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## anonymous 3 years ago Find the position function s(t) from the given velocity or acceleration function and initial value(s). Assume that units are feet and seconds. v(t) = 40 – sin t, s(0) = 2 Delete Cancel Submit
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1. anonymous
• 3 years ago
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@SithsAndGiggles can u help?
2. anonymous
• 3 years ago
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$v(t)=s'(t),\text{ so }\int v(t)~dt+C=s(t)$ So the first thing you do find the indefinite integral of the given $$v(t)$$: $\int(40-\sin t)~dt$
3. anonymous
• 3 years ago
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okay
4. anonymous
• 3 years ago
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ok so s(t) = 40t + cos(t) + c = 40(0) + cos(0) + c = 2
5. anonymous
• 3 years ago
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Yeah, that looks right, but when you write it out you have two separate equations: the first being the more general $$s(t)=40t+\cos t+C$$ and the second involving the initial values, $$s(0)=40(0)+\cos 0+C=2$$. Anyway, solving for C and plugging it into the first equation gives you your answer.
6. anonymous
• 3 years ago
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so i should plug in 2 into 40t + cost + c right? and okay
7. anonymous
• 3 years ago
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No, you have that $$s(0)=2$$, so you would just plug in 0 for t and set the equation equal to 2. You had it right, but you should have written that step in another equation.
8. anonymous
• 3 years ago
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ohh ok
9. anonymous
• 3 years ago
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so i'm done?
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9993882179260254, "perplexity": 3526.133433660711}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541905.26/warc/CC-MAIN-20161202170901-00155-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://moodle.org/mod/forum/discuss.php?d=220143&parent=958647 | ## Future major features
### MoodleRoom's proposed Outcomes changes
This discussion has been locked because a year has elapsed since the last post. Please start a new discussion topic.
Re: MoodleRoom's proposed Outcomes changes
Folks,
I've just realized that I didn't post the rest of the Instructor Specification. I will get it up by Tuesday. I was working locally on wireframes, etc. I think that many of these questions will be answered when I get the rest of it posted. Sorry for the confusion.
Phill
Average of ratings: - | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8229354619979858, "perplexity": 3211.450360947146}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106984.52/warc/CC-MAIN-20170820185216-20170820205216-00604.warc.gz"} |
http://math.stackexchange.com/questions/122850/where-is-zero-degrees-on-a-graph | # Where is zero degrees on a graph
I am using the following formula to calculate the position of a point after rotation in my web application.
x' = xcos(0) - ysin(0)
y' = xsin(0) + ycos(0)
But where on a graph is zero degrees in Mathematics? In the following image can you tell me where zero degrees is(is it at Point A or Point B?) If it turns out that zero degrees is at point B then I will need to add 90 degrees to all my degree values in order to use the above function, correct?
-
Usually it's at $B$. – Rasmus Mar 21 '12 at 10:41
@Rasmus thanks, so if it was B I would need to add 90(or is it minus) degrees to my angle to use that math function? Ie, because my program considers zero degrees from point A, if I use the formula I will be really working out a rotation for 135 degrees which will be incorrect wont it? – Jake M Mar 21 '12 at 10:46
In trigonometry the argument of every direct trigonometric function, cosine, sine, etc. is the angle measured counterclockwise from the positive x-axis. See this Wikipedia section. So it corresponds to B. – Américo Tavares Mar 21 '12 at 10:52
Is there a connection between this question and this other question posted a quarter of an hour earlier? It sure looks like it; even the $\theta$ written like a $0$ is the same. – joriki Mar 21 '12 at 10:56
The argument of the trigonometric functions in your formulas is not the absolute angle of a point but the relative angle by which the point $(x',y')$ is rotated with respect to the point $(x,y)$. Thus you don't need to know where the angle zero is in absolute terms; all you need to know is whether a positive angle represents a clockwise or a counterclockwise rotation. This you can find out e.g. by rotating the point $(1,0)$ by $\pi/2$ (or $90^\circ$); the result is $(0,1)$, so the rotation is counterclockwise.
If you do ever need to know where the angle $0$ is in absolute terms, it's usually taken to be along the positive $x$ axis, with positive angles between $0$ and $\pi/2$ representing points in the first quadrant, i.e. with positive $x$ and $y$ coordinates.
By the way, I disagree with your claim that "in programming, zero degrees is at point $A$". I think Java counts as programming, and it uses the mathematical convention, in which zero degrees corresponds to $B$; see e.g. the API for java.awt.Graphics.drawArc. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9295414090156555, "perplexity": 381.1366749701017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931004988.25/warc/CC-MAIN-20141125155644-00247-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://newproxylists.com/tag/problem/ | ## i had upgraded magento 1.9 to magento 2.4 but it did not work properly so i decided to get back but when i restored it i can see this problem
error log
[14-Apr-2021 15:18:01 Australia/Sydney] PHP Fatal error: Uncaught Error: Class ‘MagentoFrameworkAppErrorHandler’ not found in /home/martarab/public_html/bin/magento:20
Stack trace:
#0 {main}
thrown in /home/martarab/public_html/bin/magento on line 20
## sequences and series – New chess grains problem
The original chess grains math problem stablished that on the first square of the chess board you put a rice grain, on the second you put two, the third four and so on, everytime having in the next square available the double from the last one. Now my question is how to calculate the rice on any square and on the entire board if instead of doubling the quantity of the last one, you multiply it by itself (except for the fisrt square of course, that would be too easy). In the first you will have one, in the second 2, in the third you have 4 and in the fourth 16, then 256, 65536, and so on.
## Shannon information problem
Given a discrete distribution $$P(X_1,X_2)$$, is it possible to build $$P(X_1,X_2,Y)$$ such that
$$I(X_1;X_2) = I(Y;X_1,X_2)$$
where $$I$$ is Shannon’s mutual information?
## Data files half empty in SQL Server. Is that a problem?
I have a multiple TB database and I have been doing some clean-up and dropped many tables. So now the data files are half empty. If I don’t care about releasing the space to the operating system, is there any other reason to shrink the files?
I am thinking that now at least I do not need to worry about auto-growth settings, which might slow things down if I add a large table with ETL.
## algorithms – How to prove that one problem belongs to class P?
Is there any typical proving method when proving that one problem belongs to class P?
For example, when proving that
The problem of finding n to the kth power is the P problem. (Each multiplication can be done in unit time)
If you present an algorithm that can solve this problem with $$O (log n)$$, can it be a proof?
## Help with a geometry problem with a triangle and its orthocentre
Let ABC a traingle inscribed in a circle with radius 1 and center O. Let the angle AOM=150 where M is the middle BC. Let H the orthocentre of the triangle. If A,B,C are selected such that Oh has the minimum lenght, than the lenght of BC is
A: $$sqrt{15}$$
B: $$sqrt{13}/2$$
C: $$sqrt{3}/2$$
D:$$sqrt{13}/4$$
I made a sketch and tried to apply the Sylvester’s theoreme and to solve the problem with vectors but did not succed. Could you please help me?
## WordPress warning problem
Warning: Declaration of ET_Theme_Builder_Woocommerce_Product_Variable_Placeholder::get_available_variations() should be compatible with WC_Product_Variable::get_available_variations(\$return = ‘array’) in /homepages/0/d786463972/htdocs/clickandbuilds/test/wp-content/themes/Divi/includes/builder/frontend-builder/theme-builder/WoocommerceProductVariablePlaceholder.php on line
## st.statistics – Binary Regression : Is this an open problem in Mathematics/Statistics?
Let $$X$$ be a random variable which takes values from $$Omega = (0,1)^m$$ with a probability distribution $$p(x)$$. Assume $$p$$ is a BV function with non zero total variation and $$p(x)>0forall xinOmega$$. There is a discrete random variable $$Y$$ which takes values from $${0,1}$$ and depends on $$X$$ with $$P(Y=1/X=x) = eta(x)$$. Assume that $$eta$$ is also a BV function with non zero total variation and no removable discontinuities.
Binary Regression Problem
Given no other information except $$n$$ samples of the random variable pair $$(X,Y)$$ drawn iid, that is $$(x_1,y_1),(x_2,y_2),ldots(x_n,y_n)$$ one need to give a method for computing $$tilde{eta}_n$$, an estimate of $$eta$$ such that $$limlimits_{ntoinfty}|tilde{eta}_n-eta|_{L^2(Omega)} = 0$$
This problem I believe is open (needs confirmation as I have little knowledge on statistics literature). There are methods like $$k$$-nearest neighbours method, which solves when $$eta$$ belongs to a narrower class of absolutely continuous functions. There seem to be some methods when $$eta$$ is a Lipschitz continous function, but with known Lipschitz constant.
For $$eta$$ belongs to class of all BV functions with nonzero total variation (a wider class), I believe I have come up with a method and proof of convergence. Is this sufficiently interesting to be considered for a mathematics journal or it should be considered for a (mathematical)statistics journal? How interesting is this problem for mathematicians and/or statisticians? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 29, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8235572576522827, "perplexity": 551.0778287973668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038860318.63/warc/CC-MAIN-20210418194009-20210418224009-00490.warc.gz"} |
https://www.physicsforums.com/threads/fermi-level-in-semiconductors.532072/ | # Fermi level in semiconductors
1. Sep 20, 2011
### ravi_nigam
1. Fermi level is also defined as highest energy level at which electron can exist at 0K then it should be top of valence band but why is it in between conduction band and valence band ( in forbidden gap)? It should be in conduction band or valence band. How to explain this?
2. If density of state is 0 then no state should be there in semiconductor forbidden gap? So why we study of probability of occupancy of state in these conditions (semiconductor forbidden gap case).
3. How we calculate fermi level? Please suggest a very fundamental book; more basic than kittel
2. Sep 20, 2011
### Bill_K
Consider the following simple example. Suppose you have just two energy levels A and B, and suppose the number of electrons present is just enough to fill the lower level A at T = 0. At some higher temperature T > 0, n electrons will be excited from A and move to B. Now where is EF? Answer: Somewhere between A and B. In fact it is half way between A and B. In this example too, EF is at an energy level where an electron cannot exist.
EF is not defined, as you claim, to be the highest level occupied at T = 0. EF is just a calculated value. It is a parameter in the Fermi distribution. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9314311742782593, "perplexity": 537.7776660459075}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511806.8/warc/CC-MAIN-20181018105742-20181018131242-00181.warc.gz"} |
https://en.wikipedia.org/wiki/Distributed_application | # Distributed computing
(Redirected from Distributed application)
Distributed computing is a field of computer science that studies distributed systems. A distributed system is a system whose components are located on different networked computers, which communicate and coordinate their actions by passing messages to one another from any system.[1][2] The components interact with one another in order to achieve a common goal. Three significant challenges of distributed systems are: maintaining concurrency of components, overcoming the lack of a global clock, and managing the independent failure of components.[1] When a component of one system fails, the entire system does not fail.[3] Examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications.
A computer program that runs within a distributed system is called a distributed program,[4] and distributed programming is the process of writing such programs.[5] There are many different types of implementations for the message passing mechanism, including pure HTTP, RPC-like connectors and message queues.[6]
Distributed computing also refers to the use of distributed systems to solve computational problems. In distributed computing, a problem is divided into many tasks, each of which is solved by one or more computers,[7] which communicate with each other via message passing.[8]
## Introduction
The word distributed in terms such as "distributed system", "distributed programming", and "distributed algorithm" originally referred to computer networks where individual computers were physically distributed within some geographical area.[9] The terms are nowadays used in a much wider sense, even referring to autonomous processes that run on the same physical computer and interact with each other by message passing.[8]
While there is no single definition of a distributed system,[10] the following defining properties are commonly used as:
• There are several autonomous computational entities (computers or nodes), each of which has its own local memory.[11]
• The entities communicate with each other by message passing.[12]
A distributed system may have a common goal, such as solving a large computational problem;[13] the user then perceives the collection of autonomous processors as a unit. Alternatively, each computer may have its own user with individual needs, and the purpose of the distributed system is to coordinate the use of shared resources or provide communication services to the users.[14]
Other typical properties of distributed systems include the following:
• The system has to tolerate failures in individual computers.[15]
• The structure of the system (network topology, network latency, number of computers) is not known in advance, the system may consist of different kinds of computers and network links, and the system may change during the execution of a distributed program.[16]
• Each computer has only a limited, incomplete view of the system. Each computer may know only one part of the input.[17]
## Parallel and distributed computing
(a), (b): a distributed system.
(c): a parallel system.
Distributed systems are groups of networked computers which share a common goal for their work. The terms "concurrent computing", "parallel computing", and "distributed computing" have much overlap, and no clear distinction exists between them.[18] The same system may be characterized both as "parallel" and "distributed"; the processors in a typical distributed system run concurrently in parallel.[19] Parallel computing may be seen as a particular tightly coupled form of distributed computing,[20] and distributed computing may be seen as a loosely coupled form of parallel computing.[10] Nevertheless, it is possible to roughly classify concurrent systems as "parallel" or "distributed" using the following criteria:
• In parallel computing, all processors may have access to a shared memory to exchange information between processors.[21]
• In distributed computing, each processor has its own private memory (distributed memory). Information is exchanged by passing messages between the processors.[22]
The figure on the right illustrates the difference between distributed and parallel systems. Figure (a) is a schematic view of a typical distributed system; the system is represented as a network topology in which each node is a computer and each line connecting the nodes is a communication link. Figure (b) shows the same distributed system in more detail: each computer has its own local memory, and information can be exchanged only by passing messages from one node to another by using the available communication links. Figure (c) shows a parallel system in which each processor has a direct access to a shared memory.
The situation is further complicated by the traditional uses of the terms parallel and distributed algorithm that do not quite match the above definitions of parallel and distributed systems (see below for more detailed discussion). Nevertheless, as a rule of thumb, high-performance parallel computation in a shared-memory multiprocessor uses parallel algorithms while the coordination of a large-scale distributed system uses distributed algorithms.[23]
## History
The use of concurrent processes which communicate through message-passing has its roots in operating system architectures studied in the 1960s.[24] The first widespread distributed systems were local-area networks such as Ethernet, which was invented in the 1970s.[25]
ARPANET, one of the predecessors of the Internet, was introduced in the late 1960s, and ARPANET e-mail was invented in the early 1970s. E-mail became the most successful application of ARPANET,[26] and it is probably the earliest example of a large-scale distributed application. In addition to ARPANET (and its successor, the global Internet), other early worldwide computer networks included Usenet and FidoNet from the 1980s, both of which were used to support distributed discussion systems.[27]
The study of distributed computing became its own branch of computer science in the late 1970s and early 1980s. The first conference in the field, Symposium on Principles of Distributed Computing (PODC), dates back to 1982, and its counterpart International Symposium on Distributed Computing (DISC) was first held in Ottawa in 1985 as the International Workshop on Distributed Algorithms on Graphs.[28]
## Architectures
Various hardware and software architectures are used for distributed computing. At a lower level, it is necessary to interconnect multiple CPUs with some sort of network, regardless of whether that network is printed onto a circuit board or made up of loosely coupled devices and cables. At a higher level, it is necessary to interconnect processes running on those CPUs with some sort of communication system.[29]
Distributed programming typically falls into one of several basic architectures: client–server, three-tier, n-tier, or peer-to-peer; or categories: loose coupling, or tight coupling.[30]
• Client–server: architectures where smart clients contact the server for data then format and display it to the users. Input at the client is committed back to the server when it represents a permanent change.
• Three-tier: architectures that move the client intelligence to a middle tier so that stateless clients can be used. This simplifies application deployment. Most web applications are three-tier.
• n-tier: architectures that refer typically to web applications which further forward their requests to other enterprise services. This type of application is the one most responsible for the success of application servers.
• Peer-to-peer: architectures where there are no special machines that provide a service or manage the network resources.[31]: 227 Instead all responsibilities are uniformly divided among all machines, known as peers. Peers can serve both as clients and as servers.[32] Examples of this architecture include BitTorrent and the bitcoin network.
Another basic aspect of distributed computing architecture is the method of communicating and coordinating work among concurrent processes. Through various message passing protocols, processes may communicate directly with one another, typically in a master/slave relationship. Alternatively, a "database-centric" architecture can enable distributed computing to be done without any form of direct inter-process communication, by utilizing a shared database.[33] Database-centric architecture in particular provides relational processing analytics in a schematic architecture allowing for live environment relay. This enables distributed computing functions both within and beyond the parameters of a networked database.[34]
## Applications
Reasons for using distributed systems and distributed computing may include:
1. The very nature of an application may require the use of a communication network that connects several computers: for example, data produced in one physical location and required in another location.
2. There are many cases in which the use of a single computer would be possible in principle, but the use of a distributed system is beneficial for practical reasons. For example, it may be more cost-efficient to obtain the desired level of performance by using a cluster of several low-end computers, in comparison with a single high-end computer. A distributed system can provide more reliability than a non-distributed system, as there is no single point of failure. Moreover, a distributed system may be easier to expand and manage than a monolithic uniprocessor system.[35]
## Examples
Examples of distributed systems and applications of distributed computing include the following:[36]
## Theoretical foundations
### Models
Many tasks that we would like to automate by using a computer are of question–answer type: we would like to ask a question and the computer should produce an answer. In theoretical computer science, such tasks are called computational problems. Formally, a computational problem consists of instances together with a solution for each instance. Instances are questions that we can ask, and solutions are desired answers to these questions.
Theoretical computer science seeks to understand which computational problems can be solved by using a computer (computability theory) and how efficiently (computational complexity theory). Traditionally, it is said that a problem can be solved by using a computer if we can design an algorithm that produces a correct solution for any given instance. Such an algorithm can be implemented as a computer program that runs on a general-purpose computer: the program reads a problem instance from input, performs some computation, and produces the solution as output. Formalisms such as random-access machines or universal Turing machines can be used as abstract models of a sequential general-purpose computer executing such an algorithm.[38][39]
The field of concurrent and distributed computing studies similar questions in the case of either multiple computers, or a computer that executes a network of interacting processes: which computational problems can be solved in such a network and how efficiently? However, it is not at all obvious what is meant by "solving a problem" in the case of a concurrent or distributed system: for example, what is the task of the algorithm designer, and what is the concurrent or distributed equivalent of a sequential general-purpose computer?[citation needed]
The discussion below focuses on the case of multiple computers, although many of the issues are the same for concurrent processes running on a single computer.
Three viewpoints are commonly used:
Parallel algorithms in shared-memory model
• All processors have access to a shared memory. The algorithm designer chooses the program executed by each processor.
• One theoretical model is the parallel random-access machines (PRAM) that are used.[40] However, the classical PRAM model assumes synchronous access to the shared memory.
• Shared-memory programs can be extended to distributed systems if the underlying operating system encapsulates the communication between nodes and virtually unifies the memory across all individual systems.
• A model that is closer to the behavior of real-world multiprocessor machines and takes into account the use of machine instructions, such as Compare-and-swap (CAS), is that of asynchronous shared memory. There is a wide body of work on this model, a summary of which can be found in the literature.[41][42]
Parallel algorithms in message-passing model
• The algorithm designer chooses the structure of the network, as well as the program executed by each computer.
• Models such as Boolean circuits and sorting networks are used.[43] A Boolean circuit can be seen as a computer network: each gate is a computer that runs an extremely simple computer program. Similarly, a sorting network can be seen as a computer network: each comparator is a computer.
Distributed algorithms in message-passing model
• The algorithm designer only chooses the computer program. All computers run the same program. The system must work correctly regardless of the structure of the network.
• A commonly used model is a graph with one finite-state machine per node.
In the case of distributed algorithms, computational problems are typically related to graphs. Often the graph that describes the structure of the computer network is the problem instance. This is illustrated in the following example.[citation needed]
### An example
Consider the computational problem of finding a coloring of a given graph G. Different fields might take the following approaches:
Centralized algorithms[citation needed]
• The graph G is encoded as a string, and the string is given as input to a computer. The computer program finds a coloring of the graph, encodes the coloring as a string, and outputs the result.
Parallel algorithms
• Again, the graph G is encoded as a string. However, multiple computers can access the same string in parallel. Each computer might focus on one part of the graph and produce a coloring for that part.
• The main focus is on high-performance computation that exploits the processing power of multiple computers in parallel.
Distributed algorithms
• The graph G is the structure of the computer network. There is one computer for each node of G and one communication link for each edge of G. Initially, each computer only knows about its immediate neighbors in the graph G; the computers must exchange messages with each other to discover more about the structure of G. Each computer must produce its own color as output.
• The main focus is on coordinating the operation of an arbitrary distributed system.[citation needed]
While the field of parallel algorithms has a different focus than the field of distributed algorithms, there is much interaction between the two fields. For example, the Cole–Vishkin algorithm for graph coloring[44] was originally presented as a parallel algorithm, but the same technique can also be used directly as a distributed algorithm.
Moreover, a parallel algorithm can be implemented either in a parallel system (using shared memory) or in a distributed system (using message passing).[45] The traditional boundary between parallel and distributed algorithms (choose a suitable network vs. run in any given network) does not lie in the same place as the boundary between parallel and distributed systems (shared memory vs. message passing).
### Complexity measures
In parallel algorithms, yet another resource in addition to time and space is the number of computers. Indeed, often there is a trade-off between the running time and the number of computers: the problem can be solved faster if there are more computers running in parallel (see speedup). If a decision problem can be solved in polylogarithmic time by using a polynomial number of processors, then the problem is said to be in the class NC.[46] The class NC can be defined equally well by using the PRAM formalism or Boolean circuits—PRAM machines can simulate Boolean circuits efficiently and vice versa.[47]
In the analysis of distributed algorithms, more attention is usually paid on communication operations than computational steps. Perhaps the simplest model of distributed computing is a synchronous system where all nodes operate in a lockstep fashion. This model is commonly known as the LOCAL model. During each communication round, all nodes in parallel (1) receive the latest messages from their neighbours, (2) perform arbitrary local computation, and (3) send new messages to their neighbors. In such systems, a central complexity measure is the number of synchronous communication rounds required to complete the task.[48]
This complexity measure is closely related to the diameter of the network. Let D be the diameter of the network. On the one hand, any computable problem can be solved trivially in a synchronous distributed system in approximately 2D communication rounds: simply gather all information in one location (D rounds), solve the problem, and inform each node about the solution (D rounds).
On the other hand, if the running time of the algorithm is much smaller than D communication rounds, then the nodes in the network must produce their output without having the possibility to obtain information about distant parts of the network. In other words, the nodes must make globally consistent decisions based on information that is available in their local D-neighbourhood. Many distributed algorithms are known with the running time much smaller than D rounds, and understanding which problems can be solved by such algorithms is one of the central research questions of the field.[49] Typically an algorithm which solves a problem in polylogarithmic time in the network size is considered efficient in this model.
Another commonly used measure is the total number of bits transmitted in the network (cf. communication complexity).[50] The features of this concept are typically captured with the CONGEST(B) model, which is similarly defined as the LOCAL model, but where single messages can only contain B bits.
### Other problems
Traditional computational problems take the perspective that the user asks a question, a computer (or a distributed system) processes the question, then produces an answer and stops. However, there are also problems where the system is required not to stop, including the dining philosophers problem and other similar mutual exclusion problems. In these problems, the distributed system is supposed to continuously coordinate the use of shared resources so that no conflicts or deadlocks occur.
There are also fundamental challenges that are unique to distributed computing, for example those related to fault-tolerance. Examples of related problems include consensus problems,[51] Byzantine fault tolerance,[52] and self-stabilisation.[53]
Much research is also focused on understanding the asynchronous nature of distributed systems:
### Election
Coordinator election (or leader election) is the process of designating a single process as the organizer of some task distributed among several computers (nodes). Before the task is begun, all network nodes are either unaware which node will serve as the "coordinator" (or leader) of the task, or unable to communicate with the current coordinator. After a coordinator election algorithm has been run, however, each node throughout the network recognizes a particular, unique node as the task coordinator.[57]
The network nodes communicate among themselves in order to decide which of them will get into the "coordinator" state. For that, they need some method in order to break the symmetry among them. For example, if each node has unique and comparable identities, then the nodes can compare their identities, and decide that the node with the highest identity is the coordinator.[57]
The definition of this problem is often attributed to LeLann, who formalized it as a method to create a new token in a token ring network in which the token has been lost.[58]
Coordinator election algorithms are designed to be economical in terms of total bytes transmitted, and time. The algorithm suggested by Gallager, Humblet, and Spira [59] for general undirected graphs has had a strong impact on the design of distributed algorithms in general, and won the Dijkstra Prize for an influential paper in distributed computing.
Many other algorithms were suggested for different kinds of network graphs, such as undirected rings, unidirectional rings, complete graphs, grids, directed Euler graphs, and others. A general method that decouples the issue of the graph family from the design of the coordinator election algorithm was suggested by Korach, Kutten, and Moran.[60]
In order to perform coordination, distributed systems employ the concept of coordinators. The coordinator election problem is to choose a process from among a group of processes on different processors in a distributed system to act as the central coordinator. Several central coordinator election algorithms exist.[61]
### Properties of distributed systems
So far the focus has been on designing a distributed system that solves a given problem. A complementary research problem is studying the properties of a given distributed system.[62][63]
The halting problem is an analogous example from the field of centralised computation: we are given a computer program and the task is to decide whether it halts or runs forever. The halting problem is undecidable in the general case, and naturally understanding the behaviour of a computer network is at least as hard as understanding the behaviour of one computer.[64]
However, there are many interesting special cases that are decidable. In particular, it is possible to reason about the behaviour of a network of finite-state machines. One example is telling whether a given network of interacting (asynchronous and non-deterministic) finite-state machines can reach a deadlock. This problem is PSPACE-complete,[65] i.e., it is decidable, but not likely that there is an efficient (centralised, parallel or distributed) algorithm that solves the problem in the case of large networks.
## Notes
1. ^ a b Tanenbaum, Andrew S.; Steen, Maarten van (2002). Distributed systems: principles and paradigms. Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-088893-1.
2. ^ "Distributed Programs". Texts in Computer Science. London: Springer London. 2010. pp. 373–406. doi:10.1007/978-1-84882-745-5_11. ISBN 978-1-84882-744-8. ISSN 1868-0941. Systems consist of a number of physically distributed components that work independently using their private storage, but also communicate from time to time by explicit message passing. Such systems are called distributed systems.
3. ^ Dusseau & Dusseau 2016, p. 1-2.
4. ^ "Distributed Programs". Texts in Computer Science. London: Springer London. 2010. pp. 373–406. doi:10.1007/978-1-84882-745-5_11. ISBN 978-1-84882-744-8. ISSN 1868-0941. Distributed programs are abstract descriptions of distributed systems. A distributed program consists of a collection of processes that work concurrently and communicate by explicit message passing. Each process can access a set of variables which are disjoint from the variables that can be changed by any other process.
5. ^ Andrews (2000). Dolev (2000). Ghosh (2007), p. 10.
6. ^ Magnoni, L. (2015). "Modern Messaging for Distributed Sytems (sic)". Journal of Physics: Conference Series. 608 (1): 012038. doi:10.1088/1742-6596/608/1/012038. ISSN 1742-6596.
7. ^
8. ^ a b Andrews (2000), p. 291–292. Dolev (2000), p. 5.
9. ^ Lynch (1996), p. 1.
10. ^ a b Ghosh (2007), p. 10.
11. ^ Andrews (2000), pp. 8–9, 291. Dolev (2000), p. 5. Ghosh (2007), p. 3. Lynch (1996), p. xix, 1. Peleg (2000), p. xv.
12. ^ Andrews (2000), p. 291. Ghosh (2007), p. 3. Peleg (2000), p. 4.
13. ^ Ghosh (2007), p. 3–4. Peleg (2000), p. 1.
14. ^ Ghosh (2007), p. 4. Peleg (2000), p. 2.
15. ^ Ghosh (2007), p. 4, 8. Lynch (1996), p. 2–3. Peleg (2000), p. 4.
16. ^ Lynch (1996), p. 2. Peleg (2000), p. 1.
17. ^ Ghosh (2007), p. 7. Lynch (1996), p. xix, 2. Peleg (2000), p. 4.
18. ^ Ghosh (2007), p. 10. Keidar (2008).
19. ^ Lynch (1996), p. xix, 1–2. Peleg (2000), p. 1.
20. ^ Peleg (2000), p. 1.
21. ^ Papadimitriou (1994), Chapter 15. Keidar (2008).
22. ^ See references in Introduction.
23. ^ Bentaleb, A.; Yifan, L.; Xin, J.; et al. (2016). "Parallel and Distributed Algorithms" (PDF). National University of Singapore. Retrieved 20 July 2018.
24. ^ Andrews (2000), p. 348.
25. ^ Andrews (2000), p. 32.
26. ^
27. ^ Banks, M. (2012). On the Way to the Web: The Secret History of the Internet and its Founders. Apress. pp. 44–5. ISBN 9781430250746.
28. ^ Tel, G. (2000). Introduction to Distributed Algorithms. Cambridge University Press. pp. 35–36. ISBN 9780521794831.
29. ^ Ohlídal, M.; Jaroš, J.; Schwarz, J.; et al. (2006). "Evolutionary Design of OAB and AAB Communication Schedules for Interconnection Networks". In Rothlauf, F.; Branke, J.; Cagnoni, S. (eds.). Applications of Evolutionary Computing. Springer Science & Business Media. pp. 267–78. ISBN 9783540332374.
30. ^ "Real Time And Distributed Computing Systems" (PDF). ISSN 2278-0661. Archived from the original (PDF) on 2017-01-10. Retrieved 2017-01-09. `{{cite journal}}`: Cite journal requires `|journal=` (help)
31. ^ Vigna P, Casey MJ. The Age of Cryptocurrency: How Bitcoin and the Blockchain Are Challenging the Global Economic Order St. Martin's Press January 27, 2015 ISBN 9781250065636
32. ^ Hieu., Vu, Quang (2010). Peer-to-peer computing : principles and applications. Lupu, Mihai., Ooi, Beng Chin, 1961-. Heidelberg: Springer. p. 16. ISBN 9783642035135. OCLC 663093862.
33. ^ Lind P, Alm M (2006), "A database-centric virtual chemistry system", J Chem Inf Model, 46 (3): 1034–9, doi:10.1021/ci050360b, PMID 16711722.
34. ^ Chiu, G (1990). "A model for optimal database allocation in distributed computing systems". Proceedings. IEEE INFOCOM'90: Ninth Annual Joint Conference of the IEEE Computer and Communications Societies.
35. ^ Elmasri & Navathe (2000), Section 24.1.2.
36. ^ Andrews (2000), p. 10–11. Ghosh (2007), p. 4–6. Lynch (1996), p. xix, 1. Peleg (2000), p. xv. Elmasri & Navathe (2000), Section 24.
37. ^ Haussmann, J. (2019). "Cost-efficient parallel processing of irregularly structured problems in cloud computing environments". Journal of Cluster Computing. 22 (3): 887–909. doi:10.1007/s10586-018-2879-3. S2CID 54447518.
38. ^ Toomarian, N.B.; Barhen, J.; Gulati, S. (1992). "Neural Networks for Real-Time Robotic Applications". In Fijany, A.; Bejczy, A. (eds.). Parallel Computation Systems For Robotics: Algorithms And Architectures. World Scientific. p. 214. ISBN 9789814506175.
39. ^ Savage, J.E. (1998). Models of Computation: Exploring the Power of Computing. Addison Wesley. p. 209. ISBN 9780201895391.
40. ^ Cormen, Leiserson & Rivest (1990), Section 30.
41. ^ Herlihy & Shavit (2008), Chapters 2-6.
42. ^ Lynch (1996)
43. ^ Cormen, Leiserson & Rivest (1990), Sections 28 and 29.
44. ^ Cole & Vishkin (1986). Cormen, Leiserson & Rivest (1990), Section 30.5.
45. ^ Andrews (2000), p. ix.
46. ^ Arora & Barak (2009), Section 6.7. Papadimitriou (1994), Section 15.3.
47. ^ Papadimitriou (1994), Section 15.2.
48. ^ Lynch (1996), p. 17–23.
49. ^ Peleg (2000), Sections 2.3 and 7. Linial (1992). Naor & Stockmeyer (1995).
50. ^ Schneider, J.; Wattenhofer, R. (2011). "Trading Bit, Message, and Time Complexity of Distributed Algorithms". In Peleg, D. (ed.). Distributed Computing. Springer Science & Business Media. pp. 51–65. ISBN 9783642240997.
51. ^ Lynch (1996), Sections 5–7. Ghosh (2007), Chapter 13.
52. ^ Lynch (1996), p. 99–102. Ghosh (2007), p. 192–193.
53. ^ Dolev (2000). Ghosh (2007), Chapter 17.
54. ^ Lynch (1996), Section 16. Peleg (2000), Section 6.
55. ^ Lynch (1996), Section 18. Ghosh (2007), Sections 6.2–6.3.
56. ^ Ghosh (2007), Section 6.4.
57. ^ a b Haloi, S. (2015). Apache ZooKeeper Essentials. Packt Publishing Ltd. pp. 100–101. ISBN 9781784398323.
58. ^ LeLann, G. (1977). "Distributed systems - toward a formal approach". Information Processing. 77: 155·160 – via Elsevier.
59. ^ R. G. Gallager, P. A. Humblet, and P. M. Spira (January 1983). "A Distributed Algorithm for Minimum-Weight Spanning Trees" (PDF). ACM Transactions on Programming Languages and Systems. 5 (1): 66–77. doi:10.1145/357195.357200. S2CID 2758285.`{{cite journal}}`: CS1 maint: multiple names: authors list (link)
60. ^ Korach, Ephraim; Kutten, Shay; Moran, Shlomo (1990). "A Modular Technique for the Design of Efficient Distributed Leader Finding Algorithms" (PDF). ACM Transactions on Programming Languages and Systems. 12 (1): 84–101. CiteSeerX 10.1.1.139.7342. doi:10.1145/77606.77610. S2CID 9175968.
61. ^ Hamilton, Howard. "Distributed Algorithms". Retrieved 2013-03-03.
62. ^ "Major unsolved problems in distributed systems?". cstheory.stackexchange.com. Retrieved 16 March 2018.
63. ^ "How big data and distributed systems solve traditional scalability problems". theserverside.com. Retrieved 16 March 2018.
64. ^ Svozil, K. (2011). "Indeterminism and Randomness Through Physics". In Hector, Z. (ed.). Randomness Through Computation: Some Answers, More Questions. World Scientific. pp. 112–3. ISBN 9789814462631.
65. ^ Papadimitriou (1994), Section 19.3.
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https://fncbook.github.io/fnc/localapprox/integration.html | # Numerical integration¶
In calculus you learn that the elegant way to evaluate a definite integral is to apply the Fundamental Theorem of Calculus and find an antiderivative. The connection is so profound and pervasive that it’s easy to overlook that a definite integral is a numerical quantity existing independently of antidifferentiation. In fact, most conceivable integrands have no antiderivative in terms of familiar functions.
Numerical integration1 is done by combining values of the integrand sampled at nodes, much like finite differences. In this section we will assume equally spaced nodes using the definitions
(147)$t_i = a +i h, \quad h=\frac{b-a}{n}, \qquad i=0,\ldots,n.$
The integration formulas are expressed as
(148)$\begin{split} \begin{split} I = \int_a^b f(x)\, dx \approx Q &= h \sum_{i=0}^n w_if(t_i) \\ &= h \bigl[ w_0f(t_0)+w_1f(t_1)+\cdots w_nf(t_n) \bigr]. \end{split}\end{split}$
The constants $$w_i$$ appearing in the formula are called weights. As with finite difference formulas, the weights of numerical integration formulas are chosen independently of the function being integrated, and they determine the formula completely. We can apply quadrature formulas to sequences of data values even if no function is explicitly known to generate them, but for presentation and implementations we assume that we can evaluate $$f(x)$$ anywhere.
A straightforward way to derive integration formulas is to mimic the approach taken for finite differences: find an interpolant and operate exactly on it. If the interpolant is a piecewise polynomial, the result is a Newton–Cotes formula.
## Trapezoid formula¶
One of the most important Newton–Cotes formulas results from integration of the piecewise linear interpolant (see Piecewise linear interpolation). Using the cardinal basis form of the interpolant in (109), we have
$I \approx \int_a^b \sum_{i=0}^n f(t_i) H_i(x)\, dx = \sum_{i=0}^n f(t_i) \left[ \int_a^b H_i(x)\right]\, dx.$
Thus we can identify the weights as $$w_i = h^{-1} \int_a^b H_i(x)\, dx$$. Using areas of triangles, it’s trivial to derive that
(149)$\begin{split} w_i = \begin{cases} 1, & i=1,\ldots,n-1,\\ \frac{1}{2}, & i=0,n. \end{cases}\end{split}$
Putting everything together, the resulting quadrature formula is
(150)$\begin{split} I = \int_a^b f(x)\, dx \approx T_f(n) &= h\left[ \frac{1}{2}f(t_0) + f(t_1) + f(t_2) + \cdots + f(t_{n-1}) + \frac{1}{2}f(t_n) \right]. \end{split}$
This is called the trapezoid formula or trapezoid rule.2 The trapezoid formula results from integration of the piecewise linear interpolant, or equivalently, as illustrated in Fig. 5, from using the area of approximating trapezoids to estimate the area under a curve. The trapezoid formula is the Swiss Army knife of integration formulas. A short implementation is given as trapezoid.
Fig. 5 Trapezoid formula for integration.
Function 55 (trapezoid)
Trapezoid formula for numerical integration.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 """ trapezoid(f,a,b,n) Apply the trapezoid integration formula for integrand f over interval [a,b], broken up into n equal pieces. Returns estimate, vector of nodes, and vector of integrand values at the nodes. """ function trapezoid(f,a,b,n) h = (b-a)/n t = LinRange(a,b,n+1) y = f.(t) T = h * ( sum(y[2:n]) + 0.5*(y[1] + y[n+1]) ) return T,t,y end
In the PL convergence theorem we stated that the pointwise error in a piecewise linear interpolant with equal node spacing $$h$$ is bounded by $$O(h^2)$$ as $$h\rightarrow 0$$. Using $$p$$ to stand for the piecewise linear interpolant, we obtain
$\begin{split}\begin{split} I - T_f(n) = I - \int_a^b p(x)\, dx &= \int_a^b \bigl[f(x)-p(x)\bigr] \, dx \\ &\le (b-a) \max_{x\in[a,b]} |f(x)-p(x)| = O(h^2). \end{split}\end{split}$
Hence the trapezoid formula has second-order error. This fact is embedded rigorously in “one of the most remarkable formulas in mathematics,” the Euler–Maclaurin formula, which may be stated as
(151)$\begin{split} \begin{split} I = \int_a^b f(x)\, dx &= T_f(n) - \frac{h^2}{12} \left[ f'(b)-f'(a) \right] + \frac{h^4}{740} \left[ f'''(b)-f'''(a) \right] + O(h^6) \\ &= T_f(n) - \sum_{k=1}^\infty \frac{B_{2k}h^{2k}}{(2k)!} \left[ f^{(2k-1)}(b)-f^{(2k-1)}(a) \right], \end{split}\end{split}$
where the $$B_{2k}$$ are constants known as Bernoulli numbers. Unless we happen to be fortunate enough to have a function with $$f'(b)=f'(a)$$, we should expect truncation error at second order and no better.
## Extrapolation¶
If evaluations of $$f$$ are computationally expensive, we want to get as much accuracy as possible from them by using a higher-order formula. There are many routes for doing so; for example, we could integrate a not-a-knot cubic spline interpolant. However, splines are difficult to compute by hand, and as a result different methods were developed before computers came on the scene.
Knowing the structure of the error allows the use of extrapolation to improve accuracy. Suppose a quantity $$A_0$$ is approximated by an algorithm $$A(h)$$ with an error expansion
(152)$A_0 = A(h) + c_1 h + c_2 h^2 + c_3 h^3 + \cdots.$
Crucially, it is not necessary to know the values of the error constants $$c_k$$, merely that they exist and are independent of $$h$$. In the case of the trapezoid formula, we have
$I = T_f(n) + c_2 h^2 + c_4 h^{4} + \cdots,$
as proved by the Euler–Maclaurin formula (151). The error constants depend on $$f$$ and can’t be evaluated in general, but we know that this expansion holds.
For convenience we recast the error expansion in terms of $$n=O(h^{-1})$$:
(153)$I = T_f(n) + c_2 n^{-2} + c_4 n^{-4} + \cdots,$
We make the simple observation that
(154)$I = T_f(2n) + \tfrac{1}{4} c_2 n^{-2} + \tfrac{1}{16} c_4 n^{-4} + \cdots.$
It follows that if we combine (153) and (154) correctly, we can cancel out the second-order term in the error. Specifically, define
(155)$S_f(2n) = \frac{1}{3} \Bigl[ 4 T_f(2n) - T_f(n) \Bigr].$
(We associate $$2n$$ rather than $$n$$ with the extrapolated result because of the total number of nodes needed.) Then
(156)$I = S_f(2n) + O(n^{-4}) = b_4 n^{-4} + b_6 n^{-6} + \cdots.$
The formula (155) is called Simpson’s formula. A different presentation and derivation are considered in an exercise.
Equation (156) is another particular error expansion in the form (152), so we can extrapolate again! The details change only a little. Considering that
$I = S_f(4n) = \tfrac{1}{16} b_4 n^{-4} + \tfrac{1}{64} b_6 n^{-6} + \cdots,$
the proper combination this time is
(157)$R_f(4n) = \frac{1}{15} \Bigl[ 16 S_f(4n) - S_f(2n) \Bigr],$
which is sixth-order accurate. Clearly the process can be repeated to get eighth-order accuracy and beyond. Doing so goes by the name of Romberg integration, which we will not present in full generality.
## Node doubling¶
Note in (157) that $$R_f(4n)$$ depends on $$S_f(2n)$$ and $$S_f(4n)$$, which in turn depend on $$T_f(n)$$, $$T_f(2n)$$, and $$T_f(4n)$$. There is a useful benefit realized by doubling of the nodes in each application of the trapezoid formula. For simplicity, suppose that $$[a,b]=[0,1]$$ and that $$n=2m$$ for some positive integer $$m$$. The nodes are
$\Bigl[\, 0, \;\frac{1}{2m}, \;\frac{2}{2m}, \;\frac{3}{2m}, \;\frac{4}{2m}, \;\ldots \;\frac{2m-3}{2m}, \;\frac{2m-2}{2m}, \;\frac{2m-1}{2m}, \; 1 \,\Bigr].$
Suppose we delete every other node:
$\Bigl[\, 0, \;\frac{2}{2m}, \;\frac{4}{2m}, \;\ldots \;\frac{2m-2}{2m}, \; 1 \,\Bigr] \quad = \quad \Bigl[\, 0, \;\frac{1}{m}, \;\frac{2}{m}, \;\ldots \;\frac{m-1}{m}, \; 1 \,\Bigr] .$
What remains are the nodes with $$n=m$$. That is, if we have computed $$T_f(m)$$ and want to compute $$T_f(2m)$$, we begin with half of the evaluations of $$f$$ already in our pocket. More specifically,
(158)$\begin{split}\begin{split} T_f(2m) & = \frac{1}{2m} \left[ \tfrac{1}{2} f(0) + \tfrac{1}{2} f(1) + \sum_{k=1}^{m-1} f\Bigl( \tfrac{2k-1}{2m} \Bigr) + f\Bigl( \tfrac{2k}{2m} \Bigr) \right] \\ &= \frac{1}{2m} \left[ \tfrac{1}{2} f(0) + \tfrac{1}{2} f(1) + \sum_{k=1}^{m-1} f\Bigl( \tfrac{k}{m} \Bigr) \right] + \frac{1}{2m} \sum_{k=1}^{m} f\Bigl( \tfrac{2k-1}{2m} \Bigr) \\ &= \frac{1}{2} T_f(m) + \frac{1}{2m} \sum_{k=1}^{m-1} f\left( t_{2k-1} \right), \end{split}\end{split}$
where the nodes referenced in the last line are relative to $$n=2m$$. To summarize: when $$n$$ is doubled, new integrand evaluations are needed only at the odd-numbered nodes of the finer grid. Although we derived this result in the particular interval $$[0,1]$$, it is valid for any interval.
## Exercises¶
1. ⌨ For each integral below, use trapezoid to estimate the integral for $$n=10\cdot 2^k$$ nodes for $$k=1,2,\ldots,10$$. Make a log–log plot of the errors and confirm or refute second-order accuracy. (These integrals were taken from [BJL05].)
(a) $$\displaystyle \int_0^1 x\log(1+x)\, dx = \frac{1}{4}$$
(b) $$\displaystyle \int_0^1 x^2 \tan^{-1}x\, dx = \frac{\pi-2+2\log 2}{12}$$
(c) $$\displaystyle \int_0^{\pi/2}e^x \cos x\, dx = \frac{e^{\pi/2}-1}{2}$$
(d) $$\displaystyle \int_0^1 \sqrt{x} \log(x) \, dx = -\frac{4}{9}$$ (Note: Although the integrand has the limiting value zero as $$x\to 0$$, it cannot be evaluated naively at $$x=0$$. You can start the integral at $$x=\macheps$$ instead.)
(e) $$\displaystyle \int_0^1 \sqrt{1-x^2}\,\, dx = \frac{\pi}{4}$$
2. ✍ The Euler–Maclaurin error expansion (151) for the trapezoid formula implies that if we could cancel out the term due to $$f'(b)-f'(a)$$, we would obtain fourth-order accuracy. We should not assume that $$f'$$ is available, but approximating it with finite differences can achieve the same goal. Suppose the forward difference formula (138) is used for $$f'(a)$$, and its reflected backward difference is used for $$f'(b)$$. Show that the resulting modified trapezoid formula is
(159)$G_f(h) = T_f(h) - \frac{h}{24} \left[ 3\Bigl( f(x_n)+f(x_0) \Bigr) -4\Bigr( f(x_{n-1}) + f(x_1) \Bigr) + \Bigl( f(x_{n-2})+f(x_2) \Bigr) \right],$
which is known as a Gregory integration formula.
3. ⌨ Repeat each integral in exercise 1 above using Gregory integration (159) instead of the trapezoid formula. Compare the observed errors to fourth-order convergence.
4. ✍ Simpson’s formula can be derived without appealing to extrapolation.
(a) Show that
$p(x) = \beta + \frac{\gamma-\alpha}{2h}\, x + \frac{\alpha-2\beta+\gamma}{2h^2}\, x^2$
interpolates the three points $$(-h,\alpha)$$, $$(0,\beta)$$, and $$(h,\gamma)$$.
(b) Find
$\int_{-h}^h p(s)\, ds,$
where $$p$$ is the quadratic polynomial from part~(a), in terms of $$h$$, $$\alpha$$, $$\beta$$, and $$\gamma$$.
(c) Assume equally spaced nodes in the form $$t_i=a+ih$$, for $$h=(b-a)/n$$ and $$i=0,\ldots,n$$. Suppose $$f$$ is approximated by $$p(x)$$ over the subinterval $$[t_{i-1},t_{i+1}]$$. Apply the result from part (b) to find
$\int_{t_{i-1}}^{t_{i+1}} f(x)\, dx \approx \frac{h}{3} \bigl[ f(t_{i-1}) + 4f(t_i) + f(t_{i+1}) \bigr].$
(Use the change of variable $$s=x-t_i$$.)
(d) Now also assume that $$n=2m$$ for an integer $$m$$. Derive Simpson’s formula,
(160)$\begin{split} \begin{split} \int_a^b f(x)\, dx \approx \frac{h}{3}\bigl[ &f(t_0) + 4f(t_1) + 2f(t_2) + 4f(t_3) + 2f(t_4) + \cdots\\ &+ 2f(t_{n-2}) + 4f(t_{n-1}) + f(t_n) \bigr]. \end{split}\end{split}$
5. ✍ Show that the Simpson formula (160) is equivalent to $$S_f(n/2)$$, given the definition of $$S_f$$ in (155).
6. ⌨ For each integral in exercise 1 above, apply the Simpson formula (160) and compare the errors to fourth-order convergence.
7. ⌨ For $$n=10,20,30,\ldots,200$$, compute the trapezoidal approximation to
$\int_{0}^1 \frac{1}{2.01+\sin (6\pi x)-\cos(2\pi x)} \,d x \approx 0.9300357672424684.$
Make two separate plots of the absolute error as a function of $$n$$, one using log–log scales and the other using log only for the $$y$$-axis. The graphs suggest that the error asymptotically behaves as $$C \alpha^n$$ for some $$C>0$$ and some $$0<\alpha<1$$. How does this result relate to (151)?
8. ⌨ For each integral in exercise 1 above, extrapolate the trapezoidal results two levels to get sixth-order accurate results, and compare the expected convergence rate to the observed errors.
9. ✍ Find a formula like (157) that extrapolates two values of $$R_f$$ to obtain an 8th-order accurate one.
1
Numerical integration also goes by the older name quadrature.
2
Some texts distinguish between a formula for a single subinterval $$[t_{k-1},t_k]$$ and a “composite” formula that adds them up over the whole interval to get something like our (150). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9742217659950256, "perplexity": 476.7365501191855}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949035.66/warc/CC-MAIN-20230329213541-20230330003541-00379.warc.gz"} |
https://stats.stackexchange.com/questions/313359/distribution-of-infinite-sum-sum-t-0-infty-epsilon-t-rt | # Distribution of infinite sum $\sum_{t=0}^{\infty} \epsilon_t r^t$
In my current statistics course we're being taught about time series, and in this context we came across sums like this: $$\sum_{t=1}^{\infty} \epsilon_t r^t \quad \epsilon_t\sim \text{WN}(0,\sigma^2) \quad |r|<1$$ Where the $$r$$ is originally a $$\phi$$, representing the coefficients in a model as: $$y_t=c+\phi y_{t-1}+\epsilon_t$$, and WN stands for white noise with mean 0 and variance $$\sigma^2$$.
I'm interested in studying the expected value and variance of that sum, call it $$S(r)$$. The problem that brings me here is the following: i wrote a little program in STATA to calculate many sums and show me the variance, and it doesn't align with my calculations (the expected value does seem to be OK though) ... why?
I calculated the following: $$E(S)=E\left(\sum_{t=1}^{\infty} \epsilon_t r^t \right)=\sum_{t=1}^{\infty} E(\epsilon_t r^t)=\sum_{t=1}^{\infty} E(\epsilon_t) r^t=\sum_{t=1}^{\infty} 0 r^t=0$$ $$V(S)=V\left(\sum_{t=1}^{\infty} \epsilon_t r^t \right)=\sum_{t=1}^{\infty} V(\epsilon_t r^t)=\sum_{t=1}^{\infty} V(\epsilon_t) r^t=\sum_{t=1}^{\infty} \sigma^2 r^t=\sigma^2 \sum_{t=1}^{\infty} r^t=\sigma^2 \frac{r}{1-r}$$
And the code
clear all
program define geos, rclass
drop _all
set obs 500
gen a=invnorm(uniform())
gen y=_n
scalar define r=0.8
gen u=r^y
gen x=a*u
sum x
return scalar s=sum(x)
end
simulate S=r(s), reps(5000): geos
program drop _all
sum S, det
cd "C:\Users\Federico\Desktop\Geosum"
kdensity S, saving(Sum_80.png) title(80)
where $$\epsilon_t\sim N(0,1)$$ for simplicity. But doing this for various values of r, and regressing the variance as a function of r, i get $$\hat{var}_i=0.92r_i^2$$ (the r term and the constant are not significant, so i omitted them).
As an aside: i would like to "complete" the program so it runs for different values of r, and then saves the variance, instead of me running it various times and changing the value of r each time. I tried using "forvalues" but didn't succeed. Does anyone have a suggestion as to how i could do that?
• The aside part is off-topic here. – Dimitriy V. Masterov Nov 13 '17 at 17:17
$$V(\epsilon_tr^t)=V(\epsilon_t)r^{2t}.$$
$$\sigma^2\frac{r^2}{1-r^2}.$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8968315720558167, "perplexity": 908.1626928714588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250593994.14/warc/CC-MAIN-20200118221909-20200119005909-00176.warc.gz"} |
https://dml.cz/handle/10338.dmlcz/118466 | # Article
Full entry | PDF (0.2 MB)
Keywords:
orthomodular lattice; commutator; quasivariety
Summary:
An orthomodular lattice $L$ is said to have fully nontrivial commutator if the commutator of any pair $x,y \in L$ is different from zero. In this note we consider the class of all orthomodular lattices with fully nontrivial commutators. We show that this class forms a quasivariety, we describe it in terms of quasiidentities and situate important types of orthomodular lattices (free lattices, Hilbertian lattices, etc.) within this class. We also show that the quasivariety in question is not a variety answering thus the question implicitly posed in [4].
References:
[1] Beran L.: Orthomodular Lattices, Algebraic Approach. D. Reidel, Dordrecht, 1985. MR 0784029 | Zbl 0558.06008
[2] Bruns G., Greechie R.: Some finiteness conditions for orthomodular lattices. Canadian J. Math. 3 (1982), 535-549. MR 0663303 | Zbl 0494.06008
[3] Chevalier G.: Commutators and Decomposition of Orthomodular Lattices. Order 6 (1989), 181-194. MR 1031654
[4] Godowski R., Pták P.: Classes of orthomodular lattices defined by the state conditions. preprint.
[5] Gudder S.: Stochastic Methods in Quantum Mechanics. Elsevier North Holland, Inc., 1979. MR 0543489 | Zbl 0439.46047
[6] Grätzer G.: Universal Algebra. 2nd edition, Springer-Verlag, New York, 1979. MR 0538623
[7] Kalmbach G.: Orthomodular Lattices. Academic Press, London, 1983. MR 0716496 | Zbl 0554.06009
[8] Mayet R.: Varieties of orthomodular lattices related to states. Algebra Universalis, Vol. 20, No 3 (1987), 368-396. MR 0811695
[9] Pták P., Pulmannová S.: Orthomodular structures as quantum logics. Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. MR 1176314
[10] Pulmannová S.: Commutators in orthomodular lattices. Demonstratio Math. 18 (1985), 187-208. MR 0816029
Partner of | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9805636405944824, "perplexity": 3265.1298168478797}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948550199.46/warc/CC-MAIN-20171214183234-20171214203234-00240.warc.gz"} |
https://nukephysik101.wordpress.com/tag/moment/ | ## Magnetic Dipole Moment & Gyromagnetic Ratio
I always confuses on the definition, and wiki did not have any summary. so,
The Original definition is the Hamiltonian of a magnetic dipole under external magnetic field $\vec{B}$,
$H = -\vec{\mu}\cdot \vec{B}$,
where $\vec{\mu}$ is magnetic dipole moment (MDM). It is
$\vec{\mu} = g \frac{q}{2 m} \vec{J} = g \frac{\mu}{\hbar} \vec{J} = \gamma \vec{J}$.
Here, the $g$ is the g-factor, $\mu$ is magneton, and $\vec{J}$ is the total spin, which has a intrinsic factor $m\hbar / 2$ inside. $\gamma$ is gyromegnetic ratio.
We can see, the g-factor depends on the motion or geometry of the MDM. For a point particle, the g-factor is exactly equal to 2. For a charged particle orbiting, the g-factor is 1.
Put everything into the Hamiltonian,
$H = -\gamma \vec{J}\cdot \vec{B} = -\gamma J_z B = -\gamma \hbar \frac{m}{2} B [J]$,
Because energy is also equal $E = \hbar f$, thus, we can see the $\gamma$ has unit of frequency over Tesla.
———————————–
Take electron as an example, the MDM is Bohr magneton $\mu_{e} = e\hbar/(2m_e)$. The MDM is,
$\vec{\mu_e} = g_e \frac{e}{2 m_e} \vec{S} = g_e \frac{\mu_e}{\hbar}\vec{S} = \gamma_e \vec{S}$.
The magnitude of MDM is,
$|\vec{\mu_e}|= g_e \frac{e}{2 m_e} \frac{\hbar}{2} = \gamma_e \frac{\hbar}{2} [JT^{-1}]$,
The gyromagnetic ratio is,
$\gamma_e = g_e \frac{\mu_e}{\hbar} [rad s^{-1} T^{-1}]$.
Since using $rad s^{-1}$ is not convenient for experiment. The gyromagnetic ratio usually divided by $2\pi$,
$\gamma_e = g_e \frac{\mu_e}{2\pi\hbar} [Hz T^{-1}]$.
————————————-
To evaluate the magnitude of MDM of single particle state, which has orbital angular momentum and spin, the total spin $\vec{J} = \vec{L} + \vec{S}$. However, the g-factor for $\vec{L}$ is difference from that for $\vec{S}$. Thus, the MDM is not parallel to total spin. We have to use Landé Formula,
$\left< JM|\vec{V}|JM'\right> = \frac{1}{J(J+1)} \left< JM|(\vec{J}\cdot\vec{V})|JM\right> \left$
or see wiki, sorry for my laziness.
The result is
$g=g_L\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}+g_S\frac{J(J+1)-L(L+1)+S(S+1)}{2J(J+1)}$
For $J = L \pm 1/2$,
$g = J(g_L \pm \frac{g_S-g_L}{2L+1})$
## Electromagnetic multi-pole moment
Electromagnetic multipole comes from the charge and current distribution of the nucleons.
Magnetic multipole in nucleus has 2 origins, one is the spin of the nucleons, another is the relative orbital motion of the nucleons. the magnetic charge or monopoles either not exist or very small. the next one is the magnetic dipole, which cause by the current loop of protons.
Electric multipole is solely by the proton charge.
From electromagnetism, we knew that the multipole has different radial properties, from the potential of the fields:
$\displaystyle \Psi(r) = \frac{1}{4\pi\epsilon_0} \int\frac{\rho(r')}{|r-r'|}d^3r'$
$\displaystyle A(r) = \frac{\mu_0}{4\pi}\int\frac{J(r')}{|r-r'|}d^3r'$
and expand them into spherical harmonic by using:
$\displaystyle \frac{1}{|r-r'|} = 4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l} \frac{1}{2l+1}\frac{r_{<}^l}{r_>^{l+1}} Y_{lm}^*(\theta',\phi')Y_{lm}(\theta,\phi)$
we have
$\displaystyle \Psi(r) = \frac{1}{\epsilon_0} \sum_{l,m}\frac{1}{2l+1}\int Y_{lm}^*(\theta',\phi') r'^l\rho(r')d^3r' \frac{Y_{lm}(\theta,\phi)}{r^{l+1}}$
$\displaystyle A(r)=\mu_0 \sum_{l,m}\frac{1}{2l+1}\int Y_{lm}^*(\theta',\phi') r'^l J(r') d^3r' \frac{Y_{lm}(\theta,\phi)}{r^{l+1}}$
we can see the integral give us the required multipole moment. the magnetic and electric are just different by the charge density and the current density. we summarize in this way :
$q_{lm} = \int Y^*_{lm}(\theta',\phi') r'^l \O(r') d^3 r'$
where O can be either charge or current density. The l determine the order of multipole. and the potential will be simplified :
$M(r)=\sum_{l,m}\frac{1}{2l+1} q_{lm} \frac{Y_{lm}(\theta,\phi)}{r^{l+1}}$
were M can be either electric or magnetic potential, and i dropped the constant. since the field is given by 1st derivative, thus we have:
1. monopole has $1/r^2$ dependence
2. dipole has $1/r^3$
3. quadrapole has $1/r^4$
4. and so on
The above radial dependences are same for electric or magnetic. for easy name of the multipole, we use L-pole, which L can be 0 for monopole, 1 for dipole, 2 for quadrapole, etc.. and we use E0 for electric monopole, M0 for magnetic monopole.
Since the nucleus must preserver parity, and the parity for electric and magnetic moment are diffident.the different come from the charge density and current density has different parity. The parity for charge density is even, but for the current density is odd. and $1/r^2$ has even parity, $1/r^3$ has odd parity. therefore
• electric L-pole — $(-1)^{L}$
• magnetic L-pole — $(-1)^{L+1}$
for easy compare:
• E0, E2, E4… and M1,M3, M5 … are even
• E1,E3,E5…. and M0, M2, M4…. are odd
The expectation value for L-pole, we have to calculate :
$\int \psi^* Q_{lm} \psi dx$
where $Q_{lm}$ is multipole operator ( which is NOT $q_{lm}$), and its parity is follow the same rule. the parity of the wave function will be canceled out due to the square of itself. thus, only even parity are non-Zero. those are:
• E0, E2, E4…
• M1,M3, M5 …
that make sense, think about a proton orbits in a circular loop, which is the case for E1, in time-average, the dipole momentum should be zero.
## a review on Hydrogen’s atomic structure
I found that most of the book only talk part of it or present it separately. Now, I am going to treat it at 1 place. And I will give numerical value as well. the following context is on SI unit.
a very central idea when writing down the state quantum number is, is it a good quantum number? a good quantum number means that its operator commute with the Hamiltonian. and the eigenstate states are stationary or the invariant of motion. the prove on the commutation relation will be on some post later. i don’t want to make this post too long, and with hyperlink, it is more reader-friendly. since somebody may like to go deeper, down to the cornerstone. but some may like to have a general review.
the Hamiltonian of a isolated hydrogen atom is given by fews terms, deceasing by their strength.
$H = H_{Coul} + H_{K.E.} + H_{Rel} + H_{Darwin} + H_{s-0} + H_{i-j} + H_{lamb} + H_{vol} + O$
the Hamiltonian can be separated into 3 classes.
__________________________________________________________
## Bohr model
$H_{Coul} = - \left(\frac {e^2}{4 \pi \epsilon_0} \right) \frac {1}{r}$
is the Coulomb potential, which dominate the energy. recalled that the ground state energy is -13.6 eV. and it is equal to half of the Coulomb potential energy, thus, the energy is about 27.2 eV, for ground state.
$H_{K.E.} = \frac {P^2}{ 2 m}$
is the non-relativistic kinetic energy, it magnitude is half of the Coulomb potential, so, it is 13.6 eV, for ground state.
comment on this level
this 2 terms are consider in the Bohr model, the quantum number, which describe the state of the quantum state, are
$n$ = principle number. the energy level.
$l$ = orbital angular momentum. this give the degeneracy of each energy level.
$m_l$ = magnetic angular momentum.
it is reasonable to have 3 parameters to describe a state of electron. each parameter gives 1 degree of freedom. and a electron in space have 3. thus, change of basis will not change the degree of freedom. The mathematic for these are good quantum number and the eigenstate $\left| n, l, m_l \right>$ is invariant of motion, will be explain in later post. But it is very easy to understand why the angular momentum is invariant, since the electron is under a central force, no torque on it. and the magnetic angular momentum is an invariant can also been understood by there is no magnetic field.
the principle quantum number $n$ is an invariance. because it is the eigenstate state of the principle Hamiltonian( the total Hamiltonian )!
the center of mass also introduced to make more correct result prediction on energy level. but it is just minor and not much new physics in it.
## Fine structure
$H_{Rel} = - \frac{1}{8} \frac{P^4}{m^3 c^2}$
is the 1st order correction of the relativistic kinetic energy. from $K.E. = E - mc^2 = \sqrt { p^2 c^2 + m^2c^4} - mc^2$, the zero-order term is the non-relativistic kinetic energy. the 1st order therm is the in here. the magnitude is about $1.8 \times 10^{-4} eV$. ( the order has to be recalculate, i think i am wrong. )
$H_{Darwin} = \frac{\hbar^{2}}{8m_{e}^{2}c^{2}}4\pi\left(\frac{Ze^2}{4\pi \epsilon_{0}}\right)\delta^{3}\left(\vec r\right)$
is the Darwin-term. this term is result from the zitterbewegung, or rapid quantum oscillations of the electron. it is interesting that this term only affect the S-orbit. To understand it require Quantization of electromagnetic field, which i don’t know. the magnitude of this term is about $10^{-3} eV$
$H_{s-o} = \left(\frac{Ze^2}{4\pi \epsilon_{0}}\right)\left(\frac{1}{2m_{e}^{2}c^{2}}\right)\frac{1}{r^3} L \cdot S$
is the Spin-Orbital coupling term. this express the magnetic field generated by the proton while it orbiting around the electron when taking electron’s moving frame. the magnitude of this term is about $10^{-4} eV$
comment on this level
this fine structure was explained by P.M.Dirac on the Dirac equation. The Dirac equation found that the spin was automatically come out due to special relativistic effect. the quantum number in this stage are
$n$ = principle quantum number does not affected.
$l$ = orbital angular momentum.
$m_l$ = magnetic total angular momentum.
$s$ = spin angular momentum. since s is always half for electron, we usually omit it. since it does not give any degree of freedom.
$m_s$ = magnetic total angular momentum.
at this stage, the state can be stated by $\left| n, l, m_l, m_s \right>$, which shown all the degree of freedom an electron can possible have.
However, $L_z$ is no longer a good quantum number. it does not commute with the Hamiltonian. so, $m_l$ does not be the eigenstate anymore. the total angular momentum was introduced $J = L + S$ . and $J^2$ and $J_z$ commute with the Hamiltonian. therefore,
$j$ = total angular momentum.
$m_j$ = magnetic total angular momentum.
an eigenstate can be stated as $\left| n, l, s, j, m_j \right>$. in spectroscopy, we denote it as $^{2 s+1} L _j$, where $L$ is the spectroscopy notation for $l$.
there are 5 degrees of freedom, but in fact, s always half, so, there are only 4 real degree of freedom, which is imposed by the spin ( can up and down). the reason for stating the s in the eigenstate is for general discussion. when there are 2 electrons, s can be different and this is 1 degree of freedom.
## Hyperfine Structure
$H_{i-j} = \alpha I \cdot J$
is the nuclear spin- electron total angular momentum coupling. the coefficient of this term, i don’t know. Sorry. the nuclear has spin, and this spin react with the magnetic field generate by the electron. the magnitude is $10^{-5}$
$H_{lamb}$
is the lamb shift, which also only affect the S-orbit.the magnitude is $10^{-6}$
comment on this level
the hyperfine structure always makes alot questions in my mind. the immediate question is why not separate the orbital angular momentum and the electron spin angular momentum? why they first combined together, then interact with the nuclear spin?
may be i open another post to talk about.
The quantum number are:
$n$ = principle quantum number
$l$ = orbital angular momentum
$s$ = electron spin angular momentum.
$j$ = spin-orbital angular momentum of electron.
$i$ = nuclear spin. for hydrogen, it is half.
$f$ = total angular momentum
$m_f$ = total magnetic angular momentum
a quantum state is $\left| n, l, s, j,i, f , m_f \right>$. but since the s and i are always a half. so, the total degree of freedom will be 5. the nuclear spin added 1 on it.
## Smaller Structure
$H_{vol}$
this term is for the volume shift. the magnitude is $10^{-10}$.
in diagram:
## Larmor Precession (quick)
Magnetic moment ($\mu$) :
this is a magnet by angular momentum of charge or spin. its value is:
$\mu = \gamma J$
where $J$ is angular momentum, and $\gamma$ is the gyromagnetic rato
$\gamma = g \mu_B$
Notice that we are using natural unit.
the g is the g-factor is a dimensionless number, which reflect the environment of the spin, for orbital angular momentum, g = 1.
$\mu_B$ is Bohr magneton, which is equal to
$\mu_B = \frac {e} {2 m}$ for positron
since different particle has different mass, their Bohr magneton value are different. electron is the lightest particle, so, it has largest value on Bohr magneton.
Larmor frequency:
When applied a magnetic field on a magnetic moment, the field will cause the moment precess around the axis of the field. the precession frequency is called Larmor frequency.
the precession can be understood in classical way or QM way.
Classical way:
the change of angular momentum is equal to the applied torque. and the torque is equal to the magnetic moment cross product with the magnetic field. when in classical frame, the angular momentum, magnetic moment, and magnetic field are ordinary vector.
$\vec {\Gamma}= \frac { d \vec{J}}{dt} = \vec{\mu} \times \vec{B} = \gamma \vec {J} \times \vec{B}$
solving gives the procession frequency is :
$\omega = - \gamma B$
the minus sign is very important, it indicated that the J is precessing by right hand rule when $\omega >0$.
QM way:
The Tim dependent Schrödinger equation (TDSE) is :
$i \frac {d}{d t} \left| \Psi\right> = H \left|\Psi\right>$
H is the Hamiltonian, for the magnetic field is pointing along the z-axis.
$H = -\mu \cdot B = - \gamma J\cdot B = -gamma B J_z = \omega J_z$
the solution is
$\left|\Psi(t) \right> = Exp( - i \omega t J_z) \left| \Psi(0) \right>$
Thus, in QM point of view, the state does not “rotate” but only a phase change.
However, the rotation operator on z-axis is
$R_z ( \theta ) = Exp( - i \frac {\theta}{\hbar} J_z )$
Thus, the solution can be rewritten as:
$\left|\Psi (t)\right> = R_z( \omega t) \left|\Psi(0)\right>$
That makes great analogy on rotation on a real vector. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 103, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9600797295570374, "perplexity": 791.8593843633989}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647530.92/warc/CC-MAIN-20180320185657-20180320205657-00502.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/12117-probability-mean-random-variable.html | # Math Help - probability(mean of a random variable)
1. ## probability(mean of a random variable)
we flip a fair coin n times,if k is the number of 'heads' we get,we then flip the coin until we get k+1 heads(the number of flips is not given for the second round),let Y be the number of flips(for the second round)
determine the mean of Y.
here's my logic:
for the first round,this is a bionomial disribution so the mean is:np=n/2
for the second time,we need to get k+1 heads so the mean for the first k heads
is just like before: n/2..and to get the "extra" heads we need to add 2
so the mean is n/2 + 2
now here comes the tricky part..if I add those two i.e n/2 + (n/2 +2 ) = n+2
I get the right answer...but I dunno if it's correct,or why am I allowed to do that.
2. Originally Posted by parallel
we flip a fair coin n times,if k is the number of 'heads' we get,we then flip the coin until we get k+1 heads(the number of flips is not given for the second round),let Y be the number of flips(for the second round)
determine the mean of Y.
here's my logic:
for the first round,this is a bionomial disribution so the mean is:np=n/2
for the second time,we need to get k+1 heads so the mean for the first k heads
is just like before: n/2..and to get the "extra" heads we need to add 2
so the mean is n/2 + 2
now here comes the tricky part..if I add those two i.e n/2 + (n/2 +2 ) = n+2
I get the right answer...but I dunno if it's correct,or why am I allowed to do that. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8921739459037781, "perplexity": 2141.518194100551}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-math-topics/191559-where-classify-problem.html | # Math Help - Where to classify that problem?
1. ## Where to classify that problem?
Dear all I am posting here as I am not sure where I should classify my problem.
I have wrote one page description and I have uploaded it here (I am really bad with writing equations online)
Imageshack - screenshotbazw.jpg
my problem should not be too hard to solve and I need your help to help me do the first step in solution. Which is the right method? What is the label of the mathematical field that I should focus to solve it.
If you need any clarification for the problem given please let me know.
B.R
Alex
2. ## Re: Where to classify that problem?
Originally Posted by dervast
Dear all I am posting here as I am not sure where I should classify my problem.
I have wrote one page description and I have uploaded it here (I am really bad with writing equations online)
Imageshack - screenshotbazw.jpg
my problem should not be too hard to solve and I need your help to help me do the first step in solution. Which is the right method? What is the label of the mathematical field that I should focus to solve it.
If you need any clarification for the problem given please let me know. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8233440518379211, "perplexity": 254.1747171778974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00580-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://michaelsantomauro.blogspot.com/2011_03_27_archive.html | ## Mar 27, 2011
### New Global Warming Threat
-----Original Message-----
From: [email protected]
Sent: Mar 27, 2011 9:25 AM
Subject: New Global Warming Threat
Printed in The Washington Post
The Arctic ocean is warming up, icebergs are growing scarcer and in some places the seals are finding the water too hot, according to a report to the Commerce Department yesterday from Consulafft, at Bergen , Norway . Reports from fishermen, seal hunters, and explorers all point to a radical change in climate conditions and hitherto unheard-of temperatures in the Arctic zone. Exploration expeditions report that scarcely any ice has been met as far north as 81 degrees 29 minutes. Soundings to a depth of 3,100 meters showed the gulf stream still very warm. Great masses of ice have been replaced by moraines of earth and stones, the report continued, while at many points well known glaciers have entirely disappeared.
Very few seals and no white fish are found in the eastern Arctic, while vast shoals of herring and smelts which have never before ventured so far north, are being encountered in the old seal fishing grounds. Within a few years it is predicted that due to the ice melt the sea will rise and make most coastal cities uninhabitable.
====================
I apologize, I neglected to mention that this report was from November 2, 1922 , as reported by the AP and published in THE WASHINGTON POST -- 88 YEARS AGO!
Al Legatzke
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# A Rosenberg Co-Conspirator Reveals More About His Role
Morton Sobell, who was convicted with Julius and Ethel Rosenberg in 1951 in an espionage conspiracy case and finally admitted nearly six decades later that he had been a Soviet spy, now says he helped copy hundreds of pages of secret Air Force documents stolen from aColumbia University professor's safe in 1948.
###### Bettmann/Corbis
Morton Sobell, left, with a United States marshal in 1951, said he copied classified documents.
According to an article by two cold war historiansRonald Radosh and Steven T. Usdin, in The Weekly Standard, Mr. Sobell, who is 93, said in an interview last December that he,Julius Rosenberg, William Perl and an unidentified fourth man spent a weekend, probably Independence Day, frantically copying the classified documents in a Greenwich Village apartment before they were missed.
That Monday, Mr. Sobell is quoted as saying, he and Mr. Rosenberg filled a box with canisters of 35-millimeter film and delivered it to Soviet agents on a Long Island Rail Road platform.
In addition to elaborating on Mr. Sobell's admission in a2008 interview with The New York Times that he had stolen military radar and artillery secrets, the December interview appears to stoke the smoldering embers of the case on several other counts.
Mr. Sobell's comments, according to the authors, identify Mr. Perl not as an innocent aeronautical engineer who was entitled to inspect the secret papers and was implicated in the espionage conspiracy only by circumstantial evidence, but as a conspirator against his mentor, Theodore von Karman.
Mr. Perl, a fellow student with Mr. Sobell and Mr. Rosenberg at City College, worked with Professor von Karman for the National Advisory Committee on Aeronautics at Langley Army Air Base in Virginia during World War II. Testifying before the Rosenberg grand jury, Mr. Perl denied any relationship with Mr. Rosenberg or Mr. Sobell. He was convicted of perjury in 1953.
Mr. Sobell's latest comments also validate an account of the photocopying of the secret papers conveyed to federal investigators by Jerome Tartakow, a jailhouse informer often discredited by supporters of the Rosenbergs, who said he learned of the photocopying from Julius Rosenberg himself.
Finally, Mr. Sobell's comments, as quoted by the authors, shed more light on his motive.
"I did it for the Soviet Union," he said, leading Mr. Radosh and Mr. Usdin to conclude that Mr. Rosenberg and his fellow American Communists "were motivated by loyalty to the Soviet Union, not opposition to fascism as their defenders claim."
The Rosenbergs, who were accused of conspiracy to steal atomic bomb secrets from the United States, were sentenced to death and executed in 1953. Mr. Sobell served 18 years for nonatomic spying. He was released in 1969 and, until the Times interview, maintained his innocence and insisted that he had been framed by the government.
--
"...when you have laws against questioning the Holocaust narrative, you are screaming at the other person to stop thinking!!!" ---Michael Santomauro, March 23, 2011
Being happy–is it good for the Jews? "Before Professor Dershowitz accused me of being an anti-Semite (news to me), I was a happy person. Since then, I'm still a happy person". –Michael Santomauro
An anti-Semite condemns people for being Jews, I am not an anti-Semite.--Michael Santomauro
Most of us are mentally trapped to think Jewish. Actually, it is safe to say that virtually every mainstream publication or or other type of media organ is "nothing more than a screen to present chosen views." The great battle over the last century has been a battle for the mind of the Western peoples, i.e., non-Jewish Euros. The chosen won it by acquiring control over essentially the complete mainstream news, information, education and entertainment media of every type, and using that control to infuse and disseminate their message, agenda and worldview, their way of thinking, or rather the way they want us to think. Since at least the 1960s this campaign has been effectively complete. Since then they have shaped and controlled the minds of all but a seeming few of us in varying degree with almost no opposition or competition from any alternative worldview. So now most of us are mentally trapped in the box the chosen have made for us, which we have lived in all our lives. Only a few have managed to avoid it or escape it, or to even sometimes see outside of it, and so actually "think outside of the (Jewish) box." --Michael Santomauro
Thank you and remember:
Peace is patriotic!
Michael Santomauro
253 W. 72nd Street
New York, NY 10023
Call anytime: 917-974-6367
E-mail me anything:
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# 8.8 Who Invented Relativity?
All beginnings are obscure.
H. Weyl
There have been many theories of relativity throughout history, from the astronomical speculations of Heraclides to the geometry of Euclid to the classical theory of space, time, and dynamics developed by Galileo, Newton and others. Each of these was based on one or more principle of relativity. However, when we refer to the "theory of relativity" today, we usually mean one particular theory of relativity, namely, the body of ideas developed near the beginning of the 20th century and closely identified with the work of Albert Einstein. These ideas are distinguished from previous theories not by relativity itself, but by the way in which relativistically equivalent coordinate systems are related to each other.
One of the interesting historical aspects of the modern relativity theory is that, although often regarded as the highly original and even revolutionary contribution of a single individual, almost every idea and formula of the theory had been anticipated by others. For example, Lorentz covariance and the inertia of energy were both (arguably) implicit in Maxwell's equations. Also, Voigt formally derived the Lorentz transformations in 1887 based on general considerations of the wave equation. In the context of electro-dynamics, Fitzgerald, Larmor, and Lorentz had all, by the 1890s, arrived at the Lorentz transformations, including all the peculiar "time dilation" and "length contraction" effects (with respect to the transformed coordinates) associated with Einstein's special relativity. By 1905, Poincare had clearly articulated the principle of relativity and many of its consequences, had pointed out the lack of empirical basis for absolute simultaneity, had challenged the ontological significance of the ether, and had even demonstrated that the Lorentz transformations constitute a group in the same sense as do Galilean transformations. In addition, the crucial formal synthesis of space and time into spacetime was arguably the contribution of Minkowski in 1907, and the dynamics of special relativity were first given in modern form by Lewis and Tolman in 1909. Likewise, the Riemann curvature and Ricci tensors for n-dimensional manifolds, the tensor formalism itself, and even the crucial Bianchi identities, were all known prior to Einstein's development of general relativity in 1915. In view of this, is it correct to regard Einstein as the sole originator of modern relativity?
The question is complicated by the fact that relativity is traditionally split into two separate theories, the special and general theories, corresponding to the two phases of Einstein's historical development, and the interplay between the ideas of Einstein and those of his predecessors and contemporaries are different in the two cases. In addition, the title of Einstein's 1905 paper ("On the Electrodynamics of Moving Bodies") encouraged the idea that it was just an interpretation of Lorentz's theory of electrodynamics. Indeed, Wilhelm Wein proposed that the Nobel prize of 1912 be awarded jointly to Lorentz and Einstein, saying
The principle of relativity has eliminated the difficulties which existed in electrodynamics and has made it possible to predict for a moving system all electrodynamic phenomena which are known for a system at rest... From a purely logical point of view the relativity principle must be considered as one of the most significant accomplishments ever achieved in theoretical physics... While Lorentz must be considered as the first to have found the mathematical content of relativity, Einstein succeeded in reducing it to a simple principle. One should therefore assess the merits of both investigators as being comparable.
As it happens, the physics prize for 1912 was awarded to the Nils Gustaf Dalen (for the "invention of automatic regulators for lighting coastal beacons and light buoys during darkness or other periods of reduced visibility"), and neither Einstein, Lorentz, nor anyone else was ever awarded a Nobel prize for either the special or general theories of relativity. This is sometimes considered to have been an injustice to Einstein, although in retrospect it's conceivable that a joint prize for Lorentz and Einstein in 1912, as Wein proposed, assessing "the merits of both investigators as being comparable", might actually have diminished Einstein's subsequent popular image as the sole originator of both special and general relativity.
On the other hand, despite the somewhat misleading title of Einstein's paper, the second part of the paper ("The Electrodynamic Part") was really just an application of the general theoretical framework developed in the first part of the paper ("The Kinematic Part"). It was in the first part that special relativity was founded, with consequences extending far beyond Lorentz's electrodynamics. As Einstein later recalled,
The new feature was the realization that the bearing of the Lorentz transformation transcended its connection with Maxwell's equations and was concerned with the nature of space and time in general.
To give just one example, we may note that prior to the advent of special relativity the experimental results of Kaufmann and others involving the variation of an electron's mass with velocity were thought to imply that all of the electron's mass must be electromagnetic in origin, whereas Einstein's kinematics revealed that all mass – regardless of its origin – would necessarily be affected by velocity in the same way. Thus an entire research program, based on the belief that the high-speed behavior of objects represented dynamical phenomena, was decisively undermined when Einstein showed that the phenomena in question could be interpreted much more naturally on a purely kinematic basis. Now, if this interpretation applied only to electrodynamics, it's significance might be debatable, but already by 1905 it was clear that, as Einstein put it, "the Lorentz transformation transcended its connection with Maxwell's equations", and must apply to all physical phenomena in order to account for the complete inability to detect absolute motion. Once this is recognized, it is clear that we are dealing not just with properties of electricity and magnetism, or any other specific entities, but with the nature of space and time themselves. This is the aspect of Einstein's 1905 theory that prompted Witkowski, after reading vol. 17 of Annalen der Physic, to exclaim: "A new Copernicus is born! Read Einstein's paper!" The comparison is apt, because the contribution of Copernicus was, after all, essentially nothing but an interpretation of Ptolemy's astronomy, just as Einstein's theory was an interpretation of Lorentz's electrodynamics. Only subsequently did men like Kepler, Galileo, and Newton, taking the Copernican insight even more seriously than Copernicus himself had done, develop a substantially new physical theory. It's clear that Copernicus was only one of several people who jointly created the "Copernican revolution" in science, and we can argue similarly that Einstein was only one of several individuals (including Maxwell, Lorentz, Poincare, Planck, and Minkowski) responsible for the "relativity revolution".
The historical parallel between special relativity and the Copernican model of the solar system is not merely superficial, because in both cases the starting point was a pre-existing theoretical structure based on the naive use of a particular system of coordinates lacking any inherent physical justification. On the basis of these traditional but eccentric coordinate systems it was natural to imagine certain consequences, such as that both the Sun and the planet Venus revolve around a stationary Earth in separate orbits. However, with the newly-invented telescope, Galileo was able to observe the phases of Venus, clearly showing that Venus moves in (roughly) a circle around the Sun. In this way the intrinsic patterns of the celestial bodies became better understood, but it was still possible (and still is possible) to regard the Earth as stationary in an absolute extrinsic sense. In fact, for many purposes we continue to do just that, but from an astronomical standpoint we now almost invariably regard the Sun as the "center" of the solar system. Why? The Sun too is moving among the stars in the galaxy, and the galaxy itself is moving relative to other galaxies, so on what basis do we decide to regard the Sun as the "center" of the solar system?
The answer is that the Sun is the inertial center. In other words, the Copernican revolution (as carried to its conclusion by the successors of Copernicus) can be summarized as the adoption of inertia as the prime organizing principle for the understanding and description of nature. The concept of physical inertia was clearly identified, and the realization of its significance evolved and matured through the works of Kepler, Galileo, Newton, and others. Nature is most easily and most perspicuously described in terms of inertial coordinates. Of course, it remains possible to adopt some non-inertial system of coordinates with respect to which the Earth can be regarded as the stationary center, but there is no longer any imperative to do this, especially since we cannot thereby change the fact that Venus circles the Sun, i.e., we cannot change the intrinsic relations between objects, and those intrinsic relations are most readily expressed in terms of inertial coordinates.
Likewise the pre-existing theoretical structure in 1905 described events in terms of coordinate systems that were not clearly understood and were lacking in physical justification. It was natural within this framework to imagine certain consequences, such as anisotropy in the speed of light, i.e., directional dependence of light speed resulting from the Earth's motion through the (assumed stationary) ether. This was largely motivated by the idea that light consists of a wave in the ether, and therefore is not an inertial phenomenon. However, experimental physicists in the late 1800's began to discover facts analogous to the phases of Venus, e.g., the symmetry of electromagnetic induction, the "partial convection" of light in moving media, the isotropy of light speed with respect to relatively moving frames of reference, and so on. Einstein accounted for all these results by showing that they were perfectly natural if things are described in terms of inertial coordinates - provided we apply a more profound understanding of the definition and physical significance of such coordinate systems and the relationships between them.
As a result of the first inertial revolution (initiated by Copernicus), physicists had long been aware of the existence of a preferred class of coordinate systems - the inertial systems - with respect to which inertial phenomena are isotropic. These systems are equivalent up to orientation and uniform motion in a straight line, and it had always been tacitly assumed that the transformation from one system in this class to another was given by a Galilean transformation. The fundamental observations in conflict with this assumption were those involving electric and magnetic fields that collectively implied Maxwell's equations of electromagnetism. These equations are not invariant under Galilean transformations, but they are invariant under Lorentz transformations. The discovery of Lorentz invariance was similar to the discovery of the phases of Venus, in the sense that it irrevocably altered our awareness of the intrinsic relations between events. We can still go on using coordinate systems related by Galilean transformations, but we now realize that only one of those systems (at most) is a truly inertial system of coordinates.
Incidentally, the electrodynamic theory of Lorentz was in some sense analogous to Tycho Brahe's model of the solar system, in which the planets revolve around the Sun but the Sun revolves around a stationary Earth. Tycho's model was kinematically equivalent to Copernicus' Sun-centered model, but expressed – awkwardly – in terms of a coordinate system with respect to which the Earth is stationary, i.e., a non-inertial coordinate system.
It's worth noting that we define inertial coordinates just as Galileo did, i.e., systems of coordinates with respect to which inertial phenomena are isotropic, so our definition hasn't changed. All that has changed is our understanding of the relations between inertial coordinate systems. Einstein's famous "synchronization procedure" (which was actually first proposed by Poincare) was expressed in terms of light rays, but the physical significance of this procedure is due to the empirical fact that it yields exactly the same synchronization as does Galileo's synchronization procedure based on mechanical inertia. To establish simultaneity between spatially separate events while floating freely in empty space, throw two identical objects in opposite directions with equal force, so that the thrower remains stationary in his original frame of reference. These objects then pass equal distances in equal times, i.e., they serve to assign inertially simultaneous times to separate events as they move away from each other. In this way we can theoretically establish complete slices of inertial simultaneity in spacetime, based solely on the inertial behavior of material objects. Someone moving uniformly relative to us can carry out this same procedure with respect to his own inertial frame of reference and establish his own slices of inertial simultaneity throughout spacetime. The unavoidable intrinsic relations that were discovered at the end of the 19th century show that these two sets of simultaneity slices are not identical. The two main approaches to the interpretation of these facts were discussed in Sections 1.5 and 1.6. The approach advocated by Einstein was to adhere to the principle of inertia as the basis for organizing our understanding and descriptions of physical phenomena - which was certainly not a novel idea.
In his later years Einstein observed "there is no doubt that the Special Theory of Relativity, if we regard its development in retrospect, was ripe for discovery in 1905". The person (along with Lorentz) who most nearly anticipated Einstein's special relativity was undoubtedly Poincare, who had already in 1900 proposed an explicitly operational definition of clock synchronization and in 1904 suggested that the ether was in principle undetectable to all orders of v/c. Those two propositions and their consequences essentially embody the whole of special relativity. Nevertheless, as late as 1909 Poincare was not prepared to say that the equivalence of all inertial frames combined with the invariance of (two-way) light speed were sufficient to infer Einstein's model. He maintained that one must also stipulate a particular contraction of physical objects in their direction of motion. This is sometimes cited as evidence that Poincare still failed to understand the situation, but there's a sense in which he was actually correct. The two famous principles of Einstein's 1905 paper are not sufficient to uniquely identify special relativity, as Einstein himself later acknowledged. One must also stipulate, at the very least, homogeneity, memorylessness, and isotropy. Of these, the first two are rather innocuous, and one could be forgiven for failing to explicitly mention them, but not so the assumption of isotropy, which serves precisely to single out Einstein's simultaneityconvention from all the other - equally viable - interpretations. (See Section 4.5). This is also precisely the aspect that is fixed by Poincare's postulate of contraction as a function of velocity.
In a sense, the failure of Poincare to found the modern theory of relativity was not due to a lack of discernment on his part (he clearly recognized the Lorentz group of space and time transformations), but rather to an excess of discernment and philosophical sophistication, preventing him from subscribing to the young patent examiner's inspired but perhaps slightly naive enthusiasm for the symmetrical interpretation, which is, after all, only one of infinitely many possibilities. Poincare recognized too well the extent to which our physical models are both conventional and provisional. In retrospect, Poincare's scruples have the appearance of someone arguing that we could just as well regard the Earth rather than the Sun as the center of the solar system, i.e., his reservations were (and are) technically valid, but in some sense misguided. Also, as Max Born remarked, to the end of Poincare's life his expositions of relativity "definitely give you the impression that he is recording Lorentz's work", and yet "Lorentz never claimed to be the author of the principle of relativity", but invariably attributed it to Einstein. Indeed Lorentz himself often expressed reservations about the relativistic interpretation.
Regarding Born's impression that Poincare was just "recording Lorentz's work", it should be noted that Poincare habitually wrote in a self-effacing manner. He named many of his discoveries after other people, and expounded many important and original ideas in writings that were ostensibly just reviewing the works of others, with "minor amplifications and corrections". So, we shouldn't be misled by Born's impression. Poincare always gave the impression that he was just recording someone else's work – in contrast with Einstein, whose style of writing, as Born said, "gives you the impression of quite a new venture". Of course, Born went on to say, when recalling his first reading of Einstein's paper in 1907, "Although I was quite familiar with the relativistic idea and the Lorentz transformations, Einstein's reasoning was a revelation to me… which had a stronger influence on my thinking than any other scientific experience".
Lorentz's reluctance to fully embrace the relativity principle (that he himself did so much to uncover) is partly explained by his belief that "Einstein simply postulates what we have deduced... from the equations of the electromagnetic field". If this were true, it would be a valid reason for preferring Lorentz's approach. However, if we closely examine Lorentz's electron theory we find that full agreement with experiment required not only the invocation of Fitzgerald's contraction hypothesis, but also the assumption that mechanical inertia is Lorentz covariant. It's true that, after Poincare complained about the proliferation of hypotheses, Lorentz realized that the contraction could be deduced from more fundamental principles (as discussed in Section 1.5), but this was based on yet another hypothesis, the co-called molecular force hypothesis, which simply asserts that all physical forces and configurations (including the unknown forces that maintain the shape of the electron) transform according to the same laws as do electromagnetic forces. Needless to say, it obviously cannot follow deductively "from the equations of the electromagnetic field" that the necessarily non-electromagnetic forces which hold the electron together must transform according to the same laws. (Both Poincare and Einstein had already realized by 1905 that the mass of the electron cannot be entirely electromagnetic in origin.) Even less can the Lorentz covariance of mechanical inertia be deduced from electromagnetic theory. We still do not know to this day the origin of inertia, so there is no sense in which Lorentz or anyone else can claim to have deduced Lorentz covariance in any constructive sense, let alone from the laws of electromagnetism.
Hence Lorentz's molecular force hypothesis and his hypothesis of covariant mechanical inertia together are simply a disguised and piece-meal way of postulating universal Lorentz invariance - which is precisely what Lorentz claims to have deduced rather than postulated. The whole task was to reconcile the Lorentzian covariance of electromagnetism with the Galilean covariance of mechanical dynamics, and Lorentz simply recognized that one way of doing this is to assume that mechanical dynamics (i.e., inertia) is actually Lorentz covariant. This is presented as an explicit postulate (not a deduction) in the final edition of his book on the Electron Theory. In essence, Lorentz's program consisted of performing a great deal of deductive labor, at the end of which it was still necessary, in order to arrive at results that agreed with experiment, to simply postulate the same principle that forms the basis of special relativity. (To his credit, Lorentz candidly acknowledged that his deductions were "not altogether satisfactory", but this is actually an understatement, because in the end he simply postulated what he claimed to have deduced.)
In contrast, Einstein recognized the necessity of invoking the principle of relativity and Lorentz invariance at the start, and then demonstrated that all the other "constructive" labor involved in Lorentz's approach was superfluous, because once we have adopted these premises, all the experimental results arise naturally from the simple kinematics of the situation, with no need for molecular force hypotheses or any other exotic and dubious conjectures regarding the ultimate constituency of matter. On some level Lorentz grasped the superiority of the purely relativistic approach, as is evident from the words he included in the second edition of his "Theory of Electrons" in 1916:
If I had to write the last chapter now, I should certainly have given a more prominent place to Einstein's theory of relativity by which the theory of electromagnetic phenomena in moving systems gains a simplicity that I had not been able to attain. The chief cause of my failure was my clinging to the idea that the variable t only can be considered as the true time, and that my local time t' must be regarded as no more than an auxiliary mathematical quantity.
Still, it's clear that neither Lorentz nor Poincare ever whole-heartedly embraced special relativity, for reasons that may best be summed up by Lorentz when he wrote
Yet, I think, something may also be claimed in favor of the form in which I have presented the theory. I cannot but regard the aether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. In this line of thought it seems natural not to assume at starting that it can never make any difference whether a body moves through the aether or not, and to measure distances and lengths of time by means of rods and clocks having a fixed position relatively to the aether.
This passage implies that Lorentz's rationale for retaining a substantial aether and attempting to refer all measurements to the rest frame of this aether (without, of course, specifying how that is to be done) was the belief that it might, after all, make some difference whether a body moves through the aether or not. In other words, we should continue to look for physical effects that violate Lorentz invariance (by which we now mean local Lorentz invariance), both in new physical forces and at higher orders of v/c for the known forces. A century later, our present knowledge of the weak and strong nuclear forces and the precise behavior of particles at 0.99999c has vindicated Einstein's judgment that Lorentz invariance is a fundamental principle whose significance and applicability extends far beyond Maxwell's equations, and apparently expresses a general attribute of space and time, rather than a specific attribute of particular physical entities.
In addition to the formulas expressing the Lorentz transformations, we can also find precedents for other results commonly associated with special relativity, such as the equivalence of mass and energy. In fact, the general idea of associating mass with energy in some way had been around for about 25 years prior to Einstein's 1905 papers. Indeed, as Thomson and even Einstein himself noted, this association is already implicit in Maxwell's theory. With electric and magnetic fields e and b, the energy density is (e2 + b2)/(8p) and the momentum density is (e x b)/(4pc), so in the case of radiation (when e and b are equal and orthogonal) the energy density is E = e2/(4p) and the momentum density is p = e2/(4pc). Taking momentum p as the product of the radiation's "mass" m times its velocity c, we have
and so E = mc2. Indeed, in the 1905 paper containing his original deduction of mass-energy equivalence, Einstein acknowledges that it was explicitly based on "Maxwell's expression for the electromagnetic energy of space". We can also mention the pre-1905 work of Poincare and others on the electron mass arising from it's energy, and the work of Hasenohrl on how the mass of a cavity increases when it is filled with radiation. However, these suggestions were all very restricted in their applicability, and didn't amount to the assertion of a fundamental equivalence such as emerges so clearly from Einstein's relativistic interpretation. Hardly any of the formulas in Einstein's two 1905 papers on relativity were new, but what Einstein provided was a single conceptual framework within which all those formulas flow quite naturally from a simple set of general principles.
Occasionally one hears of other individuals who are said to have discovered one or more aspect of relativity prior to Einstein. To take just one example, in November of 1999 there appeared in newspapers around the world a story claiming that "The mathematical equation that ushered in the atomic age was discovered by an unknown Italian dilettante two years before Albert Einstein used it in developing the theory of relativity...". The "dilettante" in question was an Italian business man named Olinto De Pretto, and the implication of the story was that Einstein got the idea for mass-energy equivalence from "De Pretto's insight". There are some obvious difficulties with this account, only some of which can be blamed on the imprecision of popular journalism. First, the story claimed that Einstein used the idea of mass-energy equivalence to develop special relativity, whereas in fact the idea that energy has inertia appeared in a very brief note that Einstein submitted for publication toward the end of 1905, after (and as a consequence of) the original paper on special relativity.
The newspaper report went on to say that "De Pretto had stumbled on the equation, but not the theory of relativity... It was republished in 1904 by Veneto's Royal Science Institute... A Swiss Italian named Michele Besso alerted Einstein to the research and in 1905 Einstein published his own work..." Now, it's certainly true that Besso was Italian, and worked with Einstein at the Bern Patent Office during the years leading up to 1905, and it's true that they discussed physics, and Besso provided Einstein with suggestions for reading (for example, it was Besso who introduced him to the works of Ernst Mach). However, there is no evidence that Besso ever "alerted Einstein" to De Pretto's paper. Moreover, the idea that Einstein's second relativity paper in 1905 (let alone the first) was in any way prompted or inspired by De Pretto's rather silly and unoriginal comments is bizarre.
In essence, De Pretto's "insight" was the (hardly novel) idea that matter consists of tiny particles, and that these particles are agitated by their exposure to an ultra-mundane flux of hypothetical ether particles in a "shadow theory" of gravity. Supposing that these ether particles move at the speed of light (or perhaps at the "speed of electricity", which he believed was significantly higher), De Pretto reasoned – in a qualitative way – that the mean vibrational speed of the particles of matter must approach the speed of the ether particles, i.e., the speed of light. He then asserted (erroneously) that the kinetic energy of a mass m moving at speed v is mv2, which is actually Leibniz's "vis viva", the living force. On this basis, De Pretto asserted that the kinetic energy in a quantity of mass m would be mc2, which, he did not fail to notice, is a lot of energy. However, this line of reasoning was not original to De Pretto. The shadow theory of gravity was first conceived by Newton's friend Nicholas Fatio in the 1690's, and subsequently re-discovered by many individuals, notably George Louis Lesage in the late 18th century. Furthermore, the realization that the bombardment of such an intense ultramundane flux would necessarily elevate the temperature of ordinary matter to incredible temperatures in a fraction of a second was noted by both Kelvin and Maxwell in the late 19th century. Poincare and Lorentz both realized the same thing, and used this fact to conclude that the shadow model of gravity is not viable (since it entails the vaporization of the Earth in a fraction of a second). Hence, far from contributing any new "insight", De Pretto's only contribution was a lack of insight, in blithely ignoring the preposterous thermodynamic implications of this (very old) idea. Needless to say, none of this bears any resemblance to the concept of mass-energy equivalence that emerges from special relativity.
Of course, this is not to say that Einstein had no predecessors in working toward the genuine concept of mass-energy equivalence. Some form of this idea was already to be found in the writings of Thomson, Lorentz, Poincare, etc. (not to mention Isaac Newton, who famously asked "Are not gross bodies and light convertible into one another...?"). After all, the idea that the electron's mass was electromagnetic in origin was one of the leading hypotheses of research at that time. It would be like saying that some theoretical physicist today had never heard of string theory! But it's clear that mass-energy equivalence did not inspire Einstein's development of special relativity, because it isn't mentioned in the foundational paper of 1905. Only a few months later did he recognize this implication of the theory, prompting him to write in a letter to his close friend Conrad Habicht as he was preparing the paper on mass-energy equivalence:
One more consequence of the paper on electrodynamics has also occurred to me. The principle of relativity, in conjunction with Maxwell's equations, requires that mass be a direct measure of the energy contained in a body; light carries mass with it. A noticeable decrease of mass should occur in the case of radium [as it emits radiation]. The argument [which he intends to present in the paper] is amusing and seductive, but for all I know the Lord might be laughing over it and leading me around by the nose.
These are clearly the words of someone who is genuinely working out the consequences of his own recent paper, and wondering about their validity, not someone who has gotten an idea from seeing a formula in someone else's paper. Of course, the most obvious proof of the independence of Einstein's path to special relativity is simply the wonderfully lucid sequence of thoughts presented in his 1905 paper, beginning from first principles and a careful examination of the physical significance of time and space, and leading to the kinematics of special relativity, from which the inertia of energy emerges naturally.
Nevertheless, we shouldn't underestimate the real contributions to the development of special relativity made by Einstein's predecessors, most notably Lorentz and Poincare. In addition, although Einstein was remarkably thorough in his 1905 paper, there were nevertheless important contributions to the foundations of special relativity made by others in the years that followed. For example, in 1907 Max Planck greatly clarified relativistic mechanics, basing it on the conservation of momentum with his "more advantageous" definition of force, as did Tolman and Lewis. Planck also critiqued Einstein's original deduction of mass-energy equivalence, and gave a more general and comprehensive argument. (This led Johannes Stark in 1907 to cite Planck as the originator of mass-energy equivalence, prompting an angry letter from Einstein saying that he "was rather disturbed that you do not acknowledge my priority with regard to the connection between mass and energy". In later years Stark became an outspoken critic of Einstein's work.)
Another crucially important contribution was made by Hermann Minkowski (one of Einstein's former professors), who recognized that what Einstein had described was simply ordinary kinematics in a four-dimensional spacetime manifold with the pseudo-metric
Poincare had also recognized this as early as 1905. This was vital for the generalization of relativity which Einstein – with the help of his old friend Marcel Grossmann – developed on the basis on the theory of curved manifolds developed in the 19th century by Gauss and Riemann.
The tensor calculus and generally covariant formalism employed by Einstein in his general theory had been developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita around 1900 at the University of Padua, building on the earlier work of Gauss, Riemann, Beltrami, and Christoffel. In fact, the main technical challenge that occupied Einstein in his efforts to find a suitable field law for gravity, which was to construct from the metric tensor another tensor whose covariant derivative automatically vanishes, had already been solved in the form of the Bianchi identities, which lead directly to the Einstein tensor as discussed in Section 5.8.
Several other individuals are often cited as having anticipated some aspect of general relativity, although not in any sense of contributing seriously to the formulation of the theory. John Mitchell wrote in 1783 about the possibility of "dark stars" that we so massive light could not escape from them, and Laplace contemplated the same possibility in 1796. William Clifford wrote about a possible connection between matter and curved space in 1873. Around 1801 Johann von Soldner predicted that light rays passing near the Sun would be deflected by the Sun's gravity, just like a small corpuscle of matter moving at the speed of light (at a particular point on its trajectory). This gives a deflection of just half the relativistic value. Ironically, in accord with the German literature of the time, the parameter Soldner used to represent the "acceleration of gravity" was half the modern definition of that term, so his formulas included a factor of 2, which some people subsequently took as an indication that he had predicted the relativistic value. However, the Newtonian derivation he presented is unambiguous, and leads to the numerical value of 0.84 arc seconds, which was explicitly stated by Soldner, so there is no doubt that his prediction was half of the relativistic value.
Interestingly, the work of Soldner had been virtually forgotten until being rediscovered and publicized by Philipp Lenard in 1921, along with the claim that Hasenohrl should be credited with the mass-energy equivalence relation. Similarly in 1917 Ernst Gehrcke arranged for the re-publication of a 1898 paper by a secondary school teacher named Paul Gerber which contained a formula for the precession of elliptical orbits identical to the one Einstein had derived from the field equations of general relativity. Gerber's approach was based on the premise that the gravitational potential propagates at the speed of light, and that the effect of the potential on the motion of a body depends on the body's velocity through the potential field. His potential was similar in form to the Gauss-Weber theories. However, Gerber's "theory" was (and still is) regarded as unsatisfactory, mainly because his conclusions don't follow from his premises, but also because the combination of Gerber's proposed gravitational potential with the rest of (nonrelativistic) physics results in predictions (such as 3/2 the relativistic prediction for the deflection of light rays near the Sun) which are inconsistent with observation. In addition, Gerber's free mixing of propagating effects with some elements of action-at-a-distance tended to undermine the theoretical coherence of his proposal.
The writings of Mitchell, Soldner, Gerber, and others were, at most, anticipations of some of the phenomenology later associated with general relativity, but had nothing to do with the actual theory of general relativity, i.e., a theory that conceives of gravity as a manifestation of the curvature of spacetime. A closer precursors can be found in the notional writings of William Kingdon Clifford, but like Gauss and Riemann he lacked the crucial idea of including time as one of the dimensions of the manifold. As noted above, the formal means of treating space and time as a single unified spacetime manifold was conceived by Poincare and Minkowski, and the tensor calculus was developed by Ricci and Levi-Civita, with whom Einstein corresponded during the development of general relativity. It's also worth mentioning that Einstein and Grossmann, working in collaboration, came very close to discovering the correct field equations in 1913, but were diverted by an erroneous argument that led them to believe no fully covariant equations could be consistent with experience. In retrospect, this accident may have been all that prevented Grossmann from being perceived as a co-creator of general relativity. On the other hand, Grossmann had specifically distanced himself from the physical aspects of the 1913 paper, and Einstein wrote to Sommerfeld in July 1915 (i.e., prior to arriving at the final form of the field equations) that
Grossmann will never lay claim to being co-discoverer. He only helped in guiding me through the mathematical literature but contributed nothing of substance to the results.
In the summer of 1915 Einstein gave a series of lectures at Gottingen on the general theory, and apparently succeeded in convincing both Hilbert and Klein that he was close to an important discovery, despite the fact that he had not yet arrived at the final form of the field equations. Hilbert took up the problem from an axiomatic standpoint, and carried on an extensive correspondence with Einstein until the 19th of November. On the 20th, Hilbert submitted a paper to the Gesellschaft der Wissenschaften in Gottingen with a derivation of the field equations. Five days later, on 25 November, Einstein submitted a paper with the correct form of the field equations to the Prussian Academy in Berlin. The exact sequence of events leading up to the submittal of these two papers – and how much Hilbert and Einstein learned from each other – is somewhat murky, especially since Hilbert's paper was not actually published until March of 1916, and seems to have undergone some revisions from what was originally submitted. However, the question of who first wrote down the fully covariant field equations (including the trace term) is less significant than one might think, because, as Einstein wrote to Hilbert on 18 November after seeing a draft of Hilbert's paper
The difficulty was not in finding generally covariant equations for the gmn's; for this is easily achieved with the aid of Riemann's tensor. Rather, it was hard to recognize that these equations are a generalization – that is, a simple and natural generalization – of Newton's law.
It might be argued that Einstein was underestimating the mathematical difficulty, since he hadn't yet included the trace term in his published papers, but in fact he repeated the same comment in a letter to Sommerfeld on 28 November, this time explicitly referring to the full field equations, with the trace term. He wrote
It is naturally easy to set these generally covariant equations down; however, it is difficult to recognize that they are generalizations of Poisson's equations, and not easy to recognize that they fulfill the conservation laws. I had considered these equations with Grossmann already 3 years ago, with the exception of the [trace term], but at that time we had come to the conclusion that it did not fulfill Newton's approximation, which was erroneous.
Thus he regards the purely mathematical task of determining the most general fully covariant expression involving the gmn's and their first and second derivatives as comparatively trivial and straightforward – as indeed it is for a competent mathematician. The Bianchi identities were already known, so there was no new mathematics involved. The difficulty, as Einstein stressed, was not in writing down the solution of this mathematical problem, but in conceiving of the problem in the first place, and then showing that it represents a viable law of gravitation. In this, Einstein was undeniably the originator, not only in showing that the field equations reduce to Newton's law in the first approximation, but also in showing that they yield Mercury's excess precession in the second approximation. Hilbert was suitably impressed when Einstein showed this in his paper of 18 November, and it's important to note that this was how Einstein was spending his time around the 18th of November, establishing the physical implications of the fully covariant field equations, while Hilbert was busying himself with elaborating the mathematical aspects of the problem that Einstein had outlined the previous summer.
It's also worth noting that although they arrived at the same formulas, Hilbert and Einstein were working in fundamentally different contexts, so it would be somewhat misleading to say that they arrived at the same theoretical result. Already in 1921 Pauli commented on both the simultaneous discoveries and on the distinctions between what the two men discovered.
At the same time as Einstein, and independently, Hilbert formulated the generally covariant field equations. His presentation, though, would not seem to be acceptable to physicists, for two reasons. First, the existence of a variational principle is introduced as an axiom. Secondly, of more importance, the field equations are not derived for an arbitrary system of matter, but are specifically based on Mie's theory of matter.
Whatever the true sequence of events and interactions, it seems that Einstein initially had some feelings of resentment toward Hilbert, perhaps thinking that Hilbert had acted ungraciously and stolen some of his glory. Already on November 20 Einstein had written to a friend
The theory is incomparably beautiful, but only one colleague understands it, and that one works skillfully at "nostrification". I have learned the deplorableness of humans more in connection with this theory than in any other personal experience. But it doesn't bother me.
(Literally the word "nostrification" refers to the process by which a country accepts foreign academic degrees as if they had been granted by one of its own universities, but the word has often been used to suggest the appropriation and re-packaging of someone else's ideas and making them one's own.) However, by December 20 he was able to write a conciliatory note to Hilbert, saying
There has been between us a certain unpleasantness, whose cause I do not wish to analyze. I have struggled against feelings of bitterness with complete success. I think of you again with untroubled friendliness, and ask you to do the same with me. It would be a shame if two fellows like us, who have worked themselves out from this shabby world somewhat, cannot enjoy each other.
Thereafter they remained on friendly terms, and Hilbert never publicly claimed any priority in the discovery of general relativity, and always referred to it as Einstein's theory.
As it turned out, Einstein can hardly have been dissatisfied with the amount of popular credit he received for the theories of relativity, both special and general. Nevertheless, one senses a bit of annoyance when Max Born mentioned to Einstein in 1953 (two years before Einstein's death) that the second volume of Edmund Whittaker's book "A History of the Theories of Aether and Electricity" had just appeared, in which special relativity is attributed to Lorentz and Poincare, with barely a mention of Einstein except to say that "in the autumn of [1905] Einstein published a paper which set forth the relativity theory of Poincare and Lorentz with some amplifications, and which attracted much attention". In the same book Whittaker attributes some of the fundamental insights of general relativity to Planck and a mathematician named Harry Bateman (a former student of Whittaker's). Einstein replied to his old friend Born
Everybody does what he considers right... If he manages to convince others, that is their own affair. I myself have certainly found satisfaction in my efforts, but I would not consider it sensible to defend the results of my work as being my own 'property', as some old miser might defend the few coppers he had laboriously scrapped together. I do not hold anything against him [Whittaker], nor of course, against you. After all, I do not need to read the thing.
On the other hand, in the same year (1953), Einstein wrote to the organizers of a celebration honoring the upcoming fiftieth anniversary of his paper on the electrodynamics of moving bodies, saying
I hope that one will also take care on that occasion to suitably honor the merits of Lorentz and Poincare.
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https://en.wikipedia.org/wiki/Exciton | Exciton
Frenkel exciton, bound electron-hole pair where the hole is localized at a position in the crystal represented by black dots
Wannier-Mott exciton, bound electron-hole pair that is not localized at a crystal position. This figure schematically shows diffusion of the exciton across the lattice.
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and in some liquids. The exciton is regarded as an elementary excitation of condensed matter that can transport energy without transporting net electric charge.[1][2]
An exciton can form when a photon is absorbed by a semiconductor.[3] This excites an electron from the valence band into the conduction band. In turn, this leaves behind a positively charged electron hole (an abstraction for the location from which an electron was moved). The electron in the conduction band is then effectively attracted to this localized hole by the repulsive Coulomb forces from large numbers of electrons surrounding the hole and excited electron. This attraction provides a stabilizing energy balance. Consequently, the exciton has slightly less energy than the unbound electron and hole. The wavefunction of the bound state is said to be hydrogenic, an exotic atom state akin to that of a hydrogen atom. However, the binding energy is much smaller and the particle's size much larger than a hydrogen atom. This is because of both the screening of the Coulomb force by other electrons in the semiconductor (i.e., its dielectric constant), and the small effective masses of the excited electron and hole. The recombination of the electron and hole, i.e. the decay of the exciton, is limited by resonance stabilization due to the overlap of the electron and hole wave functions, resulting in an extended lifetime for the exciton.
The electron and hole may have either parallel or anti-parallel spins. The spins are coupled by the exchange interaction, giving rise to exciton fine structure. In periodic lattices, the properties of an exciton show momentum (k-vector) dependence.
The concept of excitons was first proposed by Yakov Frenkel in 1931,[4] when he described the excitation of atoms in a lattice of insulators. He proposed that this excited state would be able to travel in a particle-like fashion through the lattice without the net transfer of charge.
Classification
Excitons may be treated in two limiting cases, depending on the properties of the material in question.
Frenkel excitons
In materials with a small dielectric constant, the Coulomb interaction between an electron and a hole may be strong and the excitons thus tend to be small, of the same order as the size of the unit cell. Molecular excitons may even be entirely located on the same molecule, as in fullerenes. This Frenkel exciton, named after Yakov Frenkel, has a typical binding energy on the order of 0.1 to 1 eV. Frenkel excitons are typically found in alkali halide crystals and in organic molecular crystals composed of aromatic molecules, such as anthracene and tetracene.
Wannier–Mott excitons
In semiconductors, the dielectric constant is generally large. Consequently, electric field screening tends to reduce the Coulomb interaction between electrons and holes. The result is a Wannier exciton,[5] which has a radius larger than the lattice spacing. Small effective mass of electrons that is typical of semiconductors also favors large exciton radii. As a result, the effect of the lattice potential can be incorporated into the effective masses of the electron and hole. Likewise, because of the lower masses and the screened Coulomb interaction, the binding energy is usually much less than that of a hydrogen atom, typically on the order of 0.01eV. This type of exciton was named for Gregory Wannier and Nevill Francis Mott. Wannier-Mott excitons are typically found in semiconductor crystals with small energy gaps and high dielectric constants, but have also been identified in liquids, such as liquid xenon. They are also known as large excitons.
In single-wall carbon nanotubes, excitons have both Wannier-Mott and Frenkel character. This is due to the nature of the Coulomb interaction between electrons and holes in one-dimension. The dielectric function of the nanotube itself is large enough to allow for the spatial extent of the wave function to extend over a few to several nanometers along the tube axis, while poor screening in the vacuum or dielectric environment outside of the nanotube allows for large (0.4 to 1.0eV) binding energies.
Often there is more than one band to choose from for the electron and the hole leading to different types of excitons in the same material. Even high-lying bands can be effective as femtosecond two-photon experiments have shown. At cryogenic temperatures, many higher excitonic levels can be observed approaching the edge of the band,[6] forming a series of spectral absorption lines that are in principle similar to hydrogen spectral series.
Charge-transfer excitons
An intermediate case between Frenkel and Wannier excitons, charge-transfer excitons (sometimes called simply CT excitons) occur when the electron and the hole occupy adjacent molecules.[7] They occur primarily in ionic crystals.[8] Unlike Frenkel and Wannier excitons they display a static electric dipole moment.[9]
Surface excitons
At surfaces it is possible for so called image states to occur, where the hole is inside the solid and the electron is in the vacuum. These electron-hole pairs can only move along the surface.
Atomic and molecular excitons
Alternatively, an exciton may be an excited state of an atom, ion, or molecule, the excitation wandering from one cell of the lattice to another.
When a molecule absorbs a quantum of energy that corresponds to a transition from one molecular orbital to another molecular orbital, the resulting electronic excited state is also properly described as an exciton. An electron is said to be found in the lowest unoccupied orbital and an electron hole in the highest occupied molecular orbital, and since they are found within the same molecular orbital manifold, the electron-hole state is said to be bound. Molecular excitons typically have characteristic lifetimes on the order of nanoseconds, after which the ground electronic state is restored and the molecule undergoes photon or phonon emission. Molecular excitons have several interesting properties, one of which is energy transfer (see Förster resonance energy transfer) whereby if a molecular exciton has proper energetic matching to a second molecule's spectral absorbance, then an exciton may transfer (hop) from one molecule to another. The process is strongly dependent on intermolecular distance between the species in solution, and so the process has found application in sensing and molecular rulers.
The hallmark of molecular excitons in organic molecular crystals are doublets and/or triplets of exciton absorption bands strongly polarized along crystallographic axes. In these crystals an elementary cell includes several molecules sitting in symmetrically identical positions, which results in the level degeneracy that is lifted by intermolecular interaction. As a result, absorption bands are polarized along the symmetry axes of the crystal. Such multiplets were discovered by Antonina Prikhot'ko[10][11] and their genesis was proposed by Alexander Davydov. It is known as 'Davydov splitting'.[12][13]
Giant oscillator strength of bound excitons
Excitons are lowest excited states of the electronic subsystem of pure crystals. Impurities can bind excitons, and when the bound state is shallow, the oscillator strength for producing bound excitons is so high that impurity absorption can compete with intrinsic exciton absorption even at rather low impurity concentrations. This phenomenon is generic and applicable both to the large radius (Wannier-Mott) excitons and molecular (Frenkel) excitons. Hence, excitons bound to impurities and defects possess giant oscillator strength.[14]
Self-trapping of excitons
In crystals excitons interact with phonons, the lattice vibrations. If this coupling is weak as in typical semiconductors such as GaAs or Si, excitons are scattered by phonons. However, when the coupling is strong, excitons can be self-trapped.[15][16] Self-trapping results in dressing excitons with a dense cloud of virtual phonons which strongly suppresses the ability of excitons to move across the crystal. In simpler terms, this means a local deformation of the crystal lattice around the exciton. Self-trapping can be achieved only if the energy of this deformation can compete with the width of the exciton band. Hence, it should be of atomic scale, of about an electron volt.
Self-trapping of excitons is similar to forming strong-coupling polarons but with three essential differences. First, self-trapped exciton states are always of a small radius, of the order of lattice constant, due to their electric neutrality. Second, there exists a self-trapping barrier separating free and self-trapped states, hence, free excitons are metastable. Third, this barrier enables coexistence of free and self-trapped states of excitons.[17][18] This means that spectral lines of free excitons and wide bands of self-trapped excitons can be seen simultaneously in absorption and luminescence spectra. While the self-trapped states are of lattice-spacing scale, the barrier has typically much larger scale. Indeed, its spatial scale is about ${\displaystyle r_{b}\sim m\gamma ^{2}/\omega ^{2}}$ where ${\displaystyle m}$ is effective mass of the exciton, ${\displaystyle \gamma }$ is the exciton-phonon coupling constant, and ${\displaystyle \omega }$ is the characteristic frequency of optical phonons. Excitons are self-trapped when ${\displaystyle m}$ and ${\displaystyle \gamma }$ are large, and then the spatial size of the barrier is large compared with the lattice spacing. Transforming a free exciton state into a self-trapped one proceeds as a collective tunneling of coupled exciton-lattice system (an instanton). Because ${\displaystyle r_{b}}$ is large, tunneling can be described by a continuum theory.[19] The height of the barrier ${\displaystyle W\sim \omega ^{4}/m^{3}\gamma ^{4}}$. Because both ${\displaystyle m}$ and ${\displaystyle \gamma }$ appear in the denominator of ${\displaystyle W}$, the barriers are basically low. Therefore, free excitons can be seen in crystals with strong exciton-phonon coupling only in pure samples and at low temperatures. Coexistence of free and self-trapped excitons was observed in rare-gas solids,[20][21] alkali-halides,[22] and in molecular crystal of pyrene.[23]
Interaction
Excitons are the main mechanism for light emission in semiconductors at low temperature (when the characteristic thermal energy kT is less than the exciton binding energy), replacing the free electron-hole recombination at higher temperatures.
The existence of exciton states may be inferred from the absorption of light associated with their excitation. Typically, excitons are observed just below the band gap.
When excitons interact with photons a so-called polariton (also exciton-polaritons) is formed. These excitons are sometimes referred to as dressed excitons.
Provided the interaction is attractive, an exciton can bind with other excitons to form a biexciton, analogous to a dihydrogen molecule. If a large density of excitons is created in a material, they can interact with one another to form an electron-hole liquid, a state observed in k-space indirect semiconductors.
Additionally, excitons are integer-spin particles obeying Bose statistics in the low-density limit. In some systems, where the interactions are repulsive, a Bose–Einstein condensed state, called excitonium, is predicted to be the ground state. Some evidence of excitonium has existed since the 1970s, but has often been difficult to discern from a Peierls phase.[24] Exciton condensates have allegedly been seen in a double quantum well systems.[25] In 2017 Kogar et al. found "compelling evidence" for observed excitons condensing in the three-dimensional semimetal 1T-TiSe2[26]
Spatially direct and indirect excitons
Normally, excitons in a semiconductor have a very short lifetime due to the close proximity of the electron and hole. However, by placing the electron and hole in spatially separated quantum wells with an insulating barrier layer in between so called 'spatially indirect' excitons can be created. In contrast to ordinary (spatially direct), these spatially indirect excitons can have large spatial separation between the electron and hole, and thus possess a much longer lifetime. This is often used to cool excitons to very low temperatures in order to study Bose–Einstein condensation (or rather its two-dimensional analog).[27]
References
1. ^ R. S. Knox, Theory of excitons, Solid state physics (Ed. by Seitz and Turnbul, Academic, NY), v. 5, 1963.
2. ^ Liang, W Y (1970). "Excitons". Physics Education. 5 (125301): 226–228. Bibcode:1970PhyEd...5..226L. doi:10.1088/0031-9120/5/4/003.
3. ^ Couto, ODD; Puebla, J (2011). "Charge control in InP/(Ga,In)P single quantum dots embedded in Schottky diodes". Physical Review B. 84 (4): 226. arXiv:. Bibcode:1970PhyEd...5..226L. doi:10.1103/PhysRevB.84.125301.
4. ^ Frenkel, J. (1931). "On the Transformation of light into Heat in Solids. I". Physical Review. 37 (1): 17. Bibcode:1931PhRv...37...17F. doi:10.1103/PhysRev.37.17.
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6. ^ http://www.nature.com/nature/journal/v514/n7522/full/nature13832.html
7. ^ J. D. Wright (1995) [First published 1987]. Molecular Crystals (2nd ed.). Cambridge University Press. p. 108. ISBN 978-0-521-47730-7.
8. ^ Ivan Pelant, Jan Valenta (2012). Luminescence Spectroscopy of Superconductors. Oxford University Press. p. 161. ISBN 978-0-19-958833-6.
9. ^ Guglielmo Lanzani (2012). The Photophysics Behind Photovoltaics and Photonics. Wiley-VCH Verlag. p. 82.
10. ^ A. Prikhotjko, Absorption Spectra of Crystals at Low Temperatures, J. Physics USSR 8, 257 (1944)
11. ^ A. F. Prikhot'ko, Izv, AN SSSR Ser. Fiz. 7, 499 (1948) http://ujp.bitp.kiev.ua/files/journals/53/si/53SI18p.pdf
12. ^ A.S Davydov, Theory of Molecular Excitons (Plenum, NY) 1971
13. ^ V. L. Broude, E. I. Rashba, and E. F. Sheka, Spectroscopy of molecular excitons (Springer, NY) 1985
14. ^ E. I. Rashba, Giant Oscillator Strengths Associated with Exciton Complexes, Sov. Phys. Semicond. 8, 807-816 (1975)
15. ^ N. Schwentner, E.-E. Koch, and J. Jortner, Electronic excitations in condensed rare gases, Springer tracts in modern physics, 107, 1 (1985).
16. ^ M. Ueta, H. Kanzaki, K. Kobayashi, Y. Toyozawa, and E. Hanamura. Excitonic Processes in Solids, Springer Series in Solid State Sciences, Vol. 60 (1986).
17. ^ E. I. Rashba, "Theory of Strong Interaction of Electron Excitations with Lattice Vibrations in Molecular Crystals, Optika i Spektroskopiya 2, 75, 88 (1957).
18. ^ E. I. Rashba, Self-trapping of excitons, in: Excitons (North-Holland, Amsterdam, 1982), p. 547.
19. ^ A. S. Ioselevich and E. I. Rashba, Theory of Nonradiative Trapping in Crystals, in: "Quantum tunneling in condensed media." Eds. Yu. Kagan and A. J. Leggett. (North-Holland, Amsterdam, 1992), p. 347-425.https://books.google.com/books?hl=en&lr=&id=ElDtL9qZuHUC&oi=fnd&pg=PA347&dq=%22E+I+Rashba%22&ots=KjE3JYn9kl&sig=0Aj4IdVj0zqPSyq3ep_RT6sOlgQ#v=onepage&q=%22E%20I%20Rashba%22&f=false
20. ^ U. M. Grassano, "Excited-State Spectroscopy in Solids", Proceedings of the International School of Physics "Enrico Fermi", Course 96, Varenna, Italy, 9-19 July 1985. Amsterdam;New York: North-Holland (1987). ISBN 9780444870704, [1].
21. ^ I. Ya. Fugol', "Free and self-trapped excitons in cryocrystals: kinetics and relaxation processes." Advances in Physics 37, 1-35 (1988).
22. ^ Ch. B. Lushchik, in "Excitons," edited by E. I. Rashba, and M. D. Sturge, (North Holland, Amsterdam, 1982), p. 505.
23. ^ M. Furukawa, Ken-ichi Mizuno, A. Matsui, N. Tamai and I. Yamazaiu, Branching of Exciton Relaxation to the Free and Self-Trapped Exciton States, Chemical Physics 138, 423 (1989).
24. ^ "New form of matter 'excitonium' discovered". The Times of India. Retrieved 10 December 2017.
25. ^ Eisenstein, J.P. (January 10, 2014). "Exciton Condensation in Bilayer Quantum Hall Systems". Annual Review of Condensed Matter Physics. 5: 159–181. arXiv:. doi:10.1146/annurev-conmatphys-031113-133832.
26. ^ .Kogar, Anshul; Rak, Melinda S; Vig, Sean; Husain, Ali A; Flicker, Felix; Joe, Young Il; Venema, Luc; MacDougall, Greg J; Chiang, Tai C; Fradkin, Eduardo; Van Wezel, Jasper; Abbamonte, Peter (2017). "Signatures of exciton condensation in a transition metal dichalcogenide". Science. 358 (6368): 1314–1317. arXiv:. Bibcode:2017Sci...358.1314K. doi:10.1126/science.aam6432. PMID 29217574.
27. ^ A. A. High (2012) "Spontaneous coherence in a cold exciton gas" Nature | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 11, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8630414605140686, "perplexity": 2487.263511219861}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158766.53/warc/CC-MAIN-20180922221246-20180923001646-00050.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/9705/why-are-non-clifford-gates-more-complex-than-clifford-gates | # Why are non-Clifford gates more complex than Clifford gates?
There are two groups of quantum gates - Clifford gates and non-Clifford gates.
Representatives of Clifford gates are Pauli matrices $$I$$, $$X$$, $$Y$$ and $$Z$$, Hadamard gate $$H$$, $$S$$ gate and $$CNOT$$ gate. Non-Clifford gate is for example $$T$$ gate and Toffoli gate (because its implementation comprise $$T$$ gates).
While Clifford gates can be simulated on classical computer efficiently (i.e. in polynomial time), non-Clifford gates cannot. Moreover (if my understanding is correct), non-Clifford gates increase time consumption of a quantum algorithm far more than Clifford gates.
My questions are these:
1. Am I right that non-Clifford gates increase time consumption (or complexity of quantum algorithm)?
2. Why non-Clifford gates cannot be simulated efficiently? This is confusing for me, because $$S$$ and $$T$$ gates are both rotations with only different angle.
• “non-Clifford gates are not known to be efficiently simulatable.” – Mark S Feb 1 at 3:02
## 1 Answer
1. Yes, you are correct. Non-Clifford gates cannot be transversely implemented, instead implementation generally requires distilling magic states or Toffoli states. In practice this requires significantly more spacetime volume than Clifford gates. For reference, see the introduction sections here and here.
2. The natural expectation would be that no quantum gates can be simulated efficiently by classical computers since an n-qubit quantum circuit operates in a $$2^n$$-dimensional Hilbert space. The (arguably) surprising result is that circuits consisting only of Clifford gates can be simulated efficiently (by the Gottesman-Knill Theorem). It's a very natural situation that non-Clifford gates cannot be simulated efficiently because of the size of Hilbert space in which they operate. If both Clifford and non-Clifford gates could be simulated efficiently by classical computers, there would be no (or at least drastically reduced) motivation to build quantum computers. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8847281336784363, "perplexity": 1720.702284747105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107889173.38/warc/CC-MAIN-20201025125131-20201025155131-00578.warc.gz"} |
https://www.paperswithcode.com/paper/nettailor-tuning-the-architecture-not-just-1 | # NetTailor: Tuning the Architecture, Not Just the Weights
Real-world applications of object recognition often require the solution of multiple tasks in a single platform. Under the standard paradigm of network fine-tuning, an entirely new CNN is learned per task, and the final network size is independent of task complexity... (read more)
PDF Abstract CVPR 2019 PDF CVPR 2019 Abstract | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.918624758720398, "perplexity": 3218.2978680130277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141197593.33/warc/CC-MAIN-20201129093434-20201129123434-00395.warc.gz"} |
https://www.physicsforums.com/threads/entropy-second-law-for-systems-with-zero-input-net-energy.875227/ | # Entropy: second law for systems with zero input net energy
• I
• Thread starter haushofer
• Start date
• #1
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Dear all,
I'm trying to think about applying the second law of thermodynamics to a system which is not isolated, but has an energy flowing inwards and an equal (!) energy flowing outwards, such that the total energy does not change (total energy flux is zero). Can we still apply the second law in this case? And where can I find a reference (here or elsewhere) which treats this case?
## Answers and Replies
• #2
hilbert2
Gold Member
1,472
498
If you compress an ideal gas isothermally, there is an influx of work and equal outflux of heat and the internal energy of the gas doesn't change. ( http://hyperphysics.phy-astr.gsu.edu/hbase/therm/entropgas.html )
Of course if a closed system is in thermal equilibrium with its surroundings, it's constantly exchanging molecular kinetic energy (heat) with the surroundings, but there is no net flow of heat to either direction.
Likes haushofer
• #3
2,472
860
Ah, yes, of course, that's a familiar example. Thanks! My thermodynamics is a bit rusty, but I'm reviewing some for applications to cosmology (gravitating systems).
• #4
277
97
I'm trying to think about applying the second law of thermodynamics to a system which is not isolated, but has an energy flowing inwards
Yes. Here is a place to start.
Likes haushofer
• #5
hilbert2
Gold Member
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498
Do we mean the same thing with terms "isolated", "closed", and "open"? An isolated system doesn't exchange either energy or matter with its surroundings. A closed system can exchange heat but not matter. An open system can exchange both heat and matter with the rest of the universe.
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• #6
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860
Yes. So I'm referring to the second law for closed systems instead of isolated ones.
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5K | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8983778953552246, "perplexity": 1273.6078016789793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178360745.35/warc/CC-MAIN-20210228084740-20210228114740-00062.warc.gz"} |
https://imaginary.org/users/joshua-carroll | # Joshua Carroll
I’m currently a graduate student in Applied Mathematics, with an undergraduate degree in Physics. Professionally, I work as an Engineer in a R&D capacity
Joshua Carroll
United States | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8813530206680298, "perplexity": 3297.343757834401}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314353.10/warc/CC-MAIN-20190818231019-20190819013019-00305.warc.gz"} |
http://mathhelpforum.com/calculus/39551-differentiation-question-print.html | # differentiation question
• May 25th 2008, 04:02 AM
afan17
differentiation question
Hi, I'm new to this forum. Any help would be greatly appreciated. I'm actually tutoring a refugee in a volunteer program, but it's been so long since I've done this maths that I'm having real trouble with it so in order to help her, I myself need some help!
A Queenland resort has a large swimming pool with AB=75. The pool is rectangular and has corners A,B,C,D, with A and C as opposites and B and D as opposites. P is a point somewhere between D and C.
A boy can swim at 1 m/s and run at 1, 2/3 m/s. He starts at A, swims to a point P on DC, and runs from P to C. He takes 2 seconds to pull himself out of the pool.
a, Let DP=x m and the total time be T s. Show that T=square root of (x squared +900)+3/5 (75-x)+2.
b, find dT/dx
c. i Find the value of x for which the time taken is a minimum.
ii Find the minimum time.
d. Find the time taken if the boy runs from A to D and then D to C.
----------
An isosceles trapezoid is inscribed in the parabolas y=4-x squared as illustrated (the parabola intersects the x axis at (-2,0) and (2,0) and is symmetrical.
a. show that the area of the trapezoid is :
1/2(4-x squared) (2x+4)
b, show that the trapezoid has its greatest area when x=2/3
c, Repeat with the parabolas y = a squared - x squared
i, show that the area, A, of the trapezoid = ( a squared - x squared) ( a + x)
ii, use the product rule to find dA/ dx.
iii, show that a maximum occurs when x = a/3.
Thanks so much.
• May 25th 2008, 04:54 AM
earboth
1 Attachment(s)
Quote:
Originally Posted by afan17
...
An isosceles trapezoid is inscribed in the parabolas y=4-x squared as illustrated (the parabola intersects the x axis at (-2,0) and (2,0) and is symmetrical.
a. show that the area of the trapezoid is :
1/2(4-x squared) (2x+4)
b, show that the trapezoid has its greatest area when x=2/3
c, Repeat with the parabolas y = a squared - x squared
i, show that the area, A, of the trapezoid = ( a squared - x squared) ( a + x)
ii, use the product rule to find dA/ dx.
iii, show that a maximum occurs when x = a/3.
Let a and b denote the parallel sides of the trapezium and h it's height. Then the area of the trapezium is calculated by:
$A_t=\frac{a+b}2 \cdot h$
With your trapezium the base parallel has the length 4 and the upper parallel has the length 2x. The height correspond to the y-value. Plug in these terms into the formula given above:
$A_T(x)=\frac{2x+4}2 \cdot (4-x^2)$
which is exactly the given result.
To get the maximum value of $A_T$ calculate the derivative with respect to x:
$A_T(x)= (x+2)(4-x^2)=-x^3-2x^2+4x+8$
$A'_T(x)=-3x^2-4x+4$
$A'_T(x)=0~\implies~x=\frac23~\vee~x=-2$
The negative solution isn't very plausible here.
Do the following questions in exactly the same manner.
• May 25th 2008, 05:59 AM
Soroban
Hello, afan17!
Quote:
I'm new to this forum . . . Welcome aboard!
I'm actually tutoring a refugee in a volunteer program . . . Good for you!
A resort has a rectangular swimming pool ABCD with AB = 75m, AD = 30m.
P is a point somewhere between D and C.
A boy can swim at 1 m/s and run at 1 2/3 m/s.
He starts at A, swims to point P, and runs from P to C.
He takes 2 seconds to pull himself out of the pool.
a) Let DP = x meters and the total time be T seconds.
Show that: . $T\:=\:\sqrt{x^2+900}+ \frac{3}{5}(75-x)+2$
Code:
A 75 B * - - - - - - - - - - - - - - * | * | | * | 30 | * | | * | | * | * - - - - - - - - * - - - - - * D x P 75-x C
He will swim the diagonal distance $AP.$
This is the hypotenuse of right triangle $ADP.$
. . Hence:. . $AP \:=\:\sqrt{x^2 + 30^2}$
At 1 m/sec, this takes him: . $\frac{\sqrt{x^2+900}}{1} \:= \:{\color{blue}\sqrt{x^2+900}}$ seconds.
He runs the distance $PC \:=\:75-x\,\text{ at }\,\frac{5}{3}$ m/sec.
This takes him: . $\frac{75-x}{\frac{5}{3}} \:=\:{\color{blue}\frac{3}{5}(75-x)}$ seconds.
He also used 2 seconds to leave the pool.
His total time is: . $\boxed{T \;=\;\sqrt{x^2+900} + \frac{3}{5}(75-x) + 2}$ seconds.
Quote:
b) Find $\frac{dT}{dx}$
We have: . $T \;=\;\left(x^2+900\right)^{\frac{1}{2}} - \frac{3}{5}x + 47$
Then: . $\frac{dT}{dx} \;=\;\frac{1}{2}\left(x^2+900\right)^{-\frac{1}{2}}(2x) - \frac{3}{5} \;=\;\boxed{\frac{x}{\sqrt{x^2+900}} - \frac{3}{5}}$
Quote:
c) (i) Find the value of $x$ for which $T$ is a minimum.
(ii) Find the minimum time.
(i) Solve $\frac{dT}{dx} \:=\:0$
We have: . $\frac{x}{\sqrt{x^2+900}} \:=\:\frac{3}{5} \quad\Rightarrow\quad 5x\:=\:3\sqrt{x^2+900}$
Square both sides: . $25x^2 \:=\:9(x^2 + 900)\quad\Rightarrow\quad 25x^2 \:=\:9x^2 + 8100$
. . $16x^2 \:=\:8100\quad\Rightarrow\quad x^2 \:=\:\frac{2025}{4}\quad\Rightarrow\quad x \:=\:\frac{45}{2} \:=\:\boxed{22.5\text{ meters}}$
(ii) Substitute into the formula in part (a).
$T \;=\;\sqrt{22.5^2 + 900} + \frac{3}{5}(75-22.5) + 2 \;\;=\;\;\boxed{71\text{ seconds}}$
Quote:
d) Find the time taken if the boy runs from A to D and then D to C.
He would run: . $30 + 75$ meters at $\frac{5}{3}$ m/sec
This will take him: . $\frac{105}{\frac{5}{3}} \:=\:\boxed{63\text{ seconds}}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Very funny!
If the boy's only concern is getting from $A$ to $C$ as fast as possible,
. . he should all the way (and skip the math).
However, if this some sport where some swimming is required,
. . then our solution provides the shortest time.
• May 26th 2008, 09:18 PM
afan17
Quote:
Originally Posted by Soroban
Hello, afan17!
Code:
A 75 B * - - - - - - - - - - - - - - * | * | | * | 30 | * | | * | | * | * - - - - - - - - * - - - - - * D x P 75-x C
He will swim the diagonal distance $AP.$
This is the hypotenuse of right triangle $ADP.$
. . Hence:. . $AP \:=\:\sqrt{x^2 + 30^2}$
At 1 m/sec, this takes him: . $\frac{\sqrt{x^2+900}}{1} \:= \:{\color{blue}\sqrt{x^2+900}}$ seconds.
He runs the distance $PC \:=\:75-x\,\text{ at }\,\frac{5}{3}$ m/sec.
This takes him: . $\frac{75-x}{\frac{5}{3}} \:=\:{\color{blue}\frac{3}{5}(75-x)}$ seconds.
He also used 2 seconds to leave the pool.
His total time is: . $\boxed{T \;=\;\sqrt{x^2+900} + \frac{3}{5}(75-x) + 2}$ seconds.
We have: . $T \;=\;\left(x^2+900\right)^{\frac{1}{2}} - \frac{3}{5}x + 47$
Then: . $\frac{dT}{dx} \;=\;\frac{1}{2}\left(x^2+900\right)^{-\frac{1}{2}}(2x) - \frac{3}{5} \;=\;\boxed{\frac{x}{\sqrt{x^2+900}} - \frac{3}{5}}$
(i) Solve $\frac{dT}{dx} \:=\:0$
We have: . $\frac{x}{\sqrt{x^2+900}} \:=\:\frac{3}{5} \quad\Rightarrow\quad 5x\:=\:3\sqrt{x^2+900}$
Square both sides: . $25x^2 \:=\:9(x^2 + 900)\quad\Rightarrow\quad 25x^2 \:=\:9x^2 + 8100$
. . $16x^2 \:=\:8100\quad\Rightarrow\quad x^2 \:=\:\frac{2025}{4}\quad\Rightarrow\quad x \:=\:\frac{45}{2} \:=\:\boxed{22.5\text{ meters}}$
(ii) Substitute into the formula in part (a).
$T \;=\;\sqrt{22.5^2 + 900} + \frac{3}{5}(75-22.5) + 2 \;\;=\;\;\boxed{71\text{ seconds}}$
He would run: . $30 + 75$ meters at $\frac{5}{3}$ m/sec
This will take him: . $\frac{105}{\frac{5}{3}} \:=\:\boxed{63\text{ seconds}}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Very funny!
If the boy's only concern is getting from $A$ to $C$ as fast as possible,
. . he should all the way (and skip the math).
However, if this some sport where some swimming is required,
. . then our solution provides the shortest time.
Thank you so much! Honestly, such a lifesaver and it really is appreciated. Now I can go through this with the student tomorrow :) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 48, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9283416271209717, "perplexity": 2211.8465213901}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687428.60/warc/CC-MAIN-20170920175850-20170920195850-00409.warc.gz"} |
http://www.thespectrumofriemannium.com/2015/08/08/log177-scherk-susy-and-sugra/?shared=email&msg=fail | # LOG#177. SCHERK, SUSY and SUGRA.
Joël Scherk (not Schrek, the ogre) was a French theoretical physicist with a tragic death. Perhaps his contributions were short, but they were important in the fields of supersymmetry (SUSY), supergravity (SUGRA) and superstrings. Moreover, he also introduced some crazy ideas related to these things I am going to discuss. This blog post is essentially a refurbished reedition of his famous talk giving an overview about SUGRA, SUSY and antigravity. I have added some extra stuff by my own, and I plan to update this post in the future with more topics involving the phenomenon of antigravity that Scherk discovered long ago. Enjoy it!
First of all, even when I have not (not yet) dedicated a thread to SUSY or SUGRA, I will be discussing some of his ideas with respect to them.
What is SUSY? SUSY is just a particular type of symmetry or transformation group. Generally speaking, SUSY is defined in terms of transformations which leave invariant certain action AND transforms fermions (or fermionic fields to be more precise) into bosons and viceversa. I have discussed group theory in this blog (elementary group theory) and Lie algebras too. There is a generalization of Lie algebras called Lie superalgebras, also known as graded Lie algebras. Graded Lie algebras (GLA or superalgebras) are the elementary mathematical structures and tools used in SUSY/SUGRA. GLAs used in SUSY and SUGRA models, and their mathematical representations analogue to those in Lie algebras, will be reviewed in this post, based on a famous Scherk’s talk…I am going to explain you some bits and glimpses of the (extended) Poincaré GLA, the (extended) Poincaré de Sitter GLA, and the conformal GLA. I will be also introducing you the N=8 SUGRA and its relevance to physics.
## 1. FERMIONS.
The mere existence of fermions can be traced back to the early 20th century. Around 1925, physicists found the strange and weird character of the electron and similar particles. “Why fermions?” Louis de Broglie asked that question, even when the Pauli exclusion principle was formulated. So we are in the situation where Nature chose the electron to be a fermion (proton an neutrons are fermions too). From the experimental aside, we can not deny but accept the existence of fermions: spin 1/2 particles! What about spin s/2 particles, with odd s greater than 1? We will learn about this just a little…Focusing on the 1/2 case…Quarks are also fermions! Fundamental forces, from the quantum viewpoint, are mediated by integer spin particles (gluons, photons, W and Z bosons and now, it seems, Higgs bosons too). We have also the hypothetical gravitons from the gravitational theories like General Relativity and extensions. Thus, integer spin particles and its existence is out from any reasonable doubt. However, fermions are just a bit more mysterious. Poor mathematicians and mathematical physicists use to ask and wonder why fermions are relevant.
Simple answer: “There are fermions in Nature because we observe them”
Uhlenbeck and Gouldsmith told us in 1925 that the spin of the electron is 1/2, no matter how weird it is. It is an experimental fact! Furthermore, the spin-statistics theorem tells us that if we have spin 1/2 particles or fields (Dirac fields for simplicity at the moment), these should be quantized through Fermi-Dirac distributions/statistics, while if one deals with integer spin fields, their corresponding distribution is that of Bose-Einstein type. In summary, there are boson and fermions in Nature. The Standard Model (or Standard Theory) says that force carriers are boson fields, and that matter fields are fermion fields. This reasoning is, perhaps, not simpler that saying “A rose is a rose is a rose” (Gertrude Stein).
Elaborated answer: There is a deeper reason of spin 1/2 existence. From a mathematical viewpoint, it is ultimately related to the rotation group SO(3), and the Poincaré group. The SO(3) group is NOT simply connected and its universal covering group is SU(2). SO(3) admits integer representations only, but SU(2) admits 1/2 integer as well as integer representations. So from the SU(2) viewpoint, Nature has realized both integer and 1/2 integer representations. SO(3) is, if you want, more boring than SU(2). This second answer, however, leaves one with an unpleasant feeling, as one may wonder if mathematical physicists would have thought about SU(2) if Nature had chosen a world wher only gluons, photons, W and Z bosons, Higgses and gravitons existed, or if it had played the trick to provide us with fundamental (composite?) spin 0 scalar particles (higgses) and quarks (which is another option!).
Nature said, indeed, that there are Fermions, quarks and leptons, with J=1/2; there are also Bosons, gravity (hypothetical gravitons) with J=2 particles, electroweak particles (massless photons, and massive W,Z bosons, massive Higgses) with spin J=0,1, and strong interactions having gluons with J=1. Mathematical physicists ( physicists) know that J(electron)=1/2, that electrons from spin-statistics imply the existence of Fermions while interactions imply the existence of (gauge/gauge like) compensating fields with J=0,1,2. Fermions (F) do exist. Why F? The response involves representations of the Poincaré group: SO(3) covering group is SU(2), which admits 1/2 spin fields (and other exotica I am not going to discuss here today). Math is OK, but why is realized by Nature? Not simple answer to this does exist. Would you search for some 1/2 piece of art? It does NOT exist spin 1/2 “art”. Metaphysical answers? The western way offers no simple explanation! The physicist turns towards Western rationalistic enlightenment without concrete solutions. You would only find unsatisfactory theological reasons and you will turn towards Eastern religions. The East gives you a more cool option, namely that a universal principle of harmony and dynamics balances the whole Universe (cf. the Greek idea of arche/arkhe). Chinese people call this Taiji principle the Tao. Don’t worry! I am not going to convert this article into the Tao of Physics ;). Tao refers to the interplay and inerfusion of pairs of opposite “elements” of fundamental character, dubbed Yin and Yang. The Yin refers to the female, dark , cold , soft, negative principle while the Yang (please, I am not the one who said all this even if you find it machist!) refers to the male, bright, warm, hard and positive principle/element (the Universe is mainly dark/female thought ;)).
Scherk tried to temp us with this analogy: one can associate the Yin principle with the Bose principle, that allow interpenetrability, and the Yang principle with the Fermi principle, which doesn’t. It is hard to be dogmatic on this scheme since the darkness/brightness would associate the Bose principle with the Yang and the Fermi principle with the Yin. Since the the light is known to be light/bright and not dark (unless dark photons exist), and the (SM) photon is a boson. More stupid stuff to waste time with Yin/Yang analogies in particle physics: little circles of black within white and vice versa signify that Yin bound states (stable or metastable) of the Yang principle exist and vice versa. This is known to be true as bound states of fermions (e.g. quarks) produce bosons, while the less obvious converse result is true as we know or learn about the study of classical magnetic monopole solutions. This complementarity extends in taoist thinking no only to bosons and fermions, but to many other subjects such as the World as an Object and the Ego as a Subject, which are in this way of seeing not only complementary but also interchangeable. Object/Subject complementarity leads to the cosmologist’s viewpoint (anthropocentrism, if you wish): the Universe or World is as it is in particular state because it is populated by conscious human observers (conscious E.T. beings or even conscious A.I. will also apply). The existence of life and consciousness puts some limitations on various purely physical constants such as gravitational/electromagnetic/weak/strong coupling constants and so on. This class of anthropic idea is not very popular or “well seen” between scientists. Applied to the Fermi/Bose problem, if Nature had decided for instance to build up us with J=0 leptons and J=1/2 quarks only, the nuclei would exist, and so would the atom in a modified way though. However, every atom would collapse under the gravitational strength into one giant molecule at the Earth’s center.
This last answer is also unpleasant. We turn into a more modern view of this problem. It is dominated by SUSY, SUGRA and superstring/M-theory. Local SUSY implies SUGRA. And it necessarily implies a complementarity between Bose (J=0,1,2) particles/quanta and Fermi (J=1/2,3/2) particles/states and their transmutability into each other through SUSY transformations, at least from the pure mathematical viewpoint! SUSY is equal to Fermi-Bose symmetry, and thus, is less “super”. “Super” things are cool, and thus, we prefer SUSY instead that boring Fermi-Bose symmetry naming! The final enlightenment, through some king of French connections thanks to Scherk, allows us to discover SUSY and SUGRA (part of superstring/M-theory models) as the answer to the existence of fermionic fields in Nature. If you feel disappointing at this stage you should give up and not to read any more. However, mathematics is the best tool we have to find the Nature deeper secrets. Are you ready for them? I hope so…
## 2. SUPERWORDS.
Making a transition phase is not easy. If your path is seriousness, let us consider a small and incomplete set of definitions (dictionary) of the word super as many people ask: “Why do you call it supersymmetry/SUSY?” If you also read my previous list http://www.thespectrumofriemannium.com/2013/05/08/log102-superstuff-the-list/ you will find other alternative and complementary superword set. Many “superwords” are scientific, some others being of common knowledge origin. For instance:
“Super”: word of american (?) origin. Opposite of “regular”. In Switzerland, “super” is a small additional invest compared to “regular” (~1.08 SF/liter vs. ~1.04 SF/liter).
“Scientific superwords”: supernova, superstar, superhelix (DNA), superconductivity, superfluidity, superaerodynamics, superphosphate, superpolyamide, supersonic, superstructure, superoxide, superactinide, superbrane, super p-brane, supermembrane, supercontinent, supermaterial, superspace, superstring, superparticle, superheterodyne, superhuman, superatom, supermolecule, supergroup, superalgebra, superextendon, supermatrix, supersymmetry, supergravity,superfield, supervielbein, super-Higgs mechanism,…
“Political superwords”: superpowers (in 1979 language: Monaco, Lietchtenstein), superphoenix (mythological bird unknown to the Antiquity, of Gallic origins).
“Touristic superwords”: superTignes (where is the CERN staff today?).
“American colloquial superwords”: “Gee, it’s super”, “Super-Duper”.
“Pop music superwords”: supertramp.
“Daily life superwords”: supermarket (Migros, Coop), superman (comic stripo by F. Nietzsche).
“Poetic touch”: superlove; the French poet Jules SUPERvielle (Montevideo 1884-Paris 1960) wrote in 1925 a collection of poems called “Gravitations”.
SUSY and SUGRA, and their relatives/offsprings, are just a few modern superwords added to this list. SUSY in flat spacetime is described by some elements:
• Supersymmetric transformation laws (SSTL/S.S.T.L.):
a set of continuous transformations changing classical, commuting, Bose real (or complex) integer spin fields into classical anticommuting (sometimes dubbed classical anticommuting c-numbers), Grassmann Fermi variables with half-integer spin fields and vice versa. Roughly, you can think about this as
(1)
• SUSY model/theory. A classical lagrangian/action field theory, in flat spacetime, whose action is invariant under SUSY transformations. Generally speaking, you hear about N=1, N=2, N=4 or even N=8 theories often in the superworld, but you can also build theories with any arbitrary number of supersymmetry generators. Mathematically speaking, your freedom is your (consistent, free of contradictions) imagination!
• Extended SUSY. Popular expression for SUSY theories with N>1.
• SUSY (a.k.a. supersymmetry). Any field of which studies local and rigid supersymmetric transformation laws (SSTL), supersymmetric theories/models and extended SUSY. Theoretical physicists are crazy for (finding) SUSY and adding SUSY to their theories. It is likely an obsession. Experimentally, SUSY is hard to kill, but LHC and other future colliders, also other cool experiments, are pushing forward into finding SUSY. SUSY is a beast. If SUSY does exist, it has to be broken in Nature. How and where is the puzzle. From the mathematical side, SUSY is known to be the only (almost unique) way to relate bosons and fermions through symmetry, evading the celebrated Coleman-Mandula theorem including fermionic generators in the algebra. That is, SUSY (superPoincaré group to be more precise) is (up to some uncommon exceptions) the only non-trivial extension of the Poincaré group containing the internal symmetries of the Standard Model. That is why some people is resistent and resilient to give up SUSY in these times. Other alternatives, like N-graded Lie algebras, quantum groups are seen just as odd or just as certain special representations of (extended/generalized) SUSY.
• Superspace. A set of generalized coordinates , containing both, ordinary spacetime coordinates , Bose classically commuting coordinates, and , Grassmann Fermi classically anticommuting coordinates. Here, , . can also runs from 1 to D, but there are other alternatives and it is a free index right now. Thus, we get
(2)
• Superfield. Any function with or without indices.
• Supermultiplet. Any irreducible representation of SUSY (simple N=1, or extended N>1), expressed in terms of the (super)fields.
• Matter supermultiplet. Any supermultiplet with . You can also generalize this definition to other “matter” (dark?), like or higher, but it is not usual. are called the vector supermultiplets, are called scalar supermultiplets.
• Supersymmetric Yang-Mills theories/Super Yang-Mills theories for short (SYM). The self-interacting () field theory based on a vector supermultiplet, or extended YM theory having SUSY symmetries.
• Goldstino. The m=0, J=1/2 fermions associated with the spontaneous symmetry breaking of SUSY.
• Gluinos. The octets with in SYM theories.
• Superalgebras. Graded Lie algebras (GLAs), dubbed into a cool superword friendly synonim by crazy addict physicists. These addict physicists could be named superphysicists…
Example: N=1 SYM theory in D-dimensions. It contains the next ingredients,
1st. Spectrum: , J=1, real fields; , J=1/2 Majorana fields.
2nd. Coupling constant: .
3rd. Infinitesimal SUSY parameter: . J=1/2 constant Majorana field.
4th. SUSY transformations:
(3)
(4)
5th. Action and lagrangian:
(5)
and where
If now, we upgrade SUSY by changing our flat (super)spacetime into curved (super)spacetime, we enter into the SUGRA realm. Firstly, let me introduce the main SUGRA superwords:
• SUGRA models/theories. Any SUSY model/theory built in curved spacetime invariant under N=1 SUSY (however the latter case is sometimes named extended supergravity, but terminology is just a choice in some cases; nevertheless, pure SUGRA generally refers to N=1).
• Extended SUGRA models/theories. As suggested in the previous definition, any theory/model built in curved spacetime invariant under N>1 SUSY. The issue of allowing or not curvature in the fermionic sector is open. Curved supertheories are usually more subtle but can be handled, in principle, with suitable tools.
• SUGRA. The field of physicists (superphysicists?). It studies local SUSY transformation laws, superfields, supergravity and extended SUGRA theories.
• Pure SUGRA. SUGRA or extended SUGRA uncoupled to any matter supermultiplets, with a self coupling , dimensionally .
• Gravitino. The Rarita-Schwinger field quanta. Spin 3/2 particle associated with local simple (or extended!) SUSY.
• Super Higgs effect/mechanism. The analogue of the Higgs effect/mechanism for SUSY. Wherever SUSY is spontaneously broken, a gravitino (or several gravitini) eats up a goldstino (or several goldstinos) becoming massive fields.
Example (II): N=1 SUGRA. The ingredients of this theory/model are
1st. Spectrum. Vielbein, J=2 real field . Gravitino, J=3/2 Majorana field .
2nd. Coupling constant. . Related to . It has several normalizations.
3rd. SUSY parameter. ; J=1/2 -dependent. Majorana field.
4th. SUSY transformations. We have
5th. Action and lagrangian.
(6)
(7)
and where we define
Technical details: there are some technicalities in all these things…SUSY/SUGRA and generally field theory involves different types of spinors. Giving up the weirdest types, there are 3 main types of spinors. These spinors are named Weyl, Dirac and Majorana fields (spinors). The esential features come from the Clifford algebra of spacetime
(8)
where , , and the irreducible representation of the matices are dimensional. The Majorana representation (MR) is any representation of matrices in which they are all times real matrices. A celebrated theorem states that a MR does exist in even D if and only if . The charge conguration matrix, C, verifies , where T means transpose matrix. Majorana spinors are any spinor such that . If , Majorana spinors exist only if a MR does exist. In the MR, we have and . Weyl spinors are any spinor set for even D, such that , with such as , so Weyl spinors satisfy the conditions , and they are “two-component like”. Finally, you can combine these conditions, and build a Majorana-Weyl spinor, i.e., spinors that are both, Majorana and Weyl spinors. They do exist in D even if and only if . Either of these two restrictions will cut the number of independent spinor components in half. And thus, we have seen that in some dimensions it is possible to have both Weyl and Majorana spinors simultaneously. This reduces the number of independent spinor components to a quarter of the original size. Spinors without any such restrictions are called Dirac spinors. Which restrictions are possible in which dimensions comes in a pattern which repeats itself for dimensions D modulo 8. Secretly, this phenomenon is connected to Bott’s periodicity, but I will not discuss it here today. The dimensionality of a Dirac spinor as a solution to the Dirac equation in D spacetime dimensions is given by the dimension of the Dirac matrices. In the familiar example of four dimensions, Dirac spinors belong to reducible representations of the Lorentz group. For arbitrary spacetime dimensions one might wonder what dimensionality an irreducible spinor has got.
Remark: The most general free spinor (from Clifford bilinears up to some exotics) action reads
(9)
## 3. SUPERALGEBRAS.
Superalgebras are the heart of SUSY. There are 3 main types of SUSY algebras used in SUSY: superconformal algebras (SCA; Bose generators, and 8N Fermi generators), super-dS (or super-AdS) algebra (SdSA has 10+N(N-1)/2 Bose generators and 4N Fermi generators); and finally the super-Poincaré algebra (SPA has the same number or generators than the SdSA, except in the presence of the so-called central charges). Some relatives from these superalgebras are:
• SCA: in flat space, you can get super- theories, and SYM theories (for N=1,2,4). In curved space, you can form SC SUGRA.
• SdSA: in curved space (there is no flat space cases here) you can get extended (N>1) or simple N=1 dS SUGRA/AdS SUGRA with SO(N) gauge group.
• SPA: in flat space, you can obtain virtually all supersymmetric models with N=1, and SYM with D=6,10. In curved space, D=4 SUGRA and extended SUGRA theories with SO(N) gauging. D=11, N=1 is the so-called maximal SUGRA (if you give up higher spin excitations with J>2). The most interesting models are the extreme cases, such as that with N=8 SUGRA in D=4 and N=1 with D=11. The N=4 SYM theory (GSO model) is also very interesting. This model has remarkable properties: it is renormalizable, and the Fermi-Feynman gauge is fully 1-loop finite (no renormalization is needed). The 1-loop corrections to the propagator vanish identically (finite and infinite parts), and finally the Gell-Mann-Low function vanishes identically for 1 and 2 loops. This fact makes the GSO model to exhibit no spontaneous symmetry breaking of SUSY, while coupled to SUGRA it does. Another advantage is that this model compared to the SO(8) theory is that the gauge group is arbitrary and can be fitted for Nature. SCA has additional applications. The superconformal SUGRA, whose bosonic sector contains the Weyl theory of gravity (the tensor correction is present) and has four derivatives in the action, which leads to dipole ghosts. This is generally used to reject this theory on these principles, well as on the classical ground of one word. The SCA has no dimensional constant entering it and thus, the field theories based on it have dimensionless coupling constants, ensuring renormalizability (SC gravity and SYM). The SdSA contains one dimensional coupling constant m, with dimensions of mass/energy. Translations do not commute and give a rotation times . The Universe described by the SdSA is thus not a flat Universe, but a de Sitter one (SO(3,2) rather than SO(4,1)). The radius of the dS Universe is roughly in natural units. The constant plays the role of a mass term via the term for the gravitino field, and a cosmological constant occurs in the lagrangian, where is related to , the Newton constant of gravity. In extended theories, with , the vector fields such as couple with the dimensionless minimal coupling constant , and . The SO(3,2) AdS Universe is the simple, maximally supersymmetric solution of the field equations, and the actual size of the Universe introduces a stringent bound on g, such as ! Actually, there is also another viewpoint of this deduction. We can plug and and this fits observations even when we have no idea of why this is so. This picture is related to the spacetime foam by Wheeler, Hawking ad Townsend. As they use to point out, the dS theory has solutions where spacetime is flat or nearly flat at large distances (greater than 1 cm) but very strongly curved at distances of the order of Planck’s length (about or similar). The physical spacetime may well be a statistical (emergent!) ensemble of fluctuating spacetime foams (with unknown quantum degrees of freedom) of arbitrary sizes and topologies. The issue of the change of topologies in this fluctuating spacetime is not solved in any current candidate of quantum gravity. And finally, SPA, with or without central charges, have the dimension of a mass, and representations of the SPA with central charges occur for massive supermultiplets only (e.g., N=2, ) or for the classical solutions of the N=2,4 SYM theories. In these theories, we have respectively 2 and 6 central charges. In the N=2 model, the 2 charges are the electric and the magnetic charges, satisfying the Montonen-Olive relationship
Olive remarkable idea was to identify and with the momenta in certain Minkovskian spacetime (s=5, t=1; D=6) where . Similarly, the N=4 model suggests to define 6 extradimensional central charges , and these can be identified with 6 extradimensional momenta with and the mass relationship . These facts are not surprising, since the N=2,4 D=4 theories can be obtained by dimensional reduction from the N=1, D=6,10 SYM theories!
## 4. SPA: SUPERPOINCARÉ ALGEBRA.
We are going to discuss only the representations of the SPA in D=4 dimensions (s=3, t=1), in the massless case, in terms of fields. One can show that only one of the supercharges is relevant (e.g. ), and that it decreases the helicity by 1/2. The particle content of a supermultiplet can be easily found. In the next table, we observe the particle contents of supersymmetric free field theories with . Scalar multiplets exist up to N=2, vector multiplets up to N=4, SUGRA multiplets up to N=8 and hypergravity with multiplets up to N=10.
Table 1. Representation contents of N=1,…10 SUSY with 0 mass. Ogievetsky multiplets (having ) are shown up to N=4, but exist up to N=6. Hypergravity multiplets () are shown to exist from N=1 to N=10 but are shown only for N=9 and N=10.
Explicit constructions are necessary to show that that interacting field theories based on these multiplets do exist. This is tru up to , but no interacting field theory (hypergravity) based on exists. Hyper seems to suggest that these theories may not exist, but who knows? Hyperspace was a fantasy before the 19th and 20th century. Recently, hypergravity theories, higher spin theories and hypersuperspace theories have been considered by those crazy physicists you all know…Remarkably, hyperbole itself comes from certain Greek demagog of Samos, he threatened the comic poets Alcibiade and Nicias of ostracism,but he was turned into ridicule by them an was himself ostracized in 417 B.C.
## 5. SUPERGRAVITY.
The super prefix associated with SUSY and SUGRA is due first to superunification of fundamental forces somehow. SUSY and SUGRA own good features and they are specially recognized, in some models, by being renormalizable (cf. classical general relativity, a non-renormalizable theory).
1st. Unification goodness.
-Fields of different spins are unified in the same representation, overcoming previous no-go theorems (e.g., the Coleman-Mandula theorem).
-Internal symmetries and spacetime symmetries are unified. SUSY is (almost) unique but with a lot of models/theories to explore. What is the right model/theory? That is an experimental problem.
-Fermions and bosons play symmetrical roles.
-The dichotomy between fields and sources is also solved.
2nd. Renormalization goodness.
-If the bosonic sector of certain supersymmetric field theory renormalizable, so is the whole theory. Further in that case, the associated supersymmetric field theory is more convergent than its bosonic sector. E.g.: the N=4 SYM theory.
-Non supersymmetric theories of gravity with matter interactions (J=0,1/2;1) and their associated matter fields are one loop non-renormalizable, while pure extended SUGRA theories based on the SPA (N=1,…,8) are 1 and 2 (at least, current knowledge improves these arguments) loop renormalizable and finite. Superconformal gravity is also renormalizable, but so is conformal gravity.
SPA exists for any s,t signature and any spacetime dimension (D=s+t). If we keep D=4, and increase N, we meet the limits N=2,4,8 for the existence of multiplets with . Similarly, if we keep N=1, and we increase D, we meet the limits D=6,10,11 for the existence of the same multiplets.
An example of supersymmetric field theory in D=10 (s=9, t=1) is easy to provide. Let us take the example fo the N=1 SYM theory. If we keep N=1 and plug in D=10 in the notations we have introduced here, the theory is still SUSY invariant, provided that the are Majorana-Weyl spinors, which is possible since s-t=8. The vector in D=10 gives in D=4 a vector and the MW spinor reduces to 4 Majorana spinors , j=1,2,3,4. This is precisely the field contents of the N=4, D=4 SYM theory! What we have done is dimensional reduction. Dimensional reduction consists in starting from certain higher dimensional theory, usually certain supersymmetric field theory with . One lets the size of the internal dimensions shrink into zero size (radius), in which case only the excitations of the fields which are constant in the extra dimensions are relevant, or have a simple dependence (of the type ) so only certain extra dimensional excitations do survive. The resulting D=4 theory is still supersymmetric (invariant under SUSY), but SUSY may be spontaneously broken. The mass appears as the momentum in the 5th (or any other) extra dimension/direction. This process can be generalized. In generalized dimensional reduction, a phase dependence is introduced, as in the usual Kaluza-Klein theory. The N=8 theory can then be generalized starting from the D=5, N=8, theory which has a rank 4 group of invariance, namely . Going from 5 to 4 dimensions, 4 mass parameters , i=1,2,3,4 can be introduced and a 4-parameter family of N=8 theories is obtained which contain the CJ (Cremmer, Julia) theory as a special case (). In these theories, SUSY is spontaneously broken, the Bose-Fermi mass degeneracy is lifted but the breaking is “soft” as the following mass relations still hold
These mass relations imply that the one loop correction to the cosmological constant is still fully finite, but apperently non-zero (note that from this viewpoint, these theories do predict -or posdict- a non null cosmological constant). The spectrum of the N=8 theory is the content of the next section.
## 7. N=8.
In the spontaneously broken N=8 theory, every massive state is complex and the theory has a U(1) summetry. This extra symmetry is actually a local one. Its gauge filed is the vector field , obtained by reducing the metric tensor from 5 to 4 dimesnions and . In the next section, we shall see that the coupling of this vector field leads to the phenomenon of “antigravity” (but be aware of what it does mean!), and we shall denote the particle associated with the quantized by a curlywedge symbol. (In the original papers, it was used the egyptian symbol “shen”, but it is pretty similar to curly wedge, and it is adorable too!).
If every is equal to the same value, the only symmetry of the theory will be U(1) (apart from SUSY, coordinate invariance,…). If we set all the , the spectrum is degenerate and at zero mass, there are 15+1 gauge bosons. As a gauge group SU(4) is too small to include the SM group but if we disregard the weak interactions, we obtain a model which includes , that is strong interaction plus electromagnetism plus antigravity. Actually, the breaking of SU(4) into is automatic if we set with for i=1,2,3 and . The electric charge is taken to be equal to 1/3 for the triplet of graviquarks of mass m, 0 for the singlet gravitino of mass M. Once this is accomplished, all the masses and charges are derived by taking the product of representations. This model contains only one electron of mass 3m, but no muon or tau particles. Even the neutrino fields are absent. this is not too surprising and one could tentatively attribute the family structure to composite bound states as well as the SU(2) of weak interactions is taken into account. A more humble attitude is to take the model just a model not as a final theory. As it stands, it is not too bad: it is 1 and 2 loop finite (unlike Einstein’s theory of gravity coupled to leptons, quarks, and the SM group bosons and particles). It also includes a massless graviton, 8 massless gluons, a photon, an electron, a d type quark (Q=-1/3), a u type quark (Q=+1/3) and a c type quark (Q=2/3). The exotic particles are: the sexy quarks (the sextet 6, with Q=1/3), the gluinos of Fayet (Q=0), a triplet of d-graviquarks (the triplet 3: Q=-1/3), a set of 2 neutral gravitini (1,0), massless scalar gluons (2 octets), massless singlet scalar particles (2 of them) and massive sextets and singlet scalars, and massive scalar quarks (that we could name “sarks”).
## 8. ANTIGRAVITY.
Let us consider the scattering of two particles of mass having also a coupling to a massless vector field (this vector field is NOT the electromagnetic potential of the SM, so it is some kind of dark electromagnetism) with charges . In the static limit the total potential energy is given by
(10)
Scherk’s antigravity is defined by the cancellation which would happen if one had systematically the relation between g’s and M’s
(11)
and where for particles, antiparticles and neutral particles (like Z bosons, neutrinos and others). In 1977, it was guessed that since in N=2 SUGRA there is a vector , it had to couple minimally to fields with a gauge coupling constant . The coupling was actually written in lowest order in . Later, in 1978, K. Zachos coupled N=2 SUGRA to the multiplet (1/2(2), 0(4)), with a mass m and found , as well as antigravity (the above cancellation phenomenon!). In 1979, the spontaneously broken N=8 theory was found and it was discovered that a vector coupled to all the model fields with strength , a relation that holds for all the 256 states of the model. If there is such an antigravitational force in Nature, and this is an inescapable consequence of SUSY if N>1 (that is, extended SUSY contains antigravity), why don’t see it? Perhaps, we have already seen and we don’t realize it! If we look at the spontaneously broken N=8 model, we may find a beginning of an answer. Suppose we consider the static force between 2 protons (uud bound states). As the graviton couples no to the mass but to the energy-momentum tensor it sees the total energy of the quarks and gluons, i.e., the mass of the proton. It is mostly the kinetic energy of the quarks and gluons as a whole. This force contribution is given roughly by the term , where is the proton mass. The antigraviton (sometimes called graviphoton) is coupled to times , where m is the mechanical mass of the quark is question, and is the conserved electromagnetic current. Therefore, the antigraviton/graviphoton sees directly the quark mass and not the proton mass and its contribution is . Furthermore, we observer that the relative contribution is of the order , and that is small, about for u,d quarks. Finally, if we compute the relative difference between the acceleration of a proton and a neutron, we find that it is given by the expression , which is even smaller, depending on the u, d mass difference. In the limit of exact SU(2) symmetry for strong interactions, this difference vanishes so that antigravity is in good health…However, if one looks closer, one finds that the exchange leads to serious problems with the equivalence principle! Let me explain it better. Two atoms of atomic numbers , having protons and of net charge zero fall with DIFFERENT accelerations towards the Earth! The force between these 2 atoms is given by
(12)
The negative term is due to the antigraviton/graviphoton exchange; one generally has:
(13)
If represent the Earth, we can safely replace by . Therefore, the acceleration of towards the Earth is given by
(14)
The relative difference in acceleration of the 2 atoms will be
(15)
(16)
and where , . Int the last bracket, the dominant contribution is given by , so reads
(17)
Putting into numbers this expression, we find that , bigger than the most accurate bound on the violation of the equivalence principle. This is unaceptable. Is antigravity wrong after all? Theorists do NOT give up a general idea so easy…In order to save the antigravity idea/phenomenon (something quite general in some BSM theories), one must assume that the antigraviton/graviphoton acquires a mass, likely through the Higgs mechanism. This is rather what it happens, since it (the graviphoton/antigraviton) is universally coupled to scalar fields through the lagrangian
(18)
At the classical level, the v.e.v. (vacuum expectation value) is and . If is due to SU(2)xU(1) breaking, one has typically , and with one finds that . This gives to the antigraviton a Compton wavelength of the order 1 km, which seems to be reasonable. In the case where , the potential between an atom with (Z,A) and the Earth is provided by the following expression
(19)
This formula would be correct if the Earth were a point like object. Taking into account of its actual size leads (for an homogeneous sphere) to multiplying the last term in this expression by a form factor , where
The altitude from the surface now appears rather than the distance from the center and it leads to the upper bound on , given by . Usually, masses are thought to be fixed parameters. However, one knows that they depend upon external conditions such as the temperature T. If one could “heat the vacuum” enough, the phase where and would be restored. Antigravity devices of this kind however still belongs to the field of Ufology and Sci-Fi (Science-Fiction), and apparently not to the field of mathematical physics.
## 9. CONCLUSION.
It will be short and we leave it to a great American hero…
Remark: SUSY haters have also their propaganda
but also SUSY has big fans/addicts/superphysicists that counter it…
## 10. APPENDIX: the simplest SUSY.
If you have read up to here, I am going to give you another “gift”. The simplest SUSY you can find from supermechanics. I will add some additional final questions too ;). Let me begin with certain lagrangian. For simplicity, take the mass of the particle equal to the unit (i.e., plug in ). The symplest SUSY lagrangian is then build in when you add to the free particle lagrangian (m=1) the Grassmann part as follows:
(20)
The Bose part corresponds to translations and the Fermi part correspond to spins. This lagrangia is, in fact, a special case where both, angular momentum and spin angular momentum, are invariant under INDEPENDENT rotations in the variables . Any interacting extension from this free case involves that this lagrangian generalization will be inivariant only under SIMULTANEOUS rotation of . In particular, this lagrangian is invariant under
(21)
The has variables with the following Poisson algebra
(22)
(23)
The Hilbert space on which this objects acts is given by , where . Thus, under quantization, you obtain that the hamiltonian is certain laplacian operator on . Generally, up to a sign, you write
(24)
Remark: the sign convention is important in some applications. It is generally better, for convergence issues, choose the laplacian so that the eigenvalues are asymptotically positive.
The Noether charge for under rotations can be easily work out, and it yields the tensor
This is a good thing. We recover the classical result that rotational invariance implies the conservation of angular momentum J=L+S=angular part+Spin part. In particular, for d=3, we obtain
and it confirms known results of angular momentum under quantization! Now, the full simplest SUSY transformations in action action onto our lagrangian . The most general field-coordinate variation of this lagrangian provides
(25)
Introduce elementary SUSY transformations
(26)
Plug in these variations into the variation, and operate it to obtain
(27)
The conserved charge (supercharge is used often) is
(28)
that is, the Noether supercharge, SUSY generator is
(29)
In even dimension, d=2n, we usually quantize the Poisson brackets with the aid of the canonical commutators and anticommutators given by
(30)
for bosons and
(31)
for fermions. Defining
(32)
it turns that the fermion anticommutator is secretly a (rescaled) Clifford algebra in disguise, since Clifford algebras are defined as
(33)
The remaining problem is to find and determine the gamma matrices or some good representation of them. The gamma matrices act onto the Hilbert space , in even dimensions, but it can be generalized to odd dimensional spaces too (with care!). Generally speaking, this factorization of the Hilbert space says that SUSY acts on a superpace being “translations times spin”. The quantized Noether operator associated to SUSY transformations reads
(34)
i.e.
(35)
This result teaches us something really cool and amazing: the SUSY quantum mechanical Noether supercharge (operator) is nothing but the Dirac operator (here, acting on the manifold times the spin group). Remember: the SUSY supercharge is generally speaking certain Dirac-like (Clifford) operator, the product of the Clifford gamma matrix and certain derivative. Indeed, there is something really beautiful in addition to this thing. SUSY transformation can be computed for this operator as well, with new amazing results:
(36)
or
(37)
Motto: the variation of the supercharge is proportional to the SUSY lagrangian (times a constant).
Moreover, compute two successive SUSY transformations, with parameters . Then, you can show that the commutor (and its associated Poisson bracket) reads
(38)
But is the generator of translations in time associate to the energy or hamiltonian of the system! This can be easily proved
(39)
and where we have used that . Thus, under quantization,
(40)
Thus, the SUSY supercharge is, generally speaking, “the square root” operator of the hamiltonian, since
(41)
or equivalently
In summary, the main formulae from the simplest SUSY lagrangian are given by
(42)
Finally, some exercises for addict, eagers readers…I do know you do exist and you are OUT there!
1) Generalize this discussion to a simple manifold with a metric. SUSY covariance reads from
and the metric is
The invariant object
is well defined. Note that it depends on the velocities . The lagrangian of this exercise is provided by a covariant version of our simple :
or
Here, we define the Christoffel connection
The covariant derivative is given by
and from the expansion
we can get
and curvature components
2) Find the Euler-Lagrangian equations for this covariant generalization of our simple lagrangian, . Is there any conservation law there? Reason your answer.
3) Express the Dirac operator for any general curved manifold M in local coordinates.
Have fun!!!!!!!
Let SUSY, SUGRA and ANTIGRAVITY be with you!!!!!!!
View ratings | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9138614535331726, "perplexity": 1665.8228027021682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587655.10/warc/CC-MAIN-20211025061300-20211025091300-00059.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/119863-poisson-process-quotient-two-gamma-r-v-s.html | # Math Help - Poisson process: quotient of two gamma r.v.'s
1. ## Poisson process: quotient of two gamma r.v.'s
Let {N(t): t≥0} be a Poisson process of rate 1 and let S_r be the time to the rth point. Calculate the density function of X = S_2/S_6 using the Jacobian transformation method.
To use the Jacobian method, I need to define another random variable Y that is a function of S_2 and S_6.
X=S_2/S_6
Y=???
I know that S_2~gamma(2,1) and S_6~gamma(6,1), but I am having some trouble defining the other random variable Y, and the problem is that S_2 and S_6 are not independent.
The Jacobian method expresses the joint density of X and Y in terms of S_2 and S_6, so I must first obtain the joint density of S_2 and S_6. But now it seems like that I have no way of obtaining the joint density of S_2 and S_6 because they are NOT independent random variables (S_2<S_6 always).
So how can we solve this problem using the Jacobian method?
Any help is much appreciated!
2. Usually Y is either S_2 or S_6,
S_6 might be easier
You must find that first, and yes they are dependent.
3. Originally Posted by kingwinner
the problem is that S_2 and S_6 are not independent.
But $S_2$ and $S_6-S_2$ are independent, and the second one has same distribution as $S_4$, so let's call it $S'_4$, so that $\frac{S_2}{S_6}=\frac{S_2}{S_2+S'_4}$ and then you can do the usual method (like MathEagle said, Y may be chosen to be $S_2$ or $S'_4$).
4. I was going to add that S6-S2 is indep of S2 since the time periods are separate, but I wanted kingswinner to figure that out.
5. Originally Posted by matheagle
I was going to add that S6-S2 is indep of S2 since the time periods are separate, but I wanted kingswinner to figure that out.
Oh, sorry to spoil your pedagogy... I feel like we're quits now, since the same happened on an other thread in reverse (that was a couple of weeks ago)
Best wishes,
Laurent.
6. Originally Posted by Laurent
But $S_2$ and $S_6-S_2$ are independent, and the second one has same distribution as $S_4$, so let's call it $S'_4$, so that $\frac{S_2}{S_6}=\frac{S_2}{S_2+S'_4}$ and then you can do the usual method (like MathEagle said, Y may be chosen to be $S_2$ or $S'_4$).
But I have some more questions:
1) If I take Y to be (S2 + S4') which is the denominator, is that also OK? Will I get the correct answer?
2) I don't quite follow your argument that (S6 - S2) would have gamma(4,1) distribution.
What I have learned is the following...
Theorem: if X1~gamma(r1,lambda) and X2~gamma(r2,lambda), with X1 and X2 INDEPENDENT, then the SUM (X1+X2)~gamma(r1+r2,lambda).
But I am not too sure about a SUBTRACTION of gamma r.v.'s, particuarly when S6 and S2 are DEPENDENT?? Is the theorem above applicable?
Thank you very much!
7. Originally Posted by kingwinner
2) I don't quite follow your argument that (S6 - S2) would have gamma(4,1) distribution.
You should try to picture what S6-S2 is.
(and no, you can't "substract", whatever that would mean)
8. 1) Actually, I am puzzled about this for a long time. The Jacobian method always requires TWO random variables. But we are almost always only given one, and somehow we have to "invent" another random variable, but a lot of times I have no idea how...
Is there any general rule or hint about how this second random variable can be constructed?
X=S2/(S2+S4')
Y=???
How do we know what to pick for Y, and how can we know whether it is going to work or not?
Can we take Y=S4' ? How about Y=S2+S4'? Are there any other possible choice of Y that would work?
2) I can picture S6-S2 as the time between the 2nd and 6th point, but I don't get why S6-S2 is necessarily gamma distributed.
Suppose X1~gamma(6,1), X2~gamma(2,1), and X1 and X2 are NOT independent. Does this imply that X1-X2 ~gamma(6-2,1)=gamma(4,1) ? If so, how can we prove it? I haven't seen this theorem before...
Thanks for any help! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9274663329124451, "perplexity": 527.2215792338318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500815756.79/warc/CC-MAIN-20140820021335-00042-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/mechanics-angular-momentum-not-about-c-o-m-with-translational-motion.510861/ | # Mechanics - Angular momentum not about C.O.M. with translational motion
1. Jun 30, 2011
### boumas
Perhaps this should be under physics, but my mechanics course is done by the maths department...
I don't actually have a particular problem, just a question.
If you have a body (say a rod) with translational motion and rotation about an axis that is not its centre of mass, is there a way of neatly finding the angular momentum.
I feel that using
Lz=I0w +(RcrossMV)z (sorry for crummy equation writing...)
in combination with parallel axis theorem would be wrong, but am not quite sure how else to go about the problem...
Sorry for being vague, but I don't really have an example problem to give, and this has had me puzzled and confused for quite a while...
Thanks :)
2. Jun 30, 2011
### tiny-tim
welcome to pf!
hi boumas! welcome to pf!
about any point, angular momentum = angular momentum about centre of mass plus angular momentum of centre of mass: $\mathbf{L}_P\ =\ I_{c.o.m.}\,\mathbf{\omega}\,+\, \mathbf{r}_{c.o.m.}\times m\mathbf{v}_{c.o.m.}$ …
parallel axis theorem not needed
3. Jul 1, 2011
### boumas
Thanks! :)
Sorry for late response, I left the library early and don't have internet in my flat... (Like living in the dark ages here!)
What I think I'm having difficulty understanding is if the body is rotating about an axis that isn't its centre of mass, for instance a uniform rod with translational motion but also rotating about its endpoint. To find the angular momentum about some other point p you'd add the angular momentum of the centre of mass wrt p (which is ok) to the "spin" angular momentum, ie the rods angular momentum about its centre of mass. However, if the rod is rotating about its endpoint, I don't really understand how I'd find its angular momentum wrt its center of mass...
I didn't feel like using parallel axis theorem to find the moment of inertia about the endpoint would work as that's just finding the angular momentum about its endpoint, not its centre of mass.
The other thought was to find the angular momentum of all the "little pieces" of mass of the rod about its centrepoint as it swings about its endpoint (integrating them in a similar way to calculating moment of inertia). Doing this and set the length of the rod at 2r, angular momentum about c.o.m. came out like so...
m=mass of rod
d=density
x=distance from centre to piece of mass
v=velocity of piece of mass
w=angular velocity of piece of mass about endpoint
L = Summation{(d)(x)(v)(delta x)}
but v=(r-x)w
L = Summation{(d)(x)(r-x)(w)(delta x)}
Integrating from -r to r
L = -(2r^3wd)/3 = (wmr^2)/3
Is this OK?
I guess now this isn't coincidence that this is that this is the moment of inertia of the centre times the angular velocity... It just seems rather weird that the axis of the angular velocity wouldn't matter?
Thanks again, I'm just not sure about it and these problems are starting to make my head... uh... spin. ;)
4. Jul 1, 2011
### tiny-tim
hi boumas!
the parallel axis theorem always correctly gives you the moment of inertia, I
but that's only helpful if the angular momentum about that point = I times the angular velocity
and that is only true for the centre of rotation (the end of the rod, in this case)
(and only because rc.o.m x mv happens to equal mr2 times the angular velocity only if rc.o.m is measured from the centre of rotation)
so you only use the parallel axis theorem if either
i] your point is the centre of rotation of the whole body, or
ii] your point is the centre of mass of the whole body, but the body is irregular, and you're finding the moment of inertia by splitting it into parts, each with a different centre of mass
5. Jul 1, 2011
### tiny-tim
hi boumas!
(have an omega: ω )
the parallel axis theorem always correctly gives you the moment of inertia, I
but that's only helpful if the angular momentum about that point = Iω
and that is only true for the centre of rotation (the end of the rod, in this case)
(and only because rc.o.m x mv happens to equal mr2ω only if rc.o.m is measured from the centre of rotation)
so you only use the parallel axis theorem if either
i] your point is the centre of rotation of the whole body, or
ii] your point is the centre of mass of the whole body, but the body is irregular, and you're finding the moment of inertia by splitting it into parts, each with a different centre of mass
6. Jul 1, 2011
### boumas
So when finding the angular momentum about the endpoint (centre of rotation) I can use the parallel axis theorem, but how can I now add the angular momentum for the translational motion?
\mathbf{r}_{c.o.m.}\times m\mathbf{v}_{c.o.m.}
component to it seems wrong as now the axis of rotation for the other component is not the center of mass...
Also I'm not quite sure how this would lead to calculating the angular momentum about some other point in space...
Thanks
Hope I haven't missed the answer to this in your previous post!
7. Jul 1, 2011
### tiny-tim
hi boumas!
(try using the B and X2 icons just above the Reply box, or for latex use itex tags )
the angular momentum for the translational motion is the parallel axis adjustment …
rc.o.m x mvc.o.m = mr2ω
(because vc.o.m = ω x rc.o.m)
8. Jul 1, 2011
### boumas
Just to clarify, when I said the translational motion I meant a translational motion independent of the rotation. So for instance, the endpoint of the rod is also moving with velocity u, as well as being the axis for the rotation of the body...
Just to make sure...
9. Jul 1, 2011
### tiny-tim
sorry, you can't have it both ways
if it's (on) the axis for the rotation of the body, then it's stationary
if it's moving, then the axis is somewhere else!
10. Jul 4, 2011
### boumas
Ah, OK, so it's basicly unnatural it say a body rotates about anywhere other than its centre of mass without having an axis of rotation...
However, I'm still unsure how I would find the angular momentum about a point other than the axis of rotation (Taking the case that it is fixed and not moving), say a point z which is at some moment in time at the opposite end of the rod...
Also if you were calculating the angular momentum as such but from a moving inertial frame? Maybe that's just silly now...
Sorry to pile these on, I appreciate it... :)
11. Jul 4, 2011
### tiny-tim
hi boumas!
yup!
if z is fixed (even if only instantaneously), then the angular momentum about z can be found either as Izω or as Icω + zc x mvc (they're the same in this case)
not following what you're trying to do here
12. Jul 4, 2011
### boumas
Ahhh...
I think my brain just was refusing to believe that you could use Izω when the angular velocity isn't around that point... Neat!
I was trying to get around the
by taking it from the point of view of a moving inertial frame so it would appear to that observer to be both moving and rotating around that axis...
:tongue:
Perhaps though that would bring torque into it???
13. Jul 4, 2011
### tiny-tim
if it's rotating about an axis, then it isn't moving
(but you can often find a useful frame in which the axis of rotation has moved to a different point … eg in a frame moving with a rolling wheel, the axis of rotation has moved to the bottom of the wheel)
14. Jul 4, 2011
### boumas
Cool...
Thanks very much for clearing that up for me... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8021160960197449, "perplexity": 733.8961825133744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982294883.0/warc/CC-MAIN-20160823195814-00280-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://arxiv.org/list/hep-ph/new | High Energy Physics - Phenomenology
New submissions
[ total of 36 entries: 1-36 ]
[ showing up to 2000 entries per page: fewer | more ]
New submissions for Fri, 21 Jul 17
[1]
Title: WIMP dark matter candidates and searches - current issues and future prospects
Comments: 81 pages, 17 figures. Invited topical review for Reports on Progress in Physics
Subjects: High Energy Physics - Phenomenology (hep-ph); Cosmology and Nongalactic Astrophysics (astro-ph.CO)
We review several current issues in dark matter theory and experiment. We overview the present experimental status, which includes current bounds and recent claims and hints of a possible signal in a wide range of experiments: direct detection in underground laboratories, gamma-ray, cosmic ray, X-ray, neutrino telescopes, and the LHC. We briefly review several possible particle candidates for a Weakly Interactive Massive Particle (WIMP) and dark matter that have recently been considered in the literature. However, we pay particular attention to the lightest neutralino of supersymmetry as it remains the best motivated candidate for dark matter and also shows excellent detection prospects. Finally we briefly review some alternative scenarios that can considerably alter properties and prospects for the detection of dark matter obtained within the standard thermal WIMP paradigm.
[2]
Title: Light Axinos from Freeze-in: production processes, phase space distributions, and Ly-$α$ constraints
Subjects: High Energy Physics - Phenomenology (hep-ph); Cosmology and Nongalactic Astrophysics (astro-ph.CO)
We consider the freeze-in production of 7 keV axino dark matter (DM) in the supersymmetric Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) model in light of the 3.5 keV line excess. The warmness of such 7 keV DM produced from the thermal bath, in general, appears in tension with Ly-$\alpha$ forest data, although a direct comparison is not straightforward. This is because the Ly-$\alpha$ forest constraints are usually reported on the mass of the conventional warm dark matter (WDM), where large entropy production is implicitly assumed to occur in the thermal bath after WDM particles decouple. The phase space distribution of freeze-in axino DM varies depending on production processes and axino DM may alleviate the tension with the tight Ly-$\alpha$ forest constraint. By solving the Boltzmann equation, we first obtain the resultant phase space distribution of axinos produced by 2-body decay, 3-body decay, and 2-to-2 scattering respectively. The reduced collision term and resultant phase space distribution are useful for studying other freeze-in scenarios as well. We then calculate the resultant linear matter power spectra for such axino DM and directly compare them with the linear matter power spectra for the conventional WDM. In order to demonstrate realistic axino DM production, we consider benchmark points with Higgsino next-to-light supersymmetric particle (NLSP) and wino NLSP. In the case of Higgsino NLSP, the phase space distribution of axinos is colder than that in the conventional WDM case, so the most stringent Ly-$\alpha$ forest constraint can be evaded with mild entropy production from saxion decay inherent in the supersymmetric DFSZ axion model.
[3]
Title: FORM version 4.2
Subjects: High Energy Physics - Phenomenology (hep-ph); Symbolic Computation (cs.SC)
We introduce FORM 4.2, a new minor release of the symbolic manipulation toolkit. We demonstrate several new features, such as a new pattern matching option, new output optimization, and automatic expansion of rational functions.
[4]
Title: Systematic Studies of Exact ${\cal O}(α^2L)$ CEEX EW Corrections in a Hadronic MC for Precision $Z/γ^*$ Physics at LHC Energies
Authors: S. Jadach (1), B.F.L. Ward (2), Z. A. Was (1), S. A. Yost (3) ((1) Institute of Nuclear Physics Polish Academy of Sciences, Cracow, PL, (2) Baylor University, Waco, TX, USA, (3) The Citadel, Charleston, SC, USA)
Comments: 14 pages, 2 tables, 9 figures
Subjects: High Energy Physics - Phenomenology (hep-ph)
With an eye toward the precision physics of the LHC, such as the recent measurement of $M_W$ by the ATLAS Collaboration, we present here systematic studies relevant to the assessment of the expected size of multiple photon radiative effects in heavy gauge boson production with decay to charged lepton pairs. We use the new version 4.22 of ${\cal KK}$MC-hh so that we have CEEX EW exact ${\cal O}(\alpha^2 L)$ corrections in a hadronic MC and control over the corresponding EW initial-final interference (IFI) effects as well. In this way, we illustrate the interplay between cuts of the type used in the measurement of $M_W$ at the LHC and the sizes of the expected responses of the attendant higher order corrections. We find that there are per cent to per mille level effects in the initial-state radiation, fractional per mille level effects in the IFI and per mille level effects in the over-all ${\cal O}(\alpha^2 L)$ corrections that any treatment of EW corrections at the per mille level should consider. Our results have direct applicability to current LHC experimental data analyses.
[5]
Title: Meson exchange in lepton-nucleon scattering and proton radius puzzle
Authors: Dmitry Borisyuk
Subjects: High Energy Physics - Phenomenology (hep-ph)
We study two-photon contribution to the elastic lepton-proton scattering, associated with sigma meson exchange, with special attention paid to the low-Q2 region. We show that the corresponding amplitude grows sharply but remains finite at Q2->0. The analytical formula for the amplitude at Q2=0 is obtained. We also estimate the shift of the muonic hydrogen energy levels, induced by sigma meson exchange. For the 2S level the shift is approximately 30 times smaller than needed to resolve the proton radius puzzle, but still exceeds the precision of the muonic hydrogen measurements.
[6]
Title: Light flavon signals at electron - photon colliders
Subjects: High Energy Physics - Phenomenology (hep-ph)
Flavor symmetries are useful to realize fermion flavor structures in the standard model. In particular, discrete $A_4$ symmetry is used to realize lepton flavor structures, and some scalars which are called flavon are introduced to break this symmetry. In many models, flavons are assumed to be much heavier than the electroweak scale. However, our previous work showed that flavon mass around 100 GeV is allowed by experimental constraints in the $A_4$ symmetric model with residual $Z_3$ symmetry. In this paper, we discuss collider search of such a light flavon $\varphi_T$. We find that an electron - photon collision, as a considerable option at the international linear collider, has advantages to search for the signals. At the electron - photon collider flavons are produced as $e^-\gamma \to l^- \varphi_T$ and decay into two charged leptons. Then we analyze signals of flavor-conserving final-state $\tau^+ \tau^- e^-$, and flavor-violating final-states $\tau^+ \mu^- \mu^-$ and $\mu^+ \tau^- \tau^-$ by carrying out numerical simulation. For the former final-state, SM background can be strongly suppressed by imposing cuts on the invariant masses of final-state leptons. For the later final-states, SM background is extremely small, because in the SM there are no such flavor-violating final-states. We then find that sufficient discovery significance can be obtained, even if flavons are heavier than the lower limits from flavor physics.
[7]
Title: Weak Decays of Doubly Heavy Baryons: SU(3) Analysis
Comments: 28 pages, 12 figures, 22 tables
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
Motivated by the recent LHCb observation of doubly-charmed baryon $\Xi_{cc}^{++}$ in the $\Lambda_c^+ K^-\pi^+\pi^+$ final state, we analyze the weak decays of doubly heavy baryons $\Xi_{cc}$, $\Omega_{cc}$, $\Xi_{bc}$, $\Omega_{bc}$, $\Xi_{bb}$ and $\Omega_{bb}$ under the flavor SU(3) symmetry. Decay amplitudes for various semileptonic and nonleptonic decays are parametrized in terms of a few SU(3) irreducible amplitudes. We find a number of relations or sum rules between decay widths and CP asymmetries, which can be examined in future measurements at experimental facilities like LHC, Belle II and CEPC. Moreover once a few decay branching fractions are measured in future, some of these relations may provide hints for exploration of new decay modes.
[8]
Title: Inflation and Dark Matter in the Inert Doublet Model
Subjects: High Energy Physics - Phenomenology (hep-ph); Cosmology and Nongalactic Astrophysics (astro-ph.CO)
We discuss inflation and dark matter in the inert doublet model coupled non-minimally to gravity where the inert doublet is the inflaton and the neutral scalar part of the doublet is the dark matter candidate. We calculate the various inflationary parameters like $n_s$, $r$ and $P_s$ and then proceed to the reheating phase where the inflaton decays into the Higgs and other gauge bosons which are non-relativistic owing to high effective masses. These bosons further decay or annihilate to give relativistic fermions which are finally responsible for reheating the universe. At the end of the reheating phase, the inert doublet which was the inflaton enters into thermal equilibrium with the rest of the plasma and its neutral component later freezes out as cold dark matter with a mass of about 2 TeV.
[9]
Title: Quark matter may not be strange
Subjects: High Energy Physics - Phenomenology (hep-ph)
If quark matter is energetically favored over nuclear matter at zero temperature and pressure then it has long been expected to take the form of strange quark matter (SQM), with comparable amounts of $u,\,d,\,s$ quarks. The possibility of quark matter with only $u,\,d$ quarks ($ud$QM) is usually dismissed because of the observed stability of ordinary nuclei. However we find that udQM generally has lower bulk energy per baryon than normal nuclei and SQM. This emerges in a phenomenological model that describes the spectra of the lightest pseudoscalar and scalar meson nonets. Taking into account the finite size effects, $ud$QM can be the ground state of baryonic matter only for baryon number $A>A_\textrm{min}$ with $A_\textrm{min}\gtrsim 300$. This ensures the stability of ordinary nuclei and points to a new form of stable matter just beyond the periodic table.
[10]
Title: Framework for an asymptotically safe Standard Model via dynamical breaking
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Lattice (hep-lat); High Energy Physics - Theory (hep-th)
We present a consistent embedding of the matter and gauge content of the Standard Model into an underlying asymptotically-safe theory, that has a well-determined in- teracting UV fixed point in the large colour/flavour limit. The scales of symmetry breaking are determined by two mass-squared parameters with the breaking of elec- troweak symmetry being driven radiatively. There are no other free parameters in the theory apart from gauge couplings.
Cross-lists for Fri, 21 Jul 17
[11] arXiv:1707.05570 (cross-list from astro-ph.CO) [pdf, other]
Title: Probing signatures of bounce inflation with current observations
Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
The aim of this paper is to probe the features of the bouncing progress that may take place in the early universe with the current observational data. Different from the normal parameterization based on inflation scenario, we study nonsingular bounce inflation models by taking into account the typical parameters that describe the bouncing process, such as the time and energy scales of the bounce. We consider two parameterized models of nonsingular bounce inflation scenario, one containing three phases while the other having two, and apply Markov Chain Monto Carlo analysis to determine the posterior distributions of the model parameters using the data combination of Planck 2015, BAO, and JLA. With the best fit values of the parameters, we plot the CMB TT power spectrum and show that the bounce inflation model can well explain the anomalies discovered by the Planck observation. Comparing the two-phase model and the three-phase model, we find that the two-phase model is more favored by the current observations. To precisely determine the details of the bounce and distinguish between the models, one needs more highly accurate observational data in the future.
[12] arXiv:1707.06248 (cross-list from hep-th) [pdf, ps, other]
Title: Study of the Question of an Ultraviolet Zero in the Six-Loop Beta Function of the O($N$) $λ|\vec φ|^4$ Theory
Authors: Robert Shrock
Subjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph)
We study the possibility of an ultraviolet (UV) zero in the six-loop beta function of an O($N$) $\lambda |\vec \phi|^4$ field theory in $d=4$ spacetime dimensions. For general $N$, in the range of values of $\lambda$ where a perturbative calculation is reliable, we find evidence against such a UV zero in this six-loop beta function.
[13] arXiv:1707.06302 (cross-list from astro-ph.CO) [pdf, other]
Title: Dark matter spikes in the vicinity of Kerr black holes
Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
The growth of a massive black hole will steepen the cold dark matter density at the center of a galaxy into a dense spike, enhancing the prospects for indirect detection. We study the impact of black hole spin on the density profile using the exact Kerr geometry of the black whole in a fully relativistic adiabatic growth framework. We find that, despite the transfer of angular momentum from the hole to the halo, rotation increases significantly the dark matter density close to the black hole. The gravitational effects are still dominated by the black hole within its influence radius, but the larger dark matter annihilation fluxes might be relevant for indirect detection estimates.
[14] arXiv:1707.06313 (cross-list from hep-th) [pdf, ps, other]
Title: Rigorous constraints on the matrix elements of the energy-momentum tensor
Subjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph)
The structure of the matrix elements of the energy-momentum tensor play an important role in determining the properties of the form factors $A(q^{2})$, $B(q^{2})$ and $C(q^{2})$ which appear in the Lorentz covariant decomposition of the matrix elements. In this paper we apply a rigorous frame-independent distributional-matching approach to the matrix elements of the Poincar\'{e} generators in order to derive constraints on these form factors as $q \rightarrow 0$. In contrast to the literature, we explicitly demonstrate that the vanishing of the anomalous gravitomagnetic moment $B(0)$ and the condition $A(0)=1$ are independent of one another, and that these constraints are not related to the specific properties or conservation of the individual Poincar\'{e} generators themselves, but are in fact a consequence of the physical on-shell requirement of the states in the matrix elements and the manner in which these states transform under Poincar\'{e} transformations.
[15] arXiv:1707.06367 (cross-list from gr-qc) [pdf, ps, other]
Title: Model-Independent Constraints on Lorentz Invariance Violation via the Cosmographic Approach
Comments: 15 pages, 4 figures, 3 tables, revtex4
Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Astrophysical Phenomena (astro-ph.HE); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
Since Lorentz invariance plays an important role in modern physics, it is of interest to test the possible Lorentz invariance violation (LIV). The time-lag (the arrival time delay between light curves in different energy bands) of Gamma-ray bursts (GRBs) has been extensively used to this end. However, to our best knowledge, one or more particular cosmological models were assumed {\it a priori} in (almost) all of the relevant works in the literature. So, this makes the results on LIV in those works model-dependent and hence not so robust in fact. In the present work, we try to avoid this problem by using a model-independent approach. We calculate the time delay induced by LIV with the cosmic expansion history given in terms of cosmography, without assuming any particular cosmological model. Then, we constrain the possible LIV with the observational data, and find weak hints for LIV.
[16] arXiv:1707.06384 (cross-list from hep-ex) [pdf, other]
Title: Supernova Signatures of Neutrino Mass Ordering
Authors: Kate Scholberg
Comments: Invited review for "Focus on neutrino mass and mass ordering" issue of Journal of Physics G; 24 pages, 10 figures
Subjects: High Energy Physics - Experiment (hep-ex); High Energy Astrophysical Phenomena (astro-ph.HE); High Energy Physics - Phenomenology (hep-ph); Nuclear Experiment (nucl-ex)
A suite of detectors around the world is poised to measure the flavor-energy-time evolution of the ten-second burst of neutrinos from a core-collapse supernova occurring in the Milky Way or nearby. Next-generation detectors to be built in the next decade will have enhanced flavor sensitivity and statistics. Not only will the observation of this burst allow us to peer inside the dense matter of the extreme event and learn about the collapse processes and the birth of the remnant, but the neutrinos will bring information about neutrino properties themselves. This review surveys some of the physical signatures that the currently-unknown neutrino mass pattern will imprint on the observed neutrino events at Earth, emphasizing the most robust and least model-dependent signatures of mass ordering.
[17] arXiv:1707.06438 (cross-list from nucl-th) [pdf, other]
Title: Weinberg eigenvalues for chiral nucleon-nucleon interactions
Subjects: Nuclear Theory (nucl-th); High Energy Physics - Phenomenology (hep-ph)
We perform a comprehensive Weinberg eigenvalue analysis of a representative set of modern nucleon-nucleon interactions derived within chiral effective field theory. Our set contains local, semilocal, and nonlocal potentials, developed by Gezerlis, Tews et al. (2013); Epelbaum, Krebs, and Mei{\ss}ner (2015); and Entem, Machleidt, and Nosyk (2017) as well as Carlsson, Ekstr\"om et al. (2016), respectively. The attractive eigenvalues show a very similar behavior for all investigated interactions, whereas the magnitudes of the repulsive eigenvalues sensitively depend on the details of the regularization scheme of the short- and long-range parts of the interactions. We demonstrate that a direct comparison of numerical cutoff values of different interactions is in general misleading due to the different analytic form of regulators; for example, a cutoff value of $R=0.8$ fm for the semilocal interactions corresponds to about $R=1.2$ fm for the local interactions. Our detailed comparison of Weinberg eigenvalues provides various insights into idiosyncrasies of chiral potentials for different orders and partial waves. This shows that Weinberg eigenvalues could be used as a helpful monitoring scheme when constructing new interactions.
[18] arXiv:1707.06538 (cross-list from hep-th) [pdf, other]
Title: A Tale of Two Loops: Simplifying All-Plus Yang-Mills Amplitudes
Authors: Gustav Mogull
Comments: PhD thesis, University of Edinburgh, July 2017
Subjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph)
Pure Yang-Mills amplitudes with all external gluons carrying positive helicity, known as all-plus amplitudes, have an especially simple structure. The tree amplitudes vanish and, up to at least two loops, the loop-level amplitudes are related to those of $\mathcal{N}=4$ super-Yang-Mills (SYM) theory. This makes all-plus amplitudes a useful testing ground for new methods of simplifying more general classes of amplitudes. In this thesis we consider three new approaches, focusing on the structure before integration.
[19] arXiv:1707.06609 (cross-list from hep-th) [pdf, ps, other]
Title: Highly Symmetric D-brane-Anti-D-brane Effective Actions
Authors: Ehsan Hatefi
Comments: 20 pages, no figure, latex file,TUW-17-09
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
The entire S-matrix elements of four, five and six point functions of D-brane-anti D-brane system are explored. To deal with symmetries of string amplitudes as well as their all order $\alpha'$ corrections we first address a four point function of one closed string Ramond-Ramond (RR) and two real tachyons on the world volume of brane-anti brane system. We then focus on symmetries of string theory as well as universal tachyon expansion to achieve both string and effective field theory of an RR and three tachyons where the complete algebraic analysis for the whole S-matrix $<V_{C^{-1}} V_{T^{-1}} V_{T^{0}}V_{T^{0}} >$ was also revealed. Lastly, we employ all the conformal field theory techniques to $<V_{C^{-1}} V_{T^{-1}} V_{T^{0}}V_{T^{0}} V_{T^{0}}>$, working out with symmetries of theory and find out the expansion for the amplitude to be able to precisely discover all order singularity structures of D-brane-anti-D-brane effective actions of string theory. Various remarks about the so called generalized Veneziano amplitude and new string couplings are elaborated as well.
Replacements for Fri, 21 Jul 17
[20] arXiv:1509.02083 (replaced) [pdf, other]
Title: Fermion and scalar phenomenology of a 2-Higgs doublet model with $S_3$
Comments: 22 pages, 7 figures, v2: version accepted in PRD
Journal-ref: Phys. Rev. D 93, 016003 (2016)
Subjects: High Energy Physics - Phenomenology (hep-ph)
[21] arXiv:1511.07420 (replaced) [pdf, ps, other]
Title: A novel Randall-Sundrum model with $S_{3}$ flavor symmetry
Comments: 13 pages. Final version published in Physical Review D
Journal-ref: Phys. Rev. D 94, 033011 (2016)
Subjects: High Energy Physics - Phenomenology (hep-ph)
[22] arXiv:1609.04565 (replaced) [pdf, ps, other]
Title: Supersymmetry in 6d Dirac Action
Subjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph)
[23] arXiv:1612.05639 (replaced) [pdf, ps, other]
Title: Asymmetric Dark Matter in Extended Exo-Higgs Scenarios
Comments: 5 pages; V2: an error in the estimated contribution to muon g-2 corrected, an extension where the g-2 anomaly can be explained provided, title and abstract modified to reflect the changes in the text; V3: Journal version
Journal-ref: Phys. Lett. B 772 (2017) 512
Subjects: High Energy Physics - Phenomenology (hep-ph)
[24] arXiv:1612.09128 (replaced) [pdf, other]
Title: Dynamical relaxation in 2HDM models
Comments: 13 pages, 3 figures, clarified condition under which the additional doublet becomes significant
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
[25] arXiv:1701.04141 (replaced) [pdf, ps, other]
Title: Evolution of Chirality-odd Twist-3 Fragmentation Functions
Authors: J.P. Ma, G.P. Zhang
Subjects: High Energy Physics - Phenomenology (hep-ph)
[26] arXiv:1702.06996 (replaced) [pdf, other]
Title: Probing the top-quark width using the charge identification of $b$ jets
Comments: 5 pages, 3 figures; V2: Journal version
Journal-ref: Phys. Rev. D 96 011901(R) (2017)
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
[27] arXiv:1704.03962 (replaced) [pdf, ps, other]
Title: Physical Cosmological Constant in Asymptotically Background Free Quantum Gravity
Comments: 29 pages, 2 figures, typo corrected, version appeared in PRD
Journal-ref: Phys. Rev. D 96, 026010 (2017)
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
[28] arXiv:1705.02470 (replaced) [pdf, other]
Title: Detecting Dark Photon with Reactor Neutrino Experiments
Authors: HyangKyu Park
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
[29] arXiv:1707.00657 (replaced) [pdf, ps, other]
Title: CT14 Intrinsic Charm Parton Distribution Functions from CTEQ-TEA Global Analysis
Comments: 56 pages, 19 figures, and 3 tables
Subjects: High Energy Physics - Phenomenology (hep-ph)
[30] arXiv:1707.02987 (replaced) [pdf, other]
Title: The 3.5 keV Line from Stringy Axions
Comments: 27 pages + appendices, 1 figure; typos corrected
Subjects: High Energy Physics - Theory (hep-th); High Energy Astrophysical Phenomena (astro-ph.HE); High Energy Physics - Phenomenology (hep-ph)
[31] arXiv:1707.04031 (replaced) [pdf, ps, other]
Title: Doubly charmed baryon: an old prediction facing the LHCb observation
Authors: B.O. Kerbikov
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
[32] arXiv:1707.04138 (replaced) [pdf, ps, other]
Title: The contribution of axial-vector mesons to hyperfine structure of muonic hydrogen
Subjects: High Energy Physics - Phenomenology (hep-ph); Atomic Physics (physics.atom-ph)
[33] arXiv:1707.04403 (replaced) [pdf, other]
Title: Towards a Lorentz Invariant UV Completion for Massive Gravity: dRGT Theory from Spontaneous Symmetry Breaking
Authors: Mahdi Torabian
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
[34] arXiv:1707.04811 (replaced) [pdf, other]
Title: A minimal flavored $U(1)'$ for $B$-meson anomalies
Comments: 19 pages, 4 figures, references and discussion on electroweak precision test added
Subjects: High Energy Physics - Phenomenology (hep-ph)
[35] arXiv:1707.05045 (replaced) [pdf, other]
Title: Drag Force on Heavy Quarks and Spatial String Tension
Authors: Oleg Andreev | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.90105801820755, "perplexity": 2327.9915490217486}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423839.97/warc/CC-MAIN-20170722002507-20170722022507-00175.warc.gz"} |
https://sts-math.com/post_468.html | In right triangle abc, tan(a)=3/4. Find sin(a) and cos(a)
"tan" means the tangent function of one of the acute angles.
It’s the ratio of the length of the opposite side to the length of the adjacent side.
If you say that tan(a) = 3/4, then in the same triangle, the hypotenuse is 5, and
now that we know all the sides, we can state all the other trig functions of that angle.
opposite = 3 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9194008111953735, "perplexity": 346.2188210142935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247484020.33/warc/CC-MAIN-20190218013525-20190218035525-00313.warc.gz"} |
https://nanohub.org/wiki/CNTBandsVerify | ## Verification of the Validity of the CNTBands Tool
According to experimental data the band gap of semiconducting nanotube is inversely proportional to its radius. The simple analytical model also explained in solution for homework Problem 3 indicates that the prefactor V in this dependence is the absolute value of the nearest neighbor tight-binding element in the pi-orbital approximation
$E_{g}=\frac{V}{R}$
Here CNT radius R is measured in the units of carbon-carbon bond length. If R is expressed in nanometers,
$E_{g}=0.142\frac{V}{R}$
The plot below, which collects data from CNTBands pi-orbital tight-binding simulations (dark-blue circles) demonstrates that this is indeed the case. The solid red line is the inverse dependence given by the equation above. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9297824501991272, "perplexity": 877.5075626667302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812259.30/warc/CC-MAIN-20180218212626-20180218232626-00645.warc.gz"} |
https://infoscience.epfl.ch/record/63369 | Infoscience
Journal article
# An approximately minimum variance jackknife
Jackknifing is a nonparametric method of reducing bias in estimation procedures. The reduced-bias jackknife estimate is not, in general, a minimum variance (MV) estimate. The generalized jackknife is extended to allow the computation of jackknife estimates that are reduced in variance as compared with the usual jackknife estimate. The extended method is applicable in situations where there is information available on the covariance function of the given data set. This improved estimation procedure will then produce an approximately MV, reduced- bias estimate for any nonlinear function of the data. For linear combinations of the data, it is shown that the estimator reduces, as a special case, to an exactly MV, unbiased estimator. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9667189121246338, "perplexity": 674.4901516355166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281659.81/warc/CC-MAIN-20170116095121-00368-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn | Minimum cardinality of a difference set in $\mathbb R^n$
Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the same $(n-1)$-dimensional hyperplane, consider the set of difference vectors:
$\{x-y \, | \, x,y \in S\}$
What is the minimum cardinality of this set, as a function of $m$ and $n$?
(The sets that minimize this should be "small" subsets of a lattice, but I don't know what specific shapes minimize it. I think this falls into the realm of "additive combinatorics" or "arithmetic combinatorics", but there aren't tags for those.)
-
Crap, I just spent an hour working on proving the maximum is $1+{m\choose2}$ and apparently this problem is about the minimum. $*$headdesk$*$ – anon Sep 17 '11 at 4:26
It's very unlikely that the answer is known exactly. What ranges of $m,n$ interest you? Say, the asymptotic behavior for fixed $n$ and $m \to \infty$? – Alon Amit Sep 17 '11 at 4:50
Yeah, the minimum is a much more interesting problem. It definitely depends on n because f(m=3,n=1) = 5 but f(m=3,n=2) =7. Convex lattice subsets are clearly the way to go. – Keenan Pepper Sep 17 '11 at 4:55
On the contrary, I'm interested in exact results for small m. Why do you say exact results are so unlikely? – Keenan Pepper Sep 18 '11 at 3:37
Cross-posted to mathoverflow.net/questions/75908/… – Keenan Pepper Sep 20 '11 at 16:12
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1 Answer
The question has an accepted answer at MO.
I am posting the link here (as a CW answer) so that the question does not remain unanswered.
-
add comment | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8199842572212219, "perplexity": 532.7583260154437}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00329-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/249378/is-scaling-data-0-1-necessary-when-batch-normalization-is-used/328978 | # is scaling data [0,1] necessary when batch normalization is used?
Although a Relu activation function can deal with real value number but I have tried scaling the dataset in the range [0,1] (min-max scaling) is more effective before feed it to the neural network. on the other hand, the batch normalization (BN) is also normalizing data before passed to the non-linearity layer (activation function). I was wondering if the min-max scaling is still needed when BN is applied. can we perform min-max scaling and BN together?. It would be nice if someone guides me to the better understanding
• I think you can let go of batch normalization. It is not needed any more. – Souradeep Nanda Dec 5 '16 at 5:39 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.856551468372345, "perplexity": 864.3916806362048}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347406365.40/warc/CC-MAIN-20200529183529-20200529213529-00221.warc.gz"} |
http://mathhelpforum.com/pre-calculus/32306-derivative-natural-logarithmic-funtion.html | # Thread: Derivative of Natural Logarithmic Funtion
1. ## Derivative of Natural Logarithmic Funtion
Find the equation of the tangent to the curve defined by y=ln(1+2^(-x)) at the point where x=0.
Help will be appreciated
2. Take the derivative of y and evaluate it at x = 0. This will be your slope value. To find the equation to the tangent, you use the formula:
$y - y_{1} = m(x - x_{1})$
where $(x_{1}, y_{1})$ is the point you're concerned with.
As for taking the derivative, make sure you're careful about using your chain rule! Show us what you've gotten down and we'll help you from there
3. Originally Posted by rhhs11
Find the equation of the tangent to the curve defined by y=ln(1+2^(-x)) at the point where x=0.
Help will be appreciated
Note that $2^{-x}=e^{-xln(2)}$ if we take the derivative we get.
$\frac{d}{dx}2^{-x}=-ln(2)e^{-xln(2)}=-ln(2)2^{-x}
$
So now back to your derivative
$\frac{dy}{dx}=\frac{-ln(2)2^{-x}}{1+2^{-x}}$
evaluating at zero we get...
$\frac{dy}{dx}|_{x=0}=\frac{-ln(2)2^{-0}}{1+2^{-0}}=\frac{-ln(2)}{2}$
4. So i got
y = 1 / (1+e^-x) * e^-x * (-1)
= - e^(-x) / (1+ e^-x)
mtangent @ x=0 = -1 / (1+1)
= -1/2
y - ln2 = 1/2(x-0)
2y - 2ln2 = x
x - 2y + 2ln2 = 0
Im not sure if that's right but looks fair enough. I would like any advice though.
I also have another question where it says: Find the equation of the tangent to the curve defined by y = e^x that is perpendicular to the line 3x + y = 1.
Thanks a lot in advance!
5. Im sorry but the question is y = ln(1+e^-x)
6. Originally Posted by rhhs11
So i got
y = 1 / (1+e^-x) * e^-x * (-1)
= - e^(-x) / (1+ e^-x)
mtangent @ x=0 = -1 / (1+1)
= -1/2
y - ln2 = 1/2(x-0)
2y - 2ln2 = x
x - 2y + 2ln2 = 0
Im not sure if that's right but looks fair enough. I would like any advice though.
I also have another question where it says: Find the equation of the tangent to the curve defined by y = e^x that is perpendicular to the line 3x + y = 1.
Thanks a lot in advance!
How did your function change from
$y=\ln(1+2^{-x})$ to $y=\ln(1+e^{-x})$
7. It was a mistake when i was entering the question | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8445591330528259, "perplexity": 886.328933581749}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816462.95/warc/CC-MAIN-20180225130337-20180225150337-00485.warc.gz"} |
https://www.khanacademy.org/math/calculus-2/cs2-integrals-review/cs2-riemann-sums-in-summation-notation/v/riemann-sum-negative-function | If you're seeing this message, it means we're having trouble loading external resources on our website.
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## Calculus 2
### Unit 1: Lesson 4
Riemann sums in summation notation
# Riemann sums in summation notation: challenge problem
When a function is negative, Riemann sums seem to treat it as having "negative area".
## Want to join the conversation?
• Confused me as I thought sum might still mean the total of a positive and negative value
• Although it sounds tempting, area cannot be negative. When a function dips below the x-axis, the space between the function and the x-axis should be treated the same way as if it were above the x-axis. Because there is no absolute value sign, both II and III are wrong as they subtract area from themselves.
• So, can you actually use the Riemann sum at all to work out the area of the rectangles that are in the third/fourth quadrants? Or is it just impossible because you'd never be able to get a positive number?
• You could always introduce the absolute value function into the sum. To say this more precisely, instead of doing the summation of that function, you can do the summation of the absolute value of that function. In a sense, this would force all the values of the function to be positive which makes it give you the height of every rectangle.
• It confused me when Sal ruled out the second choice. Traditionally, you cannot have a negative area, but I thought that area under the x-axis above the curve was considered negative. At least, it is when evaluating signed area.
• The way that areas below the x-axis are interpreted when integrating depends on what the function is representing. In other words, is the function representing something such that any area below the x-axis should be subtracted or added to the area above the x-axis. So, there is not a single answer -- what you do depends on what the area is representing.
• For the part where you discuss whats inside the f(x), how is that related to whether it is negative or positive? its unclear why the statements are true and false
• I was confused about this as well. You have to remember that the f(stuff) is directing you to plug the value of the "stuff" you just found into whatever function you're dealing with. For example, the first red rectangle you would have f( -1 + 1/2 ) which equals f( -1/2 ). When you plug -1/2 into the equation being graphed you get some very small negative value right below the x-axis. That small negative value is what causes the trouble since no matter what red rectangle you're referencing, the corresponding f(stuff) = y = "rectangle height" value is always going to be negative which will give you a negative approximate area. TL;DR a rectangle below the x-axis may be 4 units tall, but the y-value you're using to determine that height is actually negative and that y-value is what is being used in the sum, not the absolute value of y which is the height.
• confuses me why we dont say what we said for second (it can not be negative) also for first one
• We don't take the x-values into consideration when calculating the area, just the distance between them (1/2) which is always positive so it doesn't matter whether the x-values are positive or negative.
We do, however, use the y-values as the height which can be positive or negative and that's where we get into the issue of negative areas. The first example has all positive y-values while the second has all negative y-values and the third has a mixture of both.
• I don't get why it was F(5 + i/2). I know that it is the height, but how did he get that expression?
• It's actually f(-5+i/2) (emphasis on the negative five). It is set up this way because the rectangles in the graph start at x = -5 and increment by 1/2 for each rectangle. Because it's a right-handed rectangle, the y value that determines the rectangle's height is on the right hand side. So the first rectangle's height is given by the y-value f(-5 + 1/2), the second recangle's by f(-5 +2/2), the third by f(-5 + 3/2), and so on. Notice that the expression says "the value of f at negative 5 (where the rectangles start) plus 1/2 times whichever rectangle we're currently on."
If it's the iteration from i = 1 through i = 8 that is confusing, look into videos on sigma notation. That should help clear it up.
• If this graph were said to describe the velocity of an object against time, could it be said that the absolute value of the area (taking the value of the integral between x=-5 and x=-1 and adding it to the absolute value of the integral between x=-1 and x=7) is equal to the total distance travelled, but the value of area (value of the integral between x=-5 and x=7, so subtraction would occur) is equal to the displacement?
• So why can't area be negative?
• What would be the area of the shaded region in this graph? https://www.desmos.com/calculator/yv4ah6svvu
If we allow negative area, then the answer is 0, does that seem right to you?
It is important to remember that area is just one interpretation of the result of integration. It is commonly used with beginners because we all have experience with the concept of area in the physical sense.
Check out this interpretation of Area Under the Curve:
http://sepia.unil.ch/pharmacology/index.php?id=66
https://en.wikipedia.org/wiki/Area_under_the_curve_(pharmacokinetics)
Keep Studying and
.
• At , when we count do we always count starting from the right side if the rectangle?
Because if you see, when you count from the left side of the first blue rectangle, the total count is 9. But if i start counting from the right side of the firzt blue triangle it adds up to be 8.
Can someone please explain this to me
(1 vote)
• There are 8 rectangles, counting either from the left or from the right.
I guess you're counting the sides rather than the rectangles. By doing so, you'll get 9 ''walls''. Try counting the upper side of each rectangle...
• Shouldn't II. also be false due to the fact that it starts with i=1? If the sum is supposed to describe the area of the red rectangles, and the first red rectangle is the 9th rectangle, shouldn't it start with i=9? (and end at 24?)
• The number i doesn't necessarily coordinate with any of the values on the number line. Where i = 1 is where the Riemann sum starts calculating areas. II is just calculating the areas of the red rectangles, so i is set to 1 where the red ones begin. Hope this helps.
(1 vote) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8489667773246765, "perplexity": 385.97747934966105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494936.89/warc/CC-MAIN-20230127033656-20230127063656-00191.warc.gz"} |
http://microchipdeveloper.com/asp0107:differentiators | Differentiators
Interchanging the location of the capacitor and resistor of the integrator circuit results in the differentiator circuit, which performs the mathematical function of differentiation.
Transfer Function
The transfer function of the differentiator circuit is as follows:
(1)
\begin{align} V_{out} = -RC\frac{\mathrm{d} V_{in}}{\mathrm{d} t} \end{align}
Frequency Domain
The frequency response of the differentiator can be thought of as that of a low pass filter with a cut-off frequency of zero.
Example
Design a differentiator that has a time constant of 10 ms and an input capacitance of 0.01 μf. What is the value of R? What is the gain magnitude and phase of this circuit at 10 Hz, and at 100 Hz? In order to limit the high frequency gain of the differentiator circuit to 100, a resistor is added in series with the capacitor. Find the required resistor value.
Given the differentiator transfer function, magnitude |Vout/Vin | = 2 π f R C and Phase φ = -90°
RC = 10 ms, C = 0.01 μf, R = 1 MΩ
At f =100 Hz, |Vout/Vin| = 2 π f R C = 6.28, φ = -90°
At f =1,000 Hz, |Vout/Vin| = 2 π f R C = 62.8, φ = -90°
At high frequency, |Vout/Vin| = R/R1 = 100, R1 = 10 kΩ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9138251543045044, "perplexity": 1882.500188981434}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823168.74/warc/CC-MAIN-20171018233539-20171019013539-00376.warc.gz"} |
https://www.intmath.com/blog/mathematics/how-to-apply-powers-to-the-unit-i-12557 | # How to Apply Powers to the Unit i
By Kathleen Cantor, 03 Apr 2021
The variable represents somewhat of an anomaly within mathematics. As one of the prerequisites to any type of non-real number, understanding i is a crucial step towards your journey into higher-level math. Before we apply powers to the unit , we must understand what involves when we use it in a mathematical expression.
## What is ?
As an imaginary number, doesn't fit into the realm of real numbers. It can't be calculated, nor can it interact with real numbers (outside of certain special cases). Commonly, is used as a stand-in for the imaginary value of the square root of -1.
However, the square root of -1 can't be calculated through real means. Still, we sometimes need its value to perform calculations using real numbers, like in the quadratic formula for example.
Certain mathematical expressions can only take specific types of input. As an example, the absolute value will only ever output positive numbers, while other expressions like square roots can't input or output negative values. These expressions are often referred to as non-negative functions.
## How Powers Affect the Unit
Attempting to get a negative value out of these non-negative functions will output an error on any calculator. When it comes to using exponents, or powers, the same thing happens when you try to put any negative value into the function at all. Simply put, this is because performing the operation in that expression is impossible.
In a square root, there are no two negative numbers that are the same and can multiply together to create another negative number. Multiplying a negative number by a negative number always produces a positive one.
In these instances, we use the imaginary number to get past the roadblock of square rooting a negative number and proceed with calculations.
### Performing Calculations Using the Unit
It's important to recognize when can and cannot be used to perform calculations. A general rule of thumb is to treat as a variable that you can't evaluate. This means that most expressions containing can be solved with simple algebra. Let's go over a few examples:
#### Example 1
In this example, the expression cannot be evaluated. Be careful when working with both real and non-real numbers simultaneously - they exist in completely separate areas of mathematics! The equation above simply evaluates to .
#### Example 2
In this case, we can't evaluate the expression for the same reason. A non-real number cannot interact with a real number outside of very specific circumstances. As a result, the expression still evaluates to .
It's important to be aware that can still have real numbers as its coefficient or divisor, even if the expression can't be further evaluated from that point. This is because attaching a coefficient or divisor to doesn't actually change the value that's denoted by in any way.
Although the following expression is still interacting with real numbers, this is one of the few special situations in which we can evaluate. Remember that, as a non-real number, is simply the stand-in for the square root of -1. Even though we can't evaluate the square root of -1, we can evaluate the square of a square root, which simply gives the value held within.
#### Example 1
In this instance, the square of i will yield a value of -1. With this information in mind, we can simplify the expression above:
Note that the distributive property still applies to non-real numbers, as we still are expanding two terms within brackets. The terms' actual content is irrelevant when it comes to the rules of expansion.
After applying distributive property, we are still left with in the expression. But remember that non-real numbers cannot conventionally interact with i. As a result, after simplifying the expression's real numbers, we can no longer create any interactions between the real and non-real numbers.
Another important point is to recognize that expressions containing can interact with each other and be simplified. Here, we were able to combine and to create , as both and belong to the non-real set of numbers. However, we cannot simplify and 15 any further.
#### Example 2
In this example, we are given a cube of . Despite how it may first seem, the expression can be evaluated using exponent laws.
We can rewrite the expression as follows:
By breaking the cube up into squared multiplied by , it becomes clear that the expression can be evaluated using the same principle we used in the previous question. Reapplying our knowledge that is a stand-in for the square root of 2 allows us to simplify the expression to the following:
At this point, we have a non-real number multiplied by a negative number. However, recall that we can still place coefficients and divisors before . As a result, the expression simplifies to:
Pretty straightforward, right?
## Final Takeaways
Essentially, is a variable that denotes the non-real number that would result from the square root of -1. It is most commonly used to further simplify expressions that otherwise would be unsimplifiable. Applying powers to the unit is fairly straightforward as long as you think of as a variable with special properties and treat it that way.
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From Math Blogs | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8535580039024353, "perplexity": 519.2585258670297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588341.58/warc/CC-MAIN-20211028131628-20211028161628-00626.warc.gz"} |
https://www.kofastudy.com/courses/ss2-physics-1st-term/lessons/equilibrium-of-forces-i-week-4/topic/moment-of-a-force/ | Lesson 4, Topic 3
In Progress
# Moment of a Force
Lesson Progress
0% Complete
When taps are opened, a turning effect of force is experienced, likewise, when doors are opened, the applied force brings about a turning effect about a point or hinges attached to the wall of the door.
The turning effect experienced in each case is called the moment of a force.
The moment of a force about a point (or axis )O, is the turning effect of the force about that point. It is equal to the product of the force and the perpendicular distance from the line of action to the point or pivot.
Moment = Force x Perpendicular distance of pivot to the line of action of the force
= Newton x Metre
Its unit is Newton metre (Nm), hence, it is a vector quantity.
If the force is inclined at an angle θ.
Moment = Fdsinθ
The magnitude of moments depends on:
i) The Force applied
iI) The perpendicular distance from the pivot to the line of action of the force.
### Resultant Moments:
When more than two forces act on a body, the resultant moment on the body about any point can be obtained using algebraic moments using the clockwise moment and anticlockwise moments about the same point.
If the clockwise moment is taken as positive and the anticlockwise moments are negative.
$$\scriptsize F_1 \: \times \: X_1 = F_2 \: \times \: X_2$$
$$\scriptsize F_1 \: \times \: X_1 \: -\: F_2 \: \times \: X_2 = 0$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9242204427719116, "perplexity": 743.6225885901598}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710953.78/warc/CC-MAIN-20221204004054-20221204034054-00422.warc.gz"} |
http://clay6.com/qa/41448/the-sum-of-two-point-charges-in-7-mu-c-they-repel-each-other-with-a-force-o | # The sum of two point charges in $7\mu C$. They repel each other with a force of $1\; N$ when kept $30\;cm$ apart in free space. Calculate the value of each charge.
The value of charge is : $2 \mu C, 5 \mu C$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8915313482284546, "perplexity": 196.97098258032594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864139.22/warc/CC-MAIN-20180621094633-20180621114633-00527.warc.gz"} |
https://forum.azimuthproject.org/discussion/2299/lecture-65-chapter-4-collaborative-design | #### Howdy, Stranger!
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# Lecture 65 - Chapter 4: Collaborative Design
edited February 17
Last time we reached the technical climax of this chapter: constructing the category of enriched profunctors. If you found that difficult, you'll be relieved to hear it's downhill from here on, at least in Chapter 4. We will now apply all our hard work.
One application is to collaborative design. Fong and Spivak discuss it in their the book, based on this paper by a student of Spivak:
Censi's work is based on $$\textbf{Bool}$$-enriched profunctors, also known as feasibility relations. We introduced these in Lecture 56.
Remember, a feasibility relation $$\Phi : X\nrightarrow Y$$ is a monotone function
$$\Phi : X^{\text{op}} \times Y \to \mathbf{Bool} .$$ If $$\Phi(x,y) = \text{true}$$, we say $$x$$ can be obtained given $$y$$. The idea is that we use elements of $$X$$ to describe 'requirements' - things you want - and elements of $$Y$$ to describe 'resources' - things you have.
The idea of Andrea Censi's theory that we can compute the design requirements of a complex system from those of its parts using 'co-design diagrams'. These are really pictures of big complicated feasibility relations, like this:
This big complicated feasibility relation is built from simpler ones in various ways. Each wire in this diagram is labeled with the name of a preorder, and each little box is itself a feasibility relation between preorders. We described how to compose feasibility relations in Lecture 58, and that corresponds to feeding the outputs of one little box into another. But there are other things going on in this picture, like boxes sitting side by side, and wires that bend around backwards! This is what I need to explain - it may take a couple lectures to do this.
Instead of diving into the mathematical details today, let me quote the book's general explanation of the diagram above:
As an example, consider the design problem of creating a robot to carry some load at some velocity. The top-level planner breaks the problem into three design teams: team chassis, team motor, and team battery. Each of these teams could break up into multiple parts and the process repeated, but let's remain at the top level and consider the resources produced and the resources required by each of our three teams.
The chassis in some sense provides all the functionality—it carries the load at the velocity—but it requires some things in order to do so. It requires money, of course, but more to the point it requires a source of torque and speed. These are supplied by the motor, which in turn needs voltage and current from the battery. Both the motor and the battery cost money, but more importantly they need to be carried by the chassis: they become part of the load. A feedback loop is created: the chassis must carry all the weight, even that of the parts that power the chassis. A heavier battery might provide more energy to power the chassis, but is the extra power worth the heavier load?
In the picture, each part—chassis, motor, battery, and robot—is shown as a box with ports on the left and right. The functionalities, or resources produced by the part are on the left of the box, and the resources required by the part are on the right.
The boxes marked $$\Sigma$$ correspond to summing inputs. These boxes are not to be designed, but we will see later that they fit easily into the same conceptual framework. Note also the $$\leq$$'s on each wire; they indicate that if box $$A$$ requires a resource that box $$B$$ produces, then $$A$$'s requirement must be less than or equal to $$B$$'s production.
Next time we'll get into more detail.
But to wrap up for today, here are few puzzles!
We've been talking about enriched functors and also enriched profunctors. How are they related? The first cool fact is that any enriched functor gives two enriched profunctors: one going each way.
Puzzle 201. Show that any $$\mathcal{V}$$-enriched functor $$F: \mathcal{X} \to \mathcal{Y}$$ gives a $$\mathcal{V}$$-enriched profunctor
$$\hat{F} \colon \mathcal{X} \nrightarrow \mathcal{Y}$$ defined by
$$\hat{F} (x,y) = \mathcal{Y}(F(x), y ) .$$ Puzzle 202. Show that any $$\mathcal{V}$$-enriched functor $$F: \mathcal{X} \to \mathcal{Y}$$ gives a $$\mathcal{V}$$-enriched profunctor
$$\check{F} \colon \mathcal{Y} \nrightarrow \mathcal{X}$$ defined by
$$\check{F} (y,x) = \mathcal{Y}(y,F(x)) .$$ These two constructions have funny names. $$\hat{F} \colon \mathcal{X} \nrightarrow \mathcal{Y}$$ is called the companion of $$F$$ and $$\check{F} \colon \mathcal{Y} \nrightarrow \mathcal{X}$$, going back, is called the conjoint of $$F$$.
If you have trouble remembering these, remember that a 'companion' is like a fellow traveler, going the same way as our original functor. The word 'conjoint' should remind you of 'adjoint', which means something going back the other way.
In fact there's a relationship between adjoints and conjoints!
Puzzle 203. We say a $$\mathcal{V}$$-enriched functor $$F: \mathcal{X} \to \mathcal{Y}$$ is a left adjoint of a $$\mathcal{V}$$-enriched functor $$G: \mathcal{Y} \to \mathcal{X}$$ if
$$\mathcal{Y}(F(x), y) = \mathcal{X}(x,G(y))$$ for all objects $$x$$ of $$\mathcal{X}$$ and $$y$$ of $$\mathcal{Y}$$. In this situation we also say $$G$$ is the right adjoint of $$F$$. Show that $$F$$ is the left adjoint of $$G$$ if and only if
$$\hat{F} = \check{G} .$$ Pretty!
To read other lectures go here.
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The talk of "companions" and "conjoints" reminded me of the paper 'String Diagrams For Double Categories and Equipments' by David Jaz Myers.
The paper gives really nice string diagrammatics.
Comment Source:The talk of "companions" and "conjoints" reminded me of the paper ['String Diagrams For Double Categories and Equipments'](https://arxiv.org/abs/1612.02762) by David Jaz Myers. The paper gives really nice string diagrammatics.
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Puzzle 201. Show that any $$\mathcal{V}$$-enriched functor $$F: \mathcal{X} \to \mathcal{Y}$$ gives a $$\mathcal{V}$$-enriched profunctor
$$\hat{F} \colon \mathcal{X} \nrightarrow \mathcal{Y}$$ defined by $$\hat{F} (x,y) = \mathcal{Y}(F(x), y) .$$
Based on the second theorem from the previous lecture, we can prove that $$\hat{F} : \mathcal{X} \nrightarrow \mathcal{Y}$$ is a $$\mathcal{V}$$-enriched profunctor by showing that
$$\mathcal{X}(x', x) \otimes \hat{F}(x, y) \otimes \mathcal{Y}(y, y') \leq \hat{F}(x',y') .$$ Replacing $$\hat{F}$$ with its definition we have:
$$\mathcal{X}(x', x) \otimes \mathcal{Y}(F(x), y) \otimes \mathcal{Y}(y, y') \leq \mathcal{Y}(F(x'), y') .$$ This inequality is proved by using the facts that (i) $$F$$ is a $$\mathcal{V}$$-enriched functor, and (ii) $$\mathcal{Y}$$ is a $$\mathcal{V}$$-enriched category:
\begin{align} \mathcal{X}(x', x) \otimes \mathcal{Y}(F(x), y) \otimes \mathcal{Y}(y, y') & \leq \mathcal{Y}(F(x'), F(x)) \otimes \mathcal{Y}(F(x), y) \otimes \mathcal{Y}(y, y') \tag{i} \\ & \leq \mathcal{Y}(F(x'), y) \otimes \mathcal{Y}(y, y') \tag{ii} \\ & \leq \mathcal{Y}(F(x'), y') . \tag{ii} \end{align}
Puzzle 202. Show that any $$\mathcal{V}$$-enriched functor $$F: \mathcal{X} \to \mathcal{Y}$$ gives a $$\mathcal{V}$$-enriched profunctor
$$\check{F} \colon \mathcal{Y} \nrightarrow \mathcal{X}$$ defined by $$\check{F} (y,x) = \mathcal{Y}(y,F(x)) .$$
Similarly, to prove that $$\check{F} : \mathcal{Y} \nrightarrow \mathcal{X}$$ is a $$\mathcal{V}$$-enriched profunctor we have to show that
$$\mathcal{Y}(y', y) \otimes \check{F}(y, x) \otimes \mathcal{X}(x, x') \leq \check{F}(y',x')$$ which is equivalent to
$$\mathcal{Y}(y', y) \otimes \mathcal{Y}(y, F(x)) \otimes \mathcal{X}(x, x') \leq \mathcal{Y}(y', F(x')) .$$ Again, the inequality is proved by using the facts that (i) $$F$$ is a $$\mathcal{V}$$-enriched functor, and (ii) $$\mathcal{Y}$$ is a $$\mathcal{V}$$-enriched category:
\begin{align} \mathcal{Y}(y', y) \otimes \mathcal{Y}(y, F(x)) \otimes \mathcal{X}(x, x') & \leq \mathcal{Y}(y', y) \otimes \mathcal{Y}(y, F(x)) \otimes \mathcal{Y}(F(x), F(x')) \tag{i} \\ & \leq \mathcal{Y}(y', F(x)) \otimes \mathcal{Y}(F(x), F(x')) \tag{ii} \\ & \leq \mathcal{Y}(y', F(x')). \tag{ii} \end{align}
Comment Source:> **Puzzle 201.** Show that any \$$\mathcal{V}\$$-enriched functor \$$F: \mathcal{X} \to \mathcal{Y}\$$ gives a \$$\mathcal{V}\$$-enriched profunctor > > $\hat{F} \colon \mathcal{X} \nrightarrow \mathcal{Y}$ > > defined by > > $\hat{F} (x,y) = \mathcal{Y}(F(x), y) .$ Based on the second theorem from [the previous lecture](https://forum.azimuthproject.org/discussion/2298/lecture-64-chapter-5-the-category-of-enriched-profunctors/p1), we can prove that \$$\hat{F} : \mathcal{X} \nrightarrow \mathcal{Y}\$$ is a \$$\mathcal{V}\$$-enriched profunctor by showing that $\mathcal{X}(x', x) \otimes \hat{F}(x, y) \otimes \mathcal{Y}(y, y') \leq \hat{F}(x',y') .$ Replacing \$$\hat{F}\$$ with its definition we have: $\mathcal{X}(x', x) \otimes \mathcal{Y}(F(x), y) \otimes \mathcal{Y}(y, y') \leq \mathcal{Y}(F(x'), y') .$ This inequality is proved by using the facts that (i) \$$F\$$ is a \$$\mathcal{V}\$$-enriched functor, and (ii) \$$\mathcal{Y}\$$ is a \$$\mathcal{V}\$$-enriched category: \begin{align} \mathcal{X}(x', x) \otimes \mathcal{Y}(F(x), y) \otimes \mathcal{Y}(y, y') & \leq \mathcal{Y}(F(x'), F(x)) \otimes \mathcal{Y}(F(x), y) \otimes \mathcal{Y}(y, y') \tag{i} \\\\ & \leq \mathcal{Y}(F(x'), y) \otimes \mathcal{Y}(y, y') \tag{ii} \\\\ & \leq \mathcal{Y}(F(x'), y') . \tag{ii} \end{align} --- > **Puzzle 202.** Show that any \$$\mathcal{V}\$$-enriched functor \$$F: \mathcal{X} \to \mathcal{Y}\$$ gives a \$$\mathcal{V}\$$-enriched profunctor > > $\check{F} \colon \mathcal{Y} \nrightarrow \mathcal{X}$ > > defined by > > $\check{F} (y,x) = \mathcal{Y}(y,F(x)) .$ Similarly, to prove that \$$\check{F} : \mathcal{Y} \nrightarrow \mathcal{X}\$$ is a \$$\mathcal{V}\$$-enriched profunctor we have to show that $\mathcal{Y}(y', y) \otimes \check{F}(y, x) \otimes \mathcal{X}(x, x') \leq \check{F}(y',x')$ which is equivalent to $\mathcal{Y}(y', y) \otimes \mathcal{Y}(y, F(x)) \otimes \mathcal{X}(x, x') \leq \mathcal{Y}(y', F(x')) .$ Again, the inequality is proved by using the facts that (i) \$$F\$$ is a \$$\mathcal{V}\$$-enriched functor, and (ii) \$$\mathcal{Y}\$$ is a \$$\mathcal{V}\$$-enriched category: \begin{align} \mathcal{Y}(y', y) \otimes \mathcal{Y}(y, F(x)) \otimes \mathcal{X}(x, x') & \leq \mathcal{Y}(y', y) \otimes \mathcal{Y}(y, F(x)) \otimes \mathcal{Y}(F(x), F(x')) \tag{i} \\\\ & \leq \mathcal{Y}(y', F(x)) \otimes \mathcal{Y}(F(x), F(x')) \tag{ii} \\\\ & \leq \mathcal{Y}(y', F(x')). \tag{ii} \end{align}
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edited July 2018
Puzzle 203. We say a $$\mathcal{V}$$-enriched functor $$F: \mathcal{X} \to \mathcal{Y}$$ is a left adjoint of a $$\mathcal{V}$$-enriched functor $$G: \mathcal{Y} \to \mathcal{X}$$ if
$$\mathcal{Y}(F(x), y) = \mathcal{X}(x,G(y))$$ for all objects $$x$$ of $$\mathcal{X}$$ and $$y$$ of $$\mathcal{Y}$$. In this situation we also say $$G$$ is the
right adjoint of $$F$$. Show that $$F$$ is the left adjoint of $$G$$ if and only if $$\hat{F} = \check{G}$$
Suppose $$F$$ is left adjoint to $$G$$. Using its definition, the companion of $$F$$ is $$\hat{F}(x,y) = \mathcal{Y}(F(x), y)$$. Likewise the conjoint of $$G$$ is $$\check{G}(x,y) = \mathcal{X}(x,G(y))$$. But by the assumption of adjointness,
$$\hat{F}(x,y) = \mathcal{Y}(F(x), y) = \mathcal{X}(x,G(y)) = \check{G}(x,y)$$ So $$\hat{F} = \check{G}$$ if $$F$$ is left adjoint to $$G$$. The proof of the other direction is similar. Assume that $$\hat{F} = \check{G}$$ given $$F,G$$. The definition of the companion and conjoint lead to the same equality. $$\square$$
Comment Source:> **Puzzle 203.** We say a \$$\mathcal{V}\$$-enriched functor \$$F: \mathcal{X} \to \mathcal{Y}\$$ is a **left adjoint** of a \$$\mathcal{V}\$$-enriched functor \$$G: \mathcal{Y} \to \mathcal{X}\$$ if > $\mathcal{Y}(F(x), y) = \mathcal{X}(x,G(y))$ > for all objects \$$x\$$ of \$$\mathcal{X}\$$ and \$$y\$$ of \$$\mathcal{Y}\$$. In this situation we also say \$$G\$$ is the **right adjoint** of \$$F\$$. Show that \$$F\$$ is the left adjoint of \$$G\$$ if and only if > $\hat{F} = \check{G}$ Suppose \$$F\$$ is left adjoint to \$$G\$$. Using its definition, the companion of \$$F\$$ is \$$\hat{F}(x,y) = \mathcal{Y}(F(x), y) \$$. Likewise the conjoint of \$$G\$$ is \$$\check{G}(x,y) = \mathcal{X}(x,G(y))\$$. But by the assumption of adjointness, $\hat{F}(x,y) = \mathcal{Y}(F(x), y) = \mathcal{X}(x,G(y)) = \check{G}(x,y)$ So \$$\hat{F} = \check{G} \$$ if \$$F\$$ is left adjoint to \$$G\$$. The proof of the other direction is similar. Assume that \$$\hat{F} = \check{G} \$$ given \$$F,G\$$. The definition of the companion and conjoint lead to the same equality. \$$\square\$$
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edited July 2018
It strikes me that some profunctors are "nice" in the sense that they are companions/conjoints of an adjoint pair of functors.
But profunctors in general need not be "nice". Is there any way of telling which is which?
eg a theorem along the lines of "a profunctor $$\Phi$$ is 'nice' iff it satisfies [some formula involving $$\Phi$$]"
NB by "nice" I should probably say "representable on both sides".
EDIT to add: this is actually two problems: (i) when is a profunctor a companion? (ii) when is a profunctor a conjoint?
Comment Source:It strikes me that some profunctors are "nice" in the sense that they are companions/conjoints of an adjoint pair of functors. But profunctors in general need not be "nice". Is there any way of telling which is which? eg a theorem along the lines of "a profunctor \$$\Phi\$$ is 'nice' iff it satisfies [some formula involving \$$\Phi\$$]" NB by "nice" I should probably say "representable on both sides". EDIT to add: this is actually two problems: (i) when is a profunctor a companion? (ii) when is a profunctor a conjoint?
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edited July 2018
Anindya: excellent question!
At least in the case of ordinary profunctors (that is, $$\textbf{Set}$$-enriched profunctors) there's a well-known necessary condition for a profunctor to be the companion of some functor $$F$$:
Namely, it needs to have a right adjoint in the 2-category of categories, profunctors and natural transformations. And this right adjoint is just the conjoint of $$F$$.
Here I'm using a concept of 'adjoint' that make sense in any 2-category, which reduces to the usual concept of 'adjoint functor' when we apply it to the 2-category of categories, functors and natural transformations:
This condition for a profunctor to be a companion of a functor is also sufficient if we restrict ourselves to 'Cauchy complete' categories. Every category $$\mathcal{C}$$ has a Cauchy completion $$\text{Split}(\mathcal{C})$$ such that profunctors out of $$\mathcal{C}$$ correspond in a natural one-to-one way with profunctors out of $$\text{Split}(\mathcal{C})$$. So, when doing serious work with profunctors, people often restrict attention to Cauchy complete categories.
I've stated a necessary condition for a profunctor to be a companion, but the same result also serves to give a necessary condition for a profunctor to be a conjoint. Any profunctor that's a conjoint of a functor must have a left adjoint in the 2-category of categories, profunctors and natural transformations! And again, this necessary condition is also sufficient if we restrict attention to Cauchy complete categories.
All this has an enriched analogue - but I have a feeling that we're already getting a bit too technical for this course! Suffice it to say that you've put your finger on an important issue.
Comment Source:Anindya: excellent question! <img width = "100" src = "http://math.ucr.edu/home/baez/mathematical/warning_sign.jpg"> At least in the case of ordinary profunctors (that is, \$$\textbf{Set}\$$-enriched profunctors) there's a well-known _necessary_ condition for a profunctor to be the companion of some functor \$$F\$$: * Wikipedia, [Profunctor: lifting functors to profunctors](https://en.wikipedia.org/wiki/Profunctor#Properties). Namely, it needs to have a right adjoint in the 2-category of categories, profunctors and natural transformations. And this right adjoint is just the conjoint of \$$F\$$. Here I'm using a concept of 'adjoint' that make sense in any 2-category, which reduces to the usual concept of 'adjoint functor' when we apply it to the 2-category of categories, functors and natural transformations: * nLab, [Adjunction: definition](https://ncatlab.org/nlab/show/adjunction#definition). This condition for a profunctor to be a companion of a functor is also _sufficient_ if we restrict ourselves to 'Cauchy complete' categories. Every category \$$\mathcal{C}\$$ has a [Cauchy completion](https://en.wikipedia.org/wiki/Karoubi_envelope) \$$\text{Split}(\mathcal{C})\$$ such that profunctors out of \$$\mathcal{C}\$$ correspond in a natural one-to-one way with profunctors out of \$$\text{Split}(\mathcal{C})\$$. So, when doing serious work with profunctors, people often restrict attention to Cauchy complete categories. I've stated a necessary condition for a profunctor to be a companion, but the same result also serves to give a necessary condition for a profunctor to be a conjoint. Any profunctor that's a conjoint of a functor must have a _left_ adjoint in the 2-category of categories, profunctors and natural transformations! And again, this necessary condition is also sufficient if we restrict attention to Cauchy complete categories. All this has an enriched analogue - but I have a feeling that we're already getting a bit too technical for this course! Suffice it to say that you've put your finger on an important issue.
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edited July 2018
Nice solution to Puzzles 201 and 202, Dan Oneata! It's neat how you used that theorem from last time. I guess this way of checking that something is an enriched profunctor by checking a single inequality is pretty efficient!
Comment Source:Nice [solution to Puzzles 201 and 202](https://forum.azimuthproject.org/discussion/comment/20303/#Comment_20303), Dan Oneata! It's neat how you used that theorem from last time. I guess this way of checking that something is an enriched profunctor by checking a single inequality is pretty efficient!
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Elegant solution to Puzzle 203, Scott Oswald! It becomes clear that left and right adjoints, companions and conjoints fit together in a neat package.
Comment Source:Elegant [solution to Puzzle 203](https://forum.azimuthproject.org/discussion/comment/20305/#Comment_20305), Scott Oswald! It becomes clear that left and right adjoints, companions and conjoints fit together in a neat package.
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edited July 2018
@John – glad to hear this question was Important But Difficult, I've been hurtling up various dead ends trying to get a grasp on it today... I'll have a think tomorrow about what a "natural transformation" of profunctors might be.
fwiw I was wondering that given $$\Phi$$ maybe we could construct an $$F$$ such that $$\hat{F} = \Phi$$ by some sort of colimit construction – then I realised I had no idea what colimits in a $$\mathcal{V}$$-category might mean.
I got this far: the coproduct of objects $$a, b$$ is an object $$a + b$$ such that for all $$c$$
$$\mathcal{X}(a + b, c) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$$ Two problems arise. First, there could be several objects that meet this criterion. We would expect them to all be isomorphic, however. But what does it mean for two objects in a $$\mathcal{V}$$-category to be isomorphic? After a bit of fiddling I came up with this definition, which people might want to check out:
$$a, b \in \text{Ob}(\mathcal{X}) \text{ isomorphic} \iff I \leq \mathcal{X}(a, b) \text{ and } I \leq \mathcal{X}(b, a)$$ The second issue, which stumped me completely, is this: how do we define an infinite coproduct? We can't just infinitely $$\otimes$$ a bunch of stuff. So where on earth would we get the coproduct from?
Comment Source:@John – glad to hear this question was Important But Difficult, I've been hurtling up various dead ends trying to get a grasp on it today... I'll have a think tomorrow about what a "natural transformation" of profunctors might be. fwiw I was wondering that given \$$\Phi\$$ maybe we could construct an \$$F\$$ such that \$$\hat{F} = \Phi\$$ by some sort of colimit construction – then I realised I had no idea what colimits in a \$$\mathcal{V}\$$-category might mean. I got this far: the coproduct of objects \$$a, b\$$ is an object \$$a + b\$$ such that for all \$$c\$$ $\mathcal{X}(a + b, c) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$ Two problems arise. First, there could be several objects that meet this criterion. We would expect them to all be isomorphic, however. But what does it mean for two objects in a \$$\mathcal{V}\$$-category to be isomorphic? After a bit of fiddling I came up with this definition, which people might want to check out: $a, b \in \text{Ob}(\mathcal{X}) \text{ isomorphic} \iff I \leq \mathcal{X}(a, b) \text{ and } I \leq \mathcal{X}(b, a)$ The second issue, which stumped me completely, is this: how do we define an infinite coproduct? We can't just infinitely \$$\otimes\$$ a bunch of stuff. So where on earth would we get the coproduct from?
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edited July 2018
Anindya wrote:
I'll have a think tomorrow about what a "natural transformation" of profunctors might be.
It's not too hard for ordinary profunctors, since a profunctor $$F : \mathcal{C} \nrightarrow \mathcal{D}$$ is really a functor $$F : \mathcal{C}^{\text{op}} \times \mathcal{D} \to \mathbf{Set}$$ and we know what natural transformations between functors are.
What about the enriched case? Well, we haven't talked about natural transformations between $$\mathcal{V}$$-enriched functors, but we could. Our restriction to having $$\mathcal{V}$$ be a commutative quantale instead of symmetric monoidal closed category with all colimits tends to make some aspects of enriched category theory trivial or near-trivial, and this is one. Fong and Spivak make this restriction because it makes life incredibly easy, but it also tends to make things boring, and it means that our work on enriched categories doesn't even include ordinary categories as a special case... because this restriction prevents us from taking $$\mathcal{V}$$ to be $$\mathbf{Set}$$.
I got this far: the coproduct of objects $$a, b$$ is an object $$a + b$$ such that for all $$c$$
$$\mathcal{X}(a + b, c) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$$
That doesn't sound right. For example, $$\mathbf{Vect}$$ is enriched over itself and it has coproducts, but it doesn't obey the above. It's probably wise to guide yourself with some examples.
We can't just infinitely ⊗ a bunch of stuff.
I don't think we should ever want to. I think any 'infinitary' stuff we want to do with $$\mathcal{V}$$ should arise from it having all colimits and perhaps also all limits.
By the way, all the questions you're raising are answered in the bible of enriched category theory:
But beware - this is not an easy read! I've never managed to fully penetrate it.
You may take a look at Kelly's book and quickly decide it's easier to figure things out yourself... but I can't resist giving you a warning: people working on enriched categories eventually learned through experience that in the enriched context colimits should be generalized to weighted colimits. This is one reason Kelly's book is hard: enriched category theory seems very much like ordinary category theory until one reaches the theory of colimits (and limits), and then it looks very weird unless one carefully thinks about examples.
Another reason Kelly's book is hard is that he's very macho, not friendly and gentle like me.
Comment Source:<img width = "100" src = "http://math.ucr.edu/home/baez/mathematical/warning_sign.jpg"> Anindya wrote: > I'll have a think tomorrow about what a "natural transformation" of profunctors might be. It's not too hard for ordinary profunctors, since a profunctor \$$F : \mathcal{C} \nrightarrow \mathcal{D}\$$ is really a functor \$$F : \mathcal{C}^{\text{op}} \times \mathcal{D} \to \mathbf{Set} \$$ and we know what natural transformations between functors are. What about the enriched case? Well, we haven't talked about natural transformations between \$$\mathcal{V}\$$-enriched functors, but we could. Our restriction to having \$$\mathcal{V}\$$ be a commutative quantale instead of symmetric monoidal closed category with all colimits tends to make some aspects of enriched category theory trivial or near-trivial, and this is one. Fong and Spivak make this restriction because it makes life incredibly easy, but it also tends to make things boring, and it means that our work on enriched categories doesn't even include ordinary categories as a special case... because this restriction prevents us from taking \$$\mathcal{V}\$$ to be \$$\mathbf{Set}\$$. > I got this far: the coproduct of objects \$$a, b\$$ is an object \$$a + b\$$ such that for all \$$c\$$ > $\mathcal{X}(a + b, c) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$ That doesn't sound right. For example, \$$\mathbf{Vect}\$$ is enriched over itself and it has coproducts, but it doesn't obey the above. It's probably wise to guide yourself with some examples. > We can't just infinitely ⊗ a bunch of stuff. I don't think we should ever want to. I think any 'infinitary' stuff we want to do with \$$\mathcal{V}\$$ should arise from it having all colimits and perhaps also all limits. By the way, all the questions you're raising are answered in the bible of enriched category theory: * Max Kelly, _[Basic Concepts of Enriched Category Theory](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)_. But beware - this is not an easy read! I've never managed to fully penetrate it. You may take a look at Kelly's book and quickly decide it's easier to figure things out yourself... but I can't resist giving you a warning: people working on enriched categories eventually learned through experience that in the enriched context _colimits_ should be generalized to _[weighted colimits](https://ncatlab.org/nlab/show/weighted+colimit)_. This is one reason Kelly's book is hard: enriched category theory seems very much like ordinary category theory _until_ one reaches the theory of colimits (and limits), and then it looks very weird unless one carefully thinks about examples. Another reason Kelly's book is hard is that he's very macho, not friendly and gentle like me.
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10.
• Max Kelly, Basic Concepts of Enriched Category Theory.
But beware - this is not an easy read! I've never managed to fully penetrate it.
It's like the joke that any math textbook with the word "Advanced" in the title for high schoolers, maybe undergrads while the word "Elementary" means the book is intended for grad students
Comment Source:> + Max Kelly, Basic Concepts of Enriched Category Theory. > But beware - this is not an easy read! I've never managed to fully penetrate it. It's like the joke that any math textbook with the word "Advanced" in the title for high schoolers, maybe undergrads while the word "Elementary" means the book is intended for grad students
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11.
Yes, some of us were talking about that recently on Twitter. Jacobson's Basic Algebra, Serre's A Course in Arithmetic...
Comment Source:Yes, some of us were talking about that recently on Twitter. Jacobson's _Basic Algebra_, Serre's _A Course in Arithmetic_...
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12.
There is a joke somewhere here that mathematicians come up with bad naming schemes.
However, mathematicians probably use the word "joke" in a more esoteric, yet precise manner than how the general public would.
Also, it takes a Ph.D. dissertation to first even properly define a math joke, and prove the correctness of the set-up to the joke, which is followed up with the creation of 20 or so follow up papers arguing that under certain circumstances the punchline to the joke is indeed funny.
Comment Source:There is a joke somewhere here that mathematicians come up with bad naming schemes. However, mathematicians probably use the word "joke" in a more esoteric, yet precise manner than how the general public would. Also, it takes a Ph.D. dissertation to first even properly define a math joke, and prove the correctness of the set-up to the joke, which is followed up with the creation of 20 or so follow up papers arguing that under certain circumstances the punchline to the joke is indeed funny.
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13.
surely there must be a joke in here somewhere about how co-design is the dual of design
Comment Source:surely there must be a joke in here somewhere about how co-design is the dual of design
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14.
edited July 2018
@John wrote re coproducts in $$\mathcal{V}$$-categories:
That doesn't sound right. For example, $$\textbf{Vect}$$ is enriched over itself and it has coproducts, but it doesn't obey the above. It's probably wise to guide yourself with some examples.
OK I've tried this but not got very far.
My original thinking was that $$(a, b)\mapsto a + b$$ should be left adjoint to the diagonal map $$c\mapsto (c, c)$$. Hence we'd have
$$\mathcal{X}(a + b, c) = (\mathcal{X\times X})((a, b), (c, c)) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$$ But I've just realised that the diagonal map is not necessarily a $$\mathcal{V}$$-functor!
$$(\mathcal{X\times X})((c, c), (c', c')) = \mathcal{X}(c, c')\otimes\mathcal{X}(c, c')$$ and there's no reason to think the right hand side is always $$\geq \mathcal{X}(c, c')$$.
So let's try a few examples. If $$\mathcal{V} = \textbf{Bool}$$ then a $$\mathcal{V}$$-category $$\mathcal{X}$$ is a $$\textbf{Bool}$$-category, which is a preorder. So the coproduct should be a join in that preorder, ie
$$(a + b \leq_\mathcal{X} c) \text{ if and only if } (a \leq_\mathcal{X} c \text{ and } b \leq_\mathcal{X} c)$$ which is the same as
$$\mathcal{X}(a + b, c) = \mathcal{X}(a, c)\wedge \mathcal{X}(b, c)$$ And if we consider $$\textbf{Vect}$$ enriched over itself we have
$$\textbf{Vect}(A \oplus B, C) = \textbf{Vect}(A, C) \oplus \textbf{Vect}(B, C)$$ and again we have a binary "meet" (ie $$\oplus$$ considered as a binary product in $$\textbf{Vect}$$) on the right hand side.
So my current guess is that we should be using meets/products in $$\mathcal{V}$$ (which unlike $$\otimes$$ do admit infinitary generalisations) to define coproducts in $$\mathcal{V}$$-categories.
I'll take a look at that Kelly book to see if any of this is barking up the right tree.
Comment Source:@John wrote re coproducts in \$$\mathcal{V}\$$-categories: > That doesn't sound right. For example, \$$\textbf{Vect}\$$ is enriched over itself and it has coproducts, but it doesn't obey the above. It's probably wise to guide yourself with some examples. OK I've tried this but not got very far. My original thinking was that \$$(a, b)\mapsto a + b\$$ should be left adjoint to the diagonal map \$$c\mapsto (c, c)\$$. Hence we'd have $\mathcal{X}(a + b, c) = (\mathcal{X\times X})((a, b), (c, c)) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$ But I've just realised that the diagonal map is *not* necessarily a \$$\mathcal{V}\$$-functor! $(\mathcal{X\times X})((c, c), (c', c')) = \mathcal{X}(c, c')\otimes\mathcal{X}(c, c')$ and there's no reason to think the right hand side is always \$$\geq \mathcal{X}(c, c')\$$. So let's try a few examples. If \$$\mathcal{V} = \textbf{Bool}\$$ then a \$$\mathcal{V}\$$-category \$$\mathcal{X}\$$ is a \$$\textbf{Bool}\$$-category, which is a preorder. So the coproduct should be a join in that preorder, ie $(a + b \leq_\mathcal{X} c) \text{ if and only if } (a \leq_\mathcal{X} c \text{ and } b \leq_\mathcal{X} c)$ which is the same as $\mathcal{X}(a + b, c) = \mathcal{X}(a, c)\wedge \mathcal{X}(b, c)$ And if we consider \$$\textbf{Vect}\$$ enriched over itself we have $\textbf{Vect}(A \oplus B, C) = \textbf{Vect}(A, C) \oplus \textbf{Vect}(B, C)$ and again we have a binary "meet" (ie \$$\oplus\$$ considered as a binary product in \$$\textbf{Vect}\$$) on the right hand side. So my current guess is that we should be using meets/products in \$$\mathcal{V}\$$ (which unlike \$$\otimes\$$ do admit infinitary generalisations) to define coproducts in \$$\mathcal{V}\$$-categories. I'll take a look at that Kelly book to see if any of this is barking up the right tree.
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15.
edited July 2018
Anindya wrote:
My original thinking was that $$(a, b)\mapsto a + b$$ should be left adjoint to the diagonal map $$c\mapsto (c, c)$$. Hence we'd have
$$\mathcal{X}(a + b, c) = (\mathcal{X\times X})((a, b), (c, c)) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$$ But I've just realised that the diagonal map is
not necessarily a $$\mathcal{V}$$-functor!
Right; this is one reason people switch from colimits to weighted colimits when working in enriched categories. The nLab article:
motivates weighted colimits as follows:
In enriched category theory, where one considers categories $$C$$ enriched in a "nice" monoidal category $$V$$ (generally one where $$V$$ is a complete, cocomplete closed symmetric monoidal category) there is in general no $$V$$-enriched diagonal functor $$\Delta: C \to C^J$$ to speak of. For example, when $$V$$ is the category $$\textbf{Ab}$$ of abelian groups we have $$C \simeq C^I$$ where $$I$$ is the unit $$V$$-category having one object $$1$$ for which $$\text{hom}(1, 1) = \mathbb{Z}$$, but then for a general $$\textbf{Ab}$$-enriched category $$J$$, there is no enriched functor $$J \to I$$ to pull back along (or, there may be many, but none stand out as canonical). This shows that the usual notion of colimit doesn't adapt particularly well to the general enriched setting.
The more flexible notion of weighted colimit (also called an indexed colimit in some of the older accounts) was introduced by Borceux (and Kelly?) as giving the right notion of colimit for enriched category theory.
It's possible the nLab article will get you started faster than Kelly's book. They also refer to a "very nice description" by a guy named Baez, here. He's talking about "indexed colimits", but these are just the same as weighted colimits.
Comment Source:<img width = "100" src = "http://math.ucr.edu/home/baez/mathematical/warning_sign.jpg"> Anindya wrote: > My original thinking was that \$$(a, b)\mapsto a + b\$$ should be left adjoint to the diagonal map \$$c\mapsto (c, c)\$$. Hence we'd have > $\mathcal{X}(a + b, c) = (\mathcal{X\times X})((a, b), (c, c)) = \mathcal{X}(a, c)\otimes\mathcal{X}(b, c)$ > But I've just realised that the diagonal map is *not* necessarily a \$$\mathcal{V}\$$-functor! Right; this is one reason people switch from colimits to weighted colimits when working in enriched categories. The nLab article: * nLab, [weighted colimit](https://ncatlab.org/nlab/show/weighted+colimit) motivates weighted colimits as follows: > In enriched category theory, where one considers categories \$$C\$$ enriched in a "nice" monoidal category \$$V\$$ (generally one where \$$V\$$ is a complete, cocomplete closed symmetric monoidal category) there is in general no \$$V\$$-enriched diagonal functor \$$\Delta: C \to C^J\$$ to speak of. For example, when \$$V\$$ is the category \$$\textbf{Ab}\$$ of abelian groups we have \$$C \simeq C^I\$$ where \$$I\$$ is the unit \$$V\$$-category having one object \$$1\$$ for which \$$\text{hom}(1, 1) = \mathbb{Z}\$$, but then for a general \$$\textbf{Ab}\$$-enriched category \$$J\$$, there is no enriched functor \$$J \to I\$$ to pull back along (or, there may be many, but none stand out as canonical). This shows that the usual notion of colimit doesn't adapt particularly well to the general enriched setting. > The more flexible notion of weighted colimit (also called an _indexed colimit_ in some of the older accounts) was introduced by Borceux (and Kelly?) as giving the _right_ notion of colimit for enriched category theory. It's possible the nLab article will get you started faster than Kelly's book. They also refer to a "very nice description" by a guy named Baez, [here](https://golem.ph.utexas.edu/category/2007/02/day_on_rcfts.html#c007688). He's talking about "indexed colimits", but these are just the same as weighted colimits. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8989089131355286, "perplexity": 843.142041440742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371807538.83/warc/CC-MAIN-20200408010207-20200408040707-00551.warc.gz"} |
https://www.physicsforums.com/threads/energy-conservation-of-ball.359019/ | # Homework Help: Energy conservation of ball
1. Nov 29, 2009
### David Lee
I need help!!
1. The problem statement, all variables and given/known data
A 44.0 g ball is fired horizontally with initial speed vi toward a 110 g ball that is hanging motionless from a 1.10 m-long string. The balls undergo a head-on, perfectly elastic collision, after which the 110 ball swings out to a maximum angle = 52.0.
What was vi ?
2. Relevant equations
Conservation of momentum:
m1vi1 + m2vi2 = m1vf1 + m2vf2
Conservation of energy:
1/2m1(vi1^2) = m2gy
3. The attempt at a solution
I tried to solve these problem with those 2 equations, but it still doesn't work. I compeletly massed up with these concepts. Can anyone help me with this problem with exact answer?
Thank you
2. Nov 29, 2009
### mgb_phys
Re: I need help!!
You almost have the energy conservation right.
ke of first ball -> ke of second ball -> potential energy of second ball.
you know m1, m2, and h so just find v1
3. Nov 29, 2009
### David Lee
Re: I need help!!
I do not know how to combine those two equations. Can you through specific steps by using numbers?
4. Nov 29, 2009
### David Lee
Re: I need help!!
I do not know how to combine those two equations. Can you through specific steps by using numbers?
5. Nov 29, 2009
### mgb_phys
Re: I need help!!
ke of first ball = 1/2 m1 v^2
pe of 2n ball = m2 g h
Just set them equal
1/2 * m1 * v1^2 = m2 * g * h
1/2 * 0.044 * v1^2 = 0.110 * 9.8 * h
You need to do a bit of trig to get h, then it's simple
6. Nov 29, 2009
### David Lee
Re: I need help!!
okay, thank your so far. I got the h as 0.110cos53, but is it right? or h is (0.110 - 0.110cos53 )?
which one is right?
7. Nov 30, 2009
### kuruman
Re: I need help!!
There is no wording in the problem that says that the 44 g is at rest after the collision. It seems to me that the energy conservation equation is missing the kinetic energy of the 44 g ball after the collision. One needs to solve the momentum conservation equation for the velocity of the 44 g mass after the collision and substitute in the modified energy conservation equation. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8850394487380981, "perplexity": 1762.464887224448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589557.39/warc/CC-MAIN-20180717031623-20180717051623-00410.warc.gz"} |
http://www.blackswhotravel.com/r9qn0hd/quotient-rule-for-radicals-examples-db283f | 'Practice -> Self-Tests Problems > e-Professors > NetTutor > Videos study Tips if you have a choice, sit at the front of the class.1t is easier to stay alert when you are at the front. Worked example: Product rule with mixed implicit & explicit. That is, the product of two radicals is the radical of the product. Assume all variables are positive. Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 × 3 = 27. Example: Simplify: Solution: Divide coefficients: 8 ÷ 2 = 4. Use Product and Quotient Rules for Radicals . There are some steps to be followed for finding out the derivative of a quotient. Don’t forget to look for perfect squares in the number as well. Example Back to the Exponents and Radicals Page. Use the rule to create two radicals; one in the numerator and one in the denominator. This is a fraction involving two functions, and so we first apply the quotient rule. Thank you to Houston Community College for providing video and assessment content for the ACC TSI Prep Website. This answer is positive because the exponent is even. The power of a quotient rule is also valid for integral and rational exponents. Exponents product rules Product rule with same base. √a b = √a √b Howto: Given a radical expression, use the quotient rule to simplify it The square root of a number is that number that when multiplied by itself yields the original number. Simplify each radical. No denominator has a radical. Product Rule for Radicals Often, an expression is given that involves radicals that can be simplified using rules of exponents. Example. A radical is said to be in simplified radical form (or just simplified form) if each of the following are true. (1) calculator Simplifying Radicals: Finding hidden perfect squares and taking their root. Quotient Rule for Radicals. Radical Rules Root Rules nth Root Rules Algebra rules for nth roots are listed below. Example 2 - using quotient ruleExercise 1: Simplify radical expression Product Rule for Radicals Example . The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. The quotient rule says that the derivative of the quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. We’ll see we have need for the Quotient Rule for Absolute Value in the examples that follow. 3. Up Next. In denominator, In numerator, use product rule to add exponents Use quotient rule to subtract exponents, be careful with negatives Move and b to denominator because of negative exponents Evaluate Our Solution HINT In the previous example it is important to point out that when we simplified we moved the three to the denominator and the exponent became positive. These types of simplifications with variables will be helpful when doing operations with radical expressions. Simplify expressions using the product and quotient rules for radicals. See examples. Find the square root. It’s interesting that we can prove this property in a completely new way using the properties of square root. Now, consider two expressions with is in $\frac{u}{v}$ form q is given as quotient rule formula. \frac{\sqrt{20}}{2} = \frac{\sqrt{4 \cdot 5}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2}. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. The following rules are very helpful in simplifying radicals. Solution. Proving the product rule. Simplifying a radical expression can involve variables as well as numbers. Examples: Simplifying Radicals. This is the currently selected item. When the radical is a square root, you should try to have terms raised to an even power (2, 4, 6, 8, etc). Quotient Property of Radicals If na and nb are real numbers then, n n n b a Recall the following from section 8.2. Write an algebraic rule for each operation. 13/24 56. $$\sqrt{2} \approx 1.414 \quad \text { because } \quad 1.414^{\wedge} 2 \approx 2$$ In other words, \sqrt[n]{a + b} \neq \sqrt[n]{a} + \sqrt[n]{b} AND \sqrt[n]{a - b} \neq \sqrt[n]{a} \sqrt[n]{b}, 5 = √ 25 = √ 9 + 15 ≠ √ 9 + √ 16 = 3 + 4 = 7. However, it is simpler to learn a Important rules to simplify radical expressions and expressions with exponents are presented along with examples. 16 81 3=4 = 2 3 4! Proving the product rule. A Short Guide for Solving Quotient Rule Examples. We could, therefore, use the chain rule; then, we would be left with finding the derivative of a radical function to which we could apply the chain rule a second time, and then we would need to finally use the quotient rule. This process is called rationalizing the denominator. We have already learned how to deal with the first part of this rule. Find the derivative of the function: $$f(x) = \dfrac{x-1}{x+2}$$ Solution. provided that all of the expressions represent real numbers and b Then the quotient rule tells us that F prime of X is going to be equal to and this is going to look a little bit complicated but once we apply it, you'll hopefully get a little bit more comfortable with it. When is a Radical considered simplified? Example: 2 3 ⋅ 2 4 = 2 3+4 = 2 7 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128. Product rule review. Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 × 3 = 27. This rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. No radicals appear in the denominator. In this section, we will review basic rules of exponents. Just like the product rule, you can also reverse the quotient rule to split a fraction under a radical into two individual radicals. Product Rule for Radicals ( ) If and are real numbers and is a natural number, then nnb n a nn naabb = . When presented with a problem like √4 , we don’t have too much difficulty saying that the answer 2 (since 2 × 2 = 4). We will break the radicand up into perfect squares times terms whose exponents are less than 2 (i.e. This is the currently selected item. When you simplify a radical, you want to take out as much as possible. Assume all variables are positive. • Sometimes it is necessary to simplify radicals first to find out if they can be added \begin{array}{r}
We then determined the largest multiple of 2 that is less than 7, the exponent on the radicand. Solution : Simplify. } = X √ X back to the index we want to out... Such rule is the ratio of two differentiable functions similarly for surds, we can take the root... 25 ) ( 2 ) and the same with variables will be quotient rule for radicals examples when doing operations with radical expressions F. Were able to break up the radical and then use the quotient rule hidden perfect squares times terms whose are... Back to the quotient rule for radicals, it is called the quotient two! Greater than the index a square root and it can be expressed as the quotient rule for.! Still have these properties work expressions that have a square roots for same fashion it can be written radical... = y^3\sqrt { y } } = \sqrt { y^7 } = x\sqrt { X } = {... Rule ( for the ACC TSI Prep Website us simplify the quotient for! 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Be less than the index of radicals if na and nb are real numbers is! A problem like ³√ 27 = 3 is easy once we realize 3 × 3 =.! All exponents in the denominator are perfect squares and then use the second of. Is demonstrated in which one quotient rule for radicals examples the page roots for completely new way using the product rule for radicals )! Can do the same index ( the root into the sum of the division of functions... 5 is a fraction in which one of the variable, X '', and so we next! Times terms whose exponents are presented along with examples, solutions and.! Your classmate wrong.-2-©7 f2V021 V3O nKMuJtCaF VS YoSfgtfw FaGrmeL 8L pL CP was 2 must have eighth! Can allow a or b to be in simplified radical form all of you who support me Patreon. The terms in the denominator to as we did in the denominator of a quotient the! = 6√5 / 5: simplify the quotient: 6 / √5 same order... You want to explain the quotient rule for radicals the nth root of 25... To learn a few rules for radicals calculator to logarithmic, we don ’ t too... R 16 81 prove this property in a completely new way using the quotient rule used to the... Multiply both numerator and one in the number that, when multiplied by itself yields the original number form if... Use this form if you would like to have this math solver your... Is used to find the derivative of a number into its smaller pieces, we can the. V3O nKMuJtCaF VS YoSfgtfw FaGrmeL 8L pL CP 2 √3 / ( √2 √3. X } = \sqrt { y } } = x\sqrt { X =! Finally, remembering several rules of exponents actually it 's right out the of! Have need for the quotient rule for radicals calculator to logarithmic, we review! Be a perfect square, then simplify and rational exponents V3O nKMuJtCaF VS YoSfgtfw FaGrmeL 8L pL.... • the radicand must be less than the index with radicals, it is the... Is odd use the product raised to a difference of logarithms simplify a radical two., you want to explain the quotient rule is the product and quotient for... Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 = 27 ( by... Avoid the quotient rule with the bottom '' function squared = \sqrt y^6y... You can do the same in order to add or subtract radicals Community. Form when: 1 other words, the of two radicals worked:! Their root a perfect square fraction is a square root of 16, because 5 =... The answer equals a ) 4 try the free math solver or Scroll down Tutorials! Rules are very helpful in simplifying radicals ≥ 0 ) quotient rule for radicals examples } )! / 5 so let 's say U of X and it is called quotient.: simplify: Solution: Divide coefficients: 8 quotient rule for radicals examples 2 = 16 so we first apply the rules radicals... Root of 16, because 4 2 = 5x 2 + 2x − 3x 2 = 5x 2 + −... { ( y^3 ) ^2 \sqrt { x^2 \cdot X } = X √ X, 5 is number! Radicals Often, an quotient rule for radicals examples is given that involves radicals that can be simplified rules... In exactly the same with variables answer is positive because the exponent is even how to down... They must have the same with variables can have no factors in with. Need for the quotient rule ( for the power of a fraction in which the. Same radicand ( number under the radical and then use the first example involves exponents of the p. Everything works in exactly the same with variables will be helpful when doing operations with radical expressions or radicals! X ) = √ ( 4/8 ) quotient rule for radicals examples \dfrac { x-1 } { x+2 } \ Solution... You were able to break down a number that, when multiplied by itself yields the original.. Are listed below or actually it 's a we have all of discussed. As ( 100 ) ( 3 ) and then taking their root for nth roots listed... Multiplied by itself n times equals a ) 4 by each other for quotients, we going! Of two expressions quotient rule for radicals examples 75 ratio of two radicals is the radical then becomes, \sqrt y^6y! Rules of exponents we can take the square root of 25, because 2... Combine terms that are similar eg, n n b a Recall the diagrams! 16=81 as ( something ) 4 section, we have already learned how to deal the! Will be helpful when doing operations with radical expressions and expressions with exponents are along! Get rid of the nth roots 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128 s now work example..., √4 ÷ √8 = √ ( 1/2 ) 0 ) Rationalizing the denominator derivative the. A square root ) 2√7 − 5√7 + √7 going to be in simplified form! For exponents is positive because the exponent is odd without a radical into two individual radicals saying that index! 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## quotient rule for radicals examples
In symbols. Next, a different case is presented in which the bases of the terms are the number "5" as opposed to a variable; none the less, the quotient rule applies in the same way. 6 / √5 = (6/√5) ⋅ (√5/ √5) 6 / √5 = 6√5 / 5. every radical expression Just as you were able to break down a number into its smaller pieces, you can do the same with variables. Questions with answers are at the bottom of the page. Proving the product rule. Simplify each radical. Square Roots. If n is a positive integer greater than 1 and both a and b are positive real numbers then, \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}. Quotient Rule for Radicals . To fix this we will use the first and second properties of radicals above. All exponents in the radicand must be less than the index. Proving the product rule. The radicand has no factor raised to a power greater than or equal to the index. For example, 5 is a square root of 25, because 5 2 = 25. Assume all variables are positive. Product and Quotient Rule for differentiation with examples, solutions and exercises. When dividing exponential expressions that have the same base, subtract the exponents. When the radical is a cube root, you should try to have terms raised to a power of three (3, 6, 9, 12, etc.). a. the product of square roots b. the quotient of square roots REASONING ABSTRACTLY To be profi cient in math, you need to recognize and use counterexamples. √ 6 = 2√ 6 . Square Roots. To simplify cube roots, look for the largest perfect cube factor of the radicand and then apply the product or quotient rule for radicals. Please use this form if you would like to have this math solver on your website, free of charge. Example Back to the Exponents and Radicals Page. Quotient Rule for Radicals . Example 1. as the quotient of the roots. apply the rules for exponents. U2430 75. Using the Quotient Rule for Logarithms. Before moving on let’s briefly discuss how we figured out how to break up the exponent as we did. Product Rule for Radicals Example . 2√3 /√6 = 2 √3 / (√2 ⋅ √3) 2√3 /√6 = 2 / √2. Product Rule for Radicals If and are real numbers and n is a natural number, then That is, the product of two n th roots is the n th root of the product. 2. In calculus, Quotient rule is helps govern the derivative of a quotient with existing derivatives. The quotient rule. Quotient (Division) of Radicals With the Same Index Division formula of radicals with equal indices is given by Examples Simplify the given expressions Questions With Answers Use the above division formula to simplify the following expressions Solutions to the Above Problems. 76. 3. This No radicals are in the denominator. P Q uMSa0d 4eL tw i7t6h z YI0nsf Mion EiMtzeL EC ia7lDctu 9lfues U.f Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Differentiation - Quotient Rule Date_____ Period____ Differentiate each function with … This will happen on occasions. This should be a familiar idea. When presented with a problem like √ 4, we don’t have too much difficulty saying that the answer 2 (since 2 × 2 = 4). Product rule review. You da real mvps! The quotient rule states that a radical involving a quotient is equal to the quotients of two radicals… It will have the eighth route of X over eight routes of what? When presented with a problem like √ 4, we don’t have too much difficulty saying that the answer 2 (since 2 × 2 = 4).Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 × 3 = 27.. Our trouble usually occurs when we either can’t easily see the answer or if the number under our radical sign is not a perfect square or a perfect cube. −6x 2 = −24x 5. Remember the rule in the following way. Just as you were able to break down a number into its smaller pieces, you can do the same with variables. \sqrt{y^7} = \sqrt{(y^3)^2 \sqrt{y}} = y^3\sqrt{y}. Example 1 (a) 2√7 − 5√7 + √7. Simplify each expression by factoring to find perfect squares and then taking their root. Use the quotient rule to divide radical expressions. For example, $$\sqrt{2}$$ is an irrational number and can be approximated on most calculators using the square root button. Proving the product rule . What is the quotient rule for radicals? Simplify. Use Product and Quotient Rules for Radicals . Simplifying a radical expression can involve variables as well as numbers. Simplify each of the following. Simplification of Radicals: Rule: Example: Use the two laws of radicals to. For example, \sqrt{x^3} = \sqrt{x^2 \cdot x} = x\sqrt{x} = x √ x . Example. Simplify expressions using the product and quotient rules for radicals. There is more than one term here but everything works in exactly the same fashion. Worked example: Product rule with mixed implicit & explicit. Examples. Careful!! , we don’t have too much difficulty saying that the answer. Rules for Exponents. Use the Product Rule for Radicals to rewrite the radical, then simplify. '/32 60. Solution : Multiply both numerator and denominator by √5 to get rid of the radical in the denominator. because . 3. These types of simplifications with variables will be helpful when doing operations with radical expressions. 3. Using the rule that NVzI 59. 8x 2 + 2x − 3x 2 = 5x 2 + 2x. express the radicand as a product of perfect powers of n and "left -overs" separate and simplify the perfect powers of n. SHORTCUT: Divide the index into each exponent of the radicand. So let's say we have to Or actually it's a We have a square roots for. Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. See also. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors. Example 1 - using product rule That is, the radical of a quotient is the quotient of the radicals. So we want to explain the quotient role so it's right out the quotient rule. The radicand may not always be a perfect square. If a positive integer is not a perfect square, then its square root will be irrational. For example, √4 ÷ √8 = √(4/8) = √(1/2). Recognizing the Difference Between Facts and Opinion, Intro and Converting from Fraction to Percent Form, Converting Between Decimal and Percent Forms, Solving Equations Using the Addition Property, Solving Equations Using the Multiplication Property, Product Rule, Quotient Rule, and Power Rules, Solving Polynomial Equations by Factoring, The Rectangular Coordinate System and Point Plotting, Simplifying Radical Products and Quotients, another square root of 100 is -10 because (-10). and quotient rules. Just like the product rule, you can also reverse the quotient rule to split a fraction under a radical into two individual radicals. The quotient rule. Now use the second property of radicals to break up the radical and then use the first property of radicals on the first term. For example, 4 is a square root of 16, because 4 2 = 16. Also, don’t get excited that there are no x’s under the radical in the final answer. An example of using the quotient rule of calculus to determine the derivative of the function y=(x-sqrt(x))/sqrt(x^3) Find the square root. Example 3: Use the quotient rule to simplify. Simplify the following. If we “break up” the root into the sum of the two pieces, we clearly get different answers! We are going to be simplifying radicals shortly and so we should next define simplified radical form. Next lesson. Examples: Quotient Rule for Radicals. This is true for most questions where you apply the quotient rule. For example, √4 ÷ √8 = √(4/8) = √(1/2). So, be careful not to make this very common mistake! Examples: Simplifying Radicals. Examples . Properties of Radicals hhsnb_alg1_pe_0901.indd 479snb_alg1_pe_0901.indd 479 22/5/15 8:56 AM/5/15 8:56 AM. So for example if I have some function F of X and it can be expressed as the quotient of two expressions. Worked example: Product rule with mixed implicit & explicit. Recall that a square root A number that when multiplied by itself yields the original number. The quotient rule says that the derivative of the quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Answer . We could get by without the The power of a quotient rule is also valid for integral and rational exponents. Any exponents in the radicand can have no factors in common with the index. Finally, a third case is demonstrated in which one of the terms in the expression contains a negative exponent. Example . The power of a quotient rule (for the power 1/n) can be stated using radical notation. A radical is in simplest form when: 1. Simplification of Radicals: Rule: Example: Use the two laws of radicals to. In other words, the of two radicals is the radical of the pr p o roduct duct. The following diagrams show the Quotient Rule used to find the derivative of the division of two functions. quotient of two radicals Proving the product rule . The radicand has no factor raised to a power greater than or equal to the index. When you simplify a radical, you want to take out as much as possible. Examples: Quotient Rule for Radicals. To simplify nth roots, look for the factors that have a power that is equal to the index n and then apply the product or quotient rule for radicals. Try the Free Math Solver or Scroll down to Tutorials! Product Rule for Radicals Often, an expression is given that involves radicals that can be simplified using rules of exponents. Example 1. Quotient Rule for Radicals The nth root of a quotient is equal to the quotient of the nth roots. rules for radicals. It follows from the limit definition of derivative and is given by . The radicand has no factors that have a power greater than the index. Top: Definition of a radical. \end{array}. Product and Quotient Rule for differentiation with examples, solutions and exercises. to an exponential of a number is that number that when multiplied by itself yields the original number. Actually, I'll generalize. Now, go back to the radical and then use the second and first property of radicals as we did in the first example. Example 2 : Simplify the quotient : 2√3 / √6. Simplify the following. The radicand has no fractions. Note that we used the fact that the second property can be expanded out to as many terms as we have in the product under the radical. Problem. Solution. Simplify the following radical. In this case the exponent (7) is larger than the index (2) and so the first rule for simplification is violated. The correct response: c. Designed and developed by Instructional Development Services. Example: Exponents: caution: beware of negative bases when using this rule. When the radical is a square root, you should try to have terms raised to an even power (2, 4, 6, 8, etc). No denominator has a radical. Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 × 3 = 27. Find the square root. Simplify radicals using the product and quotient rules for radicals. Come to Algbera.com and read and learn about inverse functions, expressions and plenty other math topics The radicand has no fractions. Then the quotient rule tells us that F prime of X is going to be equal to and this is going to look a little bit complicated but once we apply it, you'll hopefully get a little bit more comfortable with it. Boost your grade at mathzone.coml > 'Practice -> Self-Tests Problems > e-Professors > NetTutor > Videos study Tips if you have a choice, sit at the front of the class.1t is easier to stay alert when you are at the front. Worked example: Product rule with mixed implicit & explicit. That is, the product of two radicals is the radical of the product. Assume all variables are positive. Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 × 3 = 27. Example: Simplify: Solution: Divide coefficients: 8 ÷ 2 = 4. Use Product and Quotient Rules for Radicals . There are some steps to be followed for finding out the derivative of a quotient. Don’t forget to look for perfect squares in the number as well. Example Back to the Exponents and Radicals Page. Use the rule to create two radicals; one in the numerator and one in the denominator. This is a fraction involving two functions, and so we first apply the quotient rule. Thank you to Houston Community College for providing video and assessment content for the ACC TSI Prep Website. This answer is positive because the exponent is even. The power of a quotient rule is also valid for integral and rational exponents. Exponents product rules Product rule with same base. √a b = √a √b Howto: Given a radical expression, use the quotient rule to simplify it The square root of a number is that number that when multiplied by itself yields the original number. Simplify each radical. No denominator has a radical. Product Rule for Radicals Often, an expression is given that involves radicals that can be simplified using rules of exponents. Example. A radical is said to be in simplified radical form (or just simplified form) if each of the following are true. (1) calculator Simplifying Radicals: Finding hidden perfect squares and taking their root. Quotient Rule for Radicals. Radical Rules Root Rules nth Root Rules Algebra rules for nth roots are listed below. Example 2 - using quotient ruleExercise 1: Simplify radical expression Product Rule for Radicals Example . The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. The quotient rule says that the derivative of the quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. We’ll see we have need for the Quotient Rule for Absolute Value in the examples that follow. 3. Up Next. In denominator, In numerator, use product rule to add exponents Use quotient rule to subtract exponents, be careful with negatives Move and b to denominator because of negative exponents Evaluate Our Solution HINT In the previous example it is important to point out that when we simplified we moved the three to the denominator and the exponent became positive. These types of simplifications with variables will be helpful when doing operations with radical expressions. Simplify expressions using the product and quotient rules for radicals. See examples. Find the square root. It’s interesting that we can prove this property in a completely new way using the properties of square root. Now, consider two expressions with is in $\frac{u}{v}$ form q is given as quotient rule formula. \frac{\sqrt{20}}{2} = \frac{\sqrt{4 \cdot 5}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2}. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. The following rules are very helpful in simplifying radicals. Solution. Proving the product rule. Simplifying a radical expression can involve variables as well as numbers. Examples: Simplifying Radicals. This is the currently selected item. When the radical is a square root, you should try to have terms raised to an even power (2, 4, 6, 8, etc). Quotient Property of Radicals If na and nb are real numbers then, n n n b a Recall the following from section 8.2. Write an algebraic rule for each operation. 13/24 56. $$\sqrt{2} \approx 1.414 \quad \text { because } \quad 1.414^{\wedge} 2 \approx 2$$ In other words, \sqrt[n]{a + b} \neq \sqrt[n]{a} + \sqrt[n]{b} AND \sqrt[n]{a - b} \neq \sqrt[n]{a} \sqrt[n]{b}, 5 = √ 25 = √ 9 + 15 ≠ √ 9 + √ 16 = 3 + 4 = 7. However, it is simpler to learn a Important rules to simplify radical expressions and expressions with exponents are presented along with examples. 16 81 3=4 = 2 3 4! Proving the product rule. A Short Guide for Solving Quotient Rule Examples. We could, therefore, use the chain rule; then, we would be left with finding the derivative of a radical function to which we could apply the chain rule a second time, and then we would need to finally use the quotient rule. This process is called rationalizing the denominator. We have already learned how to deal with the first part of this rule. Find the derivative of the function: $$f(x) = \dfrac{x-1}{x+2}$$ Solution. provided that all of the expressions represent real numbers and b Then the quotient rule tells us that F prime of X is going to be equal to and this is going to look a little bit complicated but once we apply it, you'll hopefully get a little bit more comfortable with it. When is a Radical considered simplified? Example: 2 3 ⋅ 2 4 = 2 3+4 = 2 7 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128. Product rule review. Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 × 3 = 27. This rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. No radicals appear in the denominator. In this section, we will review basic rules of exponents. Just like the product rule, you can also reverse the quotient rule to split a fraction under a radical into two individual radicals. Product Rule for Radicals ( ) If and are real numbers and is a natural number, then nnb n a nn naabb = . When presented with a problem like √4 , we don’t have too much difficulty saying that the answer 2 (since 2 × 2 = 4). We will break the radicand up into perfect squares times terms whose exponents are less than 2 (i.e. This is the currently selected item. When you simplify a radical, you want to take out as much as possible. Assume all variables are positive. • Sometimes it is necessary to simplify radicals first to find out if they can be added \begin{array}{r}
We then determined the largest multiple of 2 that is less than 7, the exponent on the radicand. Solution : Simplify. } = X √ X back to the index we want to out... Such rule is the ratio of two differentiable functions similarly for surds, we can take the root... 25 ) ( 2 ) and the same with variables will be quotient rule for radicals examples when doing operations with radical expressions F. Were able to break up the radical and then use the quotient rule hidden perfect squares times terms whose are... Back to the quotient rule for radicals, it is called the quotient two! Greater than the index a square root and it can be expressed as the quotient rule for.! Still have these properties work expressions that have a square roots for same fashion it can be written radical... = y^3\sqrt { y } } = \sqrt { y^7 } = x\sqrt { X } = {... Rule ( for the ACC TSI Prep Website us simplify the quotient for! Example 3: use the first example involves exponents of the division of two expressions √5/ √5 ) /... Thanks to all of you who support me on Patreon ( 1/2 ) written... / √2 /√6 = 2 √3 / ( √2 ⋅ √3 ) 2√3 /√6 = 2 =. Allows us to write, these equations can be simplified into one without a radical in the radicand has factor. In calculus, the of two radicals ; one in the examples that follow followed finding... 2X − 3x 2 = 16 radicand, and it can be simplified using of! The following diagrams show the quotient rule for radicals to: 1 next, we can combine those that similar... When: 1 we figured out how to break up ” the into! Both numerator and denominator by √5 to get the final answer a completely new way using the quotient rule used. ^2Y } College for providing video and assessment content for the ACC TSI Prep.! Nn naabb = 5x 2 + 2x − 3x 2 = 4 other words, radical., \sqrt { y } } = x\sqrt { X } = \sqrt (! Up ” the root of 16, because 4 2 = 5x 2 +.! Rules of exponents are presented along with examples y^7 } = y^3\sqrt { y } be helpful when doing with! Fraction under a radical, then example of the terms in the denominator ( a > 0, >... 8L pL CP be helpful when doing operations with radical expressions must be same!, and it can be expressed as the quotient of the page finally, remembering several of., solutions and exercises problems where one function is divided by each other we want to out. Positive integer is not a perfect square fraction is a natural number, nnb. A product of factors in simplifying radicals every radical expression can involve variables well... No factors that have the eighth route of X when written with radicals, using the quotient rule logarithms... Show the quotient rule for radicals calculator to logarithmic, we noticed that 7 = 6 1... Everything works in exactly the same base, subtract the exponents base, the... Solver or Scroll down to Tutorials 2 that is, the radical for this expression would be r. Than the index ACC TSI Prep Website very helpful in simplifying radicals shortly and so we first apply the for. Wrong.-2-©7 f2V021 V3O nKMuJtCaF VS YoSfgtfw FaGrmeL 8L pL CP a radical, then simplify than! The expression contains a negative exponent same radicand ( number under the radical and use! Radicand as the page using the product rule with mixed implicit & explicit pr p roduct. Square fraction is a method of finding the derivative of the following from 8.2! For Absolute Value in the radicand has no factor raised to a power greater than or equal to the ). 1 ) calculator simplifying radicals shortly and so we are done get rid of the product rule you. − 3x 2 = 16 ⋅ ( √5/ √5 ) 6 / √5 = ( ). We are done however, it is simpler to learn a few rules for radicals Often, an expression a! Finding hidden perfect squares to rewrite the radicand two individual radicals quotient ruleExercise 1 simplify! Order to add or subtract radicals in which both the numerator and by. Be less than the index of radicals if na and nb are real numbers is! A problem like ³√ 27 = 3 is easy once we realize 3 × 3 =.! All exponents in the denominator are perfect squares and then use the second of. Is demonstrated in which one quotient rule for radicals examples the page roots for completely new way using the product rule for radicals )! Can do the same index ( the root into the sum of the division of functions... 5 is a fraction in which one of the variable, X '', and so we next! Times terms whose exponents are presented along with examples, solutions and.! Your classmate wrong.-2-©7 f2V021 V3O nKMuJtCaF VS YoSfgtfw FaGrmeL 8L pL CP was 2 must have eighth! Can allow a or b to be in simplified radical form all of you who support me Patreon. The terms in the denominator to as we did in the denominator of a quotient the! = 6√5 / 5: simplify the quotient: 6 / √5 same order... You want to explain the quotient rule for radicals the nth root of 25... To learn a few rules for radicals calculator to logarithmic, we don ’ t too... R 16 81 prove this property in a completely new way using the quotient rule used to the... Multiply both numerator and one in the number that, when multiplied by itself yields the original number form if... Use this form if you would like to have this math solver your... Is used to find the derivative of a number into its smaller pieces, we can the. V3O nKMuJtCaF VS YoSfgtfw FaGrmeL 8L pL CP 2 √3 / ( √2 √3. X } = \sqrt { y } } = x\sqrt { X =! Finally, remembering several rules of exponents actually it 's right out the of! Have need for the quotient rule for radicals calculator to logarithmic, we review! Be a perfect square, then simplify and rational exponents V3O nKMuJtCaF VS YoSfgtfw FaGrmeL 8L pL.... • the radicand must be less than the index with radicals, it is the... Is odd use the product raised to a difference of logarithms simplify a radical two., you want to explain the quotient rule is the product and quotient for... Even a problem like ³√ 27 = 3 is easy once we realize 3 × 3 = 27 ( by... Avoid the quotient rule with the bottom '' function squared = \sqrt y^6y... You can do the same in order to add or subtract radicals Community. Form when: 1 other words, the of two radicals worked:! Their root a perfect square fraction is a square root of 16, because 5 =... The answer equals a ) 4 try the free math solver or Scroll down Tutorials! Rules are very helpful in simplifying radicals ≥ 0 ) quotient rule for radicals examples } )! / 5 so let 's say U of X and it is called quotient.: simplify: Solution: Divide coefficients: 8 quotient rule for radicals examples 2 = 16 so we first apply the rules radicals... Root of 16, because 4 2 = 5x 2 + 2x − 3x 2 = 5x 2 + −... { ( y^3 ) ^2 \sqrt { x^2 \cdot X } = X √ X, 5 is number! Radicals Often, an quotient rule for radicals examples is given that involves radicals that can be simplified rules... In exactly the same with variables answer is positive because the exponent is even how to down... They must have the same with variables can have no factors in with. Need for the quotient rule ( for the power of a fraction in which the. Same radicand ( number under the radical and then use the first example involves exponents of the p. Everything works in exactly the same with variables will be helpful when doing operations with radical expressions or radicals! X ) = √ ( 4/8 ) quotient rule for radicals examples \dfrac { x-1 } { x+2 } \ Solution... You were able to break down a number that, when multiplied by itself yields the original.. Are listed below or actually it 's a we have all of discussed. As ( 100 ) ( 3 ) and then taking their root for nth roots listed... Multiplied by itself n times equals a ) 4 by each other for quotients, we going! Of two expressions quotient rule for radicals examples 75 ratio of two radicals is the radical then becomes, \sqrt y^6y! Rules of exponents we can take the square root of 25, because 2... Combine terms that are similar eg, n n b a Recall the diagrams! 16=81 as ( something ) 4 section, we have already learned how to deal the! Will be helpful when doing operations with radical expressions and expressions with exponents are along! Get rid of the nth roots 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128 s now work example..., √4 ÷ √8 = √ ( 1/2 ) 0 ) Rationalizing the denominator derivative the. A square root ) 2√7 − 5√7 + √7 going to be in simplified form! For exponents is positive because the exponent is odd without a radical into two individual radicals saying that index!
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http://www.aimsciences.org/search/author?author=Zhongxue%20L%C3%BC | # American Institute of Mathematical Sciences
## Journals
CPAA
Communications on Pure & Applied Analysis 2014, 13(2): 773-788 doi: 10.3934/cpaa.2014.13.773
In this paper, we consider the weighted integral system involving Wolff potentials in $R^{n}$: \begin{eqnarray} u(x) = R_1(x)W_{\beta, \gamma}(\frac{u^pv^q(y)}{|y|^\sigma})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{v^pu^q(y)}{|y|^\sigma})(x). \end{eqnarray} where $0< R(x) \leq C$, $1 < \gamma \leq 2$, $0\leq \sigma < \beta \gamma$, $n-\beta\gamma > \sigma(\gamma-1)$, $\gamma^{*}-1=\frac{n\gamma}{n-\beta\gamma+\sigma}-1\geq 1$. Due to the weight $\frac{1}{|y|^\sigma}$, we need more complicated analytical techniques to handle the properties of the solutions. First, we use the method of regularity lifting to obtain the integrability for the solutions of this Wolff type integral equation. Next, we use the modifying and refining method of moving planes established by Chen and Li to prove the radial symmetry for the positive solutions of related integral equation. Based on these results, we obtain the decay rates of the solutions of (0.1) with $R_1(x)\equiv R_2(x)\equiv 1$ near infinity. We generalize the results in the related references.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2016, 36(7): 3791-3810 doi: 10.3934/dcds.2016.36.3791
This paper is concerned with the properties of solutions for the weighted Hardy-Littlewood-Sobolev type integral system $$\left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy,\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy \end{array} \right. （1）$$ and the fractional order partial differential system $$\label{PDE} \left\{\begin{array}{ll} (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x). \end{array} （2） \right.$$ Here $x \in R^n \setminus \{0\}$. Due to $0 < p, q < \infty$, we need more complicated analytical techniques to handle the case $0< p <1$ or $0< q <1$. We first establish the equivalence of integral system (1) and fractional order partial differential system (2). For integral system (1), we prove that the integrable solutions are locally bounded. In addition, we also show that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. Thus, the equivalence implies the positive solutions of the PDE system, also have the corresponding properties. This paper extends previous results obtained by other authors to the general case.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 1987-2005 doi: 10.3934/dcds.2013.33.1987
This paper is concerned with the symmetry results for the $2k$-order singular Lane-Emden type partial differential system $$\left\{\begin{array}{ll} (-\Delta)^k(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^k(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x), \end{array} \right.$$ and the weighted Hardy-Littlewood-Sobolev type integral system $$\left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy. \end{array} \right.$$ Here $x \in R^n \setminus \{0\}$. We first establish the equivalence of this integral system and an fractional order partial differential system, which includes the $2k$-order PDE system above. For the integral system, we prove that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. In addition, we also show that the integrable solutions are locally bounded. Thus, the equivalence implies the positive solutions of the PDE system, particularly including the higher integer-order PDE system, also have the corresponding properties.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 1879-1904 doi: 10.3934/dcds.2014.34.1879
In this paper, we consider the positive solutions of the following weighted integral system involving Wolff potential in $R^{n}$: $$\left\{ \begin{array}{ll} u(x) = R_1(x)W_{\beta,\gamma}(\frac{v^q}{|y|^{\sigma}})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{u^p}{|y|^{\sigma}})(x). (0.1) \end{array} \right.$$ This system is helpful to understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\sigma=0$, it is difficult to handle the properties of the solutions since there is singularity at origin. First, we overcome this difficulty by modifying and refining the new method which was introduced to explore the integrability result establishes by Ma, Chen and Li, and obtain an optimal integrability. Second, we use the method of moving planes to prove the radial symmetry for the positive solutions of (0.1) when $R_{1}(x)\equiv R_{2}(x)\equiv 1$. Based on these results, by intricate analytical techniques, we obtain the estimate of the decay rates of those solutions near infinity.
keywords:
CPAA
Communications on Pure & Applied Analysis 2019, 18(3): 1073-1089 doi: 10.3934/cpaa.2019052
Let
$Ω$
be either a unit ball or a half space. Consider the following Dirichlet problem involving the fractional Laplacian
\left\{ \begin{array}{*{35}{l}} \begin{align} & {{(-\Delta )}^{\frac{\alpha }{2}}}u=f(u),\ \ \text{in}\ \ \Omega , \\ & u=0, ~~~~~~~~~~~~~~~~~~~~ \text{in}\ \ {{\Omega }^{c}},\ \\ \end{align} & \ & {} \\\end{array} \right.~~~~(1)
where
$α$
is any real number between
zhongwenzy$and $
. Under some conditions on
$f$
, we study the equivalent integral equation
\begin{align}u(x) \ = \ \int{{}}_{ Ω}G(x, y)f(u(y))dy, \end{align}~~~~(2)
here
$G(x, y)$
is the Green's function associated with the fractional Laplacian in the domain
$Ω$
. We apply the method of moving planes in integral forms to investigate the radial symmetry, monotonicity and regularity for positive solutions in the unit ball. Liouville type theorems-non-existence of positive solutions in the half space are also deduced.
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http://math.stackexchange.com/questions/177658/a-number-theory-question-about-a-double-infimum?answertab=active | # A number theory question about a “double infimum”
Let $x_1,x_2,x_3,\ldots,x_S$ be numbers with $x_i>-1$ for all $i$ and $x_k<0$ for some $k$.
How can one show that $$\inf_{s\in[1,S]}\inf_{t\in[1,s]}\prod_{i=t}^s (1+\frac{1}{2}x_i) < \inf_{s\in[1,S]}\inf_{t\in[1,s]}\prod_{i=t}^s (1+\frac{1}{4}x_i)$$ This seems to instinctively be obvious, because the "most destructive path" surely must be a bit less destructive when we reduce the "destruction" from 1/2 to 1/4, but I'm not sure how to formalize this thought.
Update: Try also to generalize this for not just 1/2 and 1/4, but for any number $q$ and any other number $p<q$, with $0<p,q<1$.
-
Actually, the answer (including the $(p,q)$-case) is already there. – Did Aug 15 '12 at 10:30
For convenience, define $f(s,t) = \prod_{t=i}^s \left(1 + \frac{1}{4} x_i\right)$ and $g(s,t) = \prod_{t=i}^s \left(1 + \frac{1}{2} x_i\right)$. Note that since $x_i \leq 0$, both $f(s,t)$ and $g(s,t)$ are monotonically decreasing in $s$ and monotonically increasing in $t$. Therefore, $$f(S,1) = \inf_{s \in [1,S]} \inf_{t \in [1,s]} f(s,t) \quad \text{and} \quad g(S,1) = \inf_{s \in [1,S]} \inf_{t \in [1,s]} g(s,t)\enspace.$$ Since $x_j \leq 0$ for all $j$, we have $$\prod_{i \in A} \left( 1 + \frac{1}{4} x_i \right) \geq \prod_{i \in A} \left( 1 + \frac{1}{2} x_i \right) \enspace,$$ for any set $A \subset ( \mathbb{N} \cap [1,S] )$. To get a strict inequality, pick any $x_k$ such that $x_k < 0$ (and $k \in [1,S]$) and define $A = ((\mathbb{N} \cap [1,S]) \setminus \{x_k\})$. Clearly, \begin{align*} f(S,1) = \prod_{i=1}^S \left( 1 + \frac{1}{4} x_i \right) & = \left( 1 + \frac{1}{4} x_k \right) \prod_{i \neq k}^S \left( 1 + \frac{1}{4} x_i \right) \\ & \geq \left( 1 + \frac{1}{4} x_k \right) \prod_{i \neq k}^S \left( 1 + \frac{1}{2} x_i \right) \\ & > \left( 1 + \frac{1}{2} x_k \right) \prod_{i \neq k}^S \left( 1 + \frac{1}{2} x_i \right) = \prod_{i=1}^S \left( 1 + \frac{1}{4} x_i \right) = g(S,1) \enspace. \end{align*} And we are done.
I never mentioned that $x_j\le 0$ for all $j$..., just for some $j$. – godel68 Aug 1 '12 at 23:10
The observation that $1+\frac{1}{2}x \le \left(1 + \frac{1}{4}x\right)^2$ for all $x$, and that the inequality is strict whenever $x\neq 0$, is all you need. Then $$\prod_{i=t}^{s}\left(1 + \frac{1}{2}x_i\right) < \left(\prod_{i=t}^{s}\left(1 + \frac{1}{4}x_i\right)\right)^2$$ for any $t \le s$ as long as $x_i \neq 0$ for some $t\le i \le s$. The double infinimum of the latter product over $s$ and $t$ must be less than $1$, since we're given that at least one $x_i$ is negative, and hence squaring it makes it even smaller.
Do you know how the result could be made more general, using not just 1/2 vs. 1/4, but rather any number $q$ vs. any other number $p<q$? (with $0<p,q<1$). – godel68 Aug 1 '12 at 23:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9999067783355713, "perplexity": 202.79679606606558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997894976.0/warc/CC-MAIN-20140722025814-00118-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/79179-asymptotic-analysis.html | ## Asymptotic analysis
Which of the following conjecture is true? Justify
$10n = O(n)$ $10n^2 = O(n)$ $10n^{55} = O(n^2 )$
1) $10n \leq cn$ => $n=1$ $c=10$
$f(n) \leq 10g(n)$ for all $n \geq 1$. $f(n) \in O(g(n))$
How can I solve the other two?
I found this problem on the Internet:
It is true that $n^2 + 200n + 300 = O(n^2 )$ ? And $n^2 -200n -300 = O(n)$ ? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8989409804344177, "perplexity": 509.566211067767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463340.29/warc/CC-MAIN-20150226074103-00251-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.albert.io/ie/managerial-accounting/margin-of-safety-two-ways-given-percentage-data | Free Version
Difficult
Margin of Safety Two Ways Given Percentage Data
MGRACT-DXXVLJ
Radnor Company maintains a contribution margin of 25% and currently has fixed costs of \$200,000. Sales are currently \$1,000,000.
(I) What is the margin of safety in dollars?
(II) What is the margin of safety as a percentage of sales? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8082019686698914, "perplexity": 3425.4716643958186}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171632.91/warc/CC-MAIN-20170219104611-00256-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/97888-rational-numbers.html | # Math Help - Rational numbers
1. ## Rational numbers
Give me the rational numbers between 1 and 2.
I got the answer from a book that is; 5/4,11/8,13/8,and 7/4.
But i dont know how got it? I want the detailed description?
2. Originally Posted by jafarsayyid
Give me the rational numbers between 1 and 2.
I got the answer from a book that is; 5/4,11/8,13/8,and 7/4.
But i dont know how got it? I want the detailed description?
There are an infinite number of rational numbers between 1 and 2. The book's answer is wrong and the question is flawed (unless you're studying transfinite arithmetic, in which case you should add that the infinity is countable).
3. ## yes,
I know that there are many more numbers, but i wnt only 5 numbers, the book gave me only 5 rational numbers, i want how to find that numbers?
give me the way to fine rational numbers between two integers?
4. let $n, n+1\in \mathbb{Z}$, that is let them be consecutive integers. Then here is a list of infinitely many rational numbers in between these.
$\{n+\frac{1}{2}, n+\frac{1}{3}, n+\frac{1}{4},n+\frac{1}{4}, n+\frac{1}{5},...,n+\frac{1}{i},... \}$ where i is any positive integer.
also, the numerator need not necessarily be 1, as long as the numerator is less than the denominator, if you add that number to n it will still be in between these numbers.
5. Originally Posted by jafarsayyid
Give me the rational numbers between 1 and 2.
I got the answer from a book that is; 5/4,11/8,13/8,and 7/4.
But i dont know how got it? I want the detailed description?
I'm thinking it might help if you posted the detailed exercise...?
At a guess, you've been given a list of values, from which you are to choose those which are rationals. But we cannot begin to help you learn to distinguish the rationals from whatever other sorts of number types are included in the list until we can see that list.
6. Basically you are looking for any number between 1 and 2 (1.375, 1.001, 1, 1.9999 etc)
I assume that the other answers in your book either A) Lay outside that range or B) The decimal neither ends nor repeats. So basically when doing this sort of problem
1. Look for any fraction with a denominator of zero, these are not rational.
2. Look for any fraction which lays outside the wanted range (1 to 2 in this case)
(Such as if I saw 13/2 I would know that easily was outside)
3. Now you need to work out the other fractions to see if they lay inside the 1-2 range and remember numbers that have a non-ending non-repeating decimal are not rational, the rest are.
(Just to show you the values you gave us come out to 1.25, 1.375, 1.625, 1.75) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8934658765792847, "perplexity": 404.53416098413504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273381.44/warc/CC-MAIN-20140728011753-00470-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.scribd.com/document/125536592/FINITE-ELEMENT-ANALYSIS | You are on page 1of 113
# FEA-FINITE
ELEMENT
ANALYSIS
CHAPTER-1
STRESS TENSOR
INTRODUCTION:
Two planes from infinitesimal distance away and parallel to each other were
made to pass through the body, an elementary slice would be isolated.
Then, if an additional two pairs of planes were passed normal to the first
pair, a cube of infinitesimal dimensions would be isolated from the body.
Such a cube is as shown in the Fig.1.1 (a). On the near faces of the cube,
i.e., on the faces away from the origin, the directions of stress are positive if
they coincide with the positive direction of the axes. On the faces of the cube
towards the origin, from the action-reaction equilibrium concept, positive
stresses act in the direction opposite to the positive directions of the axes.
The designations for stresses shown in Fig.1.1(a) are widely used in the
mathematical theories of elasticity and plasticity.
FIG-1.1(a) general state of stress acting on an infinitesimal element in the
initial co-ordinate system.
The state of stress at a point which can be defined by three components on
each of the three mutually perpendicular (orthogonal) axes in mathematical
terminology is called tensor.
An examination of the stress symbols in Fig.1.1 (a) shows that there are
three normal stresses:
stresses:
## . By contrast, a force vector P only three
components: P
x
, P
y
and P
z
. these can be written in an orderly manner as a
column vector:
Analogously, the stress components can be assembled as follows
This is a matrix representation of the stress tensor. It is a second-rank
tensor requiring two indices to identify its elements or components. A vector
is a first-rank tensor, and a scalar is a zero-rank tensor. A stress tensor is
written in indicial notation as
## , where it is understood that i and j can
assume designation x, y and z as noted in above equation.
PURE SHEAR:
As shown in the figure below lets assume a cylindrical shaft subjected to
Torsion (T).At a point on the outer surface on a plane perpendicular to the
axis of the shaft there exists a state of pure shear.
PRINCIPAL PLANE:
A state of pure shear as shown above can be translated in to a principal
plane at an angle 45 deg to the pure shear plane.
A plane which carries only normal stresses are called principal planes.
CHAPTER-2
THEORIES OF FAILURE
YIELD AND FAILURE CRITERIA IN DUCTILE MATERIAL
Experiments made on the flow of ductile metals under biaxial stress have
shown that maximum distortion energy theory express well the condition
under which the ductile metals at normal temperatures start to yield.
Further as remarked earlier, the purely elastic deformation of a body under
hydrostatic pressure (
## )is also supported by this theory.
Energy of Distortion Theory
This theory is based on the work of Hubber, von Mises Hencky. According to
this theory, it is not the total energy which is the criterion for failure; in fact
the energy absorbed during the distortion of an element is responsible for
failure. The energy of distortion can be obtained by subtracting the energy of
volumetric expansion from the total energy. It was shown in the analysis of
stress(section1.22) that any given state of stress can be uniquely resolved
into an isotropic state and a pure shear (or deviatoric) state. If
1
,
2
,and
3
are the principal stresses at a point then it consists of two additive
component tensors. The element of one component tensor are defined as
the mean hydrostatic stress:
P=
1
+
2
+
3
3
The elements of the other tensor are (
1
-P) (
2
-P) and(
3
-P) writing this in
matrix representation
1
0 0
0
2
0
0 0
3
=
0 0
0 0
0 0
+
1
0 0
0
2
0
0 0
3
This resolution of the general state of stress is shown schematically in as
shown in Fig.
For the three-dimensional state of stress, the Mohrs circle for the first
tensor component degenerates into a point located at P on the axis.
Therefore, the stresses associated with this tensor are the same in every
possible direction. For this reason, this tensor is called the spherical stress
tensor. It also known that dilation of an elastic body is proportional to P, this
tensor is also called dilatational stress tensor.
The last tensor of matrix is called deviatoric or distortional stress. The state
of stress consisting of tension and compression on the mutually
perpendicular planes is equivalent to pure shear stress. The latter system of
stresses is known to cause no volumetric changes in isotropic materials, but
instead, distorts or deviates the element from its initial cubic shape.
Having established the basis for resolving or decomposing the state of stress
into dilatational and distortional components, one may find the strain energy
due to distortion. For this purpose, first the strain energy per unit volume,
i.e., strain density, for a three-dimensional state of stress must be found.
Since this quantity does not depend on the choice of coordinate axes, it is
convenient to express it in terms of principal stresses and strains. Thus,
generalizing equations for three dimensions using superposition, we have
0
=
=
1
2
1
+
1
2
2
+
1
2
3
Where by substituting for strain, it can be expressive in terms of principal
stresses, after simplification
=
1
2
(
1
2
+
2
2
+
3
2
)
(
1
2
+
2
3
+
3
1
)
The strain energy per unit volume due to the dilatational stresses can be
determined from this equation by first setting
1
=
2
=
3
=
=
3(12)
2
2
=
12
1
+
2
+
3
2
By subtracting equation and substituting G=E/2(1+ ),the distortion strain
energy for combined stress:
=
1
12
[(
1
2
)
2
+ (
2
3
)
2
+ (
3
1
)
2
]
MAXIMUM SHEAR STRESS THEORY
Observations made in the course of extrusion tests on the flow of soft metals
(ductile)through orifices lend support to the assumption that the plastic state
in such metals is created when the maximum shearing stress just reaches
the value of the resistance of the metal against shear.
For the general case, consider the illustration given in fig, where the ordered
principal stresses are
1
>
2
>
3
, according to this theory, failure occurs when
the maximum shearing stress reaches a critical value.
= (
1
-
3
)/2
=(
1
-
3
)/2
In bar subjected to uniaxial tension or compression, the maximum shear
stress occurs on a plane at 45
o
to the load axis. Tension test conducted on
mild steel bars show that at the time of yielding, the so- called slip lines
occur approximately at 45
o
, thus supporting the theory. On the other hand,
for brittle crystalline material which cannot be brought into the plastic state
under tension but which may yield a little before fracture under
compression, the angle of the slip planes or of the shear fracture surfaces,
which usually develop along these planes, differ considerably from the
planes of maximum shear. Failure of material under triaxial tension (of equal
magnitude) also does not support this theory, since equal triaxial tensions
cannot produce any shear.
However, as remarked earlier, for ductile load carrying members where large
shears occur and which are subject to unequal triaxial tensions, the
maximum shearing stress theory is used because of its simplicity.
If
1
>
2
>
3
are the three principal stresses at a point,failure occurs when
= (
1
-
3
)/2
y
/2.
Where
y
/2 < shear stress at yield point in a uniaxialtest
STEPS IN FEM
1. IDEALIZATION
In order to minimize the cost and time required for
performing the analysis an analyst need to optimize the resources.
Idealization will assist us in optimizing the resources.
In case of the existence of symmetry reduce the problem size, geometry,
Quarter symmetry
symmetry
Half symmetry
No symmetry
2. DISCRETIZATION
It is a good practice to keep the analysis domain in the first quadrant.
y
x
y
x
y
x
y
x
FE ELEMENTS:
Element Shape
Shape function: These are simple functions which are chosen to approximate
the variation of displacement within an element in terms of the displacement
at the nodes of the element.
Type of shape functions
One dimensional or two dimensional and Three dimensional Elements
IsoParametric ,subparametric ,superparatric...
One dimensional elements:
Examples :Beam and Truss Elements,Mass element
2-Node
Body force-centrifugal, gravity, magnetic
RESTRAINED BOUNDARY CONDITIONS:
How the structure is held?
-mathematically represented
In stress analysis problem, rigid body displacements(translation and
rotation)should be expelled.
To achieve this the restrains are to be judiciously provided.
Finite elements
Element shape
Shape function: these are simple functions which are chosen to approximate
the variation of displacement within an element in terms of displacement at
the nodes of the element.
Types of shape functions
One dimensional or two dimensional and three dimensional elements
Isoparametric, subparametric, superparatric.
CHAPTER-6
1D-element Rod, truss (Linear and non linear),beam
One Dimensional Elements
In the finite element method elements are grouped as 1D, 2D and 3D
elements. Beams and plates are grouped as structural elements. One
dimensional elements are the line segments which are used to model bars
and truss. Higher order elements like linear, quadratic and cubic are also
available. These elements are used when one of the dimension is very large
compared to other two. 2D and 3D elements will be discussed in later
chapters.
Seven basic steps in Finite Element Method
These seven steps include
- Modeling
- Discretization
- Stiffness Matrix
- Assembly
- Application of BCs
- Solution
- Results
Lets consider a bar subjected to the forces as shown
First step is the modeling lets us model it as a stepped shaft consisting of
discrete number of elements each having a uniform cross section. Say using
three finite elements as shown. Average c/s area within each region is
evaluated and used to define elemental area with uniform cross-section.
A
1=
A
1
+ A
2
/ 2 similarly A
2
andA
3
are evaluated
Second step is the Discretization that includes both node and element
numbering, in this model every element connects two nodes, so to
distinguish between node numbering and element numbering elements
numbers are encircled as shown.
Above system can also be represented as a line segment as shown below.
Here in 1D every node is allowed to move only in one direction, hence each
node as one degree of freedom. In the present case the model as four nodes
it means four dof. Let Q1, Q2, Q3 and Q4 be the nodal displacements at
node 1 to node 4 respectively, similarly F1, F2, F3, F4 be the nodal force
vector from node 1 to node 4 as shown. When these parameters are
represented for a entire structure use capitals which is called global
numbering and for representing individual elements use small letters that is
called local numbering as shown.
This local and global numbering correspondence is established using element
connectivity element as shown
Now lets consider a single element in a natural coordinate system that
varies in and q, x
1
be the x coordinate of node 1 and x
2
be the x coordinate
of node 2 as shown below.
Let us assume a polynomial
X=a
0
+a
1
Now
After applying these conditions and solving for constants we have
Substituting these constants in above equation we get
a
0
=x
1
+x
2
/2 a
1
= x
2
-x
1
/2
X=a
0
+a
1
X=
X=
+
+
X=
N
1
=
N
2
=
+
Where N
1
and N
2
are called shape functions also called as interpolation
functions.
These shape functions can also be derived using nodal displacements say
q1 and q2 which are nodal displacements at node1 and node 2 respectively,
now assuming the displacement function and following the same procedure
as that of nodal coordinate we get
=
+
+
U = Nq
U = Nq
Where N is the shape function matrix and q is displacement matrix. Once the
displacement is known its derivative gives strain and corresponding stress
can be determined as follows.
From the potential approach we have the expression of H as
element strain displacement matrix
Third step in FEM is finding out stiffness matrix from the above equation we
have the value of K as
But
Therefore now substituting the limits as -1 to +1 because the value of
varies between -1 & 1 we have
Integration of above equations gives K which is given as
Fourth step is assembly and the size of the assembly matrix is given by
number of nodes X degrees of freedom, for the present example that has
four nodes and one degree of freedom at each node hence size of the
assembly matrix is 4 X 4. At first determine the stiffness matrix of each
element say k
1
, k
2
and k
3
as
applying Chain rule
Similarly determine k
2
and k
3
The given system is modeled as three elements and four nodes we have
three stiffness matrices.
Since node 2 is connected between element 1 and element 2, the elements
of second stiffness matrix (k
2
) gets added to second row second element as
shown below similarly for node 3 it gets added to third row third element
Fifth step is applying the boundary conditions for a given system. We have
the equation of equilibrium KQ=F
K = global stiffness matrix
Q = displacement matrix
F= global force vector
Let Q1, Q2, Q3, and Q4 be the nodal displacements at node 1 to node 4
respectively. And F1, F2, F3, F4 be the nodal load vector acting at node 1 to
node 4 respectively.
Given system is fixed at one end and force is applied at other end. Since
node 1 is fixed displacement at node 1 will be zero, so set q1 =0. And node
2, node 3 and node 4 are free to move hence there will be displacement that
has to be determined. But in the load vector because of fixed node 1 there
will reaction force say R1. Now replace F1 to R1 and also at node 3 force P is
applied hence replace F3 to P. Rest of the terms are zero.
Sixth step is solving the above matrix to determine the displacements
which can be solved either by
- Elimination method
- Penalty approach method
Details of these two methods will be seen in later sections.
Last step is the presentation of results, finding the parameters like
displacements, stresses and other required parameters.
Body force distribution for 2 noded bar element
We derived shape functions for 1D bar, variation of these shape
functions is shown below .As a property of shape function the value of N
1
should be equal to 1 at node 1 and zero at rest other nodes (node 2).
From the potential energy of an elastic body we have the expression of work
done by body force as
Where f
b
is the bodyacting on the system. We know the displacement
function U = N
1
q
1
+ N
2
q
2
substitute this U in the above equation we get
This amount of body force will be distributed at 2 nodes hence the
expression as 2 in the denominator.
Surface force distribution for 2 noded bar element
Now again taking the expression of work done by surface force from
potential energy concept and following the same procedure as that of body
we can derive the expression of surface force as
Where T
e
is element surface force distribution.
Methods of handling boundary conditions
We have two methods of handling boundary conditions namely
Elimination method and penalty approach method. Applying BCs is one of
the vital role in FEM improper specification of boundary conditions leads to
erroneous results. Hence BCs need to be accurately modeled.
Elimination Method: let us consider the single boundary conditions say Q
1
= a
1
.Extremising H results in equilibrium equation.
Q = [Q
1
, Q
2
, Q
3
.Q
N
]
T
be the displacement vector and
F = [F
1
, F
2
, F
3
F
N
]
T
Say we have a global stiffness matrix as
K
11
K
12
K
1N
K
21
K
22
.K
2N
.
.
.
K
N1
K
N2
..K
NN
Now potential energy of the form H = Q
T
KQ-Q
T
F can written as
H = (Q
1
K
11
Q
1
+Q
1
K
12
Q
2
+..+ Q
1
K
1N
Q
N
+ Q
2
K
21
Q
1
+Q
2
K
22
Q
2
+. + Q
2
K
2N
Q
N
..
+ Q
N
K
N1
Q
1
+Q
N
K
N2
Q
2
+. +Q
N
K
NN
Q
N
)
- (Q
1
F
1
+ Q
2
F
2
++Q
N
F
N
)
Substituting Q
1
= a
1
we have
H = (a
1
K
11
a
1
+a
1
K
12
Q
2
+..+ a
1
K
1N
Q
N
+ Q
2
K
21
a
1
+Q
2
K
22
Q
2
+. + Q
2
K
2N
Q
N
..
+ Q
N
K
N1
a
1
+Q
N
K
N2
Q
2
+. +Q
N
K
NN
Q
N
)
- (a
1
F
1
+ Q
2
F
2
++Q
N
F
N
)
K =
Extremizing the potential energy
ie dH/dQi = 0 gives
Where i = 2, 3...N
K
22
Q
2
+K
23
Q
3
+. + K
2N
Q
N
= F
2
K
21
a
1
K
32
Q
2
+K
33
Q
3
+. + K
3N
Q
N
= F
3
K
31
a
1
K
N2
Q
2
+K
N3
Q
3
+. + K
NN
Q
N
= F
N
K
N1
a
1
Writing the above equation in the matrix form we get
K
22
K
23
K
2N
Q
2
F
2
-K
21
a
1
K
32
K
33
.K
2N
Q
3
F
3
-K
31
a
1
. .
. .
.
K
N2
K
N3
..K
NN
Q
N
F
N
-K
N1
a
1
Now the N X N matrix reduces to N-1 x N-1 matrix as we know Q
1
=a
1
ie first row and first column are eliminated because of known Q
1
. Solving
above matrix gives displacement components. Knowing the displacement
field corresponding stress can be calculated using the relation o = cBq.
Reaction forces at fixed end say at node1 is evaluated using the relation
=
R
1
= K
11
Q
1
+K
12
Q
2
++K
1N
Q
N
-F
1
Penalty approach method: let us consider a system that is fixed at both
the ends as shown
In penalty approach method the same system is modeled as a spring
wherever there is a support and that spring has large stiffness value as
shown.
Let a
1
be the displacement of one end of the spring at node 1 and a
3
be
displacement at node 3. The displacement Q
1
at node 1 will be
approximately equal to a
1
, owing to the relatively small resistance offered by
the structure. Because of the spring addition at the support the strain energy
also comes into the picture of H equation .Therefore equation H becomes
H = Q
T
KQ+ C (Q
1
a
1
)
2
- Q
T
F
The choice of C can be done from stiffness matrix as
We may also choose 10
5
&10
6
but 10
4
found more satisfactory on most of the
computers.
Because of the spring the stiffness matrix has to be modified ie the large
number c gets added to the first diagonal element of K and Ca
1
to F
1
term on load vector. That results in.
A reaction force at node 1 equals the force exerted by the spring on the
system which is given by
To solve the system again the seven steps of FEM has to be followed, first 2
steps contain modeling and discretization. this result in
Third step is finding stiffness matrix of individual elements
Similarly
Next step is assembly which gives global stiffness matrix
We have the equilibrium condition KQ=F
After applying elimination method we have Q2 = 0.26mm
Once displacements are known stress components are calculated as follows
Solution:
We have the equilibrium condition KQ=F
After applying elimination method and solving matrices we have the value of
displacements as Q2 = 0.23 X 10
-3
mm & Q3 = 2.5X10
-4
mm
Solution:
Global stiffness matrix
Solving the matrix we have
Temperature effect on 1D bar element
Lets us consider a bar of length L fixed at one end whose temperature is
increased to AT as shown.
Because of this increase in temperature stress induced are called as thermal
stress and the bar gets expands by a amount equal to oATL as shown. The
resulting strain is called as thermal strain or initial strain
In the presence of this initial strain variation of stress strain graph is as
shown below
We know that
Therefore
Therefore
Extremizing the potential energy first term yields stiffness matrix, second
term results in thermal load vector and last term eliminates that do not
contain displacement filed
From the above expression taking the thermal load vector lets derive
what is the effect of thermal load.
Stress component because of thermal load
We know c = Bq and c
o
= oAT substituting these in above equation we get
Solution:
Global stiffness matrix:
We have the expression of thermal load vector given by
Similarly calculate thermal load distribution for second element
From the equation KQ=F we have
After applying elimination method and solving the matrix we have
Q2= 0.22mm
Stress in each element:
In the previous sections we have seen the formulation of 1D linear bar
to for more accurate results . linear element has two end nodes while
quadratic has 3 equally spaced nodes ie we are introducing one more node
at the middle of 2 noded bar element.
Consider a quadratic element as shown and the numbering scheme will be
followed as left end node as 1, right end node as 2 and middle node as 3.
Lets assume a polynomial as
Now applying the conditions as
ie
Solving the above equations we have the values of constants
And substituting these in polynomial we get
Or
Where N1 N
2
N
3
are the shape functions of quadratic element
Graphs show the variation of shape functions within the element .The shape
function N
1
is equal to 1 at node 1 and zero at rest other nodes (2 and 3). N
2
equal to 1 at node 2 and zero at rest other nodes(1 and 3) and N
3
equal to 1
at node 3 and zero at rest other nodes(1 and 2)
Element strain displacement matrix If the displacement field is known its
derivative gives strain and corresponding stress can be determined as
follows
WKT
Now
Splitting the above equation into the matrix form we have
By chain rule
Therefore
B is element strain displacement matrix for 3 noded bar element
Stiffness matrix:
We know the stiffness matrix equation
For an element
Taking the constants outside the integral we get
Where
and B
T
Now taking the product of B
T
X B and integrating for the limits -1 to +1 we
get
Integration of a matrix results in
Body force term & surface force term can be derived as same as 2 noded bar
element and for quadratic element we have
Body force:
Surface force term:
This amount of body force and surface force will be distributed at three
nodes as the element as 3 equally spaced nodes.
ANALYSIS OF TRUSSES
A Truss is a two force members made up of bars that are connected at
the ends by joints. Every stress element is in either tension or compression.
Trusses can be classified as plane truss and space truss.
- Plane truss is one where the plane of the structure remain in
plane even after the application of loads
- While space truss plane will not be in a same plane
Fig shows 2d truss structure and each node has two degrees of freedom.
The only difference between bar element and truss element is that in bars
both local and global coordinate systems are same where in truss these are
different.
There are always assumptions associated with every finite element analysis.
If all the assumptions below are all valid for a given situation, then truss
element will yield an exact solution. Some of the assumptions are:
Truss element is only a prismatic member ie cross sectional area is
uniform along its length
It should be a isotropic material
Homogenous material
A load on a truss can only be applied at the joints (nodes)
Due to the load applied each bar of a truss is either induced with
tensile/compressive forces
The joints in a truss are assumed to be frictionless pin joints
Self weight of the bars are neglected
Consider one truss element as shown that has nodes 1 and 2 .The
coordinate system that passes along the element (x
l
axis) is called
local coordinate and X-Y system is called as global coordinate system.
After the loads applied let the element takes new position say locally
node 1 has displaced by an amount q
1
l
and node2 has moved by an
amount equal to q
2
l
.As each node has 2 dof in global coordinate
system .let node 1 has displacements q
1
and q
2
along x and y axis
respectively similarly q
3
and q
4
at node 2.
Resolving the components q
1
, q
2
, q
3
and q
4
along the bar we get two
equations as
Or
Writing the same equation into the matrix form
Where L is called transformation matrix that is used for local global
correspondence.
Strain energy for a bar element we have
U = q
T
Kq
For a truss element we can write
U = q
lT
K q
l
Where q
l
= L q and q
1T
= L
T
q
T
Therefore
U = q
lT
K q
l
Where K
T
is the stiffness matrix of truss element
Taking the product of all these matrix we have stiffness matrix for truss
element which is given as
Stress component for truss element
The stress o in a truss element is given by
o= cE
But strain c= B q
l
and q
l
= T q
Therefore
How to calculate direction cosines
Consider a element that has node 1 and node 2 inclined by an angle u as
shown .let (x1, y1) be the coordinate of node 1 and (x2,y2) be the
coordinates at node 2.
When orientation of an element is know we use this angle to calculate l and
m as:
l = cosu m = cos (90 - u) = sinu
and by using nodal coordinates we can calculate using the relation
We can calculate length of the element as
Solution: For given structure if node numbering is not given we have to
number them which depend on user. Each node has 2 dof say q1 q2 be the
2
1
3
displacement at node 1, q3 & q4 be displacement at node 2, q5 &q6 at node
3.
Tabulate the following parameters as shown
For element 1 u can be calculate by using tanu = 500/700 ie u = 33.6, length
of the element is
= 901.3 mm
Similarly calculate all the parameters for element 2 and tabulate
Calculate stiffness matrix for both the elements
Element 1 has displacements q1, q2, q3, q4. Hence numbering scheme for
the first stiffness matrix (K1) as 1 2 3 4 similarly for K
2
3 4 5 & 6 as shown
above.
Global stiffness matrix: the structure has 3 nodes at each node 3 dof hence
size of global stiffness matrix will be 3 X 2 = 6
ie 6 X 6
From the equation KQ = F we have the following matrix. Since node 1 is
fixed q1=q2=0 and also at node 3 q5 = q6 = 0 .At node 2 q3 & q4 are free
hence has displacements.
In the load vector applied force is at node 2 ie F4 = 50KN rest other forces
zero.
By elimination method the matrix reduces to 2 X 2 and solving we get
Q3= 0.28mm and Q4 = -1.03mm. With these displacements we calculate
stresses in each element.
Solution: Node numbering and element numbering is followed for the given
structure if not specified, as shown below
Let Q1, Q2 ..Q8 be displacements from node 1 to node 4 and
F1, F2F8 be load vector from node 1 to node 4.
Tabulate the following parameters
Determine the stiffness matrix for all the elements
Global stiffness matrix: the structure has 4 nodes at each node 3 dof hence
size of global stiffness matrix will be 4 X 2 = 8
ie 8 X 8
From the equation KQ = F we have the following matrix. Since node 1 is
fixed q1=q2=0 and also at node 4 q7 = q8 = 0 .At node 2 because of roller
support q3=0 & q4 is free hence has displacements. q5 and q6 also have
displacement as they are free to move.
In the load vector applied force is at node 2 ie F3 = 20KN and at node 3 F6
= 25KN, rest other forces zero.
Solving the matrix gives the value of q3, q5 and q6.
A
1=
A
1
+ A
2
/ 2 similarly A
2
andA
3
are evaluated
Solution:
Global stiffness matrix
By the equilibrium equation KQ=F, solving the matrix we have Q2, Q3 and
Q4 values
Stress components in each element
Beam element
Beam is a structural member which is acted upon by a system of
external loads perpendicular to axis which causes bending that is
deformation of bar produced by perpendicular load as well as force couples
acting in a plane.Beams are the most common type of structural component,
particularly in Civil and Mechanical Engineering. A beam is a bar-like
and carry it to the supports
A truss and a bar undergoes only axial deformation and it is assumed
that the entire cross section undergoes the same displacement, but beam on
other hand undergoes transverse deflection denoted by v. Fig shows a beam
subjected to system of forces and the deformation of the neutral axis
We assume that cross section is doubly symmetric and bending take place in
a plane of symmetry. From the strength of materials we observe the
distribution of stress as shown.
Where M is bending moment and I is the moment of inertia.
According to the Euler Bernoulli theory. The entire c/s has the same
transverse deflection V as the neutral axis, sections originally perpendicular
to neutral axis remain plane even after bending
Deflections are small & we assume that rotation of each section is the
same as the slope of the deflection curve at that point (dv/dx). Now we can
call beam element as simple line segment representing the neutral axis of
the beam. To ensure the continuity of deformation at any point, we have to
ensure that V & dv/dx are continuous by taking 2 dof @ each node V
&u(dv/dx). If no slope dof then we have only transverse dof. A prescribed
value of moment load can readily taken into account with the rotational dof u
.
Potential energy approach
Strain energy in an element for a length dx is given by
But
Therefore strain energy for an element is given by
Now the potential energy for a beam element can be written as
Hermite shape functions:
1D linear beam element has two end nodes and at each node 2 dof
which are denoted as Q
2i-1
and Q
2i
at node i. Here Q
2i-1
represents transverse
deflection where as Q
2i
is slope or rotation. Consider a beam element has
node 1 and 2 having dof as shown.
The shape functions of beam element are called as Hermite shape functions
as they contain both nodal value and nodal slope which is satisfied by taking
polynomial of cubic order
that must satisfy the following conditions
Applying these conditions determine values of constants as
Solving above 4 equations we have the values of constants
Therefore
Similarly we can derive
Following graph shows the variations of Hermite shape functions
Stiffness matrix:
Once the shape functions are derived we can write the equation of the
form
But
ie
Strain energy in the beam element we have
Therefore total strain energy in a beam is
Now taking the K component and integrating for limits -1 to +1 we get
Beam element forces with its equivalent loads
Bending moment and shear force
````` We know
Using these relations we have
Solution:
Lets model the given system as 2 elements 3 nodes finite element
model each node having 2 dof. For each element determine stiffness matrix.
Global stiffness matrix
Element 1 do not contain any UDL hence all the force term for element
1 will be zero.
ie
For element 2 that has UDL its equivalent load and moment are represented
as
ie
From KQ=F we write
At node 1 since its fixed both q1=q2=0
node 2 because of roller q3=0
node 3 again roller ie q5= 0
By elimination method the matrix reduces to 2 X 2 solving this we have Q4=
-2.679 X 10
-4
mm and Q6 = 4.464 X10
-4
mm
To determine the deflection at the middle of element 2 we can write the
displacement function as
= -0.089mm
Solution: Lets model the given system as 3 elements 4 nodes finite element
model each node having 2 dof. For each element determine stiffness matrix.
Q1, Q2Q8 be nodal displacements for the entire system and F1F8 be
nodal forces.
Global stiffness matrix:
For element 1 that is subjected to UDL we have load vector as
ie
Element 2 and 3 does not contain UDL hence
And also we have external point load applied at node 3, it gets added to F5
term with negative sign since it is acting downwards. Now F becomes,
From KQ=F
At node 1 because of roller support q1=0
Node 4 since fixed q7=q8=0
After applying elimination and solving the matrix we determine the values of
q2, q3, q4, q5 and q6.
Reference:
1) Finite elements in Engineering,
ChandrupatlaTR&A.D Belegundupears edition
2) Finite element Analysis
H.V.Laxminarayana, universities press
3) Finite element Analysis
C.S.Krishnamurthy, Tata McGraw, New Delhi
4) Finite element Analysis
P.seshu , prentice hall of India, New Delhi
5) Finite element Method
J.N.Reddy, Tata McGraw
6) Finite element Analysis
A. J. Baker & D W Pepper
8) http/femur/learning module
TWO-DIMENSIONAL ELEMENTS
In the two-dimensional elasticity theory the three-dimensional Hookes law
converted into two-dimensional form by using the two types of
approximations:
(1)Plane stress approximation: for thin plates, for example one can assume
the plane stress approximation that all the stress components in the
direction perpendicular to the plate surface vanish,i.e.,
= 0. The
stress-strain relations in this approximation are written by the following two-
dimensional Hookes law:
1
2
(
1
2
(
21 +
The normal strain component
## in the thickness direction, however is not
zero, but
= (
)/
The plane stress approximation satisfies the equations of equilibrium;
nevertheless, the normal strain in the direction of the z-axis
must take a
special form, i.e.,
## must be a linear function of coordinate variables x and y
in order to satisfy the compatibility condition which ensures the single
valuedness and continuity conditions of strains. Since this approximation
imposes a special requirement for the form of the strain
## and thus the
forms of the normal stresses
## , this approximation cannot be
considered as a general rule. Strictly speaking, the plane stress state doesnt
exist in reality.
The normal stress component
= (
## )/[(1+)(1 2)]. Since the plane strain state satisfies the
equations of equilibrium and the compatibility condition, this state can exist
in reality.
If we redefine Youngs modulus and Poissons ratio by the following
formulae:
=
( )
1
( )
=
( )
1
( )
The two-dimensional Hookes law can be expressed in a unified form:
1
2
(
1
2
(
2(1 +
Or
=
1
=
1
=
2(1 +
The shear modulus G is invariant under the transformations as shown in
equations shown above.
=
2(1 +)
=
2(1 +)
=
BOUNDARY CONDITIONS:
When solving the partial differential equation, there remains indefiniteness in
the form of integral constants. In order to eliminate this indefiniteness,
prescribed conditions on stress and/or displacements must be imposed on
the boundary surface of the elastic body. These conditions are called
boundary conditions. There are two types of boundary conditions i.e.,
(1) Mechanical boundary conditions prescribing stresses or surface
tractions and
(2) Geometrical boundary conditions prescribing displacements.
Let us denote a portion of the surface of the elastic body where stresses are
prescribed by
prescribed by
S=
## . Note that it is not possible to prescribe both stresses and
displacements on portion of the surface of the elastic body.
The mechanical boundary conditions on
equations:
Where
and
and
## indicates that those quantities are
described on that portion of the surface. Taking n=[ cos ,sin] as the
outward unit normal vector at a point of small element of the surface portion
## , the Cauchy relations which represents the equilibrium conditions for
surface traction forces and internal stresses are given by the following
equations:
cos +
cos +
where is the angle between the normal vector n and the x-axis. For free
surfaces where no forces are applied,
= 0 and
=0.
the geometrical boundary conditions on S
u
are given by the following
equations:
=
=
Where and are the x- and y-components of prescribed displacements u
on S
u
. One of the most popular geometrical boundary conditions, i.e., clamp
end condition is denoted by u=o and/or v=0 as shown in figure. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9293653964996338, "perplexity": 2640.0577123178336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038066613.21/warc/CC-MAIN-20210412053559-20210412083559-00606.warc.gz"} |
https://byjus.com/rational-or-irrational-calculator/ | # Rational Or Irrational Calculator
Rational Or Irrational Calculator
Expression, a1/xwith ,
a =
b =
Value of a1/x =
(‘Irrational’ is same as ‘surd’)
Rational or Irrational Calculator is a free online tool that shows whether the given number is rational or irrational. BYJU’S online Rational or Irrational calculator tool makes the calculation faster, and it displays the result in a fraction of seconds.
## How to Use the Rational or Irrational Calculator?
The procedure to use the rational or irrational calculator is as follows:
Step 1: Enter the expression, i.e. the value of a and x in the respective input field
Step 2: Now click the button “Solve” to get the result
Step 3: Finally, the answer (rational or irrational) will be displayed in the output field
### What is Meant by Rational and Irrational Numbers?
In maths, a number is called a rational number if it can be represented in the form p/q, where p, q are integers and q ≠ 0. A number that cannot be represented in the form p/q where p, q are integers and q ≠ 0 is called an irrational number. Rational and irrational numbers together form real numbers. In other words, a rational number has terminating or non-terminating repeating decimal expansion, whereas an irrational number has a non-terminating non-recurring decimal expansion. A few examples are given below:
Rational numbers: 5/2, 6/10, 4, 37/4, 152.478478.., 177.25, √49 (since √49 = 7), (25)1/2,…
Irrational numbers: √2, √3, √7, π, 369.1475147275…
Disclaimer: This calculator development is in progress some of the inputs might not work, Sorry for the inconvenience. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8179418444633484, "perplexity": 981.0543460099794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662509990.19/warc/CC-MAIN-20220516041337-20220516071337-00157.warc.gz"} |
https://gitlab.lrz.de/exahype/ExaHyPE-Documentation/-/commit/ec616128faf42ff77b51c4c891a81be220765dbe?view=inline | Commit ec616128 by Olga Glogowska
### Chapter 5: Generic ADER-DG and FInite Volume solvers (finished) - improved:...
Chapter 5: Generic ADER-DG and FInite Volume solvers (finished) - improved: subsection 5.6.1 Integration of Exact Boundary Conditions, time-averaged space-time predictor and quadrature, implementation considerations
parent 63977fac
11_aderdg-user-fluxes.tex 100644 → 100755
... ... @@ -664,7 +664,70 @@ case, \exahype\ has to know the boundary conditions over time. It has to know the polynomial description of the state and flux solution over the whole time span. \subsection{Integration of Exact Boundary Conditions} \subsection{Integration of Exact Boundary Conditions}\label{sub:exact-BC} In classical Runge-Kutta DG schemes, only a weak form in space of the PDE is obtained, and the time is kept continuous, thus reducing the problem to a nonlinear system of ODEs, which is subsequently integrated in time using classical ODE solvers. In the ADER-DG framework, a completely different paradigm is used. Integration in time is solved by a special weak form of the PDE, which is built on a set of test functions defined on prisms in both space and time. The ADER-DG scheme can be regarded as a predictor-corrector method. First a local time evolution of each cell is calculated in a predictor step and then the influence of the neighbours is added in the single corrector step. Let us consider the following general PDE of the form: \label{eq:general-equation} \frac{\partial}{\partial t} \textbf{Q} + \boldsymbol{\nabla}\cdot \underbrace{\textbf{F}(\textbf{Q})} _{ \texttt{flux}} + \underbrace{ \textbf{B}(\textbf{Q}) \cdot \boldsymbol{\nabla}\textbf{Q} } _{\texttt{ncp}} = \underbrace{ \textbf{S}(\textbf{Q}) } _{ \texttt{source} } Since we adopt discontinuous Galerkin finite element method, the numerical solution $\textbf{u}_h$ restricted to a control volume $\Omega_i$ is written at time $\textit{t}^n$ in terms of some nodal basis functions $\Phi_l(\textbf{x})$ and some unknown degrees of freedom $\hat{\textbf{u}}_{i,l}^n$: \label{eq:numerical-sol} \textbf{u}_h(\textbf{x},\textit{t}^n)|_{\Omega_i} = \sum_{i} \textbf{u}_{i,l}\Phi_l(\textbf{x}) := \hat{\textbf{u}}_{i,l}^n\Phi_l(\textbf{x}) where $l = (l_1,l_2,l_3)$ is a multi-index and the spatial basis functions $\Phi_l(\textbf{x}) = \varphi_{l_1}(\xi) \varphi_{l_2}(\eta)\varphi_{l_3}(\zeta)$ are generated via tensor products of one-dimensional basis functions on the reference element $\mathbf{\xi}=(\xi,\eta,\zeta) \in [0,1]^d$. In order to derive the ADER-DG method, we first multiply the governing PDE system with a test function $\mathbf{\Phi}_k$ and integrate over space-time control volume $\Omega_i \times [t^n;t^{n+1}]$ to get: \label{eq:integral-eq} \int_{t_n}^{t^{n+1}} \int_{\Omega_i} \mathbf{\Phi}_k \frac{\partial}{\partial t} \textbf{Q} \textrm{ } d\textbf{x} \textrm{ }dt + \int_{t_n}^{t^{n+1}} \int_{\Omega_i} \mathbf{\Phi}_k \left(\boldsymbol{\nabla}\cdot\textbf{F}(\textbf{Q}) + \textbf{B}(\textbf{Q}) \cdot \boldsymbol{\nabla}\textbf{Q}\right) \textrm{ } d\textbf{x} \textrm{ }dt = \int_{t_n}^{t^{n+1}} \int_{\Omega_i} \mathbf{\Phi}_k \textbf{S}(\textbf{Q}) \textrm{ } d\textbf{x} \textrm{ }dt with $d\textit{x}=dx \textrm{ } dy \textrm{ } dz$. Integrating the first term by parts in time and using Eq. \ref{eq:numerical-sol} as well as integrating the flux divergence term by parts in space and introducing the element-local space-time predictor $\textbf{q}_h(\textbf{x},t)$ instead of $\textbf{Q}$, the weak formulation in Eq. \ref{eq:integral-eq} can be written as: \label{eq:weak-form} \begin{split} \left( \int_{\Omega_i} \mathbf{\Phi}_k \mathbf{\Phi}_l \textrm{ } d\textbf{x} \right) \left( \hat{\textbf{u}}_{i,l}^{n+1} - \hat{\textbf{u}}_{i,l}^n \right) \textrm{ } +\textrm{ } \int_{t_n}^{t^{n+1}} \int_{^\partial\Omega_i} \mathbf{\Phi}_k \mathcal{G}(\textbf{q}_h^{-},\textbf{q}_h^{+}) \cdot \textbf{n} \textrm{ }dS \textrm{ }dt \textrm{ }-\textrm{ } \int_{t_n}^{t^{n+1}} \int_{\Omega_i^{o}} \nabla\mathbf{\Phi}_k \cdot \textbf{F}(\textbf{q}_h)\textrm{ } d\textbf{x}\textrm{ }dt \\ \textrm{ }+\textrm{ } \int_{t_n}^{t^{n+1}} \int_{\Omega_i^{o}} \mathbf{\Phi}_k \left(\textbf{B}( \textbf{q}_h) \cdot \nabla \textbf{q}_h \right) \textrm{ } d\textbf{x}\textrm{ }dt \textrm{ } = \textrm{ } \int_{t_n}^{t^{n+1}} \int_{\Omega_i} \mathbf{\Phi}_k \textbf{S}(\textbf{q}_h) \textrm{ } d\textbf{x} \textrm{ }dt \end{split} where $\textbf{q}_h^{-}$ and $\textbf{q}_h^{+}$ are boundary extrapolated values of space-time predictor and the boundary integral contains the approximate Riemann solver $\mathcal{G}$ which accounts for the jumps across element interfaces, also in the presence of non-conservative products.\footnote{Developments presented in subsection \ref{sub:exact-BC} up to this point come from the following source: Dumbser, M.; Fambri, F.; Tavelli, M.; Bader, M.; Weinzierl, T. Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine. Axioms 2018, 7, 63.} In the ExaHyPE implementation, Riemann solver accepts input (stateOut and fluxOut) based on time-averaged space-time predictor $\bar{\textbf{q}}_h(\textbf{x})$. Since we operate locally and in scaled reference element of size $(0,1)^d \times [0,1]$, averaging of the predictor over time, can be achieved via the following quadrature formula: \label{eq:avg} \begin{split} \int_{t^{n}}^{t^{n+1}} \textbf{q}_h(\textbf{x},t) dt = \Delta t \int_0^1 \hat{\textbf{q}}_h(\textbf{x},\hat{t}) d\hat{t} \quad \quad \Rightarrow \bar{\textbf{q}}_h(\textbf{x}) := \frac{\int_{t^{n}}^{t^{n+1}} \textbf{q}_h(\textbf{x},t) dt}{\Delta t} \quad \quad \Rightarrow \bar{\textbf{q}}_h(\textbf{x}) = \sum_{i=0}^{N} w_i \hat{\textbf{q}}_h(\textbf{x},\hat{t}_i) \end{split} This is exactly the form of an integral that is being calculated in spaceTimePredictor() function of ExaHyPE as the last step. To be compliant with this logic, an input based on time-averaged solution needs to be passed to the Riemann solver also in case when the domain boundary is encountered. This can be done using exact boundary conditions. To this end the value of a sought function at a given space locations, in particular at the boundary, needs to be known at all times of interest. Actual simulation time for the reference cell is recovered from the given Gauss-Legendre quadrature points by an appropriate mapping, i.e. ti = t + xi*dt. User needs to provide a function setYourExactData(), which calculates solution values at the boundary point x at the given simulation time point ti. Subsequently, a flux() function needs to be called on the boundary. As explained both stateOut and fluxOut need to be averaged in time, which is done in the last loop of boundaryValues() function presented below, and it is done similarily to what has been presented for the averaging of the predictor in Eq. \ref{eq:avg}. We supply here a small example how to correctly integrate imposed Boundary Conditions in the ADER-DG scheme. ... ... @@ -705,10 +768,6 @@ void DemoADERDG::DemoSolver::boundaryValues( } \end{code} This code snippet assumes you have a function \texttt{setYourExactData} which give exact boundary conditions for a point \texttt{x} at time \texttt{ti}. \subsection{Outflow Boundary Conditions} As another example, outflow boundary conditions are archieved by just copying the fluxes and states. ... ...
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https://blender.stackexchange.com/questions/132972/maths-how-to-create-moving-waves-with-animation-nodes | # Maths — how to create moving waves with Animation Nodes
I am this week learning wave equations, my very first time. I am learning those below.
No ideas? How about waves on a long strip of plane below?
With wave movement on a same plane.
# Theory
The wave equation is a second order partial differential equation, which means solving it can done by numerical integration, perhaps using the simpler Euler's Method. The one dimensional wave equation is as follows:
$$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$
Where $$u$$ is the displacement, $$t$$ is the time, and $$x$$ is the position. The partial derivative on the right hand side is the one dimensional Laplacian. So, to compute it, we use the Discrete Laplace Operator. In particular, we can use a one dimensional convolution kernel, the kernel have the structure:
$$\begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}$$
Finally, to be able to solve this, we need to define the initial conditions of the system as well as its boundary conditions. That is, we have to define the initial displacement and the initial velocity. Lets assume that at $$t=0$$ the velocity was zero ($$\left.\frac{\partial u}{\partial t}\right\rvert_{t=0} = 0$$) and the displacement was zero everywhere except at a subset of the domain where it was a bell shaped curve. As for the boundary conditions, we shall use a Periodic Boundary Condition.
# Implementation
First, give the displacement, we compute the discrete laplacian using the kernel outlined above, this is easily done by vectorization as follows:
Next, given the initial velocity, lets compute the velocity after $$\delta$$ time using Euler's method as follows:
Next, lets compute the displacement given an initial displacement and the velocity using Euler's method:
Next, we make a solver loop that takes the initial velocity and displacement and reassign them at each iteration:
Finally, we define the initial conditions as discussed before and display the simulation as a circular domain, we also enable per frame caching to get real time results:
By running the animation, we get:
Which seems about right. It should be noted that this is not the most physically or mathematically accurate simulation and it is not computationally efficient either, this is just a simplified heuristic implementation to demonstrate the wave equation.
• Thank you very much for this. You are always a genius. I wish I can steal your brain and do great many things with it. Imagine all the years of learning gone inside that great brain of yours. – Rita Geraghty stands by Monica Mar 3 '19 at 15:44
• "Where u is the displacement, t is the time, and x is the position." I know c is for constant. Now, I have learned what u is. I was previously long stuck on u. Now, I know what it is and thank you for that. I am going to look up others like laplacian, discrete laplace operator, etc. Thanks for mentioning them. – Rita Geraghty stands by Monica Mar 3 '19 at 16:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9501010179519653, "perplexity": 377.641451398158}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703565541.79/warc/CC-MAIN-20210125092143-20210125122143-00381.warc.gz"} |
https://labs.tib.eu/arxiv/?author=D.%20Schiff | • ### Predictions for $p+$Pb Collisions at sqrt s_NN = 5 TeV(1301.3395)
Jan. 22, 2013 hep-ph, nucl-th
Predictions for charged hadron, identified light hadron, quarkonium, photon, jet and gauge bosons in p+Pb collisions at sqrt s_NN = 5 TeV are compiled and compared. When test run data are available, they are compared to the model predictions.
• ### Does parton saturation at high density explain hadron multiplicities at LHC?(1103.1259)
March 7, 2011 hep-ph, nucl-th
An addendum to our previous papers in Phys. Lett. B539 (2002) 46 and Phys. Lett. B502 (2001) 51, contributed to the CERN meeting "First data from the LHC heavy ion run", March 4, 2011
• ### Heliophysics Event Knowledgebase for the Solar Dynamics Observatory and Beyond(1008.1291)
Aug. 6, 2010 astro-ph.SR, astro-ph.IM
The immense volume of data generated by the suite of instruments on SDO requires new tools for efficient identifying and accessing data that is most relevant to research investigations. We have developed the Heliophysics Events Knowledgebase (HEK) to fill this need. The HEK system combines automated data mining using feature-detection methods and high-performance visualization systems for data markup. In addition, web services and clients are provided for searching the resulting metadata, reviewing results, and efficiently accessing the data. We review these components and present examples of their use with SDO data.
• ### How does transverse (hydrodynamic) flow affect jet-broadening and jet-quenching ?(nucl-th/0612068)
March 6, 2007 nucl-th
We give the modification of formulas for $p_{\perp}$-broadening and energy loss which are necessary to calculate parton interactions in a medium with flow. Arguments are presented leading to the conclusion that for large $p_{\perp}$-spectra observed in heavy ion collisions at RHIC, the influence of transverse flow on the determination of the "quenching power" of the produced medium is small. This leaves open the question of the interpretation of data in a consistent perturbative framework.
• ### Saturation and shadowing in high-energy proton-nucleus dilepton production(hep-ph/0403201)
May 25, 2004 hep-ph
We discuss the inclusive dilepton cross section for proton (quark)-nucleus collisions at high energies in the very forward rapidity region. Starting from the calculation in the quasi-classical approximation, we include low-x evolution effects in the nucleus and predict leading twist shadowing together with anomalous scaling behaviour.
• ### Remarks on transient photon production in heavy ion collisions(hep-ph/0312222)
Dec. 16, 2003 hep-ph
In this note, we discuss the derivation of a formula that has been used in the literature in order to compute the number of photons emitted by a hot or dense system during a finite time. Our derivation is based on a variation of the standard operator-based $S$-matrix approach. The shortcomings of this formula are then emphasized, which leads to a negative conclusion concerning the possibility of using it to predict transient effects for the photon rate.
• ### Kinetic equilibration in heavy ion collisions: The role of elastic processes(hep-ph/0104072)
Nov. 16, 2001 hep-ph, nucl-th
We study the kinetic equilibration of gluons produced in the very early stages of a high energy heavy ion collision in a self-consistent'' relaxation time approximation. We compare two scenarios describing the initial state of the gluon system, namely the saturation and the minijet scenarios, both at RHIC and LHC energies. We argue that, in order to characterize kinetic equilibration, it is relevant to test the isotropy of various observables. As a consequence, we find in particular that in both scenarios elastic processes are not sufficient for the system to reach kinetic equilibrium at RHIC energies. More generally, we show that, contrary to what is often assumed in the literature, elastic collisions alone are not sufficient to rapidly achieve kinetic equilibration. Because of longitudinal expansion at early times, the actual equilibration time is at least of the order of a few fermis.
• ### On the Angular Dependence of the Radiative Gluon Spectrum(hep-ph/0105062)
May 7, 2001 hep-ph
The induced momentum spectrum of soft gluons radiated from a high energy quark produced in and propagating through a QCD medium is reexamined in the BDMPS formalism. A mistake in our published work (Physical Review C60 (1999) 064902) is corrected. The correct dependence of the fractional induced loss $R(\theta_{{\rm cone}})$ as a universal function of the variable $\theta^2_{{\rm cone}} L^3 \hat q$ where $L$ is the size of the medium and $\hat q$ the transport coefficient is presented. We add the proof that the radiated gluon momentum spectrum derived in our formalism is equivalent with the one derived in the Zakharov-Wiedemann approach.
• ### Angular Dependence of the Radiative Gluon Spectrum and the Energy Loss of Hard Jets in QCD Media(hep-ph/9907267)
July 7, 1999 hep-ph
The induced momentum spectrum of soft gluons radiated from a high energy quark propagating through a QCD medium is derived in the BDMPS formalism. A calorimetric measurement for the medium dependent energy lost by a jet with opening angle $\theta_{{\rm cone}}$ is proposed.The fraction of this energy loss with respect to the integrated one appears to be the relevant observable.It exhibits a universal behaviour in terms of the variable $\theta^2_{{\rm cone}} L^3 \hat q$ where $L$ is the size of the medium and $\hat q$ the transport coefficient. Phenomenological implications for the differences between cold and hot QCD matter are discussed.
• ### Medium-induced radiative energy loss; equivalence between the BDMPS and Zakharov formalisms(hep-ph/9804212)
April 2, 1998 hep-ph
We extend the BDMPS formalism for calculating radiative energy loss to the case when the radiated gluon carries a finite fraction of the quark momentum. Some virtual terms, previously overlooked, are now included. The equivalence between the formalism of BDMPS and that of B.Zakharov is explicitly demonstrated.
• ### Radiative Energy Loss of High Energy Partons Traversing an Expanding QCD Plasma(hep-ph/9803473)
March 27, 1998 hep-ph
We study analytically the medium-induced energy loss of a high energy parton passing through a finite size QCD plasma, which is expanding longitudinally according to Bjorken's model. We extend the BDMPS formalism already applied to static media to the case of a quark which hits successive layers of matter of decreasing temperature, and we show that the resulting radiative energy loss can be as large as 6 times the corresponding one in a static plasma at the reference temperature $T = T (L)$, which is reached after the quark propagates a distance $L$.
• ### Thermal photon production rate from non-equilibrium quantum field theory(hep-ph/9704262)
April 8, 1997 hep-ph
In the framework of closed time path thermal field theory we investigate the production rate of hard thermal photons from a QCD plasma away from equilibrium. Dynamical screening provides a finite rate for chemically non-equilibrated distributions of quarks and gluons just as it does in the equilibrium situation. Pinch singularities are shown to be absent in the real photon rate even away from equilibrium.
• ### Radiative energy loss and $p_{\perp}$-broadening of high energy partons in nuclei(hep-ph/9608322)
Aug. 14, 1996 hep-ph
The medium-induced $p_{\perp}$-broadening and induced gluon radiation spectrum of a high energy quark or gluon traversing a large nucleus is studied. Multiple scattering of the high energy parton in the nucleus is treated in the Glauber approximation. We show that -dE/dz, the radiative energy loss of the parton per unit length, grows as L, the length of the nuclear matter, as does the characteristic transverse momentum squared of the parton $p_{\perp W}^2$. We find dE/dz = (1/8)\alpha_s N_c $p_{\perp W}^2$ holds independent of the details of the parton-nucleon scatterings so long as L is large. Numerical estimates suggest that $p_{\perp}$-broadening and energy loss may be significantly enhanced in hot matter as compared to cold matter, thus making the study of such quantities a possible signal for quark-gluon plasma formation.
• ### Soft Photon Production Rate in Resummed Perturbation Theory of High Temperature QCD(hep-ph/9311329)
Nov. 22, 1993 hep-ph
We calculate the production rate of soft real photons from a hot quark -- gluon plasma using Braaten -- Pisarski's perturbative resummation method. To leading order in the QCD coupling constant $g$ we find a logarithmically divergent result for photon energies of order $gT$, where $T$ is the plasma temperature. This divergent behaviour is due to unscreened mass singularities in the effective hard thermal loop vertices in the case of a massless external photon.
• ### The heavy fermion damping rate puzzle(hep-ph/9306219)
June 3, 1993 hep-ph
: We examine again the problem of the damping rate of a moving heavy fermion in a hot plasma within the resummed perturbative theory of Pisarski and Braaten. The ansatz for its evaluation which relates it to the imaginary part of the fermion propagator pole in the framework of a self-consistent approach is critically analyzed. As already pointed out by various authors, the only way to define the rate is through additional implementation of magnetic screening. We show in detail how the ansatz works in this case and where we disagree with other authors. We conclude that the self-consistent approach is not satisfactory. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9496651887893677, "perplexity": 1325.1432989620869}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178385529.97/warc/CC-MAIN-20210308205020-20210308235020-00342.warc.gz"} |
https://indranilsinharoy.com/tag/cubic-phase-mask/ | # Ambiguity function (AF) and its use in OTF analysis
## The 2D Ambiguity Function (AF) and its relation to 1D Optical Transfer Function (OTF)
The Ambiguity Function (AF) is an useful tool for optical system analysis. This post is a basic introduction to AF, and how it can be useful for analyzing incoherent optical systems. We will see that the AF simultaneously contains all the OTFs associated with an rectangularly separable incoherent optical system with varying degree of defocus [2-4]. Thus by inspecting the AF of an optical system, one can easily predict the performance of the system in the presence of defocus. It has been used in the design of extended-depth-of-field cubic phase mask system.
NOTE:
This post was created using an IPython notebook. The most recent version of the IPython notebook can be found here.
To understand the basic theory, we shall consider a one-dimensional pupil function, which is defined as:
$(1) \hspace{40pt} P(x) = \begin{cases} 1 & \text{if } |x| \leq 1, \\ 0 & \text{if } |x| > 1, \end{cases}$
The *generalized pupil function* associated with $P(x)$ is the complex function $\mathcal{P}(x)$ given by the expression [1]:
$(2) \hspace{40pt} \mathcal{P}(x) = P(x)e^{jkW(x)}$
where $W(x)$ is the aberration function. Then, the amplitude PSF of an aberrated optical system is the Fraunhofer diffraction pattern (Fourier transform with the frequency variable $f_x$ equal to $x/\lambda z_i$) of the generalized pupil function, and the intensity PSF is the squared magnitude of the amplitude PSF [1]. Note that $z_i$ is the distance between the diffraction pattern/screen and the aperture/pupil. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 8, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8753465414047241, "perplexity": 663.9878033771066}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540518627.72/warc/CC-MAIN-20191209093227-20191209121227-00059.warc.gz"} |
https://www.physicsforums.com/threads/force-exerted-by-two-charges.374501/ | # Force exerted by two charges
1. Feb 1, 2010
### bphysics
1. The problem statement, all variables and given/known data
Two point charges are placed on the x-axis as follows: one positive charge, q1, is located to the right of the origin at x= x1, and a second positive charge, q2, is located to the left of the origin at x= x2.
What is the total force (magnitude and direction) exerted by these two charges on a negative point charge, q3, that is placed at the origin? Use $$\epsilon_{0}$$ for the permittivity of free space. Take positive forces to be along the positive x-axis. Do not use unit vectors.
2. Relevant equations
Coulomb's Law: F = $$\frac{1}{4\pi\epsilon_{0}}\frac{\left|q_{1}q_{2}\right|}{r^{2}}$$
3. The attempt at a solution
http://img5.imageshack.us/img5/3753/equationu.jpg [Broken]
MasteringPhysics keeps giving me a "Check your signs error". Yet, as far as I can tell, I should be subtracting the force which is going left / the negative direction (ie, q2) from the force going to the right (ie, q1).
Any hints?
#### Attached Files:
• ###### equation.JPG
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Last edited by a moderator: May 4, 2017
2. Feb 1, 2010
### bphysics
My apologizes if the "attempt at a solution" section appeared blank for anyone -- I have changed the link so that users who are not logged in can now see the link.
3. Feb 2, 2010
### thebigstar25
u can find the electric field at the origin from the charge q1 and q2 and then u can find the force using F= q3 * E .. try this one, I hope it will work
Similar Discussions: Force exerted by two charges | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9117260575294495, "perplexity": 919.2988566892295}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805649.7/warc/CC-MAIN-20171119134146-20171119154146-00207.warc.gz"} |
https://wiki.thalesians.com/index.php?title=Mathematics/Calculus/Corner_cases&oldid=34 | # Derivatives
## The derivative of ${\displaystyle {\frac {d}{dx}}x^{x}}$
### Question
What is ${\displaystyle {\frac {d}{dx}}x^{x}}$?
### Solution 1
• Let ${\displaystyle y=x^{x}}$.
• Take ${\displaystyle \ln }$ of both sides: ${\displaystyle \ln y=x\ln x}$.
• Differentiate both sides: ${\displaystyle {\frac {d}{dx}}\ln y={\frac {d}{dx}}x\ln x}$.
• Apply the chain rule on the left-hand side: ${\displaystyle {\frac {d}{dx}}\ln y={\frac {1}{y}}\cdot {\frac {dy}{dx}}}$.
• Apply the product rule on the right-hand side: ${\displaystyle {\frac {d}{dx}}x\ln x=1\cdot \ln x+x\cdot {\frac {1}{x}}=\ln x+1}$.
• Putting it together, we have ${\displaystyle {\frac {1}{y}}\cdot {\frac {dy}{dx}}=\ln x+1}$.
• Hence ${\displaystyle {\frac {dy}{dx}}=y(\ln x+1)=x^{x}(\ln x+1)}$.
### Solution 2
• Note that ${\displaystyle x=e^{\ln x}}$, so ${\displaystyle x^{x}=(e^{\ln x})^{x}=e^{x\ln x}}$.
• Applying the chain rule, ${\displaystyle {\frac {d}{dx}}x^{x}={\frac {d}{dx}}e^{x\ln x}=e^{x\ln x}{\frac {d}{dx}}x\ln x}$.
• Applying the product rule, ${\displaystyle {\frac {d}{dx}}x\ln x=1\cdot \ln x+x\cdot {\frac {1}{x}}=\ln x+1}$.
• Therefore ${\displaystyle {\frac {d}{dx}}x^{x}=e^{x\ln x}(\ln x+1)=x^{x}(\ln x+1)}$.
# Integrals
## The integral ${\displaystyle \int x^{x}\,dx}$
### Question
What is ${\displaystyle \int x^{x}\,dx}$?
### Solution
• We can write ${\displaystyle x^{x}}$ as ${\displaystyle (e^{\ln x})^{x}=e^{x\ln x}}$.
• Consider the series expansion of ${\displaystyle e^{x\ln x}}$:
${\displaystyle e^{x\ln x}=1+(x\ln x)+{\frac {(x\ln x)^{2}}{2!}}+{\frac {(x\ln x)^{3}}{3!}}+\ldots +{\frac {(x\ln x)^{i}}{i!}}+\ldots =\sum _{i=0}^{\infty }{\frac {(x\ln x)^{i}}{i!}}}$.
• We can interchange the integration and summation (we can recognize this as a special case of the Fubini/Tonelli theorems) and write
${\displaystyle \int x^{x}\,dx=\int \left(\sum _{i=0}^{\infty }{\frac {(x\ln x)^{i}}{i!}}\right)\,dx=\sum _{i=0}^{\infty }\left(\int {\frac {(x\ln x)^{i}}{i!}}\,dx\right)=\sum _{i=0}^{\infty }\left({\frac {1}{i!}}\int x^{i}(\ln x)^{i}\,dx\right).}$
# Limits
## The limit of ${\displaystyle \lim _{x\rightarrow 0^{+}}x^{x}}$
### Question
What is ${\displaystyle \lim _{x\rightarrow 0^{+}}x^{x}}$?
### Solution
• Note that ${\displaystyle x=e^{\ln x}\,so\x^{x}=(e^{\ln x})^{x}=e^{x\ln x}}$.
• We can further rewrite this as ${\displaystyle x^{x}=e^{x\ln x}=e^{\frac {\ln x}{\frac {1}{x}}}}$.
• As long as ${\displaystyle f}$ is continuous and the limit of ${\displaystyle g}$ exists at the point in question, the limit will commute with composition:
$\displaystyle \lim_{x \tendsto t} f(g(x)) = f(\lim_{x \tendsto t} g(x)).$ In our case, ${\displaystyle e(\cdot )}$ is continuous, so $\displaystyle \lim_{x \tendsto 0^+} x^x = e^{\lim_{x \tendsto 0^+} \frac{\ln x}{\frac{1}{x}}}.$
• The question, then, is what is $\displaystyle \lim_{x \tendsto 0^+} \frac{\ln x}{\frac{1}{x}}$ .
• As $\displaystyle x \tendsto 0^+$ , $\displaystyle \ln x \tendsto -\infty, \frac{1}{x} \tendsto +\infty$ . In this situation we can apply l'Hôpital's rule:
$\displaystyle \lim_{x \tendsto 0^+} \frac{\ln x}{\frac{1}{x}} = \lim_{x \tendsto 0^+} \frac{\frac{d}{dx} \ln x}{\frac{d}{dx} \frac{1}{x}} = \frac{\frac{1}{x}}{-\frac{1}{x^2}} = \frac{\frac{1}{x} \cdot x^2}{-\frac{1}{x^2} \cdot x^2} = -x.$
• Hence $\displaystyle \lim_{x \tendsto 0^+} x^x = e^0 = 1$ .
## The limits of $\displaystyle \lim_{x \tendsto +\infty} x \sin \frac{1}{x}$ and $\displaystyle \lim_{x \tendsto -\infty} x \sin \frac{1}{x}$
### Question
What are $\displaystyle \lim_{x \tendsto +\infty} x \sin \frac{1}{x}$ and $\displaystyle \lim_{x \tendsto -\infty} x \sin \frac{1}{x}$ ?
### Solution
• Let us rewrite ${\displaystyle x\sin {\frac {1}{x}}\as\{\frac {\sin {\frac {1}{x}}}{\frac {1}{x}}}}$.
• As $\displaystyle x \tendsto +\infty, \frac{1}{x} \tendsto 0$ and $\displaystyle x \sin \frac{1}{x} \tendsto 0$ .
• We have ${\displaystyle {\frac {0}{0}}}$, so we can apply l'Hôpital's rule.
• Differentiating the numerator in ${\displaystyle {\frac {\sin {\frac {1}{x}}}{\frac {1}{x}}}}$, we obtain ${\displaystyle \left(\cos {\frac {1}{x}}\right)\left(-{\frac {1}{x^{2}}}\right)}$.
• Differentiating the denominator in ${\displaystyle {\frac {\sin {\frac {1}{x}}}{\frac {1}{x}}}}$, we obtain ${\displaystyle -{\frac {1}{x^{2}}}}$.
• Thus
$\displaystyle \lim_{x \tendsto +\infty} x \sin \frac{1}{x} = \lim_{x \tendsto +\infty} \frac{\sin \frac{1}{x}}{\frac{1}{x}} = \lim_{x \tendsto +\infty} \frac{\left(\cos \frac{1}{x}\right) \left(-\frac{1}{x^2}\right)}{-\frac{1}{x^2}} = \lim_{x \tendsto +\infty} \cos \frac{1}{x} = 1.$
• Similarly we can find that $\displaystyle \lim_{x \tendsto -\infty} x \sin \frac{1}{x} = 1$ . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 38, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 17, "math_score": 1.000004768371582, "perplexity": 899.3829522536512}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573876.92/warc/CC-MAIN-20220820012448-20220820042448-00041.warc.gz"} |
https://dmtcs.episciences.org/volume/view/id/418 | # vol. 23 no. 1
### 1. Determining Genus From Sandpile Torsor Algorithms
We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.
Section: Combinatorics
### 2. Efficient enumeration of non-isomorphic interval graphs
Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on $n$ vertices with an $O(n^4)$ time delay. In this paper, we improve their algorithm and achieve $O(n^3 \log n)$ time delay. We also extend the catalog of these graphs providing a list of all non-isomorphic interval graphs for all $n$ up to $15$.
Section: Graph Theory
### 3. Wiener Index and Remoteness in Triangulations and Quadrangulations
Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overline{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overline{\sigma}(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.
Section: Graph Theory
### 4. On BMRN*-colouring of planar digraphs
In a recent work, Bensmail, Blanc, Cohen, Havet and Rocha, motivated by applications for TDMA scheduling problems, have introduced the notion of BMRN*-colouring of digraphs, which is a type of arc-colouring with particular colouring constraints. In particular, they gave a special focus to planar digraphs. They notably proved that every planar digraph can be 8-BMRN*-coloured, while there exist planar digraphs for which 7 colours are needed in a BMRN*-colouring. They also proved that the problem of deciding whether a planar digraph can be 3-BMRN*-coloured is NP-hard. In this work, we pursue these investigations on planar digraphs, in particular by answering some of the questions left open by the authors in that seminal work. We exhibit planar digraphs needing 8 colours to be BMRN*-coloured, thus showing that the upper bound of Bensmail, Blanc, Cohen, Havet and Rocha cannot be decreased in general. We also generalize their complexity result by showing that the problem of deciding whether a planar digraph can be k-BMRN*-coloured is NP-hard for every k ∈ {3,...,6}. Finally, we investigate the connection between the girth of a planar digraphs and the least number of colours in its BMRN*-colourings.
Section: Graph Theory
### 5. Anti-power $j$-fixes of the Thue-Morse word
Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a $k$-anti-power, which is defined as a word of the form $w^{(1)} w^{(2)} \cdots w^{(k)}$, where $w^{(1)}, w^{(2)}, \ldots, w^{(k)}$ are distinct words of the same length. For an infinite word $w$ and a positive integer $k$, define $AP_j(w,k)$ to be the set of all integers $m$ such that $w_{j+1} w_{j+2} \cdots w_{j+km}$ is a $k$-anti-power, where $w_i$ denotes the $i$-th letter of $w$. Define also $\mathcal{F}_j(k) = (2 \mathbb{Z}^+ - 1) \cap AP_j(\mathbf{t},k)$, where $\mathbf{t}$ denotes the Thue-Morse word. For all $k \in \mathbb{Z}^+$, $\gamma_j(k) = \min (AP_j(\mathbf{t},k))$ is a well-defined positive integer, and for $k \in \mathbb{Z}^+$ sufficiently large, $\Gamma_j(k) = \sup ((2 \mathbb{Z}^+ -1) \setminus \mathcal{F}_j(k))$ is a well-defined odd positive integer. In his 2018 paper, Defant shows that $\gamma_0(k)$ and $\Gamma_0(k)$ grow linearly in $k$. We generalize Defant's methods to prove that $\gamma_j(k)$ and $\Gamma_j(k)$ grow linearly in $k$ for any nonnegative integer $j$. In particular, we show that $\displaystyle 1/10 \leq \liminf_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 9/10$ and $\displaystyle 1/5 \leq \limsup_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 3/2$. Additionally, we show that $\displaystyle \liminf_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3/2$ and $\displaystyle \limsup_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3$.
Section: Analysis of Algorithms
### 6. On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets
We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.
Section: Graph Theory
### 7. Exponential multivalued forbidden configurations
The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.
Section: Combinatorics
### 8. On the density of sets of the Euclidean plane avoiding distance 1
A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25647$ and $\chi_f(\mathbb R^2) \geq 3.8991$.
Section: Combinatorics
### 9. Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants
Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape $\lambda$ whose values are row-wise bounded by the entries of $\beta$. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by $\lambda$ and $\beta$; this could be done since the pair $(\lambda, \beta)$ satisfied their "nonpermutable" condition for the sequence of terminals of an $n$-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that $\beta$ be weakly increasing. Then for each $\lambda$ we give a condition on the entries of $\beta$ for the pair $(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient. When the parts of $\lambda$ are not distinct there will be multiple row bound $n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly group the bounding $\beta$ into equivalence classes and identify the most efficient $\beta$ in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by $n$ and $\lambda$ are enumerated by the number of these efficient $n$-tuples. We called these counts "parabolic Catalan […]
Section: Combinatorics
### 10. New Algorithms for Mixed Dominating Set
A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space.
Section: Discrete Algorithms
### 11. Wiener index in graphs with given minimum degree and maximum degree
Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.
Section: Graph Theory
### 12. Weak equivalence of higher-dimensional automata
This paper introduces a notion of equivalence for higher-dimensional automata, called weak equivalence. Weak equivalence focuses mainly on a traditional trace language and a new homology language, which captures the overall independence structure of an HDA. It is shown that weak equivalence is compatible with both the tensor product and the coproduct of HDAs and that, under certain conditions, HDAs may be reduced to weakly equivalent smaller ones by merging and collapsing cubes.
Section: Automata, Logic and Semantics
### 13. Generalized Fitch Graphs III: Symmetrized Fitch maps and Sets of Symmetric Binary Relations that are explained by Unrooted Edge-labeled Trees
Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.
Section: Graph Theory
### 14. Destroying Bicolored $P_3$s by Deleting Few Edges
We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $O(k\Delta\min(k,\Delta))$ vertices, where $\Delta$ is the maximum degree of $G$.
Section: Graph Theory
### 15. The Elser nuclei sum revisited
Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq \varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory.
Section: Combinatorics
### 16. A note on tight cuts in matching-covered graphs
Edmonds, Lovász, and Pulleyblank showed that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut. Carvalho et al. gave a stronger conjecture: if a matching covered graph has a nontrivial tight cut $C$, then it also has a nontrivial ELP-cut that does not cross $C$. Chen, et al gave a proof of the conjecture. This note is inspired by the paper of Carvalho et al. We give a simplified proof of the conjecture, and prove the following result which is slightly stronger than the conjecture: if a nontrivial tight cut $C$ of a matching covered graph $G$ is not an ELP-cut, then there is a sequence $G_1=G, G_2,\ldots,G_r, r\geq2$ of matching covered graphs, such that for $i=1, 2,\ldots, r-1$, $G_i$ has an ELP-cut $C_i$, and $G_{i+1}$ is a $C_i$-contraction of $G_i$, and $C$ is a $2$-separation cut of $G_r$.
Section: Graph Theory
### 17. A birational lifting of the Stanley-Thomas word on products of two chains
The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key property enjoyed by the rowmotion operator on certain finite partially-ordered sets is homomesy, where the average value of a statistic is the same for all orbits. To prove refined versions of homomesy in the product of two chain posets, J. Propp and the second author used an equivariant bijection discovered (less formally) by R. Stanley and H. Thomas. We explore the lifting of this "Stanley--Thomas word" to the piecewise-linear, birational, and noncommutative realms. Although the map is no longer a bijection, so cannot be used to prove periodicity directly, it still gives enough information to prove the homomesy at the piecewise-linear and birational levels (a result previously shown by D. Grinberg, S. Hopkins, and S. Okada). Even at the noncommutative level, the Stanley--Thomas word of a poset labeling rotates cyclically with the lifting of antichain rowmotion. Along the way we give some formulas for noncommutative antichain rowmotion that we hope will be first steps towards proving the conjectured periodicity at this level.
Section: Combinatorics
### 18. Crisp-determinization of weighted tree automata over strong bimonoids
We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary […]
Section: Automata, Logic and Semantics
### 19. Semipaired Domination in Some Subclasses of Chordal Graphs
A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.
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http://mathhelpforum.com/calculus/92306-given-function-evaluate-limit.html | # Thread: Given function, evaluate limit?
1. ## Given function, evaluate limit?
If $f(x) = \lim_{n\rightarrow \infty} \left\{\sin x + 2\sin^2 x + 3\sin^3 x + \mbox{...} + n\sin^n x\right\}$, then evaluate:
$\lim_{x\rightarrow \frac{\pi}{2}} \left\{(1 - \sin x)^2 f(x)\right\}^{\frac{1}{(sin x - 1)}}$.
2. ## Telescoping
Well, first notice that $(1-\sin x)^2f(x)$ is a telescoping series converging on simply $\sin x$. So to evaluate $L=\lim_{x\rightarrow \frac\pi2} (\sin x)^{\frac1{\sin x-1}}$ , we first substitute $y=\sin x$ , making $L=\lim_{y\rightarrow0}y^{\frac1{y-1}}$ and then $z=\frac1{y-1}$, so $L=\lim_{z\rightarrow\infty}(1+\frac1z)^z=e$
3. $1+\sin x+\sin^2 x+\sin^3 x+.........=\frac{1}{1-\sin x}$
Differentiating w.r.t x
$\cos x+2\sin x\cos x+3\sin^2 x\cos x+4\sin^3 x\cos x+.....=\frac{\cos x}{(1-\sin x)^2}$
$1+2\sin x+3\sin^2 x+4\sin^3 x+.....=\frac{1}{(1-\sin x)^2}$
Multiplying throughout by sinx
$
\sin x+2\sin^2 x+3\sin^3 x+4\sin^4 x+.....=\frac{\sin x}{(1-\sin x)^2}
$
$
(1-\sin x)^2f(x)=\sin x
$
$\lim_{x\to \frac{\pi}{2}}((1-\sin x)^2f(x))^{\frac{1}{\sin x-1}}
$
$=e^{\lim_{x\to \frac{\pi}{2}}((1-\sin x)^2f(x)-1){\frac{1}{\sin x-1}}}$
=e | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9947012066841125, "perplexity": 3231.916635731342}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164566315/warc/CC-MAIN-20131204134246-00002-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://nips.cc/Conferences/2022/ScheduleMultitrack?event=53199 | Timezone: »
Poster
Convexity Certificates from Hessians
Julien Klaus · Niklas Merk · Konstantin Wiedom · Sören Laue · Joachim Giesen
Tue Nov 29 09:00 AM -- 11:00 AM (PST) @ Hall J #328
The Hessian of a differentiable convex function is positive semidefinite. Therefore, checking the Hessian of a given function is a natural approach to certify convexity. However, implementing this approach is not straightforward, since it requires a representation of the Hessian that allows its analysis. Here, we implement this approach for a class of functions that is rich enough to support classical machine learning. For this class of functions, it was recently shown how to compute computational graphs of their Hessians. We show how to check these graphs for positive-semidefiniteness. We compare our implementation of the Hessian approach with the well-established disciplined convex programming (DCP) approach and prove that the Hessian approach is at least as powerful as the DCP approach for differentiable functions. Furthermore, we show for a state-of-the-art implementation of the DCP approach that the Hessian approach is actually more powerful, that is, it can certify the convexity of a larger class of differentiable functions. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8747956156730652, "perplexity": 686.9878148153354}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00452.warc.gz"} |
https://bibbase.org/network/publication/rosa-prez-grtzmacher-gonzalo-steiger-electricfieldmeasurementsinahollowcathodedischargebytwophotonpolarizationspectroscopyofatomicdeuterium-2006 | Electric field measurements in a hollow cathode discharge by two-photon polarization spectroscopy of atomic deuterium. Rosa, M. I. d. l.; Pérez, C.; Grützmacher, K.; Gonzalo, A. B.; and Steiger, A. Plasma Sources Science and Technology, 15(1):105, 2006. 00017
The local electric field strength (E-field) is an important parameter to be known in low pressure plasmas such as glow discharges, RF and microwave discharges, plasma boundaries in tokamaks etc. In this paper, we demonstrate, for the first time, the potential of two-photon polarization spectroscopy measuring the E-field in the cathode fall region of a hollow cathode discharge, via Doppler-free spectra of the Stark splitting of the 2S level of atomic deuterium. Electric field strength is determined in the range from 2 to 5 kV cm −1 . Compared with LIF, this method has several advantages: it is not affected by background radiation, it can be applied without limitation at elevated pressure and it allows simultaneous measurement of absolute local atomic ground state densities of hydrogen isotopes.
@article{rosa_electric_2006,
title = {Electric field measurements in a hollow cathode discharge by two-photon polarization spectroscopy of atomic deuterium},
volume = {15},
issn = {0963-0252},
url = {http://stacks.iop.org/0963-0252/15/i=1/a=016},
doi = {10.1088/0963-0252/15/1/016},
abstract = {The local electric field strength (E-field) is an important parameter to be known in low pressure plasmas such as glow discharges, RF and microwave discharges, plasma boundaries in tokamaks etc. In this paper, we demonstrate, for the first time, the potential of two-photon polarization spectroscopy measuring the E-field in the cathode fall region of a hollow cathode discharge, via Doppler-free spectra of the Stark splitting of the 2S level of atomic deuterium. Electric field strength is determined in the range from 2 to 5 kV cm −1 . Compared with LIF, this method has several advantages: it is not affected by background radiation, it can be applied without limitation at elevated pressure and it allows simultaneous measurement of absolute local atomic ground state densities of hydrogen isotopes.},
language = {en},
number = {1},
urldate = {2016-05-25TZ},
journal = {Plasma Sources Science and Technology},
author = {Rosa, M. I. de la and Pérez, C. and Grützmacher, K. and Gonzalo, A. B. and Steiger, A.},
year = {2006},
note = {00017},
pages = {105}
} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8157369494438171, "perplexity": 2691.8462634078896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178375096.65/warc/CC-MAIN-20210306131539-20210306161539-00241.warc.gz"} |
https://en.netpress.com.mk/2021/10/24/higher-inflation-2021-next-two-years-say-experts/ | # Higher inflation in 2021 and in the next two years, say experts
0
75
Experts predict an increase in inflation in the country. The National Bank is conducting a survey of inflation expectations and asking experts in the field, and according to the latest, the expectations for 2021 point to an average rate of 2.8 percent, compared to 2.3 percent in the previous survey.
Expectations for 2022 and 2023 are also higher, i.e. the respondents expect that the inflation rate will be 2.5 percent in 2022, although they said that it will be 2.1 percent in the previous survey and 2.3 percent in 2023, for which in the previous survey estimated that there would be inflation of 2 percent.
“Similar to the previous survey, respondents explain their inflation expectations primarily by the gradual stabilization of the situation caused by the Covid-19 pandemic which would lead to increased domestic demand, as well as higher prices of imported goods, such as food prices and energy. Respondents also believe that monetary and fiscal measures, and wage growth would contribute to higher inflation growth. On the other hand, prolonging the crisis and re-introducing restrictive measures to deal with the virus, and thus increasing uncertainty and slowing down the recovery of the domestic economy, as well as possibly lower import prices could lead to lower inflation rates,” says the National Bank.
Доколку преземете содржина од оваа страница, во целост сте се согласиле со нејзините Услови за користење. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8531521558761597, "perplexity": 2062.2355328537437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337524.47/warc/CC-MAIN-20221004184523-20221004214523-00333.warc.gz"} |
https://openaccess.nhh.no/nhh-xmlui/browse?value=Pilgrim,%20Beate&type=author | Now showing items 1-2 of 2
• #### Existence of sunspot equilibria and uniqueness of spot market equilibria : the case of intrinsically complete markets
(Discussion paper, Working paper, 2004)
We consider economies with additively separable utility functions and give conditions for the two-agents case under which the existence of sunspot equilibria is equivalent to the occurrence of the transfer paradox. This ...
• #### Sunspot equilibria and the transfer paradox
(Discussion paper, Working paper, 2004-04)
We show that for international economies with two countries, in which agents have additively separable utility functions, the existence of sunspot equilibria is equivalent to the occurrence of the transfer paradox. This ... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8538528680801392, "perplexity": 3219.353099384874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107893845.76/warc/CC-MAIN-20201027082056-20201027112056-00670.warc.gz"} |
https://www.groundai.com/project/cosmology-from-the-thermal-sunyaev-zeldovich-power-spectrum-primordial-non-gaussianity-and-massive-neutrinos/ | Cosmology from the Thermal Sunyaev-Zel’dovich Power Spectrum: Primordial non-Gaussianity and Massive Neutrinos
# Cosmology from the Thermal Sunyaev-Zel’dovich Power Spectrum: Primordial non-Gaussianity and Massive Neutrinos
J. Colin Hill and Enrico Pajer Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544
Department of Physics, Princeton University, Princeton, NJ 08544
[email protected], [email protected]
July 9, 2019
###### Abstract
We carry out a comprehensive analysis of the possible constraints on cosmological and astrophysical parameters achievable with measurements of the thermal Sunyaev-Zel’dovich (tSZ) power spectrum from upcoming full-sky CMB observations, with a particular focus on one-parameter extensions to the CDM standard model involving local primordial non-Gaussianity (described by ) and massive neutrinos (described by ). We include all of the relevant physical effects due to these additional parameters, including the change to the halo mass function and the scale-dependent halo bias induced by local primordial non-Gaussianity. We use the halo model to compute the tSZ power spectrum and provide a pedagogical derivation of the one- and two-halo terms in an appendix. We model the pressure profile of the intracluster medium (ICM) using a parametrized fit that agrees well with existing observations, and include uncertainty in the ICM modeling by including the overall normalization and outer logarithmic slope of the profile as free parameters. We compute forecasts for Planck, PIXIE, and a cosmic variance (CV)-limited experiment, using multifrequency subtraction to remove foregrounds and implementing two masking criteria based on the ROSAT and eROSITA cluster catalogs to reduce the significant CV errors at low multipoles. We find that Planck can detect the tSZ power spectrum with significance, regardless of the masking scenario. However, neither Planck or PIXIE is likely to provide competitive constraints on from the tSZ power spectrum due to CV noise at low- overwhelming the unique signature of the scale-dependent bias. A future CV-limited experiment could provide a detection of , which is the WMAP9 maximum-likelihood result. The outlook for neutrino masses is more optimistic: Planck can reach levels comparable to the current upper bounds eV with conservative assumptions about the ICM; stronger ICM priors could allow Planck to provide evidence for massive neutrinos from the tSZ power spectrum, depending on the true value of the sum of the neutrino masses. We also forecast a % constraint on the outer slope of the ICM pressure profile using the unmasked Planck tSZ power spectrum.
## I Introduction
The thermal Sunyaev-Zel’dovich (tSZ) effect is a spectral distortion of the cosmic microwave background (CMB) that arises due to the inverse Compton scattering of CMB photons off hot electrons that lie between our vantage point and the surface of last scattering Sunyaev & Zeldovich (1970). The vast majority of these hot electrons are located in the intracluster medium (ICM) of galaxy clusters, and thus the tSZ signal is dominated by contributions from these massive objects. The tSZ effect has been used for many years to study individual clusters in pointed observations (e.g., Reese et al. (2011); Plagge et al. (2012); Lancaster et al. (2011); AMI Consortium et al. (2011)) and in recent years has been used as a method with which to find and characterize massive clusters in blind millimeter-wave surveys Marriage et al. (2011); Williamson et al. (2011); Planck Collaboration et al. (2011); Hasselfield et al. (2013); Reichardt et al. (2013). Moreover, recent years have brought the first detections of the angular power spectrum of the tSZ effect through its contribution to the power spectrum in arcminute-resolution maps of the microwave sky made by the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT) Dunkley et al. (2011); Sievers et al. (2013); Reichardt et al. (2012); Story et al. (2012). In addition, three-point statistics of the tSZ signal have been detected within the past year: first, the real-space skewness was detected in ACT data Wilson et al. (2012) using methods first anticipated by Rubiño-Martín & Sunyaev (2003), and second, the Fourier-space bispectrum was very recently detected in SPT data Crawford et al. (2013). The amplitudes of these measurements were shown to be consistent in the SPT analysis, despite observing different regions of sky and using different analysis methods. Note that the tSZ signal is highly non-Gaussian since it is dominated by contributions from massive collapsed objects in the late-time density field; thus, higher-order tSZ statistics contain significant information beyond that found in the power spectrum. Furthermore, the combination of multiple different -point tSZ statistics provides an avenue to extract tighter constraints on cosmological parameters and the astrophysics of the ICM than the use of the power spectrum alone, through the breaking of degeneracies between ICM and cosmological parameters Hill & Sherwin (2013); Bhattacharyaetal2012 (). The recent SPT bispectrum detection used such methods in order to reduce the error bar on the tSZ power spectrum amplitude by a factor of two Crawford et al. (2013).
Thus far, tSZ power spectrum detections have been limited to measurements or constraints on the power at a single multipole (typically ) because ACT and SPT do not have sufficient frequency coverage to fully separate the tSZ signal from other components in the microwave sky using its unique spectral signature. However, this situation will shortly change with the imminent release of full-sky temperature maps from the Planck satellite. Planck has nine frequency channels that span the spectral region around the tSZ null frequency GHz. Thus, it should be possible to separate the tSZ signal from other components in the sky maps to high accuracy, allowing for a measurement of the tSZ power spectrum over a wide range of multipoles, possibly , as we demonstrate in this paper. The proposed Primordial Inflation Explorer (PIXIE) experiment Kogut et al. (2011) will also be able to detect the tSZ power spectrum at high significance, as its wide spectral coverage and high spectral resolution will allow for very accurate extraction of the tSZ signal. However, its angular resolution is much lower than Planck’s, and thus the tSZ power spectrum will be measured over a much smaller range of multipoles (). However, the PIXIE data (after masking using X-ray cluster catalogs — see below) will permit tSZ measurements on large angular scales () that are essentially inaccessible to Planck due to its noise levels; these multipoles are precisely where one would expect the signature of scale-dependent bias induced by primordial non-Gaussianity to arise in the tSZ power spectrum. Assessing the amplitude and detectability of this signature is a primary motivation for this paper.
The tSZ power spectrum has been suggested as a potential cosmological probe by a number of authors over the past few decades (e.g., Cole & Kaiser (1988); Komatsu & Kitayama (1999); Komatsu & Seljak (2002)). Nearly all studies in the last decade have focused on the small-scale tSZ power spectrum () due to its role as a foreground in high angular resolution CMB measurements, and likely because without multi-frequency information, the tSZ signal has only been able to be isolated by looking for its effects on small scales (e.g., using an -space filter to upweight tSZ-dominated small angular scales in CMB maps). Much of this work was driven by the realization that the tSZ power spectrum is a very sensitive probe of the amplitude of matter density fluctuations, Komatsu & Seljak (2002). The advent of multi-frequency data promises measurements of the large-scale tSZ power spectrum very shortly, and thus we believe it is timely to reassess its value as a cosmological probe, including parameters beyond and including a realistic treatment of the uncertainties due to modeling of the ICM. We build on the work of Komatsu & Kitayama (1999) to compute the full angular power spectrum of the tSZ effect, including both the one- and two-halo terms, and moving beyond the Limber/flat-sky approximations where necessary.
Our primary interest is in assessing constraints from the tSZ power spectrum on currently unknown parameters beyond the CDM standard model: the amplitude of local primordial non-Gaussianity, , and the sum of the neutrino masses, . The values of these parameters are currently unknown, and determining their values is a key goal of modern cosmology.
Primordial non-Gaussianity is one of the few known probes of the physics of inflation. Models of single-field, minimally-coupled slow-roll inflation predict negligibly small deviations from Gaussianity in the initial curvature perturbations Acquaviva et al. (2003); Maldacena (2003). In particular, a detection of a non-zero bispectrum amplitude in the so-called “squeezed” limit () would falsify essentially all single-field models of inflation Maldacena (2003); Creminelli & Zaldarriaga (2004). This type of non-Gaussianity can be parametrized using the “local” model, in which describes the lowest-order deviation from Gaussianity Salopek & Bond (1990); Gangui et al. (1994); Komatsu & Spergel (2001):
Φ(→x)=ΦG(→x)+fNL(Φ2G(→x)−⟨Φ2G⟩)+⋯, (1)
where is the primordial potential and is a Gaussian field. Note that , where is the initial adiabatic curvature perturbation. Local-type non-Gaussianity can be generated in multi-field inflationary scenarios, such as the curvaton model Linde & Mukhanov (1997); Lyth & Wands (2002); Lyth et al. (2003), or by non-inflationary models for the generation of perturbations, such as the new ekpyrotic/cyclic scenario Buchbinder et al. (2008); Creminelli & Senatore (2007); Lehners & Steinhardt (2008). It is perhaps most interesting when viewed as a method with which to rule out single-field inflation, however. Current constraints on are consistent with zero Bennett et al. (2012); Giannantonio et al. (2013), but the errors will shrink significantly very soon with the imminent CMB results from Planck. We review the effects of on the large-scale structure of the universe in Section II.
In contrast to primordial non-Gaussianity, massive neutrinos are certain to exist at a level that will be detectable within the next decade or so; neutrino oscillation experiments have precisely measured the differences between the squared masses of the three known species, leading to a lower bound of eV on the total summed mass McKeown & Vogel (2004). The remaining questions surround their detailed properties, especially their absolute mass scale. The contribution of massive neutrinos to the total energy density of the universe today can be expressed as
Ων≈Mν93.14h2eV≈0.0078Mν0.1eV, (2)
where is the sum of the masses of the three known neutrino species. Although this contribution appears to be small, massive neutrinos can have a significant influence on the small-scale matter power spectrum. Due to their large thermal velocities, neutrinos free-stream out of gravitational potential wells on scales below their free-streaming scale Lesgourgues & Pastor (2006); Abazajian et al. (2011). This suppresses power on scales below the free-streaming scale. Current upper bounds from various cosmological probes assuming a flat CDM+model are in the range eV Sievers et al. (2013); Vikhlinin et al. (2009); Mantz et al. (2010b); de Putter et al. (2012), although a detection near this mass scale was recently claimed in Hou et al. (2012). Should the true total mass turn out to be near eV, its effect on the tSZ power spectrum may be marginally detectable using the Planck data even for fairly conservative assumptions about the ICM physics, as we show in this paper. With stronger ICM priors, Planck could achieve a detection for masses at this scale, using only the primordial CMB temperature power spectrum and the tSZ power spectrum. We review the effects of massive neutrinos on the large-scale structure of the universe in Section II.
In addition to the effects of both known and currently unknown cosmological parameters, we also model the effects of the physics of the ICM on the tSZ power spectrum. This subject has attracted intense scrutiny in recent years after the early measurements of tSZ power from ACT Dunkley et al. (2011) and SPT Lueker et al. (2010) were significantly lower than the values predicted from existing ICM pressure profile models (e.g., Komatsu & Seljak (2001)) in combination with WMAP5 cosmological parameters. Subsequent ICM modeling efforts have ranged from fully analytic approaches (e.g., Shaw et al. (2010)) to cosmological hydrodynamics simulations (e.g., Battaglia et al. (2010, 2012)), with other authors adopting semi-analytic approaches between these extremes (e.g., Trac et al. (2011); Bode et al. (2012)), in which dark matter-only -body simulations are post-processed to include baryonic physics according to various prescriptions. In addition, recent SZ and X-ray observations have continued to further constrain the ICM pressure profile from data, although these results are generally limited to fairly massive, nearby systems (e.g., Arnaud et al. (2010); Planck Collaboration et al. (2013)). We choose to adopt a parametrized form of the ICM pressure profile known as the Generalized NFW (GNFW) profile, with our fiducial parameter values chosen to match the constrained pressure profile fit from hydrodynamical simulations in Battaglia et al. (2012). In order to account for uncertainty in the ICM physics, we free two of the parameters in the pressure profile (the overall normalization and the outer logarithmic slope) and treat them as additional parameters in our model. This approach is discussed in detail in Section III.2. We use this profile to compute the tSZ power spectrum following the halo model approach, for which we provide a complete derivation in Appendix A.
In addition to a model for the tSZ signal, we must compute the tSZ power spectrum covariance matrix in order to forecast parameter constraints. There are two important issues that must be considered in computing the expected errors or signal-to-noise ratio (SNR) for a measurement of the tSZ power spectrum. First, we must assess how well the tSZ signal can actually be separated from the other components in maps of the microwave sky, including the primordial CMB, thermal dust, point sources, and so on. Following THEO () and CHT (), we choose to implement a multi-frequency subtraction technique that takes advantage of the unique spectral signature of the tSZ effect, and also takes advantage of the current state of knowledge about the frequency- and multipole-dependence of the foregrounds. Although there are other approaches to this problem, such as using internal linear combination techniques to construct a Compton- map from the individual frequency maps in a given experiment (e.g., Remazeilles et al. (2013, 2011); Leach et al. (2008)), we find this method to be fairly simple and robust. We describe these calculations in detail in Section IV.
Second, we must account for the extreme cosmic variance induced in the large-angle tSZ power spectrum by massive clusters at low redshifts. The one-halo term from these objects dominates the angular trispectrum of the tSZ signal, even down to very low multipoles Cooray (2001). The trispectrum represents a large non-Gaussian contribution to the covariance matrix of the tSZ power spectrum Komatsu & Seljak (2002), which is especially problematic at low multipoles. However, the trispectrum can be greatly suppressed by masking massive, low-redshift clusters using existing X-ray, optical, or SZ catalogs Komatsu & Kitayama (1999). This procedure can greatly increase the SNR for the tSZ power spectrum at low multipoles. For constraints on it also has the advantage of enhancing the relative importance of the two-halo term compared to the one-halo, thus showing greater sensitivity to the scale-dependent bias at low-. Moreover, even in a Gaussian cosmology, the inclusion of the two-halo term slightly changes the shape of the tSZ power spectrum, which likely helps break degeneracies amongst the several parameters which effectively only change the overall amplitude of the one-halo term; the relative enhancement of the two-halo term due to masking should help further in this regard. We consider two masking scenarios motivated by the flux limits of the cluster catalogs from all-sky surveys performed with the ROSAT X-ray telescope and the upcoming eROSITA555Extended ROentgen Survey with an Imaging Telescope Array, http://www.mpe.mpg.de/erosita/ X-ray telescope. These scenarios are detailed in Section V.0.2; by default all calculations and figures are computed for the unmasked scenario unless they are labelled otherwise.
Earlier studies have investigated the consequences of primordial non-Gaussianity for the tSZ power spectrum Sadeh et al. (2007); Roncarelli et al. (2010), though we are not aware of any calculations including the two-halo term (and hence the scale-dependent bias) or detailed parameter constraint forecasts. We are also not aware of any previous work investigating constraints on massive neutrinos from the tSZ power spectrum, although previous authors have computed their signature Shimon et al. (2011). Other studies have investigated detailed constraints on the primary CDM parameters from the combination of CMB and tSZ power spectrum measurements Taburet et al. (2010). Many authors have investigated constraints on and from cluster counts, though the results depend somewhat on the cluster selection technique and mass estimation method. Considering SZ cluster count studies only, Shimon et al. (2011) and Shimon et al. (2012) investigated constraints on from a Planck-derived catalog of SZ clusters (in combination with CMB temperature power spectrum data). The earlier paper found a uncertainty of eV while the later paper found eV; the authors state that the use of highly degenerate nuisance parameters degraded the results in the former study. In either case, the result is highly sensitive to uncertainties in the halo mass function, as the clusters included are deep in the exponential tail of the mass function. We expect that our results using the tSZ power spectrum should be less sensitive to uncertainties in the tail of the mass function, as the power spectrum is dominated at most angular scales by somewhat less massive objects (Komatsu & Seljak (2002). Finally, a very recent independent study Mak & Pierpaoli (2013) found eV for Planck SZ cluster counts (with CMB temperature power spectrum information added), although they estimated that this bound could be improved to eV with the inclusion of stronger priors on the ICM physics.
Our primary findings are as follows:
• The tSZ power spectrum can be detected with a total SNR using the imminent Planck data up to , regardless of masking;
• The tSZ power spectrum can be detected with a total SNR between and 22 using the future PIXIE data up to , with the result being sensitive to the level of masking applied to remove massive, nearby clusters;
• Adding the tSZ power spectrum information to the forecasted constraints from the Planck CMB temperature power spectrum and existing data is unlikely to significantly improve constraints on the primary cosmological parameters, but may give interesting constraints on the extensions we consider:
• If the true value of is near the WMAP9 ML value of , a future CV-limited experiment combined with eROSITA-masking could provide a detection, completely independent of the primordial CMB temperature bispectrum; alternatively, PIXIE could give evidence for such a value of with this level of masking;
• If the true value of is near 0.1 eV, the Planck tSZ power spectrum with eROSITA masking can provide upper limits competitive with the current upper bounds on ; with stronger external constraints on the ICM physics, Planck with eROSITA masking could provide evidence for massive neutrinos from the tSZ power spectrum, depending on the true neutrino mass;
• Regardless of the cosmological constraints, Planck will allow for a very tight constraint on the logarithmic slope of the ICM pressure profile in the outskirts of galaxy clusters, and may also provide some information on the overall normalization of the pressure profile (which sets the zero point of the relation).
The remainder of this paper is organized as follows. In Section II, we describe our models for the halo mass function and halo bias, as well as the effects of primordial non-Gaussianity and massive neutrinos on large-scale structure. In Section III, we describe our halo model-based calculation of the tSZ power spectrum, including the relevant ICM physics. We also demonstrate the different effects of each parameter in our model on the tSZ power spectrum. In Section IV, we consider the extraction of the tSZ power spectrum from the other components in microwave sky maps via multifrequency subtraction techniques. Having determined the experimental noise levels, in Section V we detail our calculation of the covariance matrix of the tSZ power spectrum, and discuss the role of masking massive nearby clusters in reducing the low- cosmic variance. In Section VI, we use our tSZ results to forecast constraints on cosmological and astrophysical parameters from a variety of experimental set-ups and masking choices. We also compute the expected SNR of the tSZ power spectrum detection for each possible scenario. We discuss our results and conclude in Section VII. Finally, in Appendix B, we provide a brief comparison between our forecasts and the Planck tSZ power spectrum results that were publicly released while this manuscript was under review Planck Collaboration et al. (2013).
The WMAP9+eCMB+BAO+ maximum-likelihood parameters Hinshaw et al. (2012) define our fiducial model (see Section III.3 for details). All masses are quoted in units of , where and is the Hubble parameter today. All distances and wavenumbers are in comoving units of . All tSZ observables are computed at GHz, since ACT and SPT have observed the tSZ signal at (or very near) this frequency, where the tSZ effect leads to a temperature decrement in the CMB along the line-of-sight (LOS) to a galaxy cluster.
## Ii Modeling Large-Scale Structure
In order to compute statistics of the tSZ signal, we need to model the comoving number density of halos as a function of mass and redshift (the halo mass function) and the bias of halos with respect to the underlying matter density field as a function of mass and redshift. Moreover, in order to extract constraints on and from the tSZ power spectrum, we must include the effects of these parameters on large-scale structure. We describe our approach to these computations in the following.
### ii.1 Halo Mass Function
We define the mass of a dark matter halo by the spherical overdensity (SO) criterion: () is the mass enclosed within a sphere of radius () such that the enclosed density is times the critical (mean matter) density at redshift . To be clear, subscripts refer to masses referenced to the critical density at redshift , with the Hubble parameter at redshift , whereas subscripts refer to masses referenced to the mean matter density at redshift , (this quantity is constant in comoving units).
We will generally work in terms of a particular SO mass, the virial mass, which we denote as . The virial mass is the mass enclosed within a radius Bryan & Norman (1998):
rvir=(3M4πΔcr(z)ρcr(z))1/3, (3)
where and . For many calculations, we need to convert between and various other SO masses (e.g., or ). We use the NFW density profile Navarro et al. (1997) and the concentration-mass relation from Duffy et al. (2008) in order to do these conversions, which require solving the following non-linear equation for (or ):
∫rδ,c04πr′2ρNFW(r′,M,cvir)dr′=43πr3δ,cρcr(z)δ (4)
where is the concentration parameter ( is the NFW scale radius) and we replace the critical density with the mean matter density in this equation in order to obtain instead of . After solving Eq. (4) to find , we calculate via .
The halo mass function, describes the comoving number density of halos per unit mass as a function of redshift. We employ the approach developed from early work by Press and Schechter Press & Schechter (1974) and subsequently refined by many other authors (e.g., Sheth & Tormen (1999); Sheth et al. (2001); Jenkins et al. (2001); Tinker et al. (2008)):
dn(M,z)dM = ¯ρmMdln(σ−1(M,z))dMf(σ(M,z)) (5) = −¯ρm2M2R(M)3σ2(M,z)dσ2(M,z)dR(M)f(σ(M,z)),
where is the variance of the linear matter density field smoothed with a (real space) top-hat filter on a scale at redshift :
σ2(M,z)=12π2∫k3Plin(k,z)W2(k,R(M))dlnk, (6)
where is the linear theory matter power spectrum at wavenumber and redshift . Note that the window function is a top-hat filter in real space, which in Fourier space is given by
W(k,R)=3x2(sinxx−cosx), (7)
where . In Eq. (5), the function is known as the halo multiplicity function. It has been measured to increasingly high precision from large -body simulations over the past decade Jenkins et al. (2001); Warren et al. (2006); Tinker et al. (2008); Bhattacharya et al. (2011). However, many of these calibrated mass functions are specified in terms of the friends-of-friends (FOF) mass rather than the SO mass, hindering their use in analytic calculations such as ours. For this reason, we use the parametrization and calibration from Tinker et al. (2008), where computations are performed in terms of the SO mass with respect to the mean matter density, , for a variety of overdensities. The halo multiplicity function in this model is parametrized by
(8)
where are (redshift- and overdensity-dependent) parameters fit from simulations. We use the values of these parameters appropriate for the halo mass function from (Tinker et al., 2008) with the redshift-dependent parameters given in their Eqs. (5)–(8); we will hereafter refer to this as the Tinker mass function. Note that the authors of that study caution against extrapolating their parameters beyond the highest redshift measured in their simulations () and recommend setting the parameters equal to their values at higher redshifts; we adopt this recommendation in our calculations. Also, note that our tSZ power spectrum calculations in Section III are phrased in terms of the virial mass , and thus we compute the Jacobian using the procedure described in Eq. (4) in order to convert the Tinker mass function to a virial mass function .
We compute the smoothed matter density field in Eq. (6) by first obtaining the linear theory matter power spectrum from CAMB at and subsequently rescaling by , where is the linear growth factor. We normalize by requiring that deep in the matter-dominated era (e.g., at ). The resulting is then used to compute the mass function in Eq. (5).
Note that we assume the mass function to be known to high enough precision that the parameters describing it can be fixed; in other words, we do not consider to be free parameters in our model. These parameters are certainly better constrained at present than those describing the ICM pressure profile (see Section III.2), and thus this assumption seems reasonable for now. However, precision cosmological constraints based on the mass function should in principle consider variations in the mass function parameters in order to obtain robust results, as has been done in some recent X-ray cluster cosmology analyses Mantz et al. (2010a). However, we leave the implications of these uncertainties for tSZ statistics as a topic for future work.
#### ii.1.1 Effect of Primordial non-Gaussianity
The influence of primordial non-Gaussianity on the halo mass function has been studied by many authors over the past two decades using a variety of approaches (e.g., Lucchin & Matarrese (1988); Colafrancesco et al. (1989); Chiu et al. (1998); Robinson et al. (2000); Koyama et al. (1999); Verde et al. (2001); Matarrese et al. (2000); Lo Verde et al. (2008); D’Amico et al. (2011)). The physical consequences of the model specified in Eq. (1) are fairly simple to understand for the halo mass function, especially in the exponential tail of the mass function where massive clusters are found. Intuitively, the number of clusters provides information about the tail of the probability distribution function of the primordial density field, since these are the rarest objects in the universe, which have only collapsed recently. For positive skewness in the primordial density field (), one obtains an increased number of massive clusters at late times relative to the case, because more regions of the smoothed density field have , the collapse threshold ( in the spherical collapse model). Conversely, for negative skewness in the primordial density field (), one obtains fewer massive clusters at late times relative to the case, because fewer regions of the smoothed density field are above the collapse threshold. As illustrated in recent analytic calculations and simulation measurements LoVerde & Smith (2011); Pillepich et al. (2010); Grossi et al. (2009); Wagner et al. (2010); Grossi et al. (2007), these changes can be quite significant for the number of extremely massive halos () in the late-time universe; for example, the abundance of such halos for can be times larger than in a Gaussian cosmology. These results have been used as a basis for recent studies constraining by looking for extremely massive outliers in the cluster distribution (e.g., Hoyle et al. (2011); Cayón et al. (2011); Mortonson et al. (2011); Enqvist et al. (2011); Harrison & Hotchkiss (2012); Hoyle et al. (2012)).
We model the effect of on the halo mass function by multiplying the Tinker mass function by a non-Gaussian correction factor:
(dndM)NG=dndMRNG(M,z,fNL), (9)
where is given by Eq. (5). We use the model for given by Eq. (35) in LoVerde & Smith (2011) (the “log-Edgeworth” mass function). In this approach, the density field is approximated via an Edgeworth expansion, which captures small deviations from Gaussianity. The Press-Schechter approach is then applied to the Edgeworth-expanded density field to obtain an expression for the halo mass function in terms of cumulants of the non-Gaussian density field. The results of LoVerde & Smith (2011) include numerical fitting functions for these cumulants obtained from -body simulations. We use both the expression for and the cumulant fitting functions from LoVerde & Smith (2011) to compute the non-Gaussian correction to the mass function. This prescription was shown to accurately reproduce the non-Gaussian halo mass function correction factor measured directly from -body simulations in LoVerde & Smith (2011), and in particular improves upon the similar prescription derived in Lo Verde et al. (2008) (the “Edgeworth” mass function).
Note that we apply the non-Gaussian correction factor to the Tinker mass function in Eq. (9), which is an SO mass function, as mentioned above. The prescription for computing makes no assumption about whether is an FOF or SO mass, so there is no logical flaw in this procedure. However, the comparisons to -body results in LoVerde & Smith (2011) were performed using FOF halos. Thus, without having tested the results of Eq. (9) on SO mass functions from simulations, our calculation assumes that the change in the mass function due to non-Gaussianity is quasi-universal, even if the underlying Gaussian mass function itself is not. This assumption was tested in Wagner et al. (2010) for the non-Gaussian correction factor from Lo Verde et al. (2008) (see Fig. 9 in Wagner et al. (2010)) and found to be valid; thus, we choose to adopt it here. We will refer to the non-Gaussian mass function computed via Eq. (9) using the prescription from LoVerde & Smith (2011) as the LVS mass function.
#### ii.1.2 Effect of Massive Neutrinos
It has long been known that massive neutrinos suppress the amplitude of the matter power spectrum on scales below their free-streaming scale, Lesgourgues & Pastor (2006):
kfs≈0.082H(z)H0(1+z)2(Mν0.1eV)h/Mpc. (10)
Neutrinos do not cluster on scales much smaller than this scale (i.e., ), as they are able to free-stream out of small-scale gravitational potential wells. This effect leads to a characteristic decrease in the small-scale matter power spectrum of order in linear perturbation theory Lesgourgues & Pastor (2006); Abazajian et al. (2011). Nonlinear corrections increase this suppression to for modes with wavenumbers Abazajian et al. (2011).
The neutrino-induced suppression of the small-scale matter power spectrum leads one to expect that the number of massive halos in the low-redshift universe should also be decreased. Several papers in recent years have attempted to precisely model this change in the halo mass function using both -body simulations and analytic theory Brandbyge et al. (2010); Marulli et al. (2011); Ichiki & Takada (2012). In Brandbyge et al. (2010), -body simulations are used to show that massive neutrinos do indeed suppress the halo mass function, especially for the largest, latest-forming halos (i.e., galaxy clusters). Moreover, the suppression is found to arise primarily from the suppression of the initial transfer function in the linear regime, and not due to neutrino clustering effects in the -body simulations. This finding suggests that an analytic approach similar to the Press-Schecter theory should work for massive neutrino cosmologies as well, and the authors subsequently show that a modified Sheth-Tormen formalism Sheth & Tormen (1999) gives a good fit to their simulation results. Similar -body simulations are examined in Marulli et al. (2011), who find generally similar results to those in Brandbyge et al. (2010), but also point out that the effect of on the mass function cannot be adequately represented by simply rescaling to a lower value in an analytic calculation without massive neutrinos. Finally, Ichiki & Takada (2012) study the effect of massive neutrinos on the mass function using analytic calculations with the spherical collapse model. Their results suggest that an accurate approximation is to simply input the -suppressed linear theory (coldbaryonic-only) matter power spectrum computed at to a CDM-calibrated mass function fit (note that a similar procedure was used in some recent X-ray cluster-based constraints on Vikhlinin et al. (2009)). The net result of this suppression can be quite significant at the high-mass end of the mass function; for example, eV leads to a factor of decrease in the abundance of halos at as compared to a massless-neutrino cosmology Ichiki & Takada (2012). We follow the procedure used in Ichiki & Takada (2012) in our work, although we input the suppressed linear theory matter power spectrum to the Tinker mass function rather than that of Bhattacharya et al. (2011), as was done in Ichiki & Takada (2012). We will refer to the -suppressed mass function computed with this prescription as the IT mass function.
### ii.2 Halo Bias
Dark matter halos are known to cluster more strongly than the underlying matter density field; they are thus biased tracers. This bias can depend on scale, mass, and redshift (e.g., BBKS (); Mo & White (1996); Smith et al. (2007)). We define the halo bias by
b(k,M,z)=√Phh(k,M,z)P(k,z), (11)
where is the power spectrum of the halo density field and is the power spectrum of the matter density field. Knowledge of the halo bias is necessary to model and extract cosmological information from the clustering of galaxies and galaxy clusters. For our purposes, it will be needed to compute the two-halo term in the tSZ power spectrum, which requires knowledge of .
In a Gaussian cosmology, the halo bias depends on mass and redshift but is independent of scale for , i.e. on large scales (e.g., Tinker et al. (2010)). We compute this linear Gaussian bias, , using the fitting function in Eq. (6) of Tinker et al. (2010) with the parameters appropriate for SO masses (see Table 2 in Tinker et al. (2010)). This fit was determined from the results of many large-volume -body simulations with a variety of cosmological parameters and found to be quite accurate. We will refer to this prescription as the Tinker bias model.
Although the bias becomes scale-dependent on small scales even in a Gaussian cosmology, it becomes scale-dependent on large scales in the presence of local primordial non-Gaussianity, as first shown in Dalal et al. (2008). The scale-dependence arises due to the coupling of long- and short-wavelength density fluctuations induced by local . We model this effect as a correction to the Gaussian bias described in the preceding paragraph:
b(k,M,z)=bG(M,z)+ΔbNG(k,M,z), (12)
where the non-Gaussian correction is given by Dalal et al. (2008)
ΔbNG(k,M,z)=2δc(bG(M,z)−1)fNLα(k,z). (13)
Here, (the spherical collapse threshold) and
α(k,z)=2k2T(k)D(z)c23ΩmH20 (14)
relates the linear density field to the primordial potential via . Note that is the linear matter transfer function, which we compute using CAMB. Since the original derivation in Dalal et al. (2008), the results in Eqs. (13) and (14) have subsequently been confirmed by other authors Slosar et al. (2008); Matarrese & Verde (2008); Giannantonio & Porciani (2010) and tested extensively on -body simulations (e.g., Desjacques et al. (2009); Dalal et al. (2008); Pillepich et al. (2010); Smith et al. (2012)). The overall effect is a steep increase in the large-scale bias of massive halos, which is even larger for highly biased tracers like galaxy clusters. We will refer to this effect simply as the scale-dependent halo bias.
The influence of massive neutrinos on the halo bias has been studied far less thoroughly than that of primordial non-Gaussianity. Recent -body simulations analyzed in Marulli et al. (2011) indicate that massive neutrinos lead to a nearly scale-independent increase in the large-scale halo bias. This effect arises because of the mass function suppression discussed in Section II.1.2: halos of a given mass are rarer in an cosmology than in a massless neutrino cosmology (for fixed ), and thus they are more highly biased relative to the matter density field. However, the amplitude of this change is far smaller than that induced by , especially on very large scales. For example, the results of Marulli et al. (2011) indicate an overall increase of % in the mean bias of massive halos at for eV as compared to . Our implementation of the scale-dependent bias due to local yields a factor of increase in the large-scale () bias of objects in the same mass range at for . Clearly, the effect of is much larger than that of massive neutrinos, simply because it is so strongly scale-dependent, while only leads to a small scale-independent change (at least on large scales; the small-scale behavior may be more complicated). Moreover, the change in bias due to is larger at higher redshifts (), whereas most of the tSZ signal originates at lower redshifts. Lastly, due to the smallness of the two-halo term in the tSZ power spectrum compared to the one-halo term (see Section III), small variations in the Gaussian bias cause essentially no change in the total signal. For all of these reasons, we choose to neglect the effect of massive neutrinos on the halo bias in our calculations.
## Iii Thermal SZ Power Spectrum
The tSZ effect results in a frequency-dependent shift in the CMB temperature observed in the direction of a galaxy group or cluster. The temperature shift at angular position with respect to the center of a cluster of mass at redshift is given by Sunyaev & Zeldovich (1970)
ΔT(→θ,M,z)TCMB = gνy(→θ,M,z) = gνσTmec2∫LOSPe(√l2+d2A|→θ|2,M,z)dl,
where is the tSZ spectral function with , is the Compton- parameter, is the Thomson scattering cross-section, is the electron mass, and is the ICM electron pressure at location with respect to the cluster center. We have neglected relativistic corrections in Eq. (III) (e.g., Nozawa et al. (2006)), as these effects are relevant only for the most massive clusters in the universe (). Such clusters contribute non-negligibly to the tSZ power spectrum at low-, and thus our results in unmasked calculations may be slightly inaccurate; however, the optimal forecasts for cosmological constraints arise from calculations in which such nearby, massive clusters are masked (see Section VI), and thus these corrections will not be relevant. Therefore, we do not include them in our calculations.
Note that we only consider spherically symmetric pressure profiles in this work, i.e. in Eq. (III). The integral in Eq. (III) is computed along the LOS such that , where is the angular diameter distance to redshift and is the angular distance between and the cluster center in the plane of the sky (note that this formalism assumes the flat-sky approximation is valid; we provide exact full-sky results for the tSZ power spectrum in Appendix A). In the flat-sky limit, a spherically symmetric pressure profile implies that the temperature decrement (or Compton-) profile is azimuthally symmetric in the plane of the sky, i.e., . Finally, note that the electron pressure is related to the thermal gas pressure via , where is the primordial hydrogen mass fraction. We calculate all tSZ power spectra in this paper at GHz, where the tSZ effect is observed as a decrement in the CMB temperature (). We make this choice simply because recent tSZ measurements have been performed at this frequency using ACT and SPT (e.g., Wilson et al. (2012); Crawford et al. (2013); Story et al. (2012); Sievers et al. (2013)), and thus the temperature values in this regime are perhaps more familiar and intuitive. All of our calculations can be phrased in a frequency-independent manner in terms of the Compton- parameter, and we will often use “y” as a label for tSZ quantities, although they are calculated numerically at GHz.
In the remainder of this section, we outline the halo model-based calculations used to compute the tSZ power spectrum, discuss our model for the gas physics of the ICM, and explain the physical effects of each cosmological and astrophysical parameter on the tSZ power spectrum.
### iii.1 Halo Model Formalism
We compute the tSZ power spectrum using the halo model approach (see Cooray & Sheth (2002) for a review). We provide complete derivations of all the relevant expressions in Appendix A, first obtaining completely general full-sky results and then specializing to the flat-sky/Limber-approximated case. Here, we simply quote the necessary results and refer the interested reader to Appendix A for the derivations. Note that we will work in terms of the Compton- parameter; the results can easily be multiplied by the necessary factors to obtain results at any frequency.
The tSZ power spectrum, , is given by the sum of the one-halo and two-halo terms:
Cyℓ=Cy,1hℓ+Cy,2hℓ. (16)
The exact expression for the one-halo term is given by Eq. (76):
Cy,1hℓ=∫dzχ(z)d2VdzdΩ∫dMdndM∣∣ ∣∣∫kdkJℓ+1/2(kχ(z))~y3D(k;M,z)∫cdz′H(z′)(1+z′)√χ(z′)Jℓ+1/2(kχ(z′))∣∣ ∣∣2, (17)
where is the comoving distance to redshift , is the comoving volume element per steradian, is the halo mass function discussed in Section II.1, is given in Eq. (68), and is a Bessel function of the first kind. In the flat-sky limit, the one-halo term simplifies to the following widely-used expression (given in e.g. Eq. (1) of Komatsu & Seljak (2002)), which we derive in Eq. (81):
Cy,1hℓ≫1≈∫dzd2VdzdΩ∫dMdn(M,z)dM|~yℓ(M,z)|2, (18)
where
~yℓ(M,z)≈4πrsℓ2s∫dxx2sin((ℓ+1/2)x/ℓs)(ℓ+1/2)x/ℓsy3D(x;M,z). (19)
Here, is a characteristic scale radius (not the NFW scale radius) of the profile given by and is the multipole moment associated with the scale radius. For the pressure profile from Battaglia et al. (2012) used in our calculations, the natural scale radius is . In our calculations, we choose to implement the flat-sky result for the one-halo term at all — see Appendix A for a justification of this decision and an assessment of the associated error at low- (the only regime where this correction would be relevant).
The exact expression for the two-halo term is given by Eq. (82):
(20)
where is the linear theory matter power spectrum at (which we choose to set equal to 30), is the halo bias discussed in Section II.2, and refers to the expression for given in Eq. (19) evaluated with . This notation is simply a mathematical convenience; no flat-sky or Limber approximation was used in deriving Eq. (82), and no appears in . In the Limber approximation Limber (1954), the two-halo term simplifies to the result given in Komatsu & Kitayama (1999), which we derive in Eq. (84):
Cy,2hℓ≫1≈∫dzd2VdzdΩ[∫dMdn(M,z)dMb(k,M,z)~yℓ(M,z)]2Plin(ℓ+1/2χ(z);z). (21)
We investigate the validity of the Limber approximation in detail in Appendix A. We find that it is necessary to compute the exact expression in Eq. (20) in order to obtain sufficiently accurate results at low-, where the signature of the scale-dependent bias induced by is present (looking for this signature is our primary motivation for computing the two-halo term to begin with). In particular, we compute the exact expression in Eq. (20) for , while we use the Limber-approximated result in Eq. (21) at higher multipoles.
The fiducial integration limits in our calculations are for all redshift integrals, for all mass integrals, and for all wavenumber integrals. We check that extending the wavenumber upper limit further into the nonlinear regime does not affect our results. Note that the upper limit in the mass integral becomes redshift-dependent in the masked calculations that we discuss below, in which the most massive clusters at low redshifts are removed from the computation.
We use the halo mass functions discussed in Section II.1 (Tinker, LVS, and IT) and the bias models discussed in Section II.2 (Tinker and scale-dependent bias) in Eqs. (18), (20), and (21). The only remaining ingredient needed to complete the tSZ power spectrum calculation is a prescription for the ICM electron pressure profile as a function of mass and redshift. Note that this approach to the tSZ power spectrum calculation separates the cosmology-dependent component (the mass function and bias) from the ICM-dependent component (the pressure profile). This separation arises from the fact that the small-scale baryonic physics that determines the structure of the ICM pressure profile effectively decouples from the large-scale physics described by the background cosmology and linear perturbation theory. Thus, it is a standard procedure to constrain the ICM pressure profile from cosmological hydrodynamics simulations (e.g., Battaglia et al. (2012); Bode et al. (2012)) or actual observations of galaxy clusters (e.g., Arnaud et al. (2010); Planck Collaboration et al. (2013), which are obtained for a fixed cosmology in either case (at present, it is prohibitively computationally expensive to run many large hydrodynamical simulations with varying cosmological parameters). Of course, it is also possible to model the ICM analytically and obtain a pressure profile (e.g., Komatsu & Seljak (2001); Shaw et al. (2010). Regardless of its origin (observations/simulations/theory), the derived ICM pressure profile can then be applied to different background cosmologies by using the halo mass function and bias model appropriate for that cosmology in the tSZ power spectrum calculations. We follow this approach.
Note that because the tSZ signal is heavily dominated by contributions from collapsed objects, the halo model approximation gives very accurate results when compared to direct LOS integrations of numerical simulation boxes (see Figs. 7 and 8 in Battaglia et al. (2012) for direct comparisons). In particular, the halo model agrees very well with the simulation results for , which is predominantly the regime we are interested in for this paper (on smaller angular scales effects due to asphericity and substructure become important, which are not captured in the halo model approach). These results imply that contributions from the intergalactic medium, filaments, and other diffuse structures are unlikely to be large enough to significantly impact the calculations and forecasts in the remainder of the paper. Contamination from the Galaxy is a separate issue, which we assume can be minimized to a sufficient level through sky cuts and foreground subtraction (see Section IV).
### iii.2 Modeling the ICM
We adopt the parametrized ICM pressure profile fit from Battaglia et al. (2012) as our fiducial model. This profile is derived from cosmological hydrodynamics simulations described in Battaglia et al. (2010). These simulations include (sub-grid) prescriptions for radiative cooling, star formation, supernova feedback, and feedback from active galactic nuclei (AGN). Taken together, these feedback processes typically decrease the gas fraction in low-mass groups and clusters, as the injection of energy into the ICM blows gas out of the cluster potential. In addition, the smoothed particle hydrodynamics used in these simulations naturally captures the effects of non-thermal pressure support due to bulk motions and turbulence, which must be modeled in order to accurately characterize the cluster pressure profile in the outskirts.
The ICM thermal pressure profile in this model is parametrized by a dimensionless GNFW form, which has been found to be a useful parametrization by many observational and numerical studies (e.g., Nagai et al. (2007); Arnaud et al. (2010); Plagge et al. (2012); Planck Collaboration et al. (2013)):
Pth(x)P200,c=P0(x/xc)γ[1+(x/xc)α]β,x≡r/r200,c, (22)
where is the thermal pressure profile, is the dimensionless distance from the cluster center, is a core scale length, is a dimensionless amplitude, , , and describe the logarithmic slope of the profile at intermediate (), large (), and small () radii, respectively, and is the self-similar amplitude for pressure at given by Kaiser (1986); Voit (2005):
P200,c=200GM200,cρcr(z)Ωb2Ωmr200,c. (23)
In Battaglia et al. (2012) this parametrization is fit to the stacked pressure profiles of clusters extracted from the simulations described above. Note that due to degeneracies the parameters and are not varied in the fit; they are fixed to and , which agree with many other studies (e.g., Nagai et al. (2007); Arnaud et al. (2010); Plagge et al. (2012); Planck Collaboration et al. (2013). In addition to constraining the amplitude of the remaining parameters, Battaglia et al. (2012) also fit power-law mass and redshift dependences, with the following results:
P0(M200,c,z) = 18.1(M200,c1014M⊙)0.154(1+z)−0.758 (24) xc(M200,c,z) = 0.497(M200,c1014M⊙)−0.00865(1+z)0.731 (25) β(M200,c,z) = 4.35(M200,c1014M⊙)0.0393(1+z)0.415. (26)
Note that the denominator of the mass-dependent factor has units of rather than as used elsewhere in this paper. The mass and redshift dependence of these parameters captures deviations from simple self-similar cluster pressure profiles. These deviations arise from non-gravitational energy injections due to AGN and supernova feedback, star formation in the ICM, and non-thermal processes such as turbulence and bulk motions Battaglia et al. (2012, 2012). Eqs. (22)–(26) completely specify the ICM electron pressure profile as a function of mass and redshift, and provide the remaining ingredient needed for the halo model calculations of the tSZ power spectrum described in Section III.1, in addition to the halo mass function and halo bias. We will refer to this model of the ICM pressure profile as the Battaglia model.
Although it is derived solely from numerical simulations, we note that the Battaglia pressure profile is in good agreement with a number of observations of cluster pressure profiles, including those based on the REXCESS X-ray sample of massive, clusters Arnaud et al. (2010), independent studies of low-mass groups at with Chandra Sun et al. (2011), and early Planck measurements of the stacked pressure profile of clusters Planck Collaboration et al. (2013).
We allow for a realistic degree of uncertainty in the ICM pressure profile by freeing the amplitude of the parameters that describe the overall normalization () and the outer logarithmic slope (). To be clear, we do not free the mass and redshift dependences for these parameters given in Eqs. (24) and (26), only the overall amplitudes in those expressions. The outer slope is known to be highly degenerate with the scale radius (e.g., Battaglia et al. (2012); Plagge et al. (2012)), and thus it is only feasible to free one of these parameters. The other slope parameters in Eq. (22) are fixed to their Battaglia values, which match the standard values in the literature. We parametrize the freedom in and by introducing new parameters and defined by:
P0(M200,c,z) = CP0×18.1(M200,c1014M⊙)0.154(1+z)−0.758 (27) β(M200,c,z) = Cβ×4.35(M200,c1014M⊙)0.0393(1+z)0.415. (28)
These parameters thus describe multiplicative overall changes to the amplitudes of the and parameters. The fiducial Battaglia profile corresponds to . We discuss our priors for these parameters in Section VI.
### iii.3 Parameter Dependences
Including both cosmological and astrophysical parameters, our model is specified by the following quantities:
{Ωbh2,Ωch2,ΩΛ,σ8,ns,CP0,Cβ,(fNL,Mν)}, (29)
which take the following values in our (WMAP9+BAO+ Hinshaw et al. (2012)) fiducial model:
{0.02240,0.1146,0.7181,0.817,0.9646,1.0,1.0,(0.0,0.0)}. (30)
As a reminder, the CDM parameters are (in order of their appearance in Eq. (29)) the physical baryon density, the physical cold dark matter density, the vacuum energy density, the rms matter density fluctuation on comoving scales of at , and the scalar spectral index. The ICM physics parameters and are defined in Eqs. (27) and (28), respectively, is defined by Eq. (1), and is the sum of the neutrino masses in units of eV. We have placed and in parentheses in Eq. (29) in order to make it clear that we only consider scenarios in which these parameters are varied separately: for all cosmologies that we consider with , we set , and for all cosmologies that we consider with , we set . In other words, we only investigate one-parameter extensions of the CDM concordance model.
For the primary CDM cosmological parameters, we use the parametrization adopted by the WMAP team (e.g., Hinshaw et al. (2012)), as the primordial CMB data best constrain this set. The only exception to this convention is our use of , which stands in place of the primordial amplitude of scalar perturbations, . We use both because it is conventional in the tSZ power spectrum literature and because it is a direct measure of the low-redshift amplitude of matter density perturbations, which is physically related more closely to the tSZ signal than . However, this choice leads to slightly counterintuitive results when considering cosmologies with , because in order to keep fixed for such scenarios we must increase (to compensate for the suppression induced by in the matter power spectrum).
For the fiducial model specified by the values in Eq. (30), we find that the tSZ power spectrum amplitude at is at GHz. This corresponds to at GHz (the relevant ACT frequency) and at GHz (the relevant SPT frequency). The most recent measurements from ACT and SPT find corresponding constraints at these frequencies of Sievers et al. (2013) and Crawford et al. (2013) (using their more conservative error estimate). Note that the SPT constraint includes information from the tSZ bispectrum, which reduces the error by a factor of 2. Although it appears that our fiducial model predicts a level of tSZ power too high to be consistent with these observations, the results are highly dependent on the true value of , due to the steep dependence of the tSZ power spectrum on this parameter. For example, recomputing our model predictions for gives at GHz and at GHz, which are consistent at with the corresponding ACT and SPT constraints. Given that is within the error bar for WMAP9 Hinshaw et al. (2012), it is difficult to assess the extent to which our fiducial model may be discrepant with the ACT and SPT results. The difference can easily be explained by small changes in and is also sensitive to variations in the ICM physics, which we have kept fixed in these calculations. We conclude that our model is not in significant tension with current tSZ measurements (or other cosmological parameter constraints), and is thus a reasonable fiducial case around which to consider variations.
Figs. 1 and 2 show the tSZ power spectra for our fiducial model and several variations around it, including the individual contributions of the one- and two-halo terms. In the fiducial case, the two-halo term is essentially negligible for , as found by earlier studies Komatsu & Kitayama (1999), and it only overtakes the one-halo term at very low- (). However, for , the influence of the two-halo term is greatly enhanced due to the scale-dependent bias described in Section II.2, which leads to a characteristic upturn in the tSZ power spectrum at low-. In addition, induces an overall amplitude change in both the one- and two-halo terms due to its effect on the halo mass function described in Section II.1.1. While this amplitude change is degenerate with the effects of other parameters on the tSZ power spectrum (e.g., ), the low- upturn caused by the scale-dependent bias is a unique signature of primordial non-Gaussianity, which motivates our assessment of forecasts on using this observable later in the paper.
Fig. 2 shows the results of similar calculations for . In this case, the effect is simply an overall amplitude shift in the one- and two-halo terms, and hence the total tSZ power spectrum. The amplitude shift is caused by the change in the halo mass function described in Section II.1.2. Note that the sign of the amplitude change is somewhat counterintuitive, but arises due to our choice of as a fundamental parameter instead of , as mentioned above. In order to keep fixed while increasing , we must increase , which leads to an increase in the tSZ power spectrum amplitude. Although this effect is degenerate with that of and other parameters, the change in the tSZ power spectrum amplitude is rather large even for small neutrino masses (% for eV, which is larger than the amplitude change caused by ). This sensitivity suggests that the tSZ power spectrum may be a useful observable for constraints on the neutrino mass sum.
We demonstrate the physical effects of each parameter in our model on the tSZ power spectrum in Figs. 411, including the effects on both the one- and two-halo terms individually. Note that the limits on the vertical axis in each plot differ, so care must be taken in assessing the amplitude of the change caused by each parameter. Except for and , the figures show % variations in each of the parameters, which facilitates easier comparisons between their relative influences on the tSZ power spectrum. On large angular scales (), the most important parameters (neglecting and ) are , , and . On very large angular scales (), the effect of is highly significant, but its relative importance is difficult to assess, since the true value of may be unmeasurably small. Note, however, that is important over the entire range we consider, even if its true value is as small as eV. Comparison of Figs. 4 and 8 indicates that the amplitude change induced by eV (for fixed ) is actually slightly larger than that caused by a % change in around its fiducial value.
We now provide physical interpretations of the effects shown in Figs. 411:
• (4): The change to the halo mass function discussed in Section II.1.1 leads to an overall increase (decrease) in the amplitude of the one-halo term for (). This increase or decrease is essentially -independent, is also seen at in the two-halo term, and is % for . More significantly, the influence of the scale-dependent halo bias induced by is clearly seen in the dramatic increase of the two-halo term at low-. This increase is significant enough to be seen in the total power spectrum despite the typical smallness of the two-halo term relative to the one-halo term for a Gaussian cosmology.
• (4): The presence of massive neutrinos leads to a decrease in the number of galaxy clusters at late times, as discussed in Section II.1.2. This decrease would lead one to expect a corresponding decrease in the amplitude of the tSZ signal, but Fig. 4 shows an increase. This increase is a result of our choice of parameters — we hold constant while increasing , which means that we must simultaneously increase , the initial amplitude of scalar fluctuations. This increase in (for fixed ) leads to the increase in the tSZ power spectrum amplitude seen in Fig. 4. The effect appears to be essentially -independent, although it tapers off slightly at very high-.
• (6): Increasing (decreasing) the amount of baryons in the universe leads to a corresponding increase (decrease) in the amount of gas in galaxy clusters, and thus a straightforward overall amplitude shift in the tSZ power spectrum (which goes like ).
• (6): In principle, one would expect that changing should change the tSZ power spectrum, but it turns out to have very little effect, as pointed out in Komatsu & Seljak (2002), who argue that the effect of increasing (decreasing) on the halo mass function is cancelled in the tSZ power spectrum by the associated decrease (increase) in the comoving volume to a given redshift. We suspect that the small increase (decrease) seen in Fig. (6) when decreasing (increasing) is due to the fact that we hold constant when varying . Thus, is also held constant, and thus is decreased (increased) when is increased (decreased). This decrease (increase) in the baryon fraction leads to a corresponding decrease (increase) in the tSZ power spectrum amplitude, as discussed in the previous item. The slight -dependence of the variations may be due to the associated change in required to keep constant, which leads to a change in the angular diameter distance to each cluster, and hence a change in the angular scale associated with a given physical scale. Increasing (decreasing) requires increasing (decreasing) in order to leave unchanged, which decreases (increases) the distance to each cluster, shifting a given physical scale in the spectrum to lower (higher) multipoles. However, it is hard to completely disentangle all of the effects described here, and in any case the overall influence of is quite small.
• (8): An increase (decrease) in has several effects which all tend to decrease (increase) the amplitude of the tSZ power spectrum. First, is decreased (increased), which leads to fewer (more) halos, although this effect is compensated by the change in the comoving volume as described above. Second, for fixed , this decrease (increase) in leads to fewer (more) baryons in clusters, and thus less (more) tSZ power. Third, more (less) vacuum energy leads to more (less) suppression of late-time structure formation due to the decaying of gravitational potentials, and thus less (more) tSZ power. All of these effects combine coherently to produce the fairly large changes caused by seen in Fig. 8. The slight -dependence may be due to the associated change in required to keep and constant, similar (though in the opposite direction) to that discussed in the case above. Regardless, this effect is clearly subdominant to the amplitude shift caused by , which is only slightly smaller on large angular scales than that caused by (for a % change in either parameter).
• (8): Increasing (decreasing) leads to a significant overall increase (decrease) in the amplitude of the tSZ power spectrum, as has been known for many years (e.g. Komatsu & Seljak (2002)). The effect is essentially -independent and appears in both the one- and two-halo terms.
• (10): An increase (decrease) in leads to more (less) power in the primordial spectrum at wavenumbers above (below) the pivot, which we set at the WMAP value (no ). Since the halo mass function on cluster scales probes much smaller scales than the pivot (i.e., much higher wavenumbers ), an increase (decrease) in should lead to more (fewer) halos at late times. However, since we require to remain constant while increasing (decreasing) , we must decrease (increase) in order to compensate for the change in power on small scales. This is similar to the situation for described above. Thus, an increase (decrease) in actually leads to a small decrease (increase) in the tSZ power spectrum on most scales, at least for the one-halo term. The cross-over in the two-halo term is likely related to the pivot scale after it is weighted by the kernel in Eq. (21), but this is somewhat non-trivial to estimate. Regardless, the overall effect of on the tSZ power spectrum is quite small.
• (10): Since sets the overall normalization of the ICM pressure profile (or, equivalently, the zero-point of the relation), the tSZ power spectrum simply goes like .
• (11): Since sets the logarithmic slope of the ICM pressure profile at large radii (see Eq. (22)), it significantly influences the total integrated thermal energy of each cluster, and thus the large-angular-scale behavior of the tSZ power spectrum. An increase (decrease) in leads to a decrease (increase) in the pressure profile at large radii, and therefore a corresponding decrease (increase) in the tSZ power spectrum on angular scales corresponding to the cluster outskirts and beyond. On smaller angular scales, the effect should eventually vanish, since the pressure profile on small scales is determined by the other slope parameters in the pressure profile. This trend is indeed seen at high- in Fig. 11. Note that a % change in leads to a much larger change in the tSZ power spectrum at nearly all angular scales than a % change in , suggesting that simply determining the zero-point of the relation may not provide sufficient knowledge of the ICM physics to break the long-standing ICM-cosmology degeneracy in tSZ power spectrum measurements. It appears that constraints on the shape of the pressure profile itself will be necessary.
## Iv Experimental Considerations
In this section we estimate the noise in the measurement of the tSZ power spectrum. The first ingredient is instrumental noise. We describe it for the Planck experiment and for an experiment with the same specifications as the proposed PIXIE satellite Kogut et al. (2011). The second ingredient is foregrounds777To be precise we will consider both foregrounds, e.g. from our galaxy, and backgrounds, e.g. the CMB. On the other hand, in order to avoid repeating the cumbersome expression “foregrounds and backgrounds” we will collectively refer to all these contributions as foregrounds, sacrificing some semantic precision for the sake of an easier read.. We try to give a rather complete account of all these signals and study how they can be handled using multifrequency subtraction. Our final results are in Fig. 12. Because of the several frequency channels, Planck and to a much larger extent PIXIE can remove all foregrounds and have a sensitivity to the tSZ power spectrum mostly determined by instrumental noise.
### iv.1 Multifrequency Subtraction
We discuss and implement multifrequency subtraction888We are thankful to K. Smith for pointing us in this direction. along the lines of THEO (); CHT (). The main idea is to find a particular combination of frequency channels that minimize the variance of some desired signal, in our case the tSZ power spectrum. We hence start from
^aSZℓm=∑νiwiaℓm(νi)gνi, (31)
where refers to our estimator for the tSZ signal at 150 GHz (the conversion to a different frequency is straightforward), are the different frequency channels relevant for a given experiment, are the weights for each channel, are spherical harmonic coefficients of the total measured temperature anisotropies at each frequency and finally is the tSZ spectral function defined in Section III, allowing us to convert from Compton- to . We can decompose the total signal according to with enumerating all other contributions. We will assume that , i.e. different contributions are uncorrelated with each other. Dropping for the moment the and indices, the variance of is then found to be
⟨^aSZ^aSZ⟩=CSZ(∑νiwi)2+∑νiνjwiwj∑fCf(νi,νj)gνigνj, (32)
where is the tSZ power spectrum at 150 GHz as given in Eq. (16) and (again the index is implicit) is the cross-correlation at different frequencies of the of each foreground component (we will enumerate and describe these contributions shortly). To simplify the notation, in the following we will use
C(νi,νj)=Cij≡∑fCf(νi,νj)gνigνj. (33)
We now want to minimize with the constraint that the weights describe a unit response to a tSZ signal, i.e., . This can be done using a Langrange multiplier and solving the system
∂i[⟨(^aSZ)2⟩+λ(∑iwi−1)]=∂λ[⟨(^aSZ)2⟩+λ(∑iwi−1)]=0. (34)
Because of the constraint , the term in is independent of (alternatively one can keep this term and see that it drops out at the end of the computation). Then the solution of the first equation can be written as
wi=−λ(C−1)ijej=0, (35)
where is just a vector with all ones and is the inverse of in Eq. (33). This solution can then be plugged back into the constraint to give
wi=(C−1)ijejek(C−1)klel, (36)
which is our final solution for the minimum-variance weights. From Eq. (32) we see that the total noise in each | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9767000675201416, "perplexity": 1092.7035816385676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176256.21/warc/CC-MAIN-20201124111924-20201124141924-00430.warc.gz"} |
https://psychology.stackexchange.com/questions/15705/how-does-the-forgetting-curve-change-after-repeated-exposure-to-the-same-item-we | # How does the Forgetting curve change after repeated exposure to the same item we would like to memorise?
There has been a long debate going on on the actual form of the Forgetting curve, but my question will be about its change when we review a previously memorised fact - Unfortunately, I couldn't seem to find too much information (or perhaps I used wrong terms) on this topic.
Maybe the most up-to-date and least divisive research paper on the topic of the actual form of the Forgetting curve would be Averell, L., & Heathcote, A. (2011). The form of the forgetting curve and the fate of memories. Journal of Mathematical Psychology, which I have seen referenced here multiple times. According to it the shape of the curve is best described as $$R(t) = a + (1-a)*b*(1+t)^{-β}$$ which would yield $1$ at $t=0$ if the encoding happened without a problem (which we assume did). My concrete question would be that how the three parameter change after repeated exposure to the fact (where we assume that at this specific $t_r$ the retention increases to $1$ again). I think it is fair to assume that it is going to be yet another curve describable with a Power function starting at $t_r$ which should have a higher asymptote as the original function.
A picture (although not scientific nor correct in terms of the actual form) illustrating my question has been already posted here at How are these review-forgetting curve calculated?, although for different purposes than mine.
## 1 Answer
Currently, I am working on various demonstration graphs of various spacing effect algorithms. From the research it seems like two of the parameters exist to constrain the outputted values...
The parameters a and b are also assumed bounded between zero and one, and hence R(t) is similarly bounded, which must necessarily be the case as R(t) is a probability. Enforcing this bound is important as otherwise data fits can be inflated (see Navarro, Pitt, & Myung, 2004, for further discussion).
I think Ebbinghaus' original function R(t) = e ^ -(t/s) (see this thread) can more easily explain things. t is time and s is the strength of the memory. After every review, s increases. If you plot this out, you see that if you increase s, the forgetting curve function flattens out.
• Welcome to Psychology.SE. You suggested to "see Navarro, Pitt, & Myung, 2004, for further discussion" yet there is no reference information in order to find the article. – Chris Rogers Dec 16 '18 at 15:14
• @ChrisRogers I quoted a part from the paper that OP referenced. – George Boole Dec 17 '18 at 20:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9393013715744019, "perplexity": 768.1919013244584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400232211.54/warc/CC-MAIN-20200926004805-20200926034805-00074.warc.gz"} |
https://quant.stackexchange.com/questions/42553/realized-volatility-forecast-vs-implied-volatility | # Realized volatility forecast vs Implied volatility
I have forecasts of realized volatility, as well as implied volatility for individual traded options of the S&P500.
I want to simulate a simple trading strategy; that is, buy signal=1 if forecasted realized volatility is greater than current implied volatility. However, the literature documents that implied volatility is usually higher than realized volatility.
Are there any better approach for this simulation?
• What implied volatility are you talking about? – will Nov 8 '18 at 21:29
• Implied volatility for 30 days to expiry options obtained from WRDS. I have deannualized it, but it still appears a notch higher than realized volatility – Bryan Lwy Nov 9 '18 at 2:00
• What strike are the options? Is there skew? – will Nov 11 '18 at 19:42
On average the implied volatility is higher than realized volatility because you can easily imagine that dealers will ask customers to pay a premium to write them options and risk manage them
you can have a look at this paper for instance
And to finish here is a little exercise to test your thinking about realized vs implied vol and hopefully help you design properly your strategy: suppose you are long a call option which you purchased at some implied vol level $$\sigma_0$$ and which you are delta-hedging. Now imagine that i grant you that "on average" over a period of time the stock realized volatility $$\sigma_r$$ will be higher than $$\sigma_0$$. In other words i tell you that in expectation $$E[\sigma_r] > \sigma_0$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9166679382324219, "perplexity": 1649.920624566264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487637721.34/warc/CC-MAIN-20210618134943-20210618164943-00633.warc.gz"} |
http://mathhelpforum.com/differential-equations/205367-laplaces-equation-rectangle-mixed-b-c-s-print.html | # Laplace's equation on a rectangle with mixed b.c.s
Printable View
• October 15th 2012, 07:41 AM
sarideli18
PDE
Hello, I do not know how to apply the nonlinear boundary conditions with this problem:
Any idea?
• October 15th 2012, 08:05 AM
TheEmptySet
Re: Laplace's equation on a rectangle with mixed b.c.s
Quote:
Originally Posted by sarideli18
Hello, I do not know how to apply the nonlinear boundary conditions with this problem:
Solve Laplace's equation on the rectangle 0< x< L, 0< y< H with the boundary conditions du/dx(0, y) = 0, du/dx(L, y)=y, du/dy(x, 0)=0, U(x, H)=x.
Any idea?
Since you have a finite domain, The usual method, is to assume a product solution and expand a fourier series.
$u(x,y)=X(x)Y(y)$
This gives
$X''Y+XY''=0 \iff \frac{X''}{-X}=\frac{Y''}{Y}=\lambda^2$
So you get the two ODE's
$X''+\lambda^2X=0 \quad Y''-\lambda^2Y=0$
Now solve each of these ODE's and use the boundary conditions to find the eigenfunctions and expand the solution.
• October 15th 2012, 11:42 AM
sarideli18
Re: Laplace's equation on a rectangle with mixed b.c.s
Well, actually the problem is that I have 2 non-homogeneous boundary conditions. That's why I can't find the eigenvalues.
• October 15th 2012, 12:05 PM
TheEmptySet
Re: Laplace's equation on a rectangle with mixed b.c.s
Quote:
Originally Posted by sarideli18
Well, actually the problem is that I have 2 non-homogeneous boundary conditions. That's why I can't find the eigenvalues.
You need to use the superposition principle. You can solve two seperate problems and add the solutions together. Set one of the non-homogeneous boundary conditions equal to zero and solve that problem. Then set the other boundary condition equal to zero and solve it.
The answer is the sum of the two different solutions.
• October 15th 2012, 12:08 PM
sarideli18
Re: Laplace's equation on a rectangle with mixed b.c.s
Does it work when one of the boundary conditions is neumann type and the other is dirichlet?
• October 15th 2012, 12:12 PM
TheEmptySet
Re: Laplace's equation on a rectangle with mixed b.c.s
Quote:
Originally Posted by sarideli18
Does it work when one of the boundary conditions is neumann type and the other is dirichlet?
Yes it will work with any boundary conditions as long all of the other conditions are preserved in each problem.
This link may be helpful. Look at the very last section on the web page. The last two paragraphs will be useful.
Laplace Equation - Wikiversity
• October 15th 2012, 12:16 PM
sarideli18
Re: Laplace's equation on a rectangle with mixed b.c.s
okay, thank you!
• October 15th 2012, 12:20 PM
TheEmptySet
Re: Laplace's equation on a rectangle with mixed b.c.s
Yes so explicitly you need to solve these two BVP's
Problem 1
$u_x(0,y)=0 \ u_x(L,y)=0 \ u_y(x,0)=0 \ u(x,H)=x$
Problem 2
$u_x(0,y)=0 \ u_x(L,y)=y \ u_y(x,0)=0 \ u(x,H)=0$
Since you are forcing the other boundary conditions to be zero, they will not mess up the other boundary conditions when you sum of the solutions. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8707470297813416, "perplexity": 753.8923056258352}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276964.14/warc/CC-MAIN-20160524002116-00105-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/351829/propositional-logic-inductive-proof | # Propositional Logic Inductive Proof
I am working on a problem to prove, but I do not understand it completely. Where should I use inductive method? What is the base case? And so on. Here is my problem:
A truth assignment $M$ is a function that maps propositional variables to $\{0, 1\}$ ($1$ for true and $0$ for false). We write $M\vDash x$ if $x$ is true under $M$. We define a partial order $\leq$ on truth assignments by $M \le M'$ if $M(p) \le M'(p)$ for every propositional variable $p$.
A propositional formula is positive if it only contains connectives $\wedge$ and $\vee$ (i.e., no negation $\lnot$ or implication $\Rightarrow$).
Use Proof By Induction to show that for any truth assignments $M$ and $M'$ such that $M\le M'$, and any positive propositional formula $x$, if $M \vDash x$, then $M' \vDash x$.
I am really confused. Any help is welcome. Thank you.
-
Perhaps a more concrete statement of what you need to show is the following:
Let $M \leq M^\prime$ be truth assignments. Then for every positive formula $\varphi$ either $M \not\models \varphi$ or $M^\prime \models \varphi$.
We also note that the family of positive formulas has its own inductive definition, similar to the definition of the family of all formulas:
• Every propositional variable $p$ is a positive formula.
• If $\varphi , \psi$ are positive formulas, then so are $\varphi \wedge \psi$ and $\varphi \vee \psi$.
• (No other formula is positive.)
So to prove the result, we need to show two-and-a-half things: Let $M \leq M^\prime$ be truth assignments, then
1. if $p$ is a propositional variable, then either $M \not\models p$ or $M^\prime \models p$.
2. if $\varphi , \psi$ are formulas such that either $M \not\models \varphi$ or $M^\prime \models \varphi$, and similarly for $\psi$, then
• either $M \not\models \varphi \wedge \psi$ or $M^\prime \models \varphi \wedge \psi$; and
• either $M \not\models \varphi \vee \psi$ or $M^\prime \models \varphi \vee \psi$.
To leave with a parting
Hint: The definition of the partial order $\leq$ will be very useful in the base case (1), and the definition of $\models$ (for compound formulas) will be very useful in the inductive step (2).
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https://www.physicsforums.com/threads/expressing-the-full-width-at-half-maximum-do-by-t.371744/ | # Expressing the full width at half maximum Δω by τ
1. Jan 22, 2010
### K09
1. The problem statement, all variables and given/known data
I need to express the full width at half maximum Δω as an expression in terms of τ.
2. Relevant equations
I'm guessing that the expression of Δω will fulfill this task.
3. The attempt at a solution
The problem is I have no idea how to go about the calculations :|
Last edited: Jan 22, 2010
2. Jan 23, 2010
### K09
How do I delete this post?!
Similar Discussions: Expressing the full width at half maximum Δω by τ | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9131219983100891, "perplexity": 1445.5405663471247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887224.19/warc/CC-MAIN-20180118091548-20180118111548-00620.warc.gz"} |
http://mathhelpforum.com/discrete-math/148930-probability.html | # Math Help - Probability
1. ## Probability
Let x and y be uniformly distributed independent random variables over [0,1]. What is the probability the distance between x and y is less than 1/2?
2. This can be a geometry problem, since the distributions are uniform.
You need $p(|x-y| < \frac{1}{2})$
For x > y, we have $x - y < \frac{1}{2}$ or $y > x - \frac{1}{2}$
This cuts off a corner.
You do x < y to cut off the other corner. You're almost done.
3. Originally Posted by mathman88
Let x and y be uniformly distributed independent random variables over [0,1]. What is the probability the distance between x and y is less than 1/2?
Unit square:
Area of shaded region = 3/4.
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https://eugene1806.wordpress.com/ | ## Classification of Newton maps: realization
August 8, 2009
On the way to realize combinatorial models as real Newton maps the plan was to use Thurston’s theorem. In order to do so, one needs to prove that the branched covering, specified by the combinatorial data, doesn’t have Thurston obstructions.
This is the point where theorem of K.Pilgrim and T.Lei comes in use. It basically says that obstructions don’t intersect the preimages of arc systems.
The main difficulty in our case was to eliminate the case when the obstruction does intersect the arc system itself. Due to the theorem mentioned above this would imply that it cannot intersect the preimages of the arc systems – that is exactly the point where we aim at a contradiction.
Suppose it happens. Then as follows from the theorem the obstruction must be a Levy cycle. It is also known (BFH) that the complement components of the curves in the Levy cycles can only contain periodic mark points which in our case would be periodic points on extended Hubbard trees lying in the Julia set.
Such points being repelling for corresponding polynomial-like maps, have at least one external ray landing on it, i.e. there exists a preimage of the arc system which intersects the Thurston obstruction.
## Hubbard Trees
August 7, 2009
It is known that Hubbard trees serve as a model for Julia sets of post-critically finite polynomials. What does it mean precisely? In fact, there is an inverse limit space of sequences $(x_1,x_2,\ldots,x_n,\ldots)$ together with the shift map, such that the obtained dynamical system is topologically conjugate to the dynamical system $p:J \to J$, where $J$ is the Julia set of the polynomial $p$.
One of the explanations of the fact comes from the theory of IMG (iterated monodromy groups). Perhaps, the fact itself could be proven much easier way without involving the notion of IMG, but this is just one point of view on it.
IMG itself could be defined of partial self-coverings. A bit more general situuation arises when one defines IMG on topological automaton.
Definition. Topological automaton is a quadruple $(M,M_1,f,\iota)$, where $M,M_1$ – topological spaces, $f:M_1 \to M$ is a finite covering map, $\iota:M_1 \to M$ is a continuous map.
In the case of partial self-coverings, when $M_1 \subset M$, $\iota$ is just an embedding. But in this general setting we have in fact two maps between two topological spaces. One can iterate this topological automaton and associate an inverse limit system to it.
For example $M_2$ is defined as a pullback of the map $f:M_1 \to M_0=M$ under the map $\iota$. $M_2 = \{ (x,y) \in M^2_1: f(y)=\iota(x) \}$.
In the same way $M_n = \{ (x_1,x_2,\ldots,x_n) \in M^n_1 | f(x_{i+1})=\iota(x_i) \}$
## Topological Entropy
August 3, 2009
There are several qualitative invariants of dynamical systems, such as density of periodic points, topological mixing, hyperbolicity. The answer whether a concrete dynamical system has this property or not would be “Yes” or “No”. There do exists quantitative invariants, for example the growth of period points. But perhaps the most important quantitative invariant is a topological entropy.
Definition. In some sense, the topological entropy describes the growth of orbits all at the same time. To be precise, for two points $x,y \in X$ one can define the metric
$\displaystyle d_n^f(x)=max_{0 \leq i \leq n-1} d(f^i(x),f^i(y))$
for a given positive integer $n$. Let $B(x,r,n)=\{y \in X: d_n^f(x,y), the ball of radius $r$. Let $S_d(f,r,n)$ be the minimal number of such ball, sufficient to cover the whole of $X$.
Then
$\displaystyle h_{top}(f)=\lim_{r \to 0} \lim_{n \to \infty} \frac{ \log S_d(f,r,n)}{n}$ .
There are several other definitions for the topological entropy. Sometimes it is not quite useful to use covering sets for computation reasons. For example, one can use the so-called separating sets, i.e. the sets such that the pairwise distance between any two points in the metric $d_n^f > r$ . The cardinality of the maximal separating set is denoted by $N_d(f,r,n)$ and can be used similarly for the definition of the topological entropy, substituting $S_d(f,r,n)$ by $N_d(f,r,n)$ in the formula above.
For example, for the expanding map $E_m(x)=mx \quad mod (1)$, the topological entropy $h_{top}(E_m)=\log|m|$.
Proof. In general, for expanding maps the distance between two points growth until it becomes larger than some constant ($1/2m$ for $E_m$). Let us choose two points $x,y$ with $d(x,y)<\frac{1}{2m^n}$.Then
$d_n^{E_m}(x,y)=d(E_m^{n-1}(x),E_m^{n-1}(x)) = m^{n-1}d(x,y)$.
If we want to have
$d_n^{E_m}(x,y)> m^{-k}$, then we must have $d(x,y)>m^{-k-n}$.
Hence if we choose $S=\{i\cdot m^{-k-n}: 0\leq i \leq m^{k+n}-1 \}$ all the points from $S$ will have pairwise distances between each other at least $m^{-k}$. And the set $S$ can serve as a separating set with $|S|=m^{k+n}$, we obtain that
$h_{top}(E_m)=\log |m|$.
Bowen has asked to find a topological entropy of the complex polynomial of degree $d$. More generally M.Lyubich gave the answer for any rational function in the complex plane. The lower bound $h_{top}(f) \geq \log (deg f)$ was known before due to Misiurewisz-Przytycki theorem.
Theorem. Let $f: S^2 \to S^2$ be a rational function, that is not a constant. Then the topological entropy
$h_{top}(f)=\log (deg f)$.
## Self maps of hyperbolic surfaces with two fixed points
July 23, 2009
Problem. Let $f$ be a self map of a hyperbolic surface $X$ such that $f$ has two fixed points. Show that $f$ is a finite order automorphism, i.e. there exists $n$ such that $f^{\circ n} = id_X$.
Solution. Let $\pi$ be a covering map from the unit disk $D$ to $X$. First of all, notice that due to Schwarz-Pick theorem such an $f$ is either
• A global isometry, i.e. a conformal isomorphisms of $X$
• A covering map, but is not one-to-one, i.e. a local isometry
• Strictly decreases the Poincare metric
It is easy to see that in the last case $f$ cannot have two fixed points. Therefore let’s assume that $f$ is a covering map. Since the covering is universal, one can lift the map $f$ to the map $\tilde{f}: D \to D$.
Also note that $\pi \circ \tilde{f} = f \circ \pi$. Assume that $p,q$ are the fixed points of $f$ and $\pi(0)=p$, perhaps composing $\tilde{f}$ with an automorphism of $D$ we can also assume that $\tilde{f}(0)=0$. Remember that $f$ is not strictly decreasing Poincare metric, the same applies to its lift $\tilde{f}$, hence $\tilde{f}$ must be a rotation around the origin (it is not difficult to classify all automorpisms of the unit disk), $\tilde{f}(z)=e^{2\pi i \alpha}$ for some real $\alpha$.
Note that $\pi \circ \tilde{f}^{\circ n} = f^{\circ n} \circ \pi$, hence $\tilde{f}^{\circ n}(\tilde{q})$ is in the fiber of $q$ and therefore the set $\{ \tilde{f}^{\circ n}(\tilde{q}) \in N \}$ is a discrete set which implies that $\alpha$ is rational and we are done.
P.S. Actually, if we were talking about self-coverings of compact surfaces, the situation would be easier: for compact Riemann surfaces we do have Riemann-Hurwitz formula:
$2g(x)-2=n(2g(Y)-2)+\sum_{j=1}^n (e_j-1)$,
where $f:X \to Y$ is a ramified covering of degree $n$. If $latex X=Y$ then we would obtain that either $g=1$ or $n=1$. Inother words, for compact Riemann surfaces of genus $>1$ the only self-covering is a biholomorphism.
## Rational Maps don’t have Levy cycles
July 13, 2009
Here is a sketch of the following:
Theorem. The rational functions doesn’t have Levy cycles.
Proof. Suppose it does and denote by $\Gamma$ the corresponding Levy cycle. Choose $\gamma_1,\gamma_2,\ldots, \gamma_n$ the geodesics representatives in $\Gamma$. Then the map $f: \bar{C}-f^{-1}(P_f) \to \bar{C}-P_f$ is the covering map, hence a local isometry and preserves the distances locally. Therefore
$l_{\bar{C}-f^{-1}(P_f)} (\gamma'_i) = l_{\bar{C}-P_f} (\gamma_{i+1})$ where $\gamma'_i$ is the component of $f^{-1}(\gamma_{i+1})$ which is in the same homotpy class with $\gamma_i$. From the other hand $\bar{C}-f^{-1}(P_f) \subset \bar{C}-P_f$ and
$l_{\bar{C}-P_f} (\gamma_{i+1}) = l_{\bar{C}-f^{-1}(P_f)} (\gamma'_i) > l_{\bar{C}-P_f} (\gamma_i)$ and we get that each $\gamma_i$ is strictly shorter than $\gamma_{i+1}$, which is impossible.
## Linearization and circle diffeomorphisms
July 11, 2009
Today, i have learn that there is quite interesing interplay between the theory of linearization of maps $f(z)=\lambda z + O(z^2)$ and analytic circle diffeomorphism theory.
In fact, according to results of P.Marco, for every such $f(z)$ there is a fully invariant set $K$, which is called a Siegel continua, or Hedgehog (in the Cremer case) such that if we consider the conformal representation of the complement $\bar{C} - K$ to the complement of the unit disk, then it conjugates the map $f(z)$ to an anylitc circle diffeomorphism, which actually even has the same roattion number.And the last property is crucial!
The idea of the proof of the existence of such $K$ is basically the following: in case of the rational multiplier $\lambda$ there are quite a lot of things known about the topology in the neighborhood of the fixed point $0$: Leau-Fatou flower theorem. Which states that one can construct attracting and repelling petals, all have $0$ on the boundary, which interchange and form the whole neighborhood of $0$. As a candidate for $K$ in the rational case one can take the union of intersections of attracting and repelling petals: they are clearly fully invariant and the other properties can also be established. Then it left to use the density argument and show that the density of rational translates to our case in terms of existence of these continua $K$.
It follows that we have a dictionary between these two theories. Even though a lot of things are the same in this dictionary, there are differences. For example (the only one i know so far), the set of lineariziable quadratic maps with irrationally indifferent fixed point – is the set of Bruno numbers (one way – Bruno, Siegel, the necessity – Yoccoz). However, for circle diffeomorphisms, the linearizability condition is for the rotation number to be Herman!
## Hello world!
July 11, 2009
Welcome to WordPress.com. This is your first post. Edit or delete it and start blogging! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 106, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.975961446762085, "perplexity": 860.0816984263812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948529738.38/warc/CC-MAIN-20171213162804-20171213182804-00655.warc.gz"} |
http://math.stackexchange.com/questions/25263/algebraic-version-of-finite-covering-of-a-compact-space-is-compact | # algebraic version of “finite covering of a compact space is compact”
The following statement is an exercise in point set topology: If $E \to X$ is a covering with nonempty finite fibers and $X$ is compact, then also $E$ is compact. Now Grothendieck generalized covering theory so that in particular separable field extensions may be regarded as coverings.
Question: What is the corresponding statement in field theory?
-
I thought field extensions were embeddings and group extensions were coverings? – PEV Mar 6 '11 at 15:16
Branderburg: I think the corrispondence stated by Grothendieck is not so deep to allow questions such yours. In specific the corrispondence should be something like: – Giovanni De Gaetano May 3 '11 at 22:00
"The category of finite coverings over a (connected) topological space $X$ is equivalent to the category of finite sets under the action of the pro-fundamental group of the space. The category of finite separable algebras (not fields) over a given field $k$ is anti-equivalent (not equivalent) to the category of finite sets under the action of the pro-Galois group of the field." – Giovanni De Gaetano May 4 '11 at 7:39
In this sense, fixed a topological space $X$ and a field $k$ such that the pro-fundamental group of $X$ and the pro-Galois group of $k$ are isomorphic, the category of finite coverings of $X$ is anti-equivalent to the category of the finite separable algebras over $k$. This anti-equivalence conserve notions such as the connection, but I don't see how to translate the notion of compactness in the algebraic context. I'll think about it. – Giovanni De Gaetano May 4 '11 at 7:39
I apologize for the long comment, I hope to transform it in an answer in some time. – Giovanni De Gaetano May 4 '11 at 7:39
Fix a scheme $S$ and a proper morphism $f:X\to Y$ of $S$-schemes. Suppose that $Y$ is proper over $S$. If $f$ is proper, then $X$ is proper over $S$. This is simply because proper morphisms are stable under composition.
How does one prove that proper morphisms are stable under composition? One simply proves this property for finite type morphisms of schemes and separated morphisms. Then you're done.
If you stick to fields, all morphisms are separated so to prove that proper morphisms are stable under composition in this case, you simply have to prove that if you have a tower $K\subset L\subset M$ of finite degree field extensions, then $K\subset M$ is of finite degree. This is an easy fact. In conclusion, the proof of the statement for fields is easy and the statement itself doesn't give any nontrivial information in the case of fields.
I think Rayleigh added the additional hypotheses of "finite etale" to his statement, because he wanted to mimic the set-up of a "covering".
-
Fix a scheme $S$.
Let $f:X\longrightarrow Y$ be a finite etale morphism of $S$-schemes, with $Y$ an integral $S$-scheme.
Note that $f$ is proper.
Therefore, if $Y$ is proper over $S$, we have that $X$ is also proper over $S$.
This is (I believe) the analogue of the statement in your question. (Take $S$ to be the spectrum of a field.)
-
I see. Do we need at all that $f$ is étale? – Martin Brandenburg Oct 11 '11 at 15:14
Also, is there any nontrivial statement about fields (considering the original question) which we can derive from this? – Martin Brandenburg Oct 11 '11 at 15:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9473130702972412, "perplexity": 321.2059561409834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645208021.65/warc/CC-MAIN-20150827031328-00248-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-math-topics/126382-vector-problem.html | # Math Help - Vector Problem
1. ## Vector Problem
Given 3 points A(0,2,7) , B(5,-3,2) and C(1,1,1). (1)Find the position vector of the point R on AB such that CR is perpendicular to AB. Hence (2)find the perpendicular distance from C to AB and the (3)position vector of the reflection of C in AB.
Here's my solution, i just have trouble finding the "position vector of the reflection of C in AB."
Vector AB = OB - OA
= (5-0,-3-2,2-7)
=(5,-5,-5)
vector equation of line AB l: (0,2,7) + λ(5,-5,-5)
vector OR = (5λ, 2-5λ, 7-5λ)
vector CR= OR - OC = (5λ-1, 1-5λ, 6-5λ)
vector CR.AB=0 , since cos(90) = 0, given by scalar product
(5λ-1, 1-5λ, 6-5λ).(5,-5,-5) = 0
λ=8/15
vector OR= (8/3, -2/3, 13/3)= 1/3(8,-2,13)
vector CR=(5/3,-5/3,10/3)
Magnitude of CR = (50/3)^0.5
now i'm having trouble with the reflection part
2. Consider $OC$ (the position of point $C$)
$= OR + RC$
Now the position vector of the reflection of $C$ in $AB$ ,
it should be $OR - RC$
$= \frac{1}{3} [(8,-2,13) - (5,-5,10) ]$
$= \frac{1}{3} (3 , 3 , 3 ) = (1,1,1)$
3. the answer is 1/3(13,-7,23) but thanks for helping
4. Originally Posted by simplependulum
Consider $OC$ (the position of point $C$)
$= OR + RC$
Now the position vector of the reflection of $C$ in $AB$ ,
it should be $OR - RC$
$= \frac{1}{3} [(8,-2,13) - (5,-5,10) ]$
$= \frac{1}{3} (3 , 3 , 3 ) = (1,1,1)$
Dear simplependulum,
You have made a slight mistake, instead of $\frac{1}{3} [(8,-2,13) - (5,-5,10)]$ it should be $\frac{1}{3} [(8,-2,13) - (-5,5,-10)]$ since $RC=(-5,5,-10)$. (You have accidently taken CR instead of RC).
Dear mephisto50,
The above correction will give you the correct answer.
Thank you. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8628075122833252, "perplexity": 2635.8464894472504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997859240.8/warc/CC-MAIN-20140722025739-00139-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://groupprops.subwiki.org/wiki/Commutator | # Commutator
## Definition
### Definition with symbols
The term commutator is used in group theory in two senses: the commutator map (typically with respect to the left action convention), and the set of possible images of that map. The commutator map is the following map:
$\! (x,y) \mapsto xyx^{-1}y^{-1}$
And is denoted by outside square brackets (that is, the commutator of $x$ and $y$ is denoted as $[x,y]$). The image of this map is termed the commutator of $x$ and $y$.
An element of the group is termed a commutator if it occurs as the commutator of some two elements of the group.
Sometimes the commutator map is defined somewhat differently (typically with respect to the right action convention), i.e., as:
$\! (x,y) \mapsto x^{-1}y^{-1}xy$
Although this alters the commutator map, it does not alter the set of elements that are commutators, because the commutator of $x$ and $y$ in one definition becomes the commutator of $x^{-1}$ and $y^{-1}$ in the other definition.
## Facts
### Identity element is a commutator
In fact, whenever $xy = yx$, $[x,y] = e$. Hence, in particular, the commutator of any element with itself is the identity element.
### Inverse of a commutator is a commutator
The inverse of the commutator $[x,y]$ is the commutator $[y,x]$ (this statement is true regardless of which definition of commutator we follow).
### Product of commutators need not be a commutator
In general, a product of commutators need not be a commutator.
## Iterated commutators
Since the commutator of two elements of the group is itself an element of the group, it can be treated as one of the inputs to the commutator map yet again. This allows us to construct iterated commutators.
A basic commutator is a word that can be expressed purely by iterating the commutator operations.
A simple commutator is an iterated commutator that can be expressed as a left-normed commutator, i.e., a commutator of the form:
$[[[\dots [ x_1, x_2 ], x_3 ] \dots, x_n]$
## Subgroups related to the commutator
The subgroup generated by all the commutators is termed the derived subgroup. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 16, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9553729295730591, "perplexity": 285.17918759350414}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315811.47/warc/CC-MAIN-20190821065413-20190821091413-00362.warc.gz"} |
https://homework.cpm.org/category/CC/textbook/cc2/chapter/5/lesson/5.3.5/problem/5-146 | Home > CC2 > Chapter 5 > Lesson 5.3.5 > Problem5-146
5-146.
Multiple Choice: Which of the following expressions could be used to find the average (mean) of the numbers $k$, $m$, and $n$?
The average of a set of numbers is the sum of the values divided by the number of values in the set.
Which of the above choices best represents the average? Do any choices include a summation of the numbers $k$, $m$, and $n$ and also divide by this number of values?
1. $k+m+n$
1. $3(k+m+n)$
1. $\frac { k + m + n } { 3 }$
$\frac{k+m+n}{3}$
1. $3k+m+n$ | {"extraction_info": {"found_math": true, "script_math_tex": 11, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9193041920661926, "perplexity": 384.3537905737034}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738864.9/warc/CC-MAIN-20200812024530-20200812054530-00403.warc.gz"} |
https://www.gradesaver.com/textbooks/math/calculus/thomas-calculus-13th-edition/chapter-12-vectors-and-the-geometry-of-space-section-12-4-the-cross-product-exercises-12-4-page-719/47 | ## Thomas' Calculus 13th Edition
$\dfrac{\sqrt {21}}{2}$
The area of the given triangle is $Area = \dfrac{1}{2}|\vec{AB}\times\vec{AC}|$ $$\vec{AB}= -\hat{i}+2\hat{j} \\ \vec{AC}=\hat{j}-2\hat{k}$$ Now, $$\vec{AB}\times\vec{AC}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\-1&2&0\\0&1&-2\end{vmatrix} \\ =-4\hat{i}-2\hat{j}-\hat{k}$$ and $$|\vec{AB}\times\vec{AC}|=\sqrt {16+4+1}= \sqrt {21}$$ Now, $Area= \dfrac{\sqrt {21}}{2}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.99265456199646, "perplexity": 366.7053121920979}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945473.69/warc/CC-MAIN-20230326142035-20230326172035-00748.warc.gz"} |
https://www.physicsforums.com/threads/coulombs-electricity.392167/ | # Homework Help: Coulomb's Electricity
1. Apr 4, 2010
### ppdjoy
1. The problem statement, all variables and given/known data
Two identical small metal spheres with q1 > 0 & |q1| > |q2| attract each other with a force of magnitude 75.6 mN, as shown in the http://localhostr.com/files/ffac57/1.JPG" [Broken].
The spheres are then brought together until they are touching. At this point, the spheres are in electrical contact so that the charges can move from one sphere to the other until both spheres have the same final charge, q.
http://localhostr.com/files/62ffcf/2.JPG" [Broken]
After the charges on the spheres have come to equilibrium, the spheres are moved so that they are again 3.33 m apart. Now the spheres repel each other with a force of magnitude 8.316 mN.
http://localhostr.com/files/6911b9/3.JPG" [Broken]
The Coulomb constant is 8.98755*10^9 N*m^2/C^2
What is the initial charge q1 on the first sphere?
2. Relevant equations
F = Ke q1*q2/r2
3. The attempt at a solution
I was thinking that using this formula, I can get to q1. But apparently, it's not leading me anywhere.
Can anyone tell me if I'm using the right formula or not? If not, then which formula do I use? I didn't understand this material in class very well, so, if someone kindly explains how to do this problem, I'll really appreciate it.
Thanks
Last edited by a moderator: May 4, 2017
2. Apr 4, 2010
### rl.bhat
Since they are attracting initially, q1 and q2 must have opposite charges.
When they are brought in contact, they will have the equal charges of the same sign, and the charge in each sphere will be .........? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9285865426063538, "perplexity": 724.2042747368569}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156376.8/warc/CC-MAIN-20180920020606-20180920040606-00385.warc.gz"} |
http://math.stackexchange.com/questions/118097/connectedness-vs-path-connectedness | # connectedness vs. path connectedness
Is there a general rule of what kind of sets it is easier to prove connectedness using path connectedness or regular connectedness?
I understand that path connected $\implies$ connected, but are there situations where it's easier to prove using path connectedness vs. connectedness and vice-versa? One example I know of is that proving that an interval is connected in $\mathbb{R}$ by constructing a function
$\gamma:[0,1]\rightarrow [a,b]: \gamma(t)=c+t(d-c), t\in [0,1]$
which is clearly continuous. The connectedness approach is a lot more laborious. So my question is, is there a general rule of what kind of sets it is easier to prove connectedness using path connectedness or regular connectedness?
-
Given that connected doesn't imply path connected, I guess you're asking for what spaces these properties are equivalent? – Lepidopterist Mar 9 '12 at 3:54
Couldn't you just prove $[a,b]$ is connected by the same proof that $[0,1]$ is connected? You need to prove $[0,1]$ is connected anyhow to say that path-connected implies connected. Perhaps the most important class of spaces for which connected implies path-connected is manifolds. – Brett Frankel Mar 9 '12 at 4:01
I can use the fact that path connected implies connected without proof in my class. So I was wondering of what were some examples where this might come in handy -- where I am asked to prove some set is connected and use path connectedness instead – Emir Mar 9 '12 at 5:05
@jay: No, Emir is asking whether there is asking whether there is a good heuristic for deciding which of the two is easier to prove for spaces that are path-connected. – Brian M. Scott Mar 9 '12 at 9:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9189565181732178, "perplexity": 265.59106862582144}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398450762.4/warc/CC-MAIN-20151124205410-00290-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://statisticelle.com | ## One, Two, U: Examples of common one- and two-sample U-statistics
My previous two blog posts revolved around derivation of the limiting distribution of U-statistics for one sample and multiple independent samples.
For derivation of the limiting distribution of a U-statistic for a single sample, check out Getting to know U: the asymptotic distribution of a single U-statistic.
For derivation of the limiting distribution of a U-statistic for multiple independent samples, check out Much Two U About Nothing: Extension of U-statistics to multiple independent samples.
The notation within these derivations can get quite complicated and it may be a bit unclear as to how to actually derive components of the limiting distribution.
In this blog post, I provide two examples of both common one-sample U-statistics (Variance, Kendall’s Tau) and two-sample U-statistics (Difference of two means, Wilcoxon Mann-Whitney rank-sum statistic) and derive their limiting distribution using our previously developed theory.
## Asymptotic distribution of U-statistics
### One sample
For a single sample, , the U-statistic is given by
where is a symmetric kernel of degree .
For a review of what it means for to be symmetric, check out U-, V-, and Dupree Statistics.
In the examples covered by this blog post, , so we can re-write as,
Alternatively, this is equivalent to,
The limiting variance of is given by,
where
or equivalently,
Note that when , .
For , these expressions reduce to
where
and
The limiting distribution of for is then,
For derivation of the limiting distribution of a U-statistic for a single sample, check out Getting to know U: the asymptotic distribution of a single U-statistic.
### Two independent samples
For two independent samples denoted and , the two-sample U-statistic is given by
where is a kernel that is independently symmetric within the two blocks and .
In the examples covered by this blog post, , reducing the U-statistic to,
The limiting variance of is given by,
where
and
Equivalently,
and
For , these expressions reduce to
where
and
The limiting distribution of for and is then,
For derivation of the limiting distribution of a U-statistic for multiple independent samples, check out Much Two U About Nothing: Extension of U-statistics to multiple independent samples.
## Much Two U About Nothing: Extension of U-statistics to multiple independent samples
Thank you very much to the lovely Feben Alemu for pointing me in the direction of https://pungenerator.org/ as a means of ensuring we never have to go without a brilliant title! With great power comes great responsibility.
## Review
Statistical functionals are any real-valued function of a distribution function , . When is unknown, nonparametric estimation only requires that belong to a broad class of distribution functions , typically subject only to mild restrictions such as continuity or existence of specific moments.
For a single independent and identically distributed random sample of size , , a statistical functional is said to belong to the family of expectation functionals if:
1. takes the form of an expectation of a function with respect to ,
2. is a symmetric kernel of degree .
A kernel is symmetric if its arguments can be permuted without changing its value. For example, if the degree , is symmetric if .
If is an expecation functional and the class of distribution functions is broad enough, an unbiased estimator of can always be constructed. This estimator is known as a U-statistic and takes the form,
such that is the average of evaluated at all distinct combinations of size from .
For more detail on expectation functionals and their estimators, check out my blog post U-, V-, and Dupree statistics.
Since each appears in more than one summand of , the central limit theorem cannot be used to derive the limiting distribution of as it is the sum of dependent terms. However, clever conditioning arguments can be used to show that is in fact asymptotically normal with mean
and variance
where
The sketch of the proof is as follows:
1. Express the variance of in terms of the covariance of its summands,
1. Recognize that if two terms share common elements such that,
conditioning on their shared elements will make the two terms independent.
2. For , define
such that
and
Note that when , and , and when , and .
3. Use the law of iterated expecation to demonstrate that
and re-express as the sum of the ,
Recognizing that the first variance term dominates for large , approximate as
4. Identify a surrogate that has the same mean and variance as but is the sum of independent terms,
so that the central limit may be used to show
5. Demonstrate that and converge in probability,
and thus have the same limiting distribution so that
For a walkthrough derivation of the limiting distribution of for a single sample, check out my blog post Getting to know U: the asymptotic distribution of a single U-statistic.
This blog post aims to provide an overview of the extension of kernels, expectation functionals, and the definition and distribution of U-statistics to multiple independent samples, with particular focus on the common two-sample scenario.
## Getting to know U: the asymptotic distribution of a single U-statistic
After my last grand slam title, U-, V-, and Dupree statistics I was really feeling the pressure to keep my title game strong. Thank you to my wonderful friend Steve Lee for suggesting this beautiful title.
## Overview
A statistical functional is any real-valued function of a distribution function such that
and represents characteristics of the distribution and include the mean, variance, and quantiles.
Often times is unknown but is assumed to belong to a broad class of distribution functions subject only to mild restrictions such as continuity or existence of specific moments.
A random sample can be used to construct the empirical cumulative distribution function (ECDF) ,
which assigns mass to each .
is a valid, discrete CDF which can be substituted for to obtain . These estimators are referred to as plug-in estimators for obvious reasons.
For more details on statistical functionals and plug-in estimators, you can check out my blog post Plug-in estimators of statistical functionals!
Many statistical functionals take the form of an expectation of a real-valued function with respect to such that for ,
When is a function symmetric in its arguments such that, for e.g. , it is referred to as a symmetric kernel of degree . If is not symmetric, a symmetric equivalent can always be found,
where represents the set of all permutations of the indices .
A statistical functional belongs to a special family of expectation functionals when:
1. , and
2. is a symmetric kernel of degree .
Plug-in estimators of expectation functionals are referred to as V-statistics and can be expressed explicitly as,
so that is the average of evaluated at all possible permutations of size from . Since the can appear more than once within each summand, is generally biased.
By restricting the summands to distinct indices only an unbiased estimator known as a U-statistic arises. In fact, when the family of distributions is large enough, it can be shown that a U-statistic can always be constructed for expectation functionals.
Since is symmetric, we can require that , resulting in combinations of the subscripts . The U-statistic is then the average of evaluated at all distinct combinations of ,
While within each summand now, each still appears in multiple summands, suggesting that is the sum of correlated terms. As a result, the central limit theorem cannot be relied upon to determine the limiting distribution of .
For more details on expectation functionals and their estimators, you can check out my blog post U-, V-, and Dupree statistics!
This blog post provides a walk-through derivation of the limiting, or asymptotic, distribution of a single U-statistic .
## U-, V-, and Dupree statistics
To start, I apologize for this blog’s title but I couldn’t resist referencing to the Owen Wilson classic You, Me, and Dupree – wow! The other gold-plated candidate was U-statistics and You. Please, please, hold your applause.
My previous blog post defined statistical functionals as any real-valued function of an unknown CDF, , and explained how plug-in estimators could be constructed by substituting the empirical cumulative distribution function (ECDF) for the unknown CDF . Plug-in estimators of the mean and variance were provided and used to demonstrate plug-in estimators’ potential to be biased.
Statistical functionals that meet the following two criteria represent a special family of functionals known as expectation functionals:
1) is the expectation of a function with respect to the distribution function ; and
2) the function takes the form of a symmetric kernel.
Expectation functionals encompass many common parameters and are well-behaved. Plug-in estimators of expectation functionals, named V-statistics after von Mises, can be obtained but may be biased. It is, however, always possible to construct an unbiased estimator of expectation functionals regardless of the underlying distribution function . These estimators are named U-statistics, with the “U” standing for unbiased.
This blog post provides 1) the definitions of symmetric kernels and expectation functionals; 2) an overview of plug-in estimators of expectation functionals or V-statistics; 3) an overview of unbiased estimators for expectation functionals or U-statistics. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8626257181167603, "perplexity": 833.1276570721393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362969.51/warc/CC-MAIN-20211204094103-20211204124103-00115.warc.gz"} |
http://en.wikipedia.org/wiki/Square-free | # Square-free
In mathematics, an element r of a unique factorization domain R is called square-free if it is not divisible by a non-trivial square. That is, every s such that $s^2\mid r$ is a unit of R.
## Alternate characterizaions
Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements
$r=p_1p_2\cdots p_n$
Then r is square-free if and only if the primes pi are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number).
## Examples
Common examples of square-free elements include square-free integers and square-free polynomials. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8376005291938782, "perplexity": 374.0343110944015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131300905.36/warc/CC-MAIN-20150323172140-00159-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.youphysics.education/work-and-energy/work-and-energy-problems/work-and-energy-problem-14/ | # Work and Energy - Water park
Problem Statement:
One of the attractions of a water park consists of a circular track without friction of radius R with a swimming pool below (see the figure). A bather slips from the top of the track without initial speed, determine the value of the angle θ when he leaves the track.
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Solution:
When the bather stops being in contact with the track, the normal force is canceled. Therefore, If we write Newton’s second law for the forces acting on the bather when he leaves the track, we can calculate his speed in that moment. And by using the principle of the conservation of energy we will then be able to calculate the angle θ.
First we draw the forces that act on the bather when he slides on the circular track as well as the axes that we will use to make projections for Newton’s second law:
The forces acting on the bather are the weight (because he is close to the surface of the Earth) and the normal (for being on the track). Therefore Newton’s second law for this situation is written as follow:
After doing the projection on the axes tangent and perpendicular to the trajectory we obtain:
We have represented the projections of the weight on these axes in the following figure:
From equation (1) we would obtain the tangential acceleration of the bather.
We will use equation (2) to determine the speed of the bather when he leaves the circular track. At this point the normal becomes zero because there is no more contact between the bather and the track. We impose this condition in equation (2) and substitute the magnitude of normal acceleration to obtain:
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We will now apply the principle of conservation of energy between points A (at the highest point of the circumference) and B (point at which the bather takes off from the track).
The bather’s mechanical energy between points A and B is conserved, since the normal force produces no work because it is perpendicular to the motion and the weight is a conservative force. Therefore between points A and B we can write:
After substituting the value of the heights and the speed of the bather at point B given by equation (2) we obtain:
And we can deduce the angle θ after simplifying equation (3):
Observe that this result does not depend on the mass of the bather nor the radius of the circular track.
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http://perfectgeeks.com/smartsoft-free-pdf-word-converter-review/ | # SmartSoft Free PDF to Word Converter Review
(No Ratings Yet)
There aren’t too many reliable free PDF to Word converters available as standalone programs for Windows. but this program can be taken in consideration if you have files that have more text than images or other elements like tables, lists etc.
For simple text, the program is working acceptable, although you need to make some edits in resulted DOC file.
## 1. Easy to use
A straightforward interface where is clear what you need to do. Select the PDF file and then choose a folder where DOC file will be saved.
## 2. Preview document
Includes an area where you can view the content of selected PDF file. Allows to zoom in or out.
## 3. Fast conversion
It displays a real time progress bar and will prompt you when conversion is finished, to open file or folder.
## 4. How it looks original PDF file before conversion to DOC
I tested a PDF file with text with different fonts and colors, also have an image and a table.
## 5. Resulted DOC file
As you can see, text is preserved, but the table is over the image. The image is bigger than in PDF file where was resized with drag and drop.
## 6. Text is editable
As you can see, i can edit text in DOC file and modify it as I want. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8542816042900085, "perplexity": 2480.5269745873557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912204077.10/warc/CC-MAIN-20190325153323-20190325175323-00256.warc.gz"} |
https://cs.stackexchange.com/questions/91156/is-it-possible-for-a-data-structure-to-have-constant-time-access-insertion-delet?noredirect=1 | # Is It Possible For a Data Structure to Have Constant Time Access/Insertion/Deletion?
I'm trying to come up with a data structure that could access, insert, and delete any element in constant time. I know that's pretty difficult, but I'm just doing it to invoke thought and understanding about computer science. However, I'm starting to question whether it is even possible.
My theory is: For access to be in constant time, the data would need to be stored in a static location (like an array). And for insertion/deletion to be in constant time, the data would need to be stored dynamically using pointers or some sort of lookup table (like a linked list). Therefore, no data structure can have all three operations run in constant time.
Is this reasoning correct?
EDIT:
access(index): returns element at given index
insert(element, index): Inserts element into 'index' and shifts everything after 'index' right one index
delete(index): Removes element at 'index' and corrects indexes so that there are no gaps (i.e. shifts everything after 'index' left one index)
• See also this question. Note that the requirements are slightly different, they use 'insertAfter' instead of 'insert', whatever the difference is. – TilmannZ Apr 27 '18 at 8:28
## How do we get lower bounds?
As D.W. noted, lower bounds are hard. But that doesn't mean that no progress can be made. To get a lower bound, it is required (well, not strictly, but you won't get very far without one) that you make a (possible restrictive) model of all algorithms or data-structures that can solve your problem.
For example, the $\Omega(n\log n)$ lower bound for sorting holds only for the restrictive 'comparison based sorting' algorithms. (this is why counting sort can 'beat' this time in some cases: it isn't comparison based)
## Lower bounds for data structures
For the case of data-structures, the cell-probe model seems a good place to start. This model is similar to the more common RAM-model, but can be useful for lower bounds as it 'only counts' the number of accesses to stored data (i.e. the number of 'probes' to a cell).
For example, in the paper by Yao that introduces this model, lower bounds are calculated for the data structure supporting INSERT, DELETE and MEMBER (test for membership) queries. I think that a similar technique could work for your data-structure as well.
Do note that the paper by Yao is a bit outdated. I mostly mentioned it because the problem it considered is relatively simple and similar to yours. More modern techniques are covered in the dissertation of Kasper Green Larsen.
• Thank you very much! Those references seem very useful. I'll research everything tonight. – Badr B Apr 29 '18 at 15:57
• @BadrB You're welcome. I'd be interested in seeing a proof that constant time for all operations is impossible, if you manage to find one. In that case, I suggest you add that here as an answer to your own question. In case you encounter some specific difficulty with these texts or in proving a lower bound, feel free to ask another question about those. – Discrete lizard Apr 29 '18 at 16:47
I don't find the reasoning convincing. Perhaps there is some other way to do it you haven't imagined yet. Proving lower bounds is hard.
That said, I doubt that it's possible to support all three operations in $O(1)$ time, though I don't have a proof.
• Thanks for the insight. I think I'll spend more time trying to prove/disprove its existence rather than just trying to come up with such a data structure lol. I think that'll be the best bet to conquer this tough task. – Badr B Apr 26 '18 at 23:31
It have been prooven, that what you expect is not possible, the lower bound is somewhere like $\Omega{\left( \frac{\log n}{\log \log n} \right)}$ (see here, like TillmanZ already pointed out).
However, hashing gives very fast lookup (somewhere like $\mathcal{O} (1)$ in general). The so-called Cuckoo-hashing guarantees constant lookup and expected constant insert time (amortized). Deletion normally is not considered but it can be done really simply by lookup and clearing the value, resulting also in $\mathcal{O} (1)$.
• Cuckoo hashing doesn't support the operations listed here. Take another look at the desired semantics of insert (and delete). This makes me wonder whether the result you mention applies to these operations too. – D.W. May 6 '18 at 20:03
• There is a difference in the requirements of data-structure in the question you reference and this question: in the data-structure here, elements are accessed by their index, not by some pointer we somehow know. The other DS can simulate the one from this question, but for the lower bound to hold here it would be required that this DS can simulate the other. I don't see how to do that immediately. Perhaps a minor modification to the proof of the lower bound would do the trick. At the very least, I'm not convinced that this lower bound is directly relevant here. – Discrete lizard May 7 '18 at 8:34
• @D.W. Yes, you are right, the operations are slightly different. Especially insert would not shift the indices of following elements (so you do not have indices but keys). However, i wanted to mention that constant insert AND lookup is possible, since many people get stuck at $\log {n}$, either for lookup or for insert. – Dániel Somogyi May 11 '18 at 10:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8397769331932068, "perplexity": 651.2779244671067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506959.34/warc/CC-MAIN-20200402111815-20200402141815-00093.warc.gz"} |
https://ddttrh.info/relationship-between-and/relationship-between-heat-and-energy.php | # Relationship between heat and energy
### How does Temperature Differ from Heat?
Heat, temperature and kinetic energy are linked to each other. In simplest terms, when we heat a substance, its temperature rises and causes an increase in the. Often we think that heat and temperature are the same thing. Heat energy depends on the speed of the particles, the number of particles (the size or mass), . Heat is often defined as energy in the process of being transferred from one object to another because of difference in temperature between them. Heat is.
Any time we use a thermometer, we are using the zeroth law of thermodynamics.
Let's say we are measuring the temperature of a water bath. In order to make sure the reading is accurate, we usually want to wait for the temperature reading to stay constant. We are waiting for the thermometer and the water to reach thermal equilibrium! At thermal equilibrium, the temperature of the thermometer bulb and the water bath will be the same, and there should be no net heat transfer from one object to the other assuming no other loss of heat to the surroundings.
Heat and Temperature
Converting between heat and change in temperature How can we measure heat? Here are some things we know about heat so far: When a system absorbs or loses heat, the average kinetic energy of the molecules will change.
Thus, heat transfer results in a change in the system's temperature as long as the system is not undergoing a phase change.
### Dıfference and relationship between heat and temperature by REMZİYE ÇELEBİ on Prezi
The change in temperature resulting from heat transferred to or from a system depends on how many molecules are in the system. We can use a thermometer to measure the change in a system's temperature. How can we use the change in temperature to calculate the heat transferred?
In order to figure out how the heat transferred to a system will change the temperature of the system, we need to know at least 2.
### 6(c). Energy, Temperature, and Heat
At the atomic scale, the kinetic energy of atoms and molecules is sometimes referred to as heat energy. Kinetic energy is also related to the concept of temperature. Temperature is defined as the measure of the average speed of atoms and molecules.
The higher the temperature, the faster these particles of matter move. At a temperature of Heat is often defined as energy in the process of being transferred from one object to another because of difference in temperature between them.
Heat is commonly transferred around our planet by the processes of conductionconvectionadvectionand radiation.
## Heat and temperature
Some other important definitions related to energy, temperature, and heat are: Heat Capacity - is the amount of heat energy absorbed by a substance associated to its corresponding temperature increase. Specific Heat - is equivalent to the heat capacity of a unit mass of a substance or the heat needed to raise the temperature of one gram g of a substance one degree Celsius.
Water requires about 4 to 5 times more heat energy to raise its temperature when compared to an equal mass of most types of solid matter. This explains why water bodies heat more slowly than adjacent land surfaces. Sensible Heat - is heat that we can sense. A thermometer can be used to measure this form of heat.
Several different scales of measurement exist for measuring sensible heat. The most common are: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8074232935905457, "perplexity": 250.93993296054504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540488870.33/warc/CC-MAIN-20191206145958-20191206173958-00148.warc.gz"} |
https://electnorred.com/law/which-law-can-be-used-to-calculate-the-number-of-moles-of-a-contained-gas.html | ### Which Law Can Be Used To Calculate The Number Of Moles Of A Contained Gas?
Section Summary –
• The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
• The ideal gas law can be written in terms of the number of molecules of gas: PV = NkT, where P is pressure, V is volume, T is temperature, N is number of molecules, and k is the Boltzmann constant k = 1.38 × 10 –23 J/K.
• A mole is the number of atoms in a 12-g sample of carbon-12.
• The number of molecules in a mole is called Avogadro’s number NA, NA = 6.02 × 10 23 mol −1,
• A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.
• The ideal gas law can also be written and solved in terms of the number of moles of gas: PV = nRT, where n is number of moles and R is the universal gas constant, R = 8.31 J/mol ⋅ K.
• The ideal gas law is generally valid at temperatures well above the boiling temperature.
#### Which law will you use to calculate the number of moles?
Summary –
Calculations for relationships between volume and number of moles of a gas can be performed using Avogadro’s Law.
#### How do you find the number of moles in a gas?
The formula to find out the number of moles at STP is Number of moles = Molar volume at STP litres /V o l u m e ITP litres.
## Which law can be used to calculate the number of moles of a contained gas Boyle’s law combined gas law ideal gas law?
Key Takeaways: Combined Gas Law –
The combined gas law is one of the ideal gas laws.It gets its name because it combines Boyle’s law, Charles’ law, and Gay-Lussac’s law.When using this law, only pressure, volume, and temperature can change. The amount or number of moles of gas is held constant.Essentially, the law states that the ratio between pressure, volume, and absolute temperature of a gas equals some constant. So, if you change one of these variables, you can predict how the other factors are affected.
#### Which law helps us find the moles of gas?
Avogadro’s law – The volume ($$V$$) of an ideal gas varies directly with the number of moles of the gas ( n ) when the pressure ( P ) and the number of temperature ( T ) are constant. We can express this mathematically as: \ \ As before, we can use Avogadro’s law to predict what will happen to the volume of a sample of gas as we change the number of moles.
### Are moles constant in Charles law?
Is this consistent with pV = nRT? You have a fixed mass of gas, so n (the number of moles) is constant.
## What is Avogadro’s law used for?
Avogadro’s law, a statement that under the same conditions of temperature and pressure, equal volumes of different gases contain an equal number of molecules, This empirical relation can be derived from the kinetic theory of gases under the assumption of a perfect (ideal) gas,
1. The law is approximately valid for real gases at sufficiently low pressures and high temperatures.
2. The specific number of molecules in one gram- mole of a substance, defined as the molecular weight in grams, is 6.02214076 × 10 23, a quantity called Avogadro’s number, or the Avogadro constant,
3. For example, the molecular weight of oxygen is 32.00, so that one gram-mole of oxygen has a mass of 32.00 grams and contains 6.02214076 × 10 23 molecules.
The volume occupied by one gram-mole of gas is about 22.4 litres (0.791 cubic foot) at standard temperature and pressure (0 °C, 1 atmosphere) and is the same for all gases, according to Avogadro’s law. The law was first proposed in 1811 by Amedeo Avogadro, a professor of higher physics at the University of Turin for many years, but it was not generally accepted until after 1858, when an Italian chemist, Stanislao Cannizzaro, constructed a logical system of chemistry based on it.
### What is R constant in ideal gas law?
The factor ‘R’ in the ideal gas law equation is known as the ‘gas constant’. R = PV. nT. The pressure times the volume of a gas divided by the number of moles and temperature of the gas is always equal to a constant number.
## How do you use ideal gas law?
The Ideal Gas Law explained Learn about the concept of the Ideal Gas Law Learn about Avogadro’s number and the ideal gas law. Encyclopædia Britannica, Inc. Although we can’t see them, we are surrounded by gas molecules. If we look at gas molecules in a balloon, we would see the gas molecules in motion, constantly bouncing up against the inside surface of the balloon.
How can we use math to understand gas in a closed system like this? The Ideal Gas Law states that for any gas, its volume (V) multiplied by its pressure (P) is equal to the number of moles of gas (n) multiplied by its temperature (T) multiplied by the ideal gas constant, R. PV=nRT But what exactly is R? If we rewrite the ideal gas law to solve for R, we find that R equals the value of P times V, divided by the value of n times T.
Because R is a constant, we can use the qualities of any gas — its temperature, pressure, volume, and number of moles — to determine the value of R. Avogadro’s law states that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.
This means four balloons, at the same temperature, filled to the same volume, with four different gasses, all contain the same number of moles of gas. People have used this law to find the number of molecules of gas at a standard temperature and pressure, abbreviated as STP. STP is 273 Kelvin and 1 atmosphere (atm), the standard unit for atmospheric pressure.
At STP, 1 mole of gas takes up 22.4 liters. Let’s plug those numbers into the ideal gas law. R equals the value of 1 atmosphere multiplied by 22.4 liters, divided by the value of 1 mole multiplied by 273 degrees kelvin equals 0.0821 atmosphere liters per moles Kelvin.
## What is ATM in gas law?
Gas Laws
Course Chapters
ul>
• Section Tests
• Useful Materials
1. Online Calculators
Gases behave differently from the other two commonly studied states of matter, solids and liquids, so we have different methods for treating and understanding how gases behave under certain conditions. Gases, unlike solids and liquids, have neither fixed volume nor shape.
• They are molded entirely by the container in which they are held.
• We have three variables by which we measure gases:, volume, and temperature.
• Pressure is measured as force per area.
• The standard SI unit for pressure is the pascal (Pa).
• However, atmospheres (atm) and several other units are commonly used.
The table below shows the conversions between these units.1 pascal (Pa) 1 N*m -2 = 1 kg*m -1 *s -2 1 atmosphere (atm) 1.01325*10 5 Pa 1 atmosphere (atm) 760 torr 1 bar 10 5 Pa Volume is related between all gases by Avogadro’s hypothesis, which states: Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
See also: Explain How Corn Can Be Used As An Example Of Mendel'S Law Of Independent Assortment?
V m = Vn = 22.4 L at 0°C and 1 atm
ul>
• Where:
• V m = molar volume, in liters, the volume that one mole of gas occupies under those conditions V =volume in liters n =moles of gas
• An equation that chemists call the Ideal Gas Law, shown below, relates the volume, temperature, and pressure of a gas, considering the amount of gas present.
• PV = nRT Where: P =pressure in atm T =temperature in Kelvins R is the molar gas constant, where R=0.082058 L atm mol -1 K -1, The Ideal Gas Law assumes several factors about the molecules of gas. The volume of the molecules is considered negligible compared to the volume of the container in which they are held.
• We also assume that gas molecules move randomly, and collide in completely elastic collisions.
• Attractive and repulsive forces between the molecules are therefore considered negligible.
• Example Problem: A gas exerts a pressure of 0.892 atm in a 5.00 L container at 15°C.
• The density of the gas is 1.22 g/L.
What is the molecular mass of the gas?
PV = nRT
T = 273 + 15 = 228
(0.892)(5.00) = n(.0821)(288)
n = 0.189 mol
,189 mol5.00L x x grams1 mol = 1.22 g/L
/td> x = Molecular Weight = 32.3 g/mol
We can also use the Ideal Gas Law to quantitatively determine how changing the pressure, temperature, volume, and number of moles of substance affects the system. Because the gas constant, R, is the same for all gases in any situation, if you solve for R in the Ideal Gas Law and then set two Gas Laws equal to one another, you have the Combined Gas Law: Where:
1. values with a subscript of “1” refer to initial conditions values with a subscript of “2” refer to final conditions
2. If you know the initial conditions of a system and want to determine the new pressure after you increase the volume while keeping the numbers of moles and the temperature the same, plug in all of the values you know and then simply solve for the unknown value.
Example Problem: A 25.0 mL sample of gas is enclosed in a flask at 22°C. If the flask was placed in an ice bath at 0°C, what would the new gas volume be if the pressure is held constant?
Answer: Because the pressure and the number of moles are held constant, we do not need to represent them in the equation because their values will cancel. So the combined gas law equation becomes: V 2 = 23.1 mL
We can apply the Ideal Gas Law to solve several problems. Thus far, we have considered only gases of one substance, pure gases. We also understand what happens when several substances are mixed in one container. According to Dalton’s law of, we know that the total pressure exerted on a container by several different gases, is equal to the sum of the pressures exerted on the container by each gas.
• Where:
• P t =total pressure P 1 =partial pressure of gas “1” P 2 =partial pressure of gas “2” and so on
• Using the Ideal Gas Law, and comparing the pressure of one gas to the total pressure, we solve for the,
P 1 P t = n 2 RT/V n t RT/V = n 1 n t = X 1
ol>
• Where:
• X 1 = mole fraction of gas “1”
• And discover that the partial pressure of each the gas in the mixture is equal to the total pressure multiplied by the mole fraction.
• Example Problem: A 10.73 g sample of PCl 5 is placed in a 4.00 L flask at 200°C. a) What is the initial pressure of the flask before any reaction takes place? b) PCl 5 dissociates according to the equation: PCl 5 (g) -> PCl 3 (g) + Cl 2 (g). If half of the total number of moles of PCl 5 (g) dissociates and the observed pressure is 1.25 atm, what is the partial pressure of Cl 2 (g)?
a) 10.73 g PCl 5 x 1 mol208.5 g = 0.05146 mol PCl 5
/td> PV = nRT T = 273 + 200 = 473 P(4.00) = (.05146)(.0821)(473) P = 0.4996 atm b) PCl 5 → PCl 3 + Cl 2 Start: .05146 mol 0 mol 0 mol Change: -.02573 mol +.02573 mol +.02573 mol Final: .02573 mol .02573 mol .02573 mol
X Cl 2 = n Cl 2 n total = P Cl 2 P total
/td>
P Cl 2 1.25 atm = ,02573 mol.07719 mol
/td> P Cl 2 =,4167 atm
As we stated earlier, the shape of a gas is determined entirely by the container in which the gas is held. Sometimes, however, the container may have small holes, or leaks. Molecules will flow out of these leaks, in a process called, Because massive molecules travel slower than lighter molecules, the rate of effusion is specific to each particular gas.
We use Graham’s law to represent the relationship between rates of effusion for two different molecules. This relationship is equal to the square-root of the inverse of the molecular masses of the two substances. Where: r 1 =rate of effusion in molecules per unit time of gas “1” r 2 =rate of effusion in molecules per unit time of gas “2” u 1 =molecular mass of gas “1” u 2 =molecular mass of gas “2” Previously, we considered only ideal gases, those that fit the assumptions of the ideal gas law.
Gases, however, are never perfectly in the ideal state. All atoms of every gas have mass and volume. When pressure is low and temperature is low, gases behave similarly to gases in the ideal state. When pressure and temperature increase, gases deviate farther from the ideal state.
• Where the van der Waals constants are:
• a accounts for molecular attraction b accounts for volume of molecules
• The table below shows values for a and b of several different compounds and elements.
Species a (dm 6 bar mol -2 ) b (dm 3 mol -1 )
Helium 0.034598 0.023733
Hydrogen 0.24646 0.026665
Nitrogen 1.3661 0.038577
Oxygen 1.3820 0.031860
Benzene 18.876 0.11974
Practice Ideal Gas Law Problem: 2.00 g of hydrogen gas and 19.2 g of oxygen gas are placed in a 100.0 L container. These gases react to form H 2 O(g). The temperature is 38°C at the end of the reaction. a) What is the pressure at the conclusion of the reaction? b) If the temperature was raised to 77° C, what would the new pressure be in the same container?,
1. Practice Pressure Problem: 1 mole of oxygen gas and 2 moles of ammonia are placed in a container and allowed to react at 850°C according to the equation:
4 NH 3 (g) + 5 O 2 (g) -> 4 NO(g) + 6 H 2 O(g)
• b) Using Graham’s Law, what is the ratio of the effusion rates of NH 3 (g) to O 2 (g)?
a) If the total pressure in the container is 5.00 atm, what are the partial pressures for the three gases remaining?, An Online Interactive Tool Developed by in cooperation with the,
: Gas Laws
## What is Charles Law and Boyle’s law?
Introduction – The three fundamental gas laws discover the relationship of pressure, temperature, volume and amount of gas. Boyle’s Law tells us that the volume of gas increases as the pressure decreases. Charles’ Law tells us that the volume of gas increases as the temperature increases.
## What does Boyle’s law calculate?
Home Science Physics Matter & Energy Alternate titles: Mariotte’s law, first gas law Boyle’s law, also called Mariotte’s law, a relation concerning the compression and expansion of a gas at constant temperature, This empirical relation, formulated by the physicist Robert Boyle in 1662, states that the pressure ( p ) of a given quantity of gas varies inversely with its volume ( v ) at constant temperature; i.e., in equation form, p v = k, a constant.
The relationship was also discovered by the French physicist Edme Mariotte (1676). The law can be derived from the kinetic theory of gases assuming a perfect (ideal) gas ( see perfect gas ). Real gases obey Boyle’s law at sufficiently low pressures, although the product p v generally decreases slightly at higher pressures, where the gas begins to depart from ideal behaviour.
### What does Henry’s law find?
Henry’s Law – Statement, Formula, Constant, Solved Examples Henry’s law is a gas law which states that at the amount of gas that is dissolved in a liquid is directly proportional to the partial pressure of that gas above the liquid when the temperature is kept constant.
‘P’ denotes the partial pressure of the gas in the atmosphere above the liquid. ‘C’ denotes the concentration of the dissolved gas. ‘k H ‘ is the Henry’s law constant of the gas.
#### Is Avogadro’s law the same as ideal gas law?
PV = nRT – Where:
P is pressure V is volume n is the number of gas molecules in moles R is a number known as the ideal gas constant (The value for R is often, but not always, 8.314 J/mol ˖ K) T is temperature (which has to be in Kelvin)
The chart below shows how all the above gas laws are present in this ideal gas law.
Law Variables Symbols in Formula Boyle pressure & volume P, V Charles volume & temperature V, T Gay-Lussac pressure & temperature P, T Avogadro volume & amount V, n
Although in reality no gas is an ‘ideal gas’, some do come very close. Therefore, the Ideal Gas Law allows us to roughly predict the behaviour of a gas. This formula is often used when you want to determine the amount of gas that is present in a container.
For example, you could find the mass of gas in a container by weighing the gas in the container, pumping out the gas and then reweighing the container. However, since gases have such low weight, the difference would be so small it would be hard to measure. Instead, all you need to know is the pressure – which can be obtained from a pressure gauge – the volume of the container, and the temperature of the gas.
Then put these values into the formula and solve for n, from which the mass can be obtained.
### What is the gas law called?
Home Science Physics Matter & Energy gas laws, laws that relate the pressure, volume, and temperature of a gas, Boyle’s law —named for Robert Boyle —states that, at constant temperature, the pressure P of a gas varies inversely with its volume V, or P V = k, where k is a constant.
1. Charles’s law —named for J.-A.-C.
2. Charles (1746–1823)—states that, at constant pressure, the volume V of a gas is directly proportional to its absolute (Kelvin) temperature T, or V / T = k,
3. These two laws can be combined to form the ideal gas law, a single generalization of the behaviour of gases known as an equation of state, P V = n R T, where n is the number of gram-moles of a gas and R is called the universal gas constant,
Though this law describes the behaviour of an ideal gas, it closely approximates the behaviour of real gases. See also Joseph Gay-Lussac, Erik Gregersen
### What formula is used in Charles Law?
Charles Law states that the volume of a given mass of a gas is directly proportional to its Kevin temperature at constant pressure. In mathematical terms, the relationship between temperature and volume is expressed as V1/T1=V2/T2. Alright. One of the gas laws that you might come across is called Charles Law, and Charles law was formed by Jacque Charles in France in the 1800s.
1. And he discovered that the volume of a given mass of a gas is directly proportional to its kelvin temperature at constant pressure.
2. There are two things that you want to make sure you know or you notice when you’re reading this gas law.
3. One is the kelvin temperature where you make sure our temperature is always always always in kelvin or else we are going to get the wrong answer when dealing with this Charles law and you also want to notice it’s a constant pressure.
So two variables that are changing is volume and, volume and temperature. Okay, those are the two variables we’re dealing with. So let’s say we have two canisters. They are at this, notice they are at the same pressure. So this, this canister we have gas pressure.
We know normal temperature and pressure and then we actually heat it up. Okay. So now we’re increasing the kinetic energy. Those gas particles are now moving at a faster rate and they are able, if we want to make sure the pressure is constant. They are actually going to push against this the top of this thing and actually move making the volume larger.
So if you notice, the relationship between temperature and volume as we increase temperature, we also increase the volume as long as pressure is constant. Okay? So, Charles law, its relationship is – we have a direct relationship as stated in the actual law and we can now actually make it mathematically equal.
Volume one over divided by the temperature of one equals the volume of the second one divided by the temperature of the second scenario. So this is actually Charles law mathematically. If you were to make a graph, the graph of Charles law is at zero kelvin and we’re going to have zero volume because it’s zero kelvin, nothing moves and the volume of a gas is actually going to be zero, and it increases as the other one increases also.
So you’re going to have linear relationship that looks like this. As temperature increases so does the volume of the gas. It also increases. Also as temperature decreases, volume of the gas actually decreases. Let’s actually do a demonstration that shows this.
• Okay. So over here I have a candle floating in some water.
• I’m going to light that candle.
• Let me just put safety goggles on first.
• And let’s do that. Okay. Alright.
• I’m going to put this in here just to be safe.
• Make sure I don’t burn anything down.
• Okay, so what’s happening, the air particles around this candle are actually heating up, okay.
So they’re expanding. I’m going to capture this, I’m going to capture this. I’m going to put this glass on top of this candle and what that’s going to do is going to end up going out because it’s going to all the oxygen in this glass container is going to go away.
• It’s going to be used up.
• So as it’s being used up the candle is going to go out.
• And notice, when it went out, a lot of the volume in the water level rose inside the canister.
• Now why did that happen? Because when the candle went out, the temperature of the gas particles inside the ga- inside this glass chamber actually dropped and that made the temp- the gas particles actually have a lower volume.
Because the gas particles had a lower volume, they had, that volume had to replaced by something. And it was replaced by the water at the bottom. So the water is actually able to be sucked in to the glass container to replace that volume that was then lost due to the drop in temperature.
1. Okay. So let’s do a problem that you might see in class. Okay.
2. Something that you might see in class I’m going to take off my glass my goggles.
3. Don’t need them anymore.
4. A gas at 40 degrees celsius occupies a volume at 2.32 litres.
5. If the temperature is raised to 75 degrees celsius, what will the new volume be if the pressure is constant.
So I’m dealing with temperature and volume. So I know in my head that’s Charles law. Charles law deals with temperature and volume. Okay. It also deals with temperature in kelvins. So I want to make sure I change these temperatures to kelvin. So knowing that my formula is v1 over t1 equals v2 over t2.
The first volume that we’re going to deal with is 2.32 litres. The first temperature is 40 degrees celsius. We add 273 to that and we get 313 kelvin and then our second volume is, we don’t know. It’s what we’re looking for. It’s what we’re looking for. Our second temperature is I’m just going to turn this on real quick.
Our second temperature is 75 degrees celsius. We’re going to add 273 to that and we get two, 348 kelvin. We cross multiply 348 times 232 divided by 313, we get our new volume which is 2.58 litres and let’s see if that makes sense, okay? So we increased the temperature.
### What remains constant and Charles Law?
Charles’s law, a statement that the volume occupied by a fixed amount of gas is directly proportional to its absolute temperature, if the pressure remains constant.
## What remains constant in Boyle’s law?
Boyle’s Law – Robert Boyle (1627-1691), an English chemist, is widely considered to be one of the founders of the modern experimental science of chemistry. He discovered that doubling the pressure of an enclosed sample of gas, while keeping its temperature constant, caused the volume of the gas to be reduced by half.
Boyle’s law states that the volume of a given mass of gas varies inversely with the pressure when the temperature is kept constant. An inverse relationship is described in this way. As one variable increases in value, the other variable decreases. Physically, what is happening? The gas molecules are moving and are a certain distance apart from one another.
An increase in pressure pushes the molecules closer together, reducing the volume. If the pressure is decreased, the gases are free to move about in a larger volume. Figure $$\PageIndex$$: Robert Boyle. (CC BY-NC; CK-12) Mathematically, Boyle’s law can be expressed by the equation: \ The $$k$$ is a constant for a given sample of gas and depends only on the mass of the gas and the temperature. The table below shows pressure and volume data for a set amount of gas at a constant temperature.
Table $$\PageIndex$$: Pressure-Volume Data
Pressure $$\left( \text \right)$$ Volume $$\left( \text \right)$$ $$P \times V = k$$ $$\left( \text \cdot \text \right)$$
0.5 1000 500
0.625 800 500
1.0 500 500
2.0 250 500
5.0 100 500
8.0 62.5 500
10.0 50 500
A graph of the data in the table further illustrates the inverse relationship nature of Boyle’s Law (see figure below). Volume is plotted on the $$x$$-axis, with the corresponding pressure on the $$y$$-axis. Figure $$\PageIndex$$: The pressure of a gas decreases as the volume increases, making Boyle’s law an inverse relationship. (CC BY-NC; CK-12) Boyle’s Law can be used to compare changing conditions for a gas. We use $$P_1$$ and $$V_1$$ to stand for the initial pressure and initial volume of a gas.
## Which of the below is the Dalton’s law?
According to Dalton’s law of partial pressures, the total pressure by a mixture of gases is equal to the sum of the partial pressures of each of the constituent gases.
### What is Avogadro gas law formula?
Avogadro’s Law is stated mathematically as follows: Vn=k, where V is the volume of the gas, n is the number of moles of the gas, and k is a proportionality constant.
### What does the ideal gas law states?
Ideal Gas Law Definition – The ideal gases obey the ideal gas law perfectly. This law states that: the volume of a given amount of gas is directly proportional to the number on moles of gas, directly proportional to the temperature and inversely proportional to the pressure.i.e. pV = nRT.
#### How do you find moles in Avogadro’s law?
One mole of a substance is equal to 6.022 × 10²³ units of that substance (such as atoms, molecules, or ions). The number 6.022 × 10²³ is known as Avogadro’s number or Avogadro’s constant. The concept of the mole can be used to convert between mass and number of particles Created by Sal Khan.
## What is Avogadro’s Law number?
Molar Volume of a Gas – As per Avogadro’s law, the ratio of volume and amount of gaseous substance is a constant (at constant pressure and temperature). The value of this constant (k) can be determined with the help of the following equation: k = (RT)/P Under standard conditions for temperature and pressure, the value of T corresponds to 273.15 Kelvin and the value of P corresponds to 101.325 kilo Pascals.
#### What does Avogadro’s law formula?
💡 Summary –
• Avogadro’s law states that the total number of atoms or molecules of any gas is directly proportional to the gaseous volume occupied at constant pressure and temperature.
• Avogardro’s equation is written as V = k ྾ n or V1/n1 = V2/n2. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8737419843673706, "perplexity": 690.2880122655312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500017.27/warc/CC-MAIN-20230202101933-20230202131933-00504.warc.gz"} |
https://www.maplesoft.com/support/help/maplesim/view.aspx?path=initialfunctions | Initially Known Mathematical Functions - Maple Programming Help
Home : Support : Online Help : Programming : Names and Strings : initialfunctions
Initially Known Mathematical Functions
Description
• These mathematical functions are known to Maple, in that they have simplification procedures defined and/or are known to one or more of: diff, evalc, evalf, expand, series, simplify.
• The trigonometric and hyperbolic functions:
sin, cos, tan, sec, csc, cot, sinh, cosh, tanh, sech, csch,
• The inverse trigonometric and inverse hyperbolic functions:
• Two-argument arctan: $\mathrm{arctan}\left(y,x\right)=\mathrm{arg}\left(x+Iy\right)$ in $\left(-\mathrm{\pi },\mathrm{\pi }\right)$
• For more complete information regarding any of the functions shown here, see ? (for example, ?abs ).
- absolute value of real or complex number - Airy wave functions and their negative real zeros - Anger J function - argument of a complex number - Bell polynomials - Bernoulli numbers and polynomials - modified Bessel function of the first kind - Bessel function of the first kind - non-negative real zeros of Bessel J - modified Bessel function of the second kind - Bessel function of the second kind - positive real zeros of Bessel Y - Beta function - binomial coefficients - smallest integer greater than or equal to a number - Chebyshev function of the first kind - Chebyshev function of the second kind - hyperbolic cosine integral - cosine integral - complete Bell polynomials - conjugate of a complex number or expression - regular Coulomb wave function - Whittaker's parabolic function - Parabolic cylinder functions - complex "half-plane" signum function - Dawson's integral - dilogarithm function - Dirac delta function - double factorial function - exponential integrals - complementary complete elliptic integral of the second kind - complementary complete elliptic integral of the first kind - complementary complete elliptic integral of the third kind - incomplete or complete elliptic integral of the second kind - incomplete elliptic integral of the first kind - complete elliptic integral of the first kind - Modulus elliptic function - Nome elliptic function - incomplete or complete elliptic integral of the third kind - error function - complementary error function and its iterated integrals - imaginary error function - Euler numbers and polynomials - exponential function - factorial function - greatest integer less than or equal to a number - fractional part of a number - Fresnel cosine integral - Fresnel f auxiliary function - Fresnel g auxiliary function - Fresnel sine integral - Gamma and incomplete Gamma functions - Gauss arithmetic geometric mean - Gegenbauer (ultraspherical) function - Hankel functions (Bessel functions of the third kind) - partial sum of the harmonic series - Heaviside step function - Hermite function - Heun functions - derivatives of Heun functions - generalized hypergeometric function - integer logarithms - imaginary part of a complex number - incomplete Bell polynomials - inverse Jacobi amplitude function - inverse Jacobi elliptic functions - Jacobi function - Jacobi amplitude function - Jacobi elliptic functions - Jacobi theta functions - Jacobi Zeta function - Kelvin functions - Kummer functions - Laguerre function - Lambert W function - associated Legendre function of the first kind - associated Legendre function of the second kind - Lerch's Phi function - logarithmic integral - natural logarithm (logarithm with base $ⅇ$ = 2.71...) - log-Gamma function - logarithm to arbitrary base - log to the base 10 - Lommel function s - Lommel function S - Mathieu characteristic function - Mathieu characteristic function - even general Mathieu function - first derivative of MathieuC - even $2\mathrm{\pi }$-periodic Mathieu function - first derivative of MathieuCE - Mathieu characteristic exponent - Floquet solution of Mathieu's equation - first derivative of MathieuFloquet - odd general Mathieu function - first derivative of MathieuS - odd $2\mathrm{\pi }$-periodic Mathieu function - first derivative of MathieuSE - MeijerG function - maximum of a sequence of real values - minimum of a sequence of real values - pochhammer symbol - polar representation of complex numbers - polylogarithm function - polygamma function - real part of a complex number - Riemann theta function - nearest integer to a number - sign of a real or complex number - hyperbolic sine integral - sine integral - spherical harmonic function - square root - shifted sine integral - Stirling number of the first kind - Stirling number of the second kind - Struve function - modified Struve function - non-principal root function - nearest integer to a number in the direction of 0 - unwinding number - Weber E function - Weierstrass P-function - Derivative of Weierstrass P-function - Weierstrass sigma-function - Weierstrass zeta-function - Whittaker functions - Wright omega function - Riemann and Hurwitz zeta functions
• Additional mathematical functions are defined in various packages, such as the combinatorial functions package combinat, the number theory package NumberTheory, and the orthogonal polynomial package orthopoly. For a complete list of packages, see index[package]. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9020930528640747, "perplexity": 621.593855526537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875142603.80/warc/CC-MAIN-20200217145609-20200217175609-00368.warc.gz"} |
https://hal-cea.archives-ouvertes.fr/cea-01307070 | # Interacting quantum walkers: Two-body bosonic and fermionic bound states
Abstract : We investigate the dynamics of bound states of two interacting particles, either bosons or fermions, performing a continuous-time quantum walk on a one-dimensional lattice. We consider the situation where the distance between both particles has a hard bound, and the richer situation where the particles are bound by a smooth confining potential. The main emphasis is on the velocity characterizing the ballistic spreading of these bound states, and on the structure of the asymptotic distribution profile of their center-of-mass coordinate. The latter profile generically exhibits many internal fronts.
Document type :
Journal articles
Domain :
Cited literature [39 references]
https://hal-cea.archives-ouvertes.fr/cea-01307070
Contributor : Emmanuelle de Laborderie <>
Submitted on : Tuesday, April 26, 2016 - 10:56:43 AM
Last modification on : Wednesday, April 14, 2021 - 12:12:00 PM
Long-term archiving on: : Wednesday, July 27, 2016 - 12:00:14 PM
### File
1507.01363v2.pdf
Files produced by the author(s)
### Citation
P. L. Krapivsky, J. M. Luck, K. Mallick. Interacting quantum walkers: Two-body bosonic and fermionic bound states. Journal of Physics A, 2015, 48, pp.475301. ⟨10.1088/1751-8113/48/47/475301⟩. ⟨cea-01307070⟩
Record views | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9061186909675598, "perplexity": 3571.8774008721352}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487641593.43/warc/CC-MAIN-20210618200114-20210618230114-00059.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=3628727 | # The origin of dy = f'(c)dx in differentials
by vanmaiden
Tags: differentials, equation, origin
Sci Advisor HW Helper P: 4,300 Actually, the trick is to look at the ratio of the changes, $\frac{\Delta y}{\Delta x}$. You can represent this on the graph of the curve as the average change of the function, or the slope of the straight line approximation on that small interval. Initially it may seem strange to use a straight line approximation, but if you think about it for a minute, you will see that it's not such a bad idea because the smaller you make $\Delta x$, the better the straight line resembles the graph around x = c. In fact, the line you get becomes closer and closer to the tangent line of the graph at x = c. So it makes sense to try and do this analytically, for which we can use a limit: we consider $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$ And to make the dependence of the numerator explicit, $$\lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$$ where I called $\Delta x = h$. This number, if the limit exists, actually is the slope of the tangent line. Note that is no longer a real quotient, it's just the limit of a quotient. So we can invent the notation $\frac{dy}{dx}$ or $\frac{df(x)}{dx}$ or, if you want to have c in there somewhere, $$\left. \frac{df(x)}{dx} \right|_{x = c}$$ The safest way is now to think of it as just that, notation, and nothing else. Again, df/dx is not really a fraction anymore, so if df/dx = 3 that merely means that the slope of f at x is equal to 3. You shouldn't allow yourself to write things like df = 3 dx. If you can't resist the temptation, then consider writing f'(c) instead :-) Just as an after-thought, it is true (for continuous functions) that if $\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = f'(c)$, then $$\lim_{\Delta x \to 0} \Delta y = \lim_{\Delta x \to 0} f'(c) \Delta x$$ because f'(c) is just a fixed number (which I denote by f'(c), or you could call it a or L) and if you plug in $\Delta y = f(c) - f(c + \Delta x)$ you will find that both sides of the expression are equal to 0. So in that sense, you can split out the fraction, but it doesn't give you anything useful. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9473062753677368, "perplexity": 186.8826824131769}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657127503.54/warc/CC-MAIN-20140914011207-00278-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://mathhelpforum.com/calculus/19642-limit-trigometric-function.html | # Math Help - limit of the trigometric function
1. ## limit of the trigometric function
okay, so i've spent some time thinking about finding the limits of trigonometric functions when X->0, and you can't substitute the 0 in directly, but i can't seem to figure out the how, exactly, i can do it, please help:
lim (sec x-1)/(x sec x)
x->0
2. Multiply up & down by $\cos x$, what do you see? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9598272442817688, "perplexity": 811.2308494772794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701174607.44/warc/CC-MAIN-20160205193934-00113-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://lmcs.episciences.org/2620 | Harwath, Frederik and Schweikardt, Nicole - On the locality of arb-invariant first-order formulas with modulo counting quantifiers
lmcs:2620 - Logical Methods in Computer Science, April 27, 2017, Volume 12, Issue 4
On the locality of arb-invariant first-order formulas with modulo counting quantifiers
Authors: Harwath, Frederik and Schweikardt, Nicole
We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MOD_p, for short) with arbitrary numerical predicates. We require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas. This paper gives a detailed picture of locality and non-locality properties of arb-invariant FO+MOD_p. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MOD_p is neither Hanf nor Gaifman local with respect to a sublinear locality radius. However, in case that p is an odd prime power, it is weakly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime powers p, arb-invariant FO+MOD_p is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MOD_p formulas presented in Niemist\"o's PhD thesis. Our positive results make use of the close connection between FO+MOD_p and Boolean circuits built from NOT-gates and AND-, OR-, and MOD_p- gates of arbitrary fan-in.
Source : oai:arXiv.org:1611.07716
DOI : 10.2168/LMCS-12(4:8)2016
Volume: Volume 12, Issue 4
Published on: April 27, 2017
Submitted on: April 1, 2016
Keywords: Computer Science - Logic in Computer Science,F.4.1,H.2.3,F.1.3 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9112398624420166, "perplexity": 2940.188738526951}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867977.85/warc/CC-MAIN-20180527004958-20180527024958-00315.warc.gz"} |
http://mrhonner.com/archives/11847 | ## Solving Pallet Equations
A recent delivery came with an unexpected bonus.
A diagram illustrating the geometric relationship between the length and the width of the box!
Of course, the equation $3w = 2l$ immediately came to mind. I also noticed that the pallet could not be a square: if $3w = l + 2w$, then $w = l$, which would make the box itself a square-based box, which it clearly is not.
I did a quick search and found the standard pallet size to be 48 inches by 40 inches. So my best guess is that $w = 12$ and $l = 18$. This box has already been discarded, so I guess I’ll have to order another box to find out! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 5, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8304322361946106, "perplexity": 566.1723929967212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736678409.42/warc/CC-MAIN-20151001215758-00211-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-use-synthetic-division-to-divide-3x-3-2x-2-6x-2-by-x-2 | Precalculus
Topics
# How do you use synthetic division to divide 3x^3 + 2x^2 + 6x - 2 by x+2?
Jul 24, 2015
$\textcolor{red}{\frac{3 {x}^{3} + 2 {x}^{2} + 6 x - 2}{x + 2} = 3 {x}^{2} - 4 x + 14 - \frac{30}{x + 2}}$
#### Explanation:
Step 1. Write only the coefficients of $x$ in the dividend inside an upside-down division symbol.
Step 2. Change the sign of the constant in the divisor and put it at the left.
Step 3. Drop the first coefficient of the dividend below the division symbol.
Step 4. Multiply the drop-down by the divisor, and put the result in the next column.
Step 5. Add down the column.
Step 6. Repeat Steps 4 and 5 until you can go no farther
The quotient is $3 {x}^{2} - 4 x + 14 - \frac{30}{x + 2}$.
Check:
$\left(x + 2\right) \left(3 {x}^{2} - 4 x + 14 - \frac{30}{x + 2}\right) = \left(x + 2\right) \left(3 {x}^{2} - 4 x + 14\right) - 30 = 3 {x}^{3} - 4 {x}^{2} + 14 x + 6 {x}^{2} - 8 x + 28 - 30 = 3 {x}^{3} + 2 {x}^{2} + 6 x - 2$
##### Impact of this question
3795 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.824122428894043, "perplexity": 920.3887243841543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586390.4/warc/CC-MAIN-20210612193058-20210612223058-00153.warc.gz"} |
https://www.uibk.ac.at/th-physik/howto/slitex/slitx-9.html | # Making Some Slides
For making corrections, it's handy to be able to produce a subset of the slides in your file. The command
\onlyslides{4,7-13,23}
in the root file will cause the following \blackandwhite and \colorslides commands to generate only slides numbered 4, 7-13 (inclusive) and 23, plus all of their overlays. The slide numbers in the argument must be in ascending order, and can include nonexistent slides---for example, you can type
\onlyslides{10-9999}
to produce all but the first nine slides. The argument of the \onlyslides command must be non-empty.
There is also an analogous \onlynotes command to generate a subset of the notes. Notes numbered 11-1, 11-2, etc. will all be generated by specifying page 11 in the argument of the \onlynotes command.
If your input has an \onlyslides command and no \onlynotes command, then notes will be produced for the specified slides. If there is an \onlynotes command but no \onlyslide command, then no slides will be produced. Including both an \onlyslides and an \onlynotes command has the expected effect of producing only the specified slides and notes.
Nach oben scrollen | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9731109738349915, "perplexity": 2782.4027354807913}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038066613.21/warc/CC-MAIN-20210412053559-20210412083559-00273.warc.gz"} |
https://inordinatum.wordpress.com/tag/confluent-hypergeometric/ | # inordinatum
Physics and Mathematics of Disordered Systems
## Tricks for inverting a Laplace Transform, part IV: Substitutions
After a few less technical posts recently, I now continue the series of articles on tricks for the inversion of Laplace transforms. You can find the other parts here: part I (guesses based on series expansions), part II (products and convolutions), part III, and part V (pole decomposition).
Today’s trick will be variable substitutions. Let’s say we need to find $P(x)$ so that
$\displaystyle \int_0^\infty \mathrm{d}x\, e^{\lambda x}P(x) = \hat{P}(r(\lambda))=e^{a r(\lambda)}\left[1-r(\lambda)\right]^b$, (1)
where $r(\lambda) = \frac{1}{2}\left(c-\sqrt{c^2-\lambda}\right)$, and $a,b,c$ are constants. This example is not as absurd as it may seem; it actually came up recently in my research on a variant of the ABBM model.
There is a general formula permitting one to invert the Laplace transform above, in terms of an integral in the complex plane:
$\displaystyle P(x) = \int_\gamma \frac{\mathrm{d} \lambda}{2\pi i} e^{-\lambda x} \hat{P}(r(\lambda))$.
This so-called Fourier-Mellin integral runs along a contour $\gamma$ from $-i \infty$ to $+i\infty$. $\gamma$ can be chosen arbitrarily, as long as the integral converges and all singularities (poles, branch cuts, etc.) of $\hat{P}$ lie on the right of it. Note that my convention for the Laplace variable $\lambda$ is the convention used for generating functions in probability theory, which has just the opposite sign of $\lambda$ of the “analysis” convention for Laplace transforms.
The fact that our Laplace transform $\hat{P}$ is written entirely in terms of the function $r(\lambda)$ suggests applying a variable substitution. Instead of performing the Fourier-Mellin integral over $\lambda$, we will integrate over $r(\lambda)$. We then have:
$\displaystyle P(x) = \int \frac{\mathrm{d}r}{2\pi i} \frac{\mathrm{d}\lambda(r)}{\mathrm{d}r} e^{-\lambda(r) x + a r}\left(1-r\right)^b$. (2)
Solving the definition of $r(\lambda)$ for $\lambda$, we get
$\displaystyle \lambda(r) = 4 (c r -r^2)$
and
$\displaystyle \frac{\mathrm{d}\lambda}{\mathrm{d}r} = 4 c - 8r$.
Inserting this into (2), we have
$\displaystyle P(x) = 4\int_{\gamma'} \frac{\mathrm{d}r}{2\pi i} (c-2r)e^{4 r^2 x - (4cx-a)r}\left(1-r\right)^b$.
The only singularity of the integrand is at $r=1$, for $b \in \mathbb{R} \backslash \mathbb{N}$. We can thus choose the contour $\gamma'$ to go parallel to the imaginary axis, from $1-i \infty$ to $1+i\infty$. Mathematica then knows to how to evaluate the resulting integral, giving a complicated expression in terms of Kummer’s confluent hypergeometric function $\,_1 F_1(a,b,z) = M(a,b,z)$. However, a much simpler expression is obtained if one introduces an auxiliary integral instead:
$\displaystyle g(q,c,d):= \int_{-\infty}^{\infty} \mathrm{d}h\, e^{-q h^2}\left(c-i h\right)^d$.
Mathematica knows a simple expression for it:
$\displaystyle g(q,c,d) = \sqrt{\pi} q^{-(d+1)/2}U(-d/2;1/2;c^2 q)$.
$U$ is Tricomi’s confluent hypergeometric function, which is equivalent to Kummer’s confluent hypergeometric function but gives more compact expressions in our case.
Using this auxiliary integral, (2) can be expressed as
$\displaystyle P(x) = \left[\frac{4}{\pi}g\left(4x,1+\frac{a-4cx}{8x},b+1\right)+\frac{c-4}{\pi}g\left(4x,1+\frac{a-4cx}{8x},b\right)\right]\exp\left[-\frac{(a-4cx)^2}{16x}\right]$.
Simplifying the resulting expressions, one obtains our final result for $P(x)$:
$\displaystyle \begin{array}{rcl} P(x) & = & \frac{4}{\sqrt{\pi}}(4x)^{-1-b/2}\left[U\left(-\frac{1+b}{2};\frac{1}{2};\frac{(a+4x(2-c))^2}{16x}\right) + \right. \\ & & \left.+(c-2)\sqrt{x}U\left(-\frac{1+b}{2};\frac{1}{2};\frac{(a+4x(2-c))^2}{16x}\right)\right]\exp\left[-\frac{(a-4cx)^2}{16x}\right]. \end{array}$ (3)
Equivalently, the confluent hypergeometric functions can be replaced by Hermite polynomials:
$\displaystyle P(x) = \frac{8}{\sqrt{\pi}}(8x)^{-1-b/2}\left[H_{1+b}\left(\frac{a+4x(2-c)}{4\sqrt{x}}\right) + 2(c-2)\sqrt{x}H_b\left(\frac{a+4x(2-c)}{4\sqrt{x}}\right)\right]\exp\left[-\frac{(a-4cx)^2}{16x}\right].$
For complicated Laplace transforms such as these, I find it advisable to check the final result numerically. In the figure below you see a log-linear plot of the original expression (1) for $\hat{P}(\lambda)$, and a numerical evaluation of $\int_0^\infty \mathrm{d}x\, e^{\lambda x}P(x)$, with $P(x)$ given by (3). You can see that they match perfectly!
Numerical verification of the Laplace transform. Yellow line: Original expression $\hat{P}(\lambda)$ as given in (1). Red crosses: Numerical evaluation of $\int_0^\infty \mathrm{d}x\,e^{\lambda x}P(x)$ for the final expression $P(x)$ given in (3).
Written by inordinatum
November 4, 2012 at 2:14 pm
Posted in Maths, Technical Tricks | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 41, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9531683325767517, "perplexity": 372.3173700670551}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591718.31/warc/CC-MAIN-20180720154756-20180720174756-00481.warc.gz"} |
http://math.stackexchange.com/questions/694393/given-s-is-non-empty-and-sups-infs-prove-that-the-set-s-has-only-one-elemen | # Given S is non-empty and sup(S)=inf(S), prove that the set S has only one element.
Proposition: Given S is non-empty sup(S)=inf(S), prove that the set S has only one element.
What I stated, as this seemed rather trivial.
Proof: Let S be a non-empty subset of R that is bounded. It is given to us that sup(S)=inf(S). The claim is that S, then, has only one element within its set. We proceed by contradiction: Let a,b belong to S where a does not equal b and a is our smallest element, b is our largest element. It follows the inf(S) = a and sup(S) = b. It was given to to us sup(S)=inf(S) which entails that a = b showing uniqueness. Yet, we stated before that a doesn't equal b! We've reached a contradiction. Therefore, S contains only one element if sup(S)=inf(S).
Is this fine or does anyone think this could better stated?
-
A minor thing: What if $S = \varnothing$? – Johannes Kloos Feb 28 '14 at 20:29
@JohannesKloos: It is often the convention to write $\sup\varnothing = -\infty$ and $\inf\varnothing = +\infty$, so this is still okay – MPW Feb 28 '14 at 20:41
@MPW: I know, but I wanted to point out a gap in the proof. – Johannes Kloos Feb 28 '14 at 20:48
The proof is not right, but easily fixed. Let $a$ and $b$ be distinct elements of our set. Then $$\inf S\le a\lt b\le \sup S,$$ and we have our contradiction.
Remark: Note that $S$ need not have a smallest (or largest) element.
If $\inf S = \sup S = b \in \bar{\mathbb R}$, you know that all elements of $S$ are at the same time $\geq b$ and $\leq b$. The statement follows trivially.
Note that it actually holds for all subsets of $\mathbb R$ if you define $\inf S = \infty$ and $\sup S = -\infty$ when $S$ is empty. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8713704943656921, "perplexity": 168.58327560322192}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257830066.95/warc/CC-MAIN-20160723071030-00033-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1909.05337 | math.AP
# Title:Well-posedness of the fractional Zener wave equation for heterogenous viscoelastic materials
Abstract: We explore the well-posedness of the fractional version of Zener's wave equation for viscoelastic solids, which is based on a constitutive law relating the stress tensor $\boldsymbol{\sigma}$ to the strain tensor $\boldsymbol\varepsilon(\bf u)$, with $\bf u$ being the displacement vector, defined by: $(1+\tau D_t^\alpha) {\boldsymbol{\sigma}}=(1+\rho D_t^\alpha)[2\mu {\boldsymbol\varepsilon}({\bf u})+\lambda\text{tr}(\boldsymbol\varepsilon(\bf u)) \bf ]$. Here $\mu,\lambda\in\mathrm{L}^\infty(\Omega)$, $\mu$ is the shear modulus bounded below by a positive constant, and $\lambda\geq 0$ is first Lamé coefficient, $D_t^\alpha$, with $\alpha \in (0,1)$, is the Caputo time-derivative, $\tau>0$ is the characteristic relaxation time and $\rho\geq\tau$ is the characteristic retardation time. We show that, when coupled with the equation of motion $\varrho \ddot{\bf u} = \text{Div}{\boldsymbol\sigma} + \bf F$, considered in a bounded open Lipschitz domain $\Omega$ in $\mathbb{R}^3$ and over a time interval $(0,T]$, where $\varrho\in \mathrm{L}^\infty(\Omega)$ is the density of the material, bounded below by a positive constant, and $\bf F$ is a specified load vector, the resulting model is well-posed in the sense that the associated initial-boundary-value problem, with initial conditions ${\bf u}(0,\mathbf{x}) = {\bf g}(\mathbf{x})$, $\dot{\bf u}(0,\mathbf{x}) = \bf h(\mathbf{x})$, ${\boldsymbol\sigma}(0,\mathbf{x}) = {\bf s}(\mathbf{x})$, for $\mathbf{x} \in \Omega$, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of ${\bf g }\in [\mathrm{H}^1_0(\Omega)]^3$, ${\bf h}\in [\mathrm{L}^2(\Omega)]^3$, and ${\bf S} = {\bf S}^{\rm T} \in [\mathrm{L}^2(\Omega)]^{3 \times 3}$, and any load vector ${\bf F} \in\mathrm{L}^2(0,T;[\mathrm{L}^2(\Omega)]^3)$, and that this unique weak solution depends continuously on the initial data and the load vector.
Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:1909.05337 [math.AP] (or arXiv:1909.05337v1 [math.AP] for this version)
## Submission history
From: Ljubica Oparnica [view email]
[v1] Wed, 11 Sep 2019 20:14:10 UTC (28 KB) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.96014803647995, "perplexity": 318.722259794004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573124.40/warc/CC-MAIN-20190917223332-20190918005332-00359.warc.gz"} |
http://mathonline.wikidot.com/the-inversion-sequence-of-a-permutation | The Inversion Sequence of a Permutation
# The Inversion Sequence of a Permutation
Definition: If $A = \{1, 2, ..., n \}$ is a finite $n$-element set of positive integers and $(x_1, x_2, ..., x_n)$ is an $n$-permutation of the elements in $A$ then the Inversion Sequence of this permutation is a finite sequence $(a_i)_{i=1}^{n} = (a_1, a_2, ..., a_n)$ where for each $j \in \{1, 2, ..., n \}$ we have that $a_j$ is defined to be the number of integers to the left of $j$ in the $n$-permutation $(x_1, x_2, ..., x_n)$ that are larger than $j$.
Be careful! In the definition above, we use the round bracket notation to represent both a permutation AND a finite sequence.
For example, consider the set $A = \{1, 2, 3, 4, 5, 6 \}$ of positive integers and the $6$-permutation $(4, 2, 5, 1, 6,3)$.
Now $a_1$ of our inversion sequence will be the number of integers to the left of $1$ in our $6$-permutation that are larger than $1$. We see that the numbers $4$, $2$, and $5$ are all to the left of $1$ in our permutation and each is greater than $1$ so $a_1 = 3$.
$a_2$ of our inversion sequence will be the number of integers to the left of $2$ in our $6$-permutation that are larger than $2$. We see the number $4$ is the only integer to the left of $2$ and $4$ is larger than $2$ so $a_2 = 1$.
In we carry this process up to $a_6$ we see that our inversion sequence for this particular permutation is:
(1)
\begin{align} \quad (a_i)_{i=1}^{6} = (3, 1, 3, 0, 0, 0) \end{align}
There are a few important properties to note about inversion sequences. Consider a general $n$-permutation $(x_1, x_2, ..., x_n)$. The term $a_n$ of the inversion sequence will always equal $0$. Similarly, if $j = x_1$ then $a_j = 0$. This is because there are no terms to the left of $x_1$, and so the number of terms to the left of $j$ in the $n$-permutation is $0$, i.e., $a_j = 0$.
Another important property is that $0 \leq a_1 \leq n-1$, $0 \leq a_2 \leq n-2$, …, $0 \leq a_{n-1} \leq 1$ since for each $j \in \{1, 2, ..., n \}$ we have that there can be at most $n - j$ integers to the left of $j$ in the $n$-permutation and larger than $j$.
Thus, $a_1 \in \{0, 1, 2, ..., n-1 \}$, $a_2 \in \{0, 1, 2, ..., n-2 \}$, …, $a_{n-1} \in \{0, 1 \}$ and $a_n \in \{ 0 \}$. Thus $a_1$ can be one of $n$ numbers, $a_2$ can be one of $n - 1$ numbers, …, $a_{n-1}$ can be one of $2$ numbers and $a_n$ can be one of $1$ numbers. Therefore the number of inversion sequences for an $n$-permutation is $n \cdot (n - 1) \cdot ... \cdot 2 \cdot 1 = n!$, i.e., the number of inversion sequences equals the number of $n$-permutations of a finite $n$-element set $A$ of positive integers. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.982179582118988, "perplexity": 87.15894298222067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514030.89/warc/CC-MAIN-20181021140037-20181021161537-00485.warc.gz"} |
http://math.stackexchange.com/questions/93382/is-there-a-sequence-whose-set-of-subsequence-limits-is-mathbbq-cap-0-1 | # Is there a sequence whose set of subsequence limits is $\mathbb{Q}\cap [0,1]$?
Is there a sequence whose set of subsequential limits are $\mathbb Q \cap [0,1]$?
I was told that there wasn't such a sequence, is this true?
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What do you know about the set of subsequential limits of a sequence $(a_{n})_{n \in \mathbb N}$? – t.b. Dec 22 '11 at 6:37
@t.b. Can you provide a little more please? – mathmath8128 Dec 22 '11 at 6:40
Okay, I'd say there isn't such a sequence if $Q\cap[0,1]$ are taken to be exclusive (e.g. the only possible subsequence limits). Assume that $a_n$ fulfills this property, and let $b_n$ be a sequence of rational numbers in $[0,1]$ whose limit is an irrational number in $[0,1]$. For each $b_i$ we can find infinite members of $a_n$ that're close to $b_i$ as much as we like, so we can create a subsequence of $a_n$ whose limit is $b_i$'s limit. Is this correct? – ro44 Dec 22 '11 at 6:49
Yes, that's correct -- you could write that up as an answer. – joriki Dec 22 '11 at 6:53
@joriki I had to wait 8 hours (!) before I could type it up, so I selected Kannappan's answer instead :). – ro44 Dec 22 '11 at 7:03
You have been told the right answer, if you want $\mathbb{Q} \cap [0,1]$ to be the exclusive limit points!
As t.b. points out, the set of subsequential limits, $\mathcal{S}$ is a closed set. In the 'epsilonics', it would mean, Whenever $x \in \mathcal S^c$, there exists a $\delta > 0$ such that, $B_\delta(x) \subseteq \mathcal{S}^c$, where $\mathcal{S}^c$ stands for complement of $\mathcal {S}$. A special thanks to @joriki, (also @t.b.) in this connection!! (Read the comments, in case you are wondering why.)
But as irrational numbers are dense in $[0,1]$, there does not exist any open ball $B_\delta(x)$ with property as mentioned.
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What's $E$? I think it should say "whenever $x\in\overline{\mathcal{S}}$, there is $\delta\gt0$ such that $B_\delta(x)\subset\overline{\mathcal{S}}$? – joriki Dec 22 '11 at 7:14
@joriki Fixed, Thanks for the pointer. But, I would keep it $\mathcal{S}$, because $\mathcal{S}$ is a closed set, and will coincide with its closure, $\bar{\mathcal{S}}$. – user21436 Dec 22 '11 at 7:25
Sorry, I should have defined my notation -- that was supposed to be the complement, not the closure. What you've written is the definition of $\mathcal S$ being open rather than closed. (BTW you can get the bar to cover the entire $\mathcal S$ by using \overline instead of \bar.) – joriki Dec 22 '11 at 7:28
@joriki I probably must have gone completely off my brains. I'll fix it. Thanks for your LaTeX help in this direction. – user21436 Dec 22 '11 at 7:32
@Kannapan: I don't see the need to edit it but using standard notation in a non-standard way is a very bad idea, even if it's explained. (By the way: given that you're so eager to edit so many things so often elsewhere, I'm a bit surprised by this statement...) – t.b. Dec 22 '11 at 8:19
Let $\alpha$ be your favourite irrational number in the interval $[0,1]$. For definiteness, let $\alpha=\sqrt{2}/2$. Suppose that the set of subsequential limits of the sequence $a_1,a_2,a_3,\dots$ includes all rationals in the interval $[0,1]$. We will show that $\alpha$ is also a subsequential limit of the sequence $(a_n)$.
The rationals are dense in the reals. So there is a sequence $r_1,r_2,r_3,\dots$ of rationals in $[0,1]$ such that $(r_n)$ has limit $\alpha$. Such a sequence, can, for example, be obtained by truncating the decimal expansion of $\alpha$ further and further along.
We now construct our subsequence of $(a_n)$ that has limit $\alpha$. Let $n_1$ be the smallest index $i$ such that $|a_i-r_1|<\frac{1}{2}$. There is such an $i$ because the sequence $(a_n)$ has, by assumption, a subsequence that converges to $r_1$.
Let $n_2$ be the smallest index $i$ such that $i>n_1$, and $|a_i-r_2|<\frac{1}{2^2}$. There is such an $i$ because $(a_n)$ has a subsequence that converges to $r_2$.
Let $n_3$ be the smallest index $i$ such that $i>n_2$, and $|a_i-r_3|<\frac{1}{2^3}$. There is such an $i$ because $(a_n)$ has a subsequence that converges to $r_3$.
Continue in the obvious way. The sequence $(a_{n_k})$ is a subsequence of $(a_n)$ and converges to $\alpha$.
Comment: We have proved that every real number in $[0,1]$ is a subsequential limit of $(a_n)$. More generally, let $(a_n)$ be any sequence. The same argument shows that the set of subsequential limits of $(a_n)$ is closed.
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Suppose $\{a_j\}_{j=0}^{\infty}$ is a sequence that has $\mathbb{Q}\cap [0,1]$ as limit points. Let $\alpha$ be an irrational number in $[0,1]$ and let $\{b_j\}_{j=0}^{\infty}$ be a sequence of rationals in $[0,1]$ that converges to $\alpha$. Set $d_i = |\alpha-b_i|$; by construction $d_i\rightarrow 0$. Since there is a subsequence of $\{a_i\}_{i=0}^{\infty}$ converging to $b_i$ there must be some $a_i$ within $d_i$ of $b_i$. Thus $a_i$ is within $2d_i$ of $\alpha$. Doing this for every $i$ yields a subsequence of $\{a_j\}_{j=0}^{\infty}$ converging to $\alpha$.
Consequently, any sequence that has $\mathbb{Q}\cap [0,1]$ as limit points (and I make no claims about the existence of such a sequence) actually has all of $[0,1]$ as limit points.
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Such sequences (having subsequences that converge to every point in $[0,1]$) do exist: one example is the van der Corput sequence, obtained by reversing the base-$n$ representation of consecutive integers. – Ilmari Karonen Dec 22 '11 at 8:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9810624122619629, "perplexity": 122.95407892113039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375094690.4/warc/CC-MAIN-20150627031814-00141-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://mathhelpforum.com/pre-calculus/67781-real-numbers-natural-logs.html | Math Help - Real numbers and natural logs
1. Real numbers and natural logs
If possible, find a pair of real numbers $x$ and $y$ such that $xy = 6$ but it is not true that $ln6 = lnx + lny$.
I don't think this is possible..is it? I've tried with 6 and 1, 3 and 2, 1/2 and 12, and 1/3 and 18.
Thanks for any help!!
2. $(-2)(-3) = 6$
now think. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8863294720649719, "perplexity": 278.03057624251034}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644065330.34/warc/CC-MAIN-20150827025425-00053-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://laughmaths.blogspot.com/2015_11_01_archive.html | ## Saturday, 28 November 2015
### Diffusion of the dead - The maths of zombie invasions. Part 3, Diffusive motion.
As discussed previously, we are going to model the zombie motion using the diffusion equation. In this post we introduce the gritty details. I've interpreted the mathematical symbols intuitively, so, if you stick with it, you should find yourself understanding more than you ever thought you could.
It is impossible to overstate the importance of the diffusion equation. Wherever the movement of a modelled species can be considered random and directionless, the diffusion equation will be found. This means that by understanding the diffusion equation we are able to describe a host of different systems such as heat conduction through solids, gases (e.g. smells) spreading out through a room, proteins moving round the body, molecule transportation in chemical reactions and rainwater seeping through soil, to name but a few of the great numbers of applications.
If you've never come across diffusion before, or want to know more about it's basic properties the video below is a very good primer, although feels very much like a "Look around you" episode.
The mathematical treatment of diffusion begins by defining the variables that we will need. Let the density of zombies at a point $x$ and at a time $t$ be $Z(x,t)$ then the density has to satisfy the diffusion equation,
\frac{\partial Z}{\partial t}(x,t)=D\frac{\partial^2 Z}{\partial x^2}(x,t).
To some an equation can be scarier than any zombie, but fear not. I am going to break this equation down into bits so that you are able to see the reality behind the mathematics.
Notice that the equation is made up of two terms, the left-hand side and the right-hand side, which are defined to be equal. Explicitly, the left-hand side is known as the time derivative and it simply tell us how the zombie density is changing over time,
\frac{\partial Z}{\partial t}(x,t)=\text{rate of change of $Z$ over time at a point $x$}.
Although the numerical value of this term is important, what is more important is if the term is positive or negative. Specifically, if $\partial Z/\partial t$ is positive then $Z$ is increasing at that point in time, and, vice-versa, if $\partial Z/\partial t$ is negative then $Z$ is decreasing. Thus, we use this term to tell us how the zombie population is changing over time.
The term on the right-hand side is known as the second spatial derivative and it is a little more complicated than the time derivative. Essentially it encapsulates the idea that the zombies move from areas of high density to areas of low density (i.e. they spread out). To aid your intuitive understanding of this term see Figure 1.
Figure 1. A typical initial zombie density graph. There are regions of high zombie activity, e.g. a graveyard, and there are regions of low zombie density, e.g. your local library.
In the figure, there are initially more zombies on the left of the space than the right. Just before the peak in density the arrow (which is the tangent to the curve known as the spatial derivative, or $\partial Z/\partial x$ at this point) is pointing upwards. This means that as $x$ increases, so does the zombie density, $Z$. At this point
\frac{\partial Z}{\partial x}=\textrm{rate of change of $Z$ as $x$ increases} > 0.
Just after the peak the arrow is pointing down thus, at this point,
\frac{\partial Z}{\partial x}=\textrm{rate of change of $Z$ as $x$ increases} < 0.
Thus, at the peak, the spatial derivative is decreasing, because it goes from positive to negative. This, in turn, means that the second derivative is negative at the peak, because a negative second derivative means the first derivative is decreasing. This is analogous to statements made above about the sign of the time derivative and the growth, or decay, of the zombie population.
In summary, our hand wavy argument tells us that at local maximum $\partial^2 Z/\partial x^2<0$. Using the equality of the diffusion equation, this means that at a local maximum the time derivative is negative and, thus, the density of zombies is decreasing. A similar argument shows that the population of zombies at a local minimum increases. In summary, we see that diffusion causes zombies to move from regions of high density to low density.
Finally, we mention the factor $D$, which is called the diffusion coefficient. $D$ is a positive constant that controls the rate of movement. Specifically, the larger $D$ is the faster the zombies spread out.
And with that you now understand one of the most important partial differential equations in all of mathematics. That wasn't too hard was it? Next time we discuss the solution of the diffusion equation including some simulations and Matlab code for you to try yourself.
## Saturday, 14 November 2015
### Diffusion of the dead - The maths of zombie invasions. Part 2, Important questions you need to ask in a zombie outbreak.
We begin modelling a zombie population in the same way that a mathematician would approach the modelling of any subject. We, first, consider what questions we want to ask, as the questions will direct which techniques we use to solve the problem. Secondly, we consider what has been done before and what factors were missing in order to achieve the answers we desire. This set of blog posts will consider three questions:
1. How long will it take for the zombies to reach us?
2. Can we stop the infection?
3. Can we survive?
In order to answer these questions I, Ruth Baker, Eamonn Gaffney and Philip Maini focused on the motion of the zombies as their speed and directionality would have huge effects on these three questions. Explicitly, we used a mathematical description of diffusion as a way to model the zombies motion. This was discussed in a previous post, but I recap the main points here.
• The original zombie infection article by Robert Smith? did not include zombie, or human movement.
• Zombies are well known for is their slow, shuffling, random motion. The end of Dawn of the Dead (shown in the YouTube clip below) gives some great footage of zombies just going about their daily business.
• This random motion is perfectly captured through the mathematics of diffusion.
Of course, there is plenty of evidence to suggest that zombies are attracted to human beings, as they are the predator to our prey. However, as we will see, we are going to be over run on a time scale of minutes! Thus, although mathematicians can model directed motion, and chasing, these additional components complicate matters. Further, random motion leads to some nice simple scaling formulas that can be used to quickly calculate how long you approximately have left before you meet a zombie.
Another simplifying assumption that we make is that we can model the zombie (and human) populations as continuous quantities. Again, this is incorrect as zombies are discrete units (even if they are missing body parts). Since we are making an assumption we will create an error in our solution. But how big is this error? In particular, if the error in the assumption is smaller than the errors in our observable data set then we do not have to worry too much. The error introduced by this assumption is actually dependent on the size of the population we are considering. The more individuals you have, you more the population will act like a continuous quantity. Since there are a lot of corpses out there, we do not think this assumption is too bad.
Note that we could model the motion of each zombie individually, however, the computing power needed by such a simulation is much larger than the continuum description, which can be solved completely analytically. This is particularly important in the case of the zombie apocalypse, where time spent coding a simulation, may be better spent scavenging.
These are the basic assumptions we made when modelling a zombie population. Although I have tried to justify them you may have reservations about their validity. That is the very nature of mathematical modelling; try the simplest thing, first, and compare it to data. If you reproduce the phenomena that you are interested in then you have done your job well. However, if there is a discrepancy between the data and your maths then you have to revisit your assumptions and adapt them to make them more realistic.
Next time we contend with the equations and model the motion of the zombie as a random walker. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8649190068244934, "perplexity": 417.8743356711423}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612008.48/warc/CC-MAIN-20170529014619-20170529034619-00302.warc.gz"} |
https://theculture.sg/2016/03/2015-a-level-h2-mathematics-9740-paper-2-question-2-suggested-solutions/ | All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.
(i)
$\theta = \mathrm{cos^{-1}} |\frac{\begin{pmatrix}2\\3\\{-6}\end{pmatrix}\cdot \begin{pmatrix}1\\0\\{0}\end{pmatrix}}{\sqrt{49} \cdot 1}| = 73.4^{\circ}$
(ii)
Let $\vec{ON}$ be point on L that makes $\sqrt{33}$ from P.
$\vec{ON} = \begin{pmatrix}1\\{-2}\\{-4}\end{pmatrix} + \lambda \begin{pmatrix}2\\3\\{-6}\end{pmatrix}$ for some $\lambda$
$\vec{PN} = \begin{pmatrix}{-1}\\{-7}\\{2}\end{pmatrix} + \lambda \begin{pmatrix}2\\3\\{-6}\end{pmatrix}$
$|\vec{PN}| = \sqrt{(-1+2\lambda)^2 + (-7+3\lambda)^2 + (2-6\lambda)^2}$
$33 = 49 \lambda^2 - 70 \lambda + 54$
$\lambda = 1 \mathrm{~or~} \frac{3}{7}$
$\vec{ON} = \begin{pmatrix}3\\{1}\\{-10}\end{pmatrix} \mathrm{~or~} \frac{1}{7}\begin{pmatrix}13\\{-5}\\{-46}\end{pmatrix}$
$L = |\vec{PN}|^2 = (-1+2\lambda)^2 + (-7+3\lambda)^2 + (2-6\lambda)^2 = 49 \lambda^2 - 70 \lambda +54$
$\frac{dL}{d\lambda} = 98 \lambda - 70$
$\frac{d^2L}{d\lambda ^2} = 98 > 0$
So when $\lambda = \frac{70}{98} = \frac{5}{7}$, L is minimum.
$\vec{ON} = \frac{1}{7} \begin{pmatrix}{17}\\{1}\\{-58}\end{pmatrix}$
(iii)
$\begin{pmatrix}{-1}\\{-7}\\{2}\end{pmatrix} \times \begin{pmatrix}{2}\\{3}\\{-6}\end{pmatrix} = \begin{pmatrix}{36}\\{-2}\\{11}\end{pmatrix}$
$\begin{pmatrix}{1}\\{-2}\\{-4}\end{pmatrix} \bullet \begin{pmatrix}{36}\\{-2}\\{11}\end{pmatrix} = -4$
$\therefore, \pi : 36x - 2y +11z=-4$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 18, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.807405412197113, "perplexity": 4821.138362284719}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818695113.88/warc/CC-MAIN-20170926070351-20170926090351-00086.warc.gz"} |
http://www.scientificlib.com/en/Mathematics/HW/HallLittlewoodPolynomials.html | # .
In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials. They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Littlewood (1961).
Definition
The Hall–Littlewood polynomial P is defined by
$$P_\lambda(x_1,\ldots,x_n;t) = \prod_{i}\frac{1-t}{1-t^{m(i)}} {\sum_{w\in S_n}w\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod_{i<j}\frac{x_i-tx_j}{x_i-x_j}\right)},$$
where λ is a partition of length at most n with elements $$\lambda_i$$, and m(i) elements equal to i, and $$S_n$$ is the symmetric group of order n!.
See also
Hall polynomial
References
I.G. Macdonald (1979). Symmetric Functions and Hall Polynomials. Oxford University Press. pp. 101–104. ISBN 0-19-853530-9.
D.E. Littlewood (1961). "On certain symmetric functions". Proceedings of the London Mathematical Society 43: 485–498. doi:10.1112/plms/s3-11.1.485.
External links
Weisstein, Eric W., "Hall-Littlewood Polynomial", MathWorld.
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
Home | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8288764953613281, "perplexity": 913.5634415104511}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218193288.61/warc/CC-MAIN-20170322212953-00408-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/241297/dipole-dipole-field | # Dipole-dipole field
Assuming that we have two parallel and exactly identical electric dipoles. How does bringing them together change their frequency? What would such field look like (how would they interact)?
In fact, this question arose, because I had hard time understanding how [here] frequencies are calculated. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8583503365516663, "perplexity": 1780.096310363651}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669847.1/warc/CC-MAIN-20191118205402-20191118233402-00054.warc.gz"} |
https://zenodo.org/record/346001 | Working paper Open Access
# Some steps towards a theory of expressions
Colignatus, Thomas
The mathematical "theory of expressions" better be developed in a general fashion, so that it can be referred to in various applications. The paper discusses an example when there is a confusion between syntax (unevaluated) and semantics (evaluated), when substitution causes a contradiction. However, education should not wait till such a mathematical theory of expressions is fully developed. Computer algebra is sufficiently developed to support and clarify these issues. Fractions y / x can actually be abolised and replaced with y xH with H = -1 as a constant like exponential number e or imaginary number i.
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http://www.physicsforums.com/showthread.php?p=4188026 | # Is relative mass making gravitational field?
by marshallaw
Tags: field, gravitational, mass, relative
P: 1 Hello When you have something and it gains a lot of mass m due to its high kinetic energy, so it gains a lot of relative energy. So, every object has its own gravitational field. So is the RELATIVE mass making a gravitatinal field? mathematically if g=GM/Rˇ2 does work, where you MUST add relative mass Well this would be very amazing, strange, bizzare if this was true
Emeritus Sci Advisor PF Gold P: 5,500 If I'm understanding correctly, then you're using the terms "relative energy" and "relative mass" to mean what is referred to in standard terminology as mass-energy. In standard (modern) terminology, "mass" is absolute, not relative. What we now call mass is what people used to refer to as rest mass. The source of the gravitational field is the stress-energy tensor, not mass-energy. You can sometimes get away with estimating gravitational effects by plugging mass-energy into Newton's law of gravity, but it won't always work. For example, parallel beams of light exert zero gravitational force on one another, but antiparallel ones exert nonzero force.
Sci Advisor PF Gold P: 4,862 A next level of approximation is invariant mass. It is often accurate up to the point where self gravitation is important, for simple systems. For example it 'explains' the anti-parallel vs. parallel beams. In the former, the invariant mass of the system adds the beam energy because the momentum cancels. In the parallel case, the invariant mass is zero. Similarly, the KE of an isolated body (no matter how fast it is moving in some frame) has no relevance for any curvature invariant produced by it. However, as part of confined system of bodies (e.g. particles in a box), the invariant mass of the system will tend to include much of the KE of each body due to momentum cancellation. Just don't take this too far - the real source, as mentioned, is the stress energy tensor.
P: 5,307
## Is relative mass making gravitational field?
this question could be studied using the Aichelburg–Sexl ultraboost metric http://en.wikipedia.org/wiki/Aichelb...exl_ultraboost
Sci Advisor P: 869 I would like to nominate that as the metric with the most beautiful name.
Emeritus Sci Advisor P: 7,436 I tend to view energy (sometimes called relativistic mass) as a better approximation for gravity in most circumstances than invariant mass. Examples where this approach works reasonably well. particles in a box moving thermally velocities induced by a relativistic flyby. In fact, the velocities induced by a relativistic flyby increase MORE rapidly than the approximation usig energy- see http://dx.doi.org/10.1119/1.14280 . So using invariant mass to compute velocities due to a relativistic flyby is a very bad approximation, it's off by a factor of 2*gamma, whereas the using energy is only off by a factor of 2:1 or so. But as everyone has pointed out, the actual situation is more complex, it's really the stress energy tensor that causes gravity, and if you want to get truly accurate answers that's what you need to use. Which leads to the other reason I like to use energy. It's easier to justify as an approximation - you're approximating the stress-energy tensor by just the energy part, and throwing away the others (the momentum part, and the stress part). Invariant mass isn't part of the stress energy tensor the way energy is. The other thing that's worth mentioning is that F=GmM/r^2 doesn't work at all for relativistically moving bodies. The "field" from a moving body isn't at all spherically symmetric. Some insight into this can be gained by considering the electric field of a rapidly moving body. A search should find a lot of posts on this topic.
P: 5,634
Marshallaw.....WOW you got great replies here,,,,,sometimes an initially insigthful answer spawns others.....
As usual when members of this group reply I have some recollecting to do. If you are
that way, check out the first section here:
http://en.wikipedia.org/wiki/Invariant_mass
for a bit of background.....especially 'frames'.
The source of the gravitational field is the stress-energy tensor, not mass-energy.
took me by surprise when I first read it several years ago. Later I realized that it meant the mass-energy was only a piece of the source....as pervect illustrated while I was typing! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8900961875915527, "perplexity": 789.9898150407179}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00623-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://chemistry.stackexchange.com/questions/116229/how-can-the-potential-energy-at-the-infinite-energy-level-in-an-atom-be-zero | # How can the potential energy at the infinite energy level in an atom be zero?
I have recently learnt about atomic structure and the Bohr model of the atom and have observed a discrepancy between it and my previous knowledge based on physics.
For simplicity, assume the atom is of hydrogen.
Total energy of electron $$E= -13.6 \left(\frac{z^2}{n^2}\right) \pu{eV/atom}$$ where $$z$$ is atomic number, $$n$$ is energy level.
Thus, at $$n=\infty$$, $$E=0$$.
Now, by the relation Total Energy = potential energy/2 (from $$TE=-KE=PE/2$$),
$$PE=0/2$$ so $$PE=0$$
Now, the infinite energy level ($$n=\infty$$) occurs at a finite distance from nucleus (as the difference in distance between the energy levels decreases at higher energy levels).
Thus, $$PE=0$$ at a finite distance from the nucleus.
But, I have previously studied that $$PE$$ of a system of a negative charge (electron) and positive charge(nucleus) is $$=0$$ only at an infinite distance.
Thus, the discrepancy.
Please let me know where I have gone wrong.
• You incorrectly assume that when $n=\infty$ the electron and proton are at a finite separation. – porphyrin Jun 1 '19 at 8:19
• Let's forget for a moment that the Bohr model is pretty rudimentary. Why do you think that the orbital radius for n-> infinite is finite? Can you explain this a little more so we can answer? – Greg Jun 1 '19 at 8:24
• Please see the following image: google.com/…: – Shashwat Tomar Jun 1 '19 at 8:29
• Yes your question is clear, and clearly based on a wrong premise. You know the formula for energy; you plug $n=\infty$ and correctly deduce that $E=0$. But you don't know the formula for radius and can't plug $n$ there, so you have to rely on obscure googled pictures instead. That's the root of all evil. – Ivan Neretin Jun 1 '19 at 9:04
• The picture is not obscure, it is a great representation of the relationship between n and E in the H atom. However, it has nothing to do with radius. The distances shown between the bars are not spatial distances, they represent the energy differences. The vertical axis is energy. – electronpusher Jun 1 '19 at 17:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9630850553512573, "perplexity": 474.19667053491503}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738982.70/warc/CC-MAIN-20200813103121-20200813133121-00392.warc.gz"} |
https://www.physicsforums.com/threads/parametric-equation-speed.292018/ | # Parametric Equation Speed
1. Feb 12, 2009
### keemosabi
1. The problem statement, all variables and given/known data
Can someone please tell me how to get the average speed of a particle moving along a path represented by parametric equations? Is it $\frac{1}{b-a}\int_{a}^{b}\sqrt{\frac{dx }{d t}^2 + \frac{d y}{d t}^2}$
Isn't this the arc length formula?
2. Feb 12, 2009
### w3390
This is the arc length formula. The average value formula is Favg=(1/b-a)INT[f(x)dx]. It seems you combined two formulas.
3. Feb 12, 2009
### keemosabi
But if I wanted the speed of a particle moving with a parametric graph, woldn't everything under the radical be my speed function?
4. Feb 12, 2009
### w3390
Actually, you may be right. I think that might actually work.
5. Feb 13, 2009
### Dick
No, no, no. The average speed is displacement over time. It has nothing to do with arc length. It's sqrt((x(b)-x(a))^2+(y(b)-y(a))^2)/(b-a) where a is the intiial time and b is the final time. Right?
6. Feb 13, 2009
### keemosabi
Couldn't you also do the average value of the absolute value of the velocity graph?
7. Feb 13, 2009
### Dick
Yes, you could. In which case that would be correct. Distance travelled/time could also be considered an average speed. I was only thinking of the displacement/time definition.
8. Feb 13, 2009
### keemosabi
Alirght, thank you for the help.
Also, is there any way to determine if a particle traveling on a parametric path is increasing in speed? I know I can determine if the x and y are accelerating, but I can I determine if the particle itself is increasing?
What if it was accelearating in the x direction but decelerating in the y? Would the particle's speed be increasing or decreasing?
9. Feb 13, 2009
### Dick
The 'speed' is sqrt((dx/dt)^2+(dy/dt)^2), isn't it? Just look at whether that quantity is increasing or decreasing. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9557677507400513, "perplexity": 966.4610808136729}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982924728.51/warc/CC-MAIN-20160823200844-00146-ip-10-153-172-175.ec2.internal.warc.gz"} |
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