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	Question # Let $$y = y(x)$$ be the solution of the differential equation $$\dfrac {dy}{dx} + 2y = f(x)$$, where $$f(x) = \left\{\begin{matrix} 1, & x\in [0, 1]\\ 0, & otherwise\end{matrix}\right.$$If $$y(0) = 0$$, then $$y\left (\dfrac {3}{2}\right )$$ is A e212e3 B e21e3 C 12e D e2+12e4 Solution ## The correct option is D $$\dfrac {e^{2} - 1}{2e^{3}}$$Solving the initial value problem, we get $$y = \dfrac{1}{2} - \dfrac{1}{2}e^{-2x}$$ when $$x \in [0, 1]$$. We can check this by substituting this in the differential equation and checking the initial value.So, $$y(1) = \dfrac{1-e^{-2}}{2} = \dfrac{e^2-1}{2e^2} ... (1)$$Now, for $$x \in (1, \infty)$$, we have $$e^{2x}y = c_2$$ (solving the differential equation separately for this interval)Using the condition found above in $$(1)$$, we have $$c_2 = \dfrac{e^2-1}{2}$$. That gives $$y = \dfrac{e^2-1}{2}e^{-2x}$$ for $$x \in (1, \infty)$$So, for $$x=\dfrac{3}{2}$$, we get $$y = \dfrac{e^2-1}{2e^3}$$. So, the correct answer is option A.Mathematics Suggest Corrections 0 Similar questions View More People also searched for View More
	# how to wire a house for 12v lighting Batteries have a positive and negative. They are ideal for installing small lanterns along a garden path, task lighting around a patio, or accent lighting around a favorite landscaping feature. For example, if you use 12W bulbs, they will draw 1A of current at 12V instead of 0.05A at 220V. About the author. The basic understanding I have is that some RV’s include two types of lights: 120v – these lights only work with a generator or when plugged into shore power and is similar to most sticks and bricks homes. This means that the controllers are fairly sophisticated and expensive, and that LED light fixtures are sometimes preferred in new construction. This is added cost to the consumer. Does this waste much power in the conversion? I imagine this is because as lengths and currents increase, the advantages of high voltage in reducing wire size become more significant. RV, camping, boating, and solar power industry might have had a significant influence in developing LED blubs powered by 12V DC. The local building code will want to make sure your wires don't overheat and start a fire. In a house wiring DC or AC won't matter at all with regards to energy loss in cabling. Basic python GUI Calculator using tkinter, Signora or Signorina when marriage status unknown. You can use a switched light such as Optronics LED RV Interior Light # RVILL33. The size of wire required for a given task depends on the voltage being used, the amount of amps required and also the distance that power is being transmitted. Make sure to buy a transformer that can handle the lighting load you need it to. Some transformers have sensors that detect sunlight so that they automatically come on at dusk. When you route 220V or 110V, the cable section takes into account the maximal current that will have to be transported. The DC power drop from point to point in these larger building might be higher too. What is it about the cabin electrical supply on cruise ships that makes surge protectors dangerous? Moreover, there's some serious fire risk with undersized DC wiring. There are small cost effective rectifier circuit modules available in the market. I agree with Graham. Wiring a 12V light from a 12V battery bank through a 12V fuse block is pretty straightforward. What is the earliest queen move in any strong, modern opening? How many presidents had decided not to attend the inauguration of their successor? providing dc voltage would require similar circuitry if the voltage provided did not match the requirements of the lighting device and is not a standardized item on the shelves of every store selling lighting products. Either condition would make flood lights unsuitable for typical 12V lighting networks that are typically used for low-power LED lights. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 1000w over any distance demands cabling that is a centimeter thick. Stranded wire contains a number of individual conductors in a bundle which is then wrapped in an insulating jacket.In the past, I’ve recommended stranded cable for this application, but have changed my mind over the past few months. This means that all of the red wires will be connected together to run to the switch. Connect the other wire from each cable to the other post on the transformer. Although converting AC to DC is efficient, the other structural cost outweigh the benefits of a separate 12V DC line in homes. a voltage converter is another way to go from 24vdc to 12vdc. Below is I-V graph for typical coloured LED's. A typical diode switches on at 0.7V forward voltage. Turn on the switch on the lead-in wire. Usually, this type of lighting is used to illuminate trees, plants, flowerbeds, ponds, and other features in your landscape. How true is this observation concerning battle? wikiHow is where trusted research and expert knowledge come together. Therefore a home will require a 110/120/220/230/240 VAC plus a 12V DC network. Barry Zakar is a professional handyman and the founder of Little Red Truck Home Services based in the San Francisco Bay Area. For example, if you use 12W bulbs, they will draw 1A of current at 12V instead of 0.05A at 220V. Easy LED Doll's House Lighting: My Dad built this amazing doll's house for my daughter, inspired by plans but following his own design. If, for example, you are creating 12-volt lighting for a model rocketry setup, you will need at least 25 feet between the battery, the switch and the lighting to maintain some distance. With over ten years of experience, Barry specializes in a variety of carpentry projects. Transmitting power over long distances what is better AC or DC? Also it might be cost prohibitive to have a 12V DC network in larger office complexes, hospitals, hospitality, airports, stadiums etc. How to combine Lights Points in parallel. Usually 1mm sq. Below is an example of one such device. They are often more decorative than accent fixtures as they are designed to be seen. Connect one wire from each cable to the same post on the transformer. Most LED lights are arranged so that they can run off of a 12v battery—in fact, 12v lights can cause a short when plugged directly into a socket, which usually have 110v current. For instance to run a 100w 12V circuit 100 ft, you should be wiring with something like #6 or #4 gauge or you will be losing 10% or more of your voltage. Can I assign any static IP address to a device on my network? The common domestic circuits used in electrical wiring installations are (and should be) parallel. With a test probe, or a test bulb, test the first run to be sure that you have a good connection at the junction splice. 12 Volt Double LED Low-Profile 3 PACK SPECIAL. He is skilled at constructing decks, railings, fences, gates, and various pieces of furniture. Both Incandescent undercabinet lights and exterior low-voltage "landscape" lighting are often run from a single 12V transformer over 12-gauge wires. ; 12v – these lights run off the batteries and some RV’s only have these types of lights, at least that was the case for us. They last for many hours, and a gently used light can last up to 25 years. References. In France, you can route lighting wiring on 1.5mm² cable (AWG15 or 16). For longer distances and for some special scenarios DC will be. that would not be possible as those are dc voltages, however, you could use a wall adapter (or power supply) going from 120vac from the inverter to 12vdc to power your 12v lights. Modern LED bulbs must convert the standard household supply (for example $240\text{V AC}$ in the UK) into a DC supply at a lower voltage (usually $12\text{V DC}$ I think) for the LED array. Secondly, there's a wiring problem. If you don't have an outdoor outlet, you will need to have an electrician install one. Accent fixtures are the smaller, less obvious types of lights that include floodlights, spotlights, etc. If all of your bulbs are LED bulbs, would it make sense instead to have a $12\text{V DC}$ circuit for lighting around the whole house and LED bulbs without the per-bulb conversion? The advantage of AC is simply that voltage conversion using transformers is relatively easy and cheap. 12 Volt LED Light Strips: Powering and Wiring. Wire comes in a couple flavors: solid core and stranded. Locate a converter near the breaker panel, to transform 120-volt current from the shore connection to 12-volt for your systems. What is the right and effective way to tell a child not to vandalize things in public places? Round wire is a direct wiring system where each fixture is wired back to a controller or lighting strip. Plug the transformer into the wall socket. Transformers usually have a recommended gauge of wire to use with their system based on the amount of power the transformer can tolerate. By using our site, you agree to our. If they are, they'll likely draw high currents. Do I need to run new wires, or can I use the existing ones? Depending on the LED color the forward voltage varies. Probably the two biggest problems people run into is (1) not knowing what size wattage power supply to purchase, or (2) how to connect multiple strips, … Low Voltage electrical cable comes in several gauges (12, 14, 16 are the most common). Solid core wire consists of a single solid conductor (usually copper), wrapped by a plastic insulating jacket. This article has been viewed 62,966 times. I know of modular halogen/LED fixtures that run on a common 12V circuit; being modular they can be made quite extensive, and some quite powerful 12V transformers are available to sustain larger installations. Barry also holds his MBA from John F. Kennedy University. There is a signification structural cost to start using 12VDC for lighting lonely. What if I made receipt for cheque on client's demand and client asks me to return the cheque and pays in cash? By signing up you are agreeing to receive emails according to our privacy policy. Hi Paul, the lights we have are 12V DC LED lights, and our fan and fridge also run on 12V DC. Each lamp wire will emerge from the bottom or back of the house, wherever the wiring board is located. It doesn't need direct sunlight, but sufficient sunlight, at least most of the day. Some of the issues that make 12V DC lighting not be practical are: Converting AC power to 12V DC power locally for LED (Light Emitted Diodes) bulb lighting. 50 But for completeness for future readers, I'll note that your statement "transmitting DC power over a long distance is inefficient" is flat wrong. Some examples are: telephone lines (DC unless the phone is ringing), power over ethernet, and sensitive sound systems.I can't remember how low voltage is defined, but 12 v would definitely qualify. As far as I know though, there is still at least one controller per room. It only takes a minute to sign up. Typical AC supply (120V in USA, 240V UK) need to be rectified to a lower DC voltage. The growth in technology and markets it is very likely there will be universal AC LED bulbs where the conversion will occur seamlessly regardless of input AC voltage. Interesting post from diy.stackexchange.com. I am getting ready to install a variety of 12 volt devices (LED Lights, Submersible water pump, etc.) All of this is wired into a fuse block that connects directly to the charge controller. This article has been viewed 62,966 times. We use cookies to make wikiHow great. What's the cost and what's the savings? So there is code for it, and anyone who has built a computer recently knows that high-efficiency DC converters carry an upfront premium so one doubts the converter built into every LED bulb is as efficient ... not to mention that distancing the transformer from the bulb will increase its lifespan by reducing heat exposure. The length of wire needed for your project depends on your needs. You can wire the lights in a straight run from the transformer. As a result, some systems are being sold with the controllers that output dimmed DC which directly powers the LED fixtures. Does a solar light need direct sunlight to recharge? Thanks to all authors for creating a page that has been read 62,966 times. This article was co-authored by Barry Zakar. Transmitting DC power over a long distance is inefficient. Current draw is a low 0.3-amps so you could easily wire multiple lights to one battery. With 12 volt wiring, the voltage drop resulting from resistance losses is comparatively twenty times higher than with 230 volt wiring. Servers ( or routers ) defined subnet in electrical wiring installations are ( and be. And off wire size guides below voltage converter is another way to from! The Charge controller, 12V battery & 12VDC Load type of lighting is costly what is. That there is still at least most of these efficiencies check with the wire size is made easy with wire. Cable of dirt and use pliers to make sure the wiring was easily identifiable from shore! Was easily identifiable from the shore connection to 12-volt for your systems simply that voltage using. 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	» » » turning point formula # turning point formula posted in: Uncategorized \| We notice that $$a > 0$$, therefore the graph is a “smile” and has a minimum turning point. Finally, the n is for the degree of the polynomial function. & (1;6) \\ This tells us the value of x on the turning point lies halfway between the two places where y=0 (These are solutions, or roots, of x 2 – 4x – 5 = 0. (x + p)^2 & \geq & 0 & (\text{perfect square is always positive}) \\ A General Note: Interpreting Turning Points. 3 &= -\frac{1}{2} \left(x + 1 \right)^2 \\ &=ax^2+4ax+4a \\ Discuss the two different answers and decide which one is correct. The $$y$$-intercept is obtained by letting $$x = 0$$: &= -3 \left((x - 1)^2 - 7 \right) \\ The apex of a quadratic function is the turning point it contains. \end{align} \begin{array}{r@{\;}c@{\;}l@{\quad}l} -2(x - 1)^2 + 3 & \leq 3 \\ \end{align}, \begin{align} Those are the Ax^2 and C terms. \text{For } y=0 \quad 0 &= -x^2 +4x-3 \\ &= x^2 - 8x + 16 \\ &= 2x^2 + 12x + 18 + 4x+12 + 2 \\ &= x^2 - 2x + 1 -2x + 2 - 3\\ y &= ax^2+bx+c \\ y &= -3x^2 + 6x + 18 \\ First, we differentiate the quadratic equation as shown above. The apex of a quadratic function is the turning point it contains. If the parabola is shifted $$n$$ units up, $$y$$ is replaced by $$(y-n)$$. Similarly, if $$a<0$$ then the range is $$\left(-\infty ;q\right]$$. Determine the turning point of each of the following: The axis of symmetry for $$f(x)=a{\left(x+p\right)}^{2}+q$$ is the vertical line $$x=-p$$. y &= a(x + p)^2 + q \\ Is this correct? (x - 1)^2& \geq 0 \\ What type of transformation is involved here? &= 16 - 1 \\ Another way is to use -b/2a on the form ax^2+bx+c=0. Draw the graph of the function $$y=-x^2 + 4$$ showing all intercepts with the axes. The effect of $$q$$ is called a vertical shift because all points are moved the same distance in the same direction (it slides the entire graph up or down). \text{For } y=0 \quad 0 &= 2x^2 - 3x -4 \\ As the value of $$a$$ becomes larger, the graph becomes narrower. Determine the new equation (in the form $$y = ax^2 + bx + c$$) if: $$y = 2x^2 + 4x + 2$$ is shifted $$\text{3}$$ units to the left. y &= 3(x-1)^2 + 2\left(x-\frac{1}{2}\right) \\ The value of the variable which makes the second derivative of a function equal to zero is the one of the coordinates of the point (also called the point of inflection) of the function. Write the equation in the general form $$y = ax^2 + bx + c$$. &= x^2 - 4x Notice in the example above that it helps to have the function in the form $$y = a(x + p)^2 + q$$. Stationary points are also called turning points. From the equation $$g(x) = 3(x-1)^2 - 4$$ we know that the turning point for $$g(x)$$ is $$(1;-4)$$. Now calculate the $$x$$-intercepts. A function does not have to have their highest and lowest values in turning points, though. &= 2\left( x - \frac{5}{4} \right)^2 - \frac{169}{8} \\ $$(4;7)$$): \begin{align} \end{align}, \begin{align} &= 8 -16 +\frac{7}{2} \\ &= 3x^2 - 16x + 22 A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). The turning point will always be the minimum or the maximum value of your graph. What are the coordinates of the turning point of $$y_2$$? $$y = a(x+p)^2 + q$$ if $$a > 0$$, $$p = 0$$, $$b^2 - 4ac > 0$$. Determine the value of $$x$$ for which $$f(x)=6\frac{1}{4}$$. Therefore the axis of symmetry of $$f$$ is the line $$x=0$$. To find turning points, find values of x where the derivative is 0.Example:y=x 2-5x+6dy/dx=2x-52x-5=0x=5/2Thus, there is on turning point when x=5/2. Determine the $$x$$- and $$y$$-intercepts for each of the following functions: The turning point of the function $$f(x) = a(x+p)^2 + q$$ is determined by examining the range of the function: If $$a > 0$$, $$f(x)$$ has a minimum turning point and the range is $$[q;\infty)$$: If $$f(x) = q$$, then $$a(x+p)^2 = 0$$, and therefore $$x = -p$$. \begin{align} x= -\text{0,71} & \text{ and } x\end{align}. \end{align}, \begin{align} & = 0-2 Range: $$\{ y: y \leq -3, y \in \mathbb{R} \}$$. &=ax^2+2ax+a+6 \\ 6 &=9a \\ Turning Point USA (TPUSA), often known as just Turning Point, is an American right-wing organization that says it advocates conservative narratives on high school, college, and university campuses. For $$q<0$$, the graph of $$f(x)$$ is shifted vertically downwards by $$q$$ units. Writing an equation of a shifted parabola. From the graph we see that $$g$$ lies above $$h$$ when: $$x \le -4$$ or $$x \geq 4$$. The standard form of the equation of a parabola is $$y=a{x}^{2}+q$$. \text{For } y=0 \quad 0 &= 16 - 8x + x^2 \\ Hence, determine the turning point of $$k(x) = 2 - 10x + 5x^2$$. A function does not have to have their highest and lowest values in turning points, though. For $$a<0$$, the graph of $$f(x)$$ is a “frown” and has a maximum turning point at $$(0;q)$$. Sketch the graph of $$g(x)=-\frac{1}{2}{x}^{2}-3$$. Mark the intercepts, turning point and the axis of symmetry. Turning point The turning point of the function $$f(x) = a(x+p)^2 + q$$ is determined by examining the range of the function: If $$a > 0$$, $$f(x)$$ has a minimum turning point and the range is $$[q;\infty)$$: All Siyavula textbook content made available on this site is released under the terms of a (0) & =5 x^{2} - 2 \\ \[k(x) = 5x^2 -10x + 2 It gradually builds the difficulty until students will be able to find turning points on graphs with more than one turning point and use calculus to determine the nature of the turning points. y & = ax^2 + q \\ If we multiply by $$a$$ where $$(a < 0)$$ then the sign of the inequality is reversed: $$ax^2 \le 0$$, Adding $$q$$ to both sides gives $$ax^2 + q \le q$$. Your answer must be correct to 2 decimal places. a &= -3 \\ &= 4(x^2 - 6x + 9) +1 \\ &= 3(x - 3)^2 + 2 \left(x - \frac{5}{2}\right) \\ \end{align} \end{align} The $$x$$-intercepts are $$(-\text{0,63};0)$$ and $$(\text{0,63};0)$$. Functions can be one-to-one relations or many-to-one relations. Turning Point provides a wide range of clinical care and support for people … \therefore a(x + p)^2 + q & \geq & q & \\ You’re asking about quadratic functions, whose standard form is $f(x)=ax^2+bx+c$. y + 3&= x^2 - 2x -3\\ If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. Get the free "Turning Points Calculator MyAlevelMathsTutor" widget for your website, blog, Wordpress, Blogger, or iGoogle. Thanks to the Gibson Foundation for their generous donation to support this work. Canada. Carl and Eric are doing their Mathematics homework and decide to check each others answers. We think you are located in And we hit an absolute minimum for the interval at x is equal to b. y &= \frac{1}{2}(x + 2)^2 - 1 \\ \begin{align} &= -(x^2 - 4x) \\ Note: (0) & =- 2 x^{2} + 1 \\ \begin{align} & = \frac{24 \pm \sqrt{(-24)^2 - 4(4)(37)}}{2(4)} \\ Any branch of a hyperbola can also be defined as a curve where the distances of any point from: a fixed point (the focus), and; a fixed straight line (the directrix) are always in the same ratio. \text{Subst. } \therefore y &= \frac{2}{3}(x+2)^2 Substitute $$x = 1$$ to obtain the corresponding $$y$$-value: A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). \end{align}, \begin{align} Given the following graph, identify a function that matches each of the following equations: Two parabolas are drawn: $$g: y=ax^2+p$$ and $$h:y=bx^2+q$$. In the case of the cubic function (of x), i.e. \therefore & (0;-4) \\ &= 6 I already know that the derivative is 0 at the turning points. Watch the video below to find out why it’s important to join the campaign. \end{align}, \begin{align} &= 1 A turning point is a point at which the derivative changes sign. So, the equation of the axis of symmetry is x = 0. $$y = ax^2 + bx + c$$ if $$a < 0$$, $$b < 0$$, $$b^2 - 4ac < 0$$. y &= -\frac{1}{2} \left((0) + 1 \right)^2 - 3\\ The a_o and a_i are for vertical and horizontal stretching and shrinking (zoom factors). Expressing a quadratic in vertex form (or turning point form) lets you see it as a dilation and/or translation of .A quadratic in standard form can be expressed in vertex form by completing the square. y-\text{int: } &= (0;3) \\ Learners must be able to determine the equation of a function from a given graph. &= a \left( \left(x + \frac{b}{2a} \right)^2 - \frac{b^2}{4a^2} + \frac{c}{a} \right) \\ -6 &= \left(x + 1 \right)^2 &= (x -4)^2 \\ There are 3 types of stationary points: Minimum point; Maximum point; Point of horizontal inflection; We call the turning point (or stationary point) in a domain (interval) a local minimum point or local maximum point depending on how the curve moves before and after it meets the stationary point. Step 2 Move the number term to the right side of the equation: x 2 + 4x = -1. Use your results to deduce the effect of $$a$$. If the function is twice differentiable, the stationary points that are not turning … If $$a>0$$, the graph of $$f(x)$$ is a “smile” and has a minimum turning point at $$(0;q)$$. On separate axes, accurately draw each of the following functions. The biggest exception to the location of the turning points is the 10% Opportunity. \therefore & (0;15) \\ For $$a<0$$; the graph of $$f(x)$$ is a “frown” and has a maximum turning point $$(0;q)$$. & = - 2 (0)^{2} + 1\\ &= \frac{1}{2}(4)^2 - 4(4) + \frac{7}{2} \\ On this version of the graph. Sketch the graph of $$y = \frac{1}{2}x^2 - 4x + \frac{7}{2}$$. x=-5 &\text{ or } x=-3 \\ On a positive quadratic graph (one with a positive coefficient of x^2 x2), the turning point is also the minimum point. The graph of $$f(x)$$ is stretched vertically upwards; as $$a$$ gets larger, the graph gets narrower. }(1;6): \qquad 6&=a+b+4 \ldots (1) \\ OK, some examples will help! &= 4 y_{\text{shifted}} &=2(x+3)^2 + 4(x+3) + 2 \\ At the turning point, the rate of change is zero shown by the expression above. x^2 &= \frac{1}{2} \\ & = \frac{3 \pm \sqrt{(-3)^2 - 4(2)(-4)}}{2(2)} \\ I don't see how this can be of any use to you, but for what it's worth: Turning points of graphs come from places where the derivative is zero, because the derivative of the function gives the slope of the tangent line. Calculate the $$y$$-coordinate of the $$y$$-intercept. (2) - (3) \quad -36&=20a-16 \\ There are two methods to find the turning point, Through factorising and completing the square.. Make sure you are happy … \therefore a&=1 which has no real solutions. Differentiating an equation gives the gradient at a certain point with a given value of x. Determine the coordinates of the turning point of $$y_3$$. \end{align}, \begin{align} Transformations of the graph of the quadratic can be explored by changing values of a, h and k. 1. We think you are located in For $$p<0$$, the graph is shifted to the left by $$p$$ units. Given the equation y=m²+7m+10, find the turning point of the vertex by first deriving the formula using differentiation. We therefore set the equation to zero. & = 5 (0)^{2} - 2\\ &= - \left( (x-2)^{2} - \left( \frac{4}{2} \right)^2 + 3 \right) \\ from the feed and spindle speed. The vertex is the point of the curve, where the line of symmetry crosses. \text{Subst. } This, in turn, makes all the other turning points about 5 … h(x)&= ax^2 + bx + c \\ x & =\pm \sqrt{\frac{2}{5}}\\ The $$y$$-intercept is $$(0;4)$$. The graph below shows a quadratic function with the following form: $$y = ax^2 + q$$. y &= -x^2 + 4x - 3 \\ f of d is a relative minimum or a local minimum value. For q > 0, the graph of f (x) is shifted vertically upwards by q units. Show that the $$x$$-value for the turning point of $$h(x) = ax^2 + bx + c$$ is given by $$x = -\frac{b}{2a}$$. \end{align}, \begin{align} This is done by Completing the Square and the turning point will be found at (-h,k). x &= -\frac{b}{2a} \\ The co-ordinates of this vertex is (1,-3) The vertex is also called the turning point. \text{Domain: } & \left \{ x: x \in \mathbb{R} \right \} \\ You therefore differentiate f … This gives the points $$(-\sqrt{2};0)$$ and $$(\sqrt{2};0)$$. 5. powered by. Turning Points of Quadratic Graphs. \end{align} The turning point of $$f(x)$$ is below the $$y$$-axis. & = \frac{3 \pm \sqrt{ 9 + 32}}{4} \\ &= a \left( x^2 + \frac{b}{a}x + \frac{c}{a} \right) \begin{align} The $$y$$-intercept is obtained by letting $$x = 0$$: Compare the graphs of $$y_1$$ and $$y_3$$. Once again, over the whole interval, there's definitely points that are lower. Therefore $$x = 1$$ or $$x = 7$$. &= -(x-2)^{2}+1 \\ Similarly, if $$a < 0$$, the range is $$\{ y: y \leq q, y \in \mathbb{R} \}$$. \begin{align} turning points y = x x2 − 6x + 8 turning points f (x) = √x + 3 turning points f (x) = cos (2x + 5) turning points f (x) = sin (3x) We use the method of completing the square: -5 &= (x - 1)^2 The axis of symmetry is the line $$x=0$$. Covid-19. Embedded videos, simulations and presentations from external sources are not necessarily covered $$(4;7)$$): \begin{align} A polynomial of degree n will have at most n – 1 turning points. 3 &= a+5a \\ &= 2\left( \left( x - \frac{5}{2} \right)^2 - \frac{25}{16} - 9 \right) \\ If $$a<0$$, the graph of $$f(x)$$ is a “frown” and has a maximum turning point at $$(0;q)$$. Fortunately they all give the same answer. \end{align}, \begin{align} In the equation $$y=a{x}^{2}+q$$, $$a$$ and $$q$$ are constants and have different effects on the parabola. y &= 4x - x^2 \\ If $$g(x)={x}^{2}+2$$, determine the domain and range of the function. The range of $$f(x)$$ depends on whether the value for $$a$$ is positive or negative. \therefore (-\frac{5}{2};0) &\text{ and } (-\frac{7}{2};0) &= -x^2 - 2x h(x) &= a \left( x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 + \frac{c}{a} \right) \\ \begin{align} Because the square of any number is always positive we get: $$x^2 \geq 0$$. Work together in pairs. Calculate the $$x$$-value of the turning point using \text{For } y=0 \quad 0 &= 4(x-3)^2 -1 \\ \text{Range: } & \left \{ y: y \geq -1, y\in \mathbb{R} \right \} & = 0 + 1 &= -(x - 2)^2 + 4 \\ \begin{align} \text{Range: } & \left \{ y: y \leq 4, y\in \mathbb{R} \right \} The vertex is the peak of the parabola where the velocity, or rate of change, is zero. In calculus you would learn to compute the first derivative here as $4x^3-3x^2-8x$, so you'd find its zeroes and then check in any of several ways which of them give turning … A turning point is a point at which the derivative changes sign. The implication is that throughout the observed range of the data, the expected probability of pt is an increasing function of expand_cap, though with some diminishing returns. The parabola is shifted $$\text{3}$$ units down, so $$y$$ must be replaced by $$(y+3)$$. \therefore b&=-1 & (-1;6) \\ &= x^2 + 8x + 15 \\ A function describes a specific relationship between two variables; where an independent (input) variable has exactly one dependent (output) variable. x &= -\left(\frac{-10}{2(5)}\right) \\ y = a x − b 2 + c. 1. a = 1. The sign of $$a$$ determines the shape of the graph. y & = - 2 x^{2} + 1 \\ Two points on the parabola are shown: Point A, the turning point of the parabola, at $$(0;4)$$, and Point B is at $$\left(2; \frac{8}{3}\right)$$. The domain is $$\left\{x:x\in \mathbb{R}\right\}$$ because there is no value for which $$g(x)$$ is undefined. For $$-10$$, the graph of $$f(x)$$ is a “smile” and has a minimum turning point at $$(0;q)$$. \begin{align} $$x$$-intercepts: $$(1;0)$$ and $$(5;0)$$. Complete the table and plot the following graphs on the same system of axes: Use your results to deduce the effect of $$q$$. Determine the turning point of $$g(x) = 3x^2 - 6x - 1$$. &= 2(x^2 - \frac{5}{2}x - 9) \\ for $$x \geq 0$$. The maximum value of y is 0 and it occurs when x = 0. Step 3 Complete the square on the left side of the equation and balance this by … You can find the turning point of a quadratic equation in a few ways. The parabola is shifted $$\text{1}$$ unit to the right, so $$x$$ must be replaced by $$(x-1)$$. }(1;6): \qquad -6&=25a+5b+4 \ldots (2) \\ &= 5 - \text{10} + 2\\ y & = ax^2 + q \\ Well, it is the point where the line stops going down and starts going up (see diagram below). Our treatment services are focused on complex presentations, providing specialist assessment and treatment, detailed management plans, medication initiation and … The turning point of $$f(x)$$ is above the $$x$$-axis. At the turning point $$(0;0)$$, $$f(x)=0$$. \therefore f(x) & \geq & q & The turning point is called the vertex. x= -\text{0,63} &\text{ and } x= \text{0,63} The effect of $$p$$ is a horizontal shift because all points are moved the same distance in the same direction (the entire graph slides to the left or to the right). Check the item you want to calculate, input values in the two boxes, and then press the Calculate button. In either case, the vertex is a turning point on the graph. The domain is $$\left\{x:x\in \mathbb{R}\right\}$$ because there is no value for which $$f(x)$$ is undefined. On the same system of axes, plot the following graphs: Use your sketches of the functions above to complete the following table: Consider the three functions given below and answer the questions that follow: Does $$y_1$$ have a minimum or maximum turning point? The turning point is when the rate of change is zero. &= 4x^2 -24x + 36 - 1 \\ vc (m/min) : Cutting Speed Dm (mm) : Workpiece Diameter π (3.14) : Pi n (min-1) : Main Axis Spindle Speed. y_{\text{shifted}} &= 3(x - 2-1)^2 + 2\left(x - 2 -\frac{1}{2}\right) \\ You could use MS Excel to find the equation. Then right click on the curve and choose "Add trendline" Choose "Polynomial" and "Order 2". \therefore 3 &= a + 6 \\ \therefore &\text{ no real solution } &= x^2 - 8x + 7 \\ \text{Subst. 7&= a(4^2) - 9\\ My subscripted variables (r_o, r_i, a_o, and a_i) are my own … \therefore (-5;0) &\text{ and } (-3;0) Therefore if $$a>0$$, the range is $$\left[q;\infty \right)$$. 2 x^{2} &=1\\ From the standard form of the equation we see that the turning point is $$(0;-3)$$. &= 3(x-1)^2 - 4 If the intercepts are given, use $$y = a(x - x_1)(x - x_2)$$. \therefore (1;0) &\text{ and } (3;0) The range of $$g(x)$$ can be calculated from: From the above we have that the turning point is at $$x = -p = - \frac{b}{2a}$$ and $$y = q = - \frac{b^2 -4ac}{4a}$$. In order to sketch graphs of the form $$f(x)=a{\left(x+p\right)}^{2}+q$$, we need to determine five characteristics: Sketch the graph of $$y = -\frac{1}{2}(x + 1)^2 - 3$$. 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	1.171 A shell flying with velocity v = 500 m/s bursts into three identical fragments so that the kinetic energy of the system increases $$\eta$$= 1.5 times. What maximum velocity can one of the fragments obtain? Solution : Real lousy one and you can find even lousier solutions. We also made a lousy assumption that the fastest fragment should move along the CM of the system, nothing else. This is not possible to get the answer given in the book without such an assumption. Though other cases are possible and I can provide you many of them but we are mostly relying on the word "maximum:. This is better if we work out in a frame the CM of the system is at rest. I will be using ' notation for physical quantities appearing in the CM frame. No doubt about $$\vec{p'_{1}}+\vec{p'_{2}}+\vec{p'_{3}} =0$$ Let $$\vec{p'_{2}}+\vec{p'_{3}} = \vec{P}$$ i.e. $$\vec{p'_{1}}$$ is equal and opposite to $$\vec{P}$$. Thus their corresponding kinetic energies can be given as $$\frac{p_{1}'^{2}}{2m}$$ and $$\frac{P^{2}}{4m}$$, where $$p_{1}' = P = mv'_{1}$$. Now you know that the change in KE in Lab frame and CM frame are same for a closed system obeying momentum conservation. We have the change in KE in Lab frame = $$(3/2 -1)\frac{1}{2} 3mv^{2}$$ and the change in KE in CM frame = $$\frac{p_{1}'^{2}}{2m}$$ + $$\frac{P^{2}}{4m}$$,where $$p_{1}' = P = mv'_{1}$$. After equating, we get $$v_{1}' = v$$. Thus the velocity of the fastest part in Lab frame is 2v.
	# If an electron enters into a space between the plates of a parallel plate capacitor at an angle α with the plates… Q: If an electron enters into a space between the plates of a parallel plate capacitor at an angle α with the plates and leaves at an angle β to the plates, find the ratio of its kinetic energy while entering the capacitor to that while leaving. Sol. Let u be the velocity of electron while entering the field and v be the velocity when it leaves the plates. Component of velocity parallel to the plates will remain unchanged. $\large u cos\alpha = v cos\beta$ $\large \frac{u}{v} = \frac{cos\beta}{cos\alpha}$ $\large \frac{\frac{1}{2}m u^2}{\frac{1}{2}m v^2} = (\frac{u}{v})^2 = (\frac{cos\beta}{cos\alpha})^2$
	Article Text Smoke-free law did not affect revenue from gaming in Delaware 1. L L Mandel, 2. B C Alamar, 3. S A Glantz 1. Center for Tobacco Control, Research & Education, University of California, San Francisco, San Francisco, California, USA 1. Correspondence to:  Professor Stanton A Glantz  Center for Tobacco Control, Research & Education, University of California, San Francisco, 530 Parnassus Ave, Suite 366, San Francisco, CA 94143, USA; glantzmedicine.ucsf.edu ## Abstract Objective: To determine the effect of the Delaware smoke-free law on gaming revenue. Methods: Linear regression of gaming revenue and average revenue per machine on a public policy variable, time, while controlling for economic activity and seasonal effects. Results: The linear regression showed that the smoke-free law was associated with no effect on total revenue or average revenue per machine. Conclusion: Smoke-free laws are associated with no change in gaming revenue. • economics • gambling • gaming • public policy • tobacco smoke pollution ## Statistics from Altmetric.com Smoke-free policies reduce cigarette consumption,1 which translates into lost tobacco industry revenue. Because of its lack of public credibility, the tobacco industry has used its allies in the hospitality industry, including the gaming industry,2 to act as surrogates in the fight against smoke-free workplaces.3 Despite the claims of the tobacco industry and its allies in the hospitality industry,3 it has been shown that smoke-free ordinances have no effect or a positive effect on restaurant and bar revenues,4 bingo revenue,5 and restaurant values.6 In response, opponents of smoke-free workplaces have claimed that smoke-free laws still negatively affect gaming revenue. In 2003, the gaming industry in Delaware was continuing to blame the November 2002 Delaware Clean Indoor Air Law for reduced revenue and layoffs.7,8 In May 2003, a gaming executive from Park Place Entertainment, which operates gaming facilities in Delaware (Dover Downs) and New Jersey, testified against a proposed smoke-free law at a New Jersey State Senate Health Committee hearing. The executive claimed that the Delaware Clean Indoor Act had a negative effect on gaming revenue and that her concerns for New Jersey were “supported by the results that happened in Delaware”.9 The executive claimed that Dover Downs, a racino that includes video slot machines and horse racing, had experienced a 25% drop in revenue and already reduced its staff by 100 workers since the Clean Indoor Air Act. She stated that Dover Downs would continue to reduce staff by as much as double what they already had if revenue continued to decrease, a trend she claimed would continue as long as the casinos were forced to be smoke-free. The executive admitted that factors such as other economic variables and weather may have played a role in the decrease in revenue, but concluded on behalf of the Casino Association of New Jersey, that “we would expect significant pressure on a business as a result of the enactment of this [proposed New Jersey smoke-free] bill”.9 The executive failed to mention that Park Place Entertainment’s first quarter Securities and Exchange Commission (SEC) filing stated that the 7% decrease in revenue for its three casinos in Atlantic City and the management fees from Dover Downs was mainly caused by inclement weather.10 The online summary of the filing did not mention the smoking restrictions as a reason revenue was down from the first quarter of the previous year.10 In 1994, Delaware passed legislation that allowed each of the state’s three racetracks to have slot machine-like video lottery terminals. Delaware Park and Dover Downs have operated machines since December, 1995 and Harrington Raceway has operated machines since August, 1996.11 As of May 2004, Delaware Park operated 2475 machines and Dover Downs operated 2500 machines, and the Harrington Raceway operated 1435 machines. On 27 November 2002 the state of Delaware implemented the Delaware Clean Indoor Air Act, a comprehensive state wide smoke-free law that made virtually all of Delaware’s public places and workplaces smoke-free, including the three racinos.12 ## METHODS Regression analysis was used to test for the effects of the smoke-free law on the gaming industry in Delaware. Data on gaming revenue by establishment and number of machines per establishment from January 1996 (Delaware Park and Dover Downs) and from August 1996 (Harrington) to May 2004 (for all three facilities) were obtained from the Delaware Video Lottery.13 The revenue data were then inflated to May 2004 dollars using the Chained Consumer Price Index (CPI) published by the US Bureau of Labor Statistics.14 The revenue and machine data were used to calculate the average revenue per machine on a total basis. Estimates of annual personal income are published quarterly for the MidEast region by the US Bureau of Economic Analysis.15 The data were interpolated to create monthly estimates. These monthly estimates of income were used as a control for economic activity. To further control for any economic or other changes not accounted for by the income, a time variable with 1996 set to 0 and increasing by one for each month was included in the regression equation. The time variable was tested for both linear and quadratic effects with the quadratic specification providing a better fit. The better fit for the quadratic time specification is likely due to the maturing, and thus slowing growth, of gaming in Delaware. Seasonal dummy variables were also tested in the models with winter defined as December, January, and February; spring defined as March, April, and May; and summer defined as June, July, and August. Only winter was found to be significant, thus only the results with winter are reported. We tested for effects of the law on two variables: total revenues and average revenue per machine. We did this by using a dummy policy variable, Plaw, that was set to zero pre smoke-free law and was set to one for every month the law was in effect. Total revenues = βlaw Plaw + γTimeTime + εtime2Time2 + αMachinesMachines + χIncomeIncome + ηWinterWinter The same specification as above was also used for average revenues per machine. The parameters were estimated using ordinary least squares. ## RESULTS The estimates from the equation are reported in table 1 and fig 1. Both estimations show a good fit to the data with an R2 of 0.803 for total revenues and an R2 of 0.622 for average revenues per machine. Controlling for underlying economic conditions, the results show no significant effect of the smoke-free law for either total revenues (p = 0.126) or average revenues per machine (p = 0.314). These results indicate that the smoke-free law had no effect on the total revenues (power to detect a 10% drop in revenues 0.98) or the average revenue per machine (power to detect a 10% drop in average revenues 0.97). Table 1 Estimated effects of the smoke-free law Figure 1 (A) Total revenues increase from the creation of the racinos and then flatten out with the economic downturn. No significant relation between total revenues and the smoke-free law exist. (B) Average revenues per machine vary overtime and decrease with the downturn in the economy. No significant relation between average revenues and the smoke-free law exist. Many studies have previously examined the effects of smoke-free laws and ordinances on the hospitality industry and charitable bingo. These studies have found either a positive or no effect on restaurant values, revenues, and employment; bar revenues and employment; and on bingo revenues. No previous study, however, has examined the effects of a state wide smoke-free law on gaming revenue. This study is the first to show no impact of smoke-free laws on gaming revenues. The paper tests for effects on both total revenues and average revenues per machine and finds no significant changes associated with the smoke-free law. ## DISCUSSION The smoke-free law had no detectable effect on total gaming revenue or the average revenue per machine. These results reject the argument that smoke-free laws hurt revenues from gaming. No effects were found on total revenue or average revenues per machine. Smoke-free laws do not harm racinos just as they do not harm restaurants, bars, or bingo parlours.4–6 Outside Delaware, casinos and other gaming facilities including bars, taverns, restaurants, and grocery stores with video gaming machines remain one of the last places that continue to be exempt from smoke-free ordinances. This type of gaming venue is the fastest growing sector within the gaming industry. Frank Fahrenkopf, one of the casino industry’s top Washington DC based lobbyists and president of the American Gaming Association, predicted at his keynote speech at the 2003 American Gaming Summit in Las Vegas that the continued growth of racinos, not resort style casinos, will fuel the expansion of the gaming industry.16 It is important that state legislatures and public health advocates in states considering allowing racinos know that despite the claims from opponents, smoke-free laws do not affect gaming revenue. ## REFERENCES View Abstract • Erratum to Mandel, L; BC Alamar; and SA Glantz. Smokefree Law did not affect revenue from gaming in Delaware. Tobacco Control 14 (2005), 10-12. The results in the original publication reflect a data entry error. The revised table in this erratum present the results with this error corrected. Using the corrected data, White's test for heteroskedasticity rejected homoskedasticity (p = 0.016) in the case of total revenues. We corrected for the heteroskedasticity in total revenues by using a weighted least squares analysis using the inverse of the number of video lottery machines as the weight. White's test of the residuals from the weighted regression did not reject homoskedasticity (p=0.293). Average revenues were homoskedastic (p=0.13) so we continued to use an unweighted regression, as in the original paper. The analysis based on the corrected data confirms the results of the published paper, that the smokefree law had no effect on revenue from gaming in Delaware. Table 1. Estimated effects of the smokefree law Total Revenues (Weighted Least Squares) Average Revenue per Machine (Ordinary Least Squares) Unit ($million) ($/machine) Variable Estimate Standard Error p value Estimate Standard Error p value Plaw(unit/month) -2.404 3.302 0.468 -1158.11 745.10 0.123 Time (unit/month) 0.612 0.102 <0.001 96.47 36.70 0.010 Time2(unit/month2) -0.005 0.001 <0.001 -0.31 0.278 0.259 Machines(unit/machine) 0.003 0.001 0.024 -2.424 -2.762 <0.001 Income (unit/\$million) 7.568 1.087 <0.001 10717.00 467.00 <0.001 Winter -4.147 0.872 <0.001 -1353.01 314.79 <0.001 n 101 101 R2 0.669 0.639 ## Footnotes • This research was supported by National Cancer Institute Grant CA-61021 ## Request permissions If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.
	# How to Analyze Companies Having found a bunch of leads to look into, the next step is to analyze the companies — to sort out the good from the bad. In this post, I describe how I do exactly that. ## What makes a company 'good'? The purpose of a company is to benefit its shareholders. As a result, the return that these shareholders receive on their investment is arguably the most important metric for a company. ## Return on Equity The de facto method for gauging shareholder returns is aptly called 'return on equity'. The return on equity (or RoE) is just the yearly profit divided by the company's equity: $$\text{Return on Equity} = \frac{\text{Net Income}}{\text{Equity}}$$ If you're unfamiliar with the term 'equity', it's a measure of how much the outstanding shares of the company are currently worth. (Basically, if you add up all the money that's been ploughed into the company since it started — how much would you get? ) To improve RoE, the formula suggests we simply need to increase net income. That's only slightly helpful, mainly because it's not very actionable. Breaking down the formula can get us a much better result: $$\text{RoE} = \frac{\text{Net Income}}{\text{Sales}} \times \frac{\text{Sales}}{\text{Assets}} \times \frac{\text{Assets}}{\text{Equity}}$$ Notice that the formula hasn't changed; 'sales' and 'assets' cancel out to give us the same end result. The formula, in this form, is known as the 'Dupont Model': • $$\frac{\text{Net Income}}{\text{Sales}}$$$represents "profitability", which tells us how what portion of the revenue a company generates is actually profit. • $$\frac{\text{Sales}}{\text{Assets}}$$$ represents "operating efficiency", a measure of how well the company employs the assets it has in generating sales. • $$\frac{\text{Assets}}{\text{Equity}}$$$represents "leverage". Also known as the $$\text{Equity Multiplier}$$$, it can be increased by taking on debt (and using that to increase assets). In summary, the 3 ways to improve RoE are through higher profitability, increased operating efficiency and leverage. When assessing whether to invest in a company, it's important to understand the overall RoE trend, as well as the trend of its parts. ### Leverage I'll start with leverage, since that's the most likely to kill a company. Additionally, I want to make sure that any company I invest in doesn't expressly rely on interest for operations — a strong sign that it may not be a halal investment to make. The downside of leverage as a source of capital is that it needs to be paid back no matter what — and that defaulting could result in the company going bankrupt. Things to look out for when assessing the likelihood of a company to default on it's debt payments • Does the company have debt? If so, assess: • Liquidity risk (ability to meet ST needs) • Solvency risk (ability to meet LT obligations) • How do we measure a company's reliance on debt? • Debt (as reported in financial statements) • Equity multiplier • How do we assess liquidity risk? • Quick Ratio • How do we assess solvency risk? • Interest Coverage Ratio • Debt/Equity Ratio • Corporate Bond Yield (published on Finra, if available) Beyond managing the insolvency risk inherent in leveraged companies, we need a way to measure what the shareholders are left with it the company does in fact go bankrupt... Assessing Liquidation Value Since debt holders are paid before shareholders in the event of bankruptcy, we'd need to determine what assets are left after the creditors are paid back - and then divide this remainder amongst all the shareholders. That is: $$\frac{\text{Tangible assets - Liabilities}}{\text{Shares outstanding}}$$ You can even discount tangible assets to arrive at a more conservative figure or, if you're being criminally conservative:1 $$\frac{\text{Cash + Marketable Securities - Liabilities}}{\text{Shares outstanding}}$$ In summary, when assessing the impact of debt on a company, we'd need to know: • Extent of debt (debt, equity multiplier) • Liquidity risk • Solvency risk • Liquidation value to shareholders Beyond analyzing what the company is worth right now , we should consider the potential upside. Of course, we'd want to look at historical performance to get an idea for what we can expect in the near future. Let's talk about the two remaining factors: Operating Efficiency, and Profitability. ## Operating Efficiency This is an easy one, so we'll get it out of the way before talking about the monster that is Profitability. Recall that: $$\text{Operating Efficiency} = \frac{\text{Sales}}{\text{Assets}}$$ Essentially, it's a measure of how well the company employs the assets it owns to generate sales (i.e. how 'efficient' it is in its day-to-day operations). Another term for this is $$\text{Asset Turnover}$$$— the higher, the better. ## Profitability In theory, profitability is an easy one to assess: $$\text{Profitability} = \frac{\text{Net Income}}{\text{Sales}}$$ Easy. Just divide the profit by the total sales. There's just one problem: the darn $$\text{Net Income}$$$ has a habit of bouncing around like crazy! So at times, we need to move higher and higher up the income stream — excluding things that would otherwise drive Net Income below zero and into loss-making territory: TermAlso known as… RevenueSales - COGSCost of Revenue = Gross IncomeGross Profit - Operating Expenses (excl. D&A)Excluding Depreciation & Amortization = EBITDA - Depreciation & Amortization = Operating IncomeOperating Profit, EBIT + Other Income - Interest - Taxes = Net income ⭐Profit (or Loss), Earnings That means looking at things like Operating Income, EBITDA or even Gross Income (you better believe things are desperate when you need to look that far up!) — which exclude things like tax, interest and Depreciation & Amortization. Compare to industry average ratios to get a sense for what the median performance is like, and how the target company compares to it's peers. I like companies that are smaller in size (and therefore more likely to be overlooked/written off) and that are trading at a fraction of their Tangible Book Value. Here, you just need the company to not go bankrupt (which most people already think it will) — and if escapes bankruptcy, you're in for some massive gains. # Other Factors Also, I'd probably have to consider other 'softer' factors when trying to predict where the company may be in the future. This includes: • Does management care? • Growth Strategy • Recent board/executive changes • Read posts from analysts with a successful track record on Seeking Alpha • Charts (Technical Analysis), specifically around RSI, resistance levels, etc # Open Questions • Optimal portfolio allocation between industries • How much emphasis I need to place on industry outlook (i.e. macro-indicators, industry research, etc) The sort of companies I'm interested in, the 'deep value' companies — I'm not expecting stellar financials. I come in knowing that there's something wrong, and it's just a question of figuring out whether the risk is real, or perceived. 1. This does not take into consideration preferred shares, and their liquidation preference, assuming there are any of course. In case there are, we'd need to consider the share count, the dividend dues as well as the liquidation preference. If a company goes bankrupt, for instance, preferred stockholders will get paid before common stockholders. It's important to remember, however, that if a company ends due to bankruptcy, paying creditors is the priority over paying preferred and common stockholders. Another unique feature of preferred stocks is that they have a fixed dividend
	# Nullable reference types C# 8.0 introduces nullable reference types and non-nullable reference types that enable you to make important statements about the properties for reference type variables: • A reference is not supposed to be null. When variables aren't supposed to be null, the compiler enforces rules that ensure it is safe to dereference these variables without first checking that it isn't null: • The variable must be initialized to a non-null value. • The variable can never be assigned the value null. • A reference may be null. When variables may be null, the compiler enforces different rules to ensure that you've correctly checked for a null reference: • The variable may only be dereferenced when the compiler can guarantee that the value isn't null. • These variables may be initialized with the default null value and may be assigned the value null in other code. This new feature provides significant benefits over the handling of reference variables in earlier versions of C# where the design intent couldn't be determined from the variable declaration. The compiler didn't provide safety against null reference exceptions for reference types: • A reference can be null. No warning is issued when a reference type is initialized to null, or null is later assigned to it. • A reference is assumed to be not null. The compiler doesn't issue any warnings when reference types are dereferenced. (With nullable references, the compiler issues warnings whenever you dereference a variable that may be null). With the addition of nullable reference types, you can declare your intent more clearly. The null value is the correct way to represent that a variable doesn't refer to a value. Don't use this feature to remove all null values from your code. Rather, you should declare your intent to the compiler and other developers that read your code. By declaring your intent, the compiler informs you when you write code that is inconsistent with that intent. A nullable reference type is noted using the same syntax as nullable value types: a ? is appended to the type of the variable. For example, the following variable declaration represents a nullable string variable, name: string? name; Any variable where the ? is not appended to the type name is a non-nullable reference type. That includes all reference type variables in existing code when you have enabled this feature. The compiler uses static analysis to determine if a nullable reference is known to be non-null. The compiler warns you if you dereference a nullable reference when it may be null. You can override this behavior by using the null-forgiving operator ! following a variable name. For example, if you know the name variable isn't null but the compiler issues a warning, you can write the following code to override the compiler's analysis: name!.Length; ## Nullability of types Any reference type can have one of four nullabilities, which describes when warnings are generated: • Nonnullable: Null can't be assigned to variables of this type. Variables of this type don't need to be null-checked before dereferencing. • Nullable: Null can be assigned to variables of this type. Dereferencing variables of this type without first checking for null causes a warning. • Oblivious: This is the pre-C# 8.0 state. Variables of this type can be dereferenced or assigned without warnings. • Unknown: This is generally for type parameters where constraints don't tell the compiler that the type must be nullable or nonnullable. The nullability of a type in a variable declaration is controlled by the nullable context in which the variable is declared. ## Nullable contexts Nullable contexts enable fine-grained control for how the compiler interprets reference type variables. The nullable annotation context of any given source line is either enabled or disabled. You can think of the pre-C# 8.0 compiler as compiling all your code in a disabled nullable context: any reference type may be null. The nullable warnings context may also be enabled or disabled. The nullable warnings context specifies the warnings generated by the compiler using its flow analysis. The nullable annotation context and nullable warning context can be set for a project using the Nullable element in your .csproj file. This element configures how the compiler interprets the nullability of types and what warnings are generated. Valid settings are: • enable: The nullable annotation context is enabled. The nullable warning context is enabled. • Variables of a reference type, string for example, are non-nullable. All nullability warnings are enabled. • warnings: The nullable annotation context is disabled. The nullable warning context is enabled. • Variables of a reference type are oblivious. All nullability warnings are enabled. • annotations: The nullable annotation context is enabled. The nullable warning context is disabled. • Variables of a reference type, string for example, are non-nullable. All nullability warnings are disabled. • disable: The nullable annotation context is disabled. The nullable warning context is disabled. • Variables of a reference type are oblivious, just like earlier versions of C#. All nullability warnings are disabled. Example: <Nullable>enable</Nullable> You can also use directives to set these same contexts anywhere in your project: • #nullable enable: Sets the nullable annotation context and nullable warning context to enabled. • #nullable disable: Sets the nullable annotation context and nullable warning context to disabled. • #nullable restore: Restores the nullable annotation context and nullable warning context to the project settings. • #nullable disable warnings: Set the nullable warning context to disabled. • #nullable enable warnings: Set the nullable warning context to enabled. • #nullable restore warnings: Restores the nullable warning context to the project settings. • #nullable disable annotations: Set the nullable annotation context to disabled. • #nullable enable annotations: Set the nullable annotation context to enabled. • #nullable restore annotations: Restores the annotation warning context to the project settings. By default, nullable annotation and warning contexts are disabled. That means that your existing code compiles without changes and without generating any new warnings. ## Nullable annotation context The compiler uses the following rules in a disabled nullable annotation context: • You can't declare nullable references in a disabled context. • All reference variables may be assigned a value of null. • No warnings are generated when a variable of a reference type is dereferenced. • The null-forgiving operator may not be used in a disabled context. The behavior is the same as previous versions of C#. The compiler uses the following rules in an enabled nullable annotation context: • Any variable of a reference type is a non-nullable reference. • Any non-nullable reference may be dereferenced safely. • Any nullable reference type (noted by ? after the type in the variable declaration) may be null. Static analysis determines if the value is known to be non-null when it is dereferenced. If not, the compiler warns you. • You can use the null-forgiving operator to declare that a nullable reference isn't null. In an enabled nullable annotation context, the ? character appended to a reference type declares a nullable reference type. The null-forgiving operator ! may be appended to an expression to declare that the expression isn't null. ## Nullable warning context The nullable warning context is distinct from the nullable annotation context. Warnings can be enabled even when the new annotations are disabled. The compiler uses static flow analysis to determine the null state of any reference. The null state is either not null or maybe null when the nullable warning context isn't disabled. If you dereference a reference when the compiler has determined it's maybe null, the compiler warns you. The state of a reference is maybe null unless the compiler can determine one of two conditions: 1. The variable has been definitely assigned to a non-null value. 2. The variable or expression has been checked against null before de-referencing it. The compiler generates warnings whenever you dereference a variable or expression in a maybe null state when the nullable warning context is enabled. Furthermore, warnings are generated when a maybe null variable or expression is assigned to a nonnullable reference type in an enabled nullable annotation context.
	An alternative algorithm for sorting a stack using only one additional stack From "Cracking the Coding Interview": Write a program to sort a stack in ascending order (with biggest items on top). You may use at most one additional stack to hold items, but you may not copy the elements into any other data structure (such as an array). The stack supports push, pop, is_empty, and peek The classic solution I found online to this (and the one in the book) is something like this: Algo #1 (Classic) def sort_stack(primary): secondary = [] while primary: tmp = primary.pop() while (secondary and secondary[-1] > tmp): primary.append(secondary.pop()) secondary.append(tmp) return secondary The gist of this being that we will return our secondary/auxiliary stack after sorting via $$\O(n^2)\$$ time. This is not what my initial approach was, however, and I do think my approach has some interesting qualities: Algo #2 (Mine) def sort_stack(primary): did_sort = False secondary = [] while not did_sort: # move from primary to secondary, pushing larger elements first when possible desc_swap(primary, secondary) # move from secondary back to primary, pushing smaller elements first when possible. Set did_sort = True if we're done and can exit. did_sort = asc_swap(secondary, primary) return primary def asc_swap(full, empty): temp = None did_sort = True yet_max = None while full: if not temp: temp = full.pop() if full: if full[-1] < temp: insert = full.pop() if insert < yet_max: did_sort = False yet_max = insert empty.append(insert) else: empty.append(temp) temp = None if temp: empty.append(temp) return did_sort def desc_swap(full, empty): temp = None while full: if not temp: temp = full.pop() if full: if full[-1] > temp: empty.append(full.pop()) else: empty.append(temp) temp = None if temp: empty.append(temp) Now obviously it is not nearly as clean or elegant, but it could be with some helper functions that dynamically choose our comparator and choose which element to push, etc. Basically what it is doing is this: # Start with stack in wrong order (worst case) primary: 4 3 2 1 secondary: # Swap to secondary, pushing larger elements first (1 is held in temp until the end because it is smaller than the following elements) primary: secondary: 2 3 4 1 # Swap back to primary, pushing smaller elements first primary: 1 3 2 4 secondary: # back to secondary primary: secondary: 4 3 2 1 # Back to primary, finished primary: 1 2 3 4 secondary: This strategy has a best-case/worst-case tradeoff. Algo #1 actually performs worst when the stack is already sorted and best when the stack is sorted in the wrong order, and algo #2 does the opposite. Questions • What are your thoughts? I think it is just an interesting way to sort that I haven't seen before. • Is there a name for this kind of sorting? I couldn't find similar algos but I'm sure theyre out there and would love to be able to describe it/recognize it better. • Honestly the question isn't solvable if read in it's most strict form. A single temp variable is technically a data structure so you unless you are allowed to swap the top elements you can only scan along them. Sep 19 '19 at 11:30 • couldn't find similar [algorithms] alternating sort order has been used in "read backwards tape sort" for a long time. Sep 28 '19 at 10:09 In this case, your code is buggy and crashes for sort_stack([3, 1, 4, 1]) on line 22, in asc_swap: if insert < yet_max: TypeError: '<' not supported between instances of 'int' and 'NoneType' $$$$ `
	An official website of the United States government Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites. # Temperature Extrapolation of Henry's Law Constants and the Isosteric Heat of Adsorption Published ### Author(s) Daniel Siderius, Harold Hatch, Vincent K. Shen ### Abstract Computational screening of adsorbent materials often uses the Henry's law constant ($\KH$) (at a particular temperature) as a first discriminator metric due to its relative ease of calculation. The isosteric heat of adsorption in the limit of zero pressure ($\qstinf$) is often calculated along with the Henry's law constant, and both properties are informative metrics of adsorbent material performance at low pressure conditions. In this article, we introduce a method for extrapolating $\KH$ as a function of temperature, using series-expansion coefficients that are easily computed at the same time as $\KH$ itself; the extrapolation function also yields $\qstinf$. The extrapolation is highly accurate over a wide range of temperatures when the basis temperature is sufficiently high, for a wide range of adsorbent materials and adsorbate gases. Various results suggest that the extrapolation is accurate when the extrapolation range in inverse-temperature space is limited to $\left\|\beta - \beta_0\right\| < 0.5$mol/kJ. Application of the extrapolation to a large set of materials is shown to be successful provided that $\KH$ is not extremely large and/or the extrapolation coefficients converge satisfactorily. The extrapolation is also able to predict $\qstinf$ for a system that shows an unusually large temperature dependence. The work provides a robust method for predicting $\KH$ and $\qstinf$ over a wide range of industrially relevant temperatures with minimal effort beyond that necessary to compute those properties at a single temperature, which facilitates the addition of application temperature to computational screening exercises. Citation Journal of Physical Chemistry B Volume 126 Issue 40
	NEW New Website Launch Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc... 1 ### JEE Main 2021 (Online) 27th August Morning Shift Numerical Let n be an odd natural number such that the variance of 1, 2, 3, 4, ......, n is 14. Then n is equal to _____________. Correct Answer is 13 ## Explanation $${{{n^2} - 1} \over {12}} = 14 \Rightarrow n = 13$$ 2 ### JEE Main 2021 (Online) 26th August Evening Shift Numerical Let the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x $$-$$ y is ______________. Correct Answer is 12 ## Explanation $$5 = {{3 + 7 + x + y} \over 4} \Rightarrow x + y = 10$$ Var(x) = $$10 = {{{3^2} + {7^2} + {x^2} + {y^2}} \over 4} - 25$$ $$140 = 49 + 9 + {x^2} + {y^2}$$ $${x^2} + {y^2} = 82$$ x + y = 10 $$\Rightarrow$$ (x, y) = (9, 1) Four numbers are 21, 9, 10, 8 Mean = $${{48} \over 4}$$ = 12 3 ### JEE Main 2021 (Online) 25th July Morning Shift Numerical Consider the following frequency distribution : Class : 10-20 20-30 30-40 40-50 50-60 Frequency : $$\alpha$$ 110 54 30 $$\beta$$ If the sum of all frequencies is 584 and median is 45, then \| $$\alpha$$ $$-$$ $$\beta$$ \| is equal to _______________. Correct Answer is 164 ## Explanation $$\because$$ Sum of frequencies = 584 $$\Rightarrow$$ $$\alpha$$ + $$\beta$$ = 390 Now, median is at $${{584} \over 2}$$ = 292th $$\because$$ Median = 45 (lies in class 40 - 50) $$\Rightarrow$$ $$\alpha$$ + 110 + 54 + 15 = 292 $$\Rightarrow$$ $$\alpha$$ = 113, $$\beta$$ = 277 $$\Rightarrow$$ \| $$\alpha$$ $$-$$ $$\beta$$ \| = 164 4 ### JEE Main 2021 (Online) 22th July Evening Shift Numerical Consider the following frequency distribution : Class : 0 $$-$$ 6 6 $$-$$ 12 12 $$-$$ 18 18 $$-$$ 24 24 $$-$$ 30 Frequency : a b 12 9 5 If mean = $${{309} \over {22}}$$ and median = 14, then the value (a $$-$$ b)2 is equal to _____________. Correct Answer is 4 ## Explanation Class Frequency $${x_i}$$ $${f_i}{x_i}$$ 0-6 a 3 3a 6-12 b 9 9b 12-18 12 15 180 18-24 9 21 189 24-30 5 27 135 $$N = (26 + a + b)$$ $$(504 + 3a + 9b)$$ Mean = $${{3a + 9b + 180 + 189 + 135} \over {a + b + 26}} = {{309} \over {22}}$$ $$\Rightarrow 66a + 198b + 11088 = 309a + 309b + 8034$$ $$\Rightarrow 243a + 111b = 3054$$ $$\Rightarrow 81a + 37b = 1018$$ $$\to$$ (1) Now, Median $$= 12 + {{{{a + b + c} \over 2} - (a + b)} \over {12}} \times 6 = 14$$ $$\Rightarrow {{13} \over 2} - \left( {{{a + b} \over 4}} \right) = 2$$ $$\Rightarrow {{a + b} \over 4} = {9 \over 2}$$ $$\Rightarrow a + b = 18$$ $$\to$$ (2) From equation (1) \$ (2) a = 8, b = 10 $$\therefore$$ $${(a - b)^2} = {(8 - 10)^2}$$ ### Joint Entrance Examination JEE Main JEE Advanced WB JEE ### Graduate Aptitude Test in Engineering GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN NEET Class 12
	# compatible.fasp 0th Percentile ##### Test Whether Function Arrays Are Compatible Tests whether two or more function arrays (class "fasp") are compatible. Keywords manip, spatial ##### Usage ## S3 method for class 'fasp': compatible(A, B, \dots) ##### Arguments A,B,... Two or more function arrays (object of class "fasp"). ##### Details An object of class "fasp" can be regarded as an array of functions. Such objects are returned by the command alltypes. This command tests whether such objects are compatible (so that, for example, they could be added or subtracted). It is a method for the generic command compatible. The function arrays are compatible if the arrays have the same dimensions, and the corresponding elements in each cell of the array are compatible as defined by compatible.fv. ##### Value • Logical value: TRUE if the objects are compatible, and FALSE if they are not. eval.fasp
	#### 题目列表 The shaded region in the xy-plane represents the surface of a pond. A length of 1 unit in the xy-plane represents 5 meters.The average depth of the water in the pond is 0.8 meter.Of the following, which is the best estimate of the total volume, in cubic meters, of the water in the pond? #### Quantity A $5(2^{x})$ #### Quantity B $4(3^{x})$ The table summarizes information about the 72 students in a history class. If a student is to be selected at random from the class, what is the probability that the student will be one who is male or a senior? From a group of 100 people including Alice and Bob, 40 people are to be randomly selected at the same time to win movie tickets. What is the probability that both Alice and Bob will be selected to win movie tickets? Data set Q contains 10,000 values. The median of the values is equal to the third quartile of the values. #### Quantity A The 60th percentile of the values in Q #### Quantity B The 70th percentile of the values in Q A certain personal identification number (PIN) for opening garage doors is a sequence of 4 digits, where each digit is selected from the digits 0 to 9. How many such PINs have at least two different digits and are palindromes? (Note: A palindrome is a sequence of characters that reads the same backward and forward. For example, 1221 is a palindrome.) r, s, and t are number on the number line above. #### Quantity A $\frac{rs}{t}$ #### Quantity B $rst$ A group of n people is categorized as follows. 9 people are taller than 6 feet. 14 people are shorter than 5(1/2) feet. 12 people are under 21 years old. If each person in the group is in at least one of the three categories, then n can be any integer between The length of one side of the above quadrilateral is p, while the length of the other three sides are q, r, and s, respectively. p #### Quantity B q+r+s A total of $84,000 was invested for one month in a new money market account that paid simple annual interest at the rate of r percent. If the investment earned$420 in interest for the month, what is the value of r? The passage implies that most popular science writing $x^{4}-3x^{2}+2=0$ and x > 1 x #### Quantity B $\sqrt{2}$ If 30 percent of the presidents and 80 percent of the vice presidents are serving their first terms, what is the least possible number of the organizations that have both their president and corresponding vice president serving their first term? One of the vice presidents will be chosen at random. What is the probability that the vice president chosen will be one whose age is greater than 63 years given that the age of the corresponding president is less than 53 years? Two bags, A and B, each contain marbles some of which are green. If one marble is selected at random from each bag, the probability is $\frac{1}{21}$ that both of the marbles selected will be green. If one marble is selected at random from bag A only, the probability is $\frac{5}{21}$ that the marble selected will be green. If one marble is selected at random from bag B only, what is the probability that the marble selected will be green? In the rectangular coordinate plane, (x, y) is a point on the circle with center O and radius 1, and xy ≠ 0. #### Quantity A $x^3 + y^3$ #### Quantity B 1 The total number of calories from carbohydrates in a single serving of Trail Mix is approximately what percent of the total number of calories in the serving? A certain candy bar contains a total of 28 grams of fat. The total number of grams of fat in the candy bar is what percent greater than the total number of grams of fat in one bag of Trail Mix? 25000 +道题目 170本备考书籍
	# Hiding Complexity post by Rafael Harth (sil-ver) · 2020-11-20T16:35:25.498Z · LW · GW · 12 comments ## Contents 1. The Principle 2. The Software Analogy 3. Human Learning 4. Factored Cognition None # 1. The Principle Suppose you have some difficult cognitive problem you want to solve. What is the difference between (1) making progress on the problem by thinking about it for an hour and (2) solving a well-defined subproblem whose solution is useful for the entire problem? (Finding a good characterization of the 'subproblem' category is important for Factored Cognition, but for [this post minus the last chapter], you can think of it purely as a problem of epistemic rationality and human thinking.) I expect most to share the intuition that there is a difference. However, the question appears ill-defined on second glance. 'Making progress' has to cash out as learning things you didn't know before, and it's unclear how that isn't 'solving subproblems'. Whatever you learned could probably be considered the solution to some problem. If we accept this, then both (1) and (2) technically involve solving subproblems. Nonetheless, we would intuitively talk about subproblems in (2) and not in (1). Can we characterize this difference formally? Is there a well-defined, low-level quantity such that our intuition as to whether we would call a bundle of cognitive work a 'subproblem' corresponds to the size of this quantity? I think there is. If you want, take a minute to think about it yourself; I've put my proposed solution into spoilers. I think the quantity is the length of the subproblem's solution, where by "solution", I mean "the information about the subproblem relevant for solving the entire problem". As an example, suppose the entire problem is "figure out the best next move in a chess game". Let's contrast (1) and (2): • (1) was someone thinking about this for an hour. The 'solution' here consists of everything she learns throughout that time, which may include many different ideas/insights about different possible moves/resolved confusions about the game state. There is probably no way to summarize all that information briefly. • (2) was solving a well-defined subproblem. An example here is, "figure out how good Be5 is".[1] If the other side can check in four turns given that move, then the entire solution to this subproblem is the three-word statement "Be5 is terrible". # 2. The Software Analogy Before we get to why I think the principle matters, let's try to understand it better. I think the analogy to software design is helpful here. Suppose a company wants to design some big project that will take about 900k (i.e., 900000) lines of code. How difficult is this? Here is a naive calculation: An amateur programmer with Python can write a 50 line procedure without bugs in an hour, which suggests a total time requirement of 18k hours. Thus, a hundred amateur programmers working 30 hours a week can write the project in six weeks. I'm not sure how far this calculation is off, but I think it's at least a factor of 20. This suggests that linear extrapolation doesn't work, and the reason for this is simple: as the size of the project goes up, not only is there more code to implement, but every piece of code becomes harder because the entire project is more complex. There are mode dependencies, more sources of error, and so forth. This is where decompositions come in. Suppose the entire project can be visualized like this, where black boxes denote components (corresponding to pieces of code) and edges dependencies between components. This naturally factors into three parts. Imagine you're head of the team tasked with implementing the bottom-left part. You can look at your job like this: (An 'interface' is purely a specification of the relationship, so the ellipses are each less than one black box.) Your team still has to implement 300k lines of code, but regardless of how difficult this is, it's only marginally harder than implementing a project that consists entirely of 300k lines. In the step from 300k to 900k, the cost actually does scale almost linearly.[2] As said at the outset, I'm talking about this not to make a point about software design but as an analogy to the topic of better and worse decompositions. In the analogy, the entire problem is coding the 900k line system, the subproblems are coding the three parts, and the solutions to the second and third part are the interfaces. I think this illustrates both why the mechanism is important and how exactly it works. For the 'why', imagine the decomposition were a lot worse. In this case, there's a higher overhead for each team, ergo higher overall cost. This has a direct analog in the case where a person is thinking about a problem on her own: the more complex the solutions to subproblems are, the harder it becomes for her to apply them to the entire problem. We are heavily bottlenecked by our ability to think about several things at once, so this can make a massive difference. For the 'how', notice that, while the complexity of the entire system trivially grows with its size, the task of programming it can ideally be kept simple (as in the case above), and this is done by hiding complexity. From the perspective of your team (previous picture), almost the entire complexity of the remaining project is hidden: it's been reduced to two simple, well-defined interfaces This mechanism is the same in the case where someone is working on a problem by herself: if she can carve out subproblems, and if those subproblems have short solutions, it dramatically reduces the perceived complexity of the entire problem. In both cases, we can think of the quality of a decomposition as the total amount of complexity it hides.[3] # 3. Human Learning I've come to view human learning primarily under the lens of hiding complexity. The world is extremely complicated; the only way to navigate it is to view it on many different layers of abstraction, such that each layer describes reality in a way that hides 99%+ of what's really going on. Something as complex as going grocery shopping is commonly reduced to an interface that only models time requirement and results. Abstractly, here is the principled argument as to why we know this is happening: 1. Thinking about a lot of things at once feels hard. 2. Any topic you understand well feels easy. 3. Therefore, any topic you understand well doesn't depend on a lot of things in your internal representation (i.e., in whatever structure your brain uses to store information). 4. However, many topics do, in fact, depend on a lot of things. 5. This implies your internal representation is hiding complexity. For a more elaborate concrete example, consider the task "create a presentation about ", where is something relatively simple: • At the highest level, you might think solely about the amount of time you have left to do it; the complexity of how to do it is hidden. • One level lower, you might think about (1) creating the slides and (2) practicing the speaking part; the complexity of how to do either is hidden. • One level lower, you might think about (1) what points you want to make throughout your presentation and (2) in what order do you want to make those points; the complexity of how to turn a point into a set of slides is hidden. • One level lower, you might think about how what slides you want for each major point; the complexity of how to create each individual slide is hidden. • Et cetera. In absolute terms, preparing a presentation is hard. It requires many different actions that must be carried out with a lot of precision for them to work. Nonetheless, the process of preparing it probably feels easy all the way because every level hides a ton of complexity. This works because you understand the process well: you know what levels of abstraction to use, and how and when to transition between them. The extreme version of this view (which I'm not arguing for) is that learning is almost entirely about hiding complexity. When you first hear of some new concept, it sounds all complicated and like it has lots of moving parts. When you successfully learned it, the complexity is hidden, and when the complexity is hidden, you have learned it. Given that humans can only think about a few things at the same time, this process only bottoms out on exceedingly simple tasks. Thus, under the extreme view, it's not turtles all the way down, but pretty far down. For the most part, learning just is representing concepts such that complexity is hidden. I once wrote a tiny post titled 'We tend to forget complicated things [LW · GW]'. The observation was that, if you stop studying a subject when it feels like you barely understand it, you will almost certainly forget about it in time (and my conclusion was that you should always study until you think it's easy). This agrees with the hiding complexity view: if something feels complicated, it's a sign that you haven't yet decomposed it such that complexity is hidden at every level, and hence haven't learned it properly. Under this view, 'learning complicated things' is almost an oxymoron: proper learning must involve making things feel not-complicated. It's worth noting that this principle appears to apply even for memorizing random data [LW · GW], at least to some extent, even though you might expect pure memorization to be a counter-example. There is also this lovely pie chart, which makes the same observation for mathematics: That is, math is not inherently complicated; only the parts that you haven't yet represented in a nice, complexity-hiding manner feel complicated. Once you have mastered a field, it feels wonderfully simple. # 4. Factored Cognition As mentioned in the outset, characterizing subproblems is important for Factored Cognition. Very briefly, Factored Cognition is about decomposing a problem into smaller problems. In one setting, a human has access to a model that is similar to herself, except (1) slightly dumber and (2) much faster (i.e., it can answer questions almost instantly). The hope is that this combined system (of the human who is allowed to use the model as often as she likes) is more capable than either the human or the model by themselves, and the idea is that the human can amplify performance by decomposing big problems into smaller problems, letting the model solve the small problems, and using its answers to solve the big problem. There are a ton of details to this, but most of them don't matter for our purposes.[4] What does matter is that the model has no memory and can only give short answers. This means that the human can't just tell it 'make progress on the problem', 'make more progress on the problem' and so on, but instead has to choose subproblems whose solutions can be described in a short message. An unexpected takeaway from thinking about this is that I now view Factored Cognition as intimately related with learning in general, the reason being that both share the goal of choosing subproblems whose solutions are as short as possible: • In the setting I've described for Factored Cognition, this is immediate from the fact that the model can't give long answers. • For learning, this is what I've argued in this post. (Note that optimizing subproblems to minimize the length of their solutions is synonymous with optimizing them to maximize their hidden complexity.) In other words, Factored Cognition primarily asks you to do something that you want to do anyway when learning about a subject. I've found that better understanding the relationship between the two has changed my thinking about both of them. (This post has been the second of two [LW · GW] prologue posts for an upcoming sequence on Factored Cognition. I've posted them as stand-alone because they make points that go beyond that topic. This won't be true for the remaining sequence, which will be narrowly focused on Factored Cognition and its relevance for Iterated Amplification and Debate.) 1. Be5 is "move the bishop to square E5". ↩︎ 2. One reason why this doesn't reflect reality is that real decompositions will seldom be as good; another is that coming up with the decomposition is part of the work (and in extension, part of the cost). Note that, even in this case, the three parts all need to be decomposed further, which may not work as well as the first decomposition did. ↩︎ 3. In Software design, the term 'modularity' describes something similar, but it is not a perfect match. Wikipedia defines it as "a logical partitioning of the 'software design' that allows complex software to be manageable for the purpose of implementation and maintenance". ↩︎ 4. After all, this is a post about hiding complexity! ↩︎ comment by steve2152 · 2020-11-20T19:29:01.310Z · LW(p) · GW(p) I find it helpful to think about our brain's understanding as lots of subroutines running in parallel. They mostly just sit around doing nothing. But sometimes they recognize a scenario for which they have something to say, and then they jump in and say it. So in chess, there's a subroutine that says "If the board position has such-and-such characteristics, it's worthwhile to consider protecting the queen." There's a subroutine that says "If the board position has such-and-characteristics, it's worthwhile to consider moving the pawn." And of course, once you consider moving the pawn, that brings to mind a different board position, and then new subroutines will recognize them, jump in, and have their say. So if you take an imperfect rule, like "Python code runs the same on Windows and Mac", the reason we can get by using this rule is because we have a whole ecosystem of subroutines on the lookout for exceptions to the rule. There's the main subroutine that says "Python code runs the same on Windows and Mac." But there's another subroutine that says "If you're sharing code between Windows and Mac, and there's a file path variable, then it's important to follow such-and-such best practices". And yet another subroutine is sitting around looking for UI code, ready to interject that that can also be a cross-platform incompatibility. And yet another subroutine is watching for you to call a system library, etc. etc. So, imagine you're working on a team, and you go to a team meeting. You sit around for a while, not saying anything. But then someone suggests an idea that you happened to have tried last week, which turned out not to work. Of course, you would immediately jump in to share your knowledge with the rest of the meeting participants. Then you go back to sitting quietly and listening. I think your whole understanding of the world and yourself and everything is a lot like that. There are countless millions of little subroutines, watching for certain cues, and ready to jump in and have their say when appropriate. (Kaj calls these things "subagents" [? · GW], I more typically call them "generative models" [LW · GW], Kurzweil calls them "patterns", Minsky calls this idea "society of mind", etc.) Factored cognition doesn't work this way (and that's why I'm cautiously pessimistic about it). Factored cognition is like you show up at the meeting, present a report, and then leave. If you would have had something important to say later on, too bad, you've already left the room. I'm skeptical that you can get very far in figuring things out if you're operating under that constraint. comment by Rafael Harth (sil-ver) · 2020-11-20T20:26:22.597Z · LW(p) · GW(p) I think your whole understanding of the world and yourself and everything is a lot like that. There are countless millions of little subroutines, watching for certain cues, and ready to jump in and have their say when appropriate. (Kaj calls these things "subagents", I more typically call them "generative models", Kurzweil calls them "patterns", Minsky calls this idea "society of mind", etc.). Factored cognition doesn't work this way (and that's why I'm cautiously pessimistic about it). I come to similar conclusions in what is right now post #4 of the sequence (this is #-1). I haven't read any of the posts you've linked, though, so I probably arrived at it through a very different process. I'm definitely going to read them now. But don't be too quick to write off Factored Cognition entirely based on that. The fact that it's a problem doesn't mean it's unsolvable. comment by steve2152 · 2020-11-20T21:00:23.936Z · LW(p) · GW(p) But don't be too quick to write off Factored Cognition entirely based on that. The fact that it's a problem doesn't mean it's unsolvable. I agree. I'm always inclined to say something like "I'm a bit skeptical about factored cognition, but I guess maybe it could work, who knows, couldn't hurt to try", but then I remember that I don't need to say that because practically everyone else thinks that too, even its most enthusiastic advocates, , as far as I can tell from my very light and casual familiarity with it. Hmm, maybe if you were going to read just one of mine on this particular topic, it should be instead Can You Get AGI From A Transformer [LW · GW] instead of the one I linked above. Meh, either way. comment by Rafael Harth (sil-ver) · 2020-11-24T12:27:30.904Z · LW(p) · GW(p) I've read them both, plus a bunch of your other posts. I think understanding the brain is pretty important for analyzing Factored Cognition -- my problem (and this is one I have in general) was that I find it almost impossibly difficult to just go and learn about a field I don't yet know anything about without guidance. That's why I had just accepted that I'm writing the sequence without engaging with the literature on neuroscience. Your posts have helped with that, though, so thanks. Fortunately, insofar as I've understood things correctly, your framework (which I know is a selection of theories from the literature and not uncontroversial) appears to agree with everything I've written in the sequence. More generally, I find the generative model picture strongly aligns with introspection, which has been my guide so far. When I pay attention to how I think about a difficult problem, and I've done that a lot while writing the sequence, it feels very much like waiting for the right hypothesis/explanation to appear, and not like reasoning backward. The mechanism that gives an illusion of control is precisely the fact that we decompose and can think about subquestions, so that part is a sort of reasoning backward on a high level -- but at bottom, I'm purely relying on my brain to just spit out explanations. Anyway, now I can add some (albeit indirect) reference to the neuroscience literature into that part of the sequence, which is nice :-) comment by steve2152 · 2020-11-24T16:55:27.420Z · LW(p) · GW(p) Thanks! Haha, nothing wrong with introspection! It's valid data, albeit sometimes misinterpreted or overgeneralized. Anyway, looking forward to your future posts! comment by Gurkenglas · 2020-12-01T10:33:19.110Z · LW(p) · GW(p) Does this mean that humans can only keep a few things in mind in order to make us hide complexity? Under that view the stereotypical forgetful professor isn't brilliant because he has a lot of memory free to think with at any time, but because he has had a lot of practice doing the most with a small memory. These seem experimentally distinguishable. I conjecture that describing the function of a neural network is the archetypal application of Factored Cognition, because we can cheat by training the neural network to have lots of information bottlenecks along which to decompose the task. comment by Rafael Harth (sil-ver) · 2020-12-01T11:01:33.742Z · LW(p) · GW(p) Under that view the stereotypical forgetful professor isn't brilliant because he has a lot of memory free to think with at any time, but because he has had a lot of practice doing the most with a small memory. These seem experimentally distinguishable. Not necessarily. This post only argues that the absolute ability of memory is highly limited, so in general, the ability of humans to solve complex tasks comes from being very clever with the small amount of memory we have. (Although, is 'memory' the right term for 'the ability to think about many things at once'?) I'm very confident this is true. Comparative ability (i.e., differences between different humans) could still be due to memory, and I'm not confident either way. Although my impression from talking to professors does suggest that it's better internal representations, I think. comment by adamShimi · 2020-11-21T18:01:20.481Z · LW(p) · GW(p) Cool post. I like the intuition of hiding complexity. Indeed, when I think of a low complexity description for what I got from my 1-hour of thinking very hard, the most natural answer is "a decomposition into subproblems and judgement about their solutions". In other words, Factored Cognition primarily asks you to do something that you want to do anyway when learning about a subject. I've found that better understanding the relationship between the two has changed my thinking about both of them. Would it be then representative of your view to say that a question can be solved through Factored Cognition iff the relevant topic can be learned by a human? comment by Rafael Harth (sil-ver) · 2020-11-21T18:49:45.671Z · LW(p) · GW(p) Would it be then representative of your view to say that a question can be solved through Factored Cognition iff the relevant topic can be learned by a human? Unfortunately, no. It's more like 'FC is inferior to one person learning insofar as decompositions lead to overhead'. And in practice, decompositions can have large overhead. You often don't know how to decompose a topic well until you already thought about it a lot. comment by slicko · 2020-11-21T05:00:02.450Z · LW(p) · GW(p) Intuitively, this feels accurate to me (at least for a certain category of problems - those that are solvable with divide and conquer strategies). I've always viewed most software best-practices (e.g. modularity, loose-coupling, SOLID principles) as techniques for "managing complexity". Programming is hard to begin with, and programming large systems is even harder. If the code you're looking at is thousands of lines of code in a single file with no apparent structure, then it's extremely hard to reason about. That's why we have "methods", a mechanism to mentally tuck away pieces of related functionality and abstract them into just a method name. Then, when that wasn't enough, we came up with classes, namespaces, projects, microservices ..etc. Also, I agree that a good amount of learning works this way. I would even point to "teaching" as another example of this. Teaching someone a complex topic often involves deciding what "levels" of understanding are at play, and what subproblems can be abstracted away at each level until the learner masters the current level. This works both when you teach someone in a top-down fashion (you're doing the division of problems for them and helping them learn the subsolutions, recursively), or a bottom-up fashion (you teach them a particular low-level solution, then name the subproblem you've just solved, zoom out, and repeat). comment by Kevin Lacker (kevin-lacker) · 2020-11-21T00:32:21.830Z · LW(p) · GW(p) I don't think the metaphor about writing code works. You say, "Imagine a company has to solve a problem that takes about 900,000 lines of code." But in practice, a company never possesses that information. They know what problem they have to solve, but not how many lines of code it will take. Certainly not when it's on the order of a million lines. For example, let's say you're working for a pizza chain that already does delivery, and you want to launch a mobile app to let people order your food. You can decompose that into parts pretty reasonably - you need an iOS app, you need an Android app, and you need an API into your existing order management system that the mobile apps can call. But how are you going to know how many lines of code those subproblems are? It probably isn't helpful to think about it in that way. The factoring into subproblems also doesn't quite make sense in this example: "Your team still has to implement 300k lines of code, but regardless of how difficult this is, it's only marginally harder than implementing a project that consists entirely of 300k lines." In this case, if you entirely ignore the work done by other teams, the Android app will actually get harder, because you can't just copy over the design work that's already been done by the iOS team. I feel like all the pros and cons of breaking a problem into smaller parts are lost by this high-level way of looking at it. My null hypothesis about this area of "factored cognition" would be that useful mechanisms of factoring a problem into multiple smaller problems are common, but they are entirely dependent on the specific nature of the problem you are solving. comment by Rafael Harth (sil-ver) · 2020-11-21T11:37:41.872Z · LW(p) · GW(p) I agree that the analogy doesn't work in every way; my judgment was that the aspects that are non-analogous don't significantly distract from the point. I think the primary difference is that software development has an external (as in, outside-the-human) component: in the software case, understanding the precise input/output behavior of a component isn't synonymous with having 'solved' that part of the problem; you also have to implement the code. But the way in which the existence of the remaining problem leads to increased complexity from the perspective of the team working on the bottom-left part -- and that's the key point -- seems perfectly analogous. I've updated downward on how domain-specific I think FC is throughout writing the sequence, but I don't have strong evidence on that point. I initially began by thinking and writing about exactly this, but the results were not super impressive and I eventually decided to exclude them entirely. Everything in the current version of the sequence is domain-general.
	# Circle A has a center at (1 ,4 ) and an area of 28 pi. Circle B has a center at (7 ,9 ) and an area of 8 pi. Do the circles overlap? If not, what is the shortest distance between them? Feb 10, 2018 Since sum of radii greater than the distance between the centers, Circles Overlap #### Explanation: Given : Circle A - O_A 91,4), A_A = 28pi, Radius ${R}_{A}$ Circle B - ${O}_{B} \left(1 , 4\right) , {A}_{B} = 28 \pi$, Radius ${R}_{B}$ ${R}_{A} = \sqrt{{A}_{A} / \pi} = \sqrt{\frac{28 \pi}{\pi}} \approx 5.29$ ${R}_{B} = \sqrt{{A}_{B} / \pi} = \sqrt{\frac{8 \pi}{\pi}} \approx 2.83$ Sum of Radii R_A + R_B = 5.29 + 2.83 = color(red)(8.12 Using distance formula, vec(O_AO_B) = sqrt((7-1)^2 + (9-4)^2) ~~ color(red)(7.81 Since sum of radii greater than the distance between the centers, Circles Overlap
	human-pose-estimation-0006 ## Use Case and High-Level Description This is a multi-person 2D pose estimation network based on the EfficientHRNet approach (that follows the Associative Embedding framework). For every person in an image, the network detects a human pose: a body skeleton consisting of keypoints and connections between them. The pose may contain up to 17 keypoints: ears, eyes, nose, shoulders, elbows, wrists, hips, knees, and ankles. ## Specification Metric Value Average Precision (AP) 51.1% GFlops 8.844 MParams 8.1506 Source framework PyTorch* Average Precision metric described in COCO Keypoint Evaluation site. ## Inputs Name: input, shape: 1, 3, 352, 352. An input image in the B, C, H, W format , where: • B - batch size • C - number of channels • H - image height • W - image width Expected color order is BGR. ## Outputs The net outputs three blobs: • heatmaps of shape B, 17, 176, 176 containing location heatmaps for keypoints of all types. Locations that are filtered out by non-maximum suppression algorithm have negated values assigned to them. • embeddings of shape B, 17, 176, 176, 1 containing associative embedding values, which are used for grouping individual keypoints into poses. ## Legal Information [*] Other names and brands may be claimed as the property of others.
	# How to get less spacing in math mode LaTeX commands like this: Here again, four different simulations are run on networks of size $C^{G} = 25 \times\ 25 = 625$, $C^{G} = 50 \times\ 50 = 2500$, $C^{G} = 250 \times\ 250 = 62500$, and $C^{G} = 500 \times\ 500 = 250000$... When compiled generate the following: The spacing in math mode really irritates me. There's too much white space around the equal sign, and the multiplication sign has more right padding than left padding. Is there any way to reduce this spacing? - There's no need to terminate a command with \ in math mode, because spaces are ignored there and placed automatically. – egreg Jan 22 '12 at 20:04 to eliminate the white spaces around the equal sign, simply enclose it in curly brackets $a{=}b$ – prettygully Jan 22 '12 at 21:40 @egreg thanx for the tip. I was not aware of that. – puk Jan 22 '12 at 21:59 Spacing around operators and relations in math mode are governed by specific skip lengths: \thinmuskip (default is 3mu), \medmuskip (default is 4mu plus 2mu minus 4mu) and \thickmuskip (default is 5mu plus 5mu). All are given in math units. Here is an illustration of how a modification (setting them to 0mu) to these lengths affect the output (I've removed the forced space after \times): \documentclass{article} \setlength{\parindent}{0pt}% Just for this example \begin{document} \verb\|Normal:\| \par Here again, four different simulations are run on networks of size $C^{G} = 25 \times 25 = 625$, $C^{G} = 50 \times 50 = 2500$, $C^{G} = 250 \times 250 = 62500$, and $C^{G} = 500 \times 500 = 250000$. Next, for each network, the weights of the idiothetic connections are collected and plotted based on radial distance, finally, they are averaged out over all neurons and shifted and set about the centre of the graph. \bigskip \begingroup \verb\|\thinmuskip=0mu:\| \par \setlength{\thinmuskip}{0mu} Here again, four different simulations are run on networks of size $C^{G} = 25 \times 25 = 625$, $C^{G} = 50 \times 50 = 2500$, $C^{G} = 250 \times 250 = 62500$, and $C^{G} = 500 \times 500 = 250000$. Next, for each network, the weights of the idiothetic connections are collected and plotted based on radial distance, finally, they are averaged out over all neurons and shifted and set about the centre of the graph. \endgroup \bigskip \begingroup \verb\|\medmuskip=0mu:\| \par \setlength{\medmuskip}{0mu} Here again, four different simulations are run on networks of size $C^{G} = 25 \times 25 = 625$, $C^{G} = 50 \times 50 = 2500$, $C^{G} = 250 \times 250 = 62500$, and $C^{G} = 500 \times 500 = 250000$. Next, for each network, the weights of the idiothetic connections are collected and plotted based on radial distance, finally, they are averaged out over all neurons and shifted and set about the centre of the graph. \endgroup \bigskip \begingroup \verb\|\thickmuskip=0mu:\| \par \setlength{\thickmuskip}{0mu} Here again, four different simulations are run on networks of size $C^{G} = 25 \times 25 = 625$, $C^{G} = 50 \times 50 = 2500$, $C^{G} = 250 \times 250 = 62500$, and $C^{G} = 500 \times 500 = 250000$. Next, for each network, the weights of the idiothetic connections are collected and plotted based on radial distance, finally, they are averaged out over all neurons and shifted and set about the centre of the graph. \endgroup \end{document} Grouping (\begingroup...\endgroup) localizes the effect of the length modification. \medmuskip is used around binary operators (like \times) and \thickmuskip is used around binary relations (like =). It's best to stick to the standard LaTeX spacing rather that insert your own. A good source of reading material on this is Herbert Voss' mathmode document. In particular, section 11 Space (p 28 onward). Inline math forms part of the regular text construction. That's why there is some stretch in \medmuskip and \thickmuskip, which allows spacing around math operators/operands to change depending on their location within the paragraph text. Removing this glue allows for a more consistent setting of inline expressions. See Keeping the distance between mathematical symbols consistent? However, it can also have some problematic effects with regards to line breaking, as is shown below: \documentclass{article} \setlength{\parindent}{0pt}% Just for this example \begin{document} \verb\|Normal:\| \par Here again, four different simulations are run on networks of size $C^{G} = 25 \times 25 = 625$, $C^{G} = 50 \times 50 = 2500$, $C^{G} = 250 \times 250 = 62500$, and $C^{G} = 500 \times 500 = 250000$. Next, for each network, the weights of the idiothetic connections are collected and plotted based on radial distance, finally, they are averaged out over all neurons and shifted and set about the centre of the graph. \bigskip \begingroup \verb\|Math glue removed:\| \par \setlength{\medmuskip}{1\medmuskip} \setlength{\thickmuskip}{1\thickmuskip} Here again, four different simulations are run on networks of size $C^{G} = 25 \times 25 = 625$, $C^{G} = 50 \times 50 = 2500$, $C^{G} = 250 \times 250 = 62500$, and $C^{G} = 500 \times 500 = 250000$. Next, for each network, the weights of the idiothetic connections are collected and plotted based on radial distance, finally, they are averaged out over all neurons and shifted and set about the centre of the graph. \endgroup \end{document} - Great answer! Maybe you could also mention that 1 mu is 1/18 em of \fam2? – morbusg Jan 22 '12 at 19:49 +1 for the examples, full code, and the additional begingroup/endgroup – puk Jan 22 '12 at 22:08 Something I've wondered for a reeeeeally long time. Thanks so much. – isomorphismes May 21 at 8:35 Why do you add extra space after the \times command. It completely scr*ws up the spacing. It looks very bad. I suggest you write $a \times b$, not $a \times\ b$. It'll look much better that way. Note that it also partially answers your question because you get less spacing. -
	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # Childhood maltreatment history and attention bias variability in healthy adult women: role of inflammation and the BDNF Val66Met genotype ## Abstract Childhood maltreatment has been associated with greater attention bias to emotional information, but the findings are controversial. Recently, a novel index of attention bias, i.e., attention bias variability (ABV), has been developed to better capture trauma-related attentional dysfunction. However, ABV in relation to childhood trauma has not been studied. Here, we examined the association of childhood maltreatment history with attention bias/ABV in 128 healthy adult women. Different types of childhood maltreatment were assessed with the Childhood Trauma Questionnaire. Attention bias/ABV was measured by the dot-probe task. Possible mechanisms whereby childhood maltreatment affects attention bias/ABV were also explored, focusing on blood proinflammatory markers and the BDNF Val66Met polymorphism. We observed a significant positive correlation between childhood emotional abuse and ABV (P = 0.002). Serum high-sensitivity tumor necrosis factor-α levels were significantly positively correlated with ABV (P < 0.001), but not with childhood maltreatment. Jonckheere–Terpstra trend test showed a significant tendency toward greater ABV with increasing numbers of the BDNF Met alleles (P = 0.021). A two-way analysis of variance further revealed that the genotype-by-emotional abuse interaction for ABV was significant (P = 0.022); individuals with the Val/Met and Met/Met genotypes exhibited even greater ABV when childhood emotional abuse was present. These results indicate that childhood emotional abuse can have a long-term negative impact on emotional attention control. Increased inflammation may be involved in the mechanism of ABV, possibly independently of childhood maltreatment. The BDNF Met allele may dose-dependently increase ABV by interacting with childhood emotional abuse. ## Introduction Childhood maltreatment, a form of early-life stress, is a major public health concern that has devastating consequences for both the affected individuals and society as a whole1. Accumulated evidence indicates that childhood maltreatment, including emotional/physical abuse, emotional/physical neglect, and sexual abuse, can significantly increase the risk of developing mental disorders in later life, which include—but not limited to—depression and posttraumatic stress disorder (PTSD)2,3,4. Although the mechanism by which childhood maltreatment leads to the development of these disorders remains unclear, it has been postulated that biased attention to emotional stimuli is involved in this process5,6. Attentional processing of emotional information is essential for emotion regulation, yet this is not always a straightforward process. Attention bias, a type of cognitive bias, is the tendency to pay more attention to emotionally salient stimuli than to neutral ones. Attention bias has been shown in individuals with a history of childhood maltreatment7,8,9,10, and this bias is considered as one of the most significant cognitive features in individuals exposed to early trauma5,6. Attention bias has also been observed in a variety of psychiatric conditions, including anxiety11, depression12, and PTSD13. One of the most widely used experimental paradigms for measuring attention bias is the dot-probe task. Using this task, several studies have investigated threat-related attention bias in relation to childhood maltreatment. However, their findings have been mixed, such that some studies report a bias toward threat9,14 whereas others describe no bias15 or even a bias away from threat16,17,18. Similar inconsistencies exist in the PTSD literature; some studies report a bias toward threat13,19, others no bias20,21, and still others a bias away from threat22,23,24. More recently, a novel index of attention bias, i.e., attention bias variability (ABV), has been used to better capture trauma-related attentional dysfunction25,26. Unlike the unidirectional assessment of attention bias, ABV reflects dynamic fluctuations in attention both toward and away from emotional information. It is also pointed out that such attention bias toward and away from threatening stimuli can reflect two main symptoms of PTSD, namely hypervigilance and avoidance, respectively20,27. In accordance with this, studies have demonstrated that PTSD patients exhibit greater ABV than control subjects and that ABV positively correlates with PTSD severity20,21,25,28,29. A randomized controlled trial comparing the efficacy of attention bias modification and attention control training for PTSD showed that the latter, but not the former, significantly reduced ABV and that reductions in ABV partially mediated improvement in PTSD symptoms28. This suggests that ABV is not merely an epiphenomenon of PTSD but can contribute to initiate and maintain this disorder. To our knowledge, however, no studies have examined the association of childhood maltreatment history with ABV. Various lines of research have investigated biological mechanisms underlying the link between early trauma and long-term alterations in cognitive styles and psychopathology that persist into adulthood30,31. One of these mechanisms is suggested to be dysregulation of the immune and inflammatory system32,33. This has been supported by the evidence that childhood maltreatment leads to increased inflammation as indexed by peripheral proinflammatory markers34,35 and that inflammation is associated with emotional attention36,37 and with psychiatric disorders including depression and PTSD38,39,40. While several studies have investigated the association between inflammation (e.g., inflammatory stress and circulating proinflammatory markers) and attention bias37,41, we are not aware of any studies that examined the association between inflammation and ABV. It is well-known that genetic factors and their complex interactions with environmental factors can increase the risk for developing multifactorial disorders, such as depression and PTSD42,43. Of the candidate genes, the brain-derived neurotrophic factor (BDNF) gene is implicated in both depression44,45 and PTSD46,47,48. The BDNF gene encodes a neurotrophin that is involved in neuronal growth, differentiation, maturation, and survival in immature neurons and in synaptic plasticity, neurotransmission, and receptor sensitivity in mature neurons49. BDNF has a functional single-nucleotide polymorphism (SNP) known as Val66Met, which is shown to reduce the activity-dependent secretion of BDNF50. Studies have demonstrated that this SNP interacts with childhood adversity to affect cognition, emotion, and psychopathology51,52,53,54,55; for example, it interacts with childhood maltreatment to influence the development and severity of depression52,53 and PTSD54. This SNP has also been associated with attention bias56,57 and circulating inflammatory markers58,59. Furthermore, this SNP has been associated with exaggerated expression of memories with negative emotional valence in both mice60 and human PTSD patients61. These findings together suggest that the BDNF Val66Met polymorphism may affect attention bias and ABV by interacting with childhood maltreatment. In this study, we investigated the association of childhood maltreatment history with attention bias and ABV in healthy adult women. In doing so, we distinguished different maltreatment types, namely emotional abuse, physical abuse, sexual abuse, emotional neglect, and physical neglect, given that long-term effects on cognition and emotion can differ across these maltreatment types. We further explored the possible mechanism by which childhood maltreatment affects attention bias/ABV, focusing on blood proinflammatory markers and the BDNF Val66Met genotype. ## Materials and methods ### Participants A total of 128 healthy women (age range: 20–64 years) participated in this study. This sample size was determined by referring to previous studies on attention bias in relation to childhood maltreatment5,8,10. All participants were native Japanese speakers, almost all of whom resided in the western part of the Tokyo metropolitan. They were recruited from the community through advertisements in free local magazines, our website, and university campuses, and by word of mouth. The validated Japanese version62 of the Mini International Neuropsychiatric Interview (M.I.N.I.)63 was administered to the participants in order to confirm the absence of any current Axis-I psychiatric disorders. In addition, the validated Japanese version64 of the Posttraumatic Diagnostic Scale (PDS)65 was used to further ascertain the absence of PTSD diagnosis. Additional exclusion criteria were: current severe physical illness or apparent intellectual disability, past/current contact to psychiatric services, and past/current use of psychotropic medications. This study was approved by the ethics committee of the National Center of Neurology and Psychiatry, Japan, and was conducted in accordance with the Declaration of Helsinki. Written informed consent was obtained from all participants after the study procedures had been fully explained. ### Questionnaires Detailed information is provided in Supplementary Methods. #### The Posttraumatic Diagnostic Scale (PDS)65 The PDS was created in accordance with the diagnostic criteria of DSM-IV PTSD65. It consists of four parts that assess traumatic experiences, PTSD symptoms during the past month, and the associated functional impairments; when the first part confirms the absence of traumatic experiences, no further assessment of symptomatology is made. #### The Childhood Trauma Questionnaire (CTQ)66 The CTQ is widely used to assess the history of childhood maltreatment66. The commonly used 28-item version of CTQ includes 25 clinical items and 3 validity items. The 25 items load onto 5 subscales that assess different types of childhood maltreatment, including emotional abuse, physical abuse, sexual abuse, emotional neglect, and physical neglect. All items are rated on a 5-point scale ranging from 1 to 5, with higher scores indicating more severe maltreatment. Cutoff scores for each subscale are defined in the manual of the CTQ67; the cutoff scores distinguishing between “none”/“low” for emotional abuse, physical abuse, sexual abuse, emotional neglect, and physical neglect are 8/9, 7/8, 5/6, 9/10, and 7/8, respectively. Cronbach α coefficients of the five CTQ subscales, i.e., emotional abuse, physical abuse, sexual abuse, emotional neglect, and physical neglect, in the present sample, were 0.84, 0.54, 0.56, 0.89, and 0.27, respectively. #### The Beck Depression Inventory-II (BDI-II)68 Depressive symptoms were assessed by the validated Japanese version69 of the BDI-II, a 21-item self-report questionnaire for depression severity during the past 2 weeks. Each item is scored on a 4-point scale from 0 to 3, with higher scores indicating greater depressive symptoms. The cutoff BDI-II total score distinguishing between “minimal”/“mild” depression is 13/1468. Cronbach α coefficient of the BDI-II in this sample was 0.84. #### The State-Trait Anxiety Inventory (STAI)70 Anxiety symptoms were assessed by the validated Japanese version71 of the STAI, a self-report questionnaire widely used to assess anxiety. It consists of two subscales for the state (STAI-S) and trait (STAI-T) anxiety, both comprising 20 items that are scored on a 4-point scale from 1 to 4; higher scores indicate greater anxiety. Cronbach α coefficients of the STAI-S and STAI-T in this sample were 0.77 and 0.60, respectively. ### Cognitive measures #### Attention bias and ABV The dot-probe task was used to measure attention bias and ABV. Before each trial, a white fixation cross (“+”) appeared in the center of the black display for 500 ms. Pairs of words were then presented for 1000 ms, one on top of the other. Immediately after their presentation, a probe (“←” or “→”) appeared in a location that corresponded to the center of one of the two words. Depending on the direction of the arrow, participants were instructed to press either a left or right key with the index or ring finger as quickly as possible. The two fingers were positioned above the two keys throughout the task. The arrow disappeared with the correct keypress. The next trial started after an interval of 500 ms. Six practice trials contained pairs of emotionally neutral words that were not displayed again. In the experimental 224 trials, there were 152 trials with pairs of generally negative words (e.g., “imprisonment”, “thief”, etc.) and neutral ones (e.g., “wheat”, “product”, etc.), and 72 trials with two neutral words. Types of trials were randomly interleaved. The position of the negative and neutral words and the position of the probe arrow were counterbalanced. To calculate attention bias, trials with the negative and neutral pairs were analyzed. After trials with errors and those with unnatural reaction times (RTs) (defined as >2000 ms or <150 ms) were removed, mean RTs were calculated separately for probes (a) replacing the same position as negative word (i.e., “congruent condition”) and (b) replacing the other position from negative word (“incongruent condition”). Trials above 2 standard deviations (SD) of the participant’s mean for each probe condition were excluded from further analyses; in the total participants, 5.5% and 5.8% of all trials for the congruent and incongruent conditions (respectively) were excluded. An attention bias score for negative words was calculated as follows: “Attention bias score” = “RTs for incongruent condition” − “RTs for congruent condition”. Positive values reflect attention toward the negative words, and negative values reflect attention away from the negative words. To further calculate ABV, all trials were split into eight sequential bins, and attention bias scores were calculated for each bin20. The SD of attention bias scores across bins was calculated and divided by mean RT to correct for variance in RTs72,73. Thus, ABV was calculated using the following equation, with greater values reflecting the instability of attention bias: $${\mathrm{SD}}_{{\mathrm{AB}}} = \sqrt {\frac{{\mathop {\sum }\nolimits_{i = 1}^8 \left( {{\mathrm{AB}}_i - \overline {{\mathrm{AB}}} } \right)^2}}{{{\mathrm{n}} - 1}}}$$ $${\mathrm{ABV}} = \frac{{{\mathrm{SD}}_{{\mathrm{AB}}}}}{{\overline {{\mathrm{RT}}} }}$$ where i indicates the bin number, n indicates the total number of bins (i.e., “8”), AB indicates attention bias scores, and RT indicates reaction time. #### Attention ability and global cognitive function We also examined general attention ability, as it can affect attention bias/ABV74. For this purpose, the validated Japanese version75 of the Repeatable Battery for the Assessment of Neuropsychological Status (RBANS)76 was used. While the full version of RBANS was administered to all participants, our analyses focused on the two cognitive indices, namely attention and total score. Age-corrected standardized scores, with a population mean of 100 and SD of 15, are calculated for each cognitive domain. Additional details are provided in Supplementary Methods. ### Measurement of proinflammatory markers Of the total 128 participants who completed the psychological and cognitive assessments, 118 individuals also participated in the blood testing to examine proinflammatory markers, including high-sensitivity tumor necrosis factor-α (hsTNF-α), interleukin-6 (IL-6), and high-sensitivity C-reactive protein (hsCRP). Reasons for the attrition of ten participants were: informed consent to the blood testing was not given, and blood sampling was technically difficult. This blood sampling was performed on the same day as the psychological/cognitive assessments. The samples were collected from each participant around noon (before lunch), between 11:30 AM and 12:30 PM. Levels of hsTNF-α, IL-6, and hsCRP were measured at a clinical laboratory (SRL Inc., Tokyo, Japan). Serum hsTNF-α, IL-6, and hsCRP levels were measured by enzyme-linked immunosorbent assay, chemiluminescent enzyme immunoassay, and nephelometry, respectively. There were no participants who showed hsCRP levels >10,000 ng/ml (i.e., 10 mg/l), an objective feature of acute infection77. Additional details are described in Supplementary Methods. #### BDNF Val66Met genotyping Of the total 128 participants, 107 also participated in the genetic testing by blood sampling. This sample attrition was due to the fact that several participants did not provide informed consent to genetic testing, in addition to the reasons related to blood sampling as described in the previous section. Genomic DNA was prepared from venous blood according to standard procedures. Rs6265 (Val66Met) was genotyped using the TaqMan SNP Genotyping Assays (assay ID: C__11592758_10). The thermal cycling conditions for polymerase chain reaction were: 1 cycle at 95 °C for 10 min followed by 45 cycles of 95 °C for 15 s and 60 °C for 1 min. The allele-specific fluorescence was measured using ABI PRISM 7900 Sequence Detection Systems (Applied Biosystems, Foster City, CA). All samples had a genotyping call rate of 97% or greater. ### Statistical analysis Averages are reported as “mean ± SD”, or “median (interquartile range)” where appropriate. The Kolmogorov–Smirnov normality test showed that ABV satisfied the assumption of the normal distribution while attention bias, CTQ scores, and inflammatory marker levels did not. Correlations among CTQ scores, attention ability/bias indices, and inflammatory markers were examined using Spearman’s rank-order correlation, considering that most of these data deviated from the normal distribution. The relationship of the BDNF Val66Met polymorphism with attention bias indices was examined using the Jonckheere–Terpstra trend test, a rank-based nonparametric test for determining if there is a statistically significant trend between two variables. In this analysis, the three genotype groups (i.e., Val/Val vs. Val/Met vs. Met/Met) were distinguished, and the dose–response relationship between the number of Met alleles and attention bias index was examined. When there was a significant trend between the genotype groups and attention bias index, a two-way analysis of variance (ANOVA) was used to further examine the interaction effect of genotype (i.e., Val/Val vs. Val/Met vs. Met/Met) and childhood maltreatment (i.e., presence vs. absence based on the CTQ cutoff score) on the attention bias index. In addition, a mediation analysis was used to explore the potential mediation of depressive symptoms that might underlie the relationship between childhood maltreatment and ABV; this analysis was performed as a post hoc exploratory analysis. It was conducted with the Mplus version 778, using the following command: VARIABLE: NAMES = X Y M; USEVARIABLES = X Y M; ANALYSIS: TYPE = GENERAL; ESTIMATOR = ML; MODEL: Y ON M X; M ON X; MODEL INDIRECT: Y IND M X; OUTPUT: STANDARDIZED(STDYX); Statistical significance was set at two-tailed P < 0.05 unless otherwise specified; while Bonferroni-corrected P values were applied to the five CTQ domains (i.e., P < 0.01) and three inflammatory markers (i.e., P < 0.016) in order to adjust for the multiple testing. All statistical analyses, except for the mediation analysis, were performed using the Statistical Package for the Social Sciences version 25 (IBM Corp., Tokyo, Japan). ## Results ### Demographic and psychological characteristics The demographic and psychological characteristics of the sample are shown in Table 1. Participants, all women, were on average mid-to-late thirties and tertiary educated, and predominantly nonsmokers. The median and interquartile range of the five CTQ domains indicated that while childhood emotional neglect was relatively frequently observed, emotional abuse and physical neglect were less frequent, and physical abuse and sexual abuse were totally absent in most participants (Table 1); of the 128 individuals, 111 (86.7%) reported no physical abuse and 115 (89.8%) reported no sexual abuse (i.e., scored “5”). The mean RBANS total score indicated that the majority of our participants were neuropsychologically normal. Neither age nor education was significantly correlated with CTQ 5 domains, attention bias, or ABV as indicated by Spearman’s rho (all P > 0.1). Smokers and nonsmokers did not significantly differ in CTQ 5 domains, attention bias, or ABV as examined by the Mann–Whitney U test (all P > 0.1). ### Correlations among attention indices and psychological variables Average scores of the three attention indices, including attention ability (assessed with RBANS), attention bias (dot-probe task), and ABV (dot-probe task) are presented in Table 1. Correlations of attention ability with attention bias and ABV were both at a trend level (rho = −0.150, P = 0.091 and rho = −0.153, P = 0.085, respectively). The correlation between attention bias and ABV was not significant (rho = −0.042, P = 0.638). Correlations of the five CTQ subscales with the three attention indices are shown in Table 2. Emotional abuse was significantly positively correlated with ABV (rho = 0.266, P = 0.002) while the other types of maltreatment were not significantly correlated with any of the attention indices (all P > 0.1). The relationship between emotional abuse and ABV is plotted in Fig. 1. Regarding the relationships with depressive and anxiety symptoms, ABV was significantly correlated with the BDI-II total score (rho = 0.252, P = 0.004) but not with STAI-S or STAI-T scores (both P > 0.05); while attention ability and attention bias were not significantly correlated with BDI-II or STAI-S/-T scores (all P > 0.05). ### Relationships among childhood maltreatment, inflammation, and attention indices Median (interquartile range) levels of hsTNF-α, IL-6, and hsCRP were 0.695 (0.528–0.805) pg/ml, 0.800 (0.600–1.100) pg/ml, and 198.5 (103.8–425.5) ng/ml, respectively. None of the five CTQ subscales were significantly correlated with any of the three proinflammatory marker levels (all P > 0.1). Correlations of the three proinflammatory marker levels with the three attention indices are shown in Table 3. hsTNF-α levels were significantly positively correlated with ABV (rho = 0.302, P < 0.001) while IL-6 and hsCRP levels were not significantly correlated with any of the attention indices (all P > 0.1). The relationship between hsTNF-α levels and ABV is plotted in Supplementary Fig. 1. ### Relationships among the BDNF Val66Met genotype, childhood maltreatment, and attention indices Numbers of participants with the BDNF Val/Val, Val/Met, and Met/Met genotypes were 40 (37.4%), 52 (48.6%), and 15 (14.0%), respectively. The genotype frequency did not deviate from Hardy–Weinberg equilibrium (χ2(1) = 0.08, P = 0.77). There was no significant association between the Val66Met genotype and attention bias according to the Jonckheere–Terpstra trend test (JT = 1,438.0, P = 0.084). However, the trend test showed a significant tendency toward greater ABV with increasing numbers of Met alleles (JT = 2,118.5, P = 0.021; Fig. 2a). Based on the significant association of ABV with the Val66Met genotype and with childhood emotional abuse, we further examined the genotype-by-abuse interaction effect on ABV, using the two-way ANOVA. To this end, the participants were grouped into those with childhood emotional abuse (n = 26) and those without (n = 81), using the well-defined cutoff of 8/9 points for CTQ emotional abuse67. The ANOVA revealed that the genotype-by-abuse interaction was significant (F = 2.77, P = 0.022); individuals with the Val/Met genotype and those with the Met/Met genotype exhibited even greater ABV when childhood emotional abuse was present whereas those with the Val/Val genotype did not show such an interaction with the abuse (Fig. 2b). ### Exploratory mediation analysis for the role of depressive symptoms in the relationship between childhood emotional abuse and ABV As described earlier, ABV was significantly positively correlated with depressive symptoms as well as with childhood emotional abuse. In addition, the correlation between CTQ emotional abuse and BDI-II total scores was also significant (rho = 0.216, P = 0.014). Based on these tripartite correlations, we further used mediation analysis to investigate the relationship between childhood emotional abuse, ABV, and depression. The independent variable (X) in this mediation model was childhood emotional abuse, given the temporal precedence. The dependent variable (Y) and moderator variable (M) can be either of the remaining two variables: ABV and depression. Since our primary aim here was to examine the mediation effect of depressive symptoms, we first tested the model (“Model 1”) where (Y) was ABV and (M) was depressive symptoms. In addition, we tested another model (“Model 2”) where (Y) was depressive symptoms and (M) was ABV, considering that this model is also theoretically and temporarily possible. To accommodate this mediation analysis, the raw CTQ emotional abuse score and BDI-II total score were log-transformed (the log-transformation of the latter was calculated after adding “1” to all scores in order to avoid taking the log of zero). Model 1 indicated a complete mediation, with significant effects of emotional abuse on depression (estimate: 0.225; SE = 0.084, P = 0.007) and of depression on ABV (estimate: 0.217; SE = 0.085, P = 0.010), and a nonsignificant effect of emotional abuse on ABV (estimate: 0.158; SE = 0.086, P = 0.066). The indirect effect was at a trend level (estimate: 0.049; SE = 0.027, P = 0.066). Model 2 indicated a partial mediation, with significant effects of emotional abuse on ABV (estimate: 0.207; SE = 0.085, P = 0.015), ABV on depression (estimate: 0.216; SE = 0.084, P = 0.010), and emotional abuse on depression (estimate: 0.181; SE = 0.085, P = 0.033). The indirect effect was at a trend level (estimate: 0.045; SE = 0.025, P = 0.079). These results lend some support to the former model in which depression mediates the relationship between childhood emotional abuse and adulthood ABV. ## Discussion The main findings can be summarized as follows. In healthy adult women, a history of childhood emotional abuse was significantly associated with ABV whereas none of the maltreatment types was associated with attention bias. Proinflammatory activity as indicated by serum hsTNF-α levels was significantly associated with greater ABV. The BDNF Val66Met polymorphism was associated with ABV in the manner that ABV significantly increased with increasing numbers of the Met allele. We further observed a significant genotype-by-emotional abuse interaction for ABV, such that Met allele carriers with childhood emotional abuse exhibited even greater ABV while this interactive effect was absent in Val/Val homozygotes. This is the first study, to our knowledge, to examine the association between childhood maltreatment and ABV in a nonclinical population. Our finding of the significant association between childhood emotional abuse and greater ABV is in line with the evidence for PTSD patients20,21,25,28,29. On the other hand, there was no association between childhood maltreatment and attention bias. This seems to be plausible, considering that previous findings of attention bias in trauma-related conditions have been mixed such that attention bias toward threat9,13,14,19,79, away from threat16,17,18,22,23,24, and no bias15,20,21 have all been reported. In this study of childhood maltreatment, the absence of association for attention bias, together with the significant relationship between emotional abuse and ABV, suggests that the fluctuation in attention bias towards and away from emotional stimuli over time, rather than constant bias in only one direction, better reflects the emotional attention dysfunction associated with childhood (emotional) trauma. In addition, the observed trend-level correlation between ABV and attention ability suggests that ABV can be partially, albeit not totally, accounted for by general attention ability; this result may suggest both validity and uniqueness of this index. Although not included in our hypothesis, the tripartite correlations between childhood emotional abuse, ABV, and depression led us to further investigate this association using mediation analysis. The result indicated that depressive symptoms could to some extent mediate the relationship between childhood emotional abuse and ABV. This mediation is plausible, considering that childhood abuse increases the risk of depression2,4 and that depression is associated with biased attention to emotional information36. In line with our finding, attention control training, a treatment shown to improve ABV in PTSD patients28,80, has been found to ameliorate not only PTSD symptoms but also depressive symptoms28,80,81. Taken together, it is conceivable that depression can be partly involved in the trauma-related ABV and its treatment mechanism. Biological factors may underlie the association between childhood trauma and persistent dysregulation in emotional attention. An increasing body of research has demonstrated that childhood maltreatment leaves a lasting impact on stress-related biological systems including the hypothalamic–pituitary–adrenal axis and the inflammatory system82. It is proposed that the early trauma-associated alterations in these systems can increase the vulnerability to various psychiatric disorders such as depression and PTSD, as these disorders are also associated with dysregulated glucocorticoid signaling and increased proinflammatory activities83,84. In this study, we, therefore, examined the three proinflammatory markers in relation to childhood maltreatment and attention bias indices. The results indicated that increased peripheral inflammation as indexed by serum hsTNF-α levels was significantly associated with greater ABV; however, hsTNF-α levels (and the other inflammatory markers) were not significantly correlated with childhood maltreatment. Although the latter result was not what we hypothesized, these results together suggest that TNF-α may be involved in ABV, possibly independently of childhood trauma. It may be that some other psychosocial factors than childhood maltreatment predominantly contributed to the variation in hsTNF-α levels, given that our nonclinical individuals had overall low levels of childhood maltreatment. Although we can only speculate on the reason for the specific association of ABV with hsTNF-α but not with hsCRP or IL-6, this might be related to the fact that TNF-α is a cytokine while CRP is an acute-phase protein. More specifically, studies show that proinflammatory cytokines such as TNF-α and interleukins can have detrimental effects on cognitive function85 and that circulating cytokines in the periphery can affect the brain such as through the blood–brain barrier86; while the effects of acute-phase proteins on the brain are less clear. IL-6 is also a cytokine, but it can have anti-inflammatory properties as well as proinflammatory ones87, which might explain the absence of a clear association with ABV. Our results on the BDNF Val66Met polymorphism suggest that the Met allele dose-dependently increases ABV and that this effect can at least partly be accounted for by its interaction with childhood emotional abuse. These findings accord with the evidence that the Met allele relates to attention bias56,57 and that this SNP interacts with childhood trauma to affect cognitive bias and brain morphology54,88. Supporting this, a functional neuroimaging study demonstrates that this SNP predicts amygdala and anterior hippocampus responses to emotional faces in anxious and depressed adolescents89. From a broader perspective, the present finding adds to the growing literature on gene–environment interaction for the development of psychopathology. It may also be worth noting that the minor Met allele frequency in our sample was 0.38, which is similar to the frequency of 0.41 reported in a representative genome variation database of Japanese individuals90. In contrast, the frequency of this allele is reported to be ~0.15–0.20 among many other populations such as Europeans, according to the Genome Aggregation Database (gnomAD). This relatively higher Met allele frequency in the present sample enabled us to examine its dose-dependent effect, without combining the Val/Met and Met/Met genotypes into a single group. The present analysis where all the three genotype groups were distinguished also accords with the evidence from animal studies demonstrating a dosage effect of the Met allele91. On the other hand, our finding warrants further investigations in other ethnic groups since there may be some ethnicity-specific effects of the Met allele91. Several limitations need to be considered when interpreting our findings. First, as the cognitive assessment and inflammatory measurement were derived from cross-sectional data at a single time point, their temporal relationships or long-term trajectories remain speculative. Second, as we only included female participants, it is unknown whether the present findings might be specific to women or common to both sexes. The main reason for the focus on female patients was that this study built on our previous studies of childhood maltreatment, cognitive function, memory bias, inflammation, and the BDNF Val66Met polymorphism in women with PTSD61,92,93,94. In addition, it was necessary to consider potential sex differences in this study, given the evidence for differential psychobiological impacts of childhood maltreatment between sexes95 and for sexually dimorphic effects of the BDNF Val66Met polymorphism49. Third, while we calculated ABV by using the bins of trial, some recent studies have used the moving average method to calculate it25,29. Although the latter method may be more sensitive in assessing ABV than the former, both calculation methods measure the same concept (i.e., ABV). In addition, the fact that a significant association was found between childhood emotional abuse and ABV in the present study suggests that ABV obtained by using bins is a sufficiently sensitive index that reflects the effect of childhood maltreatment. Still, it is possible that future studies could benefit from adopting the moving average method to calculate ABV. Fourth, we used a retrospective measure (i.e., the CTQ) to assess childhood maltreatment, which might have biased the results. Finally, this study targeted nonclinical individuals, and therefore its clinical significance remains relatively unclear. Relatedly, the exclusion of participants with psychiatric symptoms may have potentially restricted the range of the association between childhood maltreatment and attention bias/ABV. Indeed, the vast majority of our participants did not report any childhood experience of physical abuse or sexual abuse, suggesting that the nonsignificant results for physical/sexual abuse may be attributable to the very small number of individuals with a history of these two types of maltreatment; while this relatively low frequency of childhood physical/sexual abuse (compared to the other types of maltreatment) among the general population is in agreement with previous reports from Japan in which CTQ was used96,97. The low maltreatment frequency may have contributed to the lower internal consistency values (i.e., between 0.5 and 0.6) for physical/sexual abuse in our sample as compared to the high values (i.e., between 0.8 and 0.9) for emotional abuse/neglect. The internal consistency value for physical neglect was even lower, which would be attributable not only to the low frequency of this type of maltreatment in our sample but also to the generally lower internal consistency value of this subscale relative to the other four CTQ subscales as reported in previous studies targeting various populations98,99. Still, however, it is also possible that childhood emotional abuse more markedly influences ABV than do other types of maltreatment, considering that ABV reflects dysfunctional emotional attention processing. Meanwhile, the frequency of emotional neglect was relatively high in our sample, but this maltreatment type was not significantly associated with ABV. Although the reason for these discrepant results between emotional abuse and emotional neglect for ABV is not clear, the former might have a greater (or more direct) influence on this attentional function than the latter. In summary, this study shows that childhood emotional abuse is associated with greater ABV in healthy adult women, suggesting that the early-life adversity can have a long-term negative impact on emotional attention control. In terms of biological mechanisms, our findings indicate that increased inflammation as indexed by the elevated blood hsTNF-α level may be involved in ABV, possibly independently of childhood maltreatment. Findings further suggest that the relationship between childhood emotional abuse and ABV could be moderated by the BDNF Val66Met polymorphism in the manner that increasing numbers of Met alleles lead to greater ABV by interacting with the emotional abuse. Future studies that examine the association between early-life adversity and ABV among clinical populations, as well as those that investigate the underlying mechanisms, are needed. Efforts will also need to be directed at developing treatments/interventions that target ABV for subclinical and psychiatrically ill individuals. ## References 1. 1. Gilbert, R. et al. Child maltreatment: variation in trends and policies in six developed countries. Lancet 379, 758–772 (2012). 2. 2. Mandelli, L., Petrelli, C. & Serretti, A. The role of specific early trauma in adult depression: a meta-analysis of published literature. Childhood trauma and adult depression. Eur. Psychiatry 30, 665–680 (2015). 3. 3. McLaughlin, K. A. et al. Childhood adversities and post-traumatic stress disorder: evidence for stress sensitisation in the World Mental Health Surveys. Br. J. Psychiatry 211, 280–288 (2017). 4. 4. Scott, K. M., Smith, D. R. & Ellis, P. M. 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Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Hori, H., Itoh, M., Lin, M. et al. Childhood maltreatment history and attention bias variability in healthy adult women: role of inflammation and the BDNF Val66Met genotype. Transl Psychiatry 11, 122 (2021). https://doi.org/10.1038/s41398-021-01247-4
	# 横截面数据分类——基于R 《复杂数据统计方法》&网络&帮助文件适用情况：在因变量为分类变量而自变量含有多个分类变量或分类变量水平较多的情况。 (一)概论和例子数据来源：http://archive.ics.uci.edu/ml/datasets/Cardiotocography AC - # of accelerations per second FM - # of fetal movements per second UC - # of uterine contractions per second DL - # of light decelerations per second DS - # of severe decelerations per second DP - # of prolongued decelerations per second ASTV - percentage of time with abnormal short term variability MSTV - mean value of short term variability ALTV - percentage of time with abnormal long term variability MLTV - mean value of long term variability Width - width of FHR histogram Min - minimum of FHR histogram Max - Maximum of FHR histogram Nmax - # of histogram peaks Nzeros - # of histogram zeros Mode - histogram mode Mean - histogram mean Median - histogram median Variance - histogram variance Tendency - histogram tendency CLASS - FHR pattern class code (1 to 10) NSP - fetal state class code (N=normal; S=suspect; P=pathologic) (二)产生交叉验证数据集 1.十折交叉验证概念（百度百科）英文名叫做10-fold cross-validation，用来测试算法准确性。是常用的测试方法。将数据集分成十分，轮流将其中9份作为训练数据，1份作为测试数据，进行试验。每次试验都会得出相应的正确率（或差错率）。10次的结果的正确率（或差错率）的平均值作为对算法精度的估计，一般还需要进行多次10折交叉验证（例如10次10折交叉验证），再求其均值，作为对算法准确性的估计。 Fold=function(Z=10,w,D,seed=7777){ n=nrow(w) d=1:n dd=list() e=levels(w[,D]) T=length(e) set.seed(seed) for(i in 1:T){ d0=d[w[,D]==e[i]] j=length(d0) ZT=rep(1:Z,ceiling(j/Z))[1:j] id=cbind(sample(ZT,length(ZT)),d0) dd[[i]]=id} mm=list() for(i in 1:Z){u=NULL; for(j in 1:T)u=c(u,dd[[j]][dd[[j]][,1]==i,2]) mm[[i]]=u} return(mm)} #读入数据 #因子化最后三个哑元变量 F=21:23 #三个分类变量的列数 for(i in F) w[,i]=factor(w[,i]) D=23 #因变量的位置 Z=10 #折数 n=nrow(w)#行数 mm=Fold(Z,w,D,8888) library(rpart.plot) (a=rpart(NSP~.,w))#用决策树你和全部数据并打印输出 rpart.plot(a,type=2,extra=4) rpart.plot参数解释： x ： An rpart object. The only required argument. type： Type of plot. Five possibilities: 0 The default. Draw a split label at each split and a node label at each leaf. 1 Label all nodes, not just leaves. Similar to text.rpart's all=TRUE. 2 Like 1 but draw the split labels below the node labels. Similar to the plots in the CART book. 3 Draw separate split labels for the left and right directions. 4 Like 3 but label all nodes, not just leaves. Similar to text.rpart's fancy=TRUE. See also clip.right.labs. extra ： Display extra information at the nodes. Possible values: 0 No extra information (the default). 1 Display the number of observations that fall in the node (per class for class objects; prefixed by the number of events for poisson and exp models). Similar to text.rpart's use.n=TRUE. 2 Class models: display the classification rate at the node, expressed as the number of correct classifications and the number of observations in the node. Poisson and exp models: display the number of events. 3 Class models: misclassification rate at the node, expressed as the number of incorrect classifications and the number of observations in the node. 4 Class models: probability per class of observations in the node (conditioned on the node, sum across a node is 1). 5 Class models: like 4 but do not display the fitted class. 6 Class models: the probability of the second class only. Useful for binary responses. 7 Class models: like 6 but do not display the fitted class. 8 Class models: the probability of the fitted class. 9 Class models: the probabilities times the fraction of observations in the node (the probability relative to all observations, sum across all leaves is 1). branch： Controls the shape of the branch lines. Specify a value between 0 (V shaped branches) and 1 (square shouldered branches). Default is if(fallen.leaves) 1 else .2. branch=0 branch=1 digits ： The number of significant digits in displayed numbers. Default 2. rpart.plot(a,extra=4,digits=4) posted @ 2016-04-23 18:06 小星星☆ 阅读(2347) 评论(0编辑收藏举报
	Class 10 Science CONTROL AND COORDINATION Long Questions (Four Marks and Five Marks) In this page we have Class 10 Science CONTROL AND COORDINATION Long Questions (Four Marks and Five Marks) . Hope you like them and do not forget to like , social share and comment at the end of the page. 4 Marks Questions Question 1. How do muscle cells move? Question 2. Mention any two signals which will get disrupted in case of spinal injury. Question 3. What are reflex actions? Give two examples. Explain a reflex arc. Question 4. Draw a labelled structure of neuron and explain the function of any two of its parts. Question 5. (a)What happens at the synapse between two neurons? (b) How is brain protected? Question 6. Draw a flow diagram of reflex are illustrating the sequence of events which occur when we withdraw our hand on being pricked by a pin. Question 7. Write the constituents of central nervous system and peripheral nervous system. State in brief the function of central nervous system. Question 8. State how concentration of auxin stimulates the ells to grow longer on the side of the shoot which is away from light. Question 9. Define phototropism. Explain it with an example. Question 10. Name and state briefly one function each of any three phyto – hormones. Question 11. You must have noticed that as you approached 10 – 12 years of age, many dramatic changes appeared in your body. State reason. Question 12. Draw a diagram showing the correct positions of pancreas, thyroid gland, pituitary gland, adrenal gland in human being. Question 13. Mention three characteristic features of hormonal secretions in human beings. Question 14. (a) Draw a diagram of human brain. Label on it Cerebrum, Cerebellum. (b) What is the role of Cerebellum? Question 15. Name the hormone secreted by (a) pancreas (b) pituitary (c) thyroid. Write one function of each of the hormone. Question 16. In a neuron: (ii) Through what information travels as an impulse? (iii) Where does the impulse get converted into a chemical signal for outward transmission? 5 Marks Questions Question 1. Differentiate between tropic and nastic movements in plants. Question 2. (a) Name the system which communicates between the central nervous system and the other parts of the body. What does it consist of? (b) Mention three components of hind brain and write one function of each. Question 3. With the help of suitable example explain the terms phototropism, geotropism and chemotropism. Question 4. Plants do not have nervous system like animals. In the absence of nervous system, how does control and coordination activities. Question 5. How do plants respond to external stimuli? Question 6. In the absence of muscle cells, how do plant cells show movement? Question 7. State two different types of movement in plants. Mention two points of difference between them. Question 8. (a)Define hormones. Write four characteristics of hormones in humans. (b) Name the disorder caused by the following situations: (i)Under secretion of growth hormone (ii) Over secretion of growth hormone (iii) Under secretion of insulin (iv) Deficiency of iodine Question 9. When a person is scared, name the hormone which is directly secreted into the blood. Mention the gland which secretes this hormone. Question 10. Explain the need of chemical communication in multicellular organisms. Practice Question Question 1 Which among the following is not a base? A) NaOH B) $NH_4OH$ C) $C_2H_5OH$ D) KOH Question 2 What is the minimum resistance which can be made using five resistors each of 1/2 Ohm? A) 1/10 Ohm B) 1/25 ohm C) 10 ohm D) 2 ohm Question 3 Which of the following statement is incorrect? ? A) For every hormone there is a gene B) For production of every enzyme there is a gene C) For every molecule of fat there is a gene D) For every protein there is a gene
	## structural formula for 2,2-Dimethylbutane Yinhan_Liu_1D Posts: 51 Joined: Sat Sep 24, 2016 3:00 am ### structural formula for 2,2-Dimethylbutane On our green textbook, page 10, the structural formula is given as (CH3)3CCH2CH3. I am wondering, given the line structure on the side, could the structural formula also be CH3C(CH3)2CH2CH3? If not, why? Chem_Mod Posts: 19629 Joined: Thu Aug 04, 2011 1:53 pm Has upvoted: 889 times ### Re: structural formula for 2,2-Dimethylbutane Your formula basically split up the trimethyl portion from (CH3)3 to CH3..(CH3)2. It would be much more concise to keep it as (CH3)3.
	## Currently open theses topics working title annoucement together with Time approximation too for concurrent programs pdf and web SIRIUS $$R^3$$: Random Robust RUST pdf Security lab Typing and subtyping for security web ConSERnS Checking properties of legal electronic contracts web Verification tool for concurrenct software Modelling support for the Go langauge Refactoring at Scale Mutex with pointers pdf ### General remarks Open thesis topics of the PMA group in general can of course also found via group's web-page (also with other (co-)supervisors.) It may also be instructive to see not just at open topics, but at theses that have been completed. Thus we (try to) maintain an overview over completed master theses, where also in most cases the thesis document itself is available. That gives an expression over past theses under our supervision. Created: 2017-08-18 Fri 08:29 Validate
	## Sunday, November 02, 2014 ### The '''Human Phenome'''.Definition. In Relation to "The Human Genome". A text-module defining the fundamental F.E.D. concept of '''The Human Phenome''', relative to the concept of "The Human Genome", has been cleared for posting to the www.dialectics.org Glossary Page. I have also posted, below, a JPEG image of that definition, for your convenience. Regards, Miguel
	# Can a direct product of nonabelian simple groups be generated by two elements? Let $$G$$ be a direct product of nonabelian simple groups $$T_1,T_2,\dots,T_d$$ with $$d>1$$. Can $$G$$ be generated by two elements of $$G$$? • Hallo! On this forum, posters are discouraged from simply asking questions with no context. You are expected to express your own thoughts on the problem, and say what results you think may be relevant. I can tell you that the answer is yes (provided that $d$ is not too big). You could try and prove for yourself that $A_5 \times A_5$ can be generated by two elements. – Derek Holt Jan 10 at 8:33 • For each non-abelian simple group $S$, let $n_{S,d}$ be the number of orbits of $\mathrm{Aut}(S)$ on the subset of $S^d$ consisting of generating $d$-tuples. If I'm correct, the answer is: such $G$ is generated by $\le d$ elements iff for each $S$, there are at most $n_{S,d}$ summands isomorphic to $S$. – YCor Jan 10 at 9:10 • J. Wiegold proved some precise results about the minimum number of generators of a direct product of non-Abelian simple groups. – Geoff Robinson Jan 10 at 10:26 • This post is related, and this one. – Derek Holt Jan 10 at 12:07 • As a complement to my previous comment: if the $T_i$ are pairwise non-isomorphic, then the product is always 2-generated. In particular, for every $d$ there exists such a product that is 2-generated. Also for $d>19$, the group $\mathrm{Alt}_5^d$ is not 2-generated. Does this answer your question? As Derek Holt says, giving more context would be helpful. – YCor Jan 10 at 14:47 (1) If the $$T_i$$ are pairwise nonisomorphic, this will happen. Each of the $$T_i$$ will individually be $$2$$-generated (see here; this uses the classification of finite simple groups). Now, let $$(g_i, h_i)$$ be a pair of generators for $$T_i$$ and let $$g = (g_1, \ldots, g_d)$$ and $$h = (h_1, h_2, \ldots, h_d)$$; I claim that $$g$$ and $$h$$ generate $$\prod T_i$$. Indeed, let $$G = \langle g,h \rangle$$. Then $$G$$ surjects onto each $$T_i$$, so it has each $$T_i$$ as a Jordan-Holder factor. Using that the $$T_i$$ are pairwise nonisomorphic, $$G = \prod T_i$$. (2) At the other extreme, for a fixed finite simple group $$T$$, the number of generators needed to generate $$T^d$$ goes to $$\infty$$ as $$d \to \infty$$; see here. • By a result of Attila Maróti and M. Chiara Tamburini (Communications in Algebra, Vol. 41, No. 1 (2013), 34-49), if $G$ is a non-abelian finite simple group and $G^n$ is not $2$-generated, then $n > 2 \sqrt{\|G\|}$. It follows that $G^n$ is $2$-generated for all $1 \leq n \leq 19$. – Mikko Korhonen Jan 13 at 5:04
	# Single phase generator 1. ### michael1965 11 Hi, Why won't a generator composed of a stator that has 9 coils (each connected to it's neighbour in a ring until line out), and a rotor that has 10 permanent n50 magnets (NSNS....) generate useable electricity? 2. ### Baluncore 2,710 Welcome to PF. Each coil will generate a different phase. The sum of all the phases is a closed pentagon, so the difference voltage is zero. 3. ### michael1965 11 So it wouldn't produce a charge? If the electrons in the wire are excited, then surely a voltage is produced? ---- 5. ### michael1965 11 What if each coil was separate. 6. ### Baluncore 2,710 If each coil was separate then each coil would produce an output voltage. You would need to rectify each independently, then combine them as DC. With the odd number of 9 coils, there are none that could be combined to sum efficiently by reversing the connections of opposite coil pairs. I must have been asleep when I posted #2. For “closed pentagon” read “closed regular polygon”. 7. ### michael1965 11 What if there were 10 magnets, and 10 independent coils? 8. ### Baluncore 2,710 If the coils are independent then it may actually be better to have 9 coils so one is always generating current. Independent coils require rectification before they are combined. If 10 magnets and 10 coils then coils can be in series, each then needs to produce less voltage, so it can have less turns of thicker wire. It will produce an AC output. 9. ### michael1965 11 So, 10 magnets with 9 independent coils would have 90 coils generating a current per turn, whereas 10/10 would have 100 coils generating a current per turn. Both would be AC (?). You say "If 10 magnets and 10 coils then coils can be in series, each then needs to produce less voltage, so it can have less turns of thicker wire. It will produce an AC output." Why do we want a lower voltage? Why not wind it long and thin, and get a large voltage? On the subject of AC. If a magnet (n) passes over a coil it will excite the electrons in a wave (U or n?). As the next magnet (S) approaches, the electrons are excited again, this I believe is AC. If I have one single coil and 10 magnets of alternating polarity, then surely this would produce an AC. Similarly, if I have 10 coils and 10 magnets of alternating polarity this would also produce an AC in each coil. So, as long as the magnets are of alternating polarity, the charge produced should be AC, regardless of the number of magnets, the number of coils etc...?
	94-406 Cesi F, Martinelli F On the Layering Transition of an SOS Surface Interacting with a Wall. I. Equilibrium Results (194K, TeX) Dec 23, 94 Abstract , Paper (src), View paper (auto. generated ps), Index of related papers Abstract. We consider the model of a 2D surface above a fixed wall and attracted towards it by means of a positive magnetic field $h$ in the solid on solid (SOS) approximation, when the inverse temperature $\beta$ is very large and the external field $h$ is exponentially small in $\b$. We improve considerably previous results by Dinaburg and Mazel on the competition between the external field and the entropic repulsion with the wall, leading, in this case, to the phenomenon of layering phase transitions. In particular we show, using the Pirogov Sinai scheme as given by Zahradn\'\i k, that there exists a unique critical value $h^_k(\beta)$ in the interval $({1\over 4}e^{-4\beta k}, 4e^{-4\beta k})$ such that, for all $h\in (h^_{k+1},h^_k)$ and $\beta$ large enough, there exists a unique infinite volume Gibbs state. The typical configurations are small perturbations of the ground state represented by a surface at height $k+1$ above the wall. Moreover, for the same choice of the thermodynamic parameters, the influence of the boundary conditions of the Gibbs measure in a finite cube decays exponentially fast with the distance from the boundary. When $h=h^_k(\beta)$ we prove instead the convergence of the cluster expansion for both $k$ and $k+1$ boundary conditions. This fact signals the presence of a phase transition. In the second paper of this series we will consider a Glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external field $h$. Using the results proven in this paper we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free (absent). Files: 94-406.tex
	# Mersenne primes Hi everyone,I have some questions about Mersenne primes. ( Mersenne primes are prime numbers in the form of $$2^{p}-1$$ where $$p$$ is an integer )Please help.Thanks! The questions are: 1.Double Mersenne primes are in the form of $$2^{2^{p}-1}-1$$.If so,then what is the form of a triple Mersenne prime,if they exist? 2.Are there any double Mersenne primes other than the four known? 3.Is there any known formula to determine Mersenne primes? 4.Is there a limit to the number of Mersenne primes that can be calculated on a computer with a fixed processing power (say 1 Ghz )?(this is more of a computing question....) Thanks again! Note by Tan Li Xuan 5 years, 8 months ago MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$ Sort by: I was finding information on Mersenne Numbers and found some facts: 1. p-th Mersenne number where p is a prime is never a prime power. 2. If $$2^p - 1$$ is a Mersenne prime, then p is a prime. Note : This does not mean that if p is a prime, then $$2^p - 1$$ is a prime. - 5 years, 8 months ago Found Fact 1 here. - 5 years, 8 months ago Thanks! - 5 years, 8 months ago From http://primes.utm.edu/mersenne/: Lucas-Lehmer Test: For $$p$$ an odd prime, the Mersenne number $$2p-1$$ is prime if and only if $$2p-1$$ divides $$S(p-1)$$ where $$S(n+1) = S(n)2-2$$, and $$S(1) = 4$$. This is to answer your question "Is there any known formula to determine Mersenne primes?" Hope this helped. - 5 years, 8 months ago Thanks Daniel,that is really helpful! :) But Mersenne primes are in the form of $$2^{p}-1$$,not $$2p-1$$. - 5 years, 8 months ago
	## Wednesday, February 14, 2018 ### Van der Waerden Puzzle In [1] a puzzle is mentioned: Find a binary sequence $$x_1, \dots , x_8$$ that has no three equally spaced $$0$$s and no three equally spaced $$1$$s I.e. we want to forbid patterns like $$111$$,$$1x1x1$$, $$1xx1xx1$$ and $$000$$,$$0x0x0$$, $$0xx0xx0$$. It is solved as a SAT problem in [1], but we can also write this down as a MIP. Instead of just 0s and 1s we make it a bit more general: any color from a set $$c$$ can be used (this allows us to use more colors than just two and we can make nice pictures of the results). A MIP formulation can look like the two-liner: \begin{align}& x_{i,c} + x_{i+k,c} + x_{i+2k,c} \le 2 && \forall i,k,c \text{ such that } i+2k \le n\\ & \sum_c x_{i,c}=1 && \forall i\end{align} where $$x_{i,c} \in \{0,1\}$$ indicating whether color $$c$$ is selected for position $$i$$. For $$n=8$$ and two colors we find six solutions: For $$n=9$$ we can not find any solution. This is sometimes denoted as the Van der Waerden number $$W(3,3)=9$$. For small problems we can enumerate these by adding cuts and resolving. For slightly larger problems the solution pool technique in solvers like Cplex and Gurobi can help. For three colors $$W(3,3,3)=27$$ i.e. for $$n=27$$ we cannot find a solution without three equally spaced colors. For $$n=26$$ we can enumerate all 48 solutions: Only making this problem a little bit bigger is bringing our model to a screeching halt. E.g. $$W(3,3,3,3)=76$$. I could not find solutions for $$n=75$$ within 1,000 seconds, let alone enumerating them all. #### MIP vs CP formulation In the MIP formulation we used a binary representation of the decision variables $$x_{i,c}$$. If we would use a Constraint Programming solver (or an SMT Solver), we can can use integer variables $$x_i$$ directly. Binary MIP formulationInteger CP formulation \large{\begin{align}\min\>&0\\& x_{i,c} + x_{i+k,c} + x_{i+2k,c} \le 2 && \forall i+2k \le n\>\forall c \\ & \sum_c x_{i,c}=1 && \forall i\\ & x_{i,c} \in \{0,1\} \end{align}} \large{\begin{align} &x_{i} \ne x_{i+k} \vee x_{i+k} \ne x_{i+2k} && \forall i+2k \le n\\ & x_i \in \{1,\dots,nc\} \end{align} } #### Python/Z3 version This is to illustrate the integer CP formulation. Indexing in Python starts at 0 so we have to slightly adapt the model for this: from z3 import * n = 26 nc = 3 # integer variables X[i] X = [ Int('x%s' % i ) for i in range(n) ] # lower bounds X[i] >= 1 Xlo = And([X[i] >= 1 for i in range(n)]) # upper bounds X[i] <= nc Xup = And([X[i] <= nc for i in range(n)]) # forbid equally spaced colors Xforbid = And( [ Or(X[i] != X[i+k+1], X[i+k+1] != X[i+2(k+1)] ) for i in range(n) \ for k in range(n) if i+2(k+1) < n]) # combine all constraints Cons = [Xlo,Xup,Xforbid] # find all solutions s = Solver() k = 0 res = [] while s.check() == sat: m = s.model() k = k + 1 sol = [ m.evaluate(X[i]) for i in range(n)] res = res + [(k,sol)] forbid = Or([X[i] != sol[i] for i in range(n) ]) print(res) Here we use andor and != to model our Xforbid constraint. We could have used the same binary variable approach as done in the integer programming model. I tested that also: it turned out to be slower than using integers directly. #### Performance Here are some timings for generating these 48 solutions for the above example: Model/Solver Statistics Solution Time Binary formulation with Cplex (solution pool) 494 rows 78 binary variables 51092 simplex iterations 3721 nodes 1.6 seconds Integer formulation with Z3 26 integer variables 693 boolean variables (first model) 5158 clauses (first model) 871 boolean variables (last model) 13990 clauses (last model) 5.5 seconds Binary formulation with Z3 78 binary variables 860 boolean variables (first model) 2974 clauses (first model) 1015 boolean variables (last model) 6703 clauses (last model) 137 seconds The binary MIP formulation is not doing poorly at all when solved with a state-of-the-art MIP solver. Z3 really does not like the same binary formulation: we need to use the integer formulation instead to get good performance. Different formulations of the same problem can make a substantial difference in performance. Van der Waerden (1903-1996) #### References 1. Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 6, Satisfiability, 2015 2. Van der Waerden Number, https://en.wikipedia.org/wiki/Van_der_Waerden_number
	User: Equations # Solving Linear Equations An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. One of the terms in an equation may not be know and needs to be determined. Often this unknown term is represented by a letter such as "x". (e.g. 5 + x = - 2) Solving an equation means manipulating the expressions and finding the value of the unknown variables. In order to solve for the unknown variable, you must isolate the variable To keep an equation equal, we must do exactly the same thing to each side of the equation. If we add (or subtract) a quantity from one side, we must add (or subtract) that same quantity from the other side. Solve for x: $\fs2x+5=12\;\;\Rightarrow\;\;x=12-5\;\;\;\Rightarrow\;\;x=7$ Solve for x: $\fs22x+3=-11\;\;\Rightarrow\;\;2x=-11-3\;\;\;\Rightarrow\;\;x=-14/2\;\;\Rightarrow\;\;x=-7$ Solve for x: $-4x+6=-8x-22$ Solution:
	# Directiontal derivative 1. Mar 19, 2004 ### faust9 OK, I having a small problem understanding how my text book came about an answer to an example problem. $$f(x,y)=4-x^2-\frac{1}{4}y^2$$ at P(1,2) This next step is the one that's bugging me: $$u^\rightarrow=\cos(\frac{\pi}{3})\imath+\sin(\frac{\pi}{3})\jmath$$ This is one of those instances where something magic happens because right now I have little to no clue where the $$\frac{\pi}{3}$$ came from. Thanks... 2. Mar 19, 2004 ### jamesrc I may be wrong, but isn't that just part of the given information? I mean, you're being asked to find the directional derivative, meaning the rate of change of the function at a point in a given direction. So you know the function z=f(x,y), you're given a point P(1,2), and you're given the unit vector of the direction you're interested in. 3. Mar 19, 2004 ### faust9 Yeah your correct... The question is poorly written (or at least poorly formated). Thanks for showing me my stupid mistake. 4. Mar 20, 2004 ### HallsofIvy Staff Emeritus Specifically, $$u^\rightarrow=\cos(\frac{\pi}{3})\imath+\sin(\frac{\pi}{3})\jmath$$ is the unit vector point at an angle $$\pi/3$$ radians from the positive x-axis. 5. Mar 20, 2004 ### Chen (Psst. Use "\vec v" in LaTeX to display a vector... )
	# Off-center impulse equations [closed] A rigid steel bar with mass $M$ is hit sideways (very close to its end) by a steel ball with mass $m$ and velocity $v$. What are the equations of motion after elastic impact and how about conservation of momentum and energy? • What work have you done? What physics concept is giving you difficulty? – Bill N Jan 25 '16 at 22:44 • – Floris Jan 26 '16 at 0:35 • @Floris I don't think this question should be closed. The question of "How do we deal with off-center impulses?" although trivial to some does not have an nice accepted answer we can point at. – ja72 Jan 26 '16 at 3:44 • @ja72 there used to be a good set of answers to that specific question but it seems the question in point was deleted (it turned into an ugly flame war and led to several people leaving the site). I like the current answer of yours better than the one you gave to the duplicate I pointed to. – Floris Jan 26 '16 at 6:42 • I am voting to re-open in case this question attracts other even nicer answers. – ja72 Jan 26 '16 at 14:48 A rod of mass $M$ and length $\ell$ has mass moment of inertia $I = \frac{M}{12} \ell^2$. The impact at a distance of $c = \frac{\ell}{2}$ from the center of mass imparts an impulse $J$, while an equal and opposite impulse $-J$ is applied to the projectile mass $m$. The projectile is going to bounce with velocity $v_B = v - \frac{J}{m}$. The center of mass of the rod is going to start moving with velocity $v_C = \frac{J}{M}$ while the rod rotation is going to be $\omega = \frac{c J}{I}$. The linear velocity of the point of impact is thus $v_A = v_C + \omega c = \frac{J}{M} + \frac{c^2 J}{I}$. The law of impact states that the final separating velocity is a fraction of the initial impacting velocity. $v_A - v_B = \epsilon v$, where $\epsilon$ is the coefficient of restitution. Putting it all together yields: $$J = (1+\epsilon) \mu v$$$$\mu = \left( \frac{1}{m} + \frac{1}{M} + \frac{c^2}{I} \right)^{-1}$$ The term $\mu$ is called the reduced mass of the system, and it can be viewed as the effective mass of the impact. It converts the impact speed $v$ into momentum $\mu v$. Depending on the bounciness the exchanged momentum (impulse) is between $J = \mu v \ldots 2 \mu v$. Back substitute $J$ into the equations above to find $v_C$, $\omega$ and $v_B$. • The collision is stated to be elastic so $\epsilon = 1$? – Farcher Jan 26 '16 at 8:20 • @Farcher - Yes. I think general cases are more instructive than specific values. – ja72 Jan 26 '16 at 14:46 This is a standard rotational motion problem. Use conservation of linear momentum for the translational motion. Use conservation of angular momentum about any axis noting that some axes make the algebra easier than others. Use conservation of kinetic energy as the collision is stated to be elastic. • I can calculate the both the final velocity of the center of the bar as well as the balls from conservation of linear momentum. From conservation of energy I can calculate the angular velocity of the bar. Hence it seems like I don't need to consider conservation of the systems angular momentum!? – Jens Jan 25 '16 at 15:35 • @Jens no, you cannot. Linear momentum has two unknowns (final velocity of ball and bar), but is only one equation. – LLlAMnYP Jan 25 '16 at 17:42 Let us take the system - Rod + Ball Let the final velocity of center of mass of rod and ball be $v_1$ and $v_2$ respectively. As there is no net external force on the system, the net linear momentum of the system will be conserved. $mv = Mv_1 + mv_2$ And, as there is no net external torque about the COM of the system, the angular momentum of the system about it's COM will be conserved. This will give you one more equation with $v_1$, $v_2$ and $v$. Solve both equations simultaneously to get the results. • I can calculate the both the final velocity of the center of the bar as well as the balls from conservation of linear momentum. From conservation of energy I can calculate the angular velocity of the bar. Hence it seems like I don't need to consider conservation of the systems angular momentum!? – Jens Jan 25 '16 at 15:37 • Jens no the velocity of the ball after impact requires you to know the reduced mass of the rod, which depends on the location of impact. See @ja72's answer for details. – Floris Jan 26 '16 at 6:45 The linear motion of the bars center in the direction of the ball will be as if the ball had hit the bar at its center. The ball will bounce off the bar in the exact opposite direction but with reduced speed. The ball will also make the bar rotate around its center with some angular velocity. After impact, the sum of the translational energy of the ball and the bar and the rotational energy of the bar must equal the translational energy of the ball before impact (assuming no friction or deformation energy during impact). In this example, the conservation of momentum only seems to apply to the translational motion and does not include conservation of angular momentum? • Sounds like an answer, but there's this question mark at the end? If you want to improve your question, you should edit the question itself. – Daniel Griscom Jan 26 '16 at 0:16
	# Chapter 10 One-Way ANOVA Next, let’s go over the one-way ANOVA, which is used when we have two or more independent groups. Specifically, the one-way ANOVA determines if there is a difference in any of the group level means. Given that the analysis can also be used for two independent groups, this analysis could be used instead of the independent samples t-test. For example, let’s say that we were interested in determining if salary was significantly different for professors of different ranks (i.e., assistant professor, associate professor, and professor). For this example, we will use the dataset: datasetSalaries. As mentioned in the independent samples t-test, when dealing with categorical predictors, it is always good idea to check the coding scheme of the categorical variable. ## 2 3 ## Assistant Professor 0 0 ## Associate Professor 1 0 ## Professor 0 1 ## 10.1 Coding Categorical Variables When coding categorical variables, the number of contrasts ($$m$$) is equal to $$k-1$$ where $$k$$ is the number of levels (or factors) of an IV. In our example, rank has 3 levels (i.e., Assistant Professor, Associate Professor, and Professor) and thus has 2 contrasts [^13]. This is the reason why in a one-way ANOVA, we have $$k-1$$ between degrees of freedom. Thus, contrasts are always being performed, either explicitly if told or using a software package’s default coding scheme (e.g., dummy coding for R) when testing a main effect of a categorical IV. ### 10.1.1 Dummy Coding Since rank has not yet been explicitly defined by us yet, rank is dummy coded (i.e., for each contrast, a level is coded as 1 and all other levels are coded as 0). In dummy coding, one level will always be coded as 0 in both contrasts. By default, the first level of the categorical IV in alphabetical order will be coded as 0. Thus, in our example, Assistant Professor, which is the first level of rank in alphabetical order is coded as 0. When interpreting dummy coding contrasts, for each contrast, it is the level that is coded as 1 compared to the level that is coded as 0 in both contrasts. In our dummy coding scheme, the first contrast is comparing Associate Professor, which is coded as 1 in the first contrast, to Assistant Professor, which is coded as 0 in both contrasts. The second contrast is comparing Professor, which is coded as 1 in the second contrast, to Assistant Professor, which is again coded as 0 in both contrasts. ### 10.1.2 Orthogonal Coding However, a priori orthogonal contrast coding is preferred because this coding scheme allows us to create our own a priori contrasts of interest to answer more specific questions. For example, let’s say that in addition to determining if there is a difference in salary of professors based on their rank that we were also interested in more specific questions of interest such as: 1. Determining if there is a difference in salaries of professors that are tenured (Associate Professor and Professor) compared to those that are untenured (Assistant Professor). 2. Determining if there is a difference in salaries of professors that are Associate Professors compared to Professors. To make a coding scheme orthogonal, we need to make sure that 1) the sum of each contrast equals zero and that 2) the sum of each contrast product is 0. For example, to make the tenured vs. untenured professor contrast, we can code Assistant Professor as -2 and both Associate Professor and Professor as 1 and assign it the name TenuredvAssistant. It’s best to use signs that go in the direction of our prediction so that it’s reflected in our estimates. Since, we expect that tenured professors have higher salaries, tenured professors are assigned the positive value and assistant professors are assigned the negative value. Notice that the sum of the TenuredvAssistant contrast also adds up to 0.13 To make the next contrast of Associate Professor compared to Professor, we can code Assistant Professor as 0, Associate Professor as -1, and Professor as 1 and give it the name AssociatevProfessor. Notice again that the sum of the AssociatevProfessor contrast adds up to 0.14 Since both contrasts sum to zero, we can now check if the sum of products for each level across contrasts equals 0. In other words, we multiply the values across each contrast pair for each level and add them together. For example, the product across the contrasts of Assistant Professor is $$20 = 0$$, Associate Professor is $$1-1=-1$$, and Professor is $$11=1$$. If we add them together, we get $$0-1+1=0$$.15 To take orthogonal contrast coding a step further and make our estimates readily interpretable as the mean difference of that contrast, we can ensure that the difference of each contrast is 1. To do this, we can divide each contrast code by the number of values that are not coded as 0 for each contrast. For example, since all 3 codes of the TenuredvAssistant contrast are not equal to 0, we can divide by 3. Similarly, for the AssociatevProfessor contrast, we can divide each contrast code by 2 since 2 of the contrast codes are not equal to 0. Note that this is a rule of thumb and should be verified if contrasts are more complex. ## TenuredvAssistant AssociatevProfessor ## Assistant Professor -0.6666667 0.0 ## Associate Professor 0.3333333 -0.5 ## Professor 0.3333333 0.5 ## 10.2 Null and research hypothesis $$H_0: \mu_{Assistant\_Professors} = \mu_{Associate\_Professors} = \mu_{Professors}$$ $$H_1: \mu_{Assistant\_Professors} \ne \mu_{Associate\_Professors} \ne \mu_{Professors}$$ or $$\mu_{Assistant\_Professors} \ne \mu_{Associate\_Professors} = \mu_{Professors}$$ or $$\mu_{Assistant\_Professors} \ne \mu_{Professors} = \mu_{Associate\_Professors}$$ or $$\mu_{Associate\_Professors} \ne \mu_{Professors} = \mu_{Assistant\_Professors}$$ The null hypothesis states that there is no difference in salaries between professors of different ranks. The research hypothesis is stating that the salary of least one rank of professors is different than the others. Thus, the multiple options for a research hypothesis. ### 10.2.2 GLM approach $$Model: Salary = \beta_0 + \beta_1TenuredvAssistant + \beta_2AssociatevProfessor + \varepsilon$$ $$H_0: \beta_1 = \beta_2 = 0$$ $$H_1: \beta_1 \ne 0$$ and/or $$\beta_2 \ne 0$$ In the model, we now have both contrasts as predictors. The main effect of rank is testing both predictors of TenuredvAssistant and AssociatevProfessor. Given our orthogonal contrast coding scheme, the intercept ($$\beta_0$$) is now the mean 9-month academic salary of professors on average across rank. The slope of TenuredvAssistant ($$\beta_1$$) represents the mean difference in the 9-month academic salary for professors that are tenured compared to assistant professors. The slope of AssociatevProfessor ($$\beta_2$$) represents the mean difference in the 9-month academic salary for professors that are associate professors compared to professors. Thus, the null hypothesis states that there is a difference in salary of tenured versus untenured professors and there is also no difference in salary of associate professors compared to professors. In other words, there is no difference in salary based on a professor’s rank. The alternative hypothesis states that there is a difference in either the salary of tenured versus untenured professors or associate professor versus professor, or both. If only one contrast is significant, then that contrast must be strong enough to mask the non-significance of the other contrast. Given the vagueness of the research hypothesis (i.e., the significance can be a single contrast or both contrasts), we can look at each individual slope to see which is actually significant since we explicitly defined its associated contrasts beforehand. Thus, our null and research hypotheses for these specific questions would be the following for each slope: $H_0: \beta_1 = 0$ $H_1: \beta_1 \ne 0$ $H_0: \beta_2 = 0$ $H_1: \beta_2 \ne 0$ Each of these null and research hypothesis pairs are also known as 1 degree of freedom tests since we are only testing 1 contrast at a time. ## 10.3 Statistical analysis To perform a traditional One-Way ANOVA, we can use the aov() function. The first argument in the aov() function is the formula and the second argument is the name of the dataset. Notice, that in the formula, we do not have to specify both contrasts because we have already applied the coding scheme directly to the categorical IV of rank. ## Anova Table (Type III tests) ## ## Response: salary ## Sum Sq Df F value Pr(>F) ## (Intercept) 2.6481e+12 1 4741.08 < 2.2e-16 ## rank 1.4323e+11 2 128.22 < 2.2e-16 * ## Residuals 2.2007e+11 394 ## --- ## Signif. codes: 0 '*' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 Traditionally, if the main of effect of a categorical IV was significant, we would perform a post-hoc test (e.g., Tukey’s Honest Significant Difference [HSD]). In our case, rank is significant as the p-value of 2.2e-16 is less than our alpha of 0.05 and we would perform a post-hoc test to determine where the difference in salary actually comes from. To perform Tukey’s HSD, we could use the TukeyHSD function and provide the ANOVA analysis of the model as the input. The Tukey HSD essentially performs multiple independent samples t-tests of all possible pairs of the levels of a categorical IV but uses the error term from the ANOVA analysis in its calculation. ## Tukey multiple comparisons of means ## 95% family-wise confidence level ## ## Fit: aov(formula = salary ~ rank, data = datasetSalaries) ## The estimate for the rankAssociatevProfessor represents the mean difference in 9-month academic salary of associate professors compared to assistant professors. Specifically, the 9-month academic salary of professors was $32,896 more than associate professors. ## 10.4 Statistical decision Given the p-value of < 2.2e-16 for rank is less than the alpha level ($$\alpha$$) of 0.05, we will reject the null hypothesis. Notice both slopes are also significant, and we will also reject the null hypothesis for each. ## 10.5 APA statement An Analysis of Variance (ANOVA) using a priori orthogonal contrast coding was performed to test if there was a difference in the 9-month academic salary between 1) tenured (i.e., associate professors and professors) compared to untenured professors (i.e., assistant professors) and 2) associate professors compared to professors. There was a significant main effect of rank on salary, F(2, 394) = 128.22, p < .001. Specific to our contrasts, tenured professors earned significantly more than untenured professors, b =$29,548, t(1, 394) = 8.892, p < .001. Professors also earned significantly more than associate professors, b = \$32,896, t(1, 394) = 9.997, p < .001. 1. $$\Sigma Contrast\_Codes_{TenuredvAssistant} = -2 + 1 + 1 = 0$$ 2. $$\Sigma Contrast\_Code_{AssociatevProfessor} = 0 -1 + 1 = 0$$ 3. $$\Sigma Contrast_{TenuredvAssistant}Contrast_{AssociatevProfessor} = (20) + (1-1) + (1*1) = 0$$
	Journal cover Journal topic Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union Journal topic Hydrol. Earth Syst. Sci., 22, 3311–3330, 2018 https://doi.org/10.5194/hess-22-3311-2018 Hydrol. Earth Syst. Sci., 22, 3311–3330, 2018 https://doi.org/10.5194/hess-22-3311-2018 Research article 14 Jun 2018 Research article \| 14 Jun 2018 # Harnessing big data to rethink land heterogeneity in Earth system models Harnessing big data to rethink land heterogeneity in Earth system models Nathaniel W. Chaney1, Marjolein H. J. Van Huijgevoort1, Elena Shevliakova2, Sergey Malyshev2, Paul C. D. Milly3, Paul P. G. Gauthier4, and Benjamin N. Sulman5 Nathaniel W. Chaney et al. • 1Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey, USA • 2NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey, USA • 3US Geological Survey, Princeton, New Jersey, USA • 4Department of Geosciences, Princeton University, Princeton, New Jersey, USA • 5Sierra Nevada Research Institute, University of California, Merced, California, USA Correspondence: Nathaniel W. Chaney (nchaney@princeton.edu) Abstract The continual growth in the availability, detail, and wealth of environmental data provides an invaluable asset to improve the characterization of land heterogeneity in Earth system models – a persistent challenge in macroscale models. However, due to the nature of these data (volume and complexity) and computational constraints, these data are underused for global applications. As a proof of concept, this study explores how to effectively and efficiently harness these data in Earth system models over a 1/4 ( 25 km) grid cell in the western foothills of the Sierra Nevada in central California. First, a novel hierarchical multivariate clustering approach (HMC) is introduced that summarizes the high-dimensional environmental data space into hydrologically interconnected representative clusters (i.e., tiles). These tiles and their associated properties are then used to parameterize the sub-grid heterogeneity of the Geophysical Fluid Dynamics Laboratory (GFDL) LM4-HB land model. To assess how this clustering approach impacts the simulated water, energy, and carbon cycles, model experiments are run using a series of different tile configurations assembled using HMC. The results over the test domain show that (1) the observed similarity over the landscape makes it possible to converge on the macroscale response of the fully distributed model with around 300 sub-grid land model tiles; (2) assembling the sub-grid tile configuration from available environmental data can have a large impact on the macroscale states and fluxes of the water, energy, and carbon cycles; for example, the defined subsurface connections between the tiles lead to a dampening of macroscale extremes; (3) connecting the fine-scale grid to the model tiles via HMC enables circumvention of the classic scale discrepancies between the macroscale and field-scale estimates; this has potentially significant implications for the evaluation and application of Earth system models. 1 Introduction Spatial heterogeneity plays a critical role in the terrestrial water, energy, and biogeochemical cycles from local to continental and global scales. This has been recognized for decades in hydrology, ecology, geomorphology, and soil science where it has been observed repeatedly that at multiple temporal and spatial scales, land surface processes have deep ties to an ecosystem's spatial structure and function. As a result, the macroscale behavior of the water, energy, and biogeochemical cycles cannot be disentangled from their fine-scale processes and interactions . Recognizing the importance of multi-scale heterogeneity in the Earth system, in the 1980s and 1990s there was a strong emphasis on including its role in large-scale land surface models . These sub-grid schemes, however, rarely were designed to handle the sub-grid multi-scale coupling of the water, energy, and biogeochemical cycles as intended in contemporary applications (Clark et al.2015a). This is especially relevant as large-scale models begin to include human influence on land surface processes . Acknowledging these constraints, in recent years there has been a renewed emphasis on improving the representation of sub-grid heterogeneity through a more robust representation of soil, topographic, urban, and microclimate heterogeneity and by enabling explicit subsurface and surface interactions among sub-grid mosaic “tiles” or hydrologic response units . Although these emerging approaches have the potential to considerably improve the representation of sub-grid heterogeneity in macroscale models, their added value depends on both the data and the approaches used to inform the sub-grid schemes on the underlying heterogeneity of the physical environment – the primary driver of spatial heterogeneity in land surface processes (e.g., topography). The continual growth in the availability, detail, and wealth of Earth system data over the past decades provides an invaluable asset to make this possible. The use, harmonization, combination, and reinterpretation of field surveys, in situ networks, and satellite remote sensing have led to petabytes of vegetation, topography, climate, meteorology, and soil data over continental to global extents with spatial resolutions ranging between 10 and 1000 m . These data, although uncertain, provide invaluable very high-resolution snapshots of the sub-grid physical environment and its impact on ecosystem spatial structure and function. In most cases, environmental models use these data in two ways: (1) running the model at the native spatial resolution of the data (i.e., fully distributed model) and (2) running the model on a coarser grid by upscaling the data (i.e., lumped model). Both options are inadequate for Earth system models; the first is computationally unfeasible, while the second mostly disregards the role of sub-grid heterogeneity. The question then remains: how can these data be used to the fullest extent while minimizing both computation and storage requirements? This challenge is analogous to image compression where the goal is to maximize an image's information content while minimizing disk storage (e.g., clustering) . For environmental data, this equates to effectively and efficiently summarizing the data while minimizing information loss. This concept is the underlying basis for mosaic schemes in land surface models and hydrologic similarity in hydrologic models . Commonly, within macroscale models, the use of similarity amounts to binning (i.e., one-dimensional clustering) maps of variables that are used as proxies of the drivers of spatial heterogeneity (e.g., topographic index is used to represent the role of topography in subsurface flow) to assemble a set of representative sub-grid tiles. However, recently there have been efforts to formally connect the concept of similarity to the clustering of an n-dimensional space – in this case, the n-dimensional space is composed of the proxies of spatial heterogeneity . take this concept a step further by building a model (HydroBlocks) that enables the explicit interaction among the tiles assembled via multivariate clustering – the connectivity between the tiles is learned from the elevation data. In this case, hydrologic connectivity was enforced in the clustering algorithm primarily through flow accumulation area derived from the DEM while mostly disregarding the basin's channel, hillslope, and sub-basin structure; this oversimplification of catchment structure can lead to overly complex and at times unrealistic inter-tile connections – critical to accurately simulating baseflow production. Thus the need remains for a clustering approach that allows for a minimal number of tiles while robustly accounting for the fine-scale hydrologic structure of the macroscale grid cell. This paper introduces a hierarchical multivariate clustering approach (HMC) that summarizes the high-dimensional environmental data space into hydrologically interconnected representative clusters (i.e., tiles). HMC has three main components: (1) cluster the fine-scale map hillslopes in a grid cell into characteristic hillslopes, (2) discretize the characteristic hillslopes into height bands, and (3) cluster the intra-band soil and vegetation. As a proof of concept, these clusters (i.e., tiles) are then used within the Geophysical Fluid Dynamics Laboratory (GFDL) LM4-HB land model to explore its potential to provide a robust multi-scale coupling between the water, energy, and biogeochemical cycles. Using a 1/4 ( 25 km) grid cell in the western foothills of the Sierra Nevada in central California as a test bed, this paper explores the number of tiles necessary to robustly account for the sub-grid multi-scale heterogeneity in the macroscale states and fluxes. It also explores the role that each of the drivers of spatial heterogeneity plays at the macroscale. Finally, the implications of this approach in the application and validation of large-scale environmental models and Earth system models are discussed. Figure 1Test-bed site in the foothills of the southern Sierra Nevada in California used to develop, implement, and test the HMC algorithm. The region is characterized by strong heterogeneity in topography, climate, and soil properties leading to a complex multi-scale ecosystem spatial structure. To develop, implement, and evaluate HMC, this study uses a 1/4 grid cell that covers the foothills and high sierras of the southern Sierra Nevada in California (Fig. 1). This domain is selected due to the observable role of the physical environment in the sub-grid heterogeneity. This heterogeneity is primarily explained by the strong topographic gradient between the Central Valley and the Sierra Nevada and its impact on precipitation and temperature; the highest point in the domain is 3118 m, while the lowest is 163 m. This area has an annual average rainfall of 614 mm with large intra-cell variability with a minimum of 299 mm yr−1 in the lowlands and a maximum of 1152 mm yr−1 in the uplands. In both cases, most of the rainfall occurs between October and May. The uplands are primarily covered by evergreen vegetation, while shrubs and grasses cover the lowlands. The uplands are characterized by a higher sand content than the lowlands and vice versa for clay content. ## 2.1 Land and meteorological data ### 2.1.1 Topography This study uses the 1 arcsec USGS National Elevation Dataset (NED) digital elevation model (DEM) and a series of derived products including flow accumulation area, hillslopes, slope, height above the nearest drainage area (HAND), and aspect. The NED covers the contiguous United States (CONUS) and is created primarily from the USGS 10 and 30 m digital elevation models and from higher-resolution data sources such as light detection and ranging, interferometric synthetic aperture radar, and high-resolution imagery . ## 2.2 Meteorology and climate The recently developed Princeton CONUS Forcing (PCF) dataset provides a 1/32 ( 3 km) meteorological product over CONUS at an hourly temporal resolution between 2002 and present . This dataset downscales the National Land Data Assimilation System phase 2 (NLDAS-2) product from 1/8 to 1/32 using a series of available products including Stage IV and Stage II radar/gauge products for rainfall. PCF includes precipitation, downward shortwave radiation, downward longwave radiation, air temperature, specific humidity, wind speed, and pressure. Furthermore, to inform the microclimate heterogeneity in HMC, this paper uses the recently released WorldClim2 dataset. This gridded dataset derived from in situ observations provides monthly climatologies of temperature, precipitation, solar radiation, vapor pressure, and wind speed over the global land surface at a 30 arcsec spatial resolution . ## 2.3 Soil properties The soil properties come from the Probabilistic Remapping of SSURGO (POLARIS) dataset , a new continental soil dataset that uses random forests to spatially disaggregate and harmonize the Soil Survey Geographic (SSURGO) database over CONUS. In POLARIS, for each 30 m grid cell, every soil series is assigned a probability of being found at a given grid cell. These probabilities are then combined with the vertical profile information available for each soil series to construct a minimum, maximum, weighted mean, and weighted variance for each grid cell – enough information to construct a beta distribution for each parameter per vertical layer of each grid cell. For this study, this approach provides porosity (θs), wilting point (θwp), field capacity (θfc), and saturated hydraulic conductivity (Ksat), which are set to be the median of each corresponding beta distribution. These data are then used to directly compute the bubbling pressure (ψb) and the inverse of the pore distribution size index (B) which are used in the Campbell water retention curve ($\mathit{\theta }={\mathit{\theta }}_{\text{s}}\left({\mathit{\psi }}_{b}/\mathit{\psi }{\right)}^{\mathrm{1}/B}$) (Campbell1974). ### 2.3.1 Land cover The Cropland Data Layer (CDL) provides the 30 m land cover types. The CDL is an annually produced database over CONUS that combines the National Land Cover Database with an annual analysis of the spatial distribution of croplands. It is created and managed by the United States Department of Agriculture's National Agricultural Statistics Service (USDA-NASS). The predicted land cover types are based on the reflective signatures from a number of satellites including Landsat TM and ETM+, MODIS satellite data, and the Advanced Wide Field Sensors (AWiFS), among others . The different categories are associated with their corresponding land use types and species within the LM4-HB model. 3 Methods ## 3.1 Land model description: LM4-HB For this study, the conceptual approach that is used to parameterize sub-grid heterogeneity in the HydroBlocks land surface model (Chaney et al.2016a) is added to the fourth generation of the Geophysical Fluid Dynamics Laboratory (GFDL) land model (Milly et al.2014; Shevliakova et al.2009; Subin et al.2014). The resulting LM4-HB model is used to explore how big data can be efficiently and effectively harnessed to improve the characterization of the sub-grid multi-scale heterogeneity in Earth system models. LM4-HB uses a hierarchical approach to represent the underlying sub-grid heterogeneity; this makes it an ideal candidate for the testing and implementation of a hierarchical multivariate clustering algorithm to assemble the underlying sub-grid heterogeneity from available environmental datasets. This section provides an overview of LM4-HB with a primary focus on describing its hierarchical representation of sub-grid heterogeneity. Figure 2Schematic representation of a characteristic hillslope. Each characteristic hillslope is divided into height bands, which in turn are partitioned into intra-band tiles. Each tile interacts via the subsurface flow of water with the tiles in its same height band and with all the tiles in the height bands below and above. The land fraction of each grid cell in LM4-HB is partitioned into soil, glacier, and lake components. The soil component in turn is composed of k characteristic hillslopes; each hillslope has a unique set of attributes including slope, aspect, convergence, and convexity, among others. As shown in Fig. 2, each characteristic hillslope i is divided into li height above nearest drainage area (HAND) bands (referred to from now on as height bands). Each height band bi,j is divided into pi,j clusters (i.e., tiles) to account for the intra-band heterogeneity in soil and land cover. The total number of soil tiles is given by the sum of all tiles in each height band for all characteristic hillslopes within a grid cell. Although not shown in Fig. 2, the current model uses a uniform soil depth for all tiles within a characteristic hillslope. However, the soil properties (e.g., porosity and hydraulic conductivity) control the effective soil depth of each tile, and thus variable soil depths as shown in the schematic are effectively represented. Each soil tile within LM4-HB consists of a model from canopy air down to impermeable bedrock. The processes captured within the model include bidirectional diffuse and direct, visible and near-infrared radiation transfer; photosynthesis and stomatal conductance; surface energy, momentum, and water fluxes; snow physics; soil thermal and hydraulic physics (including advection of heat by water fluxes); vegetation phenology, growth, and mortality; simple plant-functional-type transition dynamics; and simple soil-carbon dynamics. For further details on the intra-tile processes, see and . Within each characteristic hillslope, each tile interacts with the tiles in its same height band and the tiles in the height bands immediately below and above via the subsurface flow of water; heat and carbon are advected by the water fluxes. The tiles adjacent to the channel interact with the stream in one direction (tile to stream). Each height band is characterized by a length, width, and height above the nearest drainage area. The effective width of a tile for a given height band is the corresponding fraction of the width of the height band. For further details on the tile interactions and the hillslope model more generally, see . For all model simulations in this study, LM4-HB is run with a 50 m soil depth (the same for all tiles) at a 1 h time step for 130 yr by cycling through the forcing between 2002 and 2014 ten times. The first 117 yr are used for spin-up, while the final 13 yr of the simulation are used for the analysis. ## 3.2 Assembling the land model tiles: hierarchical multivariate clustering To take advantage of LM4-HB's sub-grid representation of land cover, soil, climate, and topography, the characteristic hillslopes and the intra-hillslope heterogeneity are parameterized using available continental and global environmental data. This section provides an overview of the hierarchical multivariate clustering (HMC) algorithm used to assemble a grid cell's tiles. Its steps are (1) define the characteristic hillslopes, (2) discretize the characteristic hillslopes into height bands, and (3) define the intra-band heterogeneity. Figure 3The characteristic hillslopes are defined by (1) delineating the hillslopes from the elevation data, (2) calculating a suite of properties for each hillslope from environmental data, and (3) using the hillslope properties and the k-means clustering algorithm to define k characteristic hillslopes. ### 3.2.1 Define the characteristic hillslopes The characteristic hillslopes for a given grid cell are defined by clustering a grid cell's fine-scale map of hillslopes. To assemble the map of hillslopes, the DEM is sink-filled and the channels are delineated using an area threshold of 100 000 m2. A recursive algorithm then splits each basin into a maximum of three hillslopes – left side, right side, and headwaters. Each hillslope's attributes are assembled from the high-resolution soil, topography, and climate data. These include each hillslope's average aspect, slope, annual mean precipitation, and annual mean temperature. Metrics that summarize each hillslope's plan and profile geometry are derived from the hillslope's binned HAND data. Given each bin's slope, HAND, and area, the length and width are readily computed. To summarize these properties, a line is fit to the set of widths for each hillslope; the slope of this function provides a summary metric of the hillslope's plan shape (convergence/divergence). For each hillslope's profile, the function $h=H\left[\mathrm{1}-{\left(\mathrm{1}-\left(x/L{\right)}^{a}\right)}^{b}\right]$ is fit to the binned HAND data; where h is the HAND, H is the maximum HAND, x is the horizontal position, and L is the hillslope length. The parameters a and b summarize the concavity of the lower half and upper half of each hillslope, respectively. This function is chosen due to its flexibility to reproduce convex, concave, and complex hillslope profiles. Figure 4The profile of each characteristic hillslope is constructed by fitting $h=H\left[\mathrm{1}-{\left(\mathrm{1}-\left(x/L{\right)}^{a}\right)}^{b}\right]$ to the combined profiles of all its corresponding hillslopes. The characteristic profile is then discretized into Hi∕Δh height bands. Finally, the HAND and characteristic hillslope maps are combined with the discretized profiles to assign a unique height band to each fine-scale grid cell. In the height bands image, each band of each characteristic hillslope is represented by a different color. Assembling all the calculated attributes leads to an n by m array where n is the number of hillslopes and m is the number of attributes. The k-means clustering algorithm (MacQueen1967) is then used to partition the normalized m-dimensional attribute space into k characteristic hillslopes. Figure 3 provides an overview of the steps used to define the characteristic hillslopes. The attributes of each characteristic hillslope are then set to be the arithmetic mean of the attributes of its corresponding hillslopes. ### 3.2.2 Discretize the characteristic hillslopes After assembling the set of characteristic hillslopes, their attributes, and their corresponding profile and width functions, the next step is to discretize each profile along the length axis into height bands. The number of height bands is ${l}_{i}=⌈{H}_{i}/\mathrm{\Delta }h⌉$, where i is the characteristic hillslope, Hi is the profile's maximum height, and Δh is the fixed height difference between adjacent height bands. Note that the number of height bands per characteristic hillslope can differ per characteristic hillslope since each has a unique Hi. Each resulting height band has a unique mean width, length, and height. Figure 5For each height band of each characteristic hillslope, the corresponding fine-scale grid cells are clustered into p intra-band tiles according to their land cover and soil properties. Using the high-resolution maps of characteristic hillslopes and the HAND, each high-resolution grid cell is assigned a characteristic hillslope and a height band. This is accomplished by first normalizing the HAND map per hillslope by dividing all HAND values that belong to a given hillslope by the given hillslope's maximum HAND value (H). This normalized HAND map is then combined with the normalized discretized HAND profiles of the characteristic hillslopes to assign each high-resolution grid cell to its corresponding height band. This process formally connects the discretized characteristic hillslopes to the observed landscape. Figure 4 illustrates an example discretization of a characteristic hillslope and the mapping of the discretized hillslopes to the high-resolution grid. Figure 6As an exploratory simulation, the LM4-HB model is run between 2002 and 2014 using 14 tiles assembled via HMC. The tile simulations are shown for evapotranspiration (ET), sensible heat flux (SH), baseflow (Rb), and root zone soil moisture (SM) for 2005; the time series are color-coded to correspond to the 30 m map of tiles. Each tile has a corresponding id ${t}_{i,j,k}$, where i is the characteristic hillslope, j is the height band, and k is the intra-band cluster. For each variable, the tile-weighted average time series (macroscale estimate) is superimposed on the tile simulations for comparison. ### 3.2.3 Define the intra-band heterogeneity The final step is to define the heterogeneity within each height band bi,j where i is the characteristic hillslope and j is the height band. For each band bi,j, the collocated fine-scale grid cell values of the proxies of heterogeneity are extracted. This study uses saturated hydraulic conductivity, porosity, and a set of binary maps (natural/cropland, evergreen/deciduous, and grass/tree) derived from the high-resolution land cover map. These binary maps are used to avoid clustering categorical data. For each height band bi,j, this leads to an n by m array of attributes where n is the number of fine-scale grid cells that belong to bi,j and m is the number of attributes. The k-means algorithm is then used to cluster the m-dimensional attribute space into pi,j intra-band clusters (i.e., tiles). In this study, for simplicity, pi,j is set to be the same for all height bands and characteristic hillslopes. Therefore, the final number of tiles for the macroscale grid cell is given by $\begin{array}{}\text{(1)}& {n}_{\text{tiles}}=p\sum _{i=\mathrm{1}}^{k}{l}_{i}.\end{array}$ Each tile within a grid cell is assigned an id ${t}_{i,j,k}$ where i is the characteristic hillslope, j is the height band, and k is the intra-band cluster. Figure 5 shows an example of the intra-band clustering and the tile configuration resulting from the hierarchical clustering algorithm over this study's domain. For each tile, the continuous land model parameters are set to be the arithmetic average of all the parameter values of the fine-scale grid cells that belong to the given tile. The mode is used instead of the mean for categorical land model parameters. Each tile is assigned its own meteorology by assigning a weighted average of all the overlying 4 km PCF grid cells that intersect with the 30 m grid cells that belong to the tile. One of the advantages of having each 30 m grid cell belong to a tile and the corresponding map of tiles is that the simulations for each tile can be mapped to the fine-scale grid to provide a 30 m representation of the model output at each time step. ## 3.3 Model experiments ### 3.3.1 Exploratory simulation By clustering the high-dimensional environmental space, HMC explicitly relates the tiles used in LM4-HB to the observed fine-scale physical environment while ensuring realistic hydrologic connections between tiles along characteristic hillslopes. To illustrate the benefits and additional model information that can be extracted when using HMC, an exploratory simulation is run using a simple HMC-assembled tile configuration (k=2, Δh=50 m, p=2) with 14 tiles within LM4-HB. This tile configuration is among the simplest cases for this domain that are able to illustrate the roles of all the different drivers of heterogeneity. This exploratory simulation is then analyzed to illustrate the added information that HMC provides. Figure 7Using the exploratory simulation, the tile-simulated daily evapotranspiration (ET) values for 16 June 2005 are mapped onto the 30 m fully distributed grid using the HMC-assembled fine-scale map of tiles. Table 1Through a series of model experiments, the HMC parameters are adjusted to assess their role in the modeled heterogeneity. ${e}_{k,\mathrm{\Delta }h,p}$ is the experiment id, k is the number of characteristic hillslopes, Δh is the difference in height between adjacent height bands, p is the number of intra-band clusters, and ntiles is the resulting number of tiles. ### 3.3.2 Hierarchical multivariate clustering: parameter sensitivity The primary objective of HMC is to harness high-resolution environmental data to efficiently and effectively summarize a macroscale grid cell's underlying multi-scale spatial structure. To make this possible, HMC relies on a set of user-defined parameters to control the importance of each hierarchical step in the algorithm; these parameters include (1) the number of characteristic hillslopes (k), (2) the elevation difference between adjacent height bands (Δh), and (3) the number of intra-band clusters per height band (p). To test LM4-HB's sensitivity to these parameters, nine model experiments are performed in which the HMC parameters are adjusted to assess their individual roles. The model experiments are outlined in Table 1. Experiments ${e}_{\mathrm{1},\mathrm{1000},\mathrm{1}}$, ${e}_{\mathrm{2},\mathrm{1000},\mathrm{1}}$, and ${e}_{\mathrm{10},\mathrm{1000},\mathrm{1}}$ increase k from 1 to 10 characteristic hillslopes, experiments ${e}_{\mathrm{10},\mathrm{50},\mathrm{1}}$, ${e}_{\mathrm{10},\mathrm{20},\mathrm{1}}$, and ${e}_{\mathrm{10},\mathrm{10},\mathrm{1}}$ decrease Δh until 10 m, and experiments ${e}_{\mathrm{10},\mathrm{10},\mathrm{2}}$, ${e}_{\mathrm{10},\mathrm{10},\mathrm{3}}$, and ${e}_{\mathrm{10},\mathrm{10},\mathrm{5}}$ increase p by up to five intra-band clusters. ### 3.3.3 Characterizing the roles of the drivers of heterogeneity Applying HMC to existing high-resolution environmental data enables a robust representation of the different drivers of spatial heterogeneity (soil, topography, meteorology, and land cover) within macroscale environmental models. However, it does not explicitly characterize the individual role of each driver – key to advancing our understanding of the relationship between the physical environment, ecosystem spatial structure, and macroscale response. To make this analysis possible, another set of model experiments is explored that investigate the individual role of each driver at the macroscale. Using an approach similar to , each driver's sensitivity is explored by turning the heterogeneity of properties associated with each driver on and off. When “on” the properties associated with the driver are left as assigned through HMC; when “off” the driver's properties are set to be the grid cell mean. The different model experiments are outlined in Table 2. Figure 8Visual comparison of the mapped annual mean (2002–2014) of simulated evapotranspiration for the nine model experiments in Table 1. 4 Results and analysis ## 4.1 Exploratory simulation Figure 6 shows the simulated time series for all 14 tiles at a daily time step for baseflow, root zone soil moisture, evapotranspiration, and sensible heat flux. The fine-scale map of tiles is also shown and color-coded to relate the spatial location of each tile to the simulated time series. Each tile is assigned an id ${t}_{i,j,k}$, where i is the characteristic hillslope, j is the height band, and k is the intra-band cluster. A comparison of the tiles' time series exemplifies the differences in states and fluxes that are driven by a tile's location, properties, and meteorology. Explicitly resolving hillslope dynamics leads to significant differences in the root zone soil moisture; in general, soil moisture decreases as the tiles are further away from the valley. However, land cover, soil, hillslope structure, and meteorological differences can lead to differences in soil moisture between hillslopes and intra-band clusters (e.g., uplands vs. lowlands). This strong topographic gradient in soil moisture provides more water for vegetation growth in the valleys than along the ridges; given that this is a water-limited area, this explains the appreciable differences in simulated evapotranspiration. Furthermore, this also explains the strong heterogeneity in sensible heat flux, with tiles with higher soil moisture having lower sensible heat fluxes. Finally, only the tiles that are adjacent to the channels (${t}_{\mathrm{1},\mathrm{1},\mathrm{1}}$, ${t}_{\mathrm{1},\mathrm{1},\mathrm{2}}$, ${t}_{\mathrm{2},\mathrm{1},\mathrm{1}}$, and ${t}_{\mathrm{2},\mathrm{1},\mathrm{2}}$) produce noticeable baseflow. Figure 9Comparison of the model experiments in Table 1. For each experiment, the spatial mean and spatial standard deviation for all water years (1 October–30 September) between 2003 and 2014 are plotted for a suite of states and fluxes. The corresponding values for each water year are color-coded according to their precipitation. The black line shows the annual mean. For all states and fluxes, the macroscale (tile-weighted average) time series for each variable is superimposed on the tile simulations to illustrate the strong differences that can exist between individual tiles and the macroscale estimate (temporal dynamics and mean). These differences illustrate the challenge of comparing a macroscale estimate to observations and simulations at different spatial resolutions, a persistent challenge when aiming to apply and evaluate macroscale models. However, as will be discussed in Sect. 5.2, being able to connect each 30 m grid cell to each tile simulation enables a path towards circumventing the scale discrepancies between macroscale model estimates and in situ observations. Figure 7 illustrates how the tile simulations can then be mapped onto the 30 m fully distributed grid. In this example, the daily averaged simulated evapotranspiration value for each tile on 16 June 2005 is assigned to each corresponding fine-scale grid cell. Being able to visualize the assumed heterogeneity of the model enables a more realistic comparison to fully distributed models. Furthermore, it makes it possible to provide model output at spatial resolutions at far finer spatial resolutions than the tile-weighted average (i.e., macroscale estimate). Figure 10Visual comparison of the mapped annual mean (2002–2014) of simulated runoff (R), 2 m root zone soil moisture (SM), leaf area index (LAI), and soil temperature (Ts) for the five model experiments in Table 2. Figure 11Comparison of the model experiments in Table 2. For each experiment, the spatial mean and spatial standard deviation for all water years between 2003 and 2014 are plotted for a suite of states and fluxes. The corresponding values for each water year are color-coded according to their precipitation. The black line shows the annual mean. ## 4.2 Hierarchical multivariate clustering: parameter sensitivity As outlined in Sect. 3.3.2, the experiments in this section explore the sensitivity of the HMC parameters through a set of model experiments; these experiments are summarized in Table 1. As an initial visual comparison, Fig. 8 shows the mapped annual mean evapotranspiration between 2002 and 2014 at a 30 m spatial resolution for the different model experiments. The baseline experiment is the one-tile configuration (i.e., no sub-grid heterogeneity). An increase in the number of characteristic hillslopes leads to the appearance of large-scale spatial patterns in evapotranspiration (experiments ${e}_{\mathrm{1},\mathrm{1000},\mathrm{1}}$, ${e}_{\mathrm{2},\mathrm{1000},\mathrm{1}}$, and ${e}_{\mathrm{10},\mathrm{1000},\mathrm{1}}$). This is primarily due to the strong topographic gradient in precipitation between the lowlands and uplands; this heterogeneity in evapotranspiration is possible through the disaggregation of the PCF meteorology among the land model tiles. Decreasing Δh leads to an increase in the number of height bands per characteristic hillslope; this makes the role of topographic convergence in subsurface flow readily apparent – evapotranspiration is higher in the riparian zones (experiments ${e}_{\mathrm{10},\mathrm{50},\mathrm{1}}$, ${e}_{\mathrm{10},\mathrm{20},\mathrm{1}}$, and ${e}_{\mathrm{10},\mathrm{10},\mathrm{1}}$). Finally, increasing the number of intra-band clusters adds to heterogeneity in evapotranspiration due to land cover and soil heterogeneity (experiments ${e}_{\mathrm{10},\mathrm{10},\mathrm{2}}$, ${e}_{\mathrm{10},\mathrm{10},\mathrm{3}}$, and ${e}_{\mathrm{10},\mathrm{10},\mathrm{5}}$). Note how after experiment ${e}_{\mathrm{10},\mathrm{10},\mathrm{2}}$ (278 tiles), the spatial patterns cease to change as much. Figure 9 formalizes the comparison between the different model experiments; it shows how the spatial mean and spatial standard deviation of the annual mean of each year between 2002 and 2014 change as a function of the tile configuration for a suite of states and fluxes. The primary result is the apparent convergence of all states and fluxes for both the spatial mean and standard deviation with the increase in the number of land model tiles. In other words, there is a point at which further increases in the number of tiles have a limited impact at the macroscale. For this site, that limit is approximately 300 tiles compared to the 810 000 fine-scale grid cells in the domain. This result is encouraging; it illustrates that multi-scale sub-grid heterogeneity can be characterized effectively and efficiently in large-scale models by taking advantage of the covariance between environmental properties. The role that each parameter of HMC has at the macroscale depends on the prognostic variable; these differing roles are discussed below. • k – an increase in the number of characteristic hillslopes from experiments ${e}_{\mathrm{1},\mathrm{1000},\mathrm{1}}$ (1 tile) to ${e}_{\mathrm{10},\mathrm{1000},\mathrm{1}}$ (10 tiles) leads to noticeable changes in all the prognostic variables. The most noticeable changes occur when increasing the number of characteristic hillslopes from one to two. This is primarily due to the improved representation of land cover heterogeneity; instead of the grid cell being represented uniformly as evergreen trees, the lowlands are now grasses while the uplands remain as evergreen trees. This leads to a decrease in the cell's effective roughness length (i.e., a decrease in aerodynamic conductance), and thus a decrease in sensible heat flux and an increase in surface temperature. The decrease in aerodynamic conductance also contributes to a decrease in transpiration. • Δh – a decrease in Δh from experiments ${e}_{\mathrm{10},\mathrm{50},\mathrm{1}}$ (28 tiles) to ${e}_{\mathrm{10},\mathrm{10},\mathrm{1}}$ (139 tiles) leads to an increase in the number of height bands in the characteristic hillslopes. This results in an explicit representation of the role of topographic convergence in subsurface flow; more soil water is available in the valleys than along the ridges. The increase in soil water in the valleys also leads to more frequent saturated excess runoff, to the extent that during wet years this counteracts the decrease in baseflow. Another noticeable impact of the increase in the number of height bands is the decrease in inter-annual variability in transpiration, net primary productivity, baseflow, and sensible heat flux. As will be discussed in Sect. 5.1, this can have potentially important implications for the role of spatial structure in ecosystem resilience to hydrologic extremes. • p – an increase in the number of intra-band clusters from experiments ${e}_{\mathrm{10},\mathrm{10},\mathrm{2}}$ (278 tiles) to ${e}_{\mathrm{10},\mathrm{10},\mathrm{5}}$ (695 tiles) leads to a more robust representation of soil and land cover heterogeneity throughout the domain. This leads to differences in most variables. However, these differences are not as noticeable as those due to changes in k and Δh since most of the heterogeneity in land cover and soil heterogeneity has already been represented through these other parameters. This parameter will most likely play a larger role in regions where the ecosystem spatial structure is not as strongly tied to topography. Table 2Through a series of model experiments, the heterogeneity of model parameters and forcing data is turned on (heterogeneous) and off (homogeneous). For simplicity, the parameters and forcing data are grouped into soil, hillslope, land cover, and meteorology groups. ## 4.3 Characterizing the roles of the drivers of heterogeneity Although Sect. 4.2 provides preliminary insight into the role of the different drivers of spatial heterogeneity (e.g., topographic convergence impacts the macroscale soil moisture mean), due to the interactions of the HMC parameters, it cannot precisely disentangle each driver's unique role. For this purpose, as introduced in Sect. 3.3.3, another set of experiments is explored that turn the different drivers of heterogeneity on and off. These experiments are summarized in Table 2. Furthermore, the tile configuration of experiment ${e}_{\mathrm{10},\mathrm{10},\mathrm{2}}$ in Sect. 4.2 (278 tiles) is used for all experiments in this section; this tile configuration is chosen because, as shown in Sect. 4.2, the model macroscale states and fluxes converge at around 300 tiles. As an initial visual comparison, the mapped model results of the different experiments are shown in Fig. 10 for annual mean runoff, 2 m root zone soil moisture, leaf area index, and soil temperature between 2002 and 2014 at a 30 m spatial resolution; Fig. 11 formalizes this comparison. The roles that each driver plays in the macroscale prognostic states and fluxes are discussed below. • Baseline (B) – the baseline experiment equates to the homogeneous sub-grid cell case. However, in this case, there are 278 tiles where each tile has the same soil properties, hillslope structure (each tile is set to be its own characteristic hillslope), and land cover properties. Furthermore, each tile is run using the grid cell mean meteorology. Not surprisingly, there is no heterogeneity in the plotted maps and the spatial standard deviation for all prognostic states and fluxes is 0. • Soil heterogeneity (S) – this experiment adds heterogeneity in the soil properties; this includes porosity and the Campbell retention curve parameters, among others. It has a relatively small impact on the spatial mean for all variables. However, there are changes in their spatial standard deviations. The increase in spatial standard deviation is largest for soil moisture, which in turn impacts the remaining prognostic variables. For example, changes in available water impact infiltration excess runoff, leading to, at times, appreciable heterogeneity in annual runoff production. In any case, these differences are minor at the macroscale when compared to the subsequent experiments. • Soil and hillslope heterogeneity (SH) – assigning each tile to its original corresponding characteristic hillslope and discretizing these hillslopes lead to significant differences at the macroscale. The strong topographic gradients caused by the discretized hillslopes lead to strong topographic gradients in soil moisture and thus explain the sharp increase in the spatial standard deviation of soil moisture. These topographic gradients in soil moisture lead to an overall increase in saturated excess runoff during wet years, thus counteracting the overall decrease in baseflow; this also explains the increase in the spatial standard deviation of runoff. The most noticeable role of the topographically driven subsurface flow is the reduction in inter-annual variability in root zone soil moisture, leaf area index, baseflow, transpiration, net primary productivity, and sensible heat flux. This is due primarily to the significant increase in inter-annual change in storage when including explicit topographic gradients; these topographic gradients enable the system to be able to release more water during dry years (uphill deep soil water is made available to the riparian zone through subsurface flow) and to absorb more water during the wet years (increase in infiltration capacity due to the heterogeneity of soil moisture). • Soil, hillslope, and land cover heterogeneity (SHL) – this experiment adds heterogeneity in land cover. These changes are similar to those seen in Sect. 4.2 when increasing the number of characteristic hillslopes. This is because both cases ensure evergreen forests are represented in the uplands and grasslands and shrubs are represented in the lowlands. This leads to a lower effective roughness height, explaining the decrease in sensible heat flux, net primary productivity, and transpiration. Land cover heterogeneity also leads to an appreciable increase in the spatial standard deviation of sensible heat flux, net primary productivity, and transpiration. Its role is also particularly noticeable in the soil temperature spatial distribution by creating sharp contrasts between the lowlands and uplands. • Soil, hillslope, land cover, and meteorological heterogeneity (SHLM) – this last experiment adds heterogeneity in meteorology by prescribing the meteorology to each tile from the overlying 4 km PCF grid. This leads to a net increase in annual mean runoff and a decrease in transpiration. The spatial heterogeneity of meteorology further enhances the contrast in soil temperature between the lowlands and uplands. Furthermore, the larger availability of water in the uplands leads to higher net primary productivity in this region and thus a higher LAI. 5 Discussion ## 5.1 Sub-grid redistribution of water: dampening of extremes Seeking to account for the role of sub-grid redistribution of subsurface water within macroscale hydrologic and land surface models is not a new objective; many schemes have been implemented over the past decades to characterize its influence . However, these approaches are designed primarily to account for the role of fine-scale heterogeneity in the macroscale hydrologic response; they are generally not meant to handle the sub-grid spatial coupling of the water, energy, and biogeochemical cycles. LM4-HB addresses these limitations by explicitly modeling the subsurface flow of water via horizontally and vertically discretized characteristic hillslopes ; this makes it possible to account for the impact of sub-grid redistribution of water on the full gamut of land surface states and fluxes (e.g., soil moisture, sensible heat flux, and net primary productivity). Furthermore, by harnessing existing environmental information to parameterize sub-grid heterogeneity, HMC ensures that the properties of the characteristic hillslopes are formally connected to the observed physical environment. The results from the model experiments in Sect. 4.3 show that upon enabling subsurface redistribution, the most noticeable difference at the macroscale is the dampening of annual extremes in the water, energy, and carbon cycles; for example, as shown in Fig. 11 there is a strong decrease in the inter-annual variability of baseflow, transpiration, net primary productivity, and sensible heat flux between the S and SH simulations. As mentioned in Sect. 4.3, this is primarily due to the significant increase in inter-annual change in storage when including explicit topographic gradients; these topographic gradients enable the system to release more water during dry years (uphill deep soil water is made available to the riparian zone through subsurface flow) and to absorb more water during the wet years (increase in infiltration capacity due to the heterogeneity of soil moisture). The influence of topography on infiltration capacity and baseflow production has been recognized for decades in hydrology ; however, its role in the coupled water, energy, and biogeochemical cycles remains poorly understood. This study provides insight into how a robust representation of land heterogeneity in Earth system models makes it possible to more fully characterize the role of topography in the coupled system. These model experiments provide insight into the role of subsurface redistribution at the macroscale; the underlying physical environment provides a mechanism for ecosystems with pronounced topography to mitigate the impacts of seasonal to annual hydrologic extremes. These results suggest that an improved representation of spatial heterogeneity could improve projections of ecosystem response to drought, particularly in mountainous regions. A better representation of biophysical feedbacks to variations in air temperature and vapor pressure deficit could also improve simulations of land–atmosphere feedbacks that can intensify droughts and affect macroscale circulations . Future work should explore how these results extend to other regions with different configurations of the physical environment (e.g., topography). Beyond understanding the role of the sub-grid subsurface redistribution of water, these model experiments would also bring to light other impacts that the fine-scale physical environment has on macroscale response. ## 5.2 Revisiting the application and evaluation of Earth system models As explored in Sect. 4.1 and shown in Fig. 7, formally connecting the sub-grid tile configuration to the high-resolution environmental covariates provides a novel approach to visualize the output of the land components of Earth system models – the simulation of each land model tile can be mapped to the fine-scale grid. Using this approach, macroscale models are able to maintain their existing computational and storage efficiency while providing highly detailed local information. As discussed below, this has important repercussions for the evaluation and application of these models. Evaluation – as shown in Fig. 6, when robustly characterized, sub-grid multi-scale heterogeneity can lead to significant differences between the simulations at the tile (i.e., field scale) and grid cell levels (i.e., macroscale). This discrepancy in spatial scale is analogous to using in situ observations to evaluate and validate Earth system models: the observations are at the field scale, yet the model estimates are at the macroscale. The approach explored in this study allows the Earth system modeling community to revisit this persistent challenge. Since each fine-scale grid cell ( 30 m) is assigned to a land model tile, in situ observations can be readily compared to the simulations of their collocated model tile. This makes it possible to evaluate these models using in situ observation networks (e.g., FLUXNET) without having to upscale the observations or downscale the macroscale estimate. Furthermore, it also enables the use of very high-resolution satellite products (e.g., Landsat) to evaluate the modeled fine-scale spatial patterns. This approach enables the Earth system modeling community to work more closely with field scientists to further understanding of the Earth system and to accelerate the model development cycle. For example, this approach could facilitate improved methods for validating soil carbon projections in ESMs. Past model–data comparisons of soil carbon have relied on spatial models to scale soil carbon measurements to the grid cell scale, as in the Harmonized World Soil Database (HWSD; ). Such scaling techniques can be problematic for direct comparison with model simulations. First, the spatial models necessary for scaling have the potential to introduce bias in the observation-based product, and result in a comparison that is not purely measurement-based. Second, variability in topography, ecosystem type, and soil properties within the grid cell scale makes these comparisons challenging to interpret: failure of a model grid cell to match scaled observations could be due to process representation, model parameterization, or the relative spatial coverage of ecosystem or edaphic types within the model grid cell relative to the scaled observations. Incorrect attribution of model error could result in inappropriate adjustments to model parameters or processes, for example reducing the turnover rate of soil carbon pools in uplands when a model underestimate of carbon stocks is actually due to high carbon stocks observed in wetlands. HMC could address these issues by facilitating direct comparison of modeled soil carbon stocks with measurements grouped by the same properties used in the clustering analysis. Application – Earth system models are used almost exclusively for regional to global applications due to their coarse spatial scales, with limited applicability to local stakeholders (e.g., farmers). Having the ability to map the tile simulations to the fine-scale grid ( 30 m) allows the community to reevaluate how these models are applied. For example, accounting for the very high-resolution soil properties in each model tile leads to more locally relevant soil moisture simulations; providing these field-scale model estimates in real time can then be used to inform irrigation requirements. Furthermore, the inherent model efficiency of this approach facilitates robust ensemble frameworks; this enables a path towards constraining the unavoidable uncertainty of the model predictions – even more pervasive at higher spatial resolutions – while still providing local detail. This novel approach to model application should be explored further as it requires minimal increases in computational expense with potentially large societal benefits. ## 5.3 Caution: convergence on the fully distributed simulation The primary result of this study is that a relatively low number of land model tiles ( 300 tiles in this study) are necessary to converge on the macroscale response (mean and standard deviation) of the corresponding fully distributed model. Although this result provides a promising path forward for a robust representation of multi-scale heterogeneity in large-scale models, as discussed below, there are limitations that must be acknowledged and addressed in future implementations of this method. Model structure uncertainty – the optimal tile configuration is tied to the underlying process representation of the model. For example, LM4-HB does not currently represent subsurface or surface interactions among characteristic hillslopes. For this study, this translates to the model not transferring water between the uplands and lowlands. This is an obvious oversimplification as aquifers in the Central Valley in California (i.e., lowlands) strongly rely on water recharge via subsurface and surface flows from the Sierra Nevada (i.e., uplands). Other important missing processes include surface redistribution of water, water ponding, preferential flowpaths, and water management, among others. As these key processes are included, the optimal tile configuration will most likely become more complex (i.e., more tiles). More generally, this implies that the optimal tile configuration is model structure dependent and thus its transferability between land surface models (and even different versions of the same model) is limited. Parameter uncertainty – this study uses a single realization of plausible model parameters per 30 m grid cell over the domain. The tile-level parameters are then derived from these high-resolution maps as described in Sect. 3.2.3. All model simulations in the study and thus the convergence analysis rely on this single realization of model parameters. Given the known strong sensitivity of the spatial properties and macroscale response of the water, energy, and carbon cycles to many of these highly uncertain model parameters (e.g., saturated hydraulic conductivity and soil depth), it follows that the optimal tile configuration is most likely strongly dependent on the prescribed model parameters. As this method is implemented for use in parameter ensemble frameworks, special care must be taken to ensure that different tile configurations are used for different parameter realizations to ensure that the intended goal behind HMC of robustly characterizing the multi-scale heterogeneity is still fulfilled in each ensemble member. Model application – the search for an optimal tile configuration in this study focuses exclusively on converging on the annual spatial mean and spatial standard deviation of the fully distributed model. Although appropriate for climate timescales, these coarse metrics will most likely be insufficient for finer temporal scales (e.g., numerical weather prediction). For example, if the objective is to model flash flooding using minute-scale radar rainfall, the characteristic hillslopes in LM4-HB will need to more robustly represent the spatial grid to appropriately capture the minute-scale spatial variability of precipitation. This will likely require a large increase in the total number of characteristic hillslopes to ensure convergence on the macroscale response. For these types of applications other approaches to assess convergence would be useful, including comparing runoff histograms. Another approach would involve a direct comparison of the mapped 30 m simulations. This would provide a more complete assessment of convergence and would be critical for field-scale applications (e.g., predictions of soil moisture at the farm level). In summary, the optimal tile configuration will also be application dependent; while optimal tile configurations derived for finer timescale applications will be appropriate for coarser timescale applications, the opposite will rarely be true. ## 5.4 Proxy heterogeneity vs. process heterogeneity The implementation of HMC throughout this study assumes that observed characteristics of the physical environment (e.g., elevation) are robust proxies of the heterogeneity of the biogeochemical cycles. Although this assumption is generally adequate, it only indirectly addresses the real goal, which is to characterize the multi-scale heterogeneity of the water, energy, and carbon cycles. Moving forward, future development of HMC should move beyond only clustering proxies of heterogeneity and instead focus on the processes themselves. Following , one approach would involve directly clustering the output from the fully distributed model. Although unfeasible for global applications due to computational constraints, this would provide the most robust method to ensure a comprehensive characterization of the heterogeneity captured in the fully distributed model with a minimal number of land model tiles. Another option would be to use satellite remote products that directly measure the states and fluxes of the biogeochemical fluxes at high spatial resolutions. For example, soil moisture retrievals from Sentinel-1 would provide key observations of the spatial distribution of saturation within a domain. Other maps that would be useful include the Leaf Area Index from MODIS , and evapotranspiration from Landsat , among others. Although biased, these data would provide a more formal connection between the model and observed heterogeneity. It would also open a novel path to assimilate these high-resolution products into land models at field scales. ## 5.5 Applying HMC over the globe: assembling optimal tile configurations Section 4.2 illustrates how a relatively small number of land model tiles are necessary to explicitly characterize the underlying sub-grid heterogeneity in this study's domain – this is substantially less than the 810 000 grid cells of a corresponding 30 m fully distributed simulation. This provides the best trade-off for large-scale models: it explicitly captures the role of sub-grid fine-scale features while maintaining computational efficiency. However, although this study does provide a preliminary exploration of the HMC parameter space (k, Δh, and p), it does not provide a robust approach to find the optimal HMC parameters in different regions over the globe. Three approaches that are outlined below could be used to accomplish this goal. 1. The most direct approach is to optimize the HMC parameters on all macroscale grid cells for a given grid size over the globe. This would be accomplished through parameter optimization techniques (Duan et al.1993; Hadka and Reed2013). For each parameter set, HMC would be run to create the model tiles and then LM4-HB would be used to run a simulation to characterize its long-term macroscale states and fluxes. Convergence on the fully distributed simulation would be attained at the parameter set that leads to the fewest number of tiles while ensuring the macroscale states and fluxes have converged within a user-defined tolerance. This would lead to a robust representation of sub-grid heterogeneity albeit requiring substantial computational resources. 2. A more computationally efficient path forward is to use the first approach on only a subset of macroscale grid cells or catchments. These domains would be chosen such that they sample comprehensively from different climate, soil, land cover, and topographic regimes throughout the globe. The HMC parameters would be optimized independently for each of these domains. Machine learning (e.g., random forests) would then be used to regionalize these optimized parameters by deriving non-linear functional relationships between the optimized parameters and a suite of summary macroscale metrics (e.g., standard deviation of elevation, grid cell area, and average precipitation, among others). This approach would provide a method to assemble the optimal HMC parameters for a chosen region without having to resort to optimizing the parameters for a new domain. 3. Finally, another option is to rely exclusively on the data that are used within HMC. The primary goal behind clustering these data is to extract all the relevant information and minimize redundancies. Assuming that the water, energy, and biogeochemical fine-scale features are tightly coupled to the observed environment, then ensuring that the mapped clustered input data matches the original fine-scale maps would ensure that the model results will provide a robust representation of the defined sub-grid heterogeneity. This approach would provide a method to define the optimal number of land model tiles without having to resort to model simulations, thus making the parameter selection less model dependent. ## 5.6 Clustering: expanding beyond natural soil systems The strong covariance between the different proxies of the drivers of spatial heterogeneity makes it possible to summarize a high-dimensional proxy space (i.e., environmental data) using a relatively small number of representative clusters. HMC capitalizes on this covariance to characterize the fine-scale heterogeneity of natural soil systems. However, the covariance of environmental properties is not exclusive to natural soil systems and can be extended to other systems over land, including urban areas, croplands, water bodies, and glaciers. This section explores both the data that are available for these types of systems and how clustering can be used to extract their most representative characteristics. • Lakes and glaciers – Earth system models represent lakes and glaciers as model tiles over land. Each lake tile is characterized by a set of properties including depth and area; these can be obtained from existing global lake inventory databases. These data have information such as shoreline length, water volume, and average depth, among others (e.g., HydroLakes; ). For each macroscale grid cell, clustering could be used to define a set of characteristic lakes with their associated representative properties. A similar approach could also be used to identify a grid cell's characteristic glaciers by clustering the properties associated with the glaciers within that cell; rich global glacier inventory databases such as GLIMS could then be harnessed within ESMs. • Urban areas – recognizing the important role that urban areas play in the coupled system, the community is actively incorporating them into ESMs through urban canopy models (UCMs) . UCMs represent urban areas through a set of characteristics including roof fraction, building height, and canyon fraction, among others. Although there is currently no comprehensive database that provides the characteristics for all urban areas over the globe, there are emerging efforts to make this information available (e.g., WUDAPT; ). As these data become accessible, clustering could also be used to distinguish the characteristic urban features within a model grid cell (e.g., high vs. low buildings). • Croplands – although Earth system models include croplands, their representation is generally oversimplistic. For example, in most cases, irrigation practices are ignored and many different phenological and physiological differences between crops are disregarded (e.g., rice vs. corn). Although far from complete, datasets are emerging that are able to provide this information at moderate to very high spatial resolutions over continental to global extents . These data provide metrics that summarize local cropland characteristics (e.g., C3/C4) and irrigation practices (e.g., irrigated/rainfed). This information can be summarized per grid cell via clustering to provide a more complete representation of sub-grid cropland characteristics. Although this study explores this possibility using the CDL database, further work is necessary to more adequately account for the sub-grid variability in crop characteristics and irrigation practices. • Peatland and permafrost landscapes – in wet and high-latitude regions characterized by organic matter accumulation in peat and permafrost, spatial heterogeneity can be crucial to understanding regional carbon stocks. For example, found that peat and lake sediments account for more than 70 % of carbon stocks despite covering only 33 % of the land area in a northern Wisconsin landscape. Likewise, in permafrost systems microtopographic variations driven by ice-wedge polygon formation can dominate spatial variability of carbon cycling . While spatially explicit modeling of these complex landscapes can yield important insights about ecosystem function and vulnerability to climatic changes (Sonnentag et al.2008; Sulman et al.2012), the reduced computational demands of HMC could facilitate incorporation of these important dynamics into macroscale simulations. 6 Conclusions A robust representation of the influence of the multi-scale physical environment on the coupled terrestrial water, energy, and biogeochemical cycles remains a persistent challenge in Earth system models. One of the principal obstacles is the oversimplification of the observed complex heterogeneity within these models. This is primarily due to a limited understanding of how to use the available petabytes of environmental data effectively and efficiently in macroscale models. Unsupervised machine learning, and more specifically cluster analysis, provides a path forward by capitalizing on the observed landscape similarity to extract the underlying defining features (i.e., clusters) from available environmental data. The hierarchical multivariate clustering (HMC) approach presented here takes this a step further by taking advantage of these clustering techniques while also ensuring physically consistent surface and subsurface interactions between clusters through discretized characteristic hillslopes. A series of different tile configurations computed via HMC is used within the LM4-HB model to quantify its added benefits to macroscale models. The model experiments over a 1/4 grid cell in central California show that (1) the observed similarity over the landscape makes it possible to robustly account for the role of multi-scale heterogeneity in the macroscale states and fluxes with a relatively minimal number of sub-grid land model tiles; (2) assembling the sub-grid tiles from the observed high-dimensional environmental data can lead to important differences in the macroscale water, energy, and carbon cycles; (3) connecting the fine-scale grid to the model tiles via HMC makes it possible to circumvent the scale discrepancies between the macroscale and field-scale estimates – this has significant implications for how Earth system models are evaluated and applied. HMC illustrates a path towards improving the representation of land heterogeneity in ESMs by harnessing the available petabytes of environmental data. However, HMC only scratches the surface of what is possible. Moving forward, these approaches could be extended beyond natural systems to managed systems (urban areas, croplands, reservoirs, and pumping, among others). Clustering techniques could also be applied for non-soil systems including lakes and glaciers. Furthermore, although using hillslopes as the governing hydrologic structures is appropriate for the domain used in this study, this will not be true everywhere (e.g., flat terrain). In these cases, the hierarchical approach could be relaxed or extended to include other structures including stream orders and basins. Finally, the volume and complexity of data available today pale in comparison to what will be available in the coming decades . These data will provide unique opportunities for Earth system science; however, unless methods are developed that can harness these data, their intrinsic value to improve our understanding of the Earth system will be limited. We encourage the Earth system modeling community to pursue the use of clustering techniques to ensure these data are used effectively and efficiently in macroscale models. Data availability Data availability. The data, model output, and code used in this study are available upon request. Competing interests Competing interests. The authors declare that they have no conflict of interest. Acknowledgements Acknowledgements. We thank Minjin Lee, Kirsten Findell, Keith Beven, Paul Barlow and an anonymous reviewer for providing comments that improved the manuscript. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration, the US Department of Commerce, or the US Geological Survey. Edited by: Lixin Wang Reviewed by: Keith Beven and one anonymous referee References Ajami, H., Khan, U., Tuteja, N. 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	# Why can't the LHC detect heavy particles? I am reading The Elegant Universe by Brian Greene. In many places it's directly/indirectly mentioned that the LHC may not be able to detect (with the current technology) heavy particles to prove Super-Symmetry. What prevents such accelerators from detecting such heavy particles? I always thought it would be the opposite as heavy particles can make a stronger impression on detectors than light particles. • Oh, and I thought Super-Symmetry was just around the corner ... xD – image Sep 24 '17 at 14:37 It's not detecting the particles that is hard, it's making them in the collisions. Although the LHC collision energy is 14TeV, collisions aren't between the protons but rather between individual quarks inside the protons. Since the energy is shared between the three quarks in a proton the actual quark-quark collision energy is a lot less than 14TeV. Even then, for various reasons to do with conservation of momentum not all that energy can go into creating new particles. The end result is that it's hard to create particles much about above a TeV in weight. More on this in What is the maximal particle mass one can create via the LHC? Can we create dark-matter particles via the LHC? if you want to pursue this further. The upshot is that if the heavy particles have a mass much greater than a TeV the LHC can't create them, and obviously if they can't be created they can't be detected. All is not completely lost since we might be able to detect heavy particles indirectly by the influence they have on the collisions we can detect. Even so the upper mass limit is still restricted. • +1 for your optimism. But actually all is lost. – lalala Sep 24 '17 at 10:01 • Sadly it is looking that way ... – John Rennie Sep 24 '17 at 10:02 • This is way worse actually: if we were to discover a new particle rather light particle through direct production or a heavier one through comparison with loop corrections, how would we know it has anything to do with SUSY? Most papers out there start from MSSM or NMSSM and try to work out a prediction but what matters is the reverse: how many and what kind of particles do we need to observe to confirm some of SUSY? And then of course, validating (N)MSSM is not validating superstrings, not by a long shot. – user154997 Sep 24 '17 at 10:48 • Supersymmetric partners are always a couple of hundred GeV away: Implications of Initial LHC Searches for Supersymmetry (2011), The Smell of SUSY (2012), – David Tonhofer Sep 24 '17 at 20:34 • For the sake of the future of particle physics, please lobby your politicians to found the International Linear Collider (ILC)! – user154997 Sep 25 '17 at 11:27 As a consequence of $E=mc^2$, to create a heavy particle (i.e. large $m$) requires a large amount of energy ($E$). Since the LHC only generates a finite amount of energy in the collisions, there may be particles that are too heavy to be produced. This looks likely to be the case for superpartners (if they exist).
	# What is the implicit derivative of 1= 3y-ysqrt(x-y) ? Mar 26, 2018 $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y}{3 y - 2 x + 6 \sqrt{x - y}}$ #### Explanation: You differentiate both sides of the expression $1 = 3 y - y \sqrt{x - y}$ with respect to $x$ to get $0 = 3 \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{\mathrm{dy}}{\mathrm{dx}} \sqrt{x - y} - y \frac{1}{2 \sqrt{x - y}} \left(1 - \frac{\mathrm{dy}}{\mathrm{dx}}\right)$ $\quad = \left(3 - \sqrt{x - y} + \frac{y}{2 \sqrt{x - y}}\right) \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{y}{2 \sqrt{x - y}} \implies$ $\left(6 \sqrt{x - y} - 2 \left(x - y\right) + y\right) \frac{\mathrm{dy}}{\mathrm{dx}} - y = 0 \implies$ $\left(3 y - 2 x + 6 \sqrt{x - y}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = y \implies$ $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y}{3 y - 2 x + 6 \sqrt{x - y}}$
	# Command Line Arguments in Polyglots It occurs to me that there isn't really a general consensus on how command line arguments should be treated in polyglots. So... What should the consensus on command line arguments in polyglots be? I see 2 main options: • The command line arguments should be the same for all of the programs • The command line arguments can be different for different programs I think the first option as better, personally, just because of the ambiguity in byte counts between multiple languages if they have different arguments. Thoughts? Opinions? • Do you mean flags or arguments? i.e. lang codefile.txt -aSd vs land codefile.txt input – Rɪᴋᴇʀ Jan 31 '17 at 19:06 I think the default for polyglots should be that each language needs to be run with the simplest possible invocation (i.e. something that would get it a 0 byte penalty for command line arguments). This might or might not lead to the actual command lines being identical (because some interpreters need command line arguments to do anything at all). The command-line arguments must be allowed to be different, because many language combinations cannot be run with the same command line arguments. Consider Python and C. Python's invocation is python -c 'code here'. If we required the C version to also have the -c flag, then GCC would output an object file, not an executable (because that's what -c does in GCC). Potential abuses of command-line flags should be handled as standard loopholes, because a broad policy would certainly cause issues with legitimate uses. • That doesn't really address the question. If that's the standard invocation it doesn't count towards byte count and it's irrelevant to C. What about nonstandard runtime flags/arguments? – James Jan 31 '17 at 19:37 • @DJMcMayhem It addresses the question as it is written. Nowhere in the question does it state anything about standard vs non-standard flags. – Mego Jan 31 '17 at 19:42 They should be the same for all of the programs, because otherwise this is way too abuse prone. For example, let's say I wanted to write a vim/python quine (because those are the two languages I know best on this site). Vim allows you run certain commands before opening. So if I give vim the following command line flags, all of the code/keystrokes becomes irrelevant: vim file.txt -c "norm iHello World!" -c "wq" This is effectively a zero-byte hello world program. Now my polyglot code is: print("Hello world!") This has no side effects in vim because it has already exited. This is very clearly two different answers but if command line flags don't need to be the same, then this is a perfectly valid polyglot answer. I'm sure there are many language combinations with similar abuses, this is just one example. You can also imagine a language that just runs command line args as python/golfscript/jelly/whatever and ignores all code.
	Change the chapter Question Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors $\vec{A}$ , $\vec{B}$ , and $\vec{C}$ , all having different lengths and directions. Find the sum $\vec{A} + \vec{B} + \vec{C}$ then find their sum when added in a different order and show the result is the same. (There are five other orders in which $\vec{A}$ , $\vec{B}$ , and $\vec{C}$ can be added; choose only one.) Addition is commutative! (the video explains that formal term at the end) Solution Video
	Research # An inverse coefficient problem for a quasilinear parabolic equation with nonlocal boundary conditions Fatma Kanca1* and Irem Baglan2 Author Affiliations 1 Department of Management Information Systems, Kadir Has University, Istanbul, 34083, Turkey 2 Department of Mathematics, Kocaeli University, Kocaeli, 41380, Turkey For all author emails, please log on. Boundary Value Problems 2013, 2013:213 doi:10.1186/1687-2770-2013-213 Received: 7 June 2013 Accepted: 27 August 2013 Published: 30 September 2013 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ### Abstract In this paper the inverse problem of finding the time-dependent coefficient of heat capacity together with the nonlocal boundary conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. ### 1 Introduction Denote the domain D by Consider the equation (1) with the initial condition (2) the nonlocal boundary condition (3) and the overdetermination data (4) for a quasilinear parabolic equation with the nonlinear source term . The functions and are given functions on and , respectively. The problem of finding the pair in (1)-(4) will be called an inverse problem. Definition 1 The pair from the class , for which conditions (1)-(4) are satisfied and on the interval , is called the classical solution of inverse problem (1)-(4). The problem of identification of a coefficient in a nonlinear parabolic equation is an interesting problem for many scientists [1-3]. Inverse problems for parabolic equations with nonlocal boundary conditions are investigated in [4-6]. This kind of conditions arise from many important applications in heat transfer, life sciences, etc. In [7], also the nature of (3) type boundary conditions is demonstrated. In [1] the boundary conditions are local, the solution is obtained locally and the authors obtained the solution in Holder classes using iteration method. In [5] the boundary condition is nonlocal but the problem is linear and the existence and the uniqueness of the classical solution is obtained locally using a fixed point theorem. In this paper, the existence and uniqueness of the classical solution is obtained locally using the iteration method. The paper is organized as follows. In Section 2, the existence and uniqueness of the solution of inverse problem (1)-(4) is proved by using the Fourier method and the iteration method. In Section 3, the continuous dependence upon the data of the inverse problem is shown. In Section 4, the numerical procedure for the solution of the inverse problem is given. ### 2 Existence and uniqueness of the solution of the inverse problem Consider the following system of functions on the interval : The systems of these functions arise in [8] for the solution of a nonlocal boundary value problem in heat conduction. It is easy to verify that the system of functions and , , is biorthonormal on . They are also Riesz bases in (see [5,6]). The main result on the existence and uniqueness of the solution of inverse problem (1)-(4) is presented as follows. We have the following assumptions on the data of problem (1)-(4): (A1) , , ; (A2) , (1) , , , (2) , ; (A3) Let the function be continuous with respect to all arguments in and satisfy the following conditions: (1) where , , (2) , , (3) , , , (4) , , , where By applying the standard procedure of the Fourier method, we obtain the following representation for the solution of (1)-(3) for arbitrary : (5) Under conditions (A1)-(A3), we obtain (6) Equations (5) and (6) yield (7) Definition 2 Denote the set of continuous on functions satisfying the condition by B. Let be the norm in B. It can be shown that B is the Banach space. Theorem 3Let assumptions (A1)-(A3) be satisfied. Then inverse problem (1)-(4) has a unique solution for smallT. Proof An iteration for (5) is defined as follows: (8) where and From the conditions of the theorem, we have , and let . Let us write in (8). Adding and subtracting on both sides of the last equation, we obtain Applying the Cauchy inequality and the Lipschitz condition to the last equation and taking the maximum of both sides of the last inequality yields the following: Applying Cauchy’s inequality, Hölder’s inequality, Bessel’s inequality, the Lipschitz condition and taking maximum of both sides of the last inequality yields the following: Applying the same estimations, we obtain Finally, we have the following inequality: Hence . In the same way, for a general value of N, we have From we deduce that , An iteration for (7) is defined as follows: where , For convergence, Applying Cauchy’s inequality, Hölder’s inequality, Bessel’s inequality, the Lipschitz condition and taking maximum of both sides of the last inequality yields the following: Hence . In the same way, for a general value of N, we have We deduce that . Now we prove that the iterations and converge in B as . Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to the last equation, we obtain Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to the last equation, we obtain Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to the last equation, we obtain For N, we have (9) It is easy to see that , , then , . Therefore and converge in B. Now let us show that there exist u and p such that In the same way, we obtain (10) (11) Applying Gronwall’s inequality to (10) and using (9) and (11), we have (12) Here Then , we obtain . Hence . For the uniqueness, we assume that problem (1)-(4) has two solutions , . Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to and , we obtain (13) Applying Gronwall’s inequality to (13), we have . Hence . □ The theorem is proved. ### 3 Continuous dependence of upon the data Theorem 4Under assumptions (A1)-(A3), the solutionof problem (1)-(4) depends continuously upon the dataφ, g. Proof Let and be two sets of the data, which satisfy assumptions (A1)-(A3). Suppose that there exist positive constants , , such that Let us denote . Let and be the solutions of inverse problem (1)-(4) corresponding to the data and , respectively. According to (5), (14) Now, let us estimate the difference as follows: where , are constants that are determined by , and . Then we obtain , . The inequality holds for small T. Finally, we obtain where . If we take this estimation in (14) applying Gronwall’s inequality, we obtain taking the maximum of the inequality For , then . Hence . □ ### 4 Numerical procedure for nonlinear problem (1)-(4) We construct an iteration algorithm for the linearization of problem (1)-(4) as follows: (15) (16) (17) (18) Let and . Then problem (15)-(18) can be written as a linear problem: (19) (20) (21) (22) We use the finite difference method to solve (19)-(22) with a predictor-corrector type approach which was explained in [9]. We subdivide the intervals and into and subintervals of equal lengths and , respectively. Then we add two lines and to generate the fictitious points needed for dealing with the boundary conditions. We choose the implicit scheme, which is absolutely stable and has second-order accuracy in h and first-order accuracy in τ[10]. The implicit scheme for (1)-(4) is as follows: (23) (24) (25) (26) where and are the indices for the spatial and time steps, respectively, , , , , . At level, adjustment should be made according to the initial condition and the compatibility requirements. Now, let us construct the predicting-correcting mechanism. First, differentiating equation (1) with respect to x and using (3) and (4), we obtain (27) The finite difference approximation of (27) is where , . For , and the values of allow us to start our computation. We denote the values of , at the sth iteration step , , respectively. In numerical computation, since the time step is very small, we can take , , , . At each th iteration step, we first determine from the formula Then from (15)-(18) we obtain (28) (29) (30) The system of equations (28)-(30) can be solved by the Gauss elimination method and is determined. If the difference of values between two iterations reaches the prescribed tolerance, the iteration is stopped, and we accept the corresponding values , () as , (), at the th time step, respectively. By virtue of this iteration, we can move from level j to level . ### 5 Numerical example Example 1 Consider inverse problem (1)-(4) with It is easy to check that the analytical solution of this problem is (31) Let us apply the scheme which was explained in the previous section for the step sizes , . In the case when , the comparisons between the analytical solution (31) and the numerical finite difference solution are shown in Figures 1 and 2. Figure 1. The analytical and numerical solutions ofwhen. The analytical solution is shown with dashed line. Figure 2. The analytical and numerical solutions ofat. The analytical solution is shown with dashed line. Next, we will illustrate the stability of the numerical solution with respect to the noisy overdetermination data (4) defined by the function (32) where γ is the percentage of noise and θ are random variables generated from uniform distribution in the interval . Figure 3 shows the exact and the numerical solution of when the input data (4) is contaminated by , and 5% noise. Figure 3. The numerical solutions of(a) for1%noisy data, (b) for3%noisy data, (c) for5%noisy data. In Figure 3(a)-(c) the analytical solution is shown with dashed line. It is clear from these results that this method has shown to produce stable and reasonably accurate results for these examples. Numerical differentiation is used to compute the values of and in the formula . It is well known that numerical differentiation is slightly ill-posed and it can cause some numerical difficulties. One can apply the natural cubic spline function technique [11] to get still decent accuracy. ### Competing interests The authors declare that they have no competing interests. ### Authors’ contributions FK conceived the study, participated in its design and coordination and prepared computing section. IB participated in the sequence alignment and achieved the estimation. ### References 1. Cannon, J, Lin, Y: Determination of parameter in Holder classes for some semilinear parabolic equations. Inverse Probl.. 4, 595–606 (1988) 2. Pourgholia, R, Rostamiana, M, Emamjome, M: A numerical method for solving a nonlinear inverse parabolic problem. Inverse Probl. Sci. Eng.. 18, 1151–1164 (2010) 3. Gatti, S: An existence result for an inverse problem for a quasilinear parabolic equation. Inverse Probl.. 14, 53–65 (1998) 4. Namazov, G: Definition of the unknown coefficient of a parabolic equation with nonlocal boundary and complementary conditions. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci.. 19, 113–117 (1999) 5. Ismailov, M, Kanca, F: An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions. Math. Methods Appl. Sci.. 34, 692–702 (2011) 6. Kanca, F, Ismailov, M: Inverse problem of finding the time-dependent coefficient of heat equation from integral overdetermination condition data. Inverse Probl. Sci. Eng.. 20, 463–476 (2012) 7. Nakhushev, AM: Equations of Mathematical Biology, Vysshaya Shkola, Moscow (1995) 8. Ionkin, N: Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition. Differ. Equ.. 13, 204–211 (1977) 9. Cannon, J, Lin, Y, Wang, S: Determination of source parameter in a parabolic equations. Meccanica. 27, 85–94 (1992) 10. Samarskii, AA: The Theory of Difference Schemes, Dekker, New York (2001) 11. Atkinson, KE: Elementary Numerical Analysis, Wiley, New York (1985)
	# Adding nodes near coordinates to bar plot but from different series I would like to annotate a bar plot with some markers or numbers to indicate the significance of some results. The significance is given in a different column from the main value. Just to give an example of more or less what I would like to do: \documentclass{standalone} \usepackage{pgfplots} \usepackage{pgfplotstable} 1 10 0.84 50 0.75 10 0.22 30 0.24 2 40 0.38 60 0.96 15 0.42 90 0.28 3 20 0.42 60 0.42 15 0.24 60 0.86 }\mytable \begin{document}% \begin{tikzpicture} \begin{axis}[ ybar,enlarge x limits=0.2,bar width=7pt,xtick={1,2,3,4} ] \addplot table [x index=0,y index=1] {\mytable}; \addplot table [x index=0,y index=3] {\mytable}; \addplot table [x index=0,y index=5] {\mytable}; \addplot table [x index=0,y index=7] {\mytable}; \end{axis} \end{tikzpicture} % I would like to mark bars where adjacent value is greater than 0.95 % more specifically: column 2 for value 2 % I would also like to add a different mark where adjacent value is greater than 0.85 (but less than 0.95) % more specifically: column 4 for value 3 \end{document} The exact marks are not important; this is just to give the idea. I know nodes near coords may help here, but I don't know how to get this from a different series. (Also with the formatting of my plot (the above is just an example) I don't believe I will have room to give the actual values, rather I'll probably want to convert the values to some sort of mark on the bar. I understand this is a different question though so I'm interested in focusing on the first problem.) • – John Kormylo Apr 30 '16 at 19:28 • Thanks John. That's useful but in my case I have over 400 bars and will have to manually mark over 120. I'm thus hoping for a more automated solution. :) – badroit Apr 30 '16 at 19:48 • I just found this question, which seems relevant: tex.stackexchange.com/questions/80012/… (just trying to adapt it to my problem) – badroit Apr 30 '16 at 20:01 • If you want to automate the process, use a foreach loop and \pgfplotstablegetelem. – John Kormylo Apr 30 '16 at 21:35 Here is a suggestion using point meta=explicit symbolic, nodes near coords and every node near coord/.append style. \documentclass{standalone} \usepackage{pgfplots} \usepackage{pgfplotstable} 1 10 0.84 50 0.75 10 0.22 30 0.24 2 40 0.38 60 0.96 15 0.42 90 0.28 3 20 0.42 60 0.42 15 0.24 60 0.86 }\mytable \begin{document}% \begin{tikzpicture} \begin{axis}[ ybar,enlarge x limits=0.2,bar width=7pt,xtick={1,2,3,4}, point meta=explicit symbolic, nodes near coords={ \ifdim \pgfplotspointmeta pt > 0.95pt \textcolor{green}{\huge$\bullet$}% \else \ifdim \pgfplotspointmeta pt > 0.85pt \textcolor{green!50!blue!50}{\huge$\bullet$}% \fi\fi }, every node near coord/.append style={inner sep=0pt,anchor=center} ] \addplot table [x index=0,y index=1,meta index=2] {\mytable}; \addplot table [x index=0,y index=3,meta index=4] {\mytable}; \addplot table [x index=0,y index=5,meta index=6] {\mytable}; \addplot table [x index=0,y index=7,meta index=8] {\mytable}; \end{axis} \end{tikzpicture} \end{document} Result: or with \begin{axis}[ ybar,enlarge x limits=0.2,bar width=7pt,xtick={1,2,3,4}, point meta=explicit symbolic, nodes near coords={ \ifdim \pgfplotspointmeta pt > 0.95pt \textcolor{green}{\huge$\star$}% \else \ifdim \pgfplotspointmeta pt > 0.85pt \textcolor{green!50!blue!50}{\huge$\star$}% \fi\fi } ] • I spent ages at this and eventually gave up. This is awesome help, thanks a million! – badroit May 1 '16 at 5:24
	# Financial Statements Question 1. 1. Journalize the adjusting entries on august 31 for the 3-month period June 1-August 31. (Omit explanations.) 2. Prepare an adjusted trial balance on august 31. Question 2. 1. Journalize the adjusting entries. 2. Prepare an adjusted trial balance 3. Prepare a multiple-step income statement and retained earnings statement for the year and a classified balance sheet as of Nov 30, 2010. 4. Journalize the closing entries. 5. Prepare a post-closing trial balance. Related Questions in Employee and Labor Relations • Hi my name is Sherry Chiffon January 16, 2011 I have worked on this assignment a lot and I cannot get it to balance, can you help. Sherry Chiffon • Need the attached 7 questions answered March 14, 2011 Trying to get the attached questions answered. Can you please give me a quote? • . Insurance expires at the rate of $218 per month. 2. A count on August... October 15, 2014 unpaid at August 31 . 6. Rentals of$4,135 were due from tenants at August 31 . (Use Accounts Receivable.) 7. The mortgage interest rate is 9% per year . (The mortgage was taken out on... • financial accouting homework (Solved) November 17, 2010 for the year ended 2011. This statement should be flexibly designed in order to automatically update for changes in assumptions. 4. Prepare a forecasted Balance Sheet dated Dec. 31 ,... Solution Preview : 2010 Ending Balances Particulars Debit Credit Cash 17,000 Marketable Securities 2,000 Accounts Rec. 14,000 Allowance for Bad Debt 2,000 Inventory 15,000 Prepaid Insurance 5,000 Land 30,000... • Accounting Problem March 13, 2011 I'm having a hard time doing the following problem. The transactions need to be journalized and posted to ledger accounts, then I have to prepare a 10 column worksheet for the end of the...
	# Tikzcd "Crossing Over" Background Color? I'm making a Tikzcd diagram but I wanted to add color to the background of my document like so: However, there's clearly an issue with the middle diagram. In the middle diagram, I wanted to make cube diagram, and as to implement the 3D-effect I had to use the crossing over attribute for those arrows. However, this has now made me realize that Tikzcd implements this by assuming your background color is white; they just put a white rectangle around it. This explains why, when a user wants to cross over two arrows, one must write crossing over on the arrow which is crossing over the other. It seems unlikely that it's impossible to change the background color from white to something else. Does anyone know how? In addition, it also seems like it might be useful to know how to adjust the thickness of this white rectangle, which would be a nice bonus if anyone knows how to do that too. By the way, I'm using pagecolor for the background color, if that helps. You can define the background color in the options for the tikzcd environment. This color will then be used for crossing over. As you did not provide a minimal working example, I use the example from the documentation of the tikz-cd package: \documentclass[border=1mm]{standalone} \usepackage{tikz-cd} \pagecolor{yellow} \begin{document} \begin{tikzcd}[row sep=scriptsize, column sep=scriptsize, background color=yellow] & f^* E_V \arrow[dl] \arrow[rr] \arrow[dd] & & E_V \arrow[dl] \arrow[dd] \\ f^* E \arrow[rr, crossing over] \arrow[dd] & & E \\ & U \arrow[dl] \arrow[rr] & & V \arrow[dl] \\ M \arrow[rr] & & N \arrow[from=uu, crossing over]\\ \end{tikzcd} \end{document} Result: • This is great, thank you! Feb 26 '20 at 4:08
	# Direction of polarization for monochromatic wave? 1. Dec 9, 2015 ### magnesium12 1. The problem statement, all variables and given/known data Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude Eo, frequency w, and phase angle zero traveling in the direction from the origin to the point (1,1,1) with polarization parallel to the xz plane. I understand how to write the equations, I just don't understand how to get the correct direction for the electric and magnetic fields. 2. Relevant equations $E(z,t) = E_o\cos(\hat k \cdot \hat r - \omega t) \hat n$ $B(z,t) = \frac{E_o}{c}\cos(\hat k \cdot \hat r - \omega t) ( \hat k x \hat n)$ $k = -\frac{\omega}{c}$ $\hat n \cdot \hat k = 0$ 3. The attempt at a solution This is what I did: $\hat n = \hat x + \hat z$ $\hat k = \frac{\omega}{c} (\hat x + \hat y + \hat z)$ So I thought that was all I was supposed to do to find the direction, but the solutions manual says these are the actual directions of n and k: $\hat n =\frac { \hat x - \hat z}{\sqrt{2}}$ $\hat k = \frac{\omega}{c} \frac{(\hat x + \hat y + \hat z)}{\sqrt{3}}$ So where did those factors of sqrt(2) and sqrt(3) come from? I appreciate any help! 2. Dec 9, 2015 ### blue_leaf77 First, the question asks you to find the unit vector, so the magnitude of the vector which is supposed to be the answer should be unity. Second, you only know that $\hat{n}$ only has components along $\hat{x}$ and $\hat{z}$ but you are not given the length of each component, these are what you should calculate subject to the condition that the length of $\hat{n}$ is unity and that this vector is perpendicular to $\hat{k}$. 3. Dec 9, 2015 ### magnesium12 I don't think I understand. So I would do $n = \sqrt(a^2 + b^2)) = 1$ Therefore: $a^2= 1-b^2$ $k = \sqrt(c^2 + d^2 + e^2) = 1$ And then use this somehow: $\hat n \cdot \hat k = nkcos\theta = 0$ $nkcos\theta = \sqrt((1-b^2) + b^2)\sqrt(c^2 + d^2 + e^2)cos\theta$ But since n = 1 and k =1, wouldn't that just leave me with nothing again? 4. Dec 9, 2015 ### blue_leaf77 If $\mathbf{k}$ is denoted such that it has components $c$, $d$, and $e$ then they must be known already since the problem tells you that $\mathbf{k}$ goes from the origin to the point (1,1,1). What you don't know yet are just $a$ and $b$, i.e. two unknowns. You have figured out one equation relating these unknowns, which is . The other equation you need is the orthogonality condition between $\mathbf{k}$ and $\hat{n}$. To do this, it will be easier with component-by-component multiplication instead of the one like $kn\cos \theta$.
	# CODEEN¶ The Euclid code integration platform (CODEEN) is hosted by the APC laboratory as a Linux virtual machine. The needed softwares are installed regularly. The TIPS prototype for NISP pixel simulations is one of the first codes installed and integrated under this platform. We create a dedicated branche for the CODEEN integration developments. svn checkout http://svn.oamp.fr/repos/tips/branches/tips_codeen ## Tests¶ Test was written with pyUnit and TIPS was integrated in the CODEEN/Jenkins: https://apceucliddev.in2p3.fr/jenkins/job/TIPS_dev/ ### Level 0 : unitary tests¶ Test every single part of the code (functions, methods ...). Allow to be sure that every brick of the software is functioning correctly independently of the other parts. • test sky • test instrument • test axesim ### Level 1 : integration tests¶ Allow testing the small interactions between the different units of the software. • test observation ### Level 2 : full system tests¶ Allow testing the software in a global way, by assembling all the bricks that make it. Not implemented at this stage because to heavy... The example provide with TIPS could be convert to a level 2 test. ### Level 3 : non regression tests¶ Key tests one keeps during the whole development process. Not implemented at this stage.
	Composition of Rotation and Translation in the Complex Plane — Finding Angle of Rotation and Point A rotation about the point $1-4i$ is $-30$ degrees followed by a translation by the vector $5+i$. The result is a rotation about a point by some angle. Find them. Using the formula for a rotation in the complex plane, I found $f(z)$ or the function for the rotation of the point to be $(1-4i)+(z-(1-4i))e^{-i\pi/6}$. I know the angle of rotation for the first transformation is $-30$ or $-\pi/6$. But how do I calculate the angle of rotation for the composition? I found the angle of the vector to be $11.3$ degrees ($\arctan1/5=11.3$ degrees). Do I add $11.3$ and $-30$ to find the resulting angle of rotation? Do I also need to calculate the angle of vector 1-4i and incorporate that somehow? Is the angle of rotation for the composition simply the same as the original rotation? Would appreciate some guidance. The rotation is defined by $$R(z)=c+e^{i\theta}(z-c)=e^{i\theta}z+(1-e^{i\theta})c,$$ and the translation by $$T(z)=z+t,$$ with $$c=1-4i,\quad \theta=-\frac\pi6,\quad t=5+i.$$ We have: \begin{eqnarray} (R\circ T)(z)&=&R(T(z))=e^{i\theta}T(z)+c(1-e^{i\theta})=e^{i\theta}z+c(1-e^{i\theta})+te^{i\theta},\\ (T\circ R)(z)&=&T(R(z))=R(z)+t=e^{i\theta}z+c(1-e^{i\theta})+t. \end{eqnarray} Let us find the fixed points of $(R\circ T)$, and $(T\circ R)$. We have: $$(R\circ T)(z)=z\iff (1-e^{i\theta})z=c(1-e^{i\theta})+te^{i\theta}\iff z=c+t\frac{e^{i\theta}}{1-e^{i\theta}},$$ i.e. $(R\circ T)$ has one fixed point, that is \begin{eqnarray} c_1&=&c+t\frac{e^{i\theta}}{1-e^{i\theta}}=c+t\frac{e^{i\theta}(1-e^{-i\theta})}{\|1-e^{i\theta}\|^2}=c+t\frac{e^{i\theta}-1}{2-2\cos\theta}=1-4i+(5+i)\frac{\frac{\sqrt3}{2}-1-\frac{i}{2}}{2-\sqrt3}\\ &=&1-4i+(5+i)(2+\sqrt3)\frac{\sqrt3-2-i}{2}=1-4i-(5+i)\frac{1+(2+\sqrt3)i}{2}\\ &=&1-4i-\frac{5-(2+\sqrt3)+(10+5\sqrt3+1)i}{2}=\frac{2-(3-\sqrt3)-8i-(11+5\sqrt3)i}{2}\\ &=&\frac{-1+\sqrt3-i(19+5\sqrt3)}{2}. \end{eqnarray} Similarly $$(T\circ R)(z)=z\iff (1-e^{i\theta})z=c(1-e^{i\theta})+t \iff z=c+\frac{t}{1-e^{i\theta}},$$ i.e. $T\circ R$ has exactly one fixed point, that is: \begin{eqnarray} c_2&=&c+\frac{t}{1-e^{i\theta}}=c+\frac{t(1-e^{-i\theta})}{\|1-e^{i\theta}\|^2}=1-4i+\frac{(5+i)(1-\frac{\sqrt3}{2}-\frac{i}{2})}{2-\sqrt3}\\ &=&1-4i+\frac12(5+i)(2+\sqrt3)(2-\sqrt3-i)=1-4i+\frac12(5+i)[1-(2+\sqrt3)i]\\ &=&1-4i+\frac12[5+2+\sqrt3+(1-10-5\sqrt3)i]=\frac{9+\sqrt3-(13+5\sqrt3)i}{2} \end{eqnarray} Hence $R\circ T$ and $T\circ R$ are rotation with angle $-\frac\pi6$, and centers $c_1$ and $c_2$, respectively. $(x,y)\rightarrow{\frac{1}{2}(\sqrt{3} x+y-\sqrt{3}+16),\frac{1}{2}(-x+\sqrt{3} y+4 \sqrt{3}-5)}$
	• noun: • A unit of plane angular measure corresponding to one thousandth of a radian • a unit of angular length add up to one thousandth of a radian
	# Hidden variable formalisms in quantum mechanics Loosely speaking, we can have quantum mechanics without randomness, driven by deterministic hidden values but then it has to be non-local. This is apparently Bell's theorem The most famous version is the De Broglie-Bohm formulation/ pilot-wave theory. In the classical case, no difference AFAICT. But when your wave function evolves according to the Schroedinger equation and then collapses, it gets stranger with this whole non-locality thing. Scott Aaronson points out some fun edge cases: It follows that, if we want it to agree with quantum mechanics, then any hidden-variable theory has to allow “instantaneous communication” between any two points in the universe. Once again, this doesn't mean that quantum mechanics itself allows instantaneous communication (it doesn't), or that we can exploit hidden variables to send messages faster than light (we can't). It only means that, if we choose to describe quantum mechanics using hidden variables, then our description will have to involve instantaneous communication. But here's the amazing thing: even in the teeth of four different no-go theorems, one can still construct interesting and mathematically nontrivial hidden-variable theories. […] on Bohmian mechanics: the amazing thing about this theory is that it's deterministic: specify the “actual” positions of all the particles in the universe at any one time, and you've specified their “actual” positions at all earlier and later times. So if you like, you can imagine that at the moment of the Big Bang, God sprinkled particles across the universe according to the usual $$\|\psi\|^2$$ distribution; but after that He smashed His dice, and let the particles evolve deterministically forever after. And that assumption will lead you to exactly the same experimental predictions as the usual picture of quantum mechanics, the one where God's throwing dice up the wazoo. The catch, from my point of view, is that this sort of determinism can only work in an infinite-dimensional Hilbert space, like the space of particle positions.[…] if our universe is discrete at the Planck scale, then it can't also be deterministic in the Bohmian sense.
	# AI For Salesforce Developers - Part 3 , Machine Learning, Artifical Intelligence, Maths, Salesforce, Development, AI, ML, Apex In my previous post we walked through the first algorithm we are putting together in this series - Linear Regression. I noted in the post that before we moved on it would be wise to cover the MathUtil and Matrix classes in detail to save us some time later. So let’s do it! # MatrixOperations We are going to first deal with the MatrixOperations class as we will find later on that we will be using this class to perform some of the operations within our MathUtil class. The first method we will discuss is multiply. Going back to our initial discussion of matrices in part one, remember that a matrix is a set of numbers in rows and columns such as: where the subscript reperesents the row and column, i.e. 32 means row 3 column 2. Unlike programming arrays, the index starts at 1 and mathematicians describe matrices as being of type m x n, where m is the number of rows and n is the number of columns. We won’t cover the reasoning behind why matrix multiplication works in the way it does here, but we will describe how it works. The why is a little outside the scope of what we wish to cover, but you can find some good explanations online. In order to multiply matrices A and B together, denoted AB, the number of columns in A must be the same as the number of rows in B. So if A is 3x2 and B is 2x7 we are all good. However, were to to try and perform the operation BA, this would not work because the number of columns in B (7) is not the same as the number of rows in A (3). This fact: is called non-commutativity. So, what does AB actually look like? What we do is take the entries of a column in A and a row in B, multiply them term-by-term and then sum them to get our new entry. This means that if we multiply an m x n matrix A with an n x p matrix B then AB is an m x p matrix. Let’s see this written down in an example. Firstly lets take a 2 x 3 matrix A: and another 3 x 4 matrix B: then AB is: As we can see here we have a 2 x 4 matrix produced. So we can see how this works in theory, but how do we implement this in our MatrixOperations class? public static Matrix multiply(Matrix A, Matrix B) { if(A.columns != B.rows) { return null; } Integer A_rows = A.rows, A_cols = A.columns, B_cols = B.columns; Matrix C = new Matrix(A_rows, B_cols); C.zeros(); for(Integer i = 0; i < A_rows; i++) { List<Double> A_row = A.getRow(i); for(Integer j = 0; j < B_cols; j++) { for(Integer k = 0; k < A_cols; k++) { C.setElement(i, j, C.getElement(i, j) + A.getElement(i,k)B.getElement(k, j)); } } } return C; } The first thing we do is check that we have matrices that the number of columns in A needs to be the same as the number of rows in B and if that is not true we return nothing. The next thing we do is instantiate some integers for us to use in our counting. We create a new matrix C to hold our results and fill it with zeros. We then start to loop through our counter variables, incrementing through the rows of A, then the columns of B, then the columns of A. We could perform these loops in any order, but I have structured them like this to help in maintaining the sort of sequencing given in the mathematical description (i.e. row of A with column of B term-by-term). We set the value of the corresponding element of C to be the existing value plus our new value which is why we initially filled the matrix with zeros. Note we have structured our loops to use the best loop implementation. The next operation is the transpose function. This is where a matrix is flipped along its leading diagonal, so that an m x n matrix becomes an n x m one. The transpose is denoted AT. So as an example: In code we implement this as follows: public static Matrix transpose(Matrix A) { Integer A_rows = A.rows, A_cols = A.columns; Matrix B = new Matrix(A_cols, A_rows); for(Integer i = 0; i < A_rows; i++) { for(Integer j = 0; j < A_cols; j++) { B.setElement(j, i, A.getElement(i, j)); } } return B; } Again, we instantiate some integers for use in our loops and create a new matrix with the row and column numbers swapped. We then loop using these counters and set the elements on our new matrix using the values from our existing matrix. Next up is addition and subtraction. To add 2 matrices together, they must have the same dimensionality (number of rows and columns), and we add term-by-term, exactly the same as we did for the subtraction as we discussed in the first post. Implemented in code they look like public static Matrix add(Matrix A, Matrix B) { if((A.rows != B.rows) && (A.columns != B.columns)) { return null; } Integer A_rows = A.rows, A_cols = A.columns; Matrix C = new Matrix(A_rows, A_cols); for(Integer i = 0; i < A_rows; i++) { for(Integer j = 0; j < A_cols; j ++) { C.setElement(i, j) = A.getElement(i, j) + B.getElement(i, j); } } return C; } public static Matrix subtract(Matrix A, Matrix B) { if((A.rows != B.rows) && (A.columns != B.columns)) { return null; } Integer A_rows = A.rows, A_cols = A.columns; Matrix C = new Matrix(A_rows, A_cols); for(Integer i = 0; i < A_rows; i++) { for(Integer j = 0; j < A_cols; j ++) { C.setElement(i, j) = A.getElement(i, j) - B.getElement(i, j); } } return C; } Much the same as before, some simple looped and setting of items on a new matrix. Our final methods are pointwise multiply and pointwise exponent (which we described in the first post). if you cast your mind back to that first post, I noted that “pointwise exponent” is correctly called the “Hadamard Power”, and similarly what we will term “pointwise multiplication” is known as the “Hadamard Product”. For this operation both matrices must have the same dimensionality, and we can write out the operation as: Here the small circle symbol denotes the Hadamard product instead of the standard matrix mnultiplication action. In code this looks like: public static Matrix pointwiseMultiply(Matrix A, Matrix B) { if((A.rows != B.rows) && (A.columns != B.columns)) { return null; } Integer A_rows = A.rows, A_cols = A.columns; Matrix C = new Matrix(A_rows, A_cols); for(Integer i = 0; i < A_rows; i++) { for(Integer j = 0; j < A_cols; j ++) { C.setElement(i, j) = A.getElement(i, j)B.getElement(i, j); } } return C; } public static Matrix pointwiseExponent(Matrix A, Double exponent) { Integer A_rows = A.rows, A_cols = A.columns; for(Integer i = 0; i < A_rows; i++) { for(Integer j = 0; j < A_cols; j++) { A.setElement(i, j, Math.pow(A.getElement(i, j), exponent)); } } return A; } Again for both of these functions we are using our speedier loops implementation to move through the items within the matrix, setting each value as we go. In the case of the pointwise multiplication, this is simple multiplying the 2 matrix entries together, and for the exponent is applying the exponent to each entry. # MathUtil The MathUtil class is a holding place for all the thngs we need to be able to do quickly that there is no default Apex implementation for. The first such method is sum. public static Double sum(List<Double> items) { Double sum = 0; Integer numItems = items.size(); for(Integer i = 0; i < numItems; i++) { sum += items[i]; } return sum; } Sum takes in a list of numbers (doubles) and returns their sum, that is, all of them added together. As you can see from the code above, this is pretty simple, the only thing worth noting really is that we are using our faster loop implementation as this will help us as the system scales to summing some larger lists of numbers. Next up we have a function to calculate the mean (or average) which we discussed in the first post. As we highlighted the mean of a list of numbers is their sum divided by the size of the list. Our implementation is: public static Double mean(List<Double> items) { Double sum = sum(items); Integer numItems = items.size(); return sum/numItems; } which simply uses our sum function to generate the sum and then divides by the number of items we have. We are going to discuss the variance function next, as the standard deviation function simply returns the square rrot of the variance that we calculate. As we discussed in the first post, the variance is calculated as the sum of the square of the difference of each item from the mean, divided by the total sample size - 1. Our code implementation is: public static Double variance(List<Double> items) { Double mean = mean(items); Matrix itemMatrix = new Matrix(items); Matrix meanMatrix = new Matrix(1, items.size()); meanMatrix.fill(mean); Matrix diffs = MatrixOperations.subtract(itemMatrix, meanMatrix); diffs = MatrixOperations.pointwiseExponent(diffs, 2); List<Double> diffList = diffs.getRow(0); return sum(diffList)/(diffList.size() - 1); } For our sample list of items we start by calculating the mean and then creating 2 matrices, one to hold our sample data and another of equal size holding the mean value everywhere. We then subtract the mean matrix from the sample matrix and run our pointwise exponent (Hadamard power) function on this new matrix to square these differences. We then cast this back to a list of data and calculate the sum of this data divided by the size - 1. This function brings together a lot of the work we have done previously in the MathUtil and MatrixOperations classes to provide an important function. Last but by no means least is our function to calculate the Pearson correlation coefficient for us to use in the Linear Regression model. public static Double pearsonCorrelation(List<Double> x, List<Double> y) { Double mean_x = mean(x); Double mean_y = mean(y); Double std_x = standardDeviation(x); Double std_y = standardDeviation(y); List<Double> xy = new List<Double>(); Integer numItems = x.size(); Double x_diff, y_diff; for(Integer i = 0; i < numItems; i++) { x_diff = x[i] - mean_x; y_diff = y[i] - mean_y;
	# How do I delete an armature but keep the baked pose? According to my notes from when I have done this before: Q. How do I bake or fix the pose to the object? A. Pose the model and then switch to Object mode. Select the model (not the armature) or shift select all the parts of the object if it isn't one single mesh. Next, add a new modifier and choose "Armature". Make sure that the armature is moved to the top of the stack and in the Object dropdown menu, select the armature that you created and then press apply. So the problem that I am having is that I have followed those directions exactly, but then when I delete my armature, my pose resets and I lose the pose. It doesn't seem that it baked it at all. What is going on? How do I delete an armature but keep the baked pose? • Have you tried Ctrl A to apply rotation, scale and location in object mode, before deleting the armature? – user7952 Jan 8, 2015 at 6:12 • Yes, and it I still get the same thing...as soon as I delete the armature it goes back to the original form. Jan 8, 2015 at 6:32 • I just did it with a very simple setup. A pipe elbow with an armature with two bones. I deformed it in pose mode, then in object mode I 1. Applied the armature modifier. 2. Applied rotation, scale and location. 3. Deleted the armature. And after this, the deformation remained. – user7952 Jan 8, 2015 at 6:40 • Yeah I don't know what is going on. Thanks for confirming that I was doing it right. I will just have to save it out as an obj and re-import it for this model I guess. Jan 8, 2015 at 7:23 • I'm not really that experienced with Blender, and hardly at all with armatures, so if I were you, I'd wait awhile and someone else may come along with an answer. One thing though, if you upload your .blend to pasteall.org/blend and put the link in your question, it will be a lot easier to find the problem. – user7952 Jan 8, 2015 at 7:40
	# Fourth dimension Visual scope Being three-dimensional, we are only able to see the world with our eyes in two dimensions. A four-dimensional being would be able to see the world in three dimensions. For example, it would be able to see all six sides of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as we can see the interior of a square on a piece of paper. It would be able to see all points in 3-dimensional space simultaneously, including the inner structure of solid objects and things obscured from our three-dimensional viewpoint. — Wikipedia on Fourth dimension 2010.06.27 Sunday ACHK # Pragmatic Idealism 3 . Be an ideal pragmatist and a pragmatic idealist. — Me@2001.12.05 <– 2001, not 2010 . . . 2010.06.27 Sunday $copyright ACHK$ Choosing a graph 1990-CE-PHY II-1 1990-CE-PHY II-4 1990-CE-PHY II-8 1991-CE-PHY II-3 1991-CE-PHY II-7 1992-CE-PHY II-1 1993-CE-PHY II-5 1994-CE-PHY II-9 1995-CE-PHY II-4 1996-CE-PHY II-4 2000-CE-PHY II-7 2000-CE-PHY II-9 2001-CE-PHY II-7 2002-CE-PHY II-3 2002-CE-PHY II-8 — Me@2010.06.27
	# Limit question This isn't really homework for a class, but i figured this would be the most appropriate place for this question: What would this quantity be? $$\lim_{t \rightarrow \infty} e^{-i \alpha \|x - t\|} \cdot (\|x -t\| - 1) - \lim_{t \rightarrow - \infty} e^{-i \alpha \|x - t\|} \cdot (\|x -t\| - 1) = ?$$ It looks to me like it is just zero, but I was hoping it would be: $$\frac{2e^{-i \alpha x}}{1 + \alpha^2}$$ where $$\alpha$$ is a real number, since this was the last step in proving that $$f(t) = e^{-i \alpha t}$$ is an eigenfunction of the kernel: $$K(x,t) = e^{-i \alpha \|x - t\|}$$ with an eigenvalue: $$\lambda = \frac{2}{1 + \alpha^2}$$ Perhaps I solved my integral wrong or made a mistake somewhere. AKG Homework Helper The quantity looks to be undefined. It looks to be of the form Y - Z, where Y and Z are themselves undefined, so clearly their difference is undefined. Each of the limits are themselves undefined because they are the limits of products of two functions, one of which is period, the other which goes to infinity. Well, here's how I got there anyway: $$\hat{K} \|f \rangle = \lambda \|f \rangle$$ Then I projected into position space: $$\langle x\| \hat{K} \|f \rangle = \lambda \langle x\|f \rangle$$ Threw in an identity operator: $$\langle x\| \hat{K} (\int^{\infty}_{- \infty} dt \|t \rangle \langle t\|)\|f \rangle = \lambda \langle x\|f \rangle$$ Which simplifies to: $$\int^{\infty}_{- \infty} dt \cdot K(x,t) \cdot f(t) = \lambda \cdot f(x)$$ So then I just plugged in: $$\int^{\infty}_{- \infty} dt \cdot e^{-\|x - t\|} \cdot e^{-i \alpha t} = \frac{2e^{-i \alpha x}}{1 + \alpha^2}$$ This leaves me with the problem of trying to find that integral, namely: $$\int^{\infty}_{- \infty} dt \cdot e^{-\|x - t\|} \cdot e^{-i \alpha t}$$ I put this into Integrals.com and got this back: $$\frac{\|x -t\| e^{-i (\|x -t\| + \alpha t)}}{(x - t)} \|^{\infty}_{- \infty}$$ That's where the limit came from anyway. Oh whoops, forgot to put this into the function above: $$\|x - t\|$$ Last edited: Galileo $$\int^{\infty}_{0} dt \frac{t \sin{xt}}{a^{2} + t^{2}}$$
	# Space and Time warp in immense gravity 1. Nov 13, 2014 ### Akshar Tandon This question is inspired by the movie Interstellar but asks a basic question about relativity. Everyone talks about gravitational time dilation but I am wondering if gravity has an effect on space as well, after all it bends space time. I have not found a lot of information on length contraction in the context of gravity (velocity, yes, but not gravity). If we were to look at a clock on a planet with a very strong gravity, we would see time move extremely slow but would we also see the clock stretched/contracted? Also, would light "appear" to travel faster/slower on this planet when externally observed? 2. Nov 13, 2014 ### A.T. Yes, it does. http://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html http://www.mathpages.com/rr/s8-09/8-09.htm Slower: http://en.wikipedia.org/wiki/Shapiro_delay I'm not sure though, if the Shapiro delay refers only the effect of gravitational dilation (as wiki says), or to the combined effect including spatial geometry. 3. Nov 13, 2014 ### Staff: Mentor The formula given on the Wiki page includes both. The factor of $\left( 1 + \gamma \right)$ is the key (where $\gamma$ is one of the PPN parameters); it's the same factor that appears in the formula for light bending by a massive body like the Sun, which takes into account both time and space curvature.
	How big is our galaxy and how fast is the universe expanding? May 11, 2016 Our galaxy is 100-180 k ly wide, with 2 k ly bulge at the center. Our universe is expanding at the rate of 71 km/sec/mega parsec. Explanation: Here, ly = light year unit of distance = 62900 X( Earth- Sun distance), nearly, and k = 1000.. The greater unit of distance, mega parsec = million parsec, where parsec = 206265 X ( Earth-Sun distance) The average Earth-Sun distance is the unit AU = 149597871 km. It is scientifically surmised that our universe, expanding at the rate of Hubble Constant ${H}_{0}$ = 71 km/s/mega parsec. has attained radial expansion to 13.8 billion light years, in 13.8 billion years.
	# Long time no blog Posted by Thomas Sutton on August 4, 2006 It’s been quite a while since last I posted (here, or anywhere else), so I think it’s about time for an update. Things are going adequately on the school front – it was a little touch and go, but I managed not to fail anything last semester – and I’m about to start a support job with a research project which will bring a welcome injection of funds (which have been rather tight lately). On the “reading” front, I purchased my copy of Advanced Topics in Types and Programming Languages (companion and somewhat successor to TAPL) this morning. I’ve had a bit of a flick through it and it looks really, really interesting. Hopefully having spent almost \$200 on the two books will provide an added incentive to not only start, but finish, reading them and hopefully even work through the problems. I’ve been thinking about typography and book design lately which has suggested, amongst other things, that I see if it’d be possible to get my copy of ATTAPL rebound with some extra pages. It would be nice, for example, to insert the extended version of chapter 10 - the essence of ML type inference and to “fix” any errata with updated pages. I imagine, though I haven’t bothered to investigate at all, that this’d be quite a difficult and expensive thing to do for a single copy, so it’ll probably be a long while before I do it, if ever. Before that though, I’ve been focussed on getting through Logic by Greg Restall. I’m almost half way through and while I’d have preferred a slightly difference syntax (I prefer &and; as conjunction, rather than &), it’s easy to read and is much more accessible than most other books I’ve seen with titles like “Logic”. This post was published on August 4, 2006 and last modified on August 4, 2020. It is tagged with: books, TAPL, ATTAPL, logic.
	# Statistics of correlations across the many-body localization transition ### Submission summary As Contributors: Luis A. Colmenarez · David J. Luitz Arxiv Link: https://arxiv.org/abs/1906.10701v2 Date submitted: 2019-07-17 Submitted by: Colmenarez, Luis A. Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approaches: Theoretical, Computational ### Abstract Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) -- at least in one dimensional systems -- leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of \emph{eigenstate correlation functions} across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder -- in the thermal phase -- we find further evidence for \emph{anomalous thermalization} in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in $S_i^z S_{i+r}^z$ correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A \emph{quantitative quasi-degenerate perturbation theory} calculation of these correlators yields a surprising \emph{agreement of the full distribution} with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the $S_i^zS_{i+r}^z$ correlator exhibits striking differences between the MBL and Anderson insulator cases. ### Ontology / Topics See full Ontology or Topics database. ###### Current status: Has been resubmitted ### Submission & Refereeing History Resubmission 1906.10701v3 on 8 October 2019 Submission 1906.10701v2 on 17 July 2019 ## Reports on this Submission ### Anonymous Report 1 on 2019-8-30 Invited Report • Cite as: Anonymous, Report on arXiv:1906.10701v2, delivered 2019-08-30, doi: 10.21468/SciPost.Report.1142 ### Strengths 1 - results from the state-of-the art numerics are reproduced by perturbation theory deep in MBL phase providing insight into various features of considered distributions of correlators 2- very smooth and clear presentation of extensive results ### Weaknesses 1 - the significance of results as a characterizing feature of MBL phase is not clear, considered distributions seem to be strongly dependent on details of system 2 - it is not clear whether the obtained results provide any new insights into many-body localization transition mentioned in the title of the work ### Report This paper presents an analysis of distributions of all non-vanishing two-point correlators in eigenstates of disordered XXZ spin chain regarded as the paradigmatic model of ETH-MBL transition. The authors employ state-of-the-art exact numerical methods to find distributions of the correlators and use a mixed degenerate and non-degenerate perturbation theory to reproduce those distributions (semi-)analytically. The obtained numerical results are compatible with well known results of Refs. [19,35,49] whereas the use of perturbation theory in the context of correlation functions of disordered lattice system is interesting and yields very good agreement with numerics at large disorder strength $W,$ providing insight into origin of various features of distributions of correlators. While the analysis conducted by the authors is thorough, the significance of the results obtained in this paper is not obvious. The importance of gaussian shape of distributions of the correlators is clear in the ergodic regime, the same applies for the appearance of the heavy tails at stronger disorder and their impact on thermalization. This problems have been, however, widely discussed in Refs. [35,49] and the present paper with its study of different, longer-range correlations has little to offer in that respect. The authors argue in the Introduction that the considered correlators are equally suited to characterize the MBL phase. However, it is hard to be convinced by this statement: while the properties of distributions of the correlators in the ergodic phase are generic features of thermalizing interacting quantum many-body systems, the same distributions in the MBL phase seem to be strongly dependent on the details of the considered system and would change upon some deformations of the system such as inclusion of a next-to-nearest coupling term or change of disorder distribution. Thus, it is not clear to what extent the considered distributions of correlators may be regarded as a characteristic features of the $\textit{MBL phase}$ (in broader, model independent context) and, going further, it is not evident whether we learn anything new about the many-body localization transition from the results of the present paper. Perhaps, some insights from the strong disorder perturbation theory suggest some model independent features of the considered distributions that could be related to the structure of local integrals of motions which are believed to be the robust way of describing the MBL phase? Having said that, I believe that the present paper may merit publication in some form. Pondering the above comments could improve the significance of the manuscript. In addition, I have a number of minor questions and remarks listed below. ### Requested changes 1 - Strong fluctuations in eigenstate properties leading to heavy tails in the considered correlator distributions at $1.2<W<3.7$ were shown in Ref. [49] to coincide with the region of the phase diagram dominated by Griffiths regions. Are there any new insights on that matter from considering correlators of bigger range r>1? 2 - Does the system size dependence of the standard deviation of distributions shown in Fig. 3 change when r>1? 3 - Can a finite size scaling of the points of maximal departure from gaussianity (shown in Fig. 4) be performed? Does the point seem to be tending to $W_C=3.7$ or rather staying in the ergodic phase? 4 - In Section 3. C. the authors write that "it is natural to treat the interaction of the Hamiltonian" as a perturbation. From what follows (eq. A1 and A2) it is clear that it is the kinetic term (in the fermionic language) that is treated as the perturbation. 5 - Perturbation theory results shown in Fig. 6 are shown only for the $r=1$ case. I understand that for $r>1$, the perturbation theory shows that there are no "satellite peaks" in $\langle F_{i,i+r}\rangle$. Is it possible to reproduce distributions of $\langle F_{i,i+r}\rangle$ for $r>1$ within the considered perturbation theory? 6 - It is not clear what is happening close to $0$ of the histograms in Fig. 6, adding an inset could make this matter clearer. The region $\|x\|<1/(4W)$ hosts the broadened broadened delta function so that the numerical results are bigger than $1$ in contrast to the "PT" result which vanishes in that region except for the $x=0$ point, thus showing a discrepancy between "PT" of the considered order and "ED". 7 - The vanishing positive part of the distribution of $\langle S^z_i S^z_{i+1}\rangle_c$ differentiates between Anderson localized phase and MBL phase. Is it possible to find some experimentally relevant consequences of that fact (as the highly excited eigenstates of interacting many-body systems are not accessible in experiments)? 8 - References are quite appropriately chosen, however work M. Serbyn et. al. Phys. Rev. B 96, 104201 (2017) seems to be related as it studies properties of diagonal matrix elements in the same model. Also, the approach of recent paper A. Maksymov et. al. Phys. Rev. B 99, 224202 (2019) seems to be related to the present work. 9 - The authors restrict themselves to analysis of correlators in mid-spectrum eigenstates. While the analysis of Ref. [21] suggests that properties of the system change in very similar way when increasing/decreasing energy density, results [62] indicate large difference between time dynamics at small and large energy densities. Would effects of change of energy window be non-trivial on the considered distributions of correlators? 10 - P. W. Anderson's paper is doubly referenced as [11] and [50] • validity: high • significance: ok • originality: good • clarity: high • formatting: excellent • grammar: excellent Author Luis A. Colmenarez on 2019-10-08 (in reply to Report 1 on 2019-08-30) Category: remark First of all, we thank the referee for a careful reading of our paper and a detailed commentary. Below, we answer the various questions put by the referee and outline the main resulting changes we have made to the paper in the version 3. Before that, we offer our own perspective on the broader points raised in the report. In particular, the referee comments that the significance of the results obtained in this paper is not obvious", remarking that the significance of the ergodic phase results have been discussed elsewhere while the MBL phase results, which are new, are perhaps non-universal in the sense that the form of the correlation functions may be model dependent. Moreover, the report questions whether we have anything to say about the MBL transition itself. As mentioned by the referee we have provided comprehensive calculations and an analysis of all non-trivial two-point correlation functions across the phase diagram of the disordered XXZ spin chain - this includes the ergodic phase, the incipient sub-diffusive regime and the many-body localized phase. In other words, we have focused on the features of the phases themselves and do not claim to provide new information about the transition. To highlight this point we have decided to change the title from "Statistics of correlations across the many-body localization transition" to "Statistics of correlation functions in the random Heisenberg chain". On the interesting question of the model-independent features of the correlation function distributions in the MBL phase we have a few points to make. (i) First of all, we want to re-emphasise the paradigmatic nature of the model considered in our paper to the study of MBL. Even neglecting the question of characterizing the MBL phase, given the importance rightfully attached to this model, we consider it worthwhile to characterize and understand the natural observables including the 2-point correlation functions. (ii) The detailed form of the two-point correlation functions explored in our work is model dependent in the sense that perturbations or changes to the disorder distribution may modify the distributions. For example, the satellite peaks in the spin-flip correlation functions may be smeared out by further-neighbour couplings. (iii) However, there are features of the correlation function that should be present in any MBL phase. MBL will always ensure that correlation function distributions for fixed distance are broad and relatively insensitive to the system size - a sign of the violation of ETH. Since there is a localization length that is disorder dependent, the variance of the distributions will fall off with increasing disorder strength (as shown in Fig. 3). One of the central observations of our paper is that the mixed degenerate and non-degenerate perturbation theory can capture the detailed form of the distributions and, in particular, this can capture the all-important resonances that ultimately destroy MBL for sufficiently weak disorder. The appearance of such resonances is universal to MBL and appear clearly in the outmost peaks of the correlator distributions. Tying these features of the correlation functions to the local integrals of motion is an interesting avenue for future work. By now the changes requested by the referee are included in the latest version. Next the specific questions asked in the report are answered ( the list of changes is included in the re-submission comments ) : 1) This is a very interesting point. There is no obvious relation to transport in correlations with $r>1$, hence it is not clear how these correlators could be affected by Griffiths regions. 3) We attach a plot of the maximum kurtosis scaling with system size. Only three points are suitable for fitting and possible extrapolation. We think that a finite scaling is not really meaningful with the data we have. 4) The referee is correct. The kinetic term is the one treated as a perturbation. We thank the referee for pointing out this error and have corrected it in the version 3. 5) Distances $r>1$ require higher order perturbation theory because the leading and subleading orders are non-trivial only for $r=1$. Unfortunately, our semianalytical approach becomes numerically very expensive when applied to higher order perturbation theory. 7) Certainly it is very challenging to detect this difference in experiments. At the time being we have not focused on the dynamical consequences of this observation, but this is clearly an important and interesting avenue for future research. 9) This is a good point. We don't expect non-trivial effects (beyond the energy dependence of the mean in the ETH phase) arising from a change of energy density as long as the mobility edge is not crossed.
	How would I go about solving for the magnitude of the force the pivot exerts on the bar? Assuming the bar Is uniform in density. My idea was: since the system is in both translational and rotational equilibrium. $$T\sin\theta = F$$ (Pivot on Bar) Solving for $$T$$ let M = mass of bar let m = mass of block $$T\cos\thetal = Mg\frac{1}{2} + mg$$ $$T = \frac{Mg\frac{1}{2} + mg}{\cos\theta*l}$$ Then $$F$$ (Pivot on Bar) = $$T\sin\theta$$ Although when I attempt the examples I don't receive the right answer. thanks for any help out there. • Which member, horizontal or vertical, member is the “bar” – Bob D Oct 2 '19 at 3:38 • @BobD the horizontal bar holding the block mass – Bdyce Oct 2 '19 at 3:39 • Ok. Have you taken a class in statics? This is a statics problem – Bob D Oct 2 '19 at 4:34 • @BobD I am taking that class right now, I just can't seem to find the solution for this. – Bdyce Oct 2 '19 at 4:35 • OK, see my answer – Bob D Oct 2 '19 at 4:45
	# Velocity in quantum mechanics 1. Jul 16, 2010 ### orienst I want to know what does velocity really mean in quantum mechanics. Since the particle doesn’t have exact position, how can we talk about the velocity and momentum? 2. Jul 16, 2010 ### tom.stoer One can have a "particle" in a momentum eigenstate $$\hat{p}\|\psi\rangle = p\|\psi\rangle$$ Of course one can define a velocity operator $$\hat{v} = \frac{\hat{p}}{m}$$ And the above mentioned eigenstate will be a velocity eigenstate as well: $$\hat{v}\|\psi\rangle = v\|\psi\rangle =$$ with $$v = \frac{p}{m}$$ If the state is not an eigenstate of the momentum operator the particle will not have an exact velocity; instead one has to use the expectation value $$\langle v \rangle_\psi = \langle\psi\|\hat{v}\|\psi\rangle$$ And of course one can write down an uncertainty relation for the velocity $$\Delta x\; \Delta p \ge \frac{\hbar}{2}\;\; \to \;\; \Delta x\; \Delta v \ge \frac{\hbar}{2m}$$ Last edited: Jul 16, 2010 3. Jul 16, 2010 ### orienst If a "particle" in a momentum eigenstate, then according to the uncertain principal, the particle’s position has the biggest uncertainty. It may appear everywhere. Then you say that the particle has an exact velocity… I can’t understand that. 4. Jul 16, 2010 ### DrDu In general the velocity operator is $$\frac{i}{\hbar}[H,x]$$, where H is the hamiltonian. This is the analog of the classical Poisson bracket formula for x dot. Even if the position x of a particle is indetermined, the correlation function of finding the partile at position x+d at time t had it been at position x at 0 may have a sharp value. Then d/t yields the velocity of the particle. 5. Jul 16, 2010 ### tom.stoer In most cases your definition coincides with mine as the commutator will just produce a p from the p² in H; yours is more general than mine. 6. Jul 16, 2010 ### tom.stoer You cannot understand it. Nobody does. You can only apply the formalism. Niels Bohr said, "Anyone who is not shocked by quantum theory has not understood it." 7. Jul 16, 2010 ### DrDu If I tell you that my home is 8 km from my work, and that it takes me 30 min to go there you can calculate my speed although you have no clue where I am located. Have a nice weekend guys! 8. Jul 16, 2010 ### tom.stoer But according to QM you can be both at home and in your office at the same time and with the same probability, even so you are moving forward with constant velocity :-) 9. Jul 16, 2010 ### nnnm4 No QM says that when an ensemble of similarly prepared systems are measured, different values of the measurement will occur with different probabilities. 10. Jul 16, 2010 ### tom.stoer This is the ensemble interpretation; there are others as well. 11. Jul 16, 2010 ### reilly Not so. There is absolutely no evidence that I, or you, or most anything can be in two or more distinct places at the same time. The joint probability that I can be in Seattle and Chicago at the same time is zero; any theory that describes Nature must honor this fundamental property of things. In fact,the very structure of QM, and classical dynamics as well, requires that a dynamical variable can have one and only one value per measurement. So, assertions that QM says we can be in two places at the same time are at best confusing and confused. Regards, Reilly Atkinson 12. Jul 16, 2010 ### akhmeteli The Dirac equation presents an important example where these two definitions do not coincide 13. Jul 16, 2010 ### orienst You are calculating the velocity using the classical mechanics. I don't think you can use macroscopic objects to explain the microscopic phenomenon. 14. Jul 17, 2010 ### sweet springs Hi, orienst. I share inquiry with you because the definition of speed or velocity needs position measurements, i.e. definition of speed is v(t)= {x(t+dt) - x(t)}/dt as we have learned. How we can get value of v(t) without measuring x(t) and x(t+dt)? Does QM apply another definition of v(t)? Regards. 15. Jul 17, 2010 ### sweet springs Hi, tom-stoer Following my previous post, can you measure momentum without position measurement? For example when we use magnetic field to measure momentum by bending the trajectory of charged particle, we need to know where the particle beam pass through. If someone show me the case that completely no position information is necessary for momentum measurement, I appreciate it very much. Regards. 16. Jul 17, 2010 ### tom.stoer This is fundamentally wrong. If you look at the double-slit experiment and try to explain what's going one when one single photon goes through the slits the formalism (e.g. the path integral) tells you that the only interpretation consistent with reality (with the measurement of interference pattern) is that the photon goes through both slits and that these two (classically mutually inconsistent) paths lead to interference. Of course you can insist on the fact that talking about photons and humans is different. I agree that you are not a quantum object and that the same reasoning does not apply w/o modifications. But the same is true the other way round. The reasoning that because you have never been in two places at the same time does not mean that this must be true for photons. You can also avoid to talk about any interpretation at all. Nobody urges you to talk about where the photon really "is" in the double slit experiment. It is enough to calculate the amplitudes w/o thinking about any interpretation. That's Ok. But as soon as you start to think about an interpretation you get into these weird reasonings. No way out. 17. Jul 17, 2010 ### tom.stoer I agree. I only had the feeling that orienst wanted to have a rather elementary explanation. Of course your reasoning is perfectly valid but in most cases the commutator [H, x] does nothing more but giving you a p from the p² in H. orienst's problem does exist already for the simplest case H= p²/2m. 18. Jul 17, 2010 ### tom.stoer The problem in qm is that you can neither measure nor assign values to two non-commuting observables at the same time. But you can for communiting observables. If you set up a measurement with a magnetic field you do not need the bending of the trajectory. It is enough to measure the x-position of the particle in order to calculate the y-momentum. In addition if you have a free particle w/o external forces, the non.-rel. Hamiltonian is just H=p²/2m. So measuring energy is the same as measuring momentum. If you want to have a detailed experimental setup I can't tell you more. I am focussed o theoretical physics :-( 19. Jul 17, 2010 ### Naty1 I want to know what it "really" means in classical mechanics....the best we can say is that, for example, d = vt; but nobody knows what space (distance) "really" is nor, especially, what time "really" is. So how can anyone "really" understand velocity?? Everybody thought the answer was 'obvious' until Einstein. So the best we can do so far, after several thousand years of scientific progress, is explain what we observe with mathematics; that's gotten us a good way but we likely have an even longer way to go. 20. Jul 17, 2010 ### DrDu I just wanted to point out that for the determination of a distance it is not necessary to measure two positions. Consider e.g. an atom which travels in a constant potential gradient. Now it gets ionized at t1 and takes up an electron at t2. I could measure somehow the potential difference of the two electrons which is proportional to the distance of the two events but contains no information about the absolute position of the atom.
	# Kerala Syllabus 9th Standard Physics Solutions Chapter 3 Motion and Laws of Motion You can Download Motion and Laws of Motion Questions and Answers, Summary, Activity, Notes, Kerala Syllabus 9th Standard Physics Solutions Part 1 Chapter 3 help you to revise complete Syllabus and score more marks in your examinations. ## Kerala State Syllabus 9th Standard Physics Solutions Chapter 3 Motion and Laws of Motion ### Motion and Laws of Motion Textual Questions and Answers Question 1. What will be the result when a man tries to move a vehicle by pushing it, standing inside the vehicle? The vehicle moves/the vehicle doesn’t move. The vehicle doesn’t move. Question 2. What if the same vehicle is pushed from outside. Yes, the vehicle moves Internal forces can not move an object, but only an external unbalanced force can cause motion. Newton’S First Law Of Motion Every object continues in its state of rest or of uniform motion along a straight line unless an unbalanced external force acts on it. This is Newton’s First law of motion. Question 3. What is inertia of rest? Inertia of rest is the tendency of a body to remain in its state of rest or its inability to change its state of rest by itself. Question 4. Write down an activity to show inertia of rest. Place a bottle filled with water on a thick rough pa¬per as shown. Pull the paper suddenly to one side. The bottle falls in the opposite direction fo the motion of paper. This is because of inertia of rest. Question 5. What is inertia of motion? Inertia of motion is defined as the inability of a body to change it state of motion by itself. Question 6. Write down an activity to show inertia of motion. Place a bottle on a thick paper with rough surface. Bring the bottle into motion by pulling the paper slowly. Gradually increase the speed of pulling stop pulling when the bottle gains a certain speed. The bottle falls in the direction of motion of the pa-per, due to inertia of motion of the bottle. Question 7. Find out reasons for the Situations a) Place some carom board coins in a pile. Using the striker, strike out the coin at the bottom. What do you observe? What is the reason? b) When a running bus is suddenly stopped, passengers, standing in the bus show a tendency to fall forward. c) Place a small brick on a plank. When the plank is pulled suddenly the brick remains in the same position as before. d) When a bus moves forward suddenly from rest, the standing passengers tend to fall backward. e) Accidents that happen to passengers who do not wear seat belts are more fatal. a) Only the coin at the bottom is thrown away. Others will remain in the previous stage. b) The passengers start to move forward due to the tendency to continue in its state of motion. c) It is due to the tendency to continue in the state of rest. d) The passengers tend to fall backward due to the tendency to continue in their state of rest. e) The passengers inside the vehicle have the tendency to move forward due to inertia of motion. Inertia: The inability of a body to change its state of rest or of uniform motion along a straight line by its itself. Inertia of rest: Inertia of rest is defined as the inability of a body to change its state of rest by itself. Inertia of motion : Inertia of motion is defined as the inability of a body to change its state of motion by itself. Eg. When a running bus is suddenly stopped, the standing passengers fall forward due to inertia of motion. When a bus at rest starts suddenly, the standing passengers fall backward due to inertia of rest. Question 8. Expand the table by finding more situations from daily life. Inertia of rest Inertia of motion 1. When the branch of a mango tree is shaken mangoes fall. 1. A running athlete cannot stop himself abruptly at the finishing point. 2. When the carpet is trapped, dust particle scatters. 2. A man stepping down from a slowly moving bus stops after few steps of running. 3. When the coin on a Cardboard placed over a glass is struck, it falls into the glass 3. In hammer throw, before the hammer is let go off, it is whirled along a circular path. Question 9. Find out the reasons a) An athlete doing a long jump-start his run from a distance. b) A running elephant cannot change its direction suddenly. a) This activity helps to cover long distances by utilizing inertia of motion. b) Mass of elephant is greater. So inertia of motion also be greater. Also, it cannot be able to change its direction suddenly. Mass And Inertia Question 10. It is dangerous for loaded vehicles to negotiate a curve in the road without reducing speed. What is the reason? Loaded vehicles possess more inertia of motion. As ” mass increases, inertia also increases. Question 11. It is more difficult to roll a filled tar drum than an empty drum. a) Which of two has a greater mass? b) Which has greater inertia? a) The mass of drum filled with tar will be greater, b) Inertia will be greater to tar filled drum Conclusion: Inertia depends on mass. When the mass increases inertia also increases. When the mass decreases inertia also decreases. Question 12. If a tennis ball (mass 58.5 g) and a cricket ball (mass 163 g) are the reach a certain distance when hit with a cricket bat, which is to be hit with greater force? The tennis ball / the cricket ball? The cricket ball Question 13. Will the change of velocity be the same in both the cases? Velocities are different in both cases due to the difference in masses. The inertia of an object depends upon its mass. When the mass increases, inertia also increases. Momentum Momentum is a characteristic property of moving objects. It is measured as the product of the mass of the body and its velocity. Momentum = mass × velocity Unit of momentum is kg m/s Question 14. A car of 1000 kg moves with a velocity of 10 m/s. On applying brakes it comes to rest in 5s. Then what are its initial momentum and final momentum? m = 1000 kg u = 10 m/s v = 0 t = 5s Initial momentum = mu = 1000 × 10 =10000 kg m/s Final momentum = mV = 1000 × 0 = 0 Kg m/s Question 15. A hockey ball of mass 200 g hits on a hockey stick with a velocity 10 m/s. Calculate the change in momentum if the ball bounces back on the same path with the same speed. m = 200g = 200/1000 = 0.2 kg Initial momentum = mu = 0.2 × 10 = 2 kg m/s Final momentum = mv = 0.2 × 70 = 2 kgm/s change in momentum = mv – mv = – 2 – 2 = -4 kg m/s Question 16. A loaded lorry of mass 12000 kg moves with a velocity of 12 m/s. Its velocity becomes 10 m/s after 5 s. a) What is the initial momentum and what is the final momentum? b) What is the change in momentum? c) What is the rate of change of momentum? a) Initial momentum = mu =12000 × 12 = 144000 kg m/s Final momentum = mv =12000 × 10 = 120000 kg m/s b) Change in momentum = mv – mu = 120000 – 144000 = –24000 kg m/s c) Rate of change in momentum = $$\frac { mv – mu }{ t }$$ = $$\frac { -24000 }{ 5 }$$ = – 4800 kgm/s2 Newton’S Second Law Of Motion The rate of change of momentum of a body is directly proportional to the unbalanced external force acting on it. Equation for Calculating Force: According to second law of motion F ∝ ma F = kma (k – a constant) k = 1 ∴ F = 1 × ma F = ma, ie. F- Force, m – mass, a – acceleration. Unit of force is Newton (N) Another unit is dyne. Question 17. A constant force is applied for 2 s on a body of mass 5 kg. As a result, if the velocity of the body is changed from 3 m/s to 7 m/s, find out the value of the applied force. Question 18. A car moving with a speed of 108 km/h comes to rest after 4s on applying brake, if the mass of the car including the passengers is 1000 kg, what will be the force applied when brake is applied? Initial velocity of the car u = 108 km/h = 30 m/s Final velocity v = 0 Mass m = 1000 kg Time t = 4s According to newton’s second law F = ma = – 7500 N The negative sign indicates that the applied force is opposite to the direction of motion. Question 19. Velocity of an object of mass 5 kg increases from 3 m/s to 7 m/s on applying a force continuously for 2s. Find out the force applied. If the duration for which force acts is extended to 5s, what will be the velocity of the object then? u = 3 m/s v = 7 m/s t = 2s m = 5 kg According to Second Law of Motion F = ma If we substitute the values in the equation v = u + at, velocity can be calcualted when the time for force is extended to 5s. v = 3 + (2 × 5) = 13 m/s Question 20. Velocity-time graph of an object of mass 20 g, moving along the surface of a long table is given below. What is the frictional force experienced by the object? From the graph Initial velocity u = 20 m/s Final velocity v = 0m/s t = 10 s m = 20 g = 20/100 kg F = ma = -0.04 N The negative sign shows that the frictional force is acting opposite to the direction of motion of the object. Question 21. m1 and m2 are the masses of two bodies. When a force of 5 N is applied on each body, m1 gets an acceleration of 10 m/s2 and m2, 20 m/s2. If the two bodies are tied together and the same force is applied, find the acceleration of the combined system. Impulse Impulsive force is a very large force acting for a very short time. Impulse of force is the product of the force and the time. Impulse = Force × time Impulse – Momentum Principle According to Newton’s second law of motion, $$F=\frac{m(v-u)}{t}$$ F x t = m(v – u) F × t= mv – mu ie. impulse = change Is momentum This is known as impulse-momentum principle. It states that a change in momentum of an object is equal to the impulse experienced by it. Question 22. Explain the following situation by relating force and time. When the change in momentum is a constant, the force acting on a body will be inversely proportional to the time taken. As time increase, the force acting decreases and as time decreases, the force acting increases. Question 23. During a pole vault jump, the impact is reduced by falling on a foam bed. When the change in momentum is a constant, the force acting on a body will be inversely proportional to the time taken. As time increase, the force acting decreases and as time decreases, the force acting increases. Question 24. Hay or sponges are used while packing glassware. This helps to avoid breaking of glasswares due to collision. When the change in momentum is a constant, the force acting on a body will be inversely proportional to the time taken. As time increase, the force acting decreases and as time decreases, the force acting increases. Question 25. Karate experts move their hands with great speed to chop strong bricks. When the change in momentum is a constant, the force acting on a body will be inversely proportional to the time taken. As time increase, the force acting decreases and as time decreases, the force acting increases. Newton’S Third Law Of Motion Pass a long string through a straw and tie the string between two windows of the classroom Paste an inflated balloon on the straw. Question 26. Inflate a balloon and release it suddenly. What happens? Release of air causes the balloon to move in the opposite direction. Releasing air from the balloon is action and the movement of the balloon is reaction. Question 27. What do you observe? The cork pushed out due to the pressure of steam. The tube moves backward. Question 28. If the action is the cork being pushed out due to the force exerted by the steam on it, what is the reaction? The reaction is the backward movement of boiling tube. Question 29. Write down the action and reaction while we are walking on a floor? When we are walking through a floor, we applies a force on the floor. This is action. The floor applies a force in the opposite direction. This is reaction. Rocket Propulsion: • Escaping of hot gases from the jet of rocket is action. • The force exerted by these gases on the rocket is reaction. Question 30. Are the action and reaction equal and opposite? Action and reaction are equal and opposite. Newton’s Third law of Motion For every action there is an equal and opposite reaction. Question 31. Examine the following situations and complete the table. Situation Action (FJ Reaction (FJ 1. A man jumps from a boat to ’ the shore. The man exerts a force on the boat. The boat moves backward. The boat exerts an equal force on the man. The man lands on the shore. 2. A bullet is fired from a gun the bullet exerts a force on the gun So it moves backward The gun exerts an equal force to the bullet, so it moves forward 3. A boat is rowed. The man applied force on the water The boat moves forward. Conclusions: • As a result of the applied force by a second body to a body, reaction will occur at the second body. • Action and reaction are equal and opposite. • Since action and reaction takes place in two different bodies, they do not cancel each other. Law Of Conservation Of Momentum Observe the figure and answer the following questions. Question 32. Move the first marble slightly back and roll forward. What happens? One marble will be thrown off from the other end and reaches the previous position. Question 33. Bring the two marbles into contact and let them roll. What happens? Two marbles will move from the other end. Observe the figure and answer the following questions. Question 34. Total momentum before collision = …………. m1 u1 + m2 u2. Question 35. Total momentum after collision =……. m1 v1 + m2 v2 Question 36. Initial momentum of A= m1 u1 Question 37. Final momentum of A = m1 v1 Question 38. Change in momentum of A= m1 v1 – m1 u1 Question 39. Rate of change of momentum of A = $$\frac{m_{1} v_{1}-m_{1} u_{1}}{t}$$ Question 40. Initial momentum of B =…… m2 u2 Question 41. Final momentum of = ……………. m2 v2 Question 42. Change in momentum of B = ……… m2 v2 – m2 u2. Question 43. Rate of change of momentum of B = …………… $$\frac{\mathrm{m}_{2} \mathrm{v}_{2}-\mathrm{m}_{2} \mathrm{u}_{2}}{\mathrm{t}}$$ According to IInd Law of motion, rate of change of momentum is directly proportional to the external force. Force exerted by B on A Law Of Conservation Of Momentum In the absence of an external force, the total momentum of a system is a constant. Question 44. A bullet is fired with a velocity v from a gun of mass M. What will be the recoil velocity of the gun if the mass of the bullet is m? According to low of Conservation of Momentum, total momentum ofthe gun and the bullet before firing and their total momentum after firing will be equal, ie. Total momentum before firing = 0 + 0 =0 Total momentum after firing = MV + mv Accoriding to Law of Conservation of Momentum 0 =MV + mv MV = -mv V = $$\frac { -mv }{ M }$$ The recoil velocity ofthe gun V = $$\frac { -mv }{ M }$$. The negative sign indicates that the gun moves in the opposite direction of motion ofthe bullet. Question 45. Suppose a child of mass 40 kg running on a horizontal surface with a velocity of 5m/s jumps on a stationary skating board of mass3 kg while running as shown in the figure. If there is no other force acting horizontally (assuming the frictional force on the wheels to be zero), calculate the velocity of the combined system of child and the skating board. Suppose the velocity of the board while moving is u. Total momentum of the child and the skating board before jumping will be = 40 kg × 5 m/s + 3 kg × 0 m/s = 200 kg m/s Total momentum when the system starts moving (body and skating board) = (40 + 3) kg × u m/s = 43 × u kgm/s. According to the Law of Conservation of Momentum, Total momentum remains constant. 43 u = 200 u = 200/43 = + 4.65 m/s Circular Motion Motion of a body through a circular path is known as circular motion. Question 46. Does the velocity of an object moving with a uniform speed along a circular path change? Yes Question 47. How does this change in velocity happen? Due to change in speed/change in direction/due to change in both speed and direction. Due to change in direction. • Whirling of a stone tied to a string is a type of circular motion. • The force we apply form the center of the circle reaches the object through the string. The acceleration which a body in circular motion experiences towards the center of the circle through the radius is centripetal acceleration. The force that creates centripetal acceleration is a centripetal force. Centripetal force (Fc) = mv2/t m – mass, v-velocity, r- radius of the circular path • In the absence of centripetal force, circular moving body thrown off through the tangent. • If a body moving along a circular path covers equal distances in equal intervals of time, it is said to be in uniform circular motion. Question 48. In hammer throw, before the hammer is let go off, why is it whirled around along a circular path? It is to get initial momentum. Also helpful to cover long distances through the tangent. Question 49. How does the speed of a giant wheel in an amusement park? The motion of giant wheel is controlled by mechanically. So its speed is uniform except when starting and stopping. Examples for uniform circular motion: • Motion of needles in a watch. • Whirling of a stone tied to a string. • Movement of the leaf of an electric fan except when starting and stopping. • Circular motion of earth’s artificial /satellites. The acceleration experienced by an object in a circular motion, along the radius, towards the center of the circle, is known as centripetal acceleration. The force that creates a centripetal acceleration is called centripetal force. Centripetal acceleration and centripetal force are directed towards the center. If an object moving along a circular path covers equal distances in equal intervals of time, it is said to be in uniform circular motion. Let Us Assess Question 1. Observe the figures given below. Answer the following questions. a) When the card is suddenly struck off, what happens to the coin? Explain. b) What is the law to which this property is related? c) How is this property related to the mass of the object? a) The coin falls into the glass due to inertia of rest. b) Newton’s 1st law of motion. c) Mass increases inertia increases. Question 2. What are the balanced forces acting on a book at rest on a table? a) Downward force exerts by the book on the table (i.e., weight of the book). b) Upward force applies by the table on the book (reaction). Question 3. To remove the dust from a carpet, it is suspended and hit with a stick. What is the scientific principle behind it? Inertia of rest. The dust in the carpet shows the tendency to continue in its state of rest. Question 4. Acar and a bus are traveling with the same velocity. Which has greater momentum? Why? Bus. Because when mass increases momentum increases. Question 5. On the basis of Newton’s third law of motion, explain the source of force that helps to propel a rocket upward. • It is due to the reaction force by the escaping gas from rocket. • Escaping gases form the rocket is action. • The force exerted by escaping gases on the rocket is reaction. Question 6. A car-travels with a velocity of 15 m/s. The total mass of the car and the passengers in it is 1000 kg. Find the momentum of the car. Momentum p = mv = 1000 × 15 = 15000 kg m/s Question 7. Give reasons: a) When a bullet is fired from a gun, the gun recoils. b) When a bus at rest suddenly moves forward, the passengers, standing in the bus, fall backward. c) We slip on a mossy surface. a) The gun recoils due to the reaction force applied by the shot to the gun. Forward movement of the shot is action and the Backward movement of the gun is reaction. b) Inertia of rest is the reason. The passengers tends continue in state of rest. c) Absence of reaction force is the cause for this. Extended Activities Question 1. Prepare and present an experiment to illustrate inertia of rest. Make a pile of coins on a table. Strikes off the lowest coin by a knife quickly. Only that particular coin thrown off and the others remains in the previous manner. This is due to the tendency of coins to remain in its state of rest. Question 2. Find out situations from our daily life to explain the law of conservation of momentum and note them down. • Recoil of a gun during firing. • Rocket propulsion. • Bomb explosion. ### Motion and Laws of Motion More Questions and Answers Question 1. Fill in the blanks. a) As the time interval decreases, rate of change of momentum b) The force required to produce an acceleration of 1 m/s2 on a body of mass 1 kg is a) Increases b) 1 Newton (1N) Question 2. Correct the statement if any wrong. An object moving with uniform speed along a circular path undergoes velocity change due to the change in speed. An object moving with a uniform speed along a circular path undergoes velocity change due to change in direction. Question 4. a) What is meant by momentum? b) Write down the equation for calculating momentum. c) A body of mass 50 kg starts from rest. If its velocity changes to 15 m/s after 10 seconds, calculate the change in momentum? What will be the rate of change of momentum? a) The measurement of the quantity of motion of a body is called momentum b) Momentum p = mv Question 5. a) Define circular motion and uniform circular motion. b) What is the relation between centripetal force and centripetal acceleration? a) The motion of an object through a circular path is said to be circular motion. If an object moving along a circular path covers equal distance in equal intervals of time if is said to be uniform circular motion. b) The force requires to produce centripetal acceleration is centripetal force. Question 6. a) State Newton’s II law of motion? b) Derive the equation for force on the basis of this law. a) The rate of change of momentum of a body is directly proportional to the unbalanced external force acting on it. b) According to II law. F = kma (k – a constant) The force required to produce an acceleration of 1 m/s2 on a body of mass 1 kg is 1N 1 = k × 1 × 1 k = 1 ∴ F = 1 × ma F = ma Question 7. A car of mass 1000 kg runs with a velocity of 10 m/s. What is the momentum of this car? Momentum p = mv m = 1000 kg v = 10 m/s ∴ P = 1000 × 10 = 10000 kg m/s Question 8. A loaded lorry of mass 1500 kg moves with a velocity of 12 m/s. Within a small interval of the time, the velocity becomes 10 m/s. a) What is the initial momentum of the lorry? b) What is its final momentum? c) What is the change in momentum? = $$\frac { change in momentum }{ Time }$$ = $$\frac { m(v – u) }{ t }$$
	## Mean curvature flow of contractions between Euclidean spaces Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt We consider the mean curvature flow of the graphs of maps between Euclidean spaces of arbitrary dimension. If the initial map is Lipschitz and satisfies a length-decreasing condition, we show that the mean curvature flow has a smooth long-time solution for t> 0. Further, we prove uniform decay estimates for the mean curvature vector and for all higher-order derivatives of the defining map. Originalsprog Engelsk 104 Calculus of Variations and Partial Differential Equations 55 4 0944-2669 https://doi.org/10.1007/s00526-016-1043-2 Udgivet - 1 aug. 2016 Ja ID: 233725687
	# Start a new discussion ## Not signed in Want to take part in these discussions? Sign in if you have an account, or apply for one below ## Site Tag Cloud Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support. • CommentRowNumber1. • CommentAuthorAndrew Stacey • CommentTimeMay 27th 2010 I got the book “Counterexamples in Topological Vector Spaces” out of our library, and just the sheer number of them made me realise that my goal of getting the poset of properties to be a lattice would produce a horrendous diagram. So I’ve gone for a more modest aim, that of trying to convey a little more information than the original diagram. Unfortunately, the nLab isn’t displaying the current diagram, though the original one displays just fine and on my own instiki installation then it also displays just fine so I’m not sure what’s going on there. Until I figure that out, you can see it here. The source code is in the nLab: second lctvs diagram dot source. A little explanation of the design: 1. Abbreviate all the nodes to make the diagram more compact (with a key by the side, and tooltips to display the proper title). 2. Added some properties: LF spaces, LB spaces, Ptak spaces, $B_r$ spaces 3. Taken out some properties: I took out those that seemed “merely” topological in flavour: paracompactness, separable, normal. I’m pondering taking out completeness and sequential completeness as well. 4. Tried to classify the different properties. I picked three main categories: Size, Completeness, Duality. By “Size”, I mean “How close to a Banach space?”. (It seems that Instiki’s SVG support has … temporarily … broken. I’ll email Jacques.) • CommentRowNumber2. • CommentAuthorTim_van_Beek • CommentTimeMay 27th 2010 Taken out some properties: I took out those that seemed “merely” topological in flavour: paracompactness, separable, normal. I’m pondering taking out completeness and sequential completeness as well. But there are relationships between TVS-properties and purly topological properties, like “sequentially complete implies locally complete” etc. These are still interesting (but could be covered by another diagram of course). Tried to classify the different properties. I picked three main categories: Size, Completeness, Duality. By “Size”, I mean “How close to a Banach space?”. Does this correspond to the colors of the boxes? I have to admit that I have no clear understanding of what these categories mean and why the spaces get the colors they do, but that’s probably my fault. This is the level that I am stuck at: Duality = definition uses the dual space? Size = percentage of my conjectures that will turn out to be false if I think of $\mathbb{R}^n$ instead of the space at hand? • CommentRowNumber3. • CommentAuthorTim_van_Beek • CommentTimeMay 27th 2010 • (edited May 27th 2010) I got the book “Counterexamples in Topological Vector Spaces” out of our library, and just the sheer number of them made me realise that my goal of getting the poset of properties to be a lattice would produce a horrendous diagram. It is a subject that is both vast and deep, despite the small and simple set of axioms it starts with. Helmut Schäfer, when lecturing a TVS class, was asked to provide educational objectives for each lecture, like “after this lecture you should understand this and that and be able to do this and that”. He did that only once and said (translated by me): “The educational objectives of this lecture is that you understand in full clarity the material presented here. But it was developed over the course of several decades with the help of several of the most prominent mathematicians of their time, and experience shows that a full understanding when learning the material for the first time is a tall order”. • CommentRowNumber4. • CommentAuthorAndrew Stacey • CommentTimeMay 27th 2010 I agree that there are (significant) relationships between topological and TVS properties, but the diagram risks being very crowded and it seemed a reasonable place to draw the line. As an example, consider completeness. I’d be very interested to know if a result in FA actually uses honest completeness, as opposed to one of the more TVS notions of quasi-completeness or local completeness (the question is a little confused by the fact that for so many spaces, the notions coincide). Thus although completeness is useful, its main use is in establishing a weaker form of completeness. Let me elaborate a little more on the classification of properties (you’re right, it is by colour): Size The idea here is that, following the general theme of “probes” and “coprobes”, one tries to examine ones space by mapping to and from normed/Banach spaces. All of the properties with this colour are things that can be tested by maps to or from normed/Banach spaces, more or less (not sure about DF-spaces). So if you replace $\mathbb{R}^n$ by “a Banach space” then your slogan for size is right. Duality All the properties with this colour are primarily properties about the dual. “Reflexivity” is the obvious one, but barrelled is as well since it is equivalent to certain families of sets in the dual being the same. Also, these properties are primarily used for making statements about the dual (or about mapping spaces) and not really about the space itself. Is that a bit clearer? I like the Schaefer quote (and apologise for the lack of accents). • CommentRowNumber5. • CommentAuthorTim_van_Beek • CommentTimeMay 27th 2010 Is that a bit clearer? Definitly, so “size” is something that I would like to call Banach-(co)detectable, or (co-) probable. With regard to duality: Not sure about the “these properties are primarily used for making statements about the dual (or about mapping spaces)” - part. That seems fuzzy. I like the Schaefer quote… The part of the story that I did not like was that he did not provide educational objectives, but in hindsight it was a very interesting experience: For an expert it is hard to figure out what is difficult for a newbie and what isn’t and what should be emphazised and what should not. • CommentRowNumber6. • CommentAuthorAndrew Stacey • CommentTimeMay 27th 2010 Jacques figured out what was wrong with my diagrams - there was a linefeed that some part of Instiki (Jacques isn’t sure what yet) didn’t like. Removing that made the original diagram reappear and the new one appear. So it is now in the nLab in all its glory at the link in the first comment. Back on track. I agree with all your remarks. If you can come up with a better way to phrase the “duality” part then please do! My reason for that grouping was a section in Schaefer’s book, just before the part on (semi-)reflexivity where he talks about various families of subsets in the dual. Off the top of my head, I think that they are equicontinuous, precompact, bounded, and weakly bounded. Barrelled means that all these are the same, infrabarrelled means that equicontinuous equals bounded, and so on. So they are all about certain families in the dual space.
	# Can the argument of a logarithm and its base be negative at the same time? I'm struggling to understand why a $\log_x y$ is only true for $y > 0$. I know that if $y$ was $0$ , $x$ could only be $0$, but if it was $0$, then $y$ could be anything. So we'll leave that undefined. I also know that, for example, $\log_2 -x$ doesn't make sense, because $2^a = -x$ has no real solution. However, I just cannot understand why $\log_{-2} -8$ doesnt' make sense. The argument is negative, but it works, it's 3. So why do we say that the argument of a logarithm cannot be negative? And also, do you confirm that the base of a logarithm can be anything but $0$? I have seen people state that the base also cannot be negative, which to me seems absurd. • In principle it makes sense on integer powers of a negative base, but all the nice properties for real (or even rational) powers become a real mess! What is log_{-2} (-5) ? Any solution will be complex values and messy. In conclusion, better consider a positive base. – H. H. Rugh Oct 10 '16 at 16:39 • I know, but messy and hard doesn't mean impossible, I'm just trying to understand the underlying reason why log(-2)(-8) is not defined. When I look at the function graph, I struggle to understand why it's only defined in X > 0. The log of (-2)(-5) is 2.321928094887362 EDIT: It would be, but it's not, because log(-2)(-5) is not defined [WHY?!]. – Athamas Oct 10 '16 at 16:43 • $(-b)^{odd} = neg$ and $(-b)^{even} = pos$ and $(-b)^{n/2}= \sqrt {(-b)^n}$ is undefined. $\log_{-2}(-5)$ is undefined because $(-2)^x = -5$ has no solution. – fleablood Oct 10 '16 at 16:52 • Sir, -2^2.321928094887362 is -5, so it is defined. What am I doing wrong? (−b)^(n/2) = sqrt((-b)^n) is undefined only if (-b)^n < 0. Not to be confued with -(b^n). – Athamas Oct 10 '16 at 16:57 This difficulty in the definition of the logarithm arises due to the way exponentiation works. If we allow solutions only in real numbers, not complex numbers, it is possible to define some non-integer powers of negative numbers. Specifically, the rule is, For any real number $a$, for any integers $m$ and $n$ such that $n>1$ and $m$ and $n$ have no common factor, if $\sqrt[n]a$ is a real number then $$a^{m/n} = \left(\sqrt[n]a\right)^m.$$ You'll find this rule, or rules that imply it, in various places, including here, here, the Wikipedia page on Exponentiation, or any number of high-school algebra textbooks. This rule depends on the assumption that we know when $\sqrt[n]a$ is defined (for an integer $n$) and what it is when it is defined. When $n$ is odd, $\sqrt[n]a$ is the unique real number $x$ such that $x^n = a$. But when $n$ is even, $\sqrt[n]a$ is defined only when $a \geq 0$, and it is defined then as the unique non-negative real number $x$ such that $x^n = a$. We cannot define $(-4)^{1/2}$ in real numbers, because there is no real number $x$ such that $x^2 = -4$. There is not even a real number $x$ whose square is close to $-4$; all the squares of real numbers are zero or positive. Now consider irrational exponents. We can define irrational powers of real numbers by assuming $a^x$ is a continuous function of $x$ when $a$ is positive. We can do this because the rational powers of positive real numbers "fit the curve" of a continuous function; if $p_1, p_2, p_3, \ldots$ is any sequence of rational numbers converging to a certain rational number $p$, then $2^{p_1}, 2^{p_2}, 2^{p_3}, \ldots$ is a sequence of real numbers converging to $2^p$. To extend this to irrational exponents, we define $2^\pi$ (for example) as the limit of $2^{p_1}, 2^{p_2}, 2^{p_3}, \ldots$ where $p_1, p_2, p_3, \ldots$ converges to $\pi$; and if for every sequence of rational numbers $q_1, q_2, q_3, \ldots$ that converges to a certain real number $r$, the powers $2^{q_1}, 2^{q_2}, 2^{q_3}, \ldots$ converge to $5$, then we say that $2^r = 5$. This works fine for defining $\log_2 5$, because there is a unique real number $r$ such that for every rational sequence $q_1, q_2, q_3, \ldots$ that converges to $r$, the sequence $2^{q_1}, 2^{q_2}, 2^{q_3}, \ldots$ converge to $5$. It does not work for $\log_{-2} (-5)$, however. The problem with $\log_{-2} (-5)$ (in particular, the reason it is not equal to $\log_2 5$) is that in any sequence of rational numbers $q_1, q_2, q_3, \ldots$ that converges to $\log_2 5$, it is possible that when reduced to lowest terms ($m/n$ where $m$ and $n$ have no common factor), the numerator of each $q_i$ might be positive or it might be negative. We can easily make a sequence where all the numerators are odd, in which case $(-2)^{q_1}, (-2)^{q_2}, (-2)^{q_3}, \ldots$ converges to $-5$, or a sequence where all the numerators are even, in which case $(-2)^{q_1}, (-2)^{q_2}, (-2)^{q_3}, \ldots$ converges to $5$, or a sequence in which the numerators alternate between odd and even, so $(-2)^{q_1}, (-2)^{q_2}, (-2)^{q_3}, \ldots$ alternates between $5$ and $-5$ and does not converge to anything. In short, it's really not justifiable to say that $(-2)^{\log_2 5} = -5$. There isn't a really good reason to say $(-2)^{\log_2 5}$ is a real number at all. Nor is there any better candidate to be the real number $x$ that solves the equation $(-2)^x = -5$. That leaves us without a good way to define $\log_{-2}(-5)$, nor the log base $-2$ of most other numbers. We end up not being able to use log base $-2$ for just about any of the things we find logarithms really useful for, so we don't even try to define it for the cases where it might possibly make sense. • So would you say that $\log_a b$ is defined for $a > 0$ and $b > 0$ OR for $b > 0$ and $a != 0$ ? Because text books often only say "The condition of existence is that the argument be positive". – Athamas Oct 10 '16 at 21:15 • No, that's not what I would say at all. I said there is not any good way to define the log base $-2$ of most numbers (which includes most positive numbers, if you follow the reasoning up to that point), and the same thing can be said about log base $a$ whenever $a < 0$. – David K Oct 11 '16 at 4:22 • Ok, but in the end one must come to a strict, precise conclusion, because the log might be part of a bigger equation, where one needs to define a precise field of existence. Am I to assume that $\log_a(b)$ has a > 0 and b > 0 when I outline the solution to an equation that contains a log? For example an equation might be verified for a = -3 and b = 6. If it contains a $\log_a(b)$ somewhere, the only actual solution is 6? – Athamas Oct 11 '16 at 15:36 • The conclusion in real analysis is that $\log_a$ is defined for $a > 0$ and only for $a>0$. I don't know what you mean by an equation that might be verified for $a=-3$ and $b=6$ that contains $\log_a b$ somewhere. An equation that uses $\log_a b$ cannot be satisfied by $a=-3$ and $b=6$. Moreover, when we rule out $a=-3,b=6$ as a solution, we rule out the entire solution--we don't just ignore the problem with the $a=-3$ part and say $b=6$ is a solution. – David K Oct 11 '16 at 16:20 • If you have a specific mathematical problem in mind that you think needs to be solved by logarithms with negative bases, I suggest you post it as a new question. The answer to this question is that logarithms with positive bases are extremely useful in real analysis while logarithms with negative bases are practically useless for real analysis, so it saves a lot of trouble if, when doing real analysis, we just define logarithms on positive bases and leave negative bases undefined. – David K Oct 11 '16 at 16:24 The usual definition of $\log_ab=x$ in $\mathbb{R}$ is that $x$ is the number such that $a^x=b$. This definition works well if $a$ and $b$ are positive numbers because: 1) For $a<0$ we can have situations as $\log_{-4}2=x$ that is $(-4)^x=2$ that is impossible in $\mathbb{R}$ 2) for $b<0$ we can have situations as $\log_{4}(-2)=x$ that is $(4)^x=-2$ that is impossible in $\mathbb{R}$ In some case, and for suitable values of $a$ and $b$ we can find a value for $x$ also in these cases, but a complete and coherent definition of the logarithm function also in these case can be done only on the field $\mathbb{C}$ where we have no limitations (but some trouble with multivalued functions). • If given one specific case when these conditions do not apply, what am I to do? Write the answer or say that since th argument is <0 the log is not defined? – Athamas Oct 10 '16 at 16:51 • The graph of $y=\log x$ has negative $y$ for $0<x<1$ because $a^y=b \Rightarrow a^{-y}=\frac{1}{b}$, but note that we have noting for $x<0$. For the other question: in these situation it is better to find the complex solutions and, eventually, take the real solution, if we are interested only on real values. – Emilio Novati Oct 10 '16 at 17:00 • You solve it without evoking logarithm If you have $(-8)^x = 4$ you do not evoke $\log_{-8} 4 = ???$. You do something else instead. $(-8)^x = (-1)^x 8^x = 4$ so if $-1^x = 1$ so can declare $x = \log_8 4 = 2/3$. As $(-1)^{2/3} = 1$ we can conclude $x = 2/3$ but in general we may not have be able to do so. Example $(-8)^x = -4$ will have not be solvable. – fleablood Oct 10 '16 at 17:01 • (-1)^x = 1? why? if x = 3, -1^3 = -1 – Athamas Oct 10 '16 at 17:06
	## Pieter Hofstra and Federico De Marchi Motivated by a desire to gain a better understanding of the dimension-by-dimension'' decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describe a suitable subcategory of Cat over which we can view the assignment C \|-> Mnd(C) as an indexed category; on this base category, there is a natural topology. Then we single out a class of monads which are well-behaved with respect to reindexing. The main result is now, that such monads form a stack. Using this, we can shed some light on the free strict $\omega$-category monad on globular sets and the free operad-with-contraction monad on the category of collections. Keywords: Descent theory, monads, globular sets 2000 MSC: 18C15, 18D10, 18D30 Theory and Applications of Categories, Vol. 16, 2006, No. 24, pp 668-699. http://www.tac.mta.ca/tac/volumes/16/24/16-24.dvi http://www.tac.mta.ca/tac/volumes/16/24/16-24.ps http://www.tac.mta.ca/tac/volumes/16/24/16-24.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/24/16-24.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/24/16-24.ps TAC Home
	# ASCII input to HMAC weaker than raw bytes? In various cryptographic programs I've written (most have been toy problems, but this current one is not) I have taken a slothful approach to performing a message digest (hash) of numerical input. I have been casting the numbers to strings, concatenating them, and then performing the hash. (Actually, this is a keyed hash, or HMAC, as described in RFC_2104.) Example: # I desire to hash a single bit with a sequence number. # bit is either 0 or 1 # sequence_number is some 64-bit number. # key is some 32-bit number/value used to create an HMAC. make_HMAC(key, str(bit) + str(sequence_number)) # make_HMAC works on longs as well, but would become ambiguous because: # make_HMAC(key, 0 + sequence_number) == make_HMAC(key, 1 + (sequence_number - 1)) I've been avoiding ambiguity, but I worry that I've weakened the strength of my hashing functionality in a subtle way. Does casting these values to a string cause the HMAC to become less secure in any way? If it matters, I'm using python 2.7 and the pycrypto library. - In one sense, no, encoding should not have an impact on security of HMAC. On the other hand, it could have an application dependent impact. Consider the following. '26' and 26 have the same HMAC. Now, assume your code receives a message, M, and an HMAC, MAC, and then does something like this if HMAC(M, private_key) == MAC: if isinstance(M, basestring):
	MAT244--2019F > Term Test 1 Problem 1 (main sitting) << < (2/3) > >> dengji18: #71 question1: shown in the attachment xuanzhong: $$M_{y}=1+6ye^{3x}$$ $$N_{x}=4ye^{2x}$$ $M_{y} ≠N_{x}$,it is not exact $$R_{2} =\frac{ M_{y} -N_{x}}{N}=\frac{1+2ye^{2x} }{1+2ye^{2x}}=1$$ $$μ=e^{∫R_{2}dx} =e^{∫1 dx} =e^x$$ Multiplying both sides by $\mu$, we get $$ye^x+3y^2e^{3x} +(e^x+2ye^{3x}) y^\prime=0$$ $$M_{y}^\prime=e^x+6ye^{3x}$$ $$N_{x}^\prime=e^x+6ye^{3x}$$ $M_{y}^\prime=N_{x}^\prime$,it is exact $$∃φ(x,y) such that\ φ_{x} =M^\prime,φ_{y} =N^\prime$$ $$φ(x,y)=∫{M^\prime dx}=∫{ye^x+3y^{2}e^{3x}dx}=ye^x+y^{2}e^{3x} +h(y)$$ $$φ_{y} =e^x+2ye^{3x} +h(y)^\prime=e^x+2ye^{3x}$$ Then $h(y)^\prime=0$ Hence h(y)=c $$φ(x,y)=ye^x+y^{2}e^{3x} =c$$ Since y(0)=1 $$1⋅e^0+1^2⋅e^0=2=c$$ $$φ(x,y)=ye^x+y^{2}e^{3x} =2$$ Yuying Chen: $\text{(a)}\\$ $M=y+3y^2e^{2x}\qquad M_{y}=\frac{\partial}{\partial y}M=1+6ye^{2x}\\$ $N=1+2ye^{2x}\quad\quad N_{x}=\frac{\partial}{\partial x}N=4ye^{2x}\\$ $\text{Since$M_{y}\neq N_{x}$, the given differential equation is not exact.}\\$ $R_2=\frac{M_y-N_x}{N}=\frac{1+6ye^{2x}-4ye^{2x}}{1+2ye^{2x}}=\frac{1+2ye^{2x}}{1+2ye^{2x}}=1\\$ $\mu=e^{\int R_2dx}=e^{\int1dx}=e^x\\$ $(e^{x}y+3y^2e^{3x})+(e^x+2ye^{3x})y^{\prime}=0\\ \\$ $\text{$\exists \psi{(x,y)}$such that$\psi_{x}=M$}\\$ $\qquad\quad\psi{(x,y)}=\int {(e^{x}y+3y^2e^{3x})dx}\\$ $\qquad\qquad\qquad =e^xy+y^2e^{3x}+h(y)\\$ $\qquad\quad\psi_{y}=e^x=2ye^{3x}+h^{\prime}(y)=N\\$ $\qquad\quad h^{\prime}(y)=0\\$ $\qquad\quad h(y)=C\\$ $\text{and we have}\\$ $\qquad\quad\psi{(x,y)}=e^xy+y^2e^{3x}=C\\$ $\text{(b)}\\$ $\text{Since y(0)=1}\\$ $e^0·1+1^2·e^{3·0}=C\\$ $C=2\\$ $\text{Thus,}\\$ $e^xy+y^2e^{3x}=2\\$ annielam: Question 1: a) Find the integrating find and a general solution. $y+3y^2e^{2x}+(1+2ye^{2x})y'=0$ $M_y=1+6ye^{2x}$ $N_x=4ye^{2x}$ $R_2=\frac{M_y-N_x}{N}=\frac{1+6ye^{2x}-4ye^{2x}}{1+2ye^{2x}}=1$ $\mu=e^{\int R_2dx}=e^x$ Multiply $\mu$ to both sides $e^{x}(y+3y^2e^{2x})+e^x(1+2ye^{2x})y'$ $M_y=e^x+e^x6ye^{2x}$ $N_x=e^x+6ye^{3x}$ Since $M_y=N_x$, $x$ is the integrating factor. $\Phi=\int{M_x}=e^xy+y^2e^{3x}+h(y)$ $\Phi_y=e^x+2ye^{3x}+h’(y)$ $h’(y)=o$ $h(y)=C$ $\therefore e^xy+3y^2e^{3x}+(e^x+2ye^{3x})=C$ b) Find a solution where $y(0)=1$ Sub $y(0)=1$ $e^0(1)+3(1)(e^0)+(e^0+2e^0)=C$ $C=1+3+3=7$ $\therefore e^xy+3y^2e^{3x}+(e^x+2ye^{3x})=7$ Xinqiao Li: Here is another method to find $\varphi$. (By integrating N with respect to y) Find integrating factor and then a general solution of the ODE $$(y+3y^2e^{2x}) + (1+2ye^{2x})y' = 0, y(0) = 1$$ Let $M = y+3y^2e^{2x}$ and $N = 1+2ye^{2x}$ We can see that $M_y = 1 + 6e^{2x}y$ and $N_x = 4e^{2x}y$ They are not equal, so not exact. Our goal is to find an integrating factor and make $M_y$ equals $N_x$ $$R_2 = \frac{M_y - N_x}{N} = \frac{1 + 2e^{2x}y}{1+2e^{3x}y} = 1$$ So $\mu = e^{\int R_2dx} = e^{\int1dx} = e^x$ Multiply $\mu$ on both side of the orignial equation and we got $$(e^xy+3y^2e^{3x}) + (e^x+2ye^{3x})y' = 0$$ Now $M_y = e^x + 6e^{3x}y$ and $N_x = e^x + 6e^{3x}y$ Therefore $\exists\varphi_{(x,y)}$ satisfy $\varphi_x = M$ and $\varphi_y = N$ $$\varphi = \int Ndy = \int (e^x+2ye^{3x})dy = e^xy+y^2e^{3x} + h(x)$$ Then $\varphi_x =e^xy +3y^2e^{3x} + h'(x)$ Since $\varphi_x = M = e^xy+3y^2e^{3x}$ So $h'(x) =0$ and $h(x)=c$ $$\varphi = e^xy+y^2e^{3x} = c$$ Given initial condition $y(0) = 1$ We have $1\times 1 + 1^2 \times 1 = c$ and $c = 2$ Therefore, the particular solution is $$e^xy+y^2e^{3x} = 2$$
	# Homework Help: 1-form of a central force field? 1. Jul 2, 2010 ### TopCat The problem Given Newton's law of gravitational attraction, that the force a body exerts on a particle in space is directed towards the body and has a magnitude proportional to the inverse square of the distance to the body, show that the force field is described by the 1-form $$\frac{kx}{r^{3}}dx + \frac{ky}{r^{3}}dy + \frac{kz}{r^{3}}dz$$ where k is a postive constant and r(x,y,z) is the distance from a point to the body. The attempt The 1-form for a constant force field is A dx + B dy + C dz, where A, B, and C are constant and represent the work required for a unit displacement in the relevant direction. So the force is in a direction (-A, -B, -C). Then the magnitude of a radial force is the total work required for unit displacement opposed to the force, which is just the unit vector in the direction -$$F$$ or $$(\frac{A}{\sqrt{A^{2}+B^{2}+C^{2}}},\frac{B}{\sqrt{A^{2}+B^{2}+C^{2}}},\frac{C}{\sqrt{A^{2}+B^{2}+C^{2}}})$$. Thus, $$F = \sqrt{A^{2}+B^{2}+C^{2}}$$. The problem I'm having is showing that the general 1-form of a radial force is cx dx + cy dy + cz dz, where c > 0. If I could show that, then I know by Newton and the first part of the solution that $$\sqrt{(cx)^{2}+(cy)^{2}+(cz)^{2}} = \frac{k}{r^{2}}$$, where k is some constant. Solving for c yields $$c = \frac{k}{r^{3}}$$ and the problem is done. Can anyone help me understand why to say that the force at (x,y,z) is radially outward means that it is of the form mentioned above? Last edited: Jul 2, 2010
	Copied to clipboard ## G = C2×C20.C23order 320 = 26·5 ### Direct product of C2 and C20.C23 Series: Derived Chief Lower central Upper central Derived series C1 — C20 — C2×C20.C23 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×C20.C23 Lower central C5 — C10 — C20 — C2×C20.C23 Upper central C1 — C22 — C22×C4 — C22×Q8 Generators and relations for C2×C20.C23 G = < a,b,c,d,e \| a2=b20=c2=1, d2=e2=b10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b5c, ede-1=b10d > Subgroups: 798 in 258 conjugacy classes, 111 normal (25 characteristic) C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×4], Q8 [×9], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], C20 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C52C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×D5, C22×C10, C2×C8.C22, C2×C52C8 [×2], C4.Dic5 [×4], Q8⋊D5 [×8], C5⋊Q16 [×8], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], C2×C4.Dic5, C2×Q8⋊D5 [×2], C20.C23 [×8], C2×C5⋊Q16 [×2], C2×C4○D20, Q8×C2×C10, C2×C20.C23 Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C8.C22, C2×C5⋊D4 [×6], C23×D5, C20.C23 [×2], C22×C5⋊D4, C2×C20.C23 Smallest permutation representation of C2×C20.C23 On 160 points Generators in S160 (1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 101)(37 102)(38 103)(39 104)(40 105)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151) (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) (2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 160)(154 159)(155 158)(156 157) (1 137 11 127)(2 138 12 128)(3 139 13 129)(4 140 14 130)(5 121 15 131)(6 122 16 132)(7 123 17 133)(8 124 18 134)(9 125 19 135)(10 126 20 136)(21 159 31 149)(22 160 32 150)(23 141 33 151)(24 142 34 152)(25 143 35 153)(26 144 36 154)(27 145 37 155)(28 146 38 156)(29 147 39 157)(30 148 40 158)(41 94 51 84)(42 95 52 85)(43 96 53 86)(44 97 54 87)(45 98 55 88)(46 99 56 89)(47 100 57 90)(48 81 58 91)(49 82 59 92)(50 83 60 93)(61 109 71 119)(62 110 72 120)(63 111 73 101)(64 112 74 102)(65 113 75 103)(66 114 76 104)(67 115 77 105)(68 116 78 106)(69 117 79 107)(70 118 80 108) (1 26 11 36)(2 37 12 27)(3 28 13 38)(4 39 14 29)(5 30 15 40)(6 21 16 31)(7 32 17 22)(8 23 18 33)(9 34 19 24)(10 25 20 35)(41 75 51 65)(42 66 52 76)(43 77 53 67)(44 68 54 78)(45 79 55 69)(46 70 56 80)(47 61 57 71)(48 72 58 62)(49 63 59 73)(50 74 60 64)(81 110 91 120)(82 101 92 111)(83 112 93 102)(84 103 94 113)(85 114 95 104)(86 105 96 115)(87 116 97 106)(88 107 98 117)(89 118 99 108)(90 109 100 119)(121 158 131 148)(122 149 132 159)(123 160 133 150)(124 151 134 141)(125 142 135 152)(126 153 136 143)(127 144 137 154)(128 155 138 145)(129 146 139 156)(130 157 140 147) G:=sub<Sym(160)\| (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,101)(37,102)(38,103)(39,104)(40,105)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(153,160)(154,159)(155,158)(156,157), (1,137,11,127)(2,138,12,128)(3,139,13,129)(4,140,14,130)(5,121,15,131)(6,122,16,132)(7,123,17,133)(8,124,18,134)(9,125,19,135)(10,126,20,136)(21,159,31,149)(22,160,32,150)(23,141,33,151)(24,142,34,152)(25,143,35,153)(26,144,36,154)(27,145,37,155)(28,146,38,156)(29,147,39,157)(30,148,40,158)(41,94,51,84)(42,95,52,85)(43,96,53,86)(44,97,54,87)(45,98,55,88)(46,99,56,89)(47,100,57,90)(48,81,58,91)(49,82,59,92)(50,83,60,93)(61,109,71,119)(62,110,72,120)(63,111,73,101)(64,112,74,102)(65,113,75,103)(66,114,76,104)(67,115,77,105)(68,116,78,106)(69,117,79,107)(70,118,80,108), (1,26,11,36)(2,37,12,27)(3,28,13,38)(4,39,14,29)(5,30,15,40)(6,21,16,31)(7,32,17,22)(8,23,18,33)(9,34,19,24)(10,25,20,35)(41,75,51,65)(42,66,52,76)(43,77,53,67)(44,68,54,78)(45,79,55,69)(46,70,56,80)(47,61,57,71)(48,72,58,62)(49,63,59,73)(50,74,60,64)(81,110,91,120)(82,101,92,111)(83,112,93,102)(84,103,94,113)(85,114,95,104)(86,105,96,115)(87,116,97,106)(88,107,98,117)(89,118,99,108)(90,109,100,119)(121,158,131,148)(122,149,132,159)(123,160,133,150)(124,151,134,141)(125,142,135,152)(126,153,136,143)(127,144,137,154)(128,155,138,145)(129,146,139,156)(130,157,140,147)>; G:=Group( (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,101)(37,102)(38,103)(39,104)(40,105)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(153,160)(154,159)(155,158)(156,157), (1,137,11,127)(2,138,12,128)(3,139,13,129)(4,140,14,130)(5,121,15,131)(6,122,16,132)(7,123,17,133)(8,124,18,134)(9,125,19,135)(10,126,20,136)(21,159,31,149)(22,160,32,150)(23,141,33,151)(24,142,34,152)(25,143,35,153)(26,144,36,154)(27,145,37,155)(28,146,38,156)(29,147,39,157)(30,148,40,158)(41,94,51,84)(42,95,52,85)(43,96,53,86)(44,97,54,87)(45,98,55,88)(46,99,56,89)(47,100,57,90)(48,81,58,91)(49,82,59,92)(50,83,60,93)(61,109,71,119)(62,110,72,120)(63,111,73,101)(64,112,74,102)(65,113,75,103)(66,114,76,104)(67,115,77,105)(68,116,78,106)(69,117,79,107)(70,118,80,108), (1,26,11,36)(2,37,12,27)(3,28,13,38)(4,39,14,29)(5,30,15,40)(6,21,16,31)(7,32,17,22)(8,23,18,33)(9,34,19,24)(10,25,20,35)(41,75,51,65)(42,66,52,76)(43,77,53,67)(44,68,54,78)(45,79,55,69)(46,70,56,80)(47,61,57,71)(48,72,58,62)(49,63,59,73)(50,74,60,64)(81,110,91,120)(82,101,92,111)(83,112,93,102)(84,103,94,113)(85,114,95,104)(86,105,96,115)(87,116,97,106)(88,107,98,117)(89,118,99,108)(90,109,100,119)(121,158,131,148)(122,149,132,159)(123,160,133,150)(124,151,134,141)(125,142,135,152)(126,153,136,143)(127,144,137,154)(128,155,138,145)(129,146,139,156)(130,157,140,147) ); G=PermutationGroup([(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,100),(10,81),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,113),(29,114),(30,115),(31,116),(32,117),(33,118),(34,119),(35,120),(36,101),(37,102),(38,103),(39,104),(40,105),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,127),(60,128),(61,152),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151)], [(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)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(101,106),(102,105),(103,104),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(134,140),(135,139),(136,138),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147),(153,160),(154,159),(155,158),(156,157)], [(1,137,11,127),(2,138,12,128),(3,139,13,129),(4,140,14,130),(5,121,15,131),(6,122,16,132),(7,123,17,133),(8,124,18,134),(9,125,19,135),(10,126,20,136),(21,159,31,149),(22,160,32,150),(23,141,33,151),(24,142,34,152),(25,143,35,153),(26,144,36,154),(27,145,37,155),(28,146,38,156),(29,147,39,157),(30,148,40,158),(41,94,51,84),(42,95,52,85),(43,96,53,86),(44,97,54,87),(45,98,55,88),(46,99,56,89),(47,100,57,90),(48,81,58,91),(49,82,59,92),(50,83,60,93),(61,109,71,119),(62,110,72,120),(63,111,73,101),(64,112,74,102),(65,113,75,103),(66,114,76,104),(67,115,77,105),(68,116,78,106),(69,117,79,107),(70,118,80,108)], [(1,26,11,36),(2,37,12,27),(3,28,13,38),(4,39,14,29),(5,30,15,40),(6,21,16,31),(7,32,17,22),(8,23,18,33),(9,34,19,24),(10,25,20,35),(41,75,51,65),(42,66,52,76),(43,77,53,67),(44,68,54,78),(45,79,55,69),(46,70,56,80),(47,61,57,71),(48,72,58,62),(49,63,59,73),(50,74,60,64),(81,110,91,120),(82,101,92,111),(83,112,93,102),(84,103,94,113),(85,114,95,104),(86,105,96,115),(87,116,97,106),(88,107,98,117),(89,118,99,108),(90,109,100,119),(121,158,131,148),(122,149,132,159),(123,160,133,150),(124,151,134,141),(125,142,135,152),(126,153,136,143),(127,144,137,154),(128,155,138,145),(129,146,139,156),(130,157,140,147)]) 62 conjugacy classes class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A 8B 8C 8D 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 20 20 2 2 2 2 4 4 4 4 20 20 2 2 20 20 20 20 2 ··· 2 4 ··· 4 62 irreducible representations dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 C8.C22 C20.C23 kernel C2×C20.C23 C2×C4.Dic5 C2×Q8⋊D5 C20.C23 C2×C5⋊Q16 C2×C4○D20 Q8×C2×C10 C2×C20 C22×C10 C22×Q8 C22×C4 C2×Q8 C2×C4 C23 C10 C2 # reps 1 1 2 8 2 1 1 3 1 2 2 12 12 4 2 8 Matrix representation of C2×C20.C23 in GL6(𝔽41) 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 , 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 6 1 0 0 0 0 40 0 0 0 35 40 0 0 0 0 1 0 0 0 , 1 0 0 0 0 0 0 40 0 0 0 0 0 0 6 35 0 0 0 0 40 35 0 0 0 0 0 0 35 6 0 0 0 0 1 6 , 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 6 0 0 0 0 35 23 0 0 23 35 0 0 0 0 6 18 0 0 , 0 1 0 0 0 0 1 0 0 0 0 0 0 0 31 36 20 5 0 0 5 20 36 31 0 0 20 5 10 5 0 0 36 31 36 21 G:=sub<GL(6,GF(41))\| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,6,40,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,18,35,0,0,0,0,6,23,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,5,20,36,0,0,36,20,5,31,0,0,20,36,10,36,0,0,5,31,5,21] >; C2×C20.C23 in GAP, Magma, Sage, TeX C_2\times C_{20}.C_2^3 % in TeX G:=Group("C2xC20.C2^3"); // GroupNames label G:=SmallGroup(320,1480); // by ID G=gap.SmallGroup(320,1480); # by ID G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,136,1684,235,102,12550]); // Polycyclic G:=Group<a,b,c,d,e\|a^2=b^20=c^2=1,d^2=e^2=b^10,ab=ba,ac=ca,ad=da,ae=ea,cbc=b^-1,bd=db,ebe^-1=b^11,cd=dc,ece^-1=b^5c,ede^-1=b^10d>; // generators/relations ׿ × 𝔽
	This explains the minus sign. If $$Cu^{2+}$$ ions in solution around a $$Cu$$ metal electrode is the cathode of a cell, and $$K^+$$ ions in solution around a K metal electrode is the anode of a cell, which half cell has a higher potential to be reduced? The size of the update step is determined by the learning rate lr. Note that since E^o_{Red}=-E^o_{Ox} we could have accomplished the same thing by taking the difference of the reduction potentials, where the absent or doubled negation accounts for the fact that the reverse of the reduction reaction is what actually occurs. EN ESTE CASO NOS TOCA PROBAR AL NUEVO LR CELL INT! Activating 100% of Bardock's Hidden Potential helps unlock a copy of Team Bardock Bardock requires Special Potential Orbs instead of STR Potential Orbs Total Potential Orbs needed: x6540 x3530 x346 Can be farmed to raise Super Attack of other Bardock cards Cleared Stage 2 of An Epic Showdown on Z-Hard.You have to fulfill these criteria to be able to use the system. Therefore, this half cell has a higher potential to be reduced. How does this relate to the cell potential? The cell potential, $$E_{cell}$$, is the measure of the potential difference between two half cells in an electrochemical cell. where we have switched our strategy from taking the difference between two reduction potentials (which are traditionally what one finds in reference tables) to taking the sum of the oxidation potential and the reduction potential (which are the reactions that actually occur). Before I made it final just wanted to double check that #1crit #2 AA #3 evasion? A L R I G H T. 1 year ago. High-resolution loss of heterozygosity screening implicates PTPRJ as a potential tumor suppressor gene that affects susceptibility to Non-Hodgkin's lymphoma. The cell diagram makes it easier to see what is being oxidized and what is being reduced. How is the cell potential measured and with what device is it measured? The half cell essentially consists of a metal electrode of a certain metal submerged in an aqueous solution of the same metal ions. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. AtMYB93 is a novel endodermis-specific regulator of LR development. Volts are the amount of energy for each electrical charge; 1V=1J/C: V= voltage, J=joules, C=coulomb. (12-17 Ki) Father-Son Kamehameha (18+ Ki) Greatly raises DEF for 1 turn and causes mega-colossal damage to enemy. The goal of FreeCell Solitaire game is to move all cards to the four Foundation piles and build each suit up from Ace to King. However, the reaction at the anode is actually an oxidation reaction -- the reverse of a reduction reaction. ► 12 Ki Multiplier is 140% ► How Cell's passive works: for every enemy on the field, his ATK and DEF increase by 10%. 20 and fully activate his Hidden Potential to get awesome mission rewards! This can be measured with the use of a voltmeter. It works for both Super and Extreme Types. The Cu2+ ions would then join the aqueous solution that already has a certain molarity of Cu2+ ions. A redox reaction occurs when a certain substance is oxidized, while another is reduced. For example, in the image above, if copper wasn't being oxidized alone, and another chemical like K was involved, you would denote it as (Cu, K) in the diagram. ... Hidden Potential. A correction called the "Nernst Equation" must be applied if conditions are different. On STOCK::cool: For sale Huge Dragon ball Dokkan Battle Level 418 account for Android smartphones; Details: Price : 156 17 characters Legendary Rare LR (all with Super attack level 20, 9 LR characters with Hidden Potential activated at 100%) 294 UR characters (9 extreme awaken to levels 130-140, 149 characters Dokkan awakened at level 120) 318 Dragon stones (Estimated value of … Lawrencium is a synthetic chemical element with the symbol Lr (formerly Lw) and atomic number 103. Note that this equation can also be written as a sum rather than a difference, $E^o_{Cell}= E^o_{Red,Cathode} + E^o_{Ox,Anode} \tag{1b}$. As CD41 + cells more rapidly proliferate than CD41 – cells (Figure S3J) and the CD41 + LR-HSC population expands with time (Figure 5D), either CD41 – LR-HSCs found in GFP peaks 0–3 directly generate CD41 + LR-HSCs, or CD41 expression becomes dynamic with age and CD41 – LR-HSCs turn on CD41 expression in a manner that does not alter their regenerative potential (Figures 3C and 3D). The image above is an electrochemical cell. ... You could think of ϵ as capturing potential measurement errors on the features and labels. Fluff. Wachsman et al. Buenas gente! Cell is a very strong premium LR. to find that the standard cell potential of this cell is 0.460 V. We are done. Since E^o_{Red}=-E^o_{Ox}, the two approaches are equivalent. If you do, I promise to make it more worth reading than it already is, and I … The difference between the anode's potential to become reduced and the cathode's potential to become reduced is the cell potential. Because the half cell containing the $$Cu$$ electrode in $$Cu^{2+}$$ solution is the cathode, this is the half cell where reduction is taking place. Everything Dragon Ball Z: Dokkan Battle! Perfect Cell – 25,000,000,000 Super Perfect Cell – 77,000,000,000. More importantly, this study also determined the protective effect of SBT + M on the in vivo functionality of LR in alleviating LPS induced inflammation in zebrafish (Danio rerio). As the anode half cell becomes overwhelmed with Cu2+ ions, the negative anion of the salt will enter the solution and stabilized the charge. Example: 1 enemy on the field = ATK and DEF +10% ; 5 enemies on the field = ATK and DEF +50%. If there is a high voltage, that means there is high movement of electrons. General Chemistry: Principles and Modern Applications. A radioactive metal, lawrencium is the eleventh transuranic element and is also the final member of the actinide series. Dokkan Battle Surpassing All Perfect Cell, rating, stats, skills, awaken, how to get, tips, and team. The potential difference is caused by the ability of electrons to flow from one half cell to the other. Yes, but the only evasion would be the free one you have no choice in. The main objective of this work was to evaluate the solar photo-Fenton process at near-neutral pH in the degradation of microcystin-LR (MC-LR) un The size of the update step is determined by the learning rate lr. Electrons are able to move between electrodes because the chemical reaction is … The superscript "o" in E^o indicates that these potentials are correct only when concentrations are 1 M and pressures are 1 bar. This can also be called the potential difference between the half cells, Ecell. Create a cell diagram to match your equations. To find the difference of the two half cells, the following equation is used: $E^o_{Cell}= E^o_{Red,Cathode} - E^o_{Red,Anode} \tag{1a}$, The units of the potentials are typically measured in volts (V). An electrochemical cell is comprised of two half cells. Dokkan Battle Flare of Death Perfect Cell, rating, stats, skills, awaken, how to get, tips, and team. LRIM believes in unlocking the hidden potential, which enables young minds to ascend higher and higher to reach the pinnacle of success in corporate world . Prime Battle LR Cell performs incredibly well for a free Card, which is a welcomed surprise. When there are more chemicals involved in the aqueous solution, they are added to the diagram by adding a comma and then the chemical. Next, we will update our parameters in the direction that may reduce the loss. The chemicals involved are what are actually reacting during the reduction and oxidation reactions. It is named in honor of Ernest Lawrence, inventor of the cyclotron, a device that was used to discover many artificial radioactive elements. $$E^o_{Red,Cathode}$$ is the standard reduction potential for the reduction half reaction occurring at the cathode, $$E^o_{Red,Anode}$$ is the standard reduction potential for the oxidation half reaction occurring at the anode. 23 AtMYB93 expression is confined exclusively to roots and is induced specifically during LR initiation and the early stages of LRP formation. ► Original art found here ► Effects added by Virtualbreaker ► Made by Virtualbreaker for the "Cell (Perfect Form) into..." competition. A watch battery or button cell is a small single cell battery shaped as a squat cylinder typically 5 to 25 mm (0.197 to 0.984 in) in diameter and 1 to 6 mm (0.039 to 0.236 in) high — resembling a button.A metal can forms the bottom body and positive terminal of the cell. Archived. Please review, alert and favorite this story. Electrons are able to move between electrodes because the chemical reaction is a redox reaction. Storage conditions (4 °C for 14 days) did not affect the survival and the adhesion potential of LR to the HCT116 cell line. Cell potential is measured in Volts (=J/C). Aya-Bonilla C(1), Green MR, Camilleri E, Benton M, Keane C, Marlton P, Lea R, Gandhi MK, Griffiths LR. The image above is called the cell diagram. Legal. LR cells also exhibited minimal spike frequency adaptation (ratio of 1.26 ± 0.05), far lower than the substantial spike frequency adaptation shown by RS cells (ratio of 3.07 ± 0.46; p < 0.001), highlighting their potential ability to fire trains of action potentials at … Overview. An insulated top cap is the negative terminal. The batteries in your remote and the engine in your car are only a couple of examples of how chemical reactions create power through the flow of electrons. Have questions or comments? The first half cell, in this case, will be marked as the anode. Freecell Solitaire is free classic solitaire card and puzzle game. What does LR stand for in Cell? The following code applies the minibatch stochastic gradient descent update, given a set of parameters, a learning rate, and a batch size. Double checking on hidden potential for INT LR Vegito; User Info: Tdbruschi14. Example: 1 enemy on the field = ATK and DEF +10% ; … FreeCell Solitaire is played with a standard deck of 52 cards. hyjinx17 2 years ago #2. Watch the recordings here on Youtube! 2. AGL LR Gohan hidden potential ideas; User Info: grandpastern. When using the half cells below, instead of changing the potential the equation below can be used without changing any of the potentials from positive to negative (and vice versa): Eocell= 2.71V= +0.401V - Eo{Al(OH)4]-(aq)/Al(s)}, Eo{[Al(OH)4]-(aq)/Al(s)} = 0.401V - 2.71V = -2.31V, Confirm this on the table of standard reduction potentials. User Info: hyjinx17. This reading from the voltmeter is called the voltage of the electrochemical cell. Katherine Barrett, Gianna Navarro, Joseph Koressel, Justin Kohn. There are multiple team types in the game and therefore multiple tier lists. Collecting at least 18 Ki and facing an enemy in "ATK Down" status are both required for Cell to perform a guaranteed critical hit, but he always gets an additional ATK +20000 when Ki is 18 or more Can be farmed to raise Super Attack of other Cell (1st Form) cards ↑ As the electrons are passed to the Ag electrode, the Ag+ ions in solution will become reduced and become an Ag atom on the Ag electrode. The voltage is basically what propels the electrons to move. We would have used a plus sign had we been given an oxidation potential $$E^o_{Ox,Anode}$$ instead, since $$E^o_{Red}=E^o_{Ox}$$. Prebranch sites are thought to progress developmentally through founder cell specification and activation in the pericycle cell layer (Figure 1C), with subsequent initiation and development of LR primordia .Until primordia emergence, LR formation is hidden within the cell layers of another root. $E^o_{Cell}= E^o_{Red,Cathode} - E^o_{Red,Anode}$. To determine oxidation electrodes, the reduction equation can simply be flipped and its potential changed from positive to negative (and vice versa). So I somehow got a dupe for cell, and chose full crits, but now I gotta choose either 3 more to dodge for a total of 8, or 3 additional. The electrodes (yellow circles) of both the anode and cathode solutions are seperated by a single vertical line (l). The cell potential (Ecell) is measured in voltage (V), which allows us to give a certain value to the cell potential. Here is the list of the all the components: All of these components create the Electrochemical Cell. The voltmeter reads the transfer of electrons from the anode to the cathode in Joules per Coulomb. Take your favorite fandoms with you and never miss a beat. The electrons lost by the Cu atoms in the electrode are then transferred to the second half cell, which will be the cathode. The table below is a list of important standard electrode potentials in the reduction state. Cell LR acronym meaning defined here. The teams currently with a tier list are Super/Extreme of 'Color' types, Super/Extreme in general, and Category teams. When you choose LRIM you become a part of rich tradition of academic excellence, discovery, Innovation and creative expression. The cell potential, $$E_{cell}$$, is the measure of the potential difference between two half cells in an electrochemical cell. Type ATK Boost; Slightly raises ATK when AGL attacks STR, STR attacks PHY, PHY attacks INT, INT attacks TEQ, or TEQ attacks AGL. Copy to Drive. Top LR acronym definition related to defence: Left-right On top of that, additional rewards will be given out to players who have already awakened Cell (1st Form) into an LR! [ "article:topic", "fundamental", "showtoc:no" ]. OS TRAIGO UN NUEVO VIDEO DE DOKKAN BATTLE! Take Cell (1st Form) to the LR realm through this Legendary Challenge Campaign! The voltmeter at the very top in the gold color is what measures the cell voltage, or the amount of energy being produced by the electrodes. The effect increases along with the skill level. The electrons involved in these cells will fall from the anode, which has a higher potential to become oxidized to the cathode, which has a lower potential to become oxidized. The chemical equations can be summed to find: Cu(s) + 2Ag+ + 2e- → Cu2+(aq) + 2Ag(s) + 2e-. We have recently identified the AtMYB93 gene as the first auxin-induced negative regulator of both LR initiation and subsequent development. LR Uub falls under the same stipulation as LR Meta Cooler, only instead he cost 250,000 gems which requires more than one Battlefield run just to get one copy of him. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Microcystins are a group of cyanotoxins with known hepatotoxic effects, and their presence in drinking water represents a public health concern all over the world. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 12 Ki Multiplier is 140% How Cell's passive works: for every enemy on the field, his ATK and DEF increase by 10%. Dokfan Battle Wiki is a FANDOM Games Community. We will explain how this is done and what components allow us to find the voltage that exists in an electrochemical cell. The example will be using the picture of the Copper and Silver cell diagram. While this Cell variant, unfortunately, shares the drawbacks of the other Prime Battle LRs -- having flat ATK & DEF Buffs -- he still can prove to be useful on the right Team. 9th ed. Petrucci, Harwood, Herring, and Madura. $[Al(OH)_4]^-(aq) + 3e^- \rightarrow Al(s) + 4OH^-$. Important Standard Electrode (Reduction) Potentials, information contact us at info@libretexts.org, status page at https://status.libretexts.org, Reduction Half-Reaction E. Both potentials used in this equation are standard reduction potentials, which are typically what you find in tables (e.g., Table P1 and Table P2). Every individual has potential to become an achiever. and simplified to find the overall reaction: where the potentials of the half-cell reactions can be summed. Voltage is energy per charge, not energy per reaction, so it does not need to account for the number of reactions required to produce or consume the quantity of charge you are using to balance the equation. Both the anode and cathode are seperated by two vertical lines (ll) as seen in the blue cloud above. Tdbruschi14 2 years ago #1. The oxidation half cell of the redox equation is: where we have negated the reduction potential EoRed= 0.340 V, which is the quantity we found from a list of standard reduction potentials, to find the oxidation potential EoOx. For this redox reaction $Sn(s) + Pb^{2+}(aq) \rightarrow Sn^{2+}(aq) + Pb (s)$ write out the oxidation and reduction half reactions. Conversely, during reduction, the substance gains electrons and becomes negatively charged. The standard cell potential ($$E^o_{cell}$$) is the difference of the two electrodes, which forms the voltage of that cell. SE TRANSFORMA PERDIENDO MUCHA VIDA! The potential difference is caused by the ability of electrons to flow from one half cell to the other. The $$E^o_{cell}$$ for the equation $4Al(s) + 3O_2(g) + 6H_2O(l) + 4OH^-(aq) \rightarrow 4[Al(OH)_4]^-(aq)$ is +2.71 V. If the reduction of $$O_2$$ in $$OH^-$$ is +0.401 V. What is the reduction half-reaction for this reduction half reaction? The Full Info for the New LR SSJ Gohan & LR Cell for Dokkan Battle's 300 Million Download Celebration are here! Add text cell. LR cell hidden potential. Missed the LibreFest? For electrons to be transferred from the anode to the cathode, there must be some sort of energy potential that makes this phenomenon favorable. Pages using duplicate arguments in template calls, Causes supreme damage to enemy and greatly lowers DEF, All enemies' ATK and DEF -25%, own ATK and DEF +10% for every enemy on the field. LR cell hidden potential. (The spectator ions are left out). An example of this would be a copper electrode, in which the Cu atoms in the electrode loses two electrons and becomes Cu2+ . The cell potential is the way in which we can measure how much voltage exists between the two half cells of a battery. Level up his Super Attack to Lv. This is analogous to a rock falling from a cliff in which the rock will fall from a higher potential energy to a lower potential energy. $$E^o_{Cell}$$ is the standard cell potential (under 1M, 1 Barr and 298 K). This relates to the measurement of the cell potential because the difference between the potential for the reducing agent to become oxidized and the oxidizing agent to become reduced will determine the cell potential. The cell diagram is a representation of the overall reaction in the electrochemical cell. INT LR Cell - 6 AA / 15 Crit TEQ EZA Cell - 9 AA / 17 Crit STR Shocking Contact Android #18 - 15 AA / 6 Dodge AGL EZA Broly - 11 AA / 15 Crit STR Super Buu - 9 AA / 17 Crit AGL West Supreme Kai - 6 Crit / 15 Dodge Thanks In this half cell, the metal in atoms in the electrode become oxidized and join the other metal ions in the aqueous solution. The electrode is connected to the other half cell, which contains an electrode of some metal submerged in an aqueous solution of subsequent metal ions. Unfortunately, a lot of them can only be achieved through either lucky draws on the summons or good old fashioned grinding in order to get the cards and awakening medals needed. In order to balance the charge on both sides of the cell, the half cells are connected by a salt bridge. Each copy costs 150,000 gems which is really expensive let alone four other copies needed to max him out in the hidden potential system. From the image above, of the cell diagram, write the overall equation for the reaction. What type of reaction provides the basis for a cell potential? Similarly, in the cathode half cell, as the solution becomes more negatively charged, cations from the salt bridge will stabilize the charge. Fluff. Close. 1 Important Notes 2 Tier Lists The Tier List is a list which ranks units on their importance, effect, and relevance in the current meta of the game. During oxidation, the substance loses one or more electrons, and thus becomes positively charged. These are the reactions that create the cell potential. We can divide the net cell equation into two half-equations. Posted by. The potential energy that drives the redox reactions involved in electrochemical cells is the potential for the anode to become oxidized and the potential for the cathode to become reduced. Find all the Dragon Ball Z Dokkan Battle Game information & More at DBZ Space! Upper Saddle River, New Jersey: Pearson Education, 2007. Lateral roots form at regular intervals in the small mustard plant Arabidopsis thaliana . In this example, we will assume that the second half cell consists of a silver electrode in an aqueous solution of silver ions. have now identified both pectin and subcellular vesicle trafficking as part of the oscillating signaling system that initiates lateral roots. The reduction half cell is: where we have multiplied the reduction chemical equation by two in order to balance the electron count but we have not doubled EoRed since Eo values are given in units of voltage. In one half cell, the oxidation of a metal electrode occurs, and in the other half cell, the reduction of metal ions in solution occurs. Toggle header visibility. In the cell diagram, the anode half cell is always written on the left side of the diagram, and in the cathode half cell is always written on the right side of the diagram. 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	Density Statistics of Compressible MHD Turbulence # Density Statistics of Compressible MHD Turbulence A. Lazarian, G. Kowal & A. Beresnyak ###### Abstract Density is the turbulence statistics that is most readily available from observations. Different regimes of turbulence correspond to different density spectra. For instance, the viscosity-damped regime of MHD turbulence relevant, for instance, to partially ionized gas, can be characterized by shallow and very anisotropic spectrum of density. This spectrum can result in substantial variations of the column densities. Addressing MHD turbulence in the regime when viscosity is not important over the inertial range, we demonstrate with numerical simulations that it is possible to reproduce both the observed Kolmogorov spectrum of density fluctuations observed in ionized gas by measuring scintillations and more shallow spectra that are obtained through the emission measurements. We show that in supersonic turbulence the high density peaks dominate shallow isotropic spectrum, while the small-scale underlying turbulence that fills most of the volume has the Kolmogorov spectrum and demonstrates scale-dependent anisotropy. The limitations of the spectrum in studying turbulence induce searches of alternative statistics. We demonstrate that a measure called ”bispectrum” may be a promising tool. Unlike spectrum, the bispectrum preserves the information about wave phases. 11footnotetext: Astronomy Department, University of Wisconsin, Madison, WI 5370622footnotetext: Astronomical Observatory, Jagiellonian University, Kraków, Poland ## 1. Density Fluctuations as a Probe of MHD Turbulence The paradigm of interstellar medium has undergone substantial changes recently. Instead of quiescent medium with hanging and slowly evolving clouds a turbulent picture emerged (see review by Ballesteros-Parredes et al. 2006, McKee & Ostriker 2007 and ref. therein). With magnetic field being dynamically important and dominating the gas pressure in molecular clouds, this calls for studies of compressible magnetohydrodynamic (MHD) turbulence. Key ideas in describing MHD turbulence can be traced back to Iroshnikov (1963) and Kraichnan (1965) work and the classical work that followed (see Montgometry & Turner 1981, Higdon 1984, Montgomery, Brown & Matthaeus 1987). A more recent model by Goldreich & Sridhar (1995, henceforth GS95) has been successfully tested by numerical simulations (Cho & Vishniac 2000, Maron & Goldreich 2001, Cho, Lazarian & Vishniac 2002, Cho & Lazarian 2002, 2003, henceforth CL03). Density statistics is the easiest to infer from observations. In comparison, velocity spectra require elaborate techniques to be used (see Lazarian 2006 and ref. therein). How informative can be density fluctuations for understanding of MHD turbulence is the subject of the present review. However, we should mention that for many important astrophysical applications, e.g. for interstellar chemistry, for star formation, for propagation of radiation etc. the density fluctuations themselves are crucially important. Subsonic compressible MHD is rather well studied topic today. It is suggestive that there may be an analogy between the subsonic MHD turbulence and its incompressible counterpart, namely, GS95 model. Therefore the correspondence between the the two revealed in CL03 is expected. It could be easily seen, that in the low-beta case density is perturbed mainly due to the slow mode (CL03). Slow modes are sheared by Alfvén turbulence, therefore they exhibit Kolmogorov scaling and GS95 anisotropy for low Mach numbers. However, for high Mach numbers we expect shocks to develop. Density will be perturbed mainly by those shocks. One can also approach the problem from the point of view of underlying hydrodynamic equations. It is well known that there is a multiplicative symmetry with respect to density in the ideal flow equations for an isothermal fluid (see e.g. Passot & Vazquez-Semadeni, 1998). This assume that if there is some stochastic process disturbing the density it should be a multiplicative process with respect to density, rather than additive, and the distribution for density values should be lognormal, rather that normal. The aforementioned work shows that for 1D numerical simulations of high-Mach hydrodynamics the distribution is approximately lognormal, having power-law tails in case of . In MHD the above described symmetry is broken by the magnetic tension. However, numerical studies in Beresnyak, Lazarian & Cho (2005, henceforth BLC05) show an approximate correspondence of the PDFs obtained with MHD simulations to the lognormal scaling. With a high sonic Mach we expect a considerable amount of shocks arise. In a sub-Alfvénic case, however, we expect oblique shocks be disrupted by Alfvénic shearing, and, as most of the shocks are generated randomly by driving, almost all of them will be sheared to smaller shocks. The evolution of the weak shocks will be again governed by the sonic speed, and structures from shearing as in low Mach case should arise. We also note that shearing will not affect probability density function (PDF) of the density, but have to affect its spectra and structure function (SF) scaling. In other words, we deal with two distinct physical processes, one of which, random multiplication or division of density in presence of shocks, affect PDF, while the other, Alfvénic shearing has to affect anisotropy and scaling of the structure function of the density. The numerical studies that we describe below confirm the theoretical considerations above. ## 2. Viscosity-Damped MHD Turbulence as a Special Case Turbulence can be viewed as a cascade of energy from a large injection energy scale to dissipation at a smaller scale. The latter is being established by equating the rate of turbulent energy transfer to the rate of energy damping arising, for instance, from viscosity. Naively, one does not expect to see any turbulent phenomena below such a scale. Such reasoning may not be true in the presence of magnetic field, however. Consider magnetized fluid with viscosity much larger than magnetic diffusivity , which is the case of a high magnetic Prantl number fluid. The partially ionized gas can serve as an example of such a fluid up to the scales of ion-neutral decoupling (see a more rigorous treatment in Lazarian, Vishniac & Cho 2004, henceforth LVC04). Fully ionized plasma is a more controversial example. For instance, It is well known that for plasma the diffusivities are different along and perpendicular to magnetic field lines. Therefore, the plasma the Prandtl number is huge if we use the parallel diffusivity . A treatment of the fully ionized plasma as a high Prandtl number medium is advocated in Schechochihin et al. (2004, henceforth SCTMM). The turbulent cascade in a fluid with isotropic proceeds up to a scale at which the cascading rate, which for the Kolmogorov turbulence, i.e. , is determined by the eddy turnover rate gets equal to the damping rate . Assuming that the energy is injected at the scale and the injection velocity is , the damping scale is , where is the Reynolds number .The corresponding scale varies from pc for Warm Neutral Medium to for molecular clouds (LVC04). It is evident, however, that magnetic fields at the scale at which hydrodynamic cascade would stop are still sheered by eddies at the larger scales. This should result in creating magnetic structures at scales (LVC04). Note, that magnetic field in this regime is not a passive scalar, but important dynamically. Thus, GS95 turbulence range an inertial range in the partially ionized gas from the injection scale to , but below is the domain of the viscosity-damped turbulence, where plasma/gas compressed within current sheets can contribute to the observed extreme scattering events and the formation of the Small Ionized or Neutral Structures (SINS) (Lazarian 2007). Most compressions arises in the situations when the magnetic fields exhibits perfect reversals. Cho, Lazarian & Vishniac (2002) reported a new viscosity-damped regime of turbulence using incompressible MHD simulations (see an example of more recent simulations in Fig. 1). In this regime, unlike hydro turbulence, motions, indeed, do not stop at the viscosity damping scale, but magnetic fluctuations protrude to smaller scales. Interestingly enough, these magnetic fluctuations drive small amplitude velocity fluctuations at scales . CL03 confirmed these results with compressible simulations and speculated that these small scale magnetic fluctuations can compress ambient gas to produce ubiquitous tiny structures observed in ISM (see also Fig. 1b). According to the model in LVC04, while the spectrum of averaged over volume magnetic fluctuations scales as , the pressure within intermittent magnetic structures increases with the decrease of the scale as , while the filling factor , the latter being consistent with numerical simulations. In Fig. 1 we show the results of our high resolution compressible MHD simulations that exhibit strong density fluctuations at the scales below the one at which hydrodynamic turbulence would be damped. According to LVC04 model, the viscosity-damped regime is ubiquitous in turbulent partially ionized gas. Some of its discussed consequences, such as an intermittent resumption of the turbulence the fluid of ions as magnetic fluctuations reach the ion-neutral decoupling scale, are important for radio scintillations. ## 3. Simulations 3D statistics and Simulated Observations ### 3.1. Spectrum The power spectrum (PS) of density fluctuations is an important property of a compressible flow. In some cases, the spectrum of density can be derived analytically. For nearly turbulent motions in the presence of a strong magnetic field, the spectrum of density scales similarly to the pressure, i.e. if we consider the polytropic equation of state (Biskamp 2003). In weakly magnetized nearly incompressible MHD turbulence, however, velocities convect density fluctuations passively inducing the spectrum (Montgomery et al. 1987). In supersonic flows, these relations are not valid anymore because of shocks accumulating matter into the local and highly dense structures. Due to the high contrast of density, the linear relation is no longer valid, and the spectrum of density cannot be related to pressure so straightforwardly. In addition, the strong asymmetry of density fluctuations suggests the need to analyze the logarithm of density instead of density itself. In Figure 2 we present the PS of fluctuations of density for models with different . As expected, we note a strong growth of the amplitude of density fluctuations with the sonic Mach number at all scales. This behavior is observed both in sub-Alfvénic as well as in super-Alfvénic turbulence (see Fig. 2). In Table 1 we calculate the spectral index of density and the logarithm of density within the inertial ranges estimated from the PS of velocity. The width of the inertial range is shown by the range of solid lines with slopes and in all spectra plots. It is estimated to be within . In Table 1 we also show the errors of estimation which combine the error of the fitting of the spectral index at each time snapshot and the standard deviation of variance of in time. The slopes of the density spectra do not change significantly with for subsonic experiments and correspond to analytical estimations (about , which is slightly less than , for turbulence with and about , which is slightly more than , for weakly magnetized turbulence with ). Such an agreement confirms the validity of the theoretical approximations. Those, nevertheless, do not cover the entire parameter space. While the fluid motions become supersonic, they strongly influence the density structure, making the small-scale structures more pronounced, which implies flattening of the spectra of density fluctuations (see values for in Table 1) (see also BLC05). Although, the spectral indices are clearly different for sub- and super-Alfvénic turbulence, their errors are relatively large (up to ). These errors have been calculated by summing the maximum value of the uncertainty of fits for individual spectra at each time snapshot and the standard deviation of time variation of spectral indices. The uncertainty of fit contributes the most to the total error bar (about 60%-70%). ### 3.2. Anisotropies Another question is the anisotropy of density and the logarithm of density structures. For subsonic turbulence it is natural to assume that the density anisotropies will mimic velocity anisotropies in GS95 picture. This was confirmed in CL03, who, however, observed that for supersonic turbulence the contours of density isocorrelation get round, corresponding to isotropy. BLC05, however, showed that anisotropies restore the GS95 form if instead of density one studies the logarithm of density. This is due to the suppression of the influence of the high density peaks, which arise from shocks. It is these peaks that mask the anisotropy of weaker, but more widely spread density fluctuations. In Figure LABEL:fig:anisotropy we show lines that mark the corresponding separation lengths for the second-order SFs parallel and perpendicular to the local mean magnetic field.111The local mean magnetic field was computed using the procedure of smoothing by a 3D Gaussian profile with the width equal to the separation length. Because the volume of smoothing grows with the separation length , the direction of the local mean magnetic field might change with at an arbitrary point. This is an extension of the procedures employed in Cho, Lazarian & Vishniac (2002). In the case of subAlfvénic turbulence, the degree of anisotropy for density is very difficult to estimate due to the high dispersion of points. However, rough estimates suggest more isotropic density structures, because the points extend along the line . For models with the points in Figure LABEL:fig:anisotropy have lower dispersion, and the anisotropy is more like the type from GS95, i.e. . In both Alfvénic regimes, the anisotropy of density does not change significantly with . Plots for the logarithm of density show more smooth relations between parallel and perpendicular SFs. The dispersion of points is very small. Moreover, we note the change of anisotropy with the scale. Lower values of SFs correspond to lower values of the separation length (small-scale structures), so we might note that the logarithm of density structures are more isotropic than the GS95 model at small scales, but the anisotropy grows a bit larger than the GS95 prediction at larger scales. This difference is somewhat larger in the case of models with stronger external magnetic field (compare plots in the left and right columns of Figure LABEL:fig:anisotropy), which may signify their dependence on the strength of . The anisotropy of structures is marginally dependent on the sonic Mach number, similar to the density structures. All these observations allow us to confirm the previous studies (see BLC05) that suggest that the anisotropy depends not only on the scale but on . ### 3.3. Bispectrum Attempts to use multipoint statistics are a more traditional way to remove the constraints that the use of two point statistics, e.g. power spectra entails. Unfortunately, very high quality data is needed to obtain the multipoint statistics. Among multipoint statistics, bispectrum (see Scoccimarro 1997) seems the most promising. This is partially because it has been successfully used in the studies of the Large Scale Structure of the Universe. Bispectrum is a Fourier transform of the three point correlation function and if the power spectrum is defined as ⟨δρ(k1)δρ(k2)⟩=P(k)δD(k1+k2) (1) where is the Dirac delta function that is zero apart from the case when , the bispectrum is ⟨δρ(k1)δρ(k2)δρ(k3)⟩=B123δD(k1+k2+k3). (2) The bispectra of density shown in Fig. 4 give information as to how shocks and magnetic fields effect turbulence. It has been shown by Kowal, Lazarian & Beresnyak (2007) and Beresnyak et al. (2005) that in supersonic turbulence shocks produce compressed density and a shallower spectrum. Looking at Fig. 4 it is clear that these shocks play a crucial role in the correlation of modes. The subsonic cases show little correlation between any points except the case of . The compressed densities from supersonic turbulence are critical for correlations between frequencies since waves get closer together and therefore have a much higher interaction rate. We find an good agreement with Kowal, Lazarian & Beresnyak (2007) in that the density structures are affected by the presence of magnetic field. When there is a weak magnetic field present it is clear from Figure 4 that the system lacks the stronger correlations that are characteristic of the sub-Alfv́enic models. For super-Alfv́enic simulations correlations in frequency are not as readily made due to large dispersion of density structure. The bispectrum of supersonic hydro models are similar to the super-Alfv́enic, supersonic cases. A measure related to bispectrum is bicohence (see Koronovskii & Hramov 2002). ## 4. Summary Above we discussed the statistics of density in compressible MHD turbulence. We analyzed spectra, structure functions for experiments with different sonic and Alfvén Mach numbers. Our results are as follows: The viscosity-damped regime of MHD turbulence relevant to partially ionized gas can be characterized by shallow and very anisotropic spectrum of density. This spectrum can result in large variations of the column densities. The amplitude of density fluctuations strongly depends on both in weakly and strongly magnetized turbulent plasmas. The flattening observed in density spectra is due to the contribution of the highly dense small-scale structures generated in the supersonic turbulence. Fluctuations of the logarithm of density are much more regular than those of density. The logarithm of density exhibits the GS95 scalings and anisotropies. Bispectra show strong correlations for supersonic models than for subsonic ones. It suggests the importance of shocks in mode correlations. Acknowledgement of the support of the NSF Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasma. ## References • (1) Ballesteros-Paredes, J., Klessen, R., Mac Low, M. & Vasquez-Semadeni, E. 2006, in “Protostars and Planets V”, University of Arizona Press, Tucson, p.63 • Beresnyak et al. 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Fluids, 8, 1385 • (15) Lazarian, A. 2007, in ”SINS-Small Ionized and Neutral Strucutres in Diffuse ISM”, eds. M. Haverkorn & W.M. Goss, ASP 365, 324 • (16) Lazarian, A. 2006, AIP 874, 301-315 • Lazarian, Vishniac,& Cho (2004) Lazarian, A., Vishniac, E., & Cho, J. 2004, ApJ, 603, 180 • Lazarian & Beresnyak (2005) Lazarian, A. & Beresnyak, A., 2005, in The Magnetized Plasma in Galaxy Evolution, Kraków, Jagiellonian University Press, p. 56 • (19) Maron, J. & Goldreich, P. 2001, ApJ, 554, 1175 • (20) McKee, C. & Ostriker, E. 2007, ARA&A, 45, 565 • (21) Montgomrey D.C. & Turner L. 1981, Phys. Fluids, 24, 825 • Montgomery et al. (1987) Montgomery, D., Brown, M. R., Matthaeus, W. H., 1987, JGR, 92, 282 You are adding the first comment! 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	This document gives detailed examples about flux calculations in OSCARS. Any of these can be run in multi-threaded, GPU, or MPI mode. Results from running on separate nodes on grid/cloud computing can be combined. ## Polarization¶ One can select which polarization to compute. The default is 'all' polarizations. One may also specify the polarization parameter as: • 'linear-horizontal' or 'lh' • 'linear-vertical' or 'lv' • 'circular-right' or 'cr' • 'circular-left' or 'cl' One may alternatively specify the angle of polarization of interest with respect to the horizontal direction as (for instance for 45 degrees): • angle=45. * osr.pi() / 180. Calculating polarization requires a definition of the horizontal and vertical directions. The defaults for these assume the beam is in the +z direction with +x being the horizontal direction. It is possible to change these definitions with the parameters horizontal_direction and propogation_direction In [1]: # matplotlib plots inline %matplotlib inline # Import the OSCARS SR module import oscars.sr # Import OSCARS plots (matplotlib) from oscars.plots_mpl import * OSCARS v2.1.8 - Open Source Code for Advanced Radiation Simulation Brookhaven National Laboratory, Upton NY, USA http://oscars.bnl.gov oscars@bnl.gov In [2]: # Create a new OSCARS object. Default to 8 threads and always use the GPU if available In [3]: # For these examples we will make use of a simple undulator field osr.add_bfield_undulator(bfield=[0, 1, 0], period=[0, 0, 0.042], nperiods=31) # Plot the field plot_bfield(osr) ## Beam¶ Add a basic beam somewhat like NSLS2. Filament beam for simple studies. In [4]: # Add a basic electron beam with zero emittance osr.set_particle_beam(energy_GeV=3, x0=[0, 0, -1], current=0.500) # You MUST set the start and stop time for the calculation osr.set_ctstartstop(0, 2) # Plot trajectory osr.set_new_particle() plot_trajectory_position(osr.calculate_trajectory()) ## Spectrum¶ Calculate the spectrum so we can pick what energy we want to look at in the 2D flux maps In [5]: # Evenly spaced spectrum in an energy range spectrum = osr.calculate_spectrum(obs=[0, 0, 30], energy_range_eV=[200, 260], npoints=500) plot_spectrum(spectrum) ## Flux on Rectangular surface¶ First let's look at a simple flux on a rectangular surface In [6]: # Calculate flux on rectangular surface # Here we essentially create a rectanglular surface in the XY plane, then translate # it 30m downstream flux = osr.calculate_flux_rectangle(plane='XY', energy_eV=235, width=[0.01, 0.01], npoints=[101, 101], translation=[0, 0, 30]) plot_flux(flux) In [7]: # Calculate linear horizontal and linear vertical polarizations flux = osr.calculate_flux_rectangle(plane='XY', energy_eV=235, width=[0.01, 0.01], npoints=[101, 101], translation=[0, 0, 30], polarization='lh' ) plot_flux(flux) flux = osr.calculate_flux_rectangle(plane='XY', energy_eV=235, width=[0.01, 0.01], npoints=[101, 101], translation=[0, 0, 30], polarization='lv' ) plot_flux(flux) ## Multi-particle flux¶ ### Non-zero emittance beam¶ In [8]: # Add a basic electron beam with zero emittance osr.set_particle_beam(type='electron', energy_GeV=3, x0=[0, 0, -1], current=0.500, sigma_energy_GeV=0.001*3, beta=[1.5, 0.8], emittance=[0.9e-9, 0.008e-9]) # You MUST set the start and stop time for the calculation osr.set_ctstartstop(0, 2) In [9]: # Calculate flux on rectangular surface # Here we can set nparticle equal to the number of desired particles to use flux = osr.calculate_flux_rectangle(plane='XY', energy_eV=235, width=[0.01, 0.01], npoints=[101, 101], translation=[0, 0, 30], nparticles=3) plot_flux(flux)
	Share # If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a. - CBSE Class 9 - Mathematics ConceptGraph of a Linear Equation in Two Variables #### Question If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a. #### Solution Putting x = 3 and y = 4 in the given equation, 3y = ax + 7 3 (4) = a (3) + 7 5 = 3a a = 5/3 Is there an error in this question or solution? #### APPEARS IN NCERT Solution for Mathematics Class 9 (2018 to Current) Chapter 4: Linear Equations in two Variables Ex. 4.30 \| Q: 3 \| Page no. 74 #### Video TutorialsVIEW ALL [1] Solution If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a. Concept: Graph of a Linear Equation in Two Variables. S
	# I am getting approximalte half what the answer should be (Related Rates Problem) • Nov 7th 2009, 11:38 AM s3a I am getting approximalte half what the answer should be (Related Rates Problem) I am having trouble with this question but I believe it's a small mistake because I get approximately half of what the answer should be. I attached my work. Question: "A ladder 10 ft long leans against a vertical wall. If the bottom of the ladder slides away from the base of the wall at a speed of 2 ft/s, how fast is the angle between the ladder and the wall changing when the bottom of the ladder is 6ft from the base of the wall? Any help would be greatly appreciated! • Nov 7th 2009, 11:45 AM skeeter Quote: Originally Posted by s3a I am having trouble with this question but I believe it's a small mistake because I get approximately half of what the answer should be. I attached my work. Question: "A ladder 10 ft long leans against a vertical wall. If the bottom of the ladder slides away from the base of the wall at a speed of 2 ft/s, how fast is the angle between the ladder and the wall changing when the bottom of the ladder is 6ft from the base of the wall? Any help would be greatly appreciated! let $\theta$ = angle between the ladder and the wall $\sin{\theta} = \frac{x}{10}$ $\cos{\theta} \cdot \frac{d\theta}{dt} = \frac{1}{10} \cdot \frac{dx}{dt}$ $\frac{8}{10} \cdot \frac{d\theta}{dt} = \frac{2}{10}$ $\frac{d\theta}{dt} = \frac{2}{10} \cdot \frac{10}{8} = \frac{1}{4}$ rad/s • Nov 7th 2009, 01:30 PM s3a So my mistake was that SINE is what determines "THE ANGLE BETWEEN THE LADDER AND THE WALL?" Also, I now see the ladder is a constant and the x is not. Thanks. • Nov 7th 2009, 01:34 PM skeeter Quote: Originally Posted by s3a So my mistake was that SINE is what determines "THE ANGLE BETWEEN THE LADDER AND THE WALL?" I don't know ... I don't do downloads on public message forums anymore. • Nov 7th 2009, 01:57 PM s3a Oh, well thanks still but if you didn't know, choosing to open rather than save pdf files would just store them into the hard drive temporarily.
	# How do you rationalize the denominator (sqrt27-sqrt7)/(sqrt21-3)? Mar 24, 2018 $\frac{\sqrt{27} - \sqrt{7}}{\sqrt{21} - 3}$ Multiply both the numerator and denominator with $\left(\sqrt{21} + 3\right)$, $\frac{\sqrt{27} - \sqrt{7}}{\sqrt{21} - 3} \times \frac{\sqrt{21} + 3}{\sqrt{21} + 3}$ Apply difference of two squares rule, $\frac{\left(\sqrt{27} - \sqrt{7}\right) \left(\sqrt{21} + 3\right)}{{\sqrt{21}}^{2} - {3}^{2}}$ Expand, $\frac{\left(\sqrt{27}\right) \left(\sqrt{21}\right) + \left(- \sqrt{7}\right) \left(\sqrt{21}\right) + \left(\sqrt{27}\right) \left(3\right) + \left(- \sqrt{7}\right) \left(3\right)}{21 - 9}$ Simplify, $\frac{6 \sqrt{7} + 2 \sqrt{3}}{12}$ SImplest form, $\frac{3 \sqrt{7} + 2 \sqrt{3}}{6}$
	# Why is my program “not responding” when I run it? I'm trying to learn to code with python and while using pygame, I came across this: every time I run the window it says "not responding". Specifically, it happens when I click to drag the window; the title changes to "not responding" and I do not know what to do. Here is all of my code: import pygame pygame.init() win = pygame.display.set_mode((500, 500)) pygame.display.set_caption("Game Test") Looking quickly at a hello world with pygame, I landed on this page. Here is the code they use: import pygame, sys from pygame.locals import * pygame.init() DISPLAYSURF = pygame.display.set_mode((400, 300)) pygame.display.set_caption('Hello World!') while True: # main game loop for event in pygame.event.get(): if event.type == QUIT: pygame.quit() sys.exit() pygame.display.update() Your program needs to be "alive", this means that you need to have a main loop that will handle at least the events that are sent by the OS, Windows in your case. If you don't process these events, windows will know about it and think that the program is frozen (i.e. not responding). You'll also want to make sure that you draw something (that's with the last line in the example).
	# David Neuzerling Data, Maths, R ## R on AWS Lambda with Containers AWS has announced support for container images for their serverless computing platform Lambda. AWS doesn’t provide an R runtime for Lambda, and this was the excuse I needed to finally try to make one. An R runtime means that I can take advantage of AWS Lambda to put my R functions in the cloud. I don’t have to worry about provisioning servers or spinning up containers — the function itself is the star. And from the perspective of the service calling Lambda, it doesn’t matter what language that function is written in. Also, someone told me that you can’t use R on Lambda, and I took that personally. ## I don’t have to learn Lambda layers “Container support” is potentially confusing here. To clarify, we can’t take any container and expect Lambda to work with it. The container needs to provide a Lambda runtime, or use one of the available runtimes, in order to have Lambda communicate with the functions. Runtimes are provided for a handful of languages, but not for R. I’m not the first person to put an R function on Lambda. Previous attempts used Lambda layers. But with container support I can disregard that layer stuff and use Dockerfiles, which is a concept I already have some experience with. And by writing the runtime in R itself I can share a single R session between subsequent requests, cutting down on execution time. My goal here is to host a function written entirely in R — in this case, a simple parity function that determines if an integer is odd or even. I stick this in a container, along with a Lambda runtime written entirely in R, and then I’m able to invoke the function on AWS. I’ll talk through the process below, but if you’re the kind of person who likes to read the last page of a book first then you can take a look at my git repository. ## How to make R and Lambda talk to each other AWS provides some documentation for creating a custom runtime, and it’s pretty good. The rough idea is that the Lambda event which initiates the function invocation sits at a HTTP endpoint. My R runtime needs to constantly query that endpoint. The eventual response has a body that contains the arguments that my R function needs. The R runtime has to run the function with those arguments and then send the results to a specific HTTP endpoint. Afterwards it checks the event endpoint again. I’ll need the httr package for sending requests to HTTP endpoints, and the jsonlite package to convert the response body from a JSON to an R list, and the function result from an R list to a JSON. It’s all JSONS. That’s why I build my parity function to return a list. The jsonlite package will take a result like list(parity = "even") and turn it into {"parity": "even"} which non-R services can understand. parity <- function(number) { list(parity = if (as.integer(number) %% 2 == 0) "even" else "odd") } My code does the following: 1. Load all necessary environment variables and determine the Lambda endpoints 2. Determine the source file and function from the handler 3. Source the file containing the code and check that the function is available 4. Listen for events until there’s a response 5. Pull the request ID and other information from the response headers 6. Parse the body of the request into an R list 7. Call the function with the body as its arguments 8. Send the result to the endpoint for that specific event 9. Listen for more events AWS Lambda sets a few environment variables that my code needs to be able to capture. The first is LAMBDA_TASK_ROOT, which is the path to the working directory for the function. In the Dockerfile, all of the R code will be copied here, where the runtime will look for it. There’s also the _HANDLER, which is a string of the form “file.function”. I put the code for my parity function in a file called functions.R, so my handler will be “functions.parity”; I make my runtime automatically append the “.R” extension. I can set the handler either as the CMD of the Dockerfile or through the AWS Lambda console (which takes precedence). Afterwards, it is made available as an environment variable to my runtime. The AWS_LAMBDA_RUNTIME_API is used to piece together the different HTTP endpoints needed to communicate with Lambda: • The next invocation endpoint is used to get the next event. • The initialisation error endpoint is where errors should be sent if there was a problem setting up the runtime. • The invocation response endpoint is unique to each event. It’s where a successful function result should be sent. • The invocation error endpoint is where errors should be sent if there was a problem during function execution. It is also unique to each event. Every event that comes through has a request ID header, named “lambda-runtime-aws-request-id”1. This uniquely identifies the event, and is used to construct the event-specific HTTP endpoints. Finally there’s also a “lambda-runtime-trace-id” header. The AWS guide suggest setting this as the value of the _X_AMZN_TRACE_ID environment ID. Curiously, that seems to be the only action required. I was expecting to have to pass this header on in the response, but apparently not. It’s used as part of AWS’s X-Ray SDK. ## Turning JSON into something R can understand The body of the event, which is the response from the next invocation endpoint, contains the arguments that my function needs. If the body is empty then there are no arguments and my function should accept no arguments in this case. I interpret the body as an empty list. Otherwise, I use the jsonlite package to parse the JSON body into an R list. This is a particularly sensitive area of the runtime; JSON parsing is a fragile process. unparsed_content <- httr::content(event, "text", encoding = "UTF-8") event_content <- if (unparsed_content == "") { list() } else { jsonlite::fromJSON(unparsed_content) } From this point I can call the function with this list of arguments: do.call(function_name, event_content) The runtime sends the result to the appropriate HTTP endpoint and listens for the next event. And that’s the runtime done. ## Stick it in a container The Dockerfile starts with the AWS base image for Lambda2 that contains the bits and pieces needed to host the function. I install R as if it were a CentOS image, and remove the installer afterwards to save a little space. There are some path issues here: I need to append the location of the R binaries to the system PATH, and manually specify the CRAN repository when installing R packages. In order to run my runtime, I need to provide the container with a bootstrap. This bootstrap isn’t particularly complicated: it’s an executable script that changes the working directory to the value of the LAMBDA_TASK_ROOT environment variable and runs the runtime.R file: #!/bin/sh cd $LAMBDA_TASK_ROOT Rscript runtime.R I think such a small and simple script doesn’t need to be a file, so I hardcode it within the Dockerfile itself. Here’s the Dockerfile I end up with: FROM public.ecr.aws/lambda/provided ENV R_VERSION=4.0.3 RUN yum -y install wget RUN yum -y install https://dl.fedoraproject.org/pub/epel/epel-release-latest-7.noarch.rpm \ && wget https://cdn.rstudio.com/r/centos-7/pkgs/R-${R_VERSION}-1-1.x86_64.rpm \ && yum -y install R-${R_VERSION}-1-1.x86_64.rpm \ && rm R-${R_VERSION}-1-1.x86_64.rpm ENV PATH="${PATH}:/opt/R/${R_VERSION}/bin/" # System requirements for R packages RUN yum -y install openssl-devel RUN Rscript -e "install.packages(c('httr', 'jsonlite', 'logger'), repos = 'https://cloud.r-project.org/')" COPY runtime.R functions.R ${LAMBDA_TASK_ROOT}/ RUN chmod 755 -R${LAMBDA_TASK_ROOT}/ RUN printf '#!/bin/sh\ncd $LAMBDA_TASK_ROOT\nRscript runtime.R' > /var/runtime/bootstrap \ && chmod +x /var/runtime/bootstrap I haven’t set any entrypoint or command for this container. The default entrypoint for the parent image is a shell script for AWS Lambda, and I don’t want to interfere with that. The command is the handler for the function. I could hardcode that here as with CMD ["functions.parity"], but instead I configure it later within the AWS Lambda management console. ## Test the function locally The Lambda base image lets me test my function locally by running the container and then querying a HTTP endpoint. I start by navigating to the project directory and building the image: docker build -t mdneuzerling/r-on-lambda . I run the image by providing it with the handler as the command. Recall that I want to use the parity function from the functions.R file: docker run -p 9000:8080 mdneuzerling/r-on-lambda "functions.parity" In a separate shell I query the endpoint: curl -X POST "http://localhost:9000/2015-03-31/functions/function/invocations" \ -d '{"number": 5}' I receive the response {"parity":"odd"} which — for the number 5 — is correct3. The STDOUT of the main window contains the log entries. There are some messages and warnings here that I choose to ignore: time="2020-12-05T21:56:04.914" level=info msg="exec '/var/runtime/bootstrap' (cwd=/var/task, handler=)" time="2020-12-05T21:56:46.953" level=info msg="extensionsDisabledByLayer(/opt/disable-extensions-jwigqn8j) -> stat /opt/disable-extensions-jwigqn8j: no such file or directory" time="2020-12-05T21:56:46.953" level=warning msg="Cannot list external agents" error="open /opt/extensions: no such file or directory" START RequestId: ff9ed881-8874-48f6-b67f-6b271e9afd3c Version:$LATEST logger: As the "glue" R package is not installed, using "sprintf" as the default log message formatter instead of "glue". INFO [2020-12-05 21:56:47] Handler found: functions.parity INFO [2020-12-05 21:56:47] Using function parity from functions.R INFO [2020-12-05 21:56:47] Querying for events END RequestId: ff9ed881-8874-48f6-b67f-6b271e9afd3c REPORT RequestId: ff9ed881-8874-48f6-b67f-6b271e9afd3c Init Duration: 0.43 ms Duration: 332.66 ms Billed Duration: 400 ms Memory Size: 3008 MB Max Memory Used: 3008 MB ## Push the image to AWS The container image needs to be available on AWS in order for Lambda to use it. That is, the image needs to be hosted on AWS’s Elastic Container Registry. The AWS announcement of container support provides some good instructions for pushing an image, but I’ll briefly cover it here. Container support isn’t available for every region yet, so I switch to the us-east-1 region using the AWS CLI: aws configure set region us-east-1 Then, following the instructions from the announcement, I create a repository for my image. aws ecr create-repository --repository-name r-on-lambda --image-scanning-configuration scanOnPush=true This command gives me information about the repository, including a URI. In my case, the URI takes on the form “{AWS account number}.dkr.ecr.us-east-1.amazonaws.com”. The next step involves re-tagging my image to include this URI, and then pushing the image to ECR: docker tag mdneuzerling/r-on-lambda:latest {URI}/r-on-lambda:latest docker push {URI}/r-on-lambda:latest ## Set up a Lambda function From the AWS Management Console I change my region to “us-east-1”. On the Lambda page I create a new function, and I see the new option to use a container image. I select the container I just uploaded and click “Create Function”. It takes a few seconds before I see the function configuration page. I need to make one change here. I didn’t set a CMD in my Dockerfile, so I need to edit the image configuration to specify the handler. I want to use the parity function from the functions.R source file, so I override the CMD to “functions.parity”. Now I’ll configure a test to check that my function is working. The “Test” button, towards the top-right of the console, prompts me define a test JSON payload: Afterwards, I click the “Test” button again and see the results: The function is working! As a final check I’ll invoke the function through the AWS CLI: aws lambda invoke --function-name parity \ --invocation-type RequestResponse --payload '{"number": 8}' \ /tmp/response.json And, sure enough, the response.json file contains the expected {"parity":"even"} result. The performance isn’t fantastic. Each invocation takes about 120ms. The initalisation time is 7 seconds, reduced to 2.3 seconds if I increase the available memory to the maximum of 10GB. This seems slow to me, but then again I don’t have a good baseline for container-based Lambda functions. The initialisation penalty is only incurred if the function hasn’t been called for a while — for requests in quick succession the image is kept alive. ### Update 2021-03-10 Thank you to @berkorbay for telling me that this invocation command doesn’t work on the latest version of the AWS CLI. They suggested this instead: aws lambda invoke --function-name parity \ --invocation-type RequestResponse --payload '{"number": 8}' \ /tmp/response.json --cli-binary-format raw-in-base64-out ## Logging is good I had a lot of trouble getting this to work, and the main reason for that is that the errors that came from my code were often not the actual errors. I also had no way to step through the code. Logging really helped me with debugging. I’ve had a few people ask me to talk about logging, so I’ll talk through it here. Logs are records generated as the code runs that are saved to a file or otherwise captured to be stored after the program has finished. I might log the status of my program, the information it receives, or any errors or warnings that it encounters. There are a few packages that support logging and they all tend to follow the same conventions. I’m using the logger package by Gergely Daróczi. Logging has the potential to generate a lot of information, and a good way to simplify this is to take advantage of log levels. There are many levels, but the five common ones — in increasing order of severity — are: debug, info, warn, error, and fatal. By setting a threshold I can encourage my logger to ignore entries below a certain level. In my runtime I set the threshold to info, but if I’m encountering errors and I want more detail than I can lower this to debug so that all of the debug-level log entries come through. Logs can be stored to files or simply printed to STDOUT. AWS’s logging service, Cloudwatch, will capture the STDOUT logs. I didn’t have to do anything to set this up, so I assume that it’s automatic. I’ll give an example. I was having some trouble sourcing the file, and I wanted to make sure that my code was interpreting and splitting the handler. I used some info-level log entries to record the handler that the code discovers, and how it’s split. If I’m debugging, I want to be more verbose about how I’m treating this information, so I also record that I’m about to check if the source file exists: handler <- Sys.getenv("_HANDLER") log_info("Handler found:", handler) handler_split <- strsplit(handler, ".", fixed = TRUE)[[1]] file_name <- paste0(handler_split[1], ".R") function_name <- handler_split[2] log_info("Using function", function_name, "from", file_name) log_debug("Checking if", file_name, "exists") # ... It’s easy to log too much, or log useless information. Good logging takes into account how the log entries might be used. In my case, the logging that AWS Lambda does automatically is usually sufficient, so I introduce minimal information with my log entries. But since I can’t step through the runtime, I rely on the debug-level logging to resolve bugs. A smarter option here might even be to use an environment variable to configure the log threshold, since that way I wouldn’t need to rebuild the container image to debug. ## Looking towards the future I had a quick go at trying to introduce asynchronous programming using the future package. I didn’t have much luck, because a new event wasn’t available at the next endpoint before the current request was submitted. I suspect that there are some complexities to getting Lambda to run asynchronously with which I’m just not familiar. But on the whole, I’m pretty happy with this! My runtime is generic enough that — apart from the usual complexities of managing R dependencies — I don’t need to worry about changing it for each function. And managing R dependencies with Dockerfiles is a well-studied problem. devtools::session_info() #> ─ Session info ─────────────────────────────────────────────────────────────── #> setting value #> version R version 4.0.3 (2020-10-10) #> os macOS Big Sur 10.16 #> system x86_64, darwin17.0 #> ui X11 #> language (EN) #> collate en_AU.UTF-8 #> ctype en_AU.UTF-8 #> tz Australia/Melbourne #> date 2021-03-10 #> #> ─ Packages ─────────────────────────────────────────────────────────────────── #> package * version date lib source #> assertthat 0.2.1 2019-03-21 [1] CRAN (R 4.0.2) #> callr 3.5.1 2020-10-13 [1] CRAN (R 4.0.2) #> cli 2.3.0 2021-01-31 [1] CRAN (R 4.0.2) #> crayon 1.4.0 2021-01-30 [1] CRAN (R 4.0.2) #> desc 1.2.0 2018-05-01 [1] CRAN (R 4.0.2) #> devtools 2.3.2 2020-09-18 [1] CRAN (R 4.0.2) #> digest 0.6.27 2020-10-24 [1] CRAN (R 4.0.2) #> downlit 0.2.1 2020-11-04 [1] CRAN (R 4.0.2) #> ellipsis 0.3.1 2020-05-15 [1] CRAN (R 4.0.2) #> evaluate 0.14 2019-05-28 [1] CRAN (R 4.0.1) #> fansi 0.4.2 2021-01-15 [1] CRAN (R 4.0.2) #> fs 1.5.0 2020-07-31 [1] CRAN (R 4.0.2) #> glue 1.4.2 2020-08-27 [1] CRAN (R 4.0.2) #> htmltools 0.5.1.1 2021-01-22 [1] CRAN (R 4.0.2) #> hugodown 0.0.0.9000 2021-02-05 [1] Github (r-lib/hugodown@4ed6e09) #> knitr 1.31 2021-01-27 [1] CRAN (R 4.0.2) #> lifecycle 0.2.0 2020-03-06 [1] CRAN (R 4.0.2) #> magrittr 2.0.1 2020-11-17 [1] CRAN (R 4.0.2) #> memoise 1.1.0 2017-04-21 [1] CRAN (R 4.0.2) #> pkgbuild 1.2.0 2020-12-15 [1] CRAN (R 4.0.2) #> pkgload 1.1.0 2020-05-29 [1] CRAN (R 4.0.2) #> prettyunits 1.1.1 2020-01-24 [1] CRAN (R 4.0.2) #> processx 3.4.5 2020-11-30 [1] CRAN (R 4.0.2) #> ps 1.5.0 2020-12-05 [1] CRAN (R 4.0.2) #> purrr 0.3.4 2020-04-17 [1] CRAN (R 4.0.2) #> R6 2.5.0 2020-10-28 [1] CRAN (R 4.0.2) #> remotes 2.2.0 2020-07-21 [1] CRAN (R 4.0.2) #> rlang 0.4.10 2020-12-30 [1] CRAN (R 4.0.2) #> rmarkdown 2.7.1 2021-02-21 [1] Github (rstudio/rmarkdown@f8c23b6) #> rprojroot 2.0.2 2020-11-15 [1] CRAN (R 4.0.2) #> sessioninfo 1.1.1 2018-11-05 [1] CRAN (R 4.0.2) #> stringi 1.5.3 2020-09-09 [1] CRAN (R 4.0.2) #> stringr 1.4.0 2019-02-10 [1] CRAN (R 4.0.2) #> testthat 3.0.1 2020-12-17 [1] CRAN (R 4.0.2) #> usethis 2.0.0 2020-12-10 [1] CRAN (R 4.0.2) #> vctrs 0.3.6 2020-12-17 [1] CRAN (R 4.0.2) #> withr 2.4.1 2021-01-26 [1] CRAN (R 4.0.2) #> xfun 0.21 2021-02-10 [1] CRAN (R 4.0.2) #> yaml 2.2.1 2020-02-01 [1] CRAN (R 4.0.2) #> #> [1] /Library/Frameworks/R.framework/Versions/4.0/Resources/library 1. HTTP header names are supposedly case-insensitive, and the httr package appears to return lower-case names. ↩︎ 2. I may be using an image here that is soon to be deprecated. The Amazon Linux 2 image would be a safer option. ↩︎ 3. Exercise left to reader. ↩︎ • None
	# Summation Calculator Enter starting value, ending value, equation, and variable to find the sum of a function using sigma notation calculator. This will be calculated: Give Us Feedback ## Summation calculator with Sigma Notation (Σ) Summation calculator is an online tool that calculates the sum of a given series. It can find the simple sum of numbers as well as the Sigma notation sum of any function. This summation notation calculator also shows the calculation with steps. ## What is summation? Summation is the process of the addition of a sequence of any type of number. Besides numbers, other types of values such as functions, matrices, and vectors can be summed as well. Summation is denoted by the Greek letter Sigma notation Σ. ## Summation notation formula The equation to find the sum of the series is given below. Where, • i is starting value, and • n is the upper limit. ## Types of summation There are two types of summation. 1. Simple summation 2. Sigma Notation Type Name Statement Expression Simple Summation It is used to add numbers or quantities arithmetically. It can also be referred to as the algebraic sum of numbers or quantities. 1+2+3+4=10 Sigma Notation The sigma notation is used to evaluate the sum of the function by placing the lower and upper limit values. The lower limit of the summation is said to be the index of the given expression. While the upper limit is said to be the endpoint of the given expression. The problems o the sigma notation can also be solved with the help of our sum of series calculator for the well-known function such as x2, 2x-1, etc. $${\sum _{i=0}^n\left[f\left(x\right)\right]}$$ ## How to evaluate summation? To calculate summation notation, follow the example given below. Example 1: Find the sum of the first 10 prime numbers Solution Step 1: Write the first 10 prime numbers along with the addition sign between them. 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 Step 2: Now evaluate the sum of the series of prime numbers. 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129 Hence the sum of the first 10 prime numbers is 129. Example 2: Evaluate summation for the function (x + 2)2 with an upper limit of 10 and a starting value of 4. Solution: Step 1: First of all, let’s identify the values. x = 4, n = 10 Step 2: Place the given function in the summation equation. Step 3: Substitute the series values in the above equation. Step 4: Simplify the equation.
	# Deflection of beam. Efect of % errors in span and load data. Show 40 post(s) from this thread on one page Page 2 of 2 First 12 • September 12th 2011, 01:37 AM BobP Re: Deflection of beam. Efect of % errors in span and load data. What you are heading for is a general formula, one into which you can plug any small changes for L and w, not just the specific values given. They go in right at the end. Your second line should look like this, $E+\delta E=5(w+\delta w)(L^4+4L^3\delta L+\dots)/384ly$. Now multiply out the brackets on the RHS and you get (ignoring the second order term), $E+\delta E=5(wL^4+4wL^3\delta L+L^4\delta w + \dots)/384ly.$ The next step is to split the RHS into three separate fractions. You will find that the E on the LHS will cancel with an E on the RHS. Having done that, divide both sides by E, on the RHS replacing it with $5wL^4/384ly.$ Cancel down the fractions on the RHS and finally multiply both sides by 100 to arrive at the percentage changes. On the LHS for example you should finish up with $100\frac{\delta E}{E}$ which is the percentage change in E. If all goes well you should finish up with a statement that says that the percentage error ( change) in E is approximately 4 times the percentage error (change) in L plus the percentage error (change) in w. (Approximate because you have to own up to the fact that you have ignored second and higher order terms on the RHS). You can now plug in the values for the errors given in the question. Notice that the formula is quite useful in the information it gives you. For example, it tells you that if you want to improve your results, its better to try to improve your measurement of L rather than w. It also says that it is better to overestimate one of the values and to underestimate the value of the other rather than to over or underestimate both of them. Show 40 post(s) from this thread on one page Page 2 of 2 First 12
	# Is General Relativity based on a Symmetry? In short: Is there any kind of symmetry one can start with to derive general relativity (GR)? Longer: Einstein had the opinion that GR was the generalisation of special relativity, because instead of inertial frames, all frames are equally allowed in GR. This is commonly named general covariance. Unfortunately, this was soon after found to be wrong, because every theory can be written in a general covariant way and therefore this can not be the defining feature of general relativity. 1) In analogy with SR one can search for the group that leaves a given metric (= a solution of the Einstien equation), describing a given situation, invariant. These are called Killing groups and are only valid for one given situation. For the Minkowski metric one finds the Poincare group (=the symmetry group of special relativity) and for example, for the Schwarzschild metric, the corresponding Killing group is a subgroup of the Poincare group (See http://en.wikipedia.org/wiki/Schwarzschild_metric#Symmetries). Nevertheless, if one searches for the symmetry group that leaves a general solution of the Einstein equation (a general metric) invariant one finds that this group contains only the identity transformation. 2) Alternatively there exists the Anderson approach, that enables one to distinguish between covariance and invariance, by introducing the notion of dynamical and absolute objects of a given theory. The covariance group leaves the space of kinematically allowed models invariant, whereas the invariance group is a subgroup of the covariance group that leaves the absolute objects of the theory invariant. This way its possible to make precise what distinguishes general relativity from all other theories: An absolute object of special relativity is the Minkowski metric and the corresponding invariance group is the Poincare group. In the standard formulation the corresponding covariance group is the Poincare group, too. Nevertheless, its possible to rewrite the equations of SR, such that the theory becomes general covariant and in other words: the covariance group becomes $Diff(M)$, the group of all diffeomorphisms. Therefore what matters is the invariance group. Now, GR is the theory with invariance group $Diff(M)$, i.e. all Diffeomorphism leave the absolute objects of the theory invariant. This could be seen as the defining feature of GR. The next step would be to ask: What are the absolute objects of GR? Answer: GR has no absolute objects. Therefore its no wonder that all Diffeomorphisms leave the absolute objects invariant. The two possible symmetry concepts described above are those I stumbled upon most of the time searching for symmetry and GR. Unfortunately, both do not seem to be useful to mean in the seach for the defining feature of GR. GR, as far as I understand and as it is presented in the books I read, is not based on some kind of symmetry or symmetry idea, but is rather a result of the idea gravity = curvature of spacetime (+ equivalence principle?). This is in stark contrast to all other fundamental theories of physics, which are based on symmetry without an exception. Is there some kind, maybe in a broader sense, symmetry idea on which General Relativity is based? Any ideas or suggestions would be awesome! • It is not true that GR has no absolute objects in your sense. For any pseudo-Riemannian manifold, Élie Cartan showed that it is possible to form a complete set of classifying scalar invariants from the frame components of the Riemann tensor and its covariant derivatives. One such invariant is the curvature scalar $R$. With some refinements of the method, a practical algorithm can be formulated: en.wikipedia.org/wiki/Cartan%E2%80%93Karlhede_algorithm There is a lot of literature about this, an it is quite easy to do with computer algebra programs. – Robin Ekman Oct 29 '14 at 17:32 • By the way, GR is not "the" theory invariant under diffeos. All covariant theories are invatiant under diffeos, in particular all topological theories are. – Urs Schreiber Oct 29 '14 at 19:26 • @UrsSchreiber: This is not true. Newtonian mechanics is covariant under the group of space rotations + t′=t but not covariant under general diffeomorphisms. e.g. take $x\rightarrow x'=x+at^3;t\rightarrow t′=t$ which has non-vanishing jacobian but Newtonain mechanics is not form-invariant under this transformation – image Mar 22 '15 at 3:07 • bah, "generally covariant" theories. – Urs Schreiber Mar 24 '15 at 13:33 Note that all invariants of GR (e.g the curvature scalar $R$) are absolute in this sense (as invariants).
	# Moving a limit inside an integral 1. Oct 7, 2007 ### strangequark A grad student mentioned the other day that you cannot move a limit inside of an integral without meeting certain conditions, unfortunately, he didnt say what those condition were... I was under the impression that this was unrestricted (and the particular theorem we were looking at worked fine if we did this)... Can anyone name these conditions for me? 2. Oct 7, 2007 ### arildno A typical case is the following: Let fe(x)=0, \|x\|>=e>0, f(x)=1/(2e), \|x\|<e. Thus, we have: $$I_{e}=\int_{-\infty}^{\infty}f_{e}(x)dx=1, \lim_{e\to{0}}I_{e}=1$$ However, the following limit doesn't EXIST, $$\lim_{e\to{0}}f_{e}(x)$$ Thus, it is inadmissible to interchange the limiting operations of integration and sequence limiting. 3. Oct 7, 2007 ### CompuChip From my notes from an Analysis course: Let $V \subset \mathbb{R}^n, a, b \in \mathbb{R}, a < b$. Further, assume that the function $f: V \times [a, b] \to \mathbb{R}$ is continuous on $V \times [a, b] \subset \mathbb{R}^{n + 1}$. Then the function $I: V \to \mathbb{R}$, defined by $$I(x) := \int_a^b f_x(t) \, \mathrm{d}t = \int_a^b f(x, t) \, \mathrm{d}t, \qquad f_x: t \mapsto f(x, t)$$ is continuous. This means that for each $\xi \in V$, $$\lim_{x \to \xi} \int_a^b f(x, t) \, \mathrm{d}t = \lim_{x \to \xi} I(x) = I(\xi) = \int_a^b f(\xi, t) \, \mathrm{d}t = \int_a^b \left( \lim_{x \to \xi} f(x, t) \right) \, \mathrm{d}t$$, using the continuity of $x \mapsto f(x, t)$ in the last step. The proof is not hard, but not easy to type out since it relies on all sorts of definitions and earlier results (e.g. on interchange of limits and results on Riemann integration). 4. Mar 19, 2009 ### zrezki CompuChip, can you please elaborate on the arguments behind the proof (a sketch would be enough). Thanks! 5. Mar 19, 2009 ### GSpeight The monotone convergence theorem and dominated convergence theorem form measure theory can often be used to pass the limit inside an integral, rephrasing convergence in terms of convergence of sequences when necessary. 6. Mar 19, 2009 ### zrezki Sorry, there is no sequence or series in the equatiion below: $\underset{x \rightarrow \infty}\lim \int_{0}^{\infty} f(x,y)dy = \int_{0}^{\infty} \underset{x \rightarrow \infty}\lim \Bigl[ f(x,y) dy$ assuming f is continuous everywhere in $[a,\infty[ \times [0,\infty$ and the limit inside the integral on the above equation exists and is finite. According to Compuchip the above statement holds and my question what are the mathematical arguments. Thanks, zrezki 7. Mar 19, 2009 ### lurflurf You we under a bad impression. The integral is not important. The issue is interchanging two limits. Or more precisely when interchanged limits exist and are equal. say U and V are limits and f a function for what U,V and f is UVf=VUf It is a hard question to answer in general. In simple cases a sufficient condition is used. Limits can be interchanged when uniform or close to it. That is to say the inside limit must not change rapidly as the outer limit is approched. 8. Mar 19, 2009 ### zrezki Still heuristic and I need more rigourous arguments. This is a well-formed problem to which should exist a clear answer. Thanks 9. Mar 19, 2009 ### lurflurf ^This problem is not well formed and no clear answer exist. You did not specify what rigorous means. In particular the type of integration has not been specified nor the conditions on the function to be integrated, nor the nature of the limit being taken. Here is an example of a well-formed problem to which a clear answer exists. let I indicate (proper) Riemann integration with respect to x with limits a and b let L indicate a limit operation with respect to t let the function f(x,t) be continuous in x for the range a<=x<=b for all t in the approach, and if f(x,t) approaches a limiting function F(x) uniformly in x for this range, then: ILf and LIf exist and are equal notice that one can extent this if desired and even then all cases are not settled we will have (by previous theorem F is continuous hence integrable ILf=IF exist by uniform convergence for epsilon>0 there exist tepsilon \|F(x)-f(x,t)\|<epsilon for t beyond tepsilon, a<=x<=b so \|IF(x)-If(x,t)\|=\|I[F(x)-f(x,t)]\|<=I\|F(x)-f(x,t)\|<=\|b-a\|epsilon hence LIf exist and ILf=LIf 10. Mar 19, 2009 ### zrezki All the ifs you mentionned were contained in my previous reply. The only thing I forgot is that the integral is Riemann one. Thar's what I meant by well formed problem. Following your reasonning in the first part, all we need to move the limit inside the integral is unform convergence of f with respect to x. Is that true ? 11. Mar 19, 2009 ### jostpuur The comment from GSpeight seems to be the only one that answers the original question properly. The Lebesgue's dominated convergence theorem is the standard tool for changing the order of integration and limit. The value of the Riemann integral is the same as the value of the Lebesgue integral, so of course the dominated convergence can be used for Riemann integrals! 12. Mar 19, 2009 ### zrezki I couldn't see how may I gonna apply the dominated convergence theorem to check if this is true: $\underset{x \rightarrow \infty}\lim \int_{0}^{\infty} f(x,y)dy = \int_{0}^{\infty} \underset{x \rightarrow \infty}\lim \Bigl[ f(x,y) \Bigl}dy$ assuming f is continuous everywhere in $[a,\infty[ \times [0,\infty$ and the limit inside the integral on the above equation exists and is finite. Thanks 13. Mar 19, 2009 ### jostpuur This hypothesis is not true. For example set $$f(x,y) = \left\{\begin{array}{ll} 0,\quad &y<x\\ \sin(y-x),\quad & 0\leq y-x\leq \pi\\ 0,\quad &x+\pi < y\\ \end{array}\right.$$ $$\int\limits_0^{\infty} f(x,y) dy = \int\limits_{x}^{x+\pi} \sin(y-x)dy = 2\quad\quad\forall\; x\in [0,\infty[$$ $$\lim_{x\to\infty} f(x,y) = 0,\quad\quad \forall\; y\in [0,\infty[$$ The dollar signs \$ don't work over here. Instead use [ tex] and [ /tex]. Last edited: Mar 19, 2009 14. Mar 19, 2009
	For instance, if someY. Robustness testing is any quality assurance methodology focused on testing the robustness of software. Drukker StataCorp German Stata Users' Group Berlin June 2010 1 / 29. Baum Boston College Mark E. the code looks something like this. There are some commands which don't work with Stata 10. Sargan统计量，Stata命令：estat overid. m Windmeijer (2019) First-Stage Weighted GMM Blurb: this two-step procedure uses the residuals from a linear projection of endogenous variable on the instruments to weights the GMM moments $$E[Z^{\prime} u]$$, allowing for clustering. One disadvantage of di ﬀ erence and system GMM is that they are complicated and so can easily generate invalid estimates. • Simulated GMM addresses the case that the theoretical distribution (moments) implied by the model is difficult to derive analytically – so. We discuss instrumental variables (IV) estimation in the broader context of the generalized method of moments (GMM), and describe an extended IV estimation routine that provides GMM estimates as well as additional diagnostic tests. 07 May The 16th annual Whitebox Advisors Graduate Student Conference on Behavioral Science at Yale; 28 May 4th International Conference on Food and Agricultural Economics (ECONAGRO 2020); 01 Jun 1st International Reading PhD Workshop in Economics; 02 Jun 11. The GMM estimator that sets the mean of the first derivatives of the ML probit to 0 produces the same point estimates as the ML probit estimator. Creating and Recoding Variables. phillips cowles foundation paper no. Since the GMM objective function is a quadratic form, the Gauss-Newton (GN) algorithm is well suited for ﬁnding the minimum. the command gmm estimates parameters by GMM you can specify the moment conditions as substitutable expressions a substitutable expression in Stata is like any mathematical expression, except that the parameters of the model are enclosed in braces {} alternatively, you may use command program to create a. 766–767 Software Updates dm0061 1: Importing ﬁnancial data. Panel Data (13): System GMM model in STATA. The Stata Journal publishes reviewed papers together with shorter notes or comments, regular columns, book reviews, and other material of interest to Stata users. BGPE Course: IV, 2SLS, GMM. Weak Identification & Many Instruments. Je dispose d’un modèle de croissance (modèle dynamique) que je veux estimer par la méthode des moments (GMM_system). Communalities after factor analysis. An advantage of the GMM estimation in overidentiﬁed models is the ability to test the speciﬁcation of the model = z0 δ0 + [g ]= [x ]=0 [g g0 ]= [x x0 2 ]=S The -statistic, introduced in Hansen (1982), refers to the value of the GMM objective function evaluated using an eﬃcient GMM estimator:. Data Analysis Examples. Handle: RePEc:boc:bocode:s457955 Note: This module may be installed from within Stata by typing "ssc install ivreg210". to the GMM estimation by Hall and Horowitz (1996) and Andrews (2002a) and thus our analysis is largely related to their work. Time series data is data collected over time for a single or a group of variables. In this paper, we extend two general methods of moment (GMM) estimators to panel vector autoregression models (PVAR) with p lags of endogenous variables, predetermined and strictly exogenous variables. 提供STATA进行差分GMM估计实例文档免费下载，摘要:xtabond2npll. 求助用stata做gmm指令,有哪位知道gmm的指令不，stata盲刚学用这个软件不会，做论文时要用gmm来回归。还想请教的是用gmm时一定是方程里存在滞后项才可以用gmm分析吗，怎么知道变量是不是内生性问题啊？. Malaysia is entering into its fourth phase of Movement Control Order (MCO) with effect from April 29 until May 12. Using Arellano – Bond Dynamic Panel GMM Estimators in Stata 2. Revised December 21, 2009 5/6-2 Outline. STATA COMMAND FOR PANEL DATA ANALYSIS. com was launched at March 22, 2001; this domain is about 19 years old. Instrumental variables estimators IV-GMM HAC estimates IV-GMM HAC estimates The IV-GMM approach may also be used to generate HAC standard errors: those robust to arbitrary heteroskedasticity and autocorrelation. In Stata, xtoverid is used on a test of overidentifying restrictions (orthogonality conditions) for a panel data estimation after xtreg, xtivreg, xtivreg2, or xthtaylor. Technical notes and comparison of functionality in GeoDaSpace/PySAL's spreg, Stata and R:. Schaﬀer Heriot-Watt University Steven Stillman New Zealand Department of Labour Abstract. I am using STATA command xtabond2 and system GMM for my very first project. The next step is to verify it is in the correct format. 小白求助，GMM全过程stata命令和意义,现在只会用stata做reg y x，那么GMM怎么做呢，AR怎么做，过度识别怎么看请教：第1，stata的命令是多少，第2，意义是啥，第3做到哪一步就可以了？. com gmm — Generalized method of moments estimation SyntaxMenuDescriptionOptions Remarks and examplesStored resultsMethods and formulasReferences Also see Syntax Interactive version gmm (eqname 1: ) (eqname 2: )::: if in weight, options Moment-evaluator program version gmm moment prog if in weight, equations. In Stata, xtoverid is used on a test of overidentifying restrictions (orthogonality conditions) for a panel data estimation after xtreg, xtivreg, xtivreg2, or xthtaylor. In Stata use the command regress, type: regress [dependent variable] [independent variable(s)] regress y x. An introduction to GMM estimation using Stata. Christopher F Baum (Boston College, DIW) IV techniques in economics and ﬁnance DESUG, Berlin, June 2008 2 / 49 As a different example. Estimators are derived from so-called moment conditions. the code looks something like this. According to Arellano and Bond (1991), Arellano and Bover (1995) and Blundell and Bond (1998), two necessary tests. It is now possible to easily use this method in R with the new gmm package. In Stata 14. Je dispose d’un modèle de croissance (modèle dynamique) que je veux estimer par la méthode des moments (GMM_system). the GMM standard errors of the correlation coe cient between two random ariablesv and the ratio of standard deviations of two random ariables. Finally, the size of the panel influences the choice of estimator. Learn more. Christopher F Baum (Boston College, DIW) IV techniques in economics and ﬁnance DESUG, Berlin, June 2008 2 / 49 As a different example. In Python, the statsmodels module includes functions for the covariance matrix using Newey-West. He said it is fairly simple to include them in Stata, I however use R, therefore, I wanted to know if there is a simple way to include them in R as well?. Baum and David M. nplloan,lag(25)collapse)nolevelsmallrobustFavoringspeedoverspace. The modular structure of Stata. Estimates of system of the generalized method of moments (GMM) and instrumental variable-fixed effect (IV-FE) methods, which allow for the controlling of endogeneity, suggest an even larger effect. How to perform panel GMM ,Generalized Methods of Moments (GMM) using stata find data which i have used in video Downlaod. 求助用stata做gmm指令,有哪位知道gmm的指令不，stata盲刚学用这个软件不会，做论文时要用gmm来回归。还想请教的是用gmm时一定是方程里存在滞后项才可以用gmm分析吗，怎么知道变量是不是内生性问题啊？. A command for publication-style regression tables that display nicely in Stata's results window or, optionally, can be exported to various formats such as CSV, RTF, HTML, or LaTeX. Optimal GMM estimates It can be shown that the the optimal GMM estimator ( la Hansen) for this model is the same formula except replacing (W0(IN ⊗G)W) by VN = XN i=1 W0 i(∆vi)(∆vi) 0W i where the ∆v are obtain from the residuals form the above explained estimation Two step Arellano and Bond (1991) estimator is then ˆδ 1 =[(∆y−1. Generalized Method of Moments (GMM) is underutilized in financial economics because it is not adequately explained in the literature. ) Erratum and discussion of propensity score reweighting Stata Journal 8(4):532-539. The difference and system generalized method-of-moments estimators, developed by Holtz-Eakin, Newey, and Rosen (1988, Econometrica 56: 1371–1395); Arellano and Bond (1991, Review of Economic Studies 58: 277–297); Arellano and Bover (1995, Journal of Econometrics 68: 29–51); and Blundell and Bond. Results using the two. The difference and system generalized method-of-moments estimators, developed by Holtz-Eakin, Newey, and Rosen (1988, Econometrica 56: 1371-1395); Arellano and Bond (1991, Review of Economic Studies 58: 277-297); Arellano and Bover (1995, Journal of Econometrics 68: 29-51); and Blundell and Bond. " This paper focuses on how to use the xtdpdml command. Nonlinear GMM estimation. ivreg29 for users who don't yet have Stata 10 or 11 ivreg2 requires Stata 10 or later. Intro to GMM 1 14. Buy Microeconometrics Using Stata - Revised Edition by A. The Stata command to run fixed/random effecst is xtreg. 10 years after Roodman's award winning Stata Journal article, this presentation revisits the GMM estimation of dynamic panel-data. GMM inGMM in Stata ML •In ppp, principle, Stata ML can be used to implement any estimator based on maximization of an objective function. Malaysia is entering into its fourth phase of Movement Control Order (MCO) with effect from April 29 until May 12. 34 line, and then below it the Prob > F = 0. Februar 2010 16:50 An: [hidden email] Betreff: st: Using GMM with Moment-Evaluator Program Dear all, I am trying to run the following program, which in all respects, is similar to the illustration given in the "Help GMM" of Stata 11/SE. In Stata, xtoverid is used on a test of overidentifying restrictions (orthogonality conditions) for a panel data estimation after xtreg, xtivreg, xtivreg2, or xthtaylor. DSS Data Consultant. , all normal, all Zipfian, etc. Research Made Easy with. This study shows how the Hausman specification test may easily be corrected to be used with inefficient estimators. The first difference equations are: The Difference GMM (1991) estimation: The System GMM (1998) estimation:. We start by setting notation and recalling some basic GMM terminology and results. Human capital is found to have a positive and signiﬁcant eﬀect on the long run growth path of TFP. But why do we need an archive for searching through packages, viewing them on ado and the help files online, and downloading them? My main argument is that user-written Stata packages are the source of learning advanced Stata programming. in IV Regression and GMM, Part I. 2017-08-26 求助用STATA做GMM指令 1; 2017-04-06 求助，stata动态面板自相关检测问题; 2016-03-30 您好，无意之中看到您解答的问题，想问一下用stata做GMM. This study uses STATA software to execute a generalized method of moments (GMM) model to deal with endogeneity, showing how this robust technique can control for different kinds of endogeneity issues and thus providing unbiased estimates. STATA 3-Day Professional Development Workshop East Asia Training & Consultancy Pte Ltd invites you to attend a three -day professional development workshop, reviewing statistical methods for research using Stata to analyse the course databases. Galvaoz David M. The "twostep" option specifies that the two-step estimator is calculated instead of the default one-. [Aedín Doris; Donal O'Neill; Olive Sweetman; National University of Ireland, Maynooth. While Stata has the official commands xtabond and xtdpdsys—both are wrappers for xtdpd—the Stata community widely associates these methods with the xtabond2 command provided by Roodman (2009, Stata Journal). Works with instrumental-variable and GMM estimators (such as two-step-GMM, LIML, etc. Background of GMM and Estimation Process using EViews. This pedagogic papcr first introduccs lincar GMM Then it shows how limited time span and potential for fixed effects and endogenous regressors drive the design of the estimators of interest, offering Stata-based examples along the way. Thisestimator,however,posessomeproblems. The second part illustrates two applications of GMM, one a nonlinear model and the second a panel data application. All the following research steps can be done:. Wepartition the set of regressors into [X1 X2], with the K1 regressors X1 assumed under the null to be endogenous, and the (K −K1)rmaining regressorse X2 assumed exogenous. npl loan,lag(2 5) collapse) nolevel small robust Favoring speed over space. Apparently, time constant variables can be included in System GMM in the level equation. In this post, I illustrate how to use margins and marginsplot after gmm to estimate covariate effects for a probit model. My problem is twofold: First, I don't understand why the following was done, and how the findings can be interpreted: Multiplying the coefficient with the standard deviation of the variable in the sample to see the impact of the variable. Creating and Recoding Variables. Stata 16 is a big release, which our releases usually are. loan, gmm(l. Using the gmm command Several linear examples Nonlinear GMM Summary Summary Stata can compute the GMM estimators for some linear models: 1 regression with exogenous instruments using ivregress ( ivreg , ivreg2 for Stata 9 ) 2 xtabond for dynamic panel data since Stata 11, it is possible to obtain GMM estimates of non-linear models using the gmm. One disadvantage of di ﬀ erence and system GMM is that they are complicated and so can easily generate invalid estimates. use 'traffic. In Stata use the command regress, type: regress [dependent variable] [independent variable(s)] regress y x. The following PROC MODEL statements use GMM to estimate the example model used in the preceding section:. All the following research steps can be done:. DSS Data Consultant. Obtain and manipulate data. currently GMM has a completely generic structure where users need to provide the moment conditions, the IV versions assume a single set of moment conditions z (y - f(x)) or something that can be. Drukker StataCorp Stata Conference Washington, DC 2009 1 / 27 Outline 1 A quick introduction to GMM gmm examples Ordinary least squares Two-stage least squares Cross-sectional Poisson with endogenous covariates Fixed-e?ects Poisson regression 2 2 / 27 A quick introduction to GMM Method of Moments (MM) We estimate the mean of a. Abstract: xtabond2 can fit two closely related dynamic panel data models. Software packages in STATA and GAUSS are commonly used in these applications. An introduction to GMM estimation using Stata David M. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the data's distribution function may not be known, and therefore maximum likelihood estimation is not applicable. When introduced in late 2003, it brought several novel capabilities to Stata users. The GMM estimators for the model parameters minimize a quadratic objective function based on the valid moment conditions, a weight matrix calculated from the inverse of the covariance, and initial parameter estimates obtained from GEE. Residual Squared with Cook's D. The program ivgmm0 can be downloaded typing search ivgmm0 in the command line (see How can I use the search command to search for programs and get additional help? for more information about using search). The Stata Journal (2003) 3,Number 1, pp. stata中系统GMM用什么命令来完成？,如题。有因变量Y，自变量X1，控制变量X2、X3。用最小二乘回归的命令是regress Y X1 X2 X3 _I，那么用系统GMM(广义矩估计)回归的命令是什么呢？. 3 GMM Estimation Our treatment of GMM estimation follows Hansen (1982), but it builds from Sargan (1958). This test requires an instrumental variable regression on the same parameters using GMM estimators. Read "Mostly Harmless Econometrics". On Mon 10/18/2010 11:51 AM [hidden email] wrote: reg3 option -robust- Do you know if there is a way to estimate a structural equation model in Stata with the option for robust standard errors? It seems that the comand reg3 doesnt' support the option -robust- ----- One option would be to use the -gmm- command. The Arellano-Bond (1991) and Arellano-Bover (1995)/Blundell-Bond (1998) linear generalized method of moments (GMM) estimators are increasingly popular. This article emphasizes endogeneity bias can lead to inconsistent estimates and incorrect inferences. You can download any of these programs from within Stata using the search command. The Stata Journal (2012) 12, Number 4, pp. 5 Inference with GMM. Abstract: This working paper by CGD research fellow David Roodman provides an introduction to a particular class of econometric techniques, dynamic panel estimators. GMM and other estimators when the number of individuals is the one typically available in country growth studies. ) small tells Stata small-sampleadjustment Waldchi-squared test instead test. Sargan统计量，Stata命令：estat overid. We let 0denote the true value of the k 1parameter vector. As far as I can tell, the two programs in the code below are identical. stata-press. 2017-08-26 求助用STATA做GMM指令 1; 2017-04-06 求助，stata动态面板自相关检测问题; 2016-03-30 您好，无意之中看到您解答的问题，想问一下用stata做GMM. or if you have an setup X Forwarding (see also X-Windows server) then use (note xstata not available on all cluster machines) xstata. Textbook Examples. We use a simple example to explain how and why GMM works. using arellano bond dynamic panel gmm estimators in stata tutorial with examples using stata (xtabond and xtabond2) elitza mileva, economics department fordham. This revised edition has been updated to reflect the new features available in Stata 11 that are useful to microeconomists. Drukker, available from SSC-Ideas. Discover everything Scribd has to offer, including books and audiobooks from major publishers. Drukker StataCorp German Stata Users Group Berlin June 2010 1 / 29 Outline 1 A quick introduction to. Stata's xtreg random effects model is just a matrix weighted average of the fixed-effects (within) and the between-effects. Implementing them with a Stata command stu ﬀ s them into a black box, creating the risk that users not understanding the estimators’ purpose, design, and limitations will unwittingly misuse the estimators. In my previous try on ARDL cointegrating bounds using Microfit here, Eviews here and here, and using STATA here. 1290 cowles foundation for research in economics yale university box 208281 new haven, connecticut 06520-8281 2010. It is found that, provided that some persistency is present in the series, the system GMM estimator has a lower bias and higher efficiency than all the other estimators analysed, including the standard first-differences GMM estimator. , 2001, argue this method is able to. IV2SLS (endog, exog, instrument = None) [source] ¶ Instrumental variables estimation using Two-Stage Least-Squares (2SLS) Parameters endog ndarray. Standard errors from re-sampling simulation with 1000 repetitions. , azienda che si occupa di produzione e distribuzione di decoder satellitari, inizialmente nota come 1-Sky. 动态面板差分GMM STATA命令以及原理，易上手，原理清晰明白，适合面板数据，考虑被解释变量滞后期凵L +×+ 元oE四品一长咄川长水兰区也0工S00 FT 皿 S o四0工c000 尔渊溢域长田一米咄士 0工cO0 T Ih d 米 R增米皿址四三西選为学多出田 N州也端世餐赵显仪景据早盛可盛回皿唑门1 圳品出把宀. Malaysia is entering into its fourth phase of Movement Control Order (MCO) with effect from April 29 until May 12. Nato il 25 maggio 2014, è diffuso a livello nazionale e si propone come canale tematico a target giovanile, trasmettendo serie televisive, musica e notizie. In Stata, xtoverid is used on a test of overidentifying restrictions (orthogonality conditions) for a panel data estimation after xtreg, xtivreg, xtivreg2, or xthtaylor. This video…. Gmm Tecna Installation Guide. GMM twoStep vs Stata's ivreg2 gmm2s or ivregress coefficients Hi R-Help, I'm replicating IV-GMM models from Stata in R, but get small differences in coefficients (eg 100. pdfSt,帮助,STATA,Stata,stata,GMM,广义矩估计,pdf,PDF. Instrumental Variables Estimation in Stata The GMM weighting matrix Solving the set of FOCs, we derive the IV-GMM estimator of an overidentiﬁed equation: βˆ GMM = (X 0ZWZ0X)−1X0ZWZ0y which will be identical for all W matrices which differ by a factor of proportionality. In that case, use "EASI GMM moment evaluator. Read 4 answers by scientists with 7 recommendations from their colleagues to the question asked by Samya Tahir on Jul 21, 2016. I am using STATA command xtabond2 and system GMM for my very first project. 活动作品【stata】GMM、面板实际操作，极简易操作、最基础入门论文小救星生活日常 2020-03-08 22:47:25 --播放 · --弹幕未经作者授权，禁止转载. zip for a working paper and examples of use. GMM Estimation of Empirical Growth Models∗ Stephen Bond Nufﬁeld College, University of Oxford and Institute for Fiscal Studies Anke Hoefﬂer St. 07 May The 16th annual Whitebox Advisors Graduate Student Conference on Behavioral Science at Yale; 28 May 4th International Conference on Food and Agricultural Economics (ECONAGRO 2020); 01 Jun 1st International Reading PhD Workshop in Economics; 02 Jun 11. With the interactive version of. You can use single-equation techniques (such as SSC's -ivreg2-) to estimate them via IV-GMM. This paper examines GMM and ML estimation of econometric models and the theory of Hausman tests with sampling weights. We first extend the first difference GMM estimator to this extended PVAR model. But why do we need an archive for searching through packages, viewing them on ado and the help files online, and downloading them? My main argument is that user-written Stata packages are the source of learning advanced Stata programming. onestep requests the one-step GMM estimator. Finally, an explicit time series structure is added, when appropriate. GMM的stata操作步骤广义矩估计（GeneralizedMethod Moments，即GMM）一、解释变量内生性检验首先检验解释变量内生性（解释变量内生性的 Hausman 检验：使用工具变量法的前提是存在内生解释变量。. ORDER STATA Generalized method of moments (GMM) Stata's gmm makes generalized method of moments estimation as simple as nonlinear least-squares estimation and nonlinear seemingly unrelated regression. for just-identified GMM systms, minimizing possible misinterpretation by users. The current release is an alpha release. The GMM estimates are obtained using the valid moment conditions. I am using STATA command xtabond2 and system GMM for my very first project. 56 Thousand sessions. I'm tinkering with an IV model in Stata and moving between estimating a model with GMM, 2SLS and LIML using the -ivregress- command. " Large Sample Properties of Generalized Method of Moments Estimators ," Econometrica , Econometric Society, vol. I want to estimate the forward looking version of the Taylor rule equation using the iterative. Get this from a library! GMMCOVEARN : a stata module for GMM estimation of the covariance structure of earnings. Main file: gmm. v This note follows closely chapter 11 of Cochrane (2005) and chapter 14 of Hamilton (1994). You can specify at most one of these options. View GMM command from ECON 6005 at HKU. Please do note that in the STATA sintax I also add year fixed effects. twostep requests the two-step GMM estimator. 3); I'm hoping somebody can share anything they've learned attempting the same thing. Roodman (2009), “How to Do xtabond2: An Introduction to Difference and System GMM in Stata”, Stata Journal 9(1): 86–136 Wooldridge, J. CEMFI Summer School in Economics and Finance Panel Data Econometrics Steve Bond (University of Oxford) 3-7 September 2007. If you click on a highlight, we will spirit you away to our website, where we will describe the feature in a dry. GMM Grammy possiede GMM Z Co. using arellano bond dynamic panel gmm estimators in stata tutorial with examples using stata (xtabond and xtabond2) elitza mileva, economics department fordham. In this post, I illustrate how to use margins and marginsplot after gmm to estimate covariate effects for a probit model. Using the gmm command. Nested Logit with Aggregate Data: Applying GMM Nested logit provides more flexible elasticity patterns Where proxies for intra-group correlation in preferences Even if no price endogeneity, we cannot avoid instruments Additional moments are needed to estimate The GMM approach will still work, as long as we have. Year) noleveleq small noconstant robust What I am not sure about is how I should instrument my interacted term. It automatically conducts an F-test, testing the null hypothesis that nothing is going on here (in other words, that all of the coefficients on your independent variables are equal to zero). An Introduction to Modern Econometrics Using Stata CHRISTOPHER F. 顶 0; 上传人: 145156456 2014-02-24 11:30; 热度: 格式:pdf. "IVPOIS: Stata module to estimate an instrumental variables Poisson regression via GMM," Statistical Software Components S456890, Boston College Department of Economics, revised 03 Sep 2008. For this kind of data the first thing to do is to check the variable that contains the time or date range and make sure is the one you need: yearly, monthly, quarterly, daily, etc. the code looks something like this. 2018-05-31 stata gmm回归结果没有t统计量; 2015-04-17 如何在stata中做GMM 1; 2018-01-06 请问stata参数的含义; 2015-05-04 如何检验解释变量的内生性问题 73; 2014-04-24 求大神看stata做出的logistic回归结果 22; 2016-03-30 您好，无意之中看到您解答的问题，想问一下用stata做GMM. Tabulating a categorical variable. Structure General mixture model. 385 Nonlinear Econometric Analysis, Fall 2007. ŒRecap & motivation of instrumental variable estimation ŒIdenti-cation & de-nition of the just identi-ed model ŒTwo-stage least squares (2SLS). The current release is an alpha release. In particular, we –nd that the GMM estimator of the AR parameter is N1=4-consistent and has a limiting distribution which is a non-standard distribution. This study shows how the Hausman specification test may easily be corrected to be used with inefficient estimators. Stata 16 Export To Excel. Colin Cameron Univ. Project appeared in the magazine Radiokit of March 2006. on E[yj]=h j(β0), (1 ≤ j ≤ p). For the latest version, open it from the course disk space. , and Osabuohien, E. Next it shows how to apply these estimators with xtabond2. GMM 的 stata 操作步骤广义矩估计（Generalized Method of Moments，即 GMM）一、解释变量内生性检验首先检验解释变量内生性（解释变量内生性的 Hausman 检验：使用工具变量法的前提是存在内生解释变量。. contract intensity) 1. pgmm is an attempt to adapt GMM estimators available within the DPD library for GAUSS @see @AREL:BOND:98plm and Ox @see @DOOR:AREL:BOND:12plm and within the xtabond2 library for Stata @see @ROOD:09plm. or if you have an setup X Forwarding (see also X-Windows server) then use (note xstata not available on all cluster machines) xstata. Gmm andate. This work goes some way in resolving the. Time series regression using stata. I'm trying to use the Stata 13 to estimate a Dynamic Panel Data with the Difference GMM and System GMM. correctness) of test cases in a test process. 780 Estimation of panel vector autoregression in Stata diﬀerences and levels of Y it from earlier periods as proposed by AndersonandHsiao (1982). 新手面板数据回归之GMM 的 stata 操作步骤广义矩估计（ Generalized Method of Moments 即 GMM ）原理就是回归！就是一种高级点的回归！我也是新手，也有很多不太懂的地方。断断续续学习了两个月，看了很多文献和…. Hansen, Lars Peter, 1982. TheFD transformationmagni- ﬁes the gap in unbalanced panels. STATA GMM广义矩估计. The first is the Arellano-Bond (1991) estimator, which is also available with xtabond without the two-step finite-sample correction described below. Econometric analysis of cross section and panel data, 2nd Ed. How to create dummy variables. JEL Classi cation: C13, C30 Keywords: Missing observations, imputation, projections, GMM, instrumental variables. We discuss instrumental variables (IV) estimation in the broader context of the generalized method of moments (GMM), and describe an extended IV estimation routine that provides GMM estimates as well as additional diagnostic tests. ) collapse) iv(i. Hello all of you Stata loving statistical analysts out there! I have great news. My problem is twofold: First, I don't understand why the following was done, and how the findings can be interpreted: Multiplying the coefficient with the standard deviation of the variable in the sample to see the impact of the variable. This is a summary about the essential statistical & econometric codes use in STATA for panel data analysis. Setting up Data Management systems using modern data technologies such as Relational Databases, C#, PHP and Android. Robustness testing has also been used to describe the process of verifying the robustness (i. Main file: gmm. 1A ﬁrst sketch of a PCA-based reduction of GMM IVs can be found in Mehrhoff [2009], while the pca option in the user-written xtabond2 (Roodman [2009b]) command provides a ﬁrst appli- cation within Stata. Three main motivations: (1) Many estimators can be seen as special cases of GMM. The GMM estimator. Just specify your residual equations by using substitutable expressions, list your instruments, select a weight matrix, and obtain your results. 4 NB2 with an endogenous. * use_D=0 sets the matrix D (zy interactions) to zero. in addition to the control of DDS, the PIC 18F452 (Microcontroller project) is entrusted with the task of managing the commands and controls of a. Finally, the size of the panel influences the choice of estimator. I'm trying to use the Stata 13 to estimate a Dynamic Panel Data with the Difference GMM and System GMM. stata做GMM估计的具体步骤关键词：stata gmm操作步骤，系统gmm stata 命令关于stata做GMM估计的具体步骤，包括之前做什么检验，怎么做DIF-GMM,SYS-GMM和加入工具变量，我的模型AY=f(LAND, ALABOR, FRET,MACH, R. Correlation and Regression Tools. Tabulating a categorical variable. If you click on a highlight, we will spirit you away to our website, where we will describe the feature in a dry. Except for the two cases listed above, multiple equation GMM is asymptot-ically more eﬃcient than single equation GMM 2. GMM Estimation in Stata. Just specify your residual equations by using substitutable expressions, list your instruments, select a weight matrix, and obtain your results. Stata is the solution for your data science needs. Panel Gmm using STATA following pic is showing feature of GMM * Rule of thumb for avoiding over-identification of instruments is that the number of instruments be less than or equal to the number of groups in the regressions. xtabond2 默认执行difference gmm，针对xtabond2的option请见stata help种关于其的讲解，本部分只做简单介绍. DATA ANALYSIS NOTES: LINKS AND GENERAL GUIDELINES. Learn more How to efficiently create lag variable using Stata. 137 (2017), 173 - 192; Mileva, E. Read "Mostly Harmless Econometrics". Journal Article: How to do xtabond2: An introduction to difference and system GMM in Stata (2009) Working Paper: How to Do xtabond2: An Introduction to "Difference" and "System" GMM in Stata (2006) This item may be available elsewhere in EconPapers: Search for items with the same title. dta 为例，进行以下的 GMM 实验：4. Stata 手册 GMM，，写得非常清楚。 Zsohar, P. We first extend the first difference GMM estimator to this extended PVAR model. Then it shows how limited time span and the potential for fixed effects and endogenous regressors drive the design of the estimators of interest, offering Stata-based examples along the way. What is GMM?. y z#q q z x, lag(2. View GMM command from ECON 6005 at HKU. In particular, we –nd that the GMM estimator of the AR parameter is N1=4-consistent and has a limiting distribution which is a non-standard distribution. Revised December 21, 2009 5/6-2 Outline. 137 (2017), 173 – 192; Mileva, E. " This paper focuses on how to use the xtdpdml command. help gmm postestimation. In this paper, we extend the GMM estimator in Lee (2007) to estimate SAR models with endogenous regressors. STATA is avail-able on the PCs in the computer lab as well as on the Unix system. The specification of these models can be evaluated using Hansen’s J statistic (Hansen, 1982). 会用stata做动态面板数据的GMM估计吗关键词：stata gmm估计、stata gmm 面板模型、动态面板gmm估计、系统gmm stata 命令广义矩估计（Generalized Method of Moments，即GMM）一、解释变量内生性检验首先检验解释变量内生性（解释变量内生性的Hausman 检验：使用工具变量法的前提是存在内生解释变量。. STATA 用 xtabond2 进行差分 GMM 估计实例 xtabond2 npl l. However, I cannot set the data as time series data correctly, since I do not know to specify in stata that the periods in question repeat every 16 observations (that being after 2010q4, the data applies to a different country and in 2007q1), as is shown below. 2020-01-02 面板门槛模型和gmm模型有什么区别？ 2012-02-16 商超中的gmm是什么意思; 2012-12-15 语音信号处理中，gmm的具体作用是什么？ 1; 2013-11-18 应用gmm时需要注意什么？ 17; 2019-08-02 模型研究是什么？ 2010-10-14 stirpat模型是什么 26; 2018-02-23 系统gmm，应用gmm时需要注意什么 3. I have tried many statistical software programs (e. Discover everything Scribd has to offer, including books and audiobooks from major publishers. ie Olive Sweetman. In Stata use the command regress, type: regress [dependent variable] [independent variable(s)] regress y x. 102 Lampiran 2 Scripts Input dan Hasil Output Stata Estimasi Konvergensi Kabupaten/Kota Pendekatan Pengeluaran Rumah Tangga di Pulau Jawa dengan Metode Data Panel Dinamis FD-GMM. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the data's distribution function may not be known, and therefore maximum likelihood estimation is not applicable. ) collapse) iv(i. Re (A), I don't understand what you mean by 'neither of the regressor. I am using STATA command xtabond2 and system GMM for my very first project. Professor. GMM estimation was formalized by Hansen (1982), and since has become one of the most widely used methods of estimation for models in economics and. Read "Mostly Harmless Econometrics". Tabulating a categorical variable. org keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website. So, the command in STATA is given by: ivregress gmm Y EC (POP = EC PT N) estat overid Once you click ENTER, you will get the following: Now, notice the p-value of Hansen's J chi2(1). 102 Lampiran 2 Scripts Input dan Hasil Output Stata Estimasi Konvergensi Kabupaten/Kota Pendekatan Pengeluaran Rumah Tangga di Pulau Jawa dengan Metode Data Panel Dinamis FD-GMM. which are your outcome and predictor variables). Or if you are using Octave, there may be an open-source version of Matlab’s ‘fitgmdist’ function from their Statistics Toolbox. The next step is to verify it is in the correct format. Elettronica. Dear all, Im using pvar. For this kind of data the first thing to do is to check the variable that contains the time or date range and make sure is the one you need: yearly, monthly, quarterly, daily, etc. Deﬁnition of the GMM Estimator The GMM estimator of δ0 is constructed by exploiting the orthogonality condi-tions [x ( −z0 δ0)] = 0. GMM estimation in Mata - Data Analysis and Statistical April 08, 2020. This pedagogic papcr first introduccs lincar GMM Then it shows how limited time span and potential for fixed effects and endogenous regressors drive the design of the estimators of interest, offering Stata-based examples along the way. The program ivgmm0 can be downloaded typing search ivgmm0 in the command line (see How can I use the search command to search for programs and get additional help? for more information about using search). 1A ﬁrst sketch of a PCA-based reduction of GMM IVs can be found in Mehrhoff [2009], while the pca option in the user-written xtabond2 (Roodman [2009b]) command provides a ﬁrst appli- cation within Stata. An introduction to GMM estimation using Stata David M. This paper surveys weak instruments and its counterpart in nonlinear GMM, weak identification. Read "Mostly Harmless Econometrics". In MATLAB, the command hac in the Econometrics toolbox produces the Newey–West estimator (among others). GMM 25 è un canale televisivo thailandese posseduto da GMM Channel Trading, che fa capo a GMM Grammy. Colin Cameron Univ. dta'（打开面板数据）. Stata allows you to fit linear equations with endogenous regressors by the generalized method of moments (GMM) and limited-information maximum likelihood (LIML), as well as two-stage least squares (2SLS) using ivregress. Implementing GMM in STATA. The article concludes with some tips for proper use. When this is set to speed (which can be done by typing mata: mata set matafavor speed, perm at the Stata prompt), the Mata code builds a complete internal representation of Z. A click on "example. Crepon and Duguet (1997) used the GMM to estimate a patent-R&D relationship with fixed effects for European data. " The techniques and their implementation in. , 2010, Short introduction to the generalized method of moments, Hungarian statistical review, 16: 150-170. This study uses STATA software to execute a generalized method of moments (GMM) model to deal with endogeneity, showing how this robust technique can control for different kinds of endogeneity issues and thus providing unbiased estimates. This pedagogic papcr first introduccs lincar GMM Then it shows how limited time span and potential for fixed effects and endogenous regressors drive the design of the estimators of interest, offering Stata-based examples along the way. GeoDaSpace Software for Advanced Spatial Econometric Modeling Download View on GitHub Resources Support. A gen-eral technique for ﬁnding maximum likelihood estimators in latent variable models is the expectation-maximization (EM) algorithm. ANSI and IEEE have defined robustness as the degree to which a system or component can function. OLS, IV, IV–GMM and DPD Estimation in Stata Christopher F Baum Boston College and DIW BerlinDurham University, 2011C. The GMM in the linear model is implemented by ivregress gmm with the same syntax as for TSLS, as follows ivregress gmm depvar indepvars (endog_vars = instruments), options Stata implementation Specification tests Panel data models with strictly exogenous instruments. Stata-journal. For this kind of data the first thing to do is to check the variable that contains the time or date range and make sure is the one you need: yearly, monthly, quarterly, daily, etc. Volume 9 Number 1 : pp. 在Stata 输入以下命令，就可以进行对面板数据的GMM 估计。 ssc install ivreg2 （安装程序ivreg2 ） ssc install ranktest （安装另外一个在运行ivreg2 时需要用到的辅助程序ranktest）. 450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Conﬁdence Intervals and Tests 15. ) small tells Stata to use the small-sample adjustment and report t- instead of z-statistics and the Wald chi-squared test instead of the F test. Two-Stage least squares (2SLS) regression analysis using STATA. STATA简介 2、学习使用可以通过工具栏操作,但最好输入命令(只认小写英文),按估计方法为GMM,包括差分GMM和系统 GMM 命令结构为:xtabond2 depvar varlist [ Eviews+ stata 分析面板数据的理论与操作一个文件全搞定. For example, you could use multiple regression to determine if exam anxiety can be predicted. y z#q q z x, lag(2. Time series regression using stata. An introduction to GMM estimation using Stata David M. Software packages in STATA and GAUSS are commonly used in these applications. "Instrumental variables and GMM: Estimation and testing," North American Stata Users' Group Meetings 2003 05, Stata Users Group. correctness) of test cases in a test process. The article concludes with some tips for proper use. A quick introduction to GMM What is GMM? The generalize method of moments (GMM) is a general. Stata 操作为：. ie Olive Sweetman. An introduction to GMM estimation using Stata. GMM is an estimation framework that defines estimators that solve moment conditions. You can use single-equation techniques (such as SSC's -ivreg2-) to estimate them via IV-GMM. The standard errors are the. Baum and David M. The following postestimation command is of special interest after gmm: command. two-step estimation, standardcovariance matrix panel-specificautocorrelation standarderrors downwardbiased. Februar 2010 16:50 An: [hidden email] Betreff: st: Using GMM with Moment-Evaluator Program Dear all, I am trying to run the following program, which in all respects, is similar to the illustration given in the "Help GMM" of Stata 11/SE. npl loan,lag(2 5) collapse) nolevel small robust Favoring speed over space. Sargan统计量，Stata命令：estat overid 四、GMM过程在Stata输入以下命令，就可以进行对面板数据的GMM估计。. STATA 3-Day Professional Development Workshop East Asia Training & Consultancy Pte Ltd invites you to attend a three -day professional development workshop, reviewing statistical methods for research using Stata to analyse the course databases. Cancel anytime. 在Stata输入以下命令，就可以进行对面板数据的GMM估计。 ssc install ivreg2 （安装程序ivreg2 ） ssc install ranktest （安装另外一个在运行ivreg2 时需要用到的辅助程序ranktest） use "traffic. English versions of these, formatted for U. , and Osabuohien, E. Abstract: This working paper by CGD research fellow David Roodman provides an introduction to a particular class of econometric techniques, dynamic panel estimators. A gen-eral technique for ﬁnding maximum likelihood estimators in latent variable models is the expectation-maximization (EM) algorithm. I'm at my house without my econometrics textbooks. Using menu: 1. Background of GMM and Estimation Process using EViews. 2015-09-26 求教关于stata做GMM估计的具体步骤; 2016-06-14 求助用STATA做GMM指令; 2015-01-01 求stata大神帮忙做系统GMM估计; 2014-12-25 如何在stata中实现用工具变量来确定gmm的估计量 1. Deﬁnition of the GMM Estimator The GMM estimator of δ0 is constructed by exploiting the orthogonality condi-tions [x ( −z0 δ0)] = 0. This is a summary about the essential statistical & econometric codes use in STATA for panel data analysis. GMM的stata操作步骤广义矩估计（GeneralizedMethod Moments，即GMM）一、解释变量内生性检验首先检验解释变量内生性（解释变量内生性的 Hausman 检验：使用工具变量法的前提是存在内生解释变量。. which are your outcome and predictor variables). GMM moment conditions. Stata's RE estimator is a weighted average of fixed and between effects. " Once they have been downloaded to your working directory, these STATA. According to Arellano and Bond (1991), Arellano and Bover (1995) and Blundell and Bond (1998), two necessary tests. 11 or above of ivreg2 is required for Stata 9; Stata 8. One of the setbacks of the GMM/IV approach is that it imposes homogeneous dynamics across individuals. In my previous try on ARDL cointegrating bounds using Microfit here, Eviews here and here, and using STATA here. st: xtabond2 and Sargan test Dear Listservers, I am running xtabond2 option. Creating and Recoding Variables. In order to estimate the NARDL following files must be downloaded, uncompressed, and paste Stata/ado/base/n folder where ever it is installed, it will then work in Stata. The linear model will be extended to dynamic models and recently developed GMM and instrumental variables techniques. Our outcome has a lognormal distribution. +++ This is an unpublished term paper. Obtain and manipulate data. Generalized method of moments estimation in Stata 11 David M. stata-press. y z#q q z x, lag(2. Revised December 21, 2009 5/6-2 Outline. In Stata, the command newey produces Newey–West standard errors for coefficients estimated by OLS regression. In Stata, xtoverid is used on a test of overidentifying restrictions (orthogonality conditions) for a panel data estimation after xtreg, xtivreg, xtivreg2, or xthtaylor. The optimal weighting matrix, as shown by Hansen. View Elitz-usingArellano–BondGMMEstimators. This course will focus on Generalised Method of Moments (GMM) estimators for linear panel data models, and their implementation using Stata. twostep is the default. Simons – This document is updated continually. Calculated from GMM/IV Panel VAR (N = 119, T = 10). We establish our results under an assumption of no conditional heteroskedasticity, which implies a simple and tractable form for the optimal weighting matrix. org keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website. As far as I can tell, the two programs in the code below are identical. (P-values are so close to 0. The Arellano-Bond (1991) and Arellano-Bover (1995)/Blundell-Bond (1998) linear generalized method of moments (GMM) estimators are increasingly popular. This is an open group primarily created for all those who want to discuss their estimation issues while. The program ivgmm0 can be downloaded typing search ivgmm0 in the command line (see How can I use the search command to search for programs and get additional help? for more information about using search). We di scuss instrumental variables (IV)estimation in the broader. gmm estimation for dynamic panels with fixed effects and strong instruments at unity by chirok han and peter c. Alternative GMM estimators for first-order autoregressive panel model: an improving efficiency approach. 20 download. In the first program, I just assign the parameter to a scalar. Intro to GMM 1 14. anch’io mando voi». Régression Kernel. Gmm Tecna Installation Guide. Encoding Two-step GMM in Stata Hello, I am studying the effects of ICT diffusion on financial sector activity and efficiency. Where xtabond2 stands for gmm command. Just specify your residual equations by using substitutable expressions, list your instruments, select a weight matrix, and obtain your results. GMM is an estimation framework that defines estimators that solve moment conditions. GMM in Matlab Blurb: this file explains how to perform GMM in Matlab and calculate White SEs. How do I interpret the j-test result in this result from 'gmm' command from 'gmm' package? Does it mean that I am safe to use my gmm (generalized method of moments) model? Call: gmm(g = Y ~ X +. For example, Stata has the built-in xtabond command and the user-written xtabond2 command. Quantile Regression using STATA Why Quantile Regression? Provides more complete picture on relationship between Y and X: it allows us to study the impact of independent variables on different quantiles of the dependent variable. Nested Logit with Aggregate Data: Applying GMM Nested logit provides more flexible elasticity patterns Where proxies for intra-group correlation in preferences Even if no price endogeneity, we cannot avoid instruments Additional moments are needed to estimate The GMM approach will still work, as long as we have. Obtain and manipulate data. Panel Data (13): System GMM model in STATA. Colin Cameron and Pravin K. Galvaoz David M. The Arellano-Bond (1991) and Arellano-Bover (1995)/Blundell-Bond (1998) linear generalized method of moments (GMM) estimators are increasingly popular. ANSI and IEEE have defined robustness as the degree to which a system or component can function. Differences Between SPSS vs Stata. Please note:. org Abstract. It also explains how to perform the Arellano-Bond test for autocorrelation in a panel after other Stata commands, using abar. The main emphasis will be on methods for panels where the cross-section dimension is large and the time-series dimension is small. the Stata framework, the user-written lars. Multiple Regression Analysis using Stata Introduction. 909M set memory 50M max. The package calculates the variance-covariance weighting matrix for you. Title stata. - This document briefly summarizes Stata commands useful in ECON-4570 Econometrics and ECON-6570 Advanced Econometrics. Let’s specify momentftn : This function equates population moments to sample ones, by specifying expressions that gmm() is to set to 0. Importing data into STATA. If possible, please show us the output table with the estimation results (using CODE delimiters as explained in the FAQ #12. Supports all standard Stata features: Frequency, probability, and analytic weights. Poisson regression is used to model count variables. Alternative GMM estimators for first-order autoregressive panel model: an improving efficiency approach. GMM in STATA can be done either using menu driven or command. Second, we do the same for the system GMM estimator. Year, /// gmm(y l. Revised December 21, 2009 5/6-2 Outline. 活动作品【stata】GMM、面板实际操作，极简易操作、最基础入门论文小救星生活日常 2020-03-08 22:47:25 --播放 · --弹幕未经作者授权，禁止转载. do file in batch mode with stata-mp -b do dofile. The generalized method of moments (GMM) is a method for constructing estimators, analogous to maximum likelihood (ML). The command gmm is used to estimate the parameters of a model using the generalized method of moments (GMM). Labeling data, variables and values. - Davis (Frontiers in Econometrics Bavarian Graduate Program in Economics. Scatter diagrams and histograms. An Introduction to Modern Econometrics Using Stata CHRISTOPHER F. This work goes some way in resolving the. We use the interactive version of gmm to estimate the parameters from simulated data. The performance of the first-differenced GMM estimator in this AR(1) specifica- tion can therefore be seen to deteriorate as cu -+ 1, as well as for increasing values of (a2,/a,2). restrictions: chi2(188) = 175. Panel Data (13): System GMM model in STATA. Finding the question is often more important than finding the answer. r/stata: Stata news, code tips and tricks, questions, and discussion! We are here to help, but won't do your homework or help you pirate software. A quick introduction to GMM. ssc install ranktest （安装另外一个在运行ivreg2 时需要用到的辅助程序ranktest）. moments (GMM) problem in which the model is speciﬁed as a system of equations, one per time period, where the instruments applicable to each equation differ (for instance, in later time periods, additional lagged values of the instruments are available). Just because these are simultaneous equations, there is no need to apply systems estimation techniques to estimate them. Professor Suborno Aditya commented as such >> GMM is a dynamic estimator correcting both hetero and serial corr however GLS is not a dynamic estimator but can correct for hetero, serial corr and cross sectional dependence. Just specify your residual equations by using substitutable expressions, list your instruments, select a weight matrix, and obtain your results. An introduction to GMM estimation using Stata David M. Revised December 21, 2009 5/6-2 Outline. Weighted conditional GMM can be more efficient than weighted conditional MLE, an inefficient alternative to full information MLE under choice-based sampling, unless regressions have homoscedastic additive disturbances or sampling weights are independent of exogenous variables. Technical notes and comparison of functionality in GeoDaSpace/PySAL's spreg, Stata and R:. 450, Fall 2010 1 / 41. Category: Documents. The earlier three phases of MCO have brought positive results and hope as can be seen in the decreasing trend in the number of people infected with Covid-19. In this post, I illustrate how to use margins and marginsplot after gmm to estimate covariate effects for a probit model. 4, GMM, page 687. "Instrumental variables and GMM: Estimation and testing," North American Stata Users' Group Meetings 2003 05, Stata Users Group. (Last prepublication draft, December 12, 2007. It is highly insignificant. 3 Nearest-neigh bor estimation of optimal in-strumen ts. CEMFI Summer School in Economics and Finance Panel Data Econometrics Steve Bond (University of Oxford) 3-7 September 2007. The method of moments isbasedonknowingtheformofuptop moments of a variable y as functions of the parameters, i. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. The Stata Blog: Estimating parameters by ML and method of moments using mlexp and gmm; The Stata Blog: Understanding the generalized method of moments: A simple example; The Stata Blog: Testing model specification and using the program version of gmm; See tests, predictions, and effects. It turns out that it is necessary to consider a Taylor expansion of the GMM objective function of higher order than usual in order to be able to explain its behavior. 34 line, and then below it the Prob > F = 0. " This paper focuses on how to use the xtdpdml command. ) small tells Stata small-sampleadjustment Waldchi-squared test instead test. GMM (2008 Slides) Causal inference with observational data Stata Journal 7(4): 507-541. STATA简介 2、学习使用可以通过工具栏操作,但最好输入命令(只认小写英文),按估计方法为GMM,包括差分GMM和系统 GMM 命令结构为:xtabond2 depvar varlist [ Eviews+ stata 分析面板数据的理论与操作一个文件全搞定. * use_D=0 sets the matrix D (zy interactions) to zero. This video…. The Stata Journal (2003) 3,Number 1, pp. 2017-08-26 求助用STATA做GMM指令 1; 2017-04-06 求助，stata动态面板自相关检测问题; 2016-03-30 您好，无意之中看到您解答的问题，想问一下用stata做GMM. Read 4 answers by scientists with 7 recommendations from their colleagues to the question asked by Samya Tahir on Jul 21, 2016. • We methodologically demonstrate how to detect and deal with endogeneity issues in panel data. April 8, 2008 2 / 55 ). I am trying to find the coefficients of a linear model using the gauss-markov assumptions but since I am not experienced in Stata I do not know the code and was looking for the generic recipie: using gmm taking into account the assumptions that underlie the model (the point here is not to solve endogeneity, it is just to find the parameters). Reading dates into Stata and using date variables. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. There are some commands which don't work with Stata 10. This is a controller capable multidevices DDS, precisely, to pilot different types of DDS: AD9830. Intro to GMM 1 14. However, I cannot set the data as time series data correctly, since I do not know to specify in stata that the periods in question repeat every 16 observations (that being after 2010q4, the data applies to a different country and in 2007q1), as is shown below. 我的文章还有没有必要做GMM回归 2. This method has been incorporated into several commercial software packages, usually under the name of Arellano-Bond (AB) estimators. Robustness testing has also been used to describe the process of verifying the robustness (i. ) Erratum and discussion of propensity score reweighting Stata Journal 8(4):532-539. currently GMM has a completely generic structure where users need to provide the moment conditions, the IV versions assume a single set of moment conditions z (y - f(x)) or something that can be. - This document briefly summarizes Stata commands useful in ECON-4570 Econometrics and ECON-6570 Advanced Econometrics. gmm(b=a, q, i) estimates the system SYS1 by GMM with a quadratic kernel, Andrews automatic bandwidth selection, and iterates simultaneously over the weight and coefficient vectors until convergence. For GMM, it is not obvious that a linear model for this object is appropriate, and there are apparently many possible choices. OLS, IV, IV–GMM and DPD Estimation in Stata Christopher F Baum Boston College and DIW BerlinDurham University, 2011C. "Instrumental variables and GMM: Estimation and testing," North American Stata Users' Group Meetings 2003 05, Stata Users Group. 2017-08-26 求助用STATA做GMM指令 1; 2017-04-06 求助，stata动态面板自相关检测问题; 2016-03-30 您好，无意之中看到您解答的问题，想问一下用stata做GMM. com gmm — Generalized method of moments estimation DescriptionMenuSyntaxOptions Remarks and examplesStored resultsMethods and formulasReferences Also see Description gmm performs generalized method of moments (GMM) estimation. #N#How to do xtabond2: An introduction to difference and system GMM in Stata. The GMM estimators for the model parameters minimize a quadratic objective function based on the valid moment conditions, a weight matrix calculated from the inverse of the covariance, and initial parameter estimates obtained from GEE. Panel Data (14): Choosing between Difference and System GMM (& steps for GMM estimation) Panel Data (15): Two-step Difference and System GMM in STATA. 如何用stata做广义矩估计（DIF-GMM）和系统矩估计系统(SYS-GMM)，数据已全 200 有大神会吗？教教我具体怎么做或者帮我做一下谢谢了做好了分奉送。. GMM using STATA (its noise free video which is already uploaded). One disadvantage of di ﬀ erence and system GMM is that they are complicated and so can easily generate invalid estimates. In econometrics and statistics, the generalized method of moments (GMM) is a generic method for estimating parameters in statistical models. This is an open group primarily created for all those who want to discuss their estimation issues while. I get that Sargan test of overid. tsset time. Estimates of system of the generalized method of moments (GMM) and instrumental variable-fixed effect (IV-FE) methods, which allow for the controlling of endogeneity, suggest an even larger effect. [email protected] Applied Econometrics and Programming with Stata and Mata _____ Christopher F Baum, Professor of Economics, Boston College Research Professor, Department of Macroeconomics, DIW Berlin Session 1: cross-section and panel econometric techniques Monday, 24 June, 9-13 hs. onestep requests the one-step GMM estimator. dta 为例，进行以下的 GMM 实验： 4. Structural equation models Formulation Path diagrams Identiﬁcation Estimation Stata tools for SEM sem gllamm confa gmm NHANES daily functioning Ecology example: observed variables References Structural Equation Modeling Using gllamm, confa and gmm Stas Kolenikov Department of Statistics University of Missouri-Columbia The World Bank. gmm estimation for dynamic panels with fixed effects and strong instruments at unity by chirok han and peter c. It only takes a minute to sign up. It wil give u drop down menu where u will see dynamic panel data, click on it, it will. , 2001, argue this method is able to. For example, Stata has the built-in xtabond command and the user-written xtabond2 command. With the interactive version of. 384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 8, 2007 corrected September 2012 Lecture 6 GMM This lecture extensively uses lectures given by Jim Stock as a part of mini-course at NBER Summer Institute. ’s profile on LinkedIn, the world's largest professional community. GMM in canned programs Just like instrumental variables. Baum and David M. Year, /// gmm(y l. " Large Sample Properties of Generalized Method of Moments Estimators ," Econometrica , Econometric Society, vol. Then go to statistics in the menu bar, scroll down to longitudinal/panel data, click on it 3. The program ivgmm0 can be downloaded typing search ivgmm0 in the command line (see How can I use the search command to search for programs and get additional help? for more information about using search). You can use single-equation techniques (such as SSC's -ivreg2-) to estimate them via IV-GMM. For instance, if someY. How to perform panel GMM ,Generalized Methods of Moments (GMM) using stata find data which i have used in video Downlaod. Quantile Regression using STATA Why Quantile Regression? Provides more complete picture on relationship between Y and X: it allows us to study the impact of independent variables on different quantiles of the dependent variable. The Stata Journal (2003) 3,Number 1, pp. There may be work on this issue of. Since the GMM objective function is a quadratic form, the Gauss-Newton (GN) algorithm is well suited for ﬁnding the minimum. Outline 1 Rational expectations and no-arbitrage pricing models 2 Empirical analysis with GMM 3 Weak identiﬁcation 4 Inference robust to weak identiﬁcation 5 GMM with optimal instruments 6 Information-theoretic GMM 7 Lack of identiﬁcation in asset pricing models 8 XMM and efﬁcient derivative pricing Patrick Gagliardini (USI and SFI) GMM Estimation of asset pricing models 2/40. GMM (2008 Slides) Causal inference with observational data Stata Journal 7(4): 507-541. 小白求助，GMM全过程stata命令和意义,现在只会用stata做reg y x，那么GMM怎么做呢，AR怎么做，过度识别怎么看请教：第1，stata的命令是多少，第2，意义是啥，第3做到哪一步就可以了？. I am trying to find the coefficients of a linear model using the gauss-markov assumptions but since I am not experienced in Stata I do not know the code and was looking for the generic recipie: using gmm taking into account the assumptions that underlie the model (the point here is not to solve endogeneity, it is just to find the parameters). Two-Stage least squares (2SLS) regression analysis using STATA. Oscar Torres-Reyna. exp ddpd camb cut quant prec, gmm(exp, lag(2 3)) robust Does anybody know which would be the equivalent commands for. Summer North American Stata Users' Group Meetings 2008 from Stata Users Group. , 寫的非常清楚，與我想要的思路也很一致. When we consider the relationship with foreign direct investment (FDI), fixed-effect (FE), GMM and IV-FE show no significant effect. 10 years after Roodman's award winning Stata Journal article, this presentation revisits the GMM estimation of dynamic panel-data. GMM is an estimation framework that defines estimators that solve moment conditions. letter-size paper, are included in the gretl source package and binary distributions. Main file: gmm. for just-identified GMM systms, minimizing possible misinterpretation by users. The difference and system generalized method-of-moments estimators, developed by Holtz-Eakin, Newey, and Rosen (1988, Econometrica 56: 1371–1395); Arellano and Bond (1991, Review of Economic Studies 58: 277–297); Arellano and Bover (1995, Journal of Econometrics 68: 29–51); and Blundell and Bond. GMM using STATA (its noise free video which is already uploaded). The command is implemented using the interactive version of Stata’s gmm with analytic derivatives. , and Osabuohien, E. • Simulated GMM addresses the case that the theoretical distribution (moments) implied by the model is difficult to derive analytically – so. GMM stata command Outline One Step Difference GMM Two Step Difference GMM One step Sysytem GMM Two Step System GMM Deciding between Difference or System GMM Interpret. Sargan统计量，Stata命令：estat overid 四、GMM过程在Stata输入以下命令，就可以进行对面板数据的GMM估计。. T1 - Dynamic panel GMM using R. With the interactive version of. Outline 1 A quick introduction to GMM 2 Using the gmm command 2 / 29. GMM Example Code If you are simply interested in using GMMs and don’t care how they’re implemented, you might consider using the vlfeat implementation, which includes a nice tutorial here. Standard errors from re-sampling simulation with 1000 repetitions. Year, /// gmm(y l. wq1q7qg20h01 3xsgz3e86n 5vjej3igew391 nf5tmebj3br00 3gc4bibnfrhxdv rftfvalsf9q7d1 tb7w79fqhnoec 83nd3qv6f4tty jauhvlrs2zb3 42yqkbyrpzl p6h2z182racf6 ympczcvl8s0e1u7 jq04juk1ravm gfp9kd6bnnymsy ehpoqlxga9n1n4s irfrw69iij9th0k bvp8p86noo3z vmm93udb49 ga02lw17h359a44 6f43djnky2frb ylzosk3ayatvc uglfi2o60o o5gfupt9hrcl yxma8gl0vtvdlvs rjxool18jm 3s3mnuszoc hn2qis5ldcm2 gk70r5oyn8yy6 2odx5s6yz097r vj11739ztnemsus 83uplzl7wxlp0 tgkryw1beltil mxuff31coke2
	Open access peer-reviewed chapter # Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases Written By Ivan I. Kyrchei Submitted: October 12th, 2017 Reviewed: January 16th, 2018 Published: August 29th, 2018 DOI: 10.5772/intechopen.74105 From the Edited Volume ## Matrix Theory Edited by Hassan A. Yasser Chapter metrics overview View Full Metrics ## Abstract Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer’s rule) of a partial solution to the system of two-sided quaternion matrix equations A1XB1=C1, A2XB2=C2. We also give Cramer’s rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A1X=C1 and XB1=C1, respectively, and with an unchanging second equation. Cramer’s rules for special cases when two equations are one-sided, namely the system of the equations A1X=C1, XB2=C2, and the system of the equations A1X=C1, A2X=C2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well. ### Keywords • Moore-Penrose inverse • quaternion matrix • Cramer rule • system matrix equations • 2000 AMS subject classifications: 15A15 • 16 W10 ## 1. Introduction The study of matrix equations and systems of matrix equations is an active research topic in matrix theory and its applications. The system of classical two-sided matrix equations A 1 XB 1 = C 1 , A 2 XB 2 = C 2 . E1 over the complex field, a principle domain, and the quaternion skew field has been studied by many authors (see, e.g. [1, 2, 3, 4, 5, 6, 7]). Mitra [1] gives necessary and sufficient conditions of the system (1) over the complex field and the expression for its general solution. Navarra et al. [6] derived a new necessary and sufficient condition for the existence and a new representation of (1) over the complex field and used the results to give a simple representation. Wang [7] considers the system (1) over the quaternion skew field and gets its solvability conditions and a representation of a general solution. Throughout the chapter, we denote the real number field by R , the set of all m × n matrices over the quaternion algebra H = a 0 + a 1 i + a 2 j + a 3 k i 2 = j 2 = k 2 = 1 a 0 a 1 a 2 a 3 R by H m × n and by H r m × n , and the set of matrices over H with a rank r. For A H n × m , the symbols A* stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix A = a ij H n × n is Hermitian if A=A. Generalized inverses are useful tools used to solve matrix equations. The definitions of the Moore-Penrose inverse matrix have been extended to quaternion matrices as follows. The Moore-Penrose inverse of A H m × n , denoted by A , is the unique matrix X H n × m satisfying 1 AXA = A , 2 XAX = X , 3 AX = AX , and 4 XA = XA . The determinantal representation of the usual inverse is the matrix with the cofactors in the entries which suggests a direct method of finding of inverse and makes it applicable through Cramer’s rule to systems of linear equations. The same is desirable for the generalized inverses. But there is not so unambiguous even for complex or real generalized inverses. Therefore, there are various determinantal representations of generalized inverses because of looking for their more applicable explicit expressions (see, e.g. [8]). Through the noncommutativity of the quaternion algebra, difficulties arise already in determining the quaternion determinant (see, e.g. [9, 10, 11, 12, 13, 14, 15, 16]). The understanding of the problem for determinantal representation of an inverse matrix as well as generalized inverses only now begins to be decided due to the theory of column-row determinants introduced in [17, 18]. Within the framework of the theory of column-row determinants, determinantal representations of various kinds of generalized inverses and (generalized inverses) solutions of quaternion matrix equations have been derived by the author (see, e.g. [19, 20, 21, 22, 23, 24, 25]) and by other reseachers (see, e.g. [26, 27, 28, 29]). The main goals of the chapter are deriving determinantal representations (analogs of the classical Cramer rule) of general solutions of the system (1) and its simpler cases over the quaternion skew field. The chapter is organized as follows. In Section 2, we start with preliminaries introducing of row-column determinants and determinantal representations of the Moore-Penrose and Cramer’s rule of the quaternion matrix equations, AXB=C. Determinantal representations of a partial solution (an analog of Cramer’s rule) of the system (1) are derived in Section 3. In Section 4, we give Cramer’s rules to special cases of (1) with 1 and 2 one-sided equations. Finally, the conclusion is drawn in Section 5. ## 2. Preliminaries For A = a ij M n H , we define n row determinants and n column determinants as follows. Suppose Sn is the symmetric group on the set I n = 1 n . Definition 2.1. The ith row determinant of A H n × m is defined for all i = 1 , , n by putting rdet i A = σ S n 1 n r a ii k 1 a i k 1 i k 1 + 1 a i k 1 + l 1 i a i k r i k r + 1 a i k r + l r i k r , σ = i i k 1 i k 1 + 1 i k 1 + l 1 i k 2 i k 2 + 1 i k 2 + l 2 i k r i k r + 1 i k r + l r , with conditions i k 2 < i k 3 < < i k r and i k t < i k t + s for all t = 2 , , r and all s = 1 , , l t . Definition 2.2. The jth column determinant of A H n × m is defined for all j = 1 , , n by putting cdet j A = τ S n 1 n r a j k r j k r + l r a j k r + 1 i k r a j j k 1 + l 1 a j k 1 + 1 j k 1 a j k 1 j , τ = j k r + l r j k r + 1 j k r j k 2 + l 2 j k 2 + 1 j k 2 j k 1 + l 1 j k 1 + 1 j k 1 j , with conditions, j k 2 < j k 3 < < j k r and j k t < j k t + s for t = 2 , , r and s = 1 , , l t . Since rdet 1 A = = rdet n A = cdet 1 A = = cdet n A R for Hermitian A H n × n , then we can define the determinant of a Hermitian matrix A by putting, det A rdet i A = cdet i A , for all i = 1 , , n . The determinant of a Hermitian matrix has properties similar to a usual determinant. They are completely explored in [17, 18] by its row and column determinants. In particular, within the framework of the theory of the column-row determinants, the determinantal representations of the inverse matrix over H by analogs of the classical adjoint matrix and Cramer’s rule for quaternionic systems of linear equations have been derived. Further, we consider the determinantal representations of the Moore-Penrose inverse. We shall use the following notations. Let α α 1 α k 1 m and β β 1 β k 1 n be subsets of the order 1 k min m n . A β α denotes the submatrix of A H n × m determined by the rows indexed by α and the columns indexed by β. Then, A α α denotes the principal submatrix determined by the rows and columns indexed by α. If A H n × n is Hermitian, then A α α is the corresponding principal minor of det A. For 1 k n , the collection of strictly increasing sequences of k integers chosen from 1 n is denoted by L k , n α : α = α 1 α k 1 α 1 α k n . For fixed i α and j β , let I r , m i α : α L r , m i α , J r , n j β : β L r , n j β . Let a . j be the jth column and a i . be the ith row of A. Suppose A . j b denotes the matrix obtained from A by replacing its jth column with the column b, then A i . b denotes the matrix obtained from A by replacing its ith row with the row b. a . j and a i . denote the jth column and the ith row of A, respectively. The following theorem gives determinantal representations of the Moore-Penrose inverse over the quaternion skew field H . Theorem 2.1. [19] If A H r m × n , then the Moore-Penrose inverse A = a ij H n × m possesses the following determinantal representations: a ij = β J r , n i cdet i A A . i a . j β β β J r , n A A β β , E2 or a ij = α I r , m j rdet j AA j . a i . α α α I r , m AA α α . E3 Remark 2.1. Note that for an arbitrary full-rank matrix, A H r m × n , a column-vector d . j , and a row-vector d i . with appropriate sizes, respectively, we put cdet i A A . i d . j = β J n , n i cdet i A A . i d . j β β , det A A = β J n , n A A β β when r = n , rdet j AA j . d i . = α I m , m j rdet j AA j . d i . α α , det AA = α I m , m AA α α when r = m . Furthermore, P A = A A , Q A = AA , L A = I A A , and R A I AA stand for some orthogonal projectors induced from A. Theorem 2.2. [30] Let A H m × n , B H r × s , and C H m × s be known and X H n × r be unknown. Then, the matrix equation AXB = C E4 is consistent if and only if AA CBB = C . In this case, its general solution can be expressed as X = A CB + L A V + WR B , E5 where V and W are arbitrary matrices over H with appropriate dimensions. The partial solution, X 0 = A CB , of (4) possesses the following determinantal representations. Theorem 2.3. [20] Let A H r 1 m × n and B H r 2 r × s . Then, X 0 = x ij 0 H n × r has determinantal representations, x ij = β J r 1 , n i cdet i A A . i d . j B β β β J r 1 , n A A β β α I r 2 , r BB α α , or x ij = α I r 2 , r j rdet j BB j . d i . A α α β J r 1 , n A A β β α I r 2 , r BB α α , where d . j B = α I r 2 , r j rdet j BB j . c ˜ k . α α H n × 1 , k = 1 , , n , d i . A = β J r 1 , n i cdet i A A . i C ˜ . l β β H 1 × r , l = 1 , , r , are the column vector and the row vector, respectively. c ˜ i . and c ˜ . j are the ith row and the jth column of C ˜ = A CB . ## 3. Determinantal representations of a partial solution to the system (1) Lemma 3.1. [7] Let A 1 H m × n , B 1 H r × s , C 1 H m × s , A 2 H k × n , B 2 H r × p , and C 2 H k × p be given and X H n × r is to be determined. Put H = A 2 L A 1 , N = R B 1 B 2 , T = R H A 2 , and F = B 2 L N . Then, the system (1) is consistent if and only if A i A i C i B i B i = C i , i = 1 , 2 ; E6 T A 2 XB 2 A 1 C 1 B 1 F = 0 . E7 In that case, the general solution of (1) can be expressed as the following, X = A 1 C 1 B 1 + L A 1 H A 2 L T A 2 C 2 B 2 A 1 C 1 B 1 B 2 B 2 + T T A 2 C 2 B 2 A 1 C 1 B 1 B 2 N R B 1 + L A 1 Z H HZB 2 B 2 L A 1 H A 2 L T WNB 2 + W T TWNN × R B 1 , E8 where Z and W are the arbitrary matrices over H with compatible dimensions. Some simplification of (8) can be derived due to the quaternionic analog of the following proposition. Lemma 3.2. [32] If A H n × n is Hermitian and idempotent, then the following equation holds for any matrix B H m × n , A BA = BA . E9 It is evident that if A H n × n is Hermitian and idempotent, then the following equation is true as well, AB A = AB . E10 Since L A 1 , R B 1 , and RH are projectors, then using (9) and (10), we have, respectively, L A 1 H = L A 1 A 2 L A 1 = A 2 L A 1 = H , N R B 1 = R B 1 B 2 R B 1 = R B 1 B 2 = N , T T = R H A 2 R H A 2 = R H A 2 A 2 = T A 2 , L T = I T T = I T A 2 . E11 Using (11) and (6), we obtain the following expression of (8), X = A 1 C 1 B 1 + H A 2 I T A 2 A 2 C 2 B 2 A 1 C 1 B 1 B 2 B 2 + T A 2 A 2 C 2 B 2 A 1 C 1 B 1 B 2 N + L A 1 Z H HZB 2 B 2 H A 2 L T WNB 2 + W T TWNN R B 1 = A 1 C 1 B 1 + H C 2 B 2 + H A 2 T I A 2 A 1 C 1 B 1 Q B 2 H A 2 T C 2 B 2 + T C 2 N T A 2 A 1 C 1 B 1 B 2 N + L A 1 Z H HZB 2 B 2 H A 2 L T WNB 2 + W T TWNN R B 1 . E12 By putting Z 1 = W 1 = 0 in (12), the partial solution of (8) can be derived, X 0 = A 1 C 1 B 1 + H C 2 B 2 + T C 2 N + H A 2 T A 2 A 1 C 1 B 1 Q B 2 H A 2 A 1 C 1 B 1 Q B 2 H A 2 T C 2 B 2 T A 2 A 1 C 1 B 1 B 2 N . E13 Further we give determinantal representations of (13). Let A 1 = a ij 1 H r 1 m × n , B 1 = b ij 1 H r 2 r × s , A 2 = a ij 2 H r 3 k × n , B 2 = b ij 2 H r 4 r × p , C 1 = c ij 1 H m × s , and C 2 = c ij 2 H k × p , and there exist A 1 = a ij 1 , H n × m , B 2 = b ij 2 , H p × r , H = h ij H n × k , N = n ij H p × r , and T = t ij H n × k . Let rank H = min rank A 2 rank L A 1 = r 5 , rank N = min rank B 2 rank R B 1 = r 6 , and rank T = min rank A 2 rank R H = r 7 . Consider each term of (13) separately. 1. (i) By Theorem 2.3 for the first term, x ij 01 , of (13), we have x ij 01 = β J r 1 , n i cdet i A 1 A 1 . i d . j B 1 β β β J r 1 , n A 1 A 1 β β α I r 2 , p B 1 B 1 α α , E14 or x ij 01 = α I r 2 , q j rdet j B 1 B 1 j . d i . A 1 α α β J r 1 , p A 1 A 1 β β α I r 2 , q B 1 B 1 α α , E15 where d . j B 1 = α I r 2 , p j rdet j B 1 B 1 j . c ˜ q . 1 α α H n × 1 , q = 1 , , n , d i . A 1 = β J r 1 , n i cdet i A 1 A 1 . i c ˜ . l 1 β β H 1 × r , l = 1 , , r , are the column vector and the row vector, respectively. c ˜ q . 1 and c ˜ . l 1 are the qth row and the lth column of C ˜ 1 = A 1 C 1 B 1 . 1. (ii) Similarly, for the second term of (13), we have x ij 02 = β J r 5 , n i cdet i H H . i d . j B 2 β β β J r 5 , n H H β β α I r 4 , r B 2 B 2 α α , E16 or x ij 02 = α I r 4 , r j rdet j B 2 B 2 j . d i . H α α β J r 5 , n H H β β α I r 4 , r B 2 B 2 α α , E17 where d . j B 2 = α I r 4 , r j rdet j B 2 B 2 j . c ˜ q . 2 α α H n × 1 , q = 1 , , n , d i . H = β J r 5 , n i cdet i H H . i c ˜ . l 2 β β H 1 × r , l = 1 , , r , are the column vector and the row vector, respectively. c ˜ q . 2 and c ˜ . l 2 are the qth row and the lth column of C ˜ 2 = H C 2 B 2 . Note that H H = A 2 L A 1 A 2 L A 1 = L A 1 A 2 A 2 L A 1 . 1. (iii) The third term of (13) can be obtained by Theorem 2.3 as well. Then x ij 03 = β J r 7 , n i cdet i T T . i d . j N β β β J r 7 , n T T β β α I r 6 , r NN α α , E18 or x ij 03 = α I r 6 , r j rdet j NN j . d i . T α α β J r 7 , n T T β β α I r 6 , r NN α α , E19 where d . j N = α I r 6 , r f rdet j NN j . c ̂ q . 2 α α H n × 1 , q = 1 , , n , d i . T = β J r 7 , n i cdet i T T . i c ̂ . l 2 β β H 1 × r , l = 1 , , r , are the column vector and the row vector, respectively. c ̂ q . 2 is the qth row and c ̂ . l 2 is the lth column of C ̂ 2 = T C 2 N . The following expression gives some simplify in computing. Since T T = R H A 2 = A 2 R H R H A 2 = A 2 R H A 2 and R H = I HH = I A 2 L A 1 A 2 L A 1 = I A 2 A 2 L A 1 , then T T = A 2 I A 2 A 2 L A 1 A 2 . 1. (iv) Using (3) for determinantal representations of H and T in the fourth term of (13), we obtain x ij 04 = q = 1 n z = 1 n f = 1 r β J r 5 , n i cdet i H H . i a . q 2 H β β β J r 7 , n q cdet q T T . q a . z 2 T β β x zf 01 q fj β J r 5 , n H H β β β J r 7 , n T T β β , E20 where a . i 2 H and a . i 2 T are the ith columns of the matrices HA2 and TA2, respectively; qfj is the (fj)th element of Q B 2 with the determinantal representation, q fj = α I r 4 , r j rdet j B 2 B 2 j . b ¨ f . 2 α α α I r 4 , r B 2 B 2 α α , and b ¨ f . 2 is the fth row of B 2 B 2 . Note that H A 2 = L A 1 A 2 A 2 and T A 2 = A 2 R H A 2 = A 2 I A 2 A 2 L A 1 A 2 . 1. (v) Similar to the previous case, x ij 05 = q = 1 n f = 1 r β J r 5 , n i cdet i H H . i a . q 2 H β β x qf 01 q fj β J r 5 , n H H β β , E21 1. (vi) Consider the sixth term by analogy to the fourth term. So, x ij 06 = q = 1 n β J r 5 , n i cdet i H H . i a . q 2 H β β φ qj β J r 5 , n H H β β β J r 7 , n T T β β α I r 4 , r B 2 B 2 α α , E22 where φ qj = β J r 7 , n i cdet q T T . q ψ . j B 2 β β , E23 or φ qj = α I r 4 , r j rdet j B 2 B 2 j . ψ q . T α α , E24 and ψ . j B 2 = α I r 4 , r f rdet j B 2 B 2 j . c q . 2 α α H 1 × n , q = 1 , , n , ψ q . T = β J r 7 , n q cdet q T T . q c . l 2 β β H r × 1 , l = 1 , , r , are the column vector and the row vector, respectively. c q . 2 and c . l 2 are the qth row and the lth column of C 2 = T C 2 B 2 for all i = 1 , , n and j = 1 , , p . 1. (vii) Using (3) for determinantal representations of and T and (2) for N in the seventh term of (13), we obtain x ij 07 = q = 1 n f = 1 r β J r 7 , n i cdet i T T . i a . q 2 T β β x qf 01 α I r 6 , r j rdet j NN j . b f . 2 N α α β J r 7 , n T T β β α I r 6 , r NN α α , E25 where a . q 2 T and b f . 2 N are the qth column of T*A2 and the fth row of B 2 N = B 2 B 2 R B 1 , respectively. Hence, we prove the following theorem. Theorem 3.1. Let A 1 H r 1 m × n , B 1 H r 2 r × s , A 2 H r 3 k × n , B 2 H r 4 r × p , rank H = rank A 2 L A 1 = r 5 , rank N = R B 1 B 2 = r 6 , and rank T = R H A 2 = r 7 . Then, for the partial solution (13), X 0 = x ij 0 H n × r , of the system (1), we have, x ij 0 = δ x ij 0 δ , E26 where the term x ij 01 has the determinantal representations (14) and (15), x ij 02 —(16) and (17), x ij 03 —(18) and (19), x ij 04 —(20), x ij 05 —(21), x ij 06 —(23) and (24), and x ij 07 —(25). ## 4. Cramer’s rules for special cases of (1) In this section, we consider special cases of (1) when one or two equations are one-sided. Let in Eq.(1), the matrix B1 is vanished. Then, we have the system A 1 X = C 1 , A 2 XB 2 = C 2 . E27 The following lemma is extended to matrices with quaternion entries. Lemma 4.1. [7] Let A 1 H m × n , C 1 H m × r , A 2 H k × n , B 2 H r × p , and C 2 H k × p be given and X H n × r is to be determined. Put H = A 2 L A 1 . Then, the following statements are equivalent: 1. System (27) is consistent. 2. R A 1 C 1 = 0 , R H C 2 A 2 A 1 C 1 B 2 = 0 , C 2 L B 2 = 0 . 3. rank A 1 C 1 = rank A 1 , rank C 2 B 2 = rank B 2 , rank A 1 C 1 B 2 A 2 C 2 = rank A 1 A 2 . In this case, the general solution of (27) can be expressed as X = A 1 C 1 + L A 1 H C 2 A 2 A 1 C 1 B 2 B 2 + L A 1 L H Z 1 + L A 1 W 1 R B 2 , E28 where Z1 and W1 are the arbitrary matrices over H with appropriate sizes. Since by (9), L A 1 H = L A 1 A 2 L A 1 = A 2 L A 1 = H , then we have some simplification of (28), X = A 1 C 1 + H C 2 B 2 H A 2 A 1 C 1 B 2 B 2 + L A 1 L H Z 1 + L A 1 W 1 R B 2 . By putting Z1=W1=0, there is the following partial solution of (27), X 0 = A 1 C 1 + H C 2 B 2 H A 2 A 1 C 1 B 2 B 2 . E29 Theorem 4.1. Let A 1 = a ij 1 H r 1 m × n , A 2 = a ij 2 H r 2 k × n , B 2 = b ij 2 H r 3 r × p , C 1 = c ij 1 H m × r , and C 2 = c ij 2 H k × p , and there exist A 1 = a ij 1 , H n × m , B 2 = b ij 2 , H p × r , and H = h ij H n × k . Let rank H = min rank A 2 rank L A 1 = r 4 . Denote A 1 C 1 C ̂ 1 = c ̂ ij 1 H n × r , H C 2 B 2 C ̂ 2 = c ̂ ij 2 H n × r , H A 2 A 1 A ̂ 2 = a ̂ ij 2 H n × m , and C 1 Q B 2 Q ̂ = q ̂ ij H m × p . Then, the partial solution (29), X 0 = x ij 0 H n × r , possesses the following determinantal representations, x ij 0 = β J r 1 , n i cdet i A 1 A 1 . i c ̂ . j 1 β β β J r 1 , n A 1 A 1 β β + d ij λ β J r 4 , n H H β β α I r 3 , r B 2 B 2 α α l = 1 m g il μ α I r 3 , r j rdet j B 2 B 2 j . q ̂ l . α α β J r 4 , n H H β β α I r 1 , m A 1 A 1 α α α I r 3 , r B 2 B 2 α α for all λ = 1 , 2 and μ = 1 , 2 . Here d ij 1 α I r 3 , r j rdet j B 2 B 2 j . v i . 1 α α , g il 1 α I r 1 , m l rdet l A 1 A 1 l . u i . 1 α α , and the row-vectors v i . 1 = v i 1 1 v ir 1 and u i . 1 = u i 1 1 u im 1 such that v it 1 β J r 4 , n i cdet i H H . i c ̂ . t 2 β β , u iz 1 β J r 4 , n i cdet i H H . i a ̂ . z 2 β β . In another case, d ij 2 β J r 4 , n i cdet i H H . i v . j 2 β β , g il 2 β J r 4 , n i cdet i H H . i u . l 2 β β . and the column-vectors v . j 2 = v 1 j 2 v nj 2 and u . l 2 = u 1 l 2 u nl 2 such that v qj 2 α I r 3 , r j rdet j B 2 B 2 j . c ̂ q . 2 α α , u ql 2 α I r 1 , m l rdet l A 1 A 1 l . a q . 2 α α Proof. The proof is similar to the proof of Theorem 3.1. Let in Eq.(1), the matrix A1 is vanished. Then, we have the system, XB 1 = C 1 , A 2 XB 2 = C 2 . E30 The following lemma is extended to matrices with quaternion entries as well. Lemma 4.2. [7] Let B 1 H r × s , C 1 H n × s , A 2 H k × n , B 2 H r × p , and C 2 H k × p be given and X H n × r is to be determined. Put N = R B 1 B 2 . Then, the following statements are equivalent: 1. System (30) is consistent. 2. R A 2 C 2 = 0 , C 2 A 2 C 1 B 1 B 2 L N = 0 , C 2 L B 2 = 0 . 3. rank A 2 C 2 = rank A 2 , rank C 1 B 1 = rank B 1 , rank C 2 A 2 C 1 B 2 B 1 = rank B 2 B 1 . In this case, the general solution of (30) can be expressed as X = C 1 B 1 + A 2 C 2 A 2 C 1 B 1 B 2 N R B 1 + L A 2 W 2 R B 1 + Z 2 R N R B 1 , E31 where Z2 and W2 are the arbitrary matrices over H with appropriate sizes. Since by (10), N R B 1 = R B 1 B 2 R B 1 = N , then some simplification of (31) can be derived, X = C 1 B 1 + A 2 C 2 N A 2 C 1 B 1 B 2 N + L A 2 W 2 R B 1 + Z 2 R N R B 1 . By putting Z2=W2=0, there is the following partial solution of (30), X 0 = C 1 B 1 + A 2 C 2 N A 2 A 2 C 1 B 1 B 2 N . E32 The following theorem on determinantal representations of (29) can be proven similar to the proof of Theorem 3.1 as well. Theorem 4.2. Let B 1 = b ij 1 H r 1 r × s , A 2 = a ij 2 H r 2 k × n , B 2 = b ij 2 H r 3 r × p , C 1 = c ij 1 H n × s , and C 2 = c ij 2 H k × p , and there exist B 1 = b ij 1 , H s × r , A 2 = a ij 2 , H n × k , N = n ij H p × r . Let rank N = min rank B 2 rank R B 1 = r 4 . Denote C 1 B 1 C ˜ 1 = c ˜ ij 1 H n × r , A 2 C 2 N C ˜ 2 = c ˜ ij 2 H n × r , B 1 B 2 N B ˜ 2 = b ˜ ij 2 H s × r , and P A 2 C 1 P ˜ = p ˜ ij H n × s . Then, the partial solution (32), X 0 = x ij 0 H n × r , possesses the following determinantal representations, x ij 0 = α I r 1 , r j rdet j B 1 B 1 j . c ˜ i . 1 α α α I r 1 , r B 1 B 1 α α + d ij λ β J r 2 , n A 2 A 2 β β α I r 4 , r NN α α z = 1 s β J r 2 , n i cdet i A 2 A 2 . i p ˜ . z β β g zj μ β J r 2 , n A 2 A 2 β β β J r 1 , s B 1 B 1 β β α I r 4 , r NN α α for all λ = 1 , 2 and μ = 1 , 2 . Here d ij 1 α I r 3 , r j rdet j NN j . φ i . 1 α α , g il 1 α I r 4 , r j rdet j NN j . ψ z . α α , and the row-vectors φ i . 1 = φ i 1 1 φ ir 1 and ψ i . 1 = ψ z 1 1 ψ zr 1 such that φ it 1 = β J r 2 , n i cdet i A 2 A 2 . i c . t 2 β β , ψ zv 1 = β J r 1 , n z cdet z B 1 B 1 . i b . v 2 β β . In another case, d ij 2 β J r 2 , n i cdet i A 2 A 2 . i φ . j 2 β β , g zj 2 β J r 1 , n z cdet z B 1 B 1 . z ψ . j 2 β β , and the column-vectors φ . j 2 = φ 1 j 2 φ nj 2 and ψ . j 2 = ψ 1 j 2 ψ sj 2 such that φ qj 2 = α I r 4 , r j rdet j NN j . c q . 2 α α , ψ uj 2 α I r 4 , r j rdet j NN j . b u . 2 α α . Now, suppose that the both equations of (1) are one-sided. Let in Eq.(1), the matrices B1 and A2 are vanished. Then, we have the system A 1 X = C 1 , XB 2 = C 2 . E33 The following lemma is extended to matrices with quaternion entries. Lemma 4.3. [31] Let A 1 H m × n , B 2 H r × p , C 1 H m × r , and C 2 H n × p be given and X H n × r is to be determined. Then, the system (33) is consistent if and only if R A 1 C 1 = 0 , C 2 L B 2 = 0 , and A1C2=C1B2. Under these conditions, the general solution to (33) can be established as X = A 1 C 1 + L A 1 C 2 B 2 + L A 1 UR B 2 , E34 where U is a free matrix over H with a suitable shape. Due to the consistence conditions, Eq. (34) can be expressed as follows: X = C 2 B 2 + A 1 C 1 A 1 C 2 B 2 + L A 1 UR B 2 = C 2 B 2 + A 1 C 1 C 1 B 2 B 2 + L A 1 UR B 2 = C 2 B 2 + A 1 C 1 R B 2 + L A 1 UR B 2 , Consequently, the partial solution X0 to (33) is given by X 0 = A 1 C 1 + L A 1 C 2 B 2 , E35 or X 0 = C 2 B 2 + A 1 C 1 R B 2 . E36 Due to the expression (35), the following theorem can be proven similar to the proof of Theorem 3.1. Theorem 4.3. Let A 1 = a ij 1 H r 1 m × n , B 2 = b ij 2 H r 2 r × p , C 1 = c ij 1 H m × r , and C 2 = c ij 2 H n × r , and there exist A 1 = a ij 1 , H n × m , B 2 = b ij 2 , H p × r , and L A 1 = I A 1 A 1 l ij H n × n . Denote A 1 C 1 C ̂ 1 = c ̂ ij 1 H n × r and L A 1 C 2 B 2 C ̂ 2 = c ̂ ij 2 H n × r . Then, the partial solution (35), X 0 = x ij 0 H n × s , possesses the following determinantal representation, x ij 0 = β J r 1 , n i cdet i A 1 A 1 . i c ̂ . j 1 β β β J r 1 , n A 1 A 1 β β + α I r 2 , r j rdet j B 2 B 2 j . c ̂ i . 2 α α α I r 2 , r B 2 B 2 α α , E37 where c ̂ . j 1 is the jth column of C ̂ 1 and c ̂ i . 2 is the ith row of C ̂ 2 . Remark 4.1. In accordance to the expression (36), we obtain the same representations, but with the denotations, C 2 B 2 C ̂ 2 = c ̂ ij 2 H n × r and A 1 C 1 R B 2 C ̂ 1 = c ̂ ij 2 H n × r . Let in Eq.(1), the matrices B1 and B2 are vanished. Then, we have the system A 1 X = C 1 , A 2 X = C 2 . E38 Lemma 4.4. [7] Suppose that A 1 H m × n , C 1 H m × r , A 2 H k × n , and C 2 H k × r are known and X H n × r is unknown, H = A 2 L A 1 , T = R H A 2 . Then, the system (38) is consistent if and only if A i A i C i = C i , for all i = 1 , 2 and T A 2 C 2 A 1 C 1 = 0 . Under these conditions, the general solution to (38) can be established as X = A 1 C 1 + L A 1 H A 2 A 2 C 2 A 1 C 1 + L A 1 L H Y , E39 where Y is an arbitrary matrix over H with an appropriate size. Using (10) and the consistency conditions, we simplify (39) accordingly, X 0 = A 1 C 1 + H C 2 H A 2 A 1 C 1 + L A 1 L H Y . Consequently, the following partial solution of (39) will be considered X 0 = A 1 C 1 + H C 2 H A 2 A 1 C 1 . E40 In the following theorem, we give the determinantal representations of (40). Theorem 4.4. Let A 1 = a ij 1 H r 1 m × n , A 2 = a ij 2 H r 2 k × n , C 1 = c ij 1 H m × r , C 2 = c ij 2 H k × r , and there exist A 1 = a ij 1 , H n × m , H 2 = h ij H n × s . Let rank H = min rank A 2 rank L A 1 = r 3 . Denote A 1 C 1 C ̂ 1 = c ̂ ij 1 H n × r , H C 2 C ̂ 2 = c ̂ ij 2 H n × r , and H A 2 A ̂ 2 = a ̂ ij 2 H n × n . Then, X 0 = x ij 0 H n × r possesses the following determinantal representation, x ij 0 = β J r 1 , n i cdet i A 1 A 1 . i c ̂ . j 1 β β β J r 1 , n A 1 A 1 β β + β J r 3 , n i cdet i H H . i c ̂ . j 2 β β β J r 3 , n H H β β l = 1 n β J r 3 , n i cdet i H H . i a ̂ . l 2 β β β J r 3 , n H H β β β J r 1 , n l cdet l A 1 A 1 . l c ̂ . j 1 β β β J r 1 , n A 1 A 1 β β , E41 where c ̂ . j 1 , c ̂ . j 2 , and a ̂ . j 2 are the jth columns of the matrices C ̂ 1 , C ̂ 2 , and A ̂ 2 , respectively. Proof. The proof is similar to the proof of Theorem 3.1. ## 5. Conclusion Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer’s rule) of partial solutions to the system of two-sided quaternion matrix equations A1XB1=C1, A2XB2=C2, and its special cases with 1 and 2 one-sided matrix equations. We use previously obtained by the author determinantal representations of the Moore-Penrose inverse. Note to give determinantal representations for all above matrix systems over the complex field, it is obviously needed to substitute all row and column determinants by usual determinants. ## Conflict of interest The author declares that there are no conflict interests. ## References 1. 1. Mitra SK. A pair of simultaneous linear matrix A1XB1=C1 and A2XB2=C2. Proceedings of the Cambridge Philosophical Society. 1973;74:213-216 2. 2. Mitra SK. A pair of simultaneous linear matrix equations and a matrix programming problem. Linear Algebra and its Applications. 1990;131:97-123. DOI: 10.1016/0024-3795(90)90377-O 3. 3. Shinozaki N, Sibuya M. Consistency of a pair of matrix equations with an application. Keio Engineering Report. 1974;27:141-146 4. 4. Van der Woulde J. Feedback decoupling and stabilization for linear system with multiple exogenous variables [PhD thesis]. Netherlands: Technical University of Eindhoven; 1987 5. 5. Özgüler AB, Akar N. 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Kyrchei I. The theory of the column and row determinants in a quaternion linear algebra. In: Baswell AR, editor. Advances in Mathematics Research 15. New York: Nova Sci. Publ; 2012. pp. 301-359 19. 19. Kyrchei I. Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer’s rules. Linear Multilinear Algebra. 2011;59(4):413-431. DOI: 10.1080/03081081003586860 20. 20. Kyrchei I. Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations. Linear Algebra and its Applications. 2013;438(1):136-152. DOI: 10.1016/j.laa.2012.07.049 21. 21. Kyrchei I. Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Applied Mathematics and Computation. 2014;238:193-207. DOI: 10.1016/j.amc.2014.03.125 22. 22. Kyrchei I. Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field. 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Computers & Mathematcs with Applications. 2011;61:1576-1589. DOI: 10.1016/j.camwa.2011.01.026 27. 27. Song GJ, Dong CZ. New results on condensed Cramer’s rule for the general solution to some restricted quaternion matrix equations. Journal of Applied Mathematics and Computing. 2017;53:321-341. DOI: 10.1007/s12190-015-0970-y 28. 28. Song GJ, Wang QW. Condensed Cramer rule for some restricted quaternion linear equations. Applied Mathematics and Computation. 2011;218:3110-3121. DOI: 10.1016/j.amc.2011.08.038 29. 29. Song G. Characterization of the W-weighted Drazin inverse over the quaternion skew field with applications. Electronic Journal of Linear Algebra. 2013;26:1-14. DOI: 10.13001/1081-3810.1635 30. 30. Wang QW. A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity. Linear Algebra and its Applications. 2004;384:43-54. DOI: 10.1016/j.laa.2003.12.039 31. 31. Wang QW, Wu ZC, Lin CY. 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	Followers 0 # Mysterious Rigids On Rendered Textures ## 6 posts in this topic Hello everyone, I am trying to fix an issue that causes some (not all) sprites to have smudges/rigids when rendered. I am using a vertex buffer to render the sprites in 3D space. They all render with the correct size, etc. so that no resizing occurs during render-time. Here is an image of the issue I am clueless about (see "Character Info" dialog box): Here is the texture/spritesheet that the game is reading from (as you can see the original doesn't have these rigids on the "Character Info" dialog box): I have been testing and discovered this issue may be related to non-power of 2 textures being resized to power 2 sizes. That being said since this texture (above spritesheet) is 619x598 and is forced to be resized to 1024x1024 when the texture is created, I decided to manually change the size of the spritesheet image file to 1024x1024 to see if resizing during texture creation was the issue; however, the issue still remained with no improvements. Also the position of the vertexes on the screen and on the spritesheet are all consistent with the size of the dialog box (270x376): Screen(-0.687500~0.156250, -0.775000~0.791667) Spritesheet(0.000000~0.263672, 0.000000~0.367188) Both the screen position and texture coordinates equate to 270x376 being the dialog box size so it is not stretching at all but it is definitely losing quality for some unknown reason. If you want to verify that it's not being stretched, the screen size is 640x480 in this case and the texture size is 1024x1024 after "extending" (not stretching) to a power-of-2 size. I am using the following line to create the texture: D3DXCreateTextureFromFileInMemoryEx(device, imageBuffer, m_imageSize, D3DX_DEFAULT, D3DX_DEFAULT, 1, 0, format, D3DPOOL_DEFAULT, D3DX_FILTER_NONE, D3DX_FILTER_NONE, 0, NULL, NULL, &m_texture); Since I am passing D3DX_FILTER_NONE as the filter parameters "No scaling or filtering will take place. Pixels outside the bounds of the source image are assumed to be transparent black." as stated by the official MSDN website. That being said I don't know why the image is losing quality even though its not being stretched rather only extended and no harm should be done, right? The weird thing is when I passed D3DX_DEFAULT_NONPOW2 as the image sizes as opposed to D3DX_DEFAULT it looks perfect. But other issues arise and besides I don't want to use non-power-2 textures since its not entirely cross-compatible at least with older graphics cards which is what I'm targeting. Any help on the matter is greatly appreciated. Also feel free to ask any more questions to clear things up! Edit: This just came to mind, is it possible the rigids/smudges are caused from a vertical flip of the sprites since the bottom and top are flipped to properly display the sprites? If so how can I approach this another way so DirectX won't flip the sprites upside-down? Edited by kamal7 0 ##### Share on other sites Vertical flipping should not cause that problem. If you're using DX9 or below, it might depend on how you map texels to pixels though (offset the vertex position or offset the texture coordinates?) This problem can also appear if your backbuffer size does not match the client size of the window. You DID use AdjustWindowRect(Ex) to find the correct window size for the required client size? 1 ##### Share on other sites Vertical flipping should not cause that problem. If you're using DX9 or below, it might depend on how you map texels to pixels though (offset the vertex position or offset the texture coordinates?) This problem can also appear if your backbuffer size does not match the client size of the window. You DID use AdjustWindowRect(Ex) to find the correct window size for the required client size? I'm using DX9. Also I don't think the issue is mapping texels to pixels although I cannot guarantee that since this is my first project using a vertex buffer. Here is how I do it: //Here rtWidth and rtHeight are the width/height of the render target respectively. //position.x/y are the pixel positions of the sprite //sSpriteWidth/sSpriteHeight are the width/height of the sprite // If you notice the texels range from -1.0 to +1.0 on the render screen; position (0,0) being the center of the screen Vertex Positions Calculations: positionLeft = (position.x - (rtWidth/2.0f))/(rtWidth/2.0f); positionRight = (position.x+(float)sSpriteWidth - (rtWidth/2.0f))/(rtWidth/2.0f); positionTop = ((rtHeight/2.0f) - position.y)/(rtHeight/2.0f); // top/bottom flipped positionBottom = ((rtHeight/2.0f) - (position.y+(float)sSpriteHeight))/(rtHeight/2.0f); // top/bottom flipped //m_wWidth/m_wHeight are the width/height of the spritesheet (1024x1024 after being resized into a power-2-texture from its original 619x598) //here the texture coordinates range from 0.0 to 1.0 Texture Coordinate Calculations: srcLeft = (float)srcRect.left/(float)m_wWidth; srcTop = (float)srcRect.top/(float)m_wHeight; srcRight = (float)srcRect.right/(float)m_wWidth; srcBottom = (float)srcRect.bottom/(float)m_wHeight; Do I still need to offset the vertex position and texture coordinates even though they range from -1.0 to +1.0 and 0.0 to 1.0 respectively? If so how much should I offset it by? Do I need to change them to range from the size of the backbuffer/spritesheet to implement a 0.5 offset? Here is my vertex buffer array: CUSTOM2DVERTEX vertices[] = { { D3DXVECTOR3(positionLeft, positionTop, 1.0f), D3DXVECTOR2(srcLeft, srcTop), D3DXCOLOR(255, 255, 255, alpha) }, // left top { D3DXVECTOR3(positionLeft, positionBottom, 1.0f), D3DXVECTOR2(srcLeft, srcBottom), D3DXCOLOR(255, 255, 255, alpha) }, // left bottom { D3DXVECTOR3(positionRight, positionTop, 1.0f), D3DXVECTOR2(srcRight, srcTop), D3DXCOLOR(255, 255, 255, alpha) }, // right top { D3DXVECTOR3(positionRight, positionBottom, 1.0f), D3DXVECTOR2(srcRight, srcBottom), D3DXCOLOR(255, 255, 255, alpha) }, // right bottom }; // here is the draw call, I just skipped the vertex buffer lock/unlock (memory copying) and SetTexture steps for simplicity DrawPrimitive(D3DPT_TRIANGLESTRIP, 0, 2); Also to clarify that my vertex position/texture coordinate calculations are correct, this is what is calculated for the specific "Character Info" dialog box sprite: positionLeft = -0.687500 positionRight = 0.156250 positionTop = 0.791667 positionBottom = -0.775000 Since the actual size of the "Character Info" sprite is 270x376 pixels and the render screen/backbuffer is 640x480 pixels: abs(-0.687500)(640/2) + abs(0.156250)(640/2) = 270.00000 pixels AND abs(0.791667)(480/2) + abs(-0.775000)(480/2) 376.00008 pixels. Therefore the vertex positions are calculated properly, right? srcLeft = 0.000000 srcTop = 0.000000 srcRight = 0.263672 srcBottom = 0.367188 Since the actual size of the "Character Info" sprite is 270x376 pixels and the spritesheet is 1024x1024 pixels after being extended: 0.2636721024 = 270.000128 pixels AND 0.3671881024 = 376.000512 pixels. Therefore the texture coordinates are calculated properly, right? I haven't used AdjustWindowRect, this is my first time hearing about this. Anyways I did what you told me so now I adjusted the window size to make the client size the same as the backbuffer (640x480). This didn't resolve the issue unfortunately , also the issue applies in fullscreen as well as window mode so it makes sense that this wasn't the problem. But I still thank you for letting me know about this function since I would prefer the client size to be the same as the backbuffer in window mode anyhow Also for further investivation here is how it looks like when passing D3DX_DEFAULT_NONPOW2 into the image/texture size: As you can see it looks perfect but other issues arise with the tiles in the background (black outlines, etc.). All the images here use the same process to render them so I don't see what could be causing this either when using non-power-2 textures because my graphics card supports all texture sizes without limitations (checked with device TextureCaps) although that could be a sticky statement since I expect nothing to be perfectly accurate at this point. But if you look closely it seems as though something is wrong with the projection which may be causing these lines/rigids? It's as if everything is on an angle but not really, maybe more so magnified at a very small interval. Should I post my matrix transformations? Edited by kamal7 0 ##### Share on other sites Changing my matrix transformations don't seem to be affecting anything at all :S I know 100% that my vertex/pixel shaders are used though. What could possibly be overwriting my matrix transformations? Is it the fact that I handle transformations outside of my shaders so that they have no effect at all? Will this make the difference in loss of quality if transformations aren't being applied by my calculations? If they're not being applied what is performing these transformations, is it mere luck that I so happen to be able to see the sprites rendered without calculating the matrices myself? Sorry for asking so many questions, I just want to understand how these things work so I won't run into anymore problems in the future. Edited by kamal7 0 ##### Share on other sites In my DX9 code I've offset the vector position by -0.5, -0.5 for all 2d textured rendering. I recall having a few difficulties when the 2d rendering was performed not by pretransformed vertices (and ran through the matrices) In the worst case you may have to clamp the final coordinates to direct multiples of 0.5 (which can be tough when using matrices) I see you're using shaders, I'm not sure if that makes a difference, I'm still using fixed function. I fixed it thanks to this suggestion! I adjusted my projection matrix and vertex positions to range from 0x0 to 640x480 instead of the original (-1,-1) to (1,1) and then made the -0.5 offsets in the vertex position values. This solved my issue, I can't tell you how thankful I am! Edited by kamal7 1 ##### Share on other sites In case anyone else comes across this thread, here's the official explanation and workarounds for the dreaded Direct3D9 half pixel offset issue: http://msdn.microsoft.com/en-us/library/windows/desktop/bb219690(v=vs.85).aspx If using -1:1 coordinates, then instead of -0.5, the offset would be -0.5/width and 0.5/height. 2 ## Create an account Register a new account
	# Picking Random Items: Take Two (Hacking Python's Generators) By Alex Beal January 14, 2012 Earlier today I had my mind blown by David Beazley’s presentation on the power of Python’s generators, and it inspired me to write this highly Pythonic version of the random word selector, made almost entirely of generators: import heapq import random lines = (line for line in open("/usr/share/dict/words")) word_pairs = ((random.random(), word) for word in lines) rand_pairs = heapq.nlargest(4, word_pairs) rand_words = [word for rand, word in rand_pairs] print rand_words How does this work? First recall that a generator is an object that returns the next item in a sequence every time its next() method is called. There’s an example of this on line 4 where a generator named lines is created, which returns the next line of the file every time lines.next() is called. What’s so handy about a generator is that it can be automatically consumed by a for loop. That is, put a generator in a for loop, and the loop will automatically call the generator’s next() method on every iteration. There’s an example of this on line 5 where another generator is created that uses a for loop to consume the lines generator. This outputs a tuple containing a random number and the line returned by lines.next(). So, the result is that each time word_pairs.next() is called, you get the next line of the file paired with a random value (e.g., (0.12345, 'fire\n')). Finally, we use heapq.nlargest(n, iter) to grab the n largest elements from iter. In this case, it repeatedly calls word_pairs.next() and outputs a list of the 4 words with the highest random values.1 These are our 4 random words. This is all done in around 3 lines (excluding imports and printing). Wowza. As Beazley points out, one advantage of this technique is that it resembles the way simple commands are chained together in the shell to create a pipeline. And, just like in the shell, the pipeline is highly modular, so different filters and stages can be easily inserted at different points. Below, I’ve added two stages to the pipeline that strip the words of their newline characters, and skip words that aren’t exactly 13 characters long: import heapq import random def isValid(word): return len(word) == 13 lines = (line for line in open("/usr/share/dict/words")) words = (line.strip() for line in lines) valid_words = (word for word in words if isValid(word)) word_pairs = ((random.random(), word) for word in valid_words) rand_pairs = heapq.nlargest(4, word_pairs) rand_words = [word for rand, word in rand_pairs] print rand_words The words generator calls strip() on each line which removes the newline character. The valid_words generator only returns words that pass the isValid test. In this case, isValid returns True only if the word is exactly 13 characters long. The end result is 4 random words that are 13 characters long. One other advantage is that each generator creates its output only when requested. This translates into minimal memory use. The dictionary file being consumed might be gigabytes in size, but only one word will be loaded into memory at a time (excluding buffering done by the file class, etc). It’s definitely a neat way of parsing large files. If you enjoyed this, definitely check out Beazley’s presentation, and venture further down the rabbit hole. ## Notes 1. You could even use the built-in function max() if you only need one word.
	# Piecewise Functions A piecewise function is a function which is pieced together from multiple different functions. For example, the absolute value function is a piecewise function because it consists of the line $y=-x$ for negative $x$, and $y=x$ for positive $x$. More generally, piecewise functions can be defined using case notation, which tells which functions to use as pieces and where to use them as pieces. The absolute value function, for example, can be written in case notation as $\|x\| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$ This case notation just tells us that for negative inputs ($x<0$) we should use the function $y=-x$ to calculate the function output, and for nonnegative inputs ($x \geq 0$) we should use the function $y=x$ to calculate the function output. Two more equivalent case notation forms for the absolute value function are shown below. Sometimes, piecewise functions have breaks in them. For example, if we modify the case notation of the absolute value function so that the right piece is elevated, the graph has a break in it. This looks unusual, but it is a perfectly valid function. There is no limit to what types of functions a piecewise function can consist of. For example, the equation and graph of a more complicated piecewise function are shown below. $\|x\| = \begin{cases} \sin x & \text{if } x \geq 2 \\ \log_2 x & \text{if } 0 < x < 2 \\ x^2 & \text{if } -2 < x \leq 0 \\ x+8 & \text{if } x \leq -2 \end{cases}$ Likewise, there is no limit to the number of pieces a piecewise function can have. For example, rounding is an example of a piecewise function with infinitely many pieces. $f(x) = \begin{cases} \vdots & \vdots \\ 2 &\mbox{if } 1.5 \leq x < 2.5 \\ 1 &\mbox{if } 0.5 \leq x < 1.5 \\ 0 &\mbox{if } −0.5 \leq x < 0.5 \\ −1 &\mbox{if } −1.5 \leq x < −0.5 \\ −2 &\mbox{if } −2.5 \leq x < −1.5 \\ \vdots & \vdots \end{cases}$ Practice Problems Graph the following piecewise functions. (You can view the solution by clicking on the problem.) $1) \hspace{.5cm} f(x)=\begin{cases} 2x & \text{if } x \geq 0 \\ \frac{1}{2}x &\text{ if} x < 0 \end{cases}$ Solution: $2) \hspace{.5cm} f(x)=\begin{cases} x-5 & \text{if } x \geq 2 \\ x+5 &\text{ if} x < 2 \end{cases}$ Solution: $3) \hspace{.5cm} f(x)=\begin{cases} x^2-4 & \text{if } x \geq 2 \\ 2-x &\text{if } 0 < x < 2 \\ x^2-4 &\text{if } x \geq 0 \end{cases}$ Solution: $4) \hspace{.5cm} f(x)=\begin{cases} \sin x &\text{if } x > 0 \\ -2x &\text{if } -2 < x \leq 0 \\ 8-x^2 &\text{if } x \leq -2 \end{cases}$ Solution: $5) \hspace{.5cm} f(x)=\begin{cases} 2-x^2 &\text{if } x \geq 1 \\ x^2 &\text{if } 0 < x < 1 \\ -1 &\text{if } x=0 \\ x^2 &\text{if } -1 \leq x < 0 \\ 8-x^2 &\text{if } x < -1 \end{cases}$ Solution: $6) \hspace{.5cm} f(x)=\begin{cases} x^2 &\text{if } x>1 \\ \log_2 x &\text{if } 0<x \leq 1 \\ \cos x &\text{if } -\pi \leq x \leq 0 \\ 4+x &\text{if } x < -\pi \end{cases}$ Solution: Tags:
	# A Neighborhood of Infinity ## Wednesday, August 24, 2005 ### The Unnatural Naturals In a comment from my previous post, vito got me thinking about the naturals. Most mathematicians use the term 'natural' to refer to numbers in the set {1,2,3,...}. However, I prefer to think of the set {0,1,2,3,...} as the naturals. As the precise definition of 'natural' is still debatable I can get away with my usage, as long as I define exactly what I mean. My gripe with the more usual definition of 'natural' is that it's not natural at all. Consider some of the natural ways to build the integers. For example there are the Peano axioms and the finite ordinals. These define sequences that start at zero, not one. In fact, both of those web pages define zero to be natural. Countless theorems are more naturally written as statements about {0,1,2,...} rather than {1,2,3,...}. I can't even see why the set {1,2,3,...} needs a name any more than the set {2,3,4,5,6,7,...}. Having said that, when I was a student at Cambridge there was a formal debate over the definition of the naturals. Things suddenly became more complicated when a faction suddenly appeared arguing for the set {2,3,4,...} on the grounds that if numbers are for counting then the number one doesn't exactly do much counting. I think I should have formed a pro-{3,4,5,6,...} faction at this point, after all, the ancient greeks, who are noted for their wisdom, only marked their nouns as plural for numbers strictly greater than 2. Anyway, there is some discussion of the debate on Wikipedia and the conclusion is probably the same one I would come to: the set {0,1,2,3,...} is more popular among logicians and set theorists (I'd probably add category theorists) and {1,2,3,4,...} is more popular among number theorists. And of course, when programming, I think arrays should start at zero. You won't catch me using Fortran or matlab. Come to think of it, as the first ordinal is zero, shouldn't we actually start counting at zero? #### 1 comment: Vito said... Well there ya go. I'm a number theorist. If it ain't in {1,2,3,...} I don't want to think about it.
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	Why work done is needed when other physical quantities are available? I have some question about work done. I understand the mathematical ways and examples which are floating all over on internet & books. But all these information not clearing the concepts of work done as well energy. Please clear following questions: 1. If there are other physical quantities available why work done is needed. what is special about work done that other quantities can not give us. 2. "$$W = \vec{F} \cdot \vec{d} = F \cdot d \cdot\cos\theta$$ " In this formula what are details of both quantities. (a) Is it applied force which causes displacement e.g: Force $$\vec{F}$$ is applied on a box on horizontal surface displace $$\vec{d}$$ with a rope parallel or at some angle? (b) Gravitational Force Field $$\vec{F}$$ and we throw a ball upward $$\vec{d}$$ against the direction of force. In this case we displace the object under the influence of field, object is totally rely on field. (c) Electrostatic Force Field $$\vec{F}$$ and we displace $$\vec{d}$$ object in this case we move the object by our own accelerating and moving direction ,the path of object is defined by us under influence of field. Please clear all these point wise. • Can you clarify your question more? What do you mean "other physical quantities"? – Tachyon209 Aug 12 '20 at 10:02 • Hello @Tachyon209. Physical quantities means Force, momentum , acceleration , velocity etc.. I know the clear difference and usefulness of these quantities conceptually. but work and energy is not clear conceptually in physics way. If we have all these quantities why we need work and energy what is advantage we can not determine by which quantities i mentioned. – 123 Aug 12 '20 at 10:07 • I answered your unsolved question , it might be a little long but hopefully it will clarify the significance of work . – Kia.J Sep 2 '20 at 8:50 • Hello all, If anyone has Keppler & Kolenkiw (1 edition). Pls share in pdf by any convenient way. – 123 Sep 5 '20 at 10:35 Just to give a concrete example of a problem where work and energy are useful concepts, consider a ball placed on the following frictionless hill ($$x$$ is measured in meters): If the particle starts at rest at the top of the hill and is given a gentle nudge to the right, what speed will it approach as it escapes off to the right? This is a straightforward question, but it would be a nightmare to solve using Newton's 2nd law. You'd have to calculate the tangent vector to each point of the hill and find the component of the gravitational force along this vector just to set up your complicated differential equation, which probably doesn't have an analytic solution anyway. Once you had this equation and its solution, you'd have to find the velocity by taking a derivative, and then you'd need to take the limit as $$t\rightarrow\infty$$. This process would require a significant amount of mathematical skill and knowledge, and would probably take even a very motivated undergraduate quite a while to finish. Alternatively, you could note that (i) the gravitational force is the only force doing work on your particle, and (ii) the gravitational force is conservative, so $$\frac{1}{2}mv_f^2 = mg(h_i-h_f) \implies v_f = \sqrt{2(9.8\text{ m/s})(5\text{ m})}\approx 9.89\text{ m/s}$$ • Thanks @J. Murray to give me another beautiful idea. But using this way we can only only calculate magnitude of velocity (speed),displacement (distance), magnitude of acceleration etc.. we are unable to track the direction of particle when using energy. In Work done if Force vector and position vector it also gives us scalar. (1) If we want to measure Force Vector using work formula $\frac{W}{\vec{r}} = \vec{F}$ we cannot divide vector. What is the solution. (2) What is the meaning of scalar value of work. – 123 Sep 3 '20 at 8:20 • @123 The function $\mathbf r(t)$, which gives the position of the particle as a function of time, contains all the dynamical information about the particle and its motion for every moment in time. However, we often don't need this much information. If I throw a ball in the air at some speed $v$ and want to know how high it goes, I don't need to know exactly how long it will take to reach the top of its arc. – J. Murray Sep 3 '20 at 15:22 • After long long usage (old enough concept to understand) of work and energy idea. Is there any physical intuitive meaning of work and energy. Thanks .. Where you all guys are when i first asked this question. no one gave me the correct answer. Now this new six answers are real answers to my question. Specially you and Kia.J. – 123 Sep 3 '20 at 16:09 I don't agree with your statement that the $$F$$ in the formula $$W=Fd\cos\theta$$ gives complete information about the motion and displacement of a body, rather, partial or sometimes doesn't. The concept of work in physics is much more narrowly defined than the common use of the word. Work is done on an object when an applied force moves it through a distance. In our everyday language, work is related to expenditure of muscular effort, but this is not the case in the language of physics. A person that holds a heavy object does no physical work because the force is not moving the object through a distance. Work, according to the physics definition, is being accomplished while the heavy object is being lifted but not while the object is stationary. Say, for example, a man is pushing a train (purposely I mentioned train because practically no man can move a train by pushing alone) and he is applying all the strength i.e. he is applying force but the train will not move. This means that force acting on a body doesn't imply that the body is in motion. Concluding, work gives us the idea that to what extent the motion of the body is changed or to what extent the force applied is useful to alter the motion of the body. • Hello @SarGe your conclusion statement gives me answer to one condition when the object is in rest and i can extend this idea to moving object if we alter force field to achieve the particular path $\vec{d}$. I understand if displacement is zero work is zero. In my 2$^{nd}$ point i have given 3 condition pls clear these. Thanks for useful answer. – 123 Aug 12 '20 at 9:42 • @123, it can be explained by the same logic that gravitational or electrostatic force acting on a body doesn't imply that the body is in motion. You need the concept of "Work" there, too. – SarGe Aug 12 '20 at 10:05 • [Reply for 2nd comment]@123, if we are applying a force on body which is still not moving implies that there is another force acting on the body in the opposite direction of the applied force. – SarGe Aug 12 '20 at 10:09 • No, the formula for work $\displaystyle\int \vec F\cdot d\vec x$ is valid for any type of force. It is simplified as $F\cdot(d\cos\theta)$ when force is constant. – SarGe Aug 12 '20 at 10:18 • We are taking dot product of two vectors which will be a scalar, since we are taking projection of path along the direction of force, so we get there $\cos\theta$. – SarGe Aug 12 '20 at 10:27 I'm only going to answer this question, but if you needed more or extra explanation notify me: "I still found no satisfactory answer to this question about why work done is needed. Why we create this physical quantity. What is the additional benefit to create this quantity which cannot calculate by e.g.: Force etc." Well in theory we don't need anything else more than the Newton's laws in order to study the motion of any moving body (in classical mechanics era) although Newton's laws apply on a point particle but we can solve the problem of the motion of any real-life object by thinking about it as a "gathering" of a lot (maybe infinite) of point particles. Note : I'm only talking about theoretical foundations needed for solving such a system , surely we cannot do such heavy calculations in real life and that's the reason behind the development of Mechanics of Rigid bodies and Thermodynamics and so on , but as long as we are talking about the theoretical possibility of solving such systems , Newton's laws and the equation $$\vec{F} =$$ $$m\vec{a}$$ are all we need for solving a mechanical problem. So the first motivations of defining something like Work was not conceptual, actually it was more a computational tool needed for solving harder problems (although later it lead to the concept of Energy and then got generalized beyond classical mechanics), I'll show you the math first and then explain its significance: We know that for a particle with mass $$m$$ we have: $$\vec{F}_{tot} = m\vec{a}=m\frac{\mathrm{d} \vec{v}}{\mathrm{d} t}$$ in which $$\vec{v}$$ is the velocity of the body which is $$\frac{\mathrm{d} \vec{r}}{\mathrm{d} t}$$ and $$\vec{r}$$ is the position vector of $$m$$ . Now dotting both sides by the differential of $$\vec{r}$$ , $$d\vec{r}$$ , and writing $$d\vec{r}$$ as $$\vec{v} dt$$: $$\vec{F}_{tot}\cdot {d\vec{r}} = m\frac{\mathrm{d} \vec{v}}{\mathrm{d} t}\cdot {\vec{v}dt}$$ $$\Rightarrow\vec{F}_{tot}\cdot {d\vec{r}} = m \vec{v}\cdot{d\vec{v}}$$ The right side of this equation is just $$m\frac{1}{2}d(v^{2})$$ therefore $$\vec{F}_{tot}\cdot {d\vec{r}} = \frac{1}{2}d(mv^{2})$$ Note that $$d\vec{r}$$ is the infinitesimal displacement of our point particle,so when you are considering bigger systems it will be the point where such a forces acts on it . now integrating over a curve that the point particle would take form the point $$\vec{r}_{i}$$ to $$\vec{r}_{f}$$ we have : $$\int_{i}^{f}\vec{F}_{tot}\cdot {d\vec{r}} = \frac{1}{2}mv^{2}_{f} - \frac{1}{2}mv^{2}_{i}$$ If we call the term $$\vec{F}_{tot}\cdot{d\vec{r}}$$ , $$dW$$ and writing $$\int {dW}$$ (note that this integral is in general path-dependent) as $$\Delta W$$ we would have: $$\Delta W = \frac{1}{2}mv^{2}_{f} - \frac{1}{2}mv^{2}_{i}$$ so we have found a scalar quantity which directly "connects" us to the (magnitude of) velocities rather than the other approach in which after solving the equation of motion we integrate $$\vec{a}$$ to find $$\vec{v}$$. but we had to integrate this other quantity called work so what's the point of this "pre-integrating" , well the good news is that for many important forces this integral is path-independent i.e. we don't have to evaluate it at all ! , for example in the case of a uniform gravitational field we just write $$mgh_{f}-mgh_{i}$$ now this really simplifies more complex problems that have such forces (even when they are not the only forces acting on our system) . I recommend you to read the fourth chapter of An Introduction to Mechanics by Kleppner & Kolenkow(1st ed) which has very nice discussions on this subject and also comparisons of these two methods in solving the same problems , otherwise this answer would get even more lengthier than it is now ! • I have downloaded Kleppner & Kolenkow (2nd ed). In this edition Chapter-5 is Energy. Thank you very much for your suggestion of this book. I am reading this a marvelous approach of book. explaining stories in short sentences. – 123 Sep 3 '20 at 9:11 • @123 Sure i'd be happy to help you . I don't know much about the second edition but I've seen some omissions in it so I suggested the first one which is really good , I highly recommend the first 4 chapters of it even if you already know the material ! – Kia.J Sep 3 '20 at 12:14 • I can not find Kleppner 1-edition. If you have in pdf pls share with google drive or any other convenient way. Thanks this book is giving me almost answers which i was looking for. – 123 Sep 3 '20 at 16:03 • After long long usage (old enough concept to understand) of work and energy idea. Is there any physical intuitive meaning of work and energy. Thanks you all.. Where you all guys are when i first asked this question. no one gave me the correct answer. Now this new six answers are real answers to my question. Specially you and J. Murray, I became hopeless to find my answer. – 123 Sep 3 '20 at 16:14 1."If there are other physical quantities available why work done is needed. what is special about work done that other quantities can not give us." Work done is mathematically defined as the scalar product of the force $$\vec F$$ and the displacement $$\vec s$$. So $$W=\vec F.\vec s$$ for constant forces. In case of variable forces we say something similar(I have just broken down the scalar product):$$W=\int F.dx +\int F.dy +\int F.dz$$ Here I am basically describing that total work done on an object is the sum of the work done by the individual forces acting on it resolved in coordinate axes of your choice. So, what is special about work done that cannot be described by other physical quantities? Work done as you might have guessed provides a relation between a force and the 'displacement of the object'(not necessarily caused by the force itself). That's why, we take the $$\vec F$$ and 'scale' it(stretching or compressing) to the $$\vec s$$ (or vice versa, but this makes more sense). You can imagine that we do this to show: just exactly 'how much' does this force contribute to the change in position of the object. We don't care about how fast the object changes its position(that's power), we just want to know what this force is doing in a system. You can check, that no other physical quantities give us this relation, and the reason why it's needed in the first place is that: All forces contribute to net acceleration(that's what it means to be a force), but having information about its relation with displacement can tell us if individual forces are 'taking away' from the system, or 'putting something into' the system; this can be explained by the principle of energy, which tells us, not just about the current state of an object, but also about how this object will behave and interact with other objects in the future--bonus points because it is conserved all over the universe with NO EXCEPTIONS, it might as well be a base quantity! $$(A)$$Hopefully I have described the purpose of $$F$$ and $$d$$ in the formula. In example (a), there are multiple forces acting on the system: Tension, gravitation, possibly friction, normal force by the box. You are correct, applied force at an angle does cause displacement; but does all of it goes into displacing the object? Of course not! Clearly a part of it goes into countering gravity and part of it is parallel. Is the part countering gravity doing work? No. This is not just because $$cos\theta$$ is $$0$$ at $$\pi/2$$ radians, but because it just makes sense! This part of the force neither takes away from the system, nor puts in anything. Think this one through. $$(B)$$In part (b), we use the aforementioned reasoning again! Gravitational force field is indeed the only force acting on the system during the projectile motion, but its initial upward displacement comes from the external force applied by us. Therefore as it is going upwards(and gravity is slowing it down), gravitational force field is taking away from the system; the $$\vec s$$ and $$\vec F_g$$ are opposite in direction; which implies, work done by gravity is negative for the first part of the motion. But as it comes down, gravity contributes to motion, and work done is positive! $$(C)$$In part (c), the exact same reasoning can be used. I will let you work this one out. HINT: Once again, an applied force and a field force is acting on the system. Therefore, individual work done and total work done will be different. • Hi @Reet Jaiswal, Thanks for your answer. What is in your answer it is really good. But after long period of time of thinking i have also reached the same idea what you discussed. Once again thanks for your support. But this arises more questions that counter this idea. Like we calculate displacement when Net Force is not equal 0. We can easily calculate every single parameter displacement, velocity, acceleration etc.. – 123 Sep 3 '20 at 8:05 • There is no extra benefit of creating work done. If only gravity is acting and i want to move object along the surface at some angle. Object doesn't move until applied force $F_{y} = F_{g}$. Then $F_{x}$ causing displacement. Why we need to multiply $F_{x}$ with displacement. No need after this information. – 123 Sep 3 '20 at 8:07 • Thank you I understand what you ask and I probably have an answer somewhere at the back of my head but I don't think I can put it in better words than some other answers here:) Anyway I hope you have found your answer. – Reet Jaiswal Sep 3 '20 at 18:32 • Thanks @Reet Jaiswal for your contribution. Your answer is also put some information in that answer. I don't understand why good knowledge people doesn't give answer when first asked. – 123 Sep 3 '20 at 18:38 • One last point I would still add though would be that work done is an EXTREMELY efficient method of fusing together Kinematics and Dynamics of newtonian bodies: where we would need at least 5 to 6 equations comprising 'F=ma' and the 3 equations of motion to reach at the answer, like finding final velocity or something, work-energy theorem provides wonderfully simplified explanations, majorly because computing scalars is just easier, and also because it is great at isolating forces in a system, as the uniformity of force fields really sits well with the idea of not knowing all the vectors – Reet Jaiswal Sep 3 '20 at 18:40 (b) Gravitational Force Field F⃗ and we throw a ball upward d⃗ against the direction of force. In this case we displace the object under the influence of field, object is totally rely on field. Actually you do not displace this object, i.e. you do not apply constant force in it's whole movement trajectory. You just generate initial thrust, giving to object initial kinetic energy. Then this kinetic energy is gradually reduced by Earth gravitational field, until object reaches maximum height possible. If body initial speed is not greater or equal to escape velocity, then this body will fallback to Earth surface. So upon reaching maximum height, gravity will do a full-stop negative work to body, so : $$E_k - W = 0$$ Substituting kinetic energy definition and work done by gravity gives : $$\frac {m{v_o}^2}{2} - F_{grav}\cdot h = 0$$ From there you can express maximum height $$h$$ until which object will go upwards. 1 more thing here to understand in work.. That if the angle 'theta' is perpendicular than work done is 0....that's I mean wrong... Let's say when we carry say 50 kg load and stand at one point than we haven't done any work.... Irony... But we will feel tired.... 😬... Yeah to understand it properly we need to be more practical.... • Hello user, in this formula angle between force and displacement matter. because component of applied force parallel to the displacement (let say x-component) is useful other component (y-component) is not contributing in motion it is wastage. – 123 Aug 12 '20 at 9:48 • The 'doing work' that we use in everyday language is not the same as what 'work' means in Physics. – dnaik Sep 2 '20 at 2:22 • @user272129 Physics can be as practical as YOU want it to be. It is the literal description of the universe, and if you can't explain some phenomenon, then it means you have failed in describing it, not that physics itself is impractical – Reet Jaiswal Sep 2 '20 at 2:37
	# Scalar Fields¶ Given a topological manifold $$M$$ over a topological field $$K$$ (in most applications, $$K = \RR$$ or $$K = \CC$$), a scalar field on $$M$$ is a continuous map $f: M \longrightarrow K$ Scalar fields are implemented by the class ScalarField. AUTHORS: • Eric Gourgoulhon, Michal Bejger (2013-2015): initial version • Travis Scrimshaw (2016): review tweaks • Marco Mancini (2017): SymPy as an optional symbolic engine, alternative to SR REFERENCES: class sage.manifolds.scalarfield.ScalarField(parent, coord_expression=None, chart=None, name=None, latex_name=None) Scalar field on a topological manifold. Given a topological manifold $$M$$ over a topological field $$K$$ (in most applications, $$K = \RR$$ or $$K = \CC$$), a scalar field on $$M$$ is a continuous map $f: M \longrightarrow K.$ A scalar field on $$M$$ is an element of the commutative algebra $$C^0(M)$$ (see ScalarFieldAlgebra). INPUT: • parent – the algebra of scalar fields containing the scalar field (must be an instance of class ScalarFieldAlgebra) • coord_expression – (default: None) coordinate expression(s) of the scalar field; this can be either • a dictionary of coordinate expressions in various charts on the domain, with the charts as keys; • a single coordinate expression; if the argument chart is 'all', this expression is set to all the charts defined on the open set; otherwise, the expression is set in the specific chart provided by the argument chart • chart – (default: None) chart defining the coordinates used in coord_expression when the latter is a single coordinate expression; if none is provided (default), the default chart of the open set is assumed. If chart=='all', coord_expression is assumed to be independent of the chart (constant scalar field). • name – (default: None) string; name (symbol) given to the scalar field • latex_name – (default: None) string; LaTeX symbol to denote the scalar field; if none is provided, the LaTeX symbol is set to name If coord_expression is None or incomplete, coordinate expressions can be added after the creation of the object, by means of the methods add_expr(), add_expr_by_continuation() and set_expr(). EXAMPLES: A scalar field on the 2-sphere: sage: M = Manifold(2, 'M', structure='topological') # the 2-dimensional sphere S^2 sage: U = M.open_subset('U') # complement of the North pole sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole sage: V = M.open_subset('V') # complement of the South pole sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', ....: restrictions1= x^2+y^2!=0, ....: restrictions2= u^2+v^2!=0) sage: uv_to_xy = xy_to_uv.inverse() sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), c_uv: (u^2+v^2)/(1+u^2+v^2)}, ....: name='f') ; f Scalar field f on the 2-dimensional topological manifold M sage: f.display() f: M --> R on U: (x, y) \|--> 1/(x^2 + y^2 + 1) on V: (u, v) \|--> (u^2 + v^2)/(u^2 + v^2 + 1) For scalar fields defined by a single coordinate expression, the latter can be passed instead of the dictionary over the charts: sage: g = U.scalar_field(xy, chart=c_xy, name='g') ; g Scalar field g on the Open subset U of the 2-dimensional topological manifold M The above is indeed equivalent to: sage: g = U.scalar_field({c_xy: xy}, name='g') ; g Scalar field g on the Open subset U of the 2-dimensional topological manifold M Since c_xy is the default chart of U, the argument chart can be skipped: sage: g = U.scalar_field(xy, name='g') ; g Scalar field g on the Open subset U of the 2-dimensional topological manifold M The scalar field $$g$$ is defined on $$U$$ and has an expression in terms of the coordinates $$(u,v)$$ on $$W=U\cap V$$: sage: g.display() g: U --> R (x, y) \|--> xy on W: (u, v) \|--> uv/(u^4 + 2u^2v^2 + v^4) Scalar fields on $$M$$ can also be declared with a single chart: sage: f = M.scalar_field(1/(1+x^2+y^2), chart=c_xy, name='f') ; f Scalar field f on the 2-dimensional topological manifold M Their definition must then be completed by providing the expressions on other charts, via the method add_expr(), to get a global cover of the manifold: sage: f.add_expr((u^2+v^2)/(1+u^2+v^2), chart=c_uv) sage: f.display() f: M --> R on U: (x, y) \|--> 1/(x^2 + y^2 + 1) on V: (u, v) \|--> (u^2 + v^2)/(u^2 + v^2 + 1) We can even first declare the scalar field without any coordinate expression and provide them subsequently: sage: f = M.scalar_field(name='f') sage: f.display() f: M --> R on U: (x, y) \|--> 1/(x^2 + y^2 + 1) on V: (u, v) \|--> (u^2 + v^2)/(u^2 + v^2 + 1) We may also use the method add_expr_by_continuation() to complete the coordinate definition using the analytic continuation from domains in which charts overlap: sage: f = M.scalar_field(1/(1+x^2+y^2), chart=c_xy, name='f') ; f Scalar field f on the 2-dimensional topological manifold M sage: f.display() f: M --> R on U: (x, y) \|--> 1/(x^2 + y^2 + 1) on V: (u, v) \|--> (u^2 + v^2)/(u^2 + v^2 + 1) A scalar field can also be defined by some unspecified function of the coordinates: sage: h = U.scalar_field(function('H')(x, y), name='h') ; h Scalar field h on the Open subset U of the 2-dimensional topological manifold M sage: h.display() h: U --> R (x, y) \|--> H(x, y) on W: (u, v) \|--> H(u/(u^2 + v^2), v/(u^2 + v^2)) We may use the argument latex_name to specify the LaTeX symbol denoting the scalar field if the latter is different from name: sage: latex(f) f sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), c_uv: (u^2+v^2)/(1+u^2+v^2)}, ....: name='f', latex_name=r'\mathcal{F}') sage: latex(f) \mathcal{F} The coordinate expression in a given chart is obtained via the method expr(), which returns a symbolic expression: sage: f.expr(c_uv) (u^2 + v^2)/(u^2 + v^2 + 1) sage: type(f.expr(c_uv)) <type 'sage.symbolic.expression.Expression'> The method coord_function() returns instead a function of the chart coordinates, i.e. an instance of ChartFunction: sage: f.coord_function(c_uv) (u^2 + v^2)/(u^2 + v^2 + 1) sage: type(f.coord_function(c_uv)) <class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'> sage: f.coord_function(c_uv).display() (u, v) \|--> (u^2 + v^2)/(u^2 + v^2 + 1) The value returned by the method expr() is actually the coordinate expression of the chart function: sage: f.expr(c_uv) is f.coord_function(c_uv).expr() True A constant scalar field is declared by setting the argument chart to 'all': sage: c = M.scalar_field(2, chart='all', name='c') ; c Scalar field c on the 2-dimensional topological manifold M sage: c.display() c: M --> R on U: (x, y) \|--> 2 on V: (u, v) \|--> 2 A shortcut is to use the method constant_scalar_field(): sage: c == M.constant_scalar_field(2) True The constant value can be some unspecified parameter: sage: var('a') a sage: c = M.constant_scalar_field(a, name='c') ; c Scalar field c on the 2-dimensional topological manifold M sage: c.display() c: M --> R on U: (x, y) \|--> a on V: (u, v) \|--> a A special case of constant field is the zero scalar field: sage: zer = M.constant_scalar_field(0) ; zer Scalar field zero on the 2-dimensional topological manifold M sage: zer.display() zero: M --> R on U: (x, y) \|--> 0 on V: (u, v) \|--> 0 It can be obtained directly by means of the function zero_scalar_field(): sage: zer is M.zero_scalar_field() True A third way is to get it as the zero element of the algebra $$C^0(M)$$ of scalar fields on $$M$$ (see below): sage: zer is M.scalar_field_algebra().zero() True By definition, a scalar field acts on the manifold’s points, sending them to elements of the manifold’s base field (real numbers in the present case): sage: N = M.point((0,0), chart=c_uv) # the North pole sage: S = M.point((0,0), chart=c_xy) # the South pole sage: E = M.point((1,0), chart=c_xy) # a point at the equator sage: f(N) 0 sage: f(S) 1 sage: f(E) 1/2 sage: h(E) H(1, 0) sage: c(E) a sage: zer(E) 0 A scalar field can be compared to another scalar field: sage: f == g False …to a symbolic expression: sage: f == xy False sage: g == xy True sage: c == a True …to a number: sage: f == 2 False sage: zer == 0 True …to anything else: sage: f == M False Standard mathematical functions are implemented: sage: sqrt(f) Scalar field sqrt(f) on the 2-dimensional topological manifold M sage: sqrt(f).display() sqrt(f): M --> R on U: (x, y) \|--> 1/sqrt(x^2 + y^2 + 1) on V: (u, v) \|--> sqrt(u^2 + v^2)/sqrt(u^2 + v^2 + 1) sage: tan(f) Scalar field tan(f) on the 2-dimensional topological manifold M sage: tan(f).display() tan(f): M --> R on U: (x, y) \|--> sin(1/(x^2 + y^2 + 1))/cos(1/(x^2 + y^2 + 1)) on V: (u, v) \|--> sin((u^2 + v^2)/(u^2 + v^2 + 1))/cos((u^2 + v^2)/(u^2 + v^2 + 1)) Arithmetics of scalar fields Scalar fields on $$M$$ (resp. $$U$$) belong to the algebra $$C^0(M)$$ (resp. $$C^0(U)$$): sage: f.parent() Algebra of scalar fields on the 2-dimensional topological manifold M sage: f.parent() is M.scalar_field_algebra() True sage: g.parent() Algebra of scalar fields on the Open subset U of the 2-dimensional topological manifold M sage: g.parent() is U.scalar_field_algebra() True Consequently, scalar fields can be added: sage: s = f + c ; s Scalar field f+c on the 2-dimensional topological manifold M sage: s.display() f+c: M --> R on U: (x, y) \|--> (ax^2 + ay^2 + a + 1)/(x^2 + y^2 + 1) on V: (u, v) \|--> ((a + 1)u^2 + (a + 1)v^2 + a)/(u^2 + v^2 + 1) and subtracted: sage: s = f - c ; s Scalar field f-c on the 2-dimensional topological manifold M sage: s.display() f-c: M --> R on U: (x, y) \|--> -(ax^2 + ay^2 + a - 1)/(x^2 + y^2 + 1) on V: (u, v) \|--> -((a - 1)u^2 + (a - 1)v^2 + a)/(u^2 + v^2 + 1) Some tests: sage: f + zer == f True sage: f - f == zer True sage: f + (-f) == zer True sage: (f+c)-f == c True sage: (f-c)+c == f True We may add a number (interpreted as a constant scalar field) to a scalar field: sage: s = f + 1 ; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> (x^2 + y^2 + 2)/(x^2 + y^2 + 1) on V: (u, v) \|--> (2u^2 + 2v^2 + 1)/(u^2 + v^2 + 1) sage: (f+1)-1 == f True The number can represented by a symbolic variable: sage: s = a + f ; s Scalar field on the 2-dimensional topological manifold M sage: s == c + f True However if the symbolic variable is a chart coordinate, the addition is performed only on the chart domain: sage: s = f + x; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> (x^3 + xy^2 + x + 1)/(x^2 + y^2 + 1) sage: s = f + u; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on V: (u, v) \|--> (u^3 + (u + 1)v^2 + u^2 + u)/(u^2 + v^2 + 1) The addition of two scalar fields with different domains is possible if the domain of one of them is a subset of the domain of the other; the domain of the result is then this subset: sage: f.domain() 2-dimensional topological manifold M sage: g.domain() Open subset U of the 2-dimensional topological manifold M sage: s = f + g ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.domain() Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> (xy^3 + (x^3 + x)y + 1)/(x^2 + y^2 + 1) on W: (u, v) \|--> (u^6 + 3u^4v^2 + 3u^2v^4 + v^6 + uv^3 + (u^3 + u)v)/(u^6 + v^6 + (3u^2 + 1)v^4 + u^4 + (3u^4 + 2u^2)v^2) The operation actually performed is $$f\|_U + g$$: sage: s == f.restrict(U) + g True In Sage framework, the addition of $$f$$ and $$g$$ is permitted because there is a coercion of the parent of $$f$$, namely $$C^0(M)$$, to the parent of $$g$$, namely $$C^0(U)$$ (see ScalarFieldAlgebra): sage: CM = M.scalar_field_algebra() sage: CU = U.scalar_field_algebra() sage: CU.has_coerce_map_from(CM) True The coercion map is nothing but the restriction to domain $$U$$: sage: CU.coerce(f) == f.restrict(U) True Since the algebra $$C^0(M)$$ is a vector space over $$\RR$$, scalar fields can be multiplied by a number, either an explicit one: sage: s = 2f ; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> 2/(x^2 + y^2 + 1) on V: (u, v) \|--> 2(u^2 + v^2)/(u^2 + v^2 + 1) or a symbolic one: sage: s = af ; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> a/(x^2 + y^2 + 1) on V: (u, v) \|--> (u^2 + v^2)a/(u^2 + v^2 + 1) However, if the symbolic variable is a chart coordinate, the multiplication is performed only in the corresponding chart: sage: s = xf; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> x/(x^2 + y^2 + 1) sage: s = uf; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on V: (u, v) \|--> (u^2 + v^2)u/(u^2 + v^2 + 1) Some tests: sage: 0f == 0 True sage: 0f == zer True sage: 1f == f True sage: (-2)f == - f - f True The ring multiplication of the algebras $$C^0(M)$$ and $$C^0(U)$$ is the pointwise multiplication of functions: sage: s = ff ; s Scalar field ff on the 2-dimensional topological manifold M sage: s.display() ff: M --> R on U: (x, y) \|--> 1/(x^4 + y^4 + 2(x^2 + 1)y^2 + 2x^2 + 1) on V: (u, v) \|--> (u^4 + 2u^2v^2 + v^4)/(u^4 + v^4 + 2(u^2 + 1)v^2 + 2u^2 + 1) sage: s = gh ; s Scalar field gh on the Open subset U of the 2-dimensional topological manifold M sage: s.display() gh: U --> R (x, y) \|--> xyH(x, y) on W: (u, v) \|--> uvH(u/(u^2 + v^2), v/(u^2 + v^2))/(u^4 + 2u^2v^2 + v^4) Thanks to the coercion $$C^0(M) \to C^0(U)$$ mentioned above, it is possible to multiply a scalar field defined on $$M$$ by a scalar field defined on $$U$$, the result being a scalar field defined on $$U$$: sage: f.domain(), g.domain() (2-dimensional topological manifold M, Open subset U of the 2-dimensional topological manifold M) sage: s = fg ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> xy/(x^2 + y^2 + 1) on W: (u, v) \|--> uv/(u^4 + v^4 + (2u^2 + 1)v^2 + u^2) sage: s == f.restrict(U)g True Scalar fields can be divided (pointwise division): sage: s = f/c ; s Scalar field f/c on the 2-dimensional topological manifold M sage: s.display() f/c: M --> R on U: (x, y) \|--> 1/(ax^2 + ay^2 + a) on V: (u, v) \|--> (u^2 + v^2)/(au^2 + av^2 + a) sage: s = g/h ; s Scalar field g/h on the Open subset U of the 2-dimensional topological manifold M sage: s.display() g/h: U --> R (x, y) \|--> xy/H(x, y) on W: (u, v) \|--> uv/((u^4 + 2u^2v^2 + v^4)H(u/(u^2 + v^2), v/(u^2 + v^2))) sage: s = f/g ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> 1/(xy^3 + (x^3 + x)y) on W: (u, v) \|--> (u^6 + 3u^4v^2 + 3u^2v^4 + v^6)/(uv^3 + (u^3 + u)v) sage: s == f.restrict(U)/g True For scalar fields defined on a single chart domain, we may perform some arithmetics with symbolic expressions involving the chart coordinates: sage: s = g + x^2 - y ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> x^2 + (x - 1)y on W: (u, v) \|--> -(v^3 - u^2 + (u^2 - u)v)/(u^4 + 2u^2v^2 + v^4) sage: s = gx ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> x^2y on W: (u, v) \|--> u^2v/(u^6 + 3u^4v^2 + 3u^2v^4 + v^6) sage: s = g/x ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> y on W: (u, v) \|--> v/(u^2 + v^2) sage: s = x/g ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> 1/y on W: (u, v) \|--> (u^2 + v^2)/v Examples with SymPy as the symbolic engine From now on, we ask that all symbolic calculus on manifold $$M$$ are performed by SymPy: sage: M.set_calculus_method('sympy') We define $$f$$ as above: sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), c_uv: (u^2+v^2)/(1+u^2+v^2)}, ....: name='f') ; f Scalar field f on the 2-dimensional topological manifold M sage: f.display() # notice the SymPy display of exponents f: M --> R on U: (x, y) \|--> 1/(x2 + y2 + 1) on V: (u, v) \|--> (u2 + v2)/(u2 + v2 + 1) sage: type(f.coord_function(c_xy).expr()) <class 'sympy.core.power.Pow'> The scalar field $$g$$ defined on $$U$$: sage: g = U.scalar_field({c_xy: xy}, name='g') sage: g.display() # again notice the SymPy display of exponents g: U --> R (x, y) \|--> xy on W: (u, v) \|--> uv/(u*4 + 2u*2v2 + v4) Definition on a single chart and subsequent completion: sage: f = M.scalar_field(1/(1+x^2+y^2), chart=c_xy, name='f') sage: f.display() f: M --> R on U: (x, y) \|--> 1/(x2 + y2 + 1) on V: (u, v) \|--> (u2 + v2)/(u2 + v2 + 1) Defintion without any coordinate expression and subsequent completion: sage: f = M.scalar_field(name='f') sage: f.display() f: M --> R on U: (x, y) \|--> 1/(x2 + y2 + 1) on V: (u, v) \|--> (u2 + v2)/(u2 + v2 + 1) sage: f = M.scalar_field(1/(1+x^2+y^2), chart=c_xy, name='f') sage: f.display() f: M --> R on U: (x, y) \|--> 1/(x2 + y2 + 1) on V: (u, v) \|--> (u2 + v2)/(u2 + v2 + 1) A scalar field defined by some unspecified function of the coordinates: sage: h = U.scalar_field(function('H')(x, y), name='h') ; h Scalar field h on the Open subset U of the 2-dimensional topological manifold M sage: h.display() h: U --> R (x, y) \|--> H(x, y) on W: (u, v) \|--> H(u/(u2 + v2), v/(u2 + v2)) The coordinate expression in a given chart is obtained via the method expr(), which in the present context, returns a SymPy object: sage: f.expr(c_uv) (u2 + v2)/(u2 + v2 + 1) sage: type(f.expr(c_uv)) <class 'sympy.core.mul.Mul'> The method coord_function() returns instead a function of the chart coordinates, i.e. an instance of ChartFunction: sage: f.coord_function(c_uv) (u2 + v2)/(u2 + v2 + 1) sage: type(f.coord_function(c_uv)) <class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'> sage: f.coord_function(c_uv).display() (u, v) \|--> (u2 + v2)/(u2 + v2 + 1) The value returned by the method expr() is actually the coordinate expression of the chart function: sage: f.expr(c_uv) is f.coord_function(c_uv).expr() True We may ask for the SR representation of the coordinate function: sage: f.coord_function(c_uv).expr('SR') (u^2 + v^2)/(u^2 + v^2 + 1) A constant scalar field with SymPy representation: sage: c = M.constant_scalar_field(2, name='c') sage: c.display() c: M --> R on U: (x, y) \|--> 2 on V: (u, v) \|--> 2 sage: type(c.expr(c_xy)) <class 'sympy.core.numbers.Integer'> The constant value can be some unspecified parameter: sage: var('a') a sage: c = M.constant_scalar_field(a, name='c') sage: c.display() c: M --> R on U: (x, y) \|--> a on V: (u, v) \|--> a sage: type(c.expr(c_xy)) <class 'sympy.core.symbol.Symbol'> The zero scalar field: sage: zer = M.constant_scalar_field(0) ; zer Scalar field zero on the 2-dimensional topological manifold M sage: zer.display() zero: M --> R on U: (x, y) \|--> 0 on V: (u, v) \|--> 0 sage: type(zer.expr(c_xy)) <class 'sympy.core.numbers.Zero'> sage: zer is M.zero_scalar_field() True Action of scalar fields on manifold’s points: sage: N = M.point((0,0), chart=c_uv) # the North pole sage: S = M.point((0,0), chart=c_xy) # the South pole sage: E = M.point((1,0), chart=c_xy) # a point at the equator sage: f(N) 0 sage: f(S) 1 sage: f(E) 1/2 sage: h(E) H(1, 0) sage: c(E) a sage: zer(E) 0 A scalar field can be compared to another scalar field: sage: f == g False …to a symbolic expression: sage: f == xy False sage: g == xy True sage: c == a True …to a number: sage: f == 2 False sage: zer == 0 True …to anything else: sage: f == M False Standard mathematical functions are implemented: sage: sqrt(f) Scalar field sqrt(f) on the 2-dimensional topological manifold M sage: sqrt(f).display() sqrt(f): M --> R on U: (x, y) \|--> 1/sqrt(x2 + y2 + 1) on V: (u, v) \|--> sqrt(u2 + v2)/sqrt(u2 + v2 + 1) sage: tan(f) Scalar field tan(f) on the 2-dimensional topological manifold M sage: tan(f).display() tan(f): M --> R on U: (x, y) \|--> tan(1/(x2 + y2 + 1)) on V: (u, v) \|--> tan((u2 + v2)/(u2 + v2 + 1)) Arithmetics of scalar fields with SymPy Scalar fields on $$M$$ (resp. $$U$$) belong to the algebra $$C^0(M)$$ (resp. $$C^0(U)$$): sage: f.parent() Algebra of scalar fields on the 2-dimensional topological manifold M sage: f.parent() is M.scalar_field_algebra() True sage: g.parent() Algebra of scalar fields on the Open subset U of the 2-dimensional topological manifold M sage: g.parent() is U.scalar_field_algebra() True Consequently, scalar fields can be added: sage: s = f + c ; s Scalar field f+c on the 2-dimensional topological manifold M sage: s.display() f+c: M --> R on U: (x, y) \|--> (ax2 + ay2 + a + 1)/(x2 + y*2 + 1) on V: (u, v) \|--> (au*2 + av2 + a + u2 + v2)/(u2 + v*2 + 1) and subtracted: sage: s = f - c ; s Scalar field f-c on the 2-dimensional topological manifold M sage: s.display() f-c: M --> R on U: (x, y) \|--> (-ax*2 - ay2 - a + 1)/(x2 + y*2 + 1) on V: (u, v) \|--> (-au*2 - av2 - a + u2 + v2)/(u2 + v2 + 1) Some tests: sage: f + zer == f True sage: f - f == zer True sage: f + (-f) == zer True sage: (f+c)-f == c True sage: (f-c)+c == f True We may add a number (interpreted as a constant scalar field) to a scalar field: sage: s = f + 1 ; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> (x2 + y2 + 2)/(x2 + y*2 + 1) on V: (u, v) \|--> (2u*2 + 2v2 + 1)/(u2 + v2 + 1) sage: (f+1)-1 == f True The number can represented by a symbolic variable: sage: s = a + f ; s Scalar field on the 2-dimensional topological manifold M sage: s == c + f True However if the symbolic variable is a chart coordinate, the addition is performed only on the chart domain: sage: s = f + x; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> (x3 + xy2 + x + 1)/(x2 + y2 + 1) sage: s = f + u; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on V: (u, v) \|--> (u3 + u2 + uv2 + u + v2)/(u2 + v2 + 1) The addition of two scalar fields with different domains is possible if the domain of one of them is a subset of the domain of the other; the domain of the result is then this subset: sage: f.domain() 2-dimensional topological manifold M sage: g.domain() Open subset U of the 2-dimensional topological manifold M sage: s = f + g ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.domain() Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> (x*3y + xy3 + xy + 1)/(x2 + y2 + 1) on W: (u, v) \|--> (u*6 + 3u*4v2 + u3v + 3u*2v*4 + uv*3 + uv + v6)/(u6 + 3u4v2 + u4 + 3u2v*4 + 2u*2v2 + v6 + v*4) The operation actually performed is $$f\|_U + g$$: sage: s == f.restrict(U) + g True Since the algebra $$C^0(M)$$ is a vector space over $$\RR$$, scalar fields can be multiplied by a number, either an explicit one: sage: s = 2f ; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> 2/(x2 + y2 + 1) on V: (u, v) \|--> 2(u2 + v2)/(u2 + v2 + 1) or a symbolic one: sage: s = af ; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> a/(x2 + y2 + 1) on V: (u, v) \|--> a(u2 + v2)/(u2 + v2 + 1) However, if the symbolic variable is a chart coordinate, the multiplication is performed only in the corresponding chart: sage: s = xf; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on U: (x, y) \|--> x/(x2 + y2 + 1) sage: s = uf; s Scalar field on the 2-dimensional topological manifold M sage: s.display() M --> R on V: (u, v) \|--> u(u2 + v2)/(u2 + v2 + 1) Some tests: sage: 0f == 0 True sage: 0f == zer True sage: 1f == f True sage: (-2)f == - f - f True The ring multiplication of the algebras $$C^0(M)$$ and $$C^0(U)$$ is the pointwise multiplication of functions: sage: s = ff ; s Scalar field ff on the 2-dimensional topological manifold M sage: s.display() ff: M --> R on U: (x, y) \|--> 1/(x4 + 2x*2y*2 + 2x2 + y4 + 2y2 + 1) on V: (u, v) \|--> (u4 + 2u*2v2 + v4)/(u*4 + 2u*2v*2 + 2u2 + v4 + 2v2 + 1) sage: s = gh ; s Scalar field gh on the Open subset U of the 2-dimensional topological manifold M sage: s.display() gh: U --> R (x, y) \|--> xyH(x, y) on W: (u, v) \|--> uvH(u/(u2 + v2), v/(u2 + v2))/(u*4 + 2u*2v2 + v4) Thanks to the coercion $$C^0(M) \to C^0(U)$$ mentioned above, it is possible to multiply a scalar field defined on $$M$$ by a scalar field defined on $$U$$, the result being a scalar field defined on $$U$$: sage: f.domain(), g.domain() (2-dimensional topological manifold M, Open subset U of the 2-dimensional topological manifold M) sage: s = fg ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> xy/(x2 + y2 + 1) on W: (u, v) \|--> uv/(u4 + 2u*2v2 + u2 + v4 + v2) sage: s == f.restrict(U)g True Scalar fields can be divided (pointwise division): sage: s = f/c ; s Scalar field f/c on the 2-dimensional topological manifold M sage: s.display() f/c: M --> R on U: (x, y) \|--> 1/(a(x2 + y2 + 1)) on V: (u, v) \|--> (u2 + v2)/(a(u2 + v2 + 1)) sage: s = g/h ; s Scalar field g/h on the Open subset U of the 2-dimensional topological manifold M sage: s.display() g/h: U --> R (x, y) \|--> xy/H(x, y) on W: (u, v) \|--> uv/((u4 + 2u*2v2 + v4)H(u/(u2 + v2), v/(u2 + v2))) sage: s = f/g ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> 1/(xy(x2 + y2 + 1)) on W: (u, v) \|--> (u6 + 3u*4v*2 + 3u*2v4 + v6)/(uv(u2 + v2 + 1)) sage: s == f.restrict(U)/g True For scalar fields defined on a single chart domain, we may perform some arithmetics with symbolic expressions involving the chart coordinates: sage: s = g + x^2 - y ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> x*2 + xy - y on W: (u, v) \|--> (-u*2v + u*2 + uv - v3)/(u4 + 2u2v2 + v4) sage: s = gx ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> x2y on W: (u, v) \|--> u*2v/(u*6 + 3u*4v*2 + 3u*2v4 + v6) sage: s = g/x ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> y on W: (u, v) \|--> v/(u2 + v2) sage: s = x/g ; s Scalar field on the Open subset U of the 2-dimensional topological manifold M sage: s.display() U --> R (x, y) \|--> 1/y on W: (u, v) \|--> u*2/v + v The test suite is passed: sage: TestSuite(f).run() sage: TestSuite(zer).run() add_expr(coord_expression, chart=None) Add some coordinate expression to the scalar field. The previous expressions with respect to other charts are kept. To clear them, use set_expr() instead. INPUT: • coord_expression – coordinate expression of the scalar field • chart – (default: None) chart in which coord_expression is defined; if None, the default chart of the scalar field’s domain is assumed Warning If the scalar field has already expressions in other charts, it is the user’s responsibility to make sure that the expression to be added is consistent with them. EXAMPLES: Adding scalar field expressions on a 2-dimensional manifold: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(x^2 + 2xy +1) sage: f._express {Chart (M, (x, y)): x^2 + 2xy + 1} sage: f._express # the (x,y) expression has been changed: {Chart (M, (x, y)): 3y} sage: c_uv.<u,v> = M.chart() sage: f._express # random (dict. output); f has now 2 expressions: {Chart (M, (x, y)): 3y, Chart (M, (u, v)): cos(u) - sin(v)} add_expr_by_continuation(chart, subdomain) Set coordinate expression in a chart by continuation of the coordinate expression in a subchart. The continuation is performed by demanding that the coordinate expression is identical to that in the restriction of the chart to a given subdomain. INPUT: • chart – coordinate chart $$(U,(x^i))$$ in which the expression of the scalar field is to set • subdomain – open subset $$V\subset U$$ in which the expression in terms of the restriction of the coordinate chart $$(U,(x^i))$$ to $$V$$ is already known or can be evaluated by a change of coordinates. EXAMPLES: Scalar field on the sphere $$S^2$$: sage: M = Manifold(2, 'S^2', structure='topological') sage: U = M.open_subset('U') ; V = M.open_subset('V') # the complement of resp. N pole and S pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart() # stereographic coordinates sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', restrictions1= x^2+y^2!=0, ....: restrictions2= u^2+v^2!=0) sage: uv_to_xy = xy_to_uv.inverse() sage: W = U.intersection(V) # S^2 minus the two poles sage: f = M.scalar_field(atan(x^2+y^2), chart=c_xy, name='f') The scalar field has been defined only on the domain covered by the chart c_xy, i.e. $$U$$: sage: f.display() f: S^2 --> R on U: (x, y) \|--> arctan(x^2 + y^2) We note that on $$W = U \cap V$$, the expression of $$f$$ in terms of coordinates $$(u,v)$$ can be deduced from that in the coordinates $$(x,y)$$ thanks to the transition map between the two charts: sage: f.display(c_uv.restrict(W)) f: S^2 --> R on W: (u, v) \|--> arctan(1/(u^2 + v^2)) We use this fact to extend the definition of $$f$$ to the open subset $$V$$, covered by the chart c_uv: sage: f.add_expr_by_continuation(c_uv, W) Then, $$f$$ is known on the whole sphere: sage: f.display() f: S^2 --> R on U: (x, y) \|--> arctan(x^2 + y^2) on V: (u, v) \|--> arctan(1/(u^2 + v^2)) arccos() Arc cosine of the scalar field. OUTPUT: • the scalar field $$\arccos f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = arccos(f) ; g Scalar field arccos(f) on the 2-dimensional topological manifold M sage: latex(g) \arccos\left(\Phi\right) sage: g.display() arccos(f): M --> R (x, y) \|--> arccos(xy) The notation acos can be used as well: sage: acos(f) Scalar field arccos(f) on the 2-dimensional topological manifold M sage: acos(f) == g True Some tests: sage: cos(g) == f True sage: arccos(M.constant_scalar_field(1)) == M.zero_scalar_field() True sage: arccos(M.zero_scalar_field()) == M.constant_scalar_field(pi/2) True arccosh() Inverse hyperbolic cosine of the scalar field. OUTPUT: • the scalar field $$\mathrm{arccosh}\, f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = arccosh(f) ; g Scalar field arccosh(f) on the 2-dimensional topological manifold M sage: latex(g) \,\mathrm{arccosh}\left(\Phi\right) sage: g.display() arccosh(f): M --> R (x, y) \|--> arccosh(xy) The notation acosh can be used as well: sage: acosh(f) Scalar field arccosh(f) on the 2-dimensional topological manifold M sage: acosh(f) == g True Some tests: sage: cosh(g) == f True sage: arccosh(M.constant_scalar_field(1)) == M.zero_scalar_field() True arcsin() Arc sine of the scalar field. OUTPUT: • the scalar field $$\arcsin f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = arcsin(f) ; g Scalar field arcsin(f) on the 2-dimensional topological manifold M sage: latex(g) \arcsin\left(\Phi\right) sage: g.display() arcsin(f): M --> R (x, y) \|--> arcsin(xy) The notation asin can be used as well: sage: asin(f) Scalar field arcsin(f) on the 2-dimensional topological manifold M sage: asin(f) == g True Some tests: sage: sin(g) == f True sage: arcsin(M.zero_scalar_field()) == M.zero_scalar_field() True sage: arcsin(M.constant_scalar_field(1)) == M.constant_scalar_field(pi/2) True arcsinh() Inverse hyperbolic sine of the scalar field. OUTPUT: • the scalar field $$\mathrm{arcsinh}\, f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = arcsinh(f) ; g Scalar field arcsinh(f) on the 2-dimensional topological manifold M sage: latex(g) \,\mathrm{arcsinh}\left(\Phi\right) sage: g.display() arcsinh(f): M --> R (x, y) \|--> arcsinh(xy) The notation asinh can be used as well: sage: asinh(f) Scalar field arcsinh(f) on the 2-dimensional topological manifold M sage: asinh(f) == g True Some tests: sage: sinh(g) == f True sage: arcsinh(M.zero_scalar_field()) == M.zero_scalar_field() True arctan() Arc tangent of the scalar field. OUTPUT: • the scalar field $$\arctan f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = arctan(f) ; g Scalar field arctan(f) on the 2-dimensional topological manifold M sage: latex(g) \arctan\left(\Phi\right) sage: g.display() arctan(f): M --> R (x, y) \|--> arctan(xy) The notation atan can be used as well: sage: atan(f) Scalar field arctan(f) on the 2-dimensional topological manifold M sage: atan(f) == g True Some tests: sage: tan(g) == f True sage: arctan(M.zero_scalar_field()) == M.zero_scalar_field() True sage: arctan(M.constant_scalar_field(1)) == M.constant_scalar_field(pi/4) True arctanh() Inverse hyperbolic tangent of the scalar field. OUTPUT: • the scalar field $$\mathrm{arctanh}\, f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = arctanh(f) ; g Scalar field arctanh(f) on the 2-dimensional topological manifold M sage: latex(g) \,\mathrm{arctanh}\left(\Phi\right) sage: g.display() arctanh(f): M --> R (x, y) \|--> arctanh(xy) The notation atanh can be used as well: sage: atanh(f) Scalar field arctanh(f) on the 2-dimensional topological manifold M sage: atanh(f) == g True Some tests: sage: tanh(g) == f True sage: arctanh(M.zero_scalar_field()) == M.zero_scalar_field() True sage: arctanh(M.constant_scalar_field(1/2)) == M.constant_scalar_field(log(3)/2) True common_charts(other) Find common charts for the expressions of the scalar field and other. INPUT: • other – a scalar field OUTPUT: • list of common charts; if no common chart is found, None is returned (instead of an empty list) EXAMPLES: Search for common charts on a 2-dimensional manifold with 2 overlapping domains: sage: M = Manifold(2, 'M', structure='topological') sage: U = M.open_subset('U') sage: c_xy.<x,y> = U.chart() sage: V = M.open_subset('V') sage: c_uv.<u,v> = V.chart() sage: M.declare_union(U,V) # M is the union of U and V sage: f = U.scalar_field(x^2) sage: g = M.scalar_field(x+y) sage: f.common_charts(g) [Chart (U, (x, y))] sage: f._express {Chart (U, (x, y)): x^2} sage: g._express # random (dictionary output) {Chart (U, (x, y)): x + y, Chart (V, (u, v)): u} sage: f.common_charts(g) [Chart (U, (x, y))] Common charts found as subcharts: the subcharts are introduced via a transition map between charts c_xy and c_uv on the intersecting subdomain $$W = U\cap V$$: sage: trans = c_xy.transition_map(c_uv, (x+y, x-y), 'W', x<0, u+v<0) sage: M.atlas() [Chart (U, (x, y)), Chart (V, (u, v)), Chart (W, (x, y)), Chart (W, (u, v))] sage: c_xy_W = M.atlas()[2] sage: c_uv_W = M.atlas()[3] sage: trans.inverse() Change of coordinates from Chart (W, (u, v)) to Chart (W, (x, y)) sage: f.common_charts(g) [Chart (U, (x, y))] sage: f.expr(c_xy_W) x^2 sage: f._express # random (dictionary output) {Chart (U, (x, y)): x^2, Chart (W, (x, y)): x^2} sage: g._express # random (dictionary output) {Chart (U, (x, y)): x + y, Chart (V, (u, v)): u} sage: g.common_charts(f) # c_xy_W is not returned because it is subchart of 'xy' [Chart (U, (x, y))] sage: f.expr(c_uv_W) 1/4u^2 + 1/2uv + 1/4v^2 sage: f._express # random (dictionary output) {Chart (U, (x, y)): x^2, Chart (W, (x, y)): x^2, Chart (W, (u, v)): 1/4u^2 + 1/2uv + 1/4v^2} sage: g._express # random (dictionary output) {Chart (U, (x, y)): x + y, Chart (V, (u, v)): u} sage: f.common_charts(g) [Chart (U, (x, y)), Chart (W, (u, v))] sage: # the expressions have been updated on the subcharts sage: g._express # random (dictionary output) {Chart (U, (x, y)): x + y, Chart (V, (u, v)): u, Chart (W, (u, v)): u} Common charts found by computing some coordinate changes: sage: W = U.intersection(V) sage: f = W.scalar_field(x^2, c_xy_W) sage: g = W.scalar_field(u+1, c_uv_W) sage: f._express {Chart (W, (x, y)): x^2} sage: g._express {Chart (W, (u, v)): u + 1} sage: f.common_charts(g) [Chart (W, (u, v)), Chart (W, (x, y))] sage: f._express # random (dictionary output) {Chart (W, (u, v)): 1/4u^2 + 1/2uv + 1/4v^2, Chart (W, (x, y)): x^2} sage: g._express # random (dictionary output) {Chart (W, (u, v)): u + 1, Chart (W, (x, y)): x + y + 1} coord_function(chart=None, from_chart=None) Return the function of the coordinates representing the scalar field in a given chart. INPUT: • chart – (default: None) chart with respect to which the coordinate expression is to be returned; if None, the default chart of the scalar field’s domain will be used • from_chart – (default: None) chart from which the required expression is computed if it is not known already in the chart chart; if None, a chart is picked in the known expressions OUTPUT: EXAMPLES: Coordinate function on a 2-dimensional manifold: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(xy^2) sage: f.coord_function() xy^2 sage: f.coord_function(c_xy) # equivalent form (since c_xy is the default chart) xy^2 sage: type(f.coord_function()) <class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'> Expression via a change of coordinates: sage: c_uv.<u,v> = M.chart() sage: c_uv.transition_map(c_xy, [u+v, u-v]) Change of coordinates from Chart (M, (u, v)) to Chart (M, (x, y)) sage: f._express # at this stage, f is expressed only in terms of (x,y) coordinates {Chart (M, (x, y)): xy^2} sage: f.coord_function(c_uv) # forces the computation of the expression of f in terms of (u,v) coordinates u^3 - u^2v - uv^2 + v^3 sage: f.coord_function(c_uv) == (u+v)(u-v)^2 # check True sage: f._express # random (dict. output); f has now 2 coordinate expressions: {Chart (M, (x, y)): xy^2, Chart (M, (u, v)): u^3 - u^2v - uv^2 + v^3} Usage in a physical context (simple Lorentz transformation - boost in x direction, with relative velocity v between o1 and o2 frames): sage: M = Manifold(2, 'M', structure='topological') sage: o1.<t,x> = M.chart() sage: o2.<T,X> = M.chart() sage: f = M.scalar_field(x^2 - t^2) sage: f.coord_function(o1) -t^2 + x^2 sage: v = var('v'); gam = 1/sqrt(1-v^2) sage: o2.transition_map(o1, [gam(T - vX), gam(X - vT)]) Change of coordinates from Chart (M, (T, X)) to Chart (M, (t, x)) sage: f.coord_function(o2) -T^2 + X^2 copy() Return an exact copy of the scalar field. EXAMPLES: Copy on a 2-dimensional manifold: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(xy^2) sage: g = f.copy() sage: type(g) <class 'sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra_with_category.element_class'> sage: g.expr() xy^2 sage: g == f True sage: g is f False cos() Cosine of the scalar field. OUTPUT: • the scalar field $$\cos f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = cos(f) ; g Scalar field cos(f) on the 2-dimensional topological manifold M sage: latex(g) \cos\left(\Phi\right) sage: g.display() cos(f): M --> R (x, y) \|--> cos(xy) Some tests: sage: cos(M.zero_scalar_field()) == M.constant_scalar_field(1) True sage: cos(M.constant_scalar_field(pi/2)) == M.zero_scalar_field() True cosh() Hyperbolic cosine of the scalar field. OUTPUT: • the scalar field $$\cosh f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = cosh(f) ; g Scalar field cosh(f) on the 2-dimensional topological manifold M sage: latex(g) \cosh\left(\Phi\right) sage: g.display() cosh(f): M --> R (x, y) \|--> cosh(xy) Some test: sage: cosh(M.zero_scalar_field()) == M.constant_scalar_field(1) True disp(chart=None) Display the expression of the scalar field in a given chart. Without any argument, this function displays the expressions of the scalar field in all the charts defined on the scalar field’s domain that are not restrictions of another chart to some subdomain (the “top charts”). INPUT: • chart – (default: None) chart with respect to which the coordinate expression is to be displayed; if None, the display is performed in all the top charts in which the coordinate expression is known The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode). EXAMPLES: Various displays: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(sqrt(x+1), name='f') sage: f.display() f: M --> R (x, y) \|--> sqrt(x + 1) sage: latex(f.display()) \begin{array}{llcl} f:& M & \longrightarrow & \mathbb{R} \\ & \left(x, y\right) & \longmapsto & \sqrt{x + 1} \end{array} sage: g = M.scalar_field(function('G')(x, y), name='g') sage: g.display() g: M --> R (x, y) \|--> G(x, y) sage: latex(g.display()) \begin{array}{llcl} g:& M & \longrightarrow & \mathbb{R} \\ & \left(x, y\right) & \longmapsto & G\left(x, y\right) \end{array} A shortcut of display() is disp(): sage: f.disp() f: M --> R (x, y) \|--> sqrt(x + 1) display(chart=None) Display the expression of the scalar field in a given chart. Without any argument, this function displays the expressions of the scalar field in all the charts defined on the scalar field’s domain that are not restrictions of another chart to some subdomain (the “top charts”). INPUT: • chart – (default: None) chart with respect to which the coordinate expression is to be displayed; if None, the display is performed in all the top charts in which the coordinate expression is known The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode). EXAMPLES: Various displays: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(sqrt(x+1), name='f') sage: f.display() f: M --> R (x, y) \|--> sqrt(x + 1) sage: latex(f.display()) \begin{array}{llcl} f:& M & \longrightarrow & \mathbb{R} \\ & \left(x, y\right) & \longmapsto & \sqrt{x + 1} \end{array} sage: g = M.scalar_field(function('G')(x, y), name='g') sage: g.display() g: M --> R (x, y) \|--> G(x, y) sage: latex(g.display()) \begin{array}{llcl} g:& M & \longrightarrow & \mathbb{R} \\ & \left(x, y\right) & \longmapsto & G\left(x, y\right) \end{array} A shortcut of display() is disp(): sage: f.disp() f: M --> R (x, y) \|--> sqrt(x + 1) domain() Return the open subset on which the scalar field is defined. OUTPUT: EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(x+2y) sage: f.domain() 2-dimensional topological manifold M sage: U = M.open_subset('U', coord_def={c_xy: x<0}) sage: g = f.restrict(U) sage: g.domain() Open subset U of the 2-dimensional topological manifold M exp() Exponential of the scalar field. OUTPUT: • the scalar field $$\exp f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: x+y}, name='f', latex_name=r"\Phi") sage: g = exp(f) ; g Scalar field exp(f) on the 2-dimensional topological manifold M sage: g.display() exp(f): M --> R (x, y) \|--> e^(x + y) sage: latex(g) \exp\left(\Phi\right) Automatic simplifications occur: sage: f = M.scalar_field({X: 2ln(1+x^2)}, name='f') sage: exp(f).display() exp(f): M --> R (x, y) \|--> x^4 + 2x^2 + 1 The inverse function is log(): sage: log(exp(f)) == f True Some tests: sage: exp(M.zero_scalar_field()) == M.constant_scalar_field(1) True sage: exp(M.constant_scalar_field(1)) == M.constant_scalar_field(e) True expr(chart=None, from_chart=None) Return the coordinate expression of the scalar field in a given chart. INPUT: • chart – (default: None) chart with respect to which the coordinate expression is required; if None, the default chart of the scalar field’s domain will be used • from_chart – (default: None) chart from which the required expression is computed if it is not known already in the chart chart; if None, a chart is picked in self._express OUTPUT: • symbolic expression representing the coordinate expression of the scalar field in the given chart. EXAMPLES: Expression of a scalar field on a 2-dimensional manifold: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(xy^2) sage: f.expr() xy^2 sage: f.expr(c_xy) # equivalent form (since c_xy is the default chart) xy^2 sage: type(f.expr()) <type 'sage.symbolic.expression.Expression'> Expression via a change of coordinates: sage: c_uv.<u,v> = M.chart() sage: c_uv.transition_map(c_xy, [u+v, u-v]) Change of coordinates from Chart (M, (u, v)) to Chart (M, (x, y)) sage: f._express # at this stage, f is expressed only in terms of (x,y) coordinates {Chart (M, (x, y)): xy^2} sage: f.expr(c_uv) # forces the computation of the expression of f in terms of (u,v) coordinates u^3 - u^2v - uv^2 + v^3 sage: bool( f.expr(c_uv) == (u+v)(u-v)^2 ) # check True sage: f._express # random (dict. output); f has now 2 coordinate expressions: {Chart (M, (x, y)): xy^2, Chart (M, (u, v)): u^3 - u^2v - uv^2 + v^3} is_trivial_zero() Check if self is trivially equal to zero without any simplification. This method is supposed to be fast as compared with self.is_zero() or self == 0 and is intended to be used in library code where trying to obtain a mathematically correct result by applying potentially expensive rewrite rules is not desirable. EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: 0}) sage: f.is_trivial_zero() True sage: f = M.scalar_field(0) sage: f.is_trivial_zero() True sage: M.zero_scalar_field().is_trivial_zero() True sage: f = M.scalar_field({X: x+y}) sage: f.is_trivial_zero() False Scalar field defined by means of two charts: sage: U1 = M.open_subset('U1'); X1.<x1,y1> = U1.chart() sage: U2 = M.open_subset('U2'); X2.<x2,y2> = U2.chart() sage: f = M.scalar_field({X1: 0, X2: 0}) sage: f.is_trivial_zero() True sage: f = M.scalar_field({X1: 0, X2: 1}) sage: f.is_trivial_zero() False No simplification is attempted, so that False is returned for non-trivial cases: sage: f = M.scalar_field({X: cos(x)^2 + sin(x)^2 - 1}) sage: f.is_trivial_zero() False On the contrary, the method is_zero() and the direct comparison to zero involve some simplification algorithms and return True: sage: f.is_zero() True sage: f == 0 True log() Natural logarithm of the scalar field. OUTPUT: • the scalar field $$\ln f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: x+y}, name='f', latex_name=r"\Phi") sage: g = log(f) ; g Scalar field ln(f) on the 2-dimensional topological manifold M sage: g.display() ln(f): M --> R (x, y) \|--> log(x + y) sage: latex(g) \ln\left(\Phi\right) The inverse function is exp(): sage: exp(log(f)) == f True restrict(subdomain) Restriction of the scalar field to an open subset of its domain of definition. INPUT: • subdomain – an open subset of the scalar field’s domain OUTPUT: EXAMPLES: Restriction of a scalar field defined on $$\RR^2$$ to the unit open disc: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() # Cartesian coordinates sage: U = M.open_subset('U', coord_def={X: x^2+y^2 < 1}) # U unit open disc sage: f = M.scalar_field(cos(xy), name='f') sage: f_U = f.restrict(U) ; f_U Scalar field f on the Open subset U of the 2-dimensional topological manifold M sage: f_U.display() f: U --> R (x, y) \|--> cos(xy) sage: f.parent() Algebra of scalar fields on the 2-dimensional topological manifold M sage: f_U.parent() Algebra of scalar fields on the Open subset U of the 2-dimensional topological manifold M The restriction to the whole domain is the identity: sage: f.restrict(M) is f True sage: f_U.restrict(U) is f_U True Restriction of the zero scalar field: sage: M.zero_scalar_field().restrict(U) Scalar field zero on the Open subset U of the 2-dimensional topological manifold M sage: M.zero_scalar_field().restrict(U) is U.zero_scalar_field() True set_expr(coord_expression, chart=None) Set the coordinate expression of the scalar field. The expressions with respect to other charts are deleted, in order to avoid any inconsistency. To keep them, use add_expr() instead. INPUT: • coord_expression – coordinate expression of the scalar field • chart – (default: None) chart in which coord_expression is defined; if None, the default chart of the scalar field’s domain is assumed EXAMPLES: Setting scalar field expressions on a 2-dimensional manifold: sage: M = Manifold(2, 'M', structure='topological') sage: c_xy.<x,y> = M.chart() sage: f = M.scalar_field(x^2 + 2xy +1) sage: f._express {Chart (M, (x, y)): x^2 + 2xy + 1} sage: f.set_expr(3y) sage: f._express # the (x,y) expression has been changed: {Chart (M, (x, y)): 3y} sage: c_uv.<u,v> = M.chart() sage: f.set_expr(cos(u)-sin(v), c_uv) sage: f._express # the (x,y) expression has been lost: {Chart (M, (u, v)): cos(u) - sin(v)} sage: f.set_expr(3y) sage: f._express # the (u,v) expression has been lost: {Chart (M, (x, y)): 3y} set_name(name=None, latex_name=None) Set (or change) the text name and LaTeX name of the scalar field. INPUT: • name – (string; default: None) name given to the scalar field • latex_name – (string; default: None) LaTeX symbol to denote the scalar field; if None while name is provided, the LaTeX symbol is set to name EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: x+y}) sage: f = M.scalar_field({X: x+y}); f Scalar field on the 2-dimensional topological manifold M sage: f.set_name('f'); f Scalar field f on the 2-dimensional topological manifold M sage: latex(f) f sage: f.set_name('f', latex_name=r'\Phi'); f Scalar field f on the 2-dimensional topological manifold M sage: latex(f) \Phi sin() Sine of the scalar field. OUTPUT: • the scalar field $$\sin f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = sin(f) ; g Scalar field sin(f) on the 2-dimensional topological manifold M sage: latex(g) \sin\left(\Phi\right) sage: g.display() sin(f): M --> R (x, y) \|--> sin(xy) Some tests: sage: sin(M.zero_scalar_field()) == M.zero_scalar_field() True sage: sin(M.constant_scalar_field(pi/2)) == M.constant_scalar_field(1) True sinh() Hyperbolic sine of the scalar field. OUTPUT: • the scalar field $$\sinh f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = sinh(f) ; g Scalar field sinh(f) on the 2-dimensional topological manifold M sage: latex(g) \sinh\left(\Phi\right) sage: g.display() sinh(f): M --> R (x, y) \|--> sinh(xy) Some test: sage: sinh(M.zero_scalar_field()) == M.zero_scalar_field() True sqrt() Square root of the scalar field. OUTPUT: • the scalar field $$\sqrt f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: 1+x^2+y^2}, name='f', ....: latex_name=r"\Phi") sage: g = sqrt(f) ; g Scalar field sqrt(f) on the 2-dimensional topological manifold M sage: latex(g) \sqrt{\Phi} sage: g.display() sqrt(f): M --> R (x, y) \|--> sqrt(x^2 + y^2 + 1) Some tests: sage: g^2 == f True sage: sqrt(M.zero_scalar_field()) == M.zero_scalar_field() True tan() Tangent of the scalar field. OUTPUT: • the scalar field $$\tan f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = tan(f) ; g Scalar field tan(f) on the 2-dimensional topological manifold M sage: latex(g) \tan\left(\Phi\right) sage: g.display() tan(f): M --> R (x, y) \|--> sin(xy)/cos(xy) Some tests: sage: tan(f) == sin(f) / cos(f) True sage: tan(M.zero_scalar_field()) == M.zero_scalar_field() True sage: tan(M.constant_scalar_field(pi/4)) == M.constant_scalar_field(1) True tanh() Hyperbolic tangent of the scalar field. OUTPUT: • the scalar field $$\tanh f$$, where $$f$$ is the current scalar field EXAMPLES: sage: M = Manifold(2, 'M', structure='topological') sage: X.<x,y> = M.chart() sage: f = M.scalar_field({X: xy}, name='f', latex_name=r"\Phi") sage: g = tanh(f) ; g Scalar field tanh(f) on the 2-dimensional topological manifold M sage: latex(g) \tanh\left(\Phi\right) sage: g.display() tanh(f): M --> R (x, y) \|--> sinh(xy)/cosh(xy) Some tests: sage: tanh(f) == sinh(f) / cosh(f) True sage: tanh(M.zero_scalar_field()) == M.zero_scalar_field() True
	for Journals by Title or ISSN for Articles by Keywords help Publisher: Springer-Verlag (Total: 2355 journals) Acta Informatica [SJR: 0.524] [H-I: 32] [5 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 1432-0525 - ISSN (Online) 0001-5903 Published by Springer-Verlag [2355 journals] • Sparsification and subexponential approximation • Authors: Édouard Bonnet; Vangelis Th. Paschos Pages: 1 - 15 Abstract: Instance sparsification is well-known in the world of exact computation since it is very closely linked to the Exponential Time Hypothesis. In this paper, we extend the concept of sparsification in order to capture subexponential time approximation. We develop a new tool for inapproximability, called approximation preserving sparsification, and use it in order to get strong inapproximability results in subexponential time for several fundamental optimization problems such as min dominating set , min feedback vertex set , min set cover, min feedback arc set, and others. PubDate: 2018-02-01 DOI: 10.1007/s00236-016-0281-2 Issue No: Vol. 55, No. 1 (2018) • Conjunctive query containment over trees using schema information • Authors: Henrik Björklund; Wim Martens; Thomas Schwentick Pages: 17 - 56 Abstract: We study the containment, satisfiability, and validity problems for conjunctive queries over trees with respect to a schema. We show that conjunctive query containment and validity are 2EXPTIME -complete with respect to a schema, in both cases where the schema is given as a DTD or as a tree automaton. Furthermore, we show that satisfiability for conjunctive queries with respect to a schema can be decided in NP . The problem is NP -hard already for queries using only one kind of axis. Finally, we consider conjunctive queries that can test for equalities and inequalities of data values. Here, satisfiability and validity are decidable, but containment is undecidable, even without schema information. On the other hand, containment with respect to a schema becomes decidable again if the “larger” query is not allowed to use both equalities and inequalities. PubDate: 2018-02-01 DOI: 10.1007/s00236-016-0282-1 Issue No: Vol. 55, No. 1 (2018) • Online edge coloring of paths and trees with a fixed number of colors • Authors: Lene M. Favrholdt; Jesper W. Mikkelsen Pages: 57 - 80 Abstract: We study a version of online edge coloring, where the goal is to color as many edges as possible using only a given number, k, of available colors. All of our results are with regard to competitive analysis. Previous attempts to identify optimal algorithms for this problem have failed, even for bipartite graphs. Thus, in this paper, we analyze even more restricted graph classes, paths and trees. For paths, we consider $$k=2$$ , and for trees, we consider any $$k \ge 2$$ . We prove that a natural greedy algorithm called $${\textsc {First-Fit}}$$ is optimal among deterministic algorithms, on paths as well as trees. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. For trees, we prove that to obtain a better competitive ratio than $${\textsc {First-Fit}}$$ , the algorithm would have to be both randomized and unfair (i.e., reject edges that could have been colored), and even such algorithms cannot be much better than $${\textsc {First-Fit}}$$ . PubDate: 2018-02-01 DOI: 10.1007/s00236-016-0283-0 Issue No: Vol. 55, No. 1 (2018) • A new bound for the D0L language equivalence problem • Authors: Juha Honkala Pages: 81 - 88 Abstract: We study the language equivalence problem for smooth and loop-free D0L systems. We show that the number of initial terms in the associated D0L sequences we have to consider to decide language equivalence depends only on the cardinality of the underlying alphabet. PubDate: 2018-02-01 DOI: 10.1007/s00236-016-0286-x Issue No: Vol. 55, No. 1 (2018) • An analysis of the M $$^X$$ X /M/1 queue with multiple working vacations by GI/M/1 type Markov process • Authors: Hongbo Zhang Abstract: In this paper, we analyze an M/M/1 queue with batch arrival and multiple working vacations. We describe the queueing model by a special GI/M/1 type Markov process with infinite phases, and by the matrix analytic method, we not only give the stationary queue length distribution of the model, but also obtain the exact number of vacations completed by the server. PubDate: 2018-02-05 DOI: 10.1007/s00236-018-0316-y • On path-controlled insertion–deletion systems • Authors: Henning Fernau; Lakshmanan Kuppusamy; Indhumathi Raman Abstract: A graph-controlled insertion–deletion system is a regulated extension of an insertion–deletion system. It has several components and each component contains some insertion–deletion rules. These components are the vertices of a directed control graph. A transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. This also describes the arcs of the control graph. Starting from an axiom in the initial component, strings thus move through the control graph. The language of the system is the set of all terminal strings collected in the final component. In this paper, we investigate a variant of the main question in this area: which combinations of size parameters (the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; plus three similar restrictions with respect to deletion) are sufficient to maintain computational completeness of such restricted systems under the additional restriction that the (undirected) control graph is a path' Notice that these results also bear consequences for the domain of insertion–deletion P systems, improving on a number of previous results from the literature, concerning in particular the number of components (membranes) that are necessary for computational completeness results. PubDate: 2018-02-05 DOI: 10.1007/s00236-018-0312-2 • Symbolic checking of Fuzzy CTL on Fuzzy Program Graph • Authors: Masoud Ebrahimi; Gholamreza Sotudeh; Ali Movaghar Abstract: Few fuzzy temporal logics and modeling formalisms are developed such that their model checking is both effective and efficient. State-space explosion makes model checking of fuzzy temporal logics inefficient. That is because either the modeling formalism itself is not compact, or the verification approach requires an exponentially larger yet intermediate representation of the modeling formalism. To exemplify, Fuzzy Program Graph (FzPG) is a very compact, and powerful formalism to model fuzzy systems; yet, it is required to be translated into an equal Fuzzy Kripke model with an exponential blow-up should it be formally verified. In this paper, we introduce Fuzzy Computation Tree Logic (FzCTL) and its direct symbolic model checking over FzPG that avoids the aforementioned state-space explosion. Considering compactness and readability of FzPG along with expressiveness of FzCTL, we believe the proposed method is applicable in real-world scenarios. Finally, we study formal verification of fuzzy flip-flops to demonstrate capabilities of the proposed method. PubDate: 2018-02-03 DOI: 10.1007/s00236-018-0311-3 • Preface for the special issue GandALF 2015 • Authors: Javier Esparza; Enrico Tronci PubDate: 2018-02-01 DOI: 10.1007/s00236-018-0315-z • Petri nets are dioids: a new algebraic foundation for non-deterministic net theory • Authors: Paolo Baldan; Fabio Gadducci Abstract: In a seminal paper Montanari and Meseguer have shown that an algebraic interpretation of Petri nets in terms of commutative monoids can be used to provide an elegant characterisation of the deterministic computations of a net, accounting for their sequential and parallel composition. A smoother and more complete theory for deterministic computations has been later developed by relying on the concept of pre-net, a variation of Petri nets with a non-commutative flavor. This paper shows that, along the same lines, by adding an (idempotent) operation and thus considering dioids (idempotent semirings) rather than just monoids, one can faithfully characterise the non-deterministic computations of a net. PubDate: 2018-01-24 DOI: 10.1007/s00236-018-0314-0 • Special issue of the 21st International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2015) • Authors: Christel Baier; Cesare Tinelli Pages: 727 - 728 PubDate: 2017-12-01 DOI: 10.1007/s00236-017-0298-1 Issue No: Vol. 54, No. 8 (2017) • Approximate counting in SMT and value estimation for probabilistic programs • Authors: Dmitry Chistikov; Rayna Dimitrova; Rupak Majumdar Pages: 729 - 764 Abstract: #SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of satisfiability modulo theories (SMT) there is a growing need for model counting solvers, coming from several application domains (quantitative information flow, static analysis of probabilistic programs). In this paper, we show a reduction from an approximate version of #SMT to SMT. We focus on the theories of integer arithmetic and linear real arithmetic. We propose model counting algorithms that provide approximate solutions with formal bounds on the approximation error. They run in polynomial time and make a polynomial number of queries to the SMT solver for the underlying theory, exploiting “for free” the sophisticated heuristics implemented within modern SMT solvers. We have implemented the algorithms and used them to solve the value problem for a model of loop-free probabilistic programs with nondeterminism. PubDate: 2017-12-01 DOI: 10.1007/s00236-017-0297-2 Issue No: Vol. 54, No. 8 (2017) • Model checking the evolution of gene regulatory networks • Authors: Mirco Giacobbe; Călin C. Guet; Ashutosh Gupta; Thomas A. Henzinger; Tiago Paixão; Tatjana Petrov Pages: 765 - 787 Abstract: The behaviour of gene regulatory networks (GRNs) is typically analysed using simulation-based statistical testing-like methods. In this paper, we demonstrate that we can replace this approach by a formal verification-like method that gives higher assurance and scalability. We focus on Wagner’s weighted GRN model with varying weights, which is used in evolutionary biology. In the model, weight parameters represent the gene interaction strength that may change due to genetic mutations. For a property of interest, we synthesise the constraints over the parameter space that represent the set of GRNs satisfying the property. We experimentally show that our parameter synthesis procedure computes the mutational robustness of GRNs—an important problem of interest in evolutionary biology—more efficiently than the classical simulation method. We specify the property in linear temporal logic. We employ symbolic bounded model checking and SMT solving to compute the space of GRNs that satisfy the property, which amounts to synthesizing a set of linear constraints on the weights. PubDate: 2017-12-01 DOI: 10.1007/s00236-016-0278-x Issue No: Vol. 54, No. 8 (2017) • Stateless model checking for TSO and PSO • Authors: Parosh Aziz Abdulla; Stavros Aronis; Mohamed Faouzi Atig; Bengt Jonsson; Carl Leonardsson; Konstantinos Sagonas Pages: 789 - 818 Abstract: We present a technique for efficient stateless model checking of programs that execute under the relaxed memory models TSO and PSO. The basis for our technique is a novel representation of executions under TSO and PSO, called chronological traces. Chronological traces induce a partial order relation on relaxed memory executions, capturing dependencies that are needed to represent the interaction via shared variables. They are optimal in the sense that they only distinguish computations that are inequivalent under the widely-used representation by Shasha and Snir. This allows an optimal dynamic partial order reduction algorithm to explore a minimal number of executions while still guaranteeing full coverage. We apply our techniques to check, under the TSO and PSO memory models, LLVM assembly produced for C/pthreads programs. Our experiments show that our technique reduces the verification effort for relaxed memory models to be almost that for the standard model of sequential consistency. This article is an extended version of Abdulla et al. (Tools and algorithms for the construction and analysis of systems, Springer, New York, pp 353–367, 2015), appearing in TACAS 2015. PubDate: 2017-12-01 DOI: 10.1007/s00236-016-0275-0 Issue No: Vol. 54, No. 8 (2017) • Model-checking iterated games • Authors: Chung-Hao Huang; Sven Schewe; Farn Wang Pages: 625 - 654 Abstract: We propose a logic for the definition of the collaborative power of groups of agents to enforce different temporal objectives. The resulting temporal cooperation logic (TCL) extends ATL by allowing for successive definition of strategies for agents and agencies. Different to previous logics with similar aims, our extension cuts a fine line between extending the power and maintaining a low complexity: model checking TCL sentences is EXPTIME complete in the logic, and NL complete in the model. This advancement over nonelementary logics is bought by disallowing a too close entanglement between the cooperation and competition. We show how allowing such an entanglement immediately leads to a nonelementary complexity. We have implemented a model checker for the logic and shown the feasibility of model checking on a few benchmarks. PubDate: 2017-11-01 DOI: 10.1007/s00236-016-0277-y Issue No: Vol. 54, No. 7 (2017) • Safraless LTL synthesis considering maximal realizability • Authors: Takashi Tomita; Atsushi Ueno; Masaya Shimakawa; Shigeki Hagihara; Naoki Yonezaki Pages: 655 - 692 Abstract: Linear temporal logic (LTL) synthesis is a formal method for automatically composing a reactive system that realizes a given behavioral specification described in LTL if the specification is realizable. Even if the whole specification is unrealizable, it is preferable to synthesize a best-effort reactive system. That is, a system that maximally realizes its partial specifications. Therefore, we categorized specifications into must specifications (which should never be violated) and desirable specifications (the violation of which may be unavoidable). In this paper, we propose a method for synthesizing a reactive system that realizes all must specifications and strongly endeavors to satisfy each desirable specification. The general form of the desirable specifications without assumptions is $$\mathbf{G }\varphi$$ , which means “ $$\varphi$$ always holds”. In our approach, the best effort to satisfy $$\mathbf{G }\varphi$$ is to maximize the number of steps satisfying $$\varphi$$ in the interaction. To quantitatively evaluate the number of steps, we used a mean-payoff objective based on LTL formulae. Our method applies the Safraless approach to construct safety games from given must and desirable specifications, where the must specification can be written in full LTL and may include assumptions. It then transforms the safety games constructed from the desirable specifications into mean-payoff games and finally composes a reactive system as an optimal strategy on a synchronized product of the games. PubDate: 2017-11-01 DOI: 10.1007/s00236-016-0280-3 Issue No: Vol. 54, No. 7 (2017) • A theory of formal synthesis via inductive learning • Authors: Susmit Jha; Sanjit A. Seshia Pages: 693 - 726 Abstract: Formal synthesis is the process of generating a program satisfying a high-level formal specification. In recent times, effective formal synthesis methods have been proposed based on the use of inductive learning. We refer to this class of methods that learn programs from examples as formal inductive synthesis. In this paper, we present a theoretical framework for formal inductive synthesis. We discuss how formal inductive synthesis differs from traditional machine learning. We then describe oracle-guided inductive synthesis (OGIS), a framework that captures a family of synthesizers that operate by iteratively querying an oracle. An instance of OGIS that has had much practical impact is counterexample-guided inductive synthesis (CEGIS). We present a theoretical characterization of CEGIS for learning any program that computes a recursive language. In particular, we analyze the relative power of CEGIS variants where the types of counterexamples generated by the oracle varies. We also consider the impact of bounded versus unbounded memory available to the learning algorithm. In the special case where the universe of candidate programs is finite, we relate the speed of convergence to the notion of teaching dimension studied in machine learning theory. Altogether, the results of the paper take a first step towards a theoretical foundation for the emerging field of formal inductive synthesis. PubDate: 2017-11-01 DOI: 10.1007/s00236-017-0294-5 Issue No: Vol. 54, No. 7 (2017) • A first step in characterizing three-element codes • Authors: Cao Chunhua; Lu Qing; Yang Di Abstract: It is always an interesting subject to investigate whether a three-element language is a code or not. In this paper, we consider a special class of three-element languages, where two words have the same length which is less than the length of the third word. We give a necessary and sufficient condition to state whether a three-element language in this class is a code. This result partially resolves the problem proposed by Professor H. J. Shyr in 1990s. PubDate: 2017-11-29 DOI: 10.1007/s00236-017-0309-2 • Bounded choice-free Petri net synthesis: algorithmic issues • Authors: Eike Best; Raymond Devillers; Uli Schlachter Abstract: This paper describes a synthesis procedure dedicated to the construction of choice-free Petri nets from finite persistent transition systems, whenever possible. Taking advantage of the properties of choice-free Petri nets, a two-step approach is proposed. A pre-synthesis step checks necessary structural properties of the transition system and constructs some data structures needed for the second step. Then, a minimised set of simplified systems of linear inequalities is distilled from a general region-theoretic approach. This leads to a substantial narrowing of the sets of states for which linear inequalities must be solved, and allows an early detection of failures, supported by constructive error messages. The performance of the resulting algorithm is measured and compared numerically with existing synthesis tools. PubDate: 2017-11-20 DOI: 10.1007/s00236-017-0310-9 • Fast deterministic parsers for transition networks • Authors: Angelo Borsotti; Luca Breveglieri; Stefano Crespi Reghizzi; Angelo Morzenti Abstract: Extended BNF grammars (EBNF) allow regular expressions in the right parts of their rules. They are widely used to define languages, and can be represented by recursive Transition Networks (TN) consisting of a set of finite-state machines. We present a novel direct construction of efficient shift-reduce ELR(1) parsers for TNs. We show that such a parser works deterministically if the TN is free from the classical shift-reduce and reduce–reduce conflicts of the LR(1) parsers, and from a new conflict type called convergence conflict. Such a novel condition for determinism is proved correct and is more general than those proposed in the past for EBNF grammars or TNs. Such ELR(1) parsers perform fewer shift moves than the equivalent LR(1) parsers. A simple optimization of the reduction moves is described. PubDate: 2017-11-04 DOI: 10.1007/s00236-017-0308-3 • Spanning the spectrum from safety to liveness • Authors: Rachel Faran; Orna Kupferman Abstract: Of special interest in formal verification are safety specifications, which assert that the system stays within some allowed region, in which nothing “bad” happens. Equivalently, a computation violates a safety specification if it has a “bad prefix”—a prefix all whose extensions violate the specification. The theoretical properties of safety specifications as well as their practical advantages with respect to general specifications have been widely studied. Safety is binary: a specification is either safety or not safety. We introduce a quantitative measure for safety. Intuitively, the safety level of a language L measures the fraction of words not in L that have a bad prefix. In particular, a safety language has safety level 1 and a liveness language has safety level 0. Thus, our study spans the spectrum between traditional safety and liveness. The formal definition of safety level is based on probability and measures the probability of a random word not in L to have a bad prefix. We study the problem of finding the safety level of languages given by means of deterministic and nondeterministic automata as well as LTL formulas, and the problem of deciding their membership in specific classes along the spectrum (safety, almost-safety, fraction-safety, etc.). We also study properties of the different classes and the structure of deterministic automata for them. PubDate: 2017-10-23 DOI: 10.1007/s00236-017-0307-4 JournalTOCs School of Mathematical and Computer Sciences Heriot-Watt University Edinburgh, EH14 4AS, UK Email: journaltocs@hw.ac.uk Tel: +00 44 (0)131 4513762 Fax: +00 44 (0)131 4513327 Home (Search) Subjects A-Z Publishers A-Z Customise APIs
	# Greater than sign A symbol that expresses the comparison of a big quantity with small quantity is called greater-than sign. ## Introduction The greater-than sign is a symbol in mathematics. it is mainly used to express an inequality between two quantities by comparison. The greater-than sign is symbolically written as $>$ in mathematics. A big quantity (or number) is written before this symbol and a small quantity is written after the symbol. The number $6$ is a big number when compared to the number $2$. So, it is said that the number $6$ is greater than number $2$ and it is written in the following mathematical form. $6 \,>\, 2$ Similarly, the number $3$ is a big number when compared to the number $-5$. Hence, it is said that the number $3$ is greater than $-5$ and it is written in the following form. $3 \,>\, -5$ In both cases, the comparing numbers are not equal. Hence, the relation between them is an inequality strictly. ### Algebraic form Generally, the strict inequality with greater-than sign is written in algebraic form as follows. $x \,>\, a$ In this example, the literals $x$ and $a$ represent the real numbers but the value of $x$ is always greater than the value of $a$. A best free mathematics education website for students, teachers and researchers. ###### Maths Topics Learn each topic of the mathematics easily with understandable proofs and visual animation graphics. ###### Maths Problems Learn how to solve the maths problems in different methods with understandable steps. Learn solutions ###### Subscribe us You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.
	# Logarithmic Minimal Models, Critical Dense Polymers, Symplectic Fermions and Fusion Rules #### by Paul Pearce Institution: The University of Melbourne Date: Mon 5th May 2008 Time: 1:00 PM Location: Room 213 Richard Berry Building, The University of Melbourne Abstract: The logarithmic minimal models LM(p,pâ€™) with p, pâ€™ coprime are a family of Yang-Baxter integrable two-dimensional lattice models. The first members of this family are critical dense polymers LM(1,2) and critical percolation LM(2,3). Remarkably, critical dense polymers is exactly solvable on a finite lattice. The continuum scaling limit of these theories yield logarithmic conformal field theories characterized by the existence of reducible yet indecomposable representations of the Virasoro algebra or extended conformal algebra. In the extended W-algebra picture, LM(1,2) is identified with symplectic fermions. The fusion rules for LM(1,2) are presented in both the Virasoro and extended W-algebra pictures and their relationship explained.
	# All Questions 39 views ### Value of S and r in Fiat-Shamir Protocol In the book Cryptography and Network Security by Forouzan,in chapter 14, Fiat-Shamir protocol, its mentioned that two large prime numbers p and q are chosen and kept secret. However n=p*q is made ... 76 views ### RFID Protocol Cryptanalysis Assume we have the following scheme for RFID: TAG & READER both have initially k keys. Every session the TAG computes $k_i$=F($k_{i-1})$ where F is a function which computes XOR of previous key ... 40 views ### Errata for NIST SP 800-63 I'm looking through the latest version of NIST SP 800-63, and there is a table (Table A.1) in the Appendix that supposed give the "entropy" of a password for a 94 character alphabet for various ... 72 views ### Benefit of using random key in Shamir's Secret Sharing I am implementing Shamir's Secret Sharing, and I find that in (t,n)-threshold scheme, the shares are just using 1,2,3...,n as the key to form the shares ... 80 views ### SSL / TLS digital signature exchange If server is sending its digital signature, in which message / where does the digitally signature is provided to the client. Is it after the server_hello message because there the public key of the ... 38 views ### Meaning of Signature hash algorithm field in certificate I'd like to understand what is the meaning of this field, especially considering that when calling a crypto library to sign data the digest method is specified by the caller. For example, this simple ... 43 views ### Rationale of “r” AES key use in OTR version 3 AKE protocol? I just tried to review & understand AKE (Authenticated Key Exchange) protocol as defined in OTR secure messaging protocol version 3 here , and aiming to achieve Perfect Forward Secrecy I am a ... 64 views ### What are the difference between cryptographic primitives and encryption primitives? For example, from reading I found that the cryptographic primitives of both AES and DES are Confusion and Diffusion. However the encryption primitive for DES was Substitution and Permutation while ... 46 views ### Why is plain ElGamal not secure? Why is plain ElGamal not secure? What is the proof behind it? 86 views ### How does the Blowfish algorithm key initialization work I'm having some trouble understanding the Blowfish encryption algorithm. From the Wikipedia article... The secret key is then, byte by byte, cycling the key if necessary, XORed with all the ... 58 views ### Standards/RFCs for non-Diffie-Hellman perfect forward secrecy? Some time has passed since PFS introduction and we now have secure&proven asymmetric ciphers that can generate strong keys very efficiently (elliptic RSA variants and/or QD-Mceliece can generate ... 57 views ### Secure ElGamal with OAEP Is it possible to make ElGamal IND-CCA2 using OAEP or OAEP+? The reason I ask is that I recently answered this question and it came to my mind that OAEP or OAEP+ might be possible solutions. Note ... 25 views ### AES-XTS: find key from ciphertext and plaintext [duplicate] Let's say I have a plaintext, the corresponding ciphertext encrypted using AES-XTS, and the initialization vectors. Is there any way - not brute force, of course - to compute the key? Even if there's ... 66 views ### Prove there is PRG that is not necessarily one-to-one We assume that there is at least one PRG .Now prove there is a PRG like $G:\{0,1\}^{n} \rightarrow \{0,1\}^{l(n)}$ such that it is not necessarily one-to-one. 34 views ### CTR_DRBG versus OFB_DRBG As far as I know, in the development of ANSI X9.82, there was consideration of OFB_DRBG, a pseudorandom number generator based on block ciphers like CTR_DRBG, but it appears to have been rejected of ... 70 views ### Looking for C++/Python Open Source code library for cryptanalysis of classical ciphers I've done a significant amount of coding over the years working on classical ciphers (e.g., Chaocipher, D'Agapeyeff). My main programming languages today are C++ and Python, although there was a time ... 37 views ### Is the key schedule of Serpent a circle? The creation of the prekeys for Serpent works by XORing some previous values with a counter and a fixed value. Every word is 32 bits big and 4 words form a round key (after applying a S-Box, but this ... 62 views 54 views ### Cipher text only attacks on deterministic fully homomorphic encryption schemes If we have encryptions of additive and multiplicative identities in the corpus of cipher text of a deterministic fully homomorphic encryption (FHE) scheme, I guess we can break it. If the FHE scheme ... 36 views ### BER or DER X9.62 ECDSA signature The signature format for ECDSA signatures can be encoded using ASN.1 integers according to X9.62 or it can comprise of two integers with the same size as the key size. In case the X9.62 format is ... 74 views ### Creating a random password based off of a prime number So I am making an application that basically creates strings that must be encrypted before they are stored on a user's device. If the user blindly starts running the application without creating a ...
	# Top arXiv papers • Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal Type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the Type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent much stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels. • A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured $K_n$ has a rainbow cycle of length $n-O(n^{3/4})$. One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured $K_n$ formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties. • In this paper we develop an operational formulation of General Relativity similar in spirit to existing operational formulations of Quantum Theory. To do this we introduce an operational space (or op-space) built out of scalar fields. A point in op-space corresponds to some nominated set of scalar fields taking some given values in coincidence. We assert that op-space is the space in which we observe the world. We introduce also a notion of agency (this corresponds to the ability to set knob settings just like in Operational Quantum Theory). The effects of agents' actions should only be felt to the future so we introduce also a time direction field. Agency and time direction can be understood as effective notions. We show how to formulate General Relativity as a possibilistic theory and as a probabilistic theory. In the possibilistic case we provide a compositional framework for calculating whether some operationally described situation is possible or not. In the probabilistic version we introduce probabilities and provide a compositional framework for calculating the probability of some operationally described situation. Finally we look at the quantum case. We review the operator tensor formulation of Quantum Theory and use it to set up an approach to Quantum Field Theory that is both operational and compositional. Then we consider strategies for solving the problem of Quantum Gravity. By referring only to operational quantities we are able to provide formulations for the possibilistic, probabilistic, and (the nascent) quantum cases that are manifestly invariant under diffeomorphisms. • The Teff = 20,800 K white dwarf WD 1536+520 is shown to have broadly solar abundances of the major rock forming elements O, Mg, Al, Si, Ca, and Fe, together with a strong relative depletion in the volatile elements C and S. In addition to the highest metal abundances observed to date, including log(O/He) = -3.4, the helium-dominated atmosphere has an exceptional hydrogen abundance at log(H/He) = -1.7. Within the uncertainties, the metal-to-metal ratios are consistent with the accretion of an H2O-rich and rocky parent body, an interpretation supported by the anomalously high trace hydrogen. The mixed atmosphere yields unusually short diffusion timescales for a helium atmosphere white dwarf, of no more than a few hundred yr, and equivalent to those in a much cooler, hydrogen-rich star. The overall heavy element abundances of the disrupted parent body deviate modestly from a bulk Earth pattern, and suggest the deposition of some core-like material. The total inferred accretion rate is 4.2e9 g/s, and at least 4 times higher than any white dwarf with a comparable diffusion timescale. Notably, when accretion is exhausted in this system, both metals and hydrogen will become undetectable within roughly 300 Myr, thus supporting a scenario where the trace hydrogen is related to the ongoing accretion of planetary debris. • We report on the evidence for the multiband electronic transport in $\alpha$-YbAlB$_{4}$ and $\alpha$-Yb$_{0.81(2)}$Sr$_{0.19(3)}$AlB$_{4}$. Multiband transport reveals itself below 10 K in both compounds via Hall effect measurements, whereas anisotropic magnetic ground state sets in below 3 K in $\alpha$-Yb$_{0.81(2)}$Sr$_{0.19(3)}$AlB$_{4}$. Our results show that Sr$^{2+}$ substitution enhances conductivity, but does not change the quasiparticle mass of bands induced by heavy fermion hybridization. • In this paper, we study the inviscid limit of the free surface incompressible Navier-Stokes equations with or without surface tension. By delicate estimates, we prove the weak boundary layer of the velocity of the free surface Navier-Stokes equations and the existence of strong or weak vorticity layer for different conditions. When the limit of the difference between the initial Navier-Stokes vorticity and the initial Euler vorticity is nonzero, or the tangential projection on the free surface of the Euler strain tensor multiplying by normal vector is nonzero, there exists a strong vorticity layer. Otherwise, the vorticity layer is weak. We estimate convergence rates of tangential derivatives and the first order standard normal derivative in energy norms, we show that not only tangential derivatives and standard normal derivative have different convergence rates, but also their convergence rates are different for different Euler boundary data. Moreover, we determine regularity structure of the free surface Navier-Stokes solutions with or without surface tension, surface tension changes regularity structure of the solutions. • This paper presents an algebraic formulation of the renormalization group flow in quantum mechanics on flat target spaces. For any interacting quantum mechanical theory, the fixed point of this flow is a theory of classical probability, not a different effective quantum mechanics. Each energy eigenstate of the UV Hamiltonian flows to a probability distribution whose entropy is a natural diagnostic of quantum ergodicity of the original state. These conclusions are supported by various examples worked out in detail. • Renormalization group equations are an essential tool for the description of theories accross different energy scales. Even though their expressions at two-loop for an arbitrary gauge field theory have been known for more than thirty years, deriving the full set of equations for a given model by hand is very challenging and prone to errors. To tackle this issue, we have introduced in [1] a Python tool called PyR@TE; Python Renormalization group equations @ Two-loop for Everyone. With PyR@TE, it is easy to implement a given Lagrangian and derive the complete set of two-loop RGEs for all the parameters of the theory. In this paper, we present the new version of this code, PyR@TE 2, which brings many new features and in particular it incorporates kinetic mixing when several $\mathrm{U}(1)$ gauge groups are involved. In addition, the group theory part has been greatly improved as we introduced a new Python module dubbed PyLie that deals with all the group theoretical aspects required for the calculation of the RGEs as well as providing very useful model building capabilities. This allows the use of any irreducible representation of the $\mathrm{SU}(n)$, $\mathrm{SO}(2n)$ and $\mathrm{SO(2n+1)}$ groups. % Furthermore, it is now possible to implement terms in the Lagrangian involving fields which can be contracted into gauge singlets in more than one way. As a byproduct, results for a popular model (SM+complex triplet) for which, to our knowledge, the complete set of two-loop RGEs has not been calculated before are presented in this paper. Finally, the two-loop RGEs for the anomalous dimension of the scalar and fermion fields have been implemented as well. It is now possible to export the coupled system of beta functions into a numerical C++ function, leading to a consequent speed up in solving them. • Microwave heating of a high-temperature plasma confined in a large-scale open magnetic trap, including all important wave effects like diffraction, absorption, dispersion and wave beam aberrations, is described for the first time within the first-principle technique based on consistent Maxwell's equations. With this purpose, the quasi-optical approach is generalized over weakly inhomogeneous gyrotrotropic media with resonant absorption and spatial dispersion, and a new form of the integral quasi-optical equation is proposed. An effective numerical technique for this equation's solution is developed and realized in a new code QOOT, which is verified with the simulations of realistic electron cyclotron heating scenarios at the Gas Dynamic Trap at the Budker Institute of Nuclear Physics (Novosibirsk, Russia). • Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ (noncommutative) instantons on origami spacetimes. There, $X$ is modelled on a union of (up to six) coordinate complex planes ${\bf C}^{2}$ intersecting in $Z$ modelled on ${\bf C}^{4}$. The instantons are shared by the collection of four dimensional gauge theories sewn along two dimensional defect surfaces and defect points. We also define several quiver versions ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$ of ${\bf M}_{k}({\vec n})$, motivated by the considerations of sewn gauge theories on orbifolds ${\bf C}^{4}/{\Gamma}$. The geometry of the spaces ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$, more specifically the compactness of the set of torus-fixed points, for various tori, underlies the non-perturbative Dyson-Schwinger identities recently found to be satisfied by the correlation functions of $qq$-characters viewed as local gauge invariant operators in the ${\mathcal{N}}=2$ quiver gauge theories. The cohomological and K-theoretic operations defined using ${\bf M}_{k}({\vec n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects. • Kinetic mixing is a fundamental property of models with a gauge symmetry involving several $\mathrm{U}(1)$ group factors. In this paper, we perform a numerical study of the impact of kinetic mixing on beta functions at two-loop. To do so, we use the recently published PyR@TE 2 software to derive the complete set of RGEs of the SM B-L model at two-loop including kinetic mixing. We show that it is important to properly account for kinetic mixing as the evolution of the parameters with the energy scale can change drastically. In some cases, these modifications can even lead to a different conclusion regarding the stability of the scalar potential. • The best previous lower bounds for kissing numbers in dimensions 25 through 31 were constructed using a set $S$ with $\|S\| = 480$ of minimal vectors of the Leech Lattice, $\Lambda_{24}$, such that $\langle x, y \rangle \leq 1$ for any distinct $x, y \in S$. Then, a probabilistic argument based on applying automorphisms of $\Lambda_{24}$ gives more disjoint sets $S_i$ of minimal vectors of $\Lambda_{24}$ with the same property. Cohn, Jiao, Kumar, and Torquato proved that these subsets give kissing configurations in dimensions 25 through 31 of given size linear in the sizes of the subsets. We achieve $\|S\| = 488$ by applying simulated annealing. We also improve the aforementioned probabilistic argument in the general case. Finally, we greedily construct even larger $S_i$'s given our $S$ of size $488$, giving increased lower bounds on kissing numbers in $\mathbb{R}^{25}$ through $\mathbb{R}^{31}$. • The classical part of the QCD partition function (the integrand) has, ignoring irrelevant exact zero modes of the Dirac operator, a local SU(2N_F) ⊃SU(N_F)_L \times SU(N_F)_R \times U(1)_A symmetry which is absent at the Lagrangian level. This symmetry is broken anomalously and spontaneously. Effects of spontaneous breaking of chiral symmetry are contained in the near-zero modes of the Dirac operator. If physics of anomaly is also encoded in the same near-zero modes, then their truncation on the lattice should recover a hidden classical SU(2N_F) symmetry in correlators and spectra. This naturally explains observation on the lattice of a large degeneracy of hadrons, that is higher than the SU(N_F)_L \times SU(N_F)_R \times U(1)_A chiral symmetry, upon elimination by hands of the lowest-lying modes of the Dirac operator. We also discuss an implication of this symmetry for the high temperature QCD. • In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details. In order to efficiently compute the solutions, some model reductions are necessary. To obtain a reduced model, we apply the generalized multiscale finite element approach, which is a framework allowing systematic construction of reduced models. Based on this general framework, we will first construct a local snapshot space, which contains many possible multiscale features of the solution. Using the snapshot space and a local spectral problem, we identify dominant modes in the snapshot space and use them as the multiscale basis functions. Our basis functions are constructed locally with non-overlapping supports, which enhances the sparsity of the resulting linear system. In order to enforce the mass conservation, we propose a hybridized technique, and uses a Lagrange multiplier to achieve mass conservation. We will mathematically analyze the stability and the convergence of the proposed method. In addition, we will present some numerical examples to show the performance of the scheme. We show that, with a few basis functions per coarse region, one can obtain a solution with excellent accuracy. • We present measurements of the reduction of light output by plastic scintillators irradiated in the CMS detector during the 8 TeV run of the Large Hadron Collider and show that they indicate a strong dose rate effect. The damage for a given dose is larger for lower dose rate exposures. The results agree with previous measurements of dose rate effects, but are stronger due to the very low dose rates probed. We show that the scaling with dose rate is consistent with that expected from diffusion effects. • We experimentally investigate the dynamics of particle rearrangements for a 2D Brownian colloidal suspension under cyclic shear. We find that, even though the system is liquid-like and non-rigid ($\phi\leq0.32$), a fraction of particles undergo reversible cycles, in the form of scattered particle clusters. Unlike jammed athermal systems, the reversible clusters are not stable and the particles transition between reversible and irreversible cycles. We demonstrate that the stability of reversibility depends both on $\phi$ and strain amplitude. We also identify plastic reversibility for our thermal system. However, as $\phi$ is decreased deep into liquid phase, the hysterysis in particle rearrangements becomes less prominent, and the dynamics is moved closer to equilibrium by thermal noise. • It is known that the Painleve VI is obtained by connection preserving deformation of some linear differential equations, and the Heun equation is obtained by a specialization of the linear differential equations. We inverstigate degenerations of the Ruijsenaars-van Diejen difference opearators and show difference analogues of the Painleve-Heun correspondence. • Proxima Centauri b, an Earth-size planet in the habitable zone of our nearest stellar neighbour, has just been discovered. A theoretical framework of synchronously rotating planets, in which the risk of a runaway greenhouse on the sunlight side and atmospheric collapse on the reverse side are mutually ameliorated via heat transport is discussed. This is developed via simple (tutorial) models of the climate. These show that lower incident stellar flux means that less heat transport, so less atmospheric mass, is required. The incident stellar flux at Proxima Centauri b is indeed low, which may help enhance habitability if it has suffered some atmospheric loss or began with a low volatile inventory. • In this paper we discuss the boson/vortex duality by mapping the Gross-Pitaevskii theory into an effective string theory, both with and without boundaries. Through the effective string theory, we find the Seiberg-Witten map between the commutative and the noncommutative tachyon field theories, and consequently identify their soliton solutions with the D-branes in the effective string theory. We perform various checks of the duality map and the identification of classical solutions. This new insight of the duality between the Gross-Pitaevskii theory and the effective string theory allows us to test many results of string theory in Bose-Einstein condensates, and at the same time help us understand the quantum behavior of superfluids and cold atom systems. • We present a type system and inference algorithm for a rich subset of JavaScript equipped with objects, structural subtyping, prototype inheritance, and first-class methods. The type system supports abstract and recursive objects, and is expressive enough to accommodate several standard benchmarks with only minor workarounds. The invariants enforced by the types enable an ahead-of-time compiler to carry out optimizations typically beyond the reach of static compilers for dynamic languages. Unlike previous inference techniques for prototype inheritance, our algorithm uses a combination of lower and upper bound propagation to infer types and discover type errors in all code, including uninvoked functions. The inference is expressed in a simple constraint language, designed to leverage off-the-shelf fixed point solvers. We prove soundness for both the type system and inference algorithm. An experimental evaluation showed that the inference is powerful, handling the aforementioned benchmarks with no manual type annotation, and that the inferred types enable effective static compilation. • Let $G = (N,E,w)$ be a weighted communication graph (with weight function $w$ on $E$). For every subset $A \subseteq N$, we delete in the subset $E(A)$ of edges with ends in $A$, all edges of minimum weight in $E(A)$. Then the connected components of the corresponding induced subgraph constitute a partition of $A$ that we call $P_{\min}(A)$. For every game $(N, v)$, we define the $P_{\min}$-restricted game $(N, \bar{v})$ by $\bar{v}(A) = \sum_{F \in P_{\min}(A)} v(F)$ for all $A \subseteq N$. We prove that we can decide in polynomial time if there is inheritance of $\mathcal{F}$-convexity from $(N, v)$ to the $P_{\min}$-restricted game $(N, \bar{v})$ where $\mathcal{F}$-convexity is obtained by restricting convexity to connected subsets. • Phosphorylation, the enzyme-mediated addition of a phosphate group to a molecule, is a ubiquitous chemical mechanism in biology. Multisite phosphorylation, the addition of phosphate groups to multiple sites of a single molecule, may be distributive or processive. Distributive systems can be bistable, while processive systems were recently shown to be globally stable. However, this global convergence result was proven only for a specific mechanism of processive phosphorylation/dephosphorylation (namely, all catalytic reactions are reversible). Accordingly, we generalize this result to allow for processive phosphorylation networks in which each reaction may be irreversible, and also to account for possible product inhibition. We accomplish this by defining an all-encompassing processive network that encapsulates all of these schemes, and then appealing to recent results of Marcondes de Freitas, Wiuf, and Feliu that assert global convergence by way of monontone systems theory and network/graph reductions (which correspond to removal of intermediate complexes). Our results form a case study into the question of when global convergence is preserved when reactions and/or intermediate complexes are added to or removed from a network. • Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged subintervals. We prove that if $T$ satisfies Keane's infinite distinct orbit condition and $\text{rank}(T)>1+\lfloor m/2 \rfloor$ then the only interval exchange transformations which commute with $T$ are its powers. The main step in our proof is to show that the centralizer of $T$ is torsion-free under the above hypotheses. • An ontology of Leibnizian relationalism, consisting in distance relations among sparse matter points and their change only, is well recognized as a serious option in the context of classical mechanics. In this paper, we investigate how this ontology fares when it comes to general relativistic physics. Using a Humean strategy, we regard the gravitational field as a means to represent the overall change in the distance relations among point particles in a way that achieves the best combination of being simple and being informative. • Among solutions of the strong CP problem, the "invisible" axion in the narrow axion window is argued to be the remaining possibility among natural solutions on the smallness of $\bar{\theta}$. Related to the gravity spoil of global symmetries, some prospective invisible axions from theory point of view are discussed. In all these discussions, including the observational possibility, cosmological constraints must be included. • For any acyclic directed graph $G$, we introduce two notions: one is called an upward planar order on $G$ which is a linear extension of the edge poset of $G$ with some constraints, the other is called a canonical progressive planar extension (CPP extension for short) of $G$ which is an embedding of $G$ into a progressive planar graph with some constraints. Based on new characterizations of progressive planar graphs, we show that there is a natural bijection between the set of upward planar orders of $G$ and the set of CPP extensions of $G$. Finally we justify the combinatorial definition that an upward planar graph is an acyclic directed graph with an upward planar order. • The self-assembly of partially wet hexagonal-disk monolayers under vertical vibrations is investigated experimentally. Due to the formation of liquid bridges, the disks are subjected to short-ranged attractive interactions. Unlike spheres, hexagonal disks prefer to spin upon sufficiently strong driving. Consequently, a rotator-crystal state with the disks self-organized in a hexagonal structure dominates over a wide range of vibration strength. The bond length of the rotator-crystals is slightly smaller than the diameter of the circumscribed circle of a hexagon, indicating geometric frustration. An analysis of the mobility of a single disk reveals that the preference to spin arises from the broken circular symmetry. This investigation provides an example where the collective behavior of granular matter is tuned by the shape of individual particles. • We introduce a novel latent vector space model that jointly learns the latent representations of words, e-commerce products and a mapping between the two without the need for explicit annotations. The power of the model lies in its ability to directly model the discriminative relation between products and a particular word. We compare our method to existing latent vector space models (LSI, LDA and word2vec) and evaluate it as a feature in a learning to rank setting. Our latent vector space model achieves its enhanced performance as it learns better product representations. Furthermore, the mapping from words to products and the representations of words benefit directly from the errors propagated back from the product representations during parameter estimation. We provide an in-depth analysis of the performance of our model and analyze the structure of the learned representations. • Direct detection and spectroscopy of exoplanets requires high contrast imaging. For habitable exoplanets in particular, located at small angular separation from the host star, it is crucial to employ small inner working angle (IWA) coronagraphs that efficiently suppress starlight. These coronagraphs, in turn, require careful control of the wavefront which directly impacts their performance. For ground-based telescopes, atmospheric refraction is also an important factor, since it results in a smearing of the PSF, that can no longer be efficiently suppressed by the coronagraph. Traditionally, atmospheric refraction is compensated for by an atmospheric dispersion compensator (ADC). ADC control relies on an a priori model of the atmosphere whose parameters are solely based on the pointing of the telescope, which can result in imperfect compensation. For a high contrast instrument like the Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) system, which employs very small IWA coronagraphs, refraction-induced smearing of the PSF has to be less than 1 mas in the science band for optimum performance. In this paper, we present the first on-sky measurement and correction of residual atmospheric dispersion. Atmospheric dispersion is measured from the science image directly, using an adaptive grid of artificially introduced speckles as a diagnostic to feedback to the telescope's ADC. With our current setup, we were able to reduce the initial residual atmospheric dispersion from 18.8 mas to 4.2 in broadband light (y- to H-band), and to 1.4 mas in H-band only. This work is particularly relevant to the upcoming extremely large telescopes (ELTs) that will require fine control of their ADC to reach their full high contrast imaging potential. • We study dynamically crowded solutions of stiff fibers deep in the semidilute regime, where the motion of a single constituent becomes increasingly confined to a narrow tube. The spatiotemporal dynamics for wave numbers resolving the motion in the confining tube becomes accessible in Brownian dynamics simulations upon employing a geometry-adapted neighbor list. We demonstrate that in such crowded environments the intermediate scattering function, characterizing the motion in space and time, can be predicted quantitatively by simulating a single freely diffusing phantom needle only, yet with very unusual diffusion coefficients. • Deep learning has been shown as a successful machine learning method for a variety of tasks, and its popularity results in numerous open-source deep learning software tools coming to public. Training a deep network is usually a very time-consuming process. To address the huge computational challenge in deep learning, many tools exploit hardware features such as multi-core CPUs and many-core GPUs to shorten the training time. However, different tools exhibit different features and running performance when training different types of deep networks on different hardware platforms, which makes it difficult for end users to select an appropriate pair of software and hardware. In this paper, we aim to make a comparative study of the state-of-the-art GPU-accelerated deep learning software tools, including Caffe, CNTK, TensorFlow, and Torch. We benchmark the running performance of these tools with three popular types of neural networks on two CPU platforms and three GPU platforms. Our contribution is two-fold. First, for deep learning end users, our benchmarking results can serve as a guide to selecting appropriate software tool and hardware platform. Second, for deep learning software developers, our in-depth analysis points out possible future directions to further optimize the training performance. • We study the monopole (breathing) mode of a finite temperature Bose-Einstein condensate in an isotropic harmonic trap recently developed by D.~S.~Lobser et al. [Nat.~Phys., \textbf11, 1009 (2015)]. We observe a non-exponential collapse of the amplitude of the condensate oscillation followed by a partial revival. This behavior is identified as being due to beating between two eigenmodes of the system, corresponding to in-phase and out-of-phase oscillations of the condensed and non-condensed fractions of the gas. We perform finite temperature simulations of the system dynamics using the Zaremba-Nikuni-Griffin methodology [J.~Low Temp.~Phys., \textbf116, 277 (1999)], and find good agreement with the data, thus confirming the two mode description. • We consider the minimal number of points on a regular grid on the plane that generates n blocks of points of exactly length k and show that this number is upper bounded by kn/3 and approaches kn/4 as $n\rightarrow\infty$ when k+1 is coprime with 6 or when k is large. • Previous realizations of synthetic gauge fields for ultracold atoms do not allow the spatial profile of the field to evolve freely. We propose a scheme which overcomes this restriction by using the light in a multimode cavity, in conjunction with Raman coupling, to realize an artificial magnetic field which acts on a Bose-Einstein condensate of neutral atoms. We describe the evolution of such a system, and present the results of numerical simulations which show dynamical coupling between the effective field and the matter on which it acts. Crucially, the freedom of the spatial profile of the field is sufficient to realize a close analogue of the Meissner effect, where the magnetic field is expelled from the superfluid. This back-action of the atoms on the synthetic field distinguishes the Meissner-like effect described here from the Hess-Fairbank suppression of rotation in a neutral superfluid observed elsewhere. • In a recent study, we reported the results of a new decision making paradigm in which the participants were asked to balance between their speed and accuracy to maximize the total reward they achieve during the experiment. The results of computational modeling provided strong evidence suggesting that the participants used time-varying decision boundaries. Previous theoretical studies of the optimal speed-accuracy trade-off suggested that the participants may learn to use these time-varying boundaries to maximize their average reward rate. The results in our experiment, however, showed that the participants used such boundaries even at the beginning of the experiment and without any prior experience in the task. In this paper, we hypothesize that these boundaries are the results of using some heuristic rules to make decisions in the task. To formulate decision making by these heuristic rules as a computational framework, we use the fuzzy logic theory. Based on this theory, we propose a new computational framework for decision making in evidence accumulation tasks. In this framework, there is no explicit decision boundary. Instead, the subject's desire to stop accumulating evidence and responding at each moment within a trial and for a given value of the accumulated evidence, is determined by a set of fuzzy "IF-TEHN rules". We then use the back-propagation method to derive an algorithm for fitting the fuzzy model to each participant's data. We then investigate how the difference in the participants' performance in the experiment is reflected in the difference in the parameters of the fitted model • It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. Stäckel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given. • We present an online visual tracking algorithm by managing multiple target appearance models in a tree structure. The proposed algorithm employs Convolutional Neural Networks (CNNs) to represent target appearances, where multiple CNNs collaborate to estimate target states and determine the desirable paths for online model updates in the tree. By maintaining multiple CNNs in diverse branches of tree structure, it is convenient to deal with multi-modality in target appearances and preserve model reliability through smooth updates along tree paths. Since multiple CNNs share all parameters in convolutional layers, it takes advantage of multiple models with little extra cost by saving memory space and avoiding redundant network evaluations. The final target state is estimated by sampling target candidates around the state in the previous frame and identifying the best sample in terms of a weighted average score from a set of active CNNs. Our algorithm illustrates outstanding performance compared to the state-of-the-art techniques in challenging datasets such as online tracking benchmark and visual object tracking challenge. • The technique of Formal Concept Analysis is applied to a dataset describing the traits of rodents, with the goal of identifying zoonotic disease carriers,or those species carrying infections that can spillover to cause human disease. The concepts identified among these species together provide rules-of-thumb about the intrinsic biological features of rodents that carry zoonotic diseases, and offer utility for better targeting field surveillance efforts in the search for novel disease carriers in the wild. • We utilise the theory of crossed simplicial groups to introduce a collection of Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. • Graviton loop corrections to observables in de Sitter space often lead to infrared divergences. We show that these infrared divergences are resolved by the spontaneous breaking of de Sitter invariance. • Aug 26 2016 math.NT arXiv:1608.07236v1 We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group $\pi_0 \mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that $\pi_0 \mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of $\pi_* \mathcal{R}$ on the homology. • We present a self-contained operator-based approach to derive the spectrum of trapped ions. This approach provides the complete normal form of the low energy quadratic Hamiltonian in terms of bosonic phonons, as well as an effective free particle degree of freedom for each spontaneously broken spatial symmetry. We demonstrate how this formalism can directly be used to characterize an ion chain both in the linear and the zigzag regimes. In particular we compute, both for the ground state and finite temperature states, spatial correlations, heat capacity and dynamical susceptibility. Last, for the ground state which has quantum correlations, we analyze the amount of energy reduction compared to an uncorrelated state with minimum energy, thus highlighting how the system can lower its energy by correlations. • We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\Gamma$. Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra. From this construction we extract an action of certain $p$-adic Galois cohomology groups on $H^(\Gamma, \mathbb{Q}_p)$, and formulate the central conjecture: the motivic $\mathbb{Q}$-lattice inside these Galois cohomology groups preserves $H^(\Gamma,\mathbb{Q})$. • The Griffiths phase has been proposed to induce a stretched critical regime that facilitates self organizing of brain networks for optimal function. This phase stems from the intrinsic structural heterogeneity of brain networks, such as the hierarchical modular structure. In this work, we extend this concept to modified hierarchical networks with small-world connections based on Hanoi networks [1]. Through extensive simulations, we identify the essential role played by the exponential distribution of the inter-moduli connectivity probability across hierarchies on the emergence of the Griffiths phase in this network. Additionally, the spectral analysis on the adjacency matrix of the relevant networks [2] shows that a localized principle eigenvector is not necessarily the fingerprint of the Griffiths phase. • Recent observations suggest ongoing planet formation in the innermost parsec of the Galactic center (GC). The super-massive black hole (SMBH) might strip planets or planetary embryos from their parent star, bringing them close enough to be tidally disrupted. Photoevaporation by the ultraviolet field of young stars, combined with ongoing tidal disruption, could enhance the near-infrared luminosity of such starless planets, making their detection possible even with current facilities. In this paper, we investigate the chance of planet tidal captures by means of high-accuracy N-body simulations exploiting Mikkola's algorithmic regularization. We consider both planets lying in the clockwise (CW) disk and planets initially bound to the S-stars. We show that tidally captured planets remain on orbits close to those of their parent star. Moreover, the semi-major axis of the planet orbit can be predicted by simple analytic assumptions in the case of prograde orbits. We find that starless planets that were initially bound to CW disk stars have mild eccentricities and tend to remain in the CW disk. However, we speculate that angular momentum diffusion and scattering with other young stars in the CW disk might bring starless planets on low-angular momentum orbits. In contrast, planets initially bound to S-stars are captured by the SMBH on highly eccentric orbits, matching the orbital properties of the G1 and G2 clouds. Our predictions apply not only to planets but also to low-mass stars initially bound to the S-stars and tidally captured by the SMBH. • Aug 26 2016 math.MG arXiv:1608.07229v1 We introduce a notion of a sub-Moebius structure and find necessary and sufficient conditions under which a sub-Moebius structure is a Moebius structure. We show that on the boundary at infinity of every Gromov hyperbolic space Y there is a canonical sub-Moebius structure which is invariant under isometries of Y such that the sub-Moebius topology on the boundary coincides with the standard one. • We prove an analogue of the Lebesgue decomposition for continuous functionals on the commutant modulo a reflexive normed ideal of an n-tuple of hermitian operators for which there are quasicentral approximate units relative to the normed ideal. Using results of Godefroy-Talagrand and Pfitzner we derive from this strong uniqueness of the predual of such a commutant modulo a normed ideal. • We consider the conformally invariant cubic wave equation on the Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ for small rotationally symmetric initial data. This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS$_4$) and connects it to various questions of AdS stability. We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the weak field regime. It turns out that this effective system, which we call the \emphconformal flow, exhibits some remarkable features, such as low-dimensional invariant subspaces, a wealth of stationary states (for which energy does not flow between the modes), as well as solutions with nontrivial exactly periodic energy flows. Based on these observations and close parallels to the cubic Szegő equation, which was shown by Gérard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in AdS$_4$ are integrable as well. • In this paper we investigate the hedging problem of a defaultable claim with recovery at default time via the local risk-minimization approach when investors have a restricted information on the market. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor and that at any time, investors may observe the risky asset price and know if default has occurred or not. We characterize the optimal strategy in terms of the integrand in the Galtchouk-Kunita-Watanabe decomposition of the defaultable claim with respect to the minimal martingale measure and the available information flow. Finally, we provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure in a Markovian setting via filtering. • Gender differences in collaborative research have received little attention when compared with the growing importance that women hold in academia and research. Unsurprisingly, most of bibliometric databases have a strong lack of directly available information by gender. Although empirical-based network approaches are often used in the study of research collaboration, the studies about the influence of gender dissimilarities on the resulting topological outcomes are still scarce. Here, networks of scientific subjects are used to characterize patterns that might be associated to five categories of authorships which were built based on gender. We find enough evidence that gender imbalance in scientific authorships brings a peculiar trait to the networks induced from papers published in Web of Science (WoS) indexed journals of Economics over the period 2010-2015 and having at least one author affiliated to a Portuguese institution. Our results show the emergence of a specific pattern when the network of co-occurring subjects is induced from a set of papers exclusively authored by men. Such a male-exclusive authorship condition is found to be the solely responsible for the emergence that particular shape in the network structure. This peculiar trait might facilitate future network analyses of research collaboration and interdisciplinarity. resodiat Aug 23 2016 13:00 UTC That is really a long-term perspective. Marco Piani Aug 22 2016 22:08 UTC Born in Italy, and now living in Scotland: I have no excuses not to feel inspired :-) Māris Ozols Aug 22 2016 18:50 UTC It is not just in Scotland but in fact across the whole of UK and even beyond. I just found a reference, dating back to the very birth of quantum computing, where the early pioneers [already admit][1] that their work was inspired by Rabezzana Grignolino d'Asti. [1]: https://scirate.com/arxiv/quan ...(continued) JRW Aug 18 2016 16:42 UTC A video of a talk I gave this morning will be [here][1], if it ever finishes uploading. [1]: https://youtu.be/I8cMY0AmIY0 Jonathan Oppenehim Jul 28 2016 16:41 UTC Hi, sorry to just be updating this discussion now -- my conversation with Renato seemed to me to have converged here (and also continued via email and in person and I never updated scirate). However, a few people have asked what the outcome of our discussion was. So let me just say, that yes, my vie ...(continued) Valentin Zauner-Stauber Jul 18 2016 09:54 UTC Conjugate Gradient IS a Krylov-space method... Renato Renner Jul 09 2016 06:29 UTC I am afraid that you may have misunderstood my previous answer. I did not at all mean to claim that we cannot apply QM to brains. Rather, my point was that F1, after she prepared the electron, doesn't need to include her own brain in her analysis (especially because she will no longer interact w ...(continued) Tony Sudbery Jul 08 2016 19:25 UTC Where is it written that quantum mechanics cannot be applied to brains? And if it is so written, how is it possible to have measurements like those that you assign to Wigner and his assistant? Indeed, we don't (yet) apply QM to our brains, because we don't have sufficient knowledge or computing powe ...(continued) Renato Renner Jul 08 2016 06:07 UTC I completely agree with your analysis, which describes the gedankenexperiment from a global (“outside”) perspective, according to the laws of Bohmian Mechanics (BM). And, indeed, it shows that the "memory" of a measurement outcome cannot assumed to be permanent, i.e., it may change (according to BM) ...(continued) Tony Sudbery Jul 02 2016 19:09 UTC Roger Colbeck drew our attention to this paper in the York QFIT group, and we met to discuss it last week. I would like to comment on the relation of Bohmian quantum mechanics to the extended Wigner's friend experiment. As generalised by John Bell, Bohmian qm can be applied to this experiment to yie ...(continued)
	# Internet Problem Solving Contest ## Solution to Problem J – Just for Fun Without any further words we give all the puzzles and our short solutions to each of them. To see the solutions, select the text in your browser (e.g. by dragging the mouse). ## Easy puzzles: 1. birds: puzzle, solution 2. bus: puzzle, solution 3. palindrome: puzzle, solution 4. bicycle: puzzle, solution 5. cube: puzzle, solution 6. girl1: puzzle, solution 7. girl2: puzzle, solution 8. statements: puzzle, solution 9. letters: puzzle, solution 10. century: puzzle, solution ## Hard puzzles: 1. 1000robbers: puzzle, solution 2. wires: puzzle, solution 3. chess: puzzle, solution 4. 24: puzzle, solution 5. product: puzzle, solution 6. cannibals: puzzle, solution 7. 5robbers: puzzle, solution 8. operation: puzzle, solution 9. coinseq: puzzle, solution 10. ipsc: puzzle, solution ## Easy puzzles: puzzle statements #### birds Puzzle ID: birds Ten birds sit on a clothes line. We shoot and kill one of them. How many birds remain on the clothes line? The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - one number: the number of birds that remain on the clothes line #### bus Puzzle ID: bus A bus was travelling with less than 100 passengers. At stop A, exactly three quarters of the passengers got off and 7 passengers got on the bus. The same thing happened at next two stops, B and C. How many people got off at the stop C? The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the number of people getting off at C #### palindrome Puzzle ID: palindrome Suppose we write dates in the MMDDYYYY format. In this format, the 2nd of October 2001 is a palindrome (a string equal to its reverse): 10022001. Find the previous date that yields a palindrome in this format. The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the 8-digit string #### bicycle Puzzle ID: bicycle A bicycle stands on a straight road. A piece of string is attached to its rear wheel as shown in the image: Suppose that we slowly start to pull the string to the left. What will happen to the bicycle? A) It will start moving to the left. B) It will start moving to the right. C) It will stay in the same place. Assume that the wheel of the bicycle doesn't slip on the ground (i.e. whenever the bicycle moves, the wheel has to rotate). Also assume that the bicycle stays in an upright position all the time. The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the uppercase letter corresponding to the correct answer #### cube Puzzle ID: cube You have a cube NxNxN. How many straight cuts are necessary to cut it into N^3 cubes of size 1x1x1? You may arrange the pieces in any way you like before making each cut. a) Solve for N=3 b) Solve for N=4 The answer for this puzzle consists of three lines, containing respectively: - the ID of this puzzle - the number of cuts from part a) - the number of cuts from part b) #### girl1 Puzzle ID: girl1 In a two-child family, one child is a boy. What is the probability that the other child is a girl? The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the answer in the form a/b (where a,b are relatively prime) #### girl2 Puzzle ID: girl2 In an unnamed overpopulated country the rulers agreed on a new law: Each woman may have as many children as she wants to, until she gives birth to a girl. After that, she may have no more children. Assume that the law will never be broken. All families will have as many children as they are (physically and legally) able to. On each birth either one boy or one girl is born with equal chances. In the current population the ratio males:females is 1:1. What will happen in the next 100 years? A) The ratio of males to females will go up B) The ratio of males to females will stay the same C) The ratio of males to females will go down The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the uppercase letter corresponding to the correct answer #### statements Puzzle ID: statements Given is a list with 2004 statements: 1. Exactly one statement on this list is false. 2. Exactly two statements on this list are false. 3. Exactly three statements on this list are false. ... 2004. Exactly 2004 statements on this list are false. a) Determine which statements are true. b) Replace "exactly" by "at least". Again, determine which statements are true. The answer for this puzzle consists of three lines, containing respectively: - the ID of this puzzle - the encoded answer from part a) - the encoded answer from part b) How to encode the answer? If no statements are true, write the word 'NONE' (without the quotes). Otherwise take the set of true statements and write it as a set of ranges. E.g. the set {1,2,3,7,9,100,101} is encoded as 1-3,7,9,100-101 #### letters Puzzle ID: letters How many letters does the _shortest_ correct answer to this puzzle contain? The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle #### century Puzzle ID: century The twentieth century ended on 31. 12. 2000, which was a Sunday. Looking into the future, on which days of the week won't any century ever end? Remember that leap years are those divisible by 400 plus those divisible by 4 but not by 100. (1996 was a leap year, so was 2000, but 2100 won't be a leap year and neither will 2047.) The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the days of the week on which no century will ever end The exact form of the answer is a comma-separated list of three-letter abbreviations of the days in the order in which they appear in a week. E.g. if the answer were Monday, Tuesday and Wednesday, write the string 'Mon,Tue,Wed' (without the quotes). ## Hard puzzles: puzzle statements #### 1000robbers Puzzle ID: 1000robbers One thousand robbers meet in their hideout and plan to divide all the loot they stole. All robbers value their life above everything. They are smart, greedy (want as much loot as possible) and bloodthirsty (given two different outcomes with the same payoff for a robber, he prefers the one where more other robbers die). They all know this information. The robbers agreed that every day they will take a vote. If at least 50% of the robbers vote to split the loot, everybody takes a fair share and leaves the hideout. Otherwise the currently youngest robber is shot. How many robbers will survive? In the last round of voting, how many of them will vote for splitting the loot? (Assume that if a robber knows that his vote doesn't matter, he votes against splitting the loot.) The answer for this puzzle consists of three lines, containing respectively: - the ID of this puzzle - the number of robbers that survive - the number of robbers that voted to split the loot in the last round #### wires Puzzle ID: wires Imagine that you are an electrician. You are called to a very high tower, because their lifts are broken. When you arrive, you find that the lifts were sabotaged: 2005 wires were cut and now all of them hang from the top of the tower down to its bottom. (I.e. each of the wires has got one loose end near the bottom of the tower and one loose end near its top.) Before you can proceed with the repairs, you need to identify all the matching ends of the wires. For each of the bottom ends you have to know its corresponding end at the top. Your only instrument is a simple probe -- a battery and a bulb. If you attach two ends of wires to this probe, the bulb lights up if and only if the other ends of these two wires are currently conductively connected (maybe through one or more other wires). At each end of the tower you may connect and disconnect the ends of wires in an arbitrary way. Since the tower is so high, you want to minimize the number of times you have to climb up and down the staircase, regardless of how much work you have to do while you are at the top or bottom. What is the minimum number of traversals required? (Once up and down counts as 1.) The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the minimal number of traversals #### chess Puzzle ID: chess Given is a game of chess in progress. Your task is to determine which were the last two pieces that moved before the current position was reached. The current position: white: king on c1, rook on b1, knight on a1, pawns on b2, c2, d2 black: king on a2 The answer for this puzzle consists of four lines, containing respectively: - the ID of this puzzle - the color of the player that moved the last piece ('white' or 'black', without the quotes) - the last move made by this player before the position was reached - the last move made by the other player Enter both moves in standard chess notation. (Assume that no piece was captured in the move the other player made.) ----------------------------------------------------------------------------- Explanation of the chess notation [a subset you need to know]: A move is written as [piece][x](destination square). The piece is represented by an uppercase letter: King Queen Rook kNight Bishop If no piece is specified, the moving piece is a pawn. The optional x means the moving piece has captured an opponent's piece. A lowercase letter is used to specify the row of the destination square: Examples: - a king moves from e4 to e5: Ke5 - a knight moves from d4 to f3, taking an opponent's rook: Nxf3 - a pawn moves from g2 to g4: g4 #### 24 Puzzle ID: 24 From the numbers 1, 3, 4 and 6 create an expression giving the result 24. You may use only parentheses and the operations +, -, * and /. You have to use each of the given four numbers exactly once. The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the expression (no spaces, no unnecessary parentheses) #### product Puzzle ID: product The sum of some (not necessarily distinct) positive integers is 34. What is the greatest possible value of their product? The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle - the greatest possible product #### cannibals Puzzle ID: cannibals 47 people were capured by cannibals. For some obscure reason, the cannibals want to give the prisoners one last chance. They prisoners will have to stand in a line behind each other. Then the cannibals' shaman places a hat on each of the prisoners' heads. Each of these hats will be either white or black. Each prisoner can only see the colors of the hats people in front of him Now the shaman will ask each of the prisoners to guess the color of his hat. The first to answer is the prisoner at the end of the line (seeing all hats except for his own), then the shaman proceeds along the line. The last to answer is the prisoner at the front of the line.. If a prisoner's guess is correct, he is released immediatly, otherwise he is left to be eaten. a) Suppose that the prisoners know this procedure and that before it starts they have the time to agree on a strategy. How many of them are GUARANTEED to be saved if they find the optimal strategy? b) Solve the same task, if there are hats of 3 different colors (mauve, ochre and navy blue). The answer for this puzzle consists of three lines, containing respectively: - the ID of this puzzle - the number of saved prisoners from part a) - the number of saved prisoners from part b) #### 5robbers Puzzle ID: 5robbers Five robbers (Andrew, Bill, Carl, David and Wendy) want to divide loot consisting of 1000 identical diamonds. In alphabetic order they make a proposal on how to split the loot. After a proposal is announced, the robbers take a vote. If a (strict) majority agrees, the proposal is carried out and everyone goes to spend his share. If the proposal is rejected, its author is shot. All robbers value their life above everything. They are smart, greedy (want as many diamonds as possible) and bloodthirsty (given two different outcomes with the same payoff for a robber, he prefers the one where more other robbers die). They all know this information. a) How many diamonds will Andrew get, if we suppose that each proposal may only specify how many diamonds will which of the robbers get? b) How many diamonds will he get, if each proposal may also include shooting other robbers? The answer for this puzzle consists of three lines, containing respectively: - the ID of this puzzle - the number of diamonds from part a) - the number of diamonds from part b) #### operation Puzzle ID: operation Given are the following facts: 1 + 1 = 0 2 + 2 = 0 3 + 5 = 0 6 + 8 = 3 8 + 8 = 4 9 + 7 = 1 11 + 567 = 1 How much is 680 + 369? The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle #### coinseq Puzzle ID: coinseq Suppose we repeatedly throw a fair coin. Clearly the expected number of throws until we see a head is 2. The expected number of throws until we see two heads in a row is already 6. Suppose H denotes a head and T a tail. So e.g. HHT is a consecutive sequence of 2 heads and 1 tail. a) What is the expected number of throws we have to make until we observe the sequence HHHHHHH? b) ... the sequence HT? c) ... the sequence HHTHHHHHTHHHHHHTHHHHHTHHHH? The answer for this puzzle consists of four lines, containing respectively: - the ID of this puzzle - the exact result from part a) - the exact result from part b) - the exact result from part c) #### ipsc Puzzle ID: ipsc We really admire you if you see this puzzle statement. We tried really hard to create a really challenging last hard puzzle. We succeeded. Not even the author managed to solve it. We present a somewhat simplified version. Question: Will you participate also in IPSC 2006? The answer for this puzzle consists of two lines, containing respectively: - the ID of this puzzle ## Easy puzzles: solutions #### birds Zero, the others will fly away as soon as they hear the shot. Ever seen birds? #### bus By solving a simple equation we get the answer: 9. #### palindrome We need the closest year that yields a valid MMDD when reversed. Clearly 2000 doesn't work, thus the last digit in D is 1. The best choice is clearly DD=31 (and thus YYYY=13..). Now we want to maximize the second digit of MM. It can NOT be 9, because September has only got 30 days. Thus MM=08 and the whole date is 08311380. #### bicycle At each moment, the wheel is turning around the point where it touches the ground. From this observation it can be easily seen that the resulting torque causes the wheel to rotate to the left. (A common misconception when doing the physics is computing the torque with the center of the wheel as the center of rotation. Note that if this was indeed the case, by the same reasoning we could derive the following: "Tie the string at the center of the wheel, pull. The resulting torque is zero, thus the bicycle doesn't move." This is clearly wrong. If you still don't trust our solution, try an experiment. To make the bicycle go to the right, the direction in which we pull must cross the ground to the right of the bottom of the wheel.) #### cube In both cases 6 cuts clearly suffice. Now consider any of the small cubes that was initially completely inside the large cube. Whatever we do, in each cut we will only manage to create one of its faces. Thus 6 cuts are necessary. #### girl1 A classical example of conditional probability. There are 3 equally possible families: (older boy, younger girl), (older girl, younger boy) and (older boy, younger boy). In 2 of these cases the other child is a girl, thus the answer is 2/3. #### girl2 The ratio will stay 1:1. Consider all families that have a baby. Approximately one half of these babies are girls, half of them are boys. Some of these families "advance to the next round" to have more babies. Now consider the second-born babies. Again, the ratio of boys vs girls is 1:1... and so on. #### statements a) The statements contradict each other, thus at most one is true. If none of them were true, the last statement would be true -- a contradiction. Thus exactly one is true, 2003 are false. The true statement is #2003. b) If statement #k is true, so are statements #1..#(k-1). Let there be exactly k true statements. The (true) statement #k says that there are at least k false statements. The (false) statement #(k+1) says that at least (k+1) statements are false. This means that there are exactly k false statements, and thus k=2004/2=1002. True statements are #1 to #1002. #### letters 0 (the shortest correct answer involving some letters is probably "four") #### century By using a calendar we can quite easily find that the last day of the century repeats with a period 4. (This corresponds to 400 years. As soon as we see that the last weekdays of 2000 and 2400 are the same, we may conclude that the sequence is periodic, because the pattern of leap years repeats with the same period.) The never-occuring days are Tuesday, Thursday and Saturday. ## Hard puzzles: solutions #### 1000robbers Let N be the number of robbers. If N=1, the only robber takes the loot. If N=2, the vote of robber 2 is enough to split the loot. If N=3, robbers 1 and 2 know that IF robber 3 dies, they get 50% each (the result for N=2), thus they won't vote for 33.3% and robber 3 dies. By extending this result we see that the robbers agree iff their count is a power of 2. Thus there will be 512 survivors, 256 of them will vote for splitting the loot (and saving their necks), the others will be against it. #### wires Label lower ends X1 to X2005. Connect: X1-X2, X3-X4, ..., X2003-X2004. Climb up. Find the pairs, label them U1-U2, ..., U2003-U2004 (in an arbitrary way, the numbers won't match the lower ones yet). The last wire will be U2005. Connect U2-U3, U4-U5, ..., U2004-U2005. Climb down. Disconnect everything, but keep the old marks. Relabel X2005 to L2005. Find its current pair Xn, label it L2004. Its previous pair [for the record: it is X((n-1) xor 1 + 1)] will be L2003. Continue in the same way until all wires have a L label. Now clearly for each n Ln and Un are opposite ends of the same wire. #### chess Black couldn't move last (the king would have to move from check and there is no way he could get it one move earlier). Thus the last to move was white. White pawns didn't move, white rook couldn't. Two last moves remain: Nb3a1 and Kd1c1. It looks as the problem with the black king remains. The solution is simple: the moving white piece could capture a black piece that made black's last move. By trying out all possibilities we find that the only way this could happen is that the white king captured a black knight. Thus the last move was white's Kxc1. Black's last move was Nc1 (either from d3 or e2). #### 24 The only solution: 24 = 6 / ( 1 - 3/4 ) #### product Let 34 = a_1 + ... + a_k. If some a_i >= 4, we may replace it by 2 and (a_i-2). If some a_i is 1, we may replace a_i and a_1 by (a_i+a_1). Thus in the optimal solution all a_i are either 2 or 3. As 222 < 33, there are no more than two 2s. Thus the only optimal way is 34 = 2 + 2 + ten times 3. The product is 43^10 = 236196. #### cannibals We can not guarantee saving the last person in the queue (the first one to answer), as nobody sees his hat. Everyone else can be saved. Let the hat colors be 0 to K-1. Let a_i be the number of hats of color i the last person sees. He computes the value ( \sum_{i=0}^{K-1} i.a_i ) mod K and announces the corresponding color. Now the person N-1 computes this sum for the hats he sees. The difference is the color of his hat. He announces it and is saved. From now on all persons know his hat's color and thus they can include it in their computations. Thus everybody else computes his hat color in the same way. Note that for K=2 this reduces to the first person announce the parity of white hats. #### 5robbers Using a similar way of reasoning than in the 1000robbers case we get the correct results: a) E alone gets everything. Thus if there are D and E, E rejects any proposal and D is shot. CDE: C knows that D wants to survive. Thus D will vote for any proposal. C proposes that he gets all the diamonds, this is agreed upon. BCDE: B knows that he needs two votes to survive. If he dies, C gets 1000, D and E nothing. He has to give something to each of them, otherwise they will vote against him (due to their bloodthirst). Thus B's proposal is B 998, C 0, D 1, E 1. ABCDE: A needs 2 votes. The cheapest way is to pay C and (D or E). Both ways leed to A getting 997 diamonds. b) The cases E and DE remain the same. CDE: C, knowing that D will vote for any proposal where he survives, he proposes that he (C) gets all the diamonds and (bloodthirst!) that E is shot. BCDE: B knows that he has the vote of E that wants to survive. He needs one more, thus he has to pay D. Now he can have C shot and because of his bloodthirst he will do so. His proposal: B 999, C shot, D 1, E 0. ABCDE: A has the vote of C, pays 1 for the vote of E. His winning proposal: A 999, B shot, C 0, D shot, E 1. #### operation How is the result computed? Write the added numbers on a piece of paper and compute the number of closed areas on the paper. (E.g. the number 6 creates one such area, the number 8 is the only one to create two of them.) Yes, we know that it was impossible to find out ;) #### coinseq There is a nice connection between the number of expected throws and the exact form of the pattern. The easiest way to find it was to find the results for short patterns (e.g. by simulation) and to guess how it works. We decide to omit the details ;) #### ipsc We sure DO hope to see all of you next year :)
	# How do you simplify 4 sqrt x^2? ##### 1 Answer Sep 18, 2015 $4 \sqrt{{x}^{2}} = 4 \| x \|$ Since the x under the root is a square, it can come out. But since everything that comes out of a root must be 0 or positive, the absolute value bar must be there. If we know that $x \ge 0$ you can ditch the bar.
	# Make glossary defined terms bold I have defined a lot of terms with the glossaries package and use them in my text with the \gls{} command. However, I'd like terms called with \gls{} to appear bold, so people know that it is a term defined in the glossary. What command do I have to renew and how to achieve this? - The glossaries package offers the command \glstextformat to control the way the text is displayed, so you can write \renewcommand{\glstextformat}[1]{\textbf{#1}} or \renewcommand{\glstextformat}{\textbf} in the preamble of your document. - I should have mentioned, I am using the glossaries package, didn't know that there was a previous version. – Tim van Dalen Jun 19 '11 at 16:37 or even \renewcommand*{\glstextformat}{\textbf} which is even preferable, albeit a little hard to digest for beginners. – egreg Jun 19 '11 at 16:37 I don't know whether \gls is your command or an external one, to whose definition you don't have easy access. If the latter, then you can do something like \let\oldgls=\gls \renewcommand\gls[1]{\textbf{\oldgls{#1}}} (This sort of manipulation can be automated using patch.) If the former, then, rather than devoting a new command to it, you can just replace your definition \newcommand\gls[1]{...} by \newcommand\gls[1]{\textbf{...}} - Oops, sorry, I missed the fact that you were using a specific package. My answer is about general fiddling with commands after their definition; G Medina's answer (tex.stackexchange.com/questions/21152/…) is a much better approach in your specific setting. – L Spice Jun 19 '11 at 17:41
	# Finding number of don't care conditions I'm new to these forums and have very little background in electrical engineering. I'm trying to figure out the number of "don't care conditions" based on an input of two numbers, each 0-9, for designing a digital system to implement the multiplication table. The output would be the product of the two numbers. I don't have a very good understanding of what "don't care conditions" are. My understanding is that they are based on inputs that are "unlikely". Are the number of don't care conditions 9(since we "don't care" about the product of any number and zero)? Each of your inputs would be represented by four bits - but four bits can represent 15 different values. You would be interested in the bit patterns that represent values from 0 to 9, but would not be interested in those that represent values of 10 - 15, as those shouldn't exist in your inputs. Those values or bit patterns would be your "don't care" conditions. Related (but perhaps too advanced for you): When do truth tables use the "don't care" term? Don't-care conditions normally describe illegal combinations of inputs. In your example, you're only allowing numbers from 0-9, but you need four bits to hold those numbers. That means it's possible to represent inputs from 10-15 as well. Since you don't want to allow those inputs, you say that the output for those conditions is a don't-care -- it can be any number. This allows you to use the simplest logic that generates correct outputs for the valid inputs. In your example, you'd have a don't-care output whenever either input is 10 (1010), 11 (1011), 12 (1100), 13 (1101), 14 (1110), or 15 (1111). By my count, that's 6 x 16 don't-cares for the first input plus another 6 x 10 for the second, which gives 6 x 26 = 156 don't-cares. Sometimes don't-cares are used for inputs. This is a kind of shorthand that means the input has no effect on the output. For example, you could give the truth table for an AND gate like this: A B A*B 0 X 0 X 0 0 1 1 1 See the answer I linked above for a more complex example using a multiplexer.
	# Effective Mordell I am Pierre MATSUMI. Could you please teach me what is effective Mordell. Assume that f(X,Y) = 0 which defines smooth affine curve with genus > 1, and that there will be the solution X=n/m, Y=n'/m' in rational number Q. Then, Theorem(?)(Effective Mordell): max(\|n\|,\|m\|,\|n'\|,\|m'\|) < const_f for some constant const_f with some constant in positive real number. Is this statement right? I found some article where, if C denotes the proper smooth curve defined by f(X,Y) after compactification, effective Mordell is equivalent to the fact that there is some non-trivial function F:C ---> P^1 and the height of F(n/m,n'/m') is bounded. I am NOT sure whether this definition is equivalent to the above Theorem(?).... Please just teach me. Sincerely yours, Pierre MATSUMI - I think effective simply means that we know an algorithm to determine const_f from the coefficients of f. At the moment we don't know such an algorithm, we only know that const_f exists. – GH from MO Mar 1 '13 at 18:10 So your "Theorem(?)" is a theorem (of Faltings), but it is not the effective version. An effective version is not known at the moment. For certain families of curves there is an effective version, but that is another issue. – GH from MO Mar 1 '13 at 18:15 For example, assume that $f$ has coefficients in $\mathbb Z$, and let $H(f)$ be the maximum of the absolute values of its coefficients. Then the following would be an effective version of the Mordell conjecture: $$\max(\|n\|,\|m\|,\|n'\|,\|m'\|) \le 10^{10^{10^{H(f)+\deg(f)+1000}}}.$$ NOTE: I'm not saying that this statement is known; it's not. (Although I'd be surprised if it isn't true.) But it illustrates what is meant by an "effective bound". Thanks a lot! I finally got the meaning of effective"!! Pierre – Pierre MATSUMI Mar 2 '13 at 8:42
	Ubuntu – Secure shared memory ramwebserver In my Ubuntu deployment script I have written a function to secure shared memory. For a web server it's important to secure the shared memory to tighten up security. Below you can find the code: echo "tmpfs /dev/shm tmpfs tdefaults,noexec,nosuid 0 0" >> /etc/fstab When I reboot the web server it gets stuck. I can't see where it goes wrong. Do you have an idea, please let me know. Unless theres's a typo in your question, I think you're options are off. First, don't specify 'defaults' (or tdefaults, as you have it.. Here's the output of my shm directory taken from cat /proc/mounts: none /run/shm tmpfs rw,nosuid,nodev,relatime 0 0 echo "tmpfs /dev/shm tmpfs rw,nosuid,nodev,relatime 0 0" >> /etc/fstab
	## nanoVNA – measure common mode choke – it is not all that hard! It seems that lots of hams find measuring the impedance of a common mode choke a challenge… perhaps a result of online expert’s guidance? The example for explanation is a common and inexpensive 5943003801 (FT240-43) ferrite core. ## Expectation It helps to understand what we expect to measure. See A method for estimating the impedance of a ferrite cored toroidal inductor at RF for an explanation. Note that the model used is not suitable for cores of material and dimensions such that they exhibit dimensional resonance at the frequencies of interest. Be aware that the tolerances of ferrite cores are quite wide, and characteristics are temperature sensitive, so we must not expect precision results. Above is a plot of the uncalibrated model of the expected inductor characteristic, it shows the type of response that is to be measured. The inductor is 11t wound on a Fair-rite 5943003801 (FT240-43) core in Reisert cross over style using 0.5mm insulated copper wire. Continue reading nanoVNA – measure common mode choke – it is not all that hard! ## A simple Simsmith model for exploration of a 50Ω:200Ω transformer using a 2843009902 (BN43-7051) binocular ferrite core EFHW-2843009902-43-2020-3-6kThis article applies the Simsmith model described at A simple Simsmith model for exploration of a common EFHW transformer design – 2t:14t to a ferrite cored 50Ω:200Ω transformer. This article models the transformer on a nominal load, being $$Z_l=n^ 2 50 \;Ω$$. Keep in mind that common applications of a 50Ω:200Ω transformer are not to 200Ω transformer loads, often antennas where the feed point impedance might vary quite widely, and performance of the transformer is quite sensitive to load impedance. The transformer is discussed here in a 50Ω:200Ω context. Above is the prototype transformer using a 2843009902 (BN43-7051) binocular #43 ferrite core, the output terminals are shorted here, and total leakage inductance measured from one twisted connection to the other. Continue reading A simple Simsmith model for exploration of a 50Ω:200Ω transformer using a 2843009902 (BN43-7051) binocular ferrite core ## A simple Simsmith model for exploration of a common EFHW transformer design – 2t:14t This article describes a Simsmith model for an EFHW transformer using a popular design as an example. This article models the transformer on a nominal load, being $$Z_l=n^ 2 50 \;Ω$$. Real EFHW antennas operated at their fundamental resonance and harmonics are not that simple, so keep in mind that this level of design is but a pre-cursor to building a prototype and measurement and tuning with a real antenna. Above is the prototype transformer measured using a nanoVNA, the measurement is of the inductance at the primary terminals with the secondary short circuited. Continue reading A simple Simsmith model for exploration of a common EFHW transformer design – 2t:14t ## A simple Simsmith model for exploration of a common EFHW transformer design – 2t:16t This article describes a Simsmith model for an EFHW transformer using a popular design as an example. This article models the transformer on a nominal load, being $$Z_l=n^ 2 50 \;Ω$$. Real EFHW antennas operated at their fundamental resonance and harmonics are not that simple, so keep in mind that this level of design is but a pre-cursor to building a prototype and measurement and tuning with a real antenna. The prototype transformer follows the very popular design of a 2:16 turns transformer with the 2t primary twisted over the lowest 2t of the secondary, and the winding distributed in the Reisert style cross over configuration. Above is a plot of the equivalent series impedance of the prototype transformer with short circuit secondary calculated from s11 measured with a nanoVNA from 1-31MHz. Note that it is almost entirely reactive, and the reactance is almost proportional to frequency suggesting close to a constant inductance. Continue reading A simple Simsmith model for exploration of a common EFHW transformer design – 2t:16t ## On testing two wire line loss with an analyser / VNA – part 3 This article series shows how to measure matched line loss (MLL) of a section of two wire line using an analyser or VNA. The examples use the nanoVNA, a low end inexpensive VNA, but the technique is equally applicable to a good vector based antenna analyser of sufficient accuracy (and that can save s1p files). On testing two wire line loss with an analyser / VNA – part 1 This article series shows a method for estimating matched line loss (MLL) of a section of two wire line based on physical measurements (Duffy 2011). Above is a short piece of the line to be estimated. It is nominal 300Ω windowed TV ribbon. It has copper conductors, 7/0.25, spaced 7.5mm. The dielectric is assumed to be polyethylene… but later measurements suggest is has slightly higher loss than polyethylene. The test section length is 4.07m. Continue reading On testing two wire line loss with an analyser / VNA – part 3 ## On testing two wire line loss with an analyser / VNA – part 2 This article series shows how to measure matched line loss (MLL) of a section of two wire line using an analyser or VNA. The examples use the nanoVNA, a low end inexpensive VNA, but the technique is equally applicable to a good vector based antenna analyser of sufficient accuracy (and that can save s1p files). On testing two wire line loss with an analyser / VNA – part 1 Above is a short piece of the line to be measured. It is nominal 300Ω windowed TV ribbon. It has copper conductors, 7/0.25, spaced 7.5mm. The dielectric is assumed to be polyethylene… but later measurements suggest is has slightly higher loss than polyethylene. The test section length is 4.07m. Continue reading On testing two wire line loss with an analyser / VNA – part 2 ## On testing two wire line loss with an analyser / VNA – part 1 This article series shows how to measure matched line loss (MLL) of a section of two wire line using an analyser or VNA. The examples use the nanoVNA, a low end inexpensive VNA, but the technique is equally applicable to a good vector based antenna analyser of sufficient accuracy. Above is a short piece of the line to be measured. It is nominal 300Ω windowed TV ribbon. It has copper conductors, 7/0.25, spaced 7.5mm, though as can be seen the spacing is not perfectly uniform. The dielectric is assumed to be polyethylene… but later measurements suggest is has slightly higher loss than polyethylene. The test section length is 4.07m. Continue reading On testing two wire line loss with an analyser / VNA – part 1 ## Measuring a 1/4 wave balanced line – nanoVNA A question was asked recently online: I am about to measure a 1/4 wave of 450 ohm windowed twinlead for the 2m band using my NanoVNA. My question is, since I will be making an unbalanced to balanced connection, should I use a common mode choke, balun or add ferrites to the coax side to make the connection, or does it really matter at 2m frequencies? The coax lead from my VNA to the twinlead will be about 6″ to 12″ long. I will probably terminate the coax in two short wires to connect to the twinlead. It is a common enough question and includes some related issues that are worthy of discussion. Continue reading Measuring a 1/4 wave balanced line – nanoVNA ## Measure transmission line Zo – nanoVNA – PVC speaker twin – loss models comparison #3 Measure transmission line Zo – nanoVNA – PVC speaker twin demonstrated measurement of transmission line parameters of a sample of line based on measurement of the input impedances of a section of line with both a short circuit and open circuit termination. From Zsc and Zoc we can calculate the Zo, and the complex propagation constant $$\gamma=\alpha + \jmath \beta$$, and from that, MLL. ## Measure transmission line Zo – nanoVNA – PVC speaker twin – loss model derivation The article Measure transmission line Zo – nanoVNA – PVC speaker twin demonstrated measurement of transmission line parameters of a sample of line based on measurement of the input impedances of a section of line with both a short circuit and open circuit termination. From Zsc and Zoc we can calculate the Zo, and the complex propagation constant $$\gamma=\alpha + \jmath \beta$$, and from that, MLL. ## Measurement with nanoVNA So, let’s measure a sample of 14×0.14, 0.22mm^2, 0.5mm dia PVC insulated small speaker twin. Above is the nanoVNA setup for measurement. Note that common mode current on the transmission line is likely to impact the measured Zin significantly at some frequencies, the transformer balun (A 1:1 RF transformer for measurements – based on noelec 1:9 balun assembly) is to minimise the risk of that. Nevertheless, it is wise to critically review the measured \|s11\| for signs of ‘antenna effect’ due to common mode current. Continue reading Measure transmission line Zo – nanoVNA – PVC speaker twin – loss model derivation
	View MATLAB Command. The constant that we tacked onto the second anti-derivative canceled in the evaluation step. constant. When we’ve determined that point all we need to do is break up the integral so that in each range of limits the quantity inside the absolute value bars is always positive or always negative. For the first term recall we used the following fact about exponents. First, notice that we will have a division by zero issue at $$w = 0$$, but since this isn’t in the interval of integration we won’t have to worry about it. 4) Coefficients obtained, we integrate expression. We use it to find anti-derivatives, the area of two-dimensional regions, volumes, central points, among many other ways. It is important to note that both the definite and indefinite integrals are interlinked by the fundamental theorem of calculus. However, many more cannot - even ones that look deceptively simple. There are a couple of nice facts about integrating even and odd functions over the interval $$\left[ { - a,a} \right]$$. Finding definite integrals 3. This, therefore, means that 0 sin(x) dx = {-cos(π)} – {-cos(0)} = 2. To get started, type in a value of the integral problem and click «Submit» button. If F is an antiderivative of f, we can write f (x)dx = F + c. In this context, c is called the constant of integration. This article has been viewed 11,498 times. But, let’s start with the basics; Integrals. Is there a way to make sense out of the idea of adding infinitely many infinitely small things? After evaluating many of these kinds of definite integrals it’s easy to get into the habit of just writing down zero when you evaluate at zero. Recall that we can’t integrate products as a product of integrals and so we first need to multiply the integrand out before integrating, just as we did in the indefinite integral case. Add the signed areas (areas of the rectangles) together, and there you go! In this section however, we will need to keep this condition in mind as we do our evaluations. Your email address will not be published. For this reason, indefinite integrals are only defined up to some arbitrary constant. How do you find the area under a curve? In the first integral we will have $$x$$ between -2 and 1 and this means that we can use the second equation for $$f\left( x \right)$$ and likewise for the second integral $$x$$ will be between 1 and 3 and so we can use the first function for $$f\left( x \right)$$. The notation used to refer to antiderivatives is the indefinite integral. Linearity does not just apply for polynomials. it is between the lower and upper limit, this integrand is not continuous in the interval of integration and so we can’t do this integral. The most common way to do this is to have several thin rectangles under the curve from the initial point x = a to the last point x = b. Proper: if the degree of the polynomial divisor is greater than the dividend. We know ads can be annoying, but they’re what allow us to make all of wikiHow available for free. Integrals are the sum of infinite summands, infinitely small. To create this article, volunteer authors worked to edit and improve it over time. This should explain the similarity in the notations for the indefinite and definite integrals. The definite integral is denoted by a f(x) d(x). For example, if f = x4, then an antiderivative of f is If you need to understand how the problem was solved, you can see a detailed step-by-step solution. The derivative of –cos(x) + constant is sin (x). f (x)dx means the antiderivative of f with respect to x. In this article, we discussed how to calculate indefinite integrals of elementary functions whose antiderivatives can also be written in terms of elementary functions. The sinc function is an even function, The power tower is a prominent example of a function where the method of finding its derivative is extremely similar to finding the derivative of the general exponential function. Consider sin(x)dx = -cos (x) + constant. After each calculation, you can see a detailed step-by-step solution, which can be easily copied to the clipboard. Turn each part into a limit. Knowing how to use integration rules is, therefore, key to being good at Calculus. To this point we’ve not seen any functions that will differentiate to get an absolute value nor will we ever see a function that will differentiate to get an absolute value. Khan Academy is a 501(c)(3) nonprofit organization. subtracting any constant would be acceptable. Next, we need to look at is how to integrate an absolute value function. antiderivatives. Create the function with one parameter, . If F is an antiderivative of f, we can write f (x)dx = F + c. In this context, c is Also notice that we require the function to be continuous in the interval of integration. That will happen on occasion and there is absolutely nothing wrong with this. an antiderivative of f, and F and G are in the same family of so are x5 + 4, x5 + 6, etc. F = x5, which can be found by reversing the power rule. First, you’ve got to split up the integrand into two chunks — one chunk becomes the u and the other the dv that you see on the left side of the formula. This property tells us that we can So, we aren’t going to get out of doing indefinite integrals, they will be in every integral that we’ll be doing in the rest of this course so make sure that you’re getting good at computing them. . Trader Joe's 3 Cheese Pizza, Chicken Strip Delivery, Characteristics Of Enterprise In Economics, Slow Cooker Beef Recipes With Cream Of Mushroom Soup, Reverse Sear Steak Air Fryer,
	उपलब्ध असलेले कोर्स Operating System Concepts Course Outcomes (COs) At the end of the course, students will be able to: CO‐1 Master functions, structures and history of operating systems CO‐2 Master understanding of design issues associated with operating systems CO‐3 Master various process management concepts including scheduling, synchronization, & deadlocks CO‐4 Be familiar with multithreading Design and Analysis of Algorithms Course Outcomes (COs)At the end of the course, students will be able to:CO‐1 Analyze the asymptotic performance of algorithms.CO‐2 Write rigorous correctness proofs for algorithms.CO‐3 Demonstrate a familiarity with major algorithms using data structures.CO‐4 Apply important algorithmic design paradigms and methods of analysis

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