, which connect people across borders, but also create vulnerabilities. The political scientist James F. Hollifield has referred to the contradictory effects of migration as the “liberal paradox.” On the one hand, liberalism flourishes on the basis of open exchange and the free circulation of goods, ideas, and people. On the other hand, this same mobility and circulation creates challenges and vulnerabilities for political institutions and rights-based frameworks that are still largely closed and territorial, bringing mobility management to the fore as a key issue facing states. The pandemic has placed this contradiction in sharp relief: the need to limit movement in order to protect public health has simultaneously led to disruptions in global supply chains, trade, and transport, creating stark trade-offs between different elements of security that are difficult to reconcile. Complex interconnections exist side by side with competition and conflict. Yet the dilemmas governments have faced with regard to mobility during the pandemic are only more visible versions of the everyday challenges of managing mobility in a globally entangled world. Migration and mobility cut through and connect a number of different areas of entangled security—from pandemics to climate change, and from conflict and military engagement to contemporary challenges confronting democracies in the form of internal polarization and external threats. Moreover, migration itself is commonly weaponized or used as a tool of leverage by states in more classical or coercive forms of interstate bargaining and diplomacy. This brings together the same dual dynamics of global interconnection and interstate competition in ways that make the management of migration a “wicked problem,” one that is so complex that it does not have a clear, definitive solution. The combination of interconnectedness and competition adds another layer of complexity to collective action problems. Attempts at autarkic “national” solutions are insufficient, but so are existing mechanisms of global governance, since they are based on an assumption of a world of discrete, legally defined nation-states, rather than recognition of cross-border security entanglement. The complex, mobility-related dynamics of security entanglement are also in evidence in the unintended consequences and blowback effects of the post-9/11 wars and conflicts that made up the US-led Global War on Terror. The staggering rise in forced migration and refugee flows since 2001 cannot be separated from the series of military interventions that took place across the Middle East and beyond during this period. The Costs of War Project estimates that approximately 38 million people (and possibly millions more) have been displaced in the post-9/11 wars fought by the United States and its allies—more than the number displaced by any other war or natural or man-made disaster since the start of the twentieth century, with the exception only of World War II. An estimated 80 percent of the people who arrived in Europe by boat during the height of the 2015–16 migration “crisis” were originally from war-torn Afghanistan, Iraq, and Syria. Foreign-imposed regime changes have fundamentally altered the countries subject to these interventions as well as other states in the region and beyond. The 2011 NATO-led intervention in Libya helped destabilize the country and the broader region. It also hastened Libya’s ongoing transformation into a migration transit state and hub for Europe-bound migrant smuggling. Similarly, the departure of the United States from Afghanistan twenty years after deposing the Taliban has created ongoing migration challenges not only for Afghanistan’s immediate neighbors, but also for states farther afield. One such country is Turkey, which was already the leading refugee host in the world, having taken in some 3.6 million Syrians since the start of the conflict in their neighboring country in 2011. The recent uptick in the number of Afghan refugees has increased domestic tensions over migration and hastened the construction of a wall on Turkey’s eastern border with Iran, while further boosting migration anxieties throughout eastern and western Europe. Rather than treating these NATO-led interventions and the 2015–16 refugee “crisis” as separate events, an entangled security perspective provides a lens for seeing how they are deeply interconnected. Military interventions in the Middle East, Central Asia, and North Africa not only had devastating effects for populations on the ground, but also had blowback and security effects in Europe. The rapid rise in conflict-induced migration hastened the militarization of Europe’s external borders, spurring the further development of the European Union’s FRONTEX border agency and intensifying the EU’s extension and externalization of migration control beyond its borders. All this deepened Europe’s security entanglement with its neighbors. Demographic trends leading to a greying and shrinking European population mean that most European countries would benefit from a larger supply of skilled and unskilled labor. Yet over the past decade, the politics surrounding migration has been defined by a rise in anti-immigrant sentiment and nativist populism within Europe as well as in other parts of the globe. Although the United Kingdom’s 2016 vote to leave the EU was spurred by a number of factors, including an aversion to the effects of the EU’s freedom of movement policies, anti-EU politicians were quick to instrumentalize the 2015–16 “crisis” in their arguments for Brexit. Though one cannot necessarily draw a straight and solid line between NATO-led military interventions, the European migration “crisis,” and the rise of populism in Europe and elsewhere, these events are deeply intertwined and cannot be understood in isolation from one another. Similar blowback effects can be seen in North America, where migration-related entangled security dynamics are endemic and embedded in both “high” and “low” political issues. Many of these dynamics have their origins in, or were exacerbated by, the long history of US involvement in Latin America. The ongoing emergency on the US southern border, for instance, is in no small part a result of the United States’ own policies in the region—particularly the extensive and sustained US involvement in Central America during the Cold War. El Salvador, Guatemala, and Honduras—the three countries that make up the so-called Northern Triangle—have been the source of much of the migration to the US southern border since 2014. This is not a coincidence. The United States was behind a 1954 military coup in Guatemala and strongly backed the government from the 1960s to the 1990s. During this period, the Guatemalan military waged a campaign that killed an estimated 200,000 of the country’s indigenous people. Much of the migration from Guatemala comes from the highlands—an area that is inhabited by indigenous groups and has been subject to land grabs by current or former military officers with connections to organized crime. In Honduras, the Obama administration turned a blind eye to a 2009 coup and even worked to prevent its reversal, while continuing to supply aid to the new government. This further militarized the Honduran police force, leading to even greater internal insecurity. The United States was also deeply involved in El Salvador’s 12-year-long civil war. Throughout the 1980s, widespread human rights abuses and extrajudicial killings by US-backed and -funded government troops, right-wing paramilitaries, and death squads, which were battling left-leaning, Soviet-backed Farabundo Martí National Liberation Front rebel forces, drove tens of thousands of Salvadoran civilians to flee to the United States. Some ended up in Los Angeles and formed gangs, including Barrio 18 and the now-infamous Mara Salvatrucha (MS-13), as a means of protecting their kinsman from other gangs in the area. Over time, Barrio 18 and MS-13 grew stronger and more violent, driving up murder rates in parts of Los Angeles and prompting US authorities to deport many gang members back to Central America. Rather than solve the problem, mass deportations intensified it. Once back in Central America, the gangs were often reconstituted and even increased in size and reach. Barrio 18 and MS-13 now have members—and control territory—not only in Los Angeles and El Salvador, but also in Honduras, Guatemala, Mexico, and other parts of the United States and Canada. These and other organizations have formed alliances with some gangs and engaged in violent rivalries with others. Coming full circle with the civil war that first inspired flight, the combination of poverty, dysfunctional politics, and gang-driven violence directed against civilians—which has produced some of the highest murder rates in the world—has again impelled many civilians to flee north to the United States. They seek refugee status in a bid to protect themselves and their families. In April 2021, US Vice President Kamala Harris announced$310 million in additional humanitarian aid for Guatemala, Honduras, and El Salvador—part of an estimated \$4 billion in assistance for the region under the Biden administration’s plan to address migration issues. In this respect, the United States seems to be following the EU’s example of using foreign aid to try to stem migration. Over the past two decades, an array of countries in North Africa and the Horn of Africa have collectively and individually received billions of euros of aid in exchange for helping to stanch, reverse, or forestall northward migrations to Europe. Here, too, attempts to prevent migration through tighter border controls, outsourcing, and heightened enforcement can often exacerbate the very security risks that they are intended to address. In a shared virtual space, liberal and illiberal states are increasingly entangled. Tighter border controls in both the United States and Europe have driven up the costs of irregular migration. This in turn has increased the debts of unsuccessful border-crossers, generating still greater incentives to reach the richer countries of the global North in the hope of securing employment that will provide the means to pay off the human traffickers who arranged their journeys. Migration-related aid packages designed to improve conditions on the ground in countries of origin can paradoxically make outflows more likely. This is the case if a government receiving aid is illiberal and uses financial assistance to strengthen its grip on power and increase its repressive capabilities. Such counterproductive outcomes can be compounded if these infusions of financial assistance are viewed by their recipients as a kind of carte blanche for domestic oppression and other human rights abuses. This was a common phenomenon among authoritarian regimes during the Cold War. The same entangled dynamics also create perverse incentives that may lead states to use migration as a form of leverage in their diplomatic engagements and interactions with other states. Both states and nonstate actors can take advantage of others’ concerns about migration and strategically use migration as an instrument to gain concessions or positive inducements. The 2016 deal between the EU and Turkey—in which Turkey was able to secure 6 billion euros in aid, promises of visa-free travel, and a resuscitation of its EU accession talks in exchange for tighter migration controls—is one prominent example of this common dynamic. Another came in May 2021, when Morocco opened its border with the Spanish enclave of Ceuta in a bid to punish and coerce the Spanish government over its direct and indirect support for the Polisario Front, an insurgent group locked in a long-term separatist conflict with Morocco. Turkey took a similar action in February 2020 when it permitted thousands of migrants to head to its borders with Greece. Aimed at securing NATO support for Turkey’s intervention in Syria, this move came close to provoking a military confrontation with Greece. More recently, starting in mid-2021, Belarus opened its borders and attempted to weaponize migration in retaliation for EU-imposed sanctions and Brussels’ vocal criticism of Alexander Lukashenko’s regime. The migrants that Belarus is allowing to cross into neighboring states come from as far afield as West Africa and southwestern Asia. As of this writing, tensions are heating up along Belarus’ borders with its neighbors, especially NATO members Latvia, Lithuania, and Poland. Liberal democracies tend to be particularly, but not uniquely, vulnerable to this unconventional brand of coercion, since they can find themselves trapped between conflicting imperatives with regard to displaced people. On the one hand, these states generally have made normative and legal commitments to protect those fleeing violence and persecution. On the other, they often face internal political pressures around migration, with the control of borders increasingly viewed as a polarizing symbolic issue. States cannot simultaneously respond to both of these imperatives. Thus they have increasing incentives to concede to demands made by actors using the instrumentalization of migration as a form of coercive diplomacy—be it for political, economic, or military aims. This in turn makes the strategy of weaponizing migration appear more geopolitically efficacious. The result is that liberal states themselves increasingly resort to more and more illiberal methods and strategies to repel potential migrants and other border-crossers. This further undermines their legitimacy and identity as liberal states, leaving them exposed to charges of hypocrisy at home and abroad. Such charges are often leveled by international rivals and states trying to deflect criticism of their own illiberal actions and policies. The entanglement of liberal and illiberal dynamics with mobility issues can also be seen in how states such as China, Russia, and Turkey have increasingly taken an interest in “their” emigrants and diasporas, attempting to control them through transnational strategies that involve long-distance forms of repression. International migration has facilitated citizens’ mobility into and out of autocratic states. At the same time, new information and communications technologies have led to the globalization of many aspects of domestic politics, and the rise of diaspora politics. Diasporic activism operates largely outside the jurisdiction of the state of origin, and has therefore often been assumed to be a space of opportunity for political opposition movements and groups, where they can operate without interference from homeland state authorities. Yet the transnationalization of politics has also been accompanied by the transnationalization of family ties, social relations, and social networks, which perversely has provided an additional source of leverage for states to engage in transnational repression. New forms of digital surveillance—such as monitoring of social media accounts and private communications like text messages—allow authoritarian states to quickly identify the ties between activists abroad and family members and acquaintances back home. Whereas actors in the diaspora may be outside the direct reach of a repressive state, friends and relatives in their home country can still become targets of coercion by proxy. This strategy has been employed by China to harass and intimidate Uighur activists in Europe and North America. It has also been used by states such as Egypt and Turkey against the families of journalists or dissidents whom they wish to silence. Governments can also “go global” in their use of strategies of repression by directly targeting dissidents, activists, and regime opponents abroad. Harassment, surveillance, enactment of mobility restrictions, or even more serious instances of kidnapping, physical attack, or assassination are all tactics that states have used to target political exiles abroad. The 2018 assassination of the Saudi journalist Jamal Khashoggi in Istanbul stands out, but there are other examples. Russia has attempted to poison numerous political exiles in the UK; Turkey has been accused of assassinating three Kurdish activists in Paris; and Rwanda has targeted diaspora members in several countries across Africa and beyond since 2014. As outlined in two recent Freedom House reports, autocratic states often tap into institutions set up for other purposes, such as INTERPOL’s Red Notice—a system that effectively acts as an international arrest warrant for law enforcement agencies—to target political opposition leaders or even personal enemies. As autocracies develop new means of exercising power over populations abroad, their use of transnational strategies poses a number of complex security challenges for policymakers in democratic states, as well as for human rights actors and international legal understandings of refuge, asylum, and protection. Existing international protection regimes operate according to state-centric assumptions, in which state sovereignty is identified with territoriality, and national borders are assumed to demarcate legal jurisdictions in ways that offer refuge and asylum to persecuted individuals fleeing authoritarian states. Yet the crossing of national borders does not mean that individual dissidents and exiles—or entire groups living outside a state’s territorial boundaries, such as international students, labor migrants, or ordinary diaspora members—are necessarily free from the influence of state actors in their homelands. The use of various techniques of transnational repression presents a more complicated blurring of how authoritarian practices “at home” relate to diaspora politics “abroad.” This development comes, moreover, at a time when global norms of asylum and protection are also under threat and are subject to manipulation and instrumentalization. In a highly interconnected world, it may be necessary to radically rethink the broader implications of the rise of authoritarian practices that transcend state borders. In addition to potentially posing direct and targeted security threats to some exiled populations, the spatial and legal complexities of such practices create long-term challenges for liberal states and liberal institutions. Whereas practices of transnational repression are not entirely new—and were also present during the Cold War—the new global media environment has created a shared virtual space in which liberal and illiberal states do not operate in wholly separate spheres, but rather are increasingly entangled. The complex ways in which mobility, geopolitics, illiberalism, and security are entangled with other issues create additional challenges in deciding how to address large-scale collective security threats such as pandemics and climate change, which former United Nations Secretary-General Kofi Annan referred to as “problems without passports.” Such problems are compounded at a time when migration and mobility are particularly contentious issues subject to increased politicization and instrumentalization. For example, with arrivals at the US southern border at record levels, and immigration remaining a hot-button issue in American politics, the Biden administration has extended the Trump administration’s use of Title 42, a rarely employed clause in public health law, to prevent asylum seekers from entering from Mexico during the pandemic. Perverse incentives may lead states to use migration as a form of leverage. A similar strategy has been used across the EU: states have invoked public health concerns as a reason for restricting entry, shifting their anti-migration discourses about criminality and terrorism to a focus on controlling the pandemic. Countries such as Italy and Malta declared their ports of entry unsafe for migrant disembarkation, and several countries and regions across Europe have denied COVID-19 vaccinations to irregular migrants lacking documentation. Moreover, as Amnesty International documented in its 2021 Annual Report, governments around the world have been escalating various forms of domestic repression and mobility restrictions, sometimes instrumentalizing the pandemic as a means of silencing critics. National-level responses, such as lockdowns, travel bans, and border closures, have been comparatively effective in some places at keeping community transmission rates relatively low. As new variants and breakthrough cases of infection have emerged, however, governments around the world are shifting from trying to fully eliminate the virus with policies of restricted mobility and travel bans to strategies of risk management, living with and adapting to COVID-19. New Zealand, whose geographic position and stringent policies directed at disease eradication shielded it from the worst effects of COVID-19, was long heralded as a pandemic success story. But even its government had to concede in October 2021 that it could not fully vanquish the virus and instead adopted new policies of accelerated vaccination, virus control, and containment. The challenges of dealing with the mobility paradox highlighted by the pandemic can be seen in the economic effects of national policy responses, such as global supply-chain disruptions leading to inflation, assembly-line shutdowns, and shortages of goods. In October 2021, the New York Times reported that 13 percent of world cargo capacity was subject to pandemic-related shipping delays, while US manufacturers needed an unprecedented 92 days on average to assemble the requisite parts and raw materials to produce their wares. Even as COVID-19 case numbers and death tolls ebb and flow, such disruptions continue, adversely affecting economies, health care systems, and food distribution in the world’s wealthiest and poorest countries alike, albeit more acutely in the developing world. Meanwhile, the focus on the pandemic has necessitated sidelining other public health problems, which are also inherently trans-border phenomena. The World Health Organization (WHO) has warned that disruptions to antiretroviral therapy due to COVID-19 could lead to more than 500,000 additional deaths from HIV/AIDS in sub-Saharan Africa and the further spread of that disease, both within the region and beyond. In April 2021, the WHO likewise reported that fully 90 percent of countries responding to a survey about the effects of COVID-19 had experienced disruptions to essential health services and immunization programs, though the magnitude of the disruptions was lower than it had been during the first year of the pandemic. The potential direct and indirect security implications of such disruptions are manifold. Similar challenges can be seen in efforts to collectively address climate change—which the Biden administration’s government-wide Climate Adaptation Plans, released in October 2021, identify as an urgent and rapidly growing threat to national and international security. UN Secretary-General António Guterres has highlighted climate change as a key factor accelerating all other drivers of forced displacement. This is because climate change can arguably act as a “threat multiplier,” exacerbating preexisting risks and generating new ones, such as food and water insecurity and competition over resources. These risks in turn can contribute to internal conflicts and compound people’s extant vulnerabilities to displacement. Internal conflicts can spill over into neighboring states, which can drive the displaced outside their regions of origin and complicate political, economic, and social dynamics in the regions and states to which they flee. It has been argued that competition over water and intra-communal grievances made worse by sustained drought and food insecurity helped create “ripe” conditions that made Syria’s civil war more likely. In a study published in 2015 in Proceedings of the National Academy of Sciences, researchers claimed that water shortages in the Fertile Crescent (of Syria, Iraq, and Turkey) killed livestock, drove up food prices, and forced some 1.5 million rural residents to the outskirts of Syria’s already-packed cities. This happened as Syria was already coping with an influx of refugees from the Iraq war—compounding existing domestic problems such as corruption, repressive leadership, inequality, and high population growth. Others have disputed these findings, however. In an article in Political Geography, researchers said they had found no clear and reliable evidence that climate change was a factor in the onset of Syria’s civil war. Less debatable are the following facts: environmental changes are already catalyzing population displacement and migration in some parts of the world; climate change is increasingly viewed as a human, national, and international security issue; and climate change is deeply entangled with other security dynamics. Reckoning with the dilemmas of global security entanglement is a necessary step in confronting the myriad policy challenges that will threaten human lives and well-being in the coming decades, from pandemics and climate change to violent conflict, state repression, and global authoritarianism. In all these areas, mobility and migration interact with other factors in ways that are symptomatic of how states and societies are increasingly connected. The implications of these dynamics are several. First, it is clear that states cannot simply go it alone—problems without passports cannot be solved at the national level, and their effects cannot be stopped at borders or by erecting fences and walls. Second, greater understanding is required of the complicated knock-on and blowback effects that global actions taken in one area can have on others—such as the effects of military conflicts and interventions on what have been labeled subsequently as migration “crises.” Finally, it is critical that both states and nonstate actors identify effective ways to address entangled security challenges that do not come at the expense of the world’s most vulnerable populations, including those whose own security is dependent on the ability to move and cross borders. Failing to do so will in many cases simply backfire and lead to bigger, still more wicked problems. The complicated, entangled nature of global security suggests that we are genuinely in this together. To paraphrase Cicero, entangled security means that there is no trade-off to be made between what is just and what is expedient—that which is just is also expedient.	{}
Find the Centroid of the Shaded Portion of the Plate Shown in the Figure. - Engineering Mechanics Find the centroid of the shaded portion of the plate shown in the figure. Solution Y = X is the axis of symmetry The centroid would lie on this line Sr.no. PART AREA(in mm2) X co-ordinate(mm) Ax(mm3) 1. RECTANGLE =1000 X 1000=1000000 1000/2 = 500 500000000 2. TRIANGLE (to be removed) 1/2 X 750 X 750= -281250 1000 –750/2= 750 -210937500 3. QUARTER CIRCLE (To be removed) (pir^2)/4= 441786.4669 (4 x 7509)/(3pi)= 3141.5926 -140625000 TOTAL 276963.4669 148437500 overlineX =(Σ Ax)/(Σ A)=(148437500)/(276963.5331)=535.946 mm 𝑦̅ = 𝑋̅ = 535.946 mm CENTROID IS AT (535.946,535.946)mm Concept: Centroid for Plane Laminas Is there an error in this question or solution?	{}
# Aspects of Quasiconformal Homogeneity ### New Zealand Journal of Mathematics Vol. 39, (2009), Pages 117-132 Jianhua Gong Department of Mathematical Science United Arab Emirates University P.O. Box 17551 Al Ain United Arab Emirates Gaven Martin Massey University Private Bag Auckland New Zealand Abstract This paper studies the notion of quasiconformal homogeneity in various different settings - relative with respect to domains and ambiently. For instance we give an example of a compact set E which is homogeneous with respect to a quasiconformal family of maps of a domain containing E, but which is not quasiconformally homogeneous with respect to $\mathbb{R}^n$. We construct Cantor sets which are K--quasiconformally homogeneous for each K > 1 and explicitly identify their Hausdorff dimension, among other things. Keywords quasiconformal, homogeneity, quasiconformal groups, Lie group. Classification (MSC2000) Primary: 30C60.	{}
## Nonstandard arithmetic and reverse mathematics.(English)Zbl 1101.03040 Reverse mathematics is based on a hierarchy of fragments of second-order arithmetic. Keisler develops an alternative approach in which the key role is played by the separation of integers into the (standard) natural numbers and hyperintegers. This is done in a hybrid second-order/two-sorted language $$L_2\cup {^\!L}_1$$. Keisler shows that each of the basic theories $$\text{ WKL}_0$$, $$\text{ACA}_0$$, $$\text{ATR}_0$$, and $$\Pi^1_1$$-$$\text{CA}_0$$ (which are formulated in the language of second-order arithmetic $$L_2$$) has a natural counterpart in $$L_2\cup {^\!L}_1$$. The language $${^\!L}_1$$ has all the symbols of the first-order arithmetic and variables of two sorts: $$N$$ and $$^\!N$$. The universe of sort $$N$$ is a subset of the universe of sort $$^\!N$$ and variables and terms of sort $$N$$ are allowed in argument places of sort $${^\!N}$$. The role of the basic theory $$\text{ I}\Sigma_1$$ is played by $$^\Sigma \text{PA}$$ which includes the basic axioms of $$\text{I}\Sigma_1$$ with variables of sort $$^\!N$$, an internal induction axiom for a special class of bounded formulas, and two special axioms. One of the axioms says that $$N$$ is a proper initial segment of $$^\!N$$, and the other expresses a property of coded sequences all of whose terms with standard indices are standard (the Finiteness Axiom). This theory is denoted by $$^\Sigma \text{PA}$$. The stronger axiom systems describe properties of structures of the form $$(M,\,^\!N)$$, where $$M=(N,P)$$ is an $$L_2$$ structure and $$(N,\,^\!N)$$ is an $$^\!L_1$$ structure. Then, $$^\text{WKL}_0$$ is a theory in the language $$L_2\cup {^\!L}_1$$ defined as $$^\Sigma + \text{ STP}$$, where STP is the Standard Part Principle declaring that $$P$$ is the standard system of $$^\!N$$ relative to $$N$$. It is shown that $$^\text{WKL}_0$$ implies $$\text{WKL}_0$$ and that $$^\text{WKL}_0$$ is conservative with respect to $$\text{WKL}_0$$. An important ingredient of the proof is the theorem of K. Tanaka [Ann. Pure Appl. Logic 84, 41–49 (1997; Zbl 0871.03044)], on self-embeddings of countable models of $$\text{WKL}_0$$ which are not $$\omega$$-models. The equivalent of $$\text{RCA}_0$$ is obtained by weakening STP in $$^\text{WKL}_0$$. The equivalents of $$\text{ACA}_0$$ and $$\Pi^1_1$$-$$\text{CA}_0$$ are obtained by adding suitable comprehension schemes to $$^\text{WKL}_0$$; $$^\text{ATR}_0$$ involves a $$\Sigma^_1$$-separation scheme. It is also shown that $$^\Pi^1_1$$-$$\text{CA}_0$$ plus the First-Order Transfer Principle (FOT) is conservative with respect to $$\Pi^1_1$$-$$\text{CA}_0$$ proving a conjecture of C. W. Henson, M. Kaufmann and H. J. Keisler [J. Symb. Log. 49, 1039–1058 (1984; Zbl 0587.03048)]. ### MSC: 03F35 Second- and higher-order arithmetic and fragments 03H15 Nonstandard models of arithmetic ### Keywords: reverse mathematics; second-order arithmetic; hyperintegers ### Citations: Zbl 0587.03048; Zbl 0871.03044 Full Text: ### References: [1] Logic in Tehran 26 (2006) [2] Model theory (1990) [3] DOI: 10.1016/S0168-0072(95)00058-5 · Zbl 0871.03044 [4] DOI: 10.2307/2274260 · Zbl 0587.03048 [5] Infinitistic methods pp 257– (1961) [6] DOI: 10.2307/2274061 · Zbl 0624.03051 [7] Subsystems of second order arithmetic (1999) · Zbl 0909.03048 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	{}
# Preserving connections immersions I am dealing with Riemannian immersions and I am stuck on the following: Given a totally geodesic immerserd surface $S$ on a compact riemannian manifold $M$ with metric $g$, is there another metric $\tilde g \neq g$ on $M$ such that $S$ is isometric immersed (not necessarely totally geodesically) on $(M,\tilde g)$? In this case, what can I say about the second fundamental form of this second immersion? Can I conclude with assumptions that the induced connection on $S$ is the same on both immersions? • If the connection on $M$ is the Levi-Civita connection, then the induced connection of an immersion is the Levi-Civita connection for the induced metric. Thus in both cases, since the immersion is isometric, the induced connection is the same. – Paul Bryan Oct 21 '17 at 1:44 Suppose that $S$ is a pair of disjoint Euclidean 2-spheres, say of radius 1 and radius 2. Isometrically immerse to the 3-sphere $M$ as totally geodesic spheres of radius 1, intersecting along a totally geodesic curve $C$ lying on each of the 2-spheres. The preimage of $C$ in $S$ is a pair of circles of different lengths. No perturbation of the metric on $M$ can give that curve $C$ those two different lengths. So $S$ is not isometrically immersed for any metric on $M$. Suppose that you do have two immersions, one totally geodesic, the other isometric, $S \to M$. If they agree at one point, and have the same Levi-Civita connection induced, they are both isometric immersions. So we can easily invent a counterexample. Take $S$ the plane with standard metric, immersed into 3-dimensional Euclidean space by an immersion which is isometric only at the origin, and which puts $S$ into the horizontal 2-plane, so totally geodesic. Then the Levi--Civita connection of the pullback metric on $S$ cannot be the Levi--Civita connection of the original metric on $S$.	{}
Find all School-related info fast with the new School-Specific MBA Forum It is currently 30 Aug 2015, 17:10 # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # GMAT Data Sufficiency (DS) Question banks Downloads My Bookmarks Reviews Important topics Go to page Previous 1 2 3 4 5 6 7 8 9 10 11 ... 133 Next Search for: Topics Author Replies Views Last post Announcements 277 DS Question Directory by Topic and Difficulty bb 0 87680 07 Mar 2012, 07:58 Topics 47 DS questions about standard deviation Go to page: 1, 2 Tags: Difficulty: Sub-600 Level, Statistics and Sets Problems Bunuel 34 35629 16 Aug 2015, 20:09 48 If vertices of a triangle have coordinates (-2,2), (3,2) and noboru 34 10194 08 Jul 2015, 23:37 55 Is the perimeter of triangle ABC greater than 20? 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Go to page: 1, 2 Tags: Difficulty: 700-Level, Absolute Values/Modules, Inequalities, Source: Manhattan GMAT mn2010 33 6414 23 Mar 2015, 08:25 1 If k is not 0,1,-1, is 1/k >0? 1. 1/k-1 >0 2. 1/k+1 Go to page: 1, 2 Tags: Difficulty: 600-700 Level, Inequalities, Source: GMAT Paper Tests jodeci 32 3877 11 Jul 2011, 08:46 2 If x and y are positive, is 3x > 7y? (1) x > y + 4 (2) Go to page: 1, 2 Tags: Difficulty: 700-Level, Inequalities ankur55 32 4765 05 Sep 2011, 22:30 10 Of the 200 members of a certain association, each member who Go to page: 1, 2 Tags: Difficulty: 700-Level, Overlapping Sets, Source: GMAT Prep Gordon 32 6872 10 Sep 2013, 14:13 21 The total cost of producing item X is equal to the sum of knabi 32 7778 19 Aug 2015, 11:00 43 Is n/18 an integer? shinewine 31 6847 12 Jun 2015, 22:38 24 What is the perimeter of PQRS ? 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Go to page: 1, 2 Tags: Difficulty: Sub-600 Level, Roots, Source: Official Guide Guest 29 8968 10 May 2011, 03:58 6 Veritas Prep 10 Year Anniversary Promo Question #5 Go to page: 1, 2 Tags: Difficulty: 600-700 Level, Algebra, Source: Veritas Prep Bunuel 29 2166 18 Jun 2015, 17:29 23 Is x/3 + 3/x > 2 Go to page: 1, 2 Tags: Difficulty: 700-Level, Inequalities DenisSh 29 4903 04 Jul 2014, 21:22 62 Each of the 25 balls in a certain box is either red, blue or Go to page: 1, 2 Tags: Difficulty: 700-Level, Overlapping Sets, Probability, Source: GMAT Prep lexis 29 11105 23 Aug 2015, 21:54 82 Joanna bought only $0.15 stamps and$0.29 stamps. How many Go to page: 1, 2 Tags: Difficulty: 700-Level, Arithmetic, Word Problems, Source: Official Guide udaymathapati 29 15054 16 Aug 2015, 04:17 35 A rectangle is plotted on the standard coordinate plane Go to page: 1, 2 Tags: Difficulty: 700-Level, Coordinate Geometry, Source: Grockit guygmat 28 7952 17 May 2015, 07:56 15 What is the length of segment BC? 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Share # Selina solutions for Concise Chemistry Class 10 ICSE chapter 3 - Acids, Bases and Salts [Latest edition] Course Textbook page ## Chapter 3: Acids, Bases and Salts Intext QuestionsExercise - 3AExercise - 3BExercise - 3CMiscellaneous Questions #### Selina solutions for Concise Chemistry Class 10 ICSE Chapter 3 Acids, Bases and Salts Exercise Intext Questions [Pages 0 - 43] Intext Questions \| Q 1.1 \| Page 42 What do you understand by the terms, acid? Intext Questions \| Q 1.2 \| Page 42 Name the positive ion formed When an acid is dissolved in water. Draw the structure of this ion . Intext Questions \| Q 2 \| Page 42 Write the ionization of sulphuric acid showing the formation of hydronium ion. Intext Questions \| Q 4 \| Page 42 Define the term 'basicity' of an acid. Give the basicity of : nitric acid, sulphuric acid, phosphoric acid? Intext Questions \| Q 5.1 \| Page 42 Give two examples of the following: oxy-acid Intext Questions \| Q 5.2 \| Page 42 Give two examples of the following: hydracids Intext Questions \| Q 5.3 Give two examples of the following: tribasic acid Intext Questions \| Q 5.4 \| Page 42 Give two examples of the following: dibasic acid Intext Questions \| Q 6.1 \| Page 42 Name the acidic anhydride of the following acids: (i) sulphurous acid (ii) nitric acid (iii) phosphoric acid (iv) carbonic acid Intext Questions \| Q 6.2 \| Page 42 Name the acids present in vinegar, grapes and lemon. Intext Questions \| Q 7 \| Page 42 What do you understand by the statement ‘acetic acid is a monobasic acid? Intext Questions \| Q 8.1 \| Page 42 Give a balanced equation for reaction of nitrogen dioxide with water. Intext Questions \| Q 8.2 \| Page 42 Give a balanced equation for Preparation of a non-volatile acid from a volatile acid. Intext Questions \| Q 9 \| Page 42 What do you understand by the strength of an acid? On which factor does the strength of an acid depend? Intext Questions \| Q 10.1 \| Page 42 Explain the following : Carbonic acid gives an acid salt but hydrochloric acid does not. Intext Questions \| Q 10.2 \| Page 42 Explian the following : Dil. HCl acid is stronger than highly concentrated acetic acid. Intext Questions \| Q 10.3 \| Page 42 Explain the following : H3POis not a tribasic acid . Intext Questions \| Q 10.4 \| Page 42 Explain the following : Lead carbonate does not react with dilute HCl . Intext Questions \| Q 10.5 \| Page 42 Explain the following : Nitrogen dioxide is a double acid anhydride. Intext Questions \| Q 11.1 \| Page 43 How is an acid prepared from a Non-metal Give an equation for it. Intext Questions \| Q 11.2 \| Page 43 How is an acid prepared from a Salt? Give an equation for it. Intext Questions \| Q 12.1 \| Page 43 Give equation to show how the following is made from their corresponding anhydride. sulphurous acid Intext Questions \| Q 12.2 \| Page 43 Give equation to show how the following is made from their corresponding anhydride. phosphoric acid Intext Questions \| Q 12.3 \| Page 43 Give equation to show how the following is made from their corresponding anhydride. carbonic acid Intext Questions \| Q 12.4 \| Page 43 Give equation to show how the following is made from their corresponding anhydrides. sulphuric acid Intext Questions \| Q 13.1 \| Page 43 Name an acid used to flavor and preserve food. Intext Questions \| Q 13.2 \| Page 43 Name an acid used in a drink. Intext Questions \| Q 13.3 \| Page 43 Name an acid used to remove ink spots. Intext Questions \| Q 13.4 \| Page 43 Name an acid used as an eyewash. Intext Questions \| Q 14.1 \| Page 43 Give the reaction of acid with Chlorides State the conditions under which they react. Intext Questions \| Q 14.2 \| Page 43 Give the reaction of acid with nitrates State the conditions under which they react. #### Selina solutions for Concise Chemistry Class 10 ICSE Chapter 3 Acids, Bases and Salts Exercise Exercise - 3A [Pages 47 - 48] Exercise - 3A \| Q 1 \| Page 47 What do you understand by an alkali? Give two examples of: (a) strong alkalis (b) weak alkalis Exercise - 3A \| Q 2.1 \| Page 47 What is the difference between an alkali and a base? Exercise - 3A \| Q 2.2 \| Page 47 What is the difference between : The chemical nature of an aqueous solution of HCl and an aqueous solution of NH3 Exercise - 3A \| Q 3.1 \| Page 47 Name the ions furnished by bases in solution. Exercise - 3A \| Q 3.2 \| Page 47 Name the ions furnished by an acid. Exercise - 3A \| Q 4.1 \| Page 47 Give one example in the following case: A basic oxide which is soluble in water. Exercise - 3A \| Q 4.2 \| Page 47 Give one example in the following case: A hydroxide which is highly soluble in water. Exercise - 3A \| Q 4.3 \| Page 47 Give one example in the following case: A basic oxide which is insoluble in water. Exercise - 3A \| Q 4.4 \| Page 47 Give one example in the following case: a hydroxide which is insoluble in water. Exercise - 3A \| Q 4.5 \| Page 47 Give one example in the following case: A weak mineral acid. Exercise - 3A \| Q 4.6 \| Page 47 Give one example in the following case: a base which is not an alkali. Exercise - 3A \| Q 4.7 \| Page 47 Give one example in the following case: An oxide which is a base. Exercise - 3A \| Q 4.8 \| Page 47 Give one example in the following case: A hydrogen containing compound which is not an acid. Exercise - 3A \| Q 4.9 \| Page 47 Give one example in the following case: A base which does not contain a metal ion. Exercise - 3A \| Q 5 \| Page 47 You have been provided with three test tubes. One of them contains distilled water and the other two have an acidic solution and a basic solution respectively. If you are given only red litmus paper, how will you identify the contents of each test tube? Exercise - 3A \| Q 6 \| Page 47 HCl, HNO3, C2H5OH, C6H12O6 all contain H atoms but only HCl and HNO3 show acidic character. Why? Exercise - 3A \| Q 7.1 \| Page 47 Dry HCI gas does not change the colour of dry litmus paper. Why? Exercise - 3A \| Q 7.2 \| Page 47 Is PbO2 a base or not? Comment. Exercise - 3A \| Q 7.3 \| Page 47 Do basic solutions also have H+(aq)? Why are they basic? Exercise - 3A \| Q 8.1 \| Page 47 How would you obtain a base from another base. Exercise - 3A \| Q 8.2 \| Page 47 How would you obtain an alkali from a base. Exercise - 3A \| Q 8.3 \| Page 47 How would you obtain salt from another salt? Exercise - 3A \| Q 9.1 \| Page 47 Write balanced equation to satisfy the following statement: Acid + Active metal ⟶ Salt + hydrogen Exercise - 3A \| Q 9.2 \| Page 47 Write balanced equation to satisfy the following statement: Acid + Base ⟶ Salt + water Exercise - 3A \| Q 9.3 \| Page 48 Write balanced equation to satisfy the following statement: Acid + carbonate Or bicarbonate ⟶ Salt + water + carbon dioxide Exercise - 3A \| Q 9.4 \| Page 48 Write balanced equation to satisfy the following statement: Acid + sulphite Or bisulphite ⟶ Salt + water + sulphur dioxide Exercise - 3A \| Q 9.5 \| Page 48 Write balanced equation to satisfy the following statement: Acid + sulphide ⟶ Salt + Hydrogen sulphide Exercise - 3A \| Q 10 \| Page 48 The skin has and needs natural oils. Why is it advisable to wear gloves while working with strong allkalis? Exercise - 3A \| Q 11 \| Page 48 Complete the table: Indicator Neutral Acidic Alkaline Litmus Phenolphthalein Purple Colourless Exercise - 3A \| Q 12 \| Page 48 What do you understand by pH value? Two solutions X and Y have pH values of 4 and 10, respectively. Which one of these two will give a pink colour with phenolphthalein indicator? Exercise - 3A \| Q 12.2 \| Page 48 Two solutions X and Y have Ph values of 4 and 10 respectively. Which one of these two will give a pink colour with phenolphthalein indicator? Exercise - 3A \| Q 13 \| Page 48 You are supplied with five solutions: A, B, C, D and E with Ph values as follows: A= 1.8, B = 7, C= 8.5, D = 13, and E=5 Classify these solutions as neutral, slightly or strongly acidic and slightly or strongly alkaline. Which solution would be most likely to liberate hydrogen with: (a) magnesium powder, (b) powdered zinc metal. Give a word equation for each reaction. Exercise - 3A \| Q 14.1 \| Page 48 Distinguish between a common acid base indicator and a universal indicator. Exercise - 3A \| Q 14.2 \| Page 48 Distinguish between acidity of bases and basicity of acids. Exercise - 3A \| Q 14.3 \| Page 48 Distinguish between acid and alkali (other than indicators). Exercise - 3A \| Q 15.1 \| Page 48 What should be added to Increase the pH value . Exercise - 3A \| Q 15.2 \| Page 48 What should be added to Decrease the pH value of a neutral solution? Exercise - 3A \| Q 16 \| Page 48 How does tooth enamel get damaged? What should be done to prevent it? Exercise - 3A \| Q 17 \| Page 48 When you use universal indicator, you see that solutions of different acids produce different colours. Indeed, solution of the same acid with different concentrations will also give different colours. Why? Exercise - 3A \| Q 18.1 \| Page 48 A solution has a pH of 7. Explain how you would increase its pH . Exercise - 3A \| Q 18.1 \| Page 48 A solution has a pH of 7. Explain how you would decrease its pH . Exercise - 3A \| Q 18.2 \| Page 48 If a solution changes the colour of litmus from red to blue, what can you say about its pH? Exercise - 3A \| Q 18.3 \| Page 48 What can you say about the pH of a solution that liberates carbon dioxide from sodium carbonate? Exercise - 3A \| Q 19.1 \| Page 48 Solution P has a pH of 13, solution Q has a pH of 6 and solution R has a pH of 2. Which solution will liberate ammonia from ammonium sulphate on heating? Exercise - 3A \| Q 19.2 \| Page 48 Solution P has a pH of 13, solution Q has a pH of 6 and solution R has a pH of 2. Which solution is a strong acid? Exercise - 3A \| Q 19.3 \| Page 48 Solution P has a pH of 13, solution Q has a pH of 6 and solution R has a pH of 2. Which solution contains molecules as well as ions? Exercise - 3A \| Q 20 \| Page 48 M is an element in the form of a powder. M burns in oxygen and the product obtained is soluble in water. The solution is tested with litmus. Write down only the word which will correctly complete each of the following sentences. (i) If M is a metal, then the litmus will turn _____________. (ii) If M is a non-metal, then the litmus will turn ______________. (iii) If M is a reactive metal, then _________________ will be evolved when M reacts with dilute sulphuric acid. (iv) If M is a metal, it will form ______________ oxide, which will form ____________ solution with water. (v) If M is a non-metal, it will not conduct electricity in the form of __________. #### Selina solutions for Concise Chemistry Class 10 ICSE Chapter 3 Acids, Bases and Salts Exercise Exercise - 3B [Pages 55 - 57] Exercise - 3B \| Q 1.1 \| Page 55 Define the following and give two examples in case a normal salt . Exercise - 3B \| Q 1.2 \| Page 55 Define the following and give two examples in case an acid salt . Exercise - 3B \| Q 1.3 \| Page 55 Define the following and give two examples in case a basic salt. Exercise - 3B \| Q 2.1 \| Page 55 What is a ‘salt’? Exercise - 3B \| Q 2.2 \| Page 55 Answer the following question related to salts and their preparations: What kind of salt is prepared by precipitation? Exercise - 3B \| Q 2.3 \| Page 55 Name a salt prepared by direct combination. Write an equation for the reaction that takes place in preparing the salt you have named. Exercise - 3B \| Q 2.4 \| Page 55 Name the procedure used to prepare a sodium salt such as sodium sulphate. Exercise - 3B \| Q 4.1 \| Page 55 How would you prepare : Copper sulphate crystals from a mixture of charcoal and black copper oxide. Exercise - 3B \| Q 4.2 \| Page 55 How would you prepare : zinc sulphate crystals from zinc dust (powdered zinc and zinc oxide). Exercise - 3B \| Q 4.3 \| Page 55 How would you prepare : sodium hydrogen carbonate crystals. Exercise - 3B \| Q 4.4 \| Page 55 How would you prepare : Calcium sulphate from calcium carbonate Exercise - 3B \| Q 5 \| Page 56 The following is a list of methods for the preparation of salts. A – direct combination of two elements B – reaction of a dilute acid with a metal. C – reaction of a dilute acid with an insoluble base. D – titration of a dilute acid with a solution of soluble base. E – reaction of two solutions of salts to form a precipitate. Choose from the above list A to E, the best method of preparing the following salts by giving a suitable equation in each case: 1. Anhydrous ferric chloride, 3. Sodium sulphate. 4. Copper sulphate. Exercise - 3B \| Q 6.04 \| Page 56 Name a basic salt. Exercise - 3B \| Q 6.05 \| Page 56 Name an acidic salt. Exercise - 3B \| Q 6.06 \| Page 56 Name a mixed salt. Exercise - 3B \| Q 6.07 \| Page 56 Name a complex salt. Exercise - 3B \| Q 6.08 \| Page 56 Name a double salt. Exercise - 3B \| Q 6.09 \| Page 56 Name a salts whose solubility increases with temperature. Exercise - 3B \| Q 6.1 \| Page 56 Name a salt whose solubility decreases with temperature. Exercise - 3B \| Q 7 \| Page 56 Fill in the blanks with suitable words : An acid is a compound which when dissolved in water forms hydronium ions as the only …………… ions. A base is a compound which is soluble in water and contains …………….. ions. A base reacts with an acid to form a …………….. and water only. This type of reaction is known as ……………. Exercise - 3B \| Q 8.1 \| Page 56 What would you observe when Blue litmus is introduced into a solution of hydrogen choride gas? Exercise - 3B \| Q 8.2 \| Page 56 What would you observe when Red litmus paper is introduced into a solution of ammonia in water? Exercise - 3B \| Q 8.3 \| Page 56 What would you observe when Red litmus paper is introduced in Caustic soda solution? Exercise - 3B \| Q 9.1 \| Page 56 Explain why It is necessary to find out the ration of reactants required in the preparation of sodium sulphate. Exercise - 3B \| Q 9.2 \| Page 56 Explain why, fused calcium chloride is used in the preparation of FeCI3? Exercise - 3B \| Q 9.3 \| Page 56 Explain why : Anhydrous FeCl3 cannot be prepared by heating hydrated iron (III) chloride. Exercise - 3B \| Q 10 \| Page 56 Give the preparation of the salt shown in the left column by matching with the methods given in the right column. Write a balanced equation for each preparation. Salt Method of preparation Zinc Sulphate Precipitation Ferrous sulphide Oxidation Barium Sulphate Displacement Ferric sulphate Neutralisation Sodium sulphate Synthesis Exercise - 3B \| Q 11.1 \| Page 56 Give the Ph value of pure water. Does it change if common salt it added to it? Exercise - 3B \| Q 11.2 \| Page 56 Classify the following solutions as acids, bases or salts ammonium hydroxide, barium chloride, sodium chloride, sodium hydroxide, H2SO4 and HNO3 Exercise - 3B \| Q 12.1 \| Page 56 Define the term neutralization, Give a reaction, mentioning clearly acid and base used in the reaction. Exercise - 3B \| Q 12.2 \| Page 56 Define the term neutralization, if one mole of a strong acid reacts with one mole of a strong base, the heat produced is always the same. Why? Exercise - 3B \| Q 13.1 \| Page 56 Write the balanced equation for the preparation of the following salt in the laboratory: A soluble sulphate by the action of an acid on an insoluble base. Exercise - 3B \| Q 13.2 \| Page 56 Write the balanced equation for the preparation of the following salts in the laboratory: An insoluble salt by the action of an acid on another salt. Exercise - 3B \| Q 13.3 \| Page 56 Write the balanced equation for the preparation of the following salts in the laboratory: An insoluble base by the action of a soluble base on a soluble salt Exercise - 3B \| Q 13.4 \| Page 56 Write the balanced equation for the preparation of the following salts in the laboratory: A soluble sulphate by the action of an acid on a metal. Exercise - 3B \| Q 13.4 \| Page 56 You are provided with the following chemicals : NaOH, Na2CO3, H2O, Zn(OH)2, CO2, HCI, Fe, H2SO4, CI2, Zn. Using the suitable chemicals from the given list only, state briefly how you would prepare: (a) iron (III) chloride, (b) sodium sulphate, (c) sodium zincate (d) iron (II) sulphate, (e) sodium chloride? Exercise - 3B \| Q 15 \| Page 56 For each of the salt: A, B, C and D, suggest a suitable method of its preparation. 1. A is a sodium salt. 2. B is an insoluble salt. 3. C is a soluble salt of copper. 4. D is a soluble salt of zinc. Exercise - 3B \| Q 16 \| Page 56 Choosing only substances from the list given in the box below, write equations for the reactions which you would use in the laboratory to obtain: 1. Sodiumsulphate 2. Coppersulphate 3. Iron(II)sulphate 4. Zinc carbonate Dilute sulphuric acid Copper Copper carbonate Iron Sodium carbonate Sodium Zinc Exercise - 3B \| Q 17 \| Page 57 From the formula listed below, choose one, in each case, corresponding to the salt having the given description: AgCl, CuCO3, CuSO4.5H2O, KNO3, NaCl, NaHSO4, Pb(NO3)2, ZnCO3, ZnSO4.7H2O. 1. an acid salt 2. an insoluble chloride 3. on treating with concentratedsulphuric acid, this salt changes from blue to white 4. on heating, this salt changes from green to black 5. this salt gives nitrogen dioxide on heating Exercise - 3B \| Q 18.1 \| Page 57 Ca(H2PO4)2 is an example of a compound called _______ . • acid salt • basic salt • normal salt Exercise - 3B \| Q 18.2 \| Page 57 Write the balanced equation for the reaction of : A named acid and a named alkali. Exercise - 3B \| Q 19.1 \| Page 57 State the term defined by the following sentence : A soluble base. Exercise - 3B \| Q 19.2 \| Page 57 State the term defined by the following sentence : The insoluble solid formed when two solutions are mixed together. Exercise - 3B \| Q 20 \| Page 57 Which of the following methods, (a), (b), (c), (d) or (e) is generally used for preparing the chlorides listed below from (i) to (v). Answer by writing down the chloride and the letter pertaining to the corresponding method. Each letter is to be used only once. (a) Action of an acid on a metal. (b) Action of an acid on an oxide or carbonate. (c) Direct combination. (d) Neutralization of an alkali by an acid. (e) Precipitation (double decomposition). (i) copper(II) chloride. (ii) iron(II) chloride. (iii) iron(IIl) chloride. (v) sodium chloride. Exercise - 3B \| Q 21 \| Page 57 Choose the most appropriate answer from ( SO2 , SiO2 , Al2O3 , CO , MgO , Na2O ) (a) A covalent oxide of a metalloid. (b) An oxide which when dissolved in water form acid . (c) A basic oxide (d) An amphoteric oxide. Exercise - 3B \| Q 22 \| Page 57 Complete the following table and write one equation for each to justify the statement : Reactants Products Method Soluble base + Acid (dil) Salt + water Neutralisation Titration Metal + Non-metal Salt (soluble/insoluble) …………… Insoluble base + .......... Salt (soluble) + water ……………. Active metal + Acid (dil) ………… + ………… ……………. Soluble salt solution (A) + Soluble salt solution (B) Precipitated salt + Soluble salt ……………. Carbonate/ bicarbonate + Acid (dil) Salt + ………. + ………… Decomposition of carbonate Chlorides/nitrates + Acid (conc) …………. + ………… Decomposition of chlorides and nitrates #### Selina solutions for Concise Chemistry Class 10 ICSE Chapter 3 Acids, Bases and Salts Exercise Exercise - 3C [Page 60] Exercise - 3C \| Q 1.1 \| Page 60 What do you understand by water of crystallization? Exercise - 3C \| Q 1.2 \| Page 60 Give four substances which contain water of crystallization and write their common names. Exercise - 3C \| Q 2.1 \| Page 60 Define efflorescence. Give examples. Exercise - 3C \| Q 2.2 \| Page 60 define deliquescence. Give examples. Exercise - 3C \| Q 3 \| Page 60 Answer the questions below relating your answers only to salts in the following list: Sodium chloride, anhydrous calcium chloride, copper sulphate-5-water? 1. What name is given to the water in the compound copper sulphate-5-water? 2. If copper sulphate-5-water is heated, anhydrous coppersulphate is formed. What is its colour? 3. By what means, other than healing, could you dehydrate copper sulphate-5-water and obtain anhydrous coppersulphate? 4. Which one of the salts in the given list is deliquescent? Exercise - 3C \| Q 4 \| Page 60 State your observation when the following are exposed to the atmosphere. (a) Washing soda crystals (b) Iron (III) chloride salts Exercise - 3C \| Q 5.1 \| Page 60 Give reason for the following: Sodium hydrogen sulphate is not an acid but it dissolves in water to give hydrogen ions, according to the equation NaHSO4 ⇆ H+ + Na+ + SO42- Exercise - 3C \| Q 5.2 \| Page 60 Give reason for the following: Anhydrous calcium chloride is used in a desiccator. Exercise - 3C \| Q 6 \| Page 60 Explain clearly how conc, H2SO4 is used as dehydrating as well as drying agent. Exercise - 3C \| Q 7 \| Page 60 Distinguish between drying and dehydrating agent. Exercise - 3C \| Q 8.1 \| Page 60 State whether a sample of the following would increase or decrease in a mass if exposed to air. Solid NaOH. Exercise - 3C \| Q 8.2 \| Page 60 State whether a sample of the following would increase or decrease in a mass if exposed to air. Solid CaCI2 Exercise - 3C \| Q 8.3 \| Page 60 State whether a sample of the following would increase or decrease in a mass if exposed to air. Solid Na2CO3 .10H2O Exercise - 3C \| Q 8.4 \| Page 60 State whether a sample of the following would increase or decrease in a mass if exposed to air. Conc. sulphuric acid Exercise - 3C \| Q 8.5 \| Page 60 State whether a sample of the following would increase or decrease in a mass if exposed to air. Iron (III) Chloride Exercise - 3C \| Q 9.1 \| Page 60 why does common salt get wet during the rainy season? Exercise - 3C \| Q 9.2 \| Page 60 How can this impurity be removed? Exercise - 3C \| Q 9.3 \| Page 60 Name a substance which changes the blue colour of copper sulphate crystals to white. Exercise - 3C \| Q 9.4 \| Page 60 Name two crystalline substances which do not contain water of crystallization. Exercise - 3C \| Q 10 \| Page 60 Name the salt which on hydrolysis forms (a) Acidic (b) Basic acid (c) Neutral solution. Give a balanced equation for each reaction. Exercise - 3C \| Q 11.1 \| Page 60 State the change noticed when blue litmus and red litmus is introduced in the following solution : Na2CO3 solution Exercise - 3C \| Q 11.2 \| Page 60 State the change noticed when blue litmus and red litmus is introduced in the following solution : NaCl solution Exercise - 3C \| Q 11.3 \| Page 60 State the change noticed when blue litmus and red litmus is introduced in the following solution : NH4NO3 Exercise - 3C \| Q 11.4 \| Page 60 State the change noticed when blue litmus and red litmus is introduced in the following solution : MgCl2 Solution #### Selina solutions for Concise Chemistry Class 10 ICSE Chapter 3 Acids, Bases and Salts Exercise Miscellaneous Questions [Page 61] Miscellaneous Questions \| Q 1.1 \| Page 61 Write the balanced equation for the preparation of the following compound (as major product) starting from iron and using only one other substance: Iron (II) chloride Miscellaneous Questions \| Q 1.2 \| Page 61 Write the balanced equation for the preparation of the following compound (as major product) starting from iron and using only one other substance: Iron (III) chloride Miscellaneous Questions \| Q 1.3 \| Page 61 Write the balanced equation for the preparation of the following compound (as major product) starting from iron and using only one other substance: Iron (II) sulphate Miscellaneous Questions \| Q 1.4 \| Page 61 Write the balanced equation for the preparation of the following compound (as major product) starting from iron and using only one other substance: Iron (II) sulphide Miscellaneous Questions \| Q 2 \| Page 61 Write a balanced reaction for the following conversions (A, B, C, D) $\ce{Fe→[\Delta]FeCl2 →[B]FeCO3→[C]Fe(NO3)2→[D]Fe(OH)2}$ Miscellaneous Questions \| Q 3.1 \| Page 61 What is the first step that is required to prepare Lead sulphate from Lead carbonate? Miscellaneous Questions \| Q 3.2 \| Page 61 Write the equation for the reaction that will take place when this first step is carried out. Miscellaneous Questions \| Q 3.3 \| Page 61 Why is the direct addition of dilute sulphuric acid to Lead carbonate an impractical method of preparing Lead sulphate? Miscellaneous Questions \| Q 4 \| Page 61 (a) What are the terms defined by the following? (i) A salt containing a metal ion surrounded by other ions or molecules. (ii) A base which is soluble in water. (b) Making use only of substances chosen from those given below: Dilute sulphuric acidSodium Carbonate Zinc Sodium sulphite Give equations for the reactions by which you could obtain : (i) Hydrogen (ii) Sulphur dioxide (iii) Carbon dioxide (iv) Zinc carbonate (two steps required) ## Chapter 3: Acids, Bases and Salts Intext QuestionsExercise - 3AExercise - 3BExercise - 3CMiscellaneous Questions ## Selina solutions for Concise Chemistry Class 10 ICSE chapter 3 - Acids, Bases and Salts Selina solutions for Concise Chemistry Class 10 ICSE chapter 3 (Acids, Bases and Salts) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Concise Chemistry Class 10 ICSE solutions in a manner that help students grasp basic concepts better and faster. Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Selina textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students. Concepts covered in Concise Chemistry Class 10 ICSE chapter 3 Acids, Bases and Salts are Concept of Acids and Bases, Concept of Molecule, Preparation - Laboratory Preparation of Salts (Normal and Acid Salts) – Relevant Laboratory, General Properties of Salts:, Types of Salts, Use of Litmus and pH Paper to Test for Acidity and Alkalinity, Ions Present in Mineral Acids, Alkalis and Salts and Their Solutions. Using Selina Class 10 solutions Acids, Bases and Salts exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Selina Solutions are important questions that can be asked in the final exam. Maximum students of CISCE Class 10 prefer Selina Textbook Solutions to score more in exam. Get the free view of chapter 3 Acids, Bases and Salts Class 10 extra questions for Concise Chemistry Class 10 ICSE and can use Shaalaa.com to keep it handy for your exam preparation S	{}
# (12x+12)/(4x+16)= Question Equations $$\displaystyle\frac{{{12}{x}+{12}}}{{{4}{x}+{16}}}=$$ 2021-03-05 first part: divide out the 12 from numerator 12x+12 = 12(x+1) divide out 4 from denominator 4x+16 = 4(x+4) $$\displaystyle{12}\frac{{{x}+{1}}}{{4}}{\left({x}+{4}\right)}={3}\frac{{{x}+{1}}}{{{x}+{4}}}$$ second part: x+4 = x+4, nothing to divide out $$\displaystyle{4}{x}^{{2}}-{4}={4}{\left({x}^{{2}}-{1}\right)}$$ divide out 4 from denominator 4(x^2 - 1) = 4(x+1)(x-1) multiply first and second parts together: $$\displaystyle{3}\frac{{{x}+{1}}}{{{x}+{4}}}\cdot\frac{{{x}+{4}}}{{4}}{\left({x}+{1}\right)}{\left({x}-{1}\right)}$$ x+4's cancel out x+1's cancel out answer: $$\displaystyle\frac{{3}}{{4}}{\left({x}-{1}\right)}$$ ### Relevant Questions Solve the equation. $$\displaystyle{{\log}_{{{x}}}{\left({16}-{4}{x}-{x}^{{{2}}}\right)}}={2}$$ Solve the given set of equations for value of x: x-3z=-5 2x-y+2z=16 7x-3y-5z=19 Solve the following exponential equations. $$\displaystyle{16}^{{{1}-{x}}}={2}^{{{5}{x}}}$$ Determine the solutions set of the following equation. Equation in Quadratic form $$4x+\sqrt{3}=1$$ Solve the equation $$4x^{2}=8x-7$$, inequality, or system of equations. Solve the system of equations using the addition/elimination method. 4x+3y=15 2x-5y=1 Consider the following system of linear equations: 4x+2y=25 4x-y=-5 If the value of y is 10 what is the value of X for this system: 1.1.25 2.11.25 3.1.45 4.5 $$\displaystyle{4}{x}^{{{2}}}+{5}{x}+{\frac{{{13}}}{{{8}}}}={0}$$	{}
CGAL 5.4 - 3D Envelopes CGAL::Env_sphere_traits_3< ConicTraits > Class Template Reference #include <CGAL/Env_sphere_traits_3.h> ConicTraits. ## Definition The traits class Env_sphere_traits_3 models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of spheres. Note that when projecting the intersection curve of two spheres (a circle in 3D) onto the $$xy$$-plane, the resulting curve is an ellipse. The traits class is therefore parameterized by an arrangement-traits class that is capable of handling conic curves - namely an instantiation of the Arr_conic_traits_2 class-template - and inherits from it. The conic-traits class defines a nested type named Rat_kernel, which is a geometric kernel parameterized by an exact rational type. Env_sphere_traits_3 defines its Surface_3 type to be constructible from Rat_kernel::Sphere_3. Namely, it can handle spheres whose center points have rational coordinates (i.e., of the type Rat_kernel::FT), and whose squared radius is also rational. The Surface_3 type is also convertible to a Rat_kernel::Sphere_3 object. The Xy_monotone_surface_3 type is the same as the nested Surface_3 type. The traits-class simply ignores the upper hemisphere when it computes lower envelopes, and ignores the lower hemisphere when it computes upper envelopes. Is Model Of: EnvelopeTraits_3 Examples: Envelope_3/envelope_spheres.cpp.	{}
This is a comprehensive words list for words that start with Super. All Free. Prefixes are key morphemes in English vocabulary that begin words. b (1) : exceeding or so as to exceed a norm superheat. 1 a (1) : over and above : higher in quantity, quality, or degree than : more than superhuman. See an example word page ». Sometimes you’d like to hit those kinds of charges “over” the head! As a prefix, super- originates from the Latin super, an adverb and preposition meaning above, on top of, beyond, beside. a prefix occurring originally in loanwords from Latin, with the basic meaning “above, beyond.”. From the Latin preposition super meaning above. For example, super-cooking is cooking while drunk, and super-volleyball is playing volleyball while drunk. A saccade is a twitching, jerking movement, particularly of the eyeball. Start a free trial of Quizlet Plus by Thanksgiving \| Lock in 50% off all year Try it free a prefix occurring orig. Find more ways to say super, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. ‘Hyperactive’ is a word that start with the prefix hyper. That’s just supercalifragilisticexpialidocious! If one considers the inverse function of second order tetration, called the super square root $$\text{ssrt(}x\text{)}$$, we find several other applications, mentioned below. Hey guys, want to get drunk and play super-football? A root word must have an affix or another root attached to it in order to form a true word. Words formed with super- have the following general senses: “to place or be placed above or over” … That’s the original meaning of the English prefix, too, though according to the Oxford Dictionary of English Etymology, it has taken on other senses over the centuries. Personal Learning, The word “super” comes from the Latin word super which means "above, over, beyond". You say ‘hyperactive’ about someone, who is too active and cannot relax. Definition & Meaning: Word Root Hyper Hyper- means ‘too, over, excessive, beyond’. SACCADE. Learn vocabulary, terms, and more with flashcards, games, and other study tools. School. The word ingredient Memlet, shown below, is one of many ways that a word is taught in Membean. Learn these words beginning with the prefix super, meaning "above," "over," or "beyond." https://www.thefreedictionary.com/Super+(prefix), a prefix occurring orig. You need to enable Javascript to get the best out of this site. Super-, sur-, mean over, above, beyond. Start studying super root words. Quick Summary. He is the superstar in the DC Comics line of heroes, or the star that stands “over” other heroic stars, such as Batman and Wonder Woman. Speaking of superstars, the football game that stands “over” all other football games is, you got it, the Super Bowl. The word as we first heard it was super-cadja-flawjalistic-espealedojus. 1. Rhyming reduplication form super-duper first attested 1940. super, supra & supr These ROOT-WORDS are the Prefixes SUPER & SUPR meaning OVER, ABOVE, BEYOND & GREATER IN QUALITY. The prefix super- and its variant sur- mean “over.”. The prefix hyper- is a morpheme, thus, it cannot be divided. That which is superfluous “flows over and above” what is necessary. or get it for your A vocabulary list featuring Power Prefix: super-. An easy way to remember that the prefix super-means “over” is through the comic book hero Superman, who is the man “over” all other men in terms of physical power. First, prefixes and suffixes, most of which are derived from ancient Greek or classical Latin, have a droppable -o-. This is a list of roots, suffixes, and prefixes used in medical terminology, their meanings, and their etymologies.Most of them are combining forms in New Latin and hence international scientific vocabulary.There are a few general rules about how they combine. For example, it can be said that the root of the English verb form running is run , or the root of the Spanish superlative adjective amplísimo is ampli- , since those words are clearly derived from the root forms by simple suffixes that do not alter the roots in any way. In English words from Old French, it appears as sur-. Super Bowl attested from 1966; Super Glue from 1975; as a verb by 1983. Want to increase your prefix power? This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. An easy way to remember that the prefix super- means “over” is through the comic book hero Superman, who is the man “over” all other men in terms of physical power. School systems love to have members of management who stand “over” all others, such as superintendents, who are in charge of entire school systems. super- word-forming element meaning "above, over, beyond," from Latin super (adverb and preposition) "above, over, on the top (of), beyond, besides, in addition to," from (s)uper-, variant form of PIE root uper "over." 2, it is a belt tied over a saddle. Privacy Policy. "Super"="above" is the prefix from the Greek and the root "natural" is what it sounds like so the word means "above and beyond what is considered natural". They supervise, or watch “over” the schools in their respective districts. The adjective super is an abbreviated use of the prefix super -, which comes from the Latin super -, meaning “above,” “over,” or “beyond.” Super is another way to say "the best" — you are above the rest. therm - WordReference English dictionary, questions, discussion and forums. A variant of the prefix super-, which also means “above,” is the morpheme sur-. a super accurate missile; was super careful. It was first recorded in the 1680′s, where it was used as a prefix in “superfine goods”, denoting that the goods are of the highest grade. (2) : in or to an … Root word definition: the form of a word after all affixes are removed \| Meaning, pronunciation, translations and examples All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. Since much of the English language is derived from Latin and Greek, there may be times when the root of a word isn't immediately recognizable because of its origin. The root of a word is a unit of meaning (morpheme) and, as such, it is an abstraction, though it can usually be represented alphabetically as a word. Under definition, beneath and covered by: under a table; under a tree. The primary sense seems to have shifted over time from usually meaning "beyond" to usually meaning "very much," which can be contradictory. This Prefix does not indicate greatness or superiority. Prefixes are key morphemes in English vocabulary that begin words. Learn more on how we help for The prefix super- and its variant sur- mean “over.” Examples using this prefix include superior, supervise, surname, and surface. In No. We all know that the DC Comics hero Superman is the hero who stands “over” all other men in power. Please :-). The roots of “superstition,” which appeared in English in the early 15th century, are the Latin “super” (meaning, as usual, “above”) and the participle form of “stare,” which means “to stand,” giving us a basic sense of “the act of standing over or above.” super- A prefix meaning "while drunk." super (adj.) Root word: Meanings: Origin: Examples and Definitions: a/n: ... the meaning of a word) forsaken or forfeited - completely lost; forgiven - completely given (a release of debt). There you are, on your way to having a SUPER vocabulary! The Super Bowl features the superior teams from the AFC and the NFC divisions facing off against each other, that is, the two teams that stood “over” all the rest during the football season. A surmise is a guess “sent over” available evidence, which is often not enough to prove something. in loanwords from Latin, with the basic meaning “above, beyond.” Words formed with super-have the following general senses: “to place or be placed above or over” (superimpose), “a thing placed over another” (superstructure), “situated over” (superficial) and, more figuratively, “an individual, thing, or property that exceeds customary norms or levels” (superconductivity; superman), … You'll find that the roots listed below are from Greek or Latin and can't stand alone in English; they need something joined to them to make a whole word in English. home / medterms medical dictionary a-z list / super- definition Medical Definition of Super-Medical Author: William C. Shiel Jr., MD, FACP, FACR; Super-: Prefix meaning meaning above, more than normal, or excessive. The surface of something is etymologically the face that lies “over” what it’s covering. Terms of Service. Prefixes are key morphemes in English vocabulary that begin words. Supercell definition is - an unusually large storm cell; specifically : a severe storm generated by such a cell. Another word for super. Now that you have been surrounded with superlative examples of words which contain the prefixes super- and sur-, you will never again have to look those words “over” twice before knowing what they mean. Test Prep, [9] The Oxford English Dictionary defines the word as "a nonsense word, originally used esp. And if you’re … SUPR indicates the SUPERlative. The prefix sub-, with its variants which all begin with su-, is a prolific part of the English language.Examples using this prefix include subway, suffer, supply, and suggest.An easy way to remember that the prefix sub-means “under” is through the word submarine, or a vehicle that travels “under” the sea. The prefix Super is one of the most common prefixes in English. See more. in loanwords from Latin, with the basic meaning “above, beyond.” Words formed with. Acri: bitter (ac… It is important to remember – prefixes begin words. a prefix occurring orig. When you surpass everyone else’s SAT scores at your school, you pass “over” them all, thus getting the highest score. Membean is an incredibly effective way to learn words and permanently remember them. (2) : in addition : extra supertax. For instance, a surname is that name which is “over” a family and thereby identifies it, or the family’s last name. These are vocabulary words from those roots. One who takes a survey of people wants to look “over” what they think. sur This ROOT-WORD is the Prefix SUR which means OVER, ABOVE & MORE. in loanwords from Latin, with the basic meaning “above, beyond.” Words formed with super-have the following general senses: “to place or be placed above or over” (superimpose), “a thing placed over another” (superstructure), “situated over” (superficial) and, more figuratively, “an individual, thing, or property that exceeds customary norms or levels” (superconductivity; superman), … And have you ever been hit with a surcharge on your cell phone bill, those sneaky little charges that go “over” what you are supposed to pay? aukščiausios rūšies, geriausios kokybės, aukščiausio laipsnio, Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content. The prefix super-and its variant sur-mean “over.” Examples using this prefix include superior, supervise, surname, and surface. We found 605 words starting with Super. Review the list below, as well as a few examples of English words that are based on these roots. "first-rate, excellent," 1837, from prefix in superfine (1680s), denoting "highest grade of goods," from Latin super "above, over, beyond" (see super-).Extended usage as a general term of approval is 1895 slang, revived by 1967. Roots - Word parts that hold the central meaning of a word, but cannot stand alone. As in superaspirin, superbug, superjacent, supernumerary, supersize, supertaster. All text and design are copyrighted ©2010-2021 Membean, Inc. All rights reserved. Means ‘ too, over, above, beyond & GREATER in.... Etymologically the face that lies “ over ” the schools in their respective districts over, excessive, ''. A ( 1 ): in addition: extra supertax in English, prefixes and suffixes, most which... Hold the central meaning of a word that start with the prefix super, supra & supr meaning over ''! 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Men in Power all text and design are copyrighted ©2010-2021 Membean, Inc. all rights reserved for your School means. Sur- mean “ over. ” norm superheat 1975 ; as a verb by a. Jerking movement, particularly of the most common prefixes in English the schools in their respective districts in... Is etymologically the face that lies “ over ” what they think a surmise a. Roots - word parts that hold the central meaning of a word, but can not be divided review list... Acri: bitter ( ac… definition & meaning: word root Hyper Hyper- means ‘ too, over above! Meaning over, '' or beyond., which also means “ above, beyond.., supernumerary, supersize, supertaster what is necessary in their respective....	{}
StressUpdateBase Class Referenceabstract StressUpdateBase is a material that is not called by MOOSE because of the compute=false flag set in the parameter list. More... #include <StressUpdateBase.h> Inheritance diagram for StressUpdateBase: [legend] ## Public Member Functions StressUpdateBase (const InputParameters &parameters) virtual void updateState (RankTwoTensor &strain_increment, RankTwoTensor &inelastic_strain_increment, const RankTwoTensor &rotation_increment, RankTwoTensor &stress_new, const RankTwoTensor &stress_old, const RankFourTensor &elasticity_tensor, const RankTwoTensor &elastic_strain_old, bool compute_full_tangent_operator, RankFourTensor &tangent_operator)=0 Given a strain increment that results in a trial stress, perform some procedure (such as an iterative return-mapping process) to produce an admissible stress, an elastic strain increment and an inelastic strain increment. More... void setQp (unsigned int qp) Sets the value of the global variable _qp for inheriting classes. More... virtual void propagateQpStatefulProperties () If updateState is not called during a timestep, this will be. More... virtual bool requiresIsotropicTensor ()=0 Does the model require the elasticity tensor to be isotropic? More... virtual Real computeTimeStepLimit () virtual TangentCalculationMethod getTangentCalculationMethod () void resetQpProperties () final Retained as empty methods to avoid a warning from Material.C in framework. These methods are unused in all inheriting classes and should not be overwritten. More... void resetProperties () final ## Protected Attributes const std::string _base_name Name used as a prefix for all material properties related to the stress update model. More... ## Detailed Description StressUpdateBase is a material that is not called by MOOSE because of the compute=false flag set in the parameter list. This class is a base class for materials that perform some internal computational procedure (such as an iterative return-mapping procedure) to compute an admissible state (which is usually an admissible stress that lies on or within the yield surface, as well as a set of internal parameters such as plastic strains). The computational procedure must return the admissible stress and a decomposition of the applied strain into elastic and inelastic components. All materials inheriting from this class must be called by a separate material, such as ComputeMultipleInelasticStress Definition at line 53 of file StressUpdateBase.h. ## ◆ StressUpdateBase() StressUpdateBase::StressUpdateBase ( const InputParameters & parameters ) Definition at line 35 of file StressUpdateBase.C. 36 : Material(parameters), 37 _base_name(isParamValid("base_name") ? getParam<std::string>("base_name") + "_" : "") 38 { 39 } const std::string _base_name Name used as a prefix for all material properties related to the stress update model. ## ◆ computeTimeStepLimit() Real StressUpdateBase::computeTimeStepLimit ( ) virtual Reimplemented in RadialReturnStressUpdate. Definition at line 55 of file StressUpdateBase.C. 56 { 57 return std::numeric_limits<Real>::max(); 58 } ## ◆ getTangentCalculationMethod() virtual TangentCalculationMethod StressUpdateBase::getTangentCalculationMethod ( ) inlinevirtual ## ◆ propagateQpStatefulProperties() void StressUpdateBase::propagateQpStatefulProperties ( ) virtual If updateState is not called during a timestep, this will be. This method allows derived classes to set internal parameters from their Old values, for instance Definition at line 48 of file StressUpdateBase.C. 49 { 50 mooseError( 51 "propagateQpStatefulProperties called: it needs to be implemented by your inelastic model"); 52 } ## ◆ requiresIsotropicTensor() virtual bool StressUpdateBase::requiresIsotropicTensor ( ) pure virtual Does the model require the elasticity tensor to be isotropic? Referenced by ComputeMultipleInelasticStress::initialSetup(). ## ◆ resetProperties() void StressUpdateBase::resetProperties ( ) inlinefinal Definition at line 117 of file StressUpdateBase.h. 117 {} ## ◆ resetQpProperties() void StressUpdateBase::resetQpProperties ( ) inlinefinal Retained as empty methods to avoid a warning from Material.C in framework. These methods are unused in all inheriting classes and should not be overwritten. Definition at line 116 of file StressUpdateBase.h. 116 {} ## ◆ setQp() void StressUpdateBase::setQp ( unsigned int qp ) Sets the value of the global variable _qp for inheriting classes. Definition at line 42 of file StressUpdateBase.C. 43 { 44 _qp = qp; 45 } ## ◆ updateState() virtual void StressUpdateBase::updateState ( RankTwoTensor & strain_increment, RankTwoTensor & inelastic_strain_increment, const RankTwoTensor & rotation_increment, RankTwoTensor & stress_new, const RankTwoTensor & stress_old, const RankFourTensor & elasticity_tensor, const RankTwoTensor & elastic_strain_old, bool compute_full_tangent_operator, RankFourTensor & tangent_operator ) pure virtual Given a strain increment that results in a trial stress, perform some procedure (such as an iterative return-mapping process) to produce an admissible stress, an elastic strain increment and an inelastic strain increment. If _fe_problem.currentlyComputingJacobian() = true, then updateState also computes d(stress)/d(strain) (or some approximation to it). This method is called by ComputeMultipleInelasticStress. This method is pure virutal: all inheriting classes must overwrite this method. Parameters strain_increment Upon input: the strain increment. Upon output: the elastic strain increment inelastic_strain_increment The inelastic_strain resulting from the interative procedure rotation_increment The finite-strain rotation increment stress_new Upon input: the trial stress that results from applying strain_increment as an elastic strain. Upon output: the admissible stress stress_old The old value of stress elasticity_tensor The elasticity tensor compute_full_tangent_operator The calling routine would like the full consistent tangent operator to be placed in tangent_operator, if possible. This is irrelevant if _fe_problem.currentlyComputingJacobian() = false tangent_operator d(stress)/d(strain), or some approximation to it If compute_full_tangent_operator=false, then tangent_operator=elasticity_tensor is an appropriate choice. tangent_operator is only computed if _fe_problem.currentlyComputingJacobian() = true ## ◆ _base_name const std::string StressUpdateBase::_base_name protected Name used as a prefix for all material properties related to the stress update model. Definition at line 122 of file StressUpdateBase.h. The documentation for this class was generated from the following files:	{}
# New Web Log Engine Blosxom is built on Perl. PyBlosxom is built on Python. Why put in the effort to switch? The langauge might make me pick PyBlosxom to begin with—and I did consider it back when I started this—but it’s hardly enough to switch for. PyBlosxom is maintained; Blosxom largely forks with each user. PyBlosxom has a cleanly added on method for interpreting \$variables. Blosxom appears to eval() the text. And maybe the comments. It may be that I’m mistaken about what Blosxom might do—doesn’t really matter, if it’s taking long enough for me to have to worry about it, I start to consider whether it’ll take less time to switch. As it turns out, PyBlosxom also supports the Blogger XML-RPC upload mechanism, and has better support for Technorati, TrackBack, and similar services. So this web log now runs on PyBlosxom. I’ve added redirectors at the old URLs, but this is now best accessed at http://www.evenmere.org/blog or http://www.evenmere.org/blog/index.atom.	{}
## marked as duplicate by Lord_Farin, mdp, rschwieb, draks ..., Dan RustNov 19 '13 at 22:41 • Then you would love the Crusade of Answers chatroom! That's the entire purpose of that room. Hope to see you there sometime! – rschwieb Nov 18 '13 at 21:29 • @rschwieb Thank you. I feel we have many common interests. Could you leave me a mail for contact? I would delete the post immediately. – Shuchang Nov 19 '13 at 5:25 • @ShuchangZhang I'd like to chat, but first let's talk at the Crusade of Answers. Let me know if you have trouble using the chat rooms here. – rschwieb Nov 19 '13 at 13:47 • @rschwieb I've been there. – Shuchang Nov 19 '13 at 14:07 • @ShuchangZhang It suggests that you answered many questions that were not drawing much attention. This is a commendable effort. It is not rewarded enough with upvotes, but the system is trying to compensate for it by giving you a gold badge. (I just got Tenacious badge myself, which is the silver-level version of Unsung Hero.) – user103402 Nov 20 '13 at 1:55	{}
Creasing of a material at the molecular level What exactly happens when a material (particularly paper or even cloth or a metal) is folded to form a crease? Why is it that a creased material tends to retain form, while a lightly folded one, 'might' just happen to revert to an original configuration? Are bonds being broken here, then to what extent? Are they along a single molecular line or break on a bulk basis? Also, why does a rolled sheet retain shape through a variety of radii? Is there a critical radius to this sheet regaining its flatness? Thanks. • I'm not sure whether this question belongs to the physics SE or the chem one, please let me know if this needs to be posted there. – Abhinav Apr 30 '16 at 16:48 What happens during folding is that the material undergoes plastic deformation. When a sheet of material is bent slightly that deformation is usually elastic, meaning it will return to its original shape when the deforming stress is withdrawn. But when the deformation is larger we enter the plastic zone: the material will no longer fully recover its original shape. A crease can be seen as a permanent deformation, for instance. Plastic deformation causes some permanent (irreversible) changes at the atomic/molecular level but no significant bond breaking. The excessive deformation causes layers of the material at the atomic/molecular level to slide over each other in a non-reversible manner. Precisely what slides over what depends on the micro-crystalline structure of the material: mono-crystalline (silicon chip wafers, e.g.) or multi (micro) crystalline (steel, e.g.) Re. rolled steel sheet, it's obvious that the degree of deformation depends on whether the sheet is closer to the edge or closer to the core of the roll. Undoubtedly there's a critical curvature (dependent on grade of steel) that would push the steel sheet into the plastic zone. I assume manufacturers of rolled steel sheet avoid that critical curvature by making the cores sufficiently large in radius. • Could you comment on the entropy of this system(The sheet and its immediate surroundings)? – Abhinav May 1 '16 at 5:40 • @Abhinav: in the case of crystalline materials I believe the entropic effects are minimal. But for amorphous materials entropic effects prevail. Stretching a rubber band for instance reduces its entropy by introducing some order into the system (its molecules). That's the driving force for the restoring force (elastic response) of a rubber band. – Gert May 1 '16 at 16:10 In the case of inorganic matter, such as metal sheets, folding/creasing produces a substantial bulk stress in the material which can modify the molecular structure in a large number of complex (and not fully understood) ways. For example, it can break bonds, cause amorphisation, and propagate dislocations. These same mechanisms are at play when you cut a material with a knife, see: How does a knife cut things at the atomic level? In the case of paper, however, which is an organic material, its structure consists of a complex network of microscopic fibres which are connected to each other via (weak) hydrogen bonds: When you deform this structure, you cause the fibre-fibre bonds to break, and new ones to form, causing the paper to 'remember' its new shape. I state this based on a simple duplicatable experiment, understanding thin sheet metal or paper also as a plastic material deformation. The still remaining ( or remnant) of strained edges or ridges are tell-tale permanent deformations or folds after deformation. Plate & Shell theories in continuum mechanics should perhaps suffice for an explanation. Going with mechanics of materials understanding... thin paper has some thickness. When unfolded/unravelled after creasing occurs small strain recovery is quite elastic but heavily creased edges must stay put with permanent set pinch marks. To get a physical feel of this phenomenon take a new A4 paper sheet from printer and crumple it in the hand into as small a ball as possible and then attempt to flatten it out again.. to observe (tiny shallow origamis spread uniformly) the new folds better seen when light is incident laterally. There are two types of folds discernable, schematically sketched here. Straight Ridge/crease and U-shaped semi arcs/ of local ring. When seen magnified their middle line has still retained the zero Gauss curvature $K$ geometrically. EDIT1: By virtue of Gauss Egregium theorem $K$ is conserved at its initial zero value. Gauss curvature $K= k_1\cdot k_2 = 0$ product of two curvatures. The first type is visible/recognized as generator lines of cylinders or cones with $k_1 = 0$ The second type is visible as the crown of a torus demarcating between $K<0, K>0$ regions with $k_2 = 0.$ Such folds also appear on Aluminum foil wraps. Imho we are not into the molecular level here. Please comment.	{}
# Phase difference of a standing wave between two points $x_1$ and $x_2$ I would like to refer to the answer by John Rennie to the following (previously asked) question: How to derive the phase difference of a standing wave? It's a short one so please skim thorugh that. Ok, I understand that $$\Delta \phi = k(x_2-x_1)$$, and that was what I expected. But please see the problem below with which I am struggling: I thought that obviously the answer is D, because the separation of 0.60 m is 3/4 of the wavelength and $$\Delta \phi = k\Delta x = \frac{2\pi}{\lambda} \times \frac{3}{4} \lambda = \frac{3}{2}\pi.$$ But the answer, surprisingly, was C. I do not understand how this answer is valid. One can roughly say that the phase difference of $$\pi$$ means that when one point is going up, at that instant the other point is going down. But I would like a both more physically intuitive and mathematically rigorous answer. Here are two suggestions which, combined, make a possible explanation: 1. If it's the string that has a length of $$0.80\rm\, m$$, then the length of the fundamental vibration is $$\lambda_\text{fund} = 1.6\rm\,m$$. Editors get tired, and tired people make typos. 2. Discussions of string harmonics are prone to fencepost errors. Is the fundamental frequency the same as the first harmonic, so that the $$n$$th harmonic has wavelength $$\lambda_n = \lambda_\text{fund}/n$$? Or should the "first harmonic" be the lowest excited state, distinct from the fundamental, in which case the $$n$$th harmonic has wavelength $$\lambda_n = \lambda_\text{fund}/(n+1)$$? Both kinds of labels have advantages, both appear in the literature, and some of us are even guilty of using both conventions in the same paragraph, for which we had to run a lap around the physics building as penance. In the convention where the "first harmonic" and the "fundamental" have different wavelengths, the third harmonic on a string of length $$L$$ has wavelength $$2L/4 = L/2$$, and two points separated by $$3L/4$$ are out of phase by a half-cycle. I don't know whether this is the simplest interpretation of the question or not. I hope you'll let us known once you've figured it out. the third harmonic of a wave with 80cm has wavelength 80/3, that is your fault as I read the text. • Pls read the question again. The wavelength is given by the problem by 80 cm. Aug 31 '20 at 0:31 • I think if the wavelength is 80cm why mention the third harmonic? but I agree , the text can also sustain your interpretation, So it is not clear what the exact question is, Aug 31 '20 at 17:08	{}
# chainerx.arange¶ chainerx.arange([start=0, ]stop, [step=1, ]dtype=None, device=None) Returns an array with evenly spaced values within a given interval. Values are generated within the half-open interval [start, stop). The first three arguments are mapped like the range built-in function, i.e. start and step are optional. Parameters • start – Start of the interval. • stop – End of the interval. • step – Step width between each pair of consecutive values. • dtype – Data type specifier. It is inferred from other arguments by default. • device (Device) – Device on which the array is allocated. If omitted, the default device is chosen. Returns The 1-D array of range values. Return type ndarray	{}
## an email exchange about integral representations I had an interesting email exchange [or rather exchange of emails] with a (German) reader of Introducing Monte Carlo Methods with R in the past days, as he had difficulties with the validation of the accept-reject algorithm via the integral $\mathbb{P}(Y\in \mathcal{A},U\le f(Y)/Mg(Y)) = \int_\mathcal{A} \int_0^{f(y)/Mg(y)}\,\text{d}u\,g(y)\,\text{d}y\,,$ in that it took me several iterations [as shown in the above] to realise the issue was with the notation $\int_0^a \,\text{d}u\,,$ which seemed to be missing a density term or, in other words, be different from $\int_0^1 \,\mathbb{I}_{(0,a)}(u)\,\text{d}u\,,$ What is surprising for me is that the integral $\int_0^a \,\text{d}u$ has a clear meaning as a Riemann integral, hence should be more intuitive…. This site uses Akismet to reduce spam. Learn how your comment data is processed.	{}
Posts matching the query string: Tag=Atheism. My ratings (R) are from 1 to 9. My Dates are also permalinks. Click on the column headers to sort Jots. Feel free to use the address bar like a command line interface by setting the optional query string parameters: Dtm1 (10, 20, 30, YYYYMMDDhhmmss), Dtm2 (YYYYMMDDhhmmss), IsJot (Jot or Not), Tag (zero+ times), NotTag (zero+ times), OrderBy (PostForDate, PostTitle, PostLink, PostText, PostSource, PostRating, PostSize) & Desc(Desc), and Limit(integer). 1. Today I am 40 years old TAGS: Aikido. Atheism. Chicago Swordplay Guild. Faith. Hapkido. Introversion. JavaScript. MARTIAL. Mind. My Stuff. Philosophy. Psychology. Ramblings. Science. TECH. Thoughts. Writing. 2. Anti-religion agenda among social media users TAGS: Atheism. Cyber Life. Faith. 3. Obama mentioned non-believers TAGS: Atheism. Barack Obama. Faith. News. Transcripts. USA. 4. How to respond to requests to debate creationists TAGS: Atheism. Evolution. Faith. Science. 5. Signals and the Shared Road TAGS: Atheism. Faith. Introversion. My Stuff. Philosophy. Psychology. Ramblings. Relations. Thoughts. 6. My Vices TAGS: Atheism. Faith. Food. Introversion. MARTIAL. My Stuff. Psychology. Ramblings. Relations. Self Improvement. 7. More Atheists Shout It From the Rooftops TAGS: Atheism. Culture. Faith. 8. Mitchell and Webb TAGS: Atheism. Crude. Faith. Funny. Health. Quirky. Saucy. TV. 9. Religion update TAGS: Atheism. Ethics. Faith. Mind. My Stuff. Philosophy. Psychology. 10. God is not the Creator, claims academic TAGS: Atheism. Books. Faith. Language. Literature. Text. 11. Cherrypicking Illustrated TAGS: Atheism. Faith. Images. Photos. 20080707 202113 Z Bush Edits Out Jefferson's Religious Views scienceblogs … fersons_reli.php digg.com/pol … ous_Views?OTC-ig Atheism, History, Politics, Quote, USA So typical of Bush. Aside of the religous and political implications, this is just bad quoting by Bush. Jefferson, like many of U.S. founding fathers, was a deist, which practically makes him an atheist. The article thread and digg thread also discuss Jefferson a fair bit. 20080814 142941 Z The 6 Raunchiest, Most Depraved Sex Acts (From the Bible) www.cracked. … -from-bible.html digg.com/env … ammit_pic?OTC-ig Atheism, Ethics, Faith, Funny, Saucy, Sex It's pretty funny and raunchy. It's source material is the Bible as opposed to just making stuff up. No need to make stuff given that the Bible is says some mighty strange stuff. However what gets me about this one are the various reactions from the religious. It's as if they haven't read the Bible or have some dishonest or contrived interpretations of what it says. I've noticed fallacies in the Bible since I was at least seven years old and I see from the various comments that others have too. 20081023 161949 Z "I can't imagine a President being named Obama" nz.youtube.c … ch?v=UwjlUMoLVvA digg.com/com … eing_named_Obama Atheism, Faith, Live Action, News, Politics, TV, USA, Videos Blatant numbing bigotry. I realize that she is just one vile woman but you can't help but feel that her ilk form an important part of the Republican base. 20090501 203937 Z Why I like atheists www.marshall … wordpress/?p=398 www.reddit.c … i_like_atheists/ Atheism, Ethics, Faith It's refreshing to hear a theist speak pleasantly about atheists: "Considering all that, I think atheists deserve some credit. They’re logical, moral and they stand up for what they believe in. Now if we could just get religious people to do the same". 20090811 174914 Z Her: "It's nice to meet nice young Christians like yourself." Me: "I'm actually an atheist." Her: "Excuse me!" www.reddit.c … christians_like/ Atheism, Faith, Relations, Sad Christians and Jews and so on may think they've cornered the market on being prosecuted, but we atheists face all sorts of strange hateful behavior from many so called "spiritual" folks. 20090813 183933 Z (Un)wired For God http://www.newsweek.com/id/211746 Julia Atheism, Faith, Mind, News, Philosophy "Religious beliefs may not be innate". Looking into the brain, hypothesizing, and doing experiments yields stuff that is fascinating. This may sound like religion too, but science does it in a more substantiated way that is open and repeatable. 20090917 184451 Z Man vs. God online.wsj.c … 30643556324.html Atheism, Faith, Richard Dawkins The Wall Street Journal "commissioned Karen Armstrong and Richard Dawkins to respond independently to the question "Where does evolution leave God?" Neither knew what the other would say. Here are the results." 20090918 172405 Z Charles Darwin film 'too controversial for religious America' www.telegrap … ous-America.html digg.com/wor … eligious_America Atheism, Faith, Nature, Show Biz, USA It's embarrassing that the U.S. is the most powerful country in the world and yet so obtuse in certain areas. These people of "faith" who can effectively censor SCIENCE are so cowardly, unctuous, and insecure in their faith. Darwin was human, courteous, and courageous. 20100117 174021 Z Atheist quotes http://i.imgur.com/mk9bl.jpg www.reddit.c … i_found_on_b_of/ Atheism, Images, Philosophy Of the quotes I probably like the one by Epicurus the best: "Is God willing to prevent evil, but not able? Then he is not omnipotent. Is he able, but not willing? Then he is malevolent. Is he both able and willing? Then whence cometh evil? Is he neither able nor willing? Then why call him God?" 20100130 043515 Z Richard Dawkins interviews creationist Wendy Wright (Part 1/7) www.youtube. … ch?v=US8f1w1cYvs Julia Atheism, Evolution, Live Action, Politics, Science, USA, Videos I watched all 7 painful videos. It seems that they agree on the virtue of examination of evidence. They also seem to agree on the humane treatment of people. The difference seems to be that Wright thinks a person's religious belief must influence the certainty of the theory, while Dawkins does not. Knowledge of chemistry, physics, evolution, etc. is objective, testable, open, and separate from ones religious, political, subjective, or ethical point of view. Such objective knowledge can, of course, be used for good or evil, i.e. subjectively applied and interpreted, but that does not change the knowledge itself. Even so, the emotional and spiritual feelings that people have must be dealt with. The woman is clearly pained. 20100321 033030 Z Richard Dawkins: The Greatest Show on Earth fora.tv/2009 … olocaust_Deniers digg.com/art … olocaust_Deniers Atheism, Evolution, Faith, Mind, Politics Richard Dawkins likening Creationists with Holocaust Deniers. Freedom and tolerance are good, but we should be less tolerant of religions that foster bigotry and murder. 20100505 021054 Z You Can't Derive Ought from Is blogs.discov … nt=Google+Reader Atheism, Ethics, Faith, Mind, Philosophy I'm obliged to post yet another follow up to Sam Harris' TED talk related to David Hume's Is-ought problem [W]. 20101008 192552 Z Should the Nonreligious Join in Interfaith Work www.thenewhu … -interfaith-work Julia Atheism, Faith, World Religions aren't going away anytime soon, so we should learn to live together better. 20130114 030459 Z Public Praises Science; Scientists Fault Public, Media www.people-p … lt-public-media/ Atheism, Faith, Politics, Science This survey of 2,500 scientists covers a large number of topics. Some of the findings were expected (87% believe in evolution, 84% believe in human induced climate change), but other finds were a bit more surprising (72% men, 81% white, 6% Republican, 33% believe in God). 2008-10-22t22:04:42 Z \| TAGS: Aikido, Atheism, Chicago Swordplay Guild, Faith, Hapkido, Introversion, JavaScript, MARTIAL, Mind, My Stuff, Philosophy, Psychology, Ramblings, Science, TECH, Thoughts, Writing Today I am 40 years old Today I am forty years old. The purpose of this post is to take a written snapshot of myself on this day, my 40th birthday. So far, today has been a good day to die; Yesterday, not so good. Here's what I wrote yesterday while on my lunch break: 2008-10-21 13:04 CST I think it started yesterday with Colin Powell's story about the gravestone of Khan, but whatever the cause, today I'm feeling heavy and depressed. It's like I need to cry something out. It's so heavy that even though I just made myself get lunch, I haven't touched it yet. My mind is moving in a tiny little space. It doesn't care about perspective or possibilities. I'm tired of having my time stolen from me. I'm tire of pretending. I want room. Perhaps I need to face my limitation, define them, use them, exploit them, revel in them. Although there is great power in working with others or dominating other, although there is great joy in befriending others, although there are advantages in leading, cooperating, collaborating, coordinating, and teamwork, I can barely tolerate it. I am an individual. Multiple factors distinguish today from yesterday but most of them are trivial such as the ebb and flow of my biochemical, physiological, and psychological state. Certainly my breakfast of a warm toasted "everything" bagel with a salmon shmear and a hot chocolate helped. Purposely setting myself in a more meditative with breathing, posture, and by "boxing" my thoughts helped. But happy or sad, you don't really care how you got there once you're there, although you might be interested in what to do once you're there, in which case then examining the path to the state might provide some useful info. Sometimes I like to pull back and see things from a grander perspective. In theory I could pull back and see all of time (around 13 billion years to now) and space (around 93 billion light years), but in practice we're all flashes in the pan, and I can barely comprehend the last 40 years and I often hesitate about cleaning the space of a closet. Science and engineering are wonderful things because they allow us to see further and deeper than the obvious and we can test it to make sure that we're not just making it up. But most people are not concerned about the limits of science and engineering: Most people live in the model that they and their group(s) create. So let me examine my models and their states for a moment. My family. I couldn't ask for a better wife or set of kids. Julia is spicy, complex, independent, and supportive. Connie is creative, detailed, beautiful, has strong feelings, and is becoming a fine young woman. York has his own drummer, is athletic, handsome, kind, and is learning to be man. Amy is girlish, cute, likes to tie things, and is transforming from baby to child. My finances. Financially I'm fine. My family makes more money than we spend. Our only real debt is for the house which has was not hideously overpriced like many houses were in this recent financial crisis. I work as a programmer at ICLOPS [iclops.com], a small business that I partially own, and it is doing well even though [or because?] the health care industry has serious problems in the US. Investing in gold and solar. As I've stated before: For big problems like non-hurricane proof levees, climatically dangerous environmental hazards, the health care industry, economy endangering derivatives, etc. it requires a catastrophe before action will be taken. My exercise. I am largely recovered from knee surgery (a meniscal tear) in 2007-07. My resting heart rate is 60. My weight is 74 Kg = 163 pounds. My weight went up a few pounds with the knee issue, but as I've been recovering my waist fat has gone down somewhat and I have have had noticeable muscle gain in my chest and arms. Here are my scores for the Army PFT for males of my age: 97% for 70 push ups in a row within 2 minutes; Over 100% for 84 sit ups in a row within 2 minutes; 69% for running 2 miles = 3.2 Km in 17:48 (6.75 mph = 10.86 Km/h, or 8:54 per mile). Pull ups are not in the Army PFT, but my average is 21 in a row. In addition to the martial arts classes I run 2 miles at a times, dumbbells, barbells, swimming, heavy bag, speed bag, kata, kicks, stretching, calisthenics, and a mix of other stuff. I am not as consistent with doing simple calisthenics (like these 4: push ups, crunches, spine lifts, and squats) before each shower as I would like. My diet. As usual I effectively do not drink alcohol, smoke, do drugs, or do caffeine. My diet is varied but I'm trying to shift to smaller meals as the day progresses. I'm also trying have meals and snacks after physical activity. My martial arts. I practice Western Martial Arts with the Chicago Swordplay Guild (CSG) [chicagoswordplayguild.com] 1-2 times a week. I've switched to the longsword within the past few weeks because although I achieved "scholar" in rapier in 2006, a scholar in longsword is needed to do the spear. I love the CSG but the restrictions in what one can and cannot practice are annoying. Julia, Connie, York, and I have been doing aikido with the Shinjinkai [shinjinkai.org] since 2008-08. They have a koryu ("old school") or uchideshi ("inside student") emphasis as well as share the space with a Buddhist temple. It's very much a shut up and do it school. Our family was considering Extreme KungFu Wushu Training Center [extremekungfu.com], but the location was inconvenient for us. I'm reading multiple martial arts books at the moment but the two most notable ones are Hapkido: Traditions, Philosophy, Technique (2000) by Marc Tedeschi, and The 33 Strategies of War (2006) by Robert Greene. Tedeschi's book is 1,136 pages and compiles and photographs the many techniques of hapkido. Greene's book is an all time favorite of mine and it covers military and martial principles illustrated with historical accounts from combat both physical and political. If I had to pick one thing from Greene's book it would be the concept that politics is the art of promoting and protecting your own interests, i.e. politics is not just done by politicians, but by everyone, and that it is another arena of combat. My personal martial style is more relaxed, more comprehensive than before. I've also come to the realization that I personally do it for the exercise, the variety, the play, and the fight. The traditions and historical context are interesting but I'm more interested in just doing it than being scholarly about it. I've also come to the realization that practicing martial arts is not a solo activity but a rather a very social activity because you need others to play/fight with. My science and technology. I like nanotech and sustainable/environmental/green tech (especially solar), but I don't do much other than invest. I've become quite a proponent of science as simple stuff that everyone does: exploring, seeking patterns, predicting, gathering evidence, testing, and sharing. In this sense everyone does "science". In computers I've become quite a fan of JavaScript. JavaScript is a functional programming language like Lisp, that looks like C. It is THE language because it is the language of the Web. I use JavaScript on the client-side and on the server-side. I'm still waiting for Space elevators [W] and the Technological singularity [W]. My philosophy. Same as in my About Me page: Epistemologically skeptical but hopeful. Deduction & induction, analysis & synthesis. I'm an empiricist and realist, but I love ideas and rationalizing. Nihilism is moot. Beauty is truthful. The singularity will occur because we can't help ourselves. "The Conversation" is important. We're seek to explore, clarify, and satisfy not obfuscate or lie. HOWEVER, After reading Robert Greene's book, I now see politics in a martial light and thus I now like politics and political philosophy, and see them as unavoidable and hence practical and pragmatic things. My religion. People should be free to explore, seek patterns, predict, gather evidence, test, and share they see fit, i.e. people should be free to do science. People should be free to adapt, to think for themselves. In this modern age with fast global communication, knowledge should be transparent, open, and distributed. Everyone not only lives in a model that they and their group(s) created, but they also look out for themselves and their group(s) in a political sense. In this modern age we are all become tightly knit together: The global problems in the economy and environment showcase this. Balancing, reconciling, combating, politicizing the different individuals and groups with other different individuals and groups is essential. I believe that Wikipedia and Google and YouTube are systems that demonstrate most of these qualities. I am not into Scientism [W], but I believe that non-theist leaning "religions" make themselves the most open and transparent and hence the least restrictive for "science". When you have dogma leaning religions then you tend to have groups with irreconcilable differences. Having rules, laws, courts, mechanisms, etc. that can help resolve inter-group conflicts is important. My politics. Go Obama! "Basically pro-rights tempered with sensible laws and watch-dogging." People focus so much on their differences that we forget the common ground. The government provides many necessary public services (like army, police, streets, laws, courts, etc.) and it needs to be paid for somehow (usually taxes). People should be free to do just about anything alone privately, with others privately (as long as there is consent with the others), and publicly (as long as they do not harm others). My world. Like everyone else, there is only so much time, attention, and money I can give. Like everyone else, I have been focusing on myself, my family, a few friends, a few groups, a few topics, and The World. Like everyone else, those who we do not focus on become "invisible" (hence the need to promote). So then what are MY OWN interests? How do I choose and why? Intellectually I have wide interests, but socially I'm introverted and shy. I can see the political power of dealing with people; I can see the psychological/social/political benefits of making friends. I understand the wonder of creating joy, love, and laughter. But to be honest, I'd rather be alone or silent unless we had something quite sincere (politically, emotionally, intellectually, socially, etc.) to say to each other. Although you'd think that "noise" is something we could all do without, I think for some people the "noise" is more bearable than the "silence". Some people can more comfortably tolerate "noise" (that they make or hear from others) in order to get any sort of "music". Is it hubris on my part that I can attend a social gathering and say there's just too much "noise"? I don't know. I've probably been introverted or socially inept for most of my life and now that I'm forty, I'm ready to just accept it instead of feeling guilty about it or sorry for myself or apologizing for it. Time to go home and have some cake with my family. Happy Birthday me! 2008-11-07t20:31:27 Z \| TAGS: Atheism, Cyber Life, Faith Anti-religion agenda among social media users This comment in the Digg thread expressed my opinion: I don't think Digg is anti-religion its just pro-logic. Diggers don't hate on religion, we just hate when people use religion to interfere in the liberty of others. Example: gay marriage; a religious person see a religious issue, a Digger sees a question of freedom for a social minority. Keep your religion to yourself and Diggers will keep religious barbs at bay, unless there is a good LUL to be had. It's not a matter of being anti-religious as a-religious. In a similar fashion the media is not anti-Conservative, but pro-"liberal": As in "broad-minded", as in wanting to see issues from multiple perspectives, as in a neutral point of view. Another perspective is most religions are anti-religious to other religions. EG: Christians don't believe in Zeus or Thor, i.e. to a Christian, his or her beliefs are super natural yet real, while other beliefs are superstitions. Atheists, agnostics and the like generally don't believe in the super natural. What is funny or interesting is blind to religion, race, gender, sexual orientation, or culture. The issue of taste is another story. 2009-01-21t20:44:39 Z \| TAGS: Atheism, Barack Obama, Faith, News, Transcripts, USA Obama mentioned non-believers For we know that our patchwork heritage is a strength, not a weakness. We are a nation of Christians and Muslims, Jews and Hindus — and non-believers. We are shaped by every language and culture, drawn from every end of this Earth; and because we have tasted the bitter swill of civil war and segregation, and emerged from that dark chapter stronger and more united, we cannot help but believe that the old hatreds shall someday pass; that the lines of tribe shall soon dissolve; that as the world grows smaller, our common humanity shall reveal itself; and that America must play its role in ushering in a new era of peace. 2009-02-19t19:34:13 Z \| TAGS: Atheism, Evolution, Faith, Science How to respond to requests to debate creationists Sweet. Instead of spending time on public debates, why aren't members of your institute publishing their ideas in prominent peer-reviewed journals such as Science, Nature, or the Proceedings of the National Academy of Sciences? If you want to be taken seriously by scientists and scholars, this is where you need to publish. Academic publishing is an intellectual free market, where ideas that have credible empirical support are carefully and thoroughly explored. Nothing could possibly be more exciting and electrifying to biology than scientific disproof of evolutionary theory or scientific proof of the existence of a god. That would be Nobel Prize winning work, and it would be eagerly published by any of the prominent mainstream journals. 2009-04-17t17:00:53 Z \| TAGS: Atheism, Faith, Introversion, My Stuff, Philosophy, Psychology, Ramblings, Relations, Thoughts This post is one of my occasional "brain dumps". I do it because thoughts float around in my head and if I don't do something specific with them, then they disappear, decay, or clutter up my head. So in lieu of doing something specific with these thoughts, I dump them out of my head by writing them down quickly without worrying too much about parsing them. A brain dump is about recording the "data" without worrying about distilling the "info". That is I'm supposed to be concerned about improving the signal-to-noise_ratio (SNR) [W] ("the level of a desired signal (such as music) to the level of background noise. The higher the ratio, the less obtrusive the background noise is"). ## Signals Speaking of SNR, some of my recent thinking has to do with "the signal". One of the reasons I do my brain dumps is because my head has become too noisy, but I know that there must be some signal in there, so a brain dump may help to distill the signal. However I'm just one person: There are billions of other heads on this planet, and hence billions of SNR situations. Also it's not just about thoughts, the signal may refer to any chunk of "data" that is "info". The signal is any data that is info, beautiful, or true. The idea is partially a variation of John Keats' line: "Beauty is truth, truth beauty, - that is all ye know on earth, and all ye need to know.". The signal may be an idea. The signal may be physical beauty. The signal may be a performance. The signal is both a class/concept and an instance/execution. I like using the word "signal" because there is not only the concept of the the signal (as in the info, the beauty, the truth), but also the concept of persisting, perpetuating, reproducing, permutating, sharing, and strengthening the signal, as well as the opposites: ending, reducing, hiding, and weakening the signal. Parents who miscarry only just became aware of the signal of the child before they lost the signal. Parents who lose a young child only enjoyed the new signal for a short while before it went out. When someone dies that you didn't know, it is the loss of a signal that you never heard. When someone dies that you knew well, it is the loss of a familiar signal. When someone bright and dear dies, it is the loss of a beautiful signal. We all have SNR. Some are able to share their signal externally by expressing themselves through speech, touch, writing, art. Some are able to persist their signal by the written word, photos, videos, monuments, memes. Some strengthen their signal (or the signal of others) by perpetuating, sharing, promoting, protecting, or permutating the signal. Some hide their signal. Some don't know to share their signal. Some have signals with great potential. Some have redirected signals. Some share their signal with a select group. Groups are strongest with a common signal or cause, that all in the group want support. Some blacken signals, do graffiti on signals, or diminish signals. There are signals in nature. The beauty of a bird or a cat is a signal. The wonder of a sunrise or the stars at night are signals. We seek out signals. We compare them and find ones that we attune to our own signal. We chose signals. A break up, a divorce, a separation, is a rejection of a signals, of non-harmonious signals, a devaluing of a signals. I think this usage of the word "signal" is similar to the word "soul", but the word "signal" is secular, more flexible, and yet less abstract. A friend of my recently left my company. We worked together for around 7 years. It got me to thinking of life as a shared road. The concept of life as a road or a river isn't new. Possibly the most famous usage comes from J.R.R. Tolkien who had at least 3 walking songs that start with "The Road goes ever on". See The Road Goes Ever On (song) [W]. While Tolkien's road focuses on the adventures on the road and coming home, my recent thoughts on the road have to do with sharing the road. We can make some choices on the road. Perhaps we can choose which road to start on, or who we'll travel with. But often we don't have a choice about where we get set on the road, or about the random things that may happen on the road, or who we might randomly meet on the road, or who we might have to travel with on the road. Most of the people we meet on the road are strangers. Some we have to talk with to get groceries and such. Some we have to talk with because they're classmates or co-workers or neighbors. Some of us are more extroverted and want to talk with more on the road, while some of us are more introverted and want to talk with fewer on the road. Let's assume that there are various causes for sharing the road that range from having a common "signal" (school, job, locality, political interests, etc.), to satisfying human social needs, to being attuned to the specific signal in an individual. The thing is we all get born, live, and die. that is metaphorically we all get on the road, we all travel on roads that we share with others, then we get off the road. I'd think that most people would want that shared road time to be meaningful at the least. Some may have additional requirements for that shared road time. Some may want it to be pleasurable, safe, sincere, adventurous, restful, productive, warm and fuzzy, and so on. In any case, our personal time on the road of life is finite. We spend minutes, hours, days, weeks, months, and years at a time as if time was cheap. That time you were "forced" to spend with a "nobody" or nobody? Well guess what? It cost the same time as the time you "chose" to spend with a "somebody". Some people may interpret this as "Let's hurry this up because I don't want to waste another second with a nobody". I prefer to think that regardless of what part of the road I'm on, I want to be mindful of myself, the road, and anybody (if any) that I'm sharing that road with --even if they are a "nobody". Another aspect of the shared road I've been thinking about has another temporal/spatial aspect. If you share the road with someone, and you never meet them again, then it's as if that person died. When someone dies, you never see them again in person. It's like they've permanently moved to another dimension or you permanently moved to another period in time. In one sense all the people in your past that you'll never meet again, have died. This sort of thinking can be depressing, because you might realize that all the people through out history that are dead, and that all the people in the future that you'll never meet are in a sense dead, and that all the people that you won't meet because of geography are in a sense dead, and that even in the same city all those strangers you don't meet are in a sense dead. If you don't receive a fairly regular signal from a person, then the probability is that their signal will fade to nothingness, and hence they are effectively dead. When people move away or fall out of touch, then it is as if they have died. What's sadder is a loveless relationship where you can physically get their signal, but they're not sending you a real signal, the signal you knew, so it's as if they died. The silent treatment can be very harsh. Sometimes we may be giving people the silent treatment, and not be totally aware of it. I tend to be introverted and task focused so I'm almost certainly guilty. Anyhow the point of this shared road thought is not to make you or me depressed, but to make us mindful of the shared road and to earnestly live it. It's best to physically intimately directly get their signal by taste, touch, smell, sound, and sight. However in lieu of that a phone call, a video call, a text, an email, a letter, a note in facebook, can do wonders. 2009-04-27t20:42:29 Z \| TAGS: Atheism, Faith, Food, Introversion, MARTIAL, My Stuff, Psychology, Ramblings, Relations, Self Improvement My Vices A co-worker of mine recently mentioned that I had no vices. I quickly and instinctively replied that it was a false statement because I like my salt, I like my sweets, and I procrastinate. That little exchange got me to thinking about my vices. I've explored and written about the seven deadly sins [W], virtue [W] and variations before and I may go over them a little bit today, but my emphasis for now is to cover just a little corner of the topic from my perspective, as opposed to covering the topic in general. The Seven Deadly Sins (in Latin & English) and their corresponding virtues: • Luxuria = Lust v Chastity • Gula = Gluttony v Temperance • Avaritia = Greed v Charity • Acedia = Sloth v Diligence • Ira = Anger v Patience • Invidia = Envy v Kindness • Superbia = Pride v Humility The Four Cardinal Virtues of the Ancient Greeks (and their corresponding vices): • Prudence v Folly • Temperance v Lust • Justice v Venality • Courage v Cowardice Some Christian vices of the spirit: • Blasphemy (holiness betrayed) • Apostasy (faith betrayed) • Despair (hope betrayed) • Hatred (love betrayed) • Indifference (scripturally, a "hardened heart") GLUTTONY and folly. While I don't really drink alcohol, eat too much, eat too many simple carbs, smoke, or do drugs, gluttony is still indeed a vice of mine. I'm a saltaholic and I have a fair sweet tooth, but they are also not out of control or ruining anybody's life. My blood pressure, my weight, my teeth and my budget are all good, but in the long run I will probably want modify my diet. SLOTH and folly. I have certain concrete slothful behaviours. I play WarCraft a bit too much --I've quit and restarted several times. I procrastinate on taking the trash out --Yes, I let it pile up in our back porch before I take it out. Those are all fairly small things. Having sloth and procrastination means that I don't live up to my potential in small ways (EG: Getting out of bed sooner) and big ways (EG: I don't "save the world" as much as I could). Sloth is probably my biggest vice. There are many causes for sloth: There is a fear of failing to live up to the potential, there is a fear of admitting the limits of ones potential, there is the fear of change or making decision, there is the fear of imperfection or incompleteness, etc. etc. Ignore all the noise. Screw the butterflies. Don't be too worried about mistakes or failures or looking good. Encourage action or progress no matter how small. LUST. When I was younger, my thoughts and hormones pushed that way quite a bit. Thankfully things have settled down somewhat. Every body is different so we'll see how my changes as I get older. As a species we need some lust to reproduce. Lust is spicy and natural, and there will always be some folks who will over do it, but usually some temperance fixes it. The Lust/Chastity pair is overrated. INTROVERSION. I'm on the fence on this one. People are born with physical and personality traits like black hair or introversion. While, people can easily change their hair color, it is harder to adjust a person's degree of introversion. I don't think introverts should consider themselves to have vices like sloth, folly, cowardice, despair, and indifference just because they're introverted. However we introverts do need to practice in order to increase our comfort level and our skills because communicating is important, prevalent, and often necessary. I've been thinking about Signal-to-Noise Ratio recently and an introvert is like an absence of signal, which can be disconcerting for some, and could inadvertently pass the wrong info along. MARTIAL ARTS. I do not consider martial arts to be a sin of any sort. I'm perfectly capable of doing and enjoying violence, but there is no anger or malice in it. Even ants will defend themselves. Martial arts is good fun exercise that involves diligence, patience, humility, prudence, a sense of justice, courage, hope. The physical, tactical, and strategic skills and plays involved are wide and deep, and so are the social, cultural, historical, and scientific components. ATHEISM. I do not consider atheism to be a sin of blasphemy or apostasy, any more than noting that 1 + 1 = 2, or that at standard temperature and pressure, most gases behave like an ideal gas. If I thought I was a better human being just because I'm an atheist, then that would be pride. (Although some atheists might say that we should be proud that we don't need divine wrath to make us nice to others.) If I thought everyone should be forced to immediately become atheist, than that would be folly and indifference. 2009-04-28t21:21:33 Z \| TAGS: Atheism, Culture, Faith More Atheists Shout It From the Rooftops Come out into the sunshine! -Your Friendly Neighborhood Atheist Polls show that the ranks of atheists are growing. The American Religious Identification Survey [http://livinginliminality.files.wordpress.com/2009/03/aris_report_2008.pdf], a major study released last month, found that those who claimed “no religion” were the only demographic group that grew in all 50 states in the last 18 years. Nationally, the “nones” in the population nearly doubled, to 15 percent in 2008 from 8 percent in 1990. In South Carolina, they more than tripled, to 10 percent from 3 percent. Not all the “nones” are necessarily committed atheists or agnostics, but they make up a pool of potential supporters. 2009-07-09t21:35:26 Z \| TAGS: Atheism, Crude, Faith, Funny, Health, Quirky, Saucy, TV Mitchell and Webb Who forgot to tell me about Mitchell and Webb? The British comedians are awesome! Of course, those youtube links lead you to other stuff like Storm by Tim Minchin (with text). 2009-09-24t21:09:52 Z \| TAGS: Atheism, Ethics, Faith, Mind, My Stuff, Philosophy, Psychology Religion update Religion update [http://www.georgehernandez.com/h/xzMisc/Philosophy/Religion.asp] I've updated my intro to my page on Religion so I'm posting it here for archival purposes: What is religion? There are many different definitions of religion (EGs: Religion [W] or Religion [dictionary.com/...]). Let me try a few cheesey tricks to start defining religion. One easy trick would be to use free association to come up with a list religious keywords and their complement. faith : reason spiritual : psychological supernatural : natural superstition : real intuition : experience magical thinking : scientific method mysticism : fact sacret texts : experimentation ritual : protocol ceremony : civil action dogma : chaos/freedom myth : nonsense moral : immoral mortality : immortality immortality : mortality after life : death community : individual individual : community creation : evolution cosmology : the present holy days : holidays love : hate God : Devil or nothing Obviously it's an imperfect list for several reasons. Dualism isn't everything; People may come up with other opposites; Some things go on both sides; And so on. The list is merely an exercise to think about the topic. With religion, there is not merely an epistemological issue of your knowledge of the nature of man and the universe, but a component that tells you and your community how to behave. However since the behavioural component is built upon the epistemological one, then it may be said that the epistemological encompasses the behavioural component. The epistemological component is the key. Non-religious epistemology relies on: • Logic and reasoning. • Math. • Empirical evidence. Measurement, observation, and experimentation. Rigorously and openly applied, this is the scientific method. • Intuition. Feelings and emotions. Inference without knowing how. This not only covers warm and fuzzy feelings but all satori, moments of insight and genius too. This might even be described as spiritual. Religious epistemology relies on non-religious epistemology as well, but those non-religious knowledge is trumped by mystical/supernatural insights and citations. Religions have used all manner of strong psychological, cultural, aethetic, social, and poltical devices to make their "supernatural" as real as the natural. Religious practices often include the following: • prayer; meditation; confessions; mantras; • chanting; hymns; psalms; songs; dancing; • sacrifice and burning: incense; smoking; food, drink; animals; plants; people; • rituals; pilgrimages; gatherings; ceremonies; • holidays; festivals; feasts; • temples; churches; shrines; A religion's "supernatural" aspects are clearly superstititious to those outside of the religion, but to those within the religion, their supernatural aspects are real. I personally do not believe in the supernatural, whether God or gods or demon or ghosts. However given the strong pressures that religions and their communities apply, I can empathise with those who believe. Humans are prone to religion and we love fiction. Voltair said "if God did not exist, it would be necessary to invent Him". Here are is personal view on religion in brief: • I am pragmatic and secular --not cynical or anti-religious. • I believe in nature and science. I believe that we are biologically inclined to want to believe. I am a naturalist but I acknowledge emergent experiences, complex experiences, and unexplained experiences that may truthfully feel spiritual. I believe that individuals and groups are at different stages in their own personal spiritual and emotional journeys, so I am generally tolerant and respectful of where people are. I believe that we can benefit from exploring and comparing different religions --and that it is part of our rich human heritage to do so. • I believe that there are cultural, and social and/or emotional benefits to committing to a non-secular religious system, but I also believe that humanity is very young and that we will have many faiths for a long time yet, thus it would be beneficial for people of different faiths to participate in a broader secular system. I believe that government should be neutral on religion and that religious convictions must be translated into secular and ethical terms when becoming policy. • I believe that we are individuals and social animals with a sense of self-interests and expanding group-interests. Problems arise when people cannot see or do beyond the interests of themselves or their tribe and move on to the interests of the species and the world. I believe that people will tend to be behaviorally, psychologically, and socially "good". • I was was born and raised a Catholic Christian, but I am only a nominal Catholic Christian. I believe that there are many who participate in their faith more for cultural, social, and/or emotional reasons than for literal mystical reasons. • I am an Atheist. (Yes, my mom knows.) I am not evil or immoral. I'm a human being like everyone else. 2009-10-16t18:15:14 Z \| TAGS: Atheism, Books, Faith, Language, Literature, Text God is not the Creator, claims academic This makes sense: Prof Van Wolde, 54, who will present a thesis on the subject at Radboud University in The Netherlands where she studies, said she had re-analysed the original Hebrew text and placed it in the context of the Bible as a whole, and in the context of other creation stories from ancient Mesopotamia. She said she eventually concluded the Hebrew verb "bara", which is used in the first sentence of the book of Genesis, does not mean "to create" but to "spatially separate". The first sentence should now read "in the beginning God separated the Heaven and the Earth". 2009-10-17t14:13:13 Z \| TAGS: Atheism, Faith, Images, Photos Cherrypicking Illustrated You can't tell from the context whether the person with the tatoo knows about the latter verse. It's a tattoo reading “[Thou] shall not lie with a male as one does with a woman. It is an abomination. Leviticus 18:22″. Who else sees the problem here? Leviticus also forbids tattooing. In the very next chapter. “Ye shall not make any cuttings in your flesh for the dead, nor print any marks upon you: I am the LORD. Leviticus 19:28″ I also like the fellow mentioned in the Reddit thread: Someone at my university has a tattoo that is just a chapter and verse number -- when you look it up, it's the leviticus prohibition on tattooing. Clever. Page Modified: (Hand noted: 2007-07-19 21:41:10Z) (Auto noted: 2010-12-24 22:08:01Z)	{}
Share # Solve the Following Quadratic Equations by Factorization: M/Nx^2+N/M=1-2x - CBSE Class 10 - Mathematics ConceptSolutions of Quadratic Equations by Factorization #### Question Solve the following quadratic equations by factorization: m/nx^2+n/m=1-2x #### Solution We have been given m/nx^2+n/m=1-2x (m^2x^2+n^2)/(mn)=1-2x m2x2 + 2mnx + (n2 - mn) = 0 m^2x^2+mnx+mnx+[n^2-(sqrt(mn))^2]=0 m^2x^2+mnx+mnx+(n+sqrt(mn))(n-sqrt(nm))+(msqrt(mnx)-msqrt(mnx))=0 [m^2x^2 + mnx + msqrt(mnx)]+[mnx-msqrt(mnx)+(n+sqrt(mn))(n-sqrt(mn))]=0 [m^2x^2 + mnx + msqrt(mnx)]+[(mx)(n-sqrt(mn))+(n+sqrt(mn))(n-sqrt(mn))]=0 (mx)(mx+n+sqrt(mn))+(n-sqrt(mn))(mx+n+sqrt(mn))=0 (mx+n+sqrt(mn))(mx+n-sqrt(mn))=0 Therefore, mx+n+sqrt(mn)=0 mx=-n-sqrt(mn) x=(-n-sqrt(mn))/m or mx+n-sqrt(mn)=0 mx=-n+sqrt(mn) x=(-n-sqrt(mn))/m Hence, x=(-n-sqrt(mn))/m or x=(-n-sqrt(mn))/m Is there an error in this question or solution? #### APPEARS IN Solution Solve the Following Quadratic Equations by Factorization: M/Nx^2+N/M=1-2x Concept: Solutions of Quadratic Equations by Factorization. S	{}
# Tag Info 12 Going with Patrick87's hash idea, here's a practical construction that almost meets your requirements — the probability of falsely mistaking a new value for an old one is not quite zero, but can be easily made negligibly small. Choose the parameters $n$ and $k$; practical values might be, say, $n = 128$ and $k = 16$. Let $H$ be a secure cryptographic ... 10 The second paragraph of the Wikipedia article on Bloom filters says the following, with a citation to Bloom's original 1970 paper. Bloom proposed the technique for applications where the amount of source data would require an impracticably large hash area in memory if "conventional" error-free hashing techniques were applied. He gave the example of a ... 8 No, it is not possible to have an efficient data structure with these properties, if you want to have a guarantee that the data structure will say "new" if it is really new (it'll never, ever say "not new" if it is in fact new; no false negatives allowed). Any such data structure will need to keep all of the data to ever respond "not new". See pents90's ... 7 I couldn't find the source, but the idea is simple: Use additional bloom filter to represent the set of the deletions. As this is a very simple solution, it might be considered as a folklore. Anyway, I found a short reference to this solution in the following paper (Theory and Practice of Bloom Filters for Distributed Systems): http://www.dca.fee.unicamp.... 7 I think your reasoning is in principle correct. Perfect hashing is an alternative to Bloom filters. However, classical dynamic perfect hashing is rather a theoretical result than a practical solution. Cuckoo hashing is probably the more "reasonable" alternative. Note that both dynamic perfect hashing and standard cuckoo hashing performance is only expected ... 6 Using the formula from wikipedia for Bloom filter false positives, your proposal would have a false positive probability of about 0.00726%. This assumes, among other things, that good hash functions are used. The formula is: $(1 - (1 - [1/m])^{kn})^k$ where $m$ is the number of bits in the filter, $k$ is the number of hash functions and $n$ is the number ... 6 Given that you want to insert $n$ words into the Bloom filter, and you want a false positive probability of $p$, the wikipedia page on Bloom filters gives the following formulas for choosing $m$, the number of bits in your table and $k$, the number of hash functions that you are going to use. They give $m = - \frac{n \ln p}{(\ln 2)^2}$ and $$k = \frac{m}{... 6 What about just a hash table? When you see a new item, check the hash table. If the item's spot is empty, return "new" and add the item. Otherwise, check to see if the item's spot is occupied by the item. If so, return "not new". If the spot is occupied by some other item, return "new" and overwrite the spot with the new item. You'll definitely always ... 5 A Bloom filter never gives a false negative. This is the property that makes them desirable in many situations. They are appropriate for any situation where the existence of false positives is a potential performance issue, but not a correctness issue. One obvious example is where the filter is used to avoid a more expensive check. Let's try to quantify ... 5 This balances two considerations: The more hash functions you have, the more bits will be set in the Bloom filter when you create the filter (because each element you insert causes up to k bits to be set in the filter). The more bits that are set, the higher the risk of false positives. The more hash functions you have, the less likely that one of them ... 4 In the case where the universe of items is finite, then yes: just use a bloom filter that records which elements are out of the set, rather than in the set. (I.e., use a bloom filter that represents the complement of the set of interest.) A place where this is useful is to allow a limited form of deletion. You keep two bloom filters. They start out empty.... 4 You have asked two questions. I will answer them one by one. Choosing the seed. The usual approach here is to choose the seeds randomly once and for all, and to hard-code them. If the hash family is reasonable, then all small sets of seeds should look "the same". I don't know the Murmur3 family, but it's probably reasonable enough. Note that linear ... 3 Let me simplify your third step: count the number of elements in your multiset which are not in the map, and add to it the number of elements in the map. Suppose that your elements are x_1,\ldots,x_n. For a given hash h, let x_{i_1},\ldots,x_{i_\ell} be the elements hashing to h (in order). If \ell = 1, then the elements won't be added to the map ... 3 This is how I use Bloom filter: suppose that full dictionary check is 10x slower than checking a bit in the Bloom filter. Then if you make a Bloom filter having N bits per dictionary word, then average time required for one check would be 1+10/N. For example, with N=8 (i.e., use one byte in Bloom filter per dictionary word), the avg.time will be 2.25 ... 3 Let's analyze how many hash bits you need in your new scheme versus a Bloom filter. First of all, we need to agree about terminology. I will use q to represent the probability of a false positive. For a Bloom filter the design problem of choosing m and k given that you want to hold n elements with false positive rate q is solved by k = -\lg_2 q... 3 This is explained in Wikipedia. Given n,m, the false positive probability is$$ \left(1 - \left(1 - \frac{1}{m}\right)^{kn}\right)^k. $$This is the quantity we want to minimize. While the exact expression is hard to minimize exactly, we can use the approximation$$ \left(1 - \left(1 - \frac{1}{m}\right)^{kn}\right)^k \approx (1-e^{-kn/m})^k, $$which is ... 3 If the query results are mostly false, the answer will be returned in O(1) on average. (In traditional Bloom filters, negative results are faster than positive ones.) This might be slow, however, since the lookups are random and have bad cache behavior. There are a few ways to fix this. I'd suggest: Use a blocked bloom filter (from Cache-, Hash- and ... 2 You choose the size of your Bloom filter according to the expected occupancy. If your Bloom filter is full of 1s then it's too small for the number of words you're putting it. Words in a document tend to repeat, so when estimating how big the filter should be, count the number of unique words in your document. 2 I think the Bloom filter gives you something the perfect hash function does not - it can test membership. The PHFs I know return some answer for any key you apply them to. If the key you supplied is not in your hash set, some value is still supplied. This is fine if you are storing all of the keys that are in your set somewhere and the PHF just gives a ... 2 A bit is set to 1 if it has been hit. It has k\|S\| chances of being hit, and each time it is hit with probability 1/n. Using a union bound, the probability that a bit is hit is at most k\|S\|/n. We can improve on this bound by calculating instead the probability that a bit is 0, that is, that it is not hit. For a bit not to be hit, it has to be missed ... 2 This is related to the notion of "fast path, slow path" from computer systems. In systems, one common optimization method is: if there is a common case that can be handled fast and is common, first test whether the input falls into that case and if so solve it and return immediately; otherwise, fall back to the complex computation. For instance, the slow ... 2 Your question includes the equation$$ (1 - (1 - 1/n)^{km})^{k} = (1 - e^{-k/c})^{k} $$But that is not actually an equation; it is only an approximation. 2 Build a k \times k table ans of answers, storing in each entry the smallest (according to some total order) element in that intersection or a sentinel value (e.g. -1) if the intersection is empty, and also maintain a mapping from each element to the set of all sets that contains it (e.g. using a hashtable of hashtables). When you add an element x to a ... 1 A Bloom Filter will typically be used to eliminate mismatches quickly, since it produces true negatives, but some false positives. A join (specifically, an equi-join) which is expected to have some non-matching keys can be sped up by pre-processing the valid keys from one table into a Bloom filter. The join operator tests each key from the other table ... 1 A Bloom filter doesn't have "buckets". You can make a Bloom filter of any size you want. The smaller it is, the higher the false positive rate. This should be covered in any good introduction to Bloom filters, of which there are many. 1 In the same way that 0.5 is the error of a coin flip, 0.166666 is the error of a die throw. There's nothing sacred in the probability 0.5. A Bloom filter gives you a guarantee that a coin flip cannot – it has no false negatives. Even if its false positive rate is 0.9, it still gives you some information. Whether or not a high false positive rate is ... 1 Yes, it is possible. Use$$h_2(x) = \text{hash}(\text{signature}) - h_1(x) \bmod n.$$The theory behind this: if c is a constant, the function$$f(t) = c - t \bmod n$$is an involution for any c and any n, since$$f(f(t) = c - (c-t) = t \pmod n. Therefore, we can use the hash of the signature as the constant $c$. This gives you a scheme that ... 1 Let $S,T$ be two sets of size $n$. Suppose we hash each to a $m$-bit Bloom filter, using $k$ hash functions; let $x_S$ be the $m$-bit vector corresponding to $S$, and $x_T$ the $m$-bit vector corresponding to $T$. If $S,T$ agree in a $p$ fraction of entries (i.e., $\|S \cap T\|=pn$), then the expected value of the Hamming distance between these two bit-... 1 I just want to add in here, that if you are in the fortunate situation, that you know all of the values $v_i$ that you might possibly see; then you can use a counting bloom filter. An example might be ip-addresses, and you want to know every time one appears that you have never seen before. But it is still a finite set, so you know what you can expect. The ... 1 Consider Bloom filters in which $k$ hash functions are used. If we look up a value which is in the filter, then we always conclude that it is indeed in the filter. Things are more complicated when we look up a value which is not in the table; we could erroneously conclude that it were in the table. Suppose the filter is of size $n$, and $m$ values were put ... Only top voted, non community-wiki answers of a minimum length are eligible	{}
## Calculus (3rd Edition) $$\int_0^{\pi/4}\sqrt{1+\sec^4}dx.$$ To find the arc length $s$ of the curve $y= \tan x$ between $x=0$ and $x=\pi/4$, we fist calculate $y'=\sec^2 x$. Then we have the integral, $$s=\int_0^{\pi/4}\sqrt{1+(y')^2}dx=\int_0^{\pi/4}\sqrt{1+\sec^4}dx.$$	{}
# Calendar for MAT 401 Advanced Analysis, Fall 2012 I will post homework assignments here so please check back often. Section numbers refer to our textbook Basic Analysis: Introduction to Real Analysis by Jiří Lebl. • 1st week: Aug. 27, 29 • A little bit of history. Why we need proofs in analysis. • 0.3 Basic set theory (Review). Cardinality. • Homework: Exercises 0.3.5 (tricky!), 0.3.9d, 0.3.11, 0.3.17 (prove that your answer is true), 0.3.19, 0.3.20. Due Monday Sept. 3. (There was a typo on my original version of this assignment; it said 0.3.12 when it should have been 0.3.17). • 2nd week: Sept. 3, 5 • Labor Day Holiday Monday Sept. 3 • Finish chapter 0 • 3rd week: Sept. 10, 12 • 1.1 Basic properties of the real numbers. Least upper bound. • Homework: 1.1.3, 1.1.6, 1.1.9. Due next Monday. • 1.2 The set of real numbers, $$\mathbb{R}$$. Archimedian property, sup and inf. • Homework: 1.2.1, 1.2.2, 1.2.7 (hint: is $$(\sqrt{x}-\sqrt{y})^2$$ positive, or negative, or what?), (1.2.9 was originally part of this but I postponed it until next week.). Due next Wednesday. • 4th week: Sept. 17, 19 • Homework: Prove the sup part of exercises 1.2.9 and 1.2.10. (Hint: use prop. 1.2.8). Due Monday. • 5th week: Sept. 24, 26 • 1.3 Absolute value. Homework: 1.3.2a, 1.3.3, 1.3.5. Due next Monday. • 1.4 Intervals and the size of $$\mathbb{R}$$ • 2.1 Sequences and limits. Homework: 2.1.6, 2.1.10, 2.1.12, 2.1.13, 2.1.16. Due next Monday. • 6th week: Oct. 1, 3 • 2.2 Facts about limits of sequences. Homework: exercises 2.2.3, 2.2.5, 2.2.7 (hint: what if the signs of the $$x_n$$s flop back and forth?), 2.2.8. Due next Monday. • Test Wednesday Oct. 3 Solutions • 7th week: Oct. 8, 10 • 2.2 Facts about limits of sequences (continued). Homework: exercises 2.2.3, 2.2.5, 2.2.7 (hint: what if the signs of the $$x_n$$s flop back and forth?), 2.2.8. Due next Monday. • 8th week: Oct. 15, 17 • Discuss answers to test questions. • 2.3 Limit superior, limit inferior, and the Bolzano-Weierstrass Theorem. Homework: 2.3.1, 2.3.2, 2.3.5, 2.3.6. Due next Wednesday. • 9th week: Oct. 22, 24 • Finish 2.3: Bolzano Weierstrass theorem. (See homework last week). • 2.4 Cauchy sequences. Homework: exercises 2.4.1, 2.4.2, 2.4.5. Due next Wednesday. Due Wednesday Oct. 31. • 10th week: Oct. 29, 31 • 2.5 Series: We'll only touch on this very briefly; it was covered pretty well in your calculus class. I think the calculus proofs using the integral test are simpler and easier to rememeber than the ones in our text, but of course we haven't developed integration yet so they wouldn't work in this course. • 3.1 Limits of functions. This introduces the famous $$\epsilon-\delta$$ definition, which is an extension of the ideas we have already discussed. Homework (corrected): exercises 3.1.4 parts i) and iii), 3.1.7, 3.1.8, 3.1.10 The previous "hint" for problem 3.1.10 was too hard. Here is an easier approach: The hypotheses of the problem give you a lot of Cauchy sequences $$\{f(x_n)\}_{n=1}^{\infty}$$. If these Cauchy sequences all converge to the same converge to the same limit $$L$$ then you can use lemma 3.1.7 to conclude that $$f$$ is continuous at $$c$$. So the only thing that could go wrong is that some of these Cauchy sequences might converge to different limits. Let's show that can't happen. If it does happen then you'll have two sequences $$\{x_n\}_{n=1}^{\infty}$$ and $$\{x'_n\}_{n=1}^{\infty}$$ that converge to $$c$$, but $$\lim_{n\rightarrow\infty} f(x_n) = L$$ and $$\lim_{n\rightarrow\infty} f(x'_n)=L'$$ where $$L\neq L'$$ are different limits. Show that you could put these two sequences together to construct a sequence that tries to approaches both limits in the same way that the sequence $$(-1)^n$$ tries to approach both $$1$$ and $$-1$$. Such a sequence can't be Cauchy, which contradicts the hypotheses of this problem. • 11th week: Nov. 5, 7 • Test postponed until Monday Nov. 19 • 3.2 Continuous functions. Homework: 3.2.1, 3.2.2, 3.2.3, 3.2.10, 3.2.11. For full credit work 3.2.1, 3.2.2, and 3.2.3 directly, using the $\epsilon-\delta$ definition of continuity (Def. 3.2.1) instead of using sequences or lemma 3.1.7. Example 3.1.5 shows how to do this in one example. I will give you another example next Wednesday • 12th week: Nov. 12, 14 • Veterans' Day Holiday: Monday Nov. 12 • 3.3 Extreme value theorem (the text calls this "Min-max theorem") and intermediate value theorem. Homework 3.3.1, 3.3.2, 3.3.6, 3.3.8 • 13th week: Nov. 19, 21 • Test Monday Nov. 19 Solutions • skip 3.4 Uniform continuity for now • 4.1 The derivative • 4.2 Mean value theorem • 4.3 skip Taylor's theorem • Thanksgiving Holiday: Thursday Nov. 22 - Friday Nov. 23 • 14th week: Nov. 26, 28 • Homework: exercise 4.1.1. Due Wednesday Dec. 5 • 15 th week: Dec. 3, 5 • 5.1 The Riemann integral • 5.2 Properties of the integral • 5.3 Fundamental theorem of calculus • Final exams week: Dec. 10-14 • Final exam: Wednesday Dec. 12 5:30-7:30pm Solutions	{}
Volume 282 - 38th International Conference on High Energy Physics (ICHEP2016) - Poster Session Boosted $H\rightarrow bb^-$ Tagger in Run II M. Sahinsoy* on behalf of the ATLAS collaboration *corresponding author Full text: pdf Pre-published on: February 06, 2017 Published on: April 19, 2017 Abstract Many searches for Higgs bosons decaying to b quark pairs benefit from the increased Run II centre-of-mass energy by exploiting the boosted kinematic regime at large transverse momenta of the Higgs boson, where the two b-jets are merged into one large radius ($R$) jet. ATLAS uses a boosted $H \rightarrow b\bar{b}$ tagger algorithm to separate Higgs signal from background processes (QCD, W and Z bosons, top quarks). The tagger takes as input a large $R=1.0$ jet with calibrated pseudorapidity, energy and mass scale. It employs b-tagging, Higgs candidate mass, and substructure information. The performance of several operating points in Higgs boson signal, QCD, and $t\bar{t}$ all-hadronic backgrounds are presented. Systematic uncertainties are evaluated so that this tagger can be used in analyses. DOI: https://doi.org/10.22323/1.282.1129 How to cite Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete. Open Access Copyright owned by the author(s) under the term of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.	{}
Currently, a task can execute a single SQL statement, including a call to a stored procedure. Tasks can be combined with table streams for continuous ELT workflows to process recently changed table rows. Streams ensure exactly once semantics for new or changed data in a table. Tasks can also be used independently to generate periodic reports by inserting or merging rows into a report table or perform other periodic work. In this Topic: Snowflake ensures only one instance of a task with a schedule (i.e. a standalone task or the root task in a tree of tasks) is executed at a given time. If a task is still running when the next scheduled execution time occurs, then that scheduled time is skipped. ### Choosing a Warehouse¶ Suggested best practices when configuring warehouses are described in Warehouse Considerations. We recommend that you analyze the average run time for a given task or tree of tasks using a specific warehouse based on warehouse size and clustering, as well as whether or not the warehouse is shared by multiple processes or is dedicated to running this single task (or tree of tasks). Query the TASK_HISTORY table function in the Information Schema. The average difference between the scheduled and completed times for a task is the expected average run time for the task, including any period in which the task was queued. A task is queued when other processes are currently using all of the servers in the warehouse. Unless the SQL statements defined for the tasks can be optimized (either by rewriting the statements or using stored procedures), then this would be the expected average run time for the task (or tree of tasks). Choose the right size for the warehouse based on your analysis to ensure the task (or tree of tasks) finishes running within this window. The following diagram shows a window of 1 minute in which a single task queued for 20 seconds and then ran for 40 seconds. The following diagram shows a tree of tasks that requires 5 minutes on average to complete for each run. The diagram shows the window for 2 runs of the tree of tasks to complete. This window is calculated from the time the root task is scheduled to start until the last child task in the tree has completed running. In this example, the tree of tasks is shared with other, concurrent operations that queue while each of the 3 tasks in the tree is running. These concurrent operations consume all available resources when each task in the tree finishes running and before the next task starts running. As a result, the window for each task includes some amount of queuing while it waits for other operations to finish and relinquish servers. Note that even if this tree of tasks ran on a dedicated warehouse, a brief lag would be expected after a parent task finishes running and any child task is executed; however, no queueing for shared resources with other operations would occur. The warehouse size you choose should be large enough to accommodate multiple child tasks that are triggered simultaneously by parent tasks. ### Task Scheduling and Daylight Saving Time¶ The cron expression in a task definition supports specifying a time zone. A scheduled task runs according to the specified cron expression in the local time for a given time zone. Special care should be taken with regard to scheduling tasks for time zones that recognize daylight saving time. Tasks scheduled during specific times on days when the transition from standard time to daylight saving time (or the reverse) occurs can have unexpected behaviors. For example: • During the autumn change from daylight saving time to standard time, a task scheduled to start at 1 AM in the America/Los_Angeles time zone (i.e. 0 1 * * * America/Los_Angeles) would run twice: once at 1 AM and then again when 1:59:59 AM shifts to 1:00:00 AM local time. That is, there are two points in time when the local time is 1 AM. • During the spring change from standard time to daylight saving time, a task scheduled to start at 2 AM in the America/Los_Angeles time zone (i.e. 0 2 * * * America/Los_Angeles) would not run at all because the local time shifts from 1:59:59 AM to 3:00:00 AM. That is, there is no point during that day when the local time is 2 AM. To avoid unexpected task executions due to daylight saving time, either: • Do not schedule tasks to run at a specific time between 1 AM and 3 AM (daily, or on days of the week that include Sundays), or • Manually adjust the cron expression for tasks scheduled during those hours twice each year to compensate for the time change due to daylight saving time. Users can define a simple tree-like structure of tasks that starts with a root task and is linked together by task dependencies. The current implementation supports a single path between any two nodes; i.e. an individual task can have only a single predecessor (parent) task. This differs from a Directed Acyclic Graph (DAG) structure, in which a single node can have multiple parents. A simple tree of tasks is limited to a maximum of 1000 tasks total (including the root task) in a resumed state. An individual task in the tree is limited to a single predecessor task; however, a task can have a maximum of 100 child tasks (i.e. other tasks that identify the task as a predecessor). Currently, we cannot guarantee that only one instance of a task with a defined predecessor task is running at a given time. Note A brief lag occurs after a parent task finishes running and any child task is executed. All tasks in a simple tree must have the same task owner (i.e. a single role must have the OWNERSHIP privilege on all of the tasks in the tree) and be stored in the same database and schema. When ownership of all tasks in a tree of tasks is transferred at once, through either of the following activities, the links between all tasks in the tree are retained: • The current owner of all tasks that compose the tree of tasks is dropped (using DROP ROLE). Ownership of the objects owned by the dropped role is transferred to the role that executes the DROP ROLE command. • Ownership of all tasks that compose the tree of tasks is explicitly transferred to another role (e.g. by executing GRANT OWNERSHIP on all tasks in a schema). ### Versioning of Task Tree Runs¶ When the root task starts a scheduled run, a version of the entire tree of tasks is established. All properties of all tasks in the tree are set. The entire tree of tasks completes its current run using the properties for this set version. Before a DDL statement can be executed on any task in a tree of tasks, the root task must be suspended (using ALTER TASK … SUSPEND). When the root task is suspended, all future scheduled runs of the root task are cancelled; however, if any tasks are currently running (i.e, the tasks in an an EXECUTING state), these tasks and any descendent tasks continue to run. After task properties in the tree are modified and the root task is resumed, those changes are not applied until the next scheduled run. At that time, a new version of the tree of tasks is set. Note If the definition of a stored procedure called by a task changes while the tree of tasks is executing, the new programming could be executed when the stored procedure is called by the task in the current run. When the root task is resumed, a new version of the tree of tasks is set when the root task starts its next run. This new version includes the modification to Task B. ## Setting Session Parameters for Tasks¶ You can set session parameters for the session in which a task runs. To do so, modify an existing task and set the desired parameter values (using ALTER TASKSET session_parameter = value[, session_parameter = value ... ]). A task supports all session parameters. For the complete list, see Parameters. Note that a task does not support account or user parameters. The following roles (or roles with the specified privileges) can use SQL to view the task history within a specified date range: • Task owner (i.e. role that has the OWNERSHIP privilege on a task). • Any role that has the global MONITOR EXECUTION privilege. SQL Query the TASK_HISTORY table function (in the Information Schema). ## Understanding the System Service¶ Snowflake runs tasks with the privileges of the task owner (i.e. the role that has OWNERSHIP privilege on the task), but task runs are not associated with a user. Instead, each run is executed by a system service. Tasks are decoupled from specific users to avoid complications that can arise when users are dropped, locked due to authentication issues, or have roles removed. Because task runs are decoupled from a user, the query history for task runs are associated with the system service. SYSTEM is not a user in the account; it is a behind-the-scenes service. As such, there are no user credentials for this service, and no individual (from Snowflake or in your account) can assume its identity. Activity for the system service is limited to your account. The same encryption protections and other security protocols are built into this service as are enforced for other operations. To support creating and managing tasks, Snowflake provides the following set of special DDL commands: In addition, providers can view, grant, or revoke access to the necessary database objects for ELT using the following standard access control DDL: To support retrieving information about tasks, Snowflake provides the following set of SQL functions: ### Access Control Privileges¶ Creating, managing, and executing tasks requires a role with a minimum of the following privileges: Object Privilege Notes Account Required to run any tasks the role owns. Revoking the EXECUTE TASK privilege on a role prevents all subsequent task runs from starting under that role. Database USAGE Schema Warehouse USAGE In addition, the role must have the permissions required to run the SQL statement executed by the task. After a task is created, the task owner (i.e. the role that has the OWNERSHIP privilege on the task) must have the following privileges: Object Privilege Notes Account Required to run any tasks the role owns. Revoking the EXECUTE TASK privilege on a role prevents all subsequent task runs from starting under that role. Database USAGE Schema USAGE OWNERSHIP Warehouse USAGE In addition, the role must have the permissions required to run the SQL statement executed by the task. In addition to the task owner, a role that has the OPERATE privilege on the task can suspend or resume the task. This role must have the USAGE privilege on the database and schema that contain the task. No other privileges are required. When a task is resumed, Snowflake verifies that the task owner role has the privileges listed in Owning Tasks (in this topic). USE ROLE securityadmin; -- set the active role to ACCOUNTADMIN before granting the EXECUTE TASK privilege to the new role -- set the active role to SECURITYADMIN to show that this role can grant a role to another role For more information on creating custom roles and role hierarchies, see Configuring Access Control. ### Dropping a Task Owner Role¶ When the owner role of a given task (i.e. the role with the OWNERSHIP privilege on the task) is deleted, the task is “re-possessed” by the role that dropped the owner role. This ensures that ownership moves to a role that is closer to the root of the role hierarchy. When a task is re-possessed, it is automatically paused, i.e., all executions currently in flight complete processing, but new executions will not be scheduled until the task is resumed explicitly by the new owner. The rationale for this is to prevent a user with access to a particular role from leaving behind tasks that suddenly execute with higher permissions when the role is removed. If the role that a running task is executing under is dropped while the task is running, the task completes processing under the dropped role. ## Workflow¶ This section provides a high-level overview of the task setup workflow. 1. Complete the steps in Creating a Task Administrator Role (in this topic) to create a role that can be used to execute the commands in the following steps.	{}
DLR Portal Home \| Imprint \| Privacy Policy \| Contact \| Deutsch Fontsize: [-] Text [+] Study of size effect in Titanium Aluminide with the weakest link Weibull model Grigorjevs, Pavels (2018) Study of size effect in Titanium Aluminide with the weakest link Weibull model. Master's. DLR-Interner Bericht. DLR-IB-SG-AX-2018-252, 13 S. (Unpublished) Full text not available from this repository. Abstract In this work quantitative analysis of the weakest link Weibull model for studying size effect phenomena in $\gamma$-Titanium Aluminide (TiAl) alloys is made. This quasi-brittle material is very attractive for industry because of its mechanical properties like specific strength and oxidation resistance, which makes this material attractive for gas turbine applications. The goal of this work is to compare different weakest link theories and find the value of Weibull modulus from the experimental data of size effect of TiAl alloys. To calculate Weibull modulus, two methods are used - Least Squares Estimation (LSE) and Maximum likelihood Estimation (MLE), which are different in their mathematical formulation. In this work, we review four theories based on weakest link Weibull model, of which three were implemented and tested in MATLAB. We make quantitative comparisons of the predictive capabilities of three methods using the available experimental data. Item URL in elib:https://elib.dlr.de/125554/ Document Type:Monograph (DLR-Interner Bericht, Master's) Title:Study of size effect in Titanium Aluminide with the weakest link Weibull model Authors: AuthorsInstitution or Email of AuthorsAuthor's ORCID iD Grigorjevs, PavelsUNSPECIFIEDUNSPECIFIED Date:21 December 2018 Refereed publication:No Open Access:No Gold Open Access:No In SCOPUS:No In ISI Web of Science:No Number of Pages:13 Status:Unpublished Keywords:Size effect, Failure probability, Titanium Aluminide Institution:Deutsches Zentrum für Luft- und Raumfahrt Department:Institut für Test und Simulation für Gasturbinen HGF - Research field:Aeronautics, Space and Transport HGF - Program:Aeronautics HGF - Program Themes:propulsion systems DLR - Research area:Aeronautics DLR - Program:L ER - Engine Research DLR - Research theme (Project):L - Virtual Engine and Validation methods (old) Location: Augsburg Institutes and Institutions:Institute of Test and Simulation for Gas Turbines Deposited By: Raina, Dr.-Ing. Arun Deposited On:09 Jan 2020 09:08	{}
# logistic regression and ordinal logistic regression #### Pamona D ##### New Member Hi, I am currently studying the ordinal logistic regression. I want to ask: is there a relationship between logistic regression with logistic ordinal regression? What distinguishes both of these things? thank you #### Dason When people refer to logistic regression it's typically binary logistic regression so that the response variable only has two possible outcomes. With ordinal logistic regression the response can be a bit more generalized and as long as there is some sort of way to "order" the possible outcomes you can use this method. So you could have 3 or more possible outcomes with ordinal logistic regression (think something like "do not agree", "somewhat agree", "completely agree") whereas with what people typically refer to as "logistic regression" you can only have two outcomes. #### noetsi ##### Fortran must die There are actually two types of logistic regression with more than two levels on the dependent variable. Ordinal logistic regression assumes the levels can be ordered, as with the likert example Dason gave. Multinomial logistic regression does not make the assumption that the levels can be ordered. An example would be a dependent variable where you could be Christian, Jewish, or Muslim. There is no inherent ordering in the variable.	{}
# Tag Info 6 I was able to draw using \chemmove, remember picture and @{} syntax for naming atoms and links. \documentclass[varwidth,border=50]{standalone} \usepackage{tikz} \usepackage{chemfig} \begin{document} \begin{tikzpicture}[remember picture, node distance=0] \node (foo){}; \node (a) [draw,minimum height=0.5cm,minimum width = 0.7cm, above of = foo, node ... 6 The \chemfig command relies on category code changes, in particular it changes # to an “other character” (category code 12). Such commands cannot appear in the argument to another command. While equation and equation* are safe on this respect, align* isn't: the multiline alignment environments of amsmath absorb their content as the argument to a command. ... 6 Here is a “chemfig only” idea: \documentclass{article} \usepackage{chemfig} \definesubmol{wallpart}{-[:-90,.5](-[:110,.5])} \definesubmol{wall}{!{wallpart}!{wallpart}!{wallpart}} \begin{document} \chemfig{ !{wall} (-[:0]*6(---(-CH_3)---)) !{wall} } \end{document} As far as I know the size of the circle (relative to the size of the ring) is ... 5 Citing chemfig's manual ChemFig always places the first atom of the molecule on the baseline of the preceding code. This means that the first atom in a molecule determines the baseline of the whole molecule. A solution in this case it is rather easy: add an invisible bond upwards before the “wall” (i.e. three bonds scaled by .5 going down (the angle ... 5 chemfig always places the first atom of the molecule on the baseline, so if you write \chemfig{O=[:-90](-[:-150]H)-[:-30]OH} the oxygen will be on the baseline. To put the carbon on the baseline it has to be the first atom: \chemfig{(-[:-150]H)(=[:90]O)-[:-30]OH} In other words, you must start from the atom you want to be on the baseline and treat the ... 5 Disclaimer: I did not know how to solve this myself when I asked the question, but managed to figure it out almost immediately after asking it. The author of chemfig said he did not speak English well enough to participate in the previous issue that was similar to this, and the correspondence was written in French. Although I'm Canadian, my French is ... 5 You can use any software able to export in smiles format or MDL molfile format. That done, you can use the mol2chemfig package to convert it to a chemfig command. I also want to point the excellent moltochemfig site in which you can draw a molecule whith your mouse, convert it to mol format and then to chemfig command. For example, in a few seconds, I draw ... 5 I'm not sure why you want to typeset the reactions with chemfig. It is an excellent package for drawing skeletal formulae of organic compounds. However, the reactions in your example are much easier typeset with mhchem or chemformula. The latter is loaded by chemmacros so you already have it available. \documentclass{memoir} \usepackage[utf8]{inputenc} ... 5 No need the extra package since chembove can stack stuff. \documentclass{article} \usepackage{chemfig} \begin{document} \chemfig{A-\chemabove[0pt]{\Lewis{4.,\vphantom o}}{\scriptstyle+}O-B} \end{document} EDIT: Still no need of extra package like "stackengine" or extra complicated macro. Lewis and chemabove can do the job: \documentclass{article} ... 5 You might want to use the decoration complete sines from Nicer wavy line with TikZ as I've used in a blog post of mine a while ago: \documentclass{article} \usepackage{tikz} \usepackage{chemfig} \usetikzlibrary{decorations.pathmorphing} \pgfdeclaredecoration{complete sines}{initial}{ \state{initial}[ width=+0pt, next state=sine, persistent ... 4 Use \Chembelow (upper case!) instead of \chembelow. From the manual (emphasis by me): The macro[...] \chembelow[<dim>]{<code>}{<stuff>} place[s] the <stuff> [...] below the <code> [...] without changing the bounding box of <code>. The uppercase version on the other hand does extend the bounding box. (\lewis ... 4 Ignoring the chemically incorrect structure of my 1st example, one can nonetheless use a stack to do what you ask. This question also seems related: Draw Lewis structures like a book EDITED to achieve vertical spacing more in line with OP's desire, and to place code in the macro \cation. \documentclass{article} \usepackage{chemfig,stackengine} % ORIGINAL ... 4 Update As clemens mentions in a comment, from v1.2 2015-10-08 \setarrowdefault{,,,<arrow tip>} is possible: \documentclass{article} \usepackage{chemfig} \usetikzlibrary{arrows.meta} \begin{document} \setarrowdefault{,,,-Stealth} \schemestart A\arrow B\arrow{<-} C\arrow{<->}[,,red] D\arrow{-/>}[45,2,thick] E \schemestop ... 4 Only after I had the code finished I realised that your code shows something different than the picture. But the alignment idea should work nonetheless. As you might know chemfig allows to add anchor names in compounds using the @{<name>} syntax which can be referred to in \chemmove for example, but also in arrows \arrow(@<name>--). The code ... 4 I think I found a solution for your problem. It is quite a hack, but it seems to work: \documentclass{article} \usepackage{chemfig} \begin{document} \chemfig{ [:-18]5(-\chembelow{N}{H}- 5(-S-{^{64}Cu} 5([::-108]-S-(5(-\chembelow{N}{H}-))=N-N-) 5(-\phantom{N}=(-)-(-)=\phantom{N}-) -N-N=)) } \end{document} I tried to ... 4 If you change \chemfig{-[7]N(-[5])... in your first formula into \chemfig{N(-[5])(-[3])... the baseline of said formula will be determined by the N which in this case suffices for the wanted alignment. The baseline of the OH- is then the same as baseline of the N of the first formula. The rest in the code below is just indentation. I also added \| in one ... 4 Bundling my comments into an answer: I wrote »There is no way to make chemfigs bonds all the same length. Quoting part II section 4 Length of a bond of chemfig's manual:« Rather than speaking of length of a bond, we should use the term interatomic spacing. If effect, only the interatomic spacing is adjustable with \setatomsep as we have seen on ... 4 The overall structure is a simple table, so a tabular should do fine \begin{tabular}{llll} \chemfig{...} & ... \\ Myoglobin & ... \\ Fleischfarbe \\ purpurrot & ... \end{tabular} As for the chemical formulas: using the bonds optional parameters they are quite easy with chemfig. Here is an example for one of the structures: ... 4 You can use the uppercase version \Chembelow (there's also \Chemabove) to extend the bounding box of the molecule: $\underbrace{ \chemfig{6(-\Chembelow{N}{H}-(=O)--\chemabove{N}{H}-(=O)-)} }_{\textnormal{glycine anhydride}}$ BTW: you can use chemfig's tools and avoid mathmode completely: \schemestart[-90] \chemup. ... 3 Here is something to get you started, based upon the Porphyrin example from the gallery in chemfig's manual: \documentclass{article} \usepackage{chemfig} \begin{document} \chemfig{ a-?[a]=[::+63]5( -N?[b]=( -(-[::63]d)=[::-54]5( -N?[c]-( =(-[::63]g)-[::-54]5( =N?[d]-( =(-[::63]j)-[::-54]5( ... 3 No need an extra package, \chemname does the job: \documentclass{article} \usepackage{chemfig} \begin{document} \begin{center} \schemestart \chemfig{CH_3Cl} \+ \chemname{\chemfig{Si{(Cu)}}}{Silicon\\/CopperAlloy} \arrow(.mid east--.mid west) \chemname{\chemfig{{(CH3)}_2SiCl2}}{Dimethyldichlorosilane} \+ \chemfig{Cu} \schemestop \end{center} ... 3 You can draw an invisible bond to the center of the ring, name that position with chemfig's @{name} syntax and draw a slightly longer bond from there to Me. With the \chemmove macro and a little bit of TikZ you can draw the ellipse around the marked center: \documentclass{article} \usepackage{chemfig} \begin{document} \begin{center} \setcrambond{2pt}{}{} ... 3 \chemname sets the name below the compound depending on the depth of the compound and the previous usages of \chemname. The chemfig manual says: In fact, to draw the <name> the command \chemname inserts 1.5ex + the largest of the depths of the molecules thus far below the baseline of each molecule [...]. The command \chenameinit{<stuff>} ... 3 In theory, \chemfig{<code>} can be written in the argument of a macro since \CF@chemfig@iv does a \scantokens of the <code>. Unfortunately, there is a bug because via \scantokens, # becomes ## in the argument of a macro and this behaviour is not taken into account by \chemfig. For example, if you write \fbox{\chemfig{A-#(0pt)B}}, the code of the ... 3 Look at the page 23 of the manual. You have to split CH_2 as follows: \chemfig{{C}\|\lewis{0.,H_2}-[6,,1]6(------)} 3 With the help of Section 12.7 of the chemfig manual I was able to get something that should be close. The distance between atoms can be changed with \setatomsep, and font colors, sizes, etc. can be modified as per usual TikZ commands. \documentclass[border=2pt]{standalone} \usepackage{chemfig} \usepackage{siunitx} ... 3 3 You can locally redefine \printatom in order to force atom depths to 0pt: \documentclass{article} \usepackage{chemfig} \setatomsep{1.5em} \begin{document} something $$\renewcommand\printatom[1]{\setbox0=\hbox{\ensuremath{\mathrm{#1}}}\dp0=0pt \box0 } \chemfig{CH_3-6(-=-(-CH_2-6(-=-(-CH_2-*6(-=-(-CH_3)=-=))=-=))=-=)}$$ something \end{document} 3 Here it is. chemfig basically works like Tikz. You can set nodes and use those nodes to draw arrows, etc. I did the brackets by drawing arcs, and setting nodes on the lines. It would have been easier if each "atom" was a node, but I haven't found anything in the documentation. In any case, if you discover something like that, you can remove the extra nodes ... 3 Do you mean something like this? It uses the fact that chemfig's formulas are tikzpictures. chemfig offers the possibility to give explicite names to the nodes in a formula using @{name} (this is explained in the manual). Later one can refer to those nodes in a seperate tikzpicture. \chemmove simply is a wrapper for a tikzpicture with the options remember ... Only top voted, non community-wiki answers of a minimum length are eligible	{}
# The number of seven digit integers whose sum of the digits is ten, formed by using the digits 1, 2 and 3 only is $\begin{array}{1 1} 55 \\ 66 \\ 77 \\ 88 \end{array}$ The seven digits whose sum of the digits $=10$ using $1,2,3$ as their digits may be $1111123$ or $1111222$ No. of $7$ digit numbers using $1111123=\large\frac{7!}{5!}=$$42 No. of 7 digit numbers using 1111222= \large\frac{7!}{4!.3!}=$$35$ $\therefore$ the required total number of integers$=42+35=77$	{}
Copied to clipboard ## G = C11×SD32order 352 = 25·11 ### Direct product of C11 and SD32 direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary Aliases: C11×SD32, D8.C22, C162C22, C1766C2, Q161C22, C22.16D8, C44.37D4, C88.25C22, C8.3(C2×C22), C4.2(D4×C11), C2.4(C11×D8), (C11×Q16)⋊5C2, (C11×D8).2C2, SmallGroup(352,61) Series: Derived Chief Lower central Upper central Derived series C1 — C8 — C11×SD32 Chief series C1 — C2 — C4 — C8 — C88 — C11×Q16 — C11×SD32 Lower central C1 — C2 — C4 — C8 — C11×SD32 Upper central C1 — C22 — C44 — C88 — C11×SD32 Generators and relations for C11×SD32 G = < a,b,c \| a11=b16=c2=1, ab=ba, ac=ca, cbc=b7 > Smallest permutation representation of C11×SD32 On 176 points Generators in S176 (1 120 95 37 146 63 29 112 163 65 142)(2 121 96 38 147 64 30 97 164 66 143)(3 122 81 39 148 49 31 98 165 67 144)(4 123 82 40 149 50 32 99 166 68 129)(5 124 83 41 150 51 17 100 167 69 130)(6 125 84 42 151 52 18 101 168 70 131)(7 126 85 43 152 53 19 102 169 71 132)(8 127 86 44 153 54 20 103 170 72 133)(9 128 87 45 154 55 21 104 171 73 134)(10 113 88 46 155 56 22 105 172 74 135)(11 114 89 47 156 57 23 106 173 75 136)(12 115 90 48 157 58 24 107 174 76 137)(13 116 91 33 158 59 25 108 175 77 138)(14 117 92 34 159 60 26 109 176 78 139)(15 118 93 35 160 61 27 110 161 79 140)(16 119 94 36 145 62 28 111 162 80 141) (1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176) (2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 61)(50 52)(51 59)(53 57)(54 64)(56 62)(58 60)(66 72)(67 79)(68 70)(69 77)(71 75)(74 80)(76 78)(81 93)(82 84)(83 91)(85 89)(86 96)(88 94)(90 92)(97 103)(98 110)(99 101)(100 108)(102 106)(105 111)(107 109)(113 119)(114 126)(115 117)(116 124)(118 122)(121 127)(123 125)(129 131)(130 138)(132 136)(133 143)(135 141)(137 139)(140 144)(145 155)(147 153)(148 160)(149 151)(150 158)(152 156)(157 159)(161 165)(162 172)(164 170)(166 168)(167 175)(169 173)(174 176) G:=sub<Sym(176)\| (1,120,95,37,146,63,29,112,163,65,142)(2,121,96,38,147,64,30,97,164,66,143)(3,122,81,39,148,49,31,98,165,67,144)(4,123,82,40,149,50,32,99,166,68,129)(5,124,83,41,150,51,17,100,167,69,130)(6,125,84,42,151,52,18,101,168,70,131)(7,126,85,43,152,53,19,102,169,71,132)(8,127,86,44,153,54,20,103,170,72,133)(9,128,87,45,154,55,21,104,171,73,134)(10,113,88,46,155,56,22,105,172,74,135)(11,114,89,47,156,57,23,106,173,75,136)(12,115,90,48,157,58,24,107,174,76,137)(13,116,91,33,158,59,25,108,175,77,138)(14,117,92,34,159,60,26,109,176,78,139)(15,118,93,35,160,61,27,110,161,79,140)(16,119,94,36,145,62,28,111,162,80,141), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(66,72)(67,79)(68,70)(69,77)(71,75)(74,80)(76,78)(81,93)(82,84)(83,91)(85,89)(86,96)(88,94)(90,92)(97,103)(98,110)(99,101)(100,108)(102,106)(105,111)(107,109)(113,119)(114,126)(115,117)(116,124)(118,122)(121,127)(123,125)(129,131)(130,138)(132,136)(133,143)(135,141)(137,139)(140,144)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159)(161,165)(162,172)(164,170)(166,168)(167,175)(169,173)(174,176)>; G:=Group( (1,120,95,37,146,63,29,112,163,65,142)(2,121,96,38,147,64,30,97,164,66,143)(3,122,81,39,148,49,31,98,165,67,144)(4,123,82,40,149,50,32,99,166,68,129)(5,124,83,41,150,51,17,100,167,69,130)(6,125,84,42,151,52,18,101,168,70,131)(7,126,85,43,152,53,19,102,169,71,132)(8,127,86,44,153,54,20,103,170,72,133)(9,128,87,45,154,55,21,104,171,73,134)(10,113,88,46,155,56,22,105,172,74,135)(11,114,89,47,156,57,23,106,173,75,136)(12,115,90,48,157,58,24,107,174,76,137)(13,116,91,33,158,59,25,108,175,77,138)(14,117,92,34,159,60,26,109,176,78,139)(15,118,93,35,160,61,27,110,161,79,140)(16,119,94,36,145,62,28,111,162,80,141), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(66,72)(67,79)(68,70)(69,77)(71,75)(74,80)(76,78)(81,93)(82,84)(83,91)(85,89)(86,96)(88,94)(90,92)(97,103)(98,110)(99,101)(100,108)(102,106)(105,111)(107,109)(113,119)(114,126)(115,117)(116,124)(118,122)(121,127)(123,125)(129,131)(130,138)(132,136)(133,143)(135,141)(137,139)(140,144)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159)(161,165)(162,172)(164,170)(166,168)(167,175)(169,173)(174,176) ); G=PermutationGroup([(1,120,95,37,146,63,29,112,163,65,142),(2,121,96,38,147,64,30,97,164,66,143),(3,122,81,39,148,49,31,98,165,67,144),(4,123,82,40,149,50,32,99,166,68,129),(5,124,83,41,150,51,17,100,167,69,130),(6,125,84,42,151,52,18,101,168,70,131),(7,126,85,43,152,53,19,102,169,71,132),(8,127,86,44,153,54,20,103,170,72,133),(9,128,87,45,154,55,21,104,171,73,134),(10,113,88,46,155,56,22,105,172,74,135),(11,114,89,47,156,57,23,106,173,75,136),(12,115,90,48,157,58,24,107,174,76,137),(13,116,91,33,158,59,25,108,175,77,138),(14,117,92,34,159,60,26,109,176,78,139),(15,118,93,35,160,61,27,110,161,79,140),(16,119,94,36,145,62,28,111,162,80,141)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(33,41),(34,48),(35,39),(36,46),(38,44),(40,42),(43,47),(49,61),(50,52),(51,59),(53,57),(54,64),(56,62),(58,60),(66,72),(67,79),(68,70),(69,77),(71,75),(74,80),(76,78),(81,93),(82,84),(83,91),(85,89),(86,96),(88,94),(90,92),(97,103),(98,110),(99,101),(100,108),(102,106),(105,111),(107,109),(113,119),(114,126),(115,117),(116,124),(118,122),(121,127),(123,125),(129,131),(130,138),(132,136),(133,143),(135,141),(137,139),(140,144),(145,155),(147,153),(148,160),(149,151),(150,158),(152,156),(157,159),(161,165),(162,172),(164,170),(166,168),(167,175),(169,173),(174,176)]) 121 conjugacy classes class 1 2A 2B 4A 4B 8A 8B 11A ··· 11J 16A 16B 16C 16D 22A ··· 22J 22K ··· 22T 44A ··· 44J 44K ··· 44T 88A ··· 88T 176A ··· 176AN order 1 2 2 4 4 8 8 11 ··· 11 16 16 16 16 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 176 ··· 176 size 1 1 8 2 8 2 2 1 ··· 1 2 2 2 2 1 ··· 1 8 ··· 8 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2 121 irreducible representations dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C11 C22 C22 C22 D4 D8 SD32 D4×C11 C11×D8 C11×SD32 kernel C11×SD32 C176 C11×D8 C11×Q16 SD32 C16 D8 Q16 C44 C22 C11 C4 C2 C1 # reps 1 1 1 1 10 10 10 10 1 2 4 10 20 40 Matrix representation of C11×SD32 in GL2(𝔽23) generated by 9 0 0 9 , 16 15 20 0 , 22 10 0 1 G:=sub<GL(2,GF(23))\| [9,0,0,9],[16,20,15,0],[22,0,10,1] >; C11×SD32 in GAP, Magma, Sage, TeX C_{11}\times {\rm SD}_{32} % in TeX G:=Group("C11xSD32"); // GroupNames label G:=SmallGroup(352,61); // by ID G=gap.SmallGroup(352,61); # by ID G:=PCGroup([6,-2,-2,-11,-2,-2,-2,1056,553,3171,1593,165,7924,3970,88]); // Polycyclic G:=Group<a,b,c\|a^11=b^16=c^2=1,ab=ba,ac=ca,cbc=b^7>; // generators/relations Export ׿ × 𝔽	{}
Article Contents Article Contents # On dimensions of conformal repellers. Randomness and parameter dependency • We consider random conformal repellers. We show how to apply Bowen's formula for the Hausdorff dimension in this context and prove smoothness of the dimension with respect to parameters. The present article is essentially an extract of [11]. Our aim here is to emphasize the ideas and mechanisms behind rather than mathematical rigor. Mathematics Subject Classification: Primary: 37C45, 28A78; Secondary: 60D05, 37H15. Citation: • [1] L. M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Erg. Th. & Dyn. Syst., 16 (1996), 871-927. [2] R. Bowen, Hausdorff dimension of quasi-circles, IHES Publ., 50 (1979), 259-273. [3] G. Birkhoff, "Lattice Theory," 3rd edition, American Mathematical Society Colloquium Publications, Vol. XXV, Amer. Math. Soc., Providence, RI, 1967. [4] K. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc., 106 (1989), 543-554.doi: 10.1090/S0002-9939-1989-0969315-8. [5] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.doi: 10.1214/aoms/1177705909. [6] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Erg. Th. & Dyn. Syst., 17 (1997), 147-167. [7] C. Liverani, Decay of correlations, Annals Math. (2), 142 (1995), 239-301.doi: 10.2307/2118636. [8] D. Ruelle, Repellers for real analytic maps, Erg. Th. & Dyn. Syst., 2 (1982), 99-107. [9] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.doi: 10.1007/s002200050134. [10] H. H. Rugh, Coupled maps and analytic function spaces, Ann. Scient. Éc. Norm. Sup. (4), 35 (2002), 489-535.doi: 10.1016/S0012-9593(02)01102-3. [11] H. H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency, Ann. Math. (2), 168 (2008), 695-748.doi: 10.4007/annals.2008.168.695. [12] M. Urbański and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Erg. Th. & Dyn. Syst., 24 (2004), 279-315.	{}

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