text:
Lagrangian Grassmannian

(Redirected from Maslov index)

In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is n(n+1)/2 (where the dimension of V is 2n). It may be identified with the homogeneous space

U(n)/O(n),

where U(n) is the unitary group and O(n) the orthogonal group. Following Vladimir Arnold it is denoted by Λ(n). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V.

A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space V of dimension 2n. It may be identified with the homogeneous space of complex dimension n(n+1)/2

Sp(n)/U(n),

where Sp(n) is the compact symplectic group.

Topology

The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: ${\displaystyle \Omega (Sp/U)\simeq U/O}$ , and ${\displaystyle \Omega (U/O)\simeq Z\times BO}$  – they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension).

In particular, the fundamental group of ${\displaystyle U/O}$  is infinite cyclic, with a distinguished generator given by the square of the determinant of a unitary matrix, as a mapping to the unit circle. Its first homology group is therefore also infinite cyclic, as is its first cohomology group. Arnold showed that this leads to a description of the Maslov index, introduced by V. P. Maslov.

For a Lagrangian submanifold M of V, in fact, there is a mapping

M → Λ(n)

which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in

H1(M, Z)

of the distinguished generator of

H1(Λ(n), Z).

Maslov index

A path of symplectomorphisms of a symplectic vector space may be assigned a Maslov index, named after V. P. Maslov; it will be an integer if the path is a loop, and a half-integer in general.

If this path arises from trivializing the symplectic vector bundle over a periodic orbit of a Hamiltonian vector field on a symplectic manifold or the Reeb vector field on a contact manifold, it is known as the Conley–Zehnder index. It computes the spectral flow of the Cauchy–Riemann-type operators that arise in Floer homology[citation needed].

It appeared originally in the study of the WKB approximation and appears frequently in the study of quantization, quantum chaos trace formulas, and in symplectic geometry and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.

References

• V. I. Arnold, Characteristic class entering in quantization conditions, Funktsional'nyi Analiz i Ego Prilozheniya, 1967, 1,1, 1-14, doi:10.1007/BF01075861.
• V. P. Maslov, Théorie des perturbations et méthodes asymptotiques. 1972
• Ranicki, Andrew, The Maslov index home page, archived from the original on 2015-12-01, retrieved 2009-10-23 Assorted source material relating to the Maslov index.


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# PyPy - The RPython Toolchain

This document describes the toolchain that we have developed to analyze and "compile" RPython programs (like PyPy itself) to various target platforms.

It consists of three broad sections: a slightly simplified overview, a brief introduction to each of the major components of our toolchain and then a more comprehensive section describing how the pieces fit together. If you are reading this document for the first time, the Overview is likely to be most useful, if you are trying to refresh your PyPy memory then the How It Fits Together is probably what you want.

## Overview

The job of the translation toolchain is to translate RPython programs into an efficient version of that program for one of various target platforms, generally one that is considerably lower-level than Python. It divides this task into several steps, and the purpose of this document is to introduce them.

As of the 1.2 release, RPython programs can be translated into the following languages/platforms: C/POSIX, CLI/.NET and Java/JVM.

The choice of the target platform affects the process somewhat, but to start with we describe the process of translating an RPython program into C (which is the default and original target).

The RPython translation toolchain never sees Python source code or syntax trees, but rather starts with the code objects that define the behaviour of the function objects one gives it as input. The bytecode evaluator and the Flow Object Space work through these code objects using abstract interpretation to produce a control flow graph (one per function): yet another representation of the source program, but one which is suitable for applying type inference and translation techniques and which is the fundamental data structure most of the translation steps operate on.

1. The complete program is imported, at which time arbitrary run-time initialization can be performed. Once this is done, the program must be present in memory as a form that is "static enough" in the sense of RPython.
2. The Annotator performs a global analysis starting from an specified entry point to deduce type and other information about what each variable can contain at run-time, building flow graphs using the Flow Object Space as it encounters them.
3. The RPython Typer (or RTyper) uses the high-level information inferred by the Annotator to turn the operations in the control flow graphs into low-level operations.
4. After RTyping there are two, rather different, optional transformations which can be applied -- the "backend optimizations" which are intended to make the resulting program go faster, and the "stackless transform" which transforms the program into a form of continuation passing style which allows the implementation of coroutines and other forms of non-standard control flow.
5. The next step is preparing the graphs for source generation, which involves computing the names that the various functions and types in the program will have in the final source and applying transformations which insert explicit exception handling and memory management operations.
6. The C backend (colloquially known as "GenC") produces a number of C source files (as noted above, we are ignoring the other backends for now).
7. These source files are compiled to produce an executable.

(although these steps are not quite as distinct as you might think from this presentation).

There is an interactive interface called rpython/bin/translatorshell.py_ to the translation process which allows you to interactively work through these stages.

The following figure gives a simplified overview (PDF color version):

## The Flow Model

The Flow Object Space is described in the document describing object spaces. Here we describe the data structures produced by it, which are the basic data structures of the translation process.

All these types are defined in rpython/flowspace/model.py_ (which is a rather important module in the PyPy source base, to reinforce the point).

The flow graph of a function is represented by the class FunctionGraph. It contains a reference to a collection of Blocks connected by Links.

A Block contains a list of SpaceOperations. Each SpaceOperation has an opname and a list of args and result, which are either Variables or Constants.

We have an extremely useful PyGame viewer, which allows you to visually inspect the graphs at various stages of the translation process (very useful to try to work out why things are breaking). It looks like this:

It is recommended to play with python bin/translatorshell.py on a few examples to get an idea of the structure of flow graphs. The following describes the types and their attributes in some detail:

FunctionGraph

A container for one graph (corresponding to one function).

startblock: the first block. It is where the control goes when the function is called. The input arguments of the startblock are the function's arguments. If the function takes a *args argument, the args tuple is given as the last input argument of the startblock. the (unique) block that performs a function return. It is empty, not actually containing any return operation; the return is implicit. The returned value is the unique input variable of the returnblock. the (unique) block that raises an exception out of the function. The two input variables are the exception class and the exception value, respectively. (No other block will actually link to the exceptblock if the function does not explicitly raise exceptions.)
Block

A basic block, containing a list of operations and ending in jumps to other basic blocks. All the values that are "live" during the execution of the block are stored in Variables. Each basic block uses its own distinct Variables.

inputargs: list of fresh, distinct Variables that represent all the values that can enter this block from any of the previous blocks. list of SpaceOperations. see below list of Links representing possible jumps from the end of this basic block to the beginning of other basic blocks.

Each Block ends in one of the following ways:

• unconditional jump: exitswitch is None, exits contains a single Link.
• conditional jump: exitswitch is one of the Variables that appear in the Block, and exits contains one or more Links (usually 2). Each Link's exitcase gives a concrete value. This is the equivalent of a "switch": the control follows the Link whose exitcase matches the run-time value of the exitswitch Variable. It is a run-time error if the Variable doesn't match any exitcase.
• exception catching: exitswitch is Constant(last_exception). The first Link has exitcase set to None and represents the non-exceptional path. The next Links have exitcase set to a subclass of Exception, and are taken when the last operation of the basic block raises a matching exception. (Thus the basic block must not be empty, and only the last operation is protected by the handler.)
• return or except: the returnblock and the exceptblock have operations set to an empty tuple, exitswitch to None, and exits empty.

A link from one basic block to another.

prevblock: the Block that this Link is an exit of. the target Block to which this Link points to. a list of Variables and Constants, of the same size as the target Block's inputargs, which gives all the values passed into the next block. (Note that each Variable used in the prevblock may appear zero, one or more times in the args list.) see above. None or a Variable; see below. None or a Variable; see below.

Note that args uses Variables from the prevblock, which are matched to the target block's inputargs by position, as in a tuple assignment or function call would do.

If the link is an exception-catching one, the last_exception and last_exc_value are set to two fresh Variables that are considered to be created when the link is entered; at run-time, they will hold the exception class and value, respectively. These two new variables can only be used in the same link's args list, to be passed to the next block (as usual, they may actually not appear at all, or appear several times in args).

SpaceOperation

A recorded (or otherwise generated) basic operation.

opname: the name of the operation. The Flow Space produces only operations from the list in pypy.interpreter.baseobjspace, but later the names can be changed arbitrarily. list of arguments. Each one is a Constant or a Variable seen previously in the basic block. a new Variable into which the result is to be stored.

Note that operations usually cannot implicitly raise exceptions at run-time; so for example, code generators can assume that a getitem operation on a list is safe and can be performed without bound checking. The exceptions to this rule are: (1) if the operation is the last in the block, which ends with exitswitch == Constant(last_exception), then the implicit exceptions must be checked for, generated, and caught appropriately; (2) calls to other functions, as per simple_call or call_args, can always raise whatever the called function can raise --- and such exceptions must be passed through to the parent unless they are caught as above.

Variable

A placeholder for a run-time value. There is mostly debugging stuff here.

name: it is good style to use the Variable object itself instead of its name attribute to reference a value, although the name is guaranteed unique.
Constant

A constant value used as argument to a SpaceOperation, or as value to pass across a Link to initialize an input Variable in the target Block.

value: the concrete value represented by this Constant. a hashable object representing the value.

A Constant can occasionally store a mutable Python object. It represents a static, pre-initialized, read-only version of that object. The flow graph should not attempt to actually mutate such Constants.

## The Annotation Pass

We describe briefly below how a control flow graph can be "annotated" to discover the types of the objects. This annotation pass is a form of type inference. It operates on the control flow graphs built by the Flow Object Space.

For a more comprehensive description of the annotation process, see the corresponding section of our EU report about translation.

The major goal of the annotator is to "annotate" each variable that appears in a flow graph. An "annotation" describes all the possible Python objects that this variable could contain at run-time, based on a whole-program analysis of all the flow graphs -- one per function.

An "annotation" is an instance of a subclass of SomeObject. Each subclass that represents a specific family of objects.

Here is an overview (see pypy/annotation/model/):

• SomeObject is the base class. An instance of SomeObject() represents any Python object, and as such usually means that the input program was not fully RPython.
• SomeInteger() represents any integer. SomeInteger(nonneg=True) represent a non-negative integer (>=0).
• SomeString() represents any string; SomeChar() a string of length 1.
• SomeTuple([s1,s2,..,sn]) represents a tuple of length n. The elements in this tuple are themselves constrained by the given list of annotations. For example, SomeTuple([SomeInteger(), SomeString()]) represents a tuple with two items: an integer and a string.

The result of the annotation pass is essentially a large dictionary mapping Variables to annotations.

All the SomeXxx instances are immutable. If the annotator needs to revise its belief about what a Variable can contain, it does so creating a new annotation, not mutating the existing one.

### Mutable Values and Containers

Mutable objects need special treatment during annotation, because the annotation of contained values needs to be possibly updated to account for mutation operations, and consequently the annotation information reflown through the relevant parts of the flow graphs.

• SomeList stands for a list of homogeneous type (i.e. all the elements of the list are represented by a single common SomeXxx annotation).
• SomeDict stands for a homogeneous dictionary (i.e. all keys have the same SomeXxx annotation, and so have all values).

### User-defined Classes and Instances

SomeInstance stands for an instance of the given class or any subclass of it. For each user-defined class seen by the annotator, we maintain a ClassDef (pypy.annotation.classdef) describing the attributes of the instances of the class; essentially, a ClassDef gives the set of all class-level and instance-level attributes, and for each one, a corresponding SomeXxx annotation.

Instance-level attributes are discovered progressively as the annotation progresses. Assignments like:

inst.attr = value

update the ClassDef of the given instance to record that the given attribute exists and can be as general as the given value.

For every attribute, the ClassDef also records all the positions where the attribute is read. If, at some later time, we discover an assignment that forces the annotation about the attribute to be generalized, then all the places that read the attribute so far are marked as invalid and the annotator will restart its analysis from there.

The distinction between instance-level and class-level attributes is thin; class-level attributes are essentially considered as initial values for instance-level attributes. Methods are not special in this respect, except that they are bound to the instance (i.e. self = SomeInstance(cls)) when considered as the initial value for the instance.

The inheritance rules are as follows: the union of two SomeInstance annotations is the SomeInstance of the most precise common base class. If an attribute is considered (i.e. read or written) through a SomeInstance of a parent class, then we assume that all subclasses also have the same attribute, and that the same annotation applies to them all (so code like return self.x in a method of a parent class forces the parent class and all its subclasses to have an attribute x, whose annotation is general enough to contain all the values that all the subclasses might want to store in x). However, distinct subclasses can have attributes of the same names with different, unrelated annotations if they are not used in a general way through the parent class.

## The RPython Typer

https://bitbucket.org/pypy/pypy/src/default/pypy/rpython/

The RTyper is the first place where the choice of backend makes a difference; as outlined above we are assuming that ANSI C is the target.

The RPython Typer is the bridge between the Annotator and the code generator. The information computed by the annotator is high-level, in the sense that it describe RPython types like lists or instances of user-defined classes.

To emit code we need to represent these high-level annotations in the low-level model of the target language; for C, this means structures and pointers and arrays. The Typer both determines the appropriate low-level type for each annotation and replaces each high-level operation in the control flow graphs with one or a few low-level operations. Just like low-level types, there is only a fairly restricted set of low-level operations, along the lines of reading or writing from or to a field of a structure.

In theory, this step is optional; a code generator might be able to read directly the high-level types. Our experience, however, suggests that this is very unlikely to be practical. "Compiling" high-level types into low-level ones is rather more messy than one would expect and this was the motivation for making this step explicit and isolated in a single place. After RTyping, the graphs only contain operations that already live on the level of the target language, which makes the job of the code generators much simpler.

For more detailed information, see the documentation for the RTyper.

### Example: Integer operations

Integer operations are make an easy example. Assume a graph containing the following operation:

v3 = add(v1, v2)

annotated:

v1 -> SomeInteger()
v2 -> SomeInteger()
v3 -> SomeInteger()

then obviously we want to type it and replace it with:

v3 = int_add(v1, v2)

where -- in C notation -- all three variables v1, v2 and v3 are typed int. This is done by attaching an attribute concretetype to v1, v2 and v3 (which might be instances of Variable or possibly Constant). In our model, this concretetype is pypy.rpython.lltypesystem.lltype.Signed. Of course, the purpose of replacing the operation called add with int_add is that code generators no longer have to worry about what kind of addition (or concatenation maybe?) it means.

## The Optional Transformations

Between RTyping and C source generation there are two optional transforms: the "backend optimizations" and the "stackless transform". See also D07.1 Massive Parallelism and Translation Aspects for further details.

### Backend Optimizations

The point of the backend optimizations are to make the compiled program run faster. Compared to many parts of the PyPy translator, which are very unlike a traditional compiler, most of these will be fairly familiar to people who know how compilers work.

#### Function Inlining

To reduce the overhead of the many function calls that occur when running the PyPy interpreter we implemented function inlining. This is an optimization which takes a flow graph and a callsite and inserts a copy of the flow graph into the graph of the calling function, renaming occurring variables as appropriate. This leads to problems if the original function was surrounded by a try: ... except: ... guard. In this case inlining is not always possible. If the called function is not directly raising an exception (but an exception is potentially raised by further called functions) inlining is safe, though.

In addition we also implemented heuristics which function to inline where. For this purpose we assign every function a "size". This size should roughly correspond to the increase in code-size which is to be expected should the function be inlined somewhere. This estimate is the sum of two numbers: for one every operations is assigned a specific weight, the default being a weight of one. Some operations are considered to be more effort than others, e.g. memory allocation and calls; others are considered to be no effort at all (casts...). The size estimate is for one the sum of the weights of all operations occurring in the graph. This is called the "static instruction count". The other part of the size estimate of a graph is the "median execution cost". This is again the sum of the weight of all operations in the graph, but this time weighted with a guess how often the operation is executed. To arrive at this guess we assume that at every branch we take both paths equally often, except for branches that are the end of loops, where the jump back to the end of the loop is considered more likely. This leads to a system of equations which can be solved to get approximate weights for all operations.

After the size estimate for all function has been determined, functions are being inlined into their callsites, starting from the smallest functions. Every time a function is being inlined into another function, the size of the outer function is recalculated. This is done until the remaining functions all have a size greater than a predefined limit.

#### Malloc Removal

Since RPython is a garbage collected language there is a lot of heap memory allocation going on all the time, which would either not occur at all in a more traditional explicitly managed language or results in an object which dies at a time known in advance and can thus be explicitly deallocated. For example a loop of the following form:

for i in range(n):
...

which simply iterates over all numbers from 0 to n - 1 is equivalent to the following in Python:

l = range(n)
iterator = iter(l)
try:
while 1:
i = iterator.next()
...
except StopIteration:
pass

Which means that three memory allocations are executed: The range object, the iterator for the range object and the StopIteration instance, which ends the loop.

After a small bit of inlining all these three objects are never even passed as arguments to another function and are also not stored into a globally reachable position. In such a situation the object can be removed (since it would die anyway after the function returns) and can be replaced by its contained values.

This pattern (an allocated object never leaves the current function and thus dies after the function returns) occurs quite often, especially after some inlining has happened. Therefore we implemented an optimization which "explodes" objects and thus saves one allocation in this simple (but quite common) situation.

#### Escape Analysis and Stack Allocation

Another technique to reduce the memory allocation penalty is to use stack allocation for objects that can be proved not to life longer than the stack frame they have been allocated in. This proved not to really gain us any speed, so over time it was removed again.

### The Stackless Transform

The stackless transform converts functions into a form that knows how to save the execution point and active variables into a heap structure and resume execution at that point. This was used to implement coroutines as an RPython-level feature, which in turn are used to implement coroutines, greenlets and tasklets as an application level feature for the Standard Interpreter.

The stackless transformation has been deprecated and is no longer available in trunk. It has been replaced with continulets.

## Preparation for Source Generation

This, perhaps slightly vaguely named, stage is the most recent to appear as a separate step. Its job is to make the final implementation decisions before source generation -- experience has shown that you really don't want to be doing any thinking at the same time as actually generating source code. For the C backend, this step does three things:

• inserts explicit exception handling,
• inserts explicit memory management operations,
• decides on the names functions and types will have in the final source (this mapping of objects to names is sometimes referred to as the "low-level database").

### Making Exception Handling Explicit

RPython code is free to use exceptions in much the same way as unrestricted Python, but the final result is a C program, and C has no concept of exceptions. The exception transformer implements exception handling in a similar way to CPython: exceptions are indicated by special return values and the current exception is stored in a global data structure.

In a sense the input to the exception transformer is a program in terms of the lltypesystem with exceptions and the output is a program in terms of the bare lltypesystem.

### Memory Management Details

As well as featuring exceptions, RPython is a garbage collected language; again, C is not. To square this circle, decisions about memory management must be made. In keeping with PyPy's approach to flexibility, there is freedom to change how to do it. There are three approaches implemented today:

Almost all application-level Python code allocates objects at a very fast rate; this means that the memory management implementation is critical to the performance of the PyPy interpreter.

You can choose which garbage collection strategy to use with :config:translation.gc.

## The C Back-End

https://bitbucket.org/pypy/pypy/src/default/pypy/translator/c/

GenC is usually the most actively maintained backend -- everyone working on PyPy has a C compiler, for one thing -- and is usually where new features are implemented first.

## A Historical Note

As this document has shown, the translation step is divided into more steps than one might at first expect. It is certainly divided into more steps than we expected when the project started; the very first version of GenC operated on the high-level flow graphs and the output of the annotator, and even the concept of the RTyper didn't exist yet. More recently, the fact that preparing the graphs for source generation ("databasing") and actually generating the source are best considered separately has become clear.

## Other backends

Use the :config:translation.backend option to choose which backend to use.

### The Object-Oriented Backends

The Object-Oriented backends target platforms that are less C-like and support classes, instance etc. If such a platform is targeted, the OO type system is used while rtyping. Of the OO backends, both gencli and genjava can translate the full Python interpreter.

#### GenCLI

GenCLI targets the Common Language Infrastructure, the most famous implementations of which are Microsoft's .NET and Mono.

It is the most advanced of the object oriented backends -- it can compile the PyPy interpreter as well as our two standard benchmarks, RPyStone (CPython's PyStone benchmark modified slightly to be RPython) and a RPython version of the common Richards benchmark.

It is almost entirely the work of Antonio Cuni, who started this backend as part of his Master's thesis, the Google's Summer of Code 2006 program and the Summer of PyPy program.

#### GenJVM

GenJVM targets the Java Virtual Machine: it translates RPython programs directly into Java bytecode, similarly to what GenCLI does.

So far it is the second most mature high level backend after GenCLI: it still can't translate the full Standard Interpreter, but after the Leysin sprint we were able to compile and run the rpystone and richards benchmarks.

GenJVM is almost entirely the work of Niko Matsakis, who worked on it also as part of the Summer of PyPy program.

## External Function Calls

The external function call approach is described in rffi documentation.

## How It Fits Together

As should be clear by now, the translation toolchain of PyPy is a flexible and complicated beast, formed from many separate components.

The following image summarizes the various parts of the toolchain as of the 0.9 release, with the default translation to C highlighted:

A detail that has not yet been emphasized is the interaction of the various components. It makes for a nice presentation to say that after the annotator has finished the RTyper processes the graphs and then the exception handling is made explicit and so on, but it's not entirely true. For example, the RTyper inserts calls to many low-level helpers which must first be annotated, and the GC transformer can use inlining (one of the backend optimizations) of some of its small helper functions to improve performance. The following picture attempts to summarize the components involved in performing each step of the default translation process:

A component not mentioned before is the "MixLevelAnnotator"; it provides a convenient interface for a "late" (after RTyping) translation step to declare that it needs to be able to call each of a collection of functions (which may refer to each other in a mutually recursive fashion) and annotate and rtype them all at once.

format: plain


text:
## Trigonometry (10th Edition)

1. Angle in the Quadrant I, and coterminal angle between 0 and 90. It will have positive x and y values. r is always positive. 2. Determine the sign: $\sin\theta=\frac{y}{r}= (+)$ $\cos\theta=\frac{x}{r}= (+)$ $\tan\theta=\frac{y}{x}= (+)$ 3. Remember, the reciprocal trig function will have the same sign as the original function $\csc\theta=(+)$ $\sec\theta=(+)$ $\cot\theta=(+)$


text:
Chapter 16.4, Problem 4E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.4. ∮C x2y2 dx + xy dy, C consists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and the line segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0)

(a)

To determine

To evaluate: the line integral in direct method.

Explanation

Given data:

Line integral is Cx2y2dx+xydy and curve C comprises of parabola from (0,0) to (1,1) and a line segments from (1,1) to (0,1) and from (0,1) to (0,0) .

Draw the curve C as shown in Figure 1.

From Figure 1, write the expressions for curve C1 , 0t1 .

x=t (1)

y=t2 (2)

Differentiate the equation (1) with respect to t.

dxdt=ddt(t)dxdt=1 {ddt(t)=1}dx=dt

Differentiate the equation (2) with respect to t.

dydt=dydt(t2)dydt=2t {ddt(t2)=2t}dy=2tdt

From Figure 1, write the expressions for curve C2 , 0t2 .

x=1t (3)

y=1 (4)

Differentiate the equation (3) with respect to t.

dxdt=ddt(1t)dxdt=01{ddt(k)=0,ddt(t)=1}dx=dt

Differentiate the equation (4) with respect to t.

dydt=dydt(1)dydt=0 {ddt(k)=0}dy=0dt

From Figure 1, write the expressions for curve C3 , 0t1 .

x=0 (5)

y=1t (6)

Differentiate the equation (5) with respect to t.

dxdt=ddt(0)dxdt=0{ddt(k)=0}dx=0dt

Differentiate the equation (6) with respect to t

(b)

To determine

To evaluate: The line integral using Green’s Theorem.

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# If x is positive, is x > 4?

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If x is positive, is x > 4? [#permalink]

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If x is positive, is x > 4?

(1) (4-x)^2 < 1

(2) (x-4)^2 < 1
[Reveal] Spoiler: OA

Last edited by Bunuel on 17 May 2014, 04:53, edited 1 time in total.
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If x is positive, is x > 4?

First of all notice that $$(x-4)^2 =(4-x)^2$$. So, both options give the same exact information, which implies that the answer is either D or E.

(1) (4-x)^2 < 1 --> since both sides are non-negative, then we can take the square root from the inequality: $$|4-x|<1$$ --> $$-1<4-x<1$$ --> subtract 4 from all parts: $$-5<-x<-3$$ --> multiply by -1 and flip the sign: $$5>x>3$$. So, x may or may not be greater than 4. Not sufficient.

The answer cannot be D, thus it's E.

(2) (x-4)^2 < 1. Not sufficient.

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Re: If x is positive, is x > 4? [#permalink]

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17 May 2014, 05:01
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steilbergauf wrote:
If x is positive, is x > 4?

(1) (4-x)^2 < 1

(2) (x-4)^2 < 1

Hi there,

We know x is positive and we need to know if its greater than 4

St 1 says (4-x)^2 <1 ------> Since both LHS and RHS are positive (Non-Negative), we can take a sqaure root and see what we get

$$\sqrt{(4-x)^2}$$ < 1

or |4-x|<1 or -1<4-x<1 -------> Simplify and we get range for x as 3<x<5. So x may or may not be greater than 4

St 2 By same way can be reduced $$\sqrt{(x-4)^2}$$<1 or |x-4|<1 or -1<x-4<1 ---we get same range for x 3<x<5.

For such set of questions try and build your base in equalities and modulus properties. These topics will help you dissect the question stem.

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Re: If x is positive, is x > 4? [#permalink]

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17 May 2014, 09:26
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steilbergauf wrote:
If x is positive, is x > 4?

(1) (4-x)^2 < 1

(2) (x-4)^2 < 1

As mentioned in posts above, first thing to notice is (4-x)^2 = (x-4)^2, making both the statements same.

Now lets take statement 1)

(x-4)^2 < 1
(x-4)^2 -1 < 0

using a^2-1 = (a-1)(a+1), we can write it as:

(x-4-1)(x-4+1) <0
(x-3)(x-5) <0
or 3<x<5

Hence x can be greater than 4 or less than 4. Thus, option (E) is valid.

If instead of 1 on RHS, we had any number b, we would have written a^2 - b = (a-\sqrt{b})(a+\sqrt{b})
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Re: If x is positive, is x > 4? [#permalink]

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18 Jun 2015, 04:56
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text:
# NEET Physics Alternating Current Questions Solved

A plane electromagnetic wave of frequency 25 MHz travels in free space along the x-direction. At a particular point in space and time. E= 6.3 $\stackrel{^}{j}$ v/m What is B at this point?

Explanation is a part of a Paid Course. To view Explanation Please buy the course.

Difficulty Level:


text:
# Mathematical way of determining whether a number is an integer

I'm developing a computer program, and I've run into a mathematical problem. This isn't specific to any programming language, so it isn't really appropriate to ask on stackoverflow. Is there any way to determine whether a number is an integer using a mathematical function, from which a boolean response is given.

For example:

let x equal 159 let y equal 12.5

f(x) returns 1 and f(y) returns 0

Please get back to me if you can. If it isn't possible, is there a similar way to determine whether a number is odd or even?

EDIT:

I found a solution to the problem thats to Karolis Juodelė. I'll use a floor function to round the integer down, and then subtract the output from the original number. If the output is zero, then the function returns 0.

I just need to make sure that floor is a purely mathematical function. Does anyone know?

Thanks

-
I don't think this question belongs here. Such a function must take into consideration the machine representation of the number. For determining whether a given (integer) number n is even of odd, you may compute n - 2 * int(n/2); if n is even it returns 0, otherwise 1. –  mau Mar 24 '13 at 10:56
$-4$, really? I've seen way worse questions which haven't gotten a single downvote. –  Git Gud Mar 24 '13 at 11:00
@GitGud I gave an sympathy upvote, he said why he think that it belongs here, i only see 4 downvotes, no close vote. Guys tell him what he is doing wrong, he is new here how shall he know it ? –  Dominic Michaelis Mar 24 '13 at 11:01
@Jack what is a purely mathematical function for you ? –  Dominic Michaelis Mar 25 '13 at 6:13
Stackoverflow is for things like "practical, answerable problems that are unique to the programming profession" or "a specific programming problem"; your question is appropriate. In fact, with questions like this, you will almost always get better answers from programming sites than from math sites. –  Hurkyl Mar 25 '13 at 6:21
The most basic thing you could do is check if $x = \text{floor}(x)$. Here $\text{floor}$ returns the integer part of a number (rounds down). It is present in standard libraries of most languages.


text:
# What are the memory requirements for this type of search if the hash function has an output length n bits?

Let us suppose that we perform a collision search to find a pair $x_1$, $x_2$ with $x_1 \neq x_2$ such that $h(x_1)=h(x_2)$ for a given hash function, and we would like to specify the memory requirements for this type of search.

What does memory requirements means in this question ?

## The naive approach

A naive algorithm for a research of collision is the following:

1. choose $x_1$
2. compute $h(x_1)$
3. Choose $x_2 \gets x_1 + 1$ ($x_2$ takes the value $x_1 + 1$)
4. compute $h(x_2)$ and verify if it matches $h(x_1)$
5. if it does not, do $x_2 \gets x_2 + 1$
6. go back to step 4, repeat and rinse.

How much memory do we need for this algorithm to work?
Well we need to store only $x_1$ and $x_2$. You can also store $h(x_1)$ in order to not compute it each time.

So if $h$ has a $b$ bits inputs, you need... $2 \times b$ memory to find your collision ($b$ for $x_1$ and $b$ for $x_2$). This is the memory requirement for THIS search algorithm.

However this algorithm is clearly inefficient. Assuming the output of your hash function is $n$ bits. You have $2^n$ hash possibles. This will lead to an average of $\frac{2^n}{2}$ tentative... Clearly not feasible with a complexity of $O(2^{n-1})$.

## The multi-target attack

In the previous attack we targeted a specific or fixed value ($x_1$). A more advanced attack consist of storing all the previous attempts and each time you try a new value of $x_2$, thus the multi-target.

The algorithm is the following:

1. consider an empty list $lst$.
2. choose $x$
3. add $x$ to $lst$
4. Choose $x' \gets x + 1$
5. Check if there is $x \in lst$ such that $h(x') = h(x)$
6. Add $x'$ to $lst$
7. Compute $x' \gets x' + 1$
8. go back to step 5, repeat and rinse.

I will let you guess the memory requirement for this algorithm.

In term of complexity, if you consider a list of $2^m$ elements, the complexity of this attack is $O(2^{n-m})$.

• thanks a lot I guess the memory requirement for this algorithm is number of elements in lst. – han Oct 17 '16 at 3:25
• @han the best way of thanking Biv is to accept the answer if he helped you along enough. Yeah, number of elements times $b$ of course. – Maarten Bodewes Jan 13 '17 at 1:22


text:
# Beta Function and anomalous dimension in $O(N)$ model

I am currently studying Quantum Field Theory, but I cannot find any analytic calculation for the beta fucntion in $$O(N)$$ models such as $$\phi^4$$ theory. Can anyone provide me some hints on the calculation?


text:
Now showing items 7445-7464 of 21945

• #### [2015-02-12]K-Theory and Pseudospectra for Topological Insulators [dataset]

We derive formulas and algorithms for Kitaev's invariants in the periodic table for topological insulators and superconductors for finite disordered systems on lattices with boundaries. We find that $K$-theory arises ...

• #### [2011-06-08]Keep Your Money: Let the West Pay for its Own Water Projects

The question posed here is what role the federal government should play in responding to the western water crisis, in light of the changes in the social and political landscape that have occurred in the last decade. My ...

• #### [2012-02-01]Keeping checkpoint/restart viable for exascale systems

Next-generation exascale systems, those capable of performing a quintillion operations per second, are expected to be delivered in the next 8-10 years. These systems, which will be 1,000 times faster than current systems, ...

• #### [2012-08-27]Keeping faith : factors contributing to spiritual transformation, identity, and maturity in church-attending adolescents

The current research study investigated factors associated with spiritual development among church-attending adolescents. Eighty five participants (12 – 19 years) attending church youth groups in the Albuquerque area were ...

• #### [2012-07-02]"Ye ken fine wha I mean": Variation between you know and ye ken in Scottish varieties of English

The discourse marker you know is one of the most frequently studied discourse markers, and bears the hallmarks of a typical discourse marker with its complexity and indeterminacy of function (e.g. Holmes 1986, Erman 1987, ...

• #### [2007-10-11]Killing with Kindness: The Myth of "Consumer Protection" in the Bankruptcy Abuse and Consumer Protection Act of 2005

The United States Congress considered bankruptcy reform for eight years before George W. Bush signed the Bankruptcy Abuse and Consumer Protection Act of 2005 ("BACPA") into law on April 20, 2005. Congressional proponents ...
• #### [2011-11-07]"Kindred Nature: Victorian and Edwardian Women Embrace the Living World" by Barbara T. Gates Book Review

The most famous figure in Kindred Nature is Beatrix Potter, the author of the Peter Rabbit stories. Potter's talents in mycology and scientific illustration are less well known. Moreover, why Potter turned from her ...
• #### [2011-02-08]Kinetic and Statistical Mechanical Modeling of DNA Unzipping and Kinesin Mechanochemistry

This thesis explores two topics. The first is shotgun DNA mapping (SDM). Ability to map polymerases and nucleosomes on chromatin is important for understanding the impact of chromatin remodeling on key cellular processes. ...

• #### [2007-04-13]Kinship Care, Public Policy, and the Best Interests of Children

This article characterizes kinship care arrangements, surveys the problems with which kinship caregivers contend, and considers the advantages and disadvantages of kinship care. In addition, it examines federal statutes ...

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Graduate Special Topics in Computer Science

NOTE: for descriptions of standard graduate computer science courses, see Graduate Course Descriptions.

#### G22.3033-001 Computational Systems Biology

Presently, there is no clear way to determine if the current body of biological facts is sufficient to explain phenomenology. In the biological community, it is not uncommon to assume certain biological problems to have achieved a cognitive finality without rigorous justification. In these particular cases, rigorous mathematical models with automated tools for reasoning, simulation, and computation can be of enormous help to uncover cognitive flaws, qualitative simplification or overly generalized assumptions. Some ideal candidates for such study would include: prion hypothesis, cell cycle machinery (DNA replication and repair, chromosome segregation, cell-cycle period control, spindle pole duplication, etc.), muscle contractility, processes involved in cancer (cell cycle regulation, angiogenesis, DNA repair, apoptosis, cellular senescence, tissue space modeling enzymes, etc.), signal transduction pathways, circadian rhythms (especially the effect of small molecular concentration on its robustness), and many others. We believe that the difficulty of biological modeling will become acute as biologists prepare to understand even more complex systems.

The goal of this course is to understand, design and create a large-scale computational system centered on the biology of individual cells, population of cells, intra-cellular processes, and realistic simulation and visualization of these processes at multiple spatio-temporal scales. Such a reasoning system, in the hands of a working biologist, can then be used to gain insight into the underlying biology, design refutable biological experiments, and ultimately, discover intervention schemes to suitably modify the biological processes for therapeutic purposes.

(1) Evolutionary Biology (2) Computability in Biology (3) Reconstructibility in Biology (4) Biology of Cancer (5) Biology of Aging

#### G22.3033-002 Applied Math for Algorithms

Although many mathematics courses have content that is useful in the design and analysis of algorithms, the more applicable material often comprises no more than 10% of the curriculum.

This course will endeavor to present that critical 10% as culled from a handful of math courses and other sources. We will also endeavor to show how this material is applied in the design or analysis of algorithms.

Some of the subjects that will be covered are:

Performance-related equations and their direct solution.

Recurrence equations: first order difference equations and beyond. Eigenfunction methods, multiplicity of eigenvalues, etc.

ODEs and their use in performance analyses. A limited presentation of limit theorems.

PDEs and their use in performance analyses. Limited implications of limit theorems.

Probability, and what properly educated theoreticians might wish to know. Conditional expectations are a powerful highly expressive tool with a foundation that is based on measure theory yet the concept allows us to present proofs and to acquire understandings about probabilistic processes that would otherwise be difficult to formalize. It is also worth understanding what the central limit theorem says, what it means, what it implies, and what it does not imply. Likewise, it is useful to understand the more basic probability distributions and the physical processes that they model.

Statistics is a little different from probability. One of the areas of statistical analysis that has had significant impact in TCS is the theory of Hoeffding-Chernoff bounds. It is probably a good idea to know how this material applies to more than Bernoulli Trials, and to stopping times in particular. We might also cover some issues in the computation of random variables, and at least comment on the importance of extreme points in the space of probability distributions (which is to say that we will survey majorization results).

Complex variables give valuable insights about the solution to many algorithms-based difference equations, and provide the theory necessary for computing saddle point estimates and understanding some of the limit results for PDEs.

Combinatorics is less systematic than analysis, but gives a rich set of techniques that are as varied as tableau methods (used to count seemingly complicated outcomes) and the probabilistic method (which is arguably more combinatorial that probabilistic).

Additional topics will be included as time/opportunities permit.

#### G22.3033-003 Data Mining

We live in the Age of Information. The importance of collecting data that reflects a business or scientific activity to achieve competitive advantage is widely recognized now. Advanced systems for collecting data and managing it in large databases are in place in most large and mid-range companies. However, the bottleneck of turning this data into your success is the difficulty of extracting knowledge about the system from the collected data.

What goods should be promoted to this customer? What is the probability that a certain customer will respond to a planned promotion? Can one predict the most profitable securities to buy/sell during the next trading session? Will this customer default on a loan or pay back on schedule? What medical diagnosis should be assigned to this patient? How large are the peak loads of a telephone or energy network going to be? Why does the manufacturing facility suddenly start to produce defective goods?

These are all the questions that can be answered if information hidden in a database can be found explicitly and utilized. Modeling the investigated system and discovering relations that connect variables are the subject of data mining.

The course will introduce concepts and techniques of data mining and data warehousing, including concept, principle, architecture, design, implementation, application of data warehousing and data mining. TOPICS: Data warehousing and OLAP technology for data mining Data preprocessing Descriptive data mining: characterization and comparison Association analysis Classification and prediction Cluster analysis Mining complex types of data Applications and trends in data mining

#### G22.3033-004 Web Development with Ruby on Rails

This course begins with an in-depth examination of the Ruby language and moves on to web development within the Ruby on Rails framework. An emphasis is placed on understanding the particular features of the Ruby language, how the language compares to others like Java and Python, and how it facilitates the creation of frameworks such as Ruby on Rails. This course is recommended for students with a strong interest in programming languages, web development frameworks, and software engineering. No experience with Ruby or Ruby on Rails is assumed.

#### G22.3033-005 Game Theory, Learning & Planning

There have been many instances in the recent literature in which algorithmic thinking is combined with game-theoretic, or, more specifically, socio-economic concepts to address problems arising in diverse contexts. The purpose of this course is to seize this moment and promote this interaction to create a field of Social Operating Systems (SOS).'' The emphasis will be on mathematically sophisticated techniques in the interface between algorithms, learning, optimization and game theory, as well as their applications to planning. Our primary motivation comes from our work on PLAN C (Planning with Large Agent network against Catastrophes; http://www.bioinformatics.nyu.edu/Projects/planc/index.shtml), which provides emergency management and response plans in face of man-made or natural disasters.

Topics will include some of the following (the list will crystallize gradually after initial discussions in the first class): 1: Game Theory, Nash equilibrium and General Equilibrium 2: Evolutionary game theory and Repeated Games 3: Bounded rationality: Example of Santa Fe Bar Problems 4: Social choice theory: Condorcet, Sen & Arrow 5: Elections, Technorati-Tags and Internet 6: Mechanism design 7: Multiple Objective Functions and Pareto Optimality 8: Multi-Objective Optimization Problems (MOOP) 9: Social Software: Planning (Catastrophe Response) 10: Modeling Social Networks & Bounded Rationality 11: Using Social Networks to Survey, Sample and Search 12: Collaborative Filtering: Recommender Design 13: Collaborative Filtering: Search Engines 14: Collaborative Filtering: Population Sampling and Surveys

#### G22.3033-006 Computational Photography

Course description: Computational Photography is an exciting new area at the intersection of Computer Graphics and Computer Vision. Through the use of computation, its goal is to move beyond the limitations of conventional photography to produce enhanced and novel imagery of the world around us. The main focus of the course will be on software-based methods for producing visually compelling pictures. However, it will also cover novel camera designs, for which computation is integral to their operation. The course will explain the principles behind many of the advanced tools that can be found in Adobe Photoshop, although the use of this package itself is outside the scope of the course. The course will be suitable for advanced undergraduates, masters and PhD students. A reasonable knowledge of linear algebra is required and familiarity with Matlab is desirable. Assessment will be through coursework and a course project.

#### G22.3033-007 Probabilistically Checkable Proofs and Hardness of Approximation

Many optimization problems of theoretical and practical interest are NP-complete, meaning it is impossible to compute exact solutions to these problems in polynomial time unless P = NP. A natural way to cope with this curse of NP-completeness is to seek approximate solutions instead of exact solutions. An algorithm with approximation ratio C computes, for every problem instance, a solution whose cost is within a factor C of the optimum. Optimization problems exhibit a wide range of behavior in their approximability. It is well-known that Bin-Packing has an approximation algorithm with ratio 1+\epsilon for every \epsilon > 0. In theory jargon, we say that Bin-Packing has a polynomial-time approximation scheme (PTAS). However, it wasn't known till the early 90s whether problems like MAX-3SAT, Vertex Cover, and MAX-CUT have a PTAS. A celebrated result called the PCP Theorem finally showed that these problems have no PTAS unless P = NP. Such results that rule out the possibility of good approximation algorithms (under complexity theoretic assumptions like P != NP) are called inapproximability results or hardness of approximation results.

The PCP Theorem has an equivalent formulation from the point of view of proof checking. The PCP theorem states that every NP-statement has a probabilistically checkable proof, i.e. a proof which can be "spot-checked" by reading only a constant number of bits from the proof. These bits are selected by a randomized process using a very limited amount of randomness. The checking process always accepts a correct proof of a correct statement and rejects any cheating proof of an incorrect statement with high probability. The term "holographic proof" is sometimes used to highlight this feature that a cheating proof must be wrong everywhere and therefore, can be detected by a spot-check. The discovery of the connection between proof checking and inapproximability results is one of the most exciting theoretical developments in the last decade. Since then, PCPs have led to several breakthrough results in inapproximability theory, e.g. tight hardness results for Clique, MAX-3SAT, and Set Cover.

This course will cover many of the inapproximability results and PCPs used to prove them. No prior knowledge will be assumed, except the basic theory of NP-completeness. Participants are expected to scribe notes for one lecture and/or give one presentation. No assignments/exams!


text:
# 【floyd】 HRBUST 2037 Find Your Teacher

## Description

You just finished your exam, you felt terrible because you might fail the exam. But your teacher was so kind that he might let you pass the lesson if you called him and begged for passing. Unfortunately, you didn’t have your teacher’s phone number! You began to ask whoever probably had the number, if they didn’t have, they would ask their classmates. Would you finally get the number?

## Input

The first line contains an integer T (1 <= T <= 20), indicates the number of test cases.
For each test case, the first line is two positive integers, n (n <= 50), m (m <= 2000). n is the number of people who will appear in the problem, you are indexed as 1 and your teacher is indexed as n.

Then m lines follows, each line contains two integers x, y (1 <= x, y <= n), means x have the phone number of y.

## Output

For each test case, if you could finally got your teacher’s number, output “^_^”, “T_T” the otherwise.

2
5 5
1 2
1 3
2 3
2 4
4 5
4 3
1 2
3 4
4 1

^_^

T_T

## Source

“尚学堂杯”哈尔滨理工大学第六届程序设计竞赛

## AC代码

#include<bits/stdc++.h>
using namespace std;

bool ans[55][55];

void solve(void)
{
std::ios::sync_with_stdio(false);
int t;
cin>>t;
while(t--)
{
memset(ans,false,sizeof ans);
int n,m;
cin>>n>>m;
for(int i = 1 ; i <= m ; i++)
{
int f,t;
cin>>f>>t;
ans[f][t] = true;
}
for(int k = 1 ; k <= n ; k++)
{
for(int i = 1 ; i <= n ; i++)
{
for(int j = 1 ; j <=n ; j++)
{
if(ans[i][k]==true&&ans[k][j]==true)
ans[i][j] = true;
}
}
}
if(ans[1][n]==true) cout<<"^_^"<<endl;
else cout<<"T_T"<<endl;
}
}

int main(void)
{
solve();
return 0;
}


text:
assignmentutor™您的专属作业导师

assignmentutor-lab™ 为您的留学生涯保驾护航 在代写光学Optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|光学代写Optics代考|Free Energies

Since the equilibrium configuration of a CLC is a helical structure with a pitch wave vector $q_0\left(q_0=2 \pi / p_0\right)$, its elastic free energy will necessarily reflect the presence of $q_0$. The evolution of a cholesteric to a nematic liquid crystal may be viewed as the “untwisting” of the helical structure (i.e. the twist deformation energy is involved). This is indeed rigorously demonstrated in the Frank elastic theory [1]. where consideration of the absence of mirror symmetry in cholesterics results in the addition of another factor to the twist deformation energy:
$$\begin{array}{cc} K_2(\hat{n} \cdot \nabla \times \hat{n})^2 & \rightarrow K_2\left(\hat{n} \cdot \nabla \times \hat{n}+q_0\right)^2 \ \text { nematic } & \text { cholesteric } \end{array}$$
In Figure 4.1, the director axis is described by $\hat{n}=\left(n_x, n_y, n_z\right)$, where $n_x=\cos \theta(z)$, $n_y=\sin \theta(z)$, and $n_z=0$. Note that this configuration corresponds to a state of minimum free energy $$F_2=\frac{1}{2} K_2(\hat{n} \cdot \nabla \times \hat{n})^2=\frac{1}{2} K_2\left(-\frac{\partial \theta}{\partial z}+q_0\right)^2=0 .$$
Equation (4.2) gives $q_0=-\partial \theta / \partial z$, and so we have:
$$\theta=q_0 z .$$
For a general distortion, therefore, the free energy of a CLC is given by $F_d=\frac{1}{2} K_1(\nabla \cdot \hat{n})^2+\frac{1}{2} K_2(\hat{n} \cdot \nabla \times \hat{n})^2+\frac{1}{2} K_3(\hat{n} \times \nabla \times \hat{n})^2 .$

## 物理代写|光学代写Optics代考|Field-induced Effects and Dynamics

In the purely dielectric interaction picture (i.e. no current flow), the realignment or alignment of a CLC in an applied electric or magnetic field results from the system’s tendency to minimize its total free energy. Clearly, the equilibrium configuration of the director axis depends on its initial orientation and the direction of the applied fields, as well as the signs of $\Delta \chi^m$ and $\Delta \varepsilon$. We will not delve into the various possible cases as they all involve the same basic mechanism; that is, the director axis tends to align parallel to the field for positive dielectric anisotropies, and normal to the field for negative dielectric anisotropies.

In the case of positive dielectric anisotropies, the field-induced reorientation process is analogous to that discussed for nematics (cf. Section 3.6). In cholesterics, however, the realignment of the director axis in the direction of the applied field will naturally affect the helical structure.

Figure $4.2 \mathrm{a}$ shows the unperturbed director axis configuration in the bulk of an ideal CLC. Upon the application of a magnetic field, some molecules situated in the bulk regions $A, A^{\prime}$, and so on are preferentially aligned along the direction of the field; others, situated in regions $B, B^{\prime}$, and so forth are not and would tend to reorient themselves along the field direction. As a result, the pitch of the helical structure will be increased; the helix is no longer of the ideal sinusoidal form. Finally, when the field is sufficiently high, this untwisting effect is complete as the pitch approaches infinity; that is, the CLC is said to have undergone a transition to the nematic phase [2].
This process can be described by the free-energy minimization process. The total free energy of the system is given by
$$F_{\text {total }}=\frac{1}{2} \int d z\left[K_2\left(\frac{\partial \phi}{\partial z}-q_0\right)^2-\Delta \chi^m H^2 \sin ^2 \phi\right]$$
This, upon minimization, yields the Euler equation:
$$K_2 \frac{d^2 \phi}{d z^2}+\Delta \chi^m H^2 \sin \phi \cos \phi=0$$
which is analogous to Eq. (3.75b). One can define a coherence length $\xi_H$ by
$$\xi_H=\frac{K_2}{\Delta \chi^m H^2}$$

# 光学代考

## 物理代写|光学代写Optics代考|Free Energies

$$K_2(\hat{n} \cdot \nabla \times \hat{n})^2 \rightarrow K_2\left(\hat{n} \cdot \nabla \times \hat{n}+q_0\right)^2 \text { nematic cholesteric }$$

$$F_2=\frac{1}{2} K_2(\hat{n} \cdot \nabla \times \hat{n})^2=\frac{1}{2} K_2\left(-\frac{\partial \theta}{\partial z}+q_0\right)^2=0 .$$

$$\theta=q_0 z$$

## 物理代写|光学代写Optics代考|Field-induced Effects and Dynamics

$$F_{\text {total }}=\frac{1}{2} \int d z\left[K_2\left(\frac{\partial \phi}{\partial z}-q_0\right)^2-\Delta \chi^m H^2 \sin ^2 \phi\right]$$

$$K_2 \frac{d^2 \phi}{d z^2}+\Delta \chi^m H^2 \sin \phi \cos \phi=0$$

$$\xi_H=\frac{K_2}{\Delta \chi^m H^2}$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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text:
# Nonlinear Optimal Control

#### Note: this is still a work in progress!

The ControlSystem type is an interesting system because, unlike other system types, it cannot be numerically solved on its own. Instead, it must be transformed into another system before solving. Standard methods such as the "direct method", "multiple shooting", or "discretize-then-optimize" can all be phrased as symbolic transformations to a ControlSystem: this is the strategy of this methodology.

## Defining a Nonlinear Optimal Control Problem

Here we will start by defining a classic optimal control problem. Let:

$$$x^{′′} = u^3(t)$$$

where we want to optimize our controller u(t) such that the following is minimized:

$$$L(\theta) = \sum_i \Vert 4 - x(t_i) \Vert + 2 \Vert x^\prime(t_i) \Vert + \Vert u(t_i) \Vert$$$

where $i$ is measured on (0,8) at 0.01 intervals. To do this, we rewrite the ODE in first order form:

\begin{aligned} x^\prime &= v \\ v^′ &= u^3(t) \\ \end{aligned}

and thus

$$$L(\theta) = \sum_i \Vert 4 - x(t_i) \Vert + 2 \Vert v(t_i) \Vert + \Vert u(t_i) \Vert$$$

is our loss function on the first order system.

Defining such a control system is similar to an ODESystem, except we must also specify a control variable u(t) and a loss function. Together, this problem looks as follows:

using ModelingToolkit

@variables t x(t) v(t) u(t)
@parameters p[1:2]
D = Differential(t)

loss = (4-x)^2 + 2v^2 + u^2
eqs = [
D(x) ~ v - p[2]*x
D(v) ~ p[1]*u^3 + v
]

@named sys = ControlSystem(loss,eqs,t,[x,v],[u],p)

## Solving a Control Problem via Discretize-Then-Optimize

One common way to solve nonlinear optimal control problems is by transforming them into an optimization problem by performing a Runge-Kutta discretization of the differential equation system and imposing equalities between variables in the same steps. This can be done via the runge_kutta_discretize transformation on the ControlSystem. While a tableau tab can be specified, it defaults to a 5th order RadauIIA collocation, which is a common method in the field. To perform this discretization, we simply need to give a dt and a timespan on which to discretize:

dt = 0.1
tspan = (0.0,1.0)
sys = runge_kutta_discretize(sys,dt,tspan)

Now sys is an OptimizationSystem which, when solved, gives the values of x(t), v(t), and u(t). Thus we solve the OptimizationSystem using GalacticOptim.jl:

u0 = rand(length(states(sys))) # guess for the state values
sol = solve(prob,BFGS())


text:
F. Balance the Cards
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

A balanced bracket sequence is defined as an integer sequence that can be built with the following rules:

• The empty sequence is balanced.
• If $[a_1,\ldots,a_n]$ and $[b_1,\ldots, b_m]$ are balanced, then their concatenation $[a_1,\ldots,a_n,b_1,\ldots,b_m]$ is balanced.
• If $x$ is a positive integer and $[a_1,\ldots,a_n]$ is balanced, then $[x,a_1,\ldots,a_n,-x]$ is balanced.

The positive numbers can be imagined as opening brackets and the negative numbers as closing brackets, where matching brackets must have the same type (absolute value). For example, $[1, 2, -2, -1]$ and $[1, 3, -3, 2, -2, -1]$ are balanced, but $[1, 2, -1, -2]$ and $[-1, 1]$ are not balanced.

There are $2n$ cards. Each card has a number on the front and a number on the back. Each integer $1,-1,2,-2,\ldots,n,-n$ appears exactly once on the front of some card and exactly once on the back of some (not necessarily the same) card.

You can reorder the cards however you like. You are not allowed to flip cards, so numbers cannot move between the front and back. Your task is to order the cards so that the sequences given by the front numbers and the back numbers are both balanced, or report that it is impossible.

Input

The first line contains a single integer $n$ ($1\le n\le 2\cdot 10^5$) — the number of bracket types, and half the number of cards.

The next $2n$ lines describe the cards. The $i$-th of these lines contains two integers $a_i$, $b_i$ ($-n\le a_i,b_i\le n$, $a_i\ne 0$, $b_i\ne 0$) — the numbers on the front and back of the $i$-th card, respectively. Every integer $1,-1,2,-2,\ldots,n,-n$ appears exactly once as $a_i$ and exactly once as $b_i$.

Output

On the first line, output "YES" if it's possible to reorder these cards to satisfy the condition. Otherwise, output "NO". You can print each letter in any case (upper or lower).

If it is possible, on the next $2n$ lines output the cards in an order so that the front and back are both balanced. If there are multiple solutions, you may output any.

Examples
Input
5
1 3
-3 -5
4 -3
2 2
-1 -4
-2 5
3 -1
5 1
-4 4
-5 -2

Output
YES
1 3
4 -3
-4 4
-1 -4
5 1
3 -1
2 2
-2 5
-3 -5
-5 -2

Input
2
1 1
-1 2
2 -1
-2 -2

Output
NO

Note

In the first test case, the front numbers create the balanced sequence $[1,4,-4,-1,5,3,2,-2,-3,-5]$ and the back numbers create the balanced sequence $[3,-3,4,-4,1,-1,2,5,-5,-2]$.

In the second test case, the cards are given in an order so that the front numbers are balanced, but the back numbers create the unbalanced sequence $[1,2,-1,-2]$. If we swapped the second and third cards, we would balance the back numbers and unbalance the front numbers. But there is no order that balances both.


text:
# Mistake

Nicol mistake when calculate in school. Instead of add number 20 subtract it.

What is the difference between the result and the right result?

Result

x =  -40

#### Solution:

$a = n + 20 \ \\ b = n- 20 \ \\ \ \\ x = b-a = (n - 20) - (n + 20) \ \\ x= -2\cdot 20 = -40$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
Do you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation?

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text:
# Lower bound on expectation of concave function

Let $X$ be a random variable, and let $f$ be a concave function.

Are there any known lower bounds for (or methods of lower bounding) $\mathbb{E}[f(X)]$?

Jensen's inequality only gives an upper bound on the above.

• An obvious lower bound is $\text{E}[f(X)]\geqslant \inf_{x\in S} f(x)$ where $S$ is the support of the random variable $X$. – StijnDeVuyst Nov 27 '16 at 13:03
• That bound given by @Stijn is the best possible in general: since there is no assumption connecting $X$ to $f$, $X$ could have all its probability concentrated in an interval $[\inf f, \inf f+\epsilon)$ for arbitrarily small positive $\epsilon$. – whuber Nov 27 '16 at 15:37

I don't know if this is helpful, but for any function $f$ set $Z \equiv f(X)$. Then by Jensen's Inequality
$$E(Z^2) > [E(Z)]^2 \implies \sqrt {E(Z^2)} > \big |E(Z)\big |$$
$$\implies -\sqrt {E(Z^2)} < E(Z) < \sqrt {E(Z^2)}$$
$$\implies -\sqrt {E([f(X)]^2)} < E[f(X)] < \sqrt {E([f(X)]^2)}$$


text:
# MATLAB: Confint and predint bounds trouble

boundsconfintMATLABstatistics

Hi, I'm having trouble understanding how confint and predint work. I thought that since predint takes into account the uncertanty of a predicted measurement, the predint bounds would be broader than the confint bounds (for the same confidence level, i.e. 95%). But whenever I fit simple data and I draw the confint bounds I find them broader than the predint bounds (for any type of bounds.). So what is it that I'm missing out?


text:
# Tag Info

Sysadmins have switched from one password hash algorithm to another many times. There is a standard process for this switch. The Modular Crypt Format (MCF) is the standard scheme for formatting encrypted passwords, as used by passwd, crypt, etc. (see Why are there $signs in my passwd file? ). MCF uses a single column to store the password hash algorithm ... 6 The GQ identification scheme is essentially a zero-knowledge proof of a value$x$such that$x^\mu \equiv J \pmod N$where$N$is an RSA modulus and$(\mu,N)$are system parameters and$J$is known to the verifier and$x$only known to the prover. Now your question is not directly concerned with the aforementioned proof where a user shows the possession of ... 6 A is acting as a square-root oracle in that protocol. We can use that oracle to factor$n$and break the scheme. Suppose you are an attacker that wants to impersonate A. You: Pick a random$m$; Send$m^2$to A; Compute$p = \gcd(m_1 - m, n)$, thus factoring$n$. This works with probability$1/2$for each attempt. 5 The protocol's description includes "Alice then encrypts$R_B$with her private key". This has no standard meaning. Comments have clarified it is used an "RSA encryption scheme with proper padding" and I am taking as granted that encryption of$R_B$using the private key half of$K_A$, denoted$K_A^-(R_B)$, is obtained by padding$R_B$as in encryption, then ... 5 Yes, this looks fine. I assume$A$and$B_i$are trusted parties. The protocol as I understand it looks like this:$A$,$B_1$,…,$B_n$agree on a secret key k.$A$broadcasts messages ($m_1$,MAC($m_1$,$k$)), … , ($m_j$,MAC($m_j$,$k$) which$B_1$,…,$B_n$receive and authenticate. I assume$A$and$B_i$are trusted parties, so no$B_i$will itself ... 5 Digital signatures are used to solve this type of problem. That is, a way for$A$to sign the message for$B$so that$B$is highly confident that$A$signed the message in question. There are lots of signature schemes out there, such as RSA signing, DSA, and others. A MAC is not strictly a digital signature, but has a subset of that functionality and may ... 5 As cybergibbons notes in his answer, the court decision itself is quite interesting reading. In particular, while the details of the "Megamos algorithm" itself are obviously not given in the court decision, the manner in which it is used is described in a surprisingly clear manner in paragraphs 4 and 5: "In detail the way this works is as follows: both ... 5 The usual answer is that a salt can be make public; if that was a problem, then the salt would not be called a "salt" but a "key". In some protocols, unauthenticated obtention of the salt is the norm, and is not considered to be a problem. E.g. with SRP, a password-authenticated key exchange, where any salting and hashing must necessarily occur client-side. ... 5 Authentication can either mean entity authentication or data authentication. Data authentication is a means to demonstrate that some specific data originates from a specific source and has not been modified during transmission and/or upon storage. It can be achieved by the use of digital signatures in a public key, i.e., asymmetric, setting or message ... 4 Start by naming the current password type (perhaps 3des-cfb) and storing that value on each record as an additional column in the database. You can now start updating existing records. Enumerate over the values in reasonably sized chunks and hash them securely. bcrypt is a good choice. Store the bcrypted value in the database and change the password type ... 4 An OCB like mode seems impossible with stream-ciphers. It's coupled tightly to the concept of a keyed permutation i.e. a (tweakable) block-cipher. Many authenticated encryption actually combine two distinct primitives. It's just that the specification and API only expose the combination. Essentially these xor a key-stream into the message to encrypt it ... 4 “Does anyone know what the Megamos algorithm is?” Yes. According to http://www.bicotech.com/doc/megamos_cr.pdf (PDF) — which provides the best description from my point of view — the Megamos crypto is: MEGAMOS CRYPTO Read-Write High Security Device - Memory organisation Description The MC is a high security Read-Write RFID Transponder. A ... 4 Yes, there are several ways in which Mallory could pretend to be Amy. One obvious way, which doesn't even involve Amy herself in any way, would be for Mallory to perform steps 1 and 2 of the protocol normally, as if he were Amy. Then, given Betty's nonce$n_b$, Mallory can start a second, parallel instance of the protocol, again pretending to be Amy, and ... 4 Google (and other companies) have decided to enable one-time passwords for their 2-factor authentication as a step to improve password security. Here is the webpage that explains what Google is doing in more detail (including source code): https://code.google.com/p/google-authenticator/ In a nutshell, they implement two IETF RFCs, namely RFC 6238 and RFC ... 4 That was a bad edit to Wikipedia. The phrase "Carry-forward verification" is not a standard, well-known term in the cryptographic literature. It should not have been included in Wikipedia without a reference to something more specific. But oh well, no one is perfect, sometimes these things happen. Your request for an elaborate survey of MITM defence is ... 4 For P2P authentication, you can go for web of trust concept. Simply this means, if someone is trusted by people you can trust, you can also trust that unknown person. In OpenPGP, a certificate can be signed by other users who trust the association of that public key with the person or entity listed in the certificate. So trust relationships can be ... 4 You may be interested in something like the Cryptographically Generated Address (CGA) from RFC 3972. CGA is used in Secure Neighbor Discovery Protocol (SEND) of RFC 3971 to bind a public key to an IPv6 address. The basic idea of CGA is to generate part of the IPv6 address by computing a cryptographic hash over the public key. The corresponding private key ... 4 Because$r$is not guaranteed to be a Quadratic Residue, so for random$r$there wouldn't be$m_1$such that$r \equiv m_1^2(\mod n)$, therefore authentication will be impossible in this case. 4 No. Cryptography alone cannot solve this problem. Solving this problem requires a combination of technical (e.g., cryptography, systems security) and non-technical (e.g., legal, regulatory, contractual) solutions. Even the technical part is not solely a cryptography question; it as much about systems security. 3 I think the solution to your problem is a digital signature as CodesInChaos pointed out. Here is how it would work: When a user registers with your server they are given a token which will consist of (at a minimum) the user's id. The server also uses its private key to digitally sign the token (or more likely a hash of the token). Now, say user1 and user2 ... 3 It is important to consider the model in which security is proved. In this case, the attack proceeds in two phases: first a phase where the adversary as a verifier interacts with the honest prover, then a phase where the adversary as a prover interacts with an honest verifier. The adversary wins if the honest prover accepts. The paper proves, under ... 3 Since you do not describe why TLS Handshake and IKE are appropriate in your situation, and as long as you don't describe your situation, it's hard to really help you. Also, you haven't stated if it's only IKE that's not appropriate, or if that also includes IKEv2 (which improved the IKE protocol). Therefore, I'll simply assume you meant both. As an ... 3 Yes, because Mallory can use Amy and Betty to get any encrypted nonce; Amy and Betty are oracles for Mallory. She just has to send the nonce she has to encrypt to either one of them and they perform the task for her (in another "authentication attempt", using step 1 & 2). Usually you protect against this kind of situation by performing an encryption ... 3 HMAC-SHA-256 is sufficient for up to 256 bit security. Confer e.g. NIST SP 800-107. This recommendation is based on the premise that collision attacks are infeasible against common uses of HMAC, and that you consequently only have to worry about primary pre-image attacks that attempt to recover the secret key (and use this for forging subsequent messages). ... 3 "The problem I face is that all authentication protocols require some kind of shared secret between the devices and the server." Similarly, the problem I face is that$\:1 = 0 \;\;$. Since your post includes "(eg: a digital signature of a random string by the server)", I will assume that the server has a signature key-pair and the devices alrady know ... 3 The Wikipedia article points out a good reason for using a random challenge value: preventing replay attacks. If the hash was always the same (as the hash of the symmetric key would be), then having listened in on one challenge-response cycle, a malicious listener could pass further handshake tests. 3 In addition to the earlier remarks about the missing background of your question please also consider that TLS and IKEv2 are actually not just a single authentication and key exchange protocol but rather a framework that supports many different AKA protocols. Let us use TLS as an example. In TLS you have the concept of ciphersuites and they allow you to ... 3 The probability of such a "collision" occurring randomly, with honestly-generated RSA keys, is extremely low. Mathematically justifying that assertion can be tiresome, but the idea is the following: Let$s$be a RSA signature, i.e. an integer of size$k$bits for some$k$(e.g.$k = 2048$). We consider RSA keys$(n,e)$where$n$is the modulus (of size$k\$ ...


text:
Lesson plan

# Lesson 6: Finding Side Lengths of Triangles

teaches Common Core State Standards 8.G.B http://www.corestandards.org/the-standards
teaches Common Core State Standards MP8 http://corestandards.org/Math/Practice/MP8
teaches Common Core State Standards 8.G.B.7 http://corestandards.org/Math/Content/8/G/B/7

# Lesson 6: Finding Side Lengths of Triangles

This is the first of three lessons in which students investigate relationships between the side lengths of right and non-right triangles leading to the Pythagorean Theorem.

In the warm-up for this lesson, students notice and wonder about 4 triangles. While there is a lot to notice, one important aspect is whether the triangle is a right triangle or not. This primes them to notice patterns of right and non-right triangles in the other activities in the lesson. In the next two activities, students systematically look at the side lengths of right and non-right triangles for patterns (MP8). By the end of this lesson, they see that for right triangles with legs $$a$$ and $$b$$ and hypotenuse $$c$$, the side lengths are related by $$a^2+b^2=c^2$$. In the next lesson they will prove the Pythagorean Theorem.

Lesson overview

• 6.1 Warm-up: Which One Doesn’t Belong: Triangles (5 minutes)
• 6.2 Activity: A Table of Triangles (15 minutes)
• 6.3 Activity: Meet the Pythagorean Theorem (10 minutes)
• Includes "Are you Ready for More?" extension problem
• Lesson Synthesis
• 6.4 Cool-down: Does a Squared Plus b Squared Equal c Squared? (5 minutes)

Learning goals:

• Comprehend the term “Pythagorean Theorem” (in written and spoken language) as the equation $$a^2+b^2=c^2$$ where $$a$$ and $$b$$ are the lengths of the legs and $$c$$ is the length of the hypotenuse of a right triangle.
• Describe (orally) patterns in the relationships between the side lengths of triangles.
• Determine the exact side lengths of a triangle in a coordinate grid and express them (in writing) using square root notation.

Learning goals (student facing):

• Let’s find triangle side lengths.

Learning targets (student facing):

• I can explain what the Pythagorean Theorem says.

Glossary:

• hypotenuse - The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of a right triangle. Here are some right triangles. Each hypotenuse is labeled.

• legs - The legs of a right triangle are the sides that make the right angle. The legs of a right triangle are the sides that make the right angle. Here are some right triangles. Each leg is labeled.

• Pythagorean Theorem - The Pythagorean Theorem describes the relationship between the side lengths of right triangles.The diagram shows a right triangle with squares built on each side. If we add the areas of the two small squares, we get the area of the larger square. The square of the hypotenuse is equal to the sum of the squares of the legs. This is written as $$a^2+b^2=c^2$$.

• Access the complete Grade 8 glossary.

Standards:

• This lesson builds on the standards:CCSS.5.G.B.4MS.5.G.4 CCSS.7.G.ACCSS.8.EE.A.2MS.8.EE.2
• This lesson builds towards the standards:CCSS.8.G.BCCSS.8.G.B.6MS.8.G.6 MS.8.G

M 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.


text:
Acceptable variation in accuracy of each k fold when using K-Fold Cross Validation?

I have a relatively small dataset consisting of 1432 samples.

I have trained a Random Forest Classifier and performed KFold CV. The results of running 10 Fold CV are as follows:

=== 10 Fold Cross Validation Scores ===

CVFold 1 = 90.2%
CVFold 2 = 87.6%
CVFold 3 = 86.7%
CVFold 4 = 86.7%
CVFold 5 = 83.9%
CVFold 6 = 75.8%
CVFold 7 = 87.2%
CVFold 8 = 82.8%
CVFold 9 = 86.1%
CVFold 10 = 89.3%

Mean Cross Validation Score:  85.6%

I am just not sure how to explain why there are such high variances between some of the folds, i.e from 75.8% in fold 6 to 90.2% in fold 1.

My understanding is that it is simply that the classifier found the samples in the 1st fold (90%) easier to classify than it did in fold 6 (75%) but I'm actually not entirely sure if that is the case.

I understand that each case is different but is it common to have such variances? And is it acceptable?

Edit: some details regarding my data

I have 5 classes, which are imbalanced:

class 1 - 5%

class 2 - 10%

class 3 - 15%

class 4 - 60%

class 5 - 10%

I am using SMOTE to balance the classes.

• The cross validation score variance can be quite high when the dataset has few observations. What's the size of yours (number of samples)? Dec 2 '19 at 21:12
• @RomainReboulleau I have a rather small dataset unfortunately with 1432 samples Dec 3 '19 at 9:59
• Please note, accuraccy is by far worst existing metric, you can base anything on. Please use RepeatedKFold with at least 50 cycles of training and testing. It's hard to get some insight into metrics distribution with just 10 values. Dec 3 '19 at 12:51
• What is your class balance? Dec 3 '19 at 12:51
• You apply resampling only to train split. I suggest you use stratified cv to ensure you maintain your balance when splitting. Dec 3 '19 at 13:27


text:
5

# Read the following description very carefully and then create and fully label diagram of the situation: Once you have diagram; please determine the amount that the ...

## Question

###### Read the following description very carefully and then create and fully label diagram of the situation: Once you have diagram; please determine the amount that the spring will compress in order to stop the crate from moving:Working on a completely frictionless surface, physicist decides to have some fun: The physicist sets up 5 kg bowling ball on level ground, directly in front of our favorite crate full of fruit: The stationary; 31kg crate is connected to a long; horizontal spring with k 250 N/

Read the following description very carefully and then create and fully label diagram of the situation: Once you have diagram; please determine the amount that the spring will compress in order to stop the crate from moving: Working on a completely frictionless surface, physicist decides to have some fun: The physicist sets up 5 kg bowling ball on level ground, directly in front of our favorite crate full of fruit: The stationary; 31kg crate is connected to a long; horizontal spring with k 250 N/m and the other end of the spring is connected to wall: The physicist decides to exert 2000 N horizontal force for 0.1m,at the center of the stationary bowling ball: The bowling ball was allowed to slide into the crate, creating an elastic collision with the crate and ending up with velocity of 1.44 mls in the opposite direction: The crate leaves the collision with a velocity that will eventually g0 to zero,as the spring accelerates the crate torest: Extra Credit: If this was normal surface;with friction; would the spring stretch more or less? In order to receive any credit on this part; YOU must justify your answer with conceptual and/or mathematical support A diagram wouldn't be= bad idea;either_. MacBool

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