url
stringlengths 14
2.42k
| text
stringlengths 100
1.02M
| date
stringlengths 19
19
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|---|
https://brilliant.org/problems/exhaustible/
|
# Exhaustible
Discrete Mathematics Level pending
how many subsets can be formed of a set:{1,2,3,4,5,6,7,8,9,10,11}
×
|
2018-03-22 08:19:58
|
{"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": 0.8564359545707703, "perplexity": 14015.032877769609}, "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/1521257647782.95/warc/CC-MAIN-20180322073140-20180322093140-00188.warc.gz"}
|
http://mathhelpforum.com/trigonometry/62438-i-can-t-solve-these-trig-equations.html
|
# Thread: I can't solve these Trig Equations
1. ## I can't solve these Trig Equations
Hi , out of my 100 question holiday packet, these are ones I just couldn't understand:
Solve these equations in the indicated domain
(1+cos(theta))/(sin(theta))=-1
domain: [-180,180)
cos4(theta) - sin2(theta) = 0
domain: (-90,90)
cos4(theta) - sin2(theta) = 1
domain: [-90,90)
cos3(theta) + cos5(theta) = 0
domain: (-90,90)
sin5(theta) + sin7(theta) = 0
domain: [-90,90)
cosx - root3(sinx) = 1
domain (0,2pi]
sinx - root3(cosx) = 1
domain: [-pi,pi]
[tan(10theta) + tan50(deg)]/[1-tan(10theta)tan50(deg)] = (root3)/3
domain: (0,90)
tan(theta) - tan(10theta) = 1 + tan(theta)tan10(deg)
domain: [-180,180]
tan(1/2)x + 1 = cosx
domain: [0,4pi]
2(cos^2)(1/2)x - 2 = 2cosx
domain: [-pi,pi)
2cos(theta + 30)cos(theta - 30) = 1
domain: [-180,180]
4sin(theta + 75)cos(theta - 75) = 1
domain: [-180,180)
(cos^2)(1/2)x - (1/2)cosx = 1/2
domain: real numbers
sinxtan(1/2)x = 1 - cosx
domain: real numbers
Any help is greatly appreciated!
Thanks!
2. Hello, hassapi!
Here's #8 . . .
$8)\;\;\frac{\tan 10\theta + \tan50^o}{1-(\tan 10\theta)(\tan50^o)} \:=\:\frac{\sqrt{3}}{3}\qquad\text{ domain: }(0^o,90^o)$
We're expected to recognize the left side as a Compound Angle Identity . . .
. . $\tan(10\theta + 50^o) \;=\;\frac{1}{\sqrt{3}}$
And we should know that the angle is 30° and its variations.
So we have:. . $10\theta + 50^o \;=\;\{30^o,\:210^o,\:390^o,\:570^o,\:750^o,\:930^ o\}$
. . . . . Then: . . . . . $10\theta \;=\;\{{\color{red}\rlap{/////}}-20^o,\:160^o,\:340^o,\;520^o,\:700^o,\:880^o\}$
. . Therefore: . . . . . . $\theta \;=\;\{16^o,\;34^o,\:52^o,\:70^o,\:88^o\}$
3. Originally Posted by Soroban
Hello, hassapi!
Here's #8 . . .
We're expected to recognize the left side as a Compound Angle Identity . . .
. . $\tan(10\theta + 50^o) \;=\;\frac{1}{\sqrt{3}}$
And we should know that the angle is 30° and its variations.
So we have:. . $10\theta + 50^o \;=\;\{30^o,\:210^o,\:390^o,\:570^o,\:750^o,\:930^ o\}$
. . . . . Then: . . . . . $10\theta \;=\;\{{\color{red}\rlap{/////}}-20^o,\:160^o,\:340^o,\;520^o,\:700^o,\:880^o\}$
. . Therefore: . . . . . . $\theta \;=\;\{16^o,\;34^o,\:52^o,\:70^o,\:88^o\}$
Ah, thank you very much! I don't think my teacher taught us compound angle identity...
|
2017-03-23 19:47:07
|
{"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": 9, "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": 0.9609439373016357, "perplexity": 3435.0677155476083}, "config": {"markdown_headings": true, "markdown_code": false, "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-2017-13/segments/1490218187193.40/warc/CC-MAIN-20170322212947-00119-ip-10-233-31-227.ec2.internal.warc.gz"}
|
http://www.chegg.com/homework-help/questions-and-answers/vo-vo-circuit-shown-fig-1320-nilson-bookchapter-13-problem-20-initial-energy-zero-switchcl-q615995
|
## nilson book chapter 13 problem 20
find Vo and vo in the circuit shown in FIG 13.20(nilson bookchapter 13 problem 20) if the initial energy is zero and the switchclosed at t =0
|
2013-05-21 11:05:10
|
{"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": 0.9435756206512451, "perplexity": 3603.726548508998}, "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-2013-20/segments/1368699899882/warc/CC-MAIN-20130516102459-00005-ip-10-60-113-184.ec2.internal.warc.gz"}
|
https://stacks.math.columbia.edu/tag/0BA7
|
Remark 70.10.6. Let $k$ be finite field. Let $K \supset k$ be a geometrically irreducible field extension. Then $K$ is the limit of geometrically irreducible finite type $k$-algebras $A$. Given $A$ the estimates of Lang and Weil [LW], show that for $n \gg 0$ there exists an $k$-algebra homomorphism $A \to k'$ with $k'/k$ of degree $n$. Analyzing the argument given in the proof of Lemma 70.10.5 we see that if $X$ is a quasi-separated algebraic space over $k$ and $X_ K$ is a scheme, then $X$ is a scheme. If we ever need this result we will precisely formulate it and prove it here.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
|
2020-08-08 00:57:36
|
{"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9895702600479126, "perplexity": 166.30816038553127}, "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-2020-34/segments/1596439737233.51/warc/CC-MAIN-20200807231820-20200808021820-00119.warc.gz"}
|
https://math.stackexchange.com/questions/2408528/prove-disprove-absolute-convergent
|
# prove/disprove absolute convergent
$\int_0^\infty 3^{-x}x^4cos(2x)dx$
I succeeded to prove that this integral is conditionally convergent with Dirichlet's test. I don't know how to prove/disprove absolutely convergent..
Thanks !
• What do you mean by " uniform convergence" of the integral ???? – Fred Aug 28 '17 at 9:08
• You right, fixed it. Thanks ! – Jill Aug 28 '17 at 9:20
You have that $$\lim_{x\to \infty }x^23^{-x}x^4\cos(2x)=0,$$ and thus $$3^{-x}x^4\cos(2x)=\mathcal O\left(\frac{1}{x^2}\right),$$ at the neighborhood of $+\infty$. Therefore it's absolutely integrable on $[1,+\infty )$. The integrability on $[0,1]$ is obvious. The claim follow.
• Because $$0=\lim_{x\to \infty }x^2 3^{-x}x^4\cos(2x)=\lim_{x\to \infty }\frac{3^{-x}x^4\cos(2x)}{\frac{1}{x^2}}$$ and thus $3^{-x}x^4\cos(2x)=o(1/x^2)$ which is in particular a $\mathcal O(1/x^2)$. @Jill – Surb Aug 28 '17 at 10:07
|
2019-07-17 00:58:38
|
{"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": 0.9686965942382812, "perplexity": 677.776336175342}, "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-2019-30/segments/1563195525004.24/warc/CC-MAIN-20190717001433-20190717023433-00276.warc.gz"}
|
https://www.springerprofessional.de/computer-aided-verification/2503736
|
main-content
## Über dieses Buch
The two-volume set LNCS 9206 and LNCS 9207 constitutes the refereed proceedings of the 27th International Conference on Computer Aided Verification, CAV 2015, held in San Francisco, CA, USA, in July 2015.
The total of 58 full and 11 short papers presented in the proceedings was carefully reviewed and selected from 252 submissions. The papers were organized in topical sections named: model checking and refinements; quantitative reasoning; software analysis; lightning talks; interpolation, IC3/PDR, and Invariants; SMT techniques and applications; HW verification; synthesis; termination; and concurrency.
## Inhaltsverzeichnis
### Poling: SMT Aided Linearizability Proofs
Abstract
Proofs of linearizability of concurrent data structures generally rely on identifying linearization points to establish a simulation argument between the implementation and the specification. However, for many linearizable data structure operations, the linearization points may not correspond to their internal static code locations; for example, they might reside in the code of another concurrent operation. To overcome this limitation, we identify important program patterns that expose such instances, and describe a tool (Poling) that automatically verifies the linearizability of implementations that conform to these patterns.
He Zhu, Gustavo Petri, Suresh Jagannathan
### Finding Bounded Path in Graph Using SMT for Automatic Clock Routing
Abstract
Automating the routing process is essential for the semiconductor industry to reduce time-to-market and increase productivity. This study sprang from the need to automate the following critical task in clock routing: given a set of nets, each net consisting of a driver and a receiver, connect each driver to its receiver, where the delay should be almost the same across the nets. We demonstrate that this problem can be reduced to bounded-path, that is, the NP-hard problem of finding a simple path, whose cost is bounded by a given range, connecting two given vertices in an undirected positively weighted graph. Furthermore, we show that bounded-path can be reduced to bit-vector reasoning and solved with a SAT-based bit-vector SMT solver. In order to render our solution scalable, we override the SAT solver’s decision strategy with a novel graph-aware strategy and augment conflict analysis with a graph-aware procedure. Our solution scales to graphs having millions of edges and vertices. It has been deployed at Intel for clock routing automation.
Amit Erez, Alexander Nadel
### Cutting the Mix
Abstract
While linear arithmetic has been studied in the context of SMT individually for reals and integers, mixed linear arithmetic allowing comparisons between integer and real variables has not received much attention. For linear integer arithmetic, the cuts from proofs algorithm has proven to have superior performance on many benchmarks. In this paper we extend this algorithm to the mixed case where real and integer variables occur in the same linear constraint. Our algorithm allows for an easy integration into existing SMT solvers. Experimental evaluation of our prototype implementation inside the SMT solver SMTInterpol shows that this algorithm is successful on benchmarks that are hard for all existing solvers.
Jürgen Christ, Jochen Hoenicke
### The Inez Mathematical Programming Modulo Theories Framework
Abstract
Our Mathematical Programming Modulo Theories (MPMT) constraint solving framework extends Mathematical Programming technology with techniques from the field of Automated Reasoning, e.g., solvers for first-order theories. In previous work, we used MPMT to synthesize system architectures for Boeing’s Dreamliner and we studied the theoretical aspects of MPMT by means of the Branch and Cut Modulo T ($${\text {BC}}(T)$$) transition system. $${\text {BC}}(T)$$ can be thought of as a blueprint for MPMT solvers. This paper provides a more practical and algorithmic view of $${\text {BC}}(T)$$. We elaborate on the design and features of Inez, our $${\text {BC}}(T)$$ constraint solver. Inez is an open-source, freely available superset of the OCaml programming language that uses the SCIP Branch and Cut framework to extend OCaml with MPMT capability. Inez allows users to write programs that arbitrarily interweave general computation with MPMT constraint solving.
Panagiotis Manolios, Jorge Pais, Vasilis Papavasileiou
### Using Minimal Correction Sets to More Efficiently Compute Minimal Unsatisfiable Sets
Abstract
An unsatisfiable set is a set of formulas whose conjunction is unsatisfiable. Every unsatisfiable set can be corrected, i.e., made satisfiable, by removing a subset of its members. The subset whose removal yields satisfiability is called a correction subset. Given an unsatisfiable set $${\mathcal {F}}$$ there is a well known hitting set duality between the unsatisfiable subsets of $${\mathcal {F}}$$ and the correction subsets of $${\mathcal {F}}$$: every unsatisfiable subset hits (has a non-empty intersection with) every correction subset, and, dually, every correction subset hits every unsatisfiable subset. An important problem with many applications in practice is to find a minimal unsatisfiable subset (mus) of $${\mathcal {F}}$$, i.e., an unsatisfiable subset all of whose proper subsets are satisfiable. A number of algorithms for this important problem have been proposed. In this paper we present new algorithms for finding a single mus and for finding all muses. Our algorithms exploit in a new way the duality between correction subsets and unsatisfiable subsets. We show that our algorithms advance the state of the art, enabling more effective computation of muses.
Fahiem Bacchus, George Katsirelos
### Deciding Local Theory Extensions via E-matching
Abstract
Satisfiability Modulo Theories (SMT) solvers incorporate decision procedures for theories of data types that commonly occur in software. This makes them important tools for automating verification problems. A limitation frequently encountered is that verification problems are often not fully expressible in the theories supported natively by the solvers. Many solvers allow the specification of application-specific theories as quantified axioms, but their handling is incomplete outside of narrow special cases.
In this work, we show how SMT solvers can be used to obtain complete decision procedures for local theory extensions, an important class of theories that are decidable using finite instantiation of axioms. We present an algorithm that uses E-matching to generate instances incrementally during the search, significantly reducing the number of generated instances compared to eager instantiation strategies. We have used two SMT solvers to implement this algorithm and conducted an extensive experimental evaluation on benchmarks derived from verification conditions for heap-manipulating programs. We believe that our results are of interest to both the users of SMT solvers as well as their developers.
Kshitij Bansal, Andrew Reynolds, Tim King, Clark Barrett, Thomas Wies
### Modular Deductive Verification of Multiprocessor Hardware Designs
Abstract
We present a new framework for modular verification of hardware designs in the style of the Bluespec language. That is, we formalize the idea of components in a hardware design, with well-defined input and output channels; and we show how to specify and verify components individually, with machine-checked proofs in the Coq proof assistant. As a demonstration, we verify a fairly realistic implementation of a multicore shared-memory system with two types of components: memory system and processor. Both components include nontrivial optimizations, with the memory system employing an arbitrary hierarchy of cache nodes that communicate with each other concurrently, and with the processor doing speculative execution of many concurrent read operations. Nonetheless, we prove that the combined system implements sequential consistency. To our knowledge, our memory-system proof is the first machine verification of a cache-coherence protocol parameterized over an arbitrary cache hierarchy, and our full-system proof is the first machine verification of sequential consistency for a multicore hardware design that includes caches and speculative processors.
Muralidaran Vijayaraghavan, Adam Chlipala, Arvind, Nirav Dave
### Word-Level Symbolic Trajectory Evaluation
Abstract
Symbolic trajectory evaluation (STE) is a model checking technique that has been successfully used to verify industrial designs. Existing implementations of STE, however, reason at the level of bits, allowing signals to take values in $$\{0, 1, X\}$$. This limits the amount of abstraction that can be achieved, and presents inherent limitations to scaling. The main contribution of this paper is to show how much more abstract lattices can be derived automatically from RTL descriptions, and how a model checker for the general theory of STE instantiated with such abstract lattices can be implemented in practice. This gives us the first practical word-level STE engine, called $$\mathsf {STEWord}$$. Experiments on a set of designs similar to those used in industry show that $$\mathsf {STEWord}$$ scales better than word-level BMC and also bit-level STE.
Supratik Chakraborty, Zurab Khasidashvili, Carl-Johan H. Seger, Rajkumar Gajavelly, Tanmay Haldankar, Dinesh Chhatani, Rakesh Mistry
### Verifying Linearizability of Intel® Software Guard Extensions
Abstract
Intel® Software Guard Extensions (SGX) is a collection of CPU instructions that enable an application to create secure containers that are inaccessible to untrusted entities, including the operating system and other low-level software. Establishing that the design of these instructions provides security is critical to the success of the feature, however, SGX introduces complex concurrent interactions between the instructions and the shared hardware state used to enforce security, rendering traditional approaches to validation insufficient. In this paper, we introduce Accordion, a domain specific language and compiler for automatically verifying linearizability via model checking. The compiler determines an appropriate linearization point for each instruction, computes the required linearizability assertions, and supports experimentation with a variety of model configurations across multiple model checking tools. We show that this approach to verifying linearizability works well for validating SGX and that the compiler provides improved usability over encoding the problem in a model checker directly.
Rebekah Leslie-Hurd, Dror Caspi, Matthew Fernandez
### Synthesis Through Unification
Abstract
Given a specification and a set of candidate programs (program space), the program synthesis problem is to find a candidate program that satisfies the specification. We present the synthesis through unification (STUN) approach, which is an extension of the counter-example guided inductive synthesis (CEGIS) approach. In CEGIS, the synthesizer maintains a subset S of inputs and a candidate program $$\mathtt {Prog}$$ that is correct for S. The synthesizer repeatedly checks if there exists a counterexample input c such that the execution of $$\mathtt {Prog}$$ is incorrect on c. If so, the synthesizer enlarges S to include c, and picks a program from the program space that is correct for the new set S.
The STUN approach extends CEGIS with the idea that given a program $$\mathtt {Prog}$$ that is correct for a subset of inputs, the synthesizer can try to find a program $$\mathtt {Prog}'$$ that is correct for the rest of the inputs. If $$\mathtt {Prog}$$ and $$\mathtt {Prog}'$$ can be unified into a program in the program space, then a solution has been found. We present a generic synthesis procedure based on the STUN approach and specialize it for three different domains by providing the appropriate unification operators. We implemented these specializations in prototype tools, and we show that our tools often performs significantly better on standard benchmarks than a tool based on a pure CEGIS approach.
Rajeev Alur, Pavol Černý, Arjun Radhakrishna
### From Non-preemptive to Preemptive Scheduling Using Synchronization Synthesis
Abstract
We present a computer-aided programming approach to concurrency. The approach allows programmers to program assuming a friendly, non-preemptive scheduler, and our synthesis procedure inserts synchronization to ensure that the final program works even with a preemptive scheduler. The correctness specification is implicit, inferred from the non-preemptive behavior. Let us consider sequences of calls that the program makes to an external interface. The specification requires that any such sequence produced under a preemptive scheduler should be included in the set of such sequences produced under a non-preemptive scheduler. The solution is based on a finitary abstraction, an algorithm for bounded language inclusion modulo an independence relation, and rules for inserting synchronization. We apply the approach to device-driver programming, where the driver threads call the software interface of the device and the API provided by the operating system. Our experiments demonstrate that our synthesis method is precise and efficient, and, since it does not require explicit specifications, is more practical than the conventional approach based on user-provided assertions.
Pavol Černý, Edmund M. Clarke, Thomas A. Henzinger, Arjun Radhakrishna, Leonid Ryzhyk, Roopsha Samanta, Thorsten Tarrach
### Counterexample-Guided Quantifier Instantiation for Synthesis in SMT
Abstract
We introduce the first program synthesis engine implemented inside an SMT solver. We present an approach that extracts solution functions from unsatisfiability proofs of the negated form of synthesis conjectures. We also discuss novel counterexample-guided techniques for quantifier instantiation that we use to make finding such proofs practically feasible. A particularly important class of specifications are single-invocation properties, for which we present a dedicated algorithm. To support syntax restrictions on generated solutions, our approach can transform a solution found without restrictions into the desired syntactic form. As an alternative, we show how to use evaluation function axioms to embed syntactic restrictions into constraints over algebraic datatypes, and then use an algebraic datatype decision procedure to drive synthesis. Our experimental evaluation on syntax-guided synthesis benchmarks shows that our implementation in the CVC4 SMT solver is competitive with state-of-the-art tools for synthesis.
Andrew Reynolds, Morgan Deters, Viktor Kuncak, Cesare Tinelli, Clark Barrett
### Deductive Program Repair
Abstract
We present an approach to program repair and its application to programs with recursive functions over unbounded data types. Our approach formulates program repair in the framework of deductive synthesis that uses existing program structure as a hint to guide synthesis. We introduce a new specification construct for symbolic tests. We rely on such user-specified tests as well as automatically generated ones to localize the fault and speed up synthesis. Our implementation is able to eliminate errors within seconds from a variety of functional programs, including symbolic computation code and implementations of functional data structures. The resulting programs are formally verified by the Leon system.
Etienne Kneuss, Manos Koukoutos, Viktor Kuncak
### Quantifying Conformance Using the Skorokhod Metric
Abstract
The conformance testing problem for dynamical systems asks, given two dynamical models (e.g., as Simulink diagrams), whether their behaviors are “close” to each other. In the semi-formal approach to conformance testing, the two systems are simulated on a large set of tests, and a metric, defined on pairs of real-valued, real-timed trajectories, is used to determine a lower bound on the distance. We show how the Skorokhod metric on continuous dynamical systems can be used as the foundation for conformance testing of complex dynamical models. The Skorokhod metric allows for both state value mismatches and timing distortions, and is thus well suited for checking conformance between idealized models of dynamical systems and their implementations. We demonstrate the robustness of the metric by proving a transference theorem: trajectories close under the Skorokhod metric satisfy “close” logical properties in the timed linear time logic TLTL augmented with a rich class of temporal and spatial constraint predicates. We provide an efficient window-based streaming algorithm to compute the Skorokhod metric, and use it as a basis for a conformance testing tool for Simulink. We experimentally demonstrate the effectiveness of our tool in finding discrepant behaviors on a set of control system benchmarks, including an industrial challenge problem.
Jyotirmoy V. Deshmukh, Rupak Majumdar, Vinayak S. Prabhu
### Pareto Curves of Multidimensional Mean-Payoff Games
Abstract
In this paper, we study the set of thresholds that the protagonist can force in a zero-sum two-player multidimensional mean-payoff game. The set of maximal elements of such a set is called the Pareto curve, a classical tool to analyze trade-offs. As thresholds are vectors of real numbers in multiple dimensions, there exist usually an infinite number of such maximal elements. Our main results are as follow. First, we study the geometry of this set and show that it is definable as a finite union of convex sets given by linear inequations. Second, we provide a $$\varSigma _2$$ P algorithm to decide if this set intersects a convex set defined by linear inequations, and we prove the optimality of our algorithm by providing a matching complexity lower bound for the problem. Furthermore, we show that, under natural assumptions, i.e. fixed number of dimensions and polynomially bounded weights in the game, the problem can be solved in deterministic polynomial time. Finally, we show that the Pareto curve can be effectively constructed, and under the former natural assumptions, this construction can be done in deterministic polynomial time.
Romain Brenguier, Jean-François Raskin
### Conflict-Driven Conditional Termination
Abstract
Conflict-driven learning, which is essential to the performance of sat and smt solvers, consists of a procedure that searches for a model of a formula, and refutation procedure for proving that no model exists. This paper shows that conflict-driven learning can improve the precision of a termination analysis based on abstract interpretation. We encode non-termination as satisfiability in a monadic second-order logic and use abstract interpreters to reason about the satisfiability of this formula. Our search procedure combines decisions with reachability analysis to find potentially non-terminating executions and our refutation procedure uses a conditional termination analysis. Our implementation extends the set of conditional termination arguments discovered by an existing termination analyzer.
Vijay D’Silva, Caterina Urban
### Predicate Abstraction and CEGAR for Disproving Termination of Higher-Order Functional Programs
Abstract
We propose an automated method for disproving termination of higher-order functional programs. Our method combines higher-order model checking with predicate abstraction and CEGAR. Our predicate abstraction is novel in that it computes a mixture of under- and overapproximations. For non-determinism of a source program (such as random number generation), we apply underapproximation to generate a subset of the actual branches, and check that some of the branches in the abstract program is non-terminating. For operations on infinite data domains (such as integers), we apply overapproximation to generate a superset of the actual branches, and check that every branch is non-terminating. Thus, disproving non-termination reduces to the problem of checking a certain branching property of the abstract program, which can be solved by higher-order model checking. We have implemented a prototype non-termination prover based on our method and have confirmed the effectiveness of the proposed approach through experiments.
Takuya Kuwahara, Ryosuke Sato, Hiroshi Unno, Naoki Kobayashi
### Complexity of Bradley-Manna-Sipma Lexicographic Ranking Functions
Abstract
In this paper we turn the spotlight on a class of lexicographic ranking functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper, and establish for the first time the complexity of some problems involving the inference of such functions for linear-constraint loops (without precondition). We show that finding such a function, if one exists, can be done in polynomial time in a way which is sound and complete when the variables range over the rationals (or reals). We show that when variables range over the integers, the problem is harder—deciding the existence of a ranking function is coNP-complete. Next, we study the problem of minimizing the number of components in the ranking function (a.k.a. the dimension). This number is interesting in contexts like computing iteration bounds and loop parallelization. Surprisingly, and unlike the situation for some other classes of lexicographic ranking functions, we find that even deciding whether a two-component ranking function exists is harder than the unrestricted problem: NP-complete over the rationals and $$\varSigma ^P_2$$-complete over the integers.
Amir M. Ben-Amram, Samir Genaim
### Measuring with Timed Patterns
Abstract
We propose a declarative measurement specification language for quantitative performance evaluation of hybrid (discrete-continuous) systems based on simulation traces. We use timed regular expressions with events to specify patterns that define segments of simulation traces over which measurements are to be taken. In addition, we associate measure specifications over these patterns to describe a particular type of performance evaluation (maximization, average, etc.) to be done over the matched signal segments. The resulting language enables expressive and versatile specification of measurement objectives. We develop an algorithm for our measurement framework, implement it in a prototype tool, and apply it in a case study of an automotive communication protocol. Our experiments demonstrate that the proposed technique is usable with very low overhead to a typical (computationally intensive) simulation.
Thomas Ferrère, Oded Maler, Dejan Ničković, Dogan Ulus
### Automatic Verification of Stability and Safety for Delay Differential Equations
Abstract
Delay differential equations (DDEs) arise naturally as models of, e.g., networked control systems, where the communication delay in the feedback loop cannot always be ignored. Such delays can prompt oscillations and may cause deterioration of control performance, invalidating both stability and safety properties. Nevertheless, state-exploratory automatic verification methods have until now concentrated on ordinary differential equations (and their piecewise extensions to hybrid state) only, failing to address the effects of delays on system dynamics. We overcome this problem by iterating bounded degree interval-based Taylor overapproximations of the time-wise segments of the solution to a DDE, thereby identifying and automatically analyzing the operator that yields the parameters of the Taylor overapproximation for the next temporal segment from the current one. By using constraint solving for analyzing the properties of this operator, we obtain a procedure able to provide stability and safety certificates for a simple class of DDEs.
Liang Zou, Martin Fränzle, Naijun Zhan, Peter Nazier Mosaad
### Time Robustness in MTL and Expressivity in Hybrid System Falsification
Abstract
Building on the work by Fainekos and Pappas and the one by Donzé and Maler, we introduce $$\mathbf{AvSTL }$$, an extension of metric interval temporal logic by averaged temporal operators. Its expressivity in capturing both space and time robustness helps solving falsification problems (searching for a critical path in hybrid system models); it does so by communicating a designer’s intention more faithfully to the stochastic optimization engine employed in a falsification solver. We also introduce a sliding window-like algorithm that keeps the cost of computing truth/robustness values tractable.
Takumi Akazaki, Ichiro Hasuo
### Adaptive Concretization for Parallel Program Synthesis
Abstract
Program synthesis tools work by searching for an implementation that satisfies a given specification. Two popular search strategies are symbolic search, which reduces synthesis to a formula passed to a SAT solver, and explicit search, which uses brute force or random search to find a solution. In this paper, we propose adaptive concretization, a novel synthesis algorithm that combines the best of symbolic and explicit search. Our algorithm works by partially concretizing a randomly chosen, but likely highly influential, subset of the unknowns to be synthesized. Adaptive concretization uses an online search process to find the optimal size of the concretized subset using a combination of exponential hill climbing and binary search, employing a statistical test to determine when one degree of concretization is sufficiently better than another. Moreover, our algorithm lends itself to a highly parallel implementation, further speeding up search. We implemented adaptive concretization for Sketch and evaluated it on a range of benchmarks. We found adaptive concretization is very effective, outperforming Sketch in many cases, sometimes significantly, and has good parallel scalability.
Jinseong Jeon, Xiaokang Qiu, Armando Solar-Lezama, Jeffrey S. Foster
### Automatic Completion of Distributed Protocols with Symmetry
Abstract
A distributed protocol is typically modeled as a set of communicating processes, where each process is described as an extended state machine along with fairness assumptions. Correctness is specified using safety and liveness requirements. Designing correct distributed protocols is a challenging task. Aimed at simplifying this task, we allow the designer to leave some of the guards and updates to state variables in the description of the protocol as unknown functions. The protocol completion problem then is to find interpretations for these unknown functions while guaranteeing correctness. In many distributed protocols, process behaviors are naturally symmetric, and thus, synthesized expressions are further required to obey symmetry constraints. Our counterexample-guided synthesis algorithm consists of repeatedly invoking two phases. In the first phase, candidates for unknown expressions are generated using the SMT solver Z3. This phase requires carefully orchestrating constraints to enforce the desired symmetry constraints. In the second phase, the resulting completed protocol is checked for correctness using a custom-built model checker that handles fairness assumptions, safety and liveness requirements, and exploits symmetry. When model checking fails, our tool examines a set of counterexamples to safety/liveness properties to generate constraints on unknown functions that must be satisfied by subsequent completions. For evaluation, we show that our prototype is able to automatically discover interesting missing details in distributed protocols for mutual exclusion, self stabilization, and cache coherence.
Rajeev Alur, Mukund Raghothaman, Christos Stergiou, Stavros Tripakis, Abhishek Udupa
### An Axiomatic Specification for Sequential Memory Models
Abstract
Formalizations of concurrent memory models often represent memory behavior in terms of sequences of operations, where operations are either reads, writes, or synchronizations. More concrete models of (sequential) memory behavior may include allocation and free operations, but also include details of memory layout or data representation. We present an abstract specification for sequential memory models with allocation and free operations, in the form of a set of axioms that provide enough information to reason about memory without overly constraining the behavior of implementations. We characterize a set of “well-behaved” programs that behave uniformly on all instances of the specification. We show that the specification is both feasible—the CompCert memory model implements it—and usable—we can use the axioms to prove the correctness of an optimization that changes the memory behavior of programs in an LLVM-like language.
William Mansky, Dmitri Garbuzov, Steve Zdancewic
### Approximate Synchrony: An Abstraction for Distributed Almost-Synchronous Systems
Abstract
Forms of synchrony can greatly simplify modeling, design, and verification of distributed systems. Thus, recent advances in clock synchronization protocols and their adoption hold promise for system design. However, these protocols synchronize the distributed clocks only within a certain tolerance, and there are transient phases while synchronization is still being achieved. Abstractions used for modeling and verification of such systems should accurately capture these imperfections that cause the system to only be “almost synchronized.” In this paper, we present approximate synchrony, a sound and tunable abstraction for verification of almost-synchronous systems. We show how approximate synchrony can be used for verification of both time synchronization protocols and applications running on top of them. We provide an algorithmic approach for constructing this abstraction for symmetric, almost-synchronous systems, a subclass of almost-synchronous systems. Moreover, we show how approximate synchrony also provides a useful strategy to guide state-space exploration. We have implemented approximate synchrony as a part of a model checker and used it to verify models of the Best Master Clock (BMC) algorithm, the core component of the IEEE 1588 precision time protocol, as well as the time-synchronized channel hopping protocol that is part of the IEEE 802.15.4e standard.
Ankush Desai, Sanjit A. Seshia, Shaz Qadeer, David Broman, John C. Eidson
### Automated and Modular Refinement Reasoning for Concurrent Programs
Abstract
We present civl, a language and verifier for concurrent programs based on automated and modular refinement reasoning. civl supports reasoning about a concurrent program at many levels of abstraction. Atomic actions in a high-level description are refined to fine-grain and optimized lower-level implementations. A novel combination of automata theoretic and logic-based checks is used to verify refinement. Modular specifications and proof annotations, such as location invariants and procedure pre- and post-conditions, are specified separately, independently at each level in terms of the variables visible at that level. We have implemented civl as an extension to the boogie language and verifier. We have used civl to refine a realistic concurrent garbage collection algorithm from a simple high-level specification down to a highly-concurrent implementation described in terms of individual memory accesses.
Chris Hawblitzel, Erez Petrank, Shaz Qadeer, Serdar Tasiran
### Backmatter
Weitere Informationen
|
2020-10-20 14:14:56
|
{"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": 0.5273096561431885, "perplexity": 1548.3376866741667}, "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-2020-45/segments/1603107872746.20/warc/CC-MAIN-20201020134010-20201020164010-00492.warc.gz"}
|
https://tex.stackexchange.com/questions/443887/using-package-libertine-overrides-my-sans-font-with-libertine-instead-of-bioli
|
# Using package 'libertine' overrides my sans-font with Libertine instead of Biolinum
I'm trying to use the libertine package instead of using the installed Libertine fonts on my machine (macOS). It seems that after adding the libertine package, it overrides my sans-serif font and uses Libertine instead of Biolinum.
I have the following layout:
\documentclass{book}
\usepackage{fontspec}
% \usepackage{libertine}
\setmainfont{Linux Libertine O}
\setsansfont{Linux Biolinum O}
\usepackage{titlesec}
\titleformat{\chapter}[display]
{\normalfont\sffamily\bfseries\LARGE}
{\filright \sffamily\mdseries
\fontsize{10em}{0em}\selectfont
\oldstylenums{\thechapter}}
{1em}
{\filright}
\begin{document}
\chapter{Chapter 1}
\end{document}
If I build this with xelatex, I get the following output:
If I uncomment the \usepackage{libertine}, this will be rendered instead:
The big 1 is rendered with the Libertine font instead of Biolinum, even though the \titleformat block uses \sffamily.
Is there a way I could force the sans-serif rendering?
• Try removing your \oldstylenums and loading the libertine package with osf. – TeXnician Jul 30 '18 at 19:56
• @TeXnician oh wow, removing \oldstylenums worked! – Igal Tabachnik Jul 30 '18 at 19:58
The problem is \oldstylenums which switches the font. Just use the fontspec syntax.
\documentclass{book}
\usepackage{fontspec}
\usepackage{libertine}
%\setmainfont{Linux Libertine O}
%\setsansfont{Linux Biolinum O}
\usepackage{titlesec}
\titleformat{\chapter}[display]
{\normalfont\sffamily\bfseries\LARGE}
{\filright \sffamily\mdseries
\fontsize{10em}{0em}\selectfont
{1em}
{\filright}
\begin{document}
\chapter{Chapter 1}
\end{document}
• (Still need to wait 5 minutes to accept your answer. Thanks again for the tweak, it looks even better!) – Igal Tabachnik Jul 30 '18 at 20:01
• @IgalTabachnik You're welcome. These are the parts of LaTeX's behaviour the end user will never expect :) – TeXnician Jul 30 '18 at 20:03
## The fontspec Way
The libertine package redefines \oldstylenums{} to switch the font. You can change back to a definition like the one from fontspec, which adds the OpenType font feature to the currently-selected font. Note that this is an incompatibility between the libertine and fontspec packages!
\documentclass{book}
\usepackage{fontspec}
\usepackage{libertine}
%% \oldstylenums and \liningums will change the style of the current font, as
%% in fontspec, not switch to the serif font, as in libertine.
\usepackage{titlesec}
\titleformat{\chapter}[display]
{\normalfont\sffamily\bfseries\LARGE}
{\filright \sffamily\mdseries
\fontsize{10em}{0em}\selectfont
\oldstylenums{\thechapter}}
{1em}
{\filright}
\begin{document}
\chapter{Chapter 1}
\oldstylenums{1234567890} vs. \liningnums{1234567890}.
\end{document}
You could alternatively give the command a new, unambiguous name or save the definitions after loading fontspec and before loading libertine.
## The libertine Way
The libertine package provides a second command, \oldstylenumsf{}, to switch the font to Biolinum with old-style numbers. You could simply replace \textsf{\oldstylenums{}} with \oldstylenumsf{}.
The package defines the \biolinumOsF font:
\usepackage{libertine}
and
\titleformat{\chapter}[display]
{\normalfont\sffamily\bfseries\LARGE}
{\filright\mdseries\fontsize{10em}{0em}\selectfont
\biolinumOsF\thechapter}
{1em}
{\filright}
|
2020-12-03 20:46:20
|
{"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": 0.8653707504272461, "perplexity": 12730.900987686184}, "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/1606141732696.67/warc/CC-MAIN-20201203190021-20201203220021-00305.warc.gz"}
|
https://mathoverflow.net/questions/141660/continuous-functions-as-uniformly-continuous-function
|
# Continuous functions as uniformly continuous function
Three question concerninng metrics on the real line:
Is there a metric $d$ on $\Bbb{R}$ such that a function $f : (\Bbb{R},d) \longrightarrow (\Bbb{R},d)$ ( or $f : \Bbb{R} \longrightarrow (\Bbb{R},d)$ or $f : (\Bbb{R},d) \longrightarrow \Bbb{R}$) is continuous if and only if $f : \Bbb{R} \longrightarrow \Bbb{R}$ is uniformly continuous ?
EDIT: The answer now applies to arbitrary topologies, using an idea by Pietro Majer from the comments.
Proposition: There are no topologies $\tau_0,\tau_1$ on $\mathbb R$ such that $f\colon\mathbb R\to\mathbb R$ is uniformly continuous in the Euclidean metric iff $f\colon(\mathbb R,\tau_0)\to(\mathbb R,\tau_1)$ is continuous.
Proof: $\tau_1$ cannot be indiscrete (lest all functions are uniformly continuous), hence we can fix a $\tau_1$-closed set $F$ and points $a\in F$, $b\notin F$. For every Euclidean closed set $A$ and $c>0$, let $f_c(x)=a+c\operatorname{dist}(x,A)$. Then $f_c$ is uniformly continuous, hence continuous from $(\mathbb R,\tau_0)$ to $(\mathbb R,\tau_1)$, hence the $\tau_0$-closed set $f_c^{-1}(F)$ includes $A$ and excludes all points of Euclidean distance $(b-a)/c$ from $A$. The intersection of such sets for all $c$ is just $A$. This shows that $A$ is $\tau_0$-closed, i.e., $\tau_0$ refines the Euclidean topology.
Let $f\colon\mathbb R\to\mathbb R$ be a Euclidean-continuous but not uniformly continuous function, such as $f(x)=x^2$. For every $n>0$, $f_n=f\restriction[-n,n]$ can be extended to a uniformly continuous function on $\mathbb R$. By assumption, this function is continuous from $(\mathbb R,\tau_0)$ to $(\mathbb R,\tau_1)$, hence $f_n$ is continuous from $([-n,n],\tau_0)$ to $(\mathbb R,\tau_1)$. Since $\tau_0$ refines the Euclidean topology, every point has a $\tau_0$-open neighbourhood included in some $[-n,n]$, thus $f=\bigcup_nf_n$ is continuous from $(\mathbb R,\tau_0)$ to $(\mathbb R,\tau_1)$. However, it is not uniformly continuous in the Euclidean metric, a contradiction.
• A slight variant of the second part of above argument: since the $d$-topology is finer than the Euclidean, the family of intervals $I_n:=[n,n+1]$ for $n\in\mathbb{N}$ is a locally finite closed cover of $\mathbb{R}$ in the $d$-topology. Let $f$ be $x\mapsto x^2$. On each $I_n$, $f$ coincides with some uniformly continuous function, which is $d$-continuous. So $f_{|I_n}$ is also $d$-continuous, hence $f$ is $d$-continuous, hence $U$-continuous, contradiction. This way the argument works more generally for any topology on $\mathbb{R}$, even non-metric. – Pietro Majer Sep 10 '13 at 8:33
• The first part of your proof can also be done for general topologies, that is, $C((\mathbb{R},\tau),(\mathbb{R},\tau))=UC(\mathbb{R},\mathbb{R})$ implies $\tau$ is finer than the Euclidean topology. As before such $\tau$ is not the indiscrete topology and it is translation invariant. So there is a non-empty $\tau$-open set $G$ not containing $0$. Then there is also a non-empty bounded $\tau$-open set, e.g. of the form $f^{-1}(G)$ with $f$ continuous with compact support. Since $\tau$ is also homotety invariant, this implies that $\tau$ is finer than the Euclidean topology. – Pietro Majer Sep 10 '13 at 8:43
• The first part of the proof already applies to arbitrary topologies as is (that was intentional). Thanks for the second part; I was trying in vain to show that any such topology has to coincide with the Euclidean topology, I did not realize that one could construct a non-uniform continuous counterexample regardless of that. – Emil Jeřábek Sep 10 '13 at 12:12
• Excellent. (Sorry for the variant of the first part, I had the idea while far from a computer and recalled wrongly) – Pietro Majer Sep 10 '13 at 15:26
The answer is no at least in the assumptions given in parentheses.
Assume $C(\mathbb{R}, (\mathbb{R},d)) = UC(\mathbb{R},\mathbb{R})$. Since $\operatorname{id}_\mathbb{R}$ is in $UC(\mathbb{R},\mathbb{R})$ it is also in $C(\mathbb{R}, (\mathbb{R},d))$. So any $f\in C(\mathbb{R}, \mathbb{R})$ is also in $C(\mathbb{R}, (\mathbb{R},d))$ by composition. But then $C(\mathbb{R}, \mathbb{R})\subset UC(\mathbb{R},\mathbb{R})$, a contradiction.
Assume $C((\mathbb{R},d), \mathbb{R}) = UC(\mathbb{R},\mathbb{R})$. Then $\operatorname{id}_\mathbb{R}$ is in $C^0((\mathbb{R},d), \mathbb{R})$, that is the Euclidean topology is included in the $d$-topology. On the other hand, consider any convergent sequence $x_n\to x$ in the Euclidean topology. Since the function $d(x,\cdot): (\mathbb{R},d)\to \mathbb{R}$ is continuous (it is 1-Lipschitz), it is also in $UC(\mathbb{R},\mathbb{R})$, so in particular $d(x,x_n)\to d(x,x)=0$, that is $x_n\to x$ in $d$. Therefore the two topologies coincide which leads again to the contradiction $C(\mathbb{R}, \mathbb{R}) = UC(\mathbb{R},\mathbb{R})$.
$\mathbf{Proposition}$ Suppose that $d,\rho$ are metrics on $\mathbb{R}$ that induce the Euclidean topology on $\mathbb{R}$. Then there is a bounded continuous function $f:(\mathbb{R},d)\rightarrow(\mathbb{R},\rho)$ which is not uniformly continuous.
$\mathbf{Proof}$ If $n$ is a positive integer, then there is some $\epsilon_{n}$ where $0<\epsilon_{n}<1$ and where $d(n,n+\epsilon_{n})<\frac{1}{n}$. There is a continuous function $f:\mathbb{R}\rightarrow[0,1]$ with $f(n)=0$ but $f(n+\epsilon_{n})=1$ for all $n$. However, the function $f$ cannot be uniformly continuous. If $\epsilon>0$, then $\frac{1}{n}<\epsilon$ for some $n$, and $d(n,n+\epsilon_{n})<\frac{1}{n}$, but $\rho(f(n),f(n+\epsilon_{n}))=\rho(0,1)>0$.
• I didn't see that anything in the question mandated that the metric $d$ should induce the Euclidean topology. – Todd Trimble Sep 9 '13 at 17:35
• Hmm. I must have misread the question. – Joseph Van Name Sep 10 '13 at 2:34
|
2020-09-24 08:27:44
|
{"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": 0.9732775092124939, "perplexity": 118.46607448410352}, "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-2020-40/segments/1600400214347.17/warc/CC-MAIN-20200924065412-20200924095412-00058.warc.gz"}
|
https://www.numerade.com/books/chapter/partial-derivatives-3/
|
# Calculus Early Transcendentals
## Educators
Problem 1
In Example 2 we considered the function $W=f(T, v),$ where
$W$ is the wind-chill index, $T$ is the actual temperature, and $v$ is
the wind speed. A numerical representation is given in Table 1 .
$\begin{array}{l}{\text { (a) What is the value of } f(-15,40) ? \text { What is its meaning? }} \\ {\text { (b) Describe in words the meaning of the question "For what }} \\ {\text { value of } v \text { is } f(-20, v)=-30 ?^{\circ \prime} \text { Then answer the question. }}\end{array}$
$\begin{array}{l}{\text { (c) Describe in words the meaning of the question "For what }} \\ {\text { value of } T \text { is } f(T, 20)=-49 ?^{\prime \prime} \text { Then answer the question. }} \\ {\text { (d) What is the meaning of the function } W=f(-5, v) ?} \\ {\text { Describe the behavior of this function. }}\end{array}$
$\begin{array}{l}{\text { (e) What is the meaning of the function } W=f(T, 50) ?} \\ {\text { Describe the behavior of this function. }}\end{array}$
Check back soon!
Problem 2
The temperature-humidity index $I$ (or humidex, for short) is the
perceived air temperature when the actual temperature is $T$ and
the relative humidity is $h,$ so we can write $I=f(T, h)$ . The fol-
lowing table of values of $I$ is an excerpt from a table compiled
by the National Oceanic \& Atmospheric Administration.
$$\begin{array}{l}{\text { (a) What is the value of } f(95,70) ? \text { What is its meaning? }} \\ {\text { (b) For what value of } h \text { is } f(90, h)=100 ?} \\ {\text { (c) For what value of } T \text { is } f(T, 50)=88 ?}\end{array}$$
$$\begin{array}{l}{\text { (d) What are the meanings of the functions } I=f(80, h)} \\ {\text { and } I=f(100, h) ? \text { Compare the behavior of these two }} \\ {\text { functions of } h .}\end{array}$$
Check back soon!
Problem 3
A manufacturer has modeled its yearly production function $P$
(the monetary value of its entire production in millions of
dollars) as a Cobb-Douglas function
$$P(L, K)=1.47 L^{0.65} K^{0.35}$$
where $L$ is the number of labor hours (in thousands) and $K$ is
the invested capital (in millions of dollars). Find $P(120,20)$
and interpret it.
Check back soon!
Problem 4
Verify for the Cobb-Douglas production function
$$P(L, K)=1.01 L^{\mathrm{a} \pi} K^{\mathrm{azs}}$$
discussed in Example 3 that the production will be doubled
if both the amount of labor and the amount of capital are
doubled. Determine whether this is also true for the general
production function
$$P(L, K)=b L^{\alpha} K^{1-\alpha}$$
Check back soon!
Problem 5
A model for the surface area of a human body is given by the
function
$$S=f(w, h)=0.1091 w^{0.425} h^{0.725}$$
where $w$ is the weight (in pounds), $h$ is the height (in inches),
and $S$ is measured in square feet.
$\begin{array}{l}{\text { (a) Find } f(160,70) \text { and interpret it. }} \\ {\text { (b) What is your own surface area? }}\end{array}$
Check back soon!
Problem 6
The wind-chill index $W$ discussed in Example 2 has been
modeled by the following function:
$$W(T, v)=13.12+0.6215 T-11.37 v^{0.16}+0.3965 T v^{0.16}$$
Check to see how closely this model agrees with the values in
Table 1 for a few values of $T$ and $v$ .
Check back soon!
Problem 7
The wave heights $h$ in the open sea depend on the speed $v$
of the wind and the length of time $t$ that the wind has been
blowing at that speed. Values of the function $h=f(v, t)$ are
recorded in feet in Table 4 .
$$\begin{array}{l}{\text { (a) What is the value of } f(40,15) ? \text { What is its meaning? }} \\ {\text { (b) What is the meaning of the function } h=f(30, \text { t)? Describe }} \\ {\text { the behavior of this function. }} \\ {\text { (c) What is the meaning of the function } h=f(v, 30) ? \text { Describe }} \\ {\text { the behavior of this function. }}\end{array}$$
Check back soon!
Problem 8
A company makes three sizes of cardboard boxes: small,
medium, and large. It costs $\$ 2.50$to make a small box,$\$4.00$
for a medium box, and $\$ 4.50$for a large box. Fixed costs are$\$8000 .$
$\begin{array}{l}{\text { (a) Express the cost of making } x \text { small boxes, } y \text { medium }} \\ {\text { boxes, and } z \text { large boxes as a function of three variables: }} \\ {C=f(x, y, z) .} \\ {\text { (b) Find } f(3000,5000,4000) \text { and interpret it. }} \\ {\text { (c) What is the domain of } f ?}\end{array}$
Check back soon!
Problem 9
$Let$ g(x, y)=\cos (x+2 y)$$\begin{array}{l}{\text { (a) Evaluate } g(2,-1) \text { . }} \\ {\text { (b) Find the domain of } g .} \\ {\text { (c) Find the range of } g .}\end{array} Check back soon! Problem 10 Let F(x, y)=1+\sqrt{4-y^{2}}$$\begin{array}{l}{\text { (a) Evaluate } F(3,1) \text { . }} \\ {\text { (b) Find and sketch the domain of } F .} \\ {\text { (c) Find the range of } F .}\end{array}$$Check back soon! Problem 11 Let f(x, y, z)=\sqrt{x}+\sqrt{y}+\sqrt{z}+\ln \left(4-x^{2}-y^{2}-z^{2}\right) \begin{array}{l}{\text { (a) Evaluate } f(1,1,1) \text { . }} \\ {\text { (b) Find and describe the domain of } f}\end{array} Check back soon! Problem 12 Let g(x, y, z)=x^{3} y^{2} z \sqrt{10-x-y-z}.$$\begin{array}{l}{\text { (a) Evaluate } g(1,2,3) .} \\ {\text { (b) Find and describe the domain of } g .}\end{array}$$Check back soon! Problem 13 13-22 Find and sketch the domain of the function.$$f(x, y)=\sqrt{2 x-y}$$Check back soon! Problem 14 13-22 Find and sketch the domain of the function.$$f(x, y)=\sqrt{x y}$$Check back soon! Problem 15 13-22 Find and sketch the domain of the function.$$f(x, y)=\ln \left(9-x^{2}-9 y^{2}\right)$$Check back soon! Problem 16 13-22 Find and sketch the domain of the function.$$f(x, y)=\sqrt{x^{2}-y^{2}}$$Check back soon! Problem 17 13-22 Find and sketch the domain of the function.$$f(x, y)=\sqrt{1-x^{2}}-\sqrt{1-y^{2}}$$Check back soon! Problem 18 13-22 Find and sketch the domain of the function.$$f(x, y)=\sqrt{y}+\sqrt{25-x^{2}-y^{2}}$$Check back soon! Problem 19 13-22 Find and sketch the domain of the function.$$f(x, y)=\frac{\sqrt{y-x^{2}}}{1-x^{2}}$$Check back soon! Problem 20 13-22 Find and sketch the domain of the function.$$f(x, y)=\arcsin \left(x^{2}+y^{2}-2\right)$$Check back soon! Problem 21 13-22 Find and sketch the domain of the function.$$f(x, y, z)=\sqrt{1-x^{2}-y^{2}-z^{2}}$$Check back soon! Problem 22 13-22 Find and sketch the domain of the function.$$f(x, y, z)=\ln \left(16-4 x^{2}-4 y^{2}-z^{2}\right)$$Check back soon! Problem 23 23-31 Sketch the graph of the function.$$f(x, y)=1+y$$Check back soon! Problem 24 23-31 Sketch the graph of the function.$$f(x, y)=2-x$$Check back soon! Problem 25 23-31 Sketch the graph of the function.$$f(x, y)=10-4 x-5 y$$Check back soon! Problem 26 23-31 Sketch the graph of the function.$$f(x, y)=e^{-y}$$Check back soon! Problem 27 23-31 Sketch the graph of the function.$$f(x, y)=y^{2}+1$$Check back soon! Problem 28 23-31 Sketch the graph of the function.$$f(x, y)=1+2 x^{2}+2 y^{2}$$Check back soon! Problem 29 23-31 Sketch the graph of the function.$$f(x, y)=9-x^{2}-9 y^{2}$$Check back soon! Problem 30 23-31 Sketch the graph of the function.$$f(x, y)=\sqrt{4 x^{2}+y^{2}}$$Check back soon! Problem 31 23-31 Sketch the graph of the function.$$f(x, y)=\sqrt{4-4 x^{2}-y^{2}}$$Check back soon! Problem 32 Match the function with it graph (labeled I-VI). Give reasons for your choices.$${ (a) } f(x, y)=|x|+|y| \quad \text { (b) } f(x, y)=|x y|{ (c) } ff(x, y)=\frac{1}{1+x^{2}+y^{2}} \quad \text { (d) } f(x, y)=\left(x^{2}-y^{2}\right)^{2}{ (e) } ff(x, y)=(x-y)^{2} \quad \text { (f) } f(x, y)=\sin (|x|+|y|)$$Check back soon! Problem 33 A contour map for a function f is shown. Use it to estimate the values of f(-3,3) and f(3,-2) . What can you say about the shape of the graph? Check back soon! Problem 34 Shown is a contour map of atmospheric pressure in North America on August 12, 2008. On the level curves (called isobars) the pressure is indicated in millibars (mb). \begin{array}{l}{\text { (a) Estimate the pressure at } C \text { (Chicago), } N \text { (Nashville), }} \\ {S(\text { San Francisco), and } V(\text { Vancouver) }} \\ {\text { (b) At which of these locations were the winds strongest? }}\end{array} Check back soon! Problem 35 Level curves (isothermals) are shown for the water temperature \left(\text { in }^{\circ} \mathrm{C}\right) in Long Lake (Minnesota) in 1998 as a function of depth and time of year. Estimate the temperature in the lake on June 9( day 160) at a depth of 10 \mathrm{m} and on June 29(\mathrm{day} 180) at a depth of 5 \mathrm{m} . Check back soon! Problem 36 Two contour maps are shown. One is for a function f whose graph is a cone. The other is for a function g whose graph is a paraboloid. Which is which, and why? Check back soon! Problem 37 Locate the points A and B on the map of Lonesome Mountain (Figure 12 ) . How would you describe the terrain near A ? Near B ? Check back soon! Problem 38 Make a rough sketch of a contour map for the function whose graph is shown. Check back soon! Problem 39 39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of f . GRAPH Check back soon! Problem 40 39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of f . GRAPH Check back soon! Problem 41 39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of f . GRAPH Check back soon! Problem 42 39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of f . GRAPH Check back soon! Problem 43 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=(y-2 x)^{2}$$Check back soon! Problem 44 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=x^{3}-y$$Check back soon! Problem 45 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=\sqrt{x}+y$$Check back soon! Problem 46 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=\ln \left(x^{2}+4 y^{2}\right)$$Check back soon! Problem 47 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=y e^{x}$$Check back soon! Problem 48 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=y \sec x$$Check back soon! Problem 49 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=\sqrt{y^{2}-x^{2}}$$Check back soon! Problem 50 43-50 Draw a contour map of the function showing several level curves.$$f(x, y)=y /\left(x^{2}+y^{2}\right)$$Check back soon! Problem 51 51-52 Sketch both a contour map and a graph of the function and compare them.$$f(x, y)=x^{2}+9 y^{2}$$Check back soon! Problem 52 51-52 Sketch both a contour map and a graph of the function and compare them.$$f(x, y)=\sqrt{36-9 x^{2}-4 y^{2}}$$Check back soon! Problem 53 A thin metal plate, located in the x y -plane, has temperature T(x, y) at the point (x, y) . The level curves of T are called isothermals because at all points on such a curve the tempera- ture is the same. Sketch some isothermals if the temperature function is given by$$T(x, y)=\frac{100}{1+x^{2}+2 y^{2}}$$Check back soon! Problem 54 If V(x, y) is the electric potential at a point (x, y) in the x y -plane, then the level curves of V are called equipotential curves because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if V(x, y)=c / \sqrt{r^{2}-x^{2}-y^{2}}, where c is a positive constant. Check back soon! Problem 55 55-58 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opin- ion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$$f(x, y)=x y^{2}-x^{3}(monkey saddle)$$Check back soon! Problem 56 55-58 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opin- ion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$$f(x, y)=x y^{3}-y x^{3} (dog saddle)$$Check back soon! Problem 57 55-58 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opin- ion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$$f(x, y)=e^{-\left(x^{2}+y^{2}\right) / 3}\left(\sin \left(x^{2}\right)+\cos \left(y^{2}\right)\right)$$Check back soon! Problem 58 55-58 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opin- ion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$$f(x, y)=\cos x \cos y$$Check back soon! Problem 59 59-64 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$$z=\sin (x y)$$Check back soon! Problem 60 59-64 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$$z=e^{x} \cos y$$Check back soon! Problem 61 59-64 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$$z=\sin (x-y)$$Check back soon! Problem 62 59-64 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$$z=\sin x-\sin y$$Check back soon! Problem 63 59-64 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$$z=\left(1-x^{2}\right)\left(1-y^{2}\right)$$Check back soon! Problem 64 59-64 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$$z=\frac{x-y}{1+x^{2}+y^{2}}$$Check back soon! Problem 65 65-68 Describe the level surfaces of the function.$$f(x, y, z)=x+3 y+5 z$$Check back soon! Problem 66 65-68 Describe the level surfaces of the function.$$f(x, y, z)=x^{2}+3 y^{2}+5 z^{2}$$Check back soon! Problem 67 65-68 Describe the level surfaces of the function.$$f(x, y, z)=y^{2}+z^{2}$$Check back soon! Problem 68 65-68 Describe the level surfaces of the function.$$f(x, y, z)=x^{2}-y^{2}-z^{2}$$Check back soon! Problem 69 69-70 Describe how the graph of g is obtained from the graph of f .$$\begin{array}{l}{\text { (a) } g(x, y)=f(x, y)+2} \\ {\text { (b) } g(x, y)=2 f(x, y)} \\ {\text { (c) } g(x, y)=-f(x, y)} \\ {\text { (d) } g(x, y)=2-f(x, y)}\end{array}$$Check back soon! Problem 70 69-70 Describe how the graph of g is obtained from the graph of f .$$\begin{array}{l}{\text { (a) } g(x, y)=f(x-2, y)} \\ {\text { (b) } g(x, y)=f(x, y+2)} \\ {\text { (c) } g(x, y)=f(x+3, y-4)}\end{array}$$Check back soon! Problem 71 71-72 Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maxi- mum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"?$$f(x, y)=3 x-x^{4}-4 y^{2}-10 x y$$Check back soon! Problem 72 71-72 Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maxi- mum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"?$$f(x, y)=x y e^{-x^{2}-y^{2}}$$Check back soon! Problem 73 73-74 Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both x and y become large? What happens as (x, y) approaches the origin?$$f(x, y)=\frac{x+y}{x^{2}+y^{2}}$$Check back soon! Problem 74 73-74 Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both x and y become large? What happens as (x, y) approaches the origin?$$f(x, y)=\frac{x y}{x^{2}+y^{2}}$$Check back soon! Problem 75 Use a computer to investigate the family of functions f(x, y)=e^{c x^{2}+y^{2}} . How does the shape of the graph depend on c ? Check back soon! Problem 76 Use a computer to investigate the family of surfaces$$z=\left(a x^{2}+b y^{2}\right) e^{-x^{2}-y^{2}}$$How does the shape of the graph depend on the numbers a and b ? Check back soon! Problem 77 Use a computer to investigate the family of surfaces z=x^{2}+y^{2}+c x y . In particular, you should determine the transitional values of c for which the surface changes from one type of quadric surface to another. Check back soon! Problem 78 Graph the functions$$f(x, y)=\sqrt{x^{2}+y^{2}}f(x, y)=e^{\sqrt{x^{2}+y^{2}}}f(x, y)=\ln \sqrt{x^{2}+y^{2}}f(x, y)=\sin \left(\sqrt{x^{2}+y^{2}}\right)$$and$$f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}}}$$In general, if g is a function of one variable, how is the graph of$$f(x, y)=g\left(\sqrt{x^{2}+y^{2}}\right)$$obtained from the graph of g ? Check back soon! Problem 79 \begin{array}{l}{\text { (a) Show that, by taking logarithms, the general Cobb- }} \\ {\text { Douglas function } P=b L^{L} K^{1-\alpha} \text { can be expressed as }}\end{array}$$\ln \frac{P}{K}=\ln b+\alpha \ln \frac{L}{K}
$\begin{array}{l}{\text { (b) If we let } x=\ln (L / K) \text { and } y=\ln (P / K), \text { the equation in }} \\ {\text { part (a) becomes the linear equation } y=\alpha x+\ln b \text { . Use }} \\ {\text { Table } 2 \text { (in Example } 3 \text { ) to make a table of values of }} \\ {\ln (L / K) \text { and } \ln (P / K) \text { for the years } 1899-1922 . \text { Then use a }}\end{array}$
$\begin{array}{l}{\text { graphing calculator or computer to find the least squares }} \\ {\text { regression line through the points }(\ln (L / K), \ln (P / K))}\end{array}$
$\begin{array}{l}{\text { (c) Deduce that the Cobb-Douglas production function is }} \\ {P=1.01 L^{0.75} K^{0.25}}.\end{array}$
Check back soon!
|
2020-03-29 15:34:32
|
{"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": 0.8575737476348877, "perplexity": 1568.4260614414895}, "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/1585370494349.3/warc/CC-MAIN-20200329140021-20200329170021-00422.warc.gz"}
|
https://aakashsrv1.meritnation.com/ask-answer/question/please-give-detailed-explanation-choose-correct-option/conic-sections/16911931
|
# Please give detailed explanation. Choose correct option.
Solution:
Let , CD = a and given AB = 2 CD , So
AB = 2 a
And
Radius of given circle = r , then AD = 2r ( As AD is perpendicular to AB and CD )
Now we form our diagram , As :
Here , A be at origin and AB and AD at x - axis and y - axis respectively .
So,
Coordinates of A ( 0 , 0 ) , B ( 2a , 0 ) , C ( a , 2r ) and D ( 2r , 0 )
We can see ABCD is a trapezium , and we know area of trapezium =
So,
Area of ABCD =
Now we draw a right angle triangle OBC and a perpendicular line from center to BC , As :
So,
But radius can't be negative , So we get
|
2022-05-23 23:15:39
|
{"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": 0.9674687385559082, "perplexity": 3614.210806009539}, "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/1652662562106.58/warc/CC-MAIN-20220523224456-20220524014456-00111.warc.gz"}
|
http://biblioteca.universia.net/html_bura/verColeccion/params/id/75272.html
|
## Recursos de colección
#### Project Euclid (Hosted at Cornell University Library) (203.209 recursos)
Taiwanese Journal of Mathematics
1. #### Coderivatives Related to Parametric Extended Trust Region Subproblem and Their Applications
Tran, Van Nghi
This paper deals with the Fréchet and Mordukhovich coderivatives of the normal cone mapping related to the parametric extended trust region subproblems (eTRS), in which the trust region intersects a ball with a single linear inequality constraint. We use the obtained results to investigate the Lipschitzian stability of parametric eTRS. We also propose a necessary condition for the local (or global) solution of the eTRS by using the coderivative tool.
2. #### The Spectral Method for Long-time Behavior of a Fractional Power Dissipative System
Lu, Hong; Zhang, Mingji
In this paper, we consider the fractional complex Ginzburg-Landau equation in two spatial dimensions with the dissipative effect given by a fractional Laplacian. The periodic initial value problem of the fractional complex Ginzburg-Landau equation is discretized fully by Galerkin-Fourier spectral method, and the dynamical behaviors of the discrete system are studied. The existence and convergence of global attractors of the discrete system are obtained by a priori estimates and error estimates of the discrete solution. The numerical stability and convergence of the discrete scheme are proved.
3. #### Periodic Solutions of Sublinear Impulsive Differential Equations
Niu, Yanmin; Li, Xiong
In this paper, we consider sublinear second order differential equations with impulsive effects. Basing on the Poincaré-Bohl fixed point theorem, we first will prove the existence of harmonic solutions. The existence of subharmonic solutions is also obtained by a new twist fixed point theorem recently established by Qian etc in 2015 [18].
4. #### Fixed Point Theorems via MNC in Ordered Banach Space with Application to Fractional Integro-differential Evolution Equations
Nashine, Hemant Kumar; Yang, He; Agarwal, Ravi P.
In this paper, we propose fixed point results through the notion of measure of noncompactness (MNC) in partially ordered Banach spaces. We also prove some new coupled fixed point results via MNC for more general class of function. To achieve this result, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. Further, we apply the obtained fixed point theorems to prove the existence of mild solutions for fractional integro-differential evolution equations with nonlocal conditions. At the end, an example is given to illustrate the rationality of the...
5. #### Existence of Solutions to Quasilinear Schrödinger Equations Involving Critical Sobolev Exponent
Wang, Youjun; Li, Zhouxin
By using variational approaches, we study a class of quasilinear Schrödinger equations involving critical Sobolev exponents $-\Delta u + V(x)u + \frac{1}{2} \kappa [\Delta(u^2)]u = |u|^{p-2}u + |u|^{2^*-2}u, \quad x \in \mathbb{R}^N,$ where $V(x)$ is the potential function, $\kappa \gt 0$, $\max \{ (N+3)/(N-2),2 \} \lt p \lt 2^* := 2N/(N-2)$, $N \geq 4$. If $\kappa \in [0,\overline{\kappa})$ for some $\overline{\kappa} \gt 0$, we prove the existence of a positive solution $u(x)$ satisfying $\max_{x \in \mathbb{R}^N} |u(x)| \leq \sqrt{1/(2\kappa)}$.
6. #### Toeplitz Operator for Dirichlet Space Through Sobolev Multiplier Algebra
Luo, Shuaibing; Xiao, Jie
This paper is mainly concerned with the Toeplitz operator $T_{\phi}$ over the Dirichlet space $\mathcal{D}$ with the symbol $\phi$ in the Sobolev multiplier algebra $M(W^{1,2}(\mathbb{D}))$, thereby extending several known ones in a very different manner.
7. #### Exact Controllability for Wave Equations with Switching Controls
He, Yong
In this paper, we analyze the exact controllability problem for wave equations endowed with switching controls. The goal is to control the dynamics of the system by switching among different actuators such that, in each instant of time, there are as few active actuators as possible. We prove that the system is exactly controllable under suitable geometric control conditions.
8. #### Blow-up Phenomena for a Porous Medium Equation with Time-dependent Coefficients and Inner Absorption Term Under Nonlinear Boundary Flux
Xiao, Suping; Fang, Zhong Bo
This paper deals with blow-up phenomena for an initial boundary value problem of a porous medium equation with time-dependent coefficients and inner absorption term in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficient and nonlinear functions to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, the upper and lower bounds for $t^{\ast}$ are derived in the higher dimensional space. Finally, some examples are presented to illustrate applications of our results.
9. #### Devaney's Chaos for Maps on $G$-spaces
Shah, Ekta
We study the notion of sensitivity on $G$-spaces and through examples observe that $G$-sensitivity neither implies nor is implied by sensitivity. Further, we obtain necessary and sufficient conditions for a map to be $G$-sensitive. Next, we define the notion of Devaney's chaos on $G$-space and show that $G$-sensitivity is a redundant condition in the definition.
10. #### Remark on Proper Holomorphic Maps Between Reducible Bounded Symmetric Domains
Seo, Aeryeong
In this paper we study proper holomorphic maps between bounded symmetric domains when the source domain is not irreducible. More precisely, we provide sufficient conditions for semi-product proper holomorphic maps to be product proper. As an application we characterize proper holomorphic maps between equidimensional bounded symmetric domains.
11. #### $b$-generalized $(\alpha,\beta)$-derivations and $b$-generalized $(\alpha,\beta)$-biderivations of Prime Rings
Filippis, Vincenzo De; Wei, Feng
Let $R$ be a ring, $\alpha$ and $\beta$ two automorphisms of $R$. An additive mapping $d \colon R \to R$ is called an $(\alpha,\beta)$-derivation if $d(xy) = d(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$. An additive mapping $G \colon R \to R$ is called a generalized $(\alpha,\beta)$-derivation if $G(xy) = G(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$, where $d$ is an $(\alpha,\beta)$-derivation of $R$. In this paper we introduce the definitions of $b$-generalized $(\alpha,\beta)$-derivation and $b$-generalized $(\alpha,\beta)$-biderivation. More precisely, let $d \colon R \to R$ and $G \colon R \to R$ be two additive...
12. #### Hecke Bound of Vector-valued Modular Forms and its Relationship with Cuspidality
Jin, Seokho; Lim, Jongryul; Lim, Subong
In this paper, we prove that if the Fourier coefficients of a vector-valued modular form satisfy the Hecke bound, then it is cuspidal. Furthermore, we obtain an analogous result with regard to Jacobi forms by applying an isomorphism between vector-valued modular forms and Jacobi forms. As an application, we prove a result on the growth of the number of representations of $m$ by a positive definite quadratic form $Q$.
13. #### A Note on Modularity Lifting Theorems in Higher Weights
Yu, Yih-Jeng
We follow the ideas of Khare and Ramakrishna-Khare and prove the modularity lifting theorem in higher weights. This approach somehow differs from that using Taylor-Wiles systems.
14. #### An Extending Result on Spectral Radius of Bipartite Graphs
Cheng, Yen-Jen; Fan, Feng-lei; Weng, Chih-wen
In this paper, we study the spectral radius of bipartite graphs. Let $G$ be a bipartite graph with $e$ edges without isolated vertices. It was known that the spectral radius of $G$ is at most the square root of $e$, and the upper bound is attained if and only if $G$ is a complete bipartite graph. Suppose that $G$ is not a complete bipartite graph and $(e-1,e+1)$ is not a pair of twin primes. We describe the maximal spectral radius of $G$. As a byproduct of our study, we obtain a spectral characterization of a pair $(e-1,e+1)$ of integers to...
15. #### Coalescence on Supercritical Bellman-Harris Branching Processes
Athreya, Krishna B.; Hong, Jyy-I
We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$...
16. #### Coalescence on Supercritical Bellman-Harris Branching Processes
Athreya, Krishna B.; Hong, Jyy-I
We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$...
17. #### Quantitative Recurrence Properties for Systems with Non-uniform Structure
Zhao, Cao; Chen, Ercai
Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define $R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\}$ and $R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\},$ where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and...
18. #### Quantitative Recurrence Properties for Systems with Non-uniform Structure
Zhao, Cao; Chen, Ercai
Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define $R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\}$ and $R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\},$ where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and...
19. #### General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$
Feng, Baowei
A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.
20. #### General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$
Feng, Baowei
A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.
Aviso de cookies: Usamos cookies propias y de terceros para mejorar nuestros servicios, para análisis estadístico y para mostrarle publicidad. Si continua navegando consideramos que acepta su uso en los términos establecidos en la Política de cookies.
|
2018-05-24 19:26:44
|
{"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": 2, "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": 0.7957062721252441, "perplexity": 497.54002089950546}, "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-2018-22/segments/1526794866772.91/warc/CC-MAIN-20180524190036-20180524210036-00460.warc.gz"}
|
http://vhck.cioe.pw/least-mean-square-matlab.html
|
# Least Mean Square Matlab
Least squares means are adjusted for other terms in the model (like covariates), and are less sensitive to missing data. PubMed Central. The fundamental equation is still A TAbx DA b. First of all, you need to enter MEX-Setup to determine if the compiler you want to use, follow the instructions step by step down the line. Could you please tell me how to calculate these adjusted means in MATLAB? Please consider that I have 4 groups and I should adjust for more than 2 factors. You can perform least squares fit with or without the Symbolic Math Toolbox. The inverse of a matrix does not always exist. The inverse of a square matrix A, denoted by A-1, is the matrix so that the product of A and A-1 is the Identity matrix. In reading the following, bear in mind that I used Matlab for almost 20 years before making the switch to Python in 2009, so I am intimately familiar with both. , m n, and full rank, which means that Rank(A) = n. The scope of our analysis is feasible because of our use of the particle filter. You can create plots of known, analytical functions, you can plot data from other sources such as experimental measurements, you can analyze data, perhaps by fitting it to a curve, and then plot a comparison. In Weibull++, the term rank regression is used instead of least squares, or linear regression, because the regression is performed on the rank values, more specifically, the median rank values (represented on the y-axis). Blog Making Sense of the Metadata: Clustering 4,000 Stack Overflow tags with…. least square solution 을 구하는 일반적인 방법은 다음과 같으며 아래 식의 윗 첨자 H는 hermitian transpose 를 의미합니다. Adaptive Filtering: Fundamentals of Least Mean Squares with MATLAB (R) covers the core concepts of this important field, focusing on a vital part of the statistical signal processing area-the least mean square (LMS) adaptive filter. A description can be found in Haykin, edition 4, chapter 5. Comments and Ratings (2) MATLAB Release Compatibility. If the noise is assumed to be isotropic the problem can be solved using the ‘\’ or ‘/’ operators, or the ols function. estimateGlobalMotionLeastSquares. Includes an option to give initial positive terms for x for faster solution of iterative problems using nnls. Cody is a MATLAB problem-solving game that challenges you to expand your knowledge. Mathematics of simple regression. In this submission I presented a method to estimate a linear channel in frequency domain using a least mean square (LMS) algorithm. 9% of the code I see, elementwise multiplication is what people use, both in MATLAB and Python. Find α and β by minimizing ρ = ρ(α,β). Channel Equalization using Least Mean Square (LMS) algorithm - Comparison of magnitude and phase response. And, because is a linear combination of and , it is also a random variable, and therefore has a covariance. Free PDF ebooks (user's guide, manuals, sheets) about Recursive least square matlab code ready for download I look for a PDF Ebook about : Recursive least square matlab code. How to estimate unknown parameters using Ordinary Least Squares (OLS) [18] Essential Preliminary Matrix Algebra for Signal Processing [19] Why Cholesky Decomposition ? A sample case: [20] Tests for Positive Definiteness of a Matrix [21] Solving a Triangular Matrix using Forward & Backward Substitution [22] Cholesky Factorization and Matlab code. LMS algorithm uses the estimates of the gradient vector from the available data. Least Squares with Examples in Signal Processing1 Ivan Selesnick March 7, 2013 NYU-Poly These notes address (approximate) solutions to linear equations by least squares. You can perform least squares fit with or without the Symbolic Math Toolbox. Could you please take a look and tell me if it makes sense; if it does exactly what is supposed to do? EDIT: Please, pay attention to the commented commands as well. System: 3 2 01 (1) 1 2 exx y xx. 1 Review of Least Squares Solutions to Overdetermined Systems Recall that in the last lecture we discussed the solution of overdetermined linear systems using the least squares method. This MATLAB function constructs an adaptive algorithm object based on the least mean square (LMS) algorithm with a step size of stepsize. [Schaefer et al. In this scenario you have two. The Matlab backslash operator computes a least squares solution to such a system. Locally, the deformation takes the form of either a rigid transformation or optionally a similarity. we had generated a signal and the we generated a random noise and add it to the signal,,then my using the lms algorithm we tried to cancel the effect of the noise on the signal and have the original signal back pure as possible. Last activity. You don't need pdb either since it serves for development purposes and, at this stage, you should have working code already. YellowBrickCinema - Relaxing Music 2,506,866 views. Designed and implemented Least Mean Square based adaptive filter for noise and echo cancellation in TMS320C5510 with C and Matlab. This document contain a MATLAB code of VSS-LMS for linear channel estimation. The envisaged application is the identification of an unknown system. What’s GM(1,1)? The predicted values would come from some model you have. Least Mean Square (LMS)Adaptive Learning to find the intersection between two lines. Curve Fitting with Matlab Matlab has a curve fitting toolbox (installed on machines in Hicks, but perhaps not elsewhere on campus - as of Sept. For the logged data the mean and median are 1. bird12_csm Unpublished model derived for the SCEC CSM using the method of \citebird99, available online at http://sceczero. The Normalised least mean squares filter (NLMS) is a variant of the LMS algorithm that solves this problem by normalising with the power of the input. The user controls the deformation by ma-nipulating a set of point handles. Uncertainty in the Dependent Variable, Slope, and Intercept 5. Let us consider a simple example. Just because we know 4:99 is the best value for the slope and:48 is the best value for the y-intercept doesnot mean that these are good estimates of the true values. I'm struggling to understand how to implement a least square linear classifier for my data in matlab. Distributed Average Consensus with Least-Mean-Square Deviation Lin Xiao, Stephen Boyd, and Seung-Jean Kim Abstract—We consider a stochastic model for distributed average consensus, which arises in applications such as load balancing for parallel processors, distributed coordination of mobile autonomous agents, and network synchronization. And be sure to use the curly braces for the subscript, not parentheses!. Lecture 10: Recursive Least Squares Estimation Overview † Recursive Least squares estimation; { The exponentially weighted Least squares { Recursive-in-time solution { Initialization of the algorithm { Recursion for MSE criterion † Examples: Noise canceller, Channel equalization, Echo cancellation. To illustrate the linear least-squares fitting process, suppose you have n data points that can be modeled by a first-degree polynomial. The inverse of a matrix does not always exist. We deliberately chose data of this nature to indicate the dangers in using the Method of Least Squares. An adaptive filter is a computational device that iteratively models the relationship between the input and output signals of a filter. In MATLAB, you create a matrix by entering elements in each row as comma or space delimited numbers and using semicolons to mark the end of each row. If they don't have equal variance, then you use weighted least squares. DESIGN APPROACH The method considered in this work is based on the observation that, for a length - N FIR digital, N distinct equally. The fundamental equation is still A TAbx DA b. Residuals at a point as the difference between the actual y value at a point and the estimated y value from the regression line given the x coordinate of that point. This program implements the least squares regression method, without using any of the MATLAB built-in regression tools. I need to do a least square polynomial fitting for y(i). Use features like bookmarks, note taking and highlighting while reading Adaptive Filtering: Fundamentals of Least Mean Squares with MATLAB®. Matlab Project - Download as Powerpoint Presentation (. Mean deviation is an important descriptive statistic that is not frequently encountered in mathematical statistics. The implementation was done in two steps: first the algorithm is checked using a model in Matlab. matlab,system,equation Generally this is done (if the eq is in the format you have) with an Ax=b system. x and y, p(xi) ˜ yi, in a least-squares sense. Here I’ll go over how to do Least Squares Regression, as simply as possibly, using Excel and its Solver*. I'm pretty sure there is a single multiplication I can do to accomplish this, but I can't find a formula online. Constraining W to be integers in the set [-1,1] is a nonlinear integer programming problem. 1 Least squares estimation Assume that Y i = +x i + i for i= 1 2N are independent random variables with means E(Y i)= + x i, that the collection i is a random sample from a distribution with mean 0 and standard deviation , and that all parameters (, , and ) are unknown. The picture is 2. If a matrix is given as an argument to such a function, its procedure is applied separately to each column , and a row vector of results returned. The example applica-tion is adaptive channel equalization, which has been introduced in compu-ter exercise 2. Typical applications include time-series prediction, nonlinear adaptive filtering, tracking and online learning for nonlinear regression. This MATLAB function computes a partial least-squares (PLS) regression of Y on X, using ncomp PLS components, and returns the predictor and response loadings in XL and YL, respectively. First we discuss the existence and uniqueness of LSF and. Finally, under the very specific assumptions of the classical model, by one reasonable criterion, least squares will be the most efficient use of the data. Other documents using least-squares algorithms for tting points with curve or surface structures are avail-able at the website. Last activity. Least Squares Optimization with L1-Norm Regularization Mark Schmidt CS542B Project Report December 2005 Abstract This project surveys and examines optimization ap-proaches proposed for parameter estimation in Least Squares linear regression models with an L1 penalty on the regression coefficients. Let me try and explain. The line has heights p D. pdf), Text File (. And MATLAB's use of * for matrix multiplication is a source of endless bugs. This MATLAB function solves the linear system C*x = d in the least-squares sense, subject to A*x ≤ b. Direct data domain least square algorithm requires less time to determine the weights for digital beamforming compared to recursive least mean square algorithms which is has faster processing when compared to least mean square algorithm. The terms linear regression and least squares are used synonymously in this reference. Summary A Random Variable is a variable whose possible values are numerical outcomes of a random experiment. 25, all of that over three. To compute the values of these constants we need as many equations as the number of constants in the equation. Awarded to Shujaat Khan on 24 Sep 2013. the optimal Ordinary Least Squares (OLS) estimator for model parameters is. I will base my example on two rather old but very informative papers: Gander W. This program implements the least squares regression method, without using any of the MATLAB built-in regression tools. A least mean square (LMS) algorithm in complex form is presented in this paper to estimate power system frequency where the formulated structure is very simple. In this paper, the behavior of Least Mean Square (LMS) algorithm is determined and the evaluation parameters used are number of channel taps and CIR samples of the channel. Matlab Tutorial 5: Linear Equations. hint: input the data in the matrix form, and solve the system to obtain the coefficients. Computer exercise 3: Normalized Least Mean Square This exercise is about the normalized least mean square (LMS) algorithm, a variation of the standard LMS algorithm, which has been the topic of the previous computer exercise. such as least mean square, Kalman filter, and adaptive neural network [5]. NASA Astrophysics Data System (ADS) Widodo, Achmad; Yang, Bo-Suk. LMS algorithm uses the estimates of the gradient vector from the available data. My code is below. The Adaptive Line Enhancer (ALE) is an effective learning filter for reducing Gaussian noise with a large SNR. Gavin Spring, 2015 The need to fit a curve to measured data arises in all branches of science, engineering, and economics. An introduction to least squares curve tting with Matlab 3. 이때 해를 구하는 방식이 least square solution 형태의 해를 구해주는 겁니다. Here, the errors are assumed to be following multivariate normal distribution with zero mean and standard deviation $$\sigma^2$$. This technique is the extension of the OLS method. 4 Linear Least Squares. Note: If you specify the axes position (using subplot or axes), imshow ignores any initial magnification you might have specified and defaults to the 'fit' behavior. Least Squares Calculator. 1 Least squares estimation Assume that Y i = +x i + i for i= 1 2N are independent random variables with means E(Y i)= + x i, that the collection i is a random sample from a distribution with mean 0 and standard deviation , and that all parameters (, , and ) are unknown. As a result of these recent advances, Adaptive Filtering Fundamentals Of Least Mean Squares With Matlab are becoming integrated into the daily lives of many people in professional, recreational, and education environments. This division method is an introduction to adapative gain control with the least means square algorithm which I think will shed light on the workings of how it the iterative process calculates desired gains. 1 Introduction In both ordinary least squares and maximum likelihood approaches to parameter estimation, we made the assumption of constant variance, that is the variance of an observation is the. Richter Communications Systems and Research Section While least-squares fltting procedures are commonly used in data analysis and are extensively discussed in the literature devoted to this subject, the proper as-sessment of errors resulting from such flts has received relatively little attention. Root-mean-square (rms) refersto the most common mathematical method of defining the effective voltage or current of an AC wave. (For an abundance of weather data like this check out the Oregon Climate Service) Here are the MATLAB commands to create a symbol plot with the data from PDXprecip. First we discuss the existence and uniqueness of LSF and. Graphical results of the two-stage least squareinXLSTAT: The charts which follow show the results mentioned above. Cody is a MATLAB problem-solving game that challenges you to expand your knowledge. values of a dependent variable ymeasured at. 1 Weighted Least Squares as a Solution to Heteroskedas-ticity Suppose we visit the Oracle of Regression (Figure 4), who tells us that the noise has a standard deviation that goes as 1 + x2=2. Curve fitting A weighted least squares fit for a model which is less complicated than the system that generated the data (a case of so‐called ‘undermodeling’). Most fitting algorithms implemented in ALGLIB are build on top of the linear least squares solver: Polynomial curve fitting (including linear fitting) Rational curve fitting using Floater-Hormann basis Spline curve fitting using penalized regression splines And, finally, linear least squares fitting. I am completely new to MATLAB. price, part 1: descriptive analysis · Beer sales vs. The least-squares approximate solution of Ax = y is given by xls = (ATA) 1ATy: This is the unique x 2 Rn that minimizes kAx yk. Schwartz UCLA This article presents a simple yet powerful new approach for approximating the value of America11 options by simulation. Implementation of Least Mean Square Algorithm. late a least-squares solution (i. So, let's see, this is going to be equal to square root of this is 0. Moreover, when I use curve fitting tool (available in MATLAB R2014b) with Robust fit option on, I am getting R-square of 0. matlab,system,equation Generally this is done (if the eq is in the format you have) with an Ax=b system. LMS incorporates an. values of a dependent variable ymeasured at. There is no need to use backslash here. If x is a vector, then y is a real-valued scalar. I Solving LLS with SVD-decomposition. It is used in some forms of nonlinear regression. 3*l1-4*l2=3 5*l1 -3*l2=-4 You can build the system as: x (unknowns) will be a unknowns. Poularikas. Least Squares Regression is a way of finding a straight line that best fits the data, called the "Line of Best Fit". Alternative approaches: This important special case has also given rise to many other iterative methods (or adaptive filters), such as the least mean squares filter and recursive least squares filter, that directly solves the original MSE optimization problem using stochastic gradient descents. PubMed Central. Mullette-Gillman, Department of Psychology at National University of Singapore. The MATLAB help has a list of what functions each one can do, but here is a quick summary, in roughly the order you should try them unless you already know the. Least-squares SVM regression You can find a MATLAB script for this example in n, bgenerally does not lie in span(A), so there is no exact solution to the Least Squares Problem. Least Squares Fit (1) The least squares fit is obtained by choosing the α and β so that Xm i=1 r2 i is a minimum. Inspired by: Mackey Glass Time Series Prediction Using Least Mean Square (LMS), Mackey Glass Time Series Prediction Using Fractional Least Mean Square (FLMS) Discover Live Editor Create scripts with code, output, and formatted text in a single executable document. The trust region based methods limit their step size to be more conservative. line fit by least squares is an optimal linear predictor for the dependent variable. Variable Step-Size Least Mean Square (VSS-LMS) Algorithm. In practice, least-squares lines are found by pressing a calculator button, or giving a MatLab command. hint: input the data in the matrix form, and solve the system to obtain the coefficients. That's what the Linest and Trend functions do. Plotting with MATLAB MATLAB is very useful for making scientific and engineering plots. The mean and median are 10. The terms linear regression and least squares are used synonymously in this reference. Application of least squares tting to calibration of the salinity sensor ME 121: Salinity calibration t page 1. Least Mean Square (LMS) Simulink Model. linear least squares fitting. Least squares fitting Linear least squares. The paper presents a digital implementation of the adaptive Least Mean Square (LMS) algorithm. Lectures are based on my book: "An Introduction to Numerical Computation", published by World Scientific, 2016. The least squares method is a form of mathematical regression analysis that finds the line of best fit for a dataset, providing a visual demonstration of the relationship between the data points. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. Linear Least Squares Regression¶ Here we look at the most basic linear least squares regression. 6 ounces mean dev from mean = ----- 5 = 1. Minimizes the sum of a least- squares criterion for a forward model, and the analogous criterion for a time-reversed model. 附件中的源代码是matlab编写的，实现一种对于椭圆的稳定的数据拟合算法。当然必须要提供至少5个点的数据，椭圆x，y轴的中心点，最大轴最小轴. The Normalised least mean squares filter (NLMS) is a variant of the LMS algorithm that solves this problem by normalising with the power of the input. Algorithm depends on the cost function used convergence of the algorithm : Will the coefficients of the adaptive filter converge to the desired values? Is the algorithm stable? Global convergence or local convergence? rate of convergence: This corresponds to the time required for the algorithm to converge to the optimum least squares/Wiener. Out of all possible linear fits, the least-squares regression line is the one that has the smallest possible value for the sum of the squares of the residuals. Mullette-Gillman, Department of Psychology at National University of Singapore. I also put the matlab program with which I generate the signal for the C program. values of a dependent variable ymeasured at. Gavin Department of Civil and Environmental Engineering Duke University August 3, 2019 Abstract The Levenberg-Marquardt algorithm was developed in the early 1960's to solve ne onlinear least squares problems. stage least squares Many economic models involve endogeneity: that is, a theoretical relationship does not t into the framework of y-on-X regression, in which we can assume that the yvariable is de-termined by (but does not jointly determine) X:Indeed, the simplest analytical concepts we teach in principles of economics|a demand. The fundamental equation is still A TAbx DA b. Computer exercise 5: Recursive Least Squares (RLS) This computer exercise deals with the RLS algorithm. We can then use this to improve our regression, by solving the weighted least squares problem rather than ordinary least squares (Figure 5). A "square" is determined by squaring the distance. TEST_APPROX, a MATLAB library which defines test problems for approximation, provided as a set of (x,y) data. Curve Fitting with Matlab Matlab has a curve fitting toolbox (installed on machines in Hicks, but perhaps not elsewhere on campus - as of Sept. A Matlab benchmarking toolbox for kernel adaptive filtering. Note: If you specify the axes position (using subplot or axes), imshow ignores any initial magnification you might have specified and defaults to the 'fit' behavior. In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. Suppose instead that var e s2S where s2 is unknown but S is known Š in other words we. People use * when they are multiplying a vector or matrix by a scalar, then they switch it to a variable and forget to change to. Yet in recent versions it uses more modern method called Trust Region. X = P*(R\(Q'*B)) If A is sparse, MATLAB computes a least squares solution using the sparse qr factorization of A. yˆ = b0 +b1x = 307. If the theoretical curve is simply a polynomial, the least-square approximation is a polynomial approximation. You can create plots of known, analytical functions, you can plot data from other sources such as experimental measurements, you can analyze data, perhaps by fitting it to a curve, and then plot a comparison. Here is the C program, it is something wrong in it. The LMS algorithm, as well as others related to it, is widely used in various applications of adaptive. Linear Least Squares Regression¶ Here we look at the most basic linear least squares regression. Matlab and Octave have simple built-in functions for least-squares curve fitting: polyfit and polyval. In a later chapter we will. LMS (Least Mean Squares): most basic canonical ANC algo in C; in Matlab; FxLMS (Filtered eXtended Least Mean Squares): adds an additional learned filter for the secondary path signal - signal from cancellation speakers to users ears - to account for phase problems and audio coloration added during practical noise cancellation applications in Matlab. The blue spots are the data, the green spots are the estimated nonpolynomial function. What's GM(1,1)? The predicted values would come from some model you have. hint: input the data in the matrix form, and solve the system to obtain the coefficients. If they don't have equal variance, then you use weighted least squares. [XL,YL] = plsregress(X,Y,ncomp) computes a partial least-squares (PLS) regression of Y on X, using ncomp PLS components, and returns the predictor and response loadings in XL and YL, respectively. Get this from a library! Adaptive filtering : fundamentals of least mean squares with MATLAB. The least-squares approximate solution of Ax = y is given by xls = (ATA) 1ATy: This is the unique x 2 Rn that minimizes kAx yk. Least Mean Square algorithm used to minimize ISI in communication system The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Adaptive filters are used in many diverse applications, appearing in everything from military instruments to cellphones and home appliances. Lesort Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294, USA February 1, 2008 Abstract We study theoretical and computational aspects of the least squares fit (LSF) of circles and circular arcs. A question I get asked a lot is ‘How can I do nonlinear least squares curve fitting in X?’ where X might be MATLAB, Mathematica or a whole host of alternatives. This document contain a MATLAB code of VSS-LMS for linear channel estimation. Use B for the least squares matrix in this case and c2 for the solution. Use the Least Mean Square (LMS) algorithm to subtract noise from an input signal. matlab_commandline, programs which illustrate how MATLAB can be run from the UNIX command line, that is, not with the usual MATLAB command window. My wording may have been misleading. using matlab least squares functions Hello, I have my matlab code which solves a least squares problem and gives me the right answer. (3 votes, average: 3. b], and interior breaks xi, provided xi has all its entries in (a. Assessing the fit in least-squares regression. , a system in which A is a rectangular m × n-matrix with more equations than unknowns (when m>n). 1 The recursive. We proved it two videos ago. Suppose that a matrix A is given that has more rows than columns, ie n, the number of rows, is larger than m, the number of columns. Linear versus nonlinear least squares. Engineering & Electrical Engineering Projects for $30 -$250. 5;2;1/ witherrors e D. Least-Squares Fitting of Circles and Ellipses BIT Numerical Mathematics, 34(4) pp. It is assumed that you know how to enter data or read data files which is covered in the first chapter, and it is assumed that you are familiar with the different data types. Least squares (LS)optimiza-tion problems are those in which the objective (error) function is a quadratic function of the parameter(s) being optimized. Application of least squares tting to calibration of the salinity sensor ME 121: Salinity calibration t page 1. Senior Scientist and Inventor in one of the world's 10 largest industrial corporations doing image analysis full time. Ask Question *MSE in MATLAB's nlinfit. This hand-out addresses the ordinary least-squares method of. Since this is such a common query, I thought I. This technique is the extension of the OLS method. The scope of our analysis is feasible because of our use of the particle filter. , "Distributed Beamforming for Two-Way DF Relay Cognitive Networks Under PrimarySecondary Mutual. Note these only work for linear equations! b = X\y' b2 = b(2). * , and their code silently. Matlab Tutorial 5: Linear Equations. This program implements the least squares regression method, without using any of the MATLAB built-in regression tools. Cody is a MATLAB problem-solving game that challenges you to expand your knowledge. solve a non-linear least squares problem. Implementation in C of Least Mean Square (LMS) algorithm. Remember, the square of a number is that number times itself. Use the Least Mean Square (LMS) algorithm to subtract noise from an input signal. n residual sum of squares = SUM (yi - yi_predicted)^2. (c) Within a terminal window, move to the specified directory and unpack the tar file by typing the command: tar xvf Tcodes. This is essentially because while mean deviation has a natural intuitive definition as the "mean deviation from the mean," the introduction of the absolute value makes analytical calculations using this statistic much more complicated than the standard deviation. Minimizes the sum of a least- squares criterion for a forward model, and the analogous criterion for a time-reversed model. Solve nonlinear least-squares (curve-fitting) problems in serial or parallel. All results. But yes, we should definitely have the argument ready about where popular loss functions like least-square and cross-entropy come from — at least when we try to find the most likely hypothesis for a supervised learning problem using Bayesian argument. A regression line (LSRL - Least Squares Regression Line) is a straight line that describes how a response variable y changes as an explanatory variable x changes. Least Squares Optimization with L1-Norm Regularization Mark Schmidt CS542B Project Report December 2005 Abstract This project surveys and examines optimization ap-proaches proposed for parameter estimation in Least Squares linear regression models with an L1 penalty on the regression coefficients. [X,Y] = meshgrid(x,y) transforms the domain specified by vectors x and y into arrays X and Y, which can be used to evaluate functions of two variables and three-dimensional mesh/surface plots. Poularikas. Nonlinear Least Squares. Example showing the Optimization app and linear least squares. working of Recursive least square method with an example a function in MATLAB for. This MATLAB function computes a partial least-squares (PLS) regression of Y on X, using ncomp PLS components, and returns the predictor and response loadings in XL and YL, respectively. LMS Algorithm Implementation. Matlab function for least squares fitting of X-Y data to a circle - horchler/circfit. txt) or view presentation slides online. estimate the coefficients using least squares using MATLAB's \ operator. The implementation was done in two steps: first the algorithm is checked using a model in Matlab. Since this is such a common query, I thought I. PDF [DOWNLOAD] Adaptive Filtering: Fundamentals of Least Mean Squares with MATLAB® BOOOK ONLINE. Specify the method used to calculate filter coefficients as either 'Fast transversal least-squares' or 'Sliding-window fast transversal least-squares'. Learn more about lms, optimization Optimization Toolbox. 5;2;1/ witherrors e D. Toggle Main Navigation. 285-291, (edition 3: chapter 9. , a system in which A is a rectangular m × n-matrix with more equations than unknowns (when m>n). A step by step tutorial showing how to develop a linear regression equation. Least Squares Regression can be used to match pretty much any type of function to any type of data. Uncertainty in the Dependent Variable, Slope, and Intercept 5. Curve Fitting with Matlab Matlab has a curve fitting toolbox (installed on machines in Hicks, but perhaps not elsewhere on campus - as of Sept. Use of colors and animations. Plotting with MATLAB MATLAB is very useful for making scientific and engineering plots. Gaussian distribution – how to plot it in Matlab In statistics and probability theory , the Gaussian distribution is a continuous distribution that gives a good description of data that cluster around a mean. A "square" is determined by squaring the distance. McNames Portland State University ECE 539/639 Least Squares Ver. How it works. Least Squares: A statistical method used to determine a line of best fit by minimizing the sum of squares created by a mathematical function. 1 Weighted Least Squares as a Solution to Heteroskedas-ticity Suppose we visit the Oracle of Regression (Figure 4), who tells us that the noise has a standard deviation that goes as 1 + x2=2. Many more complicated schemes use line-fitting as a foundation, and least-squares linear regression has, for years, been the workhorse technique of the field. Asked by lina. In these contexts a vector is just a convenient data structure. An iterative method is presented for solving linear systems and linear least-square systems. c Henri Gavin Department of Civil and Environmental Engineering Duke University April 13, 2011 Abstract The Levenberg-Marquardt method is a standard technique used to solve nonlinear least squares problems. Suppose that a matrix A is given that has more rows than columns, ie n, the number of rows, is larger than m, the number of columns. A question I get asked a lot is 'How can I do nonlinear least squares curve fitting in X?' where X might be MATLAB, Mathematica or a whole host of alternatives. Implementation in C of Least Mean Square (LMS) algorithm. Let me show you how to do it with a simple example of 2 eq with 2 unknowns. txt) or view presentation slides online. In a regression analysis , the goal is to determine how well a data series can be. Like us on. The LMS adaptive filter uses the reference signal on the Input port and the desired signal on the Desired port to automatically match the filter response. Channel Equalization using Least Mean Square (LMS) algorithm - Comparison of magnitude and phase response. An example of how to calculate linear regression line using least squares. [2] To View Or Download A Particular Teaching Code The name of each MATLAB Teaching Code is listed below. max nummber of iteration allowed is 600 , and the function toleranc is 10^-01 and wenn convergence is not met, i must widening the real signal that i have by one value until convergence is reached. Least squares linear regression in Excel is easy. Not sure it if it's in a toolbox or not. For a given time step t, y(t) and H(t) correspond to the Output and Regressors inports of the Recursive Least Squares Estimator block, respectively. It would have the same effect of making all of the values positive as the absolute value. Kim, Consistent normalized least mean square filtering with noisy data matrix. where A is an m x n matrix with m > n, i. MATLAB doesn't just have one ODE solver, it has eight as of the MATLAB 7. MATLAB Online uses Plotly's native web-based scientific graphing library. This technique is the extension of the OLS method. My code is below. This is described in Section 9. Least-Squares Fitting of Data with Polynomials Least-Squares Fitting of Data with B-Spline Curves. mldivide, ("\") actually does that too. The OCPLS is a powerful tool for classification problem. values of a dependent variable ymeasured at. Least Squares Fitting with Excel. Awarded to Shujaat Khan on 24 Sep 2013. Longstaff UCLA Eduardo S. Functions operate on variables within their own workspace, which is also called the local workspace , separate from the workspace you access at the MATLAB command prompt which is called the base workspace. ; Masters, Forrest J. I am basically making fitting program now. Today I would like to discuss what does it mean to fit a circle in a least square sense and if it leads to an unambiguous solution. 5;2;1/ witherrors e D. Least squares fitting with Numpy and Scipy nov 11, 2015 numerical-analysis optimization python numpy scipy. Least Squares in Matlab. MATLAB Answers. If you don't know the variances, there are methods for estimating them. This code demonstrates LMS (Least Mean Square) Filter. This MATLAB function constructs an adaptive algorithm object based on the least mean square (LMS) algorithm with a step size of stepsize. Kernel Adaptive Filtering Toolbox. Least-Squares (Model Fitting) Algorithms. sum of the squares of the errors.
|
2020-01-25 15:20:06
|
{"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": 0.5943032503128052, "perplexity": 849.321643931964}, "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-2020-05/segments/1579251672537.90/warc/CC-MAIN-20200125131641-20200125160641-00455.warc.gz"}
|
http://pdglive.lbl.gov/DataBlock.action?node=S046LGN
|
# Long-lived ${{\widetilde{\boldsymbol g}}}$ (Gluino) mass limit INSPIRE search
Limits on light gluinos (${\mathit m}_{{{\widetilde{\mathit g}}}}$ $<$ 5 GeV) were last listed in our PDG 2014 edition: K. Olive, $\mathit et~al.$ (Particle Data Group), Chinese Physics C38 070001 (2014) (http://pdg.lbl.gov).
VALUE (GeV) CL% DOCUMENT ID TECN COMMENT
$> 2060$ 95 1
2019 C
ATLS ${{\mathit R}}$-hadrons, Tglu1A, ${{\mathit \tau}}{}\geq{}$10 ns, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 1890$ 95 1
2019 C
ATLS ${{\mathit R}}$-hadrons, Tglu1A, stable
$> 2370$ 95 2
2018 S
ATLS displaced vertex + $\not E_T$, long-lived Tglu1A, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV, and ${{\mathit \tau}}$=0.17 ns
$> 1600$ 95 3
2018 AY
CMS jets+$\not E_T$, Tglu1A, c$\tau$ < 0.1 mm, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 1750$ 95 3
2018 AY
CMS jets+$\not E_T$, Tglu1A, c$\tau$ = 1 mm, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 1640$ 95 3
2018 AY
CMS jets+$\not E_T$, Tglu1A, c$\tau$ = 10 mm, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 1490$ 95 3
2018 AY
CMS jets+$\not E_T$, Tglu1A, c$\tau$ = 100 mm, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 1300$ 95 3
2018 AY
CMS jets+$\not E_T$, Tglu1A, c$\tau$ = 1 m, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 960$ 95 3
2018 AY
CMS jets+$\not E_T$, Tglu1A, c$\tau$ = 10 m, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 900$ 95 3
2018 AY
CMS jets+$\not E_T$, Tglu1A, c$\tau$ = 100 m, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 2200$ 95 4
2018 DV
CMS long-lived ${{\widetilde{\mathit g}}}$, RPV, ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\overline{\mathit t}}}{{\overline{\mathit b}}}{{\overline{\mathit s}}}$ , 0.6 mm $<$ c${{\mathit \tau}}<$ 80 mm
$>1000$ 95 5
2017 AR
CMS long-lived ${{\widetilde{\mathit g}}}$, RPV, ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit b}}}{{\overline{\mathit s}}}$ , c${{\mathit \tau}}$ = 0.3 mm
$>1300$ 95 5
2017 AR
CMS long-lived ${{\widetilde{\mathit g}}}$, RPV, ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit b}}}{{\overline{\mathit s}}}$ , c${{\mathit \tau}}$ = 1.0 mm
$>1400$ 95 5
2017 AR
CMS long-lived ${{\widetilde{\mathit g}}}$, RPV, ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit b}}}{{\overline{\mathit s}}}$ , 2 mm $<$ c${{\mathit \tau}}<$ 30 mm
$> 1580$ 95 6
2016 B
ATLS long-lived ${{\mathit R}}$-hadrons
$\text{> 740 - 1590}$ 95 7
2016 C
ATLS ${{\mathit R}}$-hadrons, Tglu1A, ${{\mathit \tau}}{}\geq{}$0.4 ns, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 1570$ 95 7
2016 C
ATLS ${{\mathit R}}$-hadrons, Tglu1A, stable
$> 1610$ 95 8
2016 BW
CMS long-lived ${{\widetilde{\mathit g}}}$ forming R-hadrons, f = 0.1, cloud interaction model
$> 1580$ 95 8
2016 BW
CMS long-lived ${{\widetilde{\mathit g}}}$ forming R-hadrons, f = 0.1, charge-suppressed interaction model
$> 1520$ 95 8
2016 BW
CMS long-lived ${{\widetilde{\mathit g}}}$ forming R-hadrons, f = 0.5, cloud interaction model
$> 1540$ 95 8
2016 BW
CMS long-lived ${{\widetilde{\mathit g}}}$ forming R-hadrons, f = 0.5, charge-suppressed interaction model
$>1270$ 95 9
2015 AE
ATLS ${{\widetilde{\mathit g}}}$ R-hadron, generic R-hadron model
$>1360$ 95 9
2015 AE
ATLS ${{\widetilde{\mathit g}}}$ decaying to 300 GeV stable sleptons, LeptoSUSY model
$>1115$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ R-hadron, stable
$>1185$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ( ${{\mathit g}}$ $/$ ${{\mathit q}}{{\overline{\mathit q}}}$) ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , lifetime 10 ns, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$>1099$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ( ${{\mathit g}}$ $/$ ${{\mathit q}}{{\overline{\mathit q}}}$) )0, lifetime 10 ns, ${\mathit m}_{{{\widetilde{\mathit g}}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$>1182$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , lifetime 10 ns, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$>1157$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , lifetime 10 ns, ${\mathit m}_{{{\widetilde{\mathit g}}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 480 GeV
$>869$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ( ${{\mathit g}}$ $/$ ${{\mathit q}}{{\overline{\mathit q}}}$) )0, lifetime 1 ns, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$>821$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ( ${{\mathit g}}$ $/$ ${{\mathit q}}{{\overline{\mathit q}}}$) )0, lifetime 1 ns, ${\mathit m}_{{{\widetilde{\mathit g}}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV ~
$>836$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , lifetime 1 ns, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$>836$ 95 10
2015 BM
ATLS ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , lifetime 10 ns, ${\mathit m}_{{{\widetilde{\mathit g}}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 480 GeV
$> 1000$ 95 11
2015 AK
CMS ${{\widetilde{\mathit g}}}$ R-hadrons, 10 ${{\mathit \mu}}$s$<{{\mathit \tau}}<$1000 s
$> 880$ 95 11
2015 AK
CMS ${{\widetilde{\mathit g}}}$ R-hadrons, 1 ${{\mathit \mu}}$s$<{{\mathit \tau}}<$1000 s
• • • We do not use the following data for averages, fits, limits, etc. • • •
$> 985$ 95 12
2013 AA
ATLS ${{\widetilde{\mathit g}}}$, ${{\mathit R}}$-hadrons, generic interaction model
$> 832$ 95 13
2013 BC
ATLS R-hadrons, ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit g}}$ $/$ ${{\mathit q}}{{\overline{\mathit q}}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , generic R-hadron model, lifetime between $10^{-5}$ and $10^{3}$ s, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 1322$ 95 14
2013 AB
CMS long-lived ${{\widetilde{\mathit g}}}$ forming R-hadrons, f = 0.1, cloud interaction model
$\text{none 200 - 341}$ 95 15
2012 P
ATLS long-lived ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit g}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV
$> 640$ 95 16
2012 AN
CMS long-lived ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit g}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$
$> 1098$ 95 17
2012 L
CMS long-lived ${{\widetilde{\mathit g}}}$ forming ${{\mathit R}}$-hadrons, f = 0.1
$> 586$ 95 18
2011 K
ATLS stable ${{\widetilde{\mathit g}}}$
$> 544$ 95 19
2011 P
ATLS stable ${{\widetilde{\mathit g}}}$, GMSB scenario, tan ${{\mathit \beta}}$=5
$> 370$ 95 20
2011
CMS long lived ${{\widetilde{\mathit g}}}$
$> 398$ 95 21
2011 C
CMS stable ${{\widetilde{\mathit g}}}$
1 AABOUD 2019C searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for metastable and stable ${{\mathit R}}$-hadrons arising as excesses in the mass distribution of reconstructed tracks with high transverse momentum and large dE/dx. Gluino ${{\mathit R}}$-hadrons with lifetimes above 10 ns are excluded at 95$\%$ C.L. with lower mass limit range between 1000 GeV and 2060 GeV, see their Figure 5(a). Masses smaller than 1290 GeV are excluded for a lifetime of 1 ns, see their Figure 6. In the case of stable ${{\mathit R}}$-hadrons, the lower mass limit is 1890 GeV, see their Figure 5(b).
2 AABOUD 2018S searched in 32.8 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for long-lived gluinos in final states with large missing transverse momentum and at least one high-mass displaced vertex with five or more tracks. The observed yield is consistent with the expected background. Exclusion limits are derived for Tglu1A models predicting the existence of long-lived gluinos reaching roughly m(${{\widetilde{\mathit g}}}$) = 2000 GeV to 2370 GeV for m(${{\widetilde{\mathit \chi}}_{{1}}^{0}}$) = 100 GeV and gluino lifetimes between 0.02 and 10 ns, see their Fig. 8. Limits are presented also as a function of the lifetime (for a fixed gluino-neutralino mass difference of 100 GeV) and of the gluino and neutralino masses (for a fixed lifetime of 1 ns). See their Fig. 9 and 10 respectively.
3 SIRUNYAN 2018AY searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events containing one or more jets and significant $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on the gluino mass in the Tglu1A, Tglu2A and Tglu3A simplified models, see their Figure 3. Limits are also set on squark, sbottom and stop masses in the Tsqk1, Tsbot1, Tstop1 and Tstop4 simplified models, see their Figure 3. Finally, limits are set on long-lived gluino masses in a Tglu1A simplified model where the gluino is metastable or long-lived with proper decay lengths in the range $10^{-3}$ mm $<$ c${{\mathit \tau}}$ $<$ $10^{5}$ mm, see their Figure 4.
4 SIRUNYAN 2018DV searched in 38.5 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for long-lived particles in events with multiple jets and two displaced vertices composed of many tracks. No events with two well-separated high-track-multiplicity vertices were observed. Limits are set on the stop and the gluino mass in RPV models of supersymmetry where the stop (gluino) is decaying solely into dijet (multijet) final states, see their Figures 6 and 7.
5 KHACHATRYAN 2017AR searched in 17.6 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for R-parity-violating SUSY in which long-lived neutralinos or gluinos decay into multijet final states. No significant excess above the Standard Model expectations is observed. Limits are set on the gluino mass for a range of mean proper decay lengths (c${{\mathit \tau}}$), see their Fig. 7. The upper limits on the production cross section times branching ratio squared (Fig. 7) are also applicable to long-lived neutralinos.
6 AABOUD 2016B searched in 3.2 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for long-lived ${{\mathit R}}$-hadrons using observables related to large ionization losses and slow propagation velocities, which are signatures of heavy charged particles traveling significantly slower than the speed of light. Exclusion limits at 95$\%$ C.L. are set on the long-lived gluino masses exceeding 1580 GeV. See their Fig. 5.
7 AABOUD 2016C searched in 3.2 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for long-lived and stable ${{\mathit R}}$-hadrons identified by anomalous specific ionization energy loss in the ATLAS Pixel detector. Gluino ${{\mathit R}}$-hadrons with lifetimes above 0.4 ns are excluded at 95$\%$ C.L. with lower mass limit range between 740 GeV and 1590 GeV. In the case of stable ${{\mathit R}}$-hadrons, the lower mass limit is 1570 GeV. See their Figs. 5 and 6.
8 KHACHATRYAN 2016BW searched in 2.5 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events with heavy stable charged particles, identified by their anomalously high energy deposits in the silicon tracker and/or long time-of-flight measurements by the muon system. No evidence for an excess over the expected background is observed. Limits are derived for pair production of gluinos as a function of mass, depending on the interaction model and on the fraction f, of produced gluinos hadronizing into a ${{\widetilde{\mathit g}}}$ - gluon state, see Fig. 4 and Table 7.
9 AAD 2015AE searched in 19.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for heavy long-lived charged particles, measured through their specific ionization energy loss in the ATLAS pixel detector or their time-of-flight in the ALTAS muon system. In the absence of an excess of events above the expected backgrounds, limits are set R-hadrons in various scenarios, see Fig. 11. Limits are also set in LeptoSUSY models where the gluino decays to stable 300 GeV leptons, see Fig. 9.
10 AAD 2015BM searched in 18.4 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for stable and metastable non-relativistic charged particles through their anomalous specific ionization energy loss in the ATLAS pixel detector. In absence of an excess of events above the expected backgrounds, limits are set within a generic R-hadron model, on stable gluino R-hadrons (see Table 5) and on metastable gluino R-hadrons decaying to ( ${{\mathit g}}$ $/$ ${{\mathit q}}{{\overline{\mathit q}}}$ ) plus a light ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ (see Fig. 7) and decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ plus a light ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ (see Fig. 9).
11 KHACHATRYAN 2015AK looked in a data set corresponding to 18.6 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV, and a search interval corresponding to 281 h of trigger lifetime, for long-lived particles that have stopped in the CMS detector. No evidence for an excess over the expected background in a cloud interaction model is observed. Assuming the decay ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit g}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ and lifetimes between 1 ${{\mathit \mu}}$s and 1000 s, limits are derived on ${{\widetilde{\mathit g}}}$ production as a function of ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$, see Figs. 4 and 6. The exclusions require that ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ is kinematically consistent with the minimum values of the jet energy thresholds used.
12 AAD 2013AA searched in 4.7 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events containing colored long-lived particles that hadronize forming ${{\mathit R}}$-hadrons. No significant excess above the expected background was found. Long-lived ${{\mathit R}}$-hadrons containing a ${{\widetilde{\mathit g}}}$ are excluded for masses up to 985 GeV at 95$\%$ C.L in a general interaction model. Also, limits independent of the fraction of ${{\mathit R}}$-hadrons that arrive charged in the muon system were derived, see Fig. 6.
13 AAD 2013BC searched in 5.0 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV and in 22.9 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for bottom squark R-hadrons that have come to rest within the ATLAS calorimeter and decay at some later time to hadronic jets and a neutralino. In absence of an excess of events above the expected backgrounds, limits are set on gluino masses for different decays, lifetimes, and neutralino masses, see their Table 6 and Fig. 10.
14 CHATRCHYAN 2013AB looked in 5.0 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV and in 18.8 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with heavy stable particles, identified by their anomalous dE/dx in the tracker or additionally requiring that it be identified as muon in the muon chambers, from pair production of ${{\widetilde{\mathit g}}}$'s. No evidence for an excess over the expected background is observed. Limits are derived for pair production of gluinos as a function of mass (see Fig. 8 and Table 5), depending on the fraction, f, of formation of ${{\widetilde{\mathit g}}}−$g (R-gluonball) states. The quoted limit is for f = 0.1, while for f = 0.5 it degrades to 1276 GeV. In the conservative scenario where every hadronic interaction causes it to become neutral, the limit decreases to 928 GeV for f = 0.1.
15 AAD 2012P looked in 31 pb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with pair production of long-lived gluinos. The hadronization of the gluinos leads to ${{\mathit R}}$-hadrons which may stop inside the detector and later decay via ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit g}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ during gaps between the proton bunches. No significant excess over the expected background is observed. From a counting experiment, a limit at 95$\%$ C.L. on the cross section as a function of ${\mathit m}_{{{\widetilde{\mathit g}}}}$ is derived for ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 100 GeV, see Fig. 4. The limit is valid for lifetimes between $10^{-5}$ and $10^{3}$ seconds and assumes the $\mathit Generic$ matter interaction model for the production cross section.
16 CHATRCHYAN 2012AN looked in 4.0 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with pair production of long-lived gluinos. The hadronization of the gluinos leads to ${{\mathit R}}$-hadrons which may stop inside the detector and later decay via ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit g}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ during gaps between the proton bunches. No significant excess over the expected background is observed. From a counting experiment, a limit at 95$\%$ C.L. on the cross section as a function of ${\mathit m}_{{{\widetilde{\mathit g}}}}$ is derived, see Fig. 3. The mass limit is valid for lifetimes between $10^{-5}$ and $10^{3}$ seconds, for what they call ''the daughter gluon energy ${{\mathit E}_{{g}}}$ $>$'' 100 GeV and assuming the $\mathit cloud$ interaction model for ${{\mathit R}}$-hadrons. Supersedes KHACHATRYAN 2011 .
17 CHATRCHYAN 2012L looked in 5.0 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with heavy stable particles, identified by their anomalous dE/dx in the tracker or additionally requiring that it be identified as muon in the muon chambers, from pair production of ${{\widetilde{\mathit g}}}$'s. No evidence for an excess over the expected background is observed. Limits are derived for pair production of gluinos as a function of mass (see Fig. 3), depending on the fraction, f, of formation of ${{\widetilde{\mathit g}}}−$g (${{\mathit R}}$-glueball) states. The quoted limit is for f = 0.1, while for f = 0.5 it degrades to 1046 GeV. In the conservative scenario where every hadronic interaction causes it to become neutral, the limit decreases to 928 GeV for f=0.1. Supersedes KHACHATRYAN 2011C.
18 AAD 2011K looked in 34 pb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with heavy stable particles, identified by their anomalous dE/dx in the tracker or time of flight in the tile calorimeter, from pair production of ${{\widetilde{\mathit g}}}$. No evidence for an excess over the SM expectation is observed. Limits are derived for pair production of gluinos as a function of mass (see Fig. 4), for a fraction, f = 10$\%$, of formation of ${{\widetilde{\mathit g}}}−{{\mathit g}}$ (R-gluonball). If instead of a phase space driven approach for the hadronic scattering of the R-hadrons, a triple-Regge model or a bag-model is used, the limit degrades to 566 and 562 GeV, respectively.
19 AAD 2011P looked in 37 pb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with heavy stable particles, reconstructed and identified by their time of flight in the Muon System. There is no requirement on their observation in the tracker to increase the sensitivity to cases where gluinos have a large fraction, f, of formation of neutral ${{\widetilde{\mathit g}}}−{{\mathit g}}$ (R-gluonball). No evidence for an excess over the SM expectation is observed. Limits are derived as a function of mass (see Fig. 4), for f=0.1. For fractions f = 0.5 and 1.0 the limit degrades to 537 and 530 GeV, respectively.
20 KHACHATRYAN 2011 looked in 10 pb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with pair production of long-lived gluinos. The hadronization of the gluinos leads to R-hadrons which may stop inside the detector and later decay via ${{\widetilde{\mathit g}}}$ $\rightarrow$ ${{\mathit g}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ during gaps between the proton bunches. No significant excess over the expected background is observed. From a counting experiment, a limit at 95$\%$ C.L. on the cross section times branching ratio is derived for ${\mathit m}_{{{\widetilde{\mathit g}}}}−{\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}>$ 100 GeV, see their Fig. 2. Assuming 100$\%$ branching ratio, lifetimes between 75 ns and $3 \times 10^{5}$ s are excluded for ${\mathit m}_{{{\widetilde{\mathit g}}}}$ = 300 GeV. The ${{\widetilde{\mathit g}}}$ mass exclusion is obtained with the same assumptions for lifetimes between 10 ${{\mathit \mu}}{{\mathit s}}$ and 1000 s, but shows some dependence on the model for R-hadron interactions with matter, illustrated in Fig. 3. From a time-profile analysis, the mass exclusion is 382 GeV for a lifetime of 10 ${{\mathit \mu}}{{\mathit s}}$ under the same assumptions as above.
21 KHACHATRYAN 2011C looked in 3.1 pb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with heavy stable particles, identified by their anomalous dE/dx in the tracker or additionally requiring that it be identified as muon in the muon chambers, from pair production of ${{\widetilde{\mathit g}}}$. No evidence for an excess over the expected background is observed. Limits are derived for pair production of gluinos as a function of mass (see Fig. 3), depending on the fraction, f, of formation of ${{\widetilde{\mathit g}}}−{{\mathit g}}$ (R-gluonball). The quoted limit is for f=0.1, while for f=0.5 it degrades to 357 GeV. In the conservative scenario where every hadronic interaction causes it to become neutral, the limit decreases to 311 GeV for f=0.1.
References:
AABOUD 2019C
PL B788 96 Search for heavy charged long-lived particles in proton-proton collisions at $\sqrt{s} = 13$ TeV using an ionisation measurement with the \mbox{ATLAS} detector
AABOUD 2018S
PR D97 052012 Search for long-lived, massive particles in events with displaced vertices and missing transverse momentum in $\sqrt{s}$ = 13 TeV $pp$ collisions with the ATLAS detector
SIRUNYAN 2018AY
JHEP 1805 025 Search for natural and split supersymmetry in proton-proton collisions at $\sqrt{s}=13$ TeV in final states with jets and missing transverse momentum
SIRUNYAN 2018DV
PR D98 092011 Search for long-lived particles with displaced vertices in multijet events in proton-proton collisions at $\sqrt{s}=$13 TeV
KHACHATRYAN 2017AR
PR D95 012009 Search for $\mathit R$-Parity Violating Supersymmetry with Displaced Vertices in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
AABOUD 2016C
PR D93 112015 Search for Metastable Heavy Charged Particles with Large Ionization Energy Loss in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 13 TeV using the ATLAS Experiment
AABOUD 2016B
PL B760 647 Search for Heavy Long-Lived Charged ${{\mathit R}}$-Hadrons with the ATLAS Detector in 3.2 ${\mathrm {fb}}{}^{-1}$ of Proton-Proton Collision Data at $\sqrt {s }$ =13 TeV
KHACHATRYAN 2016BW
PR D94 112004 Search for Long-Lived Charged Particles in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV
AAD 2015BM
EPJ C75 407 Search for Metastable Heavy Charged Particles with Large Ionisation Energy Loss in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV using the ATLAS Experiment
AAD 2015AE
JHEP 1501 068 Searches for Heavy Long-Lived Charged Particles with the ATLAS Detector in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
KHACHATRYAN 2015AK
EPJ C75 151 Search for Decays of Stopped Long-Lived Particles Produced in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
AAD 2013AA
PL B720 277 Searches for Heavy Long-Lived Sleptons and $\mathit R$-Hadrons with the ATLAS Detector in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
AAD 2013BC
PR D88 112003 Search for Long-Lived Stopped $\mathit R$-Hadrons Decaying out of Time with ${{\mathit p}}{{\mathit p}}$ Collisions using the ATLAS Detector
CHATRCHYAN 2013AB
JHEP 1307 122 Searches for Long-Lived Charged Particles in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 and 8 TeV
AAD 2012P
EPJ C72 1965 Search for Decays of Stopped, Long-lived Particles from 7 TeV ${{\mathit p}}{{\mathit p}}$ Collisions with the ATLAS Detector
CHATRCHYAN 2012L
PL B713 408 Search for Heavy Long-Lived Charged Particles in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
CHATRCHYAN 2012AN
JHEP 1208 026 Search for Stopped Long-Lived Particles Produced in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
AAD 2011P
PL B703 428 Search for Heavy Long-Lived Charged Particles with the ATLAS Detector in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
AAD 2011K
PL B701 1 Search for Stable Hadronising squarks and gluinos with the ATLAS Experiment at the LHC
KHACHATRYAN 2011C
JHEP 1103 024 Search for Heavy Stable Charged Particles in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
KHACHATRYAN 2011
PRL 106 011801 Search for Stopped Gluinos in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
|
2020-02-27 15:07:03
|
{"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": 0.908896803855896, "perplexity": 2962.3957627239083}, "config": {"markdown_headings": true, "markdown_code": false, "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-2020-10/segments/1581875146714.29/warc/CC-MAIN-20200227125512-20200227155512-00542.warc.gz"}
|
https://www.mersenneforum.org/showpost.php?s=d132a008ec64f8c97bf68f8ebd29c3e3&p=212230&postcount=4
|
View Single Post
2010-04-18, 02:22 #4
lfm
Jul 2006
Calgary
52×17 Posts
Quote:
Originally Posted by CADavis http://mersenne-aries.sili.net/digits.php calculate number of digits from an exponent i think it something above 332,000,000 for the exponent to give a 100M digit mersenne number.
Its pretty easy if you have a proper calculator. Just 1e6*ln(10)/ln(2)=3321928.09 or so.
|
2021-05-07 05:08:19
|
{"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": 0.8847456574440002, "perplexity": 8120.830399213244}, "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-2021-21/segments/1620243988774.96/warc/CC-MAIN-20210507025943-20210507055943-00033.warc.gz"}
|
https://www.albert.io/learn/electricity-and-magnetism/question/electric-fields-from-exponential-charge-distributions
|
Limited access
A sphere is centered at the origin with a radius $a$. The material is non-conducting and has a radially dependent charge density given by $\rho (r) = \rho_o e^{-r^3 / a^3}$.
What is the electric field for all $r > a$?
A
$E(r>a)=(0.37)\bigg( \cfrac{ 4 \pi \rho_o a^3}{3 \epsilon_o r^2} \bigg)$
B
$E(r>a)=(0.63)\bigg( \cfrac{4 \pi \rho_o a^3}{3 \epsilon_o r^2} \bigg)$
C
$E(r>a)=(0.37)\bigg( \cfrac{ \rho_o a^3}{3 \epsilon_o r^2} \bigg)$
D
$E(r>a)=\bigg( \cfrac{ \rho_o a^3}{3 \epsilon_o r^2} \bigg)$
E
$E(r>a)=(0.63)\bigg( \cfrac{ \rho_o a^3}{3 \epsilon_o r^2} \bigg)$
Select an assignment template
|
2017-04-30 05:08:43
|
{"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": 0.5252939462661743, "perplexity": 428.5815331002717}, "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-2017-17/segments/1492917124299.47/warc/CC-MAIN-20170423031204-00606-ip-10-145-167-34.ec2.internal.warc.gz"}
|
https://www.zbmath.org/?q=an%3A1270.81146
|
# zbMATH — the first resource for mathematics
Exploring positive monad bundles and a new heterotic standard model. (English) Zbl 1270.81146
Summary: A complete analysis of all heterotic Calabi-Yau compactifications based on positive two-term monad bundles over favourable complete intersection Calabi-Yau threefolds is performed. We show that the original data set of about 7000 models contains 91 standard-like models which we describe in detail. A closer analysis of Wilson-line breaking for these models reveals that none of them gives rise to precisely the matter field content of the standard model. We conclude that the entire set of positive two-term monads on complete intersection Calabi-Yau manifolds is ruled out on phenomenological grounds. We also take a first step in analyzing the larger class of non-positive monads. In particular, we construct a supersymmetric heterotic standard model within this class. This model has the standard model gauge group and an additional $$\text{U}(1)_{B-L}$$ symmetry, precisely three families of quarks and leptons, one pair of Higgs doublets and no anti-families or exotics of any kind.
##### MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T60 Supersymmetric field theories in quantum mechanics 81V22 Unified quantum theories 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
##### Software:
Macaulay2; SINGULAR
Full Text:
##### References:
[1] Candelas, P.; Horowitz, GT; Strominger, A.; Witten, E., Vacuum configurations for superstrings, Nucl. Phys., B 258, 46, (1985) [2] Witten, E., New issues in manifolds of SU(3) holonomy, Nucl. Phys., B 268, 79, (1986) [3] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, volume 2, Cambridge University Press, Cambridge U.K. (1987). [4] Lukas, A.; Ovrut, BA; Stelle, KS; Waldram, D., The universe as a domain wall, Phys. Rev., D 59, 086001, (1999) [5] Anderson, LB; He, Y-H; Lukas, A., Heterotic compactification, an algorithmic approach, JHEP, 07, 049, (2007) [6] Anderson, LB; He, Y-H; Lukas, A., Monad bundles in heterotic string compactifications, JHEP, 07, 104, (2008) [7] L.B. Anderson, Heterotic and M-theory Compactifications for String Phenomenology, arXiv:0808.3621 [SPIRES]. [8] Donagi, R.; He, Y-H; Ovrut, BA; Reinbacher, R., The particle spectrum of heterotic compactifications, JHEP, 12, 054, (2004) [9] He, Y-H, GUT particle spectrum from heterotic compactification, Mod. Phys. Lett., A 20, 1483, (2005) [10] Donagi, R.; He, Y-H; Ovrut, BA; Reinbacher, R., Moduli dependent spectra of heterotic compactifications, Phys. Lett., B 598, 279, (2004) [11] Braun, V.; He, Y-H; Ovrut, BA; Pantev, T., A heterotic standard model, Phys. Lett., B 618, 252, (2005) [12] Braun, V.; He, Y-H; Ovrut, BA; Pantev, T., A standard model from the $$E$$_{8} × $$E$$_{8} heterotic superstring, JHEP, 06, 039, (2005) [13] Braun, V.; He, Y-H; Ovrut, BA; Pantev, T., The exact MSSM spectrum from string theory, JHEP, 05, 043, (2006) [14] Bouchard, V.; Donagi, R., An SU(5) heterotic standard model, Phys. Lett., B 633, 783, (2006) [15] Donagi, R.; He, Y-H; Ovrut, BA; Reinbacher, R., Higgs doublets, split multiplets and heterotic SU(3)_{$$C$$} × SU(2)_{$$L$$} × U(1)_{$$Y$$} spectra, Phys. Lett., B 618, 259, (2005) [16] Donagi, R.; He, Y-H; Ovrut, BA; Reinbacher, R., The spectra of heterotic standard model vacua, JHEP, 06, 070, (2005) [17] Blumenhagen, R.; Moster, S.; Weigand, T., Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds, Nucl. Phys., B 751, 186, (2006) [18] Blumenhagen, R.; Moster, S.; Reinbacher, R.; Weigand, T., Massless spectra of three generation U(N) heterotic string vacua, JHEP, 05, 041, (2007) [19] Bouchard, V.; Cvetič, M.; Donagi, R., Tri-linear couplings in an heterotic minimal supersymmetric standard model, Nucl. Phys., B 745, 62, (2006) [20] A. Bak, V. Bouchard and R. Donagi, Exploring a new peak in the heterotic landscape, arXiv:0811.1242 [SPIRES]. [21] L.B. Anderson, J. Gray, D. Grayson, Y.-H. He and A. Lukas, Yukawa couplings in heterotic compactification, arXiv:0904.2186 [SPIRES]. [22] Y.-H. He, S.-J. Lee and A. Lukas, Heterotic models from vector bundles on toric Calabi-Yau manifolds, arXiv:0911.0865 [SPIRES]. [23] Lebedev, O.; Nilles, HP; Ramos-Sanchez, S.; Ratz, M.; Vaudrevange, PKS, Heterotic mini-landscape (II): completing the search for MSSM vacua in a $$Z$$_{6} orbifold, Phys. Lett., B 668, 331, (2008) [24] Lebedev, O.; etal., The heterotic road to the MSSM with R parity, Phys. Rev., D 77, 046013, (2008) [25] Lebedev, O.; etal., Low energy supersymmetry from the heterotic landscape, Phys. Rev. Lett., 98, 181602, (2007) [26] Lebedev, O.; etal., A mini-landscape of exact MSSM spectra in heterotic orbifolds, Phys. Lett., B 645, 88, (2007) [27] Kobayashi, T.; Raby, S.; Zhang, R-J, Constructing 5D orbifold grand unified theories from heterotic strings, Phys. Lett., B 593, 262, (2004) [28] Kobayashi, T.; Raby, S.; Zhang, R-J, Searching for realistic 4d string models with a Pati-Salam symmetry: orbifold grand unified theories from heterotic string compactification on a $$Z$$(6) orbifold, Nucl. Phys., B 704, 3, (2005) [29] Buchmüller, W.; Hamaguchi, K.; Lebedev, O.; Ratz, M., Supersymmetric standard model from the heterotic string, Phys. Rev. Lett., 96, 121602, (2006) [30] Buchmüller, W.; Hamaguchi, K.; Lebedev, O.; Ratz, M., Supersymmetric standard model from the heterotic string. II, Nucl. Phys., B 785, 149, (2007) [31] Förste, S.; Nilles, HP; Vaudrevange, PKS; Wingerter, A., Heterotic brane world, Phys. Rev., D 70, 106008, (2004) [32] Kobayashi, T.; Nilles, HP; Ploger, F.; Raby, S.; Ratz, M., Stringy origin of non-abelian discrete flavor symmetries, Nucl. Phys., B 768, 135, (2007) [33] Candelas, P.; Dale, AM; Lütken, CA; Schimmrigk, R., Complete intersection Calabi-Yau manifolds, Nucl. Phys., B 298, 493, (1988) [34] Candelas, P.; Lütken, CA; Schimmrigk, R., Complete intersection Calabi-Yau manifolds. 2. three generation manifolds, Nucl. Phys., B 306, 113, (1988) [35] Green, PS; Hubsch, T.; Lütken, CA, All Hodge numbers of all complete intersection Calabi-Yau manifolds, Class. Quant. Grav., 6, 105, (1989) [36] He, A-m; Candelas, P., On the number of complete intersection Calabi-Yau manifolds, Commun. Math. Phys., 135, 193, (1990) [37] Gagnon, M.; Ho-Kim, Q., An exhaustive List of complete intersection Calabi-Yau manifolds, Mod. Phys. Lett., A 9, 2235, (1994) [38] P. Candelas and R. Davies, New Calabi-Yau Manifolds with small Hodge numbers, arXiv:0809.4681 [SPIRES]. [39] T. Hubsch, Calabi-Yau Manifolds — A bestiary for physicists, World Scientific, Singapore (1994). [40] Distler, J.; Greene, BR, aspects of (2, 0) string compactifications, Nucl. Phys. B, 304, 1, (1988) [41] Kachru, S., some three generation (0, 2) Calabi-Yau models, Phys. Lett. B, 349, 76, (1995) [42] Douglas, MR; Zhou, C-G, Chirality change in string theory, JHEP, 06, 014, (2004) [43] L.B. Anderson, Y.H. He and A. Lukas, Algorithmic proofs of vector bundle stability, to appear. [44] Anderson, LB; Gray, J.; Lukas, A.; Ovrut, B., Stability walls in heterotic theories, JHEP, 09, 026, (2009) [45] Anderson, LB; Gray, J.; Lukas, A.; Ovrut, B., The edge of supersymmetry: stability walls in heterotic theory, Phys. Lett., B 677, 190, (2009) [46] L.B. Anderson, J. Gray, Y.H. He and A. Lukas, Compactifying on complete intersections, to appear. [47] Horrocks, G.; Mumford, D., A rank 2 vector bundle on $${\mathbb{P}^4}$$ with 15000 symmetries, Topology, 12, 63, (1973) [48] Beilinson, A., Coherent sheaves on $${\mathbb{P}^n}$$ and problems in linear algebra, Funktsional. Anal. i Prilozhen., 12, 68, (1978) [49] Maruyama, M., Moduli of stable sheaves, II, J. Math. Kyoto Univ., 18, 557, (1978) [50] C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Birkhauser Verlag, Germany (1988). [51] Fulton, W.; Lazarsfeld, R., On the connectedness of degeneracy loci and special divisors, Acta Math., 146, 271, (1981) [52] Uhlenbeck, K.; Yau, S-T, On the existence of Hermitian Yang-Mills connections in stable bundles, Comm. Pure App. Math., 39, 257, (1986) [53] Uhlenbeck, K.; Yau, S-T, A note on our previous paper: on the existence of Hermitian Yang-Mills connections in stable vector bundles, Comm. Pure App. Math., 42, 703, (1986) [54] Donaldson, S., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc., 3, 1, (1985) [55] M.C. Brambilla, Semistability of certain bundles on a quintic Calabi-Yau threefold, math.AG/0509599. [56] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete volume 2, Springer-Verlag, Berlin Germany (1994). [57] Knop, F.; Kraft, H.; Luna, D.; Vust, T., Local properties of algebraic group actions, No. 13, (1989), Basel Switzerland [58] Donagi, R.; Ovrut, BA; Pantev, T.; Reinbacher, R., SU(4) instantons on Calabi-Yau threefolds with $$Z$$_{2} × $$Z$$_{2} fundamental group, JHEP, 01, 022, (2004) [59] W. Fulton and J. Harris, Representation theory: a first course, Springer, New York U.S.A. (1991). [60] R. Hartshorne, Algebraic geometry, springer, Graduate Text in Mathematics volume 52, Springer-Verlag, Germany (1977). [61] P. Griffith and J. Harris, Principles of algebraic geometry, Wiley Interscience, U.S.A. (1978). [62] Breit, JD; Ovrut, BA; Segre, GC, $$E$$_{6} symmetry breaking in the superstring theory, Phys. Lett., B 158, 33, (1985) [63] V. Braun, Three generations on the quintic quotient, arXiv:0909.5682 [SPIRES]. [64] Candelas, P.; Ossa, X.; He, Y-H; Szendroi, B., Triadophilia: a special corner in the landscape, Adv. Theor. Math. Phys., 12, 2, (2008) [65] Braun, V.; Ovrut, BA; Pantev, T.; Reinbacher, R., Elliptic Calabi-Yau threefolds with $$Z$$(3) × $$Z$$(3) Wilson lines, JHEP, 12, 062, (2004) [66] Braun, V.; He, Y-H; Ovrut, BA; Pantev, T., Vector bundle extensions, sheaf cohomology and the heterotic standard model, Adv. Theor. Math. Phys., 10, 4, (2006) [67] Ambroso, M.; Ovrut, B., The B-L/electroweak hierarchy in heterotic string and M-theory, JHEP, 10, 011, (2009) [68] M. Ambroso and B.A. Ovrut, The B-L/electroweak hierarchy in smooth heterotic compactifications, arXiv:0910.1129 [SPIRES]. [69] V. Braun, P. Candelas and R. Davies, A three-generation Calabi-Yau manifold with small Hodge numbers, arXiv:0910.5464 [SPIRES]. [70] Braun, V.; Kreuzer, M.; Ovrut, BA; Scheidegger, E., Worldsheet instantons and torsion curves. part A: direct computation, JHEP, 10, 022, (2007) [71] G.-M. Greuel, G. Pfister and H. Schönemann, Singular: a computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2001), available at http://www.singular.uni-kl.de/. [72] D.G. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/. [73] D.J.S. Robinson, A course in the theory of groups, Graduate Texts in Mathematics, Springer, New York U.S.A. (1995).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
|
2021-07-26 14:44:10
|
{"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": 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": 0.6677273511886597, "perplexity": 14088.257814478822}, "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-2021-31/segments/1627046152129.33/warc/CC-MAIN-20210726120442-20210726150442-00467.warc.gz"}
|
https://chet-aero.com/category/mathematics/page/2/
|
Determining the Characteristic Polynomial of the Companion Matrix by Use of a Large Matrix
Most proofs of the characteristic polynomial of the companion matrix–an important specific case–proceed by induction, and start with a $2\times2$ matrix. It strikes me that an inductive proof has more force (or at least makes more sense) if a larger matrix is used. In this case we will use a “large” (numerical analysts will laugh at this characterisation) $10\times10$ matrix.
Let us begin by making a notation change. Consider the general polynomial
For this to be monic (one of the requirements for the polynomial in question) we should divide by the last coefficient, thus
Our object is thus to prove that this (or a variation of this, as we will see) is the characteristic polynomial of
The characteristic polynomial of this is the determinant of the following:
(For another application of the characteristic polynomial and the companion matrix, click here.)
To find the determinant, we expand along the first row. But then we discover that only two minors that matter: the one in the upper left corner and the one in the upper right. Breaking this up into minors and cofactors yields the following:
The second matrix, however, is an upper triangular matrix with ones for all of its diagonal entries. Its determinant, therefore, is unity. Also rewriting the coefficient of the second term, we have
or
Repeating this process for the next set of minors and cofactors yields
Note carefully the inclusion of $-\lambda$ in the second term. We can also write this as
Repeating this process until the end, it is easy to see that
or more generally
where $n$ is the degree of the polynomial (and the size of the companion matrix.) If we drop the terms we used to make the polynomial monic, we have at last
Mohr’s Circle Analysis Using Linear Algebra and Numerical Methods
Abstract
Mohr’s Circle-or more generally the stress equilibrium in solids-is a well known method to analyse the stress state of a two- or three-dimensional solid. Most engineers are exposed to its derivation using ordinary algebra, especially as it relates to the determination of principal stresses and invariants for comparison with failure criteria. In this presentation, an approach using linear algebra will be set forth, which is interesting not only from the standpoint of the stress state but from linear algebra considerations, as the problem makes an excellent illustration of linear algebra concepts from a real geometric system. A numerical implementation of the solution using Danilevsky’s Method will be detailed.
1. Introduction
The analysis of the stress state of a solid at a given point is a basic part of mechanics of materials.Although such analysis is generally associated with the theory of elasticity, in fact it is based on static equilibrium, and is also valid for the plastic region as well. In this way it is used in non-linear finite element analysis, among other applications. The usual objective of such an analysis is to determine the principal stresses at a point, which in turn can be compared to failure criteria to determine local deformation.In this approach the governing equations will be cast in a linear algebra form and the problem solved in this way, as opposed to other types of solutions given in various textbooks. Doing it in this way can have three results:
1. It allows the abstract concepts of linear algebra to be illustrated well in a physical problem.
2. It allows the physics of the determination of principal stresses to be seen in a different way with the mathematics.
3. It opens up the problem to numerical solution, as opposed to the complicated closed-form solutions usually encountered, of the invariants, principal stresses or direction cosines.
2. Two-Dimensional Analysis
2.1. Eigenvalues, Eigenvectors and Principal Stresses
The simplest way to illustrate this is to use two-dimensional analysis. Even with this simplest case, the algebra can become very difficult very quickly, and the concepts themselves obscured. Consider first the stress state shown in Figure 1, with the notation which will be used in the rest of the article.
Figure 1: Stress State Coordinates (modified from Verruijt and van Bars [8])
The theory behind this figure is described in many mechanics of materials textbooks; for this case the presentation of Jaeger and Cook [4] was used. The element is in static equilibrium along both axes. In order for the summation of moments to be zero,
$\tau_{xy}=\tau_{yx}$ (1)
The angles are done a little differently than usual; this is to allow an easier transition when three-dimensional analysis is considered. The direction cosines based on these angles are as follows:
$l=cos\alpha\$ (2)
$m=cos\beta\$ (3)
Now consider the components $p_{x},\,p_{y}$ of the stress vector p on their respective axes. Putting these into matrix form, they are computed as follows:
\left[\begin{array}{cc} \sigma_{{x}} & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}} \end{array}\right]\left[\begin{array}{c} l\\ \noalign{\medskip}m \end{array}\right]=\left[\begin{array}{c} p_{{x}}\\ \noalign{\medskip}p_{{y}} \end{array}\right] (4)
which is a classic Ax = b type of problem. At this point there are some things about the matrix in Equation 4 that need to be noted (DeRusso et al. [2]):
1. It is square.
2. It is real and symmetric. Because of these two properties:
1. The eigenvalues (and thus the principal stresses, as will be shown) are real. Since for two-dimensional space the equations for the principal stresses are quadratic, this is not a “given” from pure algebra alone.
2. The eigenvectors form an orthogonal set, which is important in the diagonalization process.
3. The sum of the diagonal entries of the matrix, referred to as the trace, is equal to the sum of the values of the eigenvalues. As will be seen, this means that, as we rotate the coordinate axes, the trace remains invariant.
At this point there are two related questions that need to be asked. The first is whether static equilibrium will hold if the coordinate axes are rotated. The obvious answer is “yes,” otherwise there would be no real static equilibrium. The stresses will obviously change in the process of rotation. These values can be found using a rotation matrix and multiplying the original matrix as follows (Strang [7]):
\left[\begin{array}{cc} cos\alpha & -sin\alpha\\ sin\alpha & cos\alpha \end{array}\right]\left[\begin{array}{cc} \sigma_{{x}} & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}} \end{array}\right]=\left[\begin{array}{cc} \sigma'_{{x}} & \tau'_{{\it xy}}\\ \noalign{\medskip}\tau'_{{\it xy}} & \sigma'_{{y}} \end{array}\right] (5)
The rotation matrix is normally associated with Wallace Givens, who taught at the University of Tennessee at Knoxville. The primed values represent the stresses in the rotated coordinate system. We can rewrite the rotation matrix as follows, to correspond with the notation given above:^
\left[\begin{array}{cc} l & -m\\ m & l \end{array}\right]\left[\begin{array}{cc} \sigma_{{x}} & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}} \end{array}\right]=\left[\begin{array}{cc} \sigma'_{{x}} & \tau'_{{\it xy}}\\ \noalign{\medskip}\tau'_{{\it xy}} & \sigma'_{{y}} \end{array}\right]\ (6)
2.1 Eigenvalues, Eigenvectors and Principal Stresses
The second question is this: is there an angle (or set of direction cosines) where the shear stresses would go away, leaving only normal stresses? Because of the properties of the matrix, the answer to this is also “yes,” and involves the process of diagonalizing the matrix. The diagonalized matrix (the matrix with only non-zero values along the diagonal) can be found if the eigenvalues of the matrix can be found, i.e., if the following equation can be solved for $\lambda$ :
\left[\begin{array}{cc} \sigma_{{x}}-\lambda & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}}-\lambda \end{array}\right]=0\ (7)
To accomplish this, we take the determinant of the left hand side of Equation 7, namely
${\lambda}^{2}-\left(\sigma_{{x}}+\sigma_{{y}}\right)\lambda-\left({\tau_{{\it xy}}}^{2}-\sigma_{x}\sigma_{y}\right)=0$ (8)
If we define
$J_{1}=\sigma_{{x}}+\sigma_{{y}}$ (9)
and
$J{}_{2}={\tau_{{\it xy}}}^{2}-\sigma_{x}\sigma_{y}$ (10)
Equation 8 can be rewritten as
$\lambda^{2}-J_{1}\lambda-J_{2}=0$ (11)
The quantities $J_{1},\,J_{2}$ are referred to as the invariants; they do not change as the axes are rotated. The first invariant is the trace of the system, which was predicted to be invariant earlier. These are very important,
especially in finite element analysis, where failure criteria are frequently computed relative to the invariants and not the principal stresses in any combination. This is discussed at length in Owen and Hinton [6].
This is the characteristic polynomial of the matrix of Equation 7. Most people generally don’t associate matrices and polynomials, but every matrix has an associated characteristic polynomial, and conversely a polynomial can have one or more corresponding matrices.The solution of Equation 8 produces the two eigenvalues of the matrix. The first eigenvalue of the matrix is
$\lambda_{1}=1/2\,\sigma_{{y}}+1/2\,\sigma_{{x}}+1/2\,\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}$ (12)
and the second
$\lambda_{1}=1/2\,\sigma_{{y}}+1/2\,\sigma_{{x}}-1/2\,\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}$ (13)
We can also determine the eigenvectors from this. Without going into the process of determining these,the first eigenvector is
$\bar{x}_{1}=\left[\begin{array}{c} {\frac{-1/2\,\sigma_{{y}}+1/2\,\sigma_{{x}}+1/2\,\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}}{\tau_{{\it xy}}}}\\ 1 \end{array}\right]\$ (14)
and the second is
$\bar{x}_{2}=\left[\begin{array}{c} {\frac{-1/2\,\sigma_{{y}}+1/2\,\sigma_{{x}}-1/2\,\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}}{\tau_{{\it xy}}}}\\ 1 \end{array}\right]$ (15)
One thing we can do with these eigenvectors is to normalize them, i.e., have it so that the sum of the squares of the entries is unity. Doing this yields
$\bar{x}_{1}=\left[\begin{array}{c} {\frac{-1/2\,\sigma_{{y}}{\tau_{{\it xy}}}^{-1}+1/2\,\sigma_{{x}}{\tau_{{\it xy}}}^{-1}+1/2\,\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}{\tau_{{\it xy}}}^{-1}}{\sqrt{1/2\,{\frac{{\sigma_{{y}}}^{2}}{{\tau_{{\it xy}}}^{2}}}-{\frac{\sigma_{{x}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}-1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{{\sigma_{{x}}}^{2}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{x}}}{{\tau_{{\it xy}}}^{2}}}+2}}}\\ {\frac{1}{\sqrt{1/2\,{\frac{{\sigma_{{y}}}^{2}}{{\tau_{{\it xy}}}^{2}}}-{\frac{\sigma_{{x}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}-1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{{\sigma_{{x}}}^{2}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{x}}}{{\tau_{{\it xy}}}^{2}}}+2}}} \end{array}\right]$ (16)
and
$\bar{x}_{2}=\left[\begin{array}{c} {\frac{-1/2\,\sigma_{{y}}{\tau_{{\it xy}}}^{-1}+1/2\,\sigma_{{x}}{\tau_{{\it xy}}}^{-1}-1/2\,\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}{\tau_{{\it xy}}}^{-1}}{\sqrt{1/2\,{\frac{{\sigma_{{y}}}^{2}}{{\tau_{{\it xy}}}^{2}}}-{\frac{\sigma_{{x}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{{\sigma_{{x}}}^{2}}{{\tau_{{\it xy}}}^{2}}}-1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{x}}}{{\tau_{{\it xy}}}^{2}}}+2}}}\\ {\frac{1}{\sqrt{1/2\,{\frac{{\sigma_{{y}}}^{2}}{{\tau_{{\it xy}}}^{2}}}-{\frac{\sigma_{{x}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{y}}}{{\tau_{{\it xy}}}^{2}}}+1/2\,{\frac{{\sigma_{{x}}}^{2}}{{\tau_{{\it xy}}}^{2}}}-1/2\,{\frac{\sqrt{{\sigma_{{y}}}^{2}-2\,\sigma_{{x}}\sigma_{{y}}+{\sigma_{{x}}}^{2}+4\,{\tau_{{\it xy}}}^{2}}\sigma_{{x}}}{{\tau_{{\it xy}}}^{2}}}+2}}} \end{array}\right]$ (17)
The eigenvectors, complicated as the algebra is, are useful in that we can diagonalize the matrix as follows:
$S^{-1}AS=\Lambda$ (18)
where
$S=\left[\begin{array}{cc} \bar{x}_{1_{1}} & \bar{x}_{2_{1}}\\ \bar{x}_{1_{2}} & \bar{x}_{2_{2}} \end{array}\right]$ (19)
This is referred to as a similarity or collinearity transformation. Similar matrices are matrices with the same eigenvalues. The matrix $\Lambda$, although simpler in form than $A$, has the same eigenvalues as the “original”matrix.
Either the original or the normalized forms of the eigenvectors can be used; the result will be the same as the scalar multiples will cancel in the matrix inversion. The normalization was done to illustrate the
relationship between the eigenvectors, the diagonalization matrix, and the Givens rotation matrix, since for the last $sin^{2}\alpha+cos^{2}\alpha=1$, an automatically normalized relationship. It is thus possible to extract the angle of the principal stresses from the eigenvectors.
Inverting the result of Equation 19 and multiplying through Equation 18 yields at last
$\Lambda=\left[\begin{array}{cc} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right]=\left[\begin{array}{cc} \sigma_{1} & 0\\ 0 & \sigma_{3} \end{array}\right]$ (20)
The two eigenvalues from Equations 12 and 13 are the principal stresses at the stress point in question.(The use of different subscripts for the eigenvalues and principal stresses comes from too many years dealing with Mohr-Coulomb failure theory.) One practical result of Equations 9 and 20 and the underlying theory is that, for any coordinate orientation,
$\sigma_{x}+\sigma_{y}=\sigma_{1}+\sigma_{3}$ (21)
This is a very handy check when working problems such as this, especially since the algebra is so involved.
2.2. Two-Dimensional Example
To see all of this “in action” consider the soil stress states shown in Figure 2.
Figure 2: Stress State Example (from Navy [5])
Consider the stress state “A.” The stresses are expressed as a ratio of a pressure P at the surface; furthermore, following geotechnical engineering convention, compressive stresses are positive and the ordinate is the “z” axis.. For this study the convention of Figure 1 will be adopted and thus $\sigma_{x}=-0.77,\,\sigma_{y}=-0.98,\,\tau_{xy}=0.05$ .By direct substitution (and carrying the results to precision unjustified in geotechnical engineering) we obtain the following:
• $\lambda_{1}=\sigma_{1}=-.7587029665P$ (Equations 12 and 20.)
• $\lambda_{2}=\sigma_{3}=-.9912970335P$ (Equations 12 and 20.) Note that the principal stresses are reversed; this is because the sign convention is reversed, and thus what was formerly the smaller of the stresses is now the larger. Also, the axis of the first principal stress changes because the first principal stress itself has changed.
• $S=\left[\begin{array}{cc} .9754128670 & -.220385433\\ .2203854365 & .9754128502 \end{array}\right]$ (Equation 19.) Note that the normalized values of the eigenvectors (Equations 16 and 17) are used. Also note the similarity between this matrix and the Givens rotation matrix of Equation 5. The diagonalization process is simply a rotation process, as the physics of the problem suggest. (The diagonal terms should be equal to each other and the absolute values ofthe off-diagonal terms should be also; they are not because of numerical errors in Maple where they were computed.
• $\alpha=12.73167230^{\circ};\,\beta=77.26832747^{\circ}$ (Equations 2, 3 and 6.)
• In both cases the trace of A and \Lambda is the same, namely $1.75P$ (Equation 21.)
The results are thus the same. The precision is obviously greater than the “slide rule era” Navy [5], but the results illustrate that numerical errors can creep in, even with digital computation.
3. Three-Dimensional Analysis
3.1. Regular Principal Stresses
It is easy to see that, although the concepts are relatively simple, the algebra is involved.It is for this reason that the two-dimensional state was illustrated. With three dimensional stresses, the principles are the same, but the algebra is even more difficult.To begin, one more direction cosine must be defined,
$n = cos\gamma$ (22)
where $\gamma$ is of course the angle of the direction of the rotated coordinate system relative to the original one. The rotated system does not have to be the principal axis system, although that one is of most interest.
For three dimensions, Equation 4 should be written as follows:
$\left[\begin{array}{ccc} \sigma_{x} & \tau_{xy} & \tau_{xz}\\ \tau_{xy} & \sigma_{y} & \tau_{yz}\\ \tau_{xz} & \tau_{yz} & \sigma_{z} \end{array}\right]\left[\begin{array}{c} l\\ m\\ n \end{array}\right]=\left[\begin{array}{c} p_{x}\\ p_{y}\\ p_{z} \end{array}\right]$ (23)
All of the moment summations of the shear stresses-which result in the symmetry of the matrix-have been included. The eigenvalues of the matrix are thus the solution of
$\left[\begin{array}{ccc} \sigma_{x}-\lambda & \tau_{xy} & \tau_{xz}\\ \tau_{xy} & \sigma_{y}-\lambda & \tau_{yz}\\ \tau_{xz} & \tau_{yz} & \sigma_{z}-\lambda \end{array}\right]=0$ (24)
The characteristic equation of this matrix-and thus the equation that solves for the eigenvalues-is
$\lambda^{3}-J_{1}\lambda^{2}-J_{2}\lambda-J_{3}=0$ (25)
where
$J_1=\sigma_{x}+\sigma_{y}+\sigma_{z}$ (26)
$J_2=\tau_{xy}^{2}+\tau_{xz}^{2}+\tau_{yz}^{2}-\left(\sigma_{x}\sigma_{y}+\sigma_{x}\sigma_{z}+\sigma_{y}\sigma_{z}\right)$ (27)
$J_3=\sigma_{x}\sigma_{y}\sigma_{z}+2\tau_{xy}\tau_{xz}\tau_{yz}-\left(\sigma_{x}\tau_{yz}^{2}+\sigma_{y}\tau_{xz}^{2}+\sigma_{z}\tau_{xy}^{2}\right)$ (28)
It is easy to show the following:
• Equation 26 reduces to Equation 9 if $\sigma_{z}=0$ .
• Equation 27 reduces to Equation 10 if additionally $\tau_{xz}=\tau_{yz}=0$ .
• For all three conditions, Equation 25 reduces to 11.
From the previous derivation, we can thus expect
$\lambda_{1}=\sigma_{1}$ (29)
$\lambda_{2}=\sigma_{2}$ (30)
$\lambda_{3}=\sigma_{3}$ (31)
Solving a cubic equation such as this can be accomplished using the Cardan Formula. Generally, it is possible that we will end up with complex roots, but because of the properties of the matrix we are promised real roots. The eigenvalues are as follows:
$\lambda_{1} = 1/6\,\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}\nonumber \\ -6\,{\frac{1/3\,J_{{2}}-1/9\,{J_{{1}}}^{2}}{\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}}}-1/3\,J_{{1}}$ (32)
$\lambda_{2} = -1/12\,\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}\nonumber \\ +3\,{\frac{1/3\,J_{{2}}-1/9\,{J_{{1}}}^{2}}{\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}}}\nonumber \\ -1/3\,J_{{1}}+1/12\,\sqrt{-3}\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}\nonumber \\ +2\sqrt{-3}\,{\frac{1/3\,J_{{2}}-1/9\,{J_{{1}}}^{2}}{\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}}}$ (33)
$\lambda_{3} = -1/12\,\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}\nonumber \\ +3\,{\frac{1/3\,J_{{2}}-1/9\,{J_{{1}}}^{2}}{\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}}}\nonumber \\ -1/3\,J_{{1}}-1/12\,\sqrt{-3}\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}\nonumber \\ -2\sqrt{-3}\,{\frac{1/3\,J_{{2}}-1/9\,{J_{{1}}}^{2}}{\sqrt[3]{36\,J_{{2}}J_{{1}}-108\,J_{{3}}-8\,{J_{{1}}}^{3}+12\,\sqrt{12\,{J_{{2}}}^{3}-3\,{J_{{2}}}^{2}{J_{{1}}}^{2}-54\,J_{{2}}J_{{1}}J_{{3}}+81\,{J_{{3}}}^{2}+12\,J_{{3}}{J_{{1}}}^{3}}}}}$ (34)
3.2. Deviatoric Principal Stresses
The algebra of Equations 32, 33 and 34 is rather involved. Is there a way to simplify it? The answer is”yes” if we use a common reduction of the Cardan Formula. Consider the following change of variables:
$\sigma_{x}'=\sigma_{x}-\frac{J_{1}}{3}$ (35)
$\sigma_{y}'=\sigma_{y}-\frac{J{}_{1}}{3}$ (36)
$\sigma_{z}'=\sigma_{z}-\frac{J{}_{1}}{3}$ (37)
Since the change is the same in all directions, we can write
$\left[\begin{array}{ccc} \sigma'_{x} & \tau_{xy} & \tau_{xz}\\ \tau_{xy} & \sigma'_{y} & \tau_{yz}\\ \tau_{xz} & \tau_{yz} & \sigma'_{z} \end{array}\right]\left[\begin{array}{c} l\\ m\\ n \end{array}\right]=\left[\begin{array}{c} p'_{x}\\ p'_{y}\\ p'_{z} \end{array}\right]$ (38)
and
$\left[\begin{array}{ccc} \sigma'_{x}-\lambda' & \tau_{xy} & \tau_{xz}\\ \tau_{xy} & \sigma'_{y}-\lambda' & \tau_{yz}\\ \tau_{xz} & \tau_{yz} & \sigma'_{z}-\lambda' \end{array}\right]=0$ (39)
The characteristic equation of the matrix in Equation 39 is
$\lambda'^{3}-J'_{2}\lambda'-J'_{3}=0$ (40)
where
$J'_{2}=\tau_{xy}^{2}+\tau_{xz}^{2}+\tau_{yz}^{2}-\left(\sigma'_{x}\sigma'_{y}+\sigma'_{x}\sigma'_{z}+\sigma'_{y}\sigma'_{z}\right)$ (41)
$J'_{3}=\sigma'_{x}\sigma'_{y}\sigma'_{z}+2\tau_{xy}\tau_{xz}\tau_{yz}-\left(\sigma'_{x}\tau_{yz}^{2}+\sigma'_{y}\tau_{xz}^{2}+\sigma'_{z}\tau_{xy}^{2}\right)$ (42)
Although our motivation for this was to simplify the characteristic equation, the deviatoric stress infact has physical significance. As Jaeger and Cook [4] point out, “…essentially ($\frac{I_{1}}{3}$ ) determines uniform compression or dilation, while the stress deviation determines distortion. Since many criteria of failure are concerned primarily with distortion, and since they must be invariant with respect to rotation of axes, it appears that the invariants of stress deviation will be involved.”
The one thing to be careful about is that the eigenvalues from Equation 24 are not the same as those from Equation 39. The latter are in fact “deviatoric eigenvalues” and are equivalent to deviatoric principal stresses, thus
$\lambda'_1 = \sigma'_1$ (43)
$\lambda'_2 = \sigma'_2$ (44)
$\lambda'_3 = \sigma'_3$ (45)
These can be converted back to full stresses using Equations 35, 36 and 37. The eigenvalues for the deviatoric stresses are
$\lambda'_{1}=1/6\,\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}+2\,{\frac{J'_{{2}}}{\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}}}$ (46)
$\lambda'_{2} = -1/12\,\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}-{\frac{J'_{{2}}}{\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}}}\nonumber \\ +1/2\,\sqrt{-1}\sqrt{3}\left(1/6\,\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}-2\,{\frac{J'_{{2}}}{\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}}}\right)$ (47)
$\lambda'_{3} = -1/12\,\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}-{\frac{J'_{{2}}}{\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}}}\nonumber \\ -1/2\,\sqrt{-1}\sqrt{3}\left(1/6\,\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}-2\,{\frac{J'_{{2}}}{\sqrt[3]{108\,J'_{{3}}+12\,\sqrt{-12\,{J'_{{2}}}^{3}+81\,{J'_{{3}}}^{2}}}}}\right)$ (48)
Solving Equations 24 and 39 will yield eigenvalues and principal stresses of one kind or another. The eigenvectors and direction cosines can be determined using the same methods used for the two-dimensional problem. It should be readily evident, however, that the explicit solution of these eivenvectors and diagonalizing rotational matrices is very involved, although using deviatoric stresses simplifies the algebra considerably. If the material is isotropic, and its properties are the same in all directions, than the direction of the principal stresses can frequently be neglected. For anisotropic materials, this is not the case.
4. Finding the Eigenvalues and Eigenvectors Using the Method of Danilevsky
For the simple 3 * 3 matrix, computing the determinant, and from it the the invariants, is not a major task. With larger matrices, it is far more difficult to find the determinant, let alone the characteristic polynomial.There have been many solutions to this problem over the years. For larger matrices the most common are Householder reflections and/or Givens rotations. We discussed the latter earlier. By doing multiple reflections and/or rotations, the matrix can be diagonalized and the eigenvalues “magically appear.” The whole problem of solving the characteristic equation is avoided, although the iterative cost in getting to the result can be considerable.
The method we’ll explore here is that of Danilevsky, first published in 1937, as presented in Faddeeva [3].
4.1. Theory
The math will be presented in $3\times3$ format, but it can be expanded to any size square matrix. The idea is, through a series of similarity transformations, to transform a $3\times3$ matrix (such as is shown in Equation 23) to the Frobenius normal form, which is
D=\left[\begin{array}{ccc} p_{{1}} & p_{{2}} & p_{{3}}\\ \noalign{\medskip}1 & 0 & 0\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right] (49)
From here we take the determinant of $D-\lambda I$ , thus
$D\left(\lambda\right)=\left|\begin{array}{ccc} p_{1}-\lambda & p_{2} & p_{3}\\ 1 & -\lambda & 0\\ 0 & 1 & -\lambda \end{array}\right|$ (50)
The determinant-and thus the characteristic polynomial-is easy to compute, and setting it equal to zero,
${\lambda}^{3}-{\lambda}^{2}p_{{1}}-p_{{2}}\lambda-p_{{3}}=0$ (51)
This is obviously the same as Equation 25, but now we can read the invariants straight out of the similar matrix.
So how do we get to Equation 49? The simple answer is through a series of row reductions. We start for the matrix $A$ by carrying the third row into the second, dividing all the elements by the element in the last row by the element $a_{3,2}$ , then subtracting the second column, multiplied by $a_{3,1},\,a_{3,2},\,a_{3,3}$ respectively from all the rest of the columns. Row and column manipulation is common with computational matrix routines, but a more straightforward way is to postmultiply $A$ by the matrix
M=\left[\begin{array}{ccc} 1 & 0 & 0\\ \noalign{\medskip}-{\frac{a_{{3,1}}}{a_{{3,2}}}} & {a_{{3,2}}}^{-1} & -{\frac{a_{{3,3}}}{a_{{3,2}}}}\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right] (52)
The result will not be similar to $A$ , but if we premultiply that result by $M^{-1}$ ,
M^{-1}=\left[\begin{array}{ccc} 1 & 0 & 0\\ \noalign{\medskip}a_{{3,1}} & a_{{3,2}} & a_{{3,3}}\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right] (53)
that result will be by Equation 19, although $M$ does not diagonalize the result the way $S$ did. Performing this series of matrix multplications yields
B=M^{-1}AM=\left[\begin{array}{ccc} a_{{1,1}}-{\frac{a_{{1,2}}a_{{3,1}}}{a_{{3,2}}}} & {\frac{a_{{1,2}}}{a_{{3,2}}}} & -{\frac{a_{{1,2}}a_{{3,3}}}{a_{{3,2}}}}+a_{{1,3}}\\ \noalign{\medskip}a_{{3,1}}\left(a_{{1,1}}-{\frac{a_{{1,2}}a_{{3,1}}}{a_{{3,2}}}}\right)+a_{{3,2}}\left(a_{{2,1}}-{\frac{a_{{2,2}}a_{{3,1}}}{a_{{3,2}}}}\right) & {\frac{a_{{1,2}}a_{{3,1}}}{a_{{3,2}}}}+a_{{2,2}}+a_{{3,3}} & a_{{3,1}}\left(-{\frac{a_{{1,2}}a_{{3,3}}}{a_{{3,2}}}}+a_{{1,3}}\right)+a_{{3,2}}\left(-{\frac{a_{{2,2}}a_{{3,3}}}{a_{{3,2}}}}+a_{{2,3}}\right)\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right] (54)
We continue this process by moving up a row, thus the new similarity transformation matrices are
N = \left[\begin{array}{ccc} {b_{{2,1}}}^{-1} & -{\frac{b_{{2,2}}}{b_{{2,1}}}} & -{\frac{b_{{2,3}}}{b_{{2,1}}}}\\ \noalign{\medskip}0 & 1 & 0\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right] (55)
N^{-1} = \left[\begin{array}{ccc} b_{{2,1}} & b_{{2,2}} & b_{{2,3}}\\ \noalign{\medskip}0 & 1 & 0\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right] (56)
Although we formed the M and N matrices first, from a conceptual and computational standpoint itis easier to form the “inverse” matrices first and then invert them. One drawback to Danilevsky’s method is that there are many opportunities for division by zero, which need to be taken into consideration when employing the method.
In any case the next step yields
C=N^{-1}BN=\left[\begin{array}{ccc} b_{{1,1}}+b_{{2,2}} & b_{{2,1}}\left(-{\frac{b_{{1,1}}b_{{2,2}}}{b_{{2,1}}}}+b_{{1,2}}\right)+b_{{2,3}} & b_{{2,1}}\left(-{\frac{b_{{1,1}}b_{{2,3}}}{b_{{2,1}}}}+b_{{1,3}}\right)\\ \noalign{\medskip}1 & 0 & 0\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right] (57)
From this Equation 49 is easily formed.
The coefficients $p_1,\,p_2$ and $p_3$ are simply the invariants, which have already been established earlier for both standard and deviatoric normal stresses. From that standpoint the method seems to be overkill.
The advantage comes when we compute the eigenvectors/direction cosines. Consider the vector
$y=\left[\begin{array}{c} y_{1}\\ y_{2}\\ y_{3} \end{array}\right]$ (58)
Premultiplying it by $D-\lambda I$ (see Equation 50) yields
\left[\begin{array}{ccc} p_{1}-\lambda & p_{2} & p_{3}\\ 1 & -\lambda & 0\\ 0 & 1 & -\lambda \end{array}\right]\left[\begin{array}{c} y_{1}\\ y_{2}\\ y_{3} \end{array}\right]=\left[\begin{array}{c} \left(p_{{1}}-\lambda\right)y_{{1,1}}+p_{{2}}y_{{2,1}}+p_{{3}}y_{{3,1}}\\ \noalign{\medskip}y_{{1,1}}-\lambda\,y_{{2,1}}\\ \noalign{\medskip}y_{{2,1}}-\lambda\,y_{{3,1}} \end{array}\right]=0 (59)
For y to be non-trivial,
$y=\left[\begin{array}{c} \lambda^{2}\\ \lambda\\ 1 \end{array}\right]$ (60)
It can be shown (Faddeeva [3]) that the eigenvectors $x_{n}$ can be found as follows (successive rotations:)
\bar{x}_{n}=MNy=\left[\begin{array}{c} {\frac{{\lambda_{n}}^{2}}{b_{{2,1}}}}-{\frac{b_{{2,2}}\lambda_{n}}{b_{{2,1}}}}-{\frac{b_{{2,3}}}{b_{{2,1}}}}\\ \noalign{\medskip}-a_{{3,1}}\left({\frac{{\lambda_{n}}^{2}}{b_{{2,1}}}}-{\frac{b_{{2,2}}\lambda_{n}}{b_{{2,1}}}}-{\frac{b_{{2,3}}}{b_{{2,1}}}}\right){a_{{3,2}}}^{-1}+{\frac{\lambda_{n}}{a_{{3,2}}}}-{\frac{a_{{3,3}}}{a_{{3,2}}}}\\ \noalign{\medskip}1 \end{array}\right] (61)
These can, as was shown earlier, be normalized to direction cosines and can form the diagonalizing matrices.
4.2. Three-Dimensional Example
An example of this is drawn from Boresi et al. [1]. It involves a stress state, which referring to Equation 23 can be written (all stresses in kPa) as
A=\left[\begin{array}{ccc} 120 & -55 & -75\\ \noalign{\medskip}-55 & 55 & 33\\ \noalign{\medskip}-75 & 33 & -85 \end{array}\right] (62)
For the first transformation, we substitute these values into Equation 54 and thus
B=\left[\begin{array}{ccc} 1 & 0 & 0\\ \noalign{\medskip}-75 & 33 & -85\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]\left[\begin{array}{ccc} 120 & -55 & -75\\ \noalign{\medskip}-55 & 55 & 33\\ \noalign{\medskip}-75 & 33 & -85 \end{array}\right]\left[\begin{array}{ccc} 1 & 0 & 0\\ \noalign{\medskip}{\frac{25}{11}} & 1/33 & {\frac{85}{33}}\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]=\left[\begin{array}{ccc} -5 & -5/3 & -{\frac{650}{3}}\\ \noalign{\medskip}2685 & 95 & 22014\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right] (63)
The second transformation, following Equation 57,
C=\left[\begin{array}{ccc} 2685 & 95 & 22014\\ \noalign{\medskip}0 & 1 & 0\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]\left[\begin{array}{ccc} -5 & -5/3 & -{\frac{650}{3}}\\ \noalign{\medskip}2685 & 95 & 22014\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right]\left[\begin{array}{ccc} {\frac{1}{2685}} & -{\frac{19}{537}} & -{\frac{7338}{895}}\\ \noalign{\medskip}0 & 1 & 0\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]=\left[\begin{array}{ccc} 90 & 18014 & -471680\\ \noalign{\medskip}1 & 0 & 0\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right] (64)
From this the characteristic equation is, from Equation 51,
${\lambda}^{3}-90\,{\lambda}^{2}-18014\,\lambda+471680=0$ (65)
which yields eigenvalues/principal stresses of 176.8, 24.06 and -110.86 kPa.
So how does this look with deviatoric stresses? Applying Equations 26, 35, 36 and 37 yields
A'=\left[\begin{array}{ccc} 90 & -55 & -75\\ \noalign{\medskip}-55 & 25 & 33\\ \noalign{\medskip}-75 & 33 & -115 \end{array}\right] (66)
It’s worth noting that, for the deviatoric stress matrix, the trace is always zero. The two similarity transformations are thus:
B' = \left[\begin{array}{ccc} 1 & 0 & 0\\ \noalign{\medskip}-75 & 33 & -115\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]\left[\begin{array}{ccc} 90 & -55 & -75\\ \noalign{\medskip}-55 & 25 & 33\\ \noalign{\medskip}-75 & 33 & -115 \end{array}\right]\left[\begin{array}{ccc} 1 & 0 & 0\\ \noalign{\medskip}{\frac{25}{11}} & 1/33 & {\frac{115}{33}}\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]=\left[\begin{array}{ccc} -35 & -5/3 & -{\frac{800}{3}}\\ \noalign{\medskip}2685 & 35 & 23964\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right] (67)
C' = \left[\begin{array}{ccc} 2685 & 35 & 23964\\ \noalign{\medskip}0 & 1 & 0\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]\left[\begin{array}{ccc} -35 & -5/3 & -{\frac{800}{3}}\\ \noalign{\medskip}2685 & 35 & 23964\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right]\left[\begin{array}{ccc} {\frac{1}{2685}} & -{\frac{7}{537}} & -{\frac{7988}{895}}\\ \noalign{\medskip}0 & 1 & 0\\ \noalign{\medskip}0 & 0 & 1 \end{array}\right]=\left[\begin{array}{ccc} 0 & 20714 & 122740\\ \noalign{\medskip}1 & 0 & 0\\ \noalign{\medskip}0 & 1 & 0 \end{array}\right] (68)
and the characteristic polynomial (missing the squared term, as expected)
$\lambda^3-20714\,\lambda-122740=0$ (69)
which yields eigenvalues/principal deviatoric stresses of 146.8, -5.94 and -140.86 kPa.
4.3. Numerical Implementation
Even with a “simple” problem such as this, the algebra of the solution can become very involved. Moreover, as is frequently the case with numerical linear algebra, there are two “loose ends” that need to be tied up: the division by zero problem in Danilevsky’s Method, and solving for the eigenvalues/principal stresses. A numerical implementation, shown below, will address both issues, if not completely.
The FORTRAN 77 code is in the appendix. It is capable of solving the problem both for standard and deviatoric stresses. The example problem given above is used. The steps used are as follows:
1. If deviatoric stresses are specified, the normal stresses are accordingly converted as shown earlier.
2. The invariants are computed. For a more general routine of Danilevsky’s Method, the matrices associated with the similarity transformations-and their inverses-would have to be computed and the matrix multiplications done. In this case the results of these multiplications-the invariants themselves-are “hard coded” into the routine, which saves a great deal of matrix multiplication. It also eliminated problems with division by zero up to this point.
3. The eigenvalues/principal stresses are computed using Newton’s Method. Obviously they could be computed analytically as shown earlier. For this type of problem, this is possible; for characteristic polynomials beyond degree four, another type of solution is necessary. The major problem with New-ton’s Method is picking initial values so as to yield distinct eigenvalues. The nature of the problem suggests that the initial normal, diagonal stresses can be used as starting values. For the sample problem this worked out very well; it was not tested on a variety of stress states.
4. Taking the three resulting eigenvalues, the eigenvectors were computed using Equation 61. Again the matrix multiplications were hard-coded into the routine. The problem of division by zero of certain values of shear stresses was not addressed. In some cases Equation 61 will result in division by zero.The eigenvectors were then normalized, which yielded three sets of direction cosines.
The results for the regular stresses are shown in Figure 3 and those for deviatoric stresses are shown in Figure 4.
Figure 3: Regular Normal Stress Results for Test Case
Figure 4: Deviatoric Normal Stress Results for Test Case
The principal stress results agree with the “hand” calculations earlier. The direction cosines-and thus the diagonalizing matrices-are the same for both cases.
5. Conclusion
We have explored the analysis of Mohr’s Circle for stress states in both two and three dimensions.The equations can be derived from strictly linear algebra considerations, the principal stresses being the eigenvalues and the direction cosines being the normalized eigenvectors. It was also shown that numericalmethods can be employed to analyze these results and the invariants, in this case using the computationally efficient Danilevsky’s Method.
Note: since we first published this on a companion site two years ago, we have made several improvements to the methods described here. These can be found in this post.
Appendix: Source Code for Numerical Implementation
! Routine to Determine Stress State from Danilevsky's Method
! Taken from Faddeeva (1959) with Example Case from Boresi et.al. (1993)
! Define Variables for Test Case
CHARACTER*12 filnam
sigmax=120.
sigmay=55.
sigmaz=-85.
tauxy=-55.
tauyz=33.
tauxz=-75.
! Set Variable for Eigenvalue Solution
! ityp = 0 Regular
! ityp = 1 Deviatoric
WRITE (6,*) 'Standard (0) or Deviatoric (1) Stress:'
IF (ityp.eq.0) filnam='danil0.txt'
IF (ityp.eq.1) filnam='danil1.txt'
! Open Output File
OPEN (7,file=filnam)
CALL danil (sigmax, sigmay, sigmaz, tauxy, tauyz, tauxz, ityp)
CLOSE (7)
STOP
END
! Subroutine danil -- Danilevsky's Method specifically written
! for 3 x 3 matrices and stress-state solution
SUBROUTINE danil (sigmax, sigmay, sigmaz, tauxy, tauyz, tauxz,
&ityp)
REAL p(3),x(3,3),sigma(3),i1,lambda
! Convert Normal Stresses to Deviatoric Stresses if ityp = 1
IF (ityp.eq.1) THEN
i1=0.3333333333*(sigmax+sigmay+sigmaz)
sigmax=sigmax-i1
sigmay=sigmay-i1
sigmaz=sigmaz-i1
END IF
! Compute Invariants
IF (ityp.eq.0) p(1)=sigmax+sigmay+sigmaz
IF (ityp.eq.1) p(1)=0.0
p(2)=-sigmax*sigmay-sigmax*sigmaz+tauxy**2+tauxz**2-sigmay*sigmaz+
&tauyz**2
p(3)=sigmax*sigmay*sigmaz-sigmax*tauyz**2+2*tauxy*tauxz*tauyz-
&tauxy**2*sigmaz-tauxz**2*sigmay
! Use Newton's Method to Determine Eigenvalues
DO 60 j=1,3,1
GO TO (10,20,30),j
10 lambda=sigmax
GO TO 40
20 lambda=sigmay
GO TO 40
30 lambda=sigmaz
40 DO 50 i=1,10,1
fun=lambda**2*p(1)-lambda**3+p(2)*lambda+p(3)
der=2*lambda*p(1)-3*lambda**2+p(2)
lambda=lambda-fun/der
delta=abs(fun/lambda)
IF (delta.lt.1.0e-5) GO TO 60
50 CONTINUE
60 sigma(j)=lambda
! Determine Eigenvectors
DO 80 i=1,3,1
lambda=sigma(i)
x(1,i)=-1/(-tauxz*sigmax*tauyz+tauxy*tauxz**2-tauxy*tauyz**2+
& tauyz*sigmay*tauxz)*tauyz*lambda**2+(tauxy*tauxz+sigmay*tauyz+
& sigmaz*tauyz)/(-tauxz*sigmax*tauyz+tauxy*tauxz**2-tauxy*tauyz**
& 2+tauyz*sigmay*tauxz)*lambda-(tauxz*tauxy*sigmaz-tauyz*tauxz**
& 2+tauyz*sigmay*sigmaz-tauyz**3)/(-tauxz*sigmax*tauyz+tauxy*
& tauxz**2-tauxy*tauyz**2+tauyz*sigmay*tauxz)
x(2,i)=-tauxz/tauyz*(-1/(-tauxz*sigmax*tauyz+tauxy*tauxz**2-
& tauxy*tauyz**2+tauyz*sigmay*tauxz)*tauyz*lambda**2+(tauxy*
& tauxz+sigmay*tauyz+sigmaz*tauyz)/(-tauxz*sigmax*tauyz+tauxy*
& tauxz**2-tauxy*tauyz**2+tauyz*sigmay*tauxz)*lambda-(tauxz*
& tauxy*sigmaz-tauyz*tauxz**2+tauyz*sigmay*sigmaz-tauyz**3)/(-
& tauxz*sigmax*tauyz+tauxy*tauxz**2-tauxy*tauyz**2+tauyz*sigmay*
& tauxz))+1/tauyz*lambda-sigmaz/tauyz
x(3,i)=1.0
! Normalise Eigenvectors
xnorm=1.0/sqrt(x(1,i)**2+x(2,i)**2+x(3,i)**2)
DO 70 j=1,3,1
70 x(j,i)=x(j,i)*xnorm
80 CONTINUE
! Write Results
WRITE (7,*) 'Stress State Using Danilevskys Method:'
IF (ityp.eq.0) WRITE (7,*) 'Regular Normal Stresses'
IF (ityp.eq.1) WRITE (7,*) 'Deviatoric Normal Stresses'
WRITE (7,*) 'Original Stress Matrix/Tensor:'
WRITE (7,100) sigmax,tauxy,tauxz
WRITE (7,100) tauxy,sigmay,tauyz
WRITE (7,100) tauxz,tauyz,sigmaz
WRITE (7,*) 'Invariants:'
WRITE (7,100) (p(j),j=1,3,1)
WRITE (7,*) 'Principal Stresses:'
WRITE (7,100) (sigma(j),j=1,3,1)
WRITE (7,*) 'Direction Cosines:'
DO 90 j=1,3,1
90 WRITE (7,110) (x(j,i),i=1,3,1)
100 FORMAT (3f15.2)
110 FORMAT (3f15.3)
RETURN
END
References
[1] Boresi, A. P., Schmidt, R. J., Sidebottom, O. M., 1993. Advanced Mechanics of Materials, 5th Edition.John Wiley & Sons, Inc.
[2] DeRusso, P., Roy, R., Close, C., 1965. State Variables for Engineers. John Wiley & Sons, Inc., New York,NY.
[3] Faddeeva, V., 1959. Computational Methods of Linear Algebra. Dover Publications, Inc.
[4] Jaeger, J., Cook, N., 1979. Fundamentals of Rock Mechanics, 3rd Edition. Chapman and Hall.
[5] Navy, U., 1986. Soil mechanics. Tech. Rep. DM 7.01.
[6] Owen, D., Hinton, E., 1980. Finite Elements in Plasticity: Theory and Practice. Pineridge Press, Swansea,Wales.
[7] Strang, G., 1993. Introduction to Linear Algebra. Wellesley-Cambridge Press.
[8] Verruijt, A., van Bars, S., 2007. Soil Mechanics. VSSD, The Netherlands.
Use of Netwon’s Method to Determine Matrix Eigenvalues
The problem here is to develop a routine that will determine one or more eigenvalues of a matrix using Newton’s method and considering the eigenvalue problem to be that of a nonlinear equation solution problem.
The simplest way to illustrate the problem and its solution is to use a 4 $\times$4 matrix.
Let us consider a square matrix $A$. The definition of an eigenvalue requires that
$Ax=\lambda x$
For our test case writing out the solutions for the left and right hand sides yields
$\left[\begin{array}{c} a_{{1,1}}x_{{1}}+a_{{1,2}}x_{{2}}+a_{{1,3}}x_{{3}}+a_{{1,4}}x_{{4}}\\ a_{{2,1}}x_{{1}}+a_{{2,2}}x_{{2}}+a_{{2,3}}x_{{3}}+a_{{2,4}}x_{{4}}\\ a_{{3,1}}x_{{1}}+a_{{3,2}}x_{{2}}+a_{{3,3}}x_{{3}}+a_{{3,4}}x_{{4}}\\ a_{{4,1}}x_{{1}}+a_{{4,2}}x_{{2}}+a_{{4,3}}x_{{3}}+a_{{4,4}}x_{{4}} \end{array}\right]=\left[\begin{array}{c} \lambda\, x_{{1}}\\ \lambda\, x_{{2}}\\ \lambda\, x_{{3}}\\ \lambda\, x_{{4}} \end{array}\right]$
The problem here is that, with the intrusion of the eigenvalue, we
have one more unknown than we have equations for. We can remedy this
by insisting that the values of $x$ be normalized, or
${x_{{1}}}^{2}+{x_{{2}}}^{2}+{x_{{3}}}^{2}+{x_{{4}}}^{2}=1$
Let us now construct a vector such that we have a closed system, all on the left hand side:
$\left[\begin{array}{c} a_{{1,1}}x_{{1}}+a_{{1,2}}x_{{2}}+a_{{1,3}}x_{{3}}+a_{{1,4}}x_{{4}}-\lambda\, x_{{1}}\\ a_{{2,1}}x_{{1}}+a_{{2,2}}x_{{2}}+a_{{2,3}}x_{{3}}+a_{{2,4}}x_{{4}}-\lambda\, x_{{2}}\\ a_{{3,1}}x_{{1}}+a_{{3,2}}x_{{2}}+a_{{3,3}}x_{{3}}+a_{{3,4}}x_{{4}}-\lambda\, x_{{3}}\\ a_{{4,1}}x_{{1}}+a_{{4,2}}x_{{2}}+a_{{4,3}}x_{{3}}+a_{{4,4}}x_{{4}}-\lambda\, x_{{4}}\\ {x_{{1}}}^{2}+{x_{{2}}}^{2}+{x_{{3}}}^{2}+{x_{{4}}}^{2}-1 \end{array}\right]=0$
Since the object of Newton’s method is to arrive at this, we need an intermediate vector $F$, or
$F=\left[\begin{array}{c} a_{{1,1}}x_{{1}}+a_{{1,2}}x_{{2}}+a_{{1,3}}x_{{3}}+a_{{1,4}}x_{{4}}-\lambda\, x_{{1}}\\ a_{{2,1}}x_{{1}}+a_{{2,2}}x_{{2}}+a_{{2,3}}x_{{3}}+a_{{2,4}}x_{{4}}-\lambda\, x_{{2}}\\ a_{{3,1}}x_{{1}}+a_{{3,2}}x_{{2}}+a_{{3,3}}x_{{3}}+a_{{3,4}}x_{{4}}-\lambda\, x_{{3}}\\ a_{{4,1}}x_{{1}}+a_{{4,2}}x_{{2}}+a_{{4,3}}x_{{3}}+a_{{4,4}}x_{{4}}-\lambda\, x_{{4}}\\ {x_{{1}}}^{2}+{x_{{2}}}^{2}+{x_{{3}}}^{2}+{x_{{4}}}^{2}-1 \end{array}\right]$
Let us make $\lambda$ the last “$x$” or
$\lambda=x_{n+1}$
The Jacobian of this vector with respect to the expanded $x$ vector is
J\left(F,x\right)=\left[\begin{array}{ccccc} a_{{1,1}}-\lambda & a_{{1,2}} & a_{{1,3}} & a_{{1,4}} & -x_{{1}}\\ \noalign{\medskip}a_{{2,1}} & a_{{2,2}}-\lambda & a_{{2,3}} & a_{{2,4}} & -x_{{2}}\\ \noalign{\medskip}a_{{3,1}} & a_{{3,2}} & a_{{3,3}}-\lambda & a_{{3,4}} & -x_{{3}}\\ \noalign{\medskip}a_{{4,1}} & a_{{4,2}} & a_{{4,3}} & a_{{4,4}}-\lambda & -x_{{4}}\\ \noalign{\medskip}2\, x_{{1}} & 2\, x_{{2}} & 2\, x_{{3}} & 2\, x_{{4}} & 0 \end{array}\right]
Now we can solve the Newton’s method equation iteratively:
$x_{m+1}=x_{m}-J\left(F,x\right)^{-1}F$
It is certainly possible to invert the Jacobian symbolically, but the algebra becomes very complicated, even for a relatively small matrix such as this one. A more sensible solution is to obtain numerical values for both Jacobian and $F$ vector, invert the former using the Gauss-Jordan technique and then multiply this by the $F$ vector before performing the Newton iteration.
Now let us consider the case of the tridiagonal matrix
A=\left[\begin{array}{cccc} 2 & -1 & 0 & 0\\ \noalign{\medskip}-1 & 2 & -1 & 0\\ \noalign{\medskip}0 & -1 & 2 & -1\\ \noalign{\medskip}0 & 0 & -1 & 2 \end{array}\right]
The eigenvalues of this matrix are
$\lambda=\frac{5}{2}\pm\frac{\sqrt{5}}{2},\frac{3}{2}\pm\frac{\sqrt{5}}{2}$
or numerically 3.618033989, 1.381966011, 2.618033989, and .381966011. The FORTRAN 77 code used for this is given at the end of the problem. Although the matrix is “hard coded” into the program, it is only necessary to change one parameter to change the matrix size.
Summary Results of Newton’s Method, 4×4 Matrix
There are two different quantities tracked here:
• The residual, i.e., the Euclidean norm of the entire $x$ vector.
• The error, i.e., the change in the eigenvalue from one step to the next.
Both these are tracked for three different starting places. For convenience,
all of the $x$ vector entries were initialized at the same value.
Some comments on the solution are as follows:
• As mentioned earlier, there were four eigenvalues to be determined. The method only returns one eigenvalue for each starting point, and that eigenvalue depends upon the starting point. The eigenvalues determined were 0.381966 (Start = 1 and Start = 2) and 2.6180339 (Start = 3.) Like the power method, this solution determines both an eigenvalue and eigenvector. Unlike the power method, it does not necessarily return the largest and/or smallest eigenvalue or even one easily predictable by the choice of starting value. With Newton’s Method, this is unsurprising; with multiple roots of an equation, which root is found very much depends upon the starting point. However, in the case where the number of roots is (in simple terms) the basic size of the matrix, this correspondence can become very complicated, and in some cases it is possible that certain eigenvalues will be missed altogether.
• As a corollary to the previous point, with some starting values the convergence almost comes apart, especially where Start = 3. With larger values than this, the program generates “nan” values and terminates abnormally. This result is a part of the method; poor initial estimates can led successive iterations “off the track” and thus fail to converge. Thus it is necessary to know the range of eigenvalues before one actually determines them, which to a great extent defeats the purpose of the method.
• The method is better at finding eigenvalues than finding eigenvectors. The residuals of the vector (which include the eigenvector plus the eigenvalue) converge much more slowly than the error. Those runs that were not limited by the termination criterion of the eigenvalue ($|\Delta\lambda|<1.0\times10^{-6}$) showed a very slow convergence continue with the eigenvectors. For the eigenvectors, the convergence rate is reasonable.
The general impression of this method, therefore, is that under proper circumstances it is capable of producing eigenvalues and (to a lesser extent) eigenvectors, but that it is necessary to a) have some idea of the values of the eigenvectors going in and b) be prepared for the method to collapse with the wrong initial guesses. One possible application (given the convergence limitations discussed above) is to use it in conjunction with, say, the Householder-Givens Method to determine the eigenvectors, which do not come out as a result of this method. The known eigenvalues give excellent starting values.
These results are confirmed when we expand the matrix to 10$\times$10.
Summary Results of Newton’s Method, 10×10 Matrix
We see that all of the errors and residuals go up and down from the initial guesses until convergence was achieved. The eigenvalues determined were 0.6902785 (Start = 1,) 1.7153704 (Start = 2,) and 1.7153703 (Start = 3.) Thus as before with three initial starting values only two eigenvalues were determined. The convergence was more lengthy than with the smaller matrix but not by much. This means that the method may be, to some extent, insensitive to matrix size, which would make it useful with larger matrices.
FORTRAN Code
The code calls several “standard” routines and includes whose significance is as follows:
• matsize.for is an include which includes a parameter statement for the actual matrix size. For the 10$\times$10 matrix, it read as follows:
• parameter (nn=10, nplus1=nn+1,mcycle=100)
• xmtinv is a subroutine to invert a matrix using Gauss-Jordan elimination.
• matvec performs matrix-vector multiplication (in that order).
• veclen computes the Euclidean norm of a vector.
c Determination of Eigenvalue(s) of Tridiagonal Matrix using
c Newton's Method
include 'matsize.for'
dimension a(nn,nn),x(nplus1),f(nplus1),fj(nplus1,nplus1),
&xt(nplus1),resid(mcycle),eigerr(mcycle)
call tstamp
c Define original matrix a
do 1 i=1,nn,1
do 1 j=1,nn,1
a(i,j)=0.0
if (i.eq.j) a(i,j) = 2.0
if (abs(i-j).eq.1) a(i,j) = -1.0
1 continue
open (unit=2,file='m5610p2d.csv')
write(2,*)'Original Matrix'
do 2 i=1,nn,1
2 write(2,3)(a(i,j),j=1,nn,1)
3 format(100(g10.3,1h,))
c Make initial guesses of eigenvalues and values of x
do 100 mm=1,3,1
vstart=float(mm)
write(2,*)'Eigenvalue Results for vstart =',vstart
do 4 i=1,nplus1,1
4 x(i)=vstart
c Cycle Newton's method
do 5 m=1,mcycle,1
eigold=x(nplus1)
c Compute Jacobian for current step
do 6 i=1,nplus1,1
do 6 j=1,nplus1,1
if(i.eq.nplus1.and.j.eq.nplus1) then
fj(nplus1,nplus1)=0.0
goto 6
endif
if(i.eq.j) then
fj(i,i)=a(i,i)-x(nplus1)
goto 6
endif
if(j.eq.nplus1) then
fj(i,nplus1)=-x(i)
goto 6
endif
if(i.eq.nplus1) then
fj(nplus1,j)=2.0*x(j)
goto 6
endif
fj(i,j)=a(i,j)
6 continue
c Output Jacobian
write(2,*)'Jacobian Matrix'
do 10 i=1,nplus1,1
10 write(2,3)(fj(i,j),j=1,nplus1,1)
c Compute Newton "F" vector
do 20 i=1,nplus1,1
if(i-nplus1)21,22,22
21 f(i)=-x(nplus1)*x(i)
do 23 j=1,nn,1
f(i)=f(i)+a(i,j)*x(j)
23 continue
goto 20
22 f(i)=-1.0
do 24 j=1,nn,1
24 f(i)=f(i)+x(i)**2
20 continue
c Invert Jacobian
call xmtinv(fj,nplus1,nplus1)
c Postmultiply Jacobian by f vector
call matvec(fj,f,xt,nplus1,nplus1)
c Update Value of x vector and output the result
write(2,*)'Result Vector for m = ',m
do 7 k=1,nplus1,1
x(k)=x(k)-xt(k)
7 write(2,*)x(k)
c Compute norm of residual and error
eigerr(m)=abs(x(nplus1)-eigold)
call veclen(xt,resid(m),nplus1)
if(resid(m).lt.1.0e-07.or.eigerr(m).lt.1.0e-07)goto 30
5 continue
c Output residual plot
30 write(2,*)'Residuals and Errors'
write(2,101)mm,mm
101 format(27hIteration,Residual (Start =,i2,
&16h),Error (Start = ,i2,1h))
do 31 i=1,m-1,1
31 write(2,32)i,resid(i),eigerr(i)
32 format(i3,2(1h,,g10.3))
100 continue
close(unit=2)
stop
end
Blessed are the Merciful
My first semester at Texas A&M University, I was required to take Analytical Geometry. My teacher was a former seminary student (come to think of it, so was my Calculus teacher!) Generally it wasn’t a difficult course but it had its moments.
One of them came with a particularly difficult problem we had for homework. I really didn’t know how to solve it, so I bluffed my way through it the best I could. At the top of the page I placed the following:
Needless to say he picked up on it immediately, and wrote “NO MERCY: 8.” Fortunately it was 8 out of 10; that result exceeded my expectations.
Evidently he was quite impressed with this show of Greek, so, in handing the papers back, he wrote my heading out on the board, pointing out that it appeared on my paper, and informing the class that he had in fact shown no mercy. One of the students, obviously unaware that any Aggie would know Greek, asked, “How did he manage to write that?”
“It was said by a very famous man,” the teacher replied.
That “very famous man,” of course, is Jesus Christ, and the passage in English is “Blessed are the merciful, for they shall obtain mercy.” (Matthew 5:7.) It is one of the Beatitudes, which open His Sermon on the Mount. The theme of being merciful and of forgiveness runs through the Gospel; in the next chapter, after the Lord’s Prayer He says again, “For and if ye shall forgive other men their trespasses, your father in heaven shall also forgive you. But and ye will not forgive men their trespasses, no more shall, your father forgive your trespasses.” (Matthew 6:14-15, Tyndale) Forgiveness is not just a nice thing to do; for the Christian, forgiveness is mandatory for eternal life.
And forgiveness is in short supply these days. Today we live in a bitter, divided society with a record incarceration rate and creeping euthanasia. Schools call the police for acts that, a generation or two ago, would occasion a call to the parents. Makes one think of Thomas Hobbes’ characterisation of life as “brutish and short.”
God’s standard for forgiveness–from Him and from us–has not changed. If we do not forgive, we are tormented in this life and the life to come. There are certainly earthly consequences for the things that people do, but these should not be confused with our response of forgiveness. And what others do should never obscure the need for us to seek forgiveness of our own sins from God. Our “zero-tolerance” society teaches us that no one is free from mistakes. But our God sent His Son to eradicate those mistakes and make us a way to eternal life.
Solving a Third-Order Differential Equation Using Simple Shooting and Regula Falsi
The object of this is to solve the differential equation
${\frac{d^{3}}{d{x}^{3}}}y(x)-\mu\,\left(1-\left(y(x)\right)^{2}\right){\frac{d^{2}}{d{x}^{2}}}y(x)+2\,\mu\, y(x)\left({\frac{d}{dx}}y(x)\right)^{2}+{\frac{d}{dx}}y(x)=0$
for the following boundary conditions and parameters:
• $\mu=\frac{1}{2}$
• $y\left(0\right)=0$
• ${\frac{d}{dx}}y(0)=\frac{1}{2}$
• $y\left(2\right)=1$
Conventional wisdom would indicate that, because of the high order of the derivatives, this problem cannot be solved using a scalar implementation of simple shooting. However, this is not the case.
To see why this is so, let us begin by implementing the following notation:
$\frac{d^{3}}{d{x}^{3}}y(x)=y_{4}\left(x\right)$
$\frac{d^{2}}{d{x}^{2}}y(x)=y_{3}\left(x\right)$
$\frac{d}{d{x}}y(x)=y_{2}\left(x\right)$
$y(x)=y_{1}\left(x\right)$
If we substitute these into the differential equation and solve for the third derivative, we have
$y_{4}\left(x\right)=\mu\,{\it y_{3}}(x)-\mu\, y_{3}(x)\left(y_{1}(x)\right)^{2}-2\,\mu\, y_{1}(x)\left(y_{2}(x)\right)^{2}-{\it y_{2}}(x)$
Now we can construct a series of first-order differential equations for our Runge-Kutta integration scheme as follows:
${\frac{d}{d{x}}}y_{1}(x)=y_{2}\left(x\right)$
${\frac{d}{d{x}}}y_{2}(x)=y_{3}\left(x\right)$
${\frac{d}{d{x}}}y_{3}(x)=\mu\,{\it y_{3}}(x)-\mu\, y_{3}(x)\left(y_{1}(x)\right)^{2}-2\,\mu\, y_{1}(x)\left(y_{2}(x)\right)^{2}-{\it y_{2}}(x)$
Now let us consider things at our first boundary point, namely
$y_{4}\left(0\right)=\mu\,{\it y_{3}}(0)-\mu\, y_{3}(0)\left(y_{1}(0)\right)^{2}-2\,\mu\, y_{1}(0)\left(y_{2}(0)\right)^{2}-{\it y_{2}}(0)$
Making the appropriate substitutions yields
$y_{4}\left(0\right)=1/2\,{\it y_{3}}(0)-1/2$
We thus see that, if we use ${\it y_{3}}(0)$ as our independent variable for shooting purposes, we also assume a value of ${\it y_{4}}(0)$ as well. Put another way, since we are given the dependent variable and its first derivative of the ODE at the first boundary, if we guess the second derivative the third derivative automatically follows in spite of the non-linearity of the problem. So we can use a scalar shooting scheme for this problem as well. The dependent variable for root-finding purposes is
$F=y_{1}\left(2\right)-y\left(2\right)$
which goes to zero as the far boundary condition is met.
To implement this we used a same simple shooting routine with a regula falsi root-finding technique. One thing that was varied was the number of integration steps in the interval; we wanted to see how this affected the convergence.
The results of this are summarized below.
The number of shooting iterations is fairly constant with the variation in integration steps; however, the final value of ${\it y_{3}}(0)$ seems to be refining itself until around 100 integration steps, at which point the accumulation of numerical error begins to affect the precision of the result.
A plot of the results is shown below.
FORTRAN 77 code is below. Some of the “includes”, subroutines and functions are as follows:
• matsize.for is a brief snippet of code which includes a parameter statement which sizes the variable size arrays.
• tstamp calls an OPEN WATCOM routine that returns the date and time and places it in the output.
• scrsho is a “line plotter” routine, very old school.
c Solution of Third-Order Ordinary Differential Equation
c Using Simple Shooting Method and Regula Falsi root-finding technique
c Solution based on Carnahan, Luther and Wilkes (1969)
include 'matsize.for'
parameter(neqs=3,nx=10)
character*40 vinput(nmax)
character*16 voutpt(nmax)
c Plot title
character*60 titlep
dimension y(neqs),dy(neqs),xyplot(0:nx,2)
c Define third derivative function
y4(y1,y2,y3,xmu)=xmu*y3-xmu*y3*y1**2-2.*xmu*y1*y2**2-y2
c Define second boundary condition function
bfin(y1,y1fin)=y1-y1fin
c Enter Boundary Conditions
data y1strt/0.0/,y2strt/0.5/,y1fin/1.0/xstrt/0.0/,xend/2.0/
c Enter initial upper and lower bounds for y(3) (Second Derivative)
data y3left/2.0/,y3rite/0.0/
c Enter number of shots
data n/50/
c Enter friction coefficient
data xmu/0.5/
c Enter nomenclature describing initial data as data statements
data vinput(1)/'Left Initial Bracket'/
data vinput(2)/'Right Initial Bracket'/
data vinput(4)/'Delta x increment'/
data vinput(5)/'First Boundary Condition'/
data vinput(6)/'Second Boundary Condition'/
data vinput(7)/'mu'/
data vinput(8)/'Number of Shots'/
c Enter nomenclature describing output for individual case
data vinput(11)/'Shot Number'/
data vinput(12)/'Value of Guess for Second Derivative'/
data voutpt(1)/'x'/
data voutpt(2)/'y(x)'/
data voutpt(3)/'y(x)'/
data voutpt(4)/'y(x)'/
data voutpt(5)/'y(x)'/
data titlep/'Plot of Final Boundary Condition vs. x'/
c Description of variables below the comment:
write(*,*)'Math 5610 Spring 2012 Final Exam'
write(*,*)'Simple Shooting Method for Third-Order ODE'
call tstamp
c Compute value of integration step size for dx
dx = (xend-xstrt)/float(nx)
write(*,*)
c Interval of uncertainty (left)
1 write(*,*)'Initial Parameters for Problem:'
write(*,200)vinput(1),y3left
c Interval of uncertainty (right)
write(*,200)vinput(2),y3rite
write(*,200)vinput(4),dx
write(*,200)vinput(5),xstrt
write(*,200)vinput(6),xend
write(*,200)vinput(7),xmu
write(*,200)vinput(8),n
do 21 iter=1,n,1
c ..... Set and print initial conditions
c ..... Since regula falsi isn't "self starting" it is necessary
c to compute the left and right values for y(3) (regula
c falsi independent variable) and compute the corresponding
c result for bfnact (regula falsi dependent variable)
c
c This is done in the first two iterations
c Iteration 1: Right Value
c Iteration 2: Left Value
c Susequent iterations adjust boundaries according to the
c method
x = xstrt
y(1)=y1strt
y(2)=y2strt
if(iter.eq.1)then
y(3) = y3rite
write(*,*)'Iteration for Right Estimate'
elseif(iter.eq.2)then
y(3) = y3left
write(*,*)'Iteration for Left Estimate'
else
y(3) = (y3left*bfinr-y3rite*bfinl)/(bfinr-bfinl)
endif
bfnold =bfnact
y3zero = y(3)
c Set print index to zero; print index changed at the first, middle
c and last iteration and printing/plotting takes place
write(*,*)
write(*,210)iter
write(*,200)vinput(1),y3left
write(*,200)vinput(12),y3zero
write(*,200)vinput(2),y3rite
write(*,*)
write(*,212)(voutpt(np),np=1,5,1)
write(*,*)
bfnact=bfin(y(1),y1fin)
write(*,202)x,y(1),y(2),y(3),bfnact
c ..... Set Selected variable for plot routine
xyplot(0,1)=x
xyplot(0,2)=bfnact
do 17 icycle=1,nx,1
c ..... Call Runge-Kutta Subroutine .....
8 if(irunge(neqs,y,dy,x,dx).ne.1)goto 10
dy(1)=y(2)
dy(2)=y(3)
dy(3)=y4(y(1),y(2),y(3),xmu)
goto 8
c ..... Print Solutions, Plot bfnact vs. x values .....
10 bfnact=bfin(y(1),y1fin)
write(*,202)x,y(1),y(2),y(3),bfnact
c ..... Set Selected variable for plot routine
nplot = icycle
xyplot(icycle,1)=x
xyplot(icycle,2)=bfnact
if(icycle.eq.nx)call scrsho(xyplot,nx,titlep)
17 continue
c ..... Ending routine if difference between current and past
c values of the boundary condition function are small enough
if(iter.eq.1)then
bfinr=bfnact
elseif(iter.eq.2)then
bfinl=bfnact
else
err = abs((bfnold-bfnact)/xmu)
if(err.lt.1.0e-06)goto 23
if(bfnact*bfinl.gt.0)then
y3left = y3zero
bfinl = bfnact
else
y3rite = y3zero
bfinr = bfnact
endif
endif
21 continue
23 continue
200 format(A40,2h ,g15.7)
202 format(1x,f7.4,2f16.7,2f16.8)
210 format(35hResults for Regula Falsi Iteration ,i2)
212 format(1x,5a16)
stop
end
function irunge(n,y,f,x,h)
c Fourth-order Runge-Kutta integration routine
c Taken from Carnahan, Luther and Wilkes (1969)
c
c The function irunge employs the fourth-order Runge-Kutta method
c with Kutta's coefficients to integrate a system of n simultaneous
c first-order ordinary differential equations f(j)=dy(j)/dx,
c (j=1,2,...,n), across one step of length h in the independent
c variable x, subject to initial conditions y(j), (j=1,2,...,n).
c Each f(j), the derivative of y(j), must be computed four times
c per integration step by the calling program. The function must
c be called five times per step (pass(1)...pass(5)) so that the
c independent variable value (x) and the solution values (y(1)...y(n))
c can be updated using the Runge-Kutta algorithm. M is the pass
c counter. Irunge returns as its value 1 to signal that all derivatives
c (the f(j)) be evaluated or 0 to signal that the integration process
c for the current step is finished. Savey(j) is used to save the
c initial value of y(j) and phi(j) is the increment function for the
c j(th) equation.
c
c The size of the savey and phi arrays depends upon the value of the parameter
c which is included.
include 'matsize.for'
dimension phi(nmax),savey(nmax),y(n),f(n)
data m/0/
m=m+1
goto(1,2,3,4,5), m
c ..... Pass 1 .....
1 irunge=1
return
c ..... Pass 2 .....
2 do 22 j=1,n,1
savey(j)=y(j)
phi(j)=f(j)
22 y(j)=savey(j)+h*f(j)/2.0
x=x+h/2.0
irunge=1
return
c ..... Pass 3 .....
3 do 33 j=1,n,1
phi(j)=phi(j)+2.0*f(j)
33 y(j) = savey(j)+0.5*h*f(j)
irunge=1
return
c ..... Pass 4 .....
4 do 44 j=1,n,1
phi(j)=phi(j)+2.0*f(j)
44 y(j)=savey(j)+h*f(j)
x=x+0.5*h
irunge=1
return
c ..... Pass 5 .....
5 do 55 j=1,n,1
55 y(j)=savey(j)+(phi(j)+f(j))*h/6.0
m=0
irunge=0
return
end
|
2018-12-14 06:44:51
|
{"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": 147, "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": 0.8403345942497253, "perplexity": 1642.8249714034025}, "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-2018-51/segments/1544376825363.58/warc/CC-MAIN-20181214044833-20181214070333-00453.warc.gz"}
|
http://physics.stackexchange.com/questions/56531/aharonov-casher-effect-for-charged-particles
|
# Aharonov-Casher effect for charged particles
All the explanations of the Aharonov-Casher effect seem to imply it only "works" for neutral particles with a magnetic moment. This seems to stem from the duality of the A-C effect with the more known Aharonov-Bohm effect. The particle being neutral is also mentioned in the original paper.
Is it strictly necessary to consider the A-C particle to be neutral? Can't the A-B and A-C phases just add up in the case of an electron in an electromagnetic field?
I know for sure that in e.g. the Pauli-Breit Hamiltonian (a non-relativistic limit of the Dirac Hamiltonian) contains a $\boldsymbol \mu\times\boldsymbol E$ A-C term, together with the $\boldsymbol A\cdot \boldsymbol p$ A-B term.
-
The Aharonov-Casher effect occurs due to a non-minimal interaction term of a spinning particle to an electromagnetic field. This interaction term can arise when the spinning particle is composite having an anomalous magnetic moment. In this case, even when the particle is neutral, it can interact with the electromagnetic field. The interaction is governed by the Pauli-Dirac equation:
$(\gamma^{\nu}\partial_{\nu}-\frac{\mu}{2}\sigma^{\alpha\beta}F_{\alpha\beta})\Psi = 0$
($\mu$ is the anomalous magnetic moment). But, if in addition the particle is charged, then it can couple minimally to the electromagnetic field in addition to possessing an anomalous magnetic moment and its equation of motion takes the form:
$(\gamma^{\nu}(\partial_{\nu}-eA_{\nu})-\frac{\mu}{2}\sigma^{\alpha\beta}F_{\alpha\beta})\Psi = 0$
Thus in principle, there can be two contributions to the topological phase, one proportional to the charge $e$ and one proportional to the anomalous magnetic moment $\mu$. However, the first contribution (proportional to $e$) is an ordinary Aharonov-Bohm phase.
Thus a composite charged particle can, in principle, have an Aharonov-Casher phase in adition to an Aharonov-Bohm phase.
It should be noticed that the conditions for the existence of the Aharonov-Casher phase are much more strict than that of the Aharonov-Bohm phase. In the case of the Aharonov-Bohm case, the particle needs to move in a force free region, i.e., in which the vector potential is locally a pure gauge $A_{\nu} = \partial_{\nu}\theta$, but its line integral over the close trajectory is nonzero due to non-simple connectedness of the configuration space. In the case of the Aharonov-Casher effect. The particle must be spinning; otherwise no anomalous magnetic moment can exist in the first place. Secondly, the trajectory must be two dimensional making the problem 2+1 dimensional. In this case only there exists a "dual potential"
$\tilde{A}^{\nu} =\epsilon^{\alpha\beta\nu}F_{\alpha\beta}$,
with respect to which the "Pauli-Dirac" equation has the form of a minimally coupled equation.
Update:
In relativistic mechanics, the anomalous magnetic moment couples differently from the standard magnetic moment. It's coupling involves the spin generators (cf. the anomalous magnetic moment term of the Dirac equation $\frac{\mu}{2}\sigma^{\alpha\beta}F_{\alpha\beta}\Psi$).
As a consequence, a scalar particle would not possess an anomalous magnetic moment. The standard magnetic moment stems from the usual minimal coupling term.
The origin of the anomalous magnetic moment is either the interaction of a fundamental particle with the quantized electromagnetic field, as in the case of the electron , or due to the internal motion of the constituents as in the case of the Proton and Neutron.
The anomalous magnetic moment adds up to the standard magnetic moment in the effect of spin precession in a magnetic field, but introduces a further interaction with the electromagnetic field proportional to the velocity of the particle (and the anomalous magnetic moment alone). This last contribution vanishes in the non-relativistic limit.
This is the reason that in the Aharonov-Casher article, in the first heuristic derivation of the interaction term they worked with a non-relativistic model, thus, the two magnetic moments appeared additively. But in their second derivation, they used the anomalous magnetic moment term. I think that they did not emphasize the fact that this interaction term is solely due to the anomalous magnetic moment because they referred to a neutral particle not possessing a standard magnetic moment.
-
If I understand correctly, you say the A-C effect "works" only on particles with anomalous magnetic moment, but reading the original A-C paper it seems any magnetic moment will do. Which is it? – rubenvb Mar 28 '13 at 14:36
@rubenvb:I have added an update answering your question. – David Bar Moshe Mar 31 '13 at 10:40
Thanks for clearing up my confusion! – rubenvb Mar 31 '13 at 11:38
|
2015-07-28 08:43:37
|
{"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": 0.903939425945282, "perplexity": 363.9973810049341}, "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-2015-32/segments/1438042981753.21/warc/CC-MAIN-20150728002301-00293-ip-10-236-191-2.ec2.internal.warc.gz"}
|
https://math.stackexchange.com/questions/1305097/geometrical-interpretation-of-sum-k-1n-k2-sum-k-1n-k3
|
Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$
Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those formulas I don't see why they should be equal.
The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)
There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.
• Who are they? Add a reference. – lhf May 30 '15 at 10:13
page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof
This identity is sometimes called Nicomachus's theorem. If you type this in google, you'll recieve numerous pictures.
1.
2.
3.
An engineering style fourier-approach. You may not consider this to be very intuitively "geometric", but I thought it could be interesting however.
Consider the time-signal $[1,2,...,k]$ (A linear function or "triangle wave"). The LHS is the square of the DC component of the signal in the temporal/spatial domain which is straight forward to calculate. The RHS is the iterated convolution of the fourier transform of [1,2,3,...,k] three times and then DC component of that.
• Could you please take a look at this – Aditya Hase Sep 18 '15 at 1:09
|
2020-02-20 12:05:29
|
{"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": 0.8527733087539673, "perplexity": 224.133062377208}, "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-2020-10/segments/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00375.warc.gz"}
|
https://www.nature.com/articles/s41598-022-12726-z
|
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Methodology for the determination of human respiration rate by using Doppler radar and Empirical Modal Decomposition
## Abstract
In this work, a methodology is presented for the determination of the respiration rate of a person under test (PUT), the detection of movements, as well as the elimination of the spurious effects produced by the movements of the PUT. The methodology is based on Empirical Modal Decomposition (EMD) applied to the phase signal obtained by means of a quadrature Doppler radar operating in S band. The EMD allows to automatically eliminate the continuos component (CC) which is present in the phase signal since one of the main characteristics of the modes generated by the EMD is that its mean is equal to zero. On the other hand, the first mode of the EMD is used for the detection of movements while the sum of the second and third modes are used for the elimination of the CC drift caused by the DC drift and the high frequency components produced by the movements of the PUT. The proposed methodology was successfully tested in a PUT at rest and performing movements of the head, arm and combination of head, arm, and torso. The average respiration rate measured was 20.78 breaths / min with a standard deviation of 2.53 breaths/min.
## Introduction
In the 70's, Doppler radar began to be used for measure respiration rate of people. Since then, numerous studies have been developed for medical applications. Doppler radar offers the possibility of monitoring respiration rate without the need to have contact with the patient, which turns out to be a great tool for all medical personnel since their exposure to sources of contagion is reduced1.
A Doppler radar system generally consists of a transmitting antenna emitting an electromagnetic wave to the chest of the PUT which modulates the signal and reflects it towards the receiving antenna. The receiving antenna leads the modulated signal to a microwave mixer to convert the received signal to intermediate frequencies (IF) which are treated at the Digital Signal Processing (DSP) stage [see Fig. 1].
Relevant challenges of Doppler radars are the null point, the DC offset, the DC drift, the phase unwrapping, the problem of separating the heart signal from the breathing signal and the problem of breathing signal harmonics. The null point problem can be solved by using an RF phase shifter2. The DC offset can be treated by using a Doppler radar architecture fed with a stepped chirp signal3. Curve fitting can be used to separate the cardiac signal from the breathing signal4 as well as remove harmonic from the breathing signal4.
On the other hand, the DC drift issue consists of low-frequency (LF) components added to signals measured at Doppler radar. These LF components are mainly caused by both RBM and the temperature variations of radar electronics circuits. It is important to mention that the DC drift must be eliminated because it can generate errors in the breathing rate measurements. The DC drift can be removed by using a DC correction algorithm, which divides the signal with DC drift into sections and removes DC present in each of them11, polynomial regression12, detrending process13.
In this work, a quadrature Doppler Radar is implemented to measure the respiration rate of people. Empirical Modal Decomposition is an adaptive technique for the analysis of non-linear and non-stationary processes, which makes it ideal for the detection of disturbances generated by random movements, due to its non-linear and non-stationary nature. In this work, a methodology is proposed for the detection of movements of the PUT and for the mitigation of their effects in determining the respiration rate.
The proposed methodology was successfully verified in experiment for the determination of the respiration rate of a PUT, the detection of movements, as well as the elimination of the spurious effects produced by the movements of the PUT. The methodology is based on EMD applied to the phase signal obtained by means of a quadrature Doppler radar operating in S band. The EMD allows to automatically eliminate the CC which is present in the phase signal since one of the main characteristics of the modes generated by the EMD is that its mean is equal to zero. On the other hand, the first mode of the EMD is used for the detection of movements while the sum of the second and third modes are used for the elimination of the CC drift and the high frequency components produced by the movements of the PUT. The proposed methodology was successfully tested in a PUT at rest and performing movements of the head, arm and combination of head, arm and torso. The average respiration rate measured was 20.78 breaths / min with a standard deviation of 2.53 breaths / min. These values were compared with a reference value of 19 breaths / min obtained through the technique of measuring the number of chest lifts of the volunteer for 1 min. It is worth mentioning that all respiration rate values fall within the respiration range of a healthy adult, which is 15 to 20 breaths / min with a range of 24 to 28 breaths / min14.
## Implementation of an S-band Doppler radar
The diagram of the S-band quadrature Doppler radar used in this work is shown in Fig. 1. From this Figure, Tx and Rx are the transmitting and receiving antenna, respectively; Gen is a continuous wave generator, DV1 and DV2 are Wilkinson power dividers, H1 and H2 are hybrid quadrature couplers, and M1 and M2 are microwave mixers. The radar operation is as follows: the microwave signal (generated by Gen) is divided by DV1 into two signals of equal amplitude and phase. The resulting signal S1 goes to the transmitting antenna (Tx) to radiate the PUT’s chest. The signal bounces off his chest and, receiving antenna captures its reflections modulated by the PUT breathing. The received signal in Rx is divided into two signals of equal amplitude and phase using the DV2 divider. An H1 hybrid coupler separates the resulting signal S2 coming out of DV1 into two signals with equal amplitude and a 90 degrees phase difference. On the heterodyne stage, the emerging signals of the hybrid coupler and the emerging DV2 signals are mixed, obtaining the IF signals, harmonics, etc. Afterwards, the IF signals are digitized by an Analog to Digital Converter (ADC) contained on the Keysight U2702A card, a sampling rate of 25 Hz was used. Finally, the data is sent to the PC and processed in MATLAB ®.
The Ettus USRP B200 was used as a continuous wave generator Gen to generate frequencies from 70 MHz to 6 GHz. A pair of ZFSC-2-10G model Mini-Circuit’s power splitters that operate from 2 to 10 GHz were used as DV1 and DV2. The AMP M/A -COM 96,341 hybrid coupler operating in the frequency range from 2 to 18 GHz was used to obtain quadrature signals in H1. Two logarithmic antennas WA5VJB operating from 850 MHz to 6.5 GHz with a typical gain of 6 dBi, were used for the transmission and reception of the microwave signal.
The ZEM-4300 model Mini Circuits’s mixers operating with RF signals from 300 MHz to 4.3 GHz were used to obtain in phase ($$B_{I}$$) and quadrature ($$B_{Q}$$) IF signals. Both signals can be modeled by Eq. (1) and Eq. (2)
$$B_{I} \left( t \right) = DC_{I} + A_{R} \cos \left( {\frac{{4\pi d_{0} }}{\lambda } + \frac{4\pi x\left( t \right)}{\lambda } + {\Delta }\varphi \left( t \right)} \right)$$
(1)
$$B_{Q} \left( t \right) = DC_{Q} + A_{R} \sin \left( {\frac{{4\pi d_{0} }}{\lambda } + \frac{4\pi x\left( t \right)}{\lambda } + {\Delta }\varphi \left( t \right)} \right)$$
(2)
where x(t) is the function of periodic object movement, d0 is the nominal distance the object, λ is the wavelength of the signal transmitted by the radar, $$\Delta \varphi$$(t) is the phase noise, t is the time, AR is the amplitude each IF signal and DCI and DCQ are amounts of DC offset caused by self-mixing of the mixers and by the reflection of static objects that is around the radar15.
From Eqs. (1) and (2), the Arctangent Demodulation, given in Eq. (3), can be obtained, which allows to measure the breathing and RBM elimination by digital processing of $${\Phi }\left( {\text{t}} \right)$$. It is important to mention that the DCI and DCQ components must be calculated before applying the arctan function; this procedure is shown in the next section.
$${\Phi }\left( t \right) = \arctan \left( {\frac{{B_{Q} - DC_{Q} }}{{B_{I} - DC_{I} }}} \right) = \frac{{4\pi d_{0} }}{\lambda } + \frac{4\pi x\left( t \right)}{\lambda } + {\Delta }\varphi \left( t \right)$$
(3)
Finally, Fig. 1 shows the experimental set up in this study, which involves the detection of RBM and the elimination of its effects in the measurement of the respiration rate of a healthy male of 28 years, 80 kg of weight and a height of 1.78 m, lying down at a distance of 29 cm from the radar. The RBMs studied in this work are head, arm and torso movements.
## Proposed methodologies
### Proposed methodology to obtain the person’s breathing rate from signals affected by RBM
Figure 2 shows the proposed methodology for RBM detection. In the beginning $$B_{I}$$ and $$B_{Q}$$ are measured by the ADC. The second stage involves the use of a 100-order low pass filter and 6 Hz cutoff frequency to eliminate high-frequency noise of BI and BQ without attenuating disturbances generated by RBM.
The third stage is to eliminate the DC offset, which is done by estimating the center of the arc formed by $$B_{I}$$ and $$B_{Q}$$, the center of which corresponds to the coordinates ($$DC_{I} , DC_{Q}$$). For center estimation, the Levenberg–Marquardt (LM) algorithm is used to fit the measured data to Eq. (4)16.
$$\left( {B_{I} - DC_{I} } \right)^{2} + \left( {B_{Q} - DC_{Q} } \right)^{2} = \left( {A_{R} } \right)^{2}$$
(4)
In the fourth stage, the phase signal given by Eq. (5) is obtained, which is based on the DACM algorithm17. This algorithm solves the phase unwrapping problem caused by the undefinition of the arctan function when BI – DCI = 0.
$${\Phi }\left[ n \right] = \mathop \sum \limits_{k = 2}^{n} \frac{{B_{I} \left[ k \right]\left\{ {B_{Q} \left[ k \right] - B_{Q} \left[ {k - 1} \right]} \right\} - \left\{ {B_{I} \left[ k \right] - B_{I} \left[ {k - 1} \right]} \right\}B_{Q} \left[ k \right]}}{{\left( {B_{I} \left[ k \right]} \right)^{2} + \left( {B_{Q} \left[ k \right]} \right)^{2} }}$$
(5)
In the fifth stage, the EMD is applied to the phase signal given in Eq. (5) to obtain the intrinsic modes functions of the phase signal. In this work the EMD is performed by using the algorithm proposed in (ref18) which is an iterative and adaptive method which was created for the analysis of nonlinear and non-stationary processes. The EMD operates under the hypothesis that any signal is composed of simple intrinsic modes of oscillation. The goal of EMD is to breakdown the original signal into its intrinsic modes functions (IMFs) through a process called sifting. Once the EMD is applied to a signal, it can be expressed as the sum of all the modes generated by the EMD and the residue r18. It is important to mention that the EMD automatically removes the DC component since one of the features of the IMFs is that they must have zero mean.
The sixth stage calculates the maximum amplitude Amax with Eq. (6) where IMF1 is the first mode of EMD which contains the oscillations caused by RBM. The experiment found that when there are disturbances in the radar signal, they are manifested in IMF1.
$$A_{max} = max\left( {IMF1} \right) - min\left( {IMF1} \right)$$
(6)
Stage 7 alerts of possible RBM signals when $$A_{max} > v_{u}$$, where vu is the threshold value defined as the maximum amplitude of IMF1 in the absence of RBM. This value is set by conducting several experiments with PUT at rest.
### Methodology for the RBM elimination and their effects on obtaining breathing rate
As mentioned before, RBM produces spurious signals at high- and low-frequencies; low-frequency ones are the cause of the undesired effect known as CC drift in the phase signal. The first stage of the proposed methodology for RBM elimination is based on the IMFs obtained in the EMD. The proposed methodology defines the spurious-free signal (IMsf) as the sum of the second mode (IMF2) and the third mode (IMF3) of the EMD. The spurious-free signal does not consider the first mode or modes greater than 3 since the goal is to eliminate the CC drift and high frequency components that are contained in the first mode.
In the second stage, the signal IMsff is obtained by filtering IMsf by means of a 20-order low pass filter and 1 Hz cutoff frequency to eliminate components outside the breathing range.
Finally, the IMsff spectrogram is obtained by using Short-time Fourier transform (STFT) to estimate the breathing rate of PUT at all times.
## Results
In this section, we show the results obtained by implementing the methodologies proposed in the previous section where BI and BQ signals are obtained with the radar operating at 4 GHz.
### Experimental results of the methodology to obtain the person’s breathing rate from signals affected by RBM
In the first stage, the capturing of BI and BQ signals were performed, subsequently, the low pass filter was implemented to eliminate high-frequency noise of BI and BQ. In the third stage, the DC offset was removed and the DACM algorithm was subsequently implemented to perform the demodulation process. Finally, the EMD, code available in (ref19), was applied to the phase signal.
Once the EMD method was applied to the signal phase and following the sixth stage of the proposed methodology, the RBM detection process was carried out. For this, the $$A_{max}$$ of the IMF1 is calculated and compared to $$v_{u} = 0.7$$ to decide whether movement exists. Figure 3a shows the phase signal for two cases where there is a head (H) movement, two cases where there is an arm (A) movement, and one case in which a movement combination of arms, legs, and torso (M) was carried out. Figure 3b shows the IMF1 of the EMD for each of the movements performed by the PUT. Figure 3b shows that IMF1 contains enough information to detect any RBM made by the PUT. It is also important to mention that $$v_{u} = 0.7$$ value was obtained by 30 measurements with the PUT at rest; 0.7 being the maximum value obtained from the measurements.
### Experimental results of both measure breathing rates methodology and CC-drift/RBM elimination methodology
This methodology uses the IMFs obtained with the EMD. Figure 4a and b show the raw experimental results of the phase signal and spectrogram obtained from the person under test (PUT) at rest. In Fig. 4a the presence of CC can be observed, which is most noticeable in the spectrogram plotted in Fig. 4b. As it can be seen, there is a concentration of power around the frequency 0 Hz at any moment of time. Figure 4c and d show the experimental results of the phase signal and spectrogram corrected with the proposed methodology. Figure 4c shows that the corrected signal does not have a CC, confirmed by its spectrogram shown in Fig. 4d. In this Figure, the components around the frequency 0 Hz have low power while there is an important power spectral density (PSD) magnitude around 0.35 Hz, corresponding to 21 breaths/min.
Figure 5 shows the results obtained after two experiments in which the PUT moves the head perpendicular to its chest. Figure 5a shows that before correction, there is a significant PSD magnitude around the frequency 0 Hz, and at 15 s, the head movement carried out adds CC drift to the phase signal; this can be observed in the spectrogram as an increase in the low-frequency PSD. It is important to mention that in the same spectrogram can be seen the generation of spurious at high frequencies, which are due to the random nature of head movement. Following the application of the proposed methodology, the spectrogram in Fig. 5b shows the mitigation of spurious signals (CC, CC drift and high frequency signals) and that the breathing rate is around 0.35 Hz, which corresponds to 20.82 breaths/min.
On the other hand, Fig. 5c and d show the spectrogram before and after correction, which are the result of the second head movement experiment. The PSD around the frequency of 0 Hz can be observed in the spectrogram in Fig. 5c. In addition, approximately after 25 s, there is a head movement, which adds CC drift and spurious of considerable power. Applying the proposed methodology, we obtain the spectrogram in Fig. 5d, which clearly shows the mitigation of spurious signals and a breathing rate of around 0.30 Hz which corresponds to 18.19 breaths/min.
Figure 6 shows the results of two experiments when PUT performs movement in the left arm until it is perpendicular to its chest. Figure 6a shows the spectrogram of the raw phase signal where it is always observed that there is a concentration of power around the frequency 0 Hz; In addition, approximately, when time is equal to 15 s, there is arm movement, which generates CC drift and spurious at high frequencies. On the other hand, Fig. 6b shows that the proposed methodology significantly mitigates spurious signals and that the breathing rate is around 0.39 Hz which corresponds to 23.44 breaths/min.
Similarly, Fig. 6c shows the spectrogram of the raw phase signal for the second experiment. As in the other cases, before correction, there is a high concentration of power at the frequency of 0 Hz, in addition, around 14 s there is CC drift and high frequency spurious caused by the movement of the arm. By applying the proposed methodology, the spectrogram in Fig. 6d was obtained, which shows the mitigation of spurious signals and a breathing rate of 0.29 Hz which corresponds to 17.62 breaths/min.
Finally, Fig. 7 shows the results of an experiment where PUT performs combined movements of arms, legs, and torso. Figure 7a shows the raw phase signal where movements are observed at 22 s. The spectrogram in Fig. 7b shows a high concentration of power at frequency 0 Hz, as well as the appearance of CC drift and spurious at high frequencies. On the other hand, Fig. 7c shows the IMsff signal where the proposed methodology eliminates CC, CC drift and spurious at high frequencies; the above is shown in the spectrogram in Fig. 7d, which made it possible to measure a breathing rate of 0.39 Hz which corresponds to 23.63 breaths/min.
Finally, when comparing our work with the work presented in (ref9) we find that both works are similar in the sense that both use EMD. However, there are significant differences. In (ref9) the authors are focused on the elimination of small random movements due to artifacts (the radar antenna) while in our work the proposed methodology is focused on the elimination of large random movements (head, arm, and torso). Another big difference is that in (ref9) the heart rate is calculated while in the present work the respiration rate is obtained. It is important to mention that the problem of DC drift is omitted in (ref9) because the antenna movements are so small that they do not add a significant DC drift. However, in our work the problem of DC drift must be considered because the movements produce significant DC drift in the radar outputs, which in turn causes the appearance of continuous component (CC) drift in the phase signal. In this work the problem of the CC drift is solved through the calculation of the first modes thrown by the EMD since being the CC drift a low frequency signal will appear in higher order modes than those calculated. Additionally, the proposed methodology in (ref9) and the proposed in this work are different because the modes of interest are not the same due to the nature of the signals and the nature of the movements.
## Conclusion
A new methodology for the determination of the respiration rate of a PUT, the detection of movements, as well as the elimination of the spurious effects produced by the movements of the PUT was proposed. The methodology is based on EMD applied to the phase signal obtained by means of a quadrature Doppler radar operating in S band. The EMD allows to automatically eliminate the CC which is present in the phase signal since one of the main characteristics of the modes generated by the EMD is that its mean is equal to zero. The first mode of the EMD is used for the detection of movements while the sum of the second and third modes are used for the elimination of the CC drift and the high frequency components produced by the movements of the PUT. The proposed methodology was successfully tested in a PUT at rest and performing movements of the head, arm and combination of head, arm, and torso.
## References
1. Mohammadzadeh, N., Gholamzadeh, M., Saeedi, S. et al. The application of wearable smart sensors for monitoring the vital signs of patients in epidemics: a systematic literature review. J. Ambient Intell. Human. Comput. (2020).
2. Pan, W., Wang, J., Huangfu, J., Li, C. & Ran, L. Null point elimination using RF phase shifter in continuous-wave Doppler radar system. Electron. Lett. 47, 1196–1198 (2011).
3. Kim, D. K. & Kim, Y. Quadrature frequency-group radar and its center estimation algorithms for small vibrational displacement. Sci Rep 9, 6763 (2019).
4. Yang, Z. K., Zhao, S., Huang, X. D. & Lu, W. Accurate Doppler radar-based heart rate measurement using matched filter. IEICE Electr. Exp. 17, 20200062 (2020).
5. Li, C. & Lin, J. Random body movement cancellation in Doppler radar vital sign detection. IEEE Trans. Microw. Theory Tech. 56, 3143–3152 (2008).
6. Gu, C., Wang, G., Li, Y., Inoue, T. & Li, C. A hybrid radar-camera sensing system with phase compensation for random body movement cancellation in Doppler vital sign detection. IEEE Trans. Microw. Theory Tech. 61, 4678–4688 (2013).
7. Lv, Q., Dong, Y., Sun, Y., Li, C., & Ran, L. Detection of bio-signals from body movement based on high-dynamic-range Doppler radar sensor. In 2015 IEEE MTT-S 2015 International Microwave Workshop Series on RF and Wireless Technologies for Biomedical and Healthcare Applications (IMWS-BIO), pp. 88–89 (2015).
8. Sekak, F. et al. Cyclostationary-based vital signs detection using microwave radar at 2.5 GHz. Sensors 20, 3396 (2020).
9. Mostafanezhad, I., Yavari, E., Boric-Lubecke, O., Lubecke, V. M. & Mandic, D. P. Cancellation of unwanted Doppler radar sensor motion using empirical mode decomposition. IEEE Sens. J. 13, 1897–1904 (2013).
10. Cardillo, E., Li, C. & Caddemi, A. Vital sign detection and radar self-motion cancellation through clutter identification. IEEE Trans. Microw. Theory Tech. 69, 1932–1942 (2021).
11. Li, Y., Wang, G., Gu, C., & Li, C. Movement-immune respiration monitoring using automatic DC-correction algorithm for CW doppler radar system. In 2014 IEEE Topical Conference on Biomedical Wireless Technologies, Networks, and Sensing Systems (BioWireleSS), 7–9 (2014).
12. Massagram, W., Hafner, N., Lubecke, V. & Boric-Lubecke, O. Tidal volume measurement through non-contact Doppler radar with DC reconstruction. IEEE Sens. J. 13, 3397–3404 (2013).
13. Zhao, H., Gao, X., Jiang, X., Hong, H., & Liu, X. Non-contact Robust Respiration Detection By Using Radar-Depth Camera Sensor Fusion. In 2020 42nd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC), 4183–4186 (2020).
14. Vitales, S. Mediciones de Signos Vitales. Available https://www.pisa.com.mx/publicidad/portal/enfermeria/manual/4_1_1.htm (2011).
15. Boric-Lubecke. et al. Doppler radar architectures and signal processing for heart rate extraction. Mikrotalasna revija15, 12–17 (2009).
16. Zakrzewski, M. Methods for Doppler radar monitoring of physiological signal, (2015).
17. Gu, C. Short-range noncontact sensors for healthcare and other emerging applications: A review. Sensors 16, 1169 (2016).
18. Ramírez-Castro, R. I., & Montejo, L. A. Transformada de Hilbert, descomposición modal empírica y sus aplicaciones en el análisis de vibraciones libres. Revista Internacional de Desastres Naturales, Accidentes e Infraestructura Civil, 11, (2011).
19. Empirical Mode Decomposition, [Online]. Available http://perso.ens-lyon.fr/patrick.flandrin/emd.html (2007).
## Author information
Authors
### Contributions
M. H. contributed to the proposed methodology, did experimental measurements, and drafted the manuscript. J. O. planned the experiment and wrote relevant text of the manuscript. A. P. planned the experiment, got financial support, and reviewed the manuscript. A. C. interpreted the results and drafted the manuscript.
### Corresponding author
Correspondence to Jose-Luis Olvera-Cervantes.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
### Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Hernandez-Aguila, M., Olvera-Cervantes, JL., Perez-Ramos, AE. et al. Methodology for the determination of human respiration rate by using Doppler radar and Empirical Modal Decomposition. Sci Rep 12, 8675 (2022). https://doi.org/10.1038/s41598-022-12726-z
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/s41598-022-12726-z
|
2022-08-09 19:16:20
|
{"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": 2, "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": 0.6611281037330627, "perplexity": 1512.7857571726727}, "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/1659882571086.77/warc/CC-MAIN-20220809185452-20220809215452-00133.warc.gz"}
|
https://www.vedantu.com/question-answer/equation-of-a-common-tangent-to-the-curves-y2-class-11-maths-cbse-5edcbad64d8add132469c8e5
|
Question
# Equation of a common tangent to the curves ${y^2} = 8x$and $xy = - 1$ is ${\text{A}}{\text{. }}3y = 9x + 2 \\ {\text{B}}{\text{. }}y = 2x + 1 \\ {\text{C}}{\text{. }}2y = x + 8 \\ {\text{D}}{\text{. }}y = x + 2 \\$
Hint:-Here, we write the equation of tangent to the parabola in slope form and then find the value of $m$ to get the equation of tangent.
Given equation of parabola ${y^2} = 8x$
And general form of parabola is ${y^2} = 4ax$
By comparing the equation we get $a = 2$
Equation of a tangent to parabola ${y^2} = 8x$ is
We know that equation of tangent of parabola in the form of slope is $y = mx + \frac{a}{m}{\text{ }}$
Put the value of $a$ in a tangent equation we get $y = mx + \frac{2}{m}{\text{ }}......{\text{(i) }}$
Now solving $({\text{i)}}$with $xy = - 1$
$x\left( {mx + \frac{2}{m}} \right) = - 1 \\ \Rightarrow m{x^2} + \left( {\frac{2}{m}} \right)x + 1 = 0 \\$
Now for the tangent to the discriminant of the above quadratic should be zero because we know for the tangent it must touch at point only. By making discriminant equal to zero we only get one point. And this is the equation of tangent.
${\left( {\frac{2}{m}} \right)^2} - 4m = 0 \\ \Rightarrow 4 - 4{m^3} = 0 \\ \Rightarrow {m^3} - 1 = 0 \\ \Rightarrow {m^3} = 1{\text{ }} \\$
$\therefore m = 1$ Only real solution
Now put the value of $m$ in equation $({\text{i)}}$
Hence required common tangent is $y = x + 2$
Here option D is the correct answer.
Note: - Whenever we face such a type of question we have to assume the tangent equation in slope form and equate it with the given equation to find the value of slope that we assumed in the tangent equation. And then by putting the value of slope we get the required tangent.
|
2021-05-06 12:30:29
|
{"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": 2, "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": 0.7472847700119019, "perplexity": 173.81411691253027}, "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-2021-21/segments/1620243988753.97/warc/CC-MAIN-20210506114045-20210506144045-00161.warc.gz"}
|
http://math.stackexchange.com/questions/245327/weak-law-of-large-numbers-for-dependent-random-variables-with-bounded-covariance
|
# Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance
I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement:
• Let $(X_n)$ be a sequence of dependent identically distributed random variables with finite variance.
• Let $\displaystyle S_n = \sum_{i=1}^n X_i$ denote the $n^\text{th}$ partial sum of the random variables $(X_n)$.
• Assume that Cov$(X_i,X_j) \leq c^{|i-j|}$ for $i, j \leq n$ where $|c| \leq 1$.
Is it possible to show that $\displaystyle \frac{S_n}{n} \rightarrow \mathbb{E}[X_1]$ in probability? In other words, is it true that given any $\epsilon>0$,
$$\lim_{n\rightarrow \infty} \mathbb{P}\bigg[\Big|\frac{S_n}{n} - \mathbb{E}[X_1]\Big| > \epsilon\bigg] = 0$$
EDIT: Following some comments, it turns out that I had the right approach so I've gone ahead and answered my own question below.
-
I should add that I've tried using the Chebyshev inequality, but can't get the right kind of bound - so I suspect that there must be another way. – Elements Nov 26 '12 at 23:44
no, that's right, you should be able to show $\sigma^2(\frac { S_n} n) \rightarrow 0$ – mike Nov 26 '12 at 23:55
@mike Hmm I suspect I may not be using the right bounds... See above – Elements Nov 27 '12 at 0:31
Fix $\epsilon > 0$ and $n \in \mathbb{N}$, then we can use Chebyshev's inequality to see that
$$\mathbb{P}\bigg[\Big|\frac{S_n}{n} - \mathbb{E}[X_1]\Big| > \epsilon\bigg] \leq \frac{\text{Var}\Big(\frac{S_n}{n}\Big)}{\epsilon^2}$$
where
$$\displaystyle \text{Var}\Big(\frac{S_n}{n}\Big)= \frac{\text{Var}(S_n)}{n^2} \leq \frac{\sum_{i=1}^n\sum_{j=1}^n \text{Cov}{(X_i,X_j)}}{n^2} \leq \frac{\sum_{i=1}^n\sum_{j=1}^n c^{|i-j|}}{n^2}$$
We can then explicitly calculate the double sum $\sum_{i=1}^n\sum_{j=1}^n c^{|i-j|}$ as follows:
\begin{align} \sum_{i=1}^n\sum_{j=1}^n c^{|i-j|} &= \sum_{i=1}^n c^{|i-i|} + 2\sum_{i=1}^n\sum_{j=1}^{i-1} c^{|i-j|} \\ &= n + 2\sum_{i=1}^n\sum_{j=1}^{i-1} c^{|i-j|} \\ &= n + 2\sum_{i=1}^n\sum_{j=1}^{i-1} c^{i-j} \\ &= n + 2\sum_{i=1}^n c^i \frac{1 - c^{-i}}{1-c^{-1}} \\ &= n + 2\sum_{i=1}^n \frac{c^i + 1}{1-c^{-1}} \\ &= n + \frac{2c}{c-1} \sum_{i=1}^n c^{i}-1 \\ &= n + \frac{2c}{c-1} \big(\frac{1-c^{n+1}}{1-c} -n \big)\\ &= n + \frac{2c}{(c-1)^2}(c^{n+1}+1) + \frac{2c}{c-1}n\\ \ \end{align}
Thus,
$$\lim_{n\rightarrow\infty} \mathbb{P}\bigg[\Big|\frac{S_n}{n} - \mathbb{E}[X_1]\Big| > \epsilon\bigg] = \lim_{n\rightarrow\infty} \frac{\text{Var}\Big(\frac{S_n}{n}\Big)}{\epsilon^2} \leq \lim_{n\rightarrow\infty} \frac{n + \frac{2c}{(c-1)^2}(c^{n+1}+1) + \frac{2c}{c-1}n}{n^2 \epsilon^2} = 0$$
Seeing how our choice of $\epsilon$ was arbitrary, the statement above holds for any $\epsilon > 0$ and shows that $\frac{S_n}{n} \rightarrow E[X_1]$ in probability, as desired.
-
|
2014-03-11 22:50:00
|
{"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.000009298324585, "perplexity": 297.5392898203648}, "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-2014-10/segments/1394011338837/warc/CC-MAIN-20140305092218-00055-ip-10-183-142-35.ec2.internal.warc.gz"}
|
https://socratic.org/earth-science
|
1
Active contributors today
## How do geologists divide Earth's past into time periods?
Lucio Margherita
Featured 5 months ago
By major worldwide geological events.
#### Explanation:
Such events however, have to leave a discernible trace of their occurrence all over the Earth's crust. The best ones, and most often used, are the rises and falls of the sea level.
When these increases and decreases of the available ocean water occur, they do simultaneously and worldwide. Yes, I know, tides are not simultaneous, but this is so because the amount of ocean water remains the same. The extra liquid available here is missing somewhere else. And, in any case, I am not referring to daily or even monthly cycles. We are speaking of geological time where the smallest time unit is closer to 1,000 years.
Over that time frame not only the phenomenon manifests itself all over the world, but it modifies the environment so as to leave a specific imprint on the subjacent rocks. What might have been a beach or a prairie becomes the seafloor with different conditions and denizens. This phenomenon is called a "transgression".
When the sea regresses under similar conditions, the movement is called a "regression"
Transgressions and regressions are not the only method to separate geological periods. The boundary between the Mesozoic and the Tertiary eons is clearly visible everywhere by a thin layer of rock rich in Iridium. This otherwise uncommon radioactive element (Ir) was possibly spread out by a planetary catastrophic event that might have caused the prompt demise of a number of animal and vegetal species, including the dinosaurs.
But a geological epoch is not only characterised by its limits. What happens between those limits is also relevant. The Tertiary was a mountain building period, the Cretaceous saw the sky full of birds, As for the inhabitants of the Earth, they may also be used to distinguish different epochs of its existence.
I am certain that you are aware that the Jurassic was the era of the dinosaurs; you might have seen fossils of trilobites, they characterise the Cambrian and Ordovician periods.
Mammals appear in the late Mesozoic and flourish in the Cainozoic; and, speaking of flowers, the angiosperms (or flourishing plants) made their appearance in the Tertiary.
Generally speaking, each geological period is delimited by major traumatic events and characterised by conditions extended over a long given period.
## What generates weather and climate patterns on Earth?
Fletcher
Featured 3 months ago
Difference in temperatures and humidity around the Earth!
#### Explanation:
Different areas of Earth get different amounts of sunlight, therefore they get different amounts of radiation and heat. This causes air currents, and the hot air tries to move to the cold air. This is what causes weather.
Our atmosphere basically becomes a heat engine. High and low pressure areas, wind, clouds, and precipitation systems are all caused, either directly or indirectly, by this uneven heating and the resulting heat redistribution processes.
This is all dependent on the availability of water. If the air is very dry the change of temperature will not result in changes to weather. Temperatures that would create a hurricane over the ocean do almost nothing over the Sahara desert.
## In what biome do we live in?
Kate M.
Featured 4 months ago
Humans can be found living in virtually all types of biomes.
#### Explanation:
Humans can be found living in virtually all types of terrestrial biomes.
Due to technological advancements, humans have been able to modify their environment and adapt to many different places. Clothing allows us to stay comfortable in areas with cooler temperatures. Building shelter allows us to stay cooler in dry, hot areas. In areas where the amount of vegetation is limited, tools have allowed us to hunt.
However, certain biomes are more popular for human habitation than others. Very few people live in the tundra . The map below shows population density from 1990 to 2015 with areas in bright green having higher density, or more people, than areas in blue.
Humans in Meghalaya, India have created living bridges out of rubber tree roots because other materials rot so quickly due to the extreme precipitation of the region (see here). The Nenets live within the Arctic Circle, herding reindeer and relying on them for food, shelter and clothing (see here)
## Why are wind speed and direction important for understanding changing weather?
James J.
Featured 2 months ago
Two reasons.
#### Explanation:
The first reason is it is if you know the wind speed and direction you will know what direction the weather is coming from and how fast it is moving. For example, if it is raining at a town 80 nautical miles (wind is measured in knots official) to the west of a second town, and the radar returns show that the precipitation is moving to the east at a speed of 20 kts, a relatively accurate prediction for the second town is that in 4 hours time it will be raining.
More importantly, wind direction and speed helps to plot the atmospheric pressure. The Buy Ballot law states that in the Northern Hemisphere, if the wind is at your back the area of low pressure is to your left. When you plot many wind points you get a pattern that illustrates the pressure pattern and when you look at speed you get an idea how far away the location is from the center of the pressure center.
http://cstar.cestm.albany.edu/PostMortems/CSTARPostMortems/2007/Mar_2_2007/2march2007storma.htm
This map shows how plotting the wind can show you the location of the pressure center. The thing to remember is that sometimes measuring stations are more than 100 miles apart, so plotting the wind like this can really help.
## How are typhoons formed?
James J.
Featured 2 months ago
The same way as a hurricane since they are the same thing.
#### Explanation:
Start with an unstable air mass formed over a dry area (like Australia or Africa)
The air mass moves over warm ocean water. Warm as in at least 26.5 C or 80 F.
As the air mass is unstable it will cause convergence in the lower part of the atmosphere., over the warm water. As the air comes together it will rise.
Condensation occurs aloft, forming convective cloud. This occurs to a great extent forming cumulonimbus clouds. The air continues to rise but because the water below is so warm, more moisture gets sucked up.
It is usually said that the winds aloft are light. This is only sort of true. What it should include is no wind sheer aloft. This means that the rising air is allow to continue to rise in the same column, over the warm water. If the winds are too strong or there is a wind sheer (change in direction and strength) the rising air will move away and the developing storm will "fall over" for a lack of a better term.
This occurs between 5 and 20 degrees latitude. At those latitudes the Coriolis effect is light but not non-existent. Light Coriolis allows the convergence to continue, whereas a stronger Coriolis will slow the convergence as it deflects the air due to the rotation of the Earth.
So the air keeps rises and forms more cloud, but because you have a whole ocean of warm water underneath the storm keeps sucking up moisture. This forms a multitude of cumulonimbus clouds that form together into a tropical storm and eventually a typhoon or hurricane.
http://expeditieaarde.blogspot.ca/2013_03_01_archive.html
This image simplifies it but it has the general point.
## What climate type is found in Los Angeles and describe its major characteristics?
Featured 1 month ago
A Mediterranean climate.
#### Explanation:
Los Angeles is classified as a warm Mediterranean climate. A Mediterranean climate is a specific type subtropical climate characterized by a dry summer, with a rainy season in the winter, and moderate changes in temperature between the seasons (you won't need a winter coat).
The summer months in LA are typically hot and very dry (it usually doesn't rain during the summer and temperatures exceed ${80}^{o} F$ [${27}^{o} C$]).
Winter months are mild and snow is incredibly rare however temperatures usually fall below freezing on at least one night per year.
Despite the fact that winter is LA's rainy season, LA averages only 15 inches (381 millimeters) of rainfall annually (to put things in perspective the average annual rainfall for all US cities is 30.2 inches [767 millimeters]).
Below is an image of an abating storm -red sky at night, sailors' delight!
Image is my own work; feel free to reuse in any way except for commercial purposes
LA also experiences a weather phenomenon known as the Santa Ana winds (the locals call them "Santa Anas").
Santa Anas are strong (40 mile per hour [or 64 kilometers per hour] plus), hot, dry winds that blow from east to west (us weather nerds like to call them katabatic winds). These winds are the result of cold air from Canada moving into the high desert regions of the Great Basin (Nevada and Utah) which displace the hot dry air to southwest into the lower-lying region of Southern California. These weather events usually last for about a week or so.
This isn't Los Angeles (it's San Diego a large city about 100 miles south of LA) but the same concept applies, this was taken during a rare wintertime Santa Ana.
Image is my own work; feel free to reuse in any way except for commercial purposes
Santa Anas typically occur, but aren't specifically relegated to, the final months of the summer and are common through the end of autumn (though they can occur at any time of year). These winds result in a dramatic increase in temperature (in the summer temperatures can reach ${110}^{o} F$ [${43}^{o} C$] and in the rare event that one occurs in the winter temps can exceed ${90}^{o} F$ [${33}^{o} C$]). Along with an increase in temperature Santa Ana's cause a drop in humidity (typically below 10%, it literally feels like you're getting sandblasted).
Another picture showing the effect of Santa Anas the change in the sky's color is actually due to the particulate matter being blown around (ironically this picture was taken on the same night as the previous one)
Image is my own work; feel free to reuse in any way except for commercial purposes
The image of the palm tree in the article I wrote here.
Thanks to the dry climate wildfires are a constant threat to Los Angeles and Southern California which can easily get out of hand. Luckily Southern California has firefighting crews who work tirelessly to combat and prevent them from occurring and spreading.
LA has variable terrain which aids in the creation numerous microclimates throughout the Los Angeles region. In the summer months the inland valleys to north (such as Laurel Canyon and Studio City) can be as much as ${30}^{o} F$ (${17}^{o} C$) warmer than the coastal regions (such as Santa Monica) due to the lack of onshore flow (winds that blow from the ocean). The converse is typically true for the winter months where the temperatures in the coastal regions are typically warmer than those in the inland valleys. These valley areas are typically dryer than the coastal regions year-round because the surrounding hills can inhibit fog, which is also common not just in LA but in all of coastal California, from settling in the region as readily as in coastal regions.
The two images below are prime examples of the oh-so-ubiquitous California fog (I wasn't driving, don't use your phone while driving, for cereal don't).
Image is my own work; feel free to reuse in any way except for commercial purposes
Image is my own work; feel free to reuse in any way except for commercial purposes
I know this is a lot of info but I hope it helps!
##### Questions
• · 50 minutes ago
• · 5 hours ago
• · Yesterday
• Yesterday
• Yesterday
• · 2 days ago
• · 2 days ago
• · 2 days ago
• · 2 days ago
• · 2 days ago
• · 2 days ago
• · 2 days ago · in Weathering
• · 2 days ago
• · 2 days ago
• · 2 days ago
• · 2 days ago
• 3 days ago · in The Atmosphere
• 3 days ago
• · 3 days ago
• · 3 days ago
• 3 days ago
• · 4 days ago
• · 4 days ago · in Ocean Movements
|
2017-04-23 17:44:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "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": 0.2815939486026764, "perplexity": 2001.4574879082638}, "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-2017-17/segments/1492917118740.31/warc/CC-MAIN-20170423031158-00346-ip-10-145-167-34.ec2.internal.warc.gz"}
|
https://www.physicsforums.com/threads/infinite-potential-well-energy-question.596964/
|
# Homework Help: Infinite potential well energy question
1. Apr 15, 2012
### phys2
1. The problem statement, all variables and given/known data
A particle of mass m is confined (in one dimension) to the region 0 ≤ x ≤ a by a potential which is zero inside the region and infinitely large outside.
If the wavefunction at time t = 0 is of the form
ψ (x,0) = Ax(a - x) inside the region
ψ (x, 0) = 0 outside the region
(a) Find the value of A to normalise the wavefunction
(b) The probability of measuring the ground state energy of the particle.
2. Relevant equations
P = integral of ψ times its complex conjugate = 1
3. The attempt at a solution
So for (a), I used the formula above and integrating with respect to x from 0 to a, I got A = square root of 30/a5
(b) Since this is an infinite potential well, the energy values would be E = ħ2n2π2 / 2ma2 so the ground state energy would be ħ2π2 / 2ma2
So am I supposed to find the probability of getting the above ground state energy function?
I was thinking that this might have something to do with energy expectation values but then that has a dψ/dt under the integral so I would end up getting zero which wouldn't make any sense.
2. Apr 16, 2012
### phys2
Does anyone have an answer?
3. Apr 16, 2012
### Steely Dan
When thinking about the probability to get a certain energy level, another way to ask the question is what is the probability that the particle will be in the state corresponding to that energy level? In other words, you know the actual wavefunction at t = 0 and you want to know what the probability is that it will be found in the state with the wavefunction corresponding to the ground state energy level. What is the process for evaluating that likelihood?
|
2018-06-19 22:50:17
|
{"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": 0.8241879940032959, "perplexity": 252.82241965862883}, "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-2018-26/segments/1529267863206.9/warc/CC-MAIN-20180619212507-20180619232507-00480.warc.gz"}
|
http://clay6.com/qa/26029/which-one-of-the-following-is-highest-melting-halide-
|
# Which one of the following is highest melting halide.
$(a)\;AgCl\qquad(b)\;AgF\qquad(c)\;AgBr\qquad(d)\;Agl$
According to Fajan, small anion is polarised to lesser extent than the larger anion. Hence will be the most ionic and has high melting point.
|
2017-11-21 15:28:14
|
{"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": 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": 0.8412127494812012, "perplexity": 2424.5416121312205}, "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-2017-47/segments/1510934806419.21/warc/CC-MAIN-20171121151133-20171121171133-00243.warc.gz"}
|
https://www.physicsforums.com/threads/imaginary-part-of-qho-solutions.953318/
|
# Imaginary Part of QHO Solutions?
• I
Gold Member
I have seen a few online lectures on solving the Schrodinger equation for the Quantum Harmonic Oscillator. The various solutions are products of the real-valued Gaussian function and the real-valued Hermite Polynomials. But I have never seen a mathematical expression for the imaginary part of those solutions. The Wikipedia entry for the QHO shows a graph for the imaginary part but no expression for it. Please help.
DrClaude
Mentor
As always in QM, two wave functions differing only by a complex phase correspond to the same state. By convention, the eigenfunctions of the time-independent Schrödinger equation are chosen to be purely real. It is only when considering the time evolution of an eigenfunction, or when considering a wave function that is a superposition of eigenstates, that the complex part is not zero.
jtbell
Mentor
The Wikipedia entry for the QHO shows a graph for the imaginary part but no expression for it.
Are you referring to the animated graphs which have the real and imaginary parts in different colors? I’m pretty sure they’re just the real solutions of the time-independent SE, multiplied by the complex time-dependent phase factor: $$e^{-iEt/\hbar} = \cos (Et/\hbar) - i \sin(Et/\hbar)$$
George Jones
Staff Emeritus
Gold Member
I have seen a few online lectures on solving the Schrodinger equation for the Quantum Harmonic Oscillator. The various solutions are products of the real-valued Gaussian function and the real-valued Hermite Polynomials. But I have never seen a mathematical expression for the imaginary part of those solutions. The Wikipedia entry for the QHO shows a graph for the imaginary part but no expression for it.
As always in QM, two wave functions differing only by a complex phase correspond to the same state. By convention, the eigenfunctions of the time-independent Schrödinger equation are chosen to be purely real. It is only when considering the time evolution of an eigenfunction, or when considering a wave function that is a superposition of eigenstates, that the complex part is not zero.
Are you referring to the animated graphs which have the real and imaginary parts in different colors? I’m pretty sure they’re just the real solutions of the time-independent SE, multiplied by the complex time-dependent phase factor: $$e^{-iEt/\hbar} = \cos (Et/\hbar) - i \sin(Et/\hbar)$$
The first four real/imaginary animations are examples of this, but the last two animations are superpositions of time-dependent stationary states, i.e., as my first quantum instructor would say, "There is (spatial) sloshing."
vanhees71
Gold Member
Are you referring to the animated graphs which have the real and imaginary parts in different colors? I’m pretty sure they’re just the real solutions of the time-independent SE, multiplied by the complex time-dependent phase factor: $$e^{-iEt/\hbar} = \cos (Et/\hbar) - i \sin(Et/\hbar)$$
Well, that may well be true, but it doesn't have to do anything with the classical counterparts plotted as A and B. The only state referring to this classical case is a coherent state (depicted as plot H). It's of course misleading to show real and imaginary parts, which both are not observable. What's observable in a statistical sense, being position-probability distributions, are the modulus squares. Than it should be immediately clear that the energy eigenstates are stationary states, i.e., nothing is moving at all!
Gold Member
Are you referring to the animated graphs which have the real and imaginary parts in different colors? I’m pretty sure they’re just the real solutions of the time-independent SE, multiplied by the complex time-dependent phase factor: $$e^{-iEt/\hbar} = \cos (Et/\hbar) - i \sin(Et/\hbar)$$
Yes, I was referring to those animated graphs. Thanks for the clarification.
hilbert2
|
2021-03-02 05:01:08
|
{"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": 0.7718185782432556, "perplexity": 293.86495768963914}, "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-2021-10/segments/1614178363217.42/warc/CC-MAIN-20210302034236-20210302064236-00483.warc.gz"}
|
http://openstudy.com/updates/50a04d08e4b05517d5365e77
|
## JerJason Group Title If the probability density of a random variable is given by: f(x)= Kx^2 for 0<x<1 0 elsewhere find the value k and the probability that the random variable takes on a value between (a) 1/4 and 3/4 (b) greater than 2/3 one year ago one year ago
1. JerJason Group Title
So that I'm understanding this correctly. I'm finding the value of "k" in order to verify that the function f(x)=kx^2 is a probability density function based on these conditions: f(x)>=0 and f(x) dx = 1 when x is between -infinity and infinity.
2. JerJason Group Title
Therefore, $f(x)=\int\limits_{0}^{1}kx^2dx=k \frac{ x^3 }{ 3 }_{0}^{1}=k/3=1$ so k=3
3. JerJason Group Title
From here I'm plugging in the limits between 1/4 (.25) and 3/4 (.75). $\int\limits_{.25}^{.75}kx^2dx=k \frac{ x^3 }{ 3 }_{.25}^{.75}=k(\frac{ .4218-.0156 }{ 3 })=.4062/3=.1354k$
4. JerJason Group Title
I have a filling I did that last step wrong.
5. hartnn Group Title
thats correct, 0.1354*3=0.4062 is the correct probability for a)
6. hartnn Group Title
you know how to do part b), right ?
7. JerJason Group Title
^ alright good. For (b) do I convert f(x) into F(x), which is the distribution function then plug in the limit .67 (or 2/3) for F(x) and then minus that from 1 like so? Find: $P(x>.67)$ $F(x)=\int\limits_{0}^{x}kx^2dx=kx^3/3$ $F(.67)=3*.67^3/3=.902289/3=.3008$ $1-.3008=.70$
8. hartnn Group Title
that is correct, but there is also another way. $$\huge \int \limits_{(2/3)}^1 kx^2dx=....$$ and u get same result 0.703
9. JerJason Group Title
yes that makes sense as well. Thanks.
10. hartnn Group Title
welcome ^_^
|
2014-08-01 14:01:57
|
{"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": 0.7900248169898987, "perplexity": 2135.495701429447}, "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/1406510274987.43/warc/CC-MAIN-20140728011754-00431-ip-10-146-231-18.ec2.internal.warc.gz"}
|
https://www.tutorialspoint.com/How-to-draw-sine-waves-with-HTML5-SVG
|
How to draw sine waves with HTML5 SVG?
SVGJavascriptWeb DevelopmentFront End Scripts
To draw sine waves with SVG, use the following that closely approximates half of a sine wave. I have used a cubic-bezier approximation. Use the <path> element.
Example
<!DOCTYPE html>
<html>
</html>
|
2021-09-27 23:18:10
|
{"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": 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": 0.6960201263427734, "perplexity": 6876.594766037967}, "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-2021-39/segments/1631780058552.54/warc/CC-MAIN-20210927211955-20210928001955-00575.warc.gz"}
|
http://wiki.cmci.info/blogtng/blogtop?btng%5Bpagination%5D%5Bstart%5D=15&btng%5Bpost%5D%5Btags%5D=
|
# Bioimage Analysis Wiki
### Sidebar
EMBL BioImage Data Analysis
EuBIAS
NEUBIAS
Popularity Ranking
RT @WimJHH: EMBL Heidelberg workshop "Cryo-EM in Industry & Academia", 6-8 February 2019. Success stories, new technologies, industry chall… About 5 hours, 19 mins ago by: Kota Miura (@cmci_) @f_levet @BertrandVernay I made it. About 6 hours, 10 mins ago by: Kota Miura (@cmci_) RT @haesleinhuepf: One more session on #ImageJ short cuts: SHIFT+U opens the command tree. I didn't know this one. Thanks @BertrandVernay a… About 6 hours, 21 mins ago by: Kota Miura (@cmci_) @ManuelTHERY @martinjones78 @NEUBIAS_COST Slowly Manu About 16 hours, 3 mins ago by: Kota Miura (@cmci_) @laure_plantard I can’t track JY himself with Trackmate. Metaphysical contradiction. About 17 hours, 55 mins ago by: Kota Miura (@cmci_) “2 chapters of my dissertation was only possible with Trackmate” About 19 hours, 23 mins ago by: Kota Miura (@cmci_)
blogtng:blogtop
Algorithm FRAP Fiji ImageJ ImageJ Plugin ImageJ Plugin 3Dviewer Imaris Java Javascript Python R bias blog dokuwiki fiji google imagej java libraries matlab meetings neubias news papers python references software webadmin
# Latex to Dokuwiki
I have been searching for a convinient way to convert Latex to Dokuwiki (also MSword doc to dowuwiki as well).
Never been really successful, though there are some Dokuwiki pluings that converts Dokuwiki to Latex and vice versa. Problem of the plugin is that latex should be installed in the webserver, but that is not the case in the EMBL. and I cannot install them as I am not the administrator.
## One way I figured out by now is the following detour.
1. Convert Latex document to Markdown using Pandoc.
2. Use markdown plugin (of Dokuwiki). Just copy and paste the markedown text in the dowkuwiki, and bound that part with <markdown> tag
Discovering the markdown plugin was a big change for this purpose. Not only latex docs, but MS doc and docx could be converted using the Pandoc and be pasted into Dokuwiki.
... so I wrote this short report using the markdown syntax. The plugin actually uses markdown extra, an enhanced version.
# ISBI 2012, Satellite Meeting
A picture with the ImageJ founder, Wayne Rusband.
A meeting titled “Bioimage analysis software: is there a future beyond ImageJ?” took place in Barcelona between April 31st and May 1st (http://bigwww.epfl.ch/eurobioimaging/). Some key presentations were given by Wayne Rusband, Johannes Schindlin, Curtis Rueden and all the famous developers among ImageJ community and more software such as Icy and BioimageXD.
Wayne talked about how he started developing NIH Image in 1987, with some pictures of initial Apple II machine he was developing with. The software was coded in pascal, and I told him that I started using it in 1993 and was fascinated by the ImageJ macro language. He also informed us with his latest update, pixel inspector that shows the distribution of values in the vicinity of the cursor location.
Intensive discussions on strengthening the community was held as well, mainly in the direction of setting up a core portal and planning periodic meetings.
## Two Lectures on Classification and Clustering, April and May
Coupled lectures on Classification and Clustering will be given in the CMCI seminar in April and May. These two approaches have been widely used for screening and system descriptions, but there seems to be more in the future in terms of image processing and analysis, as we have seen in the CMCI seminar in March. For this reason we asked for introduction for these two topics by two experts in the EMBL. Bernd Fischer will give us a lecture on Classification and Machine learning, and Jean-Karim Heriche will give a lecture on cluster analysis.
## CMCI seminar, Jan 27, 2012 (Fri) 15:00- @room 518 (Building 13)
Kota will talk about cell tracking in Drosophila embryo and Andrea Picco will talk about his work on subpixel resolution tracking of yeast endocytosis.
## Happy New Year!
Wishing you for a great year of Image Processing and Analysis!
# entry generator
## Weblog Archive
blogtng/blogtop.txt · Last modified: 2016/05/24 05:46 (external edit)
|
2018-10-20 17:12:47
|
{"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": 0.23928868770599365, "perplexity": 8851.678834112388}, "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-2018-43/segments/1539583513009.81/warc/CC-MAIN-20181020163619-20181020185119-00417.warc.gz"}
|
https://zbmath.org/?q=an:0818.11024
|
# zbMATH — the first resource for mathematics
A regularized Siegel-Weil formula: The first term identity. (English) Zbl 0818.11024
In his analysis of Siegel’s mass formula Weil introduced a unified technique based on the ‘Weil’ representation. He showed that one can form a very general class of theta functions on $$G(k_ \mathbb{A})\times H(k_ \mathbb{A})$$ where $$G$$ is a symplectic group of the form $$\text{Sp} (2n)$$ and $$H$$ is the orthogonal group of a quadratic form of degree $$m$$ and Witt index $$r$$, where $$m$$ is taken here to be even. The integral of the theta function over $$H(k) \smallsetminus H(k_ \mathbb{A})$$ converges if either $$r=0$$ or $$m-r> n+1$$. Under somewhat stronger assumptions he proved that this integral is equal to an Eisenstein series.
In a previous paper the authors proved this equality, or a modified version precisely under the conditions $$r=0$$ or $$m-r> n+1$$. In this paper they investigate what happens when this condition is not fulfilled. They define a regularization of the integral over $$H(k) \smallsetminus H(k_ \mathbb{A})$$ by using an Eisenstein series. They also make use of the general theory of the analytic continuation of Eisenstein series and they obtain relations in this case which can be regarded as extensions of the Siegel- Weil formula. In the course of this investigation they uncover interesting relations with other aspects of the theory of automorphic forms.
##### MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F27 Theta series; Weil representation; theta correspondences
Full Text:
|
2021-11-28 03:06:30
|
{"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": 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": 0.8946747183799744, "perplexity": 325.3990062433757}, "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-2021-49/segments/1637964358443.87/warc/CC-MAIN-20211128013650-20211128043650-00174.warc.gz"}
|
https://zhirenhuang.com/tags/road-traffic-data/
|
# Road traffic data
## Estimating inter-regional mobility during disruption: Comparing and combining different data sources
A quantitative understanding of people’s mobility patterns is crucial for many applications. However, it is difficult to accurately estimate mobility, in particular during disruption such as the onset of the COVID-19 pandemic. Here, we investigate …
|
2023-02-06 08:53:38
|
{"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": 0.8164312839508057, "perplexity": 1940.0058501035896}, "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-2023-06/segments/1674764500334.35/warc/CC-MAIN-20230206082428-20230206112428-00415.warc.gz"}
|
https://artofproblemsolving.com/wiki/index.php?title=1998_AIME_Problems/Problem_12&diff=15916&oldid=6860
|
# Difference between revisions of "1998 AIME Problems/Problem 12"
## Problem
Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\displaystyle \overline{DE}, \overline{EF},$ and $\overline{FD}, \displaystyle$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR}, \displaystyle$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a + b\sqrt {c}, \displaystyle$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2} + b^{2} + c^{2}$?
## Solution
Assign variables: $EP = FQ = x$ $EQ = y$ $PQ = k$ Since $\displaystyle AE = \frac {1}{2}AB$ and $AD = \frac {1}{2}AC$, $\triangle AED \sim \triangle ABC$ and $ED \parallel BC$
Alternate Interior Angles Theorem says $\displaystyle \angle PEQ = \angle BFQ$ and $\angle EPQ = \angle FBQ \displaystyle$
Vertical Angles Theorem says $\displaystyle \angle EQP = \angle FQB \displaystyle$ So $\triangle EQP \sim \triangle FQB \displaystyle$ and by CPCTC $\frac {EP}{EQ} = \frac {FB}{FQ}\Longrightarrow\frac {x}{y} = \frac {1}{x}\Longrightarrow x^{2} = y \displaystyle$
Since $\displaystyle \triangle EDF$ is equilateral, $EQ + FQ = EF = BF = 1\Longrightarrow x + y = 1$. Solving for $x$ and $y$ using $\displaystyle x^{2} = y$ and $x + y = 1$ gives $\displaystyle x = \frac {\sqrt {5} - 1}{2}$ and $\displaystyle y = \frac {3 - \sqrt {5}}{2} \displaystyle$
Using the Law of Cosines, we get
$k^{2} = x^{2} + y^{2} - 2xy\cos{\frac {\pi}{3}}$
$= \left(\frac {\sqrt {5} - 1}{2}\right)^{2} + \left(\frac {3 - \sqrt {5}}{2}\right)^{2} - 2\left(\frac {\sqrt {5} - 1}{2}\right)\left(\frac {3 - \sqrt {5}}{2}\right)\cos{\frac {\pi}{3}}$
$= 7 - 3\sqrt {5}$
We want the ratio of the squares of the sides, so $\displaystyle \frac {(2)^{2}}{k^{2}} = \frac {4}{7 - 3\sqrt {5}} = 7 + 3\sqrt {5}$ so $a^{2} + b^{2} + c^{2} = 7^{2} + 3^{2} + 5^{2} = 083$
## See also
1998 AIME (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 13 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions
Invalid username
Login to AoPS
|
2021-06-24 18:24:04
|
{"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": 46, "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": 0.5366714596748352, "perplexity": 301.992613169835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2021-25/segments/1623488556482.89/warc/CC-MAIN-20210624171713-20210624201713-00024.warc.gz"}
|
https://codereview.stackexchange.com/questions/80214/given-a-digit-sequence-print-the-possible-decodings-of-the-given-digit-sequence/80217
|
# Given a digit sequence, print the possible decodings of the given digit sequence
Examples:
12 gives:
'AB' and 'L'
and
123 gives
'ABC', 'LC' and 'AW'
Here is my attempt:
import java.util.*;
public class decoding {
static int calls = 0;
static Map<Integer, String> codes = new HashMap<Integer, String>();
private static void construct(){
codes.put(1, "A");
codes.put(2, "B");
codes.put(3, "C");
codes.put(4, "D");
codes.put(5, "E");
codes.put(6, "F");
codes.put(7, "G");
codes.put(8, "H");
codes.put(9, "I");
codes.put(10, "J");
codes.put(11, "K");
codes.put(12, "L");
codes.put(13, "M");
codes.put(14, "N");
codes.put(15, "O");
codes.put(16, "P");
codes.put(17, "Q");
codes.put(18, "R");
codes.put(19, "S");
codes.put(20, "T");
codes.put(21, "U");
codes.put(22, "V");
codes.put(23, "W");
codes.put(24, "X");
codes.put(25, "Y");
codes.put(26, "Z");
}
private static void decode(String str, String built){
construct();
int n = str.length();
if (n == 0) {
System.out.println(built);
return;
}
// If you have 0's, then they don't have a corresponding singular letter. Break off the recursion.
if (str.substring(0, 1).equals("0"))
return;
String x = codes.get(Integer.parseInt(str.substring(0, 1)));
if (x == null)
return;
decode(str.substring(1), built+x);
if (n > 1) {
// If it's more than 26, it doesn't have a corresponding letter. Break off the recursion.
if (Integer.parseInt(str.substring(0, 2)) > 26)
return;
String y = codes.get(Integer.parseInt(str.substring(0, 2)));
decode(str.substring(2), built+y);
}
}
public static void main(String[] args) {
decode(args[0], "");
}
}
I haven't done any memoization or dynamic programming in this. It's a rather crude solution with exponential run time I believe. Let me know what you think.
• Welcome to Code Review! I feel sure there is a more elegant way of coming to this solution, I hope you get some good reviews! – Phrancis Feb 11 '15 at 2:21
A few hints :
• compile your code with all warning enabled (calls doesn't seem to be used, does it ?)
• write your code in such a way that it can easily be tested. This includes taking a well-defined input and returning a well-defined value (instead of printing it).
• write tests.
An example of dynamic programming approach (but I do think it will be better for you if you try to rewrite your solution with tests first, at least you'll be able to compare results and performance): asuming you have a string s "dddddD1D2" (a string composed of any digits then digit D1 then digit D2) and you know the number of combinations for all strings "" (empty string), "d", "dd", "ddd", ..., "ddddddD1", how to you know the number of combinations for string s ?
You have two different options :
• either D2 is to be converted on its own (if D2 is positive). So if D2 is positive, you have as many combinations like this one as you have combinations for "ddddddD1" which is a known value.
• either D2 is to be converted with previous digit D1. This is possible if 1 <= 10*D1 + D2 <= 26. If it is the case, you have as many combinations as the number of combinations for "dddddd" which is a known value.
Well, I gave you a solution assuming you had a solution. How do you get such a thing working ? You just need to iterate over the string and for each string position, you keep track of the number of combinations that can be generated using characters up to that point :
• 0 character : 1 combination
• 1 character : it has to be converted on its own. Reuse the solution for 0 characters.
• 2 characters : reuse the solution for 0 and 1 characters
• n characters : reuse the solution for n-2 and n-1 characters.
This is a classic example of algorithm where we can define the number of elements with a property without having to enumerate them.
You can either save the two last results as you progress or you can keep track of the result for each index in an array. This is up to you.
• nb("") = 1
• nb("1") = nb("") = 1
• nb("12") = nb("1") + nb("") = 1 + 1 = 2
• nb("123") = nb("12") + nb("1") = 1 + 2 = 3
• nb("1234") = nb("123") = 3 because 34 cannot be converted to a letter.
Disclaimer : I haven't tried any of this but I think it kind of makes sense. You'll see if it works as you write tests and compare with the results from your current code.
• 2 years late, but thank you for your input :) I read it when you posted in 2015. – Siddhartha Oct 31 '17 at 17:44
You haven't written any unit tests, the output to system.out.println at the bottom of the recursion shows this. This is a concern because any changes you make could break the program and you might not notice.
To fix the testability issue you could move decode to an object of its own and have it accumulate the result strings in a list. This list could be printed out from main, or validated by unit tests.
|
2019-11-20 23:37:17
|
{"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": 0.3600386679172516, "perplexity": 1611.4875721081883}, "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/1573496670635.48/warc/CC-MAIN-20191120213017-20191121001017-00481.warc.gz"}
|
https://zbmath.org/0538.10034
|
## The mean twelfth power of Dirichlet L-functions on the critical line.(English)Zbl 0538.10034
D. R. Heath-Brown [Q. J. Math., Oxf. II. Ser. 29, 443-462 (1978; Zbl 0394.10020)] has shown that for $$T\geq 2$$ $(1)\quad \int^{T}_{0}| \zeta(1/2+it)|^{12}\quad dt\quad<<\quad T^ 2 \log^{17}T.$ His proof depends on Atkinson’s formula for $$\int^{T}_{0}| \zeta(1/2+it)|^ 2$$ dt. M. Jutila [J. Number Theory 18, 135-156 (1984; Zbl 0533.10034)] has established a transformation formula for $$\sum_{M_ 1\leq n\leq M_ 2}d(n)n^{-1/2- it},$$ where d(n) is the divisor function, $$M_ 1$$, $$M_ 2$$ are near t/2$$\pi$$ and $$M_ 1<t/2\pi<M_ 2$$. As he pointed out, this can be used instead of Atkinson’s formula in proving (1).
In the present paper Jutila’s formula is generalized to sums $$\sum_{M_ 1\leq n\leq M_ 2}\chi(n)d(n)n^{-1/2-it},$$ where $$\chi$$ (n) is a Dirichlet character mod q, $$M_ 1$$, $$M_ 2$$ are near qt/2$$\pi$$ and $$M_ 1<qt/2\pi<M_ 2$$. The result is applied to obtain a large values theorem for L-functions. This implies an extension of (1), viz. $\sum_{\chi \quad mod q}\int^{T}_{-T}| L(1/2+it,\chi)|^{12}\quad dt\quad<<_{\epsilon}\quad q^ 3 T^{2+\epsilon}.$ The above results are applied to deduce estimates for the zero counting function $$\sum_{\chi \quad mod q}N(\alpha,T,\chi),$$ which generalize previous estimates of D. R. Heath-Brown [J. Lond. Math. Soc., II. Ser. 19, 221-232 (1979; Zbl 0393.10043)] and are new if q is sufficiently small compared with T.
### MSC:
11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
### Citations:
Zbl 0394.10020; Zbl 0533.10034; Zbl 0393.10043
|
2023-03-30 08:58:13
|
{"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": 0.907978355884552, "perplexity": 1095.5726744057156}, "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/1679296949107.48/warc/CC-MAIN-20230330070451-20230330100451-00167.warc.gz"}
|
https://math.stackexchange.com/questions/1375932/lhospital-rule-exponental-ratio
|
# L'Hospital rule, exponental ratio
$$\lim_{x\to ∞} \frac {x^{1000000}} {e^x}$$ could anyone please provide some hits with what result I will end up?
After all applyings of L'Hospital rule, I will get $\frac {n} {e^x}$, where $n$ is large number before I got out of the $x$ powers. So, will it be the limit $0$ then? Since the infinity is nothing I have $\frac {n} {0}.$ Or will it be just the $\infty$?
• "Since the infinity is nothing": what??? – Omnomnomnom Jul 27 '15 at 18:44
• @Omnomnomnom Picky picky picky... heh – David C. Ullrich Jul 27 '15 at 18:45
• @DavidC.Ullrich I honestly don't know what that statement could mean – Omnomnomnom Jul 27 '15 at 18:47
• @Omnomnomnom Sorry. I thought it was sufficiently clear that the statement was nonsense that it would be clear I was just teasing. "what???" is exactly right, even the right number of question marks... – David C. Ullrich Jul 27 '15 at 18:50
• Simply note that $e^x$ grows faster than ANY polynomial function. – Elliot Gorokhovsky Jul 28 '15 at 3:22
The limit will be 0. Another way to see this:
Note that $$e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots + \frac{x^{1000001}}{1000001!} + \cdots > \frac{x^{1000001}}{1000001!}$$ $$0<\frac{x^{1000000}}{e^x} < \frac{1000001!}{x}$$
$$\lim_{x \to \infty} \frac{1000001!}{x} = 1000001! \lim_{x \to \infty} \frac1x = 0$$
The Squeeze Theorem will give us this result.
Repeated L'Hospital will get you to
$$\lim_{x\to \infty}\frac{10000000!}{e^x}$$
and when $x$ tends to infinity, you'll get a $0$.
• The last step in which you would "differentiate away" the constant is, strictly speaking, an incorrect application of L'Hospital. – Omnomnomnom Jul 27 '15 at 18:51
• That's true @Omnomnomnom, my mistake. – Eemil Wallin Jul 27 '15 at 18:51
A "quicker" approach: note that $$\lim_{x \to \infty} \frac{x^{1000000}}{e^x} = \left(\lim_{x \to \infty}\frac{x}{e^{x/1000000}}\right)^{1000000}$$
• Although, requires you to already know $\lim_{x \to \infty} \frac{x}{e^x} = 0$. – 6005 Jul 27 '15 at 19:47
• @6005 Or you can just use L'Hopital once, instead of 10^6 times in the other approaches. – Teepeemm Jul 27 '15 at 19:58
• @Teepeemm - I've always preferred using it once, rather than 10⁶ times. – Alec Jul 28 '15 at 12:59
The exponential beats any polynomial, so the limit is zero. If you really want to think by L'Hospital, once you differentiate $1000000$ times, the numerator will be a huge fixed number and the denominator will be $e^x$. And $\lim_{x \to +\infty} K/e^x = 0$ for any constant $K$.
If you insist on using L'Hopital, here is how you should think about it.
First remember what L'Hopital says. It says that if you have two functions of $x$, say $f(x)$ and $g(x)$, and you want to know what $f(x)/g(x)$ tends to as $x$ tends to some limit $a$, then if $f(x)$ and $g(x)$ both tend to $0$ or both tend to $\infty$, and if $f'(x)/g'(x)$ tends to a limit as $x$ tends to $a$, then these limits are the same. That is, $$\lim_{x \to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)}$$
provided they both exist. In your case we have $f(x)=x^{1000000}$ and $g(x)=e^{x}$. If we just differentiate once, then the limit of the top and the bottom is still $\infty$, so we can use L'Hopital again. In fact, if you use L'Hopital $1000000$ times, we that $$\lim_{x \to \infty} \frac{x^{1000000}}{e^{x}}=\lim_{x \to \infty}\frac{1000000!}{e^{x}}$$ Where $1000000!$ is the product of all the numbers from $1000000$ to $1$. But this is just a constant divided by $e^{x}$, so when $x$ gets big this large number stays the same, while $e^{x}$ keeps growing. And since $e^{x}$ can get as large as we could ever want, your limit has to be $0$.
This is so easy when you put $e^{x} = t$ and then $t \to \infty$ as $x \to \infty$ so that the expression changes to $f(t) = (\log t)^{a}/t$ where $a = 1000000$. This can be further rewritten as $$f(t) = \left(\frac{\log t}{t^{b}}\right)^{a} = \{g(t)\}^{a}\tag{1}$$ where $b = 1/a = 0.000001$. Now we show that $g(t) \to 0$ as $t \to \infty$ and this will imply that $f(t) \to 0$ as $t \to \infty$.
Let us choose a number $c$ such that $0 < c < b$. Since $t \to \infty$ we can assume $t > 1$ so that $t^{c} > 1$ and hence $$0 < \log t = \log\{(t^{c})^{1/c}\} = \frac{\log t^{c}}{c} \leq \frac{t^{c} - 1}{c} < \frac{t^{c}}{c}\tag{2}$$ where we have used the standard inequality $$\log x \leq x - 1$$ for $x > 1$ with $x = t^{c}$. From $(2)$ and definition of $g(t)$ we get $$0 < g(t) = \frac{\log t}{t^{b}} < \frac{t^{c}}{ct^{b}} = \frac{1}{ct^{b - c}}\tag{3}$$ for $t > 1$. Since $b - c > 0$, it follows that $t^{b - c} \to \infty$ and using Squeeze theorem on equation $(3)$ as $t \to \infty$ we see that $g(t) \to 0$ as $t \to \infty$.
|
2019-08-25 16:47:45
|
{"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": 0.9385522603988647, "perplexity": 194.95775679542453}, "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/1566027330750.45/warc/CC-MAIN-20190825151521-20190825173521-00114.warc.gz"}
|
https://www.techwhiff.com/learn/1-you-have-two-1000-resistors-two-100-f/272292
|
# 1. You have two 1000 resistors. two 100 μF capacitors and one 5 H inductor. This...
###### Question:
1. You have two 1000 resistors. two 100 μF capacitors and one 5 H inductor. This is the as many of these 5 circuit. This means you can use 1 component, 2 components All 5 components. (a) If each of this is a filter circuit then what is the lowest frequency "Low-Pass" filter you can design? (b) What is the highest frequency "high pass filter you can design. (c) Under what conditions you will have the largest amplitude output signal. (d) Draw the circuits with input and output for each of these filters
#### Similar Solved Questions
##### Write VHDL code for: Part Ta: To Create a Clock Pre-Scaling Entity that Generates a Slow...
Write VHDL code for: Part Ta: To Create a Clock Pre-Scaling Entity that Generates a Slow Clock Pulse Train from a Fast Clock Pulse Train: Obiective Create a Clock Pre-Scaling ENTITY called, PreScale, with one-bit input and 1-bit output The circuit is a 20-bit binary up-counter. Should roll back to z...
##### Review I Constants Periodic The gaseous hydrocarbon acetylene, C2H2, used in welders' torches, bums according to...
Review I Constants Periodic The gaseous hydrocarbon acetylene, C2H2, used in welders' torches, bums according to the following equation: 20,H2(g) +502(g)-4CO2(g)+ 2H20(ø) Part A What is the theoretical yield, in grams, of CO2, if 23.9 g of C2H2 completely reacts? Express your answer with ...
##### A bridge is supported by two types of rectangular cross-section piles located in a river as shown in Figure 1. The widt...
A bridge is supported by two types of rectangular cross-section piles located in a river as shown in Figure 1. The width of the piles is w 0.5 m and their lengths are eitherl 2 m or 12 2.5 m. The river of depth of about 20 m runs at 2 m/s. Water density is ρ-1000 kg/m' and its absolute visco...
##### QUESTION 1 Which of the following is not a characteristic of the monopolistic competition market structure?...
QUESTION 1 Which of the following is not a characteristic of the monopolistic competition market structure? Many sellers, each small in size relative to the overall market. Few sellers. Differentiated product. Easy, low-cost entry and exit. QUESTION 2 Which of the followi...
##### Solve: 2cos (x/3) = 2 ?
Solve: 2cos (x/3) = 2 ?...
##### Mineral: Almandine Formula: Fe3Al2(SiO4)3 -Show where each of the cations in the formula are housed &...
Mineral: Almandine Formula: Fe3Al2(SiO4)3 -Show where each of the cations in the formula are housed & describe where the M1 and M2 cations are located in this crystal system....
##### Search or type a command CENG211 homework 2 0.2 the Calculate reactions support E D b)...
Search or type a command CENG211 homework 2 0.2 the Calculate reactions support E D b) Calculate internal shear force and bending moment at points andD 1.1. 2L V2 W-ID2 (kN/m) F-2'ID2 (kN) L-IN (m) Ay В,...
##### The two pendulums shown have 14 kg balls supported by rigid mass-less rods. The pendulums rotate...
The two pendulums shown have 14 kg balls supported by rigid mass-less rods. The pendulums rotate on frictionless pivots in the same vertical plane In a circular path of radius 2 meters and collide. If each is at rest and oriented at theta = 38 degrees then released simultaneously, determine the Impu...
|
2023-02-06 21:42:48
|
{"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": 0.49761369824409485, "perplexity": 4061.6190789335074}, "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/1674764500365.52/warc/CC-MAIN-20230206212647-20230207002647-00441.warc.gz"}
|
http://mathoverflow.net/revisions/49811/list
|
Hi, I am interested in the set $\mathbb A-\mathbb A^\times$ i.e. the complement of ideles in the adele ring of a number field.
Is it measurable, and what is its volume, with respect to the standard measure of adeles?
("standard" means the same as in Tate's thesis)
Thank you.
2 added 53 characters in body
Hi, I am interested in the complement of ideles in the adele ring of a number field.
Is it measurable, and what is its volume, with respect to the standard measure of adeles?
("standard" means the same as in Tate's thesis)
Thank you.
1
# Measure of "adeles minus ideles"
Hi, I am interested in the complement of ideles in the adele ring of a number field. Is it measurable, and what is its volume, with respect to the standard measure of adeles?
Thank you.
|
2013-05-19 15:11:48
|
{"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": 0.6358051300048828, "perplexity": 479.36918993414594}, "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-2013-20/segments/1368697745221/warc/CC-MAIN-20130516094905-00066-ip-10-60-113-184.ec2.internal.warc.gz"}
|
http://efkv.icscaponnetto.it/rune-text-symbols.html
|
Rune Text Symbols
Rune was a white supremacist, a result of teenage rebellion from her immediate family (who had escaped from the Herren Clan,) leading her back into the welcoming arms of the violent clan. Full text of the Revised Common Lectionary readings for Year A - Easter - Second Sunday of Easter. Glyphs of animals were powerful. Find Norse Runes Futhark Alphabet Stones Symbols stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. Now computers are more widely established around the world the need to show other characters such as Japanese and Chinese languages along with various symbols became more important. It is possible that the runes used in the Cathedral of Dusk and the ones used by Nokris are an archaic version of the Hive runes that we see on the Dreadnaught. Just click on a line symbol to copy it to the clipboard and paste it anywhere. On these pages you will discover the symbolic meaning and hidden messages in everything that surrounds you, you will learn about your totem animals and how to interpret signs, symbols and messages from the stars, the Gods and Goddesses, the spirit guides and nature all of which are all speaking the mystical and magical language of the Universe ~ through symbols, animal encounters. Pagan runes in text (Looking for Odin's Horn) But I'm particularly looking for the Odin's Horn symbol in text if anyone could give me the Odin's Horn symbol in text it would be great and might as well post all the pagan and aryan symbols in text in this thread, would be great help to spread paganism on the internet. Just for fun, you can type in any text below and then each letter will be transformed into their nearest Rune equivalent. Runes Symbols. Historically, it appeared in Germany in many places, ranging from guidestones on the sides of roads to heraldic use in the coats of arms of various towns; there is even a German city called Wolfsangel. 54,543 downloads (29 yesterday) Free for personal use - 4 font files Download Donate to author. As well as possibly being a rune for the number zero. Sowilo is an extremely fortunate rune and is often considered the best possible rune in a reading. In this article, we unravel the mystery of runes and examine significant facts and history that provide us with a better understanding of these ancient. Yeah, yeah, we all know that's a pause symbol. process of elimination. Fade resistant UV Inks. , as magical symbols engraved in stone; they were developed into the first Rune alphabet, the “elder” Futhark (‘futhark’ being a transliteration of the first six letters), an alphabet of twenty four characters. Search Lections Texts. Maybe for this reason, the 19th rune is called Ehwaz, the horse. Nevertheless, the. best viking rune list and get free shipping. Designers need to keep a wide variety of fonts and typefaces on hand for their different projects and branding campaigns. Magical symbol - Wunjo Rune - Wekhtar Tapestry. Labyrinth A maze design of bronze age Crete that symbolizes the path of initiation. Transliteration Thanks to the wonders of modern technology, one option I have is to use actual Unicode runic text. Unicode is an attempt to standardise every character possible and the latest version (4) is shown below. There is a rune pouch with runic symbols on them and questions are posed of the person who does the rune casting. Today, it is often. top 10 black dot dress ideas and get free sh. Available in different sizes and shapes, strings come in a rainbow of colors and include features like programmable options, constant or. Screen readers and search engines see words. Another source of information about the rune names is the Gothic alphabet (not to be confused with the “Gothic” script type, a style of Latin-letter writing that was used in the middle ages). As a default, a symbol of death is triggered whenever a creature does one or more of the following, as you select: looks at the rune; reads the rune; touches the rune; passes over the rune; or passes through a portal bearing the rune. The three "punctuation marks" are three variant forms of separators found in runic inscriptions, one a single dot, one a double dot and one cross-shaped. Projects will succeed although plans cannot be rushed and patience must be observed. To start out, look for the most frequent letter (or symbol) in each cryptogram — you’ll find it’s almost always E. ⦿ DragonsBane 101. Planet's Case \$29. Our sacred symbols represent that which is best in each and every one of us and what our people are, they are our strength, and our protection which empowers our us as a people to have the. Download this Set Of Old Norse Scandinavian Runes Runic Alphabet Futhark Ancient Occult Symbols Vikings Letters On White Rune Font Vector Illustration With Light Texture Ancient Norse Letter vector illustration now. Put the felt-tip pen / paint brush aside. A unique MMO set in the vast, fantasy world of Gielinor, brimming with diverse races, guilds and ancient gods battling for dominion. Again, the birch tree is an appropriate symbol for this process of purification in preparation for new beginnings. About Viking Symbols What are Odin's symbols? There are several symbols representing Odin. The image of the Volva (seeress) sitting in a dark cave with a fire, throwing down the runes, and telling a young adventurer whether his next quest will be profitable (or even survivable), carries with us, even into today. Tolkien ᛱ ᛲ ᛳ. Put these special symbols in your chat, status, name, comments, ascii art, messages, or Twitter. Nowadays, The Devil is mostly used as a comical figure to describe human being's evil intentions itself. Looking for Runes fonts? Click to find the best 142 free fonts in the Runes style. Runes carved onto small pieces of bone or wood were thought to tell the future, heal the sick, and bless people, places, and objects. Just for fun, you can type in any text below and then each letter will be transformed into their nearest Rune equivalent. On the basis of these stressed symbols, one may by count ing arrive at a sequence of numerals which, correctly interpreted, yields a date. ©2015&Wizards&of&the&Coast&LLC& 3& RuneLore) At)1st)level,)you)learn)the)basics)ofscribing)runes, and)are)able)toactivate)amaster)rune’s)fullrange). Similar Designs More from This Artist. The runes used by the Anglo-Saxons are known as the futhorc after its first six runes, which represented f, u, th, o, r, and c. This lexicon also uses the runic script. Thousands of new, high-quality pictures added every day. 67 - Spurn Undead, Symbol of Jeron, Crusader's Purity, Silvered Fury for providing the Ancient rune text. ©2015&Wizards&of&the&Coast&LLC& 3& RuneLore) At)1st)level,)you)learn)the)basics)ofscribing)runes, and)are)able)toactivate)amaster)rune’s)fullrange). He is known to be a Demon who tempts humans to commit sins and evil deeds. Call symbols with common terms. In this article, we unravel the mystery of runes and examine significant facts and history that provide us with a better understanding of these ancient. Icons are scalable and retina-ready. He even mentioned this in the Dementia Trailer. But musicians. Inspired designs on t-shirts, posters, stickers, home decor, and more by independent artists and designers from around the world. The three "runic symbols" are the Arlaug, Tvimadur and Belgthor symbols used exclusively for enumerating years in runic calendars of the early modern period. In Celtic-based pagan traditions, it is often used as a symbol of the three realms of earth, sea, and. Ask six rune experts 'Where did the runes originate from?' and you will probably get 6 different answers. Apprentice Clerval over at the Mage Training Arena is just a bit useless, which really isn't helping him to achieve his dream of being the first wizard in centuries to build a rune guardian from scratch. Viking Rune Game 1: Print and cut out your own practice runes. The current Rune Factory 4 is the sixth installment in the Rune Factory franchise. It symbolises mental and spiritual heritage, property and old traditions and family values. It’s meaning is a bit debatable, but the most accepted meaning of the symbol is that it is a symbol of eternal life and regeneration. Who should use the Rune Translation:. For a more personalized social media experience, or generally for a more unique text format use this online Runic text converter. The following table includes only the letters. Making the web more beautiful, fast, and open through great typography. Nevertheless, thousands of stones with runic inscriptions have been found where Vikings lived. The type's ctors will throw an exception if you try to construct it from the values -1 , 0xD800 , or 0x110000 , since these are not scalar values per the Unicode specification. For my purposes, the most important part of the plot is this: A plucky American psychologist, Dr John Holden. Unfortunately, there are few remains of runic writing on paper from the Viking era. For our discussion of runes in a commemorative and funerary context, it is important to note that a rune-like “crazy paving” design on one urn from Loveden Hill (no. When triggered, a symbol of death kills one or more creatures within 60 feet of the symbol (treat as a burst) whose combined total current hit points. According to that conclusion, we accept Vegvisir as symbol of rune dance a well, case dance is movement, Vegvisir is one who will. When runes are between ( ), then they. Believe it or not, both of these day to day symbols evolved thousands of years ago in the days of the Viking from the ancient art of Runes. There is a. Unusual or foreign words and some names will be converted to alphabetic runes and appear in a contrasting colo. BERKANA - Growth, Rebirth, Birch. The institutional repository contains records of research and full text versions of open access content including articles, working papers, preprints, technical reports, conference papers and data sets in various digital formats. top 10 black dot dress ideas and get free sh. Nauthiz Rune which means confinement and pain. Another adaptation of a pagan symbol into Christianity. #2437469 - safe, artist:ficficponyfic, part of a set, cyoa:madness in mournthread, cyoa, enchanted key, key, magic key, magic runes, magnetic field, monochrome. Shyanne Dorman. Users see symbols. Put these special symbols in your chat, status, name, comments, ascii art, messages, or Twitter. The symbol that represents life in alchemic science is the Mercury Sign. It is also the emblem of a writer. Viking Rune Tattoo Designs-An alphabet of runes existed among northern Germanic tribes long before the Viking Age began, but it was the Scandinavian Vikings who, toward the end of the first millennium, left the most lasting and potent evidence of this angular set of symbols. Page once said: "We decided that on the fourth album, we would deliberately play down the group name, and there wouldn't be any information whatsoever on the outer jacket. Anglo-Saxon runes (Futhorc/Fuþorc) Notes. Two varieties of runic writing are distinguished: an early runic alphabet, called futhark, and a later runic alphabet, or Danish runes (see Figure 1). It can be used only by Reiki Masters. It is part of Viking legend that Odin's horse, Sleipnir, had runic symbols engraved upon its teeth. A shorter version of 16 runes replaces the Elder FUÞARK in the 8 th century. The image of the Volva (seeress) sitting in a dark cave with a fire, throwing down the runes, and telling a young adventurer whether his next quest will be profitable (or even survivable), carries with us, even into today. As Isaac's dementia set in, he began to hear the symbols whispering to him. Runes The most common tools of the Shadowhunter, and the source of their ability to fight the demonic Incursion at all, are the Marks of Raziel, a complex runic language given by the Angel to grant powers beyond mundanes. Rune of the possible danger of realizing this link when unprepared. Rune casting is based on the principles of the subconscious and is unrelated to fortune telling. Forerunner symbols were used by the Forerunners to convey a variety of meanings. The converter happens automatically. Divinations: Awakening, awareness, hope-happiness, the ideal, paradigm shift; or lack of vision, sleep, blindness, hopelessness, cataclysmic change. txt) or read online for free. Here’s a guide to all you need to get started with using runes. The Alchemical Symbol for death. Runes originate in the Viking period, in the time of Odin, the chief god of Norse mythology, a time when longboats sailed from fjords of Scandinavia on military missions. Get inspiration and design your own name for free. LTD and published by Marvelous AQL. Rune is an Imperial thief and a member of the Thieves Guild in Riften. Gordian Knot Hunab Ku Uraeus Flower of Life Borromean Rings Globus Cruciger Vesica Pisces The Caduceus Holy Grail Merkaba The Infinity Medicine Wheel The Labyr. A letter or mark used as mystical or magic symbol. AZfonts collection is about 100 000 font available for download, trying or purchase. Template documentation [ view ] [ edit ] [ history ] [ purge ] The above documentation is transcluded from Template:Contains Runic text/doc. Rune Nail Decals 140 Rune Symbols Eldar Futhark Runic Alphabet Waterslide Nail Art Decals Choose Either Black, SILVER, GOLD or WHITE. Who should use the Rune Translation:. This spell allows you to scribe a potent rune of power upon a surface. Rune reading, also called rune casting, is a divination tool that uses stones with symbols to answer questions about your past, present, and future. A letter or mark used as mystical or magic symbol. There are two versions of this font: BabelStone Runic: plain runes. The ancient Celts, the people who lived in Britain and Ireland. Rune Mechanics is a quest in which the player assists an apprentice at the Mage Training Arena with constructing a rune guardian. This page lists the characters in the “Runic” block of the Unicode standard, version 13. that of the 16-character rune row. Labyrinth A maze design of bronze age Crete that symbolizes the path of initiation. 0, Unicode 4. Search 123RF with an image instead of text. Maybe for this reason, the 19th rune is called Ehwaz, the horse. The other runes are represented with their more classic equivalents , for example, the letter "f" represents Fehu, etc. Used around year 1100 to 1500. Rune is a lot like Ove: he doesn't say much, he has principles, and is handy around the house. Runes can grant combat bonuses, from increasing item pickup radius to keeping enemies stunned for longer. process of elimination. Get this from a library! Runes for transformation : using ancient symbols to change your life. This is the language of. Tolkien himself. As a child, Rune washed up on shore after a shipwreck near Solitude and was found by a local fisherman. Free vector icon. Nevertheless, the. The eight Semi-Precious, natural Gemstones such as Brazil Agate, Blue Lace Agate and Tiger’s Eye are hand painted with the symbols of the witch’s magical tools needed for divination. In addition, percent encode/decode URL parameters. To start out, look for the most frequent letter (or symbol) in each cryptogram — you’ll find it’s almost always E. Algiz (or Elhaz) - Literally: “Elk” - Esoteric: Protection, Higher Self Rune of the essential link or connection with the patterns of divine or archetypal consciousness, such as the Valkyrie. The Alchemy Symbol for Fire. There is a. See more ideas about Celtic symbols, Runes, Celtic. It consisted of letters, numerals and symbols for common words or phrases. Ove sees Rune's BMW as a symbol of Rune giving up on his family and on life, and therefore takes it as an unforgivable offense. Heritage Tapestry. The process of concept becoming realized. rune-converter. Made in Haxe + OpenFL. Originally the Frisian set was created by expanding the Elder Futhark set by four runes (the first four listed below – Ac, Os, Yr and Ior). Designers need to keep a wide variety of fonts and typefaces on hand for their different projects and branding campaigns. 54,543 downloads (29 yesterday) Free for personal use - 4 font files Download Donate to author. Top free runes symbols downloads. Pronunciation: dye-ko-me-o Alias: The Master Symbol. Making the web more beautiful, fast, and open through great typography. Learn more. For gods graved Odin, for elves graved Daïn, Dvalin the Dallier for dwarfs, All-wise for Jötuns, and I, of myself, graved some for the sons of men. a small node package for converting text to the runic alphabet. Anglo Saxon Rune Symbol Alphabet The Anglo Saxon alphabet is also known as futhorc and derives from the 24 rune eldar futhark. This is the language of. For a more personalized social media experience, or generally for a more unique text format use this online Runic text converter. These are the protection symbols of Witchcraft. process of elimination. The rune reader, thus, acts as a shaman, serving as a connection between the physical and the supernatural world. 1 DE FR IT ES PT �. Write the names & explanations on the back. Raido Rune which means travel or movement. It is based on Anglo-Saxon rune and used in the novel the Hobbit. Thousands of new, high-quality pictures added every day. Sofia Metal Queen. The trajectory of Rune's health supports Ove's assessment: within a year of buying the BMW and symbolically rejecting his community, Rune is diagnosed with Alzheimer's and slowly loses his connections to his community as he loses his memory. If someone says they want your CV however in ASCII format, all this means is they want 'plain' text with no formatting such as tabs, bold or underscoring - the raw format that any computer can understand. In Scandinavia, switches of birch are used on the body to stimulate the process of purification in the sauna, and can be used in Druid sweathouse rituals too. ooo And pshht, it's almost Emoji Day. We will take a look at many of these symbols including the Eye of Horus and the All-Seeing eye in greater detail. One more quick thing about runes: Every time Dumbledore's pensieve is described, JK tells us there are runes and symbols around the outside. Dagaz - Literally: “Day” or Dawn – Esoteric: Awakening Rune of the final synthesis of consciousness and the ultimate enlightenment of mind. Runes Symbols. +++ This font has been designed to meet the highest standards of. See more ideas about Celtic symbols, Runes, Celtic. Indian currency is called Rupee or Rupiya. A number of letters resemble those used in early Greek alphabet. Rune of the possible danger of realizing this link when unprepared. 67 - Spurn Undead, Symbol of Jeron, Crusader's Purity, Silvered Fury for providing the Ancient rune text. Vector Illustration Of Runes Occult Symbols. Each letter or symbol has a meaning that corresponds to an answer to the question that has been asked. The symbol known as Odin's triangle' or the Walknot' (meaning knot of the slain) has two variants. Ancient Egyptian Symbols for Death The noun for death, is clearly depicted in the explicit determinative (at the end). The fish was used worldwide as a religious symbol associated with the pagan "Great Mother Goddess. It is NOT print. It is possible that the runes used in the Cathedral of Dusk and the ones used by Nokris are an archaic version of the Hive runes that we see on the Dreadnaught. The largest line can be assumed to read "FIRE EMBLEM. "Dethek" was also sometimes used to refer to the Dwarvish language as a whole. The original use of this symbol is place of interest on maps, but not widely known. Псевдонимы, классные шрифты, символы и теги, относящиеся к TTSulokDari. Rune symbols have been found to date back as far as 150 AD. He admits that most of the coin he makes. It is more commonly known today as the symbol of Apple keyboard's command key. Runes can grant combat bonuses, from increasing item pickup radius to keeping enemies stunned for longer. Runes played an important part in the lives of the Vikings. From what I understand, wealth was regarded by the Norse as evil in the sense that it would be evil to hoard. I want to feel like I am in. Research UNE captures, distributes and preserves digital research products created by UNE researchers. That which we can read appears to be random. Another adaptation of a pagan symbol into Christianity. These are the protection symbols of Witchcraft. From the 7th century the Latin alphabet began to replace these runes, though some runes continued to appear in Latin texts representing whole words, and the Latin alphabet was extended with the runic letters þorn and wynn. Runes (or ᚠᚢᚦᚨᚱᚲ "Futhark") are a set of alphabets developed by the Germanic tribes. For word-wraps please press 'Enter'. In Celtic-based pagan traditions, it is often used as a symbol of the three realms of earth, sea, and. First select the symbol then you can drag&drop or just copy&paste it anywhere you like. Upload an Image. Page once said: "We decided that on the fourth album, we would deliberately play down the group name, and there wouldn't be any information whatsoever on the outer jacket. BabelStone Runic is a generic Unicode runic font covering all runic characters in the Unicode Standard version 7. Download a Free Preview or High Quality Adobe Illustrator Ai, EPS, PDF and High Resolution JPEG versions. Symbols are shown along with the text describing the operation or information relating to a specific component, control, message, gauge, or indicator. Runes are sets of symbols comprising an alphabet in which each character represents a specific sound. Each rune had a name, such as 'joy' or 'ash tree'. /// means that a certain number of runes have been scratched. Ancient symbols reflected peoples' most sacred beliefs, their relation to the Universe, worshipped gods, connection to family, animals and nature. See more ideas about Celtic symbols, Runes, Celtic. Runes originate in the Viking period, in the time of Odin, the chief god of Norse mythology, a time when longboats sailed from fjords of Scandinavia on military missions. The fish was used worldwide as a religious symbol associated with the pagan "Great Mother Goddess. First select the symbol then you can drag&drop or just copy&paste it anywhere you like. ɐ Upside Down Text ⓐ Bubble Ball Text a̷͉̐ Zalgo Text Generator Heart Symbol by Cool Fancy Text Generator. Witches use different kinds of witchcraft symbols in their writing. Convert text to runes. Since it deals with the soul and our spiritual self it heals disease and illness from the original source in the aura/energy fields. Isa Rune which means a frustration or inaction. If you're into runic and Nordic and want to learn the characters then you can always use the runic generator to practice your writing skills. Sort by : Relevance. Transcriber programs take a text written in Latin letters as their input. Odin was said to have brought the runes to earth after he was hung from a tree for nine days. ELDER VIKING is almost an exact interpretation of Elder Futhark plus a few extra characters corresponding as a best fit to the basic 26 upper-case Latin characters. Despite this, there are a number of ideas and beliefs that most Druids hold in common, and that help to define the nature of Druidism today:. ☂ Symbols; Search all characters. For this article I created stylized runes on a colorful background highlighting the main power of the rune. Smileys symbol is a copy and paste text symbol that can be used in any desktop, web, or mobile applications. This lexicon also uses the runic script. I am not the creator of these symbols our ancient. 54KB Gray wolf Odin Old Norse Runes Geri and Freki, symbol free png size: 500x500px filesize: 88. See full list on omniglot. best vw b list and get free shipping. Runes can be made of various materials, but are most commonly made of stone or wood, and feature a symbol from the runic alphabet on them. Another adaptation of a pagan symbol into Christianity. Alt-Codes can be typed on Microsoft Operating Systems: First make sure that numlock is on, Then press and hold the ALT key, While keeping ALT key pressed type the code for the symbol that you want and release the ALT key. Unicode standard doesn’t freeze, it continues to evolve. Sofia Metal Queen. Runic writing has been found on everything from giant stones to tiny pieces of horn, seal tucks, metal jewelry, and weapons. Nevertheless, thousands of stones with runic inscriptions have been found where Vikings lived. The book has several color illustrations, charts of the symbols, and a clear chapterized text. There are six rune-wheels which must be matched to the correct symbol in order to remove the shield and flip the switch. Plant says the symbol he created was drawn from sacred symbols of the ancient Mu civilization, a fantastical place that supposedly existed until 14,000 years ago. The symbol may be, in fact, a rune much like The Rune of Severance and act as a means to control something such as a human being, account or object. The runes originated around 200 B. This rune reinforces your patience, now is not the time to make decisions. The lefthand edge reads: - H UMH-YKXF--XNC-KORA G_CQ-FL IT UEZF-PZL-S - The top triangle: EB G--O--PTE--E-HM --QTY. Runes can grant combat bonuses, from increasing item pickup radius to keeping enemies stunned for longer. It offends everyone who is Asatru. This is the language of. The Alchemy Symbol for Fire. Aug 9, 2018 - Explore Jessica Fitzpatrick's board "Runes and Celtic Symbols", followed by 176 people on Pinterest. The symbol that represents life in alchemic science is the Mercury Sign. The symbol is in the shape of an Odal/Othala rune in the Elder Futhark. Indian government launched a new symbol for the Indian currency on 15 July 2010. If you are solely focused on using runes in divination, this may not be ideal for you at this point in your studies, although it does give excellent perspectives on the the runes from different angles, which will improve your understanding of their meanings. runic font generator Text Padding: Nordic is a unique modern font-family based on scandinavian runes and elegant geometric forms that can fit everyday design needs. It's scary. Icelandic symbols are called Galdrastafur. Runestones were large stones carved with pictures and writing in runes. Transcriber programs take a text written in Latin letters as their input. About the symbol/letter (Theory A) @Jimmy Shelter's answer is the one that should be marked as the correct (as it answers the author's doubt), but as a comment I'd like to talk about the symbol itself. It consisted of letters, numerals and symbols for common words or phrases. Magicians and Alchemists used the Heart symbols for incantations pertaining to matters related to love and romance. Research UNE captures, distributes and preserves digital research products created by UNE researchers. top 10 black dot dress ideas and get free sh. Each rune can be equipped and placed into one of the Praetor Suit's three Rune Slots, though only one is unlocked by default. +++ This font has been designed to meet the highest standards of. The converter happens automatically. Glyphs in Elder Scrolls Online are special additions to your weapons, armor and jewelry that are created by Enchanting and boost your character capabilities. Each rune is a representation of a cosmological principle or power. "Dethek" was also sometimes used to refer to the Dwarvish language as a whole. In Britain the birch rod was used rather more ferociously to purify the criminal of their misdeeds, and earlier still in. Sort by : Relevance. Write the names & explanations on the back. Runic is a Unicode block containing runic characters. Sowilo often references health issues and healing. In this way, cars are used. Rune and Ove moved into the neighborhood with their wives on the same day. Close or ESC key Search Tips. Aug 6, 2013 - Runes are letters in a family of ancient alphabets known as runic alphabets. Who should use the Rune Translation:. rune stone, a stone bearing a runic inscription. 0 contained 81 symbols: 75 runic letters (U+16A0–U+16EA), 3 punctuation marks (Runic Single Punctuation U+16EB ᛫, Runic Multiple Punctuation U+16EC ᛬ and Runic Cross Punctuation U+16ED ᛭), and three runic symbols that are used in early modern runic calendar staves ("Golden number Runes", Runic Arlaug Symbol U+16EE ᛮ, Runic Tvimadur Symbol U+16EF ᛯ, Runic Belgthor Symbol U+16F0 ᛰ). It is believed to provide protection against evil elves, trolls and dark magic. The Disadvantages. Tammi, also known as Rune, later Scribe, is a member of The Shepherds, and formerly of Empire Eighty-Eight. Anglo-Saxon. This symbol has been found in Celtic and Nordic inscriptions and arts as well as on Germanic coins and Swedish runes as far back as the 11th century. V will appear as the runic W because V and W are interchangeable in Scandinavian languages. The Pentagram is the Sacred Symbol of The Witch Frequently confused symbols: Pentagram - The Pentacle - Inverted Pentagram - Sigil of Baphomet. Ideal for runic practices and divination, as all runes are the same size and color. Rune Stones: are used as a system of divination, decision making and communication. Rune Factory is a fantasy simulation/role-playing video game series developed by Neverland Co. rune stone, a stone bearing a runic inscription. It is also possible that this tablet is explaining how Oryx communed with the Deep itself since the Cathedral of Dusk is deep in the Dreadnaught, and Nokris also communed with the Deep to make his pact with Xol. Page once said: "We decided that on the fourth album, we would deliberately play down the group name, and there wouldn't be any information whatsoever on the outer jacket. That being said, the art and writing of the traditional Chinese culture have graceful lines which lend themselves perfectly to the art of tattooing. Load Disqus. Anglo Saxon Rune Symbol Alphabet The Anglo Saxon alphabet is also known as futhorc and derives from the 24 rune eldar futhark. You can apply the principles of letter frequency analysis to work through these puzzles. It is believed to provide protection against evil elves, trolls and dark magic. 54KB Gray wolf Odin Old Norse Runes Geri and Freki, symbol free png size: 500x500px filesize: 88. Quote Reply # Dec 19 2006 at 9:00 AM Rating: Decent __DEL__1592774965283. He also shares Ove's love of fighting bureaucracy—the two of them set. The origins are a little murky (and much debated), but the letters appear to borrow from both the Roman alphabet and the Greek alphabet. However, the. The rune set is made by a master runic practitioner with more than 10 years of experience. Runes can be made of various materials, but are most commonly made of stone or wood, and feature a symbol from the runic alphabet on them. As mentioned earlier, most text is in NFC form, where base characters and modifiers are combined into a single rune whenever possible. See more ideas about Rune symbols, Runes, Symbols and meanings. Symbols list; Symbol drawings; Train details; Optimization function tests; Teach symbols; Fork me on GitHub; Other websites. All Runes Symbol Collage Print on archival-quality professional photo paper for brighter colors and sharper whites. 60/201) is used by Fennell (1964: 363) to support the argument for some level of runic literacy in the community which used the cemetery. ELDER VIKING is almost an exact interpretation of Elder Futhark plus a few extra characters corresponding as a best fit to the basic 26 upper-case Latin characters. Marker Symbols are a set of symbols produced by the Markers. Here is the text for a. "Dethek" was also sometimes used to refer to the Dwarvish language as a whole. 0 contained 81 symbols: 75 runic letters (U+16A0–U+16EA), 3 punctuation marks (Runic Single Punctuation U+16EB ᛫, Runic Multiple Punctuation U+16EC ᛬ and Runic Cross Punctuation U+16ED ᛭), and three runic symbols that are used in early modern runic calendar staves ("Golden number Runes", Runic Arlaug Symbol U+16EE ᛮ, Runic Tvimadur Symbol U+16EF ᛯ, Runic Belgthor Symbol U+16F0 ᛰ). Runic writing has been found on everything from giant stones to tiny pieces of horn, seal tucks, metal jewelry, and weapons. They have been discovered on spear heads, charms and even on headstones in the form of spells! It is said that the word “rune” has come to mean “secret, something hidden,” and many people believe the symbols were used as a form of divination. They were also used in rituals with a goal to strengthen relationships. It would be easy to make a simple chart breaking up in to 5 sections. Transcriber programs take a text written in Latin letters as their input. It is possible that the runes used in the Cathedral of Dusk and the ones used by Nokris are an archaic version of the Hive runes that we see on the Dreadnaught. Effort reaps its reward. It has 2 variants, called Long-branch and short-twig. Completing more trials unlocks the remaining slots at four and seven trials respectively. Making the web more beautiful, fast, and open through great typography. " The text. It’s meaning is a bit debatable, but the most accepted meaning of the symbol is that it is a symbol of eternal life and regeneration. Superhero Mask Symbols Pro. This is a rune of inheritance and the discarding of traditional well worn ways in order to move forward. Some symbols present in these two crop circles appear in the new CC Sutton Hall. Analyzing the symbols carefully, we discover that this runic CC is actually a MOSAIC of symbols and runes from previous crop circles, highlighting two incredible 2009 formations illustrating the Slide, full of symbols strangers in 5 Lines of ALIENS FACE MENTAL TRANSMISSION! That's the part that interests us. It consisted of letters, numerals and symbols for common words or phrases. The runic symbols remain ambiguous — is this a good journey for Dani, or a bad one? Is there a light at the end of the tunnel, or is it winking out? Has she found a new family, or lost herself?. Topics covered include:* A brief overview of runes, from Etruscan times to the present* A summary of the myths and lore that inform runic wisdom* Definitions of the basic rune symbols* Instructions on how to read the runes and rune spreads* An introduction to. Ansuz in elder and younger futhark. This page is about the meaning, origin and characteristic of the symbol, emblem, seal, sign, logo or flag: Ansuz (rune). LATEX Mathematical Symbols The more unusual symbols are not defined in base LATEX (NFSS) and require \usepackage{amssymb} 1 Greek and Hebrew letters α \alpha κ \kappa ψ \psi z \digamma ∆ \Delta Θ \Theta β \beta λ \lambda ρ \rho ε \varepsilon Γ \Gamma Υ \Upsilon χ \chi µ \mu σ \sigma κ \varkappa Λ \Lambda Ξ \Xi. More symbols in Runic Alphabet:. Aug 5, 2020 - Explore Manjula Nebhnani-Mayank's board "Rune symbols and meanings", followed by 800 people on Pinterest. by Jenn B (Scully7491), brush category: Signs and symbols. The term Runic Alphabet can refer to any of several closely-related real-life historical writing systems. They were also used in rituals with a goal to strengthen relationships. Regardless of the trigger method or methods chosen, a creature more than 60 feet from a symbol of death can’t trigger it (even if it meets one or more of the. Runes have selection boxes with higher priority over units, meaning that if a unit is standing over a rune, the mouse cursor highlights and selects the rune first. HTML Charsets HTML ASCII HTML WIN-1252 HTML ISO-8859 HTML Symbols HTML UTF-8 HTML UTF-8 Latin Basic Latin Supplement Latin Extended A Latin Extended B Modifier Letters Diacritical Marks Greek and Coptic Cyrillic Basic Cyrillic Supplement HTML Symbols. In this article, we unravel the mystery of runes and examine significant facts and history that provide us with a better understanding of these ancient. As a default, a symbol of death is triggered whenever a creature does one or more of the following, as you select: looks at the rune; reads the rune; touches the rune; passes over the rune; or passes through a portal bearing the rune. The first of those is Valknut, also known as 'the Knot of the Slain Warriors'. BabelStone Runic is a generic Unicode runic font covering all runic characters in the Unicode Standard version 7. A set of 25 runes of the Elder Futhark are made of fine quality and color green onyx. All Runes Symbol Collage Print on archival-quality professional photo paper for brighter colors and sharper whites. There are old runic symbols on ceramic pieces, carved from wood, woven, examples from the Stone Age, on. Experts suggest that the ‘Y’ suggest that man has a choice to choose between good and evil. There are many symbols for many purposes, healing, protection, curse, luck, life love. There is a. Sowilo often references health issues and healing. Learn more. Regardless of the trigger method or methods chosen, a creature more than 60 feet from a symbol of death can’t trigger it (even if it meets one or more of the. Thousands of new, high-quality pictures added every day. As a default, a symbol of death is triggered whenever a creature does one or more of the following, as you select: looks at the rune; reads the rune; touches the rune; passes over the rune; or passes through a portal bearing the rune. This symbol text generator can convert any words and numbers you enter into symbol texts that can be used anywhere, such as facebook, youtube, twitter, etc. Kauna, Jeran, Sigel, Ehwaz, Mannaz, Ingwaz, Dagaz/Dæg, Stan, and others. It is linked to the Age of Pisces and also has associations with the Hindu deity Vishnu but more so with Dagon the fish-god. Connect with an ancient divining tradition with this deluxe polished gemstone runes set. Shyanne Dorman. It can be used only by Reiki Masters. It's arrows, stars, control characters etc. THE REIKI MASTER SYMBOL. Vikings also used runes on. Holy Earth Symbol for mother earth. This is a rune of burning light, heat, and warmth (physical and mental), and stands for an awakened and enlightened mind. For Lections search, a drop down menu will show all the available scripture citations as soon as you s. There is a rune pouch with runic symbols on them and questions are posed of the person who does the rune casting. It is neither inferred nor implied that any item sold by MOTORCYCLEiD. To meditate on these symbols is to attune to the wisdom kept in the collective store house of all existence. I'm not planning to develop this any further. The statement shows that Lindroth in fact believed that the Kensington Stone’s “rune row” was a rune row in the true sense; that these symbols have been used in other inscriptions than the Kensington Stone and that they therefore in principle have the same character as e. Yeah, yeah, we all know that's a pause symbol. Divinations: Awakening, awareness, hope-happiness, the ideal, paradigm shift; or lack of vision, sleep, blindness, hopelessness, cataclysmic change. The rune reader, thus, acts as a shaman, serving as a connection between the physical and the supernatural world. Thurisaz, conversely, requires. This is usually so they can easily import the file into their own applications without issues. Load Disqus. 54KB Gray wolf Odin Old Norse Runes Geri and Freki, symbol free png size: 500x500px filesize: 88. * There may be moving or flashing imagery. Rune Nail Decals 140 Rune Symbols Eldar Futhark Runic Alphabet Waterslide Nail Art Decals Choose Either Black, SILVER, GOLD or WHITE. Regardless of the trigger method or methods chosen, a creature more than 60 feet from a symbol of death can’t trigger it (even if it meets one or more of the. Effort reaps its reward. But the Bluetooth symbol is actually combination of Nordic runes. best vintage lace baby girl wedding ideas and get free shipping. Thousands of new, high-quality pictures added every day. This is a rune of burning light, heat, and warmth (physical and mental), and stands for an awakened and enlightened mind. Search 123RF with an image instead of text. Kaunaz Rune which means fire or shedding light on something. Rune Factory is a fantasy simulation/role-playing video game series developed by Neverland Co. ” It has been found on several stone carvings vvith funerary motifs, as is thought to signify the. Screen readers and search engines see words. Installation. It was often used in funerals and tombs in ancient Egypt. In Celtic-based pagan traditions, it is often used as a symbol of the three realms of earth, sea, and. Runes and Rune Reading: An Introduction to the Runic Symbols of Northern Europe joins a fairly large contingent of volumes contributing to this popular topic. For example:. Runes are sets of symbols comprising an alphabet in which each character represents a specific sound. Anyone who works with LaTeX knows how time-consuming it can be to find a symbol in symbols-a4. Looking for Runes fonts? Click to find the best 142 free fonts in the Runes style. Rune was a white supremacist, a result of teenage rebellion from her immediate family (who had escaped from the Herren Clan,) leading her back into the welcoming arms of the violent clan. They also made the blackletter font Wagner. But musicians. The North American version of Rune Factory 4 was published by XSeed. The original encoding of runes in UCS was based on the recommendations of the "ISO Runes Project" submitted in 1997. The sun's rays offer vital healing. Maybe for this reason, the 19th rune is called Ehwaz, the horse. The rune reader, thus, acts as a shaman, serving as a connection between the physical and the supernatural world. All humanity needs to produce high-quality text. There is a rune pouch with runic symbols on them and questions are posed of the person who does the rune casting. Viking Futhark Rune Symbols - Runes & Meanings. The ‘language’ of Symbols. Ancient Runes is a. your name) you want to write in runes and choose the font. Enter text to translate: Devon Rohlfing © 2015. Regardless of the trigger method or methods chosen, a creature more than 60 feet from a symbol of death can’t trigger it (even if it meets one or more of the. 5K Rune Essence. This is where the NFD form comes in handy. Employing shocking imagery and symbols has downsides: Many intelligent and powerful people who would make good Satanists are put off because they do not suspect Satanism contains worthwhile philosophy. The Delta Rune is an emblem representing the Dreemurr royal family. See more ideas about Rune symbols, Runes, Symbols and meanings. From the 7th century the Latin alphabet began to replace these runes, though some runes continued to appear in Latin texts representing whole words, and the Latin alphabet was extended with the runic letters þorn and wynn. How to Use Symbols. Any rune can be. Included among the graffiti scattered throughout the USG Ishimura and the Sprawl, Marker Script is readily apparent. Runic writing has been found on everything from giant stones to tiny pieces of horn, seal tucks, metal jewelry, and weapons. In fact, Mayan writing consists of symbols called glyphs. Here’s a guide to all you need to get started with using runes. More symbols in Runic Alphabet:. Basically, runes are really bad news in the 1957 film Night of the Demon (or Curse of the Demon in the US). Based on the art of rune stone casting using the symbols of an ancient Norse alphabet, Oracle of the Runes can be seen as just a bit of fun. Sowilo is an extremely fortunate rune and is often considered the best possible rune in a reading. ELDER VIKING is almost an exact interpretation of Elder Futhark plus a few extra characters corresponding as a best fit to the basic 26 upper-case Latin characters. The ancient Celts, the people who lived in Britain and Ireland. Runes were used as a method of communication across Scandinavia and in other Germanic nations from around the 3rd century CE to around the 13th century, when they were displaced by the Roman alphabet. Enter some text in the box below, then click the preview button. Yeah, yeah, we all know that's a pause symbol. Find Norse Runes Futhark Alphabet Stones Symbols stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. " Below is a table containing a full set of 24 rune (alphabet) symbols, placed within their Aett or set. For Lections search, a drop down menu will show all the available scripture citations as soon as you s. Runes originate in the Viking period, in the time of Odin, the chief god of Norse mythology, a time when longboats sailed from the fjords of Scandinavia on military missions. In Old Norse the word rune means 'letter', 'text' or. One Battle! One Victory!" In a sense, the poster is declaring that the swastika is a "sieg" rune symbol. The other runes are represented with their more classic equivalents , for example, the letter "f" represents Fehu, etc. Sowilo warns for using the powers of good to defeat evil. That being said, the art and writing of the traditional Chinese culture have graceful lines which lend themselves perfectly to the art of tattooing. #108223994 - Set of old norse scandinavian runes. The last remnants are said to be the primitive statues at Easter Island. In this way, cars are used. txt) or read online for free. Matte finish resists scratches and glare. Frozen is mi little daughter’s favourite Disney film, and I noticed the runic text in the book. Tolkien ᛱ ᛲ ᛳ. Symbols ☯ Emoji FSYMBOLS is a collection of cute and cool symbols and special text characters for your Facebook, Instagram bio, chat, posts, or some profiles. For the purpose of analyzing characters, it is often easier to handle runes after decomposition into their smallest components. Make sure the future is always new. For example the rune ᚦ makes a noise that is similar to the english written "th" and you can see we need two letter to express that sound. Originally the Frisian set was created by expanding the Elder Futhark set by four runes (the first four listed below – Ac, Os, Yr and Ior). A Little Bit of Runes An Introduction to Norse Divination (Book) : Eason, Cassandra : "Nordic runes are the most popular and easiest symbols to work with, and can effectively release energy for a positive purpose in one's life. This symbol is used to heal the soul. Runes were believed to have magical properties, and warriors used runes on their weapons to enhance their power in battle. It is the shape of an isosceles with a ‘Y’ in the middle. Maybe for this reason, the 19th rune is called Ehwaz, the horse. The symbol is in the shape of an Odal/Othala rune in the Elder Futhark. ooo And pshht, it's almost Emoji Day. Nauthiz Rune which means confinement and pain. Anyone who works with LaTeX knows how time-consuming it can be to find a symbol in symbols-a4. Claymore Symbols from Awakened by Awa303 on DeviantArt DeviantArt is the world's largest online social community for artists and art enthusiasts, allowing people to connect through the creation and sharing of art. The first of those is Valknut, also known as 'the Knot of the Slain Warriors'. Convert Unicode characters in UTF-16, UTF-8, and UTF-32 formats to their Unicode and decimal representations and vice versa. Plant’s symbol (circle around a feather) features the feather of Ma’at, the Egyptian goddess of justice and fairness. Othila : Ansuz : Gebo : Mannaz : Forward movement is an unfoldment that although it is coming from the past is unlike the past ever was. Isa Rune which means a frustration or inaction. Runes and Rune Reading: An Introduction to the Runic Symbols of Northern Europe joins a fairly large contingent of volumes contributing to this popular topic. Alt-Codes can be typed on Microsoft Operating Systems: First make sure that numlock is on, Then press and hold the ALT key, While keeping ALT key pressed type the code for the symbol that you want and release the ALT key. Adventure Time sometimes includes secret messages written in a runic code. Puzzles are a big part of Hellblade Senua’s Sacrifice. , as magical symbols engraved in stone; they were developed into the first Rune alphabet, the “elder” Futhark (‘futhark’ being a transliteration of the first six letters), an alphabet of twenty four characters. * Windows Users may have to get past SmartScreen to use. (Cookies must be enabled in your browser. You have to buy the third Rune from a black market shop. Rune was a white supremacist, a result of teenage rebellion from her immediate family (who had escaped from the Herren Clan,) leading her back into the welcoming arms of the violent clan. It consisted of letters, numerals and symbols for common words or phrases. Screen readers and search engines see words. The earliest known instance is in Fire Emblem: The Sacred Stones, where symbols from the Elder Futhark alphabet appear in the menu background. Media Source. What are Runes? View the Rune Unicode Chart. Runes are special items earned by completing Rune Trials. Occult Symbols and Meanings: Pentagon symbol is a useful witchcraft symbol. To find it, climb the stairs next to the whale carcass. Glyphs in Elder Scrolls Online are special additions to your weapons, armor and jewelry that are created by Enchanting and boost your character capabilities. Maybe for this reason, the 19th rune is called Ehwaz, the horse. Designers need to keep a wide variety of fonts and typefaces on hand for their different projects and branding campaigns. What I didn’t know is the fact it wasn’t attrezzo text, but lines with real old norse words and sentences, with sense linked to the plot of film. The runes were inspired by the Latin and Greek alphabets, but adopted for carving in wood and stone, since the vikings seldom used parchment and paper was unheard of. The runic generator will not only remove the hassle of creating runic texts but can also be used to translate runic text and writing in English to Runes. ” Origins of the Bluetooth symbol — Source: medievalarchives. In this way, cars are used. The runes originated around 200 B. rune stone, a stone bearing a runic inscription. Reference: 11096r-n4013r-runic-additions. As a Church we have a ‘language’ that helps us put expression to the moments of grace that mark our lives. Similar Designs More from This Artist. runic letter - any character from an ancient Germanic alphabet used in Scandinavia from the 3rd centur. david cofré Rate this symbol: (3. Use the text generator tool below to preview Rune font, and create awesome text-based images or logos with different colors and hundreds of text effects. Seal of the Left Hand Path Indicates Black magic and the path to Satan. The Rune Converter transforms Roman alphabet, as used in modern English, into five systems of Germanic runic writing: Elder Futhark, Anglo-Saxon runes, Long Branch Younger Futhark, Short Twig Younger Futhark and staveless runes (note that it does not translate the words themselves, it only converts letters into runes). The North American version of Rune Factory 4 was published by XSeed. A variation with three. Among the code points equivalent to rune (including rune itself), SimpleFold returns the smallest rune > r if one exists, or else the smallest rune >= 0. Each rune can be equipped and placed into one of the Praetor Suit's three Rune Slots, though only one is unlocked by default. Quote Reply # Dec 19 2006 at 9:00 AM Rating: Decent __DEL__1592774965283. Enter text to translate: Devon Rohlfing © 2015. It is also the emblem of a writer. The trajectory of Rune's health supports Ove's assessment: within a year of buying the BMW and symbolically rejecting his community, Rune is diagnosed with Alzheimer's and slowly loses his connections to his community as he loses his memory. Who should use the Rune Translation:. It would give a feeling of completion when we finally succeed. The puzzles you must solve inclu. Rune and Ove moved into the neighborhood with their wives on the same day. The best website for free high-quality Runic fonts, with 32 free Runic fonts for immediate download, and 37 professional Runic fonts for the best price on the Web. Although the members themselves never came straight out and said what they meant, many people have since research what the symbols were. And search more of iStock's library of royalty-free vector art that features Alphabet graphics available for quick and easy download. Unicode standard doesn’t freeze, it continues to evolve. The type's ctors will throw an exception if you try to construct it from the values -1 , 0xD800 , or 0x110000 , since these are not scalar values per the Unicode specification. They have been discovered on spear heads, charms and even on headstones in the form of spells! It is said that the word “rune” has come to mean “secret, something hidden,” and many people believe the symbols were used as a form of divination. Now that we a new Indian Rupee symbol, we also need to know how to type it in computer applications like MS Word, MS Excel, HTML web pages or in plain text. Literal Meanings of "Algiz" Defense, Protection, Sanctuary Description. The set comes with a regular font option, a bold option, and the complete runic alphabet. A Finnish or Scandinavian epic poem, or a division of one, especially a division of the Kalevala. Rune casting is based on the principles of the subconscious and is unrelated to fortune telling. rune n : any character from an ancient germanic alphabet used in scandinavia from the 3rd century to the middle ages; "each rune had its own magical significance" [syn: runic letter ] similar words(1). Shop Gold Valknut Symbol on Runes Pattern Duvet Cover created by LoveMalinois. " Below is a table containing a full set of 24 rune (alphabet) symbols, placed within their Aett or set. Hundreds of Mayan symbols can be found carved on stone, which allow archeologists and other researchers to gain an understanding of their culture. Symbols played a very important role in the lives of ancient people. The elements are like Earth, Wind, Fire, Water and the spirits surrounding them. ” Origins of the Bluetooth symbol — Source: medievalarchives. FISH SYMBOL - Also known as the Ichthys Symbol (Greek for fish). Galdrtanz is the way of Rune Dance and Dance is artistically, spiritual and divine way of moving, movement. this is my research I have put together and I. The ken rune, shown left, represents the flame or fire. There are two versions of this font: BabelStone Runic: plain runes. ᚠ ᚡ ᚢ ᚣ ᚤ ᚥ ᚦ ᚧ ᚨ ᚩ ᚪ ᚫ ᚬ ᚭ ᚮ ᚯ ᚰ ᚱ ᚲ ᚳ ᚴ ᚵ ᚶ ᚷ ᚸ ᚹ ᚺ ᚻ ᚼ ᚽ ᚾ ᚿ ᛀ ᛁ ᛂ ᛃ ᛄ ᛅ ᛆ ᛇ ᛈ ᛉ ᛊ ᛋ ᛌ ᛍ ᛎ ᛏ ᛐ ᛑ ᛒ ᛓ ᛔ ᛕ ᛖ ᛗ ᛘ ᛙ ᛚ ᛛ ᛜ ᛝ ᛞ ᛟ ᛠ ᛡ ᛢ ᛣ ᛤ ᛥ ᛦ ᛧ ᛨ ᛩ ᛪ. Detexify is an attempt to simplify this search. franks casket ᛴ ᛵ ᛶ ᛷ ᛸ. Rune translators have also been able to shed some more light on the game's infamous end-game mural. Thousands of new, high-quality pictures added every day. Shyanne Dorman. Without proper rendering support, you may see question marks, boxes, or other symbols instead of runes. Try dragging an image to the search box. Since the runes are a vital part of the pre-Christian northern European mythology, worldview, and spiritual practice, I thought it would be fitting … Continue reading The 10 Best Books on the. I salvage the runes with a mystic salvage kit, again, got a bunch of motes and no symbols or charms. Gibo Rune which means a blessing or partnership. V will appear as the runic W because V and W are interchangeable in Scandinavian languages. The original use of this symbol is place of interest on maps, but not widely known. FontMonger:Math Symbol. Connect with an ancient divining tradition with this deluxe polished gemstone runes set. AZfonts collection is about 100 000 font available for download, trying or purchase. Odin was said to have brought the runes to earth after he was hung from a tree for nine days. runic font generator Text Padding: Nordic is a unique modern font-family based on scandinavian runes and elegant geometric forms that can fit everyday design needs. Unfortunately, there are few remains of runic writing on paper from the Viking era. [Kaedrich Olsen] -- "This book presents an amalgam of ancient runic wisdom and modern transformative techniques such as hypnotherapy, affirmations, NLP, visualizations, meditation and a smattering of sociology. To start out, look for the most frequent letter (or symbol) in each cryptogram — you’ll find it’s almost always E. The Operator symbol can also be a possible portal for the Operator. He is known to be a Demon who tempts humans to commit sins and evil deeds. They were used in Norway, Sweden, Denmark, Germany, and Great Britain until the Latin alphabet took precedence. Duration instantaneous plus 1 minute/level (see text) Saving Throw Will negates (harmless); Spell Resistance yes (harmless) Description This spell allows you to lay your hand upon a magical glyph, symbol, or other magical spell effect (referred to in this spell description as a “rune”) and attempt to absorb the essence of its effect. This is the language of. Making the web more beautiful, fast, and open through great typography. It is linked to the Age of Pisces and also has associations with the Hindu deity Vishnu but more so with Dagon the fish-god. Size 13x13 mm Hand-engraved symbols. It's scary. I won't 'rune' this for you. Tables are in PDF format so you will need Adobe Acrobat Reader to view them. The eldest runes found in Iceland date back to the 10 th century. This book, written by an authority on divination systems, shows readers how to make their own set of runes and how to interpret them. Divinations: Connection with the gods, awakening, higher life, protection; or hidden danger, consumption by divine forces, loss of. This language is expressed in symbols and rituals (which are beyond words) and also, of course, in words that help us communicate the mystery of our relationship with the Divine.
|
2020-10-31 09:56:46
|
{"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": 0.319398432970047, "perplexity": 6804.401419297072}, "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-2020-45/segments/1603107917390.91/warc/CC-MAIN-20201031092246-20201031122246-00528.warc.gz"}
|
https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.20/share/doc/Macaulay2/Macaulay2Doc/html/_homology.html
|
# homology -- general homology functor
## Description
Most applications of this functor are dispatched through HH. If it is intended that i be of class ZZ, M be of class A, and N be of class B, then the method for computing HH_i(M,N) can be installed with code of the following form.
homology(ZZ, A, B) := opts -> (i,M,N) -> ...
|
2022-12-07 16:47:22
|
{"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": 0.5035513639450073, "perplexity": 2625.4404299021185}, "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/1669446711200.6/warc/CC-MAIN-20221207153419-20221207183419-00272.warc.gz"}
|
https://cmat.uminho.pt/events/eikonal-quasinormal-modes-and-shadow-string-corrected-d-dimensional-black-holes
|
# Eikonal quasinormal modes and shadow of string-corrected d-dimensional black holes
### online | 2021-04-21 | 14:30
Filipe Moura
ISCTE - Instituto Universitário de Lisboa e Instituto de Telecomunicações
We compute the quasinormal frequencies of $d$-dimensional spherically symmetric black holes with leading string $\alpha'$ corrections in the eikonal limit for tensorial gravitational perturbations and scalar test fields. We find that, differently than in Einstein gravity, the real parts of the frequency are no longer equal for these two cases. The corresponding imaginary parts remain equal to the principal Lyapunov exponent corresponding to circular null geodesics, to first order in $\alpha'$. We also compute the radius of the shadow cast by these black holes.
|
2022-12-10 02:34:14
|
{"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": 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": 0.7965435981750488, "perplexity": 1043.8177601802213}, "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/1669446711637.64/warc/CC-MAIN-20221210005738-20221210035738-00567.warc.gz"}
|
https://socratic.org/questions/what-is-the-electron-configuration-representing-an-atom-of-aluminum-in-an-excite
|
# What is the electron configuration representing an atom of aluminum in an excited state?
Feb 15, 2017
An example of the electron configuration of aluminum in an excited state is ${\text{1s"^2 "2s"^2 "2p"^6 "3s"^1 "3p}}^{2}$.
#### Explanation:
There are an infinite number of possible excited states for the valence electrons of aluminum, depending on how much energy the electrons have absorbed.
The ground state of aluminum is ${\text{1s"^2 "2s"^2 "2p"^6 "3s"^2 "3p}}^{1}$
In the ground state the $\text{3s}$ electrons are unable to form bonds. They must absorb energy to move to a higher energy state so that the electrons can be used to form bonds.
What happens is that the atom will promote a $\text{3s}$ electron to the empty $\text{3p}$ orbital, forming an excited state.
The atom can then form 3 ${\text{sp}}^{2}$ hybrid orbitals to form three bonds to other atoms.
This enables aluminum to achieve a more stable electron configuration like that of neon.
|
2020-07-10 18:07:29
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.22932183742523193, "perplexity": 491.686600416667}, "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-2020-29/segments/1593655911896.73/warc/CC-MAIN-20200710175432-20200710205432-00455.warc.gz"}
|
http://apmonitor.com/do/index.php/Main/IntegralObjective
|
## Integral Objective (Luus)
Objective: Set up and solve the Luus optimal control benchmark problem. Create a program to optimize and display the results. Estimated Time: 30 minutes
The Luus optimal control problem has an integral objective function. Integrals are a natural expression of many systems where the accumulation of a quantity is maximized or minimized.
$$\min_u \frac{1}{2} \int_0^2 x_1^2(t) \, dt$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=u$$ $$x_1(0) = 1$$ $$-1 \le u(t) \le 1$$
Integral expressions are reformulated in differential and algebraic form by defining a new variable.
$$x_2 = \frac{1}{2} \int_0^2 x_1^2(t) \, dt$$
The new variable and integral are differentiated and included as an additional equation. The problem then becomes a minimization of the new variable x_2 at the final time.
$$\min_u x_2\left(t_f\right)$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=u$$ $$\frac{dx_2}{dt} = \frac{1}{2} x_1^2(t)$$ $$x_1(0) = 1$$ $$x_2(0) = 0$$ $$t_f = 2$$ $$-1 \le u(t) \le 1$$
The initial condition of the integral starts at zero and becomes the integral in the time range of 0 to 2. The value that is minimized is at the final point in the time horizon of the optimal control problem.
#### Solutions
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
m = GEKKO()
nt = 101
m.time = np.linspace(0,2,nt)
# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)
u = m.Var(value=0,lb=-1,ub=1)
p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)
# Equations
m.Equation(x1.dt()==u)
m.Equation(x2.dt()==0.5*x1**2)
# Objective Function
m.Obj(x2*final)
m.options.IMODE = 6
m.solve()
# m.solve(remote=False) # for local solve
plt.figure(1)
plt.plot(m.time,x1.value,'k-',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u.value,'r--',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
# this solution uses the m.integral function and produces the same answer
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
m = GEKKO()
nt = 101
m.time = np.linspace(0,2,nt)
# Variables
x1 = m.Var(value=1)
u = m.Var(value=0,lb=-1,ub=1)
p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)
# Equations
m.Equation(x1.dt()==u)
x2 = m.Intermediate(m.integral(0.5*x1**2))
# Objective Function
m.Obj(m.integral(0.5*x1**2)*final)
m.options.IMODE = 6
m.solve()
# m.solve(remote=False) # for local solve
plt.figure(1)
plt.plot(m.time,x1.value,'k-',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,u.value,'r--',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
#### References
1. Beal, L., GEKKO for Dynamic Optimization, 2018, URL: https://gekko.readthedocs.io/en/latest/
2. Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. Article
|
2020-10-26 22:19:59
|
{"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": 0.7913489937782288, "perplexity": 3566.1816328368704}, "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/1603107892062.70/warc/CC-MAIN-20201026204531-20201026234531-00316.warc.gz"}
|
https://demo7.dspace.org/items/a35affd9-716d-4da8-8a15-b27175f619c3
|
## Massive 3D Supergravity
##### Authors
Andringa, Roel
Bergshoeff, Eric A.
de Roo, Mees
Hohm, Olaf
Sezgin, Ergin
Townsend, Paul K.
##### Description
We construct the N=1 three-dimensional supergravity theory with cosmological, Einstein-Hilbert, Lorentz Chern-Simons, and general curvature squared terms. We determine the general supersymmetric configuration, and find a family of supersymmetric adS vacua with the supersymmetric Minkowski vacuum as a limiting case. Linearizing about the Minkowski vacuum, we find three classes of unitary theories; one is the supersymmetric extension of the recently discovered massive 3D gravity'. Another is a new topologically massive supergravity' (with no Einstein-Hilbert term) that propagates a single (2,3/2) helicity supermultiplet.
Comment: 39 pages, 2 figures; v2: minor corrections, refs. added; v3: version to appear in CQG
##### Keywords
High Energy Physics - Theory
|
2022-12-06 14:19:34
|
{"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": 0.6892325282096863, "perplexity": 12036.16882488916}, "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/1669446711108.34/warc/CC-MAIN-20221206124909-20221206154909-00849.warc.gz"}
|
https://electronics.stackexchange.com/questions/380948/why-is-the-input-voltage-of-transistors-in-the-cmos-circuit-set-to-vdd-when-calc
|
# Why is the input voltage of transistors in the CMOS circuit set to Vdd when calculating the equivalent resistance?
When deriving the equivalent resistance formula of NMOS inverter the graph which is used in derivation is as shown:
$$R_{eq} = \frac{1}{-V_{dd}/2} \int_{V_{dd}}^{V_{dd}/2} \frac{V}{I_{Dsat}(1+\lambda V_{dd})} dx \approx \frac{3}{4}{V_{dd}}{I_{dsat}}(1-\frac{7}{9}\lambda V_{dd})$$ When calculating the equivalent resistances of NMOS and PMOS transistors in a CMOS inverter i was instructed to use this formula and for the saturated current which plays a part to take $$I_{Dsat}=\frac{B}{2}(V_{gs}-V_t)^2 = \frac{B}{2}(V_{dd}-V_t)^2$$ where B is a property of the transistor.
Why is Vdd taken as the gate-source voltage of both transistors for the saturated current if neither is in saturation at that point on the V(output)=V(V(input)) graph of CMOS inverter and why is it used in the derivation in the first place? How is it connected to the resistance which we want?
Edit: The resistance i'm asking for is the dynamic resistance of the transistor used when calculating the time-delay of the rising and falling edge of the graph. The equations of time-delay are given as: $$tp_{HL}=0.69R_{eqn}C_l$$ where Cl is the capacitance of the inverter and tpHL is the time for output voltage to go from logical 1 to logical 0 (highest and lowest voltage). This "discharging" of the transistor is done by the NMOS transistor so Reqn is it's dynamic resistance. Similar is with Reqp (tpLH). The formulas above are connected to this Reqn and Reqp but i don't understand the way it's calculated and why it uses Vdd for both transistors when calculating the saturated current when they are not in a state of saturation at that voltage, rather NMOS is linear and PMOS is turned off.
• Include the schematic to which this applies, yes I know the circuit of an inverter but I do not know how you're using it. I do not know what you mean by the "equivalent resistance". Maybe you mean the small signal impedance between VDD and ground when an inverter has input and output shorted. I also never like i was instructed to use this formula that makes me think the teacher has no clue/cannot explain how a circuit works. Formulas are pointless if you do not understand what happens. If you understand what happens the formulas become obvious. – Bimpelrekkie Jun 21 '18 at 13:49
• @Bimpelrekkie I tried to expand on what troubles me and added the edit with some more information. – edward_d Jun 21 '18 at 14:07
We assume that the input to a CMOS gate is driven by another CMOS gate, and that the output of a CMOS gate is either at $V_{DD}$ for a logic 1 or at ground for a logic 0. We also (usually) assume that the sources of all NMOS transistors are tied to ground and that the sources of all PMOS transistors are tied to $V_{DD}$.
Another simplifying assumption is that the inputs of the logic gate that you wish to analyze are stable and at either $V_{DD}$ or ground. If the input is at $V_{DD}$ then the PMOS transistors are cut off and we are only interested in what the NMOS transistors are doing. Since the NMOS source is at ground we use $$V_{GSN} = V_G - V_S = V_{DD} - 0 = V_{DD}$$ If you assume that the logic gate input is at ground then the NMOS is cut off and $V_{GSP} = -V_{DD}$.
Of course, that's a lot of simplifying assumptions. The dynamic behaviour is much more complex, and the effective $R_{DS}$ changes as the logic gate's output voltage (and hence the transistor's $V_{DS}$) changes. If you really want good answers, simulate in SPICE with accurate input rise/fall times and parasitic capacitances. For back-of-the envelope calculations you could approximate $R_{DS}$ with something like twice the effective $R_{DS}$ when $V_{DS} = V_{DD}$ but this would be really crude. The $R_{MID}$ in your first graph is another approximation using $I_{DS}$ when $V_{DS} = V_{DD}/2$.
|
2020-02-20 11:42:17
|
{"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": 0.8024598956108093, "perplexity": 667.1768957051394}, "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/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00277.warc.gz"}
|
https://byjus.com/questions/what-are-the-base-units-in-the-si-system-of-units/
|
# What Are The Base Units In The Si System Of Units?
The International System of Units (often abbreviated to SI from the French name ‘Le Systeme International d’Unites’) defines the standard unit of any physical quantity in terms of seven base quantities. These base quantities are:
Metre – the unit of length, denoted by the symbol m. Note that length is represented by ‘l’.
Kilogram – the unit of mass, denoted by the symbol kg. Note that mass is represented by ‘m’.
Second – the unit of time, denoted by the symbol s. Note that time is represented by ‘t’.
Ampere – the unit of electric current, denoted by the symbol A. Note that electric current is represented by ‘I’.
Kelvin – the unit of thermodynamic temperature, denoted by the symbol K. Note that the thermodynamic temperature is represented by ‘T’.
Mole – the unit for the amount of substance (or the number of particles of a given substance, denoted by the symbol mol. Note that the amount of substance is represented by ‘n’.
Candela – the unit of luminous intensity, denoted by the symbol cd. Note that luminous intensity is represented by ‘Iv’.
|
2021-09-16 12:16:35
|
{"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": 0.9598525166511536, "perplexity": 434.06617408763145}, "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-2021-39/segments/1631780053493.41/warc/CC-MAIN-20210916094919-20210916124919-00564.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/algebra/college-algebra-7th-edition/chapter-1-equations-and-graphs-section-1-7-solving-inequalities-1-7-exercises-page-149/77
|
## College Algebra 7th Edition
$x\geq\frac{c}{a}+\frac{c}{b}$
$a(bx-c)\geq bc$ We divide by $a$: $bx-c\geq\frac{bc}{a}$ Next we add $c$: $bx\geq\frac{bc}{a}+c$ And divide by $b$: $x\geq\frac{1}{b}(\frac{bc}{a}+c)$ $x\geq\frac{c}{a}+\frac{c}{b}$
|
2018-09-24 09:17:04
|
{"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": 0.5138656497001648, "perplexity": 622.6358740758793}, "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/1537267160337.78/warc/CC-MAIN-20180924090455-20180924110855-00187.warc.gz"}
|
https://support.bioconductor.org/u/1412/?page=2&sort=update&limit=all%20time&answered=all&q=
|
## User: TEXTORIS Julien
Reputation:
160
Status:
Trusted
Location:
Last seen:
6 years, 8 months ago
Joined:
14 years ago
Email:
j**************@gmail.com
#### Posts by TEXTORIS Julien
<prev • 16 results • page 2 of 2 • next >
1
372
views
1
... weinong han wrote: >Hi,all > >If i want to downweight the lower quality arrays using the arrayWeights() function in limma, how to do? please in detail. > >Thanks > >Gordon Smyth wrote: >At 08:08 AM 6/09/2005, Adaikalavan Ramasamy wrote: > > >>Gordon, this is a g ...
written 14.0 years ago by TEXTORIS Julien160
1
399
views
1
... Hi, Could someone send me an imagene file as an example ? I tried to write a read function for my quantifier software (BZSCAN), and i try to follow the read.imagene function as i also have two separate files. But as i don't know the imagene format, i don't understand all the code of the function, a ...
written 14.0 years ago by TEXTORIS Julien160 • updated 14.0 years ago by Sylvain FORET10
1
377
views
1
... Hi, Do you know if there is a mailing list, or newsgroup to talk about microarrays and statistical issues in general ? i don't want to pollute this list with general questions. Thanks Julien ...
written 14.0 years ago by TEXTORIS Julien160 • updated 14.0 years ago by STKH Steen Krogsgaard150
1
330
views
1
... >> >>As was pointed out, Bioconductor does not contain the mapping10KXBa >>cdf >>file, my library does not contain the cdf file either. I tried to >>install it by downloading the corresponding library from affy >>(10K2_libraryfile.zip, but don't know how to i ...
written 14.0 years ago by TEXTORIS Julien160
1
334
views
1
... Hi, when programming with R and objects, when you describe a new class or method, does $this or an equivalent exists ? Sometimes, there is : function(Object) (So Object is something equivalent to$this ?) function(.Object) (and .Object too ?) function(x,i,j,...) (and in this case x was equivalent ...
written 14.0 years ago by TEXTORIS Julien160 • updated 14.0 years ago by Kasper Daniel Hansen6.4k
2
440
views
2
... Hi, i've learn through the many tutorials and courses available on the net (thanks to everybody for that, it's wonderfull), and i really like working with bioc. I work in a lab where we use a cDNA, home made chip. I was wondering how i could write a package for R to describe this particular chip, ...
written 14.0 years ago by TEXTORIS Julien160 • updated 14.0 years ago by James W. MacDonald51k
#### Latest awards to TEXTORIS Julien
No awards yet. Soon to come :-)
Content
Help
Access
Use of this site constitutes acceptance of our User Agreement and Privacy Policy.
|
2019-09-21 03:18:57
|
{"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": 0.22018061578273773, "perplexity": 5615.5116973657205}, "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/1568514574182.31/warc/CC-MAIN-20190921022342-20190921044342-00243.warc.gz"}
|
http://crypto.stackexchange.com/questions?pagesize=15&sort=newest
|
# All Questions
9 views
### Attacks on elliptic-curve based cryptosystems through solving the Decisional Diffie-Hellman Problem with the Weil Pairing
Are there any examples of practical attacks on cryptosystems set over elliptic curves which utilize the easiness of DDH for certain choices of curves $E(\textbf{F}_q)$, and as such their lack of ...
14 views
### Light weight crypto library implementations [migrated]
Are there any open source crypto libraries that can be used in embedded systems or in other memory constrained places like a boot rom? I am looking for libraries that I can compile only the algorithms ...
17 views
### Basic cryptography [on hold]
{First of all I'd like to apologise about my English} I'm interested in most popular cryptography encodements, but ones, where the substitution cipher is known (for example Caesar's code), not created ...
20 views
### Help me find out what type of encryption this is. [on hold]
Alright this may not be an easy one. May not be something anyone knows. What I do know already (and I have made a password recovery tool from what I have learned) The password is translated into a ...
15 views
### What consequences do the plaintext space size has on the performances in the BGV scheme?
In the BGV paper [1], the authors say in §5.4 that you can have $\mathbb{Z}_p$ as plaintext size with a large $p$. What is the impact of the size of $p$ on the ciphertext size and computational work ...
20 views
### need help for decrypting a cipher.. i am new to this so was wondering if someone could tell me how to go about it [on hold]
how to begin to decrypt any numeric cypher? what are the basic steps to solve easy numeric cyphers?
49 views
### How to generate encryption key for block cipher
I wrote some report how to implement encryption. Now I want to address part about generating key. What are the practices to generate encryption key securely? I want to address this in my article. What ...
9 views
### where could I find info on number based ciphers specifically? [on hold]
I am trying to decode some conversations in a game and would like any help that could be offered.
39 views
### I don't know whattype of cipher this is [on hold]
I am trying to solve the following sipher Nosdhiibotidcylhrdeovedljuetre I have no experience with ciphers but substitution didn't seem to work, and since its a long string without breaks process of ...
78 views
### Is there any such thing as “proof of location”?
I'm looking for a way for someone to prove to me their geographical location without a trusted third-party. Imagine wanting to prove to someone you're actually physically in a specific place. I'd ...
32 views
### ElGamal signatures systems
Let $p$ prime number $q/p-1$ prime and $g\in (Z/pZ)^*$ element of order $q$.Also $a\in \{1,...,q-1\}$ the private key and $y\equiv g^a\pmod p$ the correspoding public key.For each of the following ...
45 views
### Attack on Double Encryption Scheme
In this protocol: A → B : { <[A, {K}pk(B)]> }pk(B) B → A : { <[B, {K}pk(A)]> }pk(A) I'd like to determine why this is insecure. How can the attacker learn the key K if A and B are both ...
29 views
### In which step does AES use the key to encrypt data?
AES has the following steps in encryption: SubBytes, ShiftRows, MixColumns, and AddRoundKey and the following steps in decryption: InvSubBytes, InvShiftRows, InvMixColumns, and Inverse of AddRoundKey. ...
115 views
### Signature and hash function properties
I understand the need for the hash function to be collision resistant and second pre-image resistant. For what reason, exactly, does a hash function need to be pre-image resistant? If this property ...
38 views
### Playfair Cipher + Vigenere Cipher?
I'm researching cryptography for a school project, came across the above two ciphers, and something occurred to me. Would combining these two schemes give a considerably stronger encryption than ...
49 views
### Library to find an addition chain for a large number? [on hold]
I need a short addition chain for a number $n>2^{100}$ in order to implement a fast exponentiation. The memory footprint is not important. Finding an optimal chain for a large number $n$ is ...
55 views
### Permuting Small Sized Set in Practice
Imagine we have a set $S$ of $m$ elements and we wants to permutes the set elements. Thus the original position of each element should be unknown after permuting. If we define a permutation function ...
38 views
### Reduction to cdp,dl or cdh?
Assuming a randomize encoding scheme that takes as input a secret key $\mathsf{sk} \in \mathbb{Z}_p$ for a large prime number $p$. Then the algorithm outputs $g^{\mathsf{sk}x}, x \in \mathbb{Z}_p^*$. ...
33 views
### Ephemeral key is held constant in ElGamal
Is there a known algorithm to recover the sender's messages in ElGamal $\mathbb{F}_p^*$ if the ephemeral key is held constant in two or more transmissions, assuming the messages are always distinct? ...
24 views
### CPA-security of a pseudorandom permutation encryption scheme
Let $F$ be a pseudorandom permutation, and define a fixed-length encryption scheme $(Gen, Enc, Dec)$ as follows: on input $m \in$ $\{0,1\}^{n/2}$ and key $k \in \{0,1\}^n$, algorithm Enc ...
34 views
### point addition in NaCl/libsodium (Curve25519)
In NaCl and libsodium, the crypto_scalarmult function implements the operation $Q = kP$ (scalar/point multiplication). There doesn't seem to be a function for point ...
53 views
### Words having weight near to minimum distance
I am studying the NP-Problem of the codes Syndrome Decoding. The formulation is show below. Input: a binary matrix $H$ of dimension $r \times n$ and a bit string $S$ of length $r$. Property: there ...
49 views
### Simulation Based Proof: What Can / Can not Simulator Do?
I have seen some examples in "Foundation of cryptography" and "Efficient two party computation", in which simulator can do some things that in the real world model the parties cannot do, for instance: ...
59 views
### Is there a signature scheme for the Cramer-Shoup cryptosystem?
I have a project that I have to use cryptosystem and signature scheme for it. I've read about Cramer-Shoup cryptosystem and I want to use it since it is more secure the ElGamal cryptosystem, but, I ...
52 views
### Is the ring of octonions “commonly” used in Cryptography?
I've recently read "Fully Homomorphic Encryption on Octonion Ring" by Yagisawa, which is based on octonion rings over finite fields. Personally I've never encountered octonion rings in cryptography ...
58 views
### Example: pre-image resistance to second pre-image resistance
It is possible to convert a pre-image resistant function $f:\{0,1\}^{n}\rightarrow \{0,1\}^{n}$ to a second-preimage resistant function? I am thinking to use a pseudo-random generator and construct ...
29 views
### Construction of ElGamal signature?
This month I am trying to study cryptography by myself. I started to read theory and problems on the internet. I found this problem on a cryptography site and I need your help for a solution or any ...
71 views
### Why is the whole initial state used in the final addition of Salsa20 and ChaCha?
Both Salsa20 and ChaCha basically work like this: Put the key, the nonce, the sequence number and a constant into a 4x4 matrix of 32-bit words. Transform the matrix invertibly with a number of ARX ...
76 views
### Show How to Efficiently Solve the Computational Diffie-Hellman Assumption given an Algorithm that Solves the Square-DH Problem
Let $q$ prime number, $G$ a cyclic group with order $q$ and let $g \in G$ be a generator of $G$. Suppose that you have an algorithm $A$ who takes input the element $g^a$ of $G$ and gives as output the ...
37 views
### search of patterns in key schedules
I am developing a new key schedule, and there is this article (Enhanced Key Expansion for AES-256 by Using Even-Odd Method) where the authors also propose a new algorithm and one of the objectives is ...
36 views
### Algorithm for factoring a number $n$ of a specific form given $n$ and $\varphi(n)$
Given the natural number $n$, which is in the form $p^2 \cdot q^2$ with $p$,$q$ prime numbers. Also $\varphi(n)$ is given. Describe a fast algorithm (polynomial time) that calculates $p$ and $q$. ...
11 views
### unable to create cms signed message [migrated]
So I'm trying to create a CMS signed message with OpenSSL and am having some difficulty. Here's what the OpenSSL docs say: ...
39 views
### Simulator in Private Outsourced Computation over Outsourced Datasets
Please, consider two honest parties $A$ and $B$ outsourced their private data to a malicious server $S$. So the parties store their data in the server. Then at a later point in time they want to ask ...
87 views
### How to prove that a function is not pseudorandom?
I am currently enrolled in a cryptography course, which uses the book by Katz and Lindell. I'm struggling with the exercies which ask for proofs, like the following one: Let G(k) be a PRG with ...
23 views
### Blinding to mask private key operations
Blinding is often used to mask private key operations when the underlying problem is integer factorization. For example, its used in both RSA and Rabin-Williams signature schemes. This presumes ...
36 views
### Encryption of byte-array with specific range?
I have a small question regarding encryption. When transferring data through the internet, before transmitting data we should put it in an array of bytes and it will be in the range [0-256]. What is ...
36 views
### El-Gamal signature with two messages
Alice uses an ElGamal signature with base the group $Z^*_{107}$ and parameter $g=3$ of order $q=53$.The private key of Alice is some $x \in \{0,1,.....,52\}$ and the public key of her is $y=10$. To ...
148 views
### Can a 1 byte difference in AES 128 bit keys make huge difference in output?
If we take some randomly generated key of AES-128 and we change any random 1 byte of that 16 byte key, will this make huge difference in the AES cipher text generated over same input string? Does ...
220 views
### Terminology: differences between the terms “pre-master secret”, “master secret”, “private key”, and “shared secret”?
Both crypto.SE and security.SE have excellent Q&As about how TLS generates session keys (I have linked some at the bottom). In reading these threads I'm having troubles with terminology since the ...
53 views
### Transforming Gaussian random bytes to uniform
I am not exactly sure if this is for math stackexchange or crypto: If a TRNG outputs a bit-string in the following format: $X_1,X_2,...X_N$ where each $X_i$ is a Gaussian Truly random BYTE and the ...
19 views
### Why is Chrome saying that “TLS_RSA_WITH_AES_128_CBC_SHA (0x2f)” is an obsolete cipher suite? [migrated]
When I go to this site, Chrome Version 44.0.2403.89 is connecting to the server with TLS_RSA_WITH_AES_128_CBC_SHA (0x2f), and it states that this is an "obsolete cipher suite". For what reason is it ...
24 views
### Proving existence of an encryption scheme that has indistinghuishable multiple encryptions in the presence of an eavesdropper, but is not CPA-secure [duplicate]
I got stuck in trying to find a solution to the 3.7 exercise of the Katz-Lindell book. The exercise also assumes the existence of a pseudorandom function. The problem is that a multiple messages ...
87 views
### How can I generate a good password from a SHA512 hash?
I have to change local administrator passwords on machines. I don't want to store password in a database. I have to generate a password that I can find later to connect to the machine again. So I ...
55 views
|
2015-07-30 02:14:47
|
{"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": 0.7849276661872864, "perplexity": 1675.2176560723715}, "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-2015-32/segments/1438042987034.19/warc/CC-MAIN-20150728002307-00044-ip-10-236-191-2.ec2.internal.warc.gz"}
|
https://math.stackexchange.com/questions/527748/why-cant-this-number-be-written-as-a-sum-of-three-squares-of-rationals
|
# Why can't this number be written as a sum of three squares of rationals?
This may be a very naive question and I apologize in advance.
Suppose that $n$ is a positive integer which cannot be written as a sum of three squares $a^2+b^2+c^2$ for integers $a,b,c\in\mathbb{Z}$.
Does it follow that $n$ cannot be written as a sum of three squares $a'^2+b'^2+c'^2$ for rationals $a',b',c'\in\mathbb{Q}$? Why?
There is a fairly easy answer; a positive integer is the sum of three squares if and only if it is not of the form $4^k (8n +7).$ Or, more useful for us, if and only if it is not $4^k m$ with $m \equiv 7 \pmod 8.$
If you have the sum of three rational squares equal to such a number, call it $t,$ then we can multiply through by the square of the least common denominator, which forces everything to be integers, so we have all integers in $x^2 + y^2 + z^2 = t L^2.$ Now, $L$ factors as some power of $2$ times an odd number, so let $L = 2^r s$ with odd $s.$ The important bit is that $s^2 \equiv 1 \pmod 8$ (CHECK THIS). So, $L^2 = 4^r s^2.$
Finally, we knew $t = 4^km$ with $m \equiv 7 \pmod 8,$ because $t$ was not integrally the sum of three squares. So $t L^2 = 4^{k+r} m s^2,$ where $m s^2 \equiv 7 \pmod 8.$ Therefore $t L^2$ is also not the sum of three integer squares and we are done.
It is true that positive integers are the sum of three integer squares if and only if they are the sum of three rational squares. This result, for a number of quadratic forms, is generally referred to as the Davenport-Cassels Theorem, but was first proved by Aubry about 1912 I think. It appears in Serre's little book, A Course in Arithmetic, with three squares on pages 45-46, and Davenport-Cassels on pages 46-47.
There are infinitely many quadratic forms with the same property, they represent a number over the integers if and only if they represent it over the rationals. The stronger property used in the Davenpost-Cassels hypothesis, for positive quadratic forms, occurs for just 70 forms of the type called "even lattices." The restriction is that the "covering radius" be strictly below $\sqrt 2.$ A full list is given by G. Nebe see http://www.math.rwth-aachen.de/~Gabriele.Nebe/pl.html and the pdf listed as Even lattices with covering radius stricktly smaller than sqrt{2}. Beiträge zur Algebra und Geometrie, Vol. 44, No. 1, 2003, 229-234 Note that I forgot the case R=A2A1A1 in Theorem 7. This root system yields two further lattices, one with covering radius =sqrt{2} and one with c.r. strictly smaller than sqrt{2}
There are 103 positive forms in three variables that have your property, that they represent an integer over the integers if and only if they represent it over the rationals. Each is summarized by six coefficients preceded by a "discriminant" I call Delta. Any primitive, positive ternary with the property is "equivalent" to one of these 103. ADC stands for Aubry-Davenport-Cassels. Let's see, only thirteen of the forms below belong on Nebe's list. Her hypotheses are far more restrictive than the adc property. Her list has 70 entries because the number of variables gets as large as ten.
$$Q(x,y,z) = a x^2 + b y^2 + c z^2 + r y z + s z x + t x y,$$ and $$\Delta = 4 a b c + r s t - a r^2 - b s^2 - c t^2$$
Delta a b c r s t
---------------------------------------------------------
2: 1 1 1 1 1 1 ADC 2 = 2
3: 1 1 1 0 0 1 ADC 3 = 3
4: 1 1 1 0 0 0 ADC 4 = 2^2
5: 1 1 2 1 1 1 ADC 5 = 5
6: 1 1 2 0 0 1 ADC 6 = 2 * 3
6: 1 1 2 1 1 0 ADC 6 = 2 * 3
7: 1 1 2 0 1 0 ADC 7 = 7
8: 1 1 2 0 0 0 ADC 8 = 2^3
9: 1 1 3 0 0 1 ADC 9 = 3^2
10: 1 1 3 1 1 0 ADC 10 = 2 * 5
10: 1 2 2 2 1 1 ADC 10 = 2 * 5
11: 1 1 3 0 1 0 ADC 11 = 11
12: 1 1 3 0 0 0 ADC 12 = 2^2 * 3
12: 1 2 2 1 1 1 ADC 12 = 2^2 * 3
12: 1 2 2 2 0 0 ADC 12 = 2^2 * 3
13: 1 2 2 1 0 1 ADC 13 = 13
14: 1 1 5 1 1 1 ADC 14 = 2 * 7
15: 1 1 4 0 1 0 ADC 15 = 3 * 5
15: 1 2 2 1 0 0 ADC 15 = 3 * 5
16: 1 2 2 0 0 0 ADC 16 = 2^4
17: 1 2 3 2 1 1 ADC 17 = 17
18: 1 2 3 2 1 0 ADC 18 = 2 * 3^2
18: 2 2 2 1 2 2 ADC 18 = 2 * 3^2
20: 1 1 5 0 0 0 ADC 20 = 2^2 * 5
20: 1 2 3 1 0 1 ADC 20 = 2^2 * 5
20: 1 2 3 2 0 0 ADC 20 = 2^2 * 5
21: 1 2 3 0 0 1 ADC 21 = 3 * 7
21: 1 2 3 1 1 0 ADC 21 = 3 * 7
22: 1 2 3 0 1 0 ADC 22 = 2 * 11
24: 1 1 6 0 0 0 ADC 24 = 2^3 * 3
24: 1 2 3 0 0 0 ADC 24 = 2^3 * 3
24: 1 2 4 2 1 1 ADC 24 = 2^3 * 3
25: 2 2 2 -1 1 1 ADC 25 = 5^2
28: 2 2 3 2 2 2 ADC 28 = 2^2 * 7
30: 1 1 10 0 0 1 ADC 30 = 2 * 3 * 5
30: 1 3 3 1 1 1 ADC 30 = 2 * 3 * 5
32: 1 2 4 0 0 0 ADC 32 = 2^5
33: 1 2 5 1 1 1 ADC 33 = 3 * 11
36: 1 2 5 2 0 0 ADC 36 = 2^2 * 3^2
36: 1 3 3 0 0 0 ADC 36 = 2^2 * 3^2
36: 1 3 4 3 1 0 ADC 36 = 2^2 * 3^2
40: 1 2 5 0 0 0 ADC 40 = 2^3 * 5
42: 1 1 11 1 1 0 ADC 42 = 2 * 3 * 7
44: 1 2 6 2 0 0 ADC 44 = 2^2 * 11
45: 2 2 3 0 0 1 ADC 45 = 3^2 * 5
46: 1 3 5 3 1 1 ADC 46 = 2 * 23
48: 1 2 6 0 0 0 ADC 48 = 2^4 * 3
49: 1 2 7 0 0 1 ADC 49 = 7^2
50: 1 4 4 3 1 1 ADC 50 = 2 * 5^2
56: 1 3 5 2 0 0 ADC 56 = 2^3 * 7
60: 2 2 5 0 0 2 ADC 60 = 2^2 * 3 * 5
60: 2 3 3 0 0 2 ADC 60 = 2^2 * 3 * 5
63: 1 3 6 3 0 0 ADC 63 = 3^2 * 7
70: 1 2 9 0 1 0 ADC 70 = 2 * 5 * 7
72: 2 2 5 1 1 1 ADC 72 = 2^3 * 3^2
72: 2 3 3 0 0 0 ADC 72 = 2^3 * 3^2
75: 1 4 5 0 0 1 ADC 75 = 3 * 5^2
78: 1 5 5 4 1 1 ADC 78 = 2 * 3 * 13
84: 1 1 21 0 0 0 ADC 84 = 2^2 * 3 * 7
90: 1 1 30 0 0 1 ADC 90 = 2 * 3^2 * 5
92: 2 3 5 2 0 2 ADC 92 = 2^2 * 23
99: 2 3 5 3 1 0 ADC 99 = 3^2 * 11
100: 2 2 7 -1 1 1 ADC 100 = 2^2 * 5^2
100: 2 3 5 0 0 2 ADC 100 = 2^2 * 5^2
112: 2 3 5 2 0 0 ADC 112 = 2^4 * 7
120: 1 3 10 0 0 0 ADC 120 = 2^3 * 3 * 5
121: 1 3 11 0 0 1 ADC 121 = 11^2
126: 3 3 5 3 3 0 ADC 126 = 2 * 3^2 * 7
140: 1 2 18 2 0 0 ADC 140 = 2^2 * 5 * 7
147: 3 3 5 -2 2 1 ADC 147 = 3 * 7^2
150: 2 5 5 5 0 0 ADC 150 = 2 * 3 * 5^2
156: 2 3 7 0 2 0 ADC 156 = 2^2 * 3 * 13
169: 2 5 5 -3 1 1 ADC 169 = 13^2
180: 2 2 15 0 0 2 ADC 180 = 2^2 * 3^2 * 5
200: 1 5 10 0 0 0 ADC 200 = 2^3 * 5^2
234: 2 3 11 3 2 0 ADC 234 = 2 * 3^2 * 13
240: 2 5 6 0 0 0 ADC 240 = 2^4 * 3 * 5
252: 3 3 7 0 0 0 ADC 252 = 2^2 * 3^2 * 7
289: 3 5 6 1 2 3 ADC 289 = 17^2
294: 5 5 5 -3 3 4 ADC 294 = 2 * 3 * 7^2
300: 1 10 10 10 0 0 ADC 300 = 2^2 * 3 * 5^2
350: 3 3 10 0 0 1 ADC 350 = 2 * 5^2 * 7
360: 1 3 30 0 0 0 ADC 360 = 2^3 * 3^2 * 5
450: 5 5 6 0 0 5 ADC 450 = 2 * 3^2 * 5^2
468: 1 6 21 6 0 0 ADC 468 = 2^2 * 3^2 * 13
490: 3 3 14 0 0 1 ADC 490 = 2 * 5 * 7^2
588: 3 7 7 0 0 0 ADC 588 = 2^2 * 3 * 7^2
600: 2 5 15 0 0 0 ADC 600 = 2^3 * 3 * 5^2
700: 5 6 6 2 0 0 ADC 700 = 2^2 * 5^2 * 7
720: 2 6 15 0 0 0 ADC 720 = 2^4 * 3^2 * 5
882: 2 11 11 1 2 2 ADC 882 = 2 * 3^2 * 7^2
900: 3 10 10 10 0 0 ADC 900 = 2^2 * 3^2 * 5^2
980: 6 6 7 0 0 2 ADC 980 = 2^2 * 5 * 7^2
1014: 1 13 23 13 1 0 ADC 1014 = 2 * 3 * 13^2
1200: 1 10 30 0 0 0 ADC 1200 = 2^4 * 3 * 5^2
1764: 1 21 21 0 0 0 ADC 1764 = 2^2 * 3^2 * 7^2
1800: 5 6 15 0 0 0 ADC 1800 = 2^3 * 3^2 * 5^2
2028: 2 7 39 0 0 2 ADC 2028 = 2^2 * 3 * 13^2
2450: 1 9 70 0 0 1 ADC 2450 = 2 * 5^2 * 7^2
3042: 3 17 17 8 3 3 ADC 3042 = 2 * 3^2 * 13^2
3600: 3 10 30 0 0 0 ADC 3600 = 2^4 * 3^2 * 5^2
4900: 2 18 35 0 0 2 ADC 4900 = 2^2 * 5^2 * 7^2
6084: 6 13 21 0 6 0 ADC 6084 = 2^2 * 3^2 * 13^2
|
2019-06-17 08:33:56
|
{"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": 0.37620440125465393, "perplexity": 66.98633032853179}, "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-2019-26/segments/1560627998462.80/warc/CC-MAIN-20190617083027-20190617105027-00030.warc.gz"}
|
https://engineering.stackexchange.com/questions/2302/mass-and-weight-of-air-in-a-room
|
# Mass and weight of air in a room
Given:
A problem in my thermodynamics text is stated as follows...
Determine the mass and the weight of the air contained in a room whose dimensions are $V=$ $15ft$ x $20ft$ x $20ft$. Assume the density of the air is $\rho=0.0724\cdot\frac{lbm}{ft^3}$.
My Solution:
First find the mass...
$$m=\rho\times V$$
$$m=0.0724\cdot\frac{lbm}{ft^3}\times 6000\cdot ft^3$$
$$=434.3\cdot lbm$$
Now find the force acting on the air due to gravity. This is the weight of the air assumed at sea-level...
$$W=m\times g$$
$$W=434.3\cdot lbm\times32.174\cdot\frac{ft}{s^2}$$
$$=13976\cdot lbf$$
Question:
I find it hard to believe that in an average size room the air weighs a whopping $14,000\cdot lbf$. Did I do something wrong in my calculations or is this correct? If this is correct perhaps we earthlings living on the surface of the earth are the real extremophiles.
• For extra fun: atmospheric air is $\sim 14.7 psi$, so the total force on the floor of the room (20 x 20) is more than 800,000 pounds!
– Dan
Apr 4 '15 at 0:57
A pound force is defined as the force required to accelerate a slug at 1 ft/s^2. The density of air is $\rho = 0.0724 \ lb_m/ft^3 = 0.0724/32.2 \ slugs/ft^3$
The weight of the air is $\rho V g = 0.0724/32.2 \ slugs/ft^3 \cdot32.2 ft/s^2\cdot 6000 ft^3 = 0.0724\cdot 6000 \ slugs\ ft/s^2 = 434.4 lb_f$
|
2022-01-19 12:05:28
|
{"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": 0.7683385014533997, "perplexity": 208.6573887736345}, "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-2022-05/segments/1642320301309.22/warc/CC-MAIN-20220119094810-20220119124810-00378.warc.gz"}
|
https://fhi-aims-club.gitlab.io/tutorials/free-atoms-calculating-atomization-energies/
|
# Home
Calculating the total energy of free atoms sounds like a simple thing to do, but in density-functional theory, it is not as simple as it may seem.
Read on on the present page for the underlying physical problem and also for an example of a control.in file that defines the problem more clearly and thus solves problems of numerical instability.
## On free atoms
Most unsuspecting users assume that free atoms are inherently spherical in their ground state and therefore, there is only one unique ground state.
THIS IS NOT THE CASE.
You may easily check this for the case of an O atom, using the PBE functional and apply what is written down below. You will find that the energy difference between a spherically symmetric state and the actual, symmetry breaking ground state amounts to 0.7 eV (written from memory). That's a lot. How come?
In DFT and/or Hartree-Fock theory, we only have a single Slater determinant. Unless all orbitals of an angular momentum shell are fully occupied, we would have to use fractional occupation numbers to define the density:
$n(r) = \sum_\text{orbitals} ( f_\text{orbital} * |\text{orbital}(r)|^2 )$
where f_orbital between 0 and 1 in the spin-polarized case. In particular, in the case of O, one of the spin channels would have to be occupied with 1/3, 1/3, 1/3 occupation in the m=-1,0,1 orbitals to create a spherically symmetric density.
This may sound ok from a formal point of view (in the basic foundations of DFT, fractional orbitals are justified as statistical mixtures of ensembles of atoms in different pure states). However, actual density functional approximation do NOT necessarily produce the average of pure states when fractional occupations are used. If you want something approaching a pure state, you would need (in the simplest case) a single determinant with only completely filled or completely empty orbitals. For something like O, this would inherently break the spherical symmetry, as only one of the p orbitals will be occupied.
This is the formal aspect. From a practical DFT point of view, all that matters is which occupation has the lowest total energy. And in practice, the lowest total energy is often given by the "pure state" occupation, i.e., f=0 or 1. So we need to break the spherical symmetry of an atom.
But this is not straightforward. You will note that since there are three p orbitals for O, there must be more than one lowest-energy state - i.e., more than one lowest-energy solution. Any electronic structure code, during its s.c.f. optimization, needs to pick one of those solutions to fall into as a stationary point of the s.c.f. problem. In other words, the s.c.f. problem, which is a nonlinear optimization problem, has more than one stationary point.
AND THIS CAN PRODUCE PROBLEMS OF NUMERICAL UNIQUENESS AND/OR STABILITY, ESPECIALLY IN MORE COMPLICATED d OR f SHELL systems.
The basic problem is that we need to start somewhere. Just like in geometry optimization, if we are searching for a local minimum, which minimum we find depends on where we start. There might be more than one local minimum in the total energy surface, one or more of which are the total energy minimum.
FHI-aims, by definition, starts with a superposition of free atom densities created from spherical free atoms with fractional occupation numbers. You will note that this may, in fact, not be a minimum of the total energy surface, but because of its symmetry, it would have to be a saddle point or a maximum of the total energy.
As an important consequence, starting from such a point WILL mean that we can fall into different local minima depending on which direction of the slope our calculation decides to go. Or, worse, since we use an optimization scheme to find a single local minimum, the density mixing scheme might be confused by the fact that more than one minimum is present and not converge at all. Even more interestingly, we use fractional occupation numbers (occupation_type) for numerical reasons in our calculation. If we introduce a large broadening, we might find fractional occupation numbers and a non-pure states. Only if we mandate a very low broadening will we safely find integer occupation numbers, but then, we face the problem of multiple allowed solutions and potential numerical instabilities.
Because, by default, we start from a saddle point or maximum, we may even fall into different local minima starting from tiny numerical changes, such as the number of MPI tasks, which changes the order of summation in the integration of matrix elements. At double precision, this leads to initial discrepancies of 1e-12 or so at some point in the s.c.f. mixing cycle, but this difference is large enough for the Pulay mixer to pick it up and determine which specific solution it will end up in after the s.c.f. cycle. But the solution(s) found are all valid from an s.c.f. point of view. It's just that there is more than one of them.
Why do we not just start from a symmetry broken state?
Because this would mean we would have to pick FOR YOU which symmetry broken state you want.
In a problem with multiple minima, we would force you to use one of them without telling you, and (a) we might not get it right or (b) you might not want the specific state we picked for you because you were interested in something else.
Either way, this is a user choice - if we were to legislate which electronic structure result you get without telling you, we would be doing something wrong.
The real way out, like in all problems of multiple minima, is that you would need to pick which minimum you want - or search through them all, systematically. This is generally difficult for any global optimization problem, but for free atoms, there is a way.
In FHI-aims, you can constrain certain occupation numbers (really, Mulliken occupations) for certain valence orbitals by hand, in control.in , using the force_occupation_basis keyword. Here is a specific example enforcing a specific occupation of 4p valence orbitals for the Se atom
force_occupation_basis 1 2 atomic 4 1 1 0. 20
force_occupation_basis 1 2 atomic 4 1 0 0. 20
Please read the manual regarding this keyword. Don't just copy these lines into every other free atom calculation - again - they enforce ONE specific shell occupation for one specific type of atom, Se. The chosen syntax for other atoms WILL BE different.
In short, we legislate Mulliken occupations of 0. in the first 20 Kohn Sham states for atom 1, spin channel 2, atomic orbital n=4 l=1 and m=1 (first line) and m=0 (second line).
This means that the electron in the 4 p spin channel 2 will have to be found completely in the m=-1 orbital.
The choice of the "20" states is also system specific. From the point of view of the mathematical constraint, we need to project the 4 p 0 and 4 p 1 orbitals out of the occupied Kohn-Sham states but not out of the unoccupied Kohn Sham states. (The basis functions define a vector space and if you calculate eigenvectors and eigenstates, each part of the underlying basis needs to go somewhere or else the dimension of your eigenspace is not the same as that of the underlying basis space.)
In longer notes, the control.in file for this particular instance of Se was this:
xc pw-lda
spin collinear
default_initial_moment hund
relativistic atomic_zora scalar
#
mixer pulay
charge_mix_param 0.05
sc_accuracy_rho 1e-5
#
occupation_type gaussian 1e-6
#
# Enforce occupation of specific atomic orbital basis functions
#
# Usage: force_occupation_basis i_atom spin basis_type basis_n basis_l basis_m occ_number max_KS_state
#
# Purpose: Flag originally programmed to compute core-hole spectroscopy simulations.
# In practice, it constrains the occupation of a specific energy level of an specific atom,
# being also able to break the symmetry of an atom.
#
#
force_occupation_basis 1 2 atomic 4 1 1 0. 20
force_occupation_basis 1 2 atomic 4 1 0 0. 20
#
################################################################################
#
# FHI-aims code project
# Volker Blum, Fritz Haber Institute Berlin, 2009
#
# Suggested "tight" defaults for Se atom (to be pasted into control.in file)
#
################################################################################
species Se
# global species definitions
nucleus 34
mass 78.96
#
l_hartree 6
#
cut_pot 8.0 2.0 1.0
basis_dep_cutoff 0.0
#
angular_grids specified
division 0.0830 110
division 0.1357 194
division 0.7377 302
division 0.8610 434
# division 0.9640 590
# division 1.0773 770
# division 2.5539 974
# outer_grid 974
outer_grid 434
################################################################################
#
# Definition of "minimal" basis
#
################################################################################
# valence basis states
valence 4 s 2.
valence 4 p 4.
valence 3 d 10.
# ion occupancy
ion_occ 4 s 1.
ion_occ 4 p 3.
ion_occ 3 d 10.
################################################################################
#
# Suggested additional basis functions. For production calculations,
# uncomment them one after another (the most important basis functions are
# listed first).
#
# Constructed for dimers: 1.85 A, 2.15 A, 2.50 A, 3.00 A, 4.00 A
#
################################################################################
# "First tier" - improvements: -336.21 meV to -36.85 meV
hydro 3 d 4.3
hydro 2 p 1.6
hydro 4 f 7.2
ionic 4 s auto
# "Second tier" - improvements: -14.39 meV to -1.49 meV
hydro 5 g 10.4
hydro 4 p 4.5
hydro 4 d 6.2
hydro 4 f 11.2
hydro 1 s 0.65
hydro 6 h 15.2
# Third tier - improvements: -0.46 meV and below
hydro 5 g 14.4
ionic 4 d auto
hydro 4 f 22.4
hydro 5 f 7.4
hydro 5 p 8
hydro 5 s 9.4
# Fourth tier - improvements: -0.12 meV and below
# hydro 5 d 11.6
# hydro 6 h 18
# hydro 4 p 4
# hydro 5 f 16
# hydro 4 s 3.9
|
2022-05-18 16:42:54
|
{"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": 0.7082259058952332, "perplexity": 1460.1273540812138}, "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-2022-21/segments/1652662522284.20/warc/CC-MAIN-20220518151003-20220518181003-00024.warc.gz"}
|
http://mathforum.in/2016/list-of-events/seminarsworkshops/talk-a-theorem-about-linear-inequalities-with-an-application-prof-k-n-raghavan-imsc
|
Talk : A theorem about linear inequalities with an application (Prof. K.N.Raghavan – IMSc)
Talk On 17th February 2016 , at 2.15p.m.
Title : A theorem about linear inequalities with an application.
Speaker: Prof.K.N.Raghavan, Institute of Mathematical Sciences(IMSc),Chennai
Venue: Auditorium , Dept.Of Mathematics
COCHIN UNIVERSITY OF SCIENCE & TECHNOLOGY
Abstract:
Let B be a m x n real matrix and B’ be its transpose. Suppose that this holds: if B’v>=0 and v>=0 then v=0. Then there exists a solution to the equation Bx>0. We will apply this result to classify “indecomposable” real square matrices satisfying these properties: (i) the non-diagonal entries are non-negative, and (ii) if the entry in position (i,j) is zero, then so is the one in position (j,i). “Indecomposable” just means that the matrix cannot be written as a direct sum in the obvious sense of two such matrices of smaller size. This classification is important in the theory of infinite dimensional Lie algebras, but should be interesting in its own right.
|
2020-02-26 12:44:40
|
{"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": 0.8992567658424377, "perplexity": 1104.8386080136027}, "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-2020-10/segments/1581875146342.41/warc/CC-MAIN-20200226115522-20200226145522-00378.warc.gz"}
|
https://physics.stackexchange.com/questions/551224/distinction-between-contraction-and-application-of-a-tensor-in-abstract-index-no
|
# Distinction between contraction and application of a tensor in abstract index notation
A somewhat similar question is this one but it is not quite the same.
I am getting used to the abstract index notation used for tensor algebra. So far so good, but the is one issue that concerns me, In General Relativity by Wald, it is discussed how the difference between two connections $$\tilde{\nabla}$$ and $$\nabla$$ induces a tensor $$C$$ of rank (1,2) in $$T_pM$$, and hence (by the index notation) we denote this tensor as $${C^c}_{ab}$$. The problem is that Wald defines this tensor by the equation $$\nabla_a\omega_b=\tilde{\nabla}_a \omega_b- {C^c}_{ab}\omega_c$$ Which seems funny, we can think of the last term in the RHS of the latter as the contraction of the (1,3) tensor $$C\otimes \omega$$ with respect to the first dual vector slot and the third vector slot. But rather than this odd way of deffining the tensor $$C$$ we could make use of the index-free notation. So that we could define the tensor $$C:V^{*}\times V\times V\to \mathbb{R}$$ as $$\omega,t,s\mapsto (\tilde{\nabla}\omega)(t,s)-(\nabla\omega )(t,s)$$ which is what would be natural way of defining a tensor "by it's action". So the question is: Is my way of understanding the abstract index notation in this case is the correct one, or rather does the definition of Wald refers to $$\sum_{k=1}^nC(e^{k*},t,s)\cdot \omega(e_{k})=(\tilde{\nabla}\omega)(t,s)-(\nabla\omega)(t,s)$$ where $$\{e_1,\ldots, e_n \}$$ is some basis of $$T_pM$$ and $$\{e^{1*},\ldots, e^{n*} \}$$ is its dual basis. Which is it? How could one know using the Abstract index notation?
Note that the first definition (the one I assume is the correct one) does indeed define a tensor, meanwhile a tensor is hardly ever characterized by a contraction. But the notation, as defined does indeed suggest the contraction interpretation.
The equation $$\nabla_a\omega_b=\tilde{\nabla}_a \omega_b- {C^c}_{ab}\omega_c$$ can be contracted with vector components $$X^a$$ and $$Y^b$$:
$$C^{c}_{ab}X^aY^b\omega_c = (\tilde{\nabla}_a\omega_b) X^aY^b - (\nabla_a\omega_b) X^a Y^b$$ which can be expressed in coordinate-free notation as
$$\mathbf C(\boldsymbol\omega,\mathbf X,\mathbf Y) = (\nabla_{\mathbf X}\boldsymbol \omega)(\mathbf Y) - (\tilde{\nabla}_{\mathbf X}\boldsymbol \omega)(\mathbf Y)$$
where $$\nabla_\mathbf{X} =X^a \nabla_a$$. Note in particular that $$\nabla_{\mathbf X}$$, which is the covariant directional derivative along $$\mathbf X$$, maps a covector field $$\boldsymbol \omega$$ to a covector field
$$\nabla_\mathbf{X} \boldsymbol \omega = (X^a\nabla_a\omega_b)\hat \epsilon^b = X^a(\partial_a \omega_b -\omega_c \Gamma^c_{ab})\hat\epsilon^b$$
You can use this to go backwards to verify that the coordinate-free expression above does indeed yield the expression you started with.
Lastly, note that the $$(0,2)$$-tensor with components $$C^c_{ab}\omega_c$$ could equivalently viewed as the $$(1,2)$$-tensor $$\mathbf C$$ with the covector $$\boldsymbol \omega$$ plugged into the first slot (i.e. $$\mathbf C(\boldsymbol \omega,\bullet,\bullet)$$ ) or as the trace over the first and fourth index of the $$(1,3)$$-tensor ($$\mathbf C \otimes \boldsymbol \omega$$). If I'm interpreting the question correctly, the answer is that your two alternate interpretations are equivalent.
• Yes. I noted that! And that was actually the answer I expected. Thanks – Victor Gustavo May May 12 at 1:20
It is possible to define the tensor $${C^c}_{ab}$$ without using contractions using the Christoffel symbols for the connections. If you remember that $$\nabla_a\omega_b=\partial_a\omega_b-\Gamma^c_{ab}\omega_c$$ $$\tilde{\nabla}_a\omega_b=\partial_a\omega_b-\tilde{\Gamma}^c_{ab}\omega_c$$ then the difference gives $$\tilde\nabla_a\omega_b-\nabla_a\omega_c=(\Gamma^c_{ab}-\tilde\Gamma^c_{ab})\omega_c\tag{1}\label{key}$$ so that what Wald has called the tensor $${C^c}_{ab}$$ is $${C^c}_{ab}=\Gamma^c_{ab}-\tilde\Gamma^c_{ab}$$
It is a general result that the difference between two $$\Gamma$$ is a tensor, even though the $$\Gamma$$'s themselves are non-tensorial. You can check reasoning as follows: if the left-hand side of \eqref{key} is tensorial, because it is the difference between to covariant derivatives; then the right-hand side must also be a tensor. This is only possible if $$\Gamma^c_{ab}-\tilde\Gamma^c_{ab}$$ is a tensor. This is not just valid for the covariant vector $$\omega_c$$, you can do this reasoning in general by applying the covariant derivatives $$\nabla$$ and $$\tilde\nabla$$ to an arbitrary rank tensor $${T^{a_1a_2...a_m}}_{b_1b_2...b_n}$$ and you will get the same result.
• Thank you for your answer, but my question was about what the translation of the definition of the tensor would be in non-index notation? I think I could have made that clearer. For that I apologize, I will up-vote your answer since it was helpful. I successfully wrote a translation in index free notation that the contraction definition really defines a unique tensor. The proof was almost immediate, but doing it made evident an important observation that I missed when getting into the Abstact Index Notation. – Victor Gustavo May May 11 at 21:41
|
2020-09-21 00:20:24
|
{"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": 45, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9637992978096008, "perplexity": 201.36607521994245}, "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-2020-40/segments/1600400198868.29/warc/CC-MAIN-20200920223634-20200921013634-00291.warc.gz"}
|
http://codeforces.com/problemset/problem/817/A
|
A. Treasure Hunt
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
Captain Bill the Hummingbird and his crew recieved an interesting challenge offer. Some stranger gave them a map, potion of teleportation and said that only this potion might help them to reach the treasure.
Bottle with potion has two values x and y written on it. These values define four moves which can be performed using the potion:
Map shows that the position of Captain Bill the Hummingbird is (x1, y1) and the position of the treasure is (x2, y2).
You task is to tell Captain Bill the Hummingbird whether he should accept this challenge or decline. If it is possible for Captain to reach the treasure using the potion then output "YES", otherwise "NO" (without quotes).
The potion can be used infinite amount of times.
Input
The first line contains four integer numbers x1, y1, x2, y2 ( - 105 ≤ x1, y1, x2, y2 ≤ 105) — positions of Captain Bill the Hummingbird and treasure respectively.
The second line contains two integer numbers x, y (1 ≤ x, y ≤ 105) — values on the potion bottle.
Output
Print "YES" if it is possible for Captain to reach the treasure using the potion, otherwise print "NO" (without quotes).
Examples
Input
0 0 0 62 3
Output
YES
Input
1 1 3 61 5
Output
NO
Note
In the first example there exists such sequence of moves:
1. — the first type of move
2. — the third type of move
|
2018-06-24 08:47:53
|
{"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": 0.29159560799598694, "perplexity": 2432.380865398786}, "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-2018-26/segments/1529267866926.96/warc/CC-MAIN-20180624083011-20180624103011-00042.warc.gz"}
|
https://msp.org/gt/2018/22-2/b11.xhtml
|
#### Volume 22, issue 2 (2018)
Affine representability results in $\mathbb{A}^1$–homotopy theory, II: Principal bundles and homogeneous spaces
|
2020-10-01 02:30:23
|
{"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": 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": 0.5306508541107178, "perplexity": 3585.562949187228}, "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-2020-40/segments/1600402130531.89/warc/CC-MAIN-20200930235415-20201001025415-00468.warc.gz"}
|
https://cs.stackexchange.com/questions/50816/conditionals-with-normal-order-evaluation/51078
|
# Conditionals with normal order evaluation
From what I've read, conditionals (like the cond statement in lisp) do not need to be primitive if normal order evaluation is used.
Using lambda calculus and normal order evaluation, how can you emulate if and cond statements?
(cond (<p1> <e1>)
(<p2> <e2>)
...
(<pn> <en>))
(if <predicate> <consequent> <alternative>)
References to SICP are welcome, though a simpler example would be helpful.
(Definitions of cond and if taken from Section 1.1.6 of SICP)
• Church encoded booleans would be the thing to look at here. Note that cond is easy to emulate using nested ifs. – Daniel Gratzer Dec 16 '15 at 17:19
Here is the church notation:
TRUE: $\lambda ab.a$
FALSE: $\lambda ab.b$
IF: $\lambda pxy.pxy$
By passing a church boolean as a predicate to the IF function, we choose one of two alternatives. TRUE chooses the first argument passed to it, and FALSE chooses the second.
We can derive COND by folding IF across a list of (condition, result) pairs
You can find wikipedia's explanation here, as well as assorted lambda calculus derivations here.
|
2020-10-25 15:59:38
|
{"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": 0.5052489042282104, "perplexity": 4357.439479768201}, "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/1603107889574.66/warc/CC-MAIN-20201025154704-20201025184704-00498.warc.gz"}
|
https://physics.meta.stackexchange.com/questions/12936/physics-has-switched-to-commonmark/12962
|
Physics has switched to CommonMark
Yesterday, I announced on Meta Stack Exchange that we'll switch all sites on the Stack Exchange network to CommonMark, a standardized and well-supported Markdown flavor.
You can read up on the details in the post on Meta.SE if you're curious.
We'll maintain an evolving migration schedule to show which site is supposed to switch over to CommonMark when.
We've got to learn and reflect as we're starting out migrating the first few sites, that's why you'll see that the migration schedule is still pretty empty except for a few sites.
Physics (both, meta and the main site) are going to be among the first sites to be migrated and we've scheduled them to be migrated on Thursday, June 4th, 2020.
We've tested the migration thoroughly on data that resembles production data of some of our communities. Still, we know that certain sites use different styles of writing and there's a chance we're going to detect some issues that we haven't found yet when running our tests.
Physics is a site that's suitable for fast feedback for two reasons: it's a site we can migrate quickly with its ~400k posts and it's a site that makes heavy use of MathJax.
Don't worry, MathJax is going to keep working after the migration. However, we want to be double-certain that we're not missing any edge cases, that's why we want to migrate a site with MathJax support first to spot edge cases early and get them out of the way.
Can this break existing posts?
We don't want to break hundreds and thousands of posts. That's why the migration will only apply updates to those posts that will look exactly the same after being updated to CommonMark. As part of the migration, we'll detect if a post changes visually after the CommonMark update. If it does, we won't update the post automatically and investigate what's going on.
What if stuff goes wrong?
If things should go horribly wrong, we've got an automated rollback in place that will undo the migration for all posts.
• Potential Problem I just played a bit with CommonMark demo, and I do see a potential problem. Usually while creating headings, I write something of the sort ##heading. But doing the same in CommonMark, doesn't yield a heading. You have to compulsorily add a <kbd>space</kbd> after the ##. So to make it work in CommonMark, I have to write ## heading. So I expect all my answers which contain the former sort of markdown to break. – user258881 Jun 3 '20 at 13:59
• So is this change going to be implemented? If yes, then what should I do about my answers? – user258881 Jun 3 '20 at 14:03
• @FakeMod you don't need to do anything about your answers. As part of the migration we're applying some automated fixes to the markdown version of a post for these well-known incompatibilities. There are a few more of those outlined in the announcement on meta.SE – Ham Vocke Jun 3 '20 at 14:09
• Possibly related: a supported HTML tag used to work, but now it doesn't. – rob Oct 7 '20 at 14:42
• @HamVocke Can you review the question rob just linked to? Hopefully you have sufficient clarity into the different systems involved to tell whether it was a result of the CommonMark migration, or something else. – Emilio Pisanty Oct 7 '20 at 15:36
Good news, then!
Suppose a user discovers a post that renders incorrectly after the transition. Should that user fix the problem themselves, or cast a custom flag for the diamond moderators, or leave a message at this question, or do something else? (possibly including nothing at all?)
• Ideally they'd go and fix it themselves. Just to be clear: incorrect rendering should only ever occur once someone edits one of those posts that couldn't be transitioned correctly with our automatic migration. – Ham Vocke Jun 2 '20 at 13:36
• @HamVocke I doubt there's an endless spring of classes of such problems; how likely do you think it is that you won't be able to catch them all (or, rather, all the ones that affect posts on the site)? – wizzwizz4 Jun 2 '20 at 18:27
• I'm certain we won't catch them all. The question is if that matters and if we want to go down the deep rabbit hole that is fixing them all preemptively. We're aiming to spot and fix non-trivial issues while accepting that the trivial ones might be left unfixed until someone comes in and edits old posts. There's a certain level of pragmatism that we have to apply to this migration in order to be able to pull this off, and for a fraction of posts this means editing could become a bit awkward. – Ham Vocke Jun 2 '20 at 19:33
Tracking the current state to keep you updated:
• physics.meta.stackexchange.com has been migrated successfully. CommonMark is now active
• I've started probing physics.stackexchange.com (no changes are being persisted). CommonMark is still disabled and I'm double-checking that MathJax rendering is going to stay sane. Will continue with the real migration as soon as I'm certain that we're good here.
• The differences we're detecting are expected but our migration script is being overly cautious. Doing a first real migration run right now. I will patch up some things to do a second run tomorrow to fix those posts that haven't been caught in the first run. That means that commonmark is enabled for physics starting now and posts are being migrated.
• Finally managed to complete the re-run for the physics main and meta sites. Everything's migrated now - calling this done. Thanks for your patience!
• Looks like everything went OK? From the community bit it looks like there were some 2.5k edits. Can you give a sense for how many are left for the second run? – Emilio Pisanty Jun 4 '20 at 20:08
• Keeping the community updated like this is exactly the kind of interaction I like to see from SE/SO as an organization. It creates a sense that community opinion matters. Thank you. – StephenG Jun 4 '20 at 21:41
• Hi @Ham, any updates on how this is going? I imagine the second run kicked up some trouble, given the delay. – Emilio Pisanty Jun 9 '20 at 10:41
• @EmilioPisanty sorry for the radio silence! The first run was just fine but left a few too many posts untouched. While working on a tweak to get those untouched posts migrated, I got in another tweak for sites that are using MathJax slightly differently than Physics and that took me longer to grok than I had hoped. I'm aiming to re-run with the new tweaks within the next 4 hours, after that we should be good to call this done. – Ham Vocke Jun 9 '20 at 10:51
• @HamVocke Thanks for the update =). – Emilio Pisanty Jun 9 '20 at 11:13
• Relative time designations, like "now", very quickly get old. Perhaps use absolute time instead? – Peter Mortensen Jun 12 '20 at 15:18
• @PeterMortensen Presumably "now" refers to the timestamp of the most recent edit, 2020-06-11 09:48Z, which is displayed just below the text of the post. However, your point is totally valid, especially if there are future edits. – rob Jun 15 '20 at 14:24
• @HamVocke Just for completeness, does this mean that every post has had its markdown edited so that it conforms to CommonMark without altering the produced html? Or is there some small population of 'sleeper' posts that might change when edited? – Emilio Pisanty Jun 16 '20 at 14:48
• @EmilioPisanty it's the latter. We've only edited a post's markdown if the resulting HTML (rendered with the new CommonMark renderer) is equivalent to the old HTML. There is going to be a small population (less than 3% of all posts approx.) that are going to look slightly different if someone was to edit them. – Ham Vocke Jun 16 '20 at 14:57
How can we help?
I'm happy to do my part to help this come off smoothly, but I'm not sure how and where we can help. The Mother-Meta announcement mentions the possibility that the migration will break (in the sense that the new html will be different to the old one) for a fraction of the posts, and that when this happens the migration will be discarded and the old html will be retained.
To me, this sounds like it calls for human intervention as a slow go-through by hand of all the failed markdown migrations $$-$$ I would much rather have a community effort to go through these and make sure everything is OK before some inattentive editor does something else on the post, fails to notice something breaking, and pushes the re-render through.
Will there be a centralized place where we can see the posts whose markdown migration has been flagged by the system, and edit them into the new renderer? If the renderer is getting bumped when you edit, will this be visible to the editor? (Say, as a banner on the Edit page saying that the renderer bumped and that extra care must be taken to ensure nothing is broken?)
As a quick partial answer -- it seems that the list of edited posts get attributed to Community ♦ and will be shown on its Revisions listing. For Meta SE they've started to come in (here), but on Physics they haven't at the time of this writing -- presumably they will show up on this listing page once posts start being edited.
Presumably this can be queried from SEDE once the data there gets updated, but in this form it cannot be used to query for failed CommonMark migrations, as the SEDE data cannot distinguish them from posts that already complied with the CommonMark spec, if I understand correctly.
• We don't know the exact amount and shape of issues we're going to run into. Judging from our test runs (based on production data) an overwhelming amount of posts will be automatically fixed or show non-visible differences only. I will take a look at migration failures and try to weed out those that are non-negotiable so that they can be fixed in an automated way (also for other sites to come). We might resort to using notifications or a sort of review queue if we can't do without manual intervention but we hope we won't even have to go there. – Ham Vocke Jun 2 '20 at 15:45
• Oh, if my comment is coming across the wrong way: I really appreciate our communities offering to step in and help resolving this, thanks for that. It's more that we need to find a solution that scales across all different sites on the SE network, some of which have millions and millions of posts. If we can minimize manual intervention and avoid having people apply mundane fixes to posts, I'll take that any day :) – Ham Vocke Jun 2 '20 at 17:55
• Sure, that makes sense. If you do need human checkers to feed data into that machine (eg by going manually over the corpus of flagged posts and taking appropriate action), then just tell us what you need. When you guys do this type of process correctly (as you've done with this migration) you have a pool of users willing to step in and donate value to the post corpus. – Emilio Pisanty Jun 2 '20 at 20:34
• In any case, it would be good to know how it's going once it gets rolling. If there's a population of posts that could be broken by editing, then please do let us know here. – Emilio Pisanty Jun 2 '20 at 20:37
• FWIW, a minor bug in the Markdown autofixer has been found, and fixed: meta.stackexchange.com/a/348897/334566 – PM 2Ring Jun 4 '20 at 8:29
• @PM2Ring Thanks for the heads-up. – Emilio Pisanty Jun 4 '20 at 10:17
More precisely, what time and date and timezone does this go live?
You said:
on Thursday, June 4th, 2020
but of course that means different things to different people, so could we get the time and date more precisely from you that the switch happens?
Also, is there a specific time when we will be presented with a completely translated site or is this translation (of old posts) being done as they are fetched by normal access or something else?
• I'm sorry about the ambiguity. I'd love to get an additional pair of eyes on this and since our engineering team is spread across multiple time zones coordination can become a little more tricky given that there are other spontaneous things that pop up for every one of us. I'm confident that I'll kick off the migration within the next 2-3 hours, it shouldn't run for much longer than 30 minutes. – Ham Vocke Jun 4 '20 at 13:50
The rendering of mathematical material has gotten way, way slower since the changeover. This makes it more of a pain to write and edit answers with mathematics in them, especially since any change to a mathematical expression causes all the math in a post to be re-rendered, which can take (I timed up) upwards of fifteen seconds. This makes even small changes very hard to verify, waiting around to see whether what you have typed was correct.
• This hasn't happened for me – BioPhysicist Jun 18 '20 at 3:45
• Unfortunately I can't reproduce what you're describing. I'm not saying this isn't happening but I also don't see why MathJax rendering would become slower after switching to CommonMark since rendering Markdown to HTML and rendering MathJax equations are two completely distinct steps. Do you have any steps that could help me reproduce this behavior? – Ham Vocke Jun 18 '20 at 6:44
|
2021-04-15 04:38:57
|
{"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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24518099427223206, "perplexity": 1485.7171713766095}, "config": {"markdown_headings": false, "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/1618038083007.51/warc/CC-MAIN-20210415035637-20210415065637-00327.warc.gz"}
|
http://en.wikipedia.org/wiki/CC_(complexity)
|
# CC (complexity)
In computational complexity theory, CC (Comparator Circuits) is the complexity class containing decision problems which can be solved by comparator circuits of polynomial size.
Comparator circuits are sorting networks in which each comparator gate is directed, each wire is initialized with an input variable, its negation, or a constant, and one of the wires is distinguished as the output wire.
The most important problem which is complete for CC is a decision variant of the stable marriage problem.
## Definition
A single comparator gate.
A comparator circuit is a network of wires and gates. Each comparator gate, which is a directed edge connecting two wires, takes its two inputs and outputs them in sorted order (the larger value ending up in the wire the edge is pointing to). The input to any wire can be either a variable, its negation, or a constant. One of the wires is designated as the output wire. The function computed by the circuit is evaluated by initializing the wires according to the input variables, executing the comparator gates in order, and outputting the value carried by the output wire.
The comparator circuit value problem (CCVP) is the problem of evaluating a comparator circuit given an encoding of the circuit and the input to the circuit. The complexity class CC is defined as the class of problems logspace reducible to CCVP.[1] An equivalent definition[2] is the class of problems AC0 reducible to CCVP.
As an example, a sorting network can be used to compute majority by designating the middle wire as an output wire:
If the middle wire is designated as output, and the wires are annotated with 16 different input variables, then the resulting comparator circuit computes majority. Since there are sorting networks which can be constructed in AC0, this shows that the majority function is in CC.
## CC-complete problems
A problem in CC is CC-complete if every problem in CC can be reduced to it using a logspace reduction. The comparator circuit value problem (CCVP) is CC-complete.
In the stable marriage problem, there is an equal number of men and women. Each person ranks all members of the opposite sex. A matching between men and women is stable if there are no unpaired man and woman who prefer each other over their current partners. A stable matching always exists. Among the stable matchings, there is one in which each woman gets the best man that she ever gets in any stable matching; this is known as the woman-optimal stable matching. The decision version of the stable matching problem is, given the rankings of all men and women, whether a given man and a given woman are matched in the woman-optimal stable matching. Although the classical Gale–Shapley algorithm cannot be implemented as a comparator circuit, Subramanian[3] came up with a different algorithm showing that the problem is in CC. The problem is also CC-complete.
Another problem which is CC-complete is lexicographically-first maximal matching.[3] In this problem, we are given a bipartite graph with an order on the vertices, and an edge. The lexicographically-first maximal matching is obtained by successively matching vertices from the first bipartition to the minimal available vertices from the second bipartition. The problem asks whether the given edge belongs to this matching.
Scott Aaronson showed that the pebbles model is CC-complete.[4] In this problem, we are given a starting number of pebbles (encoded in unary) and a description of a program which may contain only two types of instructions: combine two piles of sizes $y$ and $z$ to get a new pile of size $y+z$, or split a pile of size $y$ into piles of size $\lceil y/2 \rceil$ and $\lfloor y/2 \rfloor$. The problem is to decide whether any pebbles are present in a particular pile after executing the program. He used this to show that the problem of deciding whether any balls reach a designated sink vertex in a Digi-Comp II-like device is also CC-complete.
## Containments
The comparator circuit evaluation problem can be solved in polynomial time, and so CC is contained in P. On the other hand, comparator circuits can solve directed reachability,[3] and so CC contains NL. In the relativized world, CC and NC are incomparable,[2] and so both containments are strict.
## References
1. ^ E. W. Mayr, A. Subramanian (1992). "The complexity of circuit value and network stability". Journal of Computer and System Sciences 2 (44): 302–323.
2. ^ a b S. A. Cook, Y. Filmus, D. T. M. Le (2012). "The complexity of the comparator circuit value problem". arXiv:1208.2721.
3. ^ a b c A. Subramanian (1994). "A new approach to stable matching problems". SIAM Journal on Computing 23 (4): 671–700. doi:10.1137/s0097539789169483.
4. ^ Aaronson, Scott (4 July 2014). "The Power of the Digi-Comp II". Shtetl-Optimized. Retrieved 28 July 2014.
|
2014-09-23 04:55:01
|
{"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": 6, "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": 0.616847038269043, "perplexity": 716.9328875432051}, "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-2014-41/segments/1410657137948.80/warc/CC-MAIN-20140914011217-00111-ip-10-234-18-248.ec2.internal.warc.gz"}
|
https://www.researchaffiliates.com/publications/articles/641-pricing-stocks-and-bonds
|
The requested page you are looking for has either moved or is no longer available.
We've directed you back to our homepage so you can continue to browse the site.
# Pricing Stocks and Bonds
October 2017
Save
Key Points
• We form asset-class forecasts because they guide our ultimate aim: to fulfill our fiduciary duty of building a portfolio of assets that will meet investors’ future financial needs.
• Our process of forming expectations is based on a simple, robust framework. An understanding of this framework and its underlying assumptions informs why we focus on long-term estimates.
• Consistently forming precise forecasts is not only nearly impossible, but is also not necessary for building superior portfolios.
Nature has established patterns originating in the return of events, but only for the most part…. [N]o matter how many experiments you have done… you have not thereby imposed a limit on the nature of events so that in the future they could not vary. — Gottfried Leibniz
1
An enduring goal of asset managers, advisors, consultants, and individual investors is to seek, build, and recommend investment solutions—or portfolios of asset classes and underlying securities—that adequately meet investors’ future financial needs or aspirations. In our quest toward this universal aim, forming expectations about the risk and return of investments that we wish to hold can be a useful endeavor. A valuable input to long-term planning, this is often the first step in defining a framework for building portfolios. Interestingly, what may surprise most is that we don’t necessarily need perfectly accurate forecasts to create value-added portfolios.
In this article, we review the genesis behind the long-term framework Research Affiliates uses to generate return expectations, illustrating with a straightforward portfolio of stocks and government bonds, although the framework is in no way limited to those assets. We show a simple example of how these expectations, noisy as they can sometimes be, could have been used historically to create a portfolio able to outperform an equally simple 60/40 benchmark.
## Simplest Return Expectations
Contrary to popular belief, estimating future returns is not at all about foretelling what the future holds. Rather, the process is simply a humble attempt to quantify the returns we can expect if everything acts as it should—which almost never happens. Therefore, realized future return should always be thought of as an expected (mean) return plus unexpected variation in return that can arise from idiosyncrasies in the market, that is, shocks which should not be modeled, as well as from unknown deficiencies in the model:
Future Return = Expected Return + Unexpected Shocks
The simplest way to form a framework for estimating expected return is to focus on a scenario in which everything always happens as expected. Consider the case of purchasing a high-quality instrument (think a zero rate of default) that does not pay interim cash flows, and then holding that instrument to maturity. This could be a three-month certificate of deposit or a multi-year zero-coupon government bond hedged against default. Under this scenario, the future realized return of the investment, and additionally the expected return today, is known with complete certainty to be the purchase yield of the instrument.
Starting from the perspective of purchase yield has the benefit of certainty. When we buy an asset we know its yield without the need for fancy models. We can simply look it up! This simple valuation framework can serve as a useful approach in understanding core assumptions required to develop expected returns across a breadth of investment assets. If the answer to all of the following questions is “yes,” we know with certainty that the future return of the instrument is its yield today. We can then think of expected return as how far from these initial assumptions we need to move based on the asset in question.
The challenge arises once we price a wider variety of instruments, which inevitably forces us to answer “no” to one or more of these questions. Doing so moves the expected return away from the purchase yield and requires a more-complex model. As a result, expected return is far more difficult to estimate, and the probability of unexpected shocks increases. To help tackle this reality, we use a framework based on long-term expectations.
## Begin with a Focus on the Long Term
To reduce the impact of idiosyncratic shocks that occur in asset returns, we focus on long-term returns because, historically, the more returns that are averaged together, the tighter the distribution. My intent with this point is not to introduce a debate on mean reversion, but simply to acknowledge the data we have historically observed across markets. For example, over the last 140 years in the US equity market, Aked and Ko (2017) show that, for a 1-year horizon, the volatility of returns has been 19.2%, while at a 10-year horizon, the volatility dropped to 4.7%.3
Armed with these historical data, we can feel confident that a 10-year expected return will have a fairly tight distribution when compared to its shorter-term counterpart of a 1-year expected return. But unless an investor is comfortable accepting the historical average return as the expectation for the future, something we’ve written extensively about as being a bad idea, a tighter distribution of historical10-year returns does nothing to help us understand the expected return (mean) of this distribution. For that, we must turn to another asset characteristic, return relationship.
## Forecasting Bond Returns
As noted earlier, the process of calculating an asset’s expected return is related to how much we must relax assumptions of quality, cash flows, reinvestment rate, and holding horizon. To better understand this framework, let’s look at an example of a 10-year fixed-rate US Treasury bond (historically, without default) and compare the purchase yield to the total nominal return.4
To make a more interesting example, let’s consider a 10-year fixed-rate coupon-paying US Treasury bond, but instead of holding the bond to maturity, we create a pseudo-constant 10-year maturity bond. For this investment, we purchase a 10-year US Treasury, hold it for one year, at which point we sell the now 9-year bond to purchase a new 10-year bond. Referring to our framework, we are relaxing the hold-to-maturity and constant reinvestment rate assumptions. Doing so, we still observe a strong positive relationship, an R2 of 75% and a coefficient close to 1.
This relationship will hold for any similar security; it is not inherently unique to US Treasury bonds. In fact, the important metric is the autocorrelation in yields, which for a number of developed-market government bonds is close to 90% at a one-year horizon. This means a strong relationship exists between the yield of a bond today and the yield of a similar bond issued one year ago. Therefore, if it is not possible to hold the bond to maturity, a “similar” instrument will typically be obtained by selling the bond owned after one year and purchasing a new bond having the original maturity (creating a pseudo-constant maturity instrument); in the earlier example, this involved selling the original 10-year maturity after one year and purchasing a new 10-year maturity.
## Forecasting Stock Returns
Stocks, by their very nature, require us to relax each of our four initial assumptions or conditions. Equity holders are often wiped out by default, intermediate cash payments and reinvestment conditions change over time, and stocks do not have a predefined maturity date at which the investor’s principal is returned. For these reasons, we expect, and see, a much noisier relationship between an investor’s purchase yield and future return. At a 10-year investment horizon, however, a meaningfully positive relationship still exists, albeit with a greater amount of dispersion around the trend. Historically speaking, starting dividend yield has done a nice job of predicting future returns!
Does this mean we should rely on dividend yield alone to form our future expectation of return? Not really. Since 1871, the coefficient of dividend yield to return is, like bonds, close to 1, actually just a bit above. If we focus solely on the post-war period of January 1946 to July 2017, we see a similar relationship, but with a coefficient close to 3. Because investors don’t receive a quarterly check in the amount of $3 for every$1 of dividends paid by a company, the implication is that although simple yield is a good starting point, we should find a better way to estimate future return.
So, although dividend yields are a good starting point for deriving forward-looking estimates, we also need to consider other potentially meaningful sources of return. Through a simple decomposition of stock returns over the last two centuries, we can identify three reliable components to forecast and model (Arnott and Bernstein, 2002). In short, depending on the time span, nearly one-third to one-half of the long-term return on stocks comes from sources other than dividend yield, such as inflation, growth in dividends, and changes in valuation levels. We are able to extend this framework beyond US stocks to any asset class which relaxes the constraints in a similar way.
For those familiar with the Asset Allocation Interactive tool on our website, these three decomposition components will no doubt look familiar as the building blocks of return we incorporate in the tool. They come from a simple rearrangement of the terms in the equation used to calculate return (see Baetji and Menkhoff [2016] for a full derivation):
## More Complete Model
Adding more terms to our expected return model requires us to generate a forecasting model for each. In addition to dividend yield at each point in time, we use the long-term growth in real earnings per share to forecast cash flow growth, and the reversion in the Shiller P/E multiple for expected changes in the cash flow multiple. For expected returns, because the return from inflation is lost due to reduced purchasing power, we prefer to focus on expectations of return, net of inflation.
The Gordon Growth Model focuses on a constant-future-yield framework and only includes yield and growth as drivers of future return. Whereas the industry continues to debate the efficacy of Shiller P/E reversion, we at Research Affiliates firmly believe in mean reversion in asset prices (Brightman, Masturzo, and Treussard, 2014), although we acknowledge it to be tricky to accurately forecast. As the great Peter Bernstein (1996) once said about mean reversion:
First, it sometimes proceeds at so slow a pace that a shock will disrupt the process. Second, the regression may be so strong that matters do not come to rest once they reach the mean…. Finally, the mean itself may be unstable, so that yesterday’s normality may be supplanted today by a new normality we know nothing about. (p. 172)
For the most part, the trend in our expectations of US stock returns has been in line with subsequent realized returns; however, sometimes (especially recently) we’ve missed the mark, sometimes by a lot, even with the benefit of hindsight. Drafting expectations of return is definitely not an exact science.
Even if our models sometimes misestimate realized return does not mean we should abandon the work of modeling expectations. Certainly, the attempt to construct an accurate future expected return is one goal in the modeling process, but another immensely important goal is using the model to create portfolios with value-add. Considering only the models for stocks and bonds so far introduced in this article, we can look at their value in the context of portfolio construction.
## Building a Portfolio Using Expected Returns
Because the objective of investors is inevitably to create portfolios, expectations of risk and return can be viewed as signals to be used in achieving that end. Perhaps surprisingly, we don’t even need accurate expected returns versus future realized returns in order to generate value-add portfolios, those capable of outperforming a particular benchmark. We can demonstrate this in the context of a two-asset portfolio of US stocks and bonds. Restricting our case study to two assets is done for simplicity and to illustrate a point, not to imply that investors should ever restrict themselves simply to two assets.
A very common way to generate portfolio weights from risk and return expectations is through the use of mean-variance optimization (MVO), which aims to create the portfolios with the highest achievable return per unit of risk, also known as efficient frontier portfolios. Although the benefits of MVO are vast, the approach has a few important drawbacks, such as high sensitivity to our input expectations as well as difficulty in tying the resulting portfolio weights to the input expectations, especially when the number of assets grows. Thus, for our case study, let’s take a different, but very straightforward, approach.
Instead of comparing our risk and return expectations cross-sectionally among assets, à la MVO, let’s compare the current expected return for each asset against its own historical time series of expectations. In this way, we are attempting to neutralize some of the deficiencies (noise) in our models that we expect will be relatively constant over time. Through this comparison, we can derive a confidence score for each asset based on its expected return. For example, if an asset has a high expected return versus its expected return history, our model should show more confidence in the cheapness of that asset versus how the model has historically viewed it. It’s easy to understand why we would want to overweight that asset from its neutral position, or vice versa if the expected return is low. This approach is not new, and Asness, Ilmanen, and Maloney (2015) discuss a very similar approach.
The first step is to create a confidence score (i.e., raw weight) for each asset5 by comparing the current expected return to its historical median using a variant of standard min–max scaling. The confidence score then indicates the amount of the over- or underweight compared to the neutral position. The following equation shows that if the current expected return is higher than the historical median, the raw weight implies an overweight to the asset, or if less, an underweight:
Because we are comparing our simple strategy to the traditional 60/40 portfolio, we set the neutral weights in our portfolio to 60% for stocks and to 40% for bonds. Having tested other neutral weights against their respective benchmarks, we find they produce similar results.
Thus far, we have considered each asset in isolation, ignoring all cross-sectional relationships, which is not to imply they are not important, they definitely are. In this framework, cross-sectional relationships are introduced through weight normalization; the portfolio must be fully invested in the two assets and not take on leverage. We may want to overweight both assets, overweight one and underweight the other, or underweight both based on the assets’ current expected returns versus their respective histories. By normalizing the weights for full investment, those with higher scores will get higher allocations, and vice versa, thus introducing a cross-asset comparison of expectations.
In our example, data begin in 1910 in order to have 20 years to develop an earnings growth rate for stocks’ expected returns and another 10 years to capture the statistics necessary to calculate the raw weight. By doing so, our expected returns only contain information available at each point in history.6
Because we are interested in 10-year expected returns, it makes sense to set our rebalance period for the portfolio at 10 years. Otherwise, we would be giving up information contained in our expected return models. For shorter rebalance periods, we would use shorter-horizon expectations, although in this case, we only have long-horizon returns. Therefore, let’s “cheat just a little” and use a 5-year rebalance period, which balances the usefulness of the portfolio with consistency in the horizon of the expectations.
Over our data sample, two periods existed in which the portfolio was 100% invested in a single asset. In the early 1920s, the portfolio was 100% invested in stocks, whereas in the early part of the current century, it was 100% invested in bonds. The majority of the time, however, our portfolio has held a healthy mix of both assets. For example, starting in the 1960s and through the inflationary 1970s, the portfolio loaded up on stocks and sold bonds, before reversing course through the 1980s and 1990s. More recently, the strategy views both stocks and bonds as expensive and has been moving toward a more neutral 60/40 weight.
This simple trading strategy outperforms a 60/40 portfolio, regardless if the latter is rebalanced on a monthly basis, a five-year basis, or not at all (a pure buy-and-hold strategy). In addition, risk-adjusted outcomes improve, even while, on average, maintaining a lower exposure to US equities, the dominant risk exposure in most investors’ portfolios.
## What This Means
Our multi-question valuation framework provides a tested approach for generating future values of any asset class. Importantly, it also propels us to examine underlying core assumptions and question how asset classes inherently relax some or all of the four conditions. In this article, as we have studied and applied this framework using the two most common asset classes—stocks and bonds—to build a simple portfolio. In doing so, we are reminded of a few enduring principles.
First, our framework does not by any means imply that the expected outcomes will be close to—let alone, consistently match—future realized returns, especially over short time horizons. In our industry, a focus on achieving short-term accuracy is tempting. But deriving forecasts can give a false illusion of precision and statistical clarity. Like most managers, we have no special skill or clairvoyance in predicting the short term. As such, we take advantage of historic long-term relationships to generate our long-term expected returns.
Second, although the example in the article has focused exclusively on stocks and bonds, the concepts underlying our framework apply to any asset class. The beauty of the simplest of frameworks, such as the one we use here, is that it clearly forces us to understand how each asset relates to key assumptions. For asset classes that relax all criteria, we can then understand how to improve and expand our framework to better inform long-term expected outcomes.
Finally, even if our long-term expected returns meaningfully diverge at various times from realized returns—a fact that every investor must not only be aware of, but expect—it doesn’t mean our process is broken or unproductive. Creating a rich set of expected return models enables us to create portfolios capable of outperforming, even if the asset models themselves are operating with low accuracy over certain periods.
FEATURED TAGS
## Endnotes
1. From a letter sent in 1703 by Gottfried Leibniz (1646–1716), a German mathematician and philosopher, to Jacob Bernoulli (1654–1705), a Swiss mathematician who made significant contributions in the field of probability.
2. Many reasons exist for an investment not to be held to maturity, some due to the investor and some to the issuer. A few of these reasons include the instrument’s maturity is longer or shorter than the investment horizon, the instrument is paid back early, or the issuer recalls the instrument.
3. Aked and Ko (2017) also point out that investors should be aware that although the average volatility over a decade is low, which is meaningful for generating expectations of return, over a 10-year period investors will experience, on average, 14.9% volatility each year.
4. We use nominal returns because the bond yield is stated in nominal terms and includes an expected market inflation rate. The relationship of yield to the real return of bonds is much weaker because the market-implied inflation rate at the purchase date could be vastly different from realized inflation over the 10-year horizon. Also, as noted earlier, if the bond was a zero-coupon bond, held for 10 years, the return would be the same as the starting yield, a perfect correlation.
5. For implementation, we replace the minimum expected return with the 5th percentile rank and the maximum expected return with the 95th percentile rank, consistent with Asness, Ilmanen, and Maloney (2015).
6. Although we readily concede that no backtest can ever truly be out of sample when using data from the past known to the modeler at the time the model is created.
## References
Aked, Michael, and Amie Ko. 2017. “Time Diversification Redux.” Research Affiliates (August).
Arnott, Robert, and Peter Bernstein. 2002. “What Risk Premium Is ‘Normal’?Financial Analysts Journal, vol. 58, no. 2 (March/April):64–85.
Asness, Cliff, Antti Ilmanen, and Thomas Maloney. 2015. “Market Timing Is Back in the Hunt for Investors.” Institutional Investor (November 11).
Baetje, Fabian, and Lukas Menkhoff. 2016. “Equity Premium Prediction: Are Economic and Technical Indicators Unstable?” DIW Berlin Discussion Paper No. 1552 (February 24). Available at SSRN.
Bernstein, Peter. 1996. Against the Gods: The Remarkable Story of Risk. New York, NY: John Wiley & Sons, Inc.
Brightman, Chris, Jim Masturzo, and Jonathan Treussard. 2014. “Our Investment Beliefs.” Research Affiliates (October).
|
2021-10-20 17:20:46
|
{"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": 0.42046669125556946, "perplexity": 1513.8532658673796}, "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/1634323585322.63/warc/CC-MAIN-20211020152307-20211020182307-00425.warc.gz"}
|
https://fabricebaudoin.wordpress.com/page/2/
|
## MA3160. Fall 2017. Midterm 1 sample
Practice midterm 1
We will do the correction in class on 09/28.
## HW4. MA360 Fall 2017
Exercise 1. Two dice are rolled. Consider the events A = {sum of two dice equals 3}, B = {sum of two dice equals 7 }, and C = {at least one of the dice shows a 1}.
(a) What is P(A | C)?
(b) What is P(B | C)?
(c) Are A and C independent? What about B and C?
Exercise 2. Suppose you roll two standard, fair, 6-sided dice. What is the probability that the sum is at least 9 given that you rolled at least one 6?
Exercise 3. Color blindness is a sex-linked condition, and 5% of men and 0.25% of women are color blind. The population of the United States is 51% female. What is the probability that a color-blind American is a man?
## Lecture 7. Rough paths. Fall 2017
In the previous lecture we introduced the signature of a bounded variation path $x$ as the formal series
$\mathfrak{S} (x)_{s,t} =1 + \sum_{k=1}^{+\infty} \int_{\Delta^k [s,t]} dx^{\otimes k}.$
If now $x \in C^{p-var}([0,T],\mathbb{R}^d)$, $p \ge 1$ the iterated integrals $\int_{\Delta^k [s,t]} dx^{\otimes k}$ can only be defined as Young integrals when $p < 2$. In this lecture, we are going to derive some estimates that allow to define the signature of some (not all) paths with a finite $p$ variation when $p \ge 2$. These estimates are due to Terry Lyons in his seminal paper and this is where the rough paths theory really begins.
For $P \in \mathbb{R} [[X_1,...,X_d]]$ that can be writen as
$P=P_0+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k}a_{i_1,...,i_k} X_{i_1}...X_{i_k},$
we define
$\| P \| =|P_0|+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k}|a_{i_1,...,i_k}| \in [0,\infty].$
It is quite easy to check that for $P,Q \in \mathbb{R} [[X_1,...,X_d]]$
$\| PQ \| \le \| P \| \| Q\|.$
Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. For $p \ge 1$, we denote
$\left\| \int dx^{\otimes k}\right\|_{p-var, [s,t]}=\left( \sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{i=0}^{n-1} \left\| \int_{\Delta^k [t_i,t_{i+1}]} dx^{\otimes k} \right\|^p \right)^{1/p},$
where $\mathcal{D}[s,t]$ is the set of subdivisions of the interval $[s,t]$. Observe that for $k \ge 2$, in general
$\int_{\Delta^k [s,t]} dx^{\otimes k}+ \int_{\Delta^k [t,u]} dx^{\otimes k} \neq \int_{\Delta^k [s,u]} dx^{\otimes k}.$
Actually from the Chen’s relations we have
$\int_{\Delta^n [s,u]} dx^{\otimes n}= \int_{\Delta^n [s,t]} dx^{\otimes k}+ \int_{\Delta^n [t,u]} dx^{\otimes k} +\sum_{k=1}^{n-1} \int_{\Delta^k [s,t]} dx^{\otimes k }\int_{\Delta^{n-k} [t,u]} dx^{\otimes (n-k) }.$
It follows that $\left\| \int dx^{\otimes k}\right\|_{p-var, [s,t]}$ needs not to be the $p$-variation of $t \to \int_{\Delta^k [s,t]} dx^{\otimes k}$.
The first major result of rough paths theory is the following estimate:
Proposition: Let $p \ge 1$. There exists a constant $C \ge 0$, depending only on $p$, such that for every $x \in C^{1-var}([0,T],\mathbb{R}^d)$ and $k \ge 0$,
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\| \le \frac{C^k}{\left( \frac{k}{p}\right)!} \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^k, \quad 0 \le s \le t \le T.$
By $\left( \frac{k}{p}\right)!$, we of course mean $\Gamma \left( \frac{k}{p}+1\right)$. Some remarks are in order before we prove the result. If $p=1$, then the estimate becomes
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\| \le \frac{C^k}{k!} \| x \|_{1-var, [s,t]}^k,$
which is immediately checked because
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\|$
$\le \sum_{I \in \{1,...,d\}^k} \left\| \int_{\Delta^{k}[s,t]}dx^{I} \right\|$
$\le \sum_{I \in \{1,...,d\}^k} \int_{s \le t_1 \le t_2 \le \cdots \le t_k \le t} \| dx^{i_1}(t_1) \| \cdots \| dx^{i_k}(t_k)\|$
$\le \frac{1}{k!} \left( \sum_{j=1}^ d \| x^j \|_{1-var, [s,t]} \right)^k.$
We can also observe that for $k \le p$, the estimate is easy to obtain because
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\| \le \left\| \int dx^{\otimes k}\right\|_{\frac{p}{k}-var, [s,t]}.$
So, all the work is to prove the estimate when $k >p$. The proof is split into two lemmas. The first one is a binomial inequality which is actually quite difficult to prove:
Lemma: For $x,y >0$, $n \in \mathbb{N}, n \ge 0$, and $p \ge 1$,
$\sum_{j=0}^n \frac{x^{j/p}}{\left( \frac{j}{p}\right)!} \frac{y^{(n-j)/p}}{\left( \frac{n-j}{p}\right)!} \le p \frac{(x+y)^{n/p}}{ {\left( \frac{n}{p}\right)!}}.$
Proof: See Lemma 2.2.2 in the article by Lyons or this proof for the sharp constant $\square$
The second one is a lemma that actually already was essentially proved in the Lecture on Young’s integral, but which was not explicitly stated.
Lemma: Let $\Gamma: \{ 0 \le s \le t \le T \} \to \mathbb{R}^N$. Let us assume that:
• There exists a control $\tilde{\omega}$ such that
$\lim_{r \to 0} \sup_{(s,t), \tilde{\omega}(s,t) \le r } \frac{\| \Gamma_{s,t} \|}{r}=0;$
• There exists a control $\omega$ and $\theta >1, \xi >0$ such that for $0 \le s \le t \le u \le T$,
$\| \Gamma_{s,u} \| \le \| \Gamma_{s,t} \|+ \| \Gamma_{t,u} \| +\xi \omega(s,u)^\theta.$
Then, for all $0 \le s \le t \le T$,
$\| \Gamma_{s,t} \| \le \frac{\xi}{1-2^{1-\theta}} \omega(s,t)^\theta.$
Proof:
See the proof of the Young-Loeve estimate or Lemma 6.2 in the book by Friz-Victoir $\square$
We can now turn to the proof of the main result.
Proof:
Let us denote
$\omega(s,t)=\left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^p.$
We claim that $\omega$ is a control. Indeed for $0 \le s \le t \le u \le T$, we have from Holder’s inequality
$\omega(s,t)+\omega(t,u)$
$= \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^p+\left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [t,u]} \right)^p$
$\le \left( \sum_{j=1}^{[p]}\left( \left\| \int dx^{\otimes j}\right\|^{p/j}_{\frac{p}{j}-var, [s,t]} + \left\| \int dx^{\otimes j}\right\|^{p/j}_{\frac{p}{j}-var, [t,u]}\right)^{1/p} \right)^p$
$\le \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,u]} \right)^p =\omega(s,u).$
It is clear that for some constant $\beta > 0$ which is small enough, we have for $k \le p$,
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\| \le \frac{1}{\beta \left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}.$
Let us now consider
$\Gamma_{s,t}= \int_{\Delta^{[p]+1} [s,t]} dx^{\otimes ([p]+1)}.$
From the Chen’s relations, for $0 \le s \le t \le u \le T$,
$\Gamma_{s,u}= \Gamma_{s,t}+ \Gamma_{t,u}+\sum_{j=1}^{[p]} \int_{\Delta^j [s,t]} dx^{\otimes j }\int_{\Delta^{[p]+1-j} [t,u]} dx^{\otimes ([p]+1-j) }.$
Therefore,
$\| \Gamma_{s,u}\|$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\sum_{j=1}^{[p]} \left\| \int_{\Delta^j [s,t]} dx^{\otimes j }\right\| \left\| \int_{\Delta^{[p]+1-j} [t,u]} dx^{\otimes ([p]+1-j) }\right\|$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\frac{1}{\beta^2} \sum_{j=1}^{[p]} \frac{1}{ \left( \frac{j}{p}\right)!} \omega(s,t)^{j/p}\frac{1}{ \left( \frac{[p]+1-j}{p}\right)!} \omega(t,u)^{([p]+1-j)/p}$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\frac{1}{\beta^2} \sum_{j=0}^{[p]+1} \frac{1}{ \left( \frac{j}{p}\right)!} \omega(s,t)^{j/p}\frac{1}{ \left( \frac{[p]+1-j}{p}\right)!} \omega(t,u)^{([p]+1-j)/p}$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\frac{1}{\beta^2} p \frac{(\omega(s,t)+\omega(t,u))^{([p]+1)/p}}{ {\left( \frac{[p]+1}{p}\right)!}}$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\frac{1}{\beta^2} p \frac{\omega(s,u)^{([p]+1)/p}}{ {\left( \frac{[p]+1}{p}\right)!}}.$
On the other hand, we have
$\| \Gamma_{s,t} \| \le A \| x \|_{1-var,[s,t]}^{[p]+1}.$
We deduce from the previous lemma that
$\| \Gamma_{s,t} \| \le \frac{1}{\beta^2} \frac{p}{1-2^{1-\theta}} \frac{\omega(s,t)^{([p]+1)/p}}{ {\left( \frac{[p]+1}{p}\right)!}},$
with $\theta=\frac{[p]+1}{p}$. The general case $k \ge p$ is dealt by induction. The details are let to the reader $\square$
Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. Since
$\omega(s,t)=\left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^p$
is a control, the estimate
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\| \le \frac{C^k}{\left( \frac{k}{p}\right)!} \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^k, \quad 0 \le s \le t \le T.$
easily implies that for $k > p$,
$\left\| \int dx^{\otimes k} \right\|_{1-var, [s,t]} \le \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}.$
We stress that it does not imply a bound on the 1-variation of the path $t \to \int_{\Delta^k [0,t]} dx^{\otimes k}$. What we can get for this path, are bounds in $p$-variation:
Proposition: Let $p \ge 1$. There exists a constant $C \ge 0$, depending only on $p$, such that for every $x \in C^{1-var}([0,T],\mathbb{R}^d)$ and $k \ge 0$,
$\left\| \int_{\Delta^k [0,\cdot]} dx^{\otimes k} \right\|_{p-var, [s,t]} \le \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{1/p} \omega(0,T)^{\frac{k-1}{p}}$
where
$\omega(s,t)= \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^p, \quad 0 \le s \le t \le T.$
Proof: This is an easy consequence of the Chen’s relations. Indeed,
$\left\| \int_{\Delta^k [0,t]} dx^{\otimes k} - \int_{\Delta^k [0,s]} dx^{\otimes k} \right\|$
$=\left\| \sum_{j=1}^k \int_{\Delta^j [s,t]} dx^{\otimes j} \int_{\Delta^{j-k} [0,s]} dx^{\otimes (k-j)} \right\|$
$\le \sum_{j=1}^k \left\| \int_{\Delta^j [s,t]} dx^{\otimes j} \right\| \left\| \int_{\Delta^{j-k} [0,s]} dx^{\otimes (k-j)} \right\|$
$\le C^k \sum_{j=1}^k \frac{1}{\left( \frac{j}{p}\right)!} \omega(s,t)^{j/p} \frac{1}{\left( \frac{k-j}{p}\right)!} \omega(s,t)^{(k-j)/p}$
$\le C^k \omega(s,t)^{1/p} \sum_{j=1}^k \frac{1}{\left( \frac{j}{p}\right)!} \omega(0,T)^{(j-1)/p} \frac{1}{\left( \frac{k-j}{p}\right)!} \omega(0,T)^{(k-j)/p}$
$\le C^k \omega(s,t)^{1/p} \omega(0,T)^{(k-1)/p}\sum_{j=1}^k \frac{1}{\left( \frac{j}{p}\right)!} \frac{1}{\left( \frac{k-j}{p}\right)!}.$
and we conclude with the binomial inequality $\square$
We are now ready for a second major estimate which is the key to define iterated integrals of a path with $p$-bounded variation when $p \ge 2$.
Theorem: Let $p \ge 1$, $K > 0$ and $x,y \in C^{1-var}([0,T],\mathbb{R}^d)$ such that
$\sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \le 1,$
and
$\left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right)^p+ \left( \sum_{j=1}^{[p]} \left\| \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right)^p \le K.$
Then there exists a constant $C \ge 0$ depending only on $p$ and $K$ such that for $0\le s \le t \le T$ and $k \ge 1$
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k}- \int_{\Delta^k [s,t]} dy^{\otimes k} \right\| \le \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right) \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{k/p} ,$
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k}\right\| +\left\| \int_{\Delta^k [s,t]} dy^{\otimes k} \right\| \le \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}$
where $\omega$ is the control
$\omega(s,t)= \frac{ \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^p+ \left( \sum_{j=1}^{[p]} \left\| \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^p } { \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right)^p+ \left( \sum_{j=1}^{[p]} \left\| \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right)^p }$
$+\left( \frac{\sum_{j=1}^{[p]} \left\| \int dx^{\otimes j} - \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} }{\sum_{j=1}^{[p]} \left\| \int dx^{\otimes j} - \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} } \right)^p$
Proof: We prove by induction on $k$ that for some constants $C,\beta$,
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k}- \int_{\Delta^k [s,t]} dy^{\otimes k} \right\| \le \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right) \frac{C^k}{\beta \left( \frac{k}{p}\right)!} \omega(s,t)^{k/p},$
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k}\right\| +\left\| \int_{\Delta^k [s,t]} dy^{\otimes k} \right\| \le \frac{C^k}{\beta \left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}$
For $k \le p$, we trivially have
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k}- \int_{\Delta^k [s,t]} dy^{\otimes k} \right\|$ $\le \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right)^k \omega(s,t)^{k/p}$
$\le \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right) \omega(s,t)^{k/p}.$
and
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k}\right\| +\left\| \int_{\Delta^k [s,t]} dy^{\otimes k} \right\| \le K^{k/p} \omega(s,t)^{k/p}$.
Not let us assume that the result is true for $0 \le j \le k$ with $k > p$. Let
$\Gamma_{s,t}=\int_{\Delta^k [s,t]} dx^{\otimes (k+1)}- \int_{\Delta^k [s,t]} dy^{\otimes (k+1)}$
From the Chen’s relations, for $0 \le s \le t \le u \le T$,
$\Gamma_{s,u}= \Gamma_{s,t}+ \Gamma_{t,u}$
$+\sum_{j=1}^{k} \int_{\Delta^j [s,t]} dx^{\otimes j }\int_{\Delta^{k+1-j} [t,u]} dx^{\otimes (k+1-j) }-\sum_{j=1}^{k} \int_{\Delta^j [s,t]} dy^{\otimes j }\int_{\Delta^{k+1-j} [t,u]} dy^{\otimes (k+1-j) }.$
Therefore, from the binomial inequality
$\| \Gamma_{s,u}\|$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\sum_{j=1}^{k} \left\| \int_{\Delta^j [s,t]} dx^{\otimes j }- \int_{\Delta^j [s,t]} dy^{\otimes j } \right\| \left\| \int_{\Delta^{k+1-j} [t,u]} dx^{\otimes (k+1-j) }\right\|$
$+\sum_{j=1}^{k} \left\| \int_{\Delta^{j} [s,t]} dy^{\otimes j }\right\| \left\| \int_{\Delta^{k+1-j} [t,u]} dx^{\otimes (k+1-j) }- \int_{\Delta^{k+1-j} [t,u]} dy^{\otimes (k+1-j) } \right\|$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\frac{1}{\beta^2}\tilde{\omega}(0,T) \sum_{j=1}^{k} \frac{C^j}{\left( \frac{j}{p}\right)!} \omega(s,t)^{j/p} \frac{C^{k+1-j}}{\left( \frac{k+1-j}{p}\right)!} \omega(t,u)^{(k+1-j)/p}$
$+\frac{1}{\beta^2}\tilde{\omega}(0,T) \sum_{j=1}^{k} \frac{C^j}{\left( \frac{j}{p}\right)!} \omega(s,t)^{j/p} \frac{C^{k+1-j}}{\left( \frac{k+1-j}{p}\right)!} \omega(t,u)^{(k+1-j)/p}$
$\le \| \Gamma_{s,t} \| + \| \Gamma_{t,u} \| +\frac{2p}{\beta^2} \tilde{\omega}(0,T) C^{k+1} \frac{ \omega(s,u)^{(k+1)/p}}{\left( \frac{k+1}{p}\right)! }$
where
$\tilde{\omega}(0,T)=\sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} .$
We deduce
$\| \Gamma_{s,t} \| \le \frac{2p}{\beta^2(1-2^{1-\theta})} \tilde{\omega}(0,T) C^{k+1} \frac{ \omega(s,t)^{(k+1)/p}}{\left( \frac{k+1}{p}\right)! }$
with $\theta= \frac{k+1}{p}$. A correct choice of $\beta$ finishes the induction argument $\square$
Posted in Rough paths theory | 3 Comments
## Lecture 6. Rough paths. Fall 2017
In this lecture we introduce the central notion of the signature of a path $x \in C^{1-var}([0,T],\mathbb{R}^d)$ which is a convenient way to encode all the algebraic information on the path $x$ which is relevant to study differential equations driven by $x$. The motivation for the definition of the signature comes from formal manipulations on Taylor series.
Let us consider a differential equation
$y(t)=y(s)+\sum_{i=1}^d \int_s^t V_i (y(u) )dx^i(u),$
where the $V_i$‘s are smooth vector fields on $\mathbb{R}^n$.
If $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a $C^{\infty}$ function, by the change of variable formula,
$f(y(t))=f(y(s))+\sum^{d}_{i=1}\int^{t}_{s}V_{i}f(y(u))dx^{i}(u).$
Now, a new application of the change of variable formula to $V_{i}f(y(s))$ leads to
$f(y(t))=f(y(s))+\sum^{d}_{i=1}V_{i}f(y(s))\int^{t}_{s}dx^{i}(u)+\sum^{d}_{i,j=1}\int^{t}_{s}\int^{u}_{s} V_{j}V_{i}f(y(v))dx^{j}(v)dx^{i}(u).$
We can continue this procedure to get after $N$ steps
$f(y(t))=f(y(s))+\sum^{N}_{k=1}\sum_{I=(i_1,\cdots,i_k)}(V_{i_1}\cdots V_{i_k}f)(y(s))\int_{\Delta^{k}[s,t]}dx^{I}+R_{N}(s,t)$
for some remainder term $R_{N}(s,t)$, where we used the notations:
• $\Delta^{k}[s,t]=\{(t_1,\cdots,t_k)\in[s,t]^{k}, s\leq t_1\leq t_2\cdots\leq t_k\leq t\}$
• If $I=\left(i_1,\cdots,i_k\right)\in\{1,\cdots,d\}^k$ is a word with length $k$, $\int_{\Delta^{k}[s,t]}dx^{I}=\displaystyle \int_{s \le t_1 \le t_2 \le \cdots \le t_k \le t}dx^{i_1}(t_1)\cdots dx^{i_k}(t_k).$
If we let $N\rightarrow +\infty$, assuming $R_{N}(s,t) \to 0$ (which is by the way true for $t-s$ small enough if the $V_i$‘s are analytic), we are led to the formal expansion formula:
$f(y(t))=f(y(s))+\sum^{+\infty}_{k=1}\sum_{I=(i_1,\cdots,i_k)}(V_{i_1}\cdots V_{i_k}f)(y(s))\int_{\Delta^{k}[s,t]}dx^{I}.$
This shows, at least at the formal level, that all the information given by $x$ on $y$ is contained in the iterated integrals $\int_{\Delta^{k}[s,t]}dx^{I}$.
Let $\mathbb{R} [[X_1,...,X_d]]$ be the non commutative algebra over $\mathbb{R}$ of the formal series with $d$ indeterminates, that is the set of series
$Y=y_0+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k} a_{i_1,...,i_k} X_{i_1}...X_{i_k}.$
Definition: Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. The signature of $x$ (or Chen’s series) is the formal series:
$\mathfrak{S} (x)_{s,t} =1 + \sum_{k=1}^{+\infty} \sum_{I \in \{1,...,d\}^k} \left( \int_{\Delta^{k}[s,t]}dx^{I} \right) X_{i_1} \cdots X_{i_k}, \quad 0 \le s \le t \le T.$
As we are going to see in the next few lectures, the signature is a fascinating algebraic object. At the source of the numerous properties of the signature lie the following so-called Chen’s relations
Lemma: Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. For any word $(i_1,...,i_n) \in \{ 1, ... , d \}^n$ and any $0 \le s \le t \le u \le T$,
$\int_{\Delta^n [s,u]} dx^{(i_1,...,i_n)}=\sum_{k=0}^{n} \int_{\Delta^k [s,t]} dx^{(i_1,...,i_k)}\int_{\Delta^{n-k} [t,u]} dx^{(i_{k+1},...,i_n)},$
where we used the convention that if $I$ is a word with length 0, then $\int_{\Delta^{0} [0,t]} dx^I =1$.
Proof: It follows readily by induction on $n$ by noticing that
$\int_{\Delta^n [s,u]} dx^{(i_1,...,i_n)}=\int_s^u \left( \int_{\Delta^{n-1} [s,t_n]} dx^{(i_1,...,i_{n-1})} \right) dx^{i_n}(t_n)$ $\square$
To avoid heavy notations, it will be convenient to denote
$\int_{\Delta^k [s,t]} dx^{\otimes k} =\sum_{I \in \{1,...,d\}^k} \left( \int_{\Delta^{k}[s,t]}dx^{I} \right) X_{i_1} \cdots X_{i_k}.$
This notation actually reflects a natural algebra isomorphism between $\mathbb{R} [[X_1,...,X_d]]$ and $1\oplus_{k=1}^{+\infty} (\mathbb{R}^d)^{\otimes k}$. With this notation, observe that the signature writes then
$\mathfrak{S} (x)_{s,t} =1 + \sum_{k=1}^{+\infty} \int_{\Delta^k [s,t]} dx^{\otimes k},$
and that the Chen’s relations become
$\int_{\Delta^n [s,u]} dx^{\otimes n}=\sum_{k=0}^{n} \int_{\Delta^k [s,t]} dx^{\otimes k }\int_{\Delta^{n-k} [t,u]} dx^{\otimes (n-k) }.$
The Chen’s relations imply the following flow property for the signature:
Proposition: Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. For any $0 \le s \le t \le u \le T$,
$\mathfrak{S} (x)_{s,u} =\mathfrak{S} (x)_{s,t}\mathfrak{S} (x)_{t,u}$
Proof: Indeed,
$\mathfrak{S} (x)_{s,u}$
$=1 + \sum_{k=1}^{+\infty} \int_{\Delta^k [s,u]} dx^{\otimes k}$
$=1 + \sum_{k=1}^{+\infty}\sum_{j=0}^{k} \int_{\Delta^j [s,t]} dx^{\otimes j }\int_{\Delta^{k-j} [t,u]} dx^{\otimes (k-j) }$
$=\mathfrak{S} (x)_{s,t}\mathfrak{S} (x)_{t,u}$
$\square$
Posted in Rough paths theory | 2 Comments
## Lecture 6. Rough paths Fall 2017
In the previous lecture we defined the Young’s integral $\int y dx$ when $x \in C^{p-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{q-var} ([0,T], \mathbb{R}^{e \times d})$ with $\frac{1}{p}+\frac{1}{q} > 1$. The integral path $\int_0^t ydx$ has then a bounded $p$-variation. Now, if $V: \mathbb{R}^d \to \mathbb{R}^{d \times d}$ is a Lipschitz map, then the integral, $\int V(x) dx$ is only defined when $\frac{1}{p}+\frac{1}{p} > 1$, that is for $p < 2$. With this in mind, it is apparent that Young’s integration should be useful to solve differential equations driven by continuous paths with bounded $p$-variation for $p < 2$. If $p \ge 2$, then the Young’s integral is of no help and the rough paths theory later explained is the correct one.
The basic existence and uniqueness result is the following. Throughout this lecture, we assume that $p < 2$.
Theorem: Let $x\in C^{p-var} ([0,T], \mathbb{R}^d)$ and let $V : \mathbb{R}^e \to \mathbb{R}^{e \times d}$ be a Lipschitz continuous map, that is there exists a constant $K > 0$ such that for every $x,y \in \mathbb{R}^e$,
$\| V(x)-V(y) \| \le K \| x-y \|.$
For every $y_0 \in \mathbb{R}^e$, there is a unique solution to the differential equation:
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T.$
Moreover $y \in C^{p-var} ([0,T], \mathbb{R}^e)$.
Proof: The proof is of course based again of the fixed point theorem. Let $0 < \tau \le T$ and consider the map $\Phi$ going from the space $C^{p-var} ([0,\tau], \mathbb{R}^e)$ into itself, which is defined by
$\Phi(y)_t =y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le \tau.$
By using basic estimates on the Young’s integrals, we deduce that
$\| \Phi(y^1)-\Phi(y^2) \|_{ p-var, [0,\tau]}$
$\le C \| x \|_{p-var,[0,\tau]} ( \| V(y^1)-V(y^2) \|_{ p-var, [0,\tau]} +\| V(y^1)(0)-V(y^2)(0)\|)$
$\le CK \| x \|_{p-var,[0,\tau]}( \| y^1-y^2 \|_{ p-var, [0,\tau]}+\| y^1(0)-y^2(0)\|).$
If $\tau$ is small enough, then $CK \| x \|_{p-var,[0,\tau]} < 1$, which means that $\Phi$ is a contraction of the Banach space $C^{p-var} ([0,\tau], \mathbb{R}^e)$ endowed with the norm $\| y \|_{p-var,[0,\tau]} +\| y(0)\|$.
The fixed point of $\Phi$, let us say $y$, is the unique solution to the differential equation:
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le \tau.$
By considering then a subdivision
$\{ \tau=\tau_1 < \tau_2 <\cdots <\tau_n=T \}$
such that $C K \| x \|_{p-var,[\tau_k,\tau_{k+1}]} < 1$, we obtain a unique solution to the differential equation:
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T$ $\square$
As for the bounded variation case, the solution of a Young’s differential equation is a $C^1$ function of the initial condition,
Proposition: Let $x\in C^{p-var} ([0,T], \mathbb{R}^d)$ and let $V : \mathbb{R}^e \to \mathbb{R}^{e \times d}$ be a $C^1$ Lipschitz continuous map. Let $\pi(t,y_0)$ be the flow of the equation
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T.$
Then for every $0\le t \le T$, the map $y_0 \to \pi (t,y_0)$ is $C^1$ and the Jacobian $J_t=\frac{\partial \pi(t,y_0)}{\partial y_0}$ is the unique solution of the matrix linear equation
$J_t=Id+ \sum_{i=1}^d \int_0^t DV_i(\pi(s,y_0))J_s dx^i(s).$
As we already mentioned it before, solutions of Young’s differential equations are continuous with respect to the driving path in the $p$-variation topology
Theorem: Let $x^n \in C^{p-var} ([0,T], \mathbb{R}^d)$ and let $V : \mathbb{R}^e \to \mathbb{R}^{e\times d}$ be a Lipschitz and bounded continuous map such that for every $x,y \in \mathbb{R}^d$,
$\| V(x)-V(y) \| \le K \| x-y \|.$
Let $y^n$ be the solution of the differential equation:
$y^n(t)=y(0)+\int_0^t V(y^n(s)) dx^n(s), \quad 0\le t \le T.$
If $x^n$ converges to $x$ in $p$-variation, then $y^n$ converges in $p$-variation to the solution of the differential equation:
$y(t)=y(0)+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T.$
Proof: Let $0\le s \le t \le T$. We have
$\| y-y^n \|_{p-var,[s,t]}$
$= \left\| \int_0^\cdot V(y(u)) dx(u) -\int_0^\cdot V(y^n(u)) dx^n(u) \right\|_{p-var,[s,t]}$
$\le \left\| \int_0^\cdot (V(y(u))-V(y^n(u))) dx(u) + \int_0^\cdot V(y^n(u)) d( x(u)-x^n(u)) \right\|_{p-var,[s,t]}$
$\le \left\| \int_0^\cdot (V(y(u))-V(y^n(u))) dx(u) \right\|_{p-var,[s,t]}+\left\| \int_0^\cdot V(y^n(u)) d( x(u)-x^n(u)) \right\|_{p-var,[s,t]}$
$\le CK \| x\|_{p-var,[s,t]} \| y-y^n \|_{p-var,[s,t]}+C\| x-x^n \|_{p-var,[s,t]}(K \| y^n \|_{p-var,[s,t]}+\| V\|_{\infty, [0,T]})$
Thus, if $s,t$ is such that $CK \| x\|_{p-var,[s,t]} < 1$, we obtain
$\| y-y^n \|_{p-var,[s,t]} \le \frac{C(K \| y^n \|_{p-var,[s,t]}+\| V\|_{\infty, [0,T]})}{ 1-CK\| x\|_{p-var,[s,t]} } \| x-x^n \|_{p-var,[s,t]}.$
In the very same way, provided $CK \| x^n\|_{p-var,[s,t]} < 1$, we get
$\| y^n \|_{p-var,[s,t]} \le \frac{C\| V\|_{\infty, [0,T]}}{ 1-CK\| x^n\|_{p-var,[s,t]} }.$
Let us fix $0 < \varepsilon < 1$ and pick a sequence $0\le \tau_1 \le \cdots \le \tau_m=T$ such that $CK \| x\|_{p-var,[\tau_i,\tau_{i+1}]}+\varepsilon < 1$. Since $\| x^n\|_{p-var,[\tau_i,\tau_{i+1}]} \to \| x\|_{p-var,[\tau_i,\tau_{i+1}]}$, for $n \ge N_1$ with $N_1$ big enough, we have
$CK \| x^n\|_{p-var,[\tau_i,\tau_{i+1}]}+\frac{\varepsilon}{2} < 1.$
We deduce that for $n \ge N_1$,
$\| y^n \|_{p-var,[\tau_i,\tau_{i+1}]} \le \frac{2}{\varepsilon} C \| V\|_{\infty, [0,T]}$
and
$\| y-y^n \|_{p-var,[\tau_i,\tau_{i+1}]}$
$\le \frac{C(K \frac{2}{\varepsilon} C \| V\|_{\infty, [0,T]}+\| V\|_{\infty, [0,T]})}{ 1-CK\| x\|_{p-var,[\tau_i,\tau_{i+1}] }} \| x-x^n \|_{p-var,[\tau_i,\tau_{i+1}]}$
$\le \frac{C}{\varepsilon} \| V\|_{\infty, [0,T]} \left( \frac{2KC}{\varepsilon}+1 \right) \| x-x^n \|_{p-var,[\tau_i,\tau_{i+1}]}$
$\le \frac{C}{\varepsilon} \| V\|_{\infty, [0,T]} \left( \frac{2KC}{\varepsilon}+1 \right) \| x-x^n \|_{p-var,[0,T]}.$
For $n \ge N_2$ with $N_2 \ge N_1$ and big enough, we have
$\| x-x^n \|_{p-var,[0,T]} \le \frac{\varepsilon^3}{m},$
which implies
$\| y-y^n \|_{p-var,[0,T]} \le \frac{C}{\varepsilon} \| V\|_{\infty, [0,T]} \left( \frac{2KC}{\varepsilon}+1 \right) \varepsilon^3.$
$\square$
## HW3. MA3160 Fall 2017
Exercise 1. Two dice are simultaneously rolled. For each pair of events defined below, compute if they are independent or not.
(a) A1 ={thesumis7},B1 ={thefirstdielandsa3}.
(b) A2 = {the sum is 9}, B2 = {the second die lands a 3}.
(c) A3 = {the sum is 9}, B3 = {the first die lands even}.
(d) A4 = {the sum is 9}, B4 = {the first die is less than the second}.
(e) A5 = {two dice are equal}, B5 = {the sum is 8}.
(f) A6 = {two dice are equal}, B6 = {the first die lands even}.
(g) A7 = {two dice are not equal}, B7 = {the first die is less than the second}.
Exercise 2. Are the events A1, B1 and B3 from Exercise 1 independent?
Exercise 3. Suppose you toss a fair coin repeatedly and independently. If it comes up heads, you win a dollar, and if it comes up tails, you lose a dollar. Suppose you start with $20. What is the probability you will get to$150 before you go broke?
## Lecture 5. Rough paths. Fall 2017
In this lecture we define the Young‘s integral $\int y dx$ when $x \in C^{p-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{q-var} ([0,T], \mathbb{R}^{e \times d})$ with $\frac{1}{p}+\frac{1}{q} >1$. The cornerstone is the following Young-Loeve estimate.
Theorem: Let $x \in C^{1-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{1-var} ([0,T], \mathbb{R}^{e \times d})$. Consider now $p,q \ge 1$ with $\theta=\frac{1}{p}+\frac{1}{q} > 1$. The following estimate holds: for $0 \le s \le t \le T$,
$\left\| \int_s^t y(u)dx(u)-y(s)(x(t)-x(s)) \right\| \le \frac{1}{1-2^{1-\theta} }\| x \|_{p-var; [s,t]} \| y \|_{q-var; [s,t]}.$
Proof: For $0 \le s \le t \le T$, let us define
$\Gamma_{s,t} =\int_s^t y(u)dx(u) -y(s)(x(t)-x(s)) .$
We have for $s < t < u$,
$\Gamma_{s,u}-\Gamma_{s,t}-\Gamma_{t,u} =-y(s)(x(u)-x(s))+y(s)(x(t)-x(s))+y(t)(x(u)-x(t))= (y(s)-y(t))(x(t)-x(u)).$
As a consequence, we get
$\| \Gamma_{s,u}\|\le \| \Gamma_{s,t} \|+\| \Gamma_{t,u}\| +\| x \|_{p-var; [s,t]} \| y \|_{q-var; [t,u]}.$
Let now $\omega(s,t)=\| x \|^{1/\theta}_{p-var; [s,t]} \| y \|^{1/\theta}_{q-var; [s,t]}$. We claim that $\omega$ is a control. The continuity and the vanishing on the diagonal are obvious to check, so we just need to justify the superadditivity. Let $s < t < u$, we have from Holder’s inequality,
$\omega(s,t)+\omega(t,u)$
$=\| x \|^{1/\theta}_{p-var; [s,t]} \| y \|^{1/\theta}_{q-var; [s,t]}+\| x \|^{1/\theta}_{p-var; [t,u]} \| y \|^{1/\theta}_{q-var; [t,u]}$
$\le (\| x \|^{p}_{p-var; [s,t]} + \| x \|^{p}_{p-var; [t,u]})^{\frac{1}{p\theta}}(\| y \|^{q}_{q-var; [s,t]} + \| y \|^{q}_{q-var; [t,u]})^{\frac{1}{q\theta}}$
$\le \| x \|^{1/\theta}_{p-var; [s,u]} \| y \|^{1/\theta}_{q-var; [s,u]}=\omega(s,u).$
We have then
$\| \Gamma_{s,u}\|\le \| \Gamma_{s,t} \|+\| \Gamma_{t,u}\| +\omega(s,u)^\theta.$
For $\varepsilon > 0$, consider then the control
$\omega_\varepsilon (s,t)= \omega(s,t) +\varepsilon ( \| x \|_{1-var; [s,t]} + \| y \|_{1-var; [s,t]}).$
Define now
$\Psi(r)= \sup_{s,u, \omega_\varepsilon (s,u)\le r} \| \Gamma_{s,u}\|.$
If $s,u$ is such that $\omega_\varepsilon (s,u) \le r$, we can find a $t$ such that $\omega_\varepsilon(s,t) \le \frac{1}{2} \omega_\varepsilon(s,u)$, $\omega_\varepsilon(t,u) \le \frac{1}{2} \omega_\varepsilon(s,u)$. Indeed, the continuity of $\omega_\varepsilon$ forces the existence of a $t$ such that $\omega_\varepsilon(s,t)=\omega_\varepsilon(t,u)$. We obtain therefore
$\| \Gamma_{s,u}\|\le 2 \Psi(r/2) + r^\theta,$
which implies by maximization,
$\Psi(r)\le 2 \Psi(r/2) + r^\theta.$
By iterating $n$ times this inequality, we obtain
$\Psi(r)$
$\le 2^n \Psi\left(\frac{r}{2^n} \right) +\sum_{k=0}^{n-1} 2^{k(1-\theta)} r^\theta$
$\le 2^n \Psi\left(\frac{r}{2^n} \right) + \frac{1}{1-2^{1-\theta}} r^\theta.$
It is now clear that:
$\| \Gamma_{s,t} \|$
$\le \left\|\int_s^t (y(u)-y(s))dx(u) \right\|$
$\le \| x \|_{1-var; [s,t]} \| y-y(s) \|_{\infty; [s,t]}$
$\le ( \| x \|_{1-var; [s,t]} + \| y \|_{1-var; [s,t]})^2$
$\le \frac{1}{\varepsilon^2} \omega_\varepsilon (s,t)^2,$
Since
$\lim_{n \to \infty} 2^n \Psi\left(\frac{r}{2^n} \right) =0.$
We conclude
$\Psi(r) \le \frac{1}{1-2^{1-\theta}} r^\theta$
and thus
$\| \Gamma_{s,u}\| \le \frac{1}{1-2^{1-\theta}} \omega_\varepsilon(s,u) ^\theta$
Sending $\varepsilon \to 0$, finishes the proof $\square$
It is remarkable that the Young-Loeve estimate only involves $\| x \|_{p-var; [s,t]}$ and $\| y \|_{q-var; [s,t]}$. As a consequence, we obtain the following result whose proof is let to the reader:
Proposition: Let $x \in C^{p-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{q-var} ([0,T], \mathbb{R}^{e \times d})$ with $\theta=\frac{1}{p}+\frac{1}{q} >1$. Let us assume that there exists a sequence $x^n \in C^{1-var} ([0,T], \mathbb{R}^d)$ such that $x^n \to x$ in $C^{p-var} ([0,T], \mathbb{R}^d)$ and a sequence $y^n \in C^{1-var} ([0,T], \mathbb{R}^{e \times d})$ such that $y^n \to x$ in $C^{q-var} ([0,T], \mathbb{R}^d)$, then for every $s < t$, $\int_s^t y^n(u)dx^n(u)$ converges to a limit that we call the Young’s integral of $y$ against $x$ on the interval $[s,t]$ and denote $\int_s^t y(u)dx(u)$.
The integral $\int_s^t y(u)dx(u)$ does not depend of the sequences $x^n$ and $y^n$ and the following estimate holds: for $0 \le s \le t \le T$,
$\left\| \int_s^t y(u)dx(u)-y(s)(x(t)-x(s)) \right\| \le \frac{1}{1-2^{1-\theta} }\| x \|_{p-var; [s,t]} \| y \|_{q-var; [s,t]}.$
The closure of $C^{1-var} ([0,T], \mathbb{R}^d)$ in $C^{p-var} ([0,T], \mathbb{R}^d)$ is $C^{0, p-var} ([0,T], \mathbb{R}^d)$ and we know that $C^{p+\varepsilon-var} ([0,T], \mathbb{R}^d) \subset C^{0, p-var} ([0,T], \mathbb{R}^d)$. It is therefore obvious to extend the Young’s integral for every $x \in C^{p-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{q-var} ([0,T], \mathbb{R}^{e \times d})$ with $\theta=\frac{1}{p}+\frac{1}{q} >1$ and the Young-Loeve estimate still holds
$\left\| \int_s^t y(u)dx(u)-y(s)(x(t)-x(s)) \right\| \le \frac{1}{1-2^{1-\theta} }\| x \|_{p-var; [s,t]} \| y \|_{q-var; [s,t]}.$
From this estimate, we easily see that for $x \in C^{p-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{p-var} ([0,T], \mathbb{R}^{e \times d})$ with $\frac{1}{p}+\frac{1}{q} > 1$ the sequence of Riemann sums
$\sum_{k=0}^{n-1} y(t_i)( x_{t_{i+1}}-x_{t_i})$
will converge to $\int_s^t y(u)dx(u)$ when the mesh of the subdivision goes to 0. We record for later use the following estimate on the Young’s integral, which is also an easy consequence of the Young-Loeve estimate (see Theorem 6.8 in the book for further details).
Proposition: Let $x \in C^{p-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{q-var} ([0,T], \mathbb{R}^{e \times d})$ with $\frac{1}{p}+\frac{1}{q} > 1$. The integral path $t \to \int_0^t y(u)dx(u)$ is continuous with a finite $p$-variation and we have
$\left\|\int_0^\cdot y(u) dx(u) \right\|_{p-var, [s,t] }$
$\le C \| x \|_{p-var; [s,t]} \left( \| y \|_{q-var; [s,t]} + \| y \|_{\infty; [s,t]} \right)$
$\le 2C \| x \|_{p-var; [s,t]} \left( \| y \|_{q-var; [s,t]} + \| y(0)\| \right)$
Posted in Rough paths theory | 1 Comment
|
2017-11-22 09:15:58
|
{"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": 386, "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": 0.9519075155258179, "perplexity": 210.28767509519548}, "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-2017-47/segments/1510934806543.24/warc/CC-MAIN-20171122084446-20171122104446-00194.warc.gz"}
|
http://www.magicanoz.com/2015/09/hollow-by-menny-lindenfeld.html
|
# Hollow By Menny Lindenfeld
Phase One: The magician punches a hole through a freely chosen and signed card. He covers the hole with his finger tips and mysteriously slides the hole to a totally new location on the card! The hole is real and the spectator's can see right through it
Phase Two: At his command, without covering it, the hole visibly jumps to a third location on the card
Phase Three: The magician then takes another card, slides it over the chosen card, and the hole visibly jumps from the chosen card to the second card
Both cards can be given to the spectator as souvenirs
|
2016-12-03 09:35:47
|
{"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": 0.8714272379875183, "perplexity": 3047.0691990388927}, "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/1480698540915.89/warc/CC-MAIN-20161202170900-00330-ip-10-31-129-80.ec2.internal.warc.gz"}
|
http://aftermath.rocks/2016/03/20/equality-saturation/
|
# AfterMath
## Equality Saturation: a New Approach to Optimization
Authors: Tate, Stepp, Tatlock, Lerner
Conference: POPL 2009
A common challenge in optimizing compilers is the phase ordering problem: in which order should different optimizations be applied? The problematic issue is that applying one optimization may change the program such that some other, potentially more profitable, optimization can no longer be performed. Equality saturation is a clever idea for getting around the phase ordering problem. As an interesting side effect, an optimizer based on equality saturation can also be used for translation validation of other optimizers.
Consider the expression i * 5 where i is an integer. Multiplication instructions commonly take several CPU cycles, so compilers try to optimize such multiplications by constants to cheaper expressions. In this particular case, i << 2 + i computes the same value, but many processors can do the shift and the add in a single instruction that executes in fewer cycles than the original multiplication. Thus this replacement is an optimization.
Now consider the same expression in context, as part of a larger piece of code:
i := 0;
while (...) {
use(i * 5);
i := i + 1;
...
}
(This is a simplified version of the running example in the original paper. All omissions, misunderstandings, and other errors are entirely my fault, not the original authors’.)
This pattern of accesses to i allows the compiler perform an optimization known as strengh reduction, replacing the “strong” multiplication operation by “weaker” additions:
i := 0;
while (...) {
use(i);
i := i + 5;
...
}
This version is equivalent to the original, passing the values 0, 5, 10, … to the use operation. However, it uses a single addition per loop iteration, which is faster than both the variant with the multiplication and the variant where i * 5 is replaced by i << 2 + i.
We have seen that two optimizations apply to this piece of code and that one is better than the other. The phase ordering problem arises when, for some reason, the compiler applies the “less useful” optimization first: If we replace i * 5 by i << 2 + i, the code becomes obfuscated, and the compiler may no longer be able to detect that this is equivalent to a simple multiplication. It may thus have prevented itself from applying the more powerful strength reduction transformation.
This is where this paper’s equality saturation optimization comes in. The basic idea is simple: when finding opportunities for optimization, do not apply them destructively but simply record the information that some other variant of the program is equivalent to the existing one. So in the example, the optimizer would never replace i * 5 by i << 2 + i but only add these expressions to the same equivalence class.
The optimizer works on a representation of programs as program expression graphs (PEGs). This is a kind of cyclic data dependence graph with special θ nodes to represent loops. Here is a PEG for the looping program above:
The θ node represents the sequence of values that an expression takes in a loop. On the first iteration, it has the value of the first argument; on subsequent iterations, the second (recursive) argument is evaluated to yield the next value. In the example graph, the θ represents the value of i on subsequent iterations: 0, 1, 2, …; the multiplication by 5 yields the values for i * 5: 0, 5, 10, … .
This graph can be extended to an E-PEG (PEG with equalities) by adding more nodes and using dashed equivalence edges to connect nodes that represent the same values. Here is the extended graph that we get by optimizing the multiplication by 5:
(In this graphical representation the order of the operands of the shift operation is mixed up. The price I pay for simplifying the paper’s example is having to re-draw their graphs, and for a blog post I believe this is about good enough.)
The dashed edge connecting the two root nodes expresses the fact that i * 5 and i << 2 + i represent the same value.
The key point here is that applying this transformation only added information to the graph but did not destroy anything. If some optimization applied before, it still applies. Thus in this setting we can still perform strength reduction on the original loop because the other optimization did not destroy it.
In equality saturation, optimizations are expressed by equality axioms. Simple strength reduction can be expressed by the combination of the axioms (a + b) * m = a * m + b * m (distributivity of multiplication over addition) and θ(a, b) * m = θ(a * m, b * m) (distributivity of multiplication over θ). Whenever a node in the PEG matches one side of an equation, we may add a node representing the other side and connect the two by an equality edge. This explains the name of the approach, equality saturation: Starting with a graph, equalities are applied until the graph is saturated in that nothing more can (or should) be added. In practice, the growth of the graph must be limited to avoid uncontrollable blow-up.
Applying the two axioms above yields the following graph:
After the saturation step, the graph represents not one but many equivalent programs. The actual “optimization” consists of picking the best of these many variants. This amounts to choosing a minimal-cost subset of the nodes that together make up one of the variants. The authors encode this step as a 0-1 integer linear program and use an external solver to find a solution of minimal cost. The cost model assigns weights to nodes by operation type such that, for example, multiplications are more expensive than additions. In the running example, the solver would pick the rightmost subgraph which corresponds directly to the pseudocode after strength reduction given above.
As the authors observe, this optimization approach can also be used to perform translation validation, i.e., to ensure that the transformations performed by another optimizer are indeed valid. To perform validation, observe the original program that goes into an optimizer as well as its output, an allegedly equivalent program. Build the saturated E-PEG for the original program, yielding a representation of a large set of equivalent variants. Now, if the optimizer’s output is contained, in the E-PEG, the translation is indeed valid. Otherwise, the optimizer may have a bug; alternatively, however, the equality saturation may have been incomplete. This can be the case either because the saturation process was stopped too early, or because the set of underlying axioms is incomplete.
The authors discuss an evaluation of equality saturation for interprocedural optimization of Java bytecode. They compare their tool, Peggy, with the Soot optimization framework. They found that both optimizers give comparable results in general. However, Peggy allows users to easily specify domain-specific optimizations by adding axioms to its database; on some small examples, this allows significant speedups. Running in translation validation mode, Peggy was able to validate 98% of Soot’s interprocedural optimizations on a large benchmark set. Examining the remaining cases, they uncovered a bug in Soot that led to incorrectly optimized code.
Overall, equality saturation is a fascinating idea for structuring compilers. Unfortunately, the costs seem to prohibit its use in real-world compilers in mainstream settings.
(article written by GB)
|
2022-01-23 17:43:46
|
{"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": 0.7514848113059998, "perplexity": 909.6352744752005}, "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-2022-05/segments/1642320304309.5/warc/CC-MAIN-20220123172206-20220123202206-00703.warc.gz"}
|
https://digitalcommons.latech.edu/dissertations/809/
|
# Temperature Regime of Deep Wells During Cementing (Prediction of Static, Circulating, and Shutin Temperatures: Evaluation of the Thermal Effect of Cement Hydration)
#### Abstract
One of the important factors in a successful well completion is the proper performance of the cement slurry. The circulating temperature and test procedure used in the design, and the laboratory testing of the cement are critical to its successful performance in the well. Because of the importance of bottom-hole circulating temperatures in a well completion, a new formula has been developed to predict this temperature with greater accuracy.
The current API correlation which is used to predict the bottom-hole circulating temperature permits prediction only in wells with geothermal gradients up to $2\sp\circ{\rm F}/100$ ft; however, the empirical formula allows the prediction in wells with larger temperature gradients. In addition, the results indicate that for deep wells with high temperature gradients, the API correlation predicts bottom-hole temperatures which are too high.
The principal factor controlling the chemical reaction and the resulting performance of a cementing composition is the temperature to which it is exposed. The shut-in temperature affects how long the slurry will pump and how well it develops the strength necessary to support pipe.
A new method has been developed to calculate the bottom-hole shut-in temperature. Plots of dimensionless bottom-hole shut-in temperature versus shut-in time for various bit sizes have been constructed. Values obtained from these plots can be used in a formula to calculate the shut-in temperature.
The relation between the heat of hydration and the temperature increase inside the casing was another phase of this study. A computer program was developed to calculate the temperature increase due to cement hydration. The impact of bentonite, formation temperature, volume of the cement, and thermal properties of the formation on heat of hydration and ultimately the temperature increase inside casing is shown.
|
2019-08-24 07:45:19
|
{"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": 0.6833873987197876, "perplexity": 1753.0846905865417}, "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-2019-35/segments/1566027319915.98/warc/CC-MAIN-20190824063359-20190824085359-00459.warc.gz"}
|
http://twoqubits.wikidot.com/ex12-1
|
Ex12 1
$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 12.1}$
Show that a single-qubit gate followed by a single-qubit error is equivalent to a (possibly different) single-qubit error followed by the same gate.
|
2020-02-19 23:23:22
|
{"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": 0.726115882396698, "perplexity": 1953.8304736971397}, "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-2020-10/segments/1581875144429.5/warc/CC-MAIN-20200219214816-20200220004816-00340.warc.gz"}
|
https://zbmath.org/?q=ci%3A2031101
|
## Generalized Cayley graphs of semigroups. I.(English)Zbl 1252.20052
The study of Cayley graphs of semigroups and its valuable applications, is a very important research area which is closely related to finite state automata (see the survey by A. Kelarev, J. Ryan and J. Yearwood, [Discrete Math. 309, No. 17, 5360-5369 (2009; Zbl 1206.05050)] and Section 2.4 of the book by A. Kelarev, [Graph algebras and automata. New York: Marcel Dekker (2003; Zbl 1070.68097)]).
In this paper, the author first presents the following definition of generalized Cayley graphs of semigroups to unify the Cayley graphs of semigroups defined by right translations and left translations (note that the classical definition of Cayley graphs has no problem and the difference between the results about Cayley graphs defined by left and right actions, is because of the close relation between semigroups and these structures (similar to left and right $$S$$-acts)).
Let $$T$$ be an ideal extension of a semigroup $$S$$ and $$\rho\subseteq T^1\times T^1$$. The ‘generalized Cayley graph’ $$\text{Cay}(S,\rho)$$ of $$S$$ relative to $$\rho$$ is defined as the graph with vertex set $$S$$ and edge set $$E(\text{Cay}(S,\rho))$$ consisting of those ordered pairs $$(a,b)$$, where $$xay=b$$ for some $$(x,y)\in\rho$$. The generalized Cayley graph $$\text{Cay}(S,\omega)$$, where $$\omega=S^1\times S^1$$ is called ‘universal Cayley graph of $$S$$’. Clearly, for $$\rho_1=T^1\times \{1\}$$ and $$\rho_2=\{1\}\times T^1$$, $$\text{Cay}(S,\rho_1)$$ and $$\text{Cay}(S,\rho_2)$$ are the classical definition of Cayley graph of semigroup defined by left and right transformations, respectively. After presenting some fundamental properties and general results about generalized Cayley graphs of semigroups, the author focuses on universal Cayley graphs and after introducing some notions, she describes the universal Cayley graph of a $$\mathcal J$$-partial order of complete graphs with loops.
### MSC:
20M05 Free semigroups, generators and relations, word problems 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C20 Directed graphs (digraphs), tournaments 20M20 Semigroups of transformations, relations, partitions, etc.
### Citations:
Zbl 1206.05050; Zbl 1070.68097
Full Text:
### References:
[1] Arworn, Sr., Knauer, U., Chiangmai, N.N.: Characterization of digraphs of right (left) zero unions of groups. Thai J. Math. 1(1), 131–140 (2003) · Zbl 1055.05075 [2] Dénes, J.: Connections between transformation semigroups and graphs. In: Theory of Graphs. Gordon & Breach, New York (1967) [3] Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon, Oxford (1995) · Zbl 0835.20077 [4] Kelarev, A.V.: Graph Algebras and Automata. Dekker, New York (2003) · Zbl 1070.68097 [5] Kelarev, A.V.: On Cayley graphs of inverse semigroups. Semigroup Forum 72, 411–418 (2006) · Zbl 1107.20057 [6] Kelarev, A.V.: On undirected Cayley graphs. Australas. J. Combin. 25, 73–78 (2002) · Zbl 0993.05085 [7] Kelarev, A.V., Praeger, C.E.: On transitive Cayley graphs of groups and semigroups. Eur. J. Comb. 24(1), 59–72 (2003) · Zbl 1011.05027 [8] Kelarev, A.V., Quinn, S.J.: A combinatorial property and Cayley graphs of semigroups. Semigroup Forum 66, 89–96 (2003) · Zbl 1016.20047 [9] Kelarev, A.V., Quinn, S.J.: A combinatorial property and power graphs of semigroups. Comment. Math. Univ. Carol. 45(1), 1–7 (2004) · Zbl 1099.05042 [10] Kelarev, A.V., Quinn, S.J.: Directed graphs and combinatorial properties of semigroups. J. Algebra 251(1), 16–26 (2002) · Zbl 1005.20043 [11] Kelarev, A., Ryan, J., Yearwood, J.: Cayley graphs as classifiers for data mining: The influence of asymmetries. Discrete Math. 309, 5360–5369 (2009) · Zbl 1206.05050 [12] Panma, S., Knauer, U., Arworn, S.: On transitive Cayley graphs of right (left) groups and of Clifford semigroups. Thai J. Math. 2(1), 183–195 (2004) · Zbl 1063.05068 [13] Panma, S., Knauer, U., Arworn, S.: On transitive Cayley graphs of strong semilattices of right (left) groups. Discrete Math. 309(17), 5393–5403 (2009) · Zbl 1198.05095 [14] Panma, S., Chiangmai, N.N., Knauer, U., Arworn, S.: Characterizations of Clifford semigroup digraphs. Discrete Math. 306(12), 1247–1252 (2006) · Zbl 1096.05045 [15] Petrich, M., Reilly, N.: Completely Regular Semigroups. Wiley, New York (1999) · Zbl 0967.20034 [16] White, A.T.: Graphs, Groups and Surfaces. Amsterdam, Elsevier (2001) · Zbl 1054.05001 [17] Wilson, R.J.: Introduction to Graph Theory, 3rd edn. Longman, New York (1982) · Zbl 0489.03017 [18] Yang, D., Gao, X.: D-saturated property of the Cayley graphs of semigroups. Semigroup Forum 80, 174–180 (2010) · Zbl 1198.20046 [19] Zelinka, B.: Graphs of semigroups. Čas. Pěst. Mat. 106, 407–408 (1981) · Zbl 0479.05033 [20] Zhu, Y.: On (n,m)-semigroups. Semigroup Forum (2011). doi: 10.1007/s00233-011-9360-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
|
2022-05-25 06:02:19
|
{"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": 0.5163634419441223, "perplexity": 3445.963586422939}, "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/1652662580803.75/warc/CC-MAIN-20220525054507-20220525084507-00123.warc.gz"}
|
https://zbmath.org/?q=ut%3Adecreasing+solution
|
Found 75 Documents (Results 1–75)
100
MathJax
Full Text:
A Bowley solution with limited ceded risk for a monopolistic reinsurer. (English)Zbl 1435.91143
MSC: 91G05 91A65
Full Text:
Necessary and sufficient condition for the existence of positive solutions of a coupled system for reaction-diffusion equations. (English)Zbl 1437.34033
Reviewer: Yang Yang (Wuxi)
Full Text:
Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients. (English)Zbl 07536132
MSC: 39A22 39A12 26A12
Full Text:
Phase-shift controlling of three solitons in dispersion-decreasing fibers. (English)Zbl 1430.78006
MSC: 78A60 35Q55 74J35
Full Text:
The generalization of the Feigenbaum-Kadanoff-Shenker equation with two parameters. (Chinese. English summary)Zbl 1438.39040
MSC: 39B22 39B12 37E10
Full Text:
Factorization of the equations of marine electrodynamics. (Russian. English summary)Zbl 1393.78006
MSC: 78A25 76W05 76D05
Full Text:
Full Text:
The search of the quotient system for magnetohydrodynamics equations. (Russian. English summary)Zbl 1394.76152
MSC: 76W05 35Q35
Full Text:
New results on existence and exponential stability of the unique positive almost periodic solution for hematopoiesis model. (English)Zbl 1443.92039
MSC: 92-10 92D25
Full Text:
Full Text:
Existence and global attractivity of the unique positive periodic solution for discrete hematopoiesis model. (English)Zbl 1362.39022
MSC: 39A23 39A30
Full Text:
Exponential estimates for solutions of half-linear differential equations. (English)Zbl 1374.34102
MSC: 34C11 34D05
Full Text:
The discrete Ramsey model with decreasing population growth rate. (English)Zbl 1315.91048
MSC: 91B62 91B55
Full Text:
Full Text:
Existence and exponential stability of the unique positive almost periodic solution for the Lasota-Wazewska difference model. (English)Zbl 1417.39051
MSC: 39A24 39A30
Full Text:
On decreasing solutions of second order nearly linear differential equations. (English)Zbl 1308.34069
MSC: 34D05 26A12
Full Text:
Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations. (English)Zbl 1302.34112
MSC: 34K25 26A12
Full Text:
MSC: 34C11
Full Text:
An example of non-decreasing solution for the KdV equation posed on a bounded interval. (Un exemple de solution non décroissante de l’équation de KdV posée sur un intervalle borné.) (English. French summary)Zbl 1301.35129
MSC: 35Q53 35B10
Full Text:
Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source. (English. Russian original)Zbl 1293.35279
Proc. Steklov Inst. Math. 281, 161-178 (2013); translation from Tr. Mat. Inst. Steklova 281, 170-187 (2013); corrigendum Proc. Steklov Inst. Math. 288, 265 (2015); translation from Tr. Mat. Inst. Steklova 288, 287 (2015).
Full Text:
MSC: 39B12
Continuously decreasing solutions for polynomial-like iterative equations. (English)Zbl 1264.39020
MSC: 39B12 26A18
Full Text:
Full Text:
On integral equations of stationary distributions for biological systems. (English. Russian original)Zbl 1281.45003
J. Math. Sci., New York 171, No. 1, 34-45 (2010); translation from Sovrem. Mat., Fundam. Napravl. 36, 50-60 (2010).
MSC: 45E10
Full Text:
Full Text:
Existence and uniqueness of a monotone decreasing positive solution of a third-order two-point boundary value problem. (Chinese. English summary)Zbl 1199.34115
MSC: 34B18 47N20
Nonhomogeneous boundary value problems for equations of viscous heat-conducting gas in time-decreasing non-rectangular domains. (English)Zbl 1162.76378
MSC: 76N10 35Q30
Full Text:
Multiple positive solutions for singular three-point boundary value problem with superlinear term, sublinear term, increasing term and decreasing term. (English)Zbl 1144.34011
Reviewer: Jiaqi Mo (Wuhu)
A semi-implicit conservation element-solution element method for chemical species transport simulation to tapered ducts of internal combustion engine. (English)Zbl 1104.80015
MSC: 80A25 65D30 76M25
Full Text:
On the existence of positive solutions for 2$$n$$-order singular boundary value problems. (English)Zbl 1100.34020
Reviewer: Jiaqi Mo (Wuhu)
MSC: 34B18 34B15 34B16
Full Text:
The rapidly decreasing solution of the Cauchy problem for the Toda lattice. (English)Zbl 1178.37110
Theor. Math. Phys. 142, No. 1, 1-7 (2005); translation from Teor. Mat. Fiz. 142, No. 1, 5-12 (2005).
MSC: 37K60 37K10
Full Text:
A rapidly decreasing solution of an initial-boundary value problem for the Toda chain. (Ukrainian, English)Zbl 1095.37038
Ukr. Mat. Zh. 57, No. 8, 1144-1152 (2005); translation in Ukr. Math. J. 57, No. 8, 1350-1359 (2005).
Full Text:
Strongly decaying solutions of nonlinear forced discrete systems. (English)Zbl 1065.39016
Aulbach, Bernd (ed.) et al., New progress in difference equations. Proceedings of the 6th international conference on difference equations, Augsburg, Germany July 30–August 3, 2001. Boca Raton, FL: CRC Press (ISBN 0-415-31675-8/hbk). 493-500 (2004).
MSC: 39A11
Transformation operators for the perturbed Hill difference equation and one of their applications. (Russian, English)Zbl 1081.39021
Sib. Mat. Zh. 44, No. 4, 926-937 (2003); translation in Sib. Math. J. 44, No. 4, 729-738 (2003).
MSC: 39A70 37K60 47B36
Full Text:
On the recursive sequence $$x_{n+1}=x_{n-1}/g(x_n)$$. (English)Zbl 1019.39010
MSC: 39A11 39B05
Full Text:
Constructive approximation for a class of perturbed Hammerstein integral equations. (English)Zbl 0956.45006
MSC: 45G10 45L05
Full Text:
Singular differential equations without singular solutions. (English)Zbl 0971.34032
MSC: 34D05 34A34
The singular boundary value problems for a class of nonlinear elliptic equations. (Chinese. English summary)Zbl 0939.35059
MSC: 35J25 35B05
Explicit representation of the solution to some boundary value problem. (English)Zbl 0921.30030
Reviewer: S.Mazouzi (Annaba)
MSC: 30E25 31A05
Full Text:
Full Text:
Semilinear ODE. II: Positive solutions via super-sub-solutions method. (English)Zbl 0959.34502
Proceedings of the Prague mathematical conference 1996, PMC ’96, Prague, Czech Republic, July 8-12, 1996. In honor of the 70th birthdays of Ivo Babuška, Miroslav Fiedler, Jaroslav Kurzweil, and Vlastimil Pták. Prague: Icaris Ltd. 321-323 (1996).
MSC: 34C11
Semilinear ODE. I: Postive solutions via integral operators. (English)Zbl 0959.34501
Proceedings of the Prague mathematical conference 1996, PMC ’96, Prague, Czech Republic, July 8-12, 1996. In honor of the 70th birthdays of Ivo Babuška, Miroslav Fiedler, Jaroslav Kurzweil, and Vlastimil Pták. Prague: Icaris Ltd. 317-319 (1996).
MSC: 34C11
Oscillatory and asymptotically monotone solutions of second-order quasilinear differential equations. (English)Zbl 0872.34020
MSC: 34C10 34C11
Full Text:
Full Text:
On a nonlinear second order differential equation with oscillating coefficient. (English)Zbl 0771.34027
MSC: 34C11 34B15 34C29
Full Text:
Stable upwind schemes for hyperbolic conservation laws with source terms. (English)Zbl 0754.65077
MSC: 65M06 65M12 35L65
Full Text:
On a comparison theorem via symmetrisation. (English)Zbl 0762.35005
MSC: 35B05 35J25
Full Text:
Full Text:
Existence theorems for nonlinear dual programming. (Chinese)Zbl 0706.90072
Reviewer: Wang Shouyang
On the mechanical mathematical modelling of diffusion with memory, using FE. (Bulgarian. English, Russian summaries)Zbl 0696.76103
MSC: 76R50 76M99
High-order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws. (English)Zbl 0644.65058
Reviewer: F.v.Finckenstein
MSC: 65M06 65M12 35L65
Full Text:
Properties of generalized solutions of Dirichlet’s problem for high-order quasilinear divergence elliptic equations in the neighborhood of the boundary. (English. Russian original)Zbl 0677.35034
Differ. Equations 23, No. 2, 227-236 (1987); translation from Differ. Uravn. 23, No. 2, 308-320 (1987).
Reviewer: J.Rojtberg
MSC: 39B99
Full Text:
Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. (English)Zbl 0631.35041
Reviewer: D.T.Haimo
MSC: 35K20 35B05 35B40 35B50 35A05
Full Text:
Nonuniqueness of solutions for semilinear elliptic equations at resonance. (English)Zbl 0615.35037
Reviewer: S.M.Lenhart
MSC: 35J65 35B30 35A05
On the equation of turbulent filtration in one-dimensional porous media. (English)Zbl 0613.76102
Reviewer: D.Polisevski
Full Text:
Asymptotics of the solutions of some higher order elliptic equations in conical domains. (English. Russian original)Zbl 0609.35033
Math. USSR, Sb. 53, 89-117 (1986); translation from Mat. Sb., Nov. Ser. 125(167), No. 1, 88-116 (1984).
Reviewer: I.Diaz
Full Text:
A pointwise comparison for solutions of linear elliptic equations. (English)Zbl 0624.35025
MSC: 35J25 35B05
The classification of the conjugacy classes of the full group of homeomorphisms of an open interval and the general solution of certain functional equations. (English)Zbl 0575.39003
Reviewer: N.Ghircoiaşiu
MSC: 39B12 26A18 58D05
Full Text:
Gestion optimale d’un système de stockage à deux niveaux avec coûts concaves. (French)Zbl 0573.90027
MSC: 90B05 90C39
Comparison theorems for a class of first order Hamilton-Jacobi equations. (English)Zbl 0554.35007
MSC: 35B05 35F20 35A30
Full Text:
On the Cauchy problem for the wave equation with a singularity in the time variable. (English)Zbl 0555.35073
Reviewer: G.Hecquet
MSC: 35L15 35A05 35A20
A stability problem with regard to softening. (Russian. English summary)Zbl 0539.73138
Reviewer: J.Murzewski
Full Text:
Unbounded solutions of a nonlinear differential equation. (English)Zbl 0507.34031
MSC: 34C11 34A12 34A34
“A posteriori” evaluation of bin packing approximation algorithms. (English)Zbl 0434.68052
MSC: 68R99 68Q25
Full Text:
On the inequality $$\sum^n_{i=1}p_1 {f_i(p_i)\over f_i(q_i)}\leq 1$$. (English)Zbl 0446.39006
MSC: 39B72 26D15 94A17
Full Text:
On the oscillation of a fourth order differential equation and its adjoint. (English)Zbl 0504.34017
MSC: 34C10 34C11
Full Text:
all top 5
all top 5
all top 5
all top 3
|
2022-06-28 06:19:34
|
{"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": 0.3140605390071869, "perplexity": 4900.87706000563}, "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-2022-27/segments/1656103355949.26/warc/CC-MAIN-20220628050721-20220628080721-00295.warc.gz"}
|
http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.108.208001
|
# Synopsis: U-shaped Grains Get Clingy
Piles of staples stand up to shaking better if the staple prongs have an intermediate length.
The shape of grains in granular materials can have a large effect on their collective physics. A new study explores u-shaped grains and how they bind together through entanglements. In experiments described in Physical Review Letters, the researchers found that free-standing piles of metal staples held together longest when the staple “arms” had a particular length. To explain this optimum shape, the authors develop a model that may apply to other collections of irregular shaped objects.
Physicists have long been interested in how sand pours down a slope or how nuts pack inside a box. However, not much work has been done with “bent” or concave grains that can intertwine. Examples include polymer networks and anisotropic colloids, as well as the rafts that certain ant species form by interlocking limbs and mandibles.
Nick Gravish of the Georgia Institute of Technology in Atlanta and his colleagues decided to investigate a simple concave grain: the common staple. In their experiments, the researchers varied the length of the staple arms, while keeping the width constant. The team formed piles of uniform staples and then shook them up and down until the piles eventually collapsed. Staples with a length-to-width ratio of about $0.4$ remained upright the longest. The scientists explained their observations using simulations and theory. It turns out that lengthening the arms of a staple increases the number of entanglements with neighbors, but conversely decreases the packing density. Staples that balance these two effects create the most stable piles. – Michael Schirber
### Announcements
More Announcements »
## Subject Areas
Materials Science
## Previous Synopsis
Quantum Information
## Next Synopsis
Atomic and Molecular Physics
## Related Articles
Geophysics
### Synopsis: Acoustic Trigger For Earthquakes
Numerical simulations support the idea that acoustic waves can trigger earthquakes by reducing friction between the rocks within a fault. Read More »
Magnetism
### Synopsis: Multiferroic Surprise
Electric and magnetic polarization are spontaneously produced in an unlikely material—one with a highly symmetric crystal structure. Read More »
Soft Matter
### Synopsis: Wedged Particles Make Crystals
Rod-shaped particles in a liquid arrange into a variety of structures when subjected to confining walls, an effect that might be used to design optical materials. Read More »
|
2015-10-05 01:27:32
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.22104798257350922, "perplexity": 3874.041672626927}, "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-2015-40/segments/1443736676547.12/warc/CC-MAIN-20151001215756-00005-ip-10-137-6-227.ec2.internal.warc.gz"}
|
https://lectures.quantecon.org/py/wald_friedman.html
|
We are working to support a site-wide PDF but it is not yet available. You can download PDFs for individual lectures through the download badge on each lecture page.
Code should execute sequentially if run in a Jupyter notebook
# A Problem that Stumped Milton Friedman¶
(and that Abraham Wald solved by inventing sequential analysis)
## Contents¶
Co-authors: Chase Coleman
## Overview¶
This lecture describes a statistical decision problem encountered by Milton Friedman and W. Allen Wallis during World War II when they were analysts at the U.S. Government’s Statistical Research Group at Columbia University
This problem led Abraham Wald [Wal47] to formulate sequential analysis, an approach to statistical decision problems intimately related to dynamic programming
In this lecture, we apply dynamic programming algorithms to Friedman and Wallis and Wald’s problem
Key ideas in play will be:
• Bayes’ Law
• Dynamic programming
• Type I and type II statistical errors
• a type I error occurs when you reject a null hypothesis that is true
• a type II error is when you accept a null hypothesis that is false
• Abraham Wald’s sequential probability ratio test
• The power of a statistical test
• The critical region of a statistical test
• A uniformly most powerful test
We’ll begin with some imports
In [1]:
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import beta
import quantecon as qe
from numba import njit, prange, vectorize
from interpolation import interp
from math import gamma
## Origin of the problem¶
On pages 137-139 of his 1998 book Two Lucky People with Rose Friedman [FF98], Milton Friedman described a problem presented to him and Allen Wallis during World War II, when they worked at the US Government’s Statistical Research Group at Columbia University
Let’s listen to Milton Friedman tell us what happened
In order to understand the story, it is necessary to have an idea of a simple statistical problem, and of the standard procedure for dealing with it. The actual problem out of which sequential analysis grew will serve. The Navy has two alternative designs (say A and B) for a projectile. It wants to determine which is superior. To do so it undertakes a series of paired firings. On each round it assigns the value 1 or 0 to A accordingly as its performance is superior or inferio to that of B and conversely 0 or 1 to B. The Navy asks the statistician how to conduct the test and how to analyze the results.
The standard statistical answer was to specify a number of firings (say 1,000) and a pair of percentages (e.g., 53% and 47%) and tell the client that if A receives a 1 in more than 53% of the firings, it can be regarded as superior; if it receives a 1 in fewer than 47%, B can be regarded as superior; if the percentage is between 47% and 53%, neither can be so regarded.
When Allen Wallis was discussing such a problem with (Navy) Captain Garret L. Schyler, the captain objected that such a test, to quote from Allen’s account, may prove wasteful. If a wise and seasoned ordnance officer like Schyler were on the premises, he would see after the first few thousand or even few hundred [rounds] that the experiment need not be completed either because the new method is obviously inferior or because it is obviously superior beyond what was hoped for $\ldots$
Friedman and Wallis struggled with the problem but, after realizing that they were not able to solve it, described the problem to Abraham Wald
That started Wald on the path that led him to Sequential Analysis [Wal47]
We’ll formulate the problem using dynamic programming
## A dynamic programming approach¶
The following presentation of the problem closely follows Dmitri Berskekas’s treatment in Dynamic Programming and Stochastic Control [Ber75]
A decision maker observes iid draws of a random variable $z$
He (or she) wants to know which of two probability distributions $f_0$ or $f_1$ governs $z$
After a number of draws, also to be determined, he makes a decision as to which of the distributions is generating the draws he observes
He starts with prior
$$\pi_{-1} = \mathbb P \{ f = f_0 \mid \textrm{ no observations} \} \in (0, 1)$$
After observing $k+1$ observations $z_k, z_{k-1}, \ldots, z_0$, he updates this value to
$$\pi_k = \mathbb P \{ f = f_0 \mid z_k, z_{k-1}, \ldots, z_0 \}$$
which is calculated recursively by applying Bayes’ law:
$$\pi_{k+1} = \frac{ \pi_k f_0(z_{k+1})}{ \pi_k f_0(z_{k+1}) + (1-\pi_k) f_1 (z_{k+1}) }, \quad k = -1, 0, 1, \ldots$$
After observing $z_k, z_{k-1}, \ldots, z_0$, the decision maker believes that $z_{k+1}$ has probability distribution
$$f_{{\pi}_k} (v) = \pi_k f_0(v) + (1-\pi_k) f_1 (v)$$
This is a mixture of distributions $f_0$ and $f_1$, with the weight on $f_0$ being the posterior probability that $f = f_0$ [1]
To help illustrate this kind of distribution, let’s inspect some mixtures of beta distributions
The density of a beta probability distribution with parameters $a$ and $b$ is
$$f(z; a, b) = \frac{\Gamma(a+b) z^{a-1} (1-z)^{b-1}}{\Gamma(a) \Gamma(b)} \quad \text{where} \quad \Gamma(t) := \int_{0}^{\infty} x^{t-1} e^{-x} dx$$
The next figure shows two beta distributions in the top panel
The bottom panel presents mixtures of these distributions, with various mixing probabilities $\pi_k$
In [2]:
def beta_function_factory(a, b):
@vectorize
def p(x):
r = gamma(a + b) / (gamma(a) * gamma(b))
return r * x**(a-1) * (1 - x)**(b-1)
@njit
def p_rvs():
return np.random.beta(a, b)
return p, p_rvs
f0, _ = beta_function_factory(1, 1)
f1, _ = beta_function_factory(9, 9)
grid = np.linspace(0, 1, 50)
fig, axes = plt.subplots(2, figsize=(10, 8))
axes[0].set_title("Original Distributions")
axes[0].plot(grid, f0(grid), lw=2, label="$f_0$")
axes[0].plot(grid, f1(grid), lw=2, label="$f_1$")
axes[1].set_title("Mixtures")
for π in 0.25, 0.5, 0.75:
y = π * f0(grid) + (1 - π) * f1(grid)
axes[1].plot(y, lw=2, label=f"$\pi_k$ = {π}")
for ax in axes:
ax.legend()
ax.set(xlabel="$z$ values", ylabel="probability of $z_k$")
plt.tight_layout()
plt.show()
### Losses and costs¶
After observing $z_k, z_{k-1}, \ldots, z_0$, the decision maker chooses among three distinct actions:
• He decides that $f = f_0$ and draws no more $z$‘s
• He decides that $f = f_1$ and draws no more $z$‘s
• He postpones deciding now and instead chooses to draw a $z_{k+1}$
Associated with these three actions, the decision maker can suffer three kinds of losses:
• A loss $L_0$ if he decides $f = f_0$ when actually $f=f_1$
• A loss $L_1$ if he decides $f = f_1$ when actually $f=f_0$
• A cost $c$ if he postpones deciding and chooses instead to draw another $z$
### Digression on type I and type II errors¶
If we regard $f=f_0$ as a null hypothesis and $f=f_1$ as an alternative hypothesis, then $L_1$ and $L_0$ are losses associated with two types of statistical errors
• a type I error is an incorrect rejection of a true null hypothesis (a “false positive”)
• a type II error is a failure to reject a false null hypothesis (a “false negative”)
So when we treat $f=f_0$ as the null hypothesis
• We can think of $L_1$ as the loss associated with a type I error
• We can think of $L_0$ as the loss associated with a type II error
### Intuition¶
Let’s try to guess what an optimal decision rule might look like before we go further
Suppose at some given point in time that $\pi$ is close to 1
Then our prior beliefs and the evidence so far point strongly to $f = f_0$
If, on the other hand, $\pi$ is close to 0, then $f = f_1$ is strongly favored
Finally, if $\pi$ is in the middle of the interval $[0, 1]$, then we have little information in either direction
This reasoning suggests a decision rule such as the one shown in the figure
As we’ll see, this is indeed the correct form of the decision rule
The key problem is to determine the threshold values $\alpha, \beta$, which will depend on the parameters listed above
You might like to pause at this point and try to predict the impact of a parameter such as $c$ or $L_0$ on $\alpha$ or $\beta$
### A Bellman equation¶
Let $J(\pi)$ be the total loss for a decision maker with current belief $\pi$ who chooses optimally
With some thought, you will agree that $J$ should satisfy the Bellman equation
$$J(\pi) = \min \left\{ (1-\pi) L_0, \; \pi L_1, \; c + \mathbb E [ J (\pi') ] \right\} \tag{1}$$
where $\pi'$ is the random variable defined by
$$\pi' = \kappa(z', \pi) = \frac{ \pi f_0(z')}{ \pi f_0(z') + (1-\pi) f_1 (z') }$$
when $\pi$ is fixed and $z'$ is drawn from the current best guess, which is the distribution $f$ defined by
$$f_{\pi}(v) = \pi f_0(v) + (1-\pi) f_1 (v)$$
In the Bellman equation, minimization is over three actions:
1. Accept the hypothesis that $f = f_0$
2. Accept the hypothesis that $f = f_1$
3. Postpone deciding and draw again
We can represent the Bellman equation as
$$J(\pi) = \min \left\{ (1-\pi) L_0, \; \pi L_1, \; h(\pi) \right\} \tag{2}$$
where $\pi \in [0,1]$ and
• $(1-\pi) L_0$ is the expected loss associated with accepting $f_0$ (i.e., the cost of making a type II error)
• $\pi L_1$ is the expected loss associated with accepting $f_1$ (i.e., the cost of making a type I error)
• $h(\pi) := c + \mathbb E [J(\pi')]$ the continuation value; i.e., the expected cost associated with drawing one more $z$
The optimal decision rule is characterized by two numbers $\alpha, \beta \in (0,1) \times (0,1)$ that satisfy
$$(1- \pi) L_0 < \min \{ \pi L_1, c + \mathbb E [J(\pi')] \} \textrm { if } \pi \geq \alpha$$
and
$$\pi L_1 < \min \{ (1-\pi) L_0, c + \mathbb E [J(\pi')] \} \textrm { if } \pi \leq \beta$$
The optimal decision rule is then
$$\textrm { accept } f=f_0 \textrm{ if } \pi \geq \alpha \\ \textrm { accept } f=f_1 \textrm{ if } \pi \leq \beta \\ \textrm { draw another } z \textrm{ if } \beta \leq \pi \leq \alpha$$
Our aim is to compute the value function $J$, and from it the associated cutoffs $\alpha$ and $\beta$
To make our computations simpler, using (2), we can write the continuation value $h(\pi)$ as
\begin{align} h(\pi) &= c + \mathbb E [J(\pi')] \\ &= c + \mathbb E_{\pi'} \min \{ (1 - \pi') L_0, \pi' L_1, h(\pi') \} \\ &= c + \min \int \{ (1 - \kappa(z', \pi) ) L_0, \kappa(z', \pi) L_1, h(\kappa(z', \pi) ) \} f_0 (z') dz' \end{align} \tag{3}
The equality
$$h(\pi) = c + \min \int \{ (1 - \kappa(z', \pi) ) L_0, \kappa(z', \pi) L_1, h(\kappa(z', \pi) ) \} f_0 (z') dz' \tag{4}$$
can be understood as a functional equation, where $h$ is the unknown
Using the functional equation, (4), for the continuation value, we can back out optimal choices using the RHS of (2)
This functional equation can be solved by taking an initial guess and iterating to find the fixed point
In other words, we iterate with an operator $Q$, where
$$Q h(\pi) = c + \min \int \{ (1 - \kappa(z', \pi) ) L_0, \kappa(z', \pi) L_1, h(\kappa(z', \pi) ) \} f_0 (z') dz'$$
## Implementation¶
First we will construct a class to store the parameters of the model
In [3]:
class WaldFriedman:
def __init__(self,
c=1.25, # Cost of another draw
a0=1,
b0=1,
a1=3,
b1=1.2,
L0=25, # Cost of selecting f0 when f1 is true
L1=25, # Cost of selecting f1 when f0 is true
π_grid_size=200,
mc_size=1000):
self.c, self.π_grid_size = c, π_grid_size
self.L0, self.L1 = L0, L1
self.π_grid = np.linspace(0, 1, π_grid_size)
self.mc_size = mc_size
# Set up distributions
self.f0, self.f0_rvs = beta_function_factory(a0, b0)
self.f1, self.f1_rvs = beta_function_factory(a1, b1)
self.z0 = np.random.beta(a0, b0, mc_size)
self.z1 = np.random.beta(a1, b1, mc_size)
As in the optimal growth lecture, to approximate a continuous value function
• We iterate at a finite grid of possible values of $\pi$
• When we evaluate $\mathbb E[J(\pi')]$ between grid points, we use linear interpolation
The function operator_factory returns the operator Q
In [4]:
def operator_factory(wf, parallel_flag=True):
"""
Returns a jitted version of the Q operator.
* wf is an instance of the WaldFriedman class
"""
c, π_grid = wf.c, wf.π_grid
L0, L1 = wf.L0, wf.L1
f0, f1 = wf.f0, wf.f1
z0, z1 = wf.z0, wf.z1
mc_size = wf.mc_size
@njit
def κ(z, π):
"""
Updates π using Bayes' rule and the current observation z.
"""
π_f0, π_f1 = π * f0(z), (1 - π) * f1(z)
π_new = π_f0 / (π_f0 + π_f1)
return π_new
@njit(parallel=True)
def Q(h):
h_new = np.empty_like(π_grid)
h_func = lambda p: interp(π_grid, h, p)
for i in prange(len(π_grid)):
π = π_grid[i]
# Find the expected value of J by integrating over z
integral_f0, integral_f1 = 0, 0
for m in range(mc_size):
π_0 = κ(z0[m], π) # Draw z from f0 and update π
integral_f0 += min((1 - π_0) * L0, π_0 * L1, h_func(π_0))
π_1 = κ(z1[m], π) # Draw z from f1 and update π
integral_f1 += min((1 - π_1) * L0, π_1 * L1, h_func(π_1))
integral = (π * integral_f0 + (1 - π) * integral_f1) / mc_size
h_new[i] = c + integral
return h_new
return Q
To solve the model, we will iterate using Q to find the fixed point
In [5]:
def solve_model(wf,
use_parallel=True,
tol=1e-4,
max_iter=1000,
verbose=True,
print_skip=25):
"""
Compute the continuation value function
* wf is an instance of WaldFriedman
"""
Q = operator_factory(wf, parallel_flag=use_parallel)
# Set up loop
h = np.zeros(len(wf.π_grid))
i = 0
error = tol + 1
while i < max_iter and error > tol:
h_new = Q(h)
error = np.max(np.abs(h - h_new))
i += 1
if verbose and i % print_skip == 0:
print(f"Error at iteration {i} is {error}.")
h = h_new
if i == max_iter:
print("Failed to converge!")
if verbose and i < max_iter:
print(f"\nConverged in {i} iterations.")
return h_new
## Analysis¶
Let’s inspect the model’s solutions
We will be using the default parametization with distributions like so
In [6]:
wf = WaldFriedman()
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(wf.f0(wf.π_grid), label="$f_0$")
ax.plot(wf.f1(wf.π_grid), label="$f_1$")
ax.set(ylabel="probability of $z_k$", xlabel="$k$", title="Distributions")
ax.legend()
plt.show()
### Value Function¶
To solve the model, we will call our solve_model function
In [7]:
h_star = solve_model(wf) # solve the model
Error at iteration 25 is 0.00010029006380740668.
Converged in 26 iterations.
We will also set up a function to compute the cutoffs $\alpha$ and $\beta$ and plot these on our value function plot
In [8]:
def find_cutoff_rule(wf, h):
"""
This function takes a continuation value function and returns the corresponding
cutoffs of where you transition between continue and choosing a
specific model
"""
π_grid = wf.π_grid
L0, L1 = wf.L0, wf.L1
# Evaluate cost at all points on grid for choosing a model
payoff_f0 = (1 - π_grid) * L0
payoff_f1 = π_grid * L1
# The cutoff points can be found by differencing these costs with
# the Bellman equation (J is always less than or equal to p_c_i)
β = π_grid[np.searchsorted(payoff_f1 - h, 1e-10) - 1]
α = π_grid[np.searchsorted(h - payoff_f0, -1e-10)]
return (β, α)
β, α = find_cutoff_rule(wf, h_star)
cost_L0 = (1 - wf.π_grid) * wf.L0
cost_L1 = wf.π_grid * wf.L1
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(wf.π_grid, h_star, label='continuation value')
ax.plot(wf.π_grid, cost_L1, label='choose f1')
ax.plot(wf.π_grid, cost_L0, label='choose f0')
ax.plot(wf.π_grid, np.amin(np.column_stack([h_star, cost_L0, cost_L1]), axis=1),
lw=15, alpha=0.1, color='b', label='minimum cost')
ax.annotate(r"$\beta$", xy=(β + 0.01, 0.5), fontsize=14)
ax.annotate(r"$\alpha$", xy=(α + 0.01, 0.5), fontsize=14)
plt.vlines(β, 0, β * wf.L0, linestyle="--")
plt.vlines(α, 0, (1 - α) * wf.L1, linestyle="--")
ax.set(xlim=(0, 1), ylim=(0, 0.5 * max(wf.L0, wf.L1)), ylabel="cost",
xlabel="$\pi$", title="Value function")
plt.show()
The value function equals $\pi L_1$ for $\pi \leq \beta$, and $(1-\pi )L_0$ for $\pi \geq \alpha$
The slopes of the two linear pieces of the value function are determined by $L_1$ and $- L_0$
The value function is smooth in the interior region, where the posterior probability assigned to $f_0$ is in the indecisive region $\pi \in (\beta, \alpha)$
The decision maker continues to sample until the probability that he attaches to model $f_0$ falls below $\beta$ or above $\alpha$
### Simulations¶
The next figure shows the outcomes of 500 simulations of the decision process
On the left is a histogram of the stopping times, which equal the number of draws of $z_k$ required to make a decision
The average number of draws is around 6.6
On the right is the fraction of correct decisions at the stopping time
In this case the decision maker is correct 80% of the time
In [9]:
def simulate(wf, true_dist, h_star, π_0=0.5):
"""
This function takes an initial condition and simulates until it
stops (when a decision is made).
"""
f0, f1 = wf.f0, wf.f1
f0_rvs, f1_rvs = wf.f0_rvs, wf.f1_rvs
π_grid = wf.π_grid
def κ(z, π):
"""
Updates π using Bayes' rule and the current observation z.
"""
π_f0, π_f1 = π * f0(z), (1 - π) * f1(z)
π_new = π_f0 / (π_f0 + π_f1)
return π_new
if true_dist == "f0":
f, f_rvs = wf.f0, wf.f0_rvs
elif true_dist == "f1":
f, f_rvs = wf.f1, wf.f1_rvs
# Find cutoffs
β, α = find_cutoff_rule(wf, h_star)
# Initialize a couple useful variables
π = π_0
t = 0
# Maybe should specify which distribution is correct one so that
# the draws come from the "right" distribution
z = f_rvs()
t = t + 1
π = κ(z, π)
if π < β:
decision = 1
elif π > α:
decision = 0
if true_dist == "f0":
if decision == 0:
correct = True
else:
correct = False
elif true_dist == "f1":
if decision == 1:
correct = True
else:
correct = False
return correct, π, t
def stopping_dist(wf, h_star, ndraws=250, true_dist="f0"):
"""
Simulates repeatedly to get distributions of time needed to make a
decision and how often they are correct.
"""
tdist = np.empty(ndraws, int)
cdist = np.empty(ndraws, bool)
for i in range(ndraws):
correct, π, t = simulate(wf, true_dist, h_star)
tdist[i] = t
cdist[i] = correct
return cdist, tdist
def simulation_plot(wf):
h_star = solve_model(wf)
ndraws = 500
cdist, tdist = stopping_dist(wf, h_star, ndraws)
fig, ax = plt.subplots(1, 2, figsize=(16, 5))
ax[0].hist(tdist, bins=np.max(tdist))
ax[0].set_title(f"Stopping times over {ndraws} replications")
ax[0].set(xlabel="time", ylabel="number of stops")
ax[0].annotate(f"mean = {np.mean(tdist)}", xy=(max(tdist) / 2,
max(np.histogram(tdist, bins=max(tdist))[0]) / 2))
ax[1].hist(cdist.astype(int), bins=2)
ax[1].set_title(f"Correct decisions over {ndraws} replications")
ax[1].annotate(f"% correct = {np.mean(cdist)}",
xy=(0.05, ndraws / 2))
plt.show()
simulation_plot(wf)
Error at iteration 25 is 0.00010029006380740668.
Converged in 26 iterations.
### Comparative statics¶
Now let’s consider the following exercise
We double the cost of drawing an additional observation
Before you look, think about what will happen:
• Will the decision maker be correct more or less often?
• Will he make decisions sooner or later?
In [10]:
wf = WaldFriedman(c=2.5)
simulation_plot(wf)
Converged in 14 iterations.
Increased cost per draw has induced the decision maker to take less draws before deciding
Because he decides with less, the percentage of time he is correct drops
This leads to him having a higher expected loss when he puts equal weight on both models
### A notebook implementation¶
To facilitate comparative statics, we provide a Jupyter notebook that generates the same plots, but with sliders
With these sliders you can adjust parameters and immediately observe
• effects on the smoothness of the value function in the indecisive middle range as we increase the number of grid points in the piecewise linear approximation
• effects of different settings for the cost parameters $L_0, L_1, c$, the parameters of two beta distributions $f_0$ and $f_1$, and the number of points and linear functions $m$ to use in the piece-wise continuous approximation to the value function
• various simulations from $f_0$ and associated distributions of waiting times to making a decision
• associated histograms of correct and incorrect decisions
## Comparison with Neyman-Pearson formulation¶
For several reasons, it is useful to describe the theory underlying the test that Navy Captain G. S. Schuyler had been told to use and that led him to approach Milton Friedman and Allan Wallis to convey his conjecture that superior practical procedures existed
Evidently, the Navy had told Captail Schuyler to use what it knew to be a state-of-the-art Neyman-Pearson test
We’ll rely on Abraham Wald’s [Wal47] elegant summary of Neyman-Pearson theory
For our purposes, watch for there features of the setup:
• the assumption of a fixed sample size $n$
• the application of laws of large numbers, conditioned on alternative probability models, to interpret the probabilities $\alpha$ and $\beta$ defined in the Neyman-Pearson theory
Recall that in the sequential analytic formulation above, that
• The sample size $n$ is not fixed but rather an object to be chosen; technically $n$ is a random variable
• The parameters $\beta$ and $\alpha$ characterize cut-off rules used to determine $n$ as a random variable
• Laws of large numbers make no appearances in the sequential construction
In chapter 1 of Sequential Analysis [Wal47] Abraham Wald summarizes the Neyman-Pearson approach to hypothesis testing
Wald frames the problem as making a decision about a probability distribution that is partially known
(You have to assume that something is already known in order to state a well posed problem – usually, something means a lot)
By limiting what is unknown, Wald uses the following simple structure to illustrate the main ideas:
• a decision maker wants to decide which of two distributions $f_0$, $f_1$ govern an i.i.d. random variable $z$
• The null hypothesis $H_0$ is the statement that $f_0$ governs the data
• The alternative hypothesis $H_1$ is the statement that $f_1$ governs the data
• The problem is to devise and analyze a test of hypothesis $H_0$ against the alternative hypothesis $H_1$ on the basis of a sample of a fixed number $n$ independent observations $z_1, z_2, \ldots, z_n$ of the random variable $z$
To quote Abraham Wald,
A test procedure leading to the acceptance or rejection of the hypothesis in question is simply a rule specifying, for each possible sample of size $n$, whether the hypothesis should be accepted or rejected on the basis of the sample. This may also be expressed as follows: A test procedure is simply a subdivision of the totality of all possible samples of size $n$ into two mutually exclusive parts, say part 1 and part 2, together with the application of the rule that the hypothesis be accepted if the observed sample is contained in part 2. Part 1 is also called the critical region. Since part 2 is the totality of all samples of size 2 which are not included in part 1, part 2 is uniquely determined by part 1. Thus, choosing a test procedure is equivalent to determining a critical region.
Let’s listen to Wald longer:
As a basis for choosing among critical regions the following considerations have been advanced by Neyman and Pearson: In accepting or rejecting $H_0$ we may commit errors of two kinds. We commit an error of the first kind if we reject $H_0$ when it is true; we commit an error of the second kind if we accept $H_0$ when $H_1$ is true. After a particular critical region $W$ has been chosen, the probability of committing an error of the first kind, as well as the probability of committing an error of the second kind is uniquely determined. The probability of committing an error of the first kind is equal to the probability, determined by the assumption that $H_0$ is true, that the observed sample will be included in the critical region $W$. The probability of committing an error of the second kind is equal to the probability, determined on the assumption that $H_1$ is true, that the probability will fall outside the critical region $W$. For any given critical region $W$ we shall denote the probability of an error of the first kind by $\alpha$ and the probability of an error of the second kind by $\beta$.
Let’s listen carefully to how Wald applies a law of large numbers to interpret $\alpha$ and $\beta$:
The probabilities $\alpha$ and $\beta$ have the following important practical interpretation: Suppose that we draw a large number of samples of size $n$. Let $M$ be the number of such samples drawn. Suppose that for each of these $M$ samples we reject $H_0$ if the sample is included in $W$ and accept $H_0$ if the sample lies outside $W$. In this way we make $M$ statements of rejection or acceptance. Some of these statements will in general be wrong. If $H_0$ is true and if $M$ is large, the probability is nearly $1$ (i.e., it is practically certain) that the proportion of wrong statements (i.e., the number of wrong statements divided by $M$) will be approximately $\alpha$. If $H_1$ is true, the probability is nearly $1$ that the proportion of wrong statements will be approximately $\beta$. Thus, we can say that in the long run [ here Wald applies a law of large numbers by driving $M \rightarrow \infty$ (our comment, not Wald’s) ] the proportion of wrong statements will be $\alpha$ if $H_0$is true and $\beta$ if $H_1$ is true.
The quantity $\alpha$ is called the size of the critical region, and the quantity $1-\beta$ is called the power of the critical region
Wald notes that
one critical region $W$ is more desirable than another if it has smaller values of $\alpha$ and $\beta$. Although either $\alpha$ or $\beta$ can be made arbitrarily small by a proper choice of the critical region $W$, it is possible to make both $\alpha$ and $\beta$ arbitrarily small for a fixed value of $n$, i.e., a fixed sample size.
Wald summarizes Neyman and Pearson’s setup as follows:
Neyman and Pearson show that a region consisting of all samples $(z_1, z_2, \ldots, z_n)$ which satisfy the inequality
$$\frac{ f_1(z_1) \cdots f_1(z_n)}{f_0(z_1) \cdots f_1(z_n)} \geq k$$
is a most powerful critical region for testing the hypothesis $H_0$ against the alternative hypothesis $H_1$. The term $k$ on the right side is a constant chosen so that the region will have the required size $\alpha$.
Wald goes on to discuss Neyman and Pearson’s concept of uniformly most powerful test
Here is how Wald introduces the notion of a sequential test
A rule is given for making one of the following three decisions at any stage of the experiment (at the m th trial for each integral value of m ): (1) to accept the hypothesis H , (2) to reject the hypothesis H , (3) to continue the experiment by making an additional observation. Thus, such a test procedure is carried out sequentially. On the basis of the first observation one of the aforementioned decisions is made. If the first or second decision is made, the process is terminated. If the third decision is made, a second trial is performed. Again, on the basis of the first two observations one of the three decisions is made. If the third decision is made, a third trial is performed, and so on. The process is continued until either the first or the second decisions is made. The number n of observations required by such a test procedure is a random variable, since the value of n depends on the outcome of the observations.
Footnotes
[1] Because the decision maker believes that $z_{k+1}$ is drawn from a mixture of two i.i.d. distributions, he does not believe that the sequence $[z_{k+1}, z_{k+2}, \ldots]$ is i.i.d. Instead, he believes that it is exchangeable. See [Kre88] chapter 11, for a discussion of exchangeability.
• Share page
|
2019-02-23 10:50:25
|
{"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.772875964641571, "perplexity": 1355.0272441710672}, "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-2019-09/segments/1550249500704.80/warc/CC-MAIN-20190223102155-20190223124155-00450.warc.gz"}
|
http://en.wikipedia.org/wiki/Ordinal_analysis
|
# Ordinal analysis
Jump to: navigation, search
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof theoretic ordinal of Peano arithmetic is ε0.
## Definition
Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof theoretic ordinal of such a theory $T$ is the smallest recursive ordinal that the theory cannot prove is well founded — the supremum of all ordinals $\alpha$ for which there exists a notation $o$ in Kleene's sense such that $T$ proves that $o$ is an ordinal notation. Equivalently, it is the supremum of all ordinals $\alpha$ such that there exists a recursive relation $R$ on $\omega$ (the set of natural numbers) that well-orders it with ordinal $\alpha$ and such that $T$ proves transfinite induction of arithmetical statements for $R$.
The existence of any recursive ordinal that the theory fails to prove is well ordered follows from the $\Sigma^1_1$ bounding theorem, as the set of natural numbers that an effective theory proves to be ordinal notations is a $\Sigma^0_1$ set (see Hyperarithmetical theory). Thus the proof theoretic ordinal of a theory will always be a countable ordinal less than the Church-Kleene ordinal $\omega_1^{\mathrm{CK}}$.
In practice, the proof theoretic ordinal of a theory is a good measure of the strength of a theory. If theories have the same proof theoretic ordinal they are often equiconsistent, and if one theory has a larger proof theoretic ordinal than another it can often prove the consistency of the second theory.
## Examples
### Theories with proof theoretic ordinal ω2
• RFA, rudimentary function arithmetic.[1]
• 0, arithmetic with induction on Δ0-predicates without any axiom asserting that exponentiation is total.
### Theories with proof theoretic ordinal ω3
Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal.
### Theories with proof theoretic ordinal ωn
• 0 or EFA augmented by an axiom ensuring that each element of the n-th level $\mathcal{E}^n$ of the Grzegorczyk hierarchy is total.
### Theories with proof theoretic ordinal the Feferman-Schütte ordinal Γ0
This ordinal is sometimes considered to be the upper limit for "predicative" theories.
### Theories with larger proof theoretic ordinals
• $\Pi^1_1\mbox{-}\mathsf{CA}_0$, Π11 comprehension has a rather large proof theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ0ω) in Buchholz's notation. It is also the ordinal of $ID_{<\omega}$, the theory of finitely iterated inductive definitions.
• T0, Feferman's constructive system of explicit mathematics has a larger proof-theoretic ordinal, which is also the proof-theoretic ordinal of the KPi, Kripke-Platek Set theory with iterated admissibles and $\Sigma^1_2\mbox{-}\mathsf{AC} + \mathsf{BI}$.
• KPM, an extension of Kripke-Platek set theory based on a Mahlo cardinal, has a very large proof theoretic ordinal ϑ, which was described by Rathjen (1990).
• MLM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof theoretic ordinal ψΩ1M + ω).
Most theories capable of describing the power set of the natural numbers have proof theoretic ordinals that are so large that no explicit combinatorial description has yet (as of 2008) been given. This includes second order arithmetic and set theories with powersets. (The CZF and Kripke-Platek set theories mentioned above are weak set theories without powersets.)
## References
• Buchholz, W.; Feferman, S.; Pohlers, W.; Sieg, W. (1981), Iterated inductive definitions and sub-systems of analysis, Lecture Notes in Math. 897, Berlin: Springer-Verlag, doi:10.1007/BFb0091894, ISBN 978-3-540-11170-2
• Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics 1407, Berlin: Springer-Verlag, ISBN 3-540-51842-8, MR 1026933
• Pohlers, Wolfram (1998), "Handbook of Proof Theory", Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics (Amsterdam: Elsevier Science B. V.) 137: 210–335, ISBN 0-444-89840-9, MR 1640328 |chapter= ignored (help)
• Rathjen, Michael (1990), "Ordinal notations based on a weakly Mahlo cardinal.", Arch. Math. Logic 29 (4): 249–263, doi:10.1007/BF01651328, MR 1062729
• Rathjen, Michael (2006), "The art of ordinal analysis", International Congress of Mathematicians II, Zürich,: Eur. Math. Soc., pp. 45–69, MR 2275588
• Rose, H.E. (1984), Subrecursion. Functions and Hierarchies, Oxford logic guides 9, Oxford, New York: Clarendon Press, Oxford University Press
• Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 3-540-07911-4, MR 0505313
• Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 0-444-87943-9, MR 0882549
1. ^ Krajicek, Jan (1995). Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press. pp. 18–20. ISBN 9780521452052. defines the rudimentary sets and rudimentary functions, and proves them equivalent to the Δ0-predicates on the naturals. An ordinal analysis of the system can be found in Rose, H. E. (1984). Subrecursion: functions and hierarchies. University of Michigan: Clarendon Press. ISBN 9780198531890.
|
2015-03-05 11:13:32
|
{"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": 18, "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": 0.9440577626228333, "perplexity": 1072.301824685961}, "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-2015-11/segments/1424936464088.46/warc/CC-MAIN-20150226074104-00030-ip-10-28-5-156.ec2.internal.warc.gz"}
|
https://www.rdocumentation.org/packages/frbs/versions/3.1-0
|
# frbs v3.1-0
0
0th
Percentile
## Fuzzy Rule-Based Systems for Classification and Regression Tasks
An implementation of various learning algorithms based on fuzzy rule-based systems (FRBSs) for dealing with classification and regression tasks. Moreover, it allows to construct an FRBS model defined by human experts. FRBSs are based on the concept of fuzzy sets, proposed by Zadeh in 1965, which aims at representing the reasoning of human experts in a set of IF-THEN rules, to handle real-life problems in, e.g., control, prediction and inference, data mining, bioinformatics data processing, and robotics. FRBSs are also known as fuzzy inference systems and fuzzy models. During the modeling of an FRBS, there are two important steps that need to be conducted: structure identification and parameter estimation. Nowadays, there exists a wide variety of algorithms to generate fuzzy IF-THEN rules automatically from numerical data, covering both steps. Approaches that have been used in the past are, e.g., heuristic procedures, neuro-fuzzy techniques, clustering methods, genetic algorithms, squares methods, etc. Furthermore, in this version we provide a universal framework named 'frbsPMML', which is adopted from the Predictive Model Markup Language (PMML), for representing FRBS models. PMML is an XML-based language to provide a standard for describing models produced by data mining and machine learning algorithms. Therefore, we are allowed to export and import an FRBS model to/from 'frbsPMML'. Finally, this package aims to implement the most widely used standard procedures, thus offering a standard package for FRBS modeling to the R community.
## Functions in frbs
Name Description FRBCS.eng FRBCS: prediction phase frbsPMML The frbsPMML generator rulebase The rule checking function DM.update FIR.DM updating function ANFIS ANFIS model building FRBCS.CHI FRBCS.CHI model building GFS.GCCL GFS.GCCL model building denorm.data The data de-normalization GFS.LT.RS.test GFS.LT.RS: The prediction phase summary.frbs The summary function for frbs objects FIR.DM FIR.DM model building frbs.eng The prediction phase HGD.update FS.HGD updating function ANFIS.update ANFIS updating function defuzzifier Defuzzifier to transform from linguistic terms to crisp values data.gen3d A data generator predict.frbs The frbs prediction stage GFS.GCCL.eng GFS.GCCL.test: The prediction phase ECM Evolving Clustering Method FS.HGD FS.HGD model building GFS.Thrift GFS.Thrift model building GFS.FR.MOGUL GFS.FR.MOGUL model building frbs-package Getting started with the frbs package FRBCS.W FRBCS.W model building FH.GBML FH.GBML model building WM WM model building GFS.LT.RS GFS.LT.RS model building SLAVE.test SLAVE.test: The prediction phase inference The process of fuzzy reasoning GFS.FR.MOGUL.test GFS.FR.MOGUL: The prediction phase SBC The subtractive clustering and fuzzy c-means (SBC) model building read.frbsPMML The frbsPMML reader frbsObjectFactory The object factory for frbs objects fuzzifier Transforming from crisp set into linguistic terms SLAVE SLAVE model building frbsData Data set of the package frbs.gen The frbs model generator DENFIS DENFIS model building plotMF The plotting function norm.data The data normalization HyFIS HyFIS model building SBC.test SBC prediction phase GFS.Thrift.test GFS.Thrift: The prediction phase write.frbsPMML The frbsPMML writer DENFIS.eng DENFIS prediction function HyFIS.update HyFIS updating function frbs.learn The frbs model building function No Results!
|
2019-02-19 03:04:27
|
{"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": 0.3696293830871582, "perplexity": 13093.536609291717}, "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/1550247489304.34/warc/CC-MAIN-20190219020906-20190219042906-00063.warc.gz"}
|
http://tm.durusau.net/?cat=2061
|
## Archive for the ‘Profiling’ Category
### Watch your Python script with strace
Sunday, September 11th, 2016
Description:
Modern operating systems sandbox each process inside of a virtual memory map from which direct I/O operations are generally impossible. Instead, a process has to ask the operating system every time it wants to modify a file or communicate bytes over the network. By using operating system specific tools to watch the system calls a Python script is making — using “strace” under Linux or “truss” under Mac OS X — you can study how a program is behaving and address several different kinds of bugs.
Brandon Rhodes does a delightful presentation on using strace with Python.
Slides for Tracing Python with strace or truss.
I deeply enjoyed this presentation, which I discovered while looking at a Python regex issue.
Anticipate running strace on the Python script this week and will report back on any results or failure to obtain results! (Unlike in academic publishing, experiments and investigations do fail.)
### Debugging
Tuesday, August 23rd, 2016
Julia Evans tweeted:
It’s been two days without another suggestion.
Considering Brendan D. Gregg’s homepage, do you have another suggestion?
Too rich of a resource to not write down.
Besides, for some subjects and their relationships, you need specialized tooling to see them.
Not to mention that if you can spot patterns in subjects, detecting an unknown 0-day may be easier.
Of course, you can leave USB sticks at popular eateries near Fort Meade, MD 20755-6248, but some people prefer to work for their 0-day exploits.
😉
### PAPERS ARE AMAZING: Profiling threaded programs with Coz
Saturday, October 31st, 2015
PAPERS ARE AMAZING: Profiling threaded programs with Coz by Julia Evans.
I don’t often mention profiling at all but I mention Julia’s post because:
1. It reports a non-intuitive insight in profiling threaded programs (at least until you have seen it).
2. Julia writes a great post on new ideas with perf.
From the post:
The core idea in this paper is – if you have a line of code in a thread, and you want to know if it’s making your program slow, speed up that line of code to see if it makes the whole program faster!
Of course, you can’t actually speed up a thread. But you can slow down all other threads! So that’s what they do. The implemention here is super super super interesting – they use the perf Linux system to do this, and in particular they can do it without modifying the program’s code. So this is a) wizardry, and b) uses perf
Which are both things we love here (omg perf). I’m going to refer you to the paper for now to learn more about how they use perf to slow down threads, because I honestly don’t totally understand it myself yet. There are some difficult details like “if the thread is already waiting on another thread, should we slow it down even more?” that they get into.
The insight that slowing down all but one thread is the equivalent to speeding up the thread of interest for performance evaluation sounds obvious when mentioned. But only after it is mentioned.
I suspect the ability to have that type of insight isn’t teachable other than by demonstration across a wide range of cases. If you know of other such insights, ping me.
For those interested in “real world” application of insights, Julia mentions the use of this profiler on SQLite and Memcached.
See Julia’s post for the paper and other references.
If you aren’t already checking Julia’s blog on a regular basis you might want to start.
|
2017-12-13 10:46:51
|
{"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": 0.22640879452228546, "perplexity": 2372.0449767233918}, "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/1512948522999.27/warc/CC-MAIN-20171213104259-20171213124259-00782.warc.gz"}
|
https://hal.inria.fr/inria-00274423v2
|
# Experiments in Model-Checking Optimistic Replication Algorithms
* Auteur correspondant
2 CASSIS - Combination of approaches to the security of infinite states systems
FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies, INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : This paper describes a series of model-checking experiments to verify optimistic replication algorithms based on Operational Transformation (OT) approach used for supporting collaborative edition. We formally define, using tool UPPAAL, the behavior and the main consistency requirement (i.e. convergence property) of the collaborative editing systems, as well as the abstract behavior of the environment where these systems are supposed to operate. Due to data replication and the unpredictable nature of user interactions, such systems have infinitely many states. So, we show how to exploit some features of the UPPAAL specification language to attenuate the severe state explosion problem. Two models are proposed. The first one, called concrete model, is very close to the system implementation but runs up against a severe explosion of states. The second model, called symbolic model, aims to overcome the limitation of the concrete model by delaying the effective selection and execution of editing operations until the construction of symbolic execution traces of all sites is completed. Experimental results have shown that the symbolic model allows a significant gain in both space and time. Using the symbolic model, we have been able to show that if the number of sites exceeds $2$ then the convergence property is not satisfied for all OT algorithms considered here. A counterexample is provided for every algorithm.
Keywords :
Type de document :
Rapport
[Research Report] RR-6510, INRIA. 2008, pp.49
Domaine :
Littérature citée [14 références]
https://hal.inria.fr/inria-00274423
Contributeur : Rapport de Recherche Inria <>
Soumis le : lundi 21 avril 2008 - 10:50:42
Dernière modification le : lundi 19 mars 2018 - 22:38:02
Document(s) archivé(s) le : mardi 21 septembre 2010 - 16:37:00
### Fichiers
RR-6510.pdf
Fichiers produits par l'(les) auteur(s)
### Identifiants
• HAL Id : inria-00274423, version 2
• ARXIV : 0804.3023
### Citation
Hanifa Boucheneb, Abdessamad Imine. Experiments in Model-Checking Optimistic Replication Algorithms. [Research Report] RR-6510, INRIA. 2008, pp.49. 〈inria-00274423v2〉
### Métriques
Consultations de la notice
## 308
Téléchargements de fichiers
|
2018-04-24 16:41:50
|
{"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": 0.5060648322105408, "perplexity": 4333.161085610344}, "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-2018-17/segments/1524125946807.67/warc/CC-MAIN-20180424154911-20180424174911-00217.warc.gz"}
|
http://physics.stackexchange.com/questions/91691/is-there-is-a-reason-for-paulis-exclusion-principle
|
# Is there is a reason for Pauli's Exclusion Principle?
As a starting quantum physicist I am very interested in reasons why does Pauli's Exclusion Principle works. I mean standard explanations are not quite satisfying. Of course we can say that is because of fermionic nature of electrons - but it is just the different way to say the same thing. We can say that we need to antisymmetrize the quantum wavefunction for many electrons - well, another different way to say the same. We can say that it is because spin 1/2 of electron - but the hell, fermions has by the definition half-integral spin so it doesn't explain anything. Is the Exclusion Principle something deeper, for example in Dirac's Equation, like spin of the electron? I think it would be satisfying.
-
I think that while these "explanations" are all dancing around the same pole, they aren't created equal. I think the meat is in the fact that nature has a local Lorentz symmetry, so we expect to be able to decompose things into representations of the group $SO(3,1)$. It's a mathematical fact that this group (or it's algebra, rather) has integer and half-integer representations.
@CheshireCat Perhaps add that the last step is that the spin-statistics theorem shows that for half integer spin representations the quantum state for two particles with quantum numbers $\vec{x}$ and $\vec{y}$ (I include "position" in the quantum number vector) is antisymmetric wrt swap of arguments $\psi(\vec{x}, \vec{y}) = -\psi(\vec{x}, \vec{y})$ so that now if two particles have the same quantum numbers $\psi(\vec{x}, \vec{x}) = - \psi(\vec{x}, \vec{x})$. A further piece of trivia which I like to dwell on here: when we represent the algebra by half integer representations, we're ... – WetSavannaAnimal aka Rod Vance Dec 30 '13 at 11:15
...actually representing the double cover $PSL(2,\mathbb{C})$ of the Lorentz group $SO(3,1)$, so you could say, with a slight strech, that the Dirac belt trick "proves" there are only bosons and fermions in the world. – WetSavannaAnimal aka Rod Vance Dec 30 '13 at 11:18
|
2015-07-05 15:05:06
|
{"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": 0.7242559194564819, "perplexity": 268.42463340672276}, "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-2015-27/segments/1435375097512.42/warc/CC-MAIN-20150627031817-00274-ip-10-179-60-89.ec2.internal.warc.gz"}
|
http://mathhelpforum.com/algebra/213911-please-help-q-about-simultaneous-equations.html
|
Hi I have to solve a really hard simultaneous equation and I don't know how to do it.
Please note it has p1,p2,p3,q1,q2,q3. The numbers are subscripts.
These all represent constant values. That is they are not variables.
The two variables are x and y.
1. x(q2+q3-2q1)-y(p2+p3-2p1)+(q1p2-p1q2)+(q1p3-q3p1)=0
2. x(2q2-q1-q3)-y(2p2-p1-p3)+(p2q1-q2p1)+(p2q3-q2p3)=0
I have tried to solve this by re-arranging then subbing in but this quickly gets out of hand as the subbed in equation is very long and
Originally Posted by James7361539
Hi I have to solve a really hard simultaneous equation and I don't know how to do it.
Please note it has p1,p2,p3,q1,q2,q3. The numbers are subscripts.
These all represent constant values. That is they are not variables.
The two variables are x and y.
1. x(q2+q3-2q1)-y(p2+p3-2p1)+(q1p2-p1q2)+(q1p3-q3p1)
2. x(2q2-q1-q3)-y(2p2-p1-p3)+(p2q1-q2p1)+(p2q3-q2p3)
I have tried to solve this by re-arranging then subbing in but this quickly gets out of hand as the subbed in equation is very long and
Those aren't equations.
Originally Posted by a tutor
Those aren't equations.
Sorry I forgot the =0 at the end of them
I have edited my post now.
So, essentially, you have Ax+ By= C and Dx+ Ey= F. Multiply the first equation by E, the second equation by B, and subtract:
(AE- BD)x= CE- BF, x= (CE- BF)/(AE- BD).
Multiply the first equation by D, the second equation by A, and subtract:
(BD- AE)y= CD- AF, x= (CD- AF)/(BD- AE).
Now replace A, B, C, D, E, and F with those complicated expressions.
|
2014-03-17 07:45:52
|
{"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": 0.8748509287834167, "perplexity": 1558.5191073744338}, "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-2014-10/segments/1394678704980/warc/CC-MAIN-20140313024504-00043-ip-10-183-142-35.ec2.internal.warc.gz"}
|
https://www.physicsforums.com/threads/time-dependent-perturbation-theory.206293/
|
# Homework Help: Time-dependent perturbation theory
1. Dec 28, 2007
### ehrenfest
[SOLVED] time-dependent perturbation theory
1. The problem statement, all variables and given/known data
My book uses time-dependent perturbation theory to derive the following expression for the transition of \psi_{100} to \psi_{210} in the hydrogen atom in a uniform magnetic field with magnitude $$\mathcal{E}$$
$$\frac{131072}{59049} \frac{e^2 \mathcal{E}^2 a_o^2}{(E_2 -E_1)^2} \sin^2(\frac{E_2-E_1}{2 \hbar}t)$$
What I don't understand is why you cannot just increase $$\mathcal{E}$$ until the probability goes above 1?
2. Relevant equations
3. The attempt at a solution
2. Dec 28, 2007
### Dick
Because it's 'perturbation theory'. If the field becomes large, then whole approximation that the field can be treated as a 'perturbation' goes out the window and the formula is invalid.
|
2018-04-23 02:19:08
|
{"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": 0.5232561230659485, "perplexity": 890.1588153159422}, "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-2018-17/segments/1524125945669.54/warc/CC-MAIN-20180423011954-20180423031954-00378.warc.gz"}
|
https://artofproblemsolving.com/wiki/index.php/Rational_numbers
|
Rational number
(Redirected from Rational numbers)
A rational number is a number that can be represented as the ratio of two integers.
Examples
• All integers are rational because every integer $a$ can be represented as $a=\frac{a}{1}$
• Every number with a finite decimal expansion is rational (say, $12.345=\frac{12345}{1000}$)
• Every number with a periodic decimal expansion (e.g. 0.314314314...) is also rational.
Moreover, any rational number satisfies exactly one of the last two conditions. The same remark holds if "decimal" is replaced with any other base.
Properties
1. Rational numbers form a field. In plain English it means that you can add, subtract, multiply, and divide them (with the exception of division by $0$) and the result of each such operation is again a rational number.
2. Rational numbers are dense in the set of reals. This means that every non- empty open interval on the real line contains at least one (actually, infinitely many) rationals. Alternatively, it means that every real number can be represented as a limit of a sequence of rational numbers.
3. Despite this, the set of rational numbers is countable, i.e. the same size as the set of integers.
|
2022-08-11 23:13:21
|
{"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": 4, "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": 0.8460258841514587, "perplexity": 144.40650440370763}, "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-2022-33/segments/1659882571536.89/warc/CC-MAIN-20220811224716-20220812014716-00598.warc.gz"}
|
http://en.m.wikibooks.org/wiki/Ordinary_Differential_Equations/Glossary
|
Ordinary Differential Equations/Glossary
C
complementary function
The solution to the related homogenous equation for a nonhomogenous equation
↑Jump back a section
D
differential equation
An equation with one or more derivatives in it. $F(x,y,y',y'',y''',...)$
domain
a solution of differential equation is a function y=(x)y which, when substituted along with its derivative among the differential equation satisfies the equation from all x in some specified interval.
↑Jump back a section
F
first order equation
Any equation with a first derivative in it, but no higher derivatives. $F(x,y,y')$
↑Jump back a section
G
general solution
The solution to a differential equation in its most general form, constants included
↑Jump back a section
H
homogenous equation
Any equation that is equal to 0. In differential equations, its an equation $p_n(x)y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_{0}(x)y=0$.
↑Jump back a section
I
initial condition
A value of a function or its derivative at a particular point, used to determine the value of constants for a particular solution
initial value problem
A combination of a differential equation and an initial condition. An initial value problem is solved for a total solution including the value of all constants
integration factor
A factor a differential equation is multiplied by to discover a solution .
↑Jump back a section
L
linear equation
An equation who's terms are a linear combination of a variable and its derivatives. Such an equation is in the form $f_0(x)+f_1(x)y+f_2(x)y'+f_3(x)y''+...f_n(x)y^{n}$. No terms for y or its derivatives may be raised to a power or placed inside a function such as sin or ln
↑Jump back a section
N
nonhomogenous equation
Any equation that is not equal to 0. In differential equations, its an equation $p_n(x)y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_{0}(x)y=f(x)$, where f(x) is not 0.
non-linear equation
Any equation that is not a linear combination of a variable and its derivatives. Either one of the terms has the variable taken to a power, or is in a function such as sin or ln
↑Jump back a section
O
O.D.E
See ordinary differential equation.
order
The highest derivative found in a differntial equations. First order equations only have $y'$, second order equations have $y'$ and $y''$, etc.
ordinary differential equation
Any differential equation that has normal derivatives only
↑Jump back a section
P
partial differential equation
Any differential equation that has partial derivatives in it
particular solution
A solution to a differential equation with all constants evaluated
P.D.E
See partial differential equation.
↑Jump back a section
S
satisfy
to solve a differential equation. Used as an adjective, a solution to a differential equation satisifes that equation
second order equation
Any equation with a second derivative in it, but no higher derivatives. $F(x,y,y',y'')$
separable equation
An equation where the x and y terms are multiplied and not added. $\frac{dy}{dx}=f(x)g(y)$
substitution method
A method of turning a non-separable equation into a separable one.
↑Jump back a section
|
2013-05-19 12:24:20
|
{"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": 10, "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": 0.7724595665931702, "perplexity": 944.370001818574}, "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-2013-20/segments/1368697503739/warc/CC-MAIN-20130516094503-00040-ip-10-60-113-184.ec2.internal.warc.gz"}
|
https://freakonometrics.hypotheses.org/date/2018/11
|
Usually, when I give a course on GLMs, I try to insist on the fact that the link function is probably more important than the distribution. In order to illustrate, consider the following dataset, with 5 observations
x = c(1,2,3,4,5) y = c(1,2,4,2,6) base = data.frame(x,y)
Then consider several model, with various distributions, and either an identity link (and in that case $\mathbb{E}[Y|\mathbf{X}=\mathbf{x}]=\mathbf{x}^T\mathbf{\beta}$) or a log link function (so that $\mathbb{E}[Y|\mathbf{X}=\mathbf{x}]=e^{\mathbf{x}^T\mathbf{\beta}}$)
regNId = glm(y~x,family=gaussian(link="identity"),data=base) regNlog = glm(y~x,family=gaussian(link="log"),data=base) regPId = glm(y~x,family=poisson(link="identity"),data=base) regPlog = glm(y~x,family=poisson(link="log"),data=base) regGId = glm(y~x,family=Gamma(link="identity"),data=base) regGlog = glm(y~x,family=Gamma(link="log"),data=base) regIGId = glm(y~x,family=inverse.gaussian(link="identity"),data=base) regIGlog = glm(y~x,family=inverse.gaussian(link="log"),data=base
One can also consider some Tweedie distribution, to be even more general
library(statmod) regTwId = glm(y~x,family=tweedie(var.power=1.5,link.power=1),data=base) regTwlog = glm(y~x,family=tweedie(var.power=1.5,link.power=0),data=base)
Consider the prediction obtained in the first case, with the linear link function
library(RColorBrewer) darkcols = brewer.pal(8, "Dark2") plot(x,y,pch=19) abline(regNId,col=darkcols[1]) abline(regPId,col=darkcols[2]) abline(regGId,col=darkcols[3]) abline(regIGId,col=darkcols[4]) abline(regTwId,lty=2)
The predictions are very very close, aren’t they ? In the case of the exponential prediction, we obtain
plot(x,y,pch=19) u=seq(.8,5.2,by=.01) lines(u,predict(regNlog,newdata=data.frame(x=u),type="response"),col=darkcols[1]) lines(u,predict(regPlog,newdata=data.frame(x=u),type="response"),col=darkcols[2]) lines(u,predict(regGlog,newdata=data.frame(x=u),type="response"),col=darkcols[3]) lines(u,predict(regIGlog,newdata=data.frame(x=u),type="response"),col=darkcols[4]) lines(u,predict(regTwlog,newdata=data.frame(x=u),type="response"),lty=2)
We can actually look closer. For instance, in the linear case, consider the slope obtained with a Tweedie model (that will include all the parametric familes mentioned here, actually)
pente=function(gamma) summary(glm(y~x,family=tweedie(var.power=gamma,link.power=1),data=base))$coefficients[2,1:2] Vgamma = seq(-.5,3.5,by=.05) Vpente = Vectorize(pente)(Vgamma) plot(Vgamma,Vpente[1,],type="l",lwd=3,ylim=c(.965,1.03),xlab="power",ylab="slope") The slope here is always very very close to one ! Even more if we add a confidence interval plot(Vgamma,Vpente[1,]) lines(Vgamma,Vpente[1,]+1.96*Vpente[2,],lty=2) lines(Vgamma,Vpente[1,]-1.96*Vpente[2,],lty=2) Heuristically, for the Gamma regression, or the Inverse Gaussian one, because the variance is a power of the prediction, if the prediction is small (here on the left), the variance should be small. So, on the left of the graph, the error should be small with a higher power for the variance function. And that’s indeed what we observe here erreur=function(gamma) predict(glm(y~x,family=tweedie(var.power=gamma,link.power=1),data=base),newdata=data.frame(x=1),type="response")-y[x==1] Verreur = Vectorize(erreur)(Vgamma) plot(Vgamma,Verreur,type="l",lwd=3,ylim=c(-.1,.04),xlab="power",ylab="error") abline(h=0,lty=2) Of course, we can do the same with the exponential models pente=function(gamma) summary(glm(y~x,family=tweedie(var.power=gamma,link.power=0),data=base))$coefficients[2,1:2] Vpente = Vectorize(pente)(Vgamma) plot(Vgamma,Vpente[1,],type="l",lwd=3)
or, if we add the confidence bands, we obtain
plot(Vgamma,Vpente[1,],ylim=c(0,.8),type="l",lwd=3,xlab="power",ylab="slope") lines(Vgamma,Vpente[1,]+1.96*Vpente[2,],lty=2) lines(Vgamma,Vpente[1,]-1.96*Vpente[2,],lty=2)
So here also, the “slope” is rather similar… And if we look at the error we make on the left part of the graph, we obtain
erreur=function(gamma) predict(glm(y~x,family=tweedie(var.power=gamma,link.power=0),data=base),newdata=data.frame(x=1),type="response")-y[x==1] Verreur = Vectorize(erreur)(Vgamma) plot(Vgamma,Verreur,type="l",lwd=3,ylim=c(.001,.32),xlab="power",ylab="error")
So my point is that the distribution is usually not the most important point on GLMs, even if chapters of books on GLMs are distribution based… But as mentioned in an another post, if you consider a nonlinear transformation, like we have with GAMs, the story is more complicated…
# Bailey (1963) and Poisson regression on two factors
Consider the following dataset, from A Theory of Extramarital Affairs, by Ray Fair, published in 1978 in the Journal of Political Economy, with 563 observations, and nine variables : eight covariates, and the variable of interest, the number of extramarital affairs, over a year,
base = read.table("http://freakonometrics.free.fr/baseaffairs.txt",header=TRUE) str(base) 'data.frame': 563 obs. of 9 variables: $SEX : int 1 0 0 1 1 0 0 1 0 1 ...$ AGE : num 37 27 32 57 22 32 22 57 32 22 ... $YEARMARRIAGE: num 10 4 15 15 0.75 1.5 0.75 15 15 1.5 ...$ CHILDREN : int 0 0 1 1 0 0 0 1 1 0 ... $RELIGIOUS : int 3 4 1 5 2 2 2 2 4 4 ...$ EDUCATION : int 18 14 12 18 17 17 12 14 16 14 ... $OCCUPATION : int 7 6 1 6 6 5 1 4 1 4 ...$ SATISFACTION: int 4 4 4 5 3 5 3 4 2 5 ... $Y : int 0 0 0 0 0 0 0 0 0 0 ... Let us focus on two categorical covariates, related to the importance of religion, and the occupation df=data.frame(y=base$Y, religion=as.factor(base$RELIGIOUS), occupation=as.factor(base$OCCUPATION), expo = 1) (E=xtabs(expo~religion+occupation,data=df)) occupation religion 1 2 3 4 5 6 7 1 4 1 8 4 16 9 0 2 23 3 11 17 56 36 6 3 29 1 10 12 39 25 2 4 38 7 12 21 59 44 2 5 13 1 3 10 19 19 3 (N=xtabs(y~religion+occupation,data=df)) occupation religion 1 2 3 4 5 6 7 1 4 1 13 3 13 7 0 2 1 1 13 10 25 43 10 3 15 0 12 11 34 35 1 4 24 1 3 15 11 9 10 5 6 0 0 6 11 7 0
The two tables above are the exposure (number of observations) and the number of extramarital affairs, here as contingency tables. Without any covariate, one can assume that $N\sim\mathcal{P}(\lambda\cdot E)$, where $\lambda$ would be
sum(N)/sum(E) [1] 0.6305506
The idea with the margin method is to assume that $N_{i,j}=E_{i,j}\cdot\lambda_{i,j}$ where $\lambda_{i,j}=A_i\cdot B_j$. Bailey (1963) added two series of constraints : per row, $$\sum_j N_{i,j}=\sum_j E_{i,j}\cdot A_i\cdot B_j$$ for any $i$ and similarly, for any $j$ $$\sum_i N_{i,j}=\sum_i E_{i,j}\cdot A_i\cdot B_j$$From the first series of constraints, write $$A_i=\frac{\sum_j N_{i,j}}{\sum_j E_{i,j}\cdot B_j}$$ and use the second series to write $$B_j=\frac{\sum_i N_{i,j}}{\sum_i E_{i,j}\cdot A_i}$$Because we need $A_i$‘s to compute $B_j$‘s, and conversely, it is natural to consider some iterative procedure to solve it. Observe that we do not have unicity…
Consider here some starting values for $A_i$‘s and $B_j$‘s
A=rep(1,length(levels(df$religion))) B=rep(1,length(levels(df$occupation)))*sum(N)/sum(E) A [1] 1 1 1 1 1 B [1] 0.6305506 0.6305506 0.6305506 0.6305506 0.6305506 0.6305506 0.6305506
The predicted number of extramarital affairs would be $\hat N_{i,j}=E_{i,j}\cdot\hat A_i\cdot \hat B_j$
E * A%*%t(B) occupation religion 1 2 3 4 5 6 7 1 2.5222025 0.6305506 5.0444050 2.5222025 10.0888099 5.6749556 0.0000000 2 14.5026643 1.8916519 6.9360568 10.7193606 35.3108348 22.6998224 3.7833037 3 18.2859680 0.6305506 6.3055062 7.5666075 24.5914742 15.7637655 1.2611012 4 23.9609236 4.4138544 7.5666075 13.2415631 37.2024867 27.7442274 1.2611012 5 8.1971581 0.6305506 1.8916519 6.3055062 11.9804618 11.9804618 1.8916519 sum(B*E[1,]) [1] 26.48313 sum(B*E[2,]) [1] 95.84369 apply(t(B*t(E)),1,sum) 1 2 3 4 5 26.48313 95.84369 74.40497 115.39076 42.87744 sum(A*E[,1]) [1] 107 sum(A*E[,2]) [1] 13 apply(A*E,2,sum) 1 2 3 4 5 6 7 107 13 44 64 189 133 13
From expressions above, observe that one can very easily write expressions of $A_i$‘s and $B_j$‘s as functions of $B_j$‘s and $A_i$‘s respectively
A=apply(N,1,sum)/apply(t(B*t(E)),1,sum) B=apply(N,2,sum)/apply(A*E,2,sum)
Let it iterate one thousand times
for(i in 1:1000){ A=apply(N,1,sum)/apply(t(B*t(E)),1,sum) B=apply(N,2,sum)/apply(A*E,2,sum) }
We obtain here
A 1 2 3 4 5 1.5404346 1.0447195 1.4825650 0.6553159 0.6634763 B 1 2 3 4 5 6 7 0.4685515 0.2629769 0.8454435 0.7245310 0.4889697 0.7770553 1.6753750 E * A%*%t(B) occupation religion 1 2 3 4 5 6 7 1 2.8870914 0.4050987 10.4188024 4.4643702 12.0516123 10.7730250 0.0000000 2 11.2586111 0.8242113 9.7157637 12.8678376 28.6068235 29.2249717 10.5017811 3 20.1450811 0.3898804 12.5342484 12.8899708 28.2722423 28.8008726 4.9677044 4 11.6678702 1.2063307 6.6483904 9.9707299 18.9053460 22.4055332 2.1957997 5 4.0413463 0.1744790 1.6827951 4.8070914 6.1639760 9.7955975 3.3347148
That is our prediction, per category, of the number of affairs. Observe that here, sums per row are equal to observed numbers,
apply(N,1,sum) 1 2 3 4 5 41 103 108 73 30 apply(E * A%*%t(B),1,sum) 1 2 3 4 5 41 103 108 73 30
as well as sums per colums
apply(N,2,sum) 1 2 3 4 5 6 7 50 3 41 45 94 101 21 apply(E * A%*%t(B),2,sum) 1 2 3 4 5 6 7 50 3 41 45 94 101 21
Now, why should I mention that here, in the section on the Poisson regression in our course ? Because actually, this is exactly what we get if we run a Poisson regression on those two covariates
reg=glm(y~religion+occupation,data=df,family=poisson) summary(reg) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.32604 0.21325 -1.529 0.126285 religion2 -0.38832 0.18791 -2.066 0.038783 * religion3 -0.03829 0.18585 -0.206 0.836771 religion4 -0.85470 0.19757 -4.326 1.52e-05 *** religion5 -0.84233 0.24416 -3.450 0.000561 *** occupation2 -0.57758 0.59549 -0.970 0.332083 occupation3 0.59022 0.21349 2.765 0.005699 ** occupation4 0.43588 0.20603 2.116 0.034381 * occupation5 0.04265 0.17590 0.242 0.808399 occupation6 0.50587 0.17360 2.914 0.003569 ** occupation7 1.27415 0.26298 4.845 1.27e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
First of all, observe that the total sum of predictions equals the total sum of observations
yp = predict(reg,type="response") sum(yp) [1] 355 sum(df$y) [1] 355 But actually, the predicted number of affairs, for our 35 classes, is exactly what we got using Bailey’s technique xtabs(yp~df$religion+df$occupation) df$occupation df$religion 1 2 3 4 5 6 7 1 2.8870914 0.4050987 10.4188024 4.4643702 12.0516123 10.7730250 0.0000000 2 11.2586112 0.8242113 9.7157637 12.8678376 28.6068235 29.2249717 10.5017811 3 20.1450813 0.3898804 12.5342484 12.8899708 28.2722424 28.8008726 4.9677044 4 11.6678703 1.2063307 6.6483904 9.9707300 18.9053460 22.4055332 2.1957997 5 4.0413464 0.1744790 1.6827951 4.8070914 6.1639761 9.7955975 3.3347148 E * A%*%t(B) occupation religion 1 2 3 4 5 6 7 1 2.8870914 0.4050987 10.4188024 4.4643702 12.0516123 10.7730250 0.0000000 2 11.2586111 0.8242113 9.7157637 12.8678376 28.6068235 29.2249717 10.5017811 3 20.1450811 0.3898804 12.5342484 12.8899708 28.2722423 28.8008726 4.9677044 4 11.6678702 1.2063307 6.6483904 9.9707299 18.9053460 22.4055332 2.1957997 5 4.0413463 0.1744790 1.6827951 4.8070914 6.1639760 9.7955975 3.3347148 To be more specific, up to a multiplicate constant, series of coefficients are equal here, e.g. for $A_i$‘s a=exp(coefficients(reg)[1]+c(0,coefficients(reg)[2:5])) a/a[1] religion2 religion3 religion4 religion5 1.0000000 0.6781979 0.9624329 0.4254098 0.4307072 A/A[1] 1 2 3 4 5 1.0000000 0.6781979 0.9624329 0.4254098 0.4307072 but also for $B_j$‘s b=exp(coefficients(reg)[1]+c(0,coefficients(reg)[6:11])) b/b[1] occupation2 occupation3 occupation4 occupation5 occupation6 occupation7 1.0000000 0.5612551 1.8043769 1.5463210 1.0435773 1.6584203 3.5756477 B/B[1] 1 2 3 4 5 6 7 1.0000000 0.5612551 1.8043770 1.5463210 1.0435773 1.6584203 3.5756478 This will have major implications in non-life insurance models (for claims reserving). # Je code, donc je suis Mercredi 21 novembre, l’édition 2018 du Colloque HumanIA se tiendra à l’Agora Hydro-Québec du Complexe des sciences Pierre-Dansereau de l’UQAM dès 9h30. Dans le cadre de la semaine La France à l’UQAM, le Colloque sera suivi d’un débat sur le thème «Intelligence artificielle : l’erreur n’est-elle qu’humaine ?», dans l’après midi. J’interviendrais pour ma part dans un atelier en avant midi, sur le theme, “je code, donc je suis“. Je mets quelques liens pour alimenter la discussion, # Rencontres Mutualistes Lundi et mardi, je serais a Beaune, en Bourgogne, pour les premières rencontres mutualistes. On m’a demande d’intervenir en ouverture de la seconde journée, sur le thème “segmentation et mutualisation”. Les slides sont dès à présent en ligne. Comme j’ai peu de temps, je reviendrais sur les grands principes de la tarification et du rôle de l’actuaire. J’ai ensuite pense qu’une discussion autour du graphique suivant pourrait être intéressante, en particulier sur les deux bornes, inférieure (‘average pricing‘) et supérieure (‘perfect pricing‘) On finira avec un rapide retour sur les pricing games, pour conclure. # The “probability to win” is hard to estimate… Real-time computation (or estimation) of the “probability to win” is difficult. We’ve seem that in soccer games, in elections… but actually, as a professor, I see that frequently when I grade my students. Consider a classical multiple choice exam. After each question, imagine that you try to compute the probability that the student will pass. Consider here the case where we have 50 questions. Students pass when they have 25 correct answers, or more. Just for simulations, I will assume that students just flip a coin at each question… I have $n$ students, and 50 questions set.seed(1) n=10 M=matrix(sample(0:1,size=n*50,replace=TRUE),50,n) Let $X_{i,j}$ denote the score of student $i$ at question $j$. Let $S_{i,j}$ denote the cumulated score, i.e. $S_{i,j}=X_{i,1}+\cdots+X_{i,j}$. At step $j$, I can get some sort of prediction of the final score, using $\hat{T}_{i,j}=50\times S_{i,j}/j$. Here is the code SM=apply(M,2,cumsum) NB=SM*50/(1:50) We can actually plot it plot(NB[,1],type="s",ylim=c(0,50)) abline(h=25,col="blue") for(i in 2:n) lines(NB[,i],type="s",col="light blue") lines(NB[,3],type="s",col="red") But that’s simply the prediction of the final score, at each step. That’s not the computation of the probability to pass ! Let’s try to see how we can do it… If after $j$ questions, the students has 25 correct answer, the probability should be 1 – i.e. if $S_{i,j}\geq 25$ – since he cannot fail. Another simple case is the following : if after $j$ questions, the number of points he can get with all correct answers until the end is not sufficient, he will fail. That means if $S_{i,j}+(50-i+1)< 25$ the probability should be 0. Otherwise, to compute the probability to sucess, it is quite straightforward. It is the probability to obtain at least $25-S_{i,j}$ correct answers, out of $50-j$ questions, when the probability of success is actually $S_{i,j}/j$. We recognize the survival probability of a binomial distribution. The code is then simply PB=NB*NA for(i in 1:50){ for(j in 1:n){ if(SM[i,j]>=25) PB[i,j]=1 if(SM[i,j]+(50-i+1)<25) PB[i,j]=0 if((SM[i,j]<25)&(SM[i,j]+(50-i+1)>=25)) PB[i,j]=1-pbinom(25-SM[i,j],size=(50-i),prob=SM[i,j]/i) }} So if we plot it, we get plot(PB[,1],type="s",ylim=c(0,1)) abline(h=25,col="red") for(i in 2:n) lines(PB[,i],type="s",col="light blue") lines(PB[,3],type="s",col="red") which is much more volatile than the previous curves we obtained ! So yes, computing the “probability to win” is a complicated exercice ! Don’t blame those who try to find it hard to do ! Of course, things are slightly different if my students don’t flip a coin… this is what we obtain if half of the students are good (2/3 probability to get a question correct) and half is not good (1/3 chance), If we look at the probability to pass, we usually do not have to wait until the end (the 50 questions) to know who passed and who failed PS : I guess a less volatile solution can be obtained with a Bayesian approach… if I find some spare time this week, I will try to code it… # Big Data and Artificial Intelligence New week, I will be in France for a few days. On Monday and Tuesday, I will be in Beaune, in Burgundy, at the first “Rencontres Mutualistes” (I will upload the slides of my talk soon). And on Wednesday, I will be in Paris, at ESCP Europe Business School. I will be giving a two hour lecture on “Big Data and Artificial Intelligence”, to use some buzzwords, as asked. More honestly, it will be on (new) data and (new) algorithms for predictive modeling. Slides are now online. # Mapping cities a French version of this article is online at http://variances.eu/ Issue 53 of Insee Analyses Ile-de-France provides an analysis of “a social mosaic specific to Paris“, with the map in Figure 1. Figure 1 : INSEE, Insee Analyses 53, 2017 This map is a priori familiar to many people, in the sense that we quickly recognize the city represented, we know how to quickly find different elements, and we know how to read the information presented, almost instinctively. In urban history, the way we saw, and how we represented the maps, has often been the basis of urban planning. Changing representation has made it possible to change the structure of cities. We will take up here the two major historical turning points, mentioned in Söderström (1996), based on two recent works: the representation of Rome at the beginning of the Renaissance, and the first iconographic plans, described in Maier (2015), and the “social” or “health” maps of London of Victorian civil servants, described in Vaughan (2018). In particular, the latter are the ancestors of zoning maps, which are widely used in urban planning, but also correspond to the majority of maps produced by statisticians and economists (the INSEE map is an example). And some maps from the last century have nothing to envy to the maps produced today, in the era of big data. # Rome, Leon Battista Alberti and Leonardo Bufalini, and the unchanging motives Choay (1980) emphasizes the fundamental role in the history of urban planning of Alberti’s De Re Aedificatoria (presented in manuscript form to Pope Nicholas V in 1452, but published only in 1485). The Alberti Treaty is indeed the first text to consider construction (Alberti prefers the term “construction” – ædificatoria – to cover both architecture and urban planning) in terms of an autonomous domain to which the rational method must be applied. The history of representation sees a turning point with the Renaissance, with figurative forms to represent urban space. We will leave the medieval aesthetic with the rediscovery of perspective, which will produce a rationalization of what can be seen, even if it often induces a partial vision of the object. In his treatise, Leon Battista Alberti proposes a scientific method governing the art of building the house, but also the entire city. But it is in Descriptio urbis Romae, probably written at the same time, that he deepened the idea of urban planning, taking the particular example of Rome. In his book, Alberti does not propose any map of Rome, but a list of instructions to be followed to create one, with the coordinate tables of several important elements of the city, natural, but also artificial. The list includes the ramparts, the river (the Tiber), the city gates, more than thirty public buildings, including the Capitol, which for Alberti is the reference point of the urban plan. He proposed to represent the city by using a disc divided into 48 portions, and by using the distance to the Capitol (in addition to a compass) to place any building. All calculations are detailed in Ludi Matematici Descriptio, using triangulation techniques. In 1450, Alberti invented the geometric plan, corresponding to what we would today call the plan of a city, even if the circular shape may surprise at first sight (see Figure 2), and does not correspond to the ichnographic plan that we all use today (obtained by horizontal and geometrical projection on a plan). Figure 2 : reconstruction of Alberti’s map in Descriptio urbis Romae, by Luigi Vagnetti in Lo studio di Roma negli scritti albertiani (1974). Source: Maier (2015). Its plan corresponds to the emergence of a new mode of representation, very geometric. But it was not until Leonardo Bufalini’s plan in 1531 that the first ichnographic plan arrived (it would be unfair to forget Imola’s plan drawn in 1503 by Leonardo da Vinci). If Alberti’s plan indicated the coordinates of a building, Bufalini decided to incorporate the ground plan of the buildings into his city plan. Figure 3 : carte de Bufalini, Roma, 1551, British Library Londres. Source : Maier (2015). But if Alberti’s plan has had such an impact, it is also because it came at the time when Pope Nicholas V launched a plan to rebuild Rome, covering an entire district, from Castel Sant’Angelo to the Vatican. This is probably the first urban planning on this scale, proposing to use the urban form as an instrument of social engineering. Alberti’s representation helped this project, with a scientific vision of the map, no longer depending on the artist’s artistic skills, or to inscribe the map in a story that would give it meaning. This urban map is self-sufficient, containing the terms of its own meaning. In Latour’s (1989) terminology, these representations that can be detached from the place (or object) they represent, “while remaining immutable so that they can be moved in all directions without further distortion, loss or corruption” correspond to immutable motives. Alberti’s map is one of the first examples of these immutable mobiles. It juxtaposes the natural and the human construction, the profane and the sacred, placing measurement and position as the only values. These plans see the urban space as a whole, not offering a single point of view, such as Jacopo Filippo Foresti’s more classic maps (for the time), for example (see Figure 4). It is possible to take Foresti’s point of view to see his map. Alberti’s map exists only as an abstract object. Figure 4 : view of Rome by Jacopo Filippo Foresti, 1490. Source: Maier (2015). If Leonardo Bufalini’s map revolutionized urban mapping, and if the iconographic plan is the dominant representation today, these maps have long remained marginal, because they were exclusively reserved for administrative, military or administrative purposes. The map of Foresti has not completely disappeared: it can be found in tourist maps, for example, which are not very concerned about proportions, simply seeking to stage monuments or to indicate itineraries. We then contrast an often local, horizontal vision (on a human scale) with a vision sometimes called zenithal which proposes to conceive objects in abstract terms. It is the latter that makes it possible to represent the city in the form of different neighbourhoods, with different levels of wealth for example, resulting in geometric plans for social statistics in Victorian times, making it possible to be the subject of census, measurement and comparison. Also noteworthy is the 1748 map of Rome created by Giambattista Nolli. Previously, Leonardo Bufalini proposed to take the point of view of an eagle, flying over the city. Nolli established the now common practice of representing entire cities from above without a single focal point, each block being considered as if the cartographer were directly above it. Figure 5 : Giambattista Nolli’s map of Rome, 1748. Source: Sylvain Mottet. # London, Thomas More and Charles Booth, and the zoning maps At the end of the 19th century (from 1870 onwards) Germany saw the first “social maps”, born in the context of an increasingly dense urban population, high social tensions and deteriorating health conditions. German planners proposed a vision of the innovative city as a living organism that needed to be made to function more efficiently. In 1876, Reinhard Baumeister in Stadterweiterungen in technischer, baupolizeilicher und wirtschaftlicher Beziehung and especially Josef Stübben in Der Städtebau, in 1890, proposed the first urban planning manuals. Thus, towards the end of the first chapter, Baumeister proposes to use an urban expansion plan, a master plan to organize the future urban space. For him, it was a question of ensuring the stability and proper functioning of a city designed as a living organism to deal with the problems it faces: overcrowding in certain districts, traffic and hygiene problems, social unrest, etc. To do this, he suggests specializing the city’s sectors in functional and social terms – what we will later call a “zoning plan” (or Bauzonenplan) – and ensuring the sustainability of this specialization. However, he warns against an overly rigid and inflexible master plan: urban development cannot be planned with too much precision, and it is therefore counterproductive to want to freeze it in a totally predetermined framework. Its plan aims to provide general guidelines necessary for the cohesion of the urban organization. In particular, he notes that the more guidelines there are, the more they will have to be the subject of local plans with a limited time horizon. While the zoning plan was not originally conceived as part of the management plan, it quickly became the key document, its clearest and most effective part. The objective was to understand, at a glance, the whole city as part of an administrative project. It is not only a question of having an overall vision of the city (which the iconographic plan already allowed) but also of using colour codes that facilitate the total regulation of this city. In particular, this zoning plan made it possible to predict several years or even decades in advance what the morphological and functional characteristics of a given area would be. In particular, it allowed investors to anticipate the future of an area and guarantee a certain return on their investments. This vision proposed by Baumeister thus made it possible to see better, for example, that the most bourgeois areas were often located in the west of the cities. This position is simply because these areas are often healthier from a health point of view: the smoke and smog produced by cities are dispersed in the upper layers of the atmosphere, and when the wind comes from the west (which happens most often in most European cities) the smoke and smog are transported eastwards and towards the lower layers of the atmosphere. From this observation, it becomes natural to build factories in the east and houses in the west. Baumeister’s work was not only theoretical: he worked on the development of the city of Frankfurt in 1891, then Berlin, Cologne, Essen, etc. In Frankfurt, he thus proposed the idea of concentric zones, which was later taken up by many economists. Figure 6 shows this form of a city, in an article published in 1925 by Ernest Burgess (who would later become one of the founders of the Chicago school). At the beginning of the First World War, all German cities had a zoning plan. And in the following years, it was the United States that adopted the concept, with New York in 1916, and more than 500 cities in 1926. In that year, zoning was officially institutionalized, with the approval of the Supreme Court. In 1933, it was the Athens Charter that recognized zoning as the main and central task of urban planning. Figure 6 : the concentric city, Burgess (1925). Source : Vaugha (2018) But in parallel with German development, where civil servants imagine the instruments of contemporary urban planning, social planning in England takes place in a context of strong social tensions. The impoverishment of a large part of the population, the many very precarious housing units, the disastrous sanitary conditions and the increase in crime in large cities have made urban development management an extremely sensitive and political subject. It is not surprising to see Patrick Geddes’ work published in Edinburgh, a biologist by training (the city is seen as a living organism) and an anarchist activist, he thought of the image and cartography as a central tool in the fight against poverty. He developed and advocated the use of statistics and mapping in land use planning and urban development, probably more than anyone else at that time. But history will remember Charles Booth’s work in London from 1886 onwards. Charles Booth, who began as a merchant and shipowner, devoted himself fully to the first social surveys at the end of the 19th century, based on a precise taxonomy of social categories. He was the first to produce social maps covering the entire urban space. His investigations focused first on the East End, London’s most deprived neighbourhood, before spreading throughout the city over more than 17 years. Its objective was to provide a scientific study of the living conditions of the London population in order to put an end to the images of deprived neighbourhoods. As he said in 1902, his objective was to establish “the numerical relation which poverty, misery and depravity bear to regular earnings and comparative comfort, and to describe the general conditions under which each class lives”. Booth’s approach was based on the creation of a statistical classification of social categories, ranging from A (the lower class) to H (the upper middle class). It has therefore created, on the basis of the notes taken in the field by the inspectors, a taxonomy that distinguishes the different sectors of the social spectrum. He estimated the number of “poor” (classes A-D) at 300,000 people in the East End and 1,300,000 for the city as a whole, almost a third of the total population at the time. The impact of the figures on the public was enormous and was reinforced by the poverty maps that were included in the results volumes dealing first with the East End and then, a few years later, with the entire city, as illustrated in Figure 7. Figure 7 : Charles Booth Map Descriptive of London Poverty, in 1898. Source : Vaughan (2018). See also https://booth.lse.ac.uk/map/ The map makes it possible to move from a social logic to a spatial logic: a particular class is translated into cartographic terms, becomes a building, a block of houses, a street, an entire urban area. The social map therefore made it possible to think of the city in terms of homogeneous spatial units. This reasoning is essential for urban planning: it could not develop in the context of the complexity of the discourse, distinguishing between the different inhabitants of the same building. This social vision of mapping, with its focus on slums and poor neighbourhoods, should be brought closer to a health objective. That said, thinking about urban development in terms of health interventions to heal society from its ills is not new. In 1516, Thomas More founded one of the main forms of urban planning theory, starting with a diagnosis of the disease and then proposing a definitive solution through a total restructuring of the urban form. During the 18th century, the translation of this principle consisted in isolating particular intervention areas (characterized by their insalubrity) and removing them, sweeping away the urban past. The solution adopted at the end of the 19th century was rather to work from what already existed, and to find the most effective solutions to manage the probable future changes in the urban context. At the end of the 19th century, we also moved from “descriptive statistics” to “prescriptive statistics”, to use Ogien’s terms (2013). We no longer simply evaluate the number of smallpox patients, we begin to make the choice to vaccinate (or not) a specific population, and therefore to set up a mandatory preventive intervention (at the time the vaccine still killed about 1 person out of 300). The homme moyen (average man) by Adolphe Quetelet will launch moral statistics, with the search for people becoming the norm, the average. Diseases are also beginning to be linked to population density, poor ventilation and humidity. “Dirty, unhealthy, infectious, corrupt or simply stinking are the categories that make it possible to think what we now call pollution” in the words of Fureix and Jarrige (2015). We then move from the social map to the “moral map”, a city thought up by hygienists. Moral geography, which until then had been the subject of partial and unsystematized observations, finds in the map a (graphical) space that synthesizes and organizes it. The social map gave the globalizing vision necessary for the existence of urban planning, and for the precise location of the sites necessary for the targeted and rational functioning of its therapeutic action. In mind is Dr. John Snow’s 1854 map of the cholera epidemic, presented (and updated) in Figure 8. At the time, the dominant theory was the theory of miasmas, claiming that diseases such as plague or cholera spread in the form of bad air. In 1854, with the help of the Reverend Henry Whitehead, by interviewing local residents, he established the geographical distribution of cases, and identified the source of the epidemic: a public water pump on Broad Street. While microbial research has not scientifically established the danger of the water pump, the mapping study of the spread of the epidemic has been sufficient to convince the authorities to close it. Figure 8 : John Snow On the Mode of Communication of Cholera, in 1855. Source : https://tabsoft.co/2y82nbf However, as Vaughan (2018) points out, similar works can be found throughout England at the same time, such as Edwin Chadwick’s Sanitary Map of the Town of Leeds, shown in Figure 9. On this map, Chadwick identifies two groups of dwellings: working class houses and shops, workhouses and artisans’ houses. Colour dots, indicating contagious diseases, only seem to proliferate in poor neighbourhoods. In particular, the map noted that the patients did not live in contiguous areas, but that they are scattered around the map, while being located in poor neighbourhoods. Figure 9 : Edwin Chadwick, Sanitary Map of the Town of Leeds, 1842. Source : Vaughan (2018) et https://bit.ly/2zL3pM8 The maps had considerable public health impacts, and the zoning, formalized by Charles Booth, was the basis for spatial statistics, as it developed throughout the 20th century. If the cartography of the city is now complex and rich, it should be noted that economists have taken a long time to leave the “linear city” model, introduced in Hotelling (1929), which has been refined over time, as shown in Figure 10, pitting the residential part (RD – residential district) against the business centre (BD – business district). But that’s another story…. Figure 10 : the different forms of the linear city. Source : Fujita & Thisse (1997). References: Booth, Charles (1902) Life and Labour in London. 17 volumes. Burgess, Ernest (1925). The Growth of the City:An Introduction to a Research Project. Choay, Françoise (1980). La règle et le modèle, Paris, Seuil. Fujita, Masahisa et Thisse, Jacques-Francois. (1997), Économie géographique, Problèmes anciens et nouvelles perspectives. Annales d’Économie et de Statistique, 45, 37-87. Fureix, Emmanuel et Jarrige, François. (2015), La modernité désenchantée : relire l’histoire du XIXe siècle français, Paris, La Découverte. Hotelling, Harold (1929). Stability in Competition. The Economic Journal, 39, 41-57. Latour, Bruno (1989). La science en action. Paris, La Découverte. Maier, Jessica (2015). Rome, measured and imagined. The University of Chicago Press. Ogien, Albert (2013). Désacraliser le chiffre dans l’évaluation du secteur public, Versailles, Éditions Quæ, Söderström, Ola (1996) Paper cities : visual thinking in urban planning. Ecumene, 3, 249-281. Vaughan, Laura (2018) Mapping Society: The Spatial Dimensions of Social Cartography. UCL Press. # Variable explicative dans un intervalle Suite a une question posée ce matin en cours, je vais faire un rapide billet pour expliquer comment extraire les valeurs inférieures et supérieures quand on a des intervalles, sous R. Commençons par générer des données, n=200 set.seed(123) X=rnorm(n) Y=2+X+rnorm(n,sd = .3) Supposons maintenant que l’on n’observe plus la vraie variable $x$ mais juste une classe (on va créer huit classes, avec chacune un huitième des observations) Q=quantile(x = X,(0:8)/8) Q[1]=Q[1]-.00001 Xcut=cut(X,breaks = Q) B=data.frame(Y=Y,X=Xcut) Par exemple, pour la premiere valeur, on a as.character(Xcut[1]) [1] "(-0.626,-0.348]" Pour extraire des informations sur ces bornes, on peut utiliser le petit code suivant, qui renvoie la borne inférieure, la borne supérieur, et le milieu de l’intervalle extraire = function(x){ ax=as.character(x) lower1 = as.numeric( sub("\$$(.+),.*", "\\1", ax) ) lower2 = as.numeric( sub("\$(.+),.*", "\\1", ax) ) upper1 = as.numeric( sub("[^,]*,([^]]*)\$", "\\1", ax) ) upper2 = as.numeric( sub("[^,]*,([^]]*)\$$", "\\1", ax) ) lower = c(lower1,lower2) lower=lower[!is.na(lower)] upper = c(upper1,upper2) upper=upper[!is.na(upper)] mid = (lower+upper)/2 return(c(lower=lower,mid=mid,upper=upper)) } On peut vérifier sur notre première observation extraire(Xcut[1]) lower mid upper -0.626 -0.487 -0.348 Juste pour voir, on peut créer trois variables supplémentaires dans notre base (avec ces trois informations) B2=Vectorize(function(i) extraire(Xcut[i]))(1:n) B$lower=B2[1,] B$mid =B2[2,] B$upper=B2[3,]
et on peut comparer 4 régressions (i) on régresse sur nos 8 classes, i.e. nos 8 facteurs (ii) on régresse sur la borne inférieure de l’intervalle, (iii) sur la valeur “moyenne” de l’intervalle (iv) sur la borne supérieure
regF=lm(Y~X,data=B) regL=lm(Y~lower,data=B) regM=lm(Y~mid,data=B) regU=lm(Y~upper,data=B)
On peut comparer les prévisions avec nos quatre modeles
plot(B$Y,predict(regF),ylim=c(0,4)) points(B$Y,predict(regM),col="red") points(B$Y,predict(regU),col="blue") points(B$Y,predict(regL),col="purple") abline(a=0,b=1,lty=2)
Pour aller plus loin, on peut aussi comparer les AIC de nos modèles,
AIC(regF) [1] 204.5653 AIC(regM) [1] 201.1201 AIC(regL) [1] 266.5246 AIC(regU) [1] 255.0687
Si l’utilisation des bornes inférieures et supérieures n’est pas concluante, ici, on notera qu’utiliser la valeur moyenne de l’intervalle donne des résultats un peu meilleurs que d’utiliser 8 facteurs.
|
2021-08-01 11:43:46
|
{"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": 2, "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": 39, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.662060558795929, "perplexity": 1395.460999837959}, "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-2021-31/segments/1627046154175.76/warc/CC-MAIN-20210801092716-20210801122716-00570.warc.gz"}
|
http://mathhelpforum.com/calculus/22309-limits-indeterminate-form-print.html
|
# limits indeterminate form
• November 8th 2007, 04:15 PM
cowboys111
limits indeterminate form
$
\displaystyle\lim_{x\to\infty}
$
(1-1/x)^5x
thank you
• November 8th 2007, 04:50 PM
Soroban
Hello, cowboys111!
I believe you are expected to know that: . $\lim_{u\to\infty}\left(1 + \frac{a}{u}\right)^u \;=\;e^a$
Quote:
$\displaystyle\lim_{x\to\infty}\left(1-\frac{1}{x}\right)^{5x}$
We have: . $\lim_{x\to\infty}\left[\left(1 + \frac{\text{-}1}{x}\right)^x\right]^5 \;=\;\left[\lim_{x\to\infty}\left(1 + \frac{\text{-}1}{x}\right)^x\right]^5\;=\;\left[e^{-x}\right]^5 \;=\;e^{-5x}
$
• November 8th 2007, 05:12 PM
Plato
I sure that it is a typo.
The limit is $e^{-5}$.
• November 8th 2007, 06:13 PM
cowboys111
ok thats easy enough i just didnt know that http://www.mathhelpforum.com/math-he...965b6bd3-1.gif
thanks for the help
|
2015-04-01 22:58:23
|
{"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": 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": 0.9855381846427917, "perplexity": 12366.958215375349}, "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/1427131309963.95/warc/CC-MAIN-20150323172149-00287-ip-10-168-14-71.ec2.internal.warc.gz"}
|
https://socratic.org/questions/how-do-you-solve-1-4-x-2-5-6
|
# How do you solve 1/4 (x+2) = 5/6?
Aug 26, 2016
$1 \frac{1}{3}$
#### Explanation:
There are many methods that can be used. Today, however, I will be showing 1 method only.
The problem: $\frac{1}{4} \left(x + 2\right) = \frac{5}{6}$
• Step 1:
$\frac{1}{4} \times \left(x + 2\right) = \frac{5}{6}$
• Step 2:
$\frac{x + 2}{4} = \frac{5}{6}$
• Step 3:
Here you will cross multiply. So it becomes:
$6 \left(x + 2\right) = 20$
• Step 4:
$6 x + 12 = 20$
• Step 5:
$6 x = 8$
• Step 6:
$x = \frac{8}{6}$
0r $x = 1 \frac{1}{3}$
|
2019-12-11 22:00:08
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "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": 0.49109357595443726, "perplexity": 13833.692262577793}, "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/1575540533401.22/warc/CC-MAIN-20191211212657-20191212000657-00379.warc.gz"}
|
https://vasishth-statistics.blogspot.nl/2015/08/observed-vs-true-statistical-power-and.html
|
## Thursday, August 27, 2015
### Observed vs True Statistical Power, and the power inflation index
People (including me) routinely estimate statistical power for future studies using a pilot study's data or a previously published study's data (or perhaps using the predictions from a computational model, such as Engelmann et al 2015).
Indeed, the author of the Replicability Index has been using observed power to determine the replicability of journal articles. His observed power estimates are HUGE (in the range of 0.75) and seem totally implausible to me, given the fact that I can hardly ever replicate my studies.
This got me thinking: Gelman and Carlin have shown that when power is low, Type M error will be high. That is, the observed effects will tend to be highly exaggerated. The issue with Type M error is easy to visualize.
Suppose that a particular study has standard error 46, and sample size 37; this implies that standard deviation is $46\times \sqrt{37}= 279$. These are representative numbers from psycholinguistic studies. Suppose also that we know that the true effect (the absolute value, say on the millisecond scale for a reading study---thanks to Fred Hasselman) is D=15. Then, we can compute Type S and Type M errors for replications of this particular study by repeatedly sampling from the true distribution.
We can visualize the exaggerated effects under low power as follows (see below): On the x-axis you see the effect magnitudes, and on the y-axis is power. The red line is the power line of 0.20, which based on my own attempts at replicating my own studies (and mostly failing), I estimate to be an upper bound of the power of experiments in psycholinguistics (this is an upper bound, I think a more common value will be closer to 0.05). All those dots below the red line are exaggerated estimates in a low power situation, and if you were to use any of those points to estimate observed power, you would get a wildly optimistic power estimate which has no bearing with reality.
What does this fact about Type M error imply for Replicability Index's calculations? It implies that if power is in fact very low, and if journals are publishing larger-than-true effect sizes (and we know that they have an incentive to do so, because editors and reviewers mistakenly think that lower p-values, i.e., bigger absolute t-values, give stronger evidence for the specific alternative hypothesis of interest), then Replicability Index is possibly hugely overestimating power, and therefore hugely overestimating replicability of results.
I came up with the idea of framing this overestimation in terms of Type M error by defining something called a power inflation index. Here is how it works. For different "true" power levels, we repeatedly sample data, and compute observed power each time. Then, for each "true" power level, we can compute the ratio of the observed power to the true power in each case. The mean of this ratio is the power inflation index, and the 95% confidence interval around it gives us an indication (sorry Richard Morey! I know I am abusing the meaning of CI here and treating it like a credible interval!) of how badly we could overestimate power from a small sample study.
Here is the code for simulating and visualizing the power inflation index:
And here is the visualization:
What we see here is that if true power is as low as 0.05 (and we can never know that it is not since we never know the true effect size!), then using observed power can lead to gross overestimates by a factor of approximately 10! So, if Replicability Index reports an observed power of 0.75, what he might actually be looking at is an inflated estimate where true power is 0.08.
In summary, we can never know true power, and if we are estimating it using observed power conditional on true power being extremely low, we are likely to hugely overestimate power.
One way to test my claim is to actually try to replicate the studies that Replicability Index predicts has high replicability. My prediction is that his estimates will be wild overestimates and most studies will not replicate.
Postscript
A further thing that worries me about Replicability Index is his sloppy definitions of statistical terms. Here is how he defines power:
"Power is defined as the long-run probability of obtaining significant results in a series of exact replication studies. For example, 50% power means that a set of 100 studies is expected to produce 50 significant results and 50 non-significant results."
[Thanks to Karthik Durvasula for correcting my statement below!]
By not defining power of a test of a null hypothesis $H_0: \mu=k$, as the probability of rejecting the null hypothesis (as a function of different alternative $\mu$ such that $\mu\neq k$) when it is false, what this definition literally means is that if I sample from any distribution, including one where $\mu=0$, the probability of obtaining a significant result under repeated sampling is the power. Which of course is completely false.
Post-Post Script
Replicability Index points out in a tweet that his post-hoc power estimation corrects for inflation. But post-hoc power corrected for inflation requires knowledge of the true power, which is what we are trying to get at in the first place. How do you deflate "observed" power when you don't know what the true power is? Maybe I am missing something.
karthik durvasula said...
>>power as the probability of rejecting the null hypothesis when it is false
In my opinion, the above suggested definition is also incorrect, and quite easily leads to thinking of hypotheses as dichotomous (a view that leads to the kinds of p-value “paradoxes”, we should be guarding against).
Power is better defined with respect to a particular discrepancy from the null hypothesis. So, power is for a particular μ’. Pow(μ’) is the probability of correctly rejecting the null hypothesis when the actual discrepancy is μ’.
By, this definition, there is no *single* power calculation with respect to a null hypotheses, although there might be a worst case scenario given what the practitioner considers to be a meaningful discrepancy.
Shravan Vasishth said...
Sure; I agree that power of course has to be computed with respect to some specific mu, and so your definition is better. There is no single power calculation with respect to the null: agreed.
Rice defines power *of a test* as the probability that the null hypothesis is rejected when it is false. I should have put in *of a test* in my definition.
Mood et al provide a nice definition in terms of a power *function*, which is the point of your comment: "Given a null hypothesis, and let Gamma be the test of a null hypothesis. The power function of a test Gamma, denoted by Pi_Gamma(theta)...is the probability that H0 is rejected when the distribution from which the sample was obtained was parameterized by theta. The Mood et al definition seems better.
It's very hard to take Mayo seriously, because she writes fairly crazy stuff (such as equating p-values with something called "actual" Type I error).
karthik durvasula said...
>> nice definition in terms of a power *function*
Yes, I agree.
Also, the following is not immediately relevant to your post, so I own't waste too much ink on it. But you say:
>>It's very hard to take Mayo seriously, because she writes fairly crazy stuff (such as equating p-values with something called "actual" Type I error).
I would be very surprised if she said something like that. Do you have a reference for this particular comment?
Shravan Vasishth said...
Karthik,
Here is the comment where Mayo says that
"it would be OK so long as they reported the actual type I error, which is the P-value."
http://errorstatistics.com/2015/07/17/statistical-significance-according-to-the-u-s-dept-of-health-and-human-services-i/#comment-127764
Also see Morey's questions to Mayo in this thread and her responses.
karthik durvasula said...
When you posted that quote, it did take me by surprise. So, I decided to read the comment thread more carefully to see if this was just a terminological issue, and not really a conceptual one. And a careful reading of Mayo's response to seems to suggest to me that she intended "actual" to be in contrast to thresholded values, and not to "nominal". She was using "actual" in one of its normal English meanings, as opposed to a slightly more technical statistical sense. She writes later:
>>"I think it was my use of the word “actual” that got you confused. It only referred to the “attained” significance level or P-value, rather than a pre-designated cut-off (see the Lehmann-Romano quote).As discussed in the post that I linked to, the error probabilities associated with tests are hypothetical."
So, I am not sure that your allegation of her conceptual misunderstandings stands, though she could have shown more care in choosing her words.
There are of course other writings where she is absolutely clear about actual and nominal p-values (e.g., here), and this is true even in her correspondence with me via email. And unlike her blogposts (sometimes), her actual published work is really clear in my opinion wrt to deep/important statistical issues.
Shravan Vasishth said...
I'm happy to give her benefit of doubt as regards her understanding of these concepts. As you have seen, I also define things inaccurately from time to time.
However, there is more about her that I think hurts her credibility. She is now attacking the replication project without even acknowledging that it's a huge step forward.
karthik durvasula said...
I haven't read her blogposts on the replication project with sufficient care to be able to comment on it. But, my prior on the issue given (what I see as) her extreme clarity on other statistical/methodological issues is to grant her the benefit of the doubt.
I also wanted to thank you for maintaining this blog; it's been good fodder for the grey cells :).
|
2018-01-16 10:58:35
|
{"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": 0.7377523183822632, "perplexity": 959.4259238870738}, "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/1516084886416.17/warc/CC-MAIN-20180116105522-20180116125522-00311.warc.gz"}
|
https://www.hpmuseum.org/forum/showthread.php?tid=7918&pid=69492&mode=threaded
|
Finding polynomials from a set of coordinates
03-11-2017, 12:53 AM (This post was last modified: 03-11-2017 12:54 AM by SlideRule.)
Post: #8
SlideRule Senior Member Posts: 1,312 Joined: Dec 2013
RE: Finding polynomials from a set of coordinates
(03-10-2017 07:39 PM)tdh79 Wrote: I need to find polynomial functions from a given set of coordinates ... How can i find the function and plot for this?
Try this URL Online Curve Fitting for a free numerical / graphical (albeit PC based rather than HP-50g based) solution(s).
BEST!
SlideRule
ps: the answer is in decimal fraction versus integer fraction form.
« Next Oldest | Next Newest »
Messages In This Thread Finding polynomials from a set of coordinates - tdh79 - 03-10-2017, 07:39 PM RE: Finding polynomials from a set of coordinates - Han - 03-10-2017, 07:42 PM RE: Finding polynomials from a set of coordinates - Juan14 - 03-10-2017, 08:44 PM RE: Finding polynomials from a set of coordinates - tdh79 - 03-12-2017, 07:56 PM RE: Finding polynomials from a set of coordinates - Han - 03-10-2017, 09:09 PM RE: Finding polynomials from a set of coordinates - tdh79 - 03-10-2017, 09:13 PM RE: Finding polynomials from a set of coordinates - Han - 03-10-2017, 09:18 PM RE: Finding polynomials from a set of coordinates - tdh79 - 03-10-2017, 09:57 PM RE: Finding polynomials from a set of coordinates - SlideRule - 03-11-2017 12:53 AM RE: Finding polynomials from a set of coordinates - grsbanks - 03-11-2017, 10:25 AM RE: Finding polynomials from a set of coordinates - SlideRule - 03-12-2017, 01:49 PM RE: Finding polynomials from a set of coordinates - Csaba Tizedes - 03-12-2017, 05:58 PM RE: Finding polynomials from a set of coordinates - Dieter - 03-12-2017, 06:38 PM RE: Finding polynomials from a set of coordinates - Csaba Tizedes - 03-12-2017, 08:27 PM RE: Finding polynomials from a set of coordinates - Dieter - 03-12-2017, 11:31 PM RE: Finding polynomials from a set of coordinates - Csaba Tizedes - 03-13-2017, 03:00 PM RE: Finding polynomials from a set of coordinates - grsbanks - 03-13-2017, 03:05 PM RE: Finding polynomials from a set of coordinates - Csaba Tizedes - 03-13-2017, 03:37 PM RE: Finding polynomials from a set of coordinates - SlideRule - 03-13-2017, 02:11 AM RE: Finding polynomials from a set of coordinates - SlideRule - 03-14-2017, 02:20 AM RE: Finding polynomials from a set of coordinates - tdh79 - 03-19-2017, 10:35 AM RE: Finding polynomials from a set of coordinates - SlideRule - 03-19-2017, 11:49 PM
User(s) browsing this thread: 2 Guest(s)
|
2021-10-20 04:46:26
|
{"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": 0.835381805896759, "perplexity": 4310.011732669827}, "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-2021-43/segments/1634323585302.56/warc/CC-MAIN-20211020024111-20211020054111-00526.warc.gz"}
|
https://plainmath.net/1043/need-to-find-a-variable-ax-plus-5-equal-6x-plus-3-solve-for-x
|
Question
# Need to find a variable. ax+5=6x+3 solve for x
Upper level algebra
Need to find a variable.
$$ax+5=6x+3$$
solve for x
2020-10-29
Step 1
Consider the given linear equation,
$$ax+5-3=6x+3-3$$
$$ax+2=6x$$
Step 2
Now, add −ax both side of the equation and solve for x,
$$-ax+ax+2=6x-ax$$
$$6x-ax=2$$
$$(6-a)x=2$$
$$x=\frac{2}{6-a}$$
Where $$a\neq 6$$ because at $$a = 6$$ the value of x is not defined.
|
2021-09-23 15:38:04
|
{"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": 0.49860361218452454, "perplexity": 2788.316202933367}, "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-2021-39/segments/1631780057424.99/warc/CC-MAIN-20210923135058-20210923165058-00343.warc.gz"}
|
https://phys.wordpress.com/category/quantum-gravity/
|
## Archive for the ‘Quantum Gravity’ Category
### Inflated claims
December 2, 2009
The Institute of Physics’ Isaac Newton medal was awarded this year to Alan Guth, of inflation fame, and his talk is available at the IoP’s website. The talk is devoted to explaining how great and awesome the inflationary theory is, including the usual reasons it’s believed by many to be correct: it explains the large-scale homogeneity of the universe (by causally connecting in the past regions that are unconnected after the inflationary period) and supposedly predicts a smoothed out geometry with an average mass density close to its critical point (that is, a universe with flat spatial sections). There’s also some talk about eternal inflation and pocket universes, which i didn’t understand well enough to comment on: this article by Guth himself is a good way to learn more.
Dr Guth is extremely happy about the fact that estimates for the ratio of the observed over the critical density ($\Omega$) have jumped during the last decade from the 0.2-0.3 range to a value almost exactly equal to 1, which corresponds to the flat model. The correction comes from taking bringing dark energy into the picture, and the second half of the talk is devoted to some speculations as to its mysterious origin. To Guth, the most plausible explanation is that this dark energy corresponds to the vacuum energy density ($\Lambda$), but there’s this little problem that quantum field theory predicts an infinite value for it. Even if you try to introduce an (arbitrary) ultraviolet cut in the calculation at the usual Plank energy, the value obtained is some 120 orders of magnitude greater than the observed one. String theory and the landscape to the rescue! Which of course explains nothing, in my opinion.
Guth’s last resort is the anthropic principle, according to which $\Lambda$ is so low because it’s the only way intelligent (?) beings would be here to observe it, and then uses an analogy so broken that i’m sure i’m missing something obvious. It goes as follows: Kepler initially thought that the radius of the orbits of planets in the solar system should have values deducible from geometrical considerations alone, but, as we all know, that’s not the case: their concrete values could be different just by changing the initial conditions leading to the formation of the solar system. So it is with $\Lambda$: since we have no way of computing its unexpectedly low value, it must be that the reason for it is that we exist. This argument is so plainly wrong that, as i said, i’m sure i’m misunderstanding (perhaps one of my two or three readers will set the record straight in the comments).
Other than that, and the fact that it feels at times like a commercial (including some arguments pretty close to straw men when he shows the curves matching the observations of the microwave background (without any mention, by the way, to the possible discrepancies for low terms of the multipolar expansion)), the talk is entertaining and gives a good, if quick, overview of commonly accepted wisdom on the field these days. So, if you have a grain of salt handy and don’t mind a bit of hand-waving, just ignore my rants and go for it.
(While you’re at it, i’ll be giving a try to last year’s winner, Anton Zeilinger, and his talk on quantum information and the foundation of quantum mechanics).
### Taking issue with String Theory
October 7, 2006
With the recent publication of Lee Smolin’s and Peter Woit’s books on the troubles of our theories of everything, every blogger in town seems to be talking about the crisis of modern fundamental physics (a.k.a. string theory and, so to speak, friends). Christine has just posted a list of recent posts on the subject. Like her, i’m reading Smolin’s book, courtesy of the publisher, and a review will eventually follow (once i find something to say about it that has not already been said!). In the meantime, i just wanted to add a few links to articles that i like on this pesky matter:
• Jim’s Stab at String Theory is a very interesting discussion by Jim Weatherall on why it doesn’t really matters whether string theory is right. There you’ll find also a video interview with Peter Woit, by John Horgan (who is not specially happy with ST, either).
• Among the free contents of the latest Physics Today issue, Burton Richter takes no prisoners when it comes to describe what’s wrong with all this super stuff.
I find Richter’s stabs, er, criticisms particularly compelling: his writing is clear and to the point, and his arguments are all but crisp and pungent. It’s curious that, by contrast, Smolin’s delicacy has actually augmented my curiosity on string theory (but i’m just halfway reading his book, so let’s better wait until i’m done).
On the other hand, i’m starting to be more and more in agreement with Weatherall’s arguments on the irrelevance of this whole business. At the very least, i’m trying to keep in mind that there’s arguably much more to fundamental physics than this debate. Maybe it’s time for some fresh air.
### Nobody expects the Strings Inquisition
June 13, 2006
The recent comments by you-know-who against the positive press that Peter Woit’s book Not Even Wrong is receiving just reminded me (by some weird association of ideas) of this epoch-making gag by Monty Python. I was about to write an entry on my recently ordering this book and planning to read it on a trip next week, but now i’m scared of confessing having bought it! (and, anyway, Christine Dantas has recently put quite nicely almost my exact feelings on this matter). Imagine, i might even like it, and got immediately classified as a crackpot and a nincompoop by the theoretical physics community!
I’m trying hard to respect string theory and theorists (and even plan on reading Zee, Weinberg and Zwiebach’s books, already waiting on my shelf). Why, i even admire several string theorists. But it would help if someone told me that Motl is not the string community’s spokesman. Or to read every now and then a bit of self-criticism from said community (something in the vein of Smolin’s comments in, say, Three Roads to Quantum Gravity). For if the (so to speak) dialectic battles between those two are to be taken as the kind of discussion we theoretical physicists favour these days, poor Monty Python are just out of business. Paraphrasing Erwin Schrodinger, i wouldn’t like it, and i would be sorry i ever had anything to do with it.
Update: Christine’s again right on the spot. And this is much, much closer to the kind of discussion i was asking for!
### Leibniz space-times
May 27, 2006
More often than not, Lee Smolin’s essays are engaging and thought provoking. I specially appreciate his willingness to tackle conceptual issues, often dismissed as philosophical or uninteresting by a great deal of the physics community (which, in my opinion, should know better). Also of note are his efforts to convey to non-specialists the key ideas and problems faced by modern physics, without unduly over-simplifications or dishonest hype.
A case in point is his recent essay The Case for Background Independence, where the meaning, virtues and drawbacks of relationist theories of quantum gravity are explored in detail. More concretely, Smolin describes the close relationship between three key issues in fundamental physics, to wit:
• Must a quantum theory of gravity be background independent, or can there can be a sensible and successful background dependent approach?
• How are the parameters of the standard models of physics and cosmology to be determined?
• Can a cosmological theory be formulated in the same language we use for descriptions of subsystems of the universe, or does the extension of physics from local to cosmological require new principles or a new formulation of quantum theory?
The article begins with a brief historical review of relationism, as understood by Leibniz and summarized in his principles of sufficient reason (there’s always a rational cause for Nature’s choices) and the identity of the indiscernible (entities with exactly the same properties are to be considered the same) [1]. These principles rule out absolute space-times (like Newton’s) or a fixed Minkowskian background (like perturbative string theory), since they single out a preferred structure ‘without reason’, as do theories posing any number of free parameters (think of the much debated landscape) [2]. As is well known, Newton won the day back in the seventeenth century, until Mach’s sharp criticism marked the resurgence of relationist ideas. Mach rejected Newtonian absolute space-time, favouring a purely relational definition of inertia [3], which ultimately would inspire Einstein in his quest for the general theory of relativity [4].
Smolin’s article continues with a careful definition, in modern terms, of relational space and time, and follows with a discussion of some current theories featuring background independence: general relativity, causal sets, loop quantum gravity, causal dynamical triangulation models and background independent approaches (by Smolin himself) to M-theory. In a nutshell, it is argued that any self-respecting relational theory should comply to three principles:
• There is no background.
• The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities.
• The relationships are not fixed, but evolve according to law. Time is nothing but changes in the relationships, and consists of nothing but their ordering.
None of the theories above passes without problems this litmus test of pure relationsm. Take for instance general relativity. To begin with the dimension, topology and differential structure of space-time are givens, and thus play the role of a background. And, on the other hand, only when we apply GR to a compact universe without boundary can we aspire to a relational view, since otherwise we would have arbitrary boundary conditions (partially) determining the structure of space-time. Once you abide to these preconditions, a proper interpretation of general covariance (in which you identify space-times related by arbitrary coordinate transformations) provides a relational description of space-time (for an in-depth discussion of the subtle interplay between gauge invariance and relationsm, see also this excellent article by Lusanna and Pari, and references therein). As a second example, loop quantum gravity is also background dependent: in this case, the topological space containing the spin-networks of the theory. Other than that, loops are an almost paradigmatic case of a relational description in terms of graphs, with nodes being the entities and edges representing their relationships.
After his review of quantum gravity theories, Smolin takes issue with string theory. His subsequent train of thought heavily relies on the fact that relationism, or, more concretely, Leibniz’s principle of the indiscernible, rules out space-times with global symmetries. For if we cannot distinguish this universe from one moved 10 feet to the left, we must identify the two situations, i.e., deny any meaning or reality to the underlying, symmetric structure. But, as is happens, the M-theory programme consists, broadly speaking, in maximizing the symmetry groups of the theories embodied in the desired unified description. More concretely, in background-dependent theories, the properties of elemental entities are described in terms of representations of symmetries of the background’s vacuum state. Each of the five string theories embodied by M-string (should it exist!) has its own vacuum, related with each other via duality transformations (basically, compactifying spatial dimensions one way or the other one is able to jump from one string theory to the next). Thus, M-theory should be background independent (i.e., encompass different backgrounds), but, on the other hand, one expects that the unique unified theory will have the largest possible symmetry group consistent with the basic principles of physics, such as quantum theory and relativity. Smolin discusses some possible solutions this contradiction (which a lack, er, background to comment intelligently), including some sort of (as yet unknown) dynamical mechanism for spontaneous symmetry breaking (which would result in a Leibniz-compliant explanation for the actual properties–such as masses and coupling constants–that we find in our universe).
After all the fuss, there is disappointingly little to be said about relationist unified theories [5]. Invoking again the principle of the indiscernible, Smolin rules out symmetries that would make (unified) identities undistinguishable (if two entities have the same relationships with the rest, they are the same entity). By the same token, a universe in thermal equilibrium is out of the question. Reassuringly, our universe is not, and the negative specific heat of gravitationally bound systems precludes its evolution to such an state. The case is then made (after casting evolution theory as a relationist one, which is OK by me) for Smolin’s peculiar idea of cosmological natural selection. To my view, it is an overly speculative idea, if only for the fact that it depends on black holes giving rise to new universes when they collapse [6]. If that were the case, and provided that each new universe is created with random values for the free parameters of our theories, one would expect that a process similar to natural selection would lead to universes with its parameters tuned to favour a higher and higher number of black-holes (which seems to be the case in our universe). Nice as the idea is, i think we’re a little far from real physics here.
The article closes with a short section on the cosmological constant problem (with the interesting observation than only casual set theory has predicted so far a realistic value) and relational approaches to (cosmological) quantum theory. Again, the author adheres to non-orthodox ideas. This time, to recent proposals (see here and here) of hidden-variable theories, albeit they have far better grounds than the reproducing universes idea. The possibility of a relational hidden-variable theory is argued for with a simple and somewhat compelling line of thought. In classical physics, the phase space of a system of N particles is described by a 6N variables, while a quantum mechanical state vector would depend on 3N variables. On the other hand, in a purely relational theory one would need to use N^2 variables, as these are the number of possible relations between N particles. These would be the hidden-variables completely (and non-locally) describing our particles, which would need statistical laws when using just 3N parameters.
An amazing journey, by all accounts.
[1] See here for excellent (and free) editions of all relevant Leibniz works, including his Monadology, and here for commented excerpts of the Leibniz-Clarke correspondence.
[2] See also here for an interesting take on Leibniz’s principle under the light of Gödel’s and Turing’s incompleteness theorems as further developed by Gregory Chaitin.
[3] Julian Barbour’s “The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories” is the definitive reference to know more about the history of the absolute/relative divide. (Another amazing book by Barbour on these issues is “The End of Time : The Next Revolution in Physics”, thoroughly reviewed by Soshichi Uchii here. Smolin himself has many an interesting thing to say about Barbour’s timeless Platonia.)
[4] Barbour argues in his book that Einstein seems to have misunderstood Mach’s discussions on the concept of inertia, taking it for the dynamical quantity entering Newton’s second law instead of the inertial motion caused by space-time according to Newton’s first law.
[5] I’m also a bit surprised by Smolin’s uncritical acceptance of reductionism, which he simply considers, “to a certain degree”, as common-sense.
[6] Tellingly, the only reference where this theory is developed is Smolin’s popular science book “The Life of the Cosmos”.
Technorati Tags: , , ,
### Connes and quantum statistics
May 22, 2006
I just noticed (hat tip Not even wrong) that Alain Connes has made available his book on non-commutative geometry, one of the third roads to quantum gravity. Not an easy reading, by any account, but surely an interesting one. Not only that. He’s also making available (in his downloads page) most of his recent articles and lecture notes, which make for an impressive list. A very interesting and enjoyable reading i’ve found there is Connes’ highly original View of Mathematics (PDF), which makes for a good introduction to NCG. And there is also a brief essay, Advice to the beginner, where, besides guidelines to young mathematicians, Connes gives his particular view of the physics community:
I was asked to write some advice for young mathematicians. The first observation is that each mathematician is a special case, and in general mathematicians tend to behave like “fermions” i.e. avoid working in areas which are too trendy whereas physicists behave a lot more like “bosons” which coalesce in large packs and are often “overselling” their doings, an attitude which mathematicians despise.
The bit about overselling rings a bell, doesn’t it?
Update: Also of note is this interview with Alain Connes (PDF), mentioned, again, over at Not Even Wrong:
The interview also contains quite a few amusing stories. In one of them Connes tells about a well-known string theorist who walked out of his talk at Chicago because he wasn’t very interested, but two years later was paying rapt attention to the same talk when Connes gave it at Oxford. When Connes asked him about this, the physicist told him that the difference was that in the meantime he had heard that Witten had been seen reading Connes’s book in the library at Princeton.
### Categorical spacetime
May 17, 2006
John Baez is preparing a colloquium at the Perimeter Institute where the application of Category Theory (CT) to quantum spacetime will be discussed. This page contains links to (a few) introductory and (many) advanced papers on how the ideas of CT (and its generalization to n-categories) can be used to analyze the category of Hilbert spaces.
I’ve developed a lively interest in CT during my computer science wanderings , and even wrote an elementary introduction (geared to programmers) that you may find useful (if you’re new to the field and don’t mind digressions into computer programming).
Categories are deceptively simple: they abstract the notion of sets and their morphisms, or, if you prefer, of objects and their transformations. Thus, a category is defined as a set of objects related by arrows that are composable (in the functional sense). Composition is associative and there’s an identity arrow for each object in the category. As you can easily see, virtually any mathematical (and, by extension, physical) structure can be modeled as a category. For instance, your objects can be groups, with arrows representing group morphisms. Or you could model any discrete physical system’s evolution, by taking as objects its states and as arrows its transitions. With the latter example, we begin to see how CT relates to physics: think of arrows as processes or object transformations. CT somehow captures their essence: one moves from the basic category definition to an exploration of morphisms (called functors) between categories, and from here to constructing categories whose objects are categories, with functors as arrows. Iterate and you’re soon talking about processes consisting of processes consisting of processes consisting of… But of course that’s not all: one also studies morphisms between functors (called natural transformations), and obtains a very precise statement of how any object is equivalent to the set its transformations: objects just disappear on behalf of their relationships! This result, called Yoneda’s Lemma, is beautifully presented in Barry Mazur’s When is a thing equal to some other thing? and further explored in Brown and Porter’s Category Theory: an abstract setting for analogy and comparison (both of them being also an excellent introduction to CT). Although i’m not privy with the applications of CT to LQG and other ‘relationist’ approaches, i think that this blurring of objects in favor of their transformations are at the heart of CT’s appeal to some physicists.
Even discounting its (highly tentative) applications to physics, CT raises deep issues in pure mathematics and philosophy: for instance, this very interesting entry of the Stanford Encyclopedia of Philosophy ends its introduction with these words:
Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth. It can be applied to the study of logical systems in which case category theory is called “categorical doctrines” at the syntactic, proof-theoretic, and semantic levels. Category theory is an alternative to set theory as a foundation for mathematics. As such, it raises many issues about mathematical ontology and epistemology. Category theory thus affords philosophers and logicians much to use and reflect upon.
All that said, i’m a bit skeptical on the usefulness of CT as a means of synthesizing new laws. For what i’ve seen, it is very powerful for analyzing and discovering properties of already known structures, but i have yet to find a convincing case where it is used to reveal brand new laws. For instance, CT is used in Axiomatic QFT to formalize the theory axioms (see, e.g., Buchholz’s Algebraic quantum field theory: a status report or any other of the review papers on Local Quantum Physics), but it’s not clear to me whether CT is, in this context, more than a convenient mathematical language. Let me, however, rush to say that i’m by no means an expert, and that reading Baez’s writings above may well change my mind!
At any rate, i do like Category Theory, if only for its mathematical beauty, and i think it’s very worthwhile spending a little time learning a bit about it if you are interested in mathematical physics (see here for more of my recommended readings).
Update: Over at A Neighborhood of Infinity, there’s a discussion about “two kinds of maths” (namely, structural and content-providing) which somehow coincides with my half-baked feelings about CT (and, as a bonus, provides a link to yet another proof of Yoneda’s lemma):
Most branches of mathematics have a mixture of the two types of theorem. Typically the structure theorems are used as a tool to discover content. In some sense the content is the end and the structure is the means. But category theory seems different in this regard – it seems to be mainly about structure. Every time I read category theory I see ever more ingenious and abstract tools but I don’t really see what I think of as content. What’s hard in category theory is getting your head around the definitions and often the theorems are trivial consequences of these. For example consider the Yoneda lemma as described here. This is a pretty challenging result for beginning category theorists. And yet these notes say: “once you have thoroughly understood the statement, you should find the proof straightforward”. This exactly fits with what I’m getting at: the theorem is an immediate consequence of the definitions and the proof essentially shows that the machinery does what you’d expect it to do.
Technorati Tags: , , ,
### Boldness
May 15, 2006
This morning i’ve been a bit surprised by the number of new articles in the arXiv.org feed claiming what would be very significant advancements in our understanding of quantum gravity and related issues. Not that i’ve had the time to read them, or that i’ve got the expertise to quickly sift wheat from chaff: i’m just listing them here for those of you with better criteria (with the hope of reading some insightful comment):
Too good to be true, right?
### A problem of hierarchy
May 10, 2006
One of the many puzzles (a.k.a. Mysteries of Life) faced by modern theoretical physics is the so-called hierarchy problem: when one compares [1] the relative strength of the four fundamental forces, two widely separated scales are evident:
Interaction Coupling constant Strong 1 Electromagnetic 1/137 Weak 1/10^6 Gravitational 1/10^39
Or, as Lisa Randall puts it in this interview:
The gist of it is that the universe seems to have two entirely different mass scales, and we don’t understand why they are so different. There’s what’s called the Planck scale, which is associated with gravitational interactions. It’s a huge mass scale, but because gravitational forces are proportional to one over the mass squared, that means gravity is a very weak interaction. In units of GeV, which is how we measure masses, the Planck scale is 10 to the 19th GeV. Then there’s the electroweak scale, which sets the masses for the W and Z bosons. These are particles that are similar to the photons of electromagnetism and which we have observed and studied well. They have a mass of about 100 GeV. So the hierarchy problem, in its simplest manifestation, is how can you have these particles be so light when the other scale is so big.
As you probably know, Randall’s response to this conundrum implies a long detour through multiple dimensions, as recently reviewed over at Backreaction, which was predated by a proposal by Arkani-Hamed, Dimopoulos and Dvali, nicely explained for the rest of us in this Physics Today article. (As a warmup for higher-dimensional physics, you may find entertaining this recent pedagogical review of Kaluza-Klein theories.)
An alternative solution has been put forward by the supersymmetry proponents. As explained (hyped?) in this beatiful review of particle physics:
According to supersymmetry, every “ordinary” particle has a companion particle — differing in spin by half a unit, but with otherwise identical properties. Furthermore, the strengths of the interactions of the superpartners are identical to those of the corresponding ordinary particle. Supersymmetry so simplifies the mathematics of quantum field theory and String Theory that it allows theoriests to obtain solutions that would otherwise be far beyond their calculating ability.
For reasons too complex to explain here (even if i really understood them: see here and here for some of the nitty-gritty details), supersymmetry is claimed to lead to a unification of fundamental forces at very high energies (some 10^28K, or 10^{-39} seconds after the Big Bang), somehow making natural the wild differences in scale of the (energy-dependent) coupling constants in our current universe. As mentioned, the theory also makes easier to define renormalizable QFTs (due to some magical cancellations), and has become one of the main ingredients of String theory, although there is at least another extension to the standard model of particle physics that seem to share these magic cancellation virtues, solving in the process the hierarchy problem: this Physics World article gives an introduction to this so-called ‘little Higgs’ theory.
Personally, i find all these untestable super-theories and multiple dimensions rather unconvincing, and would prefer some old good 4-dimensional solution. Alas, no one seems to be avaialable… maybe it’s time to embrace Compactified Dementia and be done with that.
[1] The excellent little comparison of coupling constants for the fundamental forces pointed to by the above link is part of a nifty site called HyperPhysics, an amusing experiment combining HyperCards, Javascript experiments and similar online tricks with well written contents. Visit it for fun, hierarchy problem or not.
Technorati Tags: , , ,
### Arxiv Structure
May 7, 2006
I bet this is old hat for many of you, but just in case: i’ve discovered a new, nifty way of searching and browsing arxiv.org papers: Xstructure. The exciting part is browsing: articles are classified by theme and can be viewed in a variety of trees and listings. For instance, here‘s the entry page for the recently added gr-qc archive: there you’ll find submission statistics and some useful links, including Review Articles and Authority Articles, which lists the most cited ones. Interestingly, comparing the authority articles in gr-qc with those of hep-th clearly shows what we could call a quantum gravity divide: the former consists almost exclusively of papers on Loop Quantum Gravity (Smolin, Rovelli, Ashtekar and friends), while the latter is monopolized by the String and M-Theory guys (Witten, Polchinski, Randall…). Hardly surprising, i know, but still…
April 15, 2006
|
2018-12-10 04:43:44
|
{"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": 4, "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": 0.5643165707588196, "perplexity": 954.8009753873491}, "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-2018-51/segments/1544376823303.28/warc/CC-MAIN-20181210034333-20181210055833-00437.warc.gz"}
|
http://www.pifpafpuf.de/EGiaN/I.5.3.html
|
genug Unfug.
2013-08-17
# Contents
## Exercise
Given the coordinate transformation \begin{align} \label{trafo} x &= \frac{W}{2\pi}\phi\\ y &= - \frac{W}{2\pi} \log\tan(\theta/2), \label{eqy} \end{align} we are asked to prove that $$ds^2 = \Omega^2(x,y)(dx^2+dy^2)$$ and to find out $\Omega$. Further we know how $ds$ looks like in terms of $\theta$ and $\phi$: $$ds^2=d\theta^2 + \sin^2(\theta) d\phi^2$$
## Solution
We will solve for $d\theta$ and $d\phi$ and insert the result into this last equation. The easy one is $$d\phi=\frac{2\pi}{W}dx.$$ Now we differentiate the tangent first as the quotient of sine and cosine. \begin{align} \frac{d}{dq}\tan(q) &= \frac{d}{dq}\frac{\sin(q)}{\cos(q)}\\ &= \frac{\cos^2(q) + \sin^2(q)}{\cos^2(q)}\\ &= \frac{1}{\cos^2(q)} \end{align} With the logarithm in front, the whole thing looks like \begin{align} \frac{d}{dq} \log(\tan(q)) &= \frac{1}{\tan(q)}\frac{1}{\cos^2(q)}\\ &= \frac{\cos(q)}{\sin(q)}\frac{1}{\cos^2(q)}\\ &= \frac{1}{\sin(q)\cos(q)} \end{align} But we actually have $q=\theta/2$, which requires a factor $1/2$ that we shall not forget when we now differentiate the coordinate transform for $y$ of equation \ref{eqy}: \begin{align} dy &=-\frac{W}{2\pi}\frac{1}{\sin(\theta/2)\cos(\theta/2)}\frac{1}{2} d\theta \end{align} Looking up the expression involving the half angle in a collection of formulas, we find $$2 \sin(\theta/2)\cos(\theta/2) = \sin(\theta).$$ Inserting into the last equation and solving for $d\theta$ resuls in $$d\theta = -\frac{2\pi}{W}\sin(\theta) dy.$$ Plugging all this into the given equation for $ds^2$ gets us: \begin{align} ds^2 &=d\theta^2 + \sin^2(\theta) d\phi^2\\ &= (\frac{2\pi}{W}\sin(\theta) dy)^2 + \sin(\theta)^2 (\frac{2\pi}{W}dx)^2\\ &= (\frac{2\pi}{W}\sin(\theta))^2 (dx^2 + dy^2). \end{align} From the last line we see that $\Omega = \frac{2\pi}{W}\sin(\theta)$, so we are nearly there, except that we need to express the $\theta$ in terms of $x$ and $y$.
So far we found $\Omega = \frac{2\pi}{W} \sin(\theta)$ and we also know that $$y=-\frac{W}{2\pi} \log\tan(\theta/2)\,. \label{eqy2}$$ With the shortcut $b=\frac{2\pi}{W}$, \ref{eqy2} can be rearranged into $$\theta = 2 \arctan(e^{-y b}).$$ Interestingly, when I ask Wolframalpha for the sine of this, I get $$\sin(\theta)=\frac{2 e^{-y b}}{e^{-2y b}+1}.$$ For now I believe it, so that we end up with $$\Omega(x,y)=b \frac{2 e^{-y b}}{e^{-2y b}+1}$$ with $b$ as above. Why Zee keeps the dependency on $x$, I don't know. Too bad this does not look like the rather terse solution $$\Omega = \frac{b}{\cosh(b y)}$$ given in the solution section of the book. But lets insert the definition $\cosh(x) = 1/2(e^x+e^{-x})$ to get \begin{align} \Omega &= \frac{2b}{e^{by}+e^{-by}}\\ &=\frac{2b}{e^{by}+e^{-by}} \frac{e^{-by}}{e^{-by}}\\ &=\frac{2b e^{-by}}{e^{-2by}+1}, \end{align} which is indeed the solution we got. (sigh)
|
2018-08-15 16:48:03
|
{"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": 6, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 1.000004768371582, "perplexity": 627.6187557741845}, "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-2018-34/segments/1534221210243.28/warc/CC-MAIN-20180815161419-20180815181419-00688.warc.gz"}
|
https://socratic.org/questions/is-xy-1-5-a-direct-variation-inverse-variation-joint-or-neither
|
# Is xy= 1/5 a direct variation, inverse variation, joint or neither?
Given $x y = \frac{1}{5}$
multiplying $x$ by a factor of $k$
results in the need to divide $y$ by $k$ to maintain the equality.
|
2019-09-18 11:31:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "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": 0.4727562367916107, "perplexity": 859.5858453954297}, "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/1568514573284.48/warc/CC-MAIN-20190918110932-20190918132932-00341.warc.gz"}
|
https://yizhang82.dev/2/
|
# Hacking MySQL #1 - Overview, Building, and Testing
An overview of MySQL, as well as how to obtain source, build and run tests
## Overview
MySQL is one of the most widely used OpenSource relational databases and is used by many companies such as Amazon, Facebook, Google, Alibaba, etc. In my current job we deploy MySQL widely within the company, we had our MySQL 5.6 own fork and moving towards MySQL 8.0 currently in a branch. We also have an “new” storage engine built on top of RocksDB, not surprisingly called MyRocks, which lives under storage/rocksdb folder in the MySQL 5.6 fork.
On a 10000-feet view, the architecture of MySQL server looks like this:
1. Connection Management / Authentication
2. Table/Schema Management/Caching
3. SQL Parser
4. SQL Optimizer and Query Executioner
5. Execution Engine
6. Replication and logging
If you dive deeper, an execution engine itself could include following pieces:
1. Transaction / MVCC / Locking / Snapshot support
3. In-memory core database data structure (B+ tree / LSM tree / etc) and operation (insert/delete/update) for records
4. Indexing data structures, and searching/updating
5. Logging, Checkpointing & Recovery
6. Database storage persistence
7. Caching (disk blocks/pages, etc)
One of the most amazing features in MySQL is to swap the underlying storage engine while keeping the same upper layers - this way you can have different in-memory and on-disk representation of databases for different workloads, while keeping the same SQL execution functionality so the client wouldn’t even know the underlying storage engine have changed. The default storage engine is InnoDB - a B+ tree based database storage engine, and the one that we have is MyRocks which is built on top of RocksDB, a LSM-tree based database storage engine. There is an API layer called handler that storage engine need to implement/override. You can go to Comparison of MySQL database engines to see a list of common storage engines in MySQL.
Of course, the statement that they wouldn’t know the storage engine has changed is not entirely accurate. There are specific configurations you might need to tune / config the storage engine to your needs, and different storage engine has different performance / behavior / features / capabilities, so it’s not completely transparent.
## Building
You can obtain source code from MySQL website, but most folks probably would prefer a github mirror:
git clone https://github.com/mysql/mysql-server
This contains the latest MySQL 8.0.16.
In a typical Ubuntu system, you need to install following dependencies:
sudo apt install libssl-dev libzstd-dev libncurses5-dev libreadline-dev bison pkg-config
All my instructions below are tested on a Azure Linux Ubuntu 18.04 VM and on a MacBook Pro 2018. They may vary slightly due to your configuration/distribution if you are on a unix/linux system. Getting it to work on Windows requires installing OpenSSL binaries and GNU Bison. If you are using latest Visual Studio 2019, you may also need to apply a fix to boost 1.69.0 for a outdated VC workaround (a workaround for a workaround, essentially). Fortunately in most cases MySQL is pretty good about telling you exactly what is missing and where to download them.
Now let’s create a debug directory to store all our build files, and start the debug build:
mkdir debug
cd debug
make
1. WITH_DEBUG=1 requests a debug build, which makes debugger easier
2. DOWNLOAD_BOOST=1 WITH_BOOST=~/boost_1_69_0 downloads the boost at ~/boost_1_69_0 (that’s the version MySQL is asking for), and will skip the downloading if it is already there
One the build is done, you can find everything under debug/bin.
Don’t change the build directory after the fact once you done the build. The directory name is remembered and changing that naming requires a rebuild.
## Running a test
To validate that we indeed have a working MySQL build, let’s try running a quick test called select_all.
To run any test, there is a script mysql-test-run.pl located under the mysql-test directory from the build directory, and it takes a test name in the form of <testname> or <testsuite>.<testname>:
cd debug/mysql-test
./mysql-test-run.pl select_all
This runs the test under mysql-test/t/select_all.test with baseline mysql-test/r/select_all.result. It runs a simple test language containing test directives/commands and SQL commands, and compare the output with the baseline. If the output diverges from the baseline the test would fail, otherwise it would pass. Simple enough, right?
Actually, not quite. The testing of MySQL can get quite complicated when it involves multiple connections / servers communicating with each other. And stablizing the results so that they are not affected by external environment / code changes can be also an headache.
Here is what you should see:
[~/local/github/mysql-server/debug/mysql-test, 8.0, 51s, SUDO]: ./mysql-test-run.pl select_all
Logging: /home/yzha/local/github/mysql-server/mysql-test/mysql-test-run.pl select_all
MySQL Version 8.0.16
Checking supported features
- Binaries are debug compiled
Using 'all' suites
Collecting tests
Checking leftover processes
Removing old var directory
Creating var directory '/home/yzha/local/github/mysql-server/debug/mysql-test/var'
Installing system database
Using parallel: 1
==============================================================================
TEST NAME RESULT TIME (ms) COMMENT
------------------------------------------------------------------------------
[100%] main.select_all [ pass ] 36259
------------------------------------------------------------------------------
The servers were restarted 0 times
The servers were reinitialized 0 times
Spent 36.259 of 70 seconds executing testcases
## Launching and connecting
Running a test seems straight-forward enough. If you want to launch mysql server and run some SQL commands against it, it takes a bit of work.
First we need to have mysqld initializes a blank data directory:
cd debug/bin
./mysqld --initialize
In Windows you’ll need to add --console to write error output to screen. Otherwise it’s only available in the .err log file.
You should see:
2019-06-05T05:16:31.376510Z 0 [System] [MY-013169] [Server] /datadrive/github/mysql-server/debug/runtime_output_directory/mysqld (mysqld 8.0.16-debug) initializing of server in progress as process 70030
2019-06-05T05:16:44.066787Z 5 [Note] [MY-010454] [Server] A temporary password is generated for [email protected]: <.....>
2019-06-05T05:16:53.317610Z 0 [System] [MY-013170] [Server] /datadrive/github/mysql-server/debug/runtime_output_directory/mysqld (mysqld 8.0.16-debug) initializing of server has completed
Note the temporary password generated in the second line. You’ll need it later.
This means that mysqld has successfully initialized at debug/data directory:
'#innodb_temp' auto.cnf ca-key.pem ca.pem client-cert.pem client-key.pem ib_buffer_pool ib_logfile0 ib_logfile1 ibdata1 mysql mysql.ibd performance_schema private_key.pem public_key.pem server-cert.pem server-key.pem sys undo_001 undo_002
Now we can finally start the server:
cd debug/bin
./mysqld --debug
In Windows you’ll need to add --console to write error output to screen. Otherwise it’s only available in the .err log file.
--debug switch means we start the mysql server in debug mode.
Now launch another terminal / TMUX window / whatever, and connect to the mysql server:
cd debug/bin
./mysql -uroot --socket=/tmp/mysql.sock -p
You should see:
Enter password:
Welcome to the MySQL monitor. Commands end with ; or \g.
Your MySQL connection id is 28
Server version: 8.0.16-debug Source distribution
Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.
Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.
However we are not quite done yet. Mysql will ask us to change the password - you can do it by using the following:
mysql> ALTER USER 'root'@'localhost' IDENTIFIED BY "<newpassword>";
Now any future login can be done using this new password you just gave.
Finally we can run some SQL command!
mysql> SELECT @@version;
+--------------+
| @@version |
+--------------+
| 8.0.16-debug |
+--------------+
1 row in set (0.00 sec)
Before I leave, let me address a question that is absolutely going to be asked - how do I terminate the server gracefully? CTRL+C doesn’t work anymore.
The right way is to use mysqladmin:
cd debug/bin
You’ll see the server waving goodbye:
2019-06-05T05:54:09.028071Z 30 [System] [MY-013172] [Server] Received SHUTDOWN from user root. Shutting down mysqld (Version: 8.0.16-debug).
2019-06-05T05:54:10.670124Z 0 [System] [MY-010910] [Server] /datadrive/github/mysql-server/debug/runtime_output_directory/mysqld: Shutdown complete (mysqld 8.0.16-debug) Source distribution
## Debugging
In Linux/Mac debugging is relatively straightforward and isn’t any different from other applications. My personal recommendation is to use Visual Studio Code and setup lldb/gdb debugging there.
For Windows, the obvious choice is Visual Studio (2019 Community Edition is the one I’ve tested on). However it looks like mysqld is launching another mysqld instance that does the real work, so F5 debugging that mysqld from Visual Studio requires Child Process Debugging Power Tool in order for your breakpoints to hit since they need to be set in the child mysqld process, not the parent. Of course attaching to the correct mysqld would always work regardless without the help of the Child Process Debugging Power Tool.
## What’s next
I’m planning a series of articles that will go through many interesting aspects of MySQL:
1. A quick tour of the source code and important concepts in MySQL source
2. How is the parsing and AST tree generation done for complex statements
3. How are statement being executed in MySQL
5. How does MySQL optimizer / query execution work
6. How does plugin / storage engine work
7. How does system variables work
8. How does replication work
9. How does SHOW command work
10. How does binlog work
I’m also planning to write about MyRocks, as well as RocksDB / LevelDB / InnoDB, but I’ll priorize MySQL articles first as they lay down a nice foundation for rest of the stuff and this also serves as documentation when people get lost in the vast amount of MySQL source code.
Let me know what do you think about the article and/or if you are running into issues. Feel free to suggest topics as well. But I probably can’t help much with your DBA questions…
# Getting GDB to work on Mac OS X Mojave
How to get GDB to work on MacOSX Mojave
Starting from Mac OS X 10.5 (Leopard), Apple starts to lock down the system further and debuggers like GDB now have to be code signed. There is a great article describing steps to get it to work. However, there are a lot of conflicting information on the web and people are having trouble with some of those instructions, myself included. So I’d like to document what I did to get it to work, and highlight the issues I ran into:
On a high-level you need to perform these steps:
1. You need to create a certificate in System Keychain that is self-sign and always trust for code signing
2. Sign the GDB binary with the certificate. Include proper entitlements if you are on 10.14+.
3. Reboot
The article has detailed steps on these steps so I’m not going to repeat them.
A few gotchas that I ran into myself:
1. If you see this error complaining about code signing even though you had signed the GDB executable:
Starting program: /Users/yzha/github/mysql-server/debug/runtime_output_directory/mysqld
Unable to find Mach task port for process-id 55009: (os/kern) failure (0x5).
Double check if you had the proper entitlements in a XML file and pass to codesign when you are signing GDB. Many articles on the web in fact didn’t have the entitlement step as it likely is a new requirement 10.14+.
1. If you are seeing this error even if you had signed with proper entitlements:
During startup program terminated with signal ?, Unknown signal.
or this:
/Users/yizhang/github/leveldb/debug/db_test": not in executable format: file format not recognized
Make sure you stay off GDB 8.2! Upgrade to 8.3 (available in Homebrew already) or downgrade to 8.0.
# Sorting structured data using memcmp-friendly encoding part 2 - floats
Sorting structured data using memcmp-friendly encoding part 2 - sorting floats
In the last post we’ve discussed converting integers and strings into a memcmp / byte-comparable format for faster comparison (but at the trade off of doing decoding/encoding at reading/writing). In this post let’s take a look at how do we do the same for floating pointers.
# Get cherry-pick to work across file renames
Making cherry-pick work across file renames
Recently I need to port over some changes using cherry-pick and that usually works fine without any issues (except for occasional conflicts), but this time the actual file foo.cc was renamed to bar.cc. In such case git cherry-pick simply gives up and simply tells you the old file you are changing has been deleted. As far as I can tell there isn’t a good way to resolve the conflict.
There are a couple of ways to address this issue. But the easiest way I found is to just rename the file back to the original name where you had made the change on, in order to make git happy. Once that’s done, cherry-picking would work fine as usual. Now just rename the file back to the ‘new’ name. Squash the change.
This can be illustrated in following example - assuming:
2. In the target branch (that you want to cherry-pick) renames foo.cc to bar.cc
# Create the target branch as usual
git checkout -b your-target-branch
# Rename bar.cc back to foo.cc to make git cherry-pick happy
git mv bar.cc foo.cc
git commit -m "Make git happy"
# Cherry-pick as usual
git cherry-pick -x <commit>
# Rename it back
git mv foo.cc bar.cc
git commit -m "Rename back"
# Squash the 3 commits into one
In the rebase file, you’ll see:
pick 95be80db682 Make git happy
pick 3d74c6c9e13 Cherry-pick commit blah
pick 238e3c51354 Rename back
Change to:
pick 95be80db682 Make git happy
s 3d74c6c9e13 Cherry-pick commit blah
s 238e3c51354 Rename back
Here s means squash with previous commit.
Just remember in commit message deleting the first and third unrelated commit.
And now you are all set!
# Repeatable reads in InnoDB comes with a catch
A few days ago I was looking into a deadlock issue that is caused by a behavioral difference between MySQL storage engine transaction behavior in repeatable reads. This leads me to dig deeper into repeatable read behavior in InnoDB and what I found is quite interesting:
## The basics
Before we dig deeper, let’s revisit some of the basics of database isolation levels. You can refer to my earlier post for a more detailed explanation / comparison. Database isolation level defines the behavior of data read/write operations within transactions, and those can have a signficant impact to protecting the data integrity of your application. Repeatable reads guaratees that you would always observe the same value once you read it, and it would never change unless you’ve made the change yourself, giving you the illusion that it is exclusively owned by you and there is no one else. Of course, this isn’t true in practice as there are pessimistic locking and optimistic locking that defines the behavior when write conflict occurs.
# Diagnosing interesting MySQL client connection error in localhost through the source code
The art of argument parsing and policy transparency
When working with MySQL the often most frustrating part is getting strange connection errors. I’ve wasted two hours trying to connect to a MySQL server using TCP port (unix domain sockets works fine) and I’ll talk about why it didn’t work, and as usual we’ll dive into the code to understand exactly why.
To simplify the problem, let’s say I have MySQL server at port 13010 and bound to localhost, with user name root and empty password (don’t do that in production):
[~/mysql]: mysql -p 13010 -h localhost -u root
ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2)
This is typical error many people will run into and you can find many similar posts that discuss the problem but few ever got to the bottom of it. Let’s jump right in.
## -p and -P
Obviously when I write -p 13010 I meant to tell mysql client to connect to server using port 13010, but that’s not quite right:
[~/mysql]: mysql --help
-P, --port=# Port number to use for connection or 0 for default
So I actually told mysql the password is 13010 instead. Supporting both -p and -P is a apparently very bad idea.
Linux tools often have excessive amount of short options, like this one from man page for ls:
ls [[email protected]] [file …]
Personally I think they should go easy and only include the most common ones rather than using the entire alphabet.
However, the mystery is not yet solved. Note that we have been asked to enter the password, which explains why most people never suspected -p actually means password. Put in other words - if -p means password, why is this command is still asking for password?
The answer lies in the source code:
my_getopt.cc
for (optend= cur_arg; *optend; optend++)
{
opt_found= 0;
for (optp= longopts; optp->name; optp++)
{
if (optp->id && optp->id == (int) (uchar) *optend)
{
/* Option recognized. Find next what to do with it */
opt_found= 1;
if (optp->arg_type == REQUIRED_ARG ||
optp->arg_type == OPT_ARG)
{
if (*(optend + 1))
{
/* The rest of the option is option argument */
argument= optend + 1;
/* This is in effect a jump out of the outer loop */
optend= (char*) " ";
}
else
{
if (optp->arg_type == OPT_ARG)
{
if (optp->var_type == GET_BOOL)
*((my_bool*) optp->value)= (my_bool) 1;
if (get_one_option && get_one_option(optp->id, optp, argument))
return EXIT_UNSPECIFIED_ERROR;
continue;
}
/* Check if there are more arguments after this one */
argument= *++pos;
(*argc)--;
The *(optend + 1) is the most interesting part. If a short-form option is being recognized, the rest immediately following the short option is treated as argument:
if (*(optend + 1))
{
/* The rest of the option is option argument */
argument= optend + 1;
/* This is in effect a jump out of the outer loop */
optend= (char*) " ";
Given that we are not passing -p13010, the 13010 part is ignored.
But wait, why does -h localhost work fine?
Just keep looking:
if (optp->arg_type == OPT_ARG)
{
if (optp->var_type == GET_BOOL)
*((my_bool*) optp->value)= (my_bool) 1;
if (get_one_option && get_one_option(optp->id, optp, argument))
return EXIT_UNSPECIFIED_ERROR;
continue;
}
/* Check if there are more arguments after this one */
if (!pos[1])
{
return EXIT_ARGUMENT_REQUIRED;
}
argument= *++pos;
(*argc)--;
So if the argument is an optional arg, it’ll give up and only check for immediate following argument. Otherwise, for OPT_REQUIRED, it assumes the next one is the argument.
Let’s take a look at where they are defined:
{"password", 'p',
"Password to use when connecting to server. If password is not given it's asked from the tty.",
0, 0, 0, GET_PASSWORD, OPT_ARG, 0, 0, 0, 0, 0, 0},
{"host", 'h', "Connect to host.", ¤t_host,
¤t_host, 0, GET_STR_ALLOC, REQUIRED_ARG, 0, 0, 0, 0, 0, 0},
As expected, password is optional and host is required.
Also, note that how it never checked for ‘=’? So the syntax -p=abc wouldn’t work as expected as well. And hilariously =abc would become the password. For arguments with a bit more error checking like port, the error message is a bit better:
[~/mysql]: mysql -P=13010
mysql: [ERROR] Unknown suffix '=' used for variable 'port' (value '=13010')
mysql: [ERROR] mysql: Error while setting value '=13010' to 'port'
Note the ‘=13010’ part?
## Default protocol
OK. Let’s try again:
[~/mysql/mysql-fork]: mysql -P 13010 -h localhost -u root
ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2)
Still doesn’t work. We know it’s not the parsing of -P because port is OPT_REQUIRED:
{"port", 'P', "Port number to use for connection or 0 for default to, in "
"order of preference, my.cnf, \$MYSQL_TCP_PORT, "
#if MYSQL_PORT_DEFAULT == 0
"/etc/services, "
#endif
"built-in default (" STRINGIFY_ARG(MYSQL_PORT) ").",
&opt_mysql_port,
&opt_mysql_port, 0, GET_UINT, REQUIRED_ARG, 0, 0, 0, 0, 0, 0},
Note the error message socket '/var/lib/mysql/mysql.sock. This is for domain socket.
To confirm this is the issue, let’s search for the actual error message:
const char *client_errors[]=
{
"Unknown MySQL error",
"Can't create UNIX socket (%d)",
"Can't connect to local MySQL server through socket '%-.100s' (%d)",
The client_errors are looked up from error codes:
#define ER(X) (((X) >= CR_ERROR_FIRST && (X) <= CR_ERROR_LAST)? \
client_errors[(X)-CR_ERROR_FIRST]: client_errors[CR_UNKNOWN_ERROR])
And the 3rd error is CR_SOCKET_CREATE_ERROR:
#define CR_ERROR_FIRST 2000 /*Copy first error nr.*/
#define CR_UNKNOWN_ERROR 2000
#define CR_SOCKET_CREATE_ERROR 2001
Searching for that leads us back to client.cc:
if (!net->vio &&
(!mysql->options.protocol ||
mysql->options.protocol == MYSQL_PROTOCOL_SOCKET) &&
(unix_socket || mysql_unix_port) &&
(!host || !strcmp(host,LOCAL_HOST)))
{
my_socket sock= socket(AF_UNIX, SOCK_STREAM, 0);
DBUG_PRINT("info", ("Using socket"));
if (sock == SOCKET_ERROR)
{
set_mysql_extended_error(mysql, CR_SOCKET_CREATE_ERROR,
unknown_sqlstate,
ER(CR_SOCKET_CREATE_ERROR),
socket_errno);
DBUG_RETURN(STATE_MACHINE_FAILED);
}
So this means by default we are connecting using Unix domain socket, and only if host is not specifed or is localhost!
Programs should be transparent about its policies, and give information about what it is doing. If that can end up being too verbose, add a verbose option. I’ll write a separate post about this because I’ve been bitten too many times by similar issues and now my favorite past-time is to add print/printf.
So there are two ways to fix this:
1. Instead of local host, use 127.0.0.1. This fails the UNIX socket check and will fallback to TCP.
2. Use --protocol tcp to force using TCP.
So the right command would be:
mysql -P 13010 -h localhost -u root --protocol tcp
or
mysql -P 13010 -h 127.0.0.1 -u root
## Summary
These two problems can be easily avoided by adding more messages to the mysql client, such as:
Trying to connect to UNIX domain socket localhost...
Connecting to database 12310.
These would’ve avoided wasting collectively god knows how much time wasted. Maybe I should submit a patch when I get a chance.
The gotchas:
1. mysql short-option with optional args only accept arguments when they immediately follow the option, such as ‘-pmypassword’. Specifying as ‘-p blah’ and blah will be interpreted as current database. Short option with required args don’t have this problem.
2. When there is no protocol specified, mysql will try to connect as UNIX domain socket if connecting to localhost or host isn’t specified. To work around it, use IP address instead of localhost, or specify protocol explicitly using --protocol.
# Byebye Windows - going full linux
Going linux full time
In my new job, no one cares about windows.
Every single developer (with a few exceptions) use MacBook Pro, and connect to their linux VM to get work done. Some people have the trash can MacPro. You get the idea. Being in Microsoft for ~12 years, this is admittingly a interesting adventure. Even though in several Microsoft projects in the past that I have been working on had linux versions (CoreCLR, Service Fabric, etc), most development is still done in Windows, and then ported to Linux/Mac. Whenever occasionally you wonder into the no-man’s land in Linux where the project tooling / infrastructure is falling significantly behind, you want to pull your hair out. Not Linux’s fault - but a matter of priority. In some extreme cases you’d wonder how even one can put out a linux version out at all.
Not anymore. Now linux (or Mac, if you count that in) is the full time job.
After a few weeks of research and practice, I’ve been happyily chugging along with TMUX + VIM + MOSH with my custom key bindings. In this article I’ll talk about a bit of my experience of making the transition.
## I miss Visual Studio
Let’s get this one out of the way first. There is no replacement for Visual Studio. Period. The code completion (or Intelli-Sense) and debugging is simply unmatched by anything else in the market. VS Code is awesome in terms of just browsing code and doing some occasional debugging, but for writing code it is just OK as the “inteli-sense” (forgive my Microsoft VS Jargon) can be a hit or miss. Vim is good for text editing, and with plugins you can get some basic stuff to work, but again it’s no where near the quality of experience of Visual Studio. Usually it’s a love/hate relationship with Visual Studio - it’s kinda slow and some times buggy, but you can’t live without it. Well, you can, but you don’t want to.
Nowadays I use vim or VS Code / Atom for writing code, and gdb for debugging.
## Debugging using GDB is fine
Being an reasonably experienced WinDbg user, Gdb’s command line taking a bit getting used to, but that’s about it. GDB also supports a TUI mode that shows the integrated text window for source/register/etc and a command window. It’s not great as many simple key bindings stop working in that mode (taken over by the TUI component) but as long as I can see a source code “window” over SSH I’m happy.
# TMUX is awesome
TMUX is a terminal multiplexer. With TMUX you won’t lose your working state - even if you disconnect from SSH, just ‘tmux attach’ you’ll resume where you left off. In this sense it is equivalent to a Windows Remote Desktop session.
The most powerful part is that it also allow you to break the terminal into multiple panes and windows, and this way you don’t have to leave the terminal and can easily switch between many different tasks with quick shortcuts. No more need to manage windows - everything is within the terminal. It’s like a virtual desktop for terminals. It’s build in the way that you barely had to touch the mouse anymore. Well, until you move to the browser, that is.
# VIM ftw
In my Microsoft job I use vim for simple editing purposes, and I like the vim way of thinking so much that I put all my editors into vim mode / vim plugin / vim key bindings. These days I found myself spending even more time in vim over SSH and so I invested more time finding better VIM configurations and plugins.
I use junegunn/vim-plug as my VIM plugin manager. It’s pretty minimal and gets the job done.
This is the list of plugins I use:
• Command-T - blazing fast fuzzy file finder
• delimitMate - automaticlly inserting delimiters such as (), [], etc
• ack - text search tool
• vim-gitgutter - shows in leftmost column where are the git changes using +/-/~
• vim-fugitive - great git command wrappers
• vim-easytags - automated tag generation and syntax highlighting. I found the syntax highlighting can cause performance issue in large files so I turne the syntax highlighting off.
• vim-tmux-navigator - navigate between vim and tmux like they are integrated
• a - switch between header and source. Enough said.
• tcomment_vim - toggle comment/uncomment for lines
• vim-surround - easy change/add surround characters like (), [], {}
• nerdtree - navigate file/directory tree
• vim-nerdtree-tabs - making nerd-tree like an integrated panel
• vim-better-whitespace - highlight trailing whitespace characters. They are annoying for sure and lint warns about them
• lightline - a light and configurable status line for vim
• goyo - distraction free writing. Best for writing docs
## SSH is the old Remote Desktop
In my old job I usually “remote” into my development machines at office - and “remote” means “Windows Remote Desktop”. In a reasonable connection it is actually quite nice - there is little lag and you almost feel you are working on a local machine, with all the graphical UI - it’s really amazing.
With linux, you fallback to the good old text-based SSH. It’s kinda amazing in its own way that you can have text-based remote protocol for complicated full screen programs like vim. You don’t get graphical UI this way - but for the most part you don’t need to, and it’s usually blazing fast.
Mosh improves over SSH that it is async (doesn’t wait for server response) so it feels even more responsive. The trade-off is that it can get a bit jarring when you type something and it does’t react correctly initially.
## Shell matters
Windows Commmand Prompt is fine. It works. I still remember I learned my first DOS commands at a 33MHZ 386DX. But it hadn’t changed much since then. ConEmu is a popular terminal and some people (especally admins) use PowerShell as well. But none of those match the flexiblity of linux shells - they just have so much more to offer. You can switch between different shells, adding customizations, even plugins.
For now I’m using ZSH with oh-my-zsh. It has fantastic themes and plugins. My favorite features are:
• Plugins that shows me all kind of status, such as git status, any pending background process, how long the last command took, etc.
• Auto-suggestion. It automatically suggest the full command based on best match and it grays out the rest of the command that you didn’t type. It’s simple but almost feels like magic when you see it for the first time in action.
• Syntax highlighting. Enough said.
• VIM editing. Yes, you can now use VIM commands to edit your shell commands. Just think that you can easily navigate with all the muscle memory you had with vim. This should be mandatory in every thing that deal with text editing.
With all these, and throw in a few custom key bindings, the plain shell / windows command prompt just seems so boring.
# You need to work on your configurations
However, tweaking these tools so that they work for you takes time. I find myself spending quite a bit of time tweaking the configurations to make it work better for me - and the time spent paid off. All the different configuration options are indeed quite overwhelming if starting from scratch so I use Awesome dotfiles project as my starting point for tweaking and forked my own version yizhang82/dotfiles. There are a lot of things that I like about the way the things are setup:
• One script to deploy everything - TMUX/ZSH, the entire github repo containing dotfiles, and back them up
• Dotfiles are configured to include the settings/scripts from the repo at ~/dotfiles - this way things can be automatically synchronized through a git pull. This is actually quite brilliant.
• Automatically pulls the github repo every time ZSH starts - so it’s always up to date
Of course, many of the configurations there are already pretty good and is perfect as a starting point for my own configurations.
It contains all my TMUX, ZSH, VIM configurations, and by simplying cloning and running a script it goes into a new machine effortlessly. Most of these is done by the original author and I’m simply tweaking it to my needs.
## I like it
It did take a bit getting used to, but I’m happy to report that I now feel very much productive roughly on the same level of productivity when I’m working on Windows (if not more). I do miss having a fully integrated Visual Studio experience, but the command line experience (with TMUX, etc) in Linux is so much better that it more than makes up for that. Of course, at the end of the day, what matters is getting the job done - just use the right tool for the job. In a future post I can get into a bit more details with my experience with these tools and share some of my learnings/tips.
P.S. I still use Windows at home. I have custom built (by myself) PC that has i7 4770K, 32G RAM, nVidia 2080 RTX mostly for gaming. I think Windows has mostly lost the mindshare of developers these days, but it’s still the OS for gamers, and will be for quite some time.
# Fun C++ bug - transactional objects should have move semantics
Objects with transactional semantics need move support
OK. I must admit this probably isn’t the best title out there.
Let’s imagine I have an object that represents a file in a transaction. Recall that transaction needs to be all-or-nothing - if the transaction is complete the files can be kept around / moved to the final destination, otherwise they need to be deleted.
The more or less obvious idea that comes to mind is to represent this with a TxnFile class (TransactionalFile). This is the best part I love about C++, BTW - very clean scoped / destruction semantics.
class TxnFile {
public:
TxnFile(const std::string &file)
: m_file(file), m_committed(false) {
}
~TxnFile() {
if (!m_committed) {
std::remove(file);
}
}
const std::string &get_file() { return m_file; }
void commit() {
m_comitted = true;
}
}
private:
std::string m_file;
bool m_committed;
};
OK. So far so good. Let’s actually implement that business logic:
std::vector<TxnFile> txn_files;
// Collect the files
for (auto &file : some_files) {
txn_files.emplace_back(file);
}
// Do something with them. If exception is thrown we'll remove the files
for (auto txn_file : txn_files) {
do_some_work(txn_file.get_file());
}
// If all is well, commit
for (auto txn_file : txn_files) {
txn_file.commit();
}
Looks rather straight-forward, right? If you try this out yourself, you’ll soon realize something is off - the files are being deleted for no reason at all!
The problem itself is obvious-ish: it really should’ve been a auto & as otherwise are constructing copies of TxnFile and upon destruction will remove the file!
// If all is well, commit
for (auto &txn_file : txn_files) {
do_some_work(txn_file.get_file());
}
However, we are not done yet. The problem is still happening - and in some cases, the files are even removed before we actually do work!
The problem, perhaps not that surprisingly, lies with the std::vector class. When expanding size of std::vector, STL will try to create a new block of memory, and copy/move the memory to it. If the class doesn’t have a move constructor, it’ll default to copy, and destroy the old one - which isn’t unlike the auto txn_file problem we discussed earlier, though a bit more subtle to catch.
Let’s try fixing it:
class TxnFile {
public:
TxnFile(const std::string &file)
: m_file(file), m_committed(false) {
}
TxnFile(const TxnFile &&file) {
m_file = std::move(file.m_file);
m_committed = file.committed;
}
~TxnFile() {
if (!m_committed) {
std::remove(file);
}
}
const std::string &get_file() { return m_file; }
void commit() {
m_comitted = true;
}
}
private:
std::string m_file;
bool m_committed;
};
Looks reasonable, right? Actually the problem is still there! The problem is that you now have a const r-value reference const TxnFile &&. This means that even though you have a r-value reference, you can’t change it at all - and what’s the point of that if you want the move semantics? The right way is to use a regular r-value reference TxnFile &&. Keep in mind declaring move constructor disable the copy constructor so you shouldn’t run into this problem again. But just for better clarifying the intention, it’s a good practice to delete the copy constructor explicitly.
class TxnFile {
public:
TxnFile(const std::string &file)
: m_file(file), m_committed(false) {
}
TxnFile(const TxnFile &) = delete;
TxnFile &operator =(const TxnFile &) = delete;
TxnFile(TxnFile &&rhs)
: m_file(std::move(rhs.m_file), m_committed(rhs.m_committed) {
}
TxnFile &operator = (TxnFile &&rhs) {
reset();
m_file = std::move(that.m_file);
m_committed = that.m_committed;
}
~TxnFile() {
reset();
}
void reset() {
if (!m_committed) {
std::remove(file);
}
}
const std::string &get_file() { return m_file; }
void commit() {
m_comitted = true;
}
}
private:
std::string m_file;
bool m_committed;
};
## A bit of rant
OK. We are finally done. Personally I think C++ has grown to the point that it is too complicated for most people and the interaction between different features can lead to really surprising behaviors. Even if you want to write a simple class there is already too much things to consider (copy/move semantics). And don’t get me started on template meta programming. However, if you stick to a relatively sane subset of C++, maybe you’ll be fine. Just maybe. I’ve been working in large C++ codebases profesionally for 14+ years and I still make stupid mistakes.
# CoreCLR's environment is not your environment
CoreCLR maintains its own private copy of environment variables
This came up when I was helping another collegue in a previous job (where I still write C# code probably 50% of the time), diagnosing a library load failure problem inside a linux container. Internally there is this library that loads different implementations (mock implementation and real implementation) of another library based on a environment variable USE_MAGIC_TEST_LIB, and the .NET code calling that library is calling SetEnvironmentVariable to set it conditionally as part of a testing framework:
if (useTestFramework)
{
Environment.SetEnvironmentVariable("USE_MAGIC_TEST_LIB", "1");
}
This looks reasonable except that it didn’t work at all. It loaded the wrong library and things quickly went down hill after that.
We were scratching our heads for a while until we decided to add a trace to see if the environment is actually taking effect or not in the native code. Interestingly, the native code didn’t see it at all. It’s like they don’t know about each other’s environment!
Actually that observation is more or less what’s going on. Before we dig in a bit deeper, here is a bit of history of CoreCLR cross-platform implementation. Not surprisingly, .NET code started as Windows centric and all the OS calls are strictly Windows API. At some point folks decide to port it to linux/Mac as part of Rotor (later Silverlight), there are two options:
1. Design a new platform abstraction from scratch and move it to that
2. Align the API design to Windows and implement Windows API on top of Linux API
2 is obviously the cheaper solution and has the advantage that Windows code would be untouched and therefore won’t get regressions, which is super important. The caveat is that implementing Windows API using Linux API can get tricky, but is the risk people are willing to take. So the new PAL layer is introduced with “new” APIs that looks exactly like Windows APIs implemented using Linux APIs.
In the case of SetEnvironmentVariable, it is implemented in PAL/environ.cpp:
BOOL
PALAPI
SetEnvironmentVariableA(
IN LPCSTR lpName,
IN LPCSTR lpValue)
{
// ...
// All the conditions are met. Set the variable.
int iLen = strlen(lpName) + strlen(lpValue) + 2;
LPSTR string = (LPSTR) PAL_malloc(iLen);
if (string == nullptr)
{
bRet = FALSE;
ERROR("Unable to allocate memory\n");
SetLastError(ERROR_NOT_ENOUGH_MEMORY);
goto done;
}
sprintf_s(string, iLen, "%s=%s", lpName, lpValue);
nResult = EnvironPutenv(string, FALSE) ? 0 : -1;
PAL_free(string);
string = nullptr;
// If EnvironPutenv returns FALSE, it almost certainly failed to allocate memory.
if (nResult == -1)
{
bRet = FALSE;
ERROR("Unable to allocate memory\n");
SetLastError(ERROR_NOT_ENOUGH_MEMORY);
goto done;
}
This looks a bit fishy. It’s allocating its own buffer and calls into EnvironPutenv, which basically does this:
for (i = 0; palEnvironment[i] != nullptr; i++)
{
const char *existingEquals = strchr(palEnvironment[i], '=');
if (existingEquals - palEnvironment[i] == nameLength)
{
if (memcmp(entry, palEnvironment[i], nameLength) == 0)
{
free(palEnvironment[i]);
palEnvironment[i] = copy;
result = TRUE;
break;
}
}
}
if (palEnvironment[i] == nullptr)
{
_ASSERTE(i < palEnvironmentCapacity);
if (i == (palEnvironmentCapacity - 1))
{
// We found the first null, but it's the last element in our environment
// block. We need more space in our environment, so let's double its size.
int resizeRet = ResizeEnvironment(palEnvironmentCapacity * 2);
if (resizeRet != TRUE)
{
free(copy);
goto done;
}
}
_ASSERTE(copy != nullptr);
palEnvironment[i] = copy;
palEnvironment[i + 1] = nullptr;
palEnvironmentCount++;
result = TRUE;
}
So it’s basically managing its own memory in palEnvironment environment array! No wonder things don’t work.
But why go through all the trouble while Linux has its own getenv/setenv?
http://rachelbythebay.com/w/2017/01/30/env/
Modifications of environment variables are not allowed in multi-threaded programs. – the glibc manual
https://github.com/dotnet/coreclr/issues/635
From looking at the code, I suspect that the cached environment was attempt to fix thread safety or consistency problems between Environment.GetEnvironmentVariables and Environment.SetEnvironmentVariable.
The enumeration of the environment starts by reading the environ global variable, without any locks. Consider what may happen if somebody calls setenv while the enumeration is in progress.
It’s because setenv/getenv isn’t particularly thread safe - you can crash when reading environment while the environment get modifed by another thread, or two threads modifying environment at the same time can lead to leaks.
In this case, one can see a few options:
1. Do nothing - the issues are linux-specific and you should take care when calling these functions, the same way just like you call them in linux.
2. Throw PlatformNotSupported - getenv/setenv just isn’t safe
3. Adding critical section around getenv/setenv - make them safe to be called in multiple threads
4. Implement your own safe environment helpers - as a result rest of the native library won’t observe the change through getenv/setenv
1 isn’t really acceptable because .NET developers need the code to be portable - they don’t want handle the platform special oddies to make their code portable. They would like .NET platform library to be safe and reliable.
2 isn’t great either for the same reason, and also it’ll break a ton of code when ported to linux.
3 makes .NET code safe, but it wouldn’t protect against native code racing with getenv/setenv calls from within .NET code, so race conditions would still occur and .NET developer has little control.
4 is safe, but can lead to subtle breaking changes.
Unfortunately there isn’t a great option here. 1, 2, and 4 are safe option, but all of them have their downsides. At the end of the day, it comes down to compatibility/portability vs surprising behavior. .NET team favors compatibility and portability. While it can lead to sutble breaking changes, fortunately the breaking changes are consistent and therefore easier to diagnose. In our case being a container launched by another system makes the whole thing much harder to diagnose, but that’s more a problem of the container launcher itself. Even though we were bit by the very same problem, I agree it is most likely the better choice.
https://github.com/dotnet/coreclr/blob/master/src/pal/src/misc/environ.cpp#L607
And here is a simple code snippet to demonstrate the issue. I’ve tested this on my MBP.
using System;
using System.Runtime.InteropServices;
namespace set_env
{
class Program
{
[DllImport("/usr/lib/system/libsystem_c.dylib")]
static extern IntPtr getenv(string name);
[DllImport("/usr/lib/system/libsystem_c.dylib")]
static extern int setenv(string name, string value);
static void Main(string[] args)
{
string envName = "MY_ENV";
Console.WriteLine("MY_ENV={0}", Environment.GetEnvironmentVariable(envName));
Environment.SetEnvironmentVariable(envName, "~/path");
Console.WriteLine("Setting it to ~/path");
Console.WriteLine("MY_ENV={0}", Environment.GetEnvironmentVariable(envName));
IntPtr env = getenv(envName);
string envStr = Marshal.PtrToStringAnsi(env);
Console.WriteLine("getenv(MY_ENV)={0}", envStr);
Console.WriteLine("Setting it using setenv");
setenv(envName, "~/path");
env = getenv(envName);
envStr = Marshal.PtrToStringAnsi(env);
Console.WriteLine("getenv(MY_ENV)={0}", envStr);
}
}
}
Now if you are interested to dig in a bit more, here are some bonus questions for your consideration:
1. Does the same problem happen in Windows? And why/why not?
2. Why does the above code above use Marshal.PtrToStringAnsi instead of just have getenv returning the string?
That’s all for now. Thanks for reading!
# Sorting structured data in a unstructured way using memcmp-friendly encoding Part 1 - sorting integers and strings
Sorting structured data in a unstructed way using memcmp-friendly encoding part 1
In many interesting data storage applications, one often need to sort the data in order to do efficient searching - it is such a fundamental operation. The data is often structured like in databases (or it is using a database under the hood) - the application knows exactly what the data is - for example, a record/row with an integer field, a string field, as well as datetime field, etc. In those cases, you can easily sort these data by interpreting the data as what it is, and then comparing them one by one. This is usually achieved by having a base class with virtual functions, and having several derived class implementing the comparison function as well as determine the length to move the next one:
class Data {
public:
virtual int Compare(void *left, void *right) = 0;
virtual int GetLength(void *data) = 0;
};
class UInt32_Data : public Data { public:
virtual int Compare(void *left, void *right) {
auto left_int = reinterpret_cast<uint32_t *>(left);
auto right_int = reinterpret_cast<uint32_t *>(right);
if (left_int < right_int) {
return -1;
} else if (left_int == right_int) {
return 0;
} else {
return 1;
}
}
virtual int GetLength(void *data) {
// No need to read data - uint32_t is always fixed size
return sizeof(uint32_t);
}
};
You can merge these two function into one - for better efficiency. For clarity I’m keeping them separate.
Besides virtual functions, you can also implement this with a jump table that points to a list of functions, or even a switch / case, etc. They are not that fundamentally different - all of them can involve having a table of address to call/jump to, and use a memory lookup to determine the target address.
However, the additional cost of invoking the right comparing functions isn’t zero - as a matter fact it is quite significant comparing to the actual comparison function itself in the case of a virtual function call, which involves putting arguments into registry/stack, pushing return address into stack, setting up frame pointer, etc.
If you are an experienced system programmer, you might know tricks to optimize this further. For example, given this is the exact same problem as an interpreter, and people like interpreter to be fast, VM like Dalvik employed advanced techniques like writing the code in assembly, using threaded execution (which is a fancy way of saying the end of interpreter loop decodes the next instruction instead of jumping to the beginning), and using computing address instead of jump table. These are interesting topics that I might talk about at some point in the future. Anyway, those are not easy to implement and maintain.
But are there other ways to get around this? Is there a way to compare this without understanding what the data is?
The most straight-forward comparison is a byte comparison or memcmp, and this is the most universal way to compare two byte sequences. Many key/value stores (like levelDB/RocksDB) only support byte comparison and allow you to plugin a custom comparator. But before you go ahead and try to implement the custom comparator, let’s give one idea a try: what if we can represent the data somehow as a byte-comparison friendly format?
The challenge are two fold:
1. Encoding the data so that byte order is the correct sorting order
2. Support variable length data properly so that you don’t accidentally compare unrelated data
## Unsigned 32-bit integer
Let’s start with something most straight-forward: a 32-bit unsigned integer. Assume you are working on a Intel machine just like everybody else (and not something esoteric like SPARC) - those unsigned 32-bit integers are represented as little-endian, which means least significant byte will be in memory before the most significant ones. So 0x12345678 will be represented in memory as:
0x78, 0x56, 0x34, 0x12
Obviously this isn’t what we want - we need the compare the most significant byte first, which is exactly Big-Endian:
0x12, 0x34, 0x56, 0x78
Now it’s safe to do a memcmp them now - the bytes are in most-significant to least significant order, and the length is fixed 4-bytes.
Now those SPARC CPU looks pretty sweet, right?
## Signed 32-bit
Let’s make this a bit more interesting. What if the integer is signed?
For signed 32-bit, the range is -2147483648 to +2147483647. There are two cases:
1. Negative: -214783648 (0x10000000) to -1 (0xffffffff),
2. Non-Negative: 0 (0x00000000) to +2147483647 (0x7fffffff)
The non-negative case looks alright - just like the unsigned 32-bit integer case, as long as they are converted to Big-Endian.
For the negative case, the order is correct: -214783648 (0x10000000), -214783647 (0x10000001), … (0xffffffff), except the most significant bit is always one, which makes it bigger than the non-negative case. It is not hard to come up with a fix - just flip the sign bit. Now it becomes:
1. Non-Negative: -214783648 (0x00000000) to -1 (0x7fffffff),
2. Negative: 0 (0x80000000) to +2147483647 (0xffffffff)
Now this looks really nice, and these two ranges are now in the right order, and -1 (0x7ffffffff)+1 = 0 (0x80000000). Now the universe is balanced.
## 8-bit ASCII Strings
For strings, let’s again start with the easy case - 8-bit ASCII strings (we’ll refer it to ASCII string from now on, just for simplicity). For a fixed length ASCII string, it’s really easy - memcmp just works. But how about variable length ASCII?
In such case, the real problem happens when you have string A and B and A is a prefix of B:
A: A, A, A, A, A, A
B: A, A, A, A, A, A, B, B, B
What if just after A there is other data:
A: A, A, A, A, A, A, 0x43
B: A, A, A, A, A, A, B, B
In this case, 0x43 = ‘C’ which is bigger than B, even though A string is smaller than B. Oops.
The key to the problem is that you have to compare the two strings by themselves - you can’t compare other unrelated data by accident (which is our challenge #2, earlier, if you paid attention). You could pad the strings so that they are equal, if you know the maximum length ahead of time (for example, in SQL VARCHAR has max length), but that can be a waste of space.
If you dig deeper, one interesting insight is that if you can have a magic special character that is always guarantee to be smaller than any valid contents in the other string before it ends, then it’ll just work. In many cases, we had no such luxury as strings may have embedded NULLs. But that does provide some additional hint: what if we can artifically inject such marker into the string such that the one that is longer has a bigger byte marker?
A: A, A, A, A, A, A, 0x0
B: A, A, A, A, A, A, 0x1, B, B, B, 0x0
In the above case, A ends with 0x0, while B injects 0x1 as 7th char, making sure it is bigger than A when A ends. Note that 0x1 in this case means there are more data after this, so the encoder/decoder need to take that into account. This looks nice, but we do need to make sure the markers are always at the same place. In order to do that, we can pad/break the strings to split them into predictable fixed length parts with a marker at the last byte. Let’s say if we break it apart at 6 characters, it’ll be exactly like this:
A: A, A, A, A, A, A, 0x0
B: A, A, A, A, A, A, 0x1, B, B, B, 0x0, 0x0, 0x0, 0x0
Note the 4 0x0 (‘ ‘) padding in between, making sure we break the strings every 6 characters. Now, any experienced programmer will tell you that you should always ends things at power of 2 (so that it works better with cache, alignment, etc), so 8/16/32/… would be obviously a better choice. For now let’s go with 8 just to make it easier:
A: A, A, A, A, A, A, 0x0, 0x0
B: A, A, A, A, A, A, A, 0x1, B, B, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0
A bit wasteful, but much better than storing the entire string padded to max length. Also, keep in mind that this encoding supports storing NULL characters as the 0 at every 8th character has special meaning.
But we are not done yet. Do you see there is one more problem?
We are padding the strings with 0x0, and now the strings have some unwanted 0x0 characters padded which we are not able to distingush with actual spaces. Fortunately we still have plenty of run away with the encoding, we can put 1~8 there to indicate number of real characaters (not the padding):
A: A, A, A, A, A, A, 0x0, 0x0
B: A, A, A, A, A, A, A, 0x2, B, B, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0
But this isn’t quite right yet, this can easily get broken (thanks for Thief pointing it out) as the marker themselves get into comparison:
A: A, A, A, A, A, A, A, 0x3, A, A, A, 0x0, 0x0, 0x0, 0x0, 0x0
B: A, A, A, A, A, A, A, 0x2, B, B, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0
To fix this, instead of signaling the number of characters in next segment, it can represent the number of characters in the current segment:
A: A, A, A, A, A, A, A, 0x7, A, A, 0x0, 0x0, 0x0, 0x0, 0x0, 0x2
B: A, A, A, A, A, A, A, 0x7, A, A, B, 0x0, 0x0, 0x0, 0x0, 0x3
For non-NULL characters, it’ll work as any other character will be bigger. For embedded NULL characters, either the last non-NULL character would help:
A: A, A, A, A, A, A, A, 0x7, A, A, 0x0, 0x0, 0x0, 0x0, 0x0, 0x2
B: A, A, A, A, A, A, A, 0x7, A, A, 0x0, A, 0x0, 0x0, 0x0, 0x4
Or for pure NULL padding case, the last 0x2/0x4 will help disambuigate any difference.
A: A, A, A, A, A, A, A, 0x7, A, A, 0x0, 0x0, 0x0, 0x0, 0x0, 0x2
B: A, A, A, A, A, A, A, 0x7, A, A, 0x0, 0x0, 0x0, 0x0, 0x0, 0x4
This is still not quite perfect, though. If a string happens to end at N boundary:
A, A, A, A, A, A, A, 0x7, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0
The final N bytes are wasted just to provide the indicator. The fix is simple: instead of N - 1 indicating more characters, we can have two cases:
1. N - 1 : the segment is full and there are no more characters
2. N : the segment is full and there are more characters
To illustrate this idea:
A, A, A, A, A, A, A, 0x8, B, B, B, 0x0, 0x0, 0x0, 0x0, 0x3
A, A, A, A, A, A, A, 0x7
A, A, A, A, A, 0x0, 0x0, 0x5
In summary, we break down the string in the chunk of 8/16/32/… characters and using the every Nth character a special marker that indicates:
1. 1 ~ N - 1 : the number of characters in the current segment. The rest is 0x0 padding.
2. N : the current segment is full and there are more characters
We can always convert it into a case that we know about - if we can convert such string into UTF-8, which works great in byte comparison even in the case of multi-byte characters. If you haven’t looked at it yet, you should. It’s brilliant, and everyone should be talking in UTF-8 (I’m looking at you, Windows). Just go to https://en.wikipedia.org/wiki/UTF-8.
## What about sort ordering and collation?
Those cases can get really complicated depends on the encoding + collation so I won’t get into them. But the idea is always the same: transform the bytes into the correct sorting order as dicated by the character set/encoding. If the encoding byte order happens to be the right sorting order (UTF-8, for example), all the better.
If you are interested you might want to give your favorite encoding a try.
## We are not done yet
In this post I’ve discussed approaches to encode your data in a way that is sort friendly. In cases where there are lot of read/seek/lookup, it can make a really huge difference in terms of lookup performance, but in write heavy environments it may not be the right trade-off as the overhead of the encoding become bigger and the caching to offset the decoding cost become less effective. At the end of the day, there is no silver bullet and you need to pick the right solution for your scenario at hand.
However, we are not quite done yet. There is one more interesting scenario we can look at: floats. This is a non-trivial topic as we need to dive into the binary format of floats/doubles. Hope I’ll see you next time!
EDIT NOTE: This article has been updated to address some bugs in the examples and in the encoding. Thanks everyone for the suggestion/corrections!
|
2021-01-24 12:30:15
|
{"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": 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": 0.2370656132698059, "perplexity": 3818.4888639032843}, "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/1610703548716.53/warc/CC-MAIN-20210124111006-20210124141006-00351.warc.gz"}
|
http://mathhelpforum.com/discrete-math/137967-complicated-combinatorics-problem-print.html
|
# Complicated Combinatorics Problem
• April 8th 2010, 12:50 PM
CogitoErgoCogitoSum
Complicated Combinatorics Problem
In how many ways can a we arrange, in a row, six balls... when we have to choose from seven non-unique cricket balls, six non-unique tennis balls and five non-unique squash balls?
In total, 18 balls from three unique sets, each set of which contains non-unique entities, and each set is differently sized and one of which contains one fewer ball than we wish to arrange.
The answer is supposed to be 728, according to my reference. Im just not sure how to solve it. I could probably solve this by breaking it down into specific cases, and adding it all up... but Id might as well list each possible combination if Im going to do that. Im looking for a simpler method.
• April 8th 2010, 03:48 PM
awkward
Quote:
Originally Posted by CogitoErgoCogitoSum
In how many ways can a we arrange, in a row, six balls... when we have to choose from seven non-unique cricket balls, six non-unique tennis balls and five non-unique squash balls?
In total, 18 balls from three unique sets, each set of which contains non-unique entities, and each set is differently sized and one of which contains one fewer ball than we wish to arrange.
The answer is supposed to be 728, according to my reference. Im just not sure how to solve it. I could probably solve this by breaking it down into specific cases, and adding it all up... but Id might as well list each possible combination if Im going to do that. Im looking for a simpler method.
Hi Cogito,
You may or may not like the following solution, which uses exponential generating functions.
Let's say, more generally, we want to find the number of ways to select r balls from the set of cricket, tennis, and squash balls. Call this number $a_r$.
Let f be the exponential generating function of $a_r$, i.e.
$f(x) = \sum_{r=0}^{\infty} \frac{1}{r!} x^r$. Then
$f(x) = (1 + x + \frac{1}{2!}x^2 + \dots + \frac{1}{7!}x^7) \cdot (1 + x + \frac{1}{2!}x^2 + \dots + \frac{1}{6!}x^6) \cdot (1 + x + \frac{1}{2!}x^2 + \dots + \frac{1}{5!}x^5)$
by elementary properties of exponential generating functions. (It's easy to see once you know how.) The answer to your problem is $a_6$, which is the coefficient of $\frac{1}{6!} x^6$ when f is expanded.
The bad news is that finding the coefficient by pencil and paper essentially boils down to listing the cases, which you said you want to avoid. But the good news is that there are many computer algebra systems (Wolfram Alpha, for one) which will expand f for you. When this is done we find
$a_6 = 728$,
as you predicted.
|
2014-11-23 00:16:39
|
{"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": 7, "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": 0.446552574634552, "perplexity": 545.2727265389971}, "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-2014-49/segments/1416400378815.18/warc/CC-MAIN-20141119123258-00230-ip-10-235-23-156.ec2.internal.warc.gz"}
|
http://www.nicta.com.au/pub?id=769
|
# Research Publications
Detecting Regular Visit Patterns
Bojan Djordjevic, (Hans) Joachim Gudmundsson, Anh Pham, Thomas Wolle
We are given a trajectory $\T$ and an area $\A$. $\T$ might intersect $\A$ several times, and our aim is to detect whether $\T$ visits $\A$ with some regularity, e.g.~what is the longest time span that a GPS-GSM equipped elephant visited a specific lake on a daily (weekly or yearly) basis, where the elephant has to visit the lake {\em most} of the days (weeks or years), but not necessarily on {\em every} day (week or year). During the modelling of such applications, we encountered an elementary problem on bitstrings, that we call {\sc LDS (LongestDenseSubstring)}. The bits of the bitstring correspond to a sequence of regular time points, in which a bit is set to $1$ iff the trajectory $\T$ intersects the area $\A$ at the corresponding time point. For the LDS problem, we are given a string $s$ as input and want to output a longest substring of $s$, such that the ratio of $1$'s in the substring is at least a certain threshold. In our model, LDS is a core problem for many applications that aim at detecting regularity of~$\T$ intersecting $\A$. We propose an optimal algorithm to solve LDS, and also for related problems that are closer to applications, we provide efficient algorithms for detecting regularity.
## Details
Status: published Type: Conference Paper Conference Name: Algorithms - ESA 2008 Pages: 344-355 City/Country: Karlsruhe/Germany Conference URL: dx.doi.org/10.1007/978-3-540-87744-8_29 Publisher: Springer-Verlag, LNCS Volume 5193
|
2013-05-21 23:21:08
|
{"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": 0.6893608570098877, "perplexity": 1613.2315472259424}, "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-2013-20/segments/1368700842908/warc/CC-MAIN-20130516104042-00063-ip-10-60-113-184.ec2.internal.warc.gz"}
|
https://planetmath.org/CategoryOfPathsOnAGraph
|
category of paths on a graph
A nice class of illustrative examples of some notions of category theory is provided by categories of paths on a graph.
Let $G$ be an undirected graph. Denote the set of vertices of $G$ by “$V$” and denote the set of edges of $G$ by “$E$”.
A path of the graph $G$ is an ordered tuplet of vertices $(x_{1},x_{2},\ldots x_{n})$ such that, for all $i$ between $1$ and $n-1$, there exists an edge connecting $x_{i}$ and $x_{i+i}$. As a special case, we allow trivial paths which consist of a single vertex — soon we will see that these in fact play an important role as identity elements in our category.
In our category, the vertices of the graph will be the objects and the morphisms will be paths; given two of these objects $a$ and $b$, we set $\operatorname{Hom}(a,b)$ to be the set of all paths $(x_{1},x_{2},\ldots x_{n})$ such that $x_{1}=a$ and $x_{n}=b$. Given an object $a$, we set $1_{a}=(a)$, the trivial path mentioned above.
To finish specifying our category, we need to specify the composition operation. This operation will be the concatenation of paths, which is defined as follows: Given a path $(x_{1},x_{2},\ldots,x_{n})\in\operatorname{Hom}(a,b)$ and a path $(y_{1},y_{2},\ldots,y_{m})\in\operatorname{Hom}(a,b)$, we set
$a\circ b=(x_{1},x_{2},\ldots x_{n},y_{2},\ldots,y_{m}).$
(Remember that $x_{n}=y_{1}=b$.) To have a bona fide category, we need to check that this choice satisfies the defining properties (A1 - A3 in the entry http://planetmath.org/node/965category). This is rather easily verified.
A1: Given a morphism $(x_{1},x_{2},\ldots x_{n})$, it can only belong to $\operatorname{Hom}(a,b)$ if $x_{1}=a$ and $x_{n}=b$, hence $\operatorname{Hom}(a,b)\cup\operatorname{Hom}(c,d)=\emptyset$ unless $a=c$ and $b=d$.
A2: Suppose that we have four objects $a,b,c,d$ and three morphisms, $(x_{1},x_{2},\ldots x_{n})\in\operatorname{Hom}(a,b)$, $(y_{1},y_{2},\ldots y_{m})\in\operatorname{Hom}(b,c)$, and $(z_{1},z_{2},\ldots z_{k})\in\operatorname{Hom}(c,d)$. Then, by the definition of the operation $\circ$ given above,
$\displaystyle((x_{1},x_{2},\ldots,x_{n})\circ($ $\displaystyle y_{1},y_{2},\ldots,y_{m}))\circ(z_{1},z_{2},\ldots,z_{k})$ $\displaystyle=(x_{1},x_{2},\ldots,x_{n},y_{2},\ldots,y_{m})\circ(z_{1},z_{2},% \ldots,z_{k})$ $\displaystyle=(x_{1},x_{2},\ldots,x_{n},y_{2},\ldots,y_{m},z_{2},\ldots,z_{k})$ $\displaystyle(x_{1},x_{2},\ldots,x_{n})\circ($ $\displaystyle(y_{1},y_{2},\ldots,y_{m})\circ(z_{1},z_{2},\ldots,z_{k}))$ $\displaystyle=(x_{1},x_{2},\ldots,x_{n})\circ(y_{1},y_{2},\ldots,y_{m},z_{2},% \ldots,z_{k})$ $\displaystyle=(x_{1},x_{2},\ldots,x_{n},y_{2},\ldots,y_{m},z_{2},\ldots,z_{k}).$
Since these two quantities are equal, the operation is associative.
A3: It is easy enough to check that paths with a single vertex act as identity elements:
$\displaystyle(x_{1})\circ(x_{1},x_{2},\ldots,x_{n})$ $\displaystyle=(x_{1},x_{2},\ldots,x_{n})$ $\displaystyle(x_{1},x_{2},\ldots,x_{n})\circ(x_{n})$ $\displaystyle=(x_{1},x_{2},\ldots,x_{n})$
It is also possible to consider the equivalence class of paths modulo retracing. To introduce this category, we first define a binary relation $\approx$ on the class of paths as follows: Let $A$ and $B$ be any two paths such that the right endpoint of $A$ is the same as the left endpoint of $B$, i.e. $A\in\operatorname{Hom}(a,b)$ and $B\in\operatorname{Hom}(b,c)$ for some vertices $a,b,c$ of our graph. Let $d$ be any vertex which shares an edge with $d$. Then we set $A\circ B\approx A\circ(c,d,c)\circ B$.
Let $\sim$ be the smallest equivalence relations which contains $\approx$. We will call this equivalence relation retracing.
As defined above, it may not intuitively obvious what this equivalence amounts to. To this end, we may consider a different description. Define the reversal of a path to be the path obtained by reversing the order of the vertices traversed:
$(x_{1},x_{2},\ldots,x_{n-1},x_{n})^{-1}=(x_{n},x_{n-1},\ldots,x_{2},x_{1})$
Then we may show that two paths are equivalent under retracing if they may both be obtained from a third path by inserting terms of the form $XX^{-1}$. In symbols, we claim that $A\sim B$ if there exists an integer $n＞0$ and paths $X_{1},\ldots X_{n+1},Y_{1},\ldots Y_{n-1},Z_{1},\ldots Z_{n}$ such that
$A=X_{1}\circ X_{1}^{-1}\circ Z_{1}\circ X_{2}\circ X_{2}^{-1}\circ\cdots\circ X% _{n-1}\circ X_{n-1}^{-1}\circ Z_{n}\circ X_{n}\circ X_{n}^{-1}\circ Z_{n}\circ X% _{n+1}\circ X_{n+1}^{-1}$
and
$B=Y_{1}\circ Y_{1}^{-1}\circ Z_{1}\circ Y_{2}\circ Y_{2}^{-1}\circ\cdots\circ Y% _{n-1}\circ Y_{n-1}^{-1}\circ Z_{n}\circ Y_{n}\circ Y_{n}^{-1}\circ Z_{n}\circ Y% _{n+1}\circ Y_{n+1}^{-1}$
This characterization explains the choice of the term “retracing” — we do not change the equivalence class of the path if we happen to wander off somewhere in the course of following the path but then backtrack and pick the path up again where we left off on our digression.
Rather than presenting a detailed formal proof, we will sketch how the two definitions may be shown to be equivalent.
Title category of paths on a graph CategoryOfPathsOnAGraph 2013-03-22 16:45:54 2013-03-22 16:45:54 rspuzio (6075) rspuzio (6075) 18 rspuzio (6075) Example msc 20L05 msc 18B40 IndexOfCategories
|
2018-11-18 04:36:15
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 68, "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": 0.9373560547828674, "perplexity": 141.63909599426995}, "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-2018-47/segments/1542039743963.32/warc/CC-MAIN-20181118031826-20181118053826-00236.warc.gz"}
|
https://codereview.meta.stackexchange.com/questions/8999/code-review-is-getting-boring/9001
|
# Code Review is getting boring
### Boring
I find that Code Review is getting pretty boring and unprofessional. The reason why I think this way is that many good questions about real-world problems don't get very much attention. Instead, the hundredth implementation of linked-lists, heaps, project-euler, factorials, fibonacci, sortings or other programming challenge garbage get countless upvotes and answers. I'm not posting any links because I'm pretty sure you know what I mean.
### Poor quality
Consequently instead of attracting more professionals to come over to Code Review to share their code and use-cases, and who could also take a look at some other interesting problems and share their valuable knowledge, we seem to favor simple and recurring questions by not voting for the interesing ones and not encouraging their authors for providing as much context as possible so that everyone can learn from them - OP, reviewer and especially other readers.
This has also another negative side effect: there seem to be very few active users who write reviews or share their solutions to intriguing every-day matters.
### Too forgiving
I also think we are often too forgiving with questions that lack context and don't get closed. This gives a signal that poor questions are acceptable and the only beneficiary is the OP because only they know enough to be able to apply any suggestions. This makes Code Review not very useful to others.
I wish there would be some additional categorizing to easier distinguish informative questions you can learn from, from those that are pure waste of time. But since there is no such thing I usually see Code Review like this: many upvotes(7+) and answers (3+) = very likely not worth the time; few upvotes(<4) and few answers (0-2) = highly likely something interesting.
### What's next?
Is there anything we can do about? I really liked Code Review for it's every-day riddles and learnt a lot of new stuff myself either by trying to crack some of them or just by reading and seeing tools, APIs or libraries I've never seen or tried to used in a certain way before, but recently it's been somehow mostly only unexciting junk. I don't like Code Review as I used to.
This has bothered me for quite some time already and can at the same time be seen as another answer to To serve, what is it to ask for review? that tries to promote poor quality input. I 100% disagree with it. Both, well written questions and answers should be CR's target. Not questions with vague description and unclear purpose of the code.
Quite contrary to the other question I suggest we be a lot stricter about the rules and encourage people to take more time and present their code in a more attractive and informative fashion because I see Code Review primarily as a platform that should be educational to more people than just to the OP.
### Question
In other words: what can we do about under-supporting/promoting good questions and over-promoting the not so good, attractive or simply bad ones?
• "Is there anything we can do about?" About what specifically? I agree with your sentiment, but it's hard to answer the question if the question is this broad. Are you simply looking for any and all suggestions to make it less boring? Improve quality? Do you simply want the community to be less forgiving? All of the above? Again, I agree with the sentiment, but can you summarize your goal in a single question so the community knows what kind of answers you're looking for? – Mast Nov 4 '18 at 9:47
• @Mast oh, sorry; in other words I mean what can we do about not supporting/promoting good questions and over-promoting the not so good, attractive or simply bad ones? Being less forgiving for the second category could be one measure... but maybe there are other suggestions? – t3chb0t Nov 4 '18 at 9:52
• Hard to say. More participation would be a great start, but I don't have the foggiest how we'd achieve that. There are a lot of things we've already tried. Bounties, community-challenges, RoboSanta. But those don't solve the core problem of repeating questions by beginners and waning participation from people with experience. – Mast Nov 4 '18 at 13:20
• Well I don't write so many reviews like I used to write because of two reasons. One reason is that my work-load got higher, and well the other reason is you because we are targeting the same language and you are doing a pretty good job. – Heslacher Nov 5 '18 at 7:04
• @t3chb0t Let's make that better. Feel invited: codereview.meta.stackexchange.com/questions/9002/… – πάντα ῥεῖ Nov 5 '18 at 18:23
• I generally agree about the programming puzzle questions, I've never really been interested in "programming magic tricks", I'm more into hunting for memory leaks and hidden crashes. Just out of curiosity, were my questions interesting to you? – jrh Nov 5 '18 at 23:07
• @jrh yes, I've voted for every one of them and I even got two green marks ;-) – t3chb0t Nov 6 '18 at 7:02
• I'm on the same page as Heslacher: if I see a review which mentions all of the points I would make, I just upvote the answer and go on. Most of the regulars (the ones I encounter on C++ tag) usually make very dense and thorough reviews, leaving no opening for another review. – Incomputable Nov 6 '18 at 19:02
• @t3chb0t I can post more if you want, I figured there was low interest in the kind of code I made, so I stopped. Though strangely enough a whole lot of readers are apparently interested in FTP upload queues. – jrh Nov 6 '18 at 20:16
• @jrh yeah, this exactly what I mean and it shows how important it is to post questions that have high practical value! ;-) If you have more interesting stuff, freaky frameworks or crazy experiments please do post. I miss Dmitry Nogin's small utilities - they seem really odd at first but at the same time brilliant. – t3chb0t Nov 7 '18 at 20:16
• @DmitryNogin you're being mentioned - have you build anything new lately? ;-] – t3chb0t Nov 7 '18 at 20:18
• @t3chb0t sure, but keep in mind that posts with 1 score are dangerously close to getting garbage collected. – jrh Nov 7 '18 at 21:33
If you're finding Code Review boring then it's time to take a break! This is part of the natural cycle of volunteer engagement on a Stack Exchange site — it's fun to start with, but after a while you've seen all the common types of question and you don't want to have to make the same points again and again in your answers. Find something more interesting to do for a while: Code Review will still be here when or if you're ready to come back.
There are systematic reasons for the aspects of the site that you complain about:
1. Most posts will be from beginner and student programmers. That's because everyone is a beginner at some point, and beginners feel most in need of review.
2. Beginners are going to be posting beginner exercises. Everyone starts with these kinds of exercise because that's how you learn to program. Another heap implementation is boring to you because you've seen dozens of them, but for the beginner it's something new and complex to learn. (Obligatory xkcd.)
3. We're not going to see much professional code here, because of confidentiality and copyright. Even disregarding these concerns, professional programmers are better off reviewing within their organization, in order to benefit from shared knowledge and expertise.
4. We're not going to see much open source code here, because open source projects have other and better forums for review, for example through the GitHub pull request interface. (Also, copyright may be a concern too.)
5. Challenge problems are more attractive for reviewers, because the problems tend to be self-contained (don't require the reviewer to understand a lot of context), and well-specified (the reviewer doesn't need to engage in a long dialog with the poster to try to extract a specification for the code they posted).
Because these reasons are systematic, you're going to find it frustrating trying to defeat them. The systematic forces operate for everyone 100% of the time, but you only have limited time and energy. Better to spend that energy productively elsewhere.
• I cannot take a break, I have a streak of 881 consecutive days - a break would be a disaster - or I'll create a script that keeps it alive ;-P – t3chb0t Nov 8 '18 at 14:00
• ok, I thought about it and I came to the conclustion that this is not the case. I find my job very interesting and solving new problems everyday is a lot of fun. If this was the reason why I should quit Code Review (temporarily) then I would have to tell my boss to give me a looooong holiday. I'd like to once agian emphasize that I never said that beginner questions were boring. I didn't even use this word in my question. Boring questions have no practical value, they don't challange you to see things in a different or a new way or to make some research etc. [..] – t3chb0t Nov 8 '18 at 17:31
• [..] Boring questions are about problems that no-one cares about. This means I don't agree with point 1 & 2. I also don't agree with some of the the other points but will replay another time. – t3chb0t Nov 8 '18 at 17:31
I find that Code Review is getting pretty boring and unprofessional. The reason why I think this way is that many good questions about real-world problems don't get very much attention. Instead, the hundredth implementation of linked-lists, heaps, project-euler, factorials, fibonacci, sortings or other programming challenge garbage get countless upvotes and answers.
Then downvote and leave a comment stating why you find the question uninteresting. Preferably guiding them to a guide on how to ask a good question
we seem to favor simple and recurring questions by not voting for the interesting ones and not encouraging their authors for providing as much context as possible so that everyone can learn from them - OP, reviewer and especially other readers.
Share the interesting questions. Tweet them, Facebook them, discuss them in chat.
I also think we are often too forgiving with questions that lack context and don't get closed. This gives a signal that poor questions are acceptable and the only beneficiary is the OP because only they know enough to be able to apply any suggestions. This makes Code Review not very useful to others.
I disagree with you here. You might think that this gives a signal to other askers but you know what? Most people who ask poor questions don't look that much at other questions and therefore don't know anything at all about this signal. You could close all questions about anything and people will still come and ask questions about those things. Additionally, poor questions are acceptable. There is an important distinction between bad question and off-topic question. Poor questions are acceptable - not appreciated, but acceptable.
To fix this problem: Post a friendly comment stating what makes the question poor and why and how it should be improved. See Frequently Posted Comments
I wish there would be some additional categorizing to easier distinguish informative questions you can learn from
What is informative and interesting is highly subjective. My only recommendation is: Have a good title. If you see a title that is about linked lists and you don't find that topic interesting, then ignore it.
In other words: what can we do about under-supporting/promoting good questions and over-promoting the not so good, attractive or simply bad ones?
• I also think one of the contributing factors is the student population. More experienced programmers would not probably have a question about a real world application. On the other hand, a student will likely have a question about code they came across or wrote in class. – FreezePhoenix Nov 5 '18 at 17:47
• I'm not sure that downvoting questions for merely being uninteresting or repetitive is the way to go. It would make more sense to upvote and call attention to good questions. Save the downvotes for questions that are actually bad. – mdfst13 Nov 7 '18 at 14:10
• @mdfst13 I disagree with you there. The downvote button merely means "I don't like this question." and it could just as well be used for uninteresting or repetitive questions. – John Hamilton Nov 8 '18 at 6:01
• @JohnHamilton Downvoting a new user's question purely to say that you personally don't like that question has the impact of telling said new user that you don't like them whether you meant it that way or not. It's quite rude and people should avoid doing it. It is one of the reasons why Stack Exchange is considered an unfriendly place. – mdfst13 Nov 8 '18 at 18:12
• Regarding disliking linked-list questions or programming challenges, can't you set up an "ignore tags" list and the UI will de-emphasize questions with those tags? I've done this on one or two sites where there are popular categories that I don't want to see. (I wish it would just leave them out of the feed, but that's another story.) – user1118321 Nov 17 '18 at 6:39
Let me start this by providing / adding some numbers to the mix. The following are the all-time stats for questions on Code Review:
closed | deleted | count
yes | yes | 21.787
yes | no | 1.541
no | yes | *7.629
no | no | 54.699
* only 1.107 of these have a positive score
Note that our friendly neighborhood background job will automatically delete closed questions (especially those without answers) after enough time. Only a very small amount of questions seems to be deleted by the moderators and power users.
These stats suggest that we closed slightly more than 30% of all questions that were ever asked on Code Review. The question closure statistics (available to 10k+ users) state that in the last 90 days, that percentage was 37%. So at least in the last three months, we closed more questions than we'd usually do.
To some extent that can be explained by the start of a new semester and the accompanying influx of students. Most of these closures (almost two thirds) are concentrated on the "code not working as intended" reason. A quarter of closures is for "Lacks concrete context", and the rest falls between "Authorship of code" and the remaining ways to close a question.
Just looking at the closure statistics paints a rather not so useful picture of what's happening, though... As a moderator I can run some interesting analytics, though most of what I'm about to tell you can also be seen in 202_accepted's Stack Exchange Statistics Explorer
Since around a year ago, the 5-week average of questions has been sinking. Code Review peaked at around 350 questions per week in the 5-week average (caused by two weeks of almost 370 questions). Currently that average is hovering around the 250 questions per week mark.
It's then not really surprising to see answers on a similar decline. As recently as two months ago, the overall Votes also dropped by quite a bit, even though they were rather steady in contrast to the decline of questions and answers. As such I'm not too worried about that right now. It might be worth noting that the 5-week average of votes is currently around a thousand votes per week. The historical maximum in a single week is at almost 5000 votes in a single week. It's probably not a coincidence that happened around christmas ...
The clear picture that these numbers paint is that engagement is somewhat waning. So it seems to be not just you that's not quite as addicted to code review as before.
This is just to help put some numbers onto the feeling that you're having. What these numbers say about the kind of question that's recently popular is a different thing entirely.
• "The clear picture that these numbers paint is that engagement is somewhat waning." Yup. I think more regulars left than we got back, others became less regular. We lost some experience on the way too. – Mast Nov 4 '18 at 13:10
In other words: what can we do about under-supporting/promoting good questions and over-promoting the not so good, attractive or simply bad ones?
We could be more aggressive about tagging questions , , and possibly some new tag. Then you could exclude questions with those tags from your view. That would make the remaining good questions stand out more.
You also might consider browsing the Unanswered tab rather than the front page. Because what you're saying is that you prefer questions that have not received much attention.
Or you might try this advanced search. Adjust the parameters as necessary.
• This is not exactly how I ment it ;-) I don't automatically consider beginner quesitons bad and I see CR primarly as a learning platform - people are comming here to read/post something they can learn from. To me a bad question is one that is not educational to anyone, e.g by severely lacking context (aka: this is my code, how can I make it better? with a bunch of unknown variable types etc). If it's technically ok then it's just fine but it doesn't have to be good yet. Good questions are about interesting practical/strange problems that are very useful and inspirational to other people. – t3chb0t Nov 7 '18 at 20:06
I think you are right, but I don't think it's the only problem, or necessarily a fatal one. For me, the problem is that there are decent questions to answer, but they aren't easy to find and there's no real recognition for answering. Here is a reasonable question that sat around for a month. I provided what I think is a helpful answer, and have received no votes or comments in the month since. Similar story here and here. These questions were not easy to find, and now I'm not sure it was even worth it.
• Now how have received some ;-] I must have missed them. You can draw more attention to an answer if you use some more formatting or code examples ;-) Text-blobs are often difficult to comprehend on their own so probably that's why the recive less votes. There isn't anything directly visible. On the other hand, only code without comments is bad too like a large piece of code without proper explanation. – t3chb0t Nov 13 '18 at 16:50
|
2019-06-25 04:07:53
|
{"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": 0.29613807797431946, "perplexity": 1199.9426736628268}, "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-2019-26/segments/1560627999787.0/warc/CC-MAIN-20190625031825-20190625053825-00286.warc.gz"}
|
https://math.stackexchange.com/questions/2878701/does-there-exist-such-an-interval
|
# Does there exist such an interval?
If $f: \mathbb{[a,b]} \to \mathbb{R}$ and $f$ is twice differentiable at a point $c$. Does there exist an interval $[p,q]$ in $[a,b]$ where $f$ is differentiable
I think here, $f'$ is continuous at $c$.Then by the definition of continuity there must exist such an interval
• How do you define "twice differentiable at $c$" if $f$ is not differentiable on a neighborhood of $c$? – Clement C. Aug 10 '18 at 19:13
However, since $f$ is twice differentiable at $c$, that means that its derivative at point $c$ is differentiable.
Now assume that there are no intervals around $c$ that are differentiable. Then, the derivative of $f$ could not be shown to be differentiable, since $f'$ wouldn't even be continuous at $c$. Hence, the statement has been shown.
|
2019-12-08 19:30:04
|
{"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": 0.9441443681716919, "perplexity": 108.0045916913577}, "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-2019-51/segments/1575540514475.44/warc/CC-MAIN-20191208174645-20191208202645-00507.warc.gz"}
|
http://pclinuxosbrasil.com.br/npvqxb/scl4-dipole-moment-1b0189
|
Dipole Moment: The dipole moment of the molecule depends on the polarity of the bond and the molecular structure of the compound. Moment of inertia. This allows for a stronger dipole moment. Likewise, is CCl4 a dipole dipole force? Rotation-vibration spectrum. In fact, since the molecule is symmetrical, all the dipole moments will cancel each other out. Equation for dipole moment: μ = q x e x d. d : internuclear separation. It contains two pairs of electrons and two polar bonds, which confer a net dipole moment on the molecule due to their geometric arrangement. Xe in XeF2: dsp3. S in SF6: d2sp3. Molecular structure. Point group. As you go down the series there is less s-p mixing. Two identical electric point dipoles have dipole moments bar p1 = pi and bar p2 = -pi and are held on the x axis at distance. Quadrupole coupling. Nuclear quadrupole coupling. 2- In CH2Cl2 dipole moment of H-H atoms and Cl-Cl atoms do not cancel each other because angle are not 180° so they are not linear. It involves the concept of electric dipole moment, which is a measure of the separation of negative and positive charges in a system. A molecule which contains polar bonds will always have a dipole moment. Dip ole moment is measured in Debye units, which is equal to the distance between the charges multiplied by the charge (1 Debye eq uals $$3.34 \times 10^{-30}\; C\, m$$). Dipole moments occur due to the difference in electronegativity between two chemically bonded atoms. The more acute bonds would then result in a higher dipole moment. d2sp3. Cl in ClF3: dsp3. When two electrical charges, of opposite sign and equal magnitude, are separated by a distance, an electric dipole is established. 5 points for each argument. In every other case except H_2S, the polarization of charge associated with each bond is exactly cancelled by the other bonds, resulting in no net dipole moment. When a carbon atom has sp3 hybridization, it has. Sulfur dichloride (SCl2) is a polar molecule. CCl4 is an example of a nonpolar molecule. Phase equilibrium. Based on symmetry alone, we know that H_2S is the only one of these molecules that has a dipole moment. Phase diagram. The dipole moment is calculated by multiplying the distance between the hydrogen and oxygen atoms by the difference in their charge. asked Sep 11 in Physics by AmarDeep01 (50.0k points) jee main 2020; 0 votes. False. This results in bond angles closer to 90°. CCl4 > CHCl3 > CH2Cl2 > CH3Cl is the decreasing ⦠Reaction coordinate. Potential energy. Nuclear quadrupole moment. Then, the angle between the atoms is used to find the net dipole moment. CH3Cl>CH2Cl2>CHCl3>CCl4. In the case of Cl_2, the 2 atoms are identical, so no polarization of the bond is possible, and the dipole moment is zero. S in SCl4: dsp3. The hybridization of I in IF4- is. The electronegativity for C is 2.5 and Cl is 3.0, resulting in a polar covalent bond. Rotational excitation cross ⦠According to the Lewis structure, CCl4 is a tetrahedral molecule. The hybridization of the central atom in XeF5+ is: d2sp3. four Ï bonds. units of μ = Debyes where 1D = 3.336 x 10^-30 Cm e : elementary charge (1.602 x 10^-19) q : point charge. .1-This is due to in CH3Cl chlorine is EWG and it is in one direction and no other group present for cancelling/decreasing its dipole moment. Dipole moment:- The polarity of covalent bond can be conveniently measured in terms of physical quantity called dipole moment. The bond distances also increase with increasing dipole moment due larger atomic radii of the central atoms. 1 answer. Dipole Moment. The size of a dipole is measured by its dipole moment ($$\mu$$). (i) (Refer to Image 1) (ii) (Refer to Image 2) (iii) (Refer to Image 3) (iv) (Refer to Image 4) The angle formed by a water molecule is known to be 104.5° and the bond moment of the O-H bond is -1.5D. A bond dipole moment is a measure of the polarity of a chemical bond between two atoms in a molecule. Molecular dipole moment. A Ï (pi) bond is the result of the.
|
2021-02-27 19:02:22
|
{"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": 0.6939722895622253, "perplexity": 1203.2199756853606}, "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-2021-10/segments/1614178359082.48/warc/CC-MAIN-20210227174711-20210227204711-00506.warc.gz"}
|
https://www.physicsforums.com/threads/frampton-hungs-higgs-mass-ansatz.716839/
|
# Frampton & Hung's Higgs mass ansatz
Tags:
1. Oct 16, 2013
### mitchell porter
http://arxiv.org/abs/1310.3904
A Possible Reason for MH ≃ 126 GeV
Paul H. Frampton, Pham Q. Hung
(Submitted on 15 Oct 2013)
It is speculated that a possible reason for the scalar mass MH ≃ 126 GeV is equality of the lifetimes for vacuum decay and instanton-induced proton decay.
2. Oct 16, 2013
### MathematicalPhysicist
Isn't Paul Frampton the Prof who got caught with smuggling drugs in south america for a beloved model?
How can a smart man fall for such a scam?
3. Oct 16, 2013
### mitchell porter
That has nothing to do with this thread, please discuss it elsewhere.
4. Oct 17, 2013
### lpetrich
My first thought is: is there any theoretical reason why those decay lifetimes ought to be equal? Higgs vacuum instability and proton decay by instantons are two different mechanisms, so I don't see how they would be connected, let alone give the same lifetime. I read Frampton and Hung's paper, and they don't give any hint either.
Those two effects are both quantum tunneling effects, but as far as I can tell, that's most of what they have in common. A well-known bit of quantum tunneling is alpha decay of heavy nuclei, and alpha-decay lifetimes vary like crazy. So even closely-related tunneling effects can have drastically differing rates.
5. Oct 23, 2013
### mitchell porter
Another reason you wouldn't expect them to be related, is that the rate of proton decay depends on the rate at which a particular event occurs inside a proton, whereas the rate of vacuum decay depends on the rate at which a particular event occurs inside a cosmological horizon.
But perhaps that would be the point - a hypothetical BSM theory in which the Frampton-Hung relation was not just coincidence, would explain the ratio "horizon radius / proton radius".
6. Oct 23, 2013
### MathematicalPhysicist
Does this article depend on something like proton decay which as far as I can tell haven't yet been verified even?
7. Oct 23, 2013
### mitchell porter
The type of proton decay that people have unsuccessfully looked for, is based on grand unified theories in which there are new ultraheavy bosons that can directly convert a quark to a lepton. But there is another type of proton decay which is already implied by the standard model, but which is unobservably rare at the current temperature of the universe (though it may have had significant effects in the early universe).
They are talking about this other type of proton decay, that takes about a googol years to happen. It's weird, it combines an "axial anomaly" in which the overall sum of lefthanded and righthanded fermions isn't conserved, with an ultra-rare "sphaleron" configuration of the electroweak gauge fields. Somehow the consequence is that a proton goes in and antileptons come out! I want to understand it.
8. Oct 24, 2013
### MathematicalPhysicist
The question is "Is it empirically testable?"
If not then it's not any different than believing in a personal God.
9. Oct 25, 2013
### MathematicalPhysicist
BTW, is there a model or a theory that posits that protons never decay?
10. Oct 27, 2013
### lpetrich
The only proposed GUT that I know of that does not predict proton decay is trinification. It has gauge group SU(3)*SU(3)*SU(3), with one of the SU(3)'s becoming the QCD SU(3). The other two SU(3)'s become the electroweak SU(2)*U(1).
All the others do, because they have elementary-fermion multiplets that include quarks and leptons. That's what leads to baryon-number violation, and from that, protons decaying.
Every hadron can decay in proton-decay fashion, but nearly all of them decay much faster by Standard-Model interactions. However, neutrons can be stabilized against SM-interaction decay by being bound in nuclei. So proton-decay experiments also measure the decay rate of bound neutrons in otherwise-stable nuclei.
11. Oct 28, 2013
### Myslius
12. Oct 28, 2013
### torus
How is this supposed to work? I thought Instantons (and sphalerons) change baryon and lepton number only in steps of 3 (the number of generations). Therefore the proton can not decay via instantons...
Even in 't Hooft's cited paper Phys. Rev. Lett. 37, 8-11 (1976) proton decay is not considered, but rather the annihilation of proton and neutron to two leptons (he uses only two generations).
13. Oct 28, 2013
### fzero
Yes, if you just have a bare proton then energy conservation would seem to prevent it from decaying due to EW instantons. However, the 't Hooft process would cause the instability of all $A>1$ nuclei, which has many of the same consequences of proton decay.
14. Nov 1, 2013
### mitchell porter
It would be much too rare for anyone to see it happen. However, this sphaleron-induced decay is a process implied by the standard model and by our understanding of nonperturbative quantum field theory. It is not an extra postulate - I didn't make that clear.
It might be possible to indirectly test for the reality of electroweak sphalerons in cosmological data, since the sphalerons should be much more common in the high temperatures of the early universe. Actually, the challenge for the standard model here is that sphalerons might have been useful to produce the universe's baryon asymmetry (excess of matter over antimatter), except that even at those temperatures, they are still not common enough to produce the magnitude of matter excess that we see. So the standard-model sphalerons would have to be just part of a bigger, beyond-standard-model picture of all the processes occurring after the big bang.
This question gets even better when you ask it about the whole Frampton-Hung scenario. :-)
They are interested in the idea that the Higgs field is currently in a metastable state, and that it will eventually quantum-tunnel its way to the true ground state, in which the Higgs energy density becomes GUT-scale, all particles become supermassive, and atoms and nuclei as we know them would cease to exist. Their idea is that if the timescale of this vacuum decay is similar to the timescale of sphaleron-induced proton decay, that implies a Higgs boson mass that is about what we see.
This hypothesis is just a new variation on the now familiar observation that the observed Higgs boson mass is right on the edge between metastability and absolute stability of the Higgs vacuum. If we were to make the alternative hypothesis that the Higgs mass is the minimum mass for which the Higgs vacuum is absolutely stable - lasts forever - then that also lands us near the observed mass. Or if we were to suppose that the Higgs vacuum's lifetime is just on the order of ten billion years (this could be an anthropic argument), once again, that will land us near the observed mass.
My point is that there's nothing clearly special about this googol-year timescale derived from thinking about sphalerons. Any timescale from tens of billions of years, to forever, would imply a critical value of Higgs mass (well, in the case of "forever", you have to assume that it's the minimum mass that produces absolute stability, in order to get near the critical value). I should also qualify this statement - there are considerable theoretical uncertainties about exactly which values of Higgs mass and top mass, lie on which side of the boundary between metastability and absolute stability. There are competing claims which use different approximations. But they're all in the same ballpark, and the fact that it matters e.g. whether you calculate to two loops or to three loops, at least demonstrates that we are close to criticality in some sense.
So the significance of this paper by Frampton & Hung is that they are suggesting a specific new line of inquiry, regarding what could be the reason for criticality or near-criticality of the Higgs. The question is whether this new direction makes much sense, or could be made to make sense.
In that regard, the first thing to observe is that part of their scenario is that dark energy has disappeared by the time the Higgs vacuum decay occurs. I think that what is being talked about here is not just the occurrence of Higgs vacuum decay somewhere in the universe, but the domination of the true ground state throughout the universe. Recall that these decays are supposed to happen in the far future. The bubble of true vacuum (true ground state) expands at the speed of light once it appears, but in a Lambda-CDM cosmology of accelerating expansion, the universe itself is also expanding very rapidly by the time we are a googol years or more in the future; whereas, if the dark energy has disappeared, so will the acceleration, and the bubble of true vacuum can expand to reach the cosmological horizon a little more quickly.
I am assuming - I ought to check Frampton & Hung's reference 27, but for the sake of moving the discussion along, I'll just assume, and check later - that this is the reason why the lifetime of the metastable Higgs vacuum is a little longer, in the cosmology where dark energy persists.
In their paper Frampton and Hung go a little further and speculate - reasonably enough, given the connection that they wish to make - that the dark energy actually disappears when the vacuum decay occurs. This is where we can actually start to think about mechanisms. My first thought is of quintessence. The simplest idea about dark energy is that it's just vacuum energy, but the next simplest is that vacuum energy is zero and that dark energy is the energy in a new scalar field, usually called the quintessence field.
So perhaps the idea is that there's a single big scalar potential involving the Higgs scalar and the quintessence scalar, and the true ground state is one where the Higgs field takes an enormous leap in energy density, while the quintessence field drops to zero.
But this still provides no mechanism for the desired connection with sphalerons. The rate of occurrence of sphalerons, according to F&H equation 4, depends mostly on the Weinberg angle and the electromagnetic coupling constant. The rate of decay of the combined Higgs-quintessence vacuum will depend on various coefficients in the combined scalar potential. I suppose this framework would at least get us to the stage of being able to reason about the idea more concretely. Given this framework (SM + coupled Higgs/quintessence vacuum decay), you could then try to make an anthropic argument, or look for a still-deeper theory in which all these parameters were correlated in the desired way.
The other thought I've had, is that maybe F&H are thinking that the proton decay causes the vacuum decay. More precisely, that the sphaleron which causes proton decay on googol-year timescales, also causes the Higgs vacuum decay and the disappearance of dark energy. If that were so, it would explain the coincidence of timescales.
But this is problematic for two reasons. First, the sphaleron is a theoretically well-studied phenomenon of the electroweak field. One would have to suppose some novel interplay with the scalar sector, for it to have such drastic effects.
Second, what about the sphalerons in the early universe? If an electroweak sphaleron can initiate Higgs vacuum decay, and if sphalerons were relatively abundant after the big bang, then we shouldn't even be here, the universe should already be dominated by the true vacuum, with all particles supermassive. I already mentioned that there's an intriguing failed connection between sphalerons and baryon asymmetry, maybe there's some conceptual reverse backflip which somehow solves that problem and this one at the same time, but I'm failing to see it.
So the idea isn't working so far, but it's quite stimulating because of what it brings together.
|
2017-08-17 07:16:22
|
{"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": 0.5878763198852539, "perplexity": 805.3366188416636}, "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-2017-34/segments/1502886102967.65/warc/CC-MAIN-20170817053725-20170817073725-00204.warc.gz"}
|
http://mymathforum.com/algebra/53337-very-challenging-sequence-1-a.html
|
My Math Forum Very challenging sequence 1
Algebra Pre-Algebra and Basic Algebra Math Forum
April 29th, 2015, 02:15 PM #1
Senior Member
Joined: Jan 2015
From: USA
Posts: 107
Thanks: 2
Very challenging sequence 1
Hi! Does anyone know how to solve this problem! Thanx
Attached Images
Sequence 1.jpg (30.2 KB, 11 views)
April 29th, 2015, 03:16 PM #2 Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 The partial sum of a geometric series from the first term to the $n$th term is $S_n = \dfrac{a(r^n - 1)}{r - 1}$ Where $a$ is the first term and $r$ is the common ratio. In your question, you would end up with the formulae $\dfrac{a(r^x - 1)}{r - 1} = 200$ and $\dfrac{a(r^{zx} - 1)}{r - 1} = 16400$ Now solve for $a$ and $r$. Thanks from matisolla
April 29th, 2015, 03:50 PM #3 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond I believe that's "2x", not "zx". matisolla, what have you tried?
April 29th, 2015, 04:21 PM #4
Senior Member
Joined: Jan 2015
From: USA
Posts: 107
Thanks: 2
Quote:
Originally Posted by greg1313 I believe that's "2x", not "zx". matisolla, what have you tried?
Indeed my friend, it is 2X. Sorry for the resemblance in of z and 2 in my afwul calligraphy! I attach a picture of my progress so far. I dont know how to continue from there. I would appreciate your help, since you always help me in this page!
Attached Images
11198943_10153306495409700_2115045634_n.jpg (52.5 KB, 6 views)
April 29th, 2015, 04:35 PM #5 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond $\displaystyle \frac{a(r^x-1)}{r-1}=200\Rightarrow r-1=\frac{a(r^x-1)}{200}$ $\displaystyle \frac{a(r^{2x}-1)}{r-1}=\frac{200a(r^{2x}-1)}{a(r^x-1)}=16400$ $\displaystyle \frac{r^{2x}-1}{r^x-1}=82$ $\displaystyle \frac{(r^x-1)(r^x+1)}{r^x-1}=82$ $\displaystyle r^x+1=82$ $\displaystyle r^x=81$ $\displaystyle a=5,r=3\text{ (with }x=4)\text{ or }a=20,r=9\text{ (with }x=2).$ Your handwriting is difficult to read. Type out your posts!
Tags challenging, sequence
Thread Tools Display Modes Linear Mode
Similar Threads Thread Thread Starter Forum Replies Last Post matisolla Geometry 2 February 12th, 2015 02:31 PM matisolla Linear Algebra 2 February 11th, 2015 06:08 AM davedave Calculus 28 November 12th, 2012 02:36 AM matemagia Algebra 2 August 22nd, 2012 03:58 AM huu3y2 Calculus 1 April 22nd, 2011 07:12 PM
Contact - Home - Forums - Cryptocurrency Forum - Top
|
2019-04-18 18:35:16
|
{"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": 0.4717741012573242, "perplexity": 5587.325745441843}, "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-2019-18/segments/1555578526228.27/warc/CC-MAIN-20190418181435-20190418203435-00423.warc.gz"}
|
https://www.physicsforums.com/members/potatochip911.532889/recent-content
|
# Recent content by Potatochip911
6. ### Find B and H everywhere for a magnetized infinite cylinder
Thanks! This pieces everything together.
7. ### Find B and H everywhere for a magnetized infinite cylinder
Yes I agree that geometry doesn't make much sense to me either I was just copying what they had done. Setting up the problem as $$\vec{B}(2\pi s) = \mu_0(\int_s^a J_d\cdot 2\pi s\, ds - M_0\cdot 2\pi a)$$ results in the answer of $$\vec{B} = -\mu_0 M_0 (s/a)^2\hat{\phi}$$ which is just the...
8. ### Find B and H everywhere for a magnetized infinite cylinder
Ok, the difference was in the book that their ##\vec{J_b}## was a constant so they could just multiply by the area to get the result of the integral however, my ##\vec{J_b}## is linear w.r.t. s although I am still running into trouble. Defining ##dA =l \,ds\Longrightarrow \int_s^a J_d\cdot dA...
9. ### Find B and H everywhere for a magnetized infinite cylinder
Homework Statement An infinitely long cylinder of radius a has its axis along the z-direction. It has Magnetization ##M=M_0(s/a)^2\hat{\phi}## in cylindrical coordinates where ##M_0## is a constant and s is the perpendicular distance from the axis. Find the values of ##\vec{B}## and ##\vec{H}##...
10. ### Electric field inside a uniformly polarized cylinder
Am I drawing it incorrectly? I still don't find the same result as you.
11. ### Electric field inside a uniformly polarized cylinder
Homework Statement This is problem 4.13 from Griffiths. A long cylinder of radius a carries a uniform polarization P perpendicular to its axis. Find the electric field inside the cylinder. Homework Equations ##\int \vec{E}\cdot dA = q_{encl}/\varepsilon_0## The Attempt at a Solution [/B] We...
12. ### Drawing a Timing Diagram
Hmm, now I'm confused because depending on whether or not I start at the top or bottom gate outputting 0 I will end up with either ##Q=1##, ##Q'=0## or ##Q=0##, ##Q'=1##
13. ### Drawing a Timing Diagram
Homework Statement Draw the diagram for the following circuit given the following conditions: 1) X=Y=Z=1 2)X=Y=1, Z=0 3)X=Y=0, Z=1 4)X=1, Y=Z=0 Homework Equations The Attempt at a Solution [/B] ##W=XZ'+YZ##, ##V=Y'Z+XY## 1) W = 0 + 1 = 1 V = 0 + 1 = 1 and now I'm not sure how to get the...
14. ### Calculating the surface charge of a sphere and a conducting shell
So in an insulator the electrons can't flow freely therefore they won't be able to redistribute across the surface? Yes, is it just a conductor because it's metal?
15. ### Calculating the surface charge of a sphere and a conducting shell
I thought that charge only entirely resided on the surface of conductors otherwise why would they mention this as a property of conductors and not just in general? After looking around it seems like the charge will always distribute across the surface of anything in order to minimize the...
|
2022-01-17 10:46:25
|
{"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": 0.7833678722381592, "perplexity": 919.1049996988598}, "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-2022-05/segments/1642320300533.72/warc/CC-MAIN-20220117091246-20220117121246-00009.warc.gz"}
|
http://astrobunny.net/categories/aria/
|
Like lolikit says, I am a strange person because I seem to have a liking for both Haruhi and Aria. Although I am not sure what seemed to be perceived by him as two shows that are from opposite ends of some metaphorical spectrum
written by astrobunny \\ akari, aria, artbooks, beauty, books, calm, cielo, happiness, life, little, stella, things, treasures, utopia
Because I've been watching ARIA the Animation again lately and I've been feeling the nostalgia.
written by astrobunny \\ ai, akari, aria, greatest show, memorable
ARIA the Origination may have ended a year ago, but it still lives on in my heart. From today onwards, I can bring Akari with me everywhere I go. All Hail Akari! All Hail Akari! All Hail Akari!
written by astrobunny \\ akari, aria, keychain, win
Sometimes when you're sitting on a train you look at some wonderful things and wonder what Akari would say when she saw it. Well, if you own the DVDs, there are nice groups who produce anime encodes for the PSP and DS (with subtitles). Now you don't have to mutter the words "HAZUKASHII SERIFU KINSHI" to yourself while you try and imagine the scene where Akari treats herself to the amazing views of Neo-Venezia
Obviously, you'll be watching them low-res, so it would be ideal if you don't strain your eyes to see fine details, and just relax and enjoy the show.
written by astrobunny \\ anime, aria, ds, nintendo
Narrator: Oh great buddha Alicia, please guide Akari to lead us on the journey to enlightenment...
Alicia: Ara ara...
written by astrobunny \\
|
2018-12-15 06:56:26
|
{"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": 0.9808573126792908, "perplexity": 8104.0556850867715}, "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-2018-51/segments/1544376826800.31/warc/CC-MAIN-20181215061532-20181215083532-00027.warc.gz"}
|
http://dirkmittler.homeip.net/blog/archives/tag/convert-acl
|
## Managing Usershares properly, under SAMBA (meaning, under the SMB emulation given by Linux).
Just to encapsulate the subject of this posting… ‘SMB’ is a file-sharing protocol which is really Windows-owned. Granted, ‘SMB1‘ cannot be fully Windows-owned, but, on the assumption that a ‘SAMBA’ server is being used to emulate ‘SMB2′ and ‘SMB3′ (under Linux), there are many ways in which the (root-owned) configuration file at ‘/etc/samba/smb.conf’ could be faulty, and could result in weird error messages. The fixes which are highlighted in this posting worked for me, but my reader could be suffering from mistakes of an entirely different nature. I am posting this, in case the reader happens to be suffering from the same configuration mistakes which I was.
Also, my configuration issues arose partially, because I’ve switched this configuration file from a configuration in which Home Directories are being shared out in their entirety, to a configuration in which individual users can decide to share out specific folders they own. This system is one of ‘Usershares’.
What I was eventually doing was, to give the command ‘net usershare add <Share-Name> "<Path-Name>" "<Comment>" Host\\user1:f,Host\\user2:f‘. The comma and the absence of spaces in the Access Control List are important. I was getting the error-message stating, that these user-names, part of the ACL, could not be converted into SIDs. What I found was, that I had the following error in my ‘smb.conf’ file:
map to guest = bad user
guest account = nobody
(...)
server max protocol = SMB3
server min protocol = SMB2
Amazingly, the error message went away, if I changed that last detail to:
client max protocol = SMB3
server min protocol = SMB2
But, there is more to be known about my configuration:
usershare allow guests = no
usershare max shares = 10
usershare owner only = false
usershare path = /var/lib/samba/usershares
What this last set of parameters actually requests is, that individual users should not be able to grant Guest Access – unauthenticated access – to any of their folders, but potentially, to grant access which is authenticated by another SAMBA username, with their enabled password, as existing on the server.
Again to my amazement, I found that, if the server is massaged adequately, it will implement the settings exactly as requested. But it will do so with a crucial limitation. Locally on the server, the non-owner username must already have access to the exact folder named in the usershare. What level of access that username has (to the named folder itself), will cap whatever level of control is granted by way of a SAMBA client.
This observation could also be rephrased as follows: Even though (Linux) SAMBA has an impressive-looking feature in “Usershares”, while that creates rule-files in the folder ‘/var/lib/samba/usershares’, which even possess Access-Control Lists, those ACLs finally disappoint, in NOT guaranteeing what the behaviour of the SAMBA server will be, once a usershare has finally granted access (to a folder, for the client). (:2)
Usually, in order to grant such access locally on the server, some strategy with group memberships and permission-bits gets used. Which exact arrangement of groups and permission-bits gets used, is not set in stone. Any arrangement will work that grants full access. But, because it’s usually unwieldy for users to set up such local sharing of their folders, they are also not likely to succeed – without compromising their security completely – unless they also have the help of someone with root access. Therefore, the ability to give this feature to users in user-space, is theoretical at best. (:1) And, if the user wants to activate this extension, he or she must use the CLI, and cannot count on the GUI within ‘Dolphin’ to do so. But, the final command given via the CLI can be given as user.
By default, declaring a usershare ‘the easy way, via the GUI’, remains owner-only.
There is one more caveat to mention. According to my recent experiences, if ‘Dolphin’ is being used as the SAMBA client, and if it’s to authenticate with the server, using any username other than the current username on the local client-computer, then the credentials need to be specified under ‘System settings -> Network -> Connectivity’ (where the Plasma 5.8 Desktop manager puts them). If there is a set of credentials there, they will not be stored encrypted, but only stored with casual obfuscation, and Dolphin will use them. If the user wants the credentials to be stored in the ‘kwallet’ (and thus encrypted), he needs to make his username, to log into the server with, identical to his username on the local client. Otherwise, illogical error messages will appear again, this time, from the Dolphin file-browser not being consistent, in whether to try to authenticate at all. And, if an attempt is made to access the share without authenticating, then Dolphin generates an illogical error message, which can be paraphrased as a ‘File Exists’ error message.
With enough effort, and, sacrificing some security, it can all be done.
(Updated 3/29/2021, 15h10… )
|
2021-04-18 21:08:41
|
{"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": 0.4919393062591553, "perplexity": 3661.7258494346615}, "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-20210418194009-20210418224009-00208.warc.gz"}
|
http://milsabores.net/visa-d-two/equation-of-the-tangent-line-b2df02
|
# equation of the tangent line
For examples of tangent line equations, keep reading! 0000007088 00000 n Sketch the tangent line going through the given point. Support wikiHow by But what if surfaces are given as P(x,y,z)=c and Q(x,y,z)=d,with P(x,y,z) and Q(x,y,z) both differentiable how should i find equation of the tangent line … 0000000933 00000 n Tangent Line Parabola Problem: Solution: The graph of the parabola $$y=a{{x}^{2}}+bx+c$$ goes through the point $$\left( {0,1} \right)$$, and is tangent to the line $$y=4x-2$$ at the point $$\left( {1,2} \right)$$.. Find the equation of this parabola. If above line is a tangent the c = a/m. Notice we do not have a point this time, only an x-coordinate. How do I find the slop of a line and the tangent line? 0000009516 00000 n Therefore, a tangent line can be described as a linear function of the form y = ax + b. %PDF-1.4 %���� Here are the steps: Substitute the given x-value into the function to find the y-value or point. Managed to solve the problem! $$y=xe^x$$ and $$y=x$$ Finding the Tangent Line Equation with Implicit Differentiation The equation of the tangent line to 2² sin T # 0 f(x) = 0 x = 0 at r = 0 is 1 (a) y = -x (b) y = x (c) y = (d) y = 0 (e) Does not exist. Normal is a line which is perpendicular to the tangent to a curve. 0000002895 00000 n = 34, (5,3) The equation of the tangent line to the point… 0000005417 00000 n 0000005082 00000 n 0000010098 00000 n Equation of the tangent line : y-y1 = m (x-x1) y+11 = -3 (x-3) y+11 = -3x+9. Did you know you can read expert answers for this article? With over 11 years of professional tutoring experience, Jake is also the CEO of Simplifi EDU, an online tutoring service aimed at providing clients with access to a network of excellent California-based tutors. 255 0 obj <> endobj 0000002179 00000 n Then, equation of the normal will be,= Example: Consider the function,f(x) = x2 – 2x + 5. <]>> To find the equations for lines, you need to find m and c. m is the slope. Example 3 : Find a point on the curve. Asked • 06/24/20 Find an equation of the tangent line to the graph of y = g(x) at x = 2 if g(2) = −6 and g'(2) = 4. Last Updated: October 8, 2020 To find the y-coordinate, plug x = 2 into the initial function: Write the tangent line equation in point-slope form: Take the first derivative of the function to get f'(x), the equation for the tangent's slope. • The slope-intercept formula for a line is y = mx + b, where m is the slope of the line and b is the y-intercept. This gives you the slope of the tangent at (x,y). endstream endobj 256 0 obj<>/OCGs[258 0 R]>>/PieceInfo<>>>/LastModified(D:20041006101430)/MarkInfo<>>> endobj 258 0 obj<>/PageElement<>>>>> endobj 259 0 obj<>/Font<>/ProcSet[/PDF/Text]/ExtGState<>/Properties<>>>/StructParents 0>> endobj 260 0 obj<> endobj 261 0 obj<> endobj 262 0 obj<> endobj 263 0 obj<> endobj 264 0 obj<> endobj 265 0 obj<>stream 0000005938 00000 n SOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty is that we know only one point, P, on t, whereas we need two points to compute the slope. ", "It helped me with my calculus homework. Let (x, y) be the point where we draw the tangent line on the curve. Perpendicular to line means We know that the equation of the line is y = mx + c on comparing with the given equation we get the slope of line m = 3 and c = 13/5 Now, we know that the slope of the tangent at a given point to given curve is given by Given the equation of curve is … (a) Find the equation of the tangent line to the curve x=e^{t}, \quad y=e^{-t} at t=1 without eliminating the parameter. 0000007944 00000 n https://www.khanacademy.org/.../ab-2-1/v/derivative-as-slope-of-tangent-line By definition, a line is always straight and cannot be a curve. What is the equation of the tangent line at x = 1 o f of o f f (x) = x f\left(x\right)=\ \sqrt{x} f (x) = x answer choices y = 1 2 x − 1 2 y=\frac{1}{2}x-\frac{1}{2} y = 2 1 x − 2 1 See Answer. Thus, the equation of the tangent line is {eq}y = \dfrac{3}{4}x - \dfrac{{25}}{4} {/eq}. en. x�bbbebŃ3� ��� 9� How do I calculate the equation of a tangent at a point (x,y) on a circle? If you use that x and that y and the slope m, you can use algebra to find c. y=mx+c, so, c=y-mx. So to find the equation of a line that is perpendicular to the tangent line, first find the slope of the tangent line. Check out a sample Q&A here. check_circle Expert Answer. with slope -3. Unless you are given the slope of the tangent line, you'll need to find it the same way you would for any other problem: finding the derivative f'(x). To find the equation of a line you need a point and a slope. Since tangent and normal are perpendicular to each other, product of slope of the tangent and slope of the normal will be equal to -1. Find an equation of the tangent line to the graph of the following function f at the specified point. 0000005639 00000 n Want to see this answer and more? Question. The tangent line is a straight line with that slope, passing through that exact point on the graph. trailer Equation of the perpendicular tangent is 0000002425 00000 n Jake Adams is an Academic Tutor and the Owner of PCH Tutors, a Malibu, California based business offering tutors and learning resources for subject areas kindergarten-college, SAT & ACT prep, and college admissions counseling. Find f'(x), the slope of the tangent line. is x+2y+c =0 . Now you also know that f'(x) will equal 2 at the point the tangent line passes through. Two lines are perpendicular to each other if the product of their slopes is -1. How do I find the tangent line on a graph where x is 1? Meaning, we need to find the first derivative. In mathematics, a tangent line is a line that touches the graph of a certain function at one point, and has the same slope as the slope of the function at that point. If you prefer it in a different form, such as slope-intercept, you can convert into that form. at the point P(1,5). H�TQ=o� ��+nlԁ�6m"Y�,�����p�b�0���M�лww��AN���&. As wikiHow, nicely explains, to find the equation of a line tangent to a curve at a certain point, you have to find the slope of the curve at that point, which requires calculus. This article has been viewed 1,041,640 times. Write the normal equation in slope-point form. 0000012768 00000 n Trigonometric functions have their own rules for differentiation, which you can look up in your textbook or online. Differentiate to get the equation for f'(x), then set it equal to 2. 0000004389 00000 n How do I find the equation of a tangent line? This article was co-authored by Jake Adams. What Is a Tangent Line? 9/4/2020 Untitled Document Equation of a Line, Tangent Lines On this page we hope to demonstrate the H��T�n�0��+��J��r���@�Pm�f�H�\$�@����Xk�I+vgv5��W���=ϟ�p���C���d���b�f.�9�@�62D��V�/ɀ;�^p-�����5@�YK|��h&��47L�� About Pricing Login GET STARTED About Pricing Login. Use the slope-point form of the line to find the equation, with the slope you obtained earlier and the coordinates of the point. 2 5marks cFind the equation of the tangent line to the curve 2 2 5 2 2 at the from ECON 007C at DEWA Islamabad Campus Take the second derivative to get f''(x), the equation that tells you how quickly the tangent's slope is changing. To get the whole equation of the perpendicular, you need to find a point that lies on that line, call it (x°, y°). Worked example 13: Equation of a tangent to a circle The straight line \ (y = x + 4\) cuts the circle \ (x^ {2} + y^ {2} = 26\) at \ (P\) and \ (Q\). ", "Helped so much! ; The normal line is a line that is perpendicular to the tangent line and passes through the point of tangency. Read It Submit Answer . 0000001376 00000 n {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/1\/19\/Find-the-Equation-of-a-Tangent-Line-Step-1-Version-3.jpg\/v4-460px-Find-the-Equation-of-a-Tangent-Line-Step-1-Version-3.jpg","bigUrl":"\/images\/thumb\/1\/19\/Find-the-Equation-of-a-Tangent-Line-Step-1-Version-3.jpg\/aid2897229-v4-728px-Find-the-Equation-of-a-Tangent-Line-Step-1-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"
|
2021-10-24 02:13:27
|
{"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": 0.6506653428077698, "perplexity": 393.0230223368712}, "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/1634323585837.82/warc/CC-MAIN-20211024015104-20211024045104-00630.warc.gz"}
|
https://stats.stackexchange.com/questions/124819/connections-between-d-prime-d-prime-and-auc-area-under-the-roc-curve-und/169080
|
# Connections between $d^\prime$ (d-prime) and AUC (Area Under the ROC Curve); underlying assumptions
In machine learning we may use the area under the ROC curve (often abbreviated AUC, or AUROC) to summarise how well a system can discriminate between two categories. In signal detection theory often the $d'$ (sensitivity index) is used for a similar purpose. The two are closely connected, and I believe they are equivalent to each other if certain assumptions are satisfied.
The $d'$ calculation is usually presented based on assuming normal distributions for the signal distributions (see wikipedia link above, for example). The ROC curve calculation does not make this assumption: it is applicable to any classifier that outputs a continuous-valued decision criterion that can be thresholded.
Wikipedia says that $d'$ is equivalent to $2 \text{AUC} - 1$. This seems correct if the assumptions of both are satisfied; but if the assumptions are not the same it's not a universal truth.
Is it fair to characterize the difference in assumptions as "AUC makes fewer assumptions about the underlying distributions"? Or is $d'$ actually just as widely applicable as AUC, but it's just common practice that people using $d'$ tend to use the calculation that assumes normal distributions? Are there any other differences in the underlying assumptions that I've missed?
|
2019-08-19 17:06:16
|
{"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": 0.7919958829879761, "perplexity": 490.59794277956604}, "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/1566027314852.37/warc/CC-MAIN-20190819160107-20190819182107-00495.warc.gz"}
|