text:
# Uniform convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^n}{ne^{nx}}$

Prove that

$$f(x) = \sum\limits_{n=1}^\infty \frac{(-1)^n}{ne^{nx}} = \sum\limits_{n=1}^\infty \frac{(-e^{-x})^n}{n}$$

is uniform convergent for $x \in [0,\infty)$.

Attempt:

At first, this looked a lot like the alternating series. At $x=0$, it is! And for $x > 0$, $f (x)$ appears to satisfy $|f(x)| < \frac{1}{n}$, but I haven't been able to strengthen this assumption in any way. As the sum of $\frac{1}{n}$ diverges, this isn't useful to apply the M-test for uniform convergence.

Next, I've simply tried to find a way to prove pointwise convergence to $0$ for $f(x)$, but haven't been successful. If it could be shown that $f(x)$ has a pointwise limit function over an arbitrary closed interval $[a,b], 0 \le a$, then Dini's Theorem gives uniform convergence. But, I haven't been able to reach that conclusion.

• What's going on with those $n$s in the definition of $f_n$? Which indices are of the sum and which are of the function? – Julien Clancy Dec 11 '13 at 23:33
• What does Dini's theorem have to do with this? Is there a monotonic sequence somewhere? – Igor Rivin Dec 11 '13 at 23:37
• Maybe you can try to use the ratio test. Similar question: link – LorenMt Dec 12 '13 at 0:00
• Igor, examining the summand on even and odd indicies gives a decreasing and increasing sequence respectively. – user110669 Dec 12 '13 at 0:29

The series converges uniformly on $[1,\infty)$ by the Weierstrass M-Test (thanks to the exponential term). To prove it converges uniformly on $[0,1]$, use properties of convergent alternating series. For any convergent alternating series of reals with terms nonincreasing in absolute value, the absolute value of the difference between the $n$th partial sum $s_n$ and the sum $s$ is less than or equal to the absolute value of the $n+1$st term.
By the way, the series is an alternating series for all $x \geq 0$.
The proof above for $0\leq x \leq 1$ looks like it works for all $x \geq 0$, but I'm going to stick with what I wrote above.
The $n$th partial sum can be written explicitly for every $x>0.$ (think $\sum_{k=1}^n a^k/k$). Should be easy from then on.
$|\sum\limits_{n=1}^\infty \frac{(-1)^n}{ne^{nx}}-\sum\limits_{n=1}^N\frac{(-1)^n}{ne^{nx}}|<\frac{1}{(N+1)e^{(N+1)x}}$ that approaches $0$


text:
# Design¶

Summary of our design decisions and some pointers to the literature.

## SymEngine is used in different languages¶

The C++ SymEngine library doesn’t care about SymPy objects at all. We are just trying to implement things in some maintainable way, currently we settled on using Basic, Mul, Pow, … hierarchy and implement most functionality using the visitor pattern or single dispatch, so Basic doesn’t need many methods. We are keeping an option to perhaps do things differently if they turn out to be faster. Either way though, this shouldn’t matter at all for how it is actually used from Python, Ruby or Julia.

Let’s talk about just Python: the wrappers are in the symengine.py project. They are implemented using Cython, and they are free to introduce any kind of classes (including SymPy’s Expr or Sage’s Expression if needed), and the point of the wrappers is to make sure that things work out of the box from SymPy and Sage. The only job of the C++ SymEngine library is to ensure that the library’s C++ API is implemented in such a way so that the wrappers can be written to do what they need. For example, we could easily introduce SymPy’s Expr into the wrappers, by simply introducing the Expr class and make all the other classes subclass from it instead of from Basic.

That was the reason we split the wrappers, so now in the (pure) C++ symengine/symengine repository, we only have to worry about speed, correctness, maintainability and a usable API, and we can concentrate on these things without worrying or even testing any kind of wrappers. In the wrappers (symengine/symengine.py, or .jl, .rb), we simply just use the C++ (or C) API and the only thing we care is so that the (Python) wrapper can be used from sympy/Sage (and we test that in the test suite), and that it doesn’t introduce unnecessary overhead in terms of speed. Ruby or Julia (or R) wrappers then care about interoperability with other libraries in those languages.

## Reference Counted Pointers¶

### Teuchos RCP¶

Memory management is handled by RCP (reference counted pointers) from Trilinos (the module Teuchos). We have copied the relevant files into src/teuchos, so no external dependencies need to be obtained. Brief code snippets of the most frequent operations are given in our C++ Style Guide, and should be consulted if you are unsure about the syntax. In order to understand how these pointers work under the hood, read the Teuchos::RCP Beginner’s Guide (pdf). Finally, a more thorough exposition is given in Teuchos C++ Memory Management Classes, Idioms, and Related Topics — The Complete Reference (pdf).

Teuchos’ RCP implements reference counting of objects, in the exact way Python does. When an object runs out of scope, its reference count is decreased. When it is copied, its reference count is increased. When the reference count goes to zero, it is deallocated. This all happens automatically, so; as long as our C++ Style Guide is followed, things just work.

When CMAKE_BUILD_TYPE=Debug is set during the CMake build (which is the default type), then Teuchos is compiled with debugging support, which means that as long as the C++ Style Guide is followed; the C++ code should never segfault (since no access is provided to raw pointers that could segfault and Teuchos raises a nice exception with full stack trace instead of raising a segfault if there is any problem; this too, is reminiscent of very similar to Python). Use this mode when developing.

When the production build is used; by setting CMAKE_BUILD_TYPE=Release, then Teuchos is compiled without debugging support, which means that all pointer operations become either as fast as raw pointers, or very close. As such, there is pretty much zero overhead. However, in this mode, the program can segfault if you access memory incorrectly. This segfault however would be prevented under CMAKE_BUILD_TYPE=Debug, so always use the Debug build to test your code, only when all tests pass should you enable Release mode.

The Trilinos RCP pointers as described above are only used when the WITH_SYMENGINE_RCP=OFF is set in CMake. The default option is WITH_SYMENGINE_RCP=ON, which uses our own implementation of RCP (see src/symengine_rcp.h). Our implementation is faster, but it only implements a subset of all the functionality and it requires that all our objects have a refcount_ variable. Otherwise the usage of our in-house RCP and the Teuchos::RCP are identical, and continuous integration tests run both implementations of RCP to make sure the code works with both.

### Ptr and RCP¶

The RCP has an overhead with every assignment due to the change (increase/decrease) in the refcount. You can access the inner pointer as Ptr (just call .ptr()), which has the same performance as raw pointer in Release mode, yet it is 100% safe in Debug mode (i.e. if the original RCP goes out of scope and the object deallocated; then the Ptr becomes dangling, but you get a nice exception raised in Debug mode on subsequent access attempts —- in Release mode it will segfault or have undefined behavior, just like a raw pointer). The idea is that for non-ownership access — i.e. typically when you just want to read some term in Add, we should be passing around Ptr, not RCP (which has the extra refcount increment/decrement overhead, which wasteful when we do not plan to own it). Such efficient implementations for the basic classes have already been implmented in SymEngine and can be viewed in the API documentation.

### UniquePtr¶

In terms of performance, the std::unique_ptr is similar to implementations involving raw pointers and manual calls to new and delete, so there are no reasons to use such manual new/delete calls and raw pointers; instead; one should use std::unique_ptr. One issue with std::unique_ptr is that if you get access to the raw pointer using .get(), then it will segfault if this pointer becomes dangling (i.e. there are no Debug time checks implemented in the standard library for this because it returns a raw pointer, not a Ptr object). This can however be fixed by writing a new class UniquePtr that returns a Ptr instead of a raw pointer and if you want to pass this around it is 100% Debug time checked, so it cannot segfault in Debug mode. Work on the implementation of this class has been ongoing in https://github.com/certik/trilinos/pull/1. The main bottleneck is that it needs to work with custom deallocators and be a drop-in replacement for std::unique_ptr, but it will be done in due time. The beauty of this new UniquePtr class is that together with Ptr, there will be no need to ever use manual new/delete calls and raw pointers. UniquePtr has the same performance in Release mode, yet it is 100% safe in Debug mode and allows for having your cake and eating it too. It’s great.

However, the issue is that even manual new/delete patterns (or the equivalent UniquePtr) are slow, so we would like to avoid such operations, or perform them as little as possible. The feasibility of phasing out RCP in favor of UniquePtr is still under consideration. The benefits are substantial. It would mean that for example, the Add container would deallocate its contents (i.e. instead of having a hashtable full of RCP objects, it would have a hashtable full of UniquePtr objects), and on being accessed, would return a given term, which might just be a Ptr (which is very fast, but could become dangling if Add goes out of scope — which would be checked at Debug time, so no segfault, but in Release builds it would segfault, and this error propagates for example; if one does an improper access at runtime from the Python wrappers in Release mode, it would segfault, which means then that the wrappers must be made aware of this behavior and prevented and must ensure such cases are not exposed to the user). Another simpler method is to make a copy of the object in such situations. For example, if you create some symbols and then use them in expressions, then currently we just pass around as RCPs, i.e. reference counted pointers to the original symbol. With the new approach, we would need to make a copy. Since we do not want to copy the std::string from inside Symbol, we want to store the symbols in some kind of table, and only pass a simple reference to the table (and also we need to deallocate things from the table if they are not used anymore). In other words, we just reinvented RCP again. So for Symbols, it seems that the UniquePrt wouldn’t have many benefits. There might be cases where this formulation has some benefit for classes like Add. If they internally use UniquePtr, we can do an optimization in Release mode and store the contents directly in the hashtable (i.e. with no pointers at all), and still pass around the Ptr to other code (i.e in Debug mode it would use UniquePtr, thus we would make sure that things are not dangling, and in Release mode we just pass around Ptr, with the performance of a raw pointer, to the internal array), that way we avoid new/delete. Also with this one can do custom allocator, i.e. allocate a chunk of memory for the hashtable, and just do a placement new. This has been tested by the lead developer (Óndrej) and he was surprised to find that, the performance wasn’t much different to UniquePtr (for smaller objects it was a bit faster, but for larger objects — remember they are stored by value now — it was even slower). Also the creation time for each type, RCP and UniquePtr was almost the same as well. The reason is that a simple refcount initialization is negligible in terms of time compared to the new call. What is slow is if you pass around RCP instead of Ptr, because a raw pointer (which is what Ptr is in Release mode) is much faster than a refcount increase/decrease. We should still investigate if we can get rid of new/delete using the approaches from this paragraph.

### Conclusion:¶

• We should pass around Ptr instead of RCP whenever possible, and we can do this right away.

• Use UniquePtr (after it is implemented) whenever possible instead of RCP — though most places in SymEngine seem to require RCP. We should keep this in mind, there might still be one or two places where UniquePtr is the way to go.

• Never use raw new/delete calls and never use raw pointers (use UniquePtr + Ptr, and if it is not sufficient, use the slower RCP + Ptr).

• Never pass pointers to some internal data — pass Ptr and have it checked at the Debug phase using UniquePtr in Debug mode, and use the data directly in Release mode

As an example of the last point, e.g. to give access to an internal std::map (as a pointer, so that the map is not copied), here is how to do it:

class A {
private:
#ifdef DEBUG_MODE
UniquePtr<std::map<int, int> > m;
#else
std::map<int, int> m;
#endif
public:
Ptr<std::map<int, int>> get_access() {
#ifdef DEBUG_MODE
return m.ptr();
#else
return ptrFromRef(m);
#endif
}
};

That way, in debug mode, you can catch dangling references but in the optimized mode it is optimally fast.

## Object creation and is_canonical()¶

Classes like Add, Mul, Pow are initialized through their constructor using their internal representation. Add, Mul have a coeff and dict, while Pow has a base and an exp. There are restrictions on what coeff and dict can be (for example coeff cannot be zero in Mul, and if Mul is used inside Add, then Mul’s coeff must be one, etc.). All these restrictions are checked when SYMENGINE_ASSERT is enabled inside constructors using the is_canonical() method. In this manner, you don’t have to worry about creating Add/Mul/Pow with wrong arguments, as it will be caught by the tests. In the Release mode no checks are done, so you can construct Add/Mul/Pow very quickly. The idea is that depending on the algorithm, you sometimes know that things are already canonical, so you simply pass it directly to Add/Mul/Pow and you avoid expensive type checking and canonicalization. At the same time, you need to make sure that tests are still running with SYMENGINE_ASSERT enabled, so that Add/Mul/Pow is never in an inconsistent state.

The philosophy of SymEngine is that you impose as many restrictions as possible in the is_canonical() method for each class (and only check that in Debug mode), so that inside the class you can assume all those things can be taken for granted and thus call faster algorithms (e.g. in Rational you know it’s not an integer, so you don’t need to worry about that special case, at the same time if you have an integer, you are forced to use the Integer class, thus automatically using faster algorithms for just integers). Then the idea is to use the information about the algorithm to construct arguments of the SymEngine classes in canonical form and then call the constructor without any checks.

For cases where you can’t or don’t want to bother constructing in canonical form, we provide high-level functions like add, mul, pow, rational, where you just provide arguments that are not necessarily in canonical form, and these functions will check and simplify. E.g. add(x, x) will check and simplify to Mul(2, x), e.g. you never have the instance Add(x, x). In the same spirit, rational(2, 1) will check and convert to Integer(2), e.g. you never have Rational(2, 1).

Summary: always try to construct objects directly using their constructors and all the knowledge that you have for the given algorithm, that way things will be very fast. If you want slower but simpler code, you can use the add(), mul(), pow(), rational() functions that perform general and possibly slow canonicalization first.

Ideally, we would like to be able to do:

RCP<Basic> x  = make_rcp<Symbol>("x");
RCP<Basic> y  = make_rcp<Symbol>("y");
RCP<Basic> r = (x + y) + (y + x);
std::cout << r << std::endl;

But the problem is, that the +, - and << operations must be defined on the RCP objects. However, just as you should not redefine what double + double is, you should not try to redefine operator overloading for an existing type (RCP). We can override operators for Basic objects, like so:

((*x) + (*y)) + ((*y) + (*x))

But here the problem is that the + operator only gets access to Basic, but it needs to access RCP<Basic> for memory management. In order to allow for operator overloaded types that use dynamic memory allocation, we will need to create our own “handle” classes. It is hard to write a handle class in C++ that are const-correct and clean and simple to use for most C++ developers. It can be done, but it is very hard, especially since we care about performance. In our opinion, we are better off writing such a layer in Python. An example of handle classes is [2] — it is non-const correct, but should give the ok performance.

Solution: using non-member non-friend functions is much more clear and much cleaner:

add(add(x, y), add(y, x))

The function signature of add is:

inline RCP<Basic> add(const RCP<Basic> &a, const RCP<Basic> &b);

For more complicated expressions, instead of add, we might also consider using the naming scheme proposed in [1]. Another advantage of this approach is that compiler errors are much easier to understand since it either finds our function or it does not, while when overloading operators of our templated classes (and RCP), any mistake typically results in pages of compiler errors in gcc.

The Python wrappers then just call this add function and provide natural mathematical syntax (x + y) + (y + x) at the Python level.

format: plain


text:
1
JEE Main 2021 (Online) 26th August Morning Shift
+4
-1
The magnitude of vectors $$\overrightarrow {OA}$$, $$\overrightarrow {OB}$$ and $$\overrightarrow {OC}$$ in the given figure are equal. The direction of $$\overrightarrow {OA}$$ + $$\overrightarrow {OB}$$ $$-$$ $$\overrightarrow {OC}$$ with x-axis will be :

A
$${\tan ^{ - 1}}{{(1 - \sqrt 3 - \sqrt 2 )} \over {(1 + \sqrt 3 + \sqrt 2 )}}$$
B
$${\tan ^{ - 1}}{{(\sqrt 3 - 1 + \sqrt 2 )} \over {(1 + \sqrt 3 - \sqrt 2 )}}$$
C
$${\tan ^{ - 1}}{{(\sqrt 3 - 1 + \sqrt 2 )} \over {(1 - \sqrt 3 + \sqrt 2 )}}$$
D
$${\tan ^{ - 1}}{{(1 + \sqrt 3 - \sqrt 2 )} \over {(1 - \sqrt 3 - \sqrt 2 )}}$$
2
JEE Main 2021 (Online) 27th July Morning Shift
+4
-1
Assertion A : If A, B, C, D are four points on a semi-circular are with centre at 'O' such that $$\left| {\overrightarrow {AB} } \right| = \left| {\overrightarrow {BC} } \right| = \left| {\overrightarrow {CD} } \right|$$, then $$\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} = 4\overrightarrow {AO} + \overrightarrow {OB} + \overrightarrow {OC}$$

Reason R : Polygon law of vector addition yields $$\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} + \overrightarrow {AD} = 2\overrightarrow {AO}$$

In the light of the above statements, choose the most appropriate answer from the options given below :
A
A is correct but R is not correct.
B
A is not correct but R is correct.
C
Both A and R are correct and R is the correct explanation of A.
D
Both A and R are correct but R is not the correct explanation of A.
3
JEE Main 2021 (Online) 25th July Evening Shift
+4
-1
Two vectors $$\overrightarrow X$$ and $$\overrightarrow Y$$ have equal magnitude. The magnitude of ($$\overrightarrow X$$ $$-$$ $$\overrightarrow Y$$) is n times the magnitude of ($$\overrightarrow X$$ + $$\overrightarrow Y$$). The angle between $$\overrightarrow X$$ and $$\overrightarrow Y$$ is :
A
$${\cos ^{ - 1}}\left( {{{ - {n^2} - 1} \over {{n^2} - 1}}} \right)$$
B
$${\cos ^{ - 1}}\left( {{{{n^2} - 1} \over { - {n^2} - 1}}} \right)$$
C
$${\cos ^{ - 1}}\left( {{{{n^2} + 1} \over { - {n^2} - 1}}} \right)$$
D
$${\cos ^{ - 1}}\left( {{{{n^2} + 1} \over {{n^2} - 1}}} \right)$$
4
JEE Main 2021 (Online) 25th July Morning Shift
+4
-1
Match List - I with List - II

Choose the correct answer from the options given below :
A
(a) $$\to$$ (iv), (b) $$\to$$ (i), (c) $$\to$$ (iii), (d) $$\to$$ (ii)
B
(a) $$\to$$ (iv), (b) $$\to$$ (iii), (c) $$\to$$ (i), (d) $$\to$$ (ii)
C
(a) $$\to$$ (iii), (b) $$\to$$ (ii), (c) $$\to$$ (iv), (d) $$\to$$ (i)
D
(a) $$\to$$ (i), (b) $$\to$$ (iv), (c) $$\to$$ (ii), (d) $$\to$$ (iii)
JEE Main Subjects
Physics
Mechanics
Electricity
Optics
Modern Physics
Chemistry
Physical Chemistry
Inorganic Chemistry
Organic Chemistry
Mathematics
Algebra
Trigonometry
Coordinate Geometry
Calculus
EXAM MAP
Joint Entrance Examination


text:
# Tag Info

1

When opening the bottle in space, all the air that was initially in it will flow out due to the pressure difference. The inside of the bottle will then become approximatelly vacuum, so when you open it on Earth air will flow in it again. (Unless it's not sturdy enough (for example a plastic bottle), in which case it will be compressed/crumpelt before you ...

1

The bird is hovering in the box. The only way for it to hover is to increase the pressure underneath its wings and decrease the pressure above its wings. This pressure differential times the area of the bird will balance the exact weight of the bird. The pressure differential may be thought of as a net downward impulse given to the air molecules by the ...

0

This is not true, when the box is sealed you are weighing the total mass of the box and everything in it, not the weight of the bird plus the weight of the box

0

Assuming is a black box seen from the outside, consider the following situation: After a while, the bird tries to fly higher and pushes the ceiling of the box upwards. Will the weight of the box decrease? Then, after a while, the bird dies and falls to the floor. Will the weight of the box increase? If the box is sealed, it cannot change the time averaged ...

1

Let's assume that the box you have is perfectly closed and has a fixed amount of air in it. When the bird inside starts flapping it's wings it creates disturbances in the air present inside the box. The air molecules may start dancing in complex ways and it is difficult to completely describe this motion qualitatively. Consider the whole system (box + air ...

1

The weight is the same if its on your desk. It is dependent on the medium the box is in. Weight is determined by gravity and gravity itself is simply the effect of a pull on denser vs less dense molecules. If the box is sitting in air - the air has no effect on its weight because in relation to the air the box sits in there is zero weight, the same as a ...

1

The physical principle that applies here is called the Archimedes' principle. Emilio Pisanty's and other answers explains well what is going on. This answer is solely to provide the name of the underlying theory. The Archimedes' principle states basically that for a submerged object (in your case the box once with air in it, once without) the apparent ...

2

Simple answer: When you blow harder, more surrounding air gets mixed in with the stream of air from your mouth. The faster air moves, the lower pressure it has (Bernoulli's principle). So when you blow faster, your stream of air is lower pressure than the surrounding air. Thus the surrounding air fills in the stream. The surrounding air is obviously cooler ...

1

The various answers above are, essentially, different ways of describing the same things.. Boyancy explains the difference... However you could also simply note that the box with the pushed-down lid still 'contains' the air -- it's just 'outside' the lid. The weight remains the same. If you managed to add a force-field at the top of the box, and then ...

4

The key thing you're forgetting is that the atmosphere also has a vertical pressure gradient, much like any body of water. This means that, if you take your box with the lid pressed down, (*) (*) z=h +------------+ | | | | | ...

1

The way I would think about this just for a quick answer: A balloon filled with air gradually sinks. Now if you took the same balloon and made it rigid, sucked all the air out of it but it still had the same volume, it would float straight up. So I would say that the balloon filled with air weighs more. Same would go for boxes.

3

When you push the lid of the air-filled box down you are doing two things: Reducing the volume of the box; and Reducing the mass of the box and its contents (as the air is squeezed out). The weight measured on the scale is: "weight of the box and contents" - "buoyancy due to surrounding air" The "buoyancy due to surrounding air" is the weight of a ...

1

I am missing in the answers so far the consideration that air consists of single particles: all the arguments I've read so far treat it as a continuum. Sometimes particles are mentioned, but in the next step already "pressure" which is a resulting statistic phenomenon is mentioned again. The salient point is that those particles are moving in straight ...

6

Just consider the same situation in water. A box filled with water, immersed in water will have more weight on a weighing machine than an empty box (with vacuum or air)--which might even float depending on the mass and volume of the box. The only difference in this case is the density of the medium.

6

A box filled with helium would weigh less than a box filled with air, because helium is less dense than air. A rigid box containing a vacuum would weigh even less than the same box filled with helium, because it is even less dense.

3

This has everything to do with entropy: when the temperature is higher, the benefit of having more water molecules in the air (giving rise to greater entropy) become energetically more favored. This is why water "dissolves better in air" at higher temperatures. Another way of looking at this (pure statistical thermodynamics): when water is cold, few ...

1

This phenomenon has nothing to do with the properties of the air, but the properties of the water in it. Hot air means hot water in the air. Cold air means cold water in the air. Cooling water causes it to condense. This is considering a constant volume.

5

Here's a way to think about it that avoids assuming the bouyant force equation is correct. All you need to know is that air pressure increases with depth / decreases with height. First off, before we put anything at all on our scale, there's already a large force pushing down on the scale's plate. $$F_0 = P_0 A_\mathrm{plate}$$ This force gets cancelled ...

4

The internal pressure in the top of the box full of air is lower than the internal pressure at the bottom - because of the weight of the column of air. You experience this when you go up in a plane (or even an elevator) and your ears "pop". Air has mass. The box with air weighs more.

34

The buoyant force on a body immersed in a fluid is equal to the weight of the fluid it displaces. In other words, $$F_B = \rho_{\text{fluid}} V_{\text{body}} ~g$$ The force of gravity on the body is equal to $$F_g = m_{\rm body} ~g$$ The apparent weight of this body will therefore be equal to the sum of these two forces. $$W_{\rm app} = \rho_{\rm ... 3 The weight is less with vacuum inside the box. The forces on the box are the gravity acting on the mass of the box and its contents, and the buoyant force which is equal to the weight of the inner volume of the box when filled with air. Let's say the box is one cubic meter, then the air content weighs some 1200 grams depending on temperature and local ... 28 Just my two cents to complement the other answers. The mistake in your reasoning is that: the state above (where the top touches the bottom) is equivalent to having a box like A (just a box holding a vacuum). is incorrect, it rather equivalent to a box full of air (there is air in between the walls, regardeless of the vertical position of the top). In ... 13 1) Technically while you push it down it will cause an increase in pressure (so the weight will change) but assuming the box has a hole and the air can equalise then the weight of your initial and end state will be the same. This is because the air is equalised within the box at all stages and at the end state the air that was in the box is now above it. ... 7 Let me highlight lhree points: 1) The mass of a box is the sum of the mass of the box structure, and the mass of the box contents; 2) The force of gravity downward on the box depends only on the mass; (and the local acceleration of gravity) 3) The buoyant force upward depends only on the density of the surrounding medium and the volume of the box. So, ... 1 Moving fluids with screws is an ancient engineering solution. However, while it works well with water, it is not as efficient at moving air to create decent vacuums, because as @Wolphram points out, it is quite easy for backpressure to drive air the 'wrong' way. In modern scroll pumps for vacuum systems the 'scroll' action is quite different from a simple ... 0 If I did not missinterpreted the drawing, there is a spiral path through wich the air can move towards the bottom of the screw. If that happens, the presure inside the container will not increase because the air wil not be concentrated at the top, but will be free to flow anywere along the screw. -1 Air drag does not depend alone on the surface area of the object, but its value of mass/surface area.. If you can increase the surface area of the object while keeping it's mass the same, it will influence the air drag on it. But if you increase the surface area while also increasing the mass, it will go unchanged. 1 You need to measure the sound level in decibels. This is a logarithmic scale where zero decibels corresponds to a root mean square pressure of 20 micropascals, and every 20 decibels corresponds to a tenfold increase in the pressure. Once you have measured the sound level and calculated the pressure you can use the equation for the particle velocity:$$ v = ...

0

For the same reason that when you are in a train that moves at 120 km/h, you can walk down the aisle without running at 120 km/h yourself. In your case, you are walking on a floor that is moving at 120 km/h alredy. In the case of a fly, it is flying in air that is moving at 120 km/h.

1

Since warm air resides above cooler air, when a fan is switched on, it will dissipate the warm air molecules in the room thus slightly increasing the temperature and hence the ac will blow more air out. But after a time the warm air molecules loose their kinetic energy and hence their heat. This results in slight cooling of the room. This cooling, coupled ...

0

Another possibility is that a fan would "de-stratify" the air in the room. If the temp sensor was fairly high up a fan would mix the hot and cooler layers of the air in the room making the sensor a bit cooler than it would normally be.

1

A fan moves air around. It makes people feel cooler, by causing evaporation of skin moisture (sweat). A fan's motor also gets hotter. Air moving over a thermostat would have no affect (thermostats don't sweat), but the increase in temperature of the fan's motor, would increase the air temperature slightly, causing the air conditioning to work harder. If ...

1

The answers so far are OK, but at the risk of being downvoted for humor, let me suggest reading This What-if Comic If you start with an evacuated canister, things will be different from starting with a canister full of atmosphere at STP.

0

The idea is that helium weighs less than normal air. To consider a more extreme example, imagine two situations where the milk canister is underwater. In the first situation it is filled with air; in the second, lead. In the case where the milk canister is filled with air, it may even float. It would still want to go down because of gravity, but the ...

0

The difference will be : $$\Delta M=V*(\rho_{He}-\rho_{air})$$ where $\rho$ stands for the volumetric mass density and $V$ the volume inside the canister. Therefore $V*\rho_{air}$ is the mass of air inside the canister. $$M_{displayed}=V_{inside}*\rho_{gaz}+V_{canistersteel}*\rho_{steel}-V_{tot}*\rho_{air}$$ The scale measures the difference between the ...

3

I recently saw a video of a demonstration by a Japanese researcher who came up with a method that used a pair of high-powered (presumably) infrared laser beams that, where they intersected, heated the air enough to turn it into plasma, creating a pulse of white light. It works, but it's slow, low-resolution, & requires staggering amounts of power. If you ...

12

It is possible to hit spots in the air with laser pulses from multiple directions in such a way that air molecules in that spot become ionized and emit light, see the technology discussed in this article, along with this demonstration video. And if you just want a 2D screen rather than a 3D display, then of course you can also just use lasers to project it ...

6

Not really; you need to have the laser light pass through particles in a medium. Laser light is made of photons; in order to see the laser, photons must be reflected off of a something to your eyes. You cannot otherwise "see" a photon because photons don't interact via the electromagnetic force - in other words, photons don't emit photons. To see the laser ...

1

Although this is really more an engineering question, I will give a few pointers. First - the pressure equation you give relates pressure in a liquid to depth - I think. But I can't quite make out the units you use. Air pressure is "essentially constant" over most volumes - although air pressure does change with height, the density is so low that you have ...

2

For practical purposes, we can assume that the disappearance of a 1.2 cubic cm (1.2 ml) object gives a waveform that's very similar to the sudden appearance of a 1.2 ml object, except for the sign of the resulting pressure wave. Now we do have a simple means to create that effect: setting off gunpowder will suddenly produce a lot of gas. You'd need just a ...

2

The following is not my own research, but taken from Randall Munroe' wonderful what-if "Glass Half Empty" where he describes a glass of water, bottom half filled with water, top half filled with vacuum (or: nothing). (edited to exclude other two glasses) But what if the empty half of the glass were actually empty—a vacuum? (Even a vacuum arguably isn’t ...

4

I suggest a totally different approach. But it's only a partial approach with much guesswork, too. The ear is able to perceive 20 µPa. (at 2 kHz). Of course you could calculate some pressure changes at the closing void, but these actually have nothing to do with the sound pressure at your ear drum. Let's do some energy calculations. 20 µPa at 1 cm² area at ...

5

Kudos to the question-asker for thinking about everything they read! :-) I was pleased to note that the author of the previous answer mentioned "audible" means "audible" to the human ear. Note also that "audible" also depends on the frequency a bit...generally-speaking, as humans, for high-frequency sounds we need them a little more intense if we're going ...

41

Sound intensity is measured on the dB scale, which is a logarithmic scale of pressure. The "threshold of hearing" is given by the graph below: which tells you (approximately) that 0 dB is about "as low as you go" - the "threshold of hearing". Note that sound signal drops off with distance - we will have to take that into account in what follows. If you ...

1

This only a partial answer. Your understanding of acoustics needs to be enhanced a bit: for one thing, an "audible sound" means there are frequencies our ear responds to. This means roughly 20 Hz to 20 000 Hz . Now, if we assume (I know, I know :-) ) that the parchment left a vacuum in its place of some small but specified volume, and we assume a ...

5

According to the research paper linked here: http://www.researchgate.net/profile/Weicheng_Cui/publication/222221948_An_overview_of_buckling_and_ultimate_strength_of_spherical_pressure_hull_under_external_pressure/links/53f1a2950cf26b9b7dd0da3c The pressure difference which can be held by a sphere of any particular material is a function of (t/R), where t is ...

1

An interesting question, but I think this would be absolutely impossible for the following reason. Vacuum chambers, particularly large ones, need thick strong walls to prevent air pressure making them collapse. The larger the volume the greater the problem. Pressure = force over area - so force = area times pressure - $F=PA$ For area of $10 m^2$, Force ...

12

A parachute is a device specifically designed to create viscous friction. Viscous friction generates a force that: is oriented opposite to the velocity; is proportional to (a certain power of [*]) the velocity. So the falling velocity will increase until the drag force (pointing upwards) becomes equal to the weight of the falling object (pointing ...

Top 50 recent answers are included


text:
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 21

# Chaos Related Abstracts

##### 21 Drift-Wave Turbulence in a Tokamak Edge Plasma

Abstract:

Tokamak plasma is far from having a stable background. The study of turbulent transport is an important part of the current research and advanced scenarios were devised to minimize it. To do this, we used a three-wave interaction model which allows to investigate the occurrence drift-wave turbulence driven by pressure gradients in the edge plasma of a tokamak. In order to simulate the energy redistribution among different modes, the growth/decay rates for the three waves was added. After a numerical simulation, we can determine certain aspects of the temporal dynamics exhibited by the model. Indeed for a wide range of the wave decay rate, an intermittent transition from periodic behavior to chaos is observed. Then, a control strategy of chaos was introduced with the aim of reducing or eliminating the weak turbulence. Downloads 400
##### 20 An Implementation of a Dual-Spin Spacecraft Attitude Reorientation Using Properties of Its Chaotic Motion

Authors: Anton V. Doroshin

Abstract:

This article contains a description of main ideas for the attitude reorientation of spacecraft (small dual-spin spacecraft, nanosatellites) using properties of its chaotic attitude motion under the action of internal perturbations. The considering method based on intentional initiations of chaotic modes of attitude motion with big amplitudes of the nutation oscillations, and also on the redistributions of the angular momentum between coaxial bodies of the dual-spin spacecraft (DSSC), which perform in the purpose of system’s phase space changing.

Keywords: Control, Spacecraft, Chaos, Attitude Dynamics

##### 19 Feigenbaum Universality, Chaos and Fractal Dimensions in Discrete Dynamical Systems

Authors: T. K. Dutta, K. K. Das, N. Dutta

Abstract:

The salient feature of this paper is primarily concerned with Ricker’s population model: f(x)=x e^(r(1-x/k)), where r is the control parameter and k is the carrying capacity, and some fruitful results are obtained with the following objectives: 1) Determination of bifurcation values leading to a chaotic region, 2) Development of Statistical Methods and Analysis required for the measure of Fractal dimensions, 3) Calculation of various fractal dimensions. These results also help that the invariant probability distribution on the attractor, when it exists, provides detailed information about the long-term behavior of a dynamical system. At the end, some open problems are posed for further research. Downloads 143
##### 18 Small Entrepreneurs as Creators of Chaos: Increasing Returns Requires Scaling

Authors: M. B. Neace, Xin GAo

Abstract:

Small entrepreneurs are ubiquitous. Regardless of location their success depends on several behavioral characteristics and several market conditions. In this concept paper, we extend this paradigm to include elements from the science of chaos. Our observations, research findings, literature search and intuition lead us to the proposition that all entrepreneurs seek increasing returns, as did the many small entrepreneurs we have interviewed over the years. There will be a few whose initial perturbations may create tsunami-like waves of increasing returns over time resulting in very large market consequences–the butterfly impact. When small entrepreneurs perturb the market-place and their initial efforts take root a series of phase-space transitions begin to occur. They sustain the stream of increasing returns by scaling up. Chaos theory contributes to our understanding of this phenomenon. Sustaining and nourishing increasing returns of small entrepreneurs as complex adaptive systems requires scaling. In this paper we focus on the most critical element of the small entrepreneur scaling process–the mindset of the owner-operator.

Keywords: Chaos, Scaling, entrepreneur, increasing returns

##### 17 Complexity in a Leslie-Gower Delayed Prey-Predator Model

Authors: Anuraj Singh

Abstract:

The complex dynamics is explored in a prey predator system with multiple delays. The predator dynamics is governed by Leslie-Gower scheme. The existence of periodic solutions via Hopf bifurcation with respect to delay parameters is established. To substantiate analytical findings, numerical simulations are performed. The system shows rich dynamic behavior including chaos and limit cycles.

Keywords: Stability, Chaos, Hopf Bifurcation, Time Delay

##### 16 Synchronization of Chaotic T-System via Optimal Control as an Adaptive Controller

Abstract:

In this paper we study the optimal synchronization of chaotic T-system with complete uncertain parameter. Optimal control laws and parameter estimation rules are obtained by using Hamilton-Jacobi-Bellman (HJB) technique and Lyapunov stability theorem. The derived control laws are optimal adaptive control and make the states of drive and response systems asymptotically synchronized. Numerical simulation shows the effectiveness and feasibility of the proposed method. Downloads 239
##### 15 Design of Chaos Algorithm Based Optimal PID Controller for SVC

Abstract:

SVC is one of the most significant devices in FACTS technology which is used in parallel compensation, enhancing the transient stability, limiting the low frequency oscillations and etc. designing a proper controller is effective in operation of svc. In this paper the equations that describe the proposed system have been linearized and then the optimum PID controller has been designed for svc which its optimal coefficients have been earned by chaos algorithm. Quick damping of oscillations of generator is the aim of designing of optimum PID controller for svc whether the input power of generator has been changed suddenly. The system with proposed controller has been simulated for a special disturbance and the dynamic responses of generator have been presented. The simulation results showed that a system composed with proposed controller has suitable operation in fast damping of oscillations of generator.

Keywords: Chaos, PID Controller, SVC, frequency oscillation

##### 14 A Novel Parametric Chaos-Based Switching System PCSS for Image Encryption

Abstract:

In this paper, a new low-cost image encryption technique is proposed and analyzed. The developed chaos-based key generator provides complex behavior and can change it automatically via a random-like switching rule. The designed encryption scheme is called PCSS (Parametric Chaos-based Switching System). The performances of this technique were evaluated in terms of data security and privacy. Simulation results have shown the effectiveness of this technique, and it can thereafter, ready for a hardware implementation.

Keywords: Security, Image, Chaos, Encryption

##### 13 A User Interface for Easiest Way Image Encryption with Chaos

Authors: D. López-Mancilla, J. M. Roblero-Villa

Abstract:

Since 1990, the research on chaotic dynamics has received considerable attention, particularly in light of potential applications of this phenomenon in secure communications. Data encryption using chaotic systems was reported in the 90's as a new approach for signal encoding that differs from the conventional methods that use numerical algorithms as the encryption key. The algorithms for image encryption have received a lot of attention because of the need to find security on image transmission in real time over the internet and wireless networks. Known algorithms for image encryption, like the standard of data encryption (DES), have the drawback of low level of efficiency when the image is large. The encrypting based on chaos proposes a new and efficient way to get a fast and highly secure image encryption. In this work, a user interface for image encryption and a novel and easiest way to encrypt images using chaos are presented. The main idea is to reshape any image into a n-dimensional vector and combine it with vector extracted from a chaotic system, in such a way that the vector image can be hidden within the chaotic vector. Once this is done, an array is formed with the original dimensions of the image and turns again. An analysis of the security of encryption from the images using statistical analysis is made and is used a stage of optimization for image encryption security and, at the same time, the image can be accurately recovered. The user interface uses the algorithms designed for the encryption of images, allowing you to read an image from the hard drive or another external device. The user interface, encrypt the image allowing three modes of encryption. These modes are given by three different chaotic systems that the user can choose. Once encrypted image, is possible to observe the safety analysis and save it on the hard disk. The main results of this study show that this simple method of encryption, using the optimization stage, allows an encryption security, competitive with complicated encryption methods used in other works. In addition, the user interface allows encrypting image with chaos, and to submit it through any public communication channel, including internet. Downloads 355
##### 12 Chaos Fuzzy Genetic Algorithm

Abstract:

The genetic algorithms have been very successful in handling difficult optimization problems. The fundamental problem in genetic algorithms is premature convergence. This paper, present a new fuzzy genetic algorithm based on chaotic values instead of the random values in genetic algorithm processes. In this algorithm, for initial population is used chaotic sequences and then a new sexual selection proposed for selection mechanism. In this technique, the population is divided such that the male and female would be selected in an alternate way. The layout of the male and female chromosomes in each generation is different. A female chromosome is selected by tournament selection size from the female group. Then, the male chromosome is selected, in order of preference based on the maximum Hamming distance between the male chromosome and the female chromosome or The highest fitness value of male chromosome (if more than one male chromosome is having the maximum Hamming distance existed), or Random selection. The selections of crossover and mutation operators are achieved by running the fuzzy logic controllers, the crossover and mutation probabilities are varied on the basis of the phenotype and genotype characteristics of the chromosome population. Computational experiments are conducted on the proposed techniques and the results are compared with some other operators, heuristic and local search algorithms commonly used for solving p-median problems published in the literature. Downloads 282
##### 11 Playing Light Switching Games with Langton's Turmite

Authors: Crista Arangala

Abstract:

Light switching games are both popular and well studied. This paper introduces a cellular automata called Langton’s turmite to several different light switching scenarios and discusses when Langton’s turmite can solve these games. Downloads 365
##### 10 Investigation of Complexity Dynamics in a DC Glow Discharge Magnetized Plasma Using Recurrence Quantification Analysis

Authors: Vramori Mitra, Bornali Sarma, Arun K. Sarma

Abstract:

Recurrence is a ubiquitous feature of any real dynamical system. The states in phase space trajectory of a system have an inherent tendency to return to the same state or its close state after certain time laps. Recurrence quantification analysis technique, based on this fundamental feature of a dynamical system, detects evaluation of state under variation of control parameter of the system. The paper presents the investigation of nonlinear dynamical behavior of plasma floating potential fluctuations obtained by using a Langmuir probe in different magnetic field under the variation of discharge voltages. The main measures of recurrence quantification analysis are considered as determinism, linemax and entropy. The increment of the DET and linemax variables asserts that the predictability and periodicity of the system is increasing. The variable linemax indicates that the chaoticity is being diminished with the slump of magnetic field while increase of magnetic field enhancing the chaotic behavior. Fractal property of the plasma time series estimated by DFA technique (Detrended fluctuation analysis) reflects that long-range correlation of plasma fluctuations is decreasing while fractal dimension is increasing with the enhancement of magnetic field which corroborates the RQA analysis. Downloads 189
##### 9 Control of Chaotic Behaviour in Parallel-Connected DC-DC Buck-Boost Converters

Authors: Ammar Nimer Natsheh

Abstract:

Chaos control is used to design a controller that is able to eliminate the chaotic behaviour of nonlinear dynamic systems that experience such phenomena. The paper describes the control of the bifurcation behaviour of a parallel-connected DC-DC buck-boost converter used to provide an interface between energy storage batteries and photovoltaic (PV) arrays as renewable energy sources. The paper presents a delayed feedback control scheme in a module converter comprises two identical buck-boost circuits and operates in the continuous-current conduction mode (CCM). MATLAB/SIMULINK simulation results show the effectiveness and robustness of the scheme. Downloads 260
##### 8 Chaos Cryptography in Cloud Architectures with Lower Latency

Abstract:

With the rapid evolution of the internet applications, cloud computing becomes one of today’s hottest research areas due to its ability to reduce costs associated with computing. Cloud is, therefore, increasing flexibility and scalability for computing services in the internet. Cloud computing is Internet based computing due to shared resources and information which are dynamically delivered to consumers. As cloud computing share resources via the open network, hence cloud outsourcing is vulnerable to attack. Therefore, this paper will explore data security of cloud computing by implementing chaotic cryptography. The proposal scenario develops a problem transformation technique that enables customers to secretly transform their information. This work proposes the chaotic cryptographic algorithms have been applied to enhance the security of the cloud computing accessibility. However, the proposed scenario is secure, easy and straightforward process. The chaotic encryption and digital signature systems ensure the security of the proposed scenario. Though, the choice of the key size becomes crucial to prevent a brute force attack.

Keywords: Cryptography, Security, Cloud Computing, Chaos

##### 7 Chaotic Semiflows with General Acting Topological Monoids

Authors: Alica Miller

Abstract:

A semiflow is a triple consisting of a Hausdorff topological space $X$, a commutative topological monoid $T$ and a continuous monoid action of $T$ on $X$. The acting monoid $T$ is usually either the discrete monoid $\N_0$ of nonnegative integers (in which case the semiflow can be defined as a pair $(X,f)$ consisting of a phase space $X$ and a continuous function $f:X\to X$), or the monoid $\R_+$ of nonnegative real numbers (the so-called one-parameter monoid). However, it turns out that there are real-life situations where it is useful to consider the acting monoids that are a combination of discrete and continuous monoids. That, for example, happens, when we are observing certain dynamical system at discrete moments, but after some time realize that it would be beneficial to continue our observations in real time. The acting monoid in that case would be $T=\{0, t_0, 2t_0, \dots, (n-1)t_0\} \cup [nt_0,\infty)$ with the operation and topology induced from real numbers. This partly explains the motivation for the level of generality which is pursued in our research. We introduce the PSP monoids, which include all but pathological'' monoids, and most of our statements hold for them. The topic of our presentation are some recent results about chaos-related properties in semiflows, indecomposability and sensitivity of semiflows in the described general context.

Keywords: Chaos, Sensitivity, indecomposability, PSP monoids, semiflow

##### 6 Design of a Chaotic Trajectory Generator Algorithm for Mobile Robots

Abstract:

This work addresses the problem of designing an algorithm capable of generating chaotic trajectories for mobile robots. Particularly, the chaotic behavior is induced in the linear and angular velocities of a Khepera III differential mobile robot by infusing them with the states of the H´enon chaotic map. A possible application, using the properties of chaotic systems, is patrolling a work area. In this work, numerical and experimental results are reported and analyzed. In addition, two quantitative numerical tests are applied in order to measure how chaotic the generated trajectories really are. Downloads 44
##### 5 Melnikov Analysis for the Chaos of the Nonlocal Nanobeam Resting on Fractional-Order Softening Nonlinear Viscoelastic Foundations

Abstract:

In the present study, the dynamics of nanobeam resting on fractional order softening nonlinear viscoelastic pasternack foundations is studied. The Hamilton principle is used to derive the nonlinear equation of the motion. Approximate analytical solution is obtained by applying the standard averaging method. The Melnikov method is used to investigate the chaotic behaviors of device, the critical curve separating the chaotic and non-chaotic regions are found. It is shown that appearance of chaos in the system depends strongly on the fractional order parameter.

Keywords: Chaos, fractional-order, Melnikov method, nanobeam

##### 4 Chaotic Response of Electrical Insulation System with Gaseous Dielectric under High AC and DC Voltages

Authors: Arijit Basuray

Abstract:

It is well known that if an electrical insulation system is stressed under high voltage then discharge may occur in various form and if the system is made of composite dielectric having interfaces of materials having different dielectric constant discharge may occur due to gross mismatch of dielectric constant causing intense local field in the interfaces. Here author has studied, firstly, behavior of discharges in gaseous dielectric circuit under AC and DC voltages. A gaseous dielectric circuit is made such that a pair of electrode of typical geometry is used to make the discharges occur under application of AC and DC voltages. Later on, composite insulation system with air gap is also studied. Discharge response of the dielectric circuit is measured across a typically designed impedance. The time evolution of the discharge characteristics showed some interesting chaotic behavior. Author here proposed some analysis of such behavior of the discharge pattern and discussed about the possibility of presence of such discharge circuit in lumped electric circuit. Downloads 24
##### 3 Chaotic Analysis of Acid Rains with Times Series of pH Degree, Nitrate and Sulphate Concentration on Wet Samples

Abstract:

Chaos theory is one of the new paradigms of science since the last century. After determining chaos in the weather systems by Edward Lorenz the popularity of the theory was increased. Chaos is observed in many natural systems and studies continue to defect chaos to other natural systems. Acid rain is one of the environmental problems that have negative effects on environment and acid rains values are monitored continuously. In this study, we aim that analyze the chaotic behavior of acid rains in Turkey with the chaotic defecting approaches. The data of pH degree of rain waters, concentration of sulfate and nitrate data of wet rain water samples in the rain collecting stations which are located in different regions of Turkey are provided by Turkish State Meteorology Service. Lyapunov exponents, reconstruction of the phase space, power spectrums are used in this study to determine and predict the chaotic behaviors of acid rains. As a result of the analysis it is found that acid rain time series have positive Lyapunov exponents and wide power spectrums and chaotic behavior is observed in the acid rain time series. Downloads 8
##### 2 Reconsidering Taylor’s Law with Chaotic Population Dynamical Systems

Authors: Yuzuru Mitsui, Takashi Ikegami

Abstract:

The exponents of Taylor’s law in deterministic chaotic systems are computed, and their meanings are intensively discussed. Taylor’s law is the scaling relationship between the mean and variance (in both space and time) of population abundance, and this law is known to hold in a variety of ecological time series. The exponents found in the temporal Taylor’s law are different from those of the spatial Taylor’s law. The temporal Taylor’s law is calculated on the time series from the same locations (or the same initial states) of different temporal phases. However, with the spatial Taylor’s law, the mean and variance are calculated from the same temporal phase sampled from different places. Most previous studies were done with stochastic models, but we computed the temporal and spatial Taylor’s law in deterministic systems. The temporal Taylor’s law evaluated using the same initial state, and the spatial Taylor’s law was evaluated using the ensemble average and variance. There were two main discoveries from this work. First, it is often stated that deterministic systems tend to have the value two for Taylor’s exponent. However, most of the calculated exponents here were not two. Second, we investigated the relationships between chaotic features measured by the Lyapunov exponent, the correlation dimension, and other indexes with Taylor’s exponents. No strong correlations were found; however, there is some relationship in the same model, but with different parameter values, and we will discuss the meaning of those results at the end of this paper. Downloads 6
##### 1 An Efficient Discrete Chaos in Generalized Logistic Maps with Applications in Image Encryption

Authors: Ashish Ashish

Abstract:

In the last few decades, the discrete chaos of difference equations has gained a massive attention of academicians and scholars due to its tremendous applications in each and every branch of science, such as cryptography, traffic control models, secure communications, weather forecasting, and engineering. In this article, a generalized logistic discrete map is established and discrete chaos is reported through period doubling bifurcation, period three orbit and Lyapunov exponent. It is interesting to see that the generalized logistic map exhibits superior chaos due to the presence of an extra degree of freedom of an ordered parameter. The period doubling bifurcation and Lyapunov exponent are demonstrated for some particular values of parameter and the discrete chaos is determined in the sense of Devaney's definition of chaos theoretically as well as numerically. Moreover, the study discusses an extended chaos based image encryption and decryption scheme in cryptography using this novel system. Surprisingly, a larger key space for coding and more sensitive dependence on initial conditions are examined for encryption and decryption of text messages, images and videos which secure the system strongly from external cyber attacks, coding attacks, statistic attacks and differential attacks. Downloads 1


text:
# finding the difference of perfect squares

Find the difference between the smallest perfect square larger than one million and the largest perfect square smaller than one million.

I did not want to use a calculator for this question. I read this in a mathematical teaser book and was not sure how to solve it. I tried doing this with smaller numbers such as $10.$ The smallest perfect square larger than ten would would be $16$ and the smallest perfect square next to ten would be $9.$ I found the difference of those to be $7$ Then I tried this for $100.$ I found that the smallest perfect square is $81$ and the largest perfect square to be $121.$ I found the difference of those to be $40.$

However, I cannot seem to solve the one million question. Can someone help me to solve this? I was working on it for sometime now and would like to see how to solve it.

• One million is the square of one thousand... – lhf Oct 1 '14 at 3:16

We want $(x+1)^2-(x-1)^2$, where $x=1000$.

The difference is $4x$, that is, $4000$.

• Thank you. I get it now and I see how this works. I tried this with other problems like this and was able to solve this. – col Oct 1 '14 at 3:47
• Good, I am happy that you got the general structural idea. – André Nicolas Oct 1 '14 at 3:51

$a^2-b^2=(a-b)(a+b)$

Note that $100=10^2$ and thus the $a=11,b=9$ would be the values to plug into the formula above to get $(11-9)(11+9)=2*20=40$

Note that $1,000,000=1,000^2$ and using the same trick:

$(1001-999)(1001+999) = 2*2000 = 4000$ for another way to compute the value.

• +1 Thank you for you help. The explanation helped me to solve this. – col Oct 1 '14 at 3:47


text:
# How do we know that the 2012 LHC discovery is SM Higgs Boson and not a pseudo-dilaton?

To be more clear, are there any measurements or analysis done on the new particle to verify if it is SM Higgs boson and not a pseudo-dilaton.

• Yes, hundreds: a large fraction of CMS and Atlas papers do just that. How did you get the impression they did not and are not doing it as we speak? – Cosmas Zachos Oct 16 '17 at 16:46
• @CosmasZachos Your comment is a bit mean: as pointed out in my answer, John Ellis himself deemed it worth his time to address this question. The style of the question, especially the "or something completely new", is not great, agreed, but it is healthy to ask such questions and the guy is for sure not a pro, so he can't know. – user154997 Oct 20 '17 at 10:03
• @Luc Perhaps lacking in deliberative indulgence. For half a decade since the discovery of the Higgs, sustainable BSM facts have failed to surface. Has this basic fact languished unreported in the press, sending questions about it to this site? – Cosmas Zachos Oct 20 '17 at 13:32
• @Cosmas You can't expect a recent graduate to keep upto date with half a decades research work of a recent particle discovery. Also, I find it extremely hard to find resources addressing this issue without being surrounded by a sea of Jargon. Also, stack exchange provides a platform for amateurs like me to learn from experts like you guys, in order to alleviate the steep learning curve I face right now. – Invariance Oct 20 '17 at 14:28
• Well, the canonical PDG summary is not that technical: it is designed for the broadest audience. There is no hard evidence for pseudodilatons. – Cosmas Zachos Oct 20 '17 at 14:53

Interpreting pseudo-dilaton with its usual meaning in this context, I can add some information to what John Rennie rightly stated. John Ellis wrote a very nice summary [1] which dates a bit (2013) but already back then, the pseudo-dilaton model was strongly disfavoured, as detailed in section 2.2.3, based on a previous work of his [2]. His conclusion is that the pseudo-dilaton is compatible with experimental data if and only if it is fine-tuned to mimic the Standard Model exactly, i.e. the dilaton v.e.v $V$ which breaks scale invariance has to be nearly equal to the v.e.v. $v$ of the Higgs: see the two plots on Fig. 19, showing that $v/V$ has to be peaked near 1. He concludes with humour that

Clearly, there is no evidence for any significant deviation from the Standard Model, and Peter Higgs may continue to smile!

[1] Ellis, John, Higgs Physics, arXiv:1312.5672

[2] J. Ellis and T. You, JHEP 1306 (2013) 103 arXiv:1303.3879.

With any new particle we look at the things we can measure, e.g. spin, charge and decay channels, and compare them to the properties predicted for the particle from the Standard Model.

In this case we find the measurements all match the properties the Standard Model predicts for the Higgs boson. That doesn't prove it's really the Higgs boson since there could be another particle with identical values for the properties we have measured but different values for properties we can't measure yet. But we already have enough data that would make this one hell of a coincidence, so it seems pretty unlikely.


text:
# Image Viewing¶

## Overview¶

This example illustrates the basic image viewing capabilities provided by SimpleITK.

The focus of SimpleITK is on image analysis and not display. We therefor use an ad-hoc image viewing approach, relying on an external program. By default this is the Fiji/ImageJ program.

When using this functionality, SimpleITK writes the image to disc in a temporary location and then launches the viewer. Potential issues with this approach:

1. The external viewer program is not found. Either it is not installed, or not installed in the expected location, for details see Why isn’t Fiji or ImageJ found by the Show function (RuntimeError: Exception thrown…)?)

2. Writing the image to disc fails when there isn’t enough disc space.

3. The external program cannot read the image file because it was saved in a format not supported by the program.

To control viewing you have two options:

1. Procedural approach. Use the Show function, with control primarily done via environment variable settings.

2. Object oriented approach. Use the ImageViewer class, with control via the object’s interface.

One can either set the viewing application, if you just want to change the specific viewer, or set the viewing command. The latter allows you to specify viewer specific arguments such as -z for ITK-SNAP, see below, to open in a zoomed view.

## Code¶

import sys
import SimpleITK as sitk

grid_image = sitk.GridSource(outputPixelType=sitk.sitkUInt16, size=(512, 512),
sigma=(0.1, 0.1), gridSpacing=(20.0, 20.0))

# Procedural interface, using the default image viewer (Fiji/ImageJ) or
# any viewer specified by the SITK_SHOW_COMMAND environment variable.
sitk.Show(grid_image, title="grid using Show function", debugOn=True)

# Object oriented interface:
image_viewer = sitk.ImageViewer()
image_viewer.SetTitle('grid using ImageViewer class')

# Use the default image viewer.
image_viewer.Execute(grid_image)

# Change viewer, and display again.
image_viewer.SetApplication(
'/Applications/ITK-SNAP.app/Contents/MacOS/ITK-SNAP')
image_viewer.Execute(grid_image)

# Change the viewer command, (use ITK-SNAP's -z option to open the image in
# zoomed mode)
image_viewer.SetCommand(
'/Applications/ITK-SNAP.app/Contents/MacOS/ITK-SNAP -z 2')
image_viewer.Execute(grid_image)

sys.exit(0)


text:
# Re: [PATCH 00/35] treewide trivial patches converting pr_warning to pr_warn

From: Joe Perches
Date: Thu Feb 23 2017 - 12:57:10 EST

On Thu, 2017-02-23 at 17:41 +0000, Emil Velikov wrote:
> On 23 February 2017 at 17:18, Joe Perches <joe@xxxxxxxxxxx> wrote:
> > On Thu, 2017-02-23 at 09:28 -0600, Rob Herring wrote:
> > > On Fri, Feb 17, 2017 at 1:11 AM, Joe Perches <joe@xxxxxxxxxxx> wrote:
> > > > There are ~4300 uses of pr_warn and ~250 uses of the older
> > > > pr_warning in the kernel source tree.
> > > >
> > > > Make the use of pr_warn consistent across all kernel files.
> > > >
> > > > This excludes all files in tools/ as there is a separate
> > > > define pr_warning for that directory tree and pr_warn is
> > > > not used in tools/.
> > > >
> > > > Done with 'sed s/\bpr_warning\b/pr_warn/' and some emacsing.
> >
> > []
> > > Where's the removal of pr_warning so we don't have more sneak in?
> >
> > After all of these actually get applied,
> > and maybe a cycle or two later, one would
> > get sent.
> >
>
> By which point you'll get a few reincarnation of it. So you'll have to
> do the same exercise again :-(

Maybe to one or two files. Not a big deal.

> I guess the question is - are you expecting to get the series merged
> all together/via one tree ?

No. The only person that could do that effectively is Linus.

> If not, your plan is perfectly reasonable.


text:
laziness, impatience, and hubris PerlMonks

### Does Perl have if(isset?

by beanscake (Acolyte)
 on Jan 24, 2014 at 20:13 UTC Need Help??
beanscake has asked for the wisdom of the Perl Monks concerning the following question:

Please i need a solution for this, i want to use (PERL) to process a form instead of PHP, now my only problem is converting this LINE to (PERL) SCRIPT.
<?php

if(isset($_POST['comment']))$comment  = $_POST['comment']; ?> [download] Thank you. Replies are listed 'Best First'. Re: Does Perl have if(isset? by Corion (Pope) on Jan 24, 2014 at 20:29 UTC According to the PHP documentation on isset(), the function returns true if the variable has a defined value. This would be equivalent to the defined function of Perl: $comment= param('comment')
if defined param('comment');
[download]

I think the $_POST hash in PHP is populated by HTTP POST parameters - you would have to check that too. Re: Does Perl have if(isset? by Your Mother (Chancellor) on Jan 24, 2014 at 20:21 UTC You should put your code in <code> tags. And without knowing a little Perl this is unlikely to get you far but this is roughly equivalent. I didn't use the if block because it is unperly (to me) when there is no real else. use strict; # DO NOT LEAVE OUT. use warnings; # What did I just say?! use CGI ":standard"; my$comment;
$comment = param("comment") if request_method() eq "POST"; # Note,$comment could be undefined/uninitialized here
# but this is similar to your logical block.
[download]
Re: Does Perl have if(isset?
by karlgoethebier (Parson) on Jan 24, 2014 at 21:01 UTC

I couldn't do otherwise:

Best regards, Karl

«The Crux of the Biscuit is the Apostrophe»

Create A New User
Node Status?
node history
Node Type: perlquestion [id://1071990]
Approved by Corion
Front-paged by Sinistral
help
Chatterbox?
 [Corion]: Do you want to launch a script and keep the command prompt/console window open? [Corion]: Do you want to wait for a key press before closing the window? [LanX]: I want the command line in the history [tye]: -Mouse [Corion]: Option a) would mean launching cmd.exe /k c:\path\to\ batchfile- launching-perl- script.cmd. Option b) would be to add pause as the last line of said batch file. [LanX]: First day after holidays ... and already stressed by the fact that colleagues changed stuff without communication ... apparently I'm the only one trying to fight entropy [Corion]: LanX: The command is always in the history if you typed it in before. If you didn't type the command into the command line, it will not be there. I think there is doskey which can stuff command lines into the history LanX damns the cult of CB ;-) LanX WTF WTF WTF [LanX]: please forget my last 3 posts

How do I use this? | Other CB clients
Other Users?
Others making s'mores by the fire in the courtyard of the Monastery: (13)
As of 2017-03-27 15:44 GMT
Sections?
Information?
Find Nodes?
Leftovers?
Voting Booth?
Should Pluto Get Its Planethood Back?

Results (320 votes). Check out past polls.


text:
# What does $\lim\sup {a_n}$ or $\lim\inf {a_n}$ mean?

I'm working through Rudin, and the first instance I see this notation used is Definition 3.16. The definition states:

Let $\{s_n\}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ (in the extended real number system) such that $s_{n_x}\rightarrow x$ for some subsequence $\{s_{n_k}\}$. This set $E$ contains all subsequential limits as defined in definition 3.5, plus possibly the numbers $+\infty$ and $-\infty$

We now recall Definitions 1.8 and 1.23 and put:

$$s^* = \sup E$$ $$s_* = \inf E$$.

The numbers $s^*$ are called the upper and lower limits of $\{s_n\}$; we use the notation

$$\lim_{n\to\infty}\sup s_n=s^*, \lim_{n\to\infty}\inf s_n=s_*$$

The concept of $\inf$ and $\sup$ made sense to me in chapter 1/2 in the context of sets, and the first pair of definitions, with regards to the set $E$ makes sense. Based off of those definitions, I interpret $s^*$ as the largest convergent value you can create from a subsequence of $\{s_n\}$; similarly, $s_*$ is the smallest convergent value you can create from a subsequence of $\{s_n\}$ (although those definitions might be inaccurate).

The lower notation, on the other hand, confuses me. As you're taking the limit, It seems to me that you are no longer dealing with a set, just individual set elements. In that context, I can't distiguish $\lim\sup s_n$ or $\lim\inf s_n$ from simply $s_n$. How does this pair of definitions for $s^*$ and $s_*$ make sense?

Even though $\limsup$ and $\liminf$ are written with spaces, they are single symbols.

They should be read $$\limsup_{n\to\infty} s_n = s^*$$ and not $$\lim_{n\to\infty} \Big( \sup s_n \Big ) = s^*$$ which doesn't really make sense (it's trying to take the supremum of a single number at a time, which has no defined mening).

However, if you consider "$\limsup$" to be a single symbol with a definition separate from the meanings of $\lim$ and $\sup$ by their own, then it's just notation, and what you have there is a definition of what that notation means.

• Oh that does make more sense, thanks for clarifying about the notation. Jul 11, 2018 at 23:49

The confusion might dissipate if you write it out on one line. That is, $$\limsup_{n\rightarrow\infty}s_{n} \equiv\sup\left\{ s\in\mathbb{R}\colon\text{there exists a subsequence }\{s_{n_{k}}\}\text{ of }\{s_{n}\}\text{ such that }s_{n_{k}}\rightarrow s\right\} .$$ The set that we are taking the supremum of is what Rudin calls $E$. The supremum itself is what Rudin calls $s^{*}$. $\liminf$ is defined similarly. Does that help?

An equivalent definition is given by $$\limsup_{n \to \infty} s_n \equiv \lim_{n \to \infty} \sup_{k \geq n} s_k \equiv \lim_{n \to \infty} \sup \{s_k : k \geq n\} \, .$$ The right-hand side is a limit of suprema of sets (which you know), and the first two expressions are just notation. A similar definition applies for $\liminf$.

There are many equivalent definitions, one being using least upper bonds ands greatest lower bounds. You can replace these with "sup" and "inf" if you desire, as their definitions carry over to sets:

$\lim\sup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} \mbox{lub}(\{a_n,a_{n+1},...\})$

and similarly with liminf, with glb. This is closest to your desire for a sequential-limit definition, as now it's clearly a limit of a sequence of bounds.

Also, you can readily show that if $s=\lim\sup_{n\rightarrow\infty} a_n$, then there must exist a subsequence $a_{n_k}$ that converges to $s$.

Essentially $\limsup_{n \rightarrow \infty} \{a_n\}$ always exists in the extended real even if $\lim \{a_n\}$ doesn't. In particular, $\limsup$ is the largest limit a subsequence can achieve. While $\liminf$ is the smallest limit a subsequence can achieve. In particular, the limit exists if and only if $\limsup = \liminf$.


text:
## Practice Midterm W18 by Lyndon and Michael (Q 1B)

604807557
Posts: 51
Joined: Wed Nov 16, 2016 3:02 am

### Practice Midterm W18 by Lyndon and Michael (Q 1B)

I'm not sure how to approach this problem or what formula to use. Can someone please explain how to do solve this.

"Dr. Lavelle picks up the ice cream that he has just heated up (-2.8 degree C) and accidentally drops it on the ground and can’t eat it. Tears streaming down, he watches as half of it melts away when he realizes that from the moment he dropped the ice cream until now, it has received exactly 234kJ of heat. After performing quick calculations, he realizes that the enthalpy of fusion (kJ/g) for the ice cream is?"

Evelyn L 1H
Posts: 67
Joined: Fri Sep 29, 2017 7:05 am

### Re: Practice Midterm W18 by Lyndon and Michael (Q 1B)

q=m1CdeltaT + m2* delta Hfusion

q=234kj
temp: -2.8 --> 0 delta T = 2.8
m1 = 125g
m2= .5*125 = 62.5g (bc half the ice cream melted)

234000 J = 125 * 10 (found in previous problem) * 2.8 + delta H fusion * 62.5
delta H fusion = 3688 J/g --> 3.7 kj/g

Kevin Tabibian 1A
Posts: 42
Joined: Fri Sep 29, 2017 7:06 am

### Re: Practice Midterm W18 by Lyndon and Michael (Q 1B)

consider the q=mc$\Delta$T + 0.5(m$\Delta$H)
m$\Delta$H is divided by 2 because only half of the ice cream, make sure your answer is in line with the units that the question is asking

Return to “Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)”

### Who is online

Users browsing this forum: No registered users and 2 guests


text:
# Proof of Strong Duality theorem for semidefinite programs vis Farkas Lemma

I am following Lovasz' notes on semidefinite programming and I want to understand the proof of the Strong Duality theorem using the Farkas Lemma. Lovasz only proves one part of it, and I'm trying to figure out the rest.

Let $A_1,\dots,A_m$, and $C$ be $n\times n$ symmetric matrices and let $b\in\mathbb{R}^m$. Consider the semidefinite program \begin{align*} \text{maximize} \hspace{.5cm} & \mathrm{Tr}(XC)\\ \text{subject to}\hspace{.5cm}& X\succeq 0\\ & \mathrm{Tr}(XA_i)=b_i\, \forall i \end{align*} with primal optimal value given by $v_{\text{primal}}$ and its corresponding dual problem \begin{align*} \text{minimize} \hspace{.5cm} & \langle b,y\rangle\\ \text{subject to}\hspace{.5cm}& y\in\mathbb{R}^m\\ & y_1A_1+\cdots +y_mA_m-C\succeq 0 . \end{align*} with dual optimal value given by $v_{\text{dual}}$. By weak duality, it must hold that $v_{\text{primal}}\leq v_{\text{dual}}$.

The primal problem is said to be strictly feasible if there exists a positive definite $X\succ 0$ such that $\mathrm{Tr}(XA_i)=b_i$ for all $i\in\{1,\dots,m\}$. Similarly, the dual problem is strictly feasible if there exists a $y\in\mathbb{R}^m$ such that $\sum_{i=1}^n y_iA_i-C\succ 0$.

The strong duality theorem (see Theorem 3.4 in the notes) can then be stated as follows:

Theorem (Strong Duality for SDPs). Suppose that both $v_{\text{primal}}$ and $v_{\text{dual}}$ are finite. (That is, both the primal and dual problems are feasible.)

1. If the primal problem is strictly feasible, then $v_{\text{primal}}=v_{\text{dual}}$. Furthermore, there exists a $y\in\mathbb{R}^m$ such that $y_1A_1+\cdots +y_mA_m-C\succeq 0$ and $\langle b,y\rangle = v_{\text{dual}}$.

2. If the dual problem is strictly feasible, then $v_{\text{primal}}=v_{\text{dual}}$. Furthermore, there exists a $X\succeq 0$ such that $\mathrm{Tr}(XA_i)=b_i$ for all $i$ and $\mathrm{Tr}(XC) =v_{\text{primal}}$.

Lovasz essentially only proves part 2 of this theorem, where he makes use of the semidefinite Farkas Lemma:

Lemma (Farkas, nonhomogeneous semidefinite version). The following are equivalent:

(i) There exists $X\succeq 0$ such that $\mathrm{Tr}(XA_i)=b_i$ for all $i$ and $\mathrm{Tr}(XC)\geq0$.

(ii) For all $y\in\mathbb{R}^m$, it holds that $y_1A_1+\cdots+y_mA_m-C\not\succ 0$.

Lovasz then proves part 2 of the Strong Duality theorem as follows. Note that there cannot exist a $y\in\mathbb{R}^m$ such that $y_1A_1+\cdots+y_mA_m-C\succ 0$ and $\langle b,y\rangle <v_{\text{primal}}$. Hence if we define block matrices $$A_i' = \begin{pmatrix} A_i& 0\\ 0& b_i \end{pmatrix} \qquad\text{and}\qquad C' = \begin{pmatrix} C& 0\\ 0& -v_{\text{primal}} \end{pmatrix}$$ then for all $y\in\mathbb{R}^m$ it holds that $y_1A'_1+\cdots+y_mA'_m-C'\not\succ 0$. The Farkas Lemma then allows us to find an $X\succeq0$ such that $\mathrm{Tr}(XA_i)=b_i$ for all $i$ and $\mathrm{Tr}(XC)\geq v_{\text{primal}}$. This proves the desired result since $v_{\text{primal}} \geq \mathrm{Tr}(XC)$.

My question: How can we prove part (1) of the Strong Duality theorem using the Farkas Lemma? The statement of the Farkas Lemma given here doesn't help. But is there another version of Farkas Lemma that can be used to prove the other part of strong duality?

Furthermore, can anyone point out another reference that proves both parts of strong duality SDPs (not just one like Lovasz did)? I can't seem to find a nice self-contained proof.