text:
# help in Integral Maximization

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• August 23rd 2010, 07:02 AM
mido22
help in Integral Maximization
i want anyone to explain this problem to me

https://www.spoj.pl/problems/INTEGMAX/

and hints in how to solve this by math(Itwasntme)
• August 23rd 2010, 07:23 AM
Ackbeet
What ideas have you had so far?
• August 23rd 2010, 07:26 AM
mido22
Quote:

Originally Posted by Ackbeet
What ideas have you had so far?

i didin't understand the problem :D so i write i want explaination for it
• August 23rd 2010, 07:38 AM
Ackbeet
Well, you should know by now that this forum is not about simply providing answers, since that won't help you. Instead, we'll nudge you towards the answer, in hopes that you can solve it yourself. That way, you'll own the answer, and you'll really understand what's going on (see my signature).

I will happily re-state the problem and see if that gives you any ideas.

Given a set of x-coordinates $\{x_{1},\x_{2},\dots,x_{N}\}$ and a set of y-coordinates $\{y_{1},y_{2},\dots,y_{N}\},$ find a pairing such that the area under the piecewise linear function defined by the pairing is maximized.

Example:

Suppose the x-coordinates are $\{1,2,3\},$ and the y-coordinates are $\{4,5,6\}.$ One pairing would be $\{\{1,4\},\{2,5\},\{3,6\}\}.$ What would be the area under the piecewise linear functions defined by these points?

We're supposed to maximize the area under the piecewise linear functions by changing up the pairing. Another pairing in our example would be $\{\{1,5\},\{2,4\},\{3,6\}\}.$ You could calculate the area under these piecewise linear functions, and it might be different from the previous example.

So what ideas do you have now?
• August 23rd 2010, 07:44 AM
mido22
ok u meant that i must check all pairs together but how can i calculate the area this is my question

note : i knew that this forums help me only and didn't give the answer :D
• August 23rd 2010, 07:47 AM
Ackbeet
Quote:

how can i calculate the area this is my question
Well, what are the shapes with which you're dealing here? What's the area under a small straight-line segment look like?
• August 23rd 2010, 07:51 AM
mido22
it is irregular shape so i can't determine its area
• August 23rd 2010, 07:54 AM
Ackbeet
Quote:

it is irregular shape
I'm not sure I agree there. Visualize a shape with a horizontal base (the x-axis), vertical sides (the beginning point and end point of the interval), and a straight line connecting two points on the vertical lines. What shape is that? (Hint: turn your face 90 degrees and look at it sideways).
• August 23rd 2010, 07:59 AM
mido22
Quote:

Originally Posted by Ackbeet
I'm not sure I agree there. Visualize a shape with a horizontal base (the x-axis), vertical sides (the beginning point and end point of the interval), and a straight line connecting two points on the vertical lines. What shape is that? (Hint: turn your face 90 degrees and look at it sideways).

can i cut it into many shapes and calculate each area
• August 23rd 2010, 08:03 AM
Ackbeet
You could, but I don't think you'd gain anything other than complexity and possibility for errors. The shape I've described is a known shape with a known area. You don't even need to use calculus to find its area, although, of course, you can if you want to.

Put your thinking cap on! You can do this. What shape is it? I want you to figure out what shape it is. That's really the crux of this problem.
• August 23rd 2010, 08:06 AM
mido22
Quote:

Originally Posted by Ackbeet
You could, but I don't think you'd gain anything other than complexity and possibility for errors. The shape I've described is a known shape with a known area. You don't even need to use calculus to find its area, although, of course, you can if you want to.

Put your thinking cap on! You can do this. What shape is it? I want you to figure out what shape it is. That's really the crux of this problem.

i think i don't know this shape i 'm a young boy so may i didn't take it in school :D
all i know
(square , rectangle , triangle , circle , sphere , cube , cuboid , pyramid , tetrahedron )
• August 23rd 2010, 08:14 AM
Ackbeet
Ok. Try this: one of the two y-values will be smaller than the other one, in general. Divide up this figure into two pieces: everything below the lower y-value, and everything above that value that is still below the straight line. What two figures do you have now?
• August 23rd 2010, 08:18 AM
mido22
A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.

Examples:

The following are examples of polygons:
http://www.mathleague.com/help/geometry/IMG00049.GIF

so it is a ploygon

• August 23rd 2010, 08:20 AM
Ackbeet
You're correct. However, the answer is not specific enough. You're dealing with a particular kind of quadrilateral. Which one?

Incidentally, my post # 12 is an alternate route to getting the answer. It's a bit more complicated in some ways, but requires less knowledge, if that makes any sense. This approach is better because it's more scalable.
• August 23rd 2010, 08:22 AM
mido22
Quote:

Originally Posted by Ackbeet
You're correct. However, the answer is not specific enough. You're dealing with a particular kind of quadrilateral. Which one?

Incidentally, my post # 12 is an alternate route to getting the answer. It's a bit more complicated in some ways, but requires less knowledge, if that makes any sense. This approach is better because it's more scalable.

i don't know quadrilaterals except (square , rectangle , parallelogram , rhombus )
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5 Volt, 10 mA Offline Isolated Switch Mode Power Supply Design Ideas

Sensacell

Joined Jun 19, 2012
2,821
For a project I am working on, I need 5 Volts @ 10 mA maximum - to be supplied from 230 VAC 50 Hz mains.
The output needs to be galvanically isolated, but not "safety" isolated.
The output can have up to 100 mV ripple.
The solution needs to be small and cheap.

So far, my idea is to use a microchip SR087Adjustable Offline Inductorless Switching Regulator to power a small isolated 24V to 5V DC to DC converter.

This looks like a BOM cost of around $5.50 But it feels like thundering massive overkill? This could output 1 Watt, but I only need 50 Milliwatts. Ideas invited. Attachments • 1.1 MB Views: 9 • 472.3 KB Views: 7 LowQCab Joined Nov 6, 2012 516 Thread Starter Sensacell Joined Jun 19, 2012 2,821 Thanks for your design input. I cannot use a "Capacitor Dropper" configuration because the "GND" of the power supply needs to be connected to a floating node in another mains-connected circuit. The actual project involves adding DALI dimming capabilliy to existing low-cost mains-powered LED lights that are typically not dimmable. The idea is working, but I need to power the Micro and the switching MOSFET with a floating 5V supply. For my prototype, I am using the guts of a cheap USB phone charger, which is essentially an isolated 5V SMPS. The DALI interface provides isolation through optocouplers both sending and receiving data from the fixture. The optocoupler on the receiving side is powered by the DALI bus itself. LowQCab Joined Nov 6, 2012 516 Is there some reason why you can't use a Transformer ? Does the Dali-Bus run over the mains wiring, or does it use separate Low-Voltage Control Wiring ? What topology do you intend to use to Dim the cheap "non-Dimmable" LED Fixtures ? Are you intending to create a "One-Way" communication Node on the Dali-Bus ? Will the Dali-Bus continue to attempt 2-way Communication with the new One-Way Node ? Or does it have an option for a One-Way-Comm Link Output ? Some "cheap" LED Supplies will produce noise that may interfere with the data integrity of the Dali-Bus Network. A Micro Controller by its self may draw more than 10ma. Even a tiny dip in Power Supply Voltage may cause your controller to re-set. If you are PWM-ing a FET for dimming, pumping the Gate Capacitance of the FET will probably draw more than 10ma., ( certainly higher peaks ). So, your proposed Power Supply may not be over-kill at all, ( not withstanding its possibly questionable application as a solution to your problem ). Providing documentation for your current ideas regarding the solving your problem is commendable, ( and unfortunately unusual in these forums ), but you have not provided the needed details for getting from point-A to point-B. So, as always, Please clearly state the exact, underlying problem, that you are trying to solve, rather than asking for a solution to a problem that may help an odd, and possibly very impractical, workaround do the job. It sounds like you don't want to have to pay for a "Dali-Bus Compatible" Ballast/Power Supply, are they really that expensive ? If this is the case, you are adding quite a bit of complexity, and potential points of failure, to the system, and possibly spending more time and money than a "drop-in compatible", or external, Dali-Bus Compatible, LED Power Supply would cost. . . Thread Starter Sensacell Joined Jun 19, 2012 2,821 Thanks for your continued interest. The photo below shows what I have accomplished so far. I have developed a piece of hardware that accomplishes the basic goal. My hardware connects to an existing LED ballast with 3 wires and dims the LED output by shunting the current regulator with an FET. When the control FET turns on, the current continues to flow in the ballast's switching inductor, but the voltage drop over the inductor is reduced drastically so the current decays very slowly. The current regulator continues it's task without caring about the output voltage. PWM controls the brightness at around 500 Hz, which is also smoothed a bit by the existing capacitor across the LED's- so there is no visible modulation. The tricky part was eliminating the flicker caused by the asynchronous switching of the regulator and the PWM, I solved this by synching-up the PWM edges with the current regulator, since they are drastically different frequencies, it all works well The whole setup draws 3.5 Ma from the 5 volt supply, the 3.3V PIC16LF1575 is very power thrifty. The 5V supply is referenced to the MOSFET drain, which is at a voltage potential that floats relative to the mains. (the tricky problem) I want to boil this all down to a small board that can fit with the stock ballast in a small box. As I write this, it occurs to me that I might investigate ways to bootstrap the gate drive, so the PSU could reference the NEUTRAL? The big picture idea is to outfit my home with inexpensive and elegantly controllable lighting. DALI is a nice simple standard that is widely used in architectural lighting, but the fixtures are usually very expensive. I believe they are artificially expensive as the market is generally institutional and not terribly price sensitive. These standard consumer lamps are so ridiculously cheap that adding a$15.00 board that results in a $20.00 DALI dimmable lamp seems worth the effort. DALI has not caught on in the consumer market, people would rather use IoT stuff which I personally dislike, I want a simple and robust wired solution that I can understand and maintain myself. ericgibbs Joined Jan 29, 2010 12,901 hi Sensacell, I used the NMA versions of the potted DC>DC converters, usually 12V > 5V or 5V > +/-12V. On lighter loads the Vout can rise to about 10% above rated value, also the conversion efficiency drops off a low loads. For any project with low signal levels, I had to add additional +/-12Vdc filtering. E Attachments • 171.2 KB Views: 1 LowQCab Joined Nov 6, 2012 516 Well, to each his own ...... Go for it man .... But I think you are making the whole arrangement way too complex. If I were to consider doing something like this, I would skip the Dali Hardware all together and just design my own complete, centralized system. I'm not directly familiar with the Dali System, but I'd be willing-to-bet that you can get a Remote Controlled LED Power Supply that will run what ever number of fixtures that you'd like, with zero electronics in any of the fixtures. Measure to find out the average Voltage drop per fixture, and find the maximum Current Rating, then wire your fixtures in series with each other, if you get too close to your desired maximum Voltage, when calculating the number of fixtures that you want, then create 2 equal length strings of fixtures, and then wire the 2 strings in parallel. Each String of fixtures will always require the same regulated Current, but the Voltage Drop will vary with the number of fixtures in a string. 2 equal length strings, when wired in parallel, will need twice the Current. . . Thread Starter Sensacell Joined Jun 19, 2012 2,821 You cannot find fixtures that contain just LED's- a "Fixture" is typically a self-contained unit with a driver and LED's - unless you want raw LED strips or other unpackaged lights engines. Half the battle is the housing, it's gotta look good and be easy to install and not cost too much. To control a whole room full of lighting, you really need a master-slave configuration with many drivers or "ballasts" as they are sometimes called controlled from a network. This seems to be why the IoT stuff seems to have taken off in the home lighting world. The nightmarish complexity of the IoT idea makes DALI seem brutally simple by comparison! https://en.wikipedia.org/wiki/Digital_Addressable_Lighting_Interface LowQCab Joined Nov 6, 2012 516 I hear ya, but those built-in boards are super easy to remove. And there's no advantage to having an individual Power Supply inside each fixture, unless you have absolutely no access to the space above the fixture to run new wire. The fixtures probably require less than 1 amp, ( I'm guessing ), so a single ~22-gauge Wire is really all you need between each fixture in a string. ~50 Volts is probably enough to power ~10 Fixtures wired in series, depending, of course, on how the multiple LEDs are wired on the circuit board inside the Fixture, and it's really easy to make a ~50-V - 1-A Power Supply with a single SMPS-Chip, or you could probably buy ~10 of them pre-made from Ali-Express for ~$15.oo.
The entire SMPS circuit board might even fit inside a standard Electrical Junction Box.
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DickCappels

Joined Aug 21, 2008
7,426
A USB phone charger might be the ticket.

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# Fast calculations using fast-Fourier convolution

Consider an array $$X$$ with shape $$H \times W$$. Let $$Y$$ be the other array of the same shape and $$Z$$ is an array of shape $$h \times w$$.

We want to construct an array $$R$$ of shape $$(H-h + 1, W-w+ 1)$$ by the following rule:

$$\displaystyle R[k, l] = \sum_{i, j} (X[i,j] - Y[i,j])^2 +$$ $$\sum_{i=0}^{h-1}\sum_{j=0}^{w-1}[(X[k + i, l + j] - Z[i, j])^2 - (X[k + i, l + j] - Y[k + i, l + j])^2]$$

So it's better to represent it via the image: we erase some subarray of size $$h \times w$$ in $$X - Y$$, change it by $$X_{h, w} - Z$$ and compute the sum of squares.

Using dynamic programming we can solve it pretty fast, but it's still a time-consuming process. I think that it's possible to calculate such an array using fast Fourier convolutions. But we need to determine convolution arrays to do that.

Any hints?


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# MWG Proposition 2.F.1

Is homogeneity of degree zero necessary in proposition 2.F.1? It seems like the proof does not assume homogeneity of degree zero to establish the proposition. If this is true, it seems that homogeneity is not required to establish that the Slutsky matrix is negative semidefinite (only required assumptions are differentiability and Walras' law?)

Thank you so much for your help!

• Would be helpful to put here what Prop 2.F.1 is... not everybody has MWG. It'd be great if you could edit the title accordingly to help people searching for it. – Art Jan 20 '20 at 1:34

In between eq. $$(2.F.2)$$ and $$(2.F.3)$$ we read

...Walras's law tells us that $$w' = p'\cdot x(p',w')$$

Assume now that homogeneity of degree zero does not hold. Then we have, $$a>0$$

$$x(ap,aw) \neq x(p,w)$$

see definition $$2.E.1$$

Then we can set $$p'=ap,w'=aw$$ to examine the case $$x(p',w') \neq x(p,w)$$.

But then Walras' law would imply

$$w' = p'\cdot x(p',w') \implies aw = ap\cdot x(ap,aw) \implies w = p\cdot x(ap,aw)$$

while also, again by Walras' law we have

$$w = p\cdot x(p,w)$$

So under the assumption that Walras' law holds but that we do not have homogeneity of degree zero we end up saying

$$p\cdot x(ap,aw) = p\cdot x(p,w), \,\,\, x(ap,aw) \neq x(p,w)$$

I believe you can contemplate the situation further from here.

• Just for clarification -- is this supposed to imply that Walras' law implies homogeneity of degree zero? I'm wondering about a situation where p x(ap, aw) = p x(p,w) does not imply that x(ap, aw) = x(p, w)... for instance, we could have p=(1,1,1) , x(ap, aw) = (1,2,1) and x(p,w) = (1,1,2) – Jacob Bak Jan 18 '20 at 14:40
• Never mind! I see you're referring to the Weak Axiom. Thanks so much for help!!! – Jacob Bak Jan 18 '20 at 14:58
• @MichaelKim You're welcome. – Alecos Papadopoulos Jan 18 '20 at 15:39


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# How can I provide attribute extensibility in a component-based entity system?

This question and its answers move against utilizing an entity component system where the component itself is just a generic class with an dynamic container for the attributes/properties of the component.

• Entity = ID + collection of components
• Component = Container (std::map<attribute_name, attribute_value>) of attributes/properties, such as health, age, name, gender, etc.

I understand and am aware of the drawbacks (no compile-time type safety, added overhead of looking up values from the container). Nonetheless, I currently am working on/with such a system, with the following argument, which I would now like to put to the test: data-driven design.

If all the attributes of the component are in a dynamic container, it is very easy (in fact possible at all) to add new ones later on, outside of the source code. So you can easily script new attributes later on. And dumping/saving the state of the component is a piece of cake.

Compared to the proposal in the above mentioned thread, I also have an own class for each component, to implement some basic behavior and helper methods directly in them (but those also need to look up their values from the container).

Example:

My component base class has a map for all the attributes/property:

typedef std::map<std::string, PropertyBase*> PropertyMap;

So an actual component looks something like the following. In the constructor I populate the attributes I want the component to have, by adding them to the map. Later I can read and update them if needed, but always with a performance penalty, since I first have to find them in the std::map.

class Person : public ClonableComponent<Person>
{
public:
Person(ComponentFactory* const pFactory, ComponentType type, ComponentID id, GameEntity* const pEntity)
: ClonableComponent(pFactory, type, id, pEntity)
{
// General

// Feelings
}

void enterLocation(GameEntity* pLocGE)
{
if(pLocGE->hasComponent(COMP_TYPE_LOCATION))
{
updateProperty("currentLocation", pLocGE->getName());
updateProperty("casinoEnteredDateTime", (int)std::time(0));
}
}

void leaveLocation()
{
updateProperty("currentLocation", std::string("unknown"));
}
};

However, that thread made me think, and I am now doubtful that my approach will work well in the long run (maintenance, scaling and performance wise) and I am therefore considering moving to an directly typed approach with no look-up overhead. At the same time I don't want to loose that data-driven extendability.

I thought of a hybrid approach: Keep all known variables as actual language variables but also add a container (std::map) for scripted/data-driven variables. Therefore I'd know that my internal access is very fast and only the later added scripted extension would have the slow look-up and the other mentioned penalties.

Question: What do you use / would propose as an approach for a data-driven ECS in terms of attribute storage/handling? How do you keep it extendable? Would you recommend such a hybrid approach (real variables internally, container for new, external definitions)?

• make something -> make it work -> make it work better. You pick up neat tricks throughout the journey, but the important thing is having games that people play
– wes
Mar 7, 2014 at 19:32

If all the attributes are in a dynamic container, it is very easy (more correctly: possible at all) to add new ones later on, outside of the source code. So you can easily script new attributes later on.

Yes, but you can't do anything with new attributes added via scripts except via scripts without losing this "advantage." Once you need to refer to these values in source code, you should have just added a concrete field to a concrete type rather than something in a Dictionary or whatever.

If you want the ability for scripts to dynamically add new bits of data without recompilation (which is a very useful feature) than provide this functionality just to the ScriptComponent (or what have you) in your system. This sounds relatively identical to your "hybrid" approach anyway: I'd just caution that you shouldn't need the functionality anywhere but a component that is providing/running a script.

And dumping/saving the state of the component is a piece of cake.

This should not be the concern of the entity/component system. This should be the concern of the reflection/content representation system you use. It's trivial to implement without dynamic "property bag" dictionaries in languages like C# with good reflection support, and it's only slightly more work to do it in something like C++ with really poor reflection support. Consequently you shouldn't use this as a strong rationale for storing component attributes in a runtime associative array.

Would you recommend such a hybrid approach (real variables internally, container for new, external definitions)?

Basically, what I said above: don't use a runtime associative array for component attributes by default. If your scripting system needs that functionality, provide it via the components bound to the scripting system.

• I currently have this extendability everywhere since the entities are defined in XML files, where (for each component) values can be altered and completely new ones added. The latter ones are then of course only usable in scripts. But I am not really sure why I should limit that feature to only a specific component. Additionally, I will have to look into C++ reflection then, if I go with that hybrid approach then. Mar 7, 2014 at 20:00
• Allowing the attachment of arbitrary data to arbitrary types tends to result in data bloat (via potential accidental duplication), promiscuity (relying on pathological coupling to reach across to attached data on other components), and sloppiness: it doesn't make sense. What do you need to attach to a position component that isn't a position? Once you attach that data, it's not a position any more. You can achieve the same results with cleaner semantics by providing only a single defined attachment point for that data owned by the thing doing the attaching.
– user1430
Mar 7, 2014 at 20:05
• @JoshPetrie But don't you have to use associative runtime containers in order to have functionality that works on arbitrary entity types? E.g. a movement system that supports all entities with a position component. Trying to go around this problem with inheritance would have you end up at exactly the position that component-based systems try to get away from, doesn't it? Mar 7, 2014 at 22:03
• No; the question (and the linked related question) are referring to using such a container as a replacement for fields in a type. In other words, instead of "struct position { float x; float y; float z; }" you have "struct position { map<string, object> attributes; }."
– user1430
Mar 7, 2014 at 22:13

ECSs are somewhat complicated.

At first I used the following approach: Entities only contain a position vector and a set of components. A component contains a specific behavior and necessary data e.g. RenderComponent, PhysicsComponent etc. This is commonly referred to as ECS.

But this has some disadvantages: The data of an entity is closely entangled with the component functionality. Sooner or later a component needs to access and modify data stored in a different component type e.g. the MovementComponents needs to apply a force to the PhysicsComponents etc. So either directly access the other component via the entity, send an event causing the change in the other component or pull the needed value up into the entity. All bad solutions in my opinion. Also maintaining the lifecycle of each component type was quite a hassle.

Currently I'm using the following approach: An Entity consist of an unique id and a bag of keys and values (hash the key strings for ultimate performance ;-) ). Subsystems (e.g. physicsEngine, renderer) look for entities with property keys they can use and create components internally with the required behaviors. These components store the unique id of their entity and use it to read and modify the entities property values. You could even get rid of the entity class entirely, since it's only an id anyways and store the properties under the entity id in a global property store making the whole design more like a database. Jason Gregory calls this Property Centric Archictecture in his book Game Engine Architecture. Thief and Deus Ex used this approach.

It's trivial to inspect, store and load the game state. Easy to extend and I can create new behavior entirely in script (I'm using python for scripting - great language btw). Components are only known in their respective subsystem. I sacrificed compile time safety and OOP style data hiding sure and I might need some kind of templating mechanism to define and check properties. But for now this approach works suprisingly well.

• I fail to see the link to my question. You are just talking about ECS and their entity<->component<->system relationship design. I already got that covered. I am however asking about the attributes/properties/values within one component and their extendability. Mar 8, 2014 at 7:40
• You asked: What do you use / would propose as an approach for a data-driven ECS in terms of attribute storage/handling? This is what I'm using. Sorry if my answer is long winded. I'm still figuring this out. Short answer: I would move the properties from your component class to the entity class. Lookup of properties is not that expensive. I would not keep copies of variables in code. Fetch them from the store, modify them and write them back. You need to avoid allocation and copying of keys and values. Hash key strings into constants. Hope this helps. Mar 8, 2014 at 12:33
• So basically, your components just contain the behavior/logic, but no data? Isn't one of the key concepts in a "classical" ECS that each component is self-sufficient and has its own data directly accessible? Mar 8, 2014 at 17:07
• Also it seems to me as if Jason Gregory is n fact advocating distributing the data: "The properties of each game object are distributed across many data tables, one per property type [...] (rather than being centralized in a single class instance [...])." Or am I misinterpreting that part? Mar 8, 2014 at 17:17


text:
# How large are measurable cardinals of higher orders?

For each ordinal $\alpha$ define the notions of $\alpha$ - measurable cardinals and $\alpha$ - normal measures as follows:

A measure $\mu$ on a measurable cardinal $\kappa$ is a $0$-normal measure iff $\mu$ is a normal measure on $\kappa$.

A cardinal $\kappa$ is $0$-measurable iff there is a $0$-normal measure on $\kappa$.

For any ordinal $\alpha$ if the notions are defined for $\alpha$ then define them for $\alpha+1$ as follows,

A measure $\mu$ on a measurable cardinal $\kappa$ is a $(\alpha +1)$-normal measure iff $\mu$ is a normal measure on $\kappa$ such that $\mu(\{\lambda <\kappa~|~\lambda~\text{is} ~\alpha - \text{measurable}\})=1$

A cardinal $\kappa$ is $(\alpha+1)$-measurable iff there is a $(\alpha+1)$-normal measure on $\kappa$.

For any limit ordinal $\alpha$ if the notions are defined for each $\beta <\alpha$ then define them for $\alpha$ as follows,

A measure $\mu$ on a measurable cardinal $\kappa$ is a $\alpha$-normal measure iff $\mu$ is a normal measure on $\kappa$ such that for each $\beta <\alpha$ we have $\mu(\{\lambda <\kappa~|~\lambda~\text{is} ~\beta - \text{measurable}\})=1$

A cardinal $\kappa$ is $\alpha$-measurable iff there is a $\alpha$-normal measure on $\kappa$.

Question 1. How large are the $\alpha$-measurable cardinals in comparison with usual large cardinals like superstrongs, supercompacts, etc?

Question 2. Is there any ordinal $\alpha$ such that within ZFC one can prove non-existence of a $\alpha$-measurable cardinal?

-

## migrated from mathoverflow.netFeb 7 at 1:35

This question came from our site for professional mathematicians.

This looks awfully like a characterization of the Mitchell order. It should be pointed that the height of the Mitchell order is at most $(2^\kappa)^+$; and of course (as in this case) in order for $\kappa$ to be even $1$-measurable it needs to have stationary many measurable below it. –  Asaf Karagila Feb 7 at 0:17
1. Much smaller. 2. No. This is not research level, asking at the other site may be better. –  Andres Caicedo Feb 7 at 1:10

It's pretty straightforward to show that if $\kappa$ is a 2-strong cardinal$^*$, then $V_\kappa$ thinks that for all $\alpha$, there is an $\alpha$-measurable cardinal. In particular, we can show by induction that for $\alpha<\kappa$, $\kappa$ is $\alpha$-measurable.
Let $\mathcal U$ be the normal ultrafilter on $\kappa$ defined by $X\in \mathcal U$ iff $\kappa\in j(X)$. First, note that since every ultrafilter on $\kappa$ is in $V_{\kappa+2}$, $\alpha$-measurability for $\kappa$ is absolute for $M$ - that is, $M\vDash \kappa$ is $\alpha$-measurable$"$ iff $\kappa$ is $\alpha$-measurable. So, if $\kappa$ is $\alpha$-measurable for $\alpha<\kappa$, then by the definition of $\mathcal U$, $\{\beta<\kappa: \beta$ is $\alpha$-measurable$\}\in \mathcal U$. In other words, if $\kappa$ is $\alpha$-measurable (for $\alpha<\kappa$), then it's $\alpha+1$-measurable. The limit case it similar.
This also answers your second question, since $V_\kappa\vDash ZFC$.
$^*$ a cardinal $\kappa$ is 2-strong if there is $j:V\prec M$ such that $crit(j) =\kappa$ and $V_{\kappa+2}\subseteq M$.
One can prove something significantly stronger: If $\kappa$ is $2$-strong, then it has order $(2^{\kappa})^+$ -- the largest possible. This was first observed by Solovay. –  Andres Caicedo Feb 7 at 8:10


text:
# Are the norms of graphs dense in any interval?

It is known that there is a gap between 2 and the next largest norm of a graph. Is there an interval of the real line in which norms of graphs are dense?

-
What is the norm of a graph? –  Qiaochu Yuan May 9 '10 at 4:56
The norm of a graph is the largest eigenvalue of its adjacency matrix. Scott's question mathoverflow.net/questions/1822/… has some motivation for thinking about graph norms. –  Anton Geraschenko May 9 '10 at 5:35

Shearer, James B. On the distribution of the maximum eigenvalue of graphs, 1989. The theorem in this paper is that the set of largest eigenvalues of adjacency matrices of graphs is dense in the interval $\left[\sqrt{2+\sqrt{5}},\infty\right)$. Here's an online version.

Here's a related paper:

Hoffman, Alan J. On limit points of spectral radii of non-negative symmetric integral matrices, 1972. In this paper limit points less than $\sqrt{2+\sqrt{5}}$ are described. In particular, they form an increasing sequence starting at 2 and converging to $\sqrt{2+\sqrt{5}}$. Here's an online version. The author also posed the problem that led to Shearer's paper.

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These results and more are mentioned in Proposition 1.1.5 of Coxeter Graphs and Towers of Algebras by Goodman, de la Harpe, and Jones, 1989. –  Jonas Meyer May 9 '10 at 9:05
Wait, did you just tell Vaughan Jones about his own paper? MO has jumped the shark. –  Ben Webster May 9 '10 at 14:26
Perhaps I should end up with a negative score because of this... However I assure you that one does not have to be gaga to have forgotten what was in a book one wrote with 2 coauthors over 20 years ago. Looking at it it seems that the Shearer result was not yet published when our book was finished-it does not seem to be mentioned in the appendix I which is the only place I looked in the book when looking for the answer to the question. Apologies, especially to Pierre if he reads this, Vaughan Jones –  Vaughan Jones May 9 '10 at 15:52
The set of limit points was vaguely reminiscent of the set of possible Jones indices, which is why I looked. I found the proposition using the listing of $(2+5^{1/2})^{1/2}$ in the index. I then hesitated to mention it, not wanting the comment to be seen as criticism--but I had to. We haven't met, but I imagined it would be worth a laugh. No need to apologize. –  Jonas Meyer May 9 '10 at 17:34
@Everybody- No worries. It's a good thing that this tidbit is sitting somewhere on the internet, waiting to be found by google (try googling some combination of title key words, you'll see this page is quite high). I, at least, am of the opinion that anything that gets more good content on the site is a good thing. It's also good to see you here, Vaughan. –  Ben Webster May 9 '10 at 18:15


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# LHCb Theses

Последно добавени:
2021-05-06
16:03
 Search for pentaquark candidates in $B^0_{(s)}\to J/\psi p \bar{p}$ decays at LHCb / Spadaro Norella, Elisabetta This thesis presents the analysis of the $B^0_{(s)}\to J/\psi p \bar{p}$ decays, which have been performed on the data collected by the LHCb experiment at CERN, during Run~1 and Run~2 [...] CERN-THESIS-2021-040 - 199 p.

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2021-05-04
17:09
 A Selection Framework for LHCb’s Upgrade Trigger / Nolte, Niklas The LHCb experiment at the Large Hadron Collider at CERN is planning a major detector upgrade for the next data taking period [...] CERN-THESIS-2020-331 - 139 p.

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2021-04-26
13:36
 Searching for confirmation of charm CP violation in $K^0_S$ final states at LHCb / Tuci, Giulia This thesis reports a measurement of the CP asymmetry in $D^0 \to K^0_S K^0_S$ decays, based on $pp$ data collected by LHCb at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 5.7 fb${^-1}$ [...] CERN-THESIS-2020-325 - 253 p.

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2021-04-22
16:16
 Search for CP violation in the charmless decay $B^0 \to p \bar p K^+ \pi^-$ using triple product asymmetries at LHCb and feasibility studies of a SiPM-based readout system for the Upgrade II RICH detector / Bartolini, Matteo The subject of CP symmetry and its violation is often referred to as one of the least understood in particle physics [...] CERN-THESIS-2021-034 - 227 p.

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2021-04-21
11:50
 Radiation hardness of the upgraded LHCb muon detector electronics and prospects for a full angular analysis in multi-body rare charm decays / Brundu, Davide In this dissertation the results of the studies on the new readout electronics for the muon system of the LHCb experiment, and the perspectives for carrying out a full angular analysis in the rare decays $D^{0}\to \pi^+ \pi^- \mu^+ \mu^-$ and $D^{0}\to K^+ K^- \mu^+ \mu^-$ will be described [...] CERN-THESIS-2020-324 - 240 p.

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2021-04-12
14:52
 The LHCb GPU high level trigger and measurements of neutral pion and photon production with the LHCb detector / Boettcher, Thomas In Run 3 of the LHC, the LHCb experiment will collect data without a hardware trigger [...] CERN-THESIS-2021-029 - 199 p.

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2021-04-09
11:22
 An Angular Analysis of $B^+ \to K^+e^+e^-$ Decays at the LHCb Experiment / Maddrell-Mander, Samuel William Recent anomalies in the $b\to s\ell\ell$ sector point towards possible Lepton Flavour Universality (LFU) violating effects coupling to (axial-)vector currents [...] CERN-THESIS-2020-317 - 265 p.

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2021-04-06
12:26
 Searching for rare charm decays, performing alignment studies and improving the analysis ecosystem in HEP / Burr, Chris A search for rare and forbidden decays of the form D+ -> h+(-) l+ l-(+) has been performed, where h is a charged pion or kaon and l is an electron or muon, using 1.5/fb of data that was collected by the LHCb detector during 2016 [...] CERN-THESIS-2019-415 - 211 p.

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2021-04-06
11:55
 Precision measurements of indirect $CP$ violation in the charm sector with LHCb / Maguire, Kevin Scott A study of indirect $CP$ violation in $D$$^{0} mesons is presented through a measurement of the asymmetry of the effective decay widths of D$$^{0}$ and $\bar{D}$$^{0}$ mesons [...] CERN-THESIS-2018-486 - 232 p.

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2021-04-06
09:36
 Study of Beauty Baryons Decaying to $D^0pK$ and the Amplitudes of $\Lambda^0_b \to D^0pK$ in the Dalitz Plane with the LHCb Experiment / Viemann, Harald CERN-THESIS-2020-314 - 271 p.

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### Solutions to Problems to Work for Understanding

What is the solution of the stochastic differential equation
dY(t) = \mu Y(t) dt + \sigma Y(t) dX
This problem is adapted from Financial Calculus: An introduction to derivative pricing by M Baxter, and A. Rennie, Cambridge University Press, 1996, page 61. We could match both the drift and volaility terms for this SDE to the known SDE for the process \exp(\rho X(t) + \nu t) if and only if we take \sigma = \rho and \vu = \mu - (1/2)\sigma^2. That is, we guess the solution of the SDE to be
Y(t) = \exp( \sigma X(t) + (\mu - (1/2) \sigma^2)t)
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# Tone mapping

Last updated

Tone mapping is a technique used in image processing and computer graphics to map one set of colors to another to approximate the appearance of high-dynamic-range images in a medium that has a more limited dynamic range. Print-outs, CRT or LCD monitors, and projectors all have a limited dynamic range that is inadequate to reproduce the full range of light intensities present in natural scenes. Tone mapping addresses the problem of strong contrast reduction from the scene radiance to the displayable range while preserving the image details and color appearance important to appreciate the original scene content.

## Background

The introduction of film-based photography created issues since capturing the enormous dynamic range of lighting from the real world on a chemically limited negative was very difficult. Early film developers attempted to remedy this issue by designing the film stocks and the print development systems that gave a desired S-shaped tone curve with slightly enhanced contrast (about 15%) in the middle range and gradually compressed highlights and shadows . The advent of the Zone System, which bases exposure on the desired shadow tones along with varying the length of time spent in the chemical developer (thus controlling highlight tones) extended the tonal range of black and white (and later, color) negative film from its native range of about seven stops to about ten. Photographers have also used dodging and burning to overcome the limitations of the print process .

Computer graphic techniques capable of rendering high-contrast scenes shifted the focus from color to luminance as the main limiting factor of display devices. Several tone mapping operators were developed to map high dynamic range (HDR) images to standard displays. More recently, this work has branched away from utilizing luminance to extend image contrast and towards other methods such as user-assisted image reproduction. Currently, image reproduction has shifted towards display-driven solutions since displays now possess advanced image processing algorithms that help adapt rendering of the image to viewing conditions, save power, up-scale color gamut and dynamic range.

## Purpose and methods

The goals of tone mapping can be differently stated depending on the particular application. In some cases producing just aesthetically pleasing images is the main goal, while other applications might emphasize reproducing as many image details as possible, or maximizing the image contrast. The goal in realistic rendering applications might be to obtain a perceptual match between a real scene and a displayed image even though the display device is not able to reproduce the full range of luminance values.

A simple example of global tone mapping filter is ${\displaystyle V_{\text{out}}={\frac {V_{\text{in}}}{V_{\text{in}}+1}},}$ where Vin is the luminance of the original pixel and Vout is the luminance of the filtered pixel. [2] This function will map the luminance Vin in the domain ${\displaystyle [0,\infty )}$ to a displayable output range of ${\displaystyle [0,1).}$ While this filter provides a decent contrast for parts of the image with low luminance (particularly when Vin < 1), parts of the image with higher luminance will get increasingly lower contrast as the luminance of the filtered image goes to 1.

A perhaps more useful global tone mapping method is gamma compression, which has the filter ${\displaystyle V_{\text{out}}=A\,V_{\text{in}}^{\gamma },}$ where A > 0 and 0 < γ < 1. This function will map the luminance Vin in the domain ${\displaystyle [0,A^{-1/\gamma }]}$ to the output range ${\displaystyle [0,1].}$γ regulates the contrast of the image; a lower value for lower contrast. While a lower constant γ gives a lower contrast and perhaps also a duller image, it increases the exposure of underexposed parts of the image while at the same time, if A < 1, it can decrease the exposure of overexposed parts of the image enough to prevent them from being overexposed.

An even more sophisticated group of tone mapping algorithms is based on contrast or gradient domain methods, which are 'local'. Such operators concentrate on preserving contrast between neighboring regions rather than absolute value, an approach motivated by the fact that the human perception is most sensitive to contrast in images rather than absolute intensities. Those tone mapping methods usually produce very sharp images, which preserve very well small contrast details; however, this is often done at the cost of flattening an overall image contrast, and may as a side effect produce halo-like glows around dark objects. Examples of such tone mapping methods include: gradient domain high dynamic range compression and A Perceptual Framework for Contrast Processing of High Dynamic Range Images (a tone mapping is one of the applications of this framework).

One simple form of tone mapping takes a standard image (not HDR – the dynamic range already compressed) and applies unsharp masking with a large radius, which increases local contrast rather than sharpening. See unsharp masking: local contrast enhancement for details.

## Digital photography

Forms of tone mapping long precede digital photography. The manipulation of film and development process to render high contrast scenes, especially those shot in bright sunlight, on printing paper with a relatively low dynamic range, is effectively a form of tone mapping, although it is not usually called that. Local adjustment of tonality in film processing is primarily done via dodging and burning, and is particularly advocated by and associated with Ansel Adams, as described in his book The Print; see also his Zone System.

The normal process of exposure compensation, brightening shadows and altering contrast applied globally to digital images as part of a professional or serious amateur workflow is also a form of tone mapping.

However, HDR tone mapping, usually using local operators, has become increasingly popular amongst digital photographers as a post-processing technique, where several exposures at different shutter speeds are combined to produce an HDR image and a tone mapping operator is then applied to the result. There are now many examples of locally tone mapped digital images, inaccurately known as "HDR photographs", on the internet, and these are of varying quality. This popularity is partly driven by the distinctive appearance of locally tone mapped images, which many people find attractive, and partly by a desire to capture high-contrast scenes that are hard or impossible to photograph in a single exposure, and may not render attractively even when they can be captured. Although digital sensors actually capture a higher dynamic range than film, they completely lose detail in extreme highlights, clipping them to pure white, producing an unattractive result when compared with negative film, which tends to retain color and some detail in highlights.

In some cases local tone mapping is used even though the dynamic range of the source image could be captured on the target media, either to produce the distinctive appearance of a locally tone mapped image, or to produce an image closer to the photographer's artistic vision of the scene by removing sharp contrasts, which often look unattractive. In some cases, tone mapped images are produced from a single exposure which is then manipulated with conventional processing tools to produce the inputs to the HDR image generation process. This avoids the artifacts that can appear when different exposures are combined, due to moving objects in the scene or camera shake. However, when tone mapping is applied to a single exposure in this way, the intermediate image has only normal dynamic range, and the amount of shadow or highlight detail that can be rendered is only that which was captured in the original exposure.

## Display devices

One of the original goals of tone mapping was to be able to reproduce a given scene or image onto a display device such that the brightness sensation of the image to a human viewer closely matches the real-world brightness sensation. However, a perfect match for this problem is never possible and thus the output image on a display is often built from a tradeoff between different image features. Choosing between features is often based on the necessary application, and given appropriate metrics for the application, one possible solution is to treat the issue as an optimization problem .

For this method, models for the Human Visual System (HVS) and the display are first generated, along with a simple tone mapping operator. The contrast distortions are weighted according to their individual visibilities approximated by the HVS. With these models, an objective function that defines the tone curve can be created and solved using a fast quadratic solver.

With the addition of filters, this method can also be extended to videos. The filters ensure that the rapid changing of the tone-curve between frames are not salient in the final output image.

## Example of the imaging process

The images on the right show the interior of a church, a scene which has a variation in radiance much larger than that which can be displayed on a monitor or recorded by a conventional camera. The six individual exposures from the camera show the radiance of the scene in some range transformed to the range of brightnesses that can be displayed on a monitor. The range of radiances recorded in each photo is limited, so not all details can be displayed at once: for example, details of the dark church interior cannot be displayed at the same time as those of the bright stained-glass window. An algorithm is applied to the six images to recreate the high dynamic range radiance map of the original scene (a high dynamic range image). Alternatively, some higher-end consumer and specialist scientific digital cameras are able to record a high dynamic range image directly, for example with RAW images.

In the ideal case, a camera might measure luminance directly and store this in the HDR image; however, most high dynamic range images produced by cameras today are not calibrated or even proportional to luminance, due to practical reasons such as cost and time required to measure accurate luminance values it is often sufficient for artists to use multiple exposures to gain an "HDR image" which grossly approximates the true luminance signal.

The high dynamic range image is passed to a tone mapping operator, in this case a local operator, which transforms the image into a low dynamic range image suitable for viewing on a monitor. Relative to the church interior, the stained-glass window is displayed at a much lower brightness than a linear mapping between scene radiance and pixel intensity would produce. However, this inaccuracy is perceptually less important than the image detail, which can now be shown in both the window and the church interior simultaneously.

Local tone mapping technique of HDR image processing often produces a number of characteristic effects in images such as bright halos around dark objects, dark halos around bright objects, and sometimes a "cartoon-like" appearance due to extremely vivid colors and lack of large-scale color variations. These results are caused by application of geometric space distortion of captured image along with color space distortion, while only color space distortions are a tone mapping effect, and all other distortions are rather a custom filtering technique than any tone or color space mapping. So, the results of local tone mapping are often judged as perverting the nature of a documentalist photographic image and far from photographic realism.

Not all tone mapped images are visually distinctive. Reducing dynamic range with tone mapping is often useful in bright sunlit scenes, where the difference in intensity between direct illumination and shadow is great. In these cases the global contrast of the scene is reduced, but the local contrast maintained, while the image as a whole continues to look natural. Use of tone mapping in this context may not be apparent from the final image:

Tone mapping can also produce distinctive visual effects in the final image, such as the visible halo around the tower in the Cornell Law School image below. It can be used to produce these effects even when the dynamic range of the original image is not particularly high. Halos in images come about because the local tone mapping operator will brighten areas around dark objects, to maintain the local contrast in the original image, which fools the human visual system into perceiving the dark objects as being dark, even if their actual luminance is the same as that of areas of the image perceived as being bright. Usually this effect is subtle, but if the contrasts in the original image are extreme, or the photographer deliberately sets the luminance gradient to be very steep, the halos become visible.

## Related Research Articles

Gamma correction or gamma is a nonlinear operation used to encode and decode luminance or tristimulus values in video or still image systems. Gamma correction is, in the simplest cases, defined by the following power-law expression:

In photography and videography, High Dynamic Range (HDR) used to reproduce a greater range of luminosity than what is possible with photographic techniques. Standard techniques allow differentiation only within a certain range of brightness. Outside this range, no features are visible because in the brighter areas everything appears pure white, and pure black in the darker areas. The ratio between the maximum and the minimum of the tonal value in an image is known as the dynamic range. HDRI is useful for recording many real-world scenes containing very bright, direct sunlight to extreme shade, or very faint nebulae. High-dynamic-range (HDR) images are often created by capturing and then combining several different, narrower range, exposures of the same subject matter.

The Zone System is a photographic technique for determining optimal film exposure and development, formulated by Ansel Adams and Fred Archer. Adams described the Zone System as "[...] not an invention of mine; it is a codification of the principles of sensitometry, worked out by Fred Archer and myself at the Art Center School in Los Angeles, around 1939–40."

High-dynamic-range rendering, also known as high-dynamic-range lighting, is the rendering of computer graphics scenes by using lighting calculations done in high dynamic range (HDR). This allows preservation of details that may be lost due to limiting contrast ratios. Video games and computer-generated movies and special effects benefit from this as it creates more realistic scenes than with the more simplistic lighting models used.

Shadow and highlight enhancement refers to an image processing technique used to correct exposure.

High dynamic range (HDR) is a dynamic range higher than usual. The term is often used in discussing display devices, photography, 3D rendering, and sound recording including digital imaging and digital audio production. The term may apply to an analog or digitized signal, or to the means of recording, processing, and reproducing such signals.

In the theory of photography, tone reproduction is the mapping of scene luminance and color to print reflectance or display luminance, with the aim of subjectively "properly" reproducing brightness and "brightness differences".

The term post-processing is used in the video/film business for quality-improvement image processing methods used in video playback devices,, and video players software and transcoding software. It is also commonly used in real-time 3D rendering to add additional effects.

Image quality can refer to the level of accuracy in which different imaging systems capture, process, store, compress, transmit and display the signals that form an image. Another definition refers to image quality as "the weighted combination of all of the visually significant attributes of an image". The difference between the two definitions is that one focuses on the characteristics of signal processing in different imaging systems and the latter on the perceptual assessments that make an image pleasant for human viewers.

HDR PhotoStudio is a discontinued high dynamic range (HDR) graphics application developed by Unified Color for the Windows and macOS operating systems. In addition to being a HDR-merge application, HDR PhotoStudio offered a set of image editing operations that worked in its dynamic range, human color range (gamut), and in high precision. It also had a Color Integrity feature that enabled preserving an image's color tone during image editing operations — for example changing an image's contrast would not change its chromatic data. This problem is usually referred to as "color shift".

In image processing, computer graphics, and photography, exposure fusion is a technique for blending multiple exposures of the same scene (bracketing) into a single image. As in high dynamic range imaging, the goal is to capture a scene with a higher dynamic range than the camera is capable of capturing with a single exposure.

Image editing encompasses the processes of altering images, whether they are digital photographs, traditional photo-chemical photographs, or illustrations. Traditional analog image editing is known as photo retouching, using tools such as an airbrush to modify photographs or editing illustrations with any traditional art medium. Graphic software programs, which can be broadly grouped into vector graphics editors, raster graphics editors, and 3D modelers, are the primary tools with which a user may manipulate, enhance, and transform images. Many image editing programs are also used to render or create computer art from scratch.

Luminance HDR, formerly Qtpfsgui, is graphics software used for the creation and manipulation of high-dynamic-range images. Released under the terms of the GPL, it is available for Linux, Windows and Mac OS X. Luminance HDR supports several High Dynamic Range (HDR) as well as Low Dynamic Range (LDR) file formats.

A color appearance model (CAM) is a mathematical model that seeks to describe the perceptual aspects of human color vision, i.e. viewing conditions under which the appearance of a color does not tally with the corresponding physical measurement of the stimulus source.

Aurora HDR is photographic software developed by Macphun Software for Mac OS X and Windows. MacPhun changed its name to Skylum in early 2018.

EasyHDR is a High Dynamic Range (HDR) image processing software that merges differently exposed photographs into HDR radiance map and tone maps them. With a single photograph, easyHDR can be used to increase contrast and refresh colors.

JPEG XT is an image compression standard which specifies backward-compatible extensions of the base JPEG standard.

HDR10+ is a High Dynamic Range (HDR) video technology that adds dynamic metadata to HDR10 source files. HDR10+ signals the dynamic range and scene characteristics on a scene-by-scene or even frame-by-frame basis. The display device then uses the dynamic metadata to apply an appropriate tone map through the process of dynamic tone mapping. Dynamic tone mapping differs from static tone mapping by applying a different tone curve from scene-to-scene rather than use a single tone curve for an entire video. HDR10+ is the default variant of dynamic metadata as part of the HDMI 2.1 standard.

Color coding technology for data visualization

## References

1. ^ Livingstone, M. 2002. "Vision and Art: The Biology of Seeing." Harry N Abrams
2. ^ Hunt, R. 2004. "The Reproduction of Colour in Photography, Printing and Television: 6th Edition." John Wiley & Sons.
3. ^ Adams, A. 1981. "The Print, The Ansel Adams Photography Series 3." New York Graphic Society
4. ^ Land, E. H., and McCann, J. J. 1971. "Lightness and the retinex theory." Journal of the Optical Society of America 61, 1, 1–11.
5. ^ Kate Devlin, Alan Chalmers, Alexander Wilkie, Werner Purgathofer. "STAR Report on Tone Reproduction and Physically Based Spectral Rendering" in Eurographics 2002. DOI: 10.1145/1073204.1073242
6. ^ Raanan Fattal, Dani Lischinski, Michael Werman. "Gradient Domain High Dynamic Range Compression"
7. ^ Rafal Mantiuk, Karol Myszkowski, Hans-Peter Seidel. "A Perceptual Framework for Contrast Processing of High Dynamic Range Images"
8. ^ Alan Gilchrist. "An Anchoring Theory of Lightness Perception".
9. ^ Grzegorz Krawczyk, Karol Myszkowski, Hans-Peter Seidel. "Lightness Perception in Tone Reproduction for High Dynamic Range Images"
10. ^ Fairchild, M. D., Johnson, G.M.: ‘The iCAM framework for image appearance, differences and quality’. J Electron. Imaging, 2004
11. ^ Xiao, J., Li, W., Liu, G., Shaw, S., & Zhang, Y. (n.d.). Hierarchical tone mapping based on image color appearance model.
12. ^ Mantiuk, R., Daly, S., & Kerofsky, L. (n.d.). Display Adaptive Tone Mapping. http://resources.mpi-inf.mpg.de/hdr/datmo/mantiuk08datm.pdf
13. ^ https://web.archive.org/web/20150206044300/http://docs.opencv.org/trunk/doc/tutorials/photo/hdr_imaging/hdr_imaging.html
14. ^ Durand and Julie Dorsey, “Fast Bilateral Filtering for the Display of High-Dynamic-Range Images”. ACM Transactions on Graphics, 2002, 21, 3, 257 - 266. https://people.csail.mit.edu/fredo/PUBLI/Siggraph2002/DurandBilateral.pdf
1. G. Qiu et al, "Tone Mapping for HDR Image using Optimization-A New Closed Form Solution", Proc. ICPR 2006, 18th International Conference on Pattern Recognition, vol.1, pp.996-999
2. Reinhard, Erik (2002). "Photographic tone reproduction for digital images" (PDF). ACM Transactions on Graphics. 21 (3): 267–276. doi:10.1145/566654.566575.


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# Chapter 6 - Normal Probability Distributions - 6-7 Normal as Approximation to Binomial - Basic Skills and Concepts: 22

0.0107, the evidence is strong.

#### Work Step by Step

p=0.2 $q=1-p=1-0.2=0.8$ $n⋅p=50⋅0.2=10≥5.$ $n⋅q=50⋅0.8=40≥5.$ Hence, the requirements are satisfied. mean: $\mu=n\cdotp=50\cdot0.2=10.$ standard deviation: $\sigma=\sqrt{n\cdot p\cdot q}=\sqrt{50\cdot0.2\cdot0.8}=2.83.$ 3.5 is the first value more than 3, hence: $z=\frac{value-mean}{standard \ deviation}=\frac{3.5-10}{2.83}=-2.3.$ By using the table, the probability belonging to z=-2.3: 0.0107, hence the probability: 0.0107. This probability is really close to 0, hence the evidence is strong.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
We're sorry Aspose doesn't work properply without JavaScript enabled.

Resetting Numbering List in Word Document

Hi Team,

Automatically, resetting a numbering List and starts as “1” in Portrait Page when previous page as Landscape Orientation and it happened while executing “mailmerge.execute()” method to generate a document.

But, i don’t want to reset a numbering list when page orientation changes from landscape to portrait as well as vice-versa

I have attached Sample Test Package. Please have a look and provide a solution for this issue.
SampleTest_Package.zip (35.2 KB)

Document doc = new Document("C:\\Temp\\in.docx");


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Most theories and methods depend on or derive from mathematics and probability, so it is necessary to review probability before getting started on studying machine learning.

2 Review of Probability(just an outline)

2.1 Probability

Some Probability Formulas:

(1)Sum rule: $Pr[A\cup B]=Pr[A]+Pr[B]-Pr[A\cap B]$

(2)Union bound: $Pr[\cup A_i]=\sum\limits_{i=1}^nPr[A_i]$

(3)Conditional probability:

Fortunately, several days ago I received the decision letter from the Graduate School of Virginia Tech, and I was accepted by the department of Industrial and Systems Engineering(PhD program). Exciting!

Also, I definitely appreciate all the people who have ever provided me support, assistance and encouragement. Many thanks. I will cherish this precious opportunity and try my best to become one of the best students. Keep on fighting!

Thanks to my university policy, this semester I begin to take a graduate course an introduction to machine learning(ML), although I am still an undergraduate student. Before this, though I have ever applied machine learning methods like SVM to solve some real world problems  in some contest and research, this course  provides me a good opportunity to study statistical learning systematically. Thus, I believe it is worthy to make notes for this class.

1 Overview

1.1 Concepts

This afternoon I found a extremely interesting and popular game called 2048 in GitHub. The link of game is http://gabrielecirulli.github.io/2048/. To be honest, this game is not very difficult, but initially, it is very easy to make a mistake which probably will result in "game over". Anyway,  it is worthy to have a try. Additionally, here is a possible solution of this game and it is also funny. Good night.  :-D

Continue to discuss this topic about multicollinearity in regression. Firstly, it is necessary introduce how to calculate the VIF and condition number via software such as R. Of course it is really easy for us. The vif() in car and kappa() can be applied to calculate the VIF and condition number, respectively. Consider the data from the last article of this series for example

> #vif
> vif(lm(GNP~.,data=longley));
GNP.deflator   Unemployed Armed.Forces   Population         Year
81.946226    35.924858     9.406108   171.158675  1017.609561
Employed
196.247880
> #condition number
> kappa(longley[,-1]);
[1] 8521.126

From the output, it is clear that both of VIF and condition number are extremely large which means the data exist extremely multicollinearity.

2 Lasso and Least Angle Regression